Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2888,2,Mod(1,2888)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2888, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2888.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2888 = 2^{3} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2888.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: | \(23.0607961037\) |
| 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: |
\( x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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
\( x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{8} - 5\nu^{7} - 30\nu^{6} + 46\nu^{5} + 76\nu^{4} - 104\nu^{3} - 44\nu^{2} + 33\nu + 7 ) / 38 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{8} + 8\nu^{7} + 10\nu^{6} - 104\nu^{5} - 38\nu^{4} + 402\nu^{3} + 78\nu^{2} - 391\nu + 42 ) / 38 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{8} + 2\nu^{7} + 69\nu^{6} - 7\nu^{5} - 285\nu^{4} - 23\nu^{3} + 295\nu^{2} - 36\nu + 1 ) / 38 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 8\nu^{7} + 10\nu^{6} - 104\nu^{5} - 38\nu^{4} + 402\nu^{3} + 116\nu^{2} - 429\nu - 72 ) / 38 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{8} - 13\nu^{7} - 40\nu^{6} + 150\nu^{5} + 76\nu^{4} - 506\nu^{3} + 144\nu^{2} + 500\nu - 187 ) / 38 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -6\nu^{8} + 10\nu^{7} + 79\nu^{6} - 111\nu^{5} - 323\nu^{4} + 341\nu^{3} + 411\nu^{2} - 237\nu - 33 ) / 38 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{7} + 12\nu^{6} - 34\nu^{5} - 46\nu^{4} + 108\nu^{3} + 62\nu^{2} - 79\nu - 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + \beta_{4} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + 7\beta_{5} - 8\beta_{3} + \beta_{2} + 9\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - 11\beta_{7} + 9\beta_{5} + 9\beta_{4} - 4\beta_{3} + \beta_{2} + 39\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} - 2\beta_{7} - 9\beta_{6} + 48\beta_{5} + 2\beta_{4} - 59\beta_{3} + 14\beta_{2} + 70\beta _1 + 109 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14\beta_{8} - 96\beta_{7} - 2\beta_{6} + 72\beta_{5} + 68\beta_{4} - 56\beta_{3} + 18\beta_{2} + 266\beta _1 + 121 \)
|
| \(\nu^{8}\) | \(=\) |
\( 18 \beta_{8} - 46 \beta_{7} - 68 \beta_{6} + 334 \beta_{5} + 30 \beta_{4} - 434 \beta_{3} + 142 \beta_{2} + \cdots + 738 \)
|
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 |
|
0 | −2.49223 | 0 | 0.353025 | 0 | −0.687435 | 0 | 3.21120 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.95251 | 0 | 1.73942 | 0 | 1.92522 | 0 | 0.812302 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.28732 | 0 | 4.24329 | 0 | 3.37236 | 0 | −1.34280 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.0745540 | 0 | −0.900387 | 0 | −4.87539 | 0 | −2.99444 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.148922 | 0 | −3.42232 | 0 | −1.92022 | 0 | −2.97782 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.809325 | 0 | −3.44645 | 0 | 4.02567 | 0 | −2.34499 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.36188 | 0 | −2.34290 | 0 | 4.15573 | 0 | 2.57847 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.68290 | 0 | 4.09342 | 0 | 1.54115 | 0 | 4.19797 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.80359 | 0 | 2.68290 | 0 | 1.46292 | 0 | 4.86011 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
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 | 2888.2.a.y | 9 | |
| 4.b | odd | 2 | 1 | 5776.2.a.cd | 9 | ||
| 19.b | odd | 2 | 1 | 2888.2.a.x | 9 | ||
| 19.e | even | 9 | 2 | 152.2.q.c | ✓ | 18 | |
| 76.d | even | 2 | 1 | 5776.2.a.ce | 9 | ||
| 76.l | odd | 18 | 2 | 304.2.u.f | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.2.q.c | ✓ | 18 | 19.e | even | 9 | 2 | |
| 304.2.u.f | 18 | 76.l | odd | 18 | 2 | ||
| 2888.2.a.x | 9 | 19.b | odd | 2 | 1 | ||
| 2888.2.a.y | 9 | 1.a | even | 1 | 1 | trivial | |
| 5776.2.a.cd | 9 | 4.b | odd | 2 | 1 | ||
| 5776.2.a.ce | 9 | 76.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2888))\):
|
\( T_{3}^{9} - 3T_{3}^{8} - 12T_{3}^{7} + 35T_{3}^{6} + 45T_{3}^{5} - 117T_{3}^{4} - 55T_{3}^{3} + 96T_{3}^{2} - 6T_{3} - 1 \)
|
|
\( T_{5}^{9} - 3T_{5}^{8} - 33T_{5}^{7} + 83T_{5}^{6} + 369T_{5}^{5} - 717T_{5}^{4} - 1501T_{5}^{3} + 1998T_{5}^{2} + 1524T_{5} - 712 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} - 3 T^{8} + \cdots - 1 \)
|
| $5$ |
\( T^{9} - 3 T^{8} + \cdots - 712 \)
|
| $7$ |
\( T^{9} - 9 T^{8} + \cdots + 1576 \)
|
| $11$ |
\( T^{9} - 3 T^{8} + \cdots + 17929 \)
|
| $13$ |
\( T^{9} + 6 T^{8} + \cdots - 30952 \)
|
| $17$ |
\( T^{9} + 3 T^{8} + \cdots + 827 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} - 24 T^{8} + \cdots + 4544 \)
|
| $29$ |
\( T^{9} + 15 T^{8} + \cdots - 14968 \)
|
| $31$ |
\( T^{9} - 6 T^{8} + \cdots - 116856 \)
|
| $37$ |
\( T^{9} - 24 T^{8} + \cdots + 24066368 \)
|
| $41$ |
\( T^{9} + 12 T^{8} + \cdots - 1621 \)
|
| $43$ |
\( T^{9} - 9 T^{8} + \cdots - 214633 \)
|
| $47$ |
\( T^{9} + 12 T^{8} + \cdots + 9656 \)
|
| $53$ |
\( T^{9} + 18 T^{8} + \cdots + 91144 \)
|
| $59$ |
\( T^{9} - 57 T^{8} + \cdots - 7131347 \)
|
| $61$ |
\( T^{9} - 15 T^{8} + \cdots + 500264 \)
|
| $67$ |
\( T^{9} - 6 T^{8} + \cdots - 526184 \)
|
| $71$ |
\( T^{9} - 237 T^{7} + \cdots - 1710784 \)
|
| $73$ |
\( T^{9} - 36 T^{8} + \cdots - 1487296 \)
|
| $79$ |
\( T^{9} + 3 T^{8} + \cdots - 16190632 \)
|
| $83$ |
\( T^{9} - 27 T^{8} + \cdots + 24782543 \)
|
| $89$ |
\( T^{9} + 30 T^{8} + \cdots - 2317823 \)
|
| $97$ |
\( T^{9} - 3 T^{8} + \cdots + 5282353 \)
|
| show more | |
| show less |