Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,3,Mod(15,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.15");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 211 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 211.e (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: | \(5.74933357800\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Relative dimension: | \(34\) 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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −3.28947 | + | 1.89917i | 2.53131 | − | 1.46145i | 5.21373 | − | 9.03044i | 3.33934 | −5.55111 | + | 9.61480i | 2.80731 | − | 1.62080i | 24.4137i | −0.228311 | + | 0.395446i | −10.9847 | + | 6.34200i | ||||
15.2 | −3.06100 | + | 1.76727i | −3.63546 | + | 2.09894i | 4.24649 | − | 7.35513i | −3.78107 | 7.41877 | − | 12.8497i | 3.69317 | − | 2.13225i | 15.8806i | 4.31106 | − | 7.46697i | 11.5739 | − | 6.68217i | ||||
15.3 | −2.96716 | + | 1.71309i | −2.59992 | + | 1.50106i | 3.86935 | − | 6.70190i | 5.80352 | 5.14291 | − | 8.90778i | −3.19116 | + | 1.84241i | 12.8094i | 0.00637874 | − | 0.0110483i | −17.2200 | + | 9.94195i | ||||
15.4 | −2.77079 | + | 1.59971i | 1.49990 | − | 0.865968i | 3.11817 | − | 5.40083i | −5.11467 | −2.77060 | + | 4.79883i | 6.27090 | − | 3.62051i | 7.15502i | −3.00020 | + | 5.19650i | 14.1717 | − | 8.18201i | ||||
15.5 | −2.72956 | + | 1.57591i | 5.03303 | − | 2.90582i | 2.96701 | − | 5.13902i | −3.99896 | −9.15865 | + | 15.8632i | −9.94373 | + | 5.74102i | 6.09572i | 12.3876 | − | 21.4559i | 10.9154 | − | 6.30202i | ||||
15.6 | −2.45402 | + | 1.41683i | 0.156697 | − | 0.0904691i | 2.01482 | − | 3.48977i | −8.94790 | −0.256359 | + | 0.444027i | −7.84079 | + | 4.52688i | 0.0840007i | −4.48363 | + | 7.76588i | 21.9584 | − | 12.6777i | ||||
15.7 | −2.36498 | + | 1.36542i | 1.22716 | − | 0.708503i | 1.72876 | − | 2.99430i | 6.16488 | −1.93481 | + | 3.35119i | −7.88662 | + | 4.55334i | − | 1.48144i | −3.49605 | + | 6.05533i | −14.5798 | + | 8.41767i | |||
15.8 | −2.00984 | + | 1.16038i | −3.64827 | + | 2.10633i | 0.692961 | − | 1.20024i | −2.91522 | 4.88828 | − | 8.46676i | −4.31280 | + | 2.49000i | − | 6.06665i | 4.37325 | − | 7.57470i | 5.85911 | − | 3.38276i | |||
15.9 | −1.83582 | + | 1.05991i | 4.02185 | − | 2.32201i | 0.246821 | − | 0.427506i | 6.05791 | −4.92225 | + | 8.52560i | 6.83430 | − | 3.94578i | − | 7.43285i | 6.28350 | − | 10.8833i | −11.1212 | + | 6.42085i | |||
15.10 | −1.79689 | + | 1.03744i | −0.231192 | + | 0.133478i | 0.152554 | − | 0.264232i | 0.253622 | 0.276951 | − | 0.479694i | 1.69721 | − | 0.979882i | − | 7.66644i | −4.46437 | + | 7.73251i | −0.455732 | + | 0.263117i | |||
15.11 | −1.45509 | + | 0.840099i | −1.70196 | + | 0.982625i | −0.588469 | + | 1.01926i | 0.968548 | 1.65100 | − | 2.85962i | 9.80511 | − | 5.66098i | − | 8.69828i | −2.56890 | + | 4.44946i | −1.40933 | + | 0.813675i | |||
15.12 | −1.39681 | + | 0.806447i | −5.02816 | + | 2.90301i | −0.699286 | + | 1.21120i | 8.50901 | 4.68224 | − | 8.10988i | 2.45340 | − | 1.41647i | − | 8.70733i | 12.3549 | − | 21.3993i | −11.8855 | + | 6.86207i | |||
15.13 | −1.10842 | + | 0.639949i | 3.92665 | − | 2.26705i | −1.18093 | + | 2.04543i | −4.08531 | −2.90160 | + | 5.02571i | 1.75991 | − | 1.01608i | − | 8.14253i | 5.77906 | − | 10.0096i | 4.52826 | − | 2.61439i | |||
15.14 | −0.668192 | + | 0.385781i | −3.23147 | + | 1.86569i | −1.70235 | + | 2.94855i | −5.92170 | 1.43949 | − | 2.49328i | −8.58189 | + | 4.95476i | − | 5.71318i | 2.46158 | − | 4.26359i | 3.95684 | − | 2.28448i | |||
15.15 | −0.642412 | + | 0.370896i | −1.57332 | + | 0.908357i | −1.72487 | + | 2.98757i | 3.64465 | 0.673813 | − | 1.16708i | −5.13285 | + | 2.96345i | − | 5.52617i | −2.84977 | + | 4.93595i | −2.34136 | + | 1.35179i | |||
15.16 | −0.383536 | + | 0.221434i | 2.44507 | − | 1.41166i | −1.90193 | + | 3.29425i | −6.28491 | −0.625182 | + | 1.08285i | 1.84824 | − | 1.06708i | − | 3.45609i | −0.514408 | + | 0.890980i | 2.41049 | − | 1.39170i | |||
15.17 | −0.269102 | + | 0.155366i | 2.24759 | − | 1.29764i | −1.95172 | + | 3.38048i | 0.332053 | −0.403219 | + | 0.698396i | −9.33157 | + | 5.38758i | − | 2.45585i | −1.13224 | + | 1.96110i | −0.0893560 | + | 0.0515897i | |||
15.18 | −0.0594322 | + | 0.0343132i | 0.887522 | − | 0.512411i | −1.99765 | + | 3.46002i | 9.58936 | −0.0351649 | + | 0.0609074i | 5.09594 | − | 2.94214i | − | 0.548688i | −3.97487 | + | 6.88468i | −0.569917 | + | 0.329042i | |||
15.19 | −0.0226402 | + | 0.0130713i | −2.34985 | + | 1.35669i | −1.99966 | + | 3.46351i | −8.84227 | 0.0354674 | − | 0.0614314i | 9.90132 | − | 5.71653i | − | 0.209123i | −0.818790 | + | 1.41819i | 0.200191 | − | 0.115580i | |||
15.20 | 0.717030 | − | 0.413978i | 4.24124 | − | 2.44868i | −1.65725 | + | 2.87043i | 7.07258 | 2.02740 | − | 3.51156i | −6.82168 | + | 3.93850i | 6.05607i | 7.49209 | − | 12.9767i | 5.07125 | − | 2.92789i | ||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
211.e | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 211.3.e.a | ✓ | 68 |
211.e | odd | 6 | 1 | inner | 211.3.e.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
211.3.e.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
211.3.e.a | ✓ | 68 | 211.e | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(211, [\chi])\).