Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3381,2,Mod(1,3381)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, 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("3381.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3381 = 3 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3381.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: | \(26.9974209234\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.7997584.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 5x^{3} + 12x^{2} - 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 483) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 5x^{3} + 12x^{2} - 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 10\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 5\nu^{3} - 8\nu^{2} - 4\nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 4\beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + 18\beta_1 \)
|
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 |
|
−2.27267 | 1.00000 | 3.16503 | −4.15120 | −2.27267 | 0 | −2.64772 | 1.00000 | 9.43432 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83264 | 1.00000 | 1.35859 | 0.943490 | −1.83264 | 0 | 1.17548 | 1.00000 | −1.72908 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.593528 | 1.00000 | −1.64772 | 3.15120 | −0.593528 | 0 | 2.16503 | 1.00000 | −1.87033 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.289957 | 1.00000 | −1.91593 | −2.32621 | 0.289957 | 0 | −1.13545 | 1.00000 | −0.674501 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.36549 | 1.00000 | −0.135449 | 1.32621 | 1.36549 | 0 | −2.91593 | 1.00000 | 1.81092 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.04340 | 1.00000 | 2.17548 | −1.94349 | 2.04340 | 0 | 0.358585 | 1.00000 | −3.97133 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 3381.2.a.bd | 6 | |
| 7.b | odd | 2 | 1 | 3381.2.a.bc | 6 | ||
| 7.d | odd | 6 | 2 | 483.2.i.f | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.2.i.f | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 3381.2.a.bc | 6 | 7.b | odd | 2 | 1 | ||
| 3381.2.a.bd | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):
|
\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 5T_{2}^{3} + 12T_{2}^{2} + 4T_{2} - 2 \)
|
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\( T_{5}^{6} + 3T_{5}^{5} - 15T_{5}^{4} - 35T_{5}^{3} + 52T_{5}^{2} + 70T_{5} - 74 \)
|
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\( T_{11}^{6} + 14T_{11}^{5} + 52T_{11}^{4} - 24T_{11}^{3} - 320T_{11}^{2} - 128T_{11} + 256 \)
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\( T_{13}^{6} - 31T_{13}^{4} - 52T_{13}^{3} + 47T_{13}^{2} + 4T_{13} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 7 T^{4} + \cdots - 2 \)
|
| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 74 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} + 14 T^{5} + \cdots + 256 \)
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| $13$ |
\( T^{6} - 31 T^{4} + \cdots - 1 \)
|
| $17$ |
\( T^{6} + 15 T^{5} + \cdots - 18 \)
|
| $19$ |
\( T^{6} - T^{5} - 65 T^{4} + \cdots - 4 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots - 4768 \)
|
| $31$ |
\( T^{6} - 11 T^{5} + \cdots - 72 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots + 424 \)
|
| $41$ |
\( T^{6} + 18 T^{5} + \cdots + 52928 \)
|
| $43$ |
\( T^{6} + 37 T^{5} + \cdots - 19796 \)
|
| $47$ |
\( T^{6} + 3 T^{5} + \cdots - 4768 \)
|
| $53$ |
\( T^{6} + 15 T^{5} + \cdots + 73928 \)
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| $59$ |
\( T^{6} - 2 T^{5} + \cdots - 76192 \)
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| $61$ |
\( T^{6} - 12 T^{5} + \cdots + 99648 \)
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| $67$ |
\( T^{6} + 10 T^{5} + \cdots + 33799 \)
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| $71$ |
\( T^{6} + 21 T^{5} + \cdots - 3208 \)
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| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 55359 \)
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| $79$ |
\( T^{6} + 17 T^{5} + \cdots - 120548 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots - 2224 \)
|
| $89$ |
\( T^{6} + 18 T^{5} + \cdots + 5224 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 1152 \)
|
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