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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 9x^{5} + 16x^{4} + 20x^{3} - 29x^{2} - 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 9x^{5} + 16x^{4} + 20x^{3} - 29x^{2} - 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 13\nu^{3} - 20\nu^{2} + 11\nu + 9 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} - \nu^{5} - 18\nu^{4} + 2\nu^{3} + 34\nu^{2} + 8\nu - 6 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 16\nu^{3} + 20\nu^{2} - 26\nu - 12 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 11\nu^{3} + 35\nu^{2} + 22\nu - 15 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + \beta_{4} + 10\beta_{3} + 2\beta_{2} + 30\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{6} + 3\beta_{5} + 11\beta_{4} + 13\beta_{3} + 47\beta_{2} + 15\beta _1 + 91 \)
|
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 |
|
−2.76794 | 1.00000 | 5.66150 | −1.00000 | −2.76794 | −1.00000 | −10.1348 | 1.00000 | 2.76794 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88240 | 1.00000 | 1.54343 | −1.00000 | −1.88240 | −1.00000 | 0.859442 | 1.00000 | 1.88240 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.48389 | 1.00000 | 0.201931 | −1.00000 | −1.48389 | −1.00000 | 2.66814 | 1.00000 | 1.48389 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.451228 | 1.00000 | −1.79639 | −1.00000 | −0.451228 | −1.00000 | 1.71304 | 1.00000 | 0.451228 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.748534 | 1.00000 | −1.43970 | −1.00000 | 0.748534 | −1.00000 | −2.57473 | 1.00000 | −0.748534 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.43461 | 1.00000 | 0.0581079 | −1.00000 | 1.43461 | −1.00000 | −2.78586 | 1.00000 | −1.43461 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.40232 | 1.00000 | 3.77112 | −1.00000 | 2.40232 | −1.00000 | 4.25479 | 1.00000 | −2.40232 | ||||||||||||||||||||||||||||||||||||||||
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.r | ✓ | 7 |
| 3.b | odd | 2 | 1 | 7245.2.a.bn | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2415.2.a.r | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 7245.2.a.bn | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 9T_{2}^{5} - 16T_{2}^{4} + 20T_{2}^{3} + 29T_{2}^{2} - 12T_{2} - 9 \)
|
|
\( T_{11}^{7} + 2T_{11}^{6} - 55T_{11}^{5} - 64T_{11}^{4} + 831T_{11}^{3} + 674T_{11}^{2} - 3222T_{11} - 870 \)
|
|
\( T_{13}^{7} - 6T_{13}^{6} - 31T_{13}^{5} + 220T_{13}^{4} + 111T_{13}^{3} - 2006T_{13}^{2} + 2002T_{13} + 1086 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 9 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots - 870 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots + 1086 \)
|
| $17$ |
\( T^{7} - 14 T^{6} + \cdots - 16200 \)
|
| $19$ |
\( T^{7} - 6 T^{6} + \cdots + 3180 \)
|
| $23$ |
\( (T + 1)^{7} \)
|
| $29$ |
\( T^{7} + 4 T^{6} + \cdots - 192 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots + 351968 \)
|
| $37$ |
\( T^{7} - 18 T^{6} + \cdots - 43686 \)
|
| $41$ |
\( T^{7} - 10 T^{6} + \cdots - 141480 \)
|
| $43$ |
\( T^{7} - 26 T^{6} + \cdots + 137736 \)
|
| $47$ |
\( T^{7} + 4 T^{6} + \cdots + 6528 \)
|
| $53$ |
\( T^{7} - 2 T^{6} + \cdots - 96 \)
|
| $59$ |
\( T^{7} - 6 T^{6} + \cdots - 37146 \)
|
| $61$ |
\( T^{7} + 4 T^{6} + \cdots - 540 \)
|
| $67$ |
\( T^{7} - 22 T^{6} + \cdots + 25088 \)
|
| $71$ |
\( T^{7} + 8 T^{6} + \cdots + 359232 \)
|
| $73$ |
\( T^{7} - 24 T^{6} + \cdots + 43110 \)
|
| $79$ |
\( T^{7} - 2 T^{6} + \cdots + 1228320 \)
|
| $83$ |
\( T^{7} - 26 T^{6} + \cdots + 2916 \)
|
| $89$ |
\( T^{7} - 22 T^{6} + \cdots - 4800 \)
|
| $97$ |
\( T^{7} - 44 T^{6} + \cdots - 4672 \)
|
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