Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9920,2,Mod(1,9920)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9920, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9920.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9920 = 2^{6} \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9920.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: | \(79.2115988051\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8468.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 155) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta _1 + 1 \)
|
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 | −1.89122 | 0 | −1.00000 | 0 | −4.24412 | 0 | 0.576713 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.704624 | 0 | −1.00000 | 0 | 4.77104 | 0 | −2.50350 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.31743 | 0 | −1.00000 | 0 | −1.13698 | 0 | −1.26438 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.27841 | 0 | −1.00000 | 0 | −1.38995 | 0 | 2.19117 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(31\) | \(-1\) |
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 | 9920.2.a.cg | 4 | |
| 4.b | odd | 2 | 1 | 9920.2.a.cb | 4 | ||
| 8.b | even | 2 | 1 | 2480.2.a.x | 4 | ||
| 8.d | odd | 2 | 1 | 155.2.a.e | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1395.2.a.l | 4 | ||
| 40.e | odd | 2 | 1 | 775.2.a.e | 4 | ||
| 40.k | even | 4 | 2 | 775.2.b.f | 8 | ||
| 56.e | even | 2 | 1 | 7595.2.a.s | 4 | ||
| 120.m | even | 2 | 1 | 6975.2.a.bn | 4 | ||
| 248.b | even | 2 | 1 | 4805.2.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 155.2.a.e | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 775.2.a.e | 4 | 40.e | odd | 2 | 1 | ||
| 775.2.b.f | 8 | 40.k | even | 4 | 2 | ||
| 1395.2.a.l | 4 | 24.f | even | 2 | 1 | ||
| 2480.2.a.x | 4 | 8.b | even | 2 | 1 | ||
| 4805.2.a.n | 4 | 248.b | even | 2 | 1 | ||
| 6975.2.a.bn | 4 | 120.m | even | 2 | 1 | ||
| 7595.2.a.s | 4 | 56.e | even | 2 | 1 | ||
| 9920.2.a.cb | 4 | 4.b | odd | 2 | 1 | ||
| 9920.2.a.cg | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9920))\):
|
\( T_{3}^{4} - T_{3}^{3} - 5T_{3}^{2} + 3T_{3} + 4 \)
|
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\( T_{7}^{4} + 2T_{7}^{3} - 20T_{7}^{2} - 52T_{7} - 32 \)
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\( T_{11}^{4} + 4T_{11}^{3} - 8T_{11}^{2} - 12T_{11} + 16 \)
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\( T_{13}^{4} + 10T_{13}^{3} + 20T_{13}^{2} - 52T_{13} - 136 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - T^{3} - 5 T^{2} + \cdots + 4 \)
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| $5$ |
\( (T + 1)^{4} \)
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| $7$ |
\( T^{4} + 2 T^{3} + \cdots - 32 \)
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| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
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| $13$ |
\( T^{4} + 10 T^{3} + \cdots - 136 \)
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| $17$ |
\( T^{4} - 11 T^{3} + \cdots - 58 \)
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| $19$ |
\( T^{4} + 3 T^{3} + \cdots + 44 \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots - 32 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots - 584 \)
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| $31$ |
\( (T - 1)^{4} \)
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| $37$ |
\( T^{4} + 3 T^{3} + \cdots + 1538 \)
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| $41$ |
\( T^{4} + 11 T^{3} + \cdots + 506 \)
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| $43$ |
\( T^{4} - 17 T^{3} + \cdots - 236 \)
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| $47$ |
\( T^{4} - 10 T^{3} + \cdots + 1408 \)
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| $53$ |
\( T^{4} + 13 T^{3} + \cdots + 1306 \)
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| $59$ |
\( T^{4} + 3 T^{3} + \cdots - 44 \)
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| $61$ |
\( T^{4} + 22 T^{3} + \cdots + 352 \)
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| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 1296 \)
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| $71$ |
\( T^{4} - 21 T^{3} + \cdots + 584 \)
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| $73$ |
\( T^{4} - 19 T^{3} + \cdots + 34 \)
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| $79$ |
\( T^{4} - 2 T^{3} + \cdots + 6592 \)
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| $83$ |
\( T^{4} + 15 T^{3} + \cdots - 3364 \)
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| $89$ |
\( T^{4} + 10 T^{3} + \cdots + 3688 \)
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| $97$ |
\( T^{4} - 4 T^{3} + \cdots - 464 \)
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