Properties

Label 825.4.a.r.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.47528.1
Defining polynomial: \( x^{3} - x^{2} - 26x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.97123\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.97123 q^{2} -3.00000 q^{3} +16.7131 q^{4} +14.9137 q^{6} -5.48376 q^{7} -43.3148 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.97123 q^{2} -3.00000 q^{3} +16.7131 q^{4} +14.9137 q^{6} -5.48376 q^{7} -43.3148 q^{8} +9.00000 q^{9} +11.0000 q^{11} -50.1393 q^{12} -24.5738 q^{13} +27.2610 q^{14} +81.6231 q^{16} +59.3687 q^{17} -44.7411 q^{18} +5.89234 q^{19} +16.4513 q^{21} -54.6835 q^{22} +68.4570 q^{23} +129.945 q^{24} +122.162 q^{26} -27.0000 q^{27} -91.6506 q^{28} -265.022 q^{29} -196.554 q^{31} -59.2482 q^{32} -33.0000 q^{33} -295.135 q^{34} +150.418 q^{36} -166.575 q^{37} -29.2922 q^{38} +73.7214 q^{39} +424.101 q^{41} -81.7830 q^{42} +177.351 q^{43} +183.844 q^{44} -340.316 q^{46} -141.148 q^{47} -244.869 q^{48} -312.928 q^{49} -178.106 q^{51} -410.704 q^{52} +339.828 q^{53} +134.223 q^{54} +237.528 q^{56} -17.6770 q^{57} +1317.48 q^{58} +416.784 q^{59} +662.146 q^{61} +977.115 q^{62} -49.3538 q^{63} -358.448 q^{64} +164.051 q^{66} -313.713 q^{67} +992.235 q^{68} -205.371 q^{69} -153.693 q^{71} -389.834 q^{72} -153.866 q^{73} +828.084 q^{74} +98.4793 q^{76} -60.3213 q^{77} -366.486 q^{78} +403.000 q^{79} +81.0000 q^{81} -2108.30 q^{82} +652.313 q^{83} +274.952 q^{84} -881.650 q^{86} +795.065 q^{87} -476.463 q^{88} +1226.77 q^{89} +134.757 q^{91} +1144.13 q^{92} +589.662 q^{93} +701.677 q^{94} +177.745 q^{96} -959.746 q^{97} +1555.64 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 9 q^{3} + 30 q^{4} - 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 9 q^{3} + 30 q^{4} - 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} + 33 q^{11} - 90 q^{12} - 114 q^{13} - 68 q^{14} + 178 q^{16} + 104 q^{17} + 18 q^{18} - 58 q^{19} + 30 q^{21} + 22 q^{22} - 120 q^{23} + 54 q^{24} - 120 q^{26} - 81 q^{27} - 676 q^{28} - 220 q^{29} + 248 q^{31} + 258 q^{32} - 99 q^{33} - 80 q^{34} + 270 q^{36} - 838 q^{37} - 600 q^{38} + 342 q^{39} + 156 q^{41} + 204 q^{42} - 122 q^{43} + 330 q^{44} - 1256 q^{46} - 504 q^{47} - 534 q^{48} + 279 q^{49} - 312 q^{51} - 520 q^{52} - 282 q^{53} - 54 q^{54} - 1644 q^{56} + 174 q^{57} + 1644 q^{58} + 548 q^{59} + 414 q^{61} + 2448 q^{62} - 90 q^{63} - 58 q^{64} - 66 q^{66} + 428 q^{67} + 1704 q^{68} + 360 q^{69} - 912 q^{71} - 162 q^{72} - 618 q^{73} - 1612 q^{74} - 2752 q^{76} - 110 q^{77} + 360 q^{78} - 542 q^{79} + 243 q^{81} - 3372 q^{82} + 2028 q^{84} - 1548 q^{86} + 660 q^{87} - 198 q^{88} + 790 q^{89} - 772 q^{91} - 1912 q^{92} - 744 q^{93} - 424 q^{94} - 774 q^{96} - 2074 q^{97} + 3978 q^{98} + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.97123 −1.75759 −0.878797 0.477195i \(-0.841653\pi\)
−0.878797 + 0.477195i \(0.841653\pi\)
\(3\) −3.00000 −0.577350
\(4\) 16.7131 2.08914
\(5\) 0 0
\(6\) 14.9137 1.01475
\(7\) −5.48376 −0.296095 −0.148048 0.988980i \(-0.547299\pi\)
−0.148048 + 0.988980i \(0.547299\pi\)
\(8\) −43.3148 −1.91426
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) −50.1393 −1.20616
\(13\) −24.5738 −0.524272 −0.262136 0.965031i \(-0.584427\pi\)
−0.262136 + 0.965031i \(0.584427\pi\)
\(14\) 27.2610 0.520415
\(15\) 0 0
\(16\) 81.6231 1.27536
\(17\) 59.3687 0.847001 0.423501 0.905896i \(-0.360801\pi\)
0.423501 + 0.905896i \(0.360801\pi\)
\(18\) −44.7411 −0.585865
\(19\) 5.89234 0.0711471 0.0355736 0.999367i \(-0.488674\pi\)
0.0355736 + 0.999367i \(0.488674\pi\)
\(20\) 0 0
\(21\) 16.4513 0.170951
\(22\) −54.6835 −0.529935
\(23\) 68.4570 0.620621 0.310310 0.950635i \(-0.399567\pi\)
0.310310 + 0.950635i \(0.399567\pi\)
\(24\) 129.945 1.10520
\(25\) 0 0
\(26\) 122.162 0.921458
\(27\) −27.0000 −0.192450
\(28\) −91.6506 −0.618584
\(29\) −265.022 −1.69701 −0.848505 0.529187i \(-0.822497\pi\)
−0.848505 + 0.529187i \(0.822497\pi\)
\(30\) 0 0
\(31\) −196.554 −1.13878 −0.569389 0.822068i \(-0.692820\pi\)
−0.569389 + 0.822068i \(0.692820\pi\)
\(32\) −59.2482 −0.327303
\(33\) −33.0000 −0.174078
\(34\) −295.135 −1.48868
\(35\) 0 0
\(36\) 150.418 0.696379
\(37\) −166.575 −0.740131 −0.370065 0.929006i \(-0.620665\pi\)
−0.370065 + 0.929006i \(0.620665\pi\)
\(38\) −29.2922 −0.125048
\(39\) 73.7214 0.302689
\(40\) 0 0
\(41\) 424.101 1.61545 0.807725 0.589559i \(-0.200699\pi\)
0.807725 + 0.589559i \(0.200699\pi\)
\(42\) −81.7830 −0.300462
\(43\) 177.351 0.628970 0.314485 0.949262i \(-0.398168\pi\)
0.314485 + 0.949262i \(0.398168\pi\)
\(44\) 183.844 0.629899
\(45\) 0 0
\(46\) −340.316 −1.09080
\(47\) −141.148 −0.438053 −0.219026 0.975719i \(-0.570288\pi\)
−0.219026 + 0.975719i \(0.570288\pi\)
\(48\) −244.869 −0.736330
\(49\) −312.928 −0.912328
\(50\) 0 0
\(51\) −178.106 −0.489016
\(52\) −410.704 −1.09528
\(53\) 339.828 0.880736 0.440368 0.897817i \(-0.354848\pi\)
0.440368 + 0.897817i \(0.354848\pi\)
\(54\) 134.223 0.338249
\(55\) 0 0
\(56\) 237.528 0.566804
\(57\) −17.6770 −0.0410768
\(58\) 1317.48 2.98266
\(59\) 416.784 0.919672 0.459836 0.888004i \(-0.347908\pi\)
0.459836 + 0.888004i \(0.347908\pi\)
\(60\) 0 0
\(61\) 662.146 1.38982 0.694911 0.719096i \(-0.255444\pi\)
0.694911 + 0.719096i \(0.255444\pi\)
\(62\) 977.115 2.00151
\(63\) −49.3538 −0.0986984
\(64\) −358.448 −0.700094
\(65\) 0 0
\(66\) 164.051 0.305958
\(67\) −313.713 −0.572032 −0.286016 0.958225i \(-0.592331\pi\)
−0.286016 + 0.958225i \(0.592331\pi\)
\(68\) 992.235 1.76950
\(69\) −205.371 −0.358316
\(70\) 0 0
\(71\) −153.693 −0.256902 −0.128451 0.991716i \(-0.541000\pi\)
−0.128451 + 0.991716i \(0.541000\pi\)
\(72\) −389.834 −0.638088
\(73\) −153.866 −0.246694 −0.123347 0.992364i \(-0.539363\pi\)
−0.123347 + 0.992364i \(0.539363\pi\)
\(74\) 828.084 1.30085
\(75\) 0 0
\(76\) 98.4793 0.148636
\(77\) −60.3213 −0.0892760
\(78\) −366.486 −0.532004
\(79\) 403.000 0.573938 0.286969 0.957940i \(-0.407352\pi\)
0.286969 + 0.957940i \(0.407352\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −2108.30 −2.83931
\(83\) 652.313 0.862659 0.431329 0.902195i \(-0.358045\pi\)
0.431329 + 0.902195i \(0.358045\pi\)
\(84\) 274.952 0.357139
\(85\) 0 0
\(86\) −881.650 −1.10547
\(87\) 795.065 0.979769
\(88\) −476.463 −0.577172
\(89\) 1226.77 1.46109 0.730545 0.682864i \(-0.239266\pi\)
0.730545 + 0.682864i \(0.239266\pi\)
\(90\) 0 0
\(91\) 134.757 0.155234
\(92\) 1144.13 1.29656
\(93\) 589.662 0.657474
\(94\) 701.677 0.769919
\(95\) 0 0
\(96\) 177.745 0.188969
\(97\) −959.746 −1.00461 −0.502306 0.864690i \(-0.667515\pi\)
−0.502306 + 0.864690i \(0.667515\pi\)
\(98\) 1555.64 1.60350
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −1258.52 −1.23988 −0.619939 0.784650i \(-0.712843\pi\)
−0.619939 + 0.784650i \(0.712843\pi\)
\(102\) 885.406 0.859492
\(103\) 493.175 0.471786 0.235893 0.971779i \(-0.424198\pi\)
0.235893 + 0.971779i \(0.424198\pi\)
\(104\) 1064.41 1.00360
\(105\) 0 0
\(106\) −1689.36 −1.54798
\(107\) 1099.05 0.992983 0.496492 0.868042i \(-0.334621\pi\)
0.496492 + 0.868042i \(0.334621\pi\)
\(108\) −451.254 −0.402055
\(109\) 1275.04 1.12043 0.560216 0.828347i \(-0.310718\pi\)
0.560216 + 0.828347i \(0.310718\pi\)
\(110\) 0 0
\(111\) 499.726 0.427315
\(112\) −447.601 −0.377628
\(113\) 946.433 0.787902 0.393951 0.919131i \(-0.371108\pi\)
0.393951 + 0.919131i \(0.371108\pi\)
\(114\) 87.8765 0.0721964
\(115\) 0 0
\(116\) −4429.34 −3.54529
\(117\) −221.164 −0.174757
\(118\) −2071.93 −1.61641
\(119\) −325.563 −0.250793
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −3291.68 −2.44274
\(123\) −1272.30 −0.932681
\(124\) −3285.03 −2.37907
\(125\) 0 0
\(126\) 245.349 0.173472
\(127\) −2260.77 −1.57962 −0.789808 0.613354i \(-0.789820\pi\)
−0.789808 + 0.613354i \(0.789820\pi\)
\(128\) 2255.91 1.55778
\(129\) −532.052 −0.363136
\(130\) 0 0
\(131\) −57.1277 −0.0381013 −0.0190507 0.999819i \(-0.506064\pi\)
−0.0190507 + 0.999819i \(0.506064\pi\)
\(132\) −551.533 −0.363672
\(133\) −32.3122 −0.0210663
\(134\) 1559.54 1.00540
\(135\) 0 0
\(136\) −2571.54 −1.62138
\(137\) −1919.27 −1.19689 −0.598447 0.801162i \(-0.704215\pi\)
−0.598447 + 0.801162i \(0.704215\pi\)
\(138\) 1020.95 0.629774
\(139\) −1690.46 −1.03153 −0.515766 0.856729i \(-0.672493\pi\)
−0.515766 + 0.856729i \(0.672493\pi\)
\(140\) 0 0
\(141\) 423.443 0.252910
\(142\) 764.044 0.451529
\(143\) −270.312 −0.158074
\(144\) 734.608 0.425120
\(145\) 0 0
\(146\) 764.904 0.433588
\(147\) 938.785 0.526733
\(148\) −2783.99 −1.54624
\(149\) −2077.88 −1.14246 −0.571231 0.820789i \(-0.693534\pi\)
−0.571231 + 0.820789i \(0.693534\pi\)
\(150\) 0 0
\(151\) 2711.16 1.46113 0.730566 0.682842i \(-0.239256\pi\)
0.730566 + 0.682842i \(0.239256\pi\)
\(152\) −255.226 −0.136194
\(153\) 534.318 0.282334
\(154\) 299.871 0.156911
\(155\) 0 0
\(156\) 1232.11 0.632359
\(157\) −2341.23 −1.19013 −0.595066 0.803677i \(-0.702874\pi\)
−0.595066 + 0.803677i \(0.702874\pi\)
\(158\) −2003.41 −1.00875
\(159\) −1019.48 −0.508493
\(160\) 0 0
\(161\) −375.402 −0.183763
\(162\) −402.669 −0.195288
\(163\) −3523.01 −1.69291 −0.846453 0.532464i \(-0.821266\pi\)
−0.846453 + 0.532464i \(0.821266\pi\)
\(164\) 7088.05 3.37490
\(165\) 0 0
\(166\) −3242.80 −1.51620
\(167\) −4011.90 −1.85898 −0.929492 0.368842i \(-0.879754\pi\)
−0.929492 + 0.368842i \(0.879754\pi\)
\(168\) −712.584 −0.327244
\(169\) −1593.13 −0.725138
\(170\) 0 0
\(171\) 53.0310 0.0237157
\(172\) 2964.08 1.31400
\(173\) −3325.30 −1.46137 −0.730687 0.682713i \(-0.760800\pi\)
−0.730687 + 0.682713i \(0.760800\pi\)
\(174\) −3952.45 −1.72204
\(175\) 0 0
\(176\) 897.854 0.384536
\(177\) −1250.35 −0.530973
\(178\) −6098.54 −2.56800
\(179\) −26.6350 −0.0111217 −0.00556087 0.999985i \(-0.501770\pi\)
−0.00556087 + 0.999985i \(0.501770\pi\)
\(180\) 0 0
\(181\) 1934.25 0.794320 0.397160 0.917749i \(-0.369996\pi\)
0.397160 + 0.917749i \(0.369996\pi\)
\(182\) −669.906 −0.272839
\(183\) −1986.44 −0.802414
\(184\) −2965.21 −1.18803
\(185\) 0 0
\(186\) −2931.34 −1.15557
\(187\) 653.055 0.255380
\(188\) −2359.01 −0.915153
\(189\) 148.061 0.0569835
\(190\) 0 0
\(191\) −3341.59 −1.26591 −0.632956 0.774188i \(-0.718158\pi\)
−0.632956 + 0.774188i \(0.718158\pi\)
\(192\) 1075.34 0.404200
\(193\) 1293.97 0.482600 0.241300 0.970451i \(-0.422426\pi\)
0.241300 + 0.970451i \(0.422426\pi\)
\(194\) 4771.12 1.76570
\(195\) 0 0
\(196\) −5230.01 −1.90598
\(197\) 2301.17 0.832240 0.416120 0.909310i \(-0.363390\pi\)
0.416120 + 0.909310i \(0.363390\pi\)
\(198\) −492.152 −0.176645
\(199\) −3039.64 −1.08279 −0.541393 0.840770i \(-0.682103\pi\)
−0.541393 + 0.840770i \(0.682103\pi\)
\(200\) 0 0
\(201\) 941.139 0.330263
\(202\) 6256.40 2.17920
\(203\) 1453.31 0.502476
\(204\) −2976.70 −1.02162
\(205\) 0 0
\(206\) −2451.69 −0.829209
\(207\) 616.113 0.206874
\(208\) −2005.79 −0.668636
\(209\) 64.8157 0.0214517
\(210\) 0 0
\(211\) 2807.86 0.916119 0.458060 0.888921i \(-0.348545\pi\)
0.458060 + 0.888921i \(0.348545\pi\)
\(212\) 5679.58 1.83998
\(213\) 461.080 0.148322
\(214\) −5463.63 −1.74526
\(215\) 0 0
\(216\) 1169.50 0.368400
\(217\) 1077.85 0.337187
\(218\) −6338.53 −1.96926
\(219\) 461.598 0.142429
\(220\) 0 0
\(221\) −1458.91 −0.444059
\(222\) −2484.25 −0.751046
\(223\) −1443.09 −0.433348 −0.216674 0.976244i \(-0.569521\pi\)
−0.216674 + 0.976244i \(0.569521\pi\)
\(224\) 324.903 0.0969129
\(225\) 0 0
\(226\) −4704.94 −1.38481
\(227\) −3658.71 −1.06977 −0.534884 0.844926i \(-0.679645\pi\)
−0.534884 + 0.844926i \(0.679645\pi\)
\(228\) −295.438 −0.0858151
\(229\) −1695.50 −0.489265 −0.244632 0.969616i \(-0.578667\pi\)
−0.244632 + 0.969616i \(0.578667\pi\)
\(230\) 0 0
\(231\) 180.964 0.0515435
\(232\) 11479.4 3.24853
\(233\) 5782.24 1.62578 0.812890 0.582417i \(-0.197893\pi\)
0.812890 + 0.582417i \(0.197893\pi\)
\(234\) 1099.46 0.307153
\(235\) 0 0
\(236\) 6965.76 1.92132
\(237\) −1209.00 −0.331363
\(238\) 1618.45 0.440792
\(239\) −5836.16 −1.57954 −0.789770 0.613404i \(-0.789800\pi\)
−0.789770 + 0.613404i \(0.789800\pi\)
\(240\) 0 0
\(241\) −982.308 −0.262556 −0.131278 0.991346i \(-0.541908\pi\)
−0.131278 + 0.991346i \(0.541908\pi\)
\(242\) −601.519 −0.159781
\(243\) −243.000 −0.0641500
\(244\) 11066.5 2.90353
\(245\) 0 0
\(246\) 6324.91 1.63927
\(247\) −144.797 −0.0373005
\(248\) 8513.70 2.17992
\(249\) −1956.94 −0.498056
\(250\) 0 0
\(251\) 4320.23 1.08642 0.543209 0.839598i \(-0.317209\pi\)
0.543209 + 0.839598i \(0.317209\pi\)
\(252\) −824.856 −0.206195
\(253\) 753.027 0.187124
\(254\) 11238.8 2.77633
\(255\) 0 0
\(256\) −8347.07 −2.03786
\(257\) 6224.29 1.51074 0.755371 0.655298i \(-0.227457\pi\)
0.755371 + 0.655298i \(0.227457\pi\)
\(258\) 2644.95 0.638246
\(259\) 913.459 0.219149
\(260\) 0 0
\(261\) −2385.20 −0.565670
\(262\) 283.995 0.0669666
\(263\) 2588.95 0.607002 0.303501 0.952831i \(-0.401844\pi\)
0.303501 + 0.952831i \(0.401844\pi\)
\(264\) 1429.39 0.333230
\(265\) 0 0
\(266\) 160.631 0.0370260
\(267\) −3680.30 −0.843561
\(268\) −5243.12 −1.19505
\(269\) −5871.53 −1.33083 −0.665415 0.746473i \(-0.731746\pi\)
−0.665415 + 0.746473i \(0.731746\pi\)
\(270\) 0 0
\(271\) −3306.93 −0.741261 −0.370630 0.928780i \(-0.620858\pi\)
−0.370630 + 0.928780i \(0.620858\pi\)
\(272\) 4845.85 1.08023
\(273\) −404.270 −0.0896247
\(274\) 9541.14 2.10366
\(275\) 0 0
\(276\) −3432.39 −0.748571
\(277\) 2861.37 0.620660 0.310330 0.950629i \(-0.399560\pi\)
0.310330 + 0.950629i \(0.399560\pi\)
\(278\) 8403.66 1.81302
\(279\) −1768.99 −0.379593
\(280\) 0 0
\(281\) −2988.05 −0.634349 −0.317174 0.948367i \(-0.602734\pi\)
−0.317174 + 0.948367i \(0.602734\pi\)
\(282\) −2105.03 −0.444513
\(283\) −1454.60 −0.305536 −0.152768 0.988262i \(-0.548819\pi\)
−0.152768 + 0.988262i \(0.548819\pi\)
\(284\) −2568.69 −0.536704
\(285\) 0 0
\(286\) 1343.78 0.277830
\(287\) −2325.67 −0.478327
\(288\) −533.234 −0.109101
\(289\) −1388.36 −0.282589
\(290\) 0 0
\(291\) 2879.24 0.580014
\(292\) −2571.58 −0.515378
\(293\) 8586.52 1.71205 0.856023 0.516937i \(-0.172928\pi\)
0.856023 + 0.516937i \(0.172928\pi\)
\(294\) −4666.92 −0.925782
\(295\) 0 0
\(296\) 7215.19 1.41681
\(297\) −297.000 −0.0580259
\(298\) 10329.6 2.00798
\(299\) −1682.25 −0.325374
\(300\) 0 0
\(301\) −972.547 −0.186235
\(302\) −13477.8 −2.56808
\(303\) 3775.57 0.715844
\(304\) 480.951 0.0907382
\(305\) 0 0
\(306\) −2656.22 −0.496228
\(307\) −529.029 −0.0983495 −0.0491748 0.998790i \(-0.515659\pi\)
−0.0491748 + 0.998790i \(0.515659\pi\)
\(308\) −1008.16 −0.186510
\(309\) −1479.53 −0.272386
\(310\) 0 0
\(311\) −5008.00 −0.913111 −0.456555 0.889695i \(-0.650917\pi\)
−0.456555 + 0.889695i \(0.650917\pi\)
\(312\) −3193.23 −0.579426
\(313\) −7858.53 −1.41914 −0.709570 0.704635i \(-0.751111\pi\)
−0.709570 + 0.704635i \(0.751111\pi\)
\(314\) 11638.8 2.09177
\(315\) 0 0
\(316\) 6735.39 1.19904
\(317\) −7747.04 −1.37261 −0.686304 0.727315i \(-0.740768\pi\)
−0.686304 + 0.727315i \(0.740768\pi\)
\(318\) 5068.09 0.893724
\(319\) −2915.24 −0.511668
\(320\) 0 0
\(321\) −3297.15 −0.573299
\(322\) 1866.21 0.322980
\(323\) 349.820 0.0602617
\(324\) 1353.76 0.232126
\(325\) 0 0
\(326\) 17513.7 2.97544
\(327\) −3825.13 −0.646882
\(328\) −18369.9 −3.09240
\(329\) 774.019 0.129705
\(330\) 0 0
\(331\) 1355.08 0.225022 0.112511 0.993650i \(-0.464111\pi\)
0.112511 + 0.993650i \(0.464111\pi\)
\(332\) 10902.2 1.80221
\(333\) −1499.18 −0.246710
\(334\) 19944.1 3.26734
\(335\) 0 0
\(336\) 1342.80 0.218024
\(337\) −2596.97 −0.419780 −0.209890 0.977725i \(-0.567311\pi\)
−0.209890 + 0.977725i \(0.567311\pi\)
\(338\) 7919.81 1.27450
\(339\) −2839.30 −0.454896
\(340\) 0 0
\(341\) −2162.09 −0.343355
\(342\) −263.629 −0.0416826
\(343\) 3596.95 0.566231
\(344\) −7681.91 −1.20401
\(345\) 0 0
\(346\) 16530.8 2.56850
\(347\) −266.820 −0.0412785 −0.0206392 0.999787i \(-0.506570\pi\)
−0.0206392 + 0.999787i \(0.506570\pi\)
\(348\) 13288.0 2.04687
\(349\) −10549.4 −1.61804 −0.809021 0.587780i \(-0.800002\pi\)
−0.809021 + 0.587780i \(0.800002\pi\)
\(350\) 0 0
\(351\) 663.492 0.100896
\(352\) −651.730 −0.0986856
\(353\) −7152.63 −1.07846 −0.539229 0.842159i \(-0.681284\pi\)
−0.539229 + 0.842159i \(0.681284\pi\)
\(354\) 6215.79 0.933235
\(355\) 0 0
\(356\) 20503.1 3.05242
\(357\) 976.690 0.144795
\(358\) 132.409 0.0195475
\(359\) 359.182 0.0528047 0.0264024 0.999651i \(-0.491595\pi\)
0.0264024 + 0.999651i \(0.491595\pi\)
\(360\) 0 0
\(361\) −6824.28 −0.994938
\(362\) −9615.62 −1.39609
\(363\) −363.000 −0.0524864
\(364\) 2252.20 0.324306
\(365\) 0 0
\(366\) 9875.04 1.41032
\(367\) −9042.47 −1.28614 −0.643070 0.765808i \(-0.722339\pi\)
−0.643070 + 0.765808i \(0.722339\pi\)
\(368\) 5587.67 0.791515
\(369\) 3816.91 0.538483
\(370\) 0 0
\(371\) −1863.54 −0.260781
\(372\) 9855.08 1.37355
\(373\) 11929.0 1.65592 0.827960 0.560787i \(-0.189501\pi\)
0.827960 + 0.560787i \(0.189501\pi\)
\(374\) −3246.49 −0.448855
\(375\) 0 0
\(376\) 6113.78 0.838549
\(377\) 6512.59 0.889696
\(378\) −736.047 −0.100154
\(379\) 6556.08 0.888557 0.444279 0.895889i \(-0.353460\pi\)
0.444279 + 0.895889i \(0.353460\pi\)
\(380\) 0 0
\(381\) 6782.32 0.911992
\(382\) 16611.8 2.22496
\(383\) −10703.1 −1.42795 −0.713975 0.700172i \(-0.753107\pi\)
−0.713975 + 0.700172i \(0.753107\pi\)
\(384\) −6767.74 −0.899388
\(385\) 0 0
\(386\) −6432.60 −0.848214
\(387\) 1596.15 0.209657
\(388\) −16040.3 −2.09878
\(389\) −6450.30 −0.840728 −0.420364 0.907355i \(-0.638098\pi\)
−0.420364 + 0.907355i \(0.638098\pi\)
\(390\) 0 0
\(391\) 4064.20 0.525667
\(392\) 13554.4 1.74644
\(393\) 171.383 0.0219978
\(394\) −11439.6 −1.46274
\(395\) 0 0
\(396\) 1654.60 0.209966
\(397\) 12526.0 1.58353 0.791764 0.610828i \(-0.209163\pi\)
0.791764 + 0.610828i \(0.209163\pi\)
\(398\) 15110.7 1.90310
\(399\) 96.9365 0.0121626
\(400\) 0 0
\(401\) 9119.52 1.13568 0.567839 0.823139i \(-0.307779\pi\)
0.567839 + 0.823139i \(0.307779\pi\)
\(402\) −4678.62 −0.580468
\(403\) 4830.08 0.597030
\(404\) −21033.8 −2.59028
\(405\) 0 0
\(406\) −7224.76 −0.883150
\(407\) −1832.33 −0.223158
\(408\) 7714.63 0.936106
\(409\) −596.921 −0.0721658 −0.0360829 0.999349i \(-0.511488\pi\)
−0.0360829 + 0.999349i \(0.511488\pi\)
\(410\) 0 0
\(411\) 5757.82 0.691027
\(412\) 8242.49 0.985627
\(413\) −2285.54 −0.272310
\(414\) −3062.84 −0.363600
\(415\) 0 0
\(416\) 1455.95 0.171596
\(417\) 5071.38 0.595555
\(418\) −322.214 −0.0377033
\(419\) 11560.5 1.34789 0.673944 0.738782i \(-0.264599\pi\)
0.673944 + 0.738782i \(0.264599\pi\)
\(420\) 0 0
\(421\) 15182.6 1.75761 0.878805 0.477182i \(-0.158342\pi\)
0.878805 + 0.477182i \(0.158342\pi\)
\(422\) −13958.5 −1.61017
\(423\) −1270.33 −0.146018
\(424\) −14719.6 −1.68596
\(425\) 0 0
\(426\) −2292.13 −0.260691
\(427\) −3631.05 −0.411519
\(428\) 18368.5 2.07448
\(429\) 810.935 0.0912641
\(430\) 0 0
\(431\) −14902.7 −1.66551 −0.832757 0.553639i \(-0.813239\pi\)
−0.832757 + 0.553639i \(0.813239\pi\)
\(432\) −2203.82 −0.245443
\(433\) 2269.85 0.251922 0.125961 0.992035i \(-0.459799\pi\)
0.125961 + 0.992035i \(0.459799\pi\)
\(434\) −5358.26 −0.592637
\(435\) 0 0
\(436\) 21309.9 2.34074
\(437\) 403.372 0.0441554
\(438\) −2294.71 −0.250332
\(439\) −13418.7 −1.45886 −0.729431 0.684054i \(-0.760215\pi\)
−0.729431 + 0.684054i \(0.760215\pi\)
\(440\) 0 0
\(441\) −2816.36 −0.304109
\(442\) 7252.59 0.780476
\(443\) −11507.5 −1.23417 −0.617086 0.786896i \(-0.711687\pi\)
−0.617086 + 0.786896i \(0.711687\pi\)
\(444\) 8351.98 0.892719
\(445\) 0 0
\(446\) 7173.94 0.761650
\(447\) 6233.65 0.659601
\(448\) 1965.64 0.207294
\(449\) −16203.8 −1.70313 −0.851564 0.524250i \(-0.824346\pi\)
−0.851564 + 0.524250i \(0.824346\pi\)
\(450\) 0 0
\(451\) 4665.11 0.487077
\(452\) 15817.8 1.64604
\(453\) −8133.48 −0.843585
\(454\) 18188.3 1.88022
\(455\) 0 0
\(456\) 765.677 0.0786318
\(457\) −4912.79 −0.502868 −0.251434 0.967874i \(-0.580902\pi\)
−0.251434 + 0.967874i \(0.580902\pi\)
\(458\) 8428.70 0.859929
\(459\) −1602.95 −0.163005
\(460\) 0 0
\(461\) −14270.8 −1.44178 −0.720888 0.693051i \(-0.756266\pi\)
−0.720888 + 0.693051i \(0.756266\pi\)
\(462\) −899.613 −0.0905926
\(463\) −6656.46 −0.668146 −0.334073 0.942547i \(-0.608423\pi\)
−0.334073 + 0.942547i \(0.608423\pi\)
\(464\) −21631.9 −2.16430
\(465\) 0 0
\(466\) −28744.8 −2.85746
\(467\) −19347.7 −1.91714 −0.958571 0.284853i \(-0.908055\pi\)
−0.958571 + 0.284853i \(0.908055\pi\)
\(468\) −3696.34 −0.365093
\(469\) 1720.33 0.169376
\(470\) 0 0
\(471\) 7023.70 0.687123
\(472\) −18052.9 −1.76050
\(473\) 1950.86 0.189642
\(474\) 6010.22 0.582402
\(475\) 0 0
\(476\) −5441.18 −0.523941
\(477\) 3058.45 0.293579
\(478\) 29012.9 2.77619
\(479\) 9826.69 0.937355 0.468678 0.883369i \(-0.344731\pi\)
0.468678 + 0.883369i \(0.344731\pi\)
\(480\) 0 0
\(481\) 4093.39 0.388030
\(482\) 4883.28 0.461467
\(483\) 1126.21 0.106095
\(484\) 2022.29 0.189922
\(485\) 0 0
\(486\) 1208.01 0.112750
\(487\) −10278.1 −0.956353 −0.478176 0.878264i \(-0.658702\pi\)
−0.478176 + 0.878264i \(0.658702\pi\)
\(488\) −28680.7 −2.66048
\(489\) 10569.0 0.977400
\(490\) 0 0
\(491\) 11397.1 1.04755 0.523773 0.851858i \(-0.324524\pi\)
0.523773 + 0.851858i \(0.324524\pi\)
\(492\) −21264.1 −1.94850
\(493\) −15734.0 −1.43737
\(494\) 719.819 0.0655591
\(495\) 0 0
\(496\) −16043.3 −1.45235
\(497\) 842.817 0.0760674
\(498\) 9728.39 0.875381
\(499\) −9179.17 −0.823479 −0.411740 0.911302i \(-0.635079\pi\)
−0.411740 + 0.911302i \(0.635079\pi\)
\(500\) 0 0
\(501\) 12035.7 1.07329
\(502\) −21476.9 −1.90948
\(503\) −6782.80 −0.601253 −0.300626 0.953742i \(-0.597196\pi\)
−0.300626 + 0.953742i \(0.597196\pi\)
\(504\) 2137.75 0.188935
\(505\) 0 0
\(506\) −3743.47 −0.328889
\(507\) 4779.39 0.418659
\(508\) −37784.6 −3.30004
\(509\) 6814.24 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(510\) 0 0
\(511\) 843.765 0.0730449
\(512\) 23447.9 2.02395
\(513\) −159.093 −0.0136923
\(514\) −30942.4 −2.65527
\(515\) 0 0
\(516\) −8892.23 −0.758641
\(517\) −1552.62 −0.132078
\(518\) −4541.01 −0.385175
\(519\) 9975.89 0.843724
\(520\) 0 0
\(521\) 206.712 0.0173824 0.00869118 0.999962i \(-0.497233\pi\)
0.00869118 + 0.999962i \(0.497233\pi\)
\(522\) 11857.4 0.994219
\(523\) 11375.9 0.951118 0.475559 0.879684i \(-0.342246\pi\)
0.475559 + 0.879684i \(0.342246\pi\)
\(524\) −954.781 −0.0795989
\(525\) 0 0
\(526\) −12870.3 −1.06686
\(527\) −11669.1 −0.964547
\(528\) −2693.56 −0.222012
\(529\) −7480.63 −0.614830
\(530\) 0 0
\(531\) 3751.06 0.306557
\(532\) −540.036 −0.0440104
\(533\) −10421.8 −0.846936
\(534\) 18295.6 1.48264
\(535\) 0 0
\(536\) 13588.4 1.09502
\(537\) 79.9049 0.00642114
\(538\) 29188.7 2.33906
\(539\) −3442.21 −0.275077
\(540\) 0 0
\(541\) −10228.1 −0.812831 −0.406415 0.913688i \(-0.633221\pi\)
−0.406415 + 0.913688i \(0.633221\pi\)
\(542\) 16439.5 1.30284
\(543\) −5802.76 −0.458601
\(544\) −3517.49 −0.277226
\(545\) 0 0
\(546\) 2009.72 0.157524
\(547\) 15538.9 1.21461 0.607307 0.794467i \(-0.292250\pi\)
0.607307 + 0.794467i \(0.292250\pi\)
\(548\) −32077.0 −2.50048
\(549\) 5959.31 0.463274
\(550\) 0 0
\(551\) −1561.60 −0.120737
\(552\) 8895.62 0.685911
\(553\) −2209.96 −0.169940
\(554\) −14224.5 −1.09087
\(555\) 0 0
\(556\) −28252.8 −2.15501
\(557\) 5191.17 0.394896 0.197448 0.980313i \(-0.436735\pi\)
0.197448 + 0.980313i \(0.436735\pi\)
\(558\) 8794.03 0.667170
\(559\) −4358.17 −0.329752
\(560\) 0 0
\(561\) −1959.17 −0.147444
\(562\) 14854.3 1.11493
\(563\) −13722.9 −1.02727 −0.513635 0.858009i \(-0.671701\pi\)
−0.513635 + 0.858009i \(0.671701\pi\)
\(564\) 7077.04 0.528364
\(565\) 0 0
\(566\) 7231.13 0.537009
\(567\) −444.184 −0.0328995
\(568\) 6657.20 0.491778
\(569\) 23209.0 1.70997 0.854986 0.518652i \(-0.173566\pi\)
0.854986 + 0.518652i \(0.173566\pi\)
\(570\) 0 0
\(571\) −13130.5 −0.962337 −0.481168 0.876628i \(-0.659787\pi\)
−0.481168 + 0.876628i \(0.659787\pi\)
\(572\) −4517.75 −0.330239
\(573\) 10024.8 0.730874
\(574\) 11561.4 0.840705
\(575\) 0 0
\(576\) −3226.03 −0.233365
\(577\) 12261.0 0.884633 0.442317 0.896859i \(-0.354157\pi\)
0.442317 + 0.896859i \(0.354157\pi\)
\(578\) 6901.86 0.496677
\(579\) −3881.90 −0.278629
\(580\) 0 0
\(581\) −3577.13 −0.255429
\(582\) −14313.3 −1.01943
\(583\) 3738.11 0.265552
\(584\) 6664.69 0.472238
\(585\) 0 0
\(586\) −42685.5 −3.00908
\(587\) 3553.42 0.249856 0.124928 0.992166i \(-0.460130\pi\)
0.124928 + 0.992166i \(0.460130\pi\)
\(588\) 15690.0 1.10042
\(589\) −1158.16 −0.0810208
\(590\) 0 0
\(591\) −6903.50 −0.480494
\(592\) −13596.4 −0.943933
\(593\) −1942.83 −0.134540 −0.0672701 0.997735i \(-0.521429\pi\)
−0.0672701 + 0.997735i \(0.521429\pi\)
\(594\) 1476.45 0.101986
\(595\) 0 0
\(596\) −34727.9 −2.38676
\(597\) 9118.92 0.625147
\(598\) 8362.84 0.571876
\(599\) 18585.4 1.26774 0.633871 0.773439i \(-0.281465\pi\)
0.633871 + 0.773439i \(0.281465\pi\)
\(600\) 0 0
\(601\) 8010.10 0.543658 0.271829 0.962346i \(-0.412371\pi\)
0.271829 + 0.962346i \(0.412371\pi\)
\(602\) 4834.75 0.327325
\(603\) −2823.42 −0.190677
\(604\) 45311.9 3.05251
\(605\) 0 0
\(606\) −18769.2 −1.25816
\(607\) 1537.69 0.102822 0.0514111 0.998678i \(-0.483628\pi\)
0.0514111 + 0.998678i \(0.483628\pi\)
\(608\) −349.110 −0.0232867
\(609\) −4359.94 −0.290105
\(610\) 0 0
\(611\) 3468.53 0.229659
\(612\) 8930.11 0.589834
\(613\) −2298.75 −0.151461 −0.0757304 0.997128i \(-0.524129\pi\)
−0.0757304 + 0.997128i \(0.524129\pi\)
\(614\) 2629.93 0.172859
\(615\) 0 0
\(616\) 2612.81 0.170898
\(617\) −6089.51 −0.397333 −0.198666 0.980067i \(-0.563661\pi\)
−0.198666 + 0.980067i \(0.563661\pi\)
\(618\) 7355.06 0.478744
\(619\) 12255.4 0.795775 0.397887 0.917434i \(-0.369743\pi\)
0.397887 + 0.917434i \(0.369743\pi\)
\(620\) 0 0
\(621\) −1848.34 −0.119439
\(622\) 24895.9 1.60488
\(623\) −6727.29 −0.432622
\(624\) 6017.36 0.386037
\(625\) 0 0
\(626\) 39066.6 2.49427
\(627\) −194.447 −0.0123851
\(628\) −39129.3 −2.48635
\(629\) −9889.36 −0.626891
\(630\) 0 0
\(631\) −25746.5 −1.62433 −0.812165 0.583428i \(-0.801711\pi\)
−0.812165 + 0.583428i \(0.801711\pi\)
\(632\) −17455.9 −1.09867
\(633\) −8423.58 −0.528922
\(634\) 38512.3 2.41249
\(635\) 0 0
\(636\) −17038.8 −1.06231
\(637\) 7689.84 0.478308
\(638\) 14492.3 0.899305
\(639\) −1383.24 −0.0856340
\(640\) 0 0
\(641\) −598.022 −0.0368494 −0.0184247 0.999830i \(-0.505865\pi\)
−0.0184247 + 0.999830i \(0.505865\pi\)
\(642\) 16390.9 1.00763
\(643\) −14610.9 −0.896109 −0.448054 0.894006i \(-0.647883\pi\)
−0.448054 + 0.894006i \(0.647883\pi\)
\(644\) −6274.13 −0.383906
\(645\) 0 0
\(646\) −1739.04 −0.105916
\(647\) 17252.3 1.04831 0.524156 0.851622i \(-0.324381\pi\)
0.524156 + 0.851622i \(0.324381\pi\)
\(648\) −3508.50 −0.212696
\(649\) 4584.63 0.277292
\(650\) 0 0
\(651\) −3233.56 −0.194675
\(652\) −58880.5 −3.53671
\(653\) 17768.6 1.06484 0.532418 0.846481i \(-0.321283\pi\)
0.532418 + 0.846481i \(0.321283\pi\)
\(654\) 19015.6 1.13696
\(655\) 0 0
\(656\) 34616.4 2.06028
\(657\) −1384.80 −0.0822314
\(658\) −3847.82 −0.227969
\(659\) −17594.9 −1.04006 −0.520031 0.854147i \(-0.674080\pi\)
−0.520031 + 0.854147i \(0.674080\pi\)
\(660\) 0 0
\(661\) 23355.9 1.37434 0.687170 0.726497i \(-0.258853\pi\)
0.687170 + 0.726497i \(0.258853\pi\)
\(662\) −6736.44 −0.395497
\(663\) 4376.74 0.256378
\(664\) −28254.8 −1.65136
\(665\) 0 0
\(666\) 7452.76 0.433617
\(667\) −18142.6 −1.05320
\(668\) −67051.4 −3.88368
\(669\) 4329.28 0.250194
\(670\) 0 0
\(671\) 7283.61 0.419047
\(672\) −974.708 −0.0559527
\(673\) 2340.47 0.134054 0.0670271 0.997751i \(-0.478649\pi\)
0.0670271 + 0.997751i \(0.478649\pi\)
\(674\) 12910.1 0.737803
\(675\) 0 0
\(676\) −26626.1 −1.51491
\(677\) 16562.2 0.940234 0.470117 0.882604i \(-0.344212\pi\)
0.470117 + 0.882604i \(0.344212\pi\)
\(678\) 14114.8 0.799522
\(679\) 5263.01 0.297461
\(680\) 0 0
\(681\) 10976.1 0.617630
\(682\) 10748.3 0.603478
\(683\) −22303.4 −1.24951 −0.624755 0.780821i \(-0.714801\pi\)
−0.624755 + 0.780821i \(0.714801\pi\)
\(684\) 886.313 0.0495454
\(685\) 0 0
\(686\) −17881.3 −0.995204
\(687\) 5086.49 0.282477
\(688\) 14475.9 0.802163
\(689\) −8350.87 −0.461745
\(690\) 0 0
\(691\) 6244.91 0.343802 0.171901 0.985114i \(-0.445009\pi\)
0.171901 + 0.985114i \(0.445009\pi\)
\(692\) −55576.0 −3.05301
\(693\) −542.892 −0.0297587
\(694\) 1326.42 0.0725508
\(695\) 0 0
\(696\) −34438.1 −1.87554
\(697\) 25178.3 1.36829
\(698\) 52443.5 2.84386
\(699\) −17346.7 −0.938645
\(700\) 0 0
\(701\) −15277.7 −0.823154 −0.411577 0.911375i \(-0.635022\pi\)
−0.411577 + 0.911375i \(0.635022\pi\)
\(702\) −3298.37 −0.177335
\(703\) −981.519 −0.0526582
\(704\) −3942.93 −0.211086
\(705\) 0 0
\(706\) 35557.3 1.89549
\(707\) 6901.43 0.367122
\(708\) −20897.3 −1.10928
\(709\) −11732.1 −0.621449 −0.310725 0.950500i \(-0.600572\pi\)
−0.310725 + 0.950500i \(0.600572\pi\)
\(710\) 0 0
\(711\) 3627.00 0.191313
\(712\) −53137.2 −2.79691
\(713\) −13455.5 −0.706750
\(714\) −4855.35 −0.254491
\(715\) 0 0
\(716\) −445.153 −0.0232349
\(717\) 17508.5 0.911947
\(718\) −1785.57 −0.0928093
\(719\) −19006.2 −0.985828 −0.492914 0.870078i \(-0.664068\pi\)
−0.492914 + 0.870078i \(0.664068\pi\)
\(720\) 0 0
\(721\) −2704.45 −0.139694
\(722\) 33925.1 1.74870
\(723\) 2946.92 0.151587
\(724\) 32327.4 1.65945
\(725\) 0 0
\(726\) 1804.56 0.0922498
\(727\) −2650.77 −0.135229 −0.0676146 0.997712i \(-0.521539\pi\)
−0.0676146 + 0.997712i \(0.521539\pi\)
\(728\) −5836.96 −0.297160
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 10529.1 0.532738
\(732\) −33199.5 −1.67635
\(733\) 8854.22 0.446164 0.223082 0.974800i \(-0.428388\pi\)
0.223082 + 0.974800i \(0.428388\pi\)
\(734\) 44952.2 2.26051
\(735\) 0 0
\(736\) −4055.96 −0.203131
\(737\) −3450.84 −0.172474
\(738\) −18974.7 −0.946435
\(739\) 17174.8 0.854921 0.427460 0.904034i \(-0.359408\pi\)
0.427460 + 0.904034i \(0.359408\pi\)
\(740\) 0 0
\(741\) 434.391 0.0215354
\(742\) 9264.06 0.458348
\(743\) 9800.30 0.483901 0.241950 0.970289i \(-0.422213\pi\)
0.241950 + 0.970289i \(0.422213\pi\)
\(744\) −25541.1 −1.25858
\(745\) 0 0
\(746\) −59301.6 −2.91044
\(747\) 5870.82 0.287553
\(748\) 10914.6 0.533525
\(749\) −6026.92 −0.294017
\(750\) 0 0
\(751\) 2781.25 0.135139 0.0675695 0.997715i \(-0.478476\pi\)
0.0675695 + 0.997715i \(0.478476\pi\)
\(752\) −11520.9 −0.558675
\(753\) −12960.7 −0.627243
\(754\) −32375.6 −1.56372
\(755\) 0 0
\(756\) 2474.57 0.119046
\(757\) −3010.87 −0.144560 −0.0722800 0.997384i \(-0.523028\pi\)
−0.0722800 + 0.997384i \(0.523028\pi\)
\(758\) −32591.8 −1.56172
\(759\) −2259.08 −0.108036
\(760\) 0 0
\(761\) −2709.54 −0.129068 −0.0645340 0.997916i \(-0.520556\pi\)
−0.0645340 + 0.997916i \(0.520556\pi\)
\(762\) −33716.5 −1.60291
\(763\) −6992.03 −0.331754
\(764\) −55848.4 −2.64466
\(765\) 0 0
\(766\) 53207.7 2.50976
\(767\) −10242.0 −0.482159
\(768\) 25041.2 1.17656
\(769\) −28121.1 −1.31869 −0.659345 0.751841i \(-0.729166\pi\)
−0.659345 + 0.751841i \(0.729166\pi\)
\(770\) 0 0
\(771\) −18672.9 −0.872227
\(772\) 21626.2 1.00822
\(773\) −4274.61 −0.198897 −0.0994483 0.995043i \(-0.531708\pi\)
−0.0994483 + 0.995043i \(0.531708\pi\)
\(774\) −7934.85 −0.368491
\(775\) 0 0
\(776\) 41571.2 1.92309
\(777\) −2740.38 −0.126526
\(778\) 32065.9 1.47766
\(779\) 2498.95 0.114935
\(780\) 0 0
\(781\) −1690.63 −0.0774589
\(782\) −20204.1 −0.923909
\(783\) 7155.59 0.326590
\(784\) −25542.2 −1.16355
\(785\) 0 0
\(786\) −851.985 −0.0386632
\(787\) −19107.9 −0.865469 −0.432734 0.901522i \(-0.642451\pi\)
−0.432734 + 0.901522i \(0.642451\pi\)
\(788\) 38459.6 1.73866
\(789\) −7766.85 −0.350453
\(790\) 0 0
\(791\) −5190.01 −0.233294
\(792\) −4288.17 −0.192391
\(793\) −16271.4 −0.728645
\(794\) −62269.4 −2.78320
\(795\) 0 0
\(796\) −50801.9 −2.26209
\(797\) −27518.4 −1.22303 −0.611513 0.791234i \(-0.709439\pi\)
−0.611513 + 0.791234i \(0.709439\pi\)
\(798\) −481.893 −0.0213770
\(799\) −8379.74 −0.371031
\(800\) 0 0
\(801\) 11040.9 0.487030
\(802\) −45335.2 −1.99606
\(803\) −1692.53 −0.0743811
\(804\) 15729.4 0.689965
\(805\) 0 0
\(806\) −24011.4 −1.04934
\(807\) 17614.6 0.768356
\(808\) 54512.7 2.37345
\(809\) 21986.6 0.955511 0.477755 0.878493i \(-0.341450\pi\)
0.477755 + 0.878493i \(0.341450\pi\)
\(810\) 0 0
\(811\) 11835.1 0.512437 0.256218 0.966619i \(-0.417523\pi\)
0.256218 + 0.966619i \(0.417523\pi\)
\(812\) 24289.4 1.04974
\(813\) 9920.79 0.427967
\(814\) 9108.93 0.392221
\(815\) 0 0
\(816\) −14537.6 −0.623672
\(817\) 1045.01 0.0447494
\(818\) 2967.43 0.126838
\(819\) 1212.81 0.0517448
\(820\) 0 0
\(821\) −7890.58 −0.335424 −0.167712 0.985836i \(-0.553638\pi\)
−0.167712 + 0.985836i \(0.553638\pi\)
\(822\) −28623.4 −1.21455
\(823\) −1000.74 −0.0423861 −0.0211930 0.999775i \(-0.506746\pi\)
−0.0211930 + 0.999775i \(0.506746\pi\)
\(824\) −21361.8 −0.903123
\(825\) 0 0
\(826\) 11362.0 0.478611
\(827\) −22764.2 −0.957181 −0.478590 0.878038i \(-0.658852\pi\)
−0.478590 + 0.878038i \(0.658852\pi\)
\(828\) 10297.2 0.432188
\(829\) 39835.8 1.66894 0.834471 0.551051i \(-0.185773\pi\)
0.834471 + 0.551051i \(0.185773\pi\)
\(830\) 0 0
\(831\) −8584.10 −0.358338
\(832\) 8808.43 0.367040
\(833\) −18578.1 −0.772742
\(834\) −25211.0 −1.04674
\(835\) 0 0
\(836\) 1083.27 0.0448155
\(837\) 5306.96 0.219158
\(838\) −57469.7 −2.36904
\(839\) 23330.9 0.960037 0.480019 0.877258i \(-0.340630\pi\)
0.480019 + 0.877258i \(0.340630\pi\)
\(840\) 0 0
\(841\) 45847.5 1.87984
\(842\) −75476.0 −3.08916
\(843\) 8964.14 0.366241
\(844\) 46928.1 1.91390
\(845\) 0 0
\(846\) 6315.09 0.256640
\(847\) −663.535 −0.0269177
\(848\) 27737.8 1.12326
\(849\) 4363.79 0.176402
\(850\) 0 0
\(851\) −11403.3 −0.459341
\(852\) 7706.08 0.309866
\(853\) −22937.3 −0.920701 −0.460351 0.887737i \(-0.652276\pi\)
−0.460351 + 0.887737i \(0.652276\pi\)
\(854\) 18050.8 0.723284
\(855\) 0 0
\(856\) −47605.2 −1.90083
\(857\) 38472.3 1.53348 0.766738 0.641961i \(-0.221879\pi\)
0.766738 + 0.641961i \(0.221879\pi\)
\(858\) −4031.34 −0.160405
\(859\) −22970.3 −0.912384 −0.456192 0.889881i \(-0.650787\pi\)
−0.456192 + 0.889881i \(0.650787\pi\)
\(860\) 0 0
\(861\) 6977.00 0.276162
\(862\) 74084.5 2.92730
\(863\) 33587.6 1.32484 0.662420 0.749133i \(-0.269530\pi\)
0.662420 + 0.749133i \(0.269530\pi\)
\(864\) 1599.70 0.0629895
\(865\) 0 0
\(866\) −11284.0 −0.442777
\(867\) 4165.08 0.163153
\(868\) 18014.3 0.704430
\(869\) 4433.00 0.173049
\(870\) 0 0
\(871\) 7709.12 0.299901
\(872\) −55228.3 −2.14480
\(873\) −8637.71 −0.334871
\(874\) −2005.25 −0.0776073
\(875\) 0 0
\(876\) 7714.74 0.297554
\(877\) 704.130 0.0271115 0.0135558 0.999908i \(-0.495685\pi\)
0.0135558 + 0.999908i \(0.495685\pi\)
\(878\) 66707.5 2.56409
\(879\) −25759.5 −0.988450
\(880\) 0 0
\(881\) 9746.33 0.372715 0.186358 0.982482i \(-0.440332\pi\)
0.186358 + 0.982482i \(0.440332\pi\)
\(882\) 14000.7 0.534501
\(883\) 8774.24 0.334402 0.167201 0.985923i \(-0.446527\pi\)
0.167201 + 0.985923i \(0.446527\pi\)
\(884\) −24383.0 −0.927701
\(885\) 0 0
\(886\) 57206.4 2.16917
\(887\) 13505.1 0.511224 0.255612 0.966779i \(-0.417723\pi\)
0.255612 + 0.966779i \(0.417723\pi\)
\(888\) −21645.6 −0.817993
\(889\) 12397.5 0.467717
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) −24118.6 −0.905324
\(893\) −831.689 −0.0311662
\(894\) −30988.9 −1.15931
\(895\) 0 0
\(896\) −12370.9 −0.461252
\(897\) 5046.75 0.187855
\(898\) 80552.8 2.99341
\(899\) 52091.1 1.93252
\(900\) 0 0
\(901\) 20175.1 0.745984
\(902\) −23191.3 −0.856083
\(903\) 2917.64 0.107523
\(904\) −40994.6 −1.50825
\(905\) 0 0
\(906\) 40433.4 1.48268
\(907\) 2523.23 0.0923732 0.0461866 0.998933i \(-0.485293\pi\)
0.0461866 + 0.998933i \(0.485293\pi\)
\(908\) −61148.4 −2.23489
\(909\) −11326.7 −0.413293
\(910\) 0 0
\(911\) −32285.3 −1.17416 −0.587080 0.809529i \(-0.699723\pi\)
−0.587080 + 0.809529i \(0.699723\pi\)
\(912\) −1442.85 −0.0523877
\(913\) 7175.45 0.260101
\(914\) 24422.6 0.883838
\(915\) 0 0
\(916\) −28337.0 −1.02214
\(917\) 313.274 0.0112816
\(918\) 7968.65 0.286497
\(919\) −40956.5 −1.47011 −0.735055 0.678008i \(-0.762844\pi\)
−0.735055 + 0.678008i \(0.762844\pi\)
\(920\) 0 0
\(921\) 1587.09 0.0567821
\(922\) 70943.6 2.53406
\(923\) 3776.83 0.134687
\(924\) 3024.47 0.107682
\(925\) 0 0
\(926\) 33090.8 1.17433
\(927\) 4438.58 0.157262
\(928\) 15702.1 0.555437
\(929\) −6728.41 −0.237623 −0.118812 0.992917i \(-0.537908\pi\)
−0.118812 + 0.992917i \(0.537908\pi\)
\(930\) 0 0
\(931\) −1843.88 −0.0649095
\(932\) 96639.1 3.39648
\(933\) 15024.0 0.527185
\(934\) 96181.9 3.36956
\(935\) 0 0
\(936\) 9579.69 0.334532
\(937\) 36377.8 1.26831 0.634157 0.773205i \(-0.281347\pi\)
0.634157 + 0.773205i \(0.281347\pi\)
\(938\) −8552.13 −0.297694
\(939\) 23575.6 0.819340
\(940\) 0 0
\(941\) 25907.0 0.897496 0.448748 0.893658i \(-0.351870\pi\)
0.448748 + 0.893658i \(0.351870\pi\)
\(942\) −34916.4 −1.20768
\(943\) 29032.7 1.00258
\(944\) 34019.2 1.17291
\(945\) 0 0
\(946\) −9698.15 −0.333313
\(947\) −37096.3 −1.27293 −0.636466 0.771304i \(-0.719605\pi\)
−0.636466 + 0.771304i \(0.719605\pi\)
\(948\) −20206.2 −0.692263
\(949\) 3781.07 0.129335
\(950\) 0 0
\(951\) 23241.1 0.792476
\(952\) 14101.7 0.480084
\(953\) 15323.7 0.520865 0.260432 0.965492i \(-0.416135\pi\)
0.260432 + 0.965492i \(0.416135\pi\)
\(954\) −15204.3 −0.515992
\(955\) 0 0
\(956\) −97540.4 −3.29988
\(957\) 8745.72 0.295412
\(958\) −48850.7 −1.64749
\(959\) 10524.8 0.354395
\(960\) 0 0
\(961\) 8842.46 0.296817
\(962\) −20349.2 −0.682000
\(963\) 9891.45 0.330994
\(964\) −16417.4 −0.548516
\(965\) 0 0
\(966\) −5598.62 −0.186473
\(967\) 53388.0 1.77543 0.887716 0.460391i \(-0.152291\pi\)
0.887716 + 0.460391i \(0.152291\pi\)
\(968\) −5241.10 −0.174024
\(969\) −1049.46 −0.0347921
\(970\) 0 0
\(971\) −36325.7 −1.20056 −0.600281 0.799789i \(-0.704945\pi\)
−0.600281 + 0.799789i \(0.704945\pi\)
\(972\) −4061.28 −0.134018
\(973\) 9270.07 0.305432
\(974\) 51094.6 1.68088
\(975\) 0 0
\(976\) 54046.4 1.77252
\(977\) −48608.7 −1.59174 −0.795870 0.605468i \(-0.792986\pi\)
−0.795870 + 0.605468i \(0.792986\pi\)
\(978\) −52541.1 −1.71787
\(979\) 13494.4 0.440535
\(980\) 0 0
\(981\) 11475.4 0.373477
\(982\) −56657.7 −1.84116
\(983\) −31762.6 −1.03059 −0.515295 0.857013i \(-0.672318\pi\)
−0.515295 + 0.857013i \(0.672318\pi\)
\(984\) 55109.6 1.78540
\(985\) 0 0
\(986\) 78217.2 2.52631
\(987\) −2322.06 −0.0748854
\(988\) −2420.01 −0.0779258
\(989\) 12140.9 0.390352
\(990\) 0 0
\(991\) −5073.82 −0.162639 −0.0813195 0.996688i \(-0.525913\pi\)
−0.0813195 + 0.996688i \(0.525913\pi\)
\(992\) 11645.5 0.372726
\(993\) −4065.25 −0.129916
\(994\) −4189.83 −0.133696
\(995\) 0 0
\(996\) −32706.5 −1.04051
\(997\) −24607.0 −0.781657 −0.390828 0.920464i \(-0.627811\pi\)
−0.390828 + 0.920464i \(0.627811\pi\)
\(998\) 45631.8 1.44734
\(999\) 4497.54 0.142438
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 825.4.a.r.1.1 3
3.2 odd 2 2475.4.a.t.1.3 3
5.2 odd 4 825.4.c.k.199.2 6
5.3 odd 4 825.4.c.k.199.5 6
5.4 even 2 165.4.a.e.1.3 3
15.14 odd 2 495.4.a.k.1.1 3
55.54 odd 2 1815.4.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.e.1.3 3 5.4 even 2
495.4.a.k.1.1 3 15.14 odd 2
825.4.a.r.1.1 3 1.1 even 1 trivial
825.4.c.k.199.2 6 5.2 odd 4
825.4.c.k.199.5 6 5.3 odd 4
1815.4.a.r.1.1 3 55.54 odd 2
2475.4.a.t.1.3 3 3.2 odd 2