Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,6,Mod(1,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 504.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: | \(80.8334451857\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{37}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(\beta = 2\sqrt{37}\). 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 | 0 | 0 | −84.8276 | 0 | −49.0000 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 36.8276 | 0 | −49.0000 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 504.6.a.l | ✓ | 2 |
| 3.b | odd | 2 | 1 | 504.6.a.r | yes | 2 | |
| 4.b | odd | 2 | 1 | 1008.6.a.bh | 2 | ||
| 12.b | even | 2 | 1 | 1008.6.a.bs | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 504.6.a.l | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 504.6.a.r | yes | 2 | 3.b | odd | 2 | 1 | |
| 1008.6.a.bh | 2 | 4.b | odd | 2 | 1 | ||
| 1008.6.a.bs | 2 | 12.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(504))\):
|
\( T_{5}^{2} + 48T_{5} - 3124 \)
|
|
\( T_{11}^{2} + 368T_{11} - 168756 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 48T - 3124 \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} + 368T - 168756 \)
|
| $13$ |
\( T^{2} + 156T - 941116 \)
|
| $17$ |
\( T^{2} - 3312 T + 2581164 \)
|
| $19$ |
\( T^{2} - 3736 T + 3148432 \)
|
| $23$ |
\( T^{2} + 1552 T + 559404 \)
|
| $29$ |
\( T^{2} + 1728 T - 4709376 \)
|
| $31$ |
\( T^{2} + 3624 T + 3245456 \)
|
| $37$ |
\( T^{2} - 6996 T + 10379492 \)
|
| $41$ |
\( T^{2} + 26160 T + 153876812 \)
|
| $43$ |
\( T^{2} + 30184 T + 227616912 \)
|
| $47$ |
\( T^{2} - 11424 T + 32129072 \)
|
| $53$ |
\( T^{2} - 17376 T - 722487888 \)
|
| $59$ |
\( T^{2} + 15008 T - 428552784 \)
|
| $61$ |
\( T^{2} - 35564 T + 114294372 \)
|
| $67$ |
\( T^{2} + \cdots + 1203906192 \)
|
| $71$ |
\( T^{2} + 40752 T - 462343924 \)
|
| $73$ |
\( T^{2} + 53892 T + 518612228 \)
|
| $79$ |
\( T^{2} + \cdots - 1071226816 \)
|
| $83$ |
\( T^{2} + \cdots - 8758457600 \)
|
| $89$ |
\( T^{2} + \cdots - 5236430292 \)
|
| $97$ |
\( T^{2} + \cdots - 19079892924 \)
|
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