Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, 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("6039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6039.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: | \(48.2216577807\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2013) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6 \nu^{10} - 3 \nu^{9} - 80 \nu^{8} + 42 \nu^{7} + 340 \nu^{6} - 257 \nu^{5} - 506 \nu^{4} + 688 \nu^{3} + \cdots + 72 ) / 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7 \nu^{10} + 5 \nu^{9} - 99 \nu^{8} - 70 \nu^{7} + 459 \nu^{6} + 264 \nu^{5} - 834 \nu^{4} - 257 \nu^{3} + \cdots - 86 ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{10} - 9 \nu^{9} - 19 \nu^{8} + 126 \nu^{7} + 136 \nu^{6} - 584 \nu^{5} - 379 \nu^{4} + 1061 \nu^{3} + \cdots + 46 ) / 17 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 13 \nu^{10} - 2 \nu^{9} + 179 \nu^{8} + 45 \nu^{7} - 799 \nu^{6} - 211 \nu^{5} + 1323 \nu^{4} + \cdots + 31 ) / 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 20 \nu^{10} - 7 \nu^{9} + 278 \nu^{8} + 115 \nu^{7} - 1258 \nu^{6} - 475 \nu^{5} + 2157 \nu^{4} + \cdots + 66 ) / 17 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8 \nu^{10} - 21 \nu^{9} - 118 \nu^{8} + 277 \nu^{7} + 612 \nu^{6} - 1204 \nu^{5} - 1247 \nu^{4} + \cdots + 113 ) / 17 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23 \nu^{10} - 3 \nu^{9} - 318 \nu^{8} + 25 \nu^{7} + 1428 \nu^{6} - 189 \nu^{5} - 2359 \nu^{4} + \cdots + 89 ) / 17 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 20 \nu^{10} - 10 \nu^{9} - 278 \nu^{8} + 123 \nu^{7} + 1275 \nu^{6} - 613 \nu^{5} - 2225 \nu^{4} + \cdots + 36 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - 9\beta_{7} + 10\beta_{6} - 2\beta_{5} - 8\beta_{4} + 9\beta_{2} + 28\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 9 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - \beta_{4} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{10} + 13 \beta_{8} - 68 \beta_{7} + 81 \beta_{6} - 25 \beta_{5} - 54 \beta_{4} + 2 \beta_{3} + \cdots + 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 65 \beta_{10} - 25 \beta_{9} + 92 \beta_{8} - 123 \beta_{7} + 125 \beta_{6} - 108 \beta_{5} + \cdots + 434 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 20 \beta_{10} - 2 \beta_{9} + 125 \beta_{8} - 495 \beta_{7} + 612 \beta_{6} - 230 \beta_{5} + \cdots + 283 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 444 \beta_{10} - 221 \beta_{9} + 702 \beta_{8} - 1030 \beta_{7} + 1067 \beta_{6} - 870 \beta_{5} + \cdots + 2653 \)
|
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 |
|
−2.39095 | 0 | 3.71663 | 2.34211 | 0 | −0.680854 | −4.10437 | 0 | −5.59986 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.09663 | 0 | 2.39586 | 1.65820 | 0 | −1.74412 | −0.829977 | 0 | −3.47663 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.65258 | 0 | 0.731020 | −2.07534 | 0 | −0.262572 | 2.09709 | 0 | 3.42966 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.03852 | 0 | −0.921477 | 1.07279 | 0 | 1.21131 | 3.03401 | 0 | −1.11412 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0512060 | 0 | −1.99738 | −3.74211 | 0 | −4.65643 | −0.204690 | 0 | −0.191618 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.175402 | 0 | −1.96923 | 1.94648 | 0 | −2.33819 | −0.696212 | 0 | 0.341417 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.942842 | 0 | −1.11105 | −3.49194 | 0 | 1.81837 | −2.93323 | 0 | −3.29234 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.21038 | 0 | −0.534974 | 4.03730 | 0 | 0.314382 | −3.06829 | 0 | 4.88668 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.04468 | 0 | 2.18073 | 1.44717 | 0 | −4.07149 | 0.369530 | 0 | 2.95900 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.05394 | 0 | 2.21866 | 0.0248942 | 0 | 1.52855 | 0.449108 | 0 | 0.0511312 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.70023 | 0 | 5.29122 | −2.21957 | 0 | −2.11895 | 8.88702 | 0 | −5.99333 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
| \(61\) | \(-1\) |
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 | 6039.2.a.c | 11 | |
| 3.b | odd | 2 | 1 | 2013.2.a.b | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2013.2.a.b | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 6039.2.a.c | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 2 T_{2}^{10} - 14 T_{2}^{9} + 27 T_{2}^{8} + 66 T_{2}^{7} - 125 T_{2}^{6} - 115 T_{2}^{5} + \cdots - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots - 1 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} - T^{10} + \cdots - 71 \)
|
| $7$ |
\( T^{11} + 11 T^{10} + \cdots + 31 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} + 13 T^{10} + \cdots + 631 \)
|
| $17$ |
\( T^{11} - 13 T^{10} + \cdots - 626503 \)
|
| $19$ |
\( T^{11} + 12 T^{10} + \cdots - 10591 \)
|
| $23$ |
\( T^{11} - 3 T^{10} + \cdots + 28177 \)
|
| $29$ |
\( T^{11} + 2 T^{10} + \cdots - 8795203 \)
|
| $31$ |
\( T^{11} - T^{10} + \cdots + 1679 \)
|
| $37$ |
\( T^{11} + 14 T^{10} + \cdots - 7494199 \)
|
| $41$ |
\( T^{11} + 3 T^{10} + \cdots - 337149 \)
|
| $43$ |
\( T^{11} + 21 T^{10} + \cdots + 512533 \)
|
| $47$ |
\( T^{11} + \cdots + 117544331 \)
|
| $53$ |
\( T^{11} + \cdots + 508890247 \)
|
| $59$ |
\( T^{11} + 3 T^{10} + \cdots - 228413 \)
|
| $61$ |
\( (T - 1)^{11} \)
|
| $67$ |
\( T^{11} + 24 T^{10} + \cdots + 15755917 \)
|
| $71$ |
\( T^{11} + \cdots + 786742397 \)
|
| $73$ |
\( T^{11} + 42 T^{10} + \cdots - 10219489 \)
|
| $79$ |
\( T^{11} + 11 T^{10} + \cdots - 7961 \)
|
| $83$ |
\( T^{11} - 34 T^{10} + \cdots + 3176377 \)
|
| $89$ |
\( T^{11} + \cdots + 371355157 \)
|
| $97$ |
\( T^{11} + \cdots + 5102744443 \)
|
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