
// Make newform 525.2.z.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_z();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_525_2_z_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_525_z();"
function MakeCharacter_525_z()
    N := 525;
    order := 10;
    char_gens := [176, 127, 451];
    v := [10, 3, 10];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_525_z_Hecke(Kf)
    return MakeCharacter_525_z();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_525_2_z_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_525_2_z_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9025, 104500, -557025, -4262050, 51385650, -209029450, 357410150, -32659500, -324492165, -1232798050, 3258062100, 4770991600, -21272890690, 8363963750, 53977869965, -76115020990, -39162694134, 156664626810, -48305774975, -159326663300, 126859613820, 88011503200, -112308307705, -50109526670, 80207345495, 26061103670, -47753136281, -11227764380, 24188371550, 3487914580, -9847013350, -1423455350, 3859936805, 374256560, -1267171100, -94980340, 371972641, 21923010, -98056235, -4799740, 23426790, 656550, -4799685, -108220, 893475, 13200, -147136, -800, 20680, -70, -2390, 0, 245, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;