Embedded newform 7.17.d.a.5.4

• Label: 7.17.d.a.5.4
• 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			5.4
Root			\(-64.4555 - 111.640i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.5
Dual form			7.17.d.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-59.9555 + 103.846i) q^{2} +(-7943.79 + 4586.35i) q^{3} +(25578.7 + 44303.6i) q^{4} +(625327. + 361033. i) q^{5} -1.09991e6i q^{6} +(-1.45978e6 + 5.57691e6i) q^{7} -1.39928e7 q^{8} +(2.05458e7 - 3.55864e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−59.9555	+	103.846i	−0.234201	+	0.405648i	−0.959040	−	0.283270i	\(-0.908581\pi\)
							0.724839	+	0.688918i	\(0.241914\pi\)
\(3\)	−7943.79	+	4586.35i	−1.21076	+	0.699032i	−0.962925	−	0.269771i	\(-0.913052\pi\)
							−0.247834	+	0.968802i	\(0.579719\pi\)
\(5\)	625327.	+	361033.i	1.60084	+	0.924244i	0.991320	+	0.131472i	\(0.0419704\pi\)
							0.609518	+	0.792772i	\(0.291363\pi\)
\(7\)	−1.45978e6	+	5.57691e6i	−0.253223	+	0.967408i
							\(11\)	6.21540e7	+	1.07654e8i	0.289953	+	0.502213i	0.973798	−	0.227414i	\(-0.0730270\pi\)
−0.683845	+	0.729627i	\(0.739694\pi\)
\(13\)		−	2.89865e8i		−	0.355344i	−0.984090	−	0.177672i	\(-0.943143\pi\)
							0.984090	−	0.177672i	\(-0.0568566\pi\)
\(17\)	4.99734e9	−	2.88521e9i	0.716386	−	0.413606i	−0.0970349	−	0.995281i	\(-0.530936\pi\)
							0.813421	+	0.581675i	\(0.197602\pi\)
\(19\)	−7.15651e9	−	4.13181e9i	−0.421379	−	0.243283i	0.274288	−	0.961648i	\(-0.411558\pi\)
							−0.695667	+	0.718364i	\(0.744891\pi\)
\(23\)	3.77417e10	−	6.53706e10i	0.481947	−	0.834756i	−0.517839	−	0.855478i	\(-0.673263\pi\)
							0.999785	+	0.0207222i	\(0.00659654\pi\)
\(29\)	2.99884e11			0.599473			0.299737	−	0.954022i	\(-0.403101\pi\)
							0.299737	+	0.954022i	\(0.403101\pi\)
\(31\)	−9.19302e11	+	5.30759e11i	−1.07787	+	0.622306i	−0.930320	−	0.366749i	\(-0.880471\pi\)
							−0.147546	+	0.989055i	\(0.547137\pi\)
\(37\)	2.22015e12	−	3.84541e12i	0.632075	−	1.09479i	−0.355052	−	0.934846i	\(-0.615537\pi\)
							0.987127	−	0.159939i	\(-0.0511297\pi\)
\(41\)			6.89210e12i			0.863139i	0.902080	+	0.431569i	\(0.142040\pi\)
							−0.902080	+	0.431569i	\(0.857960\pi\)
\(43\)	7.25582e12			0.620781			0.310391	−	0.950609i	\(-0.399540\pi\)
							0.310391	+	0.950609i	\(0.399540\pi\)
\(47\)	1.70574e13	+	9.84807e12i	0.716356	+	0.413588i	0.813410	−	0.581691i	\(-0.197609\pi\)
							−0.0970541	+	0.995279i	\(0.530942\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.5.4	yes	20
7.2	even	3		49.17.b.a.48.13		20
7.3	odd	6	inner	7.17.d.a.3.4	✓	20
7.5	odd	6		49.17.b.a.48.14		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.4	✓	20	7.3	odd	6	inner
7.17.d.a.5.4	yes	20	1.1	even	1	trivial
49.17.b.a.48.13		20	7.2	even	3
49.17.b.a.48.14		20	7.5	odd	6