Properties

Label 4235.2.a.y.1.5
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.173513.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.293545\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.40663 q^{2} -0.321225 q^{3} +3.79186 q^{4} -1.00000 q^{5} -0.773069 q^{6} +1.00000 q^{7} +4.31233 q^{8} -2.89681 q^{9} +O(q^{10})\) \(q+2.40663 q^{2} -0.321225 q^{3} +3.79186 q^{4} -1.00000 q^{5} -0.773069 q^{6} +1.00000 q^{7} +4.31233 q^{8} -2.89681 q^{9} -2.40663 q^{10} -1.21804 q^{12} -4.76679 q^{13} +2.40663 q^{14} +0.321225 q^{15} +2.79447 q^{16} +0.284654 q^{17} -6.97155 q^{18} -7.09353 q^{19} -3.79186 q^{20} -0.321225 q^{21} +9.19849 q^{23} -1.38523 q^{24} +1.00000 q^{25} -11.4719 q^{26} +1.89420 q^{27} +3.79186 q^{28} -9.56618 q^{29} +0.773069 q^{30} -6.23683 q^{31} -1.89942 q^{32} +0.685056 q^{34} -1.00000 q^{35} -10.9843 q^{36} +2.86108 q^{37} -17.0715 q^{38} +1.53121 q^{39} -4.31233 q^{40} +3.08304 q^{41} -0.773069 q^{42} +4.95715 q^{43} +2.89681 q^{45} +22.1373 q^{46} -3.41990 q^{47} -0.897653 q^{48} +1.00000 q^{49} +2.40663 q^{50} -0.0914380 q^{51} -18.0750 q^{52} -7.44219 q^{53} +4.55865 q^{54} +4.31233 q^{56} +2.27862 q^{57} -23.0222 q^{58} -8.43413 q^{59} +1.21804 q^{60} -7.83020 q^{61} -15.0097 q^{62} -2.89681 q^{63} -10.1601 q^{64} +4.76679 q^{65} +2.33196 q^{67} +1.07937 q^{68} -2.95478 q^{69} -2.40663 q^{70} -12.7182 q^{71} -12.4920 q^{72} -0.750610 q^{73} +6.88556 q^{74} -0.321225 q^{75} -26.8976 q^{76} +3.68506 q^{78} -7.66727 q^{79} -2.79447 q^{80} +8.08198 q^{81} +7.41973 q^{82} +9.39958 q^{83} -1.21804 q^{84} -0.284654 q^{85} +11.9300 q^{86} +3.07290 q^{87} -14.7733 q^{89} +6.97155 q^{90} -4.76679 q^{91} +34.8793 q^{92} +2.00343 q^{93} -8.23043 q^{94} +7.09353 q^{95} +0.610143 q^{96} +11.9388 q^{97} +2.40663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.40663 1.70174 0.850871 0.525374i \(-0.176075\pi\)
0.850871 + 0.525374i \(0.176075\pi\)
\(3\) −0.321225 −0.185459 −0.0927297 0.995691i \(-0.529559\pi\)
−0.0927297 + 0.995691i \(0.529559\pi\)
\(4\) 3.79186 1.89593
\(5\) −1.00000 −0.447214
\(6\) −0.773069 −0.315604
\(7\) 1.00000 0.377964
\(8\) 4.31233 1.52464
\(9\) −2.89681 −0.965605
\(10\) −2.40663 −0.761043
\(11\) 0 0
\(12\) −1.21804 −0.351618
\(13\) −4.76679 −1.32207 −0.661035 0.750355i \(-0.729882\pi\)
−0.661035 + 0.750355i \(0.729882\pi\)
\(14\) 2.40663 0.643198
\(15\) 0.321225 0.0829399
\(16\) 2.79447 0.698617
\(17\) 0.284654 0.0690387 0.0345194 0.999404i \(-0.489010\pi\)
0.0345194 + 0.999404i \(0.489010\pi\)
\(18\) −6.97155 −1.64321
\(19\) −7.09353 −1.62737 −0.813683 0.581308i \(-0.802541\pi\)
−0.813683 + 0.581308i \(0.802541\pi\)
\(20\) −3.79186 −0.847885
\(21\) −0.321225 −0.0700970
\(22\) 0 0
\(23\) 9.19849 1.91802 0.959008 0.283378i \(-0.0914550\pi\)
0.959008 + 0.283378i \(0.0914550\pi\)
\(24\) −1.38523 −0.282759
\(25\) 1.00000 0.200000
\(26\) −11.4719 −2.24982
\(27\) 1.89420 0.364540
\(28\) 3.79186 0.716594
\(29\) −9.56618 −1.77639 −0.888197 0.459462i \(-0.848042\pi\)
−0.888197 + 0.459462i \(0.848042\pi\)
\(30\) 0.773069 0.141142
\(31\) −6.23683 −1.12017 −0.560084 0.828436i \(-0.689231\pi\)
−0.560084 + 0.828436i \(0.689231\pi\)
\(32\) −1.89942 −0.335774
\(33\) 0 0
\(34\) 0.685056 0.117486
\(35\) −1.00000 −0.169031
\(36\) −10.9843 −1.83072
\(37\) 2.86108 0.470359 0.235179 0.971952i \(-0.424432\pi\)
0.235179 + 0.971952i \(0.424432\pi\)
\(38\) −17.0715 −2.76936
\(39\) 1.53121 0.245190
\(40\) −4.31233 −0.681840
\(41\) 3.08304 0.481490 0.240745 0.970588i \(-0.422608\pi\)
0.240745 + 0.970588i \(0.422608\pi\)
\(42\) −0.773069 −0.119287
\(43\) 4.95715 0.755958 0.377979 0.925814i \(-0.376619\pi\)
0.377979 + 0.925814i \(0.376619\pi\)
\(44\) 0 0
\(45\) 2.89681 0.431832
\(46\) 22.1373 3.26397
\(47\) −3.41990 −0.498844 −0.249422 0.968395i \(-0.580241\pi\)
−0.249422 + 0.968395i \(0.580241\pi\)
\(48\) −0.897653 −0.129565
\(49\) 1.00000 0.142857
\(50\) 2.40663 0.340349
\(51\) −0.0914380 −0.0128039
\(52\) −18.0750 −2.50655
\(53\) −7.44219 −1.02226 −0.511131 0.859503i \(-0.670773\pi\)
−0.511131 + 0.859503i \(0.670773\pi\)
\(54\) 4.55865 0.620353
\(55\) 0 0
\(56\) 4.31233 0.576260
\(57\) 2.27862 0.301810
\(58\) −23.0222 −3.02297
\(59\) −8.43413 −1.09803 −0.549015 0.835812i \(-0.684997\pi\)
−0.549015 + 0.835812i \(0.684997\pi\)
\(60\) 1.21804 0.157248
\(61\) −7.83020 −1.00255 −0.501277 0.865287i \(-0.667136\pi\)
−0.501277 + 0.865287i \(0.667136\pi\)
\(62\) −15.0097 −1.90624
\(63\) −2.89681 −0.364964
\(64\) −10.1601 −1.27002
\(65\) 4.76679 0.591247
\(66\) 0 0
\(67\) 2.33196 0.284894 0.142447 0.989802i \(-0.454503\pi\)
0.142447 + 0.989802i \(0.454503\pi\)
\(68\) 1.07937 0.130893
\(69\) −2.95478 −0.355714
\(70\) −2.40663 −0.287647
\(71\) −12.7182 −1.50937 −0.754686 0.656086i \(-0.772211\pi\)
−0.754686 + 0.656086i \(0.772211\pi\)
\(72\) −12.4920 −1.47220
\(73\) −0.750610 −0.0878522 −0.0439261 0.999035i \(-0.513987\pi\)
−0.0439261 + 0.999035i \(0.513987\pi\)
\(74\) 6.88556 0.800430
\(75\) −0.321225 −0.0370919
\(76\) −26.8976 −3.08537
\(77\) 0 0
\(78\) 3.68506 0.417250
\(79\) −7.66727 −0.862636 −0.431318 0.902200i \(-0.641951\pi\)
−0.431318 + 0.902200i \(0.641951\pi\)
\(80\) −2.79447 −0.312431
\(81\) 8.08198 0.897998
\(82\) 7.41973 0.819372
\(83\) 9.39958 1.03174 0.515869 0.856667i \(-0.327469\pi\)
0.515869 + 0.856667i \(0.327469\pi\)
\(84\) −1.21804 −0.132899
\(85\) −0.284654 −0.0308751
\(86\) 11.9300 1.28645
\(87\) 3.07290 0.329449
\(88\) 0 0
\(89\) −14.7733 −1.56597 −0.782984 0.622042i \(-0.786303\pi\)
−0.782984 + 0.622042i \(0.786303\pi\)
\(90\) 6.97155 0.734866
\(91\) −4.76679 −0.499695
\(92\) 34.8793 3.63642
\(93\) 2.00343 0.207746
\(94\) −8.23043 −0.848904
\(95\) 7.09353 0.727781
\(96\) 0.610143 0.0622724
\(97\) 11.9388 1.21220 0.606099 0.795389i \(-0.292733\pi\)
0.606099 + 0.795389i \(0.292733\pi\)
\(98\) 2.40663 0.243106
\(99\) 0 0
\(100\) 3.79186 0.379186
\(101\) 13.7715 1.37032 0.685160 0.728393i \(-0.259732\pi\)
0.685160 + 0.728393i \(0.259732\pi\)
\(102\) −0.220057 −0.0217889
\(103\) 10.9843 1.08232 0.541158 0.840921i \(-0.317986\pi\)
0.541158 + 0.840921i \(0.317986\pi\)
\(104\) −20.5560 −2.01568
\(105\) 0.321225 0.0313484
\(106\) −17.9106 −1.73963
\(107\) −10.0331 −0.969940 −0.484970 0.874531i \(-0.661170\pi\)
−0.484970 + 0.874531i \(0.661170\pi\)
\(108\) 7.18255 0.691141
\(109\) 9.15250 0.876651 0.438325 0.898816i \(-0.355572\pi\)
0.438325 + 0.898816i \(0.355572\pi\)
\(110\) 0 0
\(111\) −0.919051 −0.0872325
\(112\) 2.79447 0.264052
\(113\) 15.4820 1.45643 0.728213 0.685351i \(-0.240351\pi\)
0.728213 + 0.685351i \(0.240351\pi\)
\(114\) 5.48379 0.513604
\(115\) −9.19849 −0.857763
\(116\) −36.2736 −3.36792
\(117\) 13.8085 1.27660
\(118\) −20.2978 −1.86857
\(119\) 0.284654 0.0260942
\(120\) 1.38523 0.126454
\(121\) 0 0
\(122\) −18.8444 −1.70609
\(123\) −0.990349 −0.0892968
\(124\) −23.6492 −2.12376
\(125\) −1.00000 −0.0894427
\(126\) −6.97155 −0.621075
\(127\) −1.07543 −0.0954293 −0.0477147 0.998861i \(-0.515194\pi\)
−0.0477147 + 0.998861i \(0.515194\pi\)
\(128\) −20.6528 −1.82547
\(129\) −1.59236 −0.140199
\(130\) 11.4719 1.00615
\(131\) 7.90703 0.690840 0.345420 0.938448i \(-0.387736\pi\)
0.345420 + 0.938448i \(0.387736\pi\)
\(132\) 0 0
\(133\) −7.09353 −0.615087
\(134\) 5.61216 0.484817
\(135\) −1.89420 −0.163027
\(136\) 1.22752 0.105259
\(137\) 6.89605 0.589169 0.294585 0.955625i \(-0.404819\pi\)
0.294585 + 0.955625i \(0.404819\pi\)
\(138\) −7.11106 −0.605334
\(139\) −21.2625 −1.80346 −0.901731 0.432297i \(-0.857703\pi\)
−0.901731 + 0.432297i \(0.857703\pi\)
\(140\) −3.79186 −0.320470
\(141\) 1.09856 0.0925153
\(142\) −30.6080 −2.56856
\(143\) 0 0
\(144\) −8.09505 −0.674588
\(145\) 9.56618 0.794428
\(146\) −1.80644 −0.149502
\(147\) −0.321225 −0.0264942
\(148\) 10.8488 0.891767
\(149\) 11.2308 0.920066 0.460033 0.887902i \(-0.347838\pi\)
0.460033 + 0.887902i \(0.347838\pi\)
\(150\) −0.773069 −0.0631208
\(151\) 0.306818 0.0249685 0.0124843 0.999922i \(-0.496026\pi\)
0.0124843 + 0.999922i \(0.496026\pi\)
\(152\) −30.5897 −2.48115
\(153\) −0.824590 −0.0666641
\(154\) 0 0
\(155\) 6.23683 0.500954
\(156\) 5.80614 0.464863
\(157\) −1.95122 −0.155724 −0.0778620 0.996964i \(-0.524809\pi\)
−0.0778620 + 0.996964i \(0.524809\pi\)
\(158\) −18.4523 −1.46798
\(159\) 2.39062 0.189588
\(160\) 1.89942 0.150163
\(161\) 9.19849 0.724942
\(162\) 19.4503 1.52816
\(163\) 17.9429 1.40539 0.702697 0.711490i \(-0.251979\pi\)
0.702697 + 0.711490i \(0.251979\pi\)
\(164\) 11.6904 0.912870
\(165\) 0 0
\(166\) 22.6213 1.75575
\(167\) −15.0799 −1.16692 −0.583460 0.812142i \(-0.698302\pi\)
−0.583460 + 0.812142i \(0.698302\pi\)
\(168\) −1.38523 −0.106873
\(169\) 9.72227 0.747867
\(170\) −0.685056 −0.0525414
\(171\) 20.5486 1.57139
\(172\) 18.7968 1.43324
\(173\) −24.7045 −1.87825 −0.939125 0.343576i \(-0.888362\pi\)
−0.939125 + 0.343576i \(0.888362\pi\)
\(174\) 7.39532 0.560638
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 2.70925 0.203640
\(178\) −35.5539 −2.66487
\(179\) −16.4156 −1.22696 −0.613479 0.789711i \(-0.710231\pi\)
−0.613479 + 0.789711i \(0.710231\pi\)
\(180\) 10.9843 0.818722
\(181\) 8.33171 0.619291 0.309646 0.950852i \(-0.399790\pi\)
0.309646 + 0.950852i \(0.399790\pi\)
\(182\) −11.4719 −0.850353
\(183\) 2.51526 0.185933
\(184\) 39.6669 2.92429
\(185\) −2.86108 −0.210351
\(186\) 4.82150 0.353529
\(187\) 0 0
\(188\) −12.9678 −0.945772
\(189\) 1.89420 0.137783
\(190\) 17.0715 1.23850
\(191\) −5.74610 −0.415773 −0.207887 0.978153i \(-0.566658\pi\)
−0.207887 + 0.978153i \(0.566658\pi\)
\(192\) 3.26369 0.235537
\(193\) 9.93884 0.715414 0.357707 0.933834i \(-0.383559\pi\)
0.357707 + 0.933834i \(0.383559\pi\)
\(194\) 28.7322 2.06285
\(195\) −1.53121 −0.109652
\(196\) 3.79186 0.270847
\(197\) 3.14958 0.224398 0.112199 0.993686i \(-0.464211\pi\)
0.112199 + 0.993686i \(0.464211\pi\)
\(198\) 0 0
\(199\) 5.31856 0.377023 0.188511 0.982071i \(-0.439634\pi\)
0.188511 + 0.982071i \(0.439634\pi\)
\(200\) 4.31233 0.304928
\(201\) −0.749084 −0.0528363
\(202\) 33.1430 2.33193
\(203\) −9.56618 −0.671414
\(204\) −0.346720 −0.0242752
\(205\) −3.08304 −0.215329
\(206\) 26.4351 1.84182
\(207\) −26.6463 −1.85205
\(208\) −13.3206 −0.923620
\(209\) 0 0
\(210\) 0.773069 0.0533468
\(211\) 1.58756 0.109292 0.0546461 0.998506i \(-0.482597\pi\)
0.0546461 + 0.998506i \(0.482597\pi\)
\(212\) −28.2197 −1.93814
\(213\) 4.08540 0.279927
\(214\) −24.1460 −1.65059
\(215\) −4.95715 −0.338075
\(216\) 8.16844 0.555792
\(217\) −6.23683 −0.423383
\(218\) 22.0267 1.49183
\(219\) 0.241115 0.0162930
\(220\) 0 0
\(221\) −1.35689 −0.0912740
\(222\) −2.21181 −0.148447
\(223\) −8.74931 −0.585897 −0.292948 0.956128i \(-0.594636\pi\)
−0.292948 + 0.956128i \(0.594636\pi\)
\(224\) −1.89942 −0.126911
\(225\) −2.89681 −0.193121
\(226\) 37.2595 2.47846
\(227\) −7.32115 −0.485922 −0.242961 0.970036i \(-0.578119\pi\)
−0.242961 + 0.970036i \(0.578119\pi\)
\(228\) 8.64020 0.572211
\(229\) 8.24158 0.544619 0.272309 0.962210i \(-0.412213\pi\)
0.272309 + 0.962210i \(0.412213\pi\)
\(230\) −22.1373 −1.45969
\(231\) 0 0
\(232\) −41.2526 −2.70836
\(233\) −13.4000 −0.877863 −0.438932 0.898520i \(-0.644643\pi\)
−0.438932 + 0.898520i \(0.644643\pi\)
\(234\) 33.2319 2.17244
\(235\) 3.41990 0.223090
\(236\) −31.9810 −2.08179
\(237\) 2.46292 0.159984
\(238\) 0.685056 0.0444056
\(239\) 3.20926 0.207589 0.103795 0.994599i \(-0.466902\pi\)
0.103795 + 0.994599i \(0.466902\pi\)
\(240\) 0.897653 0.0579432
\(241\) −6.67350 −0.429878 −0.214939 0.976627i \(-0.568955\pi\)
−0.214939 + 0.976627i \(0.568955\pi\)
\(242\) 0 0
\(243\) −8.27875 −0.531082
\(244\) −29.6910 −1.90077
\(245\) −1.00000 −0.0638877
\(246\) −2.38340 −0.151960
\(247\) 33.8133 2.15149
\(248\) −26.8953 −1.70785
\(249\) −3.01938 −0.191345
\(250\) −2.40663 −0.152209
\(251\) 14.4513 0.912159 0.456080 0.889939i \(-0.349253\pi\)
0.456080 + 0.889939i \(0.349253\pi\)
\(252\) −10.9843 −0.691946
\(253\) 0 0
\(254\) −2.58817 −0.162396
\(255\) 0.0914380 0.00572607
\(256\) −29.3834 −1.83646
\(257\) 6.68992 0.417306 0.208653 0.977990i \(-0.433092\pi\)
0.208653 + 0.977990i \(0.433092\pi\)
\(258\) −3.83222 −0.238583
\(259\) 2.86108 0.177779
\(260\) 18.0750 1.12096
\(261\) 27.7114 1.71530
\(262\) 19.0293 1.17563
\(263\) 24.5565 1.51422 0.757110 0.653287i \(-0.226610\pi\)
0.757110 + 0.653287i \(0.226610\pi\)
\(264\) 0 0
\(265\) 7.44219 0.457170
\(266\) −17.0715 −1.04672
\(267\) 4.74556 0.290423
\(268\) 8.84246 0.540139
\(269\) −24.2805 −1.48041 −0.740203 0.672384i \(-0.765271\pi\)
−0.740203 + 0.672384i \(0.765271\pi\)
\(270\) −4.55865 −0.277430
\(271\) −11.4013 −0.692580 −0.346290 0.938128i \(-0.612559\pi\)
−0.346290 + 0.938128i \(0.612559\pi\)
\(272\) 0.795456 0.0482316
\(273\) 1.53121 0.0926731
\(274\) 16.5962 1.00261
\(275\) 0 0
\(276\) −11.2041 −0.674409
\(277\) −10.2450 −0.615563 −0.307781 0.951457i \(-0.599587\pi\)
−0.307781 + 0.951457i \(0.599587\pi\)
\(278\) −51.1709 −3.06903
\(279\) 18.0669 1.08164
\(280\) −4.31233 −0.257711
\(281\) 13.9922 0.834704 0.417352 0.908745i \(-0.362958\pi\)
0.417352 + 0.908745i \(0.362958\pi\)
\(282\) 2.64382 0.157437
\(283\) 6.35828 0.377961 0.188980 0.981981i \(-0.439482\pi\)
0.188980 + 0.981981i \(0.439482\pi\)
\(284\) −48.2256 −2.86166
\(285\) −2.27862 −0.134974
\(286\) 0 0
\(287\) 3.08304 0.181986
\(288\) 5.50228 0.324225
\(289\) −16.9190 −0.995234
\(290\) 23.0222 1.35191
\(291\) −3.83503 −0.224814
\(292\) −2.84620 −0.166562
\(293\) 2.18117 0.127426 0.0637128 0.997968i \(-0.479706\pi\)
0.0637128 + 0.997968i \(0.479706\pi\)
\(294\) −0.773069 −0.0450863
\(295\) 8.43413 0.491054
\(296\) 12.3379 0.717128
\(297\) 0 0
\(298\) 27.0285 1.56572
\(299\) −43.8472 −2.53575
\(300\) −1.21804 −0.0703235
\(301\) 4.95715 0.285725
\(302\) 0.738398 0.0424900
\(303\) −4.42376 −0.254139
\(304\) −19.8226 −1.13691
\(305\) 7.83020 0.448356
\(306\) −1.98448 −0.113445
\(307\) −11.9841 −0.683966 −0.341983 0.939706i \(-0.611099\pi\)
−0.341983 + 0.939706i \(0.611099\pi\)
\(308\) 0 0
\(309\) −3.52843 −0.200726
\(310\) 15.0097 0.852495
\(311\) 9.08109 0.514941 0.257471 0.966286i \(-0.417111\pi\)
0.257471 + 0.966286i \(0.417111\pi\)
\(312\) 6.60310 0.373827
\(313\) 6.96402 0.393630 0.196815 0.980441i \(-0.436940\pi\)
0.196815 + 0.980441i \(0.436940\pi\)
\(314\) −4.69585 −0.265002
\(315\) 2.89681 0.163217
\(316\) −29.0732 −1.63550
\(317\) −22.7634 −1.27852 −0.639259 0.768992i \(-0.720759\pi\)
−0.639259 + 0.768992i \(0.720759\pi\)
\(318\) 5.75332 0.322630
\(319\) 0 0
\(320\) 10.1601 0.567969
\(321\) 3.22290 0.179884
\(322\) 22.1373 1.23367
\(323\) −2.01920 −0.112351
\(324\) 30.6457 1.70254
\(325\) −4.76679 −0.264414
\(326\) 43.1818 2.39162
\(327\) −2.94001 −0.162583
\(328\) 13.2951 0.734099
\(329\) −3.41990 −0.188545
\(330\) 0 0
\(331\) 26.5867 1.46134 0.730669 0.682732i \(-0.239208\pi\)
0.730669 + 0.682732i \(0.239208\pi\)
\(332\) 35.6419 1.95610
\(333\) −8.28802 −0.454181
\(334\) −36.2918 −1.98580
\(335\) −2.33196 −0.127409
\(336\) −0.897653 −0.0489710
\(337\) 22.3595 1.21800 0.608999 0.793171i \(-0.291571\pi\)
0.608999 + 0.793171i \(0.291571\pi\)
\(338\) 23.3979 1.27268
\(339\) −4.97321 −0.270108
\(340\) −1.07937 −0.0585369
\(341\) 0 0
\(342\) 49.4529 2.67411
\(343\) 1.00000 0.0539949
\(344\) 21.3769 1.15256
\(345\) 2.95478 0.159080
\(346\) −59.4546 −3.19630
\(347\) −17.7009 −0.950234 −0.475117 0.879923i \(-0.657594\pi\)
−0.475117 + 0.879923i \(0.657594\pi\)
\(348\) 11.6520 0.624612
\(349\) −2.85200 −0.152664 −0.0763320 0.997082i \(-0.524321\pi\)
−0.0763320 + 0.997082i \(0.524321\pi\)
\(350\) 2.40663 0.128640
\(351\) −9.02927 −0.481947
\(352\) 0 0
\(353\) 0.548804 0.0292099 0.0146049 0.999893i \(-0.495351\pi\)
0.0146049 + 0.999893i \(0.495351\pi\)
\(354\) 6.52017 0.346543
\(355\) 12.7182 0.675012
\(356\) −56.0183 −2.96896
\(357\) −0.0914380 −0.00483941
\(358\) −39.5062 −2.08797
\(359\) −23.7324 −1.25255 −0.626274 0.779603i \(-0.715421\pi\)
−0.626274 + 0.779603i \(0.715421\pi\)
\(360\) 12.4920 0.658388
\(361\) 31.3181 1.64832
\(362\) 20.0513 1.05387
\(363\) 0 0
\(364\) −18.0750 −0.947386
\(365\) 0.750610 0.0392887
\(366\) 6.05329 0.316410
\(367\) 21.5679 1.12584 0.562918 0.826513i \(-0.309679\pi\)
0.562918 + 0.826513i \(0.309679\pi\)
\(368\) 25.7049 1.33996
\(369\) −8.93099 −0.464929
\(370\) −6.88556 −0.357963
\(371\) −7.44219 −0.386379
\(372\) 7.59670 0.393871
\(373\) −9.65009 −0.499663 −0.249831 0.968289i \(-0.580375\pi\)
−0.249831 + 0.968289i \(0.580375\pi\)
\(374\) 0 0
\(375\) 0.321225 0.0165880
\(376\) −14.7478 −0.760557
\(377\) 45.5999 2.34852
\(378\) 4.55865 0.234471
\(379\) 1.91750 0.0984955 0.0492477 0.998787i \(-0.484318\pi\)
0.0492477 + 0.998787i \(0.484318\pi\)
\(380\) 26.8976 1.37982
\(381\) 0.345456 0.0176983
\(382\) −13.8287 −0.707539
\(383\) −5.38688 −0.275257 −0.137628 0.990484i \(-0.543948\pi\)
−0.137628 + 0.990484i \(0.543948\pi\)
\(384\) 6.63421 0.338550
\(385\) 0 0
\(386\) 23.9191 1.21745
\(387\) −14.3599 −0.729957
\(388\) 45.2701 2.29824
\(389\) 8.31392 0.421532 0.210766 0.977537i \(-0.432404\pi\)
0.210766 + 0.977537i \(0.432404\pi\)
\(390\) −3.68506 −0.186600
\(391\) 2.61839 0.132417
\(392\) 4.31233 0.217806
\(393\) −2.53994 −0.128123
\(394\) 7.57987 0.381868
\(395\) 7.66727 0.385782
\(396\) 0 0
\(397\) −15.8950 −0.797746 −0.398873 0.917006i \(-0.630599\pi\)
−0.398873 + 0.917006i \(0.630599\pi\)
\(398\) 12.7998 0.641596
\(399\) 2.27862 0.114074
\(400\) 2.79447 0.139723
\(401\) 34.2863 1.71218 0.856089 0.516829i \(-0.172888\pi\)
0.856089 + 0.516829i \(0.172888\pi\)
\(402\) −1.80277 −0.0899138
\(403\) 29.7296 1.48094
\(404\) 52.2197 2.59803
\(405\) −8.08198 −0.401597
\(406\) −23.0222 −1.14257
\(407\) 0 0
\(408\) −0.394311 −0.0195213
\(409\) −29.1522 −1.44148 −0.720742 0.693203i \(-0.756199\pi\)
−0.720742 + 0.693203i \(0.756199\pi\)
\(410\) −7.41973 −0.366434
\(411\) −2.21518 −0.109267
\(412\) 41.6509 2.05199
\(413\) −8.43413 −0.415017
\(414\) −64.1277 −3.15171
\(415\) −9.39958 −0.461407
\(416\) 9.05415 0.443916
\(417\) 6.83005 0.334469
\(418\) 0 0
\(419\) −19.8745 −0.970935 −0.485467 0.874255i \(-0.661351\pi\)
−0.485467 + 0.874255i \(0.661351\pi\)
\(420\) 1.21804 0.0594342
\(421\) −9.13157 −0.445046 −0.222523 0.974927i \(-0.571429\pi\)
−0.222523 + 0.974927i \(0.571429\pi\)
\(422\) 3.82067 0.185987
\(423\) 9.90682 0.481686
\(424\) −32.0932 −1.55858
\(425\) 0.284654 0.0138077
\(426\) 9.83204 0.476364
\(427\) −7.83020 −0.378930
\(428\) −38.0442 −1.83894
\(429\) 0 0
\(430\) −11.9300 −0.575316
\(431\) −6.14835 −0.296155 −0.148078 0.988976i \(-0.547309\pi\)
−0.148078 + 0.988976i \(0.547309\pi\)
\(432\) 5.29329 0.254674
\(433\) −21.4207 −1.02941 −0.514707 0.857366i \(-0.672099\pi\)
−0.514707 + 0.857366i \(0.672099\pi\)
\(434\) −15.0097 −0.720490
\(435\) −3.07290 −0.147334
\(436\) 34.7050 1.66207
\(437\) −65.2497 −3.12132
\(438\) 0.580273 0.0277265
\(439\) 23.6732 1.12986 0.564931 0.825138i \(-0.308903\pi\)
0.564931 + 0.825138i \(0.308903\pi\)
\(440\) 0 0
\(441\) −2.89681 −0.137944
\(442\) −3.26552 −0.155325
\(443\) 36.5672 1.73736 0.868680 0.495374i \(-0.164969\pi\)
0.868680 + 0.495374i \(0.164969\pi\)
\(444\) −3.48491 −0.165387
\(445\) 14.7733 0.700322
\(446\) −21.0563 −0.997046
\(447\) −3.60763 −0.170635
\(448\) −10.1601 −0.480022
\(449\) 31.3388 1.47897 0.739485 0.673173i \(-0.235069\pi\)
0.739485 + 0.673173i \(0.235069\pi\)
\(450\) −6.97155 −0.328642
\(451\) 0 0
\(452\) 58.7056 2.76128
\(453\) −0.0985578 −0.00463065
\(454\) −17.6193 −0.826914
\(455\) 4.76679 0.223470
\(456\) 9.82616 0.460152
\(457\) −15.7455 −0.736541 −0.368271 0.929719i \(-0.620050\pi\)
−0.368271 + 0.929719i \(0.620050\pi\)
\(458\) 19.8344 0.926801
\(459\) 0.539193 0.0251674
\(460\) −34.8793 −1.62626
\(461\) 14.5224 0.676374 0.338187 0.941079i \(-0.390186\pi\)
0.338187 + 0.941079i \(0.390186\pi\)
\(462\) 0 0
\(463\) −19.0465 −0.885167 −0.442583 0.896727i \(-0.645938\pi\)
−0.442583 + 0.896727i \(0.645938\pi\)
\(464\) −26.7324 −1.24102
\(465\) −2.00343 −0.0929066
\(466\) −32.2488 −1.49390
\(467\) −24.8958 −1.15204 −0.576020 0.817435i \(-0.695395\pi\)
−0.576020 + 0.817435i \(0.695395\pi\)
\(468\) 52.3599 2.42034
\(469\) 2.33196 0.107680
\(470\) 8.23043 0.379641
\(471\) 0.626780 0.0288805
\(472\) −36.3708 −1.67410
\(473\) 0 0
\(474\) 5.92733 0.272251
\(475\) −7.09353 −0.325473
\(476\) 1.07937 0.0494727
\(477\) 21.5586 0.987102
\(478\) 7.72348 0.353264
\(479\) −29.1610 −1.33240 −0.666199 0.745774i \(-0.732080\pi\)
−0.666199 + 0.745774i \(0.732080\pi\)
\(480\) −0.610143 −0.0278491
\(481\) −13.6382 −0.621847
\(482\) −16.0606 −0.731542
\(483\) −2.95478 −0.134447
\(484\) 0 0
\(485\) −11.9388 −0.542112
\(486\) −19.9239 −0.903765
\(487\) 21.2981 0.965111 0.482556 0.875865i \(-0.339709\pi\)
0.482556 + 0.875865i \(0.339709\pi\)
\(488\) −33.7664 −1.52853
\(489\) −5.76370 −0.260643
\(490\) −2.40663 −0.108720
\(491\) −21.1562 −0.954765 −0.477382 0.878696i \(-0.658414\pi\)
−0.477382 + 0.878696i \(0.658414\pi\)
\(492\) −3.75526 −0.169300
\(493\) −2.72305 −0.122640
\(494\) 81.3761 3.66129
\(495\) 0 0
\(496\) −17.4286 −0.782568
\(497\) −12.7182 −0.570489
\(498\) −7.26653 −0.325621
\(499\) −42.2019 −1.88922 −0.944608 0.328201i \(-0.893558\pi\)
−0.944608 + 0.328201i \(0.893558\pi\)
\(500\) −3.79186 −0.169577
\(501\) 4.84405 0.216416
\(502\) 34.7789 1.55226
\(503\) 1.88539 0.0840655 0.0420327 0.999116i \(-0.486617\pi\)
0.0420327 + 0.999116i \(0.486617\pi\)
\(504\) −12.4920 −0.556439
\(505\) −13.7715 −0.612825
\(506\) 0 0
\(507\) −3.12304 −0.138699
\(508\) −4.07789 −0.180927
\(509\) 36.1928 1.60422 0.802109 0.597178i \(-0.203711\pi\)
0.802109 + 0.597178i \(0.203711\pi\)
\(510\) 0.220057 0.00974430
\(511\) −0.750610 −0.0332050
\(512\) −29.4092 −1.29972
\(513\) −13.4366 −0.593240
\(514\) 16.1002 0.710147
\(515\) −10.9843 −0.484026
\(516\) −6.03800 −0.265808
\(517\) 0 0
\(518\) 6.88556 0.302534
\(519\) 7.93571 0.348339
\(520\) 20.5560 0.901439
\(521\) 24.8823 1.09012 0.545058 0.838399i \(-0.316508\pi\)
0.545058 + 0.838399i \(0.316508\pi\)
\(522\) 66.6911 2.91899
\(523\) −2.06176 −0.0901546 −0.0450773 0.998984i \(-0.514353\pi\)
−0.0450773 + 0.998984i \(0.514353\pi\)
\(524\) 29.9823 1.30978
\(525\) −0.321225 −0.0140194
\(526\) 59.0984 2.57681
\(527\) −1.77534 −0.0773349
\(528\) 0 0
\(529\) 61.6121 2.67879
\(530\) 17.9106 0.777986
\(531\) 24.4321 1.06026
\(532\) −26.8976 −1.16616
\(533\) −14.6962 −0.636563
\(534\) 11.4208 0.494226
\(535\) 10.0331 0.433770
\(536\) 10.0562 0.434361
\(537\) 5.27310 0.227551
\(538\) −58.4340 −2.51927
\(539\) 0 0
\(540\) −7.18255 −0.309088
\(541\) −21.7379 −0.934586 −0.467293 0.884103i \(-0.654771\pi\)
−0.467293 + 0.884103i \(0.654771\pi\)
\(542\) −27.4387 −1.17859
\(543\) −2.67636 −0.114853
\(544\) −0.540679 −0.0231814
\(545\) −9.15250 −0.392050
\(546\) 3.68506 0.157706
\(547\) 13.2751 0.567603 0.283802 0.958883i \(-0.408404\pi\)
0.283802 + 0.958883i \(0.408404\pi\)
\(548\) 26.1488 1.11702
\(549\) 22.6826 0.968071
\(550\) 0 0
\(551\) 67.8580 2.89085
\(552\) −12.7420 −0.542336
\(553\) −7.66727 −0.326046
\(554\) −24.6559 −1.04753
\(555\) 0.919051 0.0390115
\(556\) −80.6244 −3.41924
\(557\) 12.1905 0.516528 0.258264 0.966074i \(-0.416850\pi\)
0.258264 + 0.966074i \(0.416850\pi\)
\(558\) 43.4804 1.84067
\(559\) −23.6297 −0.999429
\(560\) −2.79447 −0.118088
\(561\) 0 0
\(562\) 33.6740 1.42045
\(563\) 19.6678 0.828899 0.414450 0.910072i \(-0.363974\pi\)
0.414450 + 0.910072i \(0.363974\pi\)
\(564\) 4.16557 0.175402
\(565\) −15.4820 −0.651334
\(566\) 15.3020 0.643192
\(567\) 8.08198 0.339411
\(568\) −54.8451 −2.30125
\(569\) 6.46635 0.271084 0.135542 0.990772i \(-0.456722\pi\)
0.135542 + 0.990772i \(0.456722\pi\)
\(570\) −5.48379 −0.229691
\(571\) 34.9789 1.46382 0.731910 0.681402i \(-0.238629\pi\)
0.731910 + 0.681402i \(0.238629\pi\)
\(572\) 0 0
\(573\) 1.84579 0.0771090
\(574\) 7.41973 0.309693
\(575\) 9.19849 0.383603
\(576\) 29.4320 1.22634
\(577\) 8.74822 0.364193 0.182097 0.983281i \(-0.441712\pi\)
0.182097 + 0.983281i \(0.441712\pi\)
\(578\) −40.7177 −1.69363
\(579\) −3.19261 −0.132680
\(580\) 36.2736 1.50618
\(581\) 9.39958 0.389960
\(582\) −9.22950 −0.382575
\(583\) 0 0
\(584\) −3.23688 −0.133943
\(585\) −13.8085 −0.570911
\(586\) 5.24927 0.216845
\(587\) 25.8465 1.06680 0.533400 0.845863i \(-0.320914\pi\)
0.533400 + 0.845863i \(0.320914\pi\)
\(588\) −1.21804 −0.0502311
\(589\) 44.2411 1.82292
\(590\) 20.2978 0.835648
\(591\) −1.01172 −0.0416168
\(592\) 7.99520 0.328601
\(593\) 27.7338 1.13889 0.569445 0.822029i \(-0.307158\pi\)
0.569445 + 0.822029i \(0.307158\pi\)
\(594\) 0 0
\(595\) −0.284654 −0.0116697
\(596\) 42.5857 1.74438
\(597\) −1.70845 −0.0699224
\(598\) −105.524 −4.31520
\(599\) 9.62585 0.393302 0.196651 0.980474i \(-0.436993\pi\)
0.196651 + 0.980474i \(0.436993\pi\)
\(600\) −1.38523 −0.0565518
\(601\) −39.4592 −1.60957 −0.804787 0.593564i \(-0.797720\pi\)
−0.804787 + 0.593564i \(0.797720\pi\)
\(602\) 11.9300 0.486231
\(603\) −6.75526 −0.275095
\(604\) 1.16341 0.0473386
\(605\) 0 0
\(606\) −10.6463 −0.432478
\(607\) 28.3346 1.15007 0.575033 0.818130i \(-0.304989\pi\)
0.575033 + 0.818130i \(0.304989\pi\)
\(608\) 13.4736 0.546427
\(609\) 3.07290 0.124520
\(610\) 18.8444 0.762987
\(611\) 16.3019 0.659506
\(612\) −3.12673 −0.126390
\(613\) −25.3797 −1.02508 −0.512538 0.858665i \(-0.671295\pi\)
−0.512538 + 0.858665i \(0.671295\pi\)
\(614\) −28.8412 −1.16394
\(615\) 0.990349 0.0399347
\(616\) 0 0
\(617\) 27.3188 1.09981 0.549907 0.835226i \(-0.314663\pi\)
0.549907 + 0.835226i \(0.314663\pi\)
\(618\) −8.49163 −0.341583
\(619\) 40.8391 1.64146 0.820731 0.571315i \(-0.193567\pi\)
0.820731 + 0.571315i \(0.193567\pi\)
\(620\) 23.6492 0.949773
\(621\) 17.4238 0.699193
\(622\) 21.8548 0.876297
\(623\) −14.7733 −0.591880
\(624\) 4.27892 0.171294
\(625\) 1.00000 0.0400000
\(626\) 16.7598 0.669857
\(627\) 0 0
\(628\) −7.39874 −0.295242
\(629\) 0.814419 0.0324730
\(630\) 6.97155 0.277753
\(631\) −15.7081 −0.625330 −0.312665 0.949863i \(-0.601222\pi\)
−0.312665 + 0.949863i \(0.601222\pi\)
\(632\) −33.0638 −1.31521
\(633\) −0.509964 −0.0202692
\(634\) −54.7829 −2.17571
\(635\) 1.07543 0.0426773
\(636\) 9.06488 0.359446
\(637\) −4.76679 −0.188867
\(638\) 0 0
\(639\) 36.8423 1.45746
\(640\) 20.6528 0.816375
\(641\) −30.9780 −1.22356 −0.611779 0.791029i \(-0.709546\pi\)
−0.611779 + 0.791029i \(0.709546\pi\)
\(642\) 7.75631 0.306117
\(643\) −21.4712 −0.846743 −0.423372 0.905956i \(-0.639154\pi\)
−0.423372 + 0.905956i \(0.639154\pi\)
\(644\) 34.8793 1.37444
\(645\) 1.59236 0.0626991
\(646\) −4.85947 −0.191193
\(647\) −15.5854 −0.612726 −0.306363 0.951915i \(-0.599112\pi\)
−0.306363 + 0.951915i \(0.599112\pi\)
\(648\) 34.8522 1.36912
\(649\) 0 0
\(650\) −11.4719 −0.449964
\(651\) 2.00343 0.0785204
\(652\) 68.0368 2.66453
\(653\) −15.7529 −0.616459 −0.308229 0.951312i \(-0.599736\pi\)
−0.308229 + 0.951312i \(0.599736\pi\)
\(654\) −7.07552 −0.276675
\(655\) −7.90703 −0.308953
\(656\) 8.61545 0.336377
\(657\) 2.17438 0.0848305
\(658\) −8.23043 −0.320856
\(659\) −9.73277 −0.379135 −0.189568 0.981868i \(-0.560709\pi\)
−0.189568 + 0.981868i \(0.560709\pi\)
\(660\) 0 0
\(661\) −30.5789 −1.18938 −0.594690 0.803955i \(-0.702725\pi\)
−0.594690 + 0.803955i \(0.702725\pi\)
\(662\) 63.9843 2.48682
\(663\) 0.435866 0.0169276
\(664\) 40.5341 1.57303
\(665\) 7.09353 0.275075
\(666\) −19.9462 −0.772899
\(667\) −87.9943 −3.40715
\(668\) −57.1810 −2.21240
\(669\) 2.81050 0.108660
\(670\) −5.61216 −0.216817
\(671\) 0 0
\(672\) 0.610143 0.0235368
\(673\) −49.9855 −1.92680 −0.963400 0.268068i \(-0.913615\pi\)
−0.963400 + 0.268068i \(0.913615\pi\)
\(674\) 53.8109 2.07272
\(675\) 1.89420 0.0729080
\(676\) 36.8654 1.41790
\(677\) 33.4873 1.28702 0.643511 0.765437i \(-0.277477\pi\)
0.643511 + 0.765437i \(0.277477\pi\)
\(678\) −11.9687 −0.459654
\(679\) 11.9388 0.458168
\(680\) −1.22752 −0.0470734
\(681\) 2.35174 0.0901188
\(682\) 0 0
\(683\) −27.2746 −1.04364 −0.521818 0.853057i \(-0.674746\pi\)
−0.521818 + 0.853057i \(0.674746\pi\)
\(684\) 77.9175 2.97925
\(685\) −6.89605 −0.263484
\(686\) 2.40663 0.0918855
\(687\) −2.64740 −0.101005
\(688\) 13.8526 0.528125
\(689\) 35.4753 1.35150
\(690\) 7.11106 0.270714
\(691\) −44.5342 −1.69416 −0.847080 0.531465i \(-0.821642\pi\)
−0.847080 + 0.531465i \(0.821642\pi\)
\(692\) −93.6760 −3.56103
\(693\) 0 0
\(694\) −42.5995 −1.61705
\(695\) 21.2625 0.806533
\(696\) 13.2514 0.502291
\(697\) 0.877599 0.0332414
\(698\) −6.86370 −0.259795
\(699\) 4.30442 0.162808
\(700\) 3.79186 0.143319
\(701\) −36.4241 −1.37572 −0.687860 0.725844i \(-0.741450\pi\)
−0.687860 + 0.725844i \(0.741450\pi\)
\(702\) −21.7301 −0.820150
\(703\) −20.2952 −0.765447
\(704\) 0 0
\(705\) −1.09856 −0.0413741
\(706\) 1.32077 0.0497077
\(707\) 13.7715 0.517932
\(708\) 10.2731 0.386087
\(709\) 3.00071 0.112694 0.0563471 0.998411i \(-0.482055\pi\)
0.0563471 + 0.998411i \(0.482055\pi\)
\(710\) 30.6080 1.14870
\(711\) 22.2107 0.832965
\(712\) −63.7074 −2.38754
\(713\) −57.3694 −2.14850
\(714\) −0.220057 −0.00823544
\(715\) 0 0
\(716\) −62.2456 −2.32623
\(717\) −1.03089 −0.0384994
\(718\) −57.1151 −2.13151
\(719\) 42.3787 1.58046 0.790230 0.612811i \(-0.209961\pi\)
0.790230 + 0.612811i \(0.209961\pi\)
\(720\) 8.09505 0.301685
\(721\) 10.9843 0.409077
\(722\) 75.3711 2.80502
\(723\) 2.14369 0.0797249
\(724\) 31.5927 1.17413
\(725\) −9.56618 −0.355279
\(726\) 0 0
\(727\) −1.27333 −0.0472253 −0.0236126 0.999721i \(-0.507517\pi\)
−0.0236126 + 0.999721i \(0.507517\pi\)
\(728\) −20.5560 −0.761855
\(729\) −21.5866 −0.799503
\(730\) 1.80644 0.0668593
\(731\) 1.41107 0.0521904
\(732\) 9.53749 0.352516
\(733\) 37.1466 1.37204 0.686020 0.727583i \(-0.259356\pi\)
0.686020 + 0.727583i \(0.259356\pi\)
\(734\) 51.9059 1.91588
\(735\) 0.321225 0.0118486
\(736\) −17.4718 −0.644020
\(737\) 0 0
\(738\) −21.4936 −0.791189
\(739\) 20.4448 0.752074 0.376037 0.926605i \(-0.377287\pi\)
0.376037 + 0.926605i \(0.377287\pi\)
\(740\) −10.8488 −0.398810
\(741\) −10.8617 −0.399014
\(742\) −17.9106 −0.657518
\(743\) −40.5600 −1.48800 −0.744000 0.668179i \(-0.767074\pi\)
−0.744000 + 0.668179i \(0.767074\pi\)
\(744\) 8.63944 0.316737
\(745\) −11.2308 −0.411466
\(746\) −23.2242 −0.850297
\(747\) −27.2288 −0.996251
\(748\) 0 0
\(749\) −10.0331 −0.366603
\(750\) 0.773069 0.0282285
\(751\) −20.4219 −0.745205 −0.372602 0.927991i \(-0.621534\pi\)
−0.372602 + 0.927991i \(0.621534\pi\)
\(752\) −9.55680 −0.348501
\(753\) −4.64212 −0.169168
\(754\) 109.742 3.99657
\(755\) −0.306818 −0.0111663
\(756\) 7.18255 0.261227
\(757\) 42.3910 1.54073 0.770364 0.637604i \(-0.220074\pi\)
0.770364 + 0.637604i \(0.220074\pi\)
\(758\) 4.61471 0.167614
\(759\) 0 0
\(760\) 30.5897 1.10960
\(761\) 4.05235 0.146898 0.0734488 0.997299i \(-0.476599\pi\)
0.0734488 + 0.997299i \(0.476599\pi\)
\(762\) 0.831385 0.0301179
\(763\) 9.15250 0.331343
\(764\) −21.7884 −0.788276
\(765\) 0.824590 0.0298131
\(766\) −12.9642 −0.468416
\(767\) 40.2037 1.45167
\(768\) 9.43868 0.340589
\(769\) −43.8660 −1.58185 −0.790924 0.611914i \(-0.790400\pi\)
−0.790924 + 0.611914i \(0.790400\pi\)
\(770\) 0 0
\(771\) −2.14897 −0.0773933
\(772\) 37.6867 1.35637
\(773\) 24.8656 0.894355 0.447177 0.894445i \(-0.352429\pi\)
0.447177 + 0.894445i \(0.352429\pi\)
\(774\) −34.5590 −1.24220
\(775\) −6.23683 −0.224033
\(776\) 51.4840 1.84817
\(777\) −0.919051 −0.0329708
\(778\) 20.0085 0.717339
\(779\) −21.8696 −0.783561
\(780\) −5.80614 −0.207893
\(781\) 0 0
\(782\) 6.30148 0.225340
\(783\) −18.1203 −0.647567
\(784\) 2.79447 0.0998024
\(785\) 1.95122 0.0696419
\(786\) −6.11268 −0.218032
\(787\) 47.0782 1.67816 0.839079 0.544009i \(-0.183094\pi\)
0.839079 + 0.544009i \(0.183094\pi\)
\(788\) 11.9428 0.425443
\(789\) −7.88817 −0.280826
\(790\) 18.4523 0.656502
\(791\) 15.4820 0.550478
\(792\) 0 0
\(793\) 37.3249 1.32545
\(794\) −38.2533 −1.35756
\(795\) −2.39062 −0.0847864
\(796\) 20.1672 0.714808
\(797\) −20.3187 −0.719725 −0.359863 0.933005i \(-0.617176\pi\)
−0.359863 + 0.933005i \(0.617176\pi\)
\(798\) 5.48379 0.194124
\(799\) −0.973489 −0.0344396
\(800\) −1.89942 −0.0671548
\(801\) 42.7955 1.51211
\(802\) 82.5144 2.91369
\(803\) 0 0
\(804\) −2.84042 −0.100174
\(805\) −9.19849 −0.324204
\(806\) 71.5482 2.52018
\(807\) 7.79949 0.274555
\(808\) 59.3875 2.08924
\(809\) −17.6907 −0.621970 −0.310985 0.950415i \(-0.600659\pi\)
−0.310985 + 0.950415i \(0.600659\pi\)
\(810\) −19.4503 −0.683414
\(811\) −25.5677 −0.897804 −0.448902 0.893581i \(-0.648185\pi\)
−0.448902 + 0.893581i \(0.648185\pi\)
\(812\) −36.2736 −1.27295
\(813\) 3.66238 0.128445
\(814\) 0 0
\(815\) −17.9429 −0.628511
\(816\) −0.255521 −0.00894501
\(817\) −35.1637 −1.23022
\(818\) −70.1585 −2.45304
\(819\) 13.8085 0.482508
\(820\) −11.6904 −0.408248
\(821\) −1.94318 −0.0678175 −0.0339087 0.999425i \(-0.510796\pi\)
−0.0339087 + 0.999425i \(0.510796\pi\)
\(822\) −5.33112 −0.185944
\(823\) 2.38290 0.0830627 0.0415313 0.999137i \(-0.486776\pi\)
0.0415313 + 0.999137i \(0.486776\pi\)
\(824\) 47.3680 1.65014
\(825\) 0 0
\(826\) −20.2978 −0.706251
\(827\) −25.6827 −0.893076 −0.446538 0.894765i \(-0.647343\pi\)
−0.446538 + 0.894765i \(0.647343\pi\)
\(828\) −101.039 −3.51135
\(829\) −45.3942 −1.57661 −0.788303 0.615288i \(-0.789040\pi\)
−0.788303 + 0.615288i \(0.789040\pi\)
\(830\) −22.6213 −0.785196
\(831\) 3.29095 0.114162
\(832\) 48.4312 1.67905
\(833\) 0.284654 0.00986268
\(834\) 16.4374 0.569180
\(835\) 15.0799 0.521863
\(836\) 0 0
\(837\) −11.8138 −0.408346
\(838\) −47.8306 −1.65228
\(839\) −33.1060 −1.14295 −0.571474 0.820620i \(-0.693628\pi\)
−0.571474 + 0.820620i \(0.693628\pi\)
\(840\) 1.38523 0.0477950
\(841\) 62.5118 2.15558
\(842\) −21.9763 −0.757353
\(843\) −4.49464 −0.154804
\(844\) 6.01980 0.207210
\(845\) −9.72227 −0.334456
\(846\) 23.8420 0.819706
\(847\) 0 0
\(848\) −20.7969 −0.714170
\(849\) −2.04244 −0.0700963
\(850\) 0.685056 0.0234972
\(851\) 26.3176 0.902156
\(852\) 15.4913 0.530722
\(853\) −18.5516 −0.635195 −0.317597 0.948226i \(-0.602876\pi\)
−0.317597 + 0.948226i \(0.602876\pi\)
\(854\) −18.8444 −0.644841
\(855\) −20.5486 −0.702748
\(856\) −43.2662 −1.47881
\(857\) −34.2369 −1.16951 −0.584755 0.811210i \(-0.698809\pi\)
−0.584755 + 0.811210i \(0.698809\pi\)
\(858\) 0 0
\(859\) −21.4685 −0.732494 −0.366247 0.930518i \(-0.619358\pi\)
−0.366247 + 0.930518i \(0.619358\pi\)
\(860\) −18.7968 −0.640965
\(861\) −0.990349 −0.0337510
\(862\) −14.7968 −0.503980
\(863\) 1.59188 0.0541881 0.0270940 0.999633i \(-0.491375\pi\)
0.0270940 + 0.999633i \(0.491375\pi\)
\(864\) −3.59790 −0.122403
\(865\) 24.7045 0.839979
\(866\) −51.5517 −1.75180
\(867\) 5.43480 0.184575
\(868\) −23.6492 −0.802705
\(869\) 0 0
\(870\) −7.39532 −0.250725
\(871\) −11.1160 −0.376650
\(872\) 39.4686 1.33658
\(873\) −34.5844 −1.17050
\(874\) −157.032 −5.31168
\(875\) −1.00000 −0.0338062
\(876\) 0.914272 0.0308904
\(877\) −23.2074 −0.783657 −0.391828 0.920038i \(-0.628157\pi\)
−0.391828 + 0.920038i \(0.628157\pi\)
\(878\) 56.9727 1.92273
\(879\) −0.700647 −0.0236323
\(880\) 0 0
\(881\) −30.9648 −1.04323 −0.521615 0.853181i \(-0.674670\pi\)
−0.521615 + 0.853181i \(0.674670\pi\)
\(882\) −6.97155 −0.234744
\(883\) −53.5660 −1.80264 −0.901319 0.433156i \(-0.857400\pi\)
−0.901319 + 0.433156i \(0.857400\pi\)
\(884\) −5.14512 −0.173049
\(885\) −2.70925 −0.0910706
\(886\) 88.0036 2.95654
\(887\) 43.8144 1.47114 0.735571 0.677447i \(-0.236914\pi\)
0.735571 + 0.677447i \(0.236914\pi\)
\(888\) −3.96326 −0.132998
\(889\) −1.07543 −0.0360689
\(890\) 35.5539 1.19177
\(891\) 0 0
\(892\) −33.1761 −1.11082
\(893\) 24.2592 0.811802
\(894\) −8.68222 −0.290377
\(895\) 16.4156 0.548712
\(896\) −20.6528 −0.689963
\(897\) 14.0848 0.470279
\(898\) 75.4209 2.51683
\(899\) 59.6626 1.98986
\(900\) −10.9843 −0.366144
\(901\) −2.11845 −0.0705758
\(902\) 0 0
\(903\) −1.59236 −0.0529904
\(904\) 66.7637 2.22053
\(905\) −8.33171 −0.276956
\(906\) −0.237192 −0.00788017
\(907\) 27.8887 0.926031 0.463015 0.886350i \(-0.346767\pi\)
0.463015 + 0.886350i \(0.346767\pi\)
\(908\) −27.7608 −0.921273
\(909\) −39.8936 −1.32319
\(910\) 11.4719 0.380289
\(911\) −31.8145 −1.05406 −0.527031 0.849846i \(-0.676695\pi\)
−0.527031 + 0.849846i \(0.676695\pi\)
\(912\) 6.36753 0.210850
\(913\) 0 0
\(914\) −37.8934 −1.25340
\(915\) −2.51526 −0.0831518
\(916\) 31.2509 1.03256
\(917\) 7.90703 0.261113
\(918\) 1.29764 0.0428284
\(919\) 0.532639 0.0175701 0.00878507 0.999961i \(-0.497204\pi\)
0.00878507 + 0.999961i \(0.497204\pi\)
\(920\) −39.6669 −1.30778
\(921\) 3.84958 0.126848
\(922\) 34.9499 1.15102
\(923\) 60.6249 1.99549
\(924\) 0 0
\(925\) 2.86108 0.0940718
\(926\) −45.8379 −1.50633
\(927\) −31.8195 −1.04509
\(928\) 18.1702 0.596467
\(929\) 14.1414 0.463964 0.231982 0.972720i \(-0.425479\pi\)
0.231982 + 0.972720i \(0.425479\pi\)
\(930\) −4.82150 −0.158103
\(931\) −7.09353 −0.232481
\(932\) −50.8109 −1.66437
\(933\) −2.91707 −0.0955007
\(934\) −59.9149 −1.96048
\(935\) 0 0
\(936\) 59.5469 1.94635
\(937\) 22.6170 0.738865 0.369432 0.929258i \(-0.379552\pi\)
0.369432 + 0.929258i \(0.379552\pi\)
\(938\) 5.61216 0.183244
\(939\) −2.23702 −0.0730023
\(940\) 12.9678 0.422962
\(941\) 9.76728 0.318404 0.159202 0.987246i \(-0.449108\pi\)
0.159202 + 0.987246i \(0.449108\pi\)
\(942\) 1.50843 0.0491472
\(943\) 28.3593 0.923505
\(944\) −23.5689 −0.767103
\(945\) −1.89420 −0.0616185
\(946\) 0 0
\(947\) −2.25459 −0.0732642 −0.0366321 0.999329i \(-0.511663\pi\)
−0.0366321 + 0.999329i \(0.511663\pi\)
\(948\) 9.33904 0.303318
\(949\) 3.57800 0.116147
\(950\) −17.0715 −0.553872
\(951\) 7.31216 0.237113
\(952\) 1.22752 0.0397843
\(953\) −34.8012 −1.12732 −0.563661 0.826006i \(-0.690608\pi\)
−0.563661 + 0.826006i \(0.690608\pi\)
\(954\) 51.8836 1.67979
\(955\) 5.74610 0.185939
\(956\) 12.1690 0.393575
\(957\) 0 0
\(958\) −70.1796 −2.26740
\(959\) 6.89605 0.222685
\(960\) −3.26369 −0.105335
\(961\) 7.89802 0.254775
\(962\) −32.8220 −1.05822
\(963\) 29.0641 0.936579
\(964\) −25.3050 −0.815018
\(965\) −9.93884 −0.319943
\(966\) −7.11106 −0.228795
\(967\) −15.6979 −0.504812 −0.252406 0.967621i \(-0.581222\pi\)
−0.252406 + 0.967621i \(0.581222\pi\)
\(968\) 0 0
\(969\) 0.648618 0.0208366
\(970\) −28.7322 −0.922535
\(971\) −40.7043 −1.30626 −0.653132 0.757244i \(-0.726545\pi\)
−0.653132 + 0.757244i \(0.726545\pi\)
\(972\) −31.3918 −1.00689
\(973\) −21.2625 −0.681645
\(974\) 51.2567 1.64237
\(975\) 1.53121 0.0490380
\(976\) −21.8812 −0.700401
\(977\) 27.2222 0.870916 0.435458 0.900209i \(-0.356586\pi\)
0.435458 + 0.900209i \(0.356586\pi\)
\(978\) −13.8711 −0.443548
\(979\) 0 0
\(980\) −3.79186 −0.121126
\(981\) −26.5131 −0.846498
\(982\) −50.9150 −1.62476
\(983\) 40.3619 1.28734 0.643672 0.765301i \(-0.277410\pi\)
0.643672 + 0.765301i \(0.277410\pi\)
\(984\) −4.27072 −0.136145
\(985\) −3.14958 −0.100354
\(986\) −6.55337 −0.208702
\(987\) 1.09856 0.0349675
\(988\) 128.215 4.07907
\(989\) 45.5982 1.44994
\(990\) 0 0
\(991\) −13.5653 −0.430916 −0.215458 0.976513i \(-0.569124\pi\)
−0.215458 + 0.976513i \(0.569124\pi\)
\(992\) 11.8464 0.376123
\(993\) −8.54031 −0.271019
\(994\) −30.6080 −0.970826
\(995\) −5.31856 −0.168610
\(996\) −11.4491 −0.362777
\(997\) −61.8983 −1.96034 −0.980170 0.198158i \(-0.936504\pi\)
−0.980170 + 0.198158i \(0.936504\pi\)
\(998\) −101.564 −3.21496
\(999\) 5.41947 0.171465
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 4235.2.a.y.1.5 5
11.10 odd 2 4235.2.a.be.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.y.1.5 5 1.1 even 1 trivial
4235.2.a.be.1.1 yes 5 11.10 odd 2