Embedded newform 273.2.bn.c.230.1

• Label: 273.2.bn.c.230.1
• Level: $273$
• Weight: $2$
• Character: 273.230
• 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.bn (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			230.1
Character	\(\chi\)	\(=\)	273.230
Dual form			273.2.bn.c.146.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.30434 - 1.33041i) q^{2} +(-0.643938 - 1.60790i) q^{3} +(2.54000 + 4.39941i) q^{4} -3.38484 q^{5} +(-0.655319 + 4.56186i) q^{6} +(-1.66594 - 2.05540i) q^{7} -8.19533i q^{8} +(-2.17069 + 2.07078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 4 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.30434	−	1.33041i	−1.62942	−	0.940744i	−0.984267	−	0.176688i	\(-0.943462\pi\)
							−0.645150	−	0.764056i	\(-0.723205\pi\)
\(3\)	−0.643938	−	1.60790i	−0.371778	−	0.928322i
							\(5\)	−3.38484			−1.51375			−0.756874	−	0.653561i	\(-0.773274\pi\)
−0.756874	+	0.653561i	\(0.773274\pi\)
\(7\)	−1.66594	−	2.05540i	−0.629665	−	0.776867i
							\(11\)	2.78853	+	1.60996i	0.840772	+	0.485420i	0.857527	−	0.514440i	\(-0.172000\pi\)
−0.0167545	+	0.999860i	\(0.505333\pi\)
\(13\)	−3.28889	+	1.47756i	−0.912175	+	0.409802i
							\(17\)	0.544347	+	0.942836i	0.132024	+	0.228671i	0.924456	−	0.381288i	\(-0.124519\pi\)
−0.792433	+	0.609959i	\(0.791186\pi\)
\(19\)	2.80314	−	1.61839i	0.643084	−	0.371285i	−0.142717	−	0.989763i	\(-0.545584\pi\)
							0.785802	+	0.618479i	\(0.212251\pi\)
\(23\)	1.75284	+	1.01200i	0.365493	+	0.211017i	0.671487	−	0.741016i	\(-0.265656\pi\)
							−0.305995	+	0.952033i	\(0.598989\pi\)
\(29\)	0.885409	+	0.511191i	0.164416	+	0.0949258i	0.579950	−	0.814652i	\(-0.303072\pi\)
							−0.415534	+	0.909578i	\(0.636405\pi\)
\(31\)		−	2.28826i		−	0.410983i	−0.978659	−	0.205492i	\(-0.934121\pi\)
							0.978659	−	0.205492i	\(-0.0658793\pi\)
\(37\)	1.22582	−	2.12317i	0.201523	−	0.349048i	−0.747496	−	0.664266i	\(-0.768744\pi\)
							0.949019	+	0.315218i	\(0.102078\pi\)
\(41\)	−1.93014	+	3.34310i	−0.301438	+	0.522105i	−0.976462	−	0.215690i	\(-0.930800\pi\)
							0.675024	+	0.737796i	\(0.264133\pi\)
\(43\)	3.01311	+	5.21886i	0.459495	+	0.795869i	0.998934	−	0.0461557i	\(-0.0146970\pi\)
							−0.539439	+	0.842025i	\(0.681364\pi\)
\(47\)	−6.68788			−0.975527			−0.487763	−	0.872976i	\(-0.662187\pi\)
							−0.487763	+	0.872976i	\(0.662187\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bn.c.230.1	yes	64
3.2	odd	2	inner	273.2.bn.c.230.31	yes	64
7.6	odd	2	inner	273.2.bn.c.230.2	yes	64
13.3	even	3	inner	273.2.bn.c.146.32	yes	64
21.20	even	2	inner	273.2.bn.c.230.32	yes	64
39.29	odd	6	inner	273.2.bn.c.146.2	yes	64
91.55	odd	6	inner	273.2.bn.c.146.31	yes	64
273.146	even	6	inner	273.2.bn.c.146.1	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bn.c.146.1	✓	64	273.146	even	6	inner
273.2.bn.c.146.2	yes	64	39.29	odd	6	inner
273.2.bn.c.146.31	yes	64	91.55	odd	6	inner
273.2.bn.c.146.32	yes	64	13.3	even	3	inner
273.2.bn.c.230.1	yes	64	1.1	even	1	trivial
273.2.bn.c.230.2	yes	64	7.6	odd	2	inner
273.2.bn.c.230.31	yes	64	3.2	odd	2	inner
273.2.bn.c.230.32	yes	64	21.20	even	2	inner