Properties

Label 825.6.a.y.1.10
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 1.10
Root \(-4.72893\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.72893 q^{2} +9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} -158.171 q^{7} -178.625 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.72893 q^{2} +9.00000 q^{3} +0.820586 q^{4} +51.5603 q^{6} -158.171 q^{7} -178.625 q^{8} +81.0000 q^{9} +121.000 q^{11} +7.38528 q^{12} -78.2418 q^{13} -906.152 q^{14} -1049.59 q^{16} +88.2340 q^{17} +464.043 q^{18} +1865.42 q^{19} -1423.54 q^{21} +693.200 q^{22} -503.122 q^{23} -1607.62 q^{24} -448.241 q^{26} +729.000 q^{27} -129.793 q^{28} +1050.22 q^{29} -9057.51 q^{31} -297.011 q^{32} +1089.00 q^{33} +505.486 q^{34} +66.4675 q^{36} +3899.39 q^{37} +10686.9 q^{38} -704.176 q^{39} -9454.73 q^{41} -8155.37 q^{42} +8918.80 q^{43} +99.2910 q^{44} -2882.35 q^{46} +9431.40 q^{47} -9446.27 q^{48} +8211.19 q^{49} +794.106 q^{51} -64.2042 q^{52} +25507.5 q^{53} +4176.39 q^{54} +28253.3 q^{56} +16788.8 q^{57} +6016.65 q^{58} +9907.66 q^{59} -15614.3 q^{61} -51889.8 q^{62} -12811.9 q^{63} +31885.2 q^{64} +6238.80 q^{66} -16879.9 q^{67} +72.4036 q^{68} -4528.10 q^{69} +71467.7 q^{71} -14468.6 q^{72} +65733.0 q^{73} +22339.3 q^{74} +1530.74 q^{76} -19138.7 q^{77} -4034.17 q^{78} +104481. q^{79} +6561.00 q^{81} -54165.4 q^{82} -15457.9 q^{83} -1168.14 q^{84} +51095.2 q^{86} +9452.01 q^{87} -21613.6 q^{88} +2774.62 q^{89} +12375.6 q^{91} -412.855 q^{92} -81517.6 q^{93} +54031.8 q^{94} -2673.10 q^{96} +125401. q^{97} +47041.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.72893 1.01274 0.506370 0.862316i \(-0.330987\pi\)
0.506370 + 0.862316i \(0.330987\pi\)
\(3\) 9.00000 0.577350
\(4\) 0.820586 0.0256433
\(5\) 0 0
\(6\) 51.5603 0.584706
\(7\) −158.171 −1.22006 −0.610032 0.792377i \(-0.708844\pi\)
−0.610032 + 0.792377i \(0.708844\pi\)
\(8\) −178.625 −0.986770
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 7.38528 0.0148052
\(13\) −78.2418 −0.128405 −0.0642023 0.997937i \(-0.520450\pi\)
−0.0642023 + 0.997937i \(0.520450\pi\)
\(14\) −906.152 −1.23561
\(15\) 0 0
\(16\) −1049.59 −1.02499
\(17\) 88.2340 0.0740480 0.0370240 0.999314i \(-0.488212\pi\)
0.0370240 + 0.999314i \(0.488212\pi\)
\(18\) 464.043 0.337580
\(19\) 1865.42 1.18548 0.592738 0.805395i \(-0.298047\pi\)
0.592738 + 0.805395i \(0.298047\pi\)
\(20\) 0 0
\(21\) −1423.54 −0.704405
\(22\) 693.200 0.305353
\(23\) −503.122 −0.198314 −0.0991571 0.995072i \(-0.531615\pi\)
−0.0991571 + 0.995072i \(0.531615\pi\)
\(24\) −1607.62 −0.569712
\(25\) 0 0
\(26\) −448.241 −0.130040
\(27\) 729.000 0.192450
\(28\) −129.793 −0.0312865
\(29\) 1050.22 0.231892 0.115946 0.993255i \(-0.463010\pi\)
0.115946 + 0.993255i \(0.463010\pi\)
\(30\) 0 0
\(31\) −9057.51 −1.69280 −0.846398 0.532550i \(-0.821234\pi\)
−0.846398 + 0.532550i \(0.821234\pi\)
\(32\) −297.011 −0.0512741
\(33\) 1089.00 0.174078
\(34\) 505.486 0.0749914
\(35\) 0 0
\(36\) 66.4675 0.00854778
\(37\) 3899.39 0.468266 0.234133 0.972205i \(-0.424775\pi\)
0.234133 + 0.972205i \(0.424775\pi\)
\(38\) 10686.9 1.20058
\(39\) −704.176 −0.0741344
\(40\) 0 0
\(41\) −9454.73 −0.878394 −0.439197 0.898391i \(-0.644737\pi\)
−0.439197 + 0.898391i \(0.644737\pi\)
\(42\) −8155.37 −0.713379
\(43\) 8918.80 0.735589 0.367795 0.929907i \(-0.380113\pi\)
0.367795 + 0.929907i \(0.380113\pi\)
\(44\) 99.2910 0.00773175
\(45\) 0 0
\(46\) −2882.35 −0.200841
\(47\) 9431.40 0.622776 0.311388 0.950283i \(-0.399206\pi\)
0.311388 + 0.950283i \(0.399206\pi\)
\(48\) −9446.27 −0.591776
\(49\) 8211.19 0.488558
\(50\) 0 0
\(51\) 794.106 0.0427516
\(52\) −64.2042 −0.00329272
\(53\) 25507.5 1.24732 0.623661 0.781695i \(-0.285645\pi\)
0.623661 + 0.781695i \(0.285645\pi\)
\(54\) 4176.39 0.194902
\(55\) 0 0
\(56\) 28253.3 1.20392
\(57\) 16788.8 0.684435
\(58\) 6016.65 0.234847
\(59\) 9907.66 0.370545 0.185273 0.982687i \(-0.440683\pi\)
0.185273 + 0.982687i \(0.440683\pi\)
\(60\) 0 0
\(61\) −15614.3 −0.537278 −0.268639 0.963241i \(-0.586574\pi\)
−0.268639 + 0.963241i \(0.586574\pi\)
\(62\) −51889.8 −1.71436
\(63\) −12811.9 −0.406688
\(64\) 31885.2 0.973058
\(65\) 0 0
\(66\) 6238.80 0.176295
\(67\) −16879.9 −0.459392 −0.229696 0.973262i \(-0.573773\pi\)
−0.229696 + 0.973262i \(0.573773\pi\)
\(68\) 72.4036 0.00189884
\(69\) −4528.10 −0.114497
\(70\) 0 0
\(71\) 71467.7 1.68253 0.841267 0.540620i \(-0.181810\pi\)
0.841267 + 0.540620i \(0.181810\pi\)
\(72\) −14468.6 −0.328923
\(73\) 65733.0 1.44370 0.721849 0.692050i \(-0.243292\pi\)
0.721849 + 0.692050i \(0.243292\pi\)
\(74\) 22339.3 0.474232
\(75\) 0 0
\(76\) 1530.74 0.0303995
\(77\) −19138.7 −0.367863
\(78\) −4034.17 −0.0750789
\(79\) 104481. 1.88352 0.941762 0.336281i \(-0.109169\pi\)
0.941762 + 0.336281i \(0.109169\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −54165.4 −0.889585
\(83\) −15457.9 −0.246294 −0.123147 0.992388i \(-0.539299\pi\)
−0.123147 + 0.992388i \(0.539299\pi\)
\(84\) −1168.14 −0.0180633
\(85\) 0 0
\(86\) 51095.2 0.744961
\(87\) 9452.01 0.133883
\(88\) −21613.6 −0.297522
\(89\) 2774.62 0.0371303 0.0185651 0.999828i \(-0.494090\pi\)
0.0185651 + 0.999828i \(0.494090\pi\)
\(90\) 0 0
\(91\) 12375.6 0.156662
\(92\) −412.855 −0.00508544
\(93\) −81517.6 −0.977336
\(94\) 54031.8 0.630710
\(95\) 0 0
\(96\) −2673.10 −0.0296031
\(97\) 125401. 1.35323 0.676614 0.736338i \(-0.263447\pi\)
0.676614 + 0.736338i \(0.263447\pi\)
\(98\) 47041.3 0.494782
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 50984.8 0.497322 0.248661 0.968591i \(-0.420010\pi\)
0.248661 + 0.968591i \(0.420010\pi\)
\(102\) 4549.37 0.0432963
\(103\) −51464.8 −0.477989 −0.238994 0.971021i \(-0.576818\pi\)
−0.238994 + 0.971021i \(0.576818\pi\)
\(104\) 13975.9 0.126706
\(105\) 0 0
\(106\) 146131. 1.26321
\(107\) 139655. 1.17923 0.589613 0.807686i \(-0.299280\pi\)
0.589613 + 0.807686i \(0.299280\pi\)
\(108\) 598.208 0.00493506
\(109\) 144034. 1.16118 0.580591 0.814195i \(-0.302822\pi\)
0.580591 + 0.814195i \(0.302822\pi\)
\(110\) 0 0
\(111\) 35094.5 0.270353
\(112\) 166014. 1.25055
\(113\) 62064.4 0.457242 0.228621 0.973515i \(-0.426578\pi\)
0.228621 + 0.973515i \(0.426578\pi\)
\(114\) 96181.7 0.693155
\(115\) 0 0
\(116\) 861.799 0.00594649
\(117\) −6337.59 −0.0428015
\(118\) 56760.3 0.375266
\(119\) −13956.1 −0.0903434
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −89453.4 −0.544124
\(123\) −85092.5 −0.507141
\(124\) −7432.47 −0.0434089
\(125\) 0 0
\(126\) −73398.3 −0.411870
\(127\) 285166. 1.56888 0.784438 0.620207i \(-0.212951\pi\)
0.784438 + 0.620207i \(0.212951\pi\)
\(128\) 192172. 1.03673
\(129\) 80269.2 0.424693
\(130\) 0 0
\(131\) −198363. −1.00991 −0.504954 0.863146i \(-0.668491\pi\)
−0.504954 + 0.863146i \(0.668491\pi\)
\(132\) 893.619 0.00446393
\(133\) −295056. −1.44636
\(134\) −96703.8 −0.465245
\(135\) 0 0
\(136\) −15760.7 −0.0730684
\(137\) −166328. −0.757120 −0.378560 0.925577i \(-0.623581\pi\)
−0.378560 + 0.925577i \(0.623581\pi\)
\(138\) −25941.1 −0.115955
\(139\) −301037. −1.32155 −0.660774 0.750585i \(-0.729772\pi\)
−0.660774 + 0.750585i \(0.729772\pi\)
\(140\) 0 0
\(141\) 84882.6 0.359560
\(142\) 409433. 1.70397
\(143\) −9467.26 −0.0387154
\(144\) −85016.4 −0.341662
\(145\) 0 0
\(146\) 376580. 1.46209
\(147\) 73900.7 0.282069
\(148\) 3199.79 0.0120079
\(149\) 107457. 0.396525 0.198263 0.980149i \(-0.436470\pi\)
0.198263 + 0.980149i \(0.436470\pi\)
\(150\) 0 0
\(151\) 24082.4 0.0859522 0.0429761 0.999076i \(-0.486316\pi\)
0.0429761 + 0.999076i \(0.486316\pi\)
\(152\) −333210. −1.16979
\(153\) 7146.95 0.0246827
\(154\) −109644. −0.372550
\(155\) 0 0
\(156\) −577.837 −0.00190105
\(157\) −479707. −1.55320 −0.776600 0.629994i \(-0.783057\pi\)
−0.776600 + 0.629994i \(0.783057\pi\)
\(158\) 598566. 1.90752
\(159\) 229568. 0.720142
\(160\) 0 0
\(161\) 79579.5 0.241956
\(162\) 37587.5 0.112527
\(163\) 437032. 1.28838 0.644190 0.764866i \(-0.277195\pi\)
0.644190 + 0.764866i \(0.277195\pi\)
\(164\) −7758.42 −0.0225249
\(165\) 0 0
\(166\) −88557.0 −0.249432
\(167\) −592592. −1.64424 −0.822119 0.569316i \(-0.807208\pi\)
−0.822119 + 0.569316i \(0.807208\pi\)
\(168\) 254280. 0.695086
\(169\) −365171. −0.983512
\(170\) 0 0
\(171\) 151099. 0.395159
\(172\) 7318.65 0.0188630
\(173\) 193454. 0.491432 0.245716 0.969342i \(-0.420977\pi\)
0.245716 + 0.969342i \(0.420977\pi\)
\(174\) 54149.9 0.135589
\(175\) 0 0
\(176\) −127000. −0.309045
\(177\) 89169.0 0.213934
\(178\) 15895.6 0.0376033
\(179\) −373735. −0.871828 −0.435914 0.899988i \(-0.643575\pi\)
−0.435914 + 0.899988i \(0.643575\pi\)
\(180\) 0 0
\(181\) −234887. −0.532921 −0.266461 0.963846i \(-0.585854\pi\)
−0.266461 + 0.963846i \(0.585854\pi\)
\(182\) 70899.0 0.158658
\(183\) −140529. −0.310198
\(184\) 89869.9 0.195691
\(185\) 0 0
\(186\) −467008. −0.989788
\(187\) 10676.3 0.0223263
\(188\) 7739.28 0.0159700
\(189\) −115307. −0.234802
\(190\) 0 0
\(191\) 261514. 0.518695 0.259348 0.965784i \(-0.416492\pi\)
0.259348 + 0.965784i \(0.416492\pi\)
\(192\) 286967. 0.561796
\(193\) 688977. 1.33141 0.665705 0.746215i \(-0.268131\pi\)
0.665705 + 0.746215i \(0.268131\pi\)
\(194\) 718412. 1.37047
\(195\) 0 0
\(196\) 6737.99 0.0125282
\(197\) 625063. 1.14752 0.573758 0.819025i \(-0.305485\pi\)
0.573758 + 0.819025i \(0.305485\pi\)
\(198\) 56149.2 0.101784
\(199\) 1.02019e6 1.82619 0.913097 0.407742i \(-0.133684\pi\)
0.913097 + 0.407742i \(0.133684\pi\)
\(200\) 0 0
\(201\) −151919. −0.265230
\(202\) 292088. 0.503658
\(203\) −166115. −0.282924
\(204\) 651.632 0.00109629
\(205\) 0 0
\(206\) −294838. −0.484079
\(207\) −40752.9 −0.0661047
\(208\) 82121.4 0.131613
\(209\) 225716. 0.357434
\(210\) 0 0
\(211\) −544322. −0.841685 −0.420843 0.907134i \(-0.638266\pi\)
−0.420843 + 0.907134i \(0.638266\pi\)
\(212\) 20931.1 0.0319855
\(213\) 643209. 0.971412
\(214\) 800073. 1.19425
\(215\) 0 0
\(216\) −130217. −0.189904
\(217\) 1.43264e6 2.06532
\(218\) 825163. 1.17598
\(219\) 591597. 0.833520
\(220\) 0 0
\(221\) −6903.58 −0.00950810
\(222\) 201054. 0.273798
\(223\) 852497. 1.14797 0.573985 0.818866i \(-0.305397\pi\)
0.573985 + 0.818866i \(0.305397\pi\)
\(224\) 46978.7 0.0625577
\(225\) 0 0
\(226\) 355562. 0.463068
\(227\) −909729. −1.17178 −0.585891 0.810390i \(-0.699256\pi\)
−0.585891 + 0.810390i \(0.699256\pi\)
\(228\) 13776.6 0.0175512
\(229\) −561762. −0.707886 −0.353943 0.935267i \(-0.615159\pi\)
−0.353943 + 0.935267i \(0.615159\pi\)
\(230\) 0 0
\(231\) −172249. −0.212386
\(232\) −187596. −0.228825
\(233\) −604654. −0.729654 −0.364827 0.931075i \(-0.618872\pi\)
−0.364827 + 0.931075i \(0.618872\pi\)
\(234\) −36307.6 −0.0433468
\(235\) 0 0
\(236\) 8130.10 0.00950201
\(237\) 940332. 1.08745
\(238\) −79953.4 −0.0914944
\(239\) −1.13697e6 −1.28752 −0.643760 0.765227i \(-0.722627\pi\)
−0.643760 + 0.765227i \(0.722627\pi\)
\(240\) 0 0
\(241\) 261399. 0.289909 0.144954 0.989438i \(-0.453696\pi\)
0.144954 + 0.989438i \(0.453696\pi\)
\(242\) 83877.2 0.0920673
\(243\) 59049.0 0.0641500
\(244\) −12812.9 −0.0137776
\(245\) 0 0
\(246\) −487489. −0.513602
\(247\) −145954. −0.152220
\(248\) 1.61789e6 1.67040
\(249\) −139121. −0.142198
\(250\) 0 0
\(251\) 565375. 0.566438 0.283219 0.959055i \(-0.408598\pi\)
0.283219 + 0.959055i \(0.408598\pi\)
\(252\) −10513.3 −0.0104288
\(253\) −60877.8 −0.0597940
\(254\) 1.63370e6 1.58886
\(255\) 0 0
\(256\) 80614.3 0.0768798
\(257\) −1.51946e6 −1.43501 −0.717507 0.696551i \(-0.754717\pi\)
−0.717507 + 0.696551i \(0.754717\pi\)
\(258\) 459856. 0.430103
\(259\) −616772. −0.571315
\(260\) 0 0
\(261\) 85068.1 0.0772975
\(262\) −1.13641e6 −1.02278
\(263\) −730370. −0.651109 −0.325554 0.945523i \(-0.605551\pi\)
−0.325554 + 0.945523i \(0.605551\pi\)
\(264\) −194522. −0.171775
\(265\) 0 0
\(266\) −1.69035e6 −1.46478
\(267\) 24971.6 0.0214372
\(268\) −13851.4 −0.0117803
\(269\) −475262. −0.400453 −0.200227 0.979750i \(-0.564168\pi\)
−0.200227 + 0.979750i \(0.564168\pi\)
\(270\) 0 0
\(271\) −961313. −0.795136 −0.397568 0.917573i \(-0.630146\pi\)
−0.397568 + 0.917573i \(0.630146\pi\)
\(272\) −92609.1 −0.0758982
\(273\) 111381. 0.0904488
\(274\) −952883. −0.766766
\(275\) 0 0
\(276\) −3715.70 −0.00293608
\(277\) 428260. 0.335358 0.167679 0.985842i \(-0.446373\pi\)
0.167679 + 0.985842i \(0.446373\pi\)
\(278\) −1.72462e6 −1.33839
\(279\) −733659. −0.564265
\(280\) 0 0
\(281\) −212752. −0.160734 −0.0803670 0.996765i \(-0.525609\pi\)
−0.0803670 + 0.996765i \(0.525609\pi\)
\(282\) 486286. 0.364141
\(283\) −563864. −0.418512 −0.209256 0.977861i \(-0.567104\pi\)
−0.209256 + 0.977861i \(0.567104\pi\)
\(284\) 58645.4 0.0431458
\(285\) 0 0
\(286\) −54237.2 −0.0392087
\(287\) 1.49547e6 1.07170
\(288\) −24057.9 −0.0170914
\(289\) −1.41207e6 −0.994517
\(290\) 0 0
\(291\) 1.12861e6 0.781287
\(292\) 53939.6 0.0370212
\(293\) 1.80961e6 1.23145 0.615723 0.787963i \(-0.288864\pi\)
0.615723 + 0.787963i \(0.288864\pi\)
\(294\) 423371. 0.285663
\(295\) 0 0
\(296\) −696527. −0.462071
\(297\) 88209.0 0.0580259
\(298\) 615616. 0.401577
\(299\) 39365.2 0.0254644
\(300\) 0 0
\(301\) −1.41070e6 −0.897466
\(302\) 137966. 0.0870473
\(303\) 458863. 0.287129
\(304\) −1.95792e6 −1.21510
\(305\) 0 0
\(306\) 40944.3 0.0249971
\(307\) −774049. −0.468730 −0.234365 0.972149i \(-0.575301\pi\)
−0.234365 + 0.972149i \(0.575301\pi\)
\(308\) −15705.0 −0.00943324
\(309\) −463184. −0.275967
\(310\) 0 0
\(311\) 1.35808e6 0.796205 0.398102 0.917341i \(-0.369669\pi\)
0.398102 + 0.917341i \(0.369669\pi\)
\(312\) 125783. 0.0731536
\(313\) 3.38315e6 1.95191 0.975956 0.217968i \(-0.0699428\pi\)
0.975956 + 0.217968i \(0.0699428\pi\)
\(314\) −2.74821e6 −1.57299
\(315\) 0 0
\(316\) 85736.0 0.0482998
\(317\) 668139. 0.373438 0.186719 0.982413i \(-0.440215\pi\)
0.186719 + 0.982413i \(0.440215\pi\)
\(318\) 1.31518e6 0.729317
\(319\) 127077. 0.0699182
\(320\) 0 0
\(321\) 1.25689e6 0.680826
\(322\) 455905. 0.245039
\(323\) 164593. 0.0877821
\(324\) 5383.87 0.00284926
\(325\) 0 0
\(326\) 2.50372e6 1.30479
\(327\) 1.29631e6 0.670409
\(328\) 1.68885e6 0.866773
\(329\) −1.49178e6 −0.759827
\(330\) 0 0
\(331\) 94884.0 0.0476018 0.0238009 0.999717i \(-0.492423\pi\)
0.0238009 + 0.999717i \(0.492423\pi\)
\(332\) −12684.5 −0.00631581
\(333\) 315851. 0.156089
\(334\) −3.39491e6 −1.66519
\(335\) 0 0
\(336\) 1.49413e6 0.722005
\(337\) 507319. 0.243336 0.121668 0.992571i \(-0.461176\pi\)
0.121668 + 0.992571i \(0.461176\pi\)
\(338\) −2.09204e6 −0.996043
\(339\) 558580. 0.263989
\(340\) 0 0
\(341\) −1.09596e6 −0.510397
\(342\) 865635. 0.400193
\(343\) 1.35961e6 0.623993
\(344\) −1.59312e6 −0.725858
\(345\) 0 0
\(346\) 1.10829e6 0.497693
\(347\) 2.91869e6 1.30126 0.650631 0.759394i \(-0.274505\pi\)
0.650631 + 0.759394i \(0.274505\pi\)
\(348\) 7756.19 0.00343321
\(349\) −1.69598e6 −0.745346 −0.372673 0.927963i \(-0.621559\pi\)
−0.372673 + 0.927963i \(0.621559\pi\)
\(350\) 0 0
\(351\) −57038.3 −0.0247115
\(352\) −35938.4 −0.0154597
\(353\) −2.54669e6 −1.08777 −0.543887 0.839159i \(-0.683048\pi\)
−0.543887 + 0.839159i \(0.683048\pi\)
\(354\) 510842. 0.216660
\(355\) 0 0
\(356\) 2276.81 0.000952144 0
\(357\) −125605. −0.0521598
\(358\) −2.14110e6 −0.882936
\(359\) −425376. −0.174196 −0.0870978 0.996200i \(-0.527759\pi\)
−0.0870978 + 0.996200i \(0.527759\pi\)
\(360\) 0 0
\(361\) 1.00369e6 0.405353
\(362\) −1.34565e6 −0.539711
\(363\) 131769. 0.0524864
\(364\) 10155.3 0.00401733
\(365\) 0 0
\(366\) −805081. −0.314150
\(367\) −1.88623e6 −0.731019 −0.365509 0.930808i \(-0.619105\pi\)
−0.365509 + 0.930808i \(0.619105\pi\)
\(368\) 528070. 0.203269
\(369\) −765833. −0.292798
\(370\) 0 0
\(371\) −4.03456e6 −1.52181
\(372\) −66892.3 −0.0250622
\(373\) 903282. 0.336164 0.168082 0.985773i \(-0.446243\pi\)
0.168082 + 0.985773i \(0.446243\pi\)
\(374\) 61163.8 0.0226108
\(375\) 0 0
\(376\) −1.68468e6 −0.614537
\(377\) −82171.3 −0.0297760
\(378\) −660585. −0.237793
\(379\) 4.94830e6 1.76953 0.884766 0.466036i \(-0.154318\pi\)
0.884766 + 0.466036i \(0.154318\pi\)
\(380\) 0 0
\(381\) 2.56650e6 0.905791
\(382\) 1.49820e6 0.525304
\(383\) −1.62974e6 −0.567703 −0.283851 0.958868i \(-0.591612\pi\)
−0.283851 + 0.958868i \(0.591612\pi\)
\(384\) 1.72955e6 0.598556
\(385\) 0 0
\(386\) 3.94710e6 1.34837
\(387\) 722423. 0.245196
\(388\) 102902. 0.0347013
\(389\) 5.33558e6 1.78775 0.893877 0.448311i \(-0.147974\pi\)
0.893877 + 0.448311i \(0.147974\pi\)
\(390\) 0 0
\(391\) −44392.4 −0.0146848
\(392\) −1.46672e6 −0.482094
\(393\) −1.78527e6 −0.583071
\(394\) 3.58094e6 1.16214
\(395\) 0 0
\(396\) 8042.57 0.00257725
\(397\) 3.89341e6 1.23981 0.619903 0.784678i \(-0.287172\pi\)
0.619903 + 0.784678i \(0.287172\pi\)
\(398\) 5.84457e6 1.84946
\(399\) −2.65550e6 −0.835055
\(400\) 0 0
\(401\) 401457. 0.124675 0.0623374 0.998055i \(-0.480145\pi\)
0.0623374 + 0.998055i \(0.480145\pi\)
\(402\) −870335. −0.268609
\(403\) 708676. 0.217363
\(404\) 41837.5 0.0127530
\(405\) 0 0
\(406\) −951662. −0.286528
\(407\) 471826. 0.141187
\(408\) −141847. −0.0421861
\(409\) −5.92484e6 −1.75133 −0.875666 0.482918i \(-0.839577\pi\)
−0.875666 + 0.482918i \(0.839577\pi\)
\(410\) 0 0
\(411\) −1.49695e6 −0.437124
\(412\) −42231.4 −0.0122572
\(413\) −1.56711e6 −0.452089
\(414\) −233470. −0.0669469
\(415\) 0 0
\(416\) 23238.7 0.00658383
\(417\) −2.70933e6 −0.762996
\(418\) 1.29311e6 0.361988
\(419\) 4.23888e6 1.17955 0.589774 0.807568i \(-0.299217\pi\)
0.589774 + 0.807568i \(0.299217\pi\)
\(420\) 0 0
\(421\) 5.04583e6 1.38748 0.693740 0.720225i \(-0.255962\pi\)
0.693740 + 0.720225i \(0.255962\pi\)
\(422\) −3.11838e6 −0.852409
\(423\) 763944. 0.207592
\(424\) −4.55627e6 −1.23082
\(425\) 0 0
\(426\) 3.68490e6 0.983788
\(427\) 2.46974e6 0.655514
\(428\) 114599. 0.0302393
\(429\) −85205.3 −0.0223524
\(430\) 0 0
\(431\) −7.38745e6 −1.91558 −0.957792 0.287461i \(-0.907189\pi\)
−0.957792 + 0.287461i \(0.907189\pi\)
\(432\) −765148. −0.197259
\(433\) 1.66687e6 0.427249 0.213625 0.976916i \(-0.431473\pi\)
0.213625 + 0.976916i \(0.431473\pi\)
\(434\) 8.20748e6 2.09163
\(435\) 0 0
\(436\) 118193. 0.0297766
\(437\) −938534. −0.235097
\(438\) 3.38922e6 0.844139
\(439\) 7.00345e6 1.73441 0.867203 0.497956i \(-0.165916\pi\)
0.867203 + 0.497956i \(0.165916\pi\)
\(440\) 0 0
\(441\) 665106. 0.162853
\(442\) −39550.1 −0.00962924
\(443\) −1.09721e6 −0.265633 −0.132816 0.991141i \(-0.542402\pi\)
−0.132816 + 0.991141i \(0.542402\pi\)
\(444\) 28798.1 0.00693276
\(445\) 0 0
\(446\) 4.88389e6 1.16260
\(447\) 967117. 0.228934
\(448\) −5.04332e6 −1.18719
\(449\) −1.62691e6 −0.380844 −0.190422 0.981702i \(-0.560986\pi\)
−0.190422 + 0.981702i \(0.560986\pi\)
\(450\) 0 0
\(451\) −1.14402e6 −0.264846
\(452\) 50929.2 0.0117252
\(453\) 216741. 0.0496245
\(454\) −5.21177e6 −1.18671
\(455\) 0 0
\(456\) −2.99889e6 −0.675380
\(457\) −857428. −0.192047 −0.0960235 0.995379i \(-0.530612\pi\)
−0.0960235 + 0.995379i \(0.530612\pi\)
\(458\) −3.21829e6 −0.716905
\(459\) 64322.6 0.0142505
\(460\) 0 0
\(461\) −5.70696e6 −1.25070 −0.625349 0.780345i \(-0.715043\pi\)
−0.625349 + 0.780345i \(0.715043\pi\)
\(462\) −986800. −0.215092
\(463\) 5.95157e6 1.29027 0.645133 0.764070i \(-0.276802\pi\)
0.645133 + 0.764070i \(0.276802\pi\)
\(464\) −1.10230e6 −0.237686
\(465\) 0 0
\(466\) −3.46402e6 −0.738950
\(467\) −5.25283e6 −1.11455 −0.557277 0.830326i \(-0.688154\pi\)
−0.557277 + 0.830326i \(0.688154\pi\)
\(468\) −5200.54 −0.00109757
\(469\) 2.66992e6 0.560488
\(470\) 0 0
\(471\) −4.31737e6 −0.896740
\(472\) −1.76975e6 −0.365643
\(473\) 1.07918e6 0.221789
\(474\) 5.38709e6 1.10131
\(475\) 0 0
\(476\) −11452.2 −0.00231670
\(477\) 2.06611e6 0.415774
\(478\) −6.51361e6 −1.30392
\(479\) 8.82420e6 1.75726 0.878631 0.477502i \(-0.158458\pi\)
0.878631 + 0.477502i \(0.158458\pi\)
\(480\) 0 0
\(481\) −305095. −0.0601275
\(482\) 1.49754e6 0.293602
\(483\) 716216. 0.139693
\(484\) 12014.2 0.00233121
\(485\) 0 0
\(486\) 338287. 0.0649673
\(487\) 5.99283e6 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(488\) 2.78911e6 0.530170
\(489\) 3.93328e6 0.743846
\(490\) 0 0
\(491\) 6.17263e6 1.15549 0.577745 0.816217i \(-0.303933\pi\)
0.577745 + 0.816217i \(0.303933\pi\)
\(492\) −69825.8 −0.0130048
\(493\) 92665.3 0.0171712
\(494\) −836159. −0.154160
\(495\) 0 0
\(496\) 9.50663e6 1.73509
\(497\) −1.13041e7 −2.05280
\(498\) −797013. −0.144010
\(499\) −2.47210e6 −0.444441 −0.222221 0.974996i \(-0.571331\pi\)
−0.222221 + 0.974996i \(0.571331\pi\)
\(500\) 0 0
\(501\) −5.33333e6 −0.949301
\(502\) 3.23899e6 0.573655
\(503\) −8.18126e6 −1.44178 −0.720892 0.693047i \(-0.756268\pi\)
−0.720892 + 0.693047i \(0.756268\pi\)
\(504\) 2.28852e6 0.401308
\(505\) 0 0
\(506\) −348764. −0.0605558
\(507\) −3.28654e6 −0.567831
\(508\) 234003. 0.0402312
\(509\) 1.19487e6 0.204421 0.102211 0.994763i \(-0.467408\pi\)
0.102211 + 0.994763i \(0.467408\pi\)
\(510\) 0 0
\(511\) −1.03971e7 −1.76141
\(512\) −5.68768e6 −0.958870
\(513\) 1.35989e6 0.228145
\(514\) −8.70486e6 −1.45330
\(515\) 0 0
\(516\) 65867.8 0.0108905
\(517\) 1.14120e6 0.187774
\(518\) −3.53344e6 −0.578593
\(519\) 1.74109e6 0.283728
\(520\) 0 0
\(521\) 1.09929e6 0.177426 0.0887128 0.996057i \(-0.471725\pi\)
0.0887128 + 0.996057i \(0.471725\pi\)
\(522\) 487349. 0.0782823
\(523\) 7.07488e6 1.13101 0.565503 0.824746i \(-0.308682\pi\)
0.565503 + 0.824746i \(0.308682\pi\)
\(524\) −162774. −0.0258974
\(525\) 0 0
\(526\) −4.18424e6 −0.659404
\(527\) −799180. −0.125348
\(528\) −1.14300e6 −0.178427
\(529\) −6.18321e6 −0.960671
\(530\) 0 0
\(531\) 802521. 0.123515
\(532\) −242119. −0.0370894
\(533\) 739755. 0.112790
\(534\) 143060. 0.0217103
\(535\) 0 0
\(536\) 3.01517e6 0.453315
\(537\) −3.36361e6 −0.503350
\(538\) −2.72274e6 −0.405555
\(539\) 993554. 0.147306
\(540\) 0 0
\(541\) −3.53440e6 −0.519186 −0.259593 0.965718i \(-0.583588\pi\)
−0.259593 + 0.965718i \(0.583588\pi\)
\(542\) −5.50729e6 −0.805267
\(543\) −2.11398e6 −0.307682
\(544\) −26206.5 −0.00379675
\(545\) 0 0
\(546\) 638091. 0.0916011
\(547\) 1.37763e6 0.196863 0.0984316 0.995144i \(-0.468617\pi\)
0.0984316 + 0.995144i \(0.468617\pi\)
\(548\) −136487. −0.0194151
\(549\) −1.26476e6 −0.179093
\(550\) 0 0
\(551\) 1.95911e6 0.274903
\(552\) 808829. 0.112982
\(553\) −1.65260e7 −2.29802
\(554\) 2.45347e6 0.339630
\(555\) 0 0
\(556\) −247027. −0.0338889
\(557\) 9.57607e6 1.30782 0.653912 0.756571i \(-0.273127\pi\)
0.653912 + 0.756571i \(0.273127\pi\)
\(558\) −4.20308e6 −0.571455
\(559\) −697823. −0.0944530
\(560\) 0 0
\(561\) 96086.8 0.0128901
\(562\) −1.21884e6 −0.162782
\(563\) 4.58027e6 0.609003 0.304502 0.952512i \(-0.401510\pi\)
0.304502 + 0.952512i \(0.401510\pi\)
\(564\) 69653.5 0.00922031
\(565\) 0 0
\(566\) −3.23033e6 −0.423844
\(567\) −1.03776e6 −0.135563
\(568\) −1.27659e7 −1.66027
\(569\) −943189. −0.122129 −0.0610644 0.998134i \(-0.519450\pi\)
−0.0610644 + 0.998134i \(0.519450\pi\)
\(570\) 0 0
\(571\) −581249. −0.0746057 −0.0373029 0.999304i \(-0.511877\pi\)
−0.0373029 + 0.999304i \(0.511877\pi\)
\(572\) −7768.70 −0.000992792 0
\(573\) 2.35363e6 0.299469
\(574\) 8.56742e6 1.08535
\(575\) 0 0
\(576\) 2.58270e6 0.324353
\(577\) −3.08163e6 −0.385338 −0.192669 0.981264i \(-0.561714\pi\)
−0.192669 + 0.981264i \(0.561714\pi\)
\(578\) −8.08965e6 −1.00719
\(579\) 6.20079e6 0.768689
\(580\) 0 0
\(581\) 2.44499e6 0.300495
\(582\) 6.46571e6 0.791241
\(583\) 3.08641e6 0.376082
\(584\) −1.17415e7 −1.42460
\(585\) 0 0
\(586\) 1.03671e7 1.24713
\(587\) 5.79374e6 0.694007 0.347004 0.937864i \(-0.387199\pi\)
0.347004 + 0.937864i \(0.387199\pi\)
\(588\) 60641.9 0.00723318
\(589\) −1.68961e7 −2.00677
\(590\) 0 0
\(591\) 5.62557e6 0.662518
\(592\) −4.09274e6 −0.479966
\(593\) −1.14174e7 −1.33331 −0.666656 0.745366i \(-0.732275\pi\)
−0.666656 + 0.745366i \(0.732275\pi\)
\(594\) 505343. 0.0587652
\(595\) 0 0
\(596\) 88178.1 0.0101682
\(597\) 9.18168e6 1.05435
\(598\) 225520. 0.0257889
\(599\) 1.33518e7 1.52045 0.760224 0.649661i \(-0.225089\pi\)
0.760224 + 0.649661i \(0.225089\pi\)
\(600\) 0 0
\(601\) 1774.23 0.000200366 0 0.000100183 1.00000i \(-0.499968\pi\)
0.000100183 1.00000i \(0.499968\pi\)
\(602\) −8.08179e6 −0.908901
\(603\) −1.36727e6 −0.153131
\(604\) 19761.7 0.00220410
\(605\) 0 0
\(606\) 2.62879e6 0.290787
\(607\) −2.48398e6 −0.273638 −0.136819 0.990596i \(-0.543688\pi\)
−0.136819 + 0.990596i \(0.543688\pi\)
\(608\) −554051. −0.0607842
\(609\) −1.49504e6 −0.163346
\(610\) 0 0
\(611\) −737930. −0.0799672
\(612\) 5864.69 0.000632946 0
\(613\) 3.19717e6 0.343649 0.171824 0.985128i \(-0.445034\pi\)
0.171824 + 0.985128i \(0.445034\pi\)
\(614\) −4.43447e6 −0.474702
\(615\) 0 0
\(616\) 3.41865e6 0.362997
\(617\) −3.43973e6 −0.363757 −0.181878 0.983321i \(-0.558218\pi\)
−0.181878 + 0.983321i \(0.558218\pi\)
\(618\) −2.65354e6 −0.279483
\(619\) −580297. −0.0608729 −0.0304364 0.999537i \(-0.509690\pi\)
−0.0304364 + 0.999537i \(0.509690\pi\)
\(620\) 0 0
\(621\) −366776. −0.0381656
\(622\) 7.78035e6 0.806349
\(623\) −438865. −0.0453013
\(624\) 739093. 0.0759867
\(625\) 0 0
\(626\) 1.93818e7 1.97678
\(627\) 2.03144e6 0.206365
\(628\) −393641. −0.0398292
\(629\) 344059. 0.0346742
\(630\) 0 0
\(631\) −1.94969e7 −1.94936 −0.974680 0.223604i \(-0.928218\pi\)
−0.974680 + 0.223604i \(0.928218\pi\)
\(632\) −1.86629e7 −1.85861
\(633\) −4.89890e6 −0.485947
\(634\) 3.82772e6 0.378196
\(635\) 0 0
\(636\) 188380. 0.0184668
\(637\) −642458. −0.0627330
\(638\) 728015. 0.0708090
\(639\) 5.78888e6 0.560845
\(640\) 0 0
\(641\) −1.50533e7 −1.44706 −0.723530 0.690293i \(-0.757482\pi\)
−0.723530 + 0.690293i \(0.757482\pi\)
\(642\) 7.20065e6 0.689500
\(643\) 1.34527e7 1.28316 0.641580 0.767056i \(-0.278279\pi\)
0.641580 + 0.767056i \(0.278279\pi\)
\(644\) 65301.9 0.00620456
\(645\) 0 0
\(646\) 942943. 0.0889005
\(647\) 1.15666e7 1.08629 0.543146 0.839638i \(-0.317233\pi\)
0.543146 + 0.839638i \(0.317233\pi\)
\(648\) −1.17196e6 −0.109641
\(649\) 1.19883e6 0.111724
\(650\) 0 0
\(651\) 1.28938e7 1.19241
\(652\) 358622. 0.0330383
\(653\) −8.60996e6 −0.790165 −0.395083 0.918646i \(-0.629284\pi\)
−0.395083 + 0.918646i \(0.629284\pi\)
\(654\) 7.42646e6 0.678950
\(655\) 0 0
\(656\) 9.92354e6 0.900341
\(657\) 5.32438e6 0.481233
\(658\) −8.54628e6 −0.769507
\(659\) −3.14337e6 −0.281957 −0.140978 0.990013i \(-0.545025\pi\)
−0.140978 + 0.990013i \(0.545025\pi\)
\(660\) 0 0
\(661\) −2.75239e6 −0.245023 −0.122511 0.992467i \(-0.539095\pi\)
−0.122511 + 0.992467i \(0.539095\pi\)
\(662\) 543584. 0.0482083
\(663\) −62132.2 −0.00548951
\(664\) 2.76116e6 0.243036
\(665\) 0 0
\(666\) 1.80948e6 0.158077
\(667\) −528390. −0.0459876
\(668\) −486273. −0.0421637
\(669\) 7.67247e6 0.662781
\(670\) 0 0
\(671\) −1.88934e6 −0.161996
\(672\) 422808. 0.0361177
\(673\) −1.66398e7 −1.41615 −0.708076 0.706136i \(-0.750437\pi\)
−0.708076 + 0.706136i \(0.750437\pi\)
\(674\) 2.90639e6 0.246436
\(675\) 0 0
\(676\) −299655. −0.0252205
\(677\) 1.05991e7 0.888789 0.444395 0.895831i \(-0.353419\pi\)
0.444395 + 0.895831i \(0.353419\pi\)
\(678\) 3.20006e6 0.267352
\(679\) −1.98348e7 −1.65103
\(680\) 0 0
\(681\) −8.18756e6 −0.676529
\(682\) −6.27867e6 −0.516900
\(683\) 544094. 0.0446296 0.0223148 0.999751i \(-0.492896\pi\)
0.0223148 + 0.999751i \(0.492896\pi\)
\(684\) 123990. 0.0101332
\(685\) 0 0
\(686\) 7.78911e6 0.631943
\(687\) −5.05586e6 −0.408698
\(688\) −9.36104e6 −0.753969
\(689\) −1.99575e6 −0.160162
\(690\) 0 0
\(691\) −7.88699e6 −0.628371 −0.314185 0.949362i \(-0.601731\pi\)
−0.314185 + 0.949362i \(0.601731\pi\)
\(692\) 158746. 0.0126020
\(693\) −1.55024e6 −0.122621
\(694\) 1.67210e7 1.31784
\(695\) 0 0
\(696\) −1.68836e6 −0.132112
\(697\) −834228. −0.0650433
\(698\) −9.71616e6 −0.754842
\(699\) −5.44188e6 −0.421266
\(700\) 0 0
\(701\) 2.22952e7 1.71363 0.856814 0.515625i \(-0.172440\pi\)
0.856814 + 0.515625i \(0.172440\pi\)
\(702\) −326768. −0.0250263
\(703\) 7.27400e6 0.555118
\(704\) 3.85811e6 0.293388
\(705\) 0 0
\(706\) −1.45898e7 −1.10163
\(707\) −8.06434e6 −0.606765
\(708\) 73170.9 0.00548599
\(709\) −8.40301e6 −0.627797 −0.313899 0.949457i \(-0.601635\pi\)
−0.313899 + 0.949457i \(0.601635\pi\)
\(710\) 0 0
\(711\) 8.46299e6 0.627841
\(712\) −495615. −0.0366390
\(713\) 4.55703e6 0.335706
\(714\) −719580. −0.0528243
\(715\) 0 0
\(716\) −306682. −0.0223566
\(717\) −1.02327e7 −0.743350
\(718\) −2.43695e6 −0.176415
\(719\) 1.39897e7 1.00922 0.504612 0.863346i \(-0.331636\pi\)
0.504612 + 0.863346i \(0.331636\pi\)
\(720\) 0 0
\(721\) 8.14027e6 0.583177
\(722\) 5.75009e6 0.410517
\(723\) 2.35259e6 0.167379
\(724\) −192745. −0.0136659
\(725\) 0 0
\(726\) 754895. 0.0531551
\(727\) −1.69990e7 −1.19286 −0.596428 0.802666i \(-0.703414\pi\)
−0.596428 + 0.802666i \(0.703414\pi\)
\(728\) −2.21059e6 −0.154589
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 786941. 0.0544689
\(732\) −115316. −0.00795450
\(733\) −9.14170e6 −0.628445 −0.314222 0.949349i \(-0.601744\pi\)
−0.314222 + 0.949349i \(0.601744\pi\)
\(734\) −1.08060e7 −0.740332
\(735\) 0 0
\(736\) 149433. 0.0101684
\(737\) −2.04247e6 −0.138512
\(738\) −4.38740e6 −0.296528
\(739\) −1.54222e7 −1.03881 −0.519405 0.854528i \(-0.673846\pi\)
−0.519405 + 0.854528i \(0.673846\pi\)
\(740\) 0 0
\(741\) −1.31358e6 −0.0878845
\(742\) −2.31137e7 −1.54120
\(743\) 2.34940e7 1.56129 0.780646 0.624973i \(-0.214890\pi\)
0.780646 + 0.624973i \(0.214890\pi\)
\(744\) 1.45610e7 0.964407
\(745\) 0 0
\(746\) 5.17484e6 0.340447
\(747\) −1.25209e6 −0.0820981
\(748\) 8760.83 0.000572521 0
\(749\) −2.20894e7 −1.43873
\(750\) 0 0
\(751\) 9.67937e6 0.626250 0.313125 0.949712i \(-0.398624\pi\)
0.313125 + 0.949712i \(0.398624\pi\)
\(752\) −9.89906e6 −0.638336
\(753\) 5.08838e6 0.327033
\(754\) −470754. −0.0301554
\(755\) 0 0
\(756\) −94619.3 −0.00602109
\(757\) 6.06490e6 0.384666 0.192333 0.981330i \(-0.438395\pi\)
0.192333 + 0.981330i \(0.438395\pi\)
\(758\) 2.83485e7 1.79208
\(759\) −547900. −0.0345221
\(760\) 0 0
\(761\) −2.42020e7 −1.51492 −0.757461 0.652880i \(-0.773561\pi\)
−0.757461 + 0.652880i \(0.773561\pi\)
\(762\) 1.47033e7 0.917331
\(763\) −2.27821e7 −1.41672
\(764\) 214595. 0.0133011
\(765\) 0 0
\(766\) −9.33665e6 −0.574936
\(767\) −775193. −0.0475797
\(768\) 725529. 0.0443866
\(769\) 3.47840e6 0.212111 0.106055 0.994360i \(-0.466178\pi\)
0.106055 + 0.994360i \(0.466178\pi\)
\(770\) 0 0
\(771\) −1.36751e7 −0.828506
\(772\) 565365. 0.0341418
\(773\) 1.02630e7 0.617769 0.308884 0.951100i \(-0.400044\pi\)
0.308884 + 0.951100i \(0.400044\pi\)
\(774\) 4.13871e6 0.248320
\(775\) 0 0
\(776\) −2.23997e7 −1.33533
\(777\) −5.55095e6 −0.329849
\(778\) 3.05672e7 1.81053
\(779\) −1.76370e7 −1.04131
\(780\) 0 0
\(781\) 8.64759e6 0.507303
\(782\) −254321. −0.0148719
\(783\) 765613. 0.0446277
\(784\) −8.61834e6 −0.500764
\(785\) 0 0
\(786\) −1.02277e7 −0.590500
\(787\) −2.49651e7 −1.43680 −0.718401 0.695629i \(-0.755126\pi\)
−0.718401 + 0.695629i \(0.755126\pi\)
\(788\) 512919. 0.0294261
\(789\) −6.57333e6 −0.375918
\(790\) 0 0
\(791\) −9.81681e6 −0.557865
\(792\) −1.75070e6 −0.0991742
\(793\) 1.22169e6 0.0689890
\(794\) 2.23050e7 1.25560
\(795\) 0 0
\(796\) 837151. 0.0468297
\(797\) 2.85688e7 1.59311 0.796557 0.604564i \(-0.206653\pi\)
0.796557 + 0.604564i \(0.206653\pi\)
\(798\) −1.52132e7 −0.845694
\(799\) 832170. 0.0461153
\(800\) 0 0
\(801\) 224744. 0.0123768
\(802\) 2.29992e6 0.126263
\(803\) 7.95370e6 0.435292
\(804\) −124663. −0.00680138
\(805\) 0 0
\(806\) 4.05995e6 0.220132
\(807\) −4.27735e6 −0.231202
\(808\) −9.10714e6 −0.490743
\(809\) 1.00028e7 0.537340 0.268670 0.963232i \(-0.413416\pi\)
0.268670 + 0.963232i \(0.413416\pi\)
\(810\) 0 0
\(811\) −7.88288e6 −0.420855 −0.210428 0.977609i \(-0.567486\pi\)
−0.210428 + 0.977609i \(0.567486\pi\)
\(812\) −136312. −0.00725511
\(813\) −8.65181e6 −0.459072
\(814\) 2.70306e6 0.142986
\(815\) 0 0
\(816\) −833482. −0.0438198
\(817\) 1.66373e7 0.872023
\(818\) −3.39430e7 −1.77364
\(819\) 1.00242e6 0.0522206
\(820\) 0 0
\(821\) 1.84331e7 0.954423 0.477211 0.878789i \(-0.341648\pi\)
0.477211 + 0.878789i \(0.341648\pi\)
\(822\) −8.57594e6 −0.442693
\(823\) −1.93527e7 −0.995961 −0.497981 0.867188i \(-0.665925\pi\)
−0.497981 + 0.867188i \(0.665925\pi\)
\(824\) 9.19288e6 0.471665
\(825\) 0 0
\(826\) −8.97785e6 −0.457849
\(827\) −1.33049e7 −0.676471 −0.338235 0.941062i \(-0.609830\pi\)
−0.338235 + 0.941062i \(0.609830\pi\)
\(828\) −33441.3 −0.00169515
\(829\) −1.61482e7 −0.816090 −0.408045 0.912962i \(-0.633789\pi\)
−0.408045 + 0.912962i \(0.633789\pi\)
\(830\) 0 0
\(831\) 3.85434e6 0.193619
\(832\) −2.49475e6 −0.124945
\(833\) 724505. 0.0361767
\(834\) −1.55216e7 −0.772717
\(835\) 0 0
\(836\) 185219. 0.00916581
\(837\) −6.60293e6 −0.325779
\(838\) 2.42842e7 1.19458
\(839\) 2.17566e7 1.06705 0.533527 0.845783i \(-0.320866\pi\)
0.533527 + 0.845783i \(0.320866\pi\)
\(840\) 0 0
\(841\) −1.94082e7 −0.946226
\(842\) 2.89072e7 1.40516
\(843\) −1.91477e6 −0.0927998
\(844\) −446663. −0.0215836
\(845\) 0 0
\(846\) 4.37658e6 0.210237
\(847\) −2.31579e6 −0.110915
\(848\) −2.67723e7 −1.27849
\(849\) −5.07478e6 −0.241628
\(850\) 0 0
\(851\) −1.96187e6 −0.0928638
\(852\) 527809. 0.0249102
\(853\) −1.29447e7 −0.609142 −0.304571 0.952490i \(-0.598513\pi\)
−0.304571 + 0.952490i \(0.598513\pi\)
\(854\) 1.41490e7 0.663866
\(855\) 0 0
\(856\) −2.49458e7 −1.16362
\(857\) 3.34203e7 1.55438 0.777192 0.629264i \(-0.216644\pi\)
0.777192 + 0.629264i \(0.216644\pi\)
\(858\) −488135. −0.0226371
\(859\) −4.22247e7 −1.95247 −0.976234 0.216719i \(-0.930464\pi\)
−0.976234 + 0.216719i \(0.930464\pi\)
\(860\) 0 0
\(861\) 1.34592e7 0.618745
\(862\) −4.23221e7 −1.93999
\(863\) −9.62825e6 −0.440069 −0.220034 0.975492i \(-0.570617\pi\)
−0.220034 + 0.975492i \(0.570617\pi\)
\(864\) −216521. −0.00986771
\(865\) 0 0
\(866\) 9.54936e6 0.432693
\(867\) −1.27086e7 −0.574185
\(868\) 1.17560e6 0.0529617
\(869\) 1.26422e7 0.567904
\(870\) 0 0
\(871\) 1.32072e6 0.0589880
\(872\) −2.57281e7 −1.14582
\(873\) 1.01575e7 0.451076
\(874\) −5.37679e6 −0.238092
\(875\) 0 0
\(876\) 485457. 0.0213742
\(877\) 1.37201e7 0.602362 0.301181 0.953567i \(-0.402619\pi\)
0.301181 + 0.953567i \(0.402619\pi\)
\(878\) 4.01222e7 1.75650
\(879\) 1.62865e7 0.710976
\(880\) 0 0
\(881\) 2.99671e7 1.30078 0.650392 0.759599i \(-0.274605\pi\)
0.650392 + 0.759599i \(0.274605\pi\)
\(882\) 3.81034e6 0.164927
\(883\) −6.57264e6 −0.283686 −0.141843 0.989889i \(-0.545303\pi\)
−0.141843 + 0.989889i \(0.545303\pi\)
\(884\) −5664.99 −0.000243819 0
\(885\) 0 0
\(886\) −6.28585e6 −0.269017
\(887\) 1.99178e7 0.850028 0.425014 0.905187i \(-0.360269\pi\)
0.425014 + 0.905187i \(0.360269\pi\)
\(888\) −6.26874e6 −0.266777
\(889\) −4.51051e7 −1.91413
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 699548. 0.0294378
\(893\) 1.75935e7 0.738285
\(894\) 5.54054e6 0.231851
\(895\) 0 0
\(896\) −3.03961e7 −1.26488
\(897\) 354287. 0.0147019
\(898\) −9.32044e6 −0.385696
\(899\) −9.51241e6 −0.392547
\(900\) 0 0
\(901\) 2.25063e6 0.0923617
\(902\) −6.55402e6 −0.268220
\(903\) −1.26963e7 −0.518152
\(904\) −1.10862e7 −0.451193
\(905\) 0 0
\(906\) 1.24170e6 0.0502568
\(907\) 4.17948e7 1.68696 0.843478 0.537163i \(-0.180504\pi\)
0.843478 + 0.537163i \(0.180504\pi\)
\(908\) −746511. −0.0300484
\(909\) 4.12977e6 0.165774
\(910\) 0 0
\(911\) −2.46957e7 −0.985882 −0.492941 0.870063i \(-0.664078\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(912\) −1.76213e7 −0.701536
\(913\) −1.87040e6 −0.0742605
\(914\) −4.91214e6 −0.194494
\(915\) 0 0
\(916\) −460974. −0.0181526
\(917\) 3.13753e7 1.23215
\(918\) 368499. 0.0144321
\(919\) −2.76370e7 −1.07945 −0.539724 0.841842i \(-0.681471\pi\)
−0.539724 + 0.841842i \(0.681471\pi\)
\(920\) 0 0
\(921\) −6.96644e6 −0.270621
\(922\) −3.26947e7 −1.26663
\(923\) −5.59176e6 −0.216045
\(924\) −141345. −0.00544628
\(925\) 0 0
\(926\) 3.40961e7 1.30670
\(927\) −4.16865e6 −0.159330
\(928\) −311928. −0.0118901
\(929\) −2.99531e7 −1.13868 −0.569340 0.822102i \(-0.692801\pi\)
−0.569340 + 0.822102i \(0.692801\pi\)
\(930\) 0 0
\(931\) 1.53173e7 0.579173
\(932\) −496171. −0.0187108
\(933\) 1.22227e7 0.459689
\(934\) −3.00931e7 −1.12875
\(935\) 0 0
\(936\) 1.13205e6 0.0422353
\(937\) 3.96082e7 1.47379 0.736896 0.676006i \(-0.236291\pi\)
0.736896 + 0.676006i \(0.236291\pi\)
\(938\) 1.52958e7 0.567629
\(939\) 3.04483e7 1.12694
\(940\) 0 0
\(941\) −3.50065e6 −0.128877 −0.0644383 0.997922i \(-0.520526\pi\)
−0.0644383 + 0.997922i \(0.520526\pi\)
\(942\) −2.47339e7 −0.908165
\(943\) 4.75688e6 0.174198
\(944\) −1.03989e7 −0.379804
\(945\) 0 0
\(946\) 6.18251e6 0.224614
\(947\) 2.56533e7 0.929540 0.464770 0.885432i \(-0.346137\pi\)
0.464770 + 0.885432i \(0.346137\pi\)
\(948\) 771624. 0.0278859
\(949\) −5.14307e6 −0.185377
\(950\) 0 0
\(951\) 6.01325e6 0.215604
\(952\) 2.49290e6 0.0891482
\(953\) −1.43726e7 −0.512629 −0.256314 0.966593i \(-0.582508\pi\)
−0.256314 + 0.966593i \(0.582508\pi\)
\(954\) 1.18366e7 0.421071
\(955\) 0 0
\(956\) −932982. −0.0330163
\(957\) 1.14369e6 0.0403673
\(958\) 5.05532e7 1.77965
\(959\) 2.63084e7 0.923735
\(960\) 0 0
\(961\) 5.34094e7 1.86556
\(962\) −1.74787e6 −0.0608935
\(963\) 1.13120e7 0.393075
\(964\) 214501. 0.00743423
\(965\) 0 0
\(966\) 4.10315e6 0.141473
\(967\) 3.41494e7 1.17440 0.587201 0.809441i \(-0.300230\pi\)
0.587201 + 0.809441i \(0.300230\pi\)
\(968\) −2.61524e6 −0.0897064
\(969\) 1.48134e6 0.0506810
\(970\) 0 0
\(971\) 8.76612e6 0.298373 0.149186 0.988809i \(-0.452335\pi\)
0.149186 + 0.988809i \(0.452335\pi\)
\(972\) 48454.8 0.00164502
\(973\) 4.76154e7 1.61237
\(974\) 3.43325e7 1.15960
\(975\) 0 0
\(976\) 1.63886e7 0.550703
\(977\) 4.92659e7 1.65124 0.825620 0.564227i \(-0.190826\pi\)
0.825620 + 0.564227i \(0.190826\pi\)
\(978\) 2.25335e7 0.753323
\(979\) 335729. 0.0111952
\(980\) 0 0
\(981\) 1.16668e7 0.387061
\(982\) 3.53625e7 1.17021
\(983\) 5.54306e7 1.82964 0.914820 0.403861i \(-0.132332\pi\)
0.914820 + 0.403861i \(0.132332\pi\)
\(984\) 1.51996e7 0.500432
\(985\) 0 0
\(986\) 530873. 0.0173899
\(987\) −1.34260e7 −0.438686
\(988\) −119768. −0.00390344
\(989\) −4.48725e6 −0.145878
\(990\) 0 0
\(991\) −1.51099e7 −0.488739 −0.244370 0.969682i \(-0.578581\pi\)
−0.244370 + 0.969682i \(0.578581\pi\)
\(992\) 2.69018e6 0.0867966
\(993\) 853956. 0.0274829
\(994\) −6.47606e7 −2.07895
\(995\) 0 0
\(996\) −114161. −0.00364643
\(997\) −3.98499e7 −1.26966 −0.634832 0.772650i \(-0.718931\pi\)
−0.634832 + 0.772650i \(0.718931\pi\)
\(998\) −1.41625e7 −0.450104
\(999\) 2.84266e6 0.0901178
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.1.10 13
5.2 odd 4 165.6.c.b.34.20 yes 26
5.3 odd 4 165.6.c.b.34.7 26
5.4 even 2 825.6.a.v.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.7 26 5.3 odd 4
165.6.c.b.34.20 yes 26 5.2 odd 4
825.6.a.v.1.4 13 5.4 even 2
825.6.a.y.1.10 13 1.1 even 1 trivial