Embedded newform 23.10.c.a.2.12

• Label: 23.10.c.a.2.12
• Level: $23$
• Weight: $10$
• Character: 23.2
• Analytic conductor: $11.846$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8458242318\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	23.2
Dual form			23.10.c.a.12.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.2974 + 9.83105i) q^{2} +(-1.23691 - 8.60291i) q^{3} +(-75.3311 - 164.952i) q^{4} +(1465.21 + 430.225i) q^{5} +(65.6541 - 143.762i) q^{6} +(-3094.73 + 3571.51i) q^{7} +(1794.27 - 12479.4i) q^{8} +(18813.2 - 5524.06i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q - 43 q^{2} - 157 q^{3} - 4619 q^{4} + 2265 q^{5} + 1532 q^{6} + 8605 q^{7} - 28412 q^{8} - 84756 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.2974	+	9.83105i	0.676057	+	0.434475i	0.833104	−	0.553116i	\(-0.186561\pi\)
							−0.157048	+	0.987591i	\(0.550198\pi\)
\(3\)	−1.23691	−	8.60291i	−0.00881643	−	0.0613196i	0.984938	−	0.172906i	\(-0.0553158\pi\)
							−0.993755	+	0.111587i	\(0.964407\pi\)
\(5\)	1465.21	+	430.225i	1.04842	+	0.307844i	0.760179	−	0.649714i	\(-0.225111\pi\)
							0.288241	+	0.957558i	\(0.406929\pi\)
\(7\)	−3094.73	+	3571.51i	−0.487171	+	0.562225i	−0.945107	−	0.326760i	\(-0.894043\pi\)
							0.457937	+	0.888985i	\(0.348589\pi\)
\(11\)	54789.3	−	35210.9i	1.12831	−	0.725121i	0.163104	−	0.986609i	\(-0.447849\pi\)
							0.965207	+	0.261488i	\(0.0842131\pi\)
\(13\)	65861.1	+	76007.8i	0.639564	+	0.738096i	0.979298	−	0.202426i	\(-0.0648825\pi\)
							−0.339734	+	0.940522i	\(0.610337\pi\)
\(17\)	−2126.35	+	4656.06i	−0.00617469	+	0.0135207i	−0.912694	−	0.408643i	\(-0.866002\pi\)
							0.906520	+	0.422164i	\(0.138729\pi\)
\(19\)	342134.	+	749169.i	0.602290	+	1.31883i	0.927725	+	0.373266i	\(0.121762\pi\)
							−0.325435	+	0.945564i	\(0.605511\pi\)
\(23\)	−344998.	−	1.29697e6i	−0.257064	−	0.966394i
							\(29\)	−476232.	+	1.04280e6i	−0.125034	+	0.273786i	−0.961789	−	0.273791i	\(-0.911722\pi\)
0.836755	+	0.547577i	\(0.184450\pi\)
\(31\)	1.23534e6	−	8.59200e6i	0.240248	−	1.67096i	−0.410647	−	0.911794i	\(-0.634697\pi\)
							0.650895	−	0.759168i	\(-0.274394\pi\)
\(37\)	−2.06373e7	+	6.05965e6i	−1.81028	+	0.531545i	−0.998616	−	0.0525866i	\(-0.983253\pi\)
							−0.811659	+	0.584131i	\(0.801435\pi\)
\(41\)	−2.52936e7	−	7.42687e6i	−1.39792	−	0.410467i	−0.505952	−	0.862562i	\(-0.668859\pi\)
							−0.891970	+	0.452094i	\(0.850677\pi\)
\(43\)	2.67347e6	+	1.85944e7i	0.119252	+	0.829418i	0.958382	+	0.285487i	\(0.0921555\pi\)
							−0.839130	+	0.543931i	\(0.816935\pi\)
\(47\)	5.25717e7			1.57149			0.785745	−	0.618551i	\(-0.212280\pi\)
							0.785745	+	0.618551i	\(0.212280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.10.c.a.2.12	✓	170
23.12	even	11	inner	23.10.c.a.12.12	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.10.c.a.2.12	✓	170	1.1	even	1	trivial
23.10.c.a.12.12	yes	170	23.12	even	11	inner