Newspace parameters
Level: | \( N \) | \(=\) | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 816.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.51579280494\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.399424.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | no (minimal twist has level 408) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2 \) |
\(\beta_{5}\) | \(=\) | \( ( 4\nu^{5} - 3\nu^{4} + 8\nu^{3} - 11\nu^{2} + 8\nu - 20 ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} + 2\beta_{3} - \beta _1 + 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - 2 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 5 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - 11\beta_{2} - 4\beta _1 + 6 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( 4\beta_{5} + 3\beta_{4} - 2\beta_{3} + 8\beta_{2} - 7\beta _1 + 7 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).
\(n\) | \(241\) | \(511\) | \(545\) | \(613\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
577.1 |
|
0 | − | 1.00000i | 0 | − | 3.48929i | 0 | − | 4.68585i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
577.2 | 0 | − | 1.00000i | 0 | − | 1.28917i | 0 | 3.62721i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
577.3 | 0 | − | 1.00000i | 0 | 1.77846i | 0 | − | 0.941367i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
577.4 | 0 | 1.00000i | 0 | − | 1.77846i | 0 | 0.941367i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
577.5 | 0 | 1.00000i | 0 | 1.28917i | 0 | − | 3.62721i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
577.6 | 0 | 1.00000i | 0 | 3.48929i | 0 | 4.68585i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 816.2.c.f | 6 | |
3.b | odd | 2 | 1 | 2448.2.c.r | 6 | ||
4.b | odd | 2 | 1 | 408.2.c.b | ✓ | 6 | |
8.b | even | 2 | 1 | 3264.2.c.n | 6 | ||
8.d | odd | 2 | 1 | 3264.2.c.o | 6 | ||
12.b | even | 2 | 1 | 1224.2.c.h | 6 | ||
17.b | even | 2 | 1 | inner | 816.2.c.f | 6 | |
51.c | odd | 2 | 1 | 2448.2.c.r | 6 | ||
68.d | odd | 2 | 1 | 408.2.c.b | ✓ | 6 | |
68.f | odd | 4 | 1 | 6936.2.a.bb | 3 | ||
68.f | odd | 4 | 1 | 6936.2.a.be | 3 | ||
136.e | odd | 2 | 1 | 3264.2.c.o | 6 | ||
136.h | even | 2 | 1 | 3264.2.c.n | 6 | ||
204.h | even | 2 | 1 | 1224.2.c.h | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
408.2.c.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
408.2.c.b | ✓ | 6 | 68.d | odd | 2 | 1 | |
816.2.c.f | 6 | 1.a | even | 1 | 1 | trivial | |
816.2.c.f | 6 | 17.b | even | 2 | 1 | inner | |
1224.2.c.h | 6 | 12.b | even | 2 | 1 | ||
1224.2.c.h | 6 | 204.h | even | 2 | 1 | ||
2448.2.c.r | 6 | 3.b | odd | 2 | 1 | ||
2448.2.c.r | 6 | 51.c | odd | 2 | 1 | ||
3264.2.c.n | 6 | 8.b | even | 2 | 1 | ||
3264.2.c.n | 6 | 136.h | even | 2 | 1 | ||
3264.2.c.o | 6 | 8.d | odd | 2 | 1 | ||
3264.2.c.o | 6 | 136.e | odd | 2 | 1 | ||
6936.2.a.bb | 3 | 68.f | odd | 4 | 1 | ||
6936.2.a.be | 3 | 68.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 17T_{5}^{4} + 64T_{5}^{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( (T^{2} + 1)^{3} \)
$5$
\( T^{6} + 17 T^{4} + 64 T^{2} + 64 \)
$7$
\( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \)
$11$
\( T^{6} + 33 T^{4} + 224 T^{2} + \cdots + 256 \)
$13$
\( (T^{3} + T^{2} - 32 T - 76)^{2} \)
$17$
\( T^{6} - 4 T^{5} - 9 T^{4} + 120 T^{3} + \cdots + 4913 \)
$19$
\( (T^{3} + 7 T^{2} - 40 T - 272)^{2} \)
$23$
\( T^{6} + 129 T^{4} + 3648 T^{2} + \cdots + 64 \)
$29$
\( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \)
$31$
\( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \)
$37$
\( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \)
$41$
\( T^{6} + 89 T^{4} + 768 T^{2} + \cdots + 1024 \)
$43$
\( (T^{3} - 3 T^{2} - 136 T + 592)^{2} \)
$47$
\( (T^{3} - 2 T^{2} - 64 T - 128)^{2} \)
$53$
\( (T - 6)^{6} \)
$59$
\( (T^{3} - 112 T + 128)^{2} \)
$61$
\( T^{6} + 68 T^{4} + 576 T^{2} + \cdots + 256 \)
$67$
\( (T - 4)^{6} \)
$71$
\( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \)
$73$
\( T^{6} + 292 T^{4} + 20480 T^{2} + \cdots + 65536 \)
$79$
\( T^{6} + 272 T^{4} + 18496 T^{2} + \cdots + 369664 \)
$83$
\( (T^{3} + 10 T^{2} - 32 T - 352)^{2} \)
$89$
\( (T^{3} + 18 T^{2} - 4 T - 584)^{2} \)
$97$
\( T^{6} + 400 T^{4} + 29696 T^{2} + \cdots + 16384 \)
show more
show less