Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7569,2,Mod(1,7569)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7569, 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("7569.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7569 = 3^{2} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7569.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: | \(60.4387692899\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.5878828125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 13x^{6} + 7x^{5} + 55x^{4} + 3x^{3} - 78x^{2} - 54x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2523) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 13x^{6} + 7x^{5} + 55x^{4} + 3x^{3} - 78x^{2} - 54x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 25\nu^{2} - 6\nu - 9 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 11\nu^{5} - 3\nu^{4} + 32\nu^{3} + 19\nu^{2} - 18\nu - 12 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 12\nu^{5} - 17\nu^{4} - 48\nu^{3} + 25\nu^{2} + 72\nu + 27 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{7} - 4\nu^{6} - 24\nu^{5} + 37\nu^{4} + 93\nu^{3} - 71\nu^{2} - 129\nu - 33 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} - 5\nu^{6} - 23\nu^{5} + 47\nu^{4} + 89\nu^{3} - 96\nu^{2} - 138\nu - 27 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 5\nu^{6} - 34\nu^{5} + 44\nu^{4} + 118\nu^{3} - 80\nu^{2} - 138\nu - 24 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{7} + \beta_{6} + 7\beta_{5} + 2\beta_{4} + 7\beta_{3} - 6\beta_{2} + 7\beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 10\beta_{6} - 8\beta_{5} + \beta_{4} + 3\beta_{3} + 6\beta_{2} + 30\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( -47\beta_{7} + 13\beta_{6} + 44\beta_{5} + 21\beta_{4} + 48\beta_{3} - 33\beta_{2} + 46\beta _1 + 129 \)
|
| \(\nu^{7}\) | \(=\) |
\( -24\beta_{7} + 81\beta_{6} - 54\beta_{5} + 17\beta_{4} + 38\beta_{3} + 35\beta_{2} + 190\beta _1 + 47 \)
|
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 |
|
−2.46784 | 0 | 4.09025 | 3.43509 | 0 | −4.60180 | −5.15841 | 0 | −8.47726 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.69344 | 0 | 0.867750 | −2.52606 | 0 | −4.38828 | 1.91740 | 0 | 4.27775 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.851203 | 0 | −1.27545 | −0.880235 | 0 | −0.914011 | 2.78808 | 0 | 0.749259 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.758720 | 0 | −1.42434 | 2.69503 | 0 | 0.874680 | 2.59812 | 0 | −2.04478 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.262852 | 0 | −1.93091 | 4.31701 | 0 | −0.159034 | 1.03325 | 0 | −1.13473 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09698 | 0 | 2.39733 | 1.13206 | 0 | −0.0912936 | 0.833198 | 0 | 2.37390 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.25879 | 0 | 3.10211 | −0.0968284 | 0 | 1.11297 | 2.48944 | 0 | −0.218715 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.67829 | 0 | 5.17326 | 0.923940 | 0 | −3.83323 | 8.49892 | 0 | 2.47458 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 7569.2.a.bh | 8 | |
| 3.b | odd | 2 | 1 | 2523.2.a.m | ✓ | 8 | |
| 29.b | even | 2 | 1 | 7569.2.a.be | 8 | ||
| 87.d | odd | 2 | 1 | 2523.2.a.n | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2523.2.a.m | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2523.2.a.n | yes | 8 | 87.d | odd | 2 | 1 | |
| 7569.2.a.be | 8 | 29.b | even | 2 | 1 | ||
| 7569.2.a.bh | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):
|
\( T_{2}^{8} - T_{2}^{7} - 13T_{2}^{6} + 7T_{2}^{5} + 55T_{2}^{4} + 3T_{2}^{3} - 78T_{2}^{2} - 54T_{2} - 9 \)
|
|
\( T_{5}^{8} - 9T_{5}^{7} + 17T_{5}^{6} + 48T_{5}^{5} - 170T_{5}^{4} + 72T_{5}^{3} + 132T_{5}^{2} - 81T_{5} - 9 \)
|
|
\( T_{7}^{8} + 12T_{7}^{7} + 43T_{7}^{6} + 19T_{7}^{5} - 115T_{7}^{4} - 46T_{7}^{3} + 63T_{7}^{2} + 17T_{7} + 1 \)
|
|
\( T_{19}^{8} + 9T_{19}^{7} - 53T_{19}^{6} - 683T_{19}^{5} - 565T_{19}^{4} + 10613T_{19}^{3} + 33402T_{19}^{2} + 30176T_{19} + 3541 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 13 T^{6} + \cdots - 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 9 T^{7} + \cdots - 9 \)
|
| $7$ |
\( T^{8} + 12 T^{7} + \cdots + 1 \)
|
| $11$ |
\( T^{8} + 12 T^{7} + \cdots + 81 \)
|
| $13$ |
\( T^{8} + 15 T^{7} + \cdots - 5 \)
|
| $17$ |
\( T^{8} + 9 T^{7} + \cdots + 531 \)
|
| $19$ |
\( T^{8} + 9 T^{7} + \cdots + 3541 \)
|
| $23$ |
\( T^{8} - 15 T^{7} + \cdots - 2655 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} + 20 T^{7} + \cdots - 8055 \)
|
| $37$ |
\( T^{8} + 17 T^{7} + \cdots + 865611 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 249525 \)
|
| $43$ |
\( T^{8} + 20 T^{7} + \cdots + 6525 \)
|
| $47$ |
\( T^{8} - 9 T^{7} + \cdots - 72909 \)
|
| $53$ |
\( T^{8} - 39 T^{7} + \cdots - 306099 \)
|
| $59$ |
\( T^{8} - 250 T^{6} + \cdots - 573975 \)
|
| $61$ |
\( T^{8} + 34 T^{7} + \cdots - 464039 \)
|
| $67$ |
\( T^{8} + 21 T^{7} + \cdots - 169679 \)
|
| $71$ |
\( T^{8} + 15 T^{7} + \cdots + 3645 \)
|
| $73$ |
\( T^{8} - 2 T^{7} + \cdots + 2369671 \)
|
| $79$ |
\( T^{8} + 6 T^{7} + \cdots - 2579879 \)
|
| $83$ |
\( T^{8} + 17 T^{7} + \cdots - 32427819 \)
|
| $89$ |
\( T^{8} + 20 T^{7} + \cdots + 291645 \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots + 40453821 \)
|
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