Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1040,6,Mod(1,1040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1040.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.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: | \(166.799172605\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 1384x^{6} + 3672x^{5} + 603912x^{4} - 998448x^{3} - 83285728x^{2} + 113377312x + 442451152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{15} \) |
| Twist minimal: | no (minimal twist has level 520) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 1384x^{6} + 3672x^{5} + 603912x^{4} - 998448x^{3} - 83285728x^{2} + 113377312x + 442451152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 347 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39865 \nu^{7} + 710912 \nu^{6} + 27403010 \nu^{5} - 565208900 \nu^{4} + \cdots + 5128090657664 ) / 51083220240 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8399 \nu^{7} - 219218 \nu^{6} + 14013580 \nu^{5} + 203951132 \nu^{4} - 7444901996 \nu^{3} + \cdots - 1473405016616 ) / 8513870040 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3688 \nu^{7} + 135573 \nu^{6} + 4144780 \nu^{5} - 144069386 \nu^{4} - 1529601584 \nu^{3} + \cdots - 1169311008944 ) / 2837956680 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7623 \nu^{7} + 93842 \nu^{6} - 14680490 \nu^{5} - 127046544 \nu^{4} + 8825418844 \nu^{3} + \cdots - 646256969696 ) / 5675913360 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 247556 \nu^{7} + 3807775 \nu^{6} + 264708040 \nu^{5} - 3592250842 \nu^{4} + \cdots - 5263998783680 ) / 25541610120 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 347 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + \beta_{6} - 5\beta_{5} - 3\beta_{4} - 11\beta_{3} + 4\beta_{2} + 520\beta _1 + 439 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 74\beta_{6} - 56\beta_{5} - 102\beta_{4} + 70\beta_{3} + 678\beta_{2} + 2302\beta _1 + 179500 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1956 \beta_{7} + 252 \beta_{6} - 4488 \beta_{5} - 3384 \beta_{4} - 12108 \beta_{3} + 3964 \beta_{2} + \cdots + 588004 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3956 \beta_{7} - 79892 \beta_{6} - 45560 \beta_{5} - 125724 \beta_{4} + 48172 \beta_{3} + \cdots + 102181380 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1436056 \beta_{7} - 177652 \beta_{6} - 3226828 \beta_{5} - 3195996 \beta_{4} - 10009060 \beta_{3} + \cdots + 546197580 \)
|
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 |
|
0 | −22.7009 | 0 | 25.0000 | 0 | 147.624 | 0 | 272.331 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −18.3575 | 0 | 25.0000 | 0 | −148.402 | 0 | 93.9965 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −11.6369 | 0 | 25.0000 | 0 | 99.8440 | 0 | −107.583 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.846664 | 0 | 25.0000 | 0 | −16.6421 | 0 | −242.283 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.74972 | 0 | 25.0000 | 0 | −177.565 | 0 | −209.941 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 21.5972 | 0 | 25.0000 | 0 | 16.0114 | 0 | 223.439 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 23.6213 | 0 | 25.0000 | 0 | 249.629 | 0 | 314.966 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 28.8804 | 0 | 25.0000 | 0 | −250.500 | 0 | 591.075 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 1040.6.a.ba | 8 | |
| 4.b | odd | 2 | 1 | 520.6.a.d | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.6.a.d | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 1040.6.a.ba | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 28 T_{3}^{7} - 1048 T_{3}^{6} + 27304 T_{3}^{5} + 354152 T_{3}^{4} - 7501648 T_{3}^{3} + \cdots - 347819184 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 28 T^{7} + \cdots - 347819184 \)
|
| $5$ |
\( (T - 25)^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 64\!\cdots\!40 \)
|
| $11$ |
\( T^{8} + \cdots - 94\!\cdots\!72 \)
|
| $13$ |
\( (T + 169)^{8} \)
|
| $17$ |
\( T^{8} + \cdots - 15\!\cdots\!60 \)
|
| $19$ |
\( T^{8} + \cdots + 79\!\cdots\!16 \)
|
| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 56\!\cdots\!96 \)
|
| $31$ |
\( T^{8} + \cdots - 49\!\cdots\!80 \)
|
| $37$ |
\( T^{8} + \cdots + 13\!\cdots\!08 \)
|
| $41$ |
\( T^{8} + \cdots + 26\!\cdots\!04 \)
|
| $43$ |
\( T^{8} + \cdots + 23\!\cdots\!08 \)
|
| $47$ |
\( T^{8} + \cdots + 13\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots + 14\!\cdots\!64 \)
|
| $59$ |
\( T^{8} + \cdots - 44\!\cdots\!68 \)
|
| $61$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 13\!\cdots\!12 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!72 \)
|
| $73$ |
\( T^{8} + \cdots + 64\!\cdots\!24 \)
|
| $79$ |
\( T^{8} + \cdots - 98\!\cdots\!32 \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots + 85\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 95\!\cdots\!20 \)
|
| show more | |
| show less |