Embedded newform 23.4.c.a.16.4

• Label: 23.4.c.a.16.4
• Level: $23$
• Weight: $4$
• Character: 23.16
• Analytic conductor: $1.357$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.35704393013\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	23.16
Dual form			23.4.c.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61249 - 1.86092i) q^{2} +(1.97417 - 1.26872i) q^{3} +(0.275643 + 1.91714i) q^{4} +(-3.07518 - 6.73370i) q^{5} +(0.822349 - 5.71956i) q^{6} +(-11.6919 + 3.43306i) q^{7} +(20.5838 + 13.2284i) q^{8} +(-8.92852 + 19.5507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61249	−	1.86092i	0.570103	−	0.657934i	−0.395344	−	0.918533i	\(-0.629375\pi\)
							0.965447	+	0.260599i	\(0.0839202\pi\)
\(3\)	1.97417	−	1.26872i	0.379928	−	0.244165i	−0.336713	−	0.941607i	\(-0.609315\pi\)
							0.716641	+	0.697442i	\(0.245679\pi\)
\(5\)	−3.07518	−	6.73370i	−0.275052	−	0.602281i	0.720812	−	0.693130i	\(-0.243769\pi\)
							−0.995865	+	0.0908496i	\(0.971042\pi\)
\(7\)	−11.6919	+	3.43306i	−0.631305	+	0.185368i	−0.581703	−	0.813401i	\(-0.697613\pi\)
							−0.0496020	+	0.998769i	\(0.515795\pi\)
\(11\)	3.34406	+	3.85925i	0.0916610	+	0.105782i	0.799727	−	0.600364i	\(-0.204978\pi\)
							−0.708066	+	0.706146i	\(0.750432\pi\)
\(13\)	−23.2908	−	6.83879i	−0.496900	−	0.145903i	0.0236737	−	0.999720i	\(-0.492464\pi\)
							−0.520574	+	0.853817i	\(0.674282\pi\)
\(17\)	9.15745	−	63.6914i	0.130647	−	0.908673i	−0.814065	−	0.580774i	\(-0.802750\pi\)
							0.944713	−	0.327900i	\(-0.106341\pi\)
\(19\)	−7.05564	−	49.0731i	−0.0851935	−	0.592534i	−0.987040	−	0.160476i	\(-0.948697\pi\)
							0.901846	−	0.432057i	\(-0.142212\pi\)
\(23\)	72.4803	−	83.1481i	0.657095	−	0.753808i
							\(29\)	−24.5481	+	170.736i	−0.157189	+	1.09327i	0.746594	+	0.665280i	\(0.231688\pi\)
−0.903783	+	0.427992i	\(0.859221\pi\)
\(31\)	−195.491	−	125.634i	−1.13262	−	0.727891i	−0.166515	−	0.986039i	\(-0.553251\pi\)
							−0.966105	+	0.258148i	\(0.916888\pi\)
\(37\)	−98.8932	+	216.546i	−0.439404	+	0.962161i	0.552303	+	0.833643i	\(0.313749\pi\)
							−0.991707	+	0.128517i	\(0.958978\pi\)
\(41\)	129.476	+	283.513i	0.493190	+	1.07994i	0.978623	+	0.205662i	\(0.0659346\pi\)
							−0.485433	+	0.874274i	\(0.661338\pi\)
\(43\)	442.252	−	284.218i	1.56844	−	1.00797i	0.588540	−	0.808468i	\(-0.299703\pi\)
							0.979897	−	0.199505i	\(-0.0639336\pi\)
\(47\)	407.467			1.26458			0.632289	−	0.774733i	\(-0.282116\pi\)
							0.632289	+	0.774733i	\(0.282116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.c.a.16.4	yes	50
3.2	odd	2		207.4.i.a.154.2		50
23.6	even	11		529.4.a.n.1.8		25
23.13	even	11	inner	23.4.c.a.13.4	✓	50
23.17	odd	22		529.4.a.m.1.8		25
69.59	odd	22		207.4.i.a.82.2		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.c.a.13.4	✓	50	23.13	even	11	inner
23.4.c.a.16.4	yes	50	1.1	even	1	trivial
207.4.i.a.82.2		50	69.59	odd	22
207.4.i.a.154.2		50	3.2	odd	2
529.4.a.m.1.8		25	23.17	odd	22
529.4.a.n.1.8		25	23.6	even	11