Newspace parameters
Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 684.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(70.7050547493\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{3}, \sqrt{19})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - 11x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{4} - 11x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{3} - 5\nu ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( -7\nu^{3} + 61\nu ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( 3\nu^{2} - 17 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{2} - 7\beta_1 ) / 48 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + 17 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( -10\beta_{2} - 61\beta_1 ) / 48 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(343\) | \(533\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
0 | 0 | 0 | −46.4618 | 0 | 20.1238 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
37.2 | 0 | 0 | 0 | −7.23174 | 0 | −93.1238 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
37.3 | 0 | 0 | 0 | 7.23174 | 0 | −93.1238 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
37.4 | 0 | 0 | 0 | 46.4618 | 0 | 20.1238 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
3.b | odd | 2 | 1 | inner |
57.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 684.5.h.d | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 684.5.h.d | ✓ | 4 |
19.b | odd | 2 | 1 | CM | 684.5.h.d | ✓ | 4 |
57.d | even | 2 | 1 | inner | 684.5.h.d | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
684.5.h.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
684.5.h.d | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
684.5.h.d | ✓ | 4 | 19.b | odd | 2 | 1 | CM |
684.5.h.d | ✓ | 4 | 57.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 2211T_{5}^{2} + 112896 \)
acting on \(S_{5}^{\mathrm{new}}(684, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 2211 T^{2} + 112896 \)
$7$
\( (T^{2} + 73 T - 1874)^{2} \)
$11$
\( T^{4} - 83571 T^{2} + \cdots + 1571963904 \)
$13$
\( T^{4} \)
$17$
\( T^{4} - 291651 T^{2} + \cdots + 1688223744 \)
$19$
\( (T + 361)^{4} \)
$23$
\( (T^{2} - 1094400)^{2} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( (T^{2} + 3527 T + 2183326)^{2} \)
$47$
\( T^{4} - 11216211 T^{2} + \cdots + 11715778900224 \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} - 3167 T - 31507634)^{2} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( (T^{2} - 10033 T + 15466366)^{2} \)
$79$
\( T^{4} \)
$83$
\( (T^{2} - 157593600)^{2} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
show more
show less