
// Make newform 6272.2.a.h in Magma, downloaded from the LMFDB on 30 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_6272_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_6272_2_a_h();".
// 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_6272_2_a_h();".
// 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_6272_a();"
function MakeCharacter_6272_a()
    N := 6272;
    order := 1;
    char_gens := [4607, 3333, 4609];
    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_6272_a_Hecke( : Kf := Kf)
    return MakeCharacter_6272_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 := [[0], [2], [2], [0], [2], [2], [2], [2], [4], [6], [0], [-10], [6], [-6], [8], [6], [14], [2], [-10], [12], [-14], [-8], [-6], [2], [2], [-6], [4], [2], [6], [2], [16], [-6], [10], [-10], [-18], [4], [18], [-2], [20], [18], [6], [2], [-16], [-2], [14], [4], [22], [0], [18], [-14], [-18], [-24], [2], [-18], [-18], [12], [10], [-8], [6], [-18], [6], [-14], [18], [28], [-10], [6], [-14], [-14], [18], [10], [-18], [-4], [8], [-10], [2], [0], [-10], [-6], [30], [-10], [26], [-34], [40], [-30], [-36], [-6], [-34], [-6], [10], [-8], [-14], [0], [-20], [10], [22], [-20], [-14], [22], [14], [-34], [38], [-2], [18], [26], [-38], [-2], [-34], [-18], [-12], [-30], [16], [-34], [-2], [46], [-44], [30], [42], [-12], [-42], [6], [34], [-2], [-22], [42], [-6], [-2], [22], [-24], [12], [-6], [-18], [44], [8], [46], [-10], [34], [-54], [-22], [18], [-6], [-18], [-10], [-28], [-22], [-14], [36], [26], [6], [-50], [-32], [22], [46], [-34], [-36], [-38], [-24], [36], [-30], [-46], [-38], [14], [58], [28], [38], [30], [-20], [16], [-54], [-34], [34], [-6], [42], [-20], [-34], [-40], [18], [-30], [-26], [52], [-46], [-32], [-30], [30], [2], [-24], [18], [46], [2], [-38], [-64], [14], [-54], [14], [26], [-34], [34], [50], [14], [-2], [-12], [-2], [40], [10], [14], [30], [42], [-64], [14], [14], [-2], [-34], [-62], [-44], [22], [-12], [-10], [-24], [2], [-12], [6], [46], [12], [30], [24], [-62], [22], [34], [-16], [-28], [26], [-74], [2], [32], [-2], [38], [8], [34], [-2], [-6], [-12], [54], [-10], [20], [14], [14], [-52], [64], [-54], [-22], [-24], [-42], [14], [-20], [-46], [-38], [-74], [46], [30], [42], [-42], [-64], [-2], [-70], [-22], [66], [-6], [-58], [-30], [50], [-26], [2], [-50], [-46], [-16], [-66], [76], [10], [-18], [-18], [66], [-32], [-4], [12], [34], [22], [40], [-2], [6], [-4], [-34], [86], [18], [-66], [-66], [-10], [66], [-64], [-14], [54], [58], [22], [30], [-24], [6], [26], [-66], [50], [-14], [-60], [-10], [56], [30], [30], [86], [-52], [18], [-22], [64], [-66], [-46], [-70], [-22], [42], [-16], [-54], [-62], [30], [-10], [32], [-82], [62], [6], [64], [42], [-70], [30], [70], [46], [-30], [-56], [-34], [74], [42], [-20], [30], [-2], [2], [18], [24], [82], [-14], [-22], [30], [-24], [-50], [-14], [-48], [6], [30], [36], [38], [-78], [72], [-14], [-14], [-34], [-94], [-20], [-70], [-58], [-50], [24], [54], [-52], [38], [-22], [0], [14], [66], [86], [58], [-6], [68], [-78], [-14], [36], [-8], [58], [82], [32], [-62], [42], [-46], [58], [-76], [10], [80], [86], [42], [54], [-54], [30], [24], [102], [66], [-60], [-2], [30], [74], [-70], [30], [-42], [-94], [-50], [62], [98], [-32], [52], [66], [12], [62], [90], [56], [-66], [-90], [18], [14], [30], [102], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6272_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6272_2_a_h();".
// 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_6272_2_a_h(:prec:=1)
    chi := MakeCharacter_6272_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_6272_2_a_h();".
// 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_6272_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6272_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<5,R![-2, 1]>,<11,R![-2, 1]>,<13,R![-2, 1]>,<23,R![-4, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;