Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2254,4,Mod(1,2254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2254, 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("2254.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2254.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: | \(132.990305153\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | −2.24264 | 4.00000 | −3.75736 | 4.48528 | 0 | −8.00000 | −21.9706 | 7.51472 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 6.24264 | 4.00000 | −12.2426 | −12.4853 | 0 | −8.00000 | 11.9706 | 24.4853 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 2254.4.a.e | 2 | |
| 7.b | odd | 2 | 1 | 322.4.a.a | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.a | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 2254.4.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} - 14 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} - 4T - 14 \)
|
| $5$ |
\( T^{2} + 16T + 46 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 60T + 612 \)
|
| $13$ |
\( T^{2} + 72T + 1288 \)
|
| $17$ |
\( T^{2} + 12T - 6926 \)
|
| $19$ |
\( T^{2} - 28T - 604 \)
|
| $23$ |
\( (T - 23)^{2} \)
|
| $29$ |
\( T^{2} + 160T + 5752 \)
|
| $31$ |
\( T^{2} + 160T - 12418 \)
|
| $37$ |
\( T^{2} - 284T + 17852 \)
|
| $41$ |
\( T^{2} + 492T + 4068 \)
|
| $43$ |
\( T^{2} - 272T - 58336 \)
|
| $47$ |
\( T^{2} + 352T - 101122 \)
|
| $53$ |
\( T^{2} - 428T + 5468 \)
|
| $59$ |
\( T^{2} + 884T + 46306 \)
|
| $61$ |
\( T^{2} + 576T + 20286 \)
|
| $67$ |
\( T^{2} - 532T + 33764 \)
|
| $71$ |
\( T^{2} - 688T - 192136 \)
|
| $73$ |
\( T^{2} + 12T - 434276 \)
|
| $79$ |
\( T^{2} + 748T - 519076 \)
|
| $83$ |
\( T^{2} - 100T - 307972 \)
|
| $89$ |
\( T^{2} + 676 T - 1347806 \)
|
| $97$ |
\( T^{2} - 124T - 116206 \)
|
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