Embedded newform 8.20.b.a.5.12

• Label: 8.20.b.a.5.12
• Level: $8$
• Weight: $20$
• Character: 8.5
• Analytic conductor: $18.305$
• Analytic rank: $0$
• Dimension: $18$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	8.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.3053357245\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial:	x^18 - 9*x^17 + 847482358*x^16 - 6779858660*x^15 + 288755694526787238*x^14 - 2021289743039984830*x^13 + 51121850635988052944521232*x^12 - 306731077539161966630890980*x^11 + 5053028465804290954773516593713297*x^10 - 25265139517319958838954290038554593*x^9 + 277525242785226225995541366728053703523138*x^8 - 1110100819550071174104182779845774739997744*x^7 + 7843581170028346695466748185748667347430921666496*x^6 - 23530739624732277774733733894667178736900394654048*x^5 + 88699644118206331836031766951870758907094189945148981184*x^4 - 177399249018514584235839520819873862547126510378177038080*x^3 + 27738447216461514660719783835924541644179956872586393406592768*x^2 - 27738358516840927193180104500114450400133860395870028292288768*x + 1155804113673024778339834102840085457131269774629483366903990104576
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	multiple of \( 2^{153}\cdot 3^{16}\cdot 5^{4}\cdot 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.12
Root			\(0.500000 + 11170.2i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.5
Dual form			8.20.b.a.5.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(255.450 + 677.520i) q^{2} -44680.9i q^{3} +(-393778. + 346145. i) q^{4} -2.26130e6i q^{5} +(3.02722e7 - 1.14138e7i) q^{6} -2.45061e7 q^{7} +(-3.35111e8 - 1.78370e8i) q^{8} -8.34122e8 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 458 q^{2} - 412108 q^{4} - 47240948 q^{6} - 80707216 q^{7} - 173313752 q^{8} - 6198727826 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	255.450	+	677.520i	0.352794	+	0.935701i
							\(3\)		−	44680.9i		−	1.31060i	−0.755369	−	0.655300i	\(-0.772542\pi\)
0.755369	−	0.655300i	\(-0.227458\pi\)
\(5\)		−	2.26130e6i		−	0.517776i	−0.965907	−	0.258888i	\(-0.916644\pi\)
							0.965907	−	0.258888i	\(-0.0833561\pi\)
\(7\)	−2.45061e7			−0.229532			−0.114766	−	0.993393i	\(-0.536612\pi\)
							−0.114766	+	0.993393i	\(0.536612\pi\)
\(11\)			1.09436e10i			1.39935i	0.714459	+	0.699677i	\(0.246673\pi\)
							−0.714459	+	0.699677i	\(0.753327\pi\)
\(13\)			4.56239e10i			1.19325i	0.802521	+	0.596624i	\(0.203492\pi\)
							−0.802521	+	0.596624i	\(0.796508\pi\)
\(17\)	−6.34826e11			−1.29835			−0.649173	−	0.760641i	\(-0.724885\pi\)
							−0.649173	+	0.760641i	\(0.724885\pi\)
\(19\)			7.57954e11i			0.538870i	0.963019	+	0.269435i	\(0.0868369\pi\)
							−0.963019	+	0.269435i	\(0.913163\pi\)
\(23\)	−9.73127e12			−1.12656			−0.563281	−	0.826265i	\(-0.690461\pi\)
							−0.563281	+	0.826265i	\(0.690461\pi\)
\(29\)			7.77941e13i			0.995785i	0.867239	+	0.497892i	\(0.165893\pi\)
							−0.867239	+	0.497892i	\(0.834107\pi\)
\(31\)	−2.14743e14			−1.45875			−0.729377	−	0.684112i	\(-0.760190\pi\)
							−0.729377	+	0.684112i	\(0.760190\pi\)
\(37\)			3.27399e14i			0.414153i	0.978325	+	0.207077i	\(0.0663949\pi\)
							−0.978325	+	0.207077i	\(0.933605\pi\)
\(41\)	1.85059e15			0.882802			0.441401	−	0.897310i	\(-0.354482\pi\)
							0.441401	+	0.897310i	\(0.354482\pi\)
\(43\)			1.28668e15i			0.390410i	0.980763	+	0.195205i	\(0.0625372\pi\)
							−0.980763	+	0.195205i	\(0.937463\pi\)
\(47\)	6.38823e15			0.832627			0.416314	−	0.909221i	\(-0.363322\pi\)
							0.416314	+	0.909221i	\(0.363322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.20.b.a.5.12	yes	18
3.2	odd	2		72.20.d.b.37.7		18
4.3	odd	2		32.20.b.a.17.15		18
8.3	odd	2		32.20.b.a.17.4		18
8.5	even	2	inner	8.20.b.a.5.11	✓	18
24.5	odd	2		72.20.d.b.37.8		18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.b.a.5.11	✓	18	8.5	even	2	inner
8.20.b.a.5.12	yes	18	1.1	even	1	trivial
32.20.b.a.17.4		18	8.3	odd	2
32.20.b.a.17.15		18	4.3	odd	2
72.20.d.b.37.7		18	3.2	odd	2
72.20.d.b.37.8		18	24.5	odd	2