Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4225,2,Mod(1,4225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4225 = 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4225.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: | \(33.7367948540\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.199374400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 8x^{4} + 10x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{6} - 8x^{4} + 10x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 4\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} + 10\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 8\beta_{4} + 38\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.54574 | −2.15293 | 4.48079 | 0 | 5.48079 | −2.93855 | −6.31544 | 1.63509 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.18733 | −0.345110 | −0.590239 | 0 | 0.409761 | −2.02956 | 3.07548 | −2.88090 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.330837 | 2.69180 | −1.89055 | 0 | −0.890547 | −3.35348 | 1.28714 | 4.24581 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.330837 | −2.69180 | −1.89055 | 0 | −0.890547 | 3.35348 | −1.28714 | 4.24581 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.18733 | 0.345110 | −0.590239 | 0 | 0.409761 | 2.02956 | −3.07548 | −2.88090 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.54574 | 2.15293 | 4.48079 | 0 | 5.48079 | 2.93855 | 6.31544 | 1.63509 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 4225.2.a.br | 6 | |
| 5.b | even | 2 | 1 | inner | 4225.2.a.br | 6 | |
| 5.c | odd | 4 | 2 | 845.2.b.d | 6 | ||
| 13.b | even | 2 | 1 | 4225.2.a.bq | 6 | ||
| 13.c | even | 3 | 2 | 325.2.e.e | 12 | ||
| 65.d | even | 2 | 1 | 4225.2.a.bq | 6 | ||
| 65.f | even | 4 | 2 | 845.2.d.d | 12 | ||
| 65.h | odd | 4 | 2 | 845.2.b.e | 6 | ||
| 65.k | even | 4 | 2 | 845.2.d.d | 12 | ||
| 65.n | even | 6 | 2 | 325.2.e.e | 12 | ||
| 65.o | even | 12 | 4 | 845.2.l.f | 24 | ||
| 65.q | odd | 12 | 4 | 65.2.n.a | ✓ | 12 | |
| 65.r | odd | 12 | 4 | 845.2.n.e | 12 | ||
| 65.t | even | 12 | 4 | 845.2.l.f | 24 | ||
| 195.bl | even | 12 | 4 | 585.2.bs.a | 12 | ||
| 260.bj | even | 12 | 4 | 1040.2.dh.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.n.a | ✓ | 12 | 65.q | odd | 12 | 4 | |
| 325.2.e.e | 12 | 13.c | even | 3 | 2 | ||
| 325.2.e.e | 12 | 65.n | even | 6 | 2 | ||
| 585.2.bs.a | 12 | 195.bl | even | 12 | 4 | ||
| 845.2.b.d | 6 | 5.c | odd | 4 | 2 | ||
| 845.2.b.e | 6 | 65.h | odd | 4 | 2 | ||
| 845.2.d.d | 12 | 65.f | even | 4 | 2 | ||
| 845.2.d.d | 12 | 65.k | even | 4 | 2 | ||
| 845.2.l.f | 24 | 65.o | even | 12 | 4 | ||
| 845.2.l.f | 24 | 65.t | even | 12 | 4 | ||
| 845.2.n.e | 12 | 65.r | odd | 12 | 4 | ||
| 1040.2.dh.a | 12 | 260.bj | even | 12 | 4 | ||
| 4225.2.a.bq | 6 | 13.b | even | 2 | 1 | ||
| 4225.2.a.bq | 6 | 65.d | even | 2 | 1 | ||
| 4225.2.a.br | 6 | 1.a | even | 1 | 1 | trivial | |
| 4225.2.a.br | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):
|
\( T_{2}^{6} - 8T_{2}^{4} + 10T_{2}^{2} - 1 \)
|
|
\( T_{3}^{6} - 12T_{3}^{4} + 35T_{3}^{2} - 4 \)
|
|
\( T_{7}^{6} - 24T_{7}^{4} + 179T_{7}^{2} - 400 \)
|
|
\( T_{11}^{3} - 13T_{11} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - 12 T^{4} + \cdots - 4 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 24 T^{4} + \cdots - 400 \)
|
| $11$ |
\( (T^{3} - 13 T - 8)^{2} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 35 T^{4} + \cdots - 169 \)
|
| $19$ |
\( (T^{3} - 6 T^{2} - T + 10)^{2} \)
|
| $23$ |
\( T^{6} - 12 T^{4} + \cdots - 4 \)
|
| $29$ |
\( (T - 3)^{6} \)
|
| $31$ |
\( (T^{3} + 4 T^{2} - 40 T + 40)^{2} \)
|
| $37$ |
\( T^{6} - 35 T^{4} + \cdots - 169 \)
|
| $41$ |
\( (T^{3} + 7 T^{2} - 29 T + 5)^{2} \)
|
| $43$ |
\( T^{6} - 80 T^{4} + \cdots - 256 \)
|
| $47$ |
\( T^{6} - 236 T^{4} + \cdots - 270400 \)
|
| $53$ |
\( T^{6} - 171 T^{4} + \cdots - 400 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} + \cdots - 136)^{2} \)
|
| $61$ |
\( (T^{3} + 3 T^{2} + \cdots - 115)^{2} \)
|
| $67$ |
\( T^{6} - 100 T^{4} + \cdots - 20164 \)
|
| $71$ |
\( (T^{3} - 6 T^{2} - T + 26)^{2} \)
|
| $73$ |
\( T^{6} - 215 T^{4} + \cdots - 250000 \)
|
| $79$ |
\( (T^{3} - 26 T^{2} + \cdots - 160)^{2} \)
|
| $83$ |
\( T^{6} - 276 T^{4} + \cdots - 640000 \)
|
| $89$ |
\( (T^{3} - 10 T^{2} + \cdots + 1586)^{2} \)
|
| $97$ |
\( T^{6} - 280 T^{4} + \cdots - 204304 \)
|
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