
// Make newform 760.2.q.f in Magma, downloaded from the LMFDB on 14 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_760_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_760_q_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_760_2_q_f();".
// 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_760_2_q_f();".
// 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 := [9, 12, 31, -14, 26, -3, 6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-1215, 14317, -3618, 3051, -911, 729, -189, -26], [1947, 483, 989, 3314, 276, 529, 20, 115], [-11694, -567, -1161, -2599, -324, -621, 161, -135], [6390, -17152, 19028, -16046, 3534, -3834, 994, -649], [6573, 16249, 338, 12064, 193, 2845, -109, 434], [-5355, 16585, -15946, 13447, -3858, 3213, -833, 514]];
    Rf_basisdens := [1, 1, 4243, 4243, 4243, 12729, 4243, 4243];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_760_q();"
function MakeCharacter_760_q()
    N := 760;
    order := 3;
    char_gens := [191, 381, 457, 401];
    v := [3, 3, 3, 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_760_q_Hecke();"
function MakeCharacter_760_q_Hecke( : Kf := false)
    N := 760;
    order := 3;
    char_gens := [191, 381, 457, 401];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, -1, -1, -1, 0, 1, 0], [-1, 0, 1, -1, 1, 0, -1, 0], [0, -1, 0, -1, 0, 0, -1, 0], [0, 0, -1, 0, 0, 3, 0, -1], [0, -2, -1, 0, 1, 1, 1, -1], [1, -3, 0, -3, -1, -1, 2, 1], [-1, 0, 0, 0, -1, 1, -2, 1], [0, 0, 2, -1, -1, 0, -2, 0], [2, 0, 1, 1, -3, 0, -1, 0], [0, 1, -1, 0, 0, -2, 0, 1], [0, -1, 2, 0, 0, -1, 0, -2], [2, -3, 0, -3, 1, -2, 1, -1], [-3, -2, 0, -2, 1, 3, 3, -1], [0, 0, 2, 0, 0, 1, 0, 3], [-1, -2, 0, -2, -2, 1, -2, 2], [-2, 0, 0, 0, 0, 2, 2, 0], [0, 4, 3, 0, 0, -2, 0, 0], [0, 1, 2, 0, 0, -5, 0, 1], [0, 1, -5, 0, 0, -1, 0, 0], [-5, 0, 3, -2, -3, 0, -3, 0], [5, 5, 0, 5, -2, -5, 3, 2], [0, -2, -2, 0, 0, 4, 0, 3], [4, -3, 0, -3, 3, -4, 2, -3], [-5, 0, -2, -3, -1, 0, 2, 0], [1, 0, -2, -3, 1, 0, 2, 0], [0, -4, -1, 0, 0, -2, 0, 6], [2, 0, -6, -4, 1, 0, 6, 0], [10, 6, 0, 6, -2, -10, 0, 2], [0, 5, 3, 0, 0, 0, 0, 3], [-1, 5, 0, 5, 2, 1, 0, -2], [2, 3, 0, 3, -2, -2, 1, 2], [0, 4, -9, 0, 0, 0, 0, -3], [-7, 0, -2, -4, 4, 0, 2, 0], [0, -4, 3, 0, 0, 7, 0, -2], [9, 0, 1, 2, 5, 0, -1, 0], [0, 1, 0, 1, 7, 0, 3, -7], [0, 7, 3, 0, 0, 2, 0, -3], [-6, 0, -3, 4, -1, 0, 3, 0], [-4, -7, 0, -7, -4, 4, -1, 4], [-1, 0, -7, -1, -4, 0, 7, 0], [0, 6, -1, 0, 0, 1, 0, -3], [-4, 0, -1, -5, -7, 0, 1, 0], [8, 8, 0, 8, -2, -8, -1, 2], [0, -1, 5, 0, 0, 11, 0, 6], [0, -6, 0, 0, 0, -6, 0, -1], [-18, 0, 2, -2, 0, 0, -2, 0], [1, 0, -1, 3, -1, 0, 1, 0], [0, 12, -1, 0, 0, 3, 0, -4], [6, 0, 0, 3, 13, 0, 0, 0], [-5, -9, 0, -9, -2, 5, -1, 2], [7, 1, 0, 1, 1, -7, 1, -1], [-4, 8, 0, 8, -7, 4, 2, 7], [0, -4, 4, 0, 0, 6, 0, -2], [0, -4, 6, 0, 0, 3, 0, 0], [0, -6, 5, 0, 0, 2, 0, -5], [-2, 0, 4, -10, -2, 0, -4, 0], [-15, 4, 0, 4, 4, 15, -1, -4], [0, 0, -5, 0, 0, 17, 0, 2], [-3, 0, 0, -7, 3, 0, 0, 0], [0, -4, -4, 0, 0, 4, 0, 5], [-18, 0, -1, -3, -3, 0, 1, 0], [7, 1, 0, 1, 11, -7, -2, -11], [1, -7, 0, -7, 0, -1, 4, 0], [-2, 0, -2, -2, -1, 0, 2, 0], [0, -4, -2, 0, 0, -16, 0, 0], [0, 1, 6, 0, 0, -1, 0, -4], [-1, 0, 3, -4, -6, 0, -3, 0], [5, 0, 3, -6, 5, 0, -3, 0], [0, 2, -3, 0, 0, 15, 0, 1], [-8, 3, 0, 3, -12, 8, 3, 12], [7, 0, 2, -5, -10, 0, -2, 0], [-10, 0, -3, 2, 10, 0, 3, 0], [0, -3, 2, 0, 0, -5, 0, 8], [11, -5, 0, -5, 1, -11, 3, -1], [0, -1, 7, 0, 0, 6, 0, -1], [0, 4, -5, 0, 0, 6, 0, 0], [-16, -6, 0, -6, -9, 16, -1, 9], [6, 0, 4, -5, -3, 0, -4, 0], [0, -7, 3, 0, 0, 7, 0, -3], [-7, -4, 0, -4, -7, 7, 10, 7], [-14, -4, 0, -4, -5, 14, 8, 5], [0, 5, 5, 0, 0, 13, 0, -3], [16, 7, 0, 7, 3, -16, -7, -3], [0, 0, -1, -10, -3, 0, 1, 0], [11, 0, -5, 0, 5, 0, 5, 0], [0, -8, 4, 0, 0, 15, 0, 5], [-1, 0, -6, 9, -7, 0, 6, 0], [24, 0, 7, 3, 2, 0, -7, 0], [6, 3, 0, 3, 4, -6, -10, -4], [-1, 0, -5, 8, 11, 0, 5, 0], [0, 12, -9, 0, 0, 4, 0, -7], [0, 5, -2, 0, 0, -5, 0, 6], [-2, -10, 0, -10, -1, 2, 2, 1], [-21, 2, 0, 2, -11, 21, 1, 11], [2, 0, 8, 9, 12, 0, -8, 0], [2, 6, 0, 6, 1, -2, 0, -1], [20, 0, 0, 0, 4, -20, -9, -4], [-7, 10, 0, 10, 7, 7, -9, -7], [3, -1, 0, -1, 4, -3, -8, -4], [-6, 0, 3, 9, 4, 0, -3, 0], [-15, 0, -1, -13, 8, 0, 1, 0], [-17, 0, 6, -6, -2, 0, -6, 0], [-5, 0, 1, 16, 5, 0, -1, 0], [0, 4, 5, 0, 0, 19, 0, -6], [-6, 2, 0, 2, -4, 6, 8, 4], [18, 1, 0, 1, 0, -18, 7, 0], [5, 0, 9, -1, -6, 0, -9, 0], [-35, 0, 0, 1, -3, 0, 0, 0], [0, -1, -2, 0, 0, 5, 0, 19], [9, 15, 0, 15, -3, -9, -4, 3], [20, 0, 2, -8, 9, 0, -2, 0], [21, 7, 0, 7, 6, -21, -1, -6], [0, 9, -14, 0, 0, 0, 0, -3], [0, -7, 5, 0, 0, -10, 0, 5], [-7, 0, 8, -5, 4, 0, -8, 0], [6, 0, -8, 10, 7, 0, 8, 0], [-9, -3, 0, -3, -17, 9, 2, 17], [-21, 0, 0, 0, -3, 21, -6, 3], [4, 0, 2, 0, 1, 0, -2, 0], [27, 0, -8, 5, 12, 0, 8, 0], [-20, 0, -8, 2, 5, 0, 8, 0], [-21, 0, -4, 0, -5, 0, 4, 0], [0, 12, -18, 0, 0, -2, 0, -3], [9, 4, 0, 4, 15, -9, 4, -15], [0, 4, 2, 0, 0, 1, 0, -10], [0, 15, 0, 0, 0, -3, 0, 2], [6, 0, -10, 16, 5, 0, 10, 0], [0, -3, 6, 0, 0, 20, 0, 5], [0, 7, -6, 0, 0, -9, 0, -2], [29, 4, 0, 4, 11, -29, -10, -11], [0, 13, -7, 0, 0, -20, 0, 8], [-17, 0, 0, 7, 12, 0, 0, 0], [20, -1, 0, -1, -7, -20, -2, 7], [9, 10, 0, 10, -2, -9, -17, 2], [11, 0, -11, 10, 7, 0, 11, 0], [-16, 0, 2, -2, -21, 0, -2, 0], [0, 0, 4, -13, -7, 0, -4, 0], [-17, -2, 0, -2, -2, 17, -4, 2], [-14, 11, 0, 11, -10, 14, 6, 10], [-4, -8, 0, -8, -4, 4, 8, 4], [-12, -4, 0, -4, 9, 12, 6, -9], [5, 0, -6, -1, 12, 0, 6, 0], [0, 0, -1, 0, 0, -9, 0, -21], [0, -3, -2, 0, 0, -1, 0, 5], [0, -13, 5, 0, 0, -2, 0, 11], [14, 2, 0, 2, 10, -14, 11, -10], [-3, 0, -6, 11, 8, 0, 6, 0], [0, 15, 7, 0, 0, 4, 0, 5], [-29, 0, -5, -5, 4, 0, 5, 0], [7, 9, 0, 9, -18, -7, -4, 18], [3, -10, 0, -10, 14, -3, 1, -14], [0, -2, 10, 0, 0, 0, 0, -4], [-15, 0, 0, 2, -2, 0, 0, 0], [9, 0, -21, -12, -4, 0, 21, 0], [0, 6, 3, 0, 0, 0, 0, 2], [-5, -2, 0, -2, -22, 5, -9, 22], [29, -1, 0, -1, 8, -29, 3, -8], [0, -14, -3, 0, 0, -15, 0, 8], [0, -8, 2, 0, 0, 0, 0, -15], [0, 8, -4, 0, 0, 10, 0, 0], [0, -4, 17, 0, 0, -1, 0, 6], [11, 0, -4, 15, -9, 0, 4, 0], [0, -1, -2, 0, 0, -3, 0, 4], [0, -10, -6, 0, 0, -16, 0, -6], [-10, -13, 0, -13, 8, 10, 17, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_760_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_760_2_q_f();".
// 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_760_2_q_f(:prec:=8)
    chi := MakeCharacter_760_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_760_2_q_f();".
// 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_760_2_q_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_760_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![36, 6, 37, 6, 31, 4, 7, -1, 1]>,<7,R![12, 23, -15, -2, 1]>],Snew);
    return Vf;
end function;