Embedded newform 354.3.h.a.5.2

• Label: 354.3.h.a.5.2
• Level: $354$
• Weight: $3$
• Character: 354.5
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	354.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(40\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(-2.92397 - 0.671113i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(-8.23136 - 3.27967i) q^{5} +(2.21976 - 3.61561i) q^{6} +(-7.65524 + 3.54169i) q^{7} +(2.34106 - 1.58728i) q^{8} +(8.09922 + 3.92463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451561	+	1.34018i	−0.225780	+	0.670092i
							\(3\)	−2.92397	−	0.671113i	−0.974657	−	0.223704i
\(5\)	−8.23136	−	3.27967i	−1.64627	−	0.655934i								−0.651556	−	0.758601i	\(-0.725883\pi\)
							−0.994716	+	0.102666i	\(0.967263\pi\)
\(7\)	−7.65524	+	3.54169i	−1.09361	+	0.505956i	−0.881928	−	0.471384i	\(-0.843755\pi\)
							−0.211677	+	0.977340i	\(0.567893\pi\)
\(11\)	−20.3084	−	5.63860i	−1.84622	−	0.512600i	−0.846416	−	0.532522i	\(-0.821244\pi\)
							−0.999802	+	0.0199228i	\(0.993658\pi\)
\(13\)	−6.43972	+	12.1466i	−0.495363	+	0.934354i	0.502211	+	0.864745i	\(0.332520\pi\)
							−0.997574	+	0.0696091i	\(0.977825\pi\)
\(17\)	12.6599	−	27.3639i	0.744701	−	1.60964i	−0.0471195	−	0.998889i	\(-0.515004\pi\)
							0.791820	−	0.610755i	\(-0.209134\pi\)
\(19\)	2.24322	−	0.493771i	0.118064	−	0.0259879i	−0.155545	−	0.987829i	\(-0.549713\pi\)
							0.273610	+	0.961841i	\(0.411782\pi\)
\(23\)	4.30170	+	0.705228i	0.187030	+	0.0306621i	0.254569	−	0.967055i	\(-0.418066\pi\)
							−0.0675387	+	0.997717i	\(0.521515\pi\)
\(29\)	11.2729	+	33.4567i	0.388720	+	1.15368i	0.946226	+	0.323507i	\(0.104862\pi\)
							−0.557506	+	0.830173i	\(0.688242\pi\)
\(31\)	10.8083	+	2.37909i	0.348656	+	0.0767450i	0.385846	−	0.922563i	\(-0.373910\pi\)
							−0.0371900	+	0.999308i	\(0.511841\pi\)
\(37\)	−9.30477	+	13.7235i	−0.251480	+	0.370906i	−0.932385	−	0.361466i	\(-0.882276\pi\)
							0.680905	+	0.732372i	\(0.261587\pi\)
\(41\)	−38.7354	+	6.35035i	−0.944766	+	0.154887i	−0.614425	−	0.788975i	\(-0.710612\pi\)
							−0.330341	+	0.943862i	\(0.607164\pi\)
\(43\)	−17.0363	−	61.3592i	−0.396193	−	1.42696i	−0.845689	−	0.533677i	\(-0.820810\pi\)
							0.449495	−	0.893283i	\(-0.351604\pi\)
\(47\)	−34.9343	+	13.9191i	−0.743283	+	0.296151i	−0.710870	−	0.703324i	\(-0.751699\pi\)
							−0.0324131	+	0.999475i	\(0.510319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.2	✓	1120
3.2	odd	2	inner	354.3.h.a.5.27	yes	1120
59.12	even	29	inner	354.3.h.a.71.27	yes	1120
177.71	odd	58	inner	354.3.h.a.71.2	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.2	✓	1120	1.1	even	1	trivial
354.3.h.a.5.27	yes	1120	3.2	odd	2	inner
354.3.h.a.71.2	yes	1120	177.71	odd	58	inner
354.3.h.a.71.27	yes	1120	59.12	even	29	inner