Newspace parameters
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(50.4735119441\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{4817})\) |
| Defining polynomial: |
\( x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of 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
\( x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 2419\nu - 1210 ) / 687 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 14456\nu - 7229 ) / 229 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 1029\nu^{2} - 2240\nu - 1242178 ) / 687 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 6\beta _1 + 21 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 28\beta_{3} + \beta_{2} + 8\beta _1 + 50673 ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 173\beta_{2} - 3096\beta _1 + 10857 ) / 6 \)
|
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 |
|
−16.0000 | −191.777 | 256.000 | −33.5898 | 3068.43 | 0 | −4096.00 | 17095.5 | 537.437 | ||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | −102.682 | 256.000 | 2094.80 | 1642.91 | 0 | −4096.00 | −9139.47 | −33516.8 | |||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 102.682 | 256.000 | −2094.80 | −1642.91 | 0 | −4096.00 | −9139.47 | 33516.8 | |||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 191.777 | 256.000 | 33.5898 | −3068.43 | 0 | −4096.00 | 17095.5 | −537.437 | |||||||||||||||||||||||||||||||
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 | 98.10.a.k | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 98.10.a.k | ✓ | 4 |
| 7.c | even | 3 | 2 | 98.10.c.m | 8 | ||
| 7.d | odd | 6 | 2 | 98.10.c.m | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.10.a.k | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 98.10.a.k | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 98.10.c.m | 8 | 7.c | even | 3 | 2 | ||
| 98.10.c.m | 8 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 47322T_{3}^{2} + 387774864 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 16)^{4} \)
$3$
\( T^{4} - 47322 T^{2} + \cdots + 387774864 \)
$5$
\( T^{4} - 4389322 T^{2} + \cdots + 4951092496 \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 4774 T - 2902464824)^{2} \)
$13$
\( T^{4} - 28441135314 T^{2} + \cdots + 20\!\cdots\!96 \)
$17$
\( T^{4} - 364177565476 T^{2} + \cdots + 32\!\cdots\!76 \)
$19$
\( T^{4} - 760834631722 T^{2} + \cdots + 42\!\cdots\!44 \)
$23$
\( (T^{2} + 1418368 T - 514307412752)^{2} \)
$29$
\( (T^{2} - 8334986 T + 14857994423056)^{2} \)
$31$
\( T^{4} - 74580188894856 T^{2} + \cdots + 12\!\cdots\!84 \)
$37$
\( (T^{2} - 1613186 T - 172221385813328)^{2} \)
$41$
\( T^{4} - 524653229945812 T^{2} + \cdots + 15\!\cdots\!36 \)
$43$
\( (T^{2} + 38240102 T + 187841499446728)^{2} \)
$47$
\( T^{4} + \cdots + 26\!\cdots\!56 \)
$53$
\( (T^{2} - 6854056 T - 71\!\cdots\!88)^{2} \)
$59$
\( T^{4} + \cdots + 12\!\cdots\!24 \)
$61$
\( T^{4} + \cdots + 24\!\cdots\!96 \)
$67$
\( (T^{2} + 407210388 T + 31\!\cdots\!44)^{2} \)
$71$
\( (T^{2} + 662334372 T + 10\!\cdots\!64)^{2} \)
$73$
\( T^{4} + \cdots + 45\!\cdots\!24 \)
$79$
\( (T^{2} + 334205588 T + 24\!\cdots\!68)^{2} \)
$83$
\( T^{4} + \cdots + 89\!\cdots\!96 \)
$89$
\( T^{4} + \cdots + 19\!\cdots\!84 \)
$97$
\( T^{4} + \cdots + 62\!\cdots\!56 \)
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