Embedded newform 5.35.c.a.3.2

• Label: 5.35.c.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	5.3
Dual form			5.35.c.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-151262. + 151262. i) q^{2} +(-1.00065e8 - 1.00065e8i) q^{3} -2.85804e10i q^{4} +(1.61018e11 + 7.45755e11i) q^{5} +3.02721e13 q^{6} +(-1.52632e14 + 1.52632e14i) q^{7} +(1.72447e15 + 1.72447e15i) q^{8} +3.34887e15i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−151262.	+	151262.i	−1.15404	+	1.15404i	−0.168301	+	0.985736i	\(0.553828\pi\)
							−0.985736	+	0.168301i	\(0.946172\pi\)
\(3\)	−1.00065e8	−	1.00065e8i	−0.774857	−	0.774857i	0.204095	−	0.978951i	\(-0.434575\pi\)
							−0.978951	+	0.204095i	\(0.934575\pi\)
\(5\)	1.61018e11	+	7.45755e11i	0.211049	+	0.977475i
							\(7\)	−1.52632e14	+	1.52632e14i	−0.656114	+	0.656114i	−0.954458	−	0.298345i	\(-0.903566\pi\)
0.298345	+	0.954458i	\(0.403566\pi\)
\(11\)	−5.40996e17			−1.07033			−0.535166	−	0.844747i	\(-0.679751\pi\)
							−0.535166	+	0.844747i	\(0.679751\pi\)
\(13\)	−1.65457e18	−	1.65457e18i	−0.191271	−	0.191271i	0.604974	−	0.796245i	\(-0.293183\pi\)
							−0.796245	+	0.604974i	\(0.793183\pi\)
\(17\)	−2.97517e20	+	2.97517e20i	−0.359650	+	0.359650i	−0.863684	−	0.504034i	\(-0.831849\pi\)
							0.504034	+	0.863684i	\(0.331849\pi\)
\(19\)		−	8.71743e21i		−	1.59066i	−0.606177	−	0.795330i	\(-0.707298\pi\)
							0.606177	−	0.795330i	\(-0.292702\pi\)
\(23\)	−1.38797e23	−	1.38797e23i	−0.984024	−	0.984024i	0.0158502	−	0.999874i	\(-0.494955\pi\)
							−0.999874	+	0.0158502i	\(0.994955\pi\)
\(29\)		−	4.12023e24i		−	0.567748i	−0.958862	−	0.283874i	\(-0.908380\pi\)
							0.958862	−	0.283874i	\(-0.0916197\pi\)
\(31\)	−2.74725e25			−1.21829			−0.609143	−	0.793060i	\(-0.708486\pi\)
							−0.609143	+	0.793060i	\(0.708486\pi\)
\(37\)	6.11857e26	−	6.11857e26i	1.34036	−	1.34036i	0.444655	−	0.895702i	\(-0.353326\pi\)
							0.895702	−	0.444655i	\(-0.146674\pi\)
\(41\)	−9.45797e26			−0.361803			−0.180902	−	0.983501i	\(-0.557902\pi\)
							−0.180902	+	0.983501i	\(0.557902\pi\)
\(43\)	4.03879e27	+	4.03879e27i	0.687523	+	0.687523i	0.961684	−	0.274161i	\(-0.0884001\pi\)
							−0.274161	+	0.961684i	\(0.588400\pi\)
\(47\)	−3.42163e28	+	3.42163e28i	−1.28401	+	1.28401i	−0.345651	+	0.938363i	\(0.612342\pi\)
							−0.938363	+	0.345651i	\(0.887658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.3.2	yes	32
5.2	odd	4	inner	5.35.c.a.2.2	✓	32
5.3	odd	4		25.35.c.b.7.15		32
5.4	even	2		25.35.c.b.18.15		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.2	✓	32	5.2	odd	4	inner
5.35.c.a.3.2	yes	32	1.1	even	1	trivial
25.35.c.b.7.15		32	5.3	odd	4
25.35.c.b.18.15		32	5.4	even	2