Newspace parameters
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(64.6637002752\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
| Defining polynomial: | \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 69) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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 \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \((\)\( -\nu^{4} + 142 \nu^{3} + 467 \nu^{2} - 37904 \nu - 76396 \)\()/1552\) |
| \(\beta_{3}\) | \(=\) | \((\)\( 11 \nu^{4} - 10 \nu^{3} - 3585 \nu^{2} + 2560 \nu + 117124 \)\()/1552\) |
| \(\beta_{4}\) | \(=\) | \((\)\( -11 \nu^{4} + 10 \nu^{3} + 5137 \nu^{2} - 5664 \nu - 399588 \)\()/1552\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{4} + \beta_{3} + 2 \beta_{1} + 182\) |
| \(\nu^{3}\) | \(=\) | \(-\beta_{4} + 11 \beta_{2} + 265 \beta_{1} + 284\) |
| \(\nu^{4}\) | \(=\) | \(325 \beta_{4} + 467 \beta_{3} + 10 \beta_{2} + 660 \beta_{1} + 48926\) |
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 |
|
−18.2528 | 0 | 205.166 | 55.2224 | 0 | −1258.24 | −1408.51 | 0 | −1007.97 | |||||||||||||||||||||||||||||||||
| 1.2 | −12.4672 | 0 | 27.4323 | 40.8788 | 0 | 1126.70 | 1253.80 | 0 | −509.647 | ||||||||||||||||||||||||||||||||||
| 1.3 | 3.24502 | 0 | −117.470 | −168.083 | 0 | 149.817 | −796.555 | 0 | −545.434 | ||||||||||||||||||||||||||||||||||
| 1.4 | 9.64907 | 0 | −34.8955 | 306.939 | 0 | 733.069 | −1571.79 | 0 | 2961.68 | ||||||||||||||||||||||||||||||||||
| 1.5 | 17.8260 | 0 | 189.767 | 31.0428 | 0 | −1247.35 | 1101.05 | 0 | 553.369 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 207.8.a.a | 5 | |
| 3.b | odd | 2 | 1 | 69.8.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.8.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 207.8.a.a | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 455 T_{2}^{3} + 474 T_{2}^{2} + 42284 T_{2} - 127016 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( -127016 + 42284 T + 474 T^{2} - 455 T^{3} + T^{5} \) |
| $3$ | \( T^{5} \) |
| $5$ | \( 3615360000 - 260643200 T + 5761760 T^{2} - 28696 T^{3} - 266 T^{4} + T^{5} \) |
| $7$ | \( -194204974150720 + 1423528608720 T - 510098608 T^{2} - 2361144 T^{3} + 496 T^{4} + T^{5} \) |
| $11$ | \( -154691619430400 + 8190801003520 T + 8448479456 T^{2} - 17900584 T^{3} - 1148 T^{4} + T^{5} \) |
| $13$ | \( 16949964907397805984 + 8797382529202512 T - 46113337584 T^{2} - 206597592 T^{3} + 642 T^{4} + T^{5} \) |
| $17$ | \( -\)\(11\!\cdots\!80\)\( + 269929356591517120 T + 11180076292112 T^{2} - 1285951984 T^{3} - 5798 T^{4} + T^{5} \) |
| $19$ | \( -\)\(14\!\cdots\!36\)\( + 185171606250158832 T + 3874831298480 T^{2} - 976972672 T^{3} + 6036 T^{4} + T^{5} \) |
| $23$ | \( ( 12167 + T )^{5} \) |
| $29$ | \( -\)\(24\!\cdots\!88\)\( - 20455752789771350832 T + 1428794454126960 T^{2} - 5923396536 T^{3} - 169162 T^{4} + T^{5} \) |
| $31$ | \( -\)\(23\!\cdots\!28\)\( - \)\(10\!\cdots\!88\)\( T - 13426584591784320 T^{2} - 43994925216 T^{3} + 199640 T^{4} + T^{5} \) |
| $37$ | \( \)\(82\!\cdots\!20\)\( + \)\(80\!\cdots\!60\)\( T - 12266792719102368 T^{2} - 105154566272 T^{3} + 202002 T^{4} + T^{5} \) |
| $41$ | \( -\)\(16\!\cdots\!52\)\( - \)\(70\!\cdots\!52\)\( T - 66617897199338800 T^{2} - 103444637496 T^{3} + 541282 T^{4} + T^{5} \) |
| $43$ | \( \)\(66\!\cdots\!00\)\( + \)\(89\!\cdots\!12\)\( T - 636892817243404752 T^{2} - 720482936448 T^{3} + 909596 T^{4} + T^{5} \) |
| $47$ | \( \)\(24\!\cdots\!00\)\( + \)\(44\!\cdots\!60\)\( T - 143091146720317440 T^{2} - 1392905639296 T^{3} + 80208 T^{4} + T^{5} \) |
| $53$ | \( -\)\(10\!\cdots\!60\)\( + \)\(79\!\cdots\!40\)\( T + 499432731442902464 T^{2} - 3057292616632 T^{3} - 278138 T^{4} + T^{5} \) |
| $59$ | \( -\)\(75\!\cdots\!64\)\( - \)\(53\!\cdots\!24\)\( T + 13664970599888805632 T^{2} - 3306712956160 T^{3} - 3177380 T^{4} + T^{5} \) |
| $61$ | \( -\)\(14\!\cdots\!84\)\( + \)\(22\!\cdots\!08\)\( T - 7818212974425020704 T^{2} - 9781093585984 T^{3} - 147782 T^{4} + T^{5} \) |
| $67$ | \( -\)\(79\!\cdots\!40\)\( + \)\(18\!\cdots\!32\)\( T - 290932021540136880 T^{2} - 27027287614400 T^{3} + 464916 T^{4} + T^{5} \) |
| $71$ | \( -\)\(49\!\cdots\!52\)\( + \)\(15\!\cdots\!48\)\( T + 179997306641763584 T^{2} - 3496700575968 T^{3} + 1576792 T^{4} + T^{5} \) |
| $73$ | \( \)\(11\!\cdots\!64\)\( + \)\(36\!\cdots\!64\)\( T - 1042990056947080848 T^{2} - 7488464098136 T^{3} + 38190 T^{4} + T^{5} \) |
| $79$ | \( -\)\(36\!\cdots\!20\)\( + \)\(60\!\cdots\!88\)\( T - 7813983134821600528 T^{2} - 20007982009272 T^{3} + 3913336 T^{4} + T^{5} \) |
| $83$ | \( -\)\(18\!\cdots\!20\)\( - \)\(12\!\cdots\!40\)\( T - \)\(15\!\cdots\!08\)\( T^{2} + 58536185513432 T^{3} + 15774716 T^{4} + T^{5} \) |
| $89$ | \( \)\(25\!\cdots\!96\)\( + \)\(95\!\cdots\!32\)\( T - \)\(40\!\cdots\!64\)\( T^{2} - 117042154036144 T^{3} + 1116482 T^{4} + T^{5} \) |
| $97$ | \( \)\(11\!\cdots\!20\)\( + \)\(21\!\cdots\!28\)\( T - \)\(62\!\cdots\!88\)\( T^{2} - 22269450612824 T^{3} + 15738566 T^{4} + T^{5} \) |
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