Embedded newform 354.2.g.a.11.2

• Label: 354.2.g.a.11.2
• Level: $354$
• Weight: $2$
• Character: 354.11
• Analytic conductor: $2.827$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.82670423155\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(10\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.a.161.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.976621 - 0.214970i) q^{2} +(-1.66116 - 0.490442i) q^{3} +(0.907575 - 0.419889i) q^{4} +(3.54823 + 0.985161i) q^{5} +(-1.72776 - 0.121874i) q^{6} +(-3.12490 + 2.96006i) q^{7} +(0.796093 - 0.605174i) q^{8} +(2.51893 + 1.62941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.976621	−	0.214970i	0.690575	−	0.152007i
							\(3\)	−1.66116	−	0.490442i	−0.959074	−	0.283157i
\(5\)	3.54823	+	0.985161i	1.58682	+	0.440577i								0.945211	−	0.326459i	\(-0.105855\pi\)
							0.641604	+	0.767036i	\(0.278269\pi\)
\(7\)	−3.12490	+	2.96006i	−1.18110	+	1.11880i	−0.190546	+	0.981678i	\(0.561026\pi\)
							−0.990555	+	0.137120i	\(0.956216\pi\)
\(11\)	3.71995	+	4.37946i	1.12161	+	1.32046i	0.941578	+	0.336796i	\(0.109343\pi\)
							0.180029	+	0.983661i	\(0.442381\pi\)
\(13\)	−1.35763	−	4.02931i	−0.376539	−	1.11753i	−0.953664	−	0.300873i	\(-0.902722\pi\)
							0.577125	−	0.816656i	\(-0.304174\pi\)
\(17\)	0.852466	−	0.899938i	0.206753	−	0.218267i	−0.614369	−	0.789019i	\(-0.710589\pi\)
							0.821123	+	0.570752i	\(0.193348\pi\)
\(19\)	−0.904314	−	2.26966i	−0.207464	−	0.520695i	0.788017	−	0.615653i	\(-0.211108\pi\)
							−0.995481	+	0.0949579i	\(0.969728\pi\)
\(23\)	4.09785	−	0.445668i	0.854460	−	0.0929282i	0.329604	−	0.944119i	\(-0.393085\pi\)
							0.524856	+	0.851191i	\(0.324119\pi\)
\(29\)	0.0866371	−	0.393596i	0.0160881	−	0.0730890i	−0.967893	−	0.251364i	\(-0.919121\pi\)
							0.983981	+	0.178275i	\(0.0570518\pi\)
\(31\)	0.750698	+	0.299105i	0.134829	+	0.0537209i	0.436577	−	0.899667i	\(-0.356191\pi\)
							−0.301748	+	0.953388i	\(0.597570\pi\)
\(37\)	−6.11423	+	8.04313i	−1.00517	+	1.32228i	−0.0591273	+	0.998250i	\(0.518832\pi\)
							−0.946046	+	0.324032i	\(0.894961\pi\)
\(41\)	0.658834	−	6.05788i	0.102893	−	0.946082i	−0.822935	−	0.568136i	\(-0.807665\pi\)
							0.925828	−	0.377946i	\(-0.123370\pi\)
\(43\)	−3.04949	−	2.59026i	−0.465042	−	0.395011i	0.384002	−	0.923332i	\(-0.374546\pi\)
							−0.849044	+	0.528322i	\(0.822821\pi\)
\(47\)	−1.44096	−	5.18987i	−0.210186	−	0.757021i	−0.991185	−	0.132488i	\(-0.957703\pi\)
							0.780999	−	0.624533i	\(-0.214710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.11.2	✓	280
3.2	odd	2		354.2.g.b.11.10	yes	280
59.43	odd	58		354.2.g.b.161.10	yes	280
177.161	even	58	inner	354.2.g.a.161.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.2	✓	280	1.1	even	1	trivial
354.2.g.a.161.2	yes	280	177.161	even	58	inner
354.2.g.b.11.10	yes	280	3.2	odd	2
354.2.g.b.161.10	yes	280	59.43	odd	58