Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5239,2,Mod(1,5239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, 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("5239.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5239 = 13^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5239.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: | \(41.8336256189\) |
| 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: |
\( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 403) |
| 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
\( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 10\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 8\nu^{4} - 6\nu^{3} - 15\nu^{2} + 9\nu + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 12\nu^{3} + 5\nu^{2} - 15\nu + 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 10\nu^{5} - \nu^{4} + 28\nu^{3} + \nu^{2} - 21\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{4} + 8\beta_{3} + 8\beta_{2} + 18\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 8\beta_{5} + 10\beta_{4} + 10\beta_{3} + 35\beta_{2} + 3\beta _1 + 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 11\beta_{6} + 11\beta_{5} + 21\beta_{4} + 53\beta_{3} + 57\beta_{2} + 89\beta _1 + 36 \)
|
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 |
|
−2.14079 | 0.217190 | 2.58298 | 0.434938 | −0.464957 | −3.26287 | −1.24803 | −2.95283 | −0.931109 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.06827 | 2.49221 | 2.27774 | −1.40222 | −5.15456 | 5.08294 | −0.574435 | 3.21112 | 2.90017 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.38904 | 1.62343 | −0.0705747 | −3.01447 | −2.25501 | −2.19250 | 2.87611 | −0.364464 | 4.18722 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.147670 | −2.43721 | −1.97819 | −4.40045 | −0.359904 | 2.90659 | −0.587461 | 2.94000 | −0.649816 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.656875 | 2.67827 | −1.56852 | 1.55873 | 1.75929 | −2.17607 | −2.34407 | 4.17313 | 1.02389 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.53950 | 0.734778 | 0.370072 | −3.72666 | 1.13119 | −3.41650 | −2.50928 | −2.46010 | −5.73720 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.71590 | −1.27835 | 0.944315 | 2.00831 | −2.19352 | 3.76625 | −1.81145 | −1.36583 | 3.44606 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.53815 | 2.96968 | 4.44218 | −2.45818 | 7.53747 | 1.29216 | 6.19862 | 5.81898 | −6.23922 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 5239.2.a.i | 8 | |
| 13.b | even | 2 | 1 | 403.2.a.e | ✓ | 8 | |
| 39.d | odd | 2 | 1 | 3627.2.a.p | 8 | ||
| 52.b | odd | 2 | 1 | 6448.2.a.bd | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.a.e | ✓ | 8 | 13.b | even | 2 | 1 | |
| 3627.2.a.p | 8 | 39.d | odd | 2 | 1 | ||
| 5239.2.a.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 6448.2.a.bd | 8 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):
|
\( T_{2}^{8} - T_{2}^{7} - 11T_{2}^{6} + 10T_{2}^{5} + 37T_{2}^{4} - 33T_{2}^{3} - 36T_{2}^{2} + 33T_{2} - 4 \)
|
|
\( T_{5}^{8} + 11T_{5}^{7} + 32T_{5}^{6} - 35T_{5}^{5} - 263T_{5}^{4} - 126T_{5}^{3} + 537T_{5}^{2} + 346T_{5} - 232 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 11 T^{6} + \cdots - 4 \)
|
| $3$ |
\( T^{8} - 7 T^{7} + \cdots + 16 \)
|
| $5$ |
\( T^{8} + 11 T^{7} + \cdots - 232 \)
|
| $7$ |
\( T^{8} - 2 T^{7} + \cdots + 3824 \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 484 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots - 9064 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 1616 \)
|
| $23$ |
\( T^{8} - 14 T^{7} + \cdots - 171104 \)
|
| $29$ |
\( T^{8} - 12 T^{7} + \cdots - 428 \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} + 2 T^{7} + \cdots - 2690 \)
|
| $41$ |
\( T^{8} + 13 T^{7} + \cdots + 214652 \)
|
| $43$ |
\( T^{8} + 5 T^{7} + \cdots - 15488 \)
|
| $47$ |
\( T^{8} + 23 T^{7} + \cdots - 7040 \)
|
| $53$ |
\( T^{8} - 25 T^{7} + \cdots - 354728 \)
|
| $59$ |
\( T^{8} - 5 T^{7} + \cdots + 3991768 \)
|
| $61$ |
\( T^{8} + 9 T^{7} + \cdots + 360296 \)
|
| $67$ |
\( T^{8} + 22 T^{7} + \cdots + 2879368 \)
|
| $71$ |
\( T^{8} - 17 T^{7} + \cdots + 183184 \)
|
| $73$ |
\( T^{8} - 27 T^{7} + \cdots - 18442 \)
|
| $79$ |
\( T^{8} + 23 T^{7} + \cdots + 4121696 \)
|
| $83$ |
\( T^{8} - 25 T^{7} + \cdots + 1073104 \)
|
| $89$ |
\( T^{8} + 2 T^{7} + \cdots - 14681734 \)
|
| $97$ |
\( T^{8} - 15 T^{7} + \cdots + 5096888 \)
|
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