
// Make newform 3042.2.a.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3042_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_3042_2_a_c();".
// 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_3042_2_a_c();".
// 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_3042_a();"
function MakeCharacter_3042_a()
    N := 3042;
    order := 1;
    char_gens := [677, 847];
    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_3042_a_Hecke(Kf)
    return MakeCharacter_3042_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], [-2], [2], [0], [0], [-2], [-6], [4], [10], [10], [-8], [-10], [-4], [-12], [6], [4], [2], [-2], [0], [4], [0], [4], [-6], [-12], [2], [16], [-8], [4], [-14], [-8], [8], [-2], [-20], [-14], [-10], [-2], [14], [-12], [-6], [0], [-22], [-12], [-16], [22], [0], [12], [-14], [-8], [-4], [-6], [16], [-20], [-28], [18], [-24], [10], [10], [2], [-10], [-4], [-14], [2], [-28], [-26], [-18], [-10], [2], [12], [16], [-26], [4], [8], [-6], [34], [4], [30], [-8], [-30], [4], [-40], [20], [20], [26], [0], [16], [-6], [28], [30], [-6], [-12], [24], [18], [-28], [14], [-24], [6], [18], [4], [20], [28], [18], [-16], [10], [28], [8], [-28], [26], [0], [22], [-32], [16], [22], [26], [-10], [-18], [6], [-32], [26], [-20], [40], [6], [-18], [-24], [-10], [22], [36], [20], [-8], [-44], [26], [-16], [-32], [-22], [30], [24], [-6], [-38], [-2], [50], [10], [-30], [-24], [48], [30], [-16], [-56], [-22], [-20], [44], [8], [42], [36], [12], [-28], [-12], [0], [6], [-2], [10], [-52], [-6], [-22], [-12], [42], [-24], [-8], [-42], [-44], [34], [-24], [0], [52], [-4], [0], [30], [10], [10], [-24], [50], [2], [32], [-6], [2], [36], [30], [2], [-14], [36], [20], [34], [44], [-28], [10], [48], [34], [40], [6], [-62], [36], [54], [-8], [-12], [50], [-36], [22], [-6], [-24], [-54], [-52], [-38], [-42], [-16], [8], [-4], [40], [-32], [-42], [-48], [-14], [-18], [-54], [-54], [-14], [-12], [10], [-34], [40], [32], [-60], [-14], [-20], [-50], [-18], [44], [-48], [36], [-54], [-40], [-12], [-76], [28], [64], [64], [-6], [20], [2], [20], [-46], [-76], [12], [30], [8], [10], [-54], [24], [2], [-22], [38], [8], [-24], [12], [44], [14], [18], [20], [-34], [-50], [34], [74], [-2], [38], [-76], [40], [-2], [-74], [32], [16], [40], [12], [76], [10], [-28], [-40], [-38], [32], [-66], [2], [46], [-70], [-42], [-28], [-6], [-60], [-66], [-10], [-48], [54], [-20], [42], [6], [78], [-80], [76], [-28], [-52], [48], [70], [24], [-74], [56], [46], [18], [-36], [28], [-10], [-24], [20], [-76], [30], [-52], [-32], [-42], [14], [66], [62], [-74], [46], [-72], [-54], [-20], [2], [-80], [56], [-30], [12], [56], [6], [20], [-48], [-64], [-38], [-6], [88], [54], [40], [2], [62], [-60], [-2], [-50], [12], [10], [96], [-10], [66], [84], [-80], [-22], [-76], [48], [-18], [72], [16], [12], [34], [-82], [-66], [-2], [-72], [20], [76], [30], [32], [-42], [-4], [-8], [-24], [-10], [22], [10], [-34], [82], [42], [6], [-24], [30], [22], [-54], [-8], [76], [-6], [-96], [-52], [-60], [-26], [94], [-90], [92], [30], [64], [54], [2], [-62], [-26], [-8], [-12], [50], [74], [24], [-34], [62], [76], [-52], [-58], [-2], [96], [-72], [58], [84], [10], [8], [72], [0], [16], [42], [-36], [26], [-90], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3042_2_a_c();".
// 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_3042_2_a_c(:prec:=1)
    chi := MakeCharacter_3042_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_3042_2_a_c();".
// 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_3042_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;