Embedded newform 273.2.y.b.101.2

• Label: 273.2.y.b.101.2
• Level: $273$
• Weight: $2$
• Character: 273.101
• 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.y (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			101.2
Character	\(\chi\)	\(=\)	273.101
Dual form			273.2.y.b.173.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30520 + 2.26068i) q^{2} +(1.39879 - 1.02146i) q^{3} +(-2.40711 - 4.16923i) q^{4} +(-1.65792 + 0.957199i) q^{5} +(0.483493 + 4.49543i) q^{6} +(2.51605 - 0.818240i) q^{7} +7.34624 q^{8} +(0.913228 - 2.85762i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 28 q^{4} - 12 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30520	+	2.26068i	−0.922917	+	1.59854i	−0.128041	+	0.991769i	\(0.540869\pi\)
							−0.794877	+	0.606771i	\(0.792464\pi\)
\(3\)	1.39879	−	1.02146i	0.807592	−	0.589742i
							\(5\)	−1.65792	+	0.957199i	−0.741443	+	0.428073i	−0.822594	−	0.568629i	\(-0.807474\pi\)
0.0811505	+	0.996702i	\(0.474141\pi\)
\(7\)	2.51605	−	0.818240i	0.950976	−	0.309266i
							\(11\)	2.78733			0.840410			0.420205	−	0.907429i	\(-0.361958\pi\)
0.420205	+	0.907429i	\(0.361958\pi\)
\(13\)	0.558739	+	3.56200i	0.154966	+	0.987920i
							\(17\)	−1.16787	−	2.02280i	−0.283249	−	0.490601i	0.688934	−	0.724824i	\(-0.258079\pi\)
−0.972183	+	0.234222i	\(0.924746\pi\)
\(19\)	7.07754			1.62370			0.811850	−	0.583866i	\(-0.198461\pi\)
							0.811850	+	0.583866i	\(0.198461\pi\)
\(23\)	3.93089	+	2.26950i	0.819648	+	0.473224i	0.850295	−	0.526306i	\(-0.176423\pi\)
							−0.0306470	+	0.999530i	\(0.509757\pi\)
\(29\)	−0.597785	+	0.345131i	−0.111006	+	0.0640893i	−0.554475	−	0.832200i	\(-0.687081\pi\)
							0.443469	+	0.896290i	\(0.353748\pi\)
\(31\)	−1.41372	+	2.44863i	−0.253911	+	0.439787i	−0.964599	−	0.263720i	\(-0.915050\pi\)
							0.710688	+	0.703507i	\(0.248384\pi\)
\(37\)	−7.57842	−	4.37540i	−1.24588	−	0.719311i	−0.275598	−	0.961273i	\(-0.588876\pi\)
							−0.970286	+	0.241962i	\(0.922209\pi\)
\(41\)	1.88876	−	1.09047i	0.294974	−	0.170303i	−0.345209	−	0.938526i	\(-0.612192\pi\)
							0.640183	+	0.768223i	\(0.278859\pi\)
\(43\)	4.93570	−	8.54888i	0.752687	−	1.30369i	−0.193829	−	0.981035i	\(-0.562091\pi\)
							0.946516	−	0.322657i	\(-0.104576\pi\)
\(47\)	−6.62901	+	3.82726i	−0.966940	+	0.558263i	−0.898302	−	0.439378i	\(-0.855199\pi\)
							−0.0686383	+	0.997642i	\(0.521865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.y.b.101.2	✓	64
3.2	odd	2	inner	273.2.y.b.101.31	yes	64
7.5	odd	6		273.2.br.b.257.31	yes	64
13.4	even	6		273.2.br.b.17.2	yes	64
21.5	even	6		273.2.br.b.257.2	yes	64
39.17	odd	6		273.2.br.b.17.31	yes	64
91.82	odd	6	inner	273.2.y.b.173.31	yes	64
273.173	even	6	inner	273.2.y.b.173.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.2	✓	64	1.1	even	1	trivial
273.2.y.b.101.31	yes	64	3.2	odd	2	inner
273.2.y.b.173.2	yes	64	273.173	even	6	inner
273.2.y.b.173.31	yes	64	91.82	odd	6	inner
273.2.br.b.17.2	yes	64	13.4	even	6
273.2.br.b.17.31	yes	64	39.17	odd	6
273.2.br.b.257.2	yes	64	21.5	even	6
273.2.br.b.257.31	yes	64	7.5	odd	6