Embedded newform 800.5.e.e.399.16

• Label: 800.5.e.e.399.16
• Level: $800$
• Weight: $5$
• Character: 800.399
• Analytic conductor: $82.696$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(82.6959704671\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.16
Character	\(\chi\)	\(=\)	800.399
Dual form			800.5.e.e.399.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+7.09911i q^{3} +2.59084 q^{7} +30.6027 q^{9} +7.95663 q^{11} +54.6467 q^{13} -28.7782i q^{17} -316.702 q^{19} +18.3927i q^{21} -846.949 q^{23} +792.279i q^{27} +766.738i q^{29} -1194.00i q^{31} +56.4850i q^{33} +2436.16 q^{37} +387.943i q^{39} +2433.43 q^{41} +2338.16i q^{43} -2192.51 q^{47} -2394.29 q^{49} +204.299 q^{51} -3041.70 q^{53} -2248.30i q^{57} +455.150 q^{59} +3920.95i q^{61} +79.2868 q^{63} +4822.82i q^{67} -6012.58i q^{69} -874.716i q^{71} +7682.80i q^{73} +20.6144 q^{77} +1475.37i q^{79} -3145.66 q^{81} +6849.50i q^{83} -5443.16 q^{87} -13113.1 q^{89} +141.581 q^{91} +8476.32 q^{93} -8603.40i q^{97} +243.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	7.09911i	0.788789i	0.918941	+	0.394395i	\(0.129046\pi\)
−0.918941	+	0.394395i				\(0.870954\pi\)
\(5\)	0	0
			\(7\)	2.59084	0.0528743	0.0264372	−	0.999650i	\(-0.491584\pi\)
0.0264372	+	0.999650i				\(0.491584\pi\)
\(11\)	7.95663	0.0657573	0.0328786	−	0.999459i	\(-0.489533\pi\)
			0.0328786	+	0.999459i	\(0.489533\pi\)
\(13\)	54.6467	0.323353	0.161677	−	0.986844i	\(-0.448310\pi\)
			0.161677	+	0.986844i	\(0.448310\pi\)
\(17\)	− 28.7782i	− 0.0995785i	−0.998760	−	0.0497892i	\(-0.984145\pi\)
			0.998760	−	0.0497892i	\(-0.0158550\pi\)
\(19\)	−316.702	−0.877291	−0.438646	−	0.898660i	\(-0.644542\pi\)
			−0.438646	+	0.898660i	\(0.644542\pi\)
\(23\)	−846.949	−1.60104	−0.800519	−	0.599308i	\(-0.795443\pi\)
			−0.800519	+	0.599308i	\(0.795443\pi\)
\(29\)	766.738i	0.911698i	0.890057	+	0.455849i	\(0.150664\pi\)
			−0.890057	+	0.455849i	\(0.849336\pi\)
\(31\)	− 1194.00i	− 1.24245i	−0.783631	−	0.621227i	\(-0.786634\pi\)
			0.783631	−	0.621227i	\(-0.213366\pi\)
\(37\)	2436.16	1.77952	0.889758	−	0.456432i	\(-0.150873\pi\)
			0.889758	+	0.456432i	\(0.150873\pi\)
\(41\)	2433.43	1.44761	0.723806	−	0.690004i	\(-0.242391\pi\)
			0.723806	+	0.690004i	\(0.242391\pi\)
\(43\)	2338.16i	1.26455i	0.774742	+	0.632277i	\(0.217880\pi\)
			−0.774742	+	0.632277i	\(0.782120\pi\)
\(47\)	−2192.51	−0.992537	−0.496269	−	0.868169i	\(-0.665297\pi\)
			−0.496269	+	0.868169i	\(0.665297\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.16		32
4.3	odd	2		200.5.e.e.99.20		32
5.2	odd	4		800.5.g.h.751.12		16
5.3	odd	4		160.5.g.a.111.6		16
5.4	even	2	inner	800.5.e.e.399.17		32
8.3	odd	2	inner	800.5.e.e.399.18		32
8.5	even	2		200.5.e.e.99.14		32
15.8	even	4		1440.5.g.a.271.5		16
20.3	even	4		40.5.g.a.11.15	✓	16
20.7	even	4		200.5.g.h.51.2		16
20.19	odd	2		200.5.e.e.99.13		32
40.3	even	4		160.5.g.a.111.5		16
40.13	odd	4		40.5.g.a.11.16	yes	16
40.19	odd	2	inner	800.5.e.e.399.15		32
40.27	even	4		800.5.g.h.751.11		16
40.29	even	2		200.5.e.e.99.19		32
40.37	odd	4		200.5.g.h.51.1		16
60.23	odd	4		360.5.g.a.91.2		16
120.53	even	4		360.5.g.a.91.1		16
120.83	odd	4		1440.5.g.a.271.12		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.15	✓	16	20.3	even	4
40.5.g.a.11.16	yes	16	40.13	odd	4
160.5.g.a.111.5		16	40.3	even	4
160.5.g.a.111.6		16	5.3	odd	4
200.5.e.e.99.13		32	20.19	odd	2
200.5.e.e.99.14		32	8.5	even	2
200.5.e.e.99.19		32	40.29	even	2
200.5.e.e.99.20		32	4.3	odd	2
200.5.g.h.51.1		16	40.37	odd	4
200.5.g.h.51.2		16	20.7	even	4
360.5.g.a.91.1		16	120.53	even	4
360.5.g.a.91.2		16	60.23	odd	4
800.5.e.e.399.15		32	40.19	odd	2	inner
800.5.e.e.399.16		32	1.1	even	1	trivial
800.5.e.e.399.17		32	5.4	even	2	inner
800.5.e.e.399.18		32	8.3	odd	2	inner
800.5.g.h.751.11		16	40.27	even	4
800.5.g.h.751.12		16	5.2	odd	4
1440.5.g.a.271.5		16	15.8	even	4
1440.5.g.a.271.12		16	120.83	odd	4