
// Make newform 370.2.g.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_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_370_2_g_d();".
// 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_370_2_g_d();".
// 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 := [160, -152, 113, 166, 24, 12, -26, -8, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-881880, 840473, -211635, 120257, 73205, -123604, 34822, -17854, 3868, 105], [-1481360957, 5881796255, 4194704142, -51534513, -422935188, -248412067, 36230609, 39766871, 27279422, 4830421], [1481360957, -720718392, -4194704142, 51534513, 422935188, 248412067, -36230609, -39766871, -27279422, -4830421], [-18775197608, -20817063063, -9133030804, 731490770, 3464752928, 577196282, 26574182, -195227485, -79772050, -12470333], [1358343716, 18569839845, -11350242598, 3432795828, -3136611584, -2353824822, 1711086296, -506483273, 281415368, -40939147], [10748, 1644771, 4140830, 821948, -229852, -433934, -247792, 81613, 420, 22047], [-13883295960, -23630326501, -17048329972, 1196070998, 1991880712, 1835553924, 565223046, -432258509, -53270874, -78590417], [27078850384, -28243314479, -19246167516, -3262671660, -910662862, 3952005236, 621147016, -234376097, -59569470, -106312323], [25888295120, -6371750919, -22346975474, -5878785400, -2322899252, 3598338044, 1324404682, -412250275, 32728460, -151550159]];
    Rf_basisdens := [1, 887851, 5161077863, 5161077863, 10322155726, 20644311452, 3551404, 10322155726, 10322155726, 10322155726];
    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_370_g();"
function MakeCharacter_370_g()
    N := 370;
    order := 4;
    char_gens := [297, 261];
    v := [3, 3];
    // 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_370_g_Hecke();"
function MakeCharacter_370_g_Hecke(Kf)
    N := 370;
    order := 4;
    char_gens := [297, 261];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 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, 0, 0, -1, 0, 0, 0], [0, 0, -1, -1, 0, 0, 0, 0, -1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 1, -1, -1], [0, 1, -2, -2, 0, 0, 0, 1, -1, 2], [1, 0, 0, -1, 0, 1, -1, 1, -1, 1], [-2, 1, 0, 0, 1, 0, 0, -1, 1, -1], [-2, -1, -3, -1, -1, 0, 0, 0, -1, 0], [-1, 0, 0, 1, 2, -1, 1, -1, 1, 1], [3, 1, -1, -1, 1, -2, 5, 1, 0, 1], [-2, 0, 0, 0, 0, 0, 2, 0, -1, -1], [2, 1, -1, -1, 3, 0, 1, 0, -1, 2], [1, -1, 3, 2, 1, 1, -1, -1, 0, -1], [0, 1, 1, 1, 1, 0, 0, -2, -1, 0], [6, 0, -1, -3, 0, 0, -4, 1, -2, 1], [-1, -1, 0, 0, 1, 0, -1, -3, 1, -1], [2, -4, 3, 5, -4, 0, -4, 3, 0, -5], [-1, 0, 1, -1, 0, 0, 3, 1, -1, 0], [2, -1, 0, 0, 1, 0, 2, 1, 2, -2], [-2, 2, -2, -4, 2, -2, 2, -2, 0, 2], [1, 4, -1, -3, 4, 0, 1, -3, -1, 2], [-2, -1, -4, -2, -1, 0, 0, 0, 0, 2], [-2, 4, -3, -3, 2, -6, 4, -3, 2, 1], [1, -2, 2, 2, -2, 4, -3, 0, -2, 0], [0, -1, -1, -1, 1, 0, 0, 0, -1, 0], [2, 1, 2, 0, -1, 2, 0, -1, -3, -1], [4, -3, 0, 4, -2, 4, -4, -1, 1, -2], [-2, 1, 6, 8, 1, 0, 0, -2, 2, -6], [1, 4, 0, -8, 4, 0, 7, 0, -2, 6], [-4, 1, -1, -3, 3, -2, 2, 0, -1, 0], [-6, -2, -1, 1, -2, 0, 4, 1, -2, -3], [-6, 2, 1, 7, 2, 0, 0, -5, 4, -3], [-1, -4, 3, 9, -4, 0, -5, 1, 3, -6], [-3, -2, 4, 3, -2, -1, 1, 7, -3, -1], [-2, -2, 4, 6, -2, -2, 0, -4, 4, -8], [-4, 0, -3, 1, 0, -4, -8, -1, 4, -1], [7, 0, 7, 1, 0, 0, -1, 3, 0, -1], [-4, 5, 0, 0, 3, 0, 0, -5, 5, -3], [4, -5, 4, 8, -6, 4, -4, 5, -1, -2], [5, 4, -2, -6, 4, 0, -1, -2, -2, 4], [0, 6, 0, 0, -6, 0, 0, 4, 0, 0], [6, -5, 4, 6, -3, 2, -2, 7, -3, -3], [-2, 4, -2, -2, 0, -4, 2, -8, 7, -5], [2, 4, -4, -6, -4, 2, 4, 2, -6, 2], [3, 7, -4, -4, 7, 0, -3, -7, -1, 3], [-2, -6, 1, 1, 4, 2, -4, 5, -4, 3], [-2, -1, 2, 8, -2, 6, -6, -3, 5, -6], [-4, 7, -3, -3, -1, -6, 2, -2, 1, 2], [-6, 4, 5, 7, -6, 2, -2, -1, 6, -1], [0, -2, -3, -3, 2, 0, 12, 5, 2, 5], [-5, -2, 10, 12, -2, 0, 3, 1, 7, -5], [-6, -2, 0, 8, -2, 0, -2, -2, 5, -3], [-3, -1, 2, 2, -3, 4, -7, 1, 1, -3], [2, 1, -7, -1, 1, 0, -8, -4, -3, -2], [-10, 11, -1, -9, 5, -8, 8, -4, 3, 4], [8, -3, -4, -8, -3, 0, -4, 5, -1, 7], [-2, -1, -2, 0, -1, -2, 4, 1, 3, -1], [-8, 0, -3, -3, -2, 0, 0, -3, 0, 5], [4, -1, 0, -4, 2, 4, 4, 5, -3, 6], [5, 0, -1, -1, 2, -2, 7, 1, 4, -3], [-6, -1, 0, 2, -1, 2, -2, -1, 1, -1], [1, 3, -2, -2, 1, -4, 5, 3, -7, 9], [-12, 2, -3, -3, 4, -6, -6, -6, 1, 2], [0, -4, -1, -5, -4, 0, 4, 6, -3, 2], [-4, -3, 0, 4, -1, -4, -14, -1, 7, -5], [-5, 7, -5, -5, 3, -10, 5, 4, -1, 6], [4, 5, -4, -4, 3, -8, 12, -5, -1, 5], [3, 6, 1, 1, -8, 2, 1, 1, -3, 2], [0, 4, -7, -7, -4, 0, 0, 3, -4, 3], [0, 7, -5, -5, -7, 0, -10, -2, -7, -2], [-2, -8, -6, 0, 0, 6, -6, -4, -2, 0], [-4, 4, -7, -3, -4, -4, -4, -1, 0, -1], [4, 6, 4, -12, 6, 0, 12, 2, -4, 8], [-3, 4, 7, 13, 4, 0, -3, -7, 8, -5], [1, -2, 6, 5, 2, 1, 11, -3, 1, -3], [-2, 0, -6, -4, 2, -2, 14, 4, 2, 6], [15, 3, 3, -1, 3, 0, -11, -1, -2, -1], [-3, 4, -6, -6, 8, -12, 9, -2, 0, 6], [-3, -6, 6, 2, -6, 0, 7, 8, -4, -6], [1, 1, 3, 3, -7, 6, -5, 6, -1, -2], [-5, 0, 2, 7, 2, -5, 9, -7, 5, -5], [-7, -5, 3, 3, -5, 0, 7, 5, 6, 3], [16, 4, 5, -9, 4, 0, -2, 3, -2, 7], [1, 7, -3, -3, -1, -6, 7, -7, 8, -5], [8, -13, 2, 2, 9, 4, 4, 2, 0, -2], [-10, 0, -12, -8, 0, 0, 6, -2, -3, 5], [7, 6, -3, -1, 6, 0, -9, -7, 0, 1], [-4, -3, 5, 9, -5, 4, -4, 4, 1, -4], [5, -8, 6, 6, -4, 12, -7, 0, -4, -2], [5, 0, 0, -5, -2, 5, 11, 5, -5, 3], [-20, 5, -7, -11, 5, -4, 4, -8, 1, 6], [12, 0, 5, -7, 0, 0, 0, 6, -3, 4], [-12, -2, 0, 0, -4, 0, 0, 2, -2, 4], [13, -1, -3, -4, 11, -1, 1, -1, -2, -7], [0, -2, -2, -2, 6, -4, 4, 8, -8, 10], [-2, -5, 2, 4, 6, -2, 14, 1, 7, 2], [4, 0, -8, -10, 12, -2, 2, -6, -2, -2], [-8, 2, -6, 2, -2, -8, -2, -4, 6, -4], [2, 7, -9, -11, 1, 2, 2, 4, -9, 12], [11, 2, 1, 1, -4, 2, 9, 1, 1, -2], [-14, 8, -11, 3, -6, -14, -6, -11, 6, -9], [10, -8, 14, 4, 1, 10, -10, 4, -2, -3], [4, 5, -2, -6, -5, 4, 4, 1, -9, 1], [3, 8, 0, 0, -8, 0, 3, 0, 2, -2], [17, -4, 8, 11, -4, 3, -3, 9, -1, -7], [-14, -14, -5, 7, -8, 12, -12, -3, -2, 1], [8, -3, 9, 1, -3, 8, -8, 2, -5, -4], [11, -11, 3, 3, 5, 6, 5, 6, -8, 5], [10, -2, 9, -1, -4, 10, -6, 3, -8, -3], [14, -1, 2, 0, -2, -2, 2, 5, -3, 2], [9, -2, 2, -7, 0, 9, -1, 9, -7, 7], [-15, 3, 5, 19, 3, 0, 1, -10, 7, -12], [1, 4, -2, -2, 0, -4, 5, -3, -1, 3], [-4, 3, -6, -14, 4, -8, 8, -1, -5, 10], [-4, 8, -3, -7, 8, 0, 8, -6, 1, 8], [7, 13, 7, 4, -5, -3, 3, -3, 10, 1], [10, -5, -10, -12, 7, -2, 2, -3, -7, 5], [-3, -1, -7, -4, -9, -3, 11, 5, 4, -5], [13, -7, 3, 6, -7, 3, -3, 7, -4, 1], [-25, 2, -2, 3, -4, 5, -5, -9, 7, 1], [-9, -1, -7, -11, -1, 0, 13, 3, -10, 1], [-5, -6, 4, 4, -2, 8, -13, -3, -7, 3], [-9, 11, -9, -9, 7, -18, 9, -4, 3, 6], [0, 7, -9, -9, -9, 0, -8, 2, -7, 0], [0, -2, 2, 2, 8, 0, -16, 0, 2, 6], [-21, 16, -13, -13, 10, -26, 5, -9, 13, 0], [-3, -1, 25, 31, -1, 0, -3, -2, 10, -21], [-8, 12, -13, -5, 0, -8, 4, -7, -4, 5], [6, -4, 4, -2, 7, 6, 26, 6, -2, 9], [-11, 2, -13, -7, 2, 0, 5, -5, 0, 7], [10, -6, 4, 2, -7, -2, 2, 12, -8, 5], [-8, 1, 0, 0, 1, 0, 8, -1, 3, 3], [6, 2, 9, 3, 2, 6, -14, -5, -8, -1], [-6, -3, 6, 20, -8, 14, -14, -5, 11, -12], [-5, 0, 6, 11, 14, -5, -7, -11, 5, 3], [-5, 12, -7, -7, 2, -14, 9, 1, -2, 9], [-9, 19, -11, -11, 3, -22, 13, -8, 7, 4], [8, 2, 23, 21, 2, 0, -6, -1, 10, -11], [1, -3, 5, 4, 1, 1, 3, -1, 2, -3], [23, -4, 3, 3, -2, 6, 17, 3, 0, -3], [-6, 6, -12, -22, 9, -10, 10, -8, -4, 13], [-8, 0, -1, -3, -2, -2, 2, 1, -2, 5], [-8, -10, 6, 6, -2, 12, -20, 16, -4, -2], [16, -11, 2, 10, -11, 8, -8, 5, -3, 1], [-3, -13, 5, 5, 3, 10, -13, 13, -8, 3], [-8, 15, -11, -15, 9, -4, 4, -22, 11, 6], [0, 11, 8, 8, -4, 0, 0, -3, 11, -4], [-6, 2, 0, -6, 2, -6, 6, 4, -4, 4], [-4, -10, 0, 0, -10, 0, 4, 10, 2, 2], [-16, 11, -12, -12, 13, -24, 8, -7, 9, 3], [9, -7, 4, 8, -7, 0, -13, 5, -7, -15], [10, -9, 10, 0, -8, 10, -16, 9, -1, -8], [-22, 3, 0, -2, -5, -2, 2, -1, 1, 7], [-8, 6, 2, 2, -10, 4, -12, -6, 4, -6], [22, -6, -9, -11, 4, -2, 2, -1, -8, 7], [-10, 12, -3, -3, -6, -6, -4, 11, -6, 9], [-26, 8, -1, -1, -6, -2, -24, -7, 2, -1], [6, -6, -18, -26, 13, -8, 8, -4, -14, 13], [17, -7, 11, 15, -7, 0, -21, 5, -4, -19], [20, 1, -10, -6, 5, 4, -4, -15, 5, 1], [6, 13, -1, 1, 7, 2, -2, -16, 15, -8], [-7, -11, 11, 21, -11, 0, -3, 6, 4, -17], [-22, 6, 2, -4, 3, -6, 6, 2, 0, 1], [-8, 7, -6, -10, 4, -4, 4, -9, 3, 6], [0, -13, 12, 12, 3, 0, 14, 1, 13, -9], [20, 2, 10, -10, 2, 0, 0, 8, -12, -2], [-20, 0, -1, 19, 0, 0, 0, -10, 13, -6], [-8, 8, -2, -6, 6, -4, 4, -6, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_g_d();".
// 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_370_2_g_d(:prec:=10)
    chi := MakeCharacter_370_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_370_2_g_d();".
// 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_370_2_g_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![72, -372, 961, -442, 74, 60, 59, -12, 2, 2, 1]>],Snew);
    return Vf;
end function;