Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2166,4,Mod(1,2166)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2166, 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("2166.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2166.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: | \(127.798137072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 603 x^{7} - 764 x^{6} + 123192 x^{5} + 325506 x^{4} - 10023031 x^{3} - 37119420 x^{2} + \cdots + 1077539768 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3\cdot 19 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 603 x^{7} - 764 x^{6} + 123192 x^{5} + 325506 x^{4} - 10023031 x^{3} - 37119420 x^{2} + \cdots + 1077539768 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 98\!\cdots\!47 \nu^{8} + \cdots - 11\!\cdots\!72 ) / 14\!\cdots\!16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 16\!\cdots\!07 \nu^{8} + \cdots - 19\!\cdots\!60 ) / 14\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23\!\cdots\!57 \nu^{8} + \cdots - 27\!\cdots\!20 ) / 71\!\cdots\!58 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17\!\cdots\!93 \nu^{8} + \cdots - 20\!\cdots\!28 ) / 47\!\cdots\!72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 72\!\cdots\!15 \nu^{8} + \cdots - 87\!\cdots\!72 ) / 14\!\cdots\!16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 40\!\cdots\!82 \nu^{8} + \cdots - 47\!\cdots\!92 ) / 71\!\cdots\!58 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 46\!\cdots\!61 \nu^{8} + \cdots + 52\!\cdots\!36 ) / 71\!\cdots\!58 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15\!\cdots\!47 \nu^{8} + \cdots + 17\!\cdots\!16 ) / 14\!\cdots\!16 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{4} - 5\beta_{2} + 19\beta_1 ) / 19 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{7} - 38\beta_{6} - 19\beta_{5} - 6\beta_{4} + 95\beta_{3} + 99\beta_{2} + 38\beta _1 + 2584 ) / 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 19 \beta_{8} + 19 \beta_{7} - 304 \beta_{6} - 19 \beta_{5} + 1166 \beta_{4} - 114 \beta_{3} + \cdots + 4902 ) / 19 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 665 \beta_{8} + 3439 \beta_{7} - 18278 \beta_{6} - 3971 \beta_{5} - 6592 \beta_{4} + 34048 \beta_{3} + \cdots + 511290 ) / 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14877 \beta_{8} + 1938 \beta_{7} - 129276 \beta_{6} + 4180 \beta_{5} + 497402 \beta_{4} + \cdots + 1450042 ) / 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 11262 \beta_{8} + 32851 \beta_{7} - 330534 \beta_{6} - 41819 \beta_{5} - 108932 \beta_{4} + \cdots + 6352184 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6797307 \beta_{8} - 1092766 \beta_{7} - 47083254 \beta_{6} + 3135608 \beta_{5} + 155725650 \beta_{4} + \cdots + 371965622 ) / 19 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 59383512 \beta_{8} + 126191426 \beta_{7} - 1952064560 \beta_{6} - 174303150 \beta_{5} + \cdots + 30854581460 ) / 19 \)
|
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 |
|
2.00000 | 3.00000 | 4.00000 | −13.1180 | 6.00000 | 16.6902 | 8.00000 | 9.00000 | −26.2360 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | −8.14090 | 6.00000 | −30.9921 | 8.00000 | 9.00000 | −16.2818 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | −5.12589 | 6.00000 | −33.6186 | 8.00000 | 9.00000 | −10.2518 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.00000 | 4.00000 | −4.54827 | 6.00000 | 10.5206 | 8.00000 | 9.00000 | −9.09654 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 3.00000 | 4.00000 | −3.33661 | 6.00000 | 33.6689 | 8.00000 | 9.00000 | −6.67322 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 3.00000 | 4.00000 | 10.1674 | 6.00000 | −32.1613 | 8.00000 | 9.00000 | 20.3348 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 3.00000 | 4.00000 | 14.5246 | 6.00000 | 22.7843 | 8.00000 | 9.00000 | 29.0491 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 3.00000 | 4.00000 | 15.1867 | 6.00000 | 0.939071 | 8.00000 | 9.00000 | 30.3733 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.00000 | 3.00000 | 4.00000 | 21.3910 | 6.00000 | 12.1690 | 8.00000 | 9.00000 | 42.7821 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 2166.4.a.bm | 9 | |
| 19.b | odd | 2 | 1 | 2166.4.a.bj | 9 | ||
| 19.e | even | 9 | 2 | 114.4.i.d | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.i.d | ✓ | 18 | 19.e | even | 9 | 2 | |
| 2166.4.a.bj | 9 | 19.b | odd | 2 | 1 | ||
| 2166.4.a.bm | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):
|
\( T_{5}^{9} - 27 T_{5}^{8} - 285 T_{5}^{7} + 9475 T_{5}^{6} + 37161 T_{5}^{5} - 1060797 T_{5}^{4} + \cdots + 398535336 \)
|
|
\( T_{13}^{9} - 99 T_{13}^{8} - 3393 T_{13}^{7} + 463647 T_{13}^{6} + 3374523 T_{13}^{5} + \cdots - 1472870495231 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{9} \)
|
| $3$ |
\( (T - 3)^{9} \)
|
| $5$ |
\( T^{9} - 27 T^{8} + \cdots + 398535336 \)
|
| $7$ |
\( T^{9} + \cdots + 51580149399 \)
|
| $11$ |
\( T^{9} + \cdots + 5252411442024 \)
|
| $13$ |
\( T^{9} + \cdots - 1472870495231 \)
|
| $17$ |
\( T^{9} + \cdots - 24\!\cdots\!16 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} + \cdots + 19\!\cdots\!48 \)
|
| $29$ |
\( T^{9} + \cdots + 48\!\cdots\!76 \)
|
| $31$ |
\( T^{9} + \cdots - 78\!\cdots\!71 \)
|
| $37$ |
\( T^{9} + \cdots + 11\!\cdots\!63 \)
|
| $41$ |
\( T^{9} + \cdots + 20\!\cdots\!00 \)
|
| $43$ |
\( T^{9} + \cdots - 82\!\cdots\!81 \)
|
| $47$ |
\( T^{9} + \cdots + 11\!\cdots\!76 \)
|
| $53$ |
\( T^{9} + \cdots + 34\!\cdots\!84 \)
|
| $59$ |
\( T^{9} + \cdots + 42\!\cdots\!44 \)
|
| $61$ |
\( T^{9} + \cdots - 11\!\cdots\!49 \)
|
| $67$ |
\( T^{9} + \cdots + 19\!\cdots\!76 \)
|
| $71$ |
\( T^{9} + \cdots + 15\!\cdots\!92 \)
|
| $73$ |
\( T^{9} + \cdots + 95\!\cdots\!19 \)
|
| $79$ |
\( T^{9} + \cdots + 70\!\cdots\!77 \)
|
| $83$ |
\( T^{9} + \cdots + 38\!\cdots\!12 \)
|
| $89$ |
\( T^{9} + \cdots + 18\!\cdots\!68 \)
|
| $97$ |
\( T^{9} + \cdots + 99\!\cdots\!19 \)
|
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