Properties

Label 405.4.a.n.1.6
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 44 x^{5} + 74 x^{4} + 479 x^{3} - 460 x^{2} - 1200 x + 288\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.26178\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.26178 q^{2} +10.1628 q^{4} +5.00000 q^{5} +30.7639 q^{7} +9.21718 q^{8} +O(q^{10})\) \(q+4.26178 q^{2} +10.1628 q^{4} +5.00000 q^{5} +30.7639 q^{7} +9.21718 q^{8} +21.3089 q^{10} -40.7146 q^{11} +63.2178 q^{13} +131.109 q^{14} -42.0204 q^{16} +6.58990 q^{17} +75.3803 q^{19} +50.8138 q^{20} -173.517 q^{22} +62.3628 q^{23} +25.0000 q^{25} +269.420 q^{26} +312.646 q^{28} -49.6084 q^{29} +103.004 q^{31} -252.819 q^{32} +28.0847 q^{34} +153.820 q^{35} -282.029 q^{37} +321.254 q^{38} +46.0859 q^{40} +157.540 q^{41} -337.814 q^{43} -413.773 q^{44} +265.776 q^{46} -44.5158 q^{47} +603.420 q^{49} +106.544 q^{50} +642.467 q^{52} -26.2752 q^{53} -203.573 q^{55} +283.557 q^{56} -211.420 q^{58} +425.926 q^{59} +850.595 q^{61} +438.981 q^{62} -741.296 q^{64} +316.089 q^{65} +96.3076 q^{67} +66.9715 q^{68} +655.545 q^{70} -952.164 q^{71} -50.8558 q^{73} -1201.95 q^{74} +766.071 q^{76} -1252.54 q^{77} -197.279 q^{79} -210.102 q^{80} +671.400 q^{82} +197.739 q^{83} +32.9495 q^{85} -1439.69 q^{86} -375.274 q^{88} -1364.54 q^{89} +1944.83 q^{91} +633.778 q^{92} -189.717 q^{94} +376.901 q^{95} -1431.16 q^{97} +2571.64 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + O(q^{10}) \) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.26178 1.50677 0.753383 0.657582i \(-0.228421\pi\)
0.753383 + 0.657582i \(0.228421\pi\)
\(3\) 0 0
\(4\) 10.1628 1.27034
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 30.7639 1.66110 0.830548 0.556947i \(-0.188027\pi\)
0.830548 + 0.556947i \(0.188027\pi\)
\(8\) 9.21718 0.407346
\(9\) 0 0
\(10\) 21.3089 0.673846
\(11\) −40.7146 −1.11599 −0.557996 0.829843i \(-0.688430\pi\)
−0.557996 + 0.829843i \(0.688430\pi\)
\(12\) 0 0
\(13\) 63.2178 1.34873 0.674364 0.738399i \(-0.264418\pi\)
0.674364 + 0.738399i \(0.264418\pi\)
\(14\) 131.109 2.50288
\(15\) 0 0
\(16\) −42.0204 −0.656569
\(17\) 6.58990 0.0940168 0.0470084 0.998894i \(-0.485031\pi\)
0.0470084 + 0.998894i \(0.485031\pi\)
\(18\) 0 0
\(19\) 75.3803 0.910180 0.455090 0.890445i \(-0.349607\pi\)
0.455090 + 0.890445i \(0.349607\pi\)
\(20\) 50.8138 0.568115
\(21\) 0 0
\(22\) −173.517 −1.68154
\(23\) 62.3628 0.565371 0.282686 0.959213i \(-0.408775\pi\)
0.282686 + 0.959213i \(0.408775\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 269.420 2.03222
\(27\) 0 0
\(28\) 312.646 2.11016
\(29\) −49.6084 −0.317657 −0.158828 0.987306i \(-0.550772\pi\)
−0.158828 + 0.987306i \(0.550772\pi\)
\(30\) 0 0
\(31\) 103.004 0.596777 0.298389 0.954444i \(-0.403551\pi\)
0.298389 + 0.954444i \(0.403551\pi\)
\(32\) −252.819 −1.39664
\(33\) 0 0
\(34\) 28.0847 0.141661
\(35\) 153.820 0.742865
\(36\) 0 0
\(37\) −282.029 −1.25312 −0.626559 0.779374i \(-0.715537\pi\)
−0.626559 + 0.779374i \(0.715537\pi\)
\(38\) 321.254 1.37143
\(39\) 0 0
\(40\) 46.0859 0.182171
\(41\) 157.540 0.600088 0.300044 0.953925i \(-0.402999\pi\)
0.300044 + 0.953925i \(0.402999\pi\)
\(42\) 0 0
\(43\) −337.814 −1.19805 −0.599025 0.800730i \(-0.704445\pi\)
−0.599025 + 0.800730i \(0.704445\pi\)
\(44\) −413.773 −1.41770
\(45\) 0 0
\(46\) 265.776 0.851882
\(47\) −44.5158 −0.138155 −0.0690777 0.997611i \(-0.522006\pi\)
−0.0690777 + 0.997611i \(0.522006\pi\)
\(48\) 0 0
\(49\) 603.420 1.75924
\(50\) 106.544 0.301353
\(51\) 0 0
\(52\) 642.467 1.71335
\(53\) −26.2752 −0.0680978 −0.0340489 0.999420i \(-0.510840\pi\)
−0.0340489 + 0.999420i \(0.510840\pi\)
\(54\) 0 0
\(55\) −203.573 −0.499087
\(56\) 283.557 0.676641
\(57\) 0 0
\(58\) −211.420 −0.478634
\(59\) 425.926 0.939845 0.469923 0.882708i \(-0.344282\pi\)
0.469923 + 0.882708i \(0.344282\pi\)
\(60\) 0 0
\(61\) 850.595 1.78537 0.892684 0.450682i \(-0.148819\pi\)
0.892684 + 0.450682i \(0.148819\pi\)
\(62\) 438.981 0.899204
\(63\) 0 0
\(64\) −741.296 −1.44784
\(65\) 316.089 0.603170
\(66\) 0 0
\(67\) 96.3076 0.175610 0.0878048 0.996138i \(-0.472015\pi\)
0.0878048 + 0.996138i \(0.472015\pi\)
\(68\) 66.9715 0.119434
\(69\) 0 0
\(70\) 655.545 1.11932
\(71\) −952.164 −1.59156 −0.795782 0.605583i \(-0.792940\pi\)
−0.795782 + 0.605583i \(0.792940\pi\)
\(72\) 0 0
\(73\) −50.8558 −0.0815373 −0.0407686 0.999169i \(-0.512981\pi\)
−0.0407686 + 0.999169i \(0.512981\pi\)
\(74\) −1201.95 −1.88815
\(75\) 0 0
\(76\) 766.071 1.15624
\(77\) −1252.54 −1.85377
\(78\) 0 0
\(79\) −197.279 −0.280957 −0.140479 0.990084i \(-0.544864\pi\)
−0.140479 + 0.990084i \(0.544864\pi\)
\(80\) −210.102 −0.293627
\(81\) 0 0
\(82\) 671.400 0.904192
\(83\) 197.739 0.261502 0.130751 0.991415i \(-0.458261\pi\)
0.130751 + 0.991415i \(0.458261\pi\)
\(84\) 0 0
\(85\) 32.9495 0.0420456
\(86\) −1439.69 −1.80518
\(87\) 0 0
\(88\) −375.274 −0.454595
\(89\) −1364.54 −1.62519 −0.812593 0.582832i \(-0.801944\pi\)
−0.812593 + 0.582832i \(0.801944\pi\)
\(90\) 0 0
\(91\) 1944.83 2.24037
\(92\) 633.778 0.718216
\(93\) 0 0
\(94\) −189.717 −0.208168
\(95\) 376.901 0.407045
\(96\) 0 0
\(97\) −1431.16 −1.49806 −0.749031 0.662535i \(-0.769481\pi\)
−0.749031 + 0.662535i \(0.769481\pi\)
\(98\) 2571.64 2.65076
\(99\) 0 0
\(100\) 254.069 0.254069
\(101\) −1107.62 −1.09121 −0.545604 0.838043i \(-0.683700\pi\)
−0.545604 + 0.838043i \(0.683700\pi\)
\(102\) 0 0
\(103\) 528.874 0.505937 0.252969 0.967475i \(-0.418593\pi\)
0.252969 + 0.967475i \(0.418593\pi\)
\(104\) 582.690 0.549399
\(105\) 0 0
\(106\) −111.979 −0.102607
\(107\) −490.910 −0.443533 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(108\) 0 0
\(109\) −351.634 −0.308994 −0.154497 0.987993i \(-0.549376\pi\)
−0.154497 + 0.987993i \(0.549376\pi\)
\(110\) −867.584 −0.752008
\(111\) 0 0
\(112\) −1292.71 −1.09063
\(113\) −1762.45 −1.46724 −0.733618 0.679563i \(-0.762170\pi\)
−0.733618 + 0.679563i \(0.762170\pi\)
\(114\) 0 0
\(115\) 311.814 0.252842
\(116\) −504.158 −0.403533
\(117\) 0 0
\(118\) 1815.20 1.41613
\(119\) 202.731 0.156171
\(120\) 0 0
\(121\) 326.681 0.245440
\(122\) 3625.05 2.69013
\(123\) 0 0
\(124\) 1046.81 0.758113
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1506.12 −1.05234 −0.526169 0.850380i \(-0.676372\pi\)
−0.526169 + 0.850380i \(0.676372\pi\)
\(128\) −1136.69 −0.784920
\(129\) 0 0
\(130\) 1347.10 0.908835
\(131\) −1275.12 −0.850444 −0.425222 0.905089i \(-0.639804\pi\)
−0.425222 + 0.905089i \(0.639804\pi\)
\(132\) 0 0
\(133\) 2318.99 1.51190
\(134\) 410.441 0.264603
\(135\) 0 0
\(136\) 60.7403 0.0382973
\(137\) −1439.42 −0.897652 −0.448826 0.893619i \(-0.648158\pi\)
−0.448826 + 0.893619i \(0.648158\pi\)
\(138\) 0 0
\(139\) −2539.50 −1.54962 −0.774811 0.632193i \(-0.782155\pi\)
−0.774811 + 0.632193i \(0.782155\pi\)
\(140\) 1563.23 0.943694
\(141\) 0 0
\(142\) −4057.91 −2.39812
\(143\) −2573.89 −1.50517
\(144\) 0 0
\(145\) −248.042 −0.142060
\(146\) −216.736 −0.122858
\(147\) 0 0
\(148\) −2866.20 −1.59189
\(149\) −93.7592 −0.0515507 −0.0257754 0.999668i \(-0.508205\pi\)
−0.0257754 + 0.999668i \(0.508205\pi\)
\(150\) 0 0
\(151\) −1068.13 −0.575650 −0.287825 0.957683i \(-0.592932\pi\)
−0.287825 + 0.957683i \(0.592932\pi\)
\(152\) 694.794 0.370758
\(153\) 0 0
\(154\) −5338.06 −2.79320
\(155\) 515.021 0.266887
\(156\) 0 0
\(157\) 181.715 0.0923721 0.0461861 0.998933i \(-0.485293\pi\)
0.0461861 + 0.998933i \(0.485293\pi\)
\(158\) −840.760 −0.423337
\(159\) 0 0
\(160\) −1264.10 −0.624598
\(161\) 1918.52 0.939136
\(162\) 0 0
\(163\) −1103.07 −0.530056 −0.265028 0.964241i \(-0.585381\pi\)
−0.265028 + 0.964241i \(0.585381\pi\)
\(164\) 1601.04 0.762319
\(165\) 0 0
\(166\) 842.718 0.394022
\(167\) 4041.70 1.87279 0.936396 0.350945i \(-0.114140\pi\)
0.936396 + 0.350945i \(0.114140\pi\)
\(168\) 0 0
\(169\) 1799.49 0.819067
\(170\) 140.423 0.0633529
\(171\) 0 0
\(172\) −3433.12 −1.52194
\(173\) 2747.76 1.20756 0.603780 0.797151i \(-0.293660\pi\)
0.603780 + 0.797151i \(0.293660\pi\)
\(174\) 0 0
\(175\) 769.098 0.332219
\(176\) 1710.85 0.732727
\(177\) 0 0
\(178\) −5815.39 −2.44877
\(179\) 2838.32 1.18517 0.592587 0.805506i \(-0.298107\pi\)
0.592587 + 0.805506i \(0.298107\pi\)
\(180\) 0 0
\(181\) 3442.25 1.41359 0.706796 0.707417i \(-0.250140\pi\)
0.706796 + 0.707417i \(0.250140\pi\)
\(182\) 8288.43 3.37571
\(183\) 0 0
\(184\) 574.809 0.230302
\(185\) −1410.15 −0.560411
\(186\) 0 0
\(187\) −268.305 −0.104922
\(188\) −452.403 −0.175505
\(189\) 0 0
\(190\) 1606.27 0.613322
\(191\) 746.295 0.282723 0.141361 0.989958i \(-0.454852\pi\)
0.141361 + 0.989958i \(0.454852\pi\)
\(192\) 0 0
\(193\) 107.908 0.0402454 0.0201227 0.999798i \(-0.493594\pi\)
0.0201227 + 0.999798i \(0.493594\pi\)
\(194\) −6099.28 −2.25723
\(195\) 0 0
\(196\) 6132.41 2.23484
\(197\) 3361.02 1.21555 0.607774 0.794110i \(-0.292063\pi\)
0.607774 + 0.794110i \(0.292063\pi\)
\(198\) 0 0
\(199\) 1368.99 0.487663 0.243831 0.969818i \(-0.421596\pi\)
0.243831 + 0.969818i \(0.421596\pi\)
\(200\) 230.430 0.0814692
\(201\) 0 0
\(202\) −4720.42 −1.64419
\(203\) −1526.15 −0.527658
\(204\) 0 0
\(205\) 787.700 0.268368
\(206\) 2253.94 0.762329
\(207\) 0 0
\(208\) −2656.44 −0.885534
\(209\) −3069.08 −1.01575
\(210\) 0 0
\(211\) 2502.55 0.816504 0.408252 0.912869i \(-0.366138\pi\)
0.408252 + 0.912869i \(0.366138\pi\)
\(212\) −267.029 −0.0865077
\(213\) 0 0
\(214\) −2092.15 −0.668300
\(215\) −1689.07 −0.535784
\(216\) 0 0
\(217\) 3168.81 0.991305
\(218\) −1498.58 −0.465582
\(219\) 0 0
\(220\) −2068.86 −0.634013
\(221\) 416.599 0.126803
\(222\) 0 0
\(223\) 1874.53 0.562904 0.281452 0.959575i \(-0.409184\pi\)
0.281452 + 0.959575i \(0.409184\pi\)
\(224\) −7777.72 −2.31996
\(225\) 0 0
\(226\) −7511.18 −2.21078
\(227\) −5078.74 −1.48497 −0.742484 0.669864i \(-0.766353\pi\)
−0.742484 + 0.669864i \(0.766353\pi\)
\(228\) 0 0
\(229\) 865.865 0.249860 0.124930 0.992166i \(-0.460129\pi\)
0.124930 + 0.992166i \(0.460129\pi\)
\(230\) 1328.88 0.380973
\(231\) 0 0
\(232\) −457.249 −0.129396
\(233\) −1142.10 −0.321122 −0.160561 0.987026i \(-0.551330\pi\)
−0.160561 + 0.987026i \(0.551330\pi\)
\(234\) 0 0
\(235\) −222.579 −0.0617849
\(236\) 4328.58 1.19393
\(237\) 0 0
\(238\) 863.995 0.235313
\(239\) 3149.37 0.852366 0.426183 0.904637i \(-0.359858\pi\)
0.426183 + 0.904637i \(0.359858\pi\)
\(240\) 0 0
\(241\) 4504.37 1.20395 0.601975 0.798515i \(-0.294381\pi\)
0.601975 + 0.798515i \(0.294381\pi\)
\(242\) 1392.24 0.369821
\(243\) 0 0
\(244\) 8644.39 2.26803
\(245\) 3017.10 0.786756
\(246\) 0 0
\(247\) 4765.38 1.22759
\(248\) 949.409 0.243095
\(249\) 0 0
\(250\) 532.722 0.134769
\(251\) −886.861 −0.223021 −0.111510 0.993763i \(-0.535569\pi\)
−0.111510 + 0.993763i \(0.535569\pi\)
\(252\) 0 0
\(253\) −2539.08 −0.630950
\(254\) −6418.77 −1.58563
\(255\) 0 0
\(256\) 1086.07 0.265153
\(257\) −2262.04 −0.549035 −0.274517 0.961582i \(-0.588518\pi\)
−0.274517 + 0.961582i \(0.588518\pi\)
\(258\) 0 0
\(259\) −8676.33 −2.08155
\(260\) 3212.34 0.766233
\(261\) 0 0
\(262\) −5434.30 −1.28142
\(263\) −7280.62 −1.70701 −0.853503 0.521088i \(-0.825526\pi\)
−0.853503 + 0.521088i \(0.825526\pi\)
\(264\) 0 0
\(265\) −131.376 −0.0304543
\(266\) 9883.04 2.27807
\(267\) 0 0
\(268\) 978.750 0.223085
\(269\) 106.781 0.0242028 0.0121014 0.999927i \(-0.496148\pi\)
0.0121014 + 0.999927i \(0.496148\pi\)
\(270\) 0 0
\(271\) 5908.85 1.32449 0.662246 0.749287i \(-0.269603\pi\)
0.662246 + 0.749287i \(0.269603\pi\)
\(272\) −276.910 −0.0617285
\(273\) 0 0
\(274\) −6134.51 −1.35255
\(275\) −1017.87 −0.223199
\(276\) 0 0
\(277\) 3290.08 0.713653 0.356827 0.934171i \(-0.383859\pi\)
0.356827 + 0.934171i \(0.383859\pi\)
\(278\) −10822.8 −2.33492
\(279\) 0 0
\(280\) 1417.78 0.302603
\(281\) −339.723 −0.0721216 −0.0360608 0.999350i \(-0.511481\pi\)
−0.0360608 + 0.999350i \(0.511481\pi\)
\(282\) 0 0
\(283\) 2599.66 0.546056 0.273028 0.962006i \(-0.411975\pi\)
0.273028 + 0.962006i \(0.411975\pi\)
\(284\) −9676.61 −2.02184
\(285\) 0 0
\(286\) −10969.3 −2.26794
\(287\) 4846.55 0.996804
\(288\) 0 0
\(289\) −4869.57 −0.991161
\(290\) −1057.10 −0.214052
\(291\) 0 0
\(292\) −516.835 −0.103580
\(293\) −178.085 −0.0355081 −0.0177540 0.999842i \(-0.505652\pi\)
−0.0177540 + 0.999842i \(0.505652\pi\)
\(294\) 0 0
\(295\) 2129.63 0.420312
\(296\) −2599.52 −0.510452
\(297\) 0 0
\(298\) −399.581 −0.0776749
\(299\) 3942.44 0.762532
\(300\) 0 0
\(301\) −10392.5 −1.99008
\(302\) −4552.13 −0.867370
\(303\) 0 0
\(304\) −3167.51 −0.597596
\(305\) 4252.97 0.798441
\(306\) 0 0
\(307\) 1537.60 0.285848 0.142924 0.989734i \(-0.454349\pi\)
0.142924 + 0.989734i \(0.454349\pi\)
\(308\) −12729.3 −2.35493
\(309\) 0 0
\(310\) 2194.91 0.402136
\(311\) −945.323 −0.172361 −0.0861807 0.996280i \(-0.527466\pi\)
−0.0861807 + 0.996280i \(0.527466\pi\)
\(312\) 0 0
\(313\) 1985.53 0.358559 0.179280 0.983798i \(-0.442623\pi\)
0.179280 + 0.983798i \(0.442623\pi\)
\(314\) 774.428 0.139183
\(315\) 0 0
\(316\) −2004.90 −0.356913
\(317\) 5183.24 0.918358 0.459179 0.888344i \(-0.348144\pi\)
0.459179 + 0.888344i \(0.348144\pi\)
\(318\) 0 0
\(319\) 2019.79 0.354503
\(320\) −3706.48 −0.647496
\(321\) 0 0
\(322\) 8176.33 1.41506
\(323\) 496.748 0.0855722
\(324\) 0 0
\(325\) 1580.45 0.269746
\(326\) −4701.05 −0.798671
\(327\) 0 0
\(328\) 1452.07 0.244443
\(329\) −1369.48 −0.229489
\(330\) 0 0
\(331\) −2173.36 −0.360903 −0.180451 0.983584i \(-0.557756\pi\)
−0.180451 + 0.983584i \(0.557756\pi\)
\(332\) 2009.57 0.332197
\(333\) 0 0
\(334\) 17224.8 2.82186
\(335\) 481.538 0.0785350
\(336\) 0 0
\(337\) −7850.31 −1.26894 −0.634472 0.772946i \(-0.718782\pi\)
−0.634472 + 0.772946i \(0.718782\pi\)
\(338\) 7669.03 1.23414
\(339\) 0 0
\(340\) 334.858 0.0534124
\(341\) −4193.78 −0.665999
\(342\) 0 0
\(343\) 8011.53 1.26117
\(344\) −3113.69 −0.488021
\(345\) 0 0
\(346\) 11710.3 1.81951
\(347\) 9372.24 1.44994 0.724969 0.688782i \(-0.241854\pi\)
0.724969 + 0.688782i \(0.241854\pi\)
\(348\) 0 0
\(349\) −1177.90 −0.180664 −0.0903321 0.995912i \(-0.528793\pi\)
−0.0903321 + 0.995912i \(0.528793\pi\)
\(350\) 3277.73 0.500577
\(351\) 0 0
\(352\) 10293.4 1.55864
\(353\) 7262.13 1.09497 0.547484 0.836816i \(-0.315586\pi\)
0.547484 + 0.836816i \(0.315586\pi\)
\(354\) 0 0
\(355\) −4760.82 −0.711769
\(356\) −13867.5 −2.06454
\(357\) 0 0
\(358\) 12096.3 1.78578
\(359\) 1939.95 0.285199 0.142600 0.989780i \(-0.454454\pi\)
0.142600 + 0.989780i \(0.454454\pi\)
\(360\) 0 0
\(361\) −1176.81 −0.171572
\(362\) 14670.1 2.12995
\(363\) 0 0
\(364\) 19764.8 2.84604
\(365\) −254.279 −0.0364646
\(366\) 0 0
\(367\) 6501.94 0.924792 0.462396 0.886674i \(-0.346990\pi\)
0.462396 + 0.886674i \(0.346990\pi\)
\(368\) −2620.51 −0.371205
\(369\) 0 0
\(370\) −6009.73 −0.844409
\(371\) −808.330 −0.113117
\(372\) 0 0
\(373\) 13318.0 1.84874 0.924372 0.381492i \(-0.124589\pi\)
0.924372 + 0.381492i \(0.124589\pi\)
\(374\) −1143.46 −0.158093
\(375\) 0 0
\(376\) −410.311 −0.0562770
\(377\) −3136.13 −0.428432
\(378\) 0 0
\(379\) −3198.42 −0.433488 −0.216744 0.976228i \(-0.569544\pi\)
−0.216744 + 0.976228i \(0.569544\pi\)
\(380\) 3830.36 0.517087
\(381\) 0 0
\(382\) 3180.55 0.425997
\(383\) 2340.50 0.312256 0.156128 0.987737i \(-0.450099\pi\)
0.156128 + 0.987737i \(0.450099\pi\)
\(384\) 0 0
\(385\) −6262.71 −0.829032
\(386\) 459.878 0.0606403
\(387\) 0 0
\(388\) −14544.5 −1.90305
\(389\) −3991.94 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(390\) 0 0
\(391\) 410.964 0.0531544
\(392\) 5561.83 0.716619
\(393\) 0 0
\(394\) 14323.9 1.83155
\(395\) −986.395 −0.125648
\(396\) 0 0
\(397\) −4960.90 −0.627155 −0.313577 0.949563i \(-0.601528\pi\)
−0.313577 + 0.949563i \(0.601528\pi\)
\(398\) 5834.31 0.734793
\(399\) 0 0
\(400\) −1050.51 −0.131314
\(401\) −827.622 −0.103066 −0.0515330 0.998671i \(-0.516411\pi\)
−0.0515330 + 0.998671i \(0.516411\pi\)
\(402\) 0 0
\(403\) 6511.70 0.804890
\(404\) −11256.4 −1.38621
\(405\) 0 0
\(406\) −6504.11 −0.795058
\(407\) 11482.7 1.39847
\(408\) 0 0
\(409\) −8367.64 −1.01162 −0.505811 0.862645i \(-0.668807\pi\)
−0.505811 + 0.862645i \(0.668807\pi\)
\(410\) 3357.00 0.404367
\(411\) 0 0
\(412\) 5374.82 0.642714
\(413\) 13103.2 1.56117
\(414\) 0 0
\(415\) 988.693 0.116947
\(416\) −15982.7 −1.88369
\(417\) 0 0
\(418\) −13079.7 −1.53050
\(419\) −4200.72 −0.489782 −0.244891 0.969551i \(-0.578752\pi\)
−0.244891 + 0.969551i \(0.578752\pi\)
\(420\) 0 0
\(421\) 6938.76 0.803265 0.401632 0.915801i \(-0.368443\pi\)
0.401632 + 0.915801i \(0.368443\pi\)
\(422\) 10665.3 1.23028
\(423\) 0 0
\(424\) −242.184 −0.0277394
\(425\) 164.747 0.0188034
\(426\) 0 0
\(427\) 26167.6 2.96567
\(428\) −4988.99 −0.563439
\(429\) 0 0
\(430\) −7198.44 −0.807301
\(431\) −6827.09 −0.762991 −0.381496 0.924371i \(-0.624591\pi\)
−0.381496 + 0.924371i \(0.624591\pi\)
\(432\) 0 0
\(433\) −2199.59 −0.244124 −0.122062 0.992522i \(-0.538951\pi\)
−0.122062 + 0.992522i \(0.538951\pi\)
\(434\) 13504.8 1.49366
\(435\) 0 0
\(436\) −3573.57 −0.392529
\(437\) 4700.92 0.514590
\(438\) 0 0
\(439\) 8324.32 0.905007 0.452504 0.891763i \(-0.350531\pi\)
0.452504 + 0.891763i \(0.350531\pi\)
\(440\) −1876.37 −0.203301
\(441\) 0 0
\(442\) 1775.45 0.191063
\(443\) 10866.8 1.16546 0.582729 0.812667i \(-0.301985\pi\)
0.582729 + 0.812667i \(0.301985\pi\)
\(444\) 0 0
\(445\) −6822.72 −0.726805
\(446\) 7988.81 0.848164
\(447\) 0 0
\(448\) −22805.2 −2.40501
\(449\) 4947.73 0.520040 0.260020 0.965603i \(-0.416271\pi\)
0.260020 + 0.965603i \(0.416271\pi\)
\(450\) 0 0
\(451\) −6414.18 −0.669694
\(452\) −17911.4 −1.86389
\(453\) 0 0
\(454\) −21644.5 −2.23750
\(455\) 9724.14 1.00192
\(456\) 0 0
\(457\) −14197.7 −1.45326 −0.726630 0.687029i \(-0.758915\pi\)
−0.726630 + 0.687029i \(0.758915\pi\)
\(458\) 3690.13 0.376481
\(459\) 0 0
\(460\) 3168.89 0.321196
\(461\) 18228.0 1.84157 0.920786 0.390069i \(-0.127549\pi\)
0.920786 + 0.390069i \(0.127549\pi\)
\(462\) 0 0
\(463\) 4341.19 0.435750 0.217875 0.975977i \(-0.430088\pi\)
0.217875 + 0.975977i \(0.430088\pi\)
\(464\) 2084.57 0.208564
\(465\) 0 0
\(466\) −4867.38 −0.483856
\(467\) −4919.63 −0.487481 −0.243740 0.969841i \(-0.578374\pi\)
−0.243740 + 0.969841i \(0.578374\pi\)
\(468\) 0 0
\(469\) 2962.80 0.291704
\(470\) −948.583 −0.0930954
\(471\) 0 0
\(472\) 3925.84 0.382842
\(473\) 13754.0 1.33702
\(474\) 0 0
\(475\) 1884.51 0.182036
\(476\) 2060.31 0.198391
\(477\) 0 0
\(478\) 13421.9 1.28432
\(479\) −4572.83 −0.436196 −0.218098 0.975927i \(-0.569985\pi\)
−0.218098 + 0.975927i \(0.569985\pi\)
\(480\) 0 0
\(481\) −17829.3 −1.69011
\(482\) 19196.6 1.81407
\(483\) 0 0
\(484\) 3319.98 0.311794
\(485\) −7155.79 −0.669954
\(486\) 0 0
\(487\) 15751.5 1.46564 0.732822 0.680421i \(-0.238203\pi\)
0.732822 + 0.680421i \(0.238203\pi\)
\(488\) 7840.09 0.727263
\(489\) 0 0
\(490\) 12858.2 1.18546
\(491\) −15308.4 −1.40704 −0.703520 0.710676i \(-0.748389\pi\)
−0.703520 + 0.710676i \(0.748389\pi\)
\(492\) 0 0
\(493\) −326.914 −0.0298650
\(494\) 20309.0 1.84968
\(495\) 0 0
\(496\) −4328.28 −0.391826
\(497\) −29292.3 −2.64374
\(498\) 0 0
\(499\) 17732.1 1.59078 0.795389 0.606099i \(-0.207267\pi\)
0.795389 + 0.606099i \(0.207267\pi\)
\(500\) 1270.34 0.113623
\(501\) 0 0
\(502\) −3779.61 −0.336040
\(503\) 10511.5 0.931775 0.465887 0.884844i \(-0.345735\pi\)
0.465887 + 0.884844i \(0.345735\pi\)
\(504\) 0 0
\(505\) −5538.08 −0.488003
\(506\) −10821.0 −0.950695
\(507\) 0 0
\(508\) −15306.4 −1.33683
\(509\) −19630.8 −1.70947 −0.854737 0.519062i \(-0.826281\pi\)
−0.854737 + 0.519062i \(0.826281\pi\)
\(510\) 0 0
\(511\) −1564.52 −0.135441
\(512\) 13722.1 1.18444
\(513\) 0 0
\(514\) −9640.30 −0.827267
\(515\) 2644.37 0.226262
\(516\) 0 0
\(517\) 1812.45 0.154180
\(518\) −36976.6 −3.13641
\(519\) 0 0
\(520\) 2913.45 0.245699
\(521\) −88.4336 −0.00743636 −0.00371818 0.999993i \(-0.501184\pi\)
−0.00371818 + 0.999993i \(0.501184\pi\)
\(522\) 0 0
\(523\) 21346.4 1.78473 0.892363 0.451317i \(-0.149046\pi\)
0.892363 + 0.451317i \(0.149046\pi\)
\(524\) −12958.8 −1.08036
\(525\) 0 0
\(526\) −31028.4 −2.57206
\(527\) 678.787 0.0561071
\(528\) 0 0
\(529\) −8277.88 −0.680355
\(530\) −559.896 −0.0458874
\(531\) 0 0
\(532\) 23567.4 1.92063
\(533\) 9959.33 0.809356
\(534\) 0 0
\(535\) −2454.55 −0.198354
\(536\) 887.684 0.0715338
\(537\) 0 0
\(538\) 455.077 0.0364679
\(539\) −24568.0 −1.96330
\(540\) 0 0
\(541\) −14432.6 −1.14696 −0.573480 0.819220i \(-0.694407\pi\)
−0.573480 + 0.819220i \(0.694407\pi\)
\(542\) 25182.2 1.99570
\(543\) 0 0
\(544\) −1666.05 −0.131308
\(545\) −1758.17 −0.138187
\(546\) 0 0
\(547\) −16569.6 −1.29518 −0.647592 0.761988i \(-0.724224\pi\)
−0.647592 + 0.761988i \(0.724224\pi\)
\(548\) −14628.5 −1.14033
\(549\) 0 0
\(550\) −4337.92 −0.336308
\(551\) −3739.49 −0.289125
\(552\) 0 0
\(553\) −6069.08 −0.466697
\(554\) 14021.6 1.07531
\(555\) 0 0
\(556\) −25808.3 −1.96855
\(557\) −11597.0 −0.882191 −0.441096 0.897460i \(-0.645410\pi\)
−0.441096 + 0.897460i \(0.645410\pi\)
\(558\) 0 0
\(559\) −21355.9 −1.61584
\(560\) −6463.57 −0.487742
\(561\) 0 0
\(562\) −1447.82 −0.108670
\(563\) 23049.3 1.72542 0.862709 0.505700i \(-0.168766\pi\)
0.862709 + 0.505700i \(0.168766\pi\)
\(564\) 0 0
\(565\) −8812.26 −0.656168
\(566\) 11079.2 0.822778
\(567\) 0 0
\(568\) −8776.27 −0.648317
\(569\) −14733.9 −1.08555 −0.542775 0.839878i \(-0.682626\pi\)
−0.542775 + 0.839878i \(0.682626\pi\)
\(570\) 0 0
\(571\) 15111.9 1.10755 0.553776 0.832666i \(-0.313186\pi\)
0.553776 + 0.832666i \(0.313186\pi\)
\(572\) −26157.8 −1.91209
\(573\) 0 0
\(574\) 20654.9 1.50195
\(575\) 1559.07 0.113074
\(576\) 0 0
\(577\) −26150.5 −1.88676 −0.943380 0.331715i \(-0.892373\pi\)
−0.943380 + 0.331715i \(0.892373\pi\)
\(578\) −20753.0 −1.49345
\(579\) 0 0
\(580\) −2520.79 −0.180466
\(581\) 6083.22 0.434379
\(582\) 0 0
\(583\) 1069.79 0.0759967
\(584\) −468.747 −0.0332139
\(585\) 0 0
\(586\) −758.961 −0.0535023
\(587\) −2091.54 −0.147065 −0.0735326 0.997293i \(-0.523427\pi\)
−0.0735326 + 0.997293i \(0.523427\pi\)
\(588\) 0 0
\(589\) 7764.49 0.543175
\(590\) 9076.02 0.633311
\(591\) 0 0
\(592\) 11851.0 0.822759
\(593\) −3260.34 −0.225778 −0.112889 0.993608i \(-0.536010\pi\)
−0.112889 + 0.993608i \(0.536010\pi\)
\(594\) 0 0
\(595\) 1013.66 0.0698417
\(596\) −952.852 −0.0654872
\(597\) 0 0
\(598\) 16801.8 1.14896
\(599\) 22999.2 1.56882 0.784410 0.620243i \(-0.212966\pi\)
0.784410 + 0.620243i \(0.212966\pi\)
\(600\) 0 0
\(601\) −14293.9 −0.970148 −0.485074 0.874473i \(-0.661207\pi\)
−0.485074 + 0.874473i \(0.661207\pi\)
\(602\) −44290.5 −2.99858
\(603\) 0 0
\(604\) −10855.1 −0.731274
\(605\) 1633.40 0.109764
\(606\) 0 0
\(607\) −16434.1 −1.09891 −0.549455 0.835523i \(-0.685165\pi\)
−0.549455 + 0.835523i \(0.685165\pi\)
\(608\) −19057.6 −1.27120
\(609\) 0 0
\(610\) 18125.2 1.20306
\(611\) −2814.19 −0.186334
\(612\) 0 0
\(613\) 3674.45 0.242104 0.121052 0.992646i \(-0.461373\pi\)
0.121052 + 0.992646i \(0.461373\pi\)
\(614\) 6552.91 0.430707
\(615\) 0 0
\(616\) −11544.9 −0.755126
\(617\) −8627.57 −0.562938 −0.281469 0.959570i \(-0.590822\pi\)
−0.281469 + 0.959570i \(0.590822\pi\)
\(618\) 0 0
\(619\) −1309.81 −0.0850496 −0.0425248 0.999095i \(-0.513540\pi\)
−0.0425248 + 0.999095i \(0.513540\pi\)
\(620\) 5234.03 0.339038
\(621\) 0 0
\(622\) −4028.76 −0.259708
\(623\) −41978.8 −2.69959
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 8461.91 0.540265
\(627\) 0 0
\(628\) 1846.72 0.117344
\(629\) −1858.54 −0.117814
\(630\) 0 0
\(631\) 14447.2 0.911463 0.455731 0.890117i \(-0.349378\pi\)
0.455731 + 0.890117i \(0.349378\pi\)
\(632\) −1818.36 −0.114447
\(633\) 0 0
\(634\) 22089.8 1.38375
\(635\) −7530.62 −0.470620
\(636\) 0 0
\(637\) 38146.9 2.37274
\(638\) 8607.88 0.534152
\(639\) 0 0
\(640\) −5683.43 −0.351027
\(641\) 20073.1 1.23688 0.618440 0.785832i \(-0.287765\pi\)
0.618440 + 0.785832i \(0.287765\pi\)
\(642\) 0 0
\(643\) −17735.9 −1.08777 −0.543885 0.839160i \(-0.683047\pi\)
−0.543885 + 0.839160i \(0.683047\pi\)
\(644\) 19497.5 1.19303
\(645\) 0 0
\(646\) 2117.03 0.128937
\(647\) 11456.8 0.696158 0.348079 0.937465i \(-0.386834\pi\)
0.348079 + 0.937465i \(0.386834\pi\)
\(648\) 0 0
\(649\) −17341.4 −1.04886
\(650\) 6735.51 0.406444
\(651\) 0 0
\(652\) −11210.2 −0.673354
\(653\) −29071.9 −1.74223 −0.871113 0.491083i \(-0.836601\pi\)
−0.871113 + 0.491083i \(0.836601\pi\)
\(654\) 0 0
\(655\) −6375.62 −0.380330
\(656\) −6619.90 −0.393999
\(657\) 0 0
\(658\) −5836.43 −0.345787
\(659\) 4355.09 0.257436 0.128718 0.991681i \(-0.458914\pi\)
0.128718 + 0.991681i \(0.458914\pi\)
\(660\) 0 0
\(661\) 27295.3 1.60615 0.803075 0.595878i \(-0.203196\pi\)
0.803075 + 0.595878i \(0.203196\pi\)
\(662\) −9262.38 −0.543796
\(663\) 0 0
\(664\) 1822.59 0.106522
\(665\) 11595.0 0.676141
\(666\) 0 0
\(667\) −3093.72 −0.179594
\(668\) 41074.8 2.37909
\(669\) 0 0
\(670\) 2052.21 0.118334
\(671\) −34631.7 −1.99246
\(672\) 0 0
\(673\) −17760.0 −1.01723 −0.508617 0.860993i \(-0.669843\pi\)
−0.508617 + 0.860993i \(0.669843\pi\)
\(674\) −33456.3 −1.91200
\(675\) 0 0
\(676\) 18287.8 1.04050
\(677\) −1937.33 −0.109982 −0.0549910 0.998487i \(-0.517513\pi\)
−0.0549910 + 0.998487i \(0.517513\pi\)
\(678\) 0 0
\(679\) −44028.0 −2.48843
\(680\) 303.702 0.0171271
\(681\) 0 0
\(682\) −17873.0 −1.00351
\(683\) 2125.71 0.119090 0.0595448 0.998226i \(-0.481035\pi\)
0.0595448 + 0.998226i \(0.481035\pi\)
\(684\) 0 0
\(685\) −7197.12 −0.401442
\(686\) 34143.4 1.90029
\(687\) 0 0
\(688\) 14195.1 0.786603
\(689\) −1661.06 −0.0918454
\(690\) 0 0
\(691\) 13826.3 0.761180 0.380590 0.924744i \(-0.375721\pi\)
0.380590 + 0.924744i \(0.375721\pi\)
\(692\) 27924.8 1.53402
\(693\) 0 0
\(694\) 39942.4 2.18472
\(695\) −12697.5 −0.693012
\(696\) 0 0
\(697\) 1038.17 0.0564183
\(698\) −5019.97 −0.272219
\(699\) 0 0
\(700\) 7816.16 0.422033
\(701\) 24464.0 1.31810 0.659052 0.752097i \(-0.270958\pi\)
0.659052 + 0.752097i \(0.270958\pi\)
\(702\) 0 0
\(703\) −21259.5 −1.14056
\(704\) 30181.6 1.61578
\(705\) 0 0
\(706\) 30949.6 1.64986
\(707\) −34074.6 −1.81260
\(708\) 0 0
\(709\) −29498.3 −1.56253 −0.781263 0.624202i \(-0.785424\pi\)
−0.781263 + 0.624202i \(0.785424\pi\)
\(710\) −20289.6 −1.07247
\(711\) 0 0
\(712\) −12577.3 −0.662012
\(713\) 6423.63 0.337401
\(714\) 0 0
\(715\) −12869.4 −0.673133
\(716\) 28845.2 1.50558
\(717\) 0 0
\(718\) 8267.63 0.429729
\(719\) 3857.66 0.200093 0.100046 0.994983i \(-0.468101\pi\)
0.100046 + 0.994983i \(0.468101\pi\)
\(720\) 0 0
\(721\) 16270.2 0.840410
\(722\) −5015.32 −0.258519
\(723\) 0 0
\(724\) 34982.7 1.79575
\(725\) −1240.21 −0.0635313
\(726\) 0 0
\(727\) 8745.10 0.446132 0.223066 0.974803i \(-0.428394\pi\)
0.223066 + 0.974803i \(0.428394\pi\)
\(728\) 17925.8 0.912604
\(729\) 0 0
\(730\) −1083.68 −0.0549436
\(731\) −2226.16 −0.112637
\(732\) 0 0
\(733\) −9418.40 −0.474593 −0.237296 0.971437i \(-0.576261\pi\)
−0.237296 + 0.971437i \(0.576261\pi\)
\(734\) 27709.8 1.39345
\(735\) 0 0
\(736\) −15766.5 −0.789621
\(737\) −3921.13 −0.195979
\(738\) 0 0
\(739\) 6219.42 0.309587 0.154794 0.987947i \(-0.450529\pi\)
0.154794 + 0.987947i \(0.450529\pi\)
\(740\) −14331.0 −0.711915
\(741\) 0 0
\(742\) −3444.92 −0.170441
\(743\) 29387.4 1.45104 0.725518 0.688203i \(-0.241600\pi\)
0.725518 + 0.688203i \(0.241600\pi\)
\(744\) 0 0
\(745\) −468.796 −0.0230542
\(746\) 56758.5 2.78563
\(747\) 0 0
\(748\) −2726.72 −0.133287
\(749\) −15102.3 −0.736751
\(750\) 0 0
\(751\) −21647.1 −1.05182 −0.525909 0.850541i \(-0.676275\pi\)
−0.525909 + 0.850541i \(0.676275\pi\)
\(752\) 1870.57 0.0907086
\(753\) 0 0
\(754\) −13365.5 −0.645547
\(755\) −5340.65 −0.257438
\(756\) 0 0
\(757\) −13907.2 −0.667722 −0.333861 0.942622i \(-0.608352\pi\)
−0.333861 + 0.942622i \(0.608352\pi\)
\(758\) −13631.0 −0.653165
\(759\) 0 0
\(760\) 3473.97 0.165808
\(761\) 23906.1 1.13876 0.569379 0.822075i \(-0.307184\pi\)
0.569379 + 0.822075i \(0.307184\pi\)
\(762\) 0 0
\(763\) −10817.6 −0.513270
\(764\) 7584.42 0.359155
\(765\) 0 0
\(766\) 9974.70 0.470497
\(767\) 26926.1 1.26760
\(768\) 0 0
\(769\) −25111.6 −1.17757 −0.588783 0.808291i \(-0.700393\pi\)
−0.588783 + 0.808291i \(0.700393\pi\)
\(770\) −26690.3 −1.24916
\(771\) 0 0
\(772\) 1096.64 0.0511255
\(773\) −14909.4 −0.693729 −0.346865 0.937915i \(-0.612754\pi\)
−0.346865 + 0.937915i \(0.612754\pi\)
\(774\) 0 0
\(775\) 2575.10 0.119355
\(776\) −13191.2 −0.610229
\(777\) 0 0
\(778\) −17012.7 −0.783980
\(779\) 11875.4 0.546188
\(780\) 0 0
\(781\) 38767.0 1.77617
\(782\) 1751.44 0.0800912
\(783\) 0 0
\(784\) −25356.0 −1.15506
\(785\) 908.574 0.0413101
\(786\) 0 0
\(787\) −2658.77 −0.120426 −0.0602128 0.998186i \(-0.519178\pi\)
−0.0602128 + 0.998186i \(0.519178\pi\)
\(788\) 34157.2 1.54416
\(789\) 0 0
\(790\) −4203.80 −0.189322
\(791\) −54220.0 −2.43722
\(792\) 0 0
\(793\) 53772.7 2.40798
\(794\) −21142.3 −0.944976
\(795\) 0 0
\(796\) 13912.7 0.619499
\(797\) 14172.2 0.629870 0.314935 0.949113i \(-0.398017\pi\)
0.314935 + 0.949113i \(0.398017\pi\)
\(798\) 0 0
\(799\) −293.355 −0.0129889
\(800\) −6320.48 −0.279329
\(801\) 0 0
\(802\) −3527.14 −0.155296
\(803\) 2070.57 0.0909950
\(804\) 0 0
\(805\) 9592.62 0.419994
\(806\) 27751.4 1.21278
\(807\) 0 0
\(808\) −10209.1 −0.444499
\(809\) 21077.7 0.916012 0.458006 0.888949i \(-0.348564\pi\)
0.458006 + 0.888949i \(0.348564\pi\)
\(810\) 0 0
\(811\) −11937.4 −0.516868 −0.258434 0.966029i \(-0.583206\pi\)
−0.258434 + 0.966029i \(0.583206\pi\)
\(812\) −15509.9 −0.670308
\(813\) 0 0
\(814\) 48936.8 2.10717
\(815\) −5515.36 −0.237048
\(816\) 0 0
\(817\) −25464.5 −1.09044
\(818\) −35661.0 −1.52428
\(819\) 0 0
\(820\) 8005.20 0.340919
\(821\) −14800.5 −0.629161 −0.314580 0.949231i \(-0.601864\pi\)
−0.314580 + 0.949231i \(0.601864\pi\)
\(822\) 0 0
\(823\) 33176.6 1.40518 0.702591 0.711594i \(-0.252026\pi\)
0.702591 + 0.711594i \(0.252026\pi\)
\(824\) 4874.73 0.206091
\(825\) 0 0
\(826\) 55842.8 2.35232
\(827\) 29001.3 1.21944 0.609718 0.792619i \(-0.291283\pi\)
0.609718 + 0.792619i \(0.291283\pi\)
\(828\) 0 0
\(829\) 13221.1 0.553904 0.276952 0.960884i \(-0.410676\pi\)
0.276952 + 0.960884i \(0.410676\pi\)
\(830\) 4213.59 0.176212
\(831\) 0 0
\(832\) −46863.1 −1.95275
\(833\) 3976.47 0.165398
\(834\) 0 0
\(835\) 20208.5 0.837538
\(836\) −31190.3 −1.29036
\(837\) 0 0
\(838\) −17902.5 −0.737987
\(839\) 42123.9 1.73335 0.866675 0.498874i \(-0.166253\pi\)
0.866675 + 0.498874i \(0.166253\pi\)
\(840\) 0 0
\(841\) −21928.0 −0.899094
\(842\) 29571.5 1.21033
\(843\) 0 0
\(844\) 25432.8 1.03724
\(845\) 8997.45 0.366298
\(846\) 0 0
\(847\) 10050.0 0.407700
\(848\) 1104.10 0.0447109
\(849\) 0 0
\(850\) 702.117 0.0283323
\(851\) −17588.1 −0.708476
\(852\) 0 0
\(853\) 14431.7 0.579287 0.289643 0.957135i \(-0.406463\pi\)
0.289643 + 0.957135i \(0.406463\pi\)
\(854\) 111521. 4.46857
\(855\) 0 0
\(856\) −4524.80 −0.180671
\(857\) −16113.4 −0.642268 −0.321134 0.947034i \(-0.604064\pi\)
−0.321134 + 0.947034i \(0.604064\pi\)
\(858\) 0 0
\(859\) −22500.4 −0.893717 −0.446858 0.894605i \(-0.647457\pi\)
−0.446858 + 0.894605i \(0.647457\pi\)
\(860\) −17165.6 −0.680630
\(861\) 0 0
\(862\) −29095.5 −1.14965
\(863\) 43396.8 1.71175 0.855877 0.517180i \(-0.173018\pi\)
0.855877 + 0.517180i \(0.173018\pi\)
\(864\) 0 0
\(865\) 13738.8 0.540038
\(866\) −9374.18 −0.367838
\(867\) 0 0
\(868\) 32203.9 1.25930
\(869\) 8032.14 0.313546
\(870\) 0 0
\(871\) 6088.35 0.236850
\(872\) −3241.07 −0.125868
\(873\) 0 0
\(874\) 20034.3 0.775366
\(875\) 3845.49 0.148573
\(876\) 0 0
\(877\) −2465.53 −0.0949317 −0.0474659 0.998873i \(-0.515115\pi\)
−0.0474659 + 0.998873i \(0.515115\pi\)
\(878\) 35476.4 1.36363
\(879\) 0 0
\(880\) 8554.23 0.327685
\(881\) −24512.9 −0.937412 −0.468706 0.883354i \(-0.655280\pi\)
−0.468706 + 0.883354i \(0.655280\pi\)
\(882\) 0 0
\(883\) 24236.1 0.923679 0.461840 0.886963i \(-0.347190\pi\)
0.461840 + 0.886963i \(0.347190\pi\)
\(884\) 4233.79 0.161084
\(885\) 0 0
\(886\) 46311.9 1.75607
\(887\) −25198.7 −0.953877 −0.476939 0.878937i \(-0.658254\pi\)
−0.476939 + 0.878937i \(0.658254\pi\)
\(888\) 0 0
\(889\) −46334.3 −1.74803
\(890\) −29076.9 −1.09512
\(891\) 0 0
\(892\) 19050.3 0.715081
\(893\) −3355.61 −0.125746
\(894\) 0 0
\(895\) 14191.6 0.530026
\(896\) −34968.9 −1.30383
\(897\) 0 0
\(898\) 21086.1 0.783579
\(899\) −5109.87 −0.189570
\(900\) 0 0
\(901\) −173.151 −0.00640233
\(902\) −27335.8 −1.00907
\(903\) 0 0
\(904\) −16244.8 −0.597672
\(905\) 17211.2 0.632178
\(906\) 0 0
\(907\) 26608.1 0.974098 0.487049 0.873375i \(-0.338073\pi\)
0.487049 + 0.873375i \(0.338073\pi\)
\(908\) −51614.0 −1.88642
\(909\) 0 0
\(910\) 41442.1 1.50966
\(911\) −24832.1 −0.903101 −0.451550 0.892246i \(-0.649129\pi\)
−0.451550 + 0.892246i \(0.649129\pi\)
\(912\) 0 0
\(913\) −8050.86 −0.291834
\(914\) −60507.4 −2.18972
\(915\) 0 0
\(916\) 8799.58 0.317409
\(917\) −39227.8 −1.41267
\(918\) 0 0
\(919\) −26107.2 −0.937102 −0.468551 0.883436i \(-0.655224\pi\)
−0.468551 + 0.883436i \(0.655224\pi\)
\(920\) 2874.05 0.102994
\(921\) 0 0
\(922\) 77683.9 2.77482
\(923\) −60193.7 −2.14659
\(924\) 0 0
\(925\) −7050.73 −0.250623
\(926\) 18501.2 0.656573
\(927\) 0 0
\(928\) 12542.0 0.443653
\(929\) 23361.5 0.825046 0.412523 0.910947i \(-0.364648\pi\)
0.412523 + 0.910947i \(0.364648\pi\)
\(930\) 0 0
\(931\) 45485.9 1.60123
\(932\) −11606.9 −0.407936
\(933\) 0 0
\(934\) −20966.4 −0.734519
\(935\) −1341.53 −0.0469226
\(936\) 0 0
\(937\) 10548.5 0.367775 0.183888 0.982947i \(-0.441132\pi\)
0.183888 + 0.982947i \(0.441132\pi\)
\(938\) 12626.8 0.439530
\(939\) 0 0
\(940\) −2262.02 −0.0784881
\(941\) 6490.14 0.224838 0.112419 0.993661i \(-0.464140\pi\)
0.112419 + 0.993661i \(0.464140\pi\)
\(942\) 0 0
\(943\) 9824.63 0.339273
\(944\) −17897.6 −0.617074
\(945\) 0 0
\(946\) 58616.4 2.01457
\(947\) −23506.0 −0.806590 −0.403295 0.915070i \(-0.632135\pi\)
−0.403295 + 0.915070i \(0.632135\pi\)
\(948\) 0 0
\(949\) −3214.99 −0.109972
\(950\) 8031.35 0.274286
\(951\) 0 0
\(952\) 1868.61 0.0636156
\(953\) −6740.26 −0.229106 −0.114553 0.993417i \(-0.536544\pi\)
−0.114553 + 0.993417i \(0.536544\pi\)
\(954\) 0 0
\(955\) 3731.48 0.126437
\(956\) 32006.2 1.08280
\(957\) 0 0
\(958\) −19488.4 −0.657245
\(959\) −44282.3 −1.49109
\(960\) 0 0
\(961\) −19181.1 −0.643857
\(962\) −75984.4 −2.54661
\(963\) 0 0
\(964\) 45776.8 1.52943
\(965\) 539.538 0.0179983
\(966\) 0 0
\(967\) 16782.5 0.558108 0.279054 0.960275i \(-0.409979\pi\)
0.279054 + 0.960275i \(0.409979\pi\)
\(968\) 3011.08 0.0999791
\(969\) 0 0
\(970\) −30496.4 −1.00946
\(971\) 46282.7 1.52964 0.764822 0.644242i \(-0.222827\pi\)
0.764822 + 0.644242i \(0.222827\pi\)
\(972\) 0 0
\(973\) −78124.9 −2.57407
\(974\) 67129.4 2.20838
\(975\) 0 0
\(976\) −35742.4 −1.17222
\(977\) −6767.95 −0.221623 −0.110812 0.993841i \(-0.535345\pi\)
−0.110812 + 0.993841i \(0.535345\pi\)
\(978\) 0 0
\(979\) 55556.9 1.81370
\(980\) 30662.0 0.999452
\(981\) 0 0
\(982\) −65240.8 −2.12008
\(983\) 50074.9 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(984\) 0 0
\(985\) 16805.1 0.543609
\(986\) −1393.24 −0.0449996
\(987\) 0 0
\(988\) 48429.3 1.55946
\(989\) −21067.0 −0.677343
\(990\) 0 0
\(991\) −2658.60 −0.0852200 −0.0426100 0.999092i \(-0.513567\pi\)
−0.0426100 + 0.999092i \(0.513567\pi\)
\(992\) −26041.4 −0.833485
\(993\) 0 0
\(994\) −124837. −3.98350
\(995\) 6844.93 0.218089
\(996\) 0 0
\(997\) −60620.1 −1.92563 −0.962817 0.270153i \(-0.912926\pi\)
−0.962817 + 0.270153i \(0.912926\pi\)
\(998\) 75570.3 2.39693
\(999\) 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 405.4.a.n.1.6 7
3.2 odd 2 405.4.a.m.1.2 7
5.4 even 2 2025.4.a.ba.1.2 7
9.2 odd 6 45.4.e.c.31.6 yes 14
9.4 even 3 135.4.e.c.46.2 14
9.5 odd 6 45.4.e.c.16.6 14
9.7 even 3 135.4.e.c.91.2 14
15.14 odd 2 2025.4.a.bb.1.6 7
45.2 even 12 225.4.k.d.49.3 28
45.14 odd 6 225.4.e.d.151.2 14
45.23 even 12 225.4.k.d.124.3 28
45.29 odd 6 225.4.e.d.76.2 14
45.32 even 12 225.4.k.d.124.12 28
45.38 even 12 225.4.k.d.49.12 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.6 14 9.5 odd 6
45.4.e.c.31.6 yes 14 9.2 odd 6
135.4.e.c.46.2 14 9.4 even 3
135.4.e.c.91.2 14 9.7 even 3
225.4.e.d.76.2 14 45.29 odd 6
225.4.e.d.151.2 14 45.14 odd 6
225.4.k.d.49.3 28 45.2 even 12
225.4.k.d.49.12 28 45.38 even 12
225.4.k.d.124.3 28 45.23 even 12
225.4.k.d.124.12 28 45.32 even 12
405.4.a.m.1.2 7 3.2 odd 2
405.4.a.n.1.6 7 1.1 even 1 trivial
2025.4.a.ba.1.2 7 5.4 even 2
2025.4.a.bb.1.6 7 15.14 odd 2