Embedded newform 7.17.d.a.5.8

• Label: 7.17.d.a.5.8
• Level: $7$
• Weight: $17$
• Character: 7.5
• Analytic conductor: $11.363$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	7.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3627180700\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^19 + 470221*x^18 - 17792382*x^17 + 149610140535*x^16 - 5885709493452*x^15 + 25413927970368382*x^14 - 921982218074544860*x^13 + 3092605513604061172669*x^12 - 78217760507911051825254*x^11 + 217775891572550635786539387*x^10 - 772894989756975230685011586*x^9 + 10848813785145958185022965942709*x^8 + 50328102663961537370536178622316*x^7 + 292280160701254761307491650568975646*x^6 + 6884264619587692201297746565224842172*x^5 + 5465648386210730671112398026225318665199*x^4 + 69498817213798305204024309903431721187110*x^3 + 44558299163904360256389313185385225104835245*x^2 - 34923314543744821707168835219945993579648950*x + 261490399505792227817024003348104130194641159225
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{38}\cdot 3^{16}\cdot 7^{22} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.8
Root			\(137.362 + 237.918i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.5
Dual form			7.17.d.a.3.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(141.862 - 245.712i) q^{2} +(1577.48 - 910.759i) q^{3} +(-7481.54 - 12958.4i) q^{4} +(402298. + 232267. i) q^{5} -516808. i q^{6} +(5.58849e6 - 1.41483e6i) q^{7} +1.43487e7 q^{8} +(-1.98644e7 + 3.44061e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	141.862	−	245.712i	0.554148	−	0.959812i	−0.443822	−	0.896115i	\(-0.646378\pi\)
							0.997969	−	0.0636967i	\(-0.0202890\pi\)
\(3\)	1577.48	−	910.759i	0.240433	−	0.138814i	−0.374943	−	0.927048i	\(-0.622338\pi\)
							0.615376	+	0.788234i	\(0.289004\pi\)
\(5\)	402298.	+	232267.i	1.02988	+	0.594603i	0.916951	−	0.398999i	\(-0.130642\pi\)
							0.112932	+	0.993603i	\(0.463976\pi\)
\(7\)	5.58849e6	−	1.41483e6i	0.969415	−	0.245427i
							\(11\)	−1.44220e8	−	2.49797e8i	−0.672799	−	1.16532i	−0.977107	−	0.212748i	\(-0.931759\pi\)
0.304308	−	0.952574i	\(-0.401575\pi\)
\(13\)			1.22881e8i			0.150640i	0.997159	+	0.0753199i	\(0.0239978\pi\)
							−0.997159	+	0.0753199i	\(0.976002\pi\)
\(17\)	8.01015e9	−	4.62466e9i	1.14828	−	0.662962i	0.199816	−	0.979833i	\(-0.435966\pi\)
							0.948469	+	0.316871i	\(0.102632\pi\)
\(19\)	−7.94437e9	−	4.58668e9i	−0.467768	−	0.270066i	0.247537	−	0.968878i	\(-0.420379\pi\)
							−0.715305	+	0.698812i	\(0.753712\pi\)
\(23\)	−6.33600e10	+	1.09743e11i	−0.809082	+	1.40137i	0.104418	+	0.994533i	\(0.466702\pi\)
							−0.913500	+	0.406838i	\(0.866631\pi\)
\(29\)	3.57973e11			0.715594			0.357797	−	0.933799i	\(-0.383528\pi\)
							0.357797	+	0.933799i	\(0.383528\pi\)
\(31\)	−4.64342e11	+	2.68088e11i	−0.544432	+	0.314328i	−0.746873	−	0.664966i	\(-0.768446\pi\)
							0.202441	+	0.979294i	\(0.435113\pi\)
\(37\)	6.19491e11	−	1.07299e12i	0.176369	−	0.305480i	−0.764265	−	0.644902i	\(-0.776898\pi\)
							0.940634	+	0.339422i	\(0.110232\pi\)
\(41\)			2.17970e12i			0.272977i	0.990642	+	0.136488i	\(0.0435816\pi\)
							−0.990642	+	0.136488i	\(0.956418\pi\)
\(43\)	−1.24768e13			−1.06747			−0.533736	−	0.845651i	\(-0.679212\pi\)
							−0.533736	+	0.845651i	\(0.679212\pi\)
\(47\)	−3.91947e13	−	2.26291e13i	−1.64605	−	0.950350i	−0.978619	−	0.205682i	\(-0.934059\pi\)
							−0.667435	−	0.744668i	\(-0.732608\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.5.8	yes	20
7.2	even	3		49.17.b.a.48.6		20
7.3	odd	6	inner	7.17.d.a.3.8	✓	20
7.5	odd	6		49.17.b.a.48.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.8	✓	20	7.3	odd	6	inner
7.17.d.a.5.8	yes	20	1.1	even	1	trivial
49.17.b.a.48.5		20	7.5	odd	6
49.17.b.a.48.6		20	7.2	even	3