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The character is determined by the two values on (the classes of) $15$ and $17$. The first element has a clear image, the second one...
sage: R.<X> = QQ[]
sage: K.<a> = NumberField( X^4 + 2*X^3 + 2*X^2 + 4*X + 4 )
sage: b = ( -a^3/2 - a^2/2 - a - 1 )
sage: b.minpoly()
x^2 - x + 1
sage: b^3
-1
sage: b^6
1
goes to a primitive 6th root of unity. This is enough to identify the character, up to conjugation. The following code creates one of the two spaces of modular symbols related to the question:
sage: D28 = DirichletGroup( 28 )
sage: D28.order()
12
sage: D28.0
Dirichlet character modulo 28 of conductor 4 mapping 15 |--> -1, 17 |--> 1
sage: D28.1
Dirichlet character modulo 28 of conductor 7 mapping 15 |--> 1, 17 |--> zeta6
sage: ch = D28.0 * D28.1
sage: ch
Dirichlet character modulo 28 of conductor 28 mapping 15 |--> -1, 17 |--> zeta6
sage: MS = ModularSymbols( ch, 2, 1 )
sage: MS
Modular Symbols space of dimension 4 and level 28, weight 2, character [-1, zeta6], sign 1, over Cyclotomic Field of order 6 and degree 2
