Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2805,2,Mod(1,2805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, 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("2805.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2805 = 3 \cdot 5 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2805.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: | \(22.3980377670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.94698456.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 10x^{2} - 3x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} - 8x^{4} + 6x^{3} + 10x^{2} - 3x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 7\nu^{3} - 5\nu^{2} - 3\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 5\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 6\nu^{2} + 3\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 13\nu^{2} - 4\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} + 7\beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 10\beta_{4} - 7\beta_{3} + \beta_{2} + 40\beta_1 \)
|
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 |
|
−2.48826 | 1.00000 | 4.19146 | −1.00000 | −2.48826 | −3.81227 | −5.45292 | 1.00000 | 2.48826 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.808070 | 1.00000 | −1.34702 | −1.00000 | −0.808070 | −3.56075 | 2.70463 | 1.00000 | 0.808070 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.544398 | 1.00000 | −1.70363 | −1.00000 | −0.544398 | 1.87448 | 2.01625 | 1.00000 | 0.544398 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.713105 | 1.00000 | −1.49148 | −1.00000 | 0.713105 | −0.878887 | −2.48979 | 1.00000 | −0.713105 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.41882 | 1.00000 | 0.0130593 | −1.00000 | 1.41882 | −3.07785 | −2.81912 | 1.00000 | −1.41882 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70880 | 1.00000 | 5.33762 | −1.00000 | 2.70880 | 2.45527 | 9.04095 | 1.00000 | −2.70880 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
| \(17\) | \(1\) |
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 | 2805.2.a.n | ✓ | 6 |
| 3.b | odd | 2 | 1 | 8415.2.a.bh | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2805.2.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8415.2.a.bh | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):
|
\( T_{2}^{6} - T_{2}^{5} - 8T_{2}^{4} + 6T_{2}^{3} + 10T_{2}^{2} - 3T_{2} - 3 \)
|
|
\( T_{7}^{6} + 7T_{7}^{5} + T_{7}^{4} - 71T_{7}^{3} - 73T_{7}^{2} + 180T_{7} + 169 \)
|
|
\( T_{19}^{6} + 11T_{19}^{5} + 13T_{19}^{4} - 245T_{19}^{3} - 1109T_{19}^{2} - 1696T_{19} - 823 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 8 T^{4} + \cdots - 3 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 169 \)
|
| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} + 2 T^{5} + \cdots - 1 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} + 11 T^{5} + \cdots - 823 \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots - 339 \)
|
| $29$ |
\( T^{6} - 16 T^{5} + \cdots + 1812 \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots - 12293 \)
|
| $37$ |
\( T^{6} - T^{5} + \cdots + 3349 \)
|
| $41$ |
\( T^{6} - 16 T^{5} + \cdots - 32232 \)
|
| $43$ |
\( T^{6} - 2 T^{5} + \cdots + 104 \)
|
| $47$ |
\( T^{6} - T^{5} + \cdots + 4128 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 1944 \)
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| $59$ |
\( T^{6} - 16 T^{5} + \cdots + 256548 \)
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| $61$ |
\( T^{6} + 9 T^{5} + \cdots + 32929 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 25363 \)
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| $71$ |
\( T^{6} - 49 T^{5} + \cdots - 22164 \)
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| $73$ |
\( T^{6} + 12 T^{5} + \cdots + 14372 \)
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| $79$ |
\( T^{6} - 14 T^{5} + \cdots - 688 \)
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| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 16719 \)
|
| $89$ |
\( T^{6} - 2 T^{5} + \cdots + 398892 \)
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| $97$ |
\( T^{6} + T^{5} + \cdots - 633989 \)
|
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