Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5950,2,Mod(1,5950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5950, 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("5950.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5950.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: | \(47.5109892027\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.267429696.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 11x^{4} + 26x^{3} + 12x^{2} - 48x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1190) |
| 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
\( x^{6} - 2x^{5} - 11x^{4} + 26x^{3} + 12x^{2} - 48x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 8\nu^{3} - 4\nu^{2} + 6\nu + 6 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 8\nu^{3} + 2\nu^{2} + 6\nu - 18 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - \nu^{4} + 11\nu^{3} + 4\nu^{2} - 24\nu + 3 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + \nu^{4} + 22\nu^{3} - 19\nu^{2} - 36\nu + 30 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 12\nu^{3} - 14\nu^{2} - 26\nu + 22 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} + 3\beta_{4} + \beta_{3} + 5\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} - 2\beta_{3} + 9\beta_{2} - 11\beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( -23\beta_{5} + 22\beta_{4} + 10\beta_{3} - 5\beta_{2} + 50\beta _1 - 42 \)
|
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 | −3.15120 | 1.00000 | 0 | 3.15120 | 1.00000 | −1.00000 | 6.93009 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.08207 | 1.00000 | 0 | 2.08207 | 1.00000 | −1.00000 | 1.33500 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.19246 | 1.00000 | 0 | 1.19246 | 1.00000 | −1.00000 | −1.57805 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.972142 | 1.00000 | 0 | −0.972142 | 1.00000 | −1.00000 | −2.05494 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.75208 | 1.00000 | 0 | −1.75208 | 1.00000 | −1.00000 | 0.0697952 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.70150 | 1.00000 | 0 | −2.70150 | 1.00000 | −1.00000 | 4.29811 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(17\) | \(1\) |
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 | 5950.2.a.bz | 6 | |
| 5.b | even | 2 | 1 | 5950.2.a.ca | 6 | ||
| 5.c | odd | 4 | 2 | 1190.2.e.f | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1190.2.e.f | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 5950.2.a.bz | 6 | 1.a | even | 1 | 1 | trivial | |
| 5950.2.a.ca | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5950))\):
|
\( T_{3}^{6} + T_{3}^{5} - 13T_{3}^{4} - 8T_{3}^{3} + 44T_{3}^{2} + 12T_{3} - 36 \)
|
|
\( T_{11}^{6} - 3T_{11}^{5} - 41T_{11}^{4} + 80T_{11}^{3} + 404T_{11}^{2} - 432T_{11} - 1128 \)
|
|
\( T_{13}^{6} - 4T_{13}^{5} - 32T_{13}^{4} + 40T_{13}^{3} + 252T_{13}^{2} + 240T_{13} + 64 \)
|
|
\( T_{19}^{6} - 7T_{19}^{5} - 15T_{19}^{4} + 168T_{19}^{3} - 164T_{19}^{2} - 348T_{19} + 284 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots - 36 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots - 1128 \)
|
| $13$ |
\( T^{6} - 4 T^{5} + \cdots + 64 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 284 \)
|
| $23$ |
\( T^{6} + 21 T^{5} + \cdots - 1296 \)
|
| $29$ |
\( T^{6} + 5 T^{5} + \cdots + 11624 \)
|
| $31$ |
\( T^{6} + 5 T^{5} + \cdots + 2592 \)
|
| $37$ |
\( T^{6} + 12 T^{5} + \cdots - 384 \)
|
| $41$ |
\( T^{6} + 29 T^{5} + \cdots - 65616 \)
|
| $43$ |
\( T^{6} + 5 T^{5} + \cdots - 26944 \)
|
| $47$ |
\( T^{6} + 25 T^{5} + \cdots + 21696 \)
|
| $53$ |
\( T^{6} + 25 T^{5} + \cdots - 2448 \)
|
| $59$ |
\( T^{6} + 15 T^{5} + \cdots + 15268 \)
|
| $61$ |
\( T^{6} + 16 T^{5} + \cdots + 85184 \)
|
| $67$ |
\( T^{6} + 19 T^{5} + \cdots - 2816 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 256 \)
|
| $73$ |
\( T^{6} + 8 T^{5} + \cdots + 4096 \)
|
| $79$ |
\( T^{6} - 26 T^{5} + \cdots - 174528 \)
|
| $83$ |
\( T^{6} + 10 T^{5} + \cdots + 3216 \)
|
| $89$ |
\( T^{6} - 18 T^{5} + \cdots + 576 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 1984 \)
|
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