Embedded newform 354.2.g.b.23.9

• Label: 354.2.g.b.23.9
• 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			23.9
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.b.77.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.725995 - 0.687699i) q^{2} +(1.69343 + 0.363726i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(-3.46404 - 0.567901i) q^{5} +(1.47956 - 0.900507i) q^{6} +(1.50148 - 2.83210i) q^{7} +(-0.647386 - 0.762162i) q^{8} +(2.73541 + 1.23189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	1.69343	+	0.363726i	0.977702	+	0.209997i
\(5\)	−3.46404	−	0.567901i	−1.54917	−	0.253973i								−0.674345	−	0.738417i	\(-0.735574\pi\)
							−0.874822	+	0.484444i	\(0.839022\pi\)
\(7\)	1.50148	−	2.83210i	0.567508	−	1.07043i	−0.419195	−	0.907896i	\(-0.637688\pi\)
							0.986703	−	0.162537i	\(-0.0519676\pi\)
\(11\)	4.00587	−	0.435665i	1.20782	−	0.131358i	0.518024	−	0.855366i	\(-0.326668\pi\)
							0.689791	+	0.724008i	\(0.257702\pi\)
\(13\)	0.769979	−	1.66428i	0.213554	−	0.461589i	−0.770963	−	0.636880i	\(-0.780225\pi\)
							0.984517	+	0.175291i	\(0.0560867\pi\)
\(17\)	4.12238	−	2.18555i	0.999825	−	0.530073i	0.113697	−	0.993515i	\(-0.463731\pi\)
							0.886127	+	0.463442i	\(0.153386\pi\)
\(19\)	−6.93590	+	4.17319i	−1.59121	+	0.957396i	−0.604177	+	0.796850i	\(0.706498\pi\)
							−0.987028	+	0.160546i	\(0.948675\pi\)
\(23\)	−1.55286	−	3.89737i	−0.323793	−	0.812658i	−0.997297	−	0.0734751i	\(-0.976591\pi\)
							0.673504	−	0.739183i	\(-0.264788\pi\)
\(29\)	−6.87054	+	7.25314i	−1.27583	+	1.34687i	−0.366710	+	0.930335i	\(0.619516\pi\)
							−0.909117	+	0.416540i	\(0.863243\pi\)
\(31\)	−1.78091	+	2.95990i	−0.319861	+	0.531614i	−0.974517	−	0.224313i	\(-0.927986\pi\)
							0.654656	+	0.755927i	\(0.272814\pi\)
\(37\)	8.09190	+	6.87332i	1.33030	+	1.12997i	0.981224	+	0.192874i	\(0.0617807\pi\)
							0.349077	+	0.937094i	\(0.386495\pi\)
\(41\)	0.174209	+	0.0694114i	0.0272070	+	0.0108402i	0.383702	−	0.923457i	\(-0.374649\pi\)
							−0.356495	+	0.934297i	\(0.616028\pi\)
\(43\)	−1.21096	+	11.1346i	−0.184670	+	1.69802i	0.424865	+	0.905257i	\(0.360322\pi\)
							−0.609535	+	0.792759i	\(0.708644\pi\)
\(47\)	−0.210521	−	1.28412i	−0.0307076	−	0.187308i	0.967019	−	0.254706i	\(-0.0819787\pi\)
							−0.997726	+	0.0673976i	\(0.978530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.23.9	yes	280
3.2	odd	2		354.2.g.a.23.9	✓	280
59.18	odd	58		354.2.g.a.77.9	yes	280
177.77	even	58	inner	354.2.g.b.77.9	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.23.9	✓	280	3.2	odd	2
354.2.g.a.77.9	yes	280	59.18	odd	58
354.2.g.b.23.9	yes	280	1.1	even	1	trivial
354.2.g.b.77.9	yes	280	177.77	even	58	inner