Embedded newform 273.2.ba.c.38.7

• Label: 273.2.ba.c.38.7
• Level: $273$
• Weight: $2$
• Character: 273.38
• 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.ba (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			38.7
Character	\(\chi\)	\(=\)	273.38
Dual form			273.2.ba.c.194.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.903010 - 1.56406i) q^{2} +(-1.69189 - 0.370830i) q^{3} +(-0.630855 + 1.09267i) q^{4} +(-1.48165 + 0.855432i) q^{5} +(0.947792 + 2.98108i) q^{6} +(-1.44874 - 2.21386i) q^{7} -1.33337 q^{8} +(2.72497 + 1.25481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.903010	−	1.56406i	−0.638525	−	1.10596i	−0.985757	−	0.168178i	\(-0.946212\pi\)
							0.347232	−	0.937779i	\(-0.387122\pi\)
\(3\)	−1.69189	−	0.370830i	−0.976812	−	0.214099i
							\(5\)	−1.48165	+	0.855432i	−0.662615	+	0.382561i	−0.793273	−	0.608867i	\(-0.791624\pi\)
0.130658	+	0.991428i	\(0.458291\pi\)
\(7\)	−1.44874	−	2.21386i	−0.547571	−	0.836759i
							\(11\)	−1.30343	+	2.25761i	−0.392999	+	0.680694i	−0.992844	−	0.119422i	\(-0.961896\pi\)
0.599844	+	0.800117i	\(0.295229\pi\)
\(13\)	3.46117	+	1.01011i	0.959955	+	0.280155i
							\(17\)	−2.88181	+	4.99143i	−0.698941	+	1.21060i	0.269894	+	0.962890i	\(0.413011\pi\)
−0.968834	+	0.247710i	\(0.920322\pi\)
\(19\)	−0.572428	−	0.991474i	−0.131324	−	0.227460i	0.792863	−	0.609400i	\(-0.208589\pi\)
							−0.924187	+	0.381940i	\(0.875256\pi\)
\(23\)	6.58054	−	3.79928i	1.37214	−	0.792204i	0.380940	−	0.924600i	\(-0.375600\pi\)
							0.991197	+	0.132396i	\(0.0422670\pi\)
\(29\)		−	2.60851i		−	0.484387i	−0.970228	−	0.242194i	\(-0.922133\pi\)
							0.970228	−	0.242194i	\(-0.0778669\pi\)
\(31\)	−4.44309	+	7.69566i	−0.798002	+	1.38218i	0.122913	+	0.992417i	\(0.460776\pi\)
							−0.920915	+	0.389763i	\(0.872557\pi\)
\(37\)	2.46567	−	1.42356i	0.405354	−	0.234031i	−0.283437	−	0.958991i	\(-0.591475\pi\)
							0.688792	+	0.724959i	\(0.258142\pi\)
\(41\)			10.1320i			1.58235i	0.611587	+	0.791177i	\(0.290531\pi\)
							−0.611587	+	0.791177i	\(0.709469\pi\)
\(43\)	−9.39075			−1.43208			−0.716038	−	0.698061i	\(-0.754046\pi\)
							−0.716038	+	0.698061i	\(0.754046\pi\)
\(47\)	−3.97126	+	2.29281i	−0.579268	+	0.334441i	−0.760843	−	0.648936i	\(-0.775214\pi\)
							0.181574	+	0.983377i	\(0.441881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.ba.c.38.7	✓	64
3.2	odd	2	inner	273.2.ba.c.38.26	yes	64
7.5	odd	6	inner	273.2.ba.c.194.8	yes	64
13.12	even	2	inner	273.2.ba.c.38.25	yes	64
21.5	even	6	inner	273.2.ba.c.194.25	yes	64
39.38	odd	2	inner	273.2.ba.c.38.8	yes	64
91.12	odd	6	inner	273.2.ba.c.194.26	yes	64
273.194	even	6	inner	273.2.ba.c.194.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.ba.c.38.7	✓	64	1.1	even	1	trivial
273.2.ba.c.38.8	yes	64	39.38	odd	2	inner
273.2.ba.c.38.25	yes	64	13.12	even	2	inner
273.2.ba.c.38.26	yes	64	3.2	odd	2	inner
273.2.ba.c.194.7	yes	64	273.194	even	6	inner
273.2.ba.c.194.8	yes	64	7.5	odd	6	inner
273.2.ba.c.194.25	yes	64	21.5	even	6	inner
273.2.ba.c.194.26	yes	64	91.12	odd	6	inner