
// Make newform 6960.2.a.k in Magma, downloaded from the LMFDB on 19 September 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6960_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_6960_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_6960_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_6960_a();"
function MakeCharacter_6960_a()
    N := 6960;
    order := 1;
    char_gens := [4351, 5221, 2321, 4177, 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_6960_a_Hecke(Kf)
    return MakeCharacter_6960_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], [-4], [-4], [-2], [6], [4], [4], [1], [-8], [2], [-6], [-4], [8], [6], [12], [6], [0], [-8], [-10], [16], [0], [-6], [-10], [14], [-4], [0], [-18], [-18], [8], [-20], [14], [-12], [6], [8], [-14], [-4], [-12], [22], [-20], [-10], [8], [14], [-18], [-16], [-4], [4], [0], [-18], [-14], [-24], [-30], [-4], [18], [8], [22], [16], [30], [-6], [16], [-6], [20], [-24], [34], [26], [-20], [-26], [-24], [14], [-6], [8], [-32], [-26], [20], [12], [-18], [-18], [-30], [-22], [36], [-2], [8], [6], [-32], [-4], [-38], [-22], [-42], [-4], [-12], [-24], [12], [-12], [-20], [-32], [-2], [10], [24], [14], [16], [-2], [-4], [-30], [-4], [-2], [-8], [-6], [-16], [-30], [-8], [46], [-2], [20], [16], [2], [8], [28], [-6], [-4], [6], [26], [42], [-16], [28], [-34], [-26], [24], [-8], [34], [20], [-48], [8], [34], [-6], [26], [-14], [-24], [-6], [26], [44], [-10], [24], [12], [-18], [-48], [10], [-22], [36], [44], [-26], [-14], [16], [-32], [28], [48], [16], [-30], [10], [-18], [-28], [-30], [-8], [20], [-30], [0], [-16], [34], [-46], [26], [60], [-2], [56], [-18], [-40], [-54], [-28], [-50], [8], [14], [-24], [28], [22], [10], [-4], [-10], [-38], [20], [10], [24], [26], [-20], [-28], [14], [-52], [-38], [-22], [-2], [-14], [-60], [30], [40], [-22], [10], [-44], [-42], [56], [-8], [26], [-60], [-26], [-10], [8], [-28], [16], [42], [60], [42], [-68], [42], [-18], [-40], [50], [-32], [48], [22], [-66], [-8], [48], [36], [26], [-44], [-32], [42], [-8], [-64], [-14], [42], [-12], [-40], [-60], [-60], [-68], [6], [10], [8], [-28], [28], [12], [-48], [2], [18], [24], [26], [34], [44], [70], [4], [-10], [-14], [48], [-36], [-58], [42], [38], [-52], [30], [50], [-4], [54], [78], [-56], [-14], [16], [46], [8], [-12], [-34], [50], [44], [-60], [-16], [4], [-58], [60], [80], [-42], [58], [-8], [-14], [-18], [16], [-6], [44], [34], [46], [56], [-26], [-84], [-12], [-50], [46], [56], [20], [44], [-38], [-36], [46], [-16], [6], [4], [6], [-30], [-88], [12], [-70], [-76], [-40], [-86], [58], [-52], [-62], [30], [16], [50], [74], [-20], [-8], [72], [22], [-10], [6], [0], [84], [44], [-56], [30], [-2], [82], [20], [74], [42], [-18], [80], [38], [-60], [-66], [-20], [56], [-22], [-52], [58], [-34], [44], [-50], [-74], [-32], [44], [30], [-84], [-10], [-38], [-48], [-76], [92], [14], [26], [-12], [-78], [-84], [-92], [-4], [-66], [-40], [-10], [-20], [72], [94], [-78], [2], [-90], [26], [-8], [-2], [-60], [-44], [16], [38], [52], [72], [2], [-42], [20], [12], [-48], [-46], [0], [-54], [36], [22], [62], [38], [-32], [-6], [22], [48], [-42], [58], [4], [28], [26], [-90], [-8], [-4], [6], [70], [-24], [52], [-42], [-48], [-2], [82], [88], [12], [-54], [30], [56], [-14], [-60], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6960_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_6960_2_a_k(:prec:=1)
    chi := MakeCharacter_6960_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_6960_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_6960_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;