Newspace parameters
| Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 143.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.43727313082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
| Defining polynomial: |
\( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{8} + 357 \nu^{7} - 350 \nu^{6} - 14878 \nu^{5} + 9822 \nu^{4} + 154641 \nu^{3} - 82789 \nu^{2} - 171682 \nu + 322364 ) / 37760 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 87 \nu^{8} - 621 \nu^{7} + 6990 \nu^{6} + 29774 \nu^{5} - 168206 \nu^{4} - 418873 \nu^{3} + 1290477 \nu^{2} + 1596466 \nu - 945052 ) / 75520 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 89 \nu^{8} + 93 \nu^{7} + 6290 \nu^{6} + 18 \nu^{5} - 148562 \nu^{4} - 109591 \nu^{3} + 1200419 \nu^{2} + 1253102 \nu - 1282084 ) / 75520 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 53 \nu^{8} - 119 \nu^{7} + 4810 \nu^{6} + 6346 \nu^{5} - 133194 \nu^{4} - 131707 \nu^{3} + 1212663 \nu^{2} + 1011254 \nu - 1429268 ) / 37760 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 283 \nu^{8} + 551 \nu^{7} + 14550 \nu^{6} - 21674 \nu^{5} - 237974 \nu^{4} + 240843 \nu^{3} + 1287513 \nu^{2} - 549206 \nu - 612268 ) / 75520 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 311 \nu^{8} + 653 \nu^{7} - 16270 \nu^{6} - 35982 \nu^{5} + 263118 \nu^{4} + 525529 \nu^{3} - 1433101 \nu^{2} - 1848178 \nu + 1310876 ) / 37760 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 145 \nu^{8} - 181 \nu^{7} - 7810 \nu^{6} + 5246 \nu^{5} + 134978 \nu^{4} - 6305 \nu^{3} - 783307 \nu^{2} - 483326 \nu + 495876 ) / 15104 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + 2\beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{2} + 19\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{8} + 2\beta_{7} - \beta_{6} - 5\beta_{5} + 28\beta_{4} - 21\beta_{3} - 29\beta_{2} + 8\beta _1 + 259 \)
|
| \(\nu^{5}\) | \(=\) |
\( 33\beta_{8} - \beta_{7} + 60\beta_{6} - 43\beta_{5} + 127\beta_{4} - 4\beta_{3} - 37\beta_{2} + 404\beta _1 + 232 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 66 \beta_{8} + 90 \beta_{7} - 2 \beta_{6} - 220 \beta_{5} + 804 \beta_{4} - 436 \beta_{3} - 804 \beta_{2} + 391 \beta _1 + 5756 \)
|
| \(\nu^{7}\) | \(=\) |
\( 928\beta_{8} + 1662\beta_{6} - 1458\beta_{5} + 4309\beta_{4} - 275\beta_{3} - 1287\beta_{2} + 9292\beta _1 + 8519 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1581 \beta_{8} + 3022 \beta_{7} + 814 \beta_{6} - 7503 \beta_{5} + 23557 \beta_{4} - 9536 \beta_{3} - 22023 \beta_{2} + 14755 \beta _1 + 138197 \)
|
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 |
|
−4.87031 | −1.15688 | 15.7199 | −10.2656 | 5.63435 | 15.3321 | −37.5984 | −25.6616 | 49.9968 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.11495 | 9.51427 | 8.93279 | 14.5196 | −39.1507 | 21.0410 | −3.83839 | 63.5214 | −59.7475 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.76323 | −4.70664 | 6.16188 | 20.6048 | 17.7122 | −28.6315 | 6.91727 | −4.84752 | −77.5406 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.62159 | −2.83913 | −5.37046 | −8.40999 | 4.60389 | −9.04976 | 21.6813 | −18.9393 | 13.6375 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.388321 | 3.09988 | −7.84921 | 16.8933 | 1.20375 | 26.1569 | −6.15457 | −17.3907 | 6.56001 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.765277 | −9.54214 | −7.41435 | −17.1562 | −7.30238 | −4.60754 | −11.7962 | 64.0525 | −13.1292 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.71870 | 7.61710 | 5.82875 | 15.9808 | 28.3257 | −21.9580 | −8.07423 | 31.0202 | 59.4279 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.08298 | 7.19985 | 8.67073 | −7.90460 | 29.3968 | 23.1330 | 2.73856 | 24.8378 | −32.2743 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.41479 | −1.18631 | 21.3199 | 5.73789 | −6.42360 | 3.58372 | 72.1247 | −25.5927 | 31.0694 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(1\) |
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 | 143.4.a.c | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1287.4.a.k | 9 | ||
| 4.b | odd | 2 | 1 | 2288.4.a.r | 9 | ||
| 11.b | odd | 2 | 1 | 1573.4.a.e | 9 | ||
| 13.b | even | 2 | 1 | 1859.4.a.d | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1287.4.a.k | 9 | 3.b | odd | 2 | 1 | ||
| 1573.4.a.e | 9 | 11.b | odd | 2 | 1 | ||
| 1859.4.a.d | 9 | 13.b | even | 2 | 1 | ||
| 2288.4.a.r | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 59T_{2}^{7} - 12T_{2}^{6} + 1144T_{2}^{5} + 345T_{2}^{4} - 7888T_{2}^{3} - 2245T_{2}^{2} + 9710T_{2} - 2988 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(143))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{9} - 59 T^{7} - 12 T^{6} + \cdots - 2988 \)
$3$
\( T^{9} - 8 T^{8} - 135 T^{7} + \cdots + 283048 \)
$5$
\( T^{9} - 30 T^{8} + \cdots - 5425892224 \)
$7$
\( T^{9} - 25 T^{8} + \cdots - 18338418984 \)
$11$
\( (T + 11)^{9} \)
$13$
\( (T + 13)^{9} \)
$17$
\( T^{9} - 53 T^{8} + \cdots + 63\!\cdots\!76 \)
$19$
\( T^{9} - 69 T^{8} + \cdots + 14\!\cdots\!00 \)
$23$
\( T^{9} - 216 T^{8} + \cdots + 22\!\cdots\!44 \)
$29$
\( T^{9} + 91 T^{8} + \cdots + 27\!\cdots\!88 \)
$31$
\( T^{9} - 636 T^{8} + \cdots + 56\!\cdots\!52 \)
$37$
\( T^{9} - 967 T^{8} + \cdots - 72\!\cdots\!76 \)
$41$
\( T^{9} + 226 T^{8} + \cdots - 58\!\cdots\!96 \)
$43$
\( T^{9} - 42 T^{8} + \cdots + 51\!\cdots\!52 \)
$47$
\( T^{9} + 269 T^{8} + \cdots - 51\!\cdots\!84 \)
$53$
\( T^{9} - 1227 T^{8} + \cdots - 12\!\cdots\!32 \)
$59$
\( T^{9} + 613 T^{8} + \cdots + 15\!\cdots\!52 \)
$61$
\( T^{9} - 427 T^{8} + \cdots + 49\!\cdots\!64 \)
$67$
\( T^{9} + 271 T^{8} + \cdots + 89\!\cdots\!72 \)
$71$
\( T^{9} - 2279 T^{8} + \cdots + 15\!\cdots\!32 \)
$73$
\( T^{9} - 3602 T^{8} + \cdots - 24\!\cdots\!24 \)
$79$
\( T^{9} + 1182 T^{8} + \cdots + 19\!\cdots\!00 \)
$83$
\( T^{9} + 1877 T^{8} + \cdots + 41\!\cdots\!88 \)
$89$
\( T^{9} - 1258 T^{8} + \cdots - 94\!\cdots\!48 \)
$97$
\( T^{9} - 4002 T^{8} + \cdots + 56\!\cdots\!64 \)
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