Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("688.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(40.5933140839\) |
| 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: |
\( x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 43) |
| 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
\( x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 24\nu^{3} - 36\nu^{2} + 103\nu + 30 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 3\nu^{3} - 17\nu^{2} - 51\nu - 16 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 3\nu^{3} + 17\nu^{2} + 59\nu + 16 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 17\nu - 19 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 26\nu^{3} + 6\nu^{2} - 153\nu - 74 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 + 22 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 2\beta_{4} + 15\beta_{3} + 17\beta_{2} - 4\beta _1 + 32 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{5} + 7\beta_{4} + 10\beta_{3} + 2\beta_{2} + 20\beta _1 + 179 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -56\beta_{5} + 64\beta_{4} + 249\beta_{3} + 289\beta_{2} - 80\beta _1 + 800 ) / 4 \)
|
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 | −9.49653 | 0 | 2.98245 | 0 | 26.0720 | 0 | 63.1842 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.20925 | 0 | 1.36370 | 0 | −13.0131 | 0 | 24.9733 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.46717 | 0 | −7.54340 | 0 | −4.58222 | 0 | −20.9131 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.43046 | 0 | 20.4116 | 0 | −29.9522 | 0 | −24.9538 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 6.49933 | 0 | 17.2665 | 0 | 23.3206 | 0 | 15.2413 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.10409 | 0 | 8.51910 | 0 | −9.84502 | 0 | 23.4681 | 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 | 688.4.a.i | 6 | |
| 4.b | odd | 2 | 1 | 43.4.a.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 387.4.a.h | 6 | ||
| 20.d | odd | 2 | 1 | 1075.4.a.b | 6 | ||
| 28.d | even | 2 | 1 | 2107.4.a.c | 6 | ||
| 172.d | even | 2 | 1 | 1849.4.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.4.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 387.4.a.h | 6 | 12.b | even | 2 | 1 | ||
| 688.4.a.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 1075.4.a.b | 6 | 20.d | odd | 2 | 1 | ||
| 1849.4.a.c | 6 | 172.d | even | 2 | 1 | ||
| 2107.4.a.c | 6 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 7T_{3}^{5} - 97T_{3}^{4} - 588T_{3}^{3} + 2140T_{3}^{2} + 11756T_{3} + 11156 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(688))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 7 T^{5} + \cdots + 11156 \)
|
| $5$ |
\( T^{6} - 43 T^{5} + \cdots - 92116 \)
|
| $7$ |
\( T^{6} + 8 T^{5} + \cdots + 10690992 \)
|
| $11$ |
\( T^{6} - 28 T^{5} + \cdots - 53187648 \)
|
| $13$ |
\( T^{6} - 56 T^{5} + \cdots - 458957340 \)
|
| $17$ |
\( T^{6} - 19 T^{5} + \cdots + 181639863 \)
|
| $19$ |
\( T^{6} - 75 T^{5} + \cdots - 673814000 \)
|
| $23$ |
\( T^{6} + \cdots - 9170218345 \)
|
| $29$ |
\( T^{6} + \cdots + 331483322700 \)
|
| $31$ |
\( T^{6} + \cdots + 8546933895145 \)
|
| $37$ |
\( T^{6} + \cdots - 15081152424000 \)
|
| $41$ |
\( T^{6} + \cdots + 84893308383715 \)
|
| $43$ |
\( (T - 43)^{6} \)
|
| $47$ |
\( T^{6} + \cdots - 68794630166960 \)
|
| $53$ |
\( T^{6} + \cdots - 27\!\cdots\!40 \)
|
| $59$ |
\( T^{6} + \cdots - 18355561214400 \)
|
| $61$ |
\( T^{6} + \cdots + 51013136843120 \)
|
| $67$ |
\( T^{6} + \cdots + 119387526352596 \)
|
| $71$ |
\( T^{6} + \cdots - 11\!\cdots\!32 \)
|
| $73$ |
\( T^{6} + \cdots + 75\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots - 24\!\cdots\!60 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots - 17\!\cdots\!80 \)
|
| $97$ |
\( T^{6} + \cdots + 16\!\cdots\!39 \)
|
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