Embedded newform 273.2.r.b.269.32

• Label: 273.2.r.b.269.32
• Level: $273$
• Weight: $2$
• Character: 273.269
• 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			269.32
Character	\(\chi\)	\(=\)	273.269
Dual form			273.2.r.b.68.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.73146i q^{2} +(-1.52761 - 0.816332i) q^{3} -5.46085 q^{4} +(0.121221 + 0.209960i) q^{5} +(2.22977 - 4.17261i) q^{6} +(-2.13335 - 1.56488i) q^{7} -9.45317i q^{8} +(1.66721 + 2.49408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.73146i			1.93143i	0.259603	+	0.965716i	\(0.416408\pi\)
							−0.259603	+	0.965716i	\(0.583592\pi\)
\(3\)	−1.52761	−	0.816332i	−0.881968	−	0.471309i
							\(5\)	0.121221	+	0.209960i	0.0542116	+	0.0938972i	0.891858	−	0.452316i	\(-0.149402\pi\)
−0.837646	+	0.546213i	\(0.816069\pi\)
\(7\)	−2.13335	−	1.56488i	−0.806329	−	0.591468i
							\(11\)	4.30944	−	2.48805i	1.29934	−	0.750176i	0.319053	−	0.947737i	\(-0.396635\pi\)
0.980291	+	0.197560i	\(0.0633018\pi\)
\(13\)	−3.55884	−	0.578487i	−0.987045	−	0.160444i
							\(17\)	−3.50135			−0.849203			−0.424601	−	0.905380i	\(-0.639586\pi\)
−0.424601	+	0.905380i	\(0.639586\pi\)
\(19\)	−1.51336	−	0.873740i	−0.347189	−	0.200450i	0.316257	−	0.948673i	\(-0.397574\pi\)
							−0.663447	+	0.748224i	\(0.730907\pi\)
\(23\)		−	2.82905i		−	0.589898i	−0.955513	−	0.294949i	\(-0.904697\pi\)
							0.955513	−	0.294949i	\(-0.0953026\pi\)
\(29\)	−0.841553	−	0.485871i	−0.156272	−	0.0902239i	0.419825	−	0.907605i	\(-0.362092\pi\)
							−0.576097	+	0.817381i	\(0.695425\pi\)
\(31\)	−4.95063	−	2.85824i	−0.889159	−	0.513356i	−0.0154914	−	0.999880i	\(-0.504931\pi\)
							−0.873667	+	0.486524i	\(0.838265\pi\)
\(37\)	−6.98982			−1.14912			−0.574560	−	0.818463i	\(-0.694827\pi\)
							−0.574560	+	0.818463i	\(0.694827\pi\)
\(41\)	−0.800378	+	1.38629i	−0.124998	+	0.216503i	−0.921732	−	0.387827i	\(-0.873226\pi\)
							0.796734	+	0.604330i	\(0.206559\pi\)
\(43\)	−1.17144	−	2.02899i	−0.178643	−	0.309418i	0.762773	−	0.646666i	\(-0.223837\pi\)
							−0.941416	+	0.337248i	\(0.890504\pi\)
\(47\)	0.0888026	+	0.153811i	0.0129532	+	0.0224356i	0.872429	−	0.488740i	\(-0.162543\pi\)
							−0.859476	+	0.511176i	\(0.829210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.269.32	yes	64
3.2	odd	2	inner	273.2.r.b.269.1	yes	64
7.5	odd	6		273.2.bf.b.152.32	yes	64
13.3	even	3		273.2.bf.b.185.1	yes	64
21.5	even	6		273.2.bf.b.152.1	yes	64
39.29	odd	6		273.2.bf.b.185.32	yes	64
91.68	odd	6	inner	273.2.r.b.68.32	yes	64
273.68	even	6	inner	273.2.r.b.68.1	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.1	✓	64	273.68	even	6	inner
273.2.r.b.68.32	yes	64	91.68	odd	6	inner
273.2.r.b.269.1	yes	64	3.2	odd	2	inner
273.2.r.b.269.32	yes	64	1.1	even	1	trivial
273.2.bf.b.152.1	yes	64	21.5	even	6
273.2.bf.b.152.32	yes	64	7.5	odd	6
273.2.bf.b.185.1	yes	64	13.3	even	3
273.2.bf.b.185.32	yes	64	39.29	odd	6