Embedded newform 23.6.c.a.2.3

• Label: 23.6.c.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	23.2
Dual form			23.6.c.a.12.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.01311 - 3.22173i) q^{2} +(-0.293619 - 2.04217i) q^{3} +(1.45843 + 3.19351i) q^{4} +(-10.1657 - 2.98491i) q^{5} +(-5.10736 + 11.1836i) q^{6} +(-159.381 + 183.936i) q^{7} +(-24.1608 + 168.042i) q^{8} +(229.073 - 67.2618i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.01311	−	3.22173i	−0.886200	−	0.569526i	0.0164674	−	0.999864i	\(-0.494758\pi\)
							−0.902668	+	0.430338i	\(0.858394\pi\)
\(3\)	−0.293619	−	2.04217i	−0.0188357	−	0.131005i	0.978234	−	0.207505i	\(-0.0665344\pi\)
							−0.997070	+	0.0765004i	\(0.975625\pi\)
\(5\)	−10.1657	−	2.98491i	−0.181849	−	0.0533956i	0.189540	−	0.981873i	\(-0.439300\pi\)
							−0.371389	+	0.928477i	\(0.621118\pi\)
\(7\)	−159.381	+	183.936i	−1.22940	+	1.41880i	−0.354102	+	0.935207i	\(0.615213\pi\)
							−0.875294	+	0.483591i	\(0.839332\pi\)
\(11\)	−420.722	+	270.382i	−1.04837	+	0.673746i	−0.947043	−	0.321108i	\(-0.895945\pi\)
							−0.101326	+	0.994853i	\(0.532308\pi\)
\(13\)	−579.397	−	668.660i	−0.950863	−	1.09735i	−0.995153	−	0.0983360i	\(-0.968648\pi\)
							0.0442900	−	0.999019i	\(-0.485897\pi\)
\(17\)	176.061	−	385.520i	0.147755	−	0.323538i	−0.821255	−	0.570562i	\(-0.806725\pi\)
							0.969009	+	0.247024i	\(0.0794527\pi\)
\(19\)	113.803	+	249.195i	0.0723222	+	0.158363i	0.942340	−	0.334656i	\(-0.108620\pi\)
							−0.870018	+	0.493020i	\(0.835893\pi\)
\(23\)	−1956.44	+	1615.14i	−0.771166	+	0.636634i
							\(29\)	−2184.46	+	4783.31i	−0.482336	+	1.05617i	0.499479	+	0.866326i	\(0.333525\pi\)
−0.981815	+	0.189842i	\(0.939202\pi\)
\(31\)	1034.80	−	7197.17i	0.193397	−	1.34511i	−0.629536	−	0.776971i	\(-0.716755\pi\)
							0.822934	−	0.568138i	\(-0.192336\pi\)
\(37\)	−4647.39	+	1364.60i	−0.558091	+	0.163870i	−0.548600	−	0.836085i	\(-0.684839\pi\)
							−0.00949038	+	0.999955i	\(0.503021\pi\)
\(41\)	10180.5	+	2989.26i	0.945821	+	0.277718i	0.718046	−	0.695996i	\(-0.245037\pi\)
							0.227775	+	0.973714i	\(0.426855\pi\)
\(43\)	775.827	+	5396.00i	0.0639873	+	0.445042i	0.996478	+	0.0838488i	\(0.0267213\pi\)
							−0.932491	+	0.361193i	\(0.882370\pi\)
\(47\)	4745.80			0.313376			0.156688	−	0.987648i	\(-0.449918\pi\)
							0.156688	+	0.987648i	\(0.449918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.6.c.a.2.3	✓	90
23.9	even	11		529.6.a.j.1.12		45
23.12	even	11	inner	23.6.c.a.12.3	yes	90
23.14	odd	22		529.6.a.k.1.12		45

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.c.a.2.3	✓	90	1.1	even	1	trivial
23.6.c.a.12.3	yes	90	23.12	even	11	inner
529.6.a.j.1.12		45	23.9	even	11
529.6.a.k.1.12		45	23.14	odd	22