Properties

Label 507.4.a.o.1.6
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.588238\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} +15.3391 q^{5} +0.641100 q^{6} -32.3928 q^{7} -3.40944 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} +15.3391 q^{5} +0.641100 q^{6} -32.3928 q^{7} -3.40944 q^{8} +9.00000 q^{9} +3.27797 q^{10} +29.5925 q^{11} -23.8630 q^{12} -6.92233 q^{14} +46.0174 q^{15} +62.9061 q^{16} -78.1958 q^{17} +1.92330 q^{18} +10.6600 q^{19} -122.013 q^{20} -97.1783 q^{21} +6.32392 q^{22} -26.8789 q^{23} -10.2283 q^{24} +110.289 q^{25} +27.0000 q^{27} +257.663 q^{28} -190.785 q^{29} +9.83392 q^{30} -128.108 q^{31} +40.7185 q^{32} +88.7776 q^{33} -16.7104 q^{34} -496.877 q^{35} -71.5890 q^{36} -379.934 q^{37} +2.27805 q^{38} -52.2979 q^{40} -464.631 q^{41} -20.7670 q^{42} +322.758 q^{43} -235.389 q^{44} +138.052 q^{45} -5.74401 q^{46} -248.529 q^{47} +188.718 q^{48} +706.292 q^{49} +23.5688 q^{50} -234.587 q^{51} +740.167 q^{53} +5.76990 q^{54} +453.924 q^{55} +110.441 q^{56} +31.9801 q^{57} -40.7706 q^{58} -340.673 q^{59} -366.038 q^{60} -590.834 q^{61} -27.3767 q^{62} -291.535 q^{63} -494.547 q^{64} +18.9718 q^{66} -340.777 q^{67} +621.995 q^{68} -80.6366 q^{69} -106.183 q^{70} -36.2243 q^{71} -30.6850 q^{72} -164.572 q^{73} -81.1919 q^{74} +330.868 q^{75} -84.7935 q^{76} -958.584 q^{77} -327.242 q^{79} +964.925 q^{80} +81.0000 q^{81} -99.2916 q^{82} -1404.48 q^{83} +772.989 q^{84} -1199.46 q^{85} +68.9734 q^{86} -572.354 q^{87} -100.894 q^{88} +736.986 q^{89} +29.5018 q^{90} +213.803 q^{92} -384.324 q^{93} -53.1107 q^{94} +163.516 q^{95} +122.156 q^{96} +1494.87 q^{97} +150.935 q^{98} +266.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213700 0.0755543 0.0377772 0.999286i \(-0.487972\pi\)
0.0377772 + 0.999286i \(0.487972\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.95433 −0.994292
\(5\) 15.3391 1.37197 0.685987 0.727614i \(-0.259371\pi\)
0.685987 + 0.727614i \(0.259371\pi\)
\(6\) 0.641100 0.0436213
\(7\) −32.3928 −1.74905 −0.874523 0.484984i \(-0.838825\pi\)
−0.874523 + 0.484984i \(0.838825\pi\)
\(8\) −3.40944 −0.150677
\(9\) 9.00000 0.333333
\(10\) 3.27797 0.103659
\(11\) 29.5925 0.811135 0.405567 0.914065i \(-0.367074\pi\)
0.405567 + 0.914065i \(0.367074\pi\)
\(12\) −23.8630 −0.574054
\(13\) 0 0
\(14\) −6.92233 −0.132148
\(15\) 46.0174 0.792110
\(16\) 62.9061 0.982907
\(17\) −78.1958 −1.11560 −0.557802 0.829974i \(-0.688355\pi\)
−0.557802 + 0.829974i \(0.688355\pi\)
\(18\) 1.92330 0.0251848
\(19\) 10.6600 0.128715 0.0643574 0.997927i \(-0.479500\pi\)
0.0643574 + 0.997927i \(0.479500\pi\)
\(20\) −122.013 −1.36414
\(21\) −97.1783 −1.00981
\(22\) 6.32392 0.0612847
\(23\) −26.8789 −0.243680 −0.121840 0.992550i \(-0.538879\pi\)
−0.121840 + 0.992550i \(0.538879\pi\)
\(24\) −10.2283 −0.0869936
\(25\) 110.289 0.882314
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 257.663 1.73906
\(29\) −190.785 −1.22165 −0.610824 0.791766i \(-0.709162\pi\)
−0.610824 + 0.791766i \(0.709162\pi\)
\(30\) 9.83392 0.0598473
\(31\) −128.108 −0.742222 −0.371111 0.928589i \(-0.621023\pi\)
−0.371111 + 0.928589i \(0.621023\pi\)
\(32\) 40.7185 0.224940
\(33\) 88.7776 0.468309
\(34\) −16.7104 −0.0842887
\(35\) −496.877 −2.39965
\(36\) −71.5890 −0.331431
\(37\) −379.934 −1.68813 −0.844065 0.536241i \(-0.819844\pi\)
−0.844065 + 0.536241i \(0.819844\pi\)
\(38\) 2.27805 0.00972496
\(39\) 0 0
\(40\) −52.2979 −0.206725
\(41\) −464.631 −1.76983 −0.884917 0.465749i \(-0.845785\pi\)
−0.884917 + 0.465749i \(0.845785\pi\)
\(42\) −20.7670 −0.0762957
\(43\) 322.758 1.14466 0.572328 0.820025i \(-0.306041\pi\)
0.572328 + 0.820025i \(0.306041\pi\)
\(44\) −235.389 −0.806504
\(45\) 138.052 0.457325
\(46\) −5.74401 −0.0184110
\(47\) −248.529 −0.771314 −0.385657 0.922642i \(-0.626025\pi\)
−0.385657 + 0.922642i \(0.626025\pi\)
\(48\) 188.718 0.567482
\(49\) 706.292 2.05916
\(50\) 23.5688 0.0666626
\(51\) −234.587 −0.644094
\(52\) 0 0
\(53\) 740.167 1.91830 0.959148 0.282904i \(-0.0912976\pi\)
0.959148 + 0.282904i \(0.0912976\pi\)
\(54\) 5.76990 0.0145404
\(55\) 453.924 1.11286
\(56\) 110.441 0.263542
\(57\) 31.9801 0.0743135
\(58\) −40.7706 −0.0923008
\(59\) −340.673 −0.751727 −0.375864 0.926675i \(-0.622654\pi\)
−0.375864 + 0.926675i \(0.622654\pi\)
\(60\) −366.038 −0.787588
\(61\) −590.834 −1.24014 −0.620070 0.784547i \(-0.712896\pi\)
−0.620070 + 0.784547i \(0.712896\pi\)
\(62\) −27.3767 −0.0560781
\(63\) −291.535 −0.583015
\(64\) −494.547 −0.965912
\(65\) 0 0
\(66\) 18.9718 0.0353828
\(67\) −340.777 −0.621382 −0.310691 0.950511i \(-0.600560\pi\)
−0.310691 + 0.950511i \(0.600560\pi\)
\(68\) 621.995 1.10924
\(69\) −80.6366 −0.140688
\(70\) −106.183 −0.181304
\(71\) −36.2243 −0.0605498 −0.0302749 0.999542i \(-0.509638\pi\)
−0.0302749 + 0.999542i \(0.509638\pi\)
\(72\) −30.6850 −0.0502258
\(73\) −164.572 −0.263859 −0.131929 0.991259i \(-0.542117\pi\)
−0.131929 + 0.991259i \(0.542117\pi\)
\(74\) −81.1919 −0.127546
\(75\) 330.868 0.509404
\(76\) −84.7935 −0.127980
\(77\) −958.584 −1.41871
\(78\) 0 0
\(79\) −327.242 −0.466046 −0.233023 0.972471i \(-0.574862\pi\)
−0.233023 + 0.972471i \(0.574862\pi\)
\(80\) 964.925 1.34852
\(81\) 81.0000 0.111111
\(82\) −99.2916 −0.133719
\(83\) −1404.48 −1.85737 −0.928684 0.370871i \(-0.879059\pi\)
−0.928684 + 0.370871i \(0.879059\pi\)
\(84\) 772.989 1.00405
\(85\) −1199.46 −1.53058
\(86\) 68.9734 0.0864837
\(87\) −572.354 −0.705319
\(88\) −100.894 −0.122220
\(89\) 736.986 0.877756 0.438878 0.898547i \(-0.355376\pi\)
0.438878 + 0.898547i \(0.355376\pi\)
\(90\) 29.5018 0.0345529
\(91\) 0 0
\(92\) 213.803 0.242289
\(93\) −384.324 −0.428522
\(94\) −53.1107 −0.0582761
\(95\) 163.516 0.176593
\(96\) 122.156 0.129869
\(97\) 1494.87 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(98\) 150.935 0.155578
\(99\) 266.333 0.270378
\(100\) −877.277 −0.877277
\(101\) −484.010 −0.476839 −0.238420 0.971162i \(-0.576629\pi\)
−0.238420 + 0.971162i \(0.576629\pi\)
\(102\) −50.1313 −0.0486641
\(103\) 214.049 0.204766 0.102383 0.994745i \(-0.467353\pi\)
0.102383 + 0.994745i \(0.467353\pi\)
\(104\) 0 0
\(105\) −1490.63 −1.38544
\(106\) 158.174 0.144936
\(107\) 148.288 0.133977 0.0669884 0.997754i \(-0.478661\pi\)
0.0669884 + 0.997754i \(0.478661\pi\)
\(108\) −214.767 −0.191351
\(109\) −504.881 −0.443659 −0.221829 0.975086i \(-0.571203\pi\)
−0.221829 + 0.975086i \(0.571203\pi\)
\(110\) 97.0035 0.0840811
\(111\) −1139.80 −0.974642
\(112\) −2037.70 −1.71915
\(113\) −1136.49 −0.946120 −0.473060 0.881030i \(-0.656851\pi\)
−0.473060 + 0.881030i \(0.656851\pi\)
\(114\) 6.83415 0.00561471
\(115\) −412.299 −0.334322
\(116\) 1517.56 1.21467
\(117\) 0 0
\(118\) −72.8019 −0.0567962
\(119\) 2532.98 1.95124
\(120\) −156.894 −0.119353
\(121\) −455.282 −0.342060
\(122\) −126.261 −0.0936979
\(123\) −1393.89 −1.02181
\(124\) 1019.01 0.737985
\(125\) −225.651 −0.161463
\(126\) −62.3010 −0.0440493
\(127\) −1286.84 −0.899122 −0.449561 0.893250i \(-0.648420\pi\)
−0.449561 + 0.893250i \(0.648420\pi\)
\(128\) −431.433 −0.297919
\(129\) 968.275 0.660867
\(130\) 0 0
\(131\) 2229.88 1.48722 0.743610 0.668614i \(-0.233112\pi\)
0.743610 + 0.668614i \(0.233112\pi\)
\(132\) −706.166 −0.465636
\(133\) −345.308 −0.225128
\(134\) −72.8241 −0.0469481
\(135\) 414.157 0.264037
\(136\) 266.604 0.168096
\(137\) 474.745 0.296060 0.148030 0.988983i \(-0.452707\pi\)
0.148030 + 0.988983i \(0.452707\pi\)
\(138\) −17.2320 −0.0106296
\(139\) 1927.02 1.17588 0.587941 0.808904i \(-0.299939\pi\)
0.587941 + 0.808904i \(0.299939\pi\)
\(140\) 3952.33 2.38595
\(141\) −745.588 −0.445318
\(142\) −7.74113 −0.00457480
\(143\) 0 0
\(144\) 566.155 0.327636
\(145\) −2926.47 −1.67607
\(146\) −35.1690 −0.0199357
\(147\) 2118.88 1.18886
\(148\) 3022.12 1.67849
\(149\) −1065.73 −0.585959 −0.292979 0.956119i \(-0.594647\pi\)
−0.292979 + 0.956119i \(0.594647\pi\)
\(150\) 70.7064 0.0384877
\(151\) −877.888 −0.473123 −0.236561 0.971617i \(-0.576020\pi\)
−0.236561 + 0.971617i \(0.576020\pi\)
\(152\) −36.3448 −0.0193944
\(153\) −703.762 −0.371868
\(154\) −204.849 −0.107190
\(155\) −1965.07 −1.01831
\(156\) 0 0
\(157\) 2314.94 1.17676 0.588382 0.808583i \(-0.299765\pi\)
0.588382 + 0.808583i \(0.299765\pi\)
\(158\) −69.9317 −0.0352118
\(159\) 2220.50 1.10753
\(160\) 624.587 0.308612
\(161\) 870.681 0.426207
\(162\) 17.3097 0.00839492
\(163\) −331.399 −0.159246 −0.0796232 0.996825i \(-0.525372\pi\)
−0.0796232 + 0.996825i \(0.525372\pi\)
\(164\) 3695.83 1.75973
\(165\) 1361.77 0.642508
\(166\) −300.137 −0.140332
\(167\) 2171.62 1.00626 0.503129 0.864212i \(-0.332182\pi\)
0.503129 + 0.864212i \(0.332182\pi\)
\(168\) 331.324 0.152156
\(169\) 0 0
\(170\) −256.324 −0.115642
\(171\) 95.9404 0.0429049
\(172\) −2567.33 −1.13812
\(173\) −2936.69 −1.29059 −0.645295 0.763933i \(-0.723266\pi\)
−0.645295 + 0.763933i \(0.723266\pi\)
\(174\) −122.312 −0.0532899
\(175\) −3572.57 −1.54321
\(176\) 1861.55 0.797270
\(177\) −1022.02 −0.434010
\(178\) 157.494 0.0663183
\(179\) 1615.57 0.674599 0.337300 0.941397i \(-0.390486\pi\)
0.337300 + 0.941397i \(0.390486\pi\)
\(180\) −1098.11 −0.454714
\(181\) 725.019 0.297736 0.148868 0.988857i \(-0.452437\pi\)
0.148868 + 0.988857i \(0.452437\pi\)
\(182\) 0 0
\(183\) −1772.50 −0.715995
\(184\) 91.6419 0.0367170
\(185\) −5827.87 −2.31607
\(186\) −82.1300 −0.0323767
\(187\) −2314.01 −0.904905
\(188\) 1976.89 0.766911
\(189\) −874.605 −0.336604
\(190\) 34.9433 0.0133424
\(191\) 1717.08 0.650490 0.325245 0.945630i \(-0.394553\pi\)
0.325245 + 0.945630i \(0.394553\pi\)
\(192\) −1483.64 −0.557670
\(193\) −435.830 −0.162548 −0.0812740 0.996692i \(-0.525899\pi\)
−0.0812740 + 0.996692i \(0.525899\pi\)
\(194\) 319.454 0.118224
\(195\) 0 0
\(196\) −5618.08 −2.04741
\(197\) −694.556 −0.251193 −0.125597 0.992081i \(-0.540085\pi\)
−0.125597 + 0.992081i \(0.540085\pi\)
\(198\) 56.9153 0.0204282
\(199\) 2899.24 1.03277 0.516386 0.856356i \(-0.327277\pi\)
0.516386 + 0.856356i \(0.327277\pi\)
\(200\) −376.024 −0.132945
\(201\) −1022.33 −0.358755
\(202\) −103.433 −0.0360273
\(203\) 6180.04 2.13672
\(204\) 1865.99 0.640417
\(205\) −7127.04 −2.42817
\(206\) 45.7423 0.0154709
\(207\) −241.910 −0.0812265
\(208\) 0 0
\(209\) 315.457 0.104405
\(210\) −318.548 −0.104676
\(211\) 5250.30 1.71301 0.856507 0.516136i \(-0.172630\pi\)
0.856507 + 0.516136i \(0.172630\pi\)
\(212\) −5887.53 −1.90735
\(213\) −108.673 −0.0349584
\(214\) 31.6891 0.0101225
\(215\) 4950.84 1.57044
\(216\) −92.0549 −0.0289979
\(217\) 4149.77 1.29818
\(218\) −107.893 −0.0335203
\(219\) −493.716 −0.152339
\(220\) −3610.66 −1.10650
\(221\) 0 0
\(222\) −243.576 −0.0736384
\(223\) 1382.46 0.415141 0.207570 0.978220i \(-0.433444\pi\)
0.207570 + 0.978220i \(0.433444\pi\)
\(224\) −1318.99 −0.393431
\(225\) 992.603 0.294105
\(226\) −242.867 −0.0714835
\(227\) −1223.57 −0.357760 −0.178880 0.983871i \(-0.557247\pi\)
−0.178880 + 0.983871i \(0.557247\pi\)
\(228\) −254.380 −0.0738893
\(229\) 4019.61 1.15993 0.579964 0.814642i \(-0.303067\pi\)
0.579964 + 0.814642i \(0.303067\pi\)
\(230\) −88.1082 −0.0252595
\(231\) −2875.75 −0.819094
\(232\) 650.468 0.184075
\(233\) 1658.40 0.466290 0.233145 0.972442i \(-0.425098\pi\)
0.233145 + 0.972442i \(0.425098\pi\)
\(234\) 0 0
\(235\) −3812.23 −1.05822
\(236\) 2709.83 0.747436
\(237\) −981.727 −0.269072
\(238\) 541.297 0.147425
\(239\) 618.554 0.167410 0.0837049 0.996491i \(-0.473325\pi\)
0.0837049 + 0.996491i \(0.473325\pi\)
\(240\) 2894.77 0.778570
\(241\) 1135.25 0.303434 0.151717 0.988424i \(-0.451520\pi\)
0.151717 + 0.988424i \(0.451520\pi\)
\(242\) −97.2938 −0.0258441
\(243\) 243.000 0.0641500
\(244\) 4699.69 1.23306
\(245\) 10833.9 2.82512
\(246\) −297.875 −0.0772025
\(247\) 0 0
\(248\) 436.777 0.111836
\(249\) −4213.44 −1.07235
\(250\) −48.2215 −0.0121992
\(251\) 2287.83 0.575326 0.287663 0.957732i \(-0.407122\pi\)
0.287663 + 0.957732i \(0.407122\pi\)
\(252\) 2318.97 0.579687
\(253\) −795.414 −0.197657
\(254\) −274.997 −0.0679326
\(255\) −3598.37 −0.883681
\(256\) 3864.18 0.943403
\(257\) −5113.49 −1.24113 −0.620566 0.784154i \(-0.713097\pi\)
−0.620566 + 0.784154i \(0.713097\pi\)
\(258\) 206.920 0.0499314
\(259\) 12307.1 2.95262
\(260\) 0 0
\(261\) −1717.06 −0.407216
\(262\) 476.526 0.112366
\(263\) 2760.17 0.647145 0.323573 0.946203i \(-0.395116\pi\)
0.323573 + 0.946203i \(0.395116\pi\)
\(264\) −302.682 −0.0705635
\(265\) 11353.5 2.63185
\(266\) −73.7923 −0.0170094
\(267\) 2210.96 0.506773
\(268\) 2710.66 0.617834
\(269\) −3310.32 −0.750312 −0.375156 0.926962i \(-0.622411\pi\)
−0.375156 + 0.926962i \(0.622411\pi\)
\(270\) 88.5053 0.0199491
\(271\) −2522.26 −0.565374 −0.282687 0.959212i \(-0.591226\pi\)
−0.282687 + 0.959212i \(0.591226\pi\)
\(272\) −4918.99 −1.09654
\(273\) 0 0
\(274\) 101.453 0.0223686
\(275\) 3263.74 0.715675
\(276\) 641.410 0.139885
\(277\) −3455.48 −0.749530 −0.374765 0.927120i \(-0.622277\pi\)
−0.374765 + 0.927120i \(0.622277\pi\)
\(278\) 411.804 0.0888430
\(279\) −1152.97 −0.247407
\(280\) 1694.07 0.361572
\(281\) 981.649 0.208400 0.104200 0.994556i \(-0.466772\pi\)
0.104200 + 0.994556i \(0.466772\pi\)
\(282\) −159.332 −0.0336457
\(283\) 5331.10 1.11979 0.559896 0.828563i \(-0.310841\pi\)
0.559896 + 0.828563i \(0.310841\pi\)
\(284\) 288.140 0.0602041
\(285\) 490.547 0.101956
\(286\) 0 0
\(287\) 15050.7 3.09552
\(288\) 366.467 0.0749801
\(289\) 1201.58 0.244572
\(290\) −625.387 −0.126634
\(291\) 4484.62 0.903412
\(292\) 1309.06 0.262353
\(293\) 2420.39 0.482597 0.241299 0.970451i \(-0.422427\pi\)
0.241299 + 0.970451i \(0.422427\pi\)
\(294\) 452.804 0.0898233
\(295\) −5225.64 −1.03135
\(296\) 1295.36 0.254363
\(297\) 798.998 0.156103
\(298\) −227.746 −0.0442717
\(299\) 0 0
\(300\) −2631.83 −0.506496
\(301\) −10455.0 −2.00205
\(302\) −187.605 −0.0357465
\(303\) −1452.03 −0.275303
\(304\) 670.581 0.126515
\(305\) −9062.88 −1.70144
\(306\) −150.394 −0.0280962
\(307\) 875.509 0.162762 0.0813810 0.996683i \(-0.474067\pi\)
0.0813810 + 0.996683i \(0.474067\pi\)
\(308\) 7624.90 1.41061
\(309\) 642.147 0.118222
\(310\) −419.935 −0.0769377
\(311\) 6419.10 1.17040 0.585199 0.810890i \(-0.301016\pi\)
0.585199 + 0.810890i \(0.301016\pi\)
\(312\) 0 0
\(313\) −5015.58 −0.905742 −0.452871 0.891576i \(-0.649600\pi\)
−0.452871 + 0.891576i \(0.649600\pi\)
\(314\) 494.701 0.0889096
\(315\) −4471.90 −0.799882
\(316\) 2602.99 0.463386
\(317\) −5228.81 −0.926433 −0.463217 0.886245i \(-0.653305\pi\)
−0.463217 + 0.886245i \(0.653305\pi\)
\(318\) 474.521 0.0836786
\(319\) −5645.80 −0.990922
\(320\) −7585.92 −1.32521
\(321\) 444.863 0.0773515
\(322\) 186.064 0.0322018
\(323\) −833.570 −0.143595
\(324\) −644.301 −0.110477
\(325\) 0 0
\(326\) −70.8199 −0.0120318
\(327\) −1514.64 −0.256146
\(328\) 1584.13 0.266674
\(329\) 8050.56 1.34906
\(330\) 291.010 0.0485442
\(331\) 3186.81 0.529193 0.264596 0.964359i \(-0.414761\pi\)
0.264596 + 0.964359i \(0.414761\pi\)
\(332\) 11171.7 1.84677
\(333\) −3419.41 −0.562710
\(334\) 464.075 0.0760271
\(335\) −5227.23 −0.852520
\(336\) −6113.11 −0.992551
\(337\) −5600.69 −0.905309 −0.452655 0.891686i \(-0.649523\pi\)
−0.452655 + 0.891686i \(0.649523\pi\)
\(338\) 0 0
\(339\) −3409.46 −0.546243
\(340\) 9540.87 1.52184
\(341\) −3791.04 −0.602042
\(342\) 20.5024 0.00324165
\(343\) −11768.0 −1.85252
\(344\) −1100.42 −0.172474
\(345\) −1236.90 −0.193021
\(346\) −627.570 −0.0975097
\(347\) −6982.33 −1.08020 −0.540102 0.841599i \(-0.681614\pi\)
−0.540102 + 0.841599i \(0.681614\pi\)
\(348\) 4552.69 0.701293
\(349\) −2872.20 −0.440531 −0.220265 0.975440i \(-0.570692\pi\)
−0.220265 + 0.975440i \(0.570692\pi\)
\(350\) −763.459 −0.116596
\(351\) 0 0
\(352\) 1204.96 0.182457
\(353\) 5104.55 0.769655 0.384827 0.922989i \(-0.374261\pi\)
0.384827 + 0.922989i \(0.374261\pi\)
\(354\) −218.406 −0.0327913
\(355\) −555.650 −0.0830728
\(356\) −5862.23 −0.872746
\(357\) 7598.94 1.12655
\(358\) 345.247 0.0509689
\(359\) 10771.0 1.58348 0.791741 0.610857i \(-0.209175\pi\)
0.791741 + 0.610857i \(0.209175\pi\)
\(360\) −470.681 −0.0689085
\(361\) −6745.36 −0.983433
\(362\) 154.936 0.0224952
\(363\) −1365.85 −0.197489
\(364\) 0 0
\(365\) −2524.39 −0.362008
\(366\) −378.783 −0.0540965
\(367\) −11380.5 −1.61868 −0.809340 0.587340i \(-0.800175\pi\)
−0.809340 + 0.587340i \(0.800175\pi\)
\(368\) −1690.84 −0.239514
\(369\) −4181.68 −0.589945
\(370\) −1245.41 −0.174989
\(371\) −23976.1 −3.35519
\(372\) 3057.04 0.426076
\(373\) 3196.75 0.443758 0.221879 0.975074i \(-0.428781\pi\)
0.221879 + 0.975074i \(0.428781\pi\)
\(374\) −494.504 −0.0683695
\(375\) −676.952 −0.0932204
\(376\) 847.346 0.116220
\(377\) 0 0
\(378\) −186.903 −0.0254319
\(379\) −2050.75 −0.277942 −0.138971 0.990296i \(-0.544379\pi\)
−0.138971 + 0.990296i \(0.544379\pi\)
\(380\) −1300.66 −0.175585
\(381\) −3860.52 −0.519108
\(382\) 366.940 0.0491473
\(383\) 3507.54 0.467955 0.233978 0.972242i \(-0.424826\pi\)
0.233978 + 0.972242i \(0.424826\pi\)
\(384\) −1294.30 −0.172004
\(385\) −14703.9 −1.94644
\(386\) −93.1369 −0.0122812
\(387\) 2904.83 0.381552
\(388\) −11890.7 −1.55582
\(389\) −6572.66 −0.856676 −0.428338 0.903618i \(-0.640901\pi\)
−0.428338 + 0.903618i \(0.640901\pi\)
\(390\) 0 0
\(391\) 2101.81 0.271850
\(392\) −2408.06 −0.310269
\(393\) 6689.65 0.858646
\(394\) −148.427 −0.0189787
\(395\) −5019.62 −0.639403
\(396\) −2118.50 −0.268835
\(397\) −5285.98 −0.668251 −0.334126 0.942529i \(-0.608441\pi\)
−0.334126 + 0.942529i \(0.608441\pi\)
\(398\) 619.567 0.0780304
\(399\) −1035.92 −0.129978
\(400\) 6937.86 0.867233
\(401\) −919.541 −0.114513 −0.0572565 0.998360i \(-0.518235\pi\)
−0.0572565 + 0.998360i \(0.518235\pi\)
\(402\) −218.472 −0.0271055
\(403\) 0 0
\(404\) 3849.97 0.474117
\(405\) 1242.47 0.152442
\(406\) 1320.67 0.161438
\(407\) −11243.2 −1.36930
\(408\) 799.811 0.0970504
\(409\) −1539.17 −0.186081 −0.0930403 0.995662i \(-0.529659\pi\)
−0.0930403 + 0.995662i \(0.529659\pi\)
\(410\) −1523.05 −0.183458
\(411\) 1424.23 0.170930
\(412\) −1702.62 −0.203597
\(413\) 11035.4 1.31481
\(414\) −51.6961 −0.00613702
\(415\) −21543.5 −2.54826
\(416\) 0 0
\(417\) 5781.06 0.678896
\(418\) 67.4132 0.00788825
\(419\) 10579.9 1.23357 0.616783 0.787133i \(-0.288436\pi\)
0.616783 + 0.787133i \(0.288436\pi\)
\(420\) 11857.0 1.37753
\(421\) 74.9351 0.00867485 0.00433743 0.999991i \(-0.498619\pi\)
0.00433743 + 0.999991i \(0.498619\pi\)
\(422\) 1121.99 0.129426
\(423\) −2236.77 −0.257105
\(424\) −2523.55 −0.289044
\(425\) −8624.15 −0.984313
\(426\) −23.2234 −0.00264126
\(427\) 19138.8 2.16906
\(428\) −1179.53 −0.133212
\(429\) 0 0
\(430\) 1057.99 0.118653
\(431\) −12165.6 −1.35962 −0.679810 0.733388i \(-0.737938\pi\)
−0.679810 + 0.733388i \(0.737938\pi\)
\(432\) 1698.46 0.189161
\(433\) −2869.23 −0.318445 −0.159222 0.987243i \(-0.550899\pi\)
−0.159222 + 0.987243i \(0.550899\pi\)
\(434\) 886.806 0.0980831
\(435\) −8779.41 −0.967680
\(436\) 4015.99 0.441126
\(437\) −286.530 −0.0313652
\(438\) −105.507 −0.0115099
\(439\) −3845.12 −0.418035 −0.209018 0.977912i \(-0.567027\pi\)
−0.209018 + 0.977912i \(0.567027\pi\)
\(440\) −1547.63 −0.167682
\(441\) 6356.63 0.686387
\(442\) 0 0
\(443\) −3858.30 −0.413799 −0.206900 0.978362i \(-0.566337\pi\)
−0.206900 + 0.978362i \(0.566337\pi\)
\(444\) 9066.37 0.969079
\(445\) 11304.7 1.20426
\(446\) 295.432 0.0313657
\(447\) −3197.18 −0.338303
\(448\) 16019.7 1.68942
\(449\) −5550.51 −0.583395 −0.291698 0.956511i \(-0.594220\pi\)
−0.291698 + 0.956511i \(0.594220\pi\)
\(450\) 212.119 0.0222209
\(451\) −13749.6 −1.43557
\(452\) 9039.99 0.940719
\(453\) −2633.66 −0.273157
\(454\) −261.478 −0.0270303
\(455\) 0 0
\(456\) −109.034 −0.0111974
\(457\) 8102.42 0.829355 0.414677 0.909968i \(-0.363894\pi\)
0.414677 + 0.909968i \(0.363894\pi\)
\(458\) 858.991 0.0876376
\(459\) −2111.29 −0.214698
\(460\) 3279.56 0.332414
\(461\) 9230.86 0.932590 0.466295 0.884629i \(-0.345588\pi\)
0.466295 + 0.884629i \(0.345588\pi\)
\(462\) −614.548 −0.0618861
\(463\) −13934.7 −1.39870 −0.699352 0.714777i \(-0.746528\pi\)
−0.699352 + 0.714777i \(0.746528\pi\)
\(464\) −12001.5 −1.20077
\(465\) −5895.20 −0.587921
\(466\) 354.400 0.0352302
\(467\) 10918.0 1.08185 0.540926 0.841070i \(-0.318074\pi\)
0.540926 + 0.841070i \(0.318074\pi\)
\(468\) 0 0
\(469\) 11038.7 1.08682
\(470\) −814.673 −0.0799533
\(471\) 6944.81 0.679405
\(472\) 1161.51 0.113268
\(473\) 9551.23 0.928470
\(474\) −209.795 −0.0203295
\(475\) 1175.69 0.113567
\(476\) −20148.2 −1.94010
\(477\) 6661.50 0.639432
\(478\) 132.185 0.0126485
\(479\) −11028.4 −1.05198 −0.525992 0.850490i \(-0.676306\pi\)
−0.525992 + 0.850490i \(0.676306\pi\)
\(480\) 1873.76 0.178177
\(481\) 0 0
\(482\) 242.602 0.0229258
\(483\) 2612.04 0.246071
\(484\) 3621.47 0.340108
\(485\) 22930.0 2.14680
\(486\) 51.9291 0.00484681
\(487\) −1078.10 −0.100315 −0.0501576 0.998741i \(-0.515972\pi\)
−0.0501576 + 0.998741i \(0.515972\pi\)
\(488\) 2014.41 0.186861
\(489\) −994.197 −0.0919410
\(490\) 2315.21 0.213450
\(491\) 6572.88 0.604134 0.302067 0.953287i \(-0.402323\pi\)
0.302067 + 0.953287i \(0.402323\pi\)
\(492\) 11087.5 1.01598
\(493\) 14918.6 1.36288
\(494\) 0 0
\(495\) 4085.32 0.370952
\(496\) −8058.77 −0.729535
\(497\) 1173.41 0.105904
\(498\) −900.411 −0.0810208
\(499\) −9956.79 −0.893241 −0.446620 0.894724i \(-0.647373\pi\)
−0.446620 + 0.894724i \(0.647373\pi\)
\(500\) 1794.90 0.160541
\(501\) 6514.86 0.580963
\(502\) 488.910 0.0434683
\(503\) −11965.0 −1.06062 −0.530310 0.847804i \(-0.677924\pi\)
−0.530310 + 0.847804i \(0.677924\pi\)
\(504\) 993.971 0.0878472
\(505\) −7424.29 −0.654211
\(506\) −169.980 −0.0149338
\(507\) 0 0
\(508\) 10235.9 0.893990
\(509\) 7160.44 0.623538 0.311769 0.950158i \(-0.399078\pi\)
0.311769 + 0.950158i \(0.399078\pi\)
\(510\) −768.971 −0.0667659
\(511\) 5330.94 0.461501
\(512\) 4277.24 0.369197
\(513\) 287.821 0.0247712
\(514\) −1092.75 −0.0937729
\(515\) 3283.33 0.280933
\(516\) −7701.98 −0.657095
\(517\) −7354.61 −0.625639
\(518\) 2630.03 0.223083
\(519\) −8810.06 −0.745123
\(520\) 0 0
\(521\) −17213.1 −1.44745 −0.723725 0.690089i \(-0.757572\pi\)
−0.723725 + 0.690089i \(0.757572\pi\)
\(522\) −366.936 −0.0307669
\(523\) −10110.1 −0.845281 −0.422640 0.906297i \(-0.638897\pi\)
−0.422640 + 0.906297i \(0.638897\pi\)
\(524\) −17737.2 −1.47873
\(525\) −10717.7 −0.890971
\(526\) 589.847 0.0488946
\(527\) 10017.5 0.828026
\(528\) 5584.65 0.460304
\(529\) −11444.5 −0.940620
\(530\) 2426.25 0.198848
\(531\) −3066.06 −0.250576
\(532\) 2746.70 0.223843
\(533\) 0 0
\(534\) 472.481 0.0382889
\(535\) 2274.61 0.183813
\(536\) 1161.86 0.0936281
\(537\) 4846.71 0.389480
\(538\) −707.416 −0.0566893
\(539\) 20901.0 1.67026
\(540\) −3294.34 −0.262529
\(541\) 5951.54 0.472970 0.236485 0.971635i \(-0.424005\pi\)
0.236485 + 0.971635i \(0.424005\pi\)
\(542\) −539.007 −0.0427165
\(543\) 2175.06 0.171898
\(544\) −3184.02 −0.250944
\(545\) −7744.43 −0.608688
\(546\) 0 0
\(547\) 5157.62 0.403152 0.201576 0.979473i \(-0.435394\pi\)
0.201576 + 0.979473i \(0.435394\pi\)
\(548\) −3776.28 −0.294370
\(549\) −5317.51 −0.413380
\(550\) 697.460 0.0540724
\(551\) −2033.77 −0.157244
\(552\) 274.926 0.0211986
\(553\) 10600.3 0.815136
\(554\) −738.436 −0.0566302
\(555\) −17483.6 −1.33718
\(556\) −15328.2 −1.16917
\(557\) −23314.4 −1.77354 −0.886772 0.462206i \(-0.847058\pi\)
−0.886772 + 0.462206i \(0.847058\pi\)
\(558\) −246.390 −0.0186927
\(559\) 0 0
\(560\) −31256.6 −2.35863
\(561\) −6942.03 −0.522447
\(562\) 209.778 0.0157455
\(563\) 2329.12 0.174353 0.0871764 0.996193i \(-0.472216\pi\)
0.0871764 + 0.996193i \(0.472216\pi\)
\(564\) 5930.66 0.442776
\(565\) −17432.7 −1.29805
\(566\) 1139.26 0.0846051
\(567\) −2623.81 −0.194338
\(568\) 123.505 0.00912348
\(569\) −17446.6 −1.28541 −0.642706 0.766113i \(-0.722188\pi\)
−0.642706 + 0.766113i \(0.722188\pi\)
\(570\) 104.830 0.00770323
\(571\) −413.526 −0.0303074 −0.0151537 0.999885i \(-0.504824\pi\)
−0.0151537 + 0.999885i \(0.504824\pi\)
\(572\) 0 0
\(573\) 5151.24 0.375560
\(574\) 3216.33 0.233880
\(575\) −2964.45 −0.215002
\(576\) −4450.92 −0.321971
\(577\) −21258.8 −1.53382 −0.766911 0.641754i \(-0.778207\pi\)
−0.766911 + 0.641754i \(0.778207\pi\)
\(578\) 256.778 0.0184785
\(579\) −1307.49 −0.0938471
\(580\) 23278.1 1.66650
\(581\) 45495.0 3.24862
\(582\) 958.362 0.0682567
\(583\) 21903.4 1.55600
\(584\) 561.098 0.0397575
\(585\) 0 0
\(586\) 517.238 0.0364623
\(587\) 2965.69 0.208530 0.104265 0.994550i \(-0.466751\pi\)
0.104265 + 0.994550i \(0.466751\pi\)
\(588\) −16854.2 −1.18207
\(589\) −1365.64 −0.0955349
\(590\) −1116.72 −0.0779230
\(591\) −2083.67 −0.145027
\(592\) −23900.2 −1.65928
\(593\) −17786.8 −1.23173 −0.615865 0.787852i \(-0.711193\pi\)
−0.615865 + 0.787852i \(0.711193\pi\)
\(594\) 170.746 0.0117943
\(595\) 38853.7 2.67705
\(596\) 8477.16 0.582614
\(597\) 8697.72 0.596271
\(598\) 0 0
\(599\) 21796.4 1.48677 0.743387 0.668861i \(-0.233218\pi\)
0.743387 + 0.668861i \(0.233218\pi\)
\(600\) −1128.07 −0.0767557
\(601\) 20468.7 1.38924 0.694622 0.719375i \(-0.255572\pi\)
0.694622 + 0.719375i \(0.255572\pi\)
\(602\) −2234.24 −0.151264
\(603\) −3066.99 −0.207127
\(604\) 6983.01 0.470422
\(605\) −6983.64 −0.469298
\(606\) −310.298 −0.0208004
\(607\) 7852.53 0.525081 0.262541 0.964921i \(-0.415440\pi\)
0.262541 + 0.964921i \(0.415440\pi\)
\(608\) 434.061 0.0289531
\(609\) 18540.1 1.23364
\(610\) −1936.74 −0.128551
\(611\) 0 0
\(612\) 5597.96 0.369745
\(613\) −2460.21 −0.162099 −0.0810497 0.996710i \(-0.525827\pi\)
−0.0810497 + 0.996710i \(0.525827\pi\)
\(614\) 187.096 0.0122974
\(615\) −21381.1 −1.40190
\(616\) 3268.23 0.213768
\(617\) −17829.3 −1.16334 −0.581670 0.813425i \(-0.697600\pi\)
−0.581670 + 0.813425i \(0.697600\pi\)
\(618\) 137.227 0.00893215
\(619\) 16901.7 1.09748 0.548739 0.835994i \(-0.315108\pi\)
0.548739 + 0.835994i \(0.315108\pi\)
\(620\) 15630.8 1.01250
\(621\) −725.729 −0.0468962
\(622\) 1371.76 0.0884286
\(623\) −23873.0 −1.53524
\(624\) 0 0
\(625\) −17247.4 −1.10384
\(626\) −1071.83 −0.0684327
\(627\) 946.372 0.0602783
\(628\) −18413.8 −1.17005
\(629\) 29709.3 1.88328
\(630\) −955.644 −0.0604345
\(631\) −1833.82 −0.115695 −0.0578473 0.998325i \(-0.518424\pi\)
−0.0578473 + 0.998325i \(0.518424\pi\)
\(632\) 1115.71 0.0702226
\(633\) 15750.9 0.989009
\(634\) −1117.40 −0.0699960
\(635\) −19739.0 −1.23357
\(636\) −17662.6 −1.10121
\(637\) 0 0
\(638\) −1206.51 −0.0748684
\(639\) −326.019 −0.0201833
\(640\) −6617.81 −0.408737
\(641\) 29172.1 1.79755 0.898773 0.438414i \(-0.144460\pi\)
0.898773 + 0.438414i \(0.144460\pi\)
\(642\) 95.0672 0.00584424
\(643\) −25103.8 −1.53965 −0.769826 0.638254i \(-0.779657\pi\)
−0.769826 + 0.638254i \(0.779657\pi\)
\(644\) −6925.69 −0.423774
\(645\) 14852.5 0.906693
\(646\) −178.134 −0.0108492
\(647\) −24471.6 −1.48698 −0.743492 0.668745i \(-0.766832\pi\)
−0.743492 + 0.668745i \(0.766832\pi\)
\(648\) −276.165 −0.0167419
\(649\) −10081.4 −0.609752
\(650\) 0 0
\(651\) 12449.3 0.749505
\(652\) 2636.06 0.158337
\(653\) 6632.68 0.397484 0.198742 0.980052i \(-0.436314\pi\)
0.198742 + 0.980052i \(0.436314\pi\)
\(654\) −323.679 −0.0193530
\(655\) 34204.5 2.04043
\(656\) −29228.1 −1.73958
\(657\) −1481.15 −0.0879529
\(658\) 1720.40 0.101928
\(659\) 26436.9 1.56272 0.781361 0.624080i \(-0.214526\pi\)
0.781361 + 0.624080i \(0.214526\pi\)
\(660\) −10832.0 −0.638840
\(661\) −27596.7 −1.62389 −0.811943 0.583737i \(-0.801590\pi\)
−0.811943 + 0.583737i \(0.801590\pi\)
\(662\) 681.020 0.0399828
\(663\) 0 0
\(664\) 4788.49 0.279863
\(665\) −5296.73 −0.308870
\(666\) −730.727 −0.0425152
\(667\) 5128.07 0.297691
\(668\) −17273.8 −1.00051
\(669\) 4147.38 0.239682
\(670\) −1117.06 −0.0644115
\(671\) −17484.3 −1.00592
\(672\) −3956.96 −0.227147
\(673\) 18860.1 1.08024 0.540122 0.841587i \(-0.318378\pi\)
0.540122 + 0.841587i \(0.318378\pi\)
\(674\) −1196.87 −0.0684000
\(675\) 2977.81 0.169801
\(676\) 0 0
\(677\) −14705.5 −0.834825 −0.417412 0.908717i \(-0.637063\pi\)
−0.417412 + 0.908717i \(0.637063\pi\)
\(678\) −728.601 −0.0412710
\(679\) −48423.1 −2.73683
\(680\) 4089.47 0.230624
\(681\) −3670.72 −0.206553
\(682\) −810.145 −0.0454869
\(683\) 17950.2 1.00563 0.502814 0.864395i \(-0.332298\pi\)
0.502814 + 0.864395i \(0.332298\pi\)
\(684\) −763.141 −0.0426600
\(685\) 7282.18 0.406186
\(686\) −2514.83 −0.139966
\(687\) 12058.8 0.669685
\(688\) 20303.5 1.12509
\(689\) 0 0
\(690\) −264.325 −0.0145836
\(691\) 8692.38 0.478544 0.239272 0.970953i \(-0.423091\pi\)
0.239272 + 0.970953i \(0.423091\pi\)
\(692\) 23359.4 1.28322
\(693\) −8627.26 −0.472904
\(694\) −1492.12 −0.0816141
\(695\) 29558.8 1.61328
\(696\) 1951.41 0.106276
\(697\) 36332.2 1.97443
\(698\) −613.789 −0.0332840
\(699\) 4975.20 0.269212
\(700\) 28417.4 1.53440
\(701\) 23275.9 1.25409 0.627047 0.778982i \(-0.284264\pi\)
0.627047 + 0.778982i \(0.284264\pi\)
\(702\) 0 0
\(703\) −4050.11 −0.217287
\(704\) −14634.9 −0.783485
\(705\) −11436.7 −0.610965
\(706\) 1090.84 0.0581507
\(707\) 15678.4 0.834014
\(708\) 8129.49 0.431532
\(709\) 34164.7 1.80971 0.904854 0.425722i \(-0.139980\pi\)
0.904854 + 0.425722i \(0.139980\pi\)
\(710\) −118.742 −0.00627651
\(711\) −2945.18 −0.155349
\(712\) −2512.71 −0.132258
\(713\) 3443.40 0.180864
\(714\) 1623.89 0.0851157
\(715\) 0 0
\(716\) −12850.8 −0.670748
\(717\) 1855.66 0.0966541
\(718\) 2301.75 0.119639
\(719\) −8391.74 −0.435270 −0.217635 0.976030i \(-0.569834\pi\)
−0.217635 + 0.976030i \(0.569834\pi\)
\(720\) 8684.32 0.449508
\(721\) −6933.64 −0.358145
\(722\) −1441.48 −0.0743026
\(723\) 3405.74 0.175188
\(724\) −5767.04 −0.296036
\(725\) −21041.5 −1.07788
\(726\) −291.881 −0.0149211
\(727\) 23500.4 1.19887 0.599436 0.800423i \(-0.295392\pi\)
0.599436 + 0.800423i \(0.295392\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −539.462 −0.0273512
\(731\) −25238.3 −1.27698
\(732\) 14099.1 0.711908
\(733\) −1289.07 −0.0649563 −0.0324781 0.999472i \(-0.510340\pi\)
−0.0324781 + 0.999472i \(0.510340\pi\)
\(734\) −2432.01 −0.122298
\(735\) 32501.7 1.63108
\(736\) −1094.47 −0.0548133
\(737\) −10084.5 −0.504024
\(738\) −893.624 −0.0445729
\(739\) 18197.8 0.905844 0.452922 0.891550i \(-0.350382\pi\)
0.452922 + 0.891550i \(0.350382\pi\)
\(740\) 46356.8 2.30285
\(741\) 0 0
\(742\) −5123.68 −0.253499
\(743\) −21277.2 −1.05059 −0.525294 0.850921i \(-0.676044\pi\)
−0.525294 + 0.850921i \(0.676044\pi\)
\(744\) 1310.33 0.0645686
\(745\) −16347.4 −0.803920
\(746\) 683.146 0.0335278
\(747\) −12640.3 −0.619123
\(748\) 18406.4 0.899739
\(749\) −4803.45 −0.234331
\(750\) −144.665 −0.00704321
\(751\) −25653.4 −1.24648 −0.623240 0.782030i \(-0.714184\pi\)
−0.623240 + 0.782030i \(0.714184\pi\)
\(752\) −15634.0 −0.758130
\(753\) 6863.50 0.332164
\(754\) 0 0
\(755\) −13466.0 −0.649112
\(756\) 6956.90 0.334682
\(757\) 26521.2 1.27336 0.636678 0.771130i \(-0.280308\pi\)
0.636678 + 0.771130i \(0.280308\pi\)
\(758\) −438.245 −0.0209997
\(759\) −2386.24 −0.114117
\(760\) −557.497 −0.0266086
\(761\) −21607.2 −1.02925 −0.514625 0.857415i \(-0.672069\pi\)
−0.514625 + 0.857415i \(0.672069\pi\)
\(762\) −824.992 −0.0392209
\(763\) 16354.5 0.775979
\(764\) −13658.2 −0.646776
\(765\) −10795.1 −0.510193
\(766\) 749.560 0.0353560
\(767\) 0 0
\(768\) 11592.5 0.544674
\(769\) 17179.2 0.805591 0.402795 0.915290i \(-0.368039\pi\)
0.402795 + 0.915290i \(0.368039\pi\)
\(770\) −3142.21 −0.147062
\(771\) −15340.5 −0.716568
\(772\) 3466.74 0.161620
\(773\) 12822.1 0.596609 0.298304 0.954471i \(-0.403579\pi\)
0.298304 + 0.954471i \(0.403579\pi\)
\(774\) 620.761 0.0288279
\(775\) −14128.9 −0.654873
\(776\) −5096.67 −0.235773
\(777\) 36921.4 1.70469
\(778\) −1404.58 −0.0647256
\(779\) −4952.98 −0.227804
\(780\) 0 0
\(781\) −1071.97 −0.0491140
\(782\) 449.157 0.0205394
\(783\) −5151.18 −0.235106
\(784\) 44430.0 2.02396
\(785\) 35509.1 1.61449
\(786\) 1429.58 0.0648744
\(787\) 35213.4 1.59495 0.797474 0.603354i \(-0.206169\pi\)
0.797474 + 0.603354i \(0.206169\pi\)
\(788\) 5524.73 0.249759
\(789\) 8280.50 0.373629
\(790\) −1072.69 −0.0483097
\(791\) 36813.9 1.65481
\(792\) −908.045 −0.0407399
\(793\) 0 0
\(794\) −1129.61 −0.0504893
\(795\) 34060.6 1.51950
\(796\) −23061.5 −1.02688
\(797\) 7500.15 0.333336 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(798\) −221.377 −0.00982038
\(799\) 19434.0 0.860481
\(800\) 4490.82 0.198468
\(801\) 6632.87 0.292585
\(802\) −196.506 −0.00865195
\(803\) −4870.10 −0.214025
\(804\) 8131.97 0.356707
\(805\) 13355.5 0.584745
\(806\) 0 0
\(807\) −9930.97 −0.433193
\(808\) 1650.20 0.0718489
\(809\) −22020.5 −0.956982 −0.478491 0.878092i \(-0.658816\pi\)
−0.478491 + 0.878092i \(0.658816\pi\)
\(810\) 265.516 0.0115176
\(811\) −20444.6 −0.885210 −0.442605 0.896717i \(-0.645946\pi\)
−0.442605 + 0.896717i \(0.645946\pi\)
\(812\) −49158.1 −2.12452
\(813\) −7566.78 −0.326419
\(814\) −2402.67 −0.103457
\(815\) −5083.37 −0.218482
\(816\) −14757.0 −0.633085
\(817\) 3440.62 0.147334
\(818\) −328.920 −0.0140592
\(819\) 0 0
\(820\) 56690.8 2.41431
\(821\) −40353.0 −1.71538 −0.857692 0.514164i \(-0.828102\pi\)
−0.857692 + 0.514164i \(0.828102\pi\)
\(822\) 304.359 0.0129145
\(823\) −33110.7 −1.40239 −0.701194 0.712971i \(-0.747349\pi\)
−0.701194 + 0.712971i \(0.747349\pi\)
\(824\) −729.787 −0.0308536
\(825\) 9791.21 0.413195
\(826\) 2358.26 0.0993392
\(827\) 43410.4 1.82531 0.912653 0.408735i \(-0.134030\pi\)
0.912653 + 0.408735i \(0.134030\pi\)
\(828\) 1924.23 0.0807629
\(829\) −5502.87 −0.230546 −0.115273 0.993334i \(-0.536774\pi\)
−0.115273 + 0.993334i \(0.536774\pi\)
\(830\) −4603.84 −0.192532
\(831\) −10366.4 −0.432741
\(832\) 0 0
\(833\) −55229.1 −2.29721
\(834\) 1235.41 0.0512935
\(835\) 33310.8 1.38056
\(836\) −2509.25 −0.103809
\(837\) −3458.92 −0.142841
\(838\) 2260.93 0.0932013
\(839\) 1698.27 0.0698819 0.0349409 0.999389i \(-0.488876\pi\)
0.0349409 + 0.999389i \(0.488876\pi\)
\(840\) 5082.22 0.208754
\(841\) 12009.8 0.492425
\(842\) 16.0136 0.000655423 0
\(843\) 2944.95 0.120320
\(844\) −41762.7 −1.70323
\(845\) 0 0
\(846\) −477.996 −0.0194254
\(847\) 14747.9 0.598279
\(848\) 46561.0 1.88551
\(849\) 15993.3 0.646512
\(850\) −1842.98 −0.0743691
\(851\) 10212.2 0.411363
\(852\) 864.421 0.0347589
\(853\) −18686.8 −0.750087 −0.375044 0.927007i \(-0.622372\pi\)
−0.375044 + 0.927007i \(0.622372\pi\)
\(854\) 4089.95 0.163882
\(855\) 1471.64 0.0588645
\(856\) −505.578 −0.0201873
\(857\) −9659.16 −0.385007 −0.192503 0.981296i \(-0.561661\pi\)
−0.192503 + 0.981296i \(0.561661\pi\)
\(858\) 0 0
\(859\) −1559.90 −0.0619595 −0.0309798 0.999520i \(-0.509863\pi\)
−0.0309798 + 0.999520i \(0.509863\pi\)
\(860\) −39380.6 −1.56147
\(861\) 45152.1 1.78720
\(862\) −2599.79 −0.102725
\(863\) 11358.7 0.448034 0.224017 0.974585i \(-0.428083\pi\)
0.224017 + 0.974585i \(0.428083\pi\)
\(864\) 1099.40 0.0432898
\(865\) −45046.3 −1.77066
\(866\) −613.155 −0.0240599
\(867\) 3604.74 0.141204
\(868\) −33008.7 −1.29077
\(869\) −9683.93 −0.378026
\(870\) −1876.16 −0.0731124
\(871\) 0 0
\(872\) 1721.36 0.0668493
\(873\) 13453.8 0.521585
\(874\) −61.2314 −0.00236977
\(875\) 7309.45 0.282405
\(876\) 3927.18 0.151469
\(877\) 33301.6 1.28223 0.641115 0.767444i \(-0.278472\pi\)
0.641115 + 0.767444i \(0.278472\pi\)
\(878\) −821.701 −0.0315844
\(879\) 7261.18 0.278628
\(880\) 28554.6 1.09383
\(881\) 28912.3 1.10565 0.552826 0.833296i \(-0.313549\pi\)
0.552826 + 0.833296i \(0.313549\pi\)
\(882\) 1358.41 0.0518595
\(883\) −49100.9 −1.87132 −0.935661 0.352901i \(-0.885195\pi\)
−0.935661 + 0.352901i \(0.885195\pi\)
\(884\) 0 0
\(885\) −15676.9 −0.595451
\(886\) −824.517 −0.0312643
\(887\) 4566.49 0.172861 0.0864304 0.996258i \(-0.472454\pi\)
0.0864304 + 0.996258i \(0.472454\pi\)
\(888\) 3886.09 0.146857
\(889\) 41684.3 1.57261
\(890\) 2415.82 0.0909870
\(891\) 2396.99 0.0901261
\(892\) −10996.5 −0.412771
\(893\) −2649.33 −0.0992795
\(894\) −683.238 −0.0255603
\(895\) 24781.4 0.925533
\(896\) 13975.3 0.521074
\(897\) 0 0
\(898\) −1186.14 −0.0440780
\(899\) 24441.0 0.906734
\(900\) −7895.49 −0.292426
\(901\) −57877.9 −2.14006
\(902\) −2938.29 −0.108464
\(903\) −31365.1 −1.15589
\(904\) 3874.78 0.142559
\(905\) 11121.2 0.408486
\(906\) −562.814 −0.0206382
\(907\) 48873.3 1.78921 0.894604 0.446860i \(-0.147458\pi\)
0.894604 + 0.446860i \(0.147458\pi\)
\(908\) 9732.72 0.355718
\(909\) −4356.09 −0.158946
\(910\) 0 0
\(911\) 31550.1 1.14742 0.573710 0.819058i \(-0.305503\pi\)
0.573710 + 0.819058i \(0.305503\pi\)
\(912\) 2011.74 0.0730433
\(913\) −41562.1 −1.50658
\(914\) 1731.49 0.0626613
\(915\) −27188.7 −0.982327
\(916\) −31973.3 −1.15331
\(917\) −72232.1 −2.60121
\(918\) −451.182 −0.0162214
\(919\) −11720.3 −0.420694 −0.210347 0.977627i \(-0.567459\pi\)
−0.210347 + 0.977627i \(0.567459\pi\)
\(920\) 1405.71 0.0503748
\(921\) 2626.53 0.0939707
\(922\) 1972.63 0.0704612
\(923\) 0 0
\(924\) 22874.7 0.814418
\(925\) −41902.7 −1.48946
\(926\) −2977.84 −0.105678
\(927\) 1926.44 0.0682553
\(928\) −7768.47 −0.274798
\(929\) 6911.23 0.244080 0.122040 0.992525i \(-0.461056\pi\)
0.122040 + 0.992525i \(0.461056\pi\)
\(930\) −1259.80 −0.0444200
\(931\) 7529.10 0.265044
\(932\) −13191.5 −0.463628
\(933\) 19257.3 0.675730
\(934\) 2333.18 0.0817386
\(935\) −35494.9 −1.24151
\(936\) 0 0
\(937\) −28673.9 −0.999717 −0.499858 0.866107i \(-0.666615\pi\)
−0.499858 + 0.866107i \(0.666615\pi\)
\(938\) 2358.97 0.0821143
\(939\) −15046.7 −0.522930
\(940\) 30323.7 1.05218
\(941\) −22463.6 −0.778207 −0.389103 0.921194i \(-0.627215\pi\)
−0.389103 + 0.921194i \(0.627215\pi\)
\(942\) 1484.10 0.0513320
\(943\) 12488.8 0.431272
\(944\) −21430.4 −0.738878
\(945\) −13415.7 −0.461812
\(946\) 2041.10 0.0701499
\(947\) −36806.6 −1.26299 −0.631497 0.775379i \(-0.717559\pi\)
−0.631497 + 0.775379i \(0.717559\pi\)
\(948\) 7808.98 0.267536
\(949\) 0 0
\(950\) 251.244 0.00858046
\(951\) −15686.4 −0.534876
\(952\) −8636.04 −0.294008
\(953\) −1781.07 −0.0605400 −0.0302700 0.999542i \(-0.509637\pi\)
−0.0302700 + 0.999542i \(0.509637\pi\)
\(954\) 1423.56 0.0483119
\(955\) 26338.5 0.892455
\(956\) −4920.19 −0.166454
\(957\) −16937.4 −0.572109
\(958\) −2356.77 −0.0794819
\(959\) −15378.3 −0.517822
\(960\) −22757.8 −0.765108
\(961\) −13379.3 −0.449107
\(962\) 0 0
\(963\) 1334.59 0.0446589
\(964\) −9030.12 −0.301702
\(965\) −6685.26 −0.223012
\(966\) 558.193 0.0185917
\(967\) −54777.2 −1.82163 −0.910815 0.412815i \(-0.864545\pi\)
−0.910815 + 0.412815i \(0.864545\pi\)
\(968\) 1552.26 0.0515408
\(969\) −2500.71 −0.0829044
\(970\) 4900.15 0.162200
\(971\) −2391.54 −0.0790403 −0.0395201 0.999219i \(-0.512583\pi\)
−0.0395201 + 0.999219i \(0.512583\pi\)
\(972\) −1932.90 −0.0637838
\(973\) −62421.5 −2.05667
\(974\) −230.391 −0.00757925
\(975\) 0 0
\(976\) −37167.0 −1.21894
\(977\) 43738.5 1.43226 0.716130 0.697967i \(-0.245912\pi\)
0.716130 + 0.697967i \(0.245912\pi\)
\(978\) −212.460 −0.00694654
\(979\) 21809.3 0.711979
\(980\) −86176.5 −2.80899
\(981\) −4543.92 −0.147886
\(982\) 1404.62 0.0456449
\(983\) −27712.4 −0.899174 −0.449587 0.893236i \(-0.648429\pi\)
−0.449587 + 0.893236i \(0.648429\pi\)
\(984\) 4752.39 0.153964
\(985\) −10653.9 −0.344631
\(986\) 3188.09 0.102971
\(987\) 24151.7 0.778882
\(988\) 0 0
\(989\) −8675.38 −0.278929
\(990\) 873.031 0.0280270
\(991\) 3550.23 0.113801 0.0569005 0.998380i \(-0.481878\pi\)
0.0569005 + 0.998380i \(0.481878\pi\)
\(992\) −5216.37 −0.166956
\(993\) 9560.42 0.305529
\(994\) 250.757 0.00800153
\(995\) 44471.8 1.41694
\(996\) 33515.1 1.06623
\(997\) −28338.4 −0.900188 −0.450094 0.892981i \(-0.648610\pi\)
−0.450094 + 0.892981i \(0.648610\pi\)
\(998\) −2127.76 −0.0674882
\(999\) −10258.2 −0.324881
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 507.4.a.o.1.6 9
3.2 odd 2 1521.4.a.bi.1.4 9
13.5 odd 4 507.4.b.k.337.9 18
13.8 odd 4 507.4.b.k.337.10 18
13.12 even 2 507.4.a.p.1.4 yes 9
39.38 odd 2 1521.4.a.bf.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.6 9 1.1 even 1 trivial
507.4.a.p.1.4 yes 9 13.12 even 2
507.4.b.k.337.9 18 13.5 odd 4
507.4.b.k.337.10 18 13.8 odd 4
1521.4.a.bf.1.6 9 39.38 odd 2
1521.4.a.bi.1.4 9 3.2 odd 2