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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.966125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 4\nu - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 5\beta_{3} + 3 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 16\beta_1 \)
|
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 |
|
−2.32142 | −1.00000 | 3.38900 | 1.00000 | 2.32142 | 0.789746 | −3.22446 | 1.00000 | −2.32142 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16309 | −1.00000 | 2.67895 | 1.00000 | 2.16309 | 3.96858 | −1.46863 | 1.00000 | −2.16309 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.12177 | −1.00000 | −0.741635 | 1.00000 | 1.12177 | −2.26042 | 3.07548 | 1.00000 | −1.12177 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.148667 | −1.00000 | −1.97790 | 1.00000 | −0.148667 | −0.316738 | −0.591381 | 1.00000 | 0.148667 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.666822 | −1.00000 | −1.55535 | 1.00000 | −0.666822 | 0.291844 | −2.37079 | 1.00000 | 0.666822 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.79079 | −1.00000 | 1.20693 | 1.00000 | −1.79079 | 1.52699 | −1.42022 | 1.00000 | 1.79079 | |||||||||||||||||||||||||||||||||||||
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.ba | ✓ | 6 |
| 19.b | odd | 2 | 1 | 5415.2.a.bj | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5415.2.a.ba | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5415.2.a.bj | yes | 6 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{6} + 3T_{2}^{5} - 3T_{2}^{4} - 12T_{2}^{3} + 7T_{2} - 1 \)
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\( T_{7}^{6} - 4T_{7}^{5} - 4T_{7}^{4} + 19T_{7}^{3} - 10T_{7}^{2} - 2T_{7} + 1 \)
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\( T_{11}^{6} - 33T_{11}^{4} - 21T_{11}^{3} + 186T_{11}^{2} + 149T_{11} + 29 \)
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\( T_{13}^{6} + 8T_{13}^{5} + T_{13}^{4} - 88T_{13}^{3} - 77T_{13}^{2} + 80T_{13} + 11 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
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| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
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| $11$ |
\( T^{6} - 33 T^{4} + \cdots + 29 \)
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| $13$ |
\( T^{6} + 8 T^{5} + \cdots + 11 \)
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| $17$ |
\( T^{6} - 3 T^{5} + \cdots - 421 \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 239 \)
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| $29$ |
\( T^{6} + 5 T^{5} + \cdots + 101 \)
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| $31$ |
\( T^{6} + 11 T^{5} + \cdots + 1255 \)
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| $37$ |
\( T^{6} - 7 T^{5} + \cdots - 199 \)
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| $41$ |
\( T^{6} + 25 T^{5} + \cdots - 71 \)
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| $43$ |
\( T^{6} + 2 T^{5} + \cdots - 19319 \)
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| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 37501 \)
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| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 1621 \)
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| $59$ |
\( T^{6} + 3 T^{5} + \cdots - 8171 \)
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| $61$ |
\( T^{6} - 4 T^{5} + \cdots - 461 \)
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| $67$ |
\( T^{6} - 3 T^{5} + \cdots - 307169 \)
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| $71$ |
\( T^{6} + 7 T^{5} + \cdots + 180055 \)
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| $73$ |
\( T^{6} + 8 T^{5} + \cdots - 256975 \)
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| $79$ |
\( T^{6} + 49 T^{5} + \cdots + 15241 \)
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| $83$ |
\( T^{6} - 19 T^{5} + \cdots + 488195 \)
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| $89$ |
\( T^{6} + 3 T^{5} + \cdots - 8806241 \)
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| $97$ |
\( T^{6} + 15 T^{5} + \cdots - 3245 \)
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