Embedded newform 460.2.w.a.163.62

• Label: 460.2.w.a.163.62
• Level: $460$
• Weight: $2$
• Character: 460.163
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $1360$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.w (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(1360\)
Relative dimension:	\(68\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			163.62
Character	\(\chi\)	\(=\)	460.163
Dual form			460.2.w.a.127.62

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34131 - 0.448218i) q^{2} +(-0.802913 - 0.601053i) q^{3} +(1.59820 - 1.20239i) q^{4} +(-2.12963 + 0.681666i) q^{5} +(-1.34636 - 0.446317i) q^{6} +(-2.03833 + 0.145784i) q^{7} +(1.60474 - 2.32912i) q^{8} +(-0.561793 - 1.91329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1360 q - 18 q^{2} - 36 q^{5} - 44 q^{6} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34131	−	0.448218i	0.948446	−	0.316938i
							\(3\)	−0.802913	−	0.601053i	−0.463562	−	0.347018i	0.341829	−	0.939762i	\(-0.388954\pi\)
−0.805391	+	0.592744i	\(0.798045\pi\)
\(5\)	−2.12963	+	0.681666i	−0.952400	+	0.304850i
							\(7\)	−2.03833	+	0.145784i	−0.770417	+	0.0551013i	−0.451013	−	0.892517i	\(-0.648937\pi\)
−0.319404	+	0.947619i	\(0.603483\pi\)
\(11\)	−3.15608	−	4.91095i	−0.951593	−	1.48071i	−0.875276	−	0.483624i	\(-0.839320\pi\)
							−0.0763176	−	0.997084i	\(-0.524316\pi\)
\(13\)	−0.0276782	+	0.386992i	−0.00767655	+	0.107332i	−0.999846	−	0.0175698i	\(-0.994407\pi\)
							0.992169	+	0.124902i	\(0.0398616\pi\)
\(17\)	−0.392425	−	1.05213i	−0.0951770	−	0.255179i	0.880633	−	0.473799i	\(-0.157118\pi\)
							−0.975810	+	0.218619i	\(0.929845\pi\)
\(19\)	0.986770	+	2.16073i	0.226381	+	0.495704i	0.988404	−	0.151845i	\(-0.0485214\pi\)
							−0.762024	+	0.647549i	\(0.775794\pi\)
\(23\)	3.78099	−	2.95027i	0.788392	−	0.615174i
							\(29\)	−0.795457	−	0.363273i	−0.147713	−	0.0674581i	0.340187	−	0.940358i	\(-0.389510\pi\)
−0.487900	+	0.872900i	\(0.662237\pi\)
\(31\)	4.64929	+	0.668467i	0.835037	+	0.120060i	0.546553	−	0.837424i	\(-0.315940\pi\)
							0.288484	+	0.957485i	\(0.406849\pi\)
\(37\)	−2.30189	+	4.21560i	−0.378429	+	0.693041i	−0.995592	−	0.0937947i	\(-0.970100\pi\)
							0.617163	+	0.786836i	\(0.288282\pi\)
\(41\)	−6.59232	−	1.93568i	−1.02955	−	0.302303i	−0.277021	−	0.960864i	\(-0.589347\pi\)
							−0.752527	+	0.658561i	\(0.771165\pi\)
\(43\)	0.431942	+	0.323348i	0.0658706	+	0.0493101i	0.631706	−	0.775208i	\(-0.282355\pi\)
							−0.565835	+	0.824519i	\(0.691446\pi\)
\(47\)	6.63170	+	6.63170i	0.967333	+	0.967333i	0.999483	−	0.0321504i	\(-0.0102355\pi\)
							−0.0321504	+	0.999483i	\(0.510236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.w.a.163.62	yes	1360
4.3	odd	2	inner	460.2.w.a.163.66	yes	1360
5.2	odd	4	inner	460.2.w.a.347.41	yes	1360
20.7	even	4	inner	460.2.w.a.347.37	yes	1360
23.12	even	11	inner	460.2.w.a.403.37	yes	1360
92.35	odd	22	inner	460.2.w.a.403.41	yes	1360
115.12	odd	44	inner	460.2.w.a.127.66	yes	1360
460.127	even	44	inner	460.2.w.a.127.62	✓	1360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.w.a.127.62	✓	1360	460.127	even	44	inner
460.2.w.a.127.66	yes	1360	115.12	odd	44	inner
460.2.w.a.163.62	yes	1360	1.1	even	1	trivial
460.2.w.a.163.66	yes	1360	4.3	odd	2	inner
460.2.w.a.347.37	yes	1360	20.7	even	4	inner
460.2.w.a.347.41	yes	1360	5.2	odd	4	inner
460.2.w.a.403.37	yes	1360	23.12	even	11	inner
460.2.w.a.403.41	yes	1360	92.35	odd	22	inner