Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9801,2,Mod(1,9801)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("9801.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9801 = 3^{4} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9801.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: | \(78.2613790211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.22545.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 99) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 6x^{2} + 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
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 |
|
−2.46934 | 0 | 4.09762 | −2.43628 | 0 | −2.33866 | −5.17972 | 0 | 6.01598 | ||||||||||||||||||||||||||||||
| 1.2 | −2.15561 | 0 | 2.64667 | 3.62393 | 0 | 2.27060 | −1.39396 | 0 | −7.81179 | |||||||||||||||||||||||||||||||
| 1.3 | 0.894434 | 0 | −1.19999 | 3.74893 | 0 | −1.45106 | −2.86218 | 0 | 3.35317 | |||||||||||||||||||||||||||||||
| 1.4 | 2.73051 | 0 | 5.45571 | −0.936586 | 0 | 0.519120 | 9.43585 | 0 | −2.55736 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
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 | 9801.2.a.bi | 4 | |
| 3.b | odd | 2 | 1 | 9801.2.a.bl | 4 | ||
| 9.c | even | 3 | 2 | 1089.2.e.i | 8 | ||
| 11.b | odd | 2 | 1 | 891.2.a.q | 4 | ||
| 33.d | even | 2 | 1 | 891.2.a.p | 4 | ||
| 99.g | even | 6 | 2 | 297.2.e.e | 8 | ||
| 99.h | odd | 6 | 2 | 99.2.e.e | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.2.e.e | ✓ | 8 | 99.h | odd | 6 | 2 | |
| 297.2.e.e | 8 | 99.g | even | 6 | 2 | ||
| 891.2.a.p | 4 | 33.d | even | 2 | 1 | ||
| 891.2.a.q | 4 | 11.b | odd | 2 | 1 | ||
| 1089.2.e.i | 8 | 9.c | even | 3 | 2 | ||
| 9801.2.a.bi | 4 | 1.a | even | 1 | 1 | trivial | |
| 9801.2.a.bl | 4 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):
|
\( T_{2}^{4} + T_{2}^{3} - 9T_{2}^{2} - 8T_{2} + 13 \)
|
|
\( T_{5}^{4} - 4T_{5}^{3} - 9T_{5}^{2} + 29T_{5} + 31 \)
|
|
\( T_{7}^{4} + T_{7}^{3} - 6T_{7}^{2} - 5T_{7} + 4 \)
|
|
\( T_{17}^{4} - 5T_{17}^{3} - 24T_{17}^{2} + 169T_{17} - 236 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 13 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 31 \)
|
| $7$ |
\( T^{4} + T^{3} - 6 T^{2} + \cdots + 4 \)
|
| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} + 7 T^{3} + \cdots - 158 \)
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| $17$ |
\( T^{4} - 5 T^{3} + \cdots - 236 \)
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| $19$ |
\( T^{4} + 9 T^{3} + \cdots - 54 \)
|
| $23$ |
\( T^{4} - 14 T^{3} + \cdots - 188 \)
|
| $29$ |
\( T^{4} - 6 T^{3} + \cdots - 150 \)
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| $31$ |
\( T^{4} + 2 T^{3} + \cdots - 59 \)
|
| $37$ |
\( T^{4} - 3 T^{3} + \cdots + 57 \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots - 26 \)
|
| $43$ |
\( T^{4} - 21 T^{3} + \cdots + 498 \)
|
| $47$ |
\( T^{4} + 7 T^{3} + \cdots + 1 \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots - 123 \)
|
| $59$ |
\( T^{4} - 2 T^{3} + \cdots + 13 \)
|
| $61$ |
\( T^{4} + 15 T^{3} + \cdots + 120 \)
|
| $67$ |
\( T^{4} - 14 T^{3} + \cdots - 2171 \)
|
| $71$ |
\( T^{4} + 3 T^{3} + \cdots - 195 \)
|
| $73$ |
\( T^{4} + 22 T^{3} + \cdots - 1028 \)
|
| $79$ |
\( T^{4} + 11 T^{3} + \cdots + 40 \)
|
| $83$ |
\( T^{4} + 18 T^{3} + \cdots + 1548 \)
|
| $89$ |
\( T^{4} + 6 T^{3} + \cdots + 2064 \)
|
| $97$ |
\( T^{4} - 26 T^{3} + \cdots - 1229 \)
|
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