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: | 6.6.435306496.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} + 22x^{2} - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 10x^{4} + 22x^{2} - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 10\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 20\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 8\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 10\beta_{4} + 30\beta_1 \)
|
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 |
|
−2.62530 | 0 | 4.89219 | −1.86348 | 0 | 5.07456 | −7.59285 | 0 | 4.89219 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.70536 | 0 | 0.908256 | −0.532588 | 0 | −2.49579 | 1.86182 | 0 | 0.908256 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.446720 | 0 | −1.80044 | 4.03036 | 0 | 1.42124 | 1.69773 | 0 | −1.80044 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.446720 | 0 | −1.80044 | −4.03036 | 0 | 1.42124 | −1.69773 | 0 | −1.80044 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.70536 | 0 | 0.908256 | 0.532588 | 0 | −2.49579 | −1.86182 | 0 | 0.908256 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.62530 | 0 | 4.89219 | 1.86348 | 0 | 5.07456 | 7.59285 | 0 | 4.89219 | |||||||||||||||||||||||||||||||||||||
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.bo | yes | 6 |
| 3.b | odd | 2 | 1 | inner | 7803.2.a.bo | yes | 6 |
| 17.b | even | 2 | 1 | 7803.2.a.bn | ✓ | 6 | |
| 51.c | odd | 2 | 1 | 7803.2.a.bn | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7803.2.a.bn | ✓ | 6 | 17.b | even | 2 | 1 | |
| 7803.2.a.bn | ✓ | 6 | 51.c | odd | 2 | 1 | |
| 7803.2.a.bo | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 7803.2.a.bo | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):
|
\( T_{2}^{6} - 10T_{2}^{4} + 22T_{2}^{2} - 4 \)
|
|
\( T_{5}^{6} - 20T_{5}^{4} + 62T_{5}^{2} - 16 \)
|
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\( T_{7}^{3} - 4T_{7}^{2} - 9T_{7} + 18 \)
|
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\( T_{11}^{6} - 30T_{11}^{4} + 256T_{11}^{2} - 576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 4 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 16 \)
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| $7$ |
\( (T^{3} - 4 T^{2} - 9 T + 18)^{2} \)
|
| $11$ |
\( T^{6} - 30 T^{4} + \cdots - 576 \)
|
| $13$ |
\( (T^{3} - 6 T^{2} + T + 16)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} - 5 T^{2} - 6 T + 18)^{2} \)
|
| $23$ |
\( T^{6} - 86 T^{4} + \cdots - 21904 \)
|
| $29$ |
\( T^{6} - 196 T^{4} + \cdots - 197136 \)
|
| $31$ |
\( (T^{3} + T^{2} - 45 T - 53)^{2} \)
|
| $37$ |
\( (T^{3} + T^{2} - 30 T - 74)^{2} \)
|
| $41$ |
\( T^{6} - 64 T^{4} + \cdots - 576 \)
|
| $43$ |
\( (T^{3} - 13 T^{2} + \cdots + 423)^{2} \)
|
| $47$ |
\( T^{6} - 174 T^{4} + \cdots - 82944 \)
|
| $53$ |
\( T^{6} - 234 T^{4} + \cdots - 39204 \)
|
| $59$ |
\( T^{6} - 364 T^{4} + \cdots - 2304 \)
|
| $61$ |
\( (T^{3} - 18 T^{2} + \cdots - 148)^{2} \)
|
| $67$ |
\( (T^{3} - 3 T^{2} - 96 T + 44)^{2} \)
|
| $71$ |
\( T^{6} - 340 T^{4} + \cdots - 85264 \)
|
| $73$ |
\( (T^{3} + 5 T^{2} - 31 T + 33)^{2} \)
|
| $79$ |
\( (T + 3)^{6} \)
|
| $83$ |
\( T^{6} - 394 T^{4} + \cdots - 401956 \)
|
| $89$ |
\( T^{6} - 122 T^{4} + \cdots - 24964 \)
|
| $97$ |
\( (T^{3} + 18 T^{2} + \cdots + 152)^{2} \)
|
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