Embedded newform 354.2.g.b.23.2

• Label: 354.2.g.b.23.2
• Level: $354$
• Weight: $2$
• Character: 354.23
• 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			23.2
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.b.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.725995 - 0.687699i) q^{2} +(-1.55051 + 0.771956i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(-0.271078 - 0.0444409i) q^{5} +(-0.594791 + 1.62672i) q^{6} +(-0.468623 + 0.883917i) q^{7} +(-0.647386 - 0.762162i) q^{8} +(1.80817 - 2.39385i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{15}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.725995	−	0.687699i	0.513356	−	0.486277i
							\(3\)	−1.55051	+	0.771956i	−0.895188	+	0.445689i
\(5\)	−0.271078	−	0.0444409i	−0.121230	−	0.0198746i								0.100862	−	0.994900i	\(-0.467840\pi\)
							−0.222091	+	0.975026i	\(0.571288\pi\)
\(7\)	−0.468623	+	0.883917i	−0.177123	+	0.334089i	−0.956138	−	0.292916i	\(-0.905374\pi\)
							0.779015	+	0.627005i	\(0.215719\pi\)
\(11\)	6.47542	−	0.704245i	1.95241	−	0.212338i	0.955764	−	0.294134i	\(-0.0950311\pi\)
							0.996649	+	0.0817958i	\(0.0260655\pi\)
\(13\)	2.31363	−	5.00082i	0.641685	−	1.38698i	−0.264620	−	0.964353i	\(-0.585246\pi\)
							0.906304	−	0.422626i	\(-0.138891\pi\)
\(17\)	−0.402783	+	0.213542i	−0.0976893	+	0.0517916i	−0.516537	−	0.856265i	\(-0.672779\pi\)
							0.418847	+	0.908057i	\(0.362434\pi\)
\(19\)	3.36816	−	2.02656i	0.772710	−	0.464924i	−0.0738160	−	0.997272i	\(-0.523518\pi\)
							0.846526	+	0.532348i	\(0.178690\pi\)
\(23\)	2.07149	+	5.19904i	0.431935	+	1.08407i	0.970416	+	0.241438i	\(0.0776189\pi\)
							−0.538481	+	0.842637i	\(0.681002\pi\)
\(29\)	−5.07522	+	5.35784i	−0.942444	+	0.994926i	−0.999994	−	0.00341655i	\(-0.998912\pi\)
							0.0575501	+	0.998343i	\(0.481671\pi\)
\(31\)	4.83154	−	8.03009i	0.867771	−	1.44225i	−0.0289164	−	0.999582i	\(-0.509206\pi\)
							0.896687	−	0.442665i	\(-0.145967\pi\)
\(37\)	−2.05596	−	1.74635i	−0.337997	−	0.287097i	0.462272	−	0.886738i	\(-0.347034\pi\)
							−0.800269	+	0.599641i	\(0.795310\pi\)
\(41\)	2.06486	+	0.822716i	0.322477	+	0.128487i	0.525759	−	0.850634i	\(-0.323781\pi\)
							−0.203282	+	0.979120i	\(0.565161\pi\)
\(43\)	−0.125425	+	1.15326i	−0.0191271	+	0.175871i	−0.999805	−	0.0197597i	\(-0.993710\pi\)
							0.980678	+	0.195631i	\(0.0626754\pi\)
\(47\)	−1.31176	−	8.00140i	−0.191340	−	1.16712i	−0.891460	−	0.453099i	\(-0.850318\pi\)
							0.700120	−	0.714025i	\(-0.253130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.23.2	yes	280
3.2	odd	2		354.2.g.a.23.1	✓	280
59.18	odd	58		354.2.g.a.77.1	yes	280
177.77	even	58	inner	354.2.g.b.77.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.23.1	✓	280	3.2	odd	2
354.2.g.a.77.1	yes	280	59.18	odd	58
354.2.g.b.23.2	yes	280	1.1	even	1	trivial
354.2.g.b.77.2	yes	280	177.77	even	58	inner