Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1815,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1815.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(107.088466660\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.7598722752.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 27x^{4} + 156x^{2} - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 27x^{4} + 156x^{2} - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 15\nu^{2} - 4 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 25\nu^{3} + 126\nu ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 25\nu^{2} - 86 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 85\nu^{3} - 528\nu ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 6\beta_{3} + 15\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 15\beta_{4} + 25\beta_{2} + 139 \)
|
| \(\nu^{5}\) | \(=\) |
\( 25\beta_{5} + 170\beta_{3} + 249\beta_1 \)
|
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 |
|
−4.60194 | −3.00000 | 13.1778 | 5.00000 | 13.8058 | −6.69261 | −23.8281 | 9.00000 | −23.0097 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.59065 | −3.00000 | −1.28852 | 5.00000 | 7.77196 | −3.97867 | 24.0633 | 9.00000 | −12.9533 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.45282 | −3.00000 | −5.88933 | 5.00000 | 4.35845 | 20.0344 | 20.1786 | 9.00000 | −7.26408 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.45282 | −3.00000 | −5.88933 | 5.00000 | −4.35845 | −20.0344 | −20.1786 | 9.00000 | 7.26408 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.59065 | −3.00000 | −1.28852 | 5.00000 | −7.77196 | 3.97867 | −24.0633 | 9.00000 | 12.9533 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.60194 | −3.00000 | 13.1778 | 5.00000 | −13.8058 | 6.69261 | 23.8281 | 9.00000 | 23.0097 | |||||||||||||||||||||||||||||||||||||
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 | 1815.4.a.z | ✓ | 6 |
| 11.b | odd | 2 | 1 | inner | 1815.4.a.z | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.4.a.z | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1815.4.a.z | ✓ | 6 | 11.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{6} - 30T_{2}^{4} + 201T_{2}^{2} - 300 \)
|
|
\( T_{7}^{6} - 462T_{7}^{4} + 25041T_{7}^{2} - 284592 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 30 T^{4} + \cdots - 300 \)
|
| $3$ |
\( (T + 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 462 T^{4} + \cdots - 284592 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 2430 T^{4} + \cdots - 159432300 \)
|
| $17$ |
\( T^{6} + \cdots - 9046312707 \)
|
| $19$ |
\( T^{6} - 1398 T^{4} + \cdots - 98361228 \)
|
| $23$ |
\( (T^{3} - 9 T^{2} + \cdots + 456126)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 2022076332 \)
|
| $31$ |
\( (T^{3} - 15 T^{2} + \cdots - 11110)^{2} \)
|
| $37$ |
\( (T^{3} + 288 T^{2} + \cdots + 500360)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 3973988650800 \)
|
| $43$ |
\( T^{6} + \cdots - 117463512332208 \)
|
| $47$ |
\( (T^{3} + 165 T^{2} + \cdots + 22960500)^{2} \)
|
| $53$ |
\( (T^{3} + 39 T^{2} + \cdots - 3130410)^{2} \)
|
| $59$ |
\( (T^{3} + 744 T^{2} + \cdots - 63682500)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} - 60 T^{2} + \cdots - 4146160)^{2} \)
|
| $71$ |
\( (T^{3} - 78 T^{2} + \cdots + 67448820)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 61\!\cdots\!08 \)
|
| $79$ |
\( T^{6} + \cdots - 39\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + 900 T^{2} + \cdots - 739699008)^{2} \)
|
| $97$ |
\( (T^{3} + 90 T^{2} + \cdots + 388505680)^{2} \)
|
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