Embedded newform 156.2.p.b.35.20

• Label: 156.2.p.b.35.20
• Level: $156$
• Weight: $2$
• Character: 156.35
• Analytic conductor: $1.246$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	156.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.24566627153\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			35.20
Character	\(\chi\)	\(=\)	156.35
Dual form			156.2.p.b.107.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40167 + 0.187933i) q^{2} +(-1.64588 - 0.539506i) q^{3} +(1.92936 + 0.526840i) q^{4} -1.98084i q^{5} +(-2.20560 - 1.06553i) q^{6} +(2.42516 - 1.40017i) q^{7} +(2.60532 + 1.10105i) q^{8} +(2.41787 + 1.77593i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\)	\(53\)	\(79\)	\(145\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40167	+	0.187933i	0.991131	+	0.132889i
							\(3\)	−1.64588	−	0.539506i	−0.950251	−	0.311484i
\(5\)		−	1.98084i		−	0.885860i								−0.896556	−	0.442930i	\(-0.853939\pi\)
							0.896556	−	0.442930i	\(-0.146061\pi\)
\(7\)	2.42516	−	1.40017i	0.916625	−	0.529214i	0.0340681	−	0.999420i	\(-0.489154\pi\)
							0.882557	+	0.470206i	\(0.155820\pi\)
\(11\)	−2.01615	+	3.49208i	−0.607893	+	1.05290i	0.383694	+	0.923460i	\(0.374652\pi\)
							−0.991587	+	0.129441i	\(0.958682\pi\)
\(13\)	0.235350	−	3.59786i	0.0652744	−	0.997867i
							\(17\)	−6.77232	+	3.91000i	−1.64253	+	0.948314i	−0.662598	+	0.748975i	\(0.730546\pi\)
−0.979931	+	0.199339i	\(0.936120\pi\)
\(19\)	−3.36616	+	1.94345i	−0.772250	+	0.445858i	−0.833677	−	0.552253i	\(-0.813768\pi\)
							0.0614269	+	0.998112i	\(0.480435\pi\)
\(23\)	0.939991	−	1.62811i	0.196002	−	0.339485i	−0.751227	−	0.660044i	\(-0.770538\pi\)
							0.947228	+	0.320559i	\(0.103871\pi\)
\(29\)	−1.74420	−	1.00701i	−0.323889	−	0.186998i	0.329236	−	0.944248i	\(-0.393209\pi\)
							−0.653125	+	0.757250i	\(0.726542\pi\)
\(31\)			4.09815i			0.736050i	0.929816	+	0.368025i	\(0.119966\pi\)
							−0.929816	+	0.368025i	\(0.880034\pi\)
\(37\)	−2.61536	+	4.52993i	−0.429962	+	0.744716i	−0.996869	−	0.0790655i	\(-0.974806\pi\)
							0.566907	+	0.823782i	\(0.308140\pi\)
\(41\)	3.61265	+	2.08577i	0.564202	+	0.325742i	0.754830	−	0.655920i	\(-0.227719\pi\)
							−0.190629	+	0.981662i	\(0.561053\pi\)
\(43\)	2.06494	−	1.19219i	0.314900	−	0.181808i	−0.334217	−	0.942496i	\(-0.608472\pi\)
							0.649117	+	0.760688i	\(0.275138\pi\)
\(47\)	2.72193			0.397035			0.198518	−	0.980097i	\(-0.436387\pi\)
							0.198518	+	0.980097i	\(0.436387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.2.p.b.35.20	yes	40
3.2	odd	2	inner	156.2.p.b.35.1	✓	40
4.3	odd	2	inner	156.2.p.b.35.14	yes	40
12.11	even	2	inner	156.2.p.b.35.7	yes	40
13.3	even	3	inner	156.2.p.b.107.7	yes	40
39.29	odd	6	inner	156.2.p.b.107.14	yes	40
52.3	odd	6	inner	156.2.p.b.107.1	yes	40
156.107	even	6	inner	156.2.p.b.107.20	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.p.b.35.1	✓	40	3.2	odd	2	inner
156.2.p.b.35.7	yes	40	12.11	even	2	inner
156.2.p.b.35.14	yes	40	4.3	odd	2	inner
156.2.p.b.35.20	yes	40	1.1	even	1	trivial
156.2.p.b.107.1	yes	40	52.3	odd	6	inner
156.2.p.b.107.7	yes	40	13.3	even	3	inner
156.2.p.b.107.14	yes	40	39.29	odd	6	inner
156.2.p.b.107.20	yes	40	156.107	even	6	inner