Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9898,2,Mod(1,9898)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, 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("9898.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9898 = 2 \cdot 7^{2} \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9898.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: | \(79.0359279207\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 17x^{5} + 6x^{4} + 25x^{3} - 8x^{2} - 8x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1414) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 17x^{5} + 6x^{4} + 25x^{3} - 8x^{2} - 8x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{6} + \nu^{5} + 34\nu^{4} + 6\nu^{3} - 40\nu^{2} - 13\nu + 9 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} - 3\nu^{5} - 50\nu^{4} + 16\nu^{3} + 59\nu^{2} - 4\nu - 9 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} - \nu^{5} - 52\nu^{4} - 16\nu^{3} + 69\nu^{2} + 16\nu - 15 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{6} + 4\nu^{5} + 120\nu^{4} + 10\nu^{3} - 159\nu^{2} - 15\nu + 40 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{6} - 3\nu^{5} - 122\nu^{4} - 26\nu^{3} + 181\nu^{2} + 34\nu - 55 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 13\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} - 19\beta_{5} - 33\beta_{4} - 18\beta_{3} - 17\beta_{2} + 9\beta _1 + 73 \)
|
| \(\nu^{5}\) | \(=\) |
\( -18\beta_{6} - 62\beta_{5} - 54\beta_{4} - 46\beta_{3} + 4\beta_{2} + 207\beta _1 + 115 \)
|
| \(\nu^{6}\) | \(=\) |
\( -46\beta_{6} - 343\beta_{5} - 554\beta_{4} - 315\beta_{3} - 265\beta_{2} + 289\beta _1 + 1215 \)
|
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 | −3.09599 | 1.00000 | −1.91816 | 3.09599 | 0 | −1.00000 | 6.58513 | 1.91816 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.88245 | 1.00000 | −3.48397 | 1.88245 | 0 | −1.00000 | 0.543617 | 3.48397 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.69824 | 1.00000 | 3.24153 | 1.69824 | 0 | −1.00000 | −0.115977 | −3.24153 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.850986 | 1.00000 | 2.28748 | −0.850986 | 0 | −1.00000 | −2.27582 | −2.28748 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.06442 | 1.00000 | −3.95085 | −1.06442 | 0 | −1.00000 | −1.86700 | 3.95085 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.36394 | 1.00000 | −0.452263 | −2.36394 | 0 | −1.00000 | 2.58820 | 0.452263 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 3.39733 | 1.00000 | 1.27624 | −3.39733 | 0 | −1.00000 | 8.54185 | −1.27624 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(101\) | \( +1 \) |
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 | 9898.2.a.v | 7 | |
| 7.b | odd | 2 | 1 | 1414.2.a.f | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1414.2.a.f | ✓ | 7 | 7.b | odd | 2 | 1 | |
| 9898.2.a.v | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):
|
\( T_{3}^{7} - T_{3}^{6} - 17T_{3}^{5} + 13T_{3}^{4} + 79T_{3}^{3} - 48T_{3}^{2} - 100T_{3} + 72 \)
|
|
\( T_{5}^{7} + 3T_{5}^{6} - 20T_{5}^{5} - 48T_{5}^{4} + 120T_{5}^{3} + 179T_{5}^{2} - 197T_{5} - 113 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( T^{7} - T^{6} + \cdots + 72 \)
|
| $5$ |
\( T^{7} + 3 T^{6} + \cdots - 113 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 6 T^{6} + \cdots + 751 \)
|
| $13$ |
\( T^{7} - 37 T^{5} + \cdots + 81 \)
|
| $17$ |
\( T^{7} - 2 T^{6} + \cdots - 8 \)
|
| $19$ |
\( T^{7} - 4 T^{6} + \cdots - 639 \)
|
| $23$ |
\( T^{7} - 6 T^{6} + \cdots - 8427 \)
|
| $29$ |
\( T^{7} - 21 T^{6} + \cdots - 5183 \)
|
| $31$ |
\( T^{7} - 8 T^{6} + \cdots - 2232 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 1720 \)
|
| $41$ |
\( T^{7} + 6 T^{6} + \cdots - 587 \)
|
| $43$ |
\( T^{7} - 98 T^{5} + \cdots - 5080 \)
|
| $47$ |
\( T^{7} - 6 T^{6} + \cdots + 73896 \)
|
| $53$ |
\( T^{7} - 31 T^{6} + \cdots + 3999 \)
|
| $59$ |
\( T^{7} + 2 T^{6} + \cdots - 31800 \)
|
| $61$ |
\( T^{7} + 14 T^{6} + \cdots + 44864 \)
|
| $67$ |
\( T^{7} - 3 T^{6} + \cdots + 5243 \)
|
| $71$ |
\( T^{7} - 13 T^{6} + \cdots + 455749 \)
|
| $73$ |
\( T^{7} - 19 T^{6} + \cdots - 212053 \)
|
| $79$ |
\( T^{7} + 3 T^{6} + \cdots - 1089353 \)
|
| $83$ |
\( T^{7} - 5 T^{6} + \cdots - 190984 \)
|
| $89$ |
\( T^{7} + 44 T^{6} + \cdots - 23409 \)
|
| $97$ |
\( T^{7} - 4 T^{6} + \cdots - 520 \)
|
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