Embedded newform 23.15.d.a.5.20

• Label: 23.15.d.a.5.20
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(133.250 + 39.1257i) q^{2} +(2339.92 - 2700.41i) q^{3} +(2441.62 + 1569.14i) q^{4} +(9176.41 + 1319.37i) q^{5} +(417450. - 268279. i) q^{6} +(-1.39751e6 + 638220. i) q^{7} +(-1.22607e6 - 1.41496e6i) q^{8} +(-1.13631e6 - 7.90321e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	133.250	+	39.1257i	1.04102	+	0.305670i	0.757182	−	0.653204i	\(-0.226576\pi\)
							0.283833	+	0.958874i	\(0.408394\pi\)
\(3\)	2339.92	−	2700.41i	1.06992	−	1.23476i	0.0990693	−	0.995081i	\(-0.468413\pi\)
							0.970853	−	0.239676i	\(-0.0770411\pi\)
\(5\)	9176.41	+	1319.37i	0.117458	+	0.0168879i	0.200793	−	0.979634i	\(-0.435648\pi\)
							−0.0833351	+	0.996522i	\(0.526557\pi\)
\(7\)	−1.39751e6	+	638220.i	−1.69694	+	0.774969i	−0.698721	+	0.715395i	\(0.746247\pi\)
							−0.998224	+	0.0595739i	\(0.981026\pi\)
\(11\)	−123425.	−	420348.i	−0.00633367	−	0.0215705i	0.956266	−	0.292500i	\(-0.0944872\pi\)
							−0.962599	+	0.270930i	\(0.912669\pi\)
\(13\)	3.97939e7	−	8.71365e7i	0.634181	−	1.38866i	−0.270561	−	0.962703i	\(-0.587209\pi\)
							0.904742	−	0.425960i	\(-0.140064\pi\)
\(17\)	−2.25387e8	−	3.50709e8i	−0.549270	−	0.854682i	0.449994	−	0.893032i	\(-0.351426\pi\)
							−0.999264	+	0.0383501i	\(0.987790\pi\)
\(19\)	−7.67139e8	+	1.19369e9i	−0.858221	+	1.33542i	0.0826241	+	0.996581i	\(0.473670\pi\)
							−0.940845	+	0.338837i	\(0.889966\pi\)
\(23\)	3.05109e9	−	1.51119e9i	0.896107	−	0.443837i
							\(29\)	4.15639e9	−	2.67115e9i	0.240952	−	0.154850i	−0.414592	−	0.910007i	\(-0.636076\pi\)
0.655544	+	0.755157i	\(0.272439\pi\)
\(31\)	1.15309e10	+	1.33074e10i	0.419115	+	0.483684i	0.925567	−	0.378584i	\(-0.123589\pi\)
							−0.506452	+	0.862268i	\(0.669043\pi\)
\(37\)	8.06289e10	−	1.15927e10i	0.849334	−	0.122116i	0.296116	−	0.955152i	\(-0.404309\pi\)
							0.553218	+	0.833036i	\(0.313400\pi\)
\(41\)	2.68976e10	−	1.87077e11i	0.138110	−	0.960578i	−0.796433	−	0.604727i	\(-0.793282\pi\)
							0.934543	−	0.355851i	\(-0.115809\pi\)
\(43\)	1.31668e11	+	1.14091e11i	0.484398	+	0.419733i	0.862520	−	0.506023i	\(-0.168885\pi\)
							−0.378123	+	0.925756i	\(0.623430\pi\)
\(47\)	−7.93021e11			−1.56531			−0.782654	−	0.622457i	\(-0.786135\pi\)
							−0.782654	+	0.622457i	\(0.786135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.20	✓	270
23.14	odd	22	inner	23.15.d.a.14.20	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.20	✓	270	1.1	even	1	trivial
23.15.d.a.14.20	yes	270	23.14	odd	22	inner