Properties

Label 825.6.a.v.1.3
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-7.57523\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.57523 q^{2} -9.00000 q^{3} +41.5346 q^{4} +77.1771 q^{6} -178.462 q^{7} -81.7612 q^{8} +81.0000 q^{9} +121.000 q^{11} -373.811 q^{12} +361.980 q^{13} +1530.36 q^{14} -627.985 q^{16} -934.004 q^{17} -694.594 q^{18} +753.929 q^{19} +1606.16 q^{21} -1037.60 q^{22} -3231.01 q^{23} +735.851 q^{24} -3104.06 q^{26} -729.000 q^{27} -7412.36 q^{28} +2606.34 q^{29} +662.215 q^{31} +8001.48 q^{32} -1089.00 q^{33} +8009.30 q^{34} +3364.30 q^{36} +12924.2 q^{37} -6465.11 q^{38} -3257.82 q^{39} -2538.56 q^{41} -13773.2 q^{42} -22022.8 q^{43} +5025.68 q^{44} +27706.7 q^{46} -20835.4 q^{47} +5651.87 q^{48} +15041.8 q^{49} +8406.03 q^{51} +15034.7 q^{52} +27450.5 q^{53} +6251.34 q^{54} +14591.3 q^{56} -6785.36 q^{57} -22350.0 q^{58} +7759.71 q^{59} +38469.2 q^{61} -5678.65 q^{62} -14455.5 q^{63} -48519.0 q^{64} +9338.43 q^{66} -35521.3 q^{67} -38793.4 q^{68} +29079.1 q^{69} -62677.1 q^{71} -6622.66 q^{72} -68950.2 q^{73} -110828. q^{74} +31314.1 q^{76} -21594.0 q^{77} +27936.5 q^{78} -17313.8 q^{79} +6561.00 q^{81} +21768.7 q^{82} +89017.9 q^{83} +66711.2 q^{84} +188850. q^{86} -23457.1 q^{87} -9893.11 q^{88} +129627. q^{89} -64599.8 q^{91} -134199. q^{92} -5959.94 q^{93} +178668. q^{94} -72013.3 q^{96} +136732. q^{97} -128987. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.57523 −1.51590 −0.757950 0.652312i \(-0.773799\pi\)
−0.757950 + 0.652312i \(0.773799\pi\)
\(3\) −9.00000 −0.577350
\(4\) 41.5346 1.29796
\(5\) 0 0
\(6\) 77.1771 0.875206
\(7\) −178.462 −1.37658 −0.688290 0.725435i \(-0.741638\pi\)
−0.688290 + 0.725435i \(0.741638\pi\)
\(8\) −81.7612 −0.451671
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −373.811 −0.749375
\(13\) 361.980 0.594054 0.297027 0.954869i \(-0.404005\pi\)
0.297027 + 0.954869i \(0.404005\pi\)
\(14\) 1530.36 2.08676
\(15\) 0 0
\(16\) −627.985 −0.613267
\(17\) −934.004 −0.783838 −0.391919 0.920000i \(-0.628189\pi\)
−0.391919 + 0.920000i \(0.628189\pi\)
\(18\) −694.594 −0.505300
\(19\) 753.929 0.479122 0.239561 0.970881i \(-0.422996\pi\)
0.239561 + 0.970881i \(0.422996\pi\)
\(20\) 0 0
\(21\) 1606.16 0.794769
\(22\) −1037.60 −0.457061
\(23\) −3231.01 −1.27356 −0.636780 0.771046i \(-0.719734\pi\)
−0.636780 + 0.771046i \(0.719734\pi\)
\(24\) 735.851 0.260773
\(25\) 0 0
\(26\) −3104.06 −0.900527
\(27\) −729.000 −0.192450
\(28\) −7412.36 −1.78674
\(29\) 2606.34 0.575488 0.287744 0.957707i \(-0.407095\pi\)
0.287744 + 0.957707i \(0.407095\pi\)
\(30\) 0 0
\(31\) 662.215 0.123764 0.0618821 0.998083i \(-0.480290\pi\)
0.0618821 + 0.998083i \(0.480290\pi\)
\(32\) 8001.48 1.38132
\(33\) −1089.00 −0.174078
\(34\) 8009.30 1.18822
\(35\) 0 0
\(36\) 3364.30 0.432652
\(37\) 12924.2 1.55203 0.776013 0.630717i \(-0.217239\pi\)
0.776013 + 0.630717i \(0.217239\pi\)
\(38\) −6465.11 −0.726302
\(39\) −3257.82 −0.342977
\(40\) 0 0
\(41\) −2538.56 −0.235846 −0.117923 0.993023i \(-0.537624\pi\)
−0.117923 + 0.993023i \(0.537624\pi\)
\(42\) −13773.2 −1.20479
\(43\) −22022.8 −1.81636 −0.908178 0.418583i \(-0.862527\pi\)
−0.908178 + 0.418583i \(0.862527\pi\)
\(44\) 5025.68 0.391348
\(45\) 0 0
\(46\) 27706.7 1.93059
\(47\) −20835.4 −1.37581 −0.687903 0.725802i \(-0.741469\pi\)
−0.687903 + 0.725802i \(0.741469\pi\)
\(48\) 5651.87 0.354070
\(49\) 15041.8 0.894974
\(50\) 0 0
\(51\) 8406.03 0.452549
\(52\) 15034.7 0.771056
\(53\) 27450.5 1.34233 0.671166 0.741307i \(-0.265794\pi\)
0.671166 + 0.741307i \(0.265794\pi\)
\(54\) 6251.34 0.291735
\(55\) 0 0
\(56\) 14591.3 0.621762
\(57\) −6785.36 −0.276621
\(58\) −22350.0 −0.872383
\(59\) 7759.71 0.290212 0.145106 0.989416i \(-0.453648\pi\)
0.145106 + 0.989416i \(0.453648\pi\)
\(60\) 0 0
\(61\) 38469.2 1.32370 0.661849 0.749638i \(-0.269772\pi\)
0.661849 + 0.749638i \(0.269772\pi\)
\(62\) −5678.65 −0.187614
\(63\) −14455.5 −0.458860
\(64\) −48519.0 −1.48068
\(65\) 0 0
\(66\) 9338.43 0.263884
\(67\) −35521.3 −0.966723 −0.483362 0.875421i \(-0.660584\pi\)
−0.483362 + 0.875421i \(0.660584\pi\)
\(68\) −38793.4 −1.01739
\(69\) 29079.1 0.735290
\(70\) 0 0
\(71\) −62677.1 −1.47558 −0.737790 0.675030i \(-0.764131\pi\)
−0.737790 + 0.675030i \(0.764131\pi\)
\(72\) −6622.66 −0.150557
\(73\) −68950.2 −1.51436 −0.757179 0.653208i \(-0.773423\pi\)
−0.757179 + 0.653208i \(0.773423\pi\)
\(74\) −110828. −2.35272
\(75\) 0 0
\(76\) 31314.1 0.621879
\(77\) −21594.0 −0.415055
\(78\) 27936.5 0.519919
\(79\) −17313.8 −0.312123 −0.156061 0.987747i \(-0.549880\pi\)
−0.156061 + 0.987747i \(0.549880\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 21768.7 0.357518
\(83\) 89017.9 1.41835 0.709173 0.705035i \(-0.249069\pi\)
0.709173 + 0.705035i \(0.249069\pi\)
\(84\) 66711.2 1.03158
\(85\) 0 0
\(86\) 188850. 2.75342
\(87\) −23457.1 −0.332258
\(88\) −9893.11 −0.136184
\(89\) 129627. 1.73468 0.867341 0.497714i \(-0.165827\pi\)
0.867341 + 0.497714i \(0.165827\pi\)
\(90\) 0 0
\(91\) −64599.8 −0.817763
\(92\) −134199. −1.65302
\(93\) −5959.94 −0.0714553
\(94\) 178668. 2.08559
\(95\) 0 0
\(96\) −72013.3 −0.797507
\(97\) 136732. 1.47551 0.737755 0.675069i \(-0.235886\pi\)
0.737755 + 0.675069i \(0.235886\pi\)
\(98\) −128987. −1.35669
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 51376.9 0.501146 0.250573 0.968098i \(-0.419381\pi\)
0.250573 + 0.968098i \(0.419381\pi\)
\(102\) −72083.7 −0.686019
\(103\) 94899.7 0.881398 0.440699 0.897655i \(-0.354731\pi\)
0.440699 + 0.897655i \(0.354731\pi\)
\(104\) −29595.9 −0.268317
\(105\) 0 0
\(106\) −235394. −2.03484
\(107\) 52232.6 0.441045 0.220522 0.975382i \(-0.429224\pi\)
0.220522 + 0.975382i \(0.429224\pi\)
\(108\) −30278.7 −0.249792
\(109\) 138553. 1.11699 0.558496 0.829507i \(-0.311378\pi\)
0.558496 + 0.829507i \(0.311378\pi\)
\(110\) 0 0
\(111\) −116318. −0.896063
\(112\) 112072. 0.844211
\(113\) 136642. 1.00667 0.503336 0.864091i \(-0.332106\pi\)
0.503336 + 0.864091i \(0.332106\pi\)
\(114\) 58186.0 0.419330
\(115\) 0 0
\(116\) 108253. 0.746958
\(117\) 29320.4 0.198018
\(118\) −66541.3 −0.439933
\(119\) 166685. 1.07902
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −329882. −2.00659
\(123\) 22847.0 0.136165
\(124\) 27504.8 0.160640
\(125\) 0 0
\(126\) 123959. 0.695587
\(127\) −23888.8 −0.131427 −0.0657136 0.997839i \(-0.520932\pi\)
−0.0657136 + 0.997839i \(0.520932\pi\)
\(128\) 160014. 0.863244
\(129\) 198205. 1.04867
\(130\) 0 0
\(131\) 67597.5 0.344154 0.172077 0.985084i \(-0.444952\pi\)
0.172077 + 0.985084i \(0.444952\pi\)
\(132\) −45231.2 −0.225945
\(133\) −134548. −0.659550
\(134\) 304604. 1.46546
\(135\) 0 0
\(136\) 76365.3 0.354037
\(137\) 205843. 0.936991 0.468496 0.883466i \(-0.344796\pi\)
0.468496 + 0.883466i \(0.344796\pi\)
\(138\) −249360. −1.11463
\(139\) 43495.6 0.190945 0.0954724 0.995432i \(-0.469564\pi\)
0.0954724 + 0.995432i \(0.469564\pi\)
\(140\) 0 0
\(141\) 187519. 0.794323
\(142\) 537470. 2.23683
\(143\) 43799.6 0.179114
\(144\) −50866.8 −0.204422
\(145\) 0 0
\(146\) 591264. 2.29562
\(147\) −135376. −0.516713
\(148\) 536801. 2.01446
\(149\) 219355. 0.809436 0.404718 0.914442i \(-0.367370\pi\)
0.404718 + 0.914442i \(0.367370\pi\)
\(150\) 0 0
\(151\) −32928.1 −0.117523 −0.0587616 0.998272i \(-0.518715\pi\)
−0.0587616 + 0.998272i \(0.518715\pi\)
\(152\) −61642.1 −0.216406
\(153\) −75654.3 −0.261279
\(154\) 185173. 0.629182
\(155\) 0 0
\(156\) −135312. −0.445169
\(157\) −457292. −1.48062 −0.740311 0.672265i \(-0.765322\pi\)
−0.740311 + 0.672265i \(0.765322\pi\)
\(158\) 148470. 0.473147
\(159\) −247054. −0.774996
\(160\) 0 0
\(161\) 576614. 1.75316
\(162\) −56262.1 −0.168433
\(163\) 52822.0 0.155721 0.0778603 0.996964i \(-0.475191\pi\)
0.0778603 + 0.996964i \(0.475191\pi\)
\(164\) −105438. −0.306117
\(165\) 0 0
\(166\) −763349. −2.15007
\(167\) 327606. 0.908993 0.454497 0.890748i \(-0.349819\pi\)
0.454497 + 0.890748i \(0.349819\pi\)
\(168\) −131322. −0.358974
\(169\) −240264. −0.647100
\(170\) 0 0
\(171\) 61068.2 0.159707
\(172\) −914707. −2.35755
\(173\) −246387. −0.625897 −0.312949 0.949770i \(-0.601317\pi\)
−0.312949 + 0.949770i \(0.601317\pi\)
\(174\) 201150. 0.503670
\(175\) 0 0
\(176\) −75986.2 −0.184907
\(177\) −69837.4 −0.167554
\(178\) −1.11158e6 −2.62961
\(179\) 302341. 0.705285 0.352643 0.935758i \(-0.385283\pi\)
0.352643 + 0.935758i \(0.385283\pi\)
\(180\) 0 0
\(181\) −465374. −1.05586 −0.527929 0.849288i \(-0.677031\pi\)
−0.527929 + 0.849288i \(0.677031\pi\)
\(182\) 553958. 1.23965
\(183\) −346223. −0.764237
\(184\) 264172. 0.575230
\(185\) 0 0
\(186\) 51107.8 0.108319
\(187\) −113014. −0.236336
\(188\) −865390. −1.78574
\(189\) 130099. 0.264923
\(190\) 0 0
\(191\) 294182. 0.583489 0.291745 0.956496i \(-0.405764\pi\)
0.291745 + 0.956496i \(0.405764\pi\)
\(192\) 436671. 0.854872
\(193\) 280529. 0.542106 0.271053 0.962564i \(-0.412628\pi\)
0.271053 + 0.962564i \(0.412628\pi\)
\(194\) −1.17251e6 −2.23673
\(195\) 0 0
\(196\) 624756. 1.16164
\(197\) 241625. 0.443585 0.221792 0.975094i \(-0.428809\pi\)
0.221792 + 0.975094i \(0.428809\pi\)
\(198\) −84045.8 −0.152354
\(199\) 621763. 1.11299 0.556496 0.830850i \(-0.312145\pi\)
0.556496 + 0.830850i \(0.312145\pi\)
\(200\) 0 0
\(201\) 319692. 0.558138
\(202\) −440569. −0.759688
\(203\) −465134. −0.792205
\(204\) 349141. 0.587389
\(205\) 0 0
\(206\) −813787. −1.33611
\(207\) −261712. −0.424520
\(208\) −227318. −0.364314
\(209\) 91225.4 0.144461
\(210\) 0 0
\(211\) −446582. −0.690550 −0.345275 0.938502i \(-0.612214\pi\)
−0.345275 + 0.938502i \(0.612214\pi\)
\(212\) 1.14014e6 1.74229
\(213\) 564094. 0.851927
\(214\) −447907. −0.668580
\(215\) 0 0
\(216\) 59603.9 0.0869242
\(217\) −118181. −0.170371
\(218\) −1.18813e6 −1.69325
\(219\) 620552. 0.874314
\(220\) 0 0
\(221\) −338090. −0.465642
\(222\) 997451. 1.35834
\(223\) −29867.7 −0.0402198 −0.0201099 0.999798i \(-0.506402\pi\)
−0.0201099 + 0.999798i \(0.506402\pi\)
\(224\) −1.42796e6 −1.90150
\(225\) 0 0
\(226\) −1.17174e6 −1.52602
\(227\) −425464. −0.548022 −0.274011 0.961726i \(-0.588351\pi\)
−0.274011 + 0.961726i \(0.588351\pi\)
\(228\) −281827. −0.359042
\(229\) −1.31835e6 −1.66127 −0.830637 0.556814i \(-0.812023\pi\)
−0.830637 + 0.556814i \(0.812023\pi\)
\(230\) 0 0
\(231\) 194346. 0.239632
\(232\) −213098. −0.259931
\(233\) 320062. 0.386228 0.193114 0.981176i \(-0.438141\pi\)
0.193114 + 0.981176i \(0.438141\pi\)
\(234\) −251429. −0.300176
\(235\) 0 0
\(236\) 322296. 0.376683
\(237\) 155825. 0.180204
\(238\) −1.42936e6 −1.63568
\(239\) 1.25534e6 1.42156 0.710781 0.703414i \(-0.248342\pi\)
0.710781 + 0.703414i \(0.248342\pi\)
\(240\) 0 0
\(241\) −1.07813e6 −1.19572 −0.597859 0.801601i \(-0.703982\pi\)
−0.597859 + 0.801601i \(0.703982\pi\)
\(242\) −125550. −0.137809
\(243\) −59049.0 −0.0641500
\(244\) 1.59780e6 1.71810
\(245\) 0 0
\(246\) −195919. −0.206413
\(247\) 272907. 0.284624
\(248\) −54143.5 −0.0559007
\(249\) −801161. −0.818882
\(250\) 0 0
\(251\) −918775. −0.920502 −0.460251 0.887789i \(-0.652241\pi\)
−0.460251 + 0.887789i \(0.652241\pi\)
\(252\) −600401. −0.595580
\(253\) −390953. −0.383993
\(254\) 204852. 0.199231
\(255\) 0 0
\(256\) 180449. 0.172089
\(257\) −797355. −0.753041 −0.376521 0.926408i \(-0.622880\pi\)
−0.376521 + 0.926408i \(0.622880\pi\)
\(258\) −1.69965e6 −1.58969
\(259\) −2.30648e6 −2.13649
\(260\) 0 0
\(261\) 211114. 0.191829
\(262\) −579664. −0.521703
\(263\) 980563. 0.874150 0.437075 0.899425i \(-0.356014\pi\)
0.437075 + 0.899425i \(0.356014\pi\)
\(264\) 89038.0 0.0786259
\(265\) 0 0
\(266\) 1.15378e6 0.999813
\(267\) −1.16664e6 −1.00152
\(268\) −1.47536e6 −1.25476
\(269\) −1.32137e6 −1.11338 −0.556691 0.830719i \(-0.687929\pi\)
−0.556691 + 0.830719i \(0.687929\pi\)
\(270\) 0 0
\(271\) −62864.5 −0.0519975 −0.0259987 0.999662i \(-0.508277\pi\)
−0.0259987 + 0.999662i \(0.508277\pi\)
\(272\) 586540. 0.480702
\(273\) 581398. 0.472136
\(274\) −1.76515e6 −1.42039
\(275\) 0 0
\(276\) 1.20779e6 0.954374
\(277\) −1.44313e6 −1.13007 −0.565037 0.825066i \(-0.691138\pi\)
−0.565037 + 0.825066i \(0.691138\pi\)
\(278\) −372984. −0.289453
\(279\) 53639.4 0.0412547
\(280\) 0 0
\(281\) −143939. −0.108746 −0.0543731 0.998521i \(-0.517316\pi\)
−0.0543731 + 0.998521i \(0.517316\pi\)
\(282\) −1.60802e6 −1.20411
\(283\) 9060.12 0.00672462 0.00336231 0.999994i \(-0.498930\pi\)
0.00336231 + 0.999994i \(0.498930\pi\)
\(284\) −2.60327e6 −1.91524
\(285\) 0 0
\(286\) −375591. −0.271519
\(287\) 453037. 0.324660
\(288\) 648120. 0.460441
\(289\) −547494. −0.385598
\(290\) 0 0
\(291\) −1.23059e6 −0.851886
\(292\) −2.86382e6 −1.96557
\(293\) 1.85674e6 1.26352 0.631759 0.775165i \(-0.282333\pi\)
0.631759 + 0.775165i \(0.282333\pi\)
\(294\) 1.16088e6 0.783286
\(295\) 0 0
\(296\) −1.05670e6 −0.701006
\(297\) −88209.0 −0.0580259
\(298\) −1.88102e6 −1.22703
\(299\) −1.16956e6 −0.756563
\(300\) 0 0
\(301\) 3.93024e6 2.50036
\(302\) 282366. 0.178154
\(303\) −462392. −0.289337
\(304\) −473456. −0.293830
\(305\) 0 0
\(306\) 648753. 0.396074
\(307\) 504688. 0.305617 0.152808 0.988256i \(-0.451168\pi\)
0.152808 + 0.988256i \(0.451168\pi\)
\(308\) −896896. −0.538723
\(309\) −854097. −0.508875
\(310\) 0 0
\(311\) −448511. −0.262950 −0.131475 0.991320i \(-0.541971\pi\)
−0.131475 + 0.991320i \(0.541971\pi\)
\(312\) 266363. 0.154913
\(313\) −825146. −0.476069 −0.238035 0.971257i \(-0.576503\pi\)
−0.238035 + 0.971257i \(0.576503\pi\)
\(314\) 3.92138e6 2.24448
\(315\) 0 0
\(316\) −719123. −0.405122
\(317\) −3.26266e6 −1.82358 −0.911789 0.410659i \(-0.865299\pi\)
−0.911789 + 0.410659i \(0.865299\pi\)
\(318\) 2.11855e6 1.17482
\(319\) 315367. 0.173516
\(320\) 0 0
\(321\) −470094. −0.254637
\(322\) −4.94460e6 −2.65761
\(323\) −704172. −0.375554
\(324\) 272508. 0.144217
\(325\) 0 0
\(326\) −452961. −0.236057
\(327\) −1.24698e6 −0.644896
\(328\) 207556. 0.106525
\(329\) 3.71834e6 1.89391
\(330\) 0 0
\(331\) −3.48175e6 −1.74674 −0.873369 0.487059i \(-0.838070\pi\)
−0.873369 + 0.487059i \(0.838070\pi\)
\(332\) 3.69732e6 1.84095
\(333\) 1.04686e6 0.517342
\(334\) −2.80930e6 −1.37794
\(335\) 0 0
\(336\) −1.00865e6 −0.487406
\(337\) 1.52759e6 0.732710 0.366355 0.930475i \(-0.380606\pi\)
0.366355 + 0.930475i \(0.380606\pi\)
\(338\) 2.06032e6 0.980939
\(339\) −1.22978e6 −0.581203
\(340\) 0 0
\(341\) 80128.0 0.0373163
\(342\) −523674. −0.242101
\(343\) 315017. 0.144577
\(344\) 1.80061e6 0.820396
\(345\) 0 0
\(346\) 2.11283e6 0.948798
\(347\) 4.01479e6 1.78994 0.894972 0.446122i \(-0.147195\pi\)
0.894972 + 0.446122i \(0.147195\pi\)
\(348\) −974279. −0.431256
\(349\) −511478. −0.224783 −0.112391 0.993664i \(-0.535851\pi\)
−0.112391 + 0.993664i \(0.535851\pi\)
\(350\) 0 0
\(351\) −263883. −0.114326
\(352\) 968179. 0.416485
\(353\) 722636. 0.308662 0.154331 0.988019i \(-0.450678\pi\)
0.154331 + 0.988019i \(0.450678\pi\)
\(354\) 598872. 0.253995
\(355\) 0 0
\(356\) 5.38400e6 2.25154
\(357\) −1.50016e6 −0.622970
\(358\) −2.59264e6 −1.06914
\(359\) 393345. 0.161078 0.0805392 0.996751i \(-0.474336\pi\)
0.0805392 + 0.996751i \(0.474336\pi\)
\(360\) 0 0
\(361\) −1.90769e6 −0.770442
\(362\) 3.99069e6 1.60058
\(363\) −131769. −0.0524864
\(364\) −2.68312e6 −1.06142
\(365\) 0 0
\(366\) 2.96894e6 1.15851
\(367\) 2.87578e6 1.11453 0.557264 0.830336i \(-0.311851\pi\)
0.557264 + 0.830336i \(0.311851\pi\)
\(368\) 2.02903e6 0.781032
\(369\) −205623. −0.0786152
\(370\) 0 0
\(371\) −4.89888e6 −1.84783
\(372\) −247543. −0.0927458
\(373\) −3.86030e6 −1.43665 −0.718323 0.695710i \(-0.755090\pi\)
−0.718323 + 0.695710i \(0.755090\pi\)
\(374\) 969125. 0.358262
\(375\) 0 0
\(376\) 1.70353e6 0.621412
\(377\) 943443. 0.341871
\(378\) −1.11563e6 −0.401597
\(379\) −1.63355e6 −0.584162 −0.292081 0.956394i \(-0.594348\pi\)
−0.292081 + 0.956394i \(0.594348\pi\)
\(380\) 0 0
\(381\) 214999. 0.0758795
\(382\) −2.52268e6 −0.884512
\(383\) −3.86858e6 −1.34758 −0.673790 0.738923i \(-0.735335\pi\)
−0.673790 + 0.738923i \(0.735335\pi\)
\(384\) −1.44013e6 −0.498394
\(385\) 0 0
\(386\) −2.40560e6 −0.821779
\(387\) −1.78385e6 −0.605452
\(388\) 5.67912e6 1.91515
\(389\) −10040.7 −0.00336428 −0.00168214 0.999999i \(-0.500535\pi\)
−0.00168214 + 0.999999i \(0.500535\pi\)
\(390\) 0 0
\(391\) 3.01778e6 0.998264
\(392\) −1.22984e6 −0.404234
\(393\) −608378. −0.198697
\(394\) −2.07199e6 −0.672431
\(395\) 0 0
\(396\) 407080. 0.130449
\(397\) 3.77944e6 1.20352 0.601758 0.798679i \(-0.294467\pi\)
0.601758 + 0.798679i \(0.294467\pi\)
\(398\) −5.33176e6 −1.68719
\(399\) 1.21093e6 0.380791
\(400\) 0 0
\(401\) −5.77840e6 −1.79451 −0.897256 0.441510i \(-0.854443\pi\)
−0.897256 + 0.441510i \(0.854443\pi\)
\(402\) −2.74143e6 −0.846082
\(403\) 239708. 0.0735226
\(404\) 2.13392e6 0.650465
\(405\) 0 0
\(406\) 3.98863e6 1.20090
\(407\) 1.56383e6 0.467954
\(408\) −687287. −0.204403
\(409\) 4.93158e6 1.45773 0.728866 0.684656i \(-0.240048\pi\)
0.728866 + 0.684656i \(0.240048\pi\)
\(410\) 0 0
\(411\) −1.85259e6 −0.540972
\(412\) 3.94162e6 1.14401
\(413\) −1.38482e6 −0.399501
\(414\) 2.24424e6 0.643530
\(415\) 0 0
\(416\) 2.89637e6 0.820580
\(417\) −391460. −0.110242
\(418\) −782278. −0.218988
\(419\) −4.38879e6 −1.22126 −0.610632 0.791914i \(-0.709085\pi\)
−0.610632 + 0.791914i \(0.709085\pi\)
\(420\) 0 0
\(421\) −6.32905e6 −1.74034 −0.870169 0.492754i \(-0.835990\pi\)
−0.870169 + 0.492754i \(0.835990\pi\)
\(422\) 3.82954e6 1.04681
\(423\) −1.68767e6 −0.458602
\(424\) −2.24438e6 −0.606293
\(425\) 0 0
\(426\) −4.83723e6 −1.29144
\(427\) −6.86531e6 −1.82218
\(428\) 2.16946e6 0.572456
\(429\) −394196. −0.103412
\(430\) 0 0
\(431\) −4.39779e6 −1.14036 −0.570179 0.821520i \(-0.693126\pi\)
−0.570179 + 0.821520i \(0.693126\pi\)
\(432\) 457801. 0.118023
\(433\) 1.57360e6 0.403344 0.201672 0.979453i \(-0.435362\pi\)
0.201672 + 0.979453i \(0.435362\pi\)
\(434\) 1.01343e6 0.258266
\(435\) 0 0
\(436\) 5.75475e6 1.44981
\(437\) −2.43595e6 −0.610190
\(438\) −5.32137e6 −1.32537
\(439\) −7.89299e6 −1.95470 −0.977351 0.211626i \(-0.932124\pi\)
−0.977351 + 0.211626i \(0.932124\pi\)
\(440\) 0 0
\(441\) 1.21839e6 0.298325
\(442\) 2.89920e6 0.705867
\(443\) 557226. 0.134903 0.0674516 0.997723i \(-0.478513\pi\)
0.0674516 + 0.997723i \(0.478513\pi\)
\(444\) −4.83121e6 −1.16305
\(445\) 0 0
\(446\) 256123. 0.0609693
\(447\) −1.97420e6 −0.467328
\(448\) 8.65881e6 2.03828
\(449\) 2.27558e6 0.532691 0.266346 0.963878i \(-0.414184\pi\)
0.266346 + 0.963878i \(0.414184\pi\)
\(450\) 0 0
\(451\) −307166. −0.0711101
\(452\) 5.67537e6 1.30662
\(453\) 296352. 0.0678521
\(454\) 3.64845e6 0.830748
\(455\) 0 0
\(456\) 554779. 0.124942
\(457\) −7.50427e6 −1.68081 −0.840404 0.541961i \(-0.817682\pi\)
−0.840404 + 0.541961i \(0.817682\pi\)
\(458\) 1.13051e7 2.51833
\(459\) 680889. 0.150850
\(460\) 0 0
\(461\) 436978. 0.0957652 0.0478826 0.998853i \(-0.484753\pi\)
0.0478826 + 0.998853i \(0.484753\pi\)
\(462\) −1.66656e6 −0.363258
\(463\) 3.11651e6 0.675642 0.337821 0.941210i \(-0.390310\pi\)
0.337821 + 0.941210i \(0.390310\pi\)
\(464\) −1.63674e6 −0.352928
\(465\) 0 0
\(466\) −2.74460e6 −0.585484
\(467\) 1.89749e6 0.402613 0.201307 0.979528i \(-0.435481\pi\)
0.201307 + 0.979528i \(0.435481\pi\)
\(468\) 1.21781e6 0.257019
\(469\) 6.33922e6 1.33077
\(470\) 0 0
\(471\) 4.11562e6 0.854837
\(472\) −634444. −0.131081
\(473\) −2.66476e6 −0.547652
\(474\) −1.33623e6 −0.273172
\(475\) 0 0
\(476\) 6.92317e6 1.40051
\(477\) 2.22349e6 0.447444
\(478\) −1.07648e7 −2.15495
\(479\) −6.10287e6 −1.21533 −0.607666 0.794193i \(-0.707894\pi\)
−0.607666 + 0.794193i \(0.707894\pi\)
\(480\) 0 0
\(481\) 4.67830e6 0.921987
\(482\) 9.24522e6 1.81259
\(483\) −5.18953e6 −1.01219
\(484\) 608108. 0.117996
\(485\) 0 0
\(486\) 506359. 0.0972451
\(487\) 7.30170e6 1.39509 0.697544 0.716542i \(-0.254276\pi\)
0.697544 + 0.716542i \(0.254276\pi\)
\(488\) −3.14529e6 −0.597876
\(489\) −475398. −0.0899053
\(490\) 0 0
\(491\) 4.40884e6 0.825317 0.412658 0.910886i \(-0.364600\pi\)
0.412658 + 0.910886i \(0.364600\pi\)
\(492\) 948942. 0.176737
\(493\) −2.43433e6 −0.451089
\(494\) −2.34024e6 −0.431462
\(495\) 0 0
\(496\) −415861. −0.0759004
\(497\) 1.11855e7 2.03125
\(498\) 6.87014e6 1.24134
\(499\) 3.45620e6 0.621367 0.310683 0.950513i \(-0.399442\pi\)
0.310683 + 0.950513i \(0.399442\pi\)
\(500\) 0 0
\(501\) −2.94845e6 −0.524808
\(502\) 7.87871e6 1.39539
\(503\) −5.07651e6 −0.894634 −0.447317 0.894375i \(-0.647620\pi\)
−0.447317 + 0.894375i \(0.647620\pi\)
\(504\) 1.18190e6 0.207254
\(505\) 0 0
\(506\) 3.35251e6 0.582095
\(507\) 2.16237e6 0.373603
\(508\) −992212. −0.170587
\(509\) 7.63894e6 1.30689 0.653445 0.756974i \(-0.273323\pi\)
0.653445 + 0.756974i \(0.273323\pi\)
\(510\) 0 0
\(511\) 1.23050e7 2.08463
\(512\) −6.66784e6 −1.12411
\(513\) −549614. −0.0922071
\(514\) 6.83750e6 1.14154
\(515\) 0 0
\(516\) 8.23237e6 1.36113
\(517\) −2.52109e6 −0.414821
\(518\) 1.97786e7 3.23871
\(519\) 2.21748e6 0.361362
\(520\) 0 0
\(521\) −9.46786e6 −1.52812 −0.764060 0.645145i \(-0.776797\pi\)
−0.764060 + 0.645145i \(0.776797\pi\)
\(522\) −1.81035e6 −0.290794
\(523\) 1.12533e6 0.179898 0.0899490 0.995946i \(-0.471330\pi\)
0.0899490 + 0.995946i \(0.471330\pi\)
\(524\) 2.80764e6 0.446696
\(525\) 0 0
\(526\) −8.40855e6 −1.32512
\(527\) −618511. −0.0970110
\(528\) 683876. 0.106756
\(529\) 4.00311e6 0.621953
\(530\) 0 0
\(531\) 628537. 0.0967374
\(532\) −5.58839e6 −0.856067
\(533\) −918907. −0.140105
\(534\) 1.00042e7 1.51820
\(535\) 0 0
\(536\) 2.90427e6 0.436641
\(537\) −2.72107e6 −0.407196
\(538\) 1.13311e7 1.68778
\(539\) 1.82006e6 0.269845
\(540\) 0 0
\(541\) 1.05433e7 1.54876 0.774379 0.632722i \(-0.218062\pi\)
0.774379 + 0.632722i \(0.218062\pi\)
\(542\) 539078. 0.0788230
\(543\) 4.18837e6 0.609600
\(544\) −7.47341e6 −1.08273
\(545\) 0 0
\(546\) −4.98562e6 −0.715711
\(547\) −9.65329e6 −1.37945 −0.689727 0.724070i \(-0.742269\pi\)
−0.689727 + 0.724070i \(0.742269\pi\)
\(548\) 8.54962e6 1.21617
\(549\) 3.11601e6 0.441232
\(550\) 0 0
\(551\) 1.96499e6 0.275729
\(552\) −2.37754e6 −0.332109
\(553\) 3.08987e6 0.429662
\(554\) 1.23752e7 1.71308
\(555\) 0 0
\(556\) 1.80657e6 0.247838
\(557\) −4.21289e6 −0.575363 −0.287682 0.957726i \(-0.592885\pi\)
−0.287682 + 0.957726i \(0.592885\pi\)
\(558\) −459970. −0.0625381
\(559\) −7.97180e6 −1.07901
\(560\) 0 0
\(561\) 1.01713e6 0.136449
\(562\) 1.23431e6 0.164848
\(563\) −9.59902e6 −1.27631 −0.638155 0.769908i \(-0.720302\pi\)
−0.638155 + 0.769908i \(0.720302\pi\)
\(564\) 7.78851e6 1.03100
\(565\) 0 0
\(566\) −77692.6 −0.0101939
\(567\) −1.17089e6 −0.152953
\(568\) 5.12455e6 0.666477
\(569\) 4.20682e6 0.544720 0.272360 0.962195i \(-0.412196\pi\)
0.272360 + 0.962195i \(0.412196\pi\)
\(570\) 0 0
\(571\) −1.22447e7 −1.57165 −0.785826 0.618447i \(-0.787762\pi\)
−0.785826 + 0.618447i \(0.787762\pi\)
\(572\) 1.81920e6 0.232482
\(573\) −2.64764e6 −0.336878
\(574\) −3.88490e6 −0.492153
\(575\) 0 0
\(576\) −3.93004e6 −0.493561
\(577\) −5.32012e6 −0.665245 −0.332623 0.943060i \(-0.607934\pi\)
−0.332623 + 0.943060i \(0.607934\pi\)
\(578\) 4.69489e6 0.584529
\(579\) −2.52476e6 −0.312985
\(580\) 0 0
\(581\) −1.58863e7 −1.95247
\(582\) 1.05526e7 1.29137
\(583\) 3.32151e6 0.404728
\(584\) 5.63745e6 0.683992
\(585\) 0 0
\(586\) −1.59220e7 −1.91537
\(587\) 5.70534e6 0.683418 0.341709 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(588\) −5.62280e6 −0.670671
\(589\) 499263. 0.0592981
\(590\) 0 0
\(591\) −2.17463e6 −0.256104
\(592\) −8.11620e6 −0.951806
\(593\) −2.15575e6 −0.251746 −0.125873 0.992046i \(-0.540173\pi\)
−0.125873 + 0.992046i \(0.540173\pi\)
\(594\) 756413. 0.0879615
\(595\) 0 0
\(596\) 9.11083e6 1.05061
\(597\) −5.59587e6 −0.642587
\(598\) 1.00293e7 1.14687
\(599\) 5.13968e6 0.585287 0.292643 0.956222i \(-0.405465\pi\)
0.292643 + 0.956222i \(0.405465\pi\)
\(600\) 0 0
\(601\) −7.48666e6 −0.845478 −0.422739 0.906252i \(-0.638931\pi\)
−0.422739 + 0.906252i \(0.638931\pi\)
\(602\) −3.37027e7 −3.79030
\(603\) −2.87723e6 −0.322241
\(604\) −1.36765e6 −0.152540
\(605\) 0 0
\(606\) 3.96512e6 0.438606
\(607\) 8.62794e6 0.950464 0.475232 0.879861i \(-0.342364\pi\)
0.475232 + 0.879861i \(0.342364\pi\)
\(608\) 6.03254e6 0.661822
\(609\) 4.18620e6 0.457380
\(610\) 0 0
\(611\) −7.54200e6 −0.817304
\(612\) −3.14227e6 −0.339129
\(613\) 1.44646e7 1.55473 0.777367 0.629047i \(-0.216555\pi\)
0.777367 + 0.629047i \(0.216555\pi\)
\(614\) −4.32782e6 −0.463285
\(615\) 0 0
\(616\) 1.76555e6 0.187468
\(617\) −5.82746e6 −0.616264 −0.308132 0.951344i \(-0.599704\pi\)
−0.308132 + 0.951344i \(0.599704\pi\)
\(618\) 7.32408e6 0.771404
\(619\) −2.32647e6 −0.244045 −0.122023 0.992527i \(-0.538938\pi\)
−0.122023 + 0.992527i \(0.538938\pi\)
\(620\) 0 0
\(621\) 2.35541e6 0.245097
\(622\) 3.84609e6 0.398605
\(623\) −2.31335e7 −2.38793
\(624\) 2.04586e6 0.210337
\(625\) 0 0
\(626\) 7.07582e6 0.721674
\(627\) −821028. −0.0834044
\(628\) −1.89934e7 −1.92178
\(629\) −1.20712e7 −1.21654
\(630\) 0 0
\(631\) −398154. −0.0398087 −0.0199043 0.999802i \(-0.506336\pi\)
−0.0199043 + 0.999802i \(0.506336\pi\)
\(632\) 1.41560e6 0.140977
\(633\) 4.01924e6 0.398689
\(634\) 2.79781e7 2.76436
\(635\) 0 0
\(636\) −1.02613e7 −1.00591
\(637\) 5.44484e6 0.531663
\(638\) −2.70435e6 −0.263033
\(639\) −5.07684e6 −0.491860
\(640\) 0 0
\(641\) 1.81586e7 1.74557 0.872784 0.488107i \(-0.162312\pi\)
0.872784 + 0.488107i \(0.162312\pi\)
\(642\) 4.03116e6 0.386005
\(643\) 4.48264e6 0.427569 0.213784 0.976881i \(-0.431421\pi\)
0.213784 + 0.976881i \(0.431421\pi\)
\(644\) 2.39494e7 2.27552
\(645\) 0 0
\(646\) 6.03844e6 0.569303
\(647\) −1.75122e7 −1.64467 −0.822336 0.569003i \(-0.807329\pi\)
−0.822336 + 0.569003i \(0.807329\pi\)
\(648\) −536435. −0.0501857
\(649\) 938925. 0.0875023
\(650\) 0 0
\(651\) 1.06362e6 0.0983639
\(652\) 2.19394e6 0.202118
\(653\) 1.89639e7 1.74039 0.870193 0.492711i \(-0.163994\pi\)
0.870193 + 0.492711i \(0.163994\pi\)
\(654\) 1.06931e7 0.977599
\(655\) 0 0
\(656\) 1.59418e6 0.144636
\(657\) −5.58496e6 −0.504786
\(658\) −3.18856e7 −2.87098
\(659\) 23732.7 0.00212879 0.00106440 0.999999i \(-0.499661\pi\)
0.00106440 + 0.999999i \(0.499661\pi\)
\(660\) 0 0
\(661\) 1.09938e7 0.978686 0.489343 0.872091i \(-0.337237\pi\)
0.489343 + 0.872091i \(0.337237\pi\)
\(662\) 2.98568e7 2.64788
\(663\) 3.04281e6 0.268839
\(664\) −7.27821e6 −0.640626
\(665\) 0 0
\(666\) −8.97706e6 −0.784239
\(667\) −8.42112e6 −0.732918
\(668\) 1.36070e7 1.17983
\(669\) 268810. 0.0232209
\(670\) 0 0
\(671\) 4.65477e6 0.399110
\(672\) 1.28517e7 1.09783
\(673\) −1.19040e6 −0.101310 −0.0506552 0.998716i \(-0.516131\pi\)
−0.0506552 + 0.998716i \(0.516131\pi\)
\(674\) −1.30994e7 −1.11072
\(675\) 0 0
\(676\) −9.97925e6 −0.839907
\(677\) 2.03890e7 1.70972 0.854860 0.518859i \(-0.173643\pi\)
0.854860 + 0.518859i \(0.173643\pi\)
\(678\) 1.05456e7 0.881046
\(679\) −2.44016e7 −2.03116
\(680\) 0 0
\(681\) 3.82918e6 0.316401
\(682\) −687116. −0.0565678
\(683\) 2.37020e7 1.94416 0.972082 0.234640i \(-0.0753911\pi\)
0.972082 + 0.234640i \(0.0753911\pi\)
\(684\) 2.53644e6 0.207293
\(685\) 0 0
\(686\) −2.70134e6 −0.219164
\(687\) 1.18651e7 0.959137
\(688\) 1.38300e7 1.11391
\(689\) 9.93651e6 0.797418
\(690\) 0 0
\(691\) −1.32012e7 −1.05176 −0.525881 0.850558i \(-0.676264\pi\)
−0.525881 + 0.850558i \(0.676264\pi\)
\(692\) −1.02336e7 −0.812387
\(693\) −1.74911e6 −0.138352
\(694\) −3.44278e7 −2.71338
\(695\) 0 0
\(696\) 1.91788e6 0.150071
\(697\) 2.37102e6 0.184865
\(698\) 4.38604e6 0.340749
\(699\) −2.88056e6 −0.222989
\(700\) 0 0
\(701\) 9.76550e6 0.750584 0.375292 0.926907i \(-0.377542\pi\)
0.375292 + 0.926907i \(0.377542\pi\)
\(702\) 2.26286e6 0.173306
\(703\) 9.74392e6 0.743610
\(704\) −5.87080e6 −0.446442
\(705\) 0 0
\(706\) −6.19677e6 −0.467900
\(707\) −9.16884e6 −0.689868
\(708\) −2.90067e6 −0.217478
\(709\) −6.77780e6 −0.506376 −0.253188 0.967417i \(-0.581479\pi\)
−0.253188 + 0.967417i \(0.581479\pi\)
\(710\) 0 0
\(711\) −1.40242e6 −0.104041
\(712\) −1.05985e7 −0.783506
\(713\) −2.13963e6 −0.157621
\(714\) 1.28642e7 0.944361
\(715\) 0 0
\(716\) 1.25576e7 0.915429
\(717\) −1.12980e7 −0.820739
\(718\) −3.37302e6 −0.244179
\(719\) −9.77686e6 −0.705305 −0.352653 0.935754i \(-0.614720\pi\)
−0.352653 + 0.935754i \(0.614720\pi\)
\(720\) 0 0
\(721\) −1.69360e7 −1.21331
\(722\) 1.63589e7 1.16791
\(723\) 9.70317e6 0.690348
\(724\) −1.93291e7 −1.37046
\(725\) 0 0
\(726\) 1.12995e6 0.0795642
\(727\) 1.13969e7 0.799745 0.399873 0.916571i \(-0.369054\pi\)
0.399873 + 0.916571i \(0.369054\pi\)
\(728\) 5.28176e6 0.369360
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.05694e7 1.42373
\(732\) −1.43802e7 −0.991946
\(733\) 6.58492e6 0.452679 0.226340 0.974048i \(-0.427324\pi\)
0.226340 + 0.974048i \(0.427324\pi\)
\(734\) −2.46605e7 −1.68951
\(735\) 0 0
\(736\) −2.58529e7 −1.75920
\(737\) −4.29808e6 −0.291478
\(738\) 1.76327e6 0.119173
\(739\) −1.51924e7 −1.02333 −0.511664 0.859186i \(-0.670971\pi\)
−0.511664 + 0.859186i \(0.670971\pi\)
\(740\) 0 0
\(741\) −2.45616e6 −0.164328
\(742\) 4.20090e7 2.80112
\(743\) 6.92543e6 0.460230 0.230115 0.973163i \(-0.426090\pi\)
0.230115 + 0.973163i \(0.426090\pi\)
\(744\) 487292. 0.0322743
\(745\) 0 0
\(746\) 3.31030e7 2.17781
\(747\) 7.21045e6 0.472782
\(748\) −4.69401e6 −0.306754
\(749\) −9.32156e6 −0.607133
\(750\) 0 0
\(751\) −1.53073e7 −0.990370 −0.495185 0.868788i \(-0.664900\pi\)
−0.495185 + 0.868788i \(0.664900\pi\)
\(752\) 1.30843e7 0.843737
\(753\) 8.26898e6 0.531452
\(754\) −8.09024e6 −0.518242
\(755\) 0 0
\(756\) 5.40361e6 0.343858
\(757\) −9.00842e6 −0.571359 −0.285679 0.958325i \(-0.592219\pi\)
−0.285679 + 0.958325i \(0.592219\pi\)
\(758\) 1.40080e7 0.885531
\(759\) 3.51857e6 0.221698
\(760\) 0 0
\(761\) 1.62103e7 1.01468 0.507339 0.861746i \(-0.330629\pi\)
0.507339 + 0.861746i \(0.330629\pi\)
\(762\) −1.84367e6 −0.115026
\(763\) −2.47265e7 −1.53763
\(764\) 1.22187e7 0.757343
\(765\) 0 0
\(766\) 3.31740e7 2.04280
\(767\) 2.80886e6 0.172402
\(768\) −1.62404e6 −0.0993558
\(769\) 1.67909e7 1.02390 0.511950 0.859015i \(-0.328923\pi\)
0.511950 + 0.859015i \(0.328923\pi\)
\(770\) 0 0
\(771\) 7.17619e6 0.434768
\(772\) 1.16516e7 0.703629
\(773\) −1.62605e7 −0.978780 −0.489390 0.872065i \(-0.662781\pi\)
−0.489390 + 0.872065i \(0.662781\pi\)
\(774\) 1.52969e7 0.917806
\(775\) 0 0
\(776\) −1.11794e7 −0.666445
\(777\) 2.07583e7 1.23350
\(778\) 86101.7 0.00509991
\(779\) −1.91389e6 −0.112999
\(780\) 0 0
\(781\) −7.58393e6 −0.444904
\(782\) −2.58781e7 −1.51327
\(783\) −1.90002e6 −0.110753
\(784\) −9.44604e6 −0.548858
\(785\) 0 0
\(786\) 5.21698e6 0.301205
\(787\) 5.01128e6 0.288411 0.144206 0.989548i \(-0.453937\pi\)
0.144206 + 0.989548i \(0.453937\pi\)
\(788\) 1.00358e7 0.575753
\(789\) −8.82506e6 −0.504691
\(790\) 0 0
\(791\) −2.43855e7 −1.38577
\(792\) −801342. −0.0453947
\(793\) 1.39251e7 0.786348
\(794\) −3.24096e7 −1.82441
\(795\) 0 0
\(796\) 2.58247e7 1.44461
\(797\) −3.07160e7 −1.71285 −0.856424 0.516273i \(-0.827319\pi\)
−0.856424 + 0.516273i \(0.827319\pi\)
\(798\) −1.03840e7 −0.577242
\(799\) 1.94604e7 1.07841
\(800\) 0 0
\(801\) 1.04998e7 0.578227
\(802\) 4.95511e7 2.72030
\(803\) −8.34297e6 −0.456596
\(804\) 1.32783e7 0.724438
\(805\) 0 0
\(806\) −2.05556e6 −0.111453
\(807\) 1.18924e7 0.642812
\(808\) −4.20064e6 −0.226353
\(809\) 6.14509e6 0.330108 0.165054 0.986284i \(-0.447220\pi\)
0.165054 + 0.986284i \(0.447220\pi\)
\(810\) 0 0
\(811\) −1.46687e7 −0.783138 −0.391569 0.920149i \(-0.628068\pi\)
−0.391569 + 0.920149i \(0.628068\pi\)
\(812\) −1.93191e7 −1.02825
\(813\) 565780. 0.0300208
\(814\) −1.34102e7 −0.709371
\(815\) 0 0
\(816\) −5.27886e6 −0.277533
\(817\) −1.66036e7 −0.870257
\(818\) −4.22894e7 −2.20978
\(819\) −5.23258e6 −0.272588
\(820\) 0 0
\(821\) −1.54951e7 −0.802300 −0.401150 0.916012i \(-0.631389\pi\)
−0.401150 + 0.916012i \(0.631389\pi\)
\(822\) 1.58864e7 0.820060
\(823\) −2.53285e7 −1.30349 −0.651747 0.758436i \(-0.725964\pi\)
−0.651747 + 0.758436i \(0.725964\pi\)
\(824\) −7.75912e6 −0.398102
\(825\) 0 0
\(826\) 1.18751e7 0.605603
\(827\) 6.01962e6 0.306059 0.153030 0.988222i \(-0.451097\pi\)
0.153030 + 0.988222i \(0.451097\pi\)
\(828\) −1.08701e7 −0.551008
\(829\) −2.70703e7 −1.36807 −0.684033 0.729451i \(-0.739775\pi\)
−0.684033 + 0.729451i \(0.739775\pi\)
\(830\) 0 0
\(831\) 1.29882e7 0.652449
\(832\) −1.75629e7 −0.879605
\(833\) −1.40491e7 −0.701514
\(834\) 3.35686e6 0.167116
\(835\) 0 0
\(836\) 3.78901e6 0.187504
\(837\) −482755. −0.0238184
\(838\) 3.76349e7 1.85132
\(839\) 2.29868e7 1.12739 0.563694 0.825984i \(-0.309380\pi\)
0.563694 + 0.825984i \(0.309380\pi\)
\(840\) 0 0
\(841\) −1.37181e7 −0.668814
\(842\) 5.42731e7 2.63818
\(843\) 1.29546e6 0.0627847
\(844\) −1.85486e7 −0.896303
\(845\) 0 0
\(846\) 1.44721e7 0.695196
\(847\) −2.61287e6 −0.125144
\(848\) −1.72385e7 −0.823208
\(849\) −81541.1 −0.00388246
\(850\) 0 0
\(851\) −4.17582e7 −1.97660
\(852\) 2.34294e7 1.10576
\(853\) 3.27587e7 1.54154 0.770768 0.637116i \(-0.219873\pi\)
0.770768 + 0.637116i \(0.219873\pi\)
\(854\) 5.88716e7 2.76224
\(855\) 0 0
\(856\) −4.27060e6 −0.199207
\(857\) 5.91710e6 0.275205 0.137603 0.990488i \(-0.456060\pi\)
0.137603 + 0.990488i \(0.456060\pi\)
\(858\) 3.38032e6 0.156762
\(859\) −3.88859e7 −1.79808 −0.899041 0.437865i \(-0.855735\pi\)
−0.899041 + 0.437865i \(0.855735\pi\)
\(860\) 0 0
\(861\) −4.07734e6 −0.187443
\(862\) 3.77121e7 1.72867
\(863\) −7.21944e6 −0.329972 −0.164986 0.986296i \(-0.552758\pi\)
−0.164986 + 0.986296i \(0.552758\pi\)
\(864\) −5.83308e6 −0.265836
\(865\) 0 0
\(866\) −1.34940e7 −0.611430
\(867\) 4.92745e6 0.222625
\(868\) −4.90858e6 −0.221134
\(869\) −2.09497e6 −0.0941086
\(870\) 0 0
\(871\) −1.28580e7 −0.574286
\(872\) −1.13283e7 −0.504514
\(873\) 1.10753e7 0.491837
\(874\) 2.08889e7 0.924988
\(875\) 0 0
\(876\) 2.57744e7 1.13482
\(877\) −2.38159e7 −1.04561 −0.522803 0.852453i \(-0.675114\pi\)
−0.522803 + 0.852453i \(0.675114\pi\)
\(878\) 6.76842e7 2.96313
\(879\) −1.67106e7 −0.729493
\(880\) 0 0
\(881\) −9.82240e6 −0.426361 −0.213181 0.977013i \(-0.568382\pi\)
−0.213181 + 0.977013i \(0.568382\pi\)
\(882\) −1.04480e7 −0.452231
\(883\) 175255. 0.00756430 0.00378215 0.999993i \(-0.498796\pi\)
0.00378215 + 0.999993i \(0.498796\pi\)
\(884\) −1.40424e7 −0.604383
\(885\) 0 0
\(886\) −4.77834e6 −0.204500
\(887\) −4.60415e7 −1.96490 −0.982450 0.186524i \(-0.940278\pi\)
−0.982450 + 0.186524i \(0.940278\pi\)
\(888\) 9.51028e6 0.404726
\(889\) 4.26325e6 0.180920
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −1.24054e6 −0.0522035
\(893\) −1.57084e7 −0.659179
\(894\) 1.69292e7 0.708423
\(895\) 0 0
\(896\) −2.85565e7 −1.18833
\(897\) 1.05261e7 0.436802
\(898\) −1.95136e7 −0.807507
\(899\) 1.72596e6 0.0712248
\(900\) 0 0
\(901\) −2.56388e7 −1.05217
\(902\) 2.63402e6 0.107796
\(903\) −3.53722e7 −1.44358
\(904\) −1.11720e7 −0.454685
\(905\) 0 0
\(906\) −2.54129e6 −0.102857
\(907\) −3.68866e7 −1.48885 −0.744425 0.667706i \(-0.767276\pi\)
−0.744425 + 0.667706i \(0.767276\pi\)
\(908\) −1.76715e7 −0.711309
\(909\) 4.16153e6 0.167049
\(910\) 0 0
\(911\) −3.44197e7 −1.37408 −0.687038 0.726622i \(-0.741089\pi\)
−0.687038 + 0.726622i \(0.741089\pi\)
\(912\) 4.26110e6 0.169643
\(913\) 1.07712e7 0.427647
\(914\) 6.43509e7 2.54794
\(915\) 0 0
\(916\) −5.47570e7 −2.15626
\(917\) −1.20636e7 −0.473755
\(918\) −5.83878e6 −0.228673
\(919\) −1.33270e7 −0.520527 −0.260263 0.965538i \(-0.583809\pi\)
−0.260263 + 0.965538i \(0.583809\pi\)
\(920\) 0 0
\(921\) −4.54220e6 −0.176448
\(922\) −3.74719e6 −0.145171
\(923\) −2.26878e7 −0.876574
\(924\) 8.07206e6 0.311032
\(925\) 0 0
\(926\) −2.67248e7 −1.02421
\(927\) 7.68688e6 0.293799
\(928\) 2.08546e7 0.794935
\(929\) 2.43196e7 0.924521 0.462260 0.886744i \(-0.347039\pi\)
0.462260 + 0.886744i \(0.347039\pi\)
\(930\) 0 0
\(931\) 1.13405e7 0.428802
\(932\) 1.32936e7 0.501307
\(933\) 4.03660e6 0.151814
\(934\) −1.62714e7 −0.610322
\(935\) 0 0
\(936\) −2.39727e6 −0.0894390
\(937\) −3.18520e7 −1.18519 −0.592595 0.805501i \(-0.701896\pi\)
−0.592595 + 0.805501i \(0.701896\pi\)
\(938\) −5.43603e7 −2.01732
\(939\) 7.42631e6 0.274859
\(940\) 0 0
\(941\) 5.35961e7 1.97315 0.986573 0.163318i \(-0.0522197\pi\)
0.986573 + 0.163318i \(0.0522197\pi\)
\(942\) −3.52924e7 −1.29585
\(943\) 8.20212e6 0.300363
\(944\) −4.87299e6 −0.177978
\(945\) 0 0
\(946\) 2.28509e7 0.830186
\(947\) −1.82063e7 −0.659700 −0.329850 0.944033i \(-0.606998\pi\)
−0.329850 + 0.944033i \(0.606998\pi\)
\(948\) 6.47211e6 0.233897
\(949\) −2.49586e7 −0.899610
\(950\) 0 0
\(951\) 2.93640e7 1.05284
\(952\) −1.36283e7 −0.487360
\(953\) 1.33257e6 0.0475289 0.0237645 0.999718i \(-0.492435\pi\)
0.0237645 + 0.999718i \(0.492435\pi\)
\(954\) −1.90669e7 −0.678281
\(955\) 0 0
\(956\) 5.21399e7 1.84512
\(957\) −2.83830e6 −0.100180
\(958\) 5.23335e7 1.84232
\(959\) −3.67353e7 −1.28984
\(960\) 0 0
\(961\) −2.81906e7 −0.984682
\(962\) −4.01175e7 −1.39764
\(963\) 4.23084e6 0.147015
\(964\) −4.47797e7 −1.55199
\(965\) 0 0
\(966\) 4.45014e7 1.53437
\(967\) 5.74029e7 1.97409 0.987047 0.160429i \(-0.0512877\pi\)
0.987047 + 0.160429i \(0.0512877\pi\)
\(968\) −1.19707e6 −0.0410610
\(969\) 6.33755e6 0.216826
\(970\) 0 0
\(971\) 3.56758e7 1.21430 0.607149 0.794588i \(-0.292313\pi\)
0.607149 + 0.794588i \(0.292313\pi\)
\(972\) −2.45258e6 −0.0832639
\(973\) −7.76232e6 −0.262851
\(974\) −6.26137e7 −2.11481
\(975\) 0 0
\(976\) −2.41581e7 −0.811780
\(977\) −5.30033e7 −1.77651 −0.888253 0.459354i \(-0.848081\pi\)
−0.888253 + 0.459354i \(0.848081\pi\)
\(978\) 4.07665e6 0.136288
\(979\) 1.56848e7 0.523026
\(980\) 0 0
\(981\) 1.12228e7 0.372331
\(982\) −3.78068e7 −1.25110
\(983\) −2.40146e7 −0.792667 −0.396333 0.918107i \(-0.629718\pi\)
−0.396333 + 0.918107i \(0.629718\pi\)
\(984\) −1.86800e6 −0.0615020
\(985\) 0 0
\(986\) 2.08750e7 0.683806
\(987\) −3.34650e7 −1.09345
\(988\) 1.13351e7 0.369430
\(989\) 7.11559e7 2.31324
\(990\) 0 0
\(991\) 5.40921e7 1.74964 0.874822 0.484444i \(-0.160978\pi\)
0.874822 + 0.484444i \(0.160978\pi\)
\(992\) 5.29870e6 0.170958
\(993\) 3.13358e7 1.00848
\(994\) −9.59182e7 −3.07918
\(995\) 0 0
\(996\) −3.32759e7 −1.06287
\(997\) −4.73500e7 −1.50863 −0.754314 0.656513i \(-0.772031\pi\)
−0.754314 + 0.656513i \(0.772031\pi\)
\(998\) −2.96378e7 −0.941931
\(999\) −9.42174e6 −0.298688
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.6.a.v.1.3 13
5.2 odd 4 165.6.c.b.34.4 26
5.3 odd 4 165.6.c.b.34.23 yes 26
5.4 even 2 825.6.a.y.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.4 26 5.2 odd 4
165.6.c.b.34.23 yes 26 5.3 odd 4
825.6.a.v.1.3 13 1.1 even 1 trivial
825.6.a.y.1.11 13 5.4 even 2