Newspace parameters
| Level: | \( N \) | \(=\) | \( 3392 = 2^{6} \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3392.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(27.0852563656\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 13x^{5} + 14x^{4} + 44x^{3} - 34x^{2} - 42x + 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1696) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 13x^{5} + 14x^{4} + 44x^{3} - 34x^{2} - 42x + 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 18\nu^{4} + 16\nu^{3} - 44\nu^{2} - 26\nu + 4 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{6} + 11\nu^{5} + 50\nu^{4} - 48\nu^{3} - 92\nu^{2} + 38\nu + 4 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 18\nu^{4} - 24\nu^{3} + 68\nu^{2} + 66\nu - 68 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{6} - 5\nu^{5} - 38\nu^{4} + 24\nu^{3} + 108\nu^{2} - 34\nu - 60 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 18\nu^{4} - 20\nu^{3} + 60\nu^{2} + 38\nu - 52 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} - 4\beta_{4} + 2\beta_{2} + 11\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{6} - 2\beta_{5} - 18\beta_{4} - 2\beta_{3} + 13\beta_{2} + 35\beta _1 + 34 \)
|
| \(\nu^{5}\) | \(=\) |
\( 56\beta_{6} - 5\beta_{5} - 69\beta_{4} - 4\beta_{3} + 38\beta_{2} + 149\beta _1 + 86 \)
|
| \(\nu^{6}\) | \(=\) |
\( 236\beta_{6} - 31\beta_{5} - 275\beta_{4} - 32\beta_{3} + 176\beta_{2} + 543\beta _1 + 418 \)
|
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 |
|
0 | −2.85690 | 0 | 1.35571 | 0 | −2.79515 | 0 | 5.16189 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.691322 | 0 | 1.07564 | 0 | −1.39470 | 0 | −2.52207 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.252083 | 0 | −3.08742 | 0 | −3.33096 | 0 | −2.93645 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.374149 | 0 | −0.724565 | 0 | 4.27701 | 0 | −2.86001 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.32103 | 0 | −0.915495 | 0 | −3.47798 | 0 | 2.38720 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.79453 | 0 | 3.98497 | 0 | 0.635141 | 0 | 4.80937 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.31060 | 0 | −2.68883 | 0 | 2.08664 | 0 | 7.96007 | 0 | ||||||||||||||||||||||||||||||||||||||||
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 | 3392.2.a.bn | 7 | |
| 4.b | odd | 2 | 1 | 3392.2.a.bm | 7 | ||
| 8.b | even | 2 | 1 | 1696.2.a.m | ✓ | 7 | |
| 8.d | odd | 2 | 1 | 1696.2.a.n | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1696.2.a.m | ✓ | 7 | 8.b | even | 2 | 1 | |
| 1696.2.a.n | yes | 7 | 8.d | odd | 2 | 1 | |
| 3392.2.a.bm | 7 | 4.b | odd | 2 | 1 | ||
| 3392.2.a.bn | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3392))\):
|
\( T_{3}^{7} - 5T_{3}^{6} - 4T_{3}^{5} + 46T_{3}^{4} - 35T_{3}^{3} - 43T_{3}^{2} + 8T_{3} + 4 \)
|
|
\( T_{5}^{7} + T_{5}^{6} - 18T_{5}^{5} - 24T_{5}^{4} + 56T_{5}^{3} + 52T_{5}^{2} - 40T_{5} - 32 \)
|
|
\( T_{7}^{7} + 4T_{7}^{6} - 20T_{7}^{5} - 96T_{7}^{4} + 28T_{7}^{3} + 408T_{7}^{2} + 160T_{7} - 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 5 T^{6} + \cdots + 4 \)
|
| $5$ |
\( T^{7} + T^{6} + \cdots - 32 \)
|
| $7$ |
\( T^{7} + 4 T^{6} + \cdots - 256 \)
|
| $11$ |
\( T^{7} - 11 T^{6} + \cdots + 1024 \)
|
| $13$ |
\( T^{7} - 9 T^{6} + \cdots - 2048 \)
|
| $17$ |
\( T^{7} - 10 T^{6} + \cdots + 634 \)
|
| $19$ |
\( T^{7} - 19 T^{6} + \cdots + 896 \)
|
| $23$ |
\( T^{7} + 6 T^{6} + \cdots - 45998 \)
|
| $29$ |
\( T^{7} + 5 T^{6} + \cdots + 124 \)
|
| $31$ |
\( T^{7} + 17 T^{6} + \cdots - 13472 \)
|
| $37$ |
\( T^{7} + T^{6} + \cdots - 220508 \)
|
| $41$ |
\( T^{7} - 10 T^{6} + \cdots - 38144 \)
|
| $43$ |
\( T^{7} - 3 T^{6} + \cdots - 313216 \)
|
| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 83968 \)
|
| $53$ |
\( (T - 1)^{7} \)
|
| $59$ |
\( T^{7} - 29 T^{6} + \cdots - 35072 \)
|
| $61$ |
\( T^{7} - 16 T^{6} + \cdots + 225856 \)
|
| $67$ |
\( T^{7} - 28 T^{6} + \cdots - 1792 \)
|
| $71$ |
\( T^{7} + 31 T^{6} + \cdots - 19144 \)
|
| $73$ |
\( T^{7} - 14 T^{6} + \cdots + 61696 \)
|
| $79$ |
\( T^{7} + 4 T^{6} + \cdots - 1834 \)
|
| $83$ |
\( T^{7} - 21 T^{6} + \cdots - 1305364 \)
|
| $89$ |
\( T^{7} + 19 T^{6} + \cdots - 225856 \)
|
| $97$ |
\( T^{7} - 26 T^{6} + \cdots - 2099938 \)
|
| show more | |
| show less |