Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8550,2,Mod(1,8550)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, 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("8550.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8550.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: | \(68.2720937282\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3356224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 8x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 1710) |
| 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
\( x^{6} - 6x^{4} + 8x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{4} - 8\nu^{2} + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( 2\nu^{5} - 10\nu^{3} + 8\nu \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 12\nu^{3} + 14\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 3\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 13 ) / 2 \)
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| \(\nu^{5}\) | \(=\) |
\( ( -5\beta_{5} + 6\beta_{4} + 11\beta_1 ) / 2 \)
|
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 |
|
1.00000 | 0 | 1.00000 | 0 | 0 | −4.05705 | 1.00000 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | 0 | 0 | −2.64265 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 0 | 0 | −0.746175 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 0 | 0 | 0.746175 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 0 | 0 | 2.64265 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 0 | 0 | 4.05705 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8550.2.a.cx | 6 | |
| 3.b | odd | 2 | 1 | 8550.2.a.cw | 6 | ||
| 5.b | even | 2 | 1 | 8550.2.a.cw | 6 | ||
| 5.c | odd | 4 | 2 | 1710.2.d.h | ✓ | 12 | |
| 15.d | odd | 2 | 1 | inner | 8550.2.a.cx | 6 | |
| 15.e | even | 4 | 2 | 1710.2.d.h | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1710.2.d.h | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 1710.2.d.h | ✓ | 12 | 15.e | even | 4 | 2 | |
| 8550.2.a.cw | 6 | 3.b | odd | 2 | 1 | ||
| 8550.2.a.cw | 6 | 5.b | even | 2 | 1 | ||
| 8550.2.a.cx | 6 | 1.a | even | 1 | 1 | trivial | |
| 8550.2.a.cx | 6 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):
|
\( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \)
|
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\( T_{11}^{6} - 44T_{11}^{4} + 304T_{11}^{2} - 64 \)
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\( T_{13}^{6} - 32T_{13}^{4} + 96T_{13}^{2} - 64 \)
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\( T_{17}^{3} - 4T_{17}^{2} - 16T_{17} + 56 \)
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\( T_{23}^{3} - 6T_{23}^{2} - 4T_{23} + 32 \)
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\( T_{53}^{3} - 10T_{53}^{2} - 52T_{53} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - 24 T^{4} + \cdots - 64 \)
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| $11$ |
\( T^{6} - 44 T^{4} + \cdots - 64 \)
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| $13$ |
\( T^{6} - 32 T^{4} + \cdots - 64 \)
|
| $17$ |
\( (T^{3} - 4 T^{2} - 16 T + 56)^{2} \)
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| $19$ |
\( (T - 1)^{6} \)
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| $23$ |
\( (T^{3} - 6 T^{2} - 4 T + 32)^{2} \)
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| $29$ |
\( T^{6} - 76 T^{4} + \cdots - 3136 \)
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| $31$ |
\( (T^{3} - 4 T^{2} - 20 T + 16)^{2} \)
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| $37$ |
\( T^{6} - 128 T^{4} + \cdots - 3136 \)
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| $41$ |
\( T^{6} - 200 T^{4} + \cdots - 179776 \)
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| $43$ |
\( T^{6} - 44 T^{4} + \cdots - 64 \)
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| $47$ |
\( (T^{3} - 10 T^{2} + \cdots + 1184)^{2} \)
|
| $53$ |
\( (T^{3} - 10 T^{2} - 52 T + 8)^{2} \)
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| $59$ |
\( T^{6} - 212 T^{4} + \cdots - 65536 \)
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| $61$ |
\( (T^{3} - 6 T^{2} + \cdots + 184)^{2} \)
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| $67$ |
\( T^{6} - 480 T^{4} + \cdots - 3936256 \)
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| $71$ |
\( T^{6} - 128 T^{4} + \cdots - 16384 \)
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| $73$ |
\( T^{6} - 368 T^{4} + \cdots - 1048576 \)
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| $79$ |
\( (T^{3} - 228 T + 416)^{2} \)
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| $83$ |
\( (T^{3} - 6 T^{2} + \cdots + 784)^{2} \)
|
| $89$ |
\( T^{6} - 200 T^{4} + \cdots - 153664 \)
|
| $97$ |
\( T^{6} - 204 T^{4} + \cdots - 87616 \)
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