
// Make newform 9600.2.a.bp in Magma, downloaded from the LMFDB on 19 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_bp();".
// 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_bp();".
// 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], [0], [2], [-4], [0], [0], [0], [2], [-10], [-4], [6], [-4], [0], [-2], [6], [6], [-4], [-8], [-14], [-2], [-4], [14], [6], [-18], [-4], [-16], [10], [-8], [12], [18], [-12], [16], [-6], [-6], [-8], [20], [-8], [-14], [-2], [-22], [12], [-6], [6], [-2], [-16], [-4], [-24], [14], [-20], [20], [-2], [6], [24], [-24], [-14], [-26], [8], [-18], [28], [6], [4], [-20], [-14], [22], [20], [34], [24], [-2], [4], [-12], [8], [-24], [-4], [-32], [-10], [28], [10], [-34], [6], [-34], [12], [2], [38], [32], [2], [-18], [-14], [16], [12], [-16], [-28], [26], [-20], [-32], [-34], [-38], [-12], [2], [4], [26], [-16], [10], [-4], [-18], [-24], [-16], [-4], [-22], [40], [4], [12], [48], [14], [-2], [36], [32], [-14], [-6], [-10], [-46], [-38], [12], [8], [30], [30], [-20], [4], [24], [36], [-32], [-26], [-48], [-38], [10], [-26], [12], [10], [-10], [20], [-6], [44], [12], [26], [-52], [28], [-16], [20], [40], [24], [14], [36], [56], [-52], [-24], [38], [-6], [-2], [46], [-12], [36], [28], [54], [16], [48], [-30], [0], [-30], [-54], [-42], [34], [-16], [-30], [-26], [54], [-40], [10], [-32], [-22], [-20], [-34], [16], [-8], [24], [54], [20], [-44], [-18], [4], [-54], [-4], [24], [42], [32], [24], [-2], [44], [48], [-32], [-18], [2], [-4], [14], [10], [-18], [34], [-64], [-42], [-56], [10], [-30], [28], [-36], [-48], [50], [8], [58], [-8], [18], [22], [70], [22], [56], [12], [-22], [40], [-60], [-40], [-2], [-12], [60], [22], [-42], [-4], [-32], [-74], [10], [-54], [48], [-8], [-20], [60], [22], [52], [32], [-16], [-58], [-4], [72], [-48], [-30], [40], [-10], [-18], [-54], [-78], [-12], [-18], [-6], [-44], [60], [22], [52], [-4], [36], [-50], [10], [12], [66], [50], [-4], [18], [-34], [82], [52], [-24], [22], [38], [-46], [56], [54], [-24], [-46], [52], [4], [70], [-6], [50], [-46], [14], [32], [-36], [-50], [44], [50], [10], [54], [10], [4], [34], [-6], [58], [-56], [-40], [-50], [28], [78], [56], [8], [40], [10], [18], [76], [-88], [50], [-34], [72], [26], [-6], [68], [50], [78], [20], [16], [-14], [4], [-28], [-32], [10], [-26], [-6], [26], [-48], [-56], [-76], [-30], [12], [34], [-20], [24], [16], [-66], [30], [6], [-14], [62], [20], [48], [86], [76], [-50], [-34], [44], [-26], [-12], [-24], [54], [-76], [-32], [24], [82], [96], [98], [-20], [-66], [-2], [8], [-10], [-30], [-60], [0], [-22], [-46], [-56], [42], [64], [-34], [74], [6], [-30], [36], [40], [36], [-44], [-96], [-2], [-88], [-52], [-32], [50], [-22], [22], [-52], [-40], [14], [14], [78], [-88], [22], [50], [-60], [92], [-84], [-6], [30], [-8], [26], [-100], [-34], [-50], [-38], [-60], [-32], [42], [18], [-8], [80], [-92], [80], [-30], [56], [-80], [30], [18], [-42], [-76], [-30], [56], [36], [-34], [78], [4], [72], [-68], [6], [-10], [70], [36], [-110], [72], [70], [10], [72], [-66], [-44], [28], [104], [78], [-26], [-28], [0], [60], [-62], [-22], [54], [-30], [-18], [-76], [24], [16], [-82], [6], [66], [108], [-22], [50], [12], [6], [68], [-20], [68], [12], [-98], [42], [80], [-54], [74], [-64], [-30], [-58], [-22], [102], [78], [-56], [-64], [30], [-90], [-44], [-6], [-28], [0], [-90], [54], [94], [78], [-92], [-98], [-94], [44], [-74], [-64], [-32], [-44], [32], [-72], [-72], [-26], [8], [-44], [-90], [-60], [78], [102], [-8], [-46], [6], [-46], [0], [88], [8], [68], [-58], [-48], [90], [-102], [-10], [78], [0], [-14], [-96], [-76]];
    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_bp();".
// 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_bp(: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_bp();".
// 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_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<17,R![0, 1]>,<29,R![-2, 1]>,<31,R![10, 1]>],Snew);
    return Vf;
end function;