Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2205,2,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, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2205.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| 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: | \(17.6070136457\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4352.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 6x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 735) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 6x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
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 |
|
−2.74912 | 0 | 5.55765 | −1.00000 | 0 | 0 | −9.78039 | 0 | 2.74912 | ||||||||||||||||||||||||||||||
| 1.2 | −2.27133 | 0 | 3.15894 | −1.00000 | 0 | 0 | −2.63234 | 0 | 2.27133 | |||||||||||||||||||||||||||||||
| 1.3 | −0.665096 | 0 | −1.55765 | −1.00000 | 0 | 0 | 2.36618 | 0 | 0.665096 | |||||||||||||||||||||||||||||||
| 1.4 | 1.68554 | 0 | 0.841058 | −1.00000 | 0 | 0 | −1.95345 | 0 | −1.68554 | |||||||||||||||||||||||||||||||
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 | 2205.2.a.bf | 4 | |
| 3.b | odd | 2 | 1 | 735.2.a.n | ✓ | 4 | |
| 7.b | odd | 2 | 1 | 2205.2.a.bg | 4 | ||
| 15.d | odd | 2 | 1 | 3675.2.a.bl | 4 | ||
| 21.c | even | 2 | 1 | 735.2.a.o | yes | 4 | |
| 21.g | even | 6 | 2 | 735.2.i.m | 8 | ||
| 21.h | odd | 6 | 2 | 735.2.i.n | 8 | ||
| 105.g | even | 2 | 1 | 3675.2.a.bk | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 735.2.a.n | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 735.2.a.o | yes | 4 | 21.c | even | 2 | 1 | |
| 735.2.i.m | 8 | 21.g | even | 6 | 2 | ||
| 735.2.i.n | 8 | 21.h | odd | 6 | 2 | ||
| 2205.2.a.bf | 4 | 1.a | even | 1 | 1 | trivial | |
| 2205.2.a.bg | 4 | 7.b | odd | 2 | 1 | ||
| 3675.2.a.bk | 4 | 105.g | even | 2 | 1 | ||
| 3675.2.a.bl | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\):
|
\( T_{2}^{4} + 4T_{2}^{3} - 12T_{2} - 7 \)
|
|
\( T_{11}^{4} + 8T_{11}^{3} - 8T_{11}^{2} - 160T_{11} - 224 \)
|
|
\( T_{13}^{4} - 32T_{13}^{2} + 64T_{13} + 16 \)
|
|
\( T_{17}^{4} - 8T_{17}^{3} - 16T_{17}^{2} + 128T_{17} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots - 7 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 8 T^{3} + \cdots - 224 \)
|
| $13$ |
\( T^{4} - 32 T^{2} + \cdots + 16 \)
|
| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 128 \)
|
| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 284 \)
|
| $23$ |
\( T^{4} - 44 T^{2} + \cdots + 196 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots - 368 \)
|
| $31$ |
\( T^{4} - 8 T^{3} + \cdots - 28 \)
|
| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 784 \)
|
| $41$ |
\( T^{4} - 56 T^{2} + \cdots + 16 \)
|
| $43$ |
\( T^{4} + 8 T^{3} + \cdots + 64 \)
|
| $47$ |
\( (T^{2} - 4 T - 68)^{2} \)
|
| $53$ |
\( T^{4} + 8 T^{3} + \cdots - 28 \)
|
| $59$ |
\( T^{4} - 16 T^{3} + \cdots - 4544 \)
|
| $61$ |
\( (T^{2} - 16 T + 62)^{2} \)
|
| $67$ |
\( T^{4} - 176 T^{2} + \cdots + 3136 \)
|
| $71$ |
\( T^{4} - 8 T^{3} + \cdots + 32 \)
|
| $73$ |
\( T^{4} - 128 T^{2} + \cdots + 3088 \)
|
| $79$ |
\( T^{4} - 184 T^{2} + \cdots + 2576 \)
|
| $83$ |
\( (T^{2} - 20 T + 92)^{2} \)
|
| $89$ |
\( (T^{2} + 4 T - 4)^{2} \)
|
| $97$ |
\( T^{4} - 192 T^{2} + \cdots - 2032 \)
|
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