Newspace parameters
Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 162.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(37.2687615464\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3^{6} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
\(\beta_{2}\) | \(=\) |
\( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \)
|
\(\beta_{3}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24} \)
|
\(\beta_{4}\) | \(=\) |
\( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \)
|
\(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \)
|
\(\beta_{6}\) | \(=\) |
\( \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
\(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} ) / 9 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( \beta_1 \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{4} + 2\beta_{2} ) / 9 \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
\(n\) | \(83\) |
\(\chi(n)\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 |
|
−4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −180.591 | − | 104.264i | 0 | −2.09808 | − | 3.63397i | 181.019i | 0 | 1179.62 | ||||||||||||||||||||||||||||||||||
53.2 | −4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −21.4919 | − | 12.4084i | 0 | 3.09808 | + | 5.36603i | 181.019i | 0 | 140.385 | |||||||||||||||||||||||||||||||||||
53.3 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 21.4919 | + | 12.4084i | 0 | 3.09808 | + | 5.36603i | − | 181.019i | 0 | 140.385 | ||||||||||||||||||||||||||||||||||
53.4 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 180.591 | + | 104.264i | 0 | −2.09808 | − | 3.63397i | − | 181.019i | 0 | 1179.62 | ||||||||||||||||||||||||||||||||||
107.1 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −180.591 | + | 104.264i | 0 | −2.09808 | + | 3.63397i | − | 181.019i | 0 | 1179.62 | ||||||||||||||||||||||||||||||||||
107.2 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −21.4919 | + | 12.4084i | 0 | 3.09808 | − | 5.36603i | − | 181.019i | 0 | 140.385 | ||||||||||||||||||||||||||||||||||
107.3 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 21.4919 | − | 12.4084i | 0 | 3.09808 | − | 5.36603i | 181.019i | 0 | 140.385 | |||||||||||||||||||||||||||||||||||
107.4 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 180.591 | − | 104.264i | 0 | −2.09808 | + | 3.63397i | 181.019i | 0 | 1179.62 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 162.7.d.f | 8 | |
3.b | odd | 2 | 1 | inner | 162.7.d.f | 8 | |
9.c | even | 3 | 1 | 162.7.b.a | ✓ | 4 | |
9.c | even | 3 | 1 | inner | 162.7.d.f | 8 | |
9.d | odd | 6 | 1 | 162.7.b.a | ✓ | 4 | |
9.d | odd | 6 | 1 | inner | 162.7.d.f | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
162.7.b.a | ✓ | 4 | 9.c | even | 3 | 1 | |
162.7.b.a | ✓ | 4 | 9.d | odd | 6 | 1 | |
162.7.d.f | 8 | 1.a | even | 1 | 1 | trivial | |
162.7.d.f | 8 | 3.b | odd | 2 | 1 | inner | |
162.7.d.f | 8 | 9.c | even | 3 | 1 | inner | |
162.7.d.f | 8 | 9.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 44100T_{5}^{6} + 1918029375T_{5}^{4} - 1181025562500T_{5}^{2} + 717201875390625 \)
acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 32 T^{2} + 1024)^{2} \)
$3$
\( T^{8} \)
$5$
\( T^{8} + \cdots + 717201875390625 \)
$7$
\( (T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676)^{2} \)
$11$
\( T^{8} - 5133564 T^{6} + \cdots + 76\!\cdots\!76 \)
$13$
\( (T^{4} + 1918 T^{3} + \cdots + 6856578909049)^{2} \)
$17$
\( (T^{4} + 15316812 T^{2} + \cdots + 44258191098489)^{2} \)
$19$
\( (T^{2} - 5122 T - 1519922)^{4} \)
$23$
\( T^{8} - 285948900 T^{6} + \cdots + 24\!\cdots\!00 \)
$29$
\( T^{8} - 2598095196 T^{6} + \cdots + 28\!\cdots\!41 \)
$31$
\( (T^{4} - 20048 T^{3} + \cdots + 13\!\cdots\!76)^{2} \)
$37$
\( (T^{2} + 10100 T - 117700343)^{4} \)
$41$
\( T^{8} - 16410776076 T^{6} + \cdots + 15\!\cdots\!96 \)
$43$
\( (T^{4} + 92470 T^{3} + \cdots + 14\!\cdots\!04)^{2} \)
$47$
\( T^{8} - 15258194352 T^{6} + \cdots + 30\!\cdots\!76 \)
$53$
\( (T^{4} + 36731822796 T^{2} + \cdots + 33\!\cdots\!16)^{2} \)
$59$
\( T^{8} - 24241511952 T^{6} + \cdots + 39\!\cdots\!16 \)
$61$
\( (T^{4} + 304528 T^{3} + \cdots + 26\!\cdots\!81)^{2} \)
$67$
\( (T^{4} - 1004486 T^{3} + \cdots + 63\!\cdots\!76)^{2} \)
$71$
\( (T^{4} + 417635798724 T^{2} + \cdots + 89\!\cdots\!36)^{2} \)
$73$
\( (T^{2} - 262912 T - 22812868139)^{4} \)
$79$
\( (T^{4} - 424358 T^{3} + \cdots + 17\!\cdots\!64)^{2} \)
$83$
\( T^{8} - 894320305488 T^{6} + \cdots + 10\!\cdots\!96 \)
$89$
\( (T^{4} + 1655675156628 T^{2} + \cdots + 64\!\cdots\!49)^{2} \)
$97$
\( (T^{4} + 1810864 T^{3} + \cdots + 33\!\cdots\!76)^{2} \)
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