Embedded newform 354.2.g.a.23.7

• Label: 354.2.g.a.23.7
• 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			23.7
Character	\(\chi\)	\(=\)	354.23
Dual form			354.2.g.a.77.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.725995 + 0.687699i) q^{2} +(1.07024 + 1.36183i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(2.51062 + 0.411595i) q^{5} +(-1.71352 - 0.252679i) q^{6} +(0.276889 - 0.522268i) q^{7} +(0.647386 + 0.762162i) q^{8} +(-0.709168 + 2.91498i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	1.07024	+	1.36183i	0.617904	+	0.786254i
\(5\)	2.51062	+	0.411595i	1.12278	+	0.184071i								0.694444	−	0.719547i	\(-0.255651\pi\)
							0.428339	+	0.903618i	\(0.359099\pi\)
\(7\)	0.276889	−	0.522268i	0.104654	−	0.197399i	−0.825655	−	0.564176i	\(-0.809194\pi\)
							0.930309	+	0.366777i	\(0.119539\pi\)
\(11\)	1.65283	−	0.179757i	0.498348	−	0.0541986i	0.144506	−	0.989504i	\(-0.453841\pi\)
							0.353842	+	0.935305i	\(0.384875\pi\)
\(13\)	0.654940	−	1.41563i	0.181648	−	0.392625i	−0.795266	−	0.606261i	\(-0.792669\pi\)
							0.976913	+	0.213636i	\(0.0685307\pi\)
\(17\)	−1.20298	+	0.637780i	−0.291766	+	0.154684i	−0.607862	−	0.794043i	\(-0.707973\pi\)
							0.316096	+	0.948727i	\(0.397628\pi\)
\(19\)	3.34245	−	2.01109i	0.766811	−	0.461375i	−0.0776574	−	0.996980i	\(-0.524744\pi\)
							0.844469	+	0.535605i	\(0.179916\pi\)
\(23\)	−0.406825	−	1.02105i	−0.0848290	−	0.212905i	0.880443	−	0.474152i	\(-0.157245\pi\)
							−0.965272	+	0.261248i	\(0.915866\pi\)
\(29\)	−5.99919	+	6.33327i	−1.11402	+	1.17606i	−0.131228	+	0.991352i	\(0.541892\pi\)
							−0.982794	+	0.184707i	\(0.940867\pi\)
\(31\)	−0.262852	+	0.436863i	−0.0472096	+	0.0784629i	−0.879563	−	0.475782i	\(-0.842165\pi\)
							0.832354	+	0.554245i	\(0.186993\pi\)
\(37\)	−0.363967	−	0.309156i	−0.0598358	−	0.0508250i	0.616969	−	0.786987i	\(-0.288360\pi\)
							−0.676805	+	0.736162i	\(0.736636\pi\)
\(41\)	−7.51709	−	2.99508i	−1.17397	−	0.467753i	−0.300099	−	0.953908i	\(-0.597020\pi\)
							−0.873873	+	0.486155i	\(0.838399\pi\)
\(43\)	0.308594	−	2.83747i	0.0470601	−	0.432711i	−0.946779	−	0.321883i	\(-0.895684\pi\)
							0.993840	−	0.110828i	\(-0.0353502\pi\)
\(47\)	1.06619	+	6.50350i	0.155520	+	0.948633i	0.943122	+	0.332447i	\(0.107874\pi\)
							−0.787601	+	0.616185i	\(0.788677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.23.7	✓	280
3.2	odd	2		354.2.g.b.23.6	yes	280
59.18	odd	58		354.2.g.b.77.6	yes	280
177.77	even	58	inner	354.2.g.a.77.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.23.7	✓	280	1.1	even	1	trivial
354.2.g.a.77.7	yes	280	177.77	even	58	inner
354.2.g.b.23.6	yes	280	3.2	odd	2
354.2.g.b.77.6	yes	280	59.18	odd	58