Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3564,2,Mod(1,3564)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3564, 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("3564.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3564.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: | \(28.4586832804\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1166692032.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 20x^{4} + 52x^{2} - 24x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{6} - 20x^{4} + 52x^{2} - 24x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{5} - \nu^{4} - 40\nu^{3} + 10\nu^{2} - 351\nu - 48 ) / 159 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -16\nu^{5} + 4\nu^{4} + 319\nu^{3} - 40\nu^{2} - 822\nu + 192 ) / 159 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\nu^{5} + 35\nu^{4} - 349\nu^{3} - 668\nu^{2} + 360\nu + 567 ) / 159 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 67\nu^{5} + 23\nu^{4} - 1306\nu^{3} - 389\nu^{2} + 2826\nu - 963 ) / 159 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 3\beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 14\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18\beta_{5} - 14\beta_{4} + 58\beta_{3} - 3\beta_{2} + 5\beta _1 + 88 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 17\beta_{3} + 79\beta_{2} + 229\beta _1 + 19 \)
|
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 | 0 | 0 | −4.06224 | 0 | −0.0652266 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.98594 | 0 | −0.575360 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −0.102381 | 0 | 3.29257 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 0.655580 | 0 | 2.68141 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 1.33036 | 0 | −4.83810 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 4.16462 | 0 | −2.49530 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
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 | 3564.2.a.o | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3564.2.a.p | yes | 6 | |
| 9.c | even | 3 | 2 | 3564.2.i.t | 12 | ||
| 9.d | odd | 6 | 2 | 3564.2.i.s | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3564.2.a.o | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3564.2.a.p | yes | 6 | 3.b | odd | 2 | 1 | |
| 3564.2.i.s | 12 | 9.d | odd | 6 | 2 | ||
| 3564.2.i.t | 12 | 9.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3564))\):
|
\( T_{5}^{6} - 20T_{5}^{4} + 52T_{5}^{2} - 24T_{5} - 3 \)
|
|
\( T_{7}^{6} + 2T_{7}^{5} - 22T_{7}^{4} - 22T_{7}^{3} + 101T_{7}^{2} + 68T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 3 \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 4 \)
|
| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 3 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 1104 \)
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| $19$ |
\( T^{6} - 95 T^{4} + \cdots - 108 \)
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| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 1296 \)
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| $29$ |
\( T^{6} - 4 T^{5} + \cdots - 1647 \)
|
| $31$ |
\( T^{6} - 6 T^{5} + \cdots + 2196 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots - 33363 \)
|
| $41$ |
\( T^{6} + 22 T^{5} + \cdots - 55152 \)
|
| $43$ |
\( T^{6} + 10 T^{5} + \cdots - 1952 \)
|
| $47$ |
\( T^{6} + 20 T^{5} + \cdots + 8784 \)
|
| $53$ |
\( T^{6} + 4 T^{5} + \cdots - 1728 \)
|
| $59$ |
\( T^{6} + 24 T^{5} + \cdots + 407268 \)
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| $61$ |
\( T^{6} + 10 T^{5} + \cdots + 253072 \)
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| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 36 \)
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| $71$ |
\( T^{6} + 40 T^{5} + \cdots + 144 \)
|
| $73$ |
\( T^{6} + 8 T^{5} + \cdots + 22653 \)
|
| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 40912 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 1387584 \)
|
| $89$ |
\( T^{6} + 24 T^{5} + \cdots + 147921 \)
|
| $97$ |
\( T^{6} - 20 T^{5} + \cdots + 455488 \)
|
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