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: | \(1\) |
| 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: |
\( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 44x^{4} - 25x^{3} - 31x^{2} + 19x - 2 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 44x^{4} - 25x^{3} - 31x^{2} + 19x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 13\nu^{5} + 3\nu^{4} + 47\nu^{3} + 26\nu^{2} - 13\nu - 10 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 9\nu^{5} - 25\nu^{4} - 23\nu^{3} + 48\nu^{2} + 15\nu - 18 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 13\nu^{5} + 3\nu^{4} + 51\nu^{3} + 22\nu^{2} - 37\nu - 6 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 26\nu^{5} - 21\nu^{4} - 102\nu^{3} + 13\nu^{2} + 75\nu - 16 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 9\nu^{6} + 61\nu^{5} - 63\nu^{4} - 227\nu^{3} + 42\nu^{2} + 149\nu - 26 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{4} + 8\beta_{2} + 12\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - \beta_{6} + 10\beta_{5} - 3\beta_{4} - 7\beta_{3} + 12\beta_{2} + 56\beta _1 + 25 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15\beta_{7} - 13\beta_{6} + 8\beta_{5} - 15\beta_{4} + 63\beta_{2} + 122\beta _1 + 127 \)
|
| \(\nu^{7}\) | \(=\) |
\( 38\beta_{7} - 23\beta_{6} + 91\beta_{5} - 51\beta_{4} - 40\beta_{3} + 122\beta_{2} + 472\beta _1 + 236 \)
|
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 | −3.34462 | 1.00000 | 2.43986 | −3.34462 | 0 | 1.00000 | 8.18651 | 2.43986 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.72024 | 1.00000 | −0.585308 | −2.72024 | 0 | 1.00000 | 4.39970 | −0.585308 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.18159 | 1.00000 | −2.50549 | −2.18159 | 0 | 1.00000 | 1.75933 | −2.50549 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.860157 | 1.00000 | −3.79513 | −0.860157 | 0 | 1.00000 | −2.26013 | −3.79513 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.551869 | 1.00000 | −2.62326 | −0.551869 | 0 | 1.00000 | −2.69544 | −2.62326 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.225609 | 1.00000 | 2.69020 | −0.225609 | 0 | 1.00000 | −2.94910 | 2.69020 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.85245 | 1.00000 | −1.79564 | 1.85245 | 0 | 1.00000 | 0.431566 | −1.79564 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.03164 | 1.00000 | −0.825229 | 2.03164 | 0 | 1.00000 | 1.12756 | −0.825229 | |||||||||||||||||||||||||||||||||||||||||||
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.w | 8 | |
| 7.b | odd | 2 | 1 | 1414.2.a.h | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1414.2.a.h | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 9898.2.a.w | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):
|
\( T_{3}^{8} + 6T_{3}^{7} + 2T_{3}^{6} - 42T_{3}^{5} - 56T_{3}^{4} + 57T_{3}^{3} + 124T_{3}^{2} + 60T_{3} + 8 \)
|
|
\( T_{5}^{8} + 7T_{5}^{7} + 2T_{5}^{6} - 80T_{5}^{5} - 154T_{5}^{4} + 139T_{5}^{3} + 611T_{5}^{2} + 531T_{5} + 142 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 6 T^{7} + \cdots + 8 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots + 142 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 106 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots - 2 \)
|
| $17$ |
\( T^{8} + 8 T^{7} + \cdots - 448 \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots + 230540 \)
|
| $23$ |
\( T^{8} - 3 T^{7} + \cdots - 109 \)
|
| $29$ |
\( T^{8} - T^{7} + \cdots + 5620 \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 39776 \)
|
| $37$ |
\( T^{8} + 17 T^{7} + \cdots - 8 \)
|
| $41$ |
\( T^{8} + 10 T^{7} + \cdots + 294622 \)
|
| $43$ |
\( T^{8} - T^{7} + \cdots + 229864 \)
|
| $47$ |
\( T^{8} + 18 T^{7} + \cdots + 1551136 \)
|
| $53$ |
\( T^{8} + 9 T^{7} + \cdots + 6537818 \)
|
| $59$ |
\( T^{8} - 3 T^{7} + \cdots + 1034840 \)
|
| $61$ |
\( T^{8} + 5 T^{7} + \cdots + 20358848 \)
|
| $67$ |
\( T^{8} - T^{7} + \cdots + 552826 \)
|
| $71$ |
\( T^{8} - 6 T^{7} + \cdots - 449057 \)
|
| $73$ |
\( T^{8} + 4 T^{7} + \cdots - 23149 \)
|
| $79$ |
\( T^{8} + 10 T^{7} + \cdots - 2365 \)
|
| $83$ |
\( T^{8} + 20 T^{7} + \cdots + 767384 \)
|
| $89$ |
\( T^{8} + 29 T^{7} + \cdots + 3938375 \)
|
| $97$ |
\( T^{8} - 2 T^{7} + \cdots + 52474192 \)
|
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