LMFDB - Newform orbit 363.2.f.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(161,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 7]))

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

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

chi := DirichletCharacter("363.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.f (of order $10$, degree $4$, not 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:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{13} - \beta_{10} + \cdots + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + \cdots - 3) q^{9} + 2 \beta_1 q^{12} + \beta_{3} q^{13} - 4 \beta_{7} q^{16}+ \cdots + (5 \beta_{12} - 5 \beta_{7} - 5 \beta_{4} + 5) q^{97}+O(q^{100}) $ q + (-b13 - b10 + b5 + b1) * q^3 + 2b4 q^4 - b14 * q^7 + (-3b12 + 3b7 + 3b4 - 3) q^9 + 2b1 q^12 + b3 * q^13 - 4b7 q^16 + (b8 + b6) * q^19 + (b11 + b9) * q^21 + 5b12 q^25 + 3b10 q^27 + (-2b14 + 2b9 - 2b6 + 2b2) * q^28 - b13 * q^31 - 6b12 q^36 + 3b5 q^37 + (-b15 + 2b14 - 2b9 + 2b6 - 2b2) * q^39 + (-b11 - 2b9) q^43 - 4b5 q^48 + (-5b10 + 7b7) * q^49 - 2b8 q^52 + (-2b3 + b2) q^57 + (2b15 + b14 - b9 + b6 - b2) q^61 - 3b6 q^63 + (8b12 - 8b7 - 8b4 + 8) q^64 - 7b1 q^67 + (-b15 + 3b14 - b11 - b8 - b3) q^73 + 5b13 q^75 + (-2b11 + 2b9) * q^76 + (-b3 - 3b2) q^79 - 9b4 q^81 + (2b15 + 2b14 + 2b11 + 2b8 + 2b3) q^84 + (9b13 - b12 + 9b10 - 9b5 - 9b1) * q^91 + 3b7 q^93 + (5b12 - 5b7 - 5b4 + 5) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) $ 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4*x^14 + 15*x^12 - 56*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{10} + 362 ) / 209 $ (v^10 + 362) / 209
$\beta_{2}$	$=$	$ ( -2\nu^{11} - 933\nu ) / 209 $ (-2v^11 - 933v) / 209
$\beta_{3}$	$=$	$ ( 3\nu^{11} + 2549\nu ) / 209 $ (3v^11 + 2549v) / 209
$\beta_{4}$	$=$	$ ( \nu^{12} + 780\nu^{2} ) / 209 $ (v^12 + 780*v^2) / 209
$\beta_{5}$	$=$	$ ( -2\nu^{12} - 1351\nu^{2} ) / 209 $ (-2v^12 - 1351v^2) / 209
$\beta_{6}$	$=$	$ ( -5\nu^{13} - 3482\nu^{3} ) / 209 $ (-5v^13 - 3482v^3) / 209
$\beta_{7}$	$=$	$ ( -4\nu^{14} - 2911\nu^{4} ) / 209 $ (-4v^14 - 2911v^4) / 209
$\beta_{8}$	$=$	$ ( -13\nu^{13} - 9513\nu^{3} ) / 209 $ (-13v^13 - 9513v^3) / 209
$\beta_{9}$	$=$	$ ( -18\nu^{15} - 12995\nu^{5} ) / 209 $ (-18v^15 - 12995v^5) / 209
$\beta_{10}$	$=$	$ ( -7\nu^{14} - 5042\nu^{4} ) / 209 $ (-7v^14 - 5042v^4) / 209
$\beta_{11}$	$=$	$ ( 49\nu^{15} + 35503\nu^{5} ) / 209 $ (49v^15 + 35503v^5) / 209
$\beta_{12}$	$=$	$ ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 $ (56v^14 - 224v^12 + 840v^10 - 3135v^8 + 11704v^6 - 3136v^4 + 840*v^2 - 224) / 209
$\beta_{13}$	$=$	$ ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 $ (104v^14 - 390v^12 + 1456v^10 - 5434v^8 + 20273v^6 - 390v^4 + 104*v^2 - 26) / 209
$\beta_{14}$	$=$	$ ( -268\nu^{15} + 1005\nu^{13} - 3752\nu^{11} + 14003\nu^{9} - 52250\nu^{7} + 1005\nu^{5} - 268\nu^{3} + 67\nu ) / 209 $ (-268v^15 + 1005v^13 - 3752v^11 + 14003v^9 - 52250v^7 + 1005v^5 - 268v^3 + 67v) / 209
$\beta_{15}$	$=$	$ ( 683 \nu^{15} - 2732 \nu^{13} + 10245 \nu^{11} - 38247 \nu^{9} + 142747 \nu^{7} - 38248 \nu^{5} + \cdots - 2732 \nu ) / 209 $ (683v^15 - 2732v^13 + 10245v^11 - 38247v^9 + 142747v^7 - 38248v^5 + 10245v^3 - 2732v) / 209

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( 2\beta_{3} + 3\beta_{2} ) / 11 $ (2b3 + 3b2) / 11
$\nu^{2}$	$=$	$ \beta_{5} + 2\beta_{4} $ b5 + 2*b4
$\nu^{3}$	$=$	$ ( -5\beta_{8} + 13\beta_{6} ) / 11 $ (-5b8 + 13b6) / 11
$\nu^{4}$	$=$	$ 4\beta_{10} - 7\beta_{7} $ 4b10 - 7b7
$\nu^{5}$	$=$	$ ( 18\beta_{11} + 49\beta_{9} ) / 11 $ (18b11 + 49b9) / 11
$\nu^{6}$	$=$	$ -15\beta_{13} + 26\beta_{12} - 26\beta_{7} - 26\beta_{4} + 26 $ -15b13 + 26b12 - 26b7 - 26b4 + 26
$\nu^{7}$	$=$	$ ( 67\beta_{15} + 183\beta_{14} + 67\beta_{11} + 67\beta_{8} + 67\beta_{3} ) / 11 $ (67b15 + 183b14 + 67b11 + 67b8 + 67*b3) / 11
$\nu^{8}$	$=$	$ -56\beta_{13} + 97\beta_{12} - 56\beta_{10} + 56\beta_{5} + 56\beta_1 $ -56b13 + 97b12 - 56b10 + 56b5 + 56*b1
$\nu^{9}$	$=$	$ ( 250\beta_{15} + 683\beta_{14} - 683\beta_{9} + 683\beta_{6} - 683\beta_{2} ) / 11 $ (250b15 + 683b14 - 683b9 + 683b6 - 683*b2) / 11
$\nu^{10}$	$=$	$ 209\beta _1 - 362 $ 209*b1 - 362
$\nu^{11}$	$=$	$ ( -933\beta_{3} - 2549\beta_{2} ) / 11 $ (-933b3 - 2549b2) / 11
$\nu^{12}$	$=$	$ -780\beta_{5} - 1351\beta_{4} $ -780b5 - 1351b4
$\nu^{13}$	$=$	$ ( 3482\beta_{8} - 9513\beta_{6} ) / 11 $ (3482b8 - 9513b6) / 11
$\nu^{14}$	$=$	$ -2911\beta_{10} + 5042\beta_{7} $ -2911b10 + 5042b7
$\nu^{15}$	$=$	$ ( -12995\beta_{11} - 35503\beta_{9} ) / 11 $ (-12995b11 - 35503b9) / 11

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$\beta_{4}$

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} $

161.1

−0.304260	+	0.418778i
0.304260	−	0.418778i
1.13551	−	1.56290i
−1.13551	+	1.56290i
−1.83730	+	0.596975i
1.83730	−	0.596975i
0.492303	−	0.159959i
−0.492303	+	0.159959i
−1.83730	−	0.596975i
1.83730	+	0.596975i
0.492303	+	0.159959i
−0.492303	−	0.159959i
−0.304260	−	0.418778i
0.304260	+	0.418778i
1.13551	+	1.56290i
−1.13551	−	1.56290i

−0.535233

1.64728i

−0.618034

−

1.90211i

−2.19769

0.714073i

−2.42705

−

1.76336i

161.2

−0.535233

1.64728i

−0.618034

−

1.90211i

2.19769

−

0.714073i

−2.42705

−

1.76336i

161.3

0.535233

−

1.64728i

−0.618034

−

1.90211i

−4.52729

1.47101i

−2.42705

−

1.76336i

161.4

0.535233

−

1.64728i

−0.618034

−

1.90211i

4.52729

−

1.47101i

−2.42705

−

1.76336i

215.1

−1.40126

−

1.01807i

1.61803

−

1.17557i

−2.79802

−

3.85115i

0.927051

2.85317i

215.2

−1.40126

−

1.01807i

1.61803

−

1.17557i

2.79802

3.85115i

0.927051

2.85317i

215.3

1.40126

1.01807i

1.61803

−

1.17557i

−1.35825

−

1.86947i

0.927051

2.85317i

215.4

1.40126

1.01807i

1.61803

−

1.17557i

1.35825

1.86947i

0.927051

2.85317i

233.1

−1.40126

1.01807i

1.61803

1.17557i

−2.79802

3.85115i

0.927051

−

2.85317i

233.2

−1.40126

1.01807i

1.61803

1.17557i

2.79802

−

3.85115i

0.927051

−

2.85317i

233.3

1.40126

−

1.01807i

1.61803

1.17557i

−1.35825

1.86947i

0.927051

−

2.85317i

233.4

1.40126

−

1.01807i

1.61803

1.17557i

1.35825

−

1.86947i

0.927051

−

2.85317i

239.1

−0.535233

−

1.64728i

−0.618034

1.90211i

−2.19769

−

0.714073i

−2.42705

1.76336i

239.2

−0.535233

−

1.64728i

−0.618034

1.90211i

2.19769

0.714073i

−2.42705

1.76336i

239.3

0.535233

1.64728i

−0.618034

1.90211i

−4.52729

−

1.47101i

−2.42705

1.76336i

239.4

0.535233

1.64728i

−0.618034

1.90211i

4.52729

1.47101i

−2.42705

1.76336i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.2.f.g		16
3.b	odd	2	1	CM	363.2.f.g		16
11.b	odd	2	1	inner	363.2.f.g		16
11.c	even	5	1		363.2.d.d	✓	4
11.c	even	5	3	inner	363.2.f.g		16
11.d	odd	10	1		363.2.d.d	✓	4
11.d	odd	10	3	inner	363.2.f.g		16
33.d	even	2	1	inner	363.2.f.g		16
33.f	even	10	1		363.2.d.d	✓	4
33.f	even	10	3	inner	363.2.f.g		16
33.h	odd	10	1		363.2.d.d	✓	4
33.h	odd	10	3	inner	363.2.f.g		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
363.2.d.d	✓	4	11.c	even	5	1
363.2.d.d	✓	4	11.d	odd	10	1
363.2.d.d	✓	4	33.f	even	10	1
363.2.d.d	✓	4	33.h	odd	10	1
363.2.f.g		16	1.a	even	1	1	trivial
363.2.f.g		16	3.b	odd	2	1	CM
363.2.f.g		16	11.b	odd	2	1	inner
363.2.f.g		16	11.c	even	5	3	inner
363.2.f.g		16	11.d	odd	10	3	inner
363.2.f.g		16	33.d	even	2	1	inner
363.2.f.g		16	33.f	even	10	3	inner
363.2.f.g		16	33.h	odd	10	3	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(363, [\chi])$:

$ T_{2} $

$ T_{5} $

$ T_{7}^{16} - 28 T_{7}^{14} + 663 T_{7}^{12} - 15176 T_{7}^{10} + 344705 T_{7}^{8} - 1836296 T_{7}^{6} + \cdots + 214358881 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} $ (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} + \cdots + 214358881 $ T^16 - 28T^14 + 663T^12 - 15176T^10 + 344705T^8 - 1836296T^6 + 9706983T^4 - 49603708*T^2 + 214358881
$11$	$ T^{16} $ T^16
$13$	$ T^{16} + \cdots + 54875873536 $ T^16 - 52T^14 + 2220T^12 - 90272T^10 + 3619664T^8 - 43691648T^6 + 520048320T^4 - 5895755008*T^2 + 54875873536
$17$	$ T^{16} $ T^16
$19$	$ T^{16} + \cdots + 214358881 $ T^16 - 76T^14 + 5655T^12 - 420584T^10 + 31280129T^8 - 50890664T^6 + 82794855T^4 - 134638636*T^2 + 214358881
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} $ (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$37$	$ (T^{8} + 27 T^{6} + \cdots + 531441)^{2} $ (T^8 + 27T^6 + 729T^4 + 19683*T^2 + 531441)^2
$41$	$ T^{16} $ T^16
$43$	$ (T^{4} + 172 T^{2} + 484)^{4} $ (T^4 + 172*T^2 + 484)^4
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + \cdots + 45\!\cdots\!61 $ T^16 - 244T^14 + 44895T^12 - 7381976T^10 + 1143894449T^8 - 108079510616T^6 + 9623641962495T^4 - 765776523919924*T^2 + 45949729863572161
$67$	$ (T^{2} - 147)^{8} $ (T^2 - 147)^8
$71$	$ T^{16} $ T^16
$73$	$ T^{16} + \cdots + 17\!\cdots\!01 $ T^16 - 292T^14 + 64815T^12 - 12954872T^10 + 2457420689T^8 - 264914177528T^6 + 27103144168815T^4 - 2496888081023908*T^2 + 174859124550883201
$79$	$ T^{16} + \cdots + 214358881 $ T^16 - 316T^14 + 99735T^12 - 31478024T^10 + 9934987649T^8 - 3808840904T^6 + 1460220135T^4 - 559813276*T^2 + 214358881
$83$	$ T^{16} $ T^16
$89$	$ T^{16} $ T^16
$97$	$ (T^{4} - 5 T^{3} + \cdots + 625)^{4} $ (T^4 - 5T^3 + 25T^2 - 125*T + 625)^4
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.f (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{13} - \beta_{10} + \cdots + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + \cdots - 3) q^{9} + 2 \beta_1 q^{12} + \beta_{3} q^{13} - 4 \beta_{7} q^{16}+ \cdots + (5 \beta_{12} - 5 \beta_{7} - 5 \beta_{4} + 5) q^{97}+O(q^{100}) \) q + (-b13 - b10 + b5 + b1) * q^3 + 2b4 q^4 - b14 * q^7 + (-3b12 + 3b7 + 3b4 - 3) q^9 + 2b1 q^12 + b3 * q^13 - 4b7 q^16 + (b8 + b6) * q^19 + (b11 + b9) * q^21 + 5b12 q^25 + 3b10 q^27 + (-2b14 + 2b9 - 2b6 + 2b2) * q^28 - b13 * q^31 - 6b12 q^36 + 3b5 q^37 + (-b15 + 2b14 - 2b9 + 2b6 - 2b2) * q^39 + (-b11 - 2b9) q^43 - 4b5 q^48 + (-5b10 + 7b7) * q^49 - 2b8 q^52 + (-2b3 + b2) q^57 + (2b15 + b14 - b9 + b6 - b2) q^61 - 3b6 q^63 + (8b12 - 8b7 - 8b4 + 8) q^64 - 7b1 q^67 + (-b15 + 3b14 - b11 - b8 - b3) q^73 + 5b13 q^75 + (-2b11 + 2b9) * q^76 + (-b3 - 3b2) q^79 - 9b4 q^81 + (2b15 + 2b14 + 2b11 + 2b8 + 2b3) q^84 + (9b13 - b12 + 9b10 - 9b5 - 9b1) * q^91 + 3b7 q^93 + (5b12 - 5b7 - 5b4 + 5) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	363.2.f.g

Level	$363$
Weight	$2$
Character orbit	363.f
Analytic conductor	$2.899$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-3
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(2.89856959337\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 11^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{10} + 362 ) / 209 \) (v^10 + 362) / 209
\(\beta_{2}\)	\(=\)	\( ( -2\nu^{11} - 933\nu ) / 209 \) (-2v^11 - 933v) / 209
\(\beta_{3}\)	\(=\)	\( ( 3\nu^{11} + 2549\nu ) / 209 \) (3v^11 + 2549v) / 209
\(\beta_{4}\)	\(=\)	\( ( \nu^{12} + 780\nu^{2} ) / 209 \) (v^12 + 780*v^2) / 209
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) (-2v^12 - 1351v^2) / 209
\(\beta_{6}\)	\(=\)	\( ( -5\nu^{13} - 3482\nu^{3} ) / 209 \) (-5v^13 - 3482v^3) / 209
\(\beta_{7}\)	\(=\)	\( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) (-4v^14 - 2911v^4) / 209
\(\beta_{8}\)	\(=\)	\( ( -13\nu^{13} - 9513\nu^{3} ) / 209 \) (-13v^13 - 9513v^3) / 209
\(\beta_{9}\)	\(=\)	\( ( -18\nu^{15} - 12995\nu^{5} ) / 209 \) (-18v^15 - 12995v^5) / 209
\(\beta_{10}\)	\(=\)	\( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) (-7v^14 - 5042v^4) / 209
\(\beta_{11}\)	\(=\)	\( ( 49\nu^{15} + 35503\nu^{5} ) / 209 \) (49v^15 + 35503v^5) / 209
\(\beta_{12}\)	\(=\)	\( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) (56v^14 - 224v^12 + 840v^10 - 3135v^8 + 11704v^6 - 3136v^4 + 840*v^2 - 224) / 209
\(\beta_{13}\)	\(=\)	\( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) (104v^14 - 390v^12 + 1456v^10 - 5434v^8 + 20273v^6 - 390v^4 + 104*v^2 - 26) / 209
\(\beta_{14}\)	\(=\)	\( ( -268\nu^{15} + 1005\nu^{13} - 3752\nu^{11} + 14003\nu^{9} - 52250\nu^{7} + 1005\nu^{5} - 268\nu^{3} + 67\nu ) / 209 \) (-268v^15 + 1005v^13 - 3752v^11 + 14003v^9 - 52250v^7 + 1005v^5 - 268v^3 + 67v) / 209
\(\beta_{15}\)	\(=\)	\( ( 683 \nu^{15} - 2732 \nu^{13} + 10245 \nu^{11} - 38247 \nu^{9} + 142747 \nu^{7} - 38248 \nu^{5} + \cdots - 2732 \nu ) / 209 \) (683v^15 - 2732v^13 + 10245v^11 - 38247v^9 + 142747v^7 - 38248v^5 + 10245v^3 - 2732v) / 209

\(\nu\)	\(=\)	\( ( 2\beta_{3} + 3\beta_{2} ) / 11 \) (2b3 + 3b2) / 11
\(\nu^{2}\)	\(=\)	\( \beta_{5} + 2\beta_{4} \) b5 + 2*b4
\(\nu^{3}\)	\(=\)	\( ( -5\beta_{8} + 13\beta_{6} ) / 11 \) (-5b8 + 13b6) / 11
\(\nu^{4}\)	\(=\)	\( 4\beta_{10} - 7\beta_{7} \) 4b10 - 7b7
\(\nu^{5}\)	\(=\)	\( ( 18\beta_{11} + 49\beta_{9} ) / 11 \) (18b11 + 49b9) / 11
\(\nu^{6}\)	\(=\)	\( -15\beta_{13} + 26\beta_{12} - 26\beta_{7} - 26\beta_{4} + 26 \) -15b13 + 26b12 - 26b7 - 26b4 + 26
\(\nu^{7}\)	\(=\)	\( ( 67\beta_{15} + 183\beta_{14} + 67\beta_{11} + 67\beta_{8} + 67\beta_{3} ) / 11 \) (67b15 + 183b14 + 67b11 + 67b8 + 67*b3) / 11
\(\nu^{8}\)	\(=\)	\( -56\beta_{13} + 97\beta_{12} - 56\beta_{10} + 56\beta_{5} + 56\beta_1 \) -56b13 + 97b12 - 56b10 + 56b5 + 56*b1
\(\nu^{9}\)	\(=\)	\( ( 250\beta_{15} + 683\beta_{14} - 683\beta_{9} + 683\beta_{6} - 683\beta_{2} ) / 11 \) (250b15 + 683b14 - 683b9 + 683b6 - 683*b2) / 11
\(\nu^{10}\)	\(=\)	\( 209\beta _1 - 362 \) 209*b1 - 362
\(\nu^{11}\)	\(=\)	\( ( -933\beta_{3} - 2549\beta_{2} ) / 11 \) (-933b3 - 2549b2) / 11
\(\nu^{12}\)	\(=\)	\( -780\beta_{5} - 1351\beta_{4} \) -780b5 - 1351b4
\(\nu^{13}\)	\(=\)	\( ( 3482\beta_{8} - 9513\beta_{6} ) / 11 \) (3482b8 - 9513b6) / 11
\(\nu^{14}\)	\(=\)	\( -2911\beta_{10} + 5042\beta_{7} \) -2911b10 + 5042b7
\(\nu^{15}\)	\(=\)	\( ( -12995\beta_{11} - 35503\beta_{9} ) / 11 \) (-12995b11 - 35503b9) / 11

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 161.4
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} + \cdots + 214358881 \) T^16 - 28T^14 + 663T^12 - 15176T^10 + 344705T^8 - 1836296T^6 + 9706983T^4 - 49603708*T^2 + 214358881
$11$	\( T^{16} \) T^16
$13$	\( T^{16} + \cdots + 54875873536 \) T^16 - 52T^14 + 2220T^12 - 90272T^10 + 3619664T^8 - 43691648T^6 + 520048320T^4 - 5895755008*T^2 + 54875873536
$17$	\( T^{16} \) T^16
$19$	\( T^{16} + \cdots + 214358881 \) T^16 - 76T^14 + 5655T^12 - 420584T^10 + 31280129T^8 - 50890664T^6 + 82794855T^4 - 134638636*T^2 + 214358881
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$37$	\( (T^{8} + 27 T^{6} + \cdots + 531441)^{2} \) (T^8 + 27T^6 + 729T^4 + 19683*T^2 + 531441)^2
$41$	\( T^{16} \) T^16
$43$	\( (T^{4} + 172 T^{2} + 484)^{4} \) (T^4 + 172*T^2 + 484)^4
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + \cdots + 45\!\cdots\!61 \) T^16 - 244T^14 + 44895T^12 - 7381976T^10 + 1143894449T^8 - 108079510616T^6 + 9623641962495T^4 - 765776523919924*T^2 + 45949729863572161
$67$	\( (T^{2} - 147)^{8} \) (T^2 - 147)^8
$71$	\( T^{16} \) T^16
$73$	\( T^{16} + \cdots + 17\!\cdots\!01 \) T^16 - 292T^14 + 64815T^12 - 12954872T^10 + 2457420689T^8 - 264914177528T^6 + 27103144168815T^4 - 2496888081023908*T^2 + 174859124550883201
$79$	\( T^{16} + \cdots + 214358881 \) T^16 - 316T^14 + 99735T^12 - 31478024T^10 + 9934987649T^8 - 3808840904T^6 + 1460220135T^4 - 559813276*T^2 + 214358881
$83$	\( T^{16} \) T^16
$89$	\( T^{16} \) T^16
$97$	\( (T^{4} - 5 T^{3} + \cdots + 625)^{4} \) (T^4 - 5T^3 + 25T^2 - 125*T + 625)^4
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\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(\beta_{4}\)