Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5041,2,Mod(1,5041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5041, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5041 = 71^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5041.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: | \(40.2525876589\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 2x^{9} - 13x^{8} + 22x^{7} + 61x^{6} - 80x^{5} - 125x^{4} + 113x^{3} + 109x^{2} - 50x - 31 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 71) |
| 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
\( x^{10} - 2x^{9} - 13x^{8} + 22x^{7} + 61x^{6} - 80x^{5} - 125x^{4} + 113x^{3} + 109x^{2} - 50x - 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 12 \nu^{9} + 78 \nu^{8} - 311 \nu^{7} - 948 \nu^{6} + 2081 \nu^{5} + 3845 \nu^{4} - 4605 \nu^{3} + \cdots + 2594 ) / 409 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15 \nu^{9} - 107 \nu^{8} - 82 \nu^{7} + 1269 \nu^{6} + 45 \nu^{5} - 5112 \nu^{4} - 337 \nu^{3} + \cdots - 3097 ) / 409 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 54 \nu^{9} + 58 \nu^{8} + 786 \nu^{7} - 642 \nu^{6} - 3843 \nu^{5} + 2125 \nu^{4} + 7021 \nu^{3} + \cdots - 630 ) / 409 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 54 \nu^{9} - 58 \nu^{8} - 786 \nu^{7} + 642 \nu^{6} + 3843 \nu^{5} - 2125 \nu^{4} - 7021 \nu^{3} + \cdots - 597 ) / 409 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 77 \nu^{9} + 113 \nu^{8} + 939 \nu^{7} - 870 \nu^{6} - 3912 \nu^{5} + 1129 \nu^{4} + 6338 \nu^{3} + \cdots - 1989 ) / 409 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 77 \nu^{9} - 113 \nu^{8} - 939 \nu^{7} + 870 \nu^{6} + 3912 \nu^{5} - 1129 \nu^{4} - 5929 \nu^{3} + \cdots + 2807 ) / 409 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 102 \nu^{9} - 155 \nu^{8} - 1212 \nu^{7} + 1349 \nu^{6} + 4805 \nu^{5} - 3105 \nu^{4} - 7036 \nu^{3} + \cdots - 37 ) / 409 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 141 \nu^{9} + 106 \nu^{8} + 1916 \nu^{7} - 722 \nu^{6} - 8603 \nu^{5} + 527 \nu^{4} + 13311 \nu^{3} + \cdots - 418 ) / 409 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 8\beta_{5} + 7\beta_{4} - \beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 8\beta_{7} + 8\beta_{6} + 10\beta_{5} + 10\beta_{4} + \beta_{3} + 2\beta_{2} + 19\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{9} + 12 \beta_{8} + 10 \beta_{7} + 13 \beta_{6} + 54 \beta_{5} + 44 \beta_{4} - 9 \beta_{3} + \cdots + 78 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{9} + 20 \beta_{8} + 54 \beta_{7} + 60 \beta_{6} + 79 \beta_{5} + 78 \beta_{4} + 6 \beta_{3} + \cdots + 85 \)
|
| \(\nu^{8}\) | \(=\) |
\( 80 \beta_{9} + 113 \beta_{8} + 79 \beta_{7} + 124 \beta_{6} + 352 \beta_{5} + 279 \beta_{4} - 65 \beta_{3} + \cdots + 471 \)
|
| \(\nu^{9}\) | \(=\) |
\( 139 \beta_{9} + 238 \beta_{8} + 352 \beta_{7} + 452 \beta_{6} + 585 \beta_{5} + 564 \beta_{4} + \cdots + 671 \)
|
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 |
|
−2.26243 | −2.82918 | 3.11859 | −3.76298 | 6.40081 | −0.305610 | −2.53073 | 5.00424 | 8.51349 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.81608 | 2.93509 | 1.29815 | −1.49619 | −5.33036 | −1.25703 | 1.27462 | 5.61477 | 2.71721 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.43725 | −0.0163835 | 0.0656852 | −2.12097 | 0.0235471 | 1.13426 | 2.78009 | −2.99973 | 3.04836 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.971890 | −0.464104 | −1.05543 | 1.70157 | 0.451058 | 2.12446 | 2.96954 | −2.78461 | −1.65374 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.449457 | 3.28531 | −1.79799 | 0.730498 | −1.47660 | −2.26797 | 1.70703 | 7.79325 | −0.328327 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.846922 | −1.23930 | −1.28272 | 1.51161 | −1.04959 | −3.53969 | −2.78021 | −1.46413 | 1.28022 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.35935 | 2.17565 | −0.152157 | 3.12954 | 2.95747 | −4.25405 | −2.92554 | 1.73343 | 4.25416 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.57334 | 1.05115 | 0.475413 | 0.479898 | 1.65382 | 5.09482 | −2.39870 | −1.89508 | 0.755044 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.45498 | −0.817458 | 4.02692 | 0.0460941 | −2.00684 | 1.07420 | 4.97604 | −2.33176 | 0.113160 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.70251 | −2.08078 | 5.30355 | −3.21907 | −5.62331 | −0.803382 | 8.92786 | 1.32963 | −8.69957 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(71\) | \(1\) |
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 | 5041.2.a.i | 10 | |
| 71.b | odd | 2 | 1 | 5041.2.a.j | 10 | ||
| 71.c | even | 5 | 2 | 71.2.c.a | ✓ | 20 | |
| 213.h | odd | 10 | 2 | 639.2.f.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.2.c.a | ✓ | 20 | 71.c | even | 5 | 2 | |
| 639.2.f.c | 20 | 213.h | odd | 10 | 2 | ||
| 5041.2.a.i | 10 | 1.a | even | 1 | 1 | trivial | |
| 5041.2.a.j | 10 | 71.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5041))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 13T_{2}^{8} + 22T_{2}^{7} + 61T_{2}^{6} - 80T_{2}^{5} - 125T_{2}^{4} + 113T_{2}^{3} + 109T_{2}^{2} - 50T_{2} - 31 \)
|
|
\( T_{7}^{10} + 3 T_{7}^{9} - 31 T_{7}^{8} - 105 T_{7}^{7} + 162 T_{7}^{6} + 624 T_{7}^{5} - 194 T_{7}^{4} + \cdots + 139 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots - 31 \)
|
| $3$ |
\( T^{10} - 2 T^{9} + \cdots + 1 \)
|
| $5$ |
\( T^{10} + 3 T^{9} + \cdots + 5 \)
|
| $7$ |
\( T^{10} + 3 T^{9} + \cdots + 139 \)
|
| $11$ |
\( T^{10} + 3 T^{9} + \cdots + 8539 \)
|
| $13$ |
\( T^{10} + T^{9} + \cdots + 166211 \)
|
| $17$ |
\( T^{10} + 16 T^{9} + \cdots + 115649 \)
|
| $19$ |
\( T^{10} - 4 T^{9} + \cdots + 19375 \)
|
| $23$ |
\( T^{10} + 11 T^{9} + \cdots + 2981269 \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots - 9211 \)
|
| $31$ |
\( T^{10} + 12 T^{9} + \cdots + 13969 \)
|
| $37$ |
\( T^{10} + 3 T^{9} + \cdots + 1974269 \)
|
| $41$ |
\( T^{10} + 30 T^{9} + \cdots + 52111 \)
|
| $43$ |
\( T^{10} - 4 T^{9} + \cdots - 16895 \)
|
| $47$ |
\( T^{10} + 48 T^{9} + \cdots - 53945 \)
|
| $53$ |
\( T^{10} + 29 T^{9} + \cdots - 505039 \)
|
| $59$ |
\( T^{10} + 17 T^{9} + \cdots - 4314151 \)
|
| $61$ |
\( T^{10} + 16 T^{9} + \cdots + 20555 \)
|
| $67$ |
\( T^{10} - 36 T^{9} + \cdots - 19318795 \)
|
| $71$ |
\( T^{10} \)
|
| $73$ |
\( T^{10} + \cdots + 480084109 \)
|
| $79$ |
\( T^{10} - 38 T^{9} + \cdots - 1927355 \)
|
| $83$ |
\( T^{10} + 10 T^{9} + \cdots + 3456029 \)
|
| $89$ |
\( T^{10} + \cdots + 106044871 \)
|
| $97$ |
\( T^{10} + \cdots + 3911261945 \)
|
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