
// Make newform 2960.2.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// 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_h();".
// 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_h();".
// 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)
    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+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], [3], [-6], [0], [0], [-2], [-6], [0], [-1], [-9], [10], [-1], [1], [0], [-12], [0], [5], [3], [-16], [-11], [0], [8], [-1], [10], [-20], [-12], [14], [1], [0], [2], [12], [-9], [8], [15], [2], [0], [-15], [-18], [17], [12], [-22], [3], [-14], [27], [-17], [8], [3], [-2], [-30], [-6], [-6], [20], [-5], [2], [13], [-8], [0], [-4], [30], [11], [-16], [-10], [6], [-6], [3], [34], [14], [4], [-19], [-16], [-11], [23], [24], [-36], [-21], [34], [-12], [15], [-32], [-26], [-19], [-4], [13], [-16], [-30], [14], [-6], [22], [26], [-16], [20], [40], [0], [-27], [7], [26], [-8], [40], [26], [38], [28], [-17], [-24], [-20], [-33], [21], [42], [-16], [23], [-3], [31], [8], [15], [34], [12], [-36], [-15], [32], [47], [37], [-26], [-20], [20], [-40], [3], [-28], [-17], [-33], [45], [-11], [10], [21], [-10], [31], [-17], [28], [-50], [-41], [-19], [-24], [2], [-12], [36], [-2], [-12], [-36], [24], [-46], [18], [16], [33], [0], [-30], [-14], [-30], [1], [30], [36], [9], [-6], [-24], [-52], [-39], [30], [42], [17], [12], [6], [46], [60], [-9], [1], [0], [16], [-30], [-17], [34], [4], [6], [-28], [-36], [24], [41], [17], [26], [-30], [-53], [-42], [-44], [-50], [-43], [51], [49], [-12], [-16], [3], [-34], [-32], [33], [22], [35], [40], [0], [-40], [-52], [-40], [-21], [-38], [-30], [-4], [-12], [56], [27], [-54], [-58], [-62], [-18], [-21], [-69], [-25], [18], [45], [-52], [-6], [15], [-49], [-38], [-25], [-61], [19], [-51], [-67], [-67], [-14], [54], [-18], [-30], [36], [-44], [-65], [30], [30], [-34], [-34], [-52], [47], [4], [-8], [-30], [48], [4], [-34], [39], [-11], [-12], [61], [-2], [20], [-48], [-10], [50], [-52], [-55], [81], [-60], [-41], [40], [-32], [-68], [-34], [-2], [-14], [36], [-28], [-32], [-13], [-54], [23], [-18], [-52], [-67], [-10], [-21], [26], [86], [-10], [-66], [-36], [-10], [-7], [17], [-87], [-65], [65], [-6], [18], [-77], [-38], [-51], [-5], [24], [12], [-18], [78], [-18], [71], [-46], [57], [85], [-87], [3], [-36], [-44], [81], [40], [73], [-30], [80], [67], [-22], [20], [1], [32], [-52], [48], [-18], [10], [-63], [92], [26], [-8], [48], [-28], [85], [19], [44], [72], [9], [-9], [-6], [52], [-30], [38], [73], [73], [8], [-1], [8], [47], [-8], [-14], [74], [6], [20], [-76], [-21], [-54], [22], [33], [94], [70], [-9], [26], [-78], [-90], [-14], [-38], [62], [0], [66], [62], [5], [-84], [-53], [-34], [42], [77], [60], [-24], [-92], [-18], [76], [-19], [-13], [-82], [-94], [-66], [43], [-38], [22], [-26], [33], [-94], [28], [49], [-9], [86], [-51], [74], [84], [10], [-18], [71], [-46], [-74], [68], [-21], [7], [27], [78], [-60], [0], [-42], [-88], [-19], [98], [24], [52], [62], [16], [-3], [-6], [3], [-60], [73], [-52], [-34]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_h();".
// 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_h(: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_h();".
// 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_h( : 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![6, 1]>],Snew);
    return Vf;
end function;