
// Make newform 3120.2.a.j in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3120_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_3120_2_a_j();".
// 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_3120_2_a_j();".
// 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_3120_a();"
function MakeCharacter_3120_a()
    N := 3120;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 2641];
    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_3120_a_Hecke(Kf)
    return MakeCharacter_3120_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], [-2], [0], [1], [0], [-2], [6], [0], [-8], [2], [6], [4], [0], [-6], [0], [14], [4], [0], [-4], [16], [12], [-6], [-4], [-12], [16], [12], [-4], [12], [16], [-6], [6], [-8], [18], [-8], [2], [16], [-12], [18], [6], [2], [0], [8], [18], [16], [4], [10], [-12], [-4], [0], [0], [26], [30], [-12], [-30], [12], [-8], [26], [-30], [28], [-6], [4], [-24], [-10], [-18], [-2], [14], [-24], [20], [6], [24], [-8], [26], [34], [-12], [36], [26], [-18], [14], [-6], [32], [-24], [14], [-32], [0], [-30], [-28], [-42], [10], [-36], [0], [-2], [-6], [-14], [18], [6], [-42], [-20], [8], [-20], [30], [-24], [6], [-32], [-16], [-12], [-6], [-24], [-10], [40], [-22], [-18], [-38], [-20], [-6], [4], [-6], [30], [18], [-16], [2], [-42], [36], [-26], [36], [20], [-12], [4], [50], [-38], [-36], [40], [-34], [6], [-22], [-6], [16], [-30], [-6], [-26], [-6], [-32], [12], [14], [48], [26], [-12], [-20], [12], [-58], [6], [28], [30], [28], [12], [16], [-54], [2], [42], [-36], [24], [-50], [30], [42], [-36], [16], [-46], [-22], [-30], [-36], [-4], [-36], [32], [40], [54], [-2], [42], [52], [-22], [46], [-54], [62], [42], [36], [48], [-34], [-44], [-34], [-24], [50], [12], [4], [42], [-24], [36], [38], [50], [-54], [-6], [30], [40], [-34], [-34], [24], [30], [64], [-48], [6], [4], [38], [-48], [28], [36], [0], [26], [-44], [-42], [-36], [-18], [-46], [-44], [54], [-74], [48], [-46], [0], [60], [28], [-24], [2], [-20], [40], [-18], [4], [36], [38], [-66], [6], [36], [36], [52], [4], [-16], [-54], [0], [-14], [60], [-2], [-54], [50], [-30], [24], [26], [-54], [36], [38], [-8], [-6], [20], [28], [-12], [-28], [14], [66], [64], [-42], [-6], [4], [78], [26], [-32], [-64], [-32], [26], [-74], [-12], [80], [-10], [78], [18], [4], [-54], [-28], [4], [-12], [38], [-54], [-20], [-54], [-24], [-12], [-6], [12], [38], [36], [64], [6], [66], [-56], [38], [42], [16], [-48], [52], [80], [12], [-22], [24], [38], [6], [30], [6], [-20], [-48], [-22], [24], [0], [-76], [-6], [-32], [-4], [-60], [-86], [-54], [14], [-62], [-44], [18], [18], [8], [-18], [-80], [-84], [46], [60], [80], [-42], [-58], [16], [-82], [60], [-6], [40], [66], [90], [-10], [28], [-48], [-90], [-38], [56], [-6], [52], [-22], [24], [0], [-72], [-48], [24], [-82], [42], [18], [60], [4], [-34], [18], [58], [74], [-6], [64], [-84], [-24], [-32], [38], [-84], [48], [-16], [6], [38], [18], [78], [-38], [-54], [-38], [-48], [-44], [-10], [28], [42], [26], [-90], [0], [28], [-96], [-34], [-56], [-30], [-68], [54], [-40], [60], [34], [-6], [-18], [-56], [2], [18], [40], [-12], [86], [-6], [24], [-8], [74], [-96], [96], [16], [42], [42], [-48], [-82], [48], [-42], [-52], [-66], [-60], [-54], [-86], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3120_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3120_2_a_j();".
// 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_3120_2_a_j(:prec:=1)
    chi := MakeCharacter_3120_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_3120_2_a_j();".
// 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_3120_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3120_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<17,R![0, 1]>,<19,R![2, 1]>,<31,R![8, 1]>],Snew);
    return Vf;
end function;