Embedded newform 8.13.d.b.3.10

• Label: 8.13.d.b.3.10
• Level: $8$
• Weight: $13$
• Character: 8.3
• Analytic conductor: $7.312$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31195053821\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 5*x^9 + 468*x^8 + 1496*x^7 + 710096*x^6 + 29155008*x^5 + 143571008*x^4 + 28213427840*x^3 + 1335549648384*x^2 + 41051831642368*x + 824967906703360
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{45}\cdot 3^{3}\cdot 23 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3.10
Root			\(32.5710 - 17.8321i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.3
Dual form			8.13.d.b.3.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(53.1419 + 35.6642i) q^{2} -1022.07 q^{3} +(1552.12 + 3790.53i) q^{4} -21249.1i q^{5} +(-54314.6 - 36451.3i) q^{6} -149114. i q^{7} +(-52703.6 + 256791. i) q^{8} +513182. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 110 q^{2} - 660 q^{3} - 2444 q^{4} - 130788 q^{6} + 27160 q^{8} + 1692798 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	53.1419	+	35.6642i	0.830342	+	0.557254i
							\(3\)	−1022.07			−1.40201			−0.701007	−	0.713154i	\(-0.747266\pi\)
−0.701007	+	0.713154i	\(0.747266\pi\)
\(5\)		−	21249.1i		−	1.35994i	−0.733239	−	0.679971i	\(-0.761992\pi\)
							0.733239	−	0.679971i	\(-0.238008\pi\)
\(7\)		−	149114.i		−	1.26745i	−0.773558	−	0.633725i	\(-0.781525\pi\)
							0.773558	−	0.633725i	\(-0.218475\pi\)
\(11\)	−2.66710e6			−1.50551			−0.752754	−	0.658302i	\(-0.771275\pi\)
							−0.752754	+	0.658302i	\(0.771275\pi\)
\(13\)		−	2.70501e6i		−	0.560413i	−0.959940	−	0.280207i	\(-0.909597\pi\)
							0.959940	−	0.280207i	\(-0.0904029\pi\)
\(17\)	8.65970e6			0.358764			0.179382	−	0.983779i	\(-0.442590\pi\)
							0.179382	+	0.983779i	\(0.442590\pi\)
\(19\)	−3.18479e7			−0.676955			−0.338477	−	0.940975i	\(-0.609912\pi\)
							−0.338477	+	0.940975i	\(0.609912\pi\)
\(23\)			7.29067e7i			0.492493i	0.969207	+	0.246247i	\(0.0791973\pi\)
							−0.969207	+	0.246247i	\(0.920803\pi\)
\(29\)		−	7.54217e8i		−	1.26797i	−0.773346	−	0.633984i	\(-0.781419\pi\)
							0.773346	−	0.633984i	\(-0.218581\pi\)
\(31\)			1.62267e8i			0.182835i	0.995813	+	0.0914176i	\(0.0291398\pi\)
							−0.995813	+	0.0914176i	\(0.970860\pi\)
\(37\)		−	2.33746e9i		−	0.911032i	−0.890228	−	0.455516i	\(-0.849455\pi\)
							0.890228	−	0.455516i	\(-0.150545\pi\)
\(41\)	5.58385e9			1.17552			0.587761	−	0.809035i	\(-0.300009\pi\)
							0.587761	+	0.809035i	\(0.300009\pi\)
\(43\)	−1.62440e9			−0.256971			−0.128485	−	0.991711i	\(-0.541012\pi\)
							−0.128485	+	0.991711i	\(0.541012\pi\)
\(47\)		−	4.97675e9i		−	0.461699i	−0.972990	−	0.230849i	\(-0.925850\pi\)
							0.972990	−	0.230849i	\(-0.0741504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.13.d.b.3.10	yes	10
3.2	odd	2		72.13.b.b.19.1		10
4.3	odd	2		32.13.d.b.15.9		10
8.3	odd	2	inner	8.13.d.b.3.9	✓	10
8.5	even	2		32.13.d.b.15.10		10
24.11	even	2		72.13.b.b.19.2		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.13.d.b.3.9	✓	10	8.3	odd	2	inner
8.13.d.b.3.10	yes	10	1.1	even	1	trivial
32.13.d.b.15.9		10	4.3	odd	2
32.13.d.b.15.10		10	8.5	even	2
72.13.b.b.19.1		10	3.2	odd	2
72.13.b.b.19.2		10	24.11	even	2