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: | \(0\) |
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 | −2.56709 | −0.178555 | 4.58996 | 0 | 0.458367 | 0.233097 | −6.64865 | −2.96812 | 0 | ||||||||||||||||||
1.2 | −2.41430 | 3.26436 | 3.82885 | 0 | −7.88114 | 1.16921 | −4.41538 | 7.65603 | 0 | ||||||||||||||||||
1.3 | −2.30206 | 0.621403 | 3.29948 | 0 | −1.43051 | 0.210935 | −2.99147 | −2.61386 | 0 | ||||||||||||||||||
1.4 | −2.25059 | −0.826129 | 3.06516 | 0 | 1.85928 | 0.537315 | −2.39723 | −2.31751 | 0 | ||||||||||||||||||
1.5 | −2.24247 | −1.80227 | 3.02866 | 0 | 4.04153 | 5.11796 | −2.30673 | 0.248182 | 0 | ||||||||||||||||||
1.6 | −2.06849 | 2.17142 | 2.27866 | 0 | −4.49155 | 2.39120 | −0.576402 | 1.71504 | 0 | ||||||||||||||||||
1.7 | −1.77260 | −0.796316 | 1.14211 | 0 | 1.41155 | −3.30701 | 1.52069 | −2.36588 | 0 | ||||||||||||||||||
1.8 | −1.44334 | −2.75896 | 0.0832441 | 0 | 3.98213 | 1.64702 | 2.76654 | 4.61187 | 0 | ||||||||||||||||||
1.9 | −1.43804 | 0.465580 | 0.0679498 | 0 | −0.669521 | 1.05447 | 2.77836 | −2.78324 | 0 | ||||||||||||||||||
1.10 | −1.38463 | −1.64656 | −0.0828005 | 0 | 2.27988 | −2.06195 | 2.88391 | −0.288826 | 0 | ||||||||||||||||||
1.11 | −1.33622 | 1.40541 | −0.214528 | 0 | −1.87793 | −0.850787 | 2.95909 | −1.02483 | 0 | ||||||||||||||||||
1.12 | −0.859912 | 1.32501 | −1.26055 | 0 | −1.13939 | 0.529848 | 2.80379 | −1.24436 | 0 | ||||||||||||||||||
1.13 | −0.810170 | 2.88822 | −1.34362 | 0 | −2.33995 | 3.73380 | 2.70891 | 5.34179 | 0 | ||||||||||||||||||
1.14 | −0.625313 | 0.793037 | −1.60898 | 0 | −0.495897 | −2.86756 | 2.25675 | −2.37109 | 0 | ||||||||||||||||||
1.15 | −0.607027 | −2.70729 | −1.63152 | 0 | 1.64340 | 4.13328 | 2.20443 | 4.32942 | 0 | ||||||||||||||||||
1.16 | −0.568441 | −1.77259 | −1.67687 | 0 | 1.00761 | −2.89075 | 2.09009 | 0.142065 | 0 | ||||||||||||||||||
1.17 | −0.447066 | 2.52995 | −1.80013 | 0 | −1.13106 | −3.76816 | 1.69891 | 3.40066 | 0 | ||||||||||||||||||
1.18 | −0.410308 | 1.14046 | −1.83165 | 0 | −0.467941 | 3.98049 | 1.57216 | −1.69935 | 0 | ||||||||||||||||||
1.19 | 0.232950 | 1.45187 | −1.94573 | 0 | 0.338212 | −4.96346 | −0.919157 | −0.892086 | 0 | ||||||||||||||||||
1.20 | 0.394755 | 2.83806 | −1.84417 | 0 | 1.12034 | −0.914717 | −1.51750 | 5.05456 | 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.o | yes | 40 |
5.b | even | 2 | 1 | 6025.2.a.l | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6025.2.a.l | ✓ | 40 | 5.b | even | 2 | 1 | |
6025.2.a.o | yes | 40 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):
\( T_{2}^{40} - 11 T_{2}^{39} + 418 T_{2}^{37} - 1090 T_{2}^{36} - 6489 T_{2}^{35} + 28938 T_{2}^{34} + 47342 T_{2}^{33} - 383558 T_{2}^{32} - 50678 T_{2}^{31} + 3167203 T_{2}^{30} - 2221315 T_{2}^{29} - 17592761 T_{2}^{28} + \cdots + 41303 \) |
\( T_{3}^{40} - 8 T_{3}^{39} - 47 T_{3}^{38} + 518 T_{3}^{37} + 724 T_{3}^{36} - 15185 T_{3}^{35} + 1998 T_{3}^{34} + 266509 T_{3}^{33} - 249314 T_{3}^{32} - 3120101 T_{3}^{31} + 4566377 T_{3}^{30} + 25690505 T_{3}^{29} + \cdots - 4723 \) |