Embedded newform 354.3.h.a.5.19

• Label: 354.3.h.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(2.89671 - 0.780416i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(4.65663 + 1.85537i) q^{5} +(-0.262140 + 4.23453i) q^{6} +(4.95113 - 2.29064i) q^{7} +(2.34106 - 1.58728i) q^{8} +(7.78190 - 4.52129i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	2.89671	−	0.780416i	0.965571	−	0.260139i
\(5\)	4.65663	+	1.85537i	0.931326	+	0.371074i								0.785941	−	0.618301i	\(-0.212179\pi\)
							0.145385	+	0.989375i	\(0.453558\pi\)
\(7\)	4.95113	−	2.29064i	0.707305	−	0.327234i	−0.0330468	−	0.999454i	\(-0.510521\pi\)
							0.740352	+	0.672220i	\(0.234659\pi\)
\(11\)	−4.70193	−	1.30548i	−0.427448	−	0.118680i	0.0471403	−	0.998888i	\(-0.484989\pi\)
							−0.474588	+	0.880208i	\(0.657403\pi\)
\(13\)	10.8382	−	20.4431i	0.833711	−	1.57254i	0.0139048	−	0.999903i	\(-0.495574\pi\)
							0.819806	−	0.572642i	\(-0.194081\pi\)
\(17\)	−6.11482	+	13.2170i	−0.359695	+	0.777468i	0.640261	+	0.768157i	\(0.278826\pi\)
							−0.999957	+	0.00931140i	\(0.997036\pi\)
\(19\)	−2.29637	+	0.505470i	−0.120862	+	0.0266037i	−0.274989	−	0.961447i	\(-0.588674\pi\)
							0.154128	+	0.988051i	\(0.450743\pi\)
\(23\)	−31.2565	−	5.12424i	−1.35898	−	0.222793i	−0.562168	−	0.827023i	\(-0.690033\pi\)
							−0.796810	+	0.604230i	\(0.793481\pi\)
\(29\)	13.4443	+	39.9013i	0.463597	+	1.37591i	0.883540	+	0.468356i	\(0.155154\pi\)
							−0.419942	+	0.907551i	\(0.637950\pi\)
\(31\)	20.2699	+	4.46175i	0.653868	+	0.143927i	0.529502	−	0.848309i	\(-0.322379\pi\)
							0.124367	+	0.992236i	\(0.460310\pi\)
\(37\)	28.7883	−	42.4596i	0.778063	−	1.14756i	−0.207498	−	0.978236i	\(-0.566532\pi\)
							0.985561	−	0.169322i	\(-0.0541577\pi\)
\(41\)	38.8246	−	6.36497i	0.946942	−	0.155243i	0.331526	−	0.943446i	\(-0.392436\pi\)
							0.615416	+	0.788203i	\(0.288988\pi\)
\(43\)	17.5336	+	63.1505i	0.407759	+	1.46862i	0.827755	+	0.561090i	\(0.189618\pi\)
							−0.419996	+	0.907526i	\(0.637968\pi\)
\(47\)	−57.2400	+	22.8065i	−1.21787	+	0.485244i	−0.888465	−	0.458944i	\(-0.848228\pi\)
							−0.329406	+	0.944188i	\(0.606849\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.19	✓	1120
3.2	odd	2	inner	354.3.h.a.5.31	yes	1120
59.12	even	29	inner	354.3.h.a.71.31	yes	1120
177.71	odd	58	inner	354.3.h.a.71.19	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.19	✓	1120	1.1	even	1	trivial
354.3.h.a.5.31	yes	1120	3.2	odd	2	inner
354.3.h.a.71.19	yes	1120	177.71	odd	58	inner
354.3.h.a.71.31	yes	1120	59.12	even	29	inner