
// Make newform 3960.2.a.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3960_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_3960_2_a_k();".
// 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_3960_2_a_k();".
// 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_3960_a();"
function MakeCharacter_3960_a()
    N := 3960;
    order := 1;
    char_gens := [991, 1981, 3521, 2377, 2521];
    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_3960_a_Hecke(Kf)
    return MakeCharacter_3960_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], [0], [1], [-2], [-1], [0], [8], [-8], [-4], [6], [0], [6], [2], [2], [4], [2], [12], [-6], [8], [0], [-8], [4], [6], [10], [-10], [2], [16], [-10], [14], [-6], [2], [8], [18], [12], [18], [12], [-14], [-4], [-10], [-12], [4], [2], [-8], [24], [20], [24], [0], [12], [-30], [6], [-20], [-12], [-10], [20], [-18], [-6], [18], [-28], [-8], [-14], [-14], [32], [10], [32], [10], [-26], [4], [-32], [30], [10], [6], [28], [28], [-4], [4], [-8], [-6], [2], [18], [10], [28], [-6], [8], [-26], [-8], [-12], [10], [28], [-38], [-20], [8], [-36], [12], [28], [-4], [-18], [18], [-22], [-18], [-34], [-26], [20], [-30], [-10], [-44], [-2], [0], [-16], [40], [-10], [18], [-4], [2], [-28], [-32], [18], [-36], [0], [18], [28], [-2], [8], [48], [24], [4], [-10], [-38], [-48], [8], [20], [-20], [-6], [-48], [18], [-26], [-42], [-22], [18], [38], [-54], [12], [-2], [-40], [14], [50], [40], [-44], [-56], [36], [32], [-44], [-26], [-20], [-42], [-32], [-40], [-8], [-18], [-52], [50], [-16], [12], [-22], [20], [-46], [-24], [56], [-36], [-34], [42], [20], [50], [4], [-44], [32], [30], [60], [-6], [-30], [14], [-32], [-20], [34], [12], [12], [22], [20], [-40], [-38], [0], [22], [-18], [4], [42], [-14], [-2], [-2], [46], [12], [-22], [46], [32], [46], [14], [28], [-66], [32], [2], [66], [28], [44], [-26], [-56], [-56], [4], [58], [70], [-58], [8], [42], [-22], [68], [6], [-64], [-10], [-26], [-54], [0], [34], [-48], [2], [-4], [8], [-22], [-24], [42], [-50], [24], [20], [-16], [36], [32], [-48], [50], [20], [-36], [4], [-12], [-60], [10], [-32], [6], [80], [26], [-28], [12], [-38], [-26], [30], [20], [-2], [10], [34], [-16], [70], [52], [-50], [-10], [10], [-32], [-34], [24], [-26], [64], [-36], [-36], [-24], [18], [-18], [-24], [-6], [40], [-22], [-2], [70], [0], [30], [16], [16], [54], [74], [-60], [-12], [32], [64], [-14], [64], [-30], [-12], [50], [4], [-44], [-56], [-48], [60], [74], [64], [54], [80], [68], [-26], [-30], [22], [-8], [-26], [-2], [44], [12], [-90], [6], [-88], [54], [6], [-60], [-4], [30], [-44], [44], [18], [52], [82], [-74], [-68], [-14], [16], [52], [78], [84], [-74], [-18], [10], [30], [-70], [80], [50], [-40], [38], [-44], [56], [42], [68], [10], [30], [2], [2], [-64], [0], [88], [20], [-40], [-40], [-42], [12], [16], [-68], [62], [-12], [14], [-34], [36], [20], [74], [50], [32], [58], [-36], [36], [84], [6], [-28], [-78], [-82], [42], [-72], [-40], [64], [-40], [-6], [46], [-8], [30], [2], [20], [16], [-64], [12], [-32], [-74], [68], [86], [22], [-54], [6], [22], [66], [4], [-14], [-42], [-56], [84], [36], [-48], [-76], [-68], [32], [-82], [-48], [-64], [30], [78], [70], [-80], [72], [-56], [34], [90], [-4], [78], [100], [-84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3960_2_a_k();".
// 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_3960_2_a_k(:prec:=1)
    chi := MakeCharacter_3960_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_3960_2_a_k();".
// 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_3960_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<13,R![0, 1]>,<17,R![-8, 1]>,<19,R![8, 1]>,<23,R![4, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;