Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7581,2,Mod(1,7581)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, 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("7581.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7581.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: | \(60.5345897723\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.130040728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 8x^{3} + 26x^{2} - 17x - 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 399) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 8x^{3} + 26x^{2} - 17x - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} - 3\nu^{2} + 12\nu + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 9\nu^{3} - 2\nu^{2} + 16\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 10\nu^{3} - 9\nu^{2} + 20\nu + 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + \beta_{3} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{4} + 9\beta_{3} + 11\beta_{2} + 20\beta _1 + 13 \)
|
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 |
|
−2.69433 | −1.00000 | 5.25941 | −0.352928 | 2.69433 | 1.00000 | −8.78192 | 1.00000 | 0.950904 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57826 | −1.00000 | 0.490907 | −2.16883 | 1.57826 | 1.00000 | 2.38174 | 1.00000 | 3.42298 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.42862 | −1.00000 | 0.0409614 | 3.16265 | 1.42862 | 1.00000 | 2.79873 | 1.00000 | −4.51823 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.487012 | −1.00000 | −1.76282 | 1.30910 | −0.487012 | 1.00000 | −1.83254 | 1.00000 | 0.637547 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.89286 | −1.00000 | 1.58292 | −1.74576 | −1.89286 | 1.00000 | −0.789480 | 1.00000 | −3.30448 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.32134 | −1.00000 | 3.38862 | 3.79577 | −2.32134 | 1.00000 | 3.22347 | 1.00000 | 8.81128 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(19\) | \(1\) |
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 | 7581.2.a.bb | 6 | |
| 19.b | odd | 2 | 1 | 7581.2.a.bd | 6 | ||
| 19.c | even | 3 | 2 | 399.2.k.d | ✓ | 12 | |
| 57.h | odd | 6 | 2 | 1197.2.k.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.k.d | ✓ | 12 | 19.c | even | 3 | 2 | |
| 1197.2.k.h | 12 | 57.h | odd | 6 | 2 | ||
| 7581.2.a.bb | 6 | 1.a | even | 1 | 1 | trivial | |
| 7581.2.a.bd | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):
|
\( T_{2}^{6} + T_{2}^{5} - 10T_{2}^{4} - 8T_{2}^{3} + 26T_{2}^{2} + 17T_{2} - 13 \)
|
|
\( T_{5}^{6} - 4T_{5}^{5} - 9T_{5}^{4} + 33T_{5}^{3} + 31T_{5}^{2} - 53T_{5} - 21 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots - 13 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots - 21 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 97 \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots + 168 \)
|
| $17$ |
\( T^{6} - 5 T^{5} + \cdots - 109 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 2748 \)
|
| $29$ |
\( T^{6} - 13 T^{5} + \cdots + 9096 \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots - 14496 \)
|
| $37$ |
\( T^{6} + 9 T^{5} + \cdots + 24683 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 3253 \)
|
| $43$ |
\( T^{6} - 13 T^{5} + \cdots + 896 \)
|
| $47$ |
\( T^{6} - 17 T^{5} + \cdots + 7336 \)
|
| $53$ |
\( T^{6} + 9 T^{5} + \cdots + 101976 \)
|
| $59$ |
\( T^{6} - 16 T^{5} + \cdots - 13864 \)
|
| $61$ |
\( T^{6} - 11 T^{5} + \cdots + 7972 \)
|
| $67$ |
\( T^{6} - 12 T^{5} + \cdots - 2488 \)
|
| $71$ |
\( T^{6} + 5 T^{5} + \cdots + 50884 \)
|
| $73$ |
\( T^{6} - 26 T^{5} + \cdots + 524 \)
|
| $79$ |
\( T^{6} + 3 T^{5} + \cdots + 532596 \)
|
| $83$ |
\( T^{6} - 3 T^{5} + \cdots - 9492 \)
|
| $89$ |
\( T^{6} - 12 T^{5} + \cdots - 6608 \)
|
| $97$ |
\( T^{6} - 7 T^{5} + \cdots + 16304 \)
|
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