Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8035,2,Mod(1,8035)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8035, 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("8035.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8035 = 5 \cdot 1607 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8035.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: | \(64.1597980241\) |
Analytic rank: | \(1\) |
Dimension: | \(140\) |
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 | −2.82578 | 1.42747 | 5.98503 | −1.00000 | −4.03370 | 3.78558 | −11.2608 | −0.962340 | 2.82578 | ||||||||||||||||||
1.2 | −2.79540 | −1.02268 | 5.81427 | −1.00000 | 2.85879 | 3.00451 | −10.6624 | −1.95413 | 2.79540 | ||||||||||||||||||
1.3 | −2.76648 | 3.20166 | 5.65342 | −1.00000 | −8.85734 | −4.04586 | −10.1071 | 7.25064 | 2.76648 | ||||||||||||||||||
1.4 | −2.72603 | −2.40516 | 5.43123 | −1.00000 | 6.55652 | −0.808497 | −9.35364 | 2.78477 | 2.72603 | ||||||||||||||||||
1.5 | −2.70698 | 2.06859 | 5.32774 | −1.00000 | −5.59964 | 3.95188 | −9.00814 | 1.27907 | 2.70698 | ||||||||||||||||||
1.6 | −2.70543 | 0.0377636 | 5.31934 | −1.00000 | −0.102167 | −3.03479 | −8.98024 | −2.99857 | 2.70543 | ||||||||||||||||||
1.7 | −2.70407 | −1.94382 | 5.31198 | −1.00000 | 5.25623 | −4.62711 | −8.95580 | 0.778451 | 2.70407 | ||||||||||||||||||
1.8 | −2.69188 | −1.94222 | 5.24620 | −1.00000 | 5.22822 | 1.57451 | −8.73838 | 0.772219 | 2.69188 | ||||||||||||||||||
1.9 | −2.67655 | −2.96805 | 5.16395 | −1.00000 | 7.94414 | 4.47636 | −8.46847 | 5.80930 | 2.67655 | ||||||||||||||||||
1.10 | −2.66914 | −3.32233 | 5.12432 | −1.00000 | 8.86776 | −1.95943 | −8.33924 | 8.03786 | 2.66914 | ||||||||||||||||||
1.11 | −2.56068 | −0.755855 | 4.55711 | −1.00000 | 1.93551 | −1.57635 | −6.54794 | −2.42868 | 2.56068 | ||||||||||||||||||
1.12 | −2.54398 | 1.20144 | 4.47181 | −1.00000 | −3.05645 | −0.416723 | −6.28824 | −1.55653 | 2.54398 | ||||||||||||||||||
1.13 | −2.54024 | 0.460298 | 4.45282 | −1.00000 | −1.16927 | −2.08486 | −6.23076 | −2.78813 | 2.54024 | ||||||||||||||||||
1.14 | −2.53622 | 2.11151 | 4.43241 | −1.00000 | −5.35524 | 2.12387 | −6.16912 | 1.45846 | 2.53622 | ||||||||||||||||||
1.15 | −2.44538 | −0.0835630 | 3.97986 | −1.00000 | 0.204343 | 5.18435 | −4.84151 | −2.99302 | 2.44538 | ||||||||||||||||||
1.16 | −2.42652 | 0.577409 | 3.88800 | −1.00000 | −1.40110 | 3.20436 | −4.58127 | −2.66660 | 2.42652 | ||||||||||||||||||
1.17 | −2.41030 | −2.44694 | 3.80955 | −1.00000 | 5.89785 | −4.60731 | −4.36157 | 2.98749 | 2.41030 | ||||||||||||||||||
1.18 | −2.40990 | 2.64302 | 3.80764 | −1.00000 | −6.36944 | −3.14385 | −4.35625 | 3.98558 | 2.40990 | ||||||||||||||||||
1.19 | −2.36612 | −3.07539 | 3.59854 | −1.00000 | 7.27675 | 3.28740 | −3.78235 | 6.45802 | 2.36612 | ||||||||||||||||||
1.20 | −2.31791 | 3.17051 | 3.37269 | −1.00000 | −7.34893 | 3.65453 | −3.18176 | 7.05211 | 2.31791 | ||||||||||||||||||
See next 80 embeddings (of 140 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(1607\) | \(1\) |
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 | 8035.2.a.d | ✓ | 140 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8035.2.a.d | ✓ | 140 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{140} + 20 T_{2}^{139} - 12 T_{2}^{138} - 2859 T_{2}^{137} - 12842 T_{2}^{136} + 182399 T_{2}^{135} + \cdots - 846834921 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8035))\).