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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 10\nu^{3} + 21\nu^{2} - 8\nu - 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 18\nu^{4} + 11\nu^{3} - 29\nu^{2} - 4\nu + 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 9\nu^{5} - 13\nu^{4} - 24\nu^{3} + 19\nu^{2} + 12\nu - 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} + 9\beta_{3} + 19\beta_{2} + 31\beta _1 + 31 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{6} + 3\beta_{5} + 14\beta_{4} + 24\beta_{3} + 61\beta_{2} + 71\beta _1 + 109 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{7} + 13\beta_{6} + 15\beta_{5} + 42\beta_{4} + 79\beta_{3} + 160\beta_{2} + 215\beta _1 + 268 \)
|
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 |
|
−1.87791 | 0 | 1.52654 | −1.00000 | 0 | −0.818685 | 0.889109 | 0 | 1.87791 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.25600 | 0 | −0.422456 | −1.00000 | 0 | −3.10169 | 3.04261 | 0 | 1.25600 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.0663929 | 0 | −1.99559 | −1.00000 | 0 | 2.56056 | 0.265279 | 0 | 0.0663929 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.285222 | 0 | −1.91865 | −1.00000 | 0 | −1.06724 | −1.11768 | 0 | −0.285222 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.64922 | 0 | 0.719922 | −1.00000 | 0 | −5.09280 | −2.11113 | 0 | −1.64922 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.86598 | 0 | 1.48188 | −1.00000 | 0 | −2.04136 | −0.966800 | 0 | −1.86598 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.57959 | 0 | 4.65426 | −1.00000 | 0 | 2.87748 | 6.84689 | 0 | −2.57959 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.82030 | 0 | 5.95409 | −1.00000 | 0 | −4.31627 | 11.1517 | 0 | −2.82030 | |||||||||||||||||||||||||||||||||||||||||||
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.bf | 8 | |
| 3.b | odd | 2 | 1 | 1045.2.a.i | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 5225.2.a.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.i | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 5225.2.a.o | 8 | 15.d | odd | 2 | 1 | ||
| 9405.2.a.bf | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):
|
\( T_{2}^{8} - 6T_{2}^{7} + 5T_{2}^{6} + 28T_{2}^{5} - 47T_{2}^{4} - 21T_{2}^{3} + 60T_{2}^{2} - 11T_{2} - 1 \)
|
|
\( T_{7}^{8} + 11T_{7}^{7} + 23T_{7}^{6} - 120T_{7}^{5} - 489T_{7}^{4} - 47T_{7}^{3} + 1792T_{7}^{2} + 2384T_{7} + 896 \)
|
|
\( T_{13}^{8} + 17T_{13}^{7} + 65T_{13}^{6} - 329T_{13}^{5} - 2544T_{13}^{4} - 1453T_{13}^{3} + 17766T_{13}^{2} + 25704T_{13} - 13392 \)
|
|
\( T_{17}^{8} - 9T_{17}^{7} - 47T_{17}^{6} + 585T_{17}^{5} - 434T_{17}^{4} - 7505T_{17}^{3} + 22974T_{17}^{2} - 23908T_{17} + 8344 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 6 T^{7} + \cdots - 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 11 T^{7} + \cdots + 896 \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} + 17 T^{7} + \cdots - 13392 \)
|
| $17$ |
\( T^{8} - 9 T^{7} + \cdots + 8344 \)
|
| $19$ |
\( (T + 1)^{8} \)
|
| $23$ |
\( T^{8} - 8 T^{7} + \cdots + 1504 \)
|
| $29$ |
\( T^{8} - 3 T^{7} + \cdots + 84152 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 35840 \)
|
| $37$ |
\( T^{8} + 17 T^{7} + \cdots - 280640 \)
|
| $41$ |
\( T^{8} - 5 T^{7} + \cdots + 30440 \)
|
| $43$ |
\( T^{8} + 21 T^{7} + \cdots - 18944 \)
|
| $47$ |
\( T^{8} - 8 T^{7} + \cdots + 29984 \)
|
| $53$ |
\( T^{8} - 19 T^{7} + \cdots + 10528 \)
|
| $59$ |
\( T^{8} - 33 T^{7} + \cdots + 6694912 \)
|
| $61$ |
\( T^{8} + T^{7} + \cdots + 25298072 \)
|
| $67$ |
\( T^{8} + 18 T^{7} + \cdots + 56432 \)
|
| $71$ |
\( T^{8} - 18 T^{7} + \cdots - 833024 \)
|
| $73$ |
\( T^{8} + 18 T^{7} + \cdots - 22472 \)
|
| $79$ |
\( T^{8} + 5 T^{7} + \cdots - 197248 \)
|
| $83$ |
\( T^{8} - 33 T^{7} + \cdots + 2043584 \)
|
| $89$ |
\( T^{8} - 20 T^{7} + \cdots + 216 \)
|
| $97$ |
\( T^{8} - 419 T^{6} + \cdots + 9664 \)
|
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