Embedded newform 23.8.c.a.12.9

• Label: 23.8.c.a.12.9
• Level: $23$
• Weight: $8$
• Character: 23.12
• Analytic conductor: $7.185$
• Analytic rank: $0$
• Dimension: $130$
• 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			12.9
Character	\(\chi\)	\(=\)	23.12
Dual form			23.8.c.a.2.9

$q$-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 \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)

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	−0.177178	−	0.984179i	\(-0.556697\pi\)
							0.821638	+	0.570010i	\(0.193061\pi\)
\(3\)	2.75598	−	19.1683i	0.0589321	−	0.409882i	−0.938907	−	0.344172i	\(-0.888160\pi\)
							0.997839	−	0.0657099i	\(-0.0209312\pi\)
\(5\)	−520.166	+	152.735i	−1.86100	+	0.546440i	−0.861770	+	0.507298i	\(0.830644\pi\)
							−0.999234	+	0.0391417i	\(0.987538\pi\)
\(7\)	−684.009	−	789.388i	−0.753735	−	0.869856i	0.241190	−	0.970478i	\(-0.422462\pi\)
							−0.994925	+	0.100622i	\(0.967917\pi\)
\(11\)	−3592.28	−	2308.62i	−0.813759	−	0.522971i	0.0663199	−	0.997798i	\(-0.478874\pi\)
							−0.880079	+	0.474827i	\(0.842511\pi\)
\(13\)	−665.546	+	768.081i	−0.0840188	+	0.0969628i	−0.796202	−	0.605030i	\(-0.793161\pi\)
							0.712184	+	0.701993i	\(0.247706\pi\)
\(17\)	−4861.97	−	10646.2i	−0.240016	−	0.525562i	0.750840	−	0.660484i	\(-0.229649\pi\)
							−0.990856	+	0.134922i	\(0.956922\pi\)
\(19\)	−7167.96	+	15695.6i	−0.239750	+	0.524978i	−0.990811	−	0.135254i	\(-0.956815\pi\)
							0.751061	+	0.660232i	\(0.229542\pi\)
\(23\)	−35012.8	+	46679.0i	−0.600039	+	0.799971i
							\(29\)	−14848.7	−	32514.2i	−0.113057	−	0.247560i	0.844643	−	0.535331i	\(-0.179813\pi\)
−0.957699	+	0.287771i	\(0.907086\pi\)
\(31\)	2234.60	+	15542.0i	0.0134720	+	0.0937000i	0.995448	−	0.0953053i	\(-0.0303827\pi\)
							−0.981976	+	0.189005i	\(0.939474\pi\)
\(37\)	−395572.	−	116150.i	−1.28386	−	0.376977i	−0.432539	−	0.901615i	\(-0.642382\pi\)
							−0.851325	+	0.524639i	\(0.824200\pi\)
\(41\)	−212845.	+	62497.0i	−0.482303	+	0.141617i	−0.513841	−	0.857885i	\(-0.671778\pi\)
							0.0315374	+	0.999503i	\(0.489960\pi\)
\(43\)	−118477.	+	824023.i	−0.227244	+	1.58052i	0.482395	+	0.875954i	\(0.339767\pi\)
							−0.709639	+	0.704565i	\(0.751142\pi\)
\(47\)	−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.12.9	yes	130
23.2	even	11	inner	23.8.c.a.2.9	✓	130
23.5	odd	22		529.8.a.j.1.44		65
23.18	even	11		529.8.a.k.1.44		65

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.c.a.2.9	✓	130	23.2	even	11	inner
23.8.c.a.12.9	yes	130	1.1	even	1	trivial
529.8.a.j.1.44		65	23.5	odd	22
529.8.a.k.1.44		65	23.18	even	11