Embedded newform 23.13.d.a.5.17

• Label: 23.13.d.a.5.17
• 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			5.17
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(53.6967 + 15.7668i) q^{2} +(228.961 - 264.235i) q^{3} +(-811.028 - 521.216i) q^{4} +(16209.3 + 2330.55i) q^{5} +(16460.6 - 10578.6i) q^{6} +(57943.8 - 26462.0i) q^{7} +(-185443. - 214013. i) q^{8} +(58234.9 + 405033. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	53.6967	+	15.7668i	0.839011	+	0.246356i	0.672884	−	0.739748i	\(-0.265055\pi\)
							0.166128	+	0.986104i	\(0.446874\pi\)
\(3\)	228.961	−	264.235i	0.314075	−	0.362462i	−0.576660	−	0.816984i	\(-0.695644\pi\)
							0.890736	+	0.454522i	\(0.150190\pi\)
\(5\)	16209.3	+	2330.55i	1.03740	+	0.149155i	0.639908	−	0.768452i	\(-0.278972\pi\)
							0.397489	+	0.917607i	\(0.369882\pi\)
\(7\)	57943.8	−	26462.0i	0.492514	−	0.224924i	−0.153645	−	0.988126i	\(-0.549101\pi\)
							0.646159	+	0.763202i	\(0.276374\pi\)
\(11\)	−24498.5	−	83434.2i	−0.0138288	−	0.0470964i	0.952290	−	0.305196i	\(-0.0987219\pi\)
							−0.966118	+	0.258099i	\(0.916904\pi\)
\(13\)	3.64213e6	−	7.97515e6i	0.754562	−	1.65226i	−0.00342660	−	0.999994i	\(-0.501091\pi\)
							0.757989	−	0.652267i	\(-0.226182\pi\)
\(17\)	2.53864e6	+	3.95021e6i	0.105174	+	0.163654i	0.889851	−	0.456250i	\(-0.150808\pi\)
							−0.784678	+	0.619904i	\(0.787171\pi\)
\(19\)	3.56830e7	−	5.55238e7i	0.758472	−	1.18021i	−0.220336	−	0.975424i	\(-0.570715\pi\)
							0.978808	−	0.204781i	\(-0.0656483\pi\)
\(23\)	1.00077e8	+	1.09084e8i	0.676031	+	0.736873i
							\(29\)	3.19404e8	−	2.05268e8i	0.536973	−	0.345091i	−0.243881	−	0.969805i	\(-0.578421\pi\)
0.780854	+	0.624714i	\(0.214784\pi\)
\(31\)	−6.06598e8	−	7.00052e8i	−0.683488	−	0.788787i	0.302935	−	0.953011i	\(-0.402034\pi\)
							−0.986423	+	0.164224i	\(0.947488\pi\)
\(37\)	2.18919e9	−	3.14757e8i	0.853242	−	0.122678i	0.298203	−	0.954502i	\(-0.403613\pi\)
							0.555038	+	0.831825i	\(0.312704\pi\)
\(41\)	−3.51662e7	+	2.44586e8i	−0.00740325	+	0.0514907i	−0.993188	−	0.116521i	\(-0.962826\pi\)
							0.985785	+	0.168012i	\(0.0537348\pi\)
\(43\)	6.19502e9	+	5.36802e9i	0.980014	+	0.849187i	0.988568	−	0.150777i	\(-0.0481774\pi\)
							−0.00855402	+	0.999963i	\(0.502723\pi\)
\(47\)	−2.12253e10			−1.96910			−0.984548	−	0.175115i	\(-0.943970\pi\)
							−0.984548	+	0.175115i	\(0.943970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.17	✓	230
23.14	odd	22	inner	23.13.d.a.14.17	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.17	✓	230	1.1	even	1	trivial
23.13.d.a.14.17	yes	230	23.14	odd	22	inner