Embedded newform 354.2.g.a.11.8

• Label: 354.2.g.a.11.8
• 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			11.8
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.a.161.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.976621 - 0.214970i) q^{2} +(1.28026 - 1.16660i) q^{3} +(0.907575 - 0.419889i) q^{4} +(-4.02621 - 1.11787i) q^{5} +(0.999541 - 1.41454i) q^{6} +(1.69268 - 1.60339i) q^{7} +(0.796093 - 0.605174i) q^{8} +(0.278110 - 2.98708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	1.28026	−	1.16660i	0.739156	−	0.673534i
\(5\)	−4.02621	−	1.11787i	−1.80057	−	0.499927i								−0.804947	−	0.593347i	\(-0.797806\pi\)
							−0.995627	+	0.0934201i	\(0.970220\pi\)
\(7\)	1.69268	−	1.60339i	0.639774	−	0.606026i	−0.297457	−	0.954735i	\(-0.596138\pi\)
							0.937231	+	0.348709i	\(0.113380\pi\)
\(11\)	−1.17336	−	1.38139i	−0.353782	−	0.416504i	0.556268	−	0.831003i	\(-0.312233\pi\)
							−0.910050	+	0.414499i	\(0.863957\pi\)
\(13\)	1.59806	+	4.74287i	0.443222	+	1.31544i	0.903853	+	0.427844i	\(0.140727\pi\)
							−0.460631	+	0.887592i	\(0.652377\pi\)
\(17\)	0.854617	−	0.902208i	0.207275	−	0.218818i	−0.614067	−	0.789254i	\(-0.710467\pi\)
							0.821342	+	0.570437i	\(0.193226\pi\)
\(19\)	−1.45218	−	3.64469i	−0.333153	−	0.836150i	−0.996161	−	0.0875383i	\(-0.972100\pi\)
							0.663008	−	0.748612i	\(-0.269279\pi\)
\(23\)	4.79396	−	0.521374i	0.999609	−	0.108714i	0.406343	−	0.913721i	\(-0.366804\pi\)
							0.593266	+	0.805007i	\(0.297838\pi\)
\(29\)	−1.57411	+	7.15124i	−0.292304	+	1.32795i	0.570035	+	0.821621i	\(0.306930\pi\)
							−0.862339	+	0.506331i	\(0.831001\pi\)
\(31\)	5.78228	+	2.30387i	1.03853	+	0.413788i	0.826180	−	0.563406i	\(-0.190509\pi\)
							0.212349	+	0.977194i	\(0.431889\pi\)
\(37\)	0.421557	−	0.554549i	0.0693036	−	0.0911673i	−0.760135	−	0.649765i	\(-0.774867\pi\)
							0.829439	+	0.558598i	\(0.188660\pi\)
\(41\)	−0.921869	+	8.47645i	−0.143972	+	1.32380i	0.669769	+	0.742570i	\(0.266393\pi\)
							−0.813740	+	0.581229i	\(0.802572\pi\)
\(43\)	−4.70158	−	3.99356i	−0.716984	−	0.609012i	0.212809	−	0.977094i	\(-0.431739\pi\)
							−0.929793	+	0.368082i	\(0.880015\pi\)
\(47\)	−1.02453	−	3.69001i	−0.149443	−	0.538244i	−0.999925	−	0.0122403i	\(-0.996104\pi\)
							0.850482	−	0.526003i	\(-0.176310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.11.8	✓	280
3.2	odd	2		354.2.g.b.11.4	yes	280
59.43	odd	58		354.2.g.b.161.4	yes	280
177.161	even	58	inner	354.2.g.a.161.8	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.8	✓	280	1.1	even	1	trivial
354.2.g.a.161.8	yes	280	177.161	even	58	inner
354.2.g.b.11.4	yes	280	3.2	odd	2
354.2.g.b.161.4	yes	280	59.43	odd	58