Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7803,2,Mod(1,7803)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, 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("7803.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7803 = 3^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7803.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: | \(62.3072686972\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1168184448.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} + 31x^{2} - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 459) |
| 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
\( x^{6} - 11x^{4} + 31x^{2} - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 10\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 11\nu^{3} - 25\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{4} + 11\beta_{3} + 30\beta_1 \)
|
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 |
|
−2.62065 | 0 | 4.86782 | 1.94310 | 0 | 3.86782 | −7.51555 | 0 | −5.09218 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83024 | 0 | 1.34976 | −3.40253 | 0 | 0.349765 | 1.19008 | 0 | 6.22743 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.884545 | 0 | −1.21758 | 1.28342 | 0 | −2.21758 | 2.84609 | 0 | −1.13524 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.884545 | 0 | −1.21758 | −1.28342 | 0 | −2.21758 | −2.84609 | 0 | −1.13524 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.83024 | 0 | 1.34976 | 3.40253 | 0 | 0.349765 | −1.19008 | 0 | 6.22743 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.62065 | 0 | 4.86782 | −1.94310 | 0 | 3.86782 | 7.51555 | 0 | −5.09218 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7803.2.a.bq | 6 | |
| 3.b | odd | 2 | 1 | inner | 7803.2.a.bq | 6 | |
| 17.b | even | 2 | 1 | 7803.2.a.bp | 6 | ||
| 17.c | even | 4 | 2 | 459.2.d.c | ✓ | 12 | |
| 51.c | odd | 2 | 1 | 7803.2.a.bp | 6 | ||
| 51.f | odd | 4 | 2 | 459.2.d.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 459.2.d.c | ✓ | 12 | 17.c | even | 4 | 2 | |
| 459.2.d.c | ✓ | 12 | 51.f | odd | 4 | 2 | |
| 7803.2.a.bp | 6 | 17.b | even | 2 | 1 | ||
| 7803.2.a.bp | 6 | 51.c | odd | 2 | 1 | ||
| 7803.2.a.bq | 6 | 1.a | even | 1 | 1 | trivial | |
| 7803.2.a.bq | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):
|
\( T_{2}^{6} - 11T_{2}^{4} + 31T_{2}^{2} - 18 \)
|
|
\( T_{5}^{6} - 17T_{5}^{4} + 69T_{5}^{2} - 72 \)
|
|
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} + 3 \)
|
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\( T_{11}^{6} - 28T_{11}^{4} + 161T_{11}^{2} - 242 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 11 T^{4} + \cdots - 18 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 17 T^{4} + \cdots - 72 \)
|
| $7$ |
\( (T^{3} - 2 T^{2} - 8 T + 3)^{2} \)
|
| $11$ |
\( T^{6} - 28 T^{4} + \cdots - 242 \)
|
| $13$ |
\( (T^{3} + 3 T^{2} - 30 T - 68)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} - 37 T^{4} + \cdots - 8 \)
|
| $29$ |
\( T^{6} - 140 T^{4} + \cdots - 50562 \)
|
| $31$ |
\( (T^{3} - 4 T^{2} + \cdots + 102)^{2} \)
|
| $37$ |
\( (T^{3} - 13 T^{2} + \cdots + 144)^{2} \)
|
| $41$ |
\( T^{6} - 97 T^{4} + \cdots - 13448 \)
|
| $43$ |
\( (T^{3} - 9 T^{2} + \cdots + 422)^{2} \)
|
| $47$ |
\( T^{6} - 11 T^{4} + \cdots - 18 \)
|
| $53$ |
\( T^{6} - 237 T^{4} + \cdots - 472392 \)
|
| $59$ |
\( T^{6} - 183 T^{4} + \cdots - 221778 \)
|
| $61$ |
\( (T^{3} - 16 T^{2} + \cdots + 39)^{2} \)
|
| $67$ |
\( (T^{3} + 10 T^{2} + 24 T + 3)^{2} \)
|
| $71$ |
\( T^{6} - 74 T^{4} + \cdots - 12168 \)
|
| $73$ |
\( (T^{3} - 9 T^{2} - 105 T + 81)^{2} \)
|
| $79$ |
\( (T^{3} - 20 T^{2} + \cdots - 159)^{2} \)
|
| $83$ |
\( T^{6} - 252 T^{4} + \cdots - 1458 \)
|
| $89$ |
\( T^{6} - 107 T^{4} + \cdots - 162 \)
|
| $97$ |
\( (T^{3} - 8 T^{2} + \cdots + 969)^{2} \)
|
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