Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4450,2,Mod(1,4450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4450.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4450 = 2 \cdot 5^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4450.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: | \(35.5334288995\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.7736352.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 13x^{3} + 15x^{2} + 32x - 40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 890) |
| 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
\( x^{5} - x^{4} - 13x^{3} + 15x^{2} + 32x - 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 11\nu^{2} - 7\nu + 22 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 13\nu^{2} - 7\nu + 32 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} - \nu^{3} + 37\nu^{2} + 7\nu - 80 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + \beta_{2} + 7\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} - 13\beta_{3} + 12\beta_{2} + 36 \)
|
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 |
|
−1.00000 | −2.96901 | 1.00000 | 0 | 2.96901 | 0.248974 | −1.00000 | 5.81500 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.65996 | 1.00000 | 0 | 1.65996 | −1.63710 | −1.00000 | −0.244538 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.38931 | 1.00000 | 0 | 1.38931 | 1.79482 | −1.00000 | −1.06981 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.83542 | 1.00000 | 0 | −1.83542 | 3.10980 | −1.00000 | 0.368748 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 3.18286 | 1.00000 | 0 | −3.18286 | −3.51649 | −1.00000 | 7.13061 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(89\) | \( -1 \) |
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 | 4450.2.a.be | 5 | |
| 5.b | even | 2 | 1 | 890.2.a.m | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 8010.2.a.bf | 5 | ||
| 20.d | odd | 2 | 1 | 7120.2.a.be | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 890.2.a.m | ✓ | 5 | 5.b | even | 2 | 1 | |
| 4450.2.a.be | 5 | 1.a | even | 1 | 1 | trivial | |
| 7120.2.a.be | 5 | 20.d | odd | 2 | 1 | ||
| 8010.2.a.bf | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4450))\):
|
\( T_{3}^{5} + T_{3}^{4} - 13T_{3}^{3} - 15T_{3}^{2} + 32T_{3} + 40 \)
|
|
\( T_{7}^{5} - 14T_{7}^{3} + 4T_{7}^{2} + 32T_{7} - 8 \)
|
|
\( T_{11}^{5} - 3T_{11}^{4} - 25T_{11}^{3} + 75T_{11}^{2} + 136T_{11} - 404 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 40 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 14 T^{3} + \cdots - 8 \)
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| $11$ |
\( T^{5} - 3 T^{4} + \cdots - 404 \)
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| $13$ |
\( T^{5} - 14 T^{3} + \cdots + 8 \)
|
| $17$ |
\( T^{5} + 6 T^{4} + \cdots - 96 \)
|
| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 3600 \)
|
| $23$ |
\( T^{5} - 8 T^{4} + \cdots + 832 \)
|
| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 472 \)
|
| $31$ |
\( T^{5} - 10 T^{4} + \cdots + 8 \)
|
| $37$ |
\( T^{5} + 14 T^{4} + \cdots + 1544 \)
|
| $41$ |
\( T^{5} - 16 T^{4} + \cdots + 1616 \)
|
| $43$ |
\( T^{5} - 7 T^{4} + \cdots - 176 \)
|
| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 184 \)
|
| $53$ |
\( T^{5} - T^{4} + \cdots - 1948 \)
|
| $59$ |
\( T^{5} - 8 T^{4} + \cdots + 192 \)
|
| $61$ |
\( T^{5} - 17 T^{4} + \cdots + 6682 \)
|
| $67$ |
\( T^{5} - 18 T^{4} + \cdots + 624 \)
|
| $71$ |
\( T^{5} - 8 T^{4} + \cdots - 4320 \)
|
| $73$ |
\( T^{5} - 160 T^{3} + \cdots - 1920 \)
|
| $79$ |
\( T^{5} + 24 T^{4} + \cdots - 2048 \)
|
| $83$ |
\( T^{5} - 11 T^{4} + \cdots - 16160 \)
|
| $89$ |
\( (T - 1)^{5} \)
|
| $97$ |
\( T^{5} + 10 T^{4} + \cdots + 170512 \)
|
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