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: | 6.6.744786576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \)
|
| 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
\( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 24\nu^{4} + \nu^{3} - 253\nu^{2} - 48\nu + 137 ) / 97 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{5} + \nu^{4} + 93\nu^{3} + 42\nu^{2} - 487\nu - 354 ) / 97 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 8\nu^{5} - 2\nu^{4} - 89\nu^{3} + 13\nu^{2} + 101\nu + 29 ) / 97 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\nu^{5} - 3\nu^{4} - 182\nu^{3} + 68\nu^{2} + 588\nu - 199 ) / 97 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + \beta_{3} + 9\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} - 12\beta_{4} + 10\beta_{3} + 4\beta_{2} - 2\beta _1 + 57 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{5} + 33\beta_{4} + 12\beta_{3} + \beta_{2} + 87\beta _1 + 12 \)
|
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 | −3.27915 | 0 | 3.53478 | 0 | 2.34459 | 0 | 7.75283 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.82559 | 0 | −0.343563 | 0 | −4.77460 | 0 | 4.98394 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.837309 | 0 | 3.44987 | 0 | −0.535976 | 0 | −2.29891 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.566270 | 0 | −3.92909 | 0 | 2.44546 | 0 | −2.67934 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.05861 | 0 | 0.680036 | 0 | −1.59123 | 0 | 1.23787 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.31717 | 0 | 1.60797 | 0 | 4.11176 | 0 | 8.00361 | 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.w | 6 | |
| 4.b | odd | 2 | 1 | 836.2.a.e | ✓ | 6 | |
| 12.b | even | 2 | 1 | 7524.2.a.q | 6 | ||
| 44.c | even | 2 | 1 | 9196.2.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 836.2.a.e | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 3344.2.a.w | 6 | 1.a | even | 1 | 1 | trivial | |
| 7524.2.a.q | 6 | 12.b | even | 2 | 1 | ||
| 9196.2.a.n | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):
|
\( T_{3}^{6} + T_{3}^{5} - 17T_{3}^{4} - 13T_{3}^{3} + 69T_{3}^{2} + 21T_{3} - 30 \)
|
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\( T_{5}^{6} - 5T_{5}^{5} - 9T_{5}^{4} + 77T_{5}^{3} - 99T_{5}^{2} + 9T_{5} + 18 \)
|
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\( T_{7}^{6} - 2T_{7}^{5} - 25T_{7}^{4} + 58T_{7}^{3} + 81T_{7}^{2} - 156T_{7} - 96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots - 30 \)
|
| $5$ |
\( T^{6} - 5 T^{5} + \cdots + 18 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 96 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots - 40 \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots - 256 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} + 5 T^{5} + \cdots - 7232 \)
|
| $29$ |
\( T^{6} - 10 T^{5} + \cdots - 11424 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 142 \)
|
| $37$ |
\( T^{6} - 7 T^{5} + \cdots - 5856 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots - 896 \)
|
| $43$ |
\( T^{6} + 16 T^{5} + \cdots + 39880 \)
|
| $47$ |
\( T^{6} - 124 T^{4} + \cdots + 10752 \)
|
| $53$ |
\( T^{6} - 20 T^{5} + \cdots + 32960 \)
|
| $59$ |
\( T^{6} + 15 T^{5} + \cdots - 28064 \)
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| $61$ |
\( T^{6} - 24 T^{5} + \cdots + 25152 \)
|
| $67$ |
\( T^{6} + 25 T^{5} + \cdots - 56022 \)
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| $71$ |
\( T^{6} - 9 T^{5} + \cdots - 82154 \)
|
| $73$ |
\( T^{6} + 26 T^{5} + \cdots + 60672 \)
|
| $79$ |
\( T^{6} - 16 T^{5} + \cdots - 4096 \)
|
| $83$ |
\( T^{6} - 2 T^{5} + \cdots - 7968 \)
|
| $89$ |
\( T^{6} - 7 T^{5} + \cdots + 1152 \)
|
| $97$ |
\( T^{6} - 37 T^{5} + \cdots + 405664 \)
|
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