
// Make newform 37.2.f.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_37_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_37_f_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_f_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_f_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, 0, -1, 0, 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_f();"
function MakeCharacter_37_f()
    N := 37;
    order := 9;
    char_gens := [2];
    v := [8];
    // 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_f_Hecke();"
function MakeCharacter_37_f_Hecke(Kf)
    N := 37;
    order := 9;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, -1]];
    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 := [[-1, 0, 0, 0, 0, 1], [0, 1, -1, 1, 0, 0], [0, 0, 0, -2, 2, 0], [0, 0, 2, 1, -1, -2], [0, 0, -1, 0, -1, 0], [-2, 1, 0, 1, 1, -1], [-2, 0, 2, 5, -1, 3], [2, -3, -4, -3, 2, 0], [-1, 1, -1, 1, 0, 0], [0, 2, -2, -5, -2, 2], [1, 2, 2, 0, -2, 0], [0, 1, 2, -2, 4, 2], [-1, -1, 0, 2, 2, 0], [-1, -4, -4, 0, -1, 5], [5, 0, 2, -5, -2, -2], [4, 1, 4, 0, 0, 0], [-1, 3, 3, 4, 0, -4], [-3, 6, 0, -3, -3, -1], [-3, -3, 1, 0, 3, -1], [4, 3, -5, 3, 4, 0], [2, -1, -1, 0, 5, -4], [3, 3, 1, 5, -8, -1], [4, 0, -4, -1, -7, 3], [1, -6, 3, 2, 0, -2], [6, 3, 1, -6, -4, -4], [-2, -2, -1, 2, 3, 3], [0, 3, 4, 5, 4, 3], [-7, 1, 0, 6, 6, -5], [1, 7, -5, 5, -7, -1], [-6, -2, -2, 2, 2, 6], [-8, 2, -4, 4, 0, -4], [-1, 0, 1, 0, -7, -1], [0, -5, 5, -10, 5, -5], [-12, 0, 12, 4, 0, -8], [-3, -10, -10, 0, 1, 9], [-9, 2, -13, 2, -9, 0], [5, 0, -5, -3, 7, 2], [-1, -2, -10, -9, 0, 9], [-2, -3, 16, -3, -2, 0], [4, -4, -5, 5, 4, -4], [5, 0, 0, 0, 8, -8], [6, -6, 0, 0, 0, 13], [8, 1, 1, 0, 10, -11], [0, -10, 2, 8, 2, -10], [-7, 6, -2, 2, -6, 7], [3, 1, 9, -3, -10, -10], [0, -2, 8, 9, 8, -2], [1, 1, 1, 0, -8, 7], [-1, -1, -1, 0, 1, 1], [6, 6, 6, 4, -10, -6], [0, 10, -10, -7, -10, 10], [0, 0, 0, -5, -4, -5], [9, 6, -7, 6, 9, 0], [0, 2, -15, 4, -15, 2], [-1, -2, 8, -8, 2, 1], [3, -1, 0, -2, -2, 9], [8, -8, 15, -8, -7, -7], [-4, -7, 7, -7, 7, 4], [-5, 2, 14, 2, -5, 0], [-6, 2, -4, 2, 0, -2], [4, 4, 0, -8, -8, 17], [3, 5, 17, 5, 3, 0], [0, 5, -11, -3, -11, 5], [-15, -3, -12, 3, 0, -3], [-9, 0, 9, 7, -6, -2], [-6, -14, -4, 2, 0, -2], [12, -8, 3, -3, 8, -12], [1, 2, 0, -3, -3, -18], [5, -9, 1, -5, 8, 8], [4, 10, -3, -7, 0, 7], [-8, 0, 8, 8, 19, 0], [0, -2, -6, 22, -6, -2], [-5, 1, -19, 1, -5, 0], [2, 5, 2, 5, 2, 0], [-11, 0, 11, 13, 3, 2], [-1, 2, 0, -1, -1, -14], [7, -10, 3, -10, 7, 0], [5, 15, -18, -5, 3, 3], [9, -1, -1, 0, -10, 11], [-4, 23, -2, 2, -23, 4], [5, -18, 9, -9, 18, -5], [-13, 0, 11, 13, -11, -11], [-19, 16, 0, 3, 3, -5], [0, -7, 9, 5, 9, -7], [-8, -8, 13, 6, 2, -13], [-15, 11, 11, 0, -11, 0], [-4, -4, -10, -14, 18, 10], [15, -7, 0, -8, -8, 13], [-11, 0, 11, 4, 0, -7], [-7, 5, -1, 5, -7, 0], [15, -6, -2, -15, 8, 8], [-10, -28, 1, -1, 28, 10], [12, -13, -13, 0, 11, 2], [2, 20, -21, -2, 1, 1], [-9, 2, 13, 2, -9, 0], [-6, 15, -5, 1, 0, -1], [7, 0, -7, -12, -12, -5], [2, 2, 0, 2, 2, 0], [0, 0, 8, 7, -7, -8], [21, -3, -7, -21, 10, 10], [0, 9, 17, -13, 17, 9], [-7, -29, -16, 16, 29, 7], [0, 10, -4, 17, -4, 10], [21, 9, -3, -21, -6, -6], [-5, -12, -3, 2, 0, -2], [-21, -6, -12, 9, 0, -9], [4, 4, -10, 1, -5, 10], [-3, 13, 13, 0, -18, 5], [-11, -11, 1, 15, -4, -1], [-11, 0, 11, -11, -10, -22], [2, -22, 10, 8, 0, -8], [22, 0, 5, -17, 0, 17], [-5, 12, -12, 12, -12, 5], [-8, 4, -20, 8, 16, 16], [-10, -17, 6, -6, 17, 10], [20, -1, 16, -16, 1, -20], [-20, 9, -21, 20, 12, 12], [-13, 18, 12, 18, -13, 0], [15, -10, 0, -5, -5, 1], [16, 16, 5, -21, 5, -5], [-10, -10, 8, 0, 10, -8], [-9, -9, -28, 2, 7, 28], [0, 4, -1, 22, -1, 4], [16, 0, -16, -9, -5, 7], [26, -7, 17, -17, 7, -26], [8, 30, 4, -4, -30, -8], [-22, 10, 10, 0, -10, 0], [-4, 35, -6, -2, 0, 2], [12, -17, 0, 5, 5, -2], [-7, -7, 7, 4, 3, -7], [-4, -7, -7, 0, 13, -6], [5, -12, -19, -12, 5, 0], [0, -2, 9, -9, 9, -2], [-7, 0, 7, -1, 16, -8], [-7, 13, 0, 7, 0, -7], [0, -1, 0, 12, 0, -1], [-1, -14, 0, 15, 15, -2], [5, 8, -19, -5, 11, 11], [-7, 0, 7, 9, 15, 2], [5, 5, -42, 1, -6, 42], [-16, -4, -4, -4, -16, 0], [23, 23, 15, -27, 4, -15], [19, 0, -19, -32, 2, -13], [24, -26, 0, 2, 2, -11], [2, -9, 9, 7, 0, -7], [-17, -3, 9, -9, 3, 17], [5, -13, 7, -7, 13, -5], [-4, -4, -4, 0, -23, 27], [0, 3, -2, 40, -2, 3], [-28, 10, -31, 31, -10, 28], [0, 19, -15, -22, -15, 19], [-10, -10, 22, 13, -3, -22], [-5, -5, 21, 1, 4, -21], [-21, 30, 30, 0, -13, -17], [-16, -16, 4, -16, -16, 0], [9, -16, 1, -9, 15, 15], [20, 6, 6, 0, 21, -27], [-4, 18, 0, -14, -14, -19], [18, -21, 11, -11, 21, -18], [10, -2, 23, 13, 0, -13], [-11, -4, -11, 0, 0, 0], [-12, 0, 12, 11, 32, -1], [-1, -1, -29, 1, 0, 29], [10, 0, -10, 16, -14, 26], [7, 28, 1, -6, 0, 6], [-2, 3, 26, 28, 0, -28], [0, 12, -17, -31, -17, 12], [-16, 4, -28, 28, -4, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_37_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_37_2_f_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_f_a(:prec:=6)
    chi := MakeCharacter_37_f();
    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_f_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_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_37_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 3, 12, 19, 15, 6, 1]>],Snew);
    return Vf;
end function;