Embedded newform 273.2.r.b.68.30

• Label: 273.2.r.b.68.30
• 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			68.30
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.39671i q^{2} +(1.36987 + 1.05994i) q^{3} -3.74420 q^{4} +(-1.10064 + 1.90637i) q^{5} +(-2.54036 + 3.28317i) q^{6} +(-1.62460 - 2.08822i) q^{7} -4.18035i q^{8} +(0.753071 + 2.90394i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.39671i			1.69473i	0.531013	+	0.847364i	\(0.321812\pi\)
							−0.531013	+	0.847364i	\(0.678188\pi\)
\(3\)	1.36987	+	1.05994i	0.790893	+	0.611954i
							\(5\)	−1.10064	+	1.90637i	−0.492223	+	0.852555i	−0.999960	−	0.00895706i	\(-0.997149\pi\)
0.507737	+	0.861512i	\(0.330482\pi\)
\(7\)	−1.62460	−	2.08822i	−0.614042	−	0.789273i
							\(11\)	−1.30172	−	0.751546i	−0.392482	−	0.226600i	0.290753	−	0.956798i	\(-0.406094\pi\)
−0.683235	+	0.730199i	\(0.739428\pi\)
\(13\)	3.47069	+	0.976871i	0.962598	+	0.270935i
							\(17\)	4.90720			1.19017			0.595085	−	0.803662i	\(-0.297118\pi\)
0.595085	+	0.803662i	\(0.297118\pi\)
\(19\)	−2.77060	+	1.59960i	−0.635618	+	0.366974i	−0.782925	−	0.622117i	\(-0.786273\pi\)
							0.147306	+	0.989091i	\(0.452940\pi\)
\(23\)			4.94443i			1.03099i	0.856894	+	0.515493i	\(0.172391\pi\)
							−0.856894	+	0.515493i	\(0.827609\pi\)
\(29\)	0.775914	−	0.447974i	0.144084	−	0.0831867i	−0.426225	−	0.904617i	\(-0.640157\pi\)
							0.570309	+	0.821430i	\(0.306824\pi\)
\(31\)	2.33226	−	1.34653i	0.418886	−	0.241844i	−0.275714	−	0.961240i	\(-0.588914\pi\)
							0.694601	+	0.719395i	\(0.255581\pi\)
\(37\)	10.0463			1.65160			0.825800	−	0.563964i	\(-0.190724\pi\)
							0.825800	+	0.563964i	\(0.190724\pi\)
\(41\)	−4.98596	−	8.63593i	−0.778676	−	1.34871i	−0.932705	−	0.360639i	\(-0.882559\pi\)
							0.154030	−	0.988066i	\(-0.450775\pi\)
\(43\)	−0.820714	+	1.42152i	−0.125158	+	0.216780i	−0.921795	−	0.387679i	\(-0.873277\pi\)
							0.796637	+	0.604458i	\(0.206610\pi\)
\(47\)	0.165217	−	0.286164i	0.0240993	−	0.0417413i	−0.853724	−	0.520725i	\(-0.825662\pi\)
							0.877824	+	0.478984i	\(0.158995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.30	yes	64
3.2	odd	2	inner	273.2.r.b.68.3	✓	64
7.3	odd	6		273.2.bf.b.185.3	yes	64
13.9	even	3		273.2.bf.b.152.30	yes	64
21.17	even	6		273.2.bf.b.185.30	yes	64
39.35	odd	6		273.2.bf.b.152.3	yes	64
91.87	odd	6	inner	273.2.r.b.269.30	yes	64
273.269	even	6	inner	273.2.r.b.269.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.3	✓	64	3.2	odd	2	inner
273.2.r.b.68.30	yes	64	1.1	even	1	trivial
273.2.r.b.269.3	yes	64	273.269	even	6	inner
273.2.r.b.269.30	yes	64	91.87	odd	6	inner
273.2.bf.b.152.3	yes	64	39.35	odd	6
273.2.bf.b.152.30	yes	64	13.9	even	3
273.2.bf.b.185.3	yes	64	7.3	odd	6
273.2.bf.b.185.30	yes	64	21.17	even	6