Newspace parameters
| Level: | \( N \) | \(=\) | \( 539 = 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 539.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.30393666895\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
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|
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| Defining polynomial: |
\( x^{3} - 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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 \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 |
|
−1.53209 | 2.87939 | 0.347296 | 2.34730 | −4.41147 | 0 | 2.53209 | 5.29086 | −3.59627 | |||||||||||||||||||||||||||
| 1.2 | −0.347296 | −0.532089 | −1.87939 | 0.120615 | 0.184793 | 0 | 1.34730 | −2.71688 | −0.0418891 | ||||||||||||||||||||||||||||
| 1.3 | 1.87939 | 0.652704 | 1.53209 | 3.53209 | 1.22668 | 0 | −0.879385 | −2.57398 | 6.63816 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 539.2.a.j | 3 | |
| 3.b | odd | 2 | 1 | 4851.2.a.bj | 3 | ||
| 4.b | odd | 2 | 1 | 8624.2.a.ch | 3 | ||
| 7.b | odd | 2 | 1 | 539.2.a.g | 3 | ||
| 7.c | even | 3 | 2 | 77.2.e.a | ✓ | 6 | |
| 7.d | odd | 6 | 2 | 539.2.e.m | 6 | ||
| 11.b | odd | 2 | 1 | 5929.2.a.x | 3 | ||
| 21.c | even | 2 | 1 | 4851.2.a.bk | 3 | ||
| 21.h | odd | 6 | 2 | 693.2.i.h | 6 | ||
| 28.d | even | 2 | 1 | 8624.2.a.co | 3 | ||
| 28.g | odd | 6 | 2 | 1232.2.q.m | 6 | ||
| 77.b | even | 2 | 1 | 5929.2.a.u | 3 | ||
| 77.h | odd | 6 | 2 | 847.2.e.c | 6 | ||
| 77.m | even | 15 | 8 | 847.2.n.g | 24 | ||
| 77.o | odd | 30 | 8 | 847.2.n.f | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.2.e.a | ✓ | 6 | 7.c | even | 3 | 2 | |
| 539.2.a.g | 3 | 7.b | odd | 2 | 1 | ||
| 539.2.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
| 539.2.e.m | 6 | 7.d | odd | 6 | 2 | ||
| 693.2.i.h | 6 | 21.h | odd | 6 | 2 | ||
| 847.2.e.c | 6 | 77.h | odd | 6 | 2 | ||
| 847.2.n.f | 24 | 77.o | odd | 30 | 8 | ||
| 847.2.n.g | 24 | 77.m | even | 15 | 8 | ||
| 1232.2.q.m | 6 | 28.g | odd | 6 | 2 | ||
| 4851.2.a.bj | 3 | 3.b | odd | 2 | 1 | ||
| 4851.2.a.bk | 3 | 21.c | even | 2 | 1 | ||
| 5929.2.a.u | 3 | 77.b | even | 2 | 1 | ||
| 5929.2.a.x | 3 | 11.b | odd | 2 | 1 | ||
| 8624.2.a.ch | 3 | 4.b | odd | 2 | 1 | ||
| 8624.2.a.co | 3 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):
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\( T_{2}^{3} - 3T_{2} - 1 \)
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\( T_{3}^{3} - 3T_{3}^{2} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3T - 1 \)
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| $3$ |
\( T^{3} - 3T^{2} + 1 \)
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| $5$ |
\( T^{3} - 6 T^{2} + \cdots - 1 \)
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| $7$ |
\( T^{3} \)
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| $11$ |
\( (T - 1)^{3} \)
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| $13$ |
\( T^{3} - 3 T^{2} + \cdots - 1 \)
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| $17$ |
\( T^{3} + 3 T^{2} + \cdots - 127 \)
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| $19$ |
\( T^{3} - 9 T^{2} + \cdots - 9 \)
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| $23$ |
\( T^{3} - 57T - 107 \)
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| $29$ |
\( T^{3} + 3 T^{2} + \cdots + 51 \)
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| $31$ |
\( T^{3} - 9 T^{2} + \cdots - 19 \)
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| $37$ |
\( T^{3} - 36T + 72 \)
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| $41$ |
\( T^{3} - 9 T^{2} + \cdots - 1 \)
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| $43$ |
\( T^{3} - 9T + 9 \)
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| $47$ |
\( T^{3} + 3 T^{2} + \cdots - 323 \)
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| $53$ |
\( T^{3} - 9 T^{2} + \cdots + 459 \)
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| $59$ |
\( T^{3} - 93T + 19 \)
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| $61$ |
\( T^{3} - 12 T^{2} + \cdots - 24 \)
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| $67$ |
\( T^{3} - 12T - 8 \)
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| $71$ |
\( T^{3} + 9 T^{2} + \cdots - 801 \)
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| $73$ |
\( T^{3} - 6 T^{2} + \cdots - 1 \)
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| $79$ |
\( T^{3} + 3 T^{2} + \cdots + 37 \)
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| $83$ |
\( T^{3} + 15 T^{2} + \cdots - 267 \)
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| $89$ |
\( T^{3} + 15 T^{2} + \cdots + 111 \)
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| $97$ |
\( T^{3} - 45 T^{2} + \cdots - 3329 \)
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