Embedded newform 10000.2.a.bi.1.8

• Label: 10000.2.a.bi.1.8
• Level: $10000$
• Weight: $2$
• Character: 10000.1
• Self dual: yes
• Analytic conductor: $79.850$
• Analytic rank: $1$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(79.8504020213\)
Analytic rank:	\(1\)
Dimension:	\(8\)
Coefficient field:	8.8.14884000000.2
Defining polynomial:	\( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			\(2.30927\) of defining polynomial
Character	\(\chi\)	\(=\)	10000.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.30927 q^{3} -4.21139 q^{7} +2.33275 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \)

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
			\(3\)	2.30927	1.33326	0.666630	−	0.745389i	\(-0.267736\pi\)
0.666630	+	0.745389i				\(0.267736\pi\)
\(5\)	0	0
			\(7\)	−4.21139	−1.59175	−0.795877	−	0.605458i	\(-0.792990\pi\)
−0.795877	+	0.605458i				\(0.792990\pi\)
\(11\)	0.558282	0.168328	0.0841641	−	0.996452i	\(-0.473178\pi\)
			0.0841641	+	0.996452i	\(0.473178\pi\)
\(13\)	−4.02999	−1.11772	−0.558859	−	0.829263i	\(-0.688761\pi\)
			−0.558859	+	0.829263i	\(0.688761\pi\)
\(17\)	3.96225	0.960987	0.480493	−	0.876998i	\(-0.340458\pi\)
			0.480493	+	0.876998i	\(0.340458\pi\)
\(19\)	6.93090	1.59006	0.795029	−	0.606572i	\(-0.207456\pi\)
			0.795029	+	0.606572i	\(0.207456\pi\)
\(23\)	−0.381287	−0.0795038	−0.0397519	−	0.999210i	\(-0.512657\pi\)
			−0.0397519	+	0.999210i	\(0.512657\pi\)
\(29\)	−8.04746	−1.49438	−0.747188	−	0.664612i	\(-0.768597\pi\)
			−0.747188	+	0.664612i	\(0.768597\pi\)
\(31\)	0.248356	0.0446061	0.0223030	−	0.999751i	\(-0.492900\pi\)
			0.0223030	+	0.999751i	\(0.492900\pi\)
\(37\)	7.17542	1.17963	0.589816	−	0.807538i	\(-0.299200\pi\)
			0.589816	+	0.807538i	\(0.299200\pi\)
\(41\)	−4.55828	−0.711884	−0.355942	−	0.934508i	\(-0.615840\pi\)
			−0.355942	+	0.934508i	\(0.615840\pi\)
\(43\)	−0.207272	−0.0316086	−0.0158043	−	0.999875i	\(-0.505031\pi\)
			−0.0158043	+	0.999875i	\(0.505031\pi\)
\(47\)	9.25893	1.35055	0.675277	−	0.737565i	\(-0.264024\pi\)
			0.675277	+	0.737565i	\(0.264024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bi.1.8		8
4.3	odd	2		2500.2.a.f.1.1		8
5.4	even	2	inner	10000.2.a.bi.1.1		8
20.3	even	4		2500.2.c.b.1249.1		8
20.7	even	4		2500.2.c.b.1249.8		8
20.19	odd	2		2500.2.a.f.1.8		8
25.8	odd	20		400.2.y.b.289.2		8
25.22	odd	20		400.2.y.b.209.2		8
100.3	even	20		500.2.i.a.49.2		8
100.19	odd	10		500.2.g.b.301.4		16
100.31	odd	10		500.2.g.b.301.1		16
100.47	even	20		100.2.i.a.9.1	✓	8
100.67	even	20		500.2.i.a.449.2		8
100.71	odd	10		500.2.g.b.201.1		16
100.79	odd	10		500.2.g.b.201.4		16
100.83	even	20		100.2.i.a.89.1	yes	8
300.47	odd	20		900.2.w.a.109.1		8
300.83	odd	20		900.2.w.a.289.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.9.1	✓	8	100.47	even	20
100.2.i.a.89.1	yes	8	100.83	even	20
400.2.y.b.209.2		8	25.22	odd	20
400.2.y.b.289.2		8	25.8	odd	20
500.2.g.b.201.1		16	100.71	odd	10
500.2.g.b.201.4		16	100.79	odd	10
500.2.g.b.301.1		16	100.31	odd	10
500.2.g.b.301.4		16	100.19	odd	10
500.2.i.a.49.2		8	100.3	even	20
500.2.i.a.449.2		8	100.67	even	20
900.2.w.a.109.1		8	300.47	odd	20
900.2.w.a.289.1		8	300.83	odd	20
2500.2.a.f.1.1		8	4.3	odd	2
2500.2.a.f.1.8		8	20.19	odd	2
2500.2.c.b.1249.1		8	20.3	even	4
2500.2.c.b.1249.8		8	20.7	even	4
10000.2.a.bi.1.1		8	5.4	even	2	inner
10000.2.a.bi.1.8		8	1.1	even	1	trivial