Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,6,Mod(1,531)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(531, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("531.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 531.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: | \(85.1638083207\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 177) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 350941 \nu^{11} - 1428132 \nu^{10} - 70855647 \nu^{9} + 91864075 \nu^{8} + \cdots - 17699688792576 ) / 277202304512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8192145 \nu^{11} - 67427634 \nu^{10} + 2640430613 \nu^{9} + 18376727045 \nu^{8} + \cdots + 10\!\cdots\!40 ) / 3049225349632 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 299377 \nu^{11} - 524416 \nu^{10} + 88399319 \nu^{9} + 187648889 \nu^{8} + \cdots + 16317679113344 ) / 108900905344 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5842853 \nu^{11} + 35029081 \nu^{10} - 1903226618 \nu^{9} - 8347251655 \nu^{8} + \cdots - 324693088418048 ) / 1524612674816 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1506791 \nu^{11} + 7464662 \nu^{10} - 465055595 \nu^{9} - 2148263267 \nu^{8} + \cdots - 94789227549184 ) / 277202304512 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 100767 \nu^{11} + 53452 \nu^{10} - 28964453 \nu^{9} - 48656271 \nu^{8} + 2821805338 \nu^{7} + \cdots - 4701282618368 ) / 11773070848 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 14438755 \nu^{11} + 3300860 \nu^{10} - 4146949925 \nu^{9} - 4582591287 \nu^{8} + \cdots - 156500452428544 ) / 1524612674816 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 24199755 \nu^{11} + 1545329 \nu^{10} - 6527695920 \nu^{9} - 8306848553 \nu^{8} + \cdots - 172136514045440 ) / 1524612674816 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 25955403 \nu^{11} + 47327234 \nu^{10} + 6681309643 \nu^{9} + 346810155 \nu^{8} + \cdots + 849314392654336 ) / 1524612674816 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} + \beta_{4} + 5\beta_{2} + 86\beta _1 + 86 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} + 6 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + \cdots + 3807 \)
|
| \(\nu^{5}\) | \(=\) |
\( 30 \beta_{11} + 18 \beta_{10} + 4 \beta_{9} + 20 \beta_{8} + 162 \beta_{7} - 42 \beta_{6} + 114 \beta_{5} + \cdots + 18767 \)
|
| \(\nu^{6}\) | \(=\) |
\( 606 \beta_{11} + 550 \beta_{10} - 40 \beta_{9} + 420 \beta_{8} + 1413 \beta_{7} - 994 \beta_{6} + \cdots + 411640 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6783 \beta_{11} + 4559 \beta_{10} + 584 \beta_{9} + 4402 \beta_{8} + 24374 \beta_{7} - 10209 \beta_{6} + \cdots + 3226133 \)
|
| \(\nu^{8}\) | \(=\) |
\( 100910 \beta_{11} + 85242 \beta_{10} - 6060 \beta_{9} + 69276 \beta_{8} + 253564 \beta_{7} + \cdots + 52195939 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1187260 \beta_{11} + 850124 \beta_{10} + 55096 \beta_{9} + 768448 \beta_{8} + 3661489 \beta_{7} + \cdots + 518319894 \)
|
| \(\nu^{10}\) | \(=\) |
\( 15917723 \beta_{11} + 12861355 \beta_{10} - 684976 \beta_{9} + 10742610 \beta_{8} + 41567962 \beta_{7} + \cdots + 7274855475 \)
|
| \(\nu^{11}\) | \(=\) |
\( 191789770 \beta_{11} + 142131150 \beta_{10} + 3961508 \beta_{9} + 124512460 \beta_{8} + \cdots + 81136464075 \)
|
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 |
|
−10.5707 | 0 | 79.7401 | 45.7686 | 0 | −1.90644 | −504.647 | 0 | −483.807 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −10.4920 | 0 | 78.0815 | −46.0665 | 0 | 183.145 | −483.486 | 0 | 483.328 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −9.23155 | 0 | 53.2215 | −89.5822 | 0 | −121.386 | −195.908 | 0 | 826.983 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −7.43261 | 0 | 23.2437 | −48.3022 | 0 | 120.542 | 65.0821 | 0 | 359.012 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −5.21799 | 0 | −4.77260 | 21.6600 | 0 | 150.168 | 191.879 | 0 | −113.022 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.68934 | 0 | −24.7675 | 83.5049 | 0 | −48.3401 | 152.667 | 0 | −224.573 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.615542 | 0 | −31.6211 | −61.9372 | 0 | −209.534 | 39.1615 | 0 | 38.1250 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.290087 | 0 | −31.9158 | −87.1048 | 0 | 167.610 | 18.5412 | 0 | 25.2680 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.10256 | 0 | −27.5792 | 81.1594 | 0 | −97.6213 | −125.269 | 0 | 170.642 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.96013 | 0 | 3.52316 | −65.6425 | 0 | 119.026 | −169.726 | 0 | −391.238 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 6.10567 | 0 | 5.27923 | 41.8018 | 0 | 208.248 | −163.148 | 0 | 255.228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 10.3715 | 0 | 75.5670 | −33.2592 | 0 | −56.9512 | 451.853 | 0 | −344.946 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 531.6.a.c | 12 | |
| 3.b | odd | 2 | 1 | 177.6.a.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.6.a.c | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 531.6.a.c | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 22 T_{2}^{11} - 49 T_{2}^{10} - 3917 T_{2}^{9} - 15182 T_{2}^{8} + 191819 T_{2}^{7} + \cdots + 15131776 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 22 T^{11} + \cdots + 15131776 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots - 65\!\cdots\!40 \)
|
| $7$ |
\( T^{12} + \cdots + 17\!\cdots\!12 \)
|
| $11$ |
\( T^{12} + \cdots - 33\!\cdots\!20 \)
|
| $13$ |
\( T^{12} + \cdots - 17\!\cdots\!60 \)
|
| $17$ |
\( T^{12} + \cdots + 68\!\cdots\!28 \)
|
| $19$ |
\( T^{12} + \cdots + 37\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots - 83\!\cdots\!52 \)
|
| $29$ |
\( T^{12} + \cdots - 34\!\cdots\!80 \)
|
| $31$ |
\( T^{12} + \cdots - 26\!\cdots\!24 \)
|
| $37$ |
\( T^{12} + \cdots + 35\!\cdots\!20 \)
|
| $41$ |
\( T^{12} + \cdots - 20\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots + 25\!\cdots\!96 \)
|
| $47$ |
\( T^{12} + \cdots - 14\!\cdots\!80 \)
|
| $53$ |
\( T^{12} + \cdots - 11\!\cdots\!68 \)
|
| $59$ |
\( (T - 3481)^{12} \)
|
| $61$ |
\( T^{12} + \cdots - 74\!\cdots\!72 \)
|
| $67$ |
\( T^{12} + \cdots + 14\!\cdots\!56 \)
|
| $71$ |
\( T^{12} + \cdots - 44\!\cdots\!60 \)
|
| $73$ |
\( T^{12} + \cdots + 51\!\cdots\!20 \)
|
| $79$ |
\( T^{12} + \cdots + 18\!\cdots\!76 \)
|
| $83$ |
\( T^{12} + \cdots + 28\!\cdots\!32 \)
|
| $89$ |
\( T^{12} + \cdots + 19\!\cdots\!44 \)
|
| $97$ |
\( T^{12} + \cdots - 10\!\cdots\!92 \)
|
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