Embedded newform 138.4.e.c.55.1

• Label: 138.4.e.c.55.1
• 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			55.1
Character	\(\chi\)	\(=\)	138.55
Dual form			138.4.e.c.133.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 - 0.563465i) q^{2} +(-1.96458 - 2.26725i) q^{3} +(3.36501 - 2.16256i) q^{4} +(-1.52360 - 10.5969i) q^{5} +(-5.04752 - 3.24384i) q^{6} +(0.567261 - 1.24213i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	−1.52360	−	10.5969i	−0.136275	−	0.947814i								−0.937136	−	0.348964i	\(-0.886534\pi\)
							0.800861	−	0.598850i	\(-0.204376\pi\)
\(7\)	0.567261	−	1.24213i	0.0306292	−	0.0670686i	−0.893699	−	0.448667i	\(-0.851899\pi\)
							0.924328	+	0.381598i	\(0.124626\pi\)
\(11\)	−29.7982	−	8.74954i	−0.816772	−	0.239826i	−0.153446	−	0.988157i	\(-0.549037\pi\)
							−0.663325	+	0.748331i	\(0.730855\pi\)
\(13\)	−8.97721	−	19.6574i	−0.191525	−	0.419382i	0.789370	−	0.613918i	\(-0.210407\pi\)
							−0.980895	+	0.194535i	\(0.937680\pi\)
\(17\)	−87.5978	−	56.2957i	−1.24974	−	0.803159i	−0.262895	−	0.964825i	\(-0.584677\pi\)
							−0.986846	+	0.161665i	\(0.948314\pi\)
\(19\)	11.8007	−	7.58384i	0.142488	−	0.0915712i	−0.467455	−	0.884017i	\(-0.654829\pi\)
							0.609942	+	0.792446i	\(0.291193\pi\)
\(23\)	101.089	−	44.1353i	0.916461	−	0.400124i
							\(29\)	24.5005	+	15.7455i	0.156884	+	0.100823i	0.616727	−	0.787177i	\(-0.288458\pi\)
−0.459844	+	0.888000i	\(0.652095\pi\)
\(31\)	44.7053	−	51.5926i	0.259010	−	0.298913i	−0.611319	−	0.791384i	\(-0.709361\pi\)
							0.870329	+	0.492471i	\(0.163906\pi\)
\(37\)	−0.861851	+	5.99430i	−0.00382939	+	0.0266340i	−0.991647	−	0.128985i	\(-0.958828\pi\)
							0.987817	+	0.155619i	\(0.0497372\pi\)
\(41\)	14.3350	+	99.7024i	0.0546038	+	0.379778i	0.998738	+	0.0502175i	\(0.0159915\pi\)
							−0.944134	+	0.329560i	\(0.893099\pi\)
\(43\)	186.175	+	214.858i	0.660267	+	0.761989i	0.982821	−	0.184562i	\(-0.0590867\pi\)
							−0.322554	+	0.946551i	\(0.604541\pi\)
\(47\)	276.855			0.859223			0.429611	−	0.903014i	\(-0.358651\pi\)
							0.429611	+	0.903014i	\(0.358651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.55.1	✓	30
23.18	even	11	inner	138.4.e.c.133.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.55.1	✓	30	1.1	even	1	trivial
138.4.e.c.133.1	yes	30	23.18	even	11	inner