Embedded newform 138.5.h.a.7.4

• Label: 138.5.h.a.7.4
• Level: $138$
• Weight: $5$
• Character: 138.7
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85223 + 2.13758i) q^{2} +(-4.37128 + 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(33.6623 - 15.3731i) q^{5} +(2.09159 - 14.5473i) q^{6} +(-9.29808 - 31.6664i) q^{7} +(19.0354 + 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85223	+	2.13758i	−0.463056	+	0.534396i
							\(3\)	−4.37128	+	2.80925i	−0.485698	+	0.312139i
\(5\)	33.6623	−	15.3731i	1.34649	−	0.614922i								0.393895	−	0.919156i	\(-0.371128\pi\)
							0.952598	+	0.304233i	\(0.0984002\pi\)
\(7\)	−9.29808	−	31.6664i	−0.189757	−	0.646252i	−0.998325	−	0.0578501i	\(-0.981575\pi\)
							0.808569	−	0.588402i	\(-0.200243\pi\)
\(11\)	−98.7920	+	85.6038i	−0.816463	+	0.707469i	−0.959334	−	0.282272i	\(-0.908912\pi\)
							0.142871	+	0.989741i	\(0.454367\pi\)
\(13\)	−187.016	−	54.9128i	−1.10660	−	0.324928i	−0.323130	−	0.946355i	\(-0.604735\pi\)
							−0.783472	+	0.621427i	\(0.786553\pi\)
\(17\)	−18.1183	−	2.60501i	−0.0626930	−	0.00901389i	0.110897	−	0.993832i	\(-0.464628\pi\)
							−0.173590	+	0.984818i	\(0.555537\pi\)
\(19\)	−152.085	+	21.8665i	−0.421287	+	0.0605719i	−0.349698	−	0.936862i	\(-0.613716\pi\)
							−0.0715884	+	0.997434i	\(0.522807\pi\)
\(23\)	−216.666	−	482.594i	−0.409576	−	0.912276i
							\(29\)	63.5801	−	442.209i	0.0756005	−	0.525813i	−0.916467	−	0.400110i	\(-0.868972\pi\)
0.992068	−	0.125704i	\(-0.0401189\pi\)
\(31\)	−692.787	−	445.227i	−0.720903	−	0.463296i	0.128048	−	0.991768i	\(-0.459129\pi\)
							−0.848951	+	0.528472i	\(0.822765\pi\)
\(37\)	−361.109	−	164.913i	−0.263776	−	0.120462i	0.279138	−	0.960251i	\(-0.409951\pi\)
							−0.542913	+	0.839789i	\(0.682679\pi\)
\(41\)	−1244.71	−	2725.53i	−0.740456	−	1.62137i	−0.782803	−	0.622269i	\(-0.786211\pi\)
							0.0423472	−	0.999103i	\(-0.486516\pi\)
\(43\)	1898.71	+	2954.45i	1.02689	+	1.59787i	0.776907	+	0.629615i	\(0.216787\pi\)
							0.249979	+	0.968251i	\(0.419576\pi\)
\(47\)	−3052.90			−1.38203			−0.691013	−	0.722842i	\(-0.742835\pi\)
							−0.691013	+	0.722842i	\(0.742835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.4	✓	160
23.10	odd	22	inner	138.5.h.a.79.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.4	✓	160	1.1	even	1	trivial
138.5.h.a.79.4	yes	160	23.10	odd	22	inner