
// Make newform 370.2.q.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_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_370_2_q_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_370_2_q_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, 0, 0, -1, 0, 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_370_q();"
function MakeCharacter_370_q()
    N := 370;
    order := 12;
    char_gens := [297, 261];
    v := [3, 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_370_q_Hecke();"
function MakeCharacter_370_q_Hecke(Kf)
    N := 370;
    order := 12;
    char_gens := [297, 261];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0, 0, 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, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 1, 1, 0], [0, -1, 1, 0, 1, 2, -1, 0], [2, 0, -2, 0, -1, -1, 1, 0], [-1, 2, 0, -2, 2, -2, 1, 0], [-1, 2, 1, -1, 1, -1, -2, -1], [2, -1, 4, 0, -1, 0, -4, 1], [1, 4, 1, 0, -1, -2, 0, 0], [-1, 1, 0, 1, 0, 2, 0, -3], [-4, 0, -2, 1, 2, 0, 4, -2], [1, 0, -4, 2, -4, 0, 1, 0], [0, 0, 0, -7, 0, 0, 0, 3], [1, 4, -2, 5, 1, 1, 0, -1], [2, -2, -6, -2, 0, -1, 3, 3], [-2, -1, -2, -5, 2, 1, 2, 10], [2, 2, -2, 0, 0, -4, 0, 6], [2, 0, 4, 0, -6, 0, -6, -2], [2, 4, 2, 5, -4, -2, 2, -5], [-2, 2, -2, -5, 0, -1, 2, 5], [-1, -3, 2, 1, 1, 2, -4, 2], [3, 3, 2, -1, -2, -3, -3, 2], [-4, 2, -4, 3, 0, -1, 4, -3], [3, -3, 3, 0, -6, 6, -6, -8], [1, -4, 1, 8, 0, 0, 0, -4], [3, -1, 0, 1, -6, 9, 6, 8], [-1, -3, 0, 3, 2, 0, 7, -3], [0, 3, 0, -3, 0, -8, 5, -5], [0, 0, 0, 3, -7, -4, -7, 3], [0, 5, -5, 0, 5, -10, 5, -4], [8, 7, 0, -2, -4, 2, 0, -7], [-7, -5, 2, -4, 9, 0, -9, 2], [5, -2, -8, 0, 3, 4, 3, 3], [3, 14, 5, 0, -5, -14, -3, 0], [0, -2, -3, 5, 2, 5, -3, -2], [1, -6, 0, 6, -2, 1, -11, -5], [-1, -1, 12, -4, -1, -3, 0, 3], [-4, -7, -4, -4, 0, 0, 0, 2], [-16, 5, 1, -1, 8, 1, -1, -5], [2, -4, 0, 2, 2, 6, 0, -6], [-8, -2, -4, 10, 4, 0, -4, -5], [-1, -5, -10, -1, 10, 5, 1, 2], [-9, 1, -1, -4, 9, 3, 2, 3], [5, -2, -3, 5, 3, 2, -5, -10], [-8, -7, 2, -7, 0, -5, -1, 12], [0, 17, 1, 0, 1, 0, -1, 0], [-8, -2, -7, 5, 7, 2, 8, -10], [5, -1, 0, -1, 0, -6, 0, 7], [5, -8, 3, 5, 3, -8, 5, 0], [2, -6, -12, -9, 2, -3, 0, 3], [-6, -5, 8, -11, -6, -6, 0, 6], [-5, -5, -17, -5, 17, 5, 5, 10], [-7, 5, -1, 0, 8, -10, 8, 0], [-1, 0, 1, -11, 1, 1, 0, -11], [-3, 2, 7, 5, 7, 2, -3, 0], [-24, 3, 1, 2, 12, -2, -1, -3], [7, 0, -7, -3, -7, 1, 0, -3], [6, -4, 0, 4, -12, -2, -4, -6], [0, -4, 1, 6, -11, 6, 1, -4], [18, -3, -4, 1, -18, 2, 8, 2], [-3, 5, 9, 2, 12, 0, -12, -1], [7, 5, -4, -2, 7, -7, 0, 7], [9, -2, 9, 6, -9, 1, 0, -6], [7, -4, -7, -9, -7, -4, 7, 0], [3, -1, 3, 0, -6, 2, -6, 16], [0, 9, 8, 0, -3, 0, 8, 9], [-2, 0, 6, 0, -4, 0, -4, -11], [0, 8, 6, 8, 6, 0, -6, -4], [-5, 2, -5, 1, 0, -1, 5, -1], [10, -7, 14, -7, 0, 14, -7, -7], [-2, -8, -5, -9, 1, 9, 5, 8], [8, -8, -9, -2, -4, 2, 9, 8], [6, 0, 0, 0, -12, 2, 2, 2], [-3, 0, 3, -5, -9, -5, -12, -5], [3, 0, -3, 0, -3, -17, 0, 0], [0, 5, -5, -11, 0, 11, 5, -5], [-6, -9, -3, -4, 6, 13, 6, 13], [2, -2, 2, 0, -18, 1, 16, 0], [3, 4, 4, -3, 4, 4, 3, 0], [11, -7, -6, 1, -6, -7, 11, 0], [5, -3, 13, 2, 8, 0, -8, -1], [4, 11, 9, 3, -2, -3, -9, -11], [13, -2, -2, 11, -2, -2, 13, 0], [0, 2, 0, -9, -10, -1, 10, 9], [3, -3, 7, 1, 7, -3, 3, 0], [-3, 0, 3, 5, 3, 5, 0, 5], [-4, -10, -4, 0, 4, 10, 4, 0], [5, -2, 5, 11, 5, 1, -10, -11], [6, -2, -19, -11, 6, -9, 0, 9], [-10, -8, -5, 4, 5, 0, -5, -2], [5, 17, 4, -11, -5, -6, -8, -6], [4, -4, 0, 4, -8, 8, 22, 4], [-10, 5, 3, 0, 7, -10, 7, 19], [0, -2, 0, 2, 0, 11, 15, 9], [-12, -11, -4, -11, 0, -11, 2, 22], [3, 0, -3, -6, -11, 26, -8, -6], [0, 10, 3, -6, -15, -6, 3, 10], [0, -4, 0, -9, 18, -9, 0, -4], [-2, -14, -4, -7, -2, 7, 0, -7], [-12, 3, -3, 12, 12, -15, 6, -15], [-1, -23, -2, 6, 2, 23, 1, -12], [5, 4, -22, 4, 0, 8, 11, -12], [0, -9, 10, 1, 12, 1, 10, -9], [-7, 6, -4, 6, 0, -23, 2, 17], [-18, -6, -11, 8, 11, 6, 18, -16], [0, -15, 10, 9, 17, 9, 10, -15], [4, -12, 22, -2, -2, 2, -22, 12], [-7, 3, 8, -7, 7, 4, -16, 4], [-1, -7, 13, 5, 13, -7, -1, 0], [-7, 10, -7, 0, -7, -10, 0, 10], [0, 9, 6, -2, 17, -2, 6, 9], [0, 0, 9, -10, -5, -10, 9, 0], [-4, -9, -4, 12, 0, 0, 0, -6], [8, -8, -10, 0, 2, 16, 2, 22], [23, -11, -8, -11, 0, 6, 4, 5], [7, 3, 10, 0, -17, -6, -17, 8], [0, 4, 8, 0, -9, 0, 8, 4], [14, 11, 0, -11, -28, -5, -20, 6], [-5, 3, -16, 1, 5, -4, 32, -4], [0, 13, 0, 12, 0, -1, 0, 1], [8, 0, -2, -2, -8, 2, 4, 2], [-7, 2, -7, 24, 5, -1, 2, -24], [-11, -10, -11, 15, 7, 5, 4, -15], [15, -2, 0, 17, 0, -2, 15, 0], [0, 9, 5, 5, -4, 5, 5, 9], [16, 11, -2, -8, -8, 8, 2, -11], [-14, 17, -12, -2, 2, 0, -2, 1], [2, 5, -18, -21, -18, 5, 2, 0], [10, 10, 6, 29, -5, -29, -6, -10], [-7, -12, 1, -11, 7, 23, -2, 23], [1, 0, -1, 4, -9, -24, -8, 4], [-4, 5, -18, 5, 0, 10, 9, -15], [3, 0, -3, 4, 5, 30, 8, 4], [2, 1, 0, -1, -4, 4, -35, 5], [-16, -3, 8, 1, 8, -1, -8, 3], [6, 2, 4, 6, -3, -6, -4, -2], [-9, 7, 16, 3, -16, -7, 9, -6], [19, 8, 19, -13, -5, -4, -14, 13], [-13, 5, 4, 13, -4, -5, 13, -26], [0, -10, -2, 10, -26, 10, -2, -10], [-6, 0, 6, 10, 18, -18, 12, 10], [-5, 28, 7, -7, 5, -21, -14, -21], [-2, 2, -14, -2, 2, 0, 28, 0], [-9, 2, 1, 0, 8, -4, 8, -25], [5, 16, 3, 21, 5, 5, 0, -5], [12, 0, -3, -16, -15, 0, 15, 8], [0, -11, 11, 23, -4, 23, 11, -11], [-18, -4, -3, -9, 9, 9, 3, 4], [10, 6, 0, -6, -20, -10, -13, -4], [-9, 0, 5, -21, 5, 0, -9, 0], [0, 12, 12, 0, -12, -24, -12, -4], [15, 16, 14, -9, -14, -16, -15, 18], [10, -8, -5, 0, 10, 8, 0, -8], [8, 17, -21, 22, 8, 5, 0, -5], [2, 1, -11, 14, -11, 1, 2, 0], [-12, 18, -22, 3, -12, -15, 0, 15], [4, -9, 5, -4, -5, 9, -4, 8], [16, 1, -7, 10, 7, -1, -16, -20], [-6, -10, 1, 18, 6, -8, -2, -8], [-9, -8, 6, 0, 15, 0, -15, 0], [0, -11, 7, 19, 3, 19, 7, -11], [-2, -15, 11, -24, 1, 24, -11, 15], [4, -4, -22, 12, -26, 0, 26, -6], [8, 24, -14, 20, 8, -4, 0, 4], [0, -9, 12, 1, 15, 1, 12, -9], [0, 9, -4, -12, 26, -12, -4, 9], [14, 19, -10, 6, -24, 0, 24, -3], [3, -3, -7, -16, -7, -3, 3, 0], [6, -1, -15, -16, -3, 16, 15, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_q_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_370_2_q_c(:prec:=8)
    chi := MakeCharacter_370_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_370_2_q_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_370_2_q_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 8, -4, 0, 14, -24, 14, -4, 1]>],Snew);
    return Vf;
end function;