Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,10,Mod(1,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(162.236288392\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - x^{7} - 2943 x^{6} + 5227 x^{5} + 2674143 x^{4} - 7013491 x^{3} - 759677093 x^{2} + \cdots + 51239796804 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 2943 x^{6} + 5227 x^{5} + 2674143 x^{4} - 7013491 x^{3} - 759677093 x^{2} + \cdots + 51239796804 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 736 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 67 \nu^{7} - 59216 \nu^{6} + 831075 \nu^{5} + 108855364 \nu^{4} - 1392176191 \nu^{3} + \cdots + 3565684744244 ) / 187504384 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1179 \nu^{7} - 7440 \nu^{6} + 2341909 \nu^{5} + 36125308 \nu^{4} - 907999177 \nu^{3} + \cdots + 5986461701612 ) / 375008768 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1179 \nu^{7} + 7440 \nu^{6} - 2341909 \nu^{5} - 36125308 \nu^{4} + 1283007945 \nu^{3} + \cdots - 5612577959916 ) / 375008768 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5569 \nu^{7} - 24496 \nu^{6} - 14004175 \nu^{5} + 41835468 \nu^{4} + 9939069163 \nu^{3} + \cdots + 626401057148 ) / 375008768 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3307 \nu^{7} + 50688 \nu^{6} - 9004117 \nu^{5} - 152876380 \nu^{4} + 7143843033 \nu^{3} + \cdots - 19198705417580 ) / 750017536 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 736 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 1125\beta _1 - 997 \)
|
| \(\nu^{4}\) | \(=\) |
\( -16\beta_{7} + 4\beta_{6} + 7\beta_{5} + 3\beta_{4} - 4\beta_{3} + 1509\beta_{2} - 1729\beta _1 + 827321 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 416 \beta_{7} + 8 \beta_{6} + 1998 \beta_{5} + 1446 \beta_{4} - 56 \beta_{3} + 1508 \beta_{2} + \cdots - 1667322 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 36480 \beta_{7} + 7568 \beta_{6} + 21110 \beta_{5} + 4390 \beta_{4} - 11488 \beta_{3} + \cdots + 1040182890 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1086368 \beta_{7} + 90696 \beta_{6} + 3279859 \beta_{5} + 1848267 \beta_{4} - 161304 \beta_{3} + \cdots - 1923352759 \)
|
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 |
|
−41.0027 | 0 | 1169.22 | 625.000 | 0 | 2401.00 | −26947.8 | 0 | −25626.7 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −34.0392 | 0 | 646.665 | 625.000 | 0 | 2401.00 | −4583.89 | 0 | −21274.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −21.5924 | 0 | −45.7679 | 625.000 | 0 | 2401.00 | 12043.6 | 0 | −13495.3 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −14.9139 | 0 | −289.575 | 625.000 | 0 | 2401.00 | 11954.6 | 0 | −9321.19 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.24526 | 0 | −484.487 | 625.000 | 0 | 2401.00 | −5226.83 | 0 | 3278.29 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 15.4124 | 0 | −274.458 | 625.000 | 0 | 2401.00 | −12121.2 | 0 | 9632.74 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 32.3875 | 0 | 536.948 | 625.000 | 0 | 2401.00 | 808.016 | 0 | 20242.2 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 33.5031 | 0 | 610.455 | 625.000 | 0 | 2401.00 | 3298.54 | 0 | 20939.4 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 315.10.a.n | ✓ | 8 |
| 3.b | odd | 2 | 1 | 315.10.a.o | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.10.a.n | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 315.10.a.o | yes | 8 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 25 T_{2}^{7} - 2670 T_{2}^{6} - 56500 T_{2}^{5} + 2205048 T_{2}^{4} + 37060000 T_{2}^{3} + \cdots + 39425937408 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 39425937408 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T - 625)^{8} \)
|
| $7$ |
\( (T - 2401)^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 16\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots - 24\!\cdots\!08 \)
|
| $19$ |
\( T^{8} + \cdots + 38\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 28\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 22\!\cdots\!76 \)
|
| $37$ |
\( T^{8} + \cdots - 17\!\cdots\!24 \)
|
| $41$ |
\( T^{8} + \cdots - 12\!\cdots\!48 \)
|
| $43$ |
\( T^{8} + \cdots - 34\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots - 31\!\cdots\!48 \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!68 \)
|
| $59$ |
\( T^{8} + \cdots + 29\!\cdots\!92 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots - 21\!\cdots\!44 \)
|
| $71$ |
\( T^{8} + \cdots + 31\!\cdots\!72 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!24 \)
|
| $79$ |
\( T^{8} + \cdots + 84\!\cdots\!84 \)
|
| $83$ |
\( T^{8} + \cdots - 20\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots - 19\!\cdots\!68 \)
|
| $97$ |
\( T^{8} + \cdots - 44\!\cdots\!96 \)
|
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