LMFDB - Newform orbit 966.2.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(85,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

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

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

chi := DirichletCharacter("966.85");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.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:	$7.71354883526$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$.

$n$	$323$	$829$	$925$
$\chi(n)$	$1$	$1$	$\zeta_{22}^{4}$

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} $

85.1

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

−0.654861

0.755750i

0.841254

−

0.540641i

−0.142315

−

0.989821i

−1.27098

−

2.78305i

−0.142315

0.989821i

0.959493

−

0.281733i

0.841254

0.540641i

0.415415

−

0.909632i

2.93560

0.861971i

127.1

0.841254

−

0.540641i

−0.142315

0.989821i

0.415415

−

0.909632i

−3.15843

0.927399i

0.415415

0.909632i

0.654861

0.755750i

−0.142315

−

0.989821i

−0.959493

−

0.281733i

−2.15565

2.48775i

169.1

−0.142315

0.989821i

0.415415

0.909632i

−0.959493

−

0.281733i

−2.42796

−

2.80202i

−0.959493

0.281733i

−0.841254

−

0.540641i

0.415415

−

0.909632i

−0.654861

0.755750i

3.11903

−

2.00448i

211.1

0.415415

0.909632i

−0.959493

0.281733i

−0.654861

0.755750i

−0.114669

−

0.0736930i

−0.654861

−

0.755750i

0.142315

0.989821i

−0.959493

−

0.281733i

0.841254

−

0.540641i

0.0193985

−

0.134919i

463.1

−0.142315

−

0.989821i

0.415415

−

0.909632i

−0.959493

0.281733i

−2.42796

2.80202i

−0.959493

−

0.281733i

−0.841254

0.540641i

0.415415

0.909632i

−0.654861

−

0.755750i

3.11903

2.00448i

547.1

−0.959493

−

0.281733i

−0.654861

0.755750i

0.841254

0.540641i

−0.0279611

0.194474i

0.841254

−

0.540641i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

−0.142315

−

0.989821i

0.0816181

−

0.178719i

673.1

0.415415

−

0.909632i

−0.959493

−

0.281733i

−0.654861

−

0.755750i

−0.114669

0.0736930i

−0.654861

0.755750i

0.142315

−

0.989821i

−0.959493

0.281733i

0.841254

0.540641i

0.0193985

0.134919i

715.1

0.841254

0.540641i

−0.142315

−

0.989821i

0.415415

0.909632i

−3.15843

−

0.927399i

0.415415

−

0.909632i

0.654861

−

0.755750i

−0.142315

0.989821i

−0.959493

0.281733i

−2.15565

−

2.48775i

841.1

−0.654861

−

0.755750i

0.841254

0.540641i

−0.142315

0.989821i

−1.27098

2.78305i

−0.142315

−

0.989821i

0.959493

0.281733i

0.841254

−

0.540641i

0.415415

0.909632i

2.93560

−

0.861971i

883.1

−0.959493

0.281733i

−0.654861

−

0.755750i

0.841254

−

0.540641i

−0.0279611

−

0.194474i

0.841254

0.540641i

−0.415415

0.909632i

−0.654861

0.755750i

−0.142315

0.989821i

0.0816181

0.178719i

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	966.2.q.a	✓	10
23.c	even	11	1	inner	966.2.q.a	✓	10

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 14 T_{5}^{9} + 97 T_{5}^{8} + 412 T_{5}^{7} + 1137 T_{5}^{6} + 2003 T_{5}^{5} + 1950 T_{5}^{4} + \cdots + 1 $ T5^10 + 14*T5^9 + 97*T5^8 + 412*T5^7 + 1137*T5^6 + 2003*T5^5 + 1950*T5^4 + 526*T5^3 + 115*T5^2 + 15*T5 + 1 acting on $S_{2}^{\mathrm{new}}(966, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 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
$3$	$ 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} + 14 T^{9} + \cdots + 1 $ T^10 + 14T^9 + 97T^8 + 412T^7 + 1137T^6 + 2003T^5 + 1950T^4 + 526T^3 + 115T^2 + 15*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} + 4 T^{9} + \cdots + 529 $ T^10 + 4T^9 + 16T^8 + 64T^7 + 311T^6 + 342T^5 + 3216T^4 + 4900T^3 + 11823T^2 + 4623*T + 529
$13$	$ T^{10} - 14 T^{9} + \cdots + 69169 $ T^10 - 14T^9 + 86T^8 - 203T^7 + 114T^6 - 1992T^5 + 5481T^4 + 21155T^3 + 52816T^2 + 70484*T + 69169
$17$	$ T^{10} - 15 T^{9} + \cdots + 124609 $ T^10 - 15T^9 + 126T^8 - 911T^7 + 5371T^6 - 14378T^5 + 9112T^4 - 121115T^3 + 657545T^2 - 253454*T + 124609
$19$	$ T^{10} + 11 T^{9} + \cdots + 121 $ T^10 + 11T^9 + 132T^8 + 803T^7 + 3003T^6 + 6072T^5 + 7986T^4 + 10527T^3 + 22869T^2 + 3146*T + 121
$23$	$ T^{10} + T^{9} + \cdots + 6436343 $ T^10 + T^9 - 21T^8 - 219T^7 + 551T^6 + 4379T^5 + 12673T^4 - 115851T^3 - 255507T^2 + 279841T + 6436343
$29$	$ T^{10} - 19 T^{9} + \cdots + 6285049 $ T^10 - 19T^9 + 207T^8 - 1821T^7 + 8584T^6 - 18864T^5 + 181470T^4 - 247062T^3 + 371497T^2 + 3547405*T + 6285049
$31$	$ T^{10} + 5 T^{9} + \cdots + 109561 $ T^10 + 5T^9 + 91T^8 - 73T^7 + 53T^6 - 17346T^5 - 14592T^4 + 229034T^3 + 866661T^2 - 401172*T + 109561
$37$	$ T^{10} + 2 T^{9} + \cdots + 80982001 $ T^10 + 2T^9 + 26T^8 + 690T^7 + 5747T^6 + 30238T^5 + 286955T^4 + 1458354T^3 + 4554256T^2 + 25098211*T + 80982001
$41$	$ T^{10} + \cdots + 1400331241 $ T^10 - 2T^9 + 92T^8 + 124T^7 - 1425T^6 - 109493T^5 + 1333660T^4 - 7330462T^3 + 39251908T^2 - 289152067*T + 1400331241
$43$	$ T^{10} - 25 T^{9} + \cdots + 73599241 $ T^10 - 25T^9 + 504T^8 - 5989T^7 + 50153T^6 - 214391T^5 + 122895T^4 + 5343318T^3 - 2934751T^2 - 15159093*T + 73599241
$47$	$ (T^{5} + 2 T^{4} + \cdots + 109)^{2} $ (T^5 + 2T^4 - 82T^3 + 75T^2 + 356T + 109)^2
$53$	$ T^{10} - 15 T^{9} + \cdots + 212521 $ T^10 - 15T^9 + 82T^8 - 119T^7 + 2522T^6 - 15599T^5 + 57402T^4 - 146811T^3 + 253691T^2 - 297806*T + 212521
$59$	$ T^{10} + 17 T^{9} + \cdots + 21557449 $ T^10 + 17T^9 + 289T^8 + 4198T^7 + 33746T^6 + 114322T^5 + 207399T^4 + 1698848T^3 + 5148906T^2 - 12146088*T + 21557449
$61$	$ T^{10} - T^{9} + \cdots + 60668521 $ T^10 - T^9 + 12T^8 + 1099T^7 + 5501T^6 + 30524T^5 + 398806T^4 + 1671471T^3 + 6734575T^2 + 25438874T + 60668521
$67$	$ T^{10} + 4 T^{9} + \cdots + 29300569 $ T^10 + 4T^9 - 28T^8 - 112T^7 + 4689T^6 - 49004T^5 - 99293T^4 + 411746T^3 + 16148702T^2 + 7161399*T + 29300569
$71$	$ T^{10} + \cdots + 460059601 $ T^10 - 6T^9 - 96T^8 - 1503T^7 + 21096T^6 + 68399T^5 + 761458T^4 + 4682483T^3 + 24902739T^2 + 9244519*T + 460059601
$73$	$ T^{10} + \cdots + 127396369 $ T^10 + 25T^9 + 317T^8 + 1501T^7 - 14681T^6 - 51611T^5 + 1370086T^4 - 5039333T^3 + 23683302T^2 - 56446287*T + 127396369
$79$	$ T^{10} - 3 T^{9} + \cdots + 978121 $ T^10 - 3T^9 - 24T^8 + 72T^7 + 961T^6 - 14994T^5 + 54189T^4 + 106372T^3 + 572632T^2 + 787244*T + 978121
$83$	$ T^{10} - 7 T^{9} + \cdots + 3659569 $ T^10 - 7T^9 + 280T^8 - 3060T^7 + 23576T^6 - 164779T^5 + 1224997T^4 - 5695234T^3 + 15919264T^2 - 13452216*T + 3659569
$89$	$ T^{10} + 41 T^{9} + \cdots + 23203489 $ T^10 + 41T^9 + 823T^8 + 11369T^7 + 132829T^6 + 1084049T^5 + 4388761T^4 + 5235353T^3 + 10827019T^2 + 13530953*T + 23203489
$97$	$ T^{10} + \cdots + 18809299609 $ T^10 + 4T^9 + 225T^8 + 2693T^7 - 5398T^6 - 125036T^5 + 3427021T^4 + 56356239T^3 + 204514198T^2 + 222863875*T + 18809299609
show more
show less

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.q (of order \(11\), degree \(10\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	966.2.q.a

Level	$966$
Weight	$2$
Character orbit	966.q
Analytic conductor	$7.714$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.71354883526\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\)	\(323\)	\(829\)	\(925\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{22}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 85.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} + 14 T^{9} + \cdots + 1 \) T^10 + 14T^9 + 97T^8 + 412T^7 + 1137T^6 + 2003T^5 + 1950T^4 + 526T^3 + 115T^2 + 15*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
$11$	\( T^{10} + 4 T^{9} + \cdots + 529 \) T^10 + 4T^9 + 16T^8 + 64T^7 + 311T^6 + 342T^5 + 3216T^4 + 4900T^3 + 11823T^2 + 4623*T + 529
$13$	\( T^{10} - 14 T^{9} + \cdots + 69169 \) T^10 - 14T^9 + 86T^8 - 203T^7 + 114T^6 - 1992T^5 + 5481T^4 + 21155T^3 + 52816T^2 + 70484*T + 69169
$17$	\( T^{10} - 15 T^{9} + \cdots + 124609 \) T^10 - 15T^9 + 126T^8 - 911T^7 + 5371T^6 - 14378T^5 + 9112T^4 - 121115T^3 + 657545T^2 - 253454*T + 124609
$19$	\( T^{10} + 11 T^{9} + \cdots + 121 \) T^10 + 11T^9 + 132T^8 + 803T^7 + 3003T^6 + 6072T^5 + 7986T^4 + 10527T^3 + 22869T^2 + 3146*T + 121
$23$	\( T^{10} + T^{9} + \cdots + 6436343 \) T^10 + T^9 - 21T^8 - 219T^7 + 551T^6 + 4379T^5 + 12673T^4 - 115851T^3 - 255507T^2 + 279841T + 6436343
$29$	\( T^{10} - 19 T^{9} + \cdots + 6285049 \) T^10 - 19T^9 + 207T^8 - 1821T^7 + 8584T^6 - 18864T^5 + 181470T^4 - 247062T^3 + 371497T^2 + 3547405*T + 6285049
$31$	\( T^{10} + 5 T^{9} + \cdots + 109561 \) T^10 + 5T^9 + 91T^8 - 73T^7 + 53T^6 - 17346T^5 - 14592T^4 + 229034T^3 + 866661T^2 - 401172*T + 109561
$37$	\( T^{10} + 2 T^{9} + \cdots + 80982001 \) T^10 + 2T^9 + 26T^8 + 690T^7 + 5747T^6 + 30238T^5 + 286955T^4 + 1458354T^3 + 4554256T^2 + 25098211*T + 80982001
$41$	\( T^{10} + \cdots + 1400331241 \) T^10 - 2T^9 + 92T^8 + 124T^7 - 1425T^6 - 109493T^5 + 1333660T^4 - 7330462T^3 + 39251908T^2 - 289152067*T + 1400331241
$43$	\( T^{10} - 25 T^{9} + \cdots + 73599241 \) T^10 - 25T^9 + 504T^8 - 5989T^7 + 50153T^6 - 214391T^5 + 122895T^4 + 5343318T^3 - 2934751T^2 - 15159093*T + 73599241
$47$	\( (T^{5} + 2 T^{4} + \cdots + 109)^{2} \) (T^5 + 2T^4 - 82T^3 + 75T^2 + 356T + 109)^2
$53$	\( T^{10} - 15 T^{9} + \cdots + 212521 \) T^10 - 15T^9 + 82T^8 - 119T^7 + 2522T^6 - 15599T^5 + 57402T^4 - 146811T^3 + 253691T^2 - 297806*T + 212521
$59$	\( T^{10} + 17 T^{9} + \cdots + 21557449 \) T^10 + 17T^9 + 289T^8 + 4198T^7 + 33746T^6 + 114322T^5 + 207399T^4 + 1698848T^3 + 5148906T^2 - 12146088*T + 21557449
$61$	\( T^{10} - T^{9} + \cdots + 60668521 \) T^10 - T^9 + 12T^8 + 1099T^7 + 5501T^6 + 30524T^5 + 398806T^4 + 1671471T^3 + 6734575T^2 + 25438874T + 60668521
$67$	\( T^{10} + 4 T^{9} + \cdots + 29300569 \) T^10 + 4T^9 - 28T^8 - 112T^7 + 4689T^6 - 49004T^5 - 99293T^4 + 411746T^3 + 16148702T^2 + 7161399*T + 29300569
$71$	\( T^{10} + \cdots + 460059601 \) T^10 - 6T^9 - 96T^8 - 1503T^7 + 21096T^6 + 68399T^5 + 761458T^4 + 4682483T^3 + 24902739T^2 + 9244519*T + 460059601
$73$	\( T^{10} + \cdots + 127396369 \) T^10 + 25T^9 + 317T^8 + 1501T^7 - 14681T^6 - 51611T^5 + 1370086T^4 - 5039333T^3 + 23683302T^2 - 56446287*T + 127396369
$79$	\( T^{10} - 3 T^{9} + \cdots + 978121 \) T^10 - 3T^9 - 24T^8 + 72T^7 + 961T^6 - 14994T^5 + 54189T^4 + 106372T^3 + 572632T^2 + 787244*T + 978121
$83$	\( T^{10} - 7 T^{9} + \cdots + 3659569 \) T^10 - 7T^9 + 280T^8 - 3060T^7 + 23576T^6 - 164779T^5 + 1224997T^4 - 5695234T^3 + 15919264T^2 - 13452216*T + 3659569
$89$	\( T^{10} + 41 T^{9} + \cdots + 23203489 \) T^10 + 41T^9 + 823T^8 + 11369T^7 + 132829T^6 + 1084049T^5 + 4388761T^4 + 5235353T^3 + 10827019T^2 + 13530953*T + 23203489
$97$	\( T^{10} + \cdots + 18809299609 \) T^10 + 4T^9 + 225T^8 + 2693T^7 - 5398T^6 - 125036T^5 + 3427021T^4 + 56356239T^3 + 204514198T^2 + 222863875*T + 18809299609
show more
show less