Embedded newform 273.2.r.b.68.13

• Label: 273.2.r.b.68.13
• Level: $273$
• Weight: $2$
• Character: 273.68
• 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.r (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			68.13
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.20

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.843135i q^{2} +(0.346738 - 1.69699i) q^{3} +1.28912 q^{4} +(-1.06280 + 1.84082i) q^{5} +(-1.43079 - 0.292347i) q^{6} +(2.34470 - 1.22572i) q^{7} -2.77318i q^{8} +(-2.75955 - 1.17682i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	0.843135i		−	0.596187i	−0.954537	−	0.298093i	\(-0.903649\pi\)
							0.954537	−	0.298093i	\(-0.0963507\pi\)
\(3\)	0.346738	−	1.69699i	0.200189	−	0.979757i
							\(5\)	−1.06280	+	1.84082i	−0.475297	+	0.823239i	−0.999600	−	0.0282929i	\(-0.990993\pi\)
0.524302	+	0.851532i	\(0.324326\pi\)
\(7\)	2.34470	−	1.22572i	0.886213	−	0.463278i
							\(11\)	−0.855509	−	0.493928i	−0.257946	−	0.148925i	0.365451	−	0.930830i	\(-0.380915\pi\)
−0.623397	+	0.781905i	\(0.714248\pi\)
\(13\)	3.08300	−	1.86952i	0.855070	−	0.518512i
							\(17\)	1.84105			0.446520			0.223260	−	0.974759i	\(-0.428330\pi\)
0.223260	+	0.974759i	\(0.428330\pi\)
\(19\)	−6.19875	+	3.57885i	−1.42209	+	0.821044i	−0.996477	−	0.0838608i	\(-0.973275\pi\)
							−0.425613	+	0.904905i	\(0.639942\pi\)
\(23\)		−	1.62700i		−	0.339254i	−0.985508	−	0.169627i	\(-0.945744\pi\)
							0.985508	−	0.169627i	\(-0.0542563\pi\)
\(29\)	−0.488355	+	0.281952i	−0.0906852	+	0.0523571i	−0.544657	−	0.838659i	\(-0.683340\pi\)
							0.453972	+	0.891016i	\(0.350007\pi\)
\(31\)	−4.23375	+	2.44435i	−0.760403	+	0.439019i	−0.829440	−	0.558595i	\(-0.811341\pi\)
							0.0690374	+	0.997614i	\(0.478007\pi\)
\(37\)	3.29157			0.541130			0.270565	−	0.962702i	\(-0.412789\pi\)
							0.270565	+	0.962702i	\(0.412789\pi\)
\(41\)	4.93559	+	8.54870i	0.770810	+	1.33508i	0.937120	+	0.349008i	\(0.113482\pi\)
							−0.166310	+	0.986074i	\(0.553185\pi\)
\(43\)	−0.214038	+	0.370726i	−0.0326406	+	0.0565351i	−0.881884	−	0.471466i	\(-0.843725\pi\)
							0.849244	+	0.528001i	\(0.177058\pi\)
\(47\)	−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.r.b.68.13	✓	64
3.2	odd	2	inner	273.2.r.b.68.20	yes	64
7.3	odd	6		273.2.bf.b.185.20	yes	64
13.9	even	3		273.2.bf.b.152.13	yes	64
21.17	even	6		273.2.bf.b.185.13	yes	64
39.35	odd	6		273.2.bf.b.152.20	yes	64
91.87	odd	6	inner	273.2.r.b.269.13	yes	64
273.269	even	6	inner	273.2.r.b.269.20	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.13	✓	64	1.1	even	1	trivial
273.2.r.b.68.20	yes	64	3.2	odd	2	inner
273.2.r.b.269.13	yes	64	91.87	odd	6	inner
273.2.r.b.269.20	yes	64	273.269	even	6	inner
273.2.bf.b.152.13	yes	64	13.9	even	3
273.2.bf.b.152.20	yes	64	39.35	odd	6
273.2.bf.b.185.13	yes	64	21.17	even	6
273.2.bf.b.185.20	yes	64	7.3	odd	6