Properties

Label 1200.4.f.b.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.b.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} +156.000i q^{33} +34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} +188.000i q^{43} +256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} -338.000i q^{53} +60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} +216.000i q^{63} -84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000i q^{73} +1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} -1212.00i q^{83} +690.000i q^{87} -330.000 q^{89} +528.000 q^{91} -864.000i q^{93} -866.000i q^{97} +468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + O(q^{10}) \) \( 2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 24.0000i − 1.29588i −0.761692 0.647939i \(-0.775631\pi\)
0.761692 0.647939i \(-0.224369\pi\)
\(8\) 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
\(13\) 22.0000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.0000i 0.199735i 0.995001 + 0.0998676i \(0.0318419\pi\)
−0.995001 + 0.0998676i \(0.968158\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) 0 0
\(23\) 168.000i 1.52306i 0.648129 + 0.761531i \(0.275552\pi\)
−0.648129 + 0.761531i \(0.724448\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −230.000 −1.47276 −0.736378 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) 288.000 1.66859 0.834296 0.551317i \(-0.185875\pi\)
0.834296 + 0.551317i \(0.185875\pi\)
\(32\) 0 0
\(33\) 156.000i 0.822913i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 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
\(43\) 188.000i 0.666738i 0.942796 + 0.333369i \(0.108185\pi\)
−0.942796 + 0.333369i \(0.891815\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 256.000i 0.794499i 0.917711 + 0.397249i \(0.130035\pi\)
−0.917711 + 0.397249i \(0.869965\pi\)
\(48\) 0 0
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) 42.0000 0.115317
\(52\) 0 0
\(53\) − 338.000i − 0.875998i −0.898976 0.437999i \(-0.855687\pi\)
0.898976 0.437999i \(-0.144313\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 60.0000i 0.139424i
\(58\) 0 0
\(59\) 100.000 0.220659 0.110330 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) 216.000i 0.431959i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 84.0000i − 0.153168i −0.997063 0.0765838i \(-0.975599\pi\)
0.997063 0.0765838i \(-0.0244013\pi\)
\(68\) 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
\(73\) − 38.0000i − 0.0609255i −0.999536 0.0304628i \(-0.990302\pi\)
0.999536 0.0304628i \(-0.00969810\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1248.00i 1.84705i
\(78\) 0 0
\(79\) −240.000 −0.341799 −0.170899 0.985288i \(-0.554667\pi\)
−0.170899 + 0.985288i \(0.554667\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 1212.00i − 1.60282i −0.598114 0.801411i \(-0.704083\pi\)
0.598114 0.801411i \(-0.295917\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 690.000i 0.850296i
\(88\) 0 0
\(89\) −330.000 −0.393033 −0.196516 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) 528.000 0.608236
\(92\) 0 0
\(93\) − 864.000i − 0.963362i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 866.000i − 0.906484i −0.891387 0.453242i \(-0.850267\pi\)
0.891387 0.453242i \(-0.149733\pi\)
\(98\) 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
\(103\) 88.0000i 0.0841835i 0.999114 + 0.0420917i \(0.0134022\pi\)
−0.999114 + 0.0420917i \(0.986598\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 36.0000i 0.0325257i 0.999868 + 0.0162629i \(0.00517686\pi\)
−0.999868 + 0.0162629i \(0.994823\pi\)
\(108\) 0 0
\(109\) 970.000 0.852378 0.426189 0.904634i \(-0.359856\pi\)
0.426189 + 0.904634i \(0.359856\pi\)
\(110\) 0 0
\(111\) 102.000 0.0872199
\(112\) 0 0
\(113\) 1042.00i 0.867461i 0.901043 + 0.433731i \(0.142803\pi\)
−0.901043 + 0.433731i \(0.857197\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 198.000i − 0.156454i
\(118\) 0 0
\(119\) 336.000 0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) 0 0
\(123\) − 366.000i − 0.268302i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1936.00i 1.35269i 0.736583 + 0.676347i \(0.236438\pi\)
−0.736583 + 0.676347i \(0.763562\pi\)
\(128\) 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
\(133\) 480.000i 0.312942i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2214.00i 1.38069i 0.723479 + 0.690346i \(0.242542\pi\)
−0.723479 + 0.690346i \(0.757458\pi\)
\(138\) 0 0
\(139\) 20.0000 0.0122042 0.00610208 0.999981i \(-0.498058\pi\)
0.00610208 + 0.999981i \(0.498058\pi\)
\(140\) 0 0
\(141\) 768.000 0.458704
\(142\) 0 0
\(143\) − 1144.00i − 0.668994i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 699.000i 0.392194i
\(148\) 0 0
\(149\) 1330.00 0.731261 0.365630 0.930760i \(-0.380853\pi\)
0.365630 + 0.930760i \(0.380853\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) − 126.000i − 0.0665784i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3514.00i 1.78629i 0.449768 + 0.893146i \(0.351507\pi\)
−0.449768 + 0.893146i \(0.648493\pi\)
\(158\) 0 0
\(159\) −1014.00 −0.505757
\(160\) 0 0
\(161\) 4032.00 1.97370
\(162\) 0 0
\(163\) 2068.00i 0.993732i 0.867827 + 0.496866i \(0.165516\pi\)
−0.867827 + 0.496866i \(0.834484\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 24.0000i − 0.0111208i −0.999985 0.00556041i \(-0.998230\pi\)
0.999985 0.00556041i \(-0.00176994\pi\)
\(168\) 0 0
\(169\) 1713.00 0.779700
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) 0 0
\(173\) − 618.000i − 0.271593i −0.990737 0.135797i \(-0.956641\pi\)
0.990737 0.135797i \(-0.0433594\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 300.000i − 0.127398i
\(178\) 0 0
\(179\) 3340.00 1.39466 0.697328 0.716752i \(-0.254372\pi\)
0.697328 + 0.716752i \(0.254372\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) 0 0
\(183\) − 2226.00i − 0.899184i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 728.000i − 0.284688i
\(188\) 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
\(193\) 1922.00i 0.716832i 0.933562 + 0.358416i \(0.116683\pi\)
−0.933562 + 0.358416i \(0.883317\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2526.00i − 0.913554i −0.889581 0.456777i \(-0.849004\pi\)
0.889581 0.456777i \(-0.150996\pi\)
\(198\) 0 0
\(199\) −1160.00 −0.413217 −0.206609 0.978424i \(-0.566243\pi\)
−0.206609 + 0.978424i \(0.566243\pi\)
\(200\) 0 0
\(201\) −252.000 −0.0884314
\(202\) 0 0
\(203\) 5520.00i 1.90851i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1512.00i − 0.507687i
\(208\) 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
\(213\) − 984.000i − 0.316538i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6912.00i − 2.16229i
\(218\) 0 0
\(219\) −114.000 −0.0351754
\(220\) 0 0
\(221\) −308.000 −0.0937481
\(222\) 0 0
\(223\) − 6032.00i − 1.81136i −0.423965 0.905678i \(-0.639362\pi\)
0.423965 0.905678i \(-0.360638\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2636.00i 0.770738i 0.922763 + 0.385369i \(0.125926\pi\)
−0.922763 + 0.385369i \(0.874074\pi\)
\(228\) 0 0
\(229\) −4830.00 −1.39378 −0.696889 0.717179i \(-0.745433\pi\)
−0.696889 + 0.717179i \(0.745433\pi\)
\(230\) 0 0
\(231\) 3744.00 1.06639
\(232\) 0 0
\(233\) 2682.00i 0.754093i 0.926194 + 0.377046i \(0.123060\pi\)
−0.926194 + 0.377046i \(0.876940\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 720.000i 0.197338i
\(238\) 0 0
\(239\) 2320.00 0.627901 0.313950 0.949439i \(-0.398347\pi\)
0.313950 + 0.949439i \(0.398347\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 440.000i − 0.113346i
\(248\) 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
\(253\) − 8736.00i − 2.17086i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7614.00i 1.84805i 0.382335 + 0.924024i \(0.375120\pi\)
−0.382335 + 0.924024i \(0.624880\pi\)
\(258\) 0 0
\(259\) 816.000 0.195767
\(260\) 0 0
\(261\) 2070.00 0.490919
\(262\) 0 0
\(263\) 4888.00i 1.14603i 0.819543 + 0.573017i \(0.194227\pi\)
−0.819543 + 0.573017i \(0.805773\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 990.000i 0.226918i
\(268\) 0 0
\(269\) −1270.00 −0.287856 −0.143928 0.989588i \(-0.545973\pi\)
−0.143928 + 0.989588i \(0.545973\pi\)
\(270\) 0 0
\(271\) −1072.00 −0.240293 −0.120146 0.992756i \(-0.538336\pi\)
−0.120146 + 0.992756i \(0.538336\pi\)
\(272\) 0 0
\(273\) − 1584.00i − 0.351165i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5394.00i 1.17001i 0.811028 + 0.585007i \(0.198908\pi\)
−0.811028 + 0.585007i \(0.801092\pi\)
\(278\) 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
\(283\) − 2772.00i − 0.582255i −0.956684 0.291128i \(-0.905970\pi\)
0.956684 0.291128i \(-0.0940305\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2928.00i − 0.602210i
\(288\) 0 0
\(289\) 4717.00 0.960106
\(290\) 0 0
\(291\) −2598.00 −0.523359
\(292\) 0 0
\(293\) 4542.00i 0.905619i 0.891607 + 0.452810i \(0.149578\pi\)
−0.891607 + 0.452810i \(0.850422\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1404.00i − 0.274304i
\(298\) 0 0
\(299\) −3696.00 −0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 0 0
\(303\) 3654.00i 0.692795i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5116.00i 0.951093i 0.879691 + 0.475546i \(0.157750\pi\)
−0.879691 + 0.475546i \(0.842250\pi\)
\(308\) 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
\(313\) − 7318.00i − 1.32153i −0.750594 0.660763i \(-0.770233\pi\)
0.750594 0.660763i \(-0.229767\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2246.00i − 0.397943i −0.980005 0.198971i \(-0.936240\pi\)
0.980005 0.198971i \(-0.0637601\pi\)
\(318\) 0 0
\(319\) 11960.0 2.09916
\(320\) 0 0
\(321\) 108.000 0.0187787
\(322\) 0 0
\(323\) − 280.000i − 0.0482341i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2910.00i − 0.492120i
\(328\) 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
\(333\) − 306.000i − 0.0503564i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11534.0i 1.86438i 0.361966 + 0.932191i \(0.382106\pi\)
−0.361966 + 0.932191i \(0.617894\pi\)
\(338\) 0 0
\(339\) 3126.00 0.500829
\(340\) 0 0
\(341\) −14976.0 −2.37829
\(342\) 0 0
\(343\) − 2640.00i − 0.415588i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11956.0i 1.84966i 0.380382 + 0.924830i \(0.375793\pi\)
−0.380382 + 0.924830i \(0.624207\pi\)
\(348\) 0 0
\(349\) −4870.00 −0.746949 −0.373474 0.927640i \(-0.621834\pi\)
−0.373474 + 0.927640i \(0.621834\pi\)
\(350\) 0 0
\(351\) −594.000 −0.0903287
\(352\) 0 0
\(353\) 10722.0i 1.61664i 0.588742 + 0.808321i \(0.299623\pi\)
−0.588742 + 0.808321i \(0.700377\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 1008.00i − 0.149437i
\(358\) 0 0
\(359\) 120.000 0.0176417 0.00882083 0.999961i \(-0.497192\pi\)
0.00882083 + 0.999961i \(0.497192\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) − 4119.00i − 0.595569i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3936.00i 0.559830i 0.960025 + 0.279915i \(0.0903063\pi\)
−0.960025 + 0.279915i \(0.909694\pi\)
\(368\) 0 0
\(369\) −1098.00 −0.154904
\(370\) 0 0
\(371\) −8112.00 −1.13519
\(372\) 0 0
\(373\) 3022.00i 0.419499i 0.977755 + 0.209750i \(0.0672649\pi\)
−0.977755 + 0.209750i \(0.932735\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5060.00i − 0.691255i
\(378\) 0 0
\(379\) −13340.0 −1.80799 −0.903997 0.427539i \(-0.859381\pi\)
−0.903997 + 0.427539i \(0.859381\pi\)
\(380\) 0 0
\(381\) 5808.00 0.780979
\(382\) 0 0
\(383\) 1008.00i 0.134481i 0.997737 + 0.0672407i \(0.0214195\pi\)
−0.997737 + 0.0672407i \(0.978580\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1692.00i − 0.222246i
\(388\) 0 0
\(389\) −9630.00 −1.25517 −0.627584 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(390\) 0 0
\(391\) −2352.00 −0.304209
\(392\) 0 0
\(393\) 2196.00i 0.281867i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 7126.00i − 0.900866i −0.892810 0.450433i \(-0.851270\pi\)
0.892810 0.450433i \(-0.148730\pi\)
\(398\) 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
\(403\) 6336.00i 0.783173i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1768.00i − 0.215323i
\(408\) 0 0
\(409\) 10870.0 1.31415 0.657074 0.753826i \(-0.271794\pi\)
0.657074 + 0.753826i \(0.271794\pi\)
\(410\) 0 0
\(411\) 6642.00 0.797143
\(412\) 0 0
\(413\) − 2400.00i − 0.285947i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 60.0000i − 0.00704607i
\(418\) 0 0
\(419\) −9700.00 −1.13097 −0.565484 0.824759i \(-0.691311\pi\)
−0.565484 + 0.824759i \(0.691311\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) 0 0
\(423\) − 2304.00i − 0.264833i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 17808.0i − 2.01824i
\(428\) 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
\(433\) 11602.0i 1.28766i 0.765169 + 0.643830i \(0.222655\pi\)
−0.765169 + 0.643830i \(0.777345\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3360.00i − 0.367805i
\(438\) 0 0
\(439\) −440.000 −0.0478361 −0.0239181 0.999714i \(-0.507614\pi\)
−0.0239181 + 0.999714i \(0.507614\pi\)
\(440\) 0 0
\(441\) 2097.00 0.226433
\(442\) 0 0
\(443\) 10188.0i 1.09266i 0.837571 + 0.546328i \(0.183975\pi\)
−0.837571 + 0.546328i \(0.816025\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3990.00i − 0.422194i
\(448\) 0 0
\(449\) 13310.0 1.39897 0.699485 0.714647i \(-0.253413\pi\)
0.699485 + 0.714647i \(0.253413\pi\)
\(450\) 0 0
\(451\) −6344.00 −0.662367
\(452\) 0 0
\(453\) − 3624.00i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3226.00i − 0.330210i −0.986276 0.165105i \(-0.947204\pi\)
0.986276 0.165105i \(-0.0527963\pi\)
\(458\) 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
\(463\) − 15072.0i − 1.51286i −0.654073 0.756431i \(-0.726941\pi\)
0.654073 0.756431i \(-0.273059\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 476.000i 0.0471663i 0.999722 + 0.0235831i \(0.00750744\pi\)
−0.999722 + 0.0235831i \(0.992493\pi\)
\(468\) 0 0
\(469\) −2016.00 −0.198487
\(470\) 0 0
\(471\) 10542.0 1.03132
\(472\) 0 0
\(473\) − 9776.00i − 0.950319i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3042.00i 0.291999i
\(478\) 0 0
\(479\) −19680.0 −1.87725 −0.938624 0.344941i \(-0.887899\pi\)
−0.938624 + 0.344941i \(0.887899\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) 0 0
\(483\) − 12096.0i − 1.13952i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5944.00i − 0.553077i −0.961003 0.276538i \(-0.910813\pi\)
0.961003 0.276538i \(-0.0891873\pi\)
\(488\) 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
\(493\) − 3220.00i − 0.294161i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 7872.00i − 0.710478i
\(498\) 0 0
\(499\) 8140.00 0.730253 0.365127 0.930958i \(-0.381026\pi\)
0.365127 + 0.930958i \(0.381026\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) 0 0
\(503\) 13768.0i 1.22045i 0.792229 + 0.610223i \(0.208920\pi\)
−0.792229 + 0.610223i \(0.791080\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 5139.00i − 0.450160i
\(508\) 0 0
\(509\) −22150.0 −1.92884 −0.964422 0.264368i \(-0.914837\pi\)
−0.964422 + 0.264368i \(0.914837\pi\)
\(510\) 0 0
\(511\) −912.000 −0.0789521
\(512\) 0 0
\(513\) − 540.000i − 0.0464748i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 13312.0i − 1.13242i
\(518\) 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
\(523\) 668.000i 0.0558501i 0.999610 + 0.0279250i \(0.00888997\pi\)
−0.999610 + 0.0279250i \(0.991110\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4032.00i 0.333276i
\(528\) 0 0
\(529\) −16057.0 −1.31972
\(530\) 0 0
\(531\) −900.000 −0.0735531
\(532\) 0 0
\(533\) 2684.00i 0.218118i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 10020.0i − 0.805205i
\(538\) 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
\(543\) 534.000i 0.0422028i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 10484.0i − 0.819494i −0.912199 0.409747i \(-0.865617\pi\)
0.912199 0.409747i \(-0.134383\pi\)
\(548\) 0 0
\(549\) −6678.00 −0.519144
\(550\) 0 0
\(551\) 4600.00 0.355656
\(552\) 0 0
\(553\) 5760.00i 0.442930i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3606.00i − 0.274311i −0.990550 0.137155i \(-0.956204\pi\)
0.990550 0.137155i \(-0.0437960\pi\)
\(558\) 0 0
\(559\) −4136.00 −0.312941
\(560\) 0 0
\(561\) −2184.00 −0.164365
\(562\) 0 0
\(563\) − 12252.0i − 0.917159i −0.888654 0.458579i \(-0.848359\pi\)
0.888654 0.458579i \(-0.151641\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1944.00i − 0.143986i
\(568\) 0 0
\(569\) 14550.0 1.07200 0.536000 0.844218i \(-0.319935\pi\)
0.536000 + 0.844218i \(0.319935\pi\)
\(570\) 0 0
\(571\) 25468.0 1.86655 0.933277 0.359157i \(-0.116936\pi\)
0.933277 + 0.359157i \(0.116936\pi\)
\(572\) 0 0
\(573\) − 5664.00i − 0.412944i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 12866.0i − 0.928282i −0.885761 0.464141i \(-0.846363\pi\)
0.885761 0.464141i \(-0.153637\pi\)
\(578\) 0 0
\(579\) 5766.00 0.413863
\(580\) 0 0
\(581\) −29088.0 −2.07706
\(582\) 0 0
\(583\) 17576.0i 1.24858i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 14844.0i − 1.04374i −0.853024 0.521872i \(-0.825234\pi\)
0.853024 0.521872i \(-0.174766\pi\)
\(588\) 0 0
\(589\) −5760.00 −0.402948
\(590\) 0 0
\(591\) −7578.00 −0.527440
\(592\) 0 0
\(593\) 20402.0i 1.41283i 0.707797 + 0.706416i \(0.249689\pi\)
−0.707797 + 0.706416i \(0.750311\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3480.00i 0.238571i
\(598\) 0 0
\(599\) 10760.0 0.733959 0.366980 0.930229i \(-0.380392\pi\)
0.366980 + 0.930229i \(0.380392\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) 0 0
\(603\) 756.000i 0.0510559i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11056.0i 0.739290i 0.929173 + 0.369645i \(0.120521\pi\)
−0.929173 + 0.369645i \(0.879479\pi\)
\(608\) 0 0
\(609\) 16560.0 1.10188
\(610\) 0 0
\(611\) −5632.00 −0.372907
\(612\) 0 0
\(613\) − 16418.0i − 1.08176i −0.841101 0.540878i \(-0.818092\pi\)
0.841101 0.540878i \(-0.181908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10374.0i 0.676891i 0.940986 + 0.338445i \(0.109901\pi\)
−0.940986 + 0.338445i \(0.890099\pi\)
\(618\) 0 0
\(619\) −5260.00 −0.341546 −0.170773 0.985310i \(-0.554627\pi\)
−0.170773 + 0.985310i \(0.554627\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 0 0
\(623\) 7920.00i 0.509323i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3120.00i − 0.198725i
\(628\) 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
\(633\) − 13404.0i − 0.841645i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 5126.00i − 0.318838i
\(638\) 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
\(643\) − 5772.00i − 0.354005i −0.984210 0.177003i \(-0.943360\pi\)
0.984210 0.177003i \(-0.0566401\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 14264.0i − 0.866732i −0.901218 0.433366i \(-0.857326\pi\)
0.901218 0.433366i \(-0.142674\pi\)
\(648\) 0 0
\(649\) −5200.00 −0.314511
\(650\) 0 0
\(651\) −20736.0 −1.24840
\(652\) 0 0
\(653\) 6902.00i 0.413623i 0.978381 + 0.206812i \(0.0663088\pi\)
−0.978381 + 0.206812i \(0.933691\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 342.000i 0.0203085i
\(658\) 0 0
\(659\) 20140.0 1.19051 0.595253 0.803539i \(-0.297052\pi\)
0.595253 + 0.803539i \(0.297052\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) 0 0
\(663\) 924.000i 0.0541255i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 38640.0i − 2.24310i
\(668\) 0 0
\(669\) −18096.0 −1.04579
\(670\) 0 0
\(671\) −38584.0 −2.21985
\(672\) 0 0
\(673\) − 7518.00i − 0.430606i −0.976547 0.215303i \(-0.930926\pi\)
0.976547 0.215303i \(-0.0690739\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18114.0i 1.02833i 0.857692 + 0.514164i \(0.171898\pi\)
−0.857692 + 0.514164i \(0.828102\pi\)
\(678\) 0 0
\(679\) −20784.0 −1.17469
\(680\) 0 0
\(681\) 7908.00 0.444986
\(682\) 0 0
\(683\) 23868.0i 1.33716i 0.743638 + 0.668582i \(0.233099\pi\)
−0.743638 + 0.668582i \(0.766901\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14490.0i 0.804699i
\(688\) 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
\(693\) − 11232.0i − 0.615683i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1708.00i 0.0928194i
\(698\) 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
\(703\) − 680.000i − 0.0364818i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29232.0i 1.55500i
\(708\) 0 0
\(709\) −3070.00 −0.162618 −0.0813091 0.996689i \(-0.525910\pi\)
−0.0813091 + 0.996689i \(0.525910\pi\)
\(710\) 0 0
\(711\) 2160.00 0.113933
\(712\) 0 0
\(713\) 48384.0i 2.54137i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6960.00i − 0.362519i
\(718\) 0 0
\(719\) 15600.0 0.809154 0.404577 0.914504i \(-0.367419\pi\)
0.404577 + 0.914504i \(0.367419\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 0 0
\(723\) − 6006.00i − 0.308943i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20696.0i 1.05581i 0.849304 + 0.527904i \(0.177022\pi\)
−0.849304 + 0.527904i \(0.822978\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −2632.00 −0.133171
\(732\) 0 0
\(733\) − 30778.0i − 1.55090i −0.631408 0.775451i \(-0.717522\pi\)
0.631408 0.775451i \(-0.282478\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4368.00i 0.218314i
\(738\) 0 0
\(739\) 11740.0 0.584388 0.292194 0.956359i \(-0.405615\pi\)
0.292194 + 0.956359i \(0.405615\pi\)
\(740\) 0 0
\(741\) −1320.00 −0.0654405
\(742\) 0 0
\(743\) − 2632.00i − 0.129958i −0.997887 0.0649789i \(-0.979302\pi\)
0.997887 0.0649789i \(-0.0206980\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10908.0i 0.534274i
\(748\) 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
\(753\) 396.000i 0.0191647i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 21646.0i − 1.03928i −0.854384 0.519642i \(-0.826066\pi\)
0.854384 0.519642i \(-0.173934\pi\)
\(758\) 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
\(763\) − 23280.0i − 1.10458i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2200.00i 0.103569i
\(768\) 0 0
\(769\) 24190.0 1.13435 0.567174 0.823598i \(-0.308037\pi\)
0.567174 + 0.823598i \(0.308037\pi\)
\(770\) 0 0
\(771\) 22842.0 1.06697
\(772\) 0 0
\(773\) − 25698.0i − 1.19572i −0.801600 0.597861i \(-0.796018\pi\)
0.801600 0.597861i \(-0.203982\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2448.00i − 0.113026i
\(778\) 0 0
\(779\) −2440.00 −0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) 0 0
\(783\) − 6210.00i − 0.283432i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33436.0i 1.51444i 0.653160 + 0.757220i \(0.273443\pi\)
−0.653160 + 0.757220i \(0.726557\pi\)
\(788\) 0 0
\(789\) 14664.0 0.661663
\(790\) 0 0
\(791\) 25008.0 1.12412
\(792\) 0 0
\(793\) 16324.0i 0.730999i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37594.0i 1.67083i 0.549623 + 0.835413i \(0.314771\pi\)
−0.549623 + 0.835413i \(0.685229\pi\)
\(798\) 0 0
\(799\) −3584.00 −0.158689
\(800\) 0 0
\(801\) 2970.00 0.131011
\(802\) 0 0
\(803\) 1976.00i 0.0868388i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3810.00i 0.166194i
\(808\) 0 0
\(809\) −4730.00 −0.205560 −0.102780 0.994704i \(-0.532774\pi\)
−0.102780 + 0.994704i \(0.532774\pi\)
\(810\) 0 0
\(811\) 8748.00 0.378772 0.189386 0.981903i \(-0.439350\pi\)
0.189386 + 0.981903i \(0.439350\pi\)
\(812\) 0 0
\(813\) 3216.00i 0.138733i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3760.00i − 0.161011i
\(818\) 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
\(823\) − 3992.00i − 0.169079i −0.996420 0.0845397i \(-0.973058\pi\)
0.996420 0.0845397i \(-0.0269420\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 14444.0i − 0.607336i −0.952778 0.303668i \(-0.901789\pi\)
0.952778 0.303668i \(-0.0982114\pi\)
\(828\) 0 0
\(829\) −42150.0 −1.76590 −0.882949 0.469468i \(-0.844446\pi\)
−0.882949 + 0.469468i \(0.844446\pi\)
\(830\) 0 0
\(831\) 16182.0 0.675508
\(832\) 0 0
\(833\) − 3262.00i − 0.135680i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7776.00i 0.321121i
\(838\) 0 0
\(839\) 13400.0 0.551394 0.275697 0.961245i \(-0.411091\pi\)
0.275697 + 0.961245i \(0.411091\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) 0 0
\(843\) − 7326.00i − 0.299313i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 32952.0i − 1.33677i
\(848\) 0 0
\(849\) −8316.00 −0.336165
\(850\) 0 0
\(851\) −5712.00 −0.230088
\(852\) 0 0
\(853\) − 8658.00i − 0.347531i −0.984787 0.173766i \(-0.944406\pi\)
0.984787 0.173766i \(-0.0555935\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42826.0i − 1.70701i −0.521084 0.853505i \(-0.674472\pi\)
0.521084 0.853505i \(-0.325528\pi\)
\(858\) 0 0
\(859\) −35900.0 −1.42595 −0.712976 0.701189i \(-0.752653\pi\)
−0.712976 + 0.701189i \(0.752653\pi\)
\(860\) 0 0
\(861\) −8784.00 −0.347686
\(862\) 0 0
\(863\) 3088.00i 0.121804i 0.998144 + 0.0609019i \(0.0193977\pi\)
−0.998144 + 0.0609019i \(0.980602\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 14151.0i − 0.554317i
\(868\) 0 0
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) 1848.00 0.0718910
\(872\) 0 0
\(873\) 7794.00i 0.302161i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35274.0i 1.35817i 0.734058 + 0.679087i \(0.237624\pi\)
−0.734058 + 0.679087i \(0.762376\pi\)
\(878\) 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
\(883\) − 12572.0i − 0.479141i −0.970879 0.239570i \(-0.922993\pi\)
0.970879 0.239570i \(-0.0770066\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 21864.0i − 0.827645i −0.910358 0.413823i \(-0.864193\pi\)
0.910358 0.413823i \(-0.135807\pi\)
\(888\) 0 0
\(889\) 46464.0 1.75293
\(890\) 0 0
\(891\) −4212.00 −0.158370
\(892\) 0 0
\(893\) − 5120.00i − 0.191864i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 11088.0i 0.412729i
\(898\) 0 0
\(899\) −66240.0 −2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) 0 0
\(903\) − 13536.0i − 0.498837i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 31236.0i 1.14352i 0.820420 + 0.571761i \(0.193740\pi\)
−0.820420 + 0.571761i \(0.806260\pi\)
\(908\) 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
\(913\) 63024.0i 2.28455i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17568.0i 0.632657i
\(918\) 0 0
\(919\) 20200.0 0.725067 0.362533 0.931971i \(-0.381912\pi\)
0.362533 + 0.931971i \(0.381912\pi\)
\(920\) 0 0
\(921\) 15348.0 0.549114
\(922\) 0 0
\(923\) 7216.00i 0.257332i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 792.000i − 0.0280612i
\(928\) 0 0
\(929\) −31010.0 −1.09516 −0.547581 0.836753i \(-0.684451\pi\)
−0.547581 + 0.836753i \(0.684451\pi\)
\(930\) 0 0
\(931\) 4660.00 0.164044
\(932\) 0 0
\(933\) − 8424.00i − 0.295594i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39174.0i 1.36580i 0.730510 + 0.682902i \(0.239283\pi\)
−0.730510 + 0.682902i \(0.760717\pi\)
\(938\) 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
\(943\) 20496.0i 0.707785i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23676.0i 0.812425i 0.913779 + 0.406213i \(0.133151\pi\)
−0.913779 + 0.406213i \(0.866849\pi\)
\(948\) 0 0
\(949\) 836.000 0.0285961
\(950\) 0 0
\(951\) −6738.00 −0.229752
\(952\) 0 0
\(953\) 18922.0i 0.643173i 0.946880 + 0.321586i \(0.104216\pi\)
−0.946880 + 0.321586i \(0.895784\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 35880.0i − 1.21195i
\(958\) 0 0
\(959\) 53136.0 1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 0 0
\(963\) − 324.000i − 0.0108419i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 39656.0i 1.31877i 0.751805 + 0.659385i \(0.229183\pi\)
−0.751805 + 0.659385i \(0.770817\pi\)
\(968\) 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
\(973\) − 480.000i − 0.0158151i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 974.000i 0.0318946i 0.999873 + 0.0159473i \(0.00507640\pi\)
−0.999873 + 0.0159473i \(0.994924\pi\)
\(978\) 0 0
\(979\) 17160.0 0.560200
\(980\) 0 0
\(981\) −8730.00 −0.284126
\(982\) 0 0
\(983\) 13608.0i 0.441534i 0.975327 + 0.220767i \(0.0708560\pi\)
−0.975327 + 0.220767i \(0.929144\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 18432.0i − 0.594425i
\(988\) 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
\(993\) 3996.00i 0.127703i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3234.00i 0.102730i 0.998680 + 0.0513650i \(0.0163572\pi\)
−0.998680 + 0.0513650i \(0.983643\pi\)
\(998\) 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.f.b.49.1 2
4.3 odd 2 75.4.b.b.49.1 2
5.2 odd 4 240.4.a.e.1.1 1
5.3 odd 4 1200.4.a.t.1.1 1
5.4 even 2 inner 1200.4.f.b.49.2 2
12.11 even 2 225.4.b.e.199.2 2
15.2 even 4 720.4.a.n.1.1 1
20.3 even 4 75.4.a.b.1.1 1
20.7 even 4 15.4.a.a.1.1 1
20.19 odd 2 75.4.b.b.49.2 2
40.27 even 4 960.4.a.b.1.1 1
40.37 odd 4 960.4.a.ba.1.1 1
60.23 odd 4 225.4.a.f.1.1 1
60.47 odd 4 45.4.a.c.1.1 1
60.59 even 2 225.4.b.e.199.1 2
140.27 odd 4 735.4.a.e.1.1 1
180.7 even 12 405.4.e.g.271.1 2
180.47 odd 12 405.4.e.i.271.1 2
180.67 even 12 405.4.e.g.136.1 2
180.167 odd 12 405.4.e.i.136.1 2
220.87 odd 4 1815.4.a.e.1.1 1
420.167 even 4 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 20.7 even 4
45.4.a.c.1.1 1 60.47 odd 4
75.4.a.b.1.1 1 20.3 even 4
75.4.b.b.49.1 2 4.3 odd 2
75.4.b.b.49.2 2 20.19 odd 2
225.4.a.f.1.1 1 60.23 odd 4
225.4.b.e.199.1 2 60.59 even 2
225.4.b.e.199.2 2 12.11 even 2
240.4.a.e.1.1 1 5.2 odd 4
405.4.e.g.136.1 2 180.67 even 12
405.4.e.g.271.1 2 180.7 even 12
405.4.e.i.136.1 2 180.167 odd 12
405.4.e.i.271.1 2 180.47 odd 12
720.4.a.n.1.1 1 15.2 even 4
735.4.a.e.1.1 1 140.27 odd 4
960.4.a.b.1.1 1 40.27 even 4
960.4.a.ba.1.1 1 40.37 odd 4
1200.4.a.t.1.1 1 5.3 odd 4
1200.4.f.b.49.1 2 1.1 even 1 trivial
1200.4.f.b.49.2 2 5.4 even 2 inner
1815.4.a.e.1.1 1 220.87 odd 4
2205.4.a.l.1.1 1 420.167 even 4