Newspace parameters
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(65.9953348299\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{16} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 37\nu^{6} + 93556\nu^{4} + 75931333\nu^{2} + 19722241824 ) / 2006320 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9859\nu^{6} - 25807324\nu^{4} - 22197698515\nu^{2} - 6262260164592 ) / 54170640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9829\nu^{6} - 27049132\nu^{4} - 24352443781\nu^{2} - 7141798486428 ) / 13542660 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1117\nu^{7} + 3077380\nu^{5} + 2773897261\nu^{3} + 814471345392\nu ) / 34160045760 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 929183\nu^{7} + 6844468748\nu^{5} + 11352744767375\nu^{3} + 5136503071908144\nu ) / 27635477019840 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 18258415\nu^{7} + 47311753516\nu^{5} + 39826591020607\nu^{3} + 10500188456688432\nu ) / 13817738509920 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -95130337\nu^{7} - 250569216868\nu^{5} - 216717212802097\nu^{3} - 61413480914212608\nu ) / 27635477019840 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{7} - 6\beta_{6} - 3\beta_{5} + 35\beta_{4} ) / 162 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 30\beta_{2} - 178\beta _1 - 136257 ) / 162 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 250\beta_{7} + 1684\beta_{6} + 1679\beta_{5} - 43459\beta_{4} ) / 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6711\beta_{3} + 57120\beta_{2} + 299596\beta _1 + 119083575 ) / 162 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 498074\beta_{7} - 4421202\beta_{6} - 6466773\beta_{5} + 237745069\beta_{4} ) / 162 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3604147\beta_{3} - 27621430\beta_{2} - 127821642\beta _1 - 35944238537 ) / 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -1776403958\beta_{7} + 4009684620\beta_{6} + 7495086327\beta_{5} - 351792378619\beta_{4} ) / 162 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
− | 11.3137i | 0 | −128.000 | − | 724.855i | 0 | −1954.36 | 1448.15i | 0 | −8200.80 | ||||||||||||||||||||||||||||||||||||||||
| 161.2 | − | 11.3137i | 0 | −128.000 | − | 618.080i | 0 | −549.440 | 1448.15i | 0 | −6992.78 | |||||||||||||||||||||||||||||||||||||||||
| 161.3 | − | 11.3137i | 0 | −128.000 | 654.335i | 0 | 1299.40 | 1448.15i | 0 | 7402.96 | ||||||||||||||||||||||||||||||||||||||||||
| 161.4 | − | 11.3137i | 0 | −128.000 | 1061.95i | 0 | −3233.60 | 1448.15i | 0 | 12014.6 | ||||||||||||||||||||||||||||||||||||||||||
| 161.5 | 11.3137i | 0 | −128.000 | − | 1061.95i | 0 | −3233.60 | − | 1448.15i | 0 | 12014.6 | |||||||||||||||||||||||||||||||||||||||||
| 161.6 | 11.3137i | 0 | −128.000 | − | 654.335i | 0 | 1299.40 | − | 1448.15i | 0 | 7402.96 | |||||||||||||||||||||||||||||||||||||||||
| 161.7 | 11.3137i | 0 | −128.000 | 618.080i | 0 | −549.440 | − | 1448.15i | 0 | −6992.78 | ||||||||||||||||||||||||||||||||||||||||||
| 161.8 | 11.3137i | 0 | −128.000 | 724.855i | 0 | −1954.36 | − | 1448.15i | 0 | −8200.80 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.9.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 162.9.b.a | ✓ | 8 |
| 9.c | even | 3 | 2 | 162.9.d.h | 16 | ||
| 9.d | odd | 6 | 2 | 162.9.d.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.9.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 162.9.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 162.9.d.h | 16 | 9.c | even | 3 | 2 | ||
| 162.9.d.h | 16 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 2463336T_{5}^{6} + 2095450167474T_{5}^{4} + 750455927372039400T_{5}^{2} + 96917652743250602030625 \)
acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 128)^{4} \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 2463336 T^{6} + \cdots + 96\!\cdots\!25 \)
$7$
\( (T^{4} + 4438 T^{3} + \cdots - 4511860509296)^{2} \)
$11$
\( T^{8} + 1083686796 T^{6} + \cdots + 28\!\cdots\!84 \)
$13$
\( (T^{4} + 58690 T^{3} + \cdots - 168608080827623)^{2} \)
$17$
\( T^{8} + 40156998768 T^{6} + \cdots + 42\!\cdots\!25 \)
$19$
\( (T^{4} - 135110 T^{3} + \cdots + 27\!\cdots\!24)^{2} \)
$23$
\( T^{8} + 283534373268 T^{6} + \cdots + 21\!\cdots\!00 \)
$29$
\( T^{8} + 1677572170848 T^{6} + \cdots + 31\!\cdots\!89 \)
$31$
\( (T^{4} + 196672 T^{3} + \cdots + 54\!\cdots\!96)^{2} \)
$37$
\( (T^{4} - 915494 T^{3} + \cdots + 16\!\cdots\!21)^{2} \)
$41$
\( T^{8} + 42402792363120 T^{6} + \cdots + 10\!\cdots\!96 \)
$43$
\( (T^{4} - 5567618 T^{3} + \cdots - 13\!\cdots\!64)^{2} \)
$47$
\( T^{8} + 62658471632112 T^{6} + \cdots + 21\!\cdots\!96 \)
$53$
\( T^{8} + 288880676034192 T^{6} + \cdots + 47\!\cdots\!24 \)
$59$
\( T^{8} + 85968765926928 T^{6} + \cdots + 11\!\cdots\!24 \)
$61$
\( (T^{4} - 6092102 T^{3} + \cdots + 12\!\cdots\!01)^{2} \)
$67$
\( (T^{4} + 40177858 T^{3} + \cdots - 44\!\cdots\!32)^{2} \)
$71$
\( T^{8} + \cdots + 34\!\cdots\!76 \)
$73$
\( (T^{4} - 98542880 T^{3} + \cdots - 56\!\cdots\!27)^{2} \)
$79$
\( (T^{4} - 42225926 T^{3} + \cdots + 52\!\cdots\!92)^{2} \)
$83$
\( T^{8} + \cdots + 15\!\cdots\!44 \)
$89$
\( T^{8} + \cdots + 39\!\cdots\!61 \)
$97$
\( (T^{4} - 170568464 T^{3} + \cdots - 41\!\cdots\!32)^{2} \)
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