Newspace parameters
| Level: | \( N \) | \(=\) | \( 9802 = 2 \cdot 13^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9802.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(78.2693640613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.226964648.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 22x^{4} + 17x^{3} + 131x^{2} - 82x - 148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 754) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 22x^{4} + 17x^{3} + 131x^{2} - 82x - 148 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 2\nu^{3} + 51\nu^{2} - 99\nu - 24 ) / 26 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 10\nu^{4} + 17\nu^{3} - 115\nu^{2} + 29\nu + 152 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{5} - 15\nu^{4} - 19\nu^{3} + 166\nu^{2} - 102\nu - 176 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{5} - 37\nu^{4} - 98\nu^{3} + 406\nu^{2} - 41\nu - 474 ) / 26 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -15\nu^{5} + 62\nu^{4} + 134\nu^{3} - 661\nu^{2} + 172\nu + 594 ) / 26 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + 16 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 7\beta_{2} - 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 28\beta_{5} + 32\beta_{4} + 19\beta_{3} + 3\beta_{2} - 2\beta _1 + 164 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 42\beta_{5} + 62\beta_{4} + 171\beta_{3} + 193\beta_{2} - 192\beta _1 + 52 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | −2.32340 | 1.00000 | −2.93190 | 2.32340 | −3.12559 | −1.00000 | 2.39821 | 2.93190 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.32340 | 1.00000 | 3.79270 | 2.32340 | 2.66299 | −1.00000 | 2.39821 | −3.79270 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.642074 | 1.00000 | −2.33727 | −0.642074 | 1.69262 | −1.00000 | −2.58774 | 2.33727 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.642074 | 1.00000 | −0.777642 | −0.642074 | −3.16546 | −1.00000 | −2.58774 | 0.777642 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.68133 | 1.00000 | −3.50932 | −2.68133 | 3.80713 | −1.00000 | 4.18953 | 3.50932 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.68133 | 1.00000 | 2.76342 | −2.68133 | −0.871698 | −1.00000 | 4.18953 | −2.76342 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( +1 \) |
| \(29\) | \( -1 \) |
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 | 9802.2.a.z | 6 | |
| 13.b | even | 2 | 1 | 754.2.a.j | ✓ | 6 | |
| 39.d | odd | 2 | 1 | 6786.2.a.bq | 6 | ||
| 52.b | odd | 2 | 1 | 6032.2.a.s | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 754.2.a.j | ✓ | 6 | 13.b | even | 2 | 1 | |
| 6032.2.a.s | 6 | 52.b | odd | 2 | 1 | ||
| 6786.2.a.bq | 6 | 39.d | odd | 2 | 1 | ||
| 9802.2.a.z | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9802))\):
|
\( T_{3}^{3} - T_{3}^{2} - 6T_{3} + 4 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} - 20T_{5}^{4} - 67T_{5}^{3} + 69T_{5}^{2} + 336T_{5} + 196 \)
|
|
\( T_{7}^{6} - T_{7}^{5} - 22T_{7}^{4} + 17T_{7}^{3} + 131T_{7}^{2} - 82T_{7} - 148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( (T^{3} - T^{2} - 6 T + 4)^{2} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 196 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots - 148 \)
|
| $11$ |
\( T^{6} + 7 T^{5} + \cdots + 1184 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 112 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 32 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 896 \)
|
| $29$ |
\( (T - 1)^{6} \)
|
| $31$ |
\( (T^{3} - 2 T^{2} + \cdots + 128)^{2} \)
|
| $37$ |
\( T^{6} + 22 T^{5} + \cdots + 82352 \)
|
| $41$ |
\( T^{6} + T^{5} + \cdots - 49672 \)
|
| $43$ |
\( T^{6} - 4 T^{5} + \cdots - 1024 \)
|
| $47$ |
\( T^{6} - 24 T^{5} + \cdots + 98048 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 49024 \)
|
| $59$ |
\( T^{6} - 10 T^{5} + \cdots + 224 \)
|
| $61$ |
\( T^{6} + 3 T^{5} + \cdots - 615592 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 363712 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 18688 \)
|
| $73$ |
\( T^{6} + T^{5} + \cdots - 68296 \)
|
| $79$ |
\( T^{6} + 39 T^{5} + \cdots - 188744 \)
|
| $83$ |
\( T^{6} + 8 T^{5} + \cdots + 10976 \)
|
| $89$ |
\( T^{6} + 5 T^{5} + \cdots + 149552 \)
|
| $97$ |
\( T^{6} + T^{5} + \cdots + 5488 \)
|
| show more | |
| show less |