Properties

Label 1200.4.a.t.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -24.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -24.0000 q^{7} +9.00000 q^{9} -52.0000 q^{11} -22.0000 q^{13} +14.0000 q^{17} +20.0000 q^{19} -72.0000 q^{21} -168.000 q^{23} +27.0000 q^{27} +230.000 q^{29} +288.000 q^{31} -156.000 q^{33} +34.0000 q^{37} -66.0000 q^{39} +122.000 q^{41} -188.000 q^{43} +256.000 q^{47} +233.000 q^{49} +42.0000 q^{51} +338.000 q^{53} +60.0000 q^{57} -100.000 q^{59} +742.000 q^{61} -216.000 q^{63} -84.0000 q^{67} -504.000 q^{69} +328.000 q^{71} +38.0000 q^{73} +1248.00 q^{77} +240.000 q^{79} +81.0000 q^{81} +1212.00 q^{83} +690.000 q^{87} +330.000 q^{89} +528.000 q^{91} +864.000 q^{93} -866.000 q^{97} -468.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −24.0000 −1.29588 −0.647939 0.761692i \(-0.724369\pi\)
−0.647939 + 0.761692i \(0.724369\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −52.0000 −1.42533 −0.712663 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 0 0
\(13\) −22.0000 −0.469362 −0.234681 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.0000 0.199735 0.0998676 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) 0 0
\(23\) −168.000 −1.52306 −0.761531 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 230.000 1.47276 0.736378 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 0 0
\(31\) 288.000 1.66859 0.834296 0.551317i \(-0.185875\pi\)
0.834296 + 0.551317i \(0.185875\pi\)
\(32\) 0 0
\(33\) −156.000 −0.822913
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000 0.151069 0.0755347 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 0 0
\(39\) −66.0000 −0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 256.000 0.794499 0.397249 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) 42.0000 0.115317
\(52\) 0 0
\(53\) 338.000 0.875998 0.437999 0.898976i \(-0.355687\pi\)
0.437999 + 0.898976i \(0.355687\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 60.0000 0.139424
\(58\) 0 0
\(59\) −100.000 −0.220659 −0.110330 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) −216.000 −0.431959
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −84.0000 −0.153168 −0.0765838 0.997063i \(-0.524401\pi\)
−0.0765838 + 0.997063i \(0.524401\pi\)
\(68\) 0 0
\(69\) −504.000 −0.879340
\(70\) 0 0
\(71\) 328.000 0.548260 0.274130 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) 38.0000 0.0609255 0.0304628 0.999536i \(-0.490302\pi\)
0.0304628 + 0.999536i \(0.490302\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1248.00 1.84705
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1212.00 1.60282 0.801411 0.598114i \(-0.204083\pi\)
0.801411 + 0.598114i \(0.204083\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 690.000 0.850296
\(88\) 0 0
\(89\) 330.000 0.393033 0.196516 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 0 0
\(91\) 528.000 0.608236
\(92\) 0 0
\(93\) 864.000 0.963362
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −866.000 −0.906484 −0.453242 0.891387i \(-0.649733\pi\)
−0.453242 + 0.891387i \(0.649733\pi\)
\(98\) 0 0
\(99\) −468.000 −0.475109
\(100\) 0 0
\(101\) −1218.00 −1.19996 −0.599978 0.800017i \(-0.704824\pi\)
−0.599978 + 0.800017i \(0.704824\pi\)
\(102\) 0 0
\(103\) −88.0000 −0.0841835 −0.0420917 0.999114i \(-0.513402\pi\)
−0.0420917 + 0.999114i \(0.513402\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.0000 0.0325257 0.0162629 0.999868i \(-0.494823\pi\)
0.0162629 + 0.999868i \(0.494823\pi\)
\(108\) 0 0
\(109\) −970.000 −0.852378 −0.426189 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(110\) 0 0
\(111\) 102.000 0.0872199
\(112\) 0 0
\(113\) −1042.00 −0.867461 −0.433731 0.901043i \(-0.642803\pi\)
−0.433731 + 0.901043i \(0.642803\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −198.000 −0.156454
\(118\) 0 0
\(119\) −336.000 −0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) 0 0
\(123\) 366.000 0.268302
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1936.00 1.35269 0.676347 0.736583i \(-0.263562\pi\)
0.676347 + 0.736583i \(0.263562\pi\)
\(128\) 0 0
\(129\) −564.000 −0.384941
\(130\) 0 0
\(131\) −732.000 −0.488207 −0.244104 0.969749i \(-0.578494\pi\)
−0.244104 + 0.969749i \(0.578494\pi\)
\(132\) 0 0
\(133\) −480.000 −0.312942
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2214.00 1.38069 0.690346 0.723479i \(-0.257458\pi\)
0.690346 + 0.723479i \(0.257458\pi\)
\(138\) 0 0
\(139\) −20.0000 −0.0122042 −0.00610208 0.999981i \(-0.501942\pi\)
−0.00610208 + 0.999981i \(0.501942\pi\)
\(140\) 0 0
\(141\) 768.000 0.458704
\(142\) 0 0
\(143\) 1144.00 0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 699.000 0.392194
\(148\) 0 0
\(149\) −1330.00 −0.731261 −0.365630 0.930760i \(-0.619147\pi\)
−0.365630 + 0.930760i \(0.619147\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 126.000 0.0665784
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3514.00 1.78629 0.893146 0.449768i \(-0.148493\pi\)
0.893146 + 0.449768i \(0.148493\pi\)
\(158\) 0 0
\(159\) 1014.00 0.505757
\(160\) 0 0
\(161\) 4032.00 1.97370
\(162\) 0 0
\(163\) −2068.00 −0.993732 −0.496866 0.867827i \(-0.665516\pi\)
−0.496866 + 0.867827i \(0.665516\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −0.0111208 −0.00556041 0.999985i \(-0.501770\pi\)
−0.00556041 + 0.999985i \(0.501770\pi\)
\(168\) 0 0
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) 0 0
\(173\) 618.000 0.271593 0.135797 0.990737i \(-0.456641\pi\)
0.135797 + 0.990737i \(0.456641\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −300.000 −0.127398
\(178\) 0 0
\(179\) −3340.00 −1.39466 −0.697328 0.716752i \(-0.745628\pi\)
−0.697328 + 0.716752i \(0.745628\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) 0 0
\(183\) 2226.00 0.899184
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −728.000 −0.284688
\(188\) 0 0
\(189\) −648.000 −0.249392
\(190\) 0 0
\(191\) 1888.00 0.715240 0.357620 0.933867i \(-0.383588\pi\)
0.357620 + 0.933867i \(0.383588\pi\)
\(192\) 0 0
\(193\) −1922.00 −0.716832 −0.358416 0.933562i \(-0.616683\pi\)
−0.358416 + 0.933562i \(0.616683\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2526.00 −0.913554 −0.456777 0.889581i \(-0.650996\pi\)
−0.456777 + 0.889581i \(0.650996\pi\)
\(198\) 0 0
\(199\) 1160.00 0.413217 0.206609 0.978424i \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(200\) 0 0
\(201\) −252.000 −0.0884314
\(202\) 0 0
\(203\) −5520.00 −1.90851
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1512.00 −0.507687
\(208\) 0 0
\(209\) −1040.00 −0.344202
\(210\) 0 0
\(211\) 4468.00 1.45777 0.728886 0.684635i \(-0.240039\pi\)
0.728886 + 0.684635i \(0.240039\pi\)
\(212\) 0 0
\(213\) 984.000 0.316538
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6912.00 −2.16229
\(218\) 0 0
\(219\) 114.000 0.0351754
\(220\) 0 0
\(221\) −308.000 −0.0937481
\(222\) 0 0
\(223\) 6032.00 1.81136 0.905678 0.423965i \(-0.139362\pi\)
0.905678 + 0.423965i \(0.139362\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2636.00 0.770738 0.385369 0.922763i \(-0.374074\pi\)
0.385369 + 0.922763i \(0.374074\pi\)
\(228\) 0 0
\(229\) 4830.00 1.39378 0.696889 0.717179i \(-0.254567\pi\)
0.696889 + 0.717179i \(0.254567\pi\)
\(230\) 0 0
\(231\) 3744.00 1.06639
\(232\) 0 0
\(233\) −2682.00 −0.754093 −0.377046 0.926194i \(-0.623060\pi\)
−0.377046 + 0.926194i \(0.623060\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 720.000 0.197338
\(238\) 0 0
\(239\) −2320.00 −0.627901 −0.313950 0.949439i \(-0.601653\pi\)
−0.313950 + 0.949439i \(0.601653\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −440.000 −0.113346
\(248\) 0 0
\(249\) 3636.00 0.925390
\(250\) 0 0
\(251\) −132.000 −0.0331943 −0.0165971 0.999862i \(-0.505283\pi\)
−0.0165971 + 0.999862i \(0.505283\pi\)
\(252\) 0 0
\(253\) 8736.00 2.17086
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7614.00 1.84805 0.924024 0.382335i \(-0.124880\pi\)
0.924024 + 0.382335i \(0.124880\pi\)
\(258\) 0 0
\(259\) −816.000 −0.195767
\(260\) 0 0
\(261\) 2070.00 0.490919
\(262\) 0 0
\(263\) −4888.00 −1.14603 −0.573017 0.819543i \(-0.694227\pi\)
−0.573017 + 0.819543i \(0.694227\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 990.000 0.226918
\(268\) 0 0
\(269\) 1270.00 0.287856 0.143928 0.989588i \(-0.454027\pi\)
0.143928 + 0.989588i \(0.454027\pi\)
\(270\) 0 0
\(271\) −1072.00 −0.240293 −0.120146 0.992756i \(-0.538336\pi\)
−0.120146 + 0.992756i \(0.538336\pi\)
\(272\) 0 0
\(273\) 1584.00 0.351165
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5394.00 1.17001 0.585007 0.811028i \(-0.301092\pi\)
0.585007 + 0.811028i \(0.301092\pi\)
\(278\) 0 0
\(279\) 2592.00 0.556197
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) 0 0
\(283\) 2772.00 0.582255 0.291128 0.956684i \(-0.405970\pi\)
0.291128 + 0.956684i \(0.405970\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2928.00 −0.602210
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) −2598.00 −0.523359
\(292\) 0 0
\(293\) −4542.00 −0.905619 −0.452810 0.891607i \(-0.649578\pi\)
−0.452810 + 0.891607i \(0.649578\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1404.00 −0.274304
\(298\) 0 0
\(299\) 3696.00 0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 0 0
\(303\) −3654.00 −0.692795
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5116.00 0.951093 0.475546 0.879691i \(-0.342250\pi\)
0.475546 + 0.879691i \(0.342250\pi\)
\(308\) 0 0
\(309\) −264.000 −0.0486034
\(310\) 0 0
\(311\) 2808.00 0.511984 0.255992 0.966679i \(-0.417598\pi\)
0.255992 + 0.966679i \(0.417598\pi\)
\(312\) 0 0
\(313\) 7318.00 1.32153 0.660763 0.750594i \(-0.270233\pi\)
0.660763 + 0.750594i \(0.270233\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2246.00 −0.397943 −0.198971 0.980005i \(-0.563760\pi\)
−0.198971 + 0.980005i \(0.563760\pi\)
\(318\) 0 0
\(319\) −11960.0 −2.09916
\(320\) 0 0
\(321\) 108.000 0.0187787
\(322\) 0 0
\(323\) 280.000 0.0482341
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2910.00 −0.492120
\(328\) 0 0
\(329\) −6144.00 −1.02957
\(330\) 0 0
\(331\) −1332.00 −0.221188 −0.110594 0.993866i \(-0.535275\pi\)
−0.110594 + 0.993866i \(0.535275\pi\)
\(332\) 0 0
\(333\) 306.000 0.0503564
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11534.0 1.86438 0.932191 0.361966i \(-0.117894\pi\)
0.932191 + 0.361966i \(0.117894\pi\)
\(338\) 0 0
\(339\) −3126.00 −0.500829
\(340\) 0 0
\(341\) −14976.0 −2.37829
\(342\) 0 0
\(343\) 2640.00 0.415588
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11956.0 1.84966 0.924830 0.380382i \(-0.124207\pi\)
0.924830 + 0.380382i \(0.124207\pi\)
\(348\) 0 0
\(349\) 4870.00 0.746949 0.373474 0.927640i \(-0.378166\pi\)
0.373474 + 0.927640i \(0.378166\pi\)
\(350\) 0 0
\(351\) −594.000 −0.0903287
\(352\) 0 0
\(353\) −10722.0 −1.61664 −0.808321 0.588742i \(-0.799623\pi\)
−0.808321 + 0.588742i \(0.799623\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1008.00 −0.149437
\(358\) 0 0
\(359\) −120.000 −0.0176417 −0.00882083 0.999961i \(-0.502808\pi\)
−0.00882083 + 0.999961i \(0.502808\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) 4119.00 0.595569
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3936.00 0.559830 0.279915 0.960025i \(-0.409694\pi\)
0.279915 + 0.960025i \(0.409694\pi\)
\(368\) 0 0
\(369\) 1098.00 0.154904
\(370\) 0 0
\(371\) −8112.00 −1.13519
\(372\) 0 0
\(373\) −3022.00 −0.419499 −0.209750 0.977755i \(-0.567265\pi\)
−0.209750 + 0.977755i \(0.567265\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5060.00 −0.691255
\(378\) 0 0
\(379\) 13340.0 1.80799 0.903997 0.427539i \(-0.140619\pi\)
0.903997 + 0.427539i \(0.140619\pi\)
\(380\) 0 0
\(381\) 5808.00 0.780979
\(382\) 0 0
\(383\) −1008.00 −0.134481 −0.0672407 0.997737i \(-0.521420\pi\)
−0.0672407 + 0.997737i \(0.521420\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1692.00 −0.222246
\(388\) 0 0
\(389\) 9630.00 1.25517 0.627584 0.778549i \(-0.284044\pi\)
0.627584 + 0.778549i \(0.284044\pi\)
\(390\) 0 0
\(391\) −2352.00 −0.304209
\(392\) 0 0
\(393\) −2196.00 −0.281867
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7126.00 −0.900866 −0.450433 0.892810i \(-0.648730\pi\)
−0.450433 + 0.892810i \(0.648730\pi\)
\(398\) 0 0
\(399\) −1440.00 −0.180677
\(400\) 0 0
\(401\) −8718.00 −1.08568 −0.542838 0.839837i \(-0.682650\pi\)
−0.542838 + 0.839837i \(0.682650\pi\)
\(402\) 0 0
\(403\) −6336.00 −0.783173
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1768.00 −0.215323
\(408\) 0 0
\(409\) −10870.0 −1.31415 −0.657074 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) 0 0
\(411\) 6642.00 0.797143
\(412\) 0 0
\(413\) 2400.00 0.285947
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −60.0000 −0.00704607
\(418\) 0 0
\(419\) 9700.00 1.13097 0.565484 0.824759i \(-0.308689\pi\)
0.565484 + 0.824759i \(0.308689\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) 0 0
\(423\) 2304.00 0.264833
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17808.0 −2.01824
\(428\) 0 0
\(429\) 3432.00 0.386244
\(430\) 0 0
\(431\) −15792.0 −1.76490 −0.882452 0.470402i \(-0.844109\pi\)
−0.882452 + 0.470402i \(0.844109\pi\)
\(432\) 0 0
\(433\) −11602.0 −1.28766 −0.643830 0.765169i \(-0.722655\pi\)
−0.643830 + 0.765169i \(0.722655\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3360.00 −0.367805
\(438\) 0 0
\(439\) 440.000 0.0478361 0.0239181 0.999714i \(-0.492386\pi\)
0.0239181 + 0.999714i \(0.492386\pi\)
\(440\) 0 0
\(441\) 2097.00 0.226433
\(442\) 0 0
\(443\) −10188.0 −1.09266 −0.546328 0.837571i \(-0.683975\pi\)
−0.546328 + 0.837571i \(0.683975\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3990.00 −0.422194
\(448\) 0 0
\(449\) −13310.0 −1.39897 −0.699485 0.714647i \(-0.746587\pi\)
−0.699485 + 0.714647i \(0.746587\pi\)
\(450\) 0 0
\(451\) −6344.00 −0.662367
\(452\) 0 0
\(453\) 3624.00 0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3226.00 −0.330210 −0.165105 0.986276i \(-0.552796\pi\)
−0.165105 + 0.986276i \(0.552796\pi\)
\(458\) 0 0
\(459\) 378.000 0.0384391
\(460\) 0 0
\(461\) 6582.00 0.664977 0.332488 0.943107i \(-0.392112\pi\)
0.332488 + 0.943107i \(0.392112\pi\)
\(462\) 0 0
\(463\) 15072.0 1.51286 0.756431 0.654073i \(-0.226941\pi\)
0.756431 + 0.654073i \(0.226941\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 476.000 0.0471663 0.0235831 0.999722i \(-0.492493\pi\)
0.0235831 + 0.999722i \(0.492493\pi\)
\(468\) 0 0
\(469\) 2016.00 0.198487
\(470\) 0 0
\(471\) 10542.0 1.03132
\(472\) 0 0
\(473\) 9776.00 0.950319
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3042.00 0.291999
\(478\) 0 0
\(479\) 19680.0 1.87725 0.938624 0.344941i \(-0.112101\pi\)
0.938624 + 0.344941i \(0.112101\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) 0 0
\(483\) 12096.0 1.13952
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −5944.00 −0.553077 −0.276538 0.961003i \(-0.589187\pi\)
−0.276538 + 0.961003i \(0.589187\pi\)
\(488\) 0 0
\(489\) −6204.00 −0.573731
\(490\) 0 0
\(491\) −10772.0 −0.990089 −0.495044 0.868868i \(-0.664848\pi\)
−0.495044 + 0.868868i \(0.664848\pi\)
\(492\) 0 0
\(493\) 3220.00 0.294161
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7872.00 −0.710478
\(498\) 0 0
\(499\) −8140.00 −0.730253 −0.365127 0.930958i \(-0.618974\pi\)
−0.365127 + 0.930958i \(0.618974\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) 0 0
\(503\) −13768.0 −1.22045 −0.610223 0.792229i \(-0.708920\pi\)
−0.610223 + 0.792229i \(0.708920\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5139.00 −0.450160
\(508\) 0 0
\(509\) 22150.0 1.92884 0.964422 0.264368i \(-0.0851633\pi\)
0.964422 + 0.264368i \(0.0851633\pi\)
\(510\) 0 0
\(511\) −912.000 −0.0789521
\(512\) 0 0
\(513\) 540.000 0.0464748
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −13312.0 −1.13242
\(518\) 0 0
\(519\) 1854.00 0.156805
\(520\) 0 0
\(521\) 1562.00 0.131348 0.0656741 0.997841i \(-0.479080\pi\)
0.0656741 + 0.997841i \(0.479080\pi\)
\(522\) 0 0
\(523\) −668.000 −0.0558501 −0.0279250 0.999610i \(-0.508890\pi\)
−0.0279250 + 0.999610i \(0.508890\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4032.00 0.333276
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) −900.000 −0.0735531
\(532\) 0 0
\(533\) −2684.00 −0.218118
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −10020.0 −0.805205
\(538\) 0 0
\(539\) −12116.0 −0.968225
\(540\) 0 0
\(541\) −6138.00 −0.487788 −0.243894 0.969802i \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) 0 0
\(543\) −534.000 −0.0422028
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −10484.0 −0.819494 −0.409747 0.912199i \(-0.634383\pi\)
−0.409747 + 0.912199i \(0.634383\pi\)
\(548\) 0 0
\(549\) 6678.00 0.519144
\(550\) 0 0
\(551\) 4600.00 0.355656
\(552\) 0 0
\(553\) −5760.00 −0.442930
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3606.00 −0.274311 −0.137155 0.990550i \(-0.543796\pi\)
−0.137155 + 0.990550i \(0.543796\pi\)
\(558\) 0 0
\(559\) 4136.00 0.312941
\(560\) 0 0
\(561\) −2184.00 −0.164365
\(562\) 0 0
\(563\) 12252.0 0.917159 0.458579 0.888654i \(-0.348359\pi\)
0.458579 + 0.888654i \(0.348359\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1944.00 −0.143986
\(568\) 0 0
\(569\) −14550.0 −1.07200 −0.536000 0.844218i \(-0.680065\pi\)
−0.536000 + 0.844218i \(0.680065\pi\)
\(570\) 0 0
\(571\) 25468.0 1.86655 0.933277 0.359157i \(-0.116936\pi\)
0.933277 + 0.359157i \(0.116936\pi\)
\(572\) 0 0
\(573\) 5664.00 0.412944
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −12866.0 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(578\) 0 0
\(579\) −5766.00 −0.413863
\(580\) 0 0
\(581\) −29088.0 −2.07706
\(582\) 0 0
\(583\) −17576.0 −1.24858
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14844.0 −1.04374 −0.521872 0.853024i \(-0.674766\pi\)
−0.521872 + 0.853024i \(0.674766\pi\)
\(588\) 0 0
\(589\) 5760.00 0.402948
\(590\) 0 0
\(591\) −7578.00 −0.527440
\(592\) 0 0
\(593\) −20402.0 −1.41283 −0.706416 0.707797i \(-0.749689\pi\)
−0.706416 + 0.707797i \(0.749689\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3480.00 0.238571
\(598\) 0 0
\(599\) −10760.0 −0.733959 −0.366980 0.930229i \(-0.619608\pi\)
−0.366980 + 0.930229i \(0.619608\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) 0 0
\(603\) −756.000 −0.0510559
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11056.0 0.739290 0.369645 0.929173i \(-0.379479\pi\)
0.369645 + 0.929173i \(0.379479\pi\)
\(608\) 0 0
\(609\) −16560.0 −1.10188
\(610\) 0 0
\(611\) −5632.00 −0.372907
\(612\) 0 0
\(613\) 16418.0 1.08176 0.540878 0.841101i \(-0.318092\pi\)
0.540878 + 0.841101i \(0.318092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10374.0 0.676891 0.338445 0.940986i \(-0.390099\pi\)
0.338445 + 0.940986i \(0.390099\pi\)
\(618\) 0 0
\(619\) 5260.00 0.341546 0.170773 0.985310i \(-0.445373\pi\)
0.170773 + 0.985310i \(0.445373\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 0 0
\(623\) −7920.00 −0.509323
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3120.00 −0.198725
\(628\) 0 0
\(629\) 476.000 0.0301739
\(630\) 0 0
\(631\) −21352.0 −1.34708 −0.673542 0.739149i \(-0.735228\pi\)
−0.673542 + 0.739149i \(0.735228\pi\)
\(632\) 0 0
\(633\) 13404.0 0.841645
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5126.00 −0.318838
\(638\) 0 0
\(639\) 2952.00 0.182753
\(640\) 0 0
\(641\) −29118.0 −1.79422 −0.897108 0.441812i \(-0.854336\pi\)
−0.897108 + 0.441812i \(0.854336\pi\)
\(642\) 0 0
\(643\) 5772.00 0.354005 0.177003 0.984210i \(-0.443360\pi\)
0.177003 + 0.984210i \(0.443360\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14264.0 −0.866732 −0.433366 0.901218i \(-0.642674\pi\)
−0.433366 + 0.901218i \(0.642674\pi\)
\(648\) 0 0
\(649\) 5200.00 0.314511
\(650\) 0 0
\(651\) −20736.0 −1.24840
\(652\) 0 0
\(653\) −6902.00 −0.413623 −0.206812 0.978381i \(-0.566309\pi\)
−0.206812 + 0.978381i \(0.566309\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 342.000 0.0203085
\(658\) 0 0
\(659\) −20140.0 −1.19051 −0.595253 0.803539i \(-0.702948\pi\)
−0.595253 + 0.803539i \(0.702948\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) 0 0
\(663\) −924.000 −0.0541255
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −38640.0 −2.24310
\(668\) 0 0
\(669\) 18096.0 1.04579
\(670\) 0 0
\(671\) −38584.0 −2.21985
\(672\) 0 0
\(673\) 7518.00 0.430606 0.215303 0.976547i \(-0.430926\pi\)
0.215303 + 0.976547i \(0.430926\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18114.0 1.02833 0.514164 0.857692i \(-0.328102\pi\)
0.514164 + 0.857692i \(0.328102\pi\)
\(678\) 0 0
\(679\) 20784.0 1.17469
\(680\) 0 0
\(681\) 7908.00 0.444986
\(682\) 0 0
\(683\) −23868.0 −1.33716 −0.668582 0.743638i \(-0.733099\pi\)
−0.668582 + 0.743638i \(0.733099\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14490.0 0.804699
\(688\) 0 0
\(689\) −7436.00 −0.411160
\(690\) 0 0
\(691\) −172.000 −0.00946916 −0.00473458 0.999989i \(-0.501507\pi\)
−0.00473458 + 0.999989i \(0.501507\pi\)
\(692\) 0 0
\(693\) 11232.0 0.615683
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1708.00 0.0928194
\(698\) 0 0
\(699\) −8046.00 −0.435376
\(700\) 0 0
\(701\) −22138.0 −1.19278 −0.596391 0.802694i \(-0.703399\pi\)
−0.596391 + 0.802694i \(0.703399\pi\)
\(702\) 0 0
\(703\) 680.000 0.0364818
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29232.0 1.55500
\(708\) 0 0
\(709\) 3070.00 0.162618 0.0813091 0.996689i \(-0.474090\pi\)
0.0813091 + 0.996689i \(0.474090\pi\)
\(710\) 0 0
\(711\) 2160.00 0.113933
\(712\) 0 0
\(713\) −48384.0 −2.54137
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6960.00 −0.362519
\(718\) 0 0
\(719\) −15600.0 −0.809154 −0.404577 0.914504i \(-0.632581\pi\)
−0.404577 + 0.914504i \(0.632581\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 0 0
\(723\) 6006.00 0.308943
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20696.0 1.05581 0.527904 0.849304i \(-0.322978\pi\)
0.527904 + 0.849304i \(0.322978\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2632.00 −0.133171
\(732\) 0 0
\(733\) 30778.0 1.55090 0.775451 0.631408i \(-0.217522\pi\)
0.775451 + 0.631408i \(0.217522\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4368.00 0.218314
\(738\) 0 0
\(739\) −11740.0 −0.584388 −0.292194 0.956359i \(-0.594385\pi\)
−0.292194 + 0.956359i \(0.594385\pi\)
\(740\) 0 0
\(741\) −1320.00 −0.0654405
\(742\) 0 0
\(743\) 2632.00 0.129958 0.0649789 0.997887i \(-0.479302\pi\)
0.0649789 + 0.997887i \(0.479302\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10908.0 0.534274
\(748\) 0 0
\(749\) −864.000 −0.0421494
\(750\) 0 0
\(751\) 20528.0 0.997440 0.498720 0.866763i \(-0.333804\pi\)
0.498720 + 0.866763i \(0.333804\pi\)
\(752\) 0 0
\(753\) −396.000 −0.0191647
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −21646.0 −1.03928 −0.519642 0.854384i \(-0.673934\pi\)
−0.519642 + 0.854384i \(0.673934\pi\)
\(758\) 0 0
\(759\) 26208.0 1.25335
\(760\) 0 0
\(761\) 18282.0 0.870857 0.435428 0.900223i \(-0.356597\pi\)
0.435428 + 0.900223i \(0.356597\pi\)
\(762\) 0 0
\(763\) 23280.0 1.10458
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2200.00 0.103569
\(768\) 0 0
\(769\) −24190.0 −1.13435 −0.567174 0.823598i \(-0.691963\pi\)
−0.567174 + 0.823598i \(0.691963\pi\)
\(770\) 0 0
\(771\) 22842.0 1.06697
\(772\) 0 0
\(773\) 25698.0 1.19572 0.597861 0.801600i \(-0.296018\pi\)
0.597861 + 0.801600i \(0.296018\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2448.00 −0.113026
\(778\) 0 0
\(779\) 2440.00 0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) 0 0
\(783\) 6210.00 0.283432
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33436.0 1.51444 0.757220 0.653160i \(-0.226557\pi\)
0.757220 + 0.653160i \(0.226557\pi\)
\(788\) 0 0
\(789\) −14664.0 −0.661663
\(790\) 0 0
\(791\) 25008.0 1.12412
\(792\) 0 0
\(793\) −16324.0 −0.730999
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37594.0 1.67083 0.835413 0.549623i \(-0.185229\pi\)
0.835413 + 0.549623i \(0.185229\pi\)
\(798\) 0 0
\(799\) 3584.00 0.158689
\(800\) 0 0
\(801\) 2970.00 0.131011
\(802\) 0 0
\(803\) −1976.00 −0.0868388
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3810.00 0.166194
\(808\) 0 0
\(809\) 4730.00 0.205560 0.102780 0.994704i \(-0.467226\pi\)
0.102780 + 0.994704i \(0.467226\pi\)
\(810\) 0 0
\(811\) 8748.00 0.378772 0.189386 0.981903i \(-0.439350\pi\)
0.189386 + 0.981903i \(0.439350\pi\)
\(812\) 0 0
\(813\) −3216.00 −0.138733
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3760.00 −0.161011
\(818\) 0 0
\(819\) 4752.00 0.202745
\(820\) 0 0
\(821\) 44142.0 1.87645 0.938226 0.346024i \(-0.112468\pi\)
0.938226 + 0.346024i \(0.112468\pi\)
\(822\) 0 0
\(823\) 3992.00 0.169079 0.0845397 0.996420i \(-0.473058\pi\)
0.0845397 + 0.996420i \(0.473058\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14444.0 −0.607336 −0.303668 0.952778i \(-0.598211\pi\)
−0.303668 + 0.952778i \(0.598211\pi\)
\(828\) 0 0
\(829\) 42150.0 1.76590 0.882949 0.469468i \(-0.155554\pi\)
0.882949 + 0.469468i \(0.155554\pi\)
\(830\) 0 0
\(831\) 16182.0 0.675508
\(832\) 0 0
\(833\) 3262.00 0.135680
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7776.00 0.321121
\(838\) 0 0
\(839\) −13400.0 −0.551394 −0.275697 0.961245i \(-0.588909\pi\)
−0.275697 + 0.961245i \(0.588909\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) 0 0
\(843\) 7326.00 0.299313
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −32952.0 −1.33677
\(848\) 0 0
\(849\) 8316.00 0.336165
\(850\) 0 0
\(851\) −5712.00 −0.230088
\(852\) 0 0
\(853\) 8658.00 0.347531 0.173766 0.984787i \(-0.444406\pi\)
0.173766 + 0.984787i \(0.444406\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42826.0 −1.70701 −0.853505 0.521084i \(-0.825528\pi\)
−0.853505 + 0.521084i \(0.825528\pi\)
\(858\) 0 0
\(859\) 35900.0 1.42595 0.712976 0.701189i \(-0.247347\pi\)
0.712976 + 0.701189i \(0.247347\pi\)
\(860\) 0 0
\(861\) −8784.00 −0.347686
\(862\) 0 0
\(863\) −3088.00 −0.121804 −0.0609019 0.998144i \(-0.519398\pi\)
−0.0609019 + 0.998144i \(0.519398\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −14151.0 −0.554317
\(868\) 0 0
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) 1848.00 0.0718910
\(872\) 0 0
\(873\) −7794.00 −0.302161
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35274.0 1.35817 0.679087 0.734058i \(-0.262376\pi\)
0.679087 + 0.734058i \(0.262376\pi\)
\(878\) 0 0
\(879\) −13626.0 −0.522860
\(880\) 0 0
\(881\) 25042.0 0.957646 0.478823 0.877911i \(-0.341064\pi\)
0.478823 + 0.877911i \(0.341064\pi\)
\(882\) 0 0
\(883\) 12572.0 0.479141 0.239570 0.970879i \(-0.422993\pi\)
0.239570 + 0.970879i \(0.422993\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21864.0 −0.827645 −0.413823 0.910358i \(-0.635807\pi\)
−0.413823 + 0.910358i \(0.635807\pi\)
\(888\) 0 0
\(889\) −46464.0 −1.75293
\(890\) 0 0
\(891\) −4212.00 −0.158370
\(892\) 0 0
\(893\) 5120.00 0.191864
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 11088.0 0.412729
\(898\) 0 0
\(899\) 66240.0 2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) 0 0
\(903\) 13536.0 0.498837
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 31236.0 1.14352 0.571761 0.820420i \(-0.306260\pi\)
0.571761 + 0.820420i \(0.306260\pi\)
\(908\) 0 0
\(909\) −10962.0 −0.399985
\(910\) 0 0
\(911\) −8272.00 −0.300838 −0.150419 0.988622i \(-0.548062\pi\)
−0.150419 + 0.988622i \(0.548062\pi\)
\(912\) 0 0
\(913\) −63024.0 −2.28455
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17568.0 0.632657
\(918\) 0 0
\(919\) −20200.0 −0.725067 −0.362533 0.931971i \(-0.618088\pi\)
−0.362533 + 0.931971i \(0.618088\pi\)
\(920\) 0 0
\(921\) 15348.0 0.549114
\(922\) 0 0
\(923\) −7216.00 −0.257332
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −792.000 −0.0280612
\(928\) 0 0
\(929\) 31010.0 1.09516 0.547581 0.836753i \(-0.315549\pi\)
0.547581 + 0.836753i \(0.315549\pi\)
\(930\) 0 0
\(931\) 4660.00 0.164044
\(932\) 0 0
\(933\) 8424.00 0.295594
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39174.0 1.36580 0.682902 0.730510i \(-0.260717\pi\)
0.682902 + 0.730510i \(0.260717\pi\)
\(938\) 0 0
\(939\) 21954.0 0.762984
\(940\) 0 0
\(941\) −4138.00 −0.143353 −0.0716764 0.997428i \(-0.522835\pi\)
−0.0716764 + 0.997428i \(0.522835\pi\)
\(942\) 0 0
\(943\) −20496.0 −0.707785
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23676.0 0.812425 0.406213 0.913779i \(-0.366849\pi\)
0.406213 + 0.913779i \(0.366849\pi\)
\(948\) 0 0
\(949\) −836.000 −0.0285961
\(950\) 0 0
\(951\) −6738.00 −0.229752
\(952\) 0 0
\(953\) −18922.0 −0.643173 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −35880.0 −1.21195
\(958\) 0 0
\(959\) −53136.0 −1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 0 0
\(963\) 324.000 0.0108419
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 39656.0 1.31877 0.659385 0.751805i \(-0.270817\pi\)
0.659385 + 0.751805i \(0.270817\pi\)
\(968\) 0 0
\(969\) 840.000 0.0278480
\(970\) 0 0
\(971\) 33228.0 1.09818 0.549092 0.835762i \(-0.314974\pi\)
0.549092 + 0.835762i \(0.314974\pi\)
\(972\) 0 0
\(973\) 480.000 0.0158151
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 974.000 0.0318946 0.0159473 0.999873i \(-0.494924\pi\)
0.0159473 + 0.999873i \(0.494924\pi\)
\(978\) 0 0
\(979\) −17160.0 −0.560200
\(980\) 0 0
\(981\) −8730.00 −0.284126
\(982\) 0 0
\(983\) −13608.0 −0.441534 −0.220767 0.975327i \(-0.570856\pi\)
−0.220767 + 0.975327i \(0.570856\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −18432.0 −0.594425
\(988\) 0 0
\(989\) 31584.0 1.01548
\(990\) 0 0
\(991\) −13472.0 −0.431839 −0.215919 0.976411i \(-0.569275\pi\)
−0.215919 + 0.976411i \(0.569275\pi\)
\(992\) 0 0
\(993\) −3996.00 −0.127703
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3234.00 0.102730 0.0513650 0.998680i \(-0.483643\pi\)
0.0513650 + 0.998680i \(0.483643\pi\)
\(998\) 0 0
\(999\) 918.000 0.0290733
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 1200.4.a.t.1.1 1
4.3 odd 2 75.4.a.b.1.1 1
5.2 odd 4 1200.4.f.b.49.1 2
5.3 odd 4 1200.4.f.b.49.2 2
5.4 even 2 240.4.a.e.1.1 1
12.11 even 2 225.4.a.f.1.1 1
15.14 odd 2 720.4.a.n.1.1 1
20.3 even 4 75.4.b.b.49.2 2
20.7 even 4 75.4.b.b.49.1 2
20.19 odd 2 15.4.a.a.1.1 1
40.19 odd 2 960.4.a.b.1.1 1
40.29 even 2 960.4.a.ba.1.1 1
60.23 odd 4 225.4.b.e.199.1 2
60.47 odd 4 225.4.b.e.199.2 2
60.59 even 2 45.4.a.c.1.1 1
140.139 even 2 735.4.a.e.1.1 1
180.59 even 6 405.4.e.i.136.1 2
180.79 odd 6 405.4.e.g.271.1 2
180.119 even 6 405.4.e.i.271.1 2
180.139 odd 6 405.4.e.g.136.1 2
220.219 even 2 1815.4.a.e.1.1 1
420.419 odd 2 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 20.19 odd 2
45.4.a.c.1.1 1 60.59 even 2
75.4.a.b.1.1 1 4.3 odd 2
75.4.b.b.49.1 2 20.7 even 4
75.4.b.b.49.2 2 20.3 even 4
225.4.a.f.1.1 1 12.11 even 2
225.4.b.e.199.1 2 60.23 odd 4
225.4.b.e.199.2 2 60.47 odd 4
240.4.a.e.1.1 1 5.4 even 2
405.4.e.g.136.1 2 180.139 odd 6
405.4.e.g.271.1 2 180.79 odd 6
405.4.e.i.136.1 2 180.59 even 6
405.4.e.i.271.1 2 180.119 even 6
720.4.a.n.1.1 1 15.14 odd 2
735.4.a.e.1.1 1 140.139 even 2
960.4.a.b.1.1 1 40.19 odd 2
960.4.a.ba.1.1 1 40.29 even 2
1200.4.a.t.1.1 1 1.1 even 1 trivial
1200.4.f.b.49.1 2 5.2 odd 4
1200.4.f.b.49.2 2 5.3 odd 4
1815.4.a.e.1.1 1 220.219 even 2
2205.4.a.l.1.1 1 420.419 odd 2