LMFDB - Newform orbit 483.2.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,2,Mod(64,483)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 12]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("483.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 483 = 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	483.q (of order $11$, degree $10$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.85677441763$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{22}^{6} + \cdots - \zeta_{22}^{2}) q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots + 1) q^{4} - \zeta_{22}^{2} q^{5} + (\zeta_{22}^{5} - \zeta_{22}^{4} + \cdots + \zeta_{22}) q^{6} + \cdots + (\zeta_{22}^{8} + \cdots + \zeta_{22}^{2}) q^{99} +O(q^{100}) $ q + (-z^6 + z^5 + z^3 - z^2) * q^2 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^3 + (-z^9 + z^8 - z^7 - z^5 + z^3 - z^2 + 1) * q^4 - z^2 * q^5 + (z^5 - z^4 - z^2 + z) * q^6 - z^5 * q^7 + (-z^9 + 2z^8 - z^7 + z^6 - 2z^5 + z^4 + z) * q^8 - z^9 * q^9 + (z^8 - z^7 - z^5 + z^4) * q^10 + (2z^9 - z^8 - z^6 + 2z^5 + z^3 - z^2 + z - 1) * q^11 + (z^9 + z^5 + z^3 - 2z^2 + 2z - 1) * q^12 + (-2z^9 + 2z^8 - 2z^7 - 2z^5 + 2z^4 - 2z^3 - 3z + 3) q^13 + (-z^9 + z^6 - z^5 + z^4 - z^3 + z^2 - z) * q^14 + z * q^15 + (-z^9 - z^8 - 2z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 - z^2 - z) q^16 + (-2z^6 - 2z^5 + z^4 + z^2 - 2z - 2) q^17 + (-z^4 + z^3 + z - 1) * q^18 + (z^9 - z^8 + z^7 - z^4 + z) * q^19 + (z^8 + z^6 - 2z^5 + 2z^4 - z^3 - z) * q^20 + z^4 * q^21 + (z^9 + 2z^8 - z^7 + z^4 - 2z^3 - z^2 + 3) * q^22 + (-2z^9 - z^7 + 2z^6 + z^5 - z^4 - 2z^3 + 3z^2 - z + 3) * q^23 + (z^8 - 2z^7 + z^6 - z^5 + 2z^4 - z^3 - 1) * q^24 - 4z^4 q^25 + (-2z^9 + 4z^8 - z^7 - 2z^6 - 3z^5 + 3z^4 + 2z^3 + z^2 - 4z + 2) q^26 + z^8 * q^27 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^28 + (2z^9 - 2z^8 - 4z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 4) * q^29 + (-z^7 + z^6 + z^4 - z^3) * q^30 + (3z^9 + z^8 + z^7 - z^6 - z^5 - 3z^4 - 2z^2 + 4z - 2) * q^31 + (z^9 - z^7 - 2z^6 + 2z^4 - 2z^3 + 2z + 1) * q^32 + (-z^9 - z^8 + z^6 - z^4 - z^3) * q^33 + (-4z^9 + 2z^8 + 3z^7 + 2z^6 - 4z^5 - 3z^3 + 5z^2 - 5z + 3) * q^34 + z^7 * q^35 + (-z^9 - z^7 + z^6 - z^5 - z^3 + z - 1) * q^36 + (-4z^5 - 3z^4 + 3z^3 + 4z^2) * q^37 + (z^9 - 3z^8 + z^7 - z^6 + 3z^5 - z^4 - z^2 + z - 1) * q^38 + (3z^9 - z^8 + z^7 - z^6 + 3z^5 - z^4 + z^3 - z^2 + 3z) q^39 + (-z^9 + z^8 + z^6 - 2z^5 + 2z^4 - 3z^3 + 2z^2 - 2z + 1) q^40 + (3z^9 - 4z^8 + 4z^7 - 3z^6 - z^4 + 2z^3 + 2z^2 + 2z - 1) q^41 + (z^8 - z^5 + z^4 - z^3 + z^2 - z + 1) * q^42 + (-z^9 - 2z^8 - 3z^7 - 3z^5 + 3z^4 + 3z^2 + 2z + 1) * q^43 + (3z^8 - 2z^7 + 4z^4 - 2z + 3) * q^44 - q^45 + (5z^9 - 7z^8 + 3z^7 - 6z^6 + 9z^5 - 7z^4 + 7z^3 - 5z^2 + 6z - 4) q^46 + (-2z^9 + 4z^7 - 4z^4 + 2z^2 - 1) * q^47 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^48 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^49 + (-4z^8 + 4z^5 - 4z^4 + 4z^3 - 4z^2 + 4z - 4) * q^50 + (-2z^9 + 2z^8 - 2z^7 + 2z^6 + 4z^4 - 3z^3 + 2z^2 - 3z + 4) * q^51 + (-5z^9 + 5z^8 - 6z^5 + z^3 - 6z) * q^52 + (2z^9 + z^8 + 3z^6 + 3z^5 + 3z^4 + z^2 + 2z) q^53 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^54 + (z^9 - z^7 - z^6 + z + 1) * q^55 + (z^9 - 2z^8 + 2z^7 - 3z^6 + 2z^5 - 2z^4 + z^3 + z - 1) q^56 + (-z^8 + z^7 - z^6 + z^3 - 1) * q^57 + (-2z^9 + 4z^8 - 2z^7 + 4z^6 - 2z^5 - 4z^3 + 4) * q^58 + (z^9 + 2z^8 - 2z^7 - 7z^6 - 2z^5 + 2z^4 + z^3 - z + 1) q^59 + (-z^7 - z^5 + 2z^4 - 2z^3 + z^2 + 1) * q^60 + (-2z^9 + z^7 - z^6 + 2z^4 + z^2 - 2z + 1) q^61 + (-3z^9 - 6z^7 + 7z^6 - 2z^5 + 7z^4 - 6z^3 - 3z) q^62 - z^3 * q^63 + (4z^9 + z^8 - z^7 - 4z^6 + z^3 - 5z^2 + z) q^64 + (2z^8 + 2z^4 + z^3 - z^2 - 2z) q^65 + (3z^9 - 4z^8 + z^7 - 2z^6 + 3z^5 - 3z^4 + 2z^3 - z^2 + 4z - 3) q^66 + (6z^7 + 4z^5 - 9z^4 + 4z^3 + 6z) q^67 + (z^9 - 5z^8 + 9z^7 - 6z^6 + 6z^5 - 9z^4 + 5z^3 - z^2 - 7) * q^68 + (3z^9 - z^8 + 3z^7 - 2z^6 + z^5 - 4z^4 + 4z^3 - z^2 - 2) q^69 + (-z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - 1) * q^70 + (6z^8 - 3z^7 + 5z^6 - 6z^5 + 4z^4 - 6z^3 + 5z^2 - 3z + 6) * q^71 + (-z^9 + z^8 - 2z^7 + 3z^6 - 2z^5 + 2z^4 - 3z^3 + 2z^2 - z + 1) * q^72 + (2z^9 - 2z^8 + 2z^7 + 5z^5 - 2z^4 + z^3 - z^2 + 2z - 5) * q^73 + (-7z^9 - 4z^8 + 4z^7 + 7z^6 - 3z^4 - z^3 + z^2 - z - 3) q^74 + 4z^3 q^75 + (3z^9 - 3z^8 + 2z^7 - 3z^6 + 4z^5 - 3z^4 + 2z^3 - 3z^2 + 3z) q^76 + (-2z^9 + z^8 - z^7 + z^6 - z^5 + 2z^4 + z^2 - 2z + 1) q^77 + (2z^9 - 2z^7 - z^6 + 4z^5 + z^4 - z^3 - 4z^2 + z + 2) * q^78 + (-3z^9 + 4z^8 - 5z^7 + 12z^6 - 5z^5 + 4z^4 - 3z^3 - 7z + 7) * q^79 + (3z^9 - 3z^8 + 3z^7 - 3z^6 + 3z^5 + 2z^3 - z^2 + z - 2) * q^80 - z^7 * q^81 + (-z^9 - 6z^8 + 8z^7 - 3z^6 + 8z^5 - 6z^4 - z^3 - z + 1) q^82 + (-2z^9 - 7z^7 + 7z^6 - 4z^5 + 7z^4 - 7z^3 + 4z^2 - 7z + 7) * q^83 + (-z^9 + z^7 - z^6 + z^4 + z^2 - z + 1) * q^84 + (2z^8 + 2z^7 - z^6 - z^4 + 2z^3 + 2z^2) * q^85 + (-z^9 + z^8 + 5z^6 - 3z^5 + 5z^4 - 9z^3 + 5z^2 - 3z + 5) * q^86 + (-4z^9 + 2z^8 - 2z^7 + 4z^6 + 2z^4 - 2z^3 + 2z^2 - 2z + 2) * q^87 + (-z^9 + z^8 - z^7 + 3z^4 + z^3 - z^2 - 3z) * q^88 + (8z^9 - 6z^8 + 4z^7 - 2z^6 + 5z^5 - 5z^4 + 2z^3 - 4z^2 + 6z - 8) q^89 + (z^6 - z^5 - z^3 + z^2) * q^90 + (2z^7 + z^6 - z^5 - 2z^4 - 2) * q^91 + (-z^9 + z^8 - 7z^7 + 4z^5 - 6z^4 + 2z^3 - 4z^2 + 4z + 3) * q^92 + (-2z^9 - z^8 - 3z^7 + z^6 - z^5 + 3z^4 + z^3 + 2z^2 - 2) * q^93 + (-10z^8 + 6z^7 - 3z^6 + 9z^5 - 12z^4 + 9z^3 - 3z^2 + 6z - 10) * q^94 + (-z^8 + z^7 + z^5 - z^4 - z^2 + z) * q^95 + (z^9 - 2z^8 + z^7 + 3z^5 - z^4 - z^3 + z^2 + z - 3) * q^96 + (-7z^9 + 2z^8 - 2z^7 + 7z^6 + 4z^4 - 5z^3 + 9z^2 - 5z + 4) * q^97 + (z^5 - z^4 - z^2 + z) * q^98 + (z^8 + z^7 - z^5 + z^3 + z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} - q^{3} + 8 q^{4} + q^{5} + 4 q^{6} - q^{7} - 7 q^{8} - q^{9} - 4 q^{10} - q^{11} - 3 q^{12} + 15 q^{13} - 7 q^{14} + q^{15} - 10 q^{16} - 24 q^{17} - 7 q^{18} + 5 q^{19} - 8 q^{20}+ \cdots - q^{99}+O(q^{100}) $ 10 * q + 4 * q^2 - q^3 + 8 * q^4 + q^5 + 4 * q^6 - q^7 - 7 * q^8 - q^9 - 4 * q^10 - q^11 - 3 * q^12 + 15 * q^13 - 7 * q^14 + q^15 - 10 * q^16 - 24 * q^17 - 7 * q^18 + 5 * q^19 - 8 * q^20 - q^21 + 26 * q^22 + 21 * q^23 - 18 * q^24 + 4 * q^25 + 6 * q^26 - q^27 - 3 * q^28 - 22 * q^29 - 4 * q^30 - 8 * q^31 + 10 * q^32 - q^33 + 8 * q^34 + q^35 - 14 * q^36 - 2 * q^37 + 2 * q^38 + 15 * q^39 - 4 * q^40 + 7 * q^41 + 4 * q^42 + q^43 + 19 * q^44 - 10 * q^45 + 15 * q^46 - 6 * q^47 + q^48 - q^49 - 16 * q^50 + 20 * q^51 - 21 * q^52 - q^53 - 7 * q^54 + 12 * q^55 + 4 * q^56 - 6 * q^57 + 22 * q^58 + 10 * q^59 + 3 * q^60 + 5 * q^61 - 34 * q^62 - q^63 + 13 * q^64 - 4 * q^65 - 7 * q^66 + 29 * q^67 - 28 * q^68 - q^69 - 4 * q^70 + 22 * q^71 - 7 * q^72 - 33 * q^73 - 36 * q^74 + 4 * q^75 + 26 * q^76 - q^77 + 28 * q^78 + 27 * q^79 - q^80 - q^81 + 38 * q^82 + 25 * q^83 + 8 * q^84 + 2 * q^85 + 18 * q^86 - 7 * q^88 - 38 * q^89 - 4 * q^90 - 18 * q^91 + 41 * q^92 - 30 * q^93 - 42 * q^94 + 6 * q^95 - 23 * q^96 - q^97 + 4 * q^98 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$.

$n$	$323$	$346$	$442$
$\chi(n)$	$1$	$1$	$\zeta_{22}^{4}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

64.1

0.654861	+	0.755750i
−0.841254	−	0.540641i
0.142315	+	0.989821i
−0.415415	+	0.909632i
0.959493	−	0.281733i
0.959493	+	0.281733i
0.142315	−	0.989821i
−0.841254	+	0.540641i
0.654861	−	0.755750i
−0.415415	−	0.909632i

−1.52977

−

0.449181i

−0.654861

0.755750i

0.455922

0.293003i

0.142315

−

0.989821i

1.34125

−

0.861971i

0.415415

0.909632i

1.52231

1.75684i

−0.142315

−

0.989821i

−0.662317

1.45027i

85.1

1.64589

−

1.89945i

0.841254

−

0.540641i

−0.614354

−

4.27292i

−0.415415

−

0.909632i

0.357685

−

2.48775i

−0.959493

0.281733i

−4.89867

−

3.14818i

0.415415

−

0.909632i

−2.41153

−

0.708089i

127.1

1.85380

−

1.19136i

−0.142315

0.989821i

1.18639

−

2.59784i

0.959493

−

0.281733i

0.915415

2.00448i

−0.654861

−

0.755750i

−0.268423

−

1.86692i

−0.959493

−

0.281733i

1.44306

−

1.66538i

169.1

−0.0681534

0.474017i

0.415415

0.909632i

1.69894

0.498853i

0.654861

0.755750i

−0.459493

0.134919i

0.841254

0.540641i

−0.750131

1.64256i

−0.654861

0.755750i

−0.402869

0.258908i

190.1

0.0982369

−

0.215109i

−0.959493

−

0.281733i

1.27310

1.46924i

−0.841254

0.540641i

−0.154861

0.178719i

−0.142315

0.989821i

0.894911

−

0.262769i

0.841254

0.540641i

0.0336545

0.234072i

211.1

0.0982369

0.215109i

−0.959493

0.281733i

1.27310

−

1.46924i

−0.841254

−

0.540641i

−0.154861

−

0.178719i

−0.142315

−

0.989821i

0.894911

0.262769i

0.841254

−

0.540641i

0.0336545

−

0.234072i

232.1

1.85380

1.19136i

−0.142315

−

0.989821i

1.18639

2.59784i

0.959493

0.281733i

0.915415

−

2.00448i

−0.654861

0.755750i

−0.268423

1.86692i

−0.959493

0.281733i

1.44306

1.66538i

358.1

1.64589

1.89945i

0.841254

0.540641i

−0.614354

4.27292i

−0.415415

0.909632i

0.357685

2.48775i

−0.959493

−

0.281733i

−4.89867

3.14818i

0.415415

0.909632i

−2.41153

0.708089i

400.1

−1.52977

0.449181i

−0.654861

−

0.755750i

0.455922

−

0.293003i

0.142315

0.989821i

1.34125

0.861971i

0.415415

−

0.909632i

1.52231

−

1.75684i

−0.142315

0.989821i

−0.662317

−

1.45027i

463.1

−0.0681534

−

0.474017i

0.415415

−

0.909632i

1.69894

−

0.498853i

0.654861

−

0.755750i

−0.459493

−

0.134919i

0.841254

−

0.540641i

−0.750131

−

1.64256i

−0.654861

−

0.755750i

−0.402869

−

0.258908i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	483.2.q.a	✓	10
23.c	even	11	1	inner	483.2.q.a	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
483.2.q.a	✓	10	1.a	even	1	1	trivial
483.2.q.a	✓	10	23.c	even	11	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 4T_{2}^{9} + 5T_{2}^{8} + 13T_{2}^{7} - 30T_{2}^{6} - T_{2}^{5} + 70T_{2}^{4} - 5T_{2}^{3} + 20T_{2}^{2} - 3T_{2} + 1 $ T2^10 - 4*T2^9 + 5*T2^8 + 13*T2^7 - 30*T2^6 - T2^5 + 70*T2^4 - 5*T2^3 + 20*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(483, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - 4 T^{9} + \cdots + 1 $ T^10 - 4T^9 + 5T^8 + 13T^7 - 30T^6 - T^5 + 70T^4 - 5T^3 + 20T^2 - 3T + 1
$3$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 - 10T^8 + T^7 + 111T^6 + 298T^5 + 496T^4 + 23T^3 + 67T^2 - 10*T + 1
$13$	$ T^{10} - 15 T^{9} + \cdots + 529 $ T^10 - 15T^9 + 137T^8 - 735T^7 + 3094T^6 - 8790T^5 + 18176T^4 - 23908T^3 + 16553T^2 - 4623*T + 529
$17$	$ T^{10} + 24 T^{9} + \cdots + 978121 $ T^10 + 24T^9 + 312T^8 + 2692T^7 + 16868T^6 + 79166T^5 + 283215T^4 + 758259T^3 + 1455237T^2 + 1756464*T + 978121
$19$	$ T^{10} - 5 T^{9} + \cdots + 1 $ T^10 - 5T^9 + 14T^8 - 37T^7 + 97T^6 - 177T^5 + 225T^4 - 212T^3 + 103T^2 + 13*T + 1
$23$	$ T^{10} - 21 T^{9} + \cdots + 6436343 $ T^10 - 21T^9 + 210T^8 - 1528T^7 + 9758T^6 - 52491T^5 + 224434T^4 - 808312T^3 + 2555070T^2 - 5876661*T + 6436343
$29$	$ T^{10} + 22 T^{9} + \cdots + 123904 $ T^10 + 22T^9 + 220T^8 + 1232T^7 + 4400T^6 + 10912T^5 + 23232T^4 + 46464T^3 + 61952T^2 + 61952*T + 123904
$31$	$ T^{10} + 8 T^{9} + \cdots + 2105401 $ T^10 + 8T^9 + 20T^8 + 457T^7 + 7253T^6 + 46584T^5 + 174738T^4 + 537165T^3 + 1352004T^2 + 1857280*T + 2105401
$37$	$ T^{10} + 2 T^{9} + \cdots + 2105401 $ T^10 + 2T^9 + 4T^8 + 965T^7 - 4450T^6 - 27556T^5 + 707617T^4 - 3935221T^3 + 9592586T^2 - 4350098*T + 2105401
$41$	$ T^{10} - 7 T^{9} + \cdots + 14645929 $ T^10 - 7T^9 + 115T^8 - 178T^7 + 2236T^6 + 10264T^5 - 86049T^4 + 297984T^3 + 1361490T^2 - 8052008*T + 14645929
$43$	$ T^{10} - T^{9} + \cdots + 6848689 $ T^10 - T^9 - 87T^8 - 551T^7 + 3928T^6 + 39104T^5 + 156190T^4 + 170114T^3 + 1989362T^2 + 607144T + 6848689
$47$	$ (T^{5} + 3 T^{4} + \cdots + 991)^{2} $ (T^5 + 3T^4 - 102T^3 + 62T^2 + 1189T + 991)^2
$53$	$ T^{10} + T^{9} + \cdots + 737881 $ T^10 + T^9 + 122T^8 + 749T^7 + 837T^6 + 20428T^5 + 196186T^4 + 656195T^3 + 1201685T^2 + 1257576T + 737881
$59$	$ T^{10} + \cdots + 229734649 $ T^10 - 10T^9 + 177T^8 - 857T^7 + 25609T^6 - 38312T^5 + 447030T^4 + 200190T^3 - 8037160T^2 - 48532714*T + 229734649
$61$	$ T^{10} - 5 T^{9} + \cdots + 212521 $ T^10 - 5T^9 + 25T^8 - 136T^7 + 581T^6 - 463T^5 + 7023T^4 - 18527T^3 + 10608T^2 - 87129*T + 212521
$67$	$ T^{10} + \cdots + 174266401 $ T^10 - 29T^9 + 247T^8 + 1329T^7 - 19005T^6 - 468159T^5 + 9452403T^4 - 63133747T^3 + 286106555T^2 - 101027253*T + 174266401
$71$	$ T^{10} - 22 T^{9} + \cdots + 33860761 $ T^10 - 22T^9 + 154T^8 - 979T^7 + 17424T^6 - 172689T^5 + 1171764T^4 - 5529821T^3 + 17986529T^2 - 27971933*T + 33860761
$73$	$ T^{10} + 33 T^{9} + \cdots + 543169 $ T^10 + 33T^9 + 561T^8 + 6094T^7 + 45474T^6 + 235928T^5 + 826551T^4 + 1807014T^3 + 2072004T^2 + 826914*T + 543169
$79$	$ T^{10} + \cdots + 867832681 $ T^10 - 27T^9 + 333T^8 + 656T^7 - 43056T^6 + 480424T^5 + 3881146T^4 - 16137729T^3 + 101363541T^2 - 181732571*T + 867832681
$83$	$ T^{10} + \cdots + 3695059369 $ T^10 - 25T^9 + 460T^8 - 7573T^7 + 107397T^6 - 1219494T^5 + 11362200T^4 - 89553803T^3 + 543086813T^2 - 2067730592*T + 3695059369
$89$	$ T^{10} + 38 T^{9} + \cdots + 14645929 $ T^10 + 38T^9 + 806T^8 + 12016T^7 + 122494T^6 + 810932T^5 + 3396376T^4 + 8137803T^3 + 8596251T^2 - 34443*T + 14645929
$97$	$ T^{10} + \cdots + 5993391889 $ T^10 + T^9 + 34T^8 + 661T^7 + 4093T^6 + 48016T^5 + 1204270T^4 + 13148719T^3 + 62311657T^2 + 421458148T + 5993391889
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.q (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{22}^{6} + \cdots - \zeta_{22}^{2}) q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \cdots + 1) q^{4} - \zeta_{22}^{2} q^{5} + (\zeta_{22}^{5} - \zeta_{22}^{4} + \cdots + \zeta_{22}) q^{6} + \cdots + (\zeta_{22}^{8} + \cdots + \zeta_{22}^{2}) q^{99} +O(q^{100}) \) q + (-z^6 + z^5 + z^3 - z^2) * q^2 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^3 + (-z^9 + z^8 - z^7 - z^5 + z^3 - z^2 + 1) * q^4 - z^2 * q^5 + (z^5 - z^4 - z^2 + z) * q^6 - z^5 * q^7 + (-z^9 + 2z^8 - z^7 + z^6 - 2z^5 + z^4 + z) * q^8 - z^9 * q^9 + (z^8 - z^7 - z^5 + z^4) * q^10 + (2z^9 - z^8 - z^6 + 2z^5 + z^3 - z^2 + z - 1) * q^11 + (z^9 + z^5 + z^3 - 2z^2 + 2z - 1) * q^12 + (-2z^9 + 2z^8 - 2z^7 - 2z^5 + 2z^4 - 2z^3 - 3z + 3) q^13 + (-z^9 + z^6 - z^5 + z^4 - z^3 + z^2 - z) * q^14 + z * q^15 + (-z^9 - z^8 - 2z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 - z^2 - z) q^16 + (-2z^6 - 2z^5 + z^4 + z^2 - 2z - 2) q^17 + (-z^4 + z^3 + z - 1) * q^18 + (z^9 - z^8 + z^7 - z^4 + z) * q^19 + (z^8 + z^6 - 2z^5 + 2z^4 - z^3 - z) * q^20 + z^4 * q^21 + (z^9 + 2z^8 - z^7 + z^4 - 2z^3 - z^2 + 3) * q^22 + (-2z^9 - z^7 + 2z^6 + z^5 - z^4 - 2z^3 + 3z^2 - z + 3) * q^23 + (z^8 - 2z^7 + z^6 - z^5 + 2z^4 - z^3 - 1) * q^24 - 4z^4 q^25 + (-2z^9 + 4z^8 - z^7 - 2z^6 - 3z^5 + 3z^4 + 2z^3 + z^2 - 4z + 2) q^26 + z^8 * q^27 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - 1) * q^28 + (2z^9 - 2z^8 - 4z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 4) * q^29 + (-z^7 + z^6 + z^4 - z^3) * q^30 + (3z^9 + z^8 + z^7 - z^6 - z^5 - 3z^4 - 2z^2 + 4z - 2) * q^31 + (z^9 - z^7 - 2z^6 + 2z^4 - 2z^3 + 2z + 1) * q^32 + (-z^9 - z^8 + z^6 - z^4 - z^3) * q^33 + (-4z^9 + 2z^8 + 3z^7 + 2z^6 - 4z^5 - 3z^3 + 5z^2 - 5z + 3) * q^34 + z^7 * q^35 + (-z^9 - z^7 + z^6 - z^5 - z^3 + z - 1) * q^36 + (-4z^5 - 3z^4 + 3z^3 + 4z^2) * q^37 + (z^9 - 3z^8 + z^7 - z^6 + 3z^5 - z^4 - z^2 + z - 1) * q^38 + (3z^9 - z^8 + z^7 - z^6 + 3z^5 - z^4 + z^3 - z^2 + 3z) q^39 + (-z^9 + z^8 + z^6 - 2z^5 + 2z^4 - 3z^3 + 2z^2 - 2z + 1) q^40 + (3z^9 - 4z^8 + 4z^7 - 3z^6 - z^4 + 2z^3 + 2z^2 + 2z - 1) q^41 + (z^8 - z^5 + z^4 - z^3 + z^2 - z + 1) * q^42 + (-z^9 - 2z^8 - 3z^7 - 3z^5 + 3z^4 + 3z^2 + 2z + 1) * q^43 + (3z^8 - 2z^7 + 4z^4 - 2z + 3) * q^44 - q^45 + (5z^9 - 7z^8 + 3z^7 - 6z^6 + 9z^5 - 7z^4 + 7z^3 - 5z^2 + 6z - 4) q^46 + (-2z^9 + 4z^7 - 4z^4 + 2z^2 - 1) * q^47 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^48 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^49 + (-4z^8 + 4z^5 - 4z^4 + 4z^3 - 4z^2 + 4z - 4) * q^50 + (-2z^9 + 2z^8 - 2z^7 + 2z^6 + 4z^4 - 3z^3 + 2z^2 - 3z + 4) * q^51 + (-5z^9 + 5z^8 - 6z^5 + z^3 - 6z) * q^52 + (2z^9 + z^8 + 3z^6 + 3z^5 + 3z^4 + z^2 + 2z) q^53 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z) * q^54 + (z^9 - z^7 - z^6 + z + 1) * q^55 + (z^9 - 2z^8 + 2z^7 - 3z^6 + 2z^5 - 2z^4 + z^3 + z - 1) q^56 + (-z^8 + z^7 - z^6 + z^3 - 1) * q^57 + (-2z^9 + 4z^8 - 2z^7 + 4z^6 - 2z^5 - 4z^3 + 4) * q^58 + (z^9 + 2z^8 - 2z^7 - 7z^6 - 2z^5 + 2z^4 + z^3 - z + 1) q^59 + (-z^7 - z^5 + 2z^4 - 2z^3 + z^2 + 1) * q^60 + (-2z^9 + z^7 - z^6 + 2z^4 + z^2 - 2z + 1) q^61 + (-3z^9 - 6z^7 + 7z^6 - 2z^5 + 7z^4 - 6z^3 - 3z) q^62 - z^3 * q^63 + (4z^9 + z^8 - z^7 - 4z^6 + z^3 - 5z^2 + z) q^64 + (2z^8 + 2z^4 + z^3 - z^2 - 2z) q^65 + (3z^9 - 4z^8 + z^7 - 2z^6 + 3z^5 - 3z^4 + 2z^3 - z^2 + 4z - 3) q^66 + (6z^7 + 4z^5 - 9z^4 + 4z^3 + 6z) q^67 + (z^9 - 5z^8 + 9z^7 - 6z^6 + 6z^5 - 9z^4 + 5z^3 - z^2 - 7) * q^68 + (3z^9 - z^8 + 3z^7 - 2z^6 + z^5 - 4z^4 + 4z^3 - z^2 - 2) q^69 + (-z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - 1) * q^70 + (6z^8 - 3z^7 + 5z^6 - 6z^5 + 4z^4 - 6z^3 + 5z^2 - 3z + 6) * q^71 + (-z^9 + z^8 - 2z^7 + 3z^6 - 2z^5 + 2z^4 - 3z^3 + 2z^2 - z + 1) * q^72 + (2z^9 - 2z^8 + 2z^7 + 5z^5 - 2z^4 + z^3 - z^2 + 2z - 5) * q^73 + (-7z^9 - 4z^8 + 4z^7 + 7z^6 - 3z^4 - z^3 + z^2 - z - 3) q^74 + 4z^3 q^75 + (3z^9 - 3z^8 + 2z^7 - 3z^6 + 4z^5 - 3z^4 + 2z^3 - 3z^2 + 3z) q^76 + (-2z^9 + z^8 - z^7 + z^6 - z^5 + 2z^4 + z^2 - 2z + 1) q^77 + (2z^9 - 2z^7 - z^6 + 4z^5 + z^4 - z^3 - 4z^2 + z + 2) * q^78 + (-3z^9 + 4z^8 - 5z^7 + 12z^6 - 5z^5 + 4z^4 - 3z^3 - 7z + 7) * q^79 + (3z^9 - 3z^8 + 3z^7 - 3z^6 + 3z^5 + 2z^3 - z^2 + z - 2) * q^80 - z^7 * q^81 + (-z^9 - 6z^8 + 8z^7 - 3z^6 + 8z^5 - 6z^4 - z^3 - z + 1) q^82 + (-2z^9 - 7z^7 + 7z^6 - 4z^5 + 7z^4 - 7z^3 + 4z^2 - 7z + 7) * q^83 + (-z^9 + z^7 - z^6 + z^4 + z^2 - z + 1) * q^84 + (2z^8 + 2z^7 - z^6 - z^4 + 2z^3 + 2z^2) * q^85 + (-z^9 + z^8 + 5z^6 - 3z^5 + 5z^4 - 9z^3 + 5z^2 - 3z + 5) * q^86 + (-4z^9 + 2z^8 - 2z^7 + 4z^6 + 2z^4 - 2z^3 + 2z^2 - 2z + 2) * q^87 + (-z^9 + z^8 - z^7 + 3z^4 + z^3 - z^2 - 3z) * q^88 + (8z^9 - 6z^8 + 4z^7 - 2z^6 + 5z^5 - 5z^4 + 2z^3 - 4z^2 + 6z - 8) q^89 + (z^6 - z^5 - z^3 + z^2) * q^90 + (2z^7 + z^6 - z^5 - 2z^4 - 2) * q^91 + (-z^9 + z^8 - 7z^7 + 4z^5 - 6z^4 + 2z^3 - 4z^2 + 4z + 3) * q^92 + (-2z^9 - z^8 - 3z^7 + z^6 - z^5 + 3z^4 + z^3 + 2z^2 - 2) * q^93 + (-10z^8 + 6z^7 - 3z^6 + 9z^5 - 12z^4 + 9z^3 - 3z^2 + 6z - 10) * q^94 + (-z^8 + z^7 + z^5 - z^4 - z^2 + z) * q^95 + (z^9 - 2z^8 + z^7 + 3z^5 - z^4 - z^3 + z^2 + z - 3) * q^96 + (-7z^9 + 2z^8 - 2z^7 + 7z^6 + 4z^4 - 5z^3 + 9z^2 - 5z + 4) * q^97 + (z^5 - z^4 - z^2 + z) * q^98 + (z^8 + z^7 - z^5 + z^3 + z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} - q^{3} + 8 q^{4} + q^{5} + 4 q^{6} - q^{7} - 7 q^{8} - q^{9} - 4 q^{10} - q^{11} - 3 q^{12} + 15 q^{13} - 7 q^{14} + q^{15} - 10 q^{16} - 24 q^{17} - 7 q^{18} + 5 q^{19} - 8 q^{20}+ \cdots - q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 - q^3 + 8 * q^4 + q^5 + 4 * q^6 - q^7 - 7 * q^8 - q^9 - 4 * q^10 - q^11 - 3 * q^12 + 15 * q^13 - 7 * q^14 + q^15 - 10 * q^16 - 24 * q^17 - 7 * q^18 + 5 * q^19 - 8 * q^20 - q^21 + 26 * q^22 + 21 * q^23 - 18 * q^24 + 4 * q^25 + 6 * q^26 - q^27 - 3 * q^28 - 22 * q^29 - 4 * q^30 - 8 * q^31 + 10 * q^32 - q^33 + 8 * q^34 + q^35 - 14 * q^36 - 2 * q^37 + 2 * q^38 + 15 * q^39 - 4 * q^40 + 7 * q^41 + 4 * q^42 + q^43 + 19 * q^44 - 10 * q^45 + 15 * q^46 - 6 * q^47 + q^48 - q^49 - 16 * q^50 + 20 * q^51 - 21 * q^52 - q^53 - 7 * q^54 + 12 * q^55 + 4 * q^56 - 6 * q^57 + 22 * q^58 + 10 * q^59 + 3 * q^60 + 5 * q^61 - 34 * q^62 - q^63 + 13 * q^64 - 4 * q^65 - 7 * q^66 + 29 * q^67 - 28 * q^68 - q^69 - 4 * q^70 + 22 * q^71 - 7 * q^72 - 33 * q^73 - 36 * q^74 + 4 * q^75 + 26 * q^76 - q^77 + 28 * q^78 + 27 * q^79 - q^80 - q^81 + 38 * q^82 + 25 * q^83 + 8 * q^84 + 2 * q^85 + 18 * q^86 - 7 * q^88 - 38 * q^89 - 4 * q^90 - 18 * q^91 + 41 * q^92 - 30 * q^93 - 42 * q^94 + 6 * q^95 - 23 * q^96 - q^97 + 4 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	483.2.q.a

Level	$483$
Weight	$2$
Character orbit	483.q
Analytic conductor	$3.857$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{22}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 64.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - 4 T^{9} + \cdots + 1 \) T^10 - 4T^9 + 5T^8 + 13T^7 - 30T^6 - T^5 + 70T^4 - 5T^3 + 20T^2 - 3T + 1
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 - 10T^8 + T^7 + 111T^6 + 298T^5 + 496T^4 + 23T^3 + 67T^2 - 10*T + 1
$13$	\( T^{10} - 15 T^{9} + \cdots + 529 \) T^10 - 15T^9 + 137T^8 - 735T^7 + 3094T^6 - 8790T^5 + 18176T^4 - 23908T^3 + 16553T^2 - 4623*T + 529
$17$	\( T^{10} + 24 T^{9} + \cdots + 978121 \) T^10 + 24T^9 + 312T^8 + 2692T^7 + 16868T^6 + 79166T^5 + 283215T^4 + 758259T^3 + 1455237T^2 + 1756464*T + 978121
$19$	\( T^{10} - 5 T^{9} + \cdots + 1 \) T^10 - 5T^9 + 14T^8 - 37T^7 + 97T^6 - 177T^5 + 225T^4 - 212T^3 + 103T^2 + 13*T + 1
$23$	\( T^{10} - 21 T^{9} + \cdots + 6436343 \) T^10 - 21T^9 + 210T^8 - 1528T^7 + 9758T^6 - 52491T^5 + 224434T^4 - 808312T^3 + 2555070T^2 - 5876661*T + 6436343
$29$	\( T^{10} + 22 T^{9} + \cdots + 123904 \) T^10 + 22T^9 + 220T^8 + 1232T^7 + 4400T^6 + 10912T^5 + 23232T^4 + 46464T^3 + 61952T^2 + 61952*T + 123904
$31$	\( T^{10} + 8 T^{9} + \cdots + 2105401 \) T^10 + 8T^9 + 20T^8 + 457T^7 + 7253T^6 + 46584T^5 + 174738T^4 + 537165T^3 + 1352004T^2 + 1857280*T + 2105401
$37$	\( T^{10} + 2 T^{9} + \cdots + 2105401 \) T^10 + 2T^9 + 4T^8 + 965T^7 - 4450T^6 - 27556T^5 + 707617T^4 - 3935221T^3 + 9592586T^2 - 4350098*T + 2105401
$41$	\( T^{10} - 7 T^{9} + \cdots + 14645929 \) T^10 - 7T^9 + 115T^8 - 178T^7 + 2236T^6 + 10264T^5 - 86049T^4 + 297984T^3 + 1361490T^2 - 8052008*T + 14645929
$43$	\( T^{10} - T^{9} + \cdots + 6848689 \) T^10 - T^9 - 87T^8 - 551T^7 + 3928T^6 + 39104T^5 + 156190T^4 + 170114T^3 + 1989362T^2 + 607144T + 6848689
$47$	\( (T^{5} + 3 T^{4} + \cdots + 991)^{2} \) (T^5 + 3T^4 - 102T^3 + 62T^2 + 1189T + 991)^2
$53$	\( T^{10} + T^{9} + \cdots + 737881 \) T^10 + T^9 + 122T^8 + 749T^7 + 837T^6 + 20428T^5 + 196186T^4 + 656195T^3 + 1201685T^2 + 1257576T + 737881
$59$	\( T^{10} + \cdots + 229734649 \) T^10 - 10T^9 + 177T^8 - 857T^7 + 25609T^6 - 38312T^5 + 447030T^4 + 200190T^3 - 8037160T^2 - 48532714*T + 229734649
$61$	\( T^{10} - 5 T^{9} + \cdots + 212521 \) T^10 - 5T^9 + 25T^8 - 136T^7 + 581T^6 - 463T^5 + 7023T^4 - 18527T^3 + 10608T^2 - 87129*T + 212521
$67$	\( T^{10} + \cdots + 174266401 \) T^10 - 29T^9 + 247T^8 + 1329T^7 - 19005T^6 - 468159T^5 + 9452403T^4 - 63133747T^3 + 286106555T^2 - 101027253*T + 174266401
$71$	\( T^{10} - 22 T^{9} + \cdots + 33860761 \) T^10 - 22T^9 + 154T^8 - 979T^7 + 17424T^6 - 172689T^5 + 1171764T^4 - 5529821T^3 + 17986529T^2 - 27971933*T + 33860761
$73$	\( T^{10} + 33 T^{9} + \cdots + 543169 \) T^10 + 33T^9 + 561T^8 + 6094T^7 + 45474T^6 + 235928T^5 + 826551T^4 + 1807014T^3 + 2072004T^2 + 826914*T + 543169
$79$	\( T^{10} + \cdots + 867832681 \) T^10 - 27T^9 + 333T^8 + 656T^7 - 43056T^6 + 480424T^5 + 3881146T^4 - 16137729T^3 + 101363541T^2 - 181732571*T + 867832681
$83$	\( T^{10} + \cdots + 3695059369 \) T^10 - 25T^9 + 460T^8 - 7573T^7 + 107397T^6 - 1219494T^5 + 11362200T^4 - 89553803T^3 + 543086813T^2 - 2067730592*T + 3695059369
$89$	\( T^{10} + 38 T^{9} + \cdots + 14645929 \) T^10 + 38T^9 + 806T^8 + 12016T^7 + 122494T^6 + 810932T^5 + 3396376T^4 + 8137803T^3 + 8596251T^2 - 34443*T + 14645929
$97$	\( T^{10} + \cdots + 5993391889 \) T^10 + T^9 + 34T^8 + 661T^7 + 4093T^6 + 48016T^5 + 1204270T^4 + 13148719T^3 + 62311657T^2 + 421458148T + 5993391889
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