Newspace parameters
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.78218718056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 14) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | 5.00000 | 4.00000 | 9.00000 | 10.0000 | 0 | 8.00000 | −2.00000 | 18.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 98.4.a.f | 1 | |
| 3.b | odd | 2 | 1 | 882.4.a.c | 1 | ||
| 4.b | odd | 2 | 1 | 784.4.a.c | 1 | ||
| 5.b | even | 2 | 1 | 2450.4.a.d | 1 | ||
| 7.b | odd | 2 | 1 | 98.4.a.d | 1 | ||
| 7.c | even | 3 | 2 | 98.4.c.a | 2 | ||
| 7.d | odd | 6 | 2 | 14.4.c.a | ✓ | 2 | |
| 21.c | even | 2 | 1 | 882.4.a.f | 1 | ||
| 21.g | even | 6 | 2 | 126.4.g.d | 2 | ||
| 21.h | odd | 6 | 2 | 882.4.g.u | 2 | ||
| 28.d | even | 2 | 1 | 784.4.a.p | 1 | ||
| 28.f | even | 6 | 2 | 112.4.i.a | 2 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.q | 1 | ||
| 35.i | odd | 6 | 2 | 350.4.e.e | 2 | ||
| 35.k | even | 12 | 4 | 350.4.j.b | 4 | ||
| 56.j | odd | 6 | 2 | 448.4.i.b | 2 | ||
| 56.m | even | 6 | 2 | 448.4.i.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.4.c.a | ✓ | 2 | 7.d | odd | 6 | 2 | |
| 98.4.a.d | 1 | 7.b | odd | 2 | 1 | ||
| 98.4.a.f | 1 | 1.a | even | 1 | 1 | trivial | |
| 98.4.c.a | 2 | 7.c | even | 3 | 2 | ||
| 112.4.i.a | 2 | 28.f | even | 6 | 2 | ||
| 126.4.g.d | 2 | 21.g | even | 6 | 2 | ||
| 350.4.e.e | 2 | 35.i | odd | 6 | 2 | ||
| 350.4.j.b | 4 | 35.k | even | 12 | 4 | ||
| 448.4.i.b | 2 | 56.j | odd | 6 | 2 | ||
| 448.4.i.e | 2 | 56.m | even | 6 | 2 | ||
| 784.4.a.c | 1 | 4.b | odd | 2 | 1 | ||
| 784.4.a.p | 1 | 28.d | even | 2 | 1 | ||
| 882.4.a.c | 1 | 3.b | odd | 2 | 1 | ||
| 882.4.a.f | 1 | 21.c | even | 2 | 1 | ||
| 882.4.g.u | 2 | 21.h | odd | 6 | 2 | ||
| 2450.4.a.d | 1 | 5.b | even | 2 | 1 | ||
| 2450.4.a.q | 1 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 5 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 2 \)
$3$
\( T - 5 \)
$5$
\( T - 9 \)
$7$
\( T \)
$11$
\( T + 57 \)
$13$
\( T - 70 \)
$17$
\( T + 51 \)
$19$
\( T + 5 \)
$23$
\( T - 69 \)
$29$
\( T - 114 \)
$31$
\( T + 23 \)
$37$
\( T + 253 \)
$41$
\( T - 42 \)
$43$
\( T + 124 \)
$47$
\( T + 201 \)
$53$
\( T + 393 \)
$59$
\( T + 219 \)
$61$
\( T - 709 \)
$67$
\( T - 419 \)
$71$
\( T + 96 \)
$73$
\( T - 313 \)
$79$
\( T - 461 \)
$83$
\( T - 588 \)
$89$
\( T - 1017 \)
$97$
\( T - 1834 \)
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