Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8967,2,Mod(1,8967)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8967, 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("8967.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8967 = 3 \cdot 7^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8967.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: | \(71.6018554925\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 13x^{8} - 2x^{7} + 54x^{6} + 14x^{5} - 81x^{4} - 26x^{3} + 30x^{2} + 12x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 13x^{8} - 2x^{7} + 54x^{6} + 14x^{5} - 81x^{4} - 26x^{3} + 30x^{2} + 12x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{9} - 13\nu^{7} - 2\nu^{6} + 54\nu^{5} + 14\nu^{4} - 81\nu^{3} - 26\nu^{2} + 30\nu + 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{8} + 4\nu^{7} + 19\nu^{6} - 36\nu^{5} - 45\nu^{4} + 80\nu^{3} + 20\nu^{2} - 29\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{9} - 4\nu^{8} - 6\nu^{7} + 38\nu^{6} - 9\nu^{5} - 94\nu^{4} + 62\nu^{3} + 53\nu^{2} - 31\nu - 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{9} - 2\nu^{8} - 23\nu^{7} + 17\nu^{6} + 81\nu^{5} - 35\nu^{4} - 100\nu^{3} + 8\nu^{2} + 33\nu + 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{9} - \nu^{8} - 25\nu^{7} + 8\nu^{6} + 99\nu^{5} - 17\nu^{4} - 141\nu^{3} + 7\nu^{2} + 51\nu + 6 \)
|
| \(\beta_{8}\) | \(=\) |
\( -2\nu^{9} + \nu^{8} + 25\nu^{7} - 7\nu^{6} - 100\nu^{5} + 8\nu^{4} + 148\nu^{3} + 11\nu^{2} - 61\nu - 9 \)
|
| \(\beta_{9}\) | \(=\) |
\( 3\nu^{9} - 3\nu^{8} - 34\nu^{7} + 25\nu^{6} + 117\nu^{5} - 49\nu^{4} - 141\nu^{3} + 8\nu^{2} + 48\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + 8\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{8} + \beta_{7} - 11\beta_{6} + 9\beta_{5} - 8\beta_{4} + 9\beta_{3} + 9\beta_{2} + 28\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{9} + 11\beta_{7} - 13\beta_{6} + 11\beta_{5} - \beta_{4} + 20\beta_{3} + 56\beta_{2} + 12\beta _1 + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( -9\beta_{8} + 13\beta_{7} - 90\beta_{6} + 67\beta_{5} - 55\beta_{4} + 69\beta_{3} + 71\beta_{2} + 170\beta _1 + 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 63 \beta_{9} + 90 \beta_{7} - 123 \beta_{6} + 94 \beta_{5} - 16 \beta_{4} + 161 \beta_{3} + \cdots + 530 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4 \beta_{9} - 63 \beta_{8} + 123 \beta_{7} - 669 \beta_{6} + 474 \beta_{5} - 366 \beta_{4} + \cdots + 560 \)
|
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 |
|
−2.66980 | 1.00000 | 5.12786 | 1.20297 | −2.66980 | 0 | −8.35076 | 1.00000 | −3.21169 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.95335 | 1.00000 | 1.81556 | −0.0503500 | −1.95335 | 0 | 0.360269 | 1.00000 | 0.0983510 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.49569 | 1.00000 | 0.237093 | 3.25737 | −1.49569 | 0 | 2.63677 | 1.00000 | −4.87202 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.790167 | 1.00000 | −1.37564 | −3.25148 | −0.790167 | 0 | 2.66732 | 1.00000 | 2.56921 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.124881 | 1.00000 | −1.98440 | 1.38970 | 0.124881 | 0 | −0.497578 | 1.00000 | 0.173548 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.261602 | 1.00000 | −1.93156 | −2.84716 | 0.261602 | 0 | −1.02850 | 1.00000 | −0.744821 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.717127 | 1.00000 | −1.48573 | −0.100922 | 0.717127 | 0 | −2.49971 | 1.00000 | −0.0723736 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.68282 | 1.00000 | 0.831883 | −1.57046 | 1.68282 | 0 | −1.96573 | 1.00000 | −2.64280 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.69584 | 1.00000 | 0.875871 | 2.84428 | 1.69584 | 0 | −1.90634 | 1.00000 | 4.82343 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.42674 | 1.00000 | 3.88907 | −0.873946 | 2.42674 | 0 | 4.58428 | 1.00000 | −2.12084 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(61\) | \( +1 \) |
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 | 8967.2.a.bb | yes | 10 |
| 7.b | odd | 2 | 1 | 8967.2.a.ba | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8967.2.a.ba | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 8967.2.a.bb | yes | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8967))\):
|
\( T_{2}^{10} - 13T_{2}^{8} + 2T_{2}^{7} + 54T_{2}^{6} - 14T_{2}^{5} - 81T_{2}^{4} + 26T_{2}^{3} + 30T_{2}^{2} - 12T_{2} + 1 \)
|
|
\( T_{5}^{10} - 22T_{5}^{8} + 150T_{5}^{6} - 328T_{5}^{4} - 4T_{5}^{3} + 202T_{5}^{2} + 30T_{5} + 1 \)
|
|
\( T_{11}^{10} + 4 T_{11}^{9} - 29 T_{11}^{8} - 108 T_{11}^{7} + 295 T_{11}^{6} + 966 T_{11}^{5} + \cdots - 1860 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 13 T^{8} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} - 22 T^{8} + \cdots + 1 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + 4 T^{9} + \cdots - 1860 \)
|
| $13$ |
\( T^{10} + 8 T^{9} + \cdots - 16500 \)
|
| $17$ |
\( T^{10} - 8 T^{9} + \cdots + 440 \)
|
| $19$ |
\( T^{10} + 20 T^{9} + \cdots - 15 \)
|
| $23$ |
\( T^{10} + 14 T^{9} + \cdots + 22036 \)
|
| $29$ |
\( T^{10} + 10 T^{9} + \cdots + 525 \)
|
| $31$ |
\( T^{10} + 8 T^{9} + \cdots - 109760 \)
|
| $37$ |
\( T^{10} - 80 T^{8} + \cdots - 100 \)
|
| $41$ |
\( T^{10} + 8 T^{9} + \cdots - 460536 \)
|
| $43$ |
\( T^{10} + 6 T^{9} + \cdots + 31495785 \)
|
| $47$ |
\( T^{10} - 6 T^{9} + \cdots - 212304 \)
|
| $53$ |
\( T^{10} + 24 T^{9} + \cdots - 2816863 \)
|
| $59$ |
\( T^{10} + 10 T^{9} + \cdots - 791451 \)
|
| $61$ |
\( (T + 1)^{10} \)
|
| $67$ |
\( T^{10} + 20 T^{9} + \cdots + 857 \)
|
| $71$ |
\( T^{10} + 24 T^{9} + \cdots - 4467652 \)
|
| $73$ |
\( T^{10} + \cdots - 139752992 \)
|
| $79$ |
\( T^{10} + 14 T^{9} + \cdots + 12681917 \)
|
| $83$ |
\( T^{10} + \cdots - 216202800 \)
|
| $89$ |
\( T^{10} + \cdots + 330685440 \)
|
| $97$ |
\( T^{10} + \cdots - 121387784 \)
|
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