# Properties

 Label 504.2.w.a Level 504 Weight 2 Character orbit 504.w Analytic conductor 4.024 Analytic rank 0 Dimension 184 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.w (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$92$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q + q^{2} + q^{4} - 2q^{6} - 2q^{7} - 8q^{8} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q + q^{2} + q^{4} - 2q^{6} - 2q^{7} - 8q^{8} - 2q^{9} + 2q^{10} - 17q^{12} - 7q^{14} - 2q^{15} + q^{16} - 4q^{17} + 13q^{18} + 6q^{20} + 2q^{22} - 4q^{23} + 12q^{24} - 156q^{25} - 4q^{26} - 8q^{28} - 18q^{30} + 2q^{31} + q^{32} + 22q^{33} - 18q^{36} + 10q^{38} - 14q^{39} + 8q^{40} - 4q^{41} + 3q^{42} + 17q^{44} - 6q^{46} + 42q^{47} - 3q^{48} - 2q^{49} - 31q^{50} - 18q^{52} - 58q^{54} + 4q^{55} - 34q^{56} - 20q^{57} - 10q^{58} + 26q^{60} + 32q^{62} + 50q^{63} - 8q^{64} + 22q^{65} + 79q^{66} + 24q^{68} + 8q^{70} - 16q^{71} - 27q^{72} - 4q^{73} - 38q^{74} - 6q^{76} - 47q^{78} + 2q^{79} - 11q^{80} + 6q^{81} + q^{84} + 46q^{86} + 4q^{87} + 14q^{88} + 4q^{89} - 35q^{90} - 48q^{92} + 9q^{94} + 22q^{95} - 16q^{96} - 4q^{97} - 83q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
205.1 −1.41403 0.0227401i 1.48240 + 0.895821i 1.99897 + 0.0643103i 3.06618i −2.07579 1.30043i −1.96423 1.77251i −2.82514 0.136393i 1.39501 + 2.65593i 0.0697253 4.33568i
205.2 −1.41269 0.0655984i −0.165594 + 1.72412i 1.99139 + 0.185341i 1.74289i 0.347032 2.42478i 1.38436 + 2.25467i −2.80107 0.392461i −2.94516 0.571007i 0.114331 2.46217i
205.3 −1.40591 0.153043i 1.60884 0.641583i 1.95316 + 0.430329i 0.532179i −2.36007 + 0.655785i 0.0251237 + 2.64563i −2.68010 0.903920i 2.17674 2.06441i 0.0814462 0.748194i
205.4 −1.40394 + 0.170165i 0.436895 1.67604i 1.94209 0.477802i 2.31532i −0.328169 + 2.42741i 1.54699 2.14635i −2.64527 + 1.00128i −2.61825 1.46451i −0.393985 3.25056i
205.5 −1.40100 0.192870i −1.38115 + 1.04519i 1.92560 + 0.540421i 2.37580i 2.13658 1.19793i 0.371350 2.61956i −2.59354 1.12852i 0.815145 2.88713i −0.458220 + 3.32850i
205.6 −1.40065 + 0.195431i −1.32655 1.11367i 1.92361 0.547458i 1.05207i 2.07567 + 1.30061i 2.60895 0.439732i −2.58731 + 1.14273i 0.519470 + 2.95468i 0.205606 + 1.47357i
205.7 −1.40063 0.195508i 1.23756 + 1.21180i 1.92355 + 0.547670i 3.52202i −1.49645 1.93924i 2.62220 0.352244i −2.58712 1.14316i 0.0630863 + 2.99934i −0.688583 + 4.93306i
205.8 −1.37849 + 0.315851i −0.628421 1.61403i 1.80048 0.870795i 3.22312i 1.37607 + 2.02644i −2.64436 + 0.0858396i −2.20690 + 1.76906i −2.21017 + 2.02858i 1.01802 + 4.44304i
205.9 −1.37435 + 0.333425i 1.66932 0.461909i 1.77765 0.916484i 3.28102i −2.14022 + 1.19142i −2.10198 1.60676i −2.13753 + 1.85228i 2.57328 1.54215i 1.09397 + 4.50925i
205.10 −1.31761 + 0.513706i −1.35345 + 1.08082i 1.47221 1.35373i 0.334033i 1.22810 2.11938i −2.60627 0.455379i −1.24439 + 2.53998i 0.663641 2.92568i −0.171595 0.440127i
205.11 −1.31691 0.515516i −0.229486 1.71678i 1.46849 + 1.35777i 1.20405i −0.582816 + 2.37914i −2.62096 + 0.361324i −1.23391 2.54509i −2.89467 + 0.787953i 0.620709 1.58563i
205.12 −1.31047 + 0.531673i −1.72905 + 0.101939i 1.43465 1.39348i 0.970877i 2.21166 1.05288i 1.53976 + 2.15154i −1.13918 + 2.58887i 2.97922 0.352514i −0.516189 1.27230i
205.13 −1.28717 0.585837i −1.65782 + 0.501625i 1.31359 + 1.50814i 4.18515i 2.42776 + 0.325538i 1.88927 1.85221i −0.807285 2.71077i 2.49674 1.66321i 2.45182 5.38698i
205.14 −1.28512 0.590314i −1.73078 0.0662226i 1.30306 + 1.51725i 2.85273i 2.18517 + 1.10681i −0.733697 + 2.54199i −0.778934 2.71906i 2.99123 + 0.229234i −1.68401 + 3.66610i
205.15 −1.22663 0.703825i −0.0860069 + 1.72991i 1.00926 + 1.72667i 0.633242i 1.32306 2.06144i −2.18459 1.49250i −0.0227161 2.82834i −2.98521 0.297569i 0.445692 0.776756i
205.16 −1.19869 0.750436i −0.854748 1.50645i 0.873692 + 1.79907i 2.34627i −0.105924 + 2.44720i 1.58767 + 2.11644i 0.302808 2.81217i −1.53881 + 2.57528i 1.76073 2.81244i
205.17 −1.18039 + 0.778903i 0.433261 + 1.67699i 0.786619 1.83881i 0.823850i −1.81763 1.64202i 0.822054 2.51480i 0.503744 + 2.78321i −2.62457 + 1.45315i 0.641700 + 0.972460i
205.18 −1.16839 + 0.796778i 1.04913 1.37816i 0.730290 1.86190i 2.16527i −0.127710 + 2.44616i −1.74348 + 1.99004i 0.630255 + 2.75731i −0.798651 2.89174i −1.72524 2.52989i
205.19 −1.12540 0.856430i 1.26962 + 1.17816i 0.533054 + 1.92765i 2.29636i −0.419826 2.41324i −2.07761 + 1.63815i 1.05100 2.62591i 0.223889 + 2.99163i −1.96667 + 2.58432i
205.20 −1.12401 0.858252i 1.72775 0.121975i 0.526807 + 1.92937i 0.697625i −2.04670 1.34574i 2.42006 1.06926i 1.06375 2.62077i 2.97024 0.421485i 0.598738 0.784139i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.92 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
63.g even 3 1 inner
504.w even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.w.a 184
7.c even 3 1 504.2.cq.a yes 184
8.b even 2 1 inner 504.2.w.a 184
9.c even 3 1 504.2.cq.a yes 184
56.p even 6 1 504.2.cq.a yes 184
63.g even 3 1 inner 504.2.w.a 184
72.n even 6 1 504.2.cq.a yes 184
504.w even 6 1 inner 504.2.w.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.w.a 184 1.a even 1 1 trivial
504.2.w.a 184 8.b even 2 1 inner
504.2.w.a 184 63.g even 3 1 inner
504.2.w.a 184 504.w even 6 1 inner
504.2.cq.a yes 184 7.c even 3 1
504.2.cq.a yes 184 9.c even 3 1
504.2.cq.a yes 184 56.p even 6 1
504.2.cq.a yes 184 72.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database