Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8022,2,Mod(1,8022)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8022, 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("8022.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8022.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: | \(64.0559925015\) |
| 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: |
\( x^{8} - x^{7} - 12x^{6} + 12x^{5} + 40x^{4} - 38x^{3} - 37x^{2} + 38x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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
\( x^{8} - x^{7} - 12x^{6} + 12x^{5} + 40x^{4} - 38x^{3} - 37x^{2} + 38x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} + 17\nu^{5} + 22\nu^{4} - 88\nu^{3} - 8\nu^{2} + 107\nu - 35 ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} - 7\nu^{6} - 102\nu^{5} + 74\nu^{4} + 316\nu^{3} - 166\nu^{2} - 317\nu + 128 ) / 34 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 6\nu^{6} + 34\nu^{5} + 61\nu^{4} - 176\nu^{3} - 135\nu^{2} + 231\nu + 15 ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{7} - \nu^{6} + 68\nu^{5} + 13\nu^{4} - 205\nu^{3} - 65\nu^{2} + 149\nu + 28 ) / 17 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} - 4\nu^{6} + 85\nu^{5} + 35\nu^{4} - 293\nu^{3} - 56\nu^{2} + 273\nu - 58 ) / 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9\nu^{7} + 7\nu^{6} + 102\nu^{5} - 74\nu^{4} - 299\nu^{3} + 166\nu^{2} + 215\nu - 94 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{2} - \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{3} + 6\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{6} - 7\beta_{5} + \beta_{4} - 9\beta_{2} - 8\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{7} - 2\beta_{6} + 2\beta_{4} + 18\beta_{3} + \beta_{2} + 40\beta _1 - 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 46\beta_{6} - 49\beta_{5} + 11\beta_{4} - 2\beta_{3} - 68\beta_{2} - 62\beta _1 + 101 \)
|
| \(\nu^{7}\) | \(=\) |
\( 79\beta_{7} - 26\beta_{6} + \beta_{5} + 23\beta_{4} + 136\beta_{3} + 14\beta_{2} + 277\beta _1 - 157 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −1.86457 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 1.86457 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −0.929436 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | 0.929436 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 0.120976 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −0.120976 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | 0.525371 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −0.525371 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | 0.965766 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −0.965766 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 2.73109 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −2.73109 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 3.39363 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −3.39363 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 4.05716 | 1.00000 | 1.00000 | −1.00000 | 1.00000 | −4.05716 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(191\) | \(1\) |
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 | 8022.2.a.n | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8022.2.a.n | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):
|
\( T_{5}^{8} - 9T_{5}^{7} + 20T_{5}^{6} + 23T_{5}^{5} - 105T_{5}^{4} + 36T_{5}^{3} + 71T_{5}^{2} - 42T_{5} + 4 \)
|
|
\( T_{11}^{8} - T_{11}^{7} - 28T_{11}^{6} + 46T_{11}^{5} + 157T_{11}^{4} - 197T_{11}^{3} - 327T_{11}^{2} + 112T_{11} + 68 \)
|
|
\( T_{13}^{8} - 6T_{13}^{7} - 30T_{13}^{6} + 163T_{13}^{5} + 179T_{13}^{4} - 478T_{13}^{3} - 233T_{13}^{2} + 198T_{13} - 17 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - 9 T^{7} + \cdots + 4 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} - T^{7} + \cdots + 68 \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots - 17 \)
|
| $17$ |
\( T^{8} - 14 T^{7} + \cdots + 2159 \)
|
| $19$ |
\( T^{8} - 6 T^{7} + \cdots - 428 \)
|
| $23$ |
\( T^{8} - 16 T^{7} + \cdots + 12604 \)
|
| $29$ |
\( T^{8} - 16 T^{7} + \cdots - 468544 \)
|
| $31$ |
\( T^{8} - 9 T^{7} + \cdots - 21481 \)
|
| $37$ |
\( T^{8} + 5 T^{7} + \cdots + 17 \)
|
| $41$ |
\( T^{8} - 10 T^{7} + \cdots - 2452 \)
|
| $43$ |
\( T^{8} - 5 T^{7} + \cdots + 10084 \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 14500 \)
|
| $53$ |
\( T^{8} - 27 T^{7} + \cdots + 155668 \)
|
| $59$ |
\( T^{8} - 18 T^{7} + \cdots + 479296 \)
|
| $61$ |
\( T^{8} + 6 T^{7} + \cdots - 8788 \)
|
| $67$ |
\( T^{8} + T^{7} + \cdots + 34000 \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots - 3356879 \)
|
| $73$ |
\( T^{8} + T^{7} + \cdots - 409 \)
|
| $79$ |
\( T^{8} + 10 T^{7} + \cdots - 7565321 \)
|
| $83$ |
\( T^{8} - 16 T^{7} + \cdots - 251 \)
|
| $89$ |
\( T^{8} - 9 T^{7} + \cdots + 23847188 \)
|
| $97$ |
\( T^{8} - 22 T^{7} + \cdots - 336596 \)
|
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