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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 1072x^{4} - 8618x^{3} + 211383x^{2} + 2371478x + 2294784 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 1072x^{4} - 8618x^{3} + 211383x^{2} + 2371478x + 2294784 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7545\nu^{5} - 378847\nu^{4} - 1508733\nu^{3} + 205092737\nu^{2} + 104079652\nu - 18153623446 ) / 222999214 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -21440\nu^{5} + 263751\nu^{4} + 20690762\nu^{3} - 80862045\nu^{2} - 4189891210\nu - 1435559084 ) / 222999214 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 129956 \nu^{5} - 2077145 \nu^{4} - 107649534 \nu^{3} + 497728421 \nu^{2} + 20870182096 \nu + 45290232432 ) / 668997642 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 294046 \nu^{5} + 5291875 \nu^{4} + 221592654 \nu^{3} - 1470082957 \nu^{2} + \cdots - 27998542464 ) / 668997642 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} - 7\beta_{4} - 5\beta_{3} + 13\beta _1 + 358 \)
|
| \(\nu^{3}\) | \(=\) |
\( -71\beta_{5} - 166\beta_{4} - 27\beta_{3} - 46\beta_{2} + 737\beta _1 + 4348 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2668\beta_{5} - 7673\beta_{4} - 3907\beta_{3} - 1708\beta_{2} + 18111\beta _1 + 243598 \)
|
| \(\nu^{5}\) | \(=\) |
\( -93797\beta_{5} - 228190\beta_{4} - 65663\beta_{3} - 65404\beta_{2} + 689589\beta _1 + 5775582 \)
|
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 | −32.6771 | 8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −5.00000 | 0 | −16.6013 | 8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | −5.00000 | 0 | 1.07447 | 8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | −5.00000 | 0 | 12.1517 | 8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 0 | 4.00000 | −5.00000 | 0 | 17.0489 | 8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 0 | 4.00000 | −5.00000 | 0 | 19.0032 | 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.bp | yes | 6 |
| 3.b | odd | 2 | 1 | 2070.4.a.bo | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2070.4.a.bo | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2070.4.a.bp | yes | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):
|
\( T_{7}^{6} - 1072T_{7}^{4} + 8618T_{7}^{3} + 211383T_{7}^{2} - 2371478T_{7} + 2294784 \)
|
|
\( T_{11}^{6} - 12T_{11}^{5} - 5410T_{11}^{4} + 14068T_{11}^{3} + 6168564T_{11}^{2} + 81399056T_{11} + 276653568 \)
|
|
\( T_{17}^{6} - 34T_{17}^{5} - 11940T_{17}^{4} + 227556T_{17}^{3} + 31224903T_{17}^{2} + 179276382T_{17} - 8465083848 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} - 1072 T^{4} + \cdots + 2294784 \)
|
| $11$ |
\( T^{6} - 12 T^{5} + \cdots + 276653568 \)
|
| $13$ |
\( T^{6} + \cdots + 7334377632 \)
|
| $17$ |
\( T^{6} + \cdots - 8465083848 \)
|
| $19$ |
\( T^{6} - 12 T^{5} + \cdots + 126998400 \)
|
| $23$ |
\( (T - 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 8905896944352 \)
|
| $31$ |
\( T^{6} + \cdots - 37033343884032 \)
|
| $37$ |
\( T^{6} + \cdots + 680528044128 \)
|
| $41$ |
\( T^{6} + \cdots + 9767141654400 \)
|
| $43$ |
\( T^{6} + \cdots - 46750417352832 \)
|
| $47$ |
\( T^{6} + \cdots + 11685801139200 \)
|
| $53$ |
\( T^{6} + \cdots + 912748668615684 \)
|
| $59$ |
\( T^{6} + \cdots + 634014669883500 \)
|
| $61$ |
\( T^{6} + \cdots + 51585265246944 \)
|
| $67$ |
\( T^{6} + \cdots + 32\!\cdots\!88 \)
|
| $71$ |
\( T^{6} + \cdots - 11\!\cdots\!40 \)
|
| $73$ |
\( T^{6} + \cdots - 10770672339888 \)
|
| $79$ |
\( T^{6} + \cdots - 31\!\cdots\!48 \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 23\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots + 45\!\cdots\!84 \)
|
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