Embedded newform 273.2.r.b.68.8

• Label: 273.2.r.b.68.8
• 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			68.8
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.25

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.85049i q^{2} +(-1.73048 + 0.0736936i) q^{3} -1.42433 q^{4} +(1.21731 - 2.10844i) q^{5} +(0.136370 + 3.20225i) q^{6} +(2.52177 - 0.800422i) q^{7} -1.06527i q^{8} +(2.98914 - 0.255051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.85049i		−	1.30850i	−0.756279	−	0.654249i	\(-0.772985\pi\)
							0.756279	−	0.654249i	\(-0.227015\pi\)
\(3\)	−1.73048	+	0.0736936i	−0.999094	+	0.0425470i
							\(5\)	1.21731	−	2.10844i	0.544398	−	0.942924i	−0.454247	−	0.890876i	\(-0.650092\pi\)
0.998645	−	0.0520486i	\(-0.0165751\pi\)
\(7\)	2.52177	−	0.800422i	0.953140	−	0.302531i
							\(11\)	−2.52051	−	1.45522i	−0.759963	−	0.438765i	0.0693192	−	0.997595i	\(-0.477917\pi\)
−0.829283	+	0.558829i	\(0.811251\pi\)
\(13\)	−2.41102	+	2.68085i	−0.668697	+	0.743535i
							\(17\)	0.935851			0.226977			0.113489	−	0.993539i	\(-0.463797\pi\)
0.113489	+	0.993539i	\(0.463797\pi\)
\(19\)	−1.91428	+	1.10521i	−0.439166	+	0.253552i	−0.703244	−	0.710949i	\(-0.748266\pi\)
							0.264078	+	0.964501i	\(0.414932\pi\)
\(23\)		−	6.24184i		−	1.30151i	−0.759287	−	0.650756i	\(-0.774452\pi\)
							0.759287	−	0.650756i	\(-0.225548\pi\)
\(29\)	−1.34529	+	0.776702i	−0.249814	+	0.144230i	−0.619679	−	0.784855i	\(-0.712737\pi\)
							0.369865	+	0.929085i	\(0.379404\pi\)
\(31\)	2.50768	−	1.44781i	0.450392	−	0.260034i	−0.257604	−	0.966251i	\(-0.582933\pi\)
							0.707996	+	0.706217i	\(0.249600\pi\)
\(37\)	8.15852			1.34125			0.670626	−	0.741796i	\(-0.266026\pi\)
							0.670626	+	0.741796i	\(0.266026\pi\)
\(41\)	4.41024	+	7.63876i	0.688764	+	1.19297i	0.972238	+	0.233994i	\(0.0751795\pi\)
							−0.283475	+	0.958980i	\(0.591487\pi\)
\(43\)	5.40029	−	9.35358i	0.823537	−	1.42641i	−0.0794962	−	0.996835i	\(-0.525331\pi\)
							0.903033	−	0.429572i	\(-0.141336\pi\)
\(47\)	−3.63232	+	6.29136i	−0.529828	+	0.917689i	0.469567	+	0.882897i	\(0.344410\pi\)
							−0.999395	+	0.0347919i	\(0.988923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.8	✓	64
3.2	odd	2	inner	273.2.r.b.68.25	yes	64
7.3	odd	6		273.2.bf.b.185.25	yes	64
13.9	even	3		273.2.bf.b.152.8	yes	64
21.17	even	6		273.2.bf.b.185.8	yes	64
39.35	odd	6		273.2.bf.b.152.25	yes	64
91.87	odd	6	inner	273.2.r.b.269.8	yes	64
273.269	even	6	inner	273.2.r.b.269.25	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.8	✓	64	1.1	even	1	trivial
273.2.r.b.68.25	yes	64	3.2	odd	2	inner
273.2.r.b.269.8	yes	64	91.87	odd	6	inner
273.2.r.b.269.25	yes	64	273.269	even	6	inner
273.2.bf.b.152.8	yes	64	13.9	even	3
273.2.bf.b.152.25	yes	64	39.35	odd	6
273.2.bf.b.185.8	yes	64	21.17	even	6
273.2.bf.b.185.25	yes	64	7.3	odd	6