Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [536,4,Mod(1,536)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(536, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("536.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 536 = 2^{3} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 536.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: | \(31.6250237631\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - x^{10} - 188 x^{9} + 164 x^{8} + 12306 x^{7} - 4717 x^{6} - 349995 x^{5} - 28469 x^{4} + \cdots + 9069327 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 188 x^{9} + 164 x^{8} + 12306 x^{7} - 4717 x^{6} - 349995 x^{5} - 28469 x^{4} + \cdots + 9069327 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 8342599 \nu^{10} - 1056876000 \nu^{9} + 2339256752 \nu^{8} + 154055019954 \nu^{7} + \cdots + 448199299516509 ) / 9833264472906 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 108350131 \nu^{10} - 359270616 \nu^{9} - 15284063060 \nu^{8} + 52751457636 \nu^{7} + \cdots + 9130463817687 ) / 19666528945812 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 37859317 \nu^{10} + 578552508 \nu^{9} + 5821793180 \nu^{8} - 74975386812 \nu^{7} + \cdots + 570982192083711 ) / 6555509648604 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14658302 \nu^{10} - 191293860 \nu^{9} + 1345754830 \nu^{8} + 26475747351 \nu^{7} + \cdots - 1131388680690 ) / 1638877412151 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 96303073 \nu^{10} - 151802814 \nu^{9} - 15248260202 \nu^{8} + 27434808846 \nu^{7} + \cdots - 172442982435075 ) / 3277754824302 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 686168569 \nu^{10} + 1270087500 \nu^{9} + 106773624272 \nu^{8} - 217360310712 \nu^{7} + \cdots + 356865446635155 ) / 19666528945812 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 120570901 \nu^{10} - 851341296 \nu^{9} - 19801086116 \nu^{8} + 127100664924 \nu^{7} + \cdots + 99058015699425 ) / 2809504135116 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 345979243 \nu^{10} - 2141713068 \nu^{9} - 49931709044 \nu^{8} + 339011651520 \nu^{7} + \cdots + 12\!\cdots\!67 ) / 6555509648604 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 714849413 \nu^{10} - 4397023698 \nu^{9} - 112542378634 \nu^{8} + 688970874054 \nu^{7} + \cdots + 11\!\cdots\!69 ) / 9833264472906 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{3} + \beta _1 + 34 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - 2\beta_{9} - 4\beta_{7} - 3\beta_{6} - 2\beta_{5} - 6\beta_{3} + 4\beta_{2} + 55\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 9 \beta_{10} + 9 \beta_{9} + 6 \beta_{8} + 95 \beta_{7} + 92 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + \cdots + 1937 \)
|
| \(\nu^{5}\) | \(=\) |
\( 155 \beta_{10} - 232 \beta_{9} - 18 \beta_{8} - 489 \beta_{7} - 454 \beta_{6} - 235 \beta_{5} + \cdots - 1674 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1210 \beta_{10} + 1310 \beta_{9} + 489 \beta_{8} + 8066 \beta_{7} + 7990 \beta_{6} + 1460 \beta_{5} + \cdots + 134186 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16720 \beta_{10} - 22028 \beta_{9} - 3228 \beta_{8} - 51428 \beta_{7} - 51022 \beta_{6} - 22280 \beta_{5} + \cdots - 335408 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 128194 \beta_{10} + 146249 \beta_{9} + 36969 \beta_{8} + 697684 \beta_{7} + 700575 \beta_{6} + \cdots + 10312519 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1614078 \beta_{10} - 2006109 \beta_{9} - 375378 \beta_{8} - 5157372 \beta_{7} - 5222700 \beta_{6} + \cdots - 43748979 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 12704670 \beta_{10} + 14933640 \beta_{9} + 3047508 \beta_{8} + 61905568 \beta_{7} + 62539774 \beta_{6} + \cdots + 845002435 \)
|
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 |
|
0 | −8.88461 | 0 | −21.4532 | 0 | 5.27280 | 0 | 51.9363 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.71618 | 0 | −0.373352 | 0 | −25.9036 | 0 | 48.9718 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −8.37300 | 0 | 19.3700 | 0 | 7.87415 | 0 | 43.1071 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.93897 | 0 | 16.0954 | 0 | −9.58478 | 0 | −2.60653 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −2.50740 | 0 | −12.9375 | 0 | 25.3941 | 0 | −20.7129 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.81384 | 0 | −7.30268 | 0 | −10.9287 | 0 | −23.7100 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.45800 | 0 | 6.85373 | 0 | 17.0604 | 0 | −20.9582 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.23549 | 0 | −6.63945 | 0 | 7.19058 | 0 | −16.5316 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 4.36869 | 0 | 2.45197 | 0 | −7.19024 | 0 | −7.91452 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 4.50382 | 0 | 13.6990 | 0 | −35.0728 | 0 | −6.71557 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 8.66800 | 0 | −12.7639 | 0 | −11.1119 | 0 | 48.1341 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(67\) | \(-1\) |
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 | 536.4.a.b | ✓ | 11 |
| 4.b | odd | 2 | 1 | 1072.4.a.k | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 536.4.a.b | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 1072.4.a.k | 11 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{11} + 12 T_{3}^{10} - 123 T_{3}^{9} - 1646 T_{3}^{8} + 4676 T_{3}^{7} + 71147 T_{3}^{6} + \cdots - 19755068 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(536))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} + 12 T^{10} + \cdots - 19755068 \)
|
| $5$ |
\( T^{11} + \cdots - 4602840236 \)
|
| $7$ |
\( T^{11} + \cdots - 983428218336 \)
|
| $11$ |
\( T^{11} + \cdots + 197235202610896 \)
|
| $13$ |
\( T^{11} + \cdots + 17\!\cdots\!16 \)
|
| $17$ |
\( T^{11} + \cdots - 10\!\cdots\!72 \)
|
| $19$ |
\( T^{11} + \cdots + 84\!\cdots\!28 \)
|
| $23$ |
\( T^{11} + \cdots - 12\!\cdots\!91 \)
|
| $29$ |
\( T^{11} + \cdots + 68\!\cdots\!96 \)
|
| $31$ |
\( T^{11} + \cdots - 30\!\cdots\!44 \)
|
| $37$ |
\( T^{11} + \cdots - 10\!\cdots\!76 \)
|
| $41$ |
\( T^{11} + \cdots + 62\!\cdots\!56 \)
|
| $43$ |
\( T^{11} + \cdots - 38\!\cdots\!48 \)
|
| $47$ |
\( T^{11} + \cdots - 52\!\cdots\!69 \)
|
| $53$ |
\( T^{11} + \cdots + 39\!\cdots\!88 \)
|
| $59$ |
\( T^{11} + \cdots + 69\!\cdots\!84 \)
|
| $61$ |
\( T^{11} + \cdots - 72\!\cdots\!32 \)
|
| $67$ |
\( (T - 67)^{11} \)
|
| $71$ |
\( T^{11} + \cdots + 18\!\cdots\!48 \)
|
| $73$ |
\( T^{11} + \cdots - 12\!\cdots\!72 \)
|
| $79$ |
\( T^{11} + \cdots + 25\!\cdots\!04 \)
|
| $83$ |
\( T^{11} + \cdots + 85\!\cdots\!08 \)
|
| $89$ |
\( T^{11} + \cdots + 27\!\cdots\!29 \)
|
| $97$ |
\( T^{11} + \cdots - 53\!\cdots\!64 \)
|
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