
// Make newform 9680.2.a.bd in Magma, downloaded from the LMFDB on 13 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9680_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_9680_2_a_bd();".
// 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_9680_2_a_bd();".
// 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_9680_a();"
function MakeCharacter_9680_a()
    N := 9680;
    order := 1;
    char_gens := [3631, 2421, 1937, 4721];
    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_9680_a_Hecke( : Kf := Kf)
    return MakeCharacter_9680_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], [3], [-1], [5], [0], [4], [-1], [-5], [-4], [-5], [1], [3], [-2], [8], [-6], [11], [0], [5], [4], [15], [-6], [8], [8], [1], [10], [-18], [6], [2], [2], [-8], [0], [7], [12], [-12], [-7], [-20], [11], [23], [-15], [-18], [2], [-8], [12], [23], [-18], [-5], [7], [-26], [18], [-8], [-9], [-22], [-2], [8], [-12], [19], [4], [-24], [14], [-18], [-28], [14], [-2], [1], [-10], [-31], [12], [-31], [18], [-6], [4], [-18], [-20], [2], [16], [34], [-8], [22], [-1], [-26], [-18], [-30], [-28], [-22], [-34], [32], [18], [17], [-23], [32], [11], [6], [32], [19], [-38], [4], [-30], [-22], [26], [19], [42], [0], [-26], [-14], [25], [-32], [5], [34], [-15], [-22], [3], [-24], [30], [-2], [19], [35], [-13], [-12], [-3], [33], [-18], [15], [10], [29], [18], [-15], [-10], [-11], [-20], [32], [16], [25], [-3], [-26], [-22], [22], [7], [-14], [-42], [-48], [17], [6], [26], [-22], [12], [-52], [-40], [11], [-36], [-42], [22], [-10], [13], [-16], [-9], [-37], [32], [-21], [-38], [3], [-7], [25], [-41], [-38], [38], [36], [-40], [-48], [-16], [-29], [-53], [32], [18], [-1], [8], [-29], [-44], [-54], [11], [37], [20], [-21], [61], [51], [0], [32], [-52], [-20], [-20], [-6], [28], [24], [50], [12], [-42], [42], [32], [30], [30], [-1], [-61], [10], [-23], [18], [26], [-9], [-8], [52], [44], [50], [1], [-44], [26], [57], [-18], [-1], [11], [0], [28], [-43], [22], [22], [-35], [-48], [-12], [25], [22], [-57], [-32], [-12], [50], [-39], [-10], [-16], [36], [7], [-5], [64], [38], [29], [53], [-28], [38], [38], [-66], [-2], [-10], [54], [-5], [3], [-12], [-54], [-22], [31], [18], [27], [-14], [-46], [30], [-43], [-52], [64], [41], [-14], [76], [-42], [30], [33], [-6], [36], [40], [-47], [66], [4], [-37], [-14], [0], [2], [-52], [69], [-67], [24], [72], [5], [-20], [-15], [-22], [-84], [63], [6], [54], [44], [71], [57], [2], [14], [-32], [-87], [-68], [54], [-39], [78], [14], [12], [50], [-16], [4], [-44], [-3], [16], [40], [36], [12], [55], [1], [-82], [-66], [-2], [-44], [72], [-63], [2], [78], [-22], [81], [87], [-10], [52], [-59], [-88], [-53], [50], [70], [78], [-25], [-69], [-2], [79], [13], [33], [74], [-48], [51], [23], [39], [-67], [10], [25], [-28], [-23], [-77], [-48], [80], [43], [-19], [37], [-31], [25], [-54], [-96], [-50], [50], [0], [-59], [39], [16], [-50], [21], [-58], [-4], [70], [-64], [41], [30], [22], [-46], [62], [-97], [78], [-11], [6], [-78], [12], [9], [12], [-28], [-40], [5], [-18], [-84], [-43], [-3], [56], [68], [24], [2], [-52], [-69], [94], [21], [99], [-24], [8], [-98], [10], [12], [-59], [-72], [25], [-54], [38], [-32], [20], [-28], [-94], [-82], [104], [-76], [-30], [29], [-56], [-10], [92], [51], [12], [-67], [-19], [-78], [22], [-24], [-31], [-97], [34], [-48], [69], [90], [30], [24], [-68], [-34], [81], [55], [21], [0], [-100], [3], [-40], [63], [-103]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9680_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9680_2_a_bd();".
// 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_9680_2_a_bd(:prec:=1)
    chi := MakeCharacter_9680_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(3169) - 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_9680_2_a_bd();".
// 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_9680_2_a_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9680_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<7,R![-5, 1]>,<13,R![-4, 1]>,<17,R![1, 1]>],Snew);
    return Vf;
end function;