
// Make newform 350.2.q.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_q_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_q_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_q_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_350_q();"
function MakeCharacter_350_q()
    N := 350;
    order := 15;
    char_gens := [127, 101];
    v := [12, 10];
    // 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_q_Hecke();"
function MakeCharacter_350_q_Hecke(Kf)
    N := 350;
    order := 15;
    char_gens := [127, 101];
    char_values := [[0, 0, -1, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0], [1, 0, -1, 1, -1, 1, 1, -1], [-1, 2, 0, 0, 2, -1, 0, 0], [-2, 0, 0, 0, 0, -3, 0, 0], [0, 1, -3, 0, 0, -3, 1, 0], [4, 0, -1, 4, 0, 0, 1, -1], [4, 2, 0, 4, -4, -2, 6, -4], [-6, 6, 0, 0, 6, -6, 0, 6], [4, -5, 0, 0, 0, 4, 0, -4], [0, 0, 3, -4, 0, 0, -4, 3], [3, 0, 0, 3, -3, 0, 3, -3], [0, -1, 0, 0, 1, 0, 0, -1], [-10, 0, 11, -10, 0, 0, -11, 11], [1, 0, -2, 2, 0, 0, 0, -2], [-2, 0, 0, 0, 2, -2, 0, 8], [-2, 0, 3, -2, 2, -2, -2, 2], [-1, 7, -6, -1, 1, 0, 6, 1], [2, 5, 0, 0, 0, 2, 0, -2], [1, -1, 0, 1, -1, 1, 0, -1], [0, 0, -3, -8, 0, 0, -8, -3], [0, -7, 1, 0, 0, 1, -7, 0], [-8, 0, 0, 0, 8, -8, 0, 4], [2, 0, 0, -7, 0, 0, 2, 0], [5, -14, 0, 0, 0, 5, 0, -5], [0, 0, 8, 2, 0, 0, 2, 8], [-6, 6, 6, -6, 6, -5, 0, 6], [-8, 0, 0, 0, 8, -8, 0, -2], [1, 9, 0, 0, 9, 1, 0, 0], [0, 2, -2, 0, 0, 0, 2, 0], [1, 0, -13, 1, 0, 0, 13, -13], [0, 0, 0, 0, 0, 0, 9, 0], [11, -11, 0, 0, 0, 11, 0, 0], [0, -4, 4, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 6, 0], [8, -2, 0, 0, -2, 8, 0, 0], [-1, 1, 1, -1, 1, 16, 0, 1], [-1, 1, 1, -1, 1, -3, 0, 1], [0, 8, 0, 0, -7, 0, 0, 8], [-3, 0, 0, 2, 0, 0, -3, 0], [-6, 0, 0, 0, 0, -6, 0, 6], [2, 0, -9, 2, -2, 2, 2, -2], [-12, 0, 0, 6, 0, 0, -12, 0], [0, 2, 0, 0, -17, 0, 0, 2], [-11, 11, 11, -11, 11, 6, 0, 11], [0, 0, -19, -3, 0, 0, -3, -19], [12, -12, -12, 12, -12, -3, 0, -12], [15, 0, -15, 0, 0, 0, 14, -15], [-1, 0, 1, 0, 0, 0, 9, 1], [0, 7, 8, 0, 0, 8, 7, 0], [-7, 0, 0, 0, 7, -7, 0, 8], [13, -13, 0, 0, 11, 13, 0, 11], [7, 0, -7, 0, 0, 0, 17, -7], [-9, 3, 6, -9, 9, 0, -6, 9], [25, 0, -10, 10, 0, 0, 0, -10], [9, 13, 0, 0, 13, 9, 0, 0], [-7, -6, 13, -7, 7, 0, -13, 7], [-14, 7, 0, -14, 14, -7, -7, 14], [-5, 0, 0, 0, 5, -5, 0, 9], [0, 13, -12, 0, 0, -12, 13, 0], [0, 0, 0, 17, 0, 0, 0, 0], [-6, -3, 0, -6, 6, 3, -9, 6], [2, 0, 5, -5, 0, 0, 0, 5], [2, 0, -18, 18, 0, 0, 0, -18], [0, -23, 12, 0, 0, 12, -23, 0], [0, -4, 0, 0, -20, 0, 0, -4], [-5, 5, 0, 0, -14, -5, 0, -14], [-1, 1, 0, 0, 18, -1, 0, 18], [-19, 0, 13, -19, 0, 0, -13, 13], [-12, 0, 12, -12, 12, -12, -12, 12], [23, 0, -1, 1, 0, 0, 0, -1], [-13, 0, 7, -13, 13, -13, -13, 13], [0, 2, 0, 0, -3, 0, 0, 2], [28, -20, 0, 28, -28, 20, 8, -28], [-14, 21, 0, 0, 0, -14, 0, 14], [0, 0, 19, -8, 0, 0, -8, 19], [20, -20, 0, 0, -20, 20, 0, -20], [0, -1, 5, 0, 0, 5, -1, 0], [22, 0, 0, 0, -22, 22, 0, -8], [-14, -4, 0, 0, -4, -14, 0, 0], [1, 4, -5, 1, -1, 0, 5, -1], [-2, 0, 0, -11, 0, 0, -2, 0], [0, 0, -16, -5, 0, 0, -5, -16], [19, -6, 0, 19, -19, 6, 13, -19], [6, 0, 0, 19, 0, 0, 6, 0], [8, -9, 0, 0, 0, 8, 0, -8], [-8, 3, 0, 0, 3, -8, 0, 0], [13, 0, 13, -13, 0, 0, 0, 13], [14, -2, 0, 0, -2, 14, 0, 0], [-26, 0, 26, 0, 0, 0, -7, 26], [-13, 0, 15, -13, 0, 0, -15, 15], [-26, 26, 0, 0, 13, -26, 0, 13], [6, 0, -4, 6, -6, 6, 6, -6], [27, -22, -5, 27, -27, 0, 5, -27], [23, 0, -26, 23, 0, 0, 26, -26], [-12, 0, 0, 0, 0, -12, 0, 0], [0, 0, 12, 4, 0, 0, 4, 12], [0, -17, 0, 0, 5, 0, 0, -17], [-1, 0, -19, -1, 1, -1, -1, 1], [-4, 17, 0, 0, 0, -4, 0, 4], [0, -2, 0, 0, -23, 0, 0, -2], [0, 0, 34, -14, 0, 0, -14, 34], [10, -10, -10, 10, -10, 22, 0, -10], [-14, -16, 30, -14, 14, 0, -30, 14], [2, -2, 0, 0, -17, 2, 0, -17], [-18, 0, 19, -18, 18, -18, -18, 18], [0, 11, -2, 0, 0, -2, 11, 0], [12, 0, -7, 12, 0, 0, 7, -7], [-29, 16, 0, 0, 16, -29, 0, 0], [16, -16, -16, 16, -16, 30, 0, -16], [1, 0, -3, 3, 0, 0, 0, -3], [-7, 11, 0, 0, 11, -7, 0, 0], [13, 0, -13, 13, -13, 0, 13, -13], [4, 0, 0, -32, 0, 0, 4, 0], [-21, 2, 0, -21, 21, -2, -19, 21], [-23, 0, 0, -12, 0, 0, -23, 0], [2, 26, -28, 2, -2, 0, 28, -2], [-7, 0, 8, -8, 0, 0, 0, 8], [21, 0, -22, 21, -21, 21, 21, -21], [11, 0, 0, 0, -11, 11, 0, -39], [23, 0, -34, 23, 0, 0, 34, -34], [0, -30, 17, 0, 0, 17, -30, 0], [24, 0, -24, 0, 0, 0, 1, -24], [6, -20, 0, 0, 0, 6, 0, -6], [3, -13, 0, 3, -3, 13, -10, -3], [0, 3, 0, 0, 11, 0, 0, 3], [-14, 0, 34, -34, 0, 0, 0, 34], [0, -12, 0, 0, -27, 0, 0, -12], [34, -34, 0, 0, -29, 34, 0, -29], [-16, 0, 16, 0, 0, 0, 8, 16], [11, -11, 0, 0, -5, 11, 0, -5], [0, -19, 32, 0, 0, 32, -19, 0], [-35, 0, 2, -2, 0, 0, 0, 2], [-5, -17, 0, 0, -17, -5, 0, 0], [21, 0, 11, -11, 0, 0, 0, 11], [-2, 15, 0, 0, 0, -2, 0, 2], [-16, 0, 0, 3, 0, 0, -16, 0], [0, 29, -35, 0, 0, -35, 29, 0], [-2, -13, 15, -2, 2, 0, -15, 2], [0, 0, 1, 28, 0, 0, 28, 1], [-24, -4, 28, -24, 24, 0, -28, 24], [-5, 0, 5, 0, 0, 0, -18, 5], [13, 0, 0, 0, -13, 13, 0, 3], [0, -27, 12, 0, 0, 12, -27, 0], [-28, 0, 28, 0, 0, 0, -35, 28], [3, 0, -30, 3, -3, 3, 3, -3], [10, 0, -10, 0, 0, 0, -6, -10], [0, 0, -8, -11, 0, 0, -11, -8], [-12, 12, 12, -12, 12, -13, 0, 12], [0, -12, 0, 0, -39, 0, 0, -12], [0, 13, 0, 0, 15, 0, 0, 13], [13, -17, 0, 0, 0, 13, 0, -13], [-6, 0, 0, -10, 0, 0, -6, 0], [15, 0, 0, 7, 0, 0, 15, 0], [0, 18, 0, 0, 40, 0, 0, 18], [-15, 15, 15, -15, 15, 10, 0, 15], [13, 0, -13, 0, 0, 0, -1, -13], [-32, 32, 0, 0, 22, -32, 0, 22], [7, 0, 0, 0, -7, 7, 0, 22], [-20, 0, -9, -20, 0, 0, 9, -9], [33, -18, -15, 33, -33, 0, 15, -33], [8, 0, 0, 0, -8, 8, 0, 36], [0, 0, -23, 27, 0, 0, 27, -23], [-22, 0, 0, -23, 0, 0, -22, 0], [28, 0, 0, 0, -28, 28, 0, -22], [0, -19, 38, 0, 0, 38, -19, 0], [0, 31, 0, 0, 0, -31, 31, 0], [19, 2, -21, 19, -19, 0, 21, -19], [-3, 0, 20, -3, 3, -3, -3, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_2_q_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_q_a(:prec:=8)
    chi := MakeCharacter_350_q();
    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_q_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_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 2, 0, 2, 9, 8, 5, 3, 1]>],Snew);
    return Vf;
end function;