Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,6,Mod(1,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("490.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.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: | \(78.5880717084\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{46}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 46 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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 \(\beta = \sqrt{46}\). 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 |
|
4.00000 | 2.21767 | 16.0000 | 25.0000 | 8.87068 | 0 | 64.0000 | −238.082 | 100.000 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 15.7823 | 16.0000 | 25.0000 | 63.1293 | 0 | 64.0000 | 6.08194 | 100.000 | |||||||||||||||||||||||||
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 | 490.6.a.w | 2 | |
| 7.b | odd | 2 | 1 | 490.6.a.t | 2 | ||
| 7.c | even | 3 | 2 | 70.6.e.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.6.e.c | ✓ | 4 | 7.c | even | 3 | 2 | |
| 490.6.a.t | 2 | 7.b | odd | 2 | 1 | ||
| 490.6.a.w | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 18T_{3} + 35 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 18T + 35 \)
|
| $5$ |
\( (T - 25)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 252T - 81460 \)
|
| $13$ |
\( T^{2} + 1588 T + 570820 \)
|
| $17$ |
\( T^{2} + 716 T - 2521436 \)
|
| $19$ |
\( T^{2} + 1032 T - 4327120 \)
|
| $23$ |
\( T^{2} + 2242 T - 830333 \)
|
| $29$ |
\( T^{2} + 8330 T + 5194209 \)
|
| $31$ |
\( T^{2} + 7652 T - 15244980 \)
|
| $37$ |
\( T^{2} + 11168 T + 31144992 \)
|
| $41$ |
\( T^{2} - 8990 T - 156148119 \)
|
| $43$ |
\( T^{2} + 6502 T - 126945285 \)
|
| $47$ |
\( T^{2} - 30364 T + 201053124 \)
|
| $53$ |
\( T^{2} + 22888 T - 438993264 \)
|
| $59$ |
\( T^{2} + 992 T - 383419008 \)
|
| $61$ |
\( T^{2} - 37242 T + 6415241 \)
|
| $67$ |
\( T^{2} + \cdots + 1038907971 \)
|
| $71$ |
\( T^{2} + \cdots + 1575329984 \)
|
| $73$ |
\( T^{2} + \cdots - 2990993660 \)
|
| $79$ |
\( T^{2} + 85136 T - 923812800 \)
|
| $83$ |
\( T^{2} + \cdots - 3228067701 \)
|
| $89$ |
\( T^{2} + \cdots + 7770771401 \)
|
| $97$ |
\( T^{2} + \cdots - 6379310140 \)
|
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