
// Make newform 37.2.e.a in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_37_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_37_e_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_37_2_e_a();".
// 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_37_2_e_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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, -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_37_e();"
function MakeCharacter_37_e()
    N := 37;
    order := 6;
    char_gens := [2];
    v := [5];
    // 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_37_e_Hecke();"
function MakeCharacter_37_e_Hecke(Kf)
    N := 37;
    order := 6;
    char_gens := [2];
    char_values := [[0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+1)]];
    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, 1, 0, 0], [0, -1, 1, -1], [-2, -2, 1, 2], [0, 2, 0, 2], [-3, 2, 0, -1], [4, 0, -2, 0], [1, -2, 1, 0], [-2, -3, 1, 3], [-1, 0, 2, 1], [-1, 0, 2, 2], [-3, 0, 6, -3], [4, 0, -7, 0], [0, 0, -3, 0], [2, 0, -4, 0], [3, 6, 0, -3], [6, 2, -6, -4], [-2, 10, -2, 0], [-2, -6, 1, 6], [0, 1, 9, 1], [0, 0, -6, 0], [0, 0, 0, 0], [-6, -9, 3, 9], [3, -3, -3, 6], [3, -4, 3, 0], [1, 0, -2, -6], [-9, 0, 0, 0], [-4, 0, 8, -6], [0, -4, 6, -4], [3, 12, 3, 0], [2, 8, 2, 0], [7, -3, -7, 6], [-9, -5, -9, 0], [-3, 4, 0, -2], [3, 9, -3, -18], [9, 4, 0, -2], [0, 9, -1, 9], [-3, -10, 3, 20], [6, -12, 6, 0], [8, -4, -4, 4], [-3, 4, 3, -8], [2, 0, -4, -2], [0, 4, -3, 4], [-3, 0, 6, -1], [-5, 0, 10, -6], [-21, 4, 21, -8], [4, 0, -8, -18], [-10, 0, 0, 0], [-11, -18, 0, 9], [14, -11, -7, 11], [0, 0, -7, 0], [3, 16, 0, -8], [4, -10, 4, 0], [-4, 12, 2, -12], [16, 0, -32, -2], [9, -4, 9, 0], [0, 6, 18, 6], [18, 4, 0, -2], [3, -5, -3, 10], [2, 12, -1, -12], [3, -28, 3, 0], [12, 0, -6, 0], [0, 8, -15, 8], [-4, 12, 0, -6], [-6, 20, -6, 0], [-9, 0, -9, 0], [9, -4, -9, 8], [12, -12, 12, 0], [0, 6, -19, 6], [1, 0, -2, 11], [-15, -4, 15, 8], [-1, -14, -1, 0], [6, -12, 0, 6], [0, -6, 10, -6], [0, -2, -21, -2], [-12, -2, 12, 4], [-16, 2, 8, -2], [6, -26, -3, 26], [3, 28, 0, -14], [6, 0, -12, -16], [5, 0, 5, 0], [0, 10, 0, -20], [-1, 0, 2, 18], [-26, -11, 13, 11], [27, 0, 0, 0], [-4, -18, 2, 18], [30, -4, 0, 2], [-28, 8, 14, -8], [-14, 24, 7, -24], [2, 20, 2, 0], [-4, 12, 2, -12], [-10, 0, 20, 14], [-10, 22, -10, 0], [17, 0, -34, 3], [33, 2, 0, -1], [18, 3, -9, -3], [-15, -17, -15, 0], [-15, -12, 15, 24], [0, 4, 0, 4], [22, 3, -11, -3], [1, 0, -2, -6], [-3, 0, 6, 9], [-1, 16, -1, 0], [9, 0, -18, 19], [-13, 0, 26, -14], [9, -13, -9, 26], [4, -6, 4, 0], [10, 25, -5, -25], [-3, -24, 0, 12], [0, -11, -27, -11], [-21, 6, 21, -12], [5, 3, 5, 0], [7, 6, -7, -12], [-6, -18, 6, 36], [-14, -24, 0, 12], [-1, -21, -1, 0], [-3, -12, 3, 24], [5, 0, -10, -3], [4, -40, -2, 40], [6, 8, -3, -8], [0, 2, -6, 2], [6, 18, -3, -18], [0, 16, 12, 16], [-27, 20, 0, -10], [-4, -38, -4, 0], [3, 11, -3, -22], [-8, -8, -8, 0], [0, 0, 0, -30], [3, 17, -3, -34], [-16, -12, 8, 12], [0, 12, -8, 12], [-29, -6, 0, 3], [0, 1, 45, 1], [-30, -24, 0, 12], [21, -6, 21, 0], [21, 10, -21, -20], [-2, 0, 4, 24], [0, -26, 3, -26], [-10, -12, 0, 6], [0, 8, 0, 0], [16, -20, -8, 20], [0, 14, 6, 14], [0, 18, -18, 18], [29, 15, -29, -30], [-8, 16, 4, -16], [2, 48, 2, 0], [21, -5, -21, 10], [7, -30, 7, 0], [1, 0, -2, 22], [2, 0, -4, -12], [-42, -2, 42, 4], [-15, -24, 0, 12], [0, 10, -15, 10], [-10, 27, 5, -27], [3, -30, 0, 15], [-2, -3, 1, 3], [-13, 0, 26, -31], [11, 0, -22, -15], [0, -2, 27, -2], [23, 6, -23, -12], [-27, -18, 27, 36], [6, 38, 6, 0], [18, 8, -18, -16], [24, -12, -12, 12], [3, -3, -3, 6], [-14, -8, -14, 0], [-45, 7, 45, -14], [-6, 0, 12, -36], [-2, -24, -2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_37_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_37_2_e_a();".
// 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_37_2_e_a(:prec:=4)
    chi := MakeCharacter_37_e();
    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_37_2_e_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_37_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_37_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;