Embedded newform 5.35.c.a.2.16

• Label: 5.35.c.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(174299. + 174299. i) q^{2} +(-1.58564e8 + 1.58564e8i) q^{3} +4.35807e10i q^{4} +(-4.99549e11 - 5.76652e11i) q^{5} -5.52754e13 q^{6} +(3.74774e13 + 3.74774e13i) q^{7} +(-4.60166e15 + 4.60166e15i) q^{8} -3.36082e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	174299.	+	174299.i	1.32980	+	1.32980i	0.905541	+	0.424258i	\(0.139465\pi\)
							0.424258	+	0.905541i	\(0.360535\pi\)
\(3\)	−1.58564e8	+	1.58564e8i	−1.22785	+	1.22785i	−0.263072	+	0.964776i	\(0.584736\pi\)
							−0.964776	+	0.263072i	\(0.915264\pi\)
\(5\)	−4.99549e11	−	5.76652e11i	−0.654769	−	0.755829i
							\(7\)	3.74774e13	+	3.74774e13i	0.161103	+	0.161103i	0.783055	−	0.621952i	\(-0.213660\pi\)
−0.621952	+	0.783055i	\(0.713660\pi\)
\(11\)	3.71687e17			0.735362			0.367681	−	0.929952i	\(-0.380152\pi\)
							0.367681	+	0.929952i	\(0.380152\pi\)
\(13\)	−5.86017e18	+	5.86017e18i	−0.677444	+	0.677444i	−0.959421	−	0.281977i	\(-0.909010\pi\)
							0.281977	+	0.959421i	\(0.409010\pi\)
\(17\)	−5.97524e20	−	5.97524e20i	−0.722310	−	0.722310i	0.246765	−	0.969075i	\(-0.420632\pi\)
							−0.969075	+	0.246765i	\(0.920632\pi\)
\(19\)			4.17693e21i			0.762159i	0.924542	+	0.381080i	\(0.124448\pi\)
							−0.924542	+	0.381080i	\(0.875552\pi\)
\(23\)	4.18287e22	−	4.18287e22i	0.296552	−	0.296552i	−0.543110	−	0.839662i	\(-0.682753\pi\)
							0.839662	+	0.543110i	\(0.182753\pi\)
\(29\)			4.95641e24i			0.682969i	0.939887	+	0.341485i	\(0.110930\pi\)
							−0.939887	+	0.341485i	\(0.889070\pi\)
\(31\)	−8.54776e24			−0.379056			−0.189528	−	0.981875i	\(-0.560696\pi\)
							−0.189528	+	0.981875i	\(0.560696\pi\)
\(37\)	−2.77942e26	−	2.77942e26i	−0.608871	−	0.608871i	0.333780	−	0.942651i	\(-0.391676\pi\)
							−0.942651	+	0.333780i	\(0.891676\pi\)
\(41\)	−3.86138e26			−0.147713			−0.0738563	−	0.997269i	\(-0.523531\pi\)
							−0.0738563	+	0.997269i	\(0.523531\pi\)
\(43\)	4.26444e27	−	4.26444e27i	0.725936	−	0.725936i	−0.243872	−	0.969807i	\(-0.578418\pi\)
							0.969807	+	0.243872i	\(0.0784176\pi\)
\(47\)	1.94063e28	+	1.94063e28i	0.728247	+	0.728247i	0.970270	−	0.242023i	\(-0.0778110\pi\)
							−0.242023	+	0.970270i	\(0.577811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.16	✓	32
5.2	odd	4		25.35.c.b.18.1		32
5.3	odd	4	inner	5.35.c.a.3.16	yes	32
5.4	even	2		25.35.c.b.7.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.16	✓	32	1.1	even	1	trivial
5.35.c.a.3.16	yes	32	5.3	odd	4	inner
25.35.c.b.7.1		32	5.4	even	2
25.35.c.b.18.1		32	5.2	odd	4