Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5952,2,Mod(1,5952)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5952, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5952.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5952 = 2^{6} \cdot 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5952.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: | \(47.5269592831\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2245192.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 10x^{3} + 7x^{2} + 3x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 2976) |
| 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
\( x^{5} - x^{4} - 10x^{3} + 7x^{2} + 3x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 11\nu^{2} - 2\nu + 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 10\nu^{2} + 7\nu + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 20\nu^{2} + 4\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( -3\nu^{4} + \nu^{3} + 31\nu^{2} - 12 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 3\beta_{3} - 2\beta_{2} - \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{4} + 6\beta_{3} - 2\beta_{2} + 5\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{4} + 35\beta_{3} - 22\beta_{2} - 7\beta _1 + 67 ) / 2 \)
|
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 |
|
0 | −1.00000 | 0 | −4.46770 | 0 | −2.96614 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.77182 | 0 | 3.87577 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −1.82988 | 0 | 1.37810 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.14892 | 0 | −3.32505 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 1.92048 | 0 | 3.03731 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 5952.2.a.ci | 5 | |
| 4.b | odd | 2 | 1 | 5952.2.a.cj | 5 | ||
| 8.b | even | 2 | 1 | 2976.2.a.t | yes | 5 | |
| 8.d | odd | 2 | 1 | 2976.2.a.s | ✓ | 5 | |
| 24.f | even | 2 | 1 | 8928.2.a.bo | 5 | ||
| 24.h | odd | 2 | 1 | 8928.2.a.bp | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2976.2.a.s | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 2976.2.a.t | yes | 5 | 8.b | even | 2 | 1 | |
| 5952.2.a.ci | 5 | 1.a | even | 1 | 1 | trivial | |
| 5952.2.a.cj | 5 | 4.b | odd | 2 | 1 | ||
| 8928.2.a.bo | 5 | 24.f | even | 2 | 1 | ||
| 8928.2.a.bp | 5 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5952))\):
|
\( T_{5}^{5} + 6T_{5}^{4} - 36T_{5}^{2} - 13T_{5} + 50 \)
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\( T_{7}^{5} - 2T_{7}^{4} - 21T_{7}^{3} + 36T_{7}^{2} + 108T_{7} - 160 \)
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\( T_{11}^{5} + 6T_{11}^{4} - 25T_{11}^{3} - 128T_{11}^{2} + 166T_{11} + 584 \)
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\( T_{13}^{5} + 10T_{13}^{4} + 9T_{13}^{3} - 194T_{13}^{2} - 722T_{13} - 740 \)
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\( T_{17}^{5} - 2T_{17}^{4} - 29T_{17}^{3} - 6T_{17}^{2} + 64T_{17} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 1)^{5} \)
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| $5$ |
\( T^{5} + 6 T^{4} + \cdots + 50 \)
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| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 160 \)
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| $11$ |
\( T^{5} + 6 T^{4} + \cdots + 584 \)
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| $13$ |
\( T^{5} + 10 T^{4} + \cdots - 740 \)
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| $17$ |
\( T^{5} - 2 T^{4} + \cdots - 32 \)
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| $19$ |
\( T^{5} - 6 T^{4} + \cdots + 80 \)
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| $23$ |
\( T^{5} - 8 T^{4} + \cdots - 1024 \)
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| $29$ |
\( T^{5} + 10 T^{4} + \cdots - 1984 \)
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| $31$ |
\( (T - 1)^{5} \)
|
| $37$ |
\( T^{5} + 10 T^{4} + \cdots + 52112 \)
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| $41$ |
\( T^{5} - 8 T^{4} + \cdots - 8 \)
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| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 10144 \)
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| $47$ |
\( T^{5} - 8 T^{4} + \cdots - 45536 \)
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| $53$ |
\( T^{5} + 14 T^{4} + \cdots + 6848 \)
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| $59$ |
\( T^{5} - 2 T^{4} + \cdots - 32 \)
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| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 14512 \)
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| $67$ |
\( T^{5} - 16 T^{4} + \cdots - 37376 \)
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| $71$ |
\( T^{5} + 6 T^{4} + \cdots + 10784 \)
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| $73$ |
\( T^{5} + 2 T^{4} + \cdots - 496 \)
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| $79$ |
\( T^{5} - 6 T^{4} + \cdots - 1696 \)
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| $83$ |
\( T^{5} + 6 T^{4} + \cdots + 80 \)
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| $89$ |
\( T^{5} + 2 T^{4} + \cdots - 98272 \)
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| $97$ |
\( T^{5} + 10 T^{4} + \cdots - 338 \)
|
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