
// Make newform 8034.2.a.e in Magma, downloaded from the LMFDB on 25 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_e();".
// 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_e();".
// 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], [0], [4], [6], [0], [10], [-6], [12], [0], [-6], [-12], [-2], [-4], [8], [-10], [16], [-12], [-14], [-14], [10], [1], [0], [2], [-2], [16], [16], [-6], [-4], [-10], [0], [-10], [-16], [0], [10], [-16], [-2], [16], [14], [-2], [0], [-20], [28], [-12], [22], [6], [-8], [-26], [28], [-26], [16], [-18], [0], [-10], [-6], [20], [6], [20], [-12], [26], [14], [4], [-14], [0], [34], [18], [8], [0], [22], [28], [24], [-30], [-38], [-30], [34], [24], [30], [16], [26], [-8], [28], [-22], [6], [-18], [-8], [24], [-32], [40], [-24], [-20], [-12], [14], [-42], [28], [-10], [-20], [-18], [36], [30], [36], [38], [-44], [-22], [-16], [2], [-8], [-2], [-38], [8], [-28], [18], [-48], [-20], [-14], [0], [18], [2], [22], [36], [28], [-42], [-10], [8], [40], [-22], [-16], [-48], [24], [6], [-6], [14], [6], [-40], [-18], [-2], [44], [-10], [24], [-36], [14], [-48], [54], [50], [28], [-24], [18], [-34], [-52], [-52], [-28], [-24], [-24], [-14], [-38], [46], [-4], [-30], [-8], [-12], [2], [16], [-40], [-10], [10], [26], [-4], [34], [-60], [-58], [-56], [-30], [-52], [38], [16], [6], [-32], [40], [-26], [-30], [8], [30], [-42], [8], [-10], [-8], [-14], [-20], [20], [-2], [28], [-6], [-46], [38], [-38], [-24], [-10], [-32], [-18], [50], [-68], [-42], [0], [12], [50], [4], [18], [-30], [-48], [60], [-8], [62], [32], [6], [-8], [30], [46], [-12], [-6], [-24], [12], [14], [22], [-12], [-48], [-4], [70], [28], [36], [-10], [-60], [24], [22], [38], [60], [40], [44], [-12], [-64], [22], [18], [-12], [-72], [44], [40], [4], [-58], [18], [-64], [10], [38], [-44], [22], [48], [30], [-6], [-40], [-48], [-30], [46], [18], [44], [-34], [-38], [68], [-54], [30], [-4], [-30], [40], [66], [4], [-60], [46], [-62], [80], [44], [-40], [56], [6], [48], [-28], [50], [-42], [20], [-2], [50], [-28], [-54], [20], [-2], [18], [-64], [-50], [-36], [0], [-22], [6], [-32], [-24], [76], [-38], [-4], [-18], [-24], [-26], [-32], [38], [14], [60], [32], [26], [-12], [72], [-26], [14], [-92], [30], [-50], [60], [-38], [-70], [-88], [-36], [-32], [38], [-10], [-30], [-16], [-60], [-4], [-60], [10], [34], [-38], [-56], [2], [-18], [-42], [48], [70], [84], [38], [-72], [80], [-18], [32], [58], [14], [-64], [14], [54], [64], [12], [62], [-56], [86], [-54], [32], [28], [-28], [10], [54], [-88], [-22], [-56], [52], [32], [-30], [0], [-58], [-92], [4], [42], [66], [-14], [62], [-6], [-44], [18], [-84], [96], [40], [70], [52], [12], [14], [6], [12], [-28], [0], [2], [-60], [70], [12], [6], [-22], [-2], [80], [-14], [38], [16], [-98], [-70], [4], [-84], [-14], [-34], [-60], [76], [90], [98], [40], [-88], [-14], [-12], [-90], [-50], [40], [40], [86], [46], [-96], [-78], [-92], [8]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : 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;