Embedded newform 273.2.bt.a.136.7

• Label: 273.2.bt.a.136.7
• 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			136.7
Character	\(\chi\)	\(=\)	273.136
Dual form			273.2.bt.a.271.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09987 - 1.09987i) q^{2} +(0.866025 - 0.500000i) q^{3} -0.419447i q^{4} +(0.745735 - 2.78312i) q^{5} +(0.402582 - 1.50246i) q^{6} +(-1.80794 + 1.93167i) q^{7} +(1.73841 + 1.73841i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{7} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09987	−	1.09987i	0.777729	−	0.777729i	−0.201716	−	0.979444i	\(-0.564652\pi\)
							0.979444	+	0.201716i	\(0.0646517\pi\)
\(3\)	0.866025	−	0.500000i	0.500000	−	0.288675i
							\(5\)	0.745735	−	2.78312i	0.333503	−	1.24465i	−0.571981	−	0.820267i	\(-0.693825\pi\)
0.905483	−	0.424382i	\(-0.139509\pi\)
\(7\)	−1.80794	+	1.93167i	−0.683336	+	0.730104i
							\(11\)	1.41711	−	5.28871i	0.427273	−	1.59461i	−0.331635	−	0.943408i	\(-0.607600\pi\)
0.758908	−	0.651198i	\(-0.225733\pi\)
\(13\)	−0.662549	+	3.54415i	−0.183758	+	0.982972i
							\(17\)	−4.36465			−1.05858			−0.529292	−	0.848440i	\(-0.677542\pi\)
−0.529292	+	0.848440i	\(0.677542\pi\)
\(19\)	−1.39486	+	0.373751i	−0.320003	+	0.0857444i	−0.415245	−	0.909710i	\(-0.636304\pi\)
							0.0952423	+	0.995454i	\(0.469637\pi\)
\(23\)			8.37225i			1.74573i	0.487958	+	0.872867i	\(0.337742\pi\)
							−0.487958	+	0.872867i	\(0.662258\pi\)
\(29\)	0.882488	−	1.52851i	0.163874	−	0.283838i	−0.772381	−	0.635160i	\(-0.780934\pi\)
							0.936255	+	0.351322i	\(0.114268\pi\)
\(31\)	0.770818	−	0.206540i	0.138443	−	0.0370957i	−0.188932	−	0.981990i	\(-0.560503\pi\)
							0.327375	+	0.944894i	\(0.393836\pi\)
\(37\)	3.86358	+	3.86358i	0.635169	+	0.635169i	0.949360	−	0.314191i	\(-0.101733\pi\)
							−0.314191	+	0.949360i	\(0.601733\pi\)
\(41\)	3.88124	−	1.03997i	0.606147	−	0.162417i	0.0573248	−	0.998356i	\(-0.481743\pi\)
							0.548823	+	0.835939i	\(0.315076\pi\)
\(43\)	−5.58033	+	3.22180i	−0.850992	+	0.491320i	−0.860985	−	0.508630i	\(-0.830152\pi\)
							0.00999354	+	0.999950i	\(0.496819\pi\)
\(47\)	−8.52304	−	2.28374i	−1.24321	−	0.333118i	−0.423502	−	0.905895i	\(-0.639200\pi\)
							−0.819711	+	0.572777i	\(0.805866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bt.a.136.7	✓	36
3.2	odd	2		819.2.et.c.136.3		36
7.5	odd	6		273.2.cg.a.19.3	yes	36
13.11	odd	12		273.2.cg.a.115.3	yes	36
21.5	even	6		819.2.gh.c.19.7		36
39.11	even	12		819.2.gh.c.388.7		36
91.89	even	12	inner	273.2.bt.a.271.7	yes	36
273.89	odd	12		819.2.et.c.271.3		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bt.a.136.7	✓	36	1.1	even	1	trivial
273.2.bt.a.271.7	yes	36	91.89	even	12	inner
273.2.cg.a.19.3	yes	36	7.5	odd	6
273.2.cg.a.115.3	yes	36	13.11	odd	12
819.2.et.c.136.3		36	3.2	odd	2
819.2.et.c.271.3		36	273.89	odd	12
819.2.gh.c.19.7		36	21.5	even	6
819.2.gh.c.388.7		36	39.11	even	12