Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5312,2,Mod(1,5312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5312 = 2^{6} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5312.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: | \(42.4165335537\) |
| 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: |
\( x^{8} - 2x^{7} - 18x^{6} + 33x^{5} + 87x^{4} - 127x^{3} - 126x^{2} + 100x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 664) |
| 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
\( x^{8} - 2x^{7} - 18x^{6} + 33x^{5} + 87x^{4} - 127x^{3} - 126x^{2} + 100x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} + 4\nu^{6} + 54\nu^{5} - 63\nu^{4} - 259\nu^{3} + 215\nu^{2} + 364\nu - 128 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{7} - 13\nu^{6} - 148\nu^{5} + 210\nu^{4} + 759\nu^{3} - 755\nu^{2} - 1223\nu + 420 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( 4\nu^{7} - 7\nu^{6} - 74\nu^{5} + 113\nu^{4} + 381\nu^{3} - 404\nu^{2} - 627\nu + 212 \)
|
| \(\beta_{5}\) | \(=\) |
\( -4\nu^{7} + 7\nu^{6} + 74\nu^{5} - 113\nu^{4} - 381\nu^{3} + 405\nu^{2} + 627\nu - 217 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{7} - 14\nu^{6} - 166\nu^{5} + 225\nu^{4} + 845\nu^{3} - 801\nu^{2} - 1340\nu + 450 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} - 19\nu^{6} - 204\nu^{5} + 307\nu^{4} + 1054\nu^{3} - 1102\nu^{2} - 1735\nu + 590 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} + 11\beta_{5} + 12\beta_{4} + 4\beta_{3} - 2\beta _1 + 43 \)
|
| \(\nu^{5}\) | \(=\) |
\( -33\beta_{7} + 13\beta_{6} - 17\beta_{5} + 16\beta_{4} + 3\beta_{3} + 28\beta_{2} + 60\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( -38\beta_{7} - 29\beta_{6} + 120\beta_{5} + 140\beta_{4} + 66\beta_{3} + 6\beta_{2} - 42\beta _1 + 427 \)
|
| \(\nu^{7}\) | \(=\) |
\( -430\beta_{7} + 151\beta_{6} - 219\beta_{5} + 208\beta_{4} + 58\beta_{3} + 338\beta_{2} + 583\beta _1 - 108 \)
|
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 | −3.37581 | 0 | 1.85525 | 0 | −2.75602 | 0 | 8.39607 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.41235 | 0 | −3.04663 | 0 | −1.34735 | 0 | 2.81942 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.33802 | 0 | −0.547163 | 0 | 3.52018 | 0 | 2.46634 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.305427 | 0 | 4.28457 | 0 | 1.13746 | 0 | −2.90671 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.297635 | 0 | −3.84665 | 0 | 1.95246 | 0 | −2.91141 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.38854 | 0 | −1.61888 | 0 | −5.02207 | 0 | −1.07195 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.98233 | 0 | 0.181960 | 0 | 2.34566 | 0 | 0.929627 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.35836 | 0 | −4.26247 | 0 | 1.16967 | 0 | 8.27861 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(83\) | \(-1\) |
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 | 5312.2.a.br | 8 | |
| 4.b | odd | 2 | 1 | 5312.2.a.bs | 8 | ||
| 8.b | even | 2 | 1 | 664.2.a.f | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 1328.2.a.n | 8 | ||
| 24.h | odd | 2 | 1 | 5976.2.a.s | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 664.2.a.f | ✓ | 8 | 8.b | even | 2 | 1 | |
| 1328.2.a.n | 8 | 8.d | odd | 2 | 1 | ||
| 5312.2.a.br | 8 | 1.a | even | 1 | 1 | trivial | |
| 5312.2.a.bs | 8 | 4.b | odd | 2 | 1 | ||
| 5976.2.a.s | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\):
|
\( T_{3}^{8} + 2T_{3}^{7} - 18T_{3}^{6} - 33T_{3}^{5} + 87T_{3}^{4} + 127T_{3}^{3} - 126T_{3}^{2} - 100T_{3} - 16 \)
|
|
\( T_{5}^{8} + 7T_{5}^{7} - 9T_{5}^{6} - 150T_{5}^{5} - 216T_{5}^{4} + 384T_{5}^{3} + 816T_{5}^{2} + 192T_{5} - 64 \)
|
|
\( T_{7}^{8} - T_{7}^{7} - 29T_{7}^{6} + 57T_{7}^{5} + 168T_{7}^{4} - 465T_{7}^{3} + 4T_{7}^{2} + 661T_{7} - 400 \)
|
|
\( T_{11}^{8} + 10T_{11}^{7} - 10T_{11}^{6} - 345T_{11}^{5} - 645T_{11}^{4} + 2467T_{11}^{3} + 9010T_{11}^{2} + 8668T_{11} + 1936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 2 T^{7} + \cdots - 16 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots - 64 \)
|
| $7$ |
\( T^{8} - T^{7} + \cdots - 400 \)
|
| $11$ |
\( T^{8} + 10 T^{7} + \cdots + 1936 \)
|
| $13$ |
\( T^{8} + 7 T^{7} + \cdots - 10432 \)
|
| $17$ |
\( T^{8} - 15 T^{7} + \cdots - 126634 \)
|
| $19$ |
\( T^{8} + 7 T^{7} + \cdots + 128 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots - 6400 \)
|
| $29$ |
\( T^{8} + 16 T^{7} + \cdots - 5920 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 1245560 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots + 808 \)
|
| $41$ |
\( T^{8} - 24 T^{7} + \cdots - 4505464 \)
|
| $43$ |
\( T^{8} - T^{7} + \cdots - 1421824 \)
|
| $47$ |
\( T^{8} + 2 T^{7} + \cdots - 8192 \)
|
| $53$ |
\( T^{8} + 7 T^{7} + \cdots - 7744 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 78704 \)
|
| $61$ |
\( T^{8} + 16 T^{7} + \cdots - 5920 \)
|
| $67$ |
\( T^{8} - 25 T^{7} + \cdots + 83584 \)
|
| $71$ |
\( T^{8} - 8 T^{7} + \cdots + 2231296 \)
|
| $73$ |
\( T^{8} - 14 T^{7} + \cdots - 172800 \)
|
| $79$ |
\( T^{8} + 4 T^{7} + \cdots - 1439744 \)
|
| $83$ |
\( (T - 1)^{8} \)
|
| $89$ |
\( T^{8} - 20 T^{7} + \cdots - 2898944 \)
|
| $97$ |
\( T^{8} - 2 T^{7} + \cdots - 1129472 \)
|
| show more | |
| show less |