Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [125,4,Mod(1,125)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, 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("125.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 125.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: | \(7.37523875072\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.497918125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{5} - 19\nu^{4} + 161\nu^{3} + 260\nu^{2} - 1344\nu - 712 ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 23\nu^{3} - 40\nu^{2} - 152\nu - 8 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 9\nu^{4} + 237\nu^{3} - 44\nu^{2} - 2048\nu - 1576 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -17\nu^{5} + 31\nu^{4} + 475\nu^{3} - 372\nu^{2} - 3200\nu - 1304 ) / 128 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{5} - 31\nu^{4} - 411\nu^{3} + 180\nu^{2} + 2624\nu + 2200 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - \beta_{3} - 3\beta_{2} - \beta _1 + 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} + \beta_{4} + 2\beta_{3} - 7\beta_{2} - 3\beta _1 + 49 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{5} + 31\beta_{4} - 3\beta_{3} - 48\beta_{2} - 18\beta _1 + 86 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -14\beta_{5} + 52\beta_{4} + 49\beta_{3} - 149\beta_{2} - 91\beta _1 + 793 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( 26\beta_{5} + 105\beta_{4} + 30\beta_{3} - 179\beta_{2} - 83\beta _1 + 441 \)
|
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 |
|
−5.19564 | 6.10690 | 18.9947 | 0 | −31.7292 | −29.3201 | −57.1243 | 10.2942 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.23279 | −5.51148 | 9.91650 | 0 | 23.3289 | −2.28738 | −8.11216 | 3.37638 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.61169 | −9.75552 | 5.04433 | 0 | 35.2339 | 14.1661 | 10.6750 | 68.1701 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.288727 | 4.57894 | −7.91664 | 0 | −1.32206 | −18.9048 | 4.59556 | −6.03327 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.95323 | −1.11531 | −4.18489 | 0 | −2.17846 | −1.57558 | −23.7999 | −25.7561 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.37562 | −8.30353 | 11.1460 | 0 | −36.3331 | −29.0783 | 13.7658 | 41.9487 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
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 | 125.4.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1125.4.a.k | 6 | ||
| 4.b | odd | 2 | 1 | 2000.4.a.n | 6 | ||
| 5.b | even | 2 | 1 | 125.4.a.c | yes | 6 | |
| 5.c | odd | 4 | 2 | 125.4.b.c | 12 | ||
| 15.d | odd | 2 | 1 | 1125.4.a.f | 6 | ||
| 20.d | odd | 2 | 1 | 2000.4.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 125.4.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 125.4.a.c | yes | 6 | 5.b | even | 2 | 1 | |
| 125.4.b.c | 12 | 5.c | odd | 4 | 2 | ||
| 1125.4.a.f | 6 | 15.d | odd | 2 | 1 | ||
| 1125.4.a.k | 6 | 3.b | odd | 2 | 1 | ||
| 2000.4.a.k | 6 | 20.d | odd | 2 | 1 | ||
| 2000.4.a.n | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{5} - 16T_{2}^{4} - 169T_{2}^{3} - 71T_{2}^{2} + 672T_{2} + 196 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(125))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 7 T^{5} + \cdots + 196 \)
|
| $3$ |
\( T^{6} + 14 T^{5} + \cdots + 13924 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 67 T^{5} + \cdots - 822879 \)
|
| $11$ |
\( T^{6} - 27 T^{5} + \cdots - 47445831 \)
|
| $13$ |
\( T^{6} + \cdots - 7178567571 \)
|
| $17$ |
\( T^{6} + \cdots + 27354562736 \)
|
| $19$ |
\( T^{6} + \cdots + 29290968875 \)
|
| $23$ |
\( T^{6} + \cdots - 53831893136 \)
|
| $29$ |
\( T^{6} + \cdots - 369868315500 \)
|
| $31$ |
\( T^{6} + \cdots + 3206882909804 \)
|
| $37$ |
\( T^{6} + \cdots - 462416505264 \)
|
| $41$ |
\( T^{6} + \cdots + 18008321955939 \)
|
| $43$ |
\( T^{6} + \cdots + 395909856491184 \)
|
| $47$ |
\( T^{6} + \cdots - 9620559208829 \)
|
| $53$ |
\( T^{6} + \cdots + 10657074605569 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!75 \)
|
| $61$ |
\( T^{6} + \cdots + 153598743998924 \)
|
| $67$ |
\( T^{6} + \cdots - 59\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 42\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots + 62\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 64\!\cdots\!64 \)
|
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