Embedded newform 5.35.c.a.2.4

• Label: 5.35.c.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-115430. - 115430. i) q^{2} +(1.39345e8 - 1.39345e8i) q^{3} +9.46824e9i q^{4} +(-7.62275e11 - 3.18319e10i) q^{5} -3.21691e13 q^{6} +(-2.97793e14 - 2.97793e14i) q^{7} +(-8.90153e14 + 8.90153e14i) q^{8} -2.21567e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−115430.	−	115430.i	−0.880660	−	0.880660i	0.112942	−	0.993602i	\(-0.463973\pi\)
							−0.993602	+	0.112942i	\(0.963973\pi\)
\(3\)	1.39345e8	−	1.39345e8i	1.07902	−	1.07902i	0.0824207	−	0.996598i	\(-0.473735\pi\)
							0.996598	−	0.0824207i	\(-0.0262651\pi\)
\(5\)	−7.62275e11	−	3.18319e10i	−0.999129	−	0.0417227i
							\(7\)	−2.97793e14	−	2.97793e14i	−1.28011	−	1.28011i	−0.940603	−	0.339509i	\(-0.889739\pi\)
−0.339509	−	0.940603i	\(-0.610261\pi\)
\(11\)	−1.99502e17			−0.394704			−0.197352	−	0.980333i	\(-0.563234\pi\)
							−0.197352	+	0.980333i	\(0.563234\pi\)
\(13\)	3.71677e18	−	3.71677e18i	0.429663	−	0.429663i	−0.458850	−	0.888514i	\(-0.651739\pi\)
							0.888514	+	0.458850i	\(0.151739\pi\)
\(17\)	−4.39500e20	−	4.39500e20i	−0.531285	−	0.531285i	0.389670	−	0.920955i	\(-0.372589\pi\)
							−0.920955	+	0.389670i	\(0.872589\pi\)
\(19\)		−	7.08301e21i		−	1.29243i	−0.763156	−	0.646214i	\(-0.776351\pi\)
							0.763156	−	0.646214i	\(-0.223649\pi\)
\(23\)	−1.16043e23	+	1.16043e23i	−0.822708	+	0.822708i	−0.986496	−	0.163788i	\(-0.947629\pi\)
							0.163788	+	0.986496i	\(0.447629\pi\)
\(29\)		−	7.43334e23i		−	0.102428i	−0.998688	−	0.0512139i	\(-0.983691\pi\)
							0.998688	−	0.0512139i	\(-0.0163090\pi\)
\(31\)	1.56192e25			0.692645			0.346322	−	0.938116i	\(-0.387430\pi\)
							0.346322	+	0.938116i	\(0.387430\pi\)
\(37\)	−4.19498e26	−	4.19498e26i	−0.918969	−	0.918969i	0.0779855	−	0.996954i	\(-0.475151\pi\)
							−0.996954	+	0.0779855i	\(0.975151\pi\)
\(41\)	4.69838e27			1.79731			0.898655	−	0.438656i	\(-0.144545\pi\)
							0.898655	+	0.438656i	\(0.144545\pi\)
\(43\)	1.38271e27	−	1.38271e27i	0.235378	−	0.235378i	−0.579555	−	0.814933i	\(-0.696774\pi\)
							0.814933	+	0.579555i	\(0.196774\pi\)
\(47\)	−1.72367e28	−	1.72367e28i	−0.646831	−	0.646831i	0.305395	−	0.952226i	\(-0.401211\pi\)
							−0.952226	+	0.305395i	\(0.901211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.4	✓	32
5.2	odd	4		25.35.c.b.18.13		32
5.3	odd	4	inner	5.35.c.a.3.4	yes	32
5.4	even	2		25.35.c.b.7.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.4	✓	32	1.1	even	1	trivial
5.35.c.a.3.4	yes	32	5.3	odd	4	inner
25.35.c.b.7.13		32	5.4	even	2
25.35.c.b.18.13		32	5.2	odd	4