Embedded newform 273.2.r.b.68.5

• Label: 273.2.r.b.68.5
• 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			68.5
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.28

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.97104i q^{2} +(1.69142 - 0.372976i) q^{3} -1.88498 q^{4} +(-0.272603 + 0.472162i) q^{5} +(-0.735148 - 3.33384i) q^{6} +(2.35644 + 1.20299i) q^{7} -0.226699i q^{8} +(2.72178 - 1.26171i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.97104i		−	1.39373i	−0.717201	−	0.696867i	\(-0.754577\pi\)
							0.717201	−	0.696867i	\(-0.245423\pi\)
\(3\)	1.69142	−	0.372976i	0.976540	−	0.215338i
							\(5\)	−0.272603	+	0.472162i	−0.121912	+	0.211157i	−0.920521	−	0.390692i	\(-0.872236\pi\)
0.798610	+	0.601849i	\(0.205569\pi\)
\(7\)	2.35644	+	1.20299i	0.890650	+	0.454689i
							\(11\)	0.383707	+	0.221534i	0.115692	+	0.0667949i	0.556729	−	0.830694i	\(-0.312056\pi\)
−0.441037	+	0.897489i	\(0.645389\pi\)
\(13\)	−2.64192	−	2.45362i	−0.732737	−	0.680512i
							\(17\)	−3.83879			−0.931044			−0.465522	−	0.885036i	\(-0.654133\pi\)
−0.465522	+	0.885036i	\(0.654133\pi\)
\(19\)	0.0717668	−	0.0414346i	0.0164644	−	0.00950575i	−0.491745	−	0.870739i	\(-0.663641\pi\)
							0.508210	+	0.861233i	\(0.330308\pi\)
\(23\)			7.16086i			1.49314i	0.665306	+	0.746571i	\(0.268301\pi\)
							−0.665306	+	0.746571i	\(0.731699\pi\)
\(29\)	−3.66493	+	2.11595i	−0.680560	+	0.392922i	−0.800066	−	0.599912i	\(-0.795202\pi\)
							0.119506	+	0.992834i	\(0.461869\pi\)
\(31\)	5.32612	−	3.07503i	0.956599	−	0.552293i	0.0614742	−	0.998109i	\(-0.480420\pi\)
							0.895125	+	0.445816i	\(0.147086\pi\)
\(37\)	−3.94217			−0.648088			−0.324044	−	0.946042i	\(-0.605043\pi\)
							−0.324044	+	0.946042i	\(0.605043\pi\)
\(41\)	−2.97274	−	5.14894i	−0.464265	−	0.804130i	0.534903	−	0.844913i	\(-0.320348\pi\)
							−0.999168	+	0.0407834i	\(0.987015\pi\)
\(43\)	−3.19077	+	5.52657i	−0.486588	+	0.842795i	−0.999881	−	0.0154185i	\(-0.995092\pi\)
							0.513293	+	0.858213i	\(0.328425\pi\)
\(47\)	−0.322950	+	0.559365i	−0.0471071	+	0.0815918i	−0.888618	−	0.458649i	\(-0.848334\pi\)
							0.841510	+	0.540241i	\(0.181667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.5	✓	64
3.2	odd	2	inner	273.2.r.b.68.28	yes	64
7.3	odd	6		273.2.bf.b.185.28	yes	64
13.9	even	3		273.2.bf.b.152.5	yes	64
21.17	even	6		273.2.bf.b.185.5	yes	64
39.35	odd	6		273.2.bf.b.152.28	yes	64
91.87	odd	6	inner	273.2.r.b.269.5	yes	64
273.269	even	6	inner	273.2.r.b.269.28	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.5	✓	64	1.1	even	1	trivial
273.2.r.b.68.28	yes	64	3.2	odd	2	inner
273.2.r.b.269.5	yes	64	91.87	odd	6	inner
273.2.r.b.269.28	yes	64	273.269	even	6	inner
273.2.bf.b.152.5	yes	64	13.9	even	3
273.2.bf.b.152.28	yes	64	39.35	odd	6
273.2.bf.b.185.5	yes	64	21.17	even	6
273.2.bf.b.185.28	yes	64	7.3	odd	6