Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3330,2,Mod(1,3330)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3330, 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("3330.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3330.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: | \(26.5901838731\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.23544108.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 20\nu^{2} - 19\nu - 22 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{4} - \nu^{3} + 37\nu^{2} - 21\nu - 36 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{4} + 4\nu^{3} - 94\nu^{2} + 33\nu + 120 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{4} - 2\nu^{3} + 77\nu^{2} - 39\nu - 96 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} - 4\beta_{2} + \beta _1 + 15 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{4} + 10\beta_{3} + 3\beta_{2} - 6\beta _1 - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{4} - 39\beta_{3} - 80\beta_{2} + 35\beta _1 + 237 ) / 2 \)
|
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 | 1.00000 | 0 | −4.23800 | 1.00000 | 0 | 1.00000 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | −1.19175 | 1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 0.647226 | 1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 3.21441 | 1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 4.56812 | 1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 3330.2.a.bl | yes | 5 |
| 3.b | odd | 2 | 1 | 3330.2.a.bk | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3330.2.a.bk | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 3330.2.a.bl | yes | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):
|
\( T_{7}^{5} - 3T_{7}^{4} - 21T_{7}^{3} + 55T_{7}^{2} + 48T_{7} - 48 \)
|
|
\( T_{11}^{5} - 5T_{11}^{4} - 29T_{11}^{3} + 111T_{11}^{2} + 210T_{11} - 588 \)
|
|
\( T_{13}^{5} - 5T_{13}^{4} - 32T_{13}^{3} + 112T_{13}^{2} + 304T_{13} - 368 \)
|
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\( T_{17}^{5} - 7T_{17}^{4} - 5T_{17}^{3} + 63T_{17}^{2} - 84 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots - 48 \)
|
| $11$ |
\( T^{5} - 5 T^{4} + \cdots - 588 \)
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| $13$ |
\( T^{5} - 5 T^{4} + \cdots - 368 \)
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| $17$ |
\( T^{5} - 7 T^{4} + \cdots - 84 \)
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| $19$ |
\( T^{5} + T^{4} + \cdots - 3632 \)
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| $23$ |
\( T^{5} - 5 T^{4} + \cdots - 384 \)
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| $29$ |
\( T^{5} - 2 T^{4} + \cdots - 96 \)
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| $31$ |
\( T^{5} - 16 T^{4} + \cdots + 3568 \)
|
| $37$ |
\( (T - 1)^{5} \)
|
| $41$ |
\( T^{5} - 2 T^{4} + \cdots + 384 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 1488 \)
|
| $47$ |
\( T^{5} - 6 T^{4} + \cdots - 12096 \)
|
| $53$ |
\( T^{5} - 3 T^{4} + \cdots + 144 \)
|
| $59$ |
\( T^{5} + 10 T^{4} + \cdots + 18816 \)
|
| $61$ |
\( T^{5} - 16 T^{4} + \cdots - 61592 \)
|
| $67$ |
\( T^{5} - 4 T^{4} + \cdots - 37376 \)
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| $71$ |
\( T^{5} + 8 T^{4} + \cdots - 4608 \)
|
| $73$ |
\( T^{5} + 3 T^{4} + \cdots - 21792 \)
|
| $79$ |
\( T^{5} - 2 T^{4} + \cdots - 16736 \)
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| $83$ |
\( T^{5} + 5 T^{4} + \cdots - 2304 \)
|
| $89$ |
\( T^{5} + 7 T^{4} + \cdots + 83856 \)
|
| $97$ |
\( T^{5} - 14 T^{4} + \cdots + 784 \)
|
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