Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 66 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.5144712945\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4862367805520722608042x + 130125819203569060903952569933488 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11\cdot 13 \) |
| 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
\( x^{3} - x^{2} - 4862367805520722608042x + 130125819203569060903952569933488 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8640\nu^{2} + 341742997270080\nu - 28007238559913276554748160 ) / 439529881 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 253376640\nu^{2} + 14192563459783927680\nu - 821340278009406951018491727360 ) / 62789983 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 205282\beta _1 + 22140518400 ) / 66421555200 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5650512521\beta_{2} + 1642657807845362\beta _1 + 30758669675914051567045988352000 ) / 9488793600 \)
|
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 |
|
4.29497e9 | −2.64736e15 | 1.84467e19 | −6.66845e22 | −1.13703e25 | 1.54537e27 | 7.92282e28 | −3.29254e30 | −2.86408e32 | |||||||||||||||||||||||||||
| 1.2 | 4.29497e9 | −5.99699e13 | 1.84467e19 | 6.16250e22 | −2.57569e23 | −2.55894e27 | 7.92282e28 | −1.02975e31 | 2.64677e32 | ||||||||||||||||||||||||||||
| 1.3 | 4.29497e9 | 5.69189e15 | 1.84467e19 | 8.97760e21 | 2.44465e25 | 5.30637e27 | 7.92282e28 | 2.20965e31 | 3.85585e31 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
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 | 2.66.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.66.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 90\!\cdots\!64 \)
acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4294967296)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 90\!\cdots\!64 \)
|
| $5$ |
\( T^{3} + \cdots + 36\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 20\!\cdots\!28 \)
|
| $11$ |
\( T^{3} + \cdots - 11\!\cdots\!48 \)
|
| $13$ |
\( T^{3} + \cdots - 54\!\cdots\!44 \)
|
| $17$ |
\( T^{3} + \cdots + 36\!\cdots\!68 \)
|
| $19$ |
\( T^{3} + \cdots - 59\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 15\!\cdots\!84 \)
|
| $29$ |
\( T^{3} + \cdots + 61\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 60\!\cdots\!12 \)
|
| $37$ |
\( T^{3} + \cdots - 19\!\cdots\!92 \)
|
| $41$ |
\( T^{3} + \cdots + 27\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots + 16\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots - 68\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots - 18\!\cdots\!84 \)
|
| $59$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 12\!\cdots\!32 \)
|
| $67$ |
\( T^{3} + \cdots + 15\!\cdots\!28 \)
|
| $71$ |
\( T^{3} + \cdots + 75\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 13\!\cdots\!04 \)
|
| $79$ |
\( T^{3} + \cdots - 90\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 16\!\cdots\!76 \)
|
| $89$ |
\( T^{3} + \cdots + 65\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 13\!\cdots\!68 \)
|
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