Embedded newform 23.13.d.a.5.18

• Label: 23.13.d.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(68.8275 + 20.2096i) q^{2} +(-392.302 + 452.741i) q^{3} +(883.025 + 567.486i) q^{4} +(18893.6 + 2716.49i) q^{5} +(-36150.9 + 23232.8i) q^{6} +(-145477. + 66436.9i) q^{7} +(-143103. - 165150. i) q^{8} +(24558.6 + 170809. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	68.8275	+	20.2096i	1.07543	+	0.315775i	0.771049	−	0.636776i	\(-0.219732\pi\)
							0.304381	+	0.952550i	\(0.401550\pi\)
\(3\)	−392.302	+	452.741i	−0.538138	+	0.621044i	−0.958078	−	0.286508i	\(-0.907506\pi\)
							0.419940	+	0.907552i	\(0.362051\pi\)
\(5\)	18893.6	+	2716.49i	1.20919	+	0.173855i	0.717276	−	0.696789i	\(-0.245389\pi\)
							0.491914	+	0.870644i	\(0.336298\pi\)
\(7\)	−145477.	+	66436.9i	−1.23653	+	0.564705i	−0.922973	−	0.384864i	\(-0.874248\pi\)
							−0.313558	+	0.949569i	\(0.601521\pi\)
\(11\)	99534.7	+	338984.i	0.0561848	+	0.191348i	0.982803	−	0.184655i	\(-0.0591166\pi\)
							−0.926619	+	0.376002i	\(0.877298\pi\)
\(13\)	−765585.	+	1.67640e6i	−0.158611	+	0.347310i	−0.972208	−	0.234120i	\(-0.924779\pi\)
							0.813597	+	0.581430i	\(0.197506\pi\)
\(17\)	−2.15260e7	−	3.34950e7i	−0.891803	−	1.38767i	−0.921610	−	0.388117i	\(-0.873126\pi\)
							0.0298070	−	0.999556i	\(-0.490511\pi\)
\(19\)	−4.05101e7	+	6.30349e7i	−0.861076	+	1.33986i	0.0782850	+	0.996931i	\(0.475056\pi\)
							−0.939361	+	0.342930i	\(0.888581\pi\)
\(23\)	−1.47616e8	−	1.11433e7i	−0.997163	−	0.0752745i
							\(29\)	2.21854e8	−	1.42577e8i	0.372975	−	0.239696i	−0.340702	−	0.940171i	\(-0.610665\pi\)
0.713677	+	0.700475i	\(0.247028\pi\)
\(31\)	5.95828e8	+	6.87622e8i	0.671353	+	0.774783i	0.984587	−	0.174893i	\(-0.0559580\pi\)
							−0.313234	+	0.949676i	\(0.601413\pi\)
\(37\)	−4.87320e8	+	7.00661e7i	−0.189935	+	0.0273085i	−0.236626	−	0.971601i	\(-0.576041\pi\)
							0.0466910	+	0.998909i	\(0.485132\pi\)
\(41\)	−7.91816e8	+	5.50720e9i	−0.166694	+	1.15939i	0.718964	+	0.695048i	\(0.244617\pi\)
							−0.885658	+	0.464338i	\(0.846292\pi\)
\(43\)	6.39354e9	+	5.54003e9i	1.01142	+	0.876399i	0.992355	−	0.123420i	\(-0.0393861\pi\)
							0.0190633	+	0.999818i	\(0.493932\pi\)
\(47\)	1.70152e10			1.57852			0.789262	−	0.614057i	\(-0.210463\pi\)
							0.789262	+	0.614057i	\(0.210463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.18	✓	230
23.14	odd	22	inner	23.13.d.a.14.18	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.18	✓	230	1.1	even	1	trivial
23.13.d.a.14.18	yes	230	23.14	odd	22	inner