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: | \(3\) |
| Coefficient field: | 3.3.238585.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 67x - 189 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 350) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 67x - 189 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 45 \)
|
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 | −6.92433 | 4.00000 | 0 | 13.8487 | 0 | −8.00000 | 20.9463 | 0 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −4.41567 | 4.00000 | 0 | 8.83134 | 0 | −8.00000 | −7.50184 | 0 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 8.34000 | 4.00000 | 0 | −16.6800 | 0 | −8.00000 | 42.5556 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 2450.4.a.cc | 3 | |
| 5.b | even | 2 | 1 | 2450.4.a.ci | 3 | ||
| 7.b | odd | 2 | 1 | 2450.4.a.cd | 3 | ||
| 7.c | even | 3 | 2 | 350.4.e.j | yes | 6 | |
| 35.c | odd | 2 | 1 | 2450.4.a.ch | 3 | ||
| 35.j | even | 6 | 2 | 350.4.e.i | ✓ | 6 | |
| 35.l | odd | 12 | 4 | 350.4.j.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.4.e.i | ✓ | 6 | 35.j | even | 6 | 2 | |
| 350.4.e.j | yes | 6 | 7.c | even | 3 | 2 | |
| 350.4.j.h | 12 | 35.l | odd | 12 | 4 | ||
| 2450.4.a.cc | 3 | 1.a | even | 1 | 1 | trivial | |
| 2450.4.a.cd | 3 | 7.b | odd | 2 | 1 | ||
| 2450.4.a.ch | 3 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.ci | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{3} + 3T_{3}^{2} - 64T_{3} - 255 \)
|
|
\( T_{11}^{3} + 26T_{11}^{2} - 1393T_{11} - 36407 \)
|
|
\( T_{19}^{3} + 18T_{19}^{2} - 12709T_{19} + 95205 \)
|
|
\( T_{23}^{3} - 53T_{23}^{2} - 14822T_{23} + 94749 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 255 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 26 T^{2} + \cdots - 36407 \)
|
| $13$ |
\( T^{3} - 80 T^{2} + \cdots + 160725 \)
|
| $17$ |
\( T^{3} - 30 T^{2} + \cdots - 6341 \)
|
| $19$ |
\( T^{3} + 18 T^{2} + \cdots + 95205 \)
|
| $23$ |
\( T^{3} - 53 T^{2} + \cdots + 94749 \)
|
| $29$ |
\( T^{3} + 404 T^{2} + \cdots + 1808541 \)
|
| $31$ |
\( T^{3} + 615 T^{2} + \cdots + 7562577 \)
|
| $37$ |
\( T^{3} + 426 T^{2} + \cdots - 11343957 \)
|
| $41$ |
\( T^{3} + 101 T^{2} + \cdots - 7167 \)
|
| $43$ |
\( T^{3} - 249 T^{2} + \cdots + 27999735 \)
|
| $47$ |
\( T^{3} - 402 T^{2} + \cdots + 52820631 \)
|
| $53$ |
\( T^{3} + 390 T^{2} + \cdots - 93437133 \)
|
| $59$ |
\( T^{3} + 91 T^{2} + \cdots - 92928465 \)
|
| $61$ |
\( T^{3} + 647 T^{2} + \cdots - 929141 \)
|
| $67$ |
\( T^{3} + 708 T^{2} + \cdots - 2673864 \)
|
| $71$ |
\( T^{3} - 533 T^{2} + \cdots + 74350109 \)
|
| $73$ |
\( T^{3} - 772 T^{2} + \cdots + 209497408 \)
|
| $79$ |
\( T^{3} - 1421 T^{2} + \cdots - 54979815 \)
|
| $83$ |
\( T^{3} + 520 T^{2} + \cdots + 294469047 \)
|
| $89$ |
\( T^{3} + 194 T^{2} + \cdots - 843604613 \)
|
| $97$ |
\( T^{3} + 1027 T^{2} + \cdots - 883021 \)
|
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