
// Make newform 8034.2.a.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_g();".
// 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_8034_2_a_g();".
// 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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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], [1], [-2], [-4], [4], [1], [2], [8], [-4], [6], [-8], [-6], [-6], [4], [-8], [2], [4], [14], [-4], [0], [-2], [16], [4], [10], [18], [-14], [-1], [8], [18], [14], [0], [8], [10], [-4], [-10], [8], [-18], [8], [0], [18], [24], [-10], [-16], [-10], [-18], [16], [4], [20], [-28], [22], [22], [-8], [-18], [4], [-10], [0], [30], [24], [-18], [2], [-4], [6], [-28], [-4], [26], [14], [-12], [-14], [24], [-14], [-22], [24], [-16], [-10], [12], [16], [10], [10], [34], [-14], [-16], [30], [-16], [26], [-8], [-12], [-30], [-2], [-18], [-16], [-32], [8], [32], [0], [-20], [28], [14], [-10], [-36], [-42], [12], [-2], [-4], [30], [-28], [-18], [36], [2], [0], [-14], [40], [-2], [18], [-32], [-20], [18], [-8], [-12], [10], [24], [18], [-30], [6], [-44], [-20], [-10], [-10], [-24], [-40], [-22], [8], [-24], [40], [-10], [-14], [-10], [6], [16], [30], [30], [-20], [22], [56], [-36], [-42], [-32], [-10], [18], [20], [-32], [-46], [-50], [12], [-12], [4], [24], [-24], [2], [-38], [14], [12], [-30], [-48], [28], [-30], [32], [-40], [62], [-38], [-14], [-36], [2], [12], [46], [-24], [2], [28], [54], [-16], [-2], [-40], [32], [-58], [42], [16], [30], [-18], [-32], [46], [40], [34], [28], [20], [-18], [36], [-38], [50], [14], [2], [8], [22], [32], [-18], [50], [12], [54], [-8], [4], [42], [20], [18], [42], [64], [-36], [-72], [-10], [-16], [-42], [-24], [30], [38], [-4], [66], [-64], [-28], [-26], [-10], [44], [-48], [-52], [14], [44], [-36], [-10], [-28], [48], [-66], [-10], [52], [-40], [-68], [28], [-16], [-10], [-46], [-36], [0], [-20], [32], [60], [-58], [-14], [40], [-54], [38], [4], [6], [8], [-2], [-6], [-24], [8], [34], [22], [-46], [-60], [62], [34], [-44], [2], [14], [-4], [18], [-24], [2], [44], [52], [46], [-46], [-24], [52], [-64], [40], [6], [56], [-36], [-46], [-42], [60], [-18], [-22], [-36], [-6], [4], [-82], [42], [-32], [-50], [-60], [-56], [-22], [70], [-32], [0], [68], [42], [-28], [-50], [8], [-2], [-64], [38], [-18], [-68], [72], [-70], [4], [-32], [30], [78], [44], [-42], [30], [-76], [26], [10], [48], [-84], [48], [54], [22], [90], [16], [-60], [12], [-12], [-38], [-78], [58], [-8], [82], [78], [-42], [-64], [-10], [44], [46], [48], [-88], [-50], [-40], [58], [-18], [16], [-58], [-26], [16], [28], [30], [-16], [-42], [10], [48], [-84], [-92], [74], [-58], [-80], [-54], [32], [60], [56], [-6], [16], [-34], [-44], [-100], [-38], [-30], [2], [30], [26], [12], [50], [-84], [-24], [80], [30], [52], [36], [86], [-90], [-84], [-76], [-96], [18], [-52], [54], [-12], [-90], [26], [14], [88], [-70], [-58], [64], [-98], [-62], [4], [-36], [18], [94], [-36], [84], [-38], [42], [40], [40], [-30], [12], [6], [-50], [-32], [-96], [-2], [78], [24], [42], [20], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_g();".
// 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_8034_2_a_g(:prec:=1)
    chi := MakeCharacter_8034_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_8034_2_a_g();".
// 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_8034_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![4, 1]>],Snew);
    return Vf;
end function;