Embedded newform 138.4.e.b.133.2

• Label: 138.4.e.b.133.2
• Level: $138$
• Weight: $4$
• Character: 138.133
• 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			133.2
Character	\(\chi\)	\(=\)	138.133
Dual form			138.4.e.b.55.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91899 - 0.563465i) q^{2} +(1.96458 - 2.26725i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.966187 + 6.71998i) q^{5} +(-5.04752 + 3.24384i) q^{6} +(5.61851 + 12.3028i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(-1.28083 - 8.90839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{6}{11}\right)\)

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
\(5\)	−0.966187	+	6.71998i	−0.0864184	+	0.601053i								0.899887	+	0.436123i	\(0.143649\pi\)
							−0.986305	+	0.164930i	\(0.947260\pi\)
\(7\)	5.61851	+	12.3028i	0.303371	+	0.664290i	0.998509	−	0.0545866i	\(-0.0173841\pi\)
							−0.695138	+	0.718876i	\(0.744657\pi\)
\(11\)	−47.7922	+	14.0330i	−1.30999	+	0.384647i	−0.860870	−	0.508825i	\(-0.830080\pi\)
							−0.449119	+	0.893472i	\(0.648262\pi\)
\(13\)	−22.9350	+	50.2207i	−0.489310	+	1.07144i	0.490488	+	0.871448i	\(0.336819\pi\)
							−0.979798	+	0.199991i	\(0.935909\pi\)
\(17\)	18.3197	−	11.7734i	0.261364	−	0.167968i	−0.403396	−	0.915026i	\(-0.632170\pi\)
							0.664760	+	0.747057i	\(0.268534\pi\)
\(19\)	10.1734	+	6.53805i	0.122839	+	0.0789438i	0.600617	−	0.799537i	\(-0.294921\pi\)
							−0.477778	+	0.878480i	\(0.658558\pi\)
\(23\)	31.4795	+	105.717i	0.285389	+	0.958412i
							\(29\)	−219.180	+	140.858i	−1.40347	+	0.901955i	−0.999915	−	0.0130008i	\(-0.995862\pi\)
−0.403554	+	0.914956i	\(0.632225\pi\)
\(31\)	146.413	+	168.969i	0.848275	+	0.978961i	0.999955	−	0.00946854i	\(-0.00301397\pi\)
							−0.151681	+	0.988430i	\(0.548469\pi\)
\(37\)	41.3192	+	287.382i	0.183590	+	1.27690i	0.848187	+	0.529696i	\(0.177694\pi\)
							−0.664597	+	0.747202i	\(0.731397\pi\)
\(41\)	41.6198	−	289.472i	0.158535	−	1.10263i	−0.742801	−	0.669512i	\(-0.766503\pi\)
							0.901336	−	0.433121i	\(-0.142588\pi\)
\(43\)	107.078	−	123.574i	0.379749	−	0.438253i	−0.533411	−	0.845857i	\(-0.679090\pi\)
							0.913159	+	0.407603i	\(0.133635\pi\)
\(47\)	−90.8415			−0.281927			−0.140964	−	0.990015i	\(-0.545020\pi\)
							−0.140964	+	0.990015i	\(0.545020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.133.2	yes	30
23.9	even	11	inner	138.4.e.b.55.2	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.55.2	✓	30	23.9	even	11	inner
138.4.e.b.133.2	yes	30	1.1	even	1	trivial