Embedded newform 10000.2.a.bi.1.3

• Label: 10000.2.a.bi.1.3
• 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			1.3
Root			\(-1.13370\) of defining polynomial
Character	\(\chi\)	\(=\)	10000.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.13370 q^{3} -0.768409 q^{7} -1.71472 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \)

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
			\(3\)	−1.13370	−0.654544	−0.327272	−	0.944930i	\(-0.606129\pi\)
−0.327272	+	0.944930i				\(0.606129\pi\)
\(5\)	0	0
			\(7\)	−0.768409	−0.290431	−0.145216	−	0.989400i	\(-0.546388\pi\)
−0.145216	+	0.989400i				\(0.546388\pi\)
\(11\)	3.05975	0.922550	0.461275	−	0.887257i	\(-0.347392\pi\)
			0.461275	+	0.887257i	\(0.347392\pi\)
\(13\)	0.225765	0.0626159	0.0313079	−	0.999510i	\(-0.490033\pi\)
			0.0313079	+	0.999510i	\(0.490033\pi\)
\(17\)	−5.86436	−1.42232	−0.711158	−	0.703032i	\(-0.751829\pi\)
			−0.711158	+	0.703032i	\(0.751829\pi\)
\(19\)	−6.16697	−1.41480	−0.707400	−	0.706814i	\(-0.750132\pi\)
			−0.707400	+	0.706814i	\(0.750132\pi\)
\(23\)	5.18957	1.08210	0.541050	−	0.840990i	\(-0.318027\pi\)
			0.541050	+	0.840990i	\(0.318027\pi\)
\(29\)	0.0474641	0.00881387	0.00440694	−	0.999990i	\(-0.498597\pi\)
			0.00440694	+	0.999990i	\(0.498597\pi\)
\(31\)	5.84181	1.04922	0.524610	−	0.851342i	\(-0.324211\pi\)
			0.524610	+	0.851342i	\(0.324211\pi\)
\(37\)	8.49052	1.39583	0.697916	−	0.716179i	\(-0.254111\pi\)
			0.697916	+	0.716179i	\(0.254111\pi\)
\(41\)	−7.05975	−1.10255	−0.551274	−	0.834324i	\(-0.685858\pi\)
			−0.551274	+	0.834324i	\(0.685858\pi\)
\(43\)	−3.14793	−0.480054	−0.240027	−	0.970766i	\(-0.577156\pi\)
			−0.240027	+	0.970766i	\(0.577156\pi\)
\(47\)	11.3868	1.66094	0.830468	−	0.557066i	\(-0.188073\pi\)
			0.830468	+	0.557066i	\(0.188073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bi.1.3		8
4.3	odd	2		2500.2.a.f.1.6		8
5.4	even	2	inner	10000.2.a.bi.1.6		8
20.3	even	4		2500.2.c.b.1249.6		8
20.7	even	4		2500.2.c.b.1249.3		8
20.19	odd	2		2500.2.a.f.1.3		8
25.8	odd	20		400.2.y.b.289.1		8
25.22	odd	20		400.2.y.b.209.1		8
100.3	even	20		500.2.i.a.49.1		8
100.19	odd	10		500.2.g.b.301.2		16
100.31	odd	10		500.2.g.b.301.3		16
100.47	even	20		100.2.i.a.9.2	✓	8
100.67	even	20		500.2.i.a.449.1		8
100.71	odd	10		500.2.g.b.201.3		16
100.79	odd	10		500.2.g.b.201.2		16
100.83	even	20		100.2.i.a.89.2	yes	8
300.47	odd	20		900.2.w.a.109.2		8
300.83	odd	20		900.2.w.a.289.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.9.2	✓	8	100.47	even	20
100.2.i.a.89.2	yes	8	100.83	even	20
400.2.y.b.209.1		8	25.22	odd	20
400.2.y.b.289.1		8	25.8	odd	20
500.2.g.b.201.2		16	100.79	odd	10
500.2.g.b.201.3		16	100.71	odd	10
500.2.g.b.301.2		16	100.19	odd	10
500.2.g.b.301.3		16	100.31	odd	10
500.2.i.a.49.1		8	100.3	even	20
500.2.i.a.449.1		8	100.67	even	20
900.2.w.a.109.2		8	300.47	odd	20
900.2.w.a.289.2		8	300.83	odd	20
2500.2.a.f.1.3		8	20.19	odd	2
2500.2.a.f.1.6		8	4.3	odd	2
2500.2.c.b.1249.3		8	20.7	even	4
2500.2.c.b.1249.6		8	20.3	even	4
10000.2.a.bi.1.3		8	1.1	even	1	trivial
10000.2.a.bi.1.6		8	5.4	even	2	inner