Embedded newform 273.2.bv.b.137.29

• Label: 273.2.bv.b.137.29
• Level: $273$
• Weight: $2$
• Character: 273.137
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17991597518\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			137.29
Character	\(\chi\)	\(=\)	273.137
Dual form			273.2.bv.b.2.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.52398 - 1.52398i) q^{2} +(0.947306 - 1.45004i) q^{3} -2.64502i q^{4} +(2.04381 + 0.547638i) q^{5} +(-0.766154 - 3.65350i) q^{6} +(-2.26018 + 1.37535i) q^{7} +(-0.982996 - 0.982996i) q^{8} +(-1.20522 - 2.74726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 2 q^{3} - 4 q^{6} - 16 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(106\)	\(157\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.52398	−	1.52398i	1.07762	−	1.07762i	0.0808926	−	0.996723i	\(-0.474223\pi\)
							0.996723	−	0.0808926i	\(-0.0257771\pi\)
\(3\)	0.947306	−	1.45004i	0.546927	−	0.837180i
							\(5\)	2.04381	+	0.547638i	0.914021	+	0.244911i	0.685028	−	0.728517i	\(-0.259790\pi\)
0.228993	+	0.973428i	\(0.426457\pi\)
\(7\)	−2.26018	+	1.37535i	−0.854268	+	0.519834i
							\(11\)	−0.845349	+	3.15489i	−0.254882	+	0.951234i	0.713274	+	0.700886i	\(0.247212\pi\)
−0.968156	+	0.250348i	\(0.919455\pi\)
\(13\)	−2.19296	+	2.86198i	−0.608218	+	0.793770i
							\(17\)	−4.74348			−1.15046			−0.575231	−	0.817991i	\(-0.695088\pi\)
−0.575231	+	0.817991i	\(0.695088\pi\)
\(19\)	3.31062	−	0.887077i	0.759508	−	0.203510i	0.141776	−	0.989899i	\(-0.454719\pi\)
							0.617731	+	0.786389i	\(0.288052\pi\)
\(23\)	0.966732			0.201577			0.100789	−	0.994908i	\(-0.467863\pi\)
							0.100789	+	0.994908i	\(0.467863\pi\)
\(29\)	−5.68894	−	3.28451i	−1.05641	−	0.609918i	−0.131972	−	0.991253i	\(-0.542131\pi\)
							−0.924437	+	0.381336i	\(0.875464\pi\)
\(31\)	−3.41266	+	0.914420i	−0.612932	+	0.164235i	−0.551913	−	0.833902i	\(-0.686102\pi\)
							−0.0610195	+	0.998137i	\(0.519435\pi\)
\(37\)	6.09685	−	6.09685i	1.00232	−	1.00232i	0.00231882	−	0.999997i	\(-0.499262\pi\)
							0.999997	−	0.00231882i	\(-0.000738104\pi\)
\(41\)	−0.196873	−	0.734739i	−0.0307463	−	0.114747i	0.948847	−	0.315736i	\(-0.102251\pi\)
							−0.979594	+	0.200989i	\(0.935585\pi\)
\(43\)	7.82734	−	4.51912i	1.19366	−	0.689159i	0.234524	−	0.972110i	\(-0.424647\pi\)
							0.959134	+	0.282951i	\(0.0913135\pi\)
\(47\)	−2.12627	+	7.93535i	−0.310148	+	1.15749i	0.618274	+	0.785963i	\(0.287832\pi\)
							−0.928422	+	0.371527i	\(0.878834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bv.b.137.29	yes	128
3.2	odd	2	inner	273.2.bv.b.137.4	yes	128
7.2	even	3		273.2.bw.b.254.4	yes	128
13.2	odd	12		273.2.bw.b.158.29	yes	128
21.2	odd	6		273.2.bw.b.254.29	yes	128
39.2	even	12		273.2.bw.b.158.4	yes	128
91.2	odd	12	inner	273.2.bv.b.2.4	✓	128
273.2	even	12	inner	273.2.bv.b.2.29	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bv.b.2.4	✓	128	91.2	odd	12	inner
273.2.bv.b.2.29	yes	128	273.2	even	12	inner
273.2.bv.b.137.4	yes	128	3.2	odd	2	inner
273.2.bv.b.137.29	yes	128	1.1	even	1	trivial
273.2.bw.b.158.4	yes	128	39.2	even	12
273.2.bw.b.158.29	yes	128	13.2	odd	12
273.2.bw.b.254.4	yes	128	7.2	even	3
273.2.bw.b.254.29	yes	128	21.2	odd	6