# Properties

 Label 63.3.r.a Level $63$ Weight $3$ Character orbit 63.r Analytic conductor $1.717$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 63.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.71662566547$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{3} + 24 q^{4} - 18 q^{5} - 14 q^{6} + 26 q^{9}+O(q^{10})$$ 24 * q + 2 * q^3 + 24 * q^4 - 18 * q^5 - 14 * q^6 + 26 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{3} + 24 q^{4} - 18 q^{5} - 14 q^{6} + 26 q^{9} - 18 q^{11} + 4 q^{12} - 10 q^{15} - 48 q^{16} - 62 q^{18} - 24 q^{19} - 18 q^{20} - 14 q^{21} - 24 q^{22} + 72 q^{23} + 54 q^{24} + 54 q^{25} - 124 q^{27} + 54 q^{29} - 212 q^{30} + 30 q^{31} + 126 q^{32} - 178 q^{33} + 60 q^{34} + 124 q^{36} + 84 q^{37} - 144 q^{38} + 92 q^{39} - 60 q^{40} + 180 q^{41} + 140 q^{42} - 60 q^{43} - 118 q^{45} - 168 q^{46} + 378 q^{47} + 436 q^{48} - 84 q^{49} - 378 q^{50} + 168 q^{51} - 18 q^{52} + 514 q^{54} - 132 q^{55} - 232 q^{57} + 90 q^{58} - 90 q^{59} + 76 q^{60} + 28 q^{63} + 324 q^{64} + 126 q^{65} + 202 q^{66} + 6 q^{67} - 738 q^{68} - 432 q^{69} - 246 q^{72} - 72 q^{73} - 792 q^{74} + 40 q^{75} + 84 q^{76} + 28 q^{78} - 6 q^{79} - 34 q^{81} - 108 q^{82} - 558 q^{83} - 322 q^{84} + 126 q^{85} + 90 q^{86} + 428 q^{87} + 168 q^{88} - 488 q^{90} + 84 q^{91} + 774 q^{92} - 738 q^{93} - 354 q^{94} + 648 q^{95} - 280 q^{96} - 270 q^{97} + 296 q^{99}+O(q^{100})$$ 24 * q + 2 * q^3 + 24 * q^4 - 18 * q^5 - 14 * q^6 + 26 * q^9 - 18 * q^11 + 4 * q^12 - 10 * q^15 - 48 * q^16 - 62 * q^18 - 24 * q^19 - 18 * q^20 - 14 * q^21 - 24 * q^22 + 72 * q^23 + 54 * q^24 + 54 * q^25 - 124 * q^27 + 54 * q^29 - 212 * q^30 + 30 * q^31 + 126 * q^32 - 178 * q^33 + 60 * q^34 + 124 * q^36 + 84 * q^37 - 144 * q^38 + 92 * q^39 - 60 * q^40 + 180 * q^41 + 140 * q^42 - 60 * q^43 - 118 * q^45 - 168 * q^46 + 378 * q^47 + 436 * q^48 - 84 * q^49 - 378 * q^50 + 168 * q^51 - 18 * q^52 + 514 * q^54 - 132 * q^55 - 232 * q^57 + 90 * q^58 - 90 * q^59 + 76 * q^60 + 28 * q^63 + 324 * q^64 + 126 * q^65 + 202 * q^66 + 6 * q^67 - 738 * q^68 - 432 * q^69 - 246 * q^72 - 72 * q^73 - 792 * q^74 + 40 * q^75 + 84 * q^76 + 28 * q^78 - 6 * q^79 - 34 * q^81 - 108 * q^82 - 558 * q^83 - 322 * q^84 + 126 * q^85 + 90 * q^86 + 428 * q^87 + 168 * q^88 - 488 * q^90 + 84 * q^91 + 774 * q^92 - 738 * q^93 - 354 * q^94 + 648 * q^95 - 280 * q^96 - 270 * q^97 + 296 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −3.28587 1.89710i −2.99829 + 0.101368i 5.19798 + 9.00316i −1.68242 + 0.971344i 10.0443 + 5.35497i 1.32288 2.29129i 24.2675i 8.97945 0.607858i 7.37094
29.2 −2.95047 1.70346i 2.25172 1.98236i 3.80352 + 6.58789i 6.84828 3.95386i −10.0205 + 2.01321i −1.32288 + 2.29129i 12.2888i 1.14047 8.92745i −26.9409
29.3 −2.50746 1.44768i 2.16846 + 2.07310i 2.19156 + 3.79590i −6.82498 + 3.94040i −2.43614 8.33747i −1.32288 + 2.29129i 1.10929i 0.404480 + 8.99091i 22.8178
29.4 −1.73625 1.00242i −0.111068 2.99794i 0.00971148 + 0.0168208i −4.27746 + 2.46959i −2.81237 + 5.31652i 1.32288 2.29129i 7.98046i −8.97533 + 0.665951i 9.90232
29.5 −0.649615 0.375055i 2.92707 + 0.657482i −1.71867 2.97682i 2.68085 1.54779i −1.65487 1.52492i 1.32288 2.29129i 5.57882i 8.13543 + 3.84899i −2.32203
29.6 0.296130 + 0.170971i −2.98677 + 0.281486i −1.94154 3.36284i −7.71344 + 4.45336i −0.932598 0.427294i −1.32288 + 2.29129i 2.69555i 8.84153 1.68146i −3.04558
29.7 0.526549 + 0.304003i 1.22623 2.73795i −1.81516 3.14396i 0.914466 0.527967i 1.47801 1.06889i −1.32288 + 2.29129i 4.63929i −5.99274 6.71469i 0.642015
29.8 0.744550 + 0.429866i −2.65028 1.40570i −1.63043 2.82399i 5.58239 3.22299i −1.36900 2.18588i 1.32288 2.29129i 6.24239i 5.04801 + 7.45102i 5.54182
29.9 1.64693 + 0.950855i 1.53158 + 2.57959i −0.191750 0.332121i 1.42048 0.820116i 0.0695945 + 5.70470i −1.32288 + 2.29129i 8.33614i −4.30852 + 7.90169i 3.11925
29.10 2.27188 + 1.31167i 2.79644 1.08624i 1.44095 + 2.49580i −7.02923 + 4.05833i 7.77795 + 1.20021i 1.32288 2.29129i 2.93316i 6.64018 6.07519i −21.2927
29.11 2.65531 + 1.53305i −2.10962 + 2.13295i 2.70046 + 4.67733i 0.225868 0.130405i −8.87162 + 2.42952i 1.32288 2.29129i 4.29534i −0.0989900 8.99946i 0.799667
29.12 2.98832 + 1.72531i −1.04547 2.81194i 3.95337 + 6.84744i 0.855181 0.493739i 1.72725 10.2067i −1.32288 + 2.29129i 13.4807i −6.81397 + 5.87961i 3.40741
50.1 −3.28587 + 1.89710i −2.99829 0.101368i 5.19798 9.00316i −1.68242 0.971344i 10.0443 5.35497i 1.32288 + 2.29129i 24.2675i 8.97945 + 0.607858i 7.37094
50.2 −2.95047 + 1.70346i 2.25172 + 1.98236i 3.80352 6.58789i 6.84828 + 3.95386i −10.0205 2.01321i −1.32288 2.29129i 12.2888i 1.14047 + 8.92745i −26.9409
50.3 −2.50746 + 1.44768i 2.16846 2.07310i 2.19156 3.79590i −6.82498 3.94040i −2.43614 + 8.33747i −1.32288 2.29129i 1.10929i 0.404480 8.99091i 22.8178
50.4 −1.73625 + 1.00242i −0.111068 + 2.99794i 0.00971148 0.0168208i −4.27746 2.46959i −2.81237 5.31652i 1.32288 + 2.29129i 7.98046i −8.97533 0.665951i 9.90232
50.5 −0.649615 + 0.375055i 2.92707 0.657482i −1.71867 + 2.97682i 2.68085 + 1.54779i −1.65487 + 1.52492i 1.32288 + 2.29129i 5.57882i 8.13543 3.84899i −2.32203
50.6 0.296130 0.170971i −2.98677 0.281486i −1.94154 + 3.36284i −7.71344 4.45336i −0.932598 + 0.427294i −1.32288 2.29129i 2.69555i 8.84153 + 1.68146i −3.04558
50.7 0.526549 0.304003i 1.22623 + 2.73795i −1.81516 + 3.14396i 0.914466 + 0.527967i 1.47801 + 1.06889i −1.32288 2.29129i 4.63929i −5.99274 + 6.71469i 0.642015
50.8 0.744550 0.429866i −2.65028 + 1.40570i −1.63043 + 2.82399i 5.58239 + 3.22299i −1.36900 + 2.18588i 1.32288 + 2.29129i 6.24239i 5.04801 7.45102i 5.54182
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 50.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.r.a 24
3.b odd 2 1 189.3.r.a 24
7.b odd 2 1 441.3.r.h 24
7.c even 3 1 441.3.j.h 24
7.c even 3 1 441.3.n.g 24
7.d odd 6 1 441.3.j.g 24
7.d odd 6 1 441.3.n.h 24
9.c even 3 1 189.3.r.a 24
9.c even 3 1 567.3.b.a 24
9.d odd 6 1 inner 63.3.r.a 24
9.d odd 6 1 567.3.b.a 24
63.i even 6 1 441.3.n.h 24
63.j odd 6 1 441.3.n.g 24
63.n odd 6 1 441.3.j.h 24
63.o even 6 1 441.3.r.h 24
63.s even 6 1 441.3.j.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.r.a 24 1.a even 1 1 trivial
63.3.r.a 24 9.d odd 6 1 inner
189.3.r.a 24 3.b odd 2 1
189.3.r.a 24 9.c even 3 1
441.3.j.g 24 7.d odd 6 1
441.3.j.g 24 63.s even 6 1
441.3.j.h 24 7.c even 3 1
441.3.j.h 24 63.n odd 6 1
441.3.n.g 24 7.c even 3 1
441.3.n.g 24 63.j odd 6 1
441.3.n.h 24 7.d odd 6 1
441.3.n.h 24 63.i even 6 1
441.3.r.h 24 7.b odd 2 1
441.3.r.h 24 63.o even 6 1
567.3.b.a 24 9.c even 3 1
567.3.b.a 24 9.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(63, [\chi])$$.