Embedded newform 354.2.g.a.101.10

• Label: 354.2.g.a.101.10
• Level: $354$
• Weight: $2$
• Character: 354.101
• 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			101.10
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.a.347.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561187 + 0.827689i) q^{2} +(1.61066 - 0.636998i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(-0.871984 + 1.88476i) q^{5} +(-0.376647 + 1.69060i) q^{6} +(-0.104911 - 0.377855i) q^{7} +(0.976621 + 0.214970i) q^{8} +(2.18847 - 2.05198i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561187	+	0.827689i	−0.396819	+	0.585265i
							\(3\)	1.61066	−	0.636998i	0.929916	−	0.367771i
\(5\)	−0.871984	+	1.88476i	−0.389963	+	0.842891i								0.608807	+	0.793318i	\(0.291648\pi\)
							−0.998770	+	0.0495732i	\(0.984214\pi\)
\(7\)	−0.104911	−	0.377855i	−0.0396526	−	0.142816i	0.941074	−	0.338201i	\(-0.109818\pi\)
							−0.980727	+	0.195385i	\(0.937404\pi\)
\(11\)	3.87828	+	3.67370i	1.16935	+	1.10766i	0.992351	+	0.123451i	\(0.0393963\pi\)
							0.176995	+	0.984212i	\(0.443362\pi\)
\(13\)	−0.113087	+	1.03982i	−0.0313648	+	0.288394i	0.967942	+	0.251173i	\(0.0808162\pi\)
							−0.999307	+	0.0372215i	\(0.988149\pi\)
\(17\)	2.34128	+	0.650054i	0.567845	+	0.157661i	0.539573	−	0.841939i	\(-0.318586\pi\)
							0.0282720	+	0.999600i	\(0.491000\pi\)
\(19\)	−1.06564	−	0.810078i	−0.244475	−	0.185845i	0.475760	−	0.879575i	\(-0.342173\pi\)
							−0.720235	+	0.693730i	\(0.755966\pi\)
\(23\)	1.14609	+	2.16176i	0.238976	+	0.450757i	0.973770	−	0.227533i	\(-0.0730660\pi\)
							−0.734794	+	0.678290i	\(0.762721\pi\)
\(29\)	2.22005	−	1.50523i	0.412253	−	0.279515i	−0.337379	−	0.941369i	\(-0.609540\pi\)
							0.749633	+	0.661854i	\(0.230230\pi\)
\(31\)	−1.27277	−	1.67430i	−0.228596	−	0.300713i	0.667467	−	0.744639i	\(-0.267378\pi\)
							−0.896063	+	0.443926i	\(0.853585\pi\)
\(37\)	−1.29612	−	5.88834i	−0.213081	−	0.968037i	−0.954782	−	0.297307i	\(-0.903912\pi\)
							0.741701	−	0.670731i	\(-0.234019\pi\)
\(41\)	−5.56206	−	2.94881i	−0.868647	−	0.460528i	−0.0264203	−	0.999651i	\(-0.508411\pi\)
							−0.842227	+	0.539123i	\(0.818756\pi\)
\(43\)	0.560650	+	0.591871i	0.0854983	+	0.0902594i	0.767341	−	0.641240i	\(-0.221580\pi\)
							−0.681842	+	0.731499i	\(0.738821\pi\)
\(47\)	−10.9793	+	5.07958i	−1.60150	+	0.740933i	−0.998390	−	0.0567237i	\(-0.981935\pi\)
							−0.603111	+	0.797657i	\(0.706073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.101.10	✓	280
3.2	odd	2		354.2.g.b.101.1	yes	280
59.52	odd	58		354.2.g.b.347.1	yes	280
177.170	even	58	inner	354.2.g.a.347.10	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.101.10	✓	280	1.1	even	1	trivial
354.2.g.a.347.10	yes	280	177.170	even	58	inner
354.2.g.b.101.1	yes	280	3.2	odd	2
354.2.g.b.347.1	yes	280	59.52	odd	58