# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 3q^{3} - 162q^{4} - 3q^{5} + 24q^{6} + 5q^{7} - 45q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 3q^{3} - 162q^{4} - 3q^{5} + 24q^{6} + 5q^{7} - 45q^{9} - 6q^{10} + 9q^{11} + 186q^{12} - 36q^{13} + 54q^{14} - 141q^{15} + 526q^{16} - 72q^{17} - 54q^{18} - 6q^{19} + 24q^{20} - 81q^{21} + 14q^{22} - 285q^{23} - 114q^{24} - 349q^{25} - 96q^{26} + 432q^{27} - 156q^{28} - 132q^{29} - 447q^{30} - 3q^{33} + 24q^{34} + 765q^{35} + 1122q^{36} + 82q^{37} - 873q^{38} + 306q^{39} + 420q^{40} + 618q^{41} - 282q^{42} + 82q^{43} + 603q^{44} + 291q^{45} + 266q^{46} - 402q^{47} - 1569q^{48} - 79q^{49} - 1845q^{50} + 453q^{51} + 189q^{52} + 564q^{53} - 2385q^{54} + 66q^{56} + 1170q^{57} + 269q^{58} + 1494q^{59} + 2265q^{60} - 2904q^{62} - 636q^{63} - 1144q^{64} - 372q^{66} - 590q^{67} + 3504q^{68} - 1005q^{69} - 105q^{70} + 1830q^{72} - 6q^{73} + 1515q^{74} - 33q^{75} - 144q^{76} - 4443q^{77} - 5985q^{78} + 1102q^{79} - 4239q^{80} - 4017q^{81} + 18q^{82} + 1830q^{83} + 3165q^{84} - 237q^{85} + 1209q^{86} + 2013q^{87} - 623q^{88} + 4266q^{89} + 9993q^{90} - 1140q^{91} + 7896q^{92} - 729q^{93} + 3975q^{96} - 792q^{97} + 5667q^{98} + 4335q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 5.38106i −0.968958 + 5.10501i −20.9559 −5.57991 9.66469i 27.4704 + 5.21402i −8.46770 + 16.4711i 69.7163i −25.1222 9.89308i −52.0063 + 30.0259i
5.2 5.07706i −4.42649 2.72143i −17.7766 4.21335 + 7.29774i −13.8169 + 22.4736i −4.29909 18.0144i 49.6363i 12.1877 + 24.0928i 37.0511 21.3915i
5.3 4.25176i 5.13732 0.779690i −10.0774 −1.48754 2.57649i −3.31505 21.8426i −13.6257 12.5435i 8.83278i 25.7842 8.01104i −10.9546 + 6.32464i
5.4 4.10714i 1.00877 5.09729i −8.86861 −3.80591 6.59204i −20.9353 4.14318i 13.1152 + 13.0764i 3.56749i −24.9647 10.2840i −27.0744 + 15.6314i
5.5 3.72062i 3.26996 + 4.03824i −5.84303 6.83336 + 11.8357i 15.0248 12.1663i 18.4941 0.984293i 8.02526i −5.61469 + 26.4098i 44.0363 25.4243i
5.6 2.91468i −4.68054 + 2.25667i −0.495360 8.60567 + 14.9055i 6.57748 + 13.6423i −4.20673 + 18.0362i 21.8736i 16.8149 21.1249i 43.4446 25.0828i
5.7 2.41653i −3.31226 + 4.00362i 2.16037 −5.35965 9.28319i 9.67487 + 8.00418i 3.86176 18.1132i 24.5529i −5.05789 26.5220i −22.4331 + 12.9518i
5.8 1.81805i −3.99842 3.31853i 4.69469 −5.16236 8.94146i −6.03325 + 7.26934i −16.8039 + 7.78656i 23.0796i 4.97478 + 26.5377i −16.2560 + 9.38543i
5.9 1.10690i 0.171156 5.19333i 6.77476 6.23688 + 10.8026i −5.74852 0.189454i 11.7098 14.3485i 16.3542i −26.9414 1.77774i 11.9574 6.90363i
5.10 0.837567i 4.56931 + 2.47415i 7.29848 −10.9584 18.9806i 2.07226 3.82711i 14.8276 + 11.0969i 12.8135i 14.7572 + 22.6103i −15.8975 + 9.17842i
5.11 0.747815i 4.77381 2.05201i 7.44077 4.35110 + 7.53632i −1.53452 3.56993i −11.6200 + 14.4213i 11.5468i 18.5785 19.5918i 5.63577 3.25382i
5.12 0.257625i 1.70621 + 4.90804i 7.93363 3.19386 + 5.53193i −1.26443 + 0.439563i −15.2112 10.5650i 4.10490i −21.1777 + 16.7483i −1.42516 + 0.822818i
5.13 1.15257i −5.00666 1.39045i 6.67158 0.137359 + 0.237913i 1.60260 5.77053i 18.5199 + 0.122959i 16.9100i 23.1333 + 13.9231i −0.274212 + 0.158316i
5.14 1.78786i 2.28100 4.66873i 4.80356 −8.47168 14.6734i 8.34703 + 4.07812i −8.67298 16.3640i 22.8910i −16.5940 21.2988i 26.2340 15.1462i
5.15 1.83815i −1.68860 + 4.91412i 4.62119 0.207277 + 0.359014i −9.03291 3.10391i 5.26194 + 17.7570i 23.1997i −21.2972 16.5960i −0.659923 + 0.381007i
5.16 3.06129i 4.98958 + 1.45054i −1.37153 2.75111 + 4.76507i −4.44054 + 15.2746i 8.35389 16.5291i 20.2917i 22.7918 + 14.4752i −14.5873 + 8.42197i
5.17 3.46625i −2.57609 4.51263i −4.01488 9.02701 + 15.6352i 15.6419 8.92935i −18.1650 + 3.61017i 13.8134i −13.7276 + 23.2498i −54.1956 + 31.2898i
5.18 3.64983i −4.97539 + 1.49850i −5.32127 −6.11568 10.5927i −5.46929 18.1593i −17.8879 4.79841i 9.77690i 22.5090 14.9113i 38.6615 22.3212i
5.19 3.72101i 3.89612 3.43806i −5.84592 1.33006 + 2.30373i 12.7931 + 14.4975i 8.98650 + 16.1939i 8.01534i 3.35949 26.7902i −8.57221 + 4.94917i
5.20 4.92859i 3.60211 + 3.74497i −16.2910 −4.62434 8.00960i −18.4575 + 17.7533i −13.0699 + 13.1217i 40.8632i −1.04964 + 26.9796i 39.4761 22.7915i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 38.22 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.i 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.i.a 44
3.b odd 2 1 189.4.i.a 44
7.d odd 6 1 63.4.s.a yes 44
9.c even 3 1 189.4.s.a 44
9.d odd 6 1 63.4.s.a yes 44
21.g even 6 1 189.4.s.a 44
63.i even 6 1 inner 63.4.i.a 44
63.t odd 6 1 189.4.i.a 44

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