
// Make newform 2960.2.a.d in Magma, downloaded from the LMFDB on 23 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_d();".
// 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_d();".
// 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], [-1], [-1], [1], [3], [-4], [0], [4], [0], [0], [-2], [1], [3], [-2], [-3], [-9], [0], [2], [4], [-15], [-7], [10], [3], [6], [-10], [3], [-14], [0], [-10], [-6], [7], [-6], [-18], [4], [15], [-8], [-7], [-8], [-6], [-9], [-12], [5], [-24], [-16], [9], [-14], [-5], [-11], [12], [-25], [-6], [6], [8], [0], [-6], [9], [-6], [-11], [-10], [30], [28], [6], [13], [12], [-10], [-18], [28], [-31], [12], [-10], [-6], [-15], [16], [-25], [19], [18], [24], [17], [-6], [32], [27], [8], [24], [-25], [16], [-9], [30], [-10], [-30], [-14], [-6], [36], [-44], [24], [-14], [-36], [21], [15], [-2], [-34], [-8], [12], [24], [30], [31], [32], [-18], [21], [33], [14], [4], [-43], [39], [-5], [-14], [-9], [-32], [-18], [-18], [21], [14], [-19], [27], [24], [-8], [0], [8], [-21], [-2], [41], [7], [-9], [-23], [8], [-15], [44], [-39], [-23], [-12], [24], [7], [-27], [40], [-48], [20], [-48], [-4], [30], [4], [-24], [14], [-6], [4], [39], [10], [-6], [4], [6], [35], [-18], [-36], [27], [22], [60], [-60], [3], [-50], [44], [-55], [24], [6], [8], [36], [29], [-11], [6], [-20], [-30], [-35], [-46], [40], [18], [-4], [0], [24], [21], [-37], [-2], [26], [-21], [32], [24], [16], [-3], [-3], [-45], [-40], [26], [-63], [42], [-48], [-35], [-58], [35], [-48], [36], [-8], [-12], [-60], [67], [-22], [12], [-38], [36], [-42], [35], [-20], [0], [60], [-6], [35], [-65], [15], [-56], [27], [-22], [-42], [-21], [13], [-36], [29], [-29], [-53], [33], [-53], [-9], [50], [18], [-54], [-6], [54], [-50], [-47], [-52], [18], [30], [-26], [36], [-5], [24], [50], [30], [24], [20], [24], [-69], [53], [28], [-9], [-34], [46], [42], [-22], [-46], [-18], [1], [-27], [18], [-59], [-54], [-22], [-20], [80], [70], [62], [40], [-36], [-16], [11], [-6], [21], [4], [72], [5], [-8], [15], [-22], [78], [-44], [36], [12], [-78], [3], [57], [-49], [-45], [-59], [-6], [-36], [37], [-58], [-69], [79], [-66], [-50], [-64], [12], [-64], [63], [38], [27], [45], [21], [1], [72], [-88], [-75], [60], [-13], [-36], [10], [-43], [54], [16], [15], [-16], [-44], [22], [42], [78], [-55], [-42], [-56], [12], [46], [-36], [53], [-39], [26], [52], [11], [-75], [48], [-50], [-36], [24], [-7], [43], [30], [69], [52], [29], [-72], [88], [-10], [30], [30], [18], [-27], [-6], [-22], [-15], [78], [66], [25], [26], [84], [16], [-28], [6], [-44], [0], [54], [58], [-1], [-24], [27], [50], [-18], [35], [84], [78], [-38], [-78], [-68], [-69], [-65], [32], [34], [60], [11], [0], [6], [-32], [93], [38], [40], [-39], [7], [42], [-31], [36], [88], [72], [60], [-29], [26], [30], [4], [-81], [-19], [-39], [-78], [28], [26], [102], [-72], [91], [102], [24], [-6], [2], [-72], [-75], [74], [-39], [-12], [-51], [76], [-48]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![-1, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;