
// Make newform 1456.2.cc.b in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1456_cc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1456_cc_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_1456_2_cc_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_1456_2_cc_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, -1, 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_1456_cc();"
function MakeCharacter_1456_cc()
    N := 1456;
    order := 6;
    char_gens := [911, 1093, 1249, 561];
    v := [6, 6, 6, 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_1456_cc_Hecke();"
function MakeCharacter_1456_cc_Hecke( : Kf := false)
    N := 1456;
    order := 6;
    char_gens := [911, 1093, 1249, 561];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 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, -1, 1], [0, 0, 0, -1], [0, 1, 0, 0], [-2, 1, 1, -1], [0, 3, 0, 1], [-4, -1, 4, 2], [2, -2, 2, 0], [0, 1, -3, 1], [0, 0, 3, 0], [-1, 0, 2, -7], [6, 0, -3, 0], [-4, 1, 2, -1], [7, 3, -7, -6], [-4, 0, 8, -4], [5, -4, 0, 2], [1, 9, 1, 0], [10, 1, -10, -2], [-10, -3, 5, 3], [8, 0, 8, 0], [-6, 0, 12, 1], [9, -6, 0, 3], [3, 0, -6, 9], [-4, 6, 2, -6], [-2, -2, -2, 0], [0, 3, -4, 3], [2, -12, 0, 6], [0, 4, 4, 4], [0, 0, 0, -10], [-1, -6, 1, 12], [0, -8, 4, -8], [12, -8, 0, 4], [-7, 6, -7, 0], [-9, 5, 9, -10], [-3, -12, -3, 0], [6, 0, -12, -2], [12, 2, 0, -1], [1, -1, 1, 0], [-2, 7, 1, -7], [-14, -4, 14, 8], [0, -6, -6, -6], [-2, 6, 0, -3], [-5, 7, 5, -14], [-6, -18, 3, 18], [-8, 14, 4, -14], [2, 2, -2, -4], [0, 1, -23, 1], [-12, -2, 6, 2], [1, -3, 1, 0], [6, 0, -12, -14], [-6, 12, 0, -6], [-13, 0, 26, 7], [-6, -5, -6, 0], [-16, -4, 16, 8], [0, -5, 10, -5], [0, -9, -5, -9], [8, 12, -8, -24], [-14, 5, 7, -5], [-1, 6, 1, -12], [5, 0, -10, -8], [0, 9, -5, 9], [6, 7, 6, 0], [-13, 0, 26, 1], [-5, -6, 0, 3], [-32, 0, 0, 0], [-11, 0, 22, -12], [5, -13, 5, 0], [11, -20, 0, 10], [3, 15, -3, -30], [0, -2, 0, 2], [-36, -5, 18, 5], [6, 0, -12, -6], [0, 19, 1, 19], [29, 2, -29, -4], [-6, -17, 3, 17], [-9, -17, -9, 0], [-7, -20, 0, 10], [-2, -26, -2, 0], [-22, -14, 11, 14], [10, 9, 10, 0], [0, 7, -7, 7], [-7, 0, 14, 2], [4, -30, -2, 30], [20, 1, -20, -2], [0, -1, 21, -1], [-18, -20, 0, 10], [0, 18, 0, 0], [2, -16, -1, 16], [10, -9, 10, 0], [11, 0, -22, 23], [-23, -2, 0, 1], [-6, -27, 3, 27], [-10, -14, -10, 0], [0, 16, 0, 16], [-1, 0, 2, -13], [19, 1, -19, -2], [-12, -17, 6, 17], [8, 42, 0, -21], [0, 2, 18, 2], [1, 0, -2, 10], [1, 34, 0, -17], [14, 14, -7, -14], [-1, 17, 1, -34], [0, -10, 6, -10], [-2, -4, 0, 2], [-18, 0, 36, 3], [24, 8, -12, -8], [-8, 0, 16, -17], [-24, -8, 0, 4], [0, 11, 4, 11], [4, -8, -4, 16], [30, -10, -15, 10], [-3, -28, -3, 0], [11, 0, -22, -15], [2, -14, 2, 0], [41, 2, -41, -4], [-16, -8, -16, 0], [0, 8, 0, 8], [0, -6, -2, -6], [-12, 12, 12, -24], [8, -5, -4, 5], [0, 12, 1, 12], [14, 16, 0, -8], [2, -10, 2, 0], [-16, 8, 8, -8], [-18, 4, 0, -2], [-13, 16, -13, 0], [1, 3, -1, -6], [40, 16, 0, -8], [2, 0, -4, 1], [4, -10, -2, 10], [-8, 12, 4, -12], [0, -17, 23, -17], [0, 6, 34, 6], [14, -6, 14, 0], [12, -10, -6, 10], [14, 22, 14, 0], [15, 19, 15, 0], [-2, -4, 2, 8], [0, 14, -17, 14], [7, 0, -14, 33], [-8, -24, 4, 24], [22, -14, -22, 28], [-17, 0, 34, -9], [0, 9, -22, 9], [-16, -4, -16, 0], [0, 0, 0, -19], [20, -22, 0, 11], [-7, 46, 0, -23], [-17, 0, 34, 15], [-7, 14, -7, 0], [0, -21, 2, -21], [-1, -26, 0, 13], [0, 4, -4, 4], [0, 11, 3, 11], [2, 44, 0, -22], [-20, 12, 20, -24], [-22, -17, -22, 0], [40, 18, 0, -9], [10, 0, -20, 6], [-14, -41, 7, 41], [6, 18, -6, -36], [13, 0, -26, 9], [-10, -6, 10, 12], [14, 12, -7, -12], [22, 0, -44, -6], [0, -21, -9, -21], [-4, 15, 4, -30], [0, 0, 0, 42], [-8, -34, 0, 17], [7, 0, -14, 3], [-10, 15, -10, 0], [48, 8, -48, -16], [7, -38, 7, 0], [15, -14, 0, 7], [-4, -13, 4, 26], [28, -14, -14, 14], [-13, 0, 26, -2], [0, 10, 14, 10], [0, 12, 22, 12], [15, 0, -30, -1], [-42, 12, 0, -6], [-49, -4, 0, 2], [10, 0, -20, 18], [40, 0, -20, 0], [-26, -21, 26, 42], [-22, -4, 0, 2], [8, 0, -16, -8], [-22, -12, 11, 12], [-16, -24, -16, 0], [4, 25, -4, -50], [9, -1, 9, 0], [8, 32, 0, -16], [-16, 2, 8, -2], [11, -1, -11, 2], [0, 0, -24, 0], [3, 0, -6, 16], [11, -12, -11, 24], [10, 0, -20, -31], [-4, -48, 0, 24], [3, 28, 3, 0], [-16, 12, 16, -24], [24, -7, -12, 7], [-2, 18, 0, -9], [-16, -8, 8, 8], [0, 9, 12, 9], [-21, 0, 42, 17], [-15, -27, 15, 54], [6, 38, -3, -38], [-10, -10, 10, 20], [0, -16, 15, -16], [6, -48, 0, 24], [0, -10, 14, -10], [34, -6, 34, 0], [-16, -8, -16, 0], [-18, 0, 36, -21], [1, 18, 0, -9], [54, -5, -54, 10], [40, 16, -20, -16], [0, 0, 0, 20], [0, -6, 37, -6], [14, 0, -28, -22], [25, 0, -50, 0], [18, 14, 18, 0], [0, -30, -6, -30], [-1, -52, 0, 26], [0, 7, -26, 7], [-12, -20, 12, 40], [14, -14, -14, 28], [6, 0, -12, -10], [0, -6, -5, -6], [0, 0, 16, 0], [-50, 17, 25, -17], [-27, 28, 0, -14], [-15, -42, 0, 21], [0, 0, 0, -48], [-16, 10, -16, 0], [-10, 12, 5, -12], [24, 20, -24, -40], [0, 9, 19, 9], [-18, 45, 9, -45], [0, -17, 13, -17], [16, 8, -16, -16], [-6, -10, 3, 10], [12, 1, 12, 0], [-7, -74, 0, 37], [-3, 69, -3, 0], [16, 4, -8, -4], [-13, 45, -13, 0], [0, -8, 0, -8], [8, 10, -4, -10], [28, -22, -14, 22], [-14, 0, 28, 42], [0, -35, 0, -35], [-54, 8, 0, -4], [-18, 10, -18, 0], [30, -16, -30, 32], [-12, 58, 6, -58], [-62, -16, 0, 8], [38, -3, 38, 0], [-50, -4, 0, 2], [0, -6, -34, -6], [6, 0, -12, -19], [0, 5, -24, 5], [12, 17, 12, 0], [7, -11, -7, 22], [-4, -34, -4, 0], [-18, 0, 36, 15], [22, -6, 22, 0], [1, 14, -1, -28], [-2, 18, 0, -9], [-10, 0, 20, 46], [-20, -43, 10, 43], [6, -6, -6, 12], [-38, -12, 38, 24], [-34, 23, 17, -23], [23, -27, 23, 0], [25, 0, -50, 10], [25, 10, 25, 0], [-14, 10, 14, -20], [0, 8, 24, 8], [-12, 26, 6, -26], [-51, -2, 0, 1], [-44, -15, 22, 15], [16, 0, -32, 4], [72, 0, 0, 0], [3, 24, 0, -12], [-7, 0, 14, -4], [-6, -18, -6, 0], [26, -1, -26, 2], [0, 0, 10, 0], [54, 10, -54, -20], [18, 42, -9, -42], [31, -33, 31, 0], [14, 42, -14, -84], [48, 6, 0, -3], [-30, -20, 0, 10], [0, 15, -8, 15], [0, 0, 48, 0], [22, 39, -11, -39], [-20, -15, 20, 30], [-7, 0, 14, -52], [0, 6, 78, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1456_cc_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1456_2_cc_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_1456_2_cc_b(:prec:=4)
    chi := MakeCharacter_1456_cc();
    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_1456_2_cc_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_1456_2_cc_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1456_cc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -4, 6, 2, 1]>],Snew);
    return Vf;
end function;