Embedded newform 500.2.e.c.307.10

• Label: 500.2.e.c.307.10
• Level: $500$
• Weight: $2$
• Character: 500.307
• 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			307.10
Character	\(\chi\)	\(=\)	500.307
Dual form			500.2.e.c.443.10

$q$-expansion

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

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.273914	−	1.38743i	0.193686	−	0.981064i
							\(3\)	−0.572461	−	0.572461i	−0.330511	−	0.330511i	0.522270	−	0.852780i	\(-0.325085\pi\)
−0.852780	+	0.522270i	\(0.825085\pi\)
\(5\)		0			0
							\(7\)	−1.95303	+	1.95303i	−0.738176	+	0.738176i	−0.972225	−	0.234049i	\(-0.924802\pi\)
0.234049	+	0.972225i	\(0.424802\pi\)
\(11\)			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	−0.991658	−	0.128896i	\(-0.958857\pi\)
							0.128896	+	0.991658i	\(0.458857\pi\)
\(17\)	−4.54058	−	4.54058i	−1.10125	−	1.10125i	−0.994260	−	0.106993i	\(-0.965878\pi\)
							−0.106993	−	0.994260i	\(-0.534122\pi\)
\(19\)	1.01176			0.232114			0.116057	−	0.993243i	\(-0.462974\pi\)
							0.116057	+	0.993243i	\(0.462974\pi\)
\(23\)	1.69974	+	1.69974i	0.354420	+	0.354420i	0.861751	−	0.507331i	\(-0.169368\pi\)
							−0.507331	+	0.861751i	\(0.669368\pi\)
\(29\)			3.45965i			0.642441i	0.947004	+	0.321220i	\(0.104093\pi\)
							−0.947004	+	0.321220i	\(0.895907\pi\)
\(31\)		−	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	0.769593	−	0.638535i	\(-0.220459\pi\)
							−0.638535	+	0.769593i	\(0.720459\pi\)
\(41\)	−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	\(-0.346830\pi\)
							−0.886440	+	0.462843i	\(0.846830\pi\)
\(47\)	−8.28301	+	8.28301i	−1.20820	+	1.20820i	−0.236592	+	0.971609i	\(0.576030\pi\)
							−0.971609	+	0.236592i	\(0.923970\pi\)

(See \(a_n\) instead)

Twists

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

    

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