
// Make newform 370.2.h.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_h_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_h_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_h_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];
        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_h();"
function MakeCharacter_370_h()
    N := 370;
    order := 4;
    char_gens := [297, 261];
    v := [1, 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_370_h_Hecke();"
function MakeCharacter_370_h_Hecke(Kf)
    N := 370;
    order := 4;
    char_gens := [297, 261];
    char_values := [[0, 0, 1, 0], [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 := [[1, 0, 0, 0], [0, 2, 0, 0], [0, -1, 0, -2], [2, -1, 2, 0], [0, -2, 1, -2], [-2, -3, 0, 3], [0, 2, 1, 2], [3, 0, -3, 4], [-4, -1, 0, 1], [6, -1, 6, 0], [-2, 0, 2, -3], [0, 1, 0, 6], [0, 0, 7, 0], [-7, 0, 0, 0], [-4, 6, -4, 0], [0, 0, 0, 3], [4, 0, -4, -4], [10, 0, -10, -1], [3, 0, -3, -6], [-6, -4, 0, 4], [-6, 0, -6, 0], [-4, 0, 4, -8], [2, 0, -2, -4], [-6, 2, -6, 0], [0, 0, -7, 0], [0, 5, -4, 5], [0, 11, 2, 11], [-2, 10, -2, 0], [-4, 0, 4, 7], [0, -2, 1, -2], [4, 2, 4, 0], [-1, 0, 1, 2], [7, -2, 7, 0], [9, 2, 0, -2], [0, 3, -4, 3], [0, 0, -12, 0], [8, 3, 8, 0], [0, -4, 9, -4], [0, -2, -4, -2], [6, 17, 6, 0], [6, -12, 6, 0], [2, -4, 0, 4], [-2, -5, -2, 0], [-12, 6, 0, -6], [-2, 0, -2, 0], [-6, -14, -6, 0], [-9, 0, 0, 0], [16, 0, -16, -3], [0, 6, 1, 6], [0, -1, 0, -1], [7, 10, 7, 0], [-12, 0, 12, 7], [5, 6, 5, 0], [-8, 0, 8, -14], [0, -2, 16, -2], [4, -15, 4, 0], [0, 14, 2, 14], [4, 5, 0, -5], [-12, -14, 0, 14], [4, 0, 4, 0], [0, 0, -28, 0], [6, 0, -6, -11], [-8, 0, 8, 14], [-4, 0, 4, -11], [-12, -8, 0, 8], [-12, 0, 12, -5], [4, 2, 4, 0], [0, 0, 0, -8], [-6, 8, 0, -8], [0, -10, -10, -10], [0, -4, -13, -4], [0, 23, 2, 23], [-6, -27, -6, 0], [8, -4, 8, 0], [0, 0, -18, 0], [-18, -1, 0, 1], [-4, 0, 4, 11], [0, 0, 0, -12], [0, 0, 0, -10], [0, -28, 0, 0], [0, -12, 10, -12], [-14, 0, 14, -16], [0, 0, 0, -29], [-9, 0, 9, 20], [-10, -11, -10, 0], [11, 8, 11, 0], [3, 0, -3, 32], [0, -8, -15, -8], [14, 5, 14, 0], [-4, -24, 0, 24], [0, -8, -9, -8], [0, 0, 0, -22], [0, -12, 10, -12], [-2, -14, 0, 14], [6, -6, 6, 0], [-34, -1, 0, 1], [-14, -9, 0, 9], [0, 16, -5, 16], [38, -4, 0, 4], [-8, 28, -8, 0], [-25, 2, 0, -2], [2, -15, 0, 15], [7, 22, 0, -22], [13, 16, 13, 0], [27, -14, 0, 14], [0, -4, 16, -4], [21, 18, 0, -18], [-12, 0, 12, -20], [0, -3, -38, -3], [-25, -6, 0, 6], [4, 27, 0, -27], [10, -3, 10, 0], [15, 0, -15, 18], [-7, 28, 0, -28], [-4, 0, 4, -21], [-3, -20, 0, 20], [0, 6, 33, 6], [16, -8, 0, 8], [0, -4, 14, -4], [6, -10, 0, 10], [-4, 0, 4, -23], [-19, 0, 19, 12], [22, 0, -22, -12], [1, -8, 0, 8], [0, -12, -7, -12], [10, 8, 10, 0], [-10, 0, 10, -1], [0, 10, -10, 10], [-28, -10, 0, 10], [-2, -35, -2, 0], [-9, -14, 0, 14], [-10, -7, -10, 0], [0, 17, 8, 17], [0, -4, 22, -4], [0, -8, -23, -8], [22, 4, 22, 0], [8, 0, -8, 3], [12, -6, 12, 0], [-6, 32, 0, -32], [-23, 24, -23, 0], [-8, 12, 0, -12], [-26, 11, 0, -11], [-12, 0, 12, -24], [0, -2, 1, -2], [0, -35, 0, 0], [10, 25, 0, -25], [0, 2, 30, 2], [0, -10, -1, -10], [21, 0, -21, 22], [20, 0, -20, 1], [-2, 17, -2, 0], [0, 6, -1, 6], [0, -16, -31, -16], [-2, 0, 2, 51], [0, 0, -14, 0], [26, 2, 26, 0], [12, -32, 12, 0], [-15, -4, 0, 4], [6, 2, 6, 0], [28, 5, 0, -5], [0, 14, 11, 14], [-26, -8, -26, 0], [0, 24, -20, 24], [-17, 26, 0, -26], [41, 6, 0, -6], [-20, 15, -20, 0], [24, 0, -24, 21], [0, -18, 12, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_h_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_h_c(:prec:=4)
    chi := MakeCharacter_370_h();
    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_h_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_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;