Embedded newform 23.11.d.a.5.8

• Label: 23.11.d.a.5.8
• Level: $23$
• Weight: $11$
• Character: 23.5
• Analytic conductor: $14.613$
• Analytic rank: $0$
• Dimension: $190$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.6132168115\)
Analytic rank:	\(0\)
Dimension:	\(190\)
Relative dimension:	\(19\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.8
Character	\(\chi\)	\(=\)	23.5
Dual form			23.11.d.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.05466 - 2.07144i) q^{2} +(17.6962 - 20.4225i) q^{3} +(-815.966 - 524.390i) q^{4} +(-5359.18 - 770.533i) q^{5} +(-167.144 + 107.417i) q^{6} +(10779.3 - 4922.74i) q^{7} +(9600.54 + 11079.6i) q^{8} +(8299.63 + 57725.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 190 q + 13 q^{2} - 71 q^{3} - 11699 q^{4} - 11 q^{5} + 3346 q^{6} - 11 q^{7} - 50582 q^{8} - 394850 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.05466	−	2.07144i	−0.220458	−	0.0647324i	0.169638	−	0.985506i	\(-0.445740\pi\)
							−0.390096	+	0.920774i	\(0.627558\pi\)
\(3\)	17.6962	−	20.4225i	0.0728238	−	0.0840432i	−0.718169	−	0.695869i	\(-0.755019\pi\)
							0.790992	+	0.611826i	\(0.209565\pi\)
\(5\)	−5359.18	−	770.533i	−1.71494	−	0.246571i	−0.786375	−	0.617750i	\(-0.788045\pi\)
							−0.928561	+	0.371179i	\(0.878954\pi\)
\(7\)	10779.3	−	4922.74i	0.641357	−	0.292898i	−0.0680659	−	0.997681i	\(-0.521683\pi\)
							0.709423	+	0.704783i	\(0.248956\pi\)
\(11\)	−40978.8	−	139561.i	−0.254446	−	0.866564i	−0.983315	−	0.181911i	\(-0.941772\pi\)
							0.728869	−	0.684653i	\(-0.240046\pi\)
\(13\)	27394.4	−	59985.4i	0.0737812	−	0.161558i	−0.869148	−	0.494553i	\(-0.835332\pi\)
							0.942929	+	0.332995i	\(0.108059\pi\)
\(17\)	1.19734e6	+	1.86310e6i	0.843281	+	1.31217i	0.948195	+	0.317690i	\(0.102907\pi\)
							−0.104914	+	0.994481i	\(0.533457\pi\)
\(19\)	1.20598e6	−	1.87654e6i	0.487049	−	0.757863i	−0.507554	−	0.861620i	\(-0.669450\pi\)
							0.994603	+	0.103757i	\(0.0330865\pi\)
\(23\)	−5.29699e6	+	3.65628e6i	−0.822981	+	0.568068i
							\(29\)	−3.87376e6	+	2.48951e6i	−0.188861	+	0.121374i	−0.631654	−	0.775250i	\(-0.717624\pi\)
0.442793	+	0.896624i	\(0.353987\pi\)
\(31\)	1.08530e7	+	1.25251e7i	0.379091	+	0.437494i	0.912945	−	0.408082i	\(-0.133802\pi\)
							−0.533854	+	0.845576i	\(0.679257\pi\)
\(37\)	1.80426e7	−	2.59414e6i	0.260191	−	0.0374098i	−0.0109850	−	0.999940i	\(-0.503497\pi\)
							0.271176	+	0.962530i	\(0.412588\pi\)
\(41\)	−6.94116e6	+	4.82768e7i	−0.0599118	+	0.416696i	0.937689	+	0.347474i	\(0.112961\pi\)
							−0.997601	+	0.0692216i	\(0.977948\pi\)
\(43\)	−5.71384e7	−	4.95107e7i	−0.388674	−	0.336788i	0.438502	−	0.898730i	\(-0.355509\pi\)
							−0.827177	+	0.561942i	\(0.810055\pi\)
\(47\)	−1.80573e8			−0.787342			−0.393671	−	0.919251i	\(-0.628795\pi\)
							−0.393671	+	0.919251i	\(0.628795\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.11.d.a.5.8	✓	190
23.14	odd	22	inner	23.11.d.a.14.8	yes	190

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.11.d.a.5.8	✓	190	1.1	even	1	trivial
23.11.d.a.14.8	yes	190	23.14	odd	22	inner