
// Make newform 370.2.m.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_m_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_m_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_370_2_m_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 := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]];
    Rf_basisdens := [1, 1, 6, 2];
    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_m();"
function MakeCharacter_370_m()
    N := 370;
    order := 6;
    char_gens := [297, 261];
    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_370_m_Hecke();"
function MakeCharacter_370_m_Hecke(Kf)
    N := 370;
    order := 6;
    char_gens := [297, 261];
    char_values := [[-1, 0, 0, 0], [1, 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 := 2;
    raw_aps := [[0, 0, 1, 0], [1, -1, 0, 0], [0, 0, -1, 1], [2, 0, 2, 0], [0, 0, 0, 0], [3, -1, -2, 2], [-1, 2, 2, -1], [0, 0, 0, 0], [2, 2, 2, 2], [-3, -1, 7, 1], [2, 3, -7, -3], [3, 0, -7, 0], [-1, 2, -1, -4], [2, -3, -3, -3], [0, 2, -2, -2], [-3, 0, 1, -1], [-10, 0, 6, 2], [0, -3, -3, 0], [2, -6, -4, 0], [4, -2, -2, 4], [-4, 0, 8, 0], [-2, 6, 4, 0], [8, 0, -6, -4], [-13, 0, 6, -1], [5, -3, -3, -3], [-1, -1, -1, -1], [-6, 4, 4, 4], [1, 7, 8, 0], [3, 0, 0, 3], [-6, 12, 0, -6], [-20, 0, 10, 0], [0, 0, -2, -4], [3, -1, -5, 1], [2, -4, 12, 2], [-13, -1, -1, -1], [-1, -5, 6, 10], [4, 0, 1, 6], [3, -6, 7, 3], [-8, 4, 4, -8], [7, 0, 2, 11], [0, 8, -8, -8], [-12, 5, 7, -10], [-4, -7, 15, 7], [9, 1, 1, 1], [12, 0, -5, 2], [-8, 9, 7, -9], [-6, 4, 4, 4], [2, 6, -10, -6], [-7, -1, 8, 2], [8, -5, -3, 10], [-7, -1, 15, 1], [4, 0, 2, 8], [-12, 0, -12, 0], [2, -10, 6, 10], [5, -10, -16, 5], [0, 4, 4, 0], [-14, 4, 4, 4], [-1, 2, -15, -1], [7, 0, -7, 0], [12, 0, -5, 2], [13, -3, -10, 6], [13, -8, 5, 0], [-16, 3, 29, -3], [-3, 0, -5, -13], [9, -18, -2, 9], [-4, 0, -3, -10], [-32, 0, 16, 0], [2, 3, 5, 0], [-2, -2, -2, -2], [1, -2, -4, 1], [9, -18, -6, 9], [-8, 1, 1, 1], [10, -6, 4, 0], [7, 0, 7, 0], [-2, 4, -4, -2], [-26, 4, 22, -8], [0, -11, -11, 0], [1, 6, -8, -6], [-4, 8, 0, -8], [34, 0, -17, 0], [8, -16, -10, 8], [-7, 9, 5, -9], [13, -11, 2, 0], [-1, 9, -7, -9], [-11, -3, -14, 0], [18, -1, -35, 1], [7, 7, 14, 0], [-4, 7, -3, -14], [18, 0, -14, -10], [18, -4, -14, 8], [20, -1, -1, -1], [17, 0, -15, -13], [2, 0, 0, 0], [4, 10, 10, 10], [8, 0, 8, 0], [-8, 16, 4, -8], [-7, 14, 2, -7], [-1, -1, 2, 2], [7, 3, -10, -6], [5, -21, 11, 21], [-20, 11, 11, 11], [4, -8, 31, 4], [-6, -6, -6, -6], [11, -10, -12, 10], [12, -24, -2, 12], [12, -24, -14, 12], [7, 7, -14, -14], [-1, 5, -3, -5], [17, -1, -16, 2], [-2, 4, -19, -2], [0, 0, -8, 0], [47, 0, -22, 3], [0, 0, 4, 8], [-10, 12, 12, 12], [-31, 0, 11, -9], [0, 0, 27, 0], [4, 5, 5, 5], [-10, -4, 14, 8], [-27, 0, 27, 0], [-28, 8, 20, -16], [22, -15, 7, 0], [-4, 0, -4, 0], [13, -7, -19, 7], [3, -6, -21, 3], [-4, 8, 0, -4], [36, 0, -20, -4], [4, 6, -14, -6], [-9, 18, -9, -9], [-8, 0, 8, 0], [-4, -12, -16, 0], [-2, 8, 8, 8], [10, -14, -4, 0], [2, 9, 9, 9], [6, -12, 7, 6], [-9, 18, -18, -9], [-12, 24, 0, -24], [-11, -14, -25, 0], [-10, 15, 5, -15], [-1, 2, 11, -1], [-13, -5, -18, 0], [10, -6, -4, 12], [12, 6, -18, -12], [-6, 0, 0, -6], [12, -12, 0, 24], [20, 0, -4, 12], [7, -14, 31, 7], [-6, 12, 19, -6], [-7, 5, 5, 5], [-2, 0, 4, 0], [18, 0, -2, 14], [17, -18, -16, 18], [-6, -3, 9, 6], [-29, -9, 38, 18], [2, -10, 6, 10], [-40, 4, 36, -8], [8, -25, 9, 25], [-2, 0, 4, 0], [-11, -8, 19, 16], [63, 0, -30, 3], [5, -10, -4, 5], [5, -10, 23, 5], [0, 0, 4, 8], [-14, -6, 20, 12], [8, -16, -10, 8], [-6, 12, -36, -6], [54, 0, -26, 2], [-8, 3, 13, -3], [-3, 6, -23, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_m_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_370_2_m_b(:prec:=4)
    chi := MakeCharacter_370_m();
    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_m_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_370_2_m_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 6, 1, -3, 1]>],Snew);
    return Vf;
end function;