Embedded newform 273.3.bo.d.244.12

• Label: 273.3.bo.d.244.12
• Level: $273$
• Weight: $3$
• Character: 273.244
• Analytic conductor: $7.439$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	273.bo (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.43871121704\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			244.12
Character	\(\chi\)	\(=\)	273.244
Dual form			273.3.bo.d.160.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15840 - 0.668803i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-1.10541 + 1.91462i) q^{4} +3.82472 q^{5} +(1.15840 - 2.00641i) q^{6} +(-6.07257 + 3.48194i) q^{7} +8.30761i q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 54 q^{3} + 44 q^{4} + 4 q^{5} - 10 q^{7} + 54 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15840	−	0.668803i	0.579200	−	0.334401i	−0.181615	−	0.983370i	\(-0.558133\pi\)
							0.760815	+	0.648968i	\(0.224799\pi\)
\(3\)	1.50000	−	0.866025i	0.500000	−	0.288675i
							\(5\)	3.82472			0.764945			0.382472	−	0.923967i	\(-0.375073\pi\)
0.382472	+	0.923967i	\(0.375073\pi\)
\(7\)	−6.07257	+	3.48194i	−0.867510	+	0.497420i
							\(11\)	17.0475	−	9.84240i	1.54978	−	0.894763i	0.551618	−	0.834097i	\(-0.314011\pi\)
0.998158	−	0.0606665i	\(-0.0193226\pi\)
\(13\)	11.9140	+	5.20156i	0.916463	+	0.400120i
							\(17\)	26.1168	+	15.0786i	1.53628	+	0.886974i	0.999052	+	0.0435393i	\(0.0138634\pi\)
0.537232	+	0.843434i	\(0.319470\pi\)
\(19\)	−7.26429	+	12.5821i	−0.382331	+	0.662217i	−0.991395	−	0.130904i	\(-0.958212\pi\)
							0.609064	+	0.793121i	\(0.291545\pi\)
\(23\)	−15.4983	−	26.8438i	−0.673838	−	1.16712i	−0.976807	−	0.214121i	\(-0.931311\pi\)
							0.302970	−	0.953000i	\(-0.402022\pi\)
\(29\)	10.7930	+	18.6941i	0.372174	+	0.644624i	0.989900	−	0.141769i	\(-0.0452792\pi\)
							−0.617726	+	0.786394i	\(0.711946\pi\)
\(31\)	−37.6446			−1.21434			−0.607170	−	0.794572i	\(-0.707695\pi\)
							−0.607170	+	0.794572i	\(0.707695\pi\)
\(37\)	38.6187	−	22.2965i	1.04375	−	0.602608i	0.122856	−	0.992425i	\(-0.460795\pi\)
							0.920893	+	0.389816i	\(0.127461\pi\)
\(41\)	−6.08738	−	10.5436i	−0.148473	−	0.257162i	0.782191	−	0.623039i	\(-0.214102\pi\)
							−0.930663	+	0.365877i	\(0.880769\pi\)
\(43\)	−5.34090	+	9.25071i	−0.124207	+	0.215133i	−0.921423	−	0.388562i	\(-0.872972\pi\)
							0.797216	+	0.603695i	\(0.206305\pi\)
\(47\)	8.11860			0.172736			0.0863681	−	0.996263i	\(-0.472474\pi\)
							0.0863681	+	0.996263i	\(0.472474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.3.bo.d.244.12	yes	36
7.6	odd	2		273.3.bo.c.244.12	yes	36
13.4	even	6		273.3.bo.c.160.12	✓	36
91.69	odd	6	inner	273.3.bo.d.160.12	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.3.bo.c.160.12	✓	36	13.4	even	6
273.3.bo.c.244.12	yes	36	7.6	odd	2
273.3.bo.d.160.12	yes	36	91.69	odd	6	inner
273.3.bo.d.244.12	yes	36	1.1	even	1	trivial