
// Make newform 9702.2.a.bu in Magma, downloaded from the LMFDB on 27 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9702_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_9702_2_a_bu();".
// 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_9702_2_a_bu();".
// 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_9702_a();"
function MakeCharacter_9702_a()
    N := 9702;
    order := 1;
    char_gens := [4313, 199, 5293];
    v := [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_9702_a_Hecke( : Kf := Kf)
    return MakeCharacter_9702_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 := [[1], [0], [0], [0], [1], [4], [-6], [4], [-6], [-6], [-8], [-10], [6], [8], [-6], [0], [0], [-8], [-4], [-6], [-2], [14], [-12], [-6], [-14], [6], [4], [12], [-4], [-18], [14], [-12], [18], [4], [6], [-10], [-2], [-4], [12], [6], [-24], [22], [-18], [14], [-6], [4], [8], [16], [-12], [22], [18], [12], [10], [0], [-30], [0], [-24], [-2], [-16], [6], [-8], [-6], [-20], [-18], [-26], [-12], [-4], [2], [-36], [4], [-6], [12], [-8], [20], [20], [6], [0], [-26], [-30], [34], [24], [-10], [0], [-26], [10], [24], [-6], [-10], [42], [-4], [-12], [-24], [20], [-12], [-4], [12], [24], [-18], [16], [20], [-28], [-18], [-12], [-18], [-28], [34], [24], [-30], [-30], [22], [-14], [-16], [30], [-44], [-16], [-6], [4], [6], [36], [-36], [22], [14], [30], [24], [-20], [6], [26], [30], [28], [4], [8], [-36], [8], [-34], [18], [34], [0], [-32], [0], [42], [-8], [18], [8], [12], [-14], [18], [-8], [54], [-20], [-42], [-52], [-30], [-28], [-36], [20], [42], [2], [30], [22], [-18], [-12], [42], [14], [0], [-54], [30], [56], [-44], [-34], [12], [12], [-50], [60], [-46], [40], [-6], [-4], [-36], [-2], [4], [32], [-60], [14], [6], [-54], [-12], [56], [4], [14], [0], [10], [-60], [-28], [48], [12], [66], [-22], [-22], [30], [-24], [54], [-2], [-26], [46], [-48], [-12], [4], [36], [6], [-44], [-10], [-12], [-64], [-12], [-12], [10], [38], [18], [18], [36], [56], [58], [-6], [8], [60], [44], [-6], [30], [-62], [36], [14], [-8], [74], [6], [52], [72], [46], [6], [-24], [-30], [12], [-56], [-32], [-70], [18], [-36], [-20], [-36], [-4], [12], [32], [6], [-42], [10], [-6], [12], [38], [-8], [-12], [-14], [-46], [-12], [4], [-44], [-18], [4], [-36], [-30], [44], [-30], [-50], [-28], [-14], [-58], [70], [-20], [12], [80], [-46], [60], [0], [20], [12], [28], [-68], [-6], [14], [-54], [28], [-30], [48], [24], [6], [-36], [44], [18], [76], [-72], [-60], [40], [10], [-18], [-46], [12], [68], [-22], [60], [-14], [30], [-4], [-36], [-84], [18], [-4], [72], [-50], [60], [-72], [-2], [-30], [-44], [38], [42], [-28], [6], [-14], [20], [28], [36], [42], [-64], [72], [-74], [84], [-88], [-36], [-70], [18], [-26], [34], [26], [42], [66], [56], [-24], [12], [-2], [20], [-36], [48], [40], [-94], [24], [-62], [-40], [-42], [-18], [60], [-78], [-42], [-16], [-6], [-30], [-36], [4], [26], [-30], [38], [14], [-24], [28], [-72], [66], [-14], [-10], [-84], [36], [46], [-66], [10], [96], [-54], [-58], [30], [-8], [-30], [80], [58], [-16], [-102], [-58], [-36], [72], [-68], [-66], [14], [-26], [-18], [-100], [90], [40], [42], [-22], [-42], [66], [-2], [38], [66], [4], [84], [-98], [78], [84], [-28], [50], [-12], [-12], [-20], [54], [36], [-12], [-32], [-102], [84], [10], [24], [60], [-30], [52], [-48], [-50], [36], [-4], [-6], [22], [78], [98], [-46], [-100], [-38], [-108], [-42], [-76], [60], [-50], [42], [-80], [-36], [-62], [76], [-88], [-102], [60], [-78], [50], [72], [68], [-36], [-68], [-66], [-52], [68], [-36], [62], [-56], [-46], [-82], [-36], [-18], [-92], [-94], [-60], [-30], [50], [36], [-8], [36], [76], [36], [72], [-38], [90], [-74], [-78], [-80], [108], [26], [-60], [-44], [26], [28], [-24], [38], [-18], [-12], [40], [-68], [-36], [-2], [-28], [-66], [-86], [30], [98], [-22], [6], [54], [-104], [116], [-8], [36], [-96], [-2], [-12], [8], [14], [36], [40], [-66], [52], [-46], [-76], [30], [-18], [-14], [12], [-14], [78], [-60], [36], [-82], [54], [50], [-48], [-2], [84], [-38], [-90], [38], [20], [-72], [48], [124], [-108], [42], [20], [-52], [84], [118], [18], [-102], [4], [-18], [-84], [0], [64], [-28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9702_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9702_2_a_bu();".
// 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_9702_2_a_bu(:prec:=1)
    chi := MakeCharacter_9702_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(4027) - 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_9702_2_a_bu();".
// 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_9702_2_a_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9702_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![-4, 1]>,<17,R![6, 1]>,<19,R![-4, 1]>,<23,R![6, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;