Embedded newform 23.13.d.a.5.6

• Label: 23.13.d.a.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-77.3817 - 22.7213i) q^{2} +(495.427 - 571.753i) q^{3} +(2025.90 + 1301.96i) q^{4} +(-13357.6 - 1920.53i) q^{5} +(-51328.0 + 32986.5i) q^{6} +(130792. - 59730.6i) q^{7} +(89139.4 + 102872. i) q^{8} +(-5821.98 - 40492.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−77.3817	−	22.7213i	−1.20909	−	0.355021i	−0.385768	−	0.922596i	\(-0.626063\pi\)
							−0.823322	+	0.567575i	\(0.807882\pi\)
\(3\)	495.427	−	571.753i	0.679598	−	0.784298i	−0.306248	−	0.951952i	\(-0.599074\pi\)
							0.985846	+	0.167654i	\(0.0536190\pi\)
\(5\)	−13357.6	−	1920.53i	−0.854884	−	0.122914i	−0.299081	−	0.954228i	\(-0.596680\pi\)
							−0.555803	+	0.831314i	\(0.687589\pi\)
\(7\)	130792.	−	59730.6i	1.11171	−	0.507701i	0.227025	−	0.973889i	\(-0.427100\pi\)
							0.884686	+	0.466187i	\(0.154373\pi\)
\(11\)	6451.53	+	21971.9i	0.00364172	+	0.0124026i	0.961294	−	0.275525i	\(-0.0888518\pi\)
							−0.957652	+	0.287928i	\(0.907034\pi\)
\(13\)	−3.57684e6	+	7.83219e6i	−0.741036	+	1.62264i	0.0408061	+	0.999167i	\(0.487007\pi\)
							−0.781842	+	0.623476i	\(0.785720\pi\)
\(17\)	1.92312e7	+	2.99244e7i	0.796734	+	1.23974i	0.967109	+	0.254363i	\(0.0818656\pi\)
							−0.170375	+	0.985379i	\(0.554498\pi\)
\(19\)	1.59638e7	−	2.48401e7i	0.339324	−	0.527998i	−0.629095	−	0.777329i	\(-0.716574\pi\)
							0.968418	+	0.249331i	\(0.0802106\pi\)
\(23\)	1.14007e8	+	9.44306e7i	0.770128	+	0.637890i
							\(29\)	5.89570e8	−	3.78893e8i	0.991168	−	0.636985i	0.0587141	−	0.998275i	\(-0.481300\pi\)
0.932453	+	0.361290i	\(0.117664\pi\)
\(31\)	−7.40235e8	−	8.54277e8i	−0.834064	−	0.962561i	0.165657	−	0.986183i	\(-0.447026\pi\)
							−0.999721	+	0.0236222i	\(0.992480\pi\)
\(37\)	3.34448e9	−	4.80864e8i	1.30352	−	0.187418i	0.544653	−	0.838662i	\(-0.316661\pi\)
							0.758869	+	0.651244i	\(0.225752\pi\)
\(41\)	2.93917e8	−	2.04424e9i	0.0618760	−	0.430357i	−0.935211	−	0.354090i	\(-0.884791\pi\)
							0.997087	−	0.0762670i	\(-0.0243002\pi\)
\(43\)	1.00345e9	+	8.69491e8i	0.158739	+	0.137548i	0.730609	−	0.682796i	\(-0.239236\pi\)
							−0.571870	+	0.820344i	\(0.693782\pi\)
\(47\)	1.35796e10			1.25979			0.629897	−	0.776678i	\(-0.283097\pi\)
							0.629897	+	0.776678i	\(0.283097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.6	✓	230
23.14	odd	22	inner	23.13.d.a.14.6	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.6	✓	230	1.1	even	1	trivial
23.13.d.a.14.6	yes	230	23.14	odd	22	inner