Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1815,2,Mod(1,1815)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, 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("1815.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1815.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: | \(14.4928479669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.437199552.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 13x^{4} + 49x^{2} - 48 \)
|
| 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
\( x^{6} - 13x^{4} + 49x^{2} - 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} + 17\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 8\beta_{2} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{4} + 9\beta_{3} + 28\beta_1 \)
|
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 |
|
−2.63162 | −1.00000 | 4.92542 | 1.00000 | 2.63162 | 4.16741 | −7.69860 | 1.00000 | −2.63162 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.13353 | −1.00000 | 2.55193 | 1.00000 | 2.13353 | −4.82155 | −1.17756 | 1.00000 | −2.13353 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23396 | −1.00000 | −0.477352 | 1.00000 | 1.23396 | −3.79281 | 3.05694 | 1.00000 | −1.23396 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.23396 | −1.00000 | −0.477352 | 1.00000 | −1.23396 | 3.79281 | −3.05694 | 1.00000 | 1.23396 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.13353 | −1.00000 | 2.55193 | 1.00000 | −2.13353 | 4.82155 | 1.17756 | 1.00000 | 2.13353 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.63162 | −1.00000 | 4.92542 | 1.00000 | −2.63162 | −4.16741 | 7.69860 | 1.00000 | 2.63162 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1815.2.a.y | ✓ | 6 |
| 3.b | odd | 2 | 1 | 5445.2.a.bz | 6 | ||
| 5.b | even | 2 | 1 | 9075.2.a.dq | 6 | ||
| 11.b | odd | 2 | 1 | inner | 1815.2.a.y | ✓ | 6 |
| 33.d | even | 2 | 1 | 5445.2.a.bz | 6 | ||
| 55.d | odd | 2 | 1 | 9075.2.a.dq | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.2.a.y | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1815.2.a.y | ✓ | 6 | 11.b | odd | 2 | 1 | inner |
| 5445.2.a.bz | 6 | 3.b | odd | 2 | 1 | ||
| 5445.2.a.bz | 6 | 33.d | even | 2 | 1 | ||
| 9075.2.a.dq | 6 | 5.b | even | 2 | 1 | ||
| 9075.2.a.dq | 6 | 55.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{6} - 13T_{2}^{4} + 49T_{2}^{2} - 48 \)
|
|
\( T_{7}^{6} - 55T_{7}^{4} + 988T_{7}^{2} - 5808 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 13 T^{4} + \cdots - 48 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 55 T^{4} + \cdots - 5808 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 52 T^{4} + \cdots - 3072 \)
|
| $17$ |
\( T^{6} - 82 T^{4} + \cdots - 16428 \)
|
| $19$ |
\( T^{6} - 115 T^{4} + \cdots - 17328 \)
|
| $23$ |
\( (T^{3} + 2 T^{2} - 29 T - 66)^{2} \)
|
| $29$ |
\( T^{6} - 52 T^{4} + \cdots - 3072 \)
|
| $31$ |
\( (T^{3} - 9 T^{2} + \cdots + 341)^{2} \)
|
| $37$ |
\( (T^{3} - 5 T^{2} - 22 T + 92)^{2} \)
|
| $41$ |
\( (T^{2} - 12)^{3} \)
|
| $43$ |
\( T^{6} - 192 T^{4} + \cdots - 15552 \)
|
| $47$ |
\( (T^{3} - 69 T + 216)^{2} \)
|
| $53$ |
\( (T^{3} + 8 T^{2} + T - 54)^{2} \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots - 912)^{2} \)
|
| $61$ |
\( T^{6} - 241 T^{4} + \cdots - 309123 \)
|
| $67$ |
\( (T^{3} - 5 T^{2} - 22 T + 92)^{2} \)
|
| $71$ |
\( (T^{3} + 2 T^{2} - 80 T + 96)^{2} \)
|
| $73$ |
\( T^{6} - 147 T^{4} + \cdots - 62208 \)
|
| $79$ |
\( T^{6} - 409 T^{4} + \cdots - 1179387 \)
|
| $83$ |
\( T^{6} - 184 T^{4} + \cdots - 110592 \)
|
| $89$ |
\( (T^{3} - 24 T^{2} + \cdots + 216)^{2} \)
|
| $97$ |
\( (T^{3} - T^{2} - 226 T + 844)^{2} \)
|
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