Embedded newform 23.13.d.a.5.13

• Label: 23.13.d.a.5.13
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.13
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.53737 + 2.50680i) q^{2} +(-912.924 + 1053.57i) q^{3} +(-3379.17 - 2171.66i) q^{4} +(21701.0 + 3120.14i) q^{5} +(-10435.0 + 6706.20i) q^{6} +(94155.6 - 42999.4i) q^{7} +(-47271.9 - 54554.7i) q^{8} +(-200948. - 1.39763e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.53737	+	2.50680i	0.133396	+	0.0391687i	0.347749	−	0.937588i	\(-0.386946\pi\)
							−0.214353	+	0.976756i	\(0.568764\pi\)
\(3\)	−912.924	+	1053.57i	−1.25230	+	1.44523i	−0.404816	+	0.914398i	\(0.632665\pi\)
							−0.847479	+	0.530828i	\(0.821881\pi\)
\(5\)	21701.0	+	3120.14i	1.38887	+	0.199689i	0.795846	−	0.605499i	\(-0.207027\pi\)
							0.593020	+	0.805188i	\(0.297936\pi\)
\(7\)	94155.6	−	42999.4i	0.800309	−	0.365489i	0.0270943	−	0.999633i	\(-0.491375\pi\)
							0.773215	+	0.634144i	\(0.218647\pi\)
\(11\)	313169.	+	1.06656e6i	0.176776	+	0.602043i	0.999438	+	0.0335130i	\(0.0106695\pi\)
							−0.822663	+	0.568530i	\(0.807512\pi\)
\(13\)	1.58717e6	−	3.47542e6i	0.328824	−	0.720025i	−0.670945	−	0.741507i	\(-0.734111\pi\)
							0.999769	+	0.0214823i	\(0.00683856\pi\)
\(17\)	1.51364e7	+	2.35527e7i	0.627090	+	0.975771i	0.998874	+	0.0474513i	\(0.0151099\pi\)
							−0.371784	+	0.928319i	\(0.621254\pi\)
\(19\)	−7.75690e6	+	1.20700e7i	−0.164879	+	0.256557i	−0.913855	−	0.406040i	\(-0.866909\pi\)
							0.748976	+	0.662597i	\(0.230546\pi\)
\(23\)	5.57309e7	−	1.37145e8i	0.376469	−	0.926429i
							\(29\)	−7.16835e8	+	4.60682e8i	−1.20512	+	0.774486i	−0.979835	−	0.199807i	\(-0.935969\pi\)
−0.225288	+	0.974292i	\(0.572332\pi\)
\(31\)	1.18766e8	+	1.37063e8i	0.133820	+	0.154437i	0.818704	−	0.574215i	\(-0.194693\pi\)
							−0.684884	+	0.728652i	\(0.740147\pi\)
\(37\)	7.93126e8	−	1.14034e8i	0.309123	−	0.0444452i	0.0139925	−	0.999902i	\(-0.495546\pi\)
							0.295131	+	0.955457i	\(0.404637\pi\)
\(41\)	1.69123e8	−	1.17628e9i	0.0356041	−	0.247632i	−0.964245	−	0.265013i	\(-0.914624\pi\)
							0.999849	+	0.0173812i	\(0.00553288\pi\)
\(43\)	5.57776e9	+	4.83316e9i	0.882366	+	0.764575i	0.972878	−	0.231321i	\(-0.0743046\pi\)
							−0.0905112	+	0.995895i	\(0.528850\pi\)
\(47\)	1.82980e10			1.69753			0.848764	−	0.528771i	\(-0.177347\pi\)
							0.848764	+	0.528771i	\(0.177347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.13	✓	230
23.14	odd	22	inner	23.13.d.a.14.13	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.13	✓	230	1.1	even	1	trivial
23.13.d.a.14.13	yes	230	23.14	odd	22	inner