
// Make newform 6026.2.a.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6026_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_6026_2_a_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_6026_2_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6026_a();"
function MakeCharacter_6026_a()
    N := 6026;
    order := 1;
    char_gens := [787, 4325];
    v := [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_6026_a_Hecke(Kf)
    return MakeCharacter_6026_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+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], [2], [-2], [-6], [2], [1], [2], [-8], [4], [6], [2], [0], [0], [-4], [12], [2], [0], [6], [-4], [-6], [18], [2], [-18], [8], [-2], [4], [10], [8], [-1], [18], [12], [4], [8], [4], [-16], [-16], [2], [0], [8], [16], [10], [10], [4], [28], [0], [-18], [0], [-6], [0], [10], [-6], [-18], [-12], [10], [16], [2], [22], [-14], [-12], [16], [24], [26], [-30], [12], [-6], [-32], [-22], [-30], [20], [8], [4], [-10], [36], [16], [22], [-6], [2], [26], [32], [0], [26], [0], [-16], [6], [-10], [-14], [32], [6], [-16], [16], [0], [-24], [28], [-14], [10], [26], [-14], [28], [-12], [18], [-18], [18], [30], [-12], [-34], [48], [38], [-32], [-12], [30], [-18], [-20], [-34], [-2], [-8], [38], [6], [24], [22], [-4], [12], [12], [44], [44], [8], [-28], [-4], [16], [40], [24], [-12], [42], [-18], [-48], [18], [-28], [-6], [4], [-10], [48], [22], [-14], [4], [-2], [10], [48], [-24], [-38], [-42], [-20], [8], [50], [-24], [4], [-18], [-14], [36], [40], [-2], [-24], [30], [-34], [4], [-24], [-46], [18], [6], [-10], [10], [-28], [22], [-40], [-22], [-32], [46], [-36], [56], [-16], [54], [-14], [-54], [-24], [-20], [-18], [18], [22], [-64], [-26], [-28], [10], [-42], [18], [-42], [6], [40], [-30], [0], [16], [-40], [-14], [14], [-6], [18], [16], [-60], [-10], [-24], [38], [-58], [-16], [-66], [0], [-26], [-32], [50], [16], [6], [30], [36], [26], [4], [-72], [6], [34], [24], [16], [-32], [58], [54], [-40], [70], [70], [40], [-10], [44], [-52], [8], [-66], [-48], [24], [-10], [2], [0], [-72], [-62], [-46], [-12], [-36], [22], [-52], [-10], [-66], [32], [20], [14], [-54], [-6], [4], [42], [50], [-44], [-62], [-58], [-24], [-42], [50], [10], [74], [66], [6], [8], [18], [-8], [28], [-10], [-2], [42], [-40], [-16], [-8], [52], [8], [48], [86], [-36], [-8], [-2], [-20], [-6], [-74], [54], [42], [-44], [-72], [-50], [0], [44], [22], [48], [-40], [4], [6], [-14], [-24], [-68], [36], [-18], [-24], [-76], [50], [-4], [-4], [-10], [-72], [24], [14], [6], [10], [-50], [-26], [64], [-54], [30], [14], [4], [56], [-76], [10], [-42], [0], [4], [22], [0], [-12], [66], [42], [4], [38], [42], [46], [48], [-24], [32], [62], [0], [-28], [-36], [48], [-42], [34], [48], [-80], [-54], [76], [46], [90], [-96], [-72], [54], [8], [42], [-60], [86], [-66], [36], [34], [72], [56], [-48], [12], [4], [46], [84], [96], [-26], [34], [-38], [-28], [-34], [56], [78], [94], [48], [-16], [-38], [30], [60], [46], [10], [-68], [-84], [-40], [34], [-20], [-50], [-18], [-2], [54], [46], [-32], [-38], [62], [24], [-28], [-18], [-54], [96], [14], [-42], [-54], [62], [-26], [74], [72], [0], [46], [0], [-36], [-8], [8], [84], [106], [78], [10], [90], [52], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6026_2_a_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_6026_2_a_a(:prec:=1)
    chi := MakeCharacter_6026_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_6026_2_a_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_6026_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>],Snew);
    return Vf;
end function;