Properties

Label 2450.2.c.t.99.1
Level $2450$
Weight $2$
Character 2450.99
Analytic conductor $19.563$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 490)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2450.99
Dual form 2450.2.c.t.99.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.41421i q^{3} -1.00000 q^{4} -3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -3.41421i q^{3} -1.00000 q^{4} -3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} -0.828427 q^{11} +3.41421i q^{12} +4.82843i q^{13} +1.00000 q^{16} +2.58579i q^{17} +8.65685i q^{18} -0.585786 q^{19} +0.828427i q^{22} +1.17157i q^{23} +3.41421 q^{24} +4.82843 q^{26} +19.3137i q^{27} +4.82843 q^{29} -2.82843 q^{31} -1.00000i q^{32} +2.82843i q^{33} +2.58579 q^{34} +8.65685 q^{36} -7.65685i q^{37} +0.585786i q^{38} +16.4853 q^{39} -3.07107 q^{41} +8.82843i q^{43} +0.828427 q^{44} +1.17157 q^{46} +5.17157i q^{47} -3.41421i q^{48} +8.82843 q^{51} -4.82843i q^{52} -6.48528i q^{53} +19.3137 q^{54} +2.00000i q^{57} -4.82843i q^{58} -8.58579 q^{59} +9.31371 q^{61} +2.82843i q^{62} -1.00000 q^{64} +2.82843 q^{66} +1.65685i q^{67} -2.58579i q^{68} +4.00000 q^{69} -4.48528 q^{71} -8.65685i q^{72} +9.41421i q^{73} -7.65685 q^{74} +0.585786 q^{76} -16.4853i q^{78} +6.82843 q^{79} +39.9706 q^{81} +3.07107i q^{82} +2.24264i q^{83} +8.82843 q^{86} -16.4853i q^{87} -0.828427i q^{88} -12.7279 q^{89} -1.17157i q^{92} +9.65685i q^{93} +5.17157 q^{94} -3.41421 q^{96} -7.75736i q^{97} +7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 8q^{6} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 8q^{6} - 12q^{9} + 8q^{11} + 4q^{16} - 8q^{19} + 8q^{24} + 8q^{26} + 8q^{29} + 16q^{34} + 12q^{36} + 32q^{39} + 16q^{41} - 8q^{44} + 16q^{46} + 24q^{51} + 32q^{54} - 40q^{59} - 8q^{61} - 4q^{64} + 16q^{69} + 16q^{71} - 8q^{74} + 8q^{76} + 16q^{79} + 92q^{81} + 24q^{86} + 32q^{94} - 8q^{96} + 40q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 − 0.707107i
\(3\) − 3.41421i − 1.97120i −0.169102 0.985599i \(-0.554087\pi\)
0.169102 0.985599i \(-0.445913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.41421 −1.39385
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −8.65685 −2.88562
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 3.41421i 0.985599i
\(13\) 4.82843i 1.33916i 0.742738 + 0.669582i \(0.233527\pi\)
−0.742738 + 0.669582i \(0.766473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.58579i 0.627145i 0.949564 + 0.313573i \(0.101526\pi\)
−0.949564 + 0.313573i \(0.898474\pi\)
\(18\) 8.65685i 2.04044i
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.828427i 0.176621i
\(23\) 1.17157i 0.244290i 0.992512 + 0.122145i \(0.0389773\pi\)
−0.992512 + 0.122145i \(0.961023\pi\)
\(24\) 3.41421 0.696923
\(25\) 0 0
\(26\) 4.82843 0.946932
\(27\) 19.3137i 3.71692i
\(28\) 0 0
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.82843i 0.492366i
\(34\) 2.58579 0.443459
\(35\) 0 0
\(36\) 8.65685 1.44281
\(37\) − 7.65685i − 1.25878i −0.777090 0.629390i \(-0.783305\pi\)
0.777090 0.629390i \(-0.216695\pi\)
\(38\) 0.585786i 0.0950271i
\(39\) 16.4853 2.63976
\(40\) 0 0
\(41\) −3.07107 −0.479620 −0.239810 0.970820i \(-0.577085\pi\)
−0.239810 + 0.970820i \(0.577085\pi\)
\(42\) 0 0
\(43\) 8.82843i 1.34632i 0.739496 + 0.673161i \(0.235064\pi\)
−0.739496 + 0.673161i \(0.764936\pi\)
\(44\) 0.828427 0.124890
\(45\) 0 0
\(46\) 1.17157 0.172739
\(47\) 5.17157i 0.754351i 0.926142 + 0.377176i \(0.123105\pi\)
−0.926142 + 0.377176i \(0.876895\pi\)
\(48\) − 3.41421i − 0.492799i
\(49\) 0 0
\(50\) 0 0
\(51\) 8.82843 1.23623
\(52\) − 4.82843i − 0.669582i
\(53\) − 6.48528i − 0.890822i −0.895326 0.445411i \(-0.853058\pi\)
0.895326 0.445411i \(-0.146942\pi\)
\(54\) 19.3137 2.62826
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) − 4.82843i − 0.634004i
\(59\) −8.58579 −1.11777 −0.558887 0.829244i \(-0.688771\pi\)
−0.558887 + 0.829244i \(0.688771\pi\)
\(60\) 0 0
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) 2.82843i 0.359211i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.82843 0.348155
\(67\) 1.65685i 0.202417i 0.994865 + 0.101208i \(0.0322709\pi\)
−0.994865 + 0.101208i \(0.967729\pi\)
\(68\) − 2.58579i − 0.313573i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −4.48528 −0.532305 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(72\) − 8.65685i − 1.02022i
\(73\) 9.41421i 1.10185i 0.834555 + 0.550925i \(0.185725\pi\)
−0.834555 + 0.550925i \(0.814275\pi\)
\(74\) −7.65685 −0.890091
\(75\) 0 0
\(76\) 0.585786 0.0671943
\(77\) 0 0
\(78\) − 16.4853i − 1.86659i
\(79\) 6.82843 0.768258 0.384129 0.923279i \(-0.374502\pi\)
0.384129 + 0.923279i \(0.374502\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 3.07107i 0.339143i
\(83\) 2.24264i 0.246162i 0.992397 + 0.123081i \(0.0392775\pi\)
−0.992397 + 0.123081i \(0.960723\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.82843 0.951994
\(87\) − 16.4853i − 1.76741i
\(88\) − 0.828427i − 0.0883106i
\(89\) −12.7279 −1.34916 −0.674579 0.738203i \(-0.735675\pi\)
−0.674579 + 0.738203i \(0.735675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1.17157i − 0.122145i
\(93\) 9.65685i 1.00137i
\(94\) 5.17157 0.533407
\(95\) 0 0
\(96\) −3.41421 −0.348462
\(97\) − 7.75736i − 0.787641i −0.919187 0.393820i \(-0.871153\pi\)
0.919187 0.393820i \(-0.128847\pi\)
\(98\) 0 0
\(99\) 7.17157 0.720770
\(100\) 0 0
\(101\) 13.3137 1.32476 0.662382 0.749166i \(-0.269546\pi\)
0.662382 + 0.749166i \(0.269546\pi\)
\(102\) − 8.82843i − 0.874145i
\(103\) 14.8284i 1.46109i 0.682865 + 0.730544i \(0.260734\pi\)
−0.682865 + 0.730544i \(0.739266\pi\)
\(104\) −4.82843 −0.473466
\(105\) 0 0
\(106\) −6.48528 −0.629906
\(107\) 9.65685i 0.933563i 0.884373 + 0.466782i \(0.154587\pi\)
−0.884373 + 0.466782i \(0.845413\pi\)
\(108\) − 19.3137i − 1.85846i
\(109\) −2.48528 −0.238047 −0.119023 0.992891i \(-0.537976\pi\)
−0.119023 + 0.992891i \(0.537976\pi\)
\(110\) 0 0
\(111\) −26.1421 −2.48130
\(112\) 0 0
\(113\) 15.3137i 1.44059i 0.693667 + 0.720296i \(0.255994\pi\)
−0.693667 + 0.720296i \(0.744006\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −4.82843 −0.448308
\(117\) − 41.7990i − 3.86432i
\(118\) 8.58579i 0.790386i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) − 9.31371i − 0.843224i
\(123\) 10.4853i 0.945426i
\(124\) 2.82843 0.254000
\(125\) 0 0
\(126\) 0 0
\(127\) 2.82843i 0.250982i 0.992095 + 0.125491i \(0.0400507\pi\)
−0.992095 + 0.125491i \(0.959949\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 30.1421 2.65387
\(130\) 0 0
\(131\) −6.24264 −0.545422 −0.272711 0.962096i \(-0.587920\pi\)
−0.272711 + 0.962096i \(0.587920\pi\)
\(132\) − 2.82843i − 0.246183i
\(133\) 0 0
\(134\) 1.65685 0.143130
\(135\) 0 0
\(136\) −2.58579 −0.221729
\(137\) − 16.0000i − 1.36697i −0.729964 0.683486i \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −19.8995 −1.68785 −0.843927 0.536459i \(-0.819762\pi\)
−0.843927 + 0.536459i \(0.819762\pi\)
\(140\) 0 0
\(141\) 17.6569 1.48698
\(142\) 4.48528i 0.376396i
\(143\) − 4.00000i − 0.334497i
\(144\) −8.65685 −0.721405
\(145\) 0 0
\(146\) 9.41421 0.779126
\(147\) 0 0
\(148\) 7.65685i 0.629390i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −11.3137 −0.920697 −0.460348 0.887738i \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) − 0.585786i − 0.0475136i
\(153\) − 22.3848i − 1.80970i
\(154\) 0 0
\(155\) 0 0
\(156\) −16.4853 −1.31988
\(157\) 6.48528i 0.517582i 0.965933 + 0.258791i \(0.0833241\pi\)
−0.965933 + 0.258791i \(0.916676\pi\)
\(158\) − 6.82843i − 0.543240i
\(159\) −22.1421 −1.75599
\(160\) 0 0
\(161\) 0 0
\(162\) − 39.9706i − 3.14038i
\(163\) 20.1421i 1.57765i 0.614615 + 0.788827i \(0.289311\pi\)
−0.614615 + 0.788827i \(0.710689\pi\)
\(164\) 3.07107 0.239810
\(165\) 0 0
\(166\) 2.24264 0.174063
\(167\) 15.7990i 1.22256i 0.791413 + 0.611281i \(0.209346\pi\)
−0.791413 + 0.611281i \(0.790654\pi\)
\(168\) 0 0
\(169\) −10.3137 −0.793362
\(170\) 0 0
\(171\) 5.07107 0.387794
\(172\) − 8.82843i − 0.673161i
\(173\) 8.82843i 0.671213i 0.942002 + 0.335606i \(0.108941\pi\)
−0.942002 + 0.335606i \(0.891059\pi\)
\(174\) −16.4853 −1.24975
\(175\) 0 0
\(176\) −0.828427 −0.0624450
\(177\) 29.3137i 2.20335i
\(178\) 12.7279i 0.953998i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.48528 −0.184730 −0.0923648 0.995725i \(-0.529443\pi\)
−0.0923648 + 0.995725i \(0.529443\pi\)
\(182\) 0 0
\(183\) − 31.7990i − 2.35065i
\(184\) −1.17157 −0.0863695
\(185\) 0 0
\(186\) 9.65685 0.708075
\(187\) − 2.14214i − 0.156648i
\(188\) − 5.17157i − 0.377176i
\(189\) 0 0
\(190\) 0 0
\(191\) 10.1421 0.733859 0.366930 0.930249i \(-0.380409\pi\)
0.366930 + 0.930249i \(0.380409\pi\)
\(192\) 3.41421i 0.246400i
\(193\) − 5.65685i − 0.407189i −0.979055 0.203595i \(-0.934738\pi\)
0.979055 0.203595i \(-0.0652625\pi\)
\(194\) −7.75736 −0.556946
\(195\) 0 0
\(196\) 0 0
\(197\) − 25.7990i − 1.83810i −0.394139 0.919051i \(-0.628957\pi\)
0.394139 0.919051i \(-0.371043\pi\)
\(198\) − 7.17157i − 0.509661i
\(199\) −16.4853 −1.16861 −0.584305 0.811534i \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) − 13.3137i − 0.936749i
\(203\) 0 0
\(204\) −8.82843 −0.618114
\(205\) 0 0
\(206\) 14.8284 1.03315
\(207\) − 10.1421i − 0.704927i
\(208\) 4.82843i 0.334791i
\(209\) 0.485281 0.0335676
\(210\) 0 0
\(211\) 18.6274 1.28236 0.641182 0.767389i \(-0.278444\pi\)
0.641182 + 0.767389i \(0.278444\pi\)
\(212\) 6.48528i 0.445411i
\(213\) 15.3137i 1.04928i
\(214\) 9.65685 0.660129
\(215\) 0 0
\(216\) −19.3137 −1.31413
\(217\) 0 0
\(218\) 2.48528i 0.168324i
\(219\) 32.1421 2.17196
\(220\) 0 0
\(221\) −12.4853 −0.839851
\(222\) 26.1421i 1.75455i
\(223\) 7.31371i 0.489762i 0.969553 + 0.244881i \(0.0787489\pi\)
−0.969553 + 0.244881i \(0.921251\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 15.3137 1.01865
\(227\) 18.2426i 1.21081i 0.795919 + 0.605403i \(0.206988\pi\)
−0.795919 + 0.605403i \(0.793012\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 16.1421 1.06670 0.533351 0.845894i \(-0.320932\pi\)
0.533351 + 0.845894i \(0.320932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.82843i 0.317002i
\(233\) − 23.3137i − 1.52733i −0.645612 0.763666i \(-0.723397\pi\)
0.645612 0.763666i \(-0.276603\pi\)
\(234\) −41.7990 −2.73249
\(235\) 0 0
\(236\) 8.58579 0.558887
\(237\) − 23.3137i − 1.51439i
\(238\) 0 0
\(239\) −1.65685 −0.107173 −0.0535865 0.998563i \(-0.517065\pi\)
−0.0535865 + 0.998563i \(0.517065\pi\)
\(240\) 0 0
\(241\) −13.4142 −0.864085 −0.432043 0.901853i \(-0.642207\pi\)
−0.432043 + 0.901853i \(0.642207\pi\)
\(242\) 10.3137i 0.662990i
\(243\) − 78.5269i − 5.03750i
\(244\) −9.31371 −0.596249
\(245\) 0 0
\(246\) 10.4853 0.668517
\(247\) − 2.82843i − 0.179969i
\(248\) − 2.82843i − 0.179605i
\(249\) 7.65685 0.485233
\(250\) 0 0
\(251\) 0.585786 0.0369745 0.0184873 0.999829i \(-0.494115\pi\)
0.0184873 + 0.999829i \(0.494115\pi\)
\(252\) 0 0
\(253\) − 0.970563i − 0.0610188i
\(254\) 2.82843 0.177471
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 9.89949i − 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) − 30.1421i − 1.87657i
\(259\) 0 0
\(260\) 0 0
\(261\) −41.7990 −2.58729
\(262\) 6.24264i 0.385672i
\(263\) − 28.0000i − 1.72655i −0.504730 0.863277i \(-0.668408\pi\)
0.504730 0.863277i \(-0.331592\pi\)
\(264\) −2.82843 −0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) 43.4558i 2.65945i
\(268\) − 1.65685i − 0.101208i
\(269\) −18.4853 −1.12707 −0.563534 0.826093i \(-0.690559\pi\)
−0.563534 + 0.826093i \(0.690559\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.58579i 0.156786i
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) − 8.14214i − 0.489214i −0.969622 0.244607i \(-0.921341\pi\)
0.969622 0.244607i \(-0.0786589\pi\)
\(278\) 19.8995i 1.19349i
\(279\) 24.4853 1.46590
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) − 17.6569i − 1.05145i
\(283\) 2.24264i 0.133311i 0.997776 + 0.0666556i \(0.0212329\pi\)
−0.997776 + 0.0666556i \(0.978767\pi\)
\(284\) 4.48528 0.266152
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 8.65685i 0.510110i
\(289\) 10.3137 0.606689
\(290\) 0 0
\(291\) −26.4853 −1.55259
\(292\) − 9.41421i − 0.550925i
\(293\) 8.34315i 0.487412i 0.969849 + 0.243706i \(0.0783632\pi\)
−0.969849 + 0.243706i \(0.921637\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.65685 0.445046
\(297\) − 16.0000i − 0.928414i
\(298\) − 6.00000i − 0.347571i
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 11.3137i 0.651031i
\(303\) − 45.4558i − 2.61137i
\(304\) −0.585786 −0.0335972
\(305\) 0 0
\(306\) −22.3848 −1.27965
\(307\) 14.9289i 0.852039i 0.904714 + 0.426020i \(0.140085\pi\)
−0.904714 + 0.426020i \(0.859915\pi\)
\(308\) 0 0
\(309\) 50.6274 2.88009
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 16.4853i 0.933295i
\(313\) − 14.3848i − 0.813076i −0.913634 0.406538i \(-0.866736\pi\)
0.913634 0.406538i \(-0.133264\pi\)
\(314\) 6.48528 0.365986
\(315\) 0 0
\(316\) −6.82843 −0.384129
\(317\) 10.4853i 0.588912i 0.955665 + 0.294456i \(0.0951385\pi\)
−0.955665 + 0.294456i \(0.904862\pi\)
\(318\) 22.1421i 1.24167i
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 32.9706 1.84024
\(322\) 0 0
\(323\) − 1.51472i − 0.0842812i
\(324\) −39.9706 −2.22059
\(325\) 0 0
\(326\) 20.1421 1.11557
\(327\) 8.48528i 0.469237i
\(328\) − 3.07107i − 0.169571i
\(329\) 0 0
\(330\) 0 0
\(331\) 33.7990 1.85776 0.928880 0.370380i \(-0.120773\pi\)
0.928880 + 0.370380i \(0.120773\pi\)
\(332\) − 2.24264i − 0.123081i
\(333\) 66.2843i 3.63236i
\(334\) 15.7990 0.864482
\(335\) 0 0
\(336\) 0 0
\(337\) 6.00000i 0.326841i 0.986557 + 0.163420i \(0.0522527\pi\)
−0.986557 + 0.163420i \(0.947747\pi\)
\(338\) 10.3137i 0.560992i
\(339\) 52.2843 2.83969
\(340\) 0 0
\(341\) 2.34315 0.126888
\(342\) − 5.07107i − 0.274212i
\(343\) 0 0
\(344\) −8.82843 −0.475997
\(345\) 0 0
\(346\) 8.82843 0.474619
\(347\) − 3.17157i − 0.170259i −0.996370 0.0851295i \(-0.972870\pi\)
0.996370 0.0851295i \(-0.0271304\pi\)
\(348\) 16.4853i 0.883704i
\(349\) −2.48528 −0.133034 −0.0665170 0.997785i \(-0.521189\pi\)
−0.0665170 + 0.997785i \(0.521189\pi\)
\(350\) 0 0
\(351\) −93.2548 −4.97757
\(352\) 0.828427i 0.0441553i
\(353\) 2.38478i 0.126929i 0.997984 + 0.0634644i \(0.0202149\pi\)
−0.997984 + 0.0634644i \(0.979785\pi\)
\(354\) 29.3137 1.55801
\(355\) 0 0
\(356\) 12.7279 0.674579
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 2.48528i 0.130623i
\(363\) 35.2132i 1.84821i
\(364\) 0 0
\(365\) 0 0
\(366\) −31.7990 −1.66216
\(367\) 24.9706i 1.30345i 0.758454 + 0.651726i \(0.225955\pi\)
−0.758454 + 0.651726i \(0.774045\pi\)
\(368\) 1.17157i 0.0610725i
\(369\) 26.5858 1.38400
\(370\) 0 0
\(371\) 0 0
\(372\) − 9.65685i − 0.500685i
\(373\) 30.4853i 1.57847i 0.614093 + 0.789234i \(0.289522\pi\)
−0.614093 + 0.789234i \(0.710478\pi\)
\(374\) −2.14214 −0.110767
\(375\) 0 0
\(376\) −5.17157 −0.266704
\(377\) 23.3137i 1.20072i
\(378\) 0 0
\(379\) −34.4853 −1.77139 −0.885695 0.464268i \(-0.846318\pi\)
−0.885695 + 0.464268i \(0.846318\pi\)
\(380\) 0 0
\(381\) 9.65685 0.494736
\(382\) − 10.1421i − 0.518917i
\(383\) 32.4853i 1.65992i 0.557823 + 0.829960i \(0.311637\pi\)
−0.557823 + 0.829960i \(0.688363\pi\)
\(384\) 3.41421 0.174231
\(385\) 0 0
\(386\) −5.65685 −0.287926
\(387\) − 76.4264i − 3.88497i
\(388\) 7.75736i 0.393820i
\(389\) −28.1421 −1.42686 −0.713431 0.700725i \(-0.752860\pi\)
−0.713431 + 0.700725i \(0.752860\pi\)
\(390\) 0 0
\(391\) −3.02944 −0.153205
\(392\) 0 0
\(393\) 21.3137i 1.07513i
\(394\) −25.7990 −1.29973
\(395\) 0 0
\(396\) −7.17157 −0.360385
\(397\) 33.7990i 1.69632i 0.529738 + 0.848161i \(0.322290\pi\)
−0.529738 + 0.848161i \(0.677710\pi\)
\(398\) 16.4853i 0.826332i
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) − 5.65685i − 0.282138i
\(403\) − 13.6569i − 0.680296i
\(404\) −13.3137 −0.662382
\(405\) 0 0
\(406\) 0 0
\(407\) 6.34315i 0.314418i
\(408\) 8.82843i 0.437072i
\(409\) 10.5858 0.523433 0.261717 0.965145i \(-0.415711\pi\)
0.261717 + 0.965145i \(0.415711\pi\)
\(410\) 0 0
\(411\) −54.6274 −2.69457
\(412\) − 14.8284i − 0.730544i
\(413\) 0 0
\(414\) −10.1421 −0.498459
\(415\) 0 0
\(416\) 4.82843 0.236733
\(417\) 67.9411i 3.32709i
\(418\) − 0.485281i − 0.0237359i
\(419\) 20.8701 1.01957 0.509785 0.860302i \(-0.329725\pi\)
0.509785 + 0.860302i \(0.329725\pi\)
\(420\) 0 0
\(421\) 17.3137 0.843819 0.421909 0.906638i \(-0.361360\pi\)
0.421909 + 0.906638i \(0.361360\pi\)
\(422\) − 18.6274i − 0.906768i
\(423\) − 44.7696i − 2.17677i
\(424\) 6.48528 0.314953
\(425\) 0 0
\(426\) 15.3137 0.741952
\(427\) 0 0
\(428\) − 9.65685i − 0.466782i
\(429\) −13.6569 −0.659359
\(430\) 0 0
\(431\) −22.3431 −1.07623 −0.538116 0.842871i \(-0.680864\pi\)
−0.538116 + 0.842871i \(0.680864\pi\)
\(432\) 19.3137i 0.929231i
\(433\) − 10.5858i − 0.508720i −0.967110 0.254360i \(-0.918135\pi\)
0.967110 0.254360i \(-0.0818649\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.48528 0.119023
\(437\) − 0.686292i − 0.0328298i
\(438\) − 32.1421i − 1.53581i
\(439\) 24.9706 1.19178 0.595890 0.803066i \(-0.296799\pi\)
0.595890 + 0.803066i \(0.296799\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.4853i 0.593864i
\(443\) − 3.02944i − 0.143933i −0.997407 0.0719665i \(-0.977073\pi\)
0.997407 0.0719665i \(-0.0229275\pi\)
\(444\) 26.1421 1.24065
\(445\) 0 0
\(446\) 7.31371 0.346314
\(447\) − 20.4853i − 0.968921i
\(448\) 0 0
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) 2.54416 0.119800
\(452\) − 15.3137i − 0.720296i
\(453\) 38.6274i 1.81487i
\(454\) 18.2426 0.856170
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 21.6569i 1.01306i 0.862221 + 0.506532i \(0.169073\pi\)
−0.862221 + 0.506532i \(0.830927\pi\)
\(458\) − 16.1421i − 0.754272i
\(459\) −49.9411 −2.33105
\(460\) 0 0
\(461\) −12.8284 −0.597479 −0.298740 0.954335i \(-0.596566\pi\)
−0.298740 + 0.954335i \(0.596566\pi\)
\(462\) 0 0
\(463\) 16.9706i 0.788689i 0.918963 + 0.394344i \(0.129028\pi\)
−0.918963 + 0.394344i \(0.870972\pi\)
\(464\) 4.82843 0.224154
\(465\) 0 0
\(466\) −23.3137 −1.07999
\(467\) − 15.8995i − 0.735741i −0.929877 0.367870i \(-0.880087\pi\)
0.929877 0.367870i \(-0.119913\pi\)
\(468\) 41.7990i 1.93216i
\(469\) 0 0
\(470\) 0 0
\(471\) 22.1421 1.02026
\(472\) − 8.58579i − 0.395193i
\(473\) − 7.31371i − 0.336285i
\(474\) −23.3137 −1.07083
\(475\) 0 0
\(476\) 0 0
\(477\) 56.1421i 2.57057i
\(478\) 1.65685i 0.0757827i
\(479\) 17.1716 0.784589 0.392295 0.919840i \(-0.371681\pi\)
0.392295 + 0.919840i \(0.371681\pi\)
\(480\) 0 0
\(481\) 36.9706 1.68571
\(482\) 13.4142i 0.611001i
\(483\) 0 0
\(484\) 10.3137 0.468805
\(485\) 0 0
\(486\) −78.5269 −3.56205
\(487\) 31.7990i 1.44095i 0.693481 + 0.720475i \(0.256076\pi\)
−0.693481 + 0.720475i \(0.743924\pi\)
\(488\) 9.31371i 0.421612i
\(489\) 68.7696 3.10987
\(490\) 0 0
\(491\) 32.2843 1.45697 0.728484 0.685062i \(-0.240225\pi\)
0.728484 + 0.685062i \(0.240225\pi\)
\(492\) − 10.4853i − 0.472713i
\(493\) 12.4853i 0.562309i
\(494\) −2.82843 −0.127257
\(495\) 0 0
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) − 7.65685i − 0.343112i
\(499\) −30.3431 −1.35835 −0.679173 0.733978i \(-0.737661\pi\)
−0.679173 + 0.733978i \(0.737661\pi\)
\(500\) 0 0
\(501\) 53.9411 2.40991
\(502\) − 0.585786i − 0.0261449i
\(503\) − 17.6569i − 0.787280i −0.919265 0.393640i \(-0.871216\pi\)
0.919265 0.393640i \(-0.128784\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.970563 −0.0431468
\(507\) 35.2132i 1.56387i
\(508\) − 2.82843i − 0.125491i
\(509\) −5.79899 −0.257036 −0.128518 0.991707i \(-0.541022\pi\)
−0.128518 + 0.991707i \(0.541022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 11.3137i − 0.499512i
\(514\) −9.89949 −0.436648
\(515\) 0 0
\(516\) −30.1421 −1.32693
\(517\) − 4.28427i − 0.188422i
\(518\) 0 0
\(519\) 30.1421 1.32309
\(520\) 0 0
\(521\) 19.0711 0.835519 0.417759 0.908558i \(-0.362816\pi\)
0.417759 + 0.908558i \(0.362816\pi\)
\(522\) 41.7990i 1.82949i
\(523\) − 23.8995i − 1.04505i −0.852623 0.522526i \(-0.824990\pi\)
0.852623 0.522526i \(-0.175010\pi\)
\(524\) 6.24264 0.272711
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) − 7.31371i − 0.318590i
\(528\) 2.82843i 0.123091i
\(529\) 21.6274 0.940322
\(530\) 0 0
\(531\) 74.3259 3.22547
\(532\) 0 0
\(533\) − 14.8284i − 0.642290i
\(534\) 43.4558 1.88052
\(535\) 0 0
\(536\) −1.65685 −0.0715652
\(537\) 13.6569i 0.589337i
\(538\) 18.4853i 0.796957i
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) − 12.0000i − 0.515444i
\(543\) 8.48528i 0.364138i
\(544\) 2.58579 0.110865
\(545\) 0 0
\(546\) 0 0
\(547\) − 10.4853i − 0.448318i −0.974553 0.224159i \(-0.928036\pi\)
0.974553 0.224159i \(-0.0719635\pi\)
\(548\) 16.0000i 0.683486i
\(549\) −80.6274 −3.44109
\(550\) 0 0
\(551\) −2.82843 −0.120495
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) −8.14214 −0.345926
\(555\) 0 0
\(556\) 19.8995 0.843927
\(557\) 15.1716i 0.642840i 0.946937 + 0.321420i \(0.104160\pi\)
−0.946937 + 0.321420i \(0.895840\pi\)
\(558\) − 24.4853i − 1.03654i
\(559\) −42.6274 −1.80295
\(560\) 0 0
\(561\) −7.31371 −0.308785
\(562\) − 8.00000i − 0.337460i
\(563\) 36.5858i 1.54191i 0.636891 + 0.770954i \(0.280220\pi\)
−0.636891 + 0.770954i \(0.719780\pi\)
\(564\) −17.6569 −0.743488
\(565\) 0 0
\(566\) 2.24264 0.0942652
\(567\) 0 0
\(568\) − 4.48528i − 0.188198i
\(569\) 29.3137 1.22889 0.614447 0.788958i \(-0.289379\pi\)
0.614447 + 0.788958i \(0.289379\pi\)
\(570\) 0 0
\(571\) −2.20101 −0.0921094 −0.0460547 0.998939i \(-0.514665\pi\)
−0.0460547 + 0.998939i \(0.514665\pi\)
\(572\) 4.00000i 0.167248i
\(573\) − 34.6274i − 1.44658i
\(574\) 0 0
\(575\) 0 0
\(576\) 8.65685 0.360702
\(577\) 6.10051i 0.253967i 0.991905 + 0.126984i \(0.0405296\pi\)
−0.991905 + 0.126984i \(0.959470\pi\)
\(578\) − 10.3137i − 0.428994i
\(579\) −19.3137 −0.802650
\(580\) 0 0
\(581\) 0 0
\(582\) 26.4853i 1.09785i
\(583\) 5.37258i 0.222510i
\(584\) −9.41421 −0.389563
\(585\) 0 0
\(586\) 8.34315 0.344652
\(587\) 17.0711i 0.704598i 0.935887 + 0.352299i \(0.114600\pi\)
−0.935887 + 0.352299i \(0.885400\pi\)
\(588\) 0 0
\(589\) 1.65685 0.0682695
\(590\) 0 0
\(591\) −88.0833 −3.62326
\(592\) − 7.65685i − 0.314695i
\(593\) − 3.27208i − 0.134368i −0.997741 0.0671841i \(-0.978599\pi\)
0.997741 0.0671841i \(-0.0214015\pi\)
\(594\) −16.0000 −0.656488
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 56.2843i 2.30356i
\(598\) 5.65685i 0.231326i
\(599\) 10.8284 0.442438 0.221219 0.975224i \(-0.428997\pi\)
0.221219 + 0.975224i \(0.428997\pi\)
\(600\) 0 0
\(601\) 6.58579 0.268640 0.134320 0.990938i \(-0.457115\pi\)
0.134320 + 0.990938i \(0.457115\pi\)
\(602\) 0 0
\(603\) − 14.3431i − 0.584098i
\(604\) 11.3137 0.460348
\(605\) 0 0
\(606\) −45.4558 −1.84652
\(607\) 16.2843i 0.660958i 0.943813 + 0.330479i \(0.107210\pi\)
−0.943813 + 0.330479i \(0.892790\pi\)
\(608\) 0.585786i 0.0237568i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.9706 −1.01020
\(612\) 22.3848i 0.904851i
\(613\) 12.3431i 0.498535i 0.968435 + 0.249267i \(0.0801898\pi\)
−0.968435 + 0.249267i \(0.919810\pi\)
\(614\) 14.9289 0.602483
\(615\) 0 0
\(616\) 0 0
\(617\) − 33.3137i − 1.34116i −0.741837 0.670580i \(-0.766045\pi\)
0.741837 0.670580i \(-0.233955\pi\)
\(618\) − 50.6274i − 2.03653i
\(619\) −29.0711 −1.16846 −0.584232 0.811586i \(-0.698604\pi\)
−0.584232 + 0.811586i \(0.698604\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) − 4.00000i − 0.160385i
\(623\) 0 0
\(624\) 16.4853 0.659939
\(625\) 0 0
\(626\) −14.3848 −0.574931
\(627\) − 1.65685i − 0.0661684i
\(628\) − 6.48528i − 0.258791i
\(629\) 19.7990 0.789437
\(630\) 0 0
\(631\) −12.4853 −0.497031 −0.248516 0.968628i \(-0.579943\pi\)
−0.248516 + 0.968628i \(0.579943\pi\)
\(632\) 6.82843i 0.271620i
\(633\) − 63.5980i − 2.52779i
\(634\) 10.4853 0.416424
\(635\) 0 0
\(636\) 22.1421 0.877993
\(637\) 0 0
\(638\) 4.00000i 0.158362i
\(639\) 38.8284 1.53603
\(640\) 0 0
\(641\) −24.6274 −0.972724 −0.486362 0.873757i \(-0.661676\pi\)
−0.486362 + 0.873757i \(0.661676\pi\)
\(642\) − 32.9706i − 1.30124i
\(643\) − 4.78680i − 0.188773i −0.995536 0.0943864i \(-0.969911\pi\)
0.995536 0.0943864i \(-0.0300889\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.51472 −0.0595958
\(647\) − 23.1127i − 0.908654i −0.890835 0.454327i \(-0.849880\pi\)
0.890835 0.454327i \(-0.150120\pi\)
\(648\) 39.9706i 1.57019i
\(649\) 7.11270 0.279198
\(650\) 0 0
\(651\) 0 0
\(652\) − 20.1421i − 0.788827i
\(653\) − 4.34315i − 0.169960i −0.996383 0.0849802i \(-0.972917\pi\)
0.996383 0.0849802i \(-0.0270827\pi\)
\(654\) 8.48528 0.331801
\(655\) 0 0
\(656\) −3.07107 −0.119905
\(657\) − 81.4975i − 3.17952i
\(658\) 0 0
\(659\) 27.1716 1.05845 0.529227 0.848480i \(-0.322482\pi\)
0.529227 + 0.848480i \(0.322482\pi\)
\(660\) 0 0
\(661\) −38.2843 −1.48909 −0.744543 0.667575i \(-0.767332\pi\)
−0.744543 + 0.667575i \(0.767332\pi\)
\(662\) − 33.7990i − 1.31364i
\(663\) 42.6274i 1.65551i
\(664\) −2.24264 −0.0870313
\(665\) 0 0
\(666\) 66.2843 2.56846
\(667\) 5.65685i 0.219034i
\(668\) − 15.7990i − 0.611281i
\(669\) 24.9706 0.965418
\(670\) 0 0
\(671\) −7.71573 −0.297862
\(672\) 0 0
\(673\) 48.0000i 1.85026i 0.379646 + 0.925132i \(0.376046\pi\)
−0.379646 + 0.925132i \(0.623954\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 10.3137 0.396681
\(677\) 39.4558i 1.51641i 0.652015 + 0.758206i \(0.273924\pi\)
−0.652015 + 0.758206i \(0.726076\pi\)
\(678\) − 52.2843i − 2.00797i
\(679\) 0 0
\(680\) 0 0
\(681\) 62.2843 2.38674
\(682\) − 2.34315i − 0.0897237i
\(683\) − 33.6569i − 1.28784i −0.765091 0.643922i \(-0.777306\pi\)
0.765091 0.643922i \(-0.222694\pi\)
\(684\) −5.07107 −0.193897
\(685\) 0 0
\(686\) 0 0
\(687\) − 55.1127i − 2.10268i
\(688\) 8.82843i 0.336581i
\(689\) 31.3137 1.19296
\(690\) 0 0
\(691\) 1.75736 0.0668531 0.0334265 0.999441i \(-0.489358\pi\)
0.0334265 + 0.999441i \(0.489358\pi\)
\(692\) − 8.82843i − 0.335606i
\(693\) 0 0
\(694\) −3.17157 −0.120391
\(695\) 0 0
\(696\) 16.4853 0.624873
\(697\) − 7.94113i − 0.300792i
\(698\) 2.48528i 0.0940693i
\(699\) −79.5980 −3.01067
\(700\) 0 0
\(701\) 2.48528 0.0938678 0.0469339 0.998898i \(-0.485055\pi\)
0.0469339 + 0.998898i \(0.485055\pi\)
\(702\) 93.2548i 3.51968i
\(703\) 4.48528i 0.169166i
\(704\) 0.828427 0.0312225
\(705\) 0 0
\(706\) 2.38478 0.0897522
\(707\) 0 0
\(708\) − 29.3137i − 1.10168i
\(709\) −45.1127 −1.69424 −0.847121 0.531399i \(-0.821666\pi\)
−0.847121 + 0.531399i \(0.821666\pi\)
\(710\) 0 0
\(711\) −59.1127 −2.21690
\(712\) − 12.7279i − 0.476999i
\(713\) − 3.31371i − 0.124099i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 5.65685i 0.211259i
\(718\) 28.2843i 1.05556i
\(719\) −41.4558 −1.54604 −0.773021 0.634380i \(-0.781255\pi\)
−0.773021 + 0.634380i \(0.781255\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.6569i 0.694336i
\(723\) 45.7990i 1.70328i
\(724\) 2.48528 0.0923648
\(725\) 0 0
\(726\) 35.2132 1.30688
\(727\) − 3.51472i − 0.130354i −0.997874 0.0651768i \(-0.979239\pi\)
0.997874 0.0651768i \(-0.0207612\pi\)
\(728\) 0 0
\(729\) −148.196 −5.48874
\(730\) 0 0
\(731\) −22.8284 −0.844340
\(732\) 31.7990i 1.17532i
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 24.9706 0.921680
\(735\) 0 0
\(736\) 1.17157 0.0431847
\(737\) − 1.37258i − 0.0505597i
\(738\) − 26.5858i − 0.978636i
\(739\) −3.17157 −0.116668 −0.0583341 0.998297i \(-0.518579\pi\)
−0.0583341 + 0.998297i \(0.518579\pi\)
\(740\) 0 0
\(741\) −9.65685 −0.354753
\(742\) 0 0
\(743\) − 51.7990i − 1.90032i −0.311762 0.950160i \(-0.600919\pi\)
0.311762 0.950160i \(-0.399081\pi\)
\(744\) −9.65685 −0.354037
\(745\) 0 0
\(746\) 30.4853 1.11615
\(747\) − 19.4142i − 0.710329i
\(748\) 2.14214i 0.0783242i
\(749\) 0 0
\(750\) 0 0
\(751\) −39.3137 −1.43458 −0.717289 0.696776i \(-0.754617\pi\)
−0.717289 + 0.696776i \(0.754617\pi\)
\(752\) 5.17157i 0.188588i
\(753\) − 2.00000i − 0.0728841i
\(754\) 23.3137 0.849035
\(755\) 0 0
\(756\) 0 0
\(757\) 3.65685i 0.132911i 0.997789 + 0.0664553i \(0.0211690\pi\)
−0.997789 + 0.0664553i \(0.978831\pi\)
\(758\) 34.4853i 1.25256i
\(759\) −3.31371 −0.120280
\(760\) 0 0
\(761\) 22.3848 0.811448 0.405724 0.913996i \(-0.367020\pi\)
0.405724 + 0.913996i \(0.367020\pi\)
\(762\) − 9.65685i − 0.349831i
\(763\) 0 0
\(764\) −10.1421 −0.366930
\(765\) 0 0
\(766\) 32.4853 1.17374
\(767\) − 41.4558i − 1.49688i
\(768\) − 3.41421i − 0.123200i
\(769\) −19.5563 −0.705220 −0.352610 0.935770i \(-0.614706\pi\)
−0.352610 + 0.935770i \(0.614706\pi\)
\(770\) 0 0
\(771\) −33.7990 −1.21724
\(772\) 5.65685i 0.203595i
\(773\) 2.00000i 0.0719350i 0.999353 + 0.0359675i \(0.0114513\pi\)
−0.999353 + 0.0359675i \(0.988549\pi\)
\(774\) −76.4264 −2.74709
\(775\) 0 0
\(776\) 7.75736 0.278473
\(777\) 0 0
\(778\) 28.1421i 1.00894i
\(779\) 1.79899 0.0644555
\(780\) 0 0
\(781\) 3.71573 0.132959
\(782\) 3.02944i 0.108332i
\(783\) 93.2548i 3.33266i
\(784\) 0 0
\(785\) 0 0
\(786\) 21.3137 0.760235
\(787\) − 1.27208i − 0.0453447i −0.999743 0.0226723i \(-0.992783\pi\)
0.999743 0.0226723i \(-0.00721745\pi\)
\(788\) 25.7990i 0.919051i
\(789\) −95.5980 −3.40338
\(790\) 0 0
\(791\) 0 0
\(792\) 7.17157i 0.254831i
\(793\) 44.9706i 1.59695i
\(794\) 33.7990 1.19948
\(795\) 0 0
\(796\) 16.4853 0.584305
\(797\) − 41.7990i − 1.48060i −0.672279 0.740298i \(-0.734684\pi\)
0.672279 0.740298i \(-0.265316\pi\)
\(798\) 0 0
\(799\) −13.3726 −0.473088
\(800\) 0 0
\(801\) 110.184 3.89315
\(802\) 6.00000i 0.211867i
\(803\) − 7.79899i − 0.275220i
\(804\) −5.65685 −0.199502
\(805\) 0 0
\(806\) −13.6569 −0.481042
\(807\) 63.1127i 2.22167i
\(808\) 13.3137i 0.468375i
\(809\) −3.02944 −0.106509 −0.0532547 0.998581i \(-0.516960\pi\)
−0.0532547 + 0.998581i \(0.516960\pi\)
\(810\) 0 0
\(811\) 32.5858 1.14424 0.572121 0.820169i \(-0.306121\pi\)
0.572121 + 0.820169i \(0.306121\pi\)
\(812\) 0 0
\(813\) − 40.9706i − 1.43690i
\(814\) 6.34315 0.222327
\(815\) 0 0
\(816\) 8.82843 0.309057
\(817\) − 5.17157i − 0.180930i
\(818\) − 10.5858i − 0.370123i
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3137 0.604253 0.302126 0.953268i \(-0.402304\pi\)
0.302126 + 0.953268i \(0.402304\pi\)
\(822\) 54.6274i 1.90535i
\(823\) − 20.2843i − 0.707065i −0.935422 0.353533i \(-0.884980\pi\)
0.935422 0.353533i \(-0.115020\pi\)
\(824\) −14.8284 −0.516573
\(825\) 0 0
\(826\) 0 0
\(827\) − 5.37258i − 0.186823i −0.995628 0.0934115i \(-0.970223\pi\)
0.995628 0.0934115i \(-0.0297772\pi\)
\(828\) 10.1421i 0.352464i
\(829\) 5.02944 0.174680 0.0873398 0.996179i \(-0.472163\pi\)
0.0873398 + 0.996179i \(0.472163\pi\)
\(830\) 0 0
\(831\) −27.7990 −0.964336
\(832\) − 4.82843i − 0.167396i
\(833\) 0 0
\(834\) 67.9411 2.35261
\(835\) 0 0
\(836\) −0.485281 −0.0167838
\(837\) − 54.6274i − 1.88820i
\(838\) − 20.8701i − 0.720944i
\(839\) 42.1421 1.45491 0.727454 0.686156i \(-0.240703\pi\)
0.727454 + 0.686156i \(0.240703\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) − 17.3137i − 0.596670i
\(843\) − 27.3137i − 0.940734i
\(844\) −18.6274 −0.641182
\(845\) 0 0
\(846\) −44.7696 −1.53921
\(847\) 0 0
\(848\) − 6.48528i − 0.222705i
\(849\) 7.65685 0.262783
\(850\) 0 0
\(851\) 8.97056 0.307507
\(852\) − 15.3137i − 0.524639i
\(853\) 43.1716i 1.47817i 0.673614 + 0.739083i \(0.264741\pi\)
−0.673614 + 0.739083i \(0.735259\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.65685 −0.330064
\(857\) − 4.92893i − 0.168369i −0.996450 0.0841846i \(-0.973171\pi\)
0.996450 0.0841846i \(-0.0268285\pi\)
\(858\) 13.6569i 0.466237i
\(859\) 7.21320 0.246111 0.123056 0.992400i \(-0.460731\pi\)
0.123056 + 0.992400i \(0.460731\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 22.3431i 0.761011i
\(863\) 4.97056i 0.169200i 0.996415 + 0.0846000i \(0.0269612\pi\)
−0.996415 + 0.0846000i \(0.973039\pi\)
\(864\) 19.3137 0.657066
\(865\) 0 0
\(866\) −10.5858 −0.359720
\(867\) − 35.2132i − 1.19590i
\(868\) 0 0
\(869\) −5.65685 −0.191896
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 2.48528i − 0.0841622i
\(873\) 67.1543i 2.27283i
\(874\) −0.686292 −0.0232142
\(875\) 0 0
\(876\) −32.1421 −1.08598
\(877\) − 30.2843i − 1.02263i −0.859394 0.511314i \(-0.829159\pi\)
0.859394 0.511314i \(-0.170841\pi\)
\(878\) − 24.9706i − 0.842716i
\(879\) 28.4853 0.960785
\(880\) 0 0
\(881\) −2.38478 −0.0803452 −0.0401726 0.999193i \(-0.512791\pi\)
−0.0401726 + 0.999193i \(0.512791\pi\)
\(882\) 0 0
\(883\) 41.6569i 1.40186i 0.713228 + 0.700932i \(0.247233\pi\)
−0.713228 + 0.700932i \(0.752767\pi\)
\(884\) 12.4853 0.419925
\(885\) 0 0
\(886\) −3.02944 −0.101776
\(887\) − 55.1127i − 1.85050i −0.379354 0.925252i \(-0.623854\pi\)
0.379354 0.925252i \(-0.376146\pi\)
\(888\) − 26.1421i − 0.877273i
\(889\) 0 0
\(890\) 0 0
\(891\) −33.1127 −1.10932
\(892\) − 7.31371i − 0.244881i
\(893\) − 3.02944i − 0.101376i
\(894\) −20.4853 −0.685130
\(895\) 0 0
\(896\) 0 0
\(897\) 19.3137i 0.644866i
\(898\) − 16.6274i − 0.554864i
\(899\) −13.6569 −0.455482
\(900\) 0 0
\(901\) 16.7696 0.558675
\(902\) − 2.54416i − 0.0847111i
\(903\) 0 0
\(904\) −15.3137 −0.509326
\(905\) 0 0
\(906\) 38.6274 1.28331
\(907\) 0.284271i 0.00943907i 0.999989 + 0.00471954i \(0.00150228\pi\)
−0.999989 + 0.00471954i \(0.998498\pi\)
\(908\) − 18.2426i − 0.605403i
\(909\) −115.255 −3.82276
\(910\) 0 0
\(911\) 36.2843 1.20215 0.601076 0.799192i \(-0.294739\pi\)
0.601076 + 0.799192i \(0.294739\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) − 1.85786i − 0.0614863i
\(914\) 21.6569 0.716345
\(915\) 0 0
\(916\) −16.1421 −0.533351
\(917\) 0 0
\(918\) 49.9411i 1.64830i
\(919\) −15.5147 −0.511783 −0.255892 0.966705i \(-0.582369\pi\)
−0.255892 + 0.966705i \(0.582369\pi\)
\(920\) 0 0
\(921\) 50.9706 1.67954
\(922\) 12.8284i 0.422482i
\(923\) − 21.6569i − 0.712844i
\(924\) 0 0
\(925\) 0 0
\(926\) 16.9706 0.557687
\(927\) − 128.368i − 4.21614i
\(928\) − 4.82843i − 0.158501i
\(929\) 17.2132 0.564747 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 23.3137i 0.763666i
\(933\) − 13.6569i − 0.447105i
\(934\) −15.8995 −0.520247
\(935\) 0 0
\(936\) 41.7990 1.36624
\(937\) 20.2426i 0.661298i 0.943754 + 0.330649i \(0.107268\pi\)
−0.943754 + 0.330649i \(0.892732\pi\)
\(938\) 0 0
\(939\) −49.1127 −1.60273
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) − 22.1421i − 0.721430i
\(943\) − 3.59798i − 0.117166i
\(944\) −8.58579 −0.279444
\(945\) 0 0
\(946\) −7.31371 −0.237789
\(947\) − 4.82843i − 0.156903i −0.996918 0.0784514i \(-0.975002\pi\)
0.996918 0.0784514i \(-0.0249975\pi\)
\(948\) 23.3137i 0.757194i
\(949\) −45.4558 −1.47556
\(950\) 0 0
\(951\) 35.7990 1.16086
\(952\) 0 0
\(953\) − 0.343146i − 0.0111156i −0.999985 0.00555779i \(-0.998231\pi\)
0.999985 0.00555779i \(-0.00176911\pi\)
\(954\) 56.1421 1.81767
\(955\) 0 0
\(956\) 1.65685 0.0535865
\(957\) 13.6569i 0.441463i
\(958\) − 17.1716i − 0.554788i
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) − 36.9706i − 1.19198i
\(963\) − 83.5980i − 2.69391i
\(964\) 13.4142 0.432043
\(965\) 0 0
\(966\) 0 0
\(967\) − 37.4558i − 1.20450i −0.798308 0.602249i \(-0.794271\pi\)
0.798308 0.602249i \(-0.205729\pi\)
\(968\) − 10.3137i − 0.331495i
\(969\) −5.17157 −0.166135
\(970\) 0 0
\(971\) −33.3553 −1.07042 −0.535212 0.844718i \(-0.679768\pi\)
−0.535212 + 0.844718i \(0.679768\pi\)
\(972\) 78.5269i 2.51875i
\(973\) 0 0
\(974\) 31.7990 1.01891
\(975\) 0 0
\(976\) 9.31371 0.298125
\(977\) − 12.6863i − 0.405870i −0.979192 0.202935i \(-0.934952\pi\)
0.979192 0.202935i \(-0.0650481\pi\)
\(978\) − 68.7696i − 2.19901i
\(979\) 10.5442 0.336993
\(980\) 0 0
\(981\) 21.5147 0.686912
\(982\) − 32.2843i − 1.03023i
\(983\) 12.2010i 0.389152i 0.980887 + 0.194576i \(0.0623331\pi\)
−0.980887 + 0.194576i \(0.937667\pi\)
\(984\) −10.4853 −0.334259
\(985\) 0 0
\(986\) 12.4853 0.397612
\(987\) 0 0
\(988\) 2.82843i 0.0899843i
\(989\) −10.3431 −0.328893
\(990\) 0 0
\(991\) 44.7696 1.42215 0.711076 0.703115i \(-0.248208\pi\)
0.711076 + 0.703115i \(0.248208\pi\)
\(992\) 2.82843i 0.0898027i
\(993\) − 115.397i − 3.66201i
\(994\) 0 0
\(995\) 0 0
\(996\) −7.65685 −0.242617
\(997\) − 18.2843i − 0.579069i −0.957168 0.289534i \(-0.906500\pi\)
0.957168 0.289534i \(-0.0935004\pi\)
\(998\) 30.3431i 0.960495i
\(999\) 147.882 4.67879
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 2450.2.c.t.99.1 4
5.2 odd 4 2450.2.a.bn.1.1 2
5.3 odd 4 490.2.a.m.1.2 yes 2
5.4 even 2 inner 2450.2.c.t.99.4 4
7.6 odd 2 2450.2.c.w.99.2 4
15.8 even 4 4410.2.a.bt.1.2 2
20.3 even 4 3920.2.a.bm.1.1 2
35.3 even 12 490.2.e.j.471.2 4
35.13 even 4 490.2.a.l.1.1 2
35.18 odd 12 490.2.e.i.471.1 4
35.23 odd 12 490.2.e.i.361.1 4
35.27 even 4 2450.2.a.bs.1.2 2
35.33 even 12 490.2.e.j.361.2 4
35.34 odd 2 2450.2.c.w.99.3 4
105.83 odd 4 4410.2.a.by.1.2 2
140.83 odd 4 3920.2.a.ca.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 35.13 even 4
490.2.a.m.1.2 yes 2 5.3 odd 4
490.2.e.i.361.1 4 35.23 odd 12
490.2.e.i.471.1 4 35.18 odd 12
490.2.e.j.361.2 4 35.33 even 12
490.2.e.j.471.2 4 35.3 even 12
2450.2.a.bn.1.1 2 5.2 odd 4
2450.2.a.bs.1.2 2 35.27 even 4
2450.2.c.t.99.1 4 1.1 even 1 trivial
2450.2.c.t.99.4 4 5.4 even 2 inner
2450.2.c.w.99.2 4 7.6 odd 2
2450.2.c.w.99.3 4 35.34 odd 2
3920.2.a.bm.1.1 2 20.3 even 4
3920.2.a.ca.1.2 2 140.83 odd 4
4410.2.a.bt.1.2 2 15.8 even 4
4410.2.a.by.1.2 2 105.83 odd 4