Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,4,Mod(271,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.271");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.e (of order \(3\), degree \(2\), 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: | \(47.7915471046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{401})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 101x^{2} + 100x + 10000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 270) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 101x^{2} + 100x + 10000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 101\nu^{2} - 101\nu + 10000 ) / 10100 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 101\nu^{2} + 30401\nu - 10000 ) / 10100 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 502 ) / 101 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 301\beta _1 - 302 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 101\beta_{3} - 502 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 271.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.50000 | + | 4.33013i | 0 | −15.2687 | − | 26.4462i | 8.00000 | 0 | 10.0000 | ||||||||||||||||||||||
| 271.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.50000 | + | 4.33013i | 0 | 14.7687 | + | 25.5802i | 8.00000 | 0 | 10.0000 | |||||||||||||||||||||||
| 541.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.50000 | − | 4.33013i | 0 | −15.2687 | + | 26.4462i | 8.00000 | 0 | 10.0000 | |||||||||||||||||||||||
| 541.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.50000 | − | 4.33013i | 0 | 14.7687 | − | 25.5802i | 8.00000 | 0 | 10.0000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 810.4.e.z | 4 | |
| 3.b | odd | 2 | 1 | 810.4.e.bd | 4 | ||
| 9.c | even | 3 | 1 | 270.4.a.n | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 810.4.e.z | 4 | |
| 9.d | odd | 6 | 1 | 270.4.a.m | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 810.4.e.bd | 4 | ||
| 36.f | odd | 6 | 1 | 2160.4.a.bb | 2 | ||
| 36.h | even | 6 | 1 | 2160.4.a.w | 2 | ||
| 45.h | odd | 6 | 1 | 1350.4.a.bm | 2 | ||
| 45.j | even | 6 | 1 | 1350.4.a.bf | 2 | ||
| 45.k | odd | 12 | 2 | 1350.4.c.bb | 4 | ||
| 45.l | even | 12 | 2 | 1350.4.c.u | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.4.a.m | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 270.4.a.n | yes | 2 | 9.c | even | 3 | 1 | |
| 810.4.e.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 810.4.e.z | 4 | 9.c | even | 3 | 1 | inner | |
| 810.4.e.bd | 4 | 3.b | odd | 2 | 1 | ||
| 810.4.e.bd | 4 | 9.d | odd | 6 | 1 | ||
| 1350.4.a.bf | 2 | 45.j | even | 6 | 1 | ||
| 1350.4.a.bm | 2 | 45.h | odd | 6 | 1 | ||
| 1350.4.c.u | 4 | 45.l | even | 12 | 2 | ||
| 1350.4.c.bb | 4 | 45.k | odd | 12 | 2 | ||
| 2160.4.a.w | 2 | 36.h | even | 6 | 1 | ||
| 2160.4.a.bb | 2 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):
|
\( T_{7}^{4} + T_{7}^{3} + 903T_{7}^{2} - 902T_{7} + 813604 \)
|
|
\( T_{11}^{4} + 33T_{11}^{3} + 1719T_{11}^{2} - 20790T_{11} + 396900 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} + T^{3} + \cdots + 813604 \)
|
| $11$ |
\( T^{4} + 33 T^{3} + \cdots + 396900 \)
|
| $13$ |
\( T^{4} + 64 T^{3} + \cdots + 6682225 \)
|
| $17$ |
\( (T^{2} - 51 T - 252)^{2} \)
|
| $19$ |
\( (T^{2} - 115 T + 2404)^{2} \)
|
| $23$ |
\( T^{4} + 21 T^{3} + \cdots + 503822916 \)
|
| $29$ |
\( T^{4} + \cdots + 1867795524 \)
|
| $31$ |
\( T^{4} + 295 T^{3} + \cdots + 640000 \)
|
| $37$ |
\( (T^{2} - 139 T - 3290)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 4978713600 \)
|
| $43$ |
\( T^{4} + 289 T^{3} + \cdots + 2808976 \)
|
| $47$ |
\( T^{4} + \cdots + 20923043904 \)
|
| $53$ |
\( (T^{2} + 366 T + 29880)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 109979783424 \)
|
| $61$ |
\( T^{4} + 85 T^{3} + \cdots + 39866596 \)
|
| $67$ |
\( T^{4} + \cdots + 54960238096 \)
|
| $71$ |
\( (T^{2} + 1494 T + 525528)^{2} \)
|
| $73$ |
\( (T^{2} + 1859 T + 855850)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 134647165249 \)
|
| $83$ |
\( T^{4} + \cdots + 172859703696 \)
|
| $89$ |
\( (T^{2} + 684 T - 1052352)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 20480472100 \)
|
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