Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3680,2,Mod(1,3680)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3680, 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("3680.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3680 = 2^{5} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3680.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: | \(29.3849479438\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.876604.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 9\nu^{2} - 6\nu + 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 20 \)
|
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 | −2.82032 | 0 | 1.00000 | 0 | −2.35036 | 0 | 4.95418 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.706809 | 0 | 1.00000 | 0 | −5.21261 | 0 | −2.50042 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.352411 | 0 | 1.00000 | 0 | 3.95190 | 0 | −2.87581 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.93128 | 0 | 1.00000 | 0 | 2.47297 | 0 | 0.729857 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.94825 | 0 | 1.00000 | 0 | 2.13809 | 0 | 5.69219 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 3680.2.a.ba | yes | 5 |
| 4.b | odd | 2 | 1 | 3680.2.a.x | ✓ | 5 | |
| 8.b | even | 2 | 1 | 7360.2.a.cl | 5 | ||
| 8.d | odd | 2 | 1 | 7360.2.a.cq | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3680.2.a.x | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 3680.2.a.ba | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 7360.2.a.cl | 5 | 8.b | even | 2 | 1 | ||
| 7360.2.a.cq | 5 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3680))\):
|
\( T_{3}^{5} - T_{3}^{4} - 10T_{3}^{3} + 7T_{3}^{2} + 15T_{3} + 4 \)
|
|
\( T_{7}^{5} - T_{7}^{4} - 29T_{7}^{3} + 52T_{7}^{2} + 130T_{7} - 256 \)
|
|
\( T_{11}^{5} - 3T_{11}^{4} - 49T_{11}^{3} + 100T_{11}^{2} + 626T_{11} - 592 \)
|
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\( T_{13}^{5} - 7T_{13}^{4} - 16T_{13}^{3} + 179T_{13}^{2} - 249T_{13} - 74 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - T^{4} - 10 T^{3} + \cdots + 4 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - T^{4} + \cdots - 256 \)
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| $11$ |
\( T^{5} - 3 T^{4} + \cdots - 592 \)
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| $13$ |
\( T^{5} - 7 T^{4} + \cdots - 74 \)
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| $17$ |
\( T^{5} - 9 T^{4} + \cdots + 16 \)
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| $19$ |
\( T^{5} - T^{4} + \cdots - 3152 \)
|
| $23$ |
\( (T + 1)^{5} \)
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| $29$ |
\( T^{5} + 10 T^{4} + \cdots - 2264 \)
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| $31$ |
\( T^{5} - 21 T^{4} + \cdots + 1132 \)
|
| $37$ |
\( T^{5} - 8 T^{4} + \cdots + 2368 \)
|
| $41$ |
\( T^{5} + 13 T^{4} + \cdots - 94 \)
|
| $43$ |
\( T^{5} + 6 T^{4} + \cdots + 33664 \)
|
| $47$ |
\( T^{5} - 55 T^{3} + \cdots + 64 \)
|
| $53$ |
\( T^{5} + 6 T^{4} + \cdots + 32 \)
|
| $59$ |
\( T^{5} - 18 T^{4} + \cdots + 64 \)
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| $61$ |
\( T^{5} + 11 T^{4} + \cdots - 148 \)
|
| $67$ |
\( T^{5} - 38 T^{4} + \cdots + 41344 \)
|
| $71$ |
\( T^{5} + 21 T^{4} + \cdots - 76 \)
|
| $73$ |
\( T^{5} + 12 T^{4} + \cdots + 712 \)
|
| $79$ |
\( T^{5} + 18 T^{4} + \cdots - 9728 \)
|
| $83$ |
\( T^{5} - 20 T^{4} + \cdots + 1024 \)
|
| $89$ |
\( T^{5} + 16 T^{4} + \cdots + 1088 \)
|
| $97$ |
\( T^{5} - 29 T^{4} + \cdots - 76 \)
|
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