Newspace parameters
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.00303247010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2636.1 |
| Defining polynomial: |
\( x^{3} - 14x - 4 \)
|
| 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
\( x^{3} - 14x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 10 \)
|
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 |
|
−5.03251 | 8.47535 | 17.3261 | 0.885690 | −42.6523 | 3.81828 | −46.9339 | 44.8316 | −4.45724 | |||||||||||||||||||||||||||
| 1.2 | 1.36122 | 3.15463 | −6.14708 | 3.03171 | 4.29415 | −7.94049 | −19.2573 | −17.0483 | 4.12682 | ||||||||||||||||||||||||||||
| 1.3 | 4.67129 | −7.62999 | 13.8209 | −11.9174 | −35.6419 | 26.1222 | 27.1912 | 31.2167 | −55.6696 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
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 | 17.4.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 153.4.a.g | 3 | ||
| 4.b | odd | 2 | 1 | 272.4.a.h | 3 | ||
| 5.b | even | 2 | 1 | 425.4.a.g | 3 | ||
| 5.c | odd | 4 | 2 | 425.4.b.f | 6 | ||
| 7.b | odd | 2 | 1 | 833.4.a.d | 3 | ||
| 8.b | even | 2 | 1 | 1088.4.a.v | 3 | ||
| 8.d | odd | 2 | 1 | 1088.4.a.x | 3 | ||
| 11.b | odd | 2 | 1 | 2057.4.a.e | 3 | ||
| 12.b | even | 2 | 1 | 2448.4.a.bi | 3 | ||
| 17.b | even | 2 | 1 | 289.4.a.b | 3 | ||
| 17.c | even | 4 | 2 | 289.4.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 153.4.a.g | 3 | 3.b | odd | 2 | 1 | ||
| 272.4.a.h | 3 | 4.b | odd | 2 | 1 | ||
| 289.4.a.b | 3 | 17.b | even | 2 | 1 | ||
| 289.4.b.b | 6 | 17.c | even | 4 | 2 | ||
| 425.4.a.g | 3 | 5.b | even | 2 | 1 | ||
| 425.4.b.f | 6 | 5.c | odd | 4 | 2 | ||
| 833.4.a.d | 3 | 7.b | odd | 2 | 1 | ||
| 1088.4.a.v | 3 | 8.b | even | 2 | 1 | ||
| 1088.4.a.x | 3 | 8.d | odd | 2 | 1 | ||
| 2057.4.a.e | 3 | 11.b | odd | 2 | 1 | ||
| 2448.4.a.bi | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 24T_{2} + 32 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 24 T + 32 \)
$3$
\( T^{3} - 4 T^{2} - 62 T + 204 \)
$5$
\( T^{3} + 8 T^{2} - 44 T + 32 \)
$7$
\( T^{3} - 22 T^{2} - 138 T + 792 \)
$11$
\( T^{3} + 28 T^{2} - 1366 T - 4692 \)
$13$
\( T^{3} - 30 T^{2} - 1472 T - 9392 \)
$17$
\( (T + 17)^{3} \)
$19$
\( T^{3} - 80 T^{2} - 4632 T + 340128 \)
$23$
\( T^{3} - 142 T^{2} - 15770 T + 1600544 \)
$29$
\( T^{3} + 456 T^{2} + 53908 T + 1518624 \)
$31$
\( T^{3} - 230 T^{2} - 11586 T - 81608 \)
$37$
\( T^{3} - 356 T^{2} - 17964 T + 6176752 \)
$41$
\( T^{3} + 294 T^{2} - 86564 T - 1638744 \)
$43$
\( T^{3} - 556 T^{2} + 51096 T + 7270272 \)
$47$
\( T^{3} - 640 T^{2} + 85328 T - 1671168 \)
$53$
\( T^{3} - 302 T^{2} + \cdots + 18162072 \)
$59$
\( T^{3} - 636 T^{2} + \cdots + 49419072 \)
$61$
\( T^{3} + 84 T^{2} - 124412 T - 6792784 \)
$67$
\( T^{3} - 1008 T^{2} + 65040 T - 765952 \)
$71$
\( T^{3} + 402 T^{2} + \cdots - 274866016 \)
$73$
\( T^{3} - 838 T^{2} + \cdots - 19957512 \)
$79$
\( T^{3} + 594 T^{2} + \cdots - 742135824 \)
$83$
\( T^{3} + 2396 T^{2} + \cdots + 142080704 \)
$89$
\( T^{3} + 170 T^{2} + \cdots - 446571376 \)
$97$
\( T^{3} + 270 T^{2} + \cdots - 206623000 \)
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