Embedded newform 273.2.bf.b.185.23

• Label: 273.2.bf.b.185.23
• 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			185.23
Character	\(\chi\)	\(=\)	273.185
Dual form			273.2.bf.b.152.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21710 + 0.702692i) q^{2} +(-1.65544 + 0.509428i) q^{3} +(-0.0124468 - 0.0215586i) q^{4} +(-1.48377 - 2.56997i) q^{5} +(-2.37281 - 0.543240i) q^{6} +(-2.30005 + 1.30759i) q^{7} -2.84575i q^{8} +(2.48097 - 1.68666i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21710	+	0.702692i	0.860619	+	0.496879i	0.864219	−	0.503115i	\(-0.167813\pi\)
							−0.00360060	+	0.999994i	\(0.501146\pi\)
\(3\)	−1.65544	+	0.509428i	−0.955769	+	0.294119i
							\(5\)	−1.48377	−	2.56997i	−0.663563	−	1.14933i	−0.979673	−	0.200603i	\(-0.935710\pi\)
0.316109	−	0.948723i	\(-0.397623\pi\)
\(7\)	−2.30005	+	1.30759i	−0.869336	+	0.494221i
							\(11\)		−	2.71980i		−	0.820052i	−0.912074	−	0.410026i	\(-0.865520\pi\)
0.912074	−	0.410026i	\(-0.134480\pi\)
\(13\)	−3.51855	+	0.787262i	−0.975871	+	0.218347i
							\(17\)	−0.233104	−	0.403748i	−0.0565360	−	0.0979232i	0.836372	−	0.548162i	\(-0.184672\pi\)
−0.892908	+	0.450239i	\(0.851339\pi\)
\(19\)		−	2.58817i		−	0.593767i	−0.954914	−	0.296884i	\(-0.904053\pi\)
							0.954914	−	0.296884i	\(-0.0959473\pi\)
\(23\)	2.09876	+	1.21172i	0.437621	+	0.252661i	0.702588	−	0.711597i	\(-0.252028\pi\)
							−0.264967	+	0.964257i	\(0.585361\pi\)
\(29\)	−7.22253	+	4.16993i	−1.34119	+	0.774336i	−0.986982	−	0.160830i	\(-0.948583\pi\)
							−0.354208	+	0.935167i	\(0.615249\pi\)
\(31\)	5.80759	+	3.35302i	1.04307	+	0.602219i	0.920703	−	0.390265i	\(-0.127617\pi\)
							0.122372	+	0.992484i	\(0.460950\pi\)
\(37\)	5.67682	−	9.83255i	0.933264	−	1.61646i	0.155563	−	0.987826i	\(-0.450281\pi\)
							0.777701	−	0.628635i	\(-0.216386\pi\)
\(41\)	−4.19212	−	7.26096i	−0.654699	−	1.13397i	−0.981969	−	0.189041i	\(-0.939462\pi\)
							0.327270	−	0.944931i	\(-0.393871\pi\)
\(43\)	−2.24397	+	3.88666i	−0.342202	+	0.592711i	−0.984841	−	0.173458i	\(-0.944506\pi\)
							0.642640	+	0.766169i	\(0.277839\pi\)
\(47\)	4.26729	+	7.39117i	0.622448	+	1.07811i	0.989028	+	0.147726i	\(0.0471953\pi\)
							−0.366580	+	0.930387i	\(0.619471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.185.23	yes	64
3.2	odd	2	inner	273.2.bf.b.185.10	yes	64
7.5	odd	6		273.2.r.b.68.10	✓	64
13.9	even	3		273.2.r.b.269.10	yes	64
21.5	even	6		273.2.r.b.68.23	yes	64
39.35	odd	6		273.2.r.b.269.23	yes	64
91.61	odd	6	inner	273.2.bf.b.152.10	yes	64
273.152	even	6	inner	273.2.bf.b.152.23	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.10	✓	64	7.5	odd	6
273.2.r.b.68.23	yes	64	21.5	even	6
273.2.r.b.269.10	yes	64	13.9	even	3
273.2.r.b.269.23	yes	64	39.35	odd	6
273.2.bf.b.152.10	yes	64	91.61	odd	6	inner
273.2.bf.b.152.23	yes	64	273.152	even	6	inner
273.2.bf.b.185.10	yes	64	3.2	odd	2	inner
273.2.bf.b.185.23	yes	64	1.1	even	1	trivial