Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,6,Mod(1,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("210.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.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: | \(33.6806021607\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{176089}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44022 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{176089}\). 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 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | 100.000 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | 49.0000 | 64.0000 | 81.0000 | 100.000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 210.6.a.m | ✓ | 2 |
| 3.b | odd | 2 | 1 | 630.6.a.t | 2 | ||
| 5.b | even | 2 | 1 | 1050.6.a.u | 2 | ||
| 5.c | odd | 4 | 2 | 1050.6.g.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.6.a.m | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 630.6.a.t | 2 | 3.b | odd | 2 | 1 | ||
| 1050.6.a.u | 2 | 5.b | even | 2 | 1 | ||
| 1050.6.g.r | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 82T_{11} - 174408 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(210))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( (T + 9)^{2} \)
|
| $5$ |
\( (T - 25)^{2} \)
|
| $7$ |
\( (T - 49)^{2} \)
|
| $11$ |
\( T^{2} - 82T - 174408 \)
|
| $13$ |
\( T^{2} - 210T - 165064 \)
|
| $17$ |
\( T^{2} + 340 T - 2788524 \)
|
| $19$ |
\( T^{2} + 506 T - 1520792 \)
|
| $23$ |
\( T^{2} - 3388 T + 2165280 \)
|
| $29$ |
\( T^{2} - 4500 T - 20294316 \)
|
| $31$ |
\( T^{2} - 10158 T + 4489472 \)
|
| $37$ |
\( T^{2} - 13464 T + 27710924 \)
|
| $41$ |
\( T^{2} - 11648 T - 169639908 \)
|
| $43$ |
\( T^{2} - 3584 T - 67224336 \)
|
| $47$ |
\( T^{2} - 14344 T - 424707072 \)
|
| $53$ |
\( T^{2} - 41406 T + 427029408 \)
|
| $59$ |
\( T^{2} + \cdots - 1690771872 \)
|
| $61$ |
\( T^{2} - 58668 T + 849213860 \)
|
| $67$ |
\( T^{2} - 56000 T + 443091696 \)
|
| $71$ |
\( T^{2} - 54134 T + 343641888 \)
|
| $73$ |
\( T^{2} + \cdots - 3909094840 \)
|
| $79$ |
\( T^{2} + 46696 T + 204220800 \)
|
| $83$ |
\( T^{2} + \cdots + 7491871152 \)
|
| $89$ |
\( T^{2} + \cdots - 7281420300 \)
|
| $97$ |
\( T^{2} + \cdots + 3072018800 \)
|
| show more | |
| show less |