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: | \(0\) |
| 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: |
\( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 836) |
| 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
\( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 12\nu^{3} - 5\nu^{2} + 28\nu - 3 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 15\nu^{3} - 25\nu^{2} - 43\nu + 48 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + \nu^{4} + 24\nu^{3} - 20\nu^{2} - 50\nu + 42 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + 7\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 10\beta_{2} - 2\beta _1 + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( -13\beta_{5} + 12\beta_{4} - 11\beta_{3} - 5\beta_{2} + 58\beta _1 - 46 \)
|
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 | −3.25580 | 0 | −2.84405 | 0 | 1.37411 | 0 | 7.60023 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.64714 | 0 | 1.84900 | 0 | −3.60453 | 0 | −0.286920 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.658537 | 0 | −2.39807 | 0 | −4.45908 | 0 | −2.56633 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.24102 | 0 | 2.83235 | 0 | 4.46564 | 0 | −1.45986 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.18868 | 0 | −3.62873 | 0 | 4.31127 | 0 | 1.79031 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.81471 | 0 | 4.18951 | 0 | −4.08740 | 0 | 4.92257 | 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.x | 6 | |
| 4.b | odd | 2 | 1 | 836.2.a.d | ✓ | 6 | |
| 12.b | even | 2 | 1 | 7524.2.a.r | 6 | ||
| 44.c | even | 2 | 1 | 9196.2.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 836.2.a.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 3344.2.a.x | 6 | 1.a | even | 1 | 1 | trivial | |
| 7524.2.a.r | 6 | 12.b | even | 2 | 1 | ||
| 9196.2.a.k | 6 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 28T_{3}^{3} + 16T_{3}^{2} - 60T_{3} + 27 \)
|
|
\( T_{5}^{6} - 28T_{5}^{4} - 6T_{5}^{3} + 228T_{5}^{2} + 48T_{5} - 543 \)
|
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\( T_{7}^{6} + 2T_{7}^{5} - 43T_{7}^{4} - 78T_{7}^{3} + 547T_{7}^{2} + 760T_{7} - 1738 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 27 \)
|
| $5$ |
\( T^{6} - 28 T^{4} + \cdots - 543 \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots - 1738 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 2462 \)
|
| $17$ |
\( T^{6} - 12 T^{5} + \cdots - 2592 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 2 T^{5} + \cdots - 1584 \)
|
| $29$ |
\( T^{6} - 4 T^{5} + \cdots - 2682 \)
|
| $31$ |
\( T^{6} - 20 T^{5} + \cdots - 7593 \)
|
| $37$ |
\( T^{6} - 22 T^{5} + \cdots + 91824 \)
|
| $41$ |
\( T^{6} + 2 T^{5} + \cdots + 24912 \)
|
| $43$ |
\( T^{6} + 10 T^{5} + \cdots + 4900 \)
|
| $47$ |
\( T^{6} + 16 T^{5} + \cdots + 328896 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots - 31104 \)
|
| $59$ |
\( T^{6} - 14 T^{5} + \cdots + 31536 \)
|
| $61$ |
\( T^{6} - 12 T^{5} + \cdots + 109728 \)
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| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 253887 \)
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| $71$ |
\( T^{6} - 30 T^{5} + \cdots + 2187 \)
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| $73$ |
\( T^{6} - 24 T^{5} + \cdots - 217728 \)
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| $79$ |
\( T^{6} - 370 T^{4} + \cdots + 428832 \)
|
| $83$ |
\( T^{6} - 14 T^{5} + \cdots - 177642 \)
|
| $89$ |
\( T^{6} + 14 T^{5} + \cdots + 19776 \)
|
| $97$ |
\( T^{6} + 46 T^{5} + \cdots + 752 \)
|
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