Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,26,Mod(1,50)]
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([0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.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: | \(197.998389976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 25534923283x^{3} - 31863478542482x^{2} + 141941149085067124800x + 2515032055818200956928000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{5}\cdot 5^{12} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 25534923283x^{3} - 31863478542482x^{2} + 141941149085067124800x + 2515032055818200956928000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 10\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2500 \nu^{4} - 209270750 \nu^{3} + 28890521825550 \nu^{2} + \cdots + 77\!\cdots\!40 ) / 1693111372659 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2500 \nu^{4} - 209270750 \nu^{3} + 47702870410650 \nu^{2} + \cdots - 11\!\cdots\!80 ) / 188123485851 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 31647500 \nu^{4} + 5650407128000 \nu^{3} + \cdots - 44\!\cdots\!40 ) / 22010447844567 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 9\beta_{2} + 18721\beta _1 + 1021396931320 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1326\beta_{4} - 311855\beta_{3} + 1515477\beta_{2} + 720874921947\beta _1 + 9559043562744600 ) / 500 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 554986029 \beta_{4} + 419429477738 \beta_{3} - 4927548353754 \beta_{2} + \cdots + 36\!\cdots\!00 ) / 2500 \)
|
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 |
|
−4096.00 | −1.36309e6 | 1.67772e7 | 0 | 5.58321e9 | −6.14282e10 | −6.87195e10 | 1.01072e12 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −4096.00 | −1.22475e6 | 1.67772e7 | 0 | 5.01659e9 | 5.40343e10 | −6.87195e10 | 6.52735e11 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4096.00 | 43423.8 | 1.67772e7 | 0 | −1.77864e8 | 1.17021e10 | −6.87195e10 | −8.45403e11 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | −4096.00 | 598018. | 1.67772e7 | 0 | −2.44948e9 | 1.45531e10 | −6.87195e10 | −4.89664e11 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | −4096.00 | 1.22241e6 | 1.67772e7 | 0 | −5.00697e9 | −6.80801e10 | −6.87195e10 | 6.46987e11 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 50.26.a.i | ✓ | 5 |
| 5.b | even | 2 | 1 | 50.26.a.j | yes | 5 | |
| 5.c | odd | 4 | 2 | 50.26.b.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.26.a.i | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 50.26.a.j | yes | 5 | 5.b | even | 2 | 1 | |
| 50.26.b.h | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 723995 T_{3}^{4} - 2343824824290 T_{3}^{3} + \cdots - 52\!\cdots\!01 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(50))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4096)^{5} \)
|
| $3$ |
\( T^{5} + \cdots - 52\!\cdots\!01 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 38\!\cdots\!32 \)
|
| $11$ |
\( T^{5} + \cdots + 14\!\cdots\!43 \)
|
| $13$ |
\( T^{5} + \cdots + 51\!\cdots\!24 \)
|
| $17$ |
\( T^{5} + \cdots + 79\!\cdots\!43 \)
|
| $19$ |
\( T^{5} + \cdots - 23\!\cdots\!25 \)
|
| $23$ |
\( T^{5} + \cdots + 65\!\cdots\!24 \)
|
| $29$ |
\( T^{5} + \cdots - 21\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 23\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots + 76\!\cdots\!68 \)
|
| $41$ |
\( T^{5} + \cdots + 55\!\cdots\!43 \)
|
| $43$ |
\( T^{5} + \cdots - 27\!\cdots\!76 \)
|
| $47$ |
\( T^{5} + \cdots + 11\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots - 90\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 43\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 42\!\cdots\!93 \)
|
| $71$ |
\( T^{5} + \cdots - 22\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots - 41\!\cdots\!51 \)
|
| $79$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 38\!\cdots\!49 \)
|
| $89$ |
\( T^{5} + \cdots + 20\!\cdots\!25 \)
|
| $97$ |
\( T^{5} + \cdots + 23\!\cdots\!68 \)
|
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