Properties

Label 855.2.cj.g
Level $855$
Weight $2$
Character orbit 855.cj
Analytic conductor $6.827$
Analytic rank $0$
Dimension $80$
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(217,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.cj (of order \(12\), degree \(4\), 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: \(20\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{5} + 8 q^{7} + 16 q^{11} + 28 q^{16} + 4 q^{17} + 80 q^{20} - 24 q^{22} - 8 q^{23} + 4 q^{25} + 32 q^{26} - 28 q^{28} - 12 q^{32} + 44 q^{35} + 96 q^{38} + 132 q^{40} - 72 q^{41} + 12 q^{47} + 36 q^{53} + 8 q^{55} + 40 q^{58} - 40 q^{61} + 4 q^{62} - 88 q^{68} - 24 q^{70} + 40 q^{73} + 24 q^{76} + 40 q^{77} - 24 q^{80} - 36 q^{82} + 8 q^{83} + 80 q^{85} - 48 q^{86} + 72 q^{91} - 12 q^{92} - 48 q^{95} + 12 q^{97} - 96 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1 −0.710164 2.65037i 0 −4.78806 + 2.76439i 0.243817 2.22274i 0 −1.05114 1.05114i 6.84655 + 6.84655i 0 −6.06422 + 0.932302i
217.2 −0.634523 2.36807i 0 −3.47309 + 2.00519i 1.77334 + 1.36209i 0 −1.80480 1.80480i 3.48510 + 3.48510i 0 2.10030 5.06366i
217.3 −0.618292 2.30750i 0 −3.21020 + 1.85341i −0.884317 2.05377i 0 2.40286 + 2.40286i 2.88317 + 2.88317i 0 −4.19231 + 3.31039i
217.4 −0.559318 2.08740i 0 −2.31237 + 1.33505i −1.67449 + 1.48192i 0 0.447425 + 0.447425i 1.02396 + 1.02396i 0 4.02994 + 2.66648i
217.5 −0.428697 1.59992i 0 −0.643905 + 0.371759i −1.12636 + 1.93166i 0 1.35415 + 1.35415i −1.47162 1.47162i 0 3.57336 + 0.973994i
217.6 −0.421488 1.57301i 0 −0.564673 + 0.326014i 2.14109 0.644768i 0 2.33530 + 2.33530i −1.55223 1.55223i 0 −1.91667 3.09621i
217.7 −0.253957 0.947782i 0 0.898255 0.518608i 1.11218 + 1.93986i 0 −3.50133 3.50133i −2.10729 2.10729i 0 1.55612 1.54674i
217.8 −0.194313 0.725187i 0 1.24391 0.718173i −0.868818 2.06038i 0 −0.801851 0.801851i −1.82427 1.82427i 0 −1.32534 + 1.03041i
217.9 −0.0833245 0.310971i 0 1.64229 0.948177i 1.96489 + 1.06733i 0 0.966282 + 0.966282i −0.886993 0.886993i 0 0.168187 0.699960i
217.10 0.0124124 + 0.0463238i 0 1.73006 0.998850i −1.58989 + 1.57234i 0 −3.34407 3.34407i 0.135567 + 0.135567i 0 −0.0925711 0.0541331i
217.11 0.124631 + 0.465131i 0 1.53124 0.884060i 0.780151 2.09556i 0 2.89550 + 2.89550i 1.28304 + 1.28304i 0 1.07194 + 0.101700i
217.12 0.209100 + 0.780371i 0 1.16679 0.673649i 0.677483 + 2.13097i 0 2.62791 + 2.62791i 1.91222 + 1.91222i 0 −1.52128 + 0.974273i
217.13 0.236978 + 0.884412i 0 1.00602 0.580828i −2.23048 + 0.157915i 0 −0.753560 0.753560i 2.04697 + 2.04697i 0 −0.668237 1.93525i
217.14 0.298381 + 1.11357i 0 0.581034 0.335460i 2.22477 0.224504i 0 −1.90219 1.90219i 2.17732 + 2.17732i 0 0.913832 + 2.41046i
217.15 0.331049 + 1.23549i 0 0.315203 0.181983i −1.16826 1.90661i 0 −1.68175 1.68175i 2.13807 + 2.13807i 0 1.96885 2.07456i
217.16 0.413118 + 1.54178i 0 −0.474360 + 0.273872i 2.22181 + 0.252125i 0 −1.02257 1.02257i 1.63910 + 1.63910i 0 0.529149 + 3.52969i
217.17 0.475128 + 1.77320i 0 −1.18645 + 0.684999i 0.205509 + 2.22660i 0 0.848094 + 0.848094i 0.817790 + 0.817790i 0 −3.85058 + 1.42233i
217.18 0.492353 + 1.83748i 0 −1.40189 + 0.809380i −0.491967 2.18128i 0 3.08353 + 3.08353i 0.512817 + 0.512817i 0 3.76584 1.97794i
217.19 0.643610 + 2.40199i 0 −3.62325 + 2.09189i −2.08186 + 0.815992i 0 −1.93358 1.93358i −3.83989 3.83989i 0 −3.29991 4.47543i
217.20 0.667315 + 2.49045i 0 −4.02500 + 2.32384i −2.22858 + 0.182854i 0 2.83578 + 2.83578i −4.82708 4.82708i 0 −1.94255 5.42815i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.cj.g 80
3.b odd 2 1 285.2.x.a 80
5.c odd 4 1 inner 855.2.cj.g 80
15.e even 4 1 285.2.x.a 80
19.d odd 6 1 inner 855.2.cj.g 80
57.f even 6 1 285.2.x.a 80
95.l even 12 1 inner 855.2.cj.g 80
285.w odd 12 1 285.2.x.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.x.a 80 3.b odd 2 1
285.2.x.a 80 15.e even 4 1
285.2.x.a 80 57.f even 6 1
285.2.x.a 80 285.w odd 12 1
855.2.cj.g 80 1.a even 1 1 trivial
855.2.cj.g 80 5.c odd 4 1 inner
855.2.cj.g 80 19.d odd 6 1 inner
855.2.cj.g 80 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):

\( T_{2}^{80} - 141 T_{2}^{76} + 60 T_{2}^{75} - 312 T_{2}^{73} + 12423 T_{2}^{72} - 8460 T_{2}^{71} + 1800 T_{2}^{70} + 33480 T_{2}^{69} - 695564 T_{2}^{68} + 678960 T_{2}^{67} - 205128 T_{2}^{66} - 2357784 T_{2}^{65} + \cdots + 160000 \) Copy content Toggle raw display
\( T_{7}^{40} - 4 T_{7}^{39} + 8 T_{7}^{38} + 12 T_{7}^{37} + 1250 T_{7}^{36} - 5284 T_{7}^{35} + 11208 T_{7}^{34} + 22932 T_{7}^{33} + 501447 T_{7}^{32} - 2193224 T_{7}^{31} + 4986832 T_{7}^{30} + \cdots + 259705859313664 \) Copy content Toggle raw display
\( T_{11}^{20} - 4 T_{11}^{19} - 88 T_{11}^{18} + 372 T_{11}^{17} + 2867 T_{11}^{16} - 13200 T_{11}^{15} - 43438 T_{11}^{14} + 228744 T_{11}^{13} + 311230 T_{11}^{12} - 2089912 T_{11}^{11} - 867984 T_{11}^{10} + 10249736 T_{11}^{9} + \cdots + 898320 \) Copy content Toggle raw display