Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4522,2,Mod(1,4522)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4522, 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("4522.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4522 = 2 \cdot 7 \cdot 17 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4522.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: | \(36.1083517940\) |
| 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: |
\( x^{11} - 5 x^{10} - 9 x^{9} + 72 x^{8} - 8 x^{7} - 342 x^{6} + 246 x^{5} + 575 x^{4} - 613 x^{3} + \cdots - 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5 x^{10} - 9 x^{9} + 72 x^{8} - 8 x^{7} - 342 x^{6} + 246 x^{5} + 575 x^{4} - 613 x^{3} + \cdots - 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 357 \nu^{10} - 1117 \nu^{9} - 5485 \nu^{8} + 17316 \nu^{7} + 26564 \nu^{6} - 89406 \nu^{5} + \cdots + 21456 ) / 4996 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 225 \nu^{10} - 683 \nu^{9} - 3289 \nu^{8} + 9528 \nu^{7} + 16784 \nu^{6} - 44656 \nu^{5} + \cdots - 2242 ) / 2498 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 607 \nu^{10} - 2431 \nu^{9} - 6919 \nu^{8} + 34564 \nu^{7} + 15792 \nu^{6} - 165114 \nu^{5} + \cdots + 69480 ) / 4996 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1279 \nu^{10} + 6003 \nu^{9} + 14011 \nu^{8} - 89200 \nu^{7} - 27884 \nu^{6} + 447434 \nu^{5} + \cdots - 214196 ) / 4996 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1417 \nu^{10} - 5889 \nu^{9} - 16761 \nu^{8} + 84852 \nu^{7} + 49236 \nu^{6} - 407810 \nu^{5} + \cdots + 142344 ) / 4996 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 927 \nu^{10} - 4013 \nu^{9} - 10653 \nu^{8} + 58440 \nu^{7} + 26784 \nu^{6} - 284802 \nu^{5} + \cdots + 131850 ) / 2498 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 744 \nu^{10} + 3241 \nu^{9} + 8744 \nu^{8} - 47643 \nu^{7} - 24807 \nu^{6} + 235309 \nu^{5} + \cdots - 89742 ) / 1249 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 9\beta_{2} + 12\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{10} + 3\beta_{8} - 2\beta_{7} - 2\beta_{6} + \beta_{4} + 13\beta_{3} + 15\beta_{2} + 58\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15 \beta_{10} + 9 \beta_{9} + 6 \beta_{8} - 17 \beta_{7} - 14 \beta_{6} + 13 \beta_{5} + \beta_{4} + \cdots + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( 37 \beta_{10} - 4 \beta_{9} + 50 \beta_{8} - 44 \beta_{7} - 31 \beta_{6} + 7 \beta_{5} + 17 \beta_{4} + \cdots + 22 \)
|
| \(\nu^{8}\) | \(=\) |
\( 183 \beta_{10} + 58 \beta_{9} + 123 \beta_{8} - 223 \beta_{7} - 151 \beta_{6} + 134 \beta_{5} + \cdots + 503 \)
|
| \(\nu^{9}\) | \(=\) |
\( 507 \beta_{10} - 73 \beta_{9} + 633 \beta_{8} - 645 \beta_{7} - 369 \beta_{6} + 156 \beta_{5} + \cdots + 174 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2084 \beta_{10} + 283 \beta_{9} + 1750 \beta_{8} - 2642 \beta_{7} - 1526 \beta_{6} + 1336 \beta_{5} + \cdots + 3034 \)
|
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 | −3.24103 | 1.00000 | −1.47048 | 3.24103 | 1.00000 | −1.00000 | 7.50430 | 1.47048 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.62716 | 1.00000 | 1.59890 | 2.62716 | 1.00000 | −1.00000 | 3.90198 | −1.59890 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.51750 | 1.00000 | 4.28314 | 2.51750 | 1.00000 | −1.00000 | 3.33779 | −4.28314 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.79665 | 1.00000 | −2.65580 | 1.79665 | 1.00000 | −1.00000 | 0.227934 | 2.65580 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.960801 | 1.00000 | −0.481825 | 0.960801 | 1.00000 | −1.00000 | −2.07686 | 0.481825 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.622180 | 1.00000 | 3.37396 | 0.622180 | 1.00000 | −1.00000 | −2.61289 | −3.37396 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −0.445823 | 1.00000 | −4.13183 | 0.445823 | 1.00000 | −1.00000 | −2.80124 | 4.13183 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0.837780 | 1.00000 | −1.01706 | −0.837780 | 1.00000 | −1.00000 | −2.29812 | 1.01706 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.79369 | 1.00000 | 1.12013 | −1.79369 | 1.00000 | −1.00000 | 0.217312 | −1.12013 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 2.05204 | 1.00000 | 0.623789 | −2.05204 | 1.00000 | −1.00000 | 1.21086 | −0.623789 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 2.52764 | 1.00000 | −3.24293 | −2.52764 | 1.00000 | −1.00000 | 3.38894 | 3.24293 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(17\) | \(1\) |
| \(19\) | \(-1\) |
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 | 4522.2.a.bc | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4522.2.a.bc | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4522))\):
|
\( T_{3}^{11} + 5 T_{3}^{10} - 9 T_{3}^{9} - 72 T_{3}^{8} - 8 T_{3}^{7} + 342 T_{3}^{6} + 246 T_{3}^{5} + \cdots + 80 \)
|
|
\( T_{5}^{11} + 2 T_{5}^{10} - 34 T_{5}^{9} - 69 T_{5}^{8} + 347 T_{5}^{7} + 688 T_{5}^{6} - 1135 T_{5}^{5} + \cdots - 414 \)
|
|
\( T_{11}^{11} + 9 T_{11}^{10} - 37 T_{11}^{9} - 393 T_{11}^{8} + 647 T_{11}^{7} + 5932 T_{11}^{6} + \cdots + 3816 \)
|
|
\( T_{13}^{11} - 78 T_{13}^{9} + 77 T_{13}^{8} + 2248 T_{13}^{7} - 4013 T_{13}^{6} - 27695 T_{13}^{5} + \cdots + 358144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{11} \)
|
| $3$ |
\( T^{11} + 5 T^{10} + \cdots + 80 \)
|
| $5$ |
\( T^{11} + 2 T^{10} + \cdots - 414 \)
|
| $7$ |
\( (T - 1)^{11} \)
|
| $11$ |
\( T^{11} + 9 T^{10} + \cdots + 3816 \)
|
| $13$ |
\( T^{11} - 78 T^{9} + \cdots + 358144 \)
|
| $17$ |
\( (T + 1)^{11} \)
|
| $19$ |
\( (T - 1)^{11} \)
|
| $23$ |
\( T^{11} + 8 T^{10} + \cdots - 12246624 \)
|
| $29$ |
\( T^{11} + 17 T^{10} + \cdots - 33300 \)
|
| $31$ |
\( T^{11} + \cdots + 313802144 \)
|
| $37$ |
\( T^{11} + 14 T^{10} + \cdots + 28798432 \)
|
| $41$ |
\( T^{11} + 16 T^{10} + \cdots + 4648608 \)
|
| $43$ |
\( T^{11} - 3 T^{10} + \cdots + 15122432 \)
|
| $47$ |
\( T^{11} + 12 T^{10} + \cdots + 4140 \)
|
| $53$ |
\( T^{11} + 34 T^{10} + \cdots - 15352704 \)
|
| $59$ |
\( T^{11} + \cdots - 435745152 \)
|
| $61$ |
\( T^{11} + \cdots - 500513110 \)
|
| $67$ |
\( T^{11} - 438 T^{9} + \cdots + 34561232 \)
|
| $71$ |
\( T^{11} + 42 T^{10} + \cdots - 398880 \)
|
| $73$ |
\( T^{11} - 14 T^{10} + \cdots - 599680 \)
|
| $79$ |
\( T^{11} + \cdots + 431554240 \)
|
| $83$ |
\( T^{11} - T^{10} + \cdots - 29955456 \)
|
| $89$ |
\( T^{11} + \cdots + 33673093884 \)
|
| $97$ |
\( T^{11} + \cdots - 14199650560 \)
|
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