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Properties

Label 855.2.l.b
Level $855$
Weight $2$
Character orbit 855.l
Analytic conductor $6.827$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(391,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 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("855.391");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.l (of order \(3\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 2 q^{3} + 80 q^{4} + 40 q^{5} + 2 q^{6} + q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 2 q^{3} + 80 q^{4} + 40 q^{5} + 2 q^{6} + q^{7} - 2 q^{9} + 2 q^{10} - q^{11} - 6 q^{12} - 18 q^{13} - q^{15} + 88 q^{16} + 6 q^{17} - 15 q^{18} + 9 q^{19} + 40 q^{20} - 4 q^{21} + 9 q^{22} - 30 q^{24} - 40 q^{25} - 8 q^{26} - 2 q^{27} + 13 q^{28} - 3 q^{29} + 7 q^{30} + 24 q^{31} + 4 q^{32} + q^{33} + q^{34} - q^{35} + 12 q^{36} - 40 q^{37} + 14 q^{38} + 23 q^{39} - 3 q^{41} - 29 q^{42} - 30 q^{43} - 34 q^{44} - q^{45} - 46 q^{46} + 6 q^{47} - 5 q^{48} - 33 q^{49} - 2 q^{50} + 37 q^{51} - 56 q^{52} - 16 q^{53} - 23 q^{54} + q^{55} + 20 q^{56} + 4 q^{58} - 2 q^{59} - 3 q^{60} + 7 q^{61} - 8 q^{62} - 41 q^{63} + 180 q^{64} - 9 q^{65} - 30 q^{66} - 102 q^{67} + 11 q^{68} - 24 q^{69} + 14 q^{71} + 22 q^{72} - 8 q^{73} + 24 q^{74} + q^{75} + 12 q^{76} + 10 q^{77} - q^{78} - 42 q^{79} + 44 q^{80} - 22 q^{81} + 5 q^{82} + 3 q^{83} - 131 q^{84} + 12 q^{85} + 14 q^{86} - 22 q^{87} + 46 q^{88} - 18 q^{89} - 15 q^{90} + 24 q^{91} - 54 q^{92} - 24 q^{93} + 42 q^{94} - 160 q^{96} - 42 q^{97} + 5 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1 −2.79898 −0.207146 + 1.71962i 5.83429 0.500000 + 0.866025i 0.579797 − 4.81318i 0.832202 + 1.44142i −10.7321 −2.91418 − 0.712424i −1.39949 − 2.42399i
391.2 −2.64837 1.01942 − 1.40028i 5.01387 0.500000 + 0.866025i −2.69979 + 3.70847i −0.602519 − 1.04359i −7.98184 −0.921583 − 2.85494i −1.32419 − 2.29356i
391.3 −2.52323 −1.51445 − 0.840498i 4.36668 0.500000 + 0.866025i 3.82131 + 2.12077i −0.953622 − 1.65172i −5.97168 1.58713 + 2.54579i −1.26161 − 2.18518i
391.4 −2.52202 1.70159 + 0.323388i 4.36059 0.500000 + 0.866025i −4.29145 − 0.815590i 1.86441 + 3.22926i −5.95344 2.79084 + 1.10055i −1.26101 − 2.18413i
391.5 −2.16307 −1.36343 + 1.06821i 2.67885 0.500000 + 0.866025i 2.94918 − 2.31060i −2.59940 − 4.50229i −1.46840 0.717865 − 2.91285i −1.08153 − 1.87327i
391.6 −2.10481 1.65239 − 0.519223i 2.43024 0.500000 + 0.866025i −3.47798 + 1.09287i −1.47954 − 2.56263i −0.905578 2.46082 − 1.71592i −1.05241 − 1.82282i
391.7 −2.02544 0.363191 + 1.69354i 2.10240 0.500000 + 0.866025i −0.735621 − 3.43017i 0.517437 + 0.896227i −0.207415 −2.73619 + 1.23016i −1.01272 − 1.75408i
391.8 −1.96735 −1.67865 + 0.426772i 1.87048 0.500000 + 0.866025i 3.30250 − 0.839611i 0.973148 + 1.68554i 0.254808 2.63573 − 1.43280i −0.983677 − 1.70378i
391.9 −1.87089 −0.923627 − 1.46523i 1.50021 0.500000 + 0.866025i 1.72800 + 2.74129i 0.563779 + 0.976493i 0.935045 −1.29383 + 2.70666i −0.935443 − 1.62023i
391.10 −1.57328 −1.70190 − 0.321745i 0.475207 0.500000 + 0.866025i 2.67757 + 0.506194i 1.95687 + 3.38939i 2.39892 2.79296 + 1.09516i −0.786640 − 1.36250i
391.11 −1.47897 −0.0923620 − 1.72959i 0.187347 0.500000 + 0.866025i 0.136600 + 2.55800i 0.821457 + 1.42281i 2.68086 −2.98294 + 0.319496i −0.739484 − 1.28082i
391.12 −1.19152 1.10342 − 1.33509i −0.580279 0.500000 + 0.866025i −1.31475 + 1.59078i −0.891236 − 1.54367i 3.07446 −0.564918 − 2.94633i −0.595760 − 1.03189i
391.13 −1.16112 1.57616 + 0.718136i −0.651797 0.500000 + 0.866025i −1.83011 − 0.833844i −0.607129 − 1.05158i 3.07906 1.96856 + 2.26380i −0.580561 − 1.00556i
391.14 −0.874146 −0.683721 + 1.59139i −1.23587 0.500000 + 0.866025i 0.597672 − 1.39111i −0.655771 − 1.13583i 2.82862 −2.06505 − 2.17613i −0.437073 − 0.757033i
391.15 −0.839542 −0.980642 + 1.42771i −1.29517 0.500000 + 0.866025i 0.823289 − 1.19862i 0.805223 + 1.39469i 2.76643 −1.07668 − 2.80013i −0.419771 − 0.727064i
391.16 −0.614024 0.922020 + 1.46625i −1.62297 0.500000 + 0.866025i −0.566142 − 0.900310i 2.40213 + 4.16062i 2.22459 −1.29976 + 2.70382i −0.307012 − 0.531760i
391.17 −0.469166 −1.65099 − 0.523685i −1.77988 0.500000 + 0.866025i 0.774587 + 0.245695i −1.69218 − 2.93093i 1.77339 2.45151 + 1.72919i −0.234583 − 0.406310i
391.18 −0.421656 1.54237 − 0.788096i −1.82221 0.500000 + 0.866025i −0.650350 + 0.332306i 1.78422 + 3.09036i 1.61166 1.75781 − 2.43107i −0.210828 − 0.365165i
391.19 −0.385961 1.31964 + 1.12186i −1.85103 0.500000 + 0.866025i −0.509328 − 0.432992i −2.43700 − 4.22101i 1.48635 0.482882 + 2.96088i −0.192981 − 0.334252i
391.20 −0.124511 −1.38476 − 1.04041i −1.98450 0.500000 + 0.866025i 0.172418 + 0.129543i −0.716284 − 1.24064i 0.496115 0.835108 + 2.88142i −0.0622557 − 0.107830i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.l.b yes 80
9.c even 3 1 855.2.j.a ✓ 80
19.c even 3 1 855.2.j.a ✓ 80
171.h even 3 1 inner 855.2.l.b yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.j.a ✓ 80 9.c even 3 1
855.2.j.a ✓ 80 19.c even 3 1
855.2.l.b yes 80 1.a even 1 1 trivial
855.2.l.b yes 80 171.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 2 T_{2}^{39} - 58 T_{2}^{38} + 116 T_{2}^{37} + 1535 T_{2}^{36} - 3072 T_{2}^{35} + \cdots + 9819 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display