Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4368,2,Mod(337,4368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4368.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4368.h (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(34.8786556029\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 2184) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5\nu^{9} - 89\nu^{7} - 296\nu^{5} + 632\nu^{3} + 1004\nu ) / 392 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{8} - 89\nu^{6} - 296\nu^{4} + 632\nu^{2} + 808 ) / 196 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{8} - 81\nu^{6} - 423\nu^{4} - 396\nu^{2} + 98\nu - 20 ) / 98 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{8} + 81\nu^{6} + 423\nu^{4} + 396\nu^{2} + 98\nu + 20 ) / 98 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\nu^{8} + 243\nu^{6} + 1318\nu^{4} + 1727\nu^{2} + 501 ) / 49 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 27\nu^{9} + 559\nu^{7} + 3186\nu^{5} + 4878\nu^{3} + 2340\nu ) / 196 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -65\nu^{8} - 1353\nu^{6} - 7768\nu^{4} - 11580\nu^{2} - 3020 ) / 196 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 65\nu^{9} + 1353\nu^{7} + 7768\nu^{5} + 11580\nu^{3} + 2628\nu ) / 392 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -289\nu^{9} - 5889\nu^{7} - 32240\nu^{5} - 42380\nu^{3} - 11588\nu ) / 392 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{9} - \beta_{8} + 6\beta_{6} - 10\beta_{4} - 10\beta_{3} - 6\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11\beta_{7} - 9\beta_{5} - 28\beta_{4} + 28\beta_{3} - 33\beta_{2} + 70 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -19\beta_{9} + 3\beta_{8} - 94\beta_{6} + 114\beta_{4} + 114\beta_{3} + 122\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 117\beta_{7} + 79\beta_{5} + 358\beta_{4} - 358\beta_{3} + 383\beta_{2} - 730 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 279\beta_{9} + 45\beta_{8} + 1274\beta_{6} - 1350\beta_{4} - 1350\beta_{3} - 1782\beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -1305\beta_{7} - 747\beta_{5} - 4462\beta_{4} + 4462\beta_{3} - 4563\beta_{2} + 8162 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -3715\beta_{9} - 1105\beta_{8} - 16354\beta_{6} + 16218\beta_{4} + 16218\beta_{3} + 23582\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).
| \(n\) | \(1093\) | \(1249\) | \(1457\) | \(2017\) | \(3823\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −1.00000 | 0 | − | 4.39335i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 337.2 | 0 | −1.00000 | 0 | − | 2.46162i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.3 | 0 | −1.00000 | 0 | − | 1.64044i | 0 | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.4 | 0 | −1.00000 | 0 | − | 0.787293i | 0 | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.5 | 0 | −1.00000 | 0 | − | 0.572766i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.6 | 0 | −1.00000 | 0 | 0.572766i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 337.7 | 0 | −1.00000 | 0 | 0.787293i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.8 | 0 | −1.00000 | 0 | 1.64044i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.9 | 0 | −1.00000 | 0 | 2.46162i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 337.10 | 0 | −1.00000 | 0 | 4.39335i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4368.2.h.s | 10 | |
| 4.b | odd | 2 | 1 | 2184.2.h.g | ✓ | 10 | |
| 13.b | even | 2 | 1 | inner | 4368.2.h.s | 10 | |
| 52.b | odd | 2 | 1 | 2184.2.h.g | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2184.2.h.g | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 2184.2.h.g | ✓ | 10 | 52.b | odd | 2 | 1 | |
| 4368.2.h.s | 10 | 1.a | even | 1 | 1 | trivial | |
| 4368.2.h.s | 10 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):
|
\( T_{5}^{10} + 29T_{5}^{8} + 212T_{5}^{6} + 496T_{5}^{4} + 336T_{5}^{2} + 64 \)
|
|
\( T_{11}^{10} + 73T_{11}^{8} + 1576T_{11}^{6} + 11200T_{11}^{4} + 12944T_{11}^{2} + 4096 \)
|
|
\( T_{17}^{5} + 11T_{17}^{4} + 18T_{17}^{3} - 96T_{17}^{2} - 128T_{17} + 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} + 29 T^{8} + \cdots + 64 \)
|
| $7$ |
\( (T^{2} + 1)^{5} \)
|
| $11$ |
\( T^{10} + 73 T^{8} + \cdots + 4096 \)
|
| $13$ |
\( T^{10} - 23 T^{8} + \cdots + 371293 \)
|
| $17$ |
\( (T^{5} + 11 T^{4} + \cdots + 256)^{2} \)
|
| $19$ |
\( T^{10} + 177 T^{8} + \cdots + 39337984 \)
|
| $23$ |
\( (T^{5} - 9 T^{4} - 8 T^{3} + \cdots - 16)^{2} \)
|
| $29$ |
\( (T^{5} + T^{4} - 30 T^{3} + \cdots + 128)^{2} \)
|
| $31$ |
\( T^{10} \)
|
| $37$ |
\( T^{10} + 109 T^{8} + \cdots + 4096 \)
|
| $41$ |
\( T^{10} + 256 T^{8} + \cdots + 6553600 \)
|
| $43$ |
\( (T^{5} - T^{4} - 148 T^{3} + \cdots + 2048)^{2} \)
|
| $47$ |
\( T^{10} + 196 T^{8} + \cdots + 16777216 \)
|
| $53$ |
\( (T^{5} - 160 T^{3} + \cdots - 9920)^{2} \)
|
| $59$ |
\( T^{10} + 316 T^{8} + \cdots + 2408704 \)
|
| $61$ |
\( (T^{5} - 3 T^{4} + \cdots - 4192)^{2} \)
|
| $67$ |
\( T^{10} + 420 T^{8} + \cdots + 262144 \)
|
| $71$ |
\( T^{10} + \cdots + 203689984 \)
|
| $73$ |
\( T^{10} + 277 T^{8} + \cdots + 50922496 \)
|
| $79$ |
\( (T^{5} + 20 T^{4} + \cdots + 5120)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 1774768384 \)
|
| $89$ |
\( T^{10} + 448 T^{8} + \cdots + 10240000 \)
|
| $97$ |
\( T^{10} + \cdots + 125440000 \)
|
| show more | |
| show less |