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: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 87) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 4\nu^{5} + 25\nu^{4} - 2\nu^{3} - 40\nu^{2} + 15\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} + 4\nu^{6} - 25\nu^{5} + 2\nu^{4} + 40\nu^{3} - 15\nu^{2} - 8\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 4\nu^{6} + 25\nu^{5} - 2\nu^{4} - 40\nu^{3} + 17\nu^{2} + 6\nu - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 6\nu^{7} - 4\nu^{6} - 31\nu^{5} + 44\nu^{4} + 34\nu^{3} - 69\nu^{2} + 16\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 8\nu^{7} + 14\nu^{6} + 33\nu^{5} - 106\nu^{4} + 2\nu^{3} + 169\nu^{2} - 84\nu - 20 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} - 22\nu^{6} + 17\nu^{5} + 106\nu^{4} - 94\nu^{3} - 147\nu^{2} + 128\nu + 12 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 5\nu^{7} + 28\nu^{5} - 23\nu^{4} - 36\nu^{3} + 40\nu^{2} - 8\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{5} + 2\beta_{4} + 3\beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{8} + \beta_{7} + \beta_{6} + 3\beta_{5} + 8\beta_{4} + 12\beta_{3} + 13\beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{8} + 4\beta_{7} + 4\beta_{6} + 14\beta_{5} + 21\beta_{4} + 37\beta_{3} - 2\beta_{2} + 48\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 41\beta_{8} + 18\beta_{7} + 20\beta_{6} + 45\beta_{5} + 69\beta_{4} + 125\beta_{3} - 4\beta_{2} + 135\beta _1 + 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( 143 \beta_{8} + 63 \beta_{7} + 71 \beta_{6} + 163 \beta_{5} + 204 \beta_{4} + 394 \beta_{3} - 22 \beta_{2} + \cdots + 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( 457 \beta_{8} + 226 \beta_{7} + 266 \beta_{6} + 528 \beta_{5} + 648 \beta_{4} + 1278 \beta_{3} + \cdots + 140 \)
|
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 |
|
−2.19178 | 0 | 2.80388 | 2.39677 | 0 | 1.35191 | −1.76193 | 0 | −5.25319 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.21072 | 0 | −0.534161 | −1.79844 | 0 | 4.27693 | 3.06816 | 0 | 2.17740 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.694939 | 0 | −1.51706 | 2.01780 | 0 | −2.39488 | 2.44414 | 0 | −1.40225 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.144378 | 0 | −1.97916 | −1.55424 | 0 | −1.52752 | 0.574502 | 0 | 0.224397 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.177073 | 0 | −1.96865 | 3.45143 | 0 | −3.79521 | −0.702739 | 0 | 0.611154 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.16391 | 0 | −0.645316 | −3.11042 | 0 | 1.74624 | −3.07891 | 0 | −3.62025 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.53422 | 0 | 4.42225 | −1.08952 | 0 | 1.73057 | 6.13851 | 0 | −2.76107 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.58860 | 0 | 4.70087 | 1.14894 | 0 | 2.76526 | 6.99148 | 0 | 2.97416 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.77801 | 0 | 5.71733 | 2.53766 | 0 | 0.846706 | 10.3268 | 0 | 7.04965 | ||||||||||||||||||||||||||||||||||||||||||||||
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.bm | 9 | |
| 3.b | odd | 2 | 1 | 2523.2.a.o | 9 | ||
| 29.b | even | 2 | 1 | 7569.2.a.bj | 9 | ||
| 29.e | even | 14 | 2 | 261.2.k.c | 18 | ||
| 87.d | odd | 2 | 1 | 2523.2.a.r | 9 | ||
| 87.h | odd | 14 | 2 | 87.2.g.a | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.2.g.a | ✓ | 18 | 87.h | odd | 14 | 2 | |
| 261.2.k.c | 18 | 29.e | even | 14 | 2 | ||
| 2523.2.a.o | 9 | 3.b | odd | 2 | 1 | ||
| 2523.2.a.r | 9 | 87.d | odd | 2 | 1 | ||
| 7569.2.a.bj | 9 | 29.b | even | 2 | 1 | ||
| 7569.2.a.bm | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):
|
\( T_{2}^{9} - 5T_{2}^{8} - 2T_{2}^{7} + 37T_{2}^{6} - 20T_{2}^{5} - 71T_{2}^{4} + 31T_{2}^{3} + 40T_{2}^{2} - 2T_{2} - 1 \)
|
|
\( T_{5}^{9} - 4T_{5}^{8} - 15T_{5}^{7} + 68T_{5}^{6} + 58T_{5}^{5} - 353T_{5}^{4} - 67T_{5}^{3} + 711T_{5}^{2} + 13T_{5} - 461 \)
|
|
\( T_{7}^{9} - 5T_{7}^{8} - 16T_{7}^{7} + 108T_{7}^{6} - 21T_{7}^{5} - 569T_{7}^{4} + 691T_{7}^{3} + 560T_{7}^{2} - 1324T_{7} + 568 \)
|
|
\( T_{19}^{9} + 9 T_{19}^{8} - 79 T_{19}^{7} - 774 T_{19}^{6} + 2044 T_{19}^{5} + 21873 T_{19}^{4} + \cdots + 767768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 5 T^{8} + \cdots - 1 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} - 4 T^{8} + \cdots - 461 \)
|
| $7$ |
\( T^{9} - 5 T^{8} + \cdots + 568 \)
|
| $11$ |
\( T^{9} + T^{8} + \cdots + 1112 \)
|
| $13$ |
\( T^{9} - T^{8} + \cdots - 46697 \)
|
| $17$ |
\( T^{9} - 2 T^{8} + \cdots - 86143 \)
|
| $19$ |
\( T^{9} + 9 T^{8} + \cdots + 767768 \)
|
| $23$ |
\( T^{9} - 4 T^{8} + \cdots - 29672 \)
|
| $29$ |
\( T^{9} \)
|
| $31$ |
\( T^{9} + 8 T^{8} + \cdots - 13336 \)
|
| $37$ |
\( T^{9} + 27 T^{8} + \cdots + 137297 \)
|
| $41$ |
\( T^{9} - 12 T^{8} + \cdots - 3560291 \)
|
| $43$ |
\( T^{9} + 16 T^{8} + \cdots + 16472 \)
|
| $47$ |
\( T^{9} + 8 T^{8} + \cdots + 904 \)
|
| $53$ |
\( T^{9} - 8 T^{8} + \cdots - 2305981 \)
|
| $59$ |
\( T^{9} - 16 T^{8} + \cdots - 897224 \)
|
| $61$ |
\( T^{9} + 21 T^{8} + \cdots - 722639 \)
|
| $67$ |
\( T^{9} - 3 T^{8} + \cdots - 45704 \)
|
| $71$ |
\( T^{9} - 33 T^{8} + \cdots - 89667752 \)
|
| $73$ |
\( T^{9} + 3 T^{8} + \cdots - 247283 \)
|
| $79$ |
\( T^{9} + 3 T^{8} + \cdots - 93512 \)
|
| $83$ |
\( T^{9} + 13 T^{8} + \cdots - 803656 \)
|
| $89$ |
\( T^{9} - 6 T^{8} + \cdots + 46711771 \)
|
| $97$ |
\( T^{9} + 4 T^{8} + \cdots - 6388081 \)
|
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