# Properties

 Label 729.2.k.a Level $729$ Weight $2$ Character orbit 729.k Analytic conductor $5.821$ Analytic rank $0$ Dimension $12960$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(4,729)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(486))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("729.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.k (of order $$243$$, degree $$162$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$12960$$ Relative dimension: $$80$$ over $$\Q(\zeta_{243})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{243}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10})$$ 12960 * q - 162 * q^2 - 162 * q^3 - 162 * q^4 - 162 * q^5 - 162 * q^6 - 162 * q^7 - 162 * q^8 - 162 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9} - 162 q^{10} - 162 q^{11} - 162 q^{12} - 162 q^{13} - 162 q^{14} - 162 q^{15} - 162 q^{16} - 162 q^{17} - 162 q^{18} - 162 q^{19} - 162 q^{20} - 162 q^{21} - 162 q^{22} - 162 q^{23} - 162 q^{24} - 162 q^{25} - 162 q^{26} - 162 q^{27} - 162 q^{28} - 162 q^{29} - 162 q^{30} - 162 q^{31} - 162 q^{32} - 162 q^{33} - 162 q^{34} - 162 q^{35} - 162 q^{36} - 162 q^{37} - 162 q^{38} - 162 q^{39} - 162 q^{40} - 162 q^{41} - 162 q^{42} - 162 q^{43} - 162 q^{44} - 162 q^{45} - 162 q^{46} - 162 q^{47} - 162 q^{48} - 162 q^{49} - 162 q^{50} - 162 q^{51} - 162 q^{52} - 162 q^{53} - 162 q^{54} - 162 q^{55} - 162 q^{56} - 162 q^{57} - 162 q^{58} - 162 q^{59} - 162 q^{60} - 162 q^{61} - 162 q^{62} - 162 q^{63} - 162 q^{64} - 162 q^{65} - 162 q^{66} - 162 q^{67} - 162 q^{68} - 162 q^{69} - 162 q^{70} - 162 q^{71} - 162 q^{72} - 162 q^{73} - 162 q^{74} - 162 q^{75} - 162 q^{76} - 162 q^{77} - 162 q^{78} - 162 q^{79} - 162 q^{80} - 162 q^{81} - 162 q^{82} - 162 q^{83} - 162 q^{84} - 162 q^{85} - 162 q^{86} - 162 q^{87} - 162 q^{88} - 162 q^{89} - 162 q^{90} - 162 q^{91} - 162 q^{92} - 162 q^{93} - 162 q^{94} - 162 q^{95} - 162 q^{96} - 162 q^{97} - 162 q^{98} - 162 q^{99}+O(q^{100})$$ 12960 * q - 162 * q^2 - 162 * q^3 - 162 * q^4 - 162 * q^5 - 162 * q^6 - 162 * q^7 - 162 * q^8 - 162 * q^9 - 162 * q^10 - 162 * q^11 - 162 * q^12 - 162 * q^13 - 162 * q^14 - 162 * q^15 - 162 * q^16 - 162 * q^17 - 162 * q^18 - 162 * q^19 - 162 * q^20 - 162 * q^21 - 162 * q^22 - 162 * q^23 - 162 * q^24 - 162 * q^25 - 162 * q^26 - 162 * q^27 - 162 * q^28 - 162 * q^29 - 162 * q^30 - 162 * q^31 - 162 * q^32 - 162 * q^33 - 162 * q^34 - 162 * q^35 - 162 * q^36 - 162 * q^37 - 162 * q^38 - 162 * q^39 - 162 * q^40 - 162 * q^41 - 162 * q^42 - 162 * q^43 - 162 * q^44 - 162 * q^45 - 162 * q^46 - 162 * q^47 - 162 * q^48 - 162 * q^49 - 162 * q^50 - 162 * q^51 - 162 * q^52 - 162 * q^53 - 162 * q^54 - 162 * q^55 - 162 * q^56 - 162 * q^57 - 162 * q^58 - 162 * q^59 - 162 * q^60 - 162 * q^61 - 162 * q^62 - 162 * q^63 - 162 * q^64 - 162 * q^65 - 162 * q^66 - 162 * q^67 - 162 * q^68 - 162 * q^69 - 162 * q^70 - 162 * q^71 - 162 * q^72 - 162 * q^73 - 162 * q^74 - 162 * q^75 - 162 * q^76 - 162 * q^77 - 162 * q^78 - 162 * q^79 - 162 * q^80 - 162 * q^81 - 162 * q^82 - 162 * q^83 - 162 * q^84 - 162 * q^85 - 162 * q^86 - 162 * q^87 - 162 * q^88 - 162 * q^89 - 162 * q^90 - 162 * q^91 - 162 * q^92 - 162 * q^93 - 162 * q^94 - 162 * q^95 - 162 * q^96 - 162 * q^97 - 162 * q^98 - 162 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.72490 0.0352304i −1.51002 0.848437i 5.42450 + 0.140291i 0.633611 + 0.194162i 4.08476 + 2.36510i 1.09035 2.71893i −9.33012 0.362051i 1.56031 + 2.56231i −1.71968 0.551395i
4.2 −2.68619 0.0347299i 0.113106 1.72835i 5.21506 + 0.134874i −1.58849 0.486773i −0.363851 + 4.63875i −0.858532 + 2.14086i −8.63515 0.335083i −2.97441 0.390976i 4.25007 + 1.36273i
4.3 −2.65153 0.0342818i 0.952364 + 1.44672i 5.03008 + 0.130090i −0.525411 0.161006i −2.47562 3.86867i −0.732244 + 1.82595i −8.03343 0.311734i −1.18601 + 2.75561i 1.38762 + 0.444923i
4.4 −2.61310 0.0337849i 1.70554 0.301886i 4.82780 + 0.124859i −2.59518 0.795260i −4.46694 + 0.731236i 0.853432 2.12814i −7.38859 0.286711i 2.81773 1.02976i 6.75458 + 2.16577i
4.5 −2.51518 0.0325190i 1.53758 0.797395i 4.32575 + 0.111875i 3.23632 + 0.991730i −3.89323 + 1.95559i −1.26529 + 3.15518i −5.84940 0.226983i 1.72832 2.45212i −8.10768 2.59962i
4.6 −2.48775 0.0321643i 0.111243 + 1.72847i 4.18852 + 0.108326i 2.61631 + 0.801736i −0.221150 4.30359i 0.558840 1.39354i −5.44433 0.211265i −2.97525 + 0.384562i −6.48293 2.07867i
4.7 −2.46704 0.0318966i −0.976238 + 1.43072i 4.08596 + 0.105673i −4.03733 1.23719i 2.45406 3.49851i −1.39476 + 3.47801i −5.14609 0.199692i −1.09392 2.79345i 9.92081 + 3.18098i
4.8 −2.38931 0.0308915i 0.801520 1.53544i 3.70850 + 0.0959110i 1.80500 + 0.553120i −1.96251 + 3.64387i 1.52892 3.81257i −4.08235 0.158414i −1.71513 2.46137i −4.29561 1.37733i
4.9 −2.38009 0.0307723i −1.09995 + 1.33795i 3.66453 + 0.0947738i 2.99225 + 0.916937i 2.65915 3.15058i −0.120994 + 0.301715i −3.96199 0.153743i −0.580208 2.94336i −7.09359 2.27447i
4.10 −2.29202 0.0296337i −1.03658 1.38762i 3.25315 + 0.0841346i 3.65994 + 1.12154i 2.33475 + 3.21117i −0.556114 + 1.38674i −2.87282 0.111478i −0.850984 + 2.87677i −8.35542 2.67906i
4.11 −2.27606 0.0294274i −1.57284 0.725368i 3.18025 + 0.0822491i −3.48668 1.06845i 3.55854 + 1.69727i 0.216487 0.539838i −2.68694 0.104265i 1.94768 + 2.28178i 7.90445 + 2.53446i
4.12 −2.08274 0.0269280i −1.71559 0.238211i 2.33777 + 0.0604606i −0.511598 0.156773i 3.56673 + 0.542331i −1.27737 + 3.18529i −0.704645 0.0273434i 2.88651 + 0.817346i 1.06131 + 0.340294i
4.13 −2.05837 0.0266128i 0.611800 + 1.62040i 2.23685 + 0.0578504i −2.65761 0.814392i −1.21619 3.35167i 1.70918 4.26207i −0.488730 0.0189650i −2.25140 + 1.98272i 5.44867 + 1.74705i
4.14 −2.03766 0.0263451i −1.63810 + 0.562705i 2.15204 + 0.0556572i 0.558373 + 0.171106i 3.35271 1.10345i 0.399839 0.997050i −0.311068 0.0120709i 2.36673 1.84353i −1.13327 0.363367i
4.15 −1.93424 0.0250080i 1.62140 + 0.609150i 1.74134 + 0.0450353i 0.582227 + 0.178416i −3.12095 1.21879i −1.25911 + 3.13976i 0.498860 + 0.0193580i 2.25787 + 1.97535i −1.12171 0.359660i
4.16 −1.90902 0.0246819i 0.0146033 1.73199i 1.64441 + 0.0425285i −2.27162 0.696109i −0.0706267 + 3.30604i 0.132583 0.330612i 0.677327 + 0.0262834i −2.99957 0.0505857i 4.31938 + 1.38495i
4.17 −1.90833 0.0246729i 1.72970 0.0902694i 1.64178 + 0.0424604i 0.325467 + 0.0997352i −3.30306 + 0.129587i 0.868282 2.16517i 0.682106 + 0.0264688i 2.98370 0.312277i −0.618636 0.198358i
4.18 −1.82547 0.0236016i 1.47646 + 0.905578i 1.33244 + 0.0344602i 2.72027 + 0.833593i −2.67385 1.68795i 1.26149 3.14568i 1.21698 + 0.0472244i 1.35986 + 2.67410i −4.94609 1.58590i
4.19 −1.68750 0.0218178i −0.499964 + 1.65832i 0.847837 + 0.0219272i 0.0353662 + 0.0108375i 0.879869 2.78751i −1.23579 + 3.08161i 1.94249 + 0.0753776i −2.50007 1.65820i −0.0594438 0.0190599i
4.20 −1.63282 0.0211109i 0.989331 1.42170i 0.666340 + 0.0172332i −1.81802 0.557111i −1.64542 + 2.30050i −0.319539 + 0.796813i 2.17581 + 0.0844315i −1.04245 2.81306i 2.95675 + 0.948044i
See next 80 embeddings (of 12960 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 727.80 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
729.k even 243 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.k.a 12960
729.k even 243 1 inner 729.2.k.a 12960

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.k.a 12960 1.a even 1 1 trivial
729.2.k.a 12960 729.k even 243 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(729, [\chi])$$.