
// Make newform 115.3.d.a in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_115_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_115_d_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_115_3_d_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_115_3_d_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 := [4725, -600, 690, 50, -34, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-3002445, -193350, 221635, 5812, -5599, 62], [-2585, 4975, 1167, -416, 172, 7], [-88575, 205185, 7360, -6668, -439, 272], [88575, 44100, -7360, 6668, 439, -272], [262830, -581535, -32506, 30371, 706, -551]];
    Rf_basisdens := [1, 249285, 16619, 249285, 249285, 249285];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_115_d();"
function MakeCharacter_115_d()
    N := 115;
    order := 2;
    char_gens := [47, 51];
    v := [2, 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_115_d_Hecke();"
function MakeCharacter_115_d_Hecke( : Kf := false)
    N := 115;
    order := 2;
    char_gens := [47, 51];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 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 := 3;
    raw_aps := [[-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 0, 0], [0, 0, -1, 1, 0, -1], [0, 0, 1, 2, 0, -2], [-1, 1, 0, 0, -1, 0], [0, 0, -1, 4, 0, 2], [0, 0, 1, 5, 0, 1], [-2, 1, 0, -3, 2, -3], [14, 0, 0, 0, -6, 0], [-1, -3, 0, 0, -3, 0], [0, 0, 2, -8, 0, 2], [26, 2, 0, 0, -5, 0], [0, 0, 2, -2, 0, 8], [20, 2, 0, 0, -2, 0], [0, 0, 0, 18, 0, 0], [38, 4, 0, 0, 8, 0], [0, 0, 3, 30, 0, -6], [0, 0, -4, -8, 0, 2], [-34, 2, 0, 0, -5, 0], [2, 0, 0, 0, -6, 0], [0, 0, -6, 30, 0, -6], [0, 0, 4, 8, 0, -14], [0, 0, -6, -12, 0, -12], [0, 0, 7, -40, 0, -2], [74, -4, 0, 0, 10, 0], [0, 0, 3, 0, 0, 18], [0, 0, -2, 62, 0, -8], [0, 0, 5, -47, 0, 5], [0, 0, -4, 10, 0, -10], [-70, -8, 0, 0, 8, 0], [44, -2, 0, 0, -16, 0], [0, 0, -13, -23, 0, -1], [104, -10, 0, 0, -26, 0], [0, 0, -9, 39, 0, -21], [-22, 10, 0, 0, -1, 0], [0, 0, -8, 8, 0, -14], [-13, -3, 0, 0, 3, 0], [14, -4, 0, 0, -8, 0], [-76, 12, 0, 0, -39, 0], [20, -2, 0, 0, -16, 0], [0, 0, 9, 45, 0, 21], [0, 0, -12, -36, 0, 12], [116, 14, 0, 0, 28, 0], [-76, -12, 0, 0, 21, 0], [0, 0, 0, 0, 0, -36], [-64, 6, 0, 0, -42, 0], [-64, -2, 0, 0, -4, 0], [0, 0, 22, -46, 0, 16], [0, 0, -20, 74, 0, -32], [200, -2, 0, 0, 2, 0], [122, -4, 0, 0, -32, 0], [0, 0, -34, 16, 0, -4], [0, 0, 21, 69, 0, -3], [-46, 4, 0, 0, 68, 0], [0, 0, 23, -32, 0, -22], [-70, 16, 0, 0, 20, 0], [-13, -11, 0, 0, -67, 0], [-82, 8, 0, 0, -68, 0], [0, 0, 18, -174, 0, 18], [0, 0, -18, -60, 0, 6], [0, 0, -2, 2, 0, -44], [-268, -24, 0, 0, -21, 0], [26, -4, 0, 0, 28, 0], [0, 0, -27, -57, 0, 33], [-283, 15, 0, 0, 57, 0], [-328, 10, 0, 0, 80, 0], [0, 0, 9, 75, 0, 57], [68, 20, 0, 0, -59, 0], [68, 6, 0, 0, -48, 0], [-148, -6, 0, 0, -18, 0], [0, 0, 6, 12, 0, -36], [0, 0, -36, -66, 0, -6], [0, 0, 14, 10, 0, -4], [0, 0, -9, -162, 0, 6], [0, 0, -4, 46, 0, -70], [0, 0, 11, 106, 0, 2], [-76, 32, 0, 0, 79, 0], [0, 0, 18, -168, 0, -36], [-37, -11, 0, 0, -7, 0], [0, 0, 12, 54, 0, 48], [0, 0, 9, -3, 0, -51], [0, 0, -8, 122, 0, 82], [0, 0, -41, -7, 0, 43], [-409, 9, 0, 0, -15, 0], [-139, -17, 0, 0, -19, 0], [-199, -25, 0, 0, 67, 0], [0, 0, 24, -54, 0, 90], [182, -4, 0, 0, 112, 0], [404, -18, 0, 0, 36, 0], [0, 0, 42, 156, 0, -54], [0, 0, 12, 102, 0, 54], [116, -22, 0, 0, 94, 0], [-22, -16, 0, 0, 52, 0], [152, 6, 0, 0, -102, 0], [0, 0, -15, -57, 0, 9], [-46, 36, 0, 0, 66, 0], [0, 0, 78, -132, 0, 36], [0, 0, -18, 72, 0, 42], [-22, -40, 0, 0, -56, 0], [329, -21, 0, 0, 33, 0], [0, 0, -14, -106, 0, -92], [0, 0, -36, 12, 0, -42], [0, 0, -42, 228, 0, -36], [0, 0, 45, -114, 0, 90], [98, -16, 0, 0, 34, 0], [584, -24, 0, 0, -3, 0], [164, -18, 0, 0, 126, 0], [62, 42, 0, 0, -15, 0], [-199, -9, 0, 0, 63, 0], [62, 12, 0, 0, 6, 0], [0, 0, 72, -216, 0, 54], [0, 0, 33, -120, 0, -42], [0, 0, 7, 239, 0, -29], [0, 0, -6, 150, 0, 150], [0, 0, -22, 184, 0, -28], [0, 0, -6, 216, 0, -18], [-106, 24, 0, 0, -192, 0], [575, 25, 0, 0, 11, 0], [0, 0, -20, -22, 0, 4], [0, 0, -3, 6, 0, -102], [-34, -36, 0, 0, 78, 0], [0, 0, -54, -300, 0, 30], [281, -5, 0, 0, 185, 0], [-262, 40, 0, 0, 44, 0], [0, 0, 71, 139, 0, -13], [0, 0, -1, -386, 0, 14], [233, -1, 0, 0, -137, 0], [0, 0, 33, 288, 0, -42], [0, 0, -30, 132, 0, -174], [-508, -6, 0, 0, -66, 0], [0, 0, -27, -33, 0, -63], [0, 0, -10, -20, 0, -16], [0, 0, -4, 304, 0, 62], [929, 27, 0, 0, 27, 0], [0, 0, -44, -190, 0, 58], [0, 0, -14, 128, 0, -158], [0, 0, 8, 16, 0, -142], [0, 0, 36, 66, 0, -132], [-94, -38, 0, 0, 41, 0], [-634, 0, 0, 0, -54, 0], [92, -58, 0, 0, 64, 0], [440, 6, 0, 0, -12, 0], [0, 0, 84, 42, 0, 48], [170, -40, 0, 0, 208, 0], [0, 0, -24, 156, 0, 168], [173, 23, 0, 0, -71, 0], [158, 28, 0, 0, 152, 0], [-352, -38, 0, 0, 164, 0], [-484, 66, 0, 0, 252, 0], [-880, 52, 0, 0, 203, 0], [0, 0, -118, 22, 0, -46], [707, -35, 0, 0, -61, 0], [434, -24, 0, 0, -48, 0], [0, 0, 28, 272, 0, -50], [0, 0, -72, -30, 0, -102], [0, 0, 6, -330, 0, -90], [194, 76, 0, 0, -4, 0], [0, 0, 79, -55, 0, -41], [0, 0, -21, 78, 0, -54], [-799, 11, 0, 0, 25, 0], [0, 0, 63, -180, 0, 222], [872, 14, 0, 0, -80, 0], [0, 0, 33, -42, 0, 90], [0, 0, -99, -105, 0, -3], [0, 0, 39, 48, 0, 18], [-46, -82, 0, 0, -191, 0], [650, 116, 0, 0, 16, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_115_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_115_3_d_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_115_3_d_a(:prec:=6)
    chi := MakeCharacter_115_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_115_3_d_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_115_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_115_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![1, 1]>],Snew);
    return Vf;
end function;