Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9196,2,Mod(1,9196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9196, 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("9196.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9196.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: | \(73.4304296988\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 16x^{8} - 3x^{7} + 84x^{6} + 16x^{5} - 174x^{4} - 16x^{3} + 122x^{2} - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 836) |
| 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
\( x^{10} - 16x^{8} - 3x^{7} + 84x^{6} + 16x^{5} - 174x^{4} - 16x^{3} + 122x^{2} - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10 \nu^{9} + 65 \nu^{8} - 159 \nu^{7} - 923 \nu^{6} + 601 \nu^{5} + 3645 \nu^{4} - 668 \nu^{3} + \cdots + 792 ) / 281 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15 \nu^{9} - 43 \nu^{8} - 98 \nu^{7} + 442 \nu^{6} - 363 \nu^{5} - 1417 \nu^{4} + 2932 \nu^{3} + \cdots + 64 ) / 281 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 58 \nu^{9} + 96 \nu^{8} - 1147 \nu^{7} - 1026 \nu^{6} + 6633 \nu^{5} + 2314 \nu^{4} - 13878 \nu^{3} + \cdots - 2319 ) / 843 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 62 \nu^{9} - 159 \nu^{8} - 761 \nu^{7} + 1752 \nu^{6} + 3108 \nu^{5} - 5782 \nu^{4} - 5097 \nu^{3} + \cdots + 21 ) / 843 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 34 \nu^{9} + 60 \nu^{8} + 372 \nu^{7} - 571 \nu^{6} - 1088 \nu^{5} + 1938 \nu^{4} + 529 \nu^{3} + \cdots + 904 ) / 281 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 38 \nu^{9} + 34 \nu^{8} + 548 \nu^{7} - 258 \nu^{6} - 2621 \nu^{5} + 480 \nu^{4} + 4674 \nu^{3} + \cdots - 312 ) / 281 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 53 \nu^{9} + 77 \nu^{8} + 646 \nu^{7} - 700 \nu^{6} - 2258 \nu^{5} + 1897 \nu^{4} + 2023 \nu^{3} + \cdots + 186 ) / 281 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 164 \nu^{9} + 339 \nu^{8} + 1877 \nu^{7} - 3465 \nu^{6} - 6372 \nu^{5} + 11596 \nu^{4} + 6684 \nu^{3} + \cdots + 162 ) / 843 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{6} + \beta_{5} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{9} + 2\beta_{8} - \beta_{7} - 7\beta_{6} + 11\beta_{5} + 2\beta_{3} + \beta_{2} + 10\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{9} + 12\beta_{8} - 9\beta_{7} - 13\beta_{6} + 19\beta_{5} + 10\beta_{3} + 2\beta_{2} + 45\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 64 \beta_{9} + 29 \beta_{8} - 13 \beta_{7} - 55 \beta_{6} + 102 \beta_{5} + 2 \beta_{4} + 27 \beta_{3} + \cdots + 94 \)
|
| \(\nu^{7}\) | \(=\) |
\( 138 \beta_{9} + 118 \beta_{8} - 73 \beta_{7} - 136 \beta_{6} + 226 \beta_{5} + \beta_{4} + 94 \beta_{3} + \cdots + 143 \)
|
| \(\nu^{8}\) | \(=\) |
\( 542 \beta_{9} + 317 \beta_{8} - 140 \beta_{7} - 472 \beta_{6} + 923 \beta_{5} + 27 \beta_{4} + \cdots + 692 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1370 \beta_{9} + 1109 \beta_{8} - 612 \beta_{7} - 1327 \beta_{6} + 2346 \beta_{5} + 25 \beta_{4} + \cdots + 1457 \)
|
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 |
|
0 | −3.05348 | 0 | 2.08326 | 0 | −3.50519 | 0 | 6.32373 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.99786 | 0 | 2.83044 | 0 | 3.85064 | 0 | 0.991458 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.43917 | 0 | −1.46040 | 0 | 2.94667 | 0 | −0.928777 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.09212 | 0 | −0.908763 | 0 | −2.29300 | 0 | −1.80726 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.295423 | 0 | 4.07897 | 0 | −1.80062 | 0 | −2.91273 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.282455 | 0 | 0.127925 | 0 | 0.161012 | 0 | −2.92022 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.12468 | 0 | 2.30867 | 0 | −1.20174 | 0 | −1.73509 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.02990 | 0 | 0.193372 | 0 | −2.66641 | 0 | 1.12049 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.16122 | 0 | 0.343496 | 0 | −0.282327 | 0 | 1.67087 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.27981 | 0 | −1.59697 | 0 | −0.209036 | 0 | 2.19752 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 9196.2.a.s | 10 | |
| 11.b | odd | 2 | 1 | 9196.2.a.t | 10 | ||
| 11.c | even | 5 | 2 | 836.2.j.b | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 836.2.j.b | ✓ | 20 | 11.c | even | 5 | 2 | |
| 9196.2.a.s | 10 | 1.a | even | 1 | 1 | trivial | |
| 9196.2.a.t | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):
|
\( T_{3}^{10} - 16T_{3}^{8} + 3T_{3}^{7} + 84T_{3}^{6} - 16T_{3}^{5} - 174T_{3}^{4} + 16T_{3}^{3} + 122T_{3}^{2} - 9 \)
|
|
\( T_{5}^{10} - 8T_{5}^{9} + 12T_{5}^{8} + 40T_{5}^{7} - 92T_{5}^{6} - 62T_{5}^{5} + 170T_{5}^{4} + 33T_{5}^{3} - 63T_{5}^{2} + 15T_{5} - 1 \)
|
|
\( T_{7}^{10} + 5 T_{7}^{9} - 14 T_{7}^{8} - 112 T_{7}^{7} - 90 T_{7}^{6} + 499 T_{7}^{5} + 1164 T_{7}^{4} + \cdots - 5 \)
|
|
\( T_{13}^{10} - 2 T_{13}^{9} - 72 T_{13}^{8} + 146 T_{13}^{7} + 1665 T_{13}^{6} - 3140 T_{13}^{5} + \cdots - 122561 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 16 T^{8} + \cdots - 9 \)
|
| $5$ |
\( T^{10} - 8 T^{9} + \cdots - 1 \)
|
| $7$ |
\( T^{10} + 5 T^{9} + \cdots - 5 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} - 2 T^{9} + \cdots - 122561 \)
|
| $17$ |
\( T^{10} + 4 T^{9} + \cdots + 45001 \)
|
| $19$ |
\( (T - 1)^{10} \)
|
| $23$ |
\( T^{10} + 8 T^{9} + \cdots - 133389 \)
|
| $29$ |
\( T^{10} - 3 T^{9} + \cdots - 9336809 \)
|
| $31$ |
\( T^{10} + 12 T^{9} + \cdots + 12595 \)
|
| $37$ |
\( T^{10} + 23 T^{9} + \cdots + 36765479 \)
|
| $41$ |
\( T^{10} - 5 T^{9} + \cdots - 75097625 \)
|
| $43$ |
\( T^{10} - 14 T^{9} + \cdots - 1045049 \)
|
| $47$ |
\( T^{10} + 27 T^{9} + \cdots + 7377655 \)
|
| $53$ |
\( T^{10} + 10 T^{9} + \cdots - 31405 \)
|
| $59$ |
\( T^{10} + 9 T^{9} + \cdots + 18438991 \)
|
| $61$ |
\( T^{10} + 4 T^{9} + \cdots - 6651 \)
|
| $67$ |
\( T^{10} + 17 T^{9} + \cdots - 4909999 \)
|
| $71$ |
\( T^{10} + 34 T^{9} + \cdots + 258655 \)
|
| $73$ |
\( T^{10} - 9 T^{9} + \cdots + 155705 \)
|
| $79$ |
\( T^{10} - 16 T^{9} + \cdots + 48879 \)
|
| $83$ |
\( T^{10} + 6 T^{9} + \cdots - 2354999 \)
|
| $89$ |
\( T^{10} + 35 T^{9} + \cdots + 19428559 \)
|
| $97$ |
\( T^{10} - 22 T^{9} + \cdots + 54515971 \)
|
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