Embedded newform 354.2.g.b.101.5

• Label: 354.2.g.b.101.5
• Level: $354$
• Weight: $2$
• Character: 354.101
• 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			101.5
Character	\(\chi\)	\(=\)	354.101
Dual form			354.2.g.b.347.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.561187 - 0.827689i) q^{2} +(-0.265467 - 1.71159i) q^{3} +(-0.370138 - 0.928977i) q^{4} +(1.40200 - 3.03036i) q^{5} +(-1.56564 - 0.740796i) q^{6} +(-0.451668 - 1.62676i) q^{7} +(-0.976621 - 0.214970i) q^{8} +(-2.85906 + 0.908738i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 10 q^{2} + q^{3} - 10 q^{4} - q^{6} + 2 q^{7} + 10 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.561187	−	0.827689i	0.396819	−	0.585265i
							\(3\)	−0.265467	−	1.71159i	−0.153267	−	0.988185i
\(5\)	1.40200	−	3.03036i	0.626991	−	1.35522i								−0.290274	−	0.956943i	\(-0.593747\pi\)
							0.917266	−	0.398276i	\(-0.130391\pi\)
\(7\)	−0.451668	−	1.62676i	−0.170715	−	0.614858i	−0.998547	−	0.0538832i	\(-0.982840\pi\)
							0.827833	−	0.560975i	\(-0.189574\pi\)
\(11\)	3.94644	+	3.73827i	1.18990	+	1.12713i	0.989061	+	0.147506i	\(0.0471244\pi\)
							0.200836	+	0.979625i	\(0.435634\pi\)
\(13\)	−0.408320	+	3.75444i	−0.113248	+	1.04129i	0.790275	+	0.612753i	\(0.209938\pi\)
							−0.903522	+	0.428541i	\(0.859028\pi\)
\(17\)	2.18672	+	0.607140i	0.530358	+	0.147253i	0.522377	−	0.852715i	\(-0.325046\pi\)
							0.00798158	+	0.999968i	\(0.497459\pi\)
\(19\)	−2.40408	−	1.82753i	−0.551533	−	0.419265i	0.291939	−	0.956437i	\(-0.405699\pi\)
							−0.843473	+	0.537172i	\(0.819493\pi\)
\(23\)	0.450040	+	0.848865i	0.0938397	+	0.177001i	0.925988	−	0.377552i	\(-0.123234\pi\)
							−0.832149	+	0.554553i	\(0.812889\pi\)
\(29\)	6.96583	−	4.72295i	1.29352	−	0.877030i	0.296567	−	0.955012i	\(-0.404158\pi\)
							0.996955	+	0.0779824i	\(0.0248478\pi\)
\(31\)	−3.00423	−	3.95200i	−0.539576	−	0.709800i	0.442820	−	0.896611i	\(-0.353978\pi\)
							−0.982396	+	0.186810i	\(0.940185\pi\)
\(37\)	1.41198	+	6.41469i	0.232128	+	1.05457i	0.938110	+	0.346338i	\(0.112575\pi\)
							−0.705982	+	0.708230i	\(0.749494\pi\)
\(41\)	−0.481006	−	0.255013i	−0.0751205	−	0.0398264i	0.430425	−	0.902626i	\(-0.358364\pi\)
							−0.505545	+	0.862800i	\(0.668709\pi\)
\(43\)	−8.43033	−	8.89980i	−1.28561	−	1.35721i	−0.900813	−	0.434206i	\(-0.857029\pi\)
							−0.384800	−	0.923000i	\(-0.625730\pi\)
\(47\)	−0.907701	+	0.419947i	−0.132402	+	0.0612556i	−0.484971	−	0.874530i	\(-0.661170\pi\)
							0.352570	+	0.935786i	\(0.385308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.g.b.101.5	yes	280
3.2	odd	2		354.2.g.a.101.5	✓	280
59.52	odd	58		354.2.g.a.347.5	yes	280
177.170	even	58	inner	354.2.g.b.347.5	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.g.a.101.5	✓	280	3.2	odd	2
354.2.g.a.347.5	yes	280	59.52	odd	58
354.2.g.b.101.5	yes	280	1.1	even	1	trivial
354.2.g.b.347.5	yes	280	177.170	even	58	inner