Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1521,4,Mod(1,1521)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1521.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1521 = 3^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1521.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: | \(89.7419051187\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 13^{2} \) |
| Twist minimal: | no (minimal twist has level 507) |
| 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
\( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{8} - 719 \nu^{7} + 1734 \nu^{6} + 27181 \nu^{5} - 27838 \nu^{4} - 334818 \nu^{3} + \cdots + 513836 ) / 198848 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3 \nu^{8} + 719 \nu^{7} - 1734 \nu^{6} - 27181 \nu^{5} + 27838 \nu^{4} + 334818 \nu^{3} + \cdots - 2502316 ) / 198848 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 127 \nu^{8} + 483 \nu^{7} - 10310 \nu^{6} - 16209 \nu^{5} + 248606 \nu^{4} + 170474 \nu^{3} + \cdots + 3265492 ) / 198848 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 171 \nu^{8} + 831 \nu^{7} - 8974 \nu^{6} - 35013 \nu^{5} + 139382 \nu^{4} + 389090 \nu^{3} + \cdots - 168892 ) / 198848 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 383 \nu^{8} - 1291 \nu^{7} + 14878 \nu^{6} + 61649 \nu^{5} - 141702 \nu^{4} - 860826 \nu^{3} + \cdots + 2264092 ) / 198848 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 743 \nu^{8} - 2293 \nu^{7} - 33670 \nu^{6} + 79991 \nu^{5} + 493806 \nu^{4} - 772630 \nu^{3} + \cdots + 2379924 ) / 198848 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 591 \nu^{8} + 1757 \nu^{7} + 24206 \nu^{6} - 67775 \nu^{5} - 300654 \nu^{4} + 771566 \nu^{3} + \cdots - 1800516 ) / 99424 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{3} + 17\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - 2\beta_{6} - 3\beta_{5} + 4\beta_{4} + 21\beta_{3} + 27\beta_{2} + \beta _1 + 161 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{8} - 15\beta_{7} - 21\beta_{6} - 76\beta_{5} + 33\beta_{4} + 44\beta_{3} + 15\beta_{2} + 313\beta _1 + 146 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{8} - 56 \beta_{7} - 87 \beta_{6} - 125 \beta_{5} + 127 \beta_{4} + 434 \beta_{3} + \cdots + 3031 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 404 \beta_{8} - 655 \beta_{7} - 449 \beta_{6} - 1646 \beta_{5} + 919 \beta_{4} + 1311 \beta_{3} + \cdots + 3912 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 714 \beta_{8} - 2012 \beta_{7} - 2778 \beta_{6} - 3688 \beta_{5} + 3420 \beta_{4} + 9290 \beta_{3} + \cdots + 61010 \)
|
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 |
|
−5.48584 | 0 | 22.0945 | −13.3185 | 0 | 21.4234 | −77.3200 | 0 | 73.0635 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.76649 | 0 | 14.7194 | −18.8390 | 0 | −23.8593 | −32.0282 | 0 | 89.7961 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.37739 | 0 | 3.40677 | 15.7127 | 0 | −17.1681 | 15.5131 | 0 | −53.0679 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.73419 | 0 | −0.524213 | −21.1246 | 0 | 25.8618 | 23.3068 | 0 | 57.7586 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.61464 | 0 | −5.39293 | −1.20859 | 0 | 28.2769 | 21.6248 | 0 | 1.95145 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.05129 | 0 | −6.89480 | −17.8886 | 0 | −30.1975 | −15.6587 | 0 | −18.8061 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.86460 | 0 | −4.52327 | 2.36060 | 0 | −4.86461 | −23.3509 | 0 | 4.40158 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.23649 | 0 | 2.47490 | 13.5815 | 0 | 1.42933 | −17.8820 | 0 | 43.9566 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.82618 | 0 | 6.63963 | −0.275426 | 0 | 0.0981245 | −5.20502 | 0 | −1.05383 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1521.4.a.be | 9 | |
| 3.b | odd | 2 | 1 | 507.4.a.q | yes | 9 | |
| 13.b | even | 2 | 1 | 1521.4.a.bj | 9 | ||
| 39.d | odd | 2 | 1 | 507.4.a.n | ✓ | 9 | |
| 39.f | even | 4 | 2 | 507.4.b.j | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 507.4.a.n | ✓ | 9 | 39.d | odd | 2 | 1 | |
| 507.4.a.q | yes | 9 | 3.b | odd | 2 | 1 | |
| 507.4.b.j | 18 | 39.f | even | 4 | 2 | ||
| 1521.4.a.be | 9 | 1.a | even | 1 | 1 | trivial | |
| 1521.4.a.bj | 9 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):
|
\( T_{2}^{9} + 8T_{2}^{8} - 20T_{2}^{7} - 251T_{2}^{6} + 8T_{2}^{5} + 2521T_{2}^{4} + 1303T_{2}^{3} - 8946T_{2}^{2} - 3640T_{2} + 9464 \)
|
|
\( T_{5}^{9} + 41 T_{5}^{8} - 28 T_{5}^{7} - 18187 T_{5}^{6} - 130261 T_{5}^{5} + 2089839 T_{5}^{4} + \cdots - 15899689 \)
|
|
\( T_{7}^{9} - T_{7}^{8} - 1864 T_{7}^{7} + 715 T_{7}^{6} + 1103305 T_{7}^{5} + 1074885 T_{7}^{4} + \cdots - 132217379 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 8 T^{8} + \cdots + 9464 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + 41 T^{8} + \cdots - 15899689 \)
|
| $7$ |
\( T^{9} - T^{8} + \cdots - 132217379 \)
|
| $11$ |
\( T^{9} + \cdots - 982063511267 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} + \cdots - 7231752990904 \)
|
| $19$ |
\( T^{9} + \cdots + 82\!\cdots\!68 \)
|
| $23$ |
\( T^{9} + \cdots - 91\!\cdots\!88 \)
|
| $29$ |
\( T^{9} + \cdots + 12\!\cdots\!83 \)
|
| $31$ |
\( T^{9} + \cdots + 68\!\cdots\!89 \)
|
| $37$ |
\( T^{9} + \cdots + 52\!\cdots\!04 \)
|
| $41$ |
\( T^{9} + \cdots - 52\!\cdots\!48 \)
|
| $43$ |
\( T^{9} + \cdots - 14\!\cdots\!52 \)
|
| $47$ |
\( T^{9} + \cdots + 26\!\cdots\!72 \)
|
| $53$ |
\( T^{9} + \cdots - 26\!\cdots\!69 \)
|
| $59$ |
\( T^{9} + \cdots - 64\!\cdots\!72 \)
|
| $61$ |
\( T^{9} + \cdots - 18\!\cdots\!04 \)
|
| $67$ |
\( T^{9} + \cdots - 38\!\cdots\!64 \)
|
| $71$ |
\( T^{9} + \cdots + 48\!\cdots\!84 \)
|
| $73$ |
\( T^{9} + \cdots - 15\!\cdots\!77 \)
|
| $79$ |
\( T^{9} + \cdots - 63\!\cdots\!89 \)
|
| $83$ |
\( T^{9} + \cdots - 17\!\cdots\!51 \)
|
| $89$ |
\( T^{9} + \cdots + 32\!\cdots\!32 \)
|
| $97$ |
\( T^{9} + \cdots + 25\!\cdots\!69 \)
|
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