Embedded newform 23.6.c.a.2.2

• Label: 23.6.c.a.2.2
• Level: $23$
• Weight: $6$
• Character: 23.2
• Analytic conductor: $3.689$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.68882785570\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(9\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	23.2
Dual form			23.6.c.a.12.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.98228 - 3.84458i) q^{2} +(-4.25176 - 29.5716i) q^{3} +(7.71364 + 16.8905i) q^{4} +(10.2972 + 3.02354i) q^{5} +(-88.2553 + 193.252i) q^{6} +(60.9133 - 70.2977i) q^{7} +(-13.5930 + 94.5412i) q^{8} +(-623.247 + 183.002i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.98228	−	3.84458i	−1.05753	−	0.679632i	−0.108267	−	0.994122i	\(-0.534530\pi\)
							−0.949261	+	0.314490i	\(0.898167\pi\)
\(3\)	−4.25176	−	29.5716i	−0.272750	−	1.89702i	−0.419349	−	0.907825i	\(-0.637742\pi\)
							0.146599	−	0.989196i	\(-0.453167\pi\)
\(5\)	10.2972	+	3.02354i	0.184202	+	0.0540867i	0.372533	−	0.928019i	\(-0.378489\pi\)
							−0.188330	+	0.982106i	\(0.560308\pi\)
\(7\)	60.9133	−	70.2977i	0.469858	−	0.542245i	−0.470514	−	0.882393i	\(-0.655931\pi\)
							0.940372	+	0.340147i	\(0.110477\pi\)
\(11\)	392.176	−	252.036i	0.977236	−	0.628031i	0.0485193	−	0.998822i	\(-0.484550\pi\)
							0.928716	+	0.370791i	\(0.120913\pi\)
\(13\)	136.496	+	157.525i	0.224007	+	0.258517i	0.856617	−	0.515953i	\(-0.172562\pi\)
							−0.632610	+	0.774470i	\(0.718017\pi\)
\(17\)	−270.936	+	593.267i	−0.227376	+	0.497884i	−0.988593	−	0.150614i	\(-0.951875\pi\)
							0.761217	+	0.648497i	\(0.224602\pi\)
\(19\)	−1104.40	−	2418.30i	−0.701848	−	1.53683i	−0.837716	−	0.546106i	\(-0.816110\pi\)
							0.135868	−	0.990727i	\(-0.456618\pi\)
\(23\)	−1274.25	+	2193.77i	−0.502266	+	0.864713i
							\(29\)	−461.937	+	1011.50i	−0.101997	+	0.223343i	−0.953750	−	0.300601i	\(-0.902813\pi\)
0.851753	+	0.523944i	\(0.175540\pi\)
\(31\)	461.014	−	3206.42i	0.0861608	−	0.599262i	−0.900301	−	0.435269i	\(-0.856653\pi\)
							0.986461	−	0.163993i	\(-0.0524375\pi\)
\(37\)	13767.3	−	4042.44i	1.65327	−	0.485444i	0.683599	−	0.729858i	\(-0.260414\pi\)
							0.969671	+	0.244414i	\(0.0785955\pi\)
\(41\)	−6591.31	−	1935.38i	−0.612368	−	0.179807i	−0.0391808	−	0.999232i	\(-0.512475\pi\)
							−0.573187	+	0.819425i	\(0.694293\pi\)
\(43\)	−236.153	−	1642.48i	−0.0194770	−	0.135466i	0.977763	−	0.209714i	\(-0.0672533\pi\)
							−0.997240	+	0.0742484i	\(0.976344\pi\)
\(47\)	12556.7			0.829144			0.414572	−	0.910017i	\(-0.363931\pi\)
							0.414572	+	0.910017i	\(0.363931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.6.c.a.2.2	✓	90
23.9	even	11		529.6.a.j.1.9		45
23.12	even	11	inner	23.6.c.a.12.2	yes	90
23.14	odd	22		529.6.a.k.1.9		45

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.c.a.2.2	✓	90	1.1	even	1	trivial
23.6.c.a.12.2	yes	90	23.12	even	11	inner
529.6.a.j.1.9		45	23.9	even	11
529.6.a.k.1.9		45	23.14	odd	22