
// Make newform 97.2.g.a in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_97_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_97_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_97_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_97_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, 0, -1, 0, 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_97_g();"
function MakeCharacter_97_g()
    N := 97;
    order := 12;
    char_gens := [5];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_97_g_Hecke();"
function MakeCharacter_97_g_Hecke( : Kf := false)
    N := 97;
    order := 12;
    char_gens := [5];
    char_values := [[0, 1, 0, 0]];
    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 := [[-2, 0, 1, 0], [-2, 1, 1, -1], [-1, -2, -1, 1], [0, 1, -1, -1], [-2, 0, -2, 0], [-1, -1, 0, 0], [-3, 3, 0, -3], [-3, -2, 2, 3], [-6, 6, 6, 0], [4, -1, -5, 5], [-6, 1, 3, -1], [3, 4, -7, -7], [-6, -3, 3, -3], [0, -3, -3, -3], [6, -8, 0, 4], [0, 3, 0, -3], [-4, 2, 2, 2], [0, 1, 0, 1], [-2, -2, 2, 2], [1, 1, 7, -8], [0, -6, -4, -6], [0, 0, 0, 6], [1, -1, 7, 8], [-2, 0, 4, -3], [-9, 0, 0, -4], [3, 8, -3, -16], [4, -6, -4, 12], [-5, -5, -5, 10], [0, 0, 0, 11], [0, 6, -6, 6], [-3, 4, -4, 3], [-11, -2, -2, -11], [-5, 4, 1, 1], [1, -4, -4, 1], [-11, -10, 10, 11], [0, 2, -6, 2], [-5, -6, 11, 11], [-10, -9, 5, 9], [-2, 0, 4, -6], [5, -5, -13, -8], [13, 13, -5, -8], [-2, 3, 5, -5], [8, 0, -4, 0], [6, 16, 0, -8], [14, 6, -7, -6], [-2, -3, -1, 1], [-7, 7, 1, -6], [-8, -5, 3, -3], [9, -18, 0, 9], [-10, 0, 10, 0], [-11, -11, 13, -2], [-3, 0, 0, 3], [21, 8, 0, -4], [-20, 7, 13, 13], [-16, -1, -1, -16], [3, 0, 0, 3], [-18, 2, 0, -1], [8, 2, -2, -8], [3, -3, 10, 13], [-3, -3, -15, 18], [10, 0, -20, 6], [-2, -6, 1, 6], [0, 6, -22, 6], [-3, -12, 12, 3], [-14, 18, 0, -9], [2, 2, -1, -1], [-20, 6, 14, 14], [7, -6, -6, 7], [10, 8, -2, 2], [-3, -3, 5, -2], [9, -4, -9, 8], [2, 13, 11, -11], [-3, 3, 1, -2], [-13, -13, 22, -9], [16, 0, -16, 0], [0, 18, 18, -18], [-15, 8, 0, -4], [6, 2, -6, -4], [-11, 2, 13, -13], [-12, 12, 1, -11], [6, -3, -3, 3], [-3, 40, 0, -20], [0, 3, 3, 3], [-5, -3, 3, 5], [20, -2, -2, 20], [-16, -4, -4, -16], [0, 7, 6, 7], [-7, -2, -2, -7], [0, -5, -6, -5], [-15, -6, 0, 3], [-17, 0, 34, -3], [-12, 6, 12, -12], [16, 12, -8, -12], [15, -11, -15, 22], [2, -2, -20, -18], [9, 0, -18, -9], [0, -14, -6, -14], [0, 5, -18, 5], [-2, 5, -3, -3], [5, 9, 4, -4], [11, 3, -11, -6], [6, -12, 6, 0], [-6, 6, -6, 6], [12, -15, -27, 27], [3, -5, 3, 0], [-9, -4, 5, -5], [-12, -15, -3, 3], [-8, 0, -8, 0], [7, -7, -11, -4], [14, -3, 3, -14], [0, 22, 0, 0], [-24, -14, 12, 14], [0, 17, 0, -34], [-26, 9, 17, 17], [18, -2, 18, 0], [3, 6, -9, -9], [0, -4, 18, -4], [1, -15, 1, 0], [5, 5, -22, 17], [-5, -22, 22, 5], [4, 0, -8, 15], [-7, 0, 7, 0], [-12, 12, 6, -12], [-48, -2, 48, 4], [-23, 0, 46, -9], [12, 22, 0, -11], [-19, -25, 25, 19], [6, 12, -18, -18], [-13, 3, -13, 0], [0, -8, -12, -8], [12, -5, -7, -7], [30, -12, 0, 6], [0, -32, 0, 0], [1, 10, -10, -1], [9, 9, -12, 3], [-9, 8, 1, 1], [38, -6, -19, 6], [-4, 36, -4, 0], [31, -31, -26, 5], [-24, -10, 0, 5], [7, -9, -7, 18], [-21, 6, -6, 21], [-24, 20, 0, -10], [-3, -18, -18, -3], [30, 6, -15, -6], [-14, 4, 4, -14], [-9, -29, 29, 9], [0, 15, 24, 15], [-26, 26, 12, -14], [-8, -8, 22, -14], [15, 10, -15, -20], [1, 0, -2, 18], [-28, -28, 4, 24], [-2, 2, 10, 8], [-2, 20, 20, -2], [16, -5, -11, -11], [-11, -10, -10, -11], [5, -17, -22, 22], [-33, 16, 0, -8], [-3, 12, 15, -15], [-3, 3, -15, -18], [5, -5, 2, 7], [-36, -12, 18, 12], [18, -4, 0, 2], [-1, -13, 14, 14], [8, -17, -25, 25], [-14, 14, -22, -36], [-9, 0, 18, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_97_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_97_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_97_2_g_a(:prec:=4)
    chi := MakeCharacter_97_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_97_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_97_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_97_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 3, 1]>],Snew);
    return Vf;
end function;