Embedded newform 273.2.r.b.269.15

• Label: 273.2.r.b.269.15
• Level: $273$
• Weight: $2$
• Character: 273.269
• 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			269.15
Character	\(\chi\)	\(=\)	273.269
Dual form			273.2.r.b.68.18

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.546208i q^{2} +(-0.761914 + 1.55547i) q^{3} +1.70166 q^{4} +(1.25657 + 2.17644i) q^{5} +(0.849611 + 0.416164i) q^{6} +(-0.612705 + 2.57383i) q^{7} -2.02188i q^{8} +(-1.83897 - 2.37027i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	0.546208i		−	0.386228i	−0.981176	−	0.193114i	\(-0.938141\pi\)
							0.981176	−	0.193114i	\(-0.0618587\pi\)
\(3\)	−0.761914	+	1.55547i	−0.439891	+	0.898051i
							\(5\)	1.25657	+	2.17644i	0.561955	+	0.973335i	0.997326	+	0.0730836i	\(0.0232840\pi\)
−0.435371	+	0.900251i	\(0.643383\pi\)
\(7\)	−0.612705	+	2.57383i	−0.231581	+	0.972816i
							\(11\)	−5.22008	+	3.01382i	−1.57391	+	0.908700i	−0.578232	+	0.815872i	\(0.696257\pi\)
−0.995682	+	0.0928275i	\(0.970409\pi\)
\(13\)	1.89716	−	3.06607i	0.526176	−	0.850376i
							\(17\)	−0.647303			−0.156994			−0.0784970	−	0.996914i	\(-0.525012\pi\)
−0.0784970	+	0.996914i	\(0.525012\pi\)
\(19\)	1.98801	+	1.14778i	0.456080	+	0.263318i	0.710394	−	0.703804i	\(-0.248517\pi\)
							−0.254315	+	0.967122i	\(0.581850\pi\)
\(23\)		−	1.03331i		−	0.215460i	−0.994180	−	0.107730i	\(-0.965642\pi\)
							0.994180	−	0.107730i	\(-0.0343582\pi\)
\(29\)	4.46255	+	2.57646i	0.828676	+	0.478436i	0.853399	−	0.521258i	\(-0.174537\pi\)
							−0.0247234	+	0.999694i	\(0.507871\pi\)
\(31\)	9.25745	+	5.34479i	1.66269	+	0.959953i	0.971424	+	0.237351i	\(0.0762792\pi\)
							0.691264	+	0.722602i	\(0.257054\pi\)
\(37\)	−3.95532			−0.650251			−0.325126	−	0.945671i	\(-0.605407\pi\)
							−0.325126	+	0.945671i	\(0.605407\pi\)
\(41\)	4.48689	−	7.77153i	0.700735	−	1.21371i	−0.267474	−	0.963565i	\(-0.586189\pi\)
							0.968209	−	0.250143i	\(-0.0804778\pi\)
\(43\)	−4.09951	−	7.10056i	−0.625169	−	1.08282i	−0.988508	−	0.151168i	\(-0.951697\pi\)
							0.363339	−	0.931657i	\(-0.381637\pi\)
\(47\)	2.09646	+	3.63118i	0.305800	+	0.529662i	0.977439	−	0.211217i	\(-0.0677426\pi\)
							−0.671639	+	0.740879i	\(0.734409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.r.b.269.15	yes	64
3.2	odd	2	inner	273.2.r.b.269.18	yes	64
7.5	odd	6		273.2.bf.b.152.15	yes	64
13.3	even	3		273.2.bf.b.185.18	yes	64
21.5	even	6		273.2.bf.b.152.18	yes	64
39.29	odd	6		273.2.bf.b.185.15	yes	64
91.68	odd	6	inner	273.2.r.b.68.15	✓	64
273.68	even	6	inner	273.2.r.b.68.18	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.15	✓	64	91.68	odd	6	inner
273.2.r.b.68.18	yes	64	273.68	even	6	inner
273.2.r.b.269.15	yes	64	1.1	even	1	trivial
273.2.r.b.269.18	yes	64	3.2	odd	2	inner
273.2.bf.b.152.15	yes	64	7.5	odd	6
273.2.bf.b.152.18	yes	64	21.5	even	6
273.2.bf.b.185.15	yes	64	39.29	odd	6
273.2.bf.b.185.18	yes	64	13.3	even	3