# Properties

 Label 405.2.r.a Level $405$ Weight $2$ Character orbit 405.r Analytic conductor $3.234$ Analytic rank $0$ Dimension $192$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(8,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([2, 27]))

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

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

chi := DirichletCharacter("405.8");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.r (of order $$36$$, degree $$12$$, not 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: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$16$$ over $$\Q(\zeta_{36})$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8}+O(q^{10})$$ 192 * q + 12 * q^2 + 12 * q^5 - 12 * q^7 + 18 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$192 q + 12 q^{2} + 12 q^{5} - 12 q^{7} + 18 q^{8} - 6 q^{10} + 36 q^{11} - 12 q^{13} - 24 q^{16} + 18 q^{17} - 36 q^{20} - 12 q^{22} + 36 q^{23} - 30 q^{25} - 24 q^{28} - 24 q^{31} + 48 q^{32} - 36 q^{35} - 6 q^{37} - 12 q^{38} - 36 q^{40} - 24 q^{41} - 12 q^{43} - 12 q^{46} + 6 q^{47} - 36 q^{50} + 12 q^{52} - 24 q^{55} - 180 q^{56} - 12 q^{58} - 60 q^{61} + 18 q^{62} + 84 q^{65} + 24 q^{67} + 60 q^{68} - 12 q^{70} + 36 q^{71} - 6 q^{73} - 72 q^{76} - 132 q^{77} - 24 q^{82} - 48 q^{83} - 12 q^{85} - 12 q^{86} - 48 q^{88} - 12 q^{91} - 258 q^{92} - 18 q^{95} + 24 q^{97} - 324 q^{98}+O(q^{100})$$ 192 * q + 12 * q^2 + 12 * q^5 - 12 * q^7 + 18 * q^8 - 6 * q^10 + 36 * q^11 - 12 * q^13 - 24 * q^16 + 18 * q^17 - 36 * q^20 - 12 * q^22 + 36 * q^23 - 30 * q^25 - 24 * q^28 - 24 * q^31 + 48 * q^32 - 36 * q^35 - 6 * q^37 - 12 * q^38 - 36 * q^40 - 24 * q^41 - 12 * q^43 - 12 * q^46 + 6 * q^47 - 36 * q^50 + 12 * q^52 - 24 * q^55 - 180 * q^56 - 12 * q^58 - 60 * q^61 + 18 * q^62 + 84 * q^65 + 24 * q^67 + 60 * q^68 - 12 * q^70 + 36 * q^71 - 6 * q^73 - 72 * q^76 - 132 * q^77 - 24 * q^82 - 48 * q^83 - 12 * q^85 - 12 * q^86 - 48 * q^88 - 12 * q^91 - 258 * q^92 - 18 * q^95 + 24 * q^97 - 324 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −2.05499 + 1.43892i 0 1.46845 4.03452i 1.82626 + 1.29027i 0 −1.14217 + 2.44940i 1.48912 + 5.55747i 0 −5.60952 0.0236494i
8.2 −1.94894 + 1.36466i 0 1.25203 3.43992i −0.278496 2.21866i 0 −1.02475 + 2.19758i 1.02263 + 3.81650i 0 3.57049 + 3.94398i
8.3 −1.65044 + 1.15565i 0 0.704386 1.93528i 0.268922 + 2.21984i 0 0.618828 1.32708i 0.0310202 + 0.115769i 0 −3.00920 3.35293i
8.4 −1.23715 + 0.866263i 0 0.0960923 0.264011i 2.23168 0.140023i 0 0.677826 1.45360i −0.671958 2.50778i 0 −2.63963 + 2.10645i
8.5 −0.865920 + 0.606324i 0 −0.301851 + 0.829330i −2.13636 + 0.660292i 0 −1.22161 + 2.61975i −0.788655 2.94330i 0 1.44956 1.86708i
8.6 −0.490069 + 0.343150i 0 −0.561625 + 1.54305i 1.25284 1.85213i 0 −0.0636379 + 0.136472i −0.563947 2.10468i 0 0.0215816 + 1.33758i
8.7 −0.263290 + 0.184358i 0 −0.648706 + 1.78231i −2.21531 0.303968i 0 2.07991 4.46037i −0.324162 1.20979i 0 0.639309 0.328378i
8.8 −0.0928047 + 0.0649826i 0 −0.679650 + 1.86732i −0.584024 + 2.15845i 0 0.344848 0.739528i −0.116914 0.436329i 0 −0.0860616 0.238266i
8.9 0.377486 0.264318i 0 −0.611409 + 1.67983i −0.538986 2.17014i 0 −1.52168 + 3.26325i 0.451753 + 1.68596i 0 −0.777066 0.676732i
8.10 0.807333 0.565301i 0 −0.351818 + 0.966613i 2.04189 + 0.911409i 0 −0.275357 + 0.590505i 0.772562 + 2.88324i 0 2.16371 0.418474i
8.11 1.08817 0.761943i 0 −0.0804885 + 0.221140i −1.23426 1.86457i 0 0.827388 1.77434i 0.768546 + 2.86825i 0 −2.76377 1.08853i
8.12 1.22359 0.856769i 0 0.0790861 0.217287i −2.10603 + 0.751431i 0 −2.03610 + 4.36642i 0.683816 + 2.55204i 0 −1.93312 + 2.72382i
8.13 1.42277 0.996232i 0 0.347747 0.955427i 0.893670 + 2.04972i 0 0.429055 0.920112i 0.442010 + 1.64960i 0 3.31348 + 2.02597i
8.14 1.66533 1.16608i 0 0.729546 2.00441i 1.48319 1.67336i 0 1.13512 2.43427i −0.0700073 0.261271i 0 0.518731 4.51621i
8.15 2.12047 1.48477i 0 1.60780 4.41741i 2.09416 0.783897i 0 −1.25154 + 2.68394i −1.80956 6.75336i 0 3.27669 4.77156i
8.16 2.18018 1.52658i 0 1.73870 4.77703i −1.64880 + 1.51045i 0 1.30062 2.78918i −2.12414 7.92739i 0 −1.28885 + 5.81007i
17.1 −2.47401 + 1.15365i 0 3.50423 4.17618i 1.69173 1.46221i 0 −0.0671403 0.00587402i −2.43862 + 9.10104i 0 −2.49848 + 5.56917i
17.2 −1.82102 + 0.849158i 0 1.30949 1.56058i 1.44997 + 1.70223i 0 3.18459 + 0.278616i −0.0193446 + 0.0721951i 0 −4.08589 1.86855i
17.3 −1.81626 + 0.846935i 0 1.29592 1.54442i −0.828498 2.07692i 0 −2.21198 0.193523i −0.00834683 + 0.0311508i 0 3.26378 + 3.07054i
17.4 −1.64734 + 0.768168i 0 0.838077 0.998782i −0.861745 + 2.06335i 0 −4.49995 0.393694i 0.327512 1.22229i 0 −0.165408 4.06100i
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.r.a 192
3.b odd 2 1 135.2.q.a 192
5.c odd 4 1 inner 405.2.r.a 192
15.d odd 2 1 675.2.ba.b 192
15.e even 4 1 135.2.q.a 192
15.e even 4 1 675.2.ba.b 192
27.e even 9 1 135.2.q.a 192
27.f odd 18 1 inner 405.2.r.a 192
135.p even 18 1 675.2.ba.b 192
135.q even 36 1 inner 405.2.r.a 192
135.r odd 36 1 135.2.q.a 192
135.r odd 36 1 675.2.ba.b 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.q.a 192 3.b odd 2 1
135.2.q.a 192 15.e even 4 1
135.2.q.a 192 27.e even 9 1
135.2.q.a 192 135.r odd 36 1
405.2.r.a 192 1.a even 1 1 trivial
405.2.r.a 192 5.c odd 4 1 inner
405.2.r.a 192 27.f odd 18 1 inner
405.2.r.a 192 135.q even 36 1 inner
675.2.ba.b 192 15.d odd 2 1
675.2.ba.b 192 15.e even 4 1
675.2.ba.b 192 135.p even 18 1
675.2.ba.b 192 135.r odd 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(405, [\chi])$$.