Embedded newform 592.2.ca.a.587.24

• Label: 592.2.ca.a.587.24
• Level: $592$
• Weight: $2$
• Character: 592.587
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $888$
• 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			587.24
Character	\(\chi\)	\(=\)	592.587
Dual form			592.2.ca.a.355.24

$q$-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 \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\)	\(113\)	\(149\)	\(223\)
\(\chi(n)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
592.2.ca.a.355.24	✓	888	592.355	even	36	inner
592.2.ca.a.587.24	yes	888	1.1	even	1	trivial
592.2.cd.a.59.1	yes	888	37.22	odd	36
592.2.cd.a.291.1	yes	888	16.3	odd	4