Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8750,2,Mod(1,8750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8750, 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("8750.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8750 = 2 \cdot 5^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8750.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: | \(69.8691017686\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 23x^{8} + 17x^{7} + 180x^{6} - 87x^{5} - 523x^{4} + 126x^{3} + 361x^{2} - 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 350) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 23x^{8} + 17x^{7} + 180x^{6} - 87x^{5} - 523x^{4} + 126x^{3} + 361x^{2} - 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 7 \nu^{9} - 71 \nu^{8} + 293 \nu^{7} + 1915 \nu^{6} - 3000 \nu^{5} - 16921 \nu^{4} + 7947 \nu^{3} + \cdots - 16950 ) / 3590 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 67 \nu^{9} - 141 \nu^{8} - 1471 \nu^{7} + 2185 \nu^{6} + 11790 \nu^{5} - 9849 \nu^{4} - 38523 \nu^{3} + \cdots - 3930 ) / 7180 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37 \nu^{9} - 35 \nu^{8} - 523 \nu^{7} + 135 \nu^{6} + 2010 \nu^{5} + 1741 \nu^{4} - 1695 \nu^{3} + \cdots - 670 ) / 3590 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 223 \nu^{9} + 405 \nu^{8} + 4103 \nu^{7} - 5665 \nu^{6} - 25310 \nu^{5} + 18421 \nu^{4} + \cdots - 10710 ) / 7180 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 305 \nu^{9} - 599 \nu^{8} - 5689 \nu^{7} + 8875 \nu^{6} + 34810 \nu^{5} - 37655 \nu^{4} - 73217 \nu^{3} + \cdots + 5150 ) / 7180 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 285 \nu^{9} + 289 \nu^{8} + 6493 \nu^{7} - 5115 \nu^{6} - 49830 \nu^{5} + 27535 \nu^{4} + \cdots + 5840 ) / 3590 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 285 \nu^{9} - 289 \nu^{8} - 6493 \nu^{7} + 5115 \nu^{6} + 49830 \nu^{5} - 27535 \nu^{4} - 137697 \nu^{3} + \cdots - 9430 ) / 3590 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{4} + 8\beta_{2} + \beta _1 + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{9} + 12\beta_{8} - 3\beta_{7} - 2\beta_{6} + 2\beta_{5} + 5\beta_{4} + \beta_{3} + 68\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17 \beta_{9} + 16 \beta_{8} - 16 \beta_{7} - 29 \beta_{6} - 13 \beta_{5} - 17 \beta_{4} + 4 \beta_{3} + \cdots + 309 \)
|
| \(\nu^{7}\) | \(=\) |
\( 125 \beta_{9} + 125 \beta_{8} - 50 \beta_{7} - 40 \beta_{6} + 35 \beta_{5} + 60 \beta_{4} + 20 \beta_{3} + \cdots + 245 \)
|
| \(\nu^{8}\) | \(=\) |
\( 217 \beta_{9} + 192 \beta_{8} - 210 \beta_{7} - 330 \beta_{6} - 100 \beta_{5} - 230 \beta_{4} + \cdots + 2602 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1257 \beta_{9} + 1237 \beta_{8} - 637 \beta_{7} - 569 \beta_{6} + 483 \beta_{5} + 468 \beta_{4} + \cdots + 2771 \)
|
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 |
|
−1.00000 | −3.20804 | 1.00000 | 0 | 3.20804 | 1.00000 | −1.00000 | 7.29151 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.99313 | 1.00000 | 0 | 2.99313 | 1.00000 | −1.00000 | 5.95884 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.33236 | 1.00000 | 0 | 2.33236 | 1.00000 | −1.00000 | 2.43992 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.06644 | 1.00000 | 0 | 1.06644 | 1.00000 | −1.00000 | −1.86270 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.235339 | 1.00000 | 0 | 0.235339 | 1.00000 | −1.00000 | −2.94462 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.260341 | 1.00000 | 0 | −0.260341 | 1.00000 | −1.00000 | −2.93222 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0.803008 | 1.00000 | 0 | −0.803008 | 1.00000 | −1.00000 | −2.35518 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.17194 | 1.00000 | 0 | −2.17194 | 1.00000 | −1.00000 | 1.71733 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 2.74101 | 1.00000 | 0 | −2.74101 | 1.00000 | −1.00000 | 4.51311 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 2.85902 | 1.00000 | 0 | −2.85902 | 1.00000 | −1.00000 | 5.17400 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
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 | 8750.2.a.w | 10 | |
| 5.b | even | 2 | 1 | 8750.2.a.x | 10 | ||
| 25.d | even | 5 | 2 | 350.2.h.d | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.2.h.d | ✓ | 20 | 25.d | even | 5 | 2 | |
| 8750.2.a.w | 10 | 1.a | even | 1 | 1 | trivial | |
| 8750.2.a.x | 10 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8750))\):
|
\( T_{3}^{10} + T_{3}^{9} - 23T_{3}^{8} - 17T_{3}^{7} + 180T_{3}^{6} + 87T_{3}^{5} - 523T_{3}^{4} - 126T_{3}^{3} + 361T_{3}^{2} - 20 \)
|
|
\( T_{11}^{10} - 3 T_{11}^{9} - 84 T_{11}^{8} + 300 T_{11}^{7} + 2135 T_{11}^{6} - 8730 T_{11}^{5} + \cdots - 113600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( T^{10} + T^{9} + \cdots - 20 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( (T - 1)^{10} \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots - 113600 \)
|
| $13$ |
\( T^{10} + 10 T^{9} + \cdots + 18911 \)
|
| $17$ |
\( T^{10} + 6 T^{9} + \cdots - 5296 \)
|
| $19$ |
\( T^{10} - 6 T^{9} + \cdots + 31680 \)
|
| $23$ |
\( T^{10} + 5 T^{9} + \cdots + 522500 \)
|
| $29$ |
\( T^{10} - 9 T^{9} + \cdots + 704176 \)
|
| $31$ |
\( T^{10} - 14 T^{9} + \cdots - 17856 \)
|
| $37$ |
\( T^{10} + 9 T^{9} + \cdots + 735920 \)
|
| $41$ |
\( T^{10} + 13 T^{9} + \cdots + 5104 \)
|
| $43$ |
\( T^{10} - T^{9} + \cdots - 2869504 \)
|
| $47$ |
\( T^{10} + 22 T^{9} + \cdots + 1024 \)
|
| $53$ |
\( T^{10} - 22 T^{9} + \cdots - 1289536 \)
|
| $59$ |
\( T^{10} - 26 T^{9} + \cdots - 38254276 \)
|
| $61$ |
\( T^{10} + \cdots + 1168476791 \)
|
| $67$ |
\( T^{10} - 2 T^{9} + \cdots + 8730304 \)
|
| $71$ |
\( T^{10} - 18 T^{9} + \cdots + 928400 \)
|
| $73$ |
\( T^{10} + 3 T^{9} + \cdots - 1607936 \)
|
| $79$ |
\( T^{10} + \cdots - 280074076 \)
|
| $83$ |
\( T^{10} + \cdots + 2621951344 \)
|
| $89$ |
\( T^{10} - 22 T^{9} + \cdots - 63711280 \)
|
| $97$ |
\( T^{10} + \cdots - 203488400 \)
|
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