Embedded newform 10000.2.a.bi.1.2

• Label: 10000.2.a.bi.1.2
• 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			1.2
Root			\(-2.08529\) of defining polynomial
Character	\(\chi\)	\(=\)	10000.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.08529 q^{3} +0.909715 q^{7} +1.34841 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \)

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
			\(3\)	−2.08529	−1.20394	−0.601970	−	0.798519i	\(-0.705617\pi\)
−0.601970	+	0.798519i				\(0.705617\pi\)
\(5\)	0	0
			\(7\)	0.909715	0.343840	0.171920	−	0.985111i	\(-0.445003\pi\)
0.171920	+	0.985111i				\(0.445003\pi\)
\(11\)	4.18178	1.26085	0.630427	−	0.776249i	\(-0.282880\pi\)
			0.630427	+	0.776249i	\(0.282880\pi\)
\(13\)	−4.84601	−1.34404	−0.672021	−	0.740532i	\(-0.734573\pi\)
			−0.672021	+	0.740532i	\(0.734573\pi\)
\(17\)	3.78365	0.917669	0.458835	−	0.888522i	\(-0.348267\pi\)
			0.458835	+	0.888522i	\(0.348267\pi\)
\(19\)	−0.0486974	−0.0111719	−0.00558597	−	0.999984i	\(-0.501778\pi\)
			−0.00558597	+	0.999984i	\(0.501778\pi\)
\(23\)	6.04216	1.25988	0.629939	−	0.776645i	\(-0.283080\pi\)
			0.629939	+	0.776645i	\(0.283080\pi\)
\(29\)	−8.31486	−1.54403	−0.772016	−	0.635604i	\(-0.780751\pi\)
			−0.772016	+	0.635604i	\(0.780751\pi\)
\(31\)	−10.3507	−1.85905	−0.929524	−	0.368761i	\(-0.879782\pi\)
			−0.929524	+	0.368761i	\(0.879782\pi\)
\(37\)	5.18183	0.851888	0.425944	−	0.904750i	\(-0.359942\pi\)
			0.425944	+	0.904750i	\(0.359942\pi\)
\(41\)	−8.18178	−1.27778	−0.638890	−	0.769298i	\(-0.720606\pi\)
			−0.638890	+	0.769298i	\(0.720606\pi\)
\(43\)	2.97442	0.453595	0.226797	−	0.973942i	\(-0.427174\pi\)
			0.226797	+	0.973942i	\(0.427174\pi\)
\(47\)	0.601677	0.0877637	0.0438818	−	0.999037i	\(-0.486027\pi\)
			0.0438818	+	0.999037i	\(0.486027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bi.1.2		8
4.3	odd	2		2500.2.a.f.1.7		8
5.4	even	2	inner	10000.2.a.bi.1.7		8
20.3	even	4		2500.2.c.b.1249.7		8
20.7	even	4		2500.2.c.b.1249.2		8
20.19	odd	2		2500.2.a.f.1.2		8
25.12	odd	20		400.2.y.b.369.1		8
25.23	odd	20		400.2.y.b.129.1		8
100.11	odd	10		500.2.g.b.101.1		16
100.23	even	20		100.2.i.a.29.2	✓	8
100.27	even	20		500.2.i.a.149.1		8
100.39	odd	10		500.2.g.b.101.4		16
100.59	odd	10		500.2.g.b.401.4		16
100.63	even	20		500.2.i.a.349.1		8
100.87	even	20		100.2.i.a.69.2	yes	8
100.91	odd	10		500.2.g.b.401.1		16
300.23	odd	20		900.2.w.a.829.2		8
300.287	odd	20		900.2.w.a.469.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.29.2	✓	8	100.23	even	20
100.2.i.a.69.2	yes	8	100.87	even	20
400.2.y.b.129.1		8	25.23	odd	20
400.2.y.b.369.1		8	25.12	odd	20
500.2.g.b.101.1		16	100.11	odd	10
500.2.g.b.101.4		16	100.39	odd	10
500.2.g.b.401.1		16	100.91	odd	10
500.2.g.b.401.4		16	100.59	odd	10
500.2.i.a.149.1		8	100.27	even	20
500.2.i.a.349.1		8	100.63	even	20
900.2.w.a.469.2		8	300.287	odd	20
900.2.w.a.829.2		8	300.23	odd	20
2500.2.a.f.1.2		8	20.19	odd	2
2500.2.a.f.1.7		8	4.3	odd	2
2500.2.c.b.1249.2		8	20.7	even	4
2500.2.c.b.1249.7		8	20.3	even	4
10000.2.a.bi.1.2		8	1.1	even	1	trivial
10000.2.a.bi.1.7		8	5.4	even	2	inner