Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2500,2,Mod(1,2500)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, 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("2500.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2500 = 2^{2} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2500.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: | \(19.9626005053\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.54296578125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 13x^{6} + 30x^{5} + 61x^{4} - 90x^{3} - 117x^{2} + 81x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 13x^{6} + 30x^{5} + 61x^{4} - 90x^{3} - 117x^{2} + 81x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 13\nu^{5} + 30\nu^{4} + 61\nu^{3} - 90\nu^{2} - 90\nu + 54 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 6\nu^{6} + 4\nu^{5} - 69\nu^{4} + 29\nu^{3} + 246\nu^{2} - 99\nu - 243 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} + 12\nu^{6} + 43\nu^{5} - 93\nu^{4} - 154\nu^{3} + 171\nu^{2} + 162\nu - 27 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 10\nu^{4} - 21\nu^{3} - 31\nu^{2} + 30\nu + 24 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 10\nu^{4} - 21\nu^{3} - 34\nu^{2} + 33\nu + 36 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 22\nu^{5} - 115\nu^{3} - 30\nu^{2} + 144\nu + 81 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} + 2\beta_{5} + 3\beta_{2} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} - 16\beta_{6} + 12\beta_{5} - 3\beta_{3} + 6\beta_{2} + 17\beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{7} - 57\beta_{6} + 35\beta_{5} - 3\beta_{4} - 9\beta_{3} + 36\beta_{2} + 78\beta _1 + 77 \)
|
| \(\nu^{6}\) | \(=\) |
\( 66\beta_{7} - 237\beta_{6} + 149\beta_{5} - 9\beta_{4} - 57\beta_{3} + 105\beta_{2} + 256\beta _1 + 326 \)
|
| \(\nu^{7}\) | \(=\) |
\( 294\beta_{7} - 879\beta_{6} + 510\beta_{5} - 66\beta_{4} - 198\beta_{3} + 447\beta_{2} + 1025\beta _1 + 1080 \)
|
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 | −2.75519 | 0 | 0 | 0 | 2.12331 | 0 | 4.59107 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.43905 | 0 | 0 | 0 | −1.43699 | 0 | −0.929147 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.571158 | 0 | 0 | 0 | −4.82694 | 0 | −2.67378 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.367190 | 0 | 0 | 0 | 2.46557 | 0 | −2.86517 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.79889 | 0 | 0 | 0 | 1.17125 | 0 | 0.236021 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.22999 | 0 | 0 | 0 | 3.47314 | 0 | 1.97285 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.90942 | 0 | 0 | 0 | 1.53239 | 0 | 5.46472 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.19428 | 0 | 0 | 0 | −4.50172 | 0 | 7.20343 | 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 | 2500.2.a.g | yes | 8 |
| 4.b | odd | 2 | 1 | 10000.2.a.bf | 8 | ||
| 5.b | even | 2 | 1 | 2500.2.a.e | ✓ | 8 | |
| 5.c | odd | 4 | 2 | 2500.2.c.d | 16 | ||
| 20.d | odd | 2 | 1 | 10000.2.a.bm | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2500.2.a.e | ✓ | 8 | 5.b | even | 2 | 1 | |
| 2500.2.a.g | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 2500.2.c.d | 16 | 5.c | odd | 4 | 2 | ||
| 10000.2.a.bf | 8 | 4.b | odd | 2 | 1 | ||
| 10000.2.a.bm | 8 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 5T_{3}^{7} - 6T_{3}^{6} + 55T_{3}^{5} - 19T_{3}^{4} - 145T_{3}^{3} + 49T_{3}^{2} + 120T_{3} + 31 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 5 T^{7} + \cdots + 31 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 36 T^{6} + \cdots - 1019 \)
|
| $11$ |
\( T^{8} + 5 T^{7} + \cdots + 2025 \)
|
| $13$ |
\( T^{8} - 10 T^{7} + \cdots - 279 \)
|
| $17$ |
\( T^{8} - 25 T^{7} + \cdots - 2349 \)
|
| $19$ |
\( T^{8} + 2 T^{7} + \cdots - 17519 \)
|
| $23$ |
\( T^{8} - 109 T^{6} + \cdots + 206631 \)
|
| $29$ |
\( T^{8} + 2 T^{7} + \cdots + 34101 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 16561 \)
|
| $37$ |
\( T^{8} - 10 T^{7} + \cdots - 551459 \)
|
| $41$ |
\( T^{8} + 12 T^{7} + \cdots + 1825011 \)
|
| $43$ |
\( T^{8} + 10 T^{7} + \cdots - 957725 \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots + 77841 \)
|
| $53$ |
\( T^{8} - 50 T^{7} + \cdots + 2400921 \)
|
| $59$ |
\( T^{8} - 31 T^{7} + \cdots + 12366351 \)
|
| $61$ |
\( T^{8} + 14 T^{7} + \cdots - 1294589 \)
|
| $67$ |
\( T^{8} - 20 T^{7} + \cdots + 82261 \)
|
| $71$ |
\( T^{8} + T^{7} + \cdots + 12857211 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots + 16831 \)
|
| $79$ |
\( T^{8} + 16 T^{7} + \cdots + 107881 \)
|
| $83$ |
\( T^{8} - 40 T^{7} + \cdots - 731349 \)
|
| $89$ |
\( T^{8} + 13 T^{7} + \cdots + 3555171 \)
|
| $97$ |
\( T^{8} - 281 T^{6} + \cdots + 22801 \)
|
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