Embedded newform 273.2.bf.b.152.2

• Label: 273.2.bf.b.152.2
• Level: $273$
• Weight: $2$
• Character: 273.152
• 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.bf (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			152.2
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25245 + 1.30045i) q^{2} +(-1.59264 + 0.680804i) q^{3} +(2.38234 - 4.12633i) q^{4} +(1.54658 - 2.67876i) q^{5} +(2.70199 - 3.60462i) q^{6} +(-2.63374 - 0.251835i) q^{7} +7.19066i q^{8} +(2.07301 - 2.16855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25245	+	1.30045i	−1.59272	+	0.919557i	−0.599881	+	0.800089i	\(0.704786\pi\)
							−0.992838	+	0.119468i	\(0.961881\pi\)
\(3\)	−1.59264	+	0.680804i	−0.919512	+	0.393062i
							\(5\)	1.54658	−	2.67876i	0.691653	−	1.19798i	−0.279643	−	0.960104i	\(-0.590216\pi\)
0.971296	−	0.237874i	\(-0.0764505\pi\)
\(7\)	−2.63374	−	0.251835i	−0.995460	−	0.0951845i
							\(11\)			3.27310i			0.986877i	0.869781	+	0.493438i	\(0.164260\pi\)
−0.869781	+	0.493438i	\(0.835740\pi\)
\(13\)	−3.57248	+	0.487210i	−0.990828	+	0.135128i
							\(17\)	−1.92219	+	3.32934i	−0.466200	+	0.807482i	−0.999255	−	0.0385984i	\(-0.987711\pi\)
0.533055	+	0.846081i	\(0.321044\pi\)
\(19\)			2.49516i			0.572429i	0.958166	+	0.286214i	\(0.0923970\pi\)
							−0.958166	+	0.286214i	\(0.907603\pi\)
\(23\)	2.95804	−	1.70783i	0.616794	−	0.356106i	−0.158826	−	0.987307i	\(-0.550771\pi\)
							0.775620	+	0.631200i	\(0.217437\pi\)
\(29\)	−3.85237	−	2.22417i	−0.715367	−	0.413017i	0.0976782	−	0.995218i	\(-0.468858\pi\)
							−0.813045	+	0.582201i	\(0.802192\pi\)
\(31\)	−5.89349	+	3.40261i	−1.05850	+	0.611126i	−0.925017	−	0.379925i	\(-0.875950\pi\)
							−0.133484	+	0.991051i	\(0.542617\pi\)
\(37\)	−0.981922	−	1.70074i	−0.161427	−	0.279600i	0.773954	−	0.633242i	\(-0.218276\pi\)
							−0.935381	+	0.353642i	\(0.884943\pi\)
\(41\)	−4.81925	+	8.34719i	−0.752641	+	1.30361i	0.193898	+	0.981022i	\(0.437887\pi\)
							−0.946539	+	0.322590i	\(0.895446\pi\)
\(43\)	3.70565	+	6.41837i	0.565106	+	0.978793i	0.997040	+	0.0768874i	\(0.0244982\pi\)
							−0.431933	+	0.901905i	\(0.642168\pi\)
\(47\)	−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.bf.b.152.2	yes	64
3.2	odd	2	inner	273.2.bf.b.152.31	yes	64
7.3	odd	6		273.2.r.b.269.2	yes	64
13.3	even	3		273.2.r.b.68.2	✓	64
21.17	even	6		273.2.r.b.269.31	yes	64
39.29	odd	6		273.2.r.b.68.31	yes	64
91.3	odd	6	inner	273.2.bf.b.185.31	yes	64
273.185	even	6	inner	273.2.bf.b.185.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.2	✓	64	13.3	even	3
273.2.r.b.68.31	yes	64	39.29	odd	6
273.2.r.b.269.2	yes	64	7.3	odd	6
273.2.r.b.269.31	yes	64	21.17	even	6
273.2.bf.b.152.2	yes	64	1.1	even	1	trivial
273.2.bf.b.152.31	yes	64	3.2	odd	2	inner
273.2.bf.b.185.2	yes	64	273.185	even	6	inner
273.2.bf.b.185.31	yes	64	91.3	odd	6	inner