Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 79x^{4} + 3x^{3} + 1374x^{2} + 154x - 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 79x^{4} + 3x^{3} + 1374x^{2} + 154x - 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 2\nu^{4} + 123\nu^{3} + 91\nu^{2} - 2851\nu - 407 ) / 275 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{5} - 6\nu^{4} - 136\nu^{3} + 168\nu^{2} + 2337\nu + 99 ) / 275 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{5} - 29\nu^{4} - 489\nu^{3} + 847\nu^{2} + 6153\nu + 176 ) / 275 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} - 16\nu^{4} - 531\nu^{3} + 688\nu^{2} + 9037\nu - 6446 ) / 275 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\nu^{5} + 2\nu^{4} - 1363\nu^{3} - 681\nu^{2} + 24796\nu + 2717 ) / 275 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 9\beta_{2} + 2\beta _1 + 1 ) / 11 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{4} - 6\beta_{3} - 6\beta_{2} + 17\beta _1 + 289 ) / 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{5} + 11\beta_{4} - 84\beta_{3} + 312\beta_{2} + 205\beta _1 + 394 ) / 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 121\beta_{5} + 506\beta_{4} - 461\beta_{3} - 109\beta_{2} + 1572\beta _1 + 13326 ) / 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1111\beta_{5} + 1342\beta_{4} - 4254\beta_{3} + 12389\beta_{2} + 14891\beta _1 + 40781 ) / 11 \)
|
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 |
|
−2.00000 | −9.34974 | 4.00000 | −5.00000 | 18.6995 | 33.2315 | −8.00000 | 60.4176 | 10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −4.46804 | 4.00000 | −5.00000 | 8.93608 | 1.96493 | −8.00000 | −7.03663 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.59586 | 4.00000 | −5.00000 | 7.19172 | −26.1921 | −8.00000 | −14.0698 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 1.48232 | 4.00000 | −5.00000 | −2.96465 | −13.5283 | −8.00000 | −24.8027 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 4.85543 | 4.00000 | −5.00000 | −9.71085 | 6.57864 | −8.00000 | −3.42483 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 6.07588 | 4.00000 | −5.00000 | −12.1518 | 22.9453 | −8.00000 | 9.91636 | 10.0000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 1210.4.a.bc | 6 | |
| 11.b | odd | 2 | 1 | 1210.4.a.bf | 6 | ||
| 11.d | odd | 10 | 2 | 110.4.g.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.b | ✓ | 12 | 11.d | odd | 10 | 2 | |
| 1210.4.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 1210.4.a.bf | 6 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{6} + 5T_{3}^{5} - 79T_{3}^{4} - 233T_{3}^{3} + 1554T_{3}^{2} + 2866T_{3} - 6569 \)
|
|
\( T_{7}^{6} - 25T_{7}^{5} - 961T_{7}^{4} + 19691T_{7}^{3} + 167079T_{7}^{2} - 2174128T_{7} + 3492544 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} + 5 T^{5} + \cdots - 6569 \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} - 25 T^{5} + \cdots + 3492544 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 1078807280 \)
|
| $17$ |
\( T^{6} + \cdots - 1220468705 \)
|
| $19$ |
\( T^{6} + \cdots + 42946896025 \)
|
| $23$ |
\( T^{6} + \cdots - 1170533509004 \)
|
| $29$ |
\( T^{6} + \cdots - 133761010900 \)
|
| $31$ |
\( T^{6} + \cdots - 764843212820 \)
|
| $37$ |
\( T^{6} + \cdots - 69437022929756 \)
|
| $41$ |
\( T^{6} + \cdots + 23833122391919 \)
|
| $43$ |
\( T^{6} + \cdots - 10\!\cdots\!05 \)
|
| $47$ |
\( T^{6} + \cdots - 336701004980 \)
|
| $53$ |
\( T^{6} + \cdots + 209095950576524 \)
|
| $59$ |
\( T^{6} + \cdots - 605963738407225 \)
|
| $61$ |
\( T^{6} + \cdots + 16\!\cdots\!80 \)
|
| $67$ |
\( T^{6} + \cdots - 55916232721561 \)
|
| $71$ |
\( T^{6} + \cdots - 58\!\cdots\!36 \)
|
| $73$ |
\( T^{6} + \cdots - 99789845937139 \)
|
| $79$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!05 \)
|
| $89$ |
\( T^{6} + \cdots + 89\!\cdots\!75 \)
|
| $97$ |
\( T^{6} + \cdots - 18\!\cdots\!79 \)
|
| show more | |
| show less |