Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8619,2,Mod(1,8619)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8619.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8619.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: | \(68.8230615021\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.15187408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 8x^{3} + 10x^{2} - 6x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 663) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 8x^{3} + 10x^{2} - 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 6\nu^{2} + 5\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 9\beta_{3} + 20\beta_{2} + 32\beta _1 + 10 \)
|
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 |
|
−2.76451 | 1.00000 | 5.64250 | 3.16934 | −2.76451 | 1.71287 | −10.0697 | 1.00000 | −8.76167 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.12890 | 1.00000 | 2.53223 | −3.11175 | −2.12890 | 2.51704 | −1.13307 | 1.00000 | 6.62462 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.47460 | 1.00000 | 0.174440 | −0.534832 | −1.47460 | −3.91274 | 2.69197 | 1.00000 | 0.788662 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0950396 | 1.00000 | −1.99097 | 1.53247 | −0.0950396 | 0.912487 | 0.379300 | 1.00000 | −0.145645 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.679939 | 1.00000 | −1.53768 | −4.08647 | 0.679939 | −3.77967 | −2.40541 | 1.00000 | −2.77855 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.78311 | 1.00000 | 1.17948 | −0.968763 | 1.78311 | 0.550014 | −1.46308 | 1.00000 | −1.72741 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
| \(17\) | \(-1\) |
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 | 8619.2.a.y | 6 | |
| 13.b | even | 2 | 1 | 663.2.a.i | ✓ | 6 | |
| 39.d | odd | 2 | 1 | 1989.2.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.a.i | ✓ | 6 | 13.b | even | 2 | 1 | |
| 1989.2.a.n | 6 | 39.d | odd | 2 | 1 | ||
| 8619.2.a.y | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
|
\( T_{2}^{6} + 4T_{2}^{5} - T_{2}^{4} - 16T_{2}^{3} - 7T_{2}^{2} + 10T_{2} + 1 \)
|
|
\( T_{5}^{6} + 4T_{5}^{5} - 12T_{5}^{4} - 48T_{5}^{3} + 16T_{5}^{2} + 80T_{5} + 32 \)
|
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\( T_{7}^{6} + 2T_{7}^{5} - 18T_{7}^{4} - 8T_{7}^{3} + 100T_{7}^{2} - 108T_{7} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 32 \)
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| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 32 \)
|
| $11$ |
\( T^{6} + 8 T^{5} + \cdots + 64 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T - 1)^{6} \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots - 1024 \)
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| $23$ |
\( T^{6} - 88 T^{4} + \cdots - 2176 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots - 416 \)
|
| $31$ |
\( T^{6} - 10 T^{5} + \cdots + 22912 \)
|
| $37$ |
\( T^{6} + 2 T^{5} + \cdots + 2344 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 1312 \)
|
| $43$ |
\( T^{6} + 16 T^{5} + \cdots + 14528 \)
|
| $47$ |
\( T^{6} + 14 T^{5} + \cdots + 256 \)
|
| $53$ |
\( T^{6} - 8 T^{5} + \cdots - 10816 \)
|
| $59$ |
\( T^{6} + 18 T^{5} + \cdots + 70736 \)
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| $61$ |
\( T^{6} + 12 T^{5} + \cdots + 647744 \)
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| $67$ |
\( T^{6} - 8 T^{5} + \cdots - 20992 \)
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| $71$ |
\( T^{6} + 12 T^{5} + \cdots + 128 \)
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| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 110888 \)
|
| $79$ |
\( T^{6} + 8 T^{5} + \cdots + 2048 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots - 194192 \)
|
| $89$ |
\( T^{6} + 26 T^{5} + \cdots + 114472 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots - 1819096 \)
|
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