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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 171x^{7} - 44x^{6} + 9767x^{5} + 2200x^{4} - 215105x^{3} - 33724x^{2} + 1380292x + 1109072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 59) |
| 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
\( x^{9} - 171x^{7} - 44x^{6} + 9767x^{5} + 2200x^{4} - 215105x^{3} - 33724x^{2} + 1380292x + 1109072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 38 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1331 \nu^{8} + 16757 \nu^{7} + 131826 \nu^{6} - 2052130 \nu^{5} - 2424511 \nu^{4} + \cdots - 277353808 ) / 4935872 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3819 \nu^{8} + 35101 \nu^{7} + 450790 \nu^{6} - 4086070 \nu^{5} - 14813507 \nu^{4} + \cdots - 857818768 ) / 9871744 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5909 \nu^{8} - 43567 \nu^{7} - 718978 \nu^{6} + 5100538 \nu^{5} + 24372309 \nu^{4} + \cdots + 1238548912 ) / 9871744 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3611 \nu^{8} - 40015 \nu^{7} - 396350 \nu^{6} + 4841506 \nu^{5} + 10692851 \nu^{4} + \cdots + 748560880 ) / 4935872 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5319 \nu^{8} - 38225 \nu^{7} - 665410 \nu^{6} + 4541090 \nu^{5} + 23670971 \nu^{4} + \cdots + 1351814544 ) / 4935872 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12293 \nu^{8} + 46991 \nu^{7} + 1583042 \nu^{6} - 4766602 \nu^{5} - 56778421 \nu^{4} + \cdots - 337773488 ) / 9871744 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 56\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{8} - 20\beta_{6} + 16\beta_{5} - 12\beta_{4} - 20\beta_{3} + 83\beta_{2} + 46\beta _1 + 2150 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12 \beta_{8} + 107 \beta_{7} - 218 \beta_{6} - 142 \beta_{5} - 146 \beta_{4} - 325 \beta_{3} + \cdots + 2890 \)
|
| \(\nu^{6}\) | \(=\) |
\( 440 \beta_{8} - 86 \beta_{7} - 2420 \beta_{6} + 2020 \beta_{5} - 1268 \beta_{4} - 2638 \beta_{3} + \cdots + 140802 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1704 \beta_{8} + 8977 \beta_{7} - 20890 \beta_{6} - 7314 \beta_{5} - 10058 \beta_{4} - 30475 \beta_{3} + \cdots + 393406 \)
|
| \(\nu^{8}\) | \(=\) |
\( 39244 \beta_{8} - 7716 \beta_{7} - 235652 \beta_{6} + 192264 \beta_{5} - 110764 \beta_{4} + \cdots + 10120654 \)
|
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 |
|
−8.39535 | 0 | 38.4819 | 72.9341 | 0 | −60.1345 | −54.4174 | 0 | −612.307 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.71539 | 0 | 0.665728 | −59.5320 | 0 | −137.853 | 179.088 | 0 | 340.249 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.80409 | 0 | −17.5289 | 42.9333 | 0 | 214.202 | 188.412 | 0 | −163.322 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.31542 | 0 | −21.0080 | 53.3327 | 0 | −148.172 | 175.744 | 0 | −176.821 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.89160 | 0 | −28.4219 | −20.9706 | 0 | −70.9632 | −114.294 | 0 | −39.6679 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.46796 | 0 | −19.9733 | −77.1268 | 0 | 107.152 | −180.241 | 0 | −267.473 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.93831 | 0 | 31.0168 | 31.0319 | 0 | 51.4370 | −7.80461 | 0 | 246.341 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.01083 | 0 | 32.1734 | −9.58620 | 0 | 72.0914 | 1.38934 | 0 | −76.7934 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.92155 | 0 | 47.5941 | 38.9836 | 0 | −235.760 | 139.124 | 0 | 347.794 | ||||||||||||||||||||||||||||||||||||||||||||||
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.a | 9 | |
| 3.b | odd | 2 | 1 | 59.6.a.a | ✓ | 9 | |
| 12.b | even | 2 | 1 | 944.6.a.e | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 59.6.a.a | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 531.6.a.a | 9 | 1.a | even | 1 | 1 | trivial | |
| 944.6.a.e | 9 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 9 T_{2}^{8} - 135 T_{2}^{7} + 1157 T_{2}^{6} + 6038 T_{2}^{5} - 44516 T_{2}^{4} + \cdots - 2252288 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 9 T^{8} + \cdots - 2252288 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} + \cdots - 186475584425516 \)
|
| $7$ |
\( T^{9} + \cdots + 17\!\cdots\!41 \)
|
| $11$ |
\( T^{9} + \cdots + 10\!\cdots\!24 \)
|
| $13$ |
\( T^{9} + \cdots + 26\!\cdots\!12 \)
|
| $17$ |
\( T^{9} + \cdots + 43\!\cdots\!48 \)
|
| $19$ |
\( T^{9} + \cdots + 32\!\cdots\!60 \)
|
| $23$ |
\( T^{9} + \cdots - 15\!\cdots\!12 \)
|
| $29$ |
\( T^{9} + \cdots - 33\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 87\!\cdots\!84 \)
|
| $37$ |
\( T^{9} + \cdots - 30\!\cdots\!24 \)
|
| $41$ |
\( T^{9} + \cdots - 10\!\cdots\!13 \)
|
| $43$ |
\( T^{9} + \cdots - 74\!\cdots\!72 \)
|
| $47$ |
\( T^{9} + \cdots + 51\!\cdots\!52 \)
|
| $53$ |
\( T^{9} + \cdots - 11\!\cdots\!88 \)
|
| $59$ |
\( (T - 3481)^{9} \)
|
| $61$ |
\( T^{9} + \cdots - 93\!\cdots\!72 \)
|
| $67$ |
\( T^{9} + \cdots - 14\!\cdots\!64 \)
|
| $71$ |
\( T^{9} + \cdots + 11\!\cdots\!04 \)
|
| $73$ |
\( T^{9} + \cdots - 22\!\cdots\!48 \)
|
| $79$ |
\( T^{9} + \cdots + 13\!\cdots\!95 \)
|
| $83$ |
\( T^{9} + \cdots + 70\!\cdots\!24 \)
|
| $89$ |
\( T^{9} + \cdots + 96\!\cdots\!20 \)
|
| $97$ |
\( T^{9} + \cdots - 15\!\cdots\!44 \)
|
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