Properties

Label 2304.3.b.l.127.2
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.l.127.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{5} +6.92820i q^{7} +O(q^{10})\) \(q-2.00000i q^{5} +6.92820i q^{7} -6.92820 q^{11} +2.00000i q^{13} -10.0000 q^{17} -20.7846 q^{19} -27.7128i q^{23} +21.0000 q^{25} +26.0000i q^{29} +6.92820i q^{31} +13.8564 q^{35} -26.0000i q^{37} +58.0000 q^{41} -48.4974 q^{43} -69.2820i q^{47} +1.00000 q^{49} -74.0000i q^{53} +13.8564i q^{55} +90.0666 q^{59} +26.0000i q^{61} +4.00000 q^{65} -6.92820 q^{67} +46.0000 q^{73} -48.0000i q^{77} +117.779i q^{79} +48.4974 q^{83} +20.0000i q^{85} +82.0000 q^{89} -13.8564 q^{91} +41.5692i q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 40q^{17} + 84q^{25} + 232q^{41} + 4q^{49} + 16q^{65} + 184q^{73} + 328q^{89} + 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.00000i − 0.400000i −0.979796 0.200000i \(-0.935906\pi\)
0.979796 0.200000i \(-0.0640942\pi\)
\(6\) 0 0
\(7\) 6.92820i 0.989743i 0.868966 + 0.494872i \(0.164785\pi\)
−0.868966 + 0.494872i \(0.835215\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.92820 −0.629837 −0.314918 0.949119i \(-0.601977\pi\)
−0.314918 + 0.949119i \(0.601977\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.153846i 0.997037 + 0.0769231i \(0.0245096\pi\)
−0.997037 + 0.0769231i \(0.975490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −10.0000 −0.588235 −0.294118 0.955769i \(-0.595026\pi\)
−0.294118 + 0.955769i \(0.595026\pi\)
\(18\) 0 0
\(19\) −20.7846 −1.09393 −0.546963 0.837157i \(-0.684216\pi\)
−0.546963 + 0.837157i \(0.684216\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 27.7128i − 1.20490i −0.798155 0.602452i \(-0.794190\pi\)
0.798155 0.602452i \(-0.205810\pi\)
\(24\) 0 0
\(25\) 21.0000 0.840000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 26.0000i 0.896552i 0.893895 + 0.448276i \(0.147962\pi\)
−0.893895 + 0.448276i \(0.852038\pi\)
\(30\) 0 0
\(31\) 6.92820i 0.223490i 0.993737 + 0.111745i \(0.0356441\pi\)
−0.993737 + 0.111745i \(0.964356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.8564 0.395897
\(36\) 0 0
\(37\) − 26.0000i − 0.702703i −0.936244 0.351351i \(-0.885722\pi\)
0.936244 0.351351i \(-0.114278\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 58.0000 1.41463 0.707317 0.706896i \(-0.249905\pi\)
0.707317 + 0.706896i \(0.249905\pi\)
\(42\) 0 0
\(43\) −48.4974 −1.12785 −0.563924 0.825827i \(-0.690709\pi\)
−0.563924 + 0.825827i \(0.690709\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 69.2820i − 1.47409i −0.675846 0.737043i \(-0.736222\pi\)
0.675846 0.737043i \(-0.263778\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 74.0000i − 1.39623i −0.715987 0.698113i \(-0.754023\pi\)
0.715987 0.698113i \(-0.245977\pi\)
\(54\) 0 0
\(55\) 13.8564i 0.251935i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 90.0666 1.52655 0.763277 0.646072i \(-0.223589\pi\)
0.763277 + 0.646072i \(0.223589\pi\)
\(60\) 0 0
\(61\) 26.0000i 0.426230i 0.977027 + 0.213115i \(0.0683608\pi\)
−0.977027 + 0.213115i \(0.931639\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.0615385
\(66\) 0 0
\(67\) −6.92820 −0.103406 −0.0517030 0.998663i \(-0.516465\pi\)
−0.0517030 + 0.998663i \(0.516465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 46.0000 0.630137 0.315068 0.949069i \(-0.397973\pi\)
0.315068 + 0.949069i \(0.397973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 48.0000i − 0.623377i
\(78\) 0 0
\(79\) 117.779i 1.49088i 0.666573 + 0.745440i \(0.267760\pi\)
−0.666573 + 0.745440i \(0.732240\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 48.4974 0.584306 0.292153 0.956372i \(-0.405628\pi\)
0.292153 + 0.956372i \(0.405628\pi\)
\(84\) 0 0
\(85\) 20.0000i 0.235294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 82.0000 0.921348 0.460674 0.887569i \(-0.347608\pi\)
0.460674 + 0.887569i \(0.347608\pi\)
\(90\) 0 0
\(91\) −13.8564 −0.152268
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 41.5692i 0.437571i
\(96\) 0 0
\(97\) 2.00000 0.0206186 0.0103093 0.999947i \(-0.496718\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 74.0000i − 0.732673i −0.930482 0.366337i \(-0.880612\pi\)
0.930482 0.366337i \(-0.119388\pi\)
\(102\) 0 0
\(103\) 76.2102i 0.739905i 0.929051 + 0.369953i \(0.120626\pi\)
−0.929051 + 0.369953i \(0.879374\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −20.7846 −0.194249 −0.0971243 0.995272i \(-0.530964\pi\)
−0.0971243 + 0.995272i \(0.530964\pi\)
\(108\) 0 0
\(109\) − 46.0000i − 0.422018i −0.977484 0.211009i \(-0.932325\pi\)
0.977484 0.211009i \(-0.0676750\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 110.000 0.973451 0.486726 0.873555i \(-0.338191\pi\)
0.486726 + 0.873555i \(0.338191\pi\)
\(114\) 0 0
\(115\) −55.4256 −0.481962
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 69.2820i − 0.582202i
\(120\) 0 0
\(121\) −73.0000 −0.603306
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 92.0000i − 0.736000i
\(126\) 0 0
\(127\) − 145.492i − 1.14561i −0.819692 0.572804i \(-0.805856\pi\)
0.819692 0.572804i \(-0.194144\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 117.779 0.899080 0.449540 0.893260i \(-0.351588\pi\)
0.449540 + 0.893260i \(0.351588\pi\)
\(132\) 0 0
\(133\) − 144.000i − 1.08271i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 0.0729927 0.0364964 0.999334i \(-0.488380\pi\)
0.0364964 + 0.999334i \(0.488380\pi\)
\(138\) 0 0
\(139\) 48.4974 0.348902 0.174451 0.984666i \(-0.444185\pi\)
0.174451 + 0.984666i \(0.444185\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 13.8564i − 0.0968979i
\(144\) 0 0
\(145\) 52.0000 0.358621
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.00000i − 0.0134228i −0.999977 0.00671141i \(-0.997864\pi\)
0.999977 0.00671141i \(-0.00213632\pi\)
\(150\) 0 0
\(151\) 90.0666i 0.596468i 0.954493 + 0.298234i \(0.0963975\pi\)
−0.954493 + 0.298234i \(0.903602\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 13.8564 0.0893962
\(156\) 0 0
\(157\) − 214.000i − 1.36306i −0.731791 0.681529i \(-0.761315\pi\)
0.731791 0.681529i \(-0.238685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 192.000 1.19255
\(162\) 0 0
\(163\) −20.7846 −0.127513 −0.0637565 0.997965i \(-0.520308\pi\)
−0.0637565 + 0.997965i \(0.520308\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 96.9948i 0.580807i 0.956904 + 0.290404i \(0.0937896\pi\)
−0.956904 + 0.290404i \(0.906210\pi\)
\(168\) 0 0
\(169\) 165.000 0.976331
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 334.000i − 1.93064i −0.261077 0.965318i \(-0.584078\pi\)
0.261077 0.965318i \(-0.415922\pi\)
\(174\) 0 0
\(175\) 145.492i 0.831384i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −187.061 −1.04504 −0.522518 0.852628i \(-0.675007\pi\)
−0.522518 + 0.852628i \(0.675007\pi\)
\(180\) 0 0
\(181\) − 2.00000i − 0.0110497i −0.999985 0.00552486i \(-0.998241\pi\)
0.999985 0.00552486i \(-0.00175863\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −52.0000 −0.281081
\(186\) 0 0
\(187\) 69.2820 0.370492
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 221.703i 1.16075i 0.814351 + 0.580373i \(0.197093\pi\)
−0.814351 + 0.580373i \(0.802907\pi\)
\(192\) 0 0
\(193\) 290.000 1.50259 0.751295 0.659966i \(-0.229429\pi\)
0.751295 + 0.659966i \(0.229429\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 26.0000i − 0.131980i −0.997820 0.0659898i \(-0.978980\pi\)
0.997820 0.0659898i \(-0.0210205\pi\)
\(198\) 0 0
\(199\) − 394.908i − 1.98446i −0.124416 0.992230i \(-0.539706\pi\)
0.124416 0.992230i \(-0.460294\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −180.133 −0.887356
\(204\) 0 0
\(205\) − 116.000i − 0.565854i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 144.000 0.688995
\(210\) 0 0
\(211\) 242.487 1.14923 0.574614 0.818425i \(-0.305152\pi\)
0.574614 + 0.818425i \(0.305152\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 96.9948i 0.451139i
\(216\) 0 0
\(217\) −48.0000 −0.221198
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 20.0000i − 0.0904977i
\(222\) 0 0
\(223\) − 339.482i − 1.52234i −0.648552 0.761170i \(-0.724625\pi\)
0.648552 0.761170i \(-0.275375\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −284.056 −1.25135 −0.625675 0.780084i \(-0.715176\pi\)
−0.625675 + 0.780084i \(0.715176\pi\)
\(228\) 0 0
\(229\) 142.000i 0.620087i 0.950722 + 0.310044i \(0.100344\pi\)
−0.950722 + 0.310044i \(0.899656\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 82.0000 0.351931 0.175966 0.984396i \(-0.443695\pi\)
0.175966 + 0.984396i \(0.443695\pi\)
\(234\) 0 0
\(235\) −138.564 −0.589634
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 387.979i − 1.62334i −0.584113 0.811672i \(-0.698558\pi\)
0.584113 0.811672i \(-0.301442\pi\)
\(240\) 0 0
\(241\) −46.0000 −0.190871 −0.0954357 0.995436i \(-0.530424\pi\)
−0.0954357 + 0.995436i \(0.530424\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.00000i − 0.00816327i
\(246\) 0 0
\(247\) − 41.5692i − 0.168296i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −145.492 −0.579650 −0.289825 0.957080i \(-0.593597\pi\)
−0.289825 + 0.957080i \(0.593597\pi\)
\(252\) 0 0
\(253\) 192.000i 0.758893i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 254.000 0.988327 0.494163 0.869369i \(-0.335474\pi\)
0.494163 + 0.869369i \(0.335474\pi\)
\(258\) 0 0
\(259\) 180.133 0.695495
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 152.420i 0.579546i 0.957095 + 0.289773i \(0.0935797\pi\)
−0.957095 + 0.289773i \(0.906420\pi\)
\(264\) 0 0
\(265\) −148.000 −0.558491
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 262.000i − 0.973978i −0.873408 0.486989i \(-0.838095\pi\)
0.873408 0.486989i \(-0.161905\pi\)
\(270\) 0 0
\(271\) 20.7846i 0.0766960i 0.999264 + 0.0383480i \(0.0122095\pi\)
−0.999264 + 0.0383480i \(0.987790\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −145.492 −0.529063
\(276\) 0 0
\(277\) − 290.000i − 1.04693i −0.852047 0.523466i \(-0.824639\pi\)
0.852047 0.523466i \(-0.175361\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 226.000 0.804270 0.402135 0.915580i \(-0.368268\pi\)
0.402135 + 0.915580i \(0.368268\pi\)
\(282\) 0 0
\(283\) 297.913 1.05270 0.526348 0.850269i \(-0.323561\pi\)
0.526348 + 0.850269i \(0.323561\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 401.836i 1.40012i
\(288\) 0 0
\(289\) −189.000 −0.653979
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 362.000i − 1.23549i −0.786377 0.617747i \(-0.788045\pi\)
0.786377 0.617747i \(-0.211955\pi\)
\(294\) 0 0
\(295\) − 180.133i − 0.610621i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 55.4256 0.185370
\(300\) 0 0
\(301\) − 336.000i − 1.11628i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 52.0000 0.170492
\(306\) 0 0
\(307\) −145.492 −0.473916 −0.236958 0.971520i \(-0.576150\pi\)
−0.236958 + 0.971520i \(0.576150\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 235.559i − 0.757424i −0.925515 0.378712i \(-0.876367\pi\)
0.925515 0.378712i \(-0.123633\pi\)
\(312\) 0 0
\(313\) 478.000 1.52716 0.763578 0.645715i \(-0.223441\pi\)
0.763578 + 0.645715i \(0.223441\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 170.000i 0.536278i 0.963380 + 0.268139i \(0.0864086\pi\)
−0.963380 + 0.268139i \(0.913591\pi\)
\(318\) 0 0
\(319\) − 180.133i − 0.564681i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 207.846 0.643486
\(324\) 0 0
\(325\) 42.0000i 0.129231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 480.000 1.45897
\(330\) 0 0
\(331\) 408.764 1.23494 0.617468 0.786596i \(-0.288158\pi\)
0.617468 + 0.786596i \(0.288158\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.8564i 0.0413624i
\(336\) 0 0
\(337\) 338.000 1.00297 0.501484 0.865167i \(-0.332788\pi\)
0.501484 + 0.865167i \(0.332788\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 48.0000i − 0.140762i
\(342\) 0 0
\(343\) 346.410i 1.00994i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −200.918 −0.579014 −0.289507 0.957176i \(-0.593491\pi\)
−0.289507 + 0.957176i \(0.593491\pi\)
\(348\) 0 0
\(349\) 506.000i 1.44986i 0.688824 + 0.724928i \(0.258127\pi\)
−0.688824 + 0.724928i \(0.741873\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −178.000 −0.504249 −0.252125 0.967695i \(-0.581129\pi\)
−0.252125 + 0.967695i \(0.581129\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 166.277i 0.463167i 0.972815 + 0.231583i \(0.0743906\pi\)
−0.972815 + 0.231583i \(0.925609\pi\)
\(360\) 0 0
\(361\) 71.0000 0.196676
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 92.0000i − 0.252055i
\(366\) 0 0
\(367\) 200.918i 0.547460i 0.961807 + 0.273730i \(0.0882575\pi\)
−0.961807 + 0.273730i \(0.911742\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 512.687 1.38191
\(372\) 0 0
\(373\) 310.000i 0.831099i 0.909571 + 0.415550i \(0.136411\pi\)
−0.909571 + 0.415550i \(0.863589\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −52.0000 −0.137931
\(378\) 0 0
\(379\) −436.477 −1.15165 −0.575827 0.817572i \(-0.695320\pi\)
−0.575827 + 0.817572i \(0.695320\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 609.682i 1.59186i 0.605390 + 0.795929i \(0.293017\pi\)
−0.605390 + 0.795929i \(0.706983\pi\)
\(384\) 0 0
\(385\) −96.0000 −0.249351
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 578.000i − 1.48586i −0.669368 0.742931i \(-0.733435\pi\)
0.669368 0.742931i \(-0.266565\pi\)
\(390\) 0 0
\(391\) 277.128i 0.708768i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 235.559 0.596352
\(396\) 0 0
\(397\) 26.0000i 0.0654912i 0.999464 + 0.0327456i \(0.0104251\pi\)
−0.999464 + 0.0327456i \(0.989575\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −250.000 −0.623441 −0.311721 0.950174i \(-0.600905\pi\)
−0.311721 + 0.950174i \(0.600905\pi\)
\(402\) 0 0
\(403\) −13.8564 −0.0343831
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 180.133i 0.442588i
\(408\) 0 0
\(409\) −290.000 −0.709046 −0.354523 0.935047i \(-0.615357\pi\)
−0.354523 + 0.935047i \(0.615357\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 624.000i 1.51090i
\(414\) 0 0
\(415\) − 96.9948i − 0.233723i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −339.482 −0.810219 −0.405110 0.914268i \(-0.632767\pi\)
−0.405110 + 0.914268i \(0.632767\pi\)
\(420\) 0 0
\(421\) − 674.000i − 1.60095i −0.599366 0.800475i \(-0.704581\pi\)
0.599366 0.800475i \(-0.295419\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −210.000 −0.494118
\(426\) 0 0
\(427\) −180.133 −0.421858
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 540.400i − 1.25383i −0.779088 0.626914i \(-0.784318\pi\)
0.779088 0.626914i \(-0.215682\pi\)
\(432\) 0 0
\(433\) −334.000 −0.771363 −0.385681 0.922632i \(-0.626034\pi\)
−0.385681 + 0.922632i \(0.626034\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 576.000i 1.31808i
\(438\) 0 0
\(439\) − 117.779i − 0.268290i −0.990962 0.134145i \(-0.957171\pi\)
0.990962 0.134145i \(-0.0428288\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 76.2102 0.172032 0.0860161 0.996294i \(-0.472586\pi\)
0.0860161 + 0.996294i \(0.472586\pi\)
\(444\) 0 0
\(445\) − 164.000i − 0.368539i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −394.000 −0.877506 −0.438753 0.898608i \(-0.644580\pi\)
−0.438753 + 0.898608i \(0.644580\pi\)
\(450\) 0 0
\(451\) −401.836 −0.890988
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 27.7128i 0.0609073i
\(456\) 0 0
\(457\) 478.000 1.04595 0.522976 0.852347i \(-0.324822\pi\)
0.522976 + 0.852347i \(0.324822\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 142.000i − 0.308026i −0.988069 0.154013i \(-0.950780\pi\)
0.988069 0.154013i \(-0.0492198\pi\)
\(462\) 0 0
\(463\) 630.466i 1.36170i 0.732423 + 0.680849i \(0.238389\pi\)
−0.732423 + 0.680849i \(0.761611\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.0445067 0.0222533 0.999752i \(-0.492916\pi\)
0.0222533 + 0.999752i \(0.492916\pi\)
\(468\) 0 0
\(469\) − 48.0000i − 0.102345i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 336.000 0.710359
\(474\) 0 0
\(475\) −436.477 −0.918899
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 734.390i − 1.53317i −0.642141 0.766586i \(-0.721954\pi\)
0.642141 0.766586i \(-0.278046\pi\)
\(480\) 0 0
\(481\) 52.0000 0.108108
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 4.00000i − 0.00824742i
\(486\) 0 0
\(487\) − 103.923i − 0.213394i −0.994292 0.106697i \(-0.965972\pi\)
0.994292 0.106697i \(-0.0340275\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 921.451 1.87668 0.938341 0.345711i \(-0.112362\pi\)
0.938341 + 0.345711i \(0.112362\pi\)
\(492\) 0 0
\(493\) − 260.000i − 0.527383i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 76.2102 0.152726 0.0763630 0.997080i \(-0.475669\pi\)
0.0763630 + 0.997080i \(0.475669\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 581.969i − 1.15700i −0.815684 0.578498i \(-0.803639\pi\)
0.815684 0.578498i \(-0.196361\pi\)
\(504\) 0 0
\(505\) −148.000 −0.293069
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 842.000i 1.65422i 0.562037 + 0.827112i \(0.310018\pi\)
−0.562037 + 0.827112i \(0.689982\pi\)
\(510\) 0 0
\(511\) 318.697i 0.623674i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 152.420 0.295962
\(516\) 0 0
\(517\) 480.000i 0.928433i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −326.000 −0.625720 −0.312860 0.949799i \(-0.601287\pi\)
−0.312860 + 0.949799i \(0.601287\pi\)
\(522\) 0 0
\(523\) 311.769 0.596117 0.298058 0.954548i \(-0.403661\pi\)
0.298058 + 0.954548i \(0.403661\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 69.2820i − 0.131465i
\(528\) 0 0
\(529\) −239.000 −0.451796
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 116.000i 0.217636i
\(534\) 0 0
\(535\) 41.5692i 0.0776995i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.92820 −0.0128538
\(540\) 0 0
\(541\) 530.000i 0.979667i 0.871816 + 0.489834i \(0.162942\pi\)
−0.871816 + 0.489834i \(0.837058\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −92.0000 −0.168807
\(546\) 0 0
\(547\) 339.482 0.620625 0.310313 0.950635i \(-0.399566\pi\)
0.310313 + 0.950635i \(0.399566\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 540.400i − 0.980762i
\(552\) 0 0
\(553\) −816.000 −1.47559
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 766.000i − 1.37522i −0.726078 0.687612i \(-0.758659\pi\)
0.726078 0.687612i \(-0.241341\pi\)
\(558\) 0 0
\(559\) − 96.9948i − 0.173515i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 491.902 0.873717 0.436858 0.899530i \(-0.356091\pi\)
0.436858 + 0.899530i \(0.356091\pi\)
\(564\) 0 0
\(565\) − 220.000i − 0.389381i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −422.000 −0.741652 −0.370826 0.928702i \(-0.620925\pi\)
−0.370826 + 0.928702i \(0.620925\pi\)
\(570\) 0 0
\(571\) −284.056 −0.497472 −0.248736 0.968571i \(-0.580015\pi\)
−0.248736 + 0.968571i \(0.580015\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 581.969i − 1.01212i
\(576\) 0 0
\(577\) −46.0000 −0.0797227 −0.0398614 0.999205i \(-0.512692\pi\)
−0.0398614 + 0.999205i \(0.512692\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 336.000i 0.578313i
\(582\) 0 0
\(583\) 512.687i 0.879395i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −630.466 −1.07405 −0.537024 0.843567i \(-0.680452\pi\)
−0.537024 + 0.843567i \(0.680452\pi\)
\(588\) 0 0
\(589\) − 144.000i − 0.244482i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −82.0000 −0.138280 −0.0691400 0.997607i \(-0.522026\pi\)
−0.0691400 + 0.997607i \(0.522026\pi\)
\(594\) 0 0
\(595\) −138.564 −0.232881
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 55.4256i 0.0925303i 0.998929 + 0.0462651i \(0.0147319\pi\)
−0.998929 + 0.0462651i \(0.985268\pi\)
\(600\) 0 0
\(601\) 334.000 0.555740 0.277870 0.960619i \(-0.410371\pi\)
0.277870 + 0.960619i \(0.410371\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 146.000i 0.241322i
\(606\) 0 0
\(607\) − 367.195i − 0.604934i −0.953160 0.302467i \(-0.902190\pi\)
0.953160 0.302467i \(-0.0978102\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 138.564 0.226782
\(612\) 0 0
\(613\) 214.000i 0.349103i 0.984648 + 0.174551i \(0.0558475\pi\)
−0.984648 + 0.174551i \(0.944152\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1118.00 −1.81199 −0.905997 0.423285i \(-0.860877\pi\)
−0.905997 + 0.423285i \(0.860877\pi\)
\(618\) 0 0
\(619\) −672.036 −1.08568 −0.542840 0.839836i \(-0.682651\pi\)
−0.542840 + 0.839836i \(0.682651\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 568.113i 0.911898i
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 260.000i 0.413355i
\(630\) 0 0
\(631\) 145.492i 0.230574i 0.993332 + 0.115287i \(0.0367788\pi\)
−0.993332 + 0.115287i \(0.963221\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −290.985 −0.458243
\(636\) 0 0
\(637\) 2.00000i 0.00313972i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0000 −0.0156006 −0.00780031 0.999970i \(-0.502483\pi\)
−0.00780031 + 0.999970i \(0.502483\pi\)
\(642\) 0 0
\(643\) −1212.44 −1.88559 −0.942796 0.333370i \(-0.891814\pi\)
−0.942796 + 0.333370i \(0.891814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 332.554i − 0.513993i −0.966412 0.256997i \(-0.917267\pi\)
0.966412 0.256997i \(-0.0827330\pi\)
\(648\) 0 0
\(649\) −624.000 −0.961479
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 670.000i − 1.02603i −0.858379 0.513017i \(-0.828528\pi\)
0.858379 0.513017i \(-0.171472\pi\)
\(654\) 0 0
\(655\) − 235.559i − 0.359632i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −824.456 −1.25107 −0.625536 0.780195i \(-0.715120\pi\)
−0.625536 + 0.780195i \(0.715120\pi\)
\(660\) 0 0
\(661\) 1222.00i 1.84871i 0.381529 + 0.924357i \(0.375398\pi\)
−0.381529 + 0.924357i \(0.624602\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −288.000 −0.433083
\(666\) 0 0
\(667\) 720.533 1.08026
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 180.133i − 0.268455i
\(672\) 0 0
\(673\) −334.000 −0.496285 −0.248143 0.968724i \(-0.579820\pi\)
−0.248143 + 0.968724i \(0.579820\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1006.00i 1.48597i 0.669309 + 0.742984i \(0.266590\pi\)
−0.669309 + 0.742984i \(0.733410\pi\)
\(678\) 0 0
\(679\) 13.8564i 0.0204071i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 187.061 0.273882 0.136941 0.990579i \(-0.456273\pi\)
0.136941 + 0.990579i \(0.456273\pi\)
\(684\) 0 0
\(685\) − 20.0000i − 0.0291971i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 148.000 0.214804
\(690\) 0 0
\(691\) −990.733 −1.43377 −0.716884 0.697193i \(-0.754432\pi\)
−0.716884 + 0.697193i \(0.754432\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 96.9948i − 0.139561i
\(696\) 0 0
\(697\) −580.000 −0.832138
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1034.00i 1.47504i 0.675328 + 0.737518i \(0.264002\pi\)
−0.675328 + 0.737518i \(0.735998\pi\)
\(702\) 0 0
\(703\) 540.400i 0.768705i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 512.687 0.725158
\(708\) 0 0
\(709\) − 530.000i − 0.747532i −0.927523 0.373766i \(-0.878066\pi\)
0.927523 0.373766i \(-0.121934\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 192.000 0.269285
\(714\) 0 0
\(715\) −27.7128 −0.0387592
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 706.677i 0.982861i 0.870917 + 0.491430i \(0.163526\pi\)
−0.870917 + 0.491430i \(0.836474\pi\)
\(720\) 0 0
\(721\) −528.000 −0.732316
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 546.000i 0.753103i
\(726\) 0 0
\(727\) − 242.487i − 0.333545i −0.985995 0.166772i \(-0.946665\pi\)
0.985995 0.166772i \(-0.0533345\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 484.974 0.663439
\(732\) 0 0
\(733\) 194.000i 0.264666i 0.991205 + 0.132333i \(0.0422468\pi\)
−0.991205 + 0.132333i \(0.957753\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 0.0651289
\(738\) 0 0
\(739\) 1351.00 1.82815 0.914073 0.405550i \(-0.132920\pi\)
0.914073 + 0.405550i \(0.132920\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 678.964i 0.913814i 0.889514 + 0.456907i \(0.151043\pi\)
−0.889514 + 0.456907i \(0.848957\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.00536913
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 144.000i − 0.192256i
\(750\) 0 0
\(751\) − 658.179i − 0.876404i −0.898877 0.438202i \(-0.855615\pi\)
0.898877 0.438202i \(-0.144385\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 180.133 0.238587
\(756\) 0 0
\(757\) 1006.00i 1.32893i 0.747319 + 0.664465i \(0.231341\pi\)
−0.747319 + 0.664465i \(0.768659\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −758.000 −0.996058 −0.498029 0.867160i \(-0.665943\pi\)
−0.498029 + 0.867160i \(0.665943\pi\)
\(762\) 0 0
\(763\) 318.697 0.417690
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 180.133i 0.234854i
\(768\) 0 0
\(769\) 2.00000 0.00260078 0.00130039 0.999999i \(-0.499586\pi\)
0.00130039 + 0.999999i \(0.499586\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 262.000i 0.338939i 0.985535 + 0.169470i \(0.0542055\pi\)
−0.985535 + 0.169470i \(0.945795\pi\)
\(774\) 0 0
\(775\) 145.492i 0.187732i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1205.51 −1.54751
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −428.000 −0.545223
\(786\) 0 0
\(787\) 1447.99 1.83989 0.919946 0.392046i \(-0.128233\pi\)
0.919946 + 0.392046i \(0.128233\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 762.102i 0.963467i
\(792\) 0 0
\(793\) −52.0000 −0.0655738
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 866.000i 1.08657i 0.839547 + 0.543287i \(0.182821\pi\)
−0.839547 + 0.543287i \(0.817179\pi\)
\(798\) 0 0
\(799\) 692.820i 0.867109i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −318.697 −0.396883
\(804\) 0 0
\(805\) − 384.000i − 0.477019i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.0000 0.0123609 0.00618047 0.999981i \(-0.498033\pi\)
0.00618047 + 0.999981i \(0.498033\pi\)
\(810\) 0 0
\(811\) −436.477 −0.538196 −0.269098 0.963113i \(-0.586726\pi\)
−0.269098 + 0.963113i \(0.586726\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 41.5692i 0.0510052i
\(816\) 0 0
\(817\) 1008.00 1.23378
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 838.000i 1.02071i 0.859965 + 0.510353i \(0.170485\pi\)
−0.859965 + 0.510353i \(0.829515\pi\)
\(822\) 0 0
\(823\) − 879.882i − 1.06912i −0.845132 0.534558i \(-0.820478\pi\)
0.845132 0.534558i \(-0.179522\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 727.461 0.879639 0.439819 0.898086i \(-0.355042\pi\)
0.439819 + 0.898086i \(0.355042\pi\)
\(828\) 0 0
\(829\) 1298.00i 1.56574i 0.622184 + 0.782871i \(0.286246\pi\)
−0.622184 + 0.782871i \(0.713754\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10.0000 −0.0120048
\(834\) 0 0
\(835\) 193.990 0.232323
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 193.990i 0.231215i 0.993295 + 0.115608i \(0.0368815\pi\)
−0.993295 + 0.115608i \(0.963118\pi\)
\(840\) 0 0
\(841\) 165.000 0.196195
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 330.000i − 0.390533i
\(846\) 0 0
\(847\) − 505.759i − 0.597118i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −720.533 −0.846690
\(852\) 0 0
\(853\) − 506.000i − 0.593200i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958529\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −998.000 −1.16453 −0.582264 0.813000i \(-0.697833\pi\)
−0.582264 + 0.813000i \(0.697833\pi\)
\(858\) 0 0
\(859\) 505.759 0.588776 0.294388 0.955686i \(-0.404884\pi\)
0.294388 + 0.955686i \(0.404884\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 166.277i − 0.192673i −0.995349 0.0963365i \(-0.969287\pi\)
0.995349 0.0963365i \(-0.0307125\pi\)
\(864\) 0 0
\(865\) −668.000 −0.772254
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 816.000i − 0.939010i
\(870\) 0 0
\(871\) − 13.8564i − 0.0159086i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 637.395 0.728451
\(876\) 0 0
\(877\) − 646.000i − 0.736602i −0.929707 0.368301i \(-0.879940\pi\)
0.929707 0.368301i \(-0.120060\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −898.000 −1.01930 −0.509648 0.860383i \(-0.670224\pi\)
−0.509648 + 0.860383i \(0.670224\pi\)
\(882\) 0 0
\(883\) 727.461 0.823852 0.411926 0.911217i \(-0.364856\pi\)
0.411926 + 0.911217i \(0.364856\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 845.241i 0.952921i 0.879196 + 0.476460i \(0.158080\pi\)
−0.879196 + 0.476460i \(0.841920\pi\)
\(888\) 0 0
\(889\) 1008.00 1.13386
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1440.00i 1.61254i
\(894\) 0 0
\(895\) 374.123i 0.418014i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −180.133 −0.200371
\(900\) 0 0
\(901\) 740.000i 0.821310i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 −0.00441989
\(906\) 0 0
\(907\) 1364.86 1.50480 0.752401 0.658705i \(-0.228895\pi\)
0.752401 + 0.658705i \(0.228895\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 387.979i 0.425883i 0.977065 + 0.212941i \(0.0683044\pi\)
−0.977065 + 0.212941i \(0.931696\pi\)
\(912\) 0 0
\(913\) −336.000 −0.368018
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 816.000i 0.889858i
\(918\) 0 0
\(919\) − 602.754i − 0.655880i −0.944699 0.327940i \(-0.893646\pi\)
0.944699 0.327940i \(-0.106354\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 546.000i − 0.590270i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1594.00 −1.71582 −0.857912 0.513797i \(-0.828238\pi\)
−0.857912 + 0.513797i \(0.828238\pi\)
\(930\) 0 0
\(931\) −20.7846 −0.0223250
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 138.564i − 0.148197i
\(936\) 0 0
\(937\) −674.000 −0.719317 −0.359658 0.933084i \(-0.617107\pi\)
−0.359658 + 0.933084i \(0.617107\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 430.000i − 0.456961i −0.973549 0.228480i \(-0.926624\pi\)
0.973549 0.228480i \(-0.0733757\pi\)
\(942\) 0 0
\(943\) − 1607.34i − 1.70450i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −76.2102 −0.0804754 −0.0402377 0.999190i \(-0.512812\pi\)
−0.0402377 + 0.999190i \(0.512812\pi\)
\(948\) 0 0
\(949\) 92.0000i 0.0969442i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 730.000 0.766002 0.383001 0.923748i \(-0.374891\pi\)
0.383001 + 0.923748i \(0.374891\pi\)
\(954\) 0 0
\(955\) 443.405 0.464298
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 69.2820i 0.0722440i
\(960\) 0 0
\(961\) 913.000 0.950052
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 580.000i − 0.601036i
\(966\) 0 0
\(967\) − 921.451i − 0.952897i −0.879202 0.476448i \(-0.841924\pi\)
0.879202 0.476448i \(-0.158076\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1475.71 −1.51978 −0.759890 0.650051i \(-0.774747\pi\)
−0.759890 + 0.650051i \(0.774747\pi\)
\(972\) 0 0
\(973\) 336.000i 0.345324i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −346.000 −0.354145 −0.177073 0.984198i \(-0.556663\pi\)
−0.177073 + 0.984198i \(0.556663\pi\)
\(978\) 0 0
\(979\) −568.113 −0.580299
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 734.390i 0.747090i 0.927612 + 0.373545i \(0.121858\pi\)
−0.927612 + 0.373545i \(0.878142\pi\)
\(984\) 0 0
\(985\) −52.0000 −0.0527919
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1344.00i 1.35895i
\(990\) 0 0
\(991\) 976.877i 0.985748i 0.870101 + 0.492874i \(0.164054\pi\)
−0.870101 + 0.492874i \(0.835946\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −789.815 −0.793784
\(996\) 0 0
\(997\) − 458.000i − 0.459378i −0.973264 0.229689i \(-0.926229\pi\)
0.973264 0.229689i \(-0.0737709\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.3.b.l.127.2 4
3.2 odd 2 768.3.b.c.127.2 4
4.3 odd 2 inner 2304.3.b.l.127.1 4
8.3 odd 2 inner 2304.3.b.l.127.3 4
8.5 even 2 inner 2304.3.b.l.127.4 4
12.11 even 2 768.3.b.c.127.4 4
16.3 odd 4 576.3.g.e.127.2 2
16.5 even 4 36.3.d.c.19.2 2
16.11 odd 4 36.3.d.c.19.1 2
16.13 even 4 576.3.g.e.127.1 2
24.5 odd 2 768.3.b.c.127.3 4
24.11 even 2 768.3.b.c.127.1 4
48.5 odd 4 12.3.d.a.7.1 2
48.11 even 4 12.3.d.a.7.2 yes 2
48.29 odd 4 192.3.g.b.127.1 2
48.35 even 4 192.3.g.b.127.2 2
80.27 even 4 900.3.f.c.199.4 4
80.37 odd 4 900.3.f.c.199.2 4
80.43 even 4 900.3.f.c.199.1 4
80.53 odd 4 900.3.f.c.199.3 4
80.59 odd 4 900.3.c.e.451.2 2
80.69 even 4 900.3.c.e.451.1 2
144.5 odd 12 324.3.f.j.55.1 2
144.11 even 12 324.3.f.j.271.1 2
144.43 odd 12 324.3.f.a.271.1 2
144.59 even 12 324.3.f.d.55.1 2
144.85 even 12 324.3.f.a.55.1 2
144.101 odd 12 324.3.f.d.271.1 2
144.133 even 12 324.3.f.g.271.1 2
144.139 odd 12 324.3.f.g.55.1 2
240.53 even 4 300.3.f.a.199.2 4
240.59 even 4 300.3.c.b.151.1 2
240.107 odd 4 300.3.f.a.199.1 4
240.149 odd 4 300.3.c.b.151.2 2
240.197 even 4 300.3.f.a.199.3 4
240.203 odd 4 300.3.f.a.199.4 4
336.251 odd 4 588.3.g.b.295.2 2
336.293 even 4 588.3.g.b.295.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.d.a.7.1 2 48.5 odd 4
12.3.d.a.7.2 yes 2 48.11 even 4
36.3.d.c.19.1 2 16.11 odd 4
36.3.d.c.19.2 2 16.5 even 4
192.3.g.b.127.1 2 48.29 odd 4
192.3.g.b.127.2 2 48.35 even 4
300.3.c.b.151.1 2 240.59 even 4
300.3.c.b.151.2 2 240.149 odd 4
300.3.f.a.199.1 4 240.107 odd 4
300.3.f.a.199.2 4 240.53 even 4
300.3.f.a.199.3 4 240.197 even 4
300.3.f.a.199.4 4 240.203 odd 4
324.3.f.a.55.1 2 144.85 even 12
324.3.f.a.271.1 2 144.43 odd 12
324.3.f.d.55.1 2 144.59 even 12
324.3.f.d.271.1 2 144.101 odd 12
324.3.f.g.55.1 2 144.139 odd 12
324.3.f.g.271.1 2 144.133 even 12
324.3.f.j.55.1 2 144.5 odd 12
324.3.f.j.271.1 2 144.11 even 12
576.3.g.e.127.1 2 16.13 even 4
576.3.g.e.127.2 2 16.3 odd 4
588.3.g.b.295.1 2 336.293 even 4
588.3.g.b.295.2 2 336.251 odd 4
768.3.b.c.127.1 4 24.11 even 2
768.3.b.c.127.2 4 3.2 odd 2
768.3.b.c.127.3 4 24.5 odd 2
768.3.b.c.127.4 4 12.11 even 2
900.3.c.e.451.1 2 80.69 even 4
900.3.c.e.451.2 2 80.59 odd 4
900.3.f.c.199.1 4 80.43 even 4
900.3.f.c.199.2 4 80.37 odd 4
900.3.f.c.199.3 4 80.53 odd 4
900.3.f.c.199.4 4 80.27 even 4
2304.3.b.l.127.1 4 4.3 odd 2 inner
2304.3.b.l.127.2 4 1.1 even 1 trivial
2304.3.b.l.127.3 4 8.3 odd 2 inner
2304.3.b.l.127.4 4 8.5 even 2 inner