Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,4,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("363.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.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: | \(21.4176933321\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 28x^{2} - 2x + 72 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 28x^{2} - 2x + 72 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 22\nu - 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 23\beta _1 + 18 \)
|
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 |
|
−5.63177 | 3.00000 | 23.7168 | 5.58204 | −16.8953 | 16.3245 | −88.5134 | 9.00000 | −31.4368 | ||||||||||||||||||||||||||||||
| 1.2 | −1.59490 | 3.00000 | −5.45630 | 8.73607 | −4.78470 | −29.0283 | 21.4614 | 9.00000 | −13.9331 | |||||||||||||||||||||||||||||||
| 1.3 | 1.81838 | 3.00000 | −4.69349 | −19.2178 | 5.45514 | 14.1043 | −23.0816 | 9.00000 | −34.9453 | |||||||||||||||||||||||||||||||
| 1.4 | 4.40829 | 3.00000 | 11.4330 | 18.8997 | 13.2249 | 18.5994 | 15.1336 | 9.00000 | 83.3152 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.4.a.q | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1089.4.a.bf | 4 | ||
| 11.b | odd | 2 | 1 | 363.4.a.s | yes | 4 | |
| 33.d | even | 2 | 1 | 1089.4.a.ba | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.4.a.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 363.4.a.s | yes | 4 | 11.b | odd | 2 | 1 | |
| 1089.4.a.ba | 4 | 33.d | even | 2 | 1 | ||
| 1089.4.a.bf | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):
|
\( T_{2}^{4} + T_{2}^{3} - 28T_{2}^{2} + 2T_{2} + 72 \)
|
|
\( T_{5}^{4} - 14T_{5}^{3} - 319T_{5}^{2} + 5216T_{5} - 17712 \)
|
|
\( T_{7}^{4} - 20T_{7}^{3} - 627T_{7}^{2} + 18830T_{7} - 124312 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 72 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 14 T^{3} + \cdots - 17712 \)
|
| $7$ |
\( T^{4} - 20 T^{3} + \cdots - 124312 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 32 T^{3} + \cdots + 1724492 \)
|
| $17$ |
\( T^{4} - 92 T^{3} + \cdots - 1870416 \)
|
| $19$ |
\( T^{4} + 34 T^{3} + \cdots + 87435072 \)
|
| $23$ |
\( T^{4} + 26 T^{3} + \cdots + 427537152 \)
|
| $29$ |
\( T^{4} - 174 T^{3} + \cdots - 111819852 \)
|
| $31$ |
\( T^{4} - 422 T^{3} + \cdots - 755977024 \)
|
| $37$ |
\( T^{4} - 518 T^{3} + \cdots - 336093777 \)
|
| $41$ |
\( T^{4} + \cdots - 1476203796 \)
|
| $43$ |
\( T^{4} + 550 T^{3} + \cdots - 474515712 \)
|
| $47$ |
\( T^{4} + \cdots - 1899747648 \)
|
| $53$ |
\( T^{4} + \cdots - 9562089852 \)
|
| $59$ |
\( T^{4} + \cdots - 2988982656 \)
|
| $61$ |
\( T^{4} + \cdots - 2291291676 \)
|
| $67$ |
\( T^{4} + \cdots - 3253123496 \)
|
| $71$ |
\( T^{4} + \cdots + 71278424064 \)
|
| $73$ |
\( T^{4} + \cdots + 5540854172 \)
|
| $79$ |
\( T^{4} + \cdots - 13109080736 \)
|
| $83$ |
\( T^{4} + \cdots - 58659282432 \)
|
| $89$ |
\( T^{4} + \cdots + 172456334064 \)
|
| $97$ |
\( T^{4} + \cdots + 840759943813 \)
|
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