Embedded newform 156.2.p.b.35.17

• Label: 156.2.p.b.35.17
• 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			35.17
Character	\(\chi\)	\(=\)	156.35
Dual form			156.2.p.b.107.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32222 - 0.501741i) q^{2} +(0.680023 + 1.59297i) q^{3} +(1.49651 - 1.32682i) q^{4} -3.37008i q^{5} +(1.69840 + 1.76506i) q^{6} +(-1.77448 + 1.02450i) q^{7} +(1.31299 - 2.50520i) q^{8} +(-2.07514 + 2.16652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32222	−	0.501741i	0.934948	−	0.354784i
							\(3\)	0.680023	+	1.59297i	0.392611	+	0.919704i
\(5\)		−	3.37008i		−	1.50714i								−0.657365	−	0.753572i	\(-0.728329\pi\)
							0.657365	−	0.753572i	\(-0.271671\pi\)
\(7\)	−1.77448	+	1.02450i	−0.670692	+	0.387224i	−0.796339	−	0.604851i	\(-0.793233\pi\)
							0.125647	+	0.992075i	\(0.459899\pi\)
\(11\)	−2.01452	+	3.48926i	−0.607401	+	1.05205i	0.384266	+	0.923223i	\(0.374455\pi\)
							−0.991667	+	0.128827i	\(0.958879\pi\)
\(13\)	−0.274038	+	3.59512i	−0.0760046	+	0.997107i
							\(17\)	0.0707999	−	0.0408763i	0.0171715	−	0.00991397i	−0.491390	−	0.870940i	\(-0.663511\pi\)
0.508561	+	0.861026i	\(0.330178\pi\)
\(19\)	0.648289	−	0.374290i	0.148728	−	0.0858679i	−0.423789	−	0.905761i	\(-0.639300\pi\)
							0.572517	+	0.819893i	\(0.305967\pi\)
\(23\)	−0.176734	+	0.306113i	−0.0368516	+	0.0638289i	−0.883863	−	0.467746i	\(-0.845066\pi\)
							0.847011	+	0.531575i	\(0.178400\pi\)
\(29\)	−0.448189	−	0.258762i	−0.0832266	−	0.0480509i	0.457809	−	0.889050i	\(-0.348634\pi\)
							−0.541036	+	0.841000i	\(0.681968\pi\)
\(31\)		−	8.36065i		−	1.50162i	−0.660520	−	0.750809i	\(-0.729664\pi\)
							0.660520	−	0.750809i	\(-0.270336\pi\)
\(37\)	5.25436	−	9.10082i	0.863811	−	1.49616i	−0.00441155	−	0.999990i	\(-0.501404\pi\)
							0.868223	−	0.496175i	\(-0.165262\pi\)
\(41\)	5.18732	+	2.99490i	0.810123	+	0.467725i	0.846999	−	0.531595i	\(-0.178407\pi\)
							−0.0368754	+	0.999320i	\(0.511740\pi\)
\(43\)	−4.33986	+	2.50562i	−0.661822	+	0.382103i	−0.792971	−	0.609260i	\(-0.791467\pi\)
							0.131149	+	0.991363i	\(0.458133\pi\)
\(47\)	7.77475			1.13406			0.567032	−	0.823696i	\(-0.308092\pi\)
							0.567032	+	0.823696i	\(0.308092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.2.p.b.35.17	yes	40
3.2	odd	2	inner	156.2.p.b.35.4	✓	40
4.3	odd	2	inner	156.2.p.b.35.11	yes	40
12.11	even	2	inner	156.2.p.b.35.10	yes	40
13.3	even	3	inner	156.2.p.b.107.10	yes	40
39.29	odd	6	inner	156.2.p.b.107.11	yes	40
52.3	odd	6	inner	156.2.p.b.107.4	yes	40
156.107	even	6	inner	156.2.p.b.107.17	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.p.b.35.4	✓	40	3.2	odd	2	inner
156.2.p.b.35.10	yes	40	12.11	even	2	inner
156.2.p.b.35.11	yes	40	4.3	odd	2	inner
156.2.p.b.35.17	yes	40	1.1	even	1	trivial
156.2.p.b.107.4	yes	40	52.3	odd	6	inner
156.2.p.b.107.10	yes	40	13.3	even	3	inner
156.2.p.b.107.11	yes	40	39.29	odd	6	inner
156.2.p.b.107.17	yes	40	156.107	even	6	inner