Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2415,2,Mod(1,2415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, 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("2415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2415.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: | \(19.2838720881\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.15751800.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 4\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - \nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 11\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{3} + 5\beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 7\beta_{2} + 17\beta _1 + 1 \)
|
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 |
|
−1.96273 | −1.00000 | 1.85229 | −1.00000 | 1.96273 | 1.00000 | 0.289915 | 1.00000 | 1.96273 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.923950 | −1.00000 | −1.14632 | −1.00000 | 0.923950 | 1.00000 | 2.90704 | 1.00000 | 0.923950 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.456154 | −1.00000 | −1.79192 | −1.00000 | −0.456154 | 1.00000 | −1.72970 | 1.00000 | −0.456154 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.646541 | −1.00000 | −1.58198 | −1.00000 | −0.646541 | 1.00000 | −2.31590 | 1.00000 | −0.646541 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.05680 | −1.00000 | 2.23042 | −1.00000 | −2.05680 | 1.00000 | 0.473937 | 1.00000 | −2.05680 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.72718 | −1.00000 | 5.43751 | −1.00000 | −2.72718 | 1.00000 | 9.37471 | 1.00000 | −2.72718 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 2415.2.a.q | ✓ | 6 |
| 3.b | odd | 2 | 1 | 7245.2.a.bj | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2415.2.a.q | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 7245.2.a.bj | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):
|
\( T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 14T_{2}^{3} - 9T_{2} + 3 \)
|
|
\( T_{11}^{6} - 6T_{11}^{5} - 15T_{11}^{4} + 118T_{11}^{3} - 45T_{11}^{2} - 360T_{11} + 270 \)
|
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\( T_{13}^{6} + 6T_{13}^{5} - 31T_{13}^{4} - 174T_{13}^{3} + 183T_{13}^{2} + 1276T_{13} + 698 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 3 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 270 \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots + 698 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 204 \)
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| $19$ |
\( T^{6} + 8 T^{5} + \cdots - 916 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots + 640 \)
|
| $31$ |
\( T^{6} + 16 T^{5} + \cdots + 35744 \)
|
| $37$ |
\( T^{6} - 8 T^{5} + \cdots - 100638 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 14884 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots + 28192 \)
|
| $47$ |
\( T^{6} - 10 T^{5} + \cdots + 10368 \)
|
| $53$ |
\( T^{6} - 28 T^{5} + \cdots + 5664 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 31534 \)
|
| $61$ |
\( T^{6} + 12 T^{5} + \cdots + 57416 \)
|
| $67$ |
\( T^{6} - 18 T^{5} + \cdots + 801072 \)
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| $71$ |
\( T^{6} - 10 T^{5} + \cdots - 84288 \)
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| $73$ |
\( T^{6} + 2 T^{5} + \cdots - 2510 \)
|
| $79$ |
\( T^{6} + 2 T^{5} + \cdots + 38944 \)
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| $83$ |
\( T^{6} - 26 T^{5} + \cdots + 1175996 \)
|
| $89$ |
\( T^{6} - 8 T^{5} + \cdots + 36096 \)
|
| $97$ |
\( T^{6} - 20 T^{5} + \cdots + 1501568 \)
|
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