Embedded newform 23.15.d.a.5.8

• Label: 23.15.d.a.5.8
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) 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			23.15.d.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-102.168 - 29.9991i) q^{2} +(2406.75 - 2777.53i) q^{3} +(-4244.84 - 2727.99i) q^{4} +(72015.7 + 10354.3i) q^{5} +(-329215. + 211573. i) q^{6} +(-51794.5 + 23653.7i) q^{7} +(1.49430e6 + 1.72452e6i) q^{8} +(-1.24157e6 - 8.63534e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−102.168	−	29.9991i	−0.798184	−	0.234368i	−0.142887	−	0.989739i	\(-0.545638\pi\)
							−0.655297	+	0.755371i	\(0.727457\pi\)
\(3\)	2406.75	−	2777.53i	1.10048	−	1.27002i	0.140458	−	0.990087i	\(-0.455142\pi\)
							0.960020	−	0.279933i	\(-0.0903122\pi\)
\(5\)	72015.7	+	10354.3i	0.921801	+	0.132535i	0.586846	−	0.809699i	\(-0.300370\pi\)
							0.334956	+	0.942234i	\(0.391279\pi\)
\(7\)	−51794.5	+	23653.7i	−0.0628923	+	0.0287219i	−0.446613	−	0.894727i	\(-0.647370\pi\)
							0.383721	+	0.923449i	\(0.374643\pi\)
\(11\)	−7.23205e6	−	2.46301e7i	−0.371119	−	1.26391i	−0.907541	−	0.419964i	\(-0.862043\pi\)
							0.536422	−	0.843950i	\(-0.319776\pi\)
\(13\)	5.66063e6	−	1.23950e7i	0.0902113	−	0.197535i	−0.859148	−	0.511727i	\(-0.829006\pi\)
							0.949360	+	0.314191i	\(0.101733\pi\)
\(17\)	−1.59561e7	−	2.48281e7i	−0.0388851	−	0.0605064i	0.821267	−	0.570544i	\(-0.193268\pi\)
							−0.860152	+	0.510038i	\(0.829631\pi\)
\(19\)	4.43562e8	−	6.90195e8i	0.496225	−	0.772141i	−0.499321	−	0.866417i	\(-0.666417\pi\)
							0.995546	+	0.0942757i	\(0.0300535\pi\)
\(23\)	3.40063e9	+	1.69016e8i	0.998767	+	0.0496402i
							\(29\)	−1.76439e10	+	1.13390e10i	−1.02284	+	0.657340i	−0.940686	−	0.339279i	\(-0.889817\pi\)
−0.0821559	+	0.996619i	\(0.526181\pi\)
\(31\)	−3.12136e10	−	3.60224e10i	−1.13452	−	1.30931i	−0.944867	−	0.327454i	\(-0.893809\pi\)
							−0.189652	−	0.981851i	\(-0.560736\pi\)
\(37\)	3.08833e10	−	4.44034e9i	0.325320	−	0.0467740i	0.0222800	−	0.999752i	\(-0.492907\pi\)
							0.303040	+	0.952978i	\(0.401998\pi\)
\(41\)	−4.00488e10	+	2.78546e11i	−0.205638	+	1.43024i	0.581541	+	0.813517i	\(0.302450\pi\)
							−0.787179	+	0.616725i	\(0.788459\pi\)
\(43\)	−2.37166e11	−	2.05505e11i	−0.872514	−	0.756038i	0.0984805	−	0.995139i	\(-0.468602\pi\)
							−0.970995	+	0.239101i	\(0.923147\pi\)
\(47\)	5.87169e11			1.15899			0.579493	−	0.814977i	\(-0.303251\pi\)
							0.579493	+	0.814977i	\(0.303251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.8	✓	270
23.14	odd	22	inner	23.15.d.a.14.8	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.8	✓	270	1.1	even	1	trivial
23.15.d.a.14.8	yes	270	23.14	odd	22	inner