Embedded newform 354.3.f.a.13.10

• Label: 354.3.f.a.13.10
• Level: $354$
• Weight: $3$
• Character: 354.13
• 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			13.10
Character	\(\chi\)	\(=\)	354.13
Dual form			354.3.f.a.109.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07786 + 0.915542i) q^{2} +(1.37887 - 1.04819i) q^{3} +(0.323564 - 1.97365i) q^{4} +(4.19506 - 7.91273i) q^{5} +(-0.526568 + 2.39222i) q^{6} +(-6.36367 + 0.692091i) q^{7} +(1.45821 + 2.42356i) q^{8} +(0.802585 - 2.89065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} - 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{45}{58}\right)\)	\(1\)

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
\(5\)	4.19506	−	7.91273i	0.839013	−	1.58255i								0.0264662	−	0.999650i	\(-0.491575\pi\)
							0.812547	−	0.582896i	\(-0.198081\pi\)
\(7\)	−6.36367	+	0.692091i	−0.909096	+	0.0988702i	−0.550704	−	0.834701i	\(-0.685641\pi\)
							−0.358393	+	0.933571i	\(0.616675\pi\)
\(11\)	−4.10878	−	12.1944i	−0.373525	−	1.10858i	−0.955402	−	0.295307i	\(-0.904578\pi\)
							0.581877	−	0.813277i	\(-0.302318\pi\)
\(13\)	1.90025	−	0.527601i	0.146173	−	0.0405847i	−0.193671	−	0.981066i	\(-0.562040\pi\)
							0.339844	+	0.940482i	\(0.389626\pi\)
\(17\)	−0.262224	−	0.0285186i	−0.0154250	−	0.00167757i	0.100403	−	0.994947i	\(-0.467987\pi\)
							−0.115828	+	0.993269i	\(0.536952\pi\)
\(19\)	0.166745	+	3.07543i	0.00877606	+	0.161865i	0.999636	+	0.0269717i	\(0.00858639\pi\)
							−0.990860	+	0.134893i	\(0.956931\pi\)
\(23\)	−17.7340	+	38.3314i	−0.771044	+	1.66658i	−0.0259146	+	0.999664i	\(0.508250\pi\)
							−0.745129	+	0.666920i	\(0.767612\pi\)
\(29\)	−5.41875	+	6.37945i	−0.186854	+	0.219981i	−0.847610	−	0.530619i	\(-0.821960\pi\)
							0.660757	+	0.750600i	\(0.270235\pi\)
\(31\)	−50.4788	−	2.73688i	−1.62835	−	0.0882866i	−0.782832	−	0.622233i	\(-0.786225\pi\)
							−0.845518	+	0.533947i	\(0.820708\pi\)
\(37\)	4.19240	−	6.96782i	0.113308	−	0.188319i	−0.795014	−	0.606591i	\(-0.792536\pi\)
							0.908322	+	0.418272i	\(0.137364\pi\)
\(41\)	57.0042	−	26.3730i	1.39035	−	0.643243i	0.425236	−	0.905083i	\(-0.360191\pi\)
							0.965111	+	0.261839i	\(0.0843291\pi\)
\(43\)	21.3694	−	63.4220i	0.496962	−	1.47493i	−0.346659	−	0.937991i	\(-0.612684\pi\)
							0.843621	−	0.536939i	\(-0.180420\pi\)
\(47\)	68.3420	−	36.2326i	1.45409	−	0.770907i	0.460900	−	0.887452i	\(-0.347527\pi\)
							0.993185	+	0.116545i	\(0.0371819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.f.a.13.10	✓	560
59.50	odd	58	inner	354.3.f.a.109.10	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.f.a.13.10	✓	560	1.1	even	1	trivial
354.3.f.a.109.10	yes	560	59.50	odd	58	inner