
// Make newform 576.2.bb.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_576_bb();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_576_2_bb_e();".
// 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_576_bb();"
function MakeCharacter_576_bb()
    N := 576;
    order := 12;
    char_gens := [127, 325, 65];
    v := [12, 3, 4];
    // 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_576_bb_Hecke(Kf)
    return MakeCharacter_576_bb();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_576_2_bb_e();".
// 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_576_2_bb_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_bb();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![65536, 8978432, 615022592, 66961801216, 4529605668864, 65234148261888, 638304994052096, 12099310841701376, 68383728502827264, -522867693995038720, -1270940444129055744, -17411820000057848320, 52519669916909275520, 362817429438562370560, 706395466379395084160, 1816677854618774416960, 3973109811737667405360, 3760234108568108074112, 3008293621653808728640, 2898867428941814917632, -633893128674998962120, -2561931499336123469440, -1455235273623812252472, -1237114307880567719028, 473404950447331448209, 989958955532405566152, 479791665250293780688, 538599259122490202056, 164870016580306565158, -78613589983483996552, -14921726447356849304, -37086914580080592148, -20879107223791719475, 4344279007693992008, 499101228449455952, 1563571872169877000, 1216211875310614698, -189546581537084768, -6925557804744304, -48670605365446072, -46279845911145158, 7250050997426568, 172437944203344, 1240988593682056, 1224421682900894, -232390112675416, 3035870852136, -28260485297716, -23026889487723, 5628465266968, -175025802064, 533946292408, 295350309958, -101695957504, 6905641872, -8254656280, -2169959238, 1185653784, -102190800, 87574008, 4482466, -9220904, 1169128, -647604, 94237, 23768, -3088, 1784, -362, -48, 8, -4, 1]>],Snew);
    return Vf;
end function;