Newspace parameters
| Level: | \( N \) | \(=\) | \( 920 = 2^{3} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 920.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.34623698596\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2597.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.95759 | 0 | 1.00000 | 0 | 3.95759 | 0 | 5.74732 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.878468 | 0 | 1.00000 | 0 | 0.121532 | 0 | −2.22829 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.07912 | 0 | 1.00000 | 0 | −2.07912 | 0 | 6.48097 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 920.2.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 8280.2.a.bj | 3 | ||
| 4.b | odd | 2 | 1 | 1840.2.a.s | 3 | ||
| 5.b | even | 2 | 1 | 4600.2.a.x | 3 | ||
| 5.c | odd | 4 | 2 | 4600.2.e.p | 6 | ||
| 8.b | even | 2 | 1 | 7360.2.a.by | 3 | ||
| 8.d | odd | 2 | 1 | 7360.2.a.cc | 3 | ||
| 20.d | odd | 2 | 1 | 9200.2.a.ce | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.2.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1840.2.a.s | 3 | 4.b | odd | 2 | 1 | ||
| 4600.2.a.x | 3 | 5.b | even | 2 | 1 | ||
| 4600.2.e.p | 6 | 5.c | odd | 4 | 2 | ||
| 7360.2.a.by | 3 | 8.b | even | 2 | 1 | ||
| 7360.2.a.cc | 3 | 8.d | odd | 2 | 1 | ||
| 8280.2.a.bj | 3 | 3.b | odd | 2 | 1 | ||
| 9200.2.a.ce | 3 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - T_{3}^{2} - 9T_{3} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(920))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - T^{2} - 9T + 8 \)
$5$
\( (T - 1)^{3} \)
$7$
\( T^{3} - 2 T^{2} - 8 T + 1 \)
$11$
\( T^{3} - 7 T^{2} + 7 T + 14 \)
$13$
\( T^{3} + T^{2} - 23 T - 50 \)
$17$
\( T^{3} - 10 T^{2} + 10 T + 83 \)
$19$
\( T^{3} + 13 T^{2} + 33 T - 62 \)
$23$
\( (T - 1)^{3} \)
$29$
\( T^{3} - 13 T^{2} + 26 T + 76 \)
$31$
\( T^{3} + 8 T^{2} + 12 T - 1 \)
$37$
\( T^{3} - 5 T^{2} - 92 T + 496 \)
$41$
\( T^{3} - 8 T^{2} - 2 T + 1 \)
$43$
\( (T - 8)^{3} \)
$47$
\( T^{3} - 2 T^{2} - 92 T + 400 \)
$53$
\( T^{3} - T^{2} - 100 T - 300 \)
$59$
\( T^{3} + 17 T^{2} + 66 T + 36 \)
$61$
\( T^{3} - 13 T^{2} - 51 T + 720 \)
$67$
\( T^{3} - 5 T^{2} - 92 T + 496 \)
$71$
\( T^{3} + 22 T^{2} + 96 T - 225 \)
$73$
\( T^{3} - 12 T^{2} - 176 T + 2120 \)
$79$
\( T^{3} + 4 T^{2} - 144 T - 512 \)
$83$
\( T^{3} + 15 T^{2} + 40 T - 36 \)
$89$
\( T^{3} + 8 T^{2} - 128 T - 64 \)
$97$
\( T^{3} + 3 T^{2} - 81 T + 50 \)
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