Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10000,2,Mod(1,10000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, 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("10000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10000 = 2^{4} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10000.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: | \(79.8504020213\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.3266578125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 5000) |
| 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
\( x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 8\nu^{5} + 15\nu^{4} + 16\nu^{3} - 29\nu^{2} - 3\nu + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{7} + 3\nu^{6} + 17\nu^{5} - 22\nu^{4} - 37\nu^{3} + 42\nu^{2} + 12\nu - 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{7} + 4\nu^{6} + 16\nu^{5} - 30\nu^{4} - 31\nu^{3} + 57\nu^{2} + 3\nu - 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( -3\nu^{7} + 5\nu^{6} + 25\nu^{5} - 36\nu^{4} - 54\nu^{3} + 66\nu^{2} + 19\nu - 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( 6\nu^{7} - 10\nu^{6} - 49\nu^{5} + 72\nu^{4} + 100\nu^{3} - 133\nu^{2} - 25\nu + 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + \beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} + 6\beta_{2} - \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + 8\beta_{5} + 16\beta_{3} + 9\beta_{2} + 11\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 10\beta_{6} + 11\beta_{5} - 9\beta_{4} + 28\beta_{3} + 36\beta_{2} - 6\beta _1 + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10\beta_{7} + 21\beta_{6} + 55\beta_{5} - 3\beta_{4} + 108\beta_{3} + 67\beta_{2} + 46\beta _1 + 93 \)
|
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 | −2.14505 | 0 | 0 | 0 | 2.05609 | 0 | 1.60124 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.70759 | 0 | 0 | 0 | 1.85864 | 0 | −0.0841263 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.470831 | 0 | 0 | 0 | −1.95931 | 0 | −2.77832 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.369714 | 0 | 0 | 0 | 3.50968 | 0 | −2.86331 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.31796 | 0 | 0 | 0 | −0.438060 | 0 | −1.26298 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.36933 | 0 | 0 | 0 | −2.47668 | 0 | −1.12493 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.67989 | 0 | 0 | 0 | 3.57735 | 0 | −0.177977 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.58658 | 0 | 0 | 0 | −4.12771 | 0 | 3.69040 | 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 | 10000.2.a.bl | 8 | |
| 4.b | odd | 2 | 1 | 5000.2.a.k | ✓ | 8 | |
| 5.b | even | 2 | 1 | 10000.2.a.bg | 8 | ||
| 20.d | odd | 2 | 1 | 5000.2.a.n | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5000.2.a.k | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 5000.2.a.n | yes | 8 | 20.d | odd | 2 | 1 | |
| 10000.2.a.bg | 8 | 5.b | even | 2 | 1 | ||
| 10000.2.a.bl | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):
|
\( T_{3}^{8} - 3T_{3}^{7} - 6T_{3}^{6} + 23T_{3}^{5} + T_{3}^{4} - 45T_{3}^{3} + 25T_{3}^{2} + 10T_{3} - 5 \)
|
|
\( T_{7}^{8} - 2T_{7}^{7} - 28T_{7}^{6} + 55T_{7}^{5} + 226T_{7}^{4} - 385T_{7}^{3} - 622T_{7}^{2} + 779T_{7} + 421 \)
|
|
\( T_{11}^{8} + 5T_{11}^{7} - 14T_{11}^{6} - 50T_{11}^{5} + 96T_{11}^{4} + 35T_{11}^{3} - 74T_{11}^{2} - 5T_{11} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 3 T^{7} + \cdots - 5 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 2 T^{7} + \cdots + 421 \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots + 1 \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots - 5939 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 7375 \)
|
| $19$ |
\( T^{8} - 10 T^{7} + \cdots + 12841 \)
|
| $23$ |
\( T^{8} + 2 T^{7} + \cdots + 14731 \)
|
| $29$ |
\( T^{8} + 10 T^{7} + \cdots - 54779 \)
|
| $31$ |
\( T^{8} - 27 T^{7} + \cdots - 155 \)
|
| $37$ |
\( T^{8} - 10 T^{7} + \cdots + 22441 \)
|
| $41$ |
\( T^{8} + 16 T^{7} + \cdots + 7471 \)
|
| $43$ |
\( T^{8} - 4 T^{7} + \cdots + 245551 \)
|
| $47$ |
\( T^{8} + 8 T^{7} + \cdots + 6301 \)
|
| $53$ |
\( T^{8} - 2 T^{7} + \cdots - 419 \)
|
| $59$ |
\( T^{8} - 39 T^{7} + \cdots + 2724451 \)
|
| $61$ |
\( T^{8} + 18 T^{7} + \cdots + 535195 \)
|
| $67$ |
\( T^{8} + 12 T^{7} + \cdots - 293099 \)
|
| $71$ |
\( T^{8} - 13 T^{7} + \cdots - 24106205 \)
|
| $73$ |
\( T^{8} - 12 T^{7} + \cdots + 2657971 \)
|
| $79$ |
\( T^{8} - 16 T^{7} + \cdots - 13311755 \)
|
| $83$ |
\( T^{8} - 64 T^{7} + \cdots - 15039569 \)
|
| $89$ |
\( T^{8} + 25 T^{7} + \cdots - 416729 \)
|
| $97$ |
\( T^{8} + 8 T^{7} + \cdots + 45223141 \)
|
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