Embedded newform 354.2.g.a.11.10

• Label: 354.2.g.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.a.161.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.976621 - 0.214970i) q^{2} +(1.57574 + 0.719053i) q^{3} +(0.907575 - 0.419889i) q^{4} +(-0.135298 - 0.0375654i) q^{5} +(1.69348 + 0.363504i) q^{6} +(-1.37447 + 1.30197i) q^{7} +(0.796093 - 0.605174i) q^{8} +(1.96593 + 2.26608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 10 q^{2} + q^{3} - 10 q^{4} + q^{6} + 2 q^{7} - 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	1.57574	+	0.719053i	0.909755	+	0.415145i
\(5\)	−0.135298	−	0.0375654i	−0.0605073	−	0.0167998i								0.237143	−	0.971475i	\(-0.423789\pi\)
							−0.297650	+	0.954675i	\(0.596203\pi\)
\(7\)	−1.37447	+	1.30197i	−0.519502	+	0.492098i	−0.901927	−	0.431888i	\(-0.857848\pi\)
							0.382425	+	0.923986i	\(0.375089\pi\)
\(11\)	1.21890	+	1.43500i	0.367511	+	0.432667i	0.914571	−	0.404426i	\(-0.132529\pi\)
							−0.547060	+	0.837094i	\(0.684253\pi\)
\(13\)	0.161505	+	0.479329i	0.0447934	+	0.132942i	0.967716	−	0.252044i	\(-0.0811028\pi\)
							−0.922922	+	0.384986i	\(0.874206\pi\)
\(17\)	2.90415	−	3.06587i	0.704359	−	0.743583i	−0.271391	−	0.962469i	\(-0.587484\pi\)
							0.975750	+	0.218886i	\(0.0702424\pi\)
\(19\)	−2.72991	−	6.85157i	−0.626285	−	1.57186i	−0.808434	−	0.588587i	\(-0.799685\pi\)
							0.182149	−	0.983271i	\(-0.441695\pi\)
\(23\)	0.393809	−	0.0428293i	0.0821148	−	0.00893052i	−0.0669694	−	0.997755i	\(-0.521333\pi\)
							0.149084	+	0.988825i	\(0.452367\pi\)
\(29\)	−1.27857	+	5.80862i	−0.237425	+	1.07863i	0.695489	+	0.718536i	\(0.255188\pi\)
							−0.932915	+	0.360097i	\(0.882743\pi\)
\(31\)	−8.73670	−	3.48102i	−1.56916	−	0.625209i	−0.586385	−	0.810032i	\(-0.699449\pi\)
							−0.982772	+	0.184823i	\(0.940829\pi\)
\(37\)	1.66192	−	2.18621i	0.273217	−	0.359411i	−0.638936	−	0.769260i	\(-0.720625\pi\)
							0.912154	+	0.409848i	\(0.134418\pi\)
\(41\)	0.129608	−	1.19173i	0.0202414	−	0.186117i	−0.979652	−	0.200703i	\(-0.935677\pi\)
							0.999894	+	0.0145859i	\(0.00464300\pi\)
\(43\)	−4.93130	−	4.18869i	−0.752017	−	0.638769i	0.187110	−	0.982339i	\(-0.440088\pi\)
							−0.939127	+	0.343570i	\(0.888364\pi\)
\(47\)	2.11648	+	7.62286i	0.308720	+	1.11191i	0.941344	+	0.337449i	\(0.109564\pi\)
							−0.632624	+	0.774459i	\(0.718022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.a.11.10	✓	280
3.2	odd	2		354.2.g.b.11.1	yes	280
59.43	odd	58		354.2.g.b.161.1	yes	280
177.161	even	58	inner	354.2.g.a.161.10	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.10	✓	280	1.1	even	1	trivial
354.2.g.a.161.10	yes	280	177.161	even	58	inner
354.2.g.b.11.1	yes	280	3.2	odd	2
354.2.g.b.161.1	yes	280	59.43	odd	58