Newspace parameters
| Level: | \( N \) | = | \( 8 = 2^{3} \) |
| Weight: | \( k \) | = | \( 21 \) |
| Character orbit: | \([\chi]\) | = | 8.d (of order \(2\) and degree \(1\)) |
Newform invariants
| Self dual: | No |
| Analytic conductor: | \(20.2811012082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} + \cdots)\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{153}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2} \) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) 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 \(x^{18} + 66008406614424 x^{16} - 170362825164232872 x^{15} + \)\(17\!\cdots\!98\)\( x^{14} - \)\(81\!\cdots\!32\)\( x^{13} + \)\(24\!\cdots\!04\)\( x^{12} - \)\(15\!\cdots\!40\)\( x^{11} + \)\(19\!\cdots\!77\)\( x^{10} - \)\(12\!\cdots\!56\)\( x^{9} + \)\(87\!\cdots\!76\)\( x^{8} - \)\(48\!\cdots\!20\)\( x^{7} + \)\(22\!\cdots\!20\)\( x^{6} - \)\(47\!\cdots\!80\)\( x^{5} + \)\(31\!\cdots\!00\)\( x^{4} + \)\(89\!\cdots\!00\)\( x^{3} + \)\(21\!\cdots\!00\)\( x^{2} + \)\(13\!\cdots\!00\)\( x + \)\(53\!\cdots\!00\)\(\):
| \(1\) | \(=\) | \(\beta_0\) |
| \(\nu\) | \(=\) | \((\)\(\beta_{4} + 2 \beta_{3} + \beta_{2} + 34 \beta_{1} + 3\)\()/4\) |
| \(\nu^{2}\) | \(=\) | \((\)\(4 \beta_{17} - 868 \beta_{16} - 2639 \beta_{15} + 5443 \beta_{14} - 60 \beta_{13} - 976 \beta_{12} + 24 \beta_{11} - 1370 \beta_{10} - 2282 \beta_{9} - 45069 \beta_{8} + 23977 \beta_{7} + 157692 \beta_{6} - 759752 \beta_{5} - 156283 \beta_{4} + 7368163 \beta_{3} + 154573730 \beta_{2} + 1767556779 \beta_{1} - 117348153374177\)\()/16\) |
| \(\nu^{3}\) | \(=\) | \((\)\(1629969 \beta_{17} + 1424850841 \beta_{16} - 55655981403 \beta_{15} - 32977917581 \beta_{14} + 5368342726 \beta_{13} - 10078546727 \beta_{12} + 2425749150 \beta_{11} - 2456646531 \beta_{10} - 10443921244 \beta_{9} + 129481030176 \beta_{8} + 2300331816283 \beta_{7} + 1592043911510 \beta_{6} + 5041570857835 \beta_{5} - 202380975665952 \beta_{4} - 572933927588211 \beta_{3} - 214120063481500 \beta_{2} + 125519286978274054 \beta_{1} + 1831369458169559968\)\()/64\) |
| \(\nu^{4}\) | \(=\) | \((\)\(-787099348314952 \beta_{17} + 11601857956326864 \beta_{16} + 285031397610029797 \beta_{15} - 944563736073735106 \beta_{14} + 707174811894028 \beta_{13} + 138509220812564676 \beta_{12} - 4397834852773200 \beta_{11} + 183898964342216426 \beta_{10} + 343122679962936614 \beta_{9} + 5738650639022817811 \beta_{8} - 4137329599191497365 \beta_{7} - 18969795394786232659 \beta_{6} + 130384700379212928676 \beta_{5} + 14615343714836581729 \beta_{4} - 533780732780652859160 \beta_{3} - 22869947056621118437833 \beta_{2} - 575093031005934617873135 \beta_{1} + 12013310827358516621954128059\)\()/128\) |
| \(\nu^{5}\) | \(=\) | \((\)\(-38178709141709832251110 \beta_{17} - 65453424728646265751338 \beta_{16} + 2636054464207271180946688 \beta_{15} + 1613909414831286600698912 \beta_{14} - 114853319862651773771943 \beta_{13} + 358450035953825924795166 \beta_{12} - 19886134874659237154500 \beta_{11} - 25182387821954988985320 \beta_{10} + 481780720145097469361272 \beta_{9} - 5350376179030541860423673 \beta_{8} - 94549041673237739297700512 \beta_{7} - 47611751921022678831468350 \beta_{6} - 247692761902209008976779118 \beta_{5} + 6112829191425608246633075173 \beta_{4} + 21251161943591362040590755782 \beta_{3} + 10998000200343310008549330028 \beta_{2} - 4503930271277169575663964245977 \beta_{1} - 110786019509274249075372283843473\)\()/128\) |
| \(\nu^{6}\) | \(=\) | \((\)\(\)\(17\!\cdots\!28\)\( \beta_{17} + \)\(89\!\cdots\!20\)\( \beta_{16} - \)\(52\!\cdots\!77\)\( \beta_{15} + \)\(18\!\cdots\!84\)\( \beta_{14} - \)\(17\!\cdots\!69\)\( \beta_{13} - \)\(21\!\cdots\!84\)\( \beta_{12} + \)\(10\!\cdots\!28\)\( \beta_{11} - \)\(35\!\cdots\!64\)\( \beta_{10} - \)\(59\!\cdots\!58\)\( \beta_{9} - \)\(88\!\cdots\!08\)\( \beta_{8} + \)\(75\!\cdots\!13\)\( \beta_{7} + \)\(20\!\cdots\!77\)\( \beta_{6} - \)\(23\!\cdots\!00\)\( \beta_{5} - \)\(25\!\cdots\!44\)\( \beta_{4} + \)\(92\!\cdots\!50\)\( \beta_{3} + \)\(41\!\cdots\!63\)\( \beta_{2} + \)\(17\!\cdots\!98\)\( \beta_{1} - \)\(18\!\cdots\!98\)\(\)\()/128\) |
| \(\nu^{7}\) | \(=\) | \((\)\(\)\(11\!\cdots\!58\)\( \beta_{17} + \)\(13\!\cdots\!74\)\( \beta_{16} - \)\(52\!\cdots\!60\)\( \beta_{15} - \)\(31\!\cdots\!86\)\( \beta_{14} + \)\(26\!\cdots\!47\)\( \beta_{13} - \)\(56\!\cdots\!62\)\( \beta_{12} - \)\(38\!\cdots\!04\)\( \beta_{11} + \)\(31\!\cdots\!74\)\( \beta_{10} - \)\(91\!\cdots\!66\)\( \beta_{9} + \)\(97\!\cdots\!23\)\( \beta_{8} + \)\(16\!\cdots\!34\)\( \beta_{7} + \)\(87\!\cdots\!78\)\( \beta_{6} + \)\(48\!\cdots\!72\)\( \beta_{5} - \)\(98\!\cdots\!39\)\( \beta_{4} - \)\(39\!\cdots\!02\)\( \beta_{3} - \)\(26\!\cdots\!46\)\( \beta_{2} + \)\(85\!\cdots\!05\)\( \beta_{1} + \)\(27\!\cdots\!67\)\(\)\()/128\) |
| \(\nu^{8}\) | \(=\) | \((\)\(-\)\(36\!\cdots\!24\)\( \beta_{17} - \)\(26\!\cdots\!52\)\( \beta_{16} + \)\(10\!\cdots\!33\)\( \beta_{15} - \)\(35\!\cdots\!34\)\( \beta_{14} + \)\(67\!\cdots\!34\)\( \beta_{13} + \)\(33\!\cdots\!84\)\( \beta_{12} - \)\(22\!\cdots\!20\)\( \beta_{11} + \)\(67\!\cdots\!26\)\( \beta_{10} + \)\(10\!\cdots\!38\)\( \beta_{9} + \)\(14\!\cdots\!97\)\( \beta_{8} - \)\(13\!\cdots\!29\)\( \beta_{7} - \)\(14\!\cdots\!95\)\( \beta_{6} + \)\(41\!\cdots\!32\)\( \beta_{5} + \)\(53\!\cdots\!11\)\( \beta_{4} - \)\(19\!\cdots\!84\)\( \beta_{3} - \)\(82\!\cdots\!61\)\( \beta_{2} - \)\(41\!\cdots\!57\)\( \beta_{1} + \)\(29\!\cdots\!37\)\(\)\()/128\) |
| \(\nu^{9}\) | \(=\) | \((\)\(-\)\(24\!\cdots\!02\)\( \beta_{17} - \)\(27\!\cdots\!94\)\( \beta_{16} + \)\(98\!\cdots\!24\)\( \beta_{15} + \)\(57\!\cdots\!44\)\( \beta_{14} + \)\(21\!\cdots\!33\)\( \beta_{13} + \)\(87\!\cdots\!54\)\( \beta_{12} + \)\(12\!\cdots\!40\)\( \beta_{11} - \)\(95\!\cdots\!96\)\( \beta_{10} + \)\(16\!\cdots\!36\)\( \beta_{9} - \)\(16\!\cdots\!29\)\( \beta_{8} - \)\(28\!\cdots\!04\)\( \beta_{7} - \)\(17\!\cdots\!58\)\( \beta_{6} - \)\(90\!\cdots\!82\)\( \beta_{5} + \)\(16\!\cdots\!73\)\( \beta_{4} + \)\(72\!\cdots\!38\)\( \beta_{3} + \)\(59\!\cdots\!40\)\( \beta_{2} - \)\(16\!\cdots\!09\)\( \beta_{1} - \)\(69\!\cdots\!97\)\(\)\()/128\) |
| \(\nu^{10}\) | \(=\) | \((\)\(\)\(75\!\cdots\!80\)\( \beta_{17} + \)\(58\!\cdots\!12\)\( \beta_{16} - \)\(19\!\cdots\!93\)\( \beta_{15} + \)\(66\!\cdots\!04\)\( \beta_{14} - \)\(16\!\cdots\!39\)\( \beta_{13} - \)\(53\!\cdots\!28\)\( \beta_{12} + \)\(48\!\cdots\!00\)\( \beta_{11} - \)\(12\!\cdots\!32\)\( \beta_{10} - \)\(17\!\cdots\!42\)\( \beta_{9} - \)\(23\!\cdots\!94\)\( \beta_{8} + \)\(23\!\cdots\!77\)\( \beta_{7} - \)\(48\!\cdots\!39\)\( \beta_{6} - \)\(71\!\cdots\!28\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{4} + \)\(40\!\cdots\!74\)\( \beta_{3} + \)\(16\!\cdots\!71\)\( \beta_{2} + \)\(87\!\cdots\!96\)\( \beta_{1} - \)\(49\!\cdots\!88\)\(\)\()/128\) |
| \(\nu^{11}\) | \(=\) | \((\)\(\)\(49\!\cdots\!90\)\( \beta_{17} + \)\(52\!\cdots\!06\)\( \beta_{16} - \)\(17\!\cdots\!52\)\( \beta_{15} - \)\(10\!\cdots\!06\)\( \beta_{14} - \)\(72\!\cdots\!13\)\( \beta_{13} - \)\(13\!\cdots\!54\)\( \beta_{12} - \)\(25\!\cdots\!56\)\( \beta_{11} + \)\(22\!\cdots\!74\)\( \beta_{10} - \)\(28\!\cdots\!98\)\( \beta_{9} + \)\(29\!\cdots\!35\)\( \beta_{8} + \)\(49\!\cdots\!14\)\( \beta_{7} + \)\(32\!\cdots\!22\)\( \beta_{6} + \)\(16\!\cdots\!20\)\( \beta_{5} - \)\(27\!\cdots\!15\)\( \beta_{4} - \)\(13\!\cdots\!78\)\( \beta_{3} - \)\(13\!\cdots\!10\)\( \beta_{2} + \)\(29\!\cdots\!73\)\( \beta_{1} + \)\(16\!\cdots\!23\)\(\)\()/128\) |
| \(\nu^{12}\) | \(=\) | \((\)\(-\)\(15\!\cdots\!88\)\( \beta_{17} - \)\(11\!\cdots\!64\)\( \beta_{16} + \)\(36\!\cdots\!17\)\( \beta_{15} - \)\(12\!\cdots\!90\)\( \beta_{14} + \)\(35\!\cdots\!56\)\( \beta_{13} + \)\(86\!\cdots\!56\)\( \beta_{12} - \)\(10\!\cdots\!92\)\( \beta_{11} + \)\(22\!\cdots\!90\)\( \beta_{10} + \)\(30\!\cdots\!82\)\( \beta_{9} + \)\(38\!\cdots\!03\)\( \beta_{8} - \)\(41\!\cdots\!29\)\( \beta_{7} + \)\(49\!\cdots\!53\)\( \beta_{6} + \)\(12\!\cdots\!32\)\( \beta_{5} + \)\(23\!\cdots\!01\)\( \beta_{4} - \)\(79\!\cdots\!52\)\( \beta_{3} - \)\(31\!\cdots\!93\)\( \beta_{2} - \)\(16\!\cdots\!51\)\( \beta_{1} + \)\(84\!\cdots\!51\)\(\)\()/128\) |
| \(\nu^{13}\) | \(=\) | \((\)\(-\)\(93\!\cdots\!74\)\( \beta_{17} - \)\(99\!\cdots\!78\)\( \beta_{16} + \)\(32\!\cdots\!24\)\( \beta_{15} + \)\(18\!\cdots\!20\)\( \beta_{14} + \)\(17\!\cdots\!61\)\( \beta_{13} + \)\(22\!\cdots\!62\)\( \beta_{12} + \)\(44\!\cdots\!60\)\( \beta_{11} - \)\(46\!\cdots\!96\)\( \beta_{10} + \)\(50\!\cdots\!92\)\( \beta_{9} - \)\(51\!\cdots\!89\)\( \beta_{8} - \)\(85\!\cdots\!48\)\( \beta_{7} - \)\(60\!\cdots\!62\)\( \beta_{6} - \)\(29\!\cdots\!34\)\( \beta_{5} + \)\(47\!\cdots\!01\)\( \beta_{4} + \)\(23\!\cdots\!74\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{2} - \)\(51\!\cdots\!69\)\( \beta_{1} - \)\(38\!\cdots\!61\)\(\)\()/128\) |
| \(\nu^{14}\) | \(=\) | \((\)\(\)\(30\!\cdots\!00\)\( \beta_{17} + \)\(21\!\cdots\!60\)\( \beta_{16} - \)\(68\!\cdots\!97\)\( \beta_{15} + \)\(22\!\cdots\!60\)\( \beta_{14} - \)\(67\!\cdots\!81\)\( \beta_{13} - \)\(14\!\cdots\!48\)\( \beta_{12} + \)\(20\!\cdots\!60\)\( \beta_{11} - \)\(38\!\cdots\!20\)\( \beta_{10} - \)\(52\!\cdots\!94\)\( \beta_{9} - \)\(63\!\cdots\!84\)\( \beta_{8} + \)\(74\!\cdots\!53\)\( \beta_{7} - \)\(14\!\cdots\!87\)\( \beta_{6} - \)\(21\!\cdots\!60\)\( \beta_{5} - \)\(48\!\cdots\!24\)\( \beta_{4} + \)\(15\!\cdots\!74\)\( \beta_{3} + \)\(60\!\cdots\!03\)\( \beta_{2} + \)\(32\!\cdots\!74\)\( \beta_{1} - \)\(14\!\cdots\!42\)\(\)\()/128\) |
| \(\nu^{15}\) | \(=\) | \((\)\(\)\(17\!\cdots\!50\)\( \beta_{17} + \)\(18\!\cdots\!26\)\( \beta_{16} - \)\(57\!\cdots\!08\)\( \beta_{15} - \)\(32\!\cdots\!06\)\( \beta_{14} - \)\(38\!\cdots\!93\)\( \beta_{13} - \)\(37\!\cdots\!62\)\( \beta_{12} - \)\(69\!\cdots\!00\)\( \beta_{11} + \)\(92\!\cdots\!90\)\( \beta_{10} - \)\(87\!\cdots\!06\)\( \beta_{9} + \)\(90\!\cdots\!19\)\( \beta_{8} + \)\(14\!\cdots\!94\)\( \beta_{7} + \)\(10\!\cdots\!38\)\( \beta_{6} + \)\(51\!\cdots\!16\)\( \beta_{5} - \)\(81\!\cdots\!27\)\( \beta_{4} - \)\(41\!\cdots\!78\)\( \beta_{3} - \)\(57\!\cdots\!02\)\( \beta_{2} + \)\(87\!\cdots\!13\)\( \beta_{1} + \)\(84\!\cdots\!75\)\(\)\()/128\) |
| \(\nu^{16}\) | \(=\) | \((\)\(-\)\(61\!\cdots\!76\)\( \beta_{17} - \)\(39\!\cdots\!32\)\( \beta_{16} + \)\(12\!\cdots\!09\)\( \beta_{15} - \)\(39\!\cdots\!90\)\( \beta_{14} + \)\(12\!\cdots\!46\)\( \beta_{13} + \)\(23\!\cdots\!96\)\( \beta_{12} - \)\(40\!\cdots\!08\)\( \beta_{11} + \)\(65\!\cdots\!50\)\( \beta_{10} + \)\(90\!\cdots\!62\)\( \beta_{9} + \)\(10\!\cdots\!57\)\( \beta_{8} - \)\(13\!\cdots\!37\)\( \beta_{7} + \)\(34\!\cdots\!41\)\( \beta_{6} + \)\(36\!\cdots\!96\)\( \beta_{5} + \)\(97\!\cdots\!91\)\( \beta_{4} - \)\(28\!\cdots\!88\)\( \beta_{3} - \)\(11\!\cdots\!09\)\( \beta_{2} - \)\(59\!\cdots\!13\)\( \beta_{1} + \)\(24\!\cdots\!77\)\(\)\()/128\) |
| \(\nu^{17}\) | \(=\) | \((\)\(-\)\(31\!\cdots\!22\)\( \beta_{17} - \)\(34\!\cdots\!70\)\( \beta_{16} + \)\(10\!\cdots\!08\)\( \beta_{15} + \)\(57\!\cdots\!12\)\( \beta_{14} + \)\(77\!\cdots\!17\)\( \beta_{13} + \)\(62\!\cdots\!82\)\( \beta_{12} + \)\(98\!\cdots\!84\)\( \beta_{11} - \)\(17\!\cdots\!64\)\( \beta_{10} + \)\(15\!\cdots\!88\)\( \beta_{9} - \)\(15\!\cdots\!89\)\( \beta_{8} - \)\(25\!\cdots\!40\)\( \beta_{7} - \)\(19\!\cdots\!14\)\( \beta_{6} - \)\(91\!\cdots\!74\)\( \beta_{5} + \)\(14\!\cdots\!73\)\( \beta_{4} + \)\(74\!\cdots\!38\)\( \beta_{3} + \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(14\!\cdots\!85\)\( \beta_{1} - \)\(18\!\cdots\!57\)\(\)\()/128\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−998.751 | − | 225.992i | 60145.3 | 946431. | + | 451420.i | 8.23733e6i | −6.00701e7 | − | 1.35924e7i | 5.26995e8i | −8.43231e8 | − | 6.64742e8i | 1.30668e8 | 1.86157e9 | − | 8.22704e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −998.751 | + | 225.992i | 60145.3 | 946431. | − | 451420.i | − | 8.23733e6i | −6.00701e7 | + | 1.35924e7i | − | 5.26995e8i | −8.43231e8 | + | 6.64742e8i | 1.30668e8 | 1.86157e9 | + | 8.22704e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −980.704 | − | 294.612i | −86392.9 | 874983. | + | 577854.i | 1.68203e7i | 8.47258e7 | + | 2.54524e7i | − | 3.64653e8i | −6.87856e8 | − | 8.24485e8i | 3.97695e9 | 4.95546e9 | − | 1.64957e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −980.704 | + | 294.612i | −86392.9 | 874983. | − | 577854.i | − | 1.68203e7i | 8.47258e7 | − | 2.54524e7i | 3.64653e8i | −6.87856e8 | + | 8.24485e8i | 3.97695e9 | 4.95546e9 | + | 1.64957e10i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | −749.926 | − | 697.272i | −16119.5 | 76200.7 | + | 1.04580e6i | − | 8.44883e6i | 1.20885e7 | + | 1.12397e7i | − | 6.17349e7i | 6.72064e8 | − | 8.37407e8i | −3.22694e9 | −5.89113e9 | + | 6.33599e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | −749.926 | + | 697.272i | −16119.5 | 76200.7 | − | 1.04580e6i | 8.44883e6i | 1.20885e7 | − | 1.12397e7i | 6.17349e7i | 6.72064e8 | + | 8.37407e8i | −3.22694e9 | −5.89113e9 | − | 6.33599e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | −340.525 | − | 965.722i | 84336.7 | −816662. | + | 657705.i | 4.86521e6i | −2.87188e7 | − | 8.14458e7i | − | 1.26224e8i | 9.13253e8 | + | 5.64703e8i | 3.62590e9 | 4.69844e9 | − | 1.65673e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | −340.525 | + | 965.722i | 84336.7 | −816662. | − | 657705.i | − | 4.86521e6i | −2.87188e7 | + | 8.14458e7i | 1.26224e8i | 9.13253e8 | − | 5.64703e8i | 3.62590e9 | 4.69844e9 | + | 1.65673e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.9 | −39.1296 | − | 1023.25i | −61558.6 | −1.04551e6 | + | 80078.8i | 2.85835e6i | 2.40876e6 | + | 6.29900e7i | 1.54680e8i | 1.22851e8 | + | 1.06669e9i | 3.02682e8 | 2.92482e9 | − | 1.11846e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.10 | −39.1296 | + | 1023.25i | −61558.6 | −1.04551e6 | − | 80078.8i | − | 2.85835e6i | 2.40876e6 | − | 6.29900e7i | − | 1.54680e8i | 1.22851e8 | − | 1.06669e9i | 3.02682e8 | 2.92482e9 | + | 1.11846e8i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.11 | 492.862 | − | 897.587i | 53283.8 | −562749. | − | 884774.i | − | 1.65732e7i | 2.62616e7 | − | 4.78269e7i | 1.30215e8i | −1.07152e9 | + | 6.90447e7i | −6.47621e8 | −1.48759e10 | − | 8.16831e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.12 | 492.862 | + | 897.587i | 53283.8 | −562749. | + | 884774.i | 1.65732e7i | 2.62616e7 | + | 4.78269e7i | − | 1.30215e8i | −1.07152e9 | − | 6.90447e7i | −6.47621e8 | −1.48759e10 | + | 8.16831e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.13 | 607.571 | − | 824.278i | 24307.8 | −310291. | − | 1.00161e6i | 1.39677e7i | 1.47687e7 | − | 2.00364e7i | − | 2.84328e8i | −1.01413e9 | − | 3.52785e8i | −2.89591e9 | 1.15133e10 | + | 8.48636e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.14 | 607.571 | + | 824.278i | 24307.8 | −310291. | + | 1.00161e6i | − | 1.39677e7i | 1.47687e7 | + | 2.00364e7i | 2.84328e8i | −1.01413e9 | + | 3.52785e8i | −2.89591e9 | 1.15133e10 | − | 8.48636e9i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.15 | 829.658 | − | 600.204i | −103279. | 328087. | − | 995927.i | − | 1.04292e7i | −8.56862e7 | + | 6.19884e7i | − | 3.68005e8i | −3.25559e8 | − | 1.02320e9i | 7.17976e9 | −6.25967e9 | − | 8.65271e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.16 | 829.658 | + | 600.204i | −103279. | 328087. | + | 995927.i | 1.04292e7i | −8.56862e7 | − | 6.19884e7i | 3.68005e8i | −3.25559e8 | + | 1.02320e9i | 7.17976e9 | −6.25967e9 | + | 8.65271e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.17 | 979.944 | − | 297.130i | −11837.6 | 872004. | − | 582341.i | 4.86160e6i | −1.16002e7 | + | 3.51730e6i | 3.96260e8i | 6.81484e8 | − | 8.29760e8i | −3.34666e9 | 1.44453e9 | + | 4.76409e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.18 | 979.944 | + | 297.130i | −11837.6 | 872004. | + | 582341.i | − | 4.86160e6i | −1.16002e7 | − | 3.51730e6i | − | 3.96260e8i | 6.81484e8 | + | 8.29760e8i | −3.34666e9 | 1.44453e9 | − | 4.76409e9i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char. orbit | Parity | Mult. | Self Twist | Proved |
|---|---|---|---|---|
| 1.a | Even | 1 | trivial | yes |
| 8.d | Odd | 1 | yes |
Hecke kernels
This newform can be constructed as the kernel of the linear operator \(T_{3}^{9} + \cdots\) acting on \(S_{21}^{\mathrm{new}}(8, [\chi])\).