Embedded newform 138.4.e.d.49.1

• Label: 138.4.e.d.49.1
• Level: $138$
• Weight: $4$
• Character: 138.49
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			49.1
Character	\(\chi\)	\(=\)	138.49
Dual form			138.4.e.d.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 + 1.97964i) q^{2} +(-1.24625 + 2.72890i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(-10.3477 + 11.9419i) q^{5} +(-5.75696 - 1.69040i) q^{6} +(-5.39451 + 3.46684i) q^{7} +(-3.32332 - 7.27706i) q^{8} +(-5.89375 - 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.284630	+	1.97964i	0.100632	+	0.699909i
							\(3\)	−1.24625	+	2.72890i	−0.239840	+	0.525176i
\(5\)	−10.3477	+	11.9419i	−0.925530	+	1.06812i								0.0719672	+	0.997407i	\(0.477072\pi\)
							−0.997497	+	0.0707111i	\(0.977473\pi\)
\(7\)	−5.39451	+	3.46684i	−0.291276	+	0.187192i	−0.678116	−	0.734955i	\(-0.737203\pi\)
							0.386840	+	0.922147i	\(0.373567\pi\)
\(11\)	8.99569	−	62.5664i	0.246573	−	1.71495i	−0.371163	−	0.928568i	\(-0.621041\pi\)
							0.617736	−	0.786385i	\(-0.288050\pi\)
\(13\)	55.2529	+	35.5089i	1.17880	+	0.757569i	0.975165	−	0.221478i	\(-0.0710881\pi\)
							0.203635	+	0.979047i	\(0.434724\pi\)
\(17\)	−42.2447	−	12.4042i	−0.602696	−	0.176968i	−0.0338713	−	0.999426i	\(-0.510784\pi\)
							−0.568825	+	0.822459i	\(0.692602\pi\)
\(19\)	−138.679	+	40.7199i	−1.67448	+	0.491672i	−0.974856	−	0.222837i	\(-0.928468\pi\)
							−0.699626	+	0.714509i	\(0.746650\pi\)
\(23\)	−57.4848	−	94.1408i	−0.521148	−	0.853466i
							\(29\)	−25.2751	−	7.42143i	−0.161843	−	0.0475215i	0.199807	−	0.979835i	\(-0.435968\pi\)
−0.361651	+	0.932314i	\(0.617787\pi\)
\(31\)	102.365	+	224.147i	0.593072	+	1.29865i	0.933568	+	0.358401i	\(0.116678\pi\)
							−0.340495	+	0.940246i	\(0.610595\pi\)
\(37\)	−251.990	−	290.812i	−1.11965	−	1.29214i	−0.951934	−	0.306303i	\(-0.900908\pi\)
							−0.167711	−	0.985836i	\(-0.553638\pi\)
\(41\)	−85.0860	+	98.1944i	−0.324102	+	0.374034i	−0.894296	−	0.447476i	\(-0.852323\pi\)
							0.570193	+	0.821511i	\(0.306868\pi\)
\(43\)	−38.6934	+	84.7267i	−0.137225	+	0.300481i	−0.965751	−	0.259469i	\(-0.916453\pi\)
							0.828526	+	0.559950i	\(0.189180\pi\)
\(47\)	73.1846			0.227129			0.113565	−	0.993531i	\(-0.463773\pi\)
							0.113565	+	0.993531i	\(0.463773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.d.49.1	yes	30
23.8	even	11	inner	138.4.e.d.31.1	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.d.31.1	✓	30	23.8	even	11	inner
138.4.e.d.49.1	yes	30	1.1	even	1	trivial