Embedded newform 460.2.j.a.47.5

• Label: 460.2.j.a.47.5
• 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			47.5
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38544 + 0.283802i) q^{2} +(2.27633 + 2.27633i) q^{3} +(1.83891 - 0.786383i) q^{4} +(2.17204 + 0.531258i) q^{5} +(-3.79975 - 2.50770i) q^{6} +(1.09581 - 1.09581i) q^{7} +(-2.32454 + 1.61138i) q^{8} +7.36333i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38544	+	0.283802i	−0.979657	+	0.200678i
							\(3\)	2.27633	+	2.27633i	1.31424	+	1.31424i	0.918260	+	0.395978i	\(0.129594\pi\)
0.395978	+	0.918260i	\(0.370406\pi\)
\(5\)	2.17204	+	0.531258i	0.971367	+	0.237586i
							\(7\)	1.09581	−	1.09581i	0.414177	−	0.414177i	−0.469014	−	0.883191i	\(-0.655390\pi\)
0.883191	+	0.469014i	\(0.155390\pi\)
\(11\)		−	2.33723i		−	0.704701i	−0.935868	−	0.352351i	\(-0.885382\pi\)
							0.935868	−	0.352351i	\(-0.114618\pi\)
\(13\)	−3.77107	+	3.77107i	−1.04591	+	1.04591i	−0.0470112	+	0.998894i	\(0.514970\pi\)
							−0.998894	+	0.0470112i	\(0.985030\pi\)
\(17\)	−3.06103	−	3.06103i	−0.742408	−	0.742408i	0.230633	−	0.973041i	\(-0.425920\pi\)
							−0.973041	+	0.230633i	\(0.925920\pi\)
\(19\)	4.41992			1.01400			0.507000	−	0.861946i	\(-0.330754\pi\)
							0.507000	+	0.861946i	\(0.330754\pi\)
\(23\)	0.707107	+	0.707107i	0.147442	+	0.147442i
							\(29\)		−	5.10491i		−	0.947959i	−0.880536	−	0.473979i	\(-0.842817\pi\)
0.880536	−	0.473979i	\(-0.157183\pi\)
\(31\)		−	8.47493i		−	1.52214i	−0.648668	−	0.761072i	\(-0.724674\pi\)
							0.648668	−	0.761072i	\(-0.275326\pi\)
\(37\)	−2.97416	−	2.97416i	−0.488949	−	0.488949i	0.419025	−	0.907974i	\(-0.362372\pi\)
							−0.907974	+	0.419025i	\(0.862372\pi\)
\(41\)	−9.92156			−1.54949			−0.774744	−	0.632275i	\(-0.782121\pi\)
							−0.774744	+	0.632275i	\(0.782121\pi\)
\(43\)	6.76603	+	6.76603i	1.03181	+	1.03181i	0.999477	+	0.0323324i	\(0.0102935\pi\)
							0.0323324	+	0.999477i	\(0.489706\pi\)
\(47\)	1.17516	−	1.17516i	0.171414	−	0.171414i	−0.616186	−	0.787600i	\(-0.711323\pi\)
							0.787600	+	0.616186i	\(0.211323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.5	✓	132
4.3	odd	2	inner	460.2.j.a.47.38	yes	132
5.3	odd	4	inner	460.2.j.a.323.38	yes	132
20.3	even	4	inner	460.2.j.a.323.5	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.5	✓	132	1.1	even	1	trivial
460.2.j.a.47.38	yes	132	4.3	odd	2	inner
460.2.j.a.323.5	yes	132	20.3	even	4	inner
460.2.j.a.323.38	yes	132	5.3	odd	4	inner