Embedded newform 273.2.r.b.68.14

• Label: 273.2.r.b.68.14
• 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			68.14
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.19

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.771110i q^{2} +(0.714765 + 1.57769i) q^{3} +1.40539 q^{4} +(0.907647 - 1.57209i) q^{5} +(1.21657 - 0.551162i) q^{6} +(0.574285 - 2.58267i) q^{7} -2.62593i q^{8} +(-1.97822 + 2.25536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	0.771110i		−	0.545257i	−0.962119	−	0.272629i	\(-0.912107\pi\)
							0.962119	−	0.272629i	\(-0.0878930\pi\)
\(3\)	0.714765	+	1.57769i	0.412670	+	0.910881i
							\(5\)	0.907647	−	1.57209i	0.405912	−	0.703060i	−0.588515	−	0.808486i	\(-0.700287\pi\)
0.994427	+	0.105426i	\(0.0336206\pi\)
\(7\)	0.574285	−	2.58267i	0.217059	−	0.976158i
							\(11\)	0.504401	+	0.291216i	0.152083	+	0.0878049i	0.574110	−	0.818778i	\(-0.305348\pi\)
−0.422028	+	0.906583i	\(0.638681\pi\)
\(13\)	−3.53127	+	0.728085i	−0.979399	+	0.201935i
							\(17\)	0.958191			0.232395			0.116198	−	0.993226i	\(-0.462929\pi\)
0.116198	+	0.993226i	\(0.462929\pi\)
\(19\)	−1.66703	+	0.962462i	−0.382444	+	0.220804i	−0.678881	−	0.734248i	\(-0.737535\pi\)
							0.296437	+	0.955052i	\(0.404201\pi\)
\(23\)			8.73145i			1.82063i	0.413912	+	0.910317i	\(0.364162\pi\)
							−0.413912	+	0.910317i	\(0.635838\pi\)
\(29\)	4.26919	−	2.46482i	0.792769	−	0.457705i	−0.0481677	−	0.998839i	\(-0.515338\pi\)
							0.840936	+	0.541134i	\(0.182005\pi\)
\(31\)	−2.83579	+	1.63724i	−0.509322	+	0.294057i	−0.732555	−	0.680708i	\(-0.761672\pi\)
							0.223233	+	0.974765i	\(0.428339\pi\)
\(37\)	−1.79997			−0.295914			−0.147957	−	0.988994i	\(-0.547270\pi\)
							−0.147957	+	0.988994i	\(0.547270\pi\)
\(41\)	−1.73088	−	2.99798i	−0.270319	−	0.468205i	0.698625	−	0.715488i	\(-0.253796\pi\)
							−0.968943	+	0.247283i	\(0.920462\pi\)
\(43\)	−0.367529	+	0.636578i	−0.0560476	+	0.0970772i	−0.892688	−	0.450675i	\(-0.851183\pi\)
							0.836640	+	0.547753i	\(0.184517\pi\)
\(47\)	−3.49201	+	6.04834i	−0.509362	+	0.882241i	0.490579	+	0.871397i	\(0.336785\pi\)
							−0.999941	+	0.0108447i	\(0.996548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.14	✓	64
3.2	odd	2	inner	273.2.r.b.68.19	yes	64
7.3	odd	6		273.2.bf.b.185.19	yes	64
13.9	even	3		273.2.bf.b.152.14	yes	64
21.17	even	6		273.2.bf.b.185.14	yes	64
39.35	odd	6		273.2.bf.b.152.19	yes	64
91.87	odd	6	inner	273.2.r.b.269.14	yes	64
273.269	even	6	inner	273.2.r.b.269.19	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.14	✓	64	1.1	even	1	trivial
273.2.r.b.68.19	yes	64	3.2	odd	2	inner
273.2.r.b.269.14	yes	64	91.87	odd	6	inner
273.2.r.b.269.19	yes	64	273.269	even	6	inner
273.2.bf.b.152.14	yes	64	13.9	even	3
273.2.bf.b.152.19	yes	64	39.35	odd	6
273.2.bf.b.185.14	yes	64	21.17	even	6
273.2.bf.b.185.19	yes	64	7.3	odd	6