Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, 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("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.5516125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 3x - 9 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 3x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 12\nu^{3} - 12\nu^{2} + 29\nu + 12 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 6\nu^{2} + 2\nu - 3 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 15\nu^{3} - 30\nu^{2} - 22\nu + 12 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} - 15\nu^{3} + 39\nu^{2} + 13\nu - 39 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{4} + 10\beta_{3} - 5\beta_{2} + 29\beta _1 + 6 \)
|
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 |
|
−2.38093 | −1.00000 | 3.66882 | −1.00000 | 2.38093 | 0.177926 | −3.97334 | 1.00000 | 2.38093 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.18631 | −1.00000 | −0.592657 | −1.00000 | 1.18631 | 4.49647 | 3.07571 | 1.00000 | 1.18631 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.796555 | −1.00000 | −1.36550 | −1.00000 | 0.796555 | −1.95426 | 2.68081 | 1.00000 | 0.796555 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.767206 | −1.00000 | −1.41139 | −1.00000 | −0.767206 | 2.66035 | −2.61724 | 1.00000 | −0.767206 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.03714 | −1.00000 | 2.14995 | −1.00000 | −2.03714 | −5.15682 | 0.305468 | 1.00000 | −2.03714 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.55945 | −1.00000 | 4.55078 | −1.00000 | −2.55945 | 3.77633 | 6.52860 | 1.00000 | −2.55945 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5415.2.a.bf | yes | 6 |
| 19.b | odd | 2 | 1 | 5415.2.a.be | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5415.2.a.be | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 5415.2.a.bf | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 6T_{2}^{3} + 20T_{2}^{2} - 3T_{2} - 9 \)
|
|
\( T_{7}^{6} - 4T_{7}^{5} - 28T_{7}^{4} + 127T_{7}^{3} + 50T_{7}^{2} - 468T_{7} + 81 \)
|
|
\( T_{11}^{6} + 14T_{11}^{5} + 45T_{11}^{4} - 141T_{11}^{3} - 942T_{11}^{2} - 1199T_{11} + 1 \)
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\( T_{13}^{6} - 8T_{13}^{5} - 15T_{13}^{4} + 136T_{13}^{3} + 211T_{13}^{2} - 256T_{13} - 389 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots - 9 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 81 \)
|
| $11$ |
\( T^{6} + 14 T^{5} + \cdots + 1 \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots - 389 \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots - 589 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 15 T^{5} + \cdots + 199 \)
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| $29$ |
\( T^{6} + 15 T^{5} + \cdots + 1 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots + 4975 \)
|
| $37$ |
\( T^{6} - 31 T^{5} + \cdots + 45 \)
|
| $41$ |
\( T^{6} - 21 T^{5} + \cdots - 21955 \)
|
| $43$ |
\( T^{6} + 2 T^{5} + \cdots + 97605 \)
|
| $47$ |
\( T^{6} + 22 T^{5} + \cdots - 151 \)
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| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 5399 \)
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| $59$ |
\( T^{6} + 7 T^{5} + \cdots + 149 \)
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| $61$ |
\( T^{6} - 4 T^{5} + \cdots - 167849 \)
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| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 975931 \)
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| $71$ |
\( T^{6} + 9 T^{5} + \cdots - 26045 \)
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| $73$ |
\( T^{6} + 2 T^{5} + \cdots - 92231 \)
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| $79$ |
\( T^{6} + T^{5} + \cdots + 110941 \)
|
| $83$ |
\( T^{6} + 27 T^{5} + \cdots - 349 \)
|
| $89$ |
\( T^{6} - 37 T^{5} + \cdots - 2830805 \)
|
| $97$ |
\( T^{6} - 23 T^{5} + \cdots + 69439 \)
|
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