Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,6,Mod(1,688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(688, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("688.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 688.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: | \(110.344068031\) |
| 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: |
\( x^{6} - x^{5} - 1079x^{4} - 228x^{3} + 318635x^{2} + 528465x - 17569161 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 86) |
| 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
\( x^{6} - x^{5} - 1079x^{4} - 228x^{3} + 318635x^{2} + 528465x - 17569161 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1033\nu^{5} - 37190\nu^{4} + 727397\nu^{3} + 24145911\nu^{2} - 78507122\nu - 2101202871 ) / 7857888 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2315\nu^{5} + 73202\nu^{4} - 793375\nu^{3} - 48607253\nu^{2} - 217742410\nu + 4039464597 ) / 15715776 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3915\nu^{5} + 44594\nu^{4} - 2665503\nu^{3} - 27428629\nu^{2} + 282211382\nu + 1851386901 ) / 15715776 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2421\nu^{5} + 51662\nu^{4} - 1395425\nu^{3} - 30774635\nu^{2} + 42420106\nu + 1718452267 ) / 5238592 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 + 361 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} - 5\beta_{4} + 16\beta_{3} + 19\beta_{2} + 477\beta _1 + 573 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1333\beta_{5} - 1498\beta_{4} - 1315\beta_{3} + 374\beta_{2} + 1623\beta _1 + 177203 \)
|
| \(\nu^{5}\) | \(=\) |
\( 871\beta_{5} + 3661\beta_{4} + 11860\beta_{3} + 15682\beta_{2} + 248203\beta _1 + 427967 \)
|
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 | −27.4353 | 0 | −11.3937 | 0 | 103.461 | 0 | 509.697 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −21.0551 | 0 | 78.6515 | 0 | −39.2999 | 0 | 200.318 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −13.2044 | 0 | −16.0709 | 0 | −171.726 | 0 | −68.6432 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 4.23106 | 0 | 92.9169 | 0 | −41.4859 | 0 | −225.098 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 20.0483 | 0 | 14.4756 | 0 | 94.4475 | 0 | 158.936 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 20.4155 | 0 | −97.5795 | 0 | −155.397 | 0 | 173.791 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 688.6.a.d | 6 | |
| 4.b | odd | 2 | 1 | 86.6.a.d | ✓ | 6 | |
| 12.b | even | 2 | 1 | 774.6.a.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 86.6.a.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 688.6.a.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 774.6.a.q | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 17T_{3}^{5} - 959T_{3}^{4} - 12726T_{3}^{3} + 259262T_{3}^{2} + 2318640T_{3} - 13209120 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 17 T^{5} + \cdots - 13209120 \)
|
| $5$ |
\( T^{6} + \cdots - 1890167832 \)
|
| $7$ |
\( T^{6} + \cdots + 425144822144 \)
|
| $11$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots - 83103173717352 \)
|
| $17$ |
\( T^{6} + \cdots - 31\!\cdots\!30 \)
|
| $19$ |
\( T^{6} + \cdots + 39\!\cdots\!96 \)
|
| $23$ |
\( T^{6} + \cdots + 55\!\cdots\!80 \)
|
| $29$ |
\( T^{6} + \cdots + 98\!\cdots\!72 \)
|
| $31$ |
\( T^{6} + \cdots + 19\!\cdots\!76 \)
|
| $37$ |
\( T^{6} + \cdots - 93\!\cdots\!68 \)
|
| $41$ |
\( T^{6} + \cdots - 19\!\cdots\!70 \)
|
| $43$ |
\( (T - 1849)^{6} \)
|
| $47$ |
\( T^{6} + \cdots - 64\!\cdots\!44 \)
|
| $53$ |
\( T^{6} + \cdots + 13\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots - 16\!\cdots\!12 \)
|
| $61$ |
\( T^{6} + \cdots + 15\!\cdots\!56 \)
|
| $67$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!88 \)
|
| $73$ |
\( T^{6} + \cdots - 29\!\cdots\!92 \)
|
| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!92 \)
|
| $83$ |
\( T^{6} + \cdots - 13\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots - 90\!\cdots\!92 \)
|
| $97$ |
\( T^{6} + \cdots - 18\!\cdots\!62 \)
|
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