
// Make newform 9600.2.a.i in Magma, downloaded from the LMFDB on 03 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9600_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_9600_2_a_i();".
// 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_9600_2_a_i();".
// 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_9600_a();"
function MakeCharacter_9600_a()
    N := 9600;
    order := 1;
    char_gens := [4351, 901, 6401, 5377];
    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_9600_a_Hecke( : Kf := Kf)
    return MakeCharacter_9600_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], [0], [-2], [6], [2], [2], [2], [6], [10], [4], [-2], [6], [0], [6], [10], [-6], [6], [-16], [-8], [-10], [0], [-12], [6], [-10], [10], [10], [4], [6], [18], [2], [-22], [2], [14], [18], [0], [-18], [-12], [-6], [-18], [-18], [-18], [-12], [22], [2], [4], [-10], [-2], [8], [-26], [26], [0], [18], [-18], [2], [14], [-6], [4], [14], [18], [8], [6], [8], [24], [6], [-14], [-30], [-18], [12], [22], [10], [-4], [-22], [38], [-18], [34], [-30], [-14], [-34], [6], [-10], [38], [28], [-18], [-4], [24], [-6], [22], [2], [6], [24], [-24], [-26], [14], [34], [-34], [-22], [34], [-16], [-26], [16], [-18], [4], [-42], [-46], [-18], [-4], [10], [-36], [10], [26], [26], [-6], [6], [-16], [18], [-12], [42], [-22], [-26], [38], [6], [-30], [-8], [-26], [10], [38], [-16], [38], [22], [2], [-50], [28], [-18], [2], [-26], [-6], [-48], [18], [42], [26], [18], [2], [-4], [38], [36], [-22], [2], [-2], [18], [46], [14], [20], [-46], [28], [20], [4], [58], [-2], [-38], [32], [2], [22], [54], [-46], [-26], [-60], [-6], [38], [-54], [18], [-2], [-48], [14], [-56], [-10], [2], [42], [58], [30], [42], [-46], [26], [26], [26], [-38], [-34], [36], [-26], [28], [-26], [0], [22], [-30], [-32], [-54], [30], [-30], [42], [22], [-6], [-60], [-6], [-62], [-38], [-66], [32], [-60], [6], [18], [-66], [-14], [-22], [-60], [68], [-2], [-30], [-18], [50], [-6], [-2], [4], [30], [-10], [56], [-10], [18], [32], [6], [30], [-14], [-14], [12], [18], [56], [-42], [18], [-54], [-22], [0], [4], [-70], [18], [14], [42], [-44], [26], [-30], [46], [-46], [66], [-14], [26], [-70], [-50], [6], [-42], [-68], [38], [14], [-26], [-8], [46], [-26], [18], [-14], [-6], [54], [-80], [38], [-74], [32], [-50], [80], [-26], [-54], [4], [22], [10], [-22], [18], [32], [58], [-58], [-20], [20], [14], [66], [-44], [-66], [-54], [-56], [42], [14], [-26], [-14], [-76], [-42], [-6], [8], [-18], [14], [0], [-36], [74], [30], [-4], [30], [-12], [-42], [-30], [-22], [14], [28], [-6], [54], [78], [28], [62], [-82], [22], [-50], [42], [46], [-14], [50], [34], [16], [30], [-22], [70], [30], [-24], [12], [10], [76], [-34], [-78], [-22], [42], [22], [-6], [-30], [-8], [-66], [-2], [-34], [-12], [4], [50], [22], [30], [34], [46], [62], [-30], [-72], [14], [-6], [-58], [58], [-42], [54], [18], [24], [-34], [-14], [18], [-34], [10], [14], [58], [50], [0], [2], [46], [76], [46], [-22], [-82], [42], [-54], [22], [82], [-14], [94], [20], [62], [-48], [62], [22], [-90], [-14], [-80], [-64], [-58], [-56], [-54], [-6], [-30], [-34], [18], [82], [-46], [-30], [-40], [-46], [-34], [-44], [-18], [-26], [-106], [64], [38], [38], [34], [80], [62], [18], [-34], [90], [-54], [-50], [18], [62], [-18], [-36], [26], [98], [-60], [10], [-94], [54], [-70], [-30], [34], [-10], [-58], [-12], [44], [8], [-6], [-10], [0], [82], [66], [-48], [86], [-26], [14], [-16], [24], [-76], [54], [-58], [42], [14], [-22], [-74], [26], [94], [-32], [-10], [-26], [52], [-26], [-68], [-24], [-18], [-26], [-10], [40], [40], [94], [-90], [-58], [66], [-20], [6], [86], [-50], [26], [14], [10], [42], [60], [-106], [98], [-50], [112], [66], [42], [-90], [90], [-90], [-50], [-20], [-62], [100], [-2], [66], [-66], [66], [-26], [-62], [-94], [-66], [20], [30], [-56], [18], [-40], [110], [-34], [-30], [78], [-54], [-26], [28], [106], [74], [-2], [-2], [2], [-10], [6], [14], [102], [-24], [90], [-106], [10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9600_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9600_2_a_i();".
// 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_9600_2_a_i(:prec:=1)
    chi := MakeCharacter_9600_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(3833) - 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_9600_2_a_i();".
// 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_9600_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-6, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<29,R![-10, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;