Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9747,2,Mod(1,9747)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9747, 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("9747.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9747 = 3^{3} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9747.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: | \(77.8301868501\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.54265221.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 4x^{3} + 12x^{2} - 3x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 513) |
| 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
\( x^{6} - x^{5} - 7x^{4} + 4x^{3} + 12x^{2} - 3x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 3\nu^{2} + 7\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 9\beta_{2} + 28\beta _1 + 2 \)
|
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 |
|
−2.49945 | 0 | 4.24724 | 1.07044 | 0 | −0.193560 | −5.61687 | 0 | −2.67551 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.58457 | 0 | 0.510849 | −1.56181 | 0 | −0.188849 | 2.35966 | 0 | 2.47479 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.657313 | 0 | −1.56794 | 4.12941 | 0 | 3.83431 | 2.34525 | 0 | −2.71431 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.433013 | 0 | −1.81250 | −2.03183 | 0 | −1.15054 | −1.65086 | 0 | −0.879808 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.38041 | 0 | −0.0944649 | 3.19283 | 0 | −5.01270 | −2.89122 | 0 | 4.40742 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.92790 | 0 | 1.71681 | 0.200956 | 0 | 3.71133 | −0.545959 | 0 | 0.387423 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 9747.2.a.bm | 6 | |
| 3.b | odd | 2 | 1 | 9747.2.a.br | 6 | ||
| 19.b | odd | 2 | 1 | 9747.2.a.bs | 6 | ||
| 19.d | odd | 6 | 2 | 513.2.f.f | ✓ | 12 | |
| 57.d | even | 2 | 1 | 9747.2.a.bl | 6 | ||
| 57.f | even | 6 | 2 | 513.2.f.h | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 513.2.f.f | ✓ | 12 | 19.d | odd | 6 | 2 | |
| 513.2.f.h | yes | 12 | 57.f | even | 6 | 2 | |
| 9747.2.a.bl | 6 | 57.d | even | 2 | 1 | ||
| 9747.2.a.bm | 6 | 1.a | even | 1 | 1 | trivial | |
| 9747.2.a.br | 6 | 3.b | odd | 2 | 1 | ||
| 9747.2.a.bs | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9747))\):
|
\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 4T_{2}^{3} + 12T_{2}^{2} + 3T_{2} - 3 \)
|
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\( T_{5}^{6} - 5T_{5}^{5} - 5T_{5}^{4} + 36T_{5}^{3} + 9T_{5}^{2} - 48T_{5} + 9 \)
|
|
\( T_{7}^{6} - T_{7}^{5} - 27T_{7}^{4} + 34T_{7}^{3} + 98T_{7}^{2} + 33T_{7} + 3 \)
|
|
\( T_{13}^{6} + 5T_{13}^{5} - 30T_{13}^{4} - 127T_{13}^{3} + 145T_{13}^{2} + 126T_{13} - 39 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 7 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 5 T^{5} + \cdots + 9 \)
|
| $7$ |
\( T^{6} - T^{5} - 27 T^{4} + \cdots + 3 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 81 \)
|
| $13$ |
\( T^{6} + 5 T^{5} + \cdots - 39 \)
|
| $17$ |
\( T^{6} - 10 T^{5} + \cdots + 27 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots - 219 \)
|
| $29$ |
\( T^{6} - 6 T^{5} + \cdots - 933 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots + 267 \)
|
| $37$ |
\( T^{6} + 13 T^{5} + \cdots - 6339 \)
|
| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 27 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots - 1271 \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots - 14823 \)
|
| $53$ |
\( T^{6} - T^{5} + \cdots + 2943 \)
|
| $59$ |
\( T^{6} + 11 T^{5} + \cdots + 16641 \)
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| $61$ |
\( T^{6} - 13 T^{5} + \cdots + 1699 \)
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| $67$ |
\( T^{6} - 9 T^{5} + \cdots + 3133 \)
|
| $71$ |
\( T^{6} - 6 T^{5} + \cdots - 21951 \)
|
| $73$ |
\( T^{6} - 15 T^{5} + \cdots + 5639 \)
|
| $79$ |
\( T^{6} + 13 T^{5} + \cdots - 122697 \)
|
| $83$ |
\( T^{6} - 8 T^{5} + \cdots - 681291 \)
|
| $89$ |
\( T^{6} + 29 T^{5} + \cdots + 81 \)
|
| $97$ |
\( T^{6} - 20 T^{5} + \cdots - 81959 \)
|
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