Embedded newform 23.13.d.a.5.4

• Label: 23.13.d.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-90.4098 - 26.5467i) q^{2} +(-761.907 + 879.288i) q^{3} +(4023.43 + 2585.70i) q^{4} +(13366.6 + 1921.82i) q^{5} +(92226.1 - 59270.1i) q^{6} +(-116992. + 53428.4i) q^{7} +(-42370.4 - 48898.1i) q^{8} +(-117012. - 813840. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−90.4098	−	26.5467i	−1.41265	−	0.414792i	−0.515643	−	0.856803i	\(-0.672447\pi\)
							−0.897010	+	0.442011i	\(0.854265\pi\)
\(3\)	−761.907	+	879.288i	−1.04514	+	1.20616i	−0.0670986	+	0.997746i	\(0.521374\pi\)
							−0.978042	+	0.208410i	\(0.933171\pi\)
\(5\)	13366.6	+	1921.82i	0.855460	+	0.122997i	0.556071	−	0.831135i	\(-0.312308\pi\)
							0.299389	+	0.954131i	\(0.403217\pi\)
\(7\)	−116992.	+	53428.4i	−0.994415	+	0.454134i	−0.845073	−	0.534651i	\(-0.820443\pi\)
							−0.149342	+	0.988786i	\(0.547716\pi\)
\(11\)	−829217.	−	2.82406e6i	−0.468072	−	1.59411i	−0.768202	−	0.640207i	\(-0.778848\pi\)
							0.300131	−	0.953898i	\(-0.402970\pi\)
\(13\)	−3.46756e6	+	7.59290e6i	−0.718396	+	1.57307i	0.0977427	+	0.995212i	\(0.468838\pi\)
							−0.816139	+	0.577856i	\(0.803890\pi\)
\(17\)	−1.27433e7	−	1.98289e7i	−0.527943	−	0.821495i	0.470191	−	0.882565i	\(-0.344185\pi\)
							−0.998133	+	0.0610698i	\(0.980549\pi\)
\(19\)	3.08804e7	−	4.80509e7i	0.656390	−	1.02136i	−0.340320	−	0.940310i	\(-0.610535\pi\)
							0.996710	−	0.0810528i	\(-0.0258282\pi\)
\(23\)	1.47266e8	−	1.50752e7i	0.994801	−	0.101835i
							\(29\)	−6.26985e7	+	4.02939e7i	−0.105407	+	0.0677409i	−0.592282	−	0.805731i	\(-0.701773\pi\)
0.486875	+	0.873471i	\(0.338137\pi\)
\(31\)	2.28277e8	+	2.63445e8i	0.257212	+	0.296838i	0.869638	−	0.493689i	\(-0.164352\pi\)
							−0.612427	+	0.790528i	\(0.709806\pi\)
\(37\)	−9.38952e8	+	1.35001e8i	−0.365959	+	0.0526170i	−0.322842	−	0.946453i	\(-0.604638\pi\)
							−0.0431177	+	0.999070i	\(0.513729\pi\)
\(41\)	−1.47060e8	+	1.02283e9i	−0.0309594	+	0.215327i	−0.999428	−	0.0338069i	\(-0.989237\pi\)
							0.968469	+	0.249134i	\(0.0801460\pi\)
\(43\)	6.30523e9	+	5.46351e9i	0.997448	+	0.864294i	0.990752	−	0.135684i	\(-0.0433231\pi\)
							0.00669602	+	0.999978i	\(0.497869\pi\)
\(47\)	−4.59974e9			−0.426723			−0.213362	−	0.976973i	\(-0.568441\pi\)
							−0.213362	+	0.976973i	\(0.568441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.4	✓	230
23.14	odd	22	inner	23.13.d.a.14.4	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.4	✓	230	1.1	even	1	trivial
23.13.d.a.14.4	yes	230	23.14	odd	22	inner