Embedded newform 273.2.bt.a.136.5

• Label: 273.2.bt.a.136.5
• Level: $273$
• Weight: $2$
• Character: 273.136
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $36$
• 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:	\(36\)
Relative dimension:	\(9\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			136.5
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.a.271.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.374685 - 0.374685i) q^{2} +(0.866025 - 0.500000i) q^{3} +1.71922i q^{4} +(-0.545981 + 2.03763i) q^{5} +(0.137144 - 0.511829i) q^{6} +(-2.03549 + 1.69021i) q^{7} +(1.39354 + 1.39354i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{7} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.374685	−	0.374685i	0.264942	−	0.264942i	−0.562116	−	0.827058i	\(-0.690013\pi\)
							0.827058	+	0.562116i	\(0.190013\pi\)
\(3\)	0.866025	−	0.500000i	0.500000	−	0.288675i
							\(5\)	−0.545981	+	2.03763i	−0.244170	+	0.911255i	0.729629	+	0.683844i	\(0.239693\pi\)
−0.973799	+	0.227412i	\(0.926974\pi\)
\(7\)	−2.03549	+	1.69021i	−0.769341	+	0.638838i
							\(11\)	−0.745933	+	2.78386i	−0.224907	+	0.839366i	0.757534	+	0.652795i	\(0.226404\pi\)
−0.982442	+	0.186570i	\(0.940263\pi\)
\(13\)	2.80107	−	2.27024i	0.776877	−	0.629652i
							\(17\)	−3.29760			−0.799786			−0.399893	−	0.916562i	\(-0.630953\pi\)
−0.399893	+	0.916562i	\(0.630953\pi\)
\(19\)	6.53354	−	1.75066i	1.49890	−	0.401628i	0.586166	−	0.810191i	\(-0.300637\pi\)
							0.912730	+	0.408563i	\(0.133970\pi\)
\(23\)		−	7.84515i		−	1.63583i	−0.575341	−	0.817914i	\(-0.695130\pi\)
							0.575341	−	0.817914i	\(-0.304870\pi\)
\(29\)	0.677462	−	1.17340i	0.125802	−	0.217895i	−0.796244	−	0.604975i	\(-0.793183\pi\)
							0.922046	+	0.387080i	\(0.126516\pi\)
\(31\)	6.38499	−	1.71085i	1.14678	−	0.307278i	0.365105	−	0.930966i	\(-0.381033\pi\)
							0.781673	+	0.623688i	\(0.214366\pi\)
\(37\)	2.87856	+	2.87856i	0.473232	+	0.473232i	0.902959	−	0.429727i	\(-0.141390\pi\)
							−0.429727	+	0.902959i	\(0.641390\pi\)
\(41\)	6.61712	−	1.77305i	1.03342	−	0.276904i	0.298036	−	0.954555i	\(-0.403668\pi\)
							0.735384	+	0.677651i	\(0.237002\pi\)
\(43\)	−4.36301	+	2.51899i	−0.665353	+	0.384142i	−0.794314	−	0.607508i	\(-0.792169\pi\)
							0.128961	+	0.991650i	\(0.458836\pi\)
\(47\)	−3.53471	−	0.947124i	−0.515591	−	0.138152i	−0.00836456	−	0.999965i	\(-0.502663\pi\)
							−0.507226	+	0.861813i	\(0.669329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.a.136.5	✓	36
3.2	odd	2		819.2.et.c.136.5		36
7.5	odd	6		273.2.cg.a.19.5	yes	36
13.11	odd	12		273.2.cg.a.115.5	yes	36
21.5	even	6		819.2.gh.c.19.5		36
39.11	even	12		819.2.gh.c.388.5		36
91.89	even	12	inner	273.2.bt.a.271.5	yes	36
273.89	odd	12		819.2.et.c.271.5		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.a.136.5	✓	36	1.1	even	1	trivial
273.2.bt.a.271.5	yes	36	91.89	even	12	inner
273.2.cg.a.19.5	yes	36	7.5	odd	6
273.2.cg.a.115.5	yes	36	13.11	odd	12
819.2.et.c.136.5		36	3.2	odd	2
819.2.et.c.271.5		36	273.89	odd	12
819.2.gh.c.19.5		36	21.5	even	6
819.2.gh.c.388.5		36	39.11	even	12