
// Make newform 350.2.h.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_h_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_350_2_h_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_350_2_h_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;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, -1, 0, 0, 0, 0, -1], [0, 1, 0, 0, -1, 0, 1, 0], [-1, 2, 0, 0, 1, -1, 1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_350_h();"
function MakeCharacter_350_h()
    N := 350;
    order := 5;
    char_gens := [127, 101];
    v := [3, 5];
    // 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_350_h_Hecke();"
function MakeCharacter_350_h_Hecke(Kf)
    N := 350;
    order := 5;
    char_gens := [127, 101];
    char_values := [[0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 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, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, -1, -1, 0, 1, 0], [-1, 1, -2, 0, 0, -1, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, -1, 0, 1, 2, -2], [2, 0, 2, -1, 0, 1, 0, 1], [4, -1, 4, 0, 3, 0, 1, 0], [-3, 0, -3, -2, -3, 0, 0, 0], [2, 1, 0, -1, 0, 3, 0, 0], [0, 0, 4, -2, 2, 1, 2, -2], [0, -1, -2, 3, -1, 0, 1, 0], [4, -2, 4, -2, 1, 3, 0, 2], [4, 1, 4, 5, 7, 2, 0, -5], [-3, 1, -2, 0, 1, -2, -4, -1], [0, 0, -1, 6, 5, -6, -6, -2], [0, 0, 1, 2, 3, -6, -2, -1], [1, -3, 1, -1, -3, -2, 0, 1], [-2, -2, -1, 2, -8, -4, -1, 1], [0, 0, -1, -4, 0, 0, 0, 0], [0, 0, -6, 3, -3, -1, -3, -3], [3, -1, 1, 1, 9, 2, 1, -1], [0, 0, 1, -4, -3, 0, 4, 3], [-5, 6, 0, 3, 1, 0, -6, 0], [-6, -3, -6, 3, -9, -9, -6, 6], [0, 0, -11, 0, -11, 0, 0, 2], [-2, -1, -5, 0, -1, -5, 1, 1], [0, 0, -3, 10, 7, -6, -10, -3], [-15, 3, -3, 0, 3, -3, -3, -3], [0, 4, 0, -6, 0, 6, 0, 6], [-7, -3, -7, 2, 4, 2, 0, -2], [-8, 3, -2, -3, -3, -5, -2, 2], [-3, -3, 1, -2, -6, 0, 3, 0], [7, -3, 7, -3, 0, 3, 0, 3], [-14, -1, -6, 1, -12, -15, -6, 6], [4, 5, 11, 0, 5, 11, 11, -5], [4, -6, -2, 0, -6, -2, 7, 6], [5, -2, 7, 0, -2, 7, 9, 2], [11, 2, 11, -1, 11, 12, 0, 1], [5, -9, -1, -4, -4, 0, 9, 0], [8, -9, 1, 9, 6, -1, 1, -1], [0, 0, -7, 3, -4, -3, -3, 8], [1, 3, -15, 2, 4, 0, -3, 0], [8, -3, 8, 4, 11, 7, 0, -4], [4, 2, -8, 0, 2, -8, -1, -2], [0, 0, 1, -9, -8, 8, 9, 0], [-3, -6, -4, 0, -6, -4, -1, 6], [-4, 6, 0, -6, -11, 2, 0, 0], [-14, 13, -2, -13, 6, -1, -2, 2], [1, 0, 4, 0, -9, 1, 4, -4], [0, 0, 0, -2, -2, 10, 2, 4], [-9, 3, 0, 8, -6, 0, -3, 0], [-5, -10, -9, 10, -18, -15, -9, 9], [8, 6, 8, 5, 15, 10, 0, -5], [-11, 6, -4, 0, 6, -4, -10, -6], [-6, -9, -12, 0, -9, -12, 1, 9], [13, 3, 13, 5, -1, -6, 0, -5], [4, 3, -9, 2, 7, 0, -3, 0], [0, 0, 9, -5, 4, -4, 5, -2], [-10, 7, -15, -7, -10, -3, -15, 15], [-4, 13, 2, 4, 9, 0, -13, 0], [-17, -5, 4, -9, -22, 0, 5, 0], [4, -10, -7, 0, -10, -7, 4, 10], [3, 6, 5, 0, 6, 5, 1, -6], [2, -3, -11, 3, 2, -1, -11, 11], [0, -11, 0, 0, 2, 2, 0, 0], [-3, 6, 1, -4, 3, 0, -6, 0], [-6, -7, -6, 4, -13, 0, 7, 0], [8, 0, 8, -4, -3, 1, 0, 4], [0, 0, -1, 0, -1, 14, 0, -8], [9, 2, 4, 0, 2, 4, 3, -2], [0, 0, -4, 2, -2, -20, -2, 0], [-2, -14, -2, -2, -16, -14, 0, 2], [14, 0, -1, 14, 14, 0, 0, 0], [1, -2, 3, 2, 3, -1, 3, -3], [0, 0, -9, -2, -11, -14, 2, -8], [6, -10, 17, 4, -4, 0, 10, 0], [-16, 10, 1, -10, -3, -6, 1, -1], [0, 0, 6, 2, 8, 15, -2, 7], [-8, 2, -9, 0, 2, -9, -19, -2], [18, -7, 18, 17, 14, -3, 0, -17], [8, -8, 0, 2, 0, 0, 8, 0], [0, 0, -6, -5, -11, -7, 5, 9], [-14, -4, -7, -5, -18, 0, 4, 0], [-4, -4, 22, -8, -8, 0, 4, 0], [-9, 10, 7, -10, -8, 1, 7, -7], [-7, -13, -6, 0, -13, -6, -3, 13], [-5, -11, -15, 0, -11, -15, -6, 11], [-3, 5, 3, 0, 5, 3, -11, -5], [18, 3, -2, -3, 6, 21, -2, 2], [4, 7, 4, -7, 13, 20, 0, 7], [12, -1, 4, -2, 11, 0, 1, 0], [0, 0, 1, 6, 7, -13, -6, 0], [7, -6, 7, -9, 7, 16, 0, 9], [8, -5, 8, -8, 9, 17, 0, 8], [7, 7, 16, 0, 7, 16, 15, -7], [0, 0, 6, -12, -6, 16, 12, 4], [-15, 9, -15, -15, -7, 8, 0, 15], [0, 0, -8, 16, 8, 4, -16, 2], [6, -6, -2, 6, 4, 0, -2, 2], [19, 1, 19, 6, 27, 21, 0, -6], [0, 0, -7, 4, -3, 3, -4, 5], [-14, 0, -12, 0, 0, -12, -16, 0], [-14, 2, -14, -4, 0, 4, 0, 4], [-13, 19, -4, 3, 6, 0, -19, 0], [0, 0, 11, -17, -6, 10, 17, 7], [-9, -4, 0, 4, -25, -13, 0, 0], [11, -1, 11, 20, 18, -2, 0, -20], [23, -8, 10, 0, -8, 10, 12, 8], [26, -6, 33, 0, -6, 33, 6, 6], [-7, 11, -1, 0, 11, -1, -7, -11], [9, 11, 1, 0, 11, 1, 9, -11], [18, 0, 18, 19, 24, 5, 0, -19], [-1, -14, -5, 5, -15, 0, 14, 0], [21, -4, 24, 8, 17, 0, 4, 0], [-17, 0, -13, 11, -17, 0, 0, 0], [10, 9, 10, 11, 6, -5, 0, -11], [-12, -3, -12, 0, -3, -12, -7, 3], [0, 0, 2, -4, -2, -2, 4, -15], [0, 0, -28, 7, -21, -34, -7, 3], [21, 4, 21, 1, 12, 11, 0, -1], [5, 0, 17, 0, 3, 5, 17, -17], [-24, 16, 0, -16, -1, -8, 0, 0], [15, -5, -10, 5, 15, 10, -10, 10], [9, -5, -11, 24, 4, 0, 5, 0], [-3, 15, -3, 3, 18, 15, 0, -3], [-5, -3, 1, 0, -3, 1, -5, 3], [1, 6, 1, -10, 3, 13, 0, 10], [-29, 9, -22, -4, -20, 0, -9, 0], [8, -15, 8, 15, 12, -7, 8, -8], [8, -8, 20, -6, 0, 0, 8, 0], [23, -8, 8, 8, 14, 15, 8, -8], [-1, 25, 16, 0, 25, 16, -10, -25], [-34, -6, -18, 0, -6, -18, 3, 6], [-1, -14, -11, 0, -14, -11, -5, 14], [27, -14, -3, 14, 14, 13, -3, 3], [-15, 4, -34, -1, -11, 0, -4, 0], [7, 3, -9, -3, 0, 10, -9, 9], [0, 4, 0, 15, 15, 0, 0, -15], [0, 0, 3, -4, -1, -6, 4, 6], [10, 20, 10, 0, 18, 18, 0, 0], [-4, 5, 7, -5, -40, 1, 7, -7], [0, 0, -5, 10, 5, -40, -10, -1], [5, -15, -17, 15, -7, -10, -17, 17], [-2, -9, 7, 9, 20, -11, 7, -7], [0, 0, 5, -7, -2, 14, 7, -15], [-16, 8, 1, -8, 22, -8, 1, -1], [0, 0, 11, -2, 9, 39, 2, -8], [-6, 12, -6, 0, 12, -6, 0, -12], [26, -2, 26, -2, 12, 14, 0, 2], [8, 8, 8, 2, 3, 1, 0, -2], [35, -11, 14, 11, 4, 24, 14, -14], [18, 11, -4, 7, 29, 0, -11, 0], [-7, 14, -24, 1, 7, 0, -14, 0], [3, -26, 3, 1, -24, -25, 0, -1], [9, -8, 0, 0, -8, 0, -11, 8], [-12, 2, -2, -2, 20, -10, -2, 2], [-25, 6, -8, -5, -19, 0, -6, 0], [0, 0, 11, -5, 6, 30, 5, 0], [-34, -3, -34, 2, -30, -32, 0, -2], [-14, 15, -14, -14, -15, -1, 0, 14], [0, 0, -10, -5, -15, -33, 5, 5], [0, 0, -5, -4, -9, 20, 4, -12], [4, 1, -19, -8, 5, 0, -1, 0], [0, 0, -17, 1, -16, -34, -1, -7], [-4, 3, 11, -3, 31, -1, 11, -11], [-14, -2, -10, -13, -16, 0, 2, 0], [-2, 24, -2, -13, 2, 15, 0, 13], [0, 0, 30, 0, 30, 10, 0, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_2_h_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_350_2_h_a(:prec:=8)
    chi := MakeCharacter_350_h();
    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_350_2_h_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_350_2_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 7, 18, -11, 15, -11, 8, -3, 1]>],Snew);
    return Vf;
end function;