Embedded newform 7.17.d.a.5.3

• Label: 7.17.d.a.5.3
• 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			5.3
Root			\(-120.823 - 209.272i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.5
Dual form			7.17.d.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-116.323 + 201.477i) q^{2} +(2610.90 - 1507.40i) q^{3} +(5705.88 + 9882.88i) q^{4} +(-301331. - 173973. i) q^{5} +701383. i q^{6} +(2.93221e6 + 4.96337e6i) q^{7} -1.79016e7 q^{8} +(-1.69788e7 + 2.94082e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−116.323	+	201.477i	−0.454387	+	0.787021i	−0.998653	−	0.0518915i	\(-0.983475\pi\)
							0.544266	+	0.838913i	\(0.316808\pi\)
\(3\)	2610.90	−	1507.40i	0.397942	−	0.229752i	−0.287653	−	0.957735i	\(-0.592875\pi\)
							0.685596	+	0.727982i	\(0.259542\pi\)
\(5\)	−301331.	−	173973.i	−0.771407	−	0.445372i	0.0619694	−	0.998078i	\(-0.480262\pi\)
							−0.833376	+	0.552706i	\(0.813595\pi\)
\(7\)	2.93221e6	+	4.96337e6i	0.508640	+	0.860979i
							\(11\)	−1.55722e8	−	2.69719e8i	−0.726456	−	1.25826i	−0.958372	−	0.285522i	\(-0.907833\pi\)
0.231916	−	0.972736i	\(-0.425500\pi\)
\(13\)		−	1.33767e9i		−	1.63985i	−0.572472	−	0.819924i	\(-0.694016\pi\)
							0.572472	−	0.819924i	\(-0.305984\pi\)
\(17\)	−6.18748e9	+	3.57235e9i	−0.886998	+	0.512109i	−0.872959	−	0.487793i	\(-0.837802\pi\)
							−0.0140387	+	0.999901i	\(0.504469\pi\)
\(19\)	−4.41495e9	−	2.54897e9i	−0.259954	−	0.150085i	0.364359	−	0.931258i	\(-0.381288\pi\)
							−0.624314	+	0.781174i	\(0.714621\pi\)
\(23\)	−1.95362e10	+	3.38377e10i	−0.249469	+	0.432093i	−0.963379	−	0.268145i	\(-0.913589\pi\)
							0.713909	+	0.700238i	\(0.246923\pi\)
\(29\)	−5.16351e11			−1.03219			−0.516097	−	0.856530i	\(-0.672616\pi\)
							−0.516097	+	0.856530i	\(0.672616\pi\)
\(31\)	4.84320e11	−	2.79622e11i	0.567857	−	0.327852i	−0.188436	−	0.982085i	\(-0.560342\pi\)
							0.756293	+	0.654233i	\(0.227009\pi\)
\(37\)	−2.36862e12	+	4.10257e12i	−0.674343	+	1.16800i	0.302317	+	0.953207i	\(0.402240\pi\)
							−0.976660	+	0.214790i	\(0.931093\pi\)
\(41\)			1.21260e13i			1.51861i	0.650734	+	0.759306i	\(0.274461\pi\)
							−0.650734	+	0.759306i	\(0.725539\pi\)
\(43\)	7.60946e12			0.651038			0.325519	−	0.945536i	\(-0.394461\pi\)
							0.325519	+	0.945536i	\(0.394461\pi\)
\(47\)	1.03624e13	+	5.98272e12i	0.435187	+	0.251256i	0.701554	−	0.712616i	\(-0.252490\pi\)
							−0.266367	+	0.963872i	\(0.585823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.5.3	yes	20
7.2	even	3		49.17.b.a.48.16		20
7.3	odd	6	inner	7.17.d.a.3.3	✓	20
7.5	odd	6		49.17.b.a.48.15		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.3	✓	20	7.3	odd	6	inner
7.17.d.a.5.3	yes	20	1.1	even	1	trivial
49.17.b.a.48.15		20	7.5	odd	6
49.17.b.a.48.16		20	7.2	even	3