Embedded newform 8.13.d.b.3.4

• Label: 8.13.d.b.3.4
• 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			3.4
Root			\(-11.5862 - 26.7344i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.3
Dual form			8.13.d.b.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-35.1724 + 53.4687i) q^{2} +1272.20 q^{3} +(-1621.80 - 3761.25i) q^{4} -19079.0i q^{5} +(-44746.4 + 68023.0i) q^{6} -136156. i q^{7} +(258152. + 45576.2i) q^{8} +1.08706e6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 110 q^{2} - 660 q^{3} - 2444 q^{4} - 130788 q^{6} + 27160 q^{8} + 1692798 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−35.1724	+	53.4687i	−0.549569	+	0.835449i
							\(3\)	1272.20			1.74513			0.872567	−	0.488495i	\(-0.162454\pi\)
0.872567	+	0.488495i	\(0.162454\pi\)
\(5\)		−	19079.0i		−	1.22106i	−0.791995	−	0.610528i	\(-0.790957\pi\)
							0.791995	−	0.610528i	\(-0.209043\pi\)
\(7\)		−	136156.i		−	1.15730i	−0.815575	−	0.578652i	\(-0.803579\pi\)
							0.815575	−	0.578652i	\(-0.196421\pi\)
\(11\)	−478611.			−0.270163			−0.135082	−	0.990834i	\(-0.543130\pi\)
							−0.135082	+	0.990834i	\(0.543130\pi\)
\(13\)			4.87546e6i			1.01008i	0.863096	+	0.505040i	\(0.168522\pi\)
							−0.863096	+	0.505040i	\(0.831478\pi\)
\(17\)	9.96078e6			0.412667			0.206334	−	0.978482i	\(-0.433847\pi\)
							0.206334	+	0.978482i	\(0.433847\pi\)
\(19\)	847952.			0.0180239			0.00901197	−	0.999959i	\(-0.497131\pi\)
							0.00901197	+	0.999959i	\(0.497131\pi\)
\(23\)			1.02872e8i			0.694915i	0.937696	+	0.347458i	\(0.112955\pi\)
							−0.937696	+	0.347458i	\(0.887045\pi\)
\(29\)		−	5.58995e8i		−	0.939766i	−0.882729	−	0.469883i	\(-0.844296\pi\)
							0.882729	−	0.469883i	\(-0.155704\pi\)
\(31\)			4.80824e8i			0.541772i	0.962611	+	0.270886i	\(0.0873166\pi\)
							−0.962611	+	0.270886i	\(0.912683\pi\)
\(37\)			2.73151e9i			1.06461i	0.846552	+	0.532306i	\(0.178675\pi\)
							−0.846552	+	0.532306i	\(0.821325\pi\)
\(41\)	−4.60219e9			−0.968861			−0.484430	−	0.874830i	\(-0.660973\pi\)
							−0.484430	+	0.874830i	\(0.660973\pi\)
\(43\)	1.16234e10			1.83875			0.919374	−	0.393385i	\(-0.128696\pi\)
							0.919374	+	0.393385i	\(0.128696\pi\)
\(47\)			5.60762e9i			0.520226i	0.965578	+	0.260113i	\(0.0837598\pi\)
							−0.965578	+	0.260113i	\(0.916240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.13.d.b.3.4	yes	10
3.2	odd	2		72.13.b.b.19.7		10
4.3	odd	2		32.13.d.b.15.1		10
8.3	odd	2	inner	8.13.d.b.3.3	✓	10
8.5	even	2		32.13.d.b.15.2		10
24.11	even	2		72.13.b.b.19.8		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.13.d.b.3.3	✓	10	8.3	odd	2	inner
8.13.d.b.3.4	yes	10	1.1	even	1	trivial
32.13.d.b.15.1		10	4.3	odd	2
32.13.d.b.15.2		10	8.5	even	2
72.13.b.b.19.7		10	3.2	odd	2
72.13.b.b.19.8		10	24.11	even	2