Properties

Label 825.6.a.y.1.11
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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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: \(0\)
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 1.11
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 \(-0.226201\pi\)
0.757950 + 0.652312i \(0.226201\pi\)
\(3\) 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 \(-0.258362\pi\)
0.688290 + 0.725435i \(0.258362\pi\)
\(8\) 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 \(-0.595995\pi\)
−0.297027 + 0.954869i \(0.595995\pi\)
\(14\) 1530.36 2.08676
\(15\) 0 0
\(16\) −627.985 −0.613267
\(17\) 934.004 0.783838 0.391919 0.920000i \(-0.371811\pi\)
0.391919 + 0.920000i \(0.371811\pi\)
\(18\) 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 \(-0.280266\pi\)
0.636780 + 0.771046i \(0.280266\pi\)
\(24\) 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 \(-0.782761\pi\)
−0.776013 + 0.630717i \(0.782761\pi\)
\(38\) 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 \(-0.137473\pi\)
0.908178 + 0.418583i \(0.137473\pi\)
\(44\) 5025.68 0.391348
\(45\) 0 0
\(46\) 27706.7 1.93059
\(47\) 20835.4 1.37581 0.687903 0.725802i \(-0.258531\pi\)
0.687903 + 0.725802i \(0.258531\pi\)
\(48\) −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 \(-0.734206\pi\)
−0.671166 + 0.741307i \(0.734206\pi\)
\(54\) 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 \(-0.339416\pi\)
0.483362 + 0.875421i \(0.339416\pi\)
\(68\) 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 \(-0.226577\pi\)
0.757179 + 0.653208i \(0.226577\pi\)
\(74\) −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 \(-0.750931\pi\)
−0.709173 + 0.705035i \(0.750931\pi\)
\(84\) 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 \(-0.764114\pi\)
−0.737755 + 0.675069i \(0.764114\pi\)
\(98\) 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 \(-0.645269\pi\)
−0.440699 + 0.897655i \(0.645269\pi\)
\(104\) −29595.9 −0.268317
\(105\) 0 0
\(106\) −235394. −2.03484
\(107\) −52232.6 −0.441045 −0.220522 0.975382i \(-0.570776\pi\)
−0.220522 + 0.975382i \(0.570776\pi\)
\(108\) 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 \(-0.667894\pi\)
−0.503336 + 0.864091i \(0.667894\pi\)
\(114\) 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 \(-0.479068\pi\)
0.0657136 + 0.997839i \(0.479068\pi\)
\(128\) −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 \(-0.655204\pi\)
−0.468496 + 0.883466i \(0.655204\pi\)
\(138\) 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 \(-0.234678\pi\)
0.740311 + 0.672265i \(0.234678\pi\)
\(158\) −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 \(-0.524809\pi\)
−0.0778603 + 0.996964i \(0.524809\pi\)
\(164\) −105438. −0.306117
\(165\) 0 0
\(166\) −763349. −2.15007
\(167\) −327606. −0.908993 −0.454497 0.890748i \(-0.650181\pi\)
−0.454497 + 0.890748i \(0.650181\pi\)
\(168\) 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 \(-0.398683\pi\)
0.312949 + 0.949770i \(0.398683\pi\)
\(174\) 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 \(-0.587372\pi\)
−0.271053 + 0.962564i \(0.587372\pi\)
\(194\) −1.17251e6 −2.23673
\(195\) 0 0
\(196\) 624756. 1.16164
\(197\) −241625. −0.443585 −0.221792 0.975094i \(-0.571191\pi\)
−0.221792 + 0.975094i \(0.571191\pi\)
\(198\) 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 \(-0.493598\pi\)
0.0201099 + 0.999798i \(0.493598\pi\)
\(224\) −1.42796e6 −1.90150
\(225\) 0 0
\(226\) −1.17174e6 −1.52602
\(227\) 425464. 0.548022 0.274011 0.961726i \(-0.411649\pi\)
0.274011 + 0.961726i \(0.411649\pi\)
\(228\) 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 \(-0.561859\pi\)
−0.193114 + 0.981176i \(0.561859\pi\)
\(234\) −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 \(-0.377120\pi\)
0.376521 + 0.926408i \(0.377120\pi\)
\(258\) 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 \(-0.643986\pi\)
−0.437075 + 0.899425i \(0.643986\pi\)
\(264\) 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 \(-0.308862\pi\)
0.565037 + 0.825066i \(0.308862\pi\)
\(278\) 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 \(-0.501070\pi\)
−0.00336231 + 0.999994i \(0.501070\pi\)
\(284\) −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 \(-0.717667\pi\)
−0.631759 + 0.775165i \(0.717667\pi\)
\(294\) 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 \(-0.548832\pi\)
−0.152808 + 0.988256i \(0.548832\pi\)
\(308\) 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 \(-0.423497\pi\)
0.238035 + 0.971257i \(0.423497\pi\)
\(314\) 3.92138e6 2.24448
\(315\) 0 0
\(316\) −719123. −0.405122
\(317\) 3.26266e6 1.82358 0.911789 0.410659i \(-0.134701\pi\)
0.911789 + 0.410659i \(0.134701\pi\)
\(318\) −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 \(-0.619394\pi\)
−0.366355 + 0.930475i \(0.619394\pi\)
\(338\) −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 \(-0.852805\pi\)
−0.894972 + 0.446122i \(0.852805\pi\)
\(348\) 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 \(-0.549322\pi\)
−0.154331 + 0.988019i \(0.549322\pi\)
\(354\) 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 \(-0.688149\pi\)
−0.557264 + 0.830336i \(0.688149\pi\)
\(368\) −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 \(-0.244910\pi\)
0.718323 + 0.695710i \(0.244910\pi\)
\(374\) 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 \(-0.264665\pi\)
0.673790 + 0.738923i \(0.264665\pi\)
\(384\) −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 \(-0.705533\pi\)
−0.601758 + 0.798679i \(0.705533\pi\)
\(398\) 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 \(-0.564638\pi\)
−0.201672 + 0.979453i \(0.564638\pi\)
\(434\) 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 \(-0.521487\pi\)
−0.0674516 + 0.997723i \(0.521487\pi\)
\(444\) −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 \(-0.182318\pi\)
0.840404 + 0.541961i \(0.182318\pi\)
\(458\) −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 \(-0.609690\pi\)
−0.337821 + 0.941210i \(0.609690\pi\)
\(464\) −1.63674e6 −0.352928
\(465\) 0 0
\(466\) −2.74460e6 −0.585484
\(467\) −1.89749e6 −0.402613 −0.201307 0.979528i \(-0.564519\pi\)
−0.201307 + 0.979528i \(0.564519\pi\)
\(468\) −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 \(-0.745724\pi\)
−0.697544 + 0.716542i \(0.745724\pi\)
\(488\) 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 \(-0.352380\pi\)
0.447317 + 0.894375i \(0.352380\pi\)
\(504\) 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 \(-0.528670\pi\)
−0.0899490 + 0.995946i \(0.528670\pi\)
\(524\) 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 \(-0.257731\pi\)
0.689727 + 0.724070i \(0.257731\pi\)
\(548\) −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 \(-0.407115\pi\)
0.287682 + 0.957726i \(0.407115\pi\)
\(558\) 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 \(-0.279698\pi\)
0.638155 + 0.769908i \(0.279698\pi\)
\(564\) 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 \(-0.392066\pi\)
0.332623 + 0.943060i \(0.392066\pi\)
\(578\) −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 \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(588\) 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 \(-0.459827\pi\)
0.125873 + 0.992046i \(0.459827\pi\)
\(594\) 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 \(-0.657636\pi\)
−0.475232 + 0.879861i \(0.657636\pi\)
\(608\) −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 \(-0.783445\pi\)
−0.777367 + 0.629047i \(0.783445\pi\)
\(614\) −4.32782e6 −0.463285
\(615\) 0 0
\(616\) 1.76555e6 0.187468
\(617\) 5.82746e6 0.616264 0.308132 0.951344i \(-0.400296\pi\)
0.308132 + 0.951344i \(0.400296\pi\)
\(618\) −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 \(-0.568579\pi\)
−0.213784 + 0.976881i \(0.568579\pi\)
\(644\) 2.39494e7 2.27552
\(645\) 0 0
\(646\) 6.03844e6 0.569303
\(647\) 1.75122e7 1.64467 0.822336 0.569003i \(-0.192671\pi\)
0.822336 + 0.569003i \(0.192671\pi\)
\(648\) 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 \(-0.836006\pi\)
−0.870193 + 0.492711i \(0.836006\pi\)
\(654\) 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 \(-0.483869\pi\)
0.0506552 + 0.998716i \(0.483869\pi\)
\(674\) −1.30994e7 −1.11072
\(675\) 0 0
\(676\) −9.97925e6 −0.839907
\(677\) −2.03890e7 −1.70972 −0.854860 0.518859i \(-0.826357\pi\)
−0.854860 + 0.518859i \(0.826357\pi\)
\(678\) −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 \(-0.924609\pi\)
−0.972082 + 0.234640i \(0.924609\pi\)
\(684\) 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 \(-0.630946\pi\)
−0.399873 + 0.916571i \(0.630946\pi\)
\(728\) −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 \(-0.572676\pi\)
−0.226340 + 0.974048i \(0.572676\pi\)
\(734\) −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 \(-0.573910\pi\)
−0.230115 + 0.973163i \(0.573910\pi\)
\(744\) 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 \(-0.407781\pi\)
0.285679 + 0.958325i \(0.407781\pi\)
\(758\) −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 \(-0.337219\pi\)
0.489390 + 0.872065i \(0.337219\pi\)
\(774\) 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 \(-0.546063\pi\)
−0.144206 + 0.989548i \(0.546063\pi\)
\(788\) −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 \(-0.172681\pi\)
0.856424 + 0.516273i \(0.172681\pi\)
\(798\) 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 \(-0.274036\pi\)
0.651747 + 0.758436i \(0.274036\pi\)
\(824\) −7.75912e6 −0.398102
\(825\) 0 0
\(826\) 1.18751e7 0.605603
\(827\) −6.01962e6 −0.306059 −0.153030 0.988222i \(-0.548903\pi\)
−0.153030 + 0.988222i \(0.548903\pi\)
\(828\) 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 \(-0.780127\pi\)
−0.770768 + 0.637116i \(0.780127\pi\)
\(854\) 5.88716e7 2.76224
\(855\) 0 0
\(856\) −4.27060e6 −0.199207
\(857\) −5.91710e6 −0.275205 −0.137603 0.990488i \(-0.543940\pi\)
−0.137603 + 0.990488i \(0.543940\pi\)
\(858\) −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 \(-0.447242\pi\)
0.164986 + 0.986296i \(0.447242\pi\)
\(864\) −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 \(-0.324886\pi\)
0.522803 + 0.852453i \(0.324886\pi\)
\(878\) −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 \(-0.501204\pi\)
−0.00378215 + 0.999993i \(0.501204\pi\)
\(884\) −1.40424e7 −0.604383
\(885\) 0 0
\(886\) −4.77834e6 −0.204500
\(887\) 4.60415e7 1.96490 0.982450 0.186524i \(-0.0597221\pi\)
0.982450 + 0.186524i \(0.0597221\pi\)
\(888\) −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 \(-0.232724\pi\)
0.744425 + 0.667706i \(0.232724\pi\)
\(908\) 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 \(-0.298104\pi\)
0.592595 + 0.805501i \(0.298104\pi\)
\(938\) 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 \(-0.393002\pi\)
0.329850 + 0.944033i \(0.393002\pi\)
\(948\) −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 \(-0.507565\pi\)
−0.0237645 + 0.999718i \(0.507565\pi\)
\(954\) −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 \(-0.948712\pi\)
−0.987047 + 0.160429i \(0.948712\pi\)
\(968\) 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 \(-0.151919\pi\)
0.888253 + 0.459354i \(0.151919\pi\)
\(978\) −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 \(-0.370282\pi\)
0.396333 + 0.918107i \(0.370282\pi\)
\(984\) −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 \(-0.227969\pi\)
0.754314 + 0.656513i \(0.227969\pi\)
\(998\) 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.y.1.11 13
5.2 odd 4 165.6.c.b.34.23 yes 26
5.3 odd 4 165.6.c.b.34.4 26
5.4 even 2 825.6.a.v.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.4 26 5.3 odd 4
165.6.c.b.34.23 yes 26 5.2 odd 4
825.6.a.v.1.3 13 5.4 even 2
825.6.a.y.1.11 13 1.1 even 1 trivial