
// Make newform 75.6.b.c in Magma, downloaded from the LMFDB on 26 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_75_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_75_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_75_6_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_75_6_b_c();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_75_b();"
function MakeCharacter_75_b()
    N := 75;
    order := 2;
    char_gens := [26, 52];
    v := [2, 1];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_75_b_Hecke();"
function MakeCharacter_75_b_Hecke( : Kf := false)
    N := 75;
    order := 2;
    char_gens := [26, 52];
    char_values := [[1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 6;
    raw_aps := [[0, 4], [0, -9], [0, 0], [0, -225], [-434, 0], [0, -613], [0, -878], [731, 0], [0, -2850], [7582, 0], [2175, 0], [0, -9310], [-12040, 0], [0, 1121], [0, 29878], [0, -5740], [5174, 0], [-38717, 0], [0, 31707], [64472, 0], [0, 19790], [105000, 0], [0, 3318], [65376, 0], [0, -89143], [-94644, 0], [0, 113956], [0, -19212], [152791, 0], [0, -4012], [0, -36368], [-43932, 0], [0, 185214], [-115916, 0], [-33122, 0], [431317, 0], [0, 40439], [0, 116299], [0, -632700], [0, -108222], [-496570, 0], [105365, 0], [427346, 0], [0, -646427], [0, -546954], [581783, 0], [-441355, 0], [0, 330457], [0, 1125080], [977379, 0], [0, -334584], [-218764, 0], [734041, 0], [1183812, 0], [0, -333732], [0, 731104], [-734254, 0], [335056, 0], [0, -1582599], [1087134, 0], [0, 607407], [0, -1233006], [0, 562753], [-2396454, 0], [0, -1528331], [0, 1000328], [2290932, 0], [0, 1289041], [0, 19102], [3246902, 0], [0, -4288038], [-88656, 0], [0, -2041839], [0, 806321], [2413145, 0], [0, 1458564], [-2656176, 0], [0, 1686643], [-2125332, 0], [-1971581, 0], [-1757444, 0], [-2839466, 0], [-4693350, 0], [0, 6713429], [-3920215, 0], [0, 6930312], [-4386052, 0], [0, 2419510], [-366228, 0], [0, -6625344], [0, 3384758], [-6619050, 0], [0, -287383], [6513368, 0], [1170371, 0], [0, 1406512], [-3656446, 0], [-2708890, 0], [0, 11401391], [6950505, 0], [0, -8085320], [0, 14602482], [0, -3546522], [14268066, 0], [-5485339, 0], [0, -2132533], [0, 2393028], [0, -1834784], [126508, 0], [10379051, 0], [0, -366608], [0, 7492718], [0, 2237484], [-6616481, 0], [-6164813, 0], [4795020, 0], [0, 1201236], [0, -10147988], [0, 19861414], [5733080, 0], [-7922438, 0], [0, 7753530], [0, -3777774], [0, 6182904], [-23847464, 0], [-5812402, 0], [7531331, 0], [8662278, 0], [0, 2944439], [0, -19236254], [26390924, 0], [0, 11516], [-24483584, 0], [0, -10138451], [-13457076, 0], [-21357347, 0], [0, 25546824], [0, 13266793], [0, -12491932], [7835140, 0], [30475077, 0], [-29032986, 0], [0, 19993453], [0, -196388], [-7050930, 0], [32611810, 0], [0, 18468729], [0, -31430972], [-28076680, 0], [0, 21195136], [0, 37091055], [-9163856, 0], [0, -13639559], [0, 29040138], [0, 21209988], [19526722, 0], [28593035, 0], [-23876894, 0], [0, 34667151], [46418750, 0], [0, -21082926], [0, -36743848], [0, -37623448], [-48920490, 0], [0, -28444958], [0, -45161100], [-28551733, 0], [0, 32652714]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_75_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_75_6_b_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_75_6_b_c(:prec:=2)
    chi := MakeCharacter_75_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_75_6_b_c();".
// 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_75_6_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_75_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![16, 0, 1]>],Snew);
    return Vf;
end function;