Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3375,2,Mod(1,3375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3375, 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("3375.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3375 = 3^{3} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3375.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: | \(26.9495106822\) |
| 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: |
\( x^{8} - 4x^{7} - 5x^{6} + 27x^{5} + 8x^{4} - 52x^{3} - 5x^{2} + 24x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{8} - 4x^{7} - 5x^{6} + 27x^{5} + 8x^{4} - 52x^{3} - 5x^{2} + 24x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 4\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 8\nu^{6} - 10\nu^{5} - 38\nu^{4} + 59\nu^{3} + 54\nu^{2} - 58\nu - 30 ) / 17 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} - 23\nu^{6} - \nu^{5} + 105\nu^{4} - 57\nu^{3} - 117\nu^{2} + 52\nu + 31 ) / 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{7} + 15\nu^{6} + 28\nu^{5} - 118\nu^{4} - 53\nu^{3} + 233\nu^{2} + 6\nu - 86 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} + 9\beta_{3} + 20\beta_{2} + 35\beta _1 + 35 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{7} + 4\beta_{6} + 8\beta_{5} + 14\beta_{4} + 23\beta_{3} + 63\beta_{2} + 90\beta _1 + 128 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14\beta_{7} + 22\beta_{6} + 37\beta_{5} + 44\beta_{4} + 77\beta_{3} + 172\beta_{2} + 281\beta _1 + 337 \)
|
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 |
|
−1.99433 | 0 | 1.97736 | 0 | 0 | 1.15949 | 0.0451562 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.77295 | 0 | 1.14337 | 0 | 0 | 2.92693 | 1.51877 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.584333 | 0 | −1.65855 | 0 | 0 | −0.345970 | 2.13782 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.181389 | 0 | −1.96710 | 0 | 0 | −3.54897 | −0.719587 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.04146 | 0 | −0.915353 | 0 | 0 | 2.09288 | −3.03623 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.82287 | 0 | 1.32284 | 0 | 0 | −3.69850 | −1.23436 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.55010 | 0 | 4.50302 | 0 | 0 | −0.470842 | 6.38295 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.75580 | 0 | 5.59442 | 0 | 0 | 3.88498 | 9.90550 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 3375.2.a.q | yes | 8 |
| 3.b | odd | 2 | 1 | 3375.2.a.g | yes | 8 | |
| 5.b | even | 2 | 1 | 3375.2.a.f | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 3375.2.a.p | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3375.2.a.f | ✓ | 8 | 5.b | even | 2 | 1 | |
| 3375.2.a.g | yes | 8 | 3.b | odd | 2 | 1 | |
| 3375.2.a.p | yes | 8 | 15.d | odd | 2 | 1 | |
| 3375.2.a.q | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3375))\):
|
\( T_{2}^{8} - 4T_{2}^{7} - 5T_{2}^{6} + 31T_{2}^{5} - 2T_{2}^{4} - 66T_{2}^{3} + 26T_{2}^{2} + 25T_{2} - 5 \)
|
|
\( T_{7}^{8} - 2T_{7}^{7} - 26T_{7}^{6} + 55T_{7}^{5} + 169T_{7}^{4} - 399T_{7}^{3} - 30T_{7}^{2} + 214T_{7} + 59 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots - 5 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 2 T^{7} + \cdots + 59 \)
|
| $11$ |
\( T^{8} - T^{7} + \cdots - 755 \)
|
| $13$ |
\( T^{8} + 7 T^{7} + \cdots + 2025 \)
|
| $17$ |
\( T^{8} - 26 T^{7} + \cdots + 5305 \)
|
| $19$ |
\( T^{8} - 2 T^{7} + \cdots + 5269 \)
|
| $23$ |
\( T^{8} - 13 T^{7} + \cdots + 32625 \)
|
| $29$ |
\( T^{8} + T^{7} + \cdots - 47205 \)
|
| $31$ |
\( T^{8} - 5 T^{7} + \cdots - 18775 \)
|
| $37$ |
\( T^{8} - 16 T^{7} + \cdots - 245 \)
|
| $41$ |
\( T^{8} + 5 T^{7} + \cdots + 88875 \)
|
| $43$ |
\( T^{8} + 11 T^{7} + \cdots + 23229 \)
|
| $47$ |
\( T^{8} - 32 T^{7} + \cdots + 15345 \)
|
| $53$ |
\( T^{8} - 19 T^{7} + \cdots - 32705 \)
|
| $59$ |
\( T^{8} + 6 T^{7} + \cdots - 6845 \)
|
| $61$ |
\( T^{8} - 243 T^{6} + \cdots + 4183219 \)
|
| $67$ |
\( T^{8} - 15 T^{7} + \cdots - 9156055 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots + 3686355 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots + 5145705 \)
|
| $79$ |
\( T^{8} + 7 T^{7} + \cdots - 41758525 \)
|
| $83$ |
\( T^{8} - 43 T^{7} + \cdots + 512195 \)
|
| $89$ |
\( T^{8} - 16 T^{7} + \cdots + 2077375 \)
|
| $97$ |
\( T^{8} + 34 T^{7} + \cdots - 42307981 \)
|
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