Embedded newform 354.2.g.b.11.6

• Label: 354.2.g.b.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	354.11
Dual form			354.2.g.b.161.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.976621 + 0.214970i) q^{2} +(-0.233636 + 1.71622i) q^{3} +(0.907575 - 0.419889i) q^{4} +(2.85285 + 0.792090i) q^{5} +(-0.140763 - 1.72632i) q^{6} +(-3.45977 + 3.27726i) q^{7} +(-0.796093 + 0.605174i) q^{8} +(-2.89083 - 0.801943i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.233636	+	1.71622i	−0.134890	+	0.990861i
\(5\)	2.85285	+	0.792090i	1.27583	+	0.354233i								0.838419	−	0.545027i	\(-0.183480\pi\)
							0.437415	+	0.899260i	\(0.355894\pi\)
\(7\)	−3.45977	+	3.27726i	−1.30767	+	1.23869i	−0.354932	+	0.934892i	\(0.615496\pi\)
							−0.952737	+	0.303797i	\(0.901745\pi\)
\(11\)	−0.467154	−	0.549977i	−0.140852	−	0.165824i	0.687213	−	0.726456i	\(-0.258834\pi\)
							−0.828065	+	0.560632i	\(0.810558\pi\)
\(13\)	0.999423	+	2.96618i	0.277190	+	0.822671i	0.992322	+	0.123681i	\(0.0394701\pi\)
							−0.715132	+	0.698990i	\(0.753633\pi\)
\(17\)	3.74279	−	3.95122i	0.907761	−	0.958311i	−0.0915363	−	0.995802i	\(-0.529178\pi\)
							0.999297	+	0.0374905i	\(0.0119364\pi\)
\(19\)	0.338571	+	0.849750i	0.0776736	+	0.194946i	0.962686	−	0.270622i	\(-0.0872294\pi\)
							−0.885012	+	0.465568i	\(0.845850\pi\)
\(23\)	−5.93899	+	0.645904i	−1.23836	+	0.134680i	−0.703770	−	0.710428i	\(-0.748501\pi\)
							−0.534595	+	0.845109i	\(0.679536\pi\)
\(29\)	−1.76426	+	8.01510i	−0.327614	+	1.48837i	0.468631	+	0.883394i	\(0.344747\pi\)
							−0.796246	+	0.604973i	\(0.793184\pi\)
\(31\)	−1.16623	−	0.464668i	−0.209461	−	0.0834568i	0.263048	−	0.964783i	\(-0.415272\pi\)
							−0.472509	+	0.881326i	\(0.656652\pi\)
\(37\)	−4.11480	+	5.41292i	−0.676468	+	0.889879i	−0.998380	−	0.0568908i	\(-0.981881\pi\)
							0.321912	+	0.946770i	\(0.395674\pi\)
\(41\)	0.819838	−	7.53829i	0.128037	−	1.17728i	−0.737113	−	0.675770i	\(-0.763811\pi\)
							0.865150	−	0.501513i	\(-0.167223\pi\)
\(43\)	5.58345	+	4.74263i	0.851468	+	0.723244i	0.962706	−	0.270550i	\(-0.0872056\pi\)
							−0.111238	+	0.993794i	\(0.535481\pi\)
\(47\)	0.560817	+	2.01988i	0.0818036	+	0.294630i	0.993124	−	0.117068i	\(-0.0373496\pi\)
							−0.911320	+	0.411698i	\(0.864936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.11.6	yes	280
3.2	odd	2		354.2.g.a.11.4	✓	280
59.43	odd	58		354.2.g.a.161.4	yes	280
177.161	even	58	inner	354.2.g.b.161.6	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.11.4	✓	280	3.2	odd	2
354.2.g.a.161.4	yes	280	59.43	odd	58
354.2.g.b.11.6	yes	280	1.1	even	1	trivial
354.2.g.b.161.6	yes	280	177.161	even	58	inner