Embedded newform 273.2.ba.c.38.5

• Label: 273.2.ba.c.38.5
• 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			38.5
Character	\(\chi\)	\(=\)	273.38
Dual form			273.2.ba.c.194.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16062 - 2.01026i) q^{2} +(-1.69161 + 0.372094i) q^{3} +(-1.69410 + 2.93426i) q^{4} +(2.66615 - 1.53930i) q^{5} +(2.71133 + 2.96871i) q^{6} +(0.235284 + 2.63527i) q^{7} +3.22233 q^{8} +(2.72309 - 1.25888i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16062	−	2.01026i	−0.820685	−	1.42147i	−0.905173	−	0.425044i	\(-0.860259\pi\)
							0.0844879	−	0.996425i	\(-0.473075\pi\)
\(3\)	−1.69161	+	0.372094i	−0.976652	+	0.214829i
							\(5\)	2.66615	−	1.53930i	1.19234	−	0.688397i	0.233502	−	0.972356i	\(-0.424981\pi\)
0.958836	+	0.283959i	\(0.0916481\pi\)
\(7\)	0.235284	+	2.63527i	0.0889289	+	0.996038i
							\(11\)	1.77607	−	3.07625i	0.535506	−	0.927524i	−0.463632	−	0.886028i	\(-0.653454\pi\)
0.999139	−	0.0414964i	\(-0.0132125\pi\)
\(13\)	3.58030	−	0.425986i	0.992996	−	0.118147i
							\(17\)	0.166325	−	0.288084i	0.0403398	−	0.0698706i	−0.845151	−	0.534528i	\(-0.820489\pi\)
0.885490	+	0.464658i	\(0.153823\pi\)
\(19\)	−2.52863	−	4.37972i	−0.580108	−	1.00478i	−0.995466	−	0.0951182i	\(-0.969677\pi\)
							0.415358	−	0.909658i	\(-0.363656\pi\)
\(23\)	−4.04017	+	2.33259i	−0.842433	+	0.486379i	−0.858090	−	0.513499i	\(-0.828349\pi\)
							0.0156577	+	0.999877i	\(0.495016\pi\)
\(29\)		−	3.95925i		−	0.735214i	−0.929981	−	0.367607i	\(-0.880177\pi\)
							0.929981	−	0.367607i	\(-0.119823\pi\)
\(31\)	1.64781	−	2.85408i	0.295955	−	0.512609i	−0.679252	−	0.733905i	\(-0.737696\pi\)
							0.975207	+	0.221297i	\(0.0710289\pi\)
\(37\)	4.60110	−	2.65645i	0.756417	−	0.436717i	−0.0715909	−	0.997434i	\(-0.522808\pi\)
							0.828008	+	0.560717i	\(0.189474\pi\)
\(41\)		−	4.01357i		−	0.626814i	−0.949619	−	0.313407i	\(-0.898530\pi\)
							0.949619	−	0.313407i	\(-0.101470\pi\)
\(43\)	10.4909			1.59985			0.799926	−	0.600099i	\(-0.204872\pi\)
							0.799926	+	0.600099i	\(0.204872\pi\)
\(47\)	0.368966	−	0.213023i	0.0538193	−	0.0310726i	−0.472849	−	0.881144i	\(-0.656774\pi\)
							0.526668	+	0.850071i	\(0.323441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.ba.c.38.5	✓	64
3.2	odd	2	inner	273.2.ba.c.38.28	yes	64
7.5	odd	6	inner	273.2.ba.c.194.6	yes	64
13.12	even	2	inner	273.2.ba.c.38.27	yes	64
21.5	even	6	inner	273.2.ba.c.194.27	yes	64
39.38	odd	2	inner	273.2.ba.c.38.6	yes	64
91.12	odd	6	inner	273.2.ba.c.194.28	yes	64
273.194	even	6	inner	273.2.ba.c.194.5	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.ba.c.38.5	✓	64	1.1	even	1	trivial
273.2.ba.c.38.6	yes	64	39.38	odd	2	inner
273.2.ba.c.38.27	yes	64	13.12	even	2	inner
273.2.ba.c.38.28	yes	64	3.2	odd	2	inner
273.2.ba.c.194.5	yes	64	273.194	even	6	inner
273.2.ba.c.194.6	yes	64	7.5	odd	6	inner
273.2.ba.c.194.27	yes	64	21.5	even	6	inner
273.2.ba.c.194.28	yes	64	91.12	odd	6	inner