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: |
\( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \)
|
| 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
\( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 33\nu^{4} - 44\nu^{3} - 37\nu^{2} + 33\nu + 5 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{7} + 9\nu^{6} - 28\nu^{5} - 25\nu^{4} + 79\nu^{3} + 20\nu^{2} - 61\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{8} - 8\nu^{7} - 26\nu^{6} + 69\nu^{5} + 67\nu^{4} - 178\nu^{3} - 41\nu^{2} + 125\nu - 15 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{8} + 16\nu^{7} + 42\nu^{6} - 143\nu^{5} - 109\nu^{4} + 382\nu^{3} + 83\nu^{2} - 267\nu + 21 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -9\nu^{8} + 26\nu^{7} + 80\nu^{6} - 233\nu^{5} - 221\nu^{4} + 624\nu^{3} + 177\nu^{2} - 447\nu + 35 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} - \beta_{7} + \beta_{6} - 4\beta_{5} + \beta_{4} + 8\beta_{3} + 9\beta_{2} + 19\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{8} - 10\beta_{7} + 4\beta_{6} - 14\beta_{5} + 9\beta_{4} + 12\beta_{3} + 36\beta_{2} + 21\beta _1 + 52 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26\beta_{8} - 13\beta_{7} + 15\beta_{6} - 48\beta_{5} + 14\beta_{4} + 57\beta_{3} + 69\beta_{2} + 102\beta _1 + 51 \)
|
| \(\nu^{8}\) | \(=\) |
\( 105 \beta_{8} - 76 \beta_{7} + 53 \beta_{6} - 135 \beta_{5} + 70 \beta_{4} + 109 \beta_{3} + 228 \beta_{2} + \cdots + 240 \)
|
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 |
|
−2.68096 | 0 | 5.18756 | −3.11062 | 0 | −2.37023 | −8.54571 | 0 | 8.33945 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.19659 | 0 | 2.82501 | 4.07769 | 0 | 3.64733 | −1.81221 | 0 | −8.95700 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.98078 | 0 | 1.92348 | −1.21902 | 0 | 0.393238 | 0.151564 | 0 | 2.41460 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.915678 | 0 | −1.16153 | −0.285007 | 0 | 5.03918 | 2.89495 | 0 | 0.260975 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.0831558 | 0 | −1.99309 | −1.93038 | 0 | 0.343925 | 0.332048 | 0 | 0.160522 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.15439 | 0 | −0.667390 | 3.44957 | 0 | 4.51754 | −3.07920 | 0 | 3.98215 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.79470 | 0 | 1.22096 | −1.34531 | 0 | −1.11399 | −1.39815 | 0 | −2.41444 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.83470 | 0 | 1.36614 | 2.25851 | 0 | −0.744275 | −1.16295 | 0 | 4.14369 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.07337 | 0 | 2.29887 | −1.89543 | 0 | −4.71271 | 0.619659 | 0 | −3.92993 | ||||||||||||||||||||||||||||||||||||||||||||||
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.bk | 9 | |
| 3.b | odd | 2 | 1 | 2523.2.a.q | 9 | ||
| 29.b | even | 2 | 1 | 7569.2.a.bl | 9 | ||
| 29.e | even | 14 | 2 | 261.2.k.b | 18 | ||
| 87.d | odd | 2 | 1 | 2523.2.a.p | 9 | ||
| 87.h | odd | 14 | 2 | 87.2.g.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.2.g.b | ✓ | 18 | 87.h | odd | 14 | 2 | |
| 261.2.k.b | 18 | 29.e | even | 14 | 2 | ||
| 2523.2.a.p | 9 | 87.d | odd | 2 | 1 | ||
| 2523.2.a.q | 9 | 3.b | odd | 2 | 1 | ||
| 7569.2.a.bk | 9 | 1.a | even | 1 | 1 | trivial | |
| 7569.2.a.bl | 9 | 29.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):
|
\( T_{2}^{9} + T_{2}^{8} - 14T_{2}^{7} - 9T_{2}^{6} + 70T_{2}^{5} + 23T_{2}^{4} - 141T_{2}^{3} - 14T_{2}^{2} + 84T_{2} + 7 \)
|
|
\( T_{5}^{9} - 27T_{5}^{7} - 24T_{5}^{6} + 222T_{5}^{5} + 383T_{5}^{4} - 383T_{5}^{3} - 1293T_{5}^{2} - 923T_{5} - 169 \)
|
|
\( T_{7}^{9} - 5T_{7}^{8} - 32T_{7}^{7} + 152T_{7}^{6} + 267T_{7}^{5} - 929T_{7}^{4} - 1129T_{7}^{3} + 376T_{7}^{2} + 340T_{7} - 104 \)
|
|
\( T_{19}^{9} + T_{19}^{8} - 27T_{19}^{7} + 2T_{19}^{6} + 116T_{19}^{5} - 7T_{19}^{4} - 145T_{19}^{3} - 2T_{19}^{2} + 40T_{19} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + T^{8} - 14 T^{7} + \cdots + 7 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} - 27 T^{7} + \cdots - 169 \)
|
| $7$ |
\( T^{9} - 5 T^{8} + \cdots - 104 \)
|
| $11$ |
\( T^{9} + 3 T^{8} + \cdots - 7496 \)
|
| $13$ |
\( T^{9} - 5 T^{8} + \cdots - 841 \)
|
| $17$ |
\( T^{9} + 16 T^{8} + \cdots - 271727 \)
|
| $19$ |
\( T^{9} + T^{8} - 27 T^{7} + \cdots - 8 \)
|
| $23$ |
\( T^{9} - 10 T^{8} + \cdots - 13448 \)
|
| $29$ |
\( T^{9} \)
|
| $31$ |
\( T^{9} + 4 T^{8} + \cdots - 465592 \)
|
| $37$ |
\( T^{9} - 25 T^{8} + \cdots + 111257 \)
|
| $41$ |
\( T^{9} + 34 T^{8} + \cdots + 583681 \)
|
| $43$ |
\( T^{9} - 12 T^{8} + \cdots + 7736 \)
|
| $47$ |
\( T^{9} + 8 T^{8} + \cdots - 7004024 \)
|
| $53$ |
\( T^{9} - 32 T^{8} + \cdots + 1284907 \)
|
| $59$ |
\( T^{9} + 10 T^{8} + \cdots + 56161784 \)
|
| $61$ |
\( T^{9} - 51 T^{8} + \cdots - 642691 \)
|
| $67$ |
\( T^{9} - 7 T^{8} + \cdots + 51076472 \)
|
| $71$ |
\( T^{9} + 7 T^{8} + \cdots - 661256 \)
|
| $73$ |
\( T^{9} - 17 T^{8} + \cdots + 1284473 \)
|
| $79$ |
\( T^{9} - 13 T^{8} + \cdots - 31304 \)
|
| $83$ |
\( T^{9} - 31 T^{8} + \cdots + 4705112 \)
|
| $89$ |
\( T^{9} - 32 T^{8} + \cdots + 237040103 \)
|
| $97$ |
\( T^{9} - 16 T^{8} + \cdots - 18124457 \)
|
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