Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2450,4,Mod(1,2450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2450.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.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: | \(144.554679514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.10197128.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 39x^{2} + 98 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 490) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 39x^{2} + 98 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 32\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} + 32\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −6.02497 | 4.00000 | 0 | −12.0499 | 0 | 8.00000 | 9.30030 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −1.64308 | 4.00000 | 0 | −3.28615 | 0 | 8.00000 | −24.3003 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 1.64308 | 4.00000 | 0 | 3.28615 | 0 | 8.00000 | −24.3003 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 6.02497 | 4.00000 | 0 | 12.0499 | 0 | 8.00000 | 9.30030 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2450.4.a.cs | 4 | |
| 5.b | even | 2 | 1 | 2450.4.a.cm | 4 | ||
| 5.c | odd | 4 | 2 | 490.4.c.d | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 2450.4.a.cs | 4 | |
| 35.c | odd | 2 | 1 | 2450.4.a.cm | 4 | ||
| 35.f | even | 4 | 2 | 490.4.c.d | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.4.c.d | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 490.4.c.d | ✓ | 8 | 35.f | even | 4 | 2 | |
| 2450.4.a.cm | 4 | 5.b | even | 2 | 1 | ||
| 2450.4.a.cm | 4 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.cs | 4 | 1.a | even | 1 | 1 | trivial | |
| 2450.4.a.cs | 4 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{4} - 39T_{3}^{2} + 98 \)
|
|
\( T_{11}^{2} + 9T_{11} - 2520 \)
|
|
\( T_{19}^{4} - 20794T_{19}^{2} + 15103008 \)
|
|
\( T_{23}^{2} - 26T_{23} - 28056 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 39T^{2} + 98 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 9 T - 2520)^{2} \)
|
| $13$ |
\( T^{4} - 463 T^{2} + 39762 \)
|
| $17$ |
\( T^{4} - 11349 T^{2} + 1648928 \)
|
| $19$ |
\( T^{4} - 20794 T^{2} + 15103008 \)
|
| $23$ |
\( (T^{2} - 26 T - 28056)^{2} \)
|
| $29$ |
\( (T^{2} + 215 T + 11274)^{2} \)
|
| $31$ |
\( T^{4} - 118648 T^{2} + 757694592 \)
|
| $37$ |
\( (T^{2} + 378 T + 25560)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 1914072192 \)
|
| $43$ |
\( (T^{2} + 112 T - 14928)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 1636148808 \)
|
| $53$ |
\( (T^{2} - 222 T + 2160)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 80763412608 \)
|
| $61$ |
\( T^{4} + \cdots + 190295143200 \)
|
| $67$ |
\( (T^{2} + 608 T + 20160)^{2} \)
|
| $71$ |
\( (T^{2} + 882 T + 103032)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 195782782752 \)
|
| $79$ |
\( (T^{2} - 771 T - 237790)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 96227090208 \)
|
| $89$ |
\( T^{4} + \cdots + 16460236800 \)
|
| $97$ |
\( T^{4} + \cdots + 711385920000 \)
|
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