Properties

Label 825.4.c.l.199.6
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2230106176.1
Defining polynomial: \(x^{6} + 41 x^{4} + 452 x^{2} + 676\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.6
Root \(4.26150i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.l.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.26150i q^{2} -3.00000i q^{3} -19.6833 q^{4} +15.7845 q^{6} +10.3207i q^{7} -61.4719i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+5.26150i q^{2} -3.00000i q^{3} -19.6833 q^{4} +15.7845 q^{6} +10.3207i q^{7} -61.4719i q^{8} -9.00000 q^{9} +11.0000 q^{11} +59.0500i q^{12} +63.9817i q^{13} -54.3024 q^{14} +165.967 q^{16} -17.1461i q^{17} -47.3535i q^{18} -90.2104 q^{19} +30.9621 q^{21} +57.8765i q^{22} -212.605i q^{23} -184.416 q^{24} -336.639 q^{26} +27.0000i q^{27} -203.146i q^{28} -57.5461 q^{29} -141.704 q^{31} +381.462i q^{32} -33.0000i q^{33} +90.2140 q^{34} +177.150 q^{36} +257.963i q^{37} -474.642i q^{38} +191.945 q^{39} -225.914 q^{41} +162.907i q^{42} -347.445i q^{43} -216.517 q^{44} +1118.62 q^{46} -404.364i q^{47} -497.902i q^{48} +236.483 q^{49} -51.4382 q^{51} -1259.37i q^{52} +259.568i q^{53} -142.060 q^{54} +634.433 q^{56} +270.631i q^{57} -302.779i q^{58} +853.067 q^{59} -203.699 q^{61} -745.573i q^{62} -92.8864i q^{63} -679.320 q^{64} +173.629 q^{66} -266.890i q^{67} +337.492i q^{68} -637.814 q^{69} +92.4460 q^{71} +553.247i q^{72} -242.026i q^{73} -1357.27 q^{74} +1775.64 q^{76} +113.528i q^{77} +1009.92i q^{78} +1021.60 q^{79} +81.0000 q^{81} -1188.65i q^{82} +706.415i q^{83} -609.438 q^{84} +1828.08 q^{86} +172.638i q^{87} -676.191i q^{88} +440.218 q^{89} -660.336 q^{91} +4184.77i q^{92} +425.111i q^{93} +2127.56 q^{94} +1144.38 q^{96} +197.761i q^{97} +1244.25i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + O(q^{10}) \) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + 66q^{11} + 112q^{14} + 100q^{16} - 292q^{19} + 24q^{21} - 288q^{24} - 1016q^{26} - 136q^{29} - 136q^{31} + 352q^{34} + 396q^{36} - 392q^{41} - 484q^{44} + 2320q^{46} + 314q^{49} + 1308q^{51} - 216q^{54} + 2736q^{56} + 2088q^{59} + 1284q^{61} - 2332q^{64} + 264q^{66} - 1200q^{69} - 1088q^{71} - 3072q^{74} + 3992q^{76} + 3172q^{79} + 486q^{81} - 2040q^{84} + 7136q^{86} + 4244q^{89} - 16q^{91} + 4304q^{94} + 4128q^{96} - 594q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.26150i 1.86022i 0.367281 + 0.930110i \(0.380289\pi\)
−0.367281 + 0.930110i \(0.619711\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −19.6833 −2.46042
\(5\) 0 0
\(6\) 15.7845 1.07400
\(7\) 10.3207i 0.557266i 0.960398 + 0.278633i \(0.0898813\pi\)
−0.960398 + 0.278633i \(0.910119\pi\)
\(8\) − 61.4719i − 2.71670i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 59.0500i 1.42052i
\(13\) 63.9817i 1.36502i 0.730874 + 0.682512i \(0.239113\pi\)
−0.730874 + 0.682512i \(0.760887\pi\)
\(14\) −54.3024 −1.03664
\(15\) 0 0
\(16\) 165.967 2.59324
\(17\) − 17.1461i − 0.244620i −0.992492 0.122310i \(-0.960970\pi\)
0.992492 0.122310i \(-0.0390301\pi\)
\(18\) − 47.3535i − 0.620073i
\(19\) −90.2104 −1.08925 −0.544623 0.838681i \(-0.683327\pi\)
−0.544623 + 0.838681i \(0.683327\pi\)
\(20\) 0 0
\(21\) 30.9621 0.321738
\(22\) 57.8765i 0.560877i
\(23\) − 212.605i − 1.92744i −0.266913 0.963721i \(-0.586004\pi\)
0.266913 0.963721i \(-0.413996\pi\)
\(24\) −184.416 −1.56849
\(25\) 0 0
\(26\) −336.639 −2.53925
\(27\) 27.0000i 0.192450i
\(28\) − 203.146i − 1.37111i
\(29\) −57.5461 −0.368484 −0.184242 0.982881i \(-0.558983\pi\)
−0.184242 + 0.982881i \(0.558983\pi\)
\(30\) 0 0
\(31\) −141.704 −0.820991 −0.410496 0.911863i \(-0.634644\pi\)
−0.410496 + 0.911863i \(0.634644\pi\)
\(32\) 381.462i 2.10730i
\(33\) − 33.0000i − 0.174078i
\(34\) 90.2140 0.455046
\(35\) 0 0
\(36\) 177.150 0.820139
\(37\) 257.963i 1.14619i 0.819490 + 0.573093i \(0.194257\pi\)
−0.819490 + 0.573093i \(0.805743\pi\)
\(38\) − 474.642i − 2.02624i
\(39\) 191.945 0.788097
\(40\) 0 0
\(41\) −225.914 −0.860533 −0.430266 0.902702i \(-0.641580\pi\)
−0.430266 + 0.902702i \(0.641580\pi\)
\(42\) 162.907i 0.598503i
\(43\) − 347.445i − 1.23221i −0.787666 0.616103i \(-0.788711\pi\)
0.787666 0.616103i \(-0.211289\pi\)
\(44\) −216.517 −0.741844
\(45\) 0 0
\(46\) 1118.62 3.58547
\(47\) − 404.364i − 1.25495i −0.778638 0.627473i \(-0.784089\pi\)
0.778638 0.627473i \(-0.215911\pi\)
\(48\) − 497.902i − 1.49721i
\(49\) 236.483 0.689455
\(50\) 0 0
\(51\) −51.4382 −0.141231
\(52\) − 1259.37i − 3.35853i
\(53\) 259.568i 0.672726i 0.941732 + 0.336363i \(0.109197\pi\)
−0.941732 + 0.336363i \(0.890803\pi\)
\(54\) −142.060 −0.358000
\(55\) 0 0
\(56\) 634.433 1.51392
\(57\) 270.631i 0.628877i
\(58\) − 302.779i − 0.685462i
\(59\) 853.067 1.88237 0.941185 0.337891i \(-0.109713\pi\)
0.941185 + 0.337891i \(0.109713\pi\)
\(60\) 0 0
\(61\) −203.699 −0.427558 −0.213779 0.976882i \(-0.568577\pi\)
−0.213779 + 0.976882i \(0.568577\pi\)
\(62\) − 745.573i − 1.52722i
\(63\) − 92.8864i − 0.185755i
\(64\) −679.320 −1.32680
\(65\) 0 0
\(66\) 173.629 0.323823
\(67\) − 266.890i − 0.486653i −0.969944 0.243327i \(-0.921761\pi\)
0.969944 0.243327i \(-0.0782387\pi\)
\(68\) 337.492i 0.601866i
\(69\) −637.814 −1.11281
\(70\) 0 0
\(71\) 92.4460 0.154526 0.0772629 0.997011i \(-0.475382\pi\)
0.0772629 + 0.997011i \(0.475382\pi\)
\(72\) 553.247i 0.905566i
\(73\) − 242.026i − 0.388040i −0.980998 0.194020i \(-0.937847\pi\)
0.980998 0.194020i \(-0.0621527\pi\)
\(74\) −1357.27 −2.13216
\(75\) 0 0
\(76\) 1775.64 2.68000
\(77\) 113.528i 0.168022i
\(78\) 1009.92i 1.46603i
\(79\) 1021.60 1.45492 0.727460 0.686150i \(-0.240701\pi\)
0.727460 + 0.686150i \(0.240701\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 1188.65i − 1.60078i
\(83\) 706.415i 0.934206i 0.884203 + 0.467103i \(0.154702\pi\)
−0.884203 + 0.467103i \(0.845298\pi\)
\(84\) −609.438 −0.791609
\(85\) 0 0
\(86\) 1828.08 2.29217
\(87\) 172.638i 0.212745i
\(88\) − 676.191i − 0.819116i
\(89\) 440.218 0.524304 0.262152 0.965027i \(-0.415568\pi\)
0.262152 + 0.965027i \(0.415568\pi\)
\(90\) 0 0
\(91\) −660.336 −0.760682
\(92\) 4184.77i 4.74231i
\(93\) 425.111i 0.473999i
\(94\) 2127.56 2.33448
\(95\) 0 0
\(96\) 1144.38 1.21665
\(97\) 197.761i 0.207006i 0.994629 + 0.103503i \(0.0330051\pi\)
−0.994629 + 0.103503i \(0.966995\pi\)
\(98\) 1244.25i 1.28254i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 1400.62 1.37987 0.689937 0.723870i \(-0.257638\pi\)
0.689937 + 0.723870i \(0.257638\pi\)
\(102\) − 270.642i − 0.262721i
\(103\) − 1345.70i − 1.28734i −0.765304 0.643669i \(-0.777411\pi\)
0.765304 0.643669i \(-0.222589\pi\)
\(104\) 3933.07 3.70836
\(105\) 0 0
\(106\) −1365.72 −1.25142
\(107\) 889.178i 0.803366i 0.915779 + 0.401683i \(0.131575\pi\)
−0.915779 + 0.401683i \(0.868425\pi\)
\(108\) − 531.450i − 0.473508i
\(109\) −1256.29 −1.10395 −0.551974 0.833861i \(-0.686125\pi\)
−0.551974 + 0.833861i \(0.686125\pi\)
\(110\) 0 0
\(111\) 773.890 0.661751
\(112\) 1712.90i 1.44512i
\(113\) − 2394.01i − 1.99301i −0.0835448 0.996504i \(-0.526624\pi\)
0.0835448 0.996504i \(-0.473376\pi\)
\(114\) −1423.93 −1.16985
\(115\) 0 0
\(116\) 1132.70 0.906626
\(117\) − 575.835i − 0.455008i
\(118\) 4488.41i 3.50162i
\(119\) 176.960 0.136318
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 1071.76i − 0.795352i
\(123\) 677.742i 0.496829i
\(124\) 2789.20 2.01998
\(125\) 0 0
\(126\) 488.721 0.345546
\(127\) − 2065.57i − 1.44322i −0.692298 0.721612i \(-0.743402\pi\)
0.692298 0.721612i \(-0.256598\pi\)
\(128\) − 522.548i − 0.360837i
\(129\) −1042.33 −0.711414
\(130\) 0 0
\(131\) 785.526 0.523907 0.261953 0.965081i \(-0.415633\pi\)
0.261953 + 0.965081i \(0.415633\pi\)
\(132\) 649.550i 0.428304i
\(133\) − 931.035i − 0.607000i
\(134\) 1404.24 0.905282
\(135\) 0 0
\(136\) −1054.00 −0.664558
\(137\) − 1276.24i − 0.795885i −0.917410 0.397942i \(-0.869724\pi\)
0.917410 0.397942i \(-0.130276\pi\)
\(138\) − 3355.86i − 2.07007i
\(139\) 2703.21 1.64952 0.824760 0.565482i \(-0.191310\pi\)
0.824760 + 0.565482i \(0.191310\pi\)
\(140\) 0 0
\(141\) −1213.09 −0.724544
\(142\) 486.405i 0.287452i
\(143\) 703.798i 0.411570i
\(144\) −1493.71 −0.864413
\(145\) 0 0
\(146\) 1273.42 0.721840
\(147\) − 709.449i − 0.398057i
\(148\) − 5077.58i − 2.82010i
\(149\) 2400.99 1.32011 0.660056 0.751217i \(-0.270533\pi\)
0.660056 + 0.751217i \(0.270533\pi\)
\(150\) 0 0
\(151\) −2517.30 −1.35665 −0.678326 0.734761i \(-0.737294\pi\)
−0.678326 + 0.734761i \(0.737294\pi\)
\(152\) 5545.40i 2.95916i
\(153\) 154.315i 0.0815399i
\(154\) −597.326 −0.312558
\(155\) 0 0
\(156\) −3778.12 −1.93905
\(157\) − 1391.42i − 0.707310i −0.935376 0.353655i \(-0.884939\pi\)
0.935376 0.353655i \(-0.115061\pi\)
\(158\) 5375.13i 2.70647i
\(159\) 778.705 0.388398
\(160\) 0 0
\(161\) 2194.23 1.07410
\(162\) 426.181i 0.206691i
\(163\) − 2720.53i − 1.30729i −0.756802 0.653644i \(-0.773239\pi\)
0.756802 0.653644i \(-0.226761\pi\)
\(164\) 4446.74 2.11727
\(165\) 0 0
\(166\) −3716.80 −1.73783
\(167\) − 2950.25i − 1.36705i −0.729927 0.683525i \(-0.760446\pi\)
0.729927 0.683525i \(-0.239554\pi\)
\(168\) − 1903.30i − 0.874064i
\(169\) −1896.65 −0.863292
\(170\) 0 0
\(171\) 811.894 0.363082
\(172\) 6838.88i 3.03174i
\(173\) 537.049i 0.236018i 0.993013 + 0.118009i \(0.0376511\pi\)
−0.993013 + 0.118009i \(0.962349\pi\)
\(174\) −908.336 −0.395752
\(175\) 0 0
\(176\) 1825.64 0.781891
\(177\) − 2559.20i − 1.08679i
\(178\) 2316.21i 0.975320i
\(179\) −2891.25 −1.20728 −0.603638 0.797259i \(-0.706283\pi\)
−0.603638 + 0.797259i \(0.706283\pi\)
\(180\) 0 0
\(181\) 435.209 0.178723 0.0893615 0.995999i \(-0.471517\pi\)
0.0893615 + 0.995999i \(0.471517\pi\)
\(182\) − 3474.36i − 1.41504i
\(183\) 611.098i 0.246851i
\(184\) −13069.2 −5.23628
\(185\) 0 0
\(186\) −2236.72 −0.881743
\(187\) − 188.607i − 0.0737556i
\(188\) 7959.23i 3.08769i
\(189\) −278.659 −0.107246
\(190\) 0 0
\(191\) −3779.49 −1.43180 −0.715901 0.698202i \(-0.753984\pi\)
−0.715901 + 0.698202i \(0.753984\pi\)
\(192\) 2037.96i 0.766027i
\(193\) 3751.91i 1.39932i 0.714476 + 0.699660i \(0.246665\pi\)
−0.714476 + 0.699660i \(0.753335\pi\)
\(194\) −1040.52 −0.385076
\(195\) 0 0
\(196\) −4654.78 −1.69635
\(197\) 3920.73i 1.41797i 0.705223 + 0.708986i \(0.250847\pi\)
−0.705223 + 0.708986i \(0.749153\pi\)
\(198\) − 520.888i − 0.186959i
\(199\) 597.084 0.212694 0.106347 0.994329i \(-0.466085\pi\)
0.106347 + 0.994329i \(0.466085\pi\)
\(200\) 0 0
\(201\) −800.669 −0.280969
\(202\) 7369.37i 2.56687i
\(203\) − 593.917i − 0.205344i
\(204\) 1012.48 0.347488
\(205\) 0 0
\(206\) 7080.40 2.39473
\(207\) 1913.44i 0.642480i
\(208\) 10618.9i 3.53984i
\(209\) −992.314 −0.328420
\(210\) 0 0
\(211\) −4384.55 −1.43054 −0.715272 0.698846i \(-0.753697\pi\)
−0.715272 + 0.698846i \(0.753697\pi\)
\(212\) − 5109.17i − 1.65519i
\(213\) − 277.338i − 0.0892155i
\(214\) −4678.41 −1.49444
\(215\) 0 0
\(216\) 1659.74 0.522829
\(217\) − 1462.48i − 0.457510i
\(218\) − 6609.95i − 2.05359i
\(219\) −726.077 −0.224035
\(220\) 0 0
\(221\) 1097.03 0.333912
\(222\) 4071.82i 1.23100i
\(223\) − 2333.03i − 0.700587i −0.936640 0.350294i \(-0.886082\pi\)
0.936640 0.350294i \(-0.113918\pi\)
\(224\) −3936.95 −1.17433
\(225\) 0 0
\(226\) 12596.1 3.70743
\(227\) 2120.00i 0.619864i 0.950759 + 0.309932i \(0.100306\pi\)
−0.950759 + 0.309932i \(0.899694\pi\)
\(228\) − 5326.93i − 1.54730i
\(229\) −2347.12 −0.677301 −0.338651 0.940912i \(-0.609970\pi\)
−0.338651 + 0.940912i \(0.609970\pi\)
\(230\) 0 0
\(231\) 340.583 0.0970075
\(232\) 3537.47i 1.00106i
\(233\) − 375.499i − 0.105578i −0.998606 0.0527891i \(-0.983189\pi\)
0.998606 0.0527891i \(-0.0168111\pi\)
\(234\) 3029.75 0.846415
\(235\) 0 0
\(236\) −16791.2 −4.63142
\(237\) − 3064.79i − 0.839998i
\(238\) 931.072i 0.253582i
\(239\) 1428.15 0.386524 0.193262 0.981147i \(-0.438093\pi\)
0.193262 + 0.981147i \(0.438093\pi\)
\(240\) 0 0
\(241\) 190.819 0.0510032 0.0255016 0.999675i \(-0.491882\pi\)
0.0255016 + 0.999675i \(0.491882\pi\)
\(242\) 636.641i 0.169111i
\(243\) − 243.000i − 0.0641500i
\(244\) 4009.49 1.05197
\(245\) 0 0
\(246\) −3565.94 −0.924211
\(247\) − 5771.81i − 1.48685i
\(248\) 8710.79i 2.23039i
\(249\) 2119.24 0.539364
\(250\) 0 0
\(251\) −6294.80 −1.58297 −0.791483 0.611191i \(-0.790691\pi\)
−0.791483 + 0.611191i \(0.790691\pi\)
\(252\) 1828.31i 0.457036i
\(253\) − 2338.65i − 0.581145i
\(254\) 10868.0 2.68471
\(255\) 0 0
\(256\) −2685.18 −0.655561
\(257\) − 4459.44i − 1.08238i −0.840900 0.541191i \(-0.817974\pi\)
0.840900 0.541191i \(-0.182026\pi\)
\(258\) − 5484.24i − 1.32339i
\(259\) −2662.36 −0.638731
\(260\) 0 0
\(261\) 517.915 0.122828
\(262\) 4133.04i 0.974581i
\(263\) − 4416.65i − 1.03552i −0.855525 0.517761i \(-0.826766\pi\)
0.855525 0.517761i \(-0.173234\pi\)
\(264\) −2028.57 −0.472917
\(265\) 0 0
\(266\) 4898.64 1.12915
\(267\) − 1320.65i − 0.302707i
\(268\) 5253.28i 1.19737i
\(269\) −1914.86 −0.434020 −0.217010 0.976169i \(-0.569630\pi\)
−0.217010 + 0.976169i \(0.569630\pi\)
\(270\) 0 0
\(271\) 6088.34 1.36472 0.682362 0.731014i \(-0.260953\pi\)
0.682362 + 0.731014i \(0.260953\pi\)
\(272\) − 2845.69i − 0.634357i
\(273\) 1981.01i 0.439180i
\(274\) 6714.91 1.48052
\(275\) 0 0
\(276\) 12554.3 2.73798
\(277\) 832.321i 0.180539i 0.995917 + 0.0902696i \(0.0287729\pi\)
−0.995917 + 0.0902696i \(0.971227\pi\)
\(278\) 14222.9i 3.06847i
\(279\) 1275.33 0.273664
\(280\) 0 0
\(281\) −2545.32 −0.540360 −0.270180 0.962810i \(-0.587083\pi\)
−0.270180 + 0.962810i \(0.587083\pi\)
\(282\) − 6382.67i − 1.34781i
\(283\) − 5911.71i − 1.24175i −0.783911 0.620874i \(-0.786778\pi\)
0.783911 0.620874i \(-0.213222\pi\)
\(284\) −1819.65 −0.380198
\(285\) 0 0
\(286\) −3703.03 −0.765611
\(287\) − 2331.59i − 0.479545i
\(288\) − 3433.15i − 0.702433i
\(289\) 4619.01 0.940161
\(290\) 0 0
\(291\) 593.282 0.119515
\(292\) 4763.87i 0.954742i
\(293\) − 6871.03i − 1.37000i −0.728543 0.685000i \(-0.759802\pi\)
0.728543 0.685000i \(-0.240198\pi\)
\(294\) 3732.76 0.740473
\(295\) 0 0
\(296\) 15857.5 3.11384
\(297\) 297.000i 0.0580259i
\(298\) 12632.8i 2.45570i
\(299\) 13602.8 2.63100
\(300\) 0 0
\(301\) 3585.88 0.686666
\(302\) − 13244.7i − 2.52367i
\(303\) − 4201.87i − 0.796670i
\(304\) −14972.0 −2.82468
\(305\) 0 0
\(306\) −811.926 −0.151682
\(307\) − 200.179i − 0.0372144i −0.999827 0.0186072i \(-0.994077\pi\)
0.999827 0.0186072i \(-0.00592320\pi\)
\(308\) − 2234.61i − 0.413404i
\(309\) −4037.10 −0.743245
\(310\) 0 0
\(311\) 5734.93 1.04565 0.522827 0.852439i \(-0.324878\pi\)
0.522827 + 0.852439i \(0.324878\pi\)
\(312\) − 11799.2i − 2.14102i
\(313\) − 3077.36i − 0.555727i −0.960621 0.277864i \(-0.910374\pi\)
0.960621 0.277864i \(-0.0896264\pi\)
\(314\) 7320.97 1.31575
\(315\) 0 0
\(316\) −20108.5 −3.57971
\(317\) − 2142.38i − 0.379584i −0.981824 0.189792i \(-0.939219\pi\)
0.981824 0.189792i \(-0.0607813\pi\)
\(318\) 4097.15i 0.722506i
\(319\) −633.007 −0.111102
\(320\) 0 0
\(321\) 2667.54 0.463823
\(322\) 11544.9i 1.99806i
\(323\) 1546.75i 0.266451i
\(324\) −1594.35 −0.273380
\(325\) 0 0
\(326\) 14314.0 2.43184
\(327\) 3768.86i 0.637365i
\(328\) 13887.4i 2.33781i
\(329\) 4173.32 0.699339
\(330\) 0 0
\(331\) −1618.23 −0.268719 −0.134359 0.990933i \(-0.542898\pi\)
−0.134359 + 0.990933i \(0.542898\pi\)
\(332\) − 13904.6i − 2.29854i
\(333\) − 2321.67i − 0.382062i
\(334\) 15522.7 2.54301
\(335\) 0 0
\(336\) 5138.70 0.834343
\(337\) − 2406.47i − 0.388988i −0.980904 0.194494i \(-0.937694\pi\)
0.980904 0.194494i \(-0.0623064\pi\)
\(338\) − 9979.23i − 1.60591i
\(339\) −7182.04 −1.15066
\(340\) 0 0
\(341\) −1558.74 −0.247538
\(342\) 4271.78i 0.675413i
\(343\) 5980.67i 0.941475i
\(344\) −21358.1 −3.34753
\(345\) 0 0
\(346\) −2825.68 −0.439045
\(347\) 6612.89i 1.02305i 0.859268 + 0.511525i \(0.170919\pi\)
−0.859268 + 0.511525i \(0.829081\pi\)
\(348\) − 3398.10i − 0.523441i
\(349\) −349.871 −0.0536623 −0.0268311 0.999640i \(-0.508542\pi\)
−0.0268311 + 0.999640i \(0.508542\pi\)
\(350\) 0 0
\(351\) −1727.50 −0.262699
\(352\) 4196.08i 0.635374i
\(353\) 1723.29i 0.259835i 0.991525 + 0.129917i \(0.0414712\pi\)
−0.991525 + 0.129917i \(0.958529\pi\)
\(354\) 13465.2 2.02166
\(355\) 0 0
\(356\) −8664.97 −1.29001
\(357\) − 530.879i − 0.0787033i
\(358\) − 15212.3i − 2.24580i
\(359\) −5875.74 −0.863816 −0.431908 0.901918i \(-0.642159\pi\)
−0.431908 + 0.901918i \(0.642159\pi\)
\(360\) 0 0
\(361\) 1278.92 0.186458
\(362\) 2289.85i 0.332464i
\(363\) − 363.000i − 0.0524864i
\(364\) 12997.6 1.87159
\(365\) 0 0
\(366\) −3215.29 −0.459197
\(367\) 5368.28i 0.763548i 0.924256 + 0.381774i \(0.124687\pi\)
−0.924256 + 0.381774i \(0.875313\pi\)
\(368\) − 35285.5i − 4.99832i
\(369\) 2033.23 0.286844
\(370\) 0 0
\(371\) −2678.93 −0.374887
\(372\) − 8367.60i − 1.16624i
\(373\) 10393.9i 1.44282i 0.692506 + 0.721412i \(0.256507\pi\)
−0.692506 + 0.721412i \(0.743493\pi\)
\(374\) 992.354 0.137202
\(375\) 0 0
\(376\) −24857.0 −3.40931
\(377\) − 3681.90i − 0.502990i
\(378\) − 1466.16i − 0.199501i
\(379\) −10918.9 −1.47986 −0.739928 0.672686i \(-0.765140\pi\)
−0.739928 + 0.672686i \(0.765140\pi\)
\(380\) 0 0
\(381\) −6196.70 −0.833246
\(382\) − 19885.8i − 2.66347i
\(383\) − 11663.6i − 1.55609i −0.628210 0.778044i \(-0.716212\pi\)
0.628210 0.778044i \(-0.283788\pi\)
\(384\) −1567.64 −0.208329
\(385\) 0 0
\(386\) −19740.7 −2.60304
\(387\) 3127.00i 0.410735i
\(388\) − 3892.59i − 0.509321i
\(389\) 5827.00 0.759487 0.379744 0.925092i \(-0.376012\pi\)
0.379744 + 0.925092i \(0.376012\pi\)
\(390\) 0 0
\(391\) −3645.34 −0.471490
\(392\) − 14537.1i − 1.87304i
\(393\) − 2356.58i − 0.302478i
\(394\) −20628.9 −2.63774
\(395\) 0 0
\(396\) 1948.65 0.247281
\(397\) − 7366.99i − 0.931332i −0.884961 0.465666i \(-0.845815\pi\)
0.884961 0.465666i \(-0.154185\pi\)
\(398\) 3141.55i 0.395658i
\(399\) −2793.11 −0.350452
\(400\) 0 0
\(401\) 14604.2 1.81870 0.909349 0.416035i \(-0.136581\pi\)
0.909349 + 0.416035i \(0.136581\pi\)
\(402\) − 4212.72i − 0.522665i
\(403\) − 9066.43i − 1.12067i
\(404\) −27569.0 −3.39507
\(405\) 0 0
\(406\) 3124.89 0.381985
\(407\) 2837.60i 0.345588i
\(408\) 3162.00i 0.383683i
\(409\) −12581.2 −1.52103 −0.760515 0.649320i \(-0.775054\pi\)
−0.760515 + 0.649320i \(0.775054\pi\)
\(410\) 0 0
\(411\) −3828.71 −0.459504
\(412\) 26487.9i 3.16739i
\(413\) 8804.26i 1.04898i
\(414\) −10067.6 −1.19516
\(415\) 0 0
\(416\) −24406.6 −2.87651
\(417\) − 8109.63i − 0.952351i
\(418\) − 5221.06i − 0.610934i
\(419\) 3776.01 0.440263 0.220131 0.975470i \(-0.429351\pi\)
0.220131 + 0.975470i \(0.429351\pi\)
\(420\) 0 0
\(421\) 12683.4 1.46830 0.734148 0.678989i \(-0.237582\pi\)
0.734148 + 0.678989i \(0.237582\pi\)
\(422\) − 23069.3i − 2.66113i
\(423\) 3639.27i 0.418316i
\(424\) 15956.2 1.82759
\(425\) 0 0
\(426\) 1459.21 0.165960
\(427\) − 2102.32i − 0.238263i
\(428\) − 17502.0i − 1.97662i
\(429\) 2111.39 0.237620
\(430\) 0 0
\(431\) 14152.6 1.58168 0.790841 0.612022i \(-0.209644\pi\)
0.790841 + 0.612022i \(0.209644\pi\)
\(432\) 4481.12i 0.499069i
\(433\) − 10950.2i − 1.21532i −0.794197 0.607661i \(-0.792108\pi\)
0.794197 0.607661i \(-0.207892\pi\)
\(434\) 7694.84 0.851070
\(435\) 0 0
\(436\) 24727.9 2.71618
\(437\) 19179.2i 2.09946i
\(438\) − 3820.25i − 0.416755i
\(439\) 11221.0 1.21993 0.609964 0.792429i \(-0.291184\pi\)
0.609964 + 0.792429i \(0.291184\pi\)
\(440\) 0 0
\(441\) −2128.35 −0.229818
\(442\) 5772.04i 0.621149i
\(443\) 9647.11i 1.03465i 0.855790 + 0.517323i \(0.173071\pi\)
−0.855790 + 0.517323i \(0.826929\pi\)
\(444\) −15232.7 −1.62818
\(445\) 0 0
\(446\) 12275.2 1.30325
\(447\) − 7202.96i − 0.762167i
\(448\) − 7011.07i − 0.739379i
\(449\) −6482.03 −0.681305 −0.340652 0.940189i \(-0.610648\pi\)
−0.340652 + 0.940189i \(0.610648\pi\)
\(450\) 0 0
\(451\) −2485.05 −0.259460
\(452\) 47122.2i 4.90363i
\(453\) 7551.89i 0.783264i
\(454\) −11154.4 −1.15308
\(455\) 0 0
\(456\) 16636.2 1.70847
\(457\) − 11319.8i − 1.15868i −0.815085 0.579342i \(-0.803310\pi\)
0.815085 0.579342i \(-0.196690\pi\)
\(458\) − 12349.4i − 1.25993i
\(459\) 462.944 0.0470771
\(460\) 0 0
\(461\) −8406.73 −0.849329 −0.424664 0.905351i \(-0.639608\pi\)
−0.424664 + 0.905351i \(0.639608\pi\)
\(462\) 1791.98i 0.180455i
\(463\) 9758.56i 0.979523i 0.871857 + 0.489761i \(0.162916\pi\)
−0.871857 + 0.489761i \(0.837084\pi\)
\(464\) −9550.78 −0.955568
\(465\) 0 0
\(466\) 1975.68 0.196399
\(467\) 16388.7i 1.62394i 0.583701 + 0.811969i \(0.301604\pi\)
−0.583701 + 0.811969i \(0.698396\pi\)
\(468\) 11334.4i 1.11951i
\(469\) 2754.49 0.271195
\(470\) 0 0
\(471\) −4174.27 −0.408365
\(472\) − 52439.6i − 5.11384i
\(473\) − 3821.89i − 0.371524i
\(474\) 16125.4 1.56258
\(475\) 0 0
\(476\) −3483.16 −0.335400
\(477\) − 2336.11i − 0.224242i
\(478\) 7514.20i 0.719020i
\(479\) −13829.0 −1.31913 −0.659567 0.751646i \(-0.729260\pi\)
−0.659567 + 0.751646i \(0.729260\pi\)
\(480\) 0 0
\(481\) −16504.9 −1.56457
\(482\) 1004.00i 0.0948771i
\(483\) − 6582.69i − 0.620130i
\(484\) −2381.68 −0.223674
\(485\) 0 0
\(486\) 1278.54 0.119333
\(487\) − 13264.4i − 1.23423i −0.786875 0.617113i \(-0.788302\pi\)
0.786875 0.617113i \(-0.211698\pi\)
\(488\) 12521.8i 1.16155i
\(489\) −8161.58 −0.754763
\(490\) 0 0
\(491\) −7468.22 −0.686428 −0.343214 0.939257i \(-0.611516\pi\)
−0.343214 + 0.939257i \(0.611516\pi\)
\(492\) − 13340.2i − 1.22241i
\(493\) 986.690i 0.0901385i
\(494\) 30368.4 2.76586
\(495\) 0 0
\(496\) −23518.2 −2.12903
\(497\) 954.109i 0.0861119i
\(498\) 11150.4i 1.00334i
\(499\) 5276.64 0.473377 0.236688 0.971586i \(-0.423938\pi\)
0.236688 + 0.971586i \(0.423938\pi\)
\(500\) 0 0
\(501\) −8850.76 −0.789267
\(502\) − 33120.1i − 2.94466i
\(503\) − 10956.8i − 0.971253i −0.874166 0.485626i \(-0.838592\pi\)
0.874166 0.485626i \(-0.161408\pi\)
\(504\) −5709.90 −0.504641
\(505\) 0 0
\(506\) 12304.8 1.08106
\(507\) 5689.96i 0.498422i
\(508\) 40657.3i 3.55093i
\(509\) −12734.4 −1.10892 −0.554462 0.832209i \(-0.687076\pi\)
−0.554462 + 0.832209i \(0.687076\pi\)
\(510\) 0 0
\(511\) 2497.87 0.216242
\(512\) − 18308.4i − 1.58032i
\(513\) − 2435.68i − 0.209626i
\(514\) 23463.3 2.01347
\(515\) 0 0
\(516\) 20516.6 1.75038
\(517\) − 4448.00i − 0.378381i
\(518\) − 14008.0i − 1.18818i
\(519\) 1611.15 0.136265
\(520\) 0 0
\(521\) −5650.70 −0.475166 −0.237583 0.971367i \(-0.576355\pi\)
−0.237583 + 0.971367i \(0.576355\pi\)
\(522\) 2725.01i 0.228487i
\(523\) 14103.2i 1.17914i 0.807716 + 0.589572i \(0.200703\pi\)
−0.807716 + 0.589572i \(0.799297\pi\)
\(524\) −15461.8 −1.28903
\(525\) 0 0
\(526\) 23238.2 1.92630
\(527\) 2429.66i 0.200830i
\(528\) − 5476.92i − 0.451425i
\(529\) −33033.8 −2.71503
\(530\) 0 0
\(531\) −7677.60 −0.627457
\(532\) 18325.9i 1.49347i
\(533\) − 14454.4i − 1.17465i
\(534\) 6948.62 0.563102
\(535\) 0 0
\(536\) −16406.2 −1.32209
\(537\) 8673.75i 0.697021i
\(538\) − 10075.0i − 0.807372i
\(539\) 2601.31 0.207878
\(540\) 0 0
\(541\) −6391.90 −0.507965 −0.253983 0.967209i \(-0.581741\pi\)
−0.253983 + 0.967209i \(0.581741\pi\)
\(542\) 32033.8i 2.53869i
\(543\) − 1305.63i − 0.103186i
\(544\) 6540.57 0.515486
\(545\) 0 0
\(546\) −10423.1 −0.816971
\(547\) − 20786.7i − 1.62482i −0.583088 0.812409i \(-0.698156\pi\)
0.583088 0.812409i \(-0.301844\pi\)
\(548\) 25120.6i 1.95821i
\(549\) 1833.29 0.142519
\(550\) 0 0
\(551\) 5191.26 0.401370
\(552\) 39207.6i 3.02317i
\(553\) 10543.6i 0.810777i
\(554\) −4379.26 −0.335843
\(555\) 0 0
\(556\) −53208.2 −4.05851
\(557\) 15125.9i 1.15064i 0.817928 + 0.575320i \(0.195123\pi\)
−0.817928 + 0.575320i \(0.804877\pi\)
\(558\) 6710.16i 0.509075i
\(559\) 22230.1 1.68199
\(560\) 0 0
\(561\) −565.820 −0.0425828
\(562\) − 13392.2i − 1.00519i
\(563\) 8706.42i 0.651744i 0.945414 + 0.325872i \(0.105658\pi\)
−0.945414 + 0.325872i \(0.894342\pi\)
\(564\) 23877.7 1.78268
\(565\) 0 0
\(566\) 31104.4 2.30992
\(567\) 835.977i 0.0619184i
\(568\) − 5682.83i − 0.419800i
\(569\) −7067.76 −0.520731 −0.260366 0.965510i \(-0.583843\pi\)
−0.260366 + 0.965510i \(0.583843\pi\)
\(570\) 0 0
\(571\) −3326.42 −0.243794 −0.121897 0.992543i \(-0.538898\pi\)
−0.121897 + 0.992543i \(0.538898\pi\)
\(572\) − 13853.1i − 1.01264i
\(573\) 11338.5i 0.826651i
\(574\) 12267.7 0.892060
\(575\) 0 0
\(576\) 6113.88 0.442266
\(577\) 3308.06i 0.238676i 0.992854 + 0.119338i \(0.0380773\pi\)
−0.992854 + 0.119338i \(0.961923\pi\)
\(578\) 24302.9i 1.74891i
\(579\) 11255.7 0.807897
\(580\) 0 0
\(581\) −7290.70 −0.520601
\(582\) 3121.55i 0.222324i
\(583\) 2855.25i 0.202834i
\(584\) −14877.8 −1.05419
\(585\) 0 0
\(586\) 36151.9 2.54850
\(587\) 5694.88i 0.400431i 0.979752 + 0.200215i \(0.0641642\pi\)
−0.979752 + 0.200215i \(0.935836\pi\)
\(588\) 13964.3i 0.979387i
\(589\) 12783.1 0.894262
\(590\) 0 0
\(591\) 11762.2 0.818666
\(592\) 42813.5i 2.97234i
\(593\) − 3907.69i − 0.270606i −0.990804 0.135303i \(-0.956799\pi\)
0.990804 0.135303i \(-0.0432008\pi\)
\(594\) −1562.66 −0.107941
\(595\) 0 0
\(596\) −47259.5 −3.24803
\(597\) − 1791.25i − 0.122799i
\(598\) 71571.1i 4.89425i
\(599\) −10727.2 −0.731720 −0.365860 0.930670i \(-0.619225\pi\)
−0.365860 + 0.930670i \(0.619225\pi\)
\(600\) 0 0
\(601\) 3348.98 0.227301 0.113650 0.993521i \(-0.463746\pi\)
0.113650 + 0.993521i \(0.463746\pi\)
\(602\) 18867.1i 1.27735i
\(603\) 2402.01i 0.162218i
\(604\) 49548.8 3.33793
\(605\) 0 0
\(606\) 22108.1 1.48198
\(607\) 21539.9i 1.44032i 0.693806 + 0.720162i \(0.255932\pi\)
−0.693806 + 0.720162i \(0.744068\pi\)
\(608\) − 34411.8i − 2.29537i
\(609\) −1781.75 −0.118555
\(610\) 0 0
\(611\) 25871.9 1.71303
\(612\) − 3037.43i − 0.200622i
\(613\) − 9284.33i − 0.611730i −0.952075 0.305865i \(-0.901054\pi\)
0.952075 0.305865i \(-0.0989457\pi\)
\(614\) 1053.24 0.0692270
\(615\) 0 0
\(616\) 6978.77 0.456465
\(617\) 20711.3i 1.35139i 0.737181 + 0.675695i \(0.236156\pi\)
−0.737181 + 0.675695i \(0.763844\pi\)
\(618\) − 21241.2i − 1.38260i
\(619\) 13282.9 0.862496 0.431248 0.902233i \(-0.358073\pi\)
0.431248 + 0.902233i \(0.358073\pi\)
\(620\) 0 0
\(621\) 5740.33 0.370936
\(622\) 30174.3i 1.94515i
\(623\) 4543.36i 0.292177i
\(624\) 31856.6 2.04373
\(625\) 0 0
\(626\) 16191.5 1.03378
\(627\) 2976.94i 0.189613i
\(628\) 27387.9i 1.74028i
\(629\) 4423.06 0.280380
\(630\) 0 0
\(631\) −14789.9 −0.933086 −0.466543 0.884498i \(-0.654501\pi\)
−0.466543 + 0.884498i \(0.654501\pi\)
\(632\) − 62799.5i − 3.95258i
\(633\) 13153.6i 0.825925i
\(634\) 11272.1 0.706109
\(635\) 0 0
\(636\) −15327.5 −0.955622
\(637\) 15130.6i 0.941123i
\(638\) − 3330.57i − 0.206675i
\(639\) −832.014 −0.0515086
\(640\) 0 0
\(641\) −5808.52 −0.357914 −0.178957 0.983857i \(-0.557272\pi\)
−0.178957 + 0.983857i \(0.557272\pi\)
\(642\) 14035.2i 0.862813i
\(643\) − 18891.2i − 1.15862i −0.815106 0.579311i \(-0.803322\pi\)
0.815106 0.579311i \(-0.196678\pi\)
\(644\) −43189.8 −2.64273
\(645\) 0 0
\(646\) −8138.24 −0.495657
\(647\) 243.046i 0.0147684i 0.999973 + 0.00738418i \(0.00235048\pi\)
−0.999973 + 0.00738418i \(0.997650\pi\)
\(648\) − 4979.22i − 0.301855i
\(649\) 9383.74 0.567556
\(650\) 0 0
\(651\) −4387.44 −0.264144
\(652\) 53549.0i 3.21648i
\(653\) − 15920.4i − 0.954081i −0.878881 0.477041i \(-0.841709\pi\)
0.878881 0.477041i \(-0.158291\pi\)
\(654\) −19829.8 −1.18564
\(655\) 0 0
\(656\) −37494.4 −2.23157
\(657\) 2178.23i 0.129347i
\(658\) 21957.9i 1.30092i
\(659\) 7476.38 0.441940 0.220970 0.975281i \(-0.429078\pi\)
0.220970 + 0.975281i \(0.429078\pi\)
\(660\) 0 0
\(661\) −14920.1 −0.877948 −0.438974 0.898500i \(-0.644658\pi\)
−0.438974 + 0.898500i \(0.644658\pi\)
\(662\) − 8514.30i − 0.499876i
\(663\) − 3291.10i − 0.192784i
\(664\) 43424.7 2.53796
\(665\) 0 0
\(666\) 12215.5 0.710720
\(667\) 12234.6i 0.710232i
\(668\) 58070.8i 3.36351i
\(669\) −6999.08 −0.404484
\(670\) 0 0
\(671\) −2240.69 −0.128914
\(672\) 11810.9i 0.677997i
\(673\) 563.692i 0.0322864i 0.999870 + 0.0161432i \(0.00513876\pi\)
−0.999870 + 0.0161432i \(0.994861\pi\)
\(674\) 12661.6 0.723602
\(675\) 0 0
\(676\) 37332.5 2.12406
\(677\) − 13280.2i − 0.753914i −0.926231 0.376957i \(-0.876970\pi\)
0.926231 0.376957i \(-0.123030\pi\)
\(678\) − 37788.3i − 2.14049i
\(679\) −2041.03 −0.115357
\(680\) 0 0
\(681\) 6359.99 0.357879
\(682\) − 8201.30i − 0.460475i
\(683\) 6856.80i 0.384141i 0.981381 + 0.192070i \(0.0615202\pi\)
−0.981381 + 0.192070i \(0.938480\pi\)
\(684\) −15980.8 −0.893334
\(685\) 0 0
\(686\) −31467.3 −1.75135
\(687\) 7041.36i 0.391040i
\(688\) − 57664.5i − 3.19541i
\(689\) −16607.6 −0.918287
\(690\) 0 0
\(691\) −28374.6 −1.56211 −0.781057 0.624460i \(-0.785319\pi\)
−0.781057 + 0.624460i \(0.785319\pi\)
\(692\) − 10570.9i − 0.580702i
\(693\) − 1021.75i − 0.0560073i
\(694\) −34793.7 −1.90310
\(695\) 0 0
\(696\) 10612.4 0.577963
\(697\) 3873.54i 0.210503i
\(698\) − 1840.84i − 0.0998237i
\(699\) −1126.50 −0.0609556
\(700\) 0 0
\(701\) 667.753 0.0359781 0.0179891 0.999838i \(-0.494274\pi\)
0.0179891 + 0.999838i \(0.494274\pi\)
\(702\) − 9089.26i − 0.488678i
\(703\) − 23271.0i − 1.24848i
\(704\) −7472.52 −0.400044
\(705\) 0 0
\(706\) −9067.10 −0.483349
\(707\) 14455.4i 0.768956i
\(708\) 50373.6i 2.67395i
\(709\) −23667.8 −1.25368 −0.626842 0.779147i \(-0.715653\pi\)
−0.626842 + 0.779147i \(0.715653\pi\)
\(710\) 0 0
\(711\) −9194.37 −0.484973
\(712\) − 27061.0i − 1.42438i
\(713\) 30126.9i 1.58241i
\(714\) 2793.22 0.146405
\(715\) 0 0
\(716\) 56909.5 2.97040
\(717\) − 4284.45i − 0.223160i
\(718\) − 30915.2i − 1.60689i
\(719\) −11835.5 −0.613896 −0.306948 0.951726i \(-0.599308\pi\)
−0.306948 + 0.951726i \(0.599308\pi\)
\(720\) 0 0
\(721\) 13888.6 0.717390
\(722\) 6729.01i 0.346853i
\(723\) − 572.458i − 0.0294467i
\(724\) −8566.38 −0.439733
\(725\) 0 0
\(726\) 1909.92 0.0976362
\(727\) − 15633.2i − 0.797530i −0.917053 0.398765i \(-0.869439\pi\)
0.917053 0.398765i \(-0.130561\pi\)
\(728\) 40592.1i 2.06654i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −5957.31 −0.301422
\(732\) − 12028.5i − 0.607356i
\(733\) − 14870.7i − 0.749335i −0.927159 0.374668i \(-0.877757\pi\)
0.927159 0.374668i \(-0.122243\pi\)
\(734\) −28245.2 −1.42037
\(735\) 0 0
\(736\) 81100.6 4.06169
\(737\) − 2935.79i − 0.146731i
\(738\) 10697.8i 0.533593i
\(739\) −20850.3 −1.03788 −0.518939 0.854812i \(-0.673673\pi\)
−0.518939 + 0.854812i \(0.673673\pi\)
\(740\) 0 0
\(741\) −17315.4 −0.858432
\(742\) − 14095.2i − 0.697372i
\(743\) − 29254.7i − 1.44448i −0.691641 0.722242i \(-0.743112\pi\)
0.691641 0.722242i \(-0.256888\pi\)
\(744\) 26132.4 1.28771
\(745\) 0 0
\(746\) −54687.2 −2.68397
\(747\) − 6357.73i − 0.311402i
\(748\) 3712.41i 0.181470i
\(749\) −9176.95 −0.447688
\(750\) 0 0
\(751\) 10936.8 0.531411 0.265705 0.964054i \(-0.414395\pi\)
0.265705 + 0.964054i \(0.414395\pi\)
\(752\) − 67111.2i − 3.25438i
\(753\) 18884.4i 0.913926i
\(754\) 19372.3 0.935672
\(755\) 0 0
\(756\) 5484.94 0.263870
\(757\) 8476.34i 0.406972i 0.979078 + 0.203486i \(0.0652272\pi\)
−0.979078 + 0.203486i \(0.934773\pi\)
\(758\) − 57449.6i − 2.75286i
\(759\) −7015.96 −0.335524
\(760\) 0 0
\(761\) 913.964 0.0435364 0.0217682 0.999763i \(-0.493070\pi\)
0.0217682 + 0.999763i \(0.493070\pi\)
\(762\) − 32603.9i − 1.55002i
\(763\) − 12965.8i − 0.615193i
\(764\) 74393.0 3.52283
\(765\) 0 0
\(766\) 61367.9 2.89466
\(767\) 54580.6i 2.56948i
\(768\) 8055.53i 0.378488i
\(769\) −32215.2 −1.51067 −0.755337 0.655337i \(-0.772527\pi\)
−0.755337 + 0.655337i \(0.772527\pi\)
\(770\) 0 0
\(771\) −13378.3 −0.624914
\(772\) − 73850.2i − 3.44291i
\(773\) 72.6900i 0.00338225i 0.999999 + 0.00169112i \(0.000538302\pi\)
−0.999999 + 0.00169112i \(0.999462\pi\)
\(774\) −16452.7 −0.764058
\(775\) 0 0
\(776\) 12156.7 0.562373
\(777\) 7987.09i 0.368771i
\(778\) 30658.7i 1.41281i
\(779\) 20379.8 0.937332
\(780\) 0 0
\(781\) 1016.91 0.0465913
\(782\) − 19179.9i − 0.877075i
\(783\) − 1553.75i − 0.0709148i
\(784\) 39248.5 1.78792
\(785\) 0 0
\(786\) 12399.1 0.562675
\(787\) − 487.318i − 0.0220724i −0.999939 0.0110362i \(-0.996487\pi\)
0.999939 0.0110362i \(-0.00351301\pi\)
\(788\) − 77173.1i − 3.48880i
\(789\) −13249.9 −0.597859
\(790\) 0 0
\(791\) 24707.9 1.11064
\(792\) 6085.72i 0.273039i
\(793\) − 13033.0i − 0.583627i
\(794\) 38761.4 1.73248
\(795\) 0 0
\(796\) −11752.6 −0.523317
\(797\) − 31379.9i − 1.39465i −0.716756 0.697324i \(-0.754374\pi\)
0.716756 0.697324i \(-0.245626\pi\)
\(798\) − 14695.9i − 0.651917i
\(799\) −6933.25 −0.306985
\(800\) 0 0
\(801\) −3961.96 −0.174768
\(802\) 76839.8i 3.38318i
\(803\) − 2662.28i − 0.116999i
\(804\) 15759.8 0.691302
\(805\) 0 0
\(806\) 47703.0 2.08470
\(807\) 5744.59i 0.250581i
\(808\) − 86099.0i − 3.74870i
\(809\) −1824.26 −0.0792802 −0.0396401 0.999214i \(-0.512621\pi\)
−0.0396401 + 0.999214i \(0.512621\pi\)
\(810\) 0 0
\(811\) −4364.52 −0.188976 −0.0944878 0.995526i \(-0.530121\pi\)
−0.0944878 + 0.995526i \(0.530121\pi\)
\(812\) 11690.3i 0.505232i
\(813\) − 18265.0i − 0.787924i
\(814\) −14930.0 −0.642870
\(815\) 0 0
\(816\) −8537.06 −0.366246
\(817\) 31343.1i 1.34218i
\(818\) − 66196.1i − 2.82945i
\(819\) 5943.02 0.253561
\(820\) 0 0
\(821\) 3306.16 0.140543 0.0702714 0.997528i \(-0.477613\pi\)
0.0702714 + 0.997528i \(0.477613\pi\)
\(822\) − 20144.7i − 0.854779i
\(823\) − 19183.7i − 0.812519i −0.913758 0.406259i \(-0.866833\pi\)
0.913758 0.406259i \(-0.133167\pi\)
\(824\) −82722.8 −3.49731
\(825\) 0 0
\(826\) −46323.6 −1.95134
\(827\) − 20646.4i − 0.868131i −0.900881 0.434066i \(-0.857079\pi\)
0.900881 0.434066i \(-0.142921\pi\)
\(828\) − 37663.0i − 1.58077i
\(829\) −5345.44 −0.223950 −0.111975 0.993711i \(-0.535718\pi\)
−0.111975 + 0.993711i \(0.535718\pi\)
\(830\) 0 0
\(831\) 2496.96 0.104234
\(832\) − 43464.0i − 1.81111i
\(833\) − 4054.75i − 0.168654i
\(834\) 42668.8 1.77158
\(835\) 0 0
\(836\) 19532.1 0.808051
\(837\) − 3826.00i − 0.158000i
\(838\) 19867.5i 0.818985i
\(839\) −29284.9 −1.20504 −0.602519 0.798105i \(-0.705836\pi\)
−0.602519 + 0.798105i \(0.705836\pi\)
\(840\) 0 0
\(841\) −21077.4 −0.864219
\(842\) 66733.8i 2.73135i
\(843\) 7635.97i 0.311977i
\(844\) 86302.6 3.51974
\(845\) 0 0
\(846\) −19148.0 −0.778159
\(847\) 1248.81i 0.0506605i
\(848\) 43079.9i 1.74454i
\(849\) −17735.1 −0.716923
\(850\) 0 0
\(851\) 54844.2 2.20921
\(852\) 5458.94i 0.219507i
\(853\) − 8070.62i − 0.323954i −0.986795 0.161977i \(-0.948213\pi\)
0.986795 0.161977i \(-0.0517870\pi\)
\(854\) 11061.4 0.443222
\(855\) 0 0
\(856\) 54659.5 2.18250
\(857\) 11344.2i 0.452169i 0.974108 + 0.226085i \(0.0725926\pi\)
−0.974108 + 0.226085i \(0.927407\pi\)
\(858\) 11109.1i 0.442026i
\(859\) −25470.6 −1.01170 −0.505848 0.862623i \(-0.668820\pi\)
−0.505848 + 0.862623i \(0.668820\pi\)
\(860\) 0 0
\(861\) −6994.78 −0.276866
\(862\) 74463.6i 2.94228i
\(863\) − 14558.4i − 0.574243i −0.957894 0.287122i \(-0.907302\pi\)
0.957894 0.287122i \(-0.0926984\pi\)
\(864\) −10299.5 −0.405550
\(865\) 0 0
\(866\) 57614.6 2.26077
\(867\) − 13857.0i − 0.542802i
\(868\) 28786.5i 1.12567i
\(869\) 11237.6 0.438675
\(870\) 0 0
\(871\) 17076.0 0.664294
\(872\) 77226.3i 2.99910i
\(873\) − 1779.85i − 0.0690019i
\(874\) −100911. −3.90546
\(875\) 0 0
\(876\) 14291.6 0.551220
\(877\) − 185.528i − 0.00714350i −0.999994 0.00357175i \(-0.998863\pi\)
0.999994 0.00357175i \(-0.00113693\pi\)
\(878\) 59039.1i 2.26933i
\(879\) −20613.1 −0.790969
\(880\) 0 0
\(881\) 18950.1 0.724681 0.362340 0.932046i \(-0.381978\pi\)
0.362340 + 0.932046i \(0.381978\pi\)
\(882\) − 11198.3i − 0.427513i
\(883\) 21258.3i 0.810189i 0.914275 + 0.405095i \(0.132761\pi\)
−0.914275 + 0.405095i \(0.867239\pi\)
\(884\) −21593.3 −0.821563
\(885\) 0 0
\(886\) −50758.2 −1.92467
\(887\) − 35707.4i − 1.35168i −0.737049 0.675839i \(-0.763781\pi\)
0.737049 0.675839i \(-0.236219\pi\)
\(888\) − 47572.5i − 1.79778i
\(889\) 21318.1 0.804259
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 45921.8i 1.72374i
\(893\) 36477.8i 1.36695i
\(894\) 37898.4 1.41780
\(895\) 0 0
\(896\) 5393.06 0.201082
\(897\) − 40808.4i − 1.51901i
\(898\) − 34105.2i − 1.26738i
\(899\) 8154.49 0.302522
\(900\) 0 0
\(901\) 4450.58 0.164562
\(902\) − 13075.1i − 0.482653i
\(903\) − 10757.6i − 0.396447i
\(904\) −147165. −5.41440
\(905\) 0 0
\(906\) −39734.2 −1.45704
\(907\) 6542.52i 0.239516i 0.992803 + 0.119758i \(0.0382118\pi\)
−0.992803 + 0.119758i \(0.961788\pi\)
\(908\) − 41728.6i − 1.52512i
\(909\) −12605.6 −0.459958
\(910\) 0 0
\(911\) 31171.2 1.13364 0.566821 0.823841i \(-0.308173\pi\)
0.566821 + 0.823841i \(0.308173\pi\)
\(912\) 44915.9i 1.63083i
\(913\) 7770.56i 0.281674i
\(914\) 59559.2 2.15541
\(915\) 0 0
\(916\) 46199.2 1.66644
\(917\) 8107.19i 0.291955i
\(918\) 2435.78i 0.0875737i
\(919\) −12031.9 −0.431877 −0.215938 0.976407i \(-0.569281\pi\)
−0.215938 + 0.976407i \(0.569281\pi\)
\(920\) 0 0
\(921\) −600.537 −0.0214857
\(922\) − 44232.0i − 1.57994i
\(923\) 5914.85i 0.210931i
\(924\) −6703.82 −0.238679
\(925\) 0 0
\(926\) −51344.7 −1.82213
\(927\) 12111.3i 0.429113i
\(928\) − 21951.6i − 0.776506i
\(929\) 12546.2 0.443085 0.221542 0.975151i \(-0.428891\pi\)
0.221542 + 0.975151i \(0.428891\pi\)
\(930\) 0 0
\(931\) −21333.2 −0.750986
\(932\) 7391.07i 0.259767i
\(933\) − 17204.8i − 0.603708i
\(934\) −86229.1 −3.02088
\(935\) 0 0
\(936\) −35397.7 −1.23612
\(937\) − 17909.8i − 0.624427i −0.950012 0.312214i \(-0.898930\pi\)
0.950012 0.312214i \(-0.101070\pi\)
\(938\) 14492.7i 0.504483i
\(939\) −9232.08 −0.320849
\(940\) 0 0
\(941\) 829.893 0.0287500 0.0143750 0.999897i \(-0.495424\pi\)
0.0143750 + 0.999897i \(0.495424\pi\)
\(942\) − 21962.9i − 0.759650i
\(943\) 48030.4i 1.65863i
\(944\) 141581. 4.88144
\(945\) 0 0
\(946\) 20108.9 0.691116
\(947\) 17654.9i 0.605814i 0.953020 + 0.302907i \(0.0979572\pi\)
−0.953020 + 0.302907i \(0.902043\pi\)
\(948\) 60325.4i 2.06675i
\(949\) 15485.2 0.529685
\(950\) 0 0
\(951\) −6427.14 −0.219153
\(952\) − 10878.0i − 0.370335i
\(953\) 30736.6i 1.04476i 0.852713 + 0.522380i \(0.174956\pi\)
−0.852713 + 0.522380i \(0.825044\pi\)
\(954\) 12291.5 0.417139
\(955\) 0 0
\(956\) −28110.8 −0.951012
\(957\) 1899.02i 0.0641449i
\(958\) − 72761.5i − 2.45388i
\(959\) 13171.7 0.443519
\(960\) 0 0
\(961\) −9711.08 −0.325974
\(962\) − 86840.6i − 2.91045i
\(963\) − 8002.61i − 0.267789i
\(964\) −3755.97 −0.125489
\(965\) 0 0
\(966\) 34634.8 1.15358
\(967\) − 23645.2i − 0.786327i −0.919469 0.393163i \(-0.871381\pi\)
0.919469 0.393163i \(-0.128619\pi\)
\(968\) − 7438.10i − 0.246973i
\(969\) 4640.26 0.153836
\(970\) 0 0
\(971\) 27402.0 0.905635 0.452818 0.891603i \(-0.350419\pi\)
0.452818 + 0.891603i \(0.350419\pi\)
\(972\) 4783.05i 0.157836i
\(973\) 27899.1i 0.919222i
\(974\) 69790.6 2.29593
\(975\) 0 0
\(976\) −33807.5 −1.10876
\(977\) 49118.7i 1.60844i 0.594332 + 0.804220i \(0.297416\pi\)
−0.594332 + 0.804220i \(0.702584\pi\)
\(978\) − 42942.1i − 1.40403i
\(979\) 4842.40 0.158084
\(980\) 0 0
\(981\) 11306.6 0.367983
\(982\) − 39294.0i − 1.27691i
\(983\) 52630.5i 1.70768i 0.520533 + 0.853842i \(0.325733\pi\)
−0.520533 + 0.853842i \(0.674267\pi\)
\(984\) 41662.1 1.34973
\(985\) 0 0
\(986\) −5191.47 −0.167677
\(987\) − 12520.0i − 0.403764i
\(988\) 113609.i 3.65827i
\(989\) −73868.4 −2.37500
\(990\) 0 0
\(991\) 45472.7 1.45761 0.728803 0.684724i \(-0.240077\pi\)
0.728803 + 0.684724i \(0.240077\pi\)
\(992\) − 54054.5i − 1.73007i
\(993\) 4854.68i 0.155145i
\(994\) −5020.04 −0.160187
\(995\) 0 0
\(996\) −41713.8 −1.32706
\(997\) − 55068.1i − 1.74927i −0.484779 0.874637i \(-0.661100\pi\)
0.484779 0.874637i \(-0.338900\pi\)
\(998\) 27763.0i 0.880585i
\(999\) −6965.01 −0.220584
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.c.l.199.6 6
5.2 odd 4 165.4.a.d.1.1 3
5.3 odd 4 825.4.a.s.1.3 3
5.4 even 2 inner 825.4.c.l.199.1 6
15.2 even 4 495.4.a.l.1.3 3
15.8 even 4 2475.4.a.s.1.1 3
55.32 even 4 1815.4.a.s.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.1 3 5.2 odd 4
495.4.a.l.1.3 3 15.2 even 4
825.4.a.s.1.3 3 5.3 odd 4
825.4.c.l.199.1 6 5.4 even 2 inner
825.4.c.l.199.6 6 1.1 even 1 trivial
1815.4.a.s.1.3 3 55.32 even 4
2475.4.a.s.1.1 3 15.8 even 4