Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(58.2193265921\) |
| 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: |
\( x^{4} - x^{3} - 80x^{2} - 30x + 1056 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} - 80x^{2} - 30x + 1056 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} - 46\nu + 130 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 5\beta_{2} + 56\beta _1 + 70 \)
|
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 |
|
−10.8809 | −9.00000 | 86.3932 | 71.5321 | 97.9278 | 101.164 | −591.845 | 81.0000 | −778.331 | ||||||||||||||||||||||||||||||
| 1.2 | −5.67080 | −9.00000 | 0.157985 | −41.2382 | 51.0372 | −17.2472 | 180.570 | 81.0000 | 233.853 | |||||||||||||||||||||||||||||||
| 1.3 | 2.79222 | −9.00000 | −24.2035 | 50.9532 | −25.1300 | 153.660 | −156.933 | 81.0000 | 142.273 | |||||||||||||||||||||||||||||||
| 1.4 | 4.75944 | −9.00000 | −9.34769 | −39.2472 | −42.8350 | −223.577 | −196.792 | 81.0000 | −186.795 | |||||||||||||||||||||||||||||||
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 | 363.6.a.m | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1089.6.a.u | 4 | ||
| 11.b | odd | 2 | 1 | 363.6.a.n | yes | 4 | |
| 33.d | even | 2 | 1 | 1089.6.a.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.6.a.m | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 363.6.a.n | yes | 4 | 11.b | odd | 2 | 1 | |
| 1089.6.a.t | 4 | 33.d | even | 2 | 1 | ||
| 1089.6.a.u | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 9T_{2}^{3} - 50T_{2}^{2} - 246T_{2} + 820 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 9 T^{3} + \cdots + 820 \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} - 42 T^{3} + \cdots + 5899024 \)
|
| $7$ |
\( T^{4} - 14 T^{3} + \cdots + 59942704 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + \cdots + 80561016724 \)
|
| $17$ |
\( T^{4} + \cdots - 1335636034544 \)
|
| $19$ |
\( T^{4} + \cdots - 25168511808 \)
|
| $23$ |
\( T^{4} + \cdots + 71886530642944 \)
|
| $29$ |
\( T^{4} + \cdots - 398306163467340 \)
|
| $31$ |
\( T^{4} + \cdots - 509465002770176 \)
|
| $37$ |
\( T^{4} + \cdots - 762131278541259 \)
|
| $41$ |
\( T^{4} + \cdots + 15\!\cdots\!92 \)
|
| $43$ |
\( T^{4} + \cdots - 32\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 38\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots - 32\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + \cdots - 42\!\cdots\!32 \)
|
| $67$ |
\( T^{4} + \cdots - 78\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots + 62\!\cdots\!48 \)
|
| $79$ |
\( T^{4} + \cdots + 21\!\cdots\!12 \)
|
| $83$ |
\( T^{4} + \cdots - 31\!\cdots\!80 \)
|
| $89$ |
\( T^{4} + \cdots + 55\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots - 12\!\cdots\!83 \)
|
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