Embedded newform 23.13.d.a.5.16

• Label: 23.13.d.a.5.16
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.16
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(47.4597 + 13.9354i) q^{2} +(72.2874 - 83.4241i) q^{3} +(-1387.55 - 891.721i) q^{4} +(-16609.8 - 2388.12i) q^{5} +(4593.29 - 2951.93i) q^{6} +(138459. - 63232.2i) q^{7} +(-186102. - 214773. i) q^{8} +(73897.8 + 513971. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	47.4597	+	13.9354i	0.741558	+	0.217741i	0.630626	−	0.776087i	\(-0.282798\pi\)
							0.110932	+	0.993828i	\(0.464616\pi\)
\(3\)	72.2874	−	83.4241i	0.0991596	−	0.114436i	−0.704001	−	0.710199i	\(-0.748605\pi\)
							0.803160	+	0.595763i	\(0.203150\pi\)
\(5\)	−16609.8	−	2388.12i	−1.06302	−	0.152840i	−0.411456	−	0.911430i	\(-0.634980\pi\)
							−0.651569	+	0.758590i	\(0.725889\pi\)
\(7\)	138459.	−	63232.2i	1.17688	−	0.537465i	0.271657	−	0.962394i	\(-0.412428\pi\)
							0.905227	+	0.424929i	\(0.139701\pi\)
\(11\)	934236.	+	3.18172e6i	0.527352	+	1.79600i	0.601651	+	0.798759i	\(0.294510\pi\)
							−0.0742990	+	0.997236i	\(0.523672\pi\)
\(13\)	−1.65966e6	+	3.63415e6i	−0.343842	+	0.752910i	−0.999999	−	0.00171535i	\(-0.999454\pi\)
							0.656156	+	0.754625i	\(0.272181\pi\)
\(17\)	3.65745e6	+	5.69111e6i	0.151525	+	0.235778i	0.908715	−	0.417417i	\(-0.137065\pi\)
							−0.757190	+	0.653195i	\(0.773428\pi\)
\(19\)	−2.42818e7	+	3.77832e7i	−0.516130	+	0.803113i	−0.997294	−	0.0735172i	\(-0.976578\pi\)
							0.481164	+	0.876630i	\(0.340214\pi\)
\(23\)	6.27994e6	−	1.47903e8i	0.0424217	−	0.999100i
							\(29\)	−3.33663e8	+	2.14432e8i	−0.560945	+	0.360498i	−0.790181	−	0.612873i	\(-0.790014\pi\)
0.229236	+	0.973371i	\(0.426377\pi\)
\(31\)	8.09847e8	+	9.34613e8i	0.912500	+	1.05308i	0.998387	+	0.0567764i	\(0.0180822\pi\)
							−0.0858868	+	0.996305i	\(0.527372\pi\)
\(37\)	1.56066e9	−	2.24388e8i	0.608271	−	0.0874561i	0.168705	−	0.985667i	\(-0.446041\pi\)
							0.439566	+	0.898210i	\(0.355132\pi\)
\(41\)	−6.01639e8	+	4.18449e9i	−0.126658	+	0.880927i	0.823090	+	0.567911i	\(0.192248\pi\)
							−0.949748	+	0.313016i	\(0.898661\pi\)
\(43\)	1.39898e9	+	1.21223e9i	0.221310	+	0.191767i	0.758458	−	0.651721i	\(-0.225953\pi\)
							−0.537148	+	0.843488i	\(0.680498\pi\)
\(47\)	−1.84251e10			−1.70932			−0.854661	−	0.519187i	\(-0.826235\pi\)
							−0.854661	+	0.519187i	\(0.826235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.16	✓	230
23.14	odd	22	inner	23.13.d.a.14.16	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.16	✓	230	1.1	even	1	trivial
23.13.d.a.14.16	yes	230	23.14	odd	22	inner