Properties

Label 4005.2.a.o.1.7
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.498937\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.75106 q^{2} +5.56834 q^{4} -1.00000 q^{5} +0.587818 q^{7} +9.81674 q^{8} +O(q^{10})\) \(q+2.75106 q^{2} +5.56834 q^{4} -1.00000 q^{5} +0.587818 q^{7} +9.81674 q^{8} -2.75106 q^{10} +0.892165 q^{11} -1.31436 q^{13} +1.61712 q^{14} +15.8698 q^{16} +6.89306 q^{17} +2.78585 q^{19} -5.56834 q^{20} +2.45440 q^{22} +0.636773 q^{23} +1.00000 q^{25} -3.61587 q^{26} +3.27317 q^{28} -5.55621 q^{29} -9.03241 q^{31} +24.0252 q^{32} +18.9632 q^{34} -0.587818 q^{35} +0.632709 q^{37} +7.66404 q^{38} -9.81674 q^{40} +7.36941 q^{41} -7.41164 q^{43} +4.96788 q^{44} +1.75180 q^{46} +6.16627 q^{47} -6.65447 q^{49} +2.75106 q^{50} -7.31878 q^{52} -3.18988 q^{53} -0.892165 q^{55} +5.77045 q^{56} -15.2855 q^{58} -5.68022 q^{59} -4.98727 q^{61} -24.8487 q^{62} +34.3554 q^{64} +1.31436 q^{65} +7.78542 q^{67} +38.3829 q^{68} -1.61712 q^{70} +14.9564 q^{71} +10.9297 q^{73} +1.74062 q^{74} +15.5126 q^{76} +0.524431 q^{77} -11.7250 q^{79} -15.8698 q^{80} +20.2737 q^{82} +14.3750 q^{83} -6.89306 q^{85} -20.3899 q^{86} +8.75815 q^{88} -1.00000 q^{89} -0.772602 q^{91} +3.54577 q^{92} +16.9638 q^{94} -2.78585 q^{95} +5.86420 q^{97} -18.3069 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75106 1.94529 0.972647 0.232287i \(-0.0746207\pi\)
0.972647 + 0.232287i \(0.0746207\pi\)
\(3\) 0 0
\(4\) 5.56834 2.78417
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.587818 0.222174 0.111087 0.993811i \(-0.464567\pi\)
0.111087 + 0.993811i \(0.464567\pi\)
\(8\) 9.81674 3.47074
\(9\) 0 0
\(10\) −2.75106 −0.869962
\(11\) 0.892165 0.268998 0.134499 0.990914i \(-0.457058\pi\)
0.134499 + 0.990914i \(0.457058\pi\)
\(12\) 0 0
\(13\) −1.31436 −0.364537 −0.182268 0.983249i \(-0.558344\pi\)
−0.182268 + 0.983249i \(0.558344\pi\)
\(14\) 1.61712 0.432195
\(15\) 0 0
\(16\) 15.8698 3.96744
\(17\) 6.89306 1.67181 0.835906 0.548872i \(-0.184943\pi\)
0.835906 + 0.548872i \(0.184943\pi\)
\(18\) 0 0
\(19\) 2.78585 0.639118 0.319559 0.947566i \(-0.396465\pi\)
0.319559 + 0.947566i \(0.396465\pi\)
\(20\) −5.56834 −1.24512
\(21\) 0 0
\(22\) 2.45440 0.523280
\(23\) 0.636773 0.132776 0.0663881 0.997794i \(-0.478852\pi\)
0.0663881 + 0.997794i \(0.478852\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.61587 −0.709131
\(27\) 0 0
\(28\) 3.27317 0.618571
\(29\) −5.55621 −1.03176 −0.515881 0.856660i \(-0.672535\pi\)
−0.515881 + 0.856660i \(0.672535\pi\)
\(30\) 0 0
\(31\) −9.03241 −1.62227 −0.811134 0.584860i \(-0.801149\pi\)
−0.811134 + 0.584860i \(0.801149\pi\)
\(32\) 24.0252 4.24710
\(33\) 0 0
\(34\) 18.9632 3.25217
\(35\) −0.587818 −0.0993594
\(36\) 0 0
\(37\) 0.632709 0.104017 0.0520084 0.998647i \(-0.483438\pi\)
0.0520084 + 0.998647i \(0.483438\pi\)
\(38\) 7.66404 1.24327
\(39\) 0 0
\(40\) −9.81674 −1.55216
\(41\) 7.36941 1.15091 0.575454 0.817834i \(-0.304825\pi\)
0.575454 + 0.817834i \(0.304825\pi\)
\(42\) 0 0
\(43\) −7.41164 −1.13026 −0.565132 0.825000i \(-0.691175\pi\)
−0.565132 + 0.825000i \(0.691175\pi\)
\(44\) 4.96788 0.748936
\(45\) 0 0
\(46\) 1.75180 0.258289
\(47\) 6.16627 0.899442 0.449721 0.893169i \(-0.351523\pi\)
0.449721 + 0.893169i \(0.351523\pi\)
\(48\) 0 0
\(49\) −6.65447 −0.950639
\(50\) 2.75106 0.389059
\(51\) 0 0
\(52\) −7.31878 −1.01493
\(53\) −3.18988 −0.438163 −0.219082 0.975707i \(-0.570306\pi\)
−0.219082 + 0.975707i \(0.570306\pi\)
\(54\) 0 0
\(55\) −0.892165 −0.120299
\(56\) 5.77045 0.771109
\(57\) 0 0
\(58\) −15.2855 −2.00708
\(59\) −5.68022 −0.739502 −0.369751 0.929131i \(-0.620557\pi\)
−0.369751 + 0.929131i \(0.620557\pi\)
\(60\) 0 0
\(61\) −4.98727 −0.638555 −0.319277 0.947661i \(-0.603440\pi\)
−0.319277 + 0.947661i \(0.603440\pi\)
\(62\) −24.8487 −3.15579
\(63\) 0 0
\(64\) 34.3554 4.29443
\(65\) 1.31436 0.163026
\(66\) 0 0
\(67\) 7.78542 0.951140 0.475570 0.879678i \(-0.342242\pi\)
0.475570 + 0.879678i \(0.342242\pi\)
\(68\) 38.3829 4.65461
\(69\) 0 0
\(70\) −1.61712 −0.193283
\(71\) 14.9564 1.77499 0.887496 0.460815i \(-0.152443\pi\)
0.887496 + 0.460815i \(0.152443\pi\)
\(72\) 0 0
\(73\) 10.9297 1.27922 0.639611 0.768698i \(-0.279095\pi\)
0.639611 + 0.768698i \(0.279095\pi\)
\(74\) 1.74062 0.202343
\(75\) 0 0
\(76\) 15.5126 1.77941
\(77\) 0.524431 0.0597644
\(78\) 0 0
\(79\) −11.7250 −1.31917 −0.659584 0.751631i \(-0.729267\pi\)
−0.659584 + 0.751631i \(0.729267\pi\)
\(80\) −15.8698 −1.77429
\(81\) 0 0
\(82\) 20.2737 2.23886
\(83\) 14.3750 1.57786 0.788930 0.614483i \(-0.210635\pi\)
0.788930 + 0.614483i \(0.210635\pi\)
\(84\) 0 0
\(85\) −6.89306 −0.747657
\(86\) −20.3899 −2.19870
\(87\) 0 0
\(88\) 8.75815 0.933622
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −0.772602 −0.0809907
\(92\) 3.54577 0.369672
\(93\) 0 0
\(94\) 16.9638 1.74968
\(95\) −2.78585 −0.285822
\(96\) 0 0
\(97\) 5.86420 0.595420 0.297710 0.954656i \(-0.403777\pi\)
0.297710 + 0.954656i \(0.403777\pi\)
\(98\) −18.3069 −1.84927
\(99\) 0 0
\(100\) 5.56834 0.556834
\(101\) 10.6416 1.05888 0.529439 0.848348i \(-0.322403\pi\)
0.529439 + 0.848348i \(0.322403\pi\)
\(102\) 0 0
\(103\) 1.82241 0.179568 0.0897839 0.995961i \(-0.471382\pi\)
0.0897839 + 0.995961i \(0.471382\pi\)
\(104\) −12.9027 −1.26521
\(105\) 0 0
\(106\) −8.77555 −0.852357
\(107\) 3.76328 0.363810 0.181905 0.983316i \(-0.441774\pi\)
0.181905 + 0.983316i \(0.441774\pi\)
\(108\) 0 0
\(109\) 5.53929 0.530568 0.265284 0.964170i \(-0.414534\pi\)
0.265284 + 0.964170i \(0.414534\pi\)
\(110\) −2.45440 −0.234018
\(111\) 0 0
\(112\) 9.32853 0.881464
\(113\) −14.1421 −1.33037 −0.665186 0.746677i \(-0.731648\pi\)
−0.665186 + 0.746677i \(0.731648\pi\)
\(114\) 0 0
\(115\) −0.636773 −0.0593794
\(116\) −30.9389 −2.87260
\(117\) 0 0
\(118\) −15.6266 −1.43855
\(119\) 4.05186 0.371434
\(120\) 0 0
\(121\) −10.2040 −0.927640
\(122\) −13.7203 −1.24218
\(123\) 0 0
\(124\) −50.2955 −4.51667
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.9731 −1.06244 −0.531219 0.847235i \(-0.678266\pi\)
−0.531219 + 0.847235i \(0.678266\pi\)
\(128\) 46.4634 4.10682
\(129\) 0 0
\(130\) 3.61587 0.317133
\(131\) 21.0827 1.84201 0.921003 0.389556i \(-0.127371\pi\)
0.921003 + 0.389556i \(0.127371\pi\)
\(132\) 0 0
\(133\) 1.63757 0.141995
\(134\) 21.4182 1.85025
\(135\) 0 0
\(136\) 67.6674 5.80243
\(137\) −21.7300 −1.85652 −0.928260 0.371931i \(-0.878696\pi\)
−0.928260 + 0.371931i \(0.878696\pi\)
\(138\) 0 0
\(139\) −7.18857 −0.609726 −0.304863 0.952396i \(-0.598611\pi\)
−0.304863 + 0.952396i \(0.598611\pi\)
\(140\) −3.27317 −0.276634
\(141\) 0 0
\(142\) 41.1459 3.45288
\(143\) −1.17262 −0.0980596
\(144\) 0 0
\(145\) 5.55621 0.461418
\(146\) 30.0682 2.48847
\(147\) 0 0
\(148\) 3.52314 0.289601
\(149\) −17.9452 −1.47013 −0.735063 0.677999i \(-0.762847\pi\)
−0.735063 + 0.677999i \(0.762847\pi\)
\(150\) 0 0
\(151\) 8.60851 0.700551 0.350275 0.936647i \(-0.386088\pi\)
0.350275 + 0.936647i \(0.386088\pi\)
\(152\) 27.3479 2.21821
\(153\) 0 0
\(154\) 1.44274 0.116259
\(155\) 9.03241 0.725500
\(156\) 0 0
\(157\) −1.31247 −0.104746 −0.0523731 0.998628i \(-0.516679\pi\)
−0.0523731 + 0.998628i \(0.516679\pi\)
\(158\) −32.2563 −2.56617
\(159\) 0 0
\(160\) −24.0252 −1.89936
\(161\) 0.374306 0.0294995
\(162\) 0 0
\(163\) −4.32734 −0.338943 −0.169472 0.985535i \(-0.554206\pi\)
−0.169472 + 0.985535i \(0.554206\pi\)
\(164\) 41.0354 3.20433
\(165\) 0 0
\(166\) 39.5465 3.06940
\(167\) −10.1225 −0.783306 −0.391653 0.920113i \(-0.628097\pi\)
−0.391653 + 0.920113i \(0.628097\pi\)
\(168\) 0 0
\(169\) −11.2725 −0.867113
\(170\) −18.9632 −1.45441
\(171\) 0 0
\(172\) −41.2705 −3.14685
\(173\) 4.46799 0.339695 0.169847 0.985470i \(-0.445673\pi\)
0.169847 + 0.985470i \(0.445673\pi\)
\(174\) 0 0
\(175\) 0.587818 0.0444349
\(176\) 14.1584 1.06723
\(177\) 0 0
\(178\) −2.75106 −0.206201
\(179\) −17.1331 −1.28059 −0.640293 0.768131i \(-0.721187\pi\)
−0.640293 + 0.768131i \(0.721187\pi\)
\(180\) 0 0
\(181\) 0.766969 0.0570083 0.0285042 0.999594i \(-0.490926\pi\)
0.0285042 + 0.999594i \(0.490926\pi\)
\(182\) −2.12548 −0.157551
\(183\) 0 0
\(184\) 6.25103 0.460832
\(185\) −0.632709 −0.0465177
\(186\) 0 0
\(187\) 6.14975 0.449714
\(188\) 34.3359 2.50420
\(189\) 0 0
\(190\) −7.66404 −0.556008
\(191\) −22.8474 −1.65318 −0.826590 0.562805i \(-0.809722\pi\)
−0.826590 + 0.562805i \(0.809722\pi\)
\(192\) 0 0
\(193\) −13.9655 −1.00526 −0.502630 0.864501i \(-0.667634\pi\)
−0.502630 + 0.864501i \(0.667634\pi\)
\(194\) 16.1328 1.15827
\(195\) 0 0
\(196\) −37.0544 −2.64674
\(197\) 14.4721 1.03110 0.515549 0.856860i \(-0.327588\pi\)
0.515549 + 0.856860i \(0.327588\pi\)
\(198\) 0 0
\(199\) 9.60667 0.680999 0.340499 0.940245i \(-0.389404\pi\)
0.340499 + 0.940245i \(0.389404\pi\)
\(200\) 9.81674 0.694148
\(201\) 0 0
\(202\) 29.2757 2.05983
\(203\) −3.26604 −0.229231
\(204\) 0 0
\(205\) −7.36941 −0.514702
\(206\) 5.01358 0.349312
\(207\) 0 0
\(208\) −20.8585 −1.44628
\(209\) 2.48544 0.171921
\(210\) 0 0
\(211\) 2.98766 0.205679 0.102839 0.994698i \(-0.467207\pi\)
0.102839 + 0.994698i \(0.467207\pi\)
\(212\) −17.7623 −1.21992
\(213\) 0 0
\(214\) 10.3530 0.707717
\(215\) 7.41164 0.505469
\(216\) 0 0
\(217\) −5.30941 −0.360426
\(218\) 15.2389 1.03211
\(219\) 0 0
\(220\) −4.96788 −0.334934
\(221\) −9.05993 −0.609437
\(222\) 0 0
\(223\) −23.4737 −1.57192 −0.785958 0.618280i \(-0.787830\pi\)
−0.785958 + 0.618280i \(0.787830\pi\)
\(224\) 14.1225 0.943597
\(225\) 0 0
\(226\) −38.9057 −2.58797
\(227\) −11.3599 −0.753982 −0.376991 0.926217i \(-0.623041\pi\)
−0.376991 + 0.926217i \(0.623041\pi\)
\(228\) 0 0
\(229\) 5.54933 0.366710 0.183355 0.983047i \(-0.441304\pi\)
0.183355 + 0.983047i \(0.441304\pi\)
\(230\) −1.75180 −0.115510
\(231\) 0 0
\(232\) −54.5439 −3.58098
\(233\) −5.52225 −0.361775 −0.180887 0.983504i \(-0.557897\pi\)
−0.180887 + 0.983504i \(0.557897\pi\)
\(234\) 0 0
\(235\) −6.16627 −0.402243
\(236\) −31.6294 −2.05890
\(237\) 0 0
\(238\) 11.1469 0.722548
\(239\) −12.7494 −0.824693 −0.412347 0.911027i \(-0.635291\pi\)
−0.412347 + 0.911027i \(0.635291\pi\)
\(240\) 0 0
\(241\) 18.6081 1.19865 0.599325 0.800506i \(-0.295436\pi\)
0.599325 + 0.800506i \(0.295436\pi\)
\(242\) −28.0720 −1.80453
\(243\) 0 0
\(244\) −27.7708 −1.77785
\(245\) 6.65447 0.425138
\(246\) 0 0
\(247\) −3.66160 −0.232982
\(248\) −88.6688 −5.63047
\(249\) 0 0
\(250\) −2.75106 −0.173992
\(251\) −10.2929 −0.649685 −0.324842 0.945768i \(-0.605311\pi\)
−0.324842 + 0.945768i \(0.605311\pi\)
\(252\) 0 0
\(253\) 0.568106 0.0357165
\(254\) −32.9386 −2.06675
\(255\) 0 0
\(256\) 59.1128 3.69455
\(257\) −9.53876 −0.595011 −0.297506 0.954720i \(-0.596155\pi\)
−0.297506 + 0.954720i \(0.596155\pi\)
\(258\) 0 0
\(259\) 0.371918 0.0231099
\(260\) 7.31878 0.453892
\(261\) 0 0
\(262\) 57.9999 3.58324
\(263\) −17.1059 −1.05480 −0.527399 0.849618i \(-0.676833\pi\)
−0.527399 + 0.849618i \(0.676833\pi\)
\(264\) 0 0
\(265\) 3.18988 0.195953
\(266\) 4.50506 0.276223
\(267\) 0 0
\(268\) 43.3519 2.64814
\(269\) −31.8968 −1.94478 −0.972391 0.233356i \(-0.925029\pi\)
−0.972391 + 0.233356i \(0.925029\pi\)
\(270\) 0 0
\(271\) −29.7027 −1.80431 −0.902154 0.431414i \(-0.858015\pi\)
−0.902154 + 0.431414i \(0.858015\pi\)
\(272\) 109.391 6.63282
\(273\) 0 0
\(274\) −59.7806 −3.61148
\(275\) 0.892165 0.0537996
\(276\) 0 0
\(277\) −11.5680 −0.695052 −0.347526 0.937670i \(-0.612978\pi\)
−0.347526 + 0.937670i \(0.612978\pi\)
\(278\) −19.7762 −1.18610
\(279\) 0 0
\(280\) −5.77045 −0.344851
\(281\) −15.3722 −0.917031 −0.458515 0.888686i \(-0.651619\pi\)
−0.458515 + 0.888686i \(0.651619\pi\)
\(282\) 0 0
\(283\) 2.75053 0.163502 0.0817509 0.996653i \(-0.473949\pi\)
0.0817509 + 0.996653i \(0.473949\pi\)
\(284\) 83.2821 4.94188
\(285\) 0 0
\(286\) −3.22596 −0.190755
\(287\) 4.33187 0.255702
\(288\) 0 0
\(289\) 30.5143 1.79496
\(290\) 15.2855 0.897594
\(291\) 0 0
\(292\) 60.8602 3.56158
\(293\) −30.7950 −1.79906 −0.899531 0.436856i \(-0.856092\pi\)
−0.899531 + 0.436856i \(0.856092\pi\)
\(294\) 0 0
\(295\) 5.68022 0.330715
\(296\) 6.21114 0.361015
\(297\) 0 0
\(298\) −49.3683 −2.85983
\(299\) −0.836946 −0.0484018
\(300\) 0 0
\(301\) −4.35669 −0.251116
\(302\) 23.6826 1.36278
\(303\) 0 0
\(304\) 44.2108 2.53566
\(305\) 4.98727 0.285570
\(306\) 0 0
\(307\) 10.6937 0.610320 0.305160 0.952301i \(-0.401290\pi\)
0.305160 + 0.952301i \(0.401290\pi\)
\(308\) 2.92021 0.166394
\(309\) 0 0
\(310\) 24.8487 1.41131
\(311\) −6.00458 −0.340488 −0.170244 0.985402i \(-0.554456\pi\)
−0.170244 + 0.985402i \(0.554456\pi\)
\(312\) 0 0
\(313\) 4.68825 0.264995 0.132498 0.991183i \(-0.457700\pi\)
0.132498 + 0.991183i \(0.457700\pi\)
\(314\) −3.61068 −0.203762
\(315\) 0 0
\(316\) −65.2889 −3.67279
\(317\) 31.9932 1.79692 0.898459 0.439057i \(-0.144687\pi\)
0.898459 + 0.439057i \(0.144687\pi\)
\(318\) 0 0
\(319\) −4.95706 −0.277542
\(320\) −34.3554 −1.92053
\(321\) 0 0
\(322\) 1.02974 0.0573852
\(323\) 19.2030 1.06848
\(324\) 0 0
\(325\) −1.31436 −0.0729073
\(326\) −11.9048 −0.659345
\(327\) 0 0
\(328\) 72.3436 3.99451
\(329\) 3.62464 0.199833
\(330\) 0 0
\(331\) −5.24309 −0.288187 −0.144093 0.989564i \(-0.546027\pi\)
−0.144093 + 0.989564i \(0.546027\pi\)
\(332\) 80.0449 4.39303
\(333\) 0 0
\(334\) −27.8477 −1.52376
\(335\) −7.78542 −0.425363
\(336\) 0 0
\(337\) 2.66495 0.145169 0.0725844 0.997362i \(-0.476875\pi\)
0.0725844 + 0.997362i \(0.476875\pi\)
\(338\) −31.0113 −1.68679
\(339\) 0 0
\(340\) −38.3829 −2.08161
\(341\) −8.05840 −0.436387
\(342\) 0 0
\(343\) −8.02634 −0.433382
\(344\) −72.7581 −3.92285
\(345\) 0 0
\(346\) 12.2917 0.660807
\(347\) 11.0892 0.595301 0.297650 0.954675i \(-0.403797\pi\)
0.297650 + 0.954675i \(0.403797\pi\)
\(348\) 0 0
\(349\) −5.81916 −0.311493 −0.155746 0.987797i \(-0.549778\pi\)
−0.155746 + 0.987797i \(0.549778\pi\)
\(350\) 1.61712 0.0864389
\(351\) 0 0
\(352\) 21.4345 1.14246
\(353\) 6.35008 0.337980 0.168990 0.985618i \(-0.445949\pi\)
0.168990 + 0.985618i \(0.445949\pi\)
\(354\) 0 0
\(355\) −14.9564 −0.793801
\(356\) −5.56834 −0.295122
\(357\) 0 0
\(358\) −47.1342 −2.49112
\(359\) 33.9604 1.79236 0.896181 0.443688i \(-0.146330\pi\)
0.896181 + 0.443688i \(0.146330\pi\)
\(360\) 0 0
\(361\) −11.2390 −0.591529
\(362\) 2.10998 0.110898
\(363\) 0 0
\(364\) −4.30211 −0.225492
\(365\) −10.9297 −0.572086
\(366\) 0 0
\(367\) −13.4491 −0.702038 −0.351019 0.936368i \(-0.614165\pi\)
−0.351019 + 0.936368i \(0.614165\pi\)
\(368\) 10.1054 0.526782
\(369\) 0 0
\(370\) −1.74062 −0.0904907
\(371\) −1.87507 −0.0973487
\(372\) 0 0
\(373\) −10.4418 −0.540658 −0.270329 0.962768i \(-0.587133\pi\)
−0.270329 + 0.962768i \(0.587133\pi\)
\(374\) 16.9183 0.874826
\(375\) 0 0
\(376\) 60.5326 3.12173
\(377\) 7.30284 0.376115
\(378\) 0 0
\(379\) 29.1034 1.49494 0.747469 0.664296i \(-0.231269\pi\)
0.747469 + 0.664296i \(0.231269\pi\)
\(380\) −15.5126 −0.795778
\(381\) 0 0
\(382\) −62.8546 −3.21592
\(383\) −26.7367 −1.36618 −0.683092 0.730333i \(-0.739365\pi\)
−0.683092 + 0.730333i \(0.739365\pi\)
\(384\) 0 0
\(385\) −0.524431 −0.0267275
\(386\) −38.4200 −1.95553
\(387\) 0 0
\(388\) 32.6539 1.65775
\(389\) 34.9180 1.77041 0.885205 0.465201i \(-0.154018\pi\)
0.885205 + 0.465201i \(0.154018\pi\)
\(390\) 0 0
\(391\) 4.38931 0.221977
\(392\) −65.3252 −3.29942
\(393\) 0 0
\(394\) 39.8138 2.00579
\(395\) 11.7250 0.589950
\(396\) 0 0
\(397\) 35.4392 1.77864 0.889322 0.457282i \(-0.151177\pi\)
0.889322 + 0.457282i \(0.151177\pi\)
\(398\) 26.4286 1.32474
\(399\) 0 0
\(400\) 15.8698 0.793488
\(401\) 22.9962 1.14837 0.574187 0.818724i \(-0.305318\pi\)
0.574187 + 0.818724i \(0.305318\pi\)
\(402\) 0 0
\(403\) 11.8718 0.591376
\(404\) 59.2560 2.94810
\(405\) 0 0
\(406\) −8.98508 −0.445922
\(407\) 0.564481 0.0279803
\(408\) 0 0
\(409\) 24.8868 1.23057 0.615286 0.788304i \(-0.289040\pi\)
0.615286 + 0.788304i \(0.289040\pi\)
\(410\) −20.2737 −1.00125
\(411\) 0 0
\(412\) 10.1478 0.499948
\(413\) −3.33894 −0.164298
\(414\) 0 0
\(415\) −14.3750 −0.705641
\(416\) −31.5777 −1.54822
\(417\) 0 0
\(418\) 6.83759 0.334438
\(419\) 38.2037 1.86637 0.933185 0.359396i \(-0.117017\pi\)
0.933185 + 0.359396i \(0.117017\pi\)
\(420\) 0 0
\(421\) 9.53347 0.464633 0.232316 0.972640i \(-0.425370\pi\)
0.232316 + 0.972640i \(0.425370\pi\)
\(422\) 8.21923 0.400106
\(423\) 0 0
\(424\) −31.3142 −1.52075
\(425\) 6.89306 0.334363
\(426\) 0 0
\(427\) −2.93161 −0.141870
\(428\) 20.9552 1.01291
\(429\) 0 0
\(430\) 20.3899 0.983287
\(431\) −15.3802 −0.740838 −0.370419 0.928865i \(-0.620786\pi\)
−0.370419 + 0.928865i \(0.620786\pi\)
\(432\) 0 0
\(433\) −17.4466 −0.838432 −0.419216 0.907887i \(-0.637695\pi\)
−0.419216 + 0.907887i \(0.637695\pi\)
\(434\) −14.6065 −0.701135
\(435\) 0 0
\(436\) 30.8447 1.47719
\(437\) 1.77395 0.0848596
\(438\) 0 0
\(439\) −15.5530 −0.742302 −0.371151 0.928573i \(-0.621037\pi\)
−0.371151 + 0.928573i \(0.621037\pi\)
\(440\) −8.75815 −0.417528
\(441\) 0 0
\(442\) −24.9244 −1.18553
\(443\) 23.9793 1.13929 0.569646 0.821890i \(-0.307080\pi\)
0.569646 + 0.821890i \(0.307080\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −64.5776 −3.05784
\(447\) 0 0
\(448\) 20.1947 0.954111
\(449\) −26.7348 −1.26170 −0.630848 0.775907i \(-0.717293\pi\)
−0.630848 + 0.775907i \(0.717293\pi\)
\(450\) 0 0
\(451\) 6.57473 0.309592
\(452\) −78.7478 −3.70399
\(453\) 0 0
\(454\) −31.2518 −1.46672
\(455\) 0.772602 0.0362201
\(456\) 0 0
\(457\) −23.7958 −1.11312 −0.556561 0.830807i \(-0.687879\pi\)
−0.556561 + 0.830807i \(0.687879\pi\)
\(458\) 15.2665 0.713359
\(459\) 0 0
\(460\) −3.54577 −0.165322
\(461\) 14.3367 0.667728 0.333864 0.942621i \(-0.391647\pi\)
0.333864 + 0.942621i \(0.391647\pi\)
\(462\) 0 0
\(463\) 5.29222 0.245950 0.122975 0.992410i \(-0.460756\pi\)
0.122975 + 0.992410i \(0.460756\pi\)
\(464\) −88.1758 −4.09346
\(465\) 0 0
\(466\) −15.1921 −0.703759
\(467\) 23.7078 1.09707 0.548534 0.836128i \(-0.315186\pi\)
0.548534 + 0.836128i \(0.315186\pi\)
\(468\) 0 0
\(469\) 4.57641 0.211319
\(470\) −16.9638 −0.782481
\(471\) 0 0
\(472\) −55.7612 −2.56662
\(473\) −6.61240 −0.304039
\(474\) 0 0
\(475\) 2.78585 0.127824
\(476\) 22.5622 1.03414
\(477\) 0 0
\(478\) −35.0745 −1.60427
\(479\) 5.23009 0.238969 0.119484 0.992836i \(-0.461876\pi\)
0.119484 + 0.992836i \(0.461876\pi\)
\(480\) 0 0
\(481\) −0.831605 −0.0379179
\(482\) 51.1919 2.33173
\(483\) 0 0
\(484\) −56.8196 −2.58271
\(485\) −5.86420 −0.266280
\(486\) 0 0
\(487\) −7.29308 −0.330481 −0.165240 0.986253i \(-0.552840\pi\)
−0.165240 + 0.986253i \(0.552840\pi\)
\(488\) −48.9587 −2.21626
\(489\) 0 0
\(490\) 18.3069 0.827020
\(491\) 17.7901 0.802857 0.401428 0.915890i \(-0.368514\pi\)
0.401428 + 0.915890i \(0.368514\pi\)
\(492\) 0 0
\(493\) −38.2993 −1.72491
\(494\) −10.0733 −0.453218
\(495\) 0 0
\(496\) −143.342 −6.43625
\(497\) 8.79161 0.394358
\(498\) 0 0
\(499\) 27.6810 1.23917 0.619586 0.784929i \(-0.287300\pi\)
0.619586 + 0.784929i \(0.287300\pi\)
\(500\) −5.56834 −0.249024
\(501\) 0 0
\(502\) −28.3165 −1.26383
\(503\) −9.64823 −0.430193 −0.215097 0.976593i \(-0.569007\pi\)
−0.215097 + 0.976593i \(0.569007\pi\)
\(504\) 0 0
\(505\) −10.6416 −0.473544
\(506\) 1.56290 0.0694792
\(507\) 0 0
\(508\) −66.6701 −2.95801
\(509\) 25.1703 1.11565 0.557826 0.829958i \(-0.311636\pi\)
0.557826 + 0.829958i \(0.311636\pi\)
\(510\) 0 0
\(511\) 6.42466 0.284210
\(512\) 69.6963 3.08017
\(513\) 0 0
\(514\) −26.2417 −1.15747
\(515\) −1.82241 −0.0803052
\(516\) 0 0
\(517\) 5.50133 0.241948
\(518\) 1.02317 0.0449555
\(519\) 0 0
\(520\) 12.9027 0.565820
\(521\) 25.9914 1.13870 0.569351 0.822094i \(-0.307195\pi\)
0.569351 + 0.822094i \(0.307195\pi\)
\(522\) 0 0
\(523\) −11.7696 −0.514650 −0.257325 0.966325i \(-0.582841\pi\)
−0.257325 + 0.966325i \(0.582841\pi\)
\(524\) 117.396 5.12846
\(525\) 0 0
\(526\) −47.0595 −2.05189
\(527\) −62.2609 −2.71213
\(528\) 0 0
\(529\) −22.5945 −0.982370
\(530\) 8.77555 0.381186
\(531\) 0 0
\(532\) 9.11856 0.395340
\(533\) −9.68603 −0.419548
\(534\) 0 0
\(535\) −3.76328 −0.162701
\(536\) 76.4274 3.30116
\(537\) 0 0
\(538\) −87.7501 −3.78318
\(539\) −5.93688 −0.255720
\(540\) 0 0
\(541\) 18.3468 0.788790 0.394395 0.918941i \(-0.370954\pi\)
0.394395 + 0.918941i \(0.370954\pi\)
\(542\) −81.7139 −3.50991
\(543\) 0 0
\(544\) 165.607 7.10036
\(545\) −5.53929 −0.237277
\(546\) 0 0
\(547\) −24.1996 −1.03470 −0.517351 0.855774i \(-0.673082\pi\)
−0.517351 + 0.855774i \(0.673082\pi\)
\(548\) −121.000 −5.16887
\(549\) 0 0
\(550\) 2.45440 0.104656
\(551\) −15.4788 −0.659417
\(552\) 0 0
\(553\) −6.89218 −0.293085
\(554\) −31.8242 −1.35208
\(555\) 0 0
\(556\) −40.0284 −1.69758
\(557\) −28.5443 −1.20946 −0.604730 0.796431i \(-0.706719\pi\)
−0.604730 + 0.796431i \(0.706719\pi\)
\(558\) 0 0
\(559\) 9.74152 0.412023
\(560\) −9.32853 −0.394202
\(561\) 0 0
\(562\) −42.2900 −1.78390
\(563\) −7.53424 −0.317531 −0.158765 0.987316i \(-0.550751\pi\)
−0.158765 + 0.987316i \(0.550751\pi\)
\(564\) 0 0
\(565\) 14.1421 0.594961
\(566\) 7.56687 0.318059
\(567\) 0 0
\(568\) 146.823 6.16054
\(569\) 13.2148 0.553994 0.276997 0.960871i \(-0.410661\pi\)
0.276997 + 0.960871i \(0.410661\pi\)
\(570\) 0 0
\(571\) 1.66699 0.0697612 0.0348806 0.999391i \(-0.488895\pi\)
0.0348806 + 0.999391i \(0.488895\pi\)
\(572\) −6.52956 −0.273015
\(573\) 0 0
\(574\) 11.9173 0.497417
\(575\) 0.636773 0.0265553
\(576\) 0 0
\(577\) 9.12732 0.379975 0.189988 0.981786i \(-0.439155\pi\)
0.189988 + 0.981786i \(0.439155\pi\)
\(578\) 83.9467 3.49172
\(579\) 0 0
\(580\) 30.9389 1.28467
\(581\) 8.44988 0.350560
\(582\) 0 0
\(583\) −2.84590 −0.117865
\(584\) 107.294 4.43985
\(585\) 0 0
\(586\) −84.7189 −3.49971
\(587\) −2.10109 −0.0867213 −0.0433607 0.999059i \(-0.513806\pi\)
−0.0433607 + 0.999059i \(0.513806\pi\)
\(588\) 0 0
\(589\) −25.1629 −1.03682
\(590\) 15.6266 0.643339
\(591\) 0 0
\(592\) 10.0409 0.412680
\(593\) −10.7003 −0.439407 −0.219704 0.975567i \(-0.570509\pi\)
−0.219704 + 0.975567i \(0.570509\pi\)
\(594\) 0 0
\(595\) −4.05186 −0.166110
\(596\) −99.9249 −4.09308
\(597\) 0 0
\(598\) −2.30249 −0.0941558
\(599\) −20.4277 −0.834654 −0.417327 0.908756i \(-0.637033\pi\)
−0.417327 + 0.908756i \(0.637033\pi\)
\(600\) 0 0
\(601\) −31.1566 −1.27090 −0.635452 0.772141i \(-0.719186\pi\)
−0.635452 + 0.772141i \(0.719186\pi\)
\(602\) −11.9855 −0.488494
\(603\) 0 0
\(604\) 47.9352 1.95045
\(605\) 10.2040 0.414853
\(606\) 0 0
\(607\) −6.90856 −0.280410 −0.140205 0.990123i \(-0.544776\pi\)
−0.140205 + 0.990123i \(0.544776\pi\)
\(608\) 66.9307 2.71440
\(609\) 0 0
\(610\) 13.7203 0.555519
\(611\) −8.10467 −0.327880
\(612\) 0 0
\(613\) 30.4773 1.23097 0.615484 0.788150i \(-0.288961\pi\)
0.615484 + 0.788150i \(0.288961\pi\)
\(614\) 29.4189 1.18725
\(615\) 0 0
\(616\) 5.14820 0.207427
\(617\) 38.7431 1.55974 0.779869 0.625943i \(-0.215286\pi\)
0.779869 + 0.625943i \(0.215286\pi\)
\(618\) 0 0
\(619\) −10.1914 −0.409626 −0.204813 0.978801i \(-0.565659\pi\)
−0.204813 + 0.978801i \(0.565659\pi\)
\(620\) 50.2955 2.01992
\(621\) 0 0
\(622\) −16.5190 −0.662350
\(623\) −0.587818 −0.0235504
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 12.8977 0.515494
\(627\) 0 0
\(628\) −7.30826 −0.291631
\(629\) 4.36130 0.173897
\(630\) 0 0
\(631\) 15.2025 0.605201 0.302600 0.953117i \(-0.402145\pi\)
0.302600 + 0.953117i \(0.402145\pi\)
\(632\) −115.101 −4.57849
\(633\) 0 0
\(634\) 88.0154 3.49554
\(635\) 11.9731 0.475136
\(636\) 0 0
\(637\) 8.74634 0.346543
\(638\) −13.6372 −0.539901
\(639\) 0 0
\(640\) −46.4634 −1.83663
\(641\) −19.5368 −0.771657 −0.385829 0.922570i \(-0.626084\pi\)
−0.385829 + 0.922570i \(0.626084\pi\)
\(642\) 0 0
\(643\) −3.76227 −0.148369 −0.0741847 0.997245i \(-0.523635\pi\)
−0.0741847 + 0.997245i \(0.523635\pi\)
\(644\) 2.08427 0.0821316
\(645\) 0 0
\(646\) 52.8287 2.07852
\(647\) 44.2880 1.74114 0.870570 0.492044i \(-0.163750\pi\)
0.870570 + 0.492044i \(0.163750\pi\)
\(648\) 0 0
\(649\) −5.06770 −0.198924
\(650\) −3.61587 −0.141826
\(651\) 0 0
\(652\) −24.0961 −0.943677
\(653\) −5.24704 −0.205333 −0.102666 0.994716i \(-0.532737\pi\)
−0.102666 + 0.994716i \(0.532737\pi\)
\(654\) 0 0
\(655\) −21.0827 −0.823770
\(656\) 116.951 4.56616
\(657\) 0 0
\(658\) 9.97162 0.388734
\(659\) 8.67677 0.337999 0.169000 0.985616i \(-0.445946\pi\)
0.169000 + 0.985616i \(0.445946\pi\)
\(660\) 0 0
\(661\) 25.1920 0.979856 0.489928 0.871763i \(-0.337023\pi\)
0.489928 + 0.871763i \(0.337023\pi\)
\(662\) −14.4241 −0.560608
\(663\) 0 0
\(664\) 141.116 5.47634
\(665\) −1.63757 −0.0635023
\(666\) 0 0
\(667\) −3.53804 −0.136994
\(668\) −56.3658 −2.18086
\(669\) 0 0
\(670\) −21.4182 −0.827456
\(671\) −4.44947 −0.171770
\(672\) 0 0
\(673\) 23.2928 0.897870 0.448935 0.893564i \(-0.351803\pi\)
0.448935 + 0.893564i \(0.351803\pi\)
\(674\) 7.33143 0.282396
\(675\) 0 0
\(676\) −62.7690 −2.41419
\(677\) −20.2570 −0.778541 −0.389270 0.921124i \(-0.627273\pi\)
−0.389270 + 0.921124i \(0.627273\pi\)
\(678\) 0 0
\(679\) 3.44708 0.132287
\(680\) −67.6674 −2.59492
\(681\) 0 0
\(682\) −22.1692 −0.848901
\(683\) 48.9807 1.87419 0.937096 0.349071i \(-0.113503\pi\)
0.937096 + 0.349071i \(0.113503\pi\)
\(684\) 0 0
\(685\) 21.7300 0.830261
\(686\) −22.0810 −0.843055
\(687\) 0 0
\(688\) −117.621 −4.48426
\(689\) 4.19263 0.159727
\(690\) 0 0
\(691\) −13.3790 −0.508959 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(692\) 24.8793 0.945769
\(693\) 0 0
\(694\) 30.5072 1.15804
\(695\) 7.18857 0.272678
\(696\) 0 0
\(697\) 50.7978 1.92410
\(698\) −16.0089 −0.605945
\(699\) 0 0
\(700\) 3.27317 0.123714
\(701\) 3.46359 0.130818 0.0654089 0.997859i \(-0.479165\pi\)
0.0654089 + 0.997859i \(0.479165\pi\)
\(702\) 0 0
\(703\) 1.76263 0.0664790
\(704\) 30.6507 1.15519
\(705\) 0 0
\(706\) 17.4695 0.657472
\(707\) 6.25532 0.235255
\(708\) 0 0
\(709\) 11.6352 0.436970 0.218485 0.975840i \(-0.429888\pi\)
0.218485 + 0.975840i \(0.429888\pi\)
\(710\) −41.1459 −1.54418
\(711\) 0 0
\(712\) −9.81674 −0.367898
\(713\) −5.75159 −0.215399
\(714\) 0 0
\(715\) 1.17262 0.0438536
\(716\) −95.4028 −3.56537
\(717\) 0 0
\(718\) 93.4272 3.48667
\(719\) 5.56903 0.207690 0.103845 0.994594i \(-0.466885\pi\)
0.103845 + 0.994594i \(0.466885\pi\)
\(720\) 0 0
\(721\) 1.07125 0.0398954
\(722\) −30.9193 −1.15070
\(723\) 0 0
\(724\) 4.27075 0.158721
\(725\) −5.55621 −0.206352
\(726\) 0 0
\(727\) −18.8775 −0.700126 −0.350063 0.936726i \(-0.613840\pi\)
−0.350063 + 0.936726i \(0.613840\pi\)
\(728\) −7.58443 −0.281098
\(729\) 0 0
\(730\) −30.0682 −1.11288
\(731\) −51.0889 −1.88959
\(732\) 0 0
\(733\) 17.6145 0.650607 0.325304 0.945610i \(-0.394534\pi\)
0.325304 + 0.945610i \(0.394534\pi\)
\(734\) −36.9994 −1.36567
\(735\) 0 0
\(736\) 15.2986 0.563914
\(737\) 6.94588 0.255855
\(738\) 0 0
\(739\) 16.1072 0.592515 0.296257 0.955108i \(-0.404261\pi\)
0.296257 + 0.955108i \(0.404261\pi\)
\(740\) −3.52314 −0.129513
\(741\) 0 0
\(742\) −5.15843 −0.189372
\(743\) 29.8955 1.09676 0.548380 0.836229i \(-0.315245\pi\)
0.548380 + 0.836229i \(0.315245\pi\)
\(744\) 0 0
\(745\) 17.9452 0.657460
\(746\) −28.7262 −1.05174
\(747\) 0 0
\(748\) 34.2439 1.25208
\(749\) 2.21212 0.0808292
\(750\) 0 0
\(751\) 9.25061 0.337559 0.168780 0.985654i \(-0.446017\pi\)
0.168780 + 0.985654i \(0.446017\pi\)
\(752\) 97.8572 3.56849
\(753\) 0 0
\(754\) 20.0906 0.731655
\(755\) −8.60851 −0.313296
\(756\) 0 0
\(757\) −32.8131 −1.19261 −0.596307 0.802756i \(-0.703366\pi\)
−0.596307 + 0.802756i \(0.703366\pi\)
\(758\) 80.0651 2.90810
\(759\) 0 0
\(760\) −27.3479 −0.992014
\(761\) −24.7217 −0.896160 −0.448080 0.893994i \(-0.647892\pi\)
−0.448080 + 0.893994i \(0.647892\pi\)
\(762\) 0 0
\(763\) 3.25610 0.117879
\(764\) −127.222 −4.60274
\(765\) 0 0
\(766\) −73.5544 −2.65763
\(767\) 7.46583 0.269576
\(768\) 0 0
\(769\) 22.3736 0.806813 0.403407 0.915021i \(-0.367826\pi\)
0.403407 + 0.915021i \(0.367826\pi\)
\(770\) −1.44274 −0.0519928
\(771\) 0 0
\(772\) −77.7649 −2.79882
\(773\) −49.9309 −1.79589 −0.897945 0.440107i \(-0.854941\pi\)
−0.897945 + 0.440107i \(0.854941\pi\)
\(774\) 0 0
\(775\) −9.03241 −0.324454
\(776\) 57.5673 2.06655
\(777\) 0 0
\(778\) 96.0615 3.44397
\(779\) 20.5301 0.735566
\(780\) 0 0
\(781\) 13.3435 0.477469
\(782\) 12.0753 0.431811
\(783\) 0 0
\(784\) −105.605 −3.77160
\(785\) 1.31247 0.0468439
\(786\) 0 0
\(787\) −35.1401 −1.25261 −0.626304 0.779579i \(-0.715433\pi\)
−0.626304 + 0.779579i \(0.715433\pi\)
\(788\) 80.5858 2.87075
\(789\) 0 0
\(790\) 32.2563 1.14763
\(791\) −8.31295 −0.295575
\(792\) 0 0
\(793\) 6.55505 0.232777
\(794\) 97.4955 3.45999
\(795\) 0 0
\(796\) 53.4933 1.89602
\(797\) 23.9334 0.847766 0.423883 0.905717i \(-0.360667\pi\)
0.423883 + 0.905717i \(0.360667\pi\)
\(798\) 0 0
\(799\) 42.5044 1.50370
\(800\) 24.0252 0.849421
\(801\) 0 0
\(802\) 63.2639 2.23393
\(803\) 9.75108 0.344108
\(804\) 0 0
\(805\) −0.374306 −0.0131926
\(806\) 32.6600 1.15040
\(807\) 0 0
\(808\) 104.466 3.67509
\(809\) 42.2870 1.48673 0.743366 0.668885i \(-0.233228\pi\)
0.743366 + 0.668885i \(0.233228\pi\)
\(810\) 0 0
\(811\) 13.8234 0.485406 0.242703 0.970101i \(-0.421966\pi\)
0.242703 + 0.970101i \(0.421966\pi\)
\(812\) −18.1864 −0.638219
\(813\) 0 0
\(814\) 1.55292 0.0544299
\(815\) 4.32734 0.151580
\(816\) 0 0
\(817\) −20.6477 −0.722372
\(818\) 68.4651 2.39383
\(819\) 0 0
\(820\) −41.0354 −1.43302
\(821\) −32.0266 −1.11774 −0.558868 0.829257i \(-0.688764\pi\)
−0.558868 + 0.829257i \(0.688764\pi\)
\(822\) 0 0
\(823\) 38.3450 1.33662 0.668311 0.743882i \(-0.267018\pi\)
0.668311 + 0.743882i \(0.267018\pi\)
\(824\) 17.8902 0.623233
\(825\) 0 0
\(826\) −9.18562 −0.319609
\(827\) 20.0855 0.698440 0.349220 0.937041i \(-0.386447\pi\)
0.349220 + 0.937041i \(0.386447\pi\)
\(828\) 0 0
\(829\) −43.3908 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(830\) −39.5465 −1.37268
\(831\) 0 0
\(832\) −45.1552 −1.56548
\(833\) −45.8697 −1.58929
\(834\) 0 0
\(835\) 10.1225 0.350305
\(836\) 13.8398 0.478658
\(837\) 0 0
\(838\) 105.101 3.63064
\(839\) −22.3152 −0.770406 −0.385203 0.922832i \(-0.625869\pi\)
−0.385203 + 0.922832i \(0.625869\pi\)
\(840\) 0 0
\(841\) 1.87148 0.0645336
\(842\) 26.2272 0.903848
\(843\) 0 0
\(844\) 16.6363 0.572645
\(845\) 11.2725 0.387785
\(846\) 0 0
\(847\) −5.99812 −0.206098
\(848\) −50.6226 −1.73839
\(849\) 0 0
\(850\) 18.9632 0.650434
\(851\) 0.402892 0.0138110
\(852\) 0 0
\(853\) −43.5152 −1.48993 −0.744965 0.667103i \(-0.767534\pi\)
−0.744965 + 0.667103i \(0.767534\pi\)
\(854\) −8.06504 −0.275980
\(855\) 0 0
\(856\) 36.9431 1.26269
\(857\) −19.0470 −0.650633 −0.325316 0.945605i \(-0.605471\pi\)
−0.325316 + 0.945605i \(0.605471\pi\)
\(858\) 0 0
\(859\) −0.896256 −0.0305799 −0.0152899 0.999883i \(-0.504867\pi\)
−0.0152899 + 0.999883i \(0.504867\pi\)
\(860\) 41.2705 1.40731
\(861\) 0 0
\(862\) −42.3119 −1.44115
\(863\) 22.6236 0.770117 0.385058 0.922892i \(-0.374181\pi\)
0.385058 + 0.922892i \(0.374181\pi\)
\(864\) 0 0
\(865\) −4.46799 −0.151916
\(866\) −47.9968 −1.63100
\(867\) 0 0
\(868\) −29.5646 −1.00349
\(869\) −10.4606 −0.354853
\(870\) 0 0
\(871\) −10.2328 −0.346725
\(872\) 54.3778 1.84146
\(873\) 0 0
\(874\) 4.88025 0.165077
\(875\) −0.587818 −0.0198719
\(876\) 0 0
\(877\) −6.52027 −0.220174 −0.110087 0.993922i \(-0.535113\pi\)
−0.110087 + 0.993922i \(0.535113\pi\)
\(878\) −42.7871 −1.44400
\(879\) 0 0
\(880\) −14.1584 −0.477281
\(881\) 22.7030 0.764884 0.382442 0.923979i \(-0.375083\pi\)
0.382442 + 0.923979i \(0.375083\pi\)
\(882\) 0 0
\(883\) −45.7804 −1.54063 −0.770317 0.637662i \(-0.779902\pi\)
−0.770317 + 0.637662i \(0.779902\pi\)
\(884\) −50.4488 −1.69678
\(885\) 0 0
\(886\) 65.9686 2.21626
\(887\) 45.0961 1.51418 0.757089 0.653311i \(-0.226621\pi\)
0.757089 + 0.653311i \(0.226621\pi\)
\(888\) 0 0
\(889\) −7.03798 −0.236046
\(890\) 2.75106 0.0922158
\(891\) 0 0
\(892\) −130.710 −4.37648
\(893\) 17.1783 0.574849
\(894\) 0 0
\(895\) 17.1331 0.572696
\(896\) 27.3120 0.912430
\(897\) 0 0
\(898\) −73.5492 −2.45437
\(899\) 50.1860 1.67380
\(900\) 0 0
\(901\) −21.9880 −0.732527
\(902\) 18.0875 0.602248
\(903\) 0 0
\(904\) −138.829 −4.61738
\(905\) −0.766969 −0.0254949
\(906\) 0 0
\(907\) 43.5083 1.44467 0.722335 0.691543i \(-0.243069\pi\)
0.722335 + 0.691543i \(0.243069\pi\)
\(908\) −63.2557 −2.09922
\(909\) 0 0
\(910\) 2.12548 0.0704588
\(911\) −14.1257 −0.468006 −0.234003 0.972236i \(-0.575183\pi\)
−0.234003 + 0.972236i \(0.575183\pi\)
\(912\) 0 0
\(913\) 12.8249 0.424441
\(914\) −65.4638 −2.16535
\(915\) 0 0
\(916\) 30.9006 1.02098
\(917\) 12.3928 0.409246
\(918\) 0 0
\(919\) 7.32776 0.241721 0.120860 0.992670i \(-0.461435\pi\)
0.120860 + 0.992670i \(0.461435\pi\)
\(920\) −6.25103 −0.206090
\(921\) 0 0
\(922\) 39.4412 1.29893
\(923\) −19.6580 −0.647050
\(924\) 0 0
\(925\) 0.632709 0.0208034
\(926\) 14.5592 0.478446
\(927\) 0 0
\(928\) −133.489 −4.38200
\(929\) −35.9672 −1.18004 −0.590022 0.807387i \(-0.700881\pi\)
−0.590022 + 0.807387i \(0.700881\pi\)
\(930\) 0 0
\(931\) −18.5383 −0.607570
\(932\) −30.7498 −1.00724
\(933\) 0 0
\(934\) 65.2217 2.13412
\(935\) −6.14975 −0.201118
\(936\) 0 0
\(937\) 34.6839 1.13307 0.566536 0.824037i \(-0.308283\pi\)
0.566536 + 0.824037i \(0.308283\pi\)
\(938\) 12.5900 0.411078
\(939\) 0 0
\(940\) −34.3359 −1.11991
\(941\) 22.1865 0.723260 0.361630 0.932322i \(-0.382220\pi\)
0.361630 + 0.932322i \(0.382220\pi\)
\(942\) 0 0
\(943\) 4.69264 0.152813
\(944\) −90.1438 −2.93393
\(945\) 0 0
\(946\) −18.1911 −0.591445
\(947\) −30.6653 −0.996488 −0.498244 0.867037i \(-0.666022\pi\)
−0.498244 + 0.867037i \(0.666022\pi\)
\(948\) 0 0
\(949\) −14.3655 −0.466323
\(950\) 7.66404 0.248654
\(951\) 0 0
\(952\) 39.7761 1.28915
\(953\) −16.1294 −0.522483 −0.261241 0.965273i \(-0.584132\pi\)
−0.261241 + 0.965273i \(0.584132\pi\)
\(954\) 0 0
\(955\) 22.8474 0.739324
\(956\) −70.9933 −2.29609
\(957\) 0 0
\(958\) 14.3883 0.464865
\(959\) −12.7733 −0.412471
\(960\) 0 0
\(961\) 50.5844 1.63175
\(962\) −2.28780 −0.0737615
\(963\) 0 0
\(964\) 103.616 3.33725
\(965\) 13.9655 0.449566
\(966\) 0 0
\(967\) 37.6822 1.21178 0.605888 0.795550i \(-0.292818\pi\)
0.605888 + 0.795550i \(0.292818\pi\)
\(968\) −100.170 −3.21960
\(969\) 0 0
\(970\) −16.1328 −0.517993
\(971\) 8.73227 0.280232 0.140116 0.990135i \(-0.455252\pi\)
0.140116 + 0.990135i \(0.455252\pi\)
\(972\) 0 0
\(973\) −4.22557 −0.135465
\(974\) −20.0637 −0.642883
\(975\) 0 0
\(976\) −79.1468 −2.53343
\(977\) −27.1633 −0.869030 −0.434515 0.900665i \(-0.643080\pi\)
−0.434515 + 0.900665i \(0.643080\pi\)
\(978\) 0 0
\(979\) −0.892165 −0.0285137
\(980\) 37.0544 1.18366
\(981\) 0 0
\(982\) 48.9417 1.56179
\(983\) 36.1763 1.15385 0.576923 0.816798i \(-0.304253\pi\)
0.576923 + 0.816798i \(0.304253\pi\)
\(984\) 0 0
\(985\) −14.4721 −0.461121
\(986\) −105.364 −3.35547
\(987\) 0 0
\(988\) −20.3890 −0.648661
\(989\) −4.71953 −0.150072
\(990\) 0 0
\(991\) 39.8979 1.26740 0.633699 0.773580i \(-0.281536\pi\)
0.633699 + 0.773580i \(0.281536\pi\)
\(992\) −217.006 −6.88994
\(993\) 0 0
\(994\) 24.1863 0.767142
\(995\) −9.60667 −0.304552
\(996\) 0 0
\(997\) −21.5114 −0.681272 −0.340636 0.940195i \(-0.610642\pi\)
−0.340636 + 0.940195i \(0.610642\pi\)
\(998\) 76.1522 2.41056
\(999\) 0 0
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 4005.2.a.o.1.7 7
3.2 odd 2 445.2.a.f.1.1 7
12.11 even 2 7120.2.a.bj.1.3 7
15.14 odd 2 2225.2.a.k.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.1 7 3.2 odd 2
2225.2.a.k.1.7 7 15.14 odd 2
4005.2.a.o.1.7 7 1.1 even 1 trivial
7120.2.a.bj.1.3 7 12.11 even 2