Properties

Label 2960.2.a.bb.1.2
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.583504.1
Defining polynomial: \(x^{5} - 8 x^{3} - 2 x^{2} + 15 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.84514\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} +O(q^{10})\) \(q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} -1.83634 q^{11} -2.76560 q^{13} -0.845141 q^{15} -6.31299 q^{17} +1.00880 q^{19} -1.81045 q^{21} +5.38831 q^{23} +1.00000 q^{25} +4.46719 q^{27} +5.38831 q^{29} +3.18704 q^{31} +1.55197 q^{33} +2.14219 q^{35} -1.00000 q^{37} +2.33732 q^{39} -1.83634 q^{41} +1.11881 q^{43} -2.28574 q^{45} +7.02338 q^{47} -2.41103 q^{49} +5.33537 q^{51} +13.9214 q^{53} -1.83634 q^{55} -0.852575 q^{57} +2.21887 q^{59} +8.69677 q^{61} -4.89648 q^{63} -2.76560 q^{65} +15.0380 q^{67} -4.55388 q^{69} +6.27408 q^{71} -7.15255 q^{73} -0.845141 q^{75} -3.93379 q^{77} +6.70065 q^{79} +3.08181 q^{81} -8.35100 q^{83} -6.31299 q^{85} -4.55388 q^{87} +2.59682 q^{89} -5.92443 q^{91} -2.69349 q^{93} +1.00880 q^{95} +13.6325 q^{97} +4.19740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + 5q^{5} + 5q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + 5q^{5} + 5q^{7} + 6q^{9} + 7q^{11} - 6q^{13} + 5q^{15} + 12q^{19} - q^{21} + 6q^{23} + 5q^{25} + 17q^{27} + 6q^{29} + 10q^{31} + 3q^{33} + 5q^{35} - 5q^{37} - 4q^{39} + 7q^{41} + 22q^{43} + 6q^{45} + 13q^{47} + 14q^{49} + 10q^{51} - 11q^{53} + 7q^{55} - 8q^{57} + 10q^{59} + 10q^{63} - 6q^{65} + 24q^{67} + 26q^{69} + 13q^{71} - 13q^{73} + 5q^{75} - 19q^{77} - 2q^{79} + q^{81} + 13q^{83} + 26q^{87} + 8q^{89} + 8q^{91} - 20q^{93} + 12q^{95} - 20q^{97} - 2q^{99} + O(q^{100}) \)

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 0
\(3\) −0.845141 −0.487942 −0.243971 0.969783i \(-0.578450\pi\)
−0.243971 + 0.969783i \(0.578450\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.14219 0.809671 0.404835 0.914390i \(-0.367329\pi\)
0.404835 + 0.914390i \(0.367329\pi\)
\(8\) 0 0
\(9\) −2.28574 −0.761912
\(10\) 0 0
\(11\) −1.83634 −0.553679 −0.276839 0.960916i \(-0.589287\pi\)
−0.276839 + 0.960916i \(0.589287\pi\)
\(12\) 0 0
\(13\) −2.76560 −0.767040 −0.383520 0.923533i \(-0.625288\pi\)
−0.383520 + 0.923533i \(0.625288\pi\)
\(14\) 0 0
\(15\) −0.845141 −0.218214
\(16\) 0 0
\(17\) −6.31299 −1.53113 −0.765563 0.643361i \(-0.777539\pi\)
−0.765563 + 0.643361i \(0.777539\pi\)
\(18\) 0 0
\(19\) 1.00880 0.231434 0.115717 0.993282i \(-0.463083\pi\)
0.115717 + 0.993282i \(0.463083\pi\)
\(20\) 0 0
\(21\) −1.81045 −0.395073
\(22\) 0 0
\(23\) 5.38831 1.12354 0.561771 0.827293i \(-0.310120\pi\)
0.561771 + 0.827293i \(0.310120\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.46719 0.859711
\(28\) 0 0
\(29\) 5.38831 1.00058 0.500292 0.865857i \(-0.333226\pi\)
0.500292 + 0.865857i \(0.333226\pi\)
\(30\) 0 0
\(31\) 3.18704 0.572409 0.286204 0.958169i \(-0.407606\pi\)
0.286204 + 0.958169i \(0.407606\pi\)
\(32\) 0 0
\(33\) 1.55197 0.270163
\(34\) 0 0
\(35\) 2.14219 0.362096
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 2.33732 0.374271
\(40\) 0 0
\(41\) −1.83634 −0.286789 −0.143394 0.989666i \(-0.545802\pi\)
−0.143394 + 0.989666i \(0.545802\pi\)
\(42\) 0 0
\(43\) 1.11881 0.170616 0.0853082 0.996355i \(-0.472813\pi\)
0.0853082 + 0.996355i \(0.472813\pi\)
\(44\) 0 0
\(45\) −2.28574 −0.340738
\(46\) 0 0
\(47\) 7.02338 1.02447 0.512233 0.858847i \(-0.328819\pi\)
0.512233 + 0.858847i \(0.328819\pi\)
\(48\) 0 0
\(49\) −2.41103 −0.344433
\(50\) 0 0
\(51\) 5.33537 0.747101
\(52\) 0 0
\(53\) 13.9214 1.91226 0.956128 0.292951i \(-0.0946372\pi\)
0.956128 + 0.292951i \(0.0946372\pi\)
\(54\) 0 0
\(55\) −1.83634 −0.247613
\(56\) 0 0
\(57\) −0.852575 −0.112926
\(58\) 0 0
\(59\) 2.21887 0.288872 0.144436 0.989514i \(-0.453863\pi\)
0.144436 + 0.989514i \(0.453863\pi\)
\(60\) 0 0
\(61\) 8.69677 1.11351 0.556754 0.830678i \(-0.312047\pi\)
0.556754 + 0.830678i \(0.312047\pi\)
\(62\) 0 0
\(63\) −4.89648 −0.616898
\(64\) 0 0
\(65\) −2.76560 −0.343031
\(66\) 0 0
\(67\) 15.0380 1.83718 0.918590 0.395212i \(-0.129329\pi\)
0.918590 + 0.395212i \(0.129329\pi\)
\(68\) 0 0
\(69\) −4.55388 −0.548223
\(70\) 0 0
\(71\) 6.27408 0.744597 0.372298 0.928113i \(-0.378570\pi\)
0.372298 + 0.928113i \(0.378570\pi\)
\(72\) 0 0
\(73\) −7.15255 −0.837143 −0.418571 0.908184i \(-0.637469\pi\)
−0.418571 + 0.908184i \(0.637469\pi\)
\(74\) 0 0
\(75\) −0.845141 −0.0975884
\(76\) 0 0
\(77\) −3.93379 −0.448297
\(78\) 0 0
\(79\) 6.70065 0.753881 0.376941 0.926237i \(-0.376976\pi\)
0.376941 + 0.926237i \(0.376976\pi\)
\(80\) 0 0
\(81\) 3.08181 0.342423
\(82\) 0 0
\(83\) −8.35100 −0.916641 −0.458321 0.888787i \(-0.651549\pi\)
−0.458321 + 0.888787i \(0.651549\pi\)
\(84\) 0 0
\(85\) −6.31299 −0.684740
\(86\) 0 0
\(87\) −4.55388 −0.488227
\(88\) 0 0
\(89\) 2.59682 0.275262 0.137631 0.990484i \(-0.456051\pi\)
0.137631 + 0.990484i \(0.456051\pi\)
\(90\) 0 0
\(91\) −5.92443 −0.621050
\(92\) 0 0
\(93\) −2.69349 −0.279302
\(94\) 0 0
\(95\) 1.00880 0.103500
\(96\) 0 0
\(97\) 13.6325 1.38417 0.692084 0.721817i \(-0.256693\pi\)
0.692084 + 0.721817i \(0.256693\pi\)
\(98\) 0 0
\(99\) 4.19740 0.421855
\(100\) 0 0
\(101\) −4.76878 −0.474511 −0.237255 0.971447i \(-0.576248\pi\)
−0.237255 + 0.971447i \(0.576248\pi\)
\(102\) 0 0
\(103\) −16.1572 −1.59202 −0.796008 0.605287i \(-0.793058\pi\)
−0.796008 + 0.605287i \(0.793058\pi\)
\(104\) 0 0
\(105\) −1.81045 −0.176682
\(106\) 0 0
\(107\) −13.6110 −1.31583 −0.657913 0.753094i \(-0.728561\pi\)
−0.657913 + 0.753094i \(0.728561\pi\)
\(108\) 0 0
\(109\) 8.57147 0.820998 0.410499 0.911861i \(-0.365354\pi\)
0.410499 + 0.911861i \(0.365354\pi\)
\(110\) 0 0
\(111\) 0.845141 0.0802172
\(112\) 0 0
\(113\) −16.9228 −1.59196 −0.795981 0.605322i \(-0.793045\pi\)
−0.795981 + 0.605322i \(0.793045\pi\)
\(114\) 0 0
\(115\) 5.38831 0.502463
\(116\) 0 0
\(117\) 6.32144 0.584417
\(118\) 0 0
\(119\) −13.5236 −1.23971
\(120\) 0 0
\(121\) −7.62784 −0.693440
\(122\) 0 0
\(123\) 1.55197 0.139936
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 21.7938 1.93389 0.966945 0.254986i \(-0.0820710\pi\)
0.966945 + 0.254986i \(0.0820710\pi\)
\(128\) 0 0
\(129\) −0.945549 −0.0832509
\(130\) 0 0
\(131\) −13.6731 −1.19462 −0.597312 0.802009i \(-0.703765\pi\)
−0.597312 + 0.802009i \(0.703765\pi\)
\(132\) 0 0
\(133\) 2.16103 0.187385
\(134\) 0 0
\(135\) 4.46719 0.384475
\(136\) 0 0
\(137\) −3.71290 −0.317214 −0.158607 0.987342i \(-0.550700\pi\)
−0.158607 + 0.987342i \(0.550700\pi\)
\(138\) 0 0
\(139\) 17.0560 1.44667 0.723334 0.690498i \(-0.242608\pi\)
0.723334 + 0.690498i \(0.242608\pi\)
\(140\) 0 0
\(141\) −5.93574 −0.499880
\(142\) 0 0
\(143\) 5.07859 0.424693
\(144\) 0 0
\(145\) 5.38831 0.447475
\(146\) 0 0
\(147\) 2.03766 0.168063
\(148\) 0 0
\(149\) 11.5007 0.942177 0.471088 0.882086i \(-0.343861\pi\)
0.471088 + 0.882086i \(0.343861\pi\)
\(150\) 0 0
\(151\) −0.736981 −0.0599747 −0.0299873 0.999550i \(-0.509547\pi\)
−0.0299873 + 0.999550i \(0.509547\pi\)
\(152\) 0 0
\(153\) 14.4298 1.16658
\(154\) 0 0
\(155\) 3.18704 0.255989
\(156\) 0 0
\(157\) 5.81100 0.463768 0.231884 0.972743i \(-0.425511\pi\)
0.231884 + 0.972743i \(0.425511\pi\)
\(158\) 0 0
\(159\) −11.7656 −0.933070
\(160\) 0 0
\(161\) 11.5428 0.909698
\(162\) 0 0
\(163\) 11.4577 0.897436 0.448718 0.893673i \(-0.351881\pi\)
0.448718 + 0.893673i \(0.351881\pi\)
\(164\) 0 0
\(165\) 1.55197 0.120821
\(166\) 0 0
\(167\) 19.3481 1.49720 0.748601 0.663021i \(-0.230726\pi\)
0.748601 + 0.663021i \(0.230726\pi\)
\(168\) 0 0
\(169\) −5.35145 −0.411650
\(170\) 0 0
\(171\) −2.30584 −0.176332
\(172\) 0 0
\(173\) −14.0805 −1.07052 −0.535261 0.844687i \(-0.679787\pi\)
−0.535261 + 0.844687i \(0.679787\pi\)
\(174\) 0 0
\(175\) 2.14219 0.161934
\(176\) 0 0
\(177\) −1.87526 −0.140953
\(178\) 0 0
\(179\) 18.3302 1.37007 0.685033 0.728512i \(-0.259788\pi\)
0.685033 + 0.728512i \(0.259788\pi\)
\(180\) 0 0
\(181\) 14.0722 1.04598 0.522989 0.852339i \(-0.324817\pi\)
0.522989 + 0.852339i \(0.324817\pi\)
\(182\) 0 0
\(183\) −7.34999 −0.543327
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 11.5928 0.847752
\(188\) 0 0
\(189\) 9.56956 0.696083
\(190\) 0 0
\(191\) 7.63049 0.552123 0.276062 0.961140i \(-0.410971\pi\)
0.276062 + 0.961140i \(0.410971\pi\)
\(192\) 0 0
\(193\) 3.18819 0.229491 0.114745 0.993395i \(-0.463395\pi\)
0.114745 + 0.993395i \(0.463395\pi\)
\(194\) 0 0
\(195\) 2.33732 0.167379
\(196\) 0 0
\(197\) −1.65654 −0.118023 −0.0590116 0.998257i \(-0.518795\pi\)
−0.0590116 + 0.998257i \(0.518795\pi\)
\(198\) 0 0
\(199\) −17.0245 −1.20683 −0.603417 0.797426i \(-0.706195\pi\)
−0.603417 + 0.797426i \(0.706195\pi\)
\(200\) 0 0
\(201\) −12.7092 −0.896438
\(202\) 0 0
\(203\) 11.5428 0.810144
\(204\) 0 0
\(205\) −1.83634 −0.128256
\(206\) 0 0
\(207\) −12.3163 −0.856040
\(208\) 0 0
\(209\) −1.85250 −0.128140
\(210\) 0 0
\(211\) 10.2460 0.705365 0.352682 0.935743i \(-0.385270\pi\)
0.352682 + 0.935743i \(0.385270\pi\)
\(212\) 0 0
\(213\) −5.30248 −0.363320
\(214\) 0 0
\(215\) 1.11881 0.0763020
\(216\) 0 0
\(217\) 6.82723 0.463463
\(218\) 0 0
\(219\) 6.04491 0.408477
\(220\) 0 0
\(221\) 17.4592 1.17443
\(222\) 0 0
\(223\) −26.8081 −1.79520 −0.897600 0.440810i \(-0.854691\pi\)
−0.897600 + 0.440810i \(0.854691\pi\)
\(224\) 0 0
\(225\) −2.28574 −0.152382
\(226\) 0 0
\(227\) 10.0644 0.667995 0.333997 0.942574i \(-0.391602\pi\)
0.333997 + 0.942574i \(0.391602\pi\)
\(228\) 0 0
\(229\) 21.0635 1.39192 0.695959 0.718081i \(-0.254979\pi\)
0.695959 + 0.718081i \(0.254979\pi\)
\(230\) 0 0
\(231\) 3.32461 0.218743
\(232\) 0 0
\(233\) 17.2123 1.12762 0.563809 0.825905i \(-0.309336\pi\)
0.563809 + 0.825905i \(0.309336\pi\)
\(234\) 0 0
\(235\) 7.02338 0.458155
\(236\) 0 0
\(237\) −5.66299 −0.367851
\(238\) 0 0
\(239\) 1.78270 0.115313 0.0576566 0.998336i \(-0.481637\pi\)
0.0576566 + 0.998336i \(0.481637\pi\)
\(240\) 0 0
\(241\) 4.28710 0.276156 0.138078 0.990421i \(-0.455907\pi\)
0.138078 + 0.990421i \(0.455907\pi\)
\(242\) 0 0
\(243\) −16.0061 −1.02679
\(244\) 0 0
\(245\) −2.41103 −0.154035
\(246\) 0 0
\(247\) −2.78993 −0.177519
\(248\) 0 0
\(249\) 7.05777 0.447268
\(250\) 0 0
\(251\) −3.55745 −0.224544 −0.112272 0.993677i \(-0.535813\pi\)
−0.112272 + 0.993677i \(0.535813\pi\)
\(252\) 0 0
\(253\) −9.89480 −0.622081
\(254\) 0 0
\(255\) 5.33537 0.334114
\(256\) 0 0
\(257\) −18.2643 −1.13930 −0.569649 0.821888i \(-0.692921\pi\)
−0.569649 + 0.821888i \(0.692921\pi\)
\(258\) 0 0
\(259\) −2.14219 −0.133109
\(260\) 0 0
\(261\) −12.3163 −0.762358
\(262\) 0 0
\(263\) −0.734714 −0.0453044 −0.0226522 0.999743i \(-0.507211\pi\)
−0.0226522 + 0.999743i \(0.507211\pi\)
\(264\) 0 0
\(265\) 13.9214 0.855186
\(266\) 0 0
\(267\) −2.19468 −0.134312
\(268\) 0 0
\(269\) 10.6357 0.648469 0.324235 0.945977i \(-0.394893\pi\)
0.324235 + 0.945977i \(0.394893\pi\)
\(270\) 0 0
\(271\) 8.23324 0.500133 0.250067 0.968229i \(-0.419547\pi\)
0.250067 + 0.968229i \(0.419547\pi\)
\(272\) 0 0
\(273\) 5.00698 0.303036
\(274\) 0 0
\(275\) −1.83634 −0.110736
\(276\) 0 0
\(277\) 14.0818 0.846094 0.423047 0.906108i \(-0.360961\pi\)
0.423047 + 0.906108i \(0.360961\pi\)
\(278\) 0 0
\(279\) −7.28473 −0.436125
\(280\) 0 0
\(281\) 17.3533 1.03521 0.517606 0.855619i \(-0.326823\pi\)
0.517606 + 0.855619i \(0.326823\pi\)
\(282\) 0 0
\(283\) −3.21736 −0.191252 −0.0956260 0.995417i \(-0.530485\pi\)
−0.0956260 + 0.995417i \(0.530485\pi\)
\(284\) 0 0
\(285\) −0.852575 −0.0505022
\(286\) 0 0
\(287\) −3.93379 −0.232205
\(288\) 0 0
\(289\) 22.8539 1.34435
\(290\) 0 0
\(291\) −11.5214 −0.675394
\(292\) 0 0
\(293\) 5.78773 0.338123 0.169061 0.985606i \(-0.445926\pi\)
0.169061 + 0.985606i \(0.445926\pi\)
\(294\) 0 0
\(295\) 2.21887 0.129188
\(296\) 0 0
\(297\) −8.20330 −0.476004
\(298\) 0 0
\(299\) −14.9019 −0.861800
\(300\) 0 0
\(301\) 2.39669 0.138143
\(302\) 0 0
\(303\) 4.03029 0.231534
\(304\) 0 0
\(305\) 8.69677 0.497976
\(306\) 0 0
\(307\) 11.0617 0.631325 0.315662 0.948872i \(-0.397773\pi\)
0.315662 + 0.948872i \(0.397773\pi\)
\(308\) 0 0
\(309\) 13.6551 0.776811
\(310\) 0 0
\(311\) 10.1319 0.574527 0.287264 0.957852i \(-0.407254\pi\)
0.287264 + 0.957852i \(0.407254\pi\)
\(312\) 0 0
\(313\) 5.24683 0.296568 0.148284 0.988945i \(-0.452625\pi\)
0.148284 + 0.988945i \(0.452625\pi\)
\(314\) 0 0
\(315\) −4.89648 −0.275885
\(316\) 0 0
\(317\) −33.3242 −1.87167 −0.935836 0.352437i \(-0.885353\pi\)
−0.935836 + 0.352437i \(0.885353\pi\)
\(318\) 0 0
\(319\) −9.89480 −0.554002
\(320\) 0 0
\(321\) 11.5032 0.642047
\(322\) 0 0
\(323\) −6.36852 −0.354354
\(324\) 0 0
\(325\) −2.76560 −0.153408
\(326\) 0 0
\(327\) −7.24410 −0.400600
\(328\) 0 0
\(329\) 15.0454 0.829480
\(330\) 0 0
\(331\) −17.6010 −0.967438 −0.483719 0.875223i \(-0.660714\pi\)
−0.483719 + 0.875223i \(0.660714\pi\)
\(332\) 0 0
\(333\) 2.28574 0.125258
\(334\) 0 0
\(335\) 15.0380 0.821612
\(336\) 0 0
\(337\) 14.3624 0.782369 0.391184 0.920312i \(-0.372065\pi\)
0.391184 + 0.920312i \(0.372065\pi\)
\(338\) 0 0
\(339\) 14.3021 0.776785
\(340\) 0 0
\(341\) −5.85250 −0.316930
\(342\) 0 0
\(343\) −20.1602 −1.08855
\(344\) 0 0
\(345\) −4.55388 −0.245173
\(346\) 0 0
\(347\) 3.69663 0.198446 0.0992228 0.995065i \(-0.468364\pi\)
0.0992228 + 0.995065i \(0.468364\pi\)
\(348\) 0 0
\(349\) 17.7954 0.952566 0.476283 0.879292i \(-0.341984\pi\)
0.476283 + 0.879292i \(0.341984\pi\)
\(350\) 0 0
\(351\) −12.3545 −0.659433
\(352\) 0 0
\(353\) −22.8156 −1.21435 −0.607177 0.794567i \(-0.707698\pi\)
−0.607177 + 0.794567i \(0.707698\pi\)
\(354\) 0 0
\(355\) 6.27408 0.332994
\(356\) 0 0
\(357\) 11.4294 0.604906
\(358\) 0 0
\(359\) −2.54296 −0.134212 −0.0671060 0.997746i \(-0.521377\pi\)
−0.0671060 + 0.997746i \(0.521377\pi\)
\(360\) 0 0
\(361\) −17.9823 −0.946438
\(362\) 0 0
\(363\) 6.44660 0.338359
\(364\) 0 0
\(365\) −7.15255 −0.374382
\(366\) 0 0
\(367\) −21.9523 −1.14590 −0.572949 0.819591i \(-0.694201\pi\)
−0.572949 + 0.819591i \(0.694201\pi\)
\(368\) 0 0
\(369\) 4.19740 0.218508
\(370\) 0 0
\(371\) 29.8223 1.54830
\(372\) 0 0
\(373\) 8.70267 0.450607 0.225304 0.974289i \(-0.427663\pi\)
0.225304 + 0.974289i \(0.427663\pi\)
\(374\) 0 0
\(375\) −0.845141 −0.0436429
\(376\) 0 0
\(377\) −14.9019 −0.767488
\(378\) 0 0
\(379\) −10.2501 −0.526510 −0.263255 0.964726i \(-0.584796\pi\)
−0.263255 + 0.964726i \(0.584796\pi\)
\(380\) 0 0
\(381\) −18.4188 −0.943626
\(382\) 0 0
\(383\) −12.8650 −0.657371 −0.328685 0.944439i \(-0.606606\pi\)
−0.328685 + 0.944439i \(0.606606\pi\)
\(384\) 0 0
\(385\) −3.93379 −0.200485
\(386\) 0 0
\(387\) −2.55730 −0.129995
\(388\) 0 0
\(389\) −14.7739 −0.749067 −0.374533 0.927213i \(-0.622197\pi\)
−0.374533 + 0.927213i \(0.622197\pi\)
\(390\) 0 0
\(391\) −34.0164 −1.72028
\(392\) 0 0
\(393\) 11.5557 0.582908
\(394\) 0 0
\(395\) 6.70065 0.337146
\(396\) 0 0
\(397\) −25.6937 −1.28953 −0.644764 0.764382i \(-0.723044\pi\)
−0.644764 + 0.764382i \(0.723044\pi\)
\(398\) 0 0
\(399\) −1.82637 −0.0914331
\(400\) 0 0
\(401\) −16.3311 −0.815538 −0.407769 0.913085i \(-0.633693\pi\)
−0.407769 + 0.913085i \(0.633693\pi\)
\(402\) 0 0
\(403\) −8.81407 −0.439060
\(404\) 0 0
\(405\) 3.08181 0.153136
\(406\) 0 0
\(407\) 1.83634 0.0910242
\(408\) 0 0
\(409\) 36.5506 1.80731 0.903655 0.428261i \(-0.140874\pi\)
0.903655 + 0.428261i \(0.140874\pi\)
\(410\) 0 0
\(411\) 3.13792 0.154782
\(412\) 0 0
\(413\) 4.75323 0.233891
\(414\) 0 0
\(415\) −8.35100 −0.409934
\(416\) 0 0
\(417\) −14.4147 −0.705891
\(418\) 0 0
\(419\) 32.4089 1.58328 0.791640 0.610988i \(-0.209228\pi\)
0.791640 + 0.610988i \(0.209228\pi\)
\(420\) 0 0
\(421\) 33.5350 1.63440 0.817199 0.576356i \(-0.195526\pi\)
0.817199 + 0.576356i \(0.195526\pi\)
\(422\) 0 0
\(423\) −16.0536 −0.780553
\(424\) 0 0
\(425\) −6.31299 −0.306225
\(426\) 0 0
\(427\) 18.6301 0.901574
\(428\) 0 0
\(429\) −4.29213 −0.207226
\(430\) 0 0
\(431\) −9.59342 −0.462099 −0.231049 0.972942i \(-0.574216\pi\)
−0.231049 + 0.972942i \(0.574216\pi\)
\(432\) 0 0
\(433\) −18.0805 −0.868894 −0.434447 0.900697i \(-0.643056\pi\)
−0.434447 + 0.900697i \(0.643056\pi\)
\(434\) 0 0
\(435\) −4.55388 −0.218342
\(436\) 0 0
\(437\) 5.43571 0.260025
\(438\) 0 0
\(439\) −8.71527 −0.415957 −0.207979 0.978133i \(-0.566688\pi\)
−0.207979 + 0.978133i \(0.566688\pi\)
\(440\) 0 0
\(441\) 5.51099 0.262428
\(442\) 0 0
\(443\) −2.08265 −0.0989495 −0.0494748 0.998775i \(-0.515755\pi\)
−0.0494748 + 0.998775i \(0.515755\pi\)
\(444\) 0 0
\(445\) 2.59682 0.123101
\(446\) 0 0
\(447\) −9.71973 −0.459728
\(448\) 0 0
\(449\) 0.135770 0.00640740 0.00320370 0.999995i \(-0.498980\pi\)
0.00320370 + 0.999995i \(0.498980\pi\)
\(450\) 0 0
\(451\) 3.37216 0.158789
\(452\) 0 0
\(453\) 0.622853 0.0292642
\(454\) 0 0
\(455\) −5.92443 −0.277742
\(456\) 0 0
\(457\) −33.8739 −1.58455 −0.792277 0.610161i \(-0.791105\pi\)
−0.792277 + 0.610161i \(0.791105\pi\)
\(458\) 0 0
\(459\) −28.2014 −1.31633
\(460\) 0 0
\(461\) −25.2123 −1.17426 −0.587128 0.809494i \(-0.699741\pi\)
−0.587128 + 0.809494i \(0.699741\pi\)
\(462\) 0 0
\(463\) 9.33652 0.433905 0.216953 0.976182i \(-0.430388\pi\)
0.216953 + 0.976182i \(0.430388\pi\)
\(464\) 0 0
\(465\) −2.69349 −0.124908
\(466\) 0 0
\(467\) −25.5507 −1.18235 −0.591173 0.806545i \(-0.701335\pi\)
−0.591173 + 0.806545i \(0.701335\pi\)
\(468\) 0 0
\(469\) 32.2141 1.48751
\(470\) 0 0
\(471\) −4.91111 −0.226292
\(472\) 0 0
\(473\) −2.05451 −0.0944666
\(474\) 0 0
\(475\) 1.00880 0.0462867
\(476\) 0 0
\(477\) −31.8207 −1.45697
\(478\) 0 0
\(479\) −0.374490 −0.0171109 −0.00855545 0.999963i \(-0.502723\pi\)
−0.00855545 + 0.999963i \(0.502723\pi\)
\(480\) 0 0
\(481\) 2.76560 0.126101
\(482\) 0 0
\(483\) −9.75527 −0.443880
\(484\) 0 0
\(485\) 13.6325 0.619019
\(486\) 0 0
\(487\) −24.5780 −1.11374 −0.556868 0.830601i \(-0.687997\pi\)
−0.556868 + 0.830601i \(0.687997\pi\)
\(488\) 0 0
\(489\) −9.68336 −0.437897
\(490\) 0 0
\(491\) 9.99671 0.451145 0.225573 0.974226i \(-0.427575\pi\)
0.225573 + 0.974226i \(0.427575\pi\)
\(492\) 0 0
\(493\) −34.0164 −1.53202
\(494\) 0 0
\(495\) 4.19740 0.188659
\(496\) 0 0
\(497\) 13.4403 0.602878
\(498\) 0 0
\(499\) 0.296002 0.0132509 0.00662543 0.999978i \(-0.497891\pi\)
0.00662543 + 0.999978i \(0.497891\pi\)
\(500\) 0 0
\(501\) −16.3519 −0.730547
\(502\) 0 0
\(503\) −0.184424 −0.00822306 −0.00411153 0.999992i \(-0.501309\pi\)
−0.00411153 + 0.999992i \(0.501309\pi\)
\(504\) 0 0
\(505\) −4.76878 −0.212208
\(506\) 0 0
\(507\) 4.52273 0.200862
\(508\) 0 0
\(509\) 0.0720659 0.00319427 0.00159713 0.999999i \(-0.499492\pi\)
0.00159713 + 0.999999i \(0.499492\pi\)
\(510\) 0 0
\(511\) −15.3221 −0.677810
\(512\) 0 0
\(513\) 4.50649 0.198966
\(514\) 0 0
\(515\) −16.1572 −0.711971
\(516\) 0 0
\(517\) −12.8973 −0.567225
\(518\) 0 0
\(519\) 11.9000 0.522352
\(520\) 0 0
\(521\) −14.4605 −0.633524 −0.316762 0.948505i \(-0.602596\pi\)
−0.316762 + 0.948505i \(0.602596\pi\)
\(522\) 0 0
\(523\) 26.2877 1.14948 0.574741 0.818336i \(-0.305103\pi\)
0.574741 + 0.818336i \(0.305103\pi\)
\(524\) 0 0
\(525\) −1.81045 −0.0790145
\(526\) 0 0
\(527\) −20.1197 −0.876430
\(528\) 0 0
\(529\) 6.03392 0.262344
\(530\) 0 0
\(531\) −5.07175 −0.220095
\(532\) 0 0
\(533\) 5.07859 0.219978
\(534\) 0 0
\(535\) −13.6110 −0.588455
\(536\) 0 0
\(537\) −15.4916 −0.668513
\(538\) 0 0
\(539\) 4.42749 0.190705
\(540\) 0 0
\(541\) 23.9637 1.03028 0.515140 0.857106i \(-0.327740\pi\)
0.515140 + 0.857106i \(0.327740\pi\)
\(542\) 0 0
\(543\) −11.8930 −0.510377
\(544\) 0 0
\(545\) 8.57147 0.367162
\(546\) 0 0
\(547\) 15.1064 0.645902 0.322951 0.946416i \(-0.395325\pi\)
0.322951 + 0.946416i \(0.395325\pi\)
\(548\) 0 0
\(549\) −19.8785 −0.848395
\(550\) 0 0
\(551\) 5.43571 0.231569
\(552\) 0 0
\(553\) 14.3540 0.610396
\(554\) 0 0
\(555\) 0.845141 0.0358742
\(556\) 0 0
\(557\) −26.9078 −1.14012 −0.570060 0.821603i \(-0.693080\pi\)
−0.570060 + 0.821603i \(0.693080\pi\)
\(558\) 0 0
\(559\) −3.09417 −0.130869
\(560\) 0 0
\(561\) −9.79757 −0.413654
\(562\) 0 0
\(563\) −31.1217 −1.31162 −0.655812 0.754924i \(-0.727674\pi\)
−0.655812 + 0.754924i \(0.727674\pi\)
\(564\) 0 0
\(565\) −16.9228 −0.711947
\(566\) 0 0
\(567\) 6.60181 0.277250
\(568\) 0 0
\(569\) 29.3657 1.23107 0.615537 0.788108i \(-0.288939\pi\)
0.615537 + 0.788108i \(0.288939\pi\)
\(570\) 0 0
\(571\) 4.29028 0.179542 0.0897712 0.995962i \(-0.471386\pi\)
0.0897712 + 0.995962i \(0.471386\pi\)
\(572\) 0 0
\(573\) −6.44884 −0.269404
\(574\) 0 0
\(575\) 5.38831 0.224708
\(576\) 0 0
\(577\) 6.48002 0.269767 0.134883 0.990861i \(-0.456934\pi\)
0.134883 + 0.990861i \(0.456934\pi\)
\(578\) 0 0
\(579\) −2.69447 −0.111978
\(580\) 0 0
\(581\) −17.8894 −0.742178
\(582\) 0 0
\(583\) −25.5645 −1.05877
\(584\) 0 0
\(585\) 6.32144 0.261359
\(586\) 0 0
\(587\) 39.1520 1.61598 0.807988 0.589199i \(-0.200557\pi\)
0.807988 + 0.589199i \(0.200557\pi\)
\(588\) 0 0
\(589\) 3.21507 0.132475
\(590\) 0 0
\(591\) 1.40001 0.0575885
\(592\) 0 0
\(593\) 4.73414 0.194408 0.0972040 0.995264i \(-0.469010\pi\)
0.0972040 + 0.995264i \(0.469010\pi\)
\(594\) 0 0
\(595\) −13.5236 −0.554414
\(596\) 0 0
\(597\) 14.3881 0.588865
\(598\) 0 0
\(599\) −6.48495 −0.264968 −0.132484 0.991185i \(-0.542295\pi\)
−0.132484 + 0.991185i \(0.542295\pi\)
\(600\) 0 0
\(601\) 39.3000 1.60308 0.801540 0.597941i \(-0.204014\pi\)
0.801540 + 0.597941i \(0.204014\pi\)
\(602\) 0 0
\(603\) −34.3728 −1.39977
\(604\) 0 0
\(605\) −7.62784 −0.310116
\(606\) 0 0
\(607\) 13.5099 0.548350 0.274175 0.961680i \(-0.411595\pi\)
0.274175 + 0.961680i \(0.411595\pi\)
\(608\) 0 0
\(609\) −9.75527 −0.395304
\(610\) 0 0
\(611\) −19.4239 −0.785806
\(612\) 0 0
\(613\) −41.8193 −1.68907 −0.844533 0.535504i \(-0.820122\pi\)
−0.844533 + 0.535504i \(0.820122\pi\)
\(614\) 0 0
\(615\) 1.55197 0.0625814
\(616\) 0 0
\(617\) −28.7404 −1.15704 −0.578522 0.815667i \(-0.696370\pi\)
−0.578522 + 0.815667i \(0.696370\pi\)
\(618\) 0 0
\(619\) 33.1568 1.33268 0.666342 0.745646i \(-0.267859\pi\)
0.666342 + 0.745646i \(0.267859\pi\)
\(620\) 0 0
\(621\) 24.0706 0.965921
\(622\) 0 0
\(623\) 5.56287 0.222872
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.56562 0.0625249
\(628\) 0 0
\(629\) 6.31299 0.251716
\(630\) 0 0
\(631\) 24.7412 0.984933 0.492466 0.870332i \(-0.336095\pi\)
0.492466 + 0.870332i \(0.336095\pi\)
\(632\) 0 0
\(633\) −8.65932 −0.344177
\(634\) 0 0
\(635\) 21.7938 0.864862
\(636\) 0 0
\(637\) 6.66795 0.264194
\(638\) 0 0
\(639\) −14.3409 −0.567317
\(640\) 0 0
\(641\) −42.9105 −1.69486 −0.847431 0.530906i \(-0.821852\pi\)
−0.847431 + 0.530906i \(0.821852\pi\)
\(642\) 0 0
\(643\) 15.4422 0.608981 0.304490 0.952515i \(-0.401514\pi\)
0.304490 + 0.952515i \(0.401514\pi\)
\(644\) 0 0
\(645\) −0.945549 −0.0372309
\(646\) 0 0
\(647\) −24.1910 −0.951045 −0.475523 0.879703i \(-0.657741\pi\)
−0.475523 + 0.879703i \(0.657741\pi\)
\(648\) 0 0
\(649\) −4.07461 −0.159942
\(650\) 0 0
\(651\) −5.76997 −0.226143
\(652\) 0 0
\(653\) −38.3080 −1.49911 −0.749554 0.661943i \(-0.769732\pi\)
−0.749554 + 0.661943i \(0.769732\pi\)
\(654\) 0 0
\(655\) −13.6731 −0.534252
\(656\) 0 0
\(657\) 16.3489 0.637830
\(658\) 0 0
\(659\) 35.4170 1.37965 0.689826 0.723975i \(-0.257687\pi\)
0.689826 + 0.723975i \(0.257687\pi\)
\(660\) 0 0
\(661\) 35.5089 1.38114 0.690568 0.723267i \(-0.257360\pi\)
0.690568 + 0.723267i \(0.257360\pi\)
\(662\) 0 0
\(663\) −14.7555 −0.573056
\(664\) 0 0
\(665\) 2.16103 0.0838012
\(666\) 0 0
\(667\) 29.0339 1.12420
\(668\) 0 0
\(669\) 22.6566 0.875954
\(670\) 0 0
\(671\) −15.9703 −0.616525
\(672\) 0 0
\(673\) −39.0126 −1.50382 −0.751912 0.659263i \(-0.770868\pi\)
−0.751912 + 0.659263i \(0.770868\pi\)
\(674\) 0 0
\(675\) 4.46719 0.171942
\(676\) 0 0
\(677\) 21.5982 0.830085 0.415042 0.909802i \(-0.363767\pi\)
0.415042 + 0.909802i \(0.363767\pi\)
\(678\) 0 0
\(679\) 29.2033 1.12072
\(680\) 0 0
\(681\) −8.50579 −0.325943
\(682\) 0 0
\(683\) 3.01886 0.115513 0.0577566 0.998331i \(-0.481605\pi\)
0.0577566 + 0.998331i \(0.481605\pi\)
\(684\) 0 0
\(685\) −3.71290 −0.141863
\(686\) 0 0
\(687\) −17.8017 −0.679176
\(688\) 0 0
\(689\) −38.5011 −1.46678
\(690\) 0 0
\(691\) −15.0805 −0.573689 −0.286844 0.957977i \(-0.592606\pi\)
−0.286844 + 0.957977i \(0.592606\pi\)
\(692\) 0 0
\(693\) 8.99162 0.341563
\(694\) 0 0
\(695\) 17.0560 0.646970
\(696\) 0 0
\(697\) 11.5928 0.439110
\(698\) 0 0
\(699\) −14.5468 −0.550212
\(700\) 0 0
\(701\) −0.0862835 −0.00325888 −0.00162944 0.999999i \(-0.500519\pi\)
−0.00162944 + 0.999999i \(0.500519\pi\)
\(702\) 0 0
\(703\) −1.00880 −0.0380475
\(704\) 0 0
\(705\) −5.93574 −0.223553
\(706\) 0 0
\(707\) −10.2156 −0.384198
\(708\) 0 0
\(709\) 1.57483 0.0591439 0.0295719 0.999563i \(-0.490586\pi\)
0.0295719 + 0.999563i \(0.490586\pi\)
\(710\) 0 0
\(711\) −15.3159 −0.574392
\(712\) 0 0
\(713\) 17.1728 0.643125
\(714\) 0 0
\(715\) 5.07859 0.189929
\(716\) 0 0
\(717\) −1.50663 −0.0562662
\(718\) 0 0
\(719\) 43.4396 1.62002 0.810011 0.586415i \(-0.199461\pi\)
0.810011 + 0.586415i \(0.199461\pi\)
\(720\) 0 0
\(721\) −34.6117 −1.28901
\(722\) 0 0
\(723\) −3.62320 −0.134748
\(724\) 0 0
\(725\) 5.38831 0.200117
\(726\) 0 0
\(727\) 25.9917 0.963978 0.481989 0.876177i \(-0.339914\pi\)
0.481989 + 0.876177i \(0.339914\pi\)
\(728\) 0 0
\(729\) 4.28201 0.158593
\(730\) 0 0
\(731\) −7.06302 −0.261235
\(732\) 0 0
\(733\) −18.7369 −0.692062 −0.346031 0.938223i \(-0.612471\pi\)
−0.346031 + 0.938223i \(0.612471\pi\)
\(734\) 0 0
\(735\) 2.03766 0.0751603
\(736\) 0 0
\(737\) −27.6149 −1.01721
\(738\) 0 0
\(739\) −0.662458 −0.0243689 −0.0121845 0.999926i \(-0.503879\pi\)
−0.0121845 + 0.999926i \(0.503879\pi\)
\(740\) 0 0
\(741\) 2.35788 0.0866189
\(742\) 0 0
\(743\) 7.98058 0.292779 0.146390 0.989227i \(-0.453235\pi\)
0.146390 + 0.989227i \(0.453235\pi\)
\(744\) 0 0
\(745\) 11.5007 0.421354
\(746\) 0 0
\(747\) 19.0882 0.698400
\(748\) 0 0
\(749\) −29.1573 −1.06539
\(750\) 0 0
\(751\) 20.1243 0.734345 0.367173 0.930153i \(-0.380326\pi\)
0.367173 + 0.930153i \(0.380326\pi\)
\(752\) 0 0
\(753\) 3.00655 0.109565
\(754\) 0 0
\(755\) −0.736981 −0.0268215
\(756\) 0 0
\(757\) −16.7104 −0.607351 −0.303676 0.952775i \(-0.598214\pi\)
−0.303676 + 0.952775i \(0.598214\pi\)
\(758\) 0 0
\(759\) 8.36250 0.303539
\(760\) 0 0
\(761\) −18.2084 −0.660053 −0.330027 0.943972i \(-0.607058\pi\)
−0.330027 + 0.943972i \(0.607058\pi\)
\(762\) 0 0
\(763\) 18.3617 0.664738
\(764\) 0 0
\(765\) 14.4298 0.521712
\(766\) 0 0
\(767\) −6.13650 −0.221576
\(768\) 0 0
\(769\) −37.9839 −1.36973 −0.684866 0.728669i \(-0.740139\pi\)
−0.684866 + 0.728669i \(0.740139\pi\)
\(770\) 0 0
\(771\) 15.4359 0.555912
\(772\) 0 0
\(773\) −20.8631 −0.750395 −0.375198 0.926945i \(-0.622425\pi\)
−0.375198 + 0.926945i \(0.622425\pi\)
\(774\) 0 0
\(775\) 3.18704 0.114482
\(776\) 0 0
\(777\) 1.81045 0.0649495
\(778\) 0 0
\(779\) −1.85250 −0.0663726
\(780\) 0 0
\(781\) −11.5214 −0.412267
\(782\) 0 0
\(783\) 24.0706 0.860214
\(784\) 0 0
\(785\) 5.81100 0.207404
\(786\) 0 0
\(787\) 0.979835 0.0349273 0.0174637 0.999847i \(-0.494441\pi\)
0.0174637 + 0.999847i \(0.494441\pi\)
\(788\) 0 0
\(789\) 0.620936 0.0221059
\(790\) 0 0
\(791\) −36.2518 −1.28897
\(792\) 0 0
\(793\) −24.0518 −0.854104
\(794\) 0 0
\(795\) −11.7656 −0.417282
\(796\) 0 0
\(797\) −6.78822 −0.240451 −0.120226 0.992747i \(-0.538362\pi\)
−0.120226 + 0.992747i \(0.538362\pi\)
\(798\) 0 0
\(799\) −44.3386 −1.56859
\(800\) 0 0
\(801\) −5.93565 −0.209726
\(802\) 0 0
\(803\) 13.1345 0.463508
\(804\) 0 0
\(805\) 11.5428 0.406829
\(806\) 0 0
\(807\) −8.98865 −0.316416
\(808\) 0 0
\(809\) 46.3317 1.62894 0.814469 0.580207i \(-0.197029\pi\)
0.814469 + 0.580207i \(0.197029\pi\)
\(810\) 0 0
\(811\) 7.08317 0.248724 0.124362 0.992237i \(-0.460312\pi\)
0.124362 + 0.992237i \(0.460312\pi\)
\(812\) 0 0
\(813\) −6.95824 −0.244036
\(814\) 0 0
\(815\) 11.4577 0.401346
\(816\) 0 0
\(817\) 1.12865 0.0394864
\(818\) 0 0
\(819\) 13.5417 0.473185
\(820\) 0 0
\(821\) 16.0721 0.560919 0.280459 0.959866i \(-0.409513\pi\)
0.280459 + 0.959866i \(0.409513\pi\)
\(822\) 0 0
\(823\) 50.4423 1.75831 0.879155 0.476536i \(-0.158108\pi\)
0.879155 + 0.476536i \(0.158108\pi\)
\(824\) 0 0
\(825\) 1.55197 0.0540326
\(826\) 0 0
\(827\) 41.1787 1.43193 0.715963 0.698139i \(-0.245988\pi\)
0.715963 + 0.698139i \(0.245988\pi\)
\(828\) 0 0
\(829\) −14.8454 −0.515601 −0.257801 0.966198i \(-0.582998\pi\)
−0.257801 + 0.966198i \(0.582998\pi\)
\(830\) 0 0
\(831\) −11.9011 −0.412845
\(832\) 0 0
\(833\) 15.2208 0.527371
\(834\) 0 0
\(835\) 19.3481 0.669569
\(836\) 0 0
\(837\) 14.2371 0.492106
\(838\) 0 0
\(839\) −32.5974 −1.12539 −0.562693 0.826666i \(-0.690235\pi\)
−0.562693 + 0.826666i \(0.690235\pi\)
\(840\) 0 0
\(841\) 0.0339222 0.00116973
\(842\) 0 0
\(843\) −14.6660 −0.505124
\(844\) 0 0
\(845\) −5.35145 −0.184096
\(846\) 0 0
\(847\) −16.3403 −0.561458
\(848\) 0 0
\(849\) 2.71912 0.0933200
\(850\) 0 0
\(851\) −5.38831 −0.184709
\(852\) 0 0
\(853\) −1.67681 −0.0574130 −0.0287065 0.999588i \(-0.509139\pi\)
−0.0287065 + 0.999588i \(0.509139\pi\)
\(854\) 0 0
\(855\) −2.30584 −0.0788582
\(856\) 0 0
\(857\) −11.4908 −0.392518 −0.196259 0.980552i \(-0.562879\pi\)
−0.196259 + 0.980552i \(0.562879\pi\)
\(858\) 0 0
\(859\) −20.8250 −0.710539 −0.355269 0.934764i \(-0.615611\pi\)
−0.355269 + 0.934764i \(0.615611\pi\)
\(860\) 0 0
\(861\) 3.32461 0.113302
\(862\) 0 0
\(863\) 40.0967 1.36491 0.682454 0.730928i \(-0.260913\pi\)
0.682454 + 0.730928i \(0.260913\pi\)
\(864\) 0 0
\(865\) −14.0805 −0.478752
\(866\) 0 0
\(867\) −19.3148 −0.655963
\(868\) 0 0
\(869\) −12.3047 −0.417408
\(870\) 0 0
\(871\) −41.5890 −1.40919
\(872\) 0 0
\(873\) −31.1603 −1.05462
\(874\) 0 0
\(875\) 2.14219 0.0724192
\(876\) 0 0
\(877\) 37.5875 1.26924 0.634619 0.772825i \(-0.281157\pi\)
0.634619 + 0.772825i \(0.281157\pi\)
\(878\) 0 0
\(879\) −4.89145 −0.164984
\(880\) 0 0
\(881\) −27.9518 −0.941721 −0.470860 0.882208i \(-0.656056\pi\)
−0.470860 + 0.882208i \(0.656056\pi\)
\(882\) 0 0
\(883\) −37.7187 −1.26933 −0.634667 0.772786i \(-0.718863\pi\)
−0.634667 + 0.772786i \(0.718863\pi\)
\(884\) 0 0
\(885\) −1.87526 −0.0630360
\(886\) 0 0
\(887\) −29.7382 −0.998510 −0.499255 0.866455i \(-0.666393\pi\)
−0.499255 + 0.866455i \(0.666393\pi\)
\(888\) 0 0
\(889\) 46.6865 1.56581
\(890\) 0 0
\(891\) −5.65926 −0.189592
\(892\) 0 0
\(893\) 7.08516 0.237096
\(894\) 0 0
\(895\) 18.3302 0.612712
\(896\) 0 0
\(897\) 12.5942 0.420509
\(898\) 0 0
\(899\) 17.1728 0.572743
\(900\) 0 0
\(901\) −87.8859 −2.92790
\(902\) 0 0
\(903\) −2.02554 −0.0674058
\(904\) 0 0
\(905\) 14.0722 0.467776
\(906\) 0 0
\(907\) 31.5583 1.04788 0.523939 0.851756i \(-0.324462\pi\)
0.523939 + 0.851756i \(0.324462\pi\)
\(908\) 0 0
\(909\) 10.9002 0.361536
\(910\) 0 0
\(911\) 21.0754 0.698258 0.349129 0.937075i \(-0.386478\pi\)
0.349129 + 0.937075i \(0.386478\pi\)
\(912\) 0 0
\(913\) 15.3353 0.507525
\(914\) 0 0
\(915\) −7.34999 −0.242983
\(916\) 0 0
\(917\) −29.2904 −0.967253
\(918\) 0 0
\(919\) −23.3636 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(920\) 0 0
\(921\) −9.34869 −0.308050
\(922\) 0 0
\(923\) −17.3516 −0.571135
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 36.9311 1.21298
\(928\) 0 0
\(929\) −47.9449 −1.57302 −0.786511 0.617577i \(-0.788114\pi\)
−0.786511 + 0.617577i \(0.788114\pi\)
\(930\) 0 0
\(931\) −2.43224 −0.0797135
\(932\) 0 0
\(933\) −8.56288 −0.280336
\(934\) 0 0
\(935\) 11.5928 0.379126
\(936\) 0 0
\(937\) 41.2487 1.34754 0.673768 0.738943i \(-0.264675\pi\)
0.673768 + 0.738943i \(0.264675\pi\)
\(938\) 0 0
\(939\) −4.43431 −0.144708
\(940\) 0 0
\(941\) −33.3222 −1.08627 −0.543136 0.839644i \(-0.682763\pi\)
−0.543136 + 0.839644i \(0.682763\pi\)
\(942\) 0 0
\(943\) −9.89480 −0.322219
\(944\) 0 0
\(945\) 9.56956 0.311298
\(946\) 0 0
\(947\) 17.4307 0.566421 0.283211 0.959058i \(-0.408600\pi\)
0.283211 + 0.959058i \(0.408600\pi\)
\(948\) 0 0
\(949\) 19.7811 0.642122
\(950\) 0 0
\(951\) 28.1636 0.913267
\(952\) 0 0
\(953\) −38.6625 −1.25240 −0.626200 0.779663i \(-0.715391\pi\)
−0.626200 + 0.779663i \(0.715391\pi\)
\(954\) 0 0
\(955\) 7.63049 0.246917
\(956\) 0 0
\(957\) 8.36250 0.270321
\(958\) 0 0
\(959\) −7.95373 −0.256839
\(960\) 0 0
\(961\) −20.8428 −0.672348
\(962\) 0 0
\(963\) 31.1112 1.00254
\(964\) 0 0
\(965\) 3.18819 0.102631
\(966\) 0 0
\(967\) −34.3441 −1.10443 −0.552216 0.833701i \(-0.686218\pi\)
−0.552216 + 0.833701i \(0.686218\pi\)
\(968\) 0 0
\(969\) 5.38230 0.172904
\(970\) 0 0
\(971\) 13.8590 0.444758 0.222379 0.974960i \(-0.428618\pi\)
0.222379 + 0.974960i \(0.428618\pi\)
\(972\) 0 0
\(973\) 36.5371 1.17133
\(974\) 0 0
\(975\) 2.33732 0.0748542
\(976\) 0 0
\(977\) 14.6157 0.467598 0.233799 0.972285i \(-0.424884\pi\)
0.233799 + 0.972285i \(0.424884\pi\)
\(978\) 0 0
\(979\) −4.76865 −0.152407
\(980\) 0 0
\(981\) −19.5921 −0.625529
\(982\) 0 0
\(983\) −47.4125 −1.51223 −0.756113 0.654442i \(-0.772904\pi\)
−0.756113 + 0.654442i \(0.772904\pi\)
\(984\) 0 0
\(985\) −1.65654 −0.0527816
\(986\) 0 0
\(987\) −12.7155 −0.404738
\(988\) 0 0
\(989\) 6.02848 0.191694
\(990\) 0 0
\(991\) 30.7010 0.975250 0.487625 0.873053i \(-0.337863\pi\)
0.487625 + 0.873053i \(0.337863\pi\)
\(992\) 0 0
\(993\) 14.8753 0.472054
\(994\) 0 0
\(995\) −17.0245 −0.539713
\(996\) 0 0
\(997\) 37.7699 1.19619 0.598093 0.801427i \(-0.295925\pi\)
0.598093 + 0.801427i \(0.295925\pi\)
\(998\) 0 0
\(999\) −4.46719 −0.141336
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 2960.2.a.bb.1.2 5
4.3 odd 2 1480.2.a.g.1.4 5
20.19 odd 2 7400.2.a.r.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.g.1.4 5 4.3 odd 2
2960.2.a.bb.1.2 5 1.1 even 1 trivial
7400.2.a.r.1.2 5 20.19 odd 2