Embedded newform 23.8.c.a.2.9

• Label: 23.8.c.a.2.9
• Level: $23$
• Weight: $8$
• Character: 23.2
• Analytic conductor: $7.185$
• Analytic rank: $0$
• Dimension: $130$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18485558613\)
Analytic rank:	\(0\)
Dimension:	\(130\)
Relative dimension:	\(13\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	23.2
Dual form			23.8.c.a.12.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.29123 + 4.68579i) q^{2} +(2.75598 + 19.1683i) q^{3} +(-21.9677 - 48.1025i) q^{4} +(-520.166 - 152.735i) q^{5} +(-69.7240 + 152.674i) q^{6} +(-684.009 + 789.388i) q^{7} +(223.109 - 1551.76i) q^{8} +(1738.58 - 510.494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 130 q + 5 q^{2} + 17 q^{3} - 907 q^{4} - 399 q^{5} + 1196 q^{6} - 301 q^{7} + 2764 q^{8} - 11694 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.29123	+	4.68579i	0.644460	+	0.414169i	0.821638	−	0.570010i	\(-0.193061\pi\)
							−0.177178	+	0.984179i	\(0.556697\pi\)
\(3\)	2.75598	+	19.1683i	0.0589321	+	0.409882i	0.997839	+	0.0657099i	\(0.0209312\pi\)
							−0.938907	+	0.344172i	\(0.888160\pi\)
\(5\)	−520.166	−	152.735i	−1.86100	−	0.546440i	−0.999234	−	0.0391417i	\(-0.987538\pi\)
							−0.861770	−	0.507298i	\(-0.830644\pi\)
\(7\)	−684.009	+	789.388i	−0.753735	+	0.869856i	−0.994925	−	0.100622i	\(-0.967917\pi\)
							0.241190	+	0.970478i	\(0.422462\pi\)
\(11\)	−3592.28	+	2308.62i	−0.813759	+	0.522971i	−0.880079	−	0.474827i	\(-0.842511\pi\)
							0.0663199	+	0.997798i	\(0.478874\pi\)
\(13\)	−665.546	−	768.081i	−0.0840188	−	0.0969628i	0.712184	−	0.701993i	\(-0.247706\pi\)
							−0.796202	+	0.605030i	\(0.793161\pi\)
\(17\)	−4861.97	+	10646.2i	−0.240016	+	0.525562i	−0.990856	−	0.134922i	\(-0.956922\pi\)
							0.750840	+	0.660484i	\(0.229649\pi\)
\(19\)	−7167.96	−	15695.6i	−0.239750	−	0.524978i	0.751061	−	0.660232i	\(-0.229542\pi\)
							−0.990811	+	0.135254i	\(0.956815\pi\)
\(23\)	−35012.8	−	46679.0i	−0.600039	−	0.799971i
							\(29\)	−14848.7	+	32514.2i	−0.113057	+	0.247560i	−0.957699	−	0.287771i	\(-0.907086\pi\)
0.844643	+	0.535331i	\(0.179813\pi\)
\(31\)	2234.60	−	15542.0i	0.0134720	−	0.0937000i	−0.981976	−	0.189005i	\(-0.939474\pi\)
							0.995448	+	0.0953053i	\(0.0303827\pi\)
\(37\)	−395572.	+	116150.i	−1.28386	+	0.376977i	−0.851325	−	0.524639i	\(-0.824200\pi\)
							−0.432539	+	0.901615i	\(0.642382\pi\)
\(41\)	−212845.	−	62497.0i	−0.482303	−	0.141617i	0.0315374	−	0.999503i	\(-0.489960\pi\)
							−0.513841	+	0.857885i	\(0.671778\pi\)
\(43\)	−118477.	−	824023.i	−0.227244	−	1.58052i	−0.709639	−	0.704565i	\(-0.751142\pi\)
							0.482395	−	0.875954i	\(-0.339767\pi\)
\(47\)	−61012.0			−0.0857181			−0.0428590	−	0.999081i	\(-0.513647\pi\)
							−0.0428590	+	0.999081i	\(0.513647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.c.a.2.9	✓	130
23.9	even	11		529.8.a.k.1.44		65
23.12	even	11	inner	23.8.c.a.12.9	yes	130
23.14	odd	22		529.8.a.j.1.44		65

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.c.a.2.9	✓	130	1.1	even	1	trivial
23.8.c.a.12.9	yes	130	23.12	even	11	inner
529.8.a.j.1.44		65	23.14	odd	22
529.8.a.k.1.44		65	23.9	even	11