Properties

Label 825.6.a.y.1.5
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 1.5
Root \(4.33747\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-3.33747 q^{2} +9.00000 q^{3} -20.8613 q^{4} -30.0373 q^{6} +103.473 q^{7} +176.423 q^{8} +81.0000 q^{9} +121.000 q^{11} -187.751 q^{12} -357.925 q^{13} -345.339 q^{14} +78.7535 q^{16} +2067.63 q^{17} -270.335 q^{18} +1320.17 q^{19} +931.258 q^{21} -403.834 q^{22} -2498.44 q^{23} +1587.81 q^{24} +1194.57 q^{26} +729.000 q^{27} -2158.58 q^{28} +1657.60 q^{29} +4690.57 q^{31} -5908.38 q^{32} +1089.00 q^{33} -6900.67 q^{34} -1689.76 q^{36} +5214.75 q^{37} -4406.02 q^{38} -3221.33 q^{39} +13577.0 q^{41} -3108.05 q^{42} +10764.0 q^{43} -2524.21 q^{44} +8338.46 q^{46} -2926.95 q^{47} +708.782 q^{48} -6100.32 q^{49} +18608.7 q^{51} +7466.78 q^{52} -21529.9 q^{53} -2433.02 q^{54} +18255.0 q^{56} +11881.5 q^{57} -5532.19 q^{58} -4541.09 q^{59} +3649.55 q^{61} -15654.6 q^{62} +8381.32 q^{63} +17198.9 q^{64} -3634.51 q^{66} -55473.7 q^{67} -43133.4 q^{68} -22485.9 q^{69} +4491.83 q^{71} +14290.3 q^{72} +33559.8 q^{73} -17404.1 q^{74} -27540.4 q^{76} +12520.2 q^{77} +10751.1 q^{78} -7912.05 q^{79} +6561.00 q^{81} -45312.9 q^{82} -20569.6 q^{83} -19427.2 q^{84} -35924.5 q^{86} +14918.4 q^{87} +21347.2 q^{88} -111119. q^{89} -37035.6 q^{91} +52120.6 q^{92} +42215.1 q^{93} +9768.61 q^{94} -53175.4 q^{96} -58271.1 q^{97} +20359.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.33747 −0.589987 −0.294994 0.955499i \(-0.595318\pi\)
−0.294994 + 0.955499i \(0.595318\pi\)
\(3\) 9.00000 0.577350
\(4\) −20.8613 −0.651915
\(5\) 0 0
\(6\) −30.0373 −0.340629
\(7\) 103.473 0.798146 0.399073 0.916919i \(-0.369332\pi\)
0.399073 + 0.916919i \(0.369332\pi\)
\(8\) 176.423 0.974609
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −187.751 −0.376383
\(13\) −357.925 −0.587400 −0.293700 0.955898i \(-0.594887\pi\)
−0.293700 + 0.955898i \(0.594887\pi\)
\(14\) −345.339 −0.470896
\(15\) 0 0
\(16\) 78.7535 0.0769078
\(17\) 2067.63 1.73521 0.867603 0.497258i \(-0.165660\pi\)
0.867603 + 0.497258i \(0.165660\pi\)
\(18\) −270.335 −0.196662
\(19\) 1320.17 0.838967 0.419484 0.907763i \(-0.362211\pi\)
0.419484 + 0.907763i \(0.362211\pi\)
\(20\) 0 0
\(21\) 931.258 0.460810
\(22\) −403.834 −0.177888
\(23\) −2498.44 −0.984802 −0.492401 0.870369i \(-0.663881\pi\)
−0.492401 + 0.870369i \(0.663881\pi\)
\(24\) 1587.81 0.562691
\(25\) 0 0
\(26\) 1194.57 0.346559
\(27\) 729.000 0.192450
\(28\) −2158.58 −0.520323
\(29\) 1657.60 0.366003 0.183002 0.983113i \(-0.441419\pi\)
0.183002 + 0.983113i \(0.441419\pi\)
\(30\) 0 0
\(31\) 4690.57 0.876640 0.438320 0.898819i \(-0.355574\pi\)
0.438320 + 0.898819i \(0.355574\pi\)
\(32\) −5908.38 −1.01998
\(33\) 1089.00 0.174078
\(34\) −6900.67 −1.02375
\(35\) 0 0
\(36\) −1689.76 −0.217305
\(37\) 5214.75 0.626223 0.313112 0.949716i \(-0.398629\pi\)
0.313112 + 0.949716i \(0.398629\pi\)
\(38\) −4406.02 −0.494980
\(39\) −3221.33 −0.339136
\(40\) 0 0
\(41\) 13577.0 1.26138 0.630688 0.776037i \(-0.282773\pi\)
0.630688 + 0.776037i \(0.282773\pi\)
\(42\) −3108.05 −0.271872
\(43\) 10764.0 0.887774 0.443887 0.896083i \(-0.353599\pi\)
0.443887 + 0.896083i \(0.353599\pi\)
\(44\) −2524.21 −0.196560
\(45\) 0 0
\(46\) 8338.46 0.581021
\(47\) −2926.95 −0.193273 −0.0966363 0.995320i \(-0.530808\pi\)
−0.0966363 + 0.995320i \(0.530808\pi\)
\(48\) 708.782 0.0444027
\(49\) −6100.32 −0.362963
\(50\) 0 0
\(51\) 18608.7 1.00182
\(52\) 7466.78 0.382935
\(53\) −21529.9 −1.05282 −0.526408 0.850232i \(-0.676462\pi\)
−0.526408 + 0.850232i \(0.676462\pi\)
\(54\) −2433.02 −0.113543
\(55\) 0 0
\(56\) 18255.0 0.777880
\(57\) 11881.5 0.484378
\(58\) −5532.19 −0.215937
\(59\) −4541.09 −0.169836 −0.0849181 0.996388i \(-0.527063\pi\)
−0.0849181 + 0.996388i \(0.527063\pi\)
\(60\) 0 0
\(61\) 3649.55 0.125578 0.0627892 0.998027i \(-0.480000\pi\)
0.0627892 + 0.998027i \(0.480000\pi\)
\(62\) −15654.6 −0.517207
\(63\) 8381.32 0.266049
\(64\) 17198.9 0.524870
\(65\) 0 0
\(66\) −3634.51 −0.102704
\(67\) −55473.7 −1.50973 −0.754867 0.655878i \(-0.772299\pi\)
−0.754867 + 0.655878i \(0.772299\pi\)
\(68\) −43133.4 −1.13121
\(69\) −22485.9 −0.568575
\(70\) 0 0
\(71\) 4491.83 0.105749 0.0528746 0.998601i \(-0.483162\pi\)
0.0528746 + 0.998601i \(0.483162\pi\)
\(72\) 14290.3 0.324870
\(73\) 33559.8 0.737076 0.368538 0.929613i \(-0.379859\pi\)
0.368538 + 0.929613i \(0.379859\pi\)
\(74\) −17404.1 −0.369464
\(75\) 0 0
\(76\) −27540.4 −0.546935
\(77\) 12520.2 0.240650
\(78\) 10751.1 0.200086
\(79\) −7912.05 −0.142634 −0.0713168 0.997454i \(-0.522720\pi\)
−0.0713168 + 0.997454i \(0.522720\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −45312.9 −0.744196
\(83\) −20569.6 −0.327741 −0.163870 0.986482i \(-0.552398\pi\)
−0.163870 + 0.986482i \(0.552398\pi\)
\(84\) −19427.2 −0.300409
\(85\) 0 0
\(86\) −35924.5 −0.523775
\(87\) 14918.4 0.211312
\(88\) 21347.2 0.293856
\(89\) −111119. −1.48701 −0.743504 0.668732i \(-0.766837\pi\)
−0.743504 + 0.668732i \(0.766837\pi\)
\(90\) 0 0
\(91\) −37035.6 −0.468831
\(92\) 52120.6 0.642007
\(93\) 42215.1 0.506128
\(94\) 9768.61 0.114028
\(95\) 0 0
\(96\) −53175.4 −0.588888
\(97\) −58271.1 −0.628816 −0.314408 0.949288i \(-0.601806\pi\)
−0.314408 + 0.949288i \(0.601806\pi\)
\(98\) 20359.7 0.214144
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −56503.6 −0.551153 −0.275577 0.961279i \(-0.588869\pi\)
−0.275577 + 0.961279i \(0.588869\pi\)
\(102\) −62106.0 −0.591062
\(103\) 95588.6 0.887796 0.443898 0.896077i \(-0.353595\pi\)
0.443898 + 0.896077i \(0.353595\pi\)
\(104\) −63146.3 −0.572486
\(105\) 0 0
\(106\) 71855.5 0.621148
\(107\) 211645. 1.78710 0.893550 0.448964i \(-0.148207\pi\)
0.893550 + 0.448964i \(0.148207\pi\)
\(108\) −15207.9 −0.125461
\(109\) 370.739 0.00298884 0.00149442 0.999999i \(-0.499524\pi\)
0.00149442 + 0.999999i \(0.499524\pi\)
\(110\) 0 0
\(111\) 46932.8 0.361550
\(112\) 8148.87 0.0613836
\(113\) −14124.9 −0.104061 −0.0520307 0.998645i \(-0.516569\pi\)
−0.0520307 + 0.998645i \(0.516569\pi\)
\(114\) −39654.2 −0.285777
\(115\) 0 0
\(116\) −34579.6 −0.238603
\(117\) −28992.0 −0.195800
\(118\) 15155.8 0.100201
\(119\) 213944. 1.38495
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −12180.3 −0.0740897
\(123\) 122193. 0.728255
\(124\) −97851.3 −0.571495
\(125\) 0 0
\(126\) −27972.4 −0.156965
\(127\) 39049.1 0.214834 0.107417 0.994214i \(-0.465742\pi\)
0.107417 + 0.994214i \(0.465742\pi\)
\(128\) 131667. 0.710317
\(129\) 96876.0 0.512556
\(130\) 0 0
\(131\) −152838. −0.778131 −0.389066 0.921210i \(-0.627202\pi\)
−0.389066 + 0.921210i \(0.627202\pi\)
\(132\) −22717.9 −0.113484
\(133\) 136602. 0.669618
\(134\) 185142. 0.890724
\(135\) 0 0
\(136\) 364778. 1.69115
\(137\) 323866. 1.47422 0.737112 0.675771i \(-0.236189\pi\)
0.737112 + 0.675771i \(0.236189\pi\)
\(138\) 75046.2 0.335452
\(139\) 195925. 0.860108 0.430054 0.902803i \(-0.358494\pi\)
0.430054 + 0.902803i \(0.358494\pi\)
\(140\) 0 0
\(141\) −26342.5 −0.111586
\(142\) −14991.4 −0.0623907
\(143\) −43309.0 −0.177108
\(144\) 6379.04 0.0256359
\(145\) 0 0
\(146\) −112005. −0.434865
\(147\) −54902.9 −0.209557
\(148\) −108786. −0.408244
\(149\) 435048. 1.60536 0.802679 0.596411i \(-0.203407\pi\)
0.802679 + 0.596411i \(0.203407\pi\)
\(150\) 0 0
\(151\) −94062.6 −0.335718 −0.167859 0.985811i \(-0.553685\pi\)
−0.167859 + 0.985811i \(0.553685\pi\)
\(152\) 232908. 0.817665
\(153\) 167478. 0.578402
\(154\) −41786.0 −0.141981
\(155\) 0 0
\(156\) 67201.0 0.221088
\(157\) −194996. −0.631360 −0.315680 0.948866i \(-0.602233\pi\)
−0.315680 + 0.948866i \(0.602233\pi\)
\(158\) 26406.3 0.0841520
\(159\) −193769. −0.607844
\(160\) 0 0
\(161\) −258521. −0.786015
\(162\) −21897.2 −0.0655542
\(163\) 154239. 0.454701 0.227350 0.973813i \(-0.426994\pi\)
0.227350 + 0.973813i \(0.426994\pi\)
\(164\) −283234. −0.822309
\(165\) 0 0
\(166\) 68650.4 0.193363
\(167\) −297517. −0.825506 −0.412753 0.910843i \(-0.635433\pi\)
−0.412753 + 0.910843i \(0.635433\pi\)
\(168\) 164295. 0.449109
\(169\) −243182. −0.654961
\(170\) 0 0
\(171\) 106934. 0.279656
\(172\) −224551. −0.578753
\(173\) −49837.8 −0.126603 −0.0633014 0.997994i \(-0.520163\pi\)
−0.0633014 + 0.997994i \(0.520163\pi\)
\(174\) −49789.8 −0.124671
\(175\) 0 0
\(176\) 9529.18 0.0231886
\(177\) −40869.8 −0.0980550
\(178\) 370856. 0.877316
\(179\) −825399. −1.92545 −0.962724 0.270487i \(-0.912815\pi\)
−0.962724 + 0.270487i \(0.912815\pi\)
\(180\) 0 0
\(181\) −631320. −1.43236 −0.716182 0.697914i \(-0.754112\pi\)
−0.716182 + 0.697914i \(0.754112\pi\)
\(182\) 123605. 0.276604
\(183\) 32846.0 0.0725027
\(184\) −440782. −0.959797
\(185\) 0 0
\(186\) −140892. −0.298609
\(187\) 250184. 0.523184
\(188\) 61059.9 0.125997
\(189\) 75431.9 0.153603
\(190\) 0 0
\(191\) 367056. 0.728030 0.364015 0.931393i \(-0.381406\pi\)
0.364015 + 0.931393i \(0.381406\pi\)
\(192\) 154790. 0.303034
\(193\) 239853. 0.463501 0.231751 0.972775i \(-0.425555\pi\)
0.231751 + 0.972775i \(0.425555\pi\)
\(194\) 194478. 0.370994
\(195\) 0 0
\(196\) 127260. 0.236621
\(197\) 889966. 1.63383 0.816917 0.576755i \(-0.195681\pi\)
0.816917 + 0.576755i \(0.195681\pi\)
\(198\) −32710.6 −0.0592960
\(199\) 2447.34 0.00438089 0.00219045 0.999998i \(-0.499303\pi\)
0.00219045 + 0.999998i \(0.499303\pi\)
\(200\) 0 0
\(201\) −499264. −0.871645
\(202\) 188579. 0.325174
\(203\) 171517. 0.292124
\(204\) −388201. −0.653102
\(205\) 0 0
\(206\) −319024. −0.523788
\(207\) −202373. −0.328267
\(208\) −28187.9 −0.0451756
\(209\) 159740. 0.252958
\(210\) 0 0
\(211\) 779165. 1.20482 0.602411 0.798186i \(-0.294207\pi\)
0.602411 + 0.798186i \(0.294207\pi\)
\(212\) 449141. 0.686347
\(213\) 40426.4 0.0610543
\(214\) −706360. −1.05437
\(215\) 0 0
\(216\) 128612. 0.187564
\(217\) 485348. 0.699687
\(218\) −1237.33 −0.00176338
\(219\) 302038. 0.425551
\(220\) 0 0
\(221\) −740058. −1.01926
\(222\) −156637. −0.213310
\(223\) 1.27044e6 1.71077 0.855387 0.517990i \(-0.173320\pi\)
0.855387 + 0.517990i \(0.173320\pi\)
\(224\) −611358. −0.814096
\(225\) 0 0
\(226\) 47141.5 0.0613949
\(227\) 1755.97 0.00226179 0.00113089 0.999999i \(-0.499640\pi\)
0.00113089 + 0.999999i \(0.499640\pi\)
\(228\) −247863. −0.315773
\(229\) −185101. −0.233249 −0.116624 0.993176i \(-0.537207\pi\)
−0.116624 + 0.993176i \(0.537207\pi\)
\(230\) 0 0
\(231\) 112682. 0.138939
\(232\) 292439. 0.356710
\(233\) 172318. 0.207941 0.103971 0.994580i \(-0.466845\pi\)
0.103971 + 0.994580i \(0.466845\pi\)
\(234\) 96759.9 0.115520
\(235\) 0 0
\(236\) 94733.0 0.110719
\(237\) −71208.5 −0.0823495
\(238\) −714033. −0.817102
\(239\) 1.58068e6 1.78999 0.894994 0.446078i \(-0.147180\pi\)
0.894994 + 0.446078i \(0.147180\pi\)
\(240\) 0 0
\(241\) 1.44227e6 1.59957 0.799785 0.600287i \(-0.204947\pi\)
0.799785 + 0.600287i \(0.204947\pi\)
\(242\) −48863.9 −0.0536352
\(243\) 59049.0 0.0641500
\(244\) −76134.3 −0.0818664
\(245\) 0 0
\(246\) −407816. −0.429662
\(247\) −472521. −0.492810
\(248\) 827525. 0.854381
\(249\) −185126. −0.189221
\(250\) 0 0
\(251\) 1.18482e6 1.18705 0.593524 0.804816i \(-0.297736\pi\)
0.593524 + 0.804816i \(0.297736\pi\)
\(252\) −174845. −0.173441
\(253\) −302311. −0.296929
\(254\) −130325. −0.126749
\(255\) 0 0
\(256\) −989801. −0.943948
\(257\) −307999. −0.290882 −0.145441 0.989367i \(-0.546460\pi\)
−0.145441 + 0.989367i \(0.546460\pi\)
\(258\) −323321. −0.302402
\(259\) 539586. 0.499818
\(260\) 0 0
\(261\) 134266. 0.122001
\(262\) 510093. 0.459088
\(263\) 1.92302e6 1.71433 0.857163 0.515045i \(-0.172225\pi\)
0.857163 + 0.515045i \(0.172225\pi\)
\(264\) 192125. 0.169658
\(265\) 0 0
\(266\) −455905. −0.395066
\(267\) −1.00007e6 −0.858524
\(268\) 1.15725e6 0.984218
\(269\) −1.14532e6 −0.965040 −0.482520 0.875885i \(-0.660278\pi\)
−0.482520 + 0.875885i \(0.660278\pi\)
\(270\) 0 0
\(271\) −737503. −0.610015 −0.305008 0.952350i \(-0.598659\pi\)
−0.305008 + 0.952350i \(0.598659\pi\)
\(272\) 162833. 0.133451
\(273\) −333321. −0.270680
\(274\) −1.08089e6 −0.869774
\(275\) 0 0
\(276\) 469085. 0.370663
\(277\) 1.91454e6 1.49922 0.749611 0.661878i \(-0.230240\pi\)
0.749611 + 0.661878i \(0.230240\pi\)
\(278\) −653895. −0.507453
\(279\) 379936. 0.292213
\(280\) 0 0
\(281\) −1.09912e6 −0.830387 −0.415193 0.909733i \(-0.636286\pi\)
−0.415193 + 0.909733i \(0.636286\pi\)
\(282\) 87917.5 0.0658343
\(283\) 371541. 0.275766 0.137883 0.990449i \(-0.455970\pi\)
0.137883 + 0.990449i \(0.455970\pi\)
\(284\) −93705.2 −0.0689395
\(285\) 0 0
\(286\) 144543. 0.104491
\(287\) 1.40485e6 1.00676
\(288\) −478578. −0.339995
\(289\) 2.85525e6 2.01094
\(290\) 0 0
\(291\) −524440. −0.363047
\(292\) −700100. −0.480511
\(293\) 2.86231e6 1.94781 0.973907 0.226948i \(-0.0728748\pi\)
0.973907 + 0.226948i \(0.0728748\pi\)
\(294\) 183237. 0.123636
\(295\) 0 0
\(296\) 920003. 0.610323
\(297\) 88209.0 0.0580259
\(298\) −1.45196e6 −0.947141
\(299\) 894254. 0.578473
\(300\) 0 0
\(301\) 1.11378e6 0.708573
\(302\) 313931. 0.198069
\(303\) −508532. −0.318209
\(304\) 103968. 0.0645231
\(305\) 0 0
\(306\) −558954. −0.341250
\(307\) −2.67276e6 −1.61851 −0.809253 0.587460i \(-0.800128\pi\)
−0.809253 + 0.587460i \(0.800128\pi\)
\(308\) −261188. −0.156883
\(309\) 860298. 0.512569
\(310\) 0 0
\(311\) 1.84386e6 1.08100 0.540500 0.841344i \(-0.318235\pi\)
0.540500 + 0.841344i \(0.318235\pi\)
\(312\) −568317. −0.330525
\(313\) 1.87741e6 1.08318 0.541588 0.840644i \(-0.317823\pi\)
0.541588 + 0.840644i \(0.317823\pi\)
\(314\) 650794. 0.372494
\(315\) 0 0
\(316\) 165056. 0.0929849
\(317\) −2.04246e6 −1.14158 −0.570789 0.821097i \(-0.693362\pi\)
−0.570789 + 0.821097i \(0.693362\pi\)
\(318\) 646699. 0.358620
\(319\) 200570. 0.110354
\(320\) 0 0
\(321\) 1.90481e6 1.03178
\(322\) 862807. 0.463739
\(323\) 2.72962e6 1.45578
\(324\) −136871. −0.0724350
\(325\) 0 0
\(326\) −514769. −0.268268
\(327\) 3336.65 0.00172561
\(328\) 2.39530e6 1.22935
\(329\) −302860. −0.154260
\(330\) 0 0
\(331\) −2.48973e6 −1.24906 −0.624528 0.781002i \(-0.714709\pi\)
−0.624528 + 0.781002i \(0.714709\pi\)
\(332\) 429108. 0.213659
\(333\) 422395. 0.208741
\(334\) 992954. 0.487038
\(335\) 0 0
\(336\) 73339.8 0.0354398
\(337\) −2.26543e6 −1.08662 −0.543308 0.839534i \(-0.682828\pi\)
−0.543308 + 0.839534i \(0.682828\pi\)
\(338\) 811615. 0.386419
\(339\) −127124. −0.0600799
\(340\) 0 0
\(341\) 567559. 0.264317
\(342\) −356888. −0.164993
\(343\) −2.37029e6 −1.08784
\(344\) 1.89902e6 0.865232
\(345\) 0 0
\(346\) 166332. 0.0746941
\(347\) 887458. 0.395662 0.197831 0.980236i \(-0.436610\pi\)
0.197831 + 0.980236i \(0.436610\pi\)
\(348\) −311217. −0.137757
\(349\) 3.37834e6 1.48470 0.742351 0.670011i \(-0.233711\pi\)
0.742351 + 0.670011i \(0.233711\pi\)
\(350\) 0 0
\(351\) −260928. −0.113045
\(352\) −714914. −0.307537
\(353\) 1.23199e6 0.526222 0.263111 0.964766i \(-0.415251\pi\)
0.263111 + 0.964766i \(0.415251\pi\)
\(354\) 136402. 0.0578512
\(355\) 0 0
\(356\) 2.31808e6 0.969402
\(357\) 1.92550e6 0.799600
\(358\) 2.75475e6 1.13599
\(359\) 1.03637e6 0.424405 0.212202 0.977226i \(-0.431936\pi\)
0.212202 + 0.977226i \(0.431936\pi\)
\(360\) 0 0
\(361\) −733257. −0.296134
\(362\) 2.10701e6 0.845076
\(363\) 131769. 0.0524864
\(364\) 772611. 0.305638
\(365\) 0 0
\(366\) −109623. −0.0427757
\(367\) 4.31603e6 1.67270 0.836352 0.548192i \(-0.184684\pi\)
0.836352 + 0.548192i \(0.184684\pi\)
\(368\) −196761. −0.0757389
\(369\) 1.09974e6 0.420458
\(370\) 0 0
\(371\) −2.22777e6 −0.840301
\(372\) −880661. −0.329953
\(373\) −2.03599e6 −0.757712 −0.378856 0.925456i \(-0.623682\pi\)
−0.378856 + 0.925456i \(0.623682\pi\)
\(374\) −834981. −0.308672
\(375\) 0 0
\(376\) −516381. −0.188365
\(377\) −593297. −0.214990
\(378\) −251752. −0.0906240
\(379\) −3.67933e6 −1.31574 −0.657872 0.753130i \(-0.728543\pi\)
−0.657872 + 0.753130i \(0.728543\pi\)
\(380\) 0 0
\(381\) 351442. 0.124034
\(382\) −1.22504e6 −0.429528
\(383\) −3.35614e6 −1.16908 −0.584539 0.811366i \(-0.698725\pi\)
−0.584539 + 0.811366i \(0.698725\pi\)
\(384\) 1.18500e6 0.410102
\(385\) 0 0
\(386\) −800501. −0.273460
\(387\) 871884. 0.295925
\(388\) 1.21561e6 0.409935
\(389\) 5.94481e6 1.99188 0.995942 0.0900021i \(-0.0286874\pi\)
0.995942 + 0.0900021i \(0.0286874\pi\)
\(390\) 0 0
\(391\) −5.16585e6 −1.70883
\(392\) −1.07624e6 −0.353747
\(393\) −1.37554e6 −0.449254
\(394\) −2.97024e6 −0.963941
\(395\) 0 0
\(396\) −204461. −0.0655199
\(397\) 4.07424e6 1.29739 0.648695 0.761048i \(-0.275315\pi\)
0.648695 + 0.761048i \(0.275315\pi\)
\(398\) −8167.95 −0.00258467
\(399\) 1.22942e6 0.386604
\(400\) 0 0
\(401\) 1.76526e6 0.548209 0.274105 0.961700i \(-0.411619\pi\)
0.274105 + 0.961700i \(0.411619\pi\)
\(402\) 1.66628e6 0.514260
\(403\) −1.67887e6 −0.514939
\(404\) 1.17874e6 0.359305
\(405\) 0 0
\(406\) −572433. −0.172349
\(407\) 630985. 0.188813
\(408\) 3.28300e6 0.976384
\(409\) −3.05109e6 −0.901877 −0.450938 0.892555i \(-0.648911\pi\)
−0.450938 + 0.892555i \(0.648911\pi\)
\(410\) 0 0
\(411\) 2.91479e6 0.851144
\(412\) −1.99410e6 −0.578767
\(413\) −469881. −0.135554
\(414\) 675416. 0.193674
\(415\) 0 0
\(416\) 2.11476e6 0.599139
\(417\) 1.76333e6 0.496584
\(418\) −533129. −0.149242
\(419\) −6.61264e6 −1.84009 −0.920046 0.391810i \(-0.871849\pi\)
−0.920046 + 0.391810i \(0.871849\pi\)
\(420\) 0 0
\(421\) 2.29303e6 0.630527 0.315263 0.949004i \(-0.397907\pi\)
0.315263 + 0.949004i \(0.397907\pi\)
\(422\) −2.60044e6 −0.710830
\(423\) −237083. −0.0644242
\(424\) −3.79837e6 −1.02608
\(425\) 0 0
\(426\) −134922. −0.0360213
\(427\) 377630. 0.100230
\(428\) −4.41519e6 −1.16504
\(429\) −389781. −0.102253
\(430\) 0 0
\(431\) 5.53759e6 1.43591 0.717956 0.696089i \(-0.245078\pi\)
0.717956 + 0.696089i \(0.245078\pi\)
\(432\) 57411.3 0.0148009
\(433\) −2.55617e6 −0.655195 −0.327598 0.944817i \(-0.606239\pi\)
−0.327598 + 0.944817i \(0.606239\pi\)
\(434\) −1.61983e6 −0.412806
\(435\) 0 0
\(436\) −7734.09 −0.00194847
\(437\) −3.29835e6 −0.826216
\(438\) −1.00804e6 −0.251070
\(439\) −696199. −0.172414 −0.0862070 0.996277i \(-0.527475\pi\)
−0.0862070 + 0.996277i \(0.527475\pi\)
\(440\) 0 0
\(441\) −494126. −0.120988
\(442\) 2.46992e6 0.601351
\(443\) 6.82989e6 1.65350 0.826750 0.562570i \(-0.190187\pi\)
0.826750 + 0.562570i \(0.190187\pi\)
\(444\) −979077. −0.235700
\(445\) 0 0
\(446\) −4.24006e6 −1.00933
\(447\) 3.91544e6 0.926854
\(448\) 1.77963e6 0.418923
\(449\) −6.68298e6 −1.56442 −0.782212 0.623013i \(-0.785909\pi\)
−0.782212 + 0.623013i \(0.785909\pi\)
\(450\) 0 0
\(451\) 1.64282e6 0.380319
\(452\) 294664. 0.0678392
\(453\) −846564. −0.193827
\(454\) −5860.49 −0.00133443
\(455\) 0 0
\(456\) 2.09617e6 0.472079
\(457\) 616838. 0.138159 0.0690797 0.997611i \(-0.477994\pi\)
0.0690797 + 0.997611i \(0.477994\pi\)
\(458\) 617769. 0.137614
\(459\) 1.50730e6 0.333941
\(460\) 0 0
\(461\) 3.55459e6 0.779001 0.389500 0.921026i \(-0.372648\pi\)
0.389500 + 0.921026i \(0.372648\pi\)
\(462\) −376074. −0.0819725
\(463\) −3.23948e6 −0.702299 −0.351150 0.936319i \(-0.614209\pi\)
−0.351150 + 0.936319i \(0.614209\pi\)
\(464\) 130542. 0.0281485
\(465\) 0 0
\(466\) −575106. −0.122683
\(467\) 7.20656e6 1.52910 0.764550 0.644564i \(-0.222961\pi\)
0.764550 + 0.644564i \(0.222961\pi\)
\(468\) 604809. 0.127645
\(469\) −5.74004e6 −1.20499
\(470\) 0 0
\(471\) −1.75497e6 −0.364516
\(472\) −801154. −0.165524
\(473\) 1.30244e6 0.267674
\(474\) 237656. 0.0485852
\(475\) 0 0
\(476\) −4.46315e6 −0.902868
\(477\) −1.74392e6 −0.350939
\(478\) −5.27549e6 −1.05607
\(479\) 772579. 0.153852 0.0769261 0.997037i \(-0.475489\pi\)
0.0769261 + 0.997037i \(0.475489\pi\)
\(480\) 0 0
\(481\) −1.86649e6 −0.367844
\(482\) −4.81353e6 −0.943726
\(483\) −2.32669e6 −0.453806
\(484\) −305430. −0.0592650
\(485\) 0 0
\(486\) −197074. −0.0378477
\(487\) −3.08873e6 −0.590143 −0.295072 0.955475i \(-0.595344\pi\)
−0.295072 + 0.955475i \(0.595344\pi\)
\(488\) 643865. 0.122390
\(489\) 1.38815e6 0.262522
\(490\) 0 0
\(491\) 4.82049e6 0.902375 0.451187 0.892429i \(-0.351001\pi\)
0.451187 + 0.892429i \(0.351001\pi\)
\(492\) −2.54910e6 −0.474761
\(493\) 3.42731e6 0.635091
\(494\) 1.57703e6 0.290751
\(495\) 0 0
\(496\) 369399. 0.0674204
\(497\) 464783. 0.0844033
\(498\) 617854. 0.111638
\(499\) −9.37761e6 −1.68594 −0.842968 0.537964i \(-0.819194\pi\)
−0.842968 + 0.537964i \(0.819194\pi\)
\(500\) 0 0
\(501\) −2.67765e6 −0.476606
\(502\) −3.95430e6 −0.700343
\(503\) 4.35545e6 0.767562 0.383781 0.923424i \(-0.374622\pi\)
0.383781 + 0.923424i \(0.374622\pi\)
\(504\) 1.47866e6 0.259293
\(505\) 0 0
\(506\) 1.00895e6 0.175184
\(507\) −2.18864e6 −0.378142
\(508\) −814615. −0.140053
\(509\) 1.05979e7 1.81311 0.906556 0.422086i \(-0.138702\pi\)
0.906556 + 0.422086i \(0.138702\pi\)
\(510\) 0 0
\(511\) 3.47253e6 0.588294
\(512\) −909912. −0.153400
\(513\) 962402. 0.161459
\(514\) 1.02794e6 0.171617
\(515\) 0 0
\(516\) −2.02096e6 −0.334143
\(517\) −354161. −0.0582739
\(518\) −1.80086e6 −0.294886
\(519\) −448540. −0.0730942
\(520\) 0 0
\(521\) 6.52699e6 1.05346 0.526731 0.850032i \(-0.323418\pi\)
0.526731 + 0.850032i \(0.323418\pi\)
\(522\) −448108. −0.0719791
\(523\) −9.35815e6 −1.49601 −0.748007 0.663690i \(-0.768989\pi\)
−0.748007 + 0.663690i \(0.768989\pi\)
\(524\) 3.18839e6 0.507275
\(525\) 0 0
\(526\) −6.41801e6 −1.01143
\(527\) 9.69837e6 1.52115
\(528\) 85762.6 0.0133879
\(529\) −194157. −0.0301657
\(530\) 0 0
\(531\) −367829. −0.0566121
\(532\) −2.84969e6 −0.436534
\(533\) −4.85955e6 −0.740932
\(534\) 3.33771e6 0.506518
\(535\) 0 0
\(536\) −9.78685e6 −1.47140
\(537\) −7.42860e6 −1.11166
\(538\) 3.82247e6 0.569361
\(539\) −738139. −0.109438
\(540\) 0 0
\(541\) 9.93079e6 1.45878 0.729392 0.684096i \(-0.239803\pi\)
0.729392 + 0.684096i \(0.239803\pi\)
\(542\) 2.46140e6 0.359901
\(543\) −5.68188e6 −0.826975
\(544\) −1.22163e7 −1.76988
\(545\) 0 0
\(546\) 1.11245e6 0.159698
\(547\) −8.79495e6 −1.25680 −0.628399 0.777891i \(-0.716289\pi\)
−0.628399 + 0.777891i \(0.716289\pi\)
\(548\) −6.75625e6 −0.961068
\(549\) 295614. 0.0418595
\(550\) 0 0
\(551\) 2.18831e6 0.307065
\(552\) −3.96704e6 −0.554139
\(553\) −818685. −0.113842
\(554\) −6.38974e6 −0.884522
\(555\) 0 0
\(556\) −4.08725e6 −0.560717
\(557\) 2.78134e6 0.379853 0.189927 0.981798i \(-0.439175\pi\)
0.189927 + 0.981798i \(0.439175\pi\)
\(558\) −1.26803e6 −0.172402
\(559\) −3.85271e6 −0.521479
\(560\) 0 0
\(561\) 2.25165e6 0.302061
\(562\) 3.66829e6 0.489918
\(563\) −460410. −0.0612172 −0.0306086 0.999531i \(-0.509745\pi\)
−0.0306086 + 0.999531i \(0.509745\pi\)
\(564\) 549539. 0.0727446
\(565\) 0 0
\(566\) −1.24001e6 −0.162698
\(567\) 678887. 0.0886829
\(568\) 792462. 0.103064
\(569\) 8.74980e6 1.13297 0.566484 0.824073i \(-0.308303\pi\)
0.566484 + 0.824073i \(0.308303\pi\)
\(570\) 0 0
\(571\) −8.95646e6 −1.14960 −0.574799 0.818295i \(-0.694920\pi\)
−0.574799 + 0.818295i \(0.694920\pi\)
\(572\) 903480. 0.115459
\(573\) 3.30351e6 0.420328
\(574\) −4.68866e6 −0.593977
\(575\) 0 0
\(576\) 1.39311e6 0.174957
\(577\) −1.14055e7 −1.42618 −0.713091 0.701071i \(-0.752706\pi\)
−0.713091 + 0.701071i \(0.752706\pi\)
\(578\) −9.52931e6 −1.18643
\(579\) 2.15867e6 0.267603
\(580\) 0 0
\(581\) −2.12840e6 −0.261585
\(582\) 1.75030e6 0.214193
\(583\) −2.60512e6 −0.317436
\(584\) 5.92072e6 0.718361
\(585\) 0 0
\(586\) −9.55288e6 −1.14919
\(587\) 1.05278e7 1.26108 0.630539 0.776157i \(-0.282834\pi\)
0.630539 + 0.776157i \(0.282834\pi\)
\(588\) 1.14534e6 0.136613
\(589\) 6.19234e6 0.735472
\(590\) 0 0
\(591\) 8.00969e6 0.943294
\(592\) 410680. 0.0481614
\(593\) 7.64531e6 0.892809 0.446404 0.894831i \(-0.352704\pi\)
0.446404 + 0.894831i \(0.352704\pi\)
\(594\) −294395. −0.0342345
\(595\) 0 0
\(596\) −9.07566e6 −1.04656
\(597\) 22026.1 0.00252931
\(598\) −2.98455e6 −0.341292
\(599\) −1.28357e7 −1.46168 −0.730838 0.682550i \(-0.760871\pi\)
−0.730838 + 0.682550i \(0.760871\pi\)
\(600\) 0 0
\(601\) −1.49774e7 −1.69142 −0.845710 0.533643i \(-0.820823\pi\)
−0.845710 + 0.533643i \(0.820823\pi\)
\(602\) −3.71722e6 −0.418049
\(603\) −4.49337e6 −0.503245
\(604\) 1.96227e6 0.218860
\(605\) 0 0
\(606\) 1.69721e6 0.187739
\(607\) −273921. −0.0301754 −0.0150877 0.999886i \(-0.504803\pi\)
−0.0150877 + 0.999886i \(0.504803\pi\)
\(608\) −7.80005e6 −0.855733
\(609\) 1.54365e6 0.168658
\(610\) 0 0
\(611\) 1.04763e6 0.113528
\(612\) −3.49381e6 −0.377069
\(613\) −3.61183e6 −0.388219 −0.194109 0.980980i \(-0.562182\pi\)
−0.194109 + 0.980980i \(0.562182\pi\)
\(614\) 8.92027e6 0.954898
\(615\) 0 0
\(616\) 2.20886e6 0.234540
\(617\) −1.11537e7 −1.17952 −0.589759 0.807579i \(-0.700777\pi\)
−0.589759 + 0.807579i \(0.700777\pi\)
\(618\) −2.87122e6 −0.302409
\(619\) −8.08693e6 −0.848314 −0.424157 0.905589i \(-0.639430\pi\)
−0.424157 + 0.905589i \(0.639430\pi\)
\(620\) 0 0
\(621\) −1.82136e6 −0.189525
\(622\) −6.15382e6 −0.637777
\(623\) −1.14978e7 −1.18685
\(624\) −253691. −0.0260822
\(625\) 0 0
\(626\) −6.26582e6 −0.639060
\(627\) 1.43766e6 0.146045
\(628\) 4.06787e6 0.411593
\(629\) 1.07822e7 1.08663
\(630\) 0 0
\(631\) −1.09030e7 −1.09012 −0.545060 0.838397i \(-0.683493\pi\)
−0.545060 + 0.838397i \(0.683493\pi\)
\(632\) −1.39587e6 −0.139012
\(633\) 7.01248e6 0.695605
\(634\) 6.81665e6 0.673516
\(635\) 0 0
\(636\) 4.04227e6 0.396262
\(637\) 2.18346e6 0.213205
\(638\) −669396. −0.0651075
\(639\) 363838. 0.0352497
\(640\) 0 0
\(641\) −1.28118e7 −1.23159 −0.615794 0.787907i \(-0.711165\pi\)
−0.615794 + 0.787907i \(0.711165\pi\)
\(642\) −6.35724e6 −0.608739
\(643\) −4.42120e6 −0.421709 −0.210854 0.977517i \(-0.567625\pi\)
−0.210854 + 0.977517i \(0.567625\pi\)
\(644\) 5.39308e6 0.512415
\(645\) 0 0
\(646\) −9.11003e6 −0.858892
\(647\) −2.82953e6 −0.265738 −0.132869 0.991134i \(-0.542419\pi\)
−0.132869 + 0.991134i \(0.542419\pi\)
\(648\) 1.15751e6 0.108290
\(649\) −549472. −0.0512076
\(650\) 0 0
\(651\) 4.36813e6 0.403964
\(652\) −3.21763e6 −0.296426
\(653\) −1.51923e7 −1.39425 −0.697127 0.716948i \(-0.745539\pi\)
−0.697127 + 0.716948i \(0.745539\pi\)
\(654\) −11136.0 −0.00101809
\(655\) 0 0
\(656\) 1.06924e6 0.0970096
\(657\) 2.71834e6 0.245692
\(658\) 1.01079e6 0.0910113
\(659\) −1.45583e7 −1.30587 −0.652933 0.757416i \(-0.726462\pi\)
−0.652933 + 0.757416i \(0.726462\pi\)
\(660\) 0 0
\(661\) 5.60047e6 0.498564 0.249282 0.968431i \(-0.419805\pi\)
0.249282 + 0.968431i \(0.419805\pi\)
\(662\) 8.30941e6 0.736928
\(663\) −6.66052e6 −0.588470
\(664\) −3.62895e6 −0.319419
\(665\) 0 0
\(666\) −1.40973e6 −0.123155
\(667\) −4.14141e6 −0.360440
\(668\) 6.20658e6 0.538159
\(669\) 1.14340e7 0.987715
\(670\) 0 0
\(671\) 441596. 0.0378633
\(672\) −5.50222e6 −0.470018
\(673\) −2.22862e6 −0.189670 −0.0948351 0.995493i \(-0.530232\pi\)
−0.0948351 + 0.995493i \(0.530232\pi\)
\(674\) 7.56081e6 0.641089
\(675\) 0 0
\(676\) 5.07310e6 0.426979
\(677\) −1.47850e7 −1.23980 −0.619898 0.784683i \(-0.712826\pi\)
−0.619898 + 0.784683i \(0.712826\pi\)
\(678\) 424274. 0.0354464
\(679\) −6.02949e6 −0.501887
\(680\) 0 0
\(681\) 15803.7 0.00130584
\(682\) −1.89421e6 −0.155944
\(683\) 415268. 0.0340625 0.0170313 0.999855i \(-0.494579\pi\)
0.0170313 + 0.999855i \(0.494579\pi\)
\(684\) −2.23077e6 −0.182312
\(685\) 0 0
\(686\) 7.91078e6 0.641814
\(687\) −1.66591e6 −0.134666
\(688\) 847703. 0.0682767
\(689\) 7.70610e6 0.618424
\(690\) 0 0
\(691\) 8.64233e6 0.688550 0.344275 0.938869i \(-0.388125\pi\)
0.344275 + 0.938869i \(0.388125\pi\)
\(692\) 1.03968e6 0.0825342
\(693\) 1.01414e6 0.0802167
\(694\) −2.96187e6 −0.233436
\(695\) 0 0
\(696\) 2.63195e6 0.205947
\(697\) 2.80723e7 2.18875
\(698\) −1.12751e7 −0.875956
\(699\) 1.55086e6 0.120055
\(700\) 0 0
\(701\) −1.30512e7 −1.00313 −0.501564 0.865121i \(-0.667242\pi\)
−0.501564 + 0.865121i \(0.667242\pi\)
\(702\) 870839. 0.0666953
\(703\) 6.88434e6 0.525381
\(704\) 2.08107e6 0.158254
\(705\) 0 0
\(706\) −4.11172e6 −0.310464
\(707\) −5.84660e6 −0.439901
\(708\) 852597. 0.0639235
\(709\) −1.56007e7 −1.16555 −0.582773 0.812635i \(-0.698032\pi\)
−0.582773 + 0.812635i \(0.698032\pi\)
\(710\) 0 0
\(711\) −640876. −0.0475445
\(712\) −1.96039e7 −1.44925
\(713\) −1.17191e7 −0.863317
\(714\) −6.42630e6 −0.471754
\(715\) 0 0
\(716\) 1.72189e7 1.25523
\(717\) 1.42262e7 1.03345
\(718\) −3.45887e6 −0.250394
\(719\) −6.44874e6 −0.465214 −0.232607 0.972571i \(-0.574726\pi\)
−0.232607 + 0.972571i \(0.574726\pi\)
\(720\) 0 0
\(721\) 9.89085e6 0.708591
\(722\) 2.44723e6 0.174715
\(723\) 1.29804e7 0.923512
\(724\) 1.31701e7 0.933779
\(725\) 0 0
\(726\) −439775. −0.0309663
\(727\) 8.36468e6 0.586967 0.293483 0.955964i \(-0.405185\pi\)
0.293483 + 0.955964i \(0.405185\pi\)
\(728\) −6.53394e6 −0.456927
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.22560e7 1.54047
\(732\) −685209. −0.0472656
\(733\) 3.08435e6 0.212033 0.106017 0.994364i \(-0.466190\pi\)
0.106017 + 0.994364i \(0.466190\pi\)
\(734\) −1.44046e7 −0.986875
\(735\) 0 0
\(736\) 1.47617e7 1.00448
\(737\) −6.71232e6 −0.455202
\(738\) −3.67034e6 −0.248065
\(739\) −2.28100e7 −1.53644 −0.768218 0.640189i \(-0.778856\pi\)
−0.768218 + 0.640189i \(0.778856\pi\)
\(740\) 0 0
\(741\) −4.25269e6 −0.284524
\(742\) 7.43511e6 0.495767
\(743\) 1.21470e7 0.807232 0.403616 0.914929i \(-0.367753\pi\)
0.403616 + 0.914929i \(0.367753\pi\)
\(744\) 7.44772e6 0.493277
\(745\) 0 0
\(746\) 6.79507e6 0.447040
\(747\) −1.66614e6 −0.109247
\(748\) −5.21915e6 −0.341072
\(749\) 2.18996e7 1.42637
\(750\) 0 0
\(751\) 2.29619e7 1.48562 0.742809 0.669503i \(-0.233493\pi\)
0.742809 + 0.669503i \(0.233493\pi\)
\(752\) −230508. −0.0148642
\(753\) 1.06634e7 0.685342
\(754\) 1.98011e6 0.126842
\(755\) 0 0
\(756\) −1.57361e6 −0.100136
\(757\) 8.97752e6 0.569399 0.284699 0.958617i \(-0.408106\pi\)
0.284699 + 0.958617i \(0.408106\pi\)
\(758\) 1.22797e7 0.776272
\(759\) −2.72080e6 −0.171432
\(760\) 0 0
\(761\) 1.81766e7 1.13776 0.568880 0.822421i \(-0.307377\pi\)
0.568880 + 0.822421i \(0.307377\pi\)
\(762\) −1.17293e6 −0.0731786
\(763\) 38361.5 0.00238553
\(764\) −7.65726e6 −0.474613
\(765\) 0 0
\(766\) 1.12010e7 0.689741
\(767\) 1.62537e6 0.0997618
\(768\) −8.90821e6 −0.544989
\(769\) −1.53679e7 −0.937127 −0.468564 0.883430i \(-0.655228\pi\)
−0.468564 + 0.883430i \(0.655228\pi\)
\(770\) 0 0
\(771\) −2.77199e6 −0.167941
\(772\) −5.00363e6 −0.302163
\(773\) 732946. 0.0441188 0.0220594 0.999757i \(-0.492978\pi\)
0.0220594 + 0.999757i \(0.492978\pi\)
\(774\) −2.90989e6 −0.174592
\(775\) 0 0
\(776\) −1.02804e7 −0.612850
\(777\) 4.85628e6 0.288570
\(778\) −1.98406e7 −1.17519
\(779\) 1.79239e7 1.05825
\(780\) 0 0
\(781\) 543511. 0.0318846
\(782\) 1.72409e7 1.00819
\(783\) 1.20839e6 0.0704373
\(784\) −480422. −0.0279147
\(785\) 0 0
\(786\) 4.59083e6 0.265054
\(787\) 2.25077e7 1.29537 0.647685 0.761908i \(-0.275737\pi\)
0.647685 + 0.761908i \(0.275737\pi\)
\(788\) −1.85658e7 −1.06512
\(789\) 1.73071e7 0.989767
\(790\) 0 0
\(791\) −1.46155e6 −0.0830562
\(792\) 1.72912e6 0.0979519
\(793\) −1.30627e6 −0.0737648
\(794\) −1.35977e7 −0.765444
\(795\) 0 0
\(796\) −51054.7 −0.00285597
\(797\) −5.07705e6 −0.283117 −0.141558 0.989930i \(-0.545211\pi\)
−0.141558 + 0.989930i \(0.545211\pi\)
\(798\) −4.10314e6 −0.228092
\(799\) −6.05185e6 −0.335368
\(800\) 0 0
\(801\) −9.00063e6 −0.495669
\(802\) −5.89149e6 −0.323437
\(803\) 4.06073e6 0.222237
\(804\) 1.04153e7 0.568238
\(805\) 0 0
\(806\) 5.60320e6 0.303807
\(807\) −1.03079e7 −0.557166
\(808\) −9.96853e6 −0.537159
\(809\) 1.15468e7 0.620282 0.310141 0.950691i \(-0.399624\pi\)
0.310141 + 0.950691i \(0.399624\pi\)
\(810\) 0 0
\(811\) −2.62543e7 −1.40168 −0.700838 0.713320i \(-0.747190\pi\)
−0.700838 + 0.713320i \(0.747190\pi\)
\(812\) −3.57806e6 −0.190440
\(813\) −6.63753e6 −0.352192
\(814\) −2.10590e6 −0.111398
\(815\) 0 0
\(816\) 1.46550e6 0.0770479
\(817\) 1.42103e7 0.744813
\(818\) 1.01829e7 0.532096
\(819\) −2.99989e6 −0.156277
\(820\) 0 0
\(821\) −3.96273e6 −0.205180 −0.102590 0.994724i \(-0.532713\pi\)
−0.102590 + 0.994724i \(0.532713\pi\)
\(822\) −9.72803e6 −0.502164
\(823\) −9.86765e6 −0.507825 −0.253913 0.967227i \(-0.581718\pi\)
−0.253913 + 0.967227i \(0.581718\pi\)
\(824\) 1.68640e7 0.865254
\(825\) 0 0
\(826\) 1.56821e6 0.0799752
\(827\) 5.90399e6 0.300180 0.150090 0.988672i \(-0.452044\pi\)
0.150090 + 0.988672i \(0.452044\pi\)
\(828\) 4.22177e6 0.214002
\(829\) 2.38090e7 1.20325 0.601624 0.798779i \(-0.294520\pi\)
0.601624 + 0.798779i \(0.294520\pi\)
\(830\) 0 0
\(831\) 1.72309e7 0.865576
\(832\) −6.15593e6 −0.308309
\(833\) −1.26132e7 −0.629816
\(834\) −5.88505e6 −0.292978
\(835\) 0 0
\(836\) −3.33239e6 −0.164907
\(837\) 3.41943e6 0.168709
\(838\) 2.20695e7 1.08563
\(839\) 1.88904e7 0.926479 0.463240 0.886233i \(-0.346687\pi\)
0.463240 + 0.886233i \(0.346687\pi\)
\(840\) 0 0
\(841\) −1.77635e7 −0.866042
\(842\) −7.65291e6 −0.372003
\(843\) −9.89210e6 −0.479424
\(844\) −1.62544e7 −0.785442
\(845\) 0 0
\(846\) 791257. 0.0380095
\(847\) 1.51495e6 0.0725587
\(848\) −1.69556e6 −0.0809697
\(849\) 3.34387e6 0.159214
\(850\) 0 0
\(851\) −1.30287e7 −0.616706
\(852\) −843347. −0.0398022
\(853\) 2.79626e7 1.31585 0.657924 0.753085i \(-0.271435\pi\)
0.657924 + 0.753085i \(0.271435\pi\)
\(854\) −1.26033e6 −0.0591344
\(855\) 0 0
\(856\) 3.73391e7 1.74172
\(857\) 2.82523e6 0.131402 0.0657009 0.997839i \(-0.479072\pi\)
0.0657009 + 0.997839i \(0.479072\pi\)
\(858\) 1.30088e6 0.0603281
\(859\) 5.68353e6 0.262806 0.131403 0.991329i \(-0.458052\pi\)
0.131403 + 0.991329i \(0.458052\pi\)
\(860\) 0 0
\(861\) 1.26437e7 0.581254
\(862\) −1.84816e7 −0.847170
\(863\) −2.23836e6 −0.102306 −0.0511532 0.998691i \(-0.516290\pi\)
−0.0511532 + 0.998691i \(0.516290\pi\)
\(864\) −4.30721e6 −0.196296
\(865\) 0 0
\(866\) 8.53116e6 0.386557
\(867\) 2.56972e7 1.16102
\(868\) −1.01250e7 −0.456136
\(869\) −957359. −0.0430056
\(870\) 0 0
\(871\) 1.98555e7 0.886818
\(872\) 65406.9 0.00291295
\(873\) −4.71996e6 −0.209605
\(874\) 1.10082e7 0.487457
\(875\) 0 0
\(876\) −6.30090e6 −0.277423
\(877\) 6.04256e6 0.265291 0.132645 0.991164i \(-0.457653\pi\)
0.132645 + 0.991164i \(0.457653\pi\)
\(878\) 2.32355e6 0.101722
\(879\) 2.57608e7 1.12457
\(880\) 0 0
\(881\) 3.77877e7 1.64025 0.820126 0.572183i \(-0.193903\pi\)
0.820126 + 0.572183i \(0.193903\pi\)
\(882\) 1.64913e6 0.0713812
\(883\) 4.27448e7 1.84494 0.922469 0.386070i \(-0.126168\pi\)
0.922469 + 0.386070i \(0.126168\pi\)
\(884\) 1.54386e7 0.664471
\(885\) 0 0
\(886\) −2.27946e7 −0.975544
\(887\) −971018. −0.0414398 −0.0207199 0.999785i \(-0.506596\pi\)
−0.0207199 + 0.999785i \(0.506596\pi\)
\(888\) 8.28002e6 0.352370
\(889\) 4.04053e6 0.171468
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −2.65030e7 −1.11528
\(893\) −3.86406e6 −0.162149
\(894\) −1.30677e7 −0.546832
\(895\) 0 0
\(896\) 1.36240e7 0.566937
\(897\) 8.04828e6 0.333981
\(898\) 2.23043e7 0.922990
\(899\) 7.77509e6 0.320853
\(900\) 0 0
\(901\) −4.45159e7 −1.82685
\(902\) −5.48286e6 −0.224383
\(903\) 1.00241e7 0.409095
\(904\) −2.49196e6 −0.101419
\(905\) 0 0
\(906\) 2.82538e6 0.114355
\(907\) 1.74008e7 0.702348 0.351174 0.936310i \(-0.385783\pi\)
0.351174 + 0.936310i \(0.385783\pi\)
\(908\) −36631.7 −0.00147449
\(909\) −4.57679e6 −0.183718
\(910\) 0 0
\(911\) −1.56534e7 −0.624903 −0.312451 0.949934i \(-0.601150\pi\)
−0.312451 + 0.949934i \(0.601150\pi\)
\(912\) 935711. 0.0372524
\(913\) −2.48892e6 −0.0988175
\(914\) −2.05868e6 −0.0815123
\(915\) 0 0
\(916\) 3.86144e6 0.152058
\(917\) −1.58146e7 −0.621062
\(918\) −5.03059e6 −0.197021
\(919\) −3.32140e7 −1.29728 −0.648639 0.761097i \(-0.724661\pi\)
−0.648639 + 0.761097i \(0.724661\pi\)
\(920\) 0 0
\(921\) −2.40549e7 −0.934445
\(922\) −1.18634e7 −0.459601
\(923\) −1.60774e6 −0.0621171
\(924\) −2.35069e6 −0.0905766
\(925\) 0 0
\(926\) 1.08117e7 0.414348
\(927\) 7.74268e6 0.295932
\(928\) −9.79372e6 −0.373317
\(929\) −7.05863e6 −0.268338 −0.134169 0.990958i \(-0.542836\pi\)
−0.134169 + 0.990958i \(0.542836\pi\)
\(930\) 0 0
\(931\) −8.05345e6 −0.304514
\(932\) −3.59477e6 −0.135560
\(933\) 1.65947e7 0.624116
\(934\) −2.40517e7 −0.902150
\(935\) 0 0
\(936\) −5.11485e6 −0.190829
\(937\) −4.85427e6 −0.180624 −0.0903120 0.995914i \(-0.528786\pi\)
−0.0903120 + 0.995914i \(0.528786\pi\)
\(938\) 1.91572e7 0.710928
\(939\) 1.68967e7 0.625372
\(940\) 0 0
\(941\) −3.79442e7 −1.39692 −0.698460 0.715649i \(-0.746131\pi\)
−0.698460 + 0.715649i \(0.746131\pi\)
\(942\) 5.85715e6 0.215060
\(943\) −3.39213e7 −1.24220
\(944\) −357627. −0.0130617
\(945\) 0 0
\(946\) −4.34687e6 −0.157924
\(947\) −1.42196e7 −0.515244 −0.257622 0.966246i \(-0.582939\pi\)
−0.257622 + 0.966246i \(0.582939\pi\)
\(948\) 1.48550e6 0.0536849
\(949\) −1.20119e7 −0.432958
\(950\) 0 0
\(951\) −1.83821e7 −0.659090
\(952\) 3.77447e7 1.34978
\(953\) −1.27436e7 −0.454529 −0.227264 0.973833i \(-0.572978\pi\)
−0.227264 + 0.973833i \(0.572978\pi\)
\(954\) 5.82030e6 0.207049
\(955\) 0 0
\(956\) −3.29751e7 −1.16692
\(957\) 1.80513e6 0.0637130
\(958\) −2.57846e6 −0.0907709
\(959\) 3.35114e7 1.17665
\(960\) 0 0
\(961\) −6.62771e6 −0.231502
\(962\) 6.22937e6 0.217023
\(963\) 1.71433e7 0.595700
\(964\) −3.00875e7 −1.04278
\(965\) 0 0
\(966\) 7.76526e6 0.267740
\(967\) 4.44841e7 1.52981 0.764906 0.644142i \(-0.222785\pi\)
0.764906 + 0.644142i \(0.222785\pi\)
\(968\) 2.58301e6 0.0886008
\(969\) 2.45666e7 0.840495
\(970\) 0 0
\(971\) −1.58495e7 −0.539469 −0.269734 0.962935i \(-0.586936\pi\)
−0.269734 + 0.962935i \(0.586936\pi\)
\(972\) −1.23184e6 −0.0418204
\(973\) 2.02730e7 0.686492
\(974\) 1.03086e7 0.348177
\(975\) 0 0
\(976\) 287415. 0.00965795
\(977\) −3.72698e7 −1.24917 −0.624584 0.780958i \(-0.714731\pi\)
−0.624584 + 0.780958i \(0.714731\pi\)
\(978\) −4.63292e6 −0.154885
\(979\) −1.34454e7 −0.448350
\(980\) 0 0
\(981\) 30029.9 0.000996279 0
\(982\) −1.60882e7 −0.532390
\(983\) 1.71461e7 0.565955 0.282978 0.959127i \(-0.408678\pi\)
0.282978 + 0.959127i \(0.408678\pi\)
\(984\) 2.15577e7 0.709764
\(985\) 0 0
\(986\) −1.14385e7 −0.374695
\(987\) −2.72574e6 −0.0890619
\(988\) 9.85740e6 0.321270
\(989\) −2.68932e7 −0.874281
\(990\) 0 0
\(991\) 3.65029e7 1.18071 0.590354 0.807144i \(-0.298988\pi\)
0.590354 + 0.807144i \(0.298988\pi\)
\(992\) −2.77137e7 −0.894159
\(993\) −2.24076e7 −0.721143
\(994\) −1.55120e6 −0.0497969
\(995\) 0 0
\(996\) 3.86197e6 0.123356
\(997\) 5.88370e6 0.187462 0.0937309 0.995598i \(-0.470121\pi\)
0.0937309 + 0.995598i \(0.470121\pi\)
\(998\) 3.12975e7 0.994681
\(999\) 3.80155e6 0.120517
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.1.5 13
5.2 odd 4 165.6.c.b.34.11 26
5.3 odd 4 165.6.c.b.34.16 yes 26
5.4 even 2 825.6.a.v.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.11 26 5.2 odd 4
165.6.c.b.34.16 yes 26 5.3 odd 4
825.6.a.v.1.9 13 5.4 even 2
825.6.a.y.1.5 13 1.1 even 1 trivial