Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,6,Mod(1,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.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: | \(93.8245345906\) |
| 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: |
\( x^{9} - 4 x^{8} - 205 x^{7} + 608 x^{6} + 13727 x^{5} - 27536 x^{4} - 346839 x^{3} + 433844 x^{2} + \cdots - 3899136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4 x^{8} - 205 x^{7} + 608 x^{6} + 13727 x^{5} - 27536 x^{4} - 346839 x^{3} + 433844 x^{2} + \cdots - 3899136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 47 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 59 \nu^{8} + 351 \nu^{7} + 10912 \nu^{6} - 55376 \nu^{5} - 601957 \nu^{4} + 2476749 \nu^{3} + \cdots + 113984 ) / 45280 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 587 \nu^{8} - 2859 \nu^{7} - 109064 \nu^{6} + 410960 \nu^{5} + 6097941 \nu^{4} - 15477745 \nu^{3} + \cdots + 182771520 ) / 271680 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 109 \nu^{8} - 1013 \nu^{7} - 17128 \nu^{6} + 162320 \nu^{5} + 669267 \nu^{4} - 7383375 \nu^{3} + \cdots - 132330560 ) / 90560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 109 \nu^{8} - 1013 \nu^{7} - 17128 \nu^{6} + 162320 \nu^{5} + 669267 \nu^{4} - 7473935 \nu^{3} + \cdots - 141024320 ) / 90560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 277 \nu^{8} + 1245 \nu^{7} + 53092 \nu^{6} - 173992 \nu^{5} - 3185691 \nu^{4} + 6123863 \nu^{3} + \cdots - 246852672 ) / 135840 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 185 \nu^{8} + 1657 \nu^{7} + 31376 \nu^{6} - 261664 \nu^{5} - 1521415 \nu^{4} + 11355611 \nu^{3} + \cdots + 109852736 ) / 90560 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 47 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 3\beta_{2} + 76\beta _1 + 45 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 3\beta_{7} - 2\beta_{6} + 4\beta_{5} - 3\beta_{4} + 117\beta_{2} + 172\beta _1 + 3583 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{8} - 11 \beta_{7} - 143 \beta_{6} + 155 \beta_{5} - 17 \beta_{4} - 14 \beta_{3} + 503 \beta_{2} + \cdots + 8013 \)
|
| \(\nu^{6}\) | \(=\) |
\( 206 \beta_{8} - 498 \beta_{7} - 450 \beta_{6} + 850 \beta_{5} - 486 \beta_{4} + 20 \beta_{3} + \cdots + 328425 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1980 \beta_{8} - 2308 \beta_{7} - 17153 \beta_{6} + 19777 \beta_{5} - 3196 \beta_{4} - 2368 \beta_{3} + \cdots + 1138793 \)
|
| \(\nu^{8}\) | \(=\) |
\( 31229 \beta_{8} - 64903 \beta_{7} - 72630 \beta_{6} + 130552 \beta_{5} - 62335 \beta_{4} + \cdots + 33126463 \)
|
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 |
|
−9.94981 | 0 | 66.9988 | −25.0000 | 0 | −113.847 | −348.231 | 0 | 248.745 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.93845 | 0 | 16.1421 | −25.0000 | 0 | −138.927 | 110.029 | 0 | 173.461 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.88132 | 0 | 2.58998 | −25.0000 | 0 | 111.916 | 172.970 | 0 | 147.033 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.31435 | 0 | −30.2725 | −25.0000 | 0 | 80.6033 | 81.8477 | 0 | 32.8587 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.926735 | 0 | −31.1412 | −25.0000 | 0 | −27.0415 | 58.5151 | 0 | 23.1684 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.49910 | 0 | −1.75989 | −25.0000 | 0 | −180.806 | −185.649 | 0 | −137.478 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.69326 | 0 | 0.413187 | −25.0000 | 0 | 161.755 | −179.832 | 0 | −142.331 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.19157 | 0 | 35.1018 | −25.0000 | 0 | −132.038 | 25.4090 | 0 | −204.789 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 10.6267 | 0 | 80.9276 | −25.0000 | 0 | 85.3847 | 519.941 | 0 | −265.669 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 585.6.a.p | yes | 9 |
| 3.b | odd | 2 | 1 | 585.6.a.o | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 585.6.a.o | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 585.6.a.p | yes | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 5 T_{2}^{8} - 201 T_{2}^{7} + 855 T_{2}^{6} + 12972 T_{2}^{5} - 42890 T_{2}^{4} + \cdots + 1347840 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(585))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 5 T^{8} + \cdots + 1347840 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( (T + 25)^{9} \)
|
| $7$ |
\( T^{9} + \cdots + 12\!\cdots\!04 \)
|
| $11$ |
\( T^{9} + \cdots + 32\!\cdots\!76 \)
|
| $13$ |
\( (T - 169)^{9} \)
|
| $17$ |
\( T^{9} + \cdots - 23\!\cdots\!44 \)
|
| $19$ |
\( T^{9} + \cdots - 22\!\cdots\!64 \)
|
| $23$ |
\( T^{9} + \cdots - 10\!\cdots\!32 \)
|
| $29$ |
\( T^{9} + \cdots - 58\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 36\!\cdots\!72 \)
|
| $37$ |
\( T^{9} + \cdots - 11\!\cdots\!04 \)
|
| $41$ |
\( T^{9} + \cdots + 79\!\cdots\!36 \)
|
| $43$ |
\( T^{9} + \cdots - 99\!\cdots\!12 \)
|
| $47$ |
\( T^{9} + \cdots + 18\!\cdots\!28 \)
|
| $53$ |
\( T^{9} + \cdots + 29\!\cdots\!00 \)
|
| $59$ |
\( T^{9} + \cdots + 78\!\cdots\!84 \)
|
| $61$ |
\( T^{9} + \cdots + 22\!\cdots\!00 \)
|
| $67$ |
\( T^{9} + \cdots + 20\!\cdots\!00 \)
|
| $71$ |
\( T^{9} + \cdots + 60\!\cdots\!48 \)
|
| $73$ |
\( T^{9} + \cdots - 69\!\cdots\!00 \)
|
| $79$ |
\( T^{9} + \cdots + 73\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots - 41\!\cdots\!60 \)
|
| $89$ |
\( T^{9} + \cdots - 33\!\cdots\!60 \)
|
| $97$ |
\( T^{9} + \cdots - 73\!\cdots\!00 \)
|
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