Embedded newform 273.2.bt.b.136.9

• Label: 273.2.bt.b.136.9
• 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			136.9
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.b.271.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.57115 - 1.57115i) q^{2} +(-0.866025 + 0.500000i) q^{3} -2.93701i q^{4} +(0.922649 - 3.44337i) q^{5} +(-0.575080 + 2.14623i) q^{6} +(-2.64488 + 0.0678683i) q^{7} +(-1.47218 - 1.47218i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{7} + 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.57115	−	1.57115i	1.11097	−	1.11097i	0.117949	−	0.993020i	\(-0.462368\pi\)
							0.993020	−	0.117949i	\(-0.0376320\pi\)
\(3\)	−0.866025	+	0.500000i	−0.500000	+	0.288675i
							\(5\)	0.922649	−	3.44337i	0.412621	−	1.53992i	−0.376931	−	0.926241i	\(-0.623021\pi\)
0.789553	−	0.613683i	\(-0.210313\pi\)
\(7\)	−2.64488	+	0.0678683i	−0.999671	+	0.0256518i
							\(11\)	−0.894327	+	3.33768i	−0.269650	+	1.00635i	0.689693	+	0.724102i	\(0.257746\pi\)
−0.959342	+	0.282245i	\(0.908921\pi\)
\(13\)	0.668132	−	3.54311i	0.185307	−	0.982681i
							\(17\)	2.52733			0.612967			0.306484	−	0.951876i	\(-0.400848\pi\)
0.306484	+	0.951876i	\(0.400848\pi\)
\(19\)	4.31297	−	1.15566i	0.989463	−	0.265126i	0.272438	−	0.962173i	\(-0.412170\pi\)
							0.717025	+	0.697047i	\(0.245503\pi\)
\(23\)			3.05290i			0.636573i	0.947995	+	0.318286i	\(0.103107\pi\)
							−0.947995	+	0.318286i	\(0.896893\pi\)
\(29\)	−1.12965	+	1.95662i	−0.209771	+	0.363335i	−0.951642	−	0.307208i	\(-0.900605\pi\)
							0.741871	+	0.670543i	\(0.233939\pi\)
\(31\)	−5.75625	+	1.54238i	−1.03385	+	0.277020i	−0.735564	−	0.677456i	\(-0.763083\pi\)
							−0.298289	+	0.954476i	\(0.596416\pi\)
\(37\)	7.41760	+	7.41760i	1.21945	+	1.21945i	0.967826	+	0.251620i	\(0.0809634\pi\)
							0.251620	+	0.967826i	\(0.419037\pi\)
\(41\)	2.06529	−	0.553392i	0.322544	−	0.0864254i	−0.0939142	−	0.995580i	\(-0.529938\pi\)
							0.416458	+	0.909155i	\(0.363271\pi\)
\(43\)	3.23658	−	1.86864i	0.493574	−	0.284965i	−0.232482	−	0.972601i	\(-0.574685\pi\)
							0.726056	+	0.687635i	\(0.241351\pi\)
\(47\)	6.55128	+	1.75541i	0.955603	+	0.256053i	0.702738	−	0.711449i	\(-0.251961\pi\)
							0.252865	+	0.967502i	\(0.418627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.b.136.9	✓	40
3.2	odd	2		819.2.et.d.136.2		40
7.5	odd	6		273.2.cg.b.19.2	yes	40
13.11	odd	12		273.2.cg.b.115.2	yes	40
21.5	even	6		819.2.gh.d.19.9		40
39.11	even	12		819.2.gh.d.388.9		40
91.89	even	12	inner	273.2.bt.b.271.9	yes	40
273.89	odd	12		819.2.et.d.271.2		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.b.136.9	✓	40	1.1	even	1	trivial
273.2.bt.b.271.9	yes	40	91.89	even	12	inner
273.2.cg.b.19.2	yes	40	7.5	odd	6
273.2.cg.b.115.2	yes	40	13.11	odd	12
819.2.et.d.136.2		40	3.2	odd	2
819.2.et.d.271.2		40	273.89	odd	12
819.2.gh.d.19.9		40	21.5	even	6
819.2.gh.d.388.9		40	39.11	even	12