Embedded newform 354.3.h.a.5.18

• Label: 354.3.h.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(2.87597 + 0.853709i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(-5.83416 - 2.32454i) q^{5} +(-2.44280 + 3.46882i) q^{6} +(6.28996 - 2.91005i) q^{7} +(2.34106 - 1.58728i) q^{8} +(7.54236 + 4.91048i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	2.87597	+	0.853709i	0.958655	+	0.284570i
\(5\)	−5.83416	−	2.32454i	−1.16683	−	0.464908i								−0.295375	−	0.955381i	\(-0.595445\pi\)
							−0.871456	+	0.490473i	\(0.836824\pi\)
\(7\)	6.28996	−	2.91005i	0.898566	−	0.415721i	0.0844530	−	0.996427i	\(-0.473086\pi\)
							0.814113	+	0.580707i	\(0.197224\pi\)
\(11\)	1.90068	+	0.527722i	0.172789	+	0.0479747i	0.352845	−	0.935682i	\(-0.385214\pi\)
							−0.180056	+	0.983656i	\(0.557628\pi\)
\(13\)	1.67994	−	3.16871i	0.129226	−	0.243747i	−0.810453	−	0.585804i	\(-0.800779\pi\)
							0.939679	+	0.342057i	\(0.111124\pi\)
\(17\)	4.38691	−	9.48215i	0.258054	−	0.557774i	−0.734637	−	0.678460i	\(-0.762648\pi\)
							0.992691	+	0.120687i	\(0.0385096\pi\)
\(19\)	14.2733	−	3.14180i	0.751228	−	0.165358i	0.177179	−	0.984179i	\(-0.443303\pi\)
							0.574049	+	0.818821i	\(0.305372\pi\)
\(23\)	34.7612	+	5.69881i	1.51136	+	0.247774i	0.859723	−	0.510760i	\(-0.170636\pi\)
							0.651633	+	0.758535i	\(0.274084\pi\)
\(29\)	−0.805635	−	2.39104i	−0.0277805	−	0.0824497i	0.932866	−	0.360224i	\(-0.117300\pi\)
							−0.960646	+	0.277774i	\(0.910403\pi\)
\(31\)	48.2639	+	10.6237i	1.55690	+	0.342700i	0.908264	−	0.418398i	\(-0.137408\pi\)
							0.648637	+	0.761098i	\(0.275339\pi\)
\(37\)	0.968740	−	1.42878i	0.0261822	−	0.0386158i	−0.814373	−	0.580342i	\(-0.802919\pi\)
							0.840555	+	0.541726i	\(0.182229\pi\)
\(41\)	−36.6745	+	6.01249i	−0.894501	+	0.146646i	−0.591463	−	0.806332i	\(-0.701449\pi\)
							−0.303038	+	0.952978i	\(0.598001\pi\)
\(43\)	−13.0341	−	46.9446i	−0.303119	−	1.09174i	−0.945573	−	0.325411i	\(-0.894497\pi\)
							0.642454	−	0.766324i	\(-0.277916\pi\)
\(47\)	8.14544	−	3.24544i	0.173307	−	0.0690519i	−0.281871	−	0.959452i	\(-0.590955\pi\)
							0.455178	+	0.890400i	\(0.349576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.18	✓	1120
3.2	odd	2	inner	354.3.h.a.5.35	yes	1120
59.12	even	29	inner	354.3.h.a.71.35	yes	1120
177.71	odd	58	inner	354.3.h.a.71.18	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.18	✓	1120	1.1	even	1	trivial
354.3.h.a.5.35	yes	1120	3.2	odd	2	inner
354.3.h.a.71.18	yes	1120	177.71	odd	58	inner
354.3.h.a.71.35	yes	1120	59.12	even	29	inner