Embedded newform 273.2.bt.a.136.9

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

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74432 - 1.74432i) q^{2} +(0.866025 - 0.500000i) q^{3} -4.08534i q^{4} +(0.130892 - 0.488495i) q^{5} +(0.638467 - 2.38279i) q^{6} +(1.09376 + 2.40909i) q^{7} +(-3.63751 - 3.63751i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{7} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.74432	−	1.74432i	1.23342	−	1.23342i	0.270784	−	0.962640i	\(-0.412717\pi\)
							0.962640	−	0.270784i	\(-0.0872829\pi\)
\(3\)	0.866025	−	0.500000i	0.500000	−	0.288675i
							\(5\)	0.130892	−	0.488495i	0.0585366	−	0.218462i	−0.930461	−	0.366390i	\(-0.880594\pi\)
0.988998	+	0.147928i	\(0.0472604\pi\)
\(7\)	1.09376	+	2.40909i	0.413402	+	0.910549i
							\(11\)	−1.54092	+	5.75079i	−0.464605	+	1.73393i	0.193592	+	0.981082i	\(0.437986\pi\)
−0.658197	+	0.752846i	\(0.728680\pi\)
\(13\)	−3.50782	−	0.833795i	−0.972894	−	0.231253i
							\(17\)	0.933228			0.226341			0.113171	−	0.993576i	\(-0.463899\pi\)
0.113171	+	0.993576i	\(0.463899\pi\)
\(19\)	−7.66303	+	2.05330i	−1.75802	+	0.471060i	−0.986308	−	0.164913i	\(-0.947266\pi\)
							−0.771711	+	0.635973i	\(0.780599\pi\)
\(23\)		−	8.12427i		−	1.69403i	−0.531571	−	0.847014i	\(-0.678398\pi\)
							0.531571	−	0.847014i	\(-0.321602\pi\)
\(29\)	1.96458	−	3.40276i	0.364814	−	0.631877i	−0.623932	−	0.781479i	\(-0.714466\pi\)
							0.988746	+	0.149602i	\(0.0477992\pi\)
\(31\)	−2.37727	+	0.636987i	−0.426970	+	0.114406i	−0.465903	−	0.884836i	\(-0.654271\pi\)
							0.0389336	+	0.999242i	\(0.487604\pi\)
\(37\)	0.859350	+	0.859350i	0.141276	+	0.141276i	0.774208	−	0.632932i	\(-0.218149\pi\)
							−0.632932	+	0.774208i	\(0.718149\pi\)
\(41\)	7.84968	−	2.10332i	1.22591	−	0.328483i	0.412927	−	0.910764i	\(-0.364506\pi\)
							0.812987	+	0.582281i	\(0.197840\pi\)
\(43\)	−0.152677	+	0.0881483i	−0.0232831	+	0.0134425i	−0.511596	−	0.859226i	\(-0.670946\pi\)
							0.488313	+	0.872668i	\(0.337612\pi\)
\(47\)	1.65836	+	0.444356i	0.241897	+	0.0648160i	0.377730	−	0.925916i	\(-0.376705\pi\)
							−0.135834	+	0.990732i	\(0.543371\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.a.136.9	✓	36
3.2	odd	2		819.2.et.c.136.1		36
7.5	odd	6		273.2.cg.a.19.1	yes	36
13.11	odd	12		273.2.cg.a.115.1	yes	36
21.5	even	6		819.2.gh.c.19.9		36
39.11	even	12		819.2.gh.c.388.9		36
91.89	even	12	inner	273.2.bt.a.271.9	yes	36
273.89	odd	12		819.2.et.c.271.1		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.a.136.9	✓	36	1.1	even	1	trivial
273.2.bt.a.271.9	yes	36	91.89	even	12	inner
273.2.cg.a.19.1	yes	36	7.5	odd	6
273.2.cg.a.115.1	yes	36	13.11	odd	12
819.2.et.c.136.1		36	3.2	odd	2
819.2.et.c.271.1		36	273.89	odd	12
819.2.gh.c.19.9		36	21.5	even	6
819.2.gh.c.388.9		36	39.11	even	12