# Properties

 Label 273.2.bf.b Level $273$ Weight $2$ Character orbit 273.bf Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(152,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 4]))

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

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

chi := DirichletCharacter("273.152");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bf (of order $$6$$, degree $$2$$, 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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 64 * q + 32 * q^4 - 12 * q^6 - 4 * q^7 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$64 q + 32 q^{4} - 12 q^{6} - 4 q^{7} + 8 q^{9} + 6 q^{12} - 12 q^{13} - 9 q^{15} - 16 q^{16} + 2 q^{18} + 10 q^{21} + 10 q^{22} - 24 q^{25} - 50 q^{28} - 16 q^{30} - 24 q^{31} - 33 q^{39} + 90 q^{40} - 48 q^{42} - 20 q^{43} - 3 q^{45} + 6 q^{48} - 10 q^{51} + 30 q^{52} - 27 q^{54} + 18 q^{55} + 4 q^{57} - 60 q^{58} + 55 q^{60} - 74 q^{63} - 84 q^{64} + 75 q^{66} - 88 q^{67} - 33 q^{69} + 20 q^{70} - 34 q^{72} + 84 q^{73} + 33 q^{75} + 18 q^{76} - 71 q^{78} + 20 q^{79} - 32 q^{81} - 6 q^{84} - 2 q^{85} + 3 q^{87} + 92 q^{88} - 76 q^{91} + 28 q^{93} + 30 q^{96} + 24 q^{97} + 22 q^{99}+O(q^{100})$$ 64 * q + 32 * q^4 - 12 * q^6 - 4 * q^7 + 8 * q^9 + 6 * q^12 - 12 * q^13 - 9 * q^15 - 16 * q^16 + 2 * q^18 + 10 * q^21 + 10 * q^22 - 24 * q^25 - 50 * q^28 - 16 * q^30 - 24 * q^31 - 33 * q^39 + 90 * q^40 - 48 * q^42 - 20 * q^43 - 3 * q^45 + 6 * q^48 - 10 * q^51 + 30 * q^52 - 27 * q^54 + 18 * q^55 + 4 * q^57 - 60 * q^58 + 55 * q^60 - 74 * q^63 - 84 * q^64 + 75 * q^66 - 88 * q^67 - 33 * q^69 + 20 * q^70 - 34 * q^72 + 84 * q^73 + 33 * q^75 + 18 * q^76 - 71 * q^78 + 20 * q^79 - 32 * q^81 - 6 * q^84 - 2 * q^85 + 3 * q^87 + 92 * q^88 - 76 * q^91 + 28 * q^93 + 30 * q^96 + 24 * q^97 + 22 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
152.1 −2.36551 + 1.36573i 1.47077 + 0.914786i 2.73043 4.72924i 0.121221 0.209960i −4.72847 0.155263i −0.288550 2.62997i 9.45317i 1.32633 + 2.69088i 0.662218i
152.2 −2.25245 + 1.30045i −1.59264 + 0.680804i 2.38234 4.12633i 1.54658 2.67876i 2.70199 3.60462i −2.63374 0.251835i 7.19066i 2.07301 2.16855i 8.04502i
152.3 −2.07561 + 1.19835i −0.232998 1.71631i 1.87210 3.24258i 1.10064 1.90637i 2.54036 + 3.28317i 2.62075 0.362837i 4.18035i −2.89142 + 0.799793i 5.27584i
152.4 −2.07482 + 1.19790i 1.22164 1.22784i 1.86992 3.23879i −1.50632 + 2.60901i −1.06387 + 4.01095i −1.19447 + 2.36077i 4.16828i −0.0151701 2.99996i 7.21764i
152.5 −1.70697 + 0.985518i −1.16871 1.27832i 0.942492 1.63244i −0.272603 + 0.472162i 3.25477 + 1.03026i −2.22004 + 1.43924i 0.226699i −0.268213 + 2.98799i 1.07462i
152.6 −1.66598 + 0.961856i −0.628043 + 1.61418i 0.850333 1.47282i −1.94643 + 3.37132i −0.506295 3.29327i −1.91023 1.83058i 0.575834i −2.21112 2.02754i 7.48874i
152.7 −1.60522 + 0.926776i 1.72065 + 0.198407i 0.717829 1.24332i −0.852778 + 1.47706i −2.94591 + 1.27617i 2.63557 0.231921i 1.04604i 2.92127 + 0.682779i 3.16134i
152.8 −1.60258 + 0.925247i 0.929062 + 1.46179i 0.712166 1.23351i 1.21731 2.10844i −2.84141 1.48302i −0.567699 + 2.58413i 1.06527i −1.27369 + 2.71620i 4.50525i
152.9 −1.33345 + 0.769868i −0.674223 + 1.59544i 0.185393 0.321109i 0.956175 1.65614i −0.329235 2.64650i 1.55646 2.13949i 2.50856i −2.09085 2.15136i 2.94451i
152.10 −1.21710 + 0.702692i 1.65544 0.509428i −0.0124468 + 0.0215586i 1.48377 2.56997i −1.65686 + 1.78329i −2.30005 1.30759i 2.84575i 2.48097 1.68666i 4.17054i
152.11 −0.913280 + 0.527282i −1.53174 0.808559i −0.443947 + 0.768938i −0.000964697 0.00167090i 1.82525 0.0692194i 1.27194 2.31995i 3.04547i 1.69246 + 2.47701i 0.00203467i
152.12 −0.765058 + 0.441707i 0.162694 1.72439i −0.609790 + 1.05619i −1.72131 + 2.98140i 0.637205 + 1.39112i −0.0494112 2.64529i 2.84422i −2.94706 0.561097i 3.04126i
152.13 −0.730177 + 0.421568i −1.64300 + 0.548211i −0.644561 + 1.11641i −1.06280 + 1.84082i 0.968576 1.09293i −0.110845 + 2.64343i 2.77318i 2.39893 1.80143i 1.79216i
152.14 −0.667801 + 0.385555i 1.00894 1.40785i −0.702694 + 1.21710i 0.907647 1.57209i −0.130967 + 1.32917i 1.94952 + 1.78868i 2.62593i −0.964085 2.84087i 1.39979i
152.15 −0.473030 + 0.273104i 0.966120 + 1.43757i −0.850828 + 1.47368i −1.25657 + 2.17644i −0.849611 0.416164i 2.53535 + 0.756296i 2.02188i −1.13323 + 2.77773i 1.37270i
152.16 −0.0436821 + 0.0252199i 1.71783 + 0.221508i −0.998728 + 1.72985i −1.19579 + 2.07117i −0.0806247 + 0.0336474i −2.29456 1.31719i 0.201631i 2.90187 + 0.761026i 0.120631i
152.17 0.0436821 0.0252199i −1.71783 + 0.221508i −0.998728 + 1.72985i 1.19579 2.07117i −0.0694519 + 0.0529993i −2.29456 1.31719i 0.201631i 2.90187 0.761026i 0.120631i
152.18 0.473030 0.273104i −0.966120 + 1.43757i −0.850828 + 1.47368i 1.25657 2.17644i −0.0643970 + 0.943866i 2.53535 + 0.756296i 2.02188i −1.13323 2.77773i 1.37270i
152.19 0.667801 0.385555i −1.00894 1.40785i −0.702694 + 1.21710i −0.907647 + 1.57209i −1.21657 0.551162i 1.94952 + 1.78868i 2.62593i −0.964085 + 2.84087i 1.39979i
152.20 0.730177 0.421568i 1.64300 + 0.548211i −0.644561 + 1.11641i 1.06280 1.84082i 1.43079 0.292347i −0.110845 + 2.64343i 2.77318i 2.39893 + 1.80143i 1.79216i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 152.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.m odd 6 1 inner
273.bf even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bf.b yes 64
3.b odd 2 1 inner 273.2.bf.b yes 64
7.d odd 6 1 273.2.r.b 64
13.c even 3 1 273.2.r.b 64
21.g even 6 1 273.2.r.b 64
39.i odd 6 1 273.2.r.b 64
91.m odd 6 1 inner 273.2.bf.b yes 64
273.bf even 6 1 inner 273.2.bf.b yes 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.r.b 64 7.d odd 6 1
273.2.r.b 64 13.c even 3 1
273.2.r.b 64 21.g even 6 1
273.2.r.b 64 39.i odd 6 1
273.2.bf.b yes 64 1.a even 1 1 trivial
273.2.bf.b yes 64 3.b odd 2 1 inner
273.2.bf.b yes 64 91.m odd 6 1 inner
273.2.bf.b yes 64 273.bf even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - 48 T_{2}^{62} + 1268 T_{2}^{60} - 23122 T_{2}^{58} + 321427 T_{2}^{56} - 3578895 T_{2}^{54} + \cdots + 134689$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.