Embedded newform 500.2.e.c.443.2

• Label: 500.2.e.c.443.2
• Level: $500$
• Weight: $2$
• Character: 500.443
• Analytic conductor: $3.993$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	500.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.99252010106\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			443.2
Character	\(\chi\)	\(=\)	500.443
Dual form			500.2.e.c.307.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38743 - 0.273914i) q^{2} +(0.572461 - 0.572461i) q^{3} +(1.84994 + 0.760074i) q^{4} +(-0.951057 + 0.637447i) q^{6} +(1.95303 + 1.95303i) q^{7} +(-2.35848 - 1.56128i) q^{8} +2.34458i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\)	\(251\)	\(377\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38743	−	0.273914i	−0.981064	−	0.193686i
							\(3\)	0.572461	−	0.572461i	0.330511	−	0.330511i	−0.522270	−	0.852780i	\(-0.674915\pi\)
0.852780	+	0.522270i	\(0.174915\pi\)
\(5\)		0			0
							\(7\)	1.95303	+	1.95303i	0.738176	+	0.738176i	0.972225	−	0.234049i	\(-0.0751975\pi\)
−0.234049	+	0.972225i	\(0.575198\pi\)
\(11\)			5.76271i			1.73752i	0.495232	+	0.868761i	\(0.335083\pi\)
							−0.495232	+	0.868761i	\(0.664917\pi\)
\(13\)	−3.11073	−	3.11073i	−0.862762	−	0.862762i	0.128896	−	0.991658i	\(-0.458857\pi\)
							−0.991658	+	0.128896i	\(0.958857\pi\)
\(17\)	−4.54058	+	4.54058i	−1.10125	+	1.10125i	−0.106993	+	0.994260i	\(0.534122\pi\)
							−0.994260	+	0.106993i	\(0.965878\pi\)
\(19\)	−1.01176			−0.232114			−0.116057	−	0.993243i	\(-0.537026\pi\)
							−0.116057	+	0.993243i	\(0.537026\pi\)
\(23\)	−1.69974	+	1.69974i	−0.354420	+	0.354420i	−0.861751	−	0.507331i	\(-0.830632\pi\)
							0.507331	+	0.861751i	\(0.330632\pi\)
\(29\)		−	3.45965i		−	0.642441i	−0.947004	−	0.321220i	\(-0.895907\pi\)
							0.947004	−	0.321220i	\(-0.104093\pi\)
\(31\)		−	1.63706i		−	0.294025i	−0.989135	−	0.147013i	\(-0.953034\pi\)
							0.989135	−	0.147013i	\(-0.0469658\pi\)
\(37\)	0.797192	−	0.797192i	0.131058	−	0.131058i	−0.638535	−	0.769593i	\(-0.720459\pi\)
							0.769593	+	0.638535i	\(0.220459\pi\)
\(41\)	−5.21586			−0.814581			−0.407291	−	0.913299i	\(-0.633526\pi\)
							−0.407291	+	0.913299i	\(0.633526\pi\)
\(43\)	2.77772	−	2.77772i	0.423598	−	0.423598i	−0.462843	−	0.886440i	\(-0.653170\pi\)
							0.886440	+	0.462843i	\(0.153170\pi\)
\(47\)	8.28301	+	8.28301i	1.20820	+	1.20820i	0.971609	+	0.236592i	\(0.0760304\pi\)
							0.236592	+	0.971609i	\(0.423970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	500.2.e.c.443.2	yes	32
4.3	odd	2	inner	500.2.e.c.443.10	yes	32
5.2	odd	4	inner	500.2.e.c.307.10	yes	32
5.3	odd	4	inner	500.2.e.c.307.7	yes	32
5.4	even	2	inner	500.2.e.c.443.15	yes	32
20.3	even	4	inner	500.2.e.c.307.15	yes	32
20.7	even	4	inner	500.2.e.c.307.2	✓	32
20.19	odd	2	inner	500.2.e.c.443.7	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.2.e.c.307.2	✓	32	20.7	even	4	inner
500.2.e.c.307.7	yes	32	5.3	odd	4	inner
500.2.e.c.307.10	yes	32	5.2	odd	4	inner
500.2.e.c.307.15	yes	32	20.3	even	4	inner
500.2.e.c.443.2	yes	32	1.1	even	1	trivial
500.2.e.c.443.7	yes	32	20.19	odd	2	inner
500.2.e.c.443.10	yes	32	4.3	odd	2	inner
500.2.e.c.443.15	yes	32	5.4	even	2	inner