Embedded newform 7.17.d.a.3.10

• Label: 7.17.d.a.3.10
• Level: $7$
• Weight: $17$
• Character: 7.3
• 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			3.10
Root			\(230.136 - 398.607i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.3
Dual form			7.17.d.a.5.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(234.636 + 406.401i) q^{2} +(-4044.92 - 2335.34i) q^{3} +(-77340.0 + 133957. i) q^{4} +(59013.4 - 34071.4i) q^{5} -2.19181e6i q^{6} +(-5.75487e6 + 338210. i) q^{7} -4.18327e7 q^{8} +(-1.06158e7 - 1.83871e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	234.636	+	406.401i	0.916546	+	1.58750i	0.804621	+	0.593788i	\(0.202368\pi\)
							0.111925	+	0.993717i	\(0.464298\pi\)
\(3\)	−4044.92	−	2335.34i	−0.616510	−	0.355942i	0.158999	−	0.987279i	\(-0.449173\pi\)
							−0.775509	+	0.631337i	\(0.782507\pi\)
\(5\)	59013.4	−	34071.4i	0.151074	−	0.0872228i	−0.422557	−	0.906336i	\(-0.638867\pi\)
							0.573632	+	0.819113i	\(0.305534\pi\)
\(7\)	−5.75487e6	+	338210.i	−0.998278	+	0.0586681i
							\(11\)	1.14580e8	−	1.98458e8i	0.534523	−	0.925821i	−0.464663	−	0.885487i	\(-0.653825\pi\)
0.999186	−	0.0403334i	\(-0.0128420\pi\)
\(13\)			1.08556e9i			1.33079i	0.746492	+	0.665394i	\(0.231736\pi\)
							−0.746492	+	0.665394i	\(0.768264\pi\)
\(17\)	−6.78285e8	−	3.91608e8i	−0.0972346	−	0.0561384i	0.450594	−	0.892729i	\(-0.351212\pi\)
							−0.547829	+	0.836590i	\(0.684545\pi\)
\(19\)	−2.45630e10	+	1.41815e10i	−1.44628	+	0.835011i	−0.998257	−	0.0590104i	\(-0.981205\pi\)
							−0.448024	+	0.894021i	\(0.647872\pi\)
\(23\)	−4.24275e10	−	7.34866e10i	−0.541782	−	0.938394i	−0.998802	−	0.0489375i	\(-0.984416\pi\)
							0.457020	−	0.889457i	\(-0.348917\pi\)
\(29\)	−5.71793e11			−1.14302			−0.571512	−	0.820594i	\(-0.693643\pi\)
							−0.571512	+	0.820594i	\(0.693643\pi\)
\(31\)	7.61056e11	+	4.39396e11i	0.892325	+	0.515184i	0.874702	−	0.484661i	\(-0.161057\pi\)
							0.0176228	+	0.999845i	\(0.494390\pi\)
\(37\)	9.66695e11	+	1.67437e12i	0.275217	+	0.476691i	0.970190	−	0.242346i	\(-0.0779168\pi\)
							−0.694973	+	0.719036i	\(0.744584\pi\)
\(41\)			4.03517e12i			0.505348i	0.967551	+	0.252674i	\(0.0813101\pi\)
							−0.967551	+	0.252674i	\(0.918690\pi\)
\(43\)	−3.48769e12			−0.298394			−0.149197	−	0.988807i	\(-0.547669\pi\)
							−0.149197	+	0.988807i	\(0.547669\pi\)
\(47\)	2.40341e13	−	1.38761e13i	1.00936	−	0.582752i	0.0983527	−	0.995152i	\(-0.468643\pi\)
							0.911003	+	0.412400i	\(0.135309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.3.10	✓	20
7.3	odd	6		49.17.b.a.48.1		20
7.4	even	3		49.17.b.a.48.2		20
7.5	odd	6	inner	7.17.d.a.5.10	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.10	✓	20	1.1	even	1	trivial
7.17.d.a.5.10	yes	20	7.5	odd	6	inner
49.17.b.a.48.1		20	7.3	odd	6
49.17.b.a.48.2		20	7.4	even	3