Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5000,2,Mod(1,5000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5000, 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("5000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5000 = 2^{3} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5000.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: | \(39.9252010106\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 16x^{6} + 22x^{5} + 86x^{4} - 60x^{3} - 155x^{2} + 40x + 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 200) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 16x^{6} + 22x^{5} + 86x^{4} - 60x^{3} - 155x^{2} + 40x + 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{2} - 3\nu + 12 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 8\nu^{2} - 9\nu - 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 6\nu^{6} - 8\nu^{5} + 70\nu^{4} + 14\nu^{3} - 228\nu^{2} + 5\nu + 164 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 12\nu^{5} + 18\nu^{4} + 42\nu^{3} - 40\nu^{2} - 39\nu + 20 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 16\nu^{5} - 18\nu^{4} - 82\nu^{3} + 20\nu^{2} + 95\nu + 12 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} - 14\nu^{6} - 72\nu^{5} + 174\nu^{4} + 310\nu^{3} - 564\nu^{2} - 311\nu + 372 ) / 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{4} + 2\beta_{3} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{7} + 10\beta_{6} - 10\beta_{4} + 10\beta_{3} - 8\beta_{2} + 13\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{7} + 17\beta_{6} + 2\beta_{5} - 15\beta_{4} + 25\beta_{3} - 5\beta_{2} + 61\beta _1 + 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( 91\beta_{7} + 93\beta_{6} + 4\beta_{5} - 99\beta_{4} + 94\beta_{3} - 62\beta_{2} + 145\beta _1 + 142 \)
|
| \(\nu^{7}\) | \(=\) |
\( 180\beta_{7} + 208\beta_{6} + 40\beta_{5} - 196\beta_{4} + 264\beta_{3} - 80\beta_{2} + 573\beta _1 + 300 \)
|
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 | −3.20817 | 0 | 0 | 0 | 2.69216 | 0 | 7.29234 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.06792 | 0 | 0 | 0 | −1.36851 | 0 | 6.41211 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.49708 | 0 | 0 | 0 | −4.94031 | 0 | −0.758738 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.961326 | 0 | 0 | 0 | 0.364298 | 0 | −2.07585 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.866733 | 0 | 0 | 0 | 3.82614 | 0 | −2.24877 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.20531 | 0 | 0 | 0 | 1.59935 | 0 | −1.54722 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.16251 | 0 | 0 | 0 | −3.20389 | 0 | 1.67645 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.49994 | 0 | 0 | 0 | −1.96923 | 0 | 3.24969 | 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 | 5000.2.a.l | 8 | |
| 4.b | odd | 2 | 1 | 10000.2.a.bk | 8 | ||
| 5.b | even | 2 | 1 | 5000.2.a.m | 8 | ||
| 20.d | odd | 2 | 1 | 10000.2.a.bh | 8 | ||
| 25.d | even | 5 | 2 | 200.2.m.c | ✓ | 16 | |
| 25.e | even | 10 | 2 | 1000.2.m.c | 16 | ||
| 25.f | odd | 20 | 4 | 1000.2.q.d | 32 | ||
| 100.j | odd | 10 | 2 | 400.2.u.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.2.m.c | ✓ | 16 | 25.d | even | 5 | 2 | |
| 400.2.u.g | 16 | 100.j | odd | 10 | 2 | ||
| 1000.2.m.c | 16 | 25.e | even | 10 | 2 | ||
| 1000.2.q.d | 32 | 25.f | odd | 20 | 4 | ||
| 5000.2.a.l | 8 | 1.a | even | 1 | 1 | trivial | |
| 5000.2.a.m | 8 | 5.b | even | 2 | 1 | ||
| 10000.2.a.bh | 8 | 20.d | odd | 2 | 1 | ||
| 10000.2.a.bk | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5000))\):
|
\( T_{3}^{8} + 2T_{3}^{7} - 16T_{3}^{6} - 22T_{3}^{5} + 86T_{3}^{4} + 60T_{3}^{3} - 155T_{3}^{2} - 40T_{3} + 80 \)
|
|
\( T_{7}^{8} + 3T_{7}^{7} - 28T_{7}^{6} - 65T_{7}^{5} + 221T_{7}^{4} + 380T_{7}^{3} - 512T_{7}^{2} - 576T_{7} + 256 \)
|
|
\( T_{11}^{8} - 5T_{11}^{7} - 34T_{11}^{6} + 195T_{11}^{5} + 261T_{11}^{4} - 2290T_{11}^{3} + 716T_{11}^{2} + 7920T_{11} - 7744 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 2 T^{7} + \cdots + 80 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 256 \)
|
| $11$ |
\( T^{8} - 5 T^{7} + \cdots - 7744 \)
|
| $13$ |
\( T^{8} + 7 T^{7} + \cdots + 131 \)
|
| $17$ |
\( T^{8} + 7 T^{7} + \cdots + 1220 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots - 64 \)
|
| $23$ |
\( T^{8} + 7 T^{7} + \cdots + 124496 \)
|
| $29$ |
\( T^{8} + 25 T^{7} + \cdots + 416161 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 320 \)
|
| $37$ |
\( T^{8} + 15 T^{7} + \cdots + 85741 \)
|
| $41$ |
\( T^{8} + 26 T^{7} + \cdots - 547964 \)
|
| $43$ |
\( T^{8} + 21 T^{7} + \cdots - 1321984 \)
|
| $47$ |
\( T^{8} - 2 T^{7} + \cdots + 9427696 \)
|
| $53$ |
\( T^{8} + 12 T^{7} + \cdots + 16736336 \)
|
| $59$ |
\( T^{8} - 36 T^{7} + \cdots - 1014464 \)
|
| $61$ |
\( T^{8} + 33 T^{7} + \cdots + 1170145 \)
|
| $67$ |
\( T^{8} + 7 T^{7} + \cdots + 178496 \)
|
| $71$ |
\( T^{8} - 12 T^{7} + \cdots - 6320 \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots - 5696764 \)
|
| $79$ |
\( T^{8} + 16 T^{7} + \cdots - 192320 \)
|
| $83$ |
\( T^{8} + T^{7} + \cdots - 794384 \)
|
| $89$ |
\( T^{8} + 30 T^{7} + \cdots - 40111424 \)
|
| $97$ |
\( T^{8} + 32 T^{7} + \cdots - 1494949 \)
|
| show more | |
| show less |