Embedded newform 8.14.b.b.5.7

• Label: 8.14.b.b.5.7
• Level: $8$
• Weight: $14$
• Character: 8.5
• Analytic conductor: $8.578$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	8.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.57847431615\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 5*x^9 + 752*x^8 + 708*x^7 - 743866*x^6 + 96647426*x^5 + 2540283092*x^4 - 180067834748*x^3 + 15101451375489*x^2 - 802096030557765*x + 31616997813655668
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{48}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.7
Root			\(27.9424 - 31.0290i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.5
Dual form			8.14.b.b.5.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(65.8848 - 62.0580i) q^{2} +1231.41i q^{3} +(489.620 - 8177.36i) q^{4} -25270.7i q^{5} +(76418.8 + 81131.3i) q^{6} +608245. q^{7} +(-475211. - 569148. i) q^{8} +77950.5 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 110 q^{2} - 4716 q^{4} - 267668 q^{6} + 586960 q^{7} - 270712 q^{8} + 2014054 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	65.8848	−	62.0580i	0.727931	−	0.685650i
							\(3\)			1231.41i			0.975247i	0.873054	+	0.487624i	\(0.162136\pi\)
−0.873054	+	0.487624i	\(0.837864\pi\)
\(5\)		−	25270.7i		−	0.723288i	−0.932316	−	0.361644i	\(-0.882216\pi\)
							0.932316	−	0.361644i	\(-0.117784\pi\)
\(7\)	608245.			1.95408			0.977038	−	0.213064i	\(-0.0683442\pi\)
							0.977038	+	0.213064i	\(0.0683442\pi\)
\(11\)		−	7.12288e6i		−	1.21228i	−0.795358	−	0.606140i	\(-0.792717\pi\)
							0.795358	−	0.606140i	\(-0.207283\pi\)
\(13\)			1.12887e7i			0.648652i	0.945945	+	0.324326i	\(0.105137\pi\)
							−0.945945	+	0.324326i	\(0.894863\pi\)
\(17\)	−5.09178e7			−0.511625			−0.255812	−	0.966726i	\(-0.582343\pi\)
							−0.255812	+	0.966726i	\(0.582343\pi\)
\(19\)			1.86178e8i			0.907881i	0.891032	+	0.453941i	\(0.149982\pi\)
							−0.891032	+	0.453941i	\(0.850018\pi\)
\(23\)	−3.07644e8			−0.433328			−0.216664	−	0.976246i	\(-0.569518\pi\)
							−0.216664	+	0.976246i	\(0.569518\pi\)
\(29\)		−	4.74147e8i		−	0.148022i	−0.997257	−	0.0740110i	\(-0.976420\pi\)
							0.997257	−	0.0740110i	\(-0.0235800\pi\)
\(31\)	−2.41919e9			−0.489574			−0.244787	−	0.969577i	\(-0.578718\pi\)
							−0.244787	+	0.969577i	\(0.578718\pi\)
\(37\)			1.55589e10i			0.996939i	0.866907	+	0.498470i	\(0.166104\pi\)
							−0.866907	+	0.498470i	\(0.833896\pi\)
\(41\)	−2.32986e10			−0.766009			−0.383005	−	0.923746i	\(-0.625111\pi\)
							−0.383005	+	0.923746i	\(0.625111\pi\)
\(43\)		−	6.55713e10i		−	1.58186i	−0.611905	−	0.790932i	\(-0.709596\pi\)
							0.611905	−	0.790932i	\(-0.290404\pi\)
\(47\)	−6.84950e10			−0.926879			−0.463440	−	0.886129i	\(-0.653385\pi\)
							−0.463440	+	0.886129i	\(0.653385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.14.b.b.5.7	✓	10
3.2	odd	2		72.14.d.c.37.4		10
4.3	odd	2		32.14.b.b.17.3		10
8.3	odd	2		32.14.b.b.17.8		10
8.5	even	2	inner	8.14.b.b.5.8	yes	10
24.5	odd	2		72.14.d.c.37.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.14.b.b.5.7	✓	10	1.1	even	1	trivial
8.14.b.b.5.8	yes	10	8.5	even	2	inner
32.14.b.b.17.3		10	4.3	odd	2
32.14.b.b.17.8		10	8.3	odd	2
72.14.d.c.37.3		10	24.5	odd	2
72.14.d.c.37.4		10	3.2	odd	2