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Properties

Label 855.2.bj.a
Level $855$
Weight $2$
Character orbit 855.bj
Analytic conductor $6.827$
Analytic rank $0$
Dimension $108$
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] = [855,2,Mod(229,855)]
 
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, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.bj (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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 54 q^{4} - q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q + 54 q^{4} - q^{5} - 4 q^{9} + 30 q^{11} - 22 q^{14} - q^{15} - 42 q^{16} - 108 q^{19} + 4 q^{20} - 16 q^{21} - 6 q^{24} + 15 q^{25} - 88 q^{26} - 54 q^{29} + 14 q^{30} - 6 q^{31} - 2 q^{35} + 92 q^{36} + 22 q^{39} + 42 q^{41} + 168 q^{44} - 22 q^{45} - 12 q^{46} + 48 q^{49} - 20 q^{50} - 8 q^{51} - 36 q^{54} - 18 q^{55} + 80 q^{56} - 48 q^{59} + 69 q^{60} + 6 q^{61} - 60 q^{64} + 2 q^{65} - 102 q^{66} - 94 q^{69} - 33 q^{70} - 148 q^{71} - 80 q^{74} + 3 q^{75} - 54 q^{76} - 18 q^{79} + 50 q^{80} + 4 q^{81} + 76 q^{84} - 12 q^{85} + 40 q^{86} + 132 q^{89} - 73 q^{90} - 12 q^{91} - 30 q^{94} + q^{95} - 38 q^{96} + 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 −2.33739 + 1.34949i −0.627589 − 1.61435i 2.64226 − 4.57652i −1.17027 − 1.90538i 3.64547 + 2.92644i 2.11886 − 1.22333i 8.86484i −2.21226 + 2.02630i 5.30667 + 2.87433i
229.2 −2.33181 + 1.34627i −1.72863 + 0.108757i 2.62488 − 4.54643i 2.05980 + 0.870190i 3.88442 − 2.58081i 1.07992 − 0.623491i 8.75013i 2.97634 − 0.376002i −5.97456 + 0.743929i
229.3 −2.30033 + 1.32810i 0.525389 + 1.65044i 2.52769 − 4.37809i 0.0597262 + 2.23527i −3.40052 − 3.09880i 1.12881 − 0.651719i 8.11568i −2.44793 + 1.73425i −3.10605 − 5.06254i
229.4 −2.13508 + 1.23269i 1.55724 − 0.758282i 2.03903 − 3.53171i −1.90202 + 1.17572i −2.39011 + 3.53859i 3.60439 − 2.08099i 5.12322i 1.85002 − 2.36166i 2.61167 − 4.85484i
229.5 −2.08196 + 1.20202i 1.73204 + 0.00449359i 1.88969 − 3.27305i 2.22938 − 0.172752i −3.61144 + 2.07259i −3.74874 + 2.16433i 4.27771i 2.99996 + 0.0155662i −4.43383 + 3.03942i
229.6 −2.03397 + 1.17431i −1.42318 − 0.987199i 1.75803 − 3.04500i −2.23350 − 0.107173i 4.05399 + 0.336675i −1.89384 + 1.09341i 3.56067i 1.05088 + 2.80992i 4.66873 − 2.40484i
229.7 −1.98761 + 1.14755i −1.05023 + 1.37732i 1.63372 − 2.82969i 0.109628 − 2.23338i 0.506903 − 3.94276i 2.29586 − 1.32551i 2.90890i −0.794037 − 2.89301i 2.34501 + 4.56488i
229.8 −1.96957 + 1.13713i 1.52332 + 0.824322i 1.58614 − 2.74728i −1.91195 + 1.15950i −3.93765 + 0.108652i −2.16426 + 1.24954i 2.66609i 1.64099 + 2.51141i 2.44723 − 4.45786i
229.9 −1.78668 + 1.03154i 1.26949 + 1.17830i 1.12815 − 1.95401i 2.14421 − 0.634323i −3.48364 − 0.795717i 3.73870 − 2.15854i 0.528769i 0.223212 + 2.99168i −3.17669 + 3.34517i
229.10 −1.68432 + 0.972440i −0.0568088 + 1.73112i 0.891278 − 1.54374i −1.69860 − 1.45422i −1.58772 − 2.97099i −2.30424 + 1.33035i − 0.422902i −2.99355 − 0.196686i 4.27512 + 0.797583i
229.11 −1.67132 + 0.964937i 0.217537 − 1.71834i 0.862209 − 1.49339i 2.12525 + 0.695196i 1.29451 + 3.08180i 0.206773 − 0.119380i − 0.531840i −2.90536 − 0.747604i −4.22280 + 0.888841i
229.12 −1.59242 + 0.919383i −0.275161 − 1.71005i 0.690531 − 1.19603i 0.659497 − 2.13660i 2.01037 + 2.47014i −3.47450 + 2.00601i − 1.13808i −2.84857 + 0.941082i 0.914160 + 4.00869i
229.13 −1.44886 + 0.836501i −1.58766 + 0.692351i 0.399468 − 0.691900i 2.22921 − 0.174981i 1.72114 − 2.33120i −0.542888 + 0.313437i − 2.00938i 2.04130 − 2.19843i −3.08345 + 2.11826i
229.14 −1.33517 + 0.770859i 0.0411841 − 1.73156i 0.188448 − 0.326401i −0.720092 + 2.11695i 1.27980 + 2.34367i −0.234735 + 0.135524i − 2.50237i −2.99661 − 0.142626i −0.670426 − 3.38157i
229.15 −1.27887 + 0.738354i −0.618632 + 1.61781i 0.0903329 − 0.156461i 0.924312 + 2.03609i −0.403365 − 2.52573i 3.62661 − 2.09383i − 2.68662i −2.23459 − 2.00165i −2.68542 − 1.92141i
229.16 −1.27162 + 0.734170i −1.70779 − 0.288878i 0.0780121 − 0.135121i −2.15601 − 0.592964i 2.38375 − 0.886467i 3.04023 − 1.75528i − 2.70758i 2.83310 + 0.986685i 3.17697 − 0.828856i
229.17 −1.18251 + 0.682722i 1.25738 + 1.19122i −0.0677800 + 0.117398i −1.20597 − 1.88298i −2.30013 − 0.550188i −0.446402 + 0.257730i − 2.91599i 0.161997 + 2.99562i 2.71163 + 1.40330i
229.18 −1.04422 + 0.602879i −1.63110 + 0.582681i −0.273074 + 0.472979i −1.04628 + 1.97618i 1.35193 − 1.59180i −1.60882 + 0.928850i − 3.07004i 2.32097 − 1.90082i −0.0988516 − 2.69434i
229.19 −0.781936 + 0.451451i 1.09298 − 1.34365i −0.592384 + 1.02604i 1.23353 − 1.86505i −0.248052 + 1.54407i 0.401142 − 0.231600i − 2.87553i −0.610778 − 2.93717i −0.122561 + 2.01523i
229.20 −0.775707 + 0.447854i 0.773675 − 1.54965i −0.598853 + 1.03724i −2.19454 + 0.428965i 0.0938741 + 1.54857i 0.879915 − 0.508019i − 2.86421i −1.80285 − 2.39786i 1.51020 − 1.31558i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.bj.a ✓ 108
5.b even 2 1 inner 855.2.bj.a ✓ 108
9.c even 3 1 inner 855.2.bj.a ✓ 108
45.j even 6 1 inner 855.2.bj.a ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.bj.a ✓ 108 1.a even 1 1 trivial
855.2.bj.a ✓ 108 5.b even 2 1 inner
855.2.bj.a ✓ 108 9.c even 3 1 inner
855.2.bj.a ✓ 108 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} - 81 T_{2}^{106} + 3480 T_{2}^{104} - 103215 T_{2}^{102} + 2348505 T_{2}^{100} + \cdots + 5887339441 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display