Newspace parameters
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(99.2412126297\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 100x^{2} - 135x + 1548 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 100x^{2} - 135x + 1548 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 6\nu^{2} - 58\nu + 198 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 9\nu^{2} + 52\nu - 348 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} + 9\beta_{2} + 70\beta _1 + 102 \)
|
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 | −8.05127 | 4.00000 | 19.0317 | 16.1025 | −3.98045 | −8.00000 | 37.8230 | −38.0634 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −7.34023 | 4.00000 | −18.8845 | 14.6805 | 33.2247 | −8.00000 | 26.8790 | 37.7690 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 2.50085 | 4.00000 | 6.39202 | −5.00170 | −1.89287 | −8.00000 | −20.7458 | −12.7840 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 8.89066 | 4.00000 | −13.5392 | −17.7813 | 11.6486 | −8.00000 | 52.0438 | 27.0785 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 1682.4.a.g | ✓ | 4 |
| 29.b | even | 2 | 1 | 1682.4.a.j | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1682.4.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1682.4.a.j | yes | 4 | 29.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} - 94T_{3}^{2} - 331T_{3} + 1314 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 1314 \)
|
| $5$ |
\( T^{4} + 7 T^{3} + \cdots + 31104 \)
|
| $7$ |
\( T^{4} - 39 T^{3} + \cdots + 2916 \)
|
| $11$ |
\( T^{4} - 14 T^{3} + \cdots + 8827542 \)
|
| $13$ |
\( T^{4} - 98 T^{3} + \cdots - 1829088 \)
|
| $17$ |
\( T^{4} - 86 T^{3} + \cdots - 2685942 \)
|
| $19$ |
\( T^{4} + 114 T^{3} + \cdots - 4967384 \)
|
| $23$ |
\( T^{4} - 283 T^{3} + \cdots - 25030296 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + \cdots - 1161813984 \)
|
| $37$ |
\( T^{4} - 354 T^{3} + \cdots - 255228624 \)
|
| $41$ |
\( T^{4} + 506 T^{3} + \cdots - 90592182 \)
|
| $43$ |
\( T^{4} + 93 T^{3} + \cdots + 11676096 \)
|
| $47$ |
\( T^{4} + \cdots + 1883605644 \)
|
| $53$ |
\( T^{4} + \cdots + 5304263616 \)
|
| $59$ |
\( T^{4} - 1067 T^{3} + \cdots - 246032616 \)
|
| $61$ |
\( T^{4} + \cdots - 43523877984 \)
|
| $67$ |
\( T^{4} + 576 T^{3} + \cdots - 866678784 \)
|
| $71$ |
\( T^{4} + \cdots - 19289176992 \)
|
| $73$ |
\( T^{4} + \cdots - 4444444536 \)
|
| $79$ |
\( T^{4} + \cdots + 152742025204 \)
|
| $83$ |
\( T^{4} + \cdots + 17589089088 \)
|
| $89$ |
\( T^{4} + \cdots - 60143265909 \)
|
| $97$ |
\( T^{4} + \cdots + 4303874709 \)
|
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