Embedded newform 273.3.bo.c.160.1

• Label: 273.3.bo.c.160.1
• Level: $273$
• Weight: $3$
• Character: 273.160
• Analytic conductor: $7.439$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• 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			160.1
Character	\(\chi\)	\(=\)	273.160
Dual form			273.3.bo.c.244.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.42934 - 1.97993i) q^{2} +(-1.50000 - 0.866025i) q^{3} +(5.84025 + 10.1156i) q^{4} +3.37699 q^{5} +(3.42934 + 5.93979i) q^{6} +(-6.27422 - 3.10389i) q^{7} -30.4138i q^{8} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 54 q^{3} + 44 q^{4} - 4 q^{5} + 10 q^{7} + 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.42934	−	1.97993i	−1.71467	−	0.989966i	−0.927990	−	0.372606i	\(-0.878464\pi\)
							−0.786681	−	0.617360i	\(-0.788202\pi\)
\(3\)	−1.50000	−	0.866025i	−0.500000	−	0.288675i
							\(5\)	3.37699			0.675398			0.337699	−	0.941254i	\(-0.390351\pi\)
0.337699	+	0.941254i	\(0.390351\pi\)
\(7\)	−6.27422	−	3.10389i	−0.896317	−	0.443413i
							\(11\)	3.88545	+	2.24326i	0.353222	+	0.203933i	0.666104	−	0.745859i	\(-0.267961\pi\)
−0.312881	+	0.949792i	\(0.601294\pi\)
\(13\)	−12.4204	+	3.83855i	−0.955413	+	0.295273i
							\(17\)	11.4347	−	6.60183i	0.672629	−	0.388343i	−0.124443	−	0.992227i	\(-0.539714\pi\)
0.797072	+	0.603884i	\(0.206381\pi\)
\(19\)	11.4026	+	19.7499i	0.600137	+	1.03947i	0.992800	+	0.119785i	\(0.0382205\pi\)
							−0.392663	+	0.919682i	\(0.628446\pi\)
\(23\)	−12.8967	+	22.3377i	−0.560724	+	0.971203i	0.436709	+	0.899603i	\(0.356144\pi\)
							−0.997433	+	0.0716004i	\(0.977189\pi\)
\(29\)	−2.16621	+	3.75198i	−0.0746968	+	0.129379i	−0.900954	−	0.433914i	\(-0.857132\pi\)
							0.826258	+	0.563292i	\(0.190466\pi\)
\(31\)	31.3506			1.01131			0.505655	−	0.862736i	\(-0.331251\pi\)
							0.505655	+	0.862736i	\(0.331251\pi\)
\(37\)	1.37617	+	0.794535i	0.0371939	+	0.0214739i	0.518482	−	0.855089i	\(-0.326497\pi\)
							−0.481288	+	0.876563i	\(0.659831\pi\)
\(41\)	34.2996	−	59.4087i	0.836576	−	1.44899i	−0.0561645	−	0.998422i	\(-0.517887\pi\)
							0.892741	−	0.450571i	\(-0.148780\pi\)
\(43\)	34.5203	+	59.7909i	0.802797	+	1.39049i	0.917768	+	0.397117i	\(0.129989\pi\)
							−0.114971	+	0.993369i	\(0.536678\pi\)
\(47\)	−3.52739			−0.0750508			−0.0375254	−	0.999296i	\(-0.511948\pi\)
							−0.0375254	+	0.999296i	\(0.511948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.3.bo.c.160.1	✓	36
7.6	odd	2		273.3.bo.d.160.1	yes	36
13.10	even	6		273.3.bo.d.244.1	yes	36
91.62	odd	6	inner	273.3.bo.c.244.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.3.bo.c.160.1	✓	36	1.1	even	1	trivial
273.3.bo.c.244.1	yes	36	91.62	odd	6	inner
273.3.bo.d.160.1	yes	36	7.6	odd	2
273.3.bo.d.244.1	yes	36	13.10	even	6