# Properties

 Label 864.2.y.b Level $864$ Weight $2$ Character orbit 864.y Analytic conductor $6.899$ Analytic rank $0$ Dimension $54$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(97,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("864.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.y (of order $$9$$, degree $$6$$, 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: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$54 q+O(q^{10})$$ 54 * q $$\operatorname{Tr}(f)(q) =$$ $$54 q - 9 q^{11} + 12 q^{17} + 18 q^{19} + 12 q^{21} - 21 q^{27} + 6 q^{29} + 36 q^{31} - 9 q^{33} + 24 q^{39} + 3 q^{41} - 21 q^{43} + 42 q^{45} - 18 q^{49} + 24 q^{51} + 36 q^{53} - 72 q^{55} + 39 q^{57} + 18 q^{59} - 18 q^{61} - 30 q^{63} + 48 q^{65} - 27 q^{67} + 24 q^{69} - 84 q^{75} + 36 q^{77} + 72 q^{79} + 36 q^{81} + 6 q^{87} + 33 q^{89} + 36 q^{91} + 72 q^{93} + 36 q^{95} + 9 q^{97} + 120 q^{99}+O(q^{100})$$ 54 * q - 9 * q^11 + 12 * q^17 + 18 * q^19 + 12 * q^21 - 21 * q^27 + 6 * q^29 + 36 * q^31 - 9 * q^33 + 24 * q^39 + 3 * q^41 - 21 * q^43 + 42 * q^45 - 18 * q^49 + 24 * q^51 + 36 * q^53 - 72 * q^55 + 39 * q^57 + 18 * q^59 - 18 * q^61 - 30 * q^63 + 48 * q^65 - 27 * q^67 + 24 * q^69 - 84 * q^75 + 36 * q^77 + 72 * q^79 + 36 * q^81 + 6 * q^87 + 33 * q^89 + 36 * q^91 + 72 * q^93 + 36 * q^95 + 9 * q^97 + 120 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −1.68626 0.395655i 0 −2.65901 0.967801i 0 −0.872194 + 4.94646i 0 2.68691 + 1.33435i 0
97.2 0 −1.59135 + 0.683824i 0 2.16034 + 0.786299i 0 0.181183 1.02754i 0 2.06477 2.17640i 0
97.3 0 −1.06169 1.36851i 0 0.217711 + 0.0792403i 0 0.399809 2.26743i 0 −0.745646 + 2.90586i 0
97.4 0 −1.00024 + 1.41404i 0 −3.61346 1.31519i 0 0.621731 3.52601i 0 −0.999043 2.82877i 0
97.5 0 −0.146186 1.72587i 0 2.31520 + 0.842664i 0 −0.690140 + 3.91398i 0 −2.95726 + 0.504595i 0
97.6 0 0.256084 + 1.71302i 0 −1.06964 0.389317i 0 −0.738542 + 4.18848i 0 −2.86884 + 0.877351i 0
97.7 0 1.11120 1.32862i 0 −0.779464 0.283702i 0 0.260701 1.47851i 0 −0.530450 2.95273i 0
97.8 0 1.53630 + 0.799856i 0 −1.92089 0.699146i 0 0.00661950 0.0375410i 0 1.72046 + 2.45764i 0
97.9 0 1.64243 + 0.549936i 0 3.46982 + 1.26291i 0 0.136240 0.772655i 0 2.39514 + 1.80646i 0
193.1 0 −1.72098 0.195544i 0 0.359864 2.04089i 0 2.05212 1.72193i 0 2.92353 + 0.673052i 0
193.2 0 −1.67597 + 0.437160i 0 −0.158606 + 0.899497i 0 −1.25116 + 1.04985i 0 2.61778 1.46534i 0
193.3 0 −0.661519 1.60075i 0 0.197119 1.11792i 0 −0.270013 + 0.226568i 0 −2.12478 + 2.11785i 0
193.4 0 −0.523926 + 1.65091i 0 −0.649143 + 3.68147i 0 1.60562 1.34727i 0 −2.45100 1.72991i 0
193.5 0 −0.0599758 + 1.73101i 0 0.392840 2.22791i 0 −3.55853 + 2.98596i 0 −2.99281 0.207638i 0
193.6 0 0.527750 1.64969i 0 −0.220114 + 1.24833i 0 −2.06659 + 1.73408i 0 −2.44296 1.74125i 0
193.7 0 1.17779 + 1.26996i 0 0.191385 1.08540i 0 2.62195 2.20007i 0 −0.225616 + 2.99150i 0
193.8 0 1.41286 1.00191i 0 0.727972 4.12853i 0 −0.397562 + 0.333594i 0 0.992364 2.83112i 0
193.9 0 1.69762 + 0.343652i 0 −0.494020 + 2.80173i 0 −1.80000 + 1.51038i 0 2.76381 + 1.16678i 0
385.1 0 −1.72098 + 0.195544i 0 0.359864 + 2.04089i 0 2.05212 + 1.72193i 0 2.92353 0.673052i 0
385.2 0 −1.67597 0.437160i 0 −0.158606 0.899497i 0 −1.25116 1.04985i 0 2.61778 + 1.46534i 0
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.b 54
4.b odd 2 1 864.2.y.c yes 54
27.e even 9 1 inner 864.2.y.b 54
108.j odd 18 1 864.2.y.c yes 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.b 54 1.a even 1 1 trivial
864.2.y.b 54 27.e even 9 1 inner
864.2.y.c yes 54 4.b odd 2 1
864.2.y.c yes 54 108.j odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{54} + 26 T_{5}^{51} + 54 T_{5}^{49} + 4271 T_{5}^{48} + 1908 T_{5}^{47} - 4941 T_{5}^{46} + 14126 T_{5}^{45} - 40653 T_{5}^{44} + 718182 T_{5}^{43} + 13296992 T_{5}^{42} - 2697336 T_{5}^{41} + \cdots + 6135275584$$ T5^54 + 26*T5^51 + 54*T5^49 + 4271*T5^48 + 1908*T5^47 - 4941*T5^46 + 14126*T5^45 - 40653*T5^44 + 718182*T5^43 + 13296992*T5^42 - 2697336*T5^41 - 35554446*T5^40 + 141061310*T5^39 + 270808920*T5^38 + 1788165054*T5^37 + 6441577406*T5^36 - 8483636034*T5^35 + 762688359*T5^34 + 62328886706*T5^33 + 97732481769*T5^32 + 488963397096*T5^31 + 1349107689617*T5^30 - 454350384630*T5^29 - 2382195346236*T5^28 + 7474354038548*T5^27 + 42690360493044*T5^26 + 117905343565482*T5^25 + 252819537069131*T5^24 + 251491739310576*T5^23 - 135230313995550*T5^22 - 622046505992596*T5^21 + 538913748918150*T5^20 + 6568715858884866*T5^19 + 21636761032335536*T5^18 + 47137235653229940*T5^17 + 79138513810887246*T5^16 + 108326844130965452*T5^15 + 123664823598339450*T5^14 + 117563339734433622*T5^13 + 92932545142593653*T5^12 + 60109943837034006*T5^11 + 30320481411615870*T5^10 + 11199074190827864*T5^9 + 3826419488486433*T5^8 + 1866109494444768*T5^7 + 1602331642769189*T5^6 + 1081964111196312*T5^5 + 33365493246636*T5^4 - 229883314988440*T5^3 + 39206249755776*T5^2 + 758017966752*T5 + 6135275584 $$T_{7}^{54} + 9 T_{7}^{52} - 106 T_{7}^{51} - 243 T_{7}^{50} - 84 T_{7}^{49} + 13673 T_{7}^{48} + 3924 T_{7}^{47} + 112917 T_{7}^{46} - 800116 T_{7}^{45} - 124686 T_{7}^{44} - 3763170 T_{7}^{43} + \cdots + 1043846369344$$ T7^54 + 9*T7^52 - 106*T7^51 - 243*T7^50 - 84*T7^49 + 13673*T7^48 + 3924*T7^47 + 112917*T7^46 - 800116*T7^45 - 124686*T7^44 - 3763170*T7^43 + 76051586*T7^42 - 241821684*T7^41 + 1225770846*T7^40 - 4672416094*T7^39 + 6834612195*T7^38 - 5038074708*T7^37 + 63701454302*T7^36 - 53246956536*T7^35 + 674718683310*T7^34 - 3143154177202*T7^33 - 169185112272*T7^32 + 23610319736802*T7^31 + 7867917985037*T7^30 - 134280503022222*T7^29 + 187502647148022*T7^28 + 211513574524496*T7^27 - 1144962048654090*T7^26 + 1790108852267880*T7^25 + 9794031111614552*T7^24 - 12068003497817844*T7^23 - 18742941435404010*T7^22 + 48929317625666462*T7^21 + 53885205590992695*T7^20 - 31067650295237364*T7^19 + 142869918867943535*T7^18 + 22684586479743978*T7^17 - 422502220761950484*T7^16 + 96258609209255114*T7^15 + 1485036003948827193*T7^14 + 1611888694963866888*T7^13 + 3356744561873973200*T7^12 + 3558092075504373438*T7^11 + 2636367448428540228*T7^10 + 1671532643850617306*T7^9 + 810319743296915220*T7^8 + 137617446154368042*T7^7 - 77892384802386607*T7^6 - 39461481058734504*T7^5 + 396321668731320*T7^4 + 3482074515105632*T7^3 + 671413897877472*T7^2 - 4364712437280*T7 + 1043846369344