Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [201,4,Mod(37,201)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("201.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 201 = 3 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 201.e (of order \(3\), 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: | \(11.8593839112\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −2.72652 | − | 4.72247i | −3.00000 | −10.8678 | + | 18.8236i | 12.9793 | 8.17955 | + | 14.1674i | −14.7538 | + | 25.5543i | 74.9007 | 9.00000 | −35.3882 | − | 61.2942i | ||||||||
37.2 | −2.52313 | − | 4.37020i | −3.00000 | −8.73242 | + | 15.1250i | 0.340792 | 7.56940 | + | 13.1106i | 12.2513 | − | 21.2198i | 47.7621 | 9.00000 | −0.859864 | − | 1.48933i | ||||||||
37.3 | −2.37227 | − | 4.10889i | −3.00000 | −7.25534 | + | 12.5666i | −19.3810 | 7.11681 | + | 12.3267i | 3.13332 | − | 5.42707i | 30.8902 | 9.00000 | 45.9769 | + | 79.6344i | ||||||||
37.4 | −1.83140 | − | 3.17208i | −3.00000 | −2.70808 | + | 4.69053i | 12.9712 | 5.49421 | + | 9.51625i | 1.64155 | − | 2.84325i | −9.46413 | 9.00000 | −23.7555 | − | 41.1456i | ||||||||
37.5 | −1.56036 | − | 2.70262i | −3.00000 | −0.869446 | + | 1.50592i | 10.0899 | 4.68108 | + | 8.10787i | −2.32642 | + | 4.02948i | −19.5392 | 9.00000 | −15.7439 | − | 27.2693i | ||||||||
37.6 | −1.55793 | − | 2.69841i | −3.00000 | −0.854281 | + | 1.47966i | −13.0874 | 4.67378 | + | 8.09523i | −12.0084 | + | 20.7992i | −19.6032 | 9.00000 | 20.3893 | + | 35.3153i | ||||||||
37.7 | −0.829978 | − | 1.43756i | −3.00000 | 2.62227 | − | 4.54191i | −7.05463 | 2.48993 | + | 4.31269i | 18.3944 | − | 31.8600i | −21.9854 | 9.00000 | 5.85519 | + | 10.1415i | ||||||||
37.8 | −0.561904 | − | 0.973245i | −3.00000 | 3.36853 | − | 5.83446i | −9.21684 | 1.68571 | + | 2.91974i | 2.79031 | − | 4.83295i | −16.5616 | 9.00000 | 5.17897 | + | 8.97024i | ||||||||
37.9 | 0.0760461 | + | 0.131716i | −3.00000 | 3.98843 | − | 6.90817i | 2.82816 | −0.228138 | − | 0.395147i | −1.28326 | + | 2.22268i | 2.42996 | 9.00000 | 0.215070 | + | 0.372513i | ||||||||
37.10 | 0.244294 | + | 0.423129i | −3.00000 | 3.88064 | − | 6.72147i | 20.3821 | −0.732881 | − | 1.26939i | 8.86889 | − | 15.3614i | 7.70076 | 9.00000 | 4.97921 | + | 8.62424i | ||||||||
37.11 | 0.824514 | + | 1.42810i | −3.00000 | 2.64035 | − | 4.57323i | −20.6834 | −2.47354 | − | 4.28430i | 1.21474 | − | 2.10399i | 21.9003 | 9.00000 | −17.0538 | − | 29.5380i | ||||||||
37.12 | 1.20764 | + | 2.09170i | −3.00000 | 1.08320 | − | 1.87615i | −7.44326 | −3.62293 | − | 6.27510i | −10.2926 | + | 17.8273i | 24.5547 | 9.00000 | −8.98880 | − | 15.5691i | ||||||||
37.13 | 1.46577 | + | 2.53879i | −3.00000 | −0.296981 | + | 0.514386i | 10.1733 | −4.39732 | − | 7.61638i | −10.4213 | + | 18.0502i | 21.7111 | 9.00000 | 14.9117 | + | 25.8278i | ||||||||
37.14 | 1.63047 | + | 2.82406i | −3.00000 | −1.31688 | + | 2.28090i | −0.0205579 | −4.89142 | − | 8.47218i | 2.93590 | − | 5.08512i | 17.4990 | 9.00000 | −0.0335191 | − | 0.0580568i | ||||||||
37.15 | 2.13007 | + | 3.68940i | −3.00000 | −5.07443 | + | 8.78916i | 13.6463 | −6.39022 | − | 11.0682i | 12.7106 | − | 22.0154i | −9.15443 | 9.00000 | 29.0676 | + | 50.3465i | ||||||||
37.16 | 2.19185 | + | 3.79640i | −3.00000 | −5.60845 | + | 9.71412i | −15.2934 | −6.57556 | − | 11.3892i | 12.5287 | − | 21.7004i | −14.1019 | 9.00000 | −33.5209 | − | 58.0599i | ||||||||
37.17 | 2.50438 | + | 4.33772i | −3.00000 | −8.54385 | + | 14.7984i | 13.1979 | −7.51314 | − | 13.0131i | −7.35263 | + | 12.7351i | −45.5181 | 9.00000 | 33.0525 | + | 57.2486i | ||||||||
37.18 | 2.68845 | + | 4.65653i | −3.00000 | −10.4555 | + | 18.1094i | −6.42823 | −8.06534 | − | 13.9696i | −7.03129 | + | 12.1786i | −69.4209 | 9.00000 | −17.2819 | − | 29.9332i | ||||||||
163.1 | −2.72652 | + | 4.72247i | −3.00000 | −10.8678 | − | 18.8236i | 12.9793 | 8.17955 | − | 14.1674i | −14.7538 | − | 25.5543i | 74.9007 | 9.00000 | −35.3882 | + | 61.2942i | ||||||||
163.2 | −2.52313 | + | 4.37020i | −3.00000 | −8.73242 | − | 15.1250i | 0.340792 | 7.56940 | − | 13.1106i | 12.2513 | + | 21.2198i | 47.7621 | 9.00000 | −0.859864 | + | 1.48933i | ||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
67.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 201.4.e.b | ✓ | 36 |
67.c | even | 3 | 1 | inner | 201.4.e.b | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
201.4.e.b | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
201.4.e.b | ✓ | 36 | 67.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{36} - 2 T_{2}^{35} + 119 T_{2}^{34} - 230 T_{2}^{33} + 8271 T_{2}^{32} - 15533 T_{2}^{31} + 384536 T_{2}^{30} - 692967 T_{2}^{29} + 13324771 T_{2}^{28} - 22941523 T_{2}^{27} + \cdots + 152371175952384 \)
acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\).