Embedded newform 35.8.f.a.13.11

• Label: 35.8.f.a.13.11
• Level: $35$
• Weight: $8$
• Character: 35.13
• Analytic conductor: $10.933$
• Analytic rank: $0$
• Dimension: $52$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	35.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9334758919\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.11
Character	\(\chi\)	\(=\)	35.13
Dual form			35.8.f.a.27.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22826 + 2.22826i) q^{2} +(-36.5217 + 36.5217i) q^{3} +118.070i q^{4} +(-253.752 - 117.196i) q^{5} -162.760i q^{6} +(-468.423 + 777.253i) q^{7} +(-548.307 - 548.307i) q^{8} -480.674i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} + 1740 q^{7} + 2512 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22826	+	2.22826i	−0.196952	+	0.196952i	−0.798692	−	0.601740i	\(-0.794474\pi\)
							0.601740	+	0.798692i	\(0.294474\pi\)
\(3\)	−36.5217	+	36.5217i	−0.780957	+	0.780957i	−0.979992	−	0.199036i	\(-0.936219\pi\)
							0.199036	+	0.979992i	\(0.436219\pi\)
\(5\)	−253.752	−	117.196i	−0.907851	−	0.419293i
							\(7\)	−468.423	+	777.253i	−0.516173	+	0.856484i
\(11\)	5022.16			1.13767										0.568835	−	0.822452i	\(-0.307394\pi\)
							0.568835	+	0.822452i	\(0.307394\pi\)
\(13\)	4527.30	−	4527.30i	0.571529	−	0.571529i	−0.361027	−	0.932555i	\(-0.617574\pi\)
							0.932555	+	0.361027i	\(0.117574\pi\)
\(17\)	−22133.3	−	22133.3i	−1.09264	−	1.09264i	−0.995246	−	0.0973892i	\(-0.968951\pi\)
							−0.0973892	−	0.995246i	\(-0.531049\pi\)
\(19\)	692.545			0.0231638			0.0115819	−	0.999933i	\(-0.496313\pi\)
							0.0115819	+	0.999933i	\(0.496313\pi\)
\(23\)	32267.8	+	32267.8i	0.552996	+	0.552996i	0.927304	−	0.374308i	\(-0.122120\pi\)
							−0.374308	+	0.927304i	\(0.622120\pi\)
\(29\)		−	159504.i		−	1.21445i	−0.794530	−	0.607225i	\(-0.792283\pi\)
							0.794530	−	0.607225i	\(-0.207717\pi\)
\(31\)			203132.i			1.22465i	0.790606	+	0.612325i	\(0.209766\pi\)
							−0.790606	+	0.612325i	\(0.790234\pi\)
\(37\)	−253251.	+	253251.i	−0.821948	+	0.821948i	−0.986387	−	0.164439i	\(-0.947419\pi\)
							0.164439	+	0.986387i	\(0.447419\pi\)
\(41\)			476030.i			1.07868i	0.842089	+	0.539338i	\(0.181325\pi\)
							−0.842089	+	0.539338i	\(0.818675\pi\)
\(43\)	−276092.	−	276092.i	−0.529559	−	0.529559i	0.390882	−	0.920441i	\(-0.372170\pi\)
							−0.920441	+	0.390882i	\(0.872170\pi\)
\(47\)	−265868.	−	265868.i	−0.373528	−	0.373528i	0.495232	−	0.868760i	\(-0.335083\pi\)
							−0.868760	+	0.495232i	\(0.835083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.8.f.a.13.11	✓	52
5.2	odd	4	inner	35.8.f.a.27.12	yes	52
7.6	odd	2	inner	35.8.f.a.13.12	yes	52
35.27	even	4	inner	35.8.f.a.27.11	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.8.f.a.13.11	✓	52	1.1	even	1	trivial
35.8.f.a.13.12	yes	52	7.6	odd	2	inner
35.8.f.a.27.11	yes	52	35.27	even	4	inner
35.8.f.a.27.12	yes	52	5.2	odd	4	inner