Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2058,2,Mod(1,2058)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2058, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2058.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2058 = 2 \cdot 3 \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2058.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: | \(16.4332127360\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.72070817.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 25x^{4} + 21x^{3} + 187x^{2} - 107x - 377 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 25x^{4} + 21x^{3} + 187x^{2} - 107x - 377 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{5} - 7\nu^{4} - 91\nu^{3} + 120\nu^{2} + 46\nu - 484 ) / 293 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{5} + 27\nu^{4} - 235\nu^{3} - 421\nu^{2} + 1162\nu + 1239 ) / 293 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{5} - 20\nu^{4} + 326\nu^{3} + 594\nu^{2} - 1208\nu - 3392 ) / 293 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\nu^{5} + 129\nu^{4} - 374\nu^{3} - 2044\nu^{2} + 1580\nu + 6408 ) / 293 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 3\beta_{2} + 10\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 16\beta_{4} + 12\beta_{3} + 14\beta_{2} + 92 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{5} - \beta_{4} - 50\beta_{3} + 68\beta_{2} + 108\beta _1 + 6 \)
|
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 |
|
1.00000 | −1.00000 | 1.00000 | −4.05574 | −1.00000 | 0 | 1.00000 | 1.00000 | −4.05574 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | −3.65204 | −1.00000 | 0 | 1.00000 | 1.00000 | −3.65204 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | −2.23949 | −1.00000 | 0 | 1.00000 | 1.00000 | −2.23949 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 0.960022 | −1.00000 | 0 | 1.00000 | 1.00000 | 0.960022 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 1.69885 | −1.00000 | 0 | 1.00000 | 1.00000 | 1.69885 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 4.28841 | −1.00000 | 0 | 1.00000 | 1.00000 | 4.28841 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 2058.2.a.m | ✓ | 6 |
| 3.b | odd | 2 | 1 | 6174.2.a.v | 6 | ||
| 7.b | odd | 2 | 1 | 2058.2.a.n | yes | 6 | |
| 7.c | even | 3 | 2 | 2058.2.e.n | 12 | ||
| 7.d | odd | 6 | 2 | 2058.2.e.m | 12 | ||
| 21.c | even | 2 | 1 | 6174.2.a.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2058.2.a.m | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2058.2.a.n | yes | 6 | 7.b | odd | 2 | 1 | |
| 2058.2.e.m | 12 | 7.d | odd | 6 | 2 | ||
| 2058.2.e.n | 12 | 7.c | even | 3 | 2 | ||
| 6174.2.a.r | 6 | 21.c | even | 2 | 1 | ||
| 6174.2.a.v | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2058))\):
|
\( T_{5}^{6} + 3T_{5}^{5} - 24T_{5}^{4} - 67T_{5}^{3} + 118T_{5}^{2} + 208T_{5} - 232 \)
|
|
\( T_{11}^{6} - 3T_{11}^{5} - 36T_{11}^{4} + 147T_{11}^{3} + 34T_{11}^{2} - 416T_{11} - 104 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 232 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots - 104 \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots - 56 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 281 \)
|
| $19$ |
\( T^{6} - 5 T^{5} + \cdots - 568 \)
|
| $23$ |
\( T^{6} - 12 T^{5} + \cdots - 4523 \)
|
| $29$ |
\( T^{6} - 21 T^{5} + \cdots - 16472 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots + 9793 \)
|
| $37$ |
\( T^{6} - 21 T^{5} + \cdots + 4696 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 301 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 104 \)
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| $47$ |
\( T^{6} - 10 T^{5} + \cdots - 19264 \)
|
| $53$ |
\( T^{6} - 19 T^{5} + \cdots + 16136 \)
|
| $59$ |
\( T^{6} - 7 T^{5} + \cdots + 4472 \)
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| $61$ |
\( T^{6} - 21 T^{5} + \cdots + 9736 \)
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| $67$ |
\( T^{6} + T^{5} + \cdots + 16792 \)
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| $71$ |
\( T^{6} - 2 T^{5} + \cdots + 20693 \)
|
| $73$ |
\( (T^{3} - 16 T^{2} + \cdots - 104)^{2} \)
|
| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 99904 \)
|
| $83$ |
\( T^{6} - 23 T^{5} + \cdots + 98728 \)
|
| $89$ |
\( (T^{3} - 3 T^{2} + \cdots - 701)^{2} \)
|
| $97$ |
\( (T^{3} - 28 T^{2} + \cdots - 448)^{2} \)
|
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