LMFDB - Newform orbit 224.3.c.b

Newspace parameters

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
Defining polynomial:	$ x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 $ x^8 + 20x^6 + 116x^4 + 180*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) $ q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9	\( q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{7} + \beta_{3}) q^{15} + \beta_1 q^{17} + (3 \beta_{7} - 3 \beta_{6} + 2 \beta_{2}) q^{19} + ( - \beta_{5} + 3 \beta_{4} + \beta_1 - 4) q^{21} + (2 \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{23} + ( - \beta_{4} - 7) q^{25} + (\beta_{7} - \beta_{6} + 9 \beta_{2}) q^{27} + (6 \beta_{4} + 2) q^{29} + ( - 5 \beta_{7} + 5 \beta_{6} + 9 \beta_{2}) q^{31} + ( - 6 \beta_{5} - \beta_1) q^{33} + (3 \beta_{7} - 6 \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{35} + ( - 14 \beta_{4} - 6) q^{37} + (3 \beta_{7} + 4 \beta_{6} - \beta_{3} - 4 \beta_{2}) q^{39} + (6 \beta_{5} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}) q^{43} + (\beta_{5} + \beta_1) q^{45} + (3 \beta_{7} - 3 \beta_{6} - 11 \beta_{2}) q^{47} + (6 \beta_{5} + 10 \beta_{4} + \beta_1 + 17) q^{49} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{2}) q^{51} + ( - 8 \beta_{4} + 34) q^{53} + ( - 7 \beta_{7} + 7 \beta_{6} - 23 \beta_{2}) q^{55} + (19 \beta_{4} - 16) q^{57} + ( - 7 \beta_{7} + 7 \beta_{6} - 6 \beta_{2}) q^{59} + 11 \beta_{5} q^{61} + (\beta_{7} - \beta_{6} - \beta_{3} + 3 \beta_{2}) q^{63} + ( - 31 \beta_{4} + 24) q^{65} + ( - 6 \beta_{7} - \beta_{6} - 5 \beta_{3} + \beta_{2}) q^{67} + ( - 10 \beta_{5} + 3 \beta_1) q^{69} + (\beta_{7} + 5 \beta_{6} - 4 \beta_{3} - 5 \beta_{2}) q^{71} + ( - 6 \beta_{5} - 3 \beta_1) q^{73} + (\beta_{7} - \beta_{6} - 8 \beta_{2}) q^{75} + ( - 10 \beta_{5} + 23 \beta_{4} - 4 \beta_1 - 26) q^{77} + ( - 5 \beta_{7} + \beta_{6} - 6 \beta_{3} - \beta_{2}) q^{79} + ( - 11 \beta_{4} - 63) q^{81} + ( - 2 \beta_{7} + 2 \beta_{6} + 23 \beta_{2}) q^{83} + (32 \beta_{4} + 8) q^{85} + ( - 6 \beta_{7} + 6 \beta_{6} + 8 \beta_{2}) q^{87} + ( - 2 \beta_{5} + 5 \beta_1) q^{89} + ( - 2 \beta_{7} + 9 \beta_{6} - 5 \beta_{3} - 34 \beta_{2}) q^{91} + ( - 44 \beta_{4} - 72) q^{93} + ( - 7 \beta_{7} - 12 \beta_{6} + 5 \beta_{3} + 12 \beta_{2}) q^{95} + ( - 4 \beta_{5} + 7 \beta_1) q^{97} + ( - 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{3} + 5 \beta_{2}) q^{99}+O(q^{100}) \) q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9 + (-b6 + b3 + b2) * q^11 + (-b5 - b1) * q^13 + (b7 + b3) * q^15 + b1 * q^17 + (3b7 - 3b6 + 2b2) q^19 + (-b5 + 3b4 + b1 - 4) q^21 + (2b7 + b6 + b3 - b2) q^23 + (-b4 - 7) * q^25 + (b7 - b6 + 9b2) q^27 + (6b4 + 2) q^29 + (-5b7 + 5b6 + 9b2) q^31 + (-6b5 - b1) q^33 + (3b7 - 6b6 + b3 - 3b2) q^35 + (-14b4 - 6) q^37 + (3b7 + 4b6 - b3 - 4b2) q^39 + (6b5 - 2b1) * q^41 + (-2b7 - b6 - b3 + b2) q^43 + (b5 + b1) * q^45 + (3b7 - 3b6 - 11b2) q^47 + (6b5 + 10b4 + b1 + 17) * q^49 + (-4b7 - 4b6 + 4b2) q^51 + (-8b4 + 34) q^53 + (-7b7 + 7b6 - 23b2) q^55 + (19b4 - 16) q^57 + (-7b7 + 7b6 - 6b2) q^59 + 11b5 q^61 + (b7 - b6 - b3 + 3b2) q^63 + (-31b4 + 24) q^65 + (-6b7 - b6 - 5b3 + b2) * q^67 + (-10b5 + 3b1) * q^69 + (b7 + 5b6 - 4b3 - 5b2) q^71 + (-6b5 - 3b1) * q^73 + (b7 - b6 - 8b2) q^75 + (-10b5 + 23b4 - 4b1 - 26) q^77 + (-5b7 + b6 - 6b3 - b2) * q^79 + (-11b4 - 63) q^81 + (-2b7 + 2b6 + 23b2) q^83 + (32b4 + 8) q^85 + (-6b7 + 6b6 + 8b2) q^87 + (-2b5 + 5b1) * q^89 + (-2b7 + 9b6 - 5b3 - 34b2) * q^91 + (-44b4 - 72) q^93 + (-7b7 - 12b6 + 5b3 + 12b2) * q^95 + (-4b5 + 7b1) * q^97 + (-2b7 - 5b6 + 3b3 + 5b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100}) $ 8 * q + 8 * q^9 - 32 * q^21 - 56 * q^25 + 16 * q^29 - 48 * q^37 + 136 * q^49 + 272 * q^53 - 128 * q^57 + 192 * q^65 - 208 * q^77 - 504 * q^81 + 64 * q^85 - 576 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 $ x^8 + 20*x^6 + 116*x^4 + 180*x^2 + 81 :

$\beta_{1}$	$=$	$ ( 4\nu^{7} + 80\nu^{5} + 464\nu^{3} + 828\nu ) / 27 $ (4v^7 + 80v^5 + 464v^3 + 828v) / 27
$\beta_{2}$	$=$	$ ( 7\nu^{7} + 131\nu^{5} + 659\nu^{3} + 567\nu ) / 27 $ (7v^7 + 131v^5 + 659v^3 + 567v) / 27
$\beta_{3}$	$=$	$ ( -5\nu^{7} - 9\nu^{6} - 91\nu^{5} - 153\nu^{4} - 427\nu^{3} - 585\nu^{2} - 261\nu - 54 ) / 27 $ (-5v^7 - 9v^6 - 91v^5 - 153v^4 - 427v^3 - 585v^2 - 261*v - 54) / 27
$\beta_{4}$	$=$	$ ( 2\nu^{6} + 40\nu^{4} + 214\nu^{2} + 180 ) / 9 $ (2v^6 + 40v^4 + 214*v^2 + 180) / 9
$\beta_{5}$	$=$	$ ( -13\nu^{7} - 251\nu^{5} - 1301\nu^{3} - 1107\nu ) / 27 $ (-13v^7 - 251v^5 - 1301v^3 - 1107v) / 27
$\beta_{6}$	$=$	$ ( 2\nu^{7} + 15\nu^{6} + 40\nu^{5} + 273\nu^{4} + 232\nu^{3} + 1335\nu^{2} + 306\nu + 1134 ) / 27 $ (2v^7 + 15v^6 + 40v^5 + 273v^4 + 232v^3 + 1335v^2 + 306*v + 1134) / 27
$\beta_{7}$	$=$	$ ( 5\nu^{7} + 15\nu^{6} + 91\nu^{5} + 273\nu^{4} + 427\nu^{3} + 1335\nu^{2} + 261\nu + 1134 ) / 27 $ (5v^7 + 15v^6 + 91v^5 + 273v^4 + 427v^3 + 1335v^2 + 261*v + 1134) / 27

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

$\nu$	$=$	$ ( \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 8 $ (b7 - b6 - b2 + b1) / 8
$\nu^{2}$	$=$	$ ( \beta_{7} - \beta_{4} + \beta_{3} - 20 ) / 4 $ (b7 - b4 + b3 - 20) / 4
$\nu^{3}$	$=$	$ ( -13\beta_{7} + 13\beta_{6} + 4\beta_{5} + 17\beta_{2} - 7\beta_1 ) / 8 $ (-13b7 + 13b6 + 4b5 + 17b2 - 7*b1) / 8
$\nu^{4}$	$=$	$ ( -6\beta_{7} - \beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 84 ) / 2 $ (-6b7 - b6 + 10b4 - 5*b3 + b2 + 84) / 2
$\nu^{5}$	$=$	$ ( 123\beta_{7} - 123\beta_{6} - 68\beta_{5} - 215\beta_{2} + 63\beta_1 ) / 8 $ (123b7 - 123b6 - 68b5 - 215b2 + 63*b1) / 8
$\nu^{6}$	$=$	$ ( 133\beta_{7} + 40\beta_{6} - 275\beta_{4} + 93\beta_{3} - 40\beta_{2} - 1580 ) / 4 $ (133b7 + 40b6 - 275b4 + 93b3 - 40*b2 - 1580) / 4
$\nu^{7}$	$=$	$ ( -1159\beta_{7} + 1159\beta_{6} + 896\beta_{5} + 2535\beta_{2} - 601\beta_1 ) / 8 $ (-1159b7 + 1159b6 + 896b5 + 2535b2 - 601*b1) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$1$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

97.1

		3.24994i
	−	0.923094i
	−	2.71359i
		1.10555i
	−	1.10555i
		2.71359i
		0.923094i
	−	3.24994i

−

3.29066i

−

5.90156i

−6.86601

1.36303i

−1.82843

97.2

−

3.29066i

5.90156i

6.86601

1.36303i

−1.82843

97.3

−

2.27411i

−

5.40107i

4.34256

−

5.49019i

3.82843

97.4

−

2.27411i

5.40107i

−4.34256

−

5.49019i

3.82843

97.5

2.27411i

−

5.40107i

−4.34256

5.49019i

3.82843

97.6

2.27411i

5.40107i

4.34256

5.49019i

3.82843

97.7

3.29066i

−

5.90156i

6.86601

−

1.36303i

−1.82843

97.8

3.29066i

5.90156i

−6.86601

−

1.36303i

−1.82843

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.3.c.b	✓	8
3.b	odd	2	1		2016.3.f.b		8
4.b	odd	2	1	inner	224.3.c.b	✓	8
7.b	odd	2	1	inner	224.3.c.b	✓	8
8.b	even	2	1		448.3.c.h		8
8.d	odd	2	1		448.3.c.h		8
12.b	even	2	1		2016.3.f.b		8
21.c	even	2	1		2016.3.f.b		8
28.d	even	2	1	inner	224.3.c.b	✓	8
56.e	even	2	1		448.3.c.h		8
56.h	odd	2	1		448.3.c.h		8
84.h	odd	2	1		2016.3.f.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.3.c.b	✓	8	1.a	even	1	1	trivial
224.3.c.b	✓	8	4.b	odd	2	1	inner
224.3.c.b	✓	8	7.b	odd	2	1	inner
224.3.c.b	✓	8	28.d	even	2	1	inner
448.3.c.h		8	8.b	even	2	1
448.3.c.h		8	8.d	odd	2	1
448.3.c.h		8	56.e	even	2	1
448.3.c.h		8	56.h	odd	2	1
2016.3.f.b		8	3.b	odd	2	1
2016.3.f.b		8	12.b	even	2	1
2016.3.f.b		8	21.c	even	2	1
2016.3.f.b		8	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 16T_{3}^{2} + 56 $ T3^4 + 16*T3^2 + 56 acting on $S_{3}^{\mathrm{new}}(224, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 16 T^{2} + 56)^{2} $ (T^4 + 16*T^2 + 56)^2
$5$	$ (T^{4} + 64 T^{2} + 1016)^{2} $ (T^4 + 64*T^2 + 1016)^2
$7$	$ T^{8} - 68 T^{6} + 2758 T^{4} + \cdots + 5764801 $ T^8 - 68T^6 + 2758T^4 - 163268*T^2 + 5764801
$11$	$ (T^{4} - 472 T^{2} + 14224)^{2} $ (T^4 - 472*T^2 + 14224)^2
$13$	$ (T^{4} + 544 T^{2} + 49784)^{2} $ (T^4 + 544*T^2 + 49784)^2
$17$	$ (T^{4} + 512 T^{2} + 65024)^{2} $ (T^4 + 512*T^2 + 65024)^2
$19$	$ (T^{4} + 1072 T^{2} + 123704)^{2} $ (T^4 + 1072*T^2 + 123704)^2
$23$	$ (T^{4} - 1112 T^{2} + 14224)^{2} $ (T^4 - 1112*T^2 + 14224)^2
$29$	$ (T^{2} - 4 T - 284)^{4} $ (T^2 - 4*T - 284)^4
$31$	$ (T^{4} + 4096 T^{2} + 1895936)^{2} $ (T^4 + 4096*T^2 + 1895936)^2
$37$	$ (T^{2} + 12 T - 1532)^{4} $ (T^2 + 12*T - 1532)^4
$41$	$ (T^{4} + 4736 T^{2} + 16256)^{2} $ (T^4 + 4736*T^2 + 16256)^2
$43$	$ (T^{4} - 1112 T^{2} + 14224)^{2} $ (T^4 - 1112*T^2 + 14224)^2
$47$	$ (T^{4} + 2944 T^{2} + 3584)^{2} $ (T^4 + 2944*T^2 + 3584)^2
$53$	$ (T^{2} - 68 T + 644)^{4} $ (T^2 - 68*T + 644)^4
$59$	$ (T^{4} + 6064 T^{2} + 2784824)^{2} $ (T^4 + 6064*T^2 + 2784824)^2
$61$	$ (T^{4} + 7744 T^{2} + 14875256)^{2} $ (T^4 + 7744*T^2 + 14875256)^2
$67$	$ (T^{4} - 15064 T^{2} + 34151824)^{2} $ (T^4 - 15064*T^2 + 34151824)^2
$71$	$ (T^{4} - 8648 T^{2} + 696976)^{2} $ (T^4 - 8648*T^2 + 696976)^2
$73$	$ (T^{4} + 6336 T^{2} + 1316736)^{2} $ (T^4 + 6336*T^2 + 1316736)^2
$79$	$ (T^{4} - 17352 T^{2} + 71703184)^{2} $ (T^4 - 17352*T^2 + 71703184)^2
$83$	$ (T^{4} + 8912 T^{2} + 9367736)^{2} $ (T^4 + 8912*T^2 + 9367736)^2
$89$	$ (T^{4} + 13376 T^{2} + 39030656)^{2} $ (T^4 + 13376*T^2 + 39030656)^2
$97$	$ (T^{4} + 27008 T^{2} + \cdots + 143638016)^{2} $ (T^4 + 27008*T^2 + 143638016)^2
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Classical	Maass
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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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9	\( q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{7} + \beta_{3}) q^{15} + \beta_1 q^{17} + (3 \beta_{7} - 3 \beta_{6} + 2 \beta_{2}) q^{19} + ( - \beta_{5} + 3 \beta_{4} + \beta_1 - 4) q^{21} + (2 \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{23} + ( - \beta_{4} - 7) q^{25} + (\beta_{7} - \beta_{6} + 9 \beta_{2}) q^{27} + (6 \beta_{4} + 2) q^{29} + ( - 5 \beta_{7} + 5 \beta_{6} + 9 \beta_{2}) q^{31} + ( - 6 \beta_{5} - \beta_1) q^{33} + (3 \beta_{7} - 6 \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{35} + ( - 14 \beta_{4} - 6) q^{37} + (3 \beta_{7} + 4 \beta_{6} - \beta_{3} - 4 \beta_{2}) q^{39} + (6 \beta_{5} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}) q^{43} + (\beta_{5} + \beta_1) q^{45} + (3 \beta_{7} - 3 \beta_{6} - 11 \beta_{2}) q^{47} + (6 \beta_{5} + 10 \beta_{4} + \beta_1 + 17) q^{49} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{2}) q^{51} + ( - 8 \beta_{4} + 34) q^{53} + ( - 7 \beta_{7} + 7 \beta_{6} - 23 \beta_{2}) q^{55} + (19 \beta_{4} - 16) q^{57} + ( - 7 \beta_{7} + 7 \beta_{6} - 6 \beta_{2}) q^{59} + 11 \beta_{5} q^{61} + (\beta_{7} - \beta_{6} - \beta_{3} + 3 \beta_{2}) q^{63} + ( - 31 \beta_{4} + 24) q^{65} + ( - 6 \beta_{7} - \beta_{6} - 5 \beta_{3} + \beta_{2}) q^{67} + ( - 10 \beta_{5} + 3 \beta_1) q^{69} + (\beta_{7} + 5 \beta_{6} - 4 \beta_{3} - 5 \beta_{2}) q^{71} + ( - 6 \beta_{5} - 3 \beta_1) q^{73} + (\beta_{7} - \beta_{6} - 8 \beta_{2}) q^{75} + ( - 10 \beta_{5} + 23 \beta_{4} - 4 \beta_1 - 26) q^{77} + ( - 5 \beta_{7} + \beta_{6} - 6 \beta_{3} - \beta_{2}) q^{79} + ( - 11 \beta_{4} - 63) q^{81} + ( - 2 \beta_{7} + 2 \beta_{6} + 23 \beta_{2}) q^{83} + (32 \beta_{4} + 8) q^{85} + ( - 6 \beta_{7} + 6 \beta_{6} + 8 \beta_{2}) q^{87} + ( - 2 \beta_{5} + 5 \beta_1) q^{89} + ( - 2 \beta_{7} + 9 \beta_{6} - 5 \beta_{3} - 34 \beta_{2}) q^{91} + ( - 44 \beta_{4} - 72) q^{93} + ( - 7 \beta_{7} - 12 \beta_{6} + 5 \beta_{3} + 12 \beta_{2}) q^{95} + ( - 4 \beta_{5} + 7 \beta_1) q^{97} + ( - 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{3} + 5 \beta_{2}) q^{99}+O(q^{100}) \) q + b2 * q^3 + b5 * q^5 + b7 * q^7 + (-b4 + 1) * q^9 + (-b6 + b3 + b2) * q^11 + (-b5 - b1) * q^13 + (b7 + b3) * q^15 + b1 * q^17 + (3b7 - 3b6 + 2b2) q^19 + (-b5 + 3b4 + b1 - 4) q^21 + (2b7 + b6 + b3 - b2) q^23 + (-b4 - 7) * q^25 + (b7 - b6 + 9b2) q^27 + (6b4 + 2) q^29 + (-5b7 + 5b6 + 9b2) q^31 + (-6b5 - b1) q^33 + (3b7 - 6b6 + b3 - 3b2) q^35 + (-14b4 - 6) q^37 + (3b7 + 4b6 - b3 - 4b2) q^39 + (6b5 - 2b1) * q^41 + (-2b7 - b6 - b3 + b2) q^43 + (b5 + b1) * q^45 + (3b7 - 3b6 - 11b2) q^47 + (6b5 + 10b4 + b1 + 17) * q^49 + (-4b7 - 4b6 + 4b2) q^51 + (-8b4 + 34) q^53 + (-7b7 + 7b6 - 23b2) q^55 + (19b4 - 16) q^57 + (-7b7 + 7b6 - 6b2) q^59 + 11b5 q^61 + (b7 - b6 - b3 + 3b2) q^63 + (-31b4 + 24) q^65 + (-6b7 - b6 - 5b3 + b2) * q^67 + (-10b5 + 3b1) * q^69 + (b7 + 5b6 - 4b3 - 5b2) q^71 + (-6b5 - 3b1) * q^73 + (b7 - b6 - 8b2) q^75 + (-10b5 + 23b4 - 4b1 - 26) q^77 + (-5b7 + b6 - 6b3 - b2) * q^79 + (-11b4 - 63) q^81 + (-2b7 + 2b6 + 23b2) q^83 + (32b4 + 8) q^85 + (-6b7 + 6b6 + 8b2) q^87 + (-2b5 + 5b1) * q^89 + (-2b7 + 9b6 - 5b3 - 34b2) * q^91 + (-44b4 - 72) q^93 + (-7b7 - 12b6 + 5b3 + 12b2) * q^95 + (-4b5 + 7b1) * q^97 + (-2b7 - 5b6 + 3b3 + 5b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100}) \) 8 * q + 8 * q^9 - 32 * q^21 - 56 * q^25 + 16 * q^29 - 48 * q^37 + 136 * q^49 + 272 * q^53 - 128 * q^57 + 192 * q^65 - 208 * q^77 - 504 * q^81 + 64 * q^85 - 576 * q^93

Introduction

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Newspace parameters

Newform invariants

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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	224.3.c.b

Level	$224$
Weight	$3$
Character orbit	224.c
Analytic conductor	$6.104$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial:	\( x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 \) x^8 + 20x^6 + 116x^4 + 180*x^2 + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{9} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 4\nu^{7} + 80\nu^{5} + 464\nu^{3} + 828\nu ) / 27 \) (4v^7 + 80v^5 + 464v^3 + 828v) / 27
\(\beta_{2}\)	\(=\)	\( ( 7\nu^{7} + 131\nu^{5} + 659\nu^{3} + 567\nu ) / 27 \) (7v^7 + 131v^5 + 659v^3 + 567v) / 27
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{7} - 9\nu^{6} - 91\nu^{5} - 153\nu^{4} - 427\nu^{3} - 585\nu^{2} - 261\nu - 54 ) / 27 \) (-5v^7 - 9v^6 - 91v^5 - 153v^4 - 427v^3 - 585v^2 - 261*v - 54) / 27
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{6} + 40\nu^{4} + 214\nu^{2} + 180 ) / 9 \) (2v^6 + 40v^4 + 214*v^2 + 180) / 9
\(\beta_{5}\)	\(=\)	\( ( -13\nu^{7} - 251\nu^{5} - 1301\nu^{3} - 1107\nu ) / 27 \) (-13v^7 - 251v^5 - 1301v^3 - 1107v) / 27
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{7} + 15\nu^{6} + 40\nu^{5} + 273\nu^{4} + 232\nu^{3} + 1335\nu^{2} + 306\nu + 1134 ) / 27 \) (2v^7 + 15v^6 + 40v^5 + 273v^4 + 232v^3 + 1335v^2 + 306*v + 1134) / 27
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{7} + 15\nu^{6} + 91\nu^{5} + 273\nu^{4} + 427\nu^{3} + 1335\nu^{2} + 261\nu + 1134 ) / 27 \) (5v^7 + 15v^6 + 91v^5 + 273v^4 + 427v^3 + 1335v^2 + 261*v + 1134) / 27

\(\nu\)	\(=\)	\( ( \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 8 \) (b7 - b6 - b2 + b1) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{7} - \beta_{4} + \beta_{3} - 20 ) / 4 \) (b7 - b4 + b3 - 20) / 4
\(\nu^{3}\)	\(=\)	\( ( -13\beta_{7} + 13\beta_{6} + 4\beta_{5} + 17\beta_{2} - 7\beta_1 ) / 8 \) (-13b7 + 13b6 + 4b5 + 17b2 - 7*b1) / 8
\(\nu^{4}\)	\(=\)	\( ( -6\beta_{7} - \beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 84 ) / 2 \) (-6b7 - b6 + 10b4 - 5*b3 + b2 + 84) / 2
\(\nu^{5}\)	\(=\)	\( ( 123\beta_{7} - 123\beta_{6} - 68\beta_{5} - 215\beta_{2} + 63\beta_1 ) / 8 \) (123b7 - 123b6 - 68b5 - 215b2 + 63*b1) / 8
\(\nu^{6}\)	\(=\)	\( ( 133\beta_{7} + 40\beta_{6} - 275\beta_{4} + 93\beta_{3} - 40\beta_{2} - 1580 ) / 4 \) (133b7 + 40b6 - 275b4 + 93b3 - 40*b2 - 1580) / 4
\(\nu^{7}\)	\(=\)	\( ( -1159\beta_{7} + 1159\beta_{6} + 896\beta_{5} + 2535\beta_{2} - 601\beta_1 ) / 8 \) (-1159b7 + 1159b6 + 896b5 + 2535b2 - 601*b1) / 8

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} + 16 T^{2} + 56)^{2} \) (T^4 + 16*T^2 + 56)^2
$5$	\( (T^{4} + 64 T^{2} + 1016)^{2} \) (T^4 + 64*T^2 + 1016)^2
$7$	\( T^{8} - 68 T^{6} + 2758 T^{4} + \cdots + 5764801 \) T^8 - 68T^6 + 2758T^4 - 163268*T^2 + 5764801
$11$	\( (T^{4} - 472 T^{2} + 14224)^{2} \) (T^4 - 472*T^2 + 14224)^2
$13$	\( (T^{4} + 544 T^{2} + 49784)^{2} \) (T^4 + 544*T^2 + 49784)^2
$17$	\( (T^{4} + 512 T^{2} + 65024)^{2} \) (T^4 + 512*T^2 + 65024)^2
$19$	\( (T^{4} + 1072 T^{2} + 123704)^{2} \) (T^4 + 1072*T^2 + 123704)^2
$23$	\( (T^{4} - 1112 T^{2} + 14224)^{2} \) (T^4 - 1112*T^2 + 14224)^2
$29$	\( (T^{2} - 4 T - 284)^{4} \) (T^2 - 4*T - 284)^4
$31$	\( (T^{4} + 4096 T^{2} + 1895936)^{2} \) (T^4 + 4096*T^2 + 1895936)^2
$37$	\( (T^{2} + 12 T - 1532)^{4} \) (T^2 + 12*T - 1532)^4
$41$	\( (T^{4} + 4736 T^{2} + 16256)^{2} \) (T^4 + 4736*T^2 + 16256)^2
$43$	\( (T^{4} - 1112 T^{2} + 14224)^{2} \) (T^4 - 1112*T^2 + 14224)^2
$47$	\( (T^{4} + 2944 T^{2} + 3584)^{2} \) (T^4 + 2944*T^2 + 3584)^2
$53$	\( (T^{2} - 68 T + 644)^{4} \) (T^2 - 68*T + 644)^4
$59$	\( (T^{4} + 6064 T^{2} + 2784824)^{2} \) (T^4 + 6064*T^2 + 2784824)^2
$61$	\( (T^{4} + 7744 T^{2} + 14875256)^{2} \) (T^4 + 7744*T^2 + 14875256)^2
$67$	\( (T^{4} - 15064 T^{2} + 34151824)^{2} \) (T^4 - 15064*T^2 + 34151824)^2
$71$	\( (T^{4} - 8648 T^{2} + 696976)^{2} \) (T^4 - 8648*T^2 + 696976)^2
$73$	\( (T^{4} + 6336 T^{2} + 1316736)^{2} \) (T^4 + 6336*T^2 + 1316736)^2
$79$	\( (T^{4} - 17352 T^{2} + 71703184)^{2} \) (T^4 - 17352*T^2 + 71703184)^2
$83$	\( (T^{4} + 8912 T^{2} + 9367736)^{2} \) (T^4 + 8912*T^2 + 9367736)^2
$89$	\( (T^{4} + 13376 T^{2} + 39030656)^{2} \) (T^4 + 13376*T^2 + 39030656)^2
$97$	\( (T^{4} + 27008 T^{2} + \cdots + 143638016)^{2} \) (T^4 + 27008*T^2 + 143638016)^2
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\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)