Newspace parameters
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.78218718056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{2}\) |
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 |
|
1.00000 | + | 1.73205i | −3.53553 | + | 6.12372i | −2.00000 | + | 3.46410i | −9.89949 | − | 17.1464i | −14.1421 | 0 | −8.00000 | −11.5000 | − | 19.9186i | 19.7990 | − | 34.2929i | ||||||||||||||||||
| 67.2 | 1.00000 | + | 1.73205i | 3.53553 | − | 6.12372i | −2.00000 | + | 3.46410i | 9.89949 | + | 17.1464i | 14.1421 | 0 | −8.00000 | −11.5000 | − | 19.9186i | −19.7990 | + | 34.2929i | |||||||||||||||||||
| 79.1 | 1.00000 | − | 1.73205i | −3.53553 | − | 6.12372i | −2.00000 | − | 3.46410i | −9.89949 | + | 17.1464i | −14.1421 | 0 | −8.00000 | −11.5000 | + | 19.9186i | 19.7990 | + | 34.2929i | |||||||||||||||||||
| 79.2 | 1.00000 | − | 1.73205i | 3.53553 | + | 6.12372i | −2.00000 | − | 3.46410i | 9.89949 | − | 17.1464i | 14.1421 | 0 | −8.00000 | −11.5000 | + | 19.9186i | −19.7990 | − | 34.2929i | |||||||||||||||||||
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 | 98.4.c.h | 4 | |
| 3.b | odd | 2 | 1 | 882.4.g.ba | 4 | ||
| 7.b | odd | 2 | 1 | inner | 98.4.c.h | 4 | |
| 7.c | even | 3 | 1 | 98.4.a.g | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 98.4.c.h | 4 | |
| 7.d | odd | 6 | 1 | 98.4.a.g | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 98.4.c.h | 4 | |
| 21.c | even | 2 | 1 | 882.4.g.ba | 4 | ||
| 21.g | even | 6 | 1 | 882.4.a.bg | 2 | ||
| 21.g | even | 6 | 1 | 882.4.g.ba | 4 | ||
| 21.h | odd | 6 | 1 | 882.4.a.bg | 2 | ||
| 21.h | odd | 6 | 1 | 882.4.g.ba | 4 | ||
| 28.f | even | 6 | 1 | 784.4.a.y | 2 | ||
| 28.g | odd | 6 | 1 | 784.4.a.y | 2 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.bx | 2 | ||
| 35.j | even | 6 | 1 | 2450.4.a.bx | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.4.a.g | ✓ | 2 | 7.c | even | 3 | 1 | |
| 98.4.a.g | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 98.4.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.4.c.h | 4 | 7.b | odd | 2 | 1 | inner | |
| 98.4.c.h | 4 | 7.c | even | 3 | 1 | inner | |
| 98.4.c.h | 4 | 7.d | odd | 6 | 1 | inner | |
| 784.4.a.y | 2 | 28.f | even | 6 | 1 | ||
| 784.4.a.y | 2 | 28.g | odd | 6 | 1 | ||
| 882.4.a.bg | 2 | 21.g | even | 6 | 1 | ||
| 882.4.a.bg | 2 | 21.h | odd | 6 | 1 | ||
| 882.4.g.ba | 4 | 3.b | odd | 2 | 1 | ||
| 882.4.g.ba | 4 | 21.c | even | 2 | 1 | ||
| 882.4.g.ba | 4 | 21.g | even | 6 | 1 | ||
| 882.4.g.ba | 4 | 21.h | odd | 6 | 1 | ||
| 2450.4.a.bx | 2 | 35.i | odd | 6 | 1 | ||
| 2450.4.a.bx | 2 | 35.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 50T_{3}^{2} + 2500 \)
acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 4)^{2} \)
$3$
\( T^{4} + 50T^{2} + 2500 \)
$5$
\( T^{4} + 392 T^{2} + 153664 \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 14 T + 196)^{2} \)
$13$
\( (T^{2} - 2592)^{2} \)
$17$
\( T^{4} + 2T^{2} + 4 \)
$19$
\( T^{4} + 2T^{2} + 4 \)
$23$
\( (T^{2} + 140 T + 19600)^{2} \)
$29$
\( (T + 286)^{4} \)
$31$
\( T^{4} + 8712 T^{2} + \cdots + 75898944 \)
$37$
\( (T^{2} - 38 T + 1444)^{2} \)
$41$
\( (T^{2} - 15842)^{2} \)
$43$
\( (T + 34)^{4} \)
$47$
\( T^{4} + 273800 T^{2} + \cdots + 74966440000 \)
$53$
\( (T^{2} - 74 T + 5476)^{2} \)
$59$
\( T^{4} + 188498 T^{2} + \cdots + 35531496004 \)
$61$
\( T^{4} + 200 T^{2} + 40000 \)
$67$
\( (T^{2} + 684 T + 467856)^{2} \)
$71$
\( (T - 588)^{4} \)
$73$
\( T^{4} + 72962 T^{2} + \cdots + 5323453444 \)
$79$
\( (T^{2} + 1220 T + 1488400)^{2} \)
$83$
\( (T^{2} - 178802)^{2} \)
$89$
\( T^{4} + 381938 T^{2} + \cdots + 145876635844 \)
$97$
\( (T^{2} - 2200802)^{2} \)
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