Embedded newform 147.3.l.a.8.11

• Label: 147.3.l.a.8.11
• Level: $147$
• Weight: $3$
• Character: 147.8
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			8.11
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45285 + 1.15861i) q^{2} +(2.57601 + 1.53758i) q^{3} +(-0.121689 + 0.533152i) q^{4} +(3.36170 + 6.98065i) q^{5} +(-5.52401 + 0.750718i) q^{6} +(6.92701 - 1.00823i) q^{7} +(-3.66600 - 7.61252i) q^{8} +(4.27169 + 7.92166i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45285	+	1.15861i	−0.726424	+	0.579304i	−0.915336	−	0.402690i	\(-0.868075\pi\)
							0.188913	+	0.981994i	\(0.439504\pi\)
\(3\)	2.57601	+	1.53758i	0.858671	+	0.512527i
							\(5\)	3.36170	+	6.98065i	0.672341	+	1.39613i	0.905776	+	0.423757i	\(0.139289\pi\)
−0.233435	+	0.972372i	\(0.574997\pi\)
\(7\)	6.92701	−	1.00823i	0.989573	−	0.144033i
							\(11\)	2.46031	−	1.96203i	0.223664	−	0.178366i	−0.505246	−	0.862975i	\(-0.668598\pi\)
0.728911	+	0.684609i	\(0.240027\pi\)
\(13\)	−10.4922	−	13.1568i	−0.807090	−	1.01206i	−0.999527	−	0.0307493i	\(-0.990211\pi\)
							0.192437	−	0.981309i	\(-0.438361\pi\)
\(17\)	3.69031	−	0.842289i	0.217077	−	0.0495464i	−0.112599	−	0.993640i	\(-0.535918\pi\)
							0.329676	+	0.944094i	\(0.393060\pi\)
\(19\)	18.5338			0.975466			0.487733	−	0.872993i	\(-0.337824\pi\)
							0.487733	+	0.872993i	\(0.337824\pi\)
\(23\)	−40.8300	−	9.31919i	−1.77522	−	0.405182i	−0.795564	−	0.605870i	\(-0.792825\pi\)
							−0.979655	+	0.200687i	\(0.935682\pi\)
\(29\)	−23.2140	+	5.29845i	−0.800484	+	0.182705i	−0.603146	−	0.797631i	\(-0.706087\pi\)
							−0.197337	+	0.980336i	\(0.563229\pi\)
\(31\)	13.2738			0.428188			0.214094	−	0.976813i	\(-0.431320\pi\)
							0.214094	+	0.976813i	\(0.431320\pi\)
\(37\)	−9.19981	−	40.3070i	−0.248643	−	1.08938i	−0.932900	−	0.360137i	\(-0.882730\pi\)
							0.684256	−	0.729242i	\(-0.260127\pi\)
\(41\)	−22.9923	−	47.7439i	−0.560787	−	1.16449i	−0.967953	−	0.251132i	\(-0.919197\pi\)
							0.407166	−	0.913354i	\(-0.366517\pi\)
\(43\)	52.2208	+	25.1482i	1.21444	+	0.584842i	0.927757	−	0.373185i	\(-0.121734\pi\)
							0.286679	+	0.958027i	\(0.407449\pi\)
\(47\)	37.9506	−	30.2646i	0.807459	−	0.643927i	−0.130198	−	0.991488i	\(-0.541561\pi\)
							0.937658	+	0.347561i	\(0.112990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.11	✓	216
3.2	odd	2	inner	147.3.l.a.8.26	yes	216
49.43	even	7	inner	147.3.l.a.92.26	yes	216
147.92	odd	14	inner	147.3.l.a.92.11	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.11	✓	216	1.1	even	1	trivial
147.3.l.a.8.26	yes	216	3.2	odd	2	inner
147.3.l.a.92.11	yes	216	147.92	odd	14	inner
147.3.l.a.92.26	yes	216	49.43	even	7	inner