Newform orbit 592.2.ce.a

• Label: 592.2.ce.a
• Level: $592$
• Weight: $2$
• Character orbit: 592.ce
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $888$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.ce (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:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 888 q - 12 q^{2} - 12 q^{3} - 18 q^{4} - 12 q^{5} - 24 q^{6} - 6 q^{8}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.41340	+	0.0479799i	−0.268085	+	0.574909i	1.99540	−	0.135630i	−3.02073	+	2.11514i	0.351326	−	0.825439i	−1.42791	−	0.251780i	−2.81378	+	0.287438i	1.66971	+	1.98988i	4.16801	−	3.13446i
53.2	−1.40611	+	0.151159i	1.18203	−	2.53487i	1.95430	−	0.425092i	0.497589	−	0.348416i	−1.27890	+	3.74299i	2.36404	+	0.416843i	−2.68371	+	0.893137i	−3.10003	−	3.69447i	−0.647000	+	0.565126i
53.3	−1.40514	−	0.159939i	0.417232	−	0.894757i	1.94884	+	0.449473i	−1.21059	+	0.847664i	−0.729376	+	1.19053i	3.38015	+	0.596011i	−2.66650	−	0.943267i	1.30186	+	1.55149i	1.83662	−	0.997467i
53.4	−1.38424	−	0.289612i	−1.08784	+	2.33288i	1.83225	+	0.801786i	−0.715745	+	0.501170i	2.18147	−	2.91422i	0.868412	+	0.153124i	−2.30407	−	1.64051i	−2.33059	−	2.77749i	1.13591	−	0.486452i
53.5	−1.38099	−	0.304723i	−0.112171	+	0.240551i	1.81429	+	0.841640i	2.04730	−	1.43353i	0.228208	−	0.298018i	−3.42353	−	0.603660i	−2.24905	−	1.71515i	1.88308	+	2.24417i	−3.26413	+	1.35584i
53.6	−1.36714	+	0.361837i	−0.435248	+	0.933393i	1.73815	−	0.989364i	3.39898	−	2.37999i	0.257310	−	1.43357i	3.65129	+	0.643820i	−2.01830	+	1.98153i	1.24658	+	1.48562i	−3.78572	+	4.48366i
53.7	−1.36005	−	0.387636i	1.12627	−	2.41528i	1.69948	+	1.05441i	−1.27557	+	0.893165i	−2.46803	+	2.84833i	−2.72292	−	0.480125i	−1.90265	−	2.09283i	−2.63676	−	3.14237i	2.08107	−	0.720292i
53.8	−1.35816	+	0.394203i	0.566544	−	1.21496i	1.68921	−	1.07078i	1.45017	−	1.01542i	−0.290519	+	1.87344i	−3.44761	−	0.607907i	−1.87211	+	2.12019i	0.773211	+	0.921477i	−1.56929	+	1.95077i
53.9	−1.32711	+	0.488639i	−1.31765	+	2.82571i	1.52246	−	1.29696i	−1.53824	+	1.07708i	0.367920	−	4.39389i	4.39038	+	0.774142i	−1.38674	+	2.46515i	−4.32007	−	5.14846i	1.51511	−	2.18106i
53.10	−1.32685	+	0.489366i	−0.458746	+	0.983785i	1.52104	−	1.29863i	0.0802757	−	0.0562096i	0.127255	−	1.52983i	−0.906841	−	0.159901i	−1.38269	+	2.46742i	1.17098	+	1.39552i	−0.0790064	+	0.113866i
53.11	−1.30060	−	0.555380i	−1.06006	+	2.27332i	1.38311	+	1.44465i	0.761611	−	0.533286i	2.64127	−	2.36793i	0.127509	+	0.0224833i	−0.996535	−	2.64706i	−2.11587	−	2.52159i	−1.28673	+	0.270607i
53.12	−1.28988	−	0.579837i	1.02134	−	2.19027i	1.32758	+	1.49584i	3.44408	−	2.41157i	−2.58740	+	2.23297i	0.715719	+	0.126201i	−0.845072	−	2.69923i	−1.82578	−	2.17588i	−5.84076	+	1.11363i
53.13	−1.18684	+	0.769038i	−1.39445	+	2.99040i	0.817160	−	1.82545i	2.73463	−	1.91481i	−0.644754	−	4.62150i	−4.01498	−	0.707949i	0.434003	+	2.79493i	−5.06967	−	6.04179i	−1.77299	+	4.37559i
53.14	−1.18555	−	0.771021i	−0.165868	+	0.355705i	0.811053	+	1.82817i	1.04174	−	0.729436i	0.470900	−	0.293818i	4.11577	+	0.725722i	0.448013	−	2.79272i	1.82935	+	2.18013i	−1.79745	+	0.0615765i
53.15	−1.11946	+	0.864185i	1.30894	−	2.80704i	0.506369	−	1.93484i	−2.76483	+	1.93596i	0.960493	+	4.27353i	1.30230	+	0.229631i	1.10520	+	2.60356i	−4.23776	−	5.05037i	1.42209	−	4.55655i
53.16	−1.08354	+	0.908808i	0.404286	−	0.866993i	0.348134	−	1.96947i	0.182595	−	0.127855i	0.349870	+	1.30684i	0.896336	+	0.158048i	1.41265	+	2.45039i	1.34013	+	1.59711i	−0.0816547	+	0.304480i
53.17	−1.04318	−	0.954869i	0.811341	−	1.73993i	0.176450	+	1.99220i	−2.86322	+	2.00485i	−2.50778	+	1.04033i	0.0647563	+	0.0114183i	1.71822	−	2.24671i	−0.440705	−	0.525212i	4.90122	+	0.642582i
53.18	−1.03567	−	0.963013i	−1.22163	+	2.61980i	0.145213	+	1.99472i	−2.40194	+	1.68186i	3.78811	−	1.53679i	−4.91748	−	0.867085i	1.77055	−	2.20571i	−3.44260	−	4.10273i	4.10726	+	0.571256i
53.19	−1.03346	+	0.965385i	−0.972700	+	2.08596i	0.136063	−	1.99537i	−2.67367	+	1.87212i	−1.00851	−	3.09478i	−2.12219	−	0.374200i	1.78568	+	2.19348i	−1.47673	−	1.75990i	0.955800	−	4.51588i
53.20	−1.01389	+	0.985913i	0.632849	−	1.35715i	0.0559521	−	1.99922i	0.965798	−	0.676259i	0.696391	+	1.99994i	2.40771	+	0.424545i	1.91432	+	2.08215i	0.487005	+	0.580391i	−0.312482	+	1.63785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner
37.f	even	9	1	inner
592.ce	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	592.2.ce.a	✓	888
16.e	even	4	1	inner	592.2.ce.a	✓	888
37.f	even	9	1	inner	592.2.ce.a	✓	888
592.ce	even	36	1	inner	592.2.ce.a	✓	888

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
592.2.ce.a	✓	888	1.a	even	1	1	trivial
592.2.ce.a	✓	888	16.e	even	4	1	inner
592.2.ce.a	✓	888	37.f	even	9	1	inner
592.2.ce.a	✓	888	592.ce	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).