Newspace parameters
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(15.7176143417\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{46})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 46x^{2} + 2116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 46x^{2} + 2116 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 46 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 46\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 23\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
2.00000 | + | 3.46410i | −6.78233 | + | 11.7473i | −8.00000 | + | 13.8564i | 47.4763 | + | 82.2314i | −54.2586 | 0 | −64.0000 | 29.5000 | + | 51.0955i | −189.905 | + | 328.926i | ||||||||||||||||||
| 67.2 | 2.00000 | + | 3.46410i | 6.78233 | − | 11.7473i | −8.00000 | + | 13.8564i | −47.4763 | − | 82.2314i | 54.2586 | 0 | −64.0000 | 29.5000 | + | 51.0955i | 189.905 | − | 328.926i | |||||||||||||||||||
| 79.1 | 2.00000 | − | 3.46410i | −6.78233 | − | 11.7473i | −8.00000 | − | 13.8564i | 47.4763 | − | 82.2314i | −54.2586 | 0 | −64.0000 | 29.5000 | − | 51.0955i | −189.905 | − | 328.926i | |||||||||||||||||||
| 79.2 | 2.00000 | − | 3.46410i | 6.78233 | + | 11.7473i | −8.00000 | − | 13.8564i | −47.4763 | + | 82.2314i | 54.2586 | 0 | −64.0000 | 29.5000 | − | 51.0955i | 189.905 | + | 328.926i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.6.c.g | 4 | |
| 7.b | odd | 2 | 1 | inner | 98.6.c.g | 4 | |
| 7.c | even | 3 | 1 | 98.6.a.e | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 98.6.c.g | 4 | |
| 7.d | odd | 6 | 1 | 98.6.a.e | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 98.6.c.g | 4 | |
| 21.g | even | 6 | 1 | 882.6.a.bo | 2 | ||
| 21.h | odd | 6 | 1 | 882.6.a.bo | 2 | ||
| 28.f | even | 6 | 1 | 784.6.a.y | 2 | ||
| 28.g | odd | 6 | 1 | 784.6.a.y | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.6.a.e | ✓ | 2 | 7.c | even | 3 | 1 | |
| 98.6.a.e | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 98.6.c.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.6.c.g | 4 | 7.b | odd | 2 | 1 | inner | |
| 98.6.c.g | 4 | 7.c | even | 3 | 1 | inner | |
| 98.6.c.g | 4 | 7.d | odd | 6 | 1 | inner | |
| 784.6.a.y | 2 | 28.f | even | 6 | 1 | ||
| 784.6.a.y | 2 | 28.g | odd | 6 | 1 | ||
| 882.6.a.bo | 2 | 21.g | even | 6 | 1 | ||
| 882.6.a.bo | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 184T_{3}^{2} + 33856 \)
acting on \(S_{6}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} + 184 T^{2} + 33856 \)
|
| $5$ |
\( T^{4} + 9016 T^{2} + 81288256 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 476 T + 226576)^{2} \)
|
| $13$ |
\( (T^{2} - 927544)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 642408662016 \)
|
| $19$ |
\( T^{4} + \cdots + 165206479936 \)
|
| $23$ |
\( (T^{2} + 3696 T + 13660416)^{2} \)
|
| $29$ |
\( (T - 1394)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 13765405950976 \)
|
| $37$ |
\( (T^{2} + 12090 T + 146168100)^{2} \)
|
| $41$ |
\( (T^{2} - 231634656)^{2} \)
|
| $43$ |
\( (T - 9724)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 73\!\cdots\!76 \)
|
| $53$ |
\( (T^{2} + 4310 T + 18576100)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots + 74\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + 20236 T + 409495696)^{2} \)
|
| $71$ |
\( (T - 29792)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 16\!\cdots\!56 \)
|
| $79$ |
\( (T^{2} - 33176 T + 1100646976)^{2} \)
|
| $83$ |
\( (T^{2} - 12534264)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $97$ |
\( (T^{2} - 315762400)^{2} \)
|
| show more | |
| show less |