
// Make newform 2340.2.bp.e in Magma, downloaded from the LMFDB on 18 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2340_bp();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2340_bp_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_2340_2_bp_e();".
// 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_2340_2_bp_e();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, 1, 1, 0], [0, 2, 0, 1], [-1, 1, -1, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    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_2340_bp();"
function MakeCharacter_2340_bp()
    N := 2340;
    order := 4;
    char_gens := [1171, 2081, 937, 1081];
    v := [4, 4, 1, 1];
    // 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_2340_bp_Hecke();"
function MakeCharacter_2340_bp_Hecke( : Kf := false)
    N := 2340;
    order := 4;
    char_gens := [1171, 2081, 937, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 0, 0], [-1, -1, 0, 1], [0, 0, 0, 0], [1, 0, -1, 1], [-3, 0, 2, 0], [1, 0, -1, -2], [-1, 0, 1, -3], [5, 1, 5, 0], [0, -1, 2, -1], [1, -3, 1, 0], [0, -3, 0, 3], [-7, -2, -7, 0], [-1, -3, -1, 0], [-4, -4, 0, 4], [5, 0, -5, -4], [-5, -1, -5, 0], [-4, -3, 0, 3], [0, -3, -10, -3], [-1, 1, -1, 0], [0, 3, -2, 3], [0, 3, 6, 3], [4, -2, 0, 2], [1, 2, 1, 0], [0, -6, -4, -6], [0, -2, -8, -2], [-1, -3, -1, 0], [1, -1, 1, 0], [5, 0, -5, -6], [-1, 0, 1, 8], [5, 0, -5, -3], [-12, -6, 0, 6], [-10, 2, 0, -2], [0, -9, -6, -9], [-7, 0, 7, 8], [-5, 0, 5, 3], [-5, 0, -5, 0], [0, -3, -14, -3], [20, 2, 0, -2], [-1, 10, -1, 0], [-12, 0, 0, 0], [0, -3, -18, -3], [12, 6, 0, -6], [0, 6, 16, 6], [0, 0, -12, 0], [-8, -6, 0, 6], [-16, -6, 0, 6], [-12, -6, 0, 6], [0, -3, -18, -3], [11, 12, 11, 0], [-1, 4, -1, 0], [-7, 0, 7, -1], [7, 0, -7, -6], [0, -5, 10, -5], [1, 0, -1, 10], [-1, 0, 1, -13], [0, 2, 20, 2], [7, 0, -7, -9], [-7, 0, 7, 6], [1, 0, -1, -2], [11, -3, 11, 0], [0, -3, 18, -3], [-12, 0, 0, 0], [0, -5, -14, -5], [-5, 0, 5, -6], [0, -3, -6, -3], [-11, 3, -11, 0], [5, 0, -5, -12], [1, 5, 1, 0], [-7, 6, -7, 0], [16, 1, 0, -1], [-5, 5, -5, 0], [7, -3, 7, 0], [19, 0, -19, -12], [-1, 0, 1, 3], [-8, -8, 0, 8], [-6, 6, 0, -6], [24, 9, 0, -9], [1, 0, -1, -2], [-13, 0, 13, 6], [0, 11, 14, 11], [13, 6, 13, 0], [-25, -5, -25, 0], [17, 0, 17, 0], [-20, 0, 0, 0], [11, 0, -11, -7], [-17, -4, -17, 0], [0, 6, -4, 6], [-1, 10, -1, 0], [0, -3, -14, -3], [-5, 0, 5, 13], [-5, -19, -5, 0], [0, 9, 14, 9], [0, 7, -2, 7], [-13, 0, 13, 21], [23, 0, -23, -7], [-13, 0, 13, 2], [-12, 9, 0, -9], [1, 0, -1, 15], [-5, 0, 5, 12], [7, 9, 7, 0], [32, -1, 0, 1], [-7, -5, -7, 0], [12, 9, 0, -9], [0, -3, -18, -3], [0, 9, 2, 9], [0, -9, -6, -9], [0, 6, 36, 6], [0, 17, 14, 17], [14, 0, 0, 0], [-5, 15, -5, 0], [-6, -6, 0, 6], [0, -6, -24, -6], [-7, -15, -7, 0], [-5, 0, 5, -15], [0, 4, -20, 4], [24, -6, 0, 6], [7, 0, -7, -5], [-13, 0, 13, 26], [0, -13, -10, -13], [1, 0, -1, 6], [-1, 18, -1, 0], [13, 8, 13, 0], [0, -9, 6, -9], [1, -21, 1, 0], [0, 10, 4, 10], [-1, -12, -1, 0], [12, -6, 0, 6], [17, 0, -17, 3], [-30, 0, 0, 0], [-7, -9, -7, 0], [-8, -14, 0, 14], [0, 9, 18, 9], [-23, -6, -23, 0], [19, 0, -19, 4], [5, 0, -5, 0], [-8, 7, 0, -7], [36, 6, 0, -6], [1, 0, -1, 16], [0, 5, -10, 5], [-11, -9, -11, 0], [5, -20, 5, 0], [11, 21, 11, 0], [-16, 2, 0, -2], [4, -3, 0, 3], [-5, 11, -5, 0], [0, 9, 34, 9], [13, 0, -13, -14], [0, 3, -42, 3], [28, -2, 0, 2], [0, -12, 8, -12], [0, 13, 22, 13], [23, -3, 23, 0], [-23, -1, -23, 0], [-19, 0, 19, 15], [-12, 12, 0, -12], [0, 9, -6, 9], [23, 0, -23, -16], [13, -12, 13, 0], [-19, -20, -19, 0], [-16, -4, 0, 4], [-31, -2, -31, 0], [0, -9, -22, -9], [0, 0, 0, 0], [0, 3, 6, 3], [0, -3, -42, -3], [-40, -6, 0, 6], [-29, 0, -29, 0], [29, 12, 29, 0], [-7, -8, -7, 0], [5, 0, -5, -19], [7, 0, -7, -12], [0, 1, 10, 1], [-24, -9, 0, 9], [-20, -18, 0, 18], [0, -4, 8, -4], [-29, 0, 29, 21], [31, 0, -31, -2], [23, 27, 23, 0], [0, -6, 24, -6], [12, 0, 0, 0], [0, 1, 46, 1], [13, 0, -13, -6], [0, -6, -24, -6], [0, 15, 6, 15], [-12, -9, 0, 9], [-13, 0, 13, -12], [24, -6, 0, 6], [-1, -6, -1, 0], [-23, 0, 23, -11], [1, 0, -1, -12], [4, 4, 0, -4], [-16, -6, 0, 6], [-11, 0, 11, -8], [31, 0, -31, -17], [-13, 4, -13, 0], [-29, -18, -29, 0], [11, -12, 11, 0], [32, 5, 0, -5], [-1, 0, 1, 5], [7, 14, 7, 0], [32, 12, 0, -12], [0, 0, 32, 0], [0, -15, -18, -15], [19, 29, 19, 0], [-29, -10, -29, 0], [-1, 0, 1, -3], [23, 0, -23, 5], [-25, 0, 25, 20], [0, -9, -54, -9], [-31, 0, 31, 12], [-12, 3, 0, -3], [-11, 0, 11, 15], [32, -10, 0, 10], [-19, 0, 19, 29], [-11, 0, 11, 12], [7, 21, 7, 0], [42, 0, 0, 0], [0, -3, 54, -3], [0, 12, 12, 12], [-4, -15, 0, 15], [-31, 9, -31, 0], [5, 0, -5, 14], [12, -12, 0, 12], [-17, 0, 17, 1], [-20, -3, 0, 3], [-7, 0, 7, -10], [0, -1, -46, -1], [17, 0, -17, -9], [-23, 0, 23, 31], [17, 18, 17, 0], [0, -9, -42, -9], [-11, -9, -11, 0], [0, -8, -8, -8], [-35, 0, 35, 15], [0, 21, 18, 21], [35, -6, 35, 0], [0, 3, 54, 3], [36, 30, 0, -30], [-48, -12, 0, 12], [-32, 4, 0, -4], [0, 9, -30, 9], [-11, 0, 11, 3], [35, 0, -35, 0], [10, 4, 0, -4], [12, 24, 0, -24], [-12, 0, 0, 0], [-23, 0, 23, -11], [35, 0, -35, -21], [-7, 7, -7, 0], [0, -3, 0, 3], [11, 4, 11, 0], [32, -10, 0, 10], [-38, -6, 0, 6], [11, 0, -11, 2], [-5, -25, -5, 0], [20, 9, 0, -9], [0, -21, -46, -21], [-29, 0, 29, 10], [0, -9, -34, -9], [-13, 3, -13, 0], [1, -31, 1, 0], [-31, 0, 31, 18], [-5, 0, 5, 36], [-40, 5, 0, -5], [0, -15, 6, -15], [-7, 0, 7, 20], [11, -8, 11, 0], [0, -3, -2, -3], [-7, -20, -7, 0], [0, 6, 0, 6], [0, -9, -34, -9], [0, -18, -8, -18], [-8, 0, 0, 0], [-5, 18, -5, 0], [0, -6, 0, 6], [0, 21, 6, 21], [23, 36, 23, 0], [25, 0, -25, -48], [0, 1, 10, 1], [-13, 0, 13, 5], [-41, 0, 41, 9], [13, 41, 13, 0], [31, 12, 31, 0], [-36, -18, 0, 18], [0, 1, -50, 1], [25, 0, -25, -24], [0, 12, 0, 12], [-7, -9, -7, 0], [36, 9, 0, -9], [-12, 3, 0, -3], [25, 35, 25, 0], [-26, -20, 0, 20], [-23, 0, 23, 1], [1, 0, -1, -18], [30, 0, 0, 0], [44, 24, 0, -24], [17, 4, 17, 0], [0, 17, 2, 17], [48, -12, 0, 12], [-31, -36, -31, 0], [-46, -16, 0, 16], [40, 12, 0, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2340_bp_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2340_2_bp_e();".
// 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_2340_2_bp_e(:prec:=4)
    chi := MakeCharacter_2340_bp();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2340_2_bp_e();".
// 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_2340_2_bp_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2340_bp();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, -12, 18, -6, 1]>],Snew);
    return Vf;
end function;