Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8100,2,Mod(1,8100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8100.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8100.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: | \(64.6788256372\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1207701504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 14x^{4} + 43x^{2} - 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 180) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 14x^{4} + 43x^{2} - 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 14\nu^{3} + 37\nu ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} + 19\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 12\nu^{2} + 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{3} + 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 12\beta_{2} + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{4} - 36\beta_{3} + 89\beta_1 \)
|
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 | 0 | 0 | −3.76963 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | −2.26496 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0 | 0 | −1.28835 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 0 | 0 | 1.28835 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 0 | 0 | 2.26496 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 0 | 0 | 3.76963 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8100.2.a.bd | 6 | |
| 3.b | odd | 2 | 1 | 8100.2.a.bc | 6 | ||
| 5.b | even | 2 | 1 | inner | 8100.2.a.bd | 6 | |
| 5.c | odd | 4 | 2 | 1620.2.d.d | 6 | ||
| 9.c | even | 3 | 2 | 2700.2.i.f | 12 | ||
| 9.d | odd | 6 | 2 | 900.2.i.f | 12 | ||
| 15.d | odd | 2 | 1 | 8100.2.a.bc | 6 | ||
| 15.e | even | 4 | 2 | 1620.2.d.c | 6 | ||
| 45.h | odd | 6 | 2 | 900.2.i.f | 12 | ||
| 45.j | even | 6 | 2 | 2700.2.i.f | 12 | ||
| 45.k | odd | 12 | 4 | 540.2.r.a | 12 | ||
| 45.l | even | 12 | 4 | 180.2.r.a | ✓ | 12 | |
| 180.v | odd | 12 | 4 | 720.2.by.e | 12 | ||
| 180.x | even | 12 | 4 | 2160.2.by.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 180.2.r.a | ✓ | 12 | 45.l | even | 12 | 4 | |
| 540.2.r.a | 12 | 45.k | odd | 12 | 4 | ||
| 720.2.by.e | 12 | 180.v | odd | 12 | 4 | ||
| 900.2.i.f | 12 | 9.d | odd | 6 | 2 | ||
| 900.2.i.f | 12 | 45.h | odd | 6 | 2 | ||
| 1620.2.d.c | 6 | 15.e | even | 4 | 2 | ||
| 1620.2.d.d | 6 | 5.c | odd | 4 | 2 | ||
| 2160.2.by.e | 12 | 180.x | even | 12 | 4 | ||
| 2700.2.i.f | 12 | 9.c | even | 3 | 2 | ||
| 2700.2.i.f | 12 | 45.j | even | 6 | 2 | ||
| 8100.2.a.bc | 6 | 3.b | odd | 2 | 1 | ||
| 8100.2.a.bc | 6 | 15.d | odd | 2 | 1 | ||
| 8100.2.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 8100.2.a.bd | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8100))\):
|
\( T_{7}^{6} - 21T_{7}^{4} + 105T_{7}^{2} - 121 \)
|
|
\( T_{11}^{3} - T_{11}^{2} - 22T_{11} + 46 \)
|
|
\( T_{17}^{6} - 36T_{17}^{4} + 108T_{17}^{2} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 21 T^{4} + \cdots - 121 \)
|
| $11$ |
\( (T^{3} - T^{2} - 22 T + 46)^{2} \)
|
| $13$ |
\( T^{6} - 39 T^{4} + \cdots - 324 \)
|
| $17$ |
\( T^{6} - 36 T^{4} + \cdots - 64 \)
|
| $19$ |
\( (T^{3} - 42 T - 72)^{2} \)
|
| $23$ |
\( T^{6} - 93 T^{4} + \cdots - 28561 \)
|
| $29$ |
\( (T - 3)^{6} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} + \cdots - 358)^{2} \)
|
| $37$ |
\( T^{6} - 60 T^{4} + \cdots - 256 \)
|
| $41$ |
\( (T^{3} - 7 T^{2} - 19 T + 97)^{2} \)
|
| $43$ |
\( T^{6} - 231 T^{4} + \cdots - 91204 \)
|
| $47$ |
\( T^{6} - 105 T^{4} + \cdots - 961 \)
|
| $53$ |
\( T^{6} - 264 T^{4} + \cdots - 331776 \)
|
| $59$ |
\( (T^{3} - 17 T^{2} + \cdots - 88)^{2} \)
|
| $61$ |
\( (T^{3} + 3 T^{2} - 39 T + 31)^{2} \)
|
| $67$ |
\( T^{6} - 105 T^{4} + \cdots - 14641 \)
|
| $71$ |
\( (T^{3} - 42 T + 72)^{2} \)
|
| $73$ |
\( T^{6} - 384 T^{4} + \cdots - 369664 \)
|
| $79$ |
\( (T^{3} + 3 T^{2} + \cdots - 108)^{2} \)
|
| $83$ |
\( T^{6} - 405 T^{4} + \cdots - 1907161 \)
|
| $89$ |
\( (T^{3} - 28 T^{2} + \cdots - 662)^{2} \)
|
| $97$ |
\( T^{6} - 339 T^{4} + \cdots - 55696 \)
|
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