Embedded newform 273.2.br.b.17.3

• Label: 273.2.br.b.17.3
• Level: $273$
• Weight: $2$
• Character: 273.17
• 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.br (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			17.3
Character	\(\chi\)	\(=\)	273.17
Dual form			273.2.br.b.257.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.36811 q^{2} +(-1.07288 + 1.35975i) q^{3} +3.60795 q^{4} +(-0.0190083 + 0.0109745i) q^{5} +(2.54070 - 3.22004i) q^{6} +(2.35552 + 1.20478i) q^{7} -3.80781 q^{8} +(-0.697847 - 2.91771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 6 q^{3} + 56 q^{4} - 6 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.36811			−1.67451			−0.837254	−	0.546814i	\(-0.815840\pi\)
							−0.837254	+	0.546814i	\(0.815840\pi\)
\(3\)	−1.07288	+	1.35975i	−0.619429	+	0.785053i
							\(5\)	−0.0190083	+	0.0109745i	−0.00850078	+	0.00490793i	−0.504244	−	0.863561i	\(-0.668229\pi\)
0.495744	+	0.868469i	\(0.334896\pi\)
\(7\)	2.35552	+	1.20478i	0.890305	+	0.455365i
							\(11\)	−0.904593	−	1.56680i	−0.272745	−	0.472409i	0.696819	−	0.717247i	\(-0.254598\pi\)
−0.969564	+	0.244839i	\(0.921265\pi\)
\(13\)	2.16898	−	2.88020i	0.601566	−	0.798823i
							\(17\)	6.47176			1.56963			0.784817	−	0.619728i	\(-0.212757\pi\)
0.784817	+	0.619728i	\(0.212757\pi\)
\(19\)	−0.875943	+	1.51718i	−0.200955	+	0.348064i	−0.948836	−	0.315768i	\(-0.897738\pi\)
							0.747881	+	0.663832i	\(0.231071\pi\)
\(23\)			5.69857i			1.18823i	0.804379	+	0.594117i	\(0.202498\pi\)
							−0.804379	+	0.594117i	\(0.797502\pi\)
\(29\)	6.52098	+	3.76489i	1.21092	+	0.699123i	0.962959	−	0.269648i	\(-0.0869075\pi\)
							0.247957	+	0.968771i	\(0.420241\pi\)
\(31\)	0.347389	−	0.601695i	0.0623929	−	0.108068i	−0.833142	−	0.553060i	\(-0.813460\pi\)
							0.895535	+	0.444992i	\(0.146793\pi\)
\(37\)		−	3.74663i		−	0.615942i	−0.951396	−	0.307971i	\(-0.900350\pi\)
							0.951396	−	0.307971i	\(-0.0996500\pi\)
\(41\)	−2.16550	−	1.25025i	−0.338194	−	0.195256i	0.321279	−	0.946985i	\(-0.395887\pi\)
							−0.659473	+	0.751728i	\(0.729221\pi\)
\(43\)	3.54180	+	6.13458i	0.540119	+	0.935514i	0.998897	+	0.0469630i	\(0.0149543\pi\)
							−0.458777	+	0.888551i	\(0.651712\pi\)
\(47\)	−6.66373	+	3.84731i	−0.972005	+	0.561188i	−0.899847	−	0.436206i	\(-0.856322\pi\)
							−0.0721583	+	0.997393i	\(0.522989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.br.b.17.3	yes	64
3.2	odd	2	inner	273.2.br.b.17.30	yes	64
7.5	odd	6		273.2.y.b.173.30	yes	64
13.10	even	6		273.2.y.b.101.3	✓	64
21.5	even	6		273.2.y.b.173.3	yes	64
39.23	odd	6		273.2.y.b.101.30	yes	64
91.75	odd	6	inner	273.2.br.b.257.30	yes	64
273.257	even	6	inner	273.2.br.b.257.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.y.b.101.3	✓	64	13.10	even	6
273.2.y.b.101.30	yes	64	39.23	odd	6
273.2.y.b.173.3	yes	64	21.5	even	6
273.2.y.b.173.30	yes	64	7.5	odd	6
273.2.br.b.17.3	yes	64	1.1	even	1	trivial
273.2.br.b.17.30	yes	64	3.2	odd	2	inner
273.2.br.b.257.3	yes	64	273.257	even	6	inner
273.2.br.b.257.30	yes	64	91.75	odd	6	inner