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.
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 |
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 \(S_{2}^{\mathrm{new}}(855, [\chi])\):
\( T_{2}^{80} - 141 T_{2}^{76} + 60 T_{2}^{75} - 312 T_{2}^{73} + 12423 T_{2}^{72} - 8460 T_{2}^{71} + 1800 T_{2}^{70} + 33480 T_{2}^{69} - 695564 T_{2}^{68} + 678960 T_{2}^{67} - 205128 T_{2}^{66} - 2357784 T_{2}^{65} + \cdots + 160000 \) |
\( T_{7}^{40} - 4 T_{7}^{39} + 8 T_{7}^{38} + 12 T_{7}^{37} + 1250 T_{7}^{36} - 5284 T_{7}^{35} + 11208 T_{7}^{34} + 22932 T_{7}^{33} + 501447 T_{7}^{32} - 2193224 T_{7}^{31} + 4986832 T_{7}^{30} + \cdots + 259705859313664 \) |
\( T_{11}^{20} - 4 T_{11}^{19} - 88 T_{11}^{18} + 372 T_{11}^{17} + 2867 T_{11}^{16} - 13200 T_{11}^{15} - 43438 T_{11}^{14} + 228744 T_{11}^{13} + 311230 T_{11}^{12} - 2089912 T_{11}^{11} - 867984 T_{11}^{10} + 10249736 T_{11}^{9} + \cdots + 898320 \) |