Properties

Label 175.4.b.b.99.2
Level $175$
Weight $4$
Character 175.99
Analytic conductor $10.325$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.b.99.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} +7.00000i q^{7} +15.0000i q^{8} +23.0000 q^{9} -8.00000 q^{11} -14.0000i q^{12} +28.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} -54.0000i q^{17} +23.0000i q^{18} +110.000 q^{19} +14.0000 q^{21} -8.00000i q^{22} +48.0000i q^{23} +30.0000 q^{24} -28.0000 q^{26} -100.000i q^{27} +49.0000i q^{28} +110.000 q^{29} +12.0000 q^{31} +161.000i q^{32} +16.0000i q^{33} +54.0000 q^{34} +161.000 q^{36} +246.000i q^{37} +110.000i q^{38} +56.0000 q^{39} +182.000 q^{41} +14.0000i q^{42} +128.000i q^{43} -56.0000 q^{44} -48.0000 q^{46} -324.000i q^{47} -82.0000i q^{48} -49.0000 q^{49} -108.000 q^{51} +196.000i q^{52} -162.000i q^{53} +100.000 q^{54} -105.000 q^{56} -220.000i q^{57} +110.000i q^{58} -810.000 q^{59} -488.000 q^{61} +12.0000i q^{62} +161.000i q^{63} +167.000 q^{64} -16.0000 q^{66} -244.000i q^{67} -378.000i q^{68} +96.0000 q^{69} -768.000 q^{71} +345.000i q^{72} -702.000i q^{73} -246.000 q^{74} +770.000 q^{76} -56.0000i q^{77} +56.0000i q^{78} -440.000 q^{79} +421.000 q^{81} +182.000i q^{82} -1302.00i q^{83} +98.0000 q^{84} -128.000 q^{86} -220.000i q^{87} -120.000i q^{88} -730.000 q^{89} -196.000 q^{91} +336.000i q^{92} -24.0000i q^{93} +324.000 q^{94} +322.000 q^{96} -294.000i q^{97} -49.0000i q^{98} -184.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{4} + 4q^{6} + 46q^{9} + O(q^{10}) \) \( 2q + 14q^{4} + 4q^{6} + 46q^{9} - 16q^{11} - 14q^{14} + 82q^{16} + 220q^{19} + 28q^{21} + 60q^{24} - 56q^{26} + 220q^{29} + 24q^{31} + 108q^{34} + 322q^{36} + 112q^{39} + 364q^{41} - 112q^{44} - 96q^{46} - 98q^{49} - 216q^{51} + 200q^{54} - 210q^{56} - 1620q^{59} - 976q^{61} + 334q^{64} - 32q^{66} + 192q^{69} - 1536q^{71} - 492q^{74} + 1540q^{76} - 880q^{79} + 842q^{81} + 196q^{84} - 256q^{86} - 1460q^{89} - 392q^{91} + 648q^{94} + 644q^{96} - 368q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.353553i 0.984251 + 0.176777i \(0.0565670\pi\)
−0.984251 + 0.176777i \(0.943433\pi\)
\(3\) − 2.00000i − 0.384900i −0.981307 0.192450i \(-0.938357\pi\)
0.981307 0.192450i \(-0.0616434\pi\)
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) 7.00000i 0.377964i
\(8\) 15.0000i 0.662913i
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) −8.00000 −0.219281 −0.109640 0.993971i \(-0.534970\pi\)
−0.109640 + 0.993971i \(0.534970\pi\)
\(12\) − 14.0000i − 0.336788i
\(13\) 28.0000i 0.597369i 0.954352 + 0.298685i \(0.0965479\pi\)
−0.954352 + 0.298685i \(0.903452\pi\)
\(14\) −7.00000 −0.133631
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) − 54.0000i − 0.770407i −0.922832 0.385204i \(-0.874131\pi\)
0.922832 0.385204i \(-0.125869\pi\)
\(18\) 23.0000i 0.301175i
\(19\) 110.000 1.32820 0.664098 0.747645i \(-0.268816\pi\)
0.664098 + 0.747645i \(0.268816\pi\)
\(20\) 0 0
\(21\) 14.0000 0.145479
\(22\) − 8.00000i − 0.0775275i
\(23\) 48.0000i 0.435161i 0.976042 + 0.217580i \(0.0698164\pi\)
−0.976042 + 0.217580i \(0.930184\pi\)
\(24\) 30.0000 0.255155
\(25\) 0 0
\(26\) −28.0000 −0.211202
\(27\) − 100.000i − 0.712778i
\(28\) 49.0000i 0.330719i
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) 0 0
\(31\) 12.0000 0.0695246 0.0347623 0.999396i \(-0.488933\pi\)
0.0347623 + 0.999396i \(0.488933\pi\)
\(32\) 161.000i 0.889408i
\(33\) 16.0000i 0.0844013i
\(34\) 54.0000 0.272380
\(35\) 0 0
\(36\) 161.000 0.745370
\(37\) 246.000i 1.09303i 0.837449 + 0.546516i \(0.184046\pi\)
−0.837449 + 0.546516i \(0.815954\pi\)
\(38\) 110.000i 0.469588i
\(39\) 56.0000 0.229928
\(40\) 0 0
\(41\) 182.000 0.693259 0.346630 0.938002i \(-0.387326\pi\)
0.346630 + 0.938002i \(0.387326\pi\)
\(42\) 14.0000i 0.0514344i
\(43\) 128.000i 0.453949i 0.973901 + 0.226975i \(0.0728834\pi\)
−0.973901 + 0.226975i \(0.927117\pi\)
\(44\) −56.0000 −0.191871
\(45\) 0 0
\(46\) −48.0000 −0.153852
\(47\) − 324.000i − 1.00554i −0.864421 0.502769i \(-0.832315\pi\)
0.864421 0.502769i \(-0.167685\pi\)
\(48\) − 82.0000i − 0.246577i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −108.000 −0.296530
\(52\) 196.000i 0.522698i
\(53\) − 162.000i − 0.419857i −0.977717 0.209928i \(-0.932677\pi\)
0.977717 0.209928i \(-0.0673231\pi\)
\(54\) 100.000 0.252005
\(55\) 0 0
\(56\) −105.000 −0.250557
\(57\) − 220.000i − 0.511223i
\(58\) 110.000i 0.249029i
\(59\) −810.000 −1.78734 −0.893670 0.448725i \(-0.851878\pi\)
−0.893670 + 0.448725i \(0.851878\pi\)
\(60\) 0 0
\(61\) −488.000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 12.0000i 0.0245807i
\(63\) 161.000i 0.321970i
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) −16.0000 −0.0298404
\(67\) − 244.000i − 0.444916i −0.974942 0.222458i \(-0.928592\pi\)
0.974942 0.222458i \(-0.0714080\pi\)
\(68\) − 378.000i − 0.674106i
\(69\) 96.0000 0.167493
\(70\) 0 0
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) 345.000i 0.564703i
\(73\) − 702.000i − 1.12552i −0.826621 0.562759i \(-0.809740\pi\)
0.826621 0.562759i \(-0.190260\pi\)
\(74\) −246.000 −0.386445
\(75\) 0 0
\(76\) 770.000 1.16217
\(77\) − 56.0000i − 0.0828804i
\(78\) 56.0000i 0.0812917i
\(79\) −440.000 −0.626631 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 182.000i 0.245104i
\(83\) − 1302.00i − 1.72184i −0.508737 0.860922i \(-0.669887\pi\)
0.508737 0.860922i \(-0.330113\pi\)
\(84\) 98.0000 0.127294
\(85\) 0 0
\(86\) −128.000 −0.160495
\(87\) − 220.000i − 0.271109i
\(88\) − 120.000i − 0.145364i
\(89\) −730.000 −0.869436 −0.434718 0.900567i \(-0.643152\pi\)
−0.434718 + 0.900567i \(0.643152\pi\)
\(90\) 0 0
\(91\) −196.000 −0.225784
\(92\) 336.000i 0.380765i
\(93\) − 24.0000i − 0.0267600i
\(94\) 324.000 0.355511
\(95\) 0 0
\(96\) 322.000 0.342333
\(97\) − 294.000i − 0.307744i −0.988091 0.153872i \(-0.950826\pi\)
0.988091 0.153872i \(-0.0491744\pi\)
\(98\) − 49.0000i − 0.0505076i
\(99\) −184.000 −0.186795
\(100\) 0 0
\(101\) −688.000 −0.677808 −0.338904 0.940821i \(-0.610056\pi\)
−0.338904 + 0.940821i \(0.610056\pi\)
\(102\) − 108.000i − 0.104839i
\(103\) 1388.00i 1.32780i 0.747820 + 0.663901i \(0.231101\pi\)
−0.747820 + 0.663901i \(0.768899\pi\)
\(104\) −420.000 −0.396004
\(105\) 0 0
\(106\) 162.000 0.148442
\(107\) − 244.000i − 0.220452i −0.993907 0.110226i \(-0.964843\pi\)
0.993907 0.110226i \(-0.0351575\pi\)
\(108\) − 700.000i − 0.623681i
\(109\) −90.0000 −0.0790866 −0.0395433 0.999218i \(-0.512590\pi\)
−0.0395433 + 0.999218i \(0.512590\pi\)
\(110\) 0 0
\(111\) 492.000 0.420708
\(112\) 287.000i 0.242133i
\(113\) 1318.00i 1.09723i 0.836075 + 0.548615i \(0.184845\pi\)
−0.836075 + 0.548615i \(0.815155\pi\)
\(114\) 220.000 0.180745
\(115\) 0 0
\(116\) 770.000 0.616316
\(117\) 644.000i 0.508870i
\(118\) − 810.000i − 0.631920i
\(119\) 378.000 0.291187
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) − 488.000i − 0.362143i
\(123\) − 364.000i − 0.266836i
\(124\) 84.0000 0.0608341
\(125\) 0 0
\(126\) −161.000 −0.113833
\(127\) 1776.00i 1.24090i 0.784245 + 0.620451i \(0.213050\pi\)
−0.784245 + 0.620451i \(0.786950\pi\)
\(128\) 1455.00i 1.00473i
\(129\) 256.000 0.174725
\(130\) 0 0
\(131\) −1118.00 −0.745650 −0.372825 0.927902i \(-0.621611\pi\)
−0.372825 + 0.927902i \(0.621611\pi\)
\(132\) 112.000i 0.0738511i
\(133\) 770.000i 0.502011i
\(134\) 244.000 0.157301
\(135\) 0 0
\(136\) 810.000 0.510713
\(137\) − 2274.00i − 1.41811i −0.705154 0.709054i \(-0.749122\pi\)
0.705154 0.709054i \(-0.250878\pi\)
\(138\) 96.0000i 0.0592178i
\(139\) 210.000 0.128144 0.0640718 0.997945i \(-0.479591\pi\)
0.0640718 + 0.997945i \(0.479591\pi\)
\(140\) 0 0
\(141\) −648.000 −0.387032
\(142\) − 768.000i − 0.453867i
\(143\) − 224.000i − 0.130992i
\(144\) 943.000 0.545718
\(145\) 0 0
\(146\) 702.000 0.397931
\(147\) 98.0000i 0.0549857i
\(148\) 1722.00i 0.956402i
\(149\) 2010.00 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(150\) 0 0
\(151\) 1112.00 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(152\) 1650.00i 0.880478i
\(153\) − 1242.00i − 0.656273i
\(154\) 56.0000 0.0293027
\(155\) 0 0
\(156\) 392.000 0.201187
\(157\) − 124.000i − 0.0630336i −0.999503 0.0315168i \(-0.989966\pi\)
0.999503 0.0315168i \(-0.0100338\pi\)
\(158\) − 440.000i − 0.221548i
\(159\) −324.000 −0.161603
\(160\) 0 0
\(161\) −336.000 −0.164475
\(162\) 421.000i 0.204178i
\(163\) 2008.00i 0.964900i 0.875924 + 0.482450i \(0.160253\pi\)
−0.875924 + 0.482450i \(0.839747\pi\)
\(164\) 1274.00 0.606602
\(165\) 0 0
\(166\) 1302.00 0.608764
\(167\) − 2884.00i − 1.33635i −0.744004 0.668176i \(-0.767076\pi\)
0.744004 0.668176i \(-0.232924\pi\)
\(168\) 210.000i 0.0964396i
\(169\) 1413.00 0.643150
\(170\) 0 0
\(171\) 2530.00 1.13143
\(172\) 896.000i 0.397206i
\(173\) 2228.00i 0.979143i 0.871963 + 0.489571i \(0.162847\pi\)
−0.871963 + 0.489571i \(0.837153\pi\)
\(174\) 220.000 0.0958515
\(175\) 0 0
\(176\) −328.000 −0.140477
\(177\) 1620.00i 0.687947i
\(178\) − 730.000i − 0.307392i
\(179\) 820.000 0.342400 0.171200 0.985236i \(-0.445236\pi\)
0.171200 + 0.985236i \(0.445236\pi\)
\(180\) 0 0
\(181\) 3892.00 1.59829 0.799144 0.601140i \(-0.205287\pi\)
0.799144 + 0.601140i \(0.205287\pi\)
\(182\) − 196.000i − 0.0798268i
\(183\) 976.000i 0.394251i
\(184\) −720.000 −0.288473
\(185\) 0 0
\(186\) 24.0000 0.00946110
\(187\) 432.000i 0.168936i
\(188\) − 2268.00i − 0.879845i
\(189\) 700.000 0.269405
\(190\) 0 0
\(191\) −5048.00 −1.91236 −0.956179 0.292782i \(-0.905419\pi\)
−0.956179 + 0.292782i \(0.905419\pi\)
\(192\) − 334.000i − 0.125544i
\(193\) − 2962.00i − 1.10471i −0.833608 0.552356i \(-0.813729\pi\)
0.833608 0.552356i \(-0.186271\pi\)
\(194\) 294.000 0.108804
\(195\) 0 0
\(196\) −343.000 −0.125000
\(197\) − 3334.00i − 1.20577i −0.797826 0.602887i \(-0.794017\pi\)
0.797826 0.602887i \(-0.205983\pi\)
\(198\) − 184.000i − 0.0660420i
\(199\) −1860.00 −0.662572 −0.331286 0.943530i \(-0.607483\pi\)
−0.331286 + 0.943530i \(0.607483\pi\)
\(200\) 0 0
\(201\) −488.000 −0.171248
\(202\) − 688.000i − 0.239641i
\(203\) 770.000i 0.266224i
\(204\) −756.000 −0.259464
\(205\) 0 0
\(206\) −1388.00 −0.469449
\(207\) 1104.00i 0.370692i
\(208\) 1148.00i 0.382690i
\(209\) −880.000 −0.291248
\(210\) 0 0
\(211\) −4268.00 −1.39252 −0.696259 0.717791i \(-0.745153\pi\)
−0.696259 + 0.717791i \(0.745153\pi\)
\(212\) − 1134.00i − 0.367375i
\(213\) 1536.00i 0.494108i
\(214\) 244.000 0.0779416
\(215\) 0 0
\(216\) 1500.00 0.472510
\(217\) 84.0000i 0.0262778i
\(218\) − 90.0000i − 0.0279613i
\(219\) −1404.00 −0.433212
\(220\) 0 0
\(221\) 1512.00 0.460218
\(222\) 492.000i 0.148743i
\(223\) − 5432.00i − 1.63118i −0.578629 0.815591i \(-0.696412\pi\)
0.578629 0.815591i \(-0.303588\pi\)
\(224\) −1127.00 −0.336165
\(225\) 0 0
\(226\) −1318.00 −0.387929
\(227\) 2046.00i 0.598228i 0.954217 + 0.299114i \(0.0966911\pi\)
−0.954217 + 0.299114i \(0.903309\pi\)
\(228\) − 1540.00i − 0.447320i
\(229\) 2980.00 0.859930 0.429965 0.902846i \(-0.358526\pi\)
0.429965 + 0.902846i \(0.358526\pi\)
\(230\) 0 0
\(231\) −112.000 −0.0319007
\(232\) 1650.00i 0.466930i
\(233\) 4458.00i 1.25345i 0.779241 + 0.626724i \(0.215605\pi\)
−0.779241 + 0.626724i \(0.784395\pi\)
\(234\) −644.000 −0.179913
\(235\) 0 0
\(236\) −5670.00 −1.56392
\(237\) 880.000i 0.241190i
\(238\) 378.000i 0.102950i
\(239\) −4440.00 −1.20167 −0.600836 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) 0 0
\(241\) 3302.00 0.882575 0.441287 0.897366i \(-0.354522\pi\)
0.441287 + 0.897366i \(0.354522\pi\)
\(242\) − 1267.00i − 0.336553i
\(243\) − 3542.00i − 0.935059i
\(244\) −3416.00 −0.896258
\(245\) 0 0
\(246\) 364.000 0.0943406
\(247\) 3080.00i 0.793424i
\(248\) 180.000i 0.0460888i
\(249\) −2604.00 −0.662738
\(250\) 0 0
\(251\) 1582.00 0.397829 0.198914 0.980017i \(-0.436258\pi\)
0.198914 + 0.980017i \(0.436258\pi\)
\(252\) 1127.00i 0.281724i
\(253\) − 384.000i − 0.0954224i
\(254\) −1776.00 −0.438725
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) − 2354.00i − 0.571356i −0.958326 0.285678i \(-0.907781\pi\)
0.958326 0.285678i \(-0.0922188\pi\)
\(258\) 256.000i 0.0617747i
\(259\) −1722.00 −0.413127
\(260\) 0 0
\(261\) 2530.00 0.600012
\(262\) − 1118.00i − 0.263627i
\(263\) − 3872.00i − 0.907824i −0.891046 0.453912i \(-0.850028\pi\)
0.891046 0.453912i \(-0.149972\pi\)
\(264\) −240.000 −0.0559507
\(265\) 0 0
\(266\) −770.000 −0.177488
\(267\) 1460.00i 0.334646i
\(268\) − 1708.00i − 0.389301i
\(269\) −180.000 −0.0407985 −0.0203992 0.999792i \(-0.506494\pi\)
−0.0203992 + 0.999792i \(0.506494\pi\)
\(270\) 0 0
\(271\) 2032.00 0.455480 0.227740 0.973722i \(-0.426866\pi\)
0.227740 + 0.973722i \(0.426866\pi\)
\(272\) − 2214.00i − 0.493542i
\(273\) 392.000i 0.0869045i
\(274\) 2274.00 0.501377
\(275\) 0 0
\(276\) 672.000 0.146557
\(277\) 5426.00i 1.17696i 0.808513 + 0.588478i \(0.200273\pi\)
−0.808513 + 0.588478i \(0.799727\pi\)
\(278\) 210.000i 0.0453056i
\(279\) 276.000 0.0592247
\(280\) 0 0
\(281\) 842.000 0.178753 0.0893764 0.995998i \(-0.471513\pi\)
0.0893764 + 0.995998i \(0.471513\pi\)
\(282\) − 648.000i − 0.136836i
\(283\) − 3782.00i − 0.794405i −0.917731 0.397202i \(-0.869981\pi\)
0.917731 0.397202i \(-0.130019\pi\)
\(284\) −5376.00 −1.12326
\(285\) 0 0
\(286\) 224.000 0.0463126
\(287\) 1274.00i 0.262027i
\(288\) 3703.00i 0.757644i
\(289\) 1997.00 0.406473
\(290\) 0 0
\(291\) −588.000 −0.118451
\(292\) − 4914.00i − 0.984829i
\(293\) − 4312.00i − 0.859760i −0.902886 0.429880i \(-0.858556\pi\)
0.902886 0.429880i \(-0.141444\pi\)
\(294\) −98.0000 −0.0194404
\(295\) 0 0
\(296\) −3690.00 −0.724584
\(297\) 800.000i 0.156299i
\(298\) 2010.00i 0.390725i
\(299\) −1344.00 −0.259952
\(300\) 0 0
\(301\) −896.000 −0.171577
\(302\) 1112.00i 0.211882i
\(303\) 1376.00i 0.260888i
\(304\) 4510.00 0.850876
\(305\) 0 0
\(306\) 1242.00 0.232027
\(307\) − 2674.00i − 0.497112i −0.968618 0.248556i \(-0.920044\pi\)
0.968618 0.248556i \(-0.0799559\pi\)
\(308\) − 392.000i − 0.0725204i
\(309\) 2776.00 0.511072
\(310\) 0 0
\(311\) −3768.00 −0.687021 −0.343511 0.939149i \(-0.611616\pi\)
−0.343511 + 0.939149i \(0.611616\pi\)
\(312\) 840.000i 0.152422i
\(313\) 2438.00i 0.440268i 0.975470 + 0.220134i \(0.0706495\pi\)
−0.975470 + 0.220134i \(0.929351\pi\)
\(314\) 124.000 0.0222857
\(315\) 0 0
\(316\) −3080.00 −0.548302
\(317\) 3186.00i 0.564491i 0.959342 + 0.282245i \(0.0910792\pi\)
−0.959342 + 0.282245i \(0.908921\pi\)
\(318\) − 324.000i − 0.0571353i
\(319\) −880.000 −0.154453
\(320\) 0 0
\(321\) −488.000 −0.0848520
\(322\) − 336.000i − 0.0581508i
\(323\) − 5940.00i − 1.02325i
\(324\) 2947.00 0.505316
\(325\) 0 0
\(326\) −2008.00 −0.341144
\(327\) 180.000i 0.0304404i
\(328\) 2730.00i 0.459570i
\(329\) 2268.00 0.380057
\(330\) 0 0
\(331\) 8672.00 1.44005 0.720025 0.693949i \(-0.244131\pi\)
0.720025 + 0.693949i \(0.244131\pi\)
\(332\) − 9114.00i − 1.50661i
\(333\) 5658.00i 0.931101i
\(334\) 2884.00 0.472471
\(335\) 0 0
\(336\) 574.000 0.0931972
\(337\) − 814.000i − 0.131577i −0.997834 0.0657884i \(-0.979044\pi\)
0.997834 0.0657884i \(-0.0209562\pi\)
\(338\) 1413.00i 0.227388i
\(339\) 2636.00 0.422324
\(340\) 0 0
\(341\) −96.0000 −0.0152454
\(342\) 2530.00i 0.400020i
\(343\) − 343.000i − 0.0539949i
\(344\) −1920.00 −0.300929
\(345\) 0 0
\(346\) −2228.00 −0.346179
\(347\) − 9344.00i − 1.44557i −0.691074 0.722784i \(-0.742862\pi\)
0.691074 0.722784i \(-0.257138\pi\)
\(348\) − 1540.00i − 0.237220i
\(349\) 5180.00 0.794496 0.397248 0.917711i \(-0.369965\pi\)
0.397248 + 0.917711i \(0.369965\pi\)
\(350\) 0 0
\(351\) 2800.00 0.425792
\(352\) − 1288.00i − 0.195030i
\(353\) 12178.0i 1.83617i 0.396379 + 0.918087i \(0.370267\pi\)
−0.396379 + 0.918087i \(0.629733\pi\)
\(354\) −1620.00 −0.243226
\(355\) 0 0
\(356\) −5110.00 −0.760757
\(357\) − 756.000i − 0.112078i
\(358\) 820.000i 0.121057i
\(359\) −440.000 −0.0646861 −0.0323431 0.999477i \(-0.510297\pi\)
−0.0323431 + 0.999477i \(0.510297\pi\)
\(360\) 0 0
\(361\) 5241.00 0.764106
\(362\) 3892.00i 0.565080i
\(363\) 2534.00i 0.366393i
\(364\) −1372.00 −0.197561
\(365\) 0 0
\(366\) −976.000 −0.139389
\(367\) 9816.00i 1.39616i 0.716019 + 0.698080i \(0.245962\pi\)
−0.716019 + 0.698080i \(0.754038\pi\)
\(368\) 1968.00i 0.278775i
\(369\) 4186.00 0.590554
\(370\) 0 0
\(371\) 1134.00 0.158691
\(372\) − 168.000i − 0.0234150i
\(373\) − 442.000i − 0.0613563i −0.999529 0.0306781i \(-0.990233\pi\)
0.999529 0.0306781i \(-0.00976669\pi\)
\(374\) −432.000 −0.0597278
\(375\) 0 0
\(376\) 4860.00 0.666583
\(377\) 3080.00i 0.420764i
\(378\) 700.000i 0.0952490i
\(379\) 3960.00 0.536706 0.268353 0.963321i \(-0.413521\pi\)
0.268353 + 0.963321i \(0.413521\pi\)
\(380\) 0 0
\(381\) 3552.00 0.477623
\(382\) − 5048.00i − 0.676121i
\(383\) 6708.00i 0.894942i 0.894298 + 0.447471i \(0.147675\pi\)
−0.894298 + 0.447471i \(0.852325\pi\)
\(384\) 2910.00 0.386720
\(385\) 0 0
\(386\) 2962.00 0.390575
\(387\) 2944.00i 0.386697i
\(388\) − 2058.00i − 0.269276i
\(389\) 13350.0 1.74003 0.870015 0.493025i \(-0.164109\pi\)
0.870015 + 0.493025i \(0.164109\pi\)
\(390\) 0 0
\(391\) 2592.00 0.335251
\(392\) − 735.000i − 0.0947018i
\(393\) 2236.00i 0.287001i
\(394\) 3334.00 0.426306
\(395\) 0 0
\(396\) −1288.00 −0.163446
\(397\) 1356.00i 0.171425i 0.996320 + 0.0857125i \(0.0273166\pi\)
−0.996320 + 0.0857125i \(0.972683\pi\)
\(398\) − 1860.00i − 0.234255i
\(399\) 1540.00 0.193224
\(400\) 0 0
\(401\) 6222.00 0.774843 0.387421 0.921903i \(-0.373366\pi\)
0.387421 + 0.921903i \(0.373366\pi\)
\(402\) − 488.000i − 0.0605453i
\(403\) 336.000i 0.0415319i
\(404\) −4816.00 −0.593082
\(405\) 0 0
\(406\) −770.000 −0.0941243
\(407\) − 1968.00i − 0.239681i
\(408\) − 1620.00i − 0.196573i
\(409\) −5150.00 −0.622619 −0.311309 0.950309i \(-0.600768\pi\)
−0.311309 + 0.950309i \(0.600768\pi\)
\(410\) 0 0
\(411\) −4548.00 −0.545830
\(412\) 9716.00i 1.16183i
\(413\) − 5670.00i − 0.675551i
\(414\) −1104.00 −0.131060
\(415\) 0 0
\(416\) −4508.00 −0.531305
\(417\) − 420.000i − 0.0493225i
\(418\) − 880.000i − 0.102972i
\(419\) −2310.00 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(420\) 0 0
\(421\) 1262.00 0.146095 0.0730476 0.997328i \(-0.476727\pi\)
0.0730476 + 0.997328i \(0.476727\pi\)
\(422\) − 4268.00i − 0.492329i
\(423\) − 7452.00i − 0.856569i
\(424\) 2430.00 0.278328
\(425\) 0 0
\(426\) −1536.00 −0.174694
\(427\) − 3416.00i − 0.387147i
\(428\) − 1708.00i − 0.192896i
\(429\) −448.000 −0.0504188
\(430\) 0 0
\(431\) −4488.00 −0.501576 −0.250788 0.968042i \(-0.580690\pi\)
−0.250788 + 0.968042i \(0.580690\pi\)
\(432\) − 4100.00i − 0.456623i
\(433\) 17038.0i 1.89098i 0.325652 + 0.945490i \(0.394416\pi\)
−0.325652 + 0.945490i \(0.605584\pi\)
\(434\) −84.0000 −0.00929062
\(435\) 0 0
\(436\) −630.000 −0.0692008
\(437\) 5280.00i 0.577979i
\(438\) − 1404.00i − 0.153164i
\(439\) −16200.0 −1.76124 −0.880619 0.473824i \(-0.842873\pi\)
−0.880619 + 0.473824i \(0.842873\pi\)
\(440\) 0 0
\(441\) −1127.00 −0.121693
\(442\) 1512.00i 0.162712i
\(443\) − 8772.00i − 0.940791i −0.882456 0.470395i \(-0.844111\pi\)
0.882456 0.470395i \(-0.155889\pi\)
\(444\) 3444.00 0.368119
\(445\) 0 0
\(446\) 5432.00 0.576710
\(447\) − 4020.00i − 0.425368i
\(448\) 1169.00i 0.123281i
\(449\) −2130.00 −0.223877 −0.111939 0.993715i \(-0.535706\pi\)
−0.111939 + 0.993715i \(0.535706\pi\)
\(450\) 0 0
\(451\) −1456.00 −0.152019
\(452\) 9226.00i 0.960076i
\(453\) − 2224.00i − 0.230668i
\(454\) −2046.00 −0.211506
\(455\) 0 0
\(456\) 3300.00 0.338896
\(457\) − 10534.0i − 1.07825i −0.842226 0.539124i \(-0.818755\pi\)
0.842226 0.539124i \(-0.181245\pi\)
\(458\) 2980.00i 0.304031i
\(459\) −5400.00 −0.549129
\(460\) 0 0
\(461\) −9268.00 −0.936342 −0.468171 0.883638i \(-0.655087\pi\)
−0.468171 + 0.883638i \(0.655087\pi\)
\(462\) − 112.000i − 0.0112786i
\(463\) − 9392.00i − 0.942728i −0.881939 0.471364i \(-0.843762\pi\)
0.881939 0.471364i \(-0.156238\pi\)
\(464\) 4510.00 0.451232
\(465\) 0 0
\(466\) −4458.00 −0.443161
\(467\) 10806.0i 1.07075i 0.844613 + 0.535377i \(0.179830\pi\)
−0.844613 + 0.535377i \(0.820170\pi\)
\(468\) 4508.00i 0.445261i
\(469\) 1708.00 0.168162
\(470\) 0 0
\(471\) −248.000 −0.0242616
\(472\) − 12150.0i − 1.18485i
\(473\) − 1024.00i − 0.0995424i
\(474\) −880.000 −0.0852737
\(475\) 0 0
\(476\) 2646.00 0.254788
\(477\) − 3726.00i − 0.357656i
\(478\) − 4440.00i − 0.424855i
\(479\) −4940.00 −0.471220 −0.235610 0.971848i \(-0.575709\pi\)
−0.235610 + 0.971848i \(0.575709\pi\)
\(480\) 0 0
\(481\) −6888.00 −0.652943
\(482\) 3302.00i 0.312037i
\(483\) 672.000i 0.0633065i
\(484\) −8869.00 −0.832926
\(485\) 0 0
\(486\) 3542.00 0.330593
\(487\) 5216.00i 0.485338i 0.970109 + 0.242669i \(0.0780229\pi\)
−0.970109 + 0.242669i \(0.921977\pi\)
\(488\) − 7320.00i − 0.679018i
\(489\) 4016.00 0.371390
\(490\) 0 0
\(491\) 4412.00 0.405521 0.202760 0.979228i \(-0.435009\pi\)
0.202760 + 0.979228i \(0.435009\pi\)
\(492\) − 2548.00i − 0.233481i
\(493\) − 5940.00i − 0.542645i
\(494\) −3080.00 −0.280518
\(495\) 0 0
\(496\) 492.000 0.0445392
\(497\) − 5376.00i − 0.485204i
\(498\) − 2604.00i − 0.234313i
\(499\) −19060.0 −1.70991 −0.854953 0.518706i \(-0.826414\pi\)
−0.854953 + 0.518706i \(0.826414\pi\)
\(500\) 0 0
\(501\) −5768.00 −0.514362
\(502\) 1582.00i 0.140654i
\(503\) 12768.0i 1.13180i 0.824473 + 0.565902i \(0.191472\pi\)
−0.824473 + 0.565902i \(0.808528\pi\)
\(504\) −2415.00 −0.213438
\(505\) 0 0
\(506\) 384.000 0.0337369
\(507\) − 2826.00i − 0.247548i
\(508\) 12432.0i 1.08579i
\(509\) 5500.00 0.478945 0.239473 0.970903i \(-0.423025\pi\)
0.239473 + 0.970903i \(0.423025\pi\)
\(510\) 0 0
\(511\) 4914.00 0.425406
\(512\) 11521.0i 0.994455i
\(513\) − 11000.0i − 0.946709i
\(514\) 2354.00 0.202005
\(515\) 0 0
\(516\) 1792.00 0.152884
\(517\) 2592.00i 0.220495i
\(518\) − 1722.00i − 0.146062i
\(519\) 4456.00 0.376872
\(520\) 0 0
\(521\) −7338.00 −0.617051 −0.308526 0.951216i \(-0.599836\pi\)
−0.308526 + 0.951216i \(0.599836\pi\)
\(522\) 2530.00i 0.212136i
\(523\) − 17582.0i − 1.46999i −0.678070 0.734997i \(-0.737183\pi\)
0.678070 0.734997i \(-0.262817\pi\)
\(524\) −7826.00 −0.652444
\(525\) 0 0
\(526\) 3872.00 0.320964
\(527\) − 648.000i − 0.0535623i
\(528\) 656.000i 0.0540696i
\(529\) 9863.00 0.810635
\(530\) 0 0
\(531\) −18630.0 −1.52255
\(532\) 5390.00i 0.439260i
\(533\) 5096.00i 0.414132i
\(534\) −1460.00 −0.118315
\(535\) 0 0
\(536\) 3660.00 0.294940
\(537\) − 1640.00i − 0.131790i
\(538\) − 180.000i − 0.0144244i
\(539\) 392.000 0.0313259
\(540\) 0 0
\(541\) −1618.00 −0.128583 −0.0642914 0.997931i \(-0.520479\pi\)
−0.0642914 + 0.997931i \(0.520479\pi\)
\(542\) 2032.00i 0.161037i
\(543\) − 7784.00i − 0.615181i
\(544\) 8694.00 0.685206
\(545\) 0 0
\(546\) −392.000 −0.0307254
\(547\) − 16144.0i − 1.26192i −0.775817 0.630958i \(-0.782662\pi\)
0.775817 0.630958i \(-0.217338\pi\)
\(548\) − 15918.0i − 1.24085i
\(549\) −11224.0 −0.872548
\(550\) 0 0
\(551\) 12100.0 0.935531
\(552\) 1440.00i 0.111033i
\(553\) − 3080.00i − 0.236844i
\(554\) −5426.00 −0.416117
\(555\) 0 0
\(556\) 1470.00 0.112126
\(557\) − 4654.00i − 0.354033i −0.984208 0.177016i \(-0.943355\pi\)
0.984208 0.177016i \(-0.0566446\pi\)
\(558\) 276.000i 0.0209391i
\(559\) −3584.00 −0.271175
\(560\) 0 0
\(561\) 864.000 0.0650234
\(562\) 842.000i 0.0631986i
\(563\) 10078.0i 0.754418i 0.926128 + 0.377209i \(0.123116\pi\)
−0.926128 + 0.377209i \(0.876884\pi\)
\(564\) −4536.00 −0.338653
\(565\) 0 0
\(566\) 3782.00 0.280865
\(567\) 2947.00i 0.218276i
\(568\) − 11520.0i − 0.851001i
\(569\) 5930.00 0.436904 0.218452 0.975848i \(-0.429899\pi\)
0.218452 + 0.975848i \(0.429899\pi\)
\(570\) 0 0
\(571\) −19048.0 −1.39603 −0.698016 0.716082i \(-0.745933\pi\)
−0.698016 + 0.716082i \(0.745933\pi\)
\(572\) − 1568.00i − 0.114618i
\(573\) 10096.0i 0.736067i
\(574\) −1274.00 −0.0926406
\(575\) 0 0
\(576\) 3841.00 0.277850
\(577\) 14366.0i 1.03651i 0.855227 + 0.518253i \(0.173418\pi\)
−0.855227 + 0.518253i \(0.826582\pi\)
\(578\) 1997.00i 0.143710i
\(579\) −5924.00 −0.425204
\(580\) 0 0
\(581\) 9114.00 0.650796
\(582\) − 588.000i − 0.0418787i
\(583\) 1296.00i 0.0920666i
\(584\) 10530.0 0.746121
\(585\) 0 0
\(586\) 4312.00 0.303971
\(587\) 3626.00i 0.254959i 0.991841 + 0.127480i \(0.0406887\pi\)
−0.991841 + 0.127480i \(0.959311\pi\)
\(588\) 686.000i 0.0481125i
\(589\) 1320.00 0.0923424
\(590\) 0 0
\(591\) −6668.00 −0.464103
\(592\) 10086.0i 0.700223i
\(593\) − 1062.00i − 0.0735432i −0.999324 0.0367716i \(-0.988293\pi\)
0.999324 0.0367716i \(-0.0117074\pi\)
\(594\) −800.000 −0.0552599
\(595\) 0 0
\(596\) 14070.0 0.966996
\(597\) 3720.00i 0.255024i
\(598\) − 1344.00i − 0.0919068i
\(599\) 10200.0 0.695761 0.347880 0.937539i \(-0.386902\pi\)
0.347880 + 0.937539i \(0.386902\pi\)
\(600\) 0 0
\(601\) −25158.0 −1.70751 −0.853757 0.520671i \(-0.825682\pi\)
−0.853757 + 0.520671i \(0.825682\pi\)
\(602\) − 896.000i − 0.0606615i
\(603\) − 5612.00i − 0.379002i
\(604\) 7784.00 0.524382
\(605\) 0 0
\(606\) −1376.00 −0.0922379
\(607\) − 25664.0i − 1.71609i −0.513570 0.858047i \(-0.671677\pi\)
0.513570 0.858047i \(-0.328323\pi\)
\(608\) 17710.0i 1.18131i
\(609\) 1540.00 0.102470
\(610\) 0 0
\(611\) 9072.00 0.600677
\(612\) − 8694.00i − 0.574239i
\(613\) 19018.0i 1.25307i 0.779395 + 0.626533i \(0.215527\pi\)
−0.779395 + 0.626533i \(0.784473\pi\)
\(614\) 2674.00 0.175755
\(615\) 0 0
\(616\) 840.000 0.0549425
\(617\) − 17334.0i − 1.13102i −0.824741 0.565511i \(-0.808679\pi\)
0.824741 0.565511i \(-0.191321\pi\)
\(618\) 2776.00i 0.180691i
\(619\) −18730.0 −1.21619 −0.608096 0.793864i \(-0.708066\pi\)
−0.608096 + 0.793864i \(0.708066\pi\)
\(620\) 0 0
\(621\) 4800.00 0.310173
\(622\) − 3768.00i − 0.242899i
\(623\) − 5110.00i − 0.328616i
\(624\) 2296.00 0.147297
\(625\) 0 0
\(626\) −2438.00 −0.155658
\(627\) 1760.00i 0.112101i
\(628\) − 868.000i − 0.0551544i
\(629\) 13284.0 0.842079
\(630\) 0 0
\(631\) −6928.00 −0.437083 −0.218541 0.975828i \(-0.570130\pi\)
−0.218541 + 0.975828i \(0.570130\pi\)
\(632\) − 6600.00i − 0.415402i
\(633\) 8536.00i 0.535980i
\(634\) −3186.00 −0.199578
\(635\) 0 0
\(636\) −2268.00 −0.141403
\(637\) − 1372.00i − 0.0853385i
\(638\) − 880.000i − 0.0546074i
\(639\) −17664.0 −1.09355
\(640\) 0 0
\(641\) 16302.0 1.00451 0.502255 0.864720i \(-0.332504\pi\)
0.502255 + 0.864720i \(0.332504\pi\)
\(642\) − 488.000i − 0.0299997i
\(643\) 4718.00i 0.289362i 0.989478 + 0.144681i \(0.0462156\pi\)
−0.989478 + 0.144681i \(0.953784\pi\)
\(644\) −2352.00 −0.143916
\(645\) 0 0
\(646\) 5940.00 0.361774
\(647\) 21436.0i 1.30253i 0.758851 + 0.651264i \(0.225761\pi\)
−0.758851 + 0.651264i \(0.774239\pi\)
\(648\) 6315.00i 0.382834i
\(649\) 6480.00 0.391930
\(650\) 0 0
\(651\) 168.000 0.0101143
\(652\) 14056.0i 0.844287i
\(653\) 4458.00i 0.267159i 0.991038 + 0.133580i \(0.0426472\pi\)
−0.991038 + 0.133580i \(0.957353\pi\)
\(654\) −180.000 −0.0107623
\(655\) 0 0
\(656\) 7462.00 0.444119
\(657\) − 16146.0i − 0.958775i
\(658\) 2268.00i 0.134371i
\(659\) 26640.0 1.57473 0.787365 0.616487i \(-0.211445\pi\)
0.787365 + 0.616487i \(0.211445\pi\)
\(660\) 0 0
\(661\) 7432.00 0.437324 0.218662 0.975801i \(-0.429831\pi\)
0.218662 + 0.975801i \(0.429831\pi\)
\(662\) 8672.00i 0.509134i
\(663\) − 3024.00i − 0.177138i
\(664\) 19530.0 1.14143
\(665\) 0 0
\(666\) −5658.00 −0.329194
\(667\) 5280.00i 0.306510i
\(668\) − 20188.0i − 1.16931i
\(669\) −10864.0 −0.627842
\(670\) 0 0
\(671\) 3904.00 0.224608
\(672\) 2254.00i 0.129390i
\(673\) 58.0000i 0.00332204i 0.999999 + 0.00166102i \(0.000528720\pi\)
−0.999999 + 0.00166102i \(0.999471\pi\)
\(674\) 814.000 0.0465194
\(675\) 0 0
\(676\) 9891.00 0.562756
\(677\) 21516.0i 1.22146i 0.791840 + 0.610729i \(0.209124\pi\)
−0.791840 + 0.610729i \(0.790876\pi\)
\(678\) 2636.00i 0.149314i
\(679\) 2058.00 0.116316
\(680\) 0 0
\(681\) 4092.00 0.230258
\(682\) − 96.0000i − 0.00539007i
\(683\) 18108.0i 1.01447i 0.861808 + 0.507235i \(0.169332\pi\)
−0.861808 + 0.507235i \(0.830668\pi\)
\(684\) 17710.0 0.989998
\(685\) 0 0
\(686\) 343.000 0.0190901
\(687\) − 5960.00i − 0.330987i
\(688\) 5248.00i 0.290811i
\(689\) 4536.00 0.250810
\(690\) 0 0
\(691\) −10078.0 −0.554827 −0.277413 0.960751i \(-0.589477\pi\)
−0.277413 + 0.960751i \(0.589477\pi\)
\(692\) 15596.0i 0.856750i
\(693\) − 1288.00i − 0.0706018i
\(694\) 9344.00 0.511086
\(695\) 0 0
\(696\) 3300.00 0.179722
\(697\) − 9828.00i − 0.534092i
\(698\) 5180.00i 0.280897i
\(699\) 8916.00 0.482452
\(700\) 0 0
\(701\) 18762.0 1.01089 0.505443 0.862860i \(-0.331329\pi\)
0.505443 + 0.862860i \(0.331329\pi\)
\(702\) 2800.00i 0.150540i
\(703\) 27060.0i 1.45176i
\(704\) −1336.00 −0.0715233
\(705\) 0 0
\(706\) −12178.0 −0.649186
\(707\) − 4816.00i − 0.256187i
\(708\) 11340.0i 0.601954i
\(709\) −6810.00 −0.360726 −0.180363 0.983600i \(-0.557727\pi\)
−0.180363 + 0.983600i \(0.557727\pi\)
\(710\) 0 0
\(711\) −10120.0 −0.533797
\(712\) − 10950.0i − 0.576360i
\(713\) 576.000i 0.0302544i
\(714\) 756.000 0.0396255
\(715\) 0 0
\(716\) 5740.00 0.299600
\(717\) 8880.00i 0.462524i
\(718\) − 440.000i − 0.0228700i
\(719\) −4860.00 −0.252083 −0.126041 0.992025i \(-0.540227\pi\)
−0.126041 + 0.992025i \(0.540227\pi\)
\(720\) 0 0
\(721\) −9716.00 −0.501862
\(722\) 5241.00i 0.270152i
\(723\) − 6604.00i − 0.339703i
\(724\) 27244.0 1.39850
\(725\) 0 0
\(726\) −2534.00 −0.129539
\(727\) 13636.0i 0.695641i 0.937561 + 0.347821i \(0.113078\pi\)
−0.937561 + 0.347821i \(0.886922\pi\)
\(728\) − 2940.00i − 0.149675i
\(729\) 4283.00 0.217599
\(730\) 0 0
\(731\) 6912.00 0.349726
\(732\) 6832.00i 0.344970i
\(733\) 2088.00i 0.105214i 0.998615 + 0.0526071i \(0.0167531\pi\)
−0.998615 + 0.0526071i \(0.983247\pi\)
\(734\) −9816.00 −0.493617
\(735\) 0 0
\(736\) −7728.00 −0.387035
\(737\) 1952.00i 0.0975615i
\(738\) 4186.00i 0.208792i
\(739\) 5160.00 0.256852 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(740\) 0 0
\(741\) 6160.00 0.305389
\(742\) 1134.00i 0.0561057i
\(743\) − 28152.0i − 1.39004i −0.718992 0.695018i \(-0.755396\pi\)
0.718992 0.695018i \(-0.244604\pi\)
\(744\) 360.000 0.0177396
\(745\) 0 0
\(746\) 442.000 0.0216927
\(747\) − 29946.0i − 1.46676i
\(748\) 3024.00i 0.147819i
\(749\) 1708.00 0.0833230
\(750\) 0 0
\(751\) −16808.0 −0.816688 −0.408344 0.912828i \(-0.633894\pi\)
−0.408344 + 0.912828i \(0.633894\pi\)
\(752\) − 13284.0i − 0.644172i
\(753\) − 3164.00i − 0.153124i
\(754\) −3080.00 −0.148763
\(755\) 0 0
\(756\) 4900.00 0.235729
\(757\) − 21674.0i − 1.04063i −0.853975 0.520314i \(-0.825815\pi\)
0.853975 0.520314i \(-0.174185\pi\)
\(758\) 3960.00i 0.189754i
\(759\) −768.000 −0.0367281
\(760\) 0 0
\(761\) 7422.00 0.353544 0.176772 0.984252i \(-0.443434\pi\)
0.176772 + 0.984252i \(0.443434\pi\)
\(762\) 3552.00i 0.168865i
\(763\) − 630.000i − 0.0298919i
\(764\) −35336.0 −1.67331
\(765\) 0 0
\(766\) −6708.00 −0.316410
\(767\) − 22680.0i − 1.06770i
\(768\) 238.000i 0.0111824i
\(769\) −13790.0 −0.646658 −0.323329 0.946287i \(-0.604802\pi\)
−0.323329 + 0.946287i \(0.604802\pi\)
\(770\) 0 0
\(771\) −4708.00 −0.219915
\(772\) − 20734.0i − 0.966623i
\(773\) − 6232.00i − 0.289973i −0.989434 0.144987i \(-0.953686\pi\)
0.989434 0.144987i \(-0.0463139\pi\)
\(774\) −2944.00 −0.136718
\(775\) 0 0
\(776\) 4410.00 0.204007
\(777\) 3444.00i 0.159013i
\(778\) 13350.0i 0.615194i
\(779\) 20020.0 0.920784
\(780\) 0 0
\(781\) 6144.00 0.281498
\(782\) 2592.00i 0.118529i
\(783\) − 11000.0i − 0.502054i
\(784\) −2009.00 −0.0915179
\(785\) 0 0
\(786\) −2236.00 −0.101470
\(787\) 1766.00i 0.0799887i 0.999200 + 0.0399943i \(0.0127340\pi\)
−0.999200 + 0.0399943i \(0.987266\pi\)
\(788\) − 23338.0i − 1.05505i
\(789\) −7744.00 −0.349422
\(790\) 0 0
\(791\) −9226.00 −0.414714
\(792\) − 2760.00i − 0.123829i
\(793\) − 13664.0i − 0.611883i
\(794\) −1356.00 −0.0606079
\(795\) 0 0
\(796\) −13020.0 −0.579751
\(797\) − 1204.00i − 0.0535105i −0.999642 0.0267552i \(-0.991483\pi\)
0.999642 0.0267552i \(-0.00851748\pi\)
\(798\) 1540.00i 0.0683150i
\(799\) −17496.0 −0.774673
\(800\) 0 0
\(801\) −16790.0 −0.740631
\(802\) 6222.00i 0.273948i
\(803\) 5616.00i 0.246805i
\(804\) −3416.00 −0.149842
\(805\) 0 0
\(806\) −336.000 −0.0146837
\(807\) 360.000i 0.0157033i
\(808\) − 10320.0i − 0.449327i
\(809\) 7050.00 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(810\) 0 0
\(811\) 23282.0 1.00807 0.504033 0.863684i \(-0.331849\pi\)
0.504033 + 0.863684i \(0.331849\pi\)
\(812\) 5390.00i 0.232946i
\(813\) − 4064.00i − 0.175315i
\(814\) 1968.00 0.0847400
\(815\) 0 0
\(816\) −4428.00 −0.189964
\(817\) 14080.0i 0.602934i
\(818\) − 5150.00i − 0.220129i
\(819\) −4508.00 −0.192335
\(820\) 0 0
\(821\) 10142.0 0.431131 0.215565 0.976489i \(-0.430841\pi\)
0.215565 + 0.976489i \(0.430841\pi\)
\(822\) − 4548.00i − 0.192980i
\(823\) − 9192.00i − 0.389323i −0.980870 0.194662i \(-0.937639\pi\)
0.980870 0.194662i \(-0.0623609\pi\)
\(824\) −20820.0 −0.880217
\(825\) 0 0
\(826\) 5670.00 0.238843
\(827\) 46716.0i 1.96430i 0.188104 + 0.982149i \(0.439766\pi\)
−0.188104 + 0.982149i \(0.560234\pi\)
\(828\) 7728.00i 0.324356i
\(829\) −11240.0 −0.470906 −0.235453 0.971886i \(-0.575657\pi\)
−0.235453 + 0.971886i \(0.575657\pi\)
\(830\) 0 0
\(831\) 10852.0 0.453010
\(832\) 4676.00i 0.194845i
\(833\) 2646.00i 0.110058i
\(834\) 420.000 0.0174381
\(835\) 0 0
\(836\) −6160.00 −0.254842
\(837\) − 1200.00i − 0.0495556i
\(838\) − 2310.00i − 0.0952239i
\(839\) −700.000 −0.0288042 −0.0144021 0.999896i \(-0.504584\pi\)
−0.0144021 + 0.999896i \(0.504584\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) 1262.00i 0.0516525i
\(843\) − 1684.00i − 0.0688019i
\(844\) −29876.0 −1.21845
\(845\) 0 0
\(846\) 7452.00 0.302843
\(847\) − 8869.00i − 0.359790i
\(848\) − 6642.00i − 0.268971i
\(849\) −7564.00 −0.305767
\(850\) 0 0
\(851\) −11808.0 −0.475644
\(852\) 10752.0i 0.432344i
\(853\) − 37492.0i − 1.50493i −0.658635 0.752463i \(-0.728866\pi\)
0.658635 0.752463i \(-0.271134\pi\)
\(854\) 3416.00 0.136877
\(855\) 0 0
\(856\) 3660.00 0.146140
\(857\) − 28894.0i − 1.15169i −0.817558 0.575846i \(-0.804673\pi\)
0.817558 0.575846i \(-0.195327\pi\)
\(858\) − 448.000i − 0.0178257i
\(859\) 2770.00 0.110025 0.0550123 0.998486i \(-0.482480\pi\)
0.0550123 + 0.998486i \(0.482480\pi\)
\(860\) 0 0
\(861\) 2548.00 0.100854
\(862\) − 4488.00i − 0.177334i
\(863\) 17688.0i 0.697690i 0.937180 + 0.348845i \(0.113426\pi\)
−0.937180 + 0.348845i \(0.886574\pi\)
\(864\) 16100.0 0.633950
\(865\) 0 0
\(866\) −17038.0 −0.668562
\(867\) − 3994.00i − 0.156451i
\(868\) 588.000i 0.0229931i
\(869\) 3520.00 0.137408
\(870\) 0 0
\(871\) 6832.00 0.265779
\(872\) − 1350.00i − 0.0524275i
\(873\) − 6762.00i − 0.262152i
\(874\) −5280.00 −0.204346
\(875\) 0 0
\(876\) −9828.00 −0.379061
\(877\) 33566.0i 1.29241i 0.763164 + 0.646205i \(0.223645\pi\)
−0.763164 + 0.646205i \(0.776355\pi\)
\(878\) − 16200.0i − 0.622692i
\(879\) −8624.00 −0.330922
\(880\) 0 0
\(881\) −16758.0 −0.640853 −0.320426 0.947273i \(-0.603826\pi\)
−0.320426 + 0.947273i \(0.603826\pi\)
\(882\) − 1127.00i − 0.0430250i
\(883\) 11468.0i 0.437066i 0.975830 + 0.218533i \(0.0701271\pi\)
−0.975830 + 0.218533i \(0.929873\pi\)
\(884\) 10584.0 0.402691
\(885\) 0 0
\(886\) 8772.00 0.332620
\(887\) 50356.0i 1.90619i 0.302674 + 0.953094i \(0.402121\pi\)
−0.302674 + 0.953094i \(0.597879\pi\)
\(888\) 7380.00i 0.278893i
\(889\) −12432.0 −0.469017
\(890\) 0 0
\(891\) −3368.00 −0.126636
\(892\) − 38024.0i − 1.42728i
\(893\) − 35640.0i − 1.33555i
\(894\) 4020.00 0.150390
\(895\) 0 0
\(896\) −10185.0 −0.379751
\(897\) 2688.00i 0.100055i
\(898\) − 2130.00i − 0.0791526i
\(899\) 1320.00 0.0489705
\(900\) 0 0
\(901\) −8748.00 −0.323461
\(902\) − 1456.00i − 0.0537467i
\(903\) 1792.00i 0.0660399i
\(904\) −19770.0 −0.727368
\(905\) 0 0
\(906\) 2224.00 0.0815535
\(907\) 8716.00i 0.319085i 0.987191 + 0.159542i \(0.0510019\pi\)
−0.987191 + 0.159542i \(0.948998\pi\)
\(908\) 14322.0i 0.523450i
\(909\) −15824.0 −0.577392
\(910\) 0 0
\(911\) 7632.00 0.277563 0.138781 0.990323i \(-0.455682\pi\)
0.138781 + 0.990323i \(0.455682\pi\)
\(912\) − 9020.00i − 0.327502i
\(913\) 10416.0i 0.377568i
\(914\) 10534.0 0.381219
\(915\) 0 0
\(916\) 20860.0 0.752439
\(917\) − 7826.00i − 0.281829i
\(918\) − 5400.00i − 0.194147i
\(919\) 23080.0 0.828443 0.414221 0.910176i \(-0.364054\pi\)
0.414221 + 0.910176i \(0.364054\pi\)
\(920\) 0 0
\(921\) −5348.00 −0.191338
\(922\) − 9268.00i − 0.331047i
\(923\) − 21504.0i − 0.766861i
\(924\) −784.000 −0.0279131
\(925\) 0 0
\(926\) 9392.00 0.333305
\(927\) 31924.0i 1.13109i
\(928\) 17710.0i 0.626465i
\(929\) −45110.0 −1.59312 −0.796561 0.604558i \(-0.793350\pi\)
−0.796561 + 0.604558i \(0.793350\pi\)
\(930\) 0 0
\(931\) −5390.00 −0.189742
\(932\) 31206.0i 1.09677i
\(933\) 7536.00i 0.264435i
\(934\) −10806.0 −0.378569
\(935\) 0 0
\(936\) −9660.00 −0.337337
\(937\) − 16674.0i − 0.581340i −0.956823 0.290670i \(-0.906122\pi\)
0.956823 0.290670i \(-0.0938782\pi\)
\(938\) 1708.00i 0.0594543i
\(939\) 4876.00 0.169459
\(940\) 0 0
\(941\) 43832.0 1.51847 0.759236 0.650815i \(-0.225573\pi\)
0.759236 + 0.650815i \(0.225573\pi\)
\(942\) − 248.000i − 0.00857779i
\(943\) 8736.00i 0.301679i
\(944\) −33210.0 −1.14501
\(945\) 0 0
\(946\) 1024.00 0.0351936
\(947\) 736.000i 0.0252553i 0.999920 + 0.0126277i \(0.00401962\pi\)
−0.999920 + 0.0126277i \(0.995980\pi\)
\(948\) 6160.00i 0.211042i
\(949\) 19656.0 0.672351
\(950\) 0 0
\(951\) 6372.00 0.217273
\(952\) 5670.00i 0.193031i
\(953\) 38138.0i 1.29634i 0.761496 + 0.648169i \(0.224465\pi\)
−0.761496 + 0.648169i \(0.775535\pi\)
\(954\) 3726.00 0.126450
\(955\) 0 0
\(956\) −31080.0 −1.05146
\(957\) 1760.00i 0.0594490i
\(958\) − 4940.00i − 0.166601i
\(959\) 15918.0 0.535995
\(960\) 0 0
\(961\) −29647.0 −0.995166
\(962\) − 6888.00i − 0.230850i
\(963\) − 5612.00i − 0.187792i
\(964\) 23114.0 0.772253
\(965\) 0 0
\(966\) −672.000 −0.0223822
\(967\) − 26224.0i − 0.872086i −0.899926 0.436043i \(-0.856380\pi\)
0.899926 0.436043i \(-0.143620\pi\)
\(968\) − 19005.0i − 0.631037i
\(969\) −11880.0 −0.393850
\(970\) 0 0
\(971\) 18762.0 0.620084 0.310042 0.950723i \(-0.399657\pi\)
0.310042 + 0.950723i \(0.399657\pi\)
\(972\) − 24794.0i − 0.818177i
\(973\) 1470.00i 0.0484337i
\(974\) −5216.00 −0.171593
\(975\) 0 0
\(976\) −20008.0 −0.656189
\(977\) − 38394.0i − 1.25725i −0.777709 0.628625i \(-0.783618\pi\)
0.777709 0.628625i \(-0.216382\pi\)
\(978\) 4016.00i 0.131306i
\(979\) 5840.00 0.190651
\(980\) 0 0
\(981\) −2070.00 −0.0673700
\(982\) 4412.00i 0.143373i
\(983\) 5388.00i 0.174822i 0.996172 + 0.0874112i \(0.0278594\pi\)
−0.996172 + 0.0874112i \(0.972141\pi\)
\(984\) 5460.00 0.176889
\(985\) 0 0
\(986\) 5940.00 0.191854
\(987\) − 4536.00i − 0.146284i
\(988\) 21560.0i 0.694246i
\(989\) −6144.00 −0.197541
\(990\) 0 0
\(991\) 25472.0 0.816493 0.408247 0.912872i \(-0.366140\pi\)
0.408247 + 0.912872i \(0.366140\pi\)
\(992\) 1932.00i 0.0618357i
\(993\) − 17344.0i − 0.554275i
\(994\) 5376.00 0.171546
\(995\) 0 0
\(996\) −18228.0 −0.579896
\(997\) 17096.0i 0.543065i 0.962429 + 0.271532i \(0.0875304\pi\)
−0.962429 + 0.271532i \(0.912470\pi\)
\(998\) − 19060.0i − 0.604543i
\(999\) 24600.0 0.779089
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 175.4.b.b.99.2 2
5.2 odd 4 7.4.a.a.1.1 1
5.3 odd 4 175.4.a.b.1.1 1
5.4 even 2 inner 175.4.b.b.99.1 2
15.2 even 4 63.4.a.b.1.1 1
15.8 even 4 1575.4.a.e.1.1 1
20.7 even 4 112.4.a.f.1.1 1
35.2 odd 12 49.4.c.c.18.1 2
35.12 even 12 49.4.c.b.18.1 2
35.13 even 4 1225.4.a.j.1.1 1
35.17 even 12 49.4.c.b.30.1 2
35.27 even 4 49.4.a.b.1.1 1
35.32 odd 12 49.4.c.c.30.1 2
40.27 even 4 448.4.a.e.1.1 1
40.37 odd 4 448.4.a.i.1.1 1
55.32 even 4 847.4.a.b.1.1 1
60.47 odd 4 1008.4.a.c.1.1 1
65.12 odd 4 1183.4.a.b.1.1 1
85.67 odd 4 2023.4.a.a.1.1 1
105.2 even 12 441.4.e.h.361.1 2
105.17 odd 12 441.4.e.e.226.1 2
105.32 even 12 441.4.e.h.226.1 2
105.47 odd 12 441.4.e.e.361.1 2
105.62 odd 4 441.4.a.i.1.1 1
140.27 odd 4 784.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 5.2 odd 4
49.4.a.b.1.1 1 35.27 even 4
49.4.c.b.18.1 2 35.12 even 12
49.4.c.b.30.1 2 35.17 even 12
49.4.c.c.18.1 2 35.2 odd 12
49.4.c.c.30.1 2 35.32 odd 12
63.4.a.b.1.1 1 15.2 even 4
112.4.a.f.1.1 1 20.7 even 4
175.4.a.b.1.1 1 5.3 odd 4
175.4.b.b.99.1 2 5.4 even 2 inner
175.4.b.b.99.2 2 1.1 even 1 trivial
441.4.a.i.1.1 1 105.62 odd 4
441.4.e.e.226.1 2 105.17 odd 12
441.4.e.e.361.1 2 105.47 odd 12
441.4.e.h.226.1 2 105.32 even 12
441.4.e.h.361.1 2 105.2 even 12
448.4.a.e.1.1 1 40.27 even 4
448.4.a.i.1.1 1 40.37 odd 4
784.4.a.g.1.1 1 140.27 odd 4
847.4.a.b.1.1 1 55.32 even 4
1008.4.a.c.1.1 1 60.47 odd 4
1183.4.a.b.1.1 1 65.12 odd 4
1225.4.a.j.1.1 1 35.13 even 4
1575.4.a.e.1.1 1 15.8 even 4
2023.4.a.a.1.1 1 85.67 odd 4