Embedded newform 138.5.g.a.29.11

• Label: 138.5.g.a.29.11
• Level: $138$
• Weight: $5$
• Character: 138.29
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			29.11
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(5.31677 - 7.26168i) q^{3} +(5.23889 + 6.04600i) q^{4} +(10.0489 + 15.6363i) q^{5} +(-22.2114 + 12.4360i) q^{6} +(-0.294426 + 2.04778i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(-24.4639 - 77.2173i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57283	−	1.17497i	−0.643207	−	0.293743i
							\(3\)	5.31677	−	7.26168i	0.590752	−	0.806853i
\(5\)	10.0489	+	15.6363i	0.401954	+	0.625453i								0.981944	−	0.189170i	\(-0.0605798\pi\)
							−0.579990	+	0.814624i	\(0.696943\pi\)
\(7\)	−0.294426	+	2.04778i	−0.00600869	+	0.0417913i	−0.992605	−	0.121387i	\(-0.961266\pi\)
							0.986597	+	0.163179i	\(0.0521747\pi\)
\(11\)	153.690	−	70.1877i	1.27016	−	0.580063i	0.337673	−	0.941263i	\(-0.390360\pi\)
							0.932488	+	0.361200i	\(0.117633\pi\)
\(13\)	44.0061	+	306.069i	0.260391	+	1.81106i	0.529900	+	0.848060i	\(0.322230\pi\)
							−0.269509	+	0.962998i	\(0.586861\pi\)
\(17\)	211.766	+	183.497i	0.732756	+	0.634936i	0.939140	−	0.343535i	\(-0.111624\pi\)
							−0.206384	+	0.978471i	\(0.566170\pi\)
\(19\)	−150.412	−	173.584i	−0.416652	−	0.480843i	0.508162	−	0.861262i	\(-0.330325\pi\)
							−0.924814	+	0.380419i	\(0.875780\pi\)
\(23\)	272.667	−	453.314i	0.515438	−	0.856927i
							\(29\)	421.140	+	364.920i	0.500761	+	0.433912i	0.868258	−	0.496113i	\(-0.165240\pi\)
−0.367497	+	0.930025i	\(0.619785\pi\)
\(31\)	994.083	−	291.889i	1.03443	−	0.303735i	0.279916	−	0.960024i	\(-0.409693\pi\)
							0.754509	+	0.656290i	\(0.227875\pi\)
\(37\)	1623.34	+	1043.26i	1.18578	+	0.762056i	0.976441	−	0.215786i	\(-0.0692314\pi\)
							0.209342	+	0.977842i	\(0.432868\pi\)
\(41\)	−1072.07	−	1668.17i	−0.637755	−	0.992366i	−0.998223	−	0.0595943i	\(-0.981019\pi\)
							0.360468	−	0.932772i	\(-0.382617\pi\)
\(43\)	−2316.94	−	680.315i	−1.25308	−	0.367937i	−0.413164	−	0.910657i	\(-0.635576\pi\)
							−0.839914	+	0.542720i	\(0.817395\pi\)
\(47\)		−	3839.26i		−	1.73801i	−0.494804	−	0.869004i	\(-0.664760\pi\)
							0.494804	−	0.869004i	\(-0.335240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.11	✓	320
3.2	odd	2	inner	138.5.g.a.29.25	yes	320
23.4	even	11	inner	138.5.g.a.119.25	yes	320
69.50	odd	22	inner	138.5.g.a.119.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.11	✓	320	1.1	even	1	trivial
138.5.g.a.29.25	yes	320	3.2	odd	2	inner
138.5.g.a.119.11	yes	320	69.50	odd	22	inner
138.5.g.a.119.25	yes	320	23.4	even	11	inner