Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5225,2,Mod(1,5225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, 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("5225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5225 = 5^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5225.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: | \(41.7218350561\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1045) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{8} + 6\nu^{7} + 47\nu^{6} - 35\nu^{5} - 110\nu^{4} + 112\nu^{3} + 26\nu^{2} - 126\nu + 13 ) / 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{8} - 12\nu^{7} - 94\nu^{6} + 41\nu^{5} + 307\nu^{4} - 50\nu^{3} - 400\nu^{2} + 78\nu + 177 ) / 29 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{8} + 22\nu^{7} + 37\nu^{6} - 196\nu^{5} - 94\nu^{4} + 488\nu^{3} + 76\nu^{2} - 288\nu + 9 ) / 29 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 16\nu^{7} + 39\nu^{6} + 103\nu^{5} - 277\nu^{4} - 202\nu^{3} + 472\nu^{2} + 162\nu - 170 ) / 29 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14\nu^{8} - 21\nu^{7} - 150\nu^{6} + 50\nu^{5} + 356\nu^{4} + 43\nu^{3} - 178\nu^{2} - 110\nu - 2 ) / 29 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -20\nu^{8} + 30\nu^{7} + 235\nu^{6} - 117\nu^{5} - 695\nu^{4} + 154\nu^{3} + 565\nu^{2} - 108\nu + 7 ) / 29 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 21\nu^{8} - 46\nu^{7} - 196\nu^{6} + 220\nu^{5} + 418\nu^{4} - 327\nu^{3} - 180\nu^{2} + 96\nu - 3 ) / 29 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16\nu^{8} - 53\nu^{7} - 130\nu^{6} + 372\nu^{5} + 324\nu^{4} - 738\nu^{3} - 249\nu^{2} + 388\nu + 35 ) / 29 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{8} - 7\beta_{7} - \beta_{6} + 9\beta_{5} + 4\beta_{4} - 9\beta_{3} + 3\beta_{2} + 6\beta _1 + 9 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -15\beta_{8} - 17\beta_{7} - 9\beta_{6} + 21\beta_{5} + 14\beta_{4} - 33\beta_{3} + 7\beta_{2} + 28\beta _1 + 57 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 51 \beta_{8} - 75 \beta_{7} - 21 \beta_{6} + 99 \beta_{5} + 51 \beta_{4} - 126 \beta_{3} + 34 \beta_{2} + \cdots + 155 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 201 \beta_{8} - 243 \beta_{7} - 99 \beta_{6} + 315 \beta_{5} + 188 \beta_{4} - 457 \beta_{3} + \cdots + 655 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 717 \beta_{8} - 939 \beta_{7} - 315 \beta_{6} + 1239 \beta_{5} + 681 \beta_{4} - 1688 \beta_{3} + \cdots + 2203 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2669 \beta_{8} - 3311 \beta_{7} - 1239 \beta_{6} + 4343 \beta_{5} + 2502 \beta_{4} - 6141 \beta_{3} + \cdots + 8381 ) / 2 \)
|
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 |
|
−2.66854 | −3.14505 | 5.12110 | 0 | 8.39270 | 4.63172 | −8.32878 | 6.89136 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.64409 | −0.185485 | 4.99120 | 0 | 0.490440 | −1.87571 | −7.90900 | −2.96560 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79978 | 2.53737 | 1.23919 | 0 | −4.56670 | −0.104026 | 1.36928 | 3.43826 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.749568 | −3.38871 | −1.43815 | 0 | 2.54007 | −1.17594 | 2.57713 | 8.48335 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.287410 | 0.930455 | −1.91740 | 0 | −0.267422 | −0.300889 | 1.12590 | −2.13425 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.0815632 | 0.583675 | −1.99335 | 0 | 0.0476064 | 3.83288 | −0.325710 | −2.65932 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.31671 | −2.24751 | −0.266275 | 0 | −2.95932 | −2.74324 | −2.98403 | 2.05132 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.53772 | 2.83656 | 0.364584 | 0 | 4.36184 | 2.01727 | −2.51481 | 5.04608 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.21339 | −0.921304 | 2.89909 | 0 | −2.03920 | 4.71795 | 1.99002 | −2.15120 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5225.2.a.q | 9 | |
| 5.b | even | 2 | 1 | 1045.2.a.j | ✓ | 9 | |
| 15.d | odd | 2 | 1 | 9405.2.a.bi | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.j | ✓ | 9 | 5.b | even | 2 | 1 | |
| 5225.2.a.q | 9 | 1.a | even | 1 | 1 | trivial | |
| 9405.2.a.bi | 9 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):
|
\( T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 27T_{2}^{6} + 25T_{2}^{5} + 70T_{2}^{4} - 21T_{2}^{3} - 51T_{2}^{2} - 8T_{2} + 1 \)
|
|
\( T_{7}^{9} - 9T_{7}^{8} + 3T_{7}^{7} + 136T_{7}^{6} - 137T_{7}^{5} - 701T_{7}^{4} + 316T_{7}^{3} + 1236T_{7}^{2} + 432T_{7} + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 3 T^{8} + \cdots + 1 \)
|
| $3$ |
\( T^{9} + 3 T^{8} + \cdots + 16 \)
|
| $5$ |
\( T^{9} \)
|
| $7$ |
\( T^{9} - 9 T^{8} + \cdots + 32 \)
|
| $11$ |
\( (T + 1)^{9} \)
|
| $13$ |
\( T^{9} - 3 T^{8} + \cdots - 4096 \)
|
| $17$ |
\( T^{9} + 5 T^{8} + \cdots + 69376 \)
|
| $19$ |
\( (T - 1)^{9} \)
|
| $23$ |
\( T^{9} + 4 T^{8} + \cdots + 4067776 \)
|
| $29$ |
\( T^{9} - 3 T^{8} + \cdots + 476032 \)
|
| $31$ |
\( T^{9} + T^{8} + \cdots + 1701376 \)
|
| $37$ |
\( T^{9} + 5 T^{8} + \cdots + 7003648 \)
|
| $41$ |
\( T^{9} + 5 T^{8} + \cdots - 2331392 \)
|
| $43$ |
\( T^{9} - 11 T^{8} + \cdots + 12703712 \)
|
| $47$ |
\( T^{9} + 30 T^{8} + \cdots - 3050368 \)
|
| $53$ |
\( T^{9} - T^{8} + \cdots + 746752 \)
|
| $59$ |
\( T^{9} - 59 T^{8} + \cdots - 7130816 \)
|
| $61$ |
\( T^{9} + 21 T^{8} + \cdots + 40008976 \)
|
| $67$ |
\( T^{9} - 2 T^{8} + \cdots + 2973104 \)
|
| $71$ |
\( T^{9} - 34 T^{8} + \cdots + 21829696 \)
|
| $73$ |
\( T^{9} - 34 T^{8} + \cdots - 14909696 \)
|
| $79$ |
\( T^{9} + 13 T^{8} + \cdots + 1052672 \)
|
| $83$ |
\( T^{9} + 51 T^{8} + \cdots - 20900704 \)
|
| $89$ |
\( T^{9} - 8 T^{8} + \cdots - 245309648 \)
|
| $97$ |
\( T^{9} - 8 T^{8} + \cdots + 50205184 \)
|
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