Embedded newform 23.6.c.a.4.2

• Label: 23.6.c.a.4.2
• Level: $23$
• Weight: $6$
• Character: 23.4
• 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			4.2
Character	\(\chi\)	\(=\)	23.4
Dual form			23.6.c.a.6.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60966 - 5.71435i) q^{2} +(-0.942040 + 0.276608i) q^{3} +(-4.88796 + 5.64100i) q^{4} +(-66.5354 - 42.7597i) q^{5} +(4.03903 + 4.66129i) q^{6} +(20.2510 + 140.849i) q^{7} +(-147.892 - 43.4250i) q^{8} +(-203.614 + 130.855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60966	−	5.71435i	−0.461326	−	1.01016i	−0.987183	−	0.159591i	\(-0.948982\pi\)
							0.525857	−	0.850573i	\(-0.323745\pi\)
\(3\)	−0.942040	+	0.276608i	−0.0604319	+	0.0177444i	−0.311809	−	0.950145i	\(-0.600935\pi\)
							0.251377	+	0.967889i	\(0.419117\pi\)
\(5\)	−66.5354	−	42.7597i	−1.19022	−	0.764909i	−0.212983	−	0.977056i	\(-0.568318\pi\)
							−0.977238	+	0.212147i	\(0.931954\pi\)
\(7\)	20.2510	+	140.849i	0.156207	+	1.08644i	0.905543	+	0.424254i	\(0.139464\pi\)
							−0.749336	+	0.662190i	\(0.769627\pi\)
\(11\)	143.447	−	314.106i	0.357446	−	0.782698i	−0.642420	−	0.766353i	\(-0.722070\pi\)
							0.999866	−	0.0163454i	\(-0.00520313\pi\)
\(13\)	122.935	−	855.031i	0.201751	−	1.40321i	−0.597333	−	0.801993i	\(-0.703773\pi\)
							0.799084	−	0.601219i	\(-0.205318\pi\)
\(17\)	−904.232	−	1043.54i	−0.758853	−	0.875763i	0.236542	−	0.971621i	\(-0.423986\pi\)
							−0.995395	+	0.0958583i	\(0.969440\pi\)
\(19\)	1026.94	−	1185.15i	0.652620	−	0.753164i	−0.328933	−	0.944353i	\(-0.606689\pi\)
							0.981553	+	0.191189i	\(0.0612344\pi\)
\(23\)	1703.24	+	1880.25i	0.671360	+	0.741132i
							\(29\)	−3832.76	−	4423.24i	−0.846285	−	0.976665i	0.153649	−	0.988125i	\(-0.450897\pi\)
−0.999934	+	0.0114603i	\(0.996352\pi\)
\(31\)	−708.736	−	208.104i	−0.132459	−	0.0388934i	0.214831	−	0.976651i	\(-0.431080\pi\)
							−0.347290	+	0.937758i	\(0.612898\pi\)
\(37\)	−1468.32	+	943.634i	−0.176326	+	0.113318i	−0.625827	−	0.779962i	\(-0.715238\pi\)
							0.449501	+	0.893280i	\(0.351602\pi\)
\(41\)	12569.8	+	8078.13i	1.16780	+	0.750501i	0.973105	−	0.230362i	\(-0.0739911\pi\)
							0.194697	+	0.980863i	\(0.437628\pi\)
\(43\)	−12487.8	+	3666.74i	−1.02994	+	0.302419i	−0.752688	−	0.658377i	\(-0.771243\pi\)
							−0.277256	+	0.960796i	\(0.589425\pi\)
\(47\)	−1456.45			−0.0961725			−0.0480862	−	0.998843i	\(-0.515312\pi\)
							−0.0480862	+	0.998843i	\(0.515312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.6.c.a.4.2	✓	90
23.6	even	11	inner	23.6.c.a.6.2	yes	90
23.11	odd	22		529.6.a.k.1.11		45
23.12	even	11		529.6.a.j.1.11		45

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.c.a.4.2	✓	90	1.1	even	1	trivial
23.6.c.a.6.2	yes	90	23.6	even	11	inner
529.6.a.j.1.11		45	23.12	even	11
529.6.a.k.1.11		45	23.11	odd	22