Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2320,2,Mod(1,2320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2320, 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("2320.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2320 = 2^{4} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2320.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: | \(18.5252932689\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.580484.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 7x^{3} - 3x^{2} + 8x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1160) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 7x^{3} - 3x^{2} + 8x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 7\nu^{2} - 3\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 4\beta_{2} + 6\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{4} + 4\beta_{3} - 4\beta_{2} + 5\beta _1 + 11 \)
|
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 | −3.04362 | 0 | 1.00000 | 0 | −3.43080 | 0 | 6.26362 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.50610 | 0 | 1.00000 | 0 | 3.62303 | 0 | 3.28054 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.857770 | 0 | 1.00000 | 0 | −2.23769 | 0 | −2.26423 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.02751 | 0 | 1.00000 | 0 | −0.377000 | 0 | −1.94422 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.37998 | 0 | 1.00000 | 0 | −4.57754 | 0 | 2.66429 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 2320.2.a.u | 5 | |
| 4.b | odd | 2 | 1 | 1160.2.a.j | ✓ | 5 | |
| 8.b | even | 2 | 1 | 9280.2.a.cl | 5 | ||
| 8.d | odd | 2 | 1 | 9280.2.a.cg | 5 | ||
| 20.d | odd | 2 | 1 | 5800.2.a.t | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1160.2.a.j | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 2320.2.a.u | 5 | 1.a | even | 1 | 1 | trivial | |
| 5800.2.a.t | 5 | 20.d | odd | 2 | 1 | ||
| 9280.2.a.cg | 5 | 8.d | odd | 2 | 1 | ||
| 9280.2.a.cl | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2320))\):
|
\( T_{3}^{5} + 3T_{3}^{4} - 7T_{3}^{3} - 20T_{3}^{2} + 8T_{3} + 16 \)
|
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\( T_{7}^{5} + 7T_{7}^{4} - T_{7}^{3} - 88T_{7}^{2} - 160T_{7} - 48 \)
|
|
\( T_{11}^{5} + 10T_{11}^{4} + 12T_{11}^{3} - 112T_{11}^{2} - 224T_{11} + 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 16 \)
|
| $5$ |
\( (T - 1)^{5} \)
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| $7$ |
\( T^{5} + 7 T^{4} + \cdots - 48 \)
|
| $11$ |
\( T^{5} + 10 T^{4} + \cdots + 64 \)
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| $13$ |
\( T^{5} - T^{4} + \cdots - 536 \)
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| $17$ |
\( T^{5} + T^{4} + \cdots - 72 \)
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| $19$ |
\( T^{5} + 10 T^{4} + \cdots + 1856 \)
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| $23$ |
\( T^{5} + T^{4} + \cdots - 16 \)
|
| $29$ |
\( (T - 1)^{5} \)
|
| $31$ |
\( T^{5} + 7 T^{4} + \cdots - 16 \)
|
| $37$ |
\( T^{5} + 8 T^{4} + \cdots + 8032 \)
|
| $41$ |
\( T^{5} + 4 T^{4} + \cdots + 992 \)
|
| $43$ |
\( T^{5} + 9 T^{4} + \cdots + 464 \)
|
| $47$ |
\( T^{5} + 8 T^{4} + \cdots - 4096 \)
|
| $53$ |
\( T^{5} + 9 T^{4} + \cdots + 29992 \)
|
| $59$ |
\( T^{5} + 29 T^{4} + \cdots - 38336 \)
|
| $61$ |
\( T^{5} - T^{4} + \cdots - 2984 \)
|
| $67$ |
\( T^{5} + 24 T^{4} + \cdots - 15616 \)
|
| $71$ |
\( T^{5} - 12 T^{4} + \cdots - 16384 \)
|
| $73$ |
\( T^{5} + 9 T^{4} + \cdots - 5832 \)
|
| $79$ |
\( T^{5} + 13 T^{4} + \cdots - 15376 \)
|
| $83$ |
\( T^{5} + 10 T^{4} + \cdots - 64 \)
|
| $89$ |
\( T^{5} - 4 T^{4} + \cdots + 50656 \)
|
| $97$ |
\( T^{5} + 15 T^{4} + \cdots + 33944 \)
|
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