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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 44x^{4} + 495x^{2} - 1200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| 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
\( x^{6} - 44x^{4} + 495x^{2} - 1200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 34\nu^{3} + 195\nu ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 44\nu^{3} - 415\nu ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 32\nu^{2} + 154 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 22\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 32\beta_{2} + 326 \)
|
| \(\nu^{5}\) | \(=\) |
\( 34\beta_{4} + 44\beta_{3} + 553\beta_1 \)
|
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 |
|
−5.26184 | 3.00000 | 19.6870 | 5.00000 | −15.7855 | 5.37309 | −61.4950 | 9.00000 | −26.3092 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.60164 | 3.00000 | 4.97180 | 5.00000 | −10.8049 | −31.6129 | 10.9065 | 9.00000 | −18.0082 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.82790 | 3.00000 | −4.65878 | 5.00000 | −5.48370 | 15.0916 | 23.1390 | 9.00000 | −9.13951 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.82790 | 3.00000 | −4.65878 | 5.00000 | 5.48370 | −15.0916 | −23.1390 | 9.00000 | 9.13951 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.60164 | 3.00000 | 4.97180 | 5.00000 | 10.8049 | 31.6129 | −10.9065 | 9.00000 | 18.0082 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.26184 | 3.00000 | 19.6870 | 5.00000 | 15.7855 | −5.37309 | 61.4950 | 9.00000 | 26.3092 | |||||||||||||||||||||||||||||||||||||
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.bb | ✓ | 6 |
| 11.b | odd | 2 | 1 | inner | 1815.4.a.bb | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.4.a.bb | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1815.4.a.bb | ✓ | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{6} - 44T_{2}^{4} + 495T_{2}^{2} - 1200 \)
|
|
\( T_{7}^{6} - 1256T_{7}^{4} + 263040T_{7}^{2} - 6571200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 44 T^{4} + \cdots - 1200 \)
|
| $3$ |
\( (T - 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 1256 T^{4} + \cdots - 6571200 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 1664 T^{4} + \cdots - 5227200 \)
|
| $17$ |
\( T^{6} + \cdots - 31678852800 \)
|
| $19$ |
\( T^{6} + \cdots - 6481171200 \)
|
| $23$ |
\( (T^{3} - 28 T^{2} + \cdots + 1938432)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 2099899468800 \)
|
| $31$ |
\( (T^{3} - 288 T^{2} + \cdots + 761600)^{2} \)
|
| $37$ |
\( (T^{3} - 166 T^{2} + \cdots + 3002872)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 12226636800 \)
|
| $43$ |
\( T^{6} + \cdots - 107322824836800 \)
|
| $47$ |
\( (T^{3} - 48 T^{2} + \cdots + 13192704)^{2} \)
|
| $53$ |
\( (T^{3} - 154 T^{2} + \cdots - 13663032)^{2} \)
|
| $59$ |
\( (T^{3} - 1040 T^{2} + \cdots + 92904000)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 49478086963200 \)
|
| $67$ |
\( (T^{3} + 584 T^{2} + \cdots - 538988864)^{2} \)
|
| $71$ |
\( (T^{3} - 532 T^{2} + \cdots - 101340672)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 86\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + 342 T^{2} + \cdots - 10996920)^{2} \)
|
| $97$ |
\( (T^{3} - 1406 T^{2} + \cdots + 927003032)^{2} \)
|
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