Embedded newform 8.13.d.b.3.6

• Label: 8.13.d.b.3.6
• Level: $8$
• Weight: $13$
• Character: 8.3
• Analytic conductor: $7.312$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31195053821\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 5*x^9 + 468*x^8 + 1496*x^7 + 710096*x^6 + 29155008*x^5 + 143571008*x^4 + 28213427840*x^3 + 1335549648384*x^2 + 41051831642368*x + 824967906703360
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{45}\cdot 3^{3}\cdot 23 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3.6
Root			\(-10.5406 - 27.3936i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.3
Dual form			8.13.d.b.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-33.0812 + 54.7872i) q^{2} -875.418 q^{3} +(-1907.27 - 3624.85i) q^{4} -2272.99i q^{5} +(28959.9 - 47961.7i) q^{6} +92079.6i q^{7} +(261690. + 15420.3i) q^{8} +234916. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 110 q^{2} - 660 q^{3} - 2444 q^{4} - 130788 q^{6} + 27160 q^{8} + 1692798 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−33.0812	+	54.7872i	−0.516893	+	0.856050i
							\(3\)	−875.418			−1.20085			−0.600424	−	0.799682i	\(-0.705002\pi\)
−0.600424	+	0.799682i	\(0.705002\pi\)
\(5\)		−	2272.99i		−	0.145471i	−0.997351	−	0.0727356i	\(-0.976827\pi\)
							0.997351	−	0.0727356i	\(-0.0231729\pi\)
\(7\)			92079.6i			0.782664i	0.920250	+	0.391332i	\(0.127986\pi\)
							−0.920250	+	0.391332i	\(0.872014\pi\)
\(11\)	2.60172e6			1.46861			0.734303	−	0.678822i	\(-0.237509\pi\)
							0.734303	+	0.678822i	\(0.237509\pi\)
\(13\)		−	8.33004e6i		−	1.72579i	−0.505387	−	0.862893i	\(-0.668650\pi\)
							0.505387	−	0.862893i	\(-0.331350\pi\)
\(17\)	−7.99706e6			−0.331312			−0.165656	−	0.986184i	\(-0.552974\pi\)
							−0.165656	+	0.986184i	\(0.552974\pi\)
\(19\)	1.09745e7			0.233272			0.116636	−	0.993175i	\(-0.462789\pi\)
							0.116636	+	0.993175i	\(0.462789\pi\)
\(23\)		−	2.00098e8i		−	1.35169i	−0.737045	−	0.675844i	\(-0.763779\pi\)
							0.737045	−	0.675844i	\(-0.236221\pi\)
\(29\)		−	1.82066e8i		−	0.306085i	−0.988220	−	0.153042i	\(-0.951093\pi\)
							0.988220	−	0.153042i	\(-0.0489071\pi\)
\(31\)			1.49678e9i			1.68650i	0.537518	+	0.843252i	\(0.319362\pi\)
							−0.537518	+	0.843252i	\(0.680638\pi\)
\(37\)		−	3.12728e9i		−	1.21887i	−0.792837	−	0.609434i	\(-0.791397\pi\)
							0.792837	−	0.609434i	\(-0.208603\pi\)
\(41\)	−2.93948e9			−0.618825			−0.309412	−	0.950928i	\(-0.600132\pi\)
							−0.309412	+	0.950928i	\(0.600132\pi\)
\(43\)	4.63601e9			0.733388			0.366694	−	0.930342i	\(-0.380490\pi\)
							0.366694	+	0.930342i	\(0.380490\pi\)
\(47\)		−	3.74487e9i		−	0.347416i	−0.984797	−	0.173708i	\(-0.944425\pi\)
							0.984797	−	0.173708i	\(-0.0555749\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.13.d.b.3.6	yes	10
3.2	odd	2		72.13.b.b.19.5		10
4.3	odd	2		32.13.d.b.15.7		10
8.3	odd	2	inner	8.13.d.b.3.5	✓	10
8.5	even	2		32.13.d.b.15.8		10
24.11	even	2		72.13.b.b.19.6		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.13.d.b.3.5	✓	10	8.3	odd	2	inner
8.13.d.b.3.6	yes	10	1.1	even	1	trivial
32.13.d.b.15.7		10	4.3	odd	2
32.13.d.b.15.8		10	8.5	even	2
72.13.b.b.19.5		10	3.2	odd	2
72.13.b.b.19.6		10	24.11	even	2