Embedded newform 273.2.ba.c.38.15

• Label: 273.2.ba.c.38.15
• 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			38.15
Character	\(\chi\)	\(=\)	273.38
Dual form			273.2.ba.c.194.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.100354 - 0.173818i) q^{2} +(-1.73110 + 0.0572655i) q^{3} +(0.979858 - 1.69716i) q^{4} +(2.67317 - 1.54335i) q^{5} +(0.183677 + 0.295150i) q^{6} +(-2.55853 - 0.673741i) q^{7} -0.794745 q^{8} +(2.99344 - 0.198265i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.100354	−	0.173818i	−0.0709608	−	0.122908i	0.828362	−	0.560193i	\(-0.189273\pi\)
							−0.899323	+	0.437286i	\(0.855940\pi\)
\(3\)	−1.73110	+	0.0572655i	−0.999453	+	0.0330623i
							\(5\)	2.67317	−	1.54335i	1.19548	−	0.690209i	0.235934	−	0.971769i	\(-0.424185\pi\)
0.959544	+	0.281560i	\(0.0908518\pi\)
\(7\)	−2.55853	−	0.673741i	−0.967033	−	0.254650i
							\(11\)	−1.43801	+	2.49070i	−0.433576	+	0.750975i	−0.997178	−	0.0750709i	\(-0.976082\pi\)
0.563602	+	0.826046i	\(0.309415\pi\)
\(13\)	−2.85209	−	2.20581i	−0.791027	−	0.611781i
							\(17\)	3.04653	−	5.27675i	0.738892	−	1.27980i	−0.214102	−	0.976811i	\(-0.568682\pi\)
0.952994	−	0.302988i	\(-0.0979842\pi\)
\(19\)	−0.575023	−	0.995969i	−0.131919	−	0.228491i	0.792497	−	0.609876i	\(-0.208781\pi\)
							−0.924416	+	0.381385i	\(0.875447\pi\)
\(23\)	3.18579	−	1.83931i	0.664282	−	0.383524i	−0.129624	−	0.991563i	\(-0.541377\pi\)
							0.793907	+	0.608040i	\(0.208044\pi\)
\(29\)			7.77891i			1.44451i	0.691629	+	0.722253i	\(0.256894\pi\)
							−0.691629	+	0.722253i	\(0.743106\pi\)
\(31\)	3.80568	−	6.59163i	0.683520	−	1.18389i	−0.290379	−	0.956912i	\(-0.593781\pi\)
							0.973899	−	0.226980i	\(-0.0728852\pi\)
\(37\)	1.11374	−	0.643021i	0.183098	−	0.105712i	−0.405649	−	0.914029i	\(-0.632955\pi\)
							0.588748	+	0.808317i	\(0.299621\pi\)
\(41\)			1.74625i			0.272719i	0.990659	+	0.136359i	\(0.0435402\pi\)
							−0.990659	+	0.136359i	\(0.956460\pi\)
\(43\)	0.749286			0.114265			0.0571325	−	0.998367i	\(-0.481804\pi\)
							0.0571325	+	0.998367i	\(0.481804\pi\)
\(47\)	−8.80645	+	5.08440i	−1.28455	+	0.741636i	−0.977677	−	0.210113i	\(-0.932617\pi\)
							−0.306875	+	0.951750i	\(0.599283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.ba.c.38.15	✓	64
3.2	odd	2	inner	273.2.ba.c.38.18	yes	64
7.5	odd	6	inner	273.2.ba.c.194.16	yes	64
13.12	even	2	inner	273.2.ba.c.38.17	yes	64
21.5	even	6	inner	273.2.ba.c.194.17	yes	64
39.38	odd	2	inner	273.2.ba.c.38.16	yes	64
91.12	odd	6	inner	273.2.ba.c.194.18	yes	64
273.194	even	6	inner	273.2.ba.c.194.15	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.ba.c.38.15	✓	64	1.1	even	1	trivial
273.2.ba.c.38.16	yes	64	39.38	odd	2	inner
273.2.ba.c.38.17	yes	64	13.12	even	2	inner
273.2.ba.c.38.18	yes	64	3.2	odd	2	inner
273.2.ba.c.194.15	yes	64	273.194	even	6	inner
273.2.ba.c.194.16	yes	64	7.5	odd	6	inner
273.2.ba.c.194.17	yes	64	21.5	even	6	inner
273.2.ba.c.194.18	yes	64	91.12	odd	6	inner