# Properties

 Label 864.2.y.d Level $864$ Weight $2$ Character orbit 864.y Analytic conductor $6.899$ Analytic rank $0$ Dimension $60$ CM no Inner twists $4$

# 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: $$60$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 12 q^{9}+O(q^{10})$$ 60 * q + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 12 q^{9} - 12 q^{17} + 24 q^{21} - 24 q^{25} + 6 q^{29} - 12 q^{33} - 30 q^{37} - 30 q^{41} - 90 q^{45} + 42 q^{49} - 36 q^{53} - 60 q^{57} + 48 q^{61} + 12 q^{65} + 78 q^{69} - 48 q^{73} - 12 q^{77} + 12 q^{81} - 102 q^{85} - 12 q^{89} - 36 q^{93} + 12 q^{97}+O(q^{100})$$ 60 * q + 12 * q^9 - 12 * q^17 + 24 * q^21 - 24 * q^25 + 6 * q^29 - 12 * q^33 - 30 * q^37 - 30 * q^41 - 90 * q^45 + 42 * q^49 - 36 * q^53 - 60 * q^57 + 48 * q^61 + 12 * q^65 + 78 * q^69 - 48 * q^73 - 12 * q^77 + 12 * q^81 - 102 * q^85 - 12 * q^89 - 36 * q^93 + 12 * q^97

## 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 −1.71652 + 0.231448i 0 −3.16823 1.15314i 0 0.327751 1.85877i 0 2.89286 0.794570i 0
97.2 0 −1.65691 + 0.504627i 0 1.76220 + 0.641388i 0 −0.498467 + 2.82694i 0 2.49070 1.67224i 0
97.3 0 −1.01321 1.40478i 0 −1.03120 0.375327i 0 −0.257635 + 1.46112i 0 −0.946819 + 2.84667i 0
97.4 0 −0.790581 1.54110i 0 3.49481 + 1.27201i 0 0.718530 4.07499i 0 −1.74996 + 2.43673i 0
97.5 0 −0.523054 + 1.65119i 0 −0.117882 0.0429055i 0 0.455772 2.58481i 0 −2.45283 1.72732i 0
97.6 0 0.523054 1.65119i 0 −0.117882 0.0429055i 0 −0.455772 + 2.58481i 0 −2.45283 1.72732i 0
97.7 0 0.790581 + 1.54110i 0 3.49481 + 1.27201i 0 −0.718530 + 4.07499i 0 −1.74996 + 2.43673i 0
97.8 0 1.01321 + 1.40478i 0 −1.03120 0.375327i 0 0.257635 1.46112i 0 −0.946819 + 2.84667i 0
97.9 0 1.65691 0.504627i 0 1.76220 + 0.641388i 0 0.498467 2.82694i 0 2.49070 1.67224i 0
97.10 0 1.71652 0.231448i 0 −3.16823 1.15314i 0 −0.327751 + 1.85877i 0 2.89286 0.794570i 0
193.1 0 −1.72168 0.189239i 0 0.679279 3.85238i 0 −2.91962 + 2.44985i 0 2.92838 + 0.651618i 0
193.2 0 −1.68152 + 0.415309i 0 −0.564773 + 3.20298i 0 −2.06064 + 1.72908i 0 2.65504 1.39670i 0
193.3 0 −1.28924 + 1.15666i 0 −0.110022 + 0.623966i 0 3.44796 2.89318i 0 0.324271 2.98242i 0
193.4 0 −0.998006 1.41562i 0 −0.476400 + 2.70180i 0 0.930891 0.781110i 0 −1.00797 + 2.82560i 0
193.5 0 −0.141380 + 1.72627i 0 0.298268 1.69156i 0 −0.0848018 + 0.0711572i 0 −2.96002 0.488119i 0
193.6 0 0.141380 1.72627i 0 0.298268 1.69156i 0 0.0848018 0.0711572i 0 −2.96002 0.488119i 0
193.7 0 0.998006 + 1.41562i 0 −0.476400 + 2.70180i 0 −0.930891 + 0.781110i 0 −1.00797 + 2.82560i 0
193.8 0 1.28924 1.15666i 0 −0.110022 + 0.623966i 0 −3.44796 + 2.89318i 0 0.324271 2.98242i 0
193.9 0 1.68152 0.415309i 0 −0.564773 + 3.20298i 0 2.06064 1.72908i 0 2.65504 1.39670i 0
193.10 0 1.72168 + 0.189239i 0 0.679279 3.85238i 0 2.91962 2.44985i 0 2.92838 + 0.651618i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.d 60
4.b odd 2 1 inner 864.2.y.d 60
27.e even 9 1 inner 864.2.y.d 60
108.j odd 18 1 inner 864.2.y.d 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.d 60 1.a even 1 1 trivial
864.2.y.d 60 4.b odd 2 1 inner
864.2.y.d 60 27.e even 9 1 inner
864.2.y.d 60 108.j odd 18 1 inner

## Hecke kernels

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

 $$T_{5}^{30} + 6 T_{5}^{28} - 19 T_{5}^{27} + 63 T_{5}^{26} - 675 T_{5}^{25} + 3456 T_{5}^{24} + 270 T_{5}^{23} + 12006 T_{5}^{22} + 32910 T_{5}^{21} - 110277 T_{5}^{20} - 36891 T_{5}^{19} + 5454057 T_{5}^{18} - 5670288 T_{5}^{17} + \cdots + 341056$$ T5^30 + 6*T5^28 - 19*T5^27 + 63*T5^26 - 675*T5^25 + 3456*T5^24 + 270*T5^23 + 12006*T5^22 + 32910*T5^21 - 110277*T5^20 - 36891*T5^19 + 5454057*T5^18 - 5670288*T5^17 + 38454480*T5^16 - 64453833*T5^15 + 6453216*T5^14 + 6415488*T5^13 - 46378302*T5^12 + 182526282*T5^11 + 648979668*T5^10 + 233741913*T5^9 - 141687648*T5^8 + 84954231*T5^7 + 290361645*T5^6 + 19060056*T5^5 + 142231644*T5^4 - 44959672*T5^3 + 5313888*T5^2 + 3861408*T5 + 341056 $$T_{7}^{60} - 21 T_{7}^{58} + 405 T_{7}^{56} + 4647 T_{7}^{54} - 122247 T_{7}^{52} + 3429648 T_{7}^{50} + 36003726 T_{7}^{48} - 665047908 T_{7}^{46} + 14409106767 T_{7}^{44} + \cdots + 10\!\cdots\!44$$ T7^60 - 21*T7^58 + 405*T7^56 + 4647*T7^54 - 122247*T7^52 + 3429648*T7^50 + 36003726*T7^48 - 665047908*T7^46 + 14409106767*T7^44 + 4135988142*T7^42 - 1031987293470*T7^40 + 20247963820056*T7^38 + 66493664344269*T7^36 - 1106804855944800*T7^34 + 9135389772541260*T7^32 + 55252118608258734*T7^30 - 325394889128996886*T7^28 + 2641688492668860483*T7^26 + 21565600611249156741*T7^24 - 81169025622683446728*T7^22 + 168171436900233937395*T7^20 + 4645449475766190655218*T7^18 + 7525600430759861469156*T7^16 - 14112603550681256806272*T7^14 + 25587942682483626677217*T7^12 + 183363582154487896665936*T7^10 + 194323612644299132283456*T7^8 + 320252144158550496143232*T7^6 + 702268816003058338845696*T7^4 - 2946339846016726901760*T7^2 + 105665856469516652544