Embedded newform 273.2.r.b.68.9

• Label: 273.2.r.b.68.9
• 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			68.9
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.24

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.53974i q^{2} +(-1.04458 - 1.38161i) q^{3} -0.370785 q^{4} +(0.956175 - 1.65614i) q^{5} +(-2.12732 + 1.60838i) q^{6} +(-2.63109 - 0.278191i) q^{7} -2.50856i q^{8} +(-0.817709 + 2.88641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.53974i		−	1.08876i	−0.838840	−	0.544379i	\(-0.816765\pi\)
							0.838840	−	0.544379i	\(-0.183235\pi\)
\(3\)	−1.04458	−	1.38161i	−0.603088	−	0.797675i
							\(5\)	0.956175	−	1.65614i	0.427614	−	0.740650i	−0.569046	−	0.822306i	\(-0.692687\pi\)
0.996661	+	0.0816556i	\(0.0260208\pi\)
\(7\)	−2.63109	−	0.278191i	−0.994457	−	0.105146i
							\(11\)	1.99849	+	1.15383i	0.602566	+	0.347892i	0.770050	−	0.637983i	\(-0.220231\pi\)
−0.167484	+	0.985875i	\(0.553564\pi\)
\(13\)	−0.963713	−	3.47437i	−0.267286	−	0.963617i
							\(17\)	3.75390			0.910453			0.455227	−	0.890376i	\(-0.349558\pi\)
0.455227	+	0.890376i	\(0.349558\pi\)
\(19\)	−4.18714	+	2.41745i	−0.960597	+	0.554601i	−0.896357	−	0.443334i	\(-0.853796\pi\)
							−0.0642400	+	0.997934i	\(0.520462\pi\)
\(23\)			0.202723i			0.0422707i	0.999777	+	0.0211353i	\(0.00672809\pi\)
							−0.999777	+	0.0211353i	\(0.993272\pi\)
\(29\)	−0.228216	+	0.131761i	−0.0423786	+	0.0244673i	−0.521040	−	0.853532i	\(-0.674456\pi\)
							0.478661	+	0.878000i	\(0.341122\pi\)
\(31\)	1.76906	−	1.02137i	0.317732	−	0.183443i	−0.332649	−	0.943051i	\(-0.607942\pi\)
							0.650381	+	0.759608i	\(0.274609\pi\)
\(37\)	4.51347			0.742010			0.371005	−	0.928631i	\(-0.379013\pi\)
							0.371005	+	0.928631i	\(0.379013\pi\)
\(41\)	−5.87191	−	10.1704i	−0.917038	−	1.58836i	−0.803889	−	0.594779i	\(-0.797240\pi\)
							−0.113149	−	0.993578i	\(-0.536094\pi\)
\(43\)	−2.00837	+	3.47860i	−0.306273	+	0.530481i	−0.977544	−	0.210731i	\(-0.932415\pi\)
							0.671271	+	0.741212i	\(0.265749\pi\)
\(47\)	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.r.b.68.9	✓	64
3.2	odd	2	inner	273.2.r.b.68.24	yes	64
7.3	odd	6		273.2.bf.b.185.24	yes	64
13.9	even	3		273.2.bf.b.152.9	yes	64
21.17	even	6		273.2.bf.b.185.9	yes	64
39.35	odd	6		273.2.bf.b.152.24	yes	64
91.87	odd	6	inner	273.2.r.b.269.9	yes	64
273.269	even	6	inner	273.2.r.b.269.24	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.9	✓	64	1.1	even	1	trivial
273.2.r.b.68.24	yes	64	3.2	odd	2	inner
273.2.r.b.269.9	yes	64	91.87	odd	6	inner
273.2.r.b.269.24	yes	64	273.269	even	6	inner
273.2.bf.b.152.9	yes	64	13.9	even	3
273.2.bf.b.152.24	yes	64	39.35	odd	6
273.2.bf.b.185.9	yes	64	21.17	even	6
273.2.bf.b.185.24	yes	64	7.3	odd	6