Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8025,2,Mod(1,8025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8025 = 3 \cdot 5^{2} \cdot 107 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8025.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: | \(64.0799476221\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
Twist minimal: | no (minimal twist has level 1605) |
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 | −2.58014 | 1.00000 | 4.65713 | 0 | −2.58014 | 1.98502 | −6.85578 | 1.00000 | 0 | ||||||||||||||||||
1.2 | −2.50926 | 1.00000 | 4.29640 | 0 | −2.50926 | 0.811131 | −5.76228 | 1.00000 | 0 | ||||||||||||||||||
1.3 | −2.25096 | 1.00000 | 3.06684 | 0 | −2.25096 | −0.523586 | −2.40142 | 1.00000 | 0 | ||||||||||||||||||
1.4 | −2.13388 | 1.00000 | 2.55344 | 0 | −2.13388 | 3.51369 | −1.18098 | 1.00000 | 0 | ||||||||||||||||||
1.5 | −1.89104 | 1.00000 | 1.57603 | 0 | −1.89104 | −3.50248 | 0.801740 | 1.00000 | 0 | ||||||||||||||||||
1.6 | −1.59994 | 1.00000 | 0.559801 | 0 | −1.59994 | 2.78827 | 2.30423 | 1.00000 | 0 | ||||||||||||||||||
1.7 | −1.30136 | 1.00000 | −0.306455 | 0 | −1.30136 | −0.670302 | 3.00153 | 1.00000 | 0 | ||||||||||||||||||
1.8 | −0.844776 | 1.00000 | −1.28635 | 0 | −0.844776 | −3.24504 | 2.77623 | 1.00000 | 0 | ||||||||||||||||||
1.9 | −0.822726 | 1.00000 | −1.32312 | 0 | −0.822726 | −1.32511 | 2.73402 | 1.00000 | 0 | ||||||||||||||||||
1.10 | −0.736346 | 1.00000 | −1.45779 | 0 | −0.736346 | 4.38760 | 2.54613 | 1.00000 | 0 | ||||||||||||||||||
1.11 | −0.363804 | 1.00000 | −1.86765 | 0 | −0.363804 | 0.0273194 | 1.40706 | 1.00000 | 0 | ||||||||||||||||||
1.12 | −0.145947 | 1.00000 | −1.97870 | 0 | −0.145947 | −4.33134 | 0.580678 | 1.00000 | 0 | ||||||||||||||||||
1.13 | 0.492378 | 1.00000 | −1.75756 | 0 | 0.492378 | 2.62877 | −1.85014 | 1.00000 | 0 | ||||||||||||||||||
1.14 | 0.811713 | 1.00000 | −1.34112 | 0 | 0.811713 | 3.87908 | −2.71203 | 1.00000 | 0 | ||||||||||||||||||
1.15 | 0.861326 | 1.00000 | −1.25812 | 0 | 0.861326 | −1.26306 | −2.80630 | 1.00000 | 0 | ||||||||||||||||||
1.16 | 0.861682 | 1.00000 | −1.25750 | 0 | 0.861682 | −0.233165 | −2.80693 | 1.00000 | 0 | ||||||||||||||||||
1.17 | 1.09634 | 1.00000 | −0.798029 | 0 | 1.09634 | −1.73554 | −3.06760 | 1.00000 | 0 | ||||||||||||||||||
1.18 | 1.27694 | 1.00000 | −0.369419 | 0 | 1.27694 | 0.700732 | −3.02561 | 1.00000 | 0 | ||||||||||||||||||
1.19 | 2.08676 | 1.00000 | 2.35456 | 0 | 2.08676 | 0.375749 | 0.739880 | 1.00000 | 0 | ||||||||||||||||||
1.20 | 2.17402 | 1.00000 | 2.72637 | 0 | 2.17402 | −4.66914 | 1.57914 | 1.00000 | 0 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(-1\) |
\(107\) | \(1\) |
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 | 8025.2.a.bo | 22 | |
5.b | even | 2 | 1 | 8025.2.a.bp | 22 | ||
5.c | odd | 4 | 2 | 1605.2.b.d | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1605.2.b.d | ✓ | 44 | 5.c | odd | 4 | 2 | |
8025.2.a.bo | 22 | 1.a | even | 1 | 1 | trivial | |
8025.2.a.bp | 22 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):
\( T_{2}^{22} + 3 T_{2}^{21} - 24 T_{2}^{20} - 74 T_{2}^{19} + 239 T_{2}^{18} + 763 T_{2}^{17} - 1281 T_{2}^{16} + \cdots + 32 \) |
\( T_{7}^{22} + 2 T_{7}^{21} - 70 T_{7}^{20} - 126 T_{7}^{19} + 1978 T_{7}^{18} + 3190 T_{7}^{17} + \cdots - 608 \) |
\( T_{11}^{22} + 16 T_{11}^{21} + 33 T_{11}^{20} - 744 T_{11}^{19} - 4172 T_{11}^{18} + 8562 T_{11}^{17} + \cdots - 141824 \) |
\( T_{13}^{22} - 139 T_{13}^{20} - 10 T_{13}^{19} + 7879 T_{13}^{18} + 1690 T_{13}^{17} - 236417 T_{13}^{16} + \cdots + 113745920 \) |