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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{22}) \) |
| Defining polynomial: |
\( x^{2} - 22 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.
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 | −9.38083 | 4.00000 | 9.38083 | −18.7617 | 0 | 8.00000 | 61.0000 | 18.7617 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 9.38083 | 4.00000 | −9.38083 | 18.7617 | 0 | 8.00000 | 61.0000 | −18.7617 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.4.a.h | ✓ | 2 |
| 3.b | odd | 2 | 1 | 882.4.a.w | 2 | ||
| 4.b | odd | 2 | 1 | 784.4.a.z | 2 | ||
| 5.b | even | 2 | 1 | 2450.4.a.bs | 2 | ||
| 7.b | odd | 2 | 1 | inner | 98.4.a.h | ✓ | 2 |
| 7.c | even | 3 | 2 | 98.4.c.g | 4 | ||
| 7.d | odd | 6 | 2 | 98.4.c.g | 4 | ||
| 21.c | even | 2 | 1 | 882.4.a.w | 2 | ||
| 21.g | even | 6 | 2 | 882.4.g.bi | 4 | ||
| 21.h | odd | 6 | 2 | 882.4.g.bi | 4 | ||
| 28.d | even | 2 | 1 | 784.4.a.z | 2 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.bs | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.4.a.h | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 98.4.a.h | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 98.4.c.g | 4 | 7.c | even | 3 | 2 | ||
| 98.4.c.g | 4 | 7.d | odd | 6 | 2 | ||
| 784.4.a.z | 2 | 4.b | odd | 2 | 1 | ||
| 784.4.a.z | 2 | 28.d | even | 2 | 1 | ||
| 882.4.a.w | 2 | 3.b | odd | 2 | 1 | ||
| 882.4.a.w | 2 | 21.c | even | 2 | 1 | ||
| 882.4.g.bi | 4 | 21.g | even | 6 | 2 | ||
| 882.4.g.bi | 4 | 21.h | odd | 6 | 2 | ||
| 2450.4.a.bs | 2 | 5.b | even | 2 | 1 | ||
| 2450.4.a.bs | 2 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 88 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 2)^{2} \)
$3$
\( T^{2} - 88 \)
$5$
\( T^{2} - 88 \)
$7$
\( T^{2} \)
$11$
\( (T - 20)^{2} \)
$13$
\( T^{2} - 4312 \)
$17$
\( T^{2} - 3168 \)
$19$
\( T^{2} - 88 \)
$23$
\( (T - 48)^{2} \)
$29$
\( (T + 166)^{2} \)
$31$
\( T^{2} - 42592 \)
$37$
\( (T + 78)^{2} \)
$41$
\( T^{2} - 155232 \)
$43$
\( (T - 436)^{2} \)
$47$
\( T^{2} - 42592 \)
$53$
\( (T - 62)^{2} \)
$59$
\( T^{2} - 443608 \)
$61$
\( T^{2} - 74008 \)
$67$
\( (T - 580)^{2} \)
$71$
\( (T + 544)^{2} \)
$73$
\( T^{2} - 360448 \)
$79$
\( (T + 680)^{2} \)
$83$
\( T^{2} - 38808 \)
$89$
\( T^{2} - 2252800 \)
$97$
\( T^{2} - 431200 \)
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