Embedded newform 273.2.r.b.68.10

• Label: 273.2.r.b.68.10
• Level: $273$
• Weight: $2$
• Character: 273.68
• 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			68.10
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.23

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.40538i q^{2} +(-0.386542 + 1.68837i) q^{3} +0.0248937 q^{4} +(1.48377 - 2.56997i) q^{5} +(2.37281 + 0.543240i) q^{6} +(0.0176221 + 2.64569i) q^{7} -2.84575i q^{8} +(-2.70117 - 1.30525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.40538i		−	0.993757i	−0.867820	−	0.496879i	\(-0.834479\pi\)
							0.867820	−	0.496879i	\(-0.165521\pi\)
\(3\)	−0.386542	+	1.68837i	−0.223170	+	0.974779i
							\(5\)	1.48377	−	2.56997i	0.663563	−	1.14933i	−0.316109	−	0.948723i	\(-0.602377\pi\)
0.979673	−	0.200603i	\(-0.0642900\pi\)
\(7\)	0.0176221	+	2.64569i	0.00666052	+	0.999978i
							\(11\)	2.35542	+	1.35990i	0.710186	+	0.410026i	0.811130	−	0.584866i	\(-0.198853\pi\)
−0.100944	+	0.994892i	\(0.532186\pi\)
\(13\)	3.51855	−	0.787262i	0.975871	−	0.218347i
							\(17\)	−0.466208			−0.113072			−0.0565360	−	0.998401i	\(-0.518006\pi\)
−0.0565360	+	0.998401i	\(0.518006\pi\)
\(19\)	2.24142	−	1.29409i	0.514218	−	0.296884i	−0.220348	−	0.975421i	\(-0.570719\pi\)
							0.734566	+	0.678538i	\(0.237386\pi\)
\(23\)		−	2.42344i		−	0.505321i	−0.967555	−	0.252661i	\(-0.918694\pi\)
							0.967555	−	0.252661i	\(-0.0813056\pi\)
\(29\)	−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	0.122372	−	0.992484i	\(-0.460950\pi\)
							0.920703	+	0.390265i	\(0.127617\pi\)
\(37\)	−11.3536			−1.86653			−0.933264	−	0.359191i	\(-0.883053\pi\)
							−0.933264	+	0.359191i	\(0.883053\pi\)
\(41\)	4.19212	+	7.26096i	0.654699	+	1.13397i	0.981969	+	0.189041i	\(0.0605379\pi\)
							−0.327270	+	0.944931i	\(0.606129\pi\)
\(43\)	−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	0.366580	+	0.930387i	\(0.380529\pi\)
							−0.989028	+	0.147726i	\(0.952805\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.10	✓	64	1.1	even	1	trivial
273.2.r.b.68.23	yes	64	3.2	odd	2	inner
273.2.r.b.269.10	yes	64	91.87	odd	6	inner
273.2.r.b.269.23	yes	64	273.269	even	6	inner
273.2.bf.b.152.10	yes	64	13.9	even	3
273.2.bf.b.152.23	yes	64	39.35	odd	6
273.2.bf.b.185.10	yes	64	21.17	even	6
273.2.bf.b.185.23	yes	64	7.3	odd	6