Embedded newform 354.2.g.b.11.3

• Label: 354.2.g.b.11.3
• Level: $354$
• Weight: $2$
• Character: 354.11
• 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			11.3
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.b.161.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976621 + 0.214970i) q^{2} +(-1.05934 + 1.37033i) q^{3} +(0.907575 - 0.419889i) q^{4} +(-3.87317 - 1.07538i) q^{5} +(0.739995 - 1.56602i) q^{6} +(-0.391370 + 0.370725i) q^{7} +(-0.796093 + 0.605174i) q^{8} +(-0.755592 - 2.90329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.976621	+	0.214970i	−0.690575	+	0.152007i
							\(3\)	−1.05934	+	1.37033i	−0.611611	+	0.791159i
\(5\)	−3.87317	−	1.07538i	−1.73213	−	0.480924i								−0.748809	−	0.662786i	\(-0.769374\pi\)
							−0.983324	+	0.181862i	\(0.941788\pi\)
\(7\)	−0.391370	+	0.370725i	−0.147924	+	0.140121i	−0.758083	−	0.652158i	\(-0.773864\pi\)
							0.610159	+	0.792279i	\(0.291105\pi\)
\(11\)	3.75836	+	4.42468i	1.13319	+	1.33409i	0.935296	+	0.353867i	\(0.115133\pi\)
							0.197891	+	0.980224i	\(0.436591\pi\)
\(13\)	−1.27962	−	3.79779i	−0.354904	−	1.05332i	−0.965283	−	0.261207i	\(-0.915880\pi\)
							0.610379	−	0.792109i	\(-0.291017\pi\)
\(17\)	4.09658	−	4.32470i	0.993566	−	1.04889i	−0.00523739	−	0.999986i	\(-0.501667\pi\)
							0.998803	−	0.0489085i	\(-0.0155743\pi\)
\(19\)	−0.599758	−	1.50528i	−0.137594	−	0.345335i	0.843992	−	0.536356i	\(-0.180199\pi\)
							−0.981586	+	0.191021i	\(0.938820\pi\)
\(23\)	1.60387	−	0.174432i	0.334430	−	0.0363715i	0.0606366	−	0.998160i	\(-0.480687\pi\)
							0.273794	+	0.961788i	\(0.411721\pi\)
\(29\)	−0.936419	+	4.25419i	−0.173889	+	0.789984i	0.806235	+	0.591596i	\(0.201502\pi\)
							−0.980124	+	0.198388i	\(0.936429\pi\)
\(31\)	2.54307	+	1.01325i	0.456749	+	0.181985i	0.587154	−	0.809475i	\(-0.300248\pi\)
							−0.130405	+	0.991461i	\(0.541628\pi\)
\(37\)	6.03077	−	7.93334i	0.991452	−	1.30423i	0.0390985	−	0.999235i	\(-0.487551\pi\)
							0.952353	−	0.304997i	\(-0.0986555\pi\)
\(41\)	1.21523	−	11.1739i	0.189787	−	1.74506i	−0.381550	−	0.924348i	\(-0.624610\pi\)
							0.571337	−	0.820716i	\(-0.306425\pi\)
\(43\)	−2.32815	−	1.97755i	−0.355040	−	0.301574i	0.452061	−	0.891987i	\(-0.350689\pi\)
							−0.807101	+	0.590413i	\(0.798965\pi\)
\(47\)	−2.73901	−	9.86501i	−0.399525	−	1.43896i	−0.840682	−	0.541529i	\(-0.817846\pi\)
							0.441157	−	0.897430i	\(-0.354568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.11.3	yes	280
3.2	odd	2		354.2.g.a.11.7	✓	280
59.43	odd	58		354.2.g.a.161.7	yes	280
177.161	even	58	inner	354.2.g.b.161.3	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.7	✓	280	3.2	odd	2
354.2.g.a.161.7	yes	280	59.43	odd	58
354.2.g.b.11.3	yes	280	1.1	even	1	trivial
354.2.g.b.161.3	yes	280	177.161	even	58	inner