Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6027,2,Mod(1,6027)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, 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("6027.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6027 = 3 \cdot 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6027.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: | \(48.1258372982\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4x^{9} - 11x^{8} + 56x^{7} + 26x^{6} - 266x^{5} + 52x^{4} + 526x^{3} - 255x^{2} - 372x + 239 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4x^{9} - 11x^{8} + 56x^{7} + 26x^{6} - 266x^{5} + 52x^{4} + 526x^{3} - 255x^{2} - 372x + 239 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 8\nu^{6} + 46\nu^{5} - 2\nu^{4} - 152\nu^{3} + 90\nu^{2} + 154\nu - 121 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - 4\nu^{8} - 8\nu^{7} + 46\nu^{6} - 4\nu^{5} - 148\nu^{4} + 110\nu^{3} + 120\nu^{2} - 163\nu + 52 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} + 5\nu^{8} + 4\nu^{7} - 54\nu^{6} + 50\nu^{5} + 148\nu^{4} - 262\nu^{3} - 46\nu^{2} + 315\nu - 153 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{8} + 4\nu^{7} + 9\nu^{6} - 47\nu^{5} - 9\nu^{4} + 160\nu^{3} - 59\nu^{2} - 165\nu + 102 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{8} - 4\nu^{7} - 9\nu^{6} + 48\nu^{5} + 8\nu^{4} - 169\nu^{3} + 67\nu^{2} + 180\nu - 114 \)
|
| \(\beta_{8}\) | \(=\) |
\( -\nu^{8} + 4\nu^{7} + 9\nu^{6} - 48\nu^{5} - 8\nu^{4} + 170\nu^{3} - 68\nu^{2} - 185\nu + 117 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -2\nu^{9} + 9\nu^{8} + 14\nu^{7} - 104\nu^{6} + 28\nu^{5} + 342\nu^{4} - 268\nu^{3} - 318\nu^{2} + 348\nu - 57 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{8} + 10\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 9\beta_{2} + 31\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + 2\beta_{6} + 12\beta_{5} + 12\beta_{4} - 10\beta_{3} + 58\beta_{2} + 13\beta _1 + 140 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{9} + 67 \beta_{8} + 82 \beta_{7} + 17 \beta_{6} + 13 \beta_{5} + 15 \beta_{4} - 10 \beta_{3} + \cdots + 95 \)
|
| \(\nu^{8}\) | \(=\) |
\( 4 \beta_{9} + 14 \beta_{8} + 36 \beta_{7} + 38 \beta_{6} + 104 \beta_{5} + 112 \beta_{4} - 74 \beta_{3} + \cdots + 951 \)
|
| \(\nu^{9}\) | \(=\) |
\( 24 \beta_{9} + 472 \beta_{8} + 638 \beta_{7} + 200 \beta_{6} + 120 \beta_{5} + 170 \beta_{4} + \cdots + 782 \)
|
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 |
|
−2.58399 | 1.00000 | 4.67700 | 0.798812 | −2.58399 | 0 | −6.91733 | 1.00000 | −2.06412 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.69562 | 1.00000 | 0.875138 | −0.871357 | −1.69562 | 0 | 1.90734 | 1.00000 | 1.47749 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.49626 | 1.00000 | 0.238800 | −0.660548 | −1.49626 | 0 | 2.63522 | 1.00000 | 0.988352 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.44855 | 1.00000 | 0.0982847 | 3.93238 | −1.44855 | 0 | 2.75472 | 1.00000 | −5.69623 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.775610 | 1.00000 | −1.39843 | 2.32372 | 0.775610 | 0 | −2.63586 | 1.00000 | 1.80230 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.32624 | 1.00000 | −0.241096 | −0.903630 | 1.32624 | 0 | −2.97222 | 1.00000 | −1.19843 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.55584 | 1.00000 | 0.420641 | −3.26436 | 1.55584 | 0 | −2.45723 | 1.00000 | −5.07882 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.13183 | 1.00000 | 2.54468 | 2.36072 | 2.13183 | 0 | 1.16116 | 1.00000 | 5.03264 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.62834 | 1.00000 | 4.90817 | −1.82844 | 2.62834 | 0 | 7.64366 | 1.00000 | −4.80575 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.80657 | 1.00000 | 5.87681 | 4.11270 | 2.80657 | 0 | 10.8805 | 1.00000 | 11.5426 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 6027.2.a.be | yes | 10 |
| 7.b | odd | 2 | 1 | 6027.2.a.bd | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6027.2.a.bd | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 6027.2.a.be | yes | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):
|
\( T_{2}^{10} - 4 T_{2}^{9} - 11 T_{2}^{8} + 56 T_{2}^{7} + 26 T_{2}^{6} - 266 T_{2}^{5} + 52 T_{2}^{4} + \cdots + 239 \)
|
|
\( T_{5}^{10} - 6 T_{5}^{9} - 12 T_{5}^{8} + 106 T_{5}^{7} + 14 T_{5}^{6} - 534 T_{5}^{5} - 16 T_{5}^{4} + \cdots - 220 \)
|
|
\( T_{13}^{10} - 82 T_{13}^{8} + 28 T_{13}^{7} + 2122 T_{13}^{6} - 1016 T_{13}^{5} - 17290 T_{13}^{4} + \cdots - 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 4 T^{9} + \cdots + 239 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} - 6 T^{9} + \cdots - 220 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + 2 T^{9} + \cdots + 320 \)
|
| $13$ |
\( T^{10} - 82 T^{8} + \cdots - 28 \)
|
| $17$ |
\( T^{10} - 8 T^{9} + \cdots - 2896 \)
|
| $19$ |
\( T^{10} - 6 T^{9} + \cdots + 3136 \)
|
| $23$ |
\( T^{10} - 132 T^{8} + \cdots - 1883344 \)
|
| $29$ |
\( T^{10} - 16 T^{9} + \cdots - 21536 \)
|
| $31$ |
\( T^{10} - 2 T^{9} + \cdots + 3760528 \)
|
| $37$ |
\( T^{10} - 24 T^{9} + \cdots + 1431556 \)
|
| $41$ |
\( (T - 1)^{10} \)
|
| $43$ |
\( T^{10} - 8 T^{9} + \cdots + 17235904 \)
|
| $47$ |
\( T^{10} + 8 T^{9} + \cdots + 2048 \)
|
| $53$ |
\( T^{10} - 24 T^{9} + \cdots - 8130800 \)
|
| $59$ |
\( T^{10} - 6 T^{9} + \cdots + 5671168 \)
|
| $61$ |
\( T^{10} + 14 T^{9} + \cdots - 889600 \)
|
| $67$ |
\( T^{10} - 26 T^{9} + \cdots - 62987776 \)
|
| $71$ |
\( T^{10} - 14 T^{9} + \cdots - 39536240 \)
|
| $73$ |
\( T^{10} + 36 T^{9} + \cdots + 72577600 \)
|
| $79$ |
\( T^{10} - 20 T^{9} + \cdots + 5070256 \)
|
| $83$ |
\( T^{10} - 40 T^{9} + \cdots + 4120576 \)
|
| $89$ |
\( T^{10} + \cdots - 174038336 \)
|
| $97$ |
\( T^{10} - 16 T^{9} + \cdots - 10394972 \)
|
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