Embedded newform 273.2.r.b.68.2

• Label: 273.2.r.b.68.2
• 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			68.2
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.31

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.60090i q^{2} +(0.206727 - 1.71967i) q^{3} -4.76468 q^{4} +(1.54658 - 2.67876i) q^{5} +(-4.47269 - 0.537676i) q^{6} +(1.09877 + 2.40680i) q^{7} +7.19066i q^{8} +(-2.91453 - 0.711004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.60090i		−	1.83911i	−0.392957	−	0.919557i	\(-0.628548\pi\)
							0.392957	−	0.919557i	\(-0.371452\pi\)
\(3\)	0.206727	−	1.71967i	0.119354	−	0.992852i
							\(5\)	1.54658	−	2.67876i	0.691653	−	1.19798i	−0.279643	−	0.960104i	\(-0.590216\pi\)
0.971296	−	0.237874i	\(-0.0764505\pi\)
\(7\)	1.09877	+	2.40680i	0.415298	+	0.909686i
							\(11\)	−2.83459	−	1.63655i	−0.854660	−	0.493438i	0.00756034	−	0.999971i	\(-0.497593\pi\)
−0.862220	+	0.506533i	\(0.830927\pi\)
\(13\)	3.57248	+	0.487210i	0.990828	+	0.135128i
							\(17\)	3.84439			0.932400			0.466200	−	0.884679i	\(-0.345623\pi\)
0.466200	+	0.884679i	\(0.345623\pi\)
\(19\)	2.16087	−	1.24758i	0.495738	−	0.286214i	−0.231214	−	0.972903i	\(-0.574270\pi\)
							0.726952	+	0.686689i	\(0.240936\pi\)
\(23\)			3.41565i			0.712212i	0.934445	+	0.356106i	\(0.115896\pi\)
							−0.934445	+	0.356106i	\(0.884104\pi\)
\(29\)	3.85237	−	2.22417i	0.715367	−	0.413017i	−0.0976782	−	0.995218i	\(-0.531142\pi\)
							0.813045	+	0.582201i	\(0.197808\pi\)
\(31\)	−5.89349	+	3.40261i	−1.05850	+	0.611126i	−0.925017	−	0.379925i	\(-0.875950\pi\)
							−0.133484	+	0.991051i	\(0.542617\pi\)
\(37\)	1.96384			0.322854			0.161427	−	0.986885i	\(-0.448390\pi\)
							0.161427	+	0.986885i	\(0.448390\pi\)
\(41\)	−4.81925	−	8.34719i	−0.752641	−	1.30361i	−0.946539	−	0.322590i	\(-0.895446\pi\)
							0.193898	−	0.981022i	\(-0.437887\pi\)
\(43\)	3.70565	−	6.41837i	0.565106	−	0.978793i	−0.431933	−	0.901905i	\(-0.642168\pi\)
							0.997040	−	0.0768874i	\(-0.0244982\pi\)
\(47\)	−1.63297	+	2.82838i	−0.238192	+	0.412561i	−0.960196	−	0.279328i	\(-0.909888\pi\)
							0.722003	+	0.691890i	\(0.243222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.2	✓	64
3.2	odd	2	inner	273.2.r.b.68.31	yes	64
7.3	odd	6		273.2.bf.b.185.31	yes	64
13.9	even	3		273.2.bf.b.152.2	yes	64
21.17	even	6		273.2.bf.b.185.2	yes	64
39.35	odd	6		273.2.bf.b.152.31	yes	64
91.87	odd	6	inner	273.2.r.b.269.2	yes	64
273.269	even	6	inner	273.2.r.b.269.31	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.2	✓	64	1.1	even	1	trivial
273.2.r.b.68.31	yes	64	3.2	odd	2	inner
273.2.r.b.269.2	yes	64	91.87	odd	6	inner
273.2.r.b.269.31	yes	64	273.269	even	6	inner
273.2.bf.b.152.2	yes	64	13.9	even	3
273.2.bf.b.152.31	yes	64	39.35	odd	6
273.2.bf.b.185.2	yes	64	21.17	even	6
273.2.bf.b.185.31	yes	64	7.3	odd	6