# Properties

 Label 63.4.g.a Level $63$ Weight $4$ Character orbit 63.g Analytic conductor $3.717$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q + q^{2} - q^{3} - 79q^{4} + 38q^{5} - 20q^{6} - 7q^{7} - 24q^{8} - 31q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + q^{2} - q^{3} - 79q^{4} + 38q^{5} - 20q^{6} - 7q^{7} - 24q^{8} - 31q^{9} - 18q^{10} - 10q^{11} - 41q^{12} - 14q^{13} - 79q^{14} + 119q^{15} - 247q^{16} - 162q^{17} + 157q^{18} + 58q^{19} - 362q^{20} + 166q^{21} - 18q^{22} + 186q^{23} + 414q^{24} + 698q^{25} - 266q^{26} + 272q^{27} - 172q^{28} + 248q^{29} + 616q^{30} + 61q^{31} - 163q^{32} + 23q^{33} + 6q^{34} + 289q^{35} - 806q^{36} - 86q^{37} + 1522q^{38} - 565q^{39} + 36q^{40} - 692q^{41} + 395q^{42} - 86q^{43} - 443q^{44} - 1483q^{45} - 270q^{46} - 1005q^{47} - 1013q^{48} - 277q^{49} + 239q^{50} - 1719q^{51} + 670q^{52} + 258q^{53} + 910q^{54} - 870q^{55} + 714q^{56} + 566q^{57} - 474q^{58} - 1665q^{59} + 4q^{60} + 439q^{61} + 1812q^{62} + 493q^{63} + 872q^{64} - 613q^{65} + 3073q^{66} + 295q^{67} + 2748q^{68} + 1389q^{69} - 1044q^{70} + 636q^{71} + 981q^{72} - 338q^{73} - 2238q^{74} - 1064q^{75} + 1006q^{76} - 2909q^{77} + 157q^{78} + 133q^{79} - 4817q^{80} + 1325q^{81} + 6q^{82} - 1356q^{83} - 7081q^{84} + 483q^{85} + 6686q^{86} + 2774q^{87} - 738q^{88} - 2200q^{89} + 2665q^{90} + 1552q^{91} - 396q^{92} + 4365q^{93} - 1191q^{94} + 3083q^{95} - 1468q^{96} - 266q^{97} + 3601q^{98} - 5395q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.71525 + 4.70295i 4.53482 2.53680i −10.7452 18.6112i 13.1170 −0.382714 + 28.2151i 12.4520 13.7094i 73.2595 14.1292 23.0079i −35.6160 + 61.6887i
4.2 −2.51592 + 4.35771i −1.54886 + 4.95994i −8.65974 14.9991i 0.150305 −17.7172 19.2283i −18.0980 + 3.93229i 46.8942 −22.2021 15.3645i −0.378156 + 0.654986i
4.3 −2.10738 + 3.65009i −5.18812 0.288776i −4.88211 8.45607i −7.82402 11.9874 18.3286i 16.1743 9.02169i 7.43579 26.8332 + 2.99641i 16.4882 28.5584i
4.4 −1.93332 + 3.34860i 4.87234 + 1.80563i −3.47543 6.01962i −13.3562 −15.4661 + 12.8247i 1.72288 + 18.4399i −4.05663 20.4794 + 17.5952i 25.8218 44.7246i
4.5 −1.83489 + 3.17813i −2.73769 4.41645i −2.73366 4.73484i 14.7742 19.0594 0.597013i 0.242329 + 18.5187i −9.29438 −12.0101 + 24.1818i −27.1090 + 46.9542i
4.6 −1.67658 + 2.90391i 0.949950 5.10858i −1.62181 2.80905i −8.70652 13.2422 + 11.3235i −14.4030 11.6428i −15.9489 −25.1952 9.70580i 14.5971 25.2830i
4.7 −1.33146 + 2.30616i 0.537708 + 5.16826i 0.454424 + 0.787086i 19.2381 −12.6348 5.64129i 18.0786 + 4.02053i −23.7236 −26.4217 + 5.55802i −25.6147 + 44.3660i
4.8 −0.724044 + 1.25408i −4.59485 + 2.42639i 2.95152 + 5.11218i 4.43275 0.283981 7.51913i −18.5126 + 0.531848i −20.1328 15.2253 22.2978i −3.20951 + 5.55903i
4.9 −0.667800 + 1.15666i 5.07151 + 1.13126i 3.10809 + 5.38337i 9.00469 −4.69524 + 5.11058i −14.2234 11.8615i −18.9871 24.4405 + 11.4744i −6.01333 + 10.4154i
4.10 −0.491847 + 0.851904i −0.512079 + 5.17086i 3.51617 + 6.09019i −18.7142 −4.15321 2.97951i 10.0203 15.5754i −14.7872 −26.4756 5.29577i 9.20454 15.9427i
4.11 −0.267129 + 0.462682i 3.78900 3.55577i 3.85728 + 6.68101i −1.39324 0.633036 + 2.70295i 16.9882 + 7.37568i −8.39564 1.71302 26.9456i 0.372176 0.644627i
4.12 0.219258 0.379765i −3.77328 3.57245i 3.90385 + 6.76167i −16.0932 −2.18401 + 0.649674i −1.97176 + 18.4150i 6.93192 1.47525 + 26.9597i −3.52855 + 6.11163i
4.13 0.295387 0.511626i −4.63295 2.35283i 3.82549 + 6.62595i 10.9845 −2.57228 + 1.67534i 1.43976 18.4642i 9.24621 15.9284 + 21.8011i 3.24467 5.61993i
4.14 0.904546 1.56672i 1.99500 + 4.79791i 2.36359 + 4.09386i −2.09779 9.32155 + 1.21433i −11.3205 + 14.6576i 23.0246 −19.0400 + 19.1437i −1.89755 + 3.28665i
4.15 1.18941 2.06012i 2.07198 4.76518i 1.17060 + 2.02754i 18.4675 −7.35242 9.93628i −16.1997 + 8.97605i 24.5999 −18.4138 19.7467i 21.9654 38.0453i
4.16 1.32738 2.29909i −4.04989 + 3.25551i 0.476130 + 0.824682i 7.35561 2.10896 + 13.6324i 16.6607 + 8.08840i 23.7661 5.80329 26.3690i 9.76368 16.9112i
4.17 1.46729 2.54141i 4.92390 + 1.65989i −0.305860 0.529765i −3.69711 11.4432 10.0781i 12.3500 13.8013i 21.6814 21.4895 + 16.3463i −5.42471 + 9.39588i
4.18 1.68671 2.92146i 0.0520064 5.19589i −1.68996 2.92710i −9.74532 −15.0919 8.91589i 0.158327 18.5196i 15.5854 −26.9946 0.540440i −16.4375 + 28.4706i
4.19 2.15287 3.72888i −4.60682 + 2.40358i −5.26968 9.12735i −15.9966 −0.955246 + 22.3529i −16.8821 7.61550i −10.9338 15.4456 22.1457i −34.4385 + 59.6493i
4.20 2.43564 4.21866i 4.95298 1.57099i −7.86471 13.6221i −6.21080 5.43624 24.7213i −4.64741 + 17.9277i −37.6522 22.0640 15.5621i −15.1273 + 26.2012i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.g.a 44
3.b odd 2 1 189.4.g.a 44
7.c even 3 1 63.4.h.a yes 44
9.c even 3 1 63.4.h.a yes 44
9.d odd 6 1 189.4.h.a 44
21.h odd 6 1 189.4.h.a 44
63.g even 3 1 inner 63.4.g.a 44
63.n odd 6 1 189.4.g.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.g.a 44 1.a even 1 1 trivial
63.4.g.a 44 63.g even 3 1 inner
63.4.h.a yes 44 7.c even 3 1
63.4.h.a yes 44 9.c even 3 1
189.4.g.a 44 3.b odd 2 1
189.4.g.a 44 63.n odd 6 1
189.4.h.a 44 9.d odd 6 1
189.4.h.a 44 21.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(63, [\chi])$$.