Embedded newform 5.35.c.a.2.13

• Label: 5.35.c.a.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(117370. + 117370. i) q^{2} +(-6.15595e7 + 6.15595e7i) q^{3} +1.03717e10i q^{4} +(3.42977e11 + 6.81501e11i) q^{5} -1.44505e13 q^{6} +(2.43299e14 + 2.43299e14i) q^{7} +(7.99081e14 - 7.99081e14i) q^{8} +9.09803e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	117370.	+	117370.i	0.895464	+	0.895464i	0.995031	−	0.0995673i	\(-0.0317459\pi\)
							−0.0995673	+	0.995031i	\(0.531746\pi\)
\(3\)	−6.15595e7	+	6.15595e7i	−0.476688	+	0.476688i	−0.904071	−	0.427383i	\(-0.859436\pi\)
							0.427383	+	0.904071i	\(0.359436\pi\)
\(5\)	3.42977e11	+	6.81501e11i	0.449547	+	0.893257i
							\(7\)	2.43299e14	+	2.43299e14i	1.04586	+	1.04586i	0.998897	+	0.0469632i	\(0.0149544\pi\)
0.0469632	+	0.998897i	\(0.485046\pi\)
\(11\)	5.26311e17			1.04128			0.520639	−	0.853777i	\(-0.325694\pi\)
							0.520639	+	0.853777i	\(0.325694\pi\)
\(13\)	8.85816e18	−	8.85816e18i	1.02402	−	1.02402i	0.0243107	−	0.999704i	\(-0.492261\pi\)
							0.999704	−	0.0243107i	\(-0.00773908\pi\)
\(17\)	−2.19521e20	−	2.19521e20i	−0.265365	−	0.265365i	0.561864	−	0.827230i	\(-0.310084\pi\)
							−0.827230	+	0.561864i	\(0.810084\pi\)
\(19\)			9.45898e21i			1.72597i	0.505230	+	0.862984i	\(0.331407\pi\)
							−0.505230	+	0.862984i	\(0.668593\pi\)
\(23\)	−6.63948e22	+	6.63948e22i	−0.470718	+	0.470718i	−0.902147	−	0.431429i	\(-0.858010\pi\)
							0.431429	+	0.902147i	\(0.358010\pi\)
\(29\)		−	1.36068e25i		−	1.87495i	−0.348048	−	0.937477i	\(-0.613156\pi\)
							0.348048	−	0.937477i	\(-0.386844\pi\)
\(31\)	2.68896e24			0.119244			0.0596218	−	0.998221i	\(-0.481011\pi\)
							0.0596218	+	0.998221i	\(0.481011\pi\)
\(37\)	−2.28900e26	−	2.28900e26i	−0.501437	−	0.501437i	0.410447	−	0.911884i	\(-0.365373\pi\)
							−0.911884	+	0.410447i	\(0.865373\pi\)
\(41\)	−2.81970e27			−1.07864			−0.539321	−	0.842100i	\(-0.681319\pi\)
							−0.539321	+	0.842100i	\(0.681319\pi\)
\(43\)	2.01928e27	−	2.01928e27i	0.343741	−	0.343741i	−0.514031	−	0.857772i	\(-0.671848\pi\)
							0.857772	+	0.514031i	\(0.171848\pi\)
\(47\)	−4.71056e27	−	4.71056e27i	−0.176770	−	0.176770i	0.613176	−	0.789946i	\(-0.289892\pi\)
							−0.789946	+	0.613176i	\(0.789892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.13	✓	32
5.2	odd	4		25.35.c.b.18.4		32
5.3	odd	4	inner	5.35.c.a.3.13	yes	32
5.4	even	2		25.35.c.b.7.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.13	✓	32	1.1	even	1	trivial
5.35.c.a.3.13	yes	32	5.3	odd	4	inner
25.35.c.b.7.4		32	5.4	even	2
25.35.c.b.18.4		32	5.2	odd	4