Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 82 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(83.1002571076\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2 x^{3} + \cdots + 15\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{45}\cdot 3^{15}\cdot 5^{5}\cdot 7^{2}\cdot 11 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of 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
\( x^{4} - 2 x^{3} + \cdots + 15\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 1920\nu - 960 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 1897922560 \nu^{3} + \cdots - 30\!\cdots\!40 ) / 35\!\cdots\!33 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 14\!\cdots\!60 \nu^{3} + \cdots - 82\!\cdots\!80 ) / 35\!\cdots\!33 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 960 ) / 1920 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 8326 \beta_{3} + 6227803771 \beta_{2} + \cdots + 34\!\cdots\!00 ) / 1843200 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 15\!\cdots\!89 \beta_{3} + \cdots + 26\!\cdots\!00 ) / 235929600 \)
|
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 |
|
1.09951e12 | −4.08130e19 | 1.20893e24 | 2.59550e28 | −4.48743e31 | 1.64669e34 | 1.32923e36 | 1.22227e39 | 2.85378e40 | ||||||||||||||||||||||||||||||
| 1.2 | 1.09951e12 | −1.09428e19 | 1.20893e24 | −1.35941e28 | −1.20318e31 | −2.01060e34 | 1.32923e36 | −3.23681e38 | −1.49469e40 | |||||||||||||||||||||||||||||||
| 1.3 | 1.09951e12 | 2.07785e19 | 1.20893e24 | −2.06533e28 | 2.28462e31 | 2.86366e34 | 1.32923e36 | −1.16794e37 | −2.27085e40 | |||||||||||||||||||||||||||||||
| 1.4 | 1.09951e12 | 2.38655e19 | 1.20893e24 | 3.32056e28 | 2.62404e31 | −6.60461e33 | 1.32923e36 | 1.26136e38 | 3.65099e40 | |||||||||||||||||||||||||||||||
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 | 2.82.a.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.82.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + \cdots + 22\!\cdots\!76 \)
acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1099511627776)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
|
| $5$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 62\!\cdots\!16 \)
|
| $11$ |
\( T^{4} + \cdots - 92\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 22\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 66\!\cdots\!04 \)
|
| $29$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 60\!\cdots\!16 \)
|
| $37$ |
\( T^{4} + \cdots - 96\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots - 23\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots - 22\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots + 32\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots - 16\!\cdots\!04 \)
|
| $59$ |
\( T^{4} + \cdots - 50\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 27\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots + 96\!\cdots\!36 \)
|
| $73$ |
\( T^{4} + \cdots + 28\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 87\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 40\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 16\!\cdots\!84 \)
|
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