Embedded newform 460.2.j.a.47.6

• Label: 460.2.j.a.47.6
• Level: $460$
• Weight: $2$
• Character: 460.47
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.6
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37758 - 0.319819i) q^{2} +(0.818006 + 0.818006i) q^{3} +(1.79543 + 0.881151i) q^{4} +(-0.0316596 + 2.23584i) q^{5} +(-0.865251 - 1.38848i) q^{6} +(-1.96436 + 1.96436i) q^{7} +(-2.19153 - 1.78807i) q^{8} -1.66173i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37758	−	0.319819i	−0.974093	−	0.226146i
							\(3\)	0.818006	+	0.818006i	0.472276	+	0.472276i	0.902650	−	0.430375i	\(-0.141619\pi\)
−0.430375	+	0.902650i	\(0.641619\pi\)
\(5\)	−0.0316596	+	2.23584i	−0.0141586	+	0.999900i
							\(7\)	−1.96436	+	1.96436i	−0.742458	+	0.742458i	−0.973051	−	0.230592i	\(-0.925934\pi\)
0.230592	+	0.973051i	\(0.425934\pi\)
\(11\)			3.99179i			1.20357i	0.798658	+	0.601785i	\(0.205544\pi\)
							−0.798658	+	0.601785i	\(0.794456\pi\)
\(13\)	−2.11586	+	2.11586i	−0.586834	+	0.586834i	−0.936773	−	0.349938i	\(-0.886202\pi\)
							0.349938	+	0.936773i	\(0.386202\pi\)
\(17\)	−3.72099	−	3.72099i	−0.902473	−	0.902473i	0.0931763	−	0.995650i	\(-0.470298\pi\)
							−0.995650	+	0.0931763i	\(0.970298\pi\)
\(19\)	0.221324			0.0507752			0.0253876	−	0.999678i	\(-0.491918\pi\)
							0.0253876	+	0.999678i	\(0.491918\pi\)
\(23\)	−0.707107	−	0.707107i	−0.147442	−	0.147442i
							\(29\)		−	6.78346i		−	1.25966i	−0.776735	−	0.629828i	\(-0.783125\pi\)
0.776735	−	0.629828i	\(-0.216875\pi\)
\(31\)			5.90961i			1.06140i	0.847560	+	0.530699i	\(0.178071\pi\)
							−0.847560	+	0.530699i	\(0.821929\pi\)
\(37\)	2.80602	+	2.80602i	0.461307	+	0.461307i	0.899084	−	0.437776i	\(-0.144234\pi\)
							−0.437776	+	0.899084i	\(0.644234\pi\)
\(41\)	−1.18357			−0.184842			−0.0924209	−	0.995720i	\(-0.529461\pi\)
							−0.0924209	+	0.995720i	\(0.529461\pi\)
\(43\)	5.71454	+	5.71454i	0.871459	+	0.871459i	0.992632	−	0.121172i	\(-0.0386653\pi\)
							−0.121172	+	0.992632i	\(0.538665\pi\)
\(47\)	−5.99928	+	5.99928i	−0.875085	+	0.875085i	−0.993021	−	0.117936i	\(-0.962372\pi\)
							0.117936	+	0.993021i	\(0.462372\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.6	✓	132
4.3	odd	2	inner	460.2.j.a.47.28	yes	132
5.3	odd	4	inner	460.2.j.a.323.28	yes	132
20.3	even	4	inner	460.2.j.a.323.6	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.6	✓	132	1.1	even	1	trivial
460.2.j.a.47.28	yes	132	4.3	odd	2	inner
460.2.j.a.323.6	yes	132	20.3	even	4	inner
460.2.j.a.323.28	yes	132	5.3	odd	4	inner