LMFDB - Newform orbit 495.3.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(406,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.406"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 $ x^8 + 30x^6 + 280x^4 + 890*x^2 + 895
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{2} - 4) q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{5} - 3 \beta_1) q^{8} - \beta_{5} q^{10} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{11} + (\beta_{6} + \beta_{5} - 2 \beta_1) q^{13}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + 25 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b2 - 4) * q^4 - b4 * q^5 + b7 * q^7 + (b6 - b5 - 3b1) q^8 - b5 * q^10 + (b7 - b5 - b4 + 3b3 - b1 - 1) q^11 + (b6 + b5 - 2b1) q^13 + (b4 - 5b3) q^14 + (4b4 + 4b3 - 4b2 + 13) q^16 + (-3b7 + 2b5 - 2b1) q^17 + (-b7 + 2b6 + b5 - 3b1) * q^19 + (b4 + 4b3 - b2 + 3) q^20 + (3b7 + 2b5 + 6b4 - b3 - 2b2 - b1 + 11) * q^22 + (8b4 - 2b3 + 2b2 - 2) q^23 + 5 * q^25 + (-6b4 - 4b3 - 5b2 + 15) q^26 + (-b7 - 4b5) q^28 + (5b7 + 2b6 + b5 - 3b1) q^29 + (-11b4 + b3 - b2 + 5) q^31 + (4b7 + 8b5 + 13b1) q^32 + (-13b4 + 7b3 + 10) * q^34 + (b7 + b6 - b1) * q^35 + (14b4 - 3b3 + 4b2 - 13) q^37 + (-8b4 + b3 - 10b2 + 25) * q^38 + (4b7 - b6 + 2b5 + 6b1) q^40 + (-4b7 + 2b6 - 4b5 + 10b1) * q^41 + (2b7 - b6 + 3b5 + 6b1) q^43 + (3b7 - 2b6 + 3b5 - 11b4 - 11b3 + b2 + 13b1 - 2) * q^44 + (-2b7 + 2b6 + 4b5 - 8b1) * q^46 + (6b3 - 2b2 + 2) * q^47 + (5b4 - b3 - b2 + 22) q^49 + 5b1 q^50 + (-4b7 - b6 - b5 + 22b1) * q^52 + (18b4 - 19b3 - 8b2 + 23) q^53 + (b7 + b6 + b5 + b4 + 3b3 + 3b2 + 4b1 - 4) q^55 + (23b4 + b3 - 4b2 + 12) * q^56 + (-2b4 - 29b3 - 10b2 + 25) q^58 + (-28b4 + 14b3 + 4b2) q^59 + (-3b7 - b5 - 23b1) * q^61 + (b7 - b6 - 9b5 + 8b1) * q^62 + (-20b4 - 36b3 + 5b2 - 76) q^64 + (4b7 - b6 + b5 - 4b1) * q^65 + (-14b4 + 20b3 - 10b2 - 6) q^67 + (-5b7 + 2b5 + 2b1) q^68 + (-5b3 - 5b2 + 10) * q^70 + (5b4 + 13b3 + 5b2 - 37) q^71 + (-4b7 - b6 - 7b5 - 8b1) q^73 + (-3b7 + 4b6 + 7b5 - 25b1) * q^74 + (-3b7 - 2b6 + 7b5 + 43b1) * q^76 + (b6 - 3b5 + 4b4 - b3 - 6b2 + 14b1 - 17) * q^77 + (8b7 + 2b6 - 12b5 - 6b1) * q^79 + (-b4 - 12b3 + 8b2 - 44) * q^80 + (14b4 + 36b3 - 2b2 - 64) q^82 + (-4b7 - b6 - 13b5 - 2b1) q^83 + (-3b7 - 3b6 + 2b5 - 7b1) * q^85 + (-12b4 - 22b3 + 13b2 - 59) q^86 + (b7 + b6 - 15b5 + 14b4 - 31b3 + 16b2 - 9b1 - 73) q^88 + (5b4 + 3b3 - 5b2 - 53) q^89 + (24b4 + 6b3 + 6b2 - 8) q^91 + (8b4 - 14b3 - 4b2 + 48) q^92 + (6b7 - 2b6 + 8b5 + 8b1) * q^94 + (7b7 - 3b6 + b5 - 2b1) q^95 + (-30b4 + 4b3 + 10b2 + 34) q^97 + (-b7 - b6 + 5b5 + 25b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 28 q^{4} - 8 q^{11} + 88 q^{16} + 20 q^{20} + 80 q^{22} - 8 q^{23} + 40 q^{25} + 100 q^{26} + 36 q^{31} + 80 q^{34} - 88 q^{37} + 160 q^{38} - 12 q^{44} + 8 q^{47} + 172 q^{49} + 152 q^{53} - 20 q^{55}+ \cdots + 312 q^{97}+O(q^{100}) $ 8 * q - 28 * q^4 - 8 * q^11 + 88 * q^16 + 20 * q^20 + 80 * q^22 - 8 * q^23 + 40 * q^25 + 100 * q^26 + 36 * q^31 + 80 * q^34 - 88 * q^37 + 160 * q^38 - 12 * q^44 + 8 * q^47 + 172 * q^49 + 152 * q^53 - 20 * q^55 + 80 * q^56 + 160 * q^58 + 16 * q^59 - 588 * q^64 - 88 * q^67 + 60 * q^70 - 276 * q^71 - 160 * q^77 - 320 * q^80 - 520 * q^82 - 420 * q^86 - 520 * q^88 - 444 * q^89 - 40 * q^91 + 368 * q^92 + 312 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 $ x^8 + 30*x^6 + 280*x^4 + 890*x^2 + 895 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 8 $ v^2 + 8
$\beta_{3}$	$=$	$ ( -\nu^{6} - 25\nu^{4} - 171\nu^{2} - 275 ) / 16 $ (-v^6 - 25v^4 - 171v^2 - 275) / 16
$\beta_{4}$	$=$	$ ( \nu^{6} + 29\nu^{4} + 235\nu^{2} + 415 ) / 16 $ (v^6 + 29v^4 + 235v^2 + 415) / 16
$\beta_{5}$	$=$	$ ( \nu^{7} + 29\nu^{5} + 235\nu^{3} + 415\nu ) / 16 $ (v^7 + 29v^5 + 235v^3 + 415*v) / 16
$\beta_{6}$	$=$	$ ( \nu^{7} + 29\nu^{5} + 251\nu^{3} + 591\nu ) / 16 $ (v^7 + 29v^5 + 251v^3 + 591*v) / 16
$\beta_{7}$	$=$	$ ( -\nu^{7} - 27\nu^{5} - 203\nu^{3} - 345\nu ) / 8 $ (-v^7 - 27v^5 - 203v^3 - 345*v) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 8 $ b2 - 8
$\nu^{3}$	$=$	$ \beta_{6} - \beta_{5} - 11\beta_1 $ b6 - b5 - 11*b1
$\nu^{4}$	$=$	$ 4\beta_{4} + 4\beta_{3} - 16\beta_{2} + 93 $ 4b4 + 4b3 - 16*b2 + 93
$\nu^{5}$	$=$	$ 4\beta_{7} - 16\beta_{6} + 24\beta_{5} + 141\beta_1 $ 4b7 - 16b6 + 24b5 + 141b1
$\nu^{6}$	$=$	$ -100\beta_{4} - 116\beta_{3} + 229\beta_{2} - 1232 $ -100b4 - 116b3 + 229*b2 - 1232
$\nu^{7}$	$=$	$ -116\beta_{7} + 229\beta_{6} - 445\beta_{5} - 1919\beta_1 $ -116b7 + 229b6 - 445b5 - 1919b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$1$

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} $

406.1

	−	3.88606i
	−	3.15955i
	−	1.66769i
	−	1.46104i
		1.46104i
		1.66769i
		3.15955i
		3.88606i

−

3.88606i

−11.1014

−2.23607

−

3.44089i

27.5966i

8.68949i

406.2

−

3.15955i

−5.98274

2.23607

5.67155i

6.26455i

−

7.06496i

406.3

−

1.66769i

1.21881

2.23607

−

6.72266i

−

8.70336i

−

3.72907i

406.4

−

1.46104i

1.86536

−2.23607

4.56066i

−

8.56953i

3.26698i

406.5

1.46104i

1.86536

−2.23607

−

4.56066i

8.56953i

−

3.26698i

406.6

1.66769i

1.21881

2.23607

6.72266i

8.70336i

3.72907i

406.7

3.15955i

−5.98274

2.23607

−

5.67155i

−

6.26455i

7.06496i

406.8

3.88606i

−11.1014

−2.23607

3.44089i

−

27.5966i

−

8.68949i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	495.3.b.a		8
3.b	odd	2	1		55.3.c.a	✓	8
11.b	odd	2	1	inner	495.3.b.a		8
12.b	even	2	1		880.3.j.a		8
15.d	odd	2	1		275.3.c.f		8
15.e	even	4	2		275.3.d.c		16
33.d	even	2	1		55.3.c.a	✓	8
132.d	odd	2	1		880.3.j.a		8
165.d	even	2	1		275.3.c.f		8
165.l	odd	4	2		275.3.d.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.3.c.a	✓	8	3.b	odd	2	1
55.3.c.a	✓	8	33.d	even	2	1
275.3.c.f		8	15.d	odd	2	1
275.3.c.f		8	165.d	even	2	1
275.3.d.c		16	15.e	even	4	2
275.3.d.c		16	165.l	odd	4	2
495.3.b.a		8	1.a	even	1	1	trivial
495.3.b.a		8	11.b	odd	2	1	inner
880.3.j.a		8	12.b	even	2	1
880.3.j.a		8	132.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 30T_{2}^{6} + 280T_{2}^{4} + 890T_{2}^{2} + 895 $ T2^8 + 30*T2^6 + 280*T2^4 + 890*T2^2 + 895 acting on $S_{3}^{\mathrm{new}}(495, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 30 T^{6} + \cdots + 895 $ T^8 + 30T^6 + 280T^4 + 890*T^2 + 895
$3$	$ T^{8} $ T^8
$5$	$ (T^{2} - 5)^{4} $ (T^2 - 5)^4
$7$	$ T^{8} + 110 T^{6} + \cdots + 358000 $ T^8 + 110T^6 + 4225T^4 + 66500*T^2 + 358000
$11$	$ T^{8} + 8 T^{7} + \cdots + 214358881 $ T^8 + 8T^7 + 48T^6 + 1816T^5 + 23870T^4 + 219736T^3 + 702768T^2 + 14172488*T + 214358881
$13$	$ T^{8} + 620 T^{6} + \cdots + 27723520 $ T^8 + 620T^6 + 109600T^4 + 4153280*T^2 + 27723520
$17$	$ T^{8} + 1270 T^{6} + \cdots + 1732720 $ T^8 + 1270T^6 + 302185T^4 + 3536100*T^2 + 1732720
$19$	$ T^{8} + \cdots + 3931928320 $ T^8 + 2090T^6 + 1199785T^4 + 217622960*T^2 + 3931928320
$23$	$ (T^{4} + 4 T^{3} + \cdots - 27584)^{2} $ (T^4 + 4T^3 - 644T^2 - 8976*T - 27584)^2
$29$	$ T^{8} + 4850 T^{6} + \cdots + 27723520 $ T^8 + 4850T^6 + 5909665T^4 + 404179520*T^2 + 27723520
$31$	$ (T^{4} - 18 T^{3} + \cdots + 178076)^{2} $ (T^4 - 18T^3 - 951T^2 + 12488*T + 178076)^2
$37$	$ (T^{4} + 44 T^{3} + \cdots - 1000204)^{2} $ (T^4 + 44T^3 - 1429T^2 - 88496*T - 1000204)^2
$41$	$ T^{8} + \cdots + 5420634603520 $ T^8 + 9520T^6 + 25555200T^4 + 21158625280*T^2 + 5420634603520
$43$	$ T^{8} + 4180 T^{6} + \cdots + 91648000 $ T^8 + 4180T^6 + 593200T^4 + 20512000*T^2 + 91648000
$47$	$ (T^{4} - 4 T^{3} + \cdots + 22336)^{2} $ (T^4 - 4T^3 - 884T^2 - 7504*T + 22336)^2
$53$	$ (T^{4} - 76 T^{3} + \cdots - 14101244)^{2} $ (T^4 - 76T^3 - 6669T^2 + 693544*T - 14101244)^2
$59$	$ (T^{4} - 8 T^{3} + \cdots - 465344)^{2} $ (T^4 - 8T^3 - 8316T^2 - 279472*T - 465344)^2
$61$	$ T^{8} + \cdots + 44785710872320 $ T^8 + 17650T^6 + 79305745T^4 + 120983959920*T^2 + 44785710872320
$67$	$ (T^{4} + 44 T^{3} + \cdots + 3497456)^{2} $ (T^4 + 44T^3 - 10784T^2 - 294896*T + 3497456)^2
$71$	$ (T^{4} + 138 T^{3} + \cdots - 22924)^{2} $ (T^4 + 138T^3 + 3169T^2 + 14172*T - 22924)^2
$73$	$ T^{8} + \cdots + 15327220913920 $ T^8 + 13900T^6 + 45636000T^4 + 48397789120*T^2 + 15327220913920
$79$	$ T^{8} + \cdots + 67909027102720 $ T^8 + 27760T^6 + 226274560T^4 + 453908807680*T^2 + 67909027102720
$83$	$ T^{8} + \cdots + 26187838846720 $ T^8 + 30700T^6 + 259522080T^4 + 400109149120*T^2 + 26187838846720
$89$	$ (T^{4} + 222 T^{3} + \cdots - 1268884)^{2} $ (T^4 + 222T^3 + 16109T^2 + 364488*T - 1268884)^2
$97$	$ (T^{4} - 156 T^{3} + \cdots - 21090064)^{2} $ (T^4 - 156T^3 - 7264T^2 + 1182384*T - 21090064)^2
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Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} - 4) q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{5} - 3 \beta_1) q^{8} - \beta_{5} q^{10} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{11} + (\beta_{6} + \beta_{5} - 2 \beta_1) q^{13}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + 25 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b2 - 4) * q^4 - b4 * q^5 + b7 * q^7 + (b6 - b5 - 3b1) q^8 - b5 * q^10 + (b7 - b5 - b4 + 3b3 - b1 - 1) q^11 + (b6 + b5 - 2b1) q^13 + (b4 - 5b3) q^14 + (4b4 + 4b3 - 4b2 + 13) q^16 + (-3b7 + 2b5 - 2b1) q^17 + (-b7 + 2b6 + b5 - 3b1) * q^19 + (b4 + 4b3 - b2 + 3) q^20 + (3b7 + 2b5 + 6b4 - b3 - 2b2 - b1 + 11) * q^22 + (8b4 - 2b3 + 2b2 - 2) q^23 + 5 * q^25 + (-6b4 - 4b3 - 5b2 + 15) q^26 + (-b7 - 4b5) q^28 + (5b7 + 2b6 + b5 - 3b1) q^29 + (-11b4 + b3 - b2 + 5) q^31 + (4b7 + 8b5 + 13b1) q^32 + (-13b4 + 7b3 + 10) * q^34 + (b7 + b6 - b1) * q^35 + (14b4 - 3b3 + 4b2 - 13) q^37 + (-8b4 + b3 - 10b2 + 25) * q^38 + (4b7 - b6 + 2b5 + 6b1) q^40 + (-4b7 + 2b6 - 4b5 + 10b1) * q^41 + (2b7 - b6 + 3b5 + 6b1) q^43 + (3b7 - 2b6 + 3b5 - 11b4 - 11b3 + b2 + 13b1 - 2) * q^44 + (-2b7 + 2b6 + 4b5 - 8b1) * q^46 + (6b3 - 2b2 + 2) * q^47 + (5b4 - b3 - b2 + 22) q^49 + 5b1 q^50 + (-4b7 - b6 - b5 + 22b1) * q^52 + (18b4 - 19b3 - 8b2 + 23) q^53 + (b7 + b6 + b5 + b4 + 3b3 + 3b2 + 4b1 - 4) q^55 + (23b4 + b3 - 4b2 + 12) * q^56 + (-2b4 - 29b3 - 10b2 + 25) q^58 + (-28b4 + 14b3 + 4b2) q^59 + (-3b7 - b5 - 23b1) * q^61 + (b7 - b6 - 9b5 + 8b1) * q^62 + (-20b4 - 36b3 + 5b2 - 76) q^64 + (4b7 - b6 + b5 - 4b1) * q^65 + (-14b4 + 20b3 - 10b2 - 6) q^67 + (-5b7 + 2b5 + 2b1) q^68 + (-5b3 - 5b2 + 10) * q^70 + (5b4 + 13b3 + 5b2 - 37) q^71 + (-4b7 - b6 - 7b5 - 8b1) q^73 + (-3b7 + 4b6 + 7b5 - 25b1) * q^74 + (-3b7 - 2b6 + 7b5 + 43b1) * q^76 + (b6 - 3b5 + 4b4 - b3 - 6b2 + 14b1 - 17) * q^77 + (8b7 + 2b6 - 12b5 - 6b1) * q^79 + (-b4 - 12b3 + 8b2 - 44) * q^80 + (14b4 + 36b3 - 2b2 - 64) q^82 + (-4b7 - b6 - 13b5 - 2b1) q^83 + (-3b7 - 3b6 + 2b5 - 7b1) * q^85 + (-12b4 - 22b3 + 13b2 - 59) q^86 + (b7 + b6 - 15b5 + 14b4 - 31b3 + 16b2 - 9b1 - 73) q^88 + (5b4 + 3b3 - 5b2 - 53) q^89 + (24b4 + 6b3 + 6b2 - 8) q^91 + (8b4 - 14b3 - 4b2 + 48) q^92 + (6b7 - 2b6 + 8b5 + 8b1) * q^94 + (7b7 - 3b6 + b5 - 2b1) q^95 + (-30b4 + 4b3 + 10b2 + 34) q^97 + (-b7 - b6 + 5b5 + 25b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 28 q^{4} - 8 q^{11} + 88 q^{16} + 20 q^{20} + 80 q^{22} - 8 q^{23} + 40 q^{25} + 100 q^{26} + 36 q^{31} + 80 q^{34} - 88 q^{37} + 160 q^{38} - 12 q^{44} + 8 q^{47} + 172 q^{49} + 152 q^{53} - 20 q^{55}+ \cdots + 312 q^{97}+O(q^{100}) \) 8 * q - 28 * q^4 - 8 * q^11 + 88 * q^16 + 20 * q^20 + 80 * q^22 - 8 * q^23 + 40 * q^25 + 100 * q^26 + 36 * q^31 + 80 * q^34 - 88 * q^37 + 160 * q^38 - 12 * q^44 + 8 * q^47 + 172 * q^49 + 152 * q^53 - 20 * q^55 + 80 * q^56 + 160 * q^58 + 16 * q^59 - 588 * q^64 - 88 * q^67 + 60 * q^70 - 276 * q^71 - 160 * q^77 - 320 * q^80 - 520 * q^82 - 420 * q^86 - 520 * q^88 - 444 * q^89 - 40 * q^91 + 368 * q^92 + 312 * q^97

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Newspace parameters

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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	495.3.b.a

Level	$495$
Weight	$3$
Character orbit	495.b
Analytic conductor	$13.488$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(13.4877730858\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 \) x^8 + 30x^6 + 280x^4 + 890*x^2 + 895
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 8 \) v^2 + 8
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} - 25\nu^{4} - 171\nu^{2} - 275 ) / 16 \) (-v^6 - 25v^4 - 171v^2 - 275) / 16
\(\beta_{4}\)	\(=\)	\( ( \nu^{6} + 29\nu^{4} + 235\nu^{2} + 415 ) / 16 \) (v^6 + 29v^4 + 235v^2 + 415) / 16
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} + 29\nu^{5} + 235\nu^{3} + 415\nu ) / 16 \) (v^7 + 29v^5 + 235v^3 + 415*v) / 16
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} + 29\nu^{5} + 251\nu^{3} + 591\nu ) / 16 \) (v^7 + 29v^5 + 251v^3 + 591*v) / 16
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} - 27\nu^{5} - 203\nu^{3} - 345\nu ) / 8 \) (-v^7 - 27v^5 - 203v^3 - 345*v) / 8

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 8 \) b2 - 8
\(\nu^{3}\)	\(=\)	\( \beta_{6} - \beta_{5} - 11\beta_1 \) b6 - b5 - 11*b1
\(\nu^{4}\)	\(=\)	\( 4\beta_{4} + 4\beta_{3} - 16\beta_{2} + 93 \) 4b4 + 4b3 - 16*b2 + 93
\(\nu^{5}\)	\(=\)	\( 4\beta_{7} - 16\beta_{6} + 24\beta_{5} + 141\beta_1 \) 4b7 - 16b6 + 24b5 + 141b1
\(\nu^{6}\)	\(=\)	\( -100\beta_{4} - 116\beta_{3} + 229\beta_{2} - 1232 \) -100b4 - 116b3 + 229*b2 - 1232
\(\nu^{7}\)	\(=\)	\( -116\beta_{7} + 229\beta_{6} - 445\beta_{5} - 1919\beta_1 \) -116b7 + 229b6 - 445b5 - 1919b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 406.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 30 T^{6} + \cdots + 895 \) T^8 + 30T^6 + 280T^4 + 890*T^2 + 895
$3$	\( T^{8} \) T^8
$5$	\( (T^{2} - 5)^{4} \) (T^2 - 5)^4
$7$	\( T^{8} + 110 T^{6} + \cdots + 358000 \) T^8 + 110T^6 + 4225T^4 + 66500*T^2 + 358000
$11$	\( T^{8} + 8 T^{7} + \cdots + 214358881 \) T^8 + 8T^7 + 48T^6 + 1816T^5 + 23870T^4 + 219736T^3 + 702768T^2 + 14172488*T + 214358881
$13$	\( T^{8} + 620 T^{6} + \cdots + 27723520 \) T^8 + 620T^6 + 109600T^4 + 4153280*T^2 + 27723520
$17$	\( T^{8} + 1270 T^{6} + \cdots + 1732720 \) T^8 + 1270T^6 + 302185T^4 + 3536100*T^2 + 1732720
$19$	\( T^{8} + \cdots + 3931928320 \) T^8 + 2090T^6 + 1199785T^4 + 217622960*T^2 + 3931928320
$23$	\( (T^{4} + 4 T^{3} + \cdots - 27584)^{2} \) (T^4 + 4T^3 - 644T^2 - 8976*T - 27584)^2
$29$	\( T^{8} + 4850 T^{6} + \cdots + 27723520 \) T^8 + 4850T^6 + 5909665T^4 + 404179520*T^2 + 27723520
$31$	\( (T^{4} - 18 T^{3} + \cdots + 178076)^{2} \) (T^4 - 18T^3 - 951T^2 + 12488*T + 178076)^2
$37$	\( (T^{4} + 44 T^{3} + \cdots - 1000204)^{2} \) (T^4 + 44T^3 - 1429T^2 - 88496*T - 1000204)^2
$41$	\( T^{8} + \cdots + 5420634603520 \) T^8 + 9520T^6 + 25555200T^4 + 21158625280*T^2 + 5420634603520
$43$	\( T^{8} + 4180 T^{6} + \cdots + 91648000 \) T^8 + 4180T^6 + 593200T^4 + 20512000*T^2 + 91648000
$47$	\( (T^{4} - 4 T^{3} + \cdots + 22336)^{2} \) (T^4 - 4T^3 - 884T^2 - 7504*T + 22336)^2
$53$	\( (T^{4} - 76 T^{3} + \cdots - 14101244)^{2} \) (T^4 - 76T^3 - 6669T^2 + 693544*T - 14101244)^2
$59$	\( (T^{4} - 8 T^{3} + \cdots - 465344)^{2} \) (T^4 - 8T^3 - 8316T^2 - 279472*T - 465344)^2
$61$	\( T^{8} + \cdots + 44785710872320 \) T^8 + 17650T^6 + 79305745T^4 + 120983959920*T^2 + 44785710872320
$67$	\( (T^{4} + 44 T^{3} + \cdots + 3497456)^{2} \) (T^4 + 44T^3 - 10784T^2 - 294896*T + 3497456)^2
$71$	\( (T^{4} + 138 T^{3} + \cdots - 22924)^{2} \) (T^4 + 138T^3 + 3169T^2 + 14172*T - 22924)^2
$73$	\( T^{8} + \cdots + 15327220913920 \) T^8 + 13900T^6 + 45636000T^4 + 48397789120*T^2 + 15327220913920
$79$	\( T^{8} + \cdots + 67909027102720 \) T^8 + 27760T^6 + 226274560T^4 + 453908807680*T^2 + 67909027102720
$83$	\( T^{8} + \cdots + 26187838846720 \) T^8 + 30700T^6 + 259522080T^4 + 400109149120*T^2 + 26187838846720
$89$	\( (T^{4} + 222 T^{3} + \cdots - 1268884)^{2} \) (T^4 + 222T^3 + 16109T^2 + 364488*T - 1268884)^2
$97$	\( (T^{4} - 156 T^{3} + \cdots - 21090064)^{2} \) (T^4 - 156T^3 - 7264T^2 + 1182384*T - 21090064)^2
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\(n\)	\(46\)	\(56\)	\(397\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)