Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,6,Mod(1,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.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: | \(128.307055850\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2x^{4} - 226x^{3} + 455x^{2} + 9816x + 4656 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{4} - 12\nu^{3} + 272\nu^{2} + 1098\nu - 96 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{4} - 12\nu^{3} + 356\nu^{2} + 1140\nu - 7782 ) / 21 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 180\nu^{3} + 568\nu^{2} - 23904\nu - 67356 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - \beta _1 + 365 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{4} - 10\beta_{3} + 11\beta_{2} + 553\beta _1 + 105 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -15\beta_{4} + 166\beta_{3} - 211\beta_{2} - 697\beta _1 + 50231 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −27.8730 | 0 | 0 | 0 | −108.742 | 0 | 533.902 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −12.8999 | 0 | 0 | 0 | 121.874 | 0 | −76.5929 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.97606 | 0 | 0 | 0 | −48.9425 | 0 | −239.095 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 19.8018 | 0 | 0 | 0 | 160.939 | 0 | 149.111 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 21.9471 | 0 | 0 | 0 | −235.128 | 0 | 238.676 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 800.6.a.s | ✓ | 5 |
| 4.b | odd | 2 | 1 | 800.6.a.v | yes | 5 | |
| 5.b | even | 2 | 1 | 800.6.a.u | yes | 5 | |
| 5.c | odd | 4 | 2 | 800.6.c.n | 10 | ||
| 20.d | odd | 2 | 1 | 800.6.a.t | yes | 5 | |
| 20.e | even | 4 | 2 | 800.6.c.o | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.6.a.s | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 800.6.a.t | yes | 5 | 20.d | odd | 2 | 1 | |
| 800.6.a.u | yes | 5 | 5.b | even | 2 | 1 | |
| 800.6.a.v | yes | 5 | 4.b | odd | 2 | 1 | |
| 800.6.c.n | 10 | 5.c | odd | 4 | 2 | ||
| 800.6.c.o | 10 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):
|
\( T_{3}^{5} + T_{3}^{4} - 910T_{3}^{3} + 914T_{3}^{2} + 161613T_{3} + 308781 \)
|
|
\( T_{11}^{5} + 5T_{11}^{4} - 581526T_{11}^{3} - 34767310T_{11}^{2} + 82493492725T_{11} + 11275947155625 \)
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\( T_{13}^{5} + 280T_{13}^{4} - 581616T_{13}^{3} - 34060800T_{13}^{2} + 49678848000T_{13} + 5363712000000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 308781 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 24544747872 \)
|
| $11$ |
\( T^{5} + \cdots + 11275947155625 \)
|
| $13$ |
\( T^{5} + \cdots + 5363712000000 \)
|
| $17$ |
\( T^{5} + \cdots + 10944985381875 \)
|
| $19$ |
\( T^{5} + \cdots - 13\!\cdots\!75 \)
|
| $23$ |
\( T^{5} + \cdots + 377296470599904 \)
|
| $29$ |
\( T^{5} + \cdots + 75\!\cdots\!64 \)
|
| $31$ |
\( T^{5} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 69\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots - 82\!\cdots\!25 \)
|
| $43$ |
\( T^{5} + \cdots - 66\!\cdots\!56 \)
|
| $47$ |
\( T^{5} + \cdots + 44\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots + 66\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 12\!\cdots\!77 \)
|
| $71$ |
\( T^{5} + \cdots - 55\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 28\!\cdots\!75 \)
|
| $79$ |
\( T^{5} + \cdots - 78\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 32\!\cdots\!75 \)
|
| $89$ |
\( T^{5} + \cdots + 16\!\cdots\!69 \)
|
| $97$ |
\( T^{5} + \cdots - 19\!\cdots\!00 \)
|
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