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: | \(1\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{20} - 44 x^{18} - 8 x^{17} + 792 x^{16} + 274 x^{15} - 7498 x^{14} - 3702 x^{13} + 39981 x^{12} + \cdots - 671 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 44 x^{18} - 8 x^{17} + 792 x^{16} + 274 x^{15} - 7498 x^{14} - 3702 x^{13} + 39981 x^{12} + \cdots - 671 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 473007594770 \nu^{19} + 51263080715 \nu^{18} - 20940016857074 \nu^{17} + \cdots - 413603723784317 ) / 270946346603208 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 616398992227 \nu^{19} - 473007594770 \nu^{18} - 27172818738703 \nu^{17} + \cdots - 10\!\cdots\!98 ) / 270946346603208 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21\!\cdots\!97 \nu^{19} + \cdots + 26\!\cdots\!15 ) / 65\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 40\!\cdots\!51 \nu^{19} + \cdots - 84\!\cdots\!01 ) / 65\!\cdots\!24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 30\!\cdots\!77 \nu^{19} + \cdots - 52\!\cdots\!44 ) / 32\!\cdots\!12 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 36\!\cdots\!86 \nu^{19} + \cdots - 58\!\cdots\!07 ) / 32\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15\!\cdots\!91 \nu^{19} + \cdots + 10\!\cdots\!84 ) / 10\!\cdots\!04 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 37\!\cdots\!59 \nu^{19} + \cdots - 91\!\cdots\!42 ) / 21\!\cdots\!08 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 15\!\cdots\!48 \nu^{19} + \cdots - 18\!\cdots\!13 ) / 65\!\cdots\!24 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 54\!\cdots\!63 \nu^{19} + \cdots + 18\!\cdots\!78 ) / 21\!\cdots\!08 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 86\!\cdots\!31 \nu^{19} + \cdots - 21\!\cdots\!28 ) / 32\!\cdots\!12 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 19\!\cdots\!76 \nu^{19} + \cdots - 26\!\cdots\!99 ) / 65\!\cdots\!24 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 20\!\cdots\!61 \nu^{19} + \cdots - 29\!\cdots\!14 ) / 65\!\cdots\!24 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 21\!\cdots\!51 \nu^{19} + \cdots + 45\!\cdots\!51 ) / 65\!\cdots\!24 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 10\!\cdots\!29 \nu^{19} + \cdots - 16\!\cdots\!56 ) / 32\!\cdots\!12 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 59\!\cdots\!97 \nu^{19} + \cdots - 47\!\cdots\!95 ) / 16\!\cdots\!56 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 39\!\cdots\!12 \nu^{19} + \cdots - 29\!\cdots\!75 ) / 65\!\cdots\!24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{18} - \beta_{17} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} + \cdots + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{18} - 2 \beta_{17} + \beta_{16} - \beta_{15} + 12 \beta_{14} + 14 \beta_{13} + 27 \beta_{12} + \cdots + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{19} + 32 \beta_{18} - 18 \beta_{17} - 2 \beta_{15} + 17 \beta_{14} + 20 \beta_{13} + 41 \beta_{12} + \cdots + 282 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{19} + 61 \beta_{18} - 42 \beta_{17} + 22 \beta_{16} - 18 \beta_{15} + 134 \beta_{14} + \cdots + 257 \)
|
| \(\nu^{8}\) | \(=\) |
\( 22 \beta_{19} + 407 \beta_{18} - 256 \beta_{17} + 10 \beta_{16} - 40 \beta_{15} + 230 \beta_{14} + \cdots + 2774 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 28 \beta_{19} + 910 \beta_{18} - 657 \beta_{17} + 333 \beta_{16} - 256 \beta_{15} + 1479 \beta_{14} + \cdots + 3592 \)
|
| \(\nu^{10}\) | \(=\) |
\( 356 \beta_{19} + 4841 \beta_{18} - 3361 \beta_{17} + 284 \beta_{16} - 624 \beta_{15} + 2917 \beta_{14} + \cdots + 28624 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 169 \beta_{19} + 12088 \beta_{18} - 9219 \beta_{17} + 4361 \beta_{16} - 3404 \beta_{15} + \cdots + 47535 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5123 \beta_{19} + 56133 \beta_{18} - 42611 \beta_{17} + 5242 \beta_{16} - 8989 \beta_{15} + \cdots + 305025 \)
|
| \(\nu^{13}\) | \(=\) |
\( 1366 \beta_{19} + 151616 \beta_{18} - 122923 \beta_{17} + 53145 \beta_{16} - 44089 \beta_{15} + \cdots + 606543 \)
|
| \(\nu^{14}\) | \(=\) |
\( 69522 \beta_{19} + 643706 \beta_{18} - 531229 \beta_{17} + 79974 \beta_{16} - 124362 \beta_{15} + \cdots + 3326412 \)
|
| \(\nu^{15}\) | \(=\) |
\( 63232 \beta_{19} + 1840617 \beta_{18} - 1594496 \beta_{17} + 621766 \beta_{16} - 563705 \beta_{15} + \cdots + 7553755 \)
|
| \(\nu^{16}\) | \(=\) |
\( 912808 \beta_{19} + 7344027 \beta_{18} - 6566100 \beta_{17} + 1097683 \beta_{16} - 1675177 \beta_{15} + \cdots + 36907607 \)
|
| \(\nu^{17}\) | \(=\) |
\( 1321644 \beta_{19} + 21900013 \beta_{18} - 20341025 \beta_{17} + 7094248 \beta_{16} - 7150319 \beta_{15} + \cdots + 92542578 \)
|
| \(\nu^{18}\) | \(=\) |
\( 11749378 \beta_{19} + 83596642 \beta_{18} - 80772598 \beta_{17} + 14108499 \beta_{16} - 22120702 \beta_{15} + \cdots + 414948427 \)
|
| \(\nu^{19}\) | \(=\) |
\( 22194732 \beta_{19} + 257205969 \beta_{18} - 256602068 \beta_{17} + 79637781 \beta_{16} + \cdots + 1121075261 \)
|
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 | −3.45150 | 1.00000 | −0.189585 | 3.45150 | −4.98643 | −1.00000 | 8.91285 | 0.189585 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.32934 | 1.00000 | −4.05088 | 3.32934 | 0.277061 | −1.00000 | 8.08448 | 4.05088 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.88841 | 1.00000 | −0.944925 | 2.88841 | 4.15798 | −1.00000 | 5.34293 | 0.944925 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −2.47627 | 1.00000 | 1.90878 | 2.47627 | 0.496785 | −1.00000 | 3.13190 | −1.90878 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −2.00128 | 1.00000 | −1.72306 | 2.00128 | −4.81413 | −1.00000 | 1.00512 | 1.72306 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.89490 | 1.00000 | −1.01839 | 1.89490 | −3.08917 | −1.00000 | 0.590646 | 1.01839 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −0.853817 | 1.00000 | 3.75256 | 0.853817 | −1.60241 | −1.00000 | −2.27100 | −3.75256 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −0.673121 | 1.00000 | −2.68765 | 0.673121 | 2.47131 | −1.00000 | −2.54691 | 2.68765 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | −0.432414 | 1.00000 | −1.71140 | 0.432414 | −0.496557 | −1.00000 | −2.81302 | 1.71140 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 0.246239 | 1.00000 | 1.33007 | −0.246239 | 4.44469 | −1.00000 | −2.93937 | −1.33007 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 0.356006 | 1.00000 | −4.34168 | −0.356006 | −4.92272 | −1.00000 | −2.87326 | 4.34168 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 0.629834 | 1.00000 | 2.23026 | −0.629834 | 2.63114 | −1.00000 | −2.60331 | −2.23026 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | −1.00000 | 0.710010 | 1.00000 | −3.59893 | −0.710010 | 4.37419 | −1.00000 | −2.49589 | 3.59893 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | −1.00000 | 1.10258 | 1.00000 | 4.38617 | −1.10258 | −3.61037 | −1.00000 | −1.78432 | −4.38617 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | −1.00000 | 1.63000 | 1.00000 | 1.99882 | −1.63000 | −0.0481933 | −1.00000 | −0.343113 | −1.99882 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | −1.00000 | 1.80057 | 1.00000 | −0.513888 | −1.80057 | −2.63291 | −1.00000 | 0.242048 | 0.513888 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | −1.00000 | 2.55124 | 1.00000 | −1.48536 | −2.55124 | −0.0617780 | −1.00000 | 3.50883 | 1.48536 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | −1.00000 | 2.94215 | 1.00000 | −3.45011 | −2.94215 | −2.67069 | −1.00000 | 5.65625 | 3.45011 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | −1.00000 | 3.01055 | 1.00000 | 1.15665 | −3.01055 | −0.562479 | −1.00000 | 6.06341 | −1.15665 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | −1.00000 | 3.02187 | 1.00000 | −3.04747 | −3.02187 | 0.644688 | −1.00000 | 6.13171 | 3.04747 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.bo | 20 | |
| 13.b | even | 2 | 1 | 9802.2.a.bp | 20 | ||
| 13.f | odd | 12 | 2 | 754.2.l.c | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 754.2.l.c | ✓ | 40 | 13.f | odd | 12 | 2 | |
| 9802.2.a.bo | 20 | 1.a | even | 1 | 1 | trivial | |
| 9802.2.a.bp | 20 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9802))\):
|
\( T_{3}^{20} - 44 T_{3}^{18} + 8 T_{3}^{17} + 792 T_{3}^{16} - 274 T_{3}^{15} - 7498 T_{3}^{14} + \cdots - 671 \)
|
|
\( T_{5}^{20} + 12 T_{5}^{19} + 4 T_{5}^{18} - 484 T_{5}^{17} - 1623 T_{5}^{16} + 5574 T_{5}^{15} + \cdots - 158259 \)
|
|
\( T_{7}^{20} + 10 T_{7}^{19} - 41 T_{7}^{18} - 678 T_{7}^{17} - 152 T_{7}^{16} + 17254 T_{7}^{15} + \cdots + 576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{20} \)
|
| $3$ |
\( T^{20} - 44 T^{18} + \cdots - 671 \)
|
| $5$ |
\( T^{20} + 12 T^{19} + \cdots - 158259 \)
|
| $7$ |
\( T^{20} + 10 T^{19} + \cdots + 576 \)
|
| $11$ |
\( T^{20} + 10 T^{19} + \cdots + 49954353 \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} + \cdots + 14959711296 \)
|
| $19$ |
\( T^{20} + 16 T^{19} + \cdots + 45046293 \)
|
| $23$ |
\( T^{20} - 10 T^{19} + \cdots - 28907712 \)
|
| $29$ |
\( (T - 1)^{20} \)
|
| $31$ |
\( T^{20} + \cdots + 5461441197 \)
|
| $37$ |
\( T^{20} + \cdots - 775208846784 \)
|
| $41$ |
\( T^{20} + \cdots + 440868691968 \)
|
| $43$ |
\( T^{20} + \cdots + 514893987796729 \)
|
| $47$ |
\( T^{20} + \cdots + 218853219135552 \)
|
| $53$ |
\( T^{20} + \cdots + 741221034768 \)
|
| $59$ |
\( T^{20} + \cdots - 2565162342336 \)
|
| $61$ |
\( T^{20} + \cdots + 65637997888 \)
|
| $67$ |
\( T^{20} + \cdots - 48\!\cdots\!96 \)
|
| $71$ |
\( T^{20} + \cdots + 43\!\cdots\!48 \)
|
| $73$ |
\( T^{20} + \cdots - 378316847960256 \)
|
| $79$ |
\( T^{20} + \cdots + 209747930521 \)
|
| $83$ |
\( T^{20} + \cdots - 11\!\cdots\!24 \)
|
| $89$ |
\( T^{20} + \cdots - 252024318134208 \)
|
| $97$ |
\( T^{20} + \cdots - 47\!\cdots\!12 \)
|
| show more | |
| show less |