
// Make newform 961.2.g.a in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_961_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_961_g_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_961_2_g_a();".
// 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_961_2_g_a();".
// 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
        poly := [1, -1, 0, 1, -1, 1, 0, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_961_g();"
function MakeCharacter_961_g()
    N := 961;
    order := 15;
    char_gens := [3];
    v := [13];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_961_g_Hecke();"
function MakeCharacter_961_g_Hecke( : Kf := false)
    N := 961;
    order := 15;
    char_gens := [3];
    char_values := [[1, 0, -1, 1, -1, 0, 1, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, -1, 0], [-2, 0, 0, 0, 2, -2, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [-2, -1, 0, 0, 0, -2, 0, 2], [0, -2, 0, 0, 0, 0, -2, 0], [2, 0, -2, 2, -2, 2, 2, -2], [0, 2, 0, 0, 4, 0, 0, 2], [-1, 1, 0, 0, -1, -1, 0, -1], [4, 0, 2, 4, 0, 0, -2, 2], [2, 0, 0, 6, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 2, 0, 0], [-7, 7, 0, -7, 7, -7, 0, 7], [0, 0, 0, 0, 2, 0, 0, 2], [0, 0, 0, 4, 0, 0, 4, 0], [-4, -4, 8, -4, 4, 0, -8, 4], [2, 0, 0, 0, -2, 2, 0, -1], [-8, 0, 10, -10, 0, 0, 0, 10], [0, 0, 0, 0, 0, 8, 0, 0], [7, -10, 3, 7, -7, 0, -3, -7], [0, -6, 4, 0, 0, 4, -6, 0], [0, 6, 0, 0, -2, 0, 0, 6], [2, -10, 0, 2, -2, 10, -8, -2], [6, 0, -6, 0, 0, 0, 8, -6], [-8, 0, 8, 0, 0, 0, -11, 8], [3, 0, -3, 3, 0, 0, 3, -3], [-3, 5, 0, -3, 3, -5, 2, 3], [0, 2, 0, 0, 9, 0, 0, 2], [0, 0, 9, -8, 0, 0, -8, 9], [0, -1, 4, 0, 0, 4, -1, 0], [4, 0, 0, 0, -4, 4, 0, 6], [0, 12, 0, 0, 0, 0, 0, 0], [6, 0, 10, 6, -6, 6, 6, -6], [0, 0, -6, 12, 0, 0, 12, -6], [-10, 0, 0, 0, 0, -10, 0, 0], [-10, 0, 10, 0, 0, 0, -8, 10], [-16, 0, 0, -5, 0, 0, -16, 0], [-1, 0, 7, -1, 0, 0, -7, 7], [-4, 4, 0, 0, 0, -4, 0, 0], [-8, 0, -2, -8, 8, -8, -8, 8], [-6, -2, 0, 0, 0, -6, 0, 6], [-10, 10, 10, -10, 10, 2, 0, 10], [-10, 10, 10, -10, 10, -13, 0, 10], [-4, 1, 0, 0, 0, -4, 0, 4], [0, -4, 12, 0, 0, 12, -4, 0], [8, 0, -14, 8, -8, 8, 8, -8], [-17, 10, 0, 0, 10, -17, 0, 0], [-4, 0, 0, 0, 0, -4, 0, 0], [0, -4, 0, 0, 0, 4, -4, 0], [-6, 6, 0, 0, -6, -6, 0, -6], [0, 0, -13, 8, 0, 0, 8, -13], [-2, -6, 8, -2, 2, 0, -8, 2], [-20, 2, 0, 0, 0, -20, 0, 20], [10, 0, 0, 0, -10, 10, 0, -18], [-11, 8, 3, -11, 11, 0, -3, 11], [0, 0, 22, -2, 0, 0, -2, 22], [24, -16, 0, 24, -24, 16, 8, -24], [-2, 0, -8, -2, 0, 0, 8, -8], [14, 0, 0, 10, 0, 0, 14, 0], [0, 0, 0, 17, 0, 0, 0, 0], [-16, 0, 16, 0, 0, 0, -4, 16], [0, 4, 0, 0, 6, 0, 0, 4], [-6, 0, 0, 0, 6, -6, 0, -19], [-13, 0, -10, 10, 0, 0, 0, -10], [-2, 0, 0, 0, 2, -2, 0, 20], [8, 9, 0, 0, 0, 8, 0, -8], [-2, 2, 0, -2, 2, -2, 0, 2], [2, 0, -2, 0, 0, 0, -16, -2], [-18, 10, 0, 0, 10, -18, 0, 0], [-18, 0, 2, -18, 0, 0, -2, 2], [0, 0, 0, 0, 12, 0, 0, 12], [-2, 0, 21, -2, 2, -2, -2, 2], [0, 0, 0, 0, 0, 18, 0, 0], [19, 0, 0, 0, 0, 0, 0, 0], [-16, -12, 0, 0, 0, -16, 0, 16], [0, 14, -16, 0, 0, -16, 14, 0], [0, 16, 0, 0, 8, 0, 0, 16], [7, 0, 0, 0, 0, 7, 0, 0], [-10, 0, 0, 22, 0, 0, -10, 0], [10, 10, 0, 0, 10, 10, 0, 0], [0, 0, -1, -18, 0, 0, -18, -1], [-20, 0, 17, -20, 20, -20, -20, 20], [0, 0, 0, 0, 0, 0, 0, 12], [-6, 0, 10, -10, 0, 0, 0, 10], [-10, 10, 10, -10, 10, 5, 0, 10], [21, 6, -27, 21, -21, 0, 27, -21], [0, 0, 14, -28, 0, 0, -28, 14], [16, 0, -8, 16, 0, 0, 8, -8], [20, 0, -20, 0, 0, 0, 22, -20], [-12, 0, 0, -22, 0, 0, -12, 0], [-6, 0, 0, -5, 0, 0, -6, 0], [0, 6, 0, 0, 33, 0, 0, 6], [-16, -2, 18, -16, 16, 0, -18, 16], [20, -20, -20, 20, -20, -8, 0, -20], [12, 0, 0, 0, -12, 12, 0, -26], [-14, 21, 0, 0, 0, -14, 0, 14], [-22, 0, 16, -22, 22, -22, -22, 22], [-2, 0, 0, 0, 0, -2, 0, 0], [-6, 0, 6, 0, 0, 0, -14, 6], [0, 20, 0, 0, 7, 0, 0, 20], [0, 1, -18, 0, 0, -18, 1, 0], [-12, 0, 0, 0, 0, 0, 0, 0], [30, -30, -30, 30, -30, 9, 0, -30], [0, -8, -4, 0, 0, -4, -8, 0], [-10, 0, 22, -10, 10, -10, -10, 10], [-14, 14, 0, 0, -10, -14, 0, -10], [4, 0, 0, 0, 0, 0, 4, 0], [4, 0, -4, 0, 0, 0, -9, -4], [-21, -1, 0, -21, 21, 1, -22, 21], [12, -12, 0, 0, 18, 12, 0, 18], [-4, 0, 20, -4, 4, -4, -4, 4], [16, -34, 0, 0, 0, 16, 0, -16], [4, 0, 0, 0, -4, 4, 0, 26], [-10, 0, 10, -10, 0, 0, 0, 10], [0, -22, 20, 0, 0, 20, -22, 0], [-12, 12, 0, -12, 12, -12, 0, 12], [-26, 0, 22, -26, 0, 0, -22, 22], [-8, 0, 8, 0, 0, 0, 4, 8], [28, 0, 0, -2, 0, 0, 28, 0], [-3, 0, 17, -3, 0, 0, -17, 17], [33, -13, 0, 33, -33, 13, 20, -33], [26, -34, 8, 26, -26, 0, -8, -26], [-40, 40, 40, -40, 40, -22, 0, 40], [-11, 0, 10, -10, 0, 0, 0, 10], [-13, 10, 3, -13, 13, 0, -3, 13], [7, -7, 0, 0, -7, 7, 0, -7], [-4, 0, 4, 0, 0, 0, 28, 4], [20, 10, 0, 0, 10, 20, 0, 0], [0, 2, 0, 0, -41, 0, 0, 2], [0, -1, 4, 0, 0, 4, -1, 0], [10, -10, -10, 10, -10, 0, 0, -10], [34, 0, 10, -10, 0, 0, 0, 10], [30, -3, 0, 0, 0, 30, 0, -30], [-14, 0, 0, -14, 14, -14, -14, 14], [0, 0, 0, 0, 2, 0, 0, 0], [35, -20, 0, 0, -20, 35, 0, 0], [18, 0, 0, 18, 0, 0, 18, 0], [16, -16, 0, 0, -30, 16, 0, -30], [-36, -12, 48, -36, 36, 0, -48, 36], [12, 0, 0, 0, -12, 12, 0, 4], [0, 0, 0, 0, 0, -28, 0, 0], [0, 0, -12, 30, 0, 0, 30, -12], [0, -6, 0, 0, -24, 0, 0, -6], [-40, 26, 0, -40, 40, -26, -14, 40], [22, 0, 2, 22, 0, 0, -2, 2], [-8, 0, 0, 16, 0, 0, -8, 0], [0, 0, -18, -12, 0, 0, -12, -18], [0, 26, -48, 0, 0, -48, 26, 0], [28, -16, -12, 28, -28, 0, 12, -28], [10, -10, -10, 10, -10, -16, 0, -10], [-31, 8, 23, -31, 31, 0, -23, 31], [20, 0, -8, 20, -20, 20, 20, -20], [0, 0, 12, -32, 0, 0, -32, 12], [11, -11, 0, 0, 15, 11, 0, 15], [21, 0, -23, 21, 0, 0, 23, -23], [12, -12, 0, 0, -2, 12, 0, -2], [0, 12, -4, 0, 0, -4, 12, 0], [-20, 0, 0, 0, 0, 0, 0, 0], [8, 14, 0, 0, 0, 8, 0, -8], [0, 38, 0, 0, 0, 0, 38, 0], [0, -8, 0, 0, -26, 0, 0, -8], [-36, 0, 42, -36, 0, 0, -42, 42], [-48, 20, 0, 0, 20, -48, 0, 0], [28, -28, 0, 28, -28, 28, 0, -28], [0, 0, 25, -36, 0, 0, -36, 25], [46, -4, -42, 46, -46, 0, 42, -46], [32, 0, -30, 30, 0, 0, 0, -30], [-20, 20, 20, -20, 20, -17, 0, 20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_961_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_961_2_g_a();".
// 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_961_2_g_a(:prec:=8)
    chi := MakeCharacter_961_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_961_2_g_a();".
// 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_961_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_961_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 2, 4, 3, 1]>,<3,R![256, 256, 0, 64, 144, 64, 20, 6, 1]>],Snew);
    return Vf;
end function;