Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3344,2,Mod(1,3344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3344.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3344 = 2^{4} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3344.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: | \(26.7019744359\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.246832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 209) |
| 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
\( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6 \)
|
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 |
|
0 | −2.26452 | 0 | 0.637602 | 0 | −2.66942 | 0 | 2.12805 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.15766 | 0 | −3.43077 | 0 | −3.93972 | 0 | 1.65548 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.121872 | 0 | −1.06025 | 0 | 3.36889 | 0 | −2.98515 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.563416 | 0 | 2.34577 | 0 | −1.69239 | 0 | −2.68256 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.98063 | 0 | −3.49235 | 0 | −1.06736 | 0 | 5.88418 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 3344.2.a.t | 5 | |
| 4.b | odd | 2 | 1 | 209.2.a.c | ✓ | 5 | |
| 12.b | even | 2 | 1 | 1881.2.a.k | 5 | ||
| 20.d | odd | 2 | 1 | 5225.2.a.h | 5 | ||
| 44.c | even | 2 | 1 | 2299.2.a.n | 5 | ||
| 76.d | even | 2 | 1 | 3971.2.a.h | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.a.c | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1881.2.a.k | 5 | 12.b | even | 2 | 1 | ||
| 2299.2.a.n | 5 | 44.c | even | 2 | 1 | ||
| 3344.2.a.t | 5 | 1.a | even | 1 | 1 | trivial | |
| 3971.2.a.h | 5 | 76.d | even | 2 | 1 | ||
| 5225.2.a.h | 5 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):
|
\( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 11T_{3}^{2} + 7T_{3} + 1 \)
|
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\( T_{5}^{5} + 5T_{5}^{4} - 3T_{5}^{3} - 33T_{5}^{2} - 9T_{5} + 19 \)
|
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\( T_{7}^{5} + 6T_{7}^{4} - T_{7}^{3} - 62T_{7}^{2} - 119T_{7} - 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + T^{4} - 9 T^{3} + \cdots + 1 \)
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| $5$ |
\( T^{5} + 5 T^{4} + \cdots + 19 \)
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| $7$ |
\( T^{5} + 6 T^{4} + \cdots - 64 \)
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| $11$ |
\( (T + 1)^{5} \)
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| $13$ |
\( T^{5} - 4 T^{4} + \cdots + 2 \)
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| $17$ |
\( T^{5} + 4 T^{4} + \cdots - 64 \)
|
| $19$ |
\( (T - 1)^{5} \)
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| $23$ |
\( T^{5} + 3 T^{4} + \cdots + 784 \)
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| $29$ |
\( T^{5} - 10 T^{4} + \cdots + 490 \)
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| $31$ |
\( T^{5} + 11 T^{4} + \cdots + 757 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 3088 \)
|
| $41$ |
\( T^{5} - 2 T^{4} + \cdots - 4112 \)
|
| $43$ |
\( T^{5} + 20 T^{4} + \cdots - 11266 \)
|
| $47$ |
\( T^{5} - 20 T^{4} + \cdots - 13184 \)
|
| $53$ |
\( T^{5} + 14 T^{4} + \cdots + 30304 \)
|
| $59$ |
\( T^{5} + 3 T^{4} + \cdots + 2000 \)
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| $61$ |
\( T^{5} + 10 T^{4} + \cdots - 736 \)
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| $67$ |
\( T^{5} + 9 T^{4} + \cdots - 17689 \)
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| $71$ |
\( T^{5} + 23 T^{4} + \cdots - 19081 \)
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| $73$ |
\( T^{5} - 340 T^{3} + \cdots + 155392 \)
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| $79$ |
\( T^{5} + 44 T^{4} + \cdots + 36800 \)
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| $83$ |
\( T^{5} - 14 T^{4} + \cdots + 3908 \)
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| $89$ |
\( T^{5} + 27 T^{4} + \cdots + 320 \)
|
| $97$ |
\( T^{5} - 15 T^{4} + \cdots - 37456 \)
|
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