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: |
\( x^{5} - 2x^{4} - 614x^{3} - 950x^{2} + 96373x + 445660 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{5} - 2x^{4} - 614x^{3} - 950x^{2} + 96373x + 445660 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 21\nu^{3} - 323\nu^{2} + 5727\nu + 27520 ) / 162 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 21\nu^{3} + 323\nu^{2} - 5241\nu - 27844 ) / 162 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 21\nu^{3} + 377\nu^{2} - 5565\nu - 41020 ) / 162 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{4} - 87\nu^{3} - 1723\nu^{2} + 21813\nu + 143702 ) / 54 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{3} - \beta_{2} + 2\beta _1 + 248 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{4} + 54\beta_{3} + 361\beta_{2} + 280\beta _1 + 4205 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 63\beta_{4} + 1347\beta_{3} + 295\beta_{2} + 859\beta _1 + 78201 \)
|
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 | −27.4065 | 8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 5.00000 | 0 | −25.4629 | 8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 5.00000 | 0 | 12.2288 | 8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | 5.00000 | 0 | 20.2597 | 8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 0 | 4.00000 | 5.00000 | 0 | 32.3809 | 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.bn | yes | 5 |
| 3.b | odd | 2 | 1 | 2070.4.a.bk | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2070.4.a.bk | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 2070.4.a.bn | yes | 5 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):
|
\( T_{7}^{5} - 12T_{7}^{4} - 1432T_{7}^{3} + 15426T_{7}^{2} + 482895T_{7} - 5598450 \)
|
|
\( T_{11}^{5} - 82T_{11}^{4} - 1242T_{11}^{3} + 195544T_{11}^{2} - 1551940T_{11} - 60466056 \)
|
|
\( T_{17}^{5} + 10T_{17}^{4} - 8772T_{17}^{3} - 18324T_{17}^{2} + 13219119T_{17} - 131982534 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 12 T^{4} + \cdots - 5598450 \)
|
| $11$ |
\( T^{5} - 82 T^{4} + \cdots - 60466056 \)
|
| $13$ |
\( T^{5} - 16 T^{4} + \cdots - 1445040 \)
|
| $17$ |
\( T^{5} + 10 T^{4} + \cdots - 131982534 \)
|
| $19$ |
\( T^{5} + 28 T^{4} + \cdots - 27245616 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 4669756650 \)
|
| $31$ |
\( T^{5} + \cdots - 10603852236 \)
|
| $37$ |
\( T^{5} + \cdots + 75662899224 \)
|
| $41$ |
\( T^{5} + \cdots + 283942799712 \)
|
| $43$ |
\( T^{5} + \cdots + 1188951433088 \)
|
| $47$ |
\( T^{5} + \cdots + 6550479515232 \)
|
| $53$ |
\( T^{5} + \cdots - 35605377198 \)
|
| $59$ |
\( T^{5} + \cdots + 30370060939950 \)
|
| $61$ |
\( T^{5} + \cdots + 11598183455800 \)
|
| $67$ |
\( T^{5} + \cdots + 2796458822820 \)
|
| $71$ |
\( T^{5} + \cdots + 2572675374564 \)
|
| $73$ |
\( T^{5} + \cdots - 31820601629544 \)
|
| $79$ |
\( T^{5} + \cdots - 93307854283776 \)
|
| $83$ |
\( T^{5} + \cdots + 52737206614812 \)
|
| $89$ |
\( T^{5} + \cdots - 68388201504000 \)
|
| $97$ |
\( T^{5} + \cdots + 58956893521440 \)
|
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