
// Make newform 372.2.j.b in Magma, downloaded from the LMFDB on 06 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_372_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_372_j_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_372_2_j_b();".
// 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_372_2_j_b();".
// 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 := [361, 209, 194, 43, 29, 1, 6, -3, 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], [55384240, 40440235, -47195553, 7497629, 2192108, -1074875, 164686, 16228], [-65903001, -49033872, -6474129, 10405212, -17987557, 3198495, 348813, -294510], [5595690, -228969, 426372, 325779, 997158, -931213, 261345, -28143], [-23973516, -5877897, -5142509, -2983130, -989293, 453785, -64855, -45075], [-119034259, 98040673, -8897747, 28199801, -8103878, 9536834, -4292904, 967254], [23665184, 8614349, 7105246, 462427, 1359136, -339265, 3158, 56305]];
    Rf_basisdens := [1, 1, 205159891, 205159891, 10797889, 10797889, 205159891, 10797889];
    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_372_j();"
function MakeCharacter_372_j()
    N := 372;
    order := 5;
    char_gens := [187, 125, 313];
    v := [5, 5, 3];
    // 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_372_j_Hecke();"
function MakeCharacter_372_j_Hecke( : Kf := false)
    N := 372;
    order := 5;
    char_gens := [187, 125, 313];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 1, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 1, 0, -1, 0, -1, 0], [-2, -1, 0, 0, 1, 1, -2, 0], [-1, 0, 0, 0, 1, 1, -1, 1], [0, -1, 2, 2, 1, 0, -1, 0], [-2, -3, 2, 1, 1, 0, -1, -1], [0, 0, -1, 0, 0, 0, 0, 1], [0, 1, 0, -2, -2, 0, 2, 2], [5, 2, -6, -5, 0, -2, 0, 1], [-3, 1, 4, 5, -1, -2, -4, -1], [-2, 0, 2, 0, 0, 2, -2, 0], [4, -1, -2, -4, 0, 1, 0, 2], [-3, -1, 1, 3, 0, 1, 0, 0], [0, 0, -4, -4, 0, 1, 4, 1], [5, 0, -5, -7, 0, 0, 7, 0], [0, -1, -1, -1, 1, -2, 6, -2], [0, 0, 1, 0, 0, -4, -1, 0], [8, 0, -4, 0, 0, 0, 4, 0], [5, -1, -5, -6, -3, 0, 6, 3], [1, 1, 0, 4, 1, 1, 1, 2], [-1, 3, 1, -4, -3, 0, 4, 3], [-4, 1, -1, 4, 0, -1, 0, 0], [8, 1, 0, -4, -3, -3, 8, -2], [-2, 1, 0, -4, 1, 1, -2, 2], [5, 1, -5, -1, -1, 0, 1, 1], [1, -2, 1, -1, 0, 2, 0, 0], [-9, 6, 9, 6, -1, 0, -6, 1], [0, 3, -4, -4, -3, -2, 10, -2], [-2, -4, 0, -6, 5, 5, -2, 1], [0, -4, -8, -8, 4, 0, 1, 0], [-3, -3, 0, -3, 3, 3, -3, 0], [0, -1, 7, 7, 1, -2, -8, -2], [0, -1, 4, 4, 1, -1, 0, -1], [10, 0, -5, 0, -1, -2, 5, 0], [6, -1, 0, -5, -2, -2, 6, -3], [3, -1, 2, -3, 0, 1, 0, 1], [-10, 0, 10, 15, -1, 0, -15, 1], [3, -2, -15, -3, 0, 2, 0, 2], [0, -2, 1, 1, 2, 3, -8, 3], [-4, -4, 0, 9, 1, 1, -4, -3], [-13, 0, 4, 0, -3, 5, -4, 0], [5, 0, 6, 0, 5, 1, -6, 0], [-8, 4, 0, -6, -1, -1, -8, 3], [4, -4, 0, 16, 2, 2, 4, -2], [0, -3, -1, -1, 3, 8, -6, 8], [-8, 0, 18, 0, 3, 4, -18, 0], [6, 0, 6, 0, 2, -1, -6, 0], [-6, -1, 8, 6, 0, 1, 0, -6], [-2, 5, -4, 2, 0, -5, 0, 0], [0, -1, -1, -1, 1, 3, -6, 3], [14, 1, -14, -19, 2, 0, 19, -2], [-7, 0, 0, 16, 3, 3, -7, 3], [0, 1, 8, 8, -1, 3, 1, 3], [1, 9, -1, -6, -6, 0, 6, 6], [0, 0, 7, 7, 0, 1, 14, 1], [-3, 0, -12, 3, 0, 0, 0, -4], [16, -2, -16, -15, -4, 0, 15, 4], [1, 6, 4, -1, 0, -6, 0, 4], [10, -3, -20, -10, 0, 3, 0, 0], [5, -1, 0, -2, 2, 2, 5, 1], [-3, -2, 3, 13, -1, 0, -13, 1], [0, 4, 5, 5, -4, 0, -7, 0], [-11, 0, 12, 0, 8, -4, -12, 0], [0, 6, -3, -3, -6, -1, 6, -1], [-2, -8, 0, 4, 1, 1, -2, -7], [-14, -5, 19, 14, 0, 5, 0, -4], [14, 2, 0, -14, -2, -2, 14, 0], [17, 0, -9, 0, -3, 7, 9, 0], [12, -6, -12, -5, 4, 0, 5, -4], [2, -6, -17, -2, 0, 6, 0, -3], [0, 1, -18, -18, -1, -1, 20, -1], [8, 0, -10, 0, -5, -10, 10, 0], [-8, 0, 10, 0, 3, 5, -10, 0], [0, 16, 0, -5, -11, -11, 0, 5], [17, 3, 0, -14, -1, -1, 17, 2], [-9, 13, 9, 1, -8, 0, -1, 8], [-9, 0, 4, 0, 0, -7, -4, 0], [-5, -6, 8, 5, 0, 6, 0, 1], [9, 0, 2, 0, -7, -1, -2, 0], [0, -4, -6, -6, 4, 1, 14, 1], [0, -4, 0, 0, 4, 10, -3, 10], [0, 6, 9, 9, -6, 0, -7, 0], [-16, 0, 0, 0, -5, 1, 0, 0], [7, 0, -5, 0, -4, 3, 5, 0], [1, -15, -1, 4, 11, 0, -4, -11], [0, 4, -2, -2, -4, 0, 27, 0], [-8, 7, 8, 5, 3, 0, -5, -3], [2, -4, 0, 3, 5, 5, 2, 1], [-3, -5, 11, 3, 0, 5, 0, 1], [-13, 3, 6, 13, 0, -3, 0, 5], [4, -6, -4, -2, 0, 0, 2, 0], [-1, 4, 1, 10, -8, 0, -10, 8], [6, 0, -18, 0, -6, -2, 18, 0], [0, 1, 12, 12, -1, -8, -3, -8], [-4, 7, 0, -23, -3, -3, -4, 4], [0, -2, -15, -15, 2, -1, -2, -1], [-1, 0, 1, 0, 2, -3, -1, 0], [0, -8, 0, 0, 9, 9, 0, 1], [14, 2, -14, 9, -4, 0, -9, 4], [-8, 4, 0, -21, -3, -3, -8, 1], [16, 0, 5, 0, 0, 5, -5, 0], [14, 0, 6, 0, -10, -2, -6, 0], [16, 9, 0, -13, -9, -9, 16, 0], [0, -2, 24, 24, 2, -2, -16, -2], [5, -2, -4, -5, 0, 2, 0, -5], [4, -3, -17, -4, 0, 3, 0, 7], [-11, 5, 0, -12, -2, -2, -11, 3], [4, 0, 9, -4, 0, 0, 0, 6], [2, -2, -15, -2, 0, 2, 0, 4], [0, -3, 9, 9, 3, -1, 15, -1], [-10, -6, 0, 28, 5, 5, -10, -1], [0, -4, 5, 5, 4, -1, -25, -1], [2, 0, -4, 0, 9, 4, 4, 0], [-20, -14, 0, -3, 9, 9, -20, -5], [-26, 2, 16, 26, 0, -2, 0, -1], [3, -10, -3, 3, 2, 0, -3, -2], [-17, 2, 0, 16, -6, -6, -17, -4], [-9, 0, 18, 9, 0, 0, 0, 6], [1, 8, -1, -10, -12, 0, 10, 12], [-14, 9, 6, 14, 0, -9, 0, -1], [6, -2, -6, -5, 2, 0, 5, -2], [9, 0, -19, 0, -2, 5, 19, 0], [7, 0, 11, 0, -10, -2, -11, 0], [25, 9, -25, -19, -6, 0, 19, 6], [-7, -7, 5, 7, 0, 7, 0, -3], [14, 11, 0, 12, -8, -8, 14, 3], [25, 0, -5, 0, 2, -11, 5, 0], [1, 6, -1, -13, -4, 0, 13, 4], [21, 8, 0, -16, -6, -6, 21, 2], [-26, 0, -4, 0, 4, 6, 4, 0], [10, 0, -10, 0, -11, -5, 10, 0], [1, -11, 0, 11, 5, 5, 1, -6], [0, -1, 12, 12, 1, -3, 14, -3], [7, -7, -7, -3, 2, 0, 3, -2], [17, 0, -28, 0, -1, 1, 28, 0], [-20, 6, 4, 20, 0, -6, 0, -16], [-19, -8, 20, 19, 0, 8, 0, -13], [-18, 4, 18, -14, 4, 0, 14, -4], [0, 0, -17, -17, 0, 3, 11, 3], [9, 0, -7, 0, 4, -8, 7, 0], [0, 6, 8, 8, -6, -5, 17, -5], [-5, 8, 5, 8, 3, 0, -8, -3], [-12, 2, -18, 12, 0, -2, 0, -1], [10, 6, -10, -8, -3, 0, 8, 3], [-13, 4, -7, 13, 0, -4, 0, -8], [0, 5, -6, -6, -5, -13, 7, -13], [-18, 0, 0, 9, 1, 1, -18, 1], [16, -6, -16, -4, 3, 0, 4, -3], [-22, 0, 7, 0, 3, 0, -7, 0], [15, -6, -15, -19, -9, 0, 19, 9], [0, -2, -4, -4, 2, 16, -3, 16], [0, 10, 4, 4, -10, 2, 18, 2], [12, -1, -9, -12, 0, 1, 0, 11], [8, -15, -8, 0, 1, 0, 0, -1], [-3, 5, -14, 3, 0, -5, 0, 2], [12, -17, 0, 1, 12, 12, 12, -5], [-11, 0, 15, 0, 10, -10, -15, 0], [15, -3, 0, -20, -2, -2, 15, -5], [4, 9, 0, 8, -4, -4, 4, 5], [-30, 2, 30, 47, 6, 0, -47, -6], [17, -13, -17, -7, -1, 0, 7, 1], [-10, 0, 9, 0, 13, 12, -9, 0], [14, -1, 13, -14, 0, 1, 0, -3], [0, -7, 18, 18, 7, 2, -11, 2], [0, -4, 0, 0, 4, 0, 0, -4], [21, 0, -10, 0, 8, -3, 10, 0], [1, 0, -2, 0, 1, 2, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_372_j_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_372_2_j_b();".
// 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_372_2_j_b(:prec:=8)
    chi := MakeCharacter_372_j();
    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_372_2_j_b();".
// 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_372_2_j_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_372_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 17, -15, -1, 1]>],Snew);
    return Vf;
end function;