Embedded newform 273.2.bf.b.152.5

• Label: 273.2.bf.b.152.5
• Level: $273$
• Weight: $2$
• Character: 273.152
• 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.bf (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			152.5
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70697 + 0.985518i) q^{2} +(-1.16871 - 1.27832i) q^{3} +(0.942492 - 1.63244i) q^{4} +(-0.272603 + 0.472162i) q^{5} +(3.25477 + 1.03026i) q^{6} +(-2.22004 + 1.43924i) q^{7} -0.226699i q^{8} +(-0.268213 + 2.98799i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70697	+	0.985518i	−1.20701	+	0.696867i	−0.962105	−	0.272681i	\(-0.912090\pi\)
							−0.244904	+	0.969547i	\(0.578756\pi\)
\(3\)	−1.16871	−	1.27832i	−0.674758	−	0.738039i
							\(5\)	−0.272603	+	0.472162i	−0.121912	+	0.211157i	−0.920521	−	0.390692i	\(-0.872236\pi\)
0.798610	+	0.601849i	\(0.205569\pi\)
\(7\)	−2.22004	+	1.43924i	−0.839098	+	0.543981i
							\(11\)		−	0.443067i		−	0.133590i	−0.997767	−	0.0667949i	\(-0.978723\pi\)
0.997767	−	0.0667949i	\(-0.0212773\pi\)
\(13\)	2.64192	−	2.45362i	0.732737	−	0.680512i
							\(17\)	1.91940	−	3.32449i	0.465522	−	0.806308i	−0.533703	−	0.845672i	\(-0.679200\pi\)
0.999225	+	0.0393642i	\(0.0125332\pi\)
\(19\)			0.0828692i			0.0190115i	0.999955	+	0.00950575i	\(0.00302582\pi\)
							−0.999955	+	0.00950575i	\(0.996974\pi\)
\(23\)	6.20148	−	3.58043i	1.29310	−	0.746571i	0.313896	−	0.949457i	\(-0.398366\pi\)
							0.979202	+	0.202887i	\(0.0650323\pi\)
\(29\)	3.66493	+	2.11595i	0.680560	+	0.392922i	0.800066	−	0.599912i	\(-0.204798\pi\)
							−0.119506	+	0.992834i	\(0.538131\pi\)
\(31\)	5.32612	−	3.07503i	0.956599	−	0.552293i	0.0614742	−	0.998109i	\(-0.480420\pi\)
							0.895125	+	0.445816i	\(0.147086\pi\)
\(37\)	1.97108	+	3.41402i	0.324044	+	0.561261i	0.981318	−	0.192391i	\(-0.0616241\pi\)
							−0.657274	+	0.753651i	\(0.728291\pi\)
\(41\)	−2.97274	+	5.14894i	−0.464265	+	0.804130i	−0.999168	−	0.0407834i	\(-0.987015\pi\)
							0.534903	+	0.844913i	\(0.320348\pi\)
\(43\)	−3.19077	−	5.52657i	−0.486588	−	0.842795i	0.513293	−	0.858213i	\(-0.328425\pi\)
							−0.999881	+	0.0154185i	\(0.995092\pi\)
\(47\)	−0.322950	+	0.559365i	−0.0471071	+	0.0815918i	−0.888618	−	0.458649i	\(-0.848334\pi\)
							0.841510	+	0.540241i	\(0.181667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.5	yes	64
3.2	odd	2	inner	273.2.bf.b.152.28	yes	64
7.3	odd	6		273.2.r.b.269.5	yes	64
13.3	even	3		273.2.r.b.68.5	✓	64
21.17	even	6		273.2.r.b.269.28	yes	64
39.29	odd	6		273.2.r.b.68.28	yes	64
91.3	odd	6	inner	273.2.bf.b.185.28	yes	64
273.185	even	6	inner	273.2.bf.b.185.5	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.5	✓	64	13.3	even	3
273.2.r.b.68.28	yes	64	39.29	odd	6
273.2.r.b.269.5	yes	64	7.3	odd	6
273.2.r.b.269.28	yes	64	21.17	even	6
273.2.bf.b.152.5	yes	64	1.1	even	1	trivial
273.2.bf.b.152.28	yes	64	3.2	odd	2	inner
273.2.bf.b.185.5	yes	64	273.185	even	6	inner
273.2.bf.b.185.28	yes	64	91.3	odd	6	inner