
// Make newform 2960.2.a.g in Magma, downloaded from the LMFDB on 22 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2960_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_2960_2_a_g();".
// 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_2960_2_a_g();".
// 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_2960_a();"
function MakeCharacter_2960_a()
    N := 2960;
    order := 1;
    char_gens := [2591, 741, 1777, 2481];
    v := [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_2960_a_Hecke( : Kf := Kf)
    return MakeCharacter_2960_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]];
    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], [0], [4], [2], [-2], [4], [0], [-6], [4], [-1], [-6], [-4], [8], [10], [-4], [10], [8], [0], [10], [4], [0], [2], [6], [6], [0], [8], [18], [14], [-8], [12], [-6], [4], [-10], [24], [-6], [4], [16], [-14], [-4], [-2], [12], [14], [2], [20], [-28], [16], [-20], [-18], [-22], [-12], [2], [-20], [14], [32], [14], [8], [-22], [2], [-4], [18], [32], [-12], [-10], [2], [-20], [34], [-4], [-18], [6], [0], [-8], [-14], [28], [24], [-14], [34], [-22], [2], [-28], [-6], [20], [-14], [20], [24], [-6], [-18], [10], [40], [-28], [-28], [16], [-20], [20], [-24], [-18], [-38], [-12], [-6], [-4], [2], [-36], [-30], [-12], [46], [12], [34], [-24], [10], [-24], [-6], [-30], [-4], [36], [2], [44], [-8], [34], [-20], [-22], [10], [2], [-36], [20], [50], [-30], [-32], [-16], [34], [-36], [24], [40], [10], [42], [-30], [-22], [-48], [-30], [26], [20], [6], [-8], [-12], [-14], [-24], [10], [-18], [-20], [-24], [42], [-30], [-20], [-56], [20], [36], [-20], [18], [-14], [-42], [4], [-6], [8], [12], [-50], [-24], [-20], [10], [2], [-22], [52], [2], [60], [10], [16], [18], [-44], [-42], [-16], [-10], [32], [-28], [26], [22], [0], [6], [-38], [36], [-14], [-32], [38], [40], [60], [-66], [24], [-30], [58], [-30], [18], [-48], [-14], [56], [66], [-62], [28], [66], [56], [16], [18], [-68], [38], [-30], [-24], [48], [28], [58], [-48], [42], [72], [42], [46], [24], [-14], [16], [16], [-38], [-62], [-16], [-64], [-60], [26], [-4], [8], [-6], [-48], [0], [66], [10], [12], [-28], [76], [36], [-16], [50], [-6], [20], [-32], [-36], [52], [24], [-62], [-62], [-8], [-6], [-46], [-12], [-2], [-72], [42], [-58], [72], [68], [6], [-46], [22], [20], [46], [-6], [-64], [-38], [-30], [52], [-10], [-44], [58], [56], [40], [34], [58], [-76], [32], [-12], [-48], [14], [60], [-24], [6], [-46], [76], [66], [66], [36], [-78], [-44], [-46], [-18], [16], [-30], [28], [80], [62], [-22], [-16], [44], [-52], [-2], [36], [26], [-32], [58], [-72], [-90], [82], [-56], [56], [-14], [-28], [28], [26], [2], [20], [2], [-78], [8], [-54], [66], [12], [-84], [-48], [26], [62], [-78], [-60], [60], [-60], [-64], [70], [-14], [-62], [-48], [-46], [82], [-30], [-28], [2], [76], [-10], [-40], [-92], [-30], [20], [-62], [-6], [-16], [-66], [-14], [-36], [-44], [-6], [-8], [-22], [-86], [16], [12], [8], [62], [82], [-8], [-22], [-28], [4], [-48], [-90], [60], [34], [60], [80], [-6], [42], [-6], [-30], [14], [-32], [58], [-60], [-72], [-80], [-86], [4], [72], [-46], [-54], [20], [-4], [-16], [10], [60], [26], [-12], [-66], [6], [6], [-48], [6], [42], [32], [-78], [50], [-56], [60], [-94], [10], [-68], [28], [-82], [6], [88], [96], [-30], [-96], [18], [26], [-104], [-28], [-78], [90], [-56], [74], [-28], [68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2960_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_g();".
// 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_2960_2_a_g(:prec:=1)
    chi := MakeCharacter_2960_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_2960_2_a_g();".
// 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_2960_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;