Embedded newform 156.2.p.b.35.9

• Label: 156.2.p.b.35.9
• 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			35.9
Character	\(\chi\)	\(=\)	156.35
Dual form			156.2.p.b.107.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.268000 + 1.38859i) q^{2} +(0.316122 + 1.70296i) q^{3} +(-1.85635 - 0.744284i) q^{4} +1.45117i q^{5} +(-2.44943 - 0.0174310i) q^{6} +(-1.17936 + 0.680902i) q^{7} +(1.53101 - 2.37824i) q^{8} +(-2.80013 + 1.07668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.268000	+	1.38859i	−0.189505	+	0.981880i
							\(3\)	0.316122	+	1.70296i	0.182513	+	0.983203i
\(5\)			1.45117i			0.648983i								0.945889	+	0.324492i	\(0.105193\pi\)
							−0.945889	+	0.324492i	\(0.894807\pi\)
\(7\)	−1.17936	+	0.680902i	−0.445755	+	0.257357i	−0.706036	−	0.708176i	\(-0.749518\pi\)
							0.260281	+	0.965533i	\(0.416185\pi\)
\(11\)	0.711538	−	1.23242i	0.214537	−	0.371589i	−0.738592	−	0.674152i	\(-0.764509\pi\)
							0.953129	+	0.302564i	\(0.0978425\pi\)
\(13\)	3.27584	+	1.50627i	0.908555	+	0.417765i
							\(17\)	−1.68294	+	0.971645i	−0.408173	+	0.235659i	−0.690004	−	0.723805i	\(-0.742391\pi\)
0.281832	+	0.959464i	\(0.409058\pi\)
\(19\)	−2.01114	+	1.16113i	−0.461387	+	0.266382i	−0.712627	−	0.701543i	\(-0.752495\pi\)
							0.251241	+	0.967925i	\(0.419161\pi\)
\(23\)	1.96157	−	3.39753i	0.409015	−	0.708435i	−0.585765	−	0.810481i	\(-0.699206\pi\)
							0.994780	+	0.102046i	\(0.0325390\pi\)
\(29\)	7.48463	+	4.32125i	1.38986	+	0.802437i	0.993299	−	0.115571i	\(-0.0368699\pi\)
							0.396562	+	0.918008i	\(0.370203\pi\)
\(31\)			6.55949i			1.17812i	0.808090	+	0.589059i	\(0.200502\pi\)
							−0.808090	+	0.589059i	\(0.799498\pi\)
\(37\)	−0.577404	+	1.00009i	−0.0949247	+	0.164414i	−0.909577	−	0.415535i	\(-0.863594\pi\)
							0.814652	+	0.579949i	\(0.196928\pi\)
\(41\)	−3.52691	−	2.03626i	−0.550811	−	0.318011i	0.198638	−	0.980073i	\(-0.436348\pi\)
							−0.749449	+	0.662062i	\(0.769681\pi\)
\(43\)	10.0505	−	5.80268i	1.53269	−	0.884900i	0.533455	−	0.845828i	\(-0.320893\pi\)
							0.999236	−	0.0390719i	\(-0.0124401\pi\)
\(47\)	9.49402			1.38485			0.692423	−	0.721492i	\(-0.256543\pi\)
							0.692423	+	0.721492i	\(0.256543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.2.p.b.35.9	yes	40
3.2	odd	2	inner	156.2.p.b.35.12	yes	40
4.3	odd	2	inner	156.2.p.b.35.16	yes	40
12.11	even	2	inner	156.2.p.b.35.5	✓	40
13.3	even	3	inner	156.2.p.b.107.5	yes	40
39.29	odd	6	inner	156.2.p.b.107.16	yes	40
52.3	odd	6	inner	156.2.p.b.107.12	yes	40
156.107	even	6	inner	156.2.p.b.107.9	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.p.b.35.5	✓	40	12.11	even	2	inner
156.2.p.b.35.9	yes	40	1.1	even	1	trivial
156.2.p.b.35.12	yes	40	3.2	odd	2	inner
156.2.p.b.35.16	yes	40	4.3	odd	2	inner
156.2.p.b.107.5	yes	40	13.3	even	3	inner
156.2.p.b.107.9	yes	40	156.107	even	6	inner
156.2.p.b.107.12	yes	40	52.3	odd	6	inner
156.2.p.b.107.16	yes	40	39.29	odd	6	inner