# Properties

 Label 63.4.h.a Level $63$ Weight $4$ Character orbit 63.h 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.h (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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 2q^{2} - q^{3} + 158q^{4} - 19q^{5} - 20q^{6} - 7q^{7} - 24q^{8} + 11q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 2q^{2} - q^{3} + 158q^{4} - 19q^{5} - 20q^{6} - 7q^{7} - 24q^{8} + 11q^{9} - 18q^{10} + 5q^{11} - 62q^{12} - 14q^{13} - 52q^{14} + 119q^{15} + 494q^{16} - 162q^{17} - 188q^{18} + 58q^{19} - 362q^{20} - 59q^{21} - 18q^{22} - 93q^{23} + 30q^{24} - 349q^{25} - 266q^{26} + 272q^{27} - 172q^{28} + 248q^{29} + 85q^{30} - 122q^{31} + 326q^{32} + 77q^{33} + 6q^{34} + 289q^{35} - 806q^{36} - 86q^{37} - 761q^{38} - 256q^{39} - 18q^{40} - 692q^{41} - 364q^{42} - 86q^{43} - 443q^{44} + 527q^{45} - 270q^{46} + 2010q^{47} - 1013q^{48} + 317q^{49} + 239q^{50} + 1209q^{51} - 335q^{52} + 258q^{53} + 577q^{54} - 870q^{55} - 1752q^{56} + 566q^{57} + 237q^{58} + 3330q^{59} + 1669q^{60} - 878q^{61} + 1812q^{62} + 2872q^{63} + 872q^{64} + 1226q^{65} + 1330q^{66} - 590q^{67} - 1374q^{68} + 1389q^{69} + 1251q^{70} + 636q^{71} - 5970q^{72} - 338q^{73} + 1119q^{74} + 2737q^{75} + 1006q^{76} + 2269q^{77} + 157q^{78} - 266q^{79} - 4817q^{80} - 505q^{81} + 6q^{82} - 1356q^{83} - 6013q^{84} + 483q^{85} - 3343q^{86} - 5755q^{87} + 369q^{88} - 2200q^{89} + 2665q^{90} + 1552q^{91} - 396q^{92} - 129q^{93} + 2382q^{94} - 6166q^{95} - 5941q^{96} - 266q^{97} + 3601q^{98} - 5395q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −5.09429 5.00043 + 1.41268i 17.9518 −3.42023 + 5.92401i −25.4737 7.19661i −0.241570 18.5187i −50.6973 23.0087 + 14.1280i 17.4236 30.1786i
25.2 −5.07998 −4.95329 + 1.57000i 17.8062 −9.23499 + 15.9955i 25.1626 7.97558i −8.82618 + 16.2818i −49.8155 22.0702 15.5533i 46.9136 81.2567i
25.3 −4.87128 −1.11598 5.07490i 15.7294 3.10540 5.37871i 5.43624 + 24.7213i 17.8495 4.93906i −37.6522 −24.5092 + 11.3269i −15.1273 + 26.2012i
25.4 −4.30573 0.221854 + 5.19141i 10.5394 7.99829 13.8535i −0.955246 22.3529i 1.84582 + 18.4280i −10.9338 −26.9016 + 2.30348i −34.4385 + 59.6493i
25.5 −3.37342 4.47377 2.64298i 3.37993 4.87266 8.43970i −15.0919 + 8.91589i −16.1176 + 9.12268i 15.5854 13.0293 23.6482i −16.4375 + 28.4706i
25.6 −2.93457 −3.89946 3.43427i 0.611720 1.84855 3.20179i 11.4432 + 10.0781i −18.1273 3.79476i 21.6814 3.41152 + 26.7836i −5.42471 + 9.39588i
25.7 −2.65476 −0.794408 + 5.13507i −0.952261 −3.67781 + 6.37015i 2.10896 13.6324i −1.32557 18.4728i 23.7661 −25.7378 8.15868i 9.76368 16.9112i
25.8 −2.37882 3.09078 4.17697i −2.34120 −9.23374 + 15.9933i −7.35242 + 9.93628i 15.8733 + 9.54133i 24.5999 −7.89419 25.8202i 21.9654 38.0453i
25.9 −1.80909 −5.15261 + 0.671239i −4.72719 1.04890 1.81674i 9.32155 1.21433i 18.3541 + 2.47507i 23.0246 26.0989 6.91727i −1.89755 + 3.28665i
25.10 −0.590775 4.35408 + 2.83584i −7.65099 −5.49223 + 9.51282i −2.57228 1.67534i −16.7104 + 7.98524i 9.24621 10.9161 + 24.6949i 3.24467 5.61993i
25.11 −0.438515 4.98047 + 1.48153i −7.80770 8.04659 13.9371i −2.18401 0.649674i 16.9337 7.49990i 6.93192 22.6101 + 14.7574i −3.52855 + 6.11163i
25.12 0.534259 1.18489 5.05925i −7.71457 0.696621 1.20658i 0.633036 2.70295i −2.10659 18.4001i −8.39564 −24.1921 11.9893i 0.372176 0.644627i
25.13 0.983694 −4.22206 + 3.02890i −7.03235 9.35711 16.2070i −4.15321 + 2.97951i −18.4989 0.890133i −14.7872 8.65150 25.5764i 9.20454 15.9427i
25.14 1.33560 −3.51545 3.82643i −6.21617 −4.50235 + 7.79829i −4.69524 5.11058i −3.16069 + 18.2486i −18.9871 −2.28318 + 26.9033i −6.01333 + 10.4154i
25.15 1.44809 0.196108 + 5.19245i −5.90304 −2.21638 + 3.83887i 0.283981 + 7.51913i 9.71690 + 15.7665i −20.1328 −26.9231 + 2.03656i −3.20951 + 5.55903i
25.16 2.66292 −4.74469 + 2.11846i −0.908849 −9.61903 + 16.6607i −12.6348 + 5.64129i −5.55741 17.6668i −23.7236 18.0243 20.1029i −25.6147 + 44.3660i
25.17 3.35315 3.94919 3.37697i 3.24362 4.35326 7.54007i 13.2422 11.3235i −2.88142 + 18.2947i −15.9489 4.19213 26.6726i 14.5971 25.2830i
25.18 3.66978 5.19361 + 0.162683i 5.46732 −7.38708 + 12.7948i 19.0594 + 0.597013i 15.9165 9.46920i −9.29438 26.9471 + 1.68983i −27.1090 + 46.9542i
25.19 3.86663 −3.99989 3.31676i 6.95086 6.67810 11.5668i −15.4661 12.8247i 15.1080 10.7120i −4.05663 4.99821 + 26.5333i 25.8218 44.7246i
25.20 4.21476 2.84415 + 4.34866i 9.76423 3.91201 6.77580i 11.9874 + 18.3286i −15.9002 9.49654i 7.43579 −10.8216 + 24.7365i 16.4882 28.5584i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 58.22 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.h 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.h.a yes 44
3.b odd 2 1 189.4.h.a 44
7.c even 3 1 63.4.g.a 44
9.c even 3 1 63.4.g.a 44
9.d odd 6 1 189.4.g.a 44
21.h odd 6 1 189.4.g.a 44
63.h even 3 1 inner 63.4.h.a yes 44
63.j odd 6 1 189.4.h.a 44

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

## Hecke kernels

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