Embedded newform 23.13.d.a.5.19

• Label: 23.13.d.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(85.4052 + 25.0772i) q^{2} +(424.703 - 490.133i) q^{3} +(3219.40 + 2068.98i) q^{4} +(-25609.1 - 3682.03i) q^{5} +(48563.0 - 31209.5i) q^{6} +(-11513.2 + 5257.91i) q^{7} +(-15685.2 - 18101.7i) q^{8} +(15774.0 + 109710. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	85.4052	+	25.0772i	1.33446	+	0.391832i	0.869688	−	0.493601i	\(-0.164320\pi\)
							0.464767	+	0.885433i	\(0.346138\pi\)
\(3\)	424.703	−	490.133i	0.582582	−	0.672336i	−0.385575	−	0.922676i	\(-0.625997\pi\)
							0.968158	+	0.250340i	\(0.0805425\pi\)
\(5\)	−25609.1	−	3682.03i	−1.63898	−	0.235650i	−0.739674	−	0.672965i	\(-0.765020\pi\)
							−0.899308	+	0.437315i	\(0.855929\pi\)
\(7\)	−11513.2	+	5257.91i	−0.0978608	+	0.0446915i	−0.463744	−	0.885969i	\(-0.653494\pi\)
							0.365883	+	0.930661i	\(0.380767\pi\)
\(11\)	−911115.	−	3.10297e6i	−0.514301	−	1.75155i	−0.649129	−	0.760679i	\(-0.724866\pi\)
							0.134828	−	0.990869i	\(-0.456952\pi\)
\(13\)	865093.	−	1.89429e6i	0.179227	−	0.392452i	−0.798602	−	0.601860i	\(-0.794427\pi\)
							0.977828	+	0.209408i	\(0.0671538\pi\)
\(17\)	−4.51503e6	−	7.02553e6i	−0.187054	−	0.291062i	0.735043	−	0.678021i	\(-0.237162\pi\)
							−0.922097	+	0.386959i	\(0.873526\pi\)
\(19\)	−2.61059e7	+	4.06215e7i	−0.554902	+	0.863445i	−0.999478	−	0.0322964i	\(-0.989718\pi\)
							0.444576	+	0.895741i	\(0.353354\pi\)
\(23\)	−1.11382e8	+	9.75120e7i	−0.752401	+	0.658705i
							\(29\)	4.71356e8	−	3.02922e8i	0.792430	−	0.509264i	−0.0807078	−	0.996738i	\(-0.525718\pi\)
0.873137	+	0.487474i	\(0.162082\pi\)
\(31\)	−6.38388e8	−	7.36739e8i	−0.719307	−	0.830125i	0.271917	−	0.962321i	\(-0.412343\pi\)
							−0.991224	+	0.132196i	\(0.957797\pi\)
\(37\)	4.86909e8	−	7.00070e7i	0.189774	−	0.0272854i	−0.0467722	−	0.998906i	\(-0.514893\pi\)
							0.236547	+	0.971620i	\(0.423984\pi\)
\(41\)	5.58351e8	−	3.88341e9i	0.117545	−	0.817543i	−0.842700	−	0.538384i	\(-0.819035\pi\)
							0.960245	−	0.279159i	\(-0.0900558\pi\)
\(43\)	5.19315e9	+	4.49989e9i	0.821523	+	0.711854i	0.960451	−	0.278449i	\(-0.0898202\pi\)
							−0.138928	+	0.990302i	\(0.544366\pi\)
\(47\)	1.42886e10			1.32557			0.662785	−	0.748810i	\(-0.269374\pi\)
							0.662785	+	0.748810i	\(0.269374\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.19	✓	230
23.14	odd	22	inner	23.13.d.a.14.19	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.19	✓	230	1.1	even	1	trivial
23.13.d.a.14.19	yes	230	23.14	odd	22	inner