Embedded newform 354.3.f.a.109.2

• Label: 354.3.f.a.109.2
• Level: $354$
• Weight: $3$
• Character: 354.109
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	354.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			109.2
Character	\(\chi\)	\(=\)	354.109
Dual form			354.3.f.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07786 - 0.915542i) q^{2} +(-1.37887 - 1.04819i) q^{3} +(0.323564 + 1.97365i) q^{4} +(-2.10279 - 3.96628i) q^{5} +(0.526568 + 2.39222i) q^{6} +(8.08885 + 0.879715i) q^{7} +(1.45821 - 2.42356i) q^{8} +(0.802585 + 2.89065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} - 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{13}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07786	−	0.915542i	−0.538930	−	0.457771i
							\(3\)	−1.37887	−	1.04819i	−0.459625	−	0.349397i
\(5\)	−2.10279	−	3.96628i	−0.420558	−	0.793256i								0.579202	−	0.815184i	\(-0.303364\pi\)
							−0.999760	+	0.0219280i	\(0.993020\pi\)
\(7\)	8.08885	+	0.879715i	1.15555	+	0.125674i	0.665759	−	0.746167i	\(-0.268108\pi\)
							0.489790	+	0.871840i	\(0.337073\pi\)
\(11\)	0.888210	−	2.63611i	0.0807464	−	0.239647i	−0.899674	−	0.436562i	\(-0.856196\pi\)
							0.980420	+	0.196915i	\(0.0630925\pi\)
\(13\)	11.4162	+	3.16970i	0.878170	+	0.243823i	0.677222	−	0.735779i	\(-0.263184\pi\)
							0.200949	+	0.979602i	\(0.435598\pi\)
\(17\)	13.0670	−	1.42112i	0.768644	−	0.0835951i	0.284609	−	0.958644i	\(-0.408136\pi\)
							0.484035	+	0.875049i	\(0.339171\pi\)
\(19\)	−0.356449	+	6.57432i	−0.0187605	+	0.346017i	0.974051	+	0.226329i	\(0.0726723\pi\)
							−0.992812	+	0.119688i	\(0.961810\pi\)
\(23\)	−5.60241	−	12.1094i	−0.243583	−	0.526496i	0.746767	−	0.665086i	\(-0.231605\pi\)
							−0.990350	+	0.138590i	\(0.955743\pi\)
\(29\)	−24.0025	−	28.2579i	−0.827672	−	0.974411i	0.172271	−	0.985050i	\(-0.444889\pi\)
							−0.999943	+	0.0106388i	\(0.996614\pi\)
\(31\)	19.0396	−	1.03230i	0.614180	−	0.0332999i	0.255575	−	0.966789i	\(-0.417735\pi\)
							0.358605	+	0.933489i	\(0.383253\pi\)
\(37\)	−16.8340	−	27.9783i	−0.454972	−	0.756170i	0.541445	−	0.840736i	\(-0.317877\pi\)
							−0.996418	+	0.0845658i	\(0.973050\pi\)
\(41\)	−13.4951	−	6.24352i	−0.329150	−	0.152281i	0.248353	−	0.968670i	\(-0.420111\pi\)
							−0.577503	+	0.816389i	\(0.695973\pi\)
\(43\)	−16.7565	−	49.7315i	−0.389686	−	1.15655i	−0.945607	−	0.325312i	\(-0.894531\pi\)
							0.555921	−	0.831235i	\(-0.312366\pi\)
\(47\)	19.3141	+	10.2397i	0.410939	+	0.217866i	0.661035	−	0.750355i	\(-0.270117\pi\)
							−0.250096	+	0.968221i	\(0.580462\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.f.a.109.2	yes	560
59.13	odd	58	inner	354.3.f.a.13.2	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.f.a.13.2	✓	560	59.13	odd	58	inner
354.3.f.a.109.2	yes	560	1.1	even	1	trivial