Embedded newform 592.2.ca.a.355.24

• Label: 592.2.ca.a.355.24
• Level: $592$
• Weight: $2$
• Character: 592.355
• 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.ca (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			355.24
Character	\(\chi\)	\(=\)	592.355
Dual form			592.2.ca.a.587.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.890246 + 1.09885i) q^{2} +(1.17835 + 0.549472i) q^{3} +(-0.414925 - 1.95649i) q^{4} +(0.175634 - 0.996072i) q^{5} +(-1.65280 + 0.805657i) q^{6} +(-0.212008 + 1.20236i) q^{7} +(2.51926 + 1.28582i) q^{8} +(-0.841780 - 1.00319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 888 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.890246	+	1.09885i	−0.629499	+	0.777001i
							\(3\)	1.17835	+	0.549472i	0.680319	+	0.317238i	0.731882	−	0.681432i	\(-0.238642\pi\)
−0.0515625	+	0.998670i	\(0.516420\pi\)
\(5\)	0.175634	−	0.996072i	0.0785461	−	0.445457i	−0.920017	−	0.391877i	\(-0.871826\pi\)
							0.998564	−	0.0535797i	\(-0.0170631\pi\)
\(7\)	−0.212008	+	1.20236i	−0.0801315	+	0.454448i	0.918170	+	0.396187i	\(0.129667\pi\)
							−0.998301	+	0.0582613i	\(0.981444\pi\)
\(11\)	3.80469	+	1.01946i	1.14716	+	0.307380i	0.781825	−	0.623498i	\(-0.214289\pi\)
							0.365331	+	0.930878i	\(0.380956\pi\)
\(13\)	−1.98342	+	2.36375i	−0.550103	+	0.655587i	−0.967421	−	0.253173i	\(-0.918526\pi\)
							0.417318	+	0.908761i	\(0.362970\pi\)
\(17\)	1.16121	+	0.101592i	0.281634	+	0.0246398i	0.227098	−	0.973872i	\(-0.427076\pi\)
							0.0545364	+	0.998512i	\(0.482632\pi\)
\(19\)	5.13938	−	1.87058i	1.17905	−	0.429141i	0.323187	−	0.946335i	\(-0.395246\pi\)
							0.855868	+	0.517194i	\(0.173024\pi\)
\(23\)	1.55001	−	0.415323i	0.323198	−	0.0866008i	−0.0935721	−	0.995613i	\(-0.529829\pi\)
							0.416770	+	0.909012i	\(0.363162\pi\)
\(29\)	4.46495	+	7.73353i	0.829121	+	1.43608i	0.898729	+	0.438506i	\(0.144492\pi\)
							−0.0696073	+	0.997574i	\(0.522175\pi\)
\(31\)	4.41586	+	4.41586i	0.793111	+	0.793111i	0.981999	−	0.188888i	\(-0.0604882\pi\)
							−0.188888	+	0.981999i	\(0.560488\pi\)
\(37\)	5.96873	+	1.17229i	0.981253	+	0.192724i
							\(41\)	−7.93462	−	6.65793i	−1.23918	−	1.03979i	−0.997588	−	0.0694166i	\(-0.977886\pi\)
−0.241591	−	0.970378i	\(-0.577669\pi\)
\(43\)		−	4.35537i		−	0.664188i	−0.943246	−	0.332094i	\(-0.892245\pi\)
							0.943246	−	0.332094i	\(-0.107755\pi\)
\(47\)	−3.15930	−	1.82402i	−0.460831	−	0.266061i	0.251562	−	0.967841i	\(-0.419056\pi\)
							−0.712394	+	0.701780i	\(0.752389\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
592.2.ca.a.355.24	✓	888	1.1	even	1	trivial
592.2.ca.a.587.24	yes	888	592.587	even	36	inner
592.2.cd.a.59.1	yes	888	16.11	odd	4
592.2.cd.a.291.1	yes	888	37.32	odd	36