Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4100,2,Mod(1,4100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4100.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4100.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: | \(32.7386648287\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3356224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 8x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 820) |
| 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
\( x^{6} - 6x^{4} + 8x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 7\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + 9\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} + 5\beta_{4} + 2\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11\beta_{5} + 21\beta_{4} + 12\beta_{3} ) / 2 \)
|
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 | −2.30725 | 0 | 0 | 0 | 1.23992 | 0 | 2.32340 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.53555 | 0 | 0 | 0 | −0.726062 | 0 | −0.642074 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.564508 | 0 | 0 | 0 | 2.22158 | 0 | −2.68133 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.564508 | 0 | 0 | 0 | −2.22158 | 0 | −2.68133 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.53555 | 0 | 0 | 0 | 0.726062 | 0 | −0.642074 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.30725 | 0 | 0 | 0 | −1.23992 | 0 | 2.32340 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(41\) | \(1\) |
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 | 4100.2.a.f | 6 | |
| 5.b | even | 2 | 1 | inner | 4100.2.a.f | 6 | |
| 5.c | odd | 4 | 2 | 820.2.d.a | ✓ | 6 | |
| 15.e | even | 4 | 2 | 7380.2.f.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 820.2.d.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 4100.2.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 4100.2.a.f | 6 | 5.b | even | 2 | 1 | inner | |
| 7380.2.f.d | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 8T_{3}^{4} + 15T_{3}^{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 8 T^{4} + \cdots - 4 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 7 T^{4} + \cdots - 4 \)
|
| $11$ |
\( (T^{3} + 2 T^{2} - 5 T - 2)^{2} \)
|
| $13$ |
\( T^{6} - 27 T^{4} + \cdots - 16 \)
|
| $17$ |
\( T^{6} - 20 T^{4} + \cdots - 256 \)
|
| $19$ |
\( (T^{3} + 3 T^{2} - T - 2)^{2} \)
|
| $23$ |
\( T^{6} - 60 T^{4} + \cdots - 3136 \)
|
| $29$ |
\( (T^{3} + 7 T^{2} - 3 T - 58)^{2} \)
|
| $31$ |
\( (T^{3} + T^{2} - 27 T - 64)^{2} \)
|
| $37$ |
\( T^{6} - 27 T^{4} + \cdots - 16 \)
|
| $41$ |
\( (T + 1)^{6} \)
|
| $43$ |
\( T^{6} - 107 T^{4} + \cdots - 35344 \)
|
| $47$ |
\( T^{6} - 25 T^{4} + \cdots - 196 \)
|
| $53$ |
\( T^{6} - 96 T^{4} + \cdots - 4096 \)
|
| $59$ |
\( (T^{3} + 7 T^{2} + \cdots - 112)^{2} \)
|
| $61$ |
\( (T^{3} + 14 T^{2} + \cdots + 74)^{2} \)
|
| $67$ |
\( T^{6} - 27 T^{4} + \cdots - 196 \)
|
| $71$ |
\( (T^{3} - 4 T^{2} + \cdots + 478)^{2} \)
|
| $73$ |
\( T^{6} - 9 T^{4} + \cdots - 16 \)
|
| $79$ |
\( (T^{3} + 11 T^{2} - 41 T - 2)^{2} \)
|
| $83$ |
\( T^{6} - 365 T^{4} + \cdots - 984064 \)
|
| $89$ |
\( (T^{3} + 6 T^{2} - 49 T - 82)^{2} \)
|
| $97$ |
\( T^{6} - 108 T^{4} + \cdots - 1024 \)
|
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