Embedded newform 8.22.b.a.5.13

• Label: 8.22.b.a.5.13
• Level: $8$
• Weight: $22$
• Character: 8.5
• Analytic conductor: $22.358$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	8.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.3581875430\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 10*x^19 + 8499037025*x^18 - 76491332940*x^17 + 29917993634631504354*x^16 - 239343947343248489484*x^15 + 56735124423752018369174556138*x^14 - 397145866777745056145685733776*x^13 + 63238887245213577325916647519604926269*x^12 - 379433318308385206734963736802075208922*x^11 + 42482893644148833600712851037696291113731507797*x^10 - 212414464742605426308676323132417376601509837748*x^9 + 16991149511760794392812139094660185274650292829286304832*x^8 - 67964596772556393289382483692574713665819457865324101248*x^7 + 3856070948945815855551883712975066709737139564290759222840809728*x^6 - 11568212608961359754849022371262813346960834248571059910113653760*x^5 + 447495609895006687050069424346090867517835281773807292283716383294251008*x^4 - 894991200509659105123235067097342555549602100555628271172491492140580864*x^3 + 21617394414514960541269362233298250396222492191038308122525553887833389622886400*x^2 - 21617393967019362214475233171141277620661834650156142656637432362426234096844800*x + 356285834965428012369716852146922959222876595571555882248526187687502452643625697280000
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	multiple of \( 2^{190}\cdot 3^{14}\cdot 5^{4}\cdot 7^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.13
Root			\(0.500000 - 6610.70i\) of defining polynomial
Character	\(\chi\)	\(=\)	8.5
Dual form			8.22.b.a.5.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(582.905 - 1325.66i) q^{2} +26442.8i q^{3} +(-1.41760e6 - 1.54547e6i) q^{4} +7.17621e6i q^{5} +(3.50542e7 + 1.54136e7i) q^{6} -5.46394e8 q^{7} +(-2.87509e9 + 9.78391e8i) q^{8} +9.76113e9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 286 q^{2} + 409876 q^{4} + 118236748 q^{6} + 564950496 q^{7} + 3649699336 q^{8} - 62762119220 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	582.905	−	1325.66i	0.402515	−	0.915413i
							\(3\)			26442.8i			0.258544i	0.991609	+	0.129272i	\(0.0412640\pi\)
−0.991609	+	0.129272i	\(0.958736\pi\)
\(5\)			7.17621e6i			0.328632i	0.986408	+	0.164316i	\(0.0525417\pi\)
							−0.986408	+	0.164316i	\(0.947458\pi\)
\(7\)	−5.46394e8			−0.731100			−0.365550	−	0.930792i	\(-0.619119\pi\)
							−0.365550	+	0.930792i	\(0.619119\pi\)
\(11\)		−	4.39803e9i		−	0.0511252i	−0.999673	−	0.0255626i	\(-0.991862\pi\)
							0.999673	−	0.0255626i	\(-0.00813771\pi\)
\(13\)			7.95447e11i			1.60032i	0.599788	+	0.800159i	\(0.295252\pi\)
							−0.599788	+	0.800159i	\(0.704748\pi\)
\(17\)	5.71073e12			0.687033			0.343517	−	0.939147i	\(-0.388382\pi\)
							0.343517	+	0.939147i	\(0.388382\pi\)
\(19\)			4.93746e12i			0.184753i	0.995724	+	0.0923763i	\(0.0294463\pi\)
							−0.995724	+	0.0923763i	\(0.970554\pi\)
\(23\)	1.23935e14			0.623812			0.311906	−	0.950113i	\(-0.399033\pi\)
							0.311906	+	0.950113i	\(0.399033\pi\)
\(29\)			1.25512e15i			0.553993i	0.960871	+	0.276997i	\(0.0893391\pi\)
							−0.960871	+	0.276997i	\(0.910661\pi\)
\(31\)	4.10812e15			0.900214			0.450107	−	0.892975i	\(-0.351386\pi\)
							0.450107	+	0.892975i	\(0.351386\pi\)
\(37\)			8.37526e15i			0.286339i	0.989698	+	0.143169i	\(0.0457294\pi\)
							−0.989698	+	0.143169i	\(0.954271\pi\)
\(41\)	−1.06322e17			−1.23707			−0.618533	−	0.785759i	\(-0.712273\pi\)
							−0.618533	+	0.785759i	\(0.712273\pi\)
\(43\)			9.16118e16i			0.646446i	0.946323	+	0.323223i	\(0.104766\pi\)
							−0.946323	+	0.323223i	\(0.895234\pi\)
\(47\)	−4.35184e17			−1.20683			−0.603414	−	0.797428i	\(-0.706193\pi\)
							−0.603414	+	0.797428i	\(0.706193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.22.b.a.5.13	✓	20
4.3	odd	2		32.22.b.a.17.9		20
8.3	odd	2		32.22.b.a.17.12		20
8.5	even	2	inner	8.22.b.a.5.14	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.b.a.5.13	✓	20	1.1	even	1	trivial
8.22.b.a.5.14	yes	20	8.5	even	2	inner
32.22.b.a.17.9		20	4.3	odd	2
32.22.b.a.17.12		20	8.3	odd	2