Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6138,2,Mod(1,6138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6138, 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("6138.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6138 = 2 \cdot 3^{2} \cdot 11 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6138.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: | \(49.0121767607\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 16x^{4} + 15x^{3} + 62x^{2} - 62x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 682) |
| 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
\( x^{6} - x^{5} - 16x^{4} + 15x^{3} + 62x^{2} - 62x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 14\nu^{3} - 15\nu^{2} + 38\nu + 26 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} - \nu^{4} + 15\nu^{3} + 13\nu^{2} - 45\nu - 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} - \nu^{4} + 15\nu^{3} + 14\nu^{2} - 45\nu - 20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{5} - 5\nu^{4} + 102\nu^{3} + 73\nu^{2} - 296\nu - 96 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - \beta_{3} + 2\beta_{2} + 7\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 12\beta_{4} - 14\beta_{3} + 3\beta_{2} + \beta _1 + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} + 31\beta_{4} - 15\beta_{3} + 27\beta_{2} + 59\beta _1 - 4 \)
|
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 | 0 | 1.00000 | −3.69147 | 0 | 3.31620 | −1.00000 | 0 | 3.69147 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −3.62760 | 0 | −3.70022 | −1.00000 | 0 | 3.62760 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −1.17437 | 0 | 4.38731 | −1.00000 | 0 | 1.17437 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −0.442053 | 0 | 1.11832 | −1.00000 | 0 | 0.442053 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | −0.141142 | 0 | −1.24892 | −1.00000 | 0 | 0.141142 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0 | 1.00000 | 4.07665 | 0 | −2.87267 | −1.00000 | 0 | −4.07665 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 6138.2.a.bq | 6 | |
| 3.b | odd | 2 | 1 | 682.2.a.i | ✓ | 6 | |
| 12.b | even | 2 | 1 | 5456.2.a.y | 6 | ||
| 33.d | even | 2 | 1 | 7502.2.a.p | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 682.2.a.i | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 5456.2.a.y | 6 | 12.b | even | 2 | 1 | ||
| 6138.2.a.bq | 6 | 1.a | even | 1 | 1 | trivial | |
| 7502.2.a.p | 6 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6138))\):
|
\( T_{5}^{6} + 5T_{5}^{5} - 10T_{5}^{4} - 81T_{5}^{3} - 108T_{5}^{2} - 42T_{5} - 4 \)
|
|
\( T_{7}^{6} - T_{7}^{5} - 27T_{7}^{4} + 12T_{7}^{3} + 192T_{7}^{2} + T_{7} - 216 \)
|
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\( T_{13}^{6} - 3T_{13}^{5} - 42T_{13}^{4} + 101T_{13}^{3} + 26T_{13}^{2} - 96T_{13} + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 5 T^{5} + \cdots - 4 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots - 216 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots + 32 \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots - 1602 \)
|
| $19$ |
\( T^{6} + 9 T^{5} + \cdots - 272 \)
|
| $23$ |
\( T^{6} + 5 T^{5} + \cdots - 184 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots + 30092 \)
|
| $31$ |
\( (T + 1)^{6} \)
|
| $37$ |
\( T^{6} - 14 T^{5} + \cdots + 250576 \)
|
| $41$ |
\( T^{6} + 14 T^{5} + \cdots - 9120 \)
|
| $43$ |
\( T^{6} + 30 T^{5} + \cdots - 95072 \)
|
| $47$ |
\( T^{6} + 14 T^{5} + \cdots - 14848 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots - 11136 \)
|
| $59$ |
\( T^{6} + 14 T^{5} + \cdots + 19680 \)
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| $61$ |
\( T^{6} + 3 T^{5} + \cdots + 40668 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 183552 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots - 109184 \)
|
| $73$ |
\( T^{6} + 7 T^{5} + \cdots - 2752 \)
|
| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 17408 \)
|
| $83$ |
\( T^{6} - 34 T^{5} + \cdots - 75104 \)
|
| $89$ |
\( T^{6} + 10 T^{5} + \cdots - 540576 \)
|
| $97$ |
\( T^{6} - 3 T^{5} + \cdots - 2045566 \)
|
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