# Properties

 Label 63.4.s.a Level $63$ Weight $4$ Character orbit 63.s 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.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 3q^{2} - 3q^{3} + 81q^{4} - 6q^{5} - 24q^{6} + 5q^{7} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 3q^{2} - 3q^{3} + 81q^{4} - 6q^{5} - 24q^{6} + 5q^{7} - 3q^{9} - 6q^{10} - 3q^{12} + 36q^{13} + 129q^{14} - 141q^{15} - 263q^{16} + 72q^{17} - 15q^{18} - 6q^{19} - 24q^{20} - 306q^{21} + 14q^{22} - 66q^{24} + 698q^{25} + 96q^{26} - 432q^{27} - 156q^{28} - 132q^{29} + 852q^{30} + 177q^{31} - 501q^{32} + 849q^{33} - 24q^{34} - 765q^{35} + 1122q^{36} + 82q^{37} - 1746q^{38} - 645q^{39} - 618q^{41} - 963q^{42} + 82q^{43} - 603q^{44} + 303q^{45} + 266q^{46} - 201q^{47} + 1569q^{48} + 515q^{49} - 1845q^{50} + 417q^{51} - 564q^{53} - 684q^{54} + 3600q^{56} + 1170q^{57} - 538q^{58} + 747q^{59} - 516q^{60} - 1209q^{61} + 2904q^{62} + 1557q^{63} - 1144q^{64} - 831q^{65} + 1029q^{66} + 295q^{67} + 7008q^{68} + 1005q^{69} - 390q^{70} - 1119q^{72} - 6q^{73} - 1788q^{75} + 144q^{76} - 1203q^{77} - 5985q^{78} - 551q^{79} + 4239q^{80} + 3741q^{81} + 18q^{82} - 1830q^{83} - 7725q^{84} - 237q^{85} - 2130q^{87} + 1246q^{88} - 4266q^{89} - 9993q^{90} - 1140q^{91} + 7896q^{92} - 1479q^{93} - 3q^{94} - 1053q^{95} + 5034q^{96} + 792q^{97} - 5667q^{98} + 4335q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −4.53659 + 2.61920i 2.85276 + 4.34301i 9.72046 16.8363i −12.2114 −24.3171 12.2305i −10.8477 15.0109i 59.9322i −10.7235 + 24.7791i 55.3983 31.9842i
47.2 −4.33849 + 2.50483i −5.18070 + 0.400483i 8.54831 14.8061i 18.5684 21.4732 14.7142i −17.9322 4.62983i 45.5709i 26.6792 4.14956i −80.5586 + 46.5105i
47.3 −4.26829 + 2.46430i −1.44219 4.99200i 8.14552 14.1085i −9.24869 18.4575 + 17.7533i 17.8986 + 4.75801i 40.8632i −22.8402 + 14.3988i 39.4761 22.7915i
47.4 −3.22249 + 1.86051i 4.92551 1.65511i 2.92296 5.06272i 2.66012 −12.7931 + 14.4975i 9.53109 15.8795i 8.01534i 21.5212 16.3045i −8.57221 + 4.94917i
47.5 −3.16085 + 1.82492i −3.78544 + 3.55956i 2.66064 4.60836i −12.2314 5.46929 18.1593i 4.78838 + 17.8905i 9.77690i 1.65907 26.9490i 38.6615 22.3212i
47.6 −3.00186 + 1.73312i 2.62001 + 4.48727i 2.00744 3.47699i 18.0540 −15.6419 8.92935i 12.2090 + 13.9263i 13.8134i −13.2711 + 23.5133i −54.1956 + 31.2898i
47.7 −2.65116 + 1.53065i 1.23858 5.04638i 0.685763 1.18778i 5.50223 4.44054 + 15.2746i −18.4916 + 1.02989i 20.2917i −23.9318 12.5007i −14.5873 + 8.42197i
47.8 −1.59189 + 0.919076i −5.10006 0.994690i −2.31060 + 4.00207i 0.414554 9.03291 3.10391i 12.7471 13.4355i 23.1997i 25.0212 + 10.1460i −0.659923 + 0.381007i
47.9 −1.54833 + 0.893930i 5.18374 + 0.358957i −2.40178 + 4.16000i −16.9434 −8.34703 + 4.07812i −9.83512 + 15.6930i 22.8910i 26.7423 + 3.72148i 26.2340 15.1462i
47.10 −0.998155 + 0.576285i −1.29916 + 5.03112i −3.33579 + 5.77776i 0.274718 −1.60260 5.77053i −9.15344 16.1001i 16.9100i −23.6244 13.0725i −0.274212 + 0.158316i
47.11 −0.223110 + 0.128812i −3.39738 3.93164i −3.96681 + 6.87072i 6.38772 1.26443 + 0.439563i −1.54394 + 18.4558i 4.10490i −3.91563 + 26.7146i −1.42516 + 0.822818i
47.12 0.647627 0.373907i 4.16400 3.10824i −3.72039 + 6.44390i 8.70220 1.53452 3.56993i 18.2993 + 2.85258i 11.5468i 7.67775 25.8854i 5.63577 3.25382i
47.13 0.725355 0.418784i 0.141982 5.19421i −3.64924 + 6.32067i −21.9169 −2.07226 3.82711i 2.19637 18.3896i 12.8135i −26.9597 1.47497i −15.8975 + 9.17842i
47.14 0.958607 0.553452i 4.58314 + 2.44844i −3.38738 + 5.86712i 12.4738 5.74852 0.189454i −18.2811 2.96677i 16.3542i 15.0103 + 22.4431i 11.9574 6.90363i
47.15 1.57448 0.909026i 0.874715 + 5.12200i −2.34735 + 4.06572i −10.3247 6.03325 + 7.26934i 15.1453 + 10.6593i 23.0796i −25.4697 + 8.96058i −16.2560 + 9.38543i
47.16 2.09278 1.20827i −5.12336 + 0.866692i −1.08019 + 1.87094i −10.7193 −9.67487 + 8.00418i −17.6173 + 5.71220i 24.5529i 25.4977 8.88075i −22.4331 + 12.9518i
47.17 2.52419 1.45734i −4.29460 + 2.92513i 0.247680 0.428994i 17.2113 −6.57748 + 13.6423i 17.7231 5.37495i 21.8736i 9.88725 25.1245i 43.4446 25.0828i
47.18 3.22215 1.86031i −1.86223 4.85099i 2.92152 5.06021i 13.6667 −15.0248 12.1663i −10.0995 15.5242i 8.02526i −20.0642 + 18.0673i 44.0363 25.4243i
47.19 3.55689 2.05357i 4.91877 + 1.67502i 4.43430 7.68044i −7.61183 20.9353 4.14318i 4.76690 17.8963i 3.56749i 21.3886 + 16.4781i −27.0744 + 15.6314i
47.20 3.68213 2.12588i 3.24389 4.05921i 5.03872 8.72732i −2.97507 3.31505 21.8426i −4.05014 + 18.0720i 8.83278i −5.95432 26.3353i −10.9546 + 6.32464i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.22 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.s even 6 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database