Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6025,2,Mod(1,6025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, 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("6025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6025 = 5^{2} \cdot 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6025.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: | \(48.1098672178\) |
Analytic rank: | \(1\) |
Dimension: | \(40\) |
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 | −2.77437 | 2.60911 | 5.69713 | 0 | −7.23863 | 1.40482 | −10.2572 | 3.80744 | 0 | ||||||||||||||||||
1.2 | −2.74359 | −0.0877195 | 5.52727 | 0 | 0.240666 | −1.38863 | −9.67736 | −2.99231 | 0 | ||||||||||||||||||
1.3 | −2.67635 | −2.41547 | 5.16285 | 0 | 6.46465 | −0.976502 | −8.46490 | 2.83451 | 0 | ||||||||||||||||||
1.4 | −2.60048 | −3.03019 | 4.76251 | 0 | 7.87996 | −1.49919 | −7.18385 | 6.18207 | 0 | ||||||||||||||||||
1.5 | −2.50972 | −0.136405 | 4.29867 | 0 | 0.342337 | −4.91163 | −5.76901 | −2.98139 | 0 | ||||||||||||||||||
1.6 | −2.49478 | 2.89336 | 4.22391 | 0 | −7.21828 | 0.185663 | −5.54817 | 5.37152 | 0 | ||||||||||||||||||
1.7 | −2.40563 | −2.30634 | 3.78704 | 0 | 5.54820 | 4.72754 | −4.29895 | 2.31922 | 0 | ||||||||||||||||||
1.8 | −2.17490 | 0.995168 | 2.73018 | 0 | −2.16439 | 3.57474 | −1.58807 | −2.00964 | 0 | ||||||||||||||||||
1.9 | −2.06779 | 0.977473 | 2.27574 | 0 | −2.02121 | −1.48726 | −0.570172 | −2.04455 | 0 | ||||||||||||||||||
1.10 | −1.92684 | −1.97893 | 1.71272 | 0 | 3.81308 | −3.74943 | 0.553547 | 0.916158 | 0 | ||||||||||||||||||
1.11 | −1.60847 | −3.19065 | 0.587161 | 0 | 5.13204 | −4.87921 | 2.27250 | 7.18022 | 0 | ||||||||||||||||||
1.12 | −1.59175 | 0.0342835 | 0.533671 | 0 | −0.0545707 | −1.01795 | 2.33403 | −2.99882 | 0 | ||||||||||||||||||
1.13 | −1.55382 | 2.23966 | 0.414344 | 0 | −3.48002 | 1.09916 | 2.46382 | 2.01608 | 0 | ||||||||||||||||||
1.14 | −1.26445 | 0.637501 | −0.401165 | 0 | −0.806088 | 3.31632 | 3.03615 | −2.59359 | 0 | ||||||||||||||||||
1.15 | −1.23402 | −3.02124 | −0.477193 | 0 | 3.72827 | 0.346766 | 3.05691 | 6.12789 | 0 | ||||||||||||||||||
1.16 | −1.19733 | −1.42352 | −0.566401 | 0 | 1.70442 | 1.72539 | 3.07283 | −0.973586 | 0 | ||||||||||||||||||
1.17 | −1.14940 | 0.361587 | −0.678878 | 0 | −0.415609 | −3.63700 | 3.07910 | −2.86925 | 0 | ||||||||||||||||||
1.18 | −0.679522 | 2.20024 | −1.53825 | 0 | −1.49511 | −5.10754 | 2.40432 | 1.84107 | 0 | ||||||||||||||||||
1.19 | −0.667714 | 2.21224 | −1.55416 | 0 | −1.47714 | −0.972247 | 2.37316 | 1.89399 | 0 | ||||||||||||||||||
1.20 | −0.599456 | 2.83593 | −1.64065 | 0 | −1.70002 | 0.360434 | 2.18241 | 5.04251 | 0 | ||||||||||||||||||
See all 40 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(241\) | \(1\) |
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 | 6025.2.a.l | ✓ | 40 |
5.b | even | 2 | 1 | 6025.2.a.o | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6025.2.a.l | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
6025.2.a.o | yes | 40 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):
\( T_{2}^{40} + 11 T_{2}^{39} - 418 T_{2}^{37} - 1090 T_{2}^{36} + 6489 T_{2}^{35} + 28938 T_{2}^{34} + \cdots + 41303 \) |
\( T_{3}^{40} + 8 T_{3}^{39} - 47 T_{3}^{38} - 518 T_{3}^{37} + 724 T_{3}^{36} + 15185 T_{3}^{35} + \cdots - 4723 \) |