Properties

Label 507.4.b.h.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-4.33039i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.h.337.8

$q$-expansion

\(f(q)\) \(=\) \(q-5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} +16.4131i q^{5} +15.9912i q^{6} +9.67968i q^{7} +66.1667i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} +16.4131i q^{5} +15.9912i q^{6} +9.67968i q^{7} +66.1667i q^{8} +9.00000 q^{9} +87.4882 q^{10} +27.5882i q^{11} +61.2393 q^{12} +51.5965 q^{14} -49.2393i q^{15} +189.390 q^{16} -107.928 q^{17} -47.9735i q^{18} +2.24723i q^{19} -335.042i q^{20} -29.0391i q^{21} +147.056 q^{22} -41.8090 q^{23} -198.500i q^{24} -144.390 q^{25} -27.0000 q^{27} -197.592i q^{28} +61.6213 q^{29} -262.465 q^{30} -191.932i q^{31} -480.187i q^{32} -82.7645i q^{33} +575.300i q^{34} -158.874 q^{35} -183.718 q^{36} +98.4236i q^{37} +11.9786 q^{38} -1086.00 q^{40} +30.7452i q^{41} -154.790 q^{42} -238.325 q^{43} -563.160i q^{44} +147.718i q^{45} +222.858i q^{46} -511.482i q^{47} -568.169 q^{48} +249.304 q^{49} +769.653i q^{50} +323.785 q^{51} +492.825 q^{53} +143.921i q^{54} -452.807 q^{55} -640.472 q^{56} -6.74170i q^{57} -328.466i q^{58} +484.179i q^{59} +1005.13i q^{60} -444.021 q^{61} -1023.07 q^{62} +87.1172i q^{63} -1044.47 q^{64} -441.167 q^{66} -190.114i q^{67} +2203.15 q^{68} +125.427 q^{69} +846.858i q^{70} -484.785i q^{71} +595.500i q^{72} -957.780i q^{73} +524.636 q^{74} +433.169 q^{75} -45.8729i q^{76} -267.045 q^{77} -375.216 q^{79} +3108.47i q^{80} +81.0000 q^{81} +163.884 q^{82} +715.765i q^{83} +592.777i q^{84} -1771.43i q^{85} +1270.37i q^{86} -184.864 q^{87} -1825.42 q^{88} -1038.15i q^{89} +787.394 q^{90} +853.451 q^{92} +575.796i q^{93} -2726.40 q^{94} -36.8840 q^{95} +1440.56i q^{96} -65.5636i q^{97} -1328.89i q^{98} +248.293i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.33039i − 1.88458i −0.334800 0.942289i \(-0.608669\pi\)
0.334800 0.942289i \(-0.391331\pi\)
\(3\) −3.00000 −0.577350
\(4\) −20.4131 −2.55164
\(5\) 16.4131i 1.46803i 0.679132 + 0.734016i \(0.262356\pi\)
−0.679132 + 0.734016i \(0.737644\pi\)
\(6\) 15.9912i 1.08806i
\(7\) 9.67968i 0.522654i 0.965250 + 0.261327i \(0.0841601\pi\)
−0.965250 + 0.261327i \(0.915840\pi\)
\(8\) 66.1667i 2.92418i
\(9\) 9.00000 0.333333
\(10\) 87.4882 2.76662
\(11\) 27.5882i 0.756195i 0.925766 + 0.378098i \(0.123422\pi\)
−0.925766 + 0.378098i \(0.876578\pi\)
\(12\) 61.2393 1.47319
\(13\) 0 0
\(14\) 51.5965 0.984982
\(15\) − 49.2393i − 0.847568i
\(16\) 189.390 2.95921
\(17\) −107.928 −1.53979 −0.769895 0.638171i \(-0.779691\pi\)
−0.769895 + 0.638171i \(0.779691\pi\)
\(18\) − 47.9735i − 0.628193i
\(19\) 2.24723i 0.0271342i 0.999908 + 0.0135671i \(0.00431868\pi\)
−0.999908 + 0.0135671i \(0.995681\pi\)
\(20\) − 335.042i − 3.74588i
\(21\) − 29.0391i − 0.301754i
\(22\) 147.056 1.42511
\(23\) −41.8090 −0.379034 −0.189517 0.981877i \(-0.560692\pi\)
−0.189517 + 0.981877i \(0.560692\pi\)
\(24\) − 198.500i − 1.68828i
\(25\) −144.390 −1.15512
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) − 197.592i − 1.33362i
\(29\) 61.6213 0.394579 0.197289 0.980345i \(-0.436786\pi\)
0.197289 + 0.980345i \(0.436786\pi\)
\(30\) −262.465 −1.59731
\(31\) − 191.932i − 1.11200i −0.831182 0.556000i \(-0.812335\pi\)
0.831182 0.556000i \(-0.187665\pi\)
\(32\) − 480.187i − 2.65269i
\(33\) − 82.7645i − 0.436589i
\(34\) 575.300i 2.90185i
\(35\) −158.874 −0.767272
\(36\) −183.718 −0.850545
\(37\) 98.4236i 0.437317i 0.975801 + 0.218659i \(0.0701681\pi\)
−0.975801 + 0.218659i \(0.929832\pi\)
\(38\) 11.9786 0.0511366
\(39\) 0 0
\(40\) −1086.00 −4.29279
\(41\) 30.7452i 0.117112i 0.998284 + 0.0585561i \(0.0186496\pi\)
−0.998284 + 0.0585561i \(0.981350\pi\)
\(42\) −154.790 −0.568680
\(43\) −238.325 −0.845216 −0.422608 0.906313i \(-0.638885\pi\)
−0.422608 + 0.906313i \(0.638885\pi\)
\(44\) − 563.160i − 1.92953i
\(45\) 147.718i 0.489344i
\(46\) 222.858i 0.714319i
\(47\) − 511.482i − 1.58739i −0.608316 0.793695i \(-0.708155\pi\)
0.608316 0.793695i \(-0.291845\pi\)
\(48\) −568.169 −1.70850
\(49\) 249.304 0.726833
\(50\) 769.653i 2.17691i
\(51\) 323.785 0.888998
\(52\) 0 0
\(53\) 492.825 1.27726 0.638630 0.769514i \(-0.279502\pi\)
0.638630 + 0.769514i \(0.279502\pi\)
\(54\) 143.921i 0.362687i
\(55\) −452.807 −1.11012
\(56\) −640.472 −1.52833
\(57\) − 6.74170i − 0.0156660i
\(58\) − 328.466i − 0.743615i
\(59\) 484.179i 1.06838i 0.845363 + 0.534192i \(0.179384\pi\)
−0.845363 + 0.534192i \(0.820616\pi\)
\(60\) 1005.13i 2.16269i
\(61\) −444.021 −0.931985 −0.465993 0.884789i \(-0.654303\pi\)
−0.465993 + 0.884789i \(0.654303\pi\)
\(62\) −1023.07 −2.09565
\(63\) 87.1172i 0.174218i
\(64\) −1044.47 −2.03998
\(65\) 0 0
\(66\) −441.167 −0.822787
\(67\) − 190.114i − 0.346658i −0.984864 0.173329i \(-0.944548\pi\)
0.984864 0.173329i \(-0.0554524\pi\)
\(68\) 2203.15 3.92898
\(69\) 125.427 0.218835
\(70\) 846.858i 1.44598i
\(71\) − 484.785i − 0.810329i −0.914244 0.405164i \(-0.867214\pi\)
0.914244 0.405164i \(-0.132786\pi\)
\(72\) 595.500i 0.974727i
\(73\) − 957.780i − 1.53561i −0.640683 0.767806i \(-0.721349\pi\)
0.640683 0.767806i \(-0.278651\pi\)
\(74\) 524.636 0.824159
\(75\) 433.169 0.666907
\(76\) − 45.8729i − 0.0692367i
\(77\) −267.045 −0.395228
\(78\) 0 0
\(79\) −375.216 −0.534368 −0.267184 0.963646i \(-0.586093\pi\)
−0.267184 + 0.963646i \(0.586093\pi\)
\(80\) 3108.47i 4.34422i
\(81\) 81.0000 0.111111
\(82\) 163.884 0.220707
\(83\) 715.765i 0.946571i 0.880909 + 0.473286i \(0.156932\pi\)
−0.880909 + 0.473286i \(0.843068\pi\)
\(84\) 592.777i 0.769967i
\(85\) − 1771.43i − 2.26046i
\(86\) 1270.37i 1.59288i
\(87\) −184.864 −0.227810
\(88\) −1825.42 −2.21125
\(89\) − 1038.15i − 1.23645i −0.786002 0.618224i \(-0.787852\pi\)
0.786002 0.618224i \(-0.212148\pi\)
\(90\) 787.394 0.922207
\(91\) 0 0
\(92\) 853.451 0.967157
\(93\) 575.796i 0.642013i
\(94\) −2726.40 −2.99156
\(95\) −36.8840 −0.0398339
\(96\) 1440.56i 1.53153i
\(97\) − 65.5636i − 0.0686286i −0.999411 0.0343143i \(-0.989075\pi\)
0.999411 0.0343143i \(-0.0109247\pi\)
\(98\) − 1328.89i − 1.36977i
\(99\) 248.293i 0.252065i
\(100\) 2947.44 2.94744
\(101\) −531.798 −0.523920 −0.261960 0.965079i \(-0.584369\pi\)
−0.261960 + 0.965079i \(0.584369\pi\)
\(102\) − 1725.90i − 1.67539i
\(103\) 735.984 0.704064 0.352032 0.935988i \(-0.385491\pi\)
0.352032 + 0.935988i \(0.385491\pi\)
\(104\) 0 0
\(105\) 476.621 0.442985
\(106\) − 2626.95i − 2.40710i
\(107\) −783.265 −0.707673 −0.353837 0.935307i \(-0.615123\pi\)
−0.353837 + 0.935307i \(0.615123\pi\)
\(108\) 551.153 0.491063
\(109\) 532.339i 0.467788i 0.972262 + 0.233894i \(0.0751468\pi\)
−0.972262 + 0.233894i \(0.924853\pi\)
\(110\) 2413.64i 2.09210i
\(111\) − 295.271i − 0.252485i
\(112\) 1833.23i 1.54664i
\(113\) −180.589 −0.150340 −0.0751699 0.997171i \(-0.523950\pi\)
−0.0751699 + 0.997171i \(0.523950\pi\)
\(114\) −35.9359 −0.0295237
\(115\) − 686.215i − 0.556434i
\(116\) −1257.88 −1.00682
\(117\) 0 0
\(118\) 2580.86 2.01346
\(119\) − 1044.71i − 0.804777i
\(120\) 3258.00 2.47844
\(121\) 569.893 0.428169
\(122\) 2366.81i 1.75640i
\(123\) − 92.2357i − 0.0676147i
\(124\) 3917.92i 2.83742i
\(125\) − 318.242i − 0.227716i
\(126\) 464.369 0.328327
\(127\) −1431.63 −1.00029 −0.500146 0.865941i \(-0.666720\pi\)
−0.500146 + 0.865941i \(0.666720\pi\)
\(128\) 1725.94i 1.19182i
\(129\) 714.976 0.487986
\(130\) 0 0
\(131\) 2067.32 1.37880 0.689400 0.724381i \(-0.257874\pi\)
0.689400 + 0.724381i \(0.257874\pi\)
\(132\) 1689.48i 1.11402i
\(133\) −21.7525 −0.0141818
\(134\) −1013.38 −0.653304
\(135\) − 443.153i − 0.282523i
\(136\) − 7141.25i − 4.50262i
\(137\) − 387.512i − 0.241660i −0.992673 0.120830i \(-0.961444\pi\)
0.992673 0.120830i \(-0.0385555\pi\)
\(138\) − 668.575i − 0.412412i
\(139\) 752.568 0.459223 0.229611 0.973282i \(-0.426254\pi\)
0.229611 + 0.973282i \(0.426254\pi\)
\(140\) 3243.10 1.95780
\(141\) 1534.45i 0.916480i
\(142\) −2584.09 −1.52713
\(143\) 0 0
\(144\) 1704.51 0.986404
\(145\) 1011.40i 0.579254i
\(146\) −5105.34 −2.89398
\(147\) −747.911 −0.419637
\(148\) − 2009.13i − 1.11587i
\(149\) 2636.72i 1.44972i 0.688895 + 0.724862i \(0.258096\pi\)
−0.688895 + 0.724862i \(0.741904\pi\)
\(150\) − 2308.96i − 1.25684i
\(151\) − 3332.42i − 1.79595i −0.440046 0.897975i \(-0.645038\pi\)
0.440046 0.897975i \(-0.354962\pi\)
\(152\) −148.692 −0.0793454
\(153\) −971.354 −0.513263
\(154\) 1423.45i 0.744839i
\(155\) 3150.20 1.63245
\(156\) 0 0
\(157\) −1625.26 −0.826179 −0.413089 0.910690i \(-0.635550\pi\)
−0.413089 + 0.910690i \(0.635550\pi\)
\(158\) 2000.05i 1.00706i
\(159\) −1478.48 −0.737426
\(160\) 7881.36 3.89423
\(161\) − 404.698i − 0.198104i
\(162\) − 431.762i − 0.209398i
\(163\) − 1835.37i − 0.881944i −0.897521 0.440972i \(-0.854634\pi\)
0.897521 0.440972i \(-0.145366\pi\)
\(164\) − 627.605i − 0.298828i
\(165\) 1358.42 0.640927
\(166\) 3815.31 1.78389
\(167\) 1945.00i 0.901248i 0.892714 + 0.450624i \(0.148798\pi\)
−0.892714 + 0.450624i \(0.851202\pi\)
\(168\) 1921.42 0.882384
\(169\) 0 0
\(170\) −9442.44 −4.26001
\(171\) 20.2251i 0.00904474i
\(172\) 4864.96 2.15668
\(173\) −2531.63 −1.11258 −0.556289 0.830989i \(-0.687775\pi\)
−0.556289 + 0.830989i \(0.687775\pi\)
\(174\) 985.397i 0.429326i
\(175\) − 1397.65i − 0.603726i
\(176\) 5224.91i 2.23774i
\(177\) − 1452.54i − 0.616832i
\(178\) −5533.76 −2.33018
\(179\) −4263.01 −1.78007 −0.890035 0.455892i \(-0.849320\pi\)
−0.890035 + 0.455892i \(0.849320\pi\)
\(180\) − 3015.38i − 1.24863i
\(181\) −3944.61 −1.61989 −0.809946 0.586504i \(-0.800504\pi\)
−0.809946 + 0.586504i \(0.800504\pi\)
\(182\) 0 0
\(183\) 1332.06 0.538082
\(184\) − 2766.36i − 1.10836i
\(185\) −1615.43 −0.641995
\(186\) 3069.22 1.20992
\(187\) − 2977.54i − 1.16438i
\(188\) 10440.9i 4.05044i
\(189\) − 261.351i − 0.100585i
\(190\) 196.606i 0.0750701i
\(191\) −214.109 −0.0811119 −0.0405559 0.999177i \(-0.512913\pi\)
−0.0405559 + 0.999177i \(0.512913\pi\)
\(192\) 3133.42 1.17779
\(193\) − 1207.19i − 0.450234i −0.974332 0.225117i \(-0.927724\pi\)
0.974332 0.225117i \(-0.0722764\pi\)
\(194\) −349.480 −0.129336
\(195\) 0 0
\(196\) −5089.06 −1.85461
\(197\) 927.631i 0.335487i 0.985831 + 0.167744i \(0.0536481\pi\)
−0.985831 + 0.167744i \(0.946352\pi\)
\(198\) 1323.50 0.475036
\(199\) −478.951 −0.170613 −0.0853064 0.996355i \(-0.527187\pi\)
−0.0853064 + 0.996355i \(0.527187\pi\)
\(200\) − 9553.77i − 3.37777i
\(201\) 570.341i 0.200143i
\(202\) 2834.69i 0.987368i
\(203\) 596.474i 0.206228i
\(204\) −6609.44 −2.26840
\(205\) −504.624 −0.171924
\(206\) − 3923.08i − 1.32686i
\(207\) −376.281 −0.126345
\(208\) 0 0
\(209\) −61.9970 −0.0205188
\(210\) − 2540.58i − 0.834840i
\(211\) −1450.95 −0.473402 −0.236701 0.971583i \(-0.576066\pi\)
−0.236701 + 0.971583i \(0.576066\pi\)
\(212\) −10060.1 −3.25910
\(213\) 1454.35i 0.467843i
\(214\) 4175.11i 1.33367i
\(215\) − 3911.66i − 1.24080i
\(216\) − 1786.50i − 0.562759i
\(217\) 1857.84 0.581191
\(218\) 2837.58 0.881583
\(219\) 2873.34i 0.886586i
\(220\) 9243.19 2.83262
\(221\) 0 0
\(222\) −1573.91 −0.475828
\(223\) 2059.79i 0.618536i 0.950975 + 0.309268i \(0.100084\pi\)
−0.950975 + 0.309268i \(0.899916\pi\)
\(224\) 4648.06 1.38644
\(225\) −1299.51 −0.385039
\(226\) 962.612i 0.283327i
\(227\) − 4482.46i − 1.31062i −0.755359 0.655311i \(-0.772537\pi\)
0.755359 0.655311i \(-0.227463\pi\)
\(228\) 137.619i 0.0399738i
\(229\) − 1630.39i − 0.470477i −0.971938 0.235239i \(-0.924413\pi\)
0.971938 0.235239i \(-0.0755872\pi\)
\(230\) −3657.80 −1.04864
\(231\) 801.134 0.228185
\(232\) 4077.27i 1.15382i
\(233\) 1903.69 0.535258 0.267629 0.963522i \(-0.413760\pi\)
0.267629 + 0.963522i \(0.413760\pi\)
\(234\) 0 0
\(235\) 8395.00 2.33034
\(236\) − 9883.58i − 2.72613i
\(237\) 1125.65 0.308517
\(238\) −5568.72 −1.51667
\(239\) − 3763.79i − 1.01866i −0.860572 0.509328i \(-0.829894\pi\)
0.860572 0.509328i \(-0.170106\pi\)
\(240\) − 9325.40i − 2.50813i
\(241\) − 3614.74i − 0.966166i −0.875575 0.483083i \(-0.839517\pi\)
0.875575 0.483083i \(-0.160483\pi\)
\(242\) − 3037.75i − 0.806918i
\(243\) −243.000 −0.0641500
\(244\) 9063.85 2.37809
\(245\) 4091.84i 1.06701i
\(246\) −491.652 −0.127425
\(247\) 0 0
\(248\) 12699.5 3.25169
\(249\) − 2147.30i − 0.546503i
\(250\) −1696.36 −0.429148
\(251\) 5729.77 1.44088 0.720438 0.693520i \(-0.243941\pi\)
0.720438 + 0.693520i \(0.243941\pi\)
\(252\) − 1778.33i − 0.444541i
\(253\) − 1153.43i − 0.286624i
\(254\) 7631.18i 1.88513i
\(255\) 5314.30i 1.30508i
\(256\) 844.191 0.206101
\(257\) 5525.79 1.34120 0.670602 0.741818i \(-0.266036\pi\)
0.670602 + 0.741818i \(0.266036\pi\)
\(258\) − 3811.10i − 0.919647i
\(259\) −952.709 −0.228565
\(260\) 0 0
\(261\) 554.591 0.131526
\(262\) − 11019.6i − 2.59846i
\(263\) 5223.21 1.22463 0.612313 0.790615i \(-0.290239\pi\)
0.612313 + 0.790615i \(0.290239\pi\)
\(264\) 5476.25 1.27667
\(265\) 8088.78i 1.87506i
\(266\) 115.949i 0.0267267i
\(267\) 3114.46i 0.713864i
\(268\) 3880.81i 0.884545i
\(269\) 7203.88 1.63282 0.816410 0.577473i \(-0.195961\pi\)
0.816410 + 0.577473i \(0.195961\pi\)
\(270\) −2362.18 −0.532436
\(271\) − 8577.69i − 1.92272i −0.275293 0.961360i \(-0.588775\pi\)
0.275293 0.961360i \(-0.411225\pi\)
\(272\) −20440.5 −4.55656
\(273\) 0 0
\(274\) −2065.59 −0.455427
\(275\) − 3983.44i − 0.873493i
\(276\) −2560.35 −0.558388
\(277\) −7169.19 −1.55507 −0.777536 0.628838i \(-0.783531\pi\)
−0.777536 + 0.628838i \(0.783531\pi\)
\(278\) − 4011.48i − 0.865442i
\(279\) − 1727.39i − 0.370667i
\(280\) − 10512.1i − 2.24364i
\(281\) 849.157i 0.180272i 0.995929 + 0.0901360i \(0.0287302\pi\)
−0.995929 + 0.0901360i \(0.971270\pi\)
\(282\) 8179.20 1.72718
\(283\) −1115.37 −0.234283 −0.117141 0.993115i \(-0.537373\pi\)
−0.117141 + 0.993115i \(0.537373\pi\)
\(284\) 9895.95i 2.06766i
\(285\) 110.652 0.0229981
\(286\) 0 0
\(287\) −297.604 −0.0612091
\(288\) − 4321.69i − 0.884229i
\(289\) 6735.49 1.37095
\(290\) 5391.14 1.09165
\(291\) 196.691i 0.0396227i
\(292\) 19551.2i 3.91832i
\(293\) − 1863.53i − 0.371565i −0.982591 0.185782i \(-0.940518\pi\)
0.982591 0.185782i \(-0.0594819\pi\)
\(294\) 3986.66i 0.790839i
\(295\) −7946.87 −1.56842
\(296\) −6512.36 −1.27879
\(297\) − 744.880i − 0.145530i
\(298\) 14054.8 2.73212
\(299\) 0 0
\(300\) −8842.31 −1.70170
\(301\) − 2306.91i − 0.441755i
\(302\) −17763.1 −3.38461
\(303\) 1595.39 0.302485
\(304\) 425.602i 0.0802959i
\(305\) − 7287.76i − 1.36818i
\(306\) 5177.70i 0.967285i
\(307\) − 6387.50i − 1.18747i −0.804660 0.593736i \(-0.797652\pi\)
0.804660 0.593736i \(-0.202348\pi\)
\(308\) 5451.21 1.00848
\(309\) −2207.95 −0.406492
\(310\) − 16791.8i − 3.07648i
\(311\) 3492.59 0.636806 0.318403 0.947955i \(-0.396853\pi\)
0.318403 + 0.947955i \(0.396853\pi\)
\(312\) 0 0
\(313\) −5912.01 −1.06762 −0.533812 0.845603i \(-0.679241\pi\)
−0.533812 + 0.845603i \(0.679241\pi\)
\(314\) 8663.29i 1.55700i
\(315\) −1429.86 −0.255757
\(316\) 7659.31 1.36351
\(317\) 1677.54i 0.297224i 0.988896 + 0.148612i \(0.0474805\pi\)
−0.988896 + 0.148612i \(0.952519\pi\)
\(318\) 7880.86i 1.38974i
\(319\) 1700.02i 0.298378i
\(320\) − 17143.0i − 2.99476i
\(321\) 2349.79 0.408575
\(322\) −2157.20 −0.373342
\(323\) − 242.540i − 0.0417810i
\(324\) −1653.46 −0.283515
\(325\) 0 0
\(326\) −9783.22 −1.66209
\(327\) − 1597.02i − 0.270077i
\(328\) −2034.31 −0.342457
\(329\) 4950.98 0.829655
\(330\) − 7240.92i − 1.20788i
\(331\) − 2010.31i − 0.333827i −0.985972 0.166913i \(-0.946620\pi\)
0.985972 0.166913i \(-0.0533800\pi\)
\(332\) − 14611.0i − 2.41531i
\(333\) 885.812i 0.145772i
\(334\) 10367.6 1.69847
\(335\) 3120.35 0.508905
\(336\) − 5499.69i − 0.892955i
\(337\) −7139.24 −1.15400 −0.577002 0.816743i \(-0.695778\pi\)
−0.577002 + 0.816743i \(0.695778\pi\)
\(338\) 0 0
\(339\) 541.768 0.0867988
\(340\) 36160.5i 5.76787i
\(341\) 5295.05 0.840889
\(342\) 107.808 0.0170455
\(343\) 5733.31i 0.902536i
\(344\) − 15769.2i − 2.47156i
\(345\) 2058.64i 0.321257i
\(346\) 13494.6i 2.09674i
\(347\) 1.13990 0.000176349 0 8.81743e−5 1.00000i \(-0.499972\pi\)
8.81743e−5 1.00000i \(0.499972\pi\)
\(348\) 3773.64 0.581289
\(349\) 12199.1i 1.87107i 0.353235 + 0.935535i \(0.385082\pi\)
−0.353235 + 0.935535i \(0.614918\pi\)
\(350\) −7450.00 −1.13777
\(351\) 0 0
\(352\) 13247.5 2.00595
\(353\) 10892.3i 1.64232i 0.570698 + 0.821160i \(0.306673\pi\)
−0.570698 + 0.821160i \(0.693327\pi\)
\(354\) −7742.59 −1.16247
\(355\) 7956.81 1.18959
\(356\) 21191.9i 3.15497i
\(357\) 3134.13i 0.464638i
\(358\) 22723.5i 3.35468i
\(359\) − 3525.78i − 0.518339i −0.965832 0.259169i \(-0.916551\pi\)
0.965832 0.259169i \(-0.0834488\pi\)
\(360\) −9773.99 −1.43093
\(361\) 6853.95 0.999264
\(362\) 21026.3i 3.05281i
\(363\) −1709.68 −0.247204
\(364\) 0 0
\(365\) 15720.1 2.25433
\(366\) − 7100.42i − 1.01406i
\(367\) 2383.75 0.339049 0.169525 0.985526i \(-0.445777\pi\)
0.169525 + 0.985526i \(0.445777\pi\)
\(368\) −7918.19 −1.12164
\(369\) 276.707i 0.0390374i
\(370\) 8610.90i 1.20989i
\(371\) 4770.39i 0.667564i
\(372\) − 11753.8i − 1.63818i
\(373\) −13282.2 −1.84377 −0.921885 0.387463i \(-0.873352\pi\)
−0.921885 + 0.387463i \(0.873352\pi\)
\(374\) −15871.5 −2.19437
\(375\) 954.727i 0.131472i
\(376\) 33843.1 4.64181
\(377\) 0 0
\(378\) −1393.11 −0.189560
\(379\) 4436.73i 0.601318i 0.953732 + 0.300659i \(0.0972065\pi\)
−0.953732 + 0.300659i \(0.902793\pi\)
\(380\) 752.917 0.101642
\(381\) 4294.90 0.577519
\(382\) 1141.28i 0.152862i
\(383\) 810.412i 0.108120i 0.998538 + 0.0540602i \(0.0172163\pi\)
−0.998538 + 0.0540602i \(0.982784\pi\)
\(384\) − 5177.83i − 0.688100i
\(385\) − 4383.03i − 0.580207i
\(386\) −6434.78 −0.848501
\(387\) −2144.93 −0.281739
\(388\) 1338.35i 0.175115i
\(389\) −3463.79 −0.451469 −0.225734 0.974189i \(-0.572478\pi\)
−0.225734 + 0.974189i \(0.572478\pi\)
\(390\) 0 0
\(391\) 4512.37 0.583633
\(392\) 16495.6i 2.12539i
\(393\) −6201.96 −0.796050
\(394\) 4944.64 0.632252
\(395\) − 6158.45i − 0.784469i
\(396\) − 5068.44i − 0.643178i
\(397\) 425.405i 0.0537796i 0.999638 + 0.0268898i \(0.00856031\pi\)
−0.999638 + 0.0268898i \(0.991440\pi\)
\(398\) 2553.00i 0.321533i
\(399\) 65.2575 0.00818787
\(400\) −27345.9 −3.41823
\(401\) − 1186.85i − 0.147801i −0.997266 0.0739007i \(-0.976455\pi\)
0.997266 0.0739007i \(-0.0235448\pi\)
\(402\) 3040.14 0.377185
\(403\) 0 0
\(404\) 10855.6 1.33685
\(405\) 1329.46i 0.163115i
\(406\) 3179.44 0.388653
\(407\) −2715.33 −0.330697
\(408\) 21423.7i 2.59959i
\(409\) − 8007.42i − 0.968071i −0.875048 0.484036i \(-0.839170\pi\)
0.875048 0.484036i \(-0.160830\pi\)
\(410\) 2689.84i 0.324005i
\(411\) 1162.54i 0.139522i
\(412\) −15023.7 −1.79652
\(413\) −4686.70 −0.558395
\(414\) 2005.73i 0.238106i
\(415\) −11747.9 −1.38960
\(416\) 0 0
\(417\) −2257.70 −0.265132
\(418\) 330.468i 0.0386692i
\(419\) 6832.46 0.796629 0.398314 0.917249i \(-0.369595\pi\)
0.398314 + 0.917249i \(0.369595\pi\)
\(420\) −9729.30 −1.13034
\(421\) − 10739.6i − 1.24326i −0.783309 0.621632i \(-0.786470\pi\)
0.783309 0.621632i \(-0.213530\pi\)
\(422\) 7734.16i 0.892163i
\(423\) − 4603.34i − 0.529130i
\(424\) 32608.6i 3.73494i
\(425\) 15583.7 1.77864
\(426\) 7752.28 0.881688
\(427\) − 4297.99i − 0.487106i
\(428\) 15988.9 1.80573
\(429\) 0 0
\(430\) −20850.7 −2.33839
\(431\) − 5214.45i − 0.582763i −0.956607 0.291382i \(-0.905885\pi\)
0.956607 0.291382i \(-0.0941150\pi\)
\(432\) −5113.52 −0.569501
\(433\) −8642.24 −0.959168 −0.479584 0.877496i \(-0.659212\pi\)
−0.479584 + 0.877496i \(0.659212\pi\)
\(434\) − 9903.02i − 1.09530i
\(435\) − 3034.19i − 0.334432i
\(436\) − 10866.7i − 1.19362i
\(437\) − 93.9545i − 0.0102848i
\(438\) 15316.0 1.67084
\(439\) 13026.2 1.41619 0.708097 0.706116i \(-0.249554\pi\)
0.708097 + 0.706116i \(0.249554\pi\)
\(440\) − 29960.7i − 3.24619i
\(441\) 2243.73 0.242278
\(442\) 0 0
\(443\) −11533.0 −1.23690 −0.618450 0.785824i \(-0.712239\pi\)
−0.618450 + 0.785824i \(0.712239\pi\)
\(444\) 6027.39i 0.644250i
\(445\) 17039.3 1.81515
\(446\) 10979.5 1.16568
\(447\) − 7910.17i − 0.836998i
\(448\) − 10110.2i − 1.06621i
\(449\) 9882.75i 1.03874i 0.854548 + 0.519372i \(0.173834\pi\)
−0.854548 + 0.519372i \(0.826166\pi\)
\(450\) 6926.88i 0.725636i
\(451\) −848.204 −0.0885596
\(452\) 3686.38 0.383613
\(453\) 9997.26i 1.03689i
\(454\) −23893.3 −2.46997
\(455\) 0 0
\(456\) 446.075 0.0458101
\(457\) 15628.1i 1.59967i 0.600218 + 0.799836i \(0.295080\pi\)
−0.600218 + 0.799836i \(0.704920\pi\)
\(458\) −8690.63 −0.886652
\(459\) 2914.06 0.296333
\(460\) 14007.8i 1.41982i
\(461\) 7747.46i 0.782723i 0.920237 + 0.391361i \(0.127996\pi\)
−0.920237 + 0.391361i \(0.872004\pi\)
\(462\) − 4270.36i − 0.430033i
\(463\) − 333.422i − 0.0334675i −0.999860 0.0167337i \(-0.994673\pi\)
0.999860 0.0167337i \(-0.00532676\pi\)
\(464\) 11670.4 1.16764
\(465\) −9450.59 −0.942496
\(466\) − 10147.4i − 1.00874i
\(467\) −8198.33 −0.812363 −0.406182 0.913792i \(-0.633140\pi\)
−0.406182 + 0.913792i \(0.633140\pi\)
\(468\) 0 0
\(469\) 1840.24 0.181182
\(470\) − 44748.7i − 4.39171i
\(471\) 4875.79 0.476995
\(472\) −32036.5 −3.12415
\(473\) − 6574.96i − 0.639148i
\(474\) − 6000.14i − 0.581425i
\(475\) − 324.477i − 0.0313432i
\(476\) 21325.8i 2.05350i
\(477\) 4435.43 0.425753
\(478\) −20062.5 −1.91974
\(479\) − 6435.88i − 0.613910i −0.951724 0.306955i \(-0.900690\pi\)
0.951724 0.306955i \(-0.0993101\pi\)
\(480\) −23644.1 −2.24833
\(481\) 0 0
\(482\) −19268.0 −1.82082
\(483\) 1214.09i 0.114375i
\(484\) −11633.3 −1.09253
\(485\) 1076.10 0.100749
\(486\) 1295.29i 0.120896i
\(487\) − 8095.37i − 0.753257i −0.926364 0.376629i \(-0.877083\pi\)
0.926364 0.376629i \(-0.122917\pi\)
\(488\) − 29379.4i − 2.72529i
\(489\) 5506.10i 0.509191i
\(490\) 21811.1 2.01087
\(491\) −5116.46 −0.470270 −0.235135 0.971963i \(-0.575553\pi\)
−0.235135 + 0.971963i \(0.575553\pi\)
\(492\) 1882.82i 0.172528i
\(493\) −6650.67 −0.607568
\(494\) 0 0
\(495\) −4075.26 −0.370039
\(496\) − 36349.9i − 3.29064i
\(497\) 4692.56 0.423521
\(498\) −11445.9 −1.02993
\(499\) 18050.7i 1.61936i 0.586870 + 0.809682i \(0.300360\pi\)
−0.586870 + 0.809682i \(0.699640\pi\)
\(500\) 6496.31i 0.581047i
\(501\) − 5834.99i − 0.520336i
\(502\) − 30541.9i − 2.71544i
\(503\) 10531.1 0.933512 0.466756 0.884386i \(-0.345423\pi\)
0.466756 + 0.884386i \(0.345423\pi\)
\(504\) −5764.25 −0.509445
\(505\) − 8728.45i − 0.769131i
\(506\) −6148.25 −0.540165
\(507\) 0 0
\(508\) 29224.1 2.55238
\(509\) 1963.31i 0.170967i 0.996340 + 0.0854834i \(0.0272435\pi\)
−0.996340 + 0.0854834i \(0.972757\pi\)
\(510\) 28327.3 2.45952
\(511\) 9271.00 0.802593
\(512\) 9307.69i 0.803409i
\(513\) − 60.6753i − 0.00522198i
\(514\) − 29454.6i − 2.52760i
\(515\) 12079.8i 1.03359i
\(516\) −14594.9 −1.24516
\(517\) 14110.9 1.20038
\(518\) 5078.31i 0.430750i
\(519\) 7594.89 0.642348
\(520\) 0 0
\(521\) −7044.93 −0.592407 −0.296203 0.955125i \(-0.595721\pi\)
−0.296203 + 0.955125i \(0.595721\pi\)
\(522\) − 2956.19i − 0.247872i
\(523\) −3213.29 −0.268657 −0.134328 0.990937i \(-0.542888\pi\)
−0.134328 + 0.990937i \(0.542888\pi\)
\(524\) −42200.4 −3.51819
\(525\) 4192.94i 0.348561i
\(526\) − 27841.7i − 2.30790i
\(527\) 20714.9i 1.71225i
\(528\) − 15674.7i − 1.29196i
\(529\) −10419.0 −0.856333
\(530\) 43116.4 3.53369
\(531\) 4357.61i 0.356128i
\(532\) 444.036 0.0361868
\(533\) 0 0
\(534\) 16601.3 1.34533
\(535\) − 12855.8i − 1.03889i
\(536\) 12579.2 1.01369
\(537\) 12789.0 1.02772
\(538\) − 38399.5i − 3.07718i
\(539\) 6877.83i 0.549627i
\(540\) 9046.13i 0.720895i
\(541\) 11251.4i 0.894150i 0.894497 + 0.447075i \(0.147534\pi\)
−0.894497 + 0.447075i \(0.852466\pi\)
\(542\) −45722.4 −3.62352
\(543\) 11833.8 0.935245
\(544\) 51825.8i 4.08458i
\(545\) −8737.33 −0.686727
\(546\) 0 0
\(547\) 1533.54 0.119871 0.0599353 0.998202i \(-0.480911\pi\)
0.0599353 + 0.998202i \(0.480911\pi\)
\(548\) 7910.31i 0.616628i
\(549\) −3996.19 −0.310662
\(550\) −21233.3 −1.64617
\(551\) 138.477i 0.0107066i
\(552\) 8299.08i 0.639914i
\(553\) − 3631.97i − 0.279289i
\(554\) 38214.6i 2.93066i
\(555\) 4846.30 0.370656
\(556\) −15362.2 −1.17177
\(557\) 16845.7i 1.28146i 0.767766 + 0.640731i \(0.221369\pi\)
−0.767766 + 0.640731i \(0.778631\pi\)
\(558\) −9207.65 −0.698550
\(559\) 0 0
\(560\) −30089.0 −2.27052
\(561\) 8932.62i 0.672256i
\(562\) 4526.34 0.339737
\(563\) 20820.1 1.55855 0.779273 0.626685i \(-0.215589\pi\)
0.779273 + 0.626685i \(0.215589\pi\)
\(564\) − 31322.8i − 2.33852i
\(565\) − 2964.03i − 0.220704i
\(566\) 5945.37i 0.441524i
\(567\) 784.054i 0.0580726i
\(568\) 32076.6 2.36955
\(569\) −23636.6 −1.74147 −0.870735 0.491752i \(-0.836357\pi\)
−0.870735 + 0.491752i \(0.836357\pi\)
\(570\) − 589.819i − 0.0433417i
\(571\) 26955.1 1.97554 0.987771 0.155913i \(-0.0498319\pi\)
0.987771 + 0.155913i \(0.0498319\pi\)
\(572\) 0 0
\(573\) 642.326 0.0468300
\(574\) 1586.35i 0.115353i
\(575\) 6036.78 0.437828
\(576\) −9400.25 −0.679995
\(577\) − 23499.8i − 1.69551i −0.530388 0.847755i \(-0.677954\pi\)
0.530388 0.847755i \(-0.322046\pi\)
\(578\) − 35902.8i − 2.58367i
\(579\) 3621.56i 0.259943i
\(580\) − 20645.7i − 1.47805i
\(581\) −6928.38 −0.494729
\(582\) 1048.44 0.0746721
\(583\) 13596.1i 0.965857i
\(584\) 63373.1 4.49040
\(585\) 0 0
\(586\) −9933.33 −0.700243
\(587\) − 4637.50i − 0.326082i −0.986619 0.163041i \(-0.947870\pi\)
0.986619 0.163041i \(-0.0521303\pi\)
\(588\) 15267.2 1.07076
\(589\) 431.316 0.0301733
\(590\) 42359.9i 2.95582i
\(591\) − 2782.89i − 0.193694i
\(592\) 18640.4i 1.29411i
\(593\) 12633.5i 0.874869i 0.899250 + 0.437434i \(0.144113\pi\)
−0.899250 + 0.437434i \(0.855887\pi\)
\(594\) −3970.51 −0.274262
\(595\) 17146.9 1.18144
\(596\) − 53823.7i − 3.69917i
\(597\) 1436.85 0.0985033
\(598\) 0 0
\(599\) −18757.1 −1.27946 −0.639730 0.768600i \(-0.720954\pi\)
−0.639730 + 0.768600i \(0.720954\pi\)
\(600\) 28661.3i 1.95016i
\(601\) −3632.98 −0.246576 −0.123288 0.992371i \(-0.539344\pi\)
−0.123288 + 0.992371i \(0.539344\pi\)
\(602\) −12296.8 −0.832522
\(603\) − 1711.02i − 0.115553i
\(604\) 68025.0i 4.58261i
\(605\) 9353.71i 0.628566i
\(606\) − 8504.08i − 0.570057i
\(607\) −12700.0 −0.849219 −0.424610 0.905377i \(-0.639589\pi\)
−0.424610 + 0.905377i \(0.639589\pi\)
\(608\) 1079.09 0.0719786
\(609\) − 1789.42i − 0.119066i
\(610\) −38846.6 −2.57845
\(611\) 0 0
\(612\) 19828.3 1.30966
\(613\) − 21640.1i − 1.42584i −0.701248 0.712918i \(-0.747373\pi\)
0.701248 0.712918i \(-0.252627\pi\)
\(614\) −34047.9 −2.23788
\(615\) 1513.87 0.0992605
\(616\) − 17669.5i − 1.15572i
\(617\) 16541.7i 1.07933i 0.841881 + 0.539663i \(0.181448\pi\)
−0.841881 + 0.539663i \(0.818552\pi\)
\(618\) 11769.2i 0.766066i
\(619\) − 21138.9i − 1.37261i −0.727316 0.686303i \(-0.759233\pi\)
0.727316 0.686303i \(-0.240767\pi\)
\(620\) −64305.2 −4.16542
\(621\) 1128.84 0.0729451
\(622\) − 18616.9i − 1.20011i
\(623\) 10049.0 0.646235
\(624\) 0 0
\(625\) −12825.4 −0.820823
\(626\) 31513.3i 2.01202i
\(627\) 185.991 0.0118465
\(628\) 33176.6 2.10811
\(629\) − 10622.7i − 0.673377i
\(630\) 7621.73i 0.481995i
\(631\) 5489.80i 0.346348i 0.984891 + 0.173174i \(0.0554023\pi\)
−0.984891 + 0.173174i \(0.944598\pi\)
\(632\) − 24826.8i − 1.56259i
\(633\) 4352.86 0.273319
\(634\) 8941.94 0.560141
\(635\) − 23497.5i − 1.46846i
\(636\) 30180.3 1.88164
\(637\) 0 0
\(638\) 9061.76 0.562318
\(639\) − 4363.06i − 0.270110i
\(640\) −28328.1 −1.74963
\(641\) −4297.04 −0.264778 −0.132389 0.991198i \(-0.542265\pi\)
−0.132389 + 0.991198i \(0.542265\pi\)
\(642\) − 12525.3i − 0.769993i
\(643\) − 25696.9i − 1.57603i −0.615655 0.788016i \(-0.711108\pi\)
0.615655 0.788016i \(-0.288892\pi\)
\(644\) 8261.14i 0.505488i
\(645\) 11735.0i 0.716378i
\(646\) −1292.83 −0.0787396
\(647\) −2174.98 −0.132160 −0.0660798 0.997814i \(-0.521049\pi\)
−0.0660798 + 0.997814i \(0.521049\pi\)
\(648\) 5359.50i 0.324909i
\(649\) −13357.6 −0.807907
\(650\) 0 0
\(651\) −5573.52 −0.335551
\(652\) 37465.5i 2.25040i
\(653\) −15454.5 −0.926160 −0.463080 0.886316i \(-0.653256\pi\)
−0.463080 + 0.886316i \(0.653256\pi\)
\(654\) −8512.73 −0.508982
\(655\) 33931.1i 2.02412i
\(656\) 5822.82i 0.346560i
\(657\) − 8620.02i − 0.511870i
\(658\) − 26390.7i − 1.56355i
\(659\) −3148.77 −0.186129 −0.0930643 0.995660i \(-0.529666\pi\)
−0.0930643 + 0.995660i \(0.529666\pi\)
\(660\) −27729.6 −1.63541
\(661\) − 2099.70i − 0.123553i −0.998090 0.0617767i \(-0.980323\pi\)
0.998090 0.0617767i \(-0.0196767\pi\)
\(662\) −10715.7 −0.629122
\(663\) 0 0
\(664\) −47359.8 −2.76795
\(665\) − 357.026i − 0.0208193i
\(666\) 4721.73 0.274720
\(667\) −2576.32 −0.149559
\(668\) − 39703.4i − 2.29966i
\(669\) − 6179.36i − 0.357112i
\(670\) − 16632.7i − 0.959071i
\(671\) − 12249.7i − 0.704763i
\(672\) −13944.2 −0.800460
\(673\) −30970.8 −1.77390 −0.886950 0.461865i \(-0.847181\pi\)
−0.886950 + 0.461865i \(0.847181\pi\)
\(674\) 38055.0i 2.17481i
\(675\) 3898.52 0.222302
\(676\) 0 0
\(677\) 14640.6 0.831141 0.415570 0.909561i \(-0.363582\pi\)
0.415570 + 0.909561i \(0.363582\pi\)
\(678\) − 2887.83i − 0.163579i
\(679\) 634.635 0.0358690
\(680\) 117210. 6.60999
\(681\) 13447.4i 0.756688i
\(682\) − 28224.7i − 1.58472i
\(683\) − 6685.83i − 0.374563i −0.982306 0.187281i \(-0.940032\pi\)
0.982306 0.187281i \(-0.0599676\pi\)
\(684\) − 412.856i − 0.0230789i
\(685\) 6360.27 0.354764
\(686\) 30560.8 1.70090
\(687\) 4891.18i 0.271630i
\(688\) −45136.3 −2.50117
\(689\) 0 0
\(690\) 10973.4 0.605434
\(691\) 30194.1i 1.66228i 0.556062 + 0.831141i \(0.312312\pi\)
−0.556062 + 0.831141i \(0.687688\pi\)
\(692\) 51678.4 2.83890
\(693\) −2403.40 −0.131743
\(694\) − 6.07611i 0 0.000332343i
\(695\) 12352.0i 0.674154i
\(696\) − 12231.8i − 0.666158i
\(697\) − 3318.28i − 0.180328i
\(698\) 65026.0 3.52618
\(699\) −5711.08 −0.309031
\(700\) 28530.3i 1.54049i
\(701\) 30300.9 1.63260 0.816298 0.577631i \(-0.196023\pi\)
0.816298 + 0.577631i \(0.196023\pi\)
\(702\) 0 0
\(703\) −221.181 −0.0118663
\(704\) − 28815.1i − 1.54263i
\(705\) −25185.0 −1.34542
\(706\) 58060.3 3.09508
\(707\) − 5147.64i − 0.273829i
\(708\) 29650.8i 1.57393i
\(709\) 26123.2i 1.38375i 0.722017 + 0.691875i \(0.243215\pi\)
−0.722017 + 0.691875i \(0.756785\pi\)
\(710\) − 42412.9i − 2.24187i
\(711\) −3376.94 −0.178123
\(712\) 68691.1 3.61560
\(713\) 8024.48i 0.421486i
\(714\) 16706.2 0.875647
\(715\) 0 0
\(716\) 87021.3 4.54209
\(717\) 11291.4i 0.588122i
\(718\) −18793.8 −0.976850
\(719\) −19325.7 −1.00240 −0.501200 0.865331i \(-0.667108\pi\)
−0.501200 + 0.865331i \(0.667108\pi\)
\(720\) 27976.2i 1.44807i
\(721\) 7124.09i 0.367982i
\(722\) − 36534.2i − 1.88319i
\(723\) 10844.2i 0.557816i
\(724\) 80521.7 4.13338
\(725\) −8897.47 −0.455784
\(726\) 9113.26i 0.465875i
\(727\) −26065.8 −1.32975 −0.664875 0.746954i \(-0.731515\pi\)
−0.664875 + 0.746954i \(0.731515\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 83794.4i − 4.24845i
\(731\) 25722.0 1.30145
\(732\) −27191.5 −1.37299
\(733\) − 1055.45i − 0.0531843i −0.999646 0.0265921i \(-0.991534\pi\)
0.999646 0.0265921i \(-0.00846554\pi\)
\(734\) − 12706.4i − 0.638965i
\(735\) − 12275.5i − 0.616041i
\(736\) 20076.2i 1.00546i
\(737\) 5244.89 0.262141
\(738\) 1474.96 0.0735690
\(739\) 9410.40i 0.468426i 0.972185 + 0.234213i \(0.0752514\pi\)
−0.972185 + 0.234213i \(0.924749\pi\)
\(740\) 32976.0 1.63814
\(741\) 0 0
\(742\) 25428.1 1.25808
\(743\) 7523.70i 0.371491i 0.982598 + 0.185746i \(0.0594700\pi\)
−0.982598 + 0.185746i \(0.940530\pi\)
\(744\) −38098.5 −1.87736
\(745\) −43276.8 −2.12824
\(746\) 70799.4i 3.47473i
\(747\) 6441.89i 0.315524i
\(748\) 60780.8i 2.97108i
\(749\) − 7581.76i − 0.369868i
\(750\) 5089.07 0.247769
\(751\) 12984.1 0.630886 0.315443 0.948945i \(-0.397847\pi\)
0.315443 + 0.948945i \(0.397847\pi\)
\(752\) − 96869.3i − 4.69742i
\(753\) −17189.3 −0.831890
\(754\) 0 0
\(755\) 54695.3 2.63651
\(756\) 5334.99i 0.256656i
\(757\) −27934.6 −1.34122 −0.670609 0.741811i \(-0.733967\pi\)
−0.670609 + 0.741811i \(0.733967\pi\)
\(758\) 23649.5 1.13323
\(759\) 3460.30i 0.165482i
\(760\) − 2440.49i − 0.116482i
\(761\) − 15519.3i − 0.739255i −0.929180 0.369627i \(-0.879485\pi\)
0.929180 0.369627i \(-0.120515\pi\)
\(762\) − 22893.5i − 1.08838i
\(763\) −5152.88 −0.244491
\(764\) 4370.62 0.206968
\(765\) − 15942.9i − 0.753487i
\(766\) 4319.82 0.203761
\(767\) 0 0
\(768\) −2532.57 −0.118993
\(769\) − 12885.2i − 0.604228i −0.953272 0.302114i \(-0.902308\pi\)
0.953272 0.302114i \(-0.0976924\pi\)
\(770\) −23363.3 −1.09345
\(771\) −16577.4 −0.774344
\(772\) 24642.4i 1.14883i
\(773\) − 5892.04i − 0.274155i −0.990560 0.137078i \(-0.956229\pi\)
0.990560 0.137078i \(-0.0437710\pi\)
\(774\) 11433.3i 0.530958i
\(775\) 27713.0i 1.28449i
\(776\) 4338.12 0.200682
\(777\) 2858.13 0.131962
\(778\) 18463.4i 0.850828i
\(779\) −69.0916 −0.00317775
\(780\) 0 0
\(781\) 13374.3 0.612766
\(782\) − 24052.7i − 1.09990i
\(783\) −1663.77 −0.0759367
\(784\) 47215.5 2.15085
\(785\) − 26675.6i − 1.21286i
\(786\) 33058.9i 1.50022i
\(787\) − 21020.4i − 0.952091i −0.879421 0.476045i \(-0.842070\pi\)
0.879421 0.476045i \(-0.157930\pi\)
\(788\) − 18935.8i − 0.856042i
\(789\) −15669.6 −0.707038
\(790\) −32827.0 −1.47839
\(791\) − 1748.05i − 0.0785757i
\(792\) −16428.7 −0.737084
\(793\) 0 0
\(794\) 2267.58 0.101352
\(795\) − 24266.4i − 1.08256i
\(796\) 9776.87 0.435342
\(797\) −31355.6 −1.39356 −0.696782 0.717283i \(-0.745386\pi\)
−0.696782 + 0.717283i \(0.745386\pi\)
\(798\) − 347.848i − 0.0154307i
\(799\) 55203.3i 2.44425i
\(800\) 69334.0i 3.06416i
\(801\) − 9343.37i − 0.412150i
\(802\) −6326.37 −0.278543
\(803\) 26423.4 1.16122
\(804\) − 11642.4i − 0.510692i
\(805\) 6642.34 0.290822
\(806\) 0 0
\(807\) −21611.7 −0.942709
\(808\) − 35187.3i − 1.53204i
\(809\) −18132.5 −0.788017 −0.394009 0.919107i \(-0.628912\pi\)
−0.394009 + 0.919107i \(0.628912\pi\)
\(810\) 7086.55 0.307402
\(811\) 24755.3i 1.07186i 0.844263 + 0.535928i \(0.180038\pi\)
−0.844263 + 0.535928i \(0.819962\pi\)
\(812\) − 12175.9i − 0.526219i
\(813\) 25733.1i 1.11008i
\(814\) 14473.8i 0.623225i
\(815\) 30124.0 1.29472
\(816\) 61321.4 2.63073
\(817\) − 535.572i − 0.0229343i
\(818\) −42682.7 −1.82441
\(819\) 0 0
\(820\) 10300.9 0.438688
\(821\) − 4082.65i − 0.173551i −0.996228 0.0867755i \(-0.972344\pi\)
0.996228 0.0867755i \(-0.0276563\pi\)
\(822\) 6196.77 0.262941
\(823\) 34327.0 1.45391 0.726954 0.686687i \(-0.240935\pi\)
0.726954 + 0.686687i \(0.240935\pi\)
\(824\) 48697.6i 2.05881i
\(825\) 11950.3i 0.504312i
\(826\) 24981.9i 1.05234i
\(827\) 3228.87i 0.135767i 0.997693 + 0.0678833i \(0.0216245\pi\)
−0.997693 + 0.0678833i \(0.978375\pi\)
\(828\) 7681.06 0.322386
\(829\) 10452.4 0.437908 0.218954 0.975735i \(-0.429735\pi\)
0.218954 + 0.975735i \(0.429735\pi\)
\(830\) 62621.0i 2.61880i
\(831\) 21507.6 0.897821
\(832\) 0 0
\(833\) −26906.9 −1.11917
\(834\) 12034.5i 0.499663i
\(835\) −31923.4 −1.32306
\(836\) 1265.55 0.0523564
\(837\) 5182.16i 0.214004i
\(838\) − 36419.7i − 1.50131i
\(839\) − 28289.0i − 1.16406i −0.813169 0.582028i \(-0.802259\pi\)
0.813169 0.582028i \(-0.197741\pi\)
\(840\) 31536.4i 1.29537i
\(841\) −20591.8 −0.844308
\(842\) −57246.1 −2.34303
\(843\) − 2547.47i − 0.104080i
\(844\) 29618.5 1.20795
\(845\) 0 0
\(846\) −24537.6 −0.997187
\(847\) 5516.39i 0.223784i
\(848\) 93335.9 3.77968
\(849\) 3346.12 0.135263
\(850\) − 83067.2i − 3.35198i
\(851\) − 4114.99i − 0.165758i
\(852\) − 29687.8i − 1.19377i
\(853\) − 26631.8i − 1.06900i −0.845170 0.534498i \(-0.820501\pi\)
0.845170 0.534498i \(-0.179499\pi\)
\(854\) −22910.0 −0.917989
\(855\) −331.956 −0.0132780
\(856\) − 51826.0i − 2.06936i
\(857\) −11796.7 −0.470209 −0.235104 0.971970i \(-0.575543\pi\)
−0.235104 + 0.971970i \(0.575543\pi\)
\(858\) 0 0
\(859\) −22672.8 −0.900567 −0.450283 0.892886i \(-0.648677\pi\)
−0.450283 + 0.892886i \(0.648677\pi\)
\(860\) 79849.0i 3.16608i
\(861\) 892.812 0.0353391
\(862\) −27795.0 −1.09826
\(863\) − 21421.1i − 0.844940i −0.906377 0.422470i \(-0.861163\pi\)
0.906377 0.422470i \(-0.138837\pi\)
\(864\) 12965.1i 0.510510i
\(865\) − 41551.9i − 1.63330i
\(866\) 46066.6i 1.80763i
\(867\) −20206.5 −0.791520
\(868\) −37924.3 −1.48299
\(869\) − 10351.5i − 0.404086i
\(870\) −16173.4 −0.630264
\(871\) 0 0
\(872\) −35223.1 −1.36790
\(873\) − 590.072i − 0.0228762i
\(874\) −500.814 −0.0193825
\(875\) 3080.48 0.119016
\(876\) − 58653.7i − 2.26224i
\(877\) 5155.20i 0.198493i 0.995063 + 0.0992466i \(0.0316433\pi\)
−0.995063 + 0.0992466i \(0.968357\pi\)
\(878\) − 69435.0i − 2.66893i
\(879\) 5590.58i 0.214523i
\(880\) −85756.9 −3.28507
\(881\) −23692.2 −0.906027 −0.453013 0.891504i \(-0.649651\pi\)
−0.453013 + 0.891504i \(0.649651\pi\)
\(882\) − 11960.0i − 0.456591i
\(883\) 14591.5 0.556108 0.278054 0.960565i \(-0.410311\pi\)
0.278054 + 0.960565i \(0.410311\pi\)
\(884\) 0 0
\(885\) 23840.6 0.905529
\(886\) 61475.2i 2.33104i
\(887\) −9722.26 −0.368029 −0.184014 0.982924i \(-0.558909\pi\)
−0.184014 + 0.982924i \(0.558909\pi\)
\(888\) 19537.1 0.738312
\(889\) − 13857.8i − 0.522806i
\(890\) − 90826.1i − 3.42078i
\(891\) 2234.64i 0.0840217i
\(892\) − 42046.6i − 1.57828i
\(893\) 1149.42 0.0430726
\(894\) −42164.3 −1.57739
\(895\) − 69969.2i − 2.61320i
\(896\) −16706.6 −0.622911
\(897\) 0 0
\(898\) 52678.9 1.95759
\(899\) − 11827.1i − 0.438771i
\(900\) 26526.9 0.982479
\(901\) −53189.7 −1.96671
\(902\) 4521.26i 0.166898i
\(903\) 6920.74i 0.255048i
\(904\) − 11949.0i − 0.439621i
\(905\) − 64743.2i − 2.37805i
\(906\) 53289.3 1.95411
\(907\) −11799.0 −0.431951 −0.215975 0.976399i \(-0.569293\pi\)
−0.215975 + 0.976399i \(0.569293\pi\)
\(908\) 91500.9i 3.34423i
\(909\) −4786.18 −0.174640
\(910\) 0 0
\(911\) −43012.4 −1.56429 −0.782143 0.623099i \(-0.785873\pi\)
−0.782143 + 0.623099i \(0.785873\pi\)
\(912\) − 1276.81i − 0.0463589i
\(913\) −19746.6 −0.715793
\(914\) 83303.8 3.01471
\(915\) 21863.3i 0.789921i
\(916\) 33281.3i 1.20049i
\(917\) 20011.0i 0.720635i
\(918\) − 15533.1i − 0.558462i
\(919\) −4951.41 −0.177728 −0.0888639 0.996044i \(-0.528324\pi\)
−0.0888639 + 0.996044i \(0.528324\pi\)
\(920\) 45404.5 1.62711
\(921\) 19162.5i 0.685587i
\(922\) 41297.0 1.47510
\(923\) 0 0
\(924\) −16353.6 −0.582245
\(925\) − 14211.3i − 0.505152i
\(926\) −1777.27 −0.0630721
\(927\) 6623.85 0.234688
\(928\) − 29589.8i − 1.04669i
\(929\) 8934.86i 0.315547i 0.987475 + 0.157774i \(0.0504316\pi\)
−0.987475 + 0.157774i \(0.949568\pi\)
\(930\) 50375.4i 1.77621i
\(931\) 560.243i 0.0197221i
\(932\) −38860.2 −1.36578
\(933\) −10477.8 −0.367660
\(934\) 43700.3i 1.53096i
\(935\) 48870.6 1.70935
\(936\) 0 0
\(937\) −13182.8 −0.459620 −0.229810 0.973235i \(-0.573811\pi\)
−0.229810 + 0.973235i \(0.573811\pi\)
\(938\) − 9809.20i − 0.341452i
\(939\) 17736.0 0.616393
\(940\) −171368. −5.94618
\(941\) 21693.7i 0.751536i 0.926714 + 0.375768i \(0.122621\pi\)
−0.926714 + 0.375768i \(0.877379\pi\)
\(942\) − 25989.9i − 0.898934i
\(943\) − 1285.43i − 0.0443895i
\(944\) 91698.4i 3.16158i
\(945\) 4289.59 0.147662
\(946\) −35047.1 −1.20452
\(947\) − 49790.0i − 1.70851i −0.519856 0.854254i \(-0.674014\pi\)
0.519856 0.854254i \(-0.325986\pi\)
\(948\) −22977.9 −0.787224
\(949\) 0 0
\(950\) −1729.59 −0.0590687
\(951\) − 5032.61i − 0.171602i
\(952\) 69125.0 2.35331
\(953\) −4217.93 −0.143371 −0.0716853 0.997427i \(-0.522838\pi\)
−0.0716853 + 0.997427i \(0.522838\pi\)
\(954\) − 23642.6i − 0.802365i
\(955\) − 3514.19i − 0.119075i
\(956\) 76830.5i 2.59924i
\(957\) − 5100.05i − 0.172269i
\(958\) −34305.8 −1.15696
\(959\) 3750.99 0.126304
\(960\) 51429.0i 1.72903i
\(961\) −7046.87 −0.236544
\(962\) 0 0
\(963\) −7049.38 −0.235891
\(964\) 73788.1i 2.46530i
\(965\) 19813.7 0.660958
\(966\) 6471.60 0.215549
\(967\) 40927.9i 1.36107i 0.732717 + 0.680534i \(0.238252\pi\)
−0.732717 + 0.680534i \(0.761748\pi\)
\(968\) 37707.9i 1.25204i
\(969\) 727.619i 0.0241223i
\(970\) − 5736.04i − 0.189869i
\(971\) −17114.8 −0.565645 −0.282822 0.959172i \(-0.591271\pi\)
−0.282822 + 0.959172i \(0.591271\pi\)
\(972\) 4960.38 0.163688
\(973\) 7284.62i 0.240015i
\(974\) −43151.5 −1.41957
\(975\) 0 0
\(976\) −84093.0 −2.75794
\(977\) 118.470i 0.00387940i 0.999998 + 0.00193970i \(0.000617427\pi\)
−0.999998 + 0.00193970i \(0.999383\pi\)
\(978\) 29349.7 0.959610
\(979\) 28640.7 0.934996
\(980\) − 83527.2i − 2.72263i
\(981\) 4791.05i 0.155929i
\(982\) 27272.8i 0.886261i
\(983\) 26002.8i 0.843705i 0.906665 + 0.421852i \(0.138620\pi\)
−0.906665 + 0.421852i \(0.861380\pi\)
\(984\) 6102.93 0.197718
\(985\) −15225.3 −0.492506
\(986\) 35450.7i 1.14501i
\(987\) −14853.0 −0.479002
\(988\) 0 0
\(989\) 9964.14 0.320365
\(990\) 21722.8i 0.697368i
\(991\) −16062.0 −0.514860 −0.257430 0.966297i \(-0.582876\pi\)
−0.257430 + 0.966297i \(0.582876\pi\)
\(992\) −92163.3 −2.94979
\(993\) 6030.93i 0.192735i
\(994\) − 25013.2i − 0.798159i
\(995\) − 7861.07i − 0.250465i
\(996\) 43832.9i 1.39448i
\(997\) 1361.54 0.0432503 0.0216251 0.999766i \(-0.493116\pi\)
0.0216251 + 0.999766i \(0.493116\pi\)
\(998\) 96217.5 3.05182
\(999\) − 2657.44i − 0.0841617i
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.b.h.337.1 8
13.5 odd 4 507.4.a.i.1.1 4
13.7 odd 12 39.4.e.c.16.1 8
13.8 odd 4 507.4.a.m.1.4 4
13.11 odd 12 39.4.e.c.22.1 yes 8
13.12 even 2 inner 507.4.b.h.337.8 8
39.5 even 4 1521.4.a.bb.1.4 4
39.8 even 4 1521.4.a.v.1.1 4
39.11 even 12 117.4.g.e.100.4 8
39.20 even 12 117.4.g.e.55.4 8
52.7 even 12 624.4.q.i.289.1 8
52.11 even 12 624.4.q.i.529.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 13.7 odd 12
39.4.e.c.22.1 yes 8 13.11 odd 12
117.4.g.e.55.4 8 39.20 even 12
117.4.g.e.100.4 8 39.11 even 12
507.4.a.i.1.1 4 13.5 odd 4
507.4.a.m.1.4 4 13.8 odd 4
507.4.b.h.337.1 8 1.1 even 1 trivial
507.4.b.h.337.8 8 13.12 even 2 inner
624.4.q.i.289.1 8 52.7 even 12
624.4.q.i.529.1 8 52.11 even 12
1521.4.a.v.1.1 4 39.8 even 4
1521.4.a.bb.1.4 4 39.5 even 4