Embedded newform 273.2.bf.b.185.32

• Label: 273.2.bf.b.185.32
• Level: $273$
• Weight: $2$
• Character: 273.185
• 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.bf (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			185.32
Character	\(\chi\)	\(=\)	273.185
Dual form			273.2.bf.b.152.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.36551 + 1.36573i) q^{2} +(-1.47077 - 0.914786i) q^{3} +(2.73043 + 4.72924i) q^{4} +(-0.121221 - 0.209960i) q^{5} +(-2.22977 - 4.17261i) q^{6} +(-0.288550 + 2.62997i) q^{7} +9.45317i q^{8} +(1.32633 + 2.69088i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.36551	+	1.36573i	1.67267	+	0.965716i	0.966135	+	0.258035i	\(0.0830751\pi\)
							0.706533	+	0.707680i	\(0.250258\pi\)
\(3\)	−1.47077	−	0.914786i	−0.849150	−	0.528152i
							\(5\)	−0.121221	−	0.209960i	−0.0542116	−	0.0938972i	0.837646	−	0.546213i	\(-0.183931\pi\)
−0.891858	+	0.452316i	\(0.850598\pi\)
\(7\)	−0.288550	+	2.62997i	−0.109062	+	0.994035i
							\(11\)		−	4.97611i		−	1.50035i	−0.661238	−	0.750176i	\(-0.729969\pi\)
0.661238	−	0.750176i	\(-0.270031\pi\)
\(13\)	3.55884	−	0.578487i	0.987045	−	0.160444i
							\(17\)	−1.75068	−	3.03226i	−0.424601	−	0.735431i	0.571782	−	0.820406i	\(-0.306252\pi\)
−0.996383	+	0.0849746i	\(0.972919\pi\)
\(19\)			1.74748i			0.400900i	0.979704	+	0.200450i	\(0.0642403\pi\)
							−0.979704	+	0.200450i	\(0.935760\pi\)
\(23\)	−2.45003	−	1.41453i	−0.510866	−	0.294949i	0.222323	−	0.974973i	\(-0.428636\pi\)
							−0.733190	+	0.680024i	\(0.761969\pi\)
\(29\)	−0.841553	+	0.485871i	−0.156272	+	0.0902239i	−0.576097	−	0.817381i	\(-0.695425\pi\)
							0.419825	+	0.907605i	\(0.362092\pi\)
\(31\)	−4.95063	−	2.85824i	−0.889159	−	0.513356i	−0.0154914	−	0.999880i	\(-0.504931\pi\)
							−0.873667	+	0.486524i	\(0.838265\pi\)
\(37\)	3.49491	−	6.05336i	0.574560	−	0.995166i	−0.421530	−	0.906815i	\(-0.638507\pi\)
							0.996089	−	0.0883518i	\(-0.0281600\pi\)
\(41\)	0.800378	+	1.38629i	0.124998	+	0.216503i	0.921732	−	0.387827i	\(-0.126774\pi\)
							−0.796734	+	0.604330i	\(0.793441\pi\)
\(43\)	−1.17144	+	2.02899i	−0.178643	+	0.309418i	−0.941416	−	0.337248i	\(-0.890504\pi\)
							0.762773	+	0.646666i	\(0.223837\pi\)
\(47\)	−0.0888026	−	0.153811i	−0.0129532	−	0.0224356i	0.859476	−	0.511176i	\(-0.170790\pi\)
							−0.872429	+	0.488740i	\(0.837457\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.1	✓	64	7.5	odd	6
273.2.r.b.68.32	yes	64	21.5	even	6
273.2.r.b.269.1	yes	64	13.9	even	3
273.2.r.b.269.32	yes	64	39.35	odd	6
273.2.bf.b.152.1	yes	64	91.61	odd	6	inner
273.2.bf.b.152.32	yes	64	273.152	even	6	inner
273.2.bf.b.185.1	yes	64	3.2	odd	2	inner
273.2.bf.b.185.32	yes	64	1.1	even	1	trivial