Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9405,2,Mod(1,9405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9405.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: | \(75.0993031010\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1045) |
| 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
\( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 21\nu^{4} - 31\nu^{3} - 12\nu^{2} + 15\nu - 1 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 32\nu^{5} - 7\nu^{4} - 59\nu^{3} + 24\nu^{2} + 25\nu - 7 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 24\nu^{4} + 8\nu^{3} - 45\nu^{2} + 19 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 16\nu^{4} - 21\nu^{3} + 31\nu^{2} + 13\nu - 15 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 30\nu^{5} - 25\nu^{4} - 47\nu^{3} + 64\nu^{2} + 15\nu - 27 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{6} - \beta_{4} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + 7\beta_{3} + \beta_{2} + 27\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{8} + \beta_{7} + 11\beta_{6} - 2\beta_{5} - 10\beta_{4} + \beta_{3} + 44\beta_{2} + 2\beta _1 + 89 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26\beta_{8} + 10\beta_{7} + 24\beta_{6} - 13\beta_{5} - 13\beta_{4} + 44\beta_{3} + 16\beta_{2} + 155\beta _1 + 85 \)
|
| \(\nu^{8}\) | \(=\) |
\( 107\beta_{8} + 13\beta_{7} + 94\beta_{6} - 28\beta_{5} - 77\beta_{4} + 16\beta_{3} + 277\beta_{2} + 36\beta _1 + 564 \)
|
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 |
|
−2.69878 | 0 | 5.28341 | −1.00000 | 0 | 0.342738 | −8.86121 | 0 | 2.69878 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.14345 | 0 | 2.59437 | −1.00000 | 0 | −1.53582 | −1.27399 | 0 | 2.14345 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.92391 | 0 | 1.70143 | −1.00000 | 0 | 5.15278 | 0.574421 | 0 | 1.92391 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.22961 | 0 | −0.488068 | −1.00000 | 0 | 1.13367 | 3.05934 | 0 | 1.22961 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.682920 | 0 | −1.53362 | −1.00000 | 0 | 0.856948 | 2.41318 | 0 | 0.682920 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.730132 | 0 | −1.46691 | −1.00000 | 0 | −1.34825 | −2.53130 | 0 | −0.730132 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.24377 | 0 | −0.453038 | −1.00000 | 0 | 3.83759 | −3.05101 | 0 | −1.24377 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.35249 | 0 | −0.170779 | −1.00000 | 0 | 2.99075 | −2.93595 | 0 | −1.35249 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.35228 | 0 | 3.53320 | −1.00000 | 0 | 1.56958 | 3.60652 | 0 | −2.35228 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 9405.2.a.bh | 9 | |
| 3.b | odd | 2 | 1 | 1045.2.a.k | ✓ | 9 | |
| 15.d | odd | 2 | 1 | 5225.2.a.p | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.k | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 5225.2.a.p | 9 | 15.d | odd | 2 | 1 | ||
| 9405.2.a.bh | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):
|
\( T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 29T_{2}^{6} + 23T_{2}^{5} + 84T_{2}^{4} - 23T_{2}^{3} - 89T_{2}^{2} + 8T_{2} + 27 \)
|
|
\( T_{7}^{9} - 13T_{7}^{8} + 55T_{7}^{7} - 56T_{7}^{6} - 169T_{7}^{5} + 389T_{7}^{4} - 56T_{7}^{3} - 408T_{7}^{2} + 320T_{7} - 64 \)
|
|
\( T_{13}^{9} - 5 T_{13}^{8} - 29 T_{13}^{7} + 125 T_{13}^{6} + 268 T_{13}^{5} - 705 T_{13}^{4} - 1080 T_{13}^{3} + \cdots - 64 \)
|
|
\( T_{17}^{9} + 13 T_{17}^{8} + 35 T_{17}^{7} - 169 T_{17}^{6} - 892 T_{17}^{5} - 213 T_{17}^{4} + \cdots - 5904 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 3 T^{8} + \cdots + 27 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( (T + 1)^{9} \)
|
| $7$ |
\( T^{9} - 13 T^{8} + \cdots - 64 \)
|
| $11$ |
\( (T + 1)^{9} \)
|
| $13$ |
\( T^{9} - 5 T^{8} + \cdots - 64 \)
|
| $17$ |
\( T^{9} + 13 T^{8} + \cdots - 5904 \)
|
| $19$ |
\( (T - 1)^{9} \)
|
| $23$ |
\( T^{9} + 8 T^{8} + \cdots + 192 \)
|
| $29$ |
\( T^{9} - 3 T^{8} + \cdots + 1680 \)
|
| $31$ |
\( T^{9} + 9 T^{8} + \cdots + 64 \)
|
| $37$ |
\( T^{9} + 7 T^{8} + \cdots - 32192 \)
|
| $41$ |
\( T^{9} + 9 T^{8} + \cdots - 1696176 \)
|
| $43$ |
\( T^{9} - 23 T^{8} + \cdots + 29632 \)
|
| $47$ |
\( T^{9} + 20 T^{8} + \cdots + 576 \)
|
| $53$ |
\( T^{9} - 5 T^{8} + \cdots - 768192 \)
|
| $59$ |
\( T^{9} + 19 T^{8} + \cdots + 3341760 \)
|
| $61$ |
\( T^{9} - T^{8} + \cdots - 23824 \)
|
| $67$ |
\( T^{9} + 10 T^{8} + \cdots + 46441456 \)
|
| $71$ |
\( T^{9} - 298 T^{7} + \cdots + 34636224 \)
|
| $73$ |
\( T^{9} - 12 T^{8} + \cdots - 160279408 \)
|
| $79$ |
\( T^{9} + 21 T^{8} + \cdots - 71054080 \)
|
| $83$ |
\( T^{9} + 47 T^{8} + \cdots - 909504 \)
|
| $89$ |
\( T^{9} - 2 T^{8} + \cdots - 37547280 \)
|
| $97$ |
\( T^{9} + 32 T^{8} + \cdots - 165140288 \)
|
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