Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1250,4,Mod(1,1250)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1250, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1250.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1250 = 2 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1250.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: | \(73.7523875072\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 145x^{6} - 180x^{5} + 5585x^{4} + 8550x^{3} - 49600x^{2} - 23400x + 24400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | no (minimal twist has level 50) |
| 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
\( x^{8} - 145x^{6} - 180x^{5} + 5585x^{4} + 8550x^{3} - 49600x^{2} - 23400x + 24400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3 \nu^{7} + 7862 \nu^{6} - 71095 \nu^{5} - 669690 \nu^{4} + 4411135 \nu^{3} + 14153360 \nu^{2} + \cdots + 19422920 ) / 25315520 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 483 \nu^{7} + 6 \nu^{6} - 54311 \nu^{5} - 229130 \nu^{4} + 1358175 \nu^{3} + 12951920 \nu^{2} + \cdots - 87980920 ) / 50631040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8957 \nu^{7} - 267418 \nu^{6} + 2569081 \nu^{5} + 31189430 \nu^{4} - 136905185 \nu^{3} + \cdots - 481941880 ) / 151893120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12677 \nu^{7} + 109910 \nu^{6} + 2122561 \nu^{5} - 11780410 \nu^{4} - 125902505 \nu^{3} + \cdots + 1204836680 ) / 151893120 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 43249 \nu^{7} + 156702 \nu^{6} + 5557613 \nu^{5} - 9725490 \nu^{4} - 198111205 \nu^{3} + \cdots - 768664920 ) / 75946560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 87161 \nu^{7} + 158322 \nu^{6} - 12903685 \nu^{5} - 33617310 \nu^{4} + 472382285 \nu^{3} + \cdots - 2650904040 ) / 151893120 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - 2\beta_{2} + 3\beta _1 + 35 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{7} + 3\beta_{6} - 5\beta_{5} - \beta_{4} - 51\beta_{3} - 24\beta_{2} + 76\beta _1 + 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( 98\beta_{7} + 86\beta_{6} - 76\beta_{5} + 82\beta_{4} - 916\beta_{3} - 258\beta_{2} + 419\beta _1 + 2050 \)
|
| \(\nu^{5}\) | \(=\) |
\( 545\beta_{7} + 485\beta_{6} - 525\beta_{5} + 105\beta_{4} - 7735\beta_{3} - 3230\beta_{2} + 6765\beta _1 + 7955 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9765 \beta_{7} + 8205 \beta_{6} - 6595 \beta_{5} + 6685 \beta_{4} - 113585 \beta_{3} - 30760 \beta_{2} + \cdots + 153215 \)
|
| \(\nu^{7}\) | \(=\) |
\( 72400 \beta_{7} + 59980 \beta_{6} - 54130 \beta_{5} + 26620 \beta_{4} - 974210 \beta_{3} - 364090 \beta_{2} + \cdots + 970920 \)
|
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 |
|
−2.00000 | −8.31354 | 4.00000 | 0 | 16.6271 | 26.4146 | −8.00000 | 42.1150 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.15719 | 4.00000 | 0 | 10.3144 | 3.36421 | −8.00000 | −0.403415 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.31006 | 4.00000 | 0 | 6.62013 | 25.0474 | −8.00000 | −16.0435 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −1.55615 | 4.00000 | 0 | 3.11230 | −19.9781 | −8.00000 | −24.5784 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −0.0915752 | 4.00000 | 0 | 0.183150 | −10.7092 | −8.00000 | −26.9916 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 4.39780 | 4.00000 | 0 | −8.79560 | −32.4633 | −8.00000 | −7.65935 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | 8.30552 | 4.00000 | 0 | −16.6110 | 11.9615 | −8.00000 | 41.9816 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | 9.72520 | 4.00000 | 0 | −19.4504 | 23.3628 | −8.00000 | 67.5796 | 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 | 1250.4.a.g | 8 | |
| 5.b | even | 2 | 1 | 1250.4.a.j | 8 | ||
| 25.d | even | 5 | 2 | 250.4.d.b | 16 | ||
| 25.e | even | 10 | 2 | 50.4.d.b | ✓ | 16 | |
| 25.f | odd | 20 | 4 | 250.4.e.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.4.d.b | ✓ | 16 | 25.e | even | 10 | 2 | |
| 250.4.d.b | 16 | 25.d | even | 5 | 2 | ||
| 250.4.e.c | 32 | 25.f | odd | 20 | 4 | ||
| 1250.4.a.g | 8 | 1.a | even | 1 | 1 | trivial | |
| 1250.4.a.j | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} - 138T_{3}^{6} + 298T_{3}^{5} + 5680T_{3}^{4} - 278T_{3}^{3} - 63483T_{3}^{2} - 84256T_{3} - 7184 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1250))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots - 7184 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 4320209664 \)
|
| $11$ |
\( T^{8} + \cdots - 1769565317824 \)
|
| $13$ |
\( T^{8} + \cdots - 519553779309 \)
|
| $17$ |
\( T^{8} + \cdots + 98261859394116 \)
|
| $19$ |
\( T^{8} + \cdots - 109942918897600 \)
|
| $23$ |
\( T^{8} + \cdots - 65\!\cdots\!64 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!25 \)
|
| $31$ |
\( T^{8} + \cdots - 55950338499264 \)
|
| $37$ |
\( T^{8} + \cdots - 15\!\cdots\!59 \)
|
| $41$ |
\( T^{8} + \cdots - 14\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots + 15\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 37\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $59$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 28\!\cdots\!61 \)
|
| $67$ |
\( T^{8} + \cdots + 20\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots - 12\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots + 79\!\cdots\!96 \)
|
| $79$ |
\( T^{8} + \cdots - 96\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 68\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 35\!\cdots\!71 \)
|
| show more | |
| show less |