Embedded newform 273.2.bv.b.2.4

• Label: 273.2.bv.b.2.4
• Level: $273$
• Weight: $2$
• Character: 273.2
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• 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			2.4
Character	\(\chi\)	\(=\)	273.2
Dual form			273.2.bv.b.137.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52398 - 1.52398i) q^{2} +(-1.72942 - 0.0953716i) q^{3} +2.64502i q^{4} +(-2.04381 + 0.547638i) q^{5} +(2.49026 + 2.78095i) q^{6} +(-2.26018 - 1.37535i) q^{7} +(0.982996 - 0.982996i) q^{8} +(2.98181 + 0.329876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 2 q^{3} - 4 q^{6} - 16 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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	−0.996723	−	0.0808926i	\(-0.974223\pi\)
							−0.0808926	−	0.996723i	\(-0.525777\pi\)
\(3\)	−1.72942	−	0.0953716i	−0.998483	−	0.0550628i
							\(5\)	−2.04381	+	0.547638i	−0.914021	+	0.244911i	−0.685028	−	0.728517i	\(-0.740210\pi\)
−0.228993	+	0.973428i	\(0.573543\pi\)
\(7\)	−2.26018	−	1.37535i	−0.854268	−	0.519834i
							\(11\)	0.845349	+	3.15489i	0.254882	+	0.951234i	0.968156	+	0.250348i	\(0.0805451\pi\)
−0.713274	+	0.700886i	\(0.752788\pi\)
\(13\)	−2.19296	−	2.86198i	−0.608218	−	0.793770i
							\(17\)	4.74348			1.15046			0.575231	−	0.817991i	\(-0.304912\pi\)
0.575231	+	0.817991i	\(0.304912\pi\)
\(19\)	3.31062	+	0.887077i	0.759508	+	0.203510i	0.617731	−	0.786389i	\(-0.288052\pi\)
							0.141776	+	0.989899i	\(0.454719\pi\)
\(23\)	−0.966732			−0.201577			−0.100789	−	0.994908i	\(-0.532137\pi\)
							−0.100789	+	0.994908i	\(0.532137\pi\)
\(29\)	5.68894	−	3.28451i	1.05641	−	0.609918i	0.131972	−	0.991253i	\(-0.457869\pi\)
							0.924437	+	0.381336i	\(0.124536\pi\)
\(31\)	−3.41266	−	0.914420i	−0.612932	−	0.164235i	−0.0610195	−	0.998137i	\(-0.519435\pi\)
							−0.551913	+	0.833902i	\(0.686102\pi\)
\(37\)	6.09685	+	6.09685i	1.00232	+	1.00232i	0.999997	+	0.00231882i	\(0.000738104\pi\)
							0.00231882	+	0.999997i	\(0.499262\pi\)
\(41\)	0.196873	−	0.734739i	0.0307463	−	0.114747i	−0.948847	−	0.315736i	\(-0.897749\pi\)
							0.979594	+	0.200989i	\(0.0644154\pi\)
\(43\)	7.82734	+	4.51912i	1.19366	+	0.689159i	0.959134	−	0.282951i	\(-0.0913135\pi\)
							0.234524	+	0.972110i	\(0.424647\pi\)
\(47\)	2.12627	+	7.93535i	0.310148	+	1.15749i	0.928422	+	0.371527i	\(0.121166\pi\)
							−0.618274	+	0.785963i	\(0.712168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bv.b.2.4	✓	128
3.2	odd	2	inner	273.2.bv.b.2.29	yes	128
7.4	even	3		273.2.bw.b.158.29	yes	128
13.7	odd	12		273.2.bw.b.254.4	yes	128
21.11	odd	6		273.2.bw.b.158.4	yes	128
39.20	even	12		273.2.bw.b.254.29	yes	128
91.46	odd	12	inner	273.2.bv.b.137.29	yes	128
273.137	even	12	inner	273.2.bv.b.137.4	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bv.b.2.4	✓	128	1.1	even	1	trivial
273.2.bv.b.2.29	yes	128	3.2	odd	2	inner
273.2.bv.b.137.4	yes	128	273.137	even	12	inner
273.2.bv.b.137.29	yes	128	91.46	odd	12	inner
273.2.bw.b.158.4	yes	128	21.11	odd	6
273.2.bw.b.158.29	yes	128	7.4	even	3
273.2.bw.b.254.4	yes	128	13.7	odd	12
273.2.bw.b.254.29	yes	128	39.20	even	12