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: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} - 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 18) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - 2x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
\(n\) | \(83\) |
\(\chi(n)\) | \(1 - \beta_{2}\) |
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 | 150.644 | + | 86.9741i | 0 | 242.000 | + | 419.156i | 181.019i | 0 | −984.000 | ||||||||||||||||||||||
53.2 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | −150.644 | − | 86.9741i | 0 | 242.000 | + | 419.156i | − | 181.019i | 0 | −984.000 | ||||||||||||||||||||||
107.1 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | 150.644 | − | 86.9741i | 0 | 242.000 | − | 419.156i | − | 181.019i | 0 | −984.000 | ||||||||||||||||||||||
107.2 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | −150.644 | + | 86.9741i | 0 | 242.000 | − | 419.156i | 181.019i | 0 | −984.000 | |||||||||||||||||||||||
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.d | 4 | |
3.b | odd | 2 | 1 | inner | 162.7.d.d | 4 | |
9.c | even | 3 | 1 | 18.7.b.a | ✓ | 2 | |
9.c | even | 3 | 1 | inner | 162.7.d.d | 4 | |
9.d | odd | 6 | 1 | 18.7.b.a | ✓ | 2 | |
9.d | odd | 6 | 1 | inner | 162.7.d.d | 4 | |
36.f | odd | 6 | 1 | 144.7.e.d | 2 | ||
36.h | even | 6 | 1 | 144.7.e.d | 2 | ||
45.h | odd | 6 | 1 | 450.7.d.a | 2 | ||
45.j | even | 6 | 1 | 450.7.d.a | 2 | ||
45.k | odd | 12 | 2 | 450.7.b.a | 4 | ||
45.l | even | 12 | 2 | 450.7.b.a | 4 | ||
72.j | odd | 6 | 1 | 576.7.e.b | 2 | ||
72.l | even | 6 | 1 | 576.7.e.k | 2 | ||
72.n | even | 6 | 1 | 576.7.e.b | 2 | ||
72.p | odd | 6 | 1 | 576.7.e.k | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.7.b.a | ✓ | 2 | 9.c | even | 3 | 1 | |
18.7.b.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
144.7.e.d | 2 | 36.f | odd | 6 | 1 | ||
144.7.e.d | 2 | 36.h | even | 6 | 1 | ||
162.7.d.d | 4 | 1.a | even | 1 | 1 | trivial | |
162.7.d.d | 4 | 3.b | odd | 2 | 1 | inner | |
162.7.d.d | 4 | 9.c | even | 3 | 1 | inner | |
162.7.d.d | 4 | 9.d | odd | 6 | 1 | inner | |
450.7.b.a | 4 | 45.k | odd | 12 | 2 | ||
450.7.b.a | 4 | 45.l | even | 12 | 2 | ||
450.7.d.a | 2 | 45.h | odd | 6 | 1 | ||
450.7.d.a | 2 | 45.j | even | 6 | 1 | ||
576.7.e.b | 2 | 72.j | odd | 6 | 1 | ||
576.7.e.b | 2 | 72.n | even | 6 | 1 | ||
576.7.e.k | 2 | 72.l | even | 6 | 1 | ||
576.7.e.k | 2 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 30258T_{5}^{2} + 915546564 \)
acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 32T^{2} + 1024 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 30258 T^{2} + \cdots + 915546564 \)
$7$
\( (T^{2} - 484 T + 234256)^{2} \)
$11$
\( T^{4} - 1797408 T^{2} + \cdots + 3230675518464 \)
$13$
\( (T^{2} + 3368 T + 11343424)^{2} \)
$17$
\( (T^{2} + 162)^{2} \)
$19$
\( (T - 5744)^{4} \)
$23$
\( T^{4} + \cdots + 130076032287744 \)
$29$
\( T^{4} - 861706098 T^{2} + \cdots + 74\!\cdots\!04 \)
$31$
\( (T^{2} - 39796 T + 1583721616)^{2} \)
$37$
\( (T - 52526)^{4} \)
$41$
\( T^{4} - 1372146498 T^{2} + \cdots + 18\!\cdots\!04 \)
$43$
\( (T^{2} + 3800 T + 14440000)^{2} \)
$47$
\( T^{4} - 5896980000 T^{2} + \cdots + 34\!\cdots\!00 \)
$53$
\( (T^{2} + 56995657938)^{2} \)
$59$
\( T^{4} - 62420337792 T^{2} + \cdots + 38\!\cdots\!64 \)
$61$
\( (T^{2} + 13250 T + 175562500)^{2} \)
$67$
\( (T^{2} + 168968 T + 28550185024)^{2} \)
$71$
\( (T^{2} + 282457292832)^{2} \)
$73$
\( (T - 236144)^{4} \)
$79$
\( (T^{2} - 35116 T + 1233133456)^{2} \)
$83$
\( T^{4} - 120559392 T^{2} + \cdots + 14\!\cdots\!64 \)
$89$
\( (T^{2} + 16725839202)^{2} \)
$97$
\( (T^{2} - 321424 T + 103313387776)^{2} \)
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