Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1040,4,Mod(1,1040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("1040.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.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: | \(61.3619864060\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{51}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 51 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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 \(\beta = \sqrt{51}\). 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 |
|
0 | −8.14143 | 0 | −5.00000 | 0 | 6.28286 | 0 | 39.2829 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 6.14143 | 0 | −5.00000 | 0 | −22.2829 | 0 | 10.7171 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 1040.4.a.h | 2 | |
| 4.b | odd | 2 | 1 | 130.4.a.f | ✓ | 2 | |
| 12.b | even | 2 | 1 | 1170.4.a.t | 2 | ||
| 20.d | odd | 2 | 1 | 650.4.a.k | 2 | ||
| 20.e | even | 4 | 2 | 650.4.b.m | 4 | ||
| 52.b | odd | 2 | 1 | 1690.4.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.4.a.f | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 650.4.a.k | 2 | 20.d | odd | 2 | 1 | ||
| 650.4.b.m | 4 | 20.e | even | 4 | 2 | ||
| 1040.4.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 1170.4.a.t | 2 | 12.b | even | 2 | 1 | ||
| 1690.4.a.o | 2 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1040))\):
|
\( T_{3}^{2} + 2T_{3} - 50 \)
|
|
\( T_{7}^{2} + 16T_{7} - 140 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T - 50 \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} + 16T - 140 \)
|
| $11$ |
\( T^{2} + 102T + 2550 \)
|
| $13$ |
\( (T - 13)^{2} \)
|
| $17$ |
\( T^{2} - 9996 \)
|
| $19$ |
\( T^{2} + 58T - 3290 \)
|
| $23$ |
\( T^{2} - 210T + 8526 \)
|
| $29$ |
\( T^{2} - 12T - 34440 \)
|
| $31$ |
\( T^{2} - 242T + 6022 \)
|
| $37$ |
\( T^{2} + 632T + 92512 \)
|
| $41$ |
\( T^{2} - 432T - 12300 \)
|
| $43$ |
\( T^{2} + 82T - 84050 \)
|
| $47$ |
\( T^{2} + 192T - 118284 \)
|
| $53$ |
\( T^{2} + 240T - 59244 \)
|
| $59$ |
\( T^{2} + 942T + 178950 \)
|
| $61$ |
\( T^{2} - 460T - 118664 \)
|
| $67$ |
\( T^{2} - 596T - 381212 \)
|
| $71$ |
\( T^{2} - 990T + 242526 \)
|
| $73$ |
\( T^{2} - 832T + 160000 \)
|
| $79$ |
\( T^{2} + 388T - 535400 \)
|
| $83$ |
\( T^{2} - 192T - 700908 \)
|
| $89$ |
\( T^{2} - 1380 T + 292500 \)
|
| $97$ |
\( T^{2} + 572 T - 1096508 \)
|
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