
// Make newform 2400.2.w.b in Magma, downloaded from the LMFDB on 18 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2400_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2400_w_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_2400_2_w_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_2400_2_w_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, 0, 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_2400_w();"
function MakeCharacter_2400_w()
    N := 2400;
    order := 4;
    char_gens := [1951, 901, 1601, 577];
    v := [2, 4, 4, 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_2400_w_Hecke();"
function MakeCharacter_2400_w_Hecke( : Kf := false)
    N := 2400;
    order := 4;
    char_gens := [1951, 901, 1601, 577];
    char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 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, -1, 0, 0], [0, 0, 0, 0], [-2, 0, 2, -2], [0, -2, 2, -2], [0, 0, 0, 2], [2, -4, 2, 0], [-2, 4, 0, -4], [-2, 0, -2, 0], [0, 4, 0, 4], [0, 0, -6, 0], [4, 2, 4, 0], [-2, 4, 0, -4], [0, 0, 0, 0], [2, 0, -2, 4], [-4, 0, 4, 4], [6, 6, 0, -6], [6, 0, 0, 0], [-4, 0, 4, 4], [0, -4, 0, -4], [0, 0, 0, -12], [-2, 8, 0, -8], [-4, 4, -4, 0], [0, 4, 6, 4], [4, -4, 4, 0], [-4, 8, 0, -8], [6, -2, 6, 0], [4, 0, -4, -4], [0, 8, 2, 8], [2, 0, -2, -12], [-6, 0, 6, 6], [0, -2, -6, -2], [-2, -12, -2, 0], [2, 0, 0, 0], [0, 4, -4, 4], [0, -8, 2, -8], [12, 2, 12, 0], [12, 0, 12, 0], [2, 0, -2, -16], [-4, 0, 4, 12], [18, -2, 0, 2], [-14, 8, 0, -8], [0, -4, 12, -4], [12, 0, -12, -8], [-8, 8, -8, 0], [14, 0, 0, 0], [0, 0, 6, 0], [2, 14, 2, 0], [4, 0, -4, 12], [0, 0, -2, 0], [-2, 0, 2, -12], [-12, -4, 0, 4], [-10, 0, 0, 0], [0, 2, -22, 2], [-10, -4, -10, 0], [-14, 8, -14, 0], [0, -4, 16, -4], [0, 0, 6, 0], [4, 22, 4, 0], [-6, 0, 0, 0], [4, -4, 4, 0], [4, 0, -4, 16], [-8, 0, 8, 20], [0, 12, 12, 12], [4, 0, -4, -16], [8, 0, 8, 0], [0, -4, 10, -4], [8, -4, 8, 0], [16, 0, -16, -4], [0, -8, -2, -8], [-18, 0, 18, -12], [-4, -16, 0, 16], [-18, 0, 18, 10], [-12, 0, 12, -10], [-22, -8, 0, 8], [6, 12, 6, 0], [0, -8, -24, -8], [8, 2, 8, 0], [-6, 12, 0, -12], [0, 0, -6, 0], [18, -6, 0, 6], [2, -8, 0, 8], [0, 8, 12, 8], [0, 0, 0, -4], [14, -16, 0, 16], [-4, -20, -4, 0], [0, -24, 6, -24], [20, -4, 20, 0], [-36, 4, 0, -4], [2, 10, 2, 0], [-16, 0, 16, -4], [-32, -4, 0, 4], [-6, 0, 6, -34], [0, 10, -14, 10], [-14, -12, 0, 12], [-10, -28, -10, 0], [0, -8, 12, -8], [-2, -16, 0, 16], [-4, -8, -4, 0], [22, -8, 0, 8], [0, 0, 0, 16], [4, -12, 4, 0], [16, 12, 16, 0], [0, 4, -6, 4], [0, 12, 10, 12], [12, -4, 12, 0], [-4, 0, 4, -28], [-2, 0, 2, 4], [-20, -8, 0, 8], [18, 0, 0, 0], [-6, 0, 6, 18], [8, 0, -8, -10], [-14, 12, -14, 0], [-2, -8, 0, 8], [0, 0, -2, 0], [18, -16, 0, 16], [12, 8, 12, 0], [-6, 0, 6, -36], [12, 0, -12, -24], [18, 2, 0, -2], [2, -24, 0, 24], [-16, 0, 16, -4], [-4, 16, -4, 0], [8, -12, 8, 0], [0, 12, 14, 12], [-12, 16, 0, -16], [0, 0, 6, 0], [-20, 8, 0, -8], [14, 0, -14, 10], [-12, 0, 12, -34], [-2, -12, 0, 12], [18, 12, 18, 0], [0, -24, -14, -24], [-4, -14, -4, 0], [-22, -8, 0, 8], [0, -8, 26, -8], [0, 0, 0, -24], [12, 0, -12, 24], [-24, -12, -24, 0], [0, 8, 14, 8], [0, 0, -6, 0], [16, 20, 0, -20], [-6, -34, -6, 0], [8, 0, -8, -12], [0, 0, -18, 0], [-36, -8, 0, 8], [8, 0, -8, -26], [26, 12, 26, 0], [6, 24, 0, -24], [6, 24, 6, 0], [12, -18, 12, 0], [30, 12, 0, -12], [-12, 16, -12, 0], [14, 0, -14, 0], [0, 0, 0, -40], [0, 16, -8, 16], [42, 0, 0, 0], [0, 8, -10, 8], [-24, 24, -24, 0], [-32, -16, 0, 16], [-28, 0, 28, 12], [-6, 0, 6, 36], [6, 0, -6, -42], [0, -2, -2, -2], [14, -36, 14, 0], [-10, -36, -10, 0], [0, 0, 30, 0], [0, -42, 0, 0], [0, 24, 18, 24], [32, 0, -32, -4], [-6, 34, 0, -34], [6, 24, 0, -24], [0, -16, -8, -16], [-36, 0, 36, 8], [2, 0, 0, 0], [0, 4, 14, 4], [0, -16, 2, -16], [28, -4, 0, 4], [-6, 22, -6, 0], [0, 32, 2, 32], [6, 0, -6, -2], [0, -2, -14, -2], [-24, 0, 24, -10], [26, 4, 26, 0], [10, 4, 10, 0], [0, 16, 24, 16], [-20, 10, -20, 0], [-12, 4, -12, 0], [0, -8, -10, -8], [0, -20, 28, -20], [-12, 0, 12, 32], [-20, -12, -20, 0], [0, -24, -22, -24], [12, -20, 0, 20], [16, 0, -16, 44], [30, 0, -30, -12], [18, 0, 0, 0], [0, 0, 0, -6], [34, 12, 34, 0], [-6, 36, -6, 0], [0, 16, 8, 16], [0, -24, 10, -24], [-16, -2, -16, 0], [0, 16, 14, 16], [6, 22, 0, -22], [4, 16, 4, 0], [22, 8, 0, -8], [32, 12, 32, 0], [0, -4, 10, -4], [0, 28, -10, 28], [-20, 32, -20, 0], [36, 20, 0, -20], [-14, -46, -14, 0], [-24, 0, 24, -12], [8, 28, 0, -28], [14, 16, 0, -16], [18, 0, -18, 30], [34, -20, 0, 20], [-18, 0, 18, 12], [32, 0, -32, -4], [-2, 32, 0, -32], [-22, 8, 0, -8], [0, 28, -18, 28], [-26, -2, -26, 0], [-8, 0, 8, -36], [0, 8, -14, 8], [6, 0, -6, 36], [-36, 16, 0, -16], [30, 0, -30, -22], [0, -30, -18, -30], [-16, 0, 16, -50], [-18, 24, 0, -24], [0, -24, -26, -24], [-54, 8, 0, -8], [4, -32, 4, 0], [-34, 0, 34, 24], [0, -16, 22, -16], [4, 0, -4, 0], [-30, -6, 0, 6], [0, -4, 40, -4], [8, -12, 8, 0], [0, 12, -42, 12], [-2, 50, -2, 0], [0, -24, 2, -24], [6, 0, -6, -12], [24, 28, 0, -28], [42, 0, -42, 6], [0, 14, 6, 14], [10, -4, 0, 4], [-6, 0, -6, 0], [-8, -38, -8, 0], [22, 8, 0, -8], [30, 0, -30, -24], [0, 0, -18, 0], [8, 0, -8, 52], [-42, 6, 0, -6], [-6, -24, 0, 24], [-12, 0, 12, -12], [32, -28, 32, 0], [8, -20, 8, 0], [30, -6, 30, 0], [12, 0, -12, 36], [0, 24, 38, 24], [4, 0, -4, 38], [-6, 12, -6, 0], [-14, 12, 0, -12], [0, 16, -16, 16], [-18, 4, 0, -4], [-4, -32, -4, 0], [40, 0, -40, -8], [2, -32, 0, 32], [8, 0, -8, -52], [12, 0, -12, -8], [-22, -16, 0, 16], [-20, -16, -20, 0], [-18, -34, -18, 0], [16, 0, -16, -12], [0, -48, 6, -48], [2, 48, 0, -48], [0, 38, -6, 38], [42, 0, 42, 0], [0, 16, -38, 16], [-2, 0, 2, 60], [10, -16, 0, 16], [24, 0, -24, -8], [0, -20, -36, -20], [8, 0, -8, 32], [-20, 8, -20, 0], [-10, -24, 0, 24], [0, -12, -42, -12], [-32, -16, 0, 16], [-16, 0, 16, 28], [14, 0, -14, -52], [0, -2, -6, -2], [-36, 0, 36, -6], [0, 4, 4, 4], [0, 24, -18, 24], [-16, 0, 16, 36], [-46, 2, 0, -2], [24, 0, -24, 16], [-12, 0, 12, -56], [8, -8, 8, 0], [26, -16, 0, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2400_w_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2400_2_w_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_2400_2_w_b(:prec:=4)
    chi := MakeCharacter_2400_w();
    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_2400_2_w_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_2400_2_w_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2400_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 32, 32, 8, 1]>,<17,R![64, 64, 32, -8, 1]>],Snew);
    return Vf;
end function;