Properties

Label 840.2.bz.b
Level $840$
Weight $2$
Character orbit 840.bz
Analytic conductor $6.707$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(19,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bz (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{3} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{3} - 48 q^{9} + 5 q^{10} + 14 q^{14} + 4 q^{16} + 22 q^{20} - 96 q^{27} - 4 q^{28} + 13 q^{30} - 30 q^{32} - 16 q^{35} - 12 q^{38} - 7 q^{40} - 2 q^{42} - 16 q^{44} - 22 q^{46} + 8 q^{48} + 12 q^{50} - 6 q^{52} + 26 q^{56} + 30 q^{58} + 48 q^{59} + 11 q^{60} + 32 q^{62} - 24 q^{64} - 60 q^{65} - 18 q^{66} + 38 q^{68} + 21 q^{70} - 10 q^{74} + 55 q^{80} - 48 q^{81} - 18 q^{82} + 10 q^{84} + 8 q^{86} + 84 q^{88} + 8 q^{90} + 16 q^{91} + 18 q^{94} - 30 q^{96} - 48 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.39955 + 0.203137i 0.500000 + 0.866025i 1.91747 0.568599i −0.835606 + 2.07407i −0.875696 1.11048i −2.10664 + 1.60064i −2.56809 + 1.18529i −0.500000 + 0.866025i 0.748152 3.07250i
19.2 −1.39446 0.235554i 0.500000 + 0.866025i 1.88903 + 0.656942i −0.911079 + 2.04204i −0.493233 1.32541i 2.41279 1.08556i −2.47943 1.36105i −0.500000 + 0.866025i 1.75147 2.63293i
19.3 −1.39442 0.235769i 0.500000 + 0.866025i 1.88883 + 0.657523i 1.85007 1.25588i −0.493029 1.32549i 2.44325 + 1.01516i −2.47880 1.36219i −0.500000 + 0.866025i −2.87588 + 1.31503i
19.4 −1.38940 0.263748i 0.500000 + 0.866025i 1.86087 + 0.732904i 0.519269 2.17494i −0.466288 1.33513i −1.58536 2.11817i −2.39220 1.50910i −0.500000 + 0.866025i −1.29511 + 2.88491i
19.5 −1.38671 + 0.277563i 0.500000 + 0.866025i 1.84592 0.769797i −1.58066 1.58162i −0.933730 1.06214i 0.458574 2.60571i −2.34608 + 1.57984i −0.500000 + 0.866025i 2.63091 + 1.75451i
19.6 −1.34731 + 0.429821i 0.500000 + 0.866025i 1.63051 1.15821i 2.17016 0.538885i −1.04589 0.951897i −2.41852 + 1.07274i −1.69898 + 2.26130i −0.500000 + 0.866025i −2.69226 + 1.65883i
19.7 −1.32928 + 0.482718i 0.500000 + 0.866025i 1.53397 1.28333i 0.805895 + 2.08579i −1.08269 0.909830i −0.514752 2.59519i −1.41958 + 2.44638i −0.500000 + 0.866025i −2.07811 2.38358i
19.8 −1.24841 0.664442i 0.500000 + 0.866025i 1.11703 + 1.65899i 1.77230 + 1.36343i −0.0487790 1.41337i −1.63954 2.07652i −0.292210 2.81329i −0.500000 + 0.866025i −1.30663 2.87971i
19.9 −1.19963 0.748930i 0.500000 + 0.866025i 0.878208 + 1.79687i −1.77230 1.36343i 0.0487790 1.41337i 1.63954 + 2.07652i 0.292210 2.81329i −0.500000 + 0.866025i 1.10498 + 2.96294i
19.10 −1.11818 + 0.865837i 0.500000 + 0.866025i 0.500651 1.93632i −2.23603 0.0125931i −1.30893 0.535453i 0.842207 + 2.50812i 1.11672 + 2.59864i −0.500000 + 0.866025i 2.51119 1.92196i
19.11 −1.09295 + 0.897480i 0.500000 + 0.866025i 0.389060 1.96179i 1.75772 + 1.38219i −1.32371 0.497779i 2.38660 + 1.14199i 1.33545 + 2.49331i −0.500000 + 0.866025i −3.16157 + 0.0668608i
19.12 −0.923113 1.07138i 0.500000 + 0.866025i −0.295723 + 1.97802i −0.519269 + 2.17494i 0.466288 1.33513i 1.58536 + 2.11817i 2.39220 1.50910i −0.500000 + 0.866025i 2.80954 1.45138i
19.13 −0.922863 + 1.07160i 0.500000 + 0.866025i −0.296648 1.97788i −2.09232 + 0.788801i −1.38946 0.263423i 0.635767 2.56823i 2.39326 + 1.50742i −0.500000 + 0.866025i 1.08564 2.97008i
19.14 −0.901393 1.08972i 0.500000 + 0.866025i −0.374981 + 1.96453i −1.85007 + 1.25588i 0.493029 1.32549i −2.44325 1.01516i 2.47880 1.36219i −0.500000 + 0.866025i 3.03620 + 0.884025i
19.15 −0.901225 1.08986i 0.500000 + 0.866025i −0.375586 + 1.96442i 0.911079 2.04204i 0.493233 1.32541i −2.41279 + 1.08556i 2.47943 1.36105i −0.500000 + 0.866025i −3.04663 + 0.847392i
19.16 −0.808138 + 1.16057i 0.500000 + 0.866025i −0.693826 1.87579i −0.750153 2.10648i −1.40915 0.119585i −2.62822 0.304041i 2.73769 + 0.710669i −0.500000 + 0.866025i 3.05094 + 0.831727i
19.17 −0.687198 + 1.23603i 0.500000 + 0.866025i −1.05552 1.69879i 0.760135 2.10290i −1.41403 + 0.0228815i 2.38144 1.15271i 2.82510 0.137241i −0.500000 + 0.866025i 2.07688 + 2.38466i
19.18 −0.523853 1.31361i 0.500000 + 0.866025i −1.45116 + 1.37628i 0.835606 2.07407i 0.875696 1.11048i 2.10664 1.60064i 2.56809 + 1.18529i −0.500000 + 0.866025i −3.16226 0.0111565i
19.19 −0.452977 1.33971i 0.500000 + 0.866025i −1.58962 + 1.21371i 1.58066 + 1.58162i 0.933730 1.06214i −0.458574 + 2.60571i 2.34608 + 1.57984i −0.500000 + 0.866025i 1.40290 2.83406i
19.20 −0.301421 1.38172i 0.500000 + 0.866025i −1.81829 + 0.832957i −2.17016 + 0.538885i 1.04589 0.951897i 2.41852 1.07274i 1.69898 + 2.26130i −0.500000 + 0.866025i 1.39872 + 2.83612i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
35.i odd 6 1 inner
280.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.bz.b yes 96
5.b even 2 1 840.2.bz.a 96
7.d odd 6 1 840.2.bz.a 96
8.d odd 2 1 inner 840.2.bz.b yes 96
35.i odd 6 1 inner 840.2.bz.b yes 96
40.e odd 2 1 840.2.bz.a 96
56.m even 6 1 840.2.bz.a 96
280.ba even 6 1 inner 840.2.bz.b yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bz.a 96 5.b even 2 1
840.2.bz.a 96 7.d odd 6 1
840.2.bz.a 96 40.e odd 2 1
840.2.bz.a 96 56.m even 6 1
840.2.bz.b yes 96 1.a even 1 1 trivial
840.2.bz.b yes 96 8.d odd 2 1 inner
840.2.bz.b yes 96 35.i odd 6 1 inner
840.2.bz.b yes 96 280.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{48} + 216 T_{17}^{46} - 176 T_{17}^{45} + 26976 T_{17}^{44} - 34492 T_{17}^{43} + 2274984 T_{17}^{42} - 3838688 T_{17}^{41} + 143959340 T_{17}^{40} - 284664672 T_{17}^{39} + 7089264040 T_{17}^{38} + \cdots + 10\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display