Embedded newform 273.2.bf.b.152.13

• Label: 273.2.bf.b.152.13
• Level: $273$
• Weight: $2$
• Character: 273.152
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			152.13
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.730177 + 0.421568i) q^{2} +(-1.64300 + 0.548211i) q^{3} +(-0.644561 + 1.11641i) q^{4} +(-1.06280 + 1.84082i) q^{5} +(0.968576 - 1.09293i) q^{6} +(-0.110845 + 2.64343i) q^{7} -2.77318i q^{8} +(2.39893 - 1.80143i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.730177	+	0.421568i	−0.516313	+	0.298093i	−0.735425	−	0.677606i	\(-0.763017\pi\)
							0.219112	+	0.975700i	\(0.429684\pi\)
\(3\)	−1.64300	+	0.548211i	−0.948589	+	0.316510i
							\(5\)	−1.06280	+	1.84082i	−0.475297	+	0.823239i	−0.999600	−	0.0282929i	\(-0.990993\pi\)
0.524302	+	0.851532i	\(0.324326\pi\)
\(7\)	−0.110845	+	2.64343i	−0.0418954	+	0.999122i
							\(11\)			0.987856i			0.297850i	0.988849	+	0.148925i	\(0.0475813\pi\)
−0.988849	+	0.148925i	\(0.952419\pi\)
\(13\)	−3.08300	−	1.86952i	−0.855070	−	0.518512i
							\(17\)	−0.920525	+	1.59440i	−0.223260	+	0.386698i	−0.955796	−	0.294030i	\(-0.905003\pi\)
0.732536	+	0.680728i	\(0.238337\pi\)
\(19\)		−	7.15770i		−	1.64209i	−0.570864	−	0.821044i	\(-0.693392\pi\)
							0.570864	−	0.821044i	\(-0.306608\pi\)
\(23\)	−1.40903	+	0.813502i	−0.293803	+	0.169627i	−0.639655	−	0.768662i	\(-0.720923\pi\)
							0.345853	+	0.938289i	\(0.387590\pi\)
\(29\)	0.488355	+	0.281952i	0.0906852	+	0.0523571i	0.544657	−	0.838659i	\(-0.316660\pi\)
							−0.453972	+	0.891016i	\(0.649993\pi\)
\(31\)	−4.23375	+	2.44435i	−0.760403	+	0.439019i	−0.829440	−	0.558595i	\(-0.811341\pi\)
							0.0690374	+	0.997614i	\(0.478007\pi\)
\(37\)	−1.64578	−	2.85058i	−0.270565	−	0.468633i	0.698442	−	0.715667i	\(-0.253877\pi\)
							−0.969007	+	0.247035i	\(0.920544\pi\)
\(41\)	4.93559	−	8.54870i	0.770810	−	1.33508i	−0.166310	−	0.986074i	\(-0.553185\pi\)
							0.937120	−	0.349008i	\(-0.113482\pi\)
\(43\)	−0.214038	−	0.370726i	−0.0326406	−	0.0565351i	0.849244	−	0.528001i	\(-0.177058\pi\)
							−0.881884	+	0.471466i	\(0.843725\pi\)
\(47\)	−6.40449	+	11.0929i	−0.934190	+	1.61807i	−0.158119	+	0.987420i	\(0.550543\pi\)
							−0.776071	+	0.630645i	\(0.782790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.13	yes	64
3.2	odd	2	inner	273.2.bf.b.152.20	yes	64
7.3	odd	6		273.2.r.b.269.13	yes	64
13.3	even	3		273.2.r.b.68.13	✓	64
21.17	even	6		273.2.r.b.269.20	yes	64
39.29	odd	6		273.2.r.b.68.20	yes	64
91.3	odd	6	inner	273.2.bf.b.185.20	yes	64
273.185	even	6	inner	273.2.bf.b.185.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.13	✓	64	13.3	even	3
273.2.r.b.68.20	yes	64	39.29	odd	6
273.2.r.b.269.13	yes	64	7.3	odd	6
273.2.r.b.269.20	yes	64	21.17	even	6
273.2.bf.b.152.13	yes	64	1.1	even	1	trivial
273.2.bf.b.152.20	yes	64	3.2	odd	2	inner
273.2.bf.b.185.13	yes	64	273.185	even	6	inner
273.2.bf.b.185.20	yes	64	91.3	odd	6	inner