Embedded newform 273.3.bo.c.160.16

• Label: 273.3.bo.c.160.16
• Level: $273$
• Weight: $3$
• Character: 273.160
• 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			160.16
Character	\(\chi\)	\(=\)	273.160
Dual form			273.3.bo.c.244.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.01987 + 1.74353i) q^{2} +(-1.50000 - 0.866025i) q^{3} +(4.07976 + 7.06636i) q^{4} +5.35167 q^{5} +(-3.01987 - 5.23058i) q^{6} +(-6.87138 + 1.33570i) q^{7} +14.5045i q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 54 q^{3} + 44 q^{4} - 4 q^{5} + 10 q^{7} + 54 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.01987	+	1.74353i	1.50994	+	0.871763i	0.999933	+	0.0115912i	\(0.00368967\pi\)
							0.510005	+	0.860172i	\(0.329644\pi\)
\(3\)	−1.50000	−	0.866025i	−0.500000	−	0.288675i
							\(5\)	5.35167			1.07033			0.535167	−	0.844747i	\(-0.320249\pi\)
0.535167	+	0.844747i	\(0.320249\pi\)
\(7\)	−6.87138	+	1.33570i	−0.981626	+	0.190815i
							\(11\)	14.7685	+	8.52662i	1.34259	+	0.775147i	0.987187	−	0.159565i	\(-0.0510092\pi\)
0.355406	+	0.934712i	\(0.384342\pi\)
\(13\)	−3.94055	+	12.3884i	−0.303119	+	0.952953i
							\(17\)	8.84365	−	5.10588i	0.520215	−	0.300346i	−0.216808	−	0.976214i	\(-0.569565\pi\)
0.737022	+	0.675868i	\(0.236231\pi\)
\(19\)	−4.19887	−	7.27265i	−0.220993	−	0.382771i	0.734117	−	0.679023i	\(-0.237596\pi\)
							−0.955110	+	0.296252i	\(0.904263\pi\)
\(23\)	8.33600	−	14.4384i	0.362435	−	0.627755i	−0.625926	−	0.779882i	\(-0.715279\pi\)
							0.988361	+	0.152127i	\(0.0486122\pi\)
\(29\)	16.5675	−	28.6958i	0.571295	−	0.989511i	−0.425139	−	0.905128i	\(-0.639775\pi\)
							0.996433	−	0.0843831i	\(-0.0268919\pi\)
\(31\)	−56.0095			−1.80676			−0.903379	−	0.428843i	\(-0.858922\pi\)
							−0.903379	+	0.428843i	\(0.858922\pi\)
\(37\)	−23.4915	−	13.5628i	−0.634906	−	0.366563i	0.147744	−	0.989026i	\(-0.452799\pi\)
							−0.782650	+	0.622462i	\(0.786132\pi\)
\(41\)	−0.675564	+	1.17011i	−0.0164772	+	0.0285393i	−0.874146	−	0.485662i	\(-0.838578\pi\)
							0.857669	+	0.514202i	\(0.171912\pi\)
\(43\)	−2.44487	−	4.23464i	−0.0568575	−	0.0984800i	0.836196	−	0.548431i	\(-0.184775\pi\)
							−0.893053	+	0.449951i	\(0.851441\pi\)
\(47\)	20.3383			0.432729			0.216365	−	0.976313i	\(-0.430580\pi\)
							0.216365	+	0.976313i	\(0.430580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.3.bo.c.160.16	✓	36
7.6	odd	2		273.3.bo.d.160.16	yes	36
13.10	even	6		273.3.bo.d.244.16	yes	36
91.62	odd	6	inner	273.3.bo.c.244.16	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.3.bo.c.160.16	✓	36	1.1	even	1	trivial
273.3.bo.c.244.16	yes	36	91.62	odd	6	inner
273.3.bo.d.160.16	yes	36	7.6	odd	2
273.3.bo.d.244.16	yes	36	13.10	even	6