Embedded newform 23.11.d.a.5.6

• Label: 23.11.d.a.5.6
• Level: $23$
• Weight: $11$
• Character: 23.5
• Analytic conductor: $14.613$
• Analytic rank: $0$
• Dimension: $190$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.6132168115\)
Analytic rank:	\(0\)
Dimension:	\(190\)
Relative dimension:	\(19\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	23.5
Dual form			23.11.d.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-25.2998 - 7.42869i) q^{2} +(-190.721 + 220.104i) q^{3} +(-276.549 - 177.727i) q^{4} +(1790.89 + 257.491i) q^{5} +(6460.29 - 4151.78i) q^{6} +(15874.6 - 7249.67i) q^{7} +(23358.1 + 26956.6i) q^{8} +(-3667.63 - 25508.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 190 q + 13 q^{2} - 71 q^{3} - 11699 q^{4} - 11 q^{5} + 3346 q^{6} - 11 q^{7} - 50582 q^{8} - 394850 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−25.2998	−	7.42869i	−0.790619	−	0.232147i	−0.138599	−	0.990349i	\(-0.544260\pi\)
							−0.652020	+	0.758202i	\(0.726078\pi\)
\(3\)	−190.721	+	220.104i	−0.784860	+	0.905777i	−0.997450	−	0.0713644i	\(-0.977265\pi\)
							0.212590	+	0.977141i	\(0.431810\pi\)
\(5\)	1790.89	+	257.491i	0.573085	+	0.0823971i	0.422763	−	0.906240i	\(-0.361060\pi\)
							0.150321	+	0.988637i	\(0.451969\pi\)
\(7\)	15874.6	−	7249.67i	0.944521	−	0.431348i	0.117219	−	0.993106i	\(-0.462602\pi\)
							0.827302	+	0.561758i	\(0.189875\pi\)
\(11\)	−24354.6	−	82944.2i	−0.151223	−	0.515018i	0.848681	−	0.528906i	\(-0.177397\pi\)
							−0.999904	+	0.0138877i	\(0.995579\pi\)
\(13\)	−46217.7	+	101203.i	−0.124478	+	0.272568i	−0.961604	−	0.274442i	\(-0.911507\pi\)
							0.837126	+	0.547010i	\(0.184234\pi\)
\(17\)	−87537.8	−	136212.i	−0.0616526	−	0.0959333i	0.809059	−	0.587728i	\(-0.199977\pi\)
							−0.870711	+	0.491795i	\(0.836341\pi\)
\(19\)	−1.90857e6	+	2.96980e6i	−0.770799	+	1.19939i	0.204580	+	0.978850i	\(0.434417\pi\)
							−0.975379	+	0.220536i	\(0.929219\pi\)
\(23\)	4.42117e6	+	4.67758e6i	0.686908	+	0.726745i
							\(29\)	−4.86881e6	+	3.12899e6i	−0.237374	+	0.152551i	−0.653920	−	0.756564i	\(-0.726877\pi\)
0.416546	+	0.909114i	\(0.363240\pi\)
\(31\)	625953.	+	722388.i	0.0218642	+	0.0252326i	0.766576	−	0.642153i	\(-0.221959\pi\)
							−0.744712	+	0.667386i	\(0.767413\pi\)
\(37\)	2.61667e7	−	3.76220e6i	0.377347	−	0.0542543i	0.0489678	−	0.998800i	\(-0.484407\pi\)
							0.328379	+	0.944546i	\(0.393498\pi\)
\(41\)	−1.48963e7	+	1.03606e8i	−0.128575	+	0.894262i	0.818787	+	0.574098i	\(0.194647\pi\)
							−0.947362	+	0.320164i	\(0.896262\pi\)
\(43\)	−1.87850e7	−	1.62773e7i	−0.127782	−	0.110724i	0.588625	−	0.808406i	\(-0.299669\pi\)
							−0.716407	+	0.697682i	\(0.754215\pi\)
\(47\)	−3.11239e8			−1.35708			−0.678538	−	0.734565i	\(-0.737386\pi\)
							−0.678538	+	0.734565i	\(0.737386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.11.d.a.5.6	✓	190
23.14	odd	22	inner	23.11.d.a.14.6	yes	190

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.11.d.a.5.6	✓	190	1.1	even	1	trivial
23.11.d.a.14.6	yes	190	23.14	odd	22	inner