Embedded newform 273.2.bf.b.152.9

• Label: 273.2.bf.b.152.9
• 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			152.9
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33345 + 0.769868i) q^{2} +(-0.674223 + 1.59544i) q^{3} +(0.185393 - 0.321109i) q^{4} +(0.956175 - 1.65614i) q^{5} +(-0.329235 - 2.64650i) q^{6} +(1.55646 - 2.13949i) q^{7} -2.50856i q^{8} +(-2.09085 - 2.15136i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33345	+	0.769868i	−0.942892	+	0.544379i	−0.890866	−	0.454267i	\(-0.849901\pi\)
							−0.0520260	+	0.998646i	\(0.516568\pi\)
\(3\)	−0.674223	+	1.59544i	−0.389263	+	0.921127i
							\(5\)	0.956175	−	1.65614i	0.427614	−	0.740650i	−0.569046	−	0.822306i	\(-0.692687\pi\)
0.996661	+	0.0816556i	\(0.0260208\pi\)
\(7\)	1.55646	−	2.13949i	0.588288	−	0.808652i
							\(11\)		−	2.30765i		−	0.695783i	−0.937535	−	0.347892i	\(-0.886898\pi\)
0.937535	−	0.347892i	\(-0.113102\pi\)
\(13\)	0.963713	−	3.47437i	0.267286	−	0.963617i
							\(17\)	−1.87695	+	3.25097i	−0.455227	+	0.788476i	−0.998701	−	0.0509499i	\(-0.983775\pi\)
0.543474	+	0.839426i	\(0.317108\pi\)
\(19\)		−	4.83490i		−	1.10920i	−0.832117	−	0.554601i	\(-0.812871\pi\)
							0.832117	−	0.554601i	\(-0.187129\pi\)
\(23\)	0.175563	−	0.101362i	0.0366075	−	0.0211353i	−0.481585	−	0.876400i	\(-0.659939\pi\)
							0.518192	+	0.855264i	\(0.326605\pi\)
\(29\)	0.228216	+	0.131761i	0.0423786	+	0.0244673i	0.521040	−	0.853532i	\(-0.325544\pi\)
							−0.478661	+	0.878000i	\(0.658878\pi\)
\(31\)	1.76906	−	1.02137i	0.317732	−	0.183443i	−0.332649	−	0.943051i	\(-0.607942\pi\)
							0.650381	+	0.759608i	\(0.274609\pi\)
\(37\)	−2.25673	−	3.90878i	−0.371005	−	0.642599i	0.618715	−	0.785615i	\(-0.287653\pi\)
							−0.989720	+	0.143016i	\(0.954320\pi\)
\(41\)	−5.87191	+	10.1704i	−0.917038	+	1.58836i	−0.113149	+	0.993578i	\(0.536094\pi\)
							−0.803889	+	0.594779i	\(0.797240\pi\)
\(43\)	−2.00837	−	3.47860i	−0.306273	−	0.530481i	0.671271	−	0.741212i	\(-0.265749\pi\)
							−0.977544	+	0.210731i	\(0.932415\pi\)
\(47\)	4.81266	−	8.33577i	0.701999	−	1.21590i	−0.265765	−	0.964038i	\(-0.585625\pi\)
							0.967764	−	0.251860i	\(-0.0810421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.9	yes	64
3.2	odd	2	inner	273.2.bf.b.152.24	yes	64
7.3	odd	6		273.2.r.b.269.9	yes	64
13.3	even	3		273.2.r.b.68.9	✓	64
21.17	even	6		273.2.r.b.269.24	yes	64
39.29	odd	6		273.2.r.b.68.24	yes	64
91.3	odd	6	inner	273.2.bf.b.185.24	yes	64
273.185	even	6	inner	273.2.bf.b.185.9	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.9	✓	64	13.3	even	3
273.2.r.b.68.24	yes	64	39.29	odd	6
273.2.r.b.269.9	yes	64	7.3	odd	6
273.2.r.b.269.24	yes	64	21.17	even	6
273.2.bf.b.152.9	yes	64	1.1	even	1	trivial
273.2.bf.b.152.24	yes	64	3.2	odd	2	inner
273.2.bf.b.185.9	yes	64	273.185	even	6	inner
273.2.bf.b.185.24	yes	64	91.3	odd	6	inner