
// Make newform 8034.2.a.b 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_b();".
// 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_b();".
// 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], [0], [-3], [-5], [1], [2], [8], [1], [0], [-4], [-7], [7], [-1], [3], [2], [8], [1], [8], [-8], [-6], [-16], [8], [12], [8], [-9], [1], [12], [13], [17], [7], [-20], [17], [12], [-15], [4], [-8], [-9], [18], [-7], [5], [-2], [-12], [11], [-12], [-21], [-11], [-21], [-28], [-20], [17], [-16], [-7], [-12], [18], [-14], [-10], [-2], [-4], [30], [-9], [26], [-4], [-15], [-35], [30], [-26], [21], [-7], [30], [18], [-14], [22], [-7], [-2], [-29], [9], [-14], [-23], [28], [-35], [8], [14], [-8], [-23], [0], [6], [-21], [3], [4], [-37], [-9], [-30], [32], [22], [-33], [1], [5], [-30], [-10], [-18], [6], [20], [-37], [0], [-39], [40], [-22], [6], [-10], [-34], [-28], [40], [4], [1], [-16], [-19], [13], [-9], [-24], [-22], [-31], [22], [-24], [-32], [-2], [4], [6], [-16], [-13], [5], [-24], [-30], [-5], [10], [-33], [21], [25], [6], [30], [-22], [17], [37], [5], [4], [26], [-8], [20], [-41], [-19], [41], [-6], [-10], [-53], [46], [30], [27], [-27], [28], [22], [-17], [-22], [-12], [32], [1], [-34], [-52], [50], [32], [-9], [14], [-23], [-40], [-21], [-4], [-40], [8], [-20], [52], [-44], [-8], [55], [-19], [-48], [24], [-24], [24], [59], [10], [60], [-18], [-36], [16], [-10], [-44], [32], [-16], [-42], [-48], [30], [-30], [10], [44], [-50], [56], [20], [-54], [-6], [8], [-2], [43], [-63], [-25], [12], [54], [-6], [2], [-21], [-24], [-45], [60], [65], [-20], [-40], [40], [34], [33], [-39], [64], [22], [0], [46], [40], [-27], [-10], [39], [13], [-66], [36], [-18], [-18], [-20], [61], [2], [-14], [60], [-50], [30], [-21], [-25], [-8], [-57], [40], [-42], [-66], [67], [-27], [4], [54], [-38], [52], [-5], [59], [54], [-15], [5], [18], [-44], [-46], [42], [35], [-78], [28], [6], [47], [73], [48], [-78], [-10], [-7], [29], [-60], [0], [18], [1], [40], [78], [15], [-80], [-54], [18], [30], [35], [12], [-45], [-45], [-60], [8], [-32], [-12], [-57], [-60], [60], [89], [-79], [38], [-58], [85], [-30], [42], [-6], [6], [-78], [36], [61], [65], [19], [-32], [46], [-55], [-71], [-7], [24], [56], [26], [-16], [75], [-80], [-10], [22], [-28], [-22], [-10], [60], [-48], [-36], [2], [74], [14], [-7], [26], [26], [-76], [56], [14], [-80], [44], [73], [-75], [-26], [-7], [66], [-36], [-10], [46], [21], [6], [-52], [-19], [-89], [-26], [-72], [-84], [57], [-20], [73], [-30], [-68], [-19], [32], [-92], [-80], [-75], [-46], [70], [-52], [13], [-60], [-52], [-58], [-38], [39], [-44], [-75], [-4], [17], [20], [-14], [-42], [68], [-41], [-6], [52], [8], [-6], [-6], [33], [54], [-92], [-7], [-62], [9], [-26], [-22], [-45], [56], [18], [-34], [80], [26], [86], [8], [-6], [-37], [-50], [35], [-44], [-83], [-33], [-17], [-32], [-84], [63], [9], [79], [-22], [71], [-66], [-74], [-102]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![3, 1]>],Snew);
    return Vf;
end function;