Newspace parameters
| Level: | \( N \) | \(=\) | \( 1233 = 3^{2} \cdot 137 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1233.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.84555456922\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{22 +2 \sqrt{5}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 137) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.09529 | 0 | −0.800331 | 1.94111 | 0 | −2.52274 | 3.06719 | 0 | −2.12608 | ||||||||||||||||||||||||||||||
| 1.2 | 0.262360 | 0 | −1.93117 | −0.0425409 | 0 | −1.64433 | −1.03138 | 0 | −0.0111610 | |||||||||||||||||||||||||||||||
| 1.3 | 1.47726 | 0 | 0.182297 | 3.53103 | 0 | −5.09529 | −2.68522 | 0 | 5.21625 | |||||||||||||||||||||||||||||||
| 1.4 | 2.35567 | 0 | 3.54920 | −3.42960 | 0 | −3.73764 | 3.64941 | 0 | −8.07901 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(137\) | \( -1 \) |
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 | 1233.2.a.d | 4 | |
| 3.b | odd | 2 | 1 | 137.2.a.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 2192.2.a.j | 4 | ||
| 15.d | odd | 2 | 1 | 3425.2.a.b | 4 | ||
| 21.c | even | 2 | 1 | 6713.2.a.b | 4 | ||
| 24.f | even | 2 | 1 | 8768.2.a.r | 4 | ||
| 24.h | odd | 2 | 1 | 8768.2.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 137.2.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1233.2.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 2192.2.a.j | 4 | 12.b | even | 2 | 1 | ||
| 3425.2.a.b | 4 | 15.d | odd | 2 | 1 | ||
| 6713.2.a.b | 4 | 21.c | even | 2 | 1 | ||
| 8768.2.a.r | 4 | 24.f | even | 2 | 1 | ||
| 8768.2.a.w | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} + 4T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1233))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} + 13 T^{3} + \cdots + 79 \)
|
| $11$ |
\( T^{4} + T^{3} + \cdots + 101 \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots - 101 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 31 \)
|
| $19$ |
\( T^{4} + 10 T^{3} + \cdots - 431 \)
|
| $23$ |
\( T^{4} - T^{3} + \cdots + 121 \)
|
| $29$ |
\( T^{4} + 11 T^{3} + \cdots - 551 \)
|
| $31$ |
\( T^{4} + 17 T^{3} + \cdots - 319 \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots - 191 \)
|
| $41$ |
\( T^{4} - 7 T^{3} + \cdots - 121 \)
|
| $43$ |
\( T^{4} + 13 T^{3} + \cdots - 191 \)
|
| $47$ |
\( T^{4} - 11 T^{3} + \cdots - 41 \)
|
| $53$ |
\( T^{4} - 2 T^{3} + \cdots - 1 \)
|
| $59$ |
\( T^{4} + 2 T^{3} + \cdots - 709 \)
|
| $61$ |
\( T^{4} - 7 T^{3} + \cdots + 11 \)
|
| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 2831 \)
|
| $71$ |
\( (T^{2} + 4 T - 16)^{2} \)
|
| $73$ |
\( T^{4} + 27 T^{3} + \cdots - 10219 \)
|
| $79$ |
\( T^{4} - 3 T^{3} + \cdots + 9329 \)
|
| $83$ |
\( T^{4} - 3 T^{3} + \cdots + 6449 \)
|
| $89$ |
\( (T^{2} + 7 T + 1)^{2} \)
|
| $97$ |
\( T^{4} + 7 T^{3} + \cdots + 211 \)
|
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