Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, 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("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8001.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: | \(63.8883066572\) |
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 | −2.76162 | 0 | 5.62653 | 2.49672 | 0 | 1.00000 | −10.0151 | 0 | −6.89498 | ||||||||||||||||||
1.2 | −2.68554 | 0 | 5.21214 | −3.72554 | 0 | 1.00000 | −8.62633 | 0 | 10.0051 | ||||||||||||||||||
1.3 | −2.66267 | 0 | 5.08984 | −1.08542 | 0 | 1.00000 | −8.22723 | 0 | 2.89013 | ||||||||||||||||||
1.4 | −2.63730 | 0 | 4.95534 | −2.98743 | 0 | 1.00000 | −7.79413 | 0 | 7.87873 | ||||||||||||||||||
1.5 | −2.52993 | 0 | 4.40057 | 4.10792 | 0 | 1.00000 | −6.07329 | 0 | −10.3928 | ||||||||||||||||||
1.6 | −2.31568 | 0 | 3.36240 | 1.76797 | 0 | 1.00000 | −3.15488 | 0 | −4.09407 | ||||||||||||||||||
1.7 | −2.26712 | 0 | 3.13982 | −1.21847 | 0 | 1.00000 | −2.58412 | 0 | 2.76242 | ||||||||||||||||||
1.8 | −2.10140 | 0 | 2.41588 | −0.560790 | 0 | 1.00000 | −0.873928 | 0 | 1.17844 | ||||||||||||||||||
1.9 | −2.09235 | 0 | 2.37795 | −3.96692 | 0 | 1.00000 | −0.790800 | 0 | 8.30020 | ||||||||||||||||||
1.10 | −1.67982 | 0 | 0.821796 | −1.48302 | 0 | 1.00000 | 1.97917 | 0 | 2.49121 | ||||||||||||||||||
1.11 | −1.50508 | 0 | 0.265269 | 2.12799 | 0 | 1.00000 | 2.61091 | 0 | −3.20280 | ||||||||||||||||||
1.12 | −1.42650 | 0 | 0.0348943 | 4.07722 | 0 | 1.00000 | 2.80322 | 0 | −5.81614 | ||||||||||||||||||
1.13 | −1.20189 | 0 | −0.555461 | 0.607848 | 0 | 1.00000 | 3.07138 | 0 | −0.730567 | ||||||||||||||||||
1.14 | −1.13347 | 0 | −0.715244 | −1.11166 | 0 | 1.00000 | 3.07765 | 0 | 1.26004 | ||||||||||||||||||
1.15 | −1.07579 | 0 | −0.842680 | 1.06884 | 0 | 1.00000 | 3.05812 | 0 | −1.14985 | ||||||||||||||||||
1.16 | −0.997785 | 0 | −1.00442 | −3.25439 | 0 | 1.00000 | 2.99777 | 0 | 3.24718 | ||||||||||||||||||
1.17 | −0.476800 | 0 | −1.77266 | −3.87912 | 0 | 1.00000 | 1.79880 | 0 | 1.84956 | ||||||||||||||||||
1.18 | −0.308449 | 0 | −1.90486 | 1.85496 | 0 | 1.00000 | 1.20445 | 0 | −0.572160 | ||||||||||||||||||
1.19 | −0.301452 | 0 | −1.90913 | −3.64431 | 0 | 1.00000 | 1.17841 | 0 | 1.09858 | ||||||||||||||||||
1.20 | −0.0449988 | 0 | −1.99798 | 2.40582 | 0 | 1.00000 | 0.179904 | 0 | −0.108259 | ||||||||||||||||||
See all 40 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(-1\) |
\(127\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8001.2.a.ba | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 8001.2.a.ba | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8001.2.a.ba | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
8001.2.a.ba | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
\( T_{2}^{40} - 67 T_{2}^{38} + 2061 T_{2}^{36} - 38594 T_{2}^{34} + 491854 T_{2}^{32} - 4518569 T_{2}^{30} + 30923256 T_{2}^{28} - 160686890 T_{2}^{26} + 640472505 T_{2}^{24} - 1965116795 T_{2}^{22} + \cdots + 1024 \) |
\( T_{5}^{40} - 142 T_{5}^{38} + 9221 T_{5}^{36} - 363032 T_{5}^{34} + 9688313 T_{5}^{32} - 185682906 T_{5}^{30} + 2642567744 T_{5}^{28} - 28492106123 T_{5}^{26} + 235485001640 T_{5}^{24} + \cdots + 1283525853184 \) |