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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{2} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{2} - x - 1961256803955162 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{4}\cdot 5\cdot 11 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 285120\sqrt{7845027215820649}\). We also show the integral \(q\)-expansion of the trace form.
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 | −2.22863e15 | 1.84467e19 | 2.44878e22 | 9.57189e24 | 8.86741e26 | −7.92282e28 | −5.33426e30 | −1.05174e32 | ||||||||||||||||||||||||
| 1.2 | −4.29497e9 | 3.37769e15 | 1.84467e19 | −9.92503e22 | −1.45071e25 | 1.82190e27 | −7.92282e28 | 1.10774e30 | 4.26277e32 | |||||||||||||||||||||||||
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.a | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.66.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 1149059960205192T_{3} - 7527621327730776295070757798384 \)
acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4294967296)^{2} \)
|
| $3$ |
\( T^{2} + \cdots - 75\!\cdots\!84 \)
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| $5$ |
\( T^{2} + \cdots - 24\!\cdots\!00 \)
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| $7$ |
\( T^{2} + \cdots + 16\!\cdots\!44 \)
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| $11$ |
\( T^{2} + \cdots - 19\!\cdots\!16 \)
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| $13$ |
\( T^{2} + \cdots - 21\!\cdots\!04 \)
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| $17$ |
\( T^{2} + \cdots - 38\!\cdots\!76 \)
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| $19$ |
\( T^{2} + \cdots - 30\!\cdots\!00 \)
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| $23$ |
\( T^{2} + \cdots + 25\!\cdots\!36 \)
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| $29$ |
\( T^{2} + \cdots + 75\!\cdots\!00 \)
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| $31$ |
\( T^{2} + \cdots + 64\!\cdots\!64 \)
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| $37$ |
\( T^{2} + \cdots + 70\!\cdots\!04 \)
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| $41$ |
\( T^{2} + \cdots - 58\!\cdots\!56 \)
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| $43$ |
\( T^{2} + \cdots + 54\!\cdots\!96 \)
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| $47$ |
\( T^{2} + \cdots + 22\!\cdots\!04 \)
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| $53$ |
\( T^{2} + \cdots - 61\!\cdots\!64 \)
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| $59$ |
\( T^{2} + \cdots + 25\!\cdots\!00 \)
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| $61$ |
\( T^{2} + \cdots + 21\!\cdots\!24 \)
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| $67$ |
\( T^{2} + \cdots - 34\!\cdots\!56 \)
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| $71$ |
\( T^{2} + \cdots - 26\!\cdots\!76 \)
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| $73$ |
\( T^{2} + \cdots - 57\!\cdots\!44 \)
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| $79$ |
\( T^{2} + \cdots + 11\!\cdots\!00 \)
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| $83$ |
\( T^{2} + \cdots + 10\!\cdots\!76 \)
|
| $89$ |
\( T^{2} + \cdots - 23\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots - 40\!\cdots\!76 \)
|
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