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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1555279308.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 59x^{2} + 268 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 59x^{2} + 268 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 50\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 50\beta_1 \)
|
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 | −7.35124 | 4.00000 | 0 | −14.7025 | 0 | 8.00000 | 27.0408 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −2.22693 | 4.00000 | 0 | −4.45386 | 0 | 8.00000 | −22.0408 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 2.22693 | 4.00000 | 0 | 4.45386 | 0 | 8.00000 | −22.0408 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 7.35124 | 4.00000 | 0 | 14.7025 | 0 | 8.00000 | 27.0408 | 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.ct | yes | 4 |
| 5.b | even | 2 | 1 | 2450.4.a.cn | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 2450.4.a.ct | yes | 4 |
| 35.c | odd | 2 | 1 | 2450.4.a.cn | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2450.4.a.cn | ✓ | 4 | 5.b | even | 2 | 1 | |
| 2450.4.a.cn | ✓ | 4 | 35.c | odd | 2 | 1 | |
| 2450.4.a.ct | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2450.4.a.ct | yes | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{4} - 59T_{3}^{2} + 268 \)
|
|
\( T_{11}^{2} - 16T_{11} - 2345 \)
|
|
\( T_{19}^{4} - 11199T_{19}^{2} + 2469888 \)
|
|
\( T_{23}^{2} - 41T_{23} - 14636 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 59T^{2} + 268 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 16 T - 2345)^{2} \)
|
| $13$ |
\( T^{4} - 12368 T^{2} + 35119792 \)
|
| $17$ |
\( T^{4} - 19559 T^{2} + 55239088 \)
|
| $19$ |
\( T^{4} - 11199 T^{2} + 2469888 \)
|
| $23$ |
\( (T^{2} - 41 T - 14636)^{2} \)
|
| $29$ |
\( (T^{2} - 15 T - 546)^{2} \)
|
| $31$ |
\( T^{4} - 24068 T^{2} + 87685312 \)
|
| $37$ |
\( (T^{2} + 13 T - 48740)^{2} \)
|
| $41$ |
\( T^{4} - 44363 T^{2} + 59959372 \)
|
| $43$ |
\( (T^{2} - 343 T - 72368)^{2} \)
|
| $47$ |
\( T^{4} - 28556 T^{2} + 62780608 \)
|
| $53$ |
\( (T^{2} + 398 T - 78440)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 119927779888 \)
|
| $61$ |
\( T^{4} + \cdots + 215093825200 \)
|
| $67$ |
\( (T^{2} - 1262 T + 388525)^{2} \)
|
| $71$ |
\( (T^{2} + 417 T + 13962)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 125234976652 \)
|
| $79$ |
\( (T^{2} - 641 T - 215870)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 478827120748 \)
|
| $89$ |
\( T^{4} + \cdots + 25138192300 \)
|
| $97$ |
\( T^{4} + \cdots + 1543680000 \)
|
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