Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4009,2,Mod(1,4009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4009 = 19 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4009.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: | \(32.0120261703\) |
Analytic rank: | \(1\) |
Dimension: | \(75\) |
Twist minimal: | yes |
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 | −2.80216 | −0.276714 | 5.85209 | −4.46085 | 0.775398 | −4.37845 | −10.7942 | −2.92343 | 12.5000 | ||||||||||||||||||
1.2 | −2.73305 | 0.112405 | 5.46956 | 0.682101 | −0.307207 | −0.658651 | −9.48248 | −2.98737 | −1.86422 | ||||||||||||||||||
1.3 | −2.71128 | −2.26408 | 5.35102 | 1.45359 | 6.13856 | 1.50475 | −9.08554 | 2.12608 | −3.94108 | ||||||||||||||||||
1.4 | −2.62786 | 2.23579 | 4.90563 | 3.65029 | −5.87534 | −3.99765 | −7.63559 | 1.99877 | −9.59245 | ||||||||||||||||||
1.5 | −2.62780 | 3.38775 | 4.90531 | −4.01450 | −8.90232 | 2.03245 | −7.63457 | 8.47685 | 10.5493 | ||||||||||||||||||
1.6 | −2.58872 | 0.285133 | 4.70149 | −3.09212 | −0.738130 | 5.20222 | −6.99340 | −2.91870 | 8.00463 | ||||||||||||||||||
1.7 | −2.52406 | −2.15147 | 4.37089 | −1.88445 | 5.43045 | −0.0300064 | −5.98429 | 1.62883 | 4.75648 | ||||||||||||||||||
1.8 | −2.42270 | 2.44086 | 3.86949 | −0.894921 | −5.91348 | 2.03761 | −4.52922 | 2.95780 | 2.16813 | ||||||||||||||||||
1.9 | −2.37585 | 1.43559 | 3.64464 | 2.52541 | −3.41075 | −1.13884 | −3.90741 | −0.939073 | −5.99997 | ||||||||||||||||||
1.10 | −2.33282 | −0.629609 | 3.44205 | 3.06029 | 1.46876 | 3.36015 | −3.36405 | −2.60359 | −7.13912 | ||||||||||||||||||
1.11 | −2.20714 | 0.00654009 | 2.87148 | −3.65103 | −0.0144349 | −1.05666 | −1.92348 | −2.99996 | 8.05835 | ||||||||||||||||||
1.12 | −2.18144 | −2.69143 | 2.75868 | 0.125286 | 5.87120 | −3.80798 | −1.65501 | 4.24381 | −0.273304 | ||||||||||||||||||
1.13 | −2.17778 | −1.25453 | 2.74274 | 2.27371 | 2.73208 | −1.62232 | −1.61752 | −1.42616 | −4.95164 | ||||||||||||||||||
1.14 | −2.06463 | 1.41935 | 2.26270 | 1.21401 | −2.93043 | 2.08294 | −0.542368 | −0.985454 | −2.50648 | ||||||||||||||||||
1.15 | −1.94602 | 3.23698 | 1.78701 | −0.488090 | −6.29925 | −2.80369 | 0.414483 | 7.47805 | 0.949835 | ||||||||||||||||||
1.16 | −1.81819 | 1.88652 | 1.30583 | −3.78374 | −3.43005 | 2.14600 | 1.26214 | 0.558942 | 6.87957 | ||||||||||||||||||
1.17 | −1.81267 | −3.27382 | 1.28578 | 1.53613 | 5.93435 | −3.38684 | 1.29465 | 7.71788 | −2.78450 | ||||||||||||||||||
1.18 | −1.72632 | 0.426642 | 0.980187 | 1.18315 | −0.736521 | −0.668631 | 1.76053 | −2.81798 | −2.04249 | ||||||||||||||||||
1.19 | −1.68291 | −2.73847 | 0.832174 | −3.05253 | 4.60859 | 2.37030 | 1.96534 | 4.49922 | 5.13711 | ||||||||||||||||||
1.20 | −1.64786 | 2.26384 | 0.715427 | 1.21347 | −3.73048 | −4.02592 | 2.11679 | 2.12497 | −1.99962 | ||||||||||||||||||
See all 75 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(19\) | \(1\) |
\(211\) | \(1\) |
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 | 4009.2.a.d | ✓ | 75 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4009.2.a.d | ✓ | 75 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{75} + 11 T_{2}^{74} - 48 T_{2}^{73} - 947 T_{2}^{72} - 62 T_{2}^{71} + 38025 T_{2}^{70} + \cdots + 28700 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).