Embedded newform 10000.2.a.bi.1.5

• Label: 10000.2.a.bi.1.5
• 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			1.5
Root			\(0.183172\) of defining polynomial
Character	\(\chi\)	\(=\)	10000.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.183172 q^{3} -1.35874 q^{7} -2.96645 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \)

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
			\(3\)	0.183172	0.105754	0.0528772	−	0.998601i	\(-0.483161\pi\)
0.0528772	+	0.998601i				\(0.483161\pi\)
\(5\)	0	0
			\(7\)	−1.35874	−0.513556	−0.256778	−	0.966470i	\(-0.582661\pi\)
−0.256778	+	0.966470i				\(0.582661\pi\)
\(11\)	−2.79981	−0.844176	−0.422088	−	0.906555i	\(-0.638703\pi\)
			−0.422088	+	0.906555i	\(0.638703\pi\)
\(13\)	2.49487	0.691953	0.345976	−	0.938243i	\(-0.387548\pi\)
			0.345976	+	0.938243i	\(0.387548\pi\)
\(17\)	−4.95922	−1.20279	−0.601394	−	0.798953i	\(-0.705388\pi\)
			−0.601394	+	0.798953i	\(0.705388\pi\)
\(19\)	5.28477	1.21241	0.606204	−	0.795309i	\(-0.292691\pi\)
			0.606204	+	0.795309i	\(0.292691\pi\)
\(23\)	7.44414	1.55221	0.776106	−	0.630603i	\(-0.217192\pi\)
			0.776106	+	0.630603i	\(0.217192\pi\)
\(29\)	0.314862	0.0584684	0.0292342	−	0.999573i	\(-0.490693\pi\)
			0.0292342	+	0.999573i	\(0.490693\pi\)
\(31\)	5.26057	0.944827	0.472413	−	0.881377i	\(-0.343383\pi\)
			0.472413	+	0.881377i	\(0.343383\pi\)
\(37\)	−0.757065	−0.124461	−0.0622304	−	0.998062i	\(-0.519821\pi\)
			−0.0622304	+	0.998062i	\(0.519821\pi\)
\(41\)	−1.20019	−0.187438	−0.0937188	−	0.995599i	\(-0.529875\pi\)
			−0.0937188	+	0.995599i	\(0.529875\pi\)
\(43\)	−10.3054	−1.57155	−0.785776	−	0.618511i	\(-0.787736\pi\)
			−0.785776	+	0.618511i	\(0.787736\pi\)
\(47\)	4.27212	0.623152	0.311576	−	0.950221i	\(-0.399143\pi\)
			0.311576	+	0.950221i	\(0.399143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bi.1.5		8
4.3	odd	2		2500.2.a.f.1.4		8
5.4	even	2	inner	10000.2.a.bi.1.4		8
20.3	even	4		2500.2.c.b.1249.4		8
20.7	even	4		2500.2.c.b.1249.5		8
20.19	odd	2		2500.2.a.f.1.5		8
25.12	odd	20		400.2.y.b.369.2		8
25.23	odd	20		400.2.y.b.129.2		8
100.11	odd	10		500.2.g.b.101.3		16
100.23	even	20		100.2.i.a.29.1	✓	8
100.27	even	20		500.2.i.a.149.2		8
100.39	odd	10		500.2.g.b.101.2		16
100.59	odd	10		500.2.g.b.401.2		16
100.63	even	20		500.2.i.a.349.2		8
100.87	even	20		100.2.i.a.69.1	yes	8
100.91	odd	10		500.2.g.b.401.3		16
300.23	odd	20		900.2.w.a.829.1		8
300.287	odd	20		900.2.w.a.469.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.29.1	✓	8	100.23	even	20
100.2.i.a.69.1	yes	8	100.87	even	20
400.2.y.b.129.2		8	25.23	odd	20
400.2.y.b.369.2		8	25.12	odd	20
500.2.g.b.101.2		16	100.39	odd	10
500.2.g.b.101.3		16	100.11	odd	10
500.2.g.b.401.2		16	100.59	odd	10
500.2.g.b.401.3		16	100.91	odd	10
500.2.i.a.149.2		8	100.27	even	20
500.2.i.a.349.2		8	100.63	even	20
900.2.w.a.469.1		8	300.287	odd	20
900.2.w.a.829.1		8	300.23	odd	20
2500.2.a.f.1.4		8	4.3	odd	2
2500.2.a.f.1.5		8	20.19	odd	2
2500.2.c.b.1249.4		8	20.3	even	4
2500.2.c.b.1249.5		8	20.7	even	4
10000.2.a.bi.1.4		8	5.4	even	2	inner
10000.2.a.bi.1.5		8	1.1	even	1	trivial