Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3610,2,Mod(1,3610)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3610, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3610.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3610.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: | \(28.8259951297\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.23153769.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} - 5x^{3} + 60x^{2} + 36x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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
\( x^{6} - 15x^{4} - 5x^{3} + 60x^{2} + 36x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 11\nu^{3} - 15\nu^{2} + 18\nu ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 11\nu^{3} - 6\nu^{2} - 27\nu ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 5\nu^{3} + 6\nu^{2} - 15\nu - 18 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 11\nu^{3} + 39\nu^{2} + 36\nu - 60 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 9\beta_{3} + 11\beta_{2} + 3\beta _1 + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 11\beta_{4} + 8\beta_{3} + 5\beta_{2} + 53\beta _1 + 38 \)
|
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 |
|
−1.00000 | −2.61722 | 1.00000 | −1.00000 | 2.61722 | 0.970436 | −1.00000 | 3.84982 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.28834 | 1.00000 | −1.00000 | 2.28834 | 2.76858 | −1.00000 | 2.23649 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.20683 | 1.00000 | −1.00000 | 1.20683 | −2.19627 | −1.00000 | −1.54356 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.408953 | 1.00000 | −1.00000 | −0.408953 | −2.30067 | −1.00000 | −2.83276 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.73892 | 1.00000 | −1.00000 | −2.73892 | 3.84897 | −1.00000 | 4.50167 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.96451 | 1.00000 | −1.00000 | −2.96451 | 2.90895 | −1.00000 | 5.78833 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 3610.2.a.bd | 6 | |
| 19.b | odd | 2 | 1 | 3610.2.a.bf | 6 | ||
| 19.f | odd | 18 | 2 | 190.2.k.c | ✓ | 12 | |
| 95.o | odd | 18 | 2 | 950.2.l.g | 12 | ||
| 95.r | even | 36 | 4 | 950.2.u.f | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.2.k.c | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 950.2.l.g | 12 | 95.o | odd | 18 | 2 | ||
| 950.2.u.f | 24 | 95.r | even | 36 | 4 | ||
| 3610.2.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 3610.2.a.bf | 6 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\):
|
\( T_{3}^{6} - 15T_{3}^{4} - 5T_{3}^{3} + 60T_{3}^{2} + 36T_{3} - 24 \)
|
|
\( T_{7}^{6} - 6T_{7}^{5} - 3T_{7}^{4} + 63T_{7}^{3} - 42T_{7}^{2} - 168T_{7} + 152 \)
|
|
\( T_{13}^{6} + 3T_{13}^{5} - 39T_{13}^{4} - 83T_{13}^{3} + 294T_{13}^{2} - 96T_{13} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 24 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 152 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 9 \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots - 8 \)
|
| $17$ |
\( T^{6} - 15 T^{5} + \cdots - 216 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 18 T^{5} + \cdots - 72 \)
|
| $29$ |
\( (T^{3} - 36 T + 72)^{2} \)
|
| $31$ |
\( T^{6} + 24 T^{5} + \cdots + 43712 \)
|
| $37$ |
\( T^{6} + 12 T^{5} + \cdots - 7624 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 42363 \)
|
| $43$ |
\( T^{6} - 9 T^{5} + \cdots - 5256 \)
|
| $47$ |
\( T^{6} - 9 T^{5} + \cdots - 198936 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 244872 \)
|
| $59$ |
\( T^{6} + 9 T^{5} + \cdots + 4293 \)
|
| $61$ |
\( T^{6} + 12 T^{5} + \cdots - 20928 \)
|
| $67$ |
\( T^{6} - 6 T^{5} + \cdots + 20928 \)
|
| $71$ |
\( T^{6} - 6 T^{5} + \cdots + 1728 \)
|
| $73$ |
\( T^{6} - 12 T^{5} + \cdots + 588088 \)
|
| $79$ |
\( T^{6} - 132 T^{4} + \cdots - 12736 \)
|
| $83$ |
\( T^{6} - 219 T^{4} + \cdots - 72 \)
|
| $89$ |
\( T^{6} + 33 T^{5} + \cdots - 10053 \)
|
| $97$ |
\( T^{6} + 18 T^{5} + \cdots + 12968 \)
|
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