Embedded newform 273.2.r.b.269.19

• Label: 273.2.r.b.269.19
• Level: $273$
• Weight: $2$
• Character: 273.269
• 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			269.19
Character	\(\chi\)	\(=\)	273.269
Dual form			273.2.r.b.68.14

$q$-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 \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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	\(0.0878930\pi\)
							−0.962119	+	0.272629i	\(0.912107\pi\)
\(3\)	0.714765	−	1.57769i	0.412670	−	0.910881i
							\(5\)	0.907647	+	1.57209i	0.405912	+	0.703060i	0.994427	−	0.105426i	\(-0.0336206\pi\)
−0.588515	+	0.808486i	\(0.700287\pi\)
\(7\)	0.574285	+	2.58267i	0.217059	+	0.976158i
							\(11\)	0.504401	−	0.291216i	0.152083	−	0.0878049i	−0.422028	−	0.906583i	\(-0.638681\pi\)
0.574110	+	0.818778i	\(0.305348\pi\)
\(13\)	−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	0.296437	−	0.955052i	\(-0.404201\pi\)
							−0.678881	+	0.734248i	\(0.737535\pi\)
\(23\)		−	8.73145i		−	1.82063i	−0.413912	−	0.910317i	\(-0.635838\pi\)
							0.413912	−	0.910317i	\(-0.364162\pi\)
\(29\)	4.26919	+	2.46482i	0.792769	+	0.457705i	0.840936	−	0.541134i	\(-0.182005\pi\)
							−0.0481677	+	0.998839i	\(0.515338\pi\)
\(31\)	−2.83579	−	1.63724i	−0.509322	−	0.294057i	0.223233	−	0.974765i	\(-0.428339\pi\)
							−0.732555	+	0.680708i	\(0.761672\pi\)
\(37\)	−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	−0.968943	−	0.247283i	\(-0.920462\pi\)
							0.698625	+	0.715488i	\(0.253796\pi\)
\(43\)	−0.367529	−	0.636578i	−0.0560476	−	0.0970772i	0.836640	−	0.547753i	\(-0.184517\pi\)
							−0.892688	+	0.450675i	\(0.851183\pi\)
\(47\)	−3.49201	−	6.04834i	−0.509362	−	0.882241i	−0.999941	−	0.0108447i	\(-0.996548\pi\)
							0.490579	−	0.871397i	\(-0.336785\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.269.19	yes	64
3.2	odd	2	inner	273.2.r.b.269.14	yes	64
7.5	odd	6		273.2.bf.b.152.19	yes	64
13.3	even	3		273.2.bf.b.185.14	yes	64
21.5	even	6		273.2.bf.b.152.14	yes	64
39.29	odd	6		273.2.bf.b.185.19	yes	64
91.68	odd	6	inner	273.2.r.b.68.19	yes	64
273.68	even	6	inner	273.2.r.b.68.14	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.14	✓	64	273.68	even	6	inner
273.2.r.b.68.19	yes	64	91.68	odd	6	inner
273.2.r.b.269.14	yes	64	3.2	odd	2	inner
273.2.r.b.269.19	yes	64	1.1	even	1	trivial
273.2.bf.b.152.14	yes	64	21.5	even	6
273.2.bf.b.152.19	yes	64	7.5	odd	6
273.2.bf.b.185.14	yes	64	13.3	even	3
273.2.bf.b.185.19	yes	64	39.29	odd	6