
// Make newform 3360.2.a.u in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3360_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_3360_2_a_u();".
// 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_3360_2_a_u();".
// 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_3360_a();"
function MakeCharacter_3360_a()
    N := 3360;
    order := 1;
    char_gens := [1471, 421, 1121, 2017, 1921];
    v := [1, 1, 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_3360_a_Hecke(Kf)
    return MakeCharacter_3360_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], [1], [1], [-1], [-4], [-2], [-2], [0], [0], [6], [-8], [2], [-6], [-4], [0], [-6], [-8], [-2], [-4], [4], [-10], [-12], [-4], [-6], [-2], [6], [16], [-12], [-10], [10], [0], [0], [2], [-8], [22], [-4], [14], [-4], [-16], [6], [-4], [22], [-20], [-6], [18], [0], [12], [-16], [-12], [6], [-6], [-12], [-22], [24], [6], [32], [14], [-16], [-6], [26], [4], [-10], [12], [0], [-2], [18], [-4], [-22], [-4], [-34], [6], [4], [24], [18], [-12], [0], [6], [14], [2], [-6], [-24], [6], [36], [-26], [-32], [28], [2], [34], [14], [-8], [12], [0], [-8], [12], [-20], [-16], [-18], [2], [36], [22], [12], [-6], [-4], [42], [-28], [-34], [-12], [46], [4], [-46], [-24], [-30], [-22], [8], [28], [-30], [44], [48], [-30], [-36], [-10], [-14], [-18], [52], [40], [22], [-26], [0], [8], [-42], [12], [24], [-12], [50], [2], [-22], [6], [28], [6], [42], [16], [46], [24], [12], [14], [-24], [-42], [-18], [40], [0], [18], [-30], [36], [24], [-28], [-44], [-20], [-6], [14], [-50], [28], [26], [-16], [-8], [-54], [-16], [12], [-2], [18], [46], [-60], [-34], [36], [34], [-16], [-54], [52], [-26], [16], [-34], [8], [32], [18], [38], [0], [-26], [26], [-44], [-22], [-16], [54], [12], [-12], [14], [4], [54], [2], [-62], [-58], [-24], [-26], [0], [-2], [-54], [-48], [-18], [-56], [-4], [-38], [-32], [66], [-10], [-16], [52], [0], [-46], [-24], [-38], [8], [2], [38], [-16], [18], [16], [-12], [30], [62], [60], [-8], [4], [2], [40], [-4], [26], [-28], [-8], [-14], [-38], [12], [-64], [-4], [0], [64], [14], [14], [-48], [8], [-72], [-68], [-48], [50], [-70], [32], [10], [-66], [-44], [-2], [-12], [-10], [46], [-64], [-52], [-58], [-74], [-42], [0], [-58], [34], [28], [-54], [-50], [28], [30], [4], [-26], [-40], [-12], [-82], [-6], [-8], [24], [60], [-56], [-10], [-4], [28], [74], [-30], [-8], [42], [-58], [-60], [-54], [-32], [-46], [-2], [-56], [14], [0], [-12], [38], [-62], [-12], [-60], [44], [2], [76], [-50], [68], [-46], [-48], [46], [66], [-68], [-48], [-62], [-8], [84], [-26], [50], [32], [58], [30], [-24], [-38], [10], [60], [-28], [8], [-62], [62], [-30], [40], [12], [-60], [-36], [70], [30], [82], [-80], [-14], [50], [6], [52], [-6], [-20], [38], [60], [-8], [6], [-72], [34], [-26], [56], [86], [-66], [72], [-64], [-46], [-16], [2], [-78], [0], [36], [-12], [-62], [6], [-72], [-6], [68], [-40], [-32], [6], [-80], [34], [0], [-12], [30], [-70], [-2], [30], [10], [64], [-54], [88], [-80], [68], [46], [-100], [72], [-78], [-42], [76], [44], [52], [-38], [80], [10], [-52], [14], [14], [-78], [-8], [-74], [6], [104], [50], [34], [-44], [96], [-10], [50], [84], [-52], [34], [78], [76], [80], [-42], [-48], [-18], [-50], [-48], [-40], [94], [-2], [-12], [-22], [16], [8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3360_2_a_u();".
// 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_3360_2_a_u(:prec:=1)
    chi := MakeCharacter_3360_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_3360_2_a_u();".
// 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_3360_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![2, 1]>,<17,R![2, 1]>,<19,R![0, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;