Embedded newform 273.2.bt.b.136.3

• Label: 273.2.bt.b.136.3
• Level: $273$
• Weight: $2$
• Character: 273.136
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.bt (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			136.3
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.b.271.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03264 + 1.03264i) q^{2} +(-0.866025 + 0.500000i) q^{3} -0.132693i q^{4} +(0.456824 - 1.70489i) q^{5} +(0.377973 - 1.41061i) q^{6} +(-2.58707 + 0.554121i) q^{7} +(-1.92826 - 1.92826i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03264	+	1.03264i	−0.730187	+	0.730187i	−0.970657	−	0.240470i	\(-0.922699\pi\)
							0.240470	+	0.970657i	\(0.422699\pi\)
\(3\)	−0.866025	+	0.500000i	−0.500000	+	0.288675i
							\(5\)	0.456824	−	1.70489i	0.204298	−	0.762450i	−0.785365	−	0.619033i	\(-0.787525\pi\)
0.989662	−	0.143416i	\(-0.0458088\pi\)
\(7\)	−2.58707	+	0.554121i	−0.977822	+	0.209438i
							\(11\)	0.583737	−	2.17854i	0.176003	−	0.656854i	−0.820375	−	0.571826i	\(-0.806235\pi\)
0.996379	−	0.0850280i	\(-0.0270980\pi\)
\(13\)	−2.06370	−	2.95654i	−0.572367	−	0.819998i
							\(17\)	1.62731			0.394680			0.197340	−	0.980335i	\(-0.436770\pi\)
0.197340	+	0.980335i	\(0.436770\pi\)
\(19\)	−6.94159	+	1.85999i	−1.59251	+	0.426712i	−0.942770	−	0.333444i	\(-0.891789\pi\)
							−0.649741	+	0.760156i	\(0.725122\pi\)
\(23\)		−	6.50210i		−	1.35578i	−0.735163	−	0.677891i	\(-0.762894\pi\)
							0.735163	−	0.677891i	\(-0.237106\pi\)
\(29\)	3.90230	−	6.75898i	0.724638	−	1.25511i	−0.234484	−	0.972120i	\(-0.575340\pi\)
							0.959123	−	0.282991i	\(-0.0913266\pi\)
\(31\)	6.30283	−	1.68884i	1.13202	−	0.303324i	0.356283	−	0.934378i	\(-0.384044\pi\)
							0.775739	+	0.631054i	\(0.217377\pi\)
\(37\)	−1.48970	−	1.48970i	−0.244905	−	0.244905i	0.573971	−	0.818876i	\(-0.305402\pi\)
							−0.818876	+	0.573971i	\(0.805402\pi\)
\(41\)	−6.84645	+	1.83450i	−1.06924	+	0.286501i	−0.750179	−	0.661235i	\(-0.770033\pi\)
							−0.319057	+	0.947736i	\(0.603366\pi\)
\(43\)	−9.48195	+	5.47441i	−1.44598	+	0.834839i	−0.998239	−	0.0593222i	\(-0.981106\pi\)
							−0.447745	+	0.894161i	\(0.647773\pi\)
\(47\)	−12.4756	−	3.34284i	−1.81976	−	0.487603i	−0.822999	−	0.568043i	\(-0.807701\pi\)
							−0.996759	+	0.0804402i	\(0.974367\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.136.3	✓	40
3.2	odd	2		819.2.et.d.136.8		40
7.5	odd	6		273.2.cg.b.19.8	yes	40
13.11	odd	12		273.2.cg.b.115.8	yes	40
21.5	even	6		819.2.gh.d.19.3		40
39.11	even	12		819.2.gh.d.388.3		40
91.89	even	12	inner	273.2.bt.b.271.3	yes	40
273.89	odd	12		819.2.et.d.271.8		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.136.3	✓	40	1.1	even	1	trivial
273.2.bt.b.271.3	yes	40	91.89	even	12	inner
273.2.cg.b.19.8	yes	40	7.5	odd	6
273.2.cg.b.115.8	yes	40	13.11	odd	12
819.2.et.d.136.8		40	3.2	odd	2
819.2.et.d.271.8		40	273.89	odd	12
819.2.gh.d.19.3		40	21.5	even	6
819.2.gh.d.388.3		40	39.11	even	12