Embedded newform 5.35.c.a.3.8

• Label: 5.35.c.a.3.8
• Level: $5$
• Weight: $35$
• Character: 5.3
• Analytic conductor: $36.613$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 35 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.6128270213\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	5.3
Dual form			5.35.c.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8626.63 - 8626.63i) q^{2} +(9.89110e7 + 9.89110e7i) q^{3} +1.70310e10i q^{4} +(-8.06672e10 - 7.58663e11i) q^{5} +1.70654e12 q^{6} +(1.10848e14 - 1.10848e14i) q^{7} +(2.95125e14 + 2.95125e14i) q^{8} +2.88960e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 131070 * q^2 - 78988260 * q^3 - 730721770860 * q^5 - 44043049959816 * q^6 + 5337954235100 * q^7 - 4536119823236220 * q^8

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8626.63	−	8626.63i	0.0658160	−	0.0658160i	−0.673433	−	0.739249i	\(-0.735181\pi\)
							0.739249	+	0.673433i	\(0.235181\pi\)
\(3\)	9.89110e7	+	9.89110e7i	0.765920	+	0.765920i	0.977385	−	0.211465i	\(-0.0678236\pi\)
							−0.211465	+	0.977385i	\(0.567824\pi\)
\(5\)	−8.06672e10	−	7.58663e11i	−0.105732	−	0.994395i
							\(7\)	1.10848e14	−	1.10848e14i	0.476498	−	0.476498i	−0.427512	−	0.904010i	\(-0.640610\pi\)
0.904010	+	0.427512i	\(0.140610\pi\)
\(11\)	−6.88211e17			−1.36159			−0.680794	−	0.732475i	\(-0.738365\pi\)
							−0.680794	+	0.732475i	\(0.738365\pi\)
\(13\)	−6.48947e18	−	6.48947e18i	−0.750191	−	0.750191i	0.224323	−	0.974515i	\(-0.427983\pi\)
							−0.974515	+	0.224323i	\(0.927983\pi\)
\(17\)	−2.93589e20	+	2.93589e20i	−0.354902	+	0.354902i	−0.861930	−	0.507028i	\(-0.830744\pi\)
							0.507028	+	0.861930i	\(0.330744\pi\)
\(19\)			1.21721e21i			0.222104i	0.993815	+	0.111052i	\(0.0354219\pi\)
							−0.993815	+	0.111052i	\(0.964578\pi\)
\(23\)	−1.01208e23	−	1.01208e23i	−0.717531	−	0.717531i	0.250568	−	0.968099i	\(-0.419383\pi\)
							−0.968099	+	0.250568i	\(0.919383\pi\)
\(29\)			1.11720e23i			0.0153945i	0.999970	+	0.00769726i	\(0.00245014\pi\)
							−0.999970	+	0.00769726i	\(0.997550\pi\)
\(31\)	5.21184e24			0.231123			0.115561	−	0.993300i	\(-0.463133\pi\)
							0.115561	+	0.993300i	\(0.463133\pi\)
\(37\)	3.43768e26	−	3.43768e26i	0.753071	−	0.753071i	−0.221980	−	0.975051i	\(-0.571252\pi\)
							0.975051	+	0.221980i	\(0.0712520\pi\)
\(41\)	−3.54150e27			−1.35476			−0.677378	−	0.735635i	\(-0.736884\pi\)
							−0.677378	+	0.735635i	\(0.736884\pi\)
\(43\)	−7.17056e27	−	7.17056e27i	−1.22065	−	1.22065i	−0.967403	−	0.253242i	\(-0.918503\pi\)
							−0.253242	−	0.967403i	\(-0.581497\pi\)
\(47\)	1.36485e28	−	1.36485e28i	0.512177	−	0.512177i	−0.403016	−	0.915193i	\(-0.632038\pi\)
							0.915193	+	0.403016i	\(0.132038\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.3.8	yes	32
5.2	odd	4	inner	5.35.c.a.2.8	✓	32
5.3	odd	4		25.35.c.b.7.9		32
5.4	even	2		25.35.c.b.18.9		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.8	✓	32	5.2	odd	4	inner
5.35.c.a.3.8	yes	32	1.1	even	1	trivial
25.35.c.b.7.9		32	5.3	odd	4
25.35.c.b.18.9		32	5.4	even	2