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: |
\( x^{8} + 11104x^{6} + 45413413x^{4} + 81011991468x^{2} + 53239758647364 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{18} \) |
| 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
\( x^{8} + 11104x^{6} + 45413413x^{4} + 81011991468x^{2} + 53239758647364 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -1323\nu^{6} - 11024262\nu^{4} - 29003973723\nu^{2} - 23874956935722 ) / 4784458360 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 381\nu^{6} - 16353614\nu^{4} - 100069066675\nu^{2} - 135571528644342 ) / 2392229180 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60309\nu^{6} + 539157111\nu^{4} + 1573880428884\nu^{2} + 1489905476964191 ) / 598057295 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3877\nu^{7} + 35753650\nu^{5} + 107261260261\nu^{3} + 103788422361222\nu ) / 5884382164680 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1411527665 \nu^{7} - 12093379188626 \nu^{5} + \cdots - 30\!\cdots\!62 \nu ) / 52\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 6873846745 \nu^{7} - 60374430342202 \nu^{5} + \cdots - 16\!\cdots\!34 \nu ) / 10\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 649357487 \nu^{7} - 5324361232926 \nu^{5} + \cdots - 10\!\cdots\!34 \nu ) / 58\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 4\beta_{6} - 13\beta_{5} + 36\beta_{4} ) / 162 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 15\beta_{2} + 738\beta _1 - 449711 ) / 162 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1673\beta_{7} - 18864\beta_{6} + 41621\beta_{5} - 459758\beta_{4} ) / 162 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11104\beta_{3} - 103125\beta_{2} - 4108806\beta _1 + 1315110043 ) / 162 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1982509\beta_{7} + 81274620\beta_{6} - 136331191\beta_{5} + 2854537534\beta_{4} ) / 162 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 48681374\beta_{3} + 530474235\beta_{2} + 17472818586\beta _1 - 4023019402499 ) / 162 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1232327353\beta_{7} - 334702500192\beta_{6} + 453769993415\beta_{5} - 14322679056422\beta_{4} ) / 162 \)
|
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 | − | 1118.47i | 0 | 2818.74 | 1448.15i | 0 | −12654.1 | ||||||||||||||||||||||||||||||||||||||||
| 161.2 | − | 11.3137i | 0 | −128.000 | − | 310.915i | 0 | −4484.76 | 1448.15i | 0 | −3517.60 | |||||||||||||||||||||||||||||||||||||||||
| 161.3 | − | 11.3137i | 0 | −128.000 | 40.9689i | 0 | 2628.61 | 1448.15i | 0 | 463.510 | ||||||||||||||||||||||||||||||||||||||||||
| 161.4 | − | 11.3137i | 0 | −128.000 | 539.890i | 0 | −216.589 | 1448.15i | 0 | 6108.16 | ||||||||||||||||||||||||||||||||||||||||||
| 161.5 | 11.3137i | 0 | −128.000 | − | 539.890i | 0 | −216.589 | − | 1448.15i | 0 | 6108.16 | |||||||||||||||||||||||||||||||||||||||||
| 161.6 | 11.3137i | 0 | −128.000 | − | 40.9689i | 0 | 2628.61 | − | 1448.15i | 0 | 463.510 | |||||||||||||||||||||||||||||||||||||||||
| 161.7 | 11.3137i | 0 | −128.000 | 310.915i | 0 | −4484.76 | − | 1448.15i | 0 | −3517.60 | ||||||||||||||||||||||||||||||||||||||||||
| 161.8 | 11.3137i | 0 | −128.000 | 1118.47i | 0 | 2818.74 | − | 1448.15i | 0 | −12654.1 | ||||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 162.9.b.b | ✓ | 8 |
| 9.c | even | 3 | 2 | 162.9.d.g | 16 | ||
| 9.d | odd | 6 | 2 | 162.9.d.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.9.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 162.9.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 162.9.d.g | 16 | 9.c | even | 3 | 2 | ||
| 162.9.d.g | 16 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 1640808T_{5}^{6} + 516495591666T_{5}^{4} + 36111140725818600T_{5}^{2} + 59163452262584150625 \)
acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 128)^{4} \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 1640808 T^{6} + \cdots + 59\!\cdots\!25 \)
$7$
\( (T^{4} - 746 T^{3} + \cdots + 7197064587664)^{2} \)
$11$
\( T^{8} + 86966028 T^{6} + \cdots + 98\!\cdots\!56 \)
$13$
\( (T^{4} - 23390 T^{3} + \cdots + 67\!\cdots\!69)^{2} \)
$17$
\( T^{8} + 50461667568 T^{6} + \cdots + 42\!\cdots\!25 \)
$19$
\( (T^{4} + 320218 T^{3} + \cdots - 10\!\cdots\!04)^{2} \)
$23$
\( T^{8} + 222640275348 T^{6} + \cdots + 10\!\cdots\!00 \)
$29$
\( T^{8} + 1329947673696 T^{6} + \cdots + 16\!\cdots\!61 \)
$31$
\( (T^{4} - 53888 T^{3} + \cdots + 14\!\cdots\!36)^{2} \)
$37$
\( (T^{4} + 729562 T^{3} + \cdots + 13\!\cdots\!53)^{2} \)
$41$
\( T^{8} + 37646502934896 T^{6} + \cdots + 11\!\cdots\!64 \)
$43$
\( (T^{4} + 4232734 T^{3} + \cdots - 29\!\cdots\!92)^{2} \)
$47$
\( T^{8} + 138220620047856 T^{6} + \cdots + 25\!\cdots\!36 \)
$53$
\( T^{8} + 18704070936720 T^{6} + \cdots + 26\!\cdots\!36 \)
$59$
\( T^{8} + 799197825064464 T^{6} + \cdots + 63\!\cdots\!96 \)
$61$
\( (T^{4} + 14168698 T^{3} + \cdots - 12\!\cdots\!83)^{2} \)
$67$
\( (T^{4} - 46506398 T^{3} + \cdots + 19\!\cdots\!56)^{2} \)
$71$
\( T^{8} + \cdots + 13\!\cdots\!04 \)
$73$
\( (T^{4} + 69655456 T^{3} + \cdots - 13\!\cdots\!11)^{2} \)
$79$
\( (T^{4} + 20445178 T^{3} + \cdots + 64\!\cdots\!84)^{2} \)
$83$
\( T^{8} + \cdots + 38\!\cdots\!96 \)
$89$
\( T^{8} + \cdots + 23\!\cdots\!21 \)
$97$
\( (T^{4} + 24721456 T^{3} + \cdots - 19\!\cdots\!92)^{2} \)
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