Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8379,2,Mod(1,8379)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8379, 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("8379.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8379.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: | \(66.9066518536\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1240016.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 2x^{2} + 16x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 399) |
| 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
\( x^{5} - x^{4} - 8x^{3} + 2x^{2} + 16x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 6\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 6 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + 14\beta_{3} + 4\beta_{2} + 9\beta _1 + 30 ) / 2 \)
|
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 |
|
−2.47552 | 0 | 4.12820 | −2.79287 | 0 | 0 | −5.26839 | 0 | 6.91381 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.91670 | 0 | 1.67374 | 2.54204 | 0 | 0 | 0.625344 | 0 | −4.87234 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.215612 | 0 | −1.95351 | 1.06804 | 0 | 0 | 0.852423 | 0 | −0.230281 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.23675 | 0 | −0.470449 | −4.29208 | 0 | 0 | −3.05533 | 0 | −5.30823 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.37108 | 0 | 3.62202 | 1.47487 | 0 | 0 | 3.84595 | 0 | 3.49704 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(19\) | \(1\) |
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 | 8379.2.a.ce | 5 | |
| 3.b | odd | 2 | 1 | 2793.2.a.be | 5 | ||
| 7.b | odd | 2 | 1 | 1197.2.a.p | 5 | ||
| 21.c | even | 2 | 1 | 399.2.a.f | ✓ | 5 | |
| 84.h | odd | 2 | 1 | 6384.2.a.cc | 5 | ||
| 105.g | even | 2 | 1 | 9975.2.a.bq | 5 | ||
| 399.h | odd | 2 | 1 | 7581.2.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.a.f | ✓ | 5 | 21.c | even | 2 | 1 | |
| 1197.2.a.p | 5 | 7.b | odd | 2 | 1 | ||
| 2793.2.a.be | 5 | 3.b | odd | 2 | 1 | ||
| 6384.2.a.cc | 5 | 84.h | odd | 2 | 1 | ||
| 7581.2.a.x | 5 | 399.h | odd | 2 | 1 | ||
| 8379.2.a.ce | 5 | 1.a | even | 1 | 1 | trivial | |
| 9975.2.a.bq | 5 | 105.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8379))\):
|
\( T_{2}^{5} + T_{2}^{4} - 8T_{2}^{3} - 6T_{2}^{2} + 13T_{2} + 3 \)
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\( T_{5}^{5} + 2T_{5}^{4} - 16T_{5}^{3} - 8T_{5}^{2} + 68T_{5} - 48 \)
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\( T_{11}^{5} - 2T_{11}^{4} - 32T_{11}^{3} + 16T_{11}^{2} + 256T_{11} + 192 \)
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\( T_{13}^{5} + 8T_{13}^{4} - 8T_{13}^{3} - 112T_{13}^{2} + 32T_{13} + 256 \)
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\( T_{17}^{5} + 2T_{17}^{4} - 72T_{17}^{3} - 176T_{17}^{2} + 1108T_{17} + 3168 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 8 T^{3} + \cdots + 3 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + 2 T^{4} + \cdots - 48 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} - 2 T^{4} + \cdots + 192 \)
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| $13$ |
\( T^{5} + 8 T^{4} + \cdots + 256 \)
|
| $17$ |
\( T^{5} + 2 T^{4} + \cdots + 3168 \)
|
| $19$ |
\( (T + 1)^{5} \)
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| $23$ |
\( T^{5} + 2 T^{4} + \cdots - 192 \)
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| $29$ |
\( T^{5} - 56 T^{3} + \cdots - 24 \)
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| $31$ |
\( T^{5} + 8 T^{4} + \cdots + 512 \)
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| $37$ |
\( T^{5} - 2 T^{4} + \cdots + 608 \)
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| $41$ |
\( T^{5} - 2 T^{4} + \cdots + 96 \)
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| $43$ |
\( T^{5} - 20 T^{4} + \cdots + 13184 \)
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| $47$ |
\( T^{5} + 26 T^{4} + \cdots - 3648 \)
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| $53$ |
\( T^{5} + 4 T^{4} + \cdots - 20376 \)
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| $59$ |
\( T^{5} + 4 T^{4} + \cdots - 6144 \)
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| $61$ |
\( T^{5} + 10 T^{4} + \cdots + 3872 \)
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| $67$ |
\( T^{5} - 10 T^{4} + \cdots - 40064 \)
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| $71$ |
\( T^{5} + 10 T^{4} + \cdots + 3888 \)
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| $73$ |
\( T^{5} + 10 T^{4} + \cdots + 32 \)
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| $79$ |
\( T^{5} - 14 T^{4} + \cdots - 4864 \)
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| $83$ |
\( T^{5} + 34 T^{4} + \cdots - 159888 \)
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| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 114336 \)
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| $97$ |
\( T^{5} - 16 T^{4} + \cdots - 194816 \)
|
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