Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2100,4,Mod(1,2100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2100.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2100.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: | \(123.904011012\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.698565.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 205x + 190 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 205x + 190 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 17\nu + 129 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} - 8\nu + 279 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + 2\beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -17\beta_{2} - 8\beta _1 + 825 ) / 6 \)
|
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 | −3.00000 | 0 | 0 | 0 | −7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | 0 | 0 | −7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 0 | 0 | −7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 2100.4.a.w | ✓ | 3 |
| 5.b | even | 2 | 1 | 2100.4.a.ba | yes | 3 | |
| 5.c | odd | 4 | 2 | 2100.4.k.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2100.4.a.w | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2100.4.a.ba | yes | 3 | 5.b | even | 2 | 1 | |
| 2100.4.k.q | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2100))\):
|
\( T_{11}^{3} - 8T_{11}^{2} - 2835T_{11} + 51894 \)
|
|
\( T_{13}^{3} + 47T_{13}^{2} - 790T_{13} - 42836 \)
|
|
\( T_{17}^{3} + 31T_{17}^{2} - 1206T_{17} + 8100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} - 8 T^{2} + \cdots + 51894 \)
|
| $13$ |
\( T^{3} + 47 T^{2} + \cdots - 42836 \)
|
| $17$ |
\( T^{3} + 31 T^{2} + \cdots + 8100 \)
|
| $19$ |
\( T^{3} - 102 T^{2} + \cdots + 471136 \)
|
| $23$ |
\( T^{3} + 35 T^{2} + \cdots + 394497 \)
|
| $29$ |
\( T^{3} - 123 T^{2} + \cdots + 2352159 \)
|
| $31$ |
\( T^{3} - 143 T^{2} + \cdots + 81980 \)
|
| $37$ |
\( T^{3} + 56 T^{2} + \cdots + 5181034 \)
|
| $41$ |
\( T^{3} - 67 T^{2} + \cdots + 5080320 \)
|
| $43$ |
\( T^{3} - 53 T^{2} + \cdots + 1224417 \)
|
| $47$ |
\( T^{3} - 34 T^{2} + \cdots + 1357344 \)
|
| $53$ |
\( T^{3} - 27 T^{2} + \cdots + 1116828 \)
|
| $59$ |
\( T^{3} - 41 T^{2} + \cdots + 6921180 \)
|
| $61$ |
\( T^{3} - 617 T^{2} + \cdots + 84831440 \)
|
| $67$ |
\( T^{3} + 928 T^{2} + \cdots - 4368084 \)
|
| $71$ |
\( T^{3} - 754 T^{2} + \cdots + 664039404 \)
|
| $73$ |
\( T^{3} + 92 T^{2} + \cdots - 338232920 \)
|
| $79$ |
\( T^{3} - 430 T^{2} + \cdots + 11601594 \)
|
| $83$ |
\( T^{3} - 897 T^{2} + \cdots + 57279636 \)
|
| $89$ |
\( T^{3} + 212 T^{2} + \cdots + 28448280 \)
|
| $97$ |
\( T^{3} + 2018 T^{2} + \cdots - 220691240 \)
|
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