Embedded newform 354.2.g.a.23.2

• Label: 354.2.g.a.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.a.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725995 + 0.687699i) q^{2} +(-1.13722 + 1.30642i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(1.48491 + 0.243438i) q^{5} +(-0.0728043 - 1.73052i) q^{6} +(-2.01420 + 3.79918i) q^{7} +(0.647386 + 0.762162i) q^{8} +(-0.413452 - 2.97137i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.13722	+	1.30642i	−0.656575	+	0.754260i
\(5\)	1.48491	+	0.243438i	0.664070	+	0.108869i								0.484384	−	0.874855i	\(-0.339044\pi\)
							0.179686	+	0.983724i	\(0.442492\pi\)
\(7\)	−2.01420	+	3.79918i	−0.761294	+	1.43595i	0.134234	+	0.990950i	\(0.457143\pi\)
							−0.895529	+	0.445004i	\(0.853202\pi\)
\(11\)	3.33575	−	0.362785i	1.00577	−	0.109384i	0.409618	−	0.912257i	\(-0.365662\pi\)
							0.596150	+	0.802873i	\(0.296697\pi\)
\(13\)	−1.13660	+	2.45671i	−0.315235	+	0.681369i	−0.998823	−	0.0484987i	\(-0.984556\pi\)
							0.683588	+	0.729868i	\(0.260418\pi\)
\(17\)	−4.98869	+	2.64484i	−1.20993	+	0.641467i	−0.946342	−	0.323167i	\(-0.895252\pi\)
							−0.263593	+	0.964634i	\(0.584908\pi\)
\(19\)	−1.99828	+	1.20232i	−0.458436	+	0.275832i	−0.725983	−	0.687713i	\(-0.758615\pi\)
							0.267546	+	0.963545i	\(0.413787\pi\)
\(23\)	−0.559800	−	1.40499i	−0.116726	−	0.292961i	0.858992	−	0.511988i	\(-0.171091\pi\)
							−0.975719	+	0.219027i	\(0.929712\pi\)
\(29\)	4.52364	−	4.77555i	0.840019	−	0.886797i	−0.154853	−	0.987937i	\(-0.549490\pi\)
							0.994872	+	0.101140i	\(0.0322491\pi\)
\(31\)	−3.61004	+	5.99993i	−0.648382	+	1.07762i	0.342058	+	0.939679i	\(0.388876\pi\)
							−0.990440	+	0.137941i	\(0.955952\pi\)
\(37\)	−8.08500	−	6.86746i	−1.32917	−	1.12900i	−0.981516	−	0.191379i	\(-0.938704\pi\)
							−0.347649	−	0.937625i	\(-0.613020\pi\)
\(41\)	−4.17876	−	1.66497i	−0.652612	−	0.260024i	0.0202470	−	0.999795i	\(-0.493555\pi\)
							−0.672859	+	0.739771i	\(0.734934\pi\)
\(43\)	−0.496374	+	4.56408i	−0.0756963	+	0.696016i	0.893388	+	0.449286i	\(0.148322\pi\)
							−0.969084	+	0.246730i	\(0.920644\pi\)
\(47\)	0.897412	+	5.47397i	0.130901	+	0.798461i	0.967915	+	0.251279i	\(0.0808511\pi\)
							−0.837014	+	0.547182i	\(0.815701\pi\)

(See \(a_n\) instead)

Twists

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

    

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