Embedded newform 273.2.cd.e.44.3

• Label: 273.2.cd.e.44.3
• Level: $273$
• Weight: $2$
• Character: 273.44
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.cd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			44.3
Character	\(\chi\)	\(=\)	273.44
Dual form			273.2.cd.e.242.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.41247 + 0.646418i) q^{2} +(0.872737 - 1.49611i) q^{3} +(3.67009 - 2.11892i) q^{4} +(-1.51647 + 0.406338i) q^{5} +(-1.13834 + 4.17346i) q^{6} +(-1.79039 - 1.94795i) q^{7} +(-3.95215 + 3.95215i) q^{8} +(-1.47666 - 2.61141i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 12 q^{3} - 8 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.41247	+	0.646418i	−1.70587	+	0.457087i	−0.974407	−	0.224793i	\(-0.927830\pi\)
							−0.731465	+	0.681879i	\(0.761163\pi\)
\(3\)	0.872737	−	1.49611i	0.503875	−	0.863777i
							\(5\)	−1.51647	+	0.406338i	−0.678188	+	0.181720i	−0.581440	−	0.813589i	\(-0.697511\pi\)
−0.0967477	+	0.995309i	\(0.530844\pi\)
\(7\)	−1.79039	−	1.94795i	−0.676704	−	0.736255i
							\(11\)	−1.16460	−	0.312052i	−0.351139	−	0.0940874i	0.0789383	−	0.996880i	\(-0.474847\pi\)
−0.430077	+	0.902792i	\(0.641514\pi\)
\(13\)	−1.67805	+	3.19126i	−0.465407	+	0.885097i
							\(17\)	−0.0180259	−	0.0312218i	−0.00437193	−	0.00757241i	0.863831	−	0.503781i	\(-0.168058\pi\)
−0.868203	+	0.496209i	\(0.834725\pi\)
\(19\)	0.354340	+	1.32242i	0.0812912	+	0.303383i	0.994586	−	0.103916i	\(-0.0331373\pi\)
							−0.913295	+	0.407299i	\(0.866471\pi\)
\(23\)	−2.87493	+	4.97952i	−0.599464	+	1.03830i	0.393436	+	0.919352i	\(0.371286\pi\)
							−0.992900	+	0.118950i	\(0.962047\pi\)
\(29\)		−	10.4846i		−	1.94695i	−0.228798	−	0.973474i	\(-0.573480\pi\)
							0.228798	−	0.973474i	\(-0.426520\pi\)
\(31\)	−1.26381	−	0.338636i	−0.226986	−	0.0608208i	0.143533	−	0.989646i	\(-0.454154\pi\)
							−0.370519	+	0.928825i	\(0.620820\pi\)
\(37\)	−9.47419	+	2.53860i	−1.55755	+	0.417343i	−0.931886	−	0.362752i	\(-0.881837\pi\)
							−0.625661	+	0.780095i	\(0.715171\pi\)
\(41\)	−2.92653	−	2.92653i	−0.457048	−	0.457048i	0.440637	−	0.897685i	\(-0.354752\pi\)
							−0.897685	+	0.440637i	\(0.854752\pi\)
\(43\)		−	4.27650i		−	0.652160i	−0.945342	−	0.326080i	\(-0.894272\pi\)
							0.945342	−	0.326080i	\(-0.105728\pi\)
\(47\)	−0.106809	−	0.398617i	−0.0155797	−	0.0581443i	0.957698	−	0.287774i	\(-0.0929152\pi\)
							−0.973278	+	0.229630i	\(0.926249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.cd.e.44.3	✓	112
3.2	odd	2	inner	273.2.cd.e.44.26	yes	112
7.4	even	3	inner	273.2.cd.e.200.26	yes	112
13.8	odd	4	inner	273.2.cd.e.86.3	yes	112
21.11	odd	6	inner	273.2.cd.e.200.3	yes	112
39.8	even	4	inner	273.2.cd.e.86.26	yes	112
91.60	odd	12	inner	273.2.cd.e.242.26	yes	112
273.242	even	12	inner	273.2.cd.e.242.3	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.cd.e.44.3	✓	112	1.1	even	1	trivial
273.2.cd.e.44.26	yes	112	3.2	odd	2	inner
273.2.cd.e.86.3	yes	112	13.8	odd	4	inner
273.2.cd.e.86.26	yes	112	39.8	even	4	inner
273.2.cd.e.200.3	yes	112	21.11	odd	6	inner
273.2.cd.e.200.26	yes	112	7.4	even	3	inner
273.2.cd.e.242.3	yes	112	273.242	even	12	inner
273.2.cd.e.242.26	yes	112	91.60	odd	12	inner