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: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 17x^{10} + 108x^{8} - 316x^{6} + 428x^{4} - 253x^{2} + 53 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 17x^{10} + 108x^{8} - 316x^{6} + 428x^{4} - 253x^{2} + 53 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} + 56\nu^{4} + \nu^{3} - 88\nu^{2} - 5\nu + 36 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{10} + 16\nu^{8} + \nu^{7} - 93\nu^{6} - 10\nu^{5} + 236\nu^{4} + 28\nu^{3} - 245\nu^{2} - 17\nu + 79 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} - 16\nu^{9} + 93\nu^{7} - 235\nu^{5} + 238\nu^{3} - \nu^{2} - 71\nu + 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} - 16\nu^{9} + 93\nu^{7} - 235\nu^{5} + 238\nu^{3} + \nu^{2} - 71\nu - 3 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{10} + 16\nu^{8} - 93\nu^{6} + \nu^{5} + 236\nu^{4} - 6\nu^{3} - 244\nu^{2} + 5\nu + 76 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{10} - 16\nu^{8} + 94\nu^{6} - 244\nu^{4} + 260\nu^{2} - 83 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{10} + 16\nu^{8} - 94\nu^{6} + 246\nu^{4} - 272\nu^{2} + 93 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{10} - 17\nu^{8} + 105\nu^{6} - 282\nu^{4} - \nu^{3} + 306\nu^{2} + 5\nu - 103 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{10} - 17\nu^{8} + 105\nu^{6} - 282\nu^{4} + \nu^{3} + 306\nu^{2} - 5\nu - 103 ) / 2 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -\nu^{11} + 17\nu^{9} - 106\nu^{7} + 291\nu^{5} + \nu^{4} - 328\nu^{3} - 6\nu^{2} + 115\nu + 5 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 6\beta_{5} - 6\beta_{4} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{10} - 5\beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 25\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{9} + 9\beta_{8} + 10\beta_{7} + 33\beta_{5} - 33\beta_{4} - \beta_{2} + 62 \)
|
| \(\nu^{7}\) | \(=\) |
\( 32 \beta_{10} - 21 \beta_{9} - 11 \beta_{8} + 20 \beta_{6} - 10 \beta_{5} + 10 \beta_{4} + 2 \beta_{3} + \cdots + 11 \)
|
| \(\nu^{8}\) | \(=\) |
\( -\beta_{10} - 12\beta_{9} + 61\beta_{8} + 74\beta_{7} + 181\beta_{5} - 181\beta_{4} - 11\beta_{2} + 306 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{11} + 170 \beta_{10} - 83 \beta_{9} - 88 \beta_{8} - \beta_{7} + 148 \beta_{6} - 73 \beta_{5} + \cdots + 87 \)
|
| \(\nu^{10}\) | \(=\) |
\( -16\beta_{10} - 98\beta_{9} + 374\beta_{8} + 490\beta_{7} + 998\beta_{5} - 998\beta_{4} - 82\beta_{2} + 1543 \)
|
| \(\nu^{11}\) | \(=\) |
\( 32 \beta_{11} + 916 \beta_{10} - 312 \beta_{9} - 620 \beta_{8} - 16 \beta_{7} + 978 \beta_{6} + \cdots + 604 \)
|
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 | −2.37692 | 1.00000 | 0.513823 | −2.37692 | 0 | 1.00000 | 2.64973 | 0.513823 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.17690 | 1.00000 | 1.91028 | −2.17690 | 0 | 1.00000 | 1.73890 | 1.91028 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.99304 | 1.00000 | −1.30496 | −1.99304 | 0 | 1.00000 | 0.972206 | −1.30496 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.21054 | 1.00000 | 1.52101 | −1.21054 | 0 | 1.00000 | −1.53458 | 1.52101 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.807951 | 1.00000 | −1.77771 | −0.807951 | 0 | 1.00000 | −2.34722 | −1.77771 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.721778 | 1.00000 | 0.799655 | −0.721778 | 0 | 1.00000 | −2.47904 | 0.799655 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0.721778 | 1.00000 | 1.67938 | 0.721778 | 0 | 1.00000 | −2.47904 | 1.67938 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0.807951 | 1.00000 | 4.12492 | 0.807951 | 0 | 1.00000 | −2.34722 | 4.12492 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 1.21054 | 1.00000 | 0.0135689 | 1.21054 | 0 | 1.00000 | −1.53458 | 0.0135689 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 1.99304 | 1.00000 | 0.332751 | 1.99304 | 0 | 1.00000 | 0.972206 | 0.332751 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 2.17690 | 1.00000 | −3.64918 | 2.17690 | 0 | 1.00000 | 1.73890 | −3.64918 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.00000 | 2.37692 | 1.00000 | −3.16355 | 2.37692 | 0 | 1.00000 | 2.64973 | −3.16355 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.bd | 12 | |
| 7.b | odd | 2 | 1 | 9898.2.a.bc | 12 | ||
| 7.d | odd | 6 | 2 | 1414.2.e.d | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1414.2.e.d | ✓ | 24 | 7.d | odd | 6 | 2 | |
| 9898.2.a.bc | 12 | 7.b | odd | 2 | 1 | ||
| 9898.2.a.bd | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):
|
\( T_{3}^{12} - 17T_{3}^{10} + 108T_{3}^{8} - 316T_{3}^{6} + 428T_{3}^{4} - 253T_{3}^{2} + 53 \)
|
|
\( T_{5}^{12} - T_{5}^{11} - 27 T_{5}^{10} + 28 T_{5}^{9} + 212 T_{5}^{8} - 291 T_{5}^{7} - 513 T_{5}^{6} + \cdots + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{12} \)
|
| $3$ |
\( T^{12} - 17 T^{10} + \cdots + 53 \)
|
| $5$ |
\( T^{12} - T^{11} + \cdots + 1 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} + 14 T^{11} + \cdots + 101344 \)
|
| $13$ |
\( T^{12} - T^{11} + \cdots + 175 \)
|
| $17$ |
\( T^{12} + T^{11} + \cdots + 13472 \)
|
| $19$ |
\( T^{12} - 63 T^{10} + \cdots + 5300 \)
|
| $23$ |
\( T^{12} + 26 T^{11} + \cdots - 246916 \)
|
| $29$ |
\( T^{12} + 27 T^{11} + \cdots + 1863436 \)
|
| $31$ |
\( T^{12} + 3 T^{11} + \cdots + 51518492 \)
|
| $37$ |
\( T^{12} + \cdots - 590412011 \)
|
| $41$ |
\( T^{12} - 258 T^{10} + \cdots + 85428052 \)
|
| $43$ |
\( T^{12} + 13 T^{11} + \cdots - 4764092 \)
|
| $47$ |
\( T^{12} + 4 T^{11} + \cdots - 8983268 \)
|
| $53$ |
\( T^{12} + 44 T^{11} + \cdots - 11634088 \)
|
| $59$ |
\( T^{12} + \cdots - 829953856 \)
|
| $61$ |
\( T^{12} - 15 T^{11} + \cdots + 7980476 \)
|
| $67$ |
\( T^{12} + \cdots + 618373159 \)
|
| $71$ |
\( T^{12} + 16 T^{11} + \cdots + 56321872 \)
|
| $73$ |
\( T^{12} + \cdots - 7795895688 \)
|
| $79$ |
\( T^{12} + 35 T^{11} + \cdots - 5604472 \)
|
| $83$ |
\( T^{12} + \cdots + 24330511079 \)
|
| $89$ |
\( T^{12} - 18 T^{11} + \cdots - 249256 \)
|
| $97$ |
\( T^{12} + \cdots - 52272028928 \)
|
| show more | |
| show less |