Newspace parameters
| Level: | \( N \) | \(=\) | \( 106 = 2 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 106.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.25420246061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.14408.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 24x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - x^{2} - 24x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 16 \)
|
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 |
|
−2.00000 | −6.78031 | 4.00000 | −9.89091 | 13.5606 | −18.2212 | −8.00000 | 18.9726 | 19.7818 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 1.27191 | 4.00000 | 11.1419 | −2.54382 | 7.73988 | −8.00000 | −25.3822 | −22.2837 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 9.50840 | 4.00000 | 17.7491 | −19.0168 | 4.48132 | −8.00000 | 63.4096 | −35.4981 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(53\) | \( +1 \) |
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 | 106.4.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 954.4.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 848.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 106.4.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 848.4.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 954.4.a.f | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 4T_{3}^{2} - 61T_{3} + 82 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(106))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} - 4 T^{2} + \cdots + 82 \)
|
| $5$ |
\( T^{3} - 19 T^{2} + \cdots + 1956 \)
|
| $7$ |
\( T^{3} + 6 T^{2} + \cdots + 632 \)
|
| $11$ |
\( T^{3} + T^{2} + \cdots + 22604 \)
|
| $13$ |
\( T^{3} + 58 T^{2} + \cdots - 72116 \)
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| $17$ |
\( T^{3} - 21 T^{2} + \cdots + 285581 \)
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| $19$ |
\( T^{3} - 222 T^{2} + \cdots - 289860 \)
|
| $23$ |
\( T^{3} - 215 T^{2} + \cdots + 332821 \)
|
| $29$ |
\( T^{3} - 204 T^{2} + \cdots + 531490 \)
|
| $31$ |
\( T^{3} + 157 T^{2} + \cdots - 2433404 \)
|
| $37$ |
\( T^{3} + 500 T^{2} + \cdots + 2260106 \)
|
| $41$ |
\( T^{3} - 354 T^{2} + \cdots + 17227592 \)
|
| $43$ |
\( T^{3} + 391 T^{2} + \cdots + 1384848 \)
|
| $47$ |
\( T^{3} - 252 T^{2} + \cdots - 547776 \)
|
| $53$ |
\( (T + 53)^{3} \)
|
| $59$ |
\( T^{3} - 981 T^{2} + \cdots - 12524220 \)
|
| $61$ |
\( T^{3} - 90 T^{2} + \cdots + 62163208 \)
|
| $67$ |
\( T^{3} + 160 T^{2} + \cdots + 93796816 \)
|
| $71$ |
\( T^{3} - 1270 T^{2} + \cdots + 257598492 \)
|
| $73$ |
\( T^{3} + 492 T^{2} + \cdots - 158400 \)
|
| $79$ |
\( T^{3} - 543 T^{2} + \cdots + 6222705 \)
|
| $83$ |
\( T^{3} - 1616 T^{2} + \cdots + 600693598 \)
|
| $89$ |
\( T^{3} - 1173 T^{2} + \cdots + 147561340 \)
|
| $97$ |
\( T^{3} + 1127 T^{2} + \cdots + 59458797 \)
|
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