Embedded newform 138.4.e.d.49.2

• Label: 138.4.e.d.49.2
• 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			49.2
Character	\(\chi\)	\(=\)	138.49
Dual form			138.4.e.d.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 + 1.97964i) q^{2} +(-1.24625 + 2.72890i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(0.957858 - 1.10543i) q^{5} +(-5.75696 - 1.69040i) q^{6} +(-12.8395 + 8.25145i) q^{7} +(-3.32332 - 7.27706i) q^{8} +(-5.89375 - 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	0.957858	−	1.10543i	0.0856734	−	0.0988724i								−0.711294	−	0.702894i	\(-0.751891\pi\)
							0.796968	+	0.604022i	\(0.206436\pi\)
\(7\)	−12.8395	+	8.25145i	−0.693268	+	0.445536i	−0.839246	−	0.543751i	\(-0.817003\pi\)
							0.145978	+	0.989288i	\(0.453367\pi\)
\(11\)	−2.48650	+	17.2940i	−0.0681554	+	0.474031i	0.926948	+	0.375190i	\(0.122423\pi\)
							−0.995103	+	0.0988410i	\(0.968486\pi\)
\(13\)	−67.9126	−	43.6448i	−1.44889	−	0.931145i	−0.999281	−	0.0379188i	\(-0.987927\pi\)
							−0.449608	−	0.893226i	\(-0.648436\pi\)
\(17\)	−41.0010	−	12.0390i	−0.584953	−	0.171758i	−0.0241519	−	0.999708i	\(-0.507689\pi\)
							−0.560801	+	0.827951i	\(0.689507\pi\)
\(19\)	69.0629	−	20.2787i	0.833901	−	0.244855i	0.163209	−	0.986592i	\(-0.447816\pi\)
							0.670692	+	0.741736i	\(0.265997\pi\)
\(23\)	−31.9031	−	105.590i	−0.289228	−	0.957260i
							\(29\)	−214.796	−	63.0699i	−1.37540	−	0.403855i	−0.491237	−	0.871026i	\(-0.663455\pi\)
−0.884167	+	0.467171i	\(0.845273\pi\)
\(31\)	−24.1601	−	52.9032i	−0.139977	−	0.306506i	0.826641	−	0.562730i	\(-0.190249\pi\)
							−0.966617	+	0.256224i	\(0.917522\pi\)
\(37\)	210.737	+	243.204i	0.936350	+	1.08061i	0.996597	+	0.0824224i	\(0.0262657\pi\)
							−0.0602472	+	0.998183i	\(0.519189\pi\)
\(41\)	−161.922	+	186.867i	−0.616778	+	0.711800i	−0.975092	−	0.221801i	\(-0.928806\pi\)
							0.358314	+	0.933601i	\(0.383352\pi\)
\(43\)	−87.7865	+	192.226i	−0.311333	+	0.681724i	−0.999019	−	0.0442843i	\(-0.985899\pi\)
							0.687686	+	0.726008i	\(0.258627\pi\)
\(47\)	−438.362			−1.36046			−0.680231	−	0.732998i	\(-0.738121\pi\)
							−0.680231	+	0.732998i	\(0.738121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.d.49.2	yes	30
23.8	even	11	inner	138.4.e.d.31.2	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.d.31.2	✓	30	23.8	even	11	inner
138.4.e.d.49.2	yes	30	1.1	even	1	trivial