Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2070,4,Mod(1,2070)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2070.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2070.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: | \(122.133953712\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 345x^{3} + 1261x^{2} + 22016x - 83268 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 345x^{3} + 1261x^{2} + 22016x - 83268 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} + 131\nu^{3} - 653\nu^{2} - 22667\nu + 21672 ) / 3114 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} - \nu^{3} + 1768\nu^{2} - 1277\nu - 128196 ) / 3114 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{4} + 89\nu^{3} - 2344\nu^{2} - 9869\nu + 81750 ) / 1038 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 5\beta_{3} - 2\beta_{2} - 3\beta _1 + 141 \)
|
| \(\nu^{3}\) | \(=\) |
\( -12\beta_{4} - 30\beta_{3} + 48\beta_{2} + 223\beta _1 - 624 \)
|
| \(\nu^{4}\) | \(=\) |
\( 445\beta_{4} + 1439\beta_{3} - 896\beta_{2} - 1701\beta _1 + 30429 \)
|
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 | 0 | 4.00000 | 5.00000 | 0 | −24.8190 | −8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | −13.7810 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | −1.97872 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | 9.49877 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | 31.0799 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 2070.4.a.bl | ✓ | 5 |
| 3.b | odd | 2 | 1 | 2070.4.a.bm | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2070.4.a.bl | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2070.4.a.bm | yes | 5 | 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):
|
\( T_{7}^{5} - 933T_{7}^{3} - 4322T_{7}^{2} + 96060T_{7} + 199800 \)
|
|
\( T_{11}^{5} + 16T_{11}^{4} - 1578T_{11}^{3} - 3696T_{11}^{2} + 663336T_{11} - 5343840 \)
|
|
\( T_{17}^{5} - 56T_{17}^{4} - 13101T_{17}^{3} + 827150T_{17}^{2} + 27038204T_{17} - 1744652040 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 933 T^{3} + \cdots + 199800 \)
|
| $11$ |
\( T^{5} + 16 T^{4} + \cdots - 5343840 \)
|
| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 811296 \)
|
| $17$ |
\( T^{5} + \cdots - 1744652040 \)
|
| $19$ |
\( T^{5} + \cdots + 4338192384 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 4403148336 \)
|
| $31$ |
\( T^{5} + \cdots - 5598781824 \)
|
| $37$ |
\( T^{5} + \cdots + 843390184064 \)
|
| $41$ |
\( T^{5} + \cdots + 1892211840 \)
|
| $43$ |
\( T^{5} + \cdots + 379092659328 \)
|
| $47$ |
\( T^{5} + \cdots - 12506112000 \)
|
| $53$ |
\( T^{5} + \cdots - 43534649448 \)
|
| $59$ |
\( T^{5} + \cdots - 9991424770200 \)
|
| $61$ |
\( T^{5} + \cdots - 38169789552 \)
|
| $67$ |
\( T^{5} + \cdots + 980998631856 \)
|
| $71$ |
\( T^{5} + \cdots + 1168952858064 \)
|
| $73$ |
\( T^{5} + \cdots - 16564651440 \)
|
| $79$ |
\( T^{5} + \cdots - 1940078592 \)
|
| $83$ |
\( T^{5} + \cdots + 47036465266272 \)
|
| $89$ |
\( T^{5} + \cdots - 656771742201600 \)
|
| $97$ |
\( T^{5} + \cdots + 3891896438784 \)
|
| show more | |
| show less |