Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(139.738672144\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 30959316x - 11291332284 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5^{4} \) |
| 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
\( x^{3} - x^{2} - 30959316x - 11291332284 \)
:
| \(\beta_{1}\) | \(=\) |
\( 30\nu - 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 100\nu^{2} - 54820\nu - 2063936160 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 10 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{2} + 5482\beta _1 + 6191863300 ) / 300 \)
|
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 |
|
−1024.00 | −176613. | 1.04858e6 | 0 | 1.80851e8 | 1.28981e9 | −1.07374e9 | 2.07317e10 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1024.00 | −26460.2 | 1.04858e6 | 0 | 2.70953e7 | −3.30980e8 | −1.07374e9 | −9.76021e9 | 0 | ||||||||||||||||||||||||||||
| 1.3 | −1024.00 | 156690. | 1.04858e6 | 0 | −1.60450e8 | −4.70592e7 | −1.07374e9 | 1.40914e10 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 50.22.a.g | ✓ | 3 |
| 5.b | even | 2 | 1 | 50.22.a.h | yes | 3 | |
| 5.c | odd | 4 | 2 | 50.22.b.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.22.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 50.22.a.h | yes | 3 | 5.b | even | 2 | 1 | |
| 50.22.b.g | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 46383T_{3}^{2} - 27146257137T_{3} - 732244559943519 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(50))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1024)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 732244559943519 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + \cdots - 20\!\cdots\!52 \)
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| $11$ |
\( T^{3} + \cdots - 77\!\cdots\!13 \)
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| $13$ |
\( T^{3} + \cdots + 13\!\cdots\!96 \)
|
| $17$ |
\( T^{3} + \cdots + 11\!\cdots\!33 \)
|
| $19$ |
\( T^{3} + \cdots - 89\!\cdots\!25 \)
|
| $23$ |
\( T^{3} + \cdots - 10\!\cdots\!04 \)
|
| $29$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 31\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots - 32\!\cdots\!32 \)
|
| $41$ |
\( T^{3} + \cdots - 23\!\cdots\!93 \)
|
| $43$ |
\( T^{3} + \cdots - 40\!\cdots\!24 \)
|
| $47$ |
\( T^{3} + \cdots + 10\!\cdots\!68 \)
|
| $53$ |
\( T^{3} + \cdots - 10\!\cdots\!24 \)
|
| $59$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 35\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots + 16\!\cdots\!23 \)
|
| $71$ |
\( T^{3} + \cdots - 16\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots + 72\!\cdots\!91 \)
|
| $79$ |
\( T^{3} + \cdots + 48\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 23\!\cdots\!29 \)
|
| $89$ |
\( T^{3} + \cdots - 19\!\cdots\!75 \)
|
| $97$ |
\( T^{3} + \cdots + 10\!\cdots\!08 \)
|
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