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: | \(25\) |
Twist minimal: | no (minimal twist has level 1205) |
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 | −2.77603 | −0.0161258 | 5.70635 | 0 | 0.0447656 | −4.43454 | −10.2889 | −2.99974 | 0 | ||||||||||||||||||
1.2 | −2.68235 | −1.41400 | 5.19500 | 0 | 3.79283 | 3.05254 | −8.57011 | −1.00061 | 0 | ||||||||||||||||||
1.3 | −2.52189 | 2.20014 | 4.35995 | 0 | −5.54852 | 1.38484 | −5.95156 | 1.84062 | 0 | ||||||||||||||||||
1.4 | −2.43298 | −3.08500 | 3.91941 | 0 | 7.50575 | 2.29332 | −4.66990 | 6.51721 | 0 | ||||||||||||||||||
1.5 | −2.31949 | −0.405385 | 3.38004 | 0 | 0.940288 | −1.18682 | −3.20098 | −2.83566 | 0 | ||||||||||||||||||
1.6 | −2.03980 | 3.04377 | 2.16080 | 0 | −6.20869 | −3.73609 | −0.327997 | 6.26453 | 0 | ||||||||||||||||||
1.7 | −1.96718 | −2.44636 | 1.86979 | 0 | 4.81242 | −2.28114 | 0.256155 | 2.98466 | 0 | ||||||||||||||||||
1.8 | −1.91059 | −2.10440 | 1.65034 | 0 | 4.02063 | −2.16427 | 0.668063 | 1.42849 | 0 | ||||||||||||||||||
1.9 | −1.31571 | 0.792486 | −0.268903 | 0 | −1.04268 | 1.49796 | 2.98522 | −2.37197 | 0 | ||||||||||||||||||
1.10 | −1.05446 | −2.79403 | −0.888118 | 0 | 2.94619 | 1.63707 | 3.04540 | 4.80663 | 0 | ||||||||||||||||||
1.11 | −0.782724 | 1.54504 | −1.38734 | 0 | −1.20934 | 2.90223 | 2.65136 | −0.612841 | 0 | ||||||||||||||||||
1.12 | −0.707449 | −1.14949 | −1.49952 | 0 | 0.813204 | −1.94091 | 2.47573 | −1.67868 | 0 | ||||||||||||||||||
1.13 | −0.362712 | −3.41577 | −1.86844 | 0 | 1.23894 | −3.34046 | 1.40313 | 8.66752 | 0 | ||||||||||||||||||
1.14 | −0.300307 | −0.482693 | −1.90982 | 0 | 0.144956 | −5.09571 | 1.17415 | −2.76701 | 0 | ||||||||||||||||||
1.15 | 0.0694400 | 2.28670 | −1.99518 | 0 | 0.158789 | −0.976003 | −0.277425 | 2.22902 | 0 | ||||||||||||||||||
1.16 | 0.566303 | 0.484852 | −1.67930 | 0 | 0.274573 | 0.0113879 | −2.08360 | −2.76492 | 0 | ||||||||||||||||||
1.17 | 1.11717 | −3.26371 | −0.751931 | 0 | −3.64612 | 3.89846 | −3.07438 | 7.65181 | 0 | ||||||||||||||||||
1.18 | 1.16007 | −2.40515 | −0.654229 | 0 | −2.79015 | −1.39900 | −3.07910 | 2.78474 | 0 | ||||||||||||||||||
1.19 | 1.25748 | 2.42991 | −0.418732 | 0 | 3.05557 | −3.44014 | −3.04152 | 2.90444 | 0 | ||||||||||||||||||
1.20 | 1.58885 | −2.11200 | 0.524458 | 0 | −3.35566 | −3.95616 | −2.34442 | 1.46053 | 0 | ||||||||||||||||||
See all 25 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.j | 25 | |
5.b | even | 2 | 1 | 1205.2.a.e | ✓ | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1205.2.a.e | ✓ | 25 | 5.b | even | 2 | 1 | |
6025.2.a.j | 25 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):
\( T_{2}^{25} + 6 T_{2}^{24} - 23 T_{2}^{23} - 197 T_{2}^{22} + 127 T_{2}^{21} + 2741 T_{2}^{20} + \cdots - 409 \) |
\( T_{3}^{25} + 15 T_{3}^{24} + 59 T_{3}^{23} - 196 T_{3}^{22} - 2010 T_{3}^{21} - 2331 T_{3}^{20} + \cdots - 832 \) |