
// Make newform 115.2.g.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_115_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_115_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_115_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_115_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, 1, -1, 1, -1, 1, -1, 1, -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_115_g();"
function MakeCharacter_115_g()
    N := 115;
    order := 11;
    char_gens := [47, 51];
    v := [11, 9];
    // 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_g_Hecke();"
function MakeCharacter_115_g_Hecke(Kf)
    N := 115;
    order := 11;
    char_gens := [47, 51];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+1)]];
    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, 1, -1, 1, 0, 1, 0, 0], [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, -1, 0, 0, 1], [0, 1, -1, 0, 0, 1, -2, 0, -2, 1], [0, 0, 0, 0, -1, -1, 0, -1, -1, 0], [-2, -1, 1, -1, 1, -1, -2, 0, 0, 0], [0, -1, 3, -3, 1, 0, 0, -2, 1, -2], [-2, 0, -2, 0, 0, 0, 1, 2, 2, -2], [4, -4, 6, -4, 6, -4, 4, 0, 5, -5], [-1, 1, -1, 0, 2, -1, 3, -3, 1, -2], [0, 0, 0, 6, -6, 0, 0, 0, 0, 1], [-1, -2, -2, -2, -1, 0, 1, 4, -4, -1], [-3, 7, -4, 4, -6, 6, -4, 4, -7, 3], [-6, 0, -4, 0, 1, 1, -1, -1, 0, 4], [0, 5, 2, 1, 1, 0, 1, 1, 2, 5], [1, -1, 0, 0, 0, 2, 8, 2, 0, 0], [0, -1, 0, 0, 0, -3, -2, 2, 3, 0], [1, -1, -4, 0, -5, 0, -4, -1, 1, 0], [-1, -4, 3, 1, 1, 1, 3, -4, -1, 0], [4, 1, 1, -1, -1, -4, 0, 2, -3, 2], [5, -5, 0, 1, 2, 3, -2, 3, 2, 1], [1, -5, 4, -1, 1, -4, 5, -1, 0, -6], [11, -6, 5, -4, 6, -6, 4, -5, 6, -11], [6, 0, 4, 0, 6, 0, 0, 1, -1, 0], [1, 4, 4, 4, -4, -4, -4, -1, 0, -6], [-5, 0, 0, 5, 0, 1, 0, -7, 0, 1], [-8, 3, -8, 0, -7, 8, -8, 8, -8, 7], [-7, 0, 2, 3, 2, 0, -7, 0, -1, 1], [-9, 5, -10, 2, -4, 2, -10, 5, -9, 0], [0, 6, -6, 0, 0, -2, -1, 4, -1, -2], [0, 4, 1, -3, 2, 10, 2, -3, 1, 4], [8, 0, 0, 4, 9, -5, 5, -9, -4, 0], [-1, 0, 10, 1, -3, 3, -3, 3, -1, -10], [9, -4, 4, -9, 0, 7, 6, -1, 6, 7], [-1, 1, 0, 10, -7, 2, -5, 2, -7, 10], [2, -4, 5, -5, 4, -2, 0, 2, 4, 2], [3, 3, -4, -2, -3, -2, -4, 3, 3, 0], [8, -6, 0, -2, 0, -6, 8, 0, 1, -1], [1, -10, 10, -1, 0, -10, 11, -9, 11, -10], [-4, 1, -15, 1, -4, 0, -1, 3, -3, 1], [3, -2, -4, 0, -4, 4, 0, 4, 2, -3], [0, 3, 6, -3, 4, -7, 4, -3, 6, 3], [4, 5, 2, -3, 3, -2, -5, -4, 0, 4], [-10, 10, 0, 8, -13, 7, -10, 7, -13, 8], [-12, -1, -12, 0, -3, 2, -1, 1, -2, 3], [-2, 8, -1, 1, -8, 2, 0, 0, -11, 0], [0, 0, -5, -1, -3, 15, -3, -1, -5, 0], [-2, -5, 4, 1, 4, -4, -1, -4, 5, 2], [-10, 0, -8, -3, 5, 1, -1, -5, 3, 8], [-3, -8, 5, -11, 1, -1, 11, -5, 8, 3], [-4, 3, 4, 6, -6, -4, -3, 4, 0, 11], [0, 8, -8, 0, 0, -8, 1, 5, 1, -8], [4, -13, 4, -1, 3, -1, 4, -13, 4, 0], [-5, 6, 8, -8, -6, 5, 0, 0, -7, 0], [-15, 15, 0, 13, -6, 10, -8, 10, -6, 13], [0, -1, -1, 10, -13, 1, -13, 10, -1, -1], [15, -5, 0, -5, 15, 0, 10, -7, 7, -10], [1, 0, 11, -11, 12, -9, 9, -12, 11, -11], [7, -13, -10, -13, 7, 0, -3, 4, -4, 3], [0, 10, -5, 10, -1, 1, -1, 10, -5, 10], [4, 6, -11, -3, -11, 6, 4, 0, 1, -1], [5, 4, 5, 0, 4, -5, 0, 0, 5, -4], [-2, -2, 2, 2, 0, -8, -13, 6, -13, -8], [8, -5, -9, 0, 0, 9, 5, -8, 0, 5], [3, -8, 12, -8, 3, 0, 1, 1, -1, -1], [7, -15, 0, -14, 14, 0, 15, -7, 0, 2], [-6, -2, -6, 0, -10, 4, 6, -6, -4, 10], [6, 3, -1, 4, 4, 4, -1, 3, 6, 0], [-13, 1, -4, 4, -1, 13, 0, 5, -6, 5], [-10, 6, -10, 0, 4, 9, -4, 4, -9, -4], [11, 8, 1, 0, 0, -1, -8, -11, 0, -5], [-7, 0, -6, 1, -12, 11, -11, 12, -1, 6], [4, -12, 18, -12, 4, 0, 13, -6, 6, -13], [7, 4, -4, -7, 0, -1, -4, 20, -4, -1], [9, -21, 9, 0, 9, -7, 13, -13, 7, -9], [-15, 4, 0, 14, -13, 14, 0, 4, -15, 0], [-5, 2, 5, -1, 5, 2, -5, 0, -3, 3], [1, -1, 0, -14, -3, 3, -10, 3, -3, -14], [-4, 5, 2, 5, -4, 0, 7, -9, 9, -7], [-16, 9, -14, 9, -16, 0, -8, 9, -9, 8], [0, 4, 11, 9, -1, 4, -1, 9, 11, 4], [2, -4, 13, -11, 13, -4, 2, 0, -2, 2], [1, -16, -11, 11, 16, -1, 0, 6, -4, 6], [-5, 17, -6, -6, -15, -6, -6, 17, -5, 0], [0, -2, 10, 2, 10, -2, 0, 0, 16, -16], [-11, 15, -15, 11, 0, 4, -6, -5, -6, 4], [-10, 3, -1, 5, -1, 1, -5, 1, -3, 10], [-8, 0, -4, -2, -4, 18, -18, 4, 2, 4], [13, -24, 15, -20, 8, -8, 20, -15, 24, -13], [0, -4, -7, 5, 1, 2, 1, 5, -7, -4], [-3, -6, -1, 1, 6, 3, 0, 4, -11, 4], [1, 5, -3, 3, -5, -1, 0, 4, -10, 4], [15, -14, 15, 0, 13, -17, 15, -15, 17, -13], [0, 7, 0, 2, 9, 2, 9, 2, 0, 7], [-16, 7, -17, 12, -11, 11, -12, 17, -7, 16], [-10, 18, -3, 0, -1, 1, 0, 3, -18, 10], [-1, 12, -1, 0, -2, -2, -1, 1, 2, 2], [17, -9, 8, -18, 8, -9, 17, 0, 1, -1], [-4, 2, -2, 4, 0, -1, 2, 12, 2, -1], [-15, 13, -27, 13, -15, 0, -17, 10, -10, 17], [19, -7, 8, -7, 19, 0, 27, -24, 24, -27], [-10, 6, -6, 10, 0, 6, -4, -3, -4, 6], [14, -15, 18, -13, 18, -15, 14, 0, 10, -10], [-19, 8, -1, 1, -8, 19, 0, 4, -13, 4], [10, -4, -6, -19, 11, -19, -6, -4, 10, 0], [1, 5, -5, -1, 0, -5, 2, 20, 2, -5], [15, -13, 25, -13, 15, 0, 11, -10, 10, -11], [-3, 0, -4, 1, 4, -1, 1, -4, -1, 4], [-13, 3, -22, 22, -12, 12, -22, 22, -3, 13], [25, -24, 11, -21, 21, -11, 24, -25, 0, -8], [11, -12, 11, 0, 6, -15, 8, -8, 15, -6], [7, -2, 4, -4, 2, -7, 0, -17, 9, -17], [15, -20, 6, -19, 5, -19, 6, -20, 15, 0], [0, 0, 0, -13, 5, -7, 9, -7, 5, -13], [12, -5, 20, -7, 4, -4, 7, -20, 5, -12], [3, 0, 16, 4, -4, -1, 1, 4, -4, -16], [-8, 9, 5, 7, 6, -6, -7, -5, -9, 8], [-7, 19, -9, -1, 1, 9, -19, 7, 0, 11], [-13, 14, -13, 0, -11, -3, -5, 5, 3, 11], [7, -21, 4, 1, 4, -21, 7, 0, 5, -5], [3, -5, 6, -10, 6, -5, 3, 0, -3, 3], [-4, 4, 0, 4, 7, 9, -19, 9, 7, 4], [0, -4, 16, -11, 6, -18, 6, -11, 16, -4], [-19, 0, 2, -5, 13, 6, -6, -13, 5, -2], [25, -7, 7, -25, 0, -9, 8, -24, 8, -9], [-6, -8, -13, 13, 8, 6, 0, 14, -10, 14], [-13, 6, -9, 2, -9, 6, -13, 0, 0, 0], [-6, 20, -12, 2, -2, 12, -20, 6, 0, 6], [14, -11, 26, -13, 20, -20, 13, -26, 11, -14], [13, -8, 19, -19, 14, -14, 19, -19, 8, -13], [0, 2, 0, -11, -6, -13, -6, -11, 0, 2], [20, 2, 20, 0, -1, 4, -5, 5, -4, 1], [-2, -1, 2, 5, 0, 5, 2, -1, -2, 0], [-8, 3, -21, 10, 2, 10, -21, 3, -8, 0], [10, -10, 0, -16, -7, -10, 11, -10, -7, -16], [13, -10, 0, 3, -3, 0, 10, -13, 0, -24], [-6, 4, -4, 4, -6, 0, 13, -11, 11, -13], [10, 11, 10, 0, -11, -6, 5, -5, 6, 11], [6, 9, 8, -8, -9, -6, 0, -10, 10, -10], [5, 24, -15, 26, -15, 24, 5, 0, -16, 16], [0, 30, -19, 32, -22, 20, -22, 32, -19, 30], [-23, 6, -17, 6, -23, 0, -5, 5, -5, 5], [-9, 0, -9, -1, -2, 10, -10, 2, 1, 9], [-2, 0, -5, -12, 0, 15, -15, 0, 12, 5], [-18, 7, -7, 18, 0, 6, -3, 10, -3, 6], [-8, 7, -6, -5, -11, -5, -6, 7, -8, 0], [19, -15, 19, -28, 19, -15, 19, 0, 11, -11], [3, 12, 3, 0, -7, 3, -8, 8, -3, 7], [-10, 10, -10, 10, 0, 7, 17, -2, 17, 7], [10, 6, 11, -2, 18, -18, 2, -11, -6, -10], [0, 14, -17, 14, -26, 0, -26, 14, -17, 14], [-17, 6, -13, 14, -14, 13, -6, 17, 0, 27], [-2, 2, 0, 10, -5, -7, -6, -7, -5, 10], [-8, 8, 0, 9, 2, 20, -6, 20, 2, 9], [4, -11, 1, -1, 1, -1, 11, -4, 0, 14], [0, 0, 2, 7, -14, -9, 9, 14, -7, -2], [3, -5, 23, 2, -2, -23, 5, -3, 0, -11], [6, -12, -1, -9, -1, -12, 6, 0, 17, -17], [0, -3, 15, 0, 22, 0, 15, -3, 0, 0], [-12, 8, -1, 1, -8, 12, 0, -7, -17, -7], [1, -1, 0, 11, -4, 8, -8, 8, -4, 11], [-17, 0, 7, -5, -15, -4, 4, 15, 5, -7], [-14, 0, -2, 0, -14, 0, -17, 2, -2, 17], [-12, 0, 0, 12, 0, 14, -9, 7, -9, 14], [1, 8, -14, 14, -14, 8, 1, 0, -9, 9], [-23, 17, -8, 19, -23, 19, -8, 17, -23, 0], [13, -11, 13, 0, 8, -14, -14, 14, 14, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_115_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_115_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_115_2_g_a(:prec:=10)
    chi := MakeCharacter_115_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_115_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_115_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_115_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 2, 125, -157, 38, 21, -2, -15, 14, -5, 1]>],Snew);
    return Vf;
end function;