Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [31,8,Mod(1,31)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.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: | \(9.68393579001\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5 x^{9} - 924 x^{8} + 1449 x^{7} + 279925 x^{6} + 265302 x^{5} - 28803712 x^{4} + \cdots + 675348992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5 x^{9} - 924 x^{8} + 1449 x^{7} + 279925 x^{6} + 265302 x^{5} - 28803712 x^{4} + \cdots + 675348992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 77265173820963 \nu^{9} + \cdots - 41\!\cdots\!04 ) / 33\!\cdots\!68 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 77265173820963 \nu^{9} + \cdots - 20\!\cdots\!76 ) / 33\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 383167735275863 \nu^{9} + \cdots - 15\!\cdots\!52 ) / 33\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 482060315563135 \nu^{9} + \cdots + 19\!\cdots\!08 ) / 33\!\cdots\!68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 627397886868445 \nu^{9} + \cdots + 19\!\cdots\!52 ) / 33\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 383367260820451 \nu^{9} + \cdots - 23\!\cdots\!20 ) / 11\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 65391135397511 \nu^{9} - 260637774577631 \nu^{8} + \cdots - 43\!\cdots\!16 ) / 10\!\cdots\!28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 786885958249789 \nu^{9} + \cdots - 51\!\cdots\!24 ) / 11\!\cdots\!56 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 185 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + \cdots + 789 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{9} - 18 \beta_{8} + 18 \beta_{7} + 2 \beta_{6} - 72 \beta_{5} - 11 \beta_{4} + 464 \beta_{3} + \cdots + 63457 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1896 \beta_{9} - 1435 \beta_{8} - 2052 \beta_{7} + 990 \beta_{6} - 1809 \beta_{5} + 1384 \beta_{4} + \cdots + 600556 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7147 \beta_{9} - 19894 \beta_{8} + 9944 \beta_{7} + 2714 \beta_{6} - 44283 \beta_{5} - 4327 \beta_{4} + \cdots + 25936523 \)
|
| \(\nu^{7}\) | \(=\) |
\( 996370 \beta_{9} - 867680 \beta_{8} - 829406 \beta_{7} + 404054 \beta_{6} - 945398 \beta_{5} + \cdots + 344385325 \)
|
| \(\nu^{8}\) | \(=\) |
\( 8278798 \beta_{9} - 15008303 \beta_{8} + 4243500 \beta_{7} + 1817794 \beta_{6} - 23042399 \beta_{5} + \cdots + 11499444326 \)
|
| \(\nu^{9}\) | \(=\) |
\( 508243647 \beta_{9} - 499059586 \beta_{8} - 305240892 \beta_{7} + 158625510 \beta_{6} + \cdots + 182494081797 \)
|
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 |
|
−20.6991 | −5.35016 | 300.454 | −186.874 | 110.744 | −1268.34 | −3569.64 | −2158.38 | 3868.12 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −18.6225 | 55.8742 | 218.799 | 386.739 | −1040.52 | −98.2676 | −1690.91 | 934.925 | −7202.07 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.67064 | −27.5523 | −52.8200 | −278.426 | 238.896 | −610.761 | 1567.83 | −1427.87 | 2414.14 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.21419 | 46.6707 | −123.097 | 165.938 | −103.338 | 798.888 | 555.977 | −8.84214 | −367.418 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.837159 | −65.0203 | −127.299 | −389.804 | −54.4323 | 1667.25 | −213.726 | 2040.64 | −326.328 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.16645 | −83.5787 | −126.639 | 203.094 | −97.4902 | −1397.82 | −297.024 | 4798.40 | 236.899 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 12.3742 | 89.8646 | 25.1201 | −126.949 | 1112.00 | 1247.44 | −1273.05 | 5888.64 | −1570.88 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 14.8300 | −49.7454 | 91.9276 | 363.504 | −737.723 | 1288.75 | −534.952 | 287.609 | 5390.75 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 15.2975 | 54.0681 | 106.014 | 508.764 | 827.106 | −1501.25 | −336.337 | 736.355 | 7782.82 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 20.7013 | 24.7693 | 300.542 | −75.9868 | 512.757 | 414.118 | 3571.84 | −1573.48 | −1573.02 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(31\) | \(1\) |
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 | 31.8.a.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 279.8.a.d | 10 | ||
| 4.b | odd | 2 | 1 | 496.8.a.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.8.a.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 279.8.a.d | 10 | 3.b | odd | 2 | 1 | ||
| 496.8.a.h | 10 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 15 T_{2}^{9} - 834 T_{2}^{8} + 13095 T_{2}^{7} + 196723 T_{2}^{6} - 3330150 T_{2}^{5} + \cdots + 419955584 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(31))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 419955584 \)
|
| $3$ |
\( T^{10} + \cdots - 12\!\cdots\!04 \)
|
| $5$ |
\( T^{10} + \cdots - 47\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 14\!\cdots\!96 \)
|
| $11$ |
\( T^{10} + \cdots + 18\!\cdots\!84 \)
|
| $13$ |
\( T^{10} + \cdots + 53\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots - 69\!\cdots\!04 \)
|
| $19$ |
\( T^{10} + \cdots + 10\!\cdots\!76 \)
|
| $23$ |
\( T^{10} + \cdots - 22\!\cdots\!56 \)
|
| $29$ |
\( T^{10} + \cdots + 25\!\cdots\!16 \)
|
| $31$ |
\( (T + 29791)^{10} \)
|
| $37$ |
\( T^{10} + \cdots - 83\!\cdots\!44 \)
|
| $41$ |
\( T^{10} + \cdots - 28\!\cdots\!76 \)
|
| $43$ |
\( T^{10} + \cdots - 81\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots + 26\!\cdots\!96 \)
|
| $53$ |
\( T^{10} + \cdots - 86\!\cdots\!84 \)
|
| $59$ |
\( T^{10} + \cdots - 62\!\cdots\!24 \)
|
| $61$ |
\( T^{10} + \cdots - 63\!\cdots\!96 \)
|
| $67$ |
\( T^{10} + \cdots - 16\!\cdots\!44 \)
|
| $71$ |
\( T^{10} + \cdots - 11\!\cdots\!64 \)
|
| $73$ |
\( T^{10} + \cdots + 73\!\cdots\!56 \)
|
| $79$ |
\( T^{10} + \cdots - 45\!\cdots\!56 \)
|
| $83$ |
\( T^{10} + \cdots + 75\!\cdots\!64 \)
|
| $89$ |
\( T^{10} + \cdots + 38\!\cdots\!76 \)
|
| $97$ |
\( T^{10} + \cdots - 50\!\cdots\!64 \)
|
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