Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,6,Mod(1,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("912.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.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: | \(146.270043669\) |
| 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: |
\( x^{5} - 2x^{4} - 7184x^{3} - 76134x^{2} + 12883743x + 275533272 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 228) |
| 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
\( x^{5} - 2x^{4} - 7184x^{3} - 76134x^{2} + 12883743x + 275533272 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 31\nu^{4} + 85\nu^{3} - 29759\nu^{2} - 3271437\nu - 242224776 ) / 2387520 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -67\nu^{4} + 4095\nu^{3} + 312483\nu^{2} - 8328551\nu - 414802968 ) / 795840 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{4} + 1237\nu^{3} + 90769\nu^{2} - 4258221\nu - 148185864 ) / 238752 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 127\nu^{4} - 2219\nu^{3} - 748319\nu^{2} + 7774707\nu + 937268856 ) / 477504 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{4} - 25\beta_{3} + 19\beta_{2} + 109\beta _1 + 17222 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 348\beta_{4} - 2149\beta_{3} + 3613\beta_{2} + 4513\beta _1 + 324122 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6714\beta_{4} - 123637\beta_{3} + 113863\beta_{2} + 659893\beta _1 + 62737118 ) / 6 \)
|
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 | −9.00000 | 0 | −91.9065 | 0 | −1.91671 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −69.8269 | 0 | 147.923 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | 19.9765 | 0 | −224.674 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.00000 | 0 | 67.0515 | 0 | −67.8869 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −9.00000 | 0 | 68.7054 | 0 | 92.5541 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(19\) | \(1\) |
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 | 912.6.a.v | 5 | |
| 4.b | odd | 2 | 1 | 228.6.a.d | ✓ | 5 | |
| 12.b | even | 2 | 1 | 684.6.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 228.6.a.d | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 684.6.a.f | 5 | 12.b | even | 2 | 1 | ||
| 912.6.a.v | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 6T_{5}^{4} - 11451T_{5}^{3} + 92232T_{5}^{2} + 32084460T_{5} - 590593248 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 9)^{5} \)
|
| $5$ |
\( T^{5} + 6 T^{4} + \cdots - 590593248 \)
|
| $7$ |
\( T^{5} + 54 T^{4} + \cdots + 400244992 \)
|
| $11$ |
\( T^{5} + \cdots + 14781508675200 \)
|
| $13$ |
\( T^{5} + \cdots - 8001780630720 \)
|
| $17$ |
\( T^{5} + \cdots + 146751918996600 \)
|
| $19$ |
\( (T + 361)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 14\!\cdots\!04 \)
|
| $29$ |
\( T^{5} + \cdots - 11\!\cdots\!92 \)
|
| $31$ |
\( T^{5} + \cdots + 10\!\cdots\!40 \)
|
| $37$ |
\( T^{5} + \cdots - 44\!\cdots\!84 \)
|
| $41$ |
\( T^{5} + \cdots - 27\!\cdots\!76 \)
|
| $43$ |
\( T^{5} + \cdots + 54\!\cdots\!16 \)
|
| $47$ |
\( T^{5} + \cdots - 12\!\cdots\!88 \)
|
| $53$ |
\( T^{5} + \cdots + 18\!\cdots\!32 \)
|
| $59$ |
\( T^{5} + \cdots + 50\!\cdots\!76 \)
|
| $61$ |
\( T^{5} + \cdots - 67\!\cdots\!80 \)
|
| $67$ |
\( T^{5} + \cdots + 96\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 91\!\cdots\!28 \)
|
| $73$ |
\( T^{5} + \cdots - 22\!\cdots\!88 \)
|
| $79$ |
\( T^{5} + \cdots - 62\!\cdots\!48 \)
|
| $83$ |
\( T^{5} + \cdots - 39\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots + 61\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots - 97\!\cdots\!00 \)
|
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