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: | \(1\) |
Dimension: | \(36\) |
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 | −5.24204 | −1.46828 | 19.4790 | 7.63398 | 7.69676 | 0 | −60.1731 | −24.8442 | −40.0176 | ||||||||||||||||||
1.2 | −5.05226 | −6.21075 | 17.5254 | −9.48476 | 31.3783 | 0 | −48.1246 | 11.5734 | 47.9195 | ||||||||||||||||||
1.3 | −4.84147 | −8.89523 | 15.4398 | 19.8253 | 43.0660 | 0 | −36.0196 | 52.1251 | −95.9835 | ||||||||||||||||||
1.4 | −4.67229 | 4.57802 | 13.8303 | −8.79022 | −21.3898 | 0 | −27.2411 | −6.04177 | 41.0705 | ||||||||||||||||||
1.5 | −4.65790 | 6.35474 | 13.6960 | 11.2439 | −29.5997 | 0 | −26.5313 | 13.3827 | −52.3728 | ||||||||||||||||||
1.6 | −4.30190 | 0.345300 | 10.5063 | −17.1344 | −1.48545 | 0 | −10.7821 | −26.8808 | 73.7105 | ||||||||||||||||||
1.7 | −3.98055 | 1.09910 | 7.84477 | −16.1194 | −4.37501 | 0 | 0.617915 | −25.7920 | 64.1642 | ||||||||||||||||||
1.8 | −3.61690 | 9.85524 | 5.08200 | 11.4029 | −35.6455 | 0 | 10.5541 | 70.1257 | −41.2432 | ||||||||||||||||||
1.9 | −3.59229 | −3.87874 | 4.90453 | 10.9640 | 13.9335 | 0 | 11.1198 | −11.9554 | −39.3859 | ||||||||||||||||||
1.10 | −2.84821 | 7.14998 | 0.112273 | −18.5031 | −20.3646 | 0 | 22.4659 | 24.1223 | 52.7005 | ||||||||||||||||||
1.11 | −2.56286 | 4.35049 | −1.43176 | 5.73556 | −11.1497 | 0 | 24.1723 | −8.07327 | −14.6994 | ||||||||||||||||||
1.12 | −2.29878 | −8.78314 | −2.71563 | −2.02920 | 20.1905 | 0 | 24.6328 | 50.1435 | 4.66467 | ||||||||||||||||||
1.13 | −2.25528 | −8.55708 | −2.91373 | 4.20502 | 19.2986 | 0 | 24.6135 | 46.2236 | −9.48347 | ||||||||||||||||||
1.14 | −1.68929 | 2.43988 | −5.14630 | −3.14907 | −4.12166 | 0 | 22.2079 | −21.0470 | 5.31970 | ||||||||||||||||||
1.15 | −1.64556 | 1.90121 | −5.29212 | 11.4725 | −3.12855 | 0 | 21.8730 | −23.3854 | −18.8787 | ||||||||||||||||||
1.16 | −1.48336 | −7.02935 | −5.79964 | −10.3252 | 10.4271 | 0 | 20.4698 | 22.4118 | 15.3160 | ||||||||||||||||||
1.17 | −0.706634 | −2.07976 | −7.50067 | 16.5960 | 1.46963 | 0 | 10.9533 | −22.6746 | −11.7273 | ||||||||||||||||||
1.18 | −0.598281 | 8.38024 | −7.64206 | 3.61677 | −5.01374 | 0 | 9.35835 | 43.2285 | −2.16385 | ||||||||||||||||||
1.19 | −0.140258 | −5.16208 | −7.98033 | 0.516531 | 0.724025 | 0 | 2.24138 | −0.352951 | −0.0724479 | ||||||||||||||||||
1.20 | 0.388024 | −0.580020 | −7.84944 | −19.1871 | −0.225062 | 0 | −6.14996 | −26.6636 | −7.44506 | ||||||||||||||||||
See all 36 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.k | 36 | |
7.b | odd | 2 | 1 | 2009.4.a.j | 36 | ||
7.d | odd | 6 | 2 | 287.4.e.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
287.4.e.a | ✓ | 72 | 7.d | odd | 6 | 2 | |
2009.4.a.j | 36 | 7.b | odd | 2 | 1 | ||
2009.4.a.k | 36 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):
\( T_{2}^{36} + 5 T_{2}^{35} - 190 T_{2}^{34} - 952 T_{2}^{33} + 16346 T_{2}^{32} + 82211 T_{2}^{31} - 842490 T_{2}^{30} - 4263555 T_{2}^{29} + 29002109 T_{2}^{28} + 148204731 T_{2}^{27} + \cdots - 32865404963328 \) |
\( T_{3}^{36} - 6 T_{3}^{35} - 586 T_{3}^{34} + 3412 T_{3}^{33} + 154674 T_{3}^{32} - 877416 T_{3}^{31} - 24330318 T_{3}^{30} + 135149378 T_{3}^{29} + 2543458024 T_{3}^{28} - 13921757022 T_{3}^{27} + \cdots - 34\!\cdots\!24 \) |