Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2205,4,Mod(1,2205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, 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("2205.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2205.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: | \(130.099211563\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.4892.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 11x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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
\( x^{3} - 11x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 7 \)
|
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 |
|
−4.11718 | 0 | 8.95114 | 5.00000 | 0 | 0 | −3.91601 | 0 | −20.5859 | |||||||||||||||||||||||||||
| 1.2 | −1.36817 | 0 | −6.12810 | 5.00000 | 0 | 0 | 19.3297 | 0 | −6.84087 | ||||||||||||||||||||||||||||
| 1.3 | 2.48535 | 0 | −1.82304 | 5.00000 | 0 | 0 | −24.4137 | 0 | 12.4267 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 2205.4.a.bj | 3 | |
| 3.b | odd | 2 | 1 | 735.4.a.s | 3 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.bi | 3 | ||
| 7.d | odd | 6 | 2 | 315.4.j.e | 6 | ||
| 21.c | even | 2 | 1 | 735.4.a.r | 3 | ||
| 21.g | even | 6 | 2 | 105.4.i.c | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.i.c | ✓ | 6 | 21.g | even | 6 | 2 | |
| 315.4.j.e | 6 | 7.d | odd | 6 | 2 | ||
| 735.4.a.r | 3 | 21.c | even | 2 | 1 | ||
| 735.4.a.s | 3 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.bi | 3 | 7.b | odd | 2 | 1 | ||
| 2205.4.a.bj | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):
|
\( T_{2}^{3} + 3T_{2}^{2} - 8T_{2} - 14 \)
|
|
\( T_{11}^{3} - T_{11}^{2} - 508T_{11} - 3780 \)
|
|
\( T_{13}^{3} - 79T_{13}^{2} - 49T_{13} + 687 \)
|
|
\( T_{17}^{3} + 72T_{17}^{2} - 104T_{17} - 19424 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3 T^{2} + \cdots - 14 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - T^{2} + \cdots - 3780 \)
|
| $13$ |
\( T^{3} - 79 T^{2} + \cdots + 687 \)
|
| $17$ |
\( T^{3} + 72 T^{2} + \cdots - 19424 \)
|
| $19$ |
\( T^{3} - 29 T^{2} + \cdots + 65521 \)
|
| $23$ |
\( T^{3} - 63 T^{2} + \cdots + 53096 \)
|
| $29$ |
\( T^{3} + 220 T^{2} + \cdots - 28096 \)
|
| $31$ |
\( T^{3} + 136 T^{2} + \cdots - 4764978 \)
|
| $37$ |
\( T^{3} - 43 T^{2} + \cdots + 2466387 \)
|
| $41$ |
\( T^{3} + 599 T^{2} + \cdots - 3130740 \)
|
| $43$ |
\( T^{3} - 170 T^{2} + \cdots + 1053316 \)
|
| $47$ |
\( T^{3} + 3 T^{2} + \cdots + 30756380 \)
|
| $53$ |
\( T^{3} + 331 T^{2} + \cdots + 10864 \)
|
| $59$ |
\( T^{3} + 1520 T^{2} + \cdots - 24592160 \)
|
| $61$ |
\( T^{3} - 1160 T^{2} + \cdots - 45301392 \)
|
| $67$ |
\( T^{3} - 806 T^{2} + \cdots + 43402660 \)
|
| $71$ |
\( T^{3} - 406 T^{2} + \cdots + 232329576 \)
|
| $73$ |
\( T^{3} - 1192 T^{2} + \cdots - 4156150 \)
|
| $79$ |
\( T^{3} + 2590 T^{2} + \cdots + 197479612 \)
|
| $83$ |
\( T^{3} + 508 T^{2} + \cdots - 30444208 \)
|
| $89$ |
\( T^{3} - 42 T^{2} + \cdots + 708440784 \)
|
| $97$ |
\( T^{3} + \cdots + 1879560768 \)
|
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