Embedded newform 273.2.bf.b.152.7

• Label: 273.2.bf.b.152.7
• 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			152.7
Character	\(\chi\)	\(=\)	273.152
Dual form			273.2.bf.b.185.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60522 + 0.926776i) q^{2} +(1.72065 + 0.198407i) q^{3} +(0.717829 - 1.24332i) q^{4} +(-0.852778 + 1.47706i) q^{5} +(-2.94591 + 1.27617i) q^{6} +(2.63557 - 0.231921i) q^{7} -1.04604i q^{8} +(2.92127 + 0.682779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60522	+	0.926776i	−1.13506	+	0.655330i	−0.945204	−	0.326481i	\(-0.894137\pi\)
							−0.189861	+	0.981811i	\(0.560804\pi\)
\(3\)	1.72065	+	0.198407i	0.993417	+	0.114551i
							\(5\)	−0.852778	+	1.47706i	−0.381374	+	0.660559i	−0.991259	−	0.131931i	\(-0.957882\pi\)
0.609885	+	0.792490i	\(0.291216\pi\)
\(7\)	2.63557	−	0.231921i	0.996151	−	0.0876579i
							\(11\)			5.89786i			1.77827i	0.457642	+	0.889137i	\(0.348694\pi\)
−0.457642	+	0.889137i	\(0.651306\pi\)
\(13\)	−1.91308	−	3.05616i	−0.530593	−	0.847626i
							\(17\)	1.01396	−	1.75623i	0.245921	−	0.425948i	−0.716469	−	0.697619i	\(-0.754243\pi\)
0.962390	+	0.271671i	\(0.0875762\pi\)
\(19\)			2.92181i			0.670308i	0.942163	+	0.335154i	\(0.108788\pi\)
							−0.942163	+	0.335154i	\(0.891212\pi\)
\(23\)	1.01362	−	0.585212i	0.211354	−	0.122025i	−0.390587	−	0.920566i	\(-0.627728\pi\)
							0.601940	+	0.798541i	\(0.294394\pi\)
\(29\)	−5.45821	−	3.15130i	−1.01356	−	0.585182i	−0.101331	−	0.994853i	\(-0.532310\pi\)
							−0.912233	+	0.409671i	\(0.865644\pi\)
\(31\)	−5.85435	+	3.38001i	−1.05147	+	0.607068i	−0.923061	−	0.384653i	\(-0.874321\pi\)
							−0.128412	+	0.991721i	\(0.540988\pi\)
\(37\)	2.24719	+	3.89225i	0.369436	+	0.639883i	0.989478	−	0.144687i	\(-0.0462174\pi\)
							−0.620041	+	0.784569i	\(0.712884\pi\)
\(41\)	−0.511145	+	0.885329i	−0.0798274	+	0.138265i	−0.903175	−	0.429272i	\(-0.858770\pi\)
							0.823348	+	0.567537i	\(0.192104\pi\)
\(43\)	−1.28948	−	2.23345i	−0.196644	−	0.340598i	0.750794	−	0.660536i	\(-0.229671\pi\)
							−0.947438	+	0.319939i	\(0.896338\pi\)
\(47\)	1.23835	−	2.14488i	0.180631	−	0.312863i	−0.761464	−	0.648207i	\(-0.775519\pi\)
							0.942096	+	0.335344i	\(0.108853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.b.152.7	yes	64
3.2	odd	2	inner	273.2.bf.b.152.26	yes	64
7.3	odd	6		273.2.r.b.269.7	yes	64
13.3	even	3		273.2.r.b.68.7	✓	64
21.17	even	6		273.2.r.b.269.26	yes	64
39.29	odd	6		273.2.r.b.68.26	yes	64
91.3	odd	6	inner	273.2.bf.b.185.26	yes	64
273.185	even	6	inner	273.2.bf.b.185.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.b.68.7	✓	64	13.3	even	3
273.2.r.b.68.26	yes	64	39.29	odd	6
273.2.r.b.269.7	yes	64	7.3	odd	6
273.2.r.b.269.26	yes	64	21.17	even	6
273.2.bf.b.152.7	yes	64	1.1	even	1	trivial
273.2.bf.b.152.26	yes	64	3.2	odd	2	inner
273.2.bf.b.185.7	yes	64	273.185	even	6	inner
273.2.bf.b.185.26	yes	64	91.3	odd	6	inner