Embedded newform 5.35.c.a.2.6

• Label: 5.35.c.a.2.6
• Level: $5$
• Weight: $35$
• Character: 5.2
• 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			2.6
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-83255.7 - 83255.7i) q^{2} +(-2.65055e7 + 2.65055e7i) q^{3} -3.31684e9i q^{4} +(5.91968e11 + 4.81301e11i) q^{5} +4.41347e12 q^{6} +(-1.23465e14 - 1.23465e14i) q^{7} +(-1.70647e15 + 1.70647e15i) q^{8} +1.52721e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−83255.7	−	83255.7i	−0.635191	−	0.635191i	0.314174	−	0.949365i	\(-0.398272\pi\)
							−0.949365	+	0.314174i	\(0.898272\pi\)
\(3\)	−2.65055e7	+	2.65055e7i	−0.205246	+	0.205246i	−0.802243	−	0.596997i	\(-0.796360\pi\)
							0.596997	+	0.802243i	\(0.296360\pi\)
\(5\)	5.91968e11	+	4.81301e11i	0.775904	+	0.630850i
							\(7\)	−1.23465e14	−	1.23465e14i	−0.530734	−	0.530734i	0.390057	−	0.920791i	\(-0.372455\pi\)
−0.920791	+	0.390057i	\(0.872455\pi\)
\(11\)	1.69267e17			0.334886			0.167443	−	0.985882i	\(-0.446449\pi\)
							0.167443	+	0.985882i	\(0.446449\pi\)
\(13\)	3.25688e18	−	3.25688e18i	0.376500	−	0.376500i	−0.493338	−	0.869838i	\(-0.664223\pi\)
							0.869838	+	0.493338i	\(0.164223\pi\)
\(17\)	−2.13866e20	−	2.13866e20i	−0.258529	−	0.258529i	0.565927	−	0.824456i	\(-0.308519\pi\)
							−0.824456	+	0.565927i	\(0.808519\pi\)
\(19\)		−	3.34669e21i		−	0.610668i	−0.952245	−	0.305334i	\(-0.901232\pi\)
							0.952245	−	0.305334i	\(-0.0987680\pi\)
\(23\)	1.84413e23	−	1.84413e23i	1.30743	−	1.30743i	0.384170	−	0.923262i	\(-0.374488\pi\)
							0.923262	−	0.384170i	\(-0.125512\pi\)
\(29\)			5.36217e24i			0.738881i	0.929254	+	0.369440i	\(0.120451\pi\)
							−0.929254	+	0.369440i	\(0.879549\pi\)
\(31\)	−3.82453e25			−1.69602			−0.848008	−	0.529984i	\(-0.822198\pi\)
							−0.848008	+	0.529984i	\(0.822198\pi\)
\(37\)	−3.49895e26	−	3.49895e26i	−0.766493	−	0.766493i	0.210994	−	0.977487i	\(-0.432330\pi\)
							−0.977487	+	0.210994i	\(0.932330\pi\)
\(41\)	−2.58920e27			−0.990468			−0.495234	−	0.868760i	\(-0.664918\pi\)
							−0.495234	+	0.868760i	\(0.664918\pi\)
\(43\)	−5.57253e27	+	5.57253e27i	−0.948613	+	0.948613i	−0.998743	−	0.0501298i	\(-0.984037\pi\)
							0.0501298	+	0.998743i	\(0.484037\pi\)
\(47\)	−4.46164e27	−	4.46164e27i	−0.167429	−	0.167429i	0.618419	−	0.785848i	\(-0.287773\pi\)
							−0.785848	+	0.618419i	\(0.787773\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.6	✓	32
5.2	odd	4		25.35.c.b.18.11		32
5.3	odd	4	inner	5.35.c.a.3.6	yes	32
5.4	even	2		25.35.c.b.7.11		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.6	✓	32	1.1	even	1	trivial
5.35.c.a.3.6	yes	32	5.3	odd	4	inner
25.35.c.b.7.11		32	5.4	even	2
25.35.c.b.18.11		32	5.2	odd	4