Embedded newform 273.2.u.c.62.1

• Label: 273.2.u.c.62.1
• Level: $273$
• Weight: $2$
• Character: 273.62
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.u (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			62.1
Character	\(\chi\)	\(=\)	273.62
Dual form			273.2.u.c.251.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35301 - 2.34347i) q^{2} +(-1.31833 + 1.12339i) q^{3} +(-2.66125 + 4.60942i) q^{4} +0.967216i q^{5} +(4.41634 + 1.56953i) q^{6} +(-2.52716 + 0.783246i) q^{7} +8.99072 q^{8} +(0.476002 - 2.96200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 40 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35301	−	2.34347i	−0.956720	−	1.65709i	−0.730383	−	0.683038i	\(-0.760658\pi\)
							−0.226337	−	0.974049i	\(-0.572675\pi\)
\(3\)	−1.31833	+	1.12339i	−0.761140	+	0.648588i
							\(5\)			0.967216i			0.432552i	0.976332	+	0.216276i	\(0.0693911\pi\)
−0.976332	+	0.216276i	\(0.930609\pi\)
\(7\)	−2.52716	+	0.783246i	−0.955176	+	0.296039i
							\(11\)	−1.41869	−	2.45724i	−0.427750	−	0.740885i	0.568923	−	0.822391i	\(-0.307360\pi\)
−0.996673	+	0.0815059i	\(0.974027\pi\)
\(13\)	3.02075	−	1.96852i	0.837806	−	0.545968i
							\(17\)	3.38765	−	5.86757i	0.821625	−	1.42310i	−0.0828470	−	0.996562i	\(-0.526401\pi\)
0.904472	−	0.426534i	\(-0.140265\pi\)
\(19\)	−1.63797	+	2.83704i	−0.375775	+	0.650862i	−0.990443	−	0.137924i	\(-0.955957\pi\)
							0.614667	+	0.788786i	\(0.289290\pi\)
\(23\)	2.67158	−	1.54243i	0.557062	−	0.321620i	−0.194903	−	0.980822i	\(-0.562439\pi\)
							0.751965	+	0.659203i	\(0.229106\pi\)
\(29\)	1.27685	−	0.737187i	0.237104	−	0.136892i	−0.376741	−	0.926319i	\(-0.622955\pi\)
							0.613845	+	0.789426i	\(0.289622\pi\)
\(31\)	2.59492			0.466061			0.233030	−	0.972469i	\(-0.425136\pi\)
							0.233030	+	0.972469i	\(0.425136\pi\)
\(37\)	−1.12653	+	0.650404i	−0.185201	+	0.106926i	−0.589734	−	0.807598i	\(-0.700767\pi\)
							0.404533	+	0.914523i	\(0.367434\pi\)
\(41\)	−7.00735	+	4.04570i	−1.09436	+	0.631831i	−0.934735	−	0.355346i	\(-0.884363\pi\)
							−0.159629	+	0.987177i	\(0.551030\pi\)
\(43\)	2.55507	−	4.42551i	0.389645	−	0.674884i	−0.602757	−	0.797925i	\(-0.705931\pi\)
							0.992402	+	0.123041i	\(0.0392645\pi\)
\(47\)		−	6.46207i		−	0.942590i	−0.881976	−	0.471295i	\(-0.843787\pi\)
							0.881976	−	0.471295i	\(-0.156213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.u.c.62.1	✓	64
3.2	odd	2	inner	273.2.u.c.62.31	yes	64
7.6	odd	2	inner	273.2.u.c.62.2	yes	64
13.4	even	6	inner	273.2.u.c.251.32	yes	64
21.20	even	2	inner	273.2.u.c.62.32	yes	64
39.17	odd	6	inner	273.2.u.c.251.2	yes	64
91.69	odd	6	inner	273.2.u.c.251.31	yes	64
273.251	even	6	inner	273.2.u.c.251.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.u.c.62.1	✓	64	1.1	even	1	trivial
273.2.u.c.62.2	yes	64	7.6	odd	2	inner
273.2.u.c.62.31	yes	64	3.2	odd	2	inner
273.2.u.c.62.32	yes	64	21.20	even	2	inner
273.2.u.c.251.1	yes	64	273.251	even	6	inner
273.2.u.c.251.2	yes	64	39.17	odd	6	inner
273.2.u.c.251.31	yes	64	91.69	odd	6	inner
273.2.u.c.251.32	yes	64	13.4	even	6	inner