Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,4,Mod(7,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 54.e (of order \(9\), degree \(6\), 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: | \(3.18610314031\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | 1.87939 | − | 0.684040i | −5.00873 | − | 1.38297i | 3.06418 | − | 2.57115i | −2.36712 | − | 13.4246i | −10.3593 | + | 0.827035i | −10.7136 | − | 8.98975i | 4.00000 | − | 6.92820i | 23.1748 | + | 13.8539i | −13.6317 | − | 23.6108i |
7.2 | 1.87939 | − | 0.684040i | −4.02802 | + | 3.28253i | 3.06418 | − | 2.57115i | 2.39299 | + | 13.5713i | −5.32482 | + | 8.92448i | 20.6081 | + | 17.2922i | 4.00000 | − | 6.92820i | 5.44995 | − | 26.4442i | 13.7807 | + | 23.8688i |
7.3 | 1.87939 | − | 0.684040i | 2.51736 | − | 4.54565i | 3.06418 | − | 2.57115i | 0.127307 | + | 0.721992i | 1.62168 | − | 10.2650i | 1.18538 | + | 0.994655i | 4.00000 | − | 6.92820i | −14.3258 | − | 22.8860i | 0.733130 | + | 1.26982i |
7.4 | 1.87939 | − | 0.684040i | 3.83244 | + | 3.50891i | 3.06418 | − | 2.57115i | −3.01146 | − | 17.0788i | 9.60286 | + | 3.97304i | 16.7015 | + | 14.0142i | 4.00000 | − | 6.92820i | 2.37516 | + | 26.8953i | −17.3423 | − | 30.0378i |
7.5 | 1.87939 | − | 0.684040i | 4.39270 | + | 2.77565i | 3.06418 | − | 2.57115i | 3.05422 | + | 17.3214i | 10.1542 | + | 2.21173i | −19.9833 | − | 16.7680i | 4.00000 | − | 6.92820i | 11.5916 | + | 24.3851i | 17.5886 | + | 30.4643i |
13.1 | −0.347296 | − | 1.96962i | −4.63338 | + | 2.35197i | −3.75877 | + | 1.36808i | 6.77242 | + | 5.68273i | 6.24164 | + | 8.30915i | 30.7336 | + | 11.1861i | 4.00000 | + | 6.92820i | 15.9365 | − | 21.7952i | 8.84076 | − | 15.3127i |
13.2 | −0.347296 | − | 1.96962i | −3.45245 | − | 3.88337i | −3.75877 | + | 1.36808i | 0.657259 | + | 0.551505i | −6.44973 | + | 8.14868i | −20.5288 | − | 7.47187i | 4.00000 | + | 6.92820i | −3.16120 | + | 26.8143i | 0.857990 | − | 1.48608i |
13.3 | −0.347296 | − | 1.96962i | −1.19733 | + | 5.05632i | −3.75877 | + | 1.36808i | −9.84993 | − | 8.26508i | 10.3748 | + | 0.602240i | −26.8516 | − | 9.77320i | 4.00000 | + | 6.92820i | −24.1328 | − | 12.1082i | −12.8582 | + | 22.2710i |
13.4 | −0.347296 | − | 1.96962i | 3.93867 | + | 3.38924i | −3.75877 | + | 1.36808i | 7.73816 | + | 6.49309i | 5.30761 | − | 8.93473i | 7.54967 | + | 2.74785i | 4.00000 | + | 6.92820i | 4.02617 | + | 26.6981i | 10.1015 | − | 17.4962i |
13.5 | −0.347296 | − | 1.96962i | 4.23116 | − | 3.01618i | −3.75877 | + | 1.36808i | −14.9069 | − | 12.5084i | −7.41017 | − | 7.28624i | 11.7781 | + | 4.28689i | 4.00000 | + | 6.92820i | 8.80536 | − | 25.5238i | −19.4596 | + | 33.7050i |
25.1 | −0.347296 | + | 1.96962i | −4.63338 | − | 2.35197i | −3.75877 | − | 1.36808i | 6.77242 | − | 5.68273i | 6.24164 | − | 8.30915i | 30.7336 | − | 11.1861i | 4.00000 | − | 6.92820i | 15.9365 | + | 21.7952i | 8.84076 | + | 15.3127i |
25.2 | −0.347296 | + | 1.96962i | −3.45245 | + | 3.88337i | −3.75877 | − | 1.36808i | 0.657259 | − | 0.551505i | −6.44973 | − | 8.14868i | −20.5288 | + | 7.47187i | 4.00000 | − | 6.92820i | −3.16120 | − | 26.8143i | 0.857990 | + | 1.48608i |
25.3 | −0.347296 | + | 1.96962i | −1.19733 | − | 5.05632i | −3.75877 | − | 1.36808i | −9.84993 | + | 8.26508i | 10.3748 | − | 0.602240i | −26.8516 | + | 9.77320i | 4.00000 | − | 6.92820i | −24.1328 | + | 12.1082i | −12.8582 | − | 22.2710i |
25.4 | −0.347296 | + | 1.96962i | 3.93867 | − | 3.38924i | −3.75877 | − | 1.36808i | 7.73816 | − | 6.49309i | 5.30761 | + | 8.93473i | 7.54967 | − | 2.74785i | 4.00000 | − | 6.92820i | 4.02617 | − | 26.6981i | 10.1015 | + | 17.4962i |
25.5 | −0.347296 | + | 1.96962i | 4.23116 | + | 3.01618i | −3.75877 | − | 1.36808i | −14.9069 | + | 12.5084i | −7.41017 | + | 7.28624i | 11.7781 | − | 4.28689i | 4.00000 | − | 6.92820i | 8.80536 | + | 25.5238i | −19.4596 | − | 33.7050i |
31.1 | 1.87939 | + | 0.684040i | −5.00873 | + | 1.38297i | 3.06418 | + | 2.57115i | −2.36712 | + | 13.4246i | −10.3593 | − | 0.827035i | −10.7136 | + | 8.98975i | 4.00000 | + | 6.92820i | 23.1748 | − | 13.8539i | −13.6317 | + | 23.6108i |
31.2 | 1.87939 | + | 0.684040i | −4.02802 | − | 3.28253i | 3.06418 | + | 2.57115i | 2.39299 | − | 13.5713i | −5.32482 | − | 8.92448i | 20.6081 | − | 17.2922i | 4.00000 | + | 6.92820i | 5.44995 | + | 26.4442i | 13.7807 | − | 23.8688i |
31.3 | 1.87939 | + | 0.684040i | 2.51736 | + | 4.54565i | 3.06418 | + | 2.57115i | 0.127307 | − | 0.721992i | 1.62168 | + | 10.2650i | 1.18538 | − | 0.994655i | 4.00000 | + | 6.92820i | −14.3258 | + | 22.8860i | 0.733130 | − | 1.26982i |
31.4 | 1.87939 | + | 0.684040i | 3.83244 | − | 3.50891i | 3.06418 | + | 2.57115i | −3.01146 | + | 17.0788i | 9.60286 | − | 3.97304i | 16.7015 | − | 14.0142i | 4.00000 | + | 6.92820i | 2.37516 | − | 26.8953i | −17.3423 | + | 30.0378i |
31.5 | 1.87939 | + | 0.684040i | 4.39270 | − | 2.77565i | 3.06418 | + | 2.57115i | 3.05422 | − | 17.3214i | 10.1542 | − | 2.21173i | −19.9833 | + | 16.7680i | 4.00000 | + | 6.92820i | 11.5916 | − | 24.3851i | 17.5886 | − | 30.4643i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 54.4.e.b | ✓ | 30 |
3.b | odd | 2 | 1 | 162.4.e.b | 30 | ||
27.e | even | 9 | 1 | inner | 54.4.e.b | ✓ | 30 |
27.e | even | 9 | 1 | 1458.4.a.i | 15 | ||
27.f | odd | 18 | 1 | 162.4.e.b | 30 | ||
27.f | odd | 18 | 1 | 1458.4.a.j | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.4.e.b | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
54.4.e.b | ✓ | 30 | 27.e | even | 9 | 1 | inner |
162.4.e.b | 30 | 3.b | odd | 2 | 1 | ||
162.4.e.b | 30 | 27.f | odd | 18 | 1 | ||
1458.4.a.i | 15 | 27.e | even | 9 | 1 | ||
1458.4.a.j | 15 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{30} + 12 T_{5}^{29} + 288 T_{5}^{28} + 1077 T_{5}^{27} - 7965 T_{5}^{26} - 1743363 T_{5}^{25} + 7544988 T_{5}^{24} - 107176932 T_{5}^{23} + 5239574748 T_{5}^{22} + 56186887842 T_{5}^{21} + \cdots + 54\!\cdots\!96 \)
acting on \(S_{4}^{\mathrm{new}}(54, [\chi])\).