Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2499,2,Mod(1,2499)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2499, 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("2499.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2499.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: | \(19.9546154651\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1383597.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 7x^{3} + 14x^{2} + 7x - 15 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 357) |
| 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
\( x^{5} - 2x^{4} - 7x^{3} + 14x^{2} + 7x - 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 7\nu^{2} - 5\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} + 20 \)
|
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 |
|
−2.49001 | 1.00000 | 4.20014 | 2.33064 | −2.49001 | 0 | −5.47838 | 1.00000 | −5.80332 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.80754 | 1.00000 | 1.26719 | −2.65545 | −1.80754 | 0 | 1.32458 | 1.00000 | 4.79983 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.22966 | 1.00000 | −0.487931 | 2.41946 | −1.22966 | 0 | 3.05931 | 1.00000 | −2.97512 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.13115 | 1.00000 | −0.720502 | 1.26998 | 1.13115 | 0 | −3.07729 | 1.00000 | 1.43654 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.39606 | 1.00000 | 3.74110 | −4.36463 | 2.39606 | 0 | 4.17178 | 1.00000 | −10.4579 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2499.2.a.bb | 5 | |
| 3.b | odd | 2 | 1 | 7497.2.a.bw | 5 | ||
| 7.b | odd | 2 | 1 | 2499.2.a.ba | 5 | ||
| 7.d | odd | 6 | 2 | 357.2.i.f | ✓ | 10 | |
| 21.c | even | 2 | 1 | 7497.2.a.bv | 5 | ||
| 21.g | even | 6 | 2 | 1071.2.i.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.i.f | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 1071.2.i.g | 10 | 21.g | even | 6 | 2 | ||
| 2499.2.a.ba | 5 | 7.b | odd | 2 | 1 | ||
| 2499.2.a.bb | 5 | 1.a | even | 1 | 1 | trivial | |
| 7497.2.a.bv | 5 | 21.c | even | 2 | 1 | ||
| 7497.2.a.bw | 5 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2499))\):
|
\( T_{2}^{5} + 2T_{2}^{4} - 7T_{2}^{3} - 14T_{2}^{2} + 7T_{2} + 15 \)
|
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\( T_{5}^{5} + T_{5}^{4} - 19T_{5}^{3} + 5T_{5}^{2} + 85T_{5} - 83 \)
|
|
\( T_{11}^{5} + 11T_{11}^{4} + 26T_{11}^{3} - 92T_{11}^{2} - 461T_{11} - 498 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 15 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} + T^{4} + \cdots - 83 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 11 T^{4} + \cdots - 498 \)
|
| $13$ |
\( T^{5} + 7 T^{4} + \cdots + 189 \)
|
| $17$ |
\( (T + 1)^{5} \)
|
| $19$ |
\( T^{5} + 9 T^{4} + \cdots - 108 \)
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| $23$ |
\( T^{5} + 23 T^{4} + \cdots + 944 \)
|
| $29$ |
\( T^{5} + 18 T^{4} + \cdots - 6441 \)
|
| $31$ |
\( T^{5} + 9 T^{4} + \cdots + 5047 \)
|
| $37$ |
\( T^{5} - 122 T^{3} + \cdots - 12484 \)
|
| $41$ |
\( T^{5} + 3 T^{4} + \cdots - 315 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots + 972 \)
|
| $47$ |
\( T^{5} + 11 T^{4} + \cdots + 3 \)
|
| $53$ |
\( T^{5} + 3 T^{4} + \cdots - 1328 \)
|
| $59$ |
\( T^{5} - 14 T^{4} + \cdots + 997 \)
|
| $61$ |
\( T^{5} + 29 T^{4} + \cdots + 984 \)
|
| $67$ |
\( T^{5} - 16 T^{4} + \cdots + 15876 \)
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| $71$ |
\( T^{5} + 19 T^{4} + \cdots - 108 \)
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| $73$ |
\( T^{5} + 11 T^{4} + \cdots - 2082 \)
|
| $79$ |
\( T^{5} - T^{4} + \cdots + 1152 \)
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| $83$ |
\( T^{5} + 5 T^{4} + \cdots - 3701 \)
|
| $89$ |
\( T^{5} + 8 T^{4} + \cdots - 10 \)
|
| $97$ |
\( T^{5} + 19 T^{4} + \cdots + 284264 \)
|
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