Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2009,4,Mod(1,2009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2009.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2009 = 7^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2009.a (trivial) |
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: | yes |
Analytic conductor: | \(118.534837202\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Twist minimal: | no (minimal twist has level 287) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −5.56779 | 8.65991 | 23.0003 | −13.7471 | −48.2165 | 0 | −83.5184 | 47.9940 | 76.5410 | ||||||||||||||||||
1.2 | −5.52545 | −4.18525 | 22.5307 | −6.77679 | 23.1254 | 0 | −80.2885 | −9.48369 | 37.4448 | ||||||||||||||||||
1.3 | −5.30261 | 1.50705 | 20.1177 | 18.8661 | −7.99129 | 0 | −64.2554 | −24.7288 | −100.039 | ||||||||||||||||||
1.4 | −5.12991 | 4.65672 | 18.3160 | −8.04761 | −23.8886 | 0 | −52.9201 | −5.31492 | 41.2835 | ||||||||||||||||||
1.5 | −4.68637 | −9.79636 | 13.9621 | 2.50471 | 45.9094 | 0 | −27.9405 | 68.9687 | −11.7380 | ||||||||||||||||||
1.6 | −4.44628 | 3.60545 | 11.7694 | 7.35509 | −16.0309 | 0 | −16.7599 | −14.0007 | −32.7028 | ||||||||||||||||||
1.7 | −4.31737 | 8.67456 | 10.6397 | −0.0539338 | −37.4513 | 0 | −11.3965 | 48.2480 | 0.232852 | ||||||||||||||||||
1.8 | −4.11365 | −8.35880 | 8.92214 | −18.0628 | 34.3852 | 0 | −3.79335 | 42.8695 | 74.3042 | ||||||||||||||||||
1.9 | −4.01943 | −5.51783 | 8.15585 | 10.7255 | 22.1786 | 0 | −0.626446 | 3.44645 | −43.1104 | ||||||||||||||||||
1.10 | −3.66338 | −1.94510 | 5.42033 | −2.50423 | 7.12562 | 0 | 9.45029 | −23.2166 | 9.17393 | ||||||||||||||||||
1.11 | −3.42310 | −0.926042 | 3.71763 | 9.50387 | 3.16994 | 0 | 14.6590 | −26.1424 | −32.5327 | ||||||||||||||||||
1.12 | −3.25509 | −4.79420 | 2.59562 | −8.80361 | 15.6056 | 0 | 17.5917 | −4.01566 | 28.6566 | ||||||||||||||||||
1.13 | −2.71328 | 8.93459 | −0.638133 | −12.0134 | −24.2420 | 0 | 23.4376 | 52.8269 | 32.5957 | ||||||||||||||||||
1.14 | −2.57834 | 5.14811 | −1.35214 | 22.1686 | −13.2736 | 0 | 24.1130 | −0.496954 | −57.1582 | ||||||||||||||||||
1.15 | −2.25123 | 5.39880 | −2.93196 | −4.87704 | −12.1539 | 0 | 24.6104 | 2.14704 | 10.9793 | ||||||||||||||||||
1.16 | −1.75078 | 0.529464 | −4.93476 | −5.78068 | −0.926975 | 0 | 22.6459 | −26.7197 | 10.1207 | ||||||||||||||||||
1.17 | −1.74060 | −6.03699 | −4.97031 | 20.3587 | 10.5080 | 0 | 22.5761 | 9.44523 | −35.4364 | ||||||||||||||||||
1.18 | −1.38854 | −1.80099 | −6.07196 | −19.5363 | 2.50075 | 0 | 19.5395 | −23.7564 | 27.1269 | ||||||||||||||||||
1.19 | −1.08367 | −5.78762 | −6.82566 | −13.9821 | 6.27188 | 0 | 16.0661 | 6.49657 | 15.1520 | ||||||||||||||||||
1.20 | −0.149425 | 6.65948 | −7.97767 | −12.7223 | −0.995092 | 0 | 2.38746 | 17.3487 | 1.90102 | ||||||||||||||||||
See all 44 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(-1\) |
\(41\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2009.4.a.m | 44 | |
7.b | odd | 2 | 1 | 2009.4.a.l | 44 | ||
7.d | odd | 6 | 2 | 287.4.e.b | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
287.4.e.b | ✓ | 88 | 7.d | odd | 6 | 2 | |
2009.4.a.l | 44 | 7.b | odd | 2 | 1 | ||
2009.4.a.m | 44 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):
\( T_{2}^{44} - 3 T_{2}^{43} - 278 T_{2}^{42} + 808 T_{2}^{41} + 35734 T_{2}^{40} - 100321 T_{2}^{39} + \cdots + 422240214581248 \) |
\( T_{3}^{44} - 6 T_{3}^{43} - 802 T_{3}^{42} + 4792 T_{3}^{41} + 295326 T_{3}^{40} - 1744424 T_{3}^{39} + \cdots - 49\!\cdots\!16 \) |