Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5054,2,Mod(1,5054)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5054, 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("5054.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5054 = 2 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5054.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: | \(40.3563931816\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1528713.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 3x^{4} + 7x^{3} + 3x^{2} - 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 266) |
| 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
\( x^{6} - 3x^{5} - 3x^{4} + 7x^{3} + 3x^{2} - 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 4\nu^{2} + \nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 5\nu^{2} - \nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} + 9\nu^{2} - 2\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} - 2\beta_{3} - \beta_{2} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} + 11\beta_{4} - 8\beta_{3} - 2\beta_{2} + 21\beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{5} + 35\beta_{4} - 23\beta_{3} - 8\beta_{2} + 68\beta _1 + 33 \)
|
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 | −2.44476 | 1.00000 | −1.45177 | −2.44476 | −1.00000 | 1.00000 | 2.97685 | −1.45177 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.11161 | 1.00000 | −3.59028 | −2.11161 | −1.00000 | 1.00000 | 1.45891 | −3.59028 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.10878 | 1.00000 | 3.89136 | −1.10878 | −1.00000 | 1.00000 | −1.77060 | 3.89136 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.108781 | 1.00000 | −1.47989 | 0.108781 | −1.00000 | 1.00000 | −2.98817 | −1.47989 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.11161 | 1.00000 | 0.363595 | 1.11161 | −1.00000 | 1.00000 | −1.76432 | 0.363595 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.44476 | 1.00000 | −0.733019 | 1.44476 | −1.00000 | 1.00000 | −0.912670 | −0.733019 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5054.2.a.bc | 6 | |
| 19.b | odd | 2 | 1 | 5054.2.a.bb | 6 | ||
| 19.f | odd | 18 | 2 | 266.2.u.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 266.2.u.b | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 5054.2.a.bb | 6 | 19.b | odd | 2 | 1 | ||
| 5054.2.a.bc | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):
|
\( T_{3}^{6} + 3T_{3}^{5} - 3T_{3}^{4} - 11T_{3}^{3} + 3T_{3}^{2} + 9T_{3} - 1 \)
|
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\( T_{5}^{6} + 3T_{5}^{5} - 12T_{5}^{4} - 47T_{5}^{3} - 42T_{5}^{2} + 8 \)
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\( T_{13}^{6} + 6T_{13}^{5} + 3T_{13}^{4} - 17T_{13}^{3} - 12T_{13}^{2} + 12T_{13} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 8 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots + 8 \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 1 \)
|
| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 60 T^{4} + \cdots + 1216 \)
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| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 296 \)
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| $31$ |
\( T^{6} + 3 T^{5} + \cdots + 2368 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots - 296 \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 9 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots - 43919 \)
|
| $47$ |
\( T^{6} - 9 T^{5} + \cdots + 1864 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 49832 \)
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| $59$ |
\( T^{6} + 15 T^{5} + \cdots + 41887 \)
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| $61$ |
\( T^{6} + 30 T^{5} + \cdots + 35648 \)
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| $67$ |
\( T^{6} + 15 T^{5} + \cdots - 47843 \)
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| $71$ |
\( T^{6} + 21 T^{5} + \cdots + 9784 \)
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| $73$ |
\( T^{6} + 33 T^{5} + \cdots + 73511 \)
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| $79$ |
\( T^{6} - 30 T^{5} + \cdots + 17704 \)
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| $83$ |
\( T^{6} + 33 T^{5} + \cdots + 9343 \)
|
| $89$ |
\( T^{6} - 12 T^{5} + \cdots - 5013 \)
|
| $97$ |
\( T^{6} - 18 T^{5} + \cdots + 14912 \)
|
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