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: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2778460829 \nu^{10} - 187297311739 \nu^{9} - 3169078538 \nu^{8} + 37422999014637 \nu^{7} + \cdots - 46\!\cdots\!20 ) / 42\!\cdots\!52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 14193213478 \nu^{10} + 162977595845 \nu^{9} + 2207753333560 \nu^{8} + \cdots + 49\!\cdots\!76 ) / 42\!\cdots\!52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 47676035497 \nu^{10} + 351855671964 \nu^{9} + 9131611480310 \nu^{8} + \cdots + 21\!\cdots\!64 ) / 42\!\cdots\!52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 28858453840 \nu^{10} + 377512348731 \nu^{9} + 4935210416071 \nu^{8} + \cdots + 82\!\cdots\!16 ) / 21\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 15568094860 \nu^{10} + 166942662163 \nu^{9} + 2866062504113 \nu^{8} + \cdots + 33\!\cdots\!72 ) / 10\!\cdots\!88 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 70699645381 \nu^{10} + 724092912446 \nu^{9} + 12116309216206 \nu^{8} + \cdots - 53\!\cdots\!64 ) / 42\!\cdots\!52 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 76278029747 \nu^{10} - 810327211110 \nu^{9} - 13430535220258 \nu^{8} + \cdots - 28\!\cdots\!88 ) / 42\!\cdots\!52 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 314311388515 \nu^{10} + 3040305309961 \nu^{9} + 56749599552340 \nu^{8} + \cdots + 94\!\cdots\!44 ) / 84\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots + 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{10} + 3 \beta_{9} - 4 \beta_{8} - 7 \beta_{7} + \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + \cdots + 2969 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 250 \beta_{10} - 115 \beta_{9} + 108 \beta_{8} + 127 \beta_{7} - 69 \beta_{6} + 342 \beta_{5} + \cdots + 1288 \)
|
| \(\nu^{6}\) | \(=\) |
\( 204 \beta_{10} + 584 \beta_{9} - 421 \beta_{8} - 708 \beta_{7} + 198 \beta_{6} + 1255 \beta_{5} + \cdots + 218504 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 25176 \beta_{10} - 10300 \beta_{9} + 9842 \beta_{8} + 12148 \beta_{7} - 1988 \beta_{6} + \cdots + 116322 \)
|
| \(\nu^{8}\) | \(=\) |
\( 14096 \beta_{10} + 78956 \beta_{9} - 36456 \beta_{8} - 50500 \beta_{7} + 27296 \beta_{6} + \cdots + 16953209 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2379226 \beta_{10} - 853361 \beta_{9} + 854257 \beta_{8} + 1090113 \beta_{7} + 210331 \beta_{6} + \cdots + 10355213 \)
|
| \(\nu^{10}\) | \(=\) |
\( 632954 \beta_{10} + 9039827 \beta_{9} - 2990368 \beta_{8} - 2910255 \beta_{7} + 3281173 \beta_{6} + \cdots + 1353173089 \)
|
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 |
|
−8.42442 | 0 | 38.9709 | 30.3784 | 0 | −197.653 | −58.7254 | 0 | −255.921 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.66878 | 0 | 26.8102 | 109.801 | 0 | 156.301 | 39.7996 | 0 | −842.036 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.91273 | 0 | 15.7858 | 11.1343 | 0 | −193.283 | 112.084 | 0 | −76.9686 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.75393 | 0 | −9.40015 | 8.06966 | 0 | −28.8581 | 196.813 | 0 | −38.3626 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.20625 | 0 | −21.7200 | −26.7258 | 0 | 39.0273 | 172.240 | 0 | 85.6897 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.21624 | 0 | −30.5208 | −0.914506 | 0 | 61.7584 | −76.0403 | 0 | −1.11226 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.75662 | 0 | −24.4011 | −5.39522 | 0 | −153.490 | −155.476 | 0 | −14.8726 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 6.62527 | 0 | 11.8942 | 85.2025 | 0 | −103.010 | −133.206 | 0 | 564.490 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 6.70379 | 0 | 12.9408 | −105.016 | 0 | −129.262 | −127.769 | 0 | −704.003 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 9.44473 | 0 | 57.2030 | −13.7903 | 0 | 67.4858 | 238.036 | 0 | −130.245 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 10.2194 | 0 | 72.4370 | 99.2561 | 0 | 109.985 | 413.244 | 0 | 1014.34 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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.b | 11 | |
| 3.b | odd | 2 | 1 | 177.6.a.a | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.6.a.a | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 531.6.a.b | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 6 T_{2}^{10} - 233 T_{2}^{9} + 1135 T_{2}^{8} + 20480 T_{2}^{7} - 75693 T_{2}^{6} + \cdots + 97836992 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 6 T^{10} + \cdots + 97836992 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + \cdots + 484006946487552 \)
|
| $7$ |
\( T^{11} + \cdots - 63\!\cdots\!60 \)
|
| $11$ |
\( T^{11} + \cdots + 75\!\cdots\!92 \)
|
| $13$ |
\( T^{11} + \cdots + 86\!\cdots\!44 \)
|
| $17$ |
\( T^{11} + \cdots + 45\!\cdots\!24 \)
|
| $19$ |
\( T^{11} + \cdots - 93\!\cdots\!28 \)
|
| $23$ |
\( T^{11} + \cdots - 86\!\cdots\!84 \)
|
| $29$ |
\( T^{11} + \cdots + 27\!\cdots\!08 \)
|
| $31$ |
\( T^{11} + \cdots - 75\!\cdots\!48 \)
|
| $37$ |
\( T^{11} + \cdots - 16\!\cdots\!40 \)
|
| $41$ |
\( T^{11} + \cdots - 38\!\cdots\!56 \)
|
| $43$ |
\( T^{11} + \cdots + 12\!\cdots\!40 \)
|
| $47$ |
\( T^{11} + \cdots - 13\!\cdots\!00 \)
|
| $53$ |
\( T^{11} + \cdots + 90\!\cdots\!56 \)
|
| $59$ |
\( (T + 3481)^{11} \)
|
| $61$ |
\( T^{11} + \cdots - 18\!\cdots\!64 \)
|
| $67$ |
\( T^{11} + \cdots - 78\!\cdots\!40 \)
|
| $71$ |
\( T^{11} + \cdots - 44\!\cdots\!24 \)
|
| $73$ |
\( T^{11} + \cdots - 58\!\cdots\!00 \)
|
| $79$ |
\( T^{11} + \cdots - 92\!\cdots\!20 \)
|
| $83$ |
\( T^{11} + \cdots - 16\!\cdots\!84 \)
|
| $89$ |
\( T^{11} + \cdots + 19\!\cdots\!16 \)
|
| $97$ |
\( T^{11} + \cdots - 55\!\cdots\!76 \)
|
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