Embedded newform 273.2.r.b.68.6

• Label: 273.2.r.b.68.6
• 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			68.6
Character	\(\chi\)	\(=\)	273.68
Dual form			273.2.r.b.269.27

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.92371i q^{2} +(-1.08390 - 1.35099i) q^{3} -1.70067 q^{4} +(-1.94643 + 3.37132i) q^{5} +(-2.59891 + 2.08510i) q^{6} +(-0.630216 + 2.56960i) q^{7} -0.575834i q^{8} +(-0.650341 + 2.92866i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.92371i		−	1.36027i	−0.733087	−	0.680135i	\(-0.761921\pi\)
							0.733087	−	0.680135i	\(-0.238079\pi\)
\(3\)	−1.08390	−	1.35099i	−0.625787	−	0.779994i
							\(5\)	−1.94643	+	3.37132i	−0.870470	+	1.50770i	−0.00895828	+	0.999960i	\(0.502852\pi\)
−0.861512	+	0.507738i	\(0.830482\pi\)
\(7\)	−0.630216	+	2.56960i	−0.238199	+	0.971216i
							\(11\)	−2.13331	−	1.23167i	−0.643217	−	0.371362i	0.142636	−	0.989775i	\(-0.454442\pi\)
−0.785853	+	0.618414i	\(0.787776\pi\)
\(13\)	−0.534698	+	3.56568i	−0.148299	+	0.988943i
							\(17\)	−6.77955			−1.64428			−0.822141	−	0.569283i	\(-0.807221\pi\)
−0.822141	+	0.569283i	\(0.807221\pi\)
\(19\)	2.32508	−	1.34238i	0.533409	−	0.307964i	−0.208995	−	0.977917i	\(-0.567019\pi\)
							0.742403	+	0.669953i	\(0.233686\pi\)
\(23\)		−	2.78942i		−	0.581634i	−0.956779	−	0.290817i	\(-0.906073\pi\)
							0.956779	−	0.290817i	\(-0.0939271\pi\)
\(29\)	−0.0349680	+	0.0201888i	−0.00649340	+	0.00374897i	−0.503243	−	0.864145i	\(-0.667860\pi\)
							0.496750	+	0.867894i	\(0.334527\pi\)
\(31\)	−0.270921	+	0.156417i	−0.0486589	+	0.0280932i	−0.524132	−	0.851637i	\(-0.675610\pi\)
							0.475473	+	0.879730i	\(0.342277\pi\)
\(37\)	4.07862			0.670522			0.335261	−	0.942125i	\(-0.391176\pi\)
							0.335261	+	0.942125i	\(0.391176\pi\)
\(41\)	1.73777	+	3.00991i	0.271394	+	0.470068i	0.969219	−	0.246200i	\(-0.0791820\pi\)
							−0.697825	+	0.716268i	\(0.745849\pi\)
\(43\)	−5.35117	+	9.26849i	−0.816045	+	1.41343i	0.0925293	+	0.995710i	\(0.470505\pi\)
							−0.908575	+	0.417722i	\(0.862829\pi\)
\(47\)	−2.67357	+	4.63075i	−0.389980	+	0.675465i	−0.992446	−	0.122679i	\(-0.960851\pi\)
							0.602466	+	0.798144i	\(0.294185\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.68.6	✓	64
3.2	odd	2	inner	273.2.r.b.68.27	yes	64
7.3	odd	6		273.2.bf.b.185.27	yes	64
13.9	even	3		273.2.bf.b.152.6	yes	64
21.17	even	6		273.2.bf.b.185.6	yes	64
39.35	odd	6		273.2.bf.b.152.27	yes	64
91.87	odd	6	inner	273.2.r.b.269.6	yes	64
273.269	even	6	inner	273.2.r.b.269.27	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.6	✓	64	1.1	even	1	trivial
273.2.r.b.68.27	yes	64	3.2	odd	2	inner
273.2.r.b.269.6	yes	64	91.87	odd	6	inner
273.2.r.b.269.27	yes	64	273.269	even	6	inner
273.2.bf.b.152.6	yes	64	13.9	even	3
273.2.bf.b.152.27	yes	64	39.35	odd	6
273.2.bf.b.185.6	yes	64	21.17	even	6
273.2.bf.b.185.27	yes	64	7.3	odd	6