Properties

Label 4235.2.a.y.1.4
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 1.4
Root \(0.859039\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.164091 q^{2} -3.27813 q^{3} -1.97307 q^{4} -1.00000 q^{5} -0.537912 q^{6} +1.00000 q^{7} -0.651947 q^{8} +7.74611 q^{9} +O(q^{10})\) \(q+0.164091 q^{2} -3.27813 q^{3} -1.97307 q^{4} -1.00000 q^{5} -0.537912 q^{6} +1.00000 q^{7} -0.651947 q^{8} +7.74611 q^{9} -0.164091 q^{10} +6.46799 q^{12} -3.42008 q^{13} +0.164091 q^{14} +3.27813 q^{15} +3.83917 q^{16} -7.07103 q^{17} +1.27107 q^{18} -3.96406 q^{19} +1.97307 q^{20} -3.27813 q^{21} +1.19102 q^{23} +2.13717 q^{24} +1.00000 q^{25} -0.561206 q^{26} -15.5584 q^{27} -1.97307 q^{28} -1.37724 q^{29} +0.537912 q^{30} +6.97897 q^{31} +1.93387 q^{32} -1.16030 q^{34} -1.00000 q^{35} -15.2837 q^{36} +4.23612 q^{37} -0.650467 q^{38} +11.2115 q^{39} +0.651947 q^{40} +10.2573 q^{41} -0.537912 q^{42} -6.46699 q^{43} -7.74611 q^{45} +0.195436 q^{46} -0.387258 q^{47} -12.5853 q^{48} +1.00000 q^{49} +0.164091 q^{50} +23.1797 q^{51} +6.74807 q^{52} +9.52227 q^{53} -2.55299 q^{54} -0.651947 q^{56} +12.9947 q^{57} -0.225994 q^{58} -5.87905 q^{59} -6.46799 q^{60} +3.14306 q^{61} +1.14519 q^{62} +7.74611 q^{63} -7.36101 q^{64} +3.42008 q^{65} +14.1685 q^{67} +13.9517 q^{68} -3.90431 q^{69} -0.164091 q^{70} +0.339839 q^{71} -5.05006 q^{72} -11.7433 q^{73} +0.695111 q^{74} -3.27813 q^{75} +7.82138 q^{76} +1.83970 q^{78} +10.0204 q^{79} -3.83917 q^{80} +27.7639 q^{81} +1.68313 q^{82} +0.194278 q^{83} +6.46799 q^{84} +7.07103 q^{85} -1.06118 q^{86} +4.51478 q^{87} +12.6852 q^{89} -1.27107 q^{90} -3.42008 q^{91} -2.34997 q^{92} -22.8780 q^{93} -0.0635457 q^{94} +3.96406 q^{95} -6.33947 q^{96} -18.4750 q^{97} +0.164091 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} - 5q^{6} + 5q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 5q - 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} - 5q^{6} + 5q^{7} + 6q^{8} + 3q^{9} + 2q^{10} + 11q^{12} - 12q^{13} - 2q^{14} + 2q^{15} + 2q^{16} - 14q^{17} - 7q^{18} - 9q^{19} - 4q^{20} - 2q^{21} + 17q^{23} - 6q^{24} + 5q^{25} - 11q^{26} - 11q^{27} + 4q^{28} - 3q^{29} + 5q^{30} + 2q^{31} + 5q^{32} + 16q^{34} - 5q^{35} - 15q^{36} + 4q^{37} - 11q^{38} - 2q^{39} - 6q^{40} + 15q^{41} - 5q^{42} - 4q^{43} - 3q^{45} + 10q^{46} - 2q^{47} - 10q^{48} + 5q^{49} - 2q^{50} + 18q^{51} - 4q^{52} + 6q^{53} - 4q^{54} + 6q^{56} - 32q^{58} - 6q^{59} - 11q^{60} - 20q^{61} - 21q^{62} + 3q^{63} - 26q^{64} + 12q^{65} + 3q^{67} - 5q^{68} + 2q^{70} - 6q^{71} - 34q^{72} - 11q^{73} + 15q^{74} - 2q^{75} - 47q^{76} + 31q^{78} - 19q^{79} - 2q^{80} + 33q^{81} - 8q^{83} + 11q^{84} + 14q^{85} + 27q^{86} + 30q^{87} + q^{89} + 7q^{90} - 12q^{91} + 44q^{92} + 3q^{93} - 28q^{94} + 9q^{95} + 4q^{96} - 7q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.164091 0.116030 0.0580151 0.998316i \(-0.481523\pi\)
0.0580151 + 0.998316i \(0.481523\pi\)
\(3\) −3.27813 −1.89263 −0.946314 0.323250i \(-0.895224\pi\)
−0.946314 + 0.323250i \(0.895224\pi\)
\(4\) −1.97307 −0.986537
\(5\) −1.00000 −0.447214
\(6\) −0.537912 −0.219602
\(7\) 1.00000 0.377964
\(8\) −0.651947 −0.230498
\(9\) 7.74611 2.58204
\(10\) −0.164091 −0.0518903
\(11\) 0 0
\(12\) 6.46799 1.86715
\(13\) −3.42008 −0.948560 −0.474280 0.880374i \(-0.657292\pi\)
−0.474280 + 0.880374i \(0.657292\pi\)
\(14\) 0.164091 0.0438553
\(15\) 3.27813 0.846409
\(16\) 3.83917 0.959792
\(17\) −7.07103 −1.71498 −0.857489 0.514502i \(-0.827977\pi\)
−0.857489 + 0.514502i \(0.827977\pi\)
\(18\) 1.27107 0.299594
\(19\) −3.96406 −0.909417 −0.454708 0.890640i \(-0.650257\pi\)
−0.454708 + 0.890640i \(0.650257\pi\)
\(20\) 1.97307 0.441193
\(21\) −3.27813 −0.715346
\(22\) 0 0
\(23\) 1.19102 0.248344 0.124172 0.992261i \(-0.460372\pi\)
0.124172 + 0.992261i \(0.460372\pi\)
\(24\) 2.13717 0.436247
\(25\) 1.00000 0.200000
\(26\) −0.561206 −0.110062
\(27\) −15.5584 −2.99421
\(28\) −1.97307 −0.372876
\(29\) −1.37724 −0.255748 −0.127874 0.991790i \(-0.540815\pi\)
−0.127874 + 0.991790i \(0.540815\pi\)
\(30\) 0.537912 0.0982089
\(31\) 6.97897 1.25346 0.626730 0.779236i \(-0.284393\pi\)
0.626730 + 0.779236i \(0.284393\pi\)
\(32\) 1.93387 0.341863
\(33\) 0 0
\(34\) −1.16030 −0.198989
\(35\) −1.00000 −0.169031
\(36\) −15.2837 −2.54728
\(37\) 4.23612 0.696414 0.348207 0.937418i \(-0.386791\pi\)
0.348207 + 0.937418i \(0.386791\pi\)
\(38\) −0.650467 −0.105520
\(39\) 11.2115 1.79527
\(40\) 0.651947 0.103082
\(41\) 10.2573 1.60192 0.800958 0.598721i \(-0.204324\pi\)
0.800958 + 0.598721i \(0.204324\pi\)
\(42\) −0.537912 −0.0830017
\(43\) −6.46699 −0.986207 −0.493103 0.869971i \(-0.664138\pi\)
−0.493103 + 0.869971i \(0.664138\pi\)
\(44\) 0 0
\(45\) −7.74611 −1.15472
\(46\) 0.195436 0.0288154
\(47\) −0.387258 −0.0564873 −0.0282437 0.999601i \(-0.508991\pi\)
−0.0282437 + 0.999601i \(0.508991\pi\)
\(48\) −12.5853 −1.81653
\(49\) 1.00000 0.142857
\(50\) 0.164091 0.0232060
\(51\) 23.1797 3.24581
\(52\) 6.74807 0.935790
\(53\) 9.52227 1.30798 0.653992 0.756501i \(-0.273093\pi\)
0.653992 + 0.756501i \(0.273093\pi\)
\(54\) −2.55299 −0.347418
\(55\) 0 0
\(56\) −0.651947 −0.0871201
\(57\) 12.9947 1.72119
\(58\) −0.225994 −0.0296745
\(59\) −5.87905 −0.765387 −0.382694 0.923875i \(-0.625003\pi\)
−0.382694 + 0.923875i \(0.625003\pi\)
\(60\) −6.46799 −0.835013
\(61\) 3.14306 0.402428 0.201214 0.979547i \(-0.435511\pi\)
0.201214 + 0.979547i \(0.435511\pi\)
\(62\) 1.14519 0.145439
\(63\) 7.74611 0.975918
\(64\) −7.36101 −0.920126
\(65\) 3.42008 0.424209
\(66\) 0 0
\(67\) 14.1685 1.73095 0.865476 0.500950i \(-0.167016\pi\)
0.865476 + 0.500950i \(0.167016\pi\)
\(68\) 13.9517 1.69189
\(69\) −3.90431 −0.470023
\(70\) −0.164091 −0.0196127
\(71\) 0.339839 0.0403315 0.0201657 0.999797i \(-0.493581\pi\)
0.0201657 + 0.999797i \(0.493581\pi\)
\(72\) −5.05006 −0.595155
\(73\) −11.7433 −1.37445 −0.687225 0.726444i \(-0.741171\pi\)
−0.687225 + 0.726444i \(0.741171\pi\)
\(74\) 0.695111 0.0808050
\(75\) −3.27813 −0.378525
\(76\) 7.82138 0.897173
\(77\) 0 0
\(78\) 1.83970 0.208305
\(79\) 10.0204 1.12739 0.563694 0.825984i \(-0.309380\pi\)
0.563694 + 0.825984i \(0.309380\pi\)
\(80\) −3.83917 −0.429232
\(81\) 27.7639 3.08488
\(82\) 1.68313 0.185870
\(83\) 0.194278 0.0213248 0.0106624 0.999943i \(-0.496606\pi\)
0.0106624 + 0.999943i \(0.496606\pi\)
\(84\) 6.46799 0.705715
\(85\) 7.07103 0.766961
\(86\) −1.06118 −0.114430
\(87\) 4.51478 0.484035
\(88\) 0 0
\(89\) 12.6852 1.34463 0.672314 0.740266i \(-0.265300\pi\)
0.672314 + 0.740266i \(0.265300\pi\)
\(90\) −1.27107 −0.133983
\(91\) −3.42008 −0.358522
\(92\) −2.34997 −0.245001
\(93\) −22.8780 −2.37233
\(94\) −0.0635457 −0.00655423
\(95\) 3.96406 0.406704
\(96\) −6.33947 −0.647019
\(97\) −18.4750 −1.87585 −0.937926 0.346836i \(-0.887256\pi\)
−0.937926 + 0.346836i \(0.887256\pi\)
\(98\) 0.164091 0.0165757
\(99\) 0 0
\(100\) −1.97307 −0.197307
\(101\) 1.83395 0.182484 0.0912422 0.995829i \(-0.470916\pi\)
0.0912422 + 0.995829i \(0.470916\pi\)
\(102\) 3.80360 0.376612
\(103\) 15.2837 1.50594 0.752972 0.658053i \(-0.228620\pi\)
0.752972 + 0.658053i \(0.228620\pi\)
\(104\) 2.22971 0.218641
\(105\) 3.27813 0.319912
\(106\) 1.56252 0.151766
\(107\) 1.17954 0.114031 0.0570154 0.998373i \(-0.481842\pi\)
0.0570154 + 0.998373i \(0.481842\pi\)
\(108\) 30.6978 2.95390
\(109\) −3.75666 −0.359823 −0.179911 0.983683i \(-0.557581\pi\)
−0.179911 + 0.983683i \(0.557581\pi\)
\(110\) 0 0
\(111\) −13.8865 −1.31805
\(112\) 3.83917 0.362767
\(113\) 10.3037 0.969289 0.484645 0.874711i \(-0.338949\pi\)
0.484645 + 0.874711i \(0.338949\pi\)
\(114\) 2.13231 0.199710
\(115\) −1.19102 −0.111063
\(116\) 2.71741 0.252305
\(117\) −26.4923 −2.44922
\(118\) −0.964701 −0.0888080
\(119\) −7.07103 −0.648201
\(120\) −2.13717 −0.195096
\(121\) 0 0
\(122\) 0.515750 0.0466938
\(123\) −33.6246 −3.03183
\(124\) −13.7700 −1.23659
\(125\) −1.00000 −0.0894427
\(126\) 1.27107 0.113236
\(127\) 7.15239 0.634672 0.317336 0.948313i \(-0.397212\pi\)
0.317336 + 0.948313i \(0.397212\pi\)
\(128\) −5.07562 −0.448625
\(129\) 21.1996 1.86652
\(130\) 0.561206 0.0492210
\(131\) 19.4758 1.70161 0.850803 0.525484i \(-0.176116\pi\)
0.850803 + 0.525484i \(0.176116\pi\)
\(132\) 0 0
\(133\) −3.96406 −0.343727
\(134\) 2.32492 0.200843
\(135\) 15.5584 1.33905
\(136\) 4.60994 0.395299
\(137\) −9.59810 −0.820021 −0.410010 0.912081i \(-0.634475\pi\)
−0.410010 + 0.912081i \(0.634475\pi\)
\(138\) −0.640663 −0.0545368
\(139\) −14.3907 −1.22060 −0.610300 0.792170i \(-0.708951\pi\)
−0.610300 + 0.792170i \(0.708951\pi\)
\(140\) 1.97307 0.166755
\(141\) 1.26948 0.106909
\(142\) 0.0557646 0.00467966
\(143\) 0 0
\(144\) 29.7386 2.47822
\(145\) 1.37724 0.114374
\(146\) −1.92698 −0.159478
\(147\) −3.27813 −0.270375
\(148\) −8.35818 −0.687038
\(149\) −21.4554 −1.75770 −0.878849 0.477100i \(-0.841688\pi\)
−0.878849 + 0.477100i \(0.841688\pi\)
\(150\) −0.537912 −0.0439204
\(151\) 0.0822054 0.00668978 0.00334489 0.999994i \(-0.498935\pi\)
0.00334489 + 0.999994i \(0.498935\pi\)
\(152\) 2.58436 0.209619
\(153\) −54.7730 −4.42814
\(154\) 0 0
\(155\) −6.97897 −0.560565
\(156\) −22.1210 −1.77110
\(157\) 20.0790 1.60248 0.801239 0.598345i \(-0.204175\pi\)
0.801239 + 0.598345i \(0.204175\pi\)
\(158\) 1.64427 0.130811
\(159\) −31.2152 −2.47553
\(160\) −1.93387 −0.152886
\(161\) 1.19102 0.0938653
\(162\) 4.55582 0.357939
\(163\) −17.7257 −1.38838 −0.694191 0.719791i \(-0.744238\pi\)
−0.694191 + 0.719791i \(0.744238\pi\)
\(164\) −20.2383 −1.58035
\(165\) 0 0
\(166\) 0.0318793 0.00247432
\(167\) 22.9221 1.77376 0.886881 0.461998i \(-0.152867\pi\)
0.886881 + 0.461998i \(0.152867\pi\)
\(168\) 2.13717 0.164886
\(169\) −1.30304 −0.100234
\(170\) 1.16030 0.0889906
\(171\) −30.7060 −2.34815
\(172\) 12.7599 0.972930
\(173\) −19.6082 −1.49078 −0.745392 0.666626i \(-0.767738\pi\)
−0.745392 + 0.666626i \(0.767738\pi\)
\(174\) 0.740837 0.0561627
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 19.2723 1.44859
\(178\) 2.08153 0.156017
\(179\) 7.83308 0.585472 0.292736 0.956193i \(-0.405434\pi\)
0.292736 + 0.956193i \(0.405434\pi\)
\(180\) 15.2837 1.13918
\(181\) 0.984920 0.0732085 0.0366043 0.999330i \(-0.488346\pi\)
0.0366043 + 0.999330i \(0.488346\pi\)
\(182\) −0.561206 −0.0415994
\(183\) −10.3034 −0.761646
\(184\) −0.776480 −0.0572429
\(185\) −4.23612 −0.311446
\(186\) −3.75408 −0.275262
\(187\) 0 0
\(188\) 0.764088 0.0557269
\(189\) −15.5584 −1.13170
\(190\) 0.650467 0.0471899
\(191\) −0.106330 −0.00769380 −0.00384690 0.999993i \(-0.501225\pi\)
−0.00384690 + 0.999993i \(0.501225\pi\)
\(192\) 24.1303 1.74146
\(193\) −3.99558 −0.287608 −0.143804 0.989606i \(-0.545934\pi\)
−0.143804 + 0.989606i \(0.545934\pi\)
\(194\) −3.03159 −0.217655
\(195\) −11.2115 −0.802869
\(196\) −1.97307 −0.140934
\(197\) −5.82520 −0.415028 −0.207514 0.978232i \(-0.566537\pi\)
−0.207514 + 0.978232i \(0.566537\pi\)
\(198\) 0 0
\(199\) −7.39859 −0.524472 −0.262236 0.965004i \(-0.584460\pi\)
−0.262236 + 0.965004i \(0.584460\pi\)
\(200\) −0.651947 −0.0460996
\(201\) −46.4460 −3.27605
\(202\) 0.300935 0.0211737
\(203\) −1.37724 −0.0966636
\(204\) −45.7354 −3.20212
\(205\) −10.2573 −0.716398
\(206\) 2.50792 0.174735
\(207\) 9.22575 0.641234
\(208\) −13.1303 −0.910421
\(209\) 0 0
\(210\) 0.537912 0.0371195
\(211\) −12.9364 −0.890581 −0.445290 0.895386i \(-0.646900\pi\)
−0.445290 + 0.895386i \(0.646900\pi\)
\(212\) −18.7881 −1.29038
\(213\) −1.11403 −0.0763324
\(214\) 0.193553 0.0132310
\(215\) 6.46699 0.441045
\(216\) 10.1432 0.690159
\(217\) 6.97897 0.473764
\(218\) −0.616436 −0.0417503
\(219\) 38.4961 2.60132
\(220\) 0 0
\(221\) 24.1835 1.62676
\(222\) −2.27866 −0.152934
\(223\) −19.6080 −1.31305 −0.656523 0.754306i \(-0.727973\pi\)
−0.656523 + 0.754306i \(0.727973\pi\)
\(224\) 1.93387 0.129212
\(225\) 7.74611 0.516407
\(226\) 1.69075 0.112467
\(227\) 6.20128 0.411594 0.205797 0.978595i \(-0.434021\pi\)
0.205797 + 0.978595i \(0.434021\pi\)
\(228\) −25.6395 −1.69801
\(229\) −15.5651 −1.02857 −0.514286 0.857619i \(-0.671943\pi\)
−0.514286 + 0.857619i \(0.671943\pi\)
\(230\) −0.195436 −0.0128866
\(231\) 0 0
\(232\) 0.897891 0.0589494
\(233\) −26.0819 −1.70869 −0.854343 0.519710i \(-0.826040\pi\)
−0.854343 + 0.519710i \(0.826040\pi\)
\(234\) −4.34716 −0.284183
\(235\) 0.387258 0.0252619
\(236\) 11.5998 0.755083
\(237\) −32.8483 −2.13372
\(238\) −1.16030 −0.0752108
\(239\) −1.21063 −0.0783091 −0.0391545 0.999233i \(-0.512466\pi\)
−0.0391545 + 0.999233i \(0.512466\pi\)
\(240\) 12.5853 0.812376
\(241\) 18.7671 1.20889 0.604447 0.796645i \(-0.293394\pi\)
0.604447 + 0.796645i \(0.293394\pi\)
\(242\) 0 0
\(243\) −44.3386 −2.84432
\(244\) −6.20150 −0.397010
\(245\) −1.00000 −0.0638877
\(246\) −5.51751 −0.351783
\(247\) 13.5574 0.862636
\(248\) −4.54992 −0.288920
\(249\) −0.636867 −0.0403598
\(250\) −0.164091 −0.0103781
\(251\) −21.8153 −1.37697 −0.688485 0.725250i \(-0.741724\pi\)
−0.688485 + 0.725250i \(0.741724\pi\)
\(252\) −15.2837 −0.962780
\(253\) 0 0
\(254\) 1.17364 0.0736410
\(255\) −23.1797 −1.45157
\(256\) 13.8891 0.868072
\(257\) −11.9068 −0.742725 −0.371362 0.928488i \(-0.621109\pi\)
−0.371362 + 0.928488i \(0.621109\pi\)
\(258\) 3.47867 0.216573
\(259\) 4.23612 0.263220
\(260\) −6.74807 −0.418498
\(261\) −10.6683 −0.660351
\(262\) 3.19581 0.197438
\(263\) 10.2101 0.629583 0.314792 0.949161i \(-0.398065\pi\)
0.314792 + 0.949161i \(0.398065\pi\)
\(264\) 0 0
\(265\) −9.52227 −0.584948
\(266\) −0.650467 −0.0398827
\(267\) −41.5837 −2.54488
\(268\) −27.9554 −1.70765
\(269\) 10.4517 0.637253 0.318626 0.947880i \(-0.396778\pi\)
0.318626 + 0.947880i \(0.396778\pi\)
\(270\) 2.55299 0.155370
\(271\) −24.2173 −1.47110 −0.735548 0.677472i \(-0.763075\pi\)
−0.735548 + 0.677472i \(0.763075\pi\)
\(272\) −27.1469 −1.64602
\(273\) 11.2115 0.678548
\(274\) −1.57496 −0.0951471
\(275\) 0 0
\(276\) 7.70348 0.463695
\(277\) −7.96360 −0.478486 −0.239243 0.970960i \(-0.576899\pi\)
−0.239243 + 0.970960i \(0.576899\pi\)
\(278\) −2.36138 −0.141626
\(279\) 54.0599 3.23648
\(280\) 0.651947 0.0389613
\(281\) 1.17820 0.0702857 0.0351429 0.999382i \(-0.488811\pi\)
0.0351429 + 0.999382i \(0.488811\pi\)
\(282\) 0.208311 0.0124047
\(283\) 11.5877 0.688818 0.344409 0.938820i \(-0.388079\pi\)
0.344409 + 0.938820i \(0.388079\pi\)
\(284\) −0.670527 −0.0397885
\(285\) −12.9947 −0.769738
\(286\) 0 0
\(287\) 10.2573 0.605467
\(288\) 14.9800 0.882703
\(289\) 32.9995 1.94115
\(290\) 0.225994 0.0132708
\(291\) 60.5634 3.55029
\(292\) 23.1704 1.35595
\(293\) 2.54149 0.148476 0.0742378 0.997241i \(-0.476348\pi\)
0.0742378 + 0.997241i \(0.476348\pi\)
\(294\) −0.537912 −0.0313717
\(295\) 5.87905 0.342291
\(296\) −2.76173 −0.160522
\(297\) 0 0
\(298\) −3.52065 −0.203946
\(299\) −4.07338 −0.235569
\(300\) 6.46799 0.373429
\(301\) −6.46699 −0.372751
\(302\) 0.0134892 0.000776216 0
\(303\) −6.01191 −0.345375
\(304\) −15.2187 −0.872851
\(305\) −3.14306 −0.179971
\(306\) −8.98778 −0.513797
\(307\) 2.89989 0.165505 0.0827526 0.996570i \(-0.473629\pi\)
0.0827526 + 0.996570i \(0.473629\pi\)
\(308\) 0 0
\(309\) −50.1017 −2.85019
\(310\) −1.14519 −0.0650424
\(311\) −17.3344 −0.982942 −0.491471 0.870894i \(-0.663541\pi\)
−0.491471 + 0.870894i \(0.663541\pi\)
\(312\) −7.30928 −0.413807
\(313\) −0.201307 −0.0113785 −0.00568927 0.999984i \(-0.501811\pi\)
−0.00568927 + 0.999984i \(0.501811\pi\)
\(314\) 3.29479 0.185936
\(315\) −7.74611 −0.436444
\(316\) −19.7711 −1.11221
\(317\) 0.108622 0.00610082 0.00305041 0.999995i \(-0.499029\pi\)
0.00305041 + 0.999995i \(0.499029\pi\)
\(318\) −5.12215 −0.287236
\(319\) 0 0
\(320\) 7.36101 0.411493
\(321\) −3.86669 −0.215818
\(322\) 0.195436 0.0108912
\(323\) 28.0300 1.55963
\(324\) −54.7803 −3.04335
\(325\) −3.42008 −0.189712
\(326\) −2.90863 −0.161094
\(327\) 12.3148 0.681010
\(328\) −6.68719 −0.369238
\(329\) −0.387258 −0.0213502
\(330\) 0 0
\(331\) 2.63909 0.145057 0.0725287 0.997366i \(-0.476893\pi\)
0.0725287 + 0.997366i \(0.476893\pi\)
\(332\) −0.383325 −0.0210377
\(333\) 32.8135 1.79817
\(334\) 3.76131 0.205810
\(335\) −14.1685 −0.774106
\(336\) −12.5853 −0.686583
\(337\) −12.5232 −0.682184 −0.341092 0.940030i \(-0.610797\pi\)
−0.341092 + 0.940030i \(0.610797\pi\)
\(338\) −0.213818 −0.0116301
\(339\) −33.7768 −1.83450
\(340\) −13.9517 −0.756636
\(341\) 0 0
\(342\) −5.03859 −0.272456
\(343\) 1.00000 0.0539949
\(344\) 4.21614 0.227319
\(345\) 3.90431 0.210201
\(346\) −3.21754 −0.172976
\(347\) −27.1671 −1.45841 −0.729203 0.684297i \(-0.760109\pi\)
−0.729203 + 0.684297i \(0.760109\pi\)
\(348\) −8.90800 −0.477519
\(349\) 14.0130 0.750100 0.375050 0.927005i \(-0.377626\pi\)
0.375050 + 0.927005i \(0.377626\pi\)
\(350\) 0.164091 0.00877105
\(351\) 53.2108 2.84018
\(352\) 0 0
\(353\) 15.7525 0.838423 0.419212 0.907889i \(-0.362307\pi\)
0.419212 + 0.907889i \(0.362307\pi\)
\(354\) 3.16241 0.168080
\(355\) −0.339839 −0.0180368
\(356\) −25.0288 −1.32653
\(357\) 23.1797 1.22680
\(358\) 1.28534 0.0679324
\(359\) 27.9672 1.47605 0.738027 0.674772i \(-0.235758\pi\)
0.738027 + 0.674772i \(0.235758\pi\)
\(360\) 5.05006 0.266161
\(361\) −3.28626 −0.172961
\(362\) 0.161617 0.00849440
\(363\) 0 0
\(364\) 6.74807 0.353695
\(365\) 11.7433 0.614673
\(366\) −1.69069 −0.0883739
\(367\) 3.15007 0.164433 0.0822163 0.996615i \(-0.473800\pi\)
0.0822163 + 0.996615i \(0.473800\pi\)
\(368\) 4.57252 0.238359
\(369\) 79.4539 4.13620
\(370\) −0.695111 −0.0361371
\(371\) 9.52227 0.494372
\(372\) 45.1399 2.34039
\(373\) −25.6739 −1.32935 −0.664673 0.747135i \(-0.731429\pi\)
−0.664673 + 0.747135i \(0.731429\pi\)
\(374\) 0 0
\(375\) 3.27813 0.169282
\(376\) 0.252472 0.0130202
\(377\) 4.71029 0.242592
\(378\) −2.55299 −0.131312
\(379\) −4.43236 −0.227675 −0.113838 0.993499i \(-0.536314\pi\)
−0.113838 + 0.993499i \(0.536314\pi\)
\(380\) −7.82138 −0.401228
\(381\) −23.4464 −1.20120
\(382\) −0.0174479 −0.000892713 0
\(383\) 7.49597 0.383026 0.191513 0.981490i \(-0.438661\pi\)
0.191513 + 0.981490i \(0.438661\pi\)
\(384\) 16.6385 0.849080
\(385\) 0 0
\(386\) −0.655641 −0.0333712
\(387\) −50.0940 −2.54642
\(388\) 36.4525 1.85060
\(389\) −22.4902 −1.14030 −0.570148 0.821542i \(-0.693114\pi\)
−0.570148 + 0.821542i \(0.693114\pi\)
\(390\) −1.83970 −0.0931570
\(391\) −8.42172 −0.425905
\(392\) −0.651947 −0.0329283
\(393\) −63.8441 −3.22051
\(394\) −0.955865 −0.0481558
\(395\) −10.0204 −0.504183
\(396\) 0 0
\(397\) −4.29363 −0.215491 −0.107746 0.994179i \(-0.534363\pi\)
−0.107746 + 0.994179i \(0.534363\pi\)
\(398\) −1.21405 −0.0608546
\(399\) 12.9947 0.650547
\(400\) 3.83917 0.191958
\(401\) 25.1851 1.25768 0.628842 0.777533i \(-0.283529\pi\)
0.628842 + 0.777533i \(0.283529\pi\)
\(402\) −7.62139 −0.380120
\(403\) −23.8687 −1.18898
\(404\) −3.61851 −0.180028
\(405\) −27.7639 −1.37960
\(406\) −0.225994 −0.0112159
\(407\) 0 0
\(408\) −15.1120 −0.748154
\(409\) −2.35178 −0.116288 −0.0581440 0.998308i \(-0.518518\pi\)
−0.0581440 + 0.998308i \(0.518518\pi\)
\(410\) −1.68313 −0.0831238
\(411\) 31.4638 1.55199
\(412\) −30.1558 −1.48567
\(413\) −5.87905 −0.289289
\(414\) 1.51387 0.0744025
\(415\) −0.194278 −0.00953673
\(416\) −6.61399 −0.324278
\(417\) 47.1744 2.31014
\(418\) 0 0
\(419\) 9.08678 0.443918 0.221959 0.975056i \(-0.428755\pi\)
0.221959 + 0.975056i \(0.428755\pi\)
\(420\) −6.46799 −0.315605
\(421\) 20.7124 1.00946 0.504731 0.863277i \(-0.331592\pi\)
0.504731 + 0.863277i \(0.331592\pi\)
\(422\) −2.12276 −0.103334
\(423\) −2.99974 −0.145852
\(424\) −6.20802 −0.301488
\(425\) −7.07103 −0.342996
\(426\) −0.182804 −0.00885686
\(427\) 3.14306 0.152104
\(428\) −2.32733 −0.112496
\(429\) 0 0
\(430\) 1.06118 0.0511745
\(431\) −6.97733 −0.336086 −0.168043 0.985780i \(-0.553745\pi\)
−0.168043 + 0.985780i \(0.553745\pi\)
\(432\) −59.7312 −2.87382
\(433\) 5.68801 0.273348 0.136674 0.990616i \(-0.456359\pi\)
0.136674 + 0.990616i \(0.456359\pi\)
\(434\) 1.14519 0.0549708
\(435\) −4.51478 −0.216467
\(436\) 7.41217 0.354979
\(437\) −4.72126 −0.225848
\(438\) 6.31687 0.301832
\(439\) 38.3629 1.83096 0.915482 0.402360i \(-0.131810\pi\)
0.915482 + 0.402360i \(0.131810\pi\)
\(440\) 0 0
\(441\) 7.74611 0.368862
\(442\) 3.96831 0.188753
\(443\) 14.6210 0.694667 0.347333 0.937742i \(-0.387087\pi\)
0.347333 + 0.937742i \(0.387087\pi\)
\(444\) 27.3992 1.30031
\(445\) −12.6852 −0.601336
\(446\) −3.21750 −0.152353
\(447\) 70.3336 3.32667
\(448\) −7.36101 −0.347775
\(449\) −30.4767 −1.43829 −0.719143 0.694862i \(-0.755466\pi\)
−0.719143 + 0.694862i \(0.755466\pi\)
\(450\) 1.27107 0.0599188
\(451\) 0 0
\(452\) −20.3299 −0.956240
\(453\) −0.269480 −0.0126613
\(454\) 1.01758 0.0477573
\(455\) 3.42008 0.160336
\(456\) −8.47184 −0.396730
\(457\) 1.90600 0.0891590 0.0445795 0.999006i \(-0.485805\pi\)
0.0445795 + 0.999006i \(0.485805\pi\)
\(458\) −2.55410 −0.119345
\(459\) 110.014 5.13500
\(460\) 2.34997 0.109568
\(461\) 18.6043 0.866490 0.433245 0.901276i \(-0.357368\pi\)
0.433245 + 0.901276i \(0.357368\pi\)
\(462\) 0 0
\(463\) 6.73166 0.312847 0.156423 0.987690i \(-0.450004\pi\)
0.156423 + 0.987690i \(0.450004\pi\)
\(464\) −5.28747 −0.245465
\(465\) 22.8780 1.06094
\(466\) −4.27982 −0.198259
\(467\) 10.7816 0.498915 0.249457 0.968386i \(-0.419748\pi\)
0.249457 + 0.968386i \(0.419748\pi\)
\(468\) 52.2713 2.41624
\(469\) 14.1685 0.654239
\(470\) 0.0635457 0.00293114
\(471\) −65.8214 −3.03289
\(472\) 3.83283 0.176420
\(473\) 0 0
\(474\) −5.39012 −0.247576
\(475\) −3.96406 −0.181883
\(476\) 13.9517 0.639474
\(477\) 73.7606 3.37726
\(478\) −0.198654 −0.00908621
\(479\) 22.6770 1.03614 0.518069 0.855339i \(-0.326651\pi\)
0.518069 + 0.855339i \(0.326651\pi\)
\(480\) 6.33947 0.289356
\(481\) −14.4879 −0.660590
\(482\) 3.07952 0.140268
\(483\) −3.90431 −0.177652
\(484\) 0 0
\(485\) 18.4750 0.838906
\(486\) −7.27558 −0.330027
\(487\) −21.6859 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(488\) −2.04911 −0.0927589
\(489\) 58.1070 2.62769
\(490\) −0.164091 −0.00741289
\(491\) −2.54893 −0.115031 −0.0575157 0.998345i \(-0.518318\pi\)
−0.0575157 + 0.998345i \(0.518318\pi\)
\(492\) 66.3438 2.99101
\(493\) 9.73854 0.438602
\(494\) 2.22465 0.100092
\(495\) 0 0
\(496\) 26.7935 1.20306
\(497\) 0.339839 0.0152439
\(498\) −0.104504 −0.00468296
\(499\) 24.1867 1.08274 0.541372 0.840783i \(-0.317905\pi\)
0.541372 + 0.840783i \(0.317905\pi\)
\(500\) 1.97307 0.0882386
\(501\) −75.1414 −3.35707
\(502\) −3.57971 −0.159770
\(503\) 39.3976 1.75665 0.878327 0.478060i \(-0.158660\pi\)
0.878327 + 0.478060i \(0.158660\pi\)
\(504\) −5.05006 −0.224947
\(505\) −1.83395 −0.0816095
\(506\) 0 0
\(507\) 4.27153 0.189705
\(508\) −14.1122 −0.626127
\(509\) −16.1856 −0.717415 −0.358707 0.933450i \(-0.616782\pi\)
−0.358707 + 0.933450i \(0.616782\pi\)
\(510\) −3.80360 −0.168426
\(511\) −11.7433 −0.519494
\(512\) 12.4303 0.549348
\(513\) 61.6742 2.72298
\(514\) −1.95380 −0.0861784
\(515\) −15.2837 −0.673478
\(516\) −41.8284 −1.84139
\(517\) 0 0
\(518\) 0.695111 0.0305414
\(519\) 64.2782 2.82150
\(520\) −2.22971 −0.0977794
\(521\) −44.2132 −1.93701 −0.968507 0.248986i \(-0.919903\pi\)
−0.968507 + 0.248986i \(0.919903\pi\)
\(522\) −1.75057 −0.0766206
\(523\) −26.7584 −1.17006 −0.585031 0.811011i \(-0.698918\pi\)
−0.585031 + 0.811011i \(0.698918\pi\)
\(524\) −38.4272 −1.67870
\(525\) −3.27813 −0.143069
\(526\) 1.67539 0.0730506
\(527\) −49.3486 −2.14966
\(528\) 0 0
\(529\) −21.5815 −0.938325
\(530\) −1.56252 −0.0678716
\(531\) −45.5398 −1.97626
\(532\) 7.82138 0.339100
\(533\) −35.0807 −1.51951
\(534\) −6.82353 −0.295283
\(535\) −1.17954 −0.0509961
\(536\) −9.23709 −0.398981
\(537\) −25.6778 −1.10808
\(538\) 1.71504 0.0739405
\(539\) 0 0
\(540\) −30.6978 −1.32102
\(541\) 34.7014 1.49193 0.745964 0.665986i \(-0.231989\pi\)
0.745964 + 0.665986i \(0.231989\pi\)
\(542\) −3.97385 −0.170692
\(543\) −3.22869 −0.138556
\(544\) −13.6745 −0.586287
\(545\) 3.75666 0.160918
\(546\) 1.83970 0.0787321
\(547\) −28.9159 −1.23635 −0.618176 0.786039i \(-0.712128\pi\)
−0.618176 + 0.786039i \(0.712128\pi\)
\(548\) 18.9378 0.808981
\(549\) 24.3465 1.03908
\(550\) 0 0
\(551\) 5.45947 0.232581
\(552\) 2.54540 0.108339
\(553\) 10.0204 0.426113
\(554\) −1.30676 −0.0555188
\(555\) 13.8865 0.589451
\(556\) 28.3938 1.20417
\(557\) −11.3349 −0.480275 −0.240138 0.970739i \(-0.577193\pi\)
−0.240138 + 0.970739i \(0.577193\pi\)
\(558\) 8.87076 0.375529
\(559\) 22.1176 0.935476
\(560\) −3.83917 −0.162235
\(561\) 0 0
\(562\) 0.193333 0.00815526
\(563\) −28.6998 −1.20955 −0.604776 0.796396i \(-0.706737\pi\)
−0.604776 + 0.796396i \(0.706737\pi\)
\(564\) −2.50478 −0.105470
\(565\) −10.3037 −0.433479
\(566\) 1.90144 0.0799236
\(567\) 27.7639 1.16597
\(568\) −0.221557 −0.00929633
\(569\) −7.31957 −0.306853 −0.153426 0.988160i \(-0.549031\pi\)
−0.153426 + 0.988160i \(0.549031\pi\)
\(570\) −2.13231 −0.0893128
\(571\) −32.3161 −1.35239 −0.676193 0.736724i \(-0.736372\pi\)
−0.676193 + 0.736724i \(0.736372\pi\)
\(572\) 0 0
\(573\) 0.348565 0.0145615
\(574\) 1.68313 0.0702524
\(575\) 1.19102 0.0496689
\(576\) −57.0192 −2.37580
\(577\) −28.4485 −1.18433 −0.592163 0.805818i \(-0.701726\pi\)
−0.592163 + 0.805818i \(0.701726\pi\)
\(578\) 5.41494 0.225232
\(579\) 13.0980 0.544335
\(580\) −2.71741 −0.112834
\(581\) 0.194278 0.00806000
\(582\) 9.93793 0.411940
\(583\) 0 0
\(584\) 7.65602 0.316808
\(585\) 26.4923 1.09532
\(586\) 0.417037 0.0172276
\(587\) −25.0706 −1.03477 −0.517387 0.855752i \(-0.673095\pi\)
−0.517387 + 0.855752i \(0.673095\pi\)
\(588\) 6.46799 0.266735
\(589\) −27.6650 −1.13992
\(590\) 0.964701 0.0397161
\(591\) 19.0957 0.785493
\(592\) 16.2632 0.668413
\(593\) −39.6026 −1.62629 −0.813143 0.582064i \(-0.802245\pi\)
−0.813143 + 0.582064i \(0.802245\pi\)
\(594\) 0 0
\(595\) 7.07103 0.289884
\(596\) 42.3332 1.73403
\(597\) 24.2535 0.992631
\(598\) −0.668406 −0.0273332
\(599\) 5.99181 0.244819 0.122409 0.992480i \(-0.460938\pi\)
0.122409 + 0.992480i \(0.460938\pi\)
\(600\) 2.13717 0.0872494
\(601\) −45.3928 −1.85161 −0.925806 0.378000i \(-0.876612\pi\)
−0.925806 + 0.378000i \(0.876612\pi\)
\(602\) −1.06118 −0.0432504
\(603\) 109.750 4.46939
\(604\) −0.162197 −0.00659972
\(605\) 0 0
\(606\) −0.986502 −0.0400739
\(607\) 22.3454 0.906973 0.453487 0.891263i \(-0.350180\pi\)
0.453487 + 0.891263i \(0.350180\pi\)
\(608\) −7.66596 −0.310896
\(609\) 4.51478 0.182948
\(610\) −0.515750 −0.0208821
\(611\) 1.32445 0.0535816
\(612\) 108.071 4.36852
\(613\) 27.4246 1.10767 0.553835 0.832627i \(-0.313164\pi\)
0.553835 + 0.832627i \(0.313164\pi\)
\(614\) 0.475847 0.0192036
\(615\) 33.6246 1.35587
\(616\) 0 0
\(617\) 1.53475 0.0617866 0.0308933 0.999523i \(-0.490165\pi\)
0.0308933 + 0.999523i \(0.490165\pi\)
\(618\) −8.22126 −0.330708
\(619\) −8.61965 −0.346453 −0.173226 0.984882i \(-0.555419\pi\)
−0.173226 + 0.984882i \(0.555419\pi\)
\(620\) 13.7700 0.553018
\(621\) −18.5303 −0.743594
\(622\) −2.84442 −0.114051
\(623\) 12.6852 0.508222
\(624\) 43.0427 1.72309
\(625\) 1.00000 0.0400000
\(626\) −0.0330328 −0.00132025
\(627\) 0 0
\(628\) −39.6173 −1.58090
\(629\) −29.9538 −1.19433
\(630\) −1.27107 −0.0506407
\(631\) 11.5656 0.460420 0.230210 0.973141i \(-0.426059\pi\)
0.230210 + 0.973141i \(0.426059\pi\)
\(632\) −6.53280 −0.259861
\(633\) 42.4073 1.68554
\(634\) 0.0178239 0.000707879 0
\(635\) −7.15239 −0.283834
\(636\) 61.5899 2.44220
\(637\) −3.42008 −0.135509
\(638\) 0 0
\(639\) 2.63243 0.104137
\(640\) 5.07562 0.200631
\(641\) 12.2408 0.483484 0.241742 0.970341i \(-0.422281\pi\)
0.241742 + 0.970341i \(0.422281\pi\)
\(642\) −0.634491 −0.0250414
\(643\) −23.5116 −0.927206 −0.463603 0.886043i \(-0.653444\pi\)
−0.463603 + 0.886043i \(0.653444\pi\)
\(644\) −2.34997 −0.0926016
\(645\) −21.1996 −0.834734
\(646\) 4.59948 0.180964
\(647\) 27.9447 1.09862 0.549310 0.835619i \(-0.314891\pi\)
0.549310 + 0.835619i \(0.314891\pi\)
\(648\) −18.1006 −0.711059
\(649\) 0 0
\(650\) −0.561206 −0.0220123
\(651\) −22.8780 −0.896658
\(652\) 34.9741 1.36969
\(653\) 2.10740 0.0824688 0.0412344 0.999150i \(-0.486871\pi\)
0.0412344 + 0.999150i \(0.486871\pi\)
\(654\) 2.02075 0.0790177
\(655\) −19.4758 −0.760982
\(656\) 39.3794 1.53751
\(657\) −90.9650 −3.54888
\(658\) −0.0635457 −0.00247727
\(659\) 18.8837 0.735606 0.367803 0.929904i \(-0.380110\pi\)
0.367803 + 0.929904i \(0.380110\pi\)
\(660\) 0 0
\(661\) 27.9541 1.08729 0.543644 0.839316i \(-0.317044\pi\)
0.543644 + 0.839316i \(0.317044\pi\)
\(662\) 0.433052 0.0168310
\(663\) −79.2766 −3.07885
\(664\) −0.126659 −0.00491532
\(665\) 3.96406 0.153719
\(666\) 5.38441 0.208642
\(667\) −1.64032 −0.0635135
\(668\) −45.2269 −1.74988
\(669\) 64.2773 2.48511
\(670\) −2.32492 −0.0898196
\(671\) 0 0
\(672\) −6.33947 −0.244550
\(673\) −4.43986 −0.171144 −0.0855720 0.996332i \(-0.527272\pi\)
−0.0855720 + 0.996332i \(0.527272\pi\)
\(674\) −2.05496 −0.0791539
\(675\) −15.5584 −0.598841
\(676\) 2.57099 0.0988844
\(677\) 44.2511 1.70071 0.850354 0.526210i \(-0.176388\pi\)
0.850354 + 0.526210i \(0.176388\pi\)
\(678\) −5.54248 −0.212858
\(679\) −18.4750 −0.709005
\(680\) −4.60994 −0.176783
\(681\) −20.3286 −0.778993
\(682\) 0 0
\(683\) −0.738650 −0.0282637 −0.0141318 0.999900i \(-0.504498\pi\)
−0.0141318 + 0.999900i \(0.504498\pi\)
\(684\) 60.5852 2.31653
\(685\) 9.59810 0.366724
\(686\) 0.164091 0.00626504
\(687\) 51.0244 1.94670
\(688\) −24.8279 −0.946554
\(689\) −32.5669 −1.24070
\(690\) 0.640663 0.0243896
\(691\) −27.6402 −1.05148 −0.525742 0.850644i \(-0.676212\pi\)
−0.525742 + 0.850644i \(0.676212\pi\)
\(692\) 38.6884 1.47071
\(693\) 0 0
\(694\) −4.45789 −0.169219
\(695\) 14.3907 0.545869
\(696\) −2.94340 −0.111569
\(697\) −72.5295 −2.74725
\(698\) 2.29942 0.0870341
\(699\) 85.4999 3.23390
\(700\) −1.97307 −0.0745752
\(701\) −21.9762 −0.830030 −0.415015 0.909814i \(-0.636224\pi\)
−0.415015 + 0.909814i \(0.636224\pi\)
\(702\) 8.73144 0.329547
\(703\) −16.7922 −0.633330
\(704\) 0 0
\(705\) −1.26948 −0.0478114
\(706\) 2.58486 0.0972824
\(707\) 1.83395 0.0689726
\(708\) −38.0256 −1.42909
\(709\) −14.9149 −0.560141 −0.280071 0.959979i \(-0.590358\pi\)
−0.280071 + 0.959979i \(0.590358\pi\)
\(710\) −0.0557646 −0.00209281
\(711\) 77.6195 2.91096
\(712\) −8.27008 −0.309934
\(713\) 8.31208 0.311290
\(714\) 3.80360 0.142346
\(715\) 0 0
\(716\) −15.4552 −0.577590
\(717\) 3.96859 0.148210
\(718\) 4.58918 0.171267
\(719\) 1.21369 0.0452628 0.0226314 0.999744i \(-0.492796\pi\)
0.0226314 + 0.999744i \(0.492796\pi\)
\(720\) −29.7386 −1.10829
\(721\) 15.2837 0.569193
\(722\) −0.539248 −0.0200687
\(723\) −61.5209 −2.28799
\(724\) −1.94332 −0.0722229
\(725\) −1.37724 −0.0511496
\(726\) 0 0
\(727\) −6.85349 −0.254182 −0.127091 0.991891i \(-0.540564\pi\)
−0.127091 + 0.991891i \(0.540564\pi\)
\(728\) 2.22971 0.0826387
\(729\) 62.0557 2.29836
\(730\) 1.92698 0.0713206
\(731\) 45.7283 1.69132
\(732\) 20.3293 0.751392
\(733\) −39.3168 −1.45220 −0.726100 0.687589i \(-0.758669\pi\)
−0.726100 + 0.687589i \(0.758669\pi\)
\(734\) 0.516900 0.0190791
\(735\) 3.27813 0.120916
\(736\) 2.30327 0.0848997
\(737\) 0 0
\(738\) 13.0377 0.479924
\(739\) 10.2900 0.378523 0.189261 0.981927i \(-0.439391\pi\)
0.189261 + 0.981927i \(0.439391\pi\)
\(740\) 8.35818 0.307253
\(741\) −44.4429 −1.63265
\(742\) 1.56252 0.0573620
\(743\) −41.1973 −1.51138 −0.755691 0.654928i \(-0.772699\pi\)
−0.755691 + 0.654928i \(0.772699\pi\)
\(744\) 14.9152 0.546818
\(745\) 21.4554 0.786067
\(746\) −4.21287 −0.154244
\(747\) 1.50490 0.0550613
\(748\) 0 0
\(749\) 1.17954 0.0430996
\(750\) 0.537912 0.0196418
\(751\) 20.0630 0.732110 0.366055 0.930593i \(-0.380708\pi\)
0.366055 + 0.930593i \(0.380708\pi\)
\(752\) −1.48675 −0.0542161
\(753\) 71.5134 2.60609
\(754\) 0.772918 0.0281480
\(755\) −0.0822054 −0.00299176
\(756\) 30.6978 1.11647
\(757\) −15.7342 −0.571869 −0.285934 0.958249i \(-0.592304\pi\)
−0.285934 + 0.958249i \(0.592304\pi\)
\(758\) −0.727312 −0.0264172
\(759\) 0 0
\(760\) −2.58436 −0.0937444
\(761\) 34.1599 1.23829 0.619147 0.785275i \(-0.287479\pi\)
0.619147 + 0.785275i \(0.287479\pi\)
\(762\) −3.84736 −0.139375
\(763\) −3.75666 −0.136000
\(764\) 0.209798 0.00759022
\(765\) 54.7730 1.98032
\(766\) 1.23002 0.0444426
\(767\) 20.1068 0.726016
\(768\) −45.5304 −1.64294
\(769\) −50.2610 −1.81246 −0.906229 0.422786i \(-0.861052\pi\)
−0.906229 + 0.422786i \(0.861052\pi\)
\(770\) 0 0
\(771\) 39.0319 1.40570
\(772\) 7.88358 0.283736
\(773\) −41.0757 −1.47739 −0.738696 0.674039i \(-0.764558\pi\)
−0.738696 + 0.674039i \(0.764558\pi\)
\(774\) −8.22000 −0.295462
\(775\) 6.97897 0.250692
\(776\) 12.0447 0.432380
\(777\) −13.8865 −0.498177
\(778\) −3.69044 −0.132309
\(779\) −40.6604 −1.45681
\(780\) 22.1210 0.792060
\(781\) 0 0
\(782\) −1.38193 −0.0494178
\(783\) 21.4277 0.765762
\(784\) 3.83917 0.137113
\(785\) −20.0790 −0.716650
\(786\) −10.4763 −0.373676
\(787\) 2.28478 0.0814435 0.0407217 0.999171i \(-0.487034\pi\)
0.0407217 + 0.999171i \(0.487034\pi\)
\(788\) 11.4935 0.409441
\(789\) −33.4701 −1.19157
\(790\) −1.64427 −0.0585004
\(791\) 10.3037 0.366357
\(792\) 0 0
\(793\) −10.7495 −0.381727
\(794\) −0.704547 −0.0250034
\(795\) 31.2152 1.10709
\(796\) 14.5980 0.517411
\(797\) −40.5602 −1.43672 −0.718358 0.695673i \(-0.755106\pi\)
−0.718358 + 0.695673i \(0.755106\pi\)
\(798\) 2.13231 0.0754831
\(799\) 2.73831 0.0968745
\(800\) 1.93387 0.0683726
\(801\) 98.2610 3.47188
\(802\) 4.13266 0.145929
\(803\) 0 0
\(804\) 91.6414 3.23194
\(805\) −1.19102 −0.0419778
\(806\) −3.91664 −0.137958
\(807\) −34.2621 −1.20608
\(808\) −1.19564 −0.0420623
\(809\) −3.69566 −0.129932 −0.0649662 0.997887i \(-0.520694\pi\)
−0.0649662 + 0.997887i \(0.520694\pi\)
\(810\) −4.55582 −0.160075
\(811\) −36.6961 −1.28858 −0.644288 0.764783i \(-0.722846\pi\)
−0.644288 + 0.764783i \(0.722846\pi\)
\(812\) 2.71741 0.0953622
\(813\) 79.3874 2.78424
\(814\) 0 0
\(815\) 17.7257 0.620904
\(816\) 88.9910 3.11531
\(817\) 25.6355 0.896873
\(818\) −0.385907 −0.0134929
\(819\) −26.4923 −0.925717
\(820\) 20.2383 0.706753
\(821\) −1.93727 −0.0676112 −0.0338056 0.999428i \(-0.510763\pi\)
−0.0338056 + 0.999428i \(0.510763\pi\)
\(822\) 5.16293 0.180078
\(823\) 5.94215 0.207130 0.103565 0.994623i \(-0.466975\pi\)
0.103565 + 0.994623i \(0.466975\pi\)
\(824\) −9.96414 −0.347117
\(825\) 0 0
\(826\) −0.964701 −0.0335663
\(827\) 47.7378 1.66001 0.830004 0.557758i \(-0.188338\pi\)
0.830004 + 0.557758i \(0.188338\pi\)
\(828\) −18.2031 −0.632601
\(829\) 26.5410 0.921807 0.460903 0.887450i \(-0.347525\pi\)
0.460903 + 0.887450i \(0.347525\pi\)
\(830\) −0.0318793 −0.00110655
\(831\) 26.1057 0.905596
\(832\) 25.1752 0.872795
\(833\) −7.07103 −0.244997
\(834\) 7.74091 0.268046
\(835\) −22.9221 −0.793251
\(836\) 0 0
\(837\) −108.581 −3.75312
\(838\) 1.49106 0.0515079
\(839\) 8.91496 0.307779 0.153889 0.988088i \(-0.450820\pi\)
0.153889 + 0.988088i \(0.450820\pi\)
\(840\) −2.13717 −0.0737392
\(841\) −27.1032 −0.934593
\(842\) 3.39873 0.117128
\(843\) −3.86230 −0.133025
\(844\) 25.5245 0.878591
\(845\) 1.30304 0.0448259
\(846\) −0.492232 −0.0169233
\(847\) 0 0
\(848\) 36.5576 1.25539
\(849\) −37.9860 −1.30368
\(850\) −1.16030 −0.0397978
\(851\) 5.04529 0.172950
\(852\) 2.19807 0.0753048
\(853\) −11.0187 −0.377274 −0.188637 0.982047i \(-0.560407\pi\)
−0.188637 + 0.982047i \(0.560407\pi\)
\(854\) 0.515750 0.0176486
\(855\) 30.7060 1.05012
\(856\) −0.769000 −0.0262839
\(857\) −20.1527 −0.688402 −0.344201 0.938896i \(-0.611850\pi\)
−0.344201 + 0.938896i \(0.611850\pi\)
\(858\) 0 0
\(859\) −28.8976 −0.985975 −0.492987 0.870037i \(-0.664095\pi\)
−0.492987 + 0.870037i \(0.664095\pi\)
\(860\) −12.7599 −0.435107
\(861\) −33.6246 −1.14592
\(862\) −1.14492 −0.0389961
\(863\) 8.28391 0.281987 0.140994 0.990010i \(-0.454970\pi\)
0.140994 + 0.990010i \(0.454970\pi\)
\(864\) −30.0878 −1.02361
\(865\) 19.6082 0.666699
\(866\) 0.933353 0.0317166
\(867\) −108.177 −3.67387
\(868\) −13.7700 −0.467385
\(869\) 0 0
\(870\) −0.740837 −0.0251167
\(871\) −48.4573 −1.64191
\(872\) 2.44914 0.0829385
\(873\) −143.109 −4.84352
\(874\) −0.774718 −0.0262052
\(875\) −1.00000 −0.0338062
\(876\) −75.9556 −2.56630
\(877\) 30.9005 1.04344 0.521718 0.853118i \(-0.325291\pi\)
0.521718 + 0.853118i \(0.325291\pi\)
\(878\) 6.29503 0.212447
\(879\) −8.33133 −0.281009
\(880\) 0 0
\(881\) −42.5576 −1.43380 −0.716901 0.697175i \(-0.754440\pi\)
−0.716901 + 0.697175i \(0.754440\pi\)
\(882\) 1.27107 0.0427992
\(883\) 23.4043 0.787616 0.393808 0.919193i \(-0.371157\pi\)
0.393808 + 0.919193i \(0.371157\pi\)
\(884\) −47.7159 −1.60486
\(885\) −19.2723 −0.647830
\(886\) 2.39919 0.0806023
\(887\) −34.6100 −1.16209 −0.581044 0.813872i \(-0.697356\pi\)
−0.581044 + 0.813872i \(0.697356\pi\)
\(888\) 9.05329 0.303809
\(889\) 7.15239 0.239883
\(890\) −2.08153 −0.0697731
\(891\) 0 0
\(892\) 38.6879 1.29537
\(893\) 1.53511 0.0513705
\(894\) 11.5411 0.385994
\(895\) −7.83308 −0.261831
\(896\) −5.07562 −0.169564
\(897\) 13.3530 0.445845
\(898\) −5.00097 −0.166885
\(899\) −9.61175 −0.320570
\(900\) −15.2837 −0.509455
\(901\) −67.3323 −2.24316
\(902\) 0 0
\(903\) 21.1996 0.705479
\(904\) −6.71746 −0.223419
\(905\) −0.984920 −0.0327399
\(906\) −0.0442193 −0.00146909
\(907\) −46.2557 −1.53589 −0.767947 0.640513i \(-0.778722\pi\)
−0.767947 + 0.640513i \(0.778722\pi\)
\(908\) −12.2356 −0.406052
\(909\) 14.2060 0.471182
\(910\) 0.561206 0.0186038
\(911\) 36.1685 1.19832 0.599158 0.800631i \(-0.295502\pi\)
0.599158 + 0.800631i \(0.295502\pi\)
\(912\) 49.8888 1.65198
\(913\) 0 0
\(914\) 0.312758 0.0103451
\(915\) 10.3034 0.340619
\(916\) 30.7111 1.01472
\(917\) 19.4758 0.643147
\(918\) 18.0523 0.595815
\(919\) 50.5931 1.66891 0.834457 0.551074i \(-0.185782\pi\)
0.834457 + 0.551074i \(0.185782\pi\)
\(920\) 0.776480 0.0255998
\(921\) −9.50620 −0.313240
\(922\) 3.05281 0.100539
\(923\) −1.16228 −0.0382568
\(924\) 0 0
\(925\) 4.23612 0.139283
\(926\) 1.10461 0.0362996
\(927\) 118.389 3.88840
\(928\) −2.66341 −0.0874307
\(929\) −27.2117 −0.892787 −0.446394 0.894837i \(-0.647292\pi\)
−0.446394 + 0.894837i \(0.647292\pi\)
\(930\) 3.75408 0.123101
\(931\) −3.96406 −0.129917
\(932\) 51.4616 1.68568
\(933\) 56.8243 1.86034
\(934\) 1.76917 0.0578891
\(935\) 0 0
\(936\) 17.2716 0.564540
\(937\) 7.87042 0.257115 0.128558 0.991702i \(-0.458965\pi\)
0.128558 + 0.991702i \(0.458965\pi\)
\(938\) 2.32492 0.0759114
\(939\) 0.659910 0.0215354
\(940\) −0.764088 −0.0249218
\(941\) −14.5219 −0.473401 −0.236700 0.971583i \(-0.576066\pi\)
−0.236700 + 0.971583i \(0.576066\pi\)
\(942\) −10.8007 −0.351907
\(943\) 12.2166 0.397826
\(944\) −22.5707 −0.734613
\(945\) 15.5584 0.506113
\(946\) 0 0
\(947\) 49.1404 1.59685 0.798424 0.602095i \(-0.205667\pi\)
0.798424 + 0.602095i \(0.205667\pi\)
\(948\) 64.8121 2.10500
\(949\) 40.1631 1.30375
\(950\) −0.650467 −0.0211039
\(951\) −0.356077 −0.0115466
\(952\) 4.60994 0.149409
\(953\) 17.4892 0.566532 0.283266 0.959041i \(-0.408582\pi\)
0.283266 + 0.959041i \(0.408582\pi\)
\(954\) 12.1035 0.391864
\(955\) 0.106330 0.00344077
\(956\) 2.38866 0.0772548
\(957\) 0 0
\(958\) 3.72110 0.120223
\(959\) −9.59810 −0.309939
\(960\) −24.1303 −0.778802
\(961\) 17.7061 0.571163
\(962\) −2.37734 −0.0766484
\(963\) 9.13687 0.294432
\(964\) −37.0289 −1.19262
\(965\) 3.99558 0.128622
\(966\) −0.640663 −0.0206130
\(967\) −12.9466 −0.416334 −0.208167 0.978093i \(-0.566750\pi\)
−0.208167 + 0.978093i \(0.566750\pi\)
\(968\) 0 0
\(969\) −91.8858 −2.95180
\(970\) 3.03159 0.0973384
\(971\) 13.5026 0.433318 0.216659 0.976247i \(-0.430484\pi\)
0.216659 + 0.976247i \(0.430484\pi\)
\(972\) 87.4833 2.80603
\(973\) −14.3907 −0.461343
\(974\) −3.55847 −0.114021
\(975\) 11.2115 0.359054
\(976\) 12.0668 0.386247
\(977\) −45.9006 −1.46849 −0.734245 0.678884i \(-0.762464\pi\)
−0.734245 + 0.678884i \(0.762464\pi\)
\(978\) 9.53486 0.304891
\(979\) 0 0
\(980\) 1.97307 0.0630275
\(981\) −29.0995 −0.929076
\(982\) −0.418257 −0.0133471
\(983\) 31.0288 0.989666 0.494833 0.868988i \(-0.335229\pi\)
0.494833 + 0.868988i \(0.335229\pi\)
\(984\) 21.9215 0.698831
\(985\) 5.82520 0.185606
\(986\) 1.59801 0.0508910
\(987\) 1.26948 0.0404080
\(988\) −26.7497 −0.851023
\(989\) −7.70230 −0.244919
\(990\) 0 0
\(991\) −20.5880 −0.654000 −0.327000 0.945024i \(-0.606038\pi\)
−0.327000 + 0.945024i \(0.606038\pi\)
\(992\) 13.4964 0.428512
\(993\) −8.65126 −0.274540
\(994\) 0.0557646 0.00176875
\(995\) 7.39859 0.234551
\(996\) 1.25659 0.0398165
\(997\) −43.6146 −1.38129 −0.690644 0.723195i \(-0.742673\pi\)
−0.690644 + 0.723195i \(0.742673\pi\)
\(998\) 3.96883 0.125631
\(999\) −65.9071 −2.08521
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.1.4 5
11.10 odd 2 4235.2.a.be.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.y.1.4 5 1.1 even 1 trivial
4235.2.a.be.1.2 yes 5 11.10 odd 2