Newspace parameters
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 29.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.65113077458\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
| Defining polynomial: | \(x^{7} - 3 x^{6} - 184 x^{5} + 584 x^{4} + 10145 x^{3} - 34491 x^{2} - 149754 x + 524902\) |
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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 \(x^{7} - 3 x^{6} - 184 x^{5} + 584 x^{4} + 10145 x^{3} - 34491 x^{2} - 149754 x + 524902\):
| \(\beta_{0}\) | \(=\) | \( 1 \) |
| \(\beta_{1}\) | \(=\) | \( \nu \) |
| \(\beta_{2}\) | \(=\) | \( \nu^{2} + \nu - 54 \) |
| \(\beta_{3}\) | \(=\) | \((\)\( \nu^{6} - 6 \nu^{5} - 114 \nu^{4} + 618 \nu^{3} + 2727 \nu^{2} - 10724 \nu - 11462 \)\()/240\) |
| \(\beta_{4}\) | \(=\) | \((\)\( \nu^{5} - 5 \nu^{4} - 119 \nu^{3} + 547 \nu^{2} + 2890 \nu - 11242 \)\()/96\) |
| \(\beta_{5}\) | \(=\) | \((\)\( \nu^{6} + 10 \nu^{5} - 218 \nu^{4} - 1094 \nu^{3} + 13231 \nu^{2} + 21356 \nu - 185766 \)\()/960\) |
| \(\beta_{6}\) | \(=\) | \((\)\( \nu^{6} + 2 \nu^{5} - 226 \nu^{4} - 238 \nu^{3} + 14279 \nu^{2} + 5916 \nu - 208054 \)\()/960\) |
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \(\beta_{1}\) |
| \(\nu^{2}\) | \(=\) | \(\beta_{2} - \beta_{1} + 54\) |
| \(\nu^{3}\) | \(=\) | \(-2 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} - \beta_{3} + 4 \beta_{2} + 71 \beta_{1} - 41\) |
| \(\nu^{4}\) | \(=\) | \(-16 \beta_{6} + 8 \beta_{5} + 2 \beta_{3} + 105 \beta_{2} - 95 \beta_{1} + 3846\) |
| \(\nu^{5}\) | \(=\) | \(-318 \beta_{6} + 754 \beta_{5} - 856 \beta_{4} - 109 \beta_{3} + 454 \beta_{2} + 5631 \beta_{1} - 3945\) |
| \(\nu^{6}\) | \(=\) | \(-2496 \beta_{6} + 1728 \beta_{5} - 192 \beta_{4} + 432 \beta_{3} + 9495 \beta_{2} - 7471 \beta_{1} + 304316\) |
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 |
|
−8.56883 | 18.3274 | 41.4249 | −84.3249 | −157.045 | 216.816 | −80.7606 | 92.8952 | 722.566 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.83842 | −24.5656 | 14.7640 | −54.3066 | 167.990 | −37.6697 | 117.867 | 360.469 | 371.372 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.90786 | 29.3989 | −16.7287 | 64.0801 | −114.887 | −138.793 | 190.425 | 621.298 | −250.416 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.60554 | −13.2844 | −25.2112 | 69.0035 | 34.6131 | 156.573 | 149.066 | −66.5241 | −179.792 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.83960 | 15.9679 | 2.10095 | 31.5616 | 93.2461 | 106.304 | −174.599 | 11.9736 | 184.307 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 9.92709 | 15.4219 | 66.5471 | −58.0818 | 153.094 | −210.388 | 342.952 | −5.16616 | −576.583 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.1540 | −15.2661 | 71.1029 | 64.0682 | −155.012 | 91.1564 | 397.049 | −9.94539 | 650.545 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(-1\) |
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 | 29.6.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 261.6.a.e | 7 | ||
| 4.b | odd | 2 | 1 | 464.6.a.k | 7 | ||
| 5.b | even | 2 | 1 | 725.6.a.b | 7 | ||
| 29.b | even | 2 | 1 | 841.6.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.6.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 261.6.a.e | 7 | 3.b | odd | 2 | 1 | ||
| 464.6.a.k | 7 | 4.b | odd | 2 | 1 | ||
| 725.6.a.b | 7 | 5.b | even | 2 | 1 | ||
| 841.6.a.b | 7 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4 T_{2}^{6} - 181 T_{2}^{5} + 346 T_{2}^{4} + 10616 T_{2}^{3} + 2416 T_{2}^{2} - 186896 T_{2} - 351200 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( -351200 - 186896 T + 2416 T^{2} + 10616 T^{3} + 346 T^{4} - 181 T^{5} - 4 T^{6} + T^{7} \) |
| $3$ | \( 661023756 - 22844001 T - 7496290 T^{2} + 280279 T^{3} + 26056 T^{4} - 1015 T^{5} - 26 T^{6} + T^{7} \) |
| $5$ | \( 2378174390186 - 71107482707 T - 1882536724 T^{2} + 54141495 T^{3} + 456170 T^{4} - 13189 T^{5} - 32 T^{6} + T^{7} \) |
| $7$ | \( 361848785235968 + 2579467705856 T - 180798693888 T^{2} + 679110080 T^{3} + 12004416 T^{4} - 61112 T^{5} - 184 T^{6} + T^{7} \) |
| $11$ | \( 515865611322413044 + 3479126648563423 T - 17338617711626 T^{2} - 73256939689 T^{3} + 323786496 T^{4} + 15177 T^{5} - 1106 T^{6} + T^{7} \) |
| $13$ | \( 10011422243374034 + 35719316041077 T - 6217250606588 T^{2} - 8368783321 T^{3} + 449335138 T^{4} - 1099773 T^{5} - 408 T^{6} + T^{7} \) |
| $17$ | \( -\)\(17\!\cdots\!28\)\( - 4762080661877200832 T + 4400022913305568 T^{2} + 11750005682096 T^{3} - 3880384296 T^{4} - 6477972 T^{5} + 874 T^{6} + T^{7} \) |
| $19$ | \( \)\(15\!\cdots\!92\)\( - 384177380729022144 T - 22726079248952448 T^{2} - 2802099264912 T^{3} + 19511173648 T^{4} - 1439924 T^{5} - 4288 T^{6} + T^{7} \) |
| $23$ | \( -1541996530463904256 - 37102640165875136 T + 139547314402368 T^{2} + 1885764458736 T^{3} - 7019218112 T^{4} + 2107804 T^{5} + 4532 T^{6} + T^{7} \) |
| $29$ | \( ( -841 + T )^{7} \) |
| $31$ | \( \)\(64\!\cdots\!48\)\( - \)\(87\!\cdots\!01\)\( T - 2325896554397613794 T^{2} + 2013745795872983 T^{3} + 398022767396 T^{4} - 75300839 T^{5} - 7794 T^{6} + T^{7} \) |
| $37$ | \( -\)\(16\!\cdots\!56\)\( - \)\(12\!\cdots\!48\)\( T + 30949163252985802240 T^{2} + 20164120745134336 T^{3} + 445110854496 T^{4} - 273984608 T^{5} - 5086 T^{6} + T^{7} \) |
| $41$ | \( -\)\(56\!\cdots\!72\)\( - \)\(80\!\cdots\!60\)\( T - \)\(18\!\cdots\!92\)\( T^{2} + 15992490422438592 T^{3} + 4322757774032 T^{4} - 217946616 T^{5} - 19826 T^{6} + T^{7} \) |
| $43$ | \( -\)\(58\!\cdots\!60\)\( - \)\(11\!\cdots\!41\)\( T - \)\(26\!\cdots\!82\)\( T^{2} + 25211595416791175 T^{3} + 5437038516432 T^{4} - 316513743 T^{5} - 19498 T^{6} + T^{7} \) |
| $47$ | \( -\)\(14\!\cdots\!36\)\( - \)\(31\!\cdots\!69\)\( T + 37913247078045875370 T^{2} + 9699630620043943 T^{3} - 1210354130884 T^{4} - 403592367 T^{5} - 14278 T^{6} + T^{7} \) |
| $53$ | \( \)\(84\!\cdots\!94\)\( - \)\(61\!\cdots\!75\)\( T + \)\(86\!\cdots\!32\)\( T^{2} - 4330722850397705 T^{3} - 39591011680062 T^{4} - 55959141 T^{5} + 58644 T^{6} + T^{7} \) |
| $59$ | \( \)\(15\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( T - \)\(13\!\cdots\!80\)\( T^{2} + 1325580183274664304 T^{3} + 24300942873424 T^{4} - 2216676980 T^{5} - 12888 T^{6} + T^{7} \) |
| $61$ | \( \)\(45\!\cdots\!80\)\( - \)\(25\!\cdots\!08\)\( T + \)\(46\!\cdots\!72\)\( T^{2} - 3470412083411730640 T^{3} + 73658693252072 T^{4} + 2180773148 T^{5} - 102866 T^{6} + T^{7} \) |
| $67$ | \( -\)\(11\!\cdots\!60\)\( + \)\(23\!\cdots\!68\)\( T - \)\(67\!\cdots\!24\)\( T^{2} - 7624284713722041344 T^{3} + 328821833213824 T^{4} - 647318752 T^{5} - 102996 T^{6} + T^{7} \) |
| $71$ | \( -\)\(20\!\cdots\!52\)\( + \)\(47\!\cdots\!44\)\( T + \)\(29\!\cdots\!88\)\( T^{2} - 5025269352663123344 T^{3} - 372698633777536 T^{4} - 5405760892 T^{5} + 51596 T^{6} + T^{7} \) |
| $73$ | \( \)\(43\!\cdots\!96\)\( + \)\(24\!\cdots\!32\)\( T + \)\(27\!\cdots\!20\)\( T^{2} + 8121415099482870016 T^{3} - 149044648658016 T^{4} - 6051106208 T^{5} + 17566 T^{6} + T^{7} \) |
| $79$ | \( -\)\(28\!\cdots\!00\)\( + \)\(15\!\cdots\!99\)\( T + \)\(37\!\cdots\!62\)\( T^{2} - 29939303716339944633 T^{3} + 170160611431108 T^{4} + 12042119713 T^{5} - 212058 T^{6} + T^{7} \) |
| $83$ | \( -\)\(97\!\cdots\!52\)\( + \)\(31\!\cdots\!56\)\( T + \)\(95\!\cdots\!52\)\( T^{2} + 42724404510698882672 T^{3} - 2173461254983760 T^{4} - 15465968916 T^{5} + 122928 T^{6} + T^{7} \) |
| $89$ | \( -\)\(79\!\cdots\!20\)\( - \)\(17\!\cdots\!88\)\( T + \)\(75\!\cdots\!92\)\( T^{2} + \)\(15\!\cdots\!96\)\( T^{3} - 1270895508816816 T^{4} - 23989383192 T^{5} + 66510 T^{6} + T^{7} \) |
| $97$ | \( \)\(62\!\cdots\!52\)\( + \)\(96\!\cdots\!96\)\( T + \)\(35\!\cdots\!72\)\( T^{2} + 3500459353391389888 T^{3} - 1404596434864624 T^{4} - 10035608344 T^{5} + 118182 T^{6} + T^{7} \) |
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