Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,6,Mod(1,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, 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("324.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.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: | \(51.9643576194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 110x^{3} + 39x^{2} + 2214x - 1944 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{7} \) |
| Twist minimal: | no (minimal twist has level 36) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 110x^{3} + 39x^{2} + 2214x - 1944 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 20\nu^{3} + 74\nu^{2} - 1209\nu - 1674 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} - 46\nu^{3} - 208\nu^{2} + 1941\nu - 3186 ) / 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 92\nu^{2} + 195\nu - 1056 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 92\nu^{2} - 33\nu + 993 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + 21 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + 7\beta_{3} + 27\beta_{2} + 27\beta _1 + 4830 ) / 108 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 155\beta_{4} + 283\beta_{3} + 27\beta_{2} + 189\beta _1 + 11814 ) / 108 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 304\beta_{4} + 671\beta_{3} + 1269\beta_{2} + 1431\beta _1 + 181065 ) / 54 \)
|
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 |
|
0 | 0 | 0 | −81.4540 | 0 | −179.262 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −26.3205 | 0 | 63.2575 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −9.76844 | 0 | 136.668 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 28.1436 | 0 | −151.408 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 110.399 | 0 | 101.745 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
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 | 324.6.a.e | 5 | |
| 3.b | odd | 2 | 1 | 324.6.a.d | 5 | ||
| 9.c | even | 3 | 2 | 36.6.e.a | ✓ | 10 | |
| 9.d | odd | 6 | 2 | 108.6.e.a | 10 | ||
| 36.f | odd | 6 | 2 | 144.6.i.d | 10 | ||
| 36.h | even | 6 | 2 | 432.6.i.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.6.e.a | ✓ | 10 | 9.c | even | 3 | 2 | |
| 108.6.e.a | 10 | 9.d | odd | 6 | 2 | ||
| 144.6.i.d | 10 | 36.f | odd | 6 | 2 | ||
| 324.6.a.d | 5 | 3.b | odd | 2 | 1 | ||
| 324.6.a.e | 5 | 1.a | even | 1 | 1 | trivial | |
| 432.6.i.d | 10 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 21T_{5}^{4} - 9981T_{5}^{3} - 56727T_{5}^{2} + 7030800T_{5} + 65069568 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 21 T^{4} + \cdots + 65069568 \)
|
| $7$ |
\( T^{5} + \cdots - 23874172652 \)
|
| $11$ |
\( T^{5} + \cdots + 1341086443965 \)
|
| $13$ |
\( T^{5} + \cdots - 37769624404556 \)
|
| $17$ |
\( T^{5} + \cdots + 113704762586184 \)
|
| $19$ |
\( T^{5} + \cdots - 97\!\cdots\!92 \)
|
| $23$ |
\( T^{5} + \cdots + 21\!\cdots\!36 \)
|
| $29$ |
\( T^{5} + \cdots - 31\!\cdots\!56 \)
|
| $31$ |
\( T^{5} + \cdots + 26\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots - 92\!\cdots\!47 \)
|
| $43$ |
\( T^{5} + \cdots - 25\!\cdots\!11 \)
|
| $47$ |
\( T^{5} + \cdots + 43\!\cdots\!32 \)
|
| $53$ |
\( T^{5} + \cdots - 86\!\cdots\!28 \)
|
| $59$ |
\( T^{5} + \cdots + 22\!\cdots\!59 \)
|
| $61$ |
\( T^{5} + \cdots + 67\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots - 14\!\cdots\!57 \)
|
| $71$ |
\( T^{5} + \cdots + 18\!\cdots\!64 \)
|
| $73$ |
\( T^{5} + \cdots - 17\!\cdots\!80 \)
|
| $79$ |
\( T^{5} + \cdots + 36\!\cdots\!80 \)
|
| $83$ |
\( T^{5} + \cdots + 12\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots - 18\!\cdots\!72 \)
|
| $97$ |
\( T^{5} + \cdots - 13\!\cdots\!75 \)
|
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