Embedded newform 23.8.c.a.2.12

• Label: 23.8.c.a.2.12
• Level: $23$
• Weight: $8$
• Character: 23.2
• Analytic conductor: $7.185$
• Analytic rank: $0$
• Dimension: $130$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	23.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18485558613\)
Analytic rank:	\(0\)
Dimension:	\(130\)
Relative dimension:	\(13\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	23.2
Dual form			23.8.c.a.12.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.7906 + 8.86268i) q^{2} +(-11.8811 - 82.6349i) q^{3} +(58.4604 + 128.010i) q^{4} +(-324.875 - 95.3920i) q^{5} +(568.519 - 1244.88i) q^{6} +(482.425 - 556.748i) q^{7} +(-29.6934 + 206.522i) q^{8} +(-4588.96 + 1347.44i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 130 q + 5 q^{2} + 17 q^{3} - 907 q^{4} - 399 q^{5} + 1196 q^{6} - 301 q^{7} + 2764 q^{8} - 11694 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.7906	+	8.86268i	1.21893	+	0.783358i	0.982130	−	0.188201i	\(-0.0602657\pi\)
							0.236798	+	0.971559i	\(0.423902\pi\)
\(3\)	−11.8811	−	82.6349i	−0.254058	−	1.76701i	−0.573308	−	0.819340i	\(-0.694340\pi\)
							0.319250	−	0.947671i	\(-0.396569\pi\)
\(5\)	−324.875	−	95.3920i	−1.16231	−	0.341285i	−0.356980	−	0.934112i	\(-0.616194\pi\)
							−0.805330	+	0.592827i	\(0.798012\pi\)
\(7\)	482.425	−	556.748i	0.531602	−	0.613502i	−0.424895	−	0.905243i	\(-0.639689\pi\)
							0.956497	+	0.291741i	\(0.0942345\pi\)
\(11\)	2700.79	−	1735.69i	0.611810	−	0.393187i	−0.197725	−	0.980257i	\(-0.563355\pi\)
							0.809536	+	0.587071i	\(0.199719\pi\)
\(13\)	5942.71	+	6858.25i	0.750209	+	0.865788i	0.994589	−	0.103892i	\(-0.0331296\pi\)
							−0.244379	+	0.969680i	\(0.578584\pi\)
\(17\)	8999.77	−	19706.7i	0.444283	−	0.972845i	−0.546509	−	0.837453i	\(-0.684044\pi\)
							0.990792	−	0.135391i	\(-0.0432292\pi\)
\(19\)	17238.7	+	37747.4i	0.576589	+	1.26255i	0.943214	+	0.332185i	\(0.107786\pi\)
							−0.366625	+	0.930369i	\(0.619487\pi\)
\(23\)	−56626.9	+	14079.2i	−0.970454	+	0.241285i
							\(29\)	77365.7	−	169407.i	0.589054	−	1.28985i	−0.346958	−	0.937881i	\(-0.612785\pi\)
0.936012	−	0.351968i	\(-0.114488\pi\)
\(31\)	3838.12	−	26694.7i	0.0231394	−	0.160938i	−0.974975	−	0.222313i	\(-0.928639\pi\)
							0.998115	+	0.0613746i	\(0.0195484\pi\)
\(37\)	−321727.	+	94467.6i	−1.04419	+	0.306603i	−0.758470	−	0.651708i	\(-0.774053\pi\)
							−0.285725	+	0.958312i	\(0.592234\pi\)
\(41\)	271136.	+	79612.8i	0.614390	+	0.180401i	0.574098	−	0.818787i	\(-0.305353\pi\)
							0.0402923	+	0.999188i	\(0.487171\pi\)
\(43\)	−33504.5	−	233029.i	−0.0642634	−	0.446962i	−0.996395	−	0.0848410i	\(-0.972962\pi\)
							0.932131	−	0.362121i	\(-0.117947\pi\)
\(47\)	1.13207e6			1.59049			0.795243	−	0.606291i	\(-0.207343\pi\)
							0.795243	+	0.606291i	\(0.207343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.c.a.2.12	✓	130
23.9	even	11		529.8.a.k.1.55		65
23.12	even	11	inner	23.8.c.a.12.12	yes	130
23.14	odd	22		529.8.a.j.1.55		65

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.c.a.2.12	✓	130	1.1	even	1	trivial
23.8.c.a.12.12	yes	130	23.12	even	11	inner
529.8.a.j.1.55		65	23.14	odd	22
529.8.a.k.1.55		65	23.9	even	11