Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7840,2,Mod(1,7840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7840, 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("7840.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7840.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: | \(62.6027151847\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.170145936.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} + 22x^{2} - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1120) |
| 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
\( x^{6} - 10x^{4} + 22x^{2} - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + \nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 19\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{4} + 9\nu^{2} - 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} + 9\beta_{4} + 32 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{3} + 20\beta_{2} + 41\beta_1 ) / 2 \)
|
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 | −3.39276 | 0 | −1.00000 | 0 | 0 | 0 | 8.51082 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.52994 | 0 | −1.00000 | 0 | 0 | 0 | −0.659278 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.385303 | 0 | −1.00000 | 0 | 0 | 0 | −2.85154 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.385303 | 0 | −1.00000 | 0 | 0 | 0 | −2.85154 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.52994 | 0 | −1.00000 | 0 | 0 | 0 | −0.659278 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.39276 | 0 | −1.00000 | 0 | 0 | 0 | 8.51082 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7840.2.a.cd | 6 | |
| 4.b | odd | 2 | 1 | inner | 7840.2.a.cd | 6 | |
| 7.b | odd | 2 | 1 | 7840.2.a.ce | 6 | ||
| 7.d | odd | 6 | 2 | 1120.2.q.k | ✓ | 12 | |
| 28.d | even | 2 | 1 | 7840.2.a.ce | 6 | ||
| 28.f | even | 6 | 2 | 1120.2.q.k | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1120.2.q.k | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 1120.2.q.k | ✓ | 12 | 28.f | even | 6 | 2 | |
| 7840.2.a.cd | 6 | 1.a | even | 1 | 1 | trivial | |
| 7840.2.a.cd | 6 | 4.b | odd | 2 | 1 | inner | |
| 7840.2.a.ce | 6 | 7.b | odd | 2 | 1 | ||
| 7840.2.a.ce | 6 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7840))\):
|
\( T_{3}^{6} - 14T_{3}^{4} + 29T_{3}^{2} - 4 \)
|
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\( T_{11}^{6} - 29T_{11}^{4} + 56T_{11}^{2} - 16 \)
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\( T_{13}^{3} + T_{13}^{2} - 34T_{13} - 52 \)
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\( T_{19}^{6} - 129T_{19}^{4} + 5220T_{19}^{2} - 63504 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 14 T^{4} + \cdots - 4 \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 29 T^{4} + \cdots - 16 \)
|
| $13$ |
\( (T^{3} + T^{2} - 34 T - 52)^{2} \)
|
| $17$ |
\( (T^{3} - 4 T^{2} + \cdots + 136)^{2} \)
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| $19$ |
\( T^{6} - 129 T^{4} + \cdots - 63504 \)
|
| $23$ |
\( T^{6} - 47 T^{4} + \cdots - 1369 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 33 T - 18)^{2} \)
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| $31$ |
\( T^{6} - 116 T^{4} + \cdots - 50176 \)
|
| $37$ |
\( (T^{3} - 5 T^{2} - 28 T - 16)^{2} \)
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| $41$ |
\( (T - 1)^{6} \)
|
| $43$ |
\( T^{6} - 166 T^{4} + \cdots - 142884 \)
|
| $47$ |
\( T^{6} - 29 T^{4} + \cdots - 16 \)
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| $53$ |
\( (T^{3} - 7 T^{2} + \cdots + 108)^{2} \)
|
| $59$ |
\( T^{6} - 116 T^{4} + \cdots - 256 \)
|
| $61$ |
\( (T^{3} + 12 T^{2} + \cdots - 776)^{2} \)
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| $67$ |
\( T^{6} - 314 T^{4} + \cdots - 891136 \)
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| $71$ |
\( T^{6} - 340 T^{4} + \cdots - 1327104 \)
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| $73$ |
\( (T^{3} - 6 T^{2} + \cdots + 432)^{2} \)
|
| $79$ |
\( T^{6} - 248 T^{4} + \cdots - 3136 \)
|
| $83$ |
\( T^{6} - 242 T^{4} + \cdots - 12544 \)
|
| $89$ |
\( (T^{3} + 20 T^{2} + \cdots + 126)^{2} \)
|
| $97$ |
\( (T^{3} - 10 T^{2} + \cdots - 56)^{2} \)
|
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