Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5776,2,Mod(1,5776)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5776, 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("5776.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5776 = 2^{4} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5776.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: | \(46.1215922075\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3022625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2888) |
| 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
\( x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 10\nu^{2} + 7\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 10\nu^{2} + 9\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 8\nu^{2} + 10\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 5\beta_{4} - 6\beta_{3} - \beta_{2} + 3\beta _1 + 12 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{5} + 9\beta_{4} - 15\beta_{3} - 4\beta_{2} + 8\beta _1 + 37 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{5} + 15\beta_{4} - 20\beta_{3} - 3\beta_{2} + 13\beta _1 + 46 \)
|
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 | −2.29838 | 0 | 2.83118 | 0 | 4.51153 | 0 | 2.28257 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.60721 | 0 | −1.92601 | 0 | 1.29923 | 0 | −0.416870 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.354424 | 0 | −1.55742 | 0 | −1.82103 | 0 | −2.87438 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.45214 | 0 | 3.79176 | 0 | 3.95766 | 0 | −0.891299 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.77311 | 0 | −3.10181 | 0 | −4.25689 | 0 | 4.69013 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.03477 | 0 | 1.96231 | 0 | −1.69050 | 0 | 6.20985 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5776.2.a.bz | 6 | |
| 4.b | odd | 2 | 1 | 2888.2.a.t | ✓ | 6 | |
| 19.b | odd | 2 | 1 | 5776.2.a.bx | 6 | ||
| 76.d | even | 2 | 1 | 2888.2.a.u | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2888.2.a.t | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 2888.2.a.u | yes | 6 | 76.d | even | 2 | 1 | |
| 5776.2.a.bx | 6 | 19.b | odd | 2 | 1 | ||
| 5776.2.a.bz | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5776))\):
|
\( T_{3}^{6} - 3T_{3}^{5} - 9T_{3}^{4} + 24T_{3}^{3} + 24T_{3}^{2} - 40T_{3} - 16 \)
|
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\( T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 26T_{5}^{3} + 109T_{5}^{2} - 70T_{5} - 196 \)
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\( T_{7}^{6} - 2T_{7}^{5} - 29T_{7}^{4} + 38T_{7}^{3} + 212T_{7}^{2} - 40T_{7} - 304 \)
|
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\( T_{11}^{6} - 5T_{11}^{5} - 29T_{11}^{4} + 166T_{11}^{3} - 116T_{11}^{2} - 152T_{11} - 16 \)
|
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\( T_{13}^{6} + 10T_{13}^{5} + 6T_{13}^{4} - 194T_{13}^{3} - 686T_{13}^{2} - 822T_{13} - 311 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots - 16 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 196 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 304 \)
|
| $11$ |
\( T^{6} - 5 T^{5} + \cdots - 16 \)
|
| $13$ |
\( T^{6} + 10 T^{5} + \cdots - 311 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots - 76 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots - 6224 \)
|
| $29$ |
\( T^{6} - 7 T^{5} + \cdots + 63541 \)
|
| $31$ |
\( T^{6} - 5 T^{5} + \cdots + 2000 \)
|
| $37$ |
\( T^{6} + 17 T^{5} + \cdots + 379 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 12211 \)
|
| $43$ |
\( T^{6} - 11 T^{5} + \cdots - 64976 \)
|
| $47$ |
\( T^{6} + 18 T^{5} + \cdots + 496 \)
|
| $53$ |
\( T^{6} - 7 T^{5} + \cdots + 1621 \)
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| $59$ |
\( T^{6} - T^{5} + \cdots - 1264 \)
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| $61$ |
\( T^{6} + 4 T^{5} + \cdots + 1319 \)
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| $67$ |
\( T^{6} - 20 T^{5} + \cdots + 13264 \)
|
| $71$ |
\( T^{6} + 6 T^{5} + \cdots + 17744 \)
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| $73$ |
\( T^{6} + 2 T^{5} + \cdots + 71 \)
|
| $79$ |
\( T^{6} - 8 T^{5} + \cdots - 61504 \)
|
| $83$ |
\( T^{6} - 22 T^{5} + \cdots - 41024 \)
|
| $89$ |
\( T^{6} + 14 T^{5} + \cdots - 1769 \)
|
| $97$ |
\( T^{6} + 41 T^{5} + \cdots + 324539 \)
|
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