
// Make newform 6370.2.a.z in Magma, downloaded from the LMFDB on 03 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_6370_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6370_2_a_z();".
// 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_6370_2_a_z();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6370_a();"
function MakeCharacter_6370_a()
    N := 6370;
    order := 1;
    char_gens := [5097, 5981, 1471];
    v := [1, 1, 1];
    // 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;

function MakeCharacter_6370_a_Hecke( : Kf := Kf)
    return MakeCharacter_6370_a();
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 := [[1], [2], [1], [0], [0], [-1], [0], [-2], [6], [6], [4], [2], [0], [8], [0], [12], [-6], [10], [-4], [6], [-2], [8], [-12], [0], [10], [18], [4], [0], [-16], [-6], [2], [-12], [-6], [4], [0], [-10], [-14], [20], [0], [-6], [-12], [-2], [-12], [-22], [-6], [16], [-4], [-8], [-12], [-14], [6], [6], [28], [-24], [-24], [-18], [-18], [-20], [8], [30], [-14], [6], [16], [0], [-8], [-18], [8], [14], [-12], [10], [18], [6], [-8], [-4], [20], [0], [6], [-2], [-30], [-32], [36], [8], [-30], [16], [-8], [0], [6], [-10], [18], [32], [-6], [0], [-16], [36], [-40], [12], [-6], [-30], [34], [-16], [-28], [42], [-6], [-6], [-4], [-2], [12], [-6], [0], [46], [4], [-34], [-42], [-26], [-34], [30], [4], [-24], [24], [36], [22], [26], [18], [36], [10], [18], [8], [24], [-8], [-50], [-16], [-36], [-40], [-16], [36], [40], [18], [40], [-30], [42], [-2], [-24], [14], [-36], [-2], [-36], [10], [-24], [40], [24], [-22], [6], [20], [48], [-28], [-24], [20], [12], [52], [-18], [-60], [6], [-40], [24], [-42], [48], [-16], [-26], [38], [42], [24], [58], [36], [-46], [-32], [-30], [-28], [12], [-56], [10], [8], [-48], [-40], [-6], [-24], [42], [56], [28], [50], [-36], [16], [-36], [20], [-30], [36], [-12], [-58], [44], [42], [48], [24], [-8], [-14], [-26], [-30], [-42], [-56], [-36], [-30], [4], [2], [-42], [62], [48], [12], [-8], [-22], [-66], [36], [42], [-22], [4], [-42], [20], [-18], [26], [60], [36], [-44], [-72], [-4], [-56], [14], [54], [-26], [-24], [40], [-18], [-60], [24], [-36], [-56], [76], [20], [6], [0], [16], [6], [68], [30], [50], [36], [36], [-26], [-42], [-24], [-70], [-44], [-18], [-74], [-10], [-12], [22], [-2], [-6], [28], [-48], [-12], [-4], [24], [-50], [20], [-62], [-4], [-32], [-56], [-36], [44], [-70], [12], [-48], [14], [24], [-26], [76], [60], [-34], [42], [-8], [66], [18], [18], [6], [-30], [8], [54], [-8], [-6], [36], [-8], [76], [6], [-16], [-48], [-76], [38], [24], [-86], [-30], [56], [84], [-48], [18], [-40], [-48], [70], [54], [-54], [46], [-18], [-44], [-22], [-6], [20], [6], [34], [-52], [-44], [-78], [12], [8], [-72], [-80], [60], [-88], [72], [-88], [-78], [4], [-20], [62], [90], [66], [-64], [78], [-36], [-38], [-28], [84], [-66], [-86], [38], [0], [-44], [14], [72], [-72], [-90], [-42], [-24], [-82], [78], [78], [-60], [-38], [-10], [18], [-16], [38], [-60], [16], [12], [-42], [-32], [92], [-18], [-12], [-2], [78], [28], [66], [-6], [56], [6], [-86], [-96], [26], [94], [92], [24], [-10], [6], [-24], [-62], [78], [50], [-56], [42], [-76], [12], [22], [54], [68], [42], [18], [16], [62], [-18], [40], [78], [64], [48], [60], [44], [14], [-42], [-78], [-8], [-30], [72], [18], [70], [-12], [12], [70], [66], [36], [54], [-38], [-48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6370_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6370_2_a_z();".
// 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_6370_2_a_z(:prec:=1)
    chi := MakeCharacter_6370_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6370_2_a_z();".
// 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_6370_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6370_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![0, 1]>,<17,R![0, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;