Embedded newform 156.2.p.b.35.3

• Label: 156.2.p.b.35.3
• 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			35.3
Character	\(\chi\)	\(=\)	156.35
Dual form			156.2.p.b.107.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36641 - 0.364576i) q^{2} +(-1.10270 + 1.33569i) q^{3} +(1.73417 + 0.996323i) q^{4} +0.171572i q^{5} +(1.99370 - 1.42308i) q^{6} +(-2.51891 + 1.45429i) q^{7} +(-2.00636 - 1.99362i) q^{8} +(-0.568109 - 2.94572i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36641	−	0.364576i	−0.966200	−	0.257794i
							\(3\)	−1.10270	+	1.33569i	−0.636644	+	0.771158i
\(5\)			0.171572i			0.0767293i								0.999264	+	0.0383647i	\(0.0122149\pi\)
							−0.999264	+	0.0383647i	\(0.987785\pi\)
\(7\)	−2.51891	+	1.45429i	−0.952057	+	0.549670i	−0.893719	−	0.448627i	\(-0.851913\pi\)
							−0.0583377	+	0.998297i	\(0.518580\pi\)
\(11\)	−0.674360	+	1.16803i	−0.203327	+	0.352173i	−0.949598	−	0.313469i	\(-0.898509\pi\)
							0.746271	+	0.665642i	\(0.231842\pi\)
\(13\)	−3.29935	+	1.45406i	−0.915075	+	0.403283i
							\(17\)	−4.48929	+	2.59189i	−1.08881	+	0.628627i	−0.933260	−	0.359201i	\(-0.883049\pi\)
−0.155553	+	0.987828i	\(0.549716\pi\)
\(19\)	−4.90054	+	2.82933i	−1.12426	+	0.649092i	−0.942485	−	0.334248i	\(-0.891518\pi\)
							−0.181775	+	0.983340i	\(0.558184\pi\)
\(23\)	1.40931	−	2.44099i	0.293861	−	0.508983i	−0.680858	−	0.732415i	\(-0.738393\pi\)
							0.974719	+	0.223433i	\(0.0717263\pi\)
\(29\)	2.51772	+	1.45360i	0.467528	+	0.269927i	0.715204	−	0.698915i	\(-0.246334\pi\)
							−0.247676	+	0.968843i	\(0.579667\pi\)
\(31\)		−	3.62947i		−	0.651871i	−0.945392	−	0.325936i	\(-0.894321\pi\)
							0.945392	−	0.325936i	\(-0.105679\pi\)
\(37\)	−1.64578	+	2.85058i	−0.270565	+	0.468632i	−0.969007	−	0.247035i	\(-0.920544\pi\)
							0.698442	+	0.715667i	\(0.253877\pi\)
\(41\)	4.85479	+	2.80291i	0.758191	+	0.437742i	0.828646	−	0.559773i	\(-0.189112\pi\)
							−0.0704550	+	0.997515i	\(0.522445\pi\)
\(43\)	6.79514	−	3.92318i	1.03625	−	0.598279i	0.117481	−	0.993075i	\(-0.462518\pi\)
							0.918769	+	0.394796i	\(0.129185\pi\)
\(47\)	−7.05455			−1.02901			−0.514506	−	0.857487i	\(-0.672025\pi\)
							−0.514506	+	0.857487i	\(0.672025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.2.p.b.35.3	✓	40
3.2	odd	2	inner	156.2.p.b.35.18	yes	40
4.3	odd	2	inner	156.2.p.b.35.6	yes	40
12.11	even	2	inner	156.2.p.b.35.15	yes	40
13.3	even	3	inner	156.2.p.b.107.15	yes	40
39.29	odd	6	inner	156.2.p.b.107.6	yes	40
52.3	odd	6	inner	156.2.p.b.107.18	yes	40
156.107	even	6	inner	156.2.p.b.107.3	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.p.b.35.3	✓	40	1.1	even	1	trivial
156.2.p.b.35.6	yes	40	4.3	odd	2	inner
156.2.p.b.35.15	yes	40	12.11	even	2	inner
156.2.p.b.35.18	yes	40	3.2	odd	2	inner
156.2.p.b.107.3	yes	40	156.107	even	6	inner
156.2.p.b.107.6	yes	40	39.29	odd	6	inner
156.2.p.b.107.15	yes	40	13.3	even	3	inner
156.2.p.b.107.18	yes	40	52.3	odd	6	inner