Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7360,2,Mod(1,7360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, 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("7360.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7360 = 2^{6} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7360.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: | \(58.7698958877\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1143052.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 9x^{3} + 11x^{2} + 6x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3680) |
| 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
\( x^{5} - x^{4} - 9x^{3} + 11x^{2} + 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 7\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{2} + 4\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 9\nu^{2} + 2\nu + 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{4} + 9\beta_{3} - 11\beta _1 + 28 \)
|
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 | −3.29640 | 0 | 1.00000 | 0 | 4.53739 | 0 | 7.86626 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.530415 | 0 | 1.00000 | 0 | −2.35235 | 0 | −2.71866 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.503031 | 0 | 1.00000 | 0 | −3.96918 | 0 | −2.74696 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.80707 | 0 | 1.00000 | 0 | 4.65156 | 0 | 0.265511 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.51671 | 0 | 1.00000 | 0 | −1.86742 | 0 | 3.33385 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 7360.2.a.cr | 5 | |
| 4.b | odd | 2 | 1 | 7360.2.a.cn | 5 | ||
| 8.b | even | 2 | 1 | 3680.2.a.v | ✓ | 5 | |
| 8.d | odd | 2 | 1 | 3680.2.a.z | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3680.2.a.v | ✓ | 5 | 8.b | even | 2 | 1 | |
| 3680.2.a.z | yes | 5 | 8.d | odd | 2 | 1 | |
| 7360.2.a.cn | 5 | 4.b | odd | 2 | 1 | ||
| 7360.2.a.cr | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):
|
\( T_{3}^{5} - T_{3}^{4} - 10T_{3}^{3} + 15T_{3}^{2} + 3T_{3} - 4 \)
|
|
\( T_{7}^{5} - T_{7}^{4} - 33T_{7}^{3} - 4T_{7}^{2} + 286T_{7} + 368 \)
|
|
\( T_{11}^{5} - 9T_{11}^{4} - T_{11}^{3} + 208T_{11}^{2} - 626T_{11} + 536 \)
|
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\( T_{13}^{5} + 3T_{13}^{4} - 68T_{13}^{3} - 155T_{13}^{2} + 1131T_{13} + 1514 \)
|
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\( T_{17}^{5} + 21T_{17}^{4} + 145T_{17}^{3} + 322T_{17}^{2} - 48T_{17} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - T^{4} - 10 T^{3} + \cdots - 4 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - T^{4} + \cdots + 368 \)
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| $11$ |
\( T^{5} - 9 T^{4} + \cdots + 536 \)
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| $13$ |
\( T^{5} + 3 T^{4} + \cdots + 1514 \)
|
| $17$ |
\( T^{5} + 21 T^{4} + \cdots - 16 \)
|
| $19$ |
\( T^{5} - 7 T^{4} + \cdots - 8 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( T^{5} - 6 T^{4} + \cdots - 776 \)
|
| $31$ |
\( T^{5} + T^{4} + \cdots + 1052 \)
|
| $37$ |
\( T^{5} + 8 T^{4} + \cdots - 1024 \)
|
| $41$ |
\( T^{5} - 19 T^{4} + \cdots + 482 \)
|
| $43$ |
\( T^{5} - 14 T^{4} + \cdots - 256 \)
|
| $47$ |
\( T^{5} + 8 T^{4} + \cdots + 23456 \)
|
| $53$ |
\( T^{5} - 6 T^{4} + \cdots + 5344 \)
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| $59$ |
\( T^{5} - 6 T^{4} + \cdots - 21568 \)
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| $61$ |
\( T^{5} + 23 T^{4} + \cdots - 5132 \)
|
| $67$ |
\( T^{5} + 2 T^{4} + \cdots + 1024 \)
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| $71$ |
\( T^{5} - 21 T^{4} + \cdots + 50636 \)
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| $73$ |
\( T^{5} - 235 T^{3} + \cdots - 35096 \)
|
| $79$ |
\( T^{5} - 42 T^{4} + \cdots + 242944 \)
|
| $83$ |
\( T^{5} - 28 T^{4} + \cdots + 25600 \)
|
| $89$ |
\( T^{5} - 298 T^{3} + \cdots - 31936 \)
|
| $97$ |
\( T^{5} + 17 T^{4} + \cdots + 33148 \)
|
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