
// Make newform 6031.2.a.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6031_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_6031_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_6031_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_6031_a();"
function MakeCharacter_6031_a()
    N := 6031;
    order := 1;
    char_gens := [816, 5218];
    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_6031_a_Hecke(Kf)
    return MakeCharacter_6031_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 := [[0], [3], [4], [1], [1], [4], [-2], [0], [-8], [0], [2], [1], [3], [4], [-9], [9], [-6], [-6], [8], [13], [5], [-14], [9], [-12], [12], [-3], [-14], [4], [-10], [8], [15], [-16], [6], [-12], [-11], [-8], [-7], [-1], [-12], [9], [-2], [-9], [-6], [-6], [19], [4], [1], [7], [18], [9], [22], [6], [-10], [2], [-8], [-9], [-18], [-17], [22], [24], [-14], [-6], [31], [-8], [8], [18], [2], [13], [-6], [-6], [-6], [-3], [-8], [13], [3], [-16], [-16], [23], [38], [-22], [-29], [-16], [40], [35], [-26], [-1], [-28], [-16], [-12], [-10], [-8], [30], [12], [28], [-34], [6], [-5], [21], [-20], [30], [12], [-8], [2], [6], [-23], [-18], [40], [-13], [-9], [34], [-32], [25], [15], [1], [-16], [-37], [26], [24], [12], [-17], [18], [-13], [-43], [28], [-20], [-2], [-10], [-51], [-14], [39], [-29], [-15], [15], [-50], [-21], [4], [-41], [35], [-16], [10], [29], [17], [-8], [10], [42], [40], [56], [32], [-14], [-48], [14], [-18], [-32], [47], [-32], [-18], [-38], [-18], [-13], [30], [22], [-9], [-54], [12], [38], [35], [8], [-20], [7], [52], [-24], [-50], [-18], [-61], [49], [46], [-10], [10], [31], [30], [-38], [-6], [16], [6], [-36], [-7], [47], [-28], [-4], [47], [34], [-60], [28], [29], [-53], [63], [60], [4], [-31], [4], [64], [-23], [-62], [-39], [56], [36], [-24], [-68], [36], [-49], [-18], [-26], [28], [48], [10], [37], [24], [0], [-20], [-30], [-17], [19], [-19], [46], [15], [-44], [-26], [-41], [-53], [-14], [-19], [11], [-57], [-55], [-27], [27], [-14], [-26], [-44], [-24], [-38], [34], [-35], [22], [14], [-72], [-70], [-64], [41], [-4], [-72], [-66], [64], [56], [66], [-33], [43], [52], [-41], [-56], [-22], [36], [70], [-70], [-52], [-39], [47], [-36], [47], [64], [-40], [8], [-44], [-46], [-18], [0], [12], [-70], [-83], [-36], [57], [62], [40], [83], [32], [69], [68], [-46], [66], [36], [-42], [32], [47], [7], [-87], [-57], [-31], [-62], [-16], [-19], [60], [-63], [47], [36], [72], [48], [-8], [-54], [55], [26], [31], [-3], [79], [41], [-18], [-20], [9], [42], [-65], [-20], [52], [5], [36], [16], [61], [-16], [48], [-4], [-28], [-34], [71], [6], [6], [36], [86], [-12], [65], [41], [-18], [-40], [19], [-85], [14], [-76], [80], [72], [35], [53], [-78], [1], [-28], [-41], [-60], [86], [-22], [66], [-66], [-72], [3], [32], [20], [-57], [-44], [-6], [-73], [74], [96], [-92], [92], [-18], [-28], [44], [-22], [-28], [59], [80], [-57], [-26], [10], [61], [38], [-90], [44], [-38], [86], [-27], [-75], [-70], [88], [0], [-85], [-8], [-14], [-6], [63], [-2], [76], [-7], [-23], [66], [31], [-32], [-38], [-4], [26], [-43], [-50], [78], [8], [81], [91], [41], [72], [34], [-34], [30], [-8], [7], [46], [8], [-90], [-40], [64], [75], [74], [-19], [24], [105], [-16], [6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6031_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6031_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_6031_2_a_a(:prec:=1)
    chi := MakeCharacter_6031_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_6031_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_6031_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6031_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;