
// Make newform 350.4.j.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_j_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_4_j_b();".
// 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_4_j_b();".
// 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_350_j();"
function MakeCharacter_350_j()
    N := 350;
    order := 6;
    char_gens := [127, 101];
    v := [3, 2];
    // 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_j_Hecke();"
function MakeCharacter_350_j_Hecke(Kf)
    N := 350;
    order := 6;
    char_gens := [127, 101];
    char_values := [[-1, 0, 0, 0], [0, 0, -1, 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 := 4;
    raw_aps := [[0, -2, 0, 2], [0, 5, 0, 0], [0, 0, 0, 0], [0, 14, 0, 7], [57, 0, -57, 0], [0, 0, 0, -70], [0, 51, 0, 0], [0, 0, 5, 0], [0, 69, 0, -69], [-114, 0, 0, 0], [-23, 0, 23, 0], [0, 253, 0, -253], [-42, 0, 0, 0], [0, 0, 0, -124], [0, -201, 0, 201], [0, 393, 0, 0], [219, 0, -219, 0], [0, 0, 709, 0], [0, 419, 0, 0], [-96, 0, 0, 0], [0, 313, 0, 0], [0, 0, 461, 0], [0, 0, 0, -588], [0, 0, -1017, 0], [0, 0, 0, 1834], [285, 0, -285, 0], [0, -499, 0, 499], [0, 1107, 0, -1107], [923, 0, -923, 0], [0, 0, 0, 1542], [0, 0, 0, 2056], [0, 0, -2049, 0], [0, -141, 0, 0], [-1484, 0, 0, 0], [0, 0, -57, 0], [-839, 0, 839, 0], [0, -2833, 0, 0], [0, -2311, 0, 2311], [0, 0, 0, -1260], [0, 3267, 0, -3267], [1287, 0, -1287, 0], [-2674, 0, 0, 0], [0, 0, -4185, 0], [0, 85, 0, 0], [0, 0, 0, 390], [-2833, 0, 2833, 0], [-124, 0, 0, 0], [0, 0, 0, 56], [0, -3057, 0, 0], [0, 0, -961, 0], [0, -2829, 0, 2829], [3540, 0, 0, 0], [-5231, 0, 5231, 0], [5040, 0, 0, 0], [0, 1437, 0, -1437], [0, 2325, 0, 0], [-2385, 0, 2385, 0], [0, 0, 331, 0], [0, 4871, 0, 0], [-7026, 0, 0, 0], [0, 5353, 0, 0], [0, 0, 0, 4158], [0, 0, 0, 9604], [-10131, 0, 10131, 0], [0, 10799, 0, -10799], [0, -531, 0, 531], [0, 0, 7015, 0], [0, 0, 0, -8990], [0, -8709, 0, 0], [-6482, 0, 0, 0], [0, 2133, 0, 0], [0, 0, 3849, 0], [0, 6491, 0, 0], [0, 923, 0, -923], [-6344, 0, 0, 0], [0, -5007, 0, 5007], [12291, 0, -12291, 0], [0, -887, 0, 887], [0, 0, -11955, 0], [-3421, 0, 3421, 0], [5460, 0, 0, 0], [7730, 0, 0, 0], [11313, 0, -11313, 0], [0, 0, 0, 4214], [0, 0, 16553, 0], [0, -16395, 0, 16395], [15090, 0, 0, 0], [0, 14785, 0, -14785], [2898, 0, 0, 0], [0, 0, 0, 464], [0, -4233, 0, 4233], [2739, 0, -2739, 0], [0, 17051, 0, 0], [-4296, 0, 0, 0], [0, 0, 3401, 0], [0, 0, 0, 16800], [0, 0, 1839, 0], [-303, 0, 303, 0], [0, -21667, 0, 21667], [0, 0, -5039, 0], [0, 0, 0, 2392], [0, -22149, 0, 0], [0, 8349, 0, 0], [0, 0, -15345, 0], [11593, 0, -11593, 0], [0, -14593, 0, 0], [0, 0, 0, 15372], [0, -14373, 0, 14373], [2547, 0, -2547, 0], [-7042, 0, 0, 0], [0, 22591, 0, -22591], [0, 8485, 0, 0], [0, 0, 0, 18282], [2291, 0, -2291, 0], [-6928, 0, 0, 0], [-24975, 0, 24975, 0], [0, 0, 0, 9548], [0, 10131, 0, 0], [0, 16659, 0, -16659], [-29556, 0, 0, 0], [-191, 0, 191, 0], [0, 0, 0, 2606], [0, 4209, 0, -4209], [0, -24303, 0, 0], [0, 0, -15041, 0], [24726, 0, 0, 0], [0, 0, -4957, 0], [0, 0, 27669, 0], [0, 0, 0, 13888], [0, 14243, 0, -14243], [36959, 0, -36959, 0], [0, 0, 0, -12528], [0, 0, 17767, 0], [0, 0, 0, 28726], [0, 0, 26469, 0], [-5054, 0, 0, 0], [0, 35565, 0, 0], [0, -8629, 0, 0], [0, 0, 0, -20706], [-16185, 0, 16185, 0], [-11788, 0, 0, 0], [0, 0, 29793, 0], [0, -30323, 0, 0], [0, 0, 0, -21156], [-5269, 0, 5269, 0], [39816, 0, 0, 0], [0, 0, 0, 14546], [0, -31449, 0, 0], [0, 0, -24523, 0], [0, -8163, 0, 8163], [0, -4367, 0, 4367], [-50190, 0, 0, 0], [0, 0, 0, 12308], [0, -31617, 0, 31617], [0, -13525, 0, 0], [-19248, 0, 0, 0], [0, 0, -8695, 0], [0, 0, 19479, 0], [0, 0, 0, 12502], [15993, 0, -15993, 0], [0, -44001, 0, 44001], [0, 0, 0, -4002], [0, 0, 0, -10544], [0, 0, 6183, 0], [0, 3723, 0, 0], [0, 45897, 0, 0], [-6467, 0, 6467, 0], [0, 23039, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_4_j_b();".
// 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_4_j_b(:prec:=4)
    chi := MakeCharacter_350_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_j_b();".
// 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_4_j_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![625, 0, -25, 0, 1]>,<11,R![3249, -57, 1]>],Snew);
    return Vf;
end function;