Embedded newform 138.4.e.d.55.3

• Label: 138.4.e.d.55.3
• Level: $138$
• Weight: $4$
• Character: 138.55
• 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			55.3
Character	\(\chi\)	\(=\)	138.55
Dual form			138.4.e.d.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 - 0.563465i) q^{2} +(1.96458 + 2.26725i) q^{3} +(3.36501 - 2.16256i) q^{4} +(1.91350 + 13.3087i) q^{5} +(5.04752 + 3.24384i) q^{6} +(-11.4577 + 25.0888i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.91899	−	0.563465i	0.678464	−	0.199215i
							\(3\)	1.96458	+	2.26725i	0.378084	+	0.436332i
\(5\)	1.91350	+	13.3087i	0.171149	+	1.19037i								0.876463	+	0.481470i	\(0.159897\pi\)
							−0.705314	+	0.708895i	\(0.749194\pi\)
\(7\)	−11.4577	+	25.0888i	−0.618656	+	1.35467i	0.297837	+	0.954617i	\(0.403735\pi\)
							−0.916493	+	0.400051i	\(0.868992\pi\)
\(11\)	−49.7764	−	14.6157i	−1.36438	−	0.400617i	−0.484073	−	0.875028i	\(-0.660843\pi\)
							−0.880304	+	0.474411i	\(0.842661\pi\)
\(13\)	−8.02479	−	17.5718i	−0.171206	−	0.374888i	0.804507	−	0.593944i	\(-0.202430\pi\)
							−0.975712	+	0.219055i	\(0.929703\pi\)
\(17\)	99.0825	+	63.6764i	1.41359	+	0.908459i	0.999998	−	0.00200350i	\(-0.000637735\pi\)
							0.413592	+	0.910462i	\(0.364274\pi\)
\(19\)	75.4651	−	48.4985i	0.911204	−	0.585595i	0.00111095	−	0.999999i	\(-0.499646\pi\)
							0.910093	+	0.414404i	\(0.136010\pi\)
\(23\)	60.7031	−	92.0985i	0.550325	−	0.834951i
							\(29\)	188.487	+	121.133i	1.20693	+	0.775649i	0.980142	−	0.198294i	\(-0.0635402\pi\)
0.226791	+	0.973943i	\(0.427177\pi\)
\(31\)	−42.6897	+	49.2665i	−0.247332	+	0.285436i	−0.865818	−	0.500360i	\(-0.833201\pi\)
							0.618486	+	0.785796i	\(0.287747\pi\)
\(37\)	−7.10735	+	49.4327i	−0.0315795	+	0.219640i	−0.999500	−	0.0316160i	\(-0.989935\pi\)
							0.967921	+	0.251256i	\(0.0808437\pi\)
\(41\)	−64.9397	−	451.665i	−0.247363	−	1.72045i	−0.613337	−	0.789821i	\(-0.710173\pi\)
							0.365975	−	0.930625i	\(-0.380736\pi\)
\(43\)	269.811	+	311.378i	0.956877	+	1.10430i	0.994472	+	0.105002i	\(0.0334850\pi\)
							−0.0375945	+	0.999293i	\(0.511970\pi\)
\(47\)	347.548			1.07862			0.539310	−	0.842107i	\(-0.318685\pi\)
							0.539310	+	0.842107i	\(0.318685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.d.55.3	✓	30
23.18	even	11	inner	138.4.e.d.133.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.d.55.3	✓	30	1.1	even	1	trivial
138.4.e.d.133.3	yes	30	23.18	even	11	inner