
// Make newform 370.2.b.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_b_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_b_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_b_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 := [8, 4, 1, -46, 154, -124, 51, -4, 2, -2, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [81466, 989985, -5967211, 3781077, -1171720, -261048, -48258, 63073, -27007, -8189], [-972956, 2894160, -3989713, -260445, 1401884, -566796, -56736, 10432, 29441, -9773], [-346156, 926646, -1662629, 2027820, -1213433, 540120, -71109, 30482, -16670, 11204], [-912988, -1180074, 6038593, -12786054, 10297489, -3969330, 373635, -151780, 174376, -76468], [-1174924, 4399206, 9536722, -22392627, 16117294, -5058726, 400662, -307876, 280360, -93757], [2825248, -3152958, 3241250, -14749095, 12501560, -5584197, 396594, -195959, 194594, -109244], [123896, 5464986, -8939543, -7007874, 10028530, -5536122, 264366, -66016, 142891, -116740], [-1400668, -5693844, 8386744, -21830091, 16479244, -7542114, 635400, -304354, 250360, -150175]];
    Rf_basisdens := [1, 1, 1402866, 1402866, 467622, 1402866, 1402866, 1402866, 1402866, 1402866];
    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_b();"
function MakeCharacter_370_b()
    N := 370;
    order := 2;
    char_gens := [297, 261];
    v := [1, 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_370_b_Hecke();"
function MakeCharacter_370_b_Hecke(Kf)
    N := 370;
    order := 2;
    char_gens := [297, 261];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [1, -1, 0, -1, 1, -1, 0, 0, 0, 1], [0, 1, 1, -1, -1, 1, 0, -1, 0, -1], [1, -2, 0, -2, 1, -1, 0, 0, 0, 0], [0, -1, 0, -1, 2, -3, 2, 0, -2, 0], [0, 1, -1, -1, 1, -1, 0, -1, 0, -1], [0, 0, 0, 0, 0, -2, 0, -2, 0, 2], [0, 1, 0, 1, 0, 1, -2, 0, 2, 0], [-3, -1, 0, -2, -1, -2, 1, 0, 1, 0], [3, 2, 0, 3, -2, 3, -1, 3, -1, -3], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 2, 2, 1, 0, 1, 0], [0, 1, -5, 1, 1, 1, -2, 1, 2, 1], [0, -2, -4, 2, 0, 0, 0, 2, 0, 2], [0, -1, 1, -3, 3, -3, 4, 1, -4, 1], [-4, 0, 0, 2, 0, 2, -2, 2, -2, -2], [-1, 0, 0, -1, 2, 1, 1, -1, 1, 1], [0, 2, -4, 1, -2, 4, -3, 0, 3, 0], [-2, 4, 0, 4, -4, 4, 0, 4, 0, -4], [0, -4, -2, 3, -4, 0, 1, 0, -1, 0], [-4, 1, 0, 1, -2, -3, 0, -2, 0, 2], [0, 0, 4, -2, -2, 2, 2, 0, -2, 0], [-4, 2, 0, 0, 0, 4, 2, 2, 2, -2], [0, 1, -3, 1, 3, 1, -2, 3, 2, 3], [2, 4, 0, 6, -2, 4, -2, 2, -2, -2], [0, 2, -8, 0, 1, 2, -2, 1, 2, 1], [0, -2, 2, 2, 5, -5, 0, 2, 0, 2], [1, -2, 0, 1, -1, -1, -3, 2, -3, -2], [0, -1, 3, 3, -3, 3, -2, 1, 2, 1], [0, 6, 0, -2, 2, 2, -4, -2, 4, -2], [-6, 4, 0, 4, -2, 4, 0, 2, 0, -2], [0, 0, 2, 1, -4, 4, -1, 0, 1, 0], [-9, 6, 0, 5, -2, 8, 1, 4, 1, -4], [4, 4, 0, 4, -6, -2, 0, 0, 0, 0], [6, -2, 0, -4, -2, -2, 2, 2, 2, -2], [0, -5, -1, 1, 3, -9, 4, -1, -4, -1], [0, -7, -5, 3, -7, -1, 4, -1, -4, -1], [0, 0, 6, -2, 5, -4, 2, 1, -2, 1], [0, 3, 1, -5, 5, -3, 2, -1, -2, -1], [6, 0, 0, 2, 4, -2, -2, -6, -2, 6], [4, -8, 0, -10, 2, -8, 2, -2, 2, 2], [-3, 2, 0, 1, 2, 1, 1, -3, 1, 3], [0, -2, -14, 2, -2, 2, 0, 2, 0, 2], [0, 0, -6, 2, -2, 4, -2, 2, 2, 2], [0, 2, 0, 2, 2, 6, 0, 2, 0, -2], [3, -2, 0, 0, -1, -3, -2, 0, -2, 0], [0, 3, -3, 1, 1, -1, -4, -3, 4, -3], [0, -3, 7, 3, -5, 1, 0, -1, 0, -1], [-4, 2, 0, -2, -4, 0, 4, 2, 4, -2], [0, 4, 6, -8, 7, -5, 4, -2, -4, -2], [3, 2, 0, -1, -4, -1, 3, 1, 3, -1], [14, 2, 0, 0, 0, -2, 2, -4, 2, 4], [0, 2, 0, 6, 2, 8, -4, 4, -4, -4], [0, 2, 2, -2, 6, -4, 0, 0, 0, 0], [0, -3, 9, 1, -3, -5, 2, -5, -2, -5], [-4, 4, 0, -2, -6, -4, 6, -2, 6, 2], [10, -2, 0, -2, -2, -2, 0, 2, 0, -2], [0, 0, 4, 4, -5, 4, -4, -1, 4, -1], [-18, 8, 0, 10, -2, 6, -2, 0, -2, 0], [0, 0, 8, -2, -4, 4, 2, 0, -2, 0], [0, 1, 3, 3, 7, -3, -4, 3, 4, 3], [0, 0, 8, -8, 11, -11, 8, 0, -8, 0], [1, 7, 0, 6, 1, 4, 1, -4, 1, 4], [0, 4, 2, -2, -4, 4, -2, -4, 2, -4], [0, 3, -9, 5, -1, 5, -8, 1, 8, 1], [-10, -8, 0, -4, 0, -2, -4, 6, -4, -6], [0, 6, -16, -3, 2, 6, -3, 2, 3, 2], [0, 6, 12, -4, -2, 4, -2, -4, 2, -4], [-2, -2, 0, -2, 0, 2, 0, 4, 0, -4], [0, -3, 1, 1, 5, -5, 2, 3, -2, 3], [14, 2, 0, 4, 4, 8, -2, 2, -2, -2], [0, -3, -3, -3, -3, 1, 6, 1, -6, 1], [0, -6, 10, 0, 2, -2, 6, 6, -6, 6], [12, -4, 0, -10, 1, -5, 6, -2, 6, 2], [0, 0, 12, -4, 4, -4, 4, 0, -4, 0], [-1, -7, 0, -6, 5, 6, -1, 8, -1, -8], [0, 0, -26, 0, 2, -2, 0, 0, 0, 0], [-10, 4, 0, -4, -6, -4, 8, -2, 8, 2], [-4, 8, 0, 10, -4, -4, -2, -8, -2, 8], [-4, -4, 0, -2, 7, 1, -2, -2, -2, 2], [-6, 3, 0, 1, -6, 3, 2, 6, 2, -6], [-7, 0, 0, 3, 1, 1, -3, 0, -3, 0], [0, 2, -4, 2, -11, 13, -4, 0, 4, 0], [3, -3, 0, -6, 7, -4, 3, -8, 3, 8], [0, 0, -10, 7, -4, 6, -7, 2, 7, 2], [-4, -6, 0, -6, -2, -2, 0, 6, 0, -6], [0, 1, -7, 1, 1, 1, -2, 1, 2, 1], [1, -10, 0, -7, 7, 3, -3, 6, -3, -6], [0, 1, 12, -7, 12, -11, 6, 0, -6, 0], [0, -1, -3, 11, -11, 11, -10, 1, 10, 1], [24, 1, 0, 1, 6, -3, 0, -10, 0, 10], [0, 4, 0, 4, 0, 4, -8, 0, 8, 0], [-4, 0, 0, -2, -3, -1, 2, 2, 2, -2], [-18, 2, 0, -4, -6, -4, 6, 0, 6, 0], [0, -4, 18, -6, -5, 0, 10, -1, -10, -1], [2, 8, 0, 2, 0, 4, 6, -4, 6, 4], [-3, -6, 0, -9, 9, 1, 3, -2, 3, 2], [0, 12, 4, 0, -6, 14, -12, -4, 12, -4], [2, 2, 0, 2, -7, -6, 0, -1, 0, 1], [0, -9, -3, 5, 1, -5, 4, 5, -4, 5], [0, -2, -6, 6, -3, -2, -4, -3, 4, -3], [0, 3, -11, 5, -9, 9, -8, -3, 8, -3], [2, 2, 0, -2, 2, 10, 4, 6, 4, -6], [15, -6, 0, 1, 8, 2, -7, 0, -7, 0], [0, -2, -14, 4, 2, -4, -2, 0, 2, 0], [0, 5, 9, 1, 3, 1, -6, -1, 6, -1], [0, 2, 16, -7, -6, 6, 5, -2, -5, -2], [12, -14, 0, -14, 10, -10, 0, -6, 0, 6], [17, -6, 0, -2, 1, -7, -4, -2, -4, 2], [0, 3, -10, -1, 10, -9, -2, -2, 2, -2], [0, 7, -13, 3, -5, 13, -10, 1, 10, 1], [0, 8, -6, -4, -7, 17, -4, 2, 4, 2], [-1, -14, 0, -8, 7, -3, -6, 4, -6, -4], [15, -1, 0, 4, 7, -2, -5, -8, -5, 8], [3, -16, 0, -13, 4, -16, -3, -4, -3, 4], [0, 1, 7, -5, 15, -9, 4, 5, -4, 5], [0, 5, 2, -7, 6, 1, 2, 2, -2, 2], [0, 9, -2, -3, -8, 19, -6, 2, 6, 2], [0, 0, 0, 10, -5, 1, -10, 6, -10, -6], [25, 1, 0, 4, -1, -4, -3, -4, -3, 4], [0, -6, -4, -6, 17, -23, 12, 0, -12, 0], [0, -2, 14, 0, 2, -4, 2, 0, -2, 0], [0, -5, -19, 3, -7, -1, 2, -3, -2, -3], [-1, 10, 0, 7, -11, 5, 3, 6, 3, -6], [-6, -2, 0, -2, 3, -10, 0, -11, 0, 11], [-3, 0, 0, 5, -4, -1, -5, 3, -5, -3], [-16, -6, 0, -8, 4, -8, 2, -6, 2, 6], [0, -4, -2, 10, -5, 0, -6, -1, 6, -1], [0, -5, 19, 1, -5, -1, 4, -1, -4, -1], [-13, -2, 0, -2, -1, -7, 0, -4, 0, 4], [0, -3, 37, 1, -3, 3, 2, 3, -2, 3], [8, 22, 0, 18, -2, 26, 4, 6, 4, -6], [0, 4, -20, 4, -3, 12, -8, 5, 8, 5], [19, -12, 0, -5, 10, -14, -7, -12, -7, 12], [-20, 10, 0, 6, -6, 6, 4, 2, 4, -2], [0, 7, -25, 5, 3, 1, -12, -3, 12, -3], [0, 6, 6, 0, -12, 12, -6, -6, 6, -6], [0, 9, 8, -11, 14, -9, 2, -4, -2, -4], [-28, 14, 0, 14, -8, 10, 0, 4, 0, -4], [4, -8, 0, -11, 6, -8, 3, -6, 3, 6], [2, 6, 0, 8, 0, 4, -2, -2, -2, 2], [0, 8, 8, -4, 4, 8, -4, 4, 4, 4], [0, 5, 27, -5, 15, -5, 0, 5, 0, 5], [1, -23, 0, -20, 11, -16, -3, -4, -3, 4], [8, 2, 0, 8, -2, -8, -6, -8, -6, 8], [0, -14, -2, 14, -15, 2, 0, 1, 0, 1], [0, 7, -19, -1, 1, 9, -6, 3, 6, 3], [-32, 10, 0, 10, -6, 10, 0, 6, 0, -6], [0, 7, -1, -3, 11, -3, -4, 1, 4, 1], [0, -7, -5, -1, 13, -9, 8, 11, -8, 11], [-1, 14, 0, 8, -9, 3, 6, -2, 6, 2], [0, 9, 13, -9, 9, -5, 0, -5, 0, -5], [0, 7, 7, -7, -1, 3, 0, -5, 0, -5], [0, -8, 12, -6, 2, -14, 14, -4, -14, -4], [4, -2, 0, 6, -2, 6, -8, 10, -8, -10], [-24, 6, 0, -2, -2, 14, 8, 10, 8, -10], [23, 2, 0, 2, 3, 11, 0, 6, 0, -6], [0, 10, 4, -2, -5, 15, -8, 0, 8, 0], [14, -4, 0, -14, -4, -10, 10, -2, 10, 2], [0, 1, 3, -3, -5, 13, 2, 7, -2, 7], [0, -4, 10, -3, 8, -16, 7, -4, -7, -4], [0, 4, -6, -6, -1, 4, 2, -1, -2, -1], [3, -18, 0, -16, -1, -23, -2, -4, -2, 4], [0, -7, 11, 11, 1, -5, -4, 3, 4, 3], [0, -5, -13, -1, 5, -11, 6, -1, -6, -1], [1, 11, 0, 14, -3, 0, -3, -8, -3, 8], [0, -2, 4, 10, -18, 10, -8, -6, 8, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_b_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_b_d(:prec:=10)
    chi := MakeCharacter_370_b();
    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_b_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_b_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 169, 0, 234, 0, 107, 0, 18, 0, 1]>,<7,R![160000, 0, 92800, 0, 18096, 0, 1592, 0, 65, 0, 1]>],Snew);
    return Vf;
end function;