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: | \(0\) |
| 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: |
\( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \)
|
| 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
\( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4276 \nu^{8} + 993 \nu^{7} - 231050 \nu^{6} - 177378 \nu^{5} + 3665268 \nu^{4} + 4377940 \nu^{3} + \cdots - 1151901 ) / 1169766 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 64999 \nu^{8} + 190713 \nu^{7} - 3890801 \nu^{6} - 11363973 \nu^{5} + 68463819 \nu^{4} + \cdots - 40902156 ) / 15206958 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 24987 \nu^{8} - 7166 \nu^{7} - 1397001 \nu^{6} - 297443 \nu^{5} + 23628597 \nu^{4} + \cdots + 3317895 ) / 5068986 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 90230 \nu^{8} + 32184 \nu^{7} - 5333239 \nu^{6} - 3836655 \nu^{5} + 94118139 \nu^{4} + \cdots + 6113457 ) / 15206958 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 135002 \nu^{8} - 25263 \nu^{7} - 7390255 \nu^{6} - 2116755 \nu^{5} + 120654441 \nu^{4} + \cdots - 154170810 ) / 15206958 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 279719 \nu^{8} - 175299 \nu^{7} - 15606016 \nu^{6} + 2115444 \nu^{5} + 265872168 \nu^{4} + \cdots + 313226181 ) / 15206958 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 450169 \nu^{8} - 76689 \nu^{7} - 24966176 \nu^{6} - 9175896 \nu^{5} + 417633228 \nu^{4} + \cdots + 15335469 ) / 15206958 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 735613 \nu^{8} - 134067 \nu^{7} - 41110133 \nu^{6} - 11744913 \nu^{5} + 695505123 \nu^{4} + \cdots + 129805956 ) / 15206958 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 12\beta_{3} - \beta _1 - 5 ) / 13 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{7} - 3\beta_{6} + 10\beta_{5} - 10\beta_{3} - 20\beta _1 + 147 ) / 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 13 \beta_{8} + 28 \beta_{7} + 36 \beta_{6} + 49 \beta_{5} + 19 \beta_{4} - 258 \beta_{3} + 13 \beta_{2} + \cdots - 18 ) / 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13 \beta_{8} + 104 \beta_{7} - 53 \beta_{6} + 233 \beta_{5} - 46 \beta_{4} - 494 \beta_{3} + \cdots + 3056 ) / 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 377 \beta_{8} + 633 \beta_{7} + 987 \beta_{6} + 1585 \beta_{5} + 407 \beta_{4} - 6283 \beta_{3} + \cdots + 1824 ) / 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 507 \beta_{8} + 3033 \beta_{7} - 872 \beta_{6} + 6044 \beta_{5} - 2007 \beta_{4} - 17455 \beta_{3} + \cdots + 71121 ) / 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 8788 \beta_{8} + 14072 \beta_{7} + 25574 \beta_{6} + 46062 \beta_{5} + 8136 \beta_{4} - 162988 \beta_{3} + \cdots + 94223 ) / 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 15964 \beta_{8} + 82361 \beta_{7} - 9523 \beta_{6} + 167537 \beta_{5} - 69817 \beta_{4} + \cdots + 1753215 ) / 13 \)
|
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 |
|
−5.52257 | 0 | 22.4988 | 6.08065 | 0 | 20.2718 | −80.0709 | 0 | −33.5808 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.69820 | 0 | 14.0731 | −4.47249 | 0 | 27.2096 | −28.5326 | 0 | 21.0127 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.03025 | 0 | 8.24289 | −8.08864 | 0 | 5.95078 | −0.978887 | 0 | 32.5992 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.34727 | 0 | −2.49032 | −15.3991 | 0 | −10.1317 | 24.6236 | 0 | 36.1458 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.447278 | 0 | −7.79994 | −1.93073 | 0 | −8.14537 | 7.06697 | 0 | 0.863573 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.213700 | 0 | −7.95433 | 15.3391 | 0 | 32.3928 | −3.40944 | 0 | 3.27797 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.82093 | 0 | −0.0423641 | 3.41089 | 0 | 13.3442 | −22.6869 | 0 | 9.62187 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.17344 | 0 | 2.07074 | −6.74147 | 0 | −14.1726 | −18.8162 | 0 | −21.3937 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.83750 | 0 | 15.4014 | −21.1983 | 0 | 16.2806 | 35.8043 | 0 | −102.547 | ||||||||||||||||||||||||||||||||||||||||||||||
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.bf | 9 | |
| 3.b | odd | 2 | 1 | 507.4.a.p | yes | 9 | |
| 13.b | even | 2 | 1 | 1521.4.a.bi | 9 | ||
| 39.d | odd | 2 | 1 | 507.4.a.o | ✓ | 9 | |
| 39.f | even | 4 | 2 | 507.4.b.k | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 507.4.a.o | ✓ | 9 | 39.d | odd | 2 | 1 | |
| 507.4.a.p | yes | 9 | 3.b | odd | 2 | 1 | |
| 507.4.b.k | 18 | 39.f | even | 4 | 2 | ||
| 1521.4.a.bf | 9 | 1.a | even | 1 | 1 | trivial | |
| 1521.4.a.bi | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):
|
\( T_{2}^{9} + 6T_{2}^{8} - 40T_{2}^{7} - 251T_{2}^{6} + 452T_{2}^{5} + 3075T_{2}^{4} - 1401T_{2}^{3} - 11386T_{2}^{2} - 2288T_{2} + 1016 \)
|
|
\( T_{5}^{9} + 33 T_{5}^{8} - 8 T_{5}^{7} - 8831 T_{5}^{6} - 66645 T_{5}^{5} + 239411 T_{5}^{4} + \cdots - 48900601 \)
|
|
\( T_{7}^{9} - 83 T_{7}^{8} + 1920 T_{7}^{7} + 9589 T_{7}^{6} - 809119 T_{7}^{5} + 4484863 T_{7}^{4} + \cdots + 27017466139 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 6 T^{8} + \cdots + 1016 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + 33 T^{8} + \cdots - 48900601 \)
|
| $7$ |
\( T^{9} + \cdots + 27017466139 \)
|
| $11$ |
\( T^{9} + \cdots - 1288234570811 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} + \cdots + 25\!\cdots\!76 \)
|
| $19$ |
\( T^{9} + \cdots - 17\!\cdots\!76 \)
|
| $23$ |
\( T^{9} + \cdots - 60\!\cdots\!44 \)
|
| $29$ |
\( T^{9} + \cdots - 34\!\cdots\!27 \)
|
| $31$ |
\( T^{9} + \cdots - 40\!\cdots\!53 \)
|
| $37$ |
\( T^{9} + \cdots - 80\!\cdots\!56 \)
|
| $41$ |
\( T^{9} + \cdots - 36\!\cdots\!68 \)
|
| $43$ |
\( T^{9} + \cdots + 31\!\cdots\!88 \)
|
| $47$ |
\( T^{9} + \cdots - 38\!\cdots\!12 \)
|
| $53$ |
\( T^{9} + \cdots - 65\!\cdots\!51 \)
|
| $59$ |
\( T^{9} + \cdots + 14\!\cdots\!76 \)
|
| $61$ |
\( T^{9} + \cdots - 61\!\cdots\!68 \)
|
| $67$ |
\( T^{9} + \cdots + 81\!\cdots\!92 \)
|
| $71$ |
\( T^{9} + \cdots + 71\!\cdots\!12 \)
|
| $73$ |
\( T^{9} + \cdots + 19\!\cdots\!77 \)
|
| $79$ |
\( T^{9} + \cdots + 11\!\cdots\!83 \)
|
| $83$ |
\( T^{9} + \cdots + 20\!\cdots\!97 \)
|
| $89$ |
\( T^{9} + \cdots - 55\!\cdots\!72 \)
|
| $97$ |
\( T^{9} + \cdots + 31\!\cdots\!39 \)
|
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