Embedded newform 273.2.bf.b.152.20

• Label: 273.2.bf.b.152.20
• 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			152.20
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.730177 - 0.421568i) q^{2} +(1.64300 + 0.548211i) q^{3} +(-0.644561 + 1.11641i) q^{4} +(1.06280 - 1.84082i) q^{5} +(1.43079 - 0.292347i) q^{6} +(-0.110845 + 2.64343i) q^{7} +2.77318i q^{8} +(2.39893 + 1.80143i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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	−0.219112	−	0.975700i	\(-0.570316\pi\)
							0.735425	+	0.677606i	\(0.236983\pi\)
\(3\)	1.64300	+	0.548211i	0.948589	+	0.316510i
							\(5\)	1.06280	−	1.84082i	0.475297	−	0.823239i	−0.524302	−	0.851532i	\(-0.675674\pi\)
0.999600	+	0.0282929i	\(0.00900711\pi\)
\(7\)	−0.110845	+	2.64343i	−0.0418954	+	0.999122i
							\(11\)		−	0.987856i		−	0.297850i	−0.988849	−	0.148925i	\(-0.952419\pi\)
0.988849	−	0.148925i	\(-0.0475813\pi\)
\(13\)	−3.08300	−	1.86952i	−0.855070	−	0.518512i
							\(17\)	0.920525	−	1.59440i	0.223260	−	0.386698i	−0.732536	−	0.680728i	\(-0.761663\pi\)
0.955796	+	0.294030i	\(0.0949967\pi\)
\(19\)		−	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	−0.345853	−	0.938289i	\(-0.612410\pi\)
							0.639655	+	0.768662i	\(0.279077\pi\)
\(29\)	−0.488355	−	0.281952i	−0.0906852	−	0.0523571i	0.453972	−	0.891016i	\(-0.350007\pi\)
							−0.544657	+	0.838659i	\(0.683340\pi\)
\(31\)	−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	\(0.446815\pi\)
							−0.937120	+	0.349008i	\(0.886518\pi\)
\(43\)	−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	\(-0.449457\pi\)
							0.776071	−	0.630645i	\(-0.217210\pi\)

(See \(a_n\) instead)

Twists

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

    

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