Embedded newform 354.3.h.a.5.14

• Label: 354.3.h.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(1.42759 + 2.63856i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(5.73580 + 2.28535i) q^{5} +(-4.18080 + 0.721757i) q^{6} +(-0.923037 + 0.427042i) q^{7} +(2.34106 - 1.58728i) q^{8} +(-4.92400 + 7.53354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	1.42759	+	2.63856i	0.475862	+	0.879520i
\(5\)	5.73580	+	2.28535i	1.14716	+	0.457071i								0.864740	−	0.502220i	\(-0.167483\pi\)
							0.282421	+	0.959291i	\(0.408863\pi\)
\(7\)	−0.923037	+	0.427042i	−0.131862	+	0.0610060i	−0.484711	−	0.874674i	\(-0.661075\pi\)
							0.352848	+	0.935680i	\(0.385213\pi\)
\(11\)	17.2771	+	4.79696i	1.57065	+	0.436087i	0.940333	−	0.340255i	\(-0.110513\pi\)
							0.630312	+	0.776342i	\(0.282927\pi\)
\(13\)	6.00401	−	11.3248i	0.461847	−	0.871136i	−0.537760	−	0.843098i	\(-0.680729\pi\)
							0.999607	−	0.0280381i	\(-0.00892597\pi\)
\(17\)	−9.57004	+	20.6853i	−0.562944	+	1.21678i	0.391992	+	0.919969i	\(0.371786\pi\)
							−0.954935	+	0.296814i	\(0.904076\pi\)
\(19\)	−22.0212	+	4.84723i	−1.15901	+	0.255117i	−0.752539	−	0.658547i	\(-0.771171\pi\)
							−0.406470	+	0.913664i	\(0.633240\pi\)
\(23\)	39.4352	+	6.46508i	1.71458	+	0.281090i	0.937416	−	0.348213i	\(-0.113211\pi\)
							0.777160	+	0.629303i	\(0.216660\pi\)
\(29\)	−10.3165	−	30.6182i	−0.355741	−	1.05580i	−0.964870	−	0.262729i	\(-0.915377\pi\)
							0.609129	−	0.793071i	\(-0.291519\pi\)
\(31\)	4.55891	+	1.00349i	0.147061	+	0.0323707i	0.287891	−	0.957663i	\(-0.407046\pi\)
							−0.140830	+	0.990034i	\(0.544977\pi\)
\(37\)	13.7690	−	20.3078i	0.372136	−	0.548859i	−0.594802	−	0.803872i	\(-0.702769\pi\)
							0.966938	+	0.255013i	\(0.0820798\pi\)
\(41\)	−6.52336	+	1.06945i	−0.159106	+	0.0260842i	−0.240809	−	0.970573i	\(-0.577413\pi\)
							0.0817022	+	0.996657i	\(0.473964\pi\)
\(43\)	−8.13038	−	29.2830i	−0.189079	−	0.681000i	−0.995921	−	0.0902327i	\(-0.971239\pi\)
							0.806842	−	0.590767i	\(-0.201175\pi\)
\(47\)	−76.9875	+	30.6746i	−1.63803	+	0.652652i	−0.993698	−	0.112092i	\(-0.964245\pi\)
							−0.644335	+	0.764743i	\(0.722866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.14	✓	1120
3.2	odd	2	inner	354.3.h.a.5.40	yes	1120
59.12	even	29	inner	354.3.h.a.71.40	yes	1120
177.71	odd	58	inner	354.3.h.a.71.14	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.14	✓	1120	1.1	even	1	trivial
354.3.h.a.5.40	yes	1120	3.2	odd	2	inner
354.3.h.a.71.14	yes	1120	177.71	odd	58	inner
354.3.h.a.71.40	yes	1120	59.12	even	29	inner