Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,4,Mod(449,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.449");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(47.2015280046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.2068430400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 23x^{4} + 133x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 23x^{4} + 133x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{3} + \nu ) / 30 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{3} + 61\nu ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} + 24\nu^{2} + 7 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} - 44\nu^{2} - 162 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 37\nu^{3} + 291\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - \beta_{3} - 31 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - 29\beta_{2} + 23\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{4} + 11\beta_{3} + 179 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -17\beta_{5} + 491\beta_{2} - 269\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 8.19070i | 0 | 0 | 0 | − | 21.7061i | 0 | −40.0875 | 0 | ||||||||||||||||||||||||||||||||||
| 449.2 | 0 | − | 5.25747i | 0 | 0 | 0 | 9.15590i | 0 | −0.640965 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.3 | 0 | − | 2.06677i | 0 | 0 | 0 | 28.8620i | 0 | 22.7285 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 2.06677i | 0 | 0 | 0 | − | 28.8620i | 0 | 22.7285 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.5 | 0 | 5.25747i | 0 | 0 | 0 | − | 9.15590i | 0 | −0.640965 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.6 | 0 | 8.19070i | 0 | 0 | 0 | 21.7061i | 0 | −40.0875 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 800.4.c.n | 6 | |
| 4.b | odd | 2 | 1 | 800.4.c.m | 6 | ||
| 5.b | even | 2 | 1 | inner | 800.4.c.n | 6 | |
| 5.c | odd | 4 | 1 | 800.4.a.v | yes | 3 | |
| 5.c | odd | 4 | 1 | 800.4.a.x | yes | 3 | |
| 20.d | odd | 2 | 1 | 800.4.c.m | 6 | ||
| 20.e | even | 4 | 1 | 800.4.a.u | ✓ | 3 | |
| 20.e | even | 4 | 1 | 800.4.a.w | yes | 3 | |
| 40.i | odd | 4 | 1 | 1600.4.a.cq | 3 | ||
| 40.i | odd | 4 | 1 | 1600.4.a.cs | 3 | ||
| 40.k | even | 4 | 1 | 1600.4.a.cr | 3 | ||
| 40.k | even | 4 | 1 | 1600.4.a.ct | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.4.a.u | ✓ | 3 | 20.e | even | 4 | 1 | |
| 800.4.a.v | yes | 3 | 5.c | odd | 4 | 1 | |
| 800.4.a.w | yes | 3 | 20.e | even | 4 | 1 | |
| 800.4.a.x | yes | 3 | 5.c | odd | 4 | 1 | |
| 800.4.c.m | 6 | 4.b | odd | 2 | 1 | ||
| 800.4.c.m | 6 | 20.d | odd | 2 | 1 | ||
| 800.4.c.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 800.4.c.n | 6 | 5.b | even | 2 | 1 | inner | |
| 1600.4.a.cq | 3 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.cr | 3 | 40.k | even | 4 | 1 | ||
| 1600.4.a.cs | 3 | 40.i | odd | 4 | 1 | ||
| 1600.4.a.ct | 3 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):
|
\( T_{3}^{6} + 99T_{3}^{4} + 2259T_{3}^{2} + 7921 \)
|
|
\( T_{11}^{3} - 31T_{11}^{2} - 485T_{11} + 8475 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 99 T^{4} + \cdots + 7921 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1388 T^{4} + \cdots + 32901696 \)
|
| $11$ |
\( (T^{3} - 31 T^{2} + \cdots + 8475)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 36864000000 \)
|
| $17$ |
\( T^{6} + \cdots + 240811025625 \)
|
| $19$ |
\( (T^{3} + 87 T^{2} + \cdots + 7925)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 4734540864 \)
|
| $29$ |
\( (T^{3} + 84 T^{2} + \cdots - 278528)^{2} \)
|
| $31$ |
\( (T^{3} - 294 T^{2} + \cdots + 1628600)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 110195406760000 \)
|
| $41$ |
\( (T^{3} + 345 T^{2} + \cdots - 14242325)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 84656487202816 \)
|
| $47$ |
\( T^{6} + \cdots + 130195037372416 \)
|
| $53$ |
\( T^{6} + \cdots + 5867560000 \)
|
| $59$ |
\( (T^{3} + 1040 T^{2} + \cdots - 140160000)^{2} \)
|
| $61$ |
\( (T^{3} - 482 T^{2} + \cdots + 80633480)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 58\!\cdots\!89 \)
|
| $71$ |
\( (T^{3} - 2048 T^{2} + \cdots - 220992000)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 355167004515625 \)
|
| $79$ |
\( (T^{3} + 998 T^{2} + \cdots - 361942200)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!25 \)
|
| $89$ |
\( (T^{3} + 1189 T^{2} + \cdots - 641970729)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
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