Properties

Label 2475.2.a.w.1.1
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.41421 q^{7} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.41421 q^{7} +1.58579 q^{8} +1.00000 q^{11} -2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} -0.414214 q^{17} +3.58579 q^{19} -0.414214 q^{22} +1.00000 q^{23} +1.17157 q^{26} -4.41421 q^{28} -6.82843 q^{29} +8.48528 q^{31} -4.41421 q^{32} +0.171573 q^{34} +5.82843 q^{37} -1.48528 q^{38} -8.89949 q^{41} +0.343146 q^{43} -1.82843 q^{44} -0.414214 q^{46} +9.48528 q^{47} -1.17157 q^{49} +5.17157 q^{52} -3.65685 q^{53} +3.82843 q^{56} +2.82843 q^{58} -11.0000 q^{59} +3.17157 q^{61} -3.51472 q^{62} -4.17157 q^{64} +11.6569 q^{67} +0.757359 q^{68} -2.17157 q^{71} +3.17157 q^{73} -2.41421 q^{74} -6.55635 q^{76} +2.41421 q^{77} +4.75736 q^{79} +3.68629 q^{82} +12.4853 q^{83} -0.142136 q^{86} +1.58579 q^{88} +7.65685 q^{89} -6.82843 q^{91} -1.82843 q^{92} -3.92893 q^{94} -0.171573 q^{97} +0.485281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{7} + 6q^{8} + 2q^{11} - 2q^{14} + 6q^{16} + 2q^{17} + 10q^{19} + 2q^{22} + 2q^{23} + 8q^{26} - 6q^{28} - 8q^{29} - 6q^{32} + 6q^{34} + 6q^{37} + 14q^{38} + 2q^{41} + 12q^{43} + 2q^{44} + 2q^{46} + 2q^{47} - 8q^{49} + 16q^{52} + 4q^{53} + 2q^{56} - 22q^{59} + 12q^{61} - 24q^{62} - 14q^{64} + 12q^{67} + 10q^{68} - 10q^{71} + 12q^{73} - 2q^{74} + 18q^{76} + 2q^{77} + 18q^{79} + 30q^{82} + 8q^{83} + 28q^{86} + 6q^{88} + 4q^{89} - 8q^{91} + 2q^{92} - 22q^{94} - 6q^{97} - 16q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −0.414214 −0.100462 −0.0502308 0.998738i \(-0.515996\pi\)
−0.0502308 + 0.998738i \(0.515996\pi\)
\(18\) 0 0
\(19\) 3.58579 0.822636 0.411318 0.911492i \(-0.365069\pi\)
0.411318 + 0.911492i \(0.365069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.414214 −0.0883106
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.17157 0.229764
\(27\) 0 0
\(28\) −4.41421 −0.834208
\(29\) −6.82843 −1.26801 −0.634004 0.773330i \(-0.718590\pi\)
−0.634004 + 0.773330i \(0.718590\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) 0.171573 0.0294245
\(35\) 0 0
\(36\) 0 0
\(37\) 5.82843 0.958188 0.479094 0.877764i \(-0.340965\pi\)
0.479094 + 0.877764i \(0.340965\pi\)
\(38\) −1.48528 −0.240944
\(39\) 0 0
\(40\) 0 0
\(41\) −8.89949 −1.38987 −0.694934 0.719074i \(-0.744566\pi\)
−0.694934 + 0.719074i \(0.744566\pi\)
\(42\) 0 0
\(43\) 0.343146 0.0523292 0.0261646 0.999658i \(-0.491671\pi\)
0.0261646 + 0.999658i \(0.491671\pi\)
\(44\) −1.82843 −0.275646
\(45\) 0 0
\(46\) −0.414214 −0.0610725
\(47\) 9.48528 1.38357 0.691785 0.722103i \(-0.256824\pi\)
0.691785 + 0.722103i \(0.256824\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) 0 0
\(52\) 5.17157 0.717168
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.82843 0.511595
\(57\) 0 0
\(58\) 2.82843 0.371391
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 3.17157 0.406078 0.203039 0.979171i \(-0.434918\pi\)
0.203039 + 0.979171i \(0.434918\pi\)
\(62\) −3.51472 −0.446370
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) 11.6569 1.42411 0.712056 0.702123i \(-0.247764\pi\)
0.712056 + 0.702123i \(0.247764\pi\)
\(68\) 0.757359 0.0918433
\(69\) 0 0
\(70\) 0 0
\(71\) −2.17157 −0.257718 −0.128859 0.991663i \(-0.541132\pi\)
−0.128859 + 0.991663i \(0.541132\pi\)
\(72\) 0 0
\(73\) 3.17157 0.371205 0.185602 0.982625i \(-0.440576\pi\)
0.185602 + 0.982625i \(0.440576\pi\)
\(74\) −2.41421 −0.280647
\(75\) 0 0
\(76\) −6.55635 −0.752065
\(77\) 2.41421 0.275125
\(78\) 0 0
\(79\) 4.75736 0.535245 0.267622 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.68629 0.407083
\(83\) 12.4853 1.37044 0.685219 0.728337i \(-0.259707\pi\)
0.685219 + 0.728337i \(0.259707\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.142136 −0.0153269
\(87\) 0 0
\(88\) 1.58579 0.169045
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) −6.82843 −0.715814
\(92\) −1.82843 −0.190627
\(93\) 0 0
\(94\) −3.92893 −0.405238
\(95\) 0 0
\(96\) 0 0
\(97\) −0.171573 −0.0174206 −0.00871029 0.999962i \(-0.502773\pi\)
−0.00871029 + 0.999962i \(0.502773\pi\)
\(98\) 0.485281 0.0490208
\(99\) 0 0
\(100\) 0 0
\(101\) 4.89949 0.487518 0.243759 0.969836i \(-0.421619\pi\)
0.243759 + 0.969836i \(0.421619\pi\)
\(102\) 0 0
\(103\) 2.34315 0.230877 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(104\) −4.48528 −0.439818
\(105\) 0 0
\(106\) 1.51472 0.147122
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) 0 0
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.24264 0.684365
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.4853 1.15923
\(117\) 0 0
\(118\) 4.55635 0.419446
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.31371 −0.118938
\(123\) 0 0
\(124\) −15.5147 −1.39326
\(125\) 0 0
\(126\) 0 0
\(127\) 1.24264 0.110267 0.0551333 0.998479i \(-0.482442\pi\)
0.0551333 + 0.998479i \(0.482442\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) −1.17157 −0.102361 −0.0511804 0.998689i \(-0.516298\pi\)
−0.0511804 + 0.998689i \(0.516298\pi\)
\(132\) 0 0
\(133\) 8.65685 0.750644
\(134\) −4.82843 −0.417113
\(135\) 0 0
\(136\) −0.656854 −0.0563248
\(137\) 16.1421 1.37912 0.689558 0.724231i \(-0.257805\pi\)
0.689558 + 0.724231i \(0.257805\pi\)
\(138\) 0 0
\(139\) 14.9706 1.26979 0.634893 0.772600i \(-0.281044\pi\)
0.634893 + 0.772600i \(0.281044\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.899495 0.0754839
\(143\) −2.82843 −0.236525
\(144\) 0 0
\(145\) 0 0
\(146\) −1.31371 −0.108723
\(147\) 0 0
\(148\) −10.6569 −0.875988
\(149\) 17.7279 1.45233 0.726164 0.687522i \(-0.241301\pi\)
0.726164 + 0.687522i \(0.241301\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 5.68629 0.461219
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −1.97056 −0.156770
\(159\) 0 0
\(160\) 0 0
\(161\) 2.41421 0.190267
\(162\) 0 0
\(163\) 23.7990 1.86408 0.932040 0.362354i \(-0.118027\pi\)
0.932040 + 0.362354i \(0.118027\pi\)
\(164\) 16.2721 1.27064
\(165\) 0 0
\(166\) −5.17157 −0.401392
\(167\) −17.7990 −1.37733 −0.688664 0.725081i \(-0.741802\pi\)
−0.688664 + 0.725081i \(0.741802\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) −0.627417 −0.0478401
\(173\) 18.5563 1.41081 0.705407 0.708803i \(-0.250764\pi\)
0.705407 + 0.708803i \(0.250764\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −3.17157 −0.237719
\(179\) 22.7990 1.70408 0.852038 0.523480i \(-0.175366\pi\)
0.852038 + 0.523480i \(0.175366\pi\)
\(180\) 0 0
\(181\) 11.9706 0.889765 0.444882 0.895589i \(-0.353245\pi\)
0.444882 + 0.895589i \(0.353245\pi\)
\(182\) 2.82843 0.209657
\(183\) 0 0
\(184\) 1.58579 0.116906
\(185\) 0 0
\(186\) 0 0
\(187\) −0.414214 −0.0302903
\(188\) −17.3431 −1.26488
\(189\) 0 0
\(190\) 0 0
\(191\) −11.8284 −0.855875 −0.427937 0.903808i \(-0.640760\pi\)
−0.427937 + 0.903808i \(0.640760\pi\)
\(192\) 0 0
\(193\) 19.3137 1.39023 0.695116 0.718898i \(-0.255353\pi\)
0.695116 + 0.718898i \(0.255353\pi\)
\(194\) 0.0710678 0.00510237
\(195\) 0 0
\(196\) 2.14214 0.153010
\(197\) 13.2426 0.943499 0.471750 0.881733i \(-0.343623\pi\)
0.471750 + 0.881733i \(0.343623\pi\)
\(198\) 0 0
\(199\) 5.17157 0.366603 0.183302 0.983057i \(-0.441322\pi\)
0.183302 + 0.983057i \(0.441322\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.02944 −0.142791
\(203\) −16.4853 −1.15704
\(204\) 0 0
\(205\) 0 0
\(206\) −0.970563 −0.0676223
\(207\) 0 0
\(208\) −8.48528 −0.588348
\(209\) 3.58579 0.248034
\(210\) 0 0
\(211\) 9.31371 0.641182 0.320591 0.947218i \(-0.396118\pi\)
0.320591 + 0.947218i \(0.396118\pi\)
\(212\) 6.68629 0.459216
\(213\) 0 0
\(214\) −7.17157 −0.490239
\(215\) 0 0
\(216\) 0 0
\(217\) 20.4853 1.39063
\(218\) 7.17157 0.485720
\(219\) 0 0
\(220\) 0 0
\(221\) 1.17157 0.0788085
\(222\) 0 0
\(223\) 26.8284 1.79656 0.898282 0.439419i \(-0.144816\pi\)
0.898282 + 0.439419i \(0.144816\pi\)
\(224\) −10.6569 −0.712041
\(225\) 0 0
\(226\) 4.14214 0.275531
\(227\) −1.51472 −0.100535 −0.0502677 0.998736i \(-0.516007\pi\)
−0.0502677 + 0.998736i \(0.516007\pi\)
\(228\) 0 0
\(229\) −19.4853 −1.28762 −0.643812 0.765184i \(-0.722648\pi\)
−0.643812 + 0.765184i \(0.722648\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.8284 −0.710921
\(233\) −14.5563 −0.953618 −0.476809 0.879007i \(-0.658207\pi\)
−0.476809 + 0.879007i \(0.658207\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 20.1127 1.30923
\(237\) 0 0
\(238\) 0.414214 0.0268495
\(239\) −12.3431 −0.798412 −0.399206 0.916861i \(-0.630714\pi\)
−0.399206 + 0.916861i \(0.630714\pi\)
\(240\) 0 0
\(241\) −14.1421 −0.910975 −0.455488 0.890242i \(-0.650535\pi\)
−0.455488 + 0.890242i \(0.650535\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 0 0
\(244\) −5.79899 −0.371242
\(245\) 0 0
\(246\) 0 0
\(247\) −10.1421 −0.645329
\(248\) 13.4558 0.854447
\(249\) 0 0
\(250\) 0 0
\(251\) 24.9706 1.57613 0.788064 0.615593i \(-0.211084\pi\)
0.788064 + 0.615593i \(0.211084\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) −0.514719 −0.0322963
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −13.3137 −0.830486 −0.415243 0.909710i \(-0.636304\pi\)
−0.415243 + 0.909710i \(0.636304\pi\)
\(258\) 0 0
\(259\) 14.0711 0.874334
\(260\) 0 0
\(261\) 0 0
\(262\) 0.485281 0.0299808
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.58579 −0.219859
\(267\) 0 0
\(268\) −21.3137 −1.30194
\(269\) −23.7990 −1.45105 −0.725525 0.688196i \(-0.758403\pi\)
−0.725525 + 0.688196i \(0.758403\pi\)
\(270\) 0 0
\(271\) −10.8995 −0.662097 −0.331049 0.943614i \(-0.607402\pi\)
−0.331049 + 0.943614i \(0.607402\pi\)
\(272\) −1.24264 −0.0753462
\(273\) 0 0
\(274\) −6.68629 −0.403934
\(275\) 0 0
\(276\) 0 0
\(277\) 0.828427 0.0497754 0.0248877 0.999690i \(-0.492077\pi\)
0.0248877 + 0.999690i \(0.492077\pi\)
\(278\) −6.20101 −0.371912
\(279\) 0 0
\(280\) 0 0
\(281\) −17.9289 −1.06955 −0.534775 0.844994i \(-0.679604\pi\)
−0.534775 + 0.844994i \(0.679604\pi\)
\(282\) 0 0
\(283\) −18.8995 −1.12346 −0.561729 0.827321i \(-0.689864\pi\)
−0.561729 + 0.827321i \(0.689864\pi\)
\(284\) 3.97056 0.235610
\(285\) 0 0
\(286\) 1.17157 0.0692766
\(287\) −21.4853 −1.26824
\(288\) 0 0
\(289\) −16.8284 −0.989907
\(290\) 0 0
\(291\) 0 0
\(292\) −5.79899 −0.339360
\(293\) 20.4142 1.19261 0.596306 0.802758i \(-0.296635\pi\)
0.596306 + 0.802758i \(0.296635\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9.24264 0.537218
\(297\) 0 0
\(298\) −7.34315 −0.425377
\(299\) −2.82843 −0.163572
\(300\) 0 0
\(301\) 0.828427 0.0477497
\(302\) −5.79899 −0.333694
\(303\) 0 0
\(304\) 10.7574 0.616977
\(305\) 0 0
\(306\) 0 0
\(307\) −29.3137 −1.67302 −0.836511 0.547950i \(-0.815408\pi\)
−0.836511 + 0.547950i \(0.815408\pi\)
\(308\) −4.41421 −0.251523
\(309\) 0 0
\(310\) 0 0
\(311\) −2.34315 −0.132868 −0.0664338 0.997791i \(-0.521162\pi\)
−0.0664338 + 0.997791i \(0.521162\pi\)
\(312\) 0 0
\(313\) −1.14214 −0.0645573 −0.0322787 0.999479i \(-0.510276\pi\)
−0.0322787 + 0.999479i \(0.510276\pi\)
\(314\) 2.48528 0.140253
\(315\) 0 0
\(316\) −8.69848 −0.489328
\(317\) 25.1716 1.41378 0.706888 0.707325i \(-0.250098\pi\)
0.706888 + 0.707325i \(0.250098\pi\)
\(318\) 0 0
\(319\) −6.82843 −0.382319
\(320\) 0 0
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −1.48528 −0.0826433
\(324\) 0 0
\(325\) 0 0
\(326\) −9.85786 −0.545977
\(327\) 0 0
\(328\) −14.1127 −0.779243
\(329\) 22.8995 1.26249
\(330\) 0 0
\(331\) −3.85786 −0.212047 −0.106024 0.994364i \(-0.533812\pi\)
−0.106024 + 0.994364i \(0.533812\pi\)
\(332\) −22.8284 −1.25287
\(333\) 0 0
\(334\) 7.37258 0.403410
\(335\) 0 0
\(336\) 0 0
\(337\) 24.1421 1.31511 0.657553 0.753408i \(-0.271592\pi\)
0.657553 + 0.753408i \(0.271592\pi\)
\(338\) 2.07107 0.112651
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528 0.459504
\(342\) 0 0
\(343\) −19.7279 −1.06521
\(344\) 0.544156 0.0293389
\(345\) 0 0
\(346\) −7.68629 −0.413218
\(347\) −26.8284 −1.44023 −0.720113 0.693857i \(-0.755910\pi\)
−0.720113 + 0.693857i \(0.755910\pi\)
\(348\) 0 0
\(349\) 14.4853 0.775379 0.387690 0.921790i \(-0.373273\pi\)
0.387690 + 0.921790i \(0.373273\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.41421 −0.235278
\(353\) 12.4853 0.664524 0.332262 0.943187i \(-0.392188\pi\)
0.332262 + 0.943187i \(0.392188\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −9.44365 −0.499112
\(359\) −32.4853 −1.71451 −0.857254 0.514894i \(-0.827831\pi\)
−0.857254 + 0.514894i \(0.827831\pi\)
\(360\) 0 0
\(361\) −6.14214 −0.323270
\(362\) −4.95837 −0.260606
\(363\) 0 0
\(364\) 12.4853 0.654407
\(365\) 0 0
\(366\) 0 0
\(367\) 21.3137 1.11257 0.556283 0.830993i \(-0.312227\pi\)
0.556283 + 0.830993i \(0.312227\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) −8.82843 −0.458349
\(372\) 0 0
\(373\) 12.3431 0.639104 0.319552 0.947569i \(-0.396468\pi\)
0.319552 + 0.947569i \(0.396468\pi\)
\(374\) 0.171573 0.00887182
\(375\) 0 0
\(376\) 15.0416 0.775713
\(377\) 19.3137 0.994707
\(378\) 0 0
\(379\) 14.8284 0.761685 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.89949 0.250680
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 0 0
\(388\) 0.313708 0.0159261
\(389\) 6.34315 0.321610 0.160805 0.986986i \(-0.448591\pi\)
0.160805 + 0.986986i \(0.448591\pi\)
\(390\) 0 0
\(391\) −0.414214 −0.0209477
\(392\) −1.85786 −0.0938363
\(393\) 0 0
\(394\) −5.48528 −0.276344
\(395\) 0 0
\(396\) 0 0
\(397\) −31.9411 −1.60308 −0.801540 0.597942i \(-0.795985\pi\)
−0.801540 + 0.597942i \(0.795985\pi\)
\(398\) −2.14214 −0.107376
\(399\) 0 0
\(400\) 0 0
\(401\) 7.79899 0.389463 0.194731 0.980857i \(-0.437616\pi\)
0.194731 + 0.980857i \(0.437616\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −8.95837 −0.445696
\(405\) 0 0
\(406\) 6.82843 0.338889
\(407\) 5.82843 0.288904
\(408\) 0 0
\(409\) 24.1421 1.19375 0.596876 0.802334i \(-0.296408\pi\)
0.596876 + 0.802334i \(0.296408\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.28427 −0.211071
\(413\) −26.5563 −1.30675
\(414\) 0 0
\(415\) 0 0
\(416\) 12.4853 0.612141
\(417\) 0 0
\(418\) −1.48528 −0.0726475
\(419\) 8.51472 0.415971 0.207986 0.978132i \(-0.433309\pi\)
0.207986 + 0.978132i \(0.433309\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) −3.85786 −0.187798
\(423\) 0 0
\(424\) −5.79899 −0.281624
\(425\) 0 0
\(426\) 0 0
\(427\) 7.65685 0.370541
\(428\) −31.6569 −1.53019
\(429\) 0 0
\(430\) 0 0
\(431\) −6.82843 −0.328914 −0.164457 0.986384i \(-0.552587\pi\)
−0.164457 + 0.986384i \(0.552587\pi\)
\(432\) 0 0
\(433\) −9.31371 −0.447588 −0.223794 0.974636i \(-0.571844\pi\)
−0.223794 + 0.974636i \(0.571844\pi\)
\(434\) −8.48528 −0.407307
\(435\) 0 0
\(436\) 31.6569 1.51609
\(437\) 3.58579 0.171531
\(438\) 0 0
\(439\) 27.7279 1.32338 0.661691 0.749777i \(-0.269839\pi\)
0.661691 + 0.749777i \(0.269839\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.485281 −0.0230825
\(443\) 25.9706 1.23390 0.616949 0.787003i \(-0.288368\pi\)
0.616949 + 0.787003i \(0.288368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.1127 −0.526202
\(447\) 0 0
\(448\) −10.0711 −0.475813
\(449\) −10.4853 −0.494831 −0.247416 0.968909i \(-0.579581\pi\)
−0.247416 + 0.968909i \(0.579581\pi\)
\(450\) 0 0
\(451\) −8.89949 −0.419061
\(452\) 18.2843 0.860020
\(453\) 0 0
\(454\) 0.627417 0.0294461
\(455\) 0 0
\(456\) 0 0
\(457\) −32.1421 −1.50355 −0.751773 0.659422i \(-0.770801\pi\)
−0.751773 + 0.659422i \(0.770801\pi\)
\(458\) 8.07107 0.377136
\(459\) 0 0
\(460\) 0 0
\(461\) −40.7696 −1.89883 −0.949414 0.314028i \(-0.898321\pi\)
−0.949414 + 0.314028i \(0.898321\pi\)
\(462\) 0 0
\(463\) −1.02944 −0.0478420 −0.0239210 0.999714i \(-0.507615\pi\)
−0.0239210 + 0.999714i \(0.507615\pi\)
\(464\) −20.4853 −0.951005
\(465\) 0 0
\(466\) 6.02944 0.279308
\(467\) −34.6274 −1.60237 −0.801183 0.598420i \(-0.795796\pi\)
−0.801183 + 0.598420i \(0.795796\pi\)
\(468\) 0 0
\(469\) 28.1421 1.29948
\(470\) 0 0
\(471\) 0 0
\(472\) −17.4437 −0.802909
\(473\) 0.343146 0.0157779
\(474\) 0 0
\(475\) 0 0
\(476\) 1.82843 0.0838058
\(477\) 0 0
\(478\) 5.11270 0.233849
\(479\) 7.51472 0.343356 0.171678 0.985153i \(-0.445081\pi\)
0.171678 + 0.985153i \(0.445081\pi\)
\(480\) 0 0
\(481\) −16.4853 −0.751664
\(482\) 5.85786 0.266818
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) 0 0
\(487\) −10.4853 −0.475133 −0.237567 0.971371i \(-0.576350\pi\)
−0.237567 + 0.971371i \(0.576350\pi\)
\(488\) 5.02944 0.227672
\(489\) 0 0
\(490\) 0 0
\(491\) −4.14214 −0.186932 −0.0934660 0.995622i \(-0.529795\pi\)
−0.0934660 + 0.995622i \(0.529795\pi\)
\(492\) 0 0
\(493\) 2.82843 0.127386
\(494\) 4.20101 0.189012
\(495\) 0 0
\(496\) 25.4558 1.14300
\(497\) −5.24264 −0.235165
\(498\) 0 0
\(499\) 40.8284 1.82773 0.913866 0.406017i \(-0.133082\pi\)
0.913866 + 0.406017i \(0.133082\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −10.3431 −0.461637
\(503\) 22.2843 0.993607 0.496803 0.867863i \(-0.334507\pi\)
0.496803 + 0.867863i \(0.334507\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.414214 −0.0184140
\(507\) 0 0
\(508\) −2.27208 −0.100807
\(509\) 40.6274 1.80078 0.900389 0.435085i \(-0.143282\pi\)
0.900389 + 0.435085i \(0.143282\pi\)
\(510\) 0 0
\(511\) 7.65685 0.338719
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) 5.51472 0.243244
\(515\) 0 0
\(516\) 0 0
\(517\) 9.48528 0.417162
\(518\) −5.82843 −0.256086
\(519\) 0 0
\(520\) 0 0
\(521\) 7.85786 0.344259 0.172130 0.985074i \(-0.444935\pi\)
0.172130 + 0.985074i \(0.444935\pi\)
\(522\) 0 0
\(523\) 0.213203 0.00932274 0.00466137 0.999989i \(-0.498516\pi\)
0.00466137 + 0.999989i \(0.498516\pi\)
\(524\) 2.14214 0.0935796
\(525\) 0 0
\(526\) 4.54416 0.198135
\(527\) −3.51472 −0.153104
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −15.8284 −0.686249
\(533\) 25.1716 1.09030
\(534\) 0 0
\(535\) 0 0
\(536\) 18.4853 0.798443
\(537\) 0 0
\(538\) 9.85786 0.425003
\(539\) −1.17157 −0.0504632
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) 4.51472 0.193924
\(543\) 0 0
\(544\) 1.82843 0.0783932
\(545\) 0 0
\(546\) 0 0
\(547\) 17.8701 0.764068 0.382034 0.924148i \(-0.375224\pi\)
0.382034 + 0.924148i \(0.375224\pi\)
\(548\) −29.5147 −1.26081
\(549\) 0 0
\(550\) 0 0
\(551\) −24.4853 −1.04311
\(552\) 0 0
\(553\) 11.4853 0.488404
\(554\) −0.343146 −0.0145789
\(555\) 0 0
\(556\) −27.3726 −1.16086
\(557\) −10.8284 −0.458815 −0.229408 0.973330i \(-0.573679\pi\)
−0.229408 + 0.973330i \(0.573679\pi\)
\(558\) 0 0
\(559\) −0.970563 −0.0410504
\(560\) 0 0
\(561\) 0 0
\(562\) 7.42641 0.313264
\(563\) −7.31371 −0.308236 −0.154118 0.988052i \(-0.549254\pi\)
−0.154118 + 0.988052i \(0.549254\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.82843 0.329053
\(567\) 0 0
\(568\) −3.44365 −0.144492
\(569\) −6.75736 −0.283283 −0.141642 0.989918i \(-0.545238\pi\)
−0.141642 + 0.989918i \(0.545238\pi\)
\(570\) 0 0
\(571\) −42.9706 −1.79826 −0.899131 0.437680i \(-0.855800\pi\)
−0.899131 + 0.437680i \(0.855800\pi\)
\(572\) 5.17157 0.216234
\(573\) 0 0
\(574\) 8.89949 0.371458
\(575\) 0 0
\(576\) 0 0
\(577\) 9.97056 0.415080 0.207540 0.978227i \(-0.433454\pi\)
0.207540 + 0.978227i \(0.433454\pi\)
\(578\) 6.97056 0.289937
\(579\) 0 0
\(580\) 0 0
\(581\) 30.1421 1.25051
\(582\) 0 0
\(583\) −3.65685 −0.151451
\(584\) 5.02944 0.208120
\(585\) 0 0
\(586\) −8.45584 −0.349308
\(587\) −25.3431 −1.04602 −0.523012 0.852325i \(-0.675192\pi\)
−0.523012 + 0.852325i \(0.675192\pi\)
\(588\) 0 0
\(589\) 30.4264 1.25370
\(590\) 0 0
\(591\) 0 0
\(592\) 17.4853 0.718641
\(593\) −35.7990 −1.47009 −0.735044 0.678019i \(-0.762839\pi\)
−0.735044 + 0.678019i \(0.762839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −32.4142 −1.32774
\(597\) 0 0
\(598\) 1.17157 0.0479092
\(599\) −13.6863 −0.559207 −0.279603 0.960116i \(-0.590203\pi\)
−0.279603 + 0.960116i \(0.590203\pi\)
\(600\) 0 0
\(601\) −9.17157 −0.374116 −0.187058 0.982349i \(-0.559895\pi\)
−0.187058 + 0.982349i \(0.559895\pi\)
\(602\) −0.343146 −0.0139856
\(603\) 0 0
\(604\) −25.5980 −1.04157
\(605\) 0 0
\(606\) 0 0
\(607\) −30.9706 −1.25706 −0.628528 0.777787i \(-0.716342\pi\)
−0.628528 + 0.777787i \(0.716342\pi\)
\(608\) −15.8284 −0.641927
\(609\) 0 0
\(610\) 0 0
\(611\) −26.8284 −1.08536
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 12.1421 0.490017
\(615\) 0 0
\(616\) 3.82843 0.154252
\(617\) −38.1421 −1.53554 −0.767772 0.640723i \(-0.778635\pi\)
−0.767772 + 0.640723i \(0.778635\pi\)
\(618\) 0 0
\(619\) −6.62742 −0.266378 −0.133189 0.991091i \(-0.542522\pi\)
−0.133189 + 0.991091i \(0.542522\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.970563 0.0389160
\(623\) 18.4853 0.740597
\(624\) 0 0
\(625\) 0 0
\(626\) 0.473088 0.0189084
\(627\) 0 0
\(628\) 10.9706 0.437773
\(629\) −2.41421 −0.0962610
\(630\) 0 0
\(631\) −18.6274 −0.741546 −0.370773 0.928724i \(-0.620907\pi\)
−0.370773 + 0.928724i \(0.620907\pi\)
\(632\) 7.54416 0.300090
\(633\) 0 0
\(634\) −10.4264 −0.414086
\(635\) 0 0
\(636\) 0 0
\(637\) 3.31371 0.131294
\(638\) 2.82843 0.111979
\(639\) 0 0
\(640\) 0 0
\(641\) 25.5147 1.00777 0.503885 0.863771i \(-0.331903\pi\)
0.503885 + 0.863771i \(0.331903\pi\)
\(642\) 0 0
\(643\) −0.970563 −0.0382753 −0.0191376 0.999817i \(-0.506092\pi\)
−0.0191376 + 0.999817i \(0.506092\pi\)
\(644\) −4.41421 −0.173944
\(645\) 0 0
\(646\) 0.615224 0.0242057
\(647\) 28.6569 1.12662 0.563309 0.826247i \(-0.309528\pi\)
0.563309 + 0.826247i \(0.309528\pi\)
\(648\) 0 0
\(649\) −11.0000 −0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) −43.5147 −1.70417
\(653\) −23.5147 −0.920202 −0.460101 0.887867i \(-0.652187\pi\)
−0.460101 + 0.887867i \(0.652187\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −26.6985 −1.04240
\(657\) 0 0
\(658\) −9.48528 −0.369775
\(659\) 47.1127 1.83525 0.917625 0.397447i \(-0.130104\pi\)
0.917625 + 0.397447i \(0.130104\pi\)
\(660\) 0 0
\(661\) −23.3431 −0.907943 −0.453972 0.891016i \(-0.649993\pi\)
−0.453972 + 0.891016i \(0.649993\pi\)
\(662\) 1.59798 0.0621072
\(663\) 0 0
\(664\) 19.7990 0.768350
\(665\) 0 0
\(666\) 0 0
\(667\) −6.82843 −0.264398
\(668\) 32.5442 1.25917
\(669\) 0 0
\(670\) 0 0
\(671\) 3.17157 0.122437
\(672\) 0 0
\(673\) −0.343146 −0.0132273 −0.00661365 0.999978i \(-0.502105\pi\)
−0.00661365 + 0.999978i \(0.502105\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) 9.14214 0.351621
\(677\) 23.5147 0.903744 0.451872 0.892083i \(-0.350756\pi\)
0.451872 + 0.892083i \(0.350756\pi\)
\(678\) 0 0
\(679\) −0.414214 −0.0158961
\(680\) 0 0
\(681\) 0 0
\(682\) −3.51472 −0.134586
\(683\) 12.5147 0.478862 0.239431 0.970913i \(-0.423039\pi\)
0.239431 + 0.970913i \(0.423039\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.17157 0.311992
\(687\) 0 0
\(688\) 1.02944 0.0392469
\(689\) 10.3431 0.394042
\(690\) 0 0
\(691\) −35.4558 −1.34880 −0.674402 0.738364i \(-0.735598\pi\)
−0.674402 + 0.738364i \(0.735598\pi\)
\(692\) −33.9289 −1.28978
\(693\) 0 0
\(694\) 11.1127 0.421832
\(695\) 0 0
\(696\) 0 0
\(697\) 3.68629 0.139628
\(698\) −6.00000 −0.227103
\(699\) 0 0
\(700\) 0 0
\(701\) 41.3848 1.56308 0.781541 0.623854i \(-0.214434\pi\)
0.781541 + 0.623854i \(0.214434\pi\)
\(702\) 0 0
\(703\) 20.8995 0.788239
\(704\) −4.17157 −0.157222
\(705\) 0 0
\(706\) −5.17157 −0.194635
\(707\) 11.8284 0.444854
\(708\) 0 0
\(709\) −29.1421 −1.09446 −0.547228 0.836984i \(-0.684317\pi\)
−0.547228 + 0.836984i \(0.684317\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.1421 0.455046
\(713\) 8.48528 0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) −41.6863 −1.55789
\(717\) 0 0
\(718\) 13.4558 0.502168
\(719\) −9.65685 −0.360140 −0.180070 0.983654i \(-0.557632\pi\)
−0.180070 + 0.983654i \(0.557632\pi\)
\(720\) 0 0
\(721\) 5.65685 0.210672
\(722\) 2.54416 0.0946837
\(723\) 0 0
\(724\) −21.8873 −0.813435
\(725\) 0 0
\(726\) 0 0
\(727\) −16.9706 −0.629403 −0.314702 0.949191i \(-0.601904\pi\)
−0.314702 + 0.949191i \(0.601904\pi\)
\(728\) −10.8284 −0.401328
\(729\) 0 0
\(730\) 0 0
\(731\) −0.142136 −0.00525708
\(732\) 0 0
\(733\) −32.1421 −1.18720 −0.593598 0.804761i \(-0.702293\pi\)
−0.593598 + 0.804761i \(0.702293\pi\)
\(734\) −8.82843 −0.325863
\(735\) 0 0
\(736\) −4.41421 −0.162710
\(737\) 11.6569 0.429386
\(738\) 0 0
\(739\) 20.4142 0.750949 0.375474 0.926833i \(-0.377480\pi\)
0.375474 + 0.926833i \(0.377480\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.65685 0.134247
\(743\) −31.1127 −1.14141 −0.570707 0.821154i \(-0.693331\pi\)
−0.570707 + 0.821154i \(0.693331\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.11270 −0.187189
\(747\) 0 0
\(748\) 0.757359 0.0276918
\(749\) 41.7990 1.52730
\(750\) 0 0
\(751\) 31.5980 1.15303 0.576513 0.817088i \(-0.304413\pi\)
0.576513 + 0.817088i \(0.304413\pi\)
\(752\) 28.4558 1.03768
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 10.6863 0.388400 0.194200 0.980962i \(-0.437789\pi\)
0.194200 + 0.980962i \(0.437789\pi\)
\(758\) −6.14214 −0.223092
\(759\) 0 0
\(760\) 0 0
\(761\) 5.17157 0.187469 0.0937347 0.995597i \(-0.470119\pi\)
0.0937347 + 0.995597i \(0.470119\pi\)
\(762\) 0 0
\(763\) −41.7990 −1.51323
\(764\) 21.6274 0.782452
\(765\) 0 0
\(766\) −8.28427 −0.299323
\(767\) 31.1127 1.12341
\(768\) 0 0
\(769\) −7.65685 −0.276113 −0.138057 0.990424i \(-0.544086\pi\)
−0.138057 + 0.990424i \(0.544086\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35.3137 −1.27097
\(773\) −0.828427 −0.0297965 −0.0148982 0.999889i \(-0.504742\pi\)
−0.0148982 + 0.999889i \(0.504742\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.272078 −0.00976703
\(777\) 0 0
\(778\) −2.62742 −0.0941975
\(779\) −31.9117 −1.14335
\(780\) 0 0
\(781\) −2.17157 −0.0777050
\(782\) 0.171573 0.00613543
\(783\) 0 0
\(784\) −3.51472 −0.125526
\(785\) 0 0
\(786\) 0 0
\(787\) 7.92893 0.282636 0.141318 0.989964i \(-0.454866\pi\)
0.141318 + 0.989964i \(0.454866\pi\)
\(788\) −24.2132 −0.862560
\(789\) 0 0
\(790\) 0 0
\(791\) −24.1421 −0.858396
\(792\) 0 0
\(793\) −8.97056 −0.318554
\(794\) 13.2304 0.469531
\(795\) 0 0
\(796\) −9.45584 −0.335154
\(797\) 40.9706 1.45125 0.725626 0.688089i \(-0.241550\pi\)
0.725626 + 0.688089i \(0.241550\pi\)
\(798\) 0 0
\(799\) −3.92893 −0.138996
\(800\) 0 0
\(801\) 0 0
\(802\) −3.23045 −0.114071
\(803\) 3.17157 0.111922
\(804\) 0 0
\(805\) 0 0
\(806\) 9.94113 0.350161
\(807\) 0 0
\(808\) 7.76955 0.273332
\(809\) −7.72792 −0.271699 −0.135850 0.990729i \(-0.543376\pi\)
−0.135850 + 0.990729i \(0.543376\pi\)
\(810\) 0 0
\(811\) 10.2132 0.358634 0.179317 0.983791i \(-0.442611\pi\)
0.179317 + 0.983791i \(0.442611\pi\)
\(812\) 30.1421 1.05778
\(813\) 0 0
\(814\) −2.41421 −0.0846181
\(815\) 0 0
\(816\) 0 0
\(817\) 1.23045 0.0430479
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) −15.7990 −0.551389 −0.275694 0.961245i \(-0.588908\pi\)
−0.275694 + 0.961245i \(0.588908\pi\)
\(822\) 0 0
\(823\) 14.9706 0.521841 0.260921 0.965360i \(-0.415974\pi\)
0.260921 + 0.965360i \(0.415974\pi\)
\(824\) 3.71573 0.129444
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) −12.6863 −0.441146 −0.220573 0.975371i \(-0.570793\pi\)
−0.220573 + 0.975371i \(0.570793\pi\)
\(828\) 0 0
\(829\) −47.9411 −1.66506 −0.832532 0.553977i \(-0.813110\pi\)
−0.832532 + 0.553977i \(0.813110\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 11.7990 0.409056
\(833\) 0.485281 0.0168140
\(834\) 0 0
\(835\) 0 0
\(836\) −6.55635 −0.226756
\(837\) 0 0
\(838\) −3.52691 −0.121835
\(839\) 47.3137 1.63345 0.816725 0.577027i \(-0.195787\pi\)
0.816725 + 0.577027i \(0.195787\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) 11.1838 0.385418
\(843\) 0 0
\(844\) −17.0294 −0.586177
\(845\) 0 0
\(846\) 0 0
\(847\) 2.41421 0.0829534
\(848\) −10.9706 −0.376731
\(849\) 0 0
\(850\) 0 0
\(851\) 5.82843 0.199796
\(852\) 0 0
\(853\) 19.1716 0.656422 0.328211 0.944604i \(-0.393554\pi\)
0.328211 + 0.944604i \(0.393554\pi\)
\(854\) −3.17157 −0.108529
\(855\) 0 0
\(856\) 27.4558 0.938421
\(857\) −36.6985 −1.25360 −0.626798 0.779182i \(-0.715635\pi\)
−0.626798 + 0.779182i \(0.715635\pi\)
\(858\) 0 0
\(859\) −20.4853 −0.698949 −0.349474 0.936946i \(-0.613640\pi\)
−0.349474 + 0.936946i \(0.613640\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.82843 0.0963366
\(863\) −28.6863 −0.976493 −0.488246 0.872706i \(-0.662363\pi\)
−0.488246 + 0.872706i \(0.662363\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.85786 0.131096
\(867\) 0 0
\(868\) −37.4558 −1.27133
\(869\) 4.75736 0.161382
\(870\) 0 0
\(871\) −32.9706 −1.11716
\(872\) −27.4558 −0.929772
\(873\) 0 0
\(874\) −1.48528 −0.0502404
\(875\) 0 0
\(876\) 0 0
\(877\) −15.1127 −0.510320 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(878\) −11.4853 −0.387609
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) −38.6274 −1.29992 −0.649958 0.759970i \(-0.725214\pi\)
−0.649958 + 0.759970i \(0.725214\pi\)
\(884\) −2.14214 −0.0720478
\(885\) 0 0
\(886\) −10.7574 −0.361401
\(887\) −22.1421 −0.743460 −0.371730 0.928341i \(-0.621235\pi\)
−0.371730 + 0.928341i \(0.621235\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) −49.0538 −1.64244
\(893\) 34.0122 1.13817
\(894\) 0 0
\(895\) 0 0
\(896\) 25.4853 0.851403
\(897\) 0 0
\(898\) 4.34315 0.144933
\(899\) −57.9411 −1.93244
\(900\) 0 0
\(901\) 1.51472 0.0504626
\(902\) 3.68629 0.122740
\(903\) 0 0
\(904\) −15.8579 −0.527425
\(905\) 0 0
\(906\) 0 0
\(907\) 2.48528 0.0825224 0.0412612 0.999148i \(-0.486862\pi\)
0.0412612 + 0.999148i \(0.486862\pi\)
\(908\) 2.76955 0.0919108
\(909\) 0 0
\(910\) 0 0
\(911\) −28.5147 −0.944735 −0.472367 0.881402i \(-0.656600\pi\)
−0.472367 + 0.881402i \(0.656600\pi\)
\(912\) 0 0
\(913\) 12.4853 0.413203
\(914\) 13.3137 0.440378
\(915\) 0 0
\(916\) 35.6274 1.17716
\(917\) −2.82843 −0.0934029
\(918\) 0 0
\(919\) 1.78680 0.0589410 0.0294705 0.999566i \(-0.490618\pi\)
0.0294705 + 0.999566i \(0.490618\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 16.8873 0.556154
\(923\) 6.14214 0.202171
\(924\) 0 0
\(925\) 0 0
\(926\) 0.426407 0.0140126
\(927\) 0 0
\(928\) 30.1421 0.989464
\(929\) 13.7990 0.452730 0.226365 0.974043i \(-0.427316\pi\)
0.226365 + 0.974043i \(0.427316\pi\)
\(930\) 0 0
\(931\) −4.20101 −0.137683
\(932\) 26.6152 0.871811
\(933\) 0 0
\(934\) 14.3431 0.469322
\(935\) 0 0
\(936\) 0 0
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) −11.6569 −0.380610
\(939\) 0 0
\(940\) 0 0
\(941\) 22.8995 0.746502 0.373251 0.927730i \(-0.378243\pi\)
0.373251 + 0.927730i \(0.378243\pi\)
\(942\) 0 0
\(943\) −8.89949 −0.289807
\(944\) −33.0000 −1.07406
\(945\) 0 0
\(946\) −0.142136 −0.00462123
\(947\) −36.7990 −1.19581 −0.597903 0.801568i \(-0.703999\pi\)
−0.597903 + 0.801568i \(0.703999\pi\)
\(948\) 0 0
\(949\) −8.97056 −0.291197
\(950\) 0 0
\(951\) 0 0
\(952\) −1.58579 −0.0513956
\(953\) −29.0416 −0.940751 −0.470375 0.882466i \(-0.655881\pi\)
−0.470375 + 0.882466i \(0.655881\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 22.5685 0.729919
\(957\) 0 0
\(958\) −3.11270 −0.100567
\(959\) 38.9706 1.25843
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 6.82843 0.220157
\(963\) 0 0
\(964\) 25.8579 0.832826
\(965\) 0 0
\(966\) 0 0
\(967\) −38.0000 −1.22200 −0.610999 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) 1.58579 0.0509691
\(969\) 0 0
\(970\) 0 0
\(971\) 50.3137 1.61464 0.807322 0.590111i \(-0.200916\pi\)
0.807322 + 0.590111i \(0.200916\pi\)
\(972\) 0 0
\(973\) 36.1421 1.15866
\(974\) 4.34315 0.139163
\(975\) 0 0
\(976\) 9.51472 0.304559
\(977\) 60.5685 1.93776 0.968880 0.247532i \(-0.0796196\pi\)
0.968880 + 0.247532i \(0.0796196\pi\)
\(978\) 0 0
\(979\) 7.65685 0.244714
\(980\) 0 0
\(981\) 0 0
\(982\) 1.71573 0.0547511
\(983\) 51.2843 1.63571 0.817857 0.575421i \(-0.195162\pi\)
0.817857 + 0.575421i \(0.195162\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.17157 −0.0373105
\(987\) 0 0
\(988\) 18.5442 0.589968
\(989\) 0.343146 0.0109114
\(990\) 0 0
\(991\) −42.2843 −1.34320 −0.671602 0.740912i \(-0.734394\pi\)
−0.671602 + 0.740912i \(0.734394\pi\)
\(992\) −37.4558 −1.18922
\(993\) 0 0
\(994\) 2.17157 0.0688781
\(995\) 0 0
\(996\) 0 0
\(997\) 47.2548 1.49658 0.748288 0.663374i \(-0.230876\pi\)
0.748288 + 0.663374i \(0.230876\pi\)
\(998\) −16.9117 −0.535330
\(999\) 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 2475.2.a.w.1.1 2
3.2 odd 2 825.2.a.d.1.2 2
5.2 odd 4 2475.2.c.o.199.2 4
5.3 odd 4 2475.2.c.o.199.3 4
5.4 even 2 2475.2.a.l.1.2 2
15.2 even 4 825.2.c.d.199.3 4
15.8 even 4 825.2.c.d.199.2 4
15.14 odd 2 825.2.a.f.1.1 yes 2
33.32 even 2 9075.2.a.ca.1.1 2
165.164 even 2 9075.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.2 2 3.2 odd 2
825.2.a.f.1.1 yes 2 15.14 odd 2
825.2.c.d.199.2 4 15.8 even 4
825.2.c.d.199.3 4 15.2 even 4
2475.2.a.l.1.2 2 5.4 even 2
2475.2.a.w.1.1 2 1.1 even 1 trivial
2475.2.c.o.199.2 4 5.2 odd 4
2475.2.c.o.199.3 4 5.3 odd 4
9075.2.a.w.1.2 2 165.164 even 2
9075.2.a.ca.1.1 2 33.32 even 2