Embedded newform 490.2.w.a.353.14

• Label: 490.2.w.a.353.14
• Level: $490$
• Weight: $2$
• Character: 490.353
• Analytic conductor: $3.913$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.w (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.91266969904\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(28\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			353.14
Character	\(\chi\)	\(=\)	490.353
Dual form			490.2.w.a.397.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.804598 + 0.593820i) q^{2} +(0.554697 - 2.93165i) q^{3} +(0.294755 - 0.955573i) q^{4} +(-1.98786 - 1.02393i) q^{5} +(1.29456 + 2.68819i) q^{6} +(-0.393112 - 2.61638i) q^{7} +(0.330279 + 0.943883i) q^{8} +(-5.49424 - 2.15633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.804598	+	0.593820i	−0.568937	+	0.419894i
							\(3\)	0.554697	−	2.93165i	0.320255	−	1.69259i	−0.339097	−	0.940751i	\(-0.610121\pi\)
0.659351	−	0.751835i	\(-0.270831\pi\)
\(5\)	−1.98786	−	1.02393i	−0.888997	−	0.457914i
							\(7\)	−0.393112	−	2.61638i	−0.148583	−	0.988900i
\(11\)	−0.180293	−	0.459379i	−0.0543603	−	0.138508i								0.901114	−	0.433582i	\(-0.142751\pi\)
							−0.955474	+	0.295075i	\(0.904655\pi\)
\(13\)	0.126207	+	1.12012i	0.0350034	+	0.310664i	0.998998	+	0.0447439i	\(0.0142472\pi\)
							−0.963995	+	0.265920i	\(0.914324\pi\)
\(17\)	0.235465	+	6.29294i	0.0571087	+	1.52626i	0.673716	+	0.738991i	\(0.264697\pi\)
							−0.616607	+	0.787271i	\(0.711493\pi\)
\(19\)	−3.57699	−	6.19554i	−0.820619	−	1.42135i	−0.905222	−	0.424939i	\(-0.860296\pi\)
							0.0846035	−	0.996415i	\(-0.473038\pi\)
\(23\)	8.44878	+	0.316131i	1.76169	+	0.0659179i	0.898504	−	0.438965i	\(-0.144655\pi\)
							0.863188	+	0.504883i	\(0.168464\pi\)
\(29\)	2.67019	+	0.609454i	0.495843	+	0.113173i	0.463127	−	0.886292i	\(-0.346727\pi\)
							0.0327155	+	0.999465i	\(0.489584\pi\)
\(31\)	−6.23402	−	3.59921i	−1.11966	−	0.646438i	−0.178348	−	0.983968i	\(-0.557075\pi\)
							−0.941315	+	0.337530i	\(0.890409\pi\)
\(37\)	−3.76526	+	7.12421i	−0.619004	+	1.17121i	0.352861	+	0.935676i	\(0.385209\pi\)
							−0.971865	+	0.235538i	\(0.924315\pi\)
\(41\)	0.120482	−	0.250183i	0.0188161	−	0.0390720i	−0.891349	−	0.453318i	\(-0.850240\pi\)
							0.910165	+	0.414246i	\(0.135955\pi\)
\(43\)	−0.334921	−	0.117194i	−0.0510749	−	0.0178719i	0.304621	−	0.952474i	\(-0.401470\pi\)
							−0.355696	+	0.934602i	\(0.615756\pi\)
\(47\)	−5.09182	−	6.89917i	−0.742719	−	1.00635i	−0.999208	−	0.0398008i	\(-0.987328\pi\)
							0.256489	−	0.966547i	\(-0.417434\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.353.14	yes	672
5.2	odd	4	inner	490.2.w.a.157.1	yes	672
49.5	odd	42	inner	490.2.w.a.103.1	✓	672
245.152	even	84	inner	490.2.w.a.397.14	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.1	✓	672	49.5	odd	42	inner
490.2.w.a.157.1	yes	672	5.2	odd	4	inner
490.2.w.a.353.14	yes	672	1.1	even	1	trivial
490.2.w.a.397.14	yes	672	245.152	even	84	inner