Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5566,2,Mod(1,5566)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5566, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5566.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5566 = 2 \cdot 11^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5566.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: | \(44.4447337650\) |
| 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: |
\( x^{10} - 4x^{9} - 14x^{8} + 50x^{7} + 85x^{6} - 188x^{5} - 248x^{4} + 186x^{3} + 260x^{2} + 52x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 506) |
| 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
\( x^{10} - 4x^{9} - 14x^{8} + 50x^{7} + 85x^{6} - 188x^{5} - 248x^{4} + 186x^{3} + 260x^{2} + 52x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{9} - 17\nu^{8} + 28\nu^{7} + 97\nu^{6} - 246\nu^{5} - 130\nu^{4} + 294\nu^{3} + 156\nu^{2} + 228\nu - 20 ) / 164 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{9} + 22 \nu^{8} + 53 \nu^{7} - 222 \nu^{6} - 328 \nu^{5} + 694 \nu^{4} + 1110 \nu^{3} + \cdots - 196 ) / 164 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{9} - 22 \nu^{8} - 53 \nu^{7} + 222 \nu^{6} + 328 \nu^{5} - 612 \nu^{4} - 1356 \nu^{3} + \cdots + 524 ) / 164 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29 \nu^{9} - 103 \nu^{8} - 537 \nu^{7} + 1673 \nu^{6} + 3526 \nu^{5} - 7830 \nu^{4} - 10046 \nu^{3} + \cdots + 1924 ) / 656 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 35 \nu^{9} - 195 \nu^{8} - 207 \nu^{7} + 2169 \nu^{6} - 82 \nu^{5} - 7810 \nu^{4} + 1578 \nu^{3} + \cdots - 1580 ) / 656 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 51 \nu^{9} + 249 \nu^{8} + 475 \nu^{7} - 2863 \nu^{6} - 1722 \nu^{5} + 10162 \nu^{4} + 3614 \nu^{3} + \cdots - 556 ) / 328 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 93 \nu^{9} - 442 \nu^{8} - 953 \nu^{7} + 5228 \nu^{6} + 4100 \nu^{5} - 19370 \nu^{4} - 10068 \nu^{3} + \cdots - 520 ) / 328 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 105 \nu^{9} + 462 \nu^{8} + 1277 \nu^{7} - 5728 \nu^{6} - 6560 \nu^{5} + 22118 \nu^{4} + 16668 \nu^{3} + \cdots - 16 ) / 328 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{8} + \beta_{7} - 4\beta_{6} - 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{8} + 9\beta_{7} - 18\beta_{6} - 12\beta_{5} - \beta_{4} + 17\beta_{3} + 3\beta_{2} + 15\beta _1 + 50 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{9} + 34 \beta_{8} + 21 \beta_{7} - 66 \beta_{6} - 36 \beta_{5} - 9 \beta_{4} + 61 \beta_{3} + \cdots + 140 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{9} + 142 \beta_{8} + 101 \beta_{7} - 256 \beta_{6} - 158 \beta_{5} - 13 \beta_{4} + \cdots + 614 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 72 \beta_{9} + 458 \beta_{8} + 315 \beta_{7} - 940 \beta_{6} - 558 \beta_{5} - 61 \beta_{4} + \cdots + 2080 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 352 \beta_{9} + 1710 \beta_{8} + 1265 \beta_{7} - 3532 \beta_{6} - 2234 \beta_{5} - 73 \beta_{4} + \cdots + 8202 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1648 \beta_{9} + 5826 \beta_{8} + 4387 \beta_{7} - 13068 \beta_{6} - 8350 \beta_{5} - 161 \beta_{4} + \cdots + 29732 \)
|
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 |
|
−1.00000 | −2.81018 | 1.00000 | 2.25015 | 2.81018 | 3.09785 | −1.00000 | 4.89711 | −2.25015 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.32445 | 1.00000 | 4.41136 | 2.32445 | −3.63075 | −1.00000 | 2.40306 | −4.41136 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.12936 | 1.00000 | −3.00387 | 1.12936 | −1.41127 | −1.00000 | −1.72455 | 3.00387 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.452922 | 1.00000 | 3.47624 | 0.452922 | 1.39404 | −1.00000 | −2.79486 | −3.47624 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.941096 | 1.00000 | −1.56994 | −0.941096 | 3.96075 | −1.00000 | −2.11434 | 1.56994 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.37730 | 1.00000 | −3.06976 | −1.37730 | −1.21819 | −1.00000 | −1.10305 | 3.06976 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.71308 | 1.00000 | 1.76497 | −1.71308 | −3.74883 | −1.00000 | −0.0653420 | −1.76497 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.40934 | 1.00000 | 1.21899 | −2.40934 | −4.62522 | −1.00000 | 2.80492 | −1.21899 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 3.10337 | 1.00000 | 2.70739 | −3.10337 | 1.47043 | −1.00000 | 6.63091 | −2.70739 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 3.17272 | 1.00000 | 3.81446 | −3.17272 | 0.711206 | −1.00000 | 7.06614 | −3.81446 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(23\) | \(1\) |
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 | 5566.2.a.bt | 10 | |
| 11.b | odd | 2 | 1 | 5566.2.a.bu | 10 | ||
| 11.d | odd | 10 | 2 | 506.2.e.h | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 506.2.e.h | ✓ | 20 | 11.d | odd | 10 | 2 | |
| 5566.2.a.bt | 10 | 1.a | even | 1 | 1 | trivial | |
| 5566.2.a.bu | 10 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5566))\):
|
\( T_{3}^{10} - 6 T_{3}^{9} - 5 T_{3}^{8} + 86 T_{3}^{7} - 83 T_{3}^{6} - 336 T_{3}^{5} + 563 T_{3}^{4} + \cdots + 176 \)
|
|
\( T_{5}^{10} - 12 T_{5}^{9} + 30 T_{5}^{8} + 162 T_{5}^{7} - 881 T_{5}^{6} + 294 T_{5}^{5} + 5323 T_{5}^{4} + \cdots - 11099 \)
|
|
\( T_{7}^{10} + 4 T_{7}^{9} - 33 T_{7}^{8} - 118 T_{7}^{7} + 369 T_{7}^{6} + 1030 T_{7}^{5} - 1729 T_{7}^{4} + \cdots - 1936 \)
|
|
\( T_{13}^{10} - 3 T_{13}^{9} - 42 T_{13}^{8} + 159 T_{13}^{7} + 264 T_{13}^{6} - 1455 T_{13}^{5} + \cdots + 1936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( T^{10} - 6 T^{9} + \cdots + 176 \)
|
| $5$ |
\( T^{10} - 12 T^{9} + \cdots - 11099 \)
|
| $7$ |
\( T^{10} + 4 T^{9} + \cdots - 1936 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} - 3 T^{9} + \cdots + 1936 \)
|
| $17$ |
\( T^{10} - 4 T^{9} + \cdots - 48400 \)
|
| $19$ |
\( T^{10} + 8 T^{9} + \cdots - 132121 \)
|
| $23$ |
\( (T + 1)^{10} \)
|
| $29$ |
\( T^{10} - 15 T^{9} + \cdots + 3175664 \)
|
| $31$ |
\( T^{10} - 165 T^{8} + \cdots + 438064 \)
|
| $37$ |
\( T^{10} - 18 T^{9} + \cdots + 8080 \)
|
| $41$ |
\( T^{10} + 3 T^{9} + \cdots - 4189 \)
|
| $43$ |
\( T^{10} + \cdots - 402684496 \)
|
| $47$ |
\( T^{10} - 42 T^{9} + \cdots - 12827536 \)
|
| $53$ |
\( T^{10} - 11 T^{9} + \cdots - 9676909 \)
|
| $59$ |
\( T^{10} - 54 T^{9} + \cdots + 64784 \)
|
| $61$ |
\( T^{10} + 6 T^{9} + \cdots + 3605419 \)
|
| $67$ |
\( T^{10} - 24 T^{9} + \cdots + 84428096 \)
|
| $71$ |
\( T^{10} + \cdots - 107024399 \)
|
| $73$ |
\( T^{10} - 42 T^{9} + \cdots + 74951 \)
|
| $79$ |
\( T^{10} + \cdots - 393081280 \)
|
| $83$ |
\( T^{10} + 21 T^{9} + \cdots + 67031824 \)
|
| $89$ |
\( T^{10} + \cdots + 2135104144 \)
|
| $97$ |
\( T^{10} + \cdots - 1780772720 \)
|
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