Embedded newform 273.2.ba.c.38.1

• Label: 273.2.ba.c.38.1
• Level: $273$
• Weight: $2$
• Character: 273.38
• 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.ba (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			38.1
Character	\(\chi\)	\(=\)	273.38
Dual form			273.2.ba.c.194.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32290 - 2.29133i) q^{2} +(-1.08139 + 1.35300i) q^{3} +(-2.50014 + 4.33037i) q^{4} +(-2.63347 + 1.52043i) q^{5} +(4.53074 + 0.687935i) q^{6} +(2.25766 - 1.37948i) q^{7} +7.93816 q^{8} +(-0.661205 - 2.92623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32290	−	2.29133i	−0.935433	−	1.62022i	−0.773859	−	0.633357i	\(-0.781676\pi\)
							−0.161574	−	0.986861i	\(-0.551657\pi\)
\(3\)	−1.08139	+	1.35300i	−0.624339	+	0.781153i
							\(5\)	−2.63347	+	1.52043i	−1.17772	+	0.679959i	−0.955487	−	0.295034i	\(-0.904669\pi\)
−0.222236	+	0.974993i	\(0.571336\pi\)
\(7\)	2.25766	−	1.37948i	0.853315	−	0.521395i
							\(11\)	0.962914	−	1.66782i	0.290329	−	0.502865i	−0.683558	−	0.729896i	\(-0.739568\pi\)
0.973888	+	0.227031i	\(0.0729018\pi\)
\(13\)	−3.30166	+	1.44880i	−0.915717	+	0.401824i
							\(17\)	1.25588	−	2.17524i	0.304595	−	0.527574i	−0.672576	−	0.740028i	\(-0.734812\pi\)
0.977171	+	0.212454i	\(0.0681455\pi\)
\(19\)	−1.25405	−	2.17208i	−0.287699	−	0.498309i	0.685561	−	0.728015i	\(-0.259557\pi\)
							−0.973260	+	0.229706i	\(0.926224\pi\)
\(23\)	6.29765	−	3.63595i	1.31315	−	0.758148i	0.330534	−	0.943794i	\(-0.392771\pi\)
							0.982617	+	0.185647i	\(0.0594380\pi\)
\(29\)		−	3.34366i		−	0.620902i	−0.950589	−	0.310451i	\(-0.899520\pi\)
							0.950589	−	0.310451i	\(-0.100480\pi\)
\(31\)	2.39107	−	4.14146i	0.429450	−	0.743829i	−0.567375	−	0.823460i	\(-0.692041\pi\)
							0.996824	+	0.0796310i	\(0.0253742\pi\)
\(37\)	6.24946	−	3.60813i	1.02741	−	0.593173i	0.111165	−	0.993802i	\(-0.464542\pi\)
							0.916240	+	0.400629i	\(0.131208\pi\)
\(41\)		−	4.50913i		−	0.704208i	−0.935961	−	0.352104i	\(-0.885466\pi\)
							0.935961	−	0.352104i	\(-0.114534\pi\)
\(43\)	−2.54962			−0.388814			−0.194407	−	0.980921i	\(-0.562278\pi\)
							−0.194407	+	0.980921i	\(0.562278\pi\)
\(47\)	−8.40648	+	4.85348i	−1.22621	+	0.707953i	−0.966235	−	0.257662i	\(-0.917048\pi\)
							−0.259976	+	0.965615i	\(0.583715\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.ba.c.38.1	✓	64
3.2	odd	2	inner	273.2.ba.c.38.32	yes	64
7.5	odd	6	inner	273.2.ba.c.194.2	yes	64
13.12	even	2	inner	273.2.ba.c.38.31	yes	64
21.5	even	6	inner	273.2.ba.c.194.31	yes	64
39.38	odd	2	inner	273.2.ba.c.38.2	yes	64
91.12	odd	6	inner	273.2.ba.c.194.32	yes	64
273.194	even	6	inner	273.2.ba.c.194.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.ba.c.38.1	✓	64	1.1	even	1	trivial
273.2.ba.c.38.2	yes	64	39.38	odd	2	inner
273.2.ba.c.38.31	yes	64	13.12	even	2	inner
273.2.ba.c.38.32	yes	64	3.2	odd	2	inner
273.2.ba.c.194.1	yes	64	273.194	even	6	inner
273.2.ba.c.194.2	yes	64	7.5	odd	6	inner
273.2.ba.c.194.31	yes	64	21.5	even	6	inner
273.2.ba.c.194.32	yes	64	91.12	odd	6	inner