Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,26,Mod(49,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.49");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(197.998389976\) |
| 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: |
\( x^{10} + 5211818408605 x^{8} + \cdots + 28\!\cdots\!01 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{38}\cdot 3^{10}\cdot 5^{24} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 5211818408605 x^{8} + \cdots + 28\!\cdots\!01 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1161020069065 \nu^{8} + \cdots - 24\!\cdots\!69 ) / 16\!\cdots\!76 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 25\!\cdots\!31 \nu^{8} + \cdots + 89\!\cdots\!63 ) / 10\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\!\cdots\!03 \nu^{8} + \cdots + 20\!\cdots\!59 ) / 82\!\cdots\!96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 26\!\cdots\!39 \nu^{8} + \cdots - 55\!\cdots\!47 ) / 42\!\cdots\!92 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15\!\cdots\!65 \nu^{9} + \cdots + 80\!\cdots\!85 \nu ) / 33\!\cdots\!12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 22\!\cdots\!35 \nu^{9} + \cdots + 15\!\cdots\!27 \nu ) / 33\!\cdots\!12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!97 \nu^{9} + \cdots + 25\!\cdots\!93 \nu ) / 46\!\cdots\!08 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15\!\cdots\!93 \nu^{9} + \cdots + 11\!\cdots\!51 \nu ) / 65\!\cdots\!08 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 22\!\cdots\!57 \nu^{9} + \cdots + 23\!\cdots\!61 \nu ) / 84\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_{6} - 144799\beta_{5} \)
|
| \(\nu^{2}\) | \(=\) |
\( -9\beta_{3} + \beta_{2} - 308319\beta _1 - 1042363681721 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2652\beta_{9} + 189313\beta_{8} - 878619\beta_{7} - 1512782441334\beta_{6} + 465845814391417639\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( - 683916324 \beta_{4} + 19086881503686 \beta_{3} - 1442268076198 \beta_{2} + \cdots + 16\!\cdots\!21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 57\!\cdots\!00 \beta_{9} + \cdots - 93\!\cdots\!99 \beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\!\cdots\!80 \beta_{4} + \cdots - 26\!\cdots\!01 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 10\!\cdots\!80 \beta_{9} + \cdots + 17\!\cdots\!79 \beta_{5} \)
|
| \(\nu^{8}\) | \(=\) |
\( 18\!\cdots\!68 \beta_{4} + \cdots + 43\!\cdots\!61 \)
|
| \(\nu^{9}\) | \(=\) |
\( 19\!\cdots\!84 \beta_{9} + \cdots - 32\!\cdots\!79 \beta_{5} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 4096.00i | − | 1.22241e6i | −1.67772e7 | 0 | −5.00697e9 | − | 6.80801e10i | 6.87195e10i | −6.46987e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 4096.00i | − | 598018.i | −1.67772e7 | 0 | −2.44948e9 | 1.45531e10i | 6.87195e10i | 4.89664e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 4096.00i | − | 43423.8i | −1.67772e7 | 0 | −1.77864e8 | 1.17021e10i | 6.87195e10i | 8.45403e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 4096.00i | 1.22475e6i | −1.67772e7 | 0 | 5.01659e9 | 5.40343e10i | 6.87195e10i | −6.52735e11 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | − | 4096.00i | 1.36309e6i | −1.67772e7 | 0 | 5.58321e9 | − | 6.14282e10i | 6.87195e10i | −1.01072e12 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 4096.00i | − | 1.36309e6i | −1.67772e7 | 0 | 5.58321e9 | 6.14282e10i | − | 6.87195e10i | −1.01072e12 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 4096.00i | − | 1.22475e6i | −1.67772e7 | 0 | 5.01659e9 | − | 5.40343e10i | − | 6.87195e10i | −6.52735e11 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 4096.00i | 43423.8i | −1.67772e7 | 0 | −1.77864e8 | − | 1.17021e10i | − | 6.87195e10i | 8.45403e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 4096.00i | 598018.i | −1.67772e7 | 0 | −2.44948e9 | − | 1.45531e10i | − | 6.87195e10i | 4.89664e11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 4096.00i | 1.22241e6i | −1.67772e7 | 0 | −5.00697e9 | 6.80801e10i | − | 6.87195e10i | −6.46987e11 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.26.b.h | 10 | |
| 5.b | even | 2 | 1 | inner | 50.26.b.h | 10 | |
| 5.c | odd | 4 | 1 | 50.26.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 50.26.a.j | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.26.a.i | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 50.26.a.j | yes | 5 | 5.c | odd | 4 | 1 | |
| 50.26.b.h | 10 | 1.a | even | 1 | 1 | trivial | |
| 50.26.b.h | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 5211818408605 T_{3}^{8} + \cdots + 28\!\cdots\!01 \)
acting on \(S_{26}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16777216)^{5} \)
|
| $3$ |
\( T^{10} + \cdots + 28\!\cdots\!01 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 14\!\cdots\!24 \)
|
| $11$ |
\( (T^{5} + \cdots + 14\!\cdots\!43)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 26\!\cdots\!76 \)
|
| $17$ |
\( T^{10} + \cdots + 62\!\cdots\!49 \)
|
| $19$ |
\( (T^{5} + \cdots + 23\!\cdots\!25)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 43\!\cdots\!76 \)
|
| $29$ |
\( (T^{5} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots + 23\!\cdots\!68)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 58\!\cdots\!24 \)
|
| $41$ |
\( (T^{5} + \cdots + 55\!\cdots\!43)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 74\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots + 14\!\cdots\!24 \)
|
| $53$ |
\( T^{10} + \cdots + 81\!\cdots\!76 \)
|
| $59$ |
\( (T^{5} + \cdots - 27\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 43\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 17\!\cdots\!49 \)
|
| $71$ |
\( (T^{5} + \cdots - 22\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 17\!\cdots\!01 \)
|
| $79$ |
\( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 14\!\cdots\!01 \)
|
| $89$ |
\( (T^{5} + \cdots - 20\!\cdots\!25)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 53\!\cdots\!24 \)
|
| show more | |
| show less |