Embedded newform 592.2.cd.a.59.1

• Label: 592.2.cd.a.59.1
• Level: $592$
• Weight: $2$
• Character: 592.59
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $888$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.cd (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72714379966\)
Analytic rank:	\(0\)
Dimension:	\(888\)
Relative dimension:	\(74\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			59.1
Character	\(\chi\)	\(=\)	592.59
Dual form			592.2.cd.a.291.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41400 - 0.0243566i) q^{2} +(-0.549472 + 1.17835i) q^{3} +(1.99881 + 0.0688807i) q^{4} +(0.996072 + 0.175634i) q^{5} +(0.805657 - 1.65280i) q^{6} +(-0.212008 + 1.20236i) q^{7} +(-2.82465 - 0.146082i) q^{8} +(0.841780 + 1.00319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 888 q - 18 q^{2} - 12 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41400	−	0.0243566i	−0.999852	−	0.0172227i
							\(3\)	−0.549472	+	1.17835i	−0.317238	+	0.680319i	−0.998670	−	0.0515625i	\(-0.983580\pi\)
0.681432	+	0.731882i	\(0.261358\pi\)
\(5\)	0.996072	+	0.175634i	0.445457	+	0.0785461i	0.391877	−	0.920017i	\(-0.371826\pi\)
							0.0535797	+	0.998564i	\(0.482937\pi\)
\(7\)	−0.212008	+	1.20236i	−0.0801315	+	0.454448i	0.918170	+	0.396187i	\(0.129667\pi\)
							−0.998301	+	0.0582613i	\(0.981444\pi\)
\(11\)	1.01946	−	3.80469i	0.307380	−	1.14716i	−0.623498	−	0.781825i	\(-0.714289\pi\)
							0.930878	−	0.365331i	\(-0.119044\pi\)
\(13\)	2.36375	+	1.98342i	0.655587	+	0.550103i	0.908761	−	0.417318i	\(-0.137030\pi\)
							−0.253173	+	0.967421i	\(0.581474\pi\)
\(17\)	1.16121	+	0.101592i	0.281634	+	0.0246398i	0.227098	−	0.973872i	\(-0.427076\pi\)
							0.0545364	+	0.998512i	\(0.482632\pi\)
\(19\)	1.87058	+	5.13938i	0.429141	+	1.17905i	0.946335	+	0.323187i	\(0.104754\pi\)
							−0.517194	+	0.855868i	\(0.673024\pi\)
\(23\)	1.55001	−	0.415323i	0.323198	−	0.0866008i	−0.0935721	−	0.995613i	\(-0.529829\pi\)
							0.416770	+	0.909012i	\(0.363162\pi\)
\(29\)	7.73353	−	4.46495i	1.43608	−	0.829121i	0.438506	−	0.898729i	\(-0.355508\pi\)
							0.997574	+	0.0696073i	\(0.0221746\pi\)
\(31\)	−4.41586	−	4.41586i	−0.793111	−	0.793111i	0.188888	−	0.981999i	\(-0.439512\pi\)
							−0.981999	+	0.188888i	\(0.939512\pi\)
\(37\)	−1.17229	+	5.96873i	−0.192724	+	0.981253i
							\(41\)	7.93462	+	6.65793i	1.23918	+	1.03979i	0.997588	+	0.0694166i	\(0.0221138\pi\)
0.241591	+	0.970378i	\(0.422331\pi\)
\(43\)	−4.35537			−0.664188			−0.332094	−	0.943246i	\(-0.607755\pi\)
							−0.332094	+	0.943246i	\(0.607755\pi\)
\(47\)	3.15930	+	1.82402i	0.460831	+	0.266061i	0.712394	−	0.701780i	\(-0.247611\pi\)
							−0.251562	+	0.967841i	\(0.580944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.2.cd.a.59.1	yes	888
16.3	odd	4		592.2.ca.a.355.24	✓	888
37.32	odd	36		592.2.ca.a.587.24	yes	888
592.291	even	36	inner	592.2.cd.a.291.1	yes	888

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
592.2.ca.a.355.24	✓	888	16.3	odd	4
592.2.ca.a.587.24	yes	888	37.32	odd	36
592.2.cd.a.59.1	yes	888	1.1	even	1	trivial
592.2.cd.a.291.1	yes	888	592.291	even	36	inner