Embedded newform 7.17.d.a.3.6

• Label: 7.17.d.a.3.6
• 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			3.6
Root			\(45.0238 - 77.9836i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.3
Dual form			7.17.d.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(49.5238 + 85.7778i) q^{2} +(289.874 + 167.359i) q^{3} +(27862.8 - 48259.7i) q^{4} +(-159884. + 92309.1i) q^{5} +33153.0i q^{6} +(-5.76147e6 + 195847. i) q^{7} +1.20107e7 q^{8} +(-2.14673e7 - 3.71825e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	49.5238	+	85.7778i	0.193452	+	0.335070i	0.946392	−	0.323020i	\(-0.104698\pi\)
							−0.752940	+	0.658090i	\(0.771365\pi\)
\(3\)	289.874	+	167.359i	0.0441813	+	0.0255081i	0.521928	−	0.852990i	\(-0.325213\pi\)
							−0.477747	+	0.878498i	\(0.658546\pi\)
\(5\)	−159884.	+	92309.1i	−0.409303	+	0.236311i	−0.690490	−	0.723342i	\(-0.742605\pi\)
							0.281187	+	0.959653i	\(0.409272\pi\)
\(7\)	−5.76147e6	+	195847.i	−0.999423	+	0.0339729i
							\(11\)	−2.23546e7	+	3.87193e7i	−0.104286	+	0.180628i	−0.913446	−	0.406960i	\(-0.866589\pi\)
0.809160	+	0.587588i	\(0.199922\pi\)
\(13\)		−	1.46560e9i		−	1.79667i	−0.439312	−	0.898335i	\(-0.644778\pi\)
							0.439312	−	0.898335i	\(-0.355222\pi\)
\(17\)	−8.32802e9	−	4.80818e9i	−1.19385	−	0.689271i	−0.234674	−	0.972074i	\(-0.575402\pi\)
							−0.959178	+	0.282804i	\(0.908736\pi\)
\(19\)	8.57378e9	−	4.95007e9i	0.504828	−	0.291463i	−0.225877	−	0.974156i	\(-0.572525\pi\)
							0.730705	+	0.682693i	\(0.239191\pi\)
\(23\)	6.13458e10	+	1.06254e11i	0.783362	+	1.35682i	0.929973	+	0.367628i	\(0.119830\pi\)
							−0.146611	+	0.989194i	\(0.546837\pi\)
\(29\)	8.48876e10			0.169692			0.0848458	−	0.996394i	\(-0.472960\pi\)
							0.0848458	+	0.996394i	\(0.472960\pi\)
\(31\)	−5.65540e11	−	3.26515e11i	−0.663086	−	0.382833i	0.130366	−	0.991466i	\(-0.458385\pi\)
							−0.793452	+	0.608633i	\(0.791718\pi\)
\(37\)	−8.15307e11	−	1.41215e12i	−0.232117	−	0.402039i	0.726314	−	0.687363i	\(-0.241232\pi\)
							−0.958431	+	0.285324i	\(0.907899\pi\)
\(41\)			8.18870e12i			1.02552i	0.858532	+	0.512760i	\(0.171377\pi\)
							−0.858532	+	0.512760i	\(0.828623\pi\)
\(43\)	1.12773e13			0.964845			0.482422	−	0.875939i	\(-0.339757\pi\)
							0.482422	+	0.875939i	\(0.339757\pi\)
\(47\)	−1.26377e13	+	7.29639e12i	−0.530745	+	0.306426i	−0.741320	−	0.671152i	\(-0.765800\pi\)
							0.210575	+	0.977578i	\(0.432466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.3.6	✓	20
7.3	odd	6		49.17.b.a.48.10		20
7.4	even	3		49.17.b.a.48.9		20
7.5	odd	6	inner	7.17.d.a.5.6	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.6	✓	20	1.1	even	1	trivial
7.17.d.a.5.6	yes	20	7.5	odd	6	inner
49.17.b.a.48.9		20	7.4	even	3
49.17.b.a.48.10		20	7.3	odd	6