Properties

Label 2352.4.a.bq.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1345}) \)
Defining polynomial: \(x^{2} - x - 336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(18.8371\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -15.8371 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -15.8371 q^{5} +9.00000 q^{9} -51.8371 q^{11} +38.8371 q^{13} +47.5114 q^{15} +27.3485 q^{17} -76.5114 q^{19} -147.348 q^{23} +125.814 q^{25} -27.0000 q^{27} +240.208 q^{29} +296.674 q^{31} +155.511 q^{33} -161.534 q^{37} -116.511 q^{39} -102.977 q^{41} +328.557 q^{43} -142.534 q^{45} -67.9546 q^{47} -82.0454 q^{51} -66.4886 q^{53} +820.951 q^{55} +229.534 q^{57} +461.928 q^{59} +185.348 q^{61} -615.068 q^{65} -545.208 q^{67} +442.045 q^{69} +130.742 q^{71} +181.299 q^{73} -377.443 q^{75} +409.697 q^{79} +81.0000 q^{81} -347.928 q^{83} -433.121 q^{85} -720.625 q^{87} +1157.16 q^{89} -890.023 q^{93} +1211.72 q^{95} +1618.30 q^{97} -466.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 5q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 5q^{5} + 18q^{9} - 67q^{11} + 41q^{13} - 15q^{15} - 92q^{17} - 43q^{19} - 148q^{23} + 435q^{25} - 54q^{27} + 77q^{29} + 520q^{31} + 201q^{33} + 7q^{37} - 123q^{39} - 426q^{41} + 107q^{43} + 45q^{45} - 576q^{47} + 276q^{51} - 243q^{53} + 505q^{55} + 129q^{57} + 7q^{59} + 224q^{61} - 570q^{65} - 687q^{67} + 444q^{69} - 472q^{71} - 921q^{73} - 1305q^{75} + 526q^{79} + 162q^{81} + 221q^{83} - 2920q^{85} - 231q^{87} + 774q^{89} - 1560q^{93} + 1910q^{95} + 1953q^{97} - 603q^{99} + O(q^{100}) \)

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
\(5\) −15.8371 −1.41652 −0.708258 0.705954i \(-0.750518\pi\)
−0.708258 + 0.705954i \(0.750518\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −51.8371 −1.42086 −0.710431 0.703767i \(-0.751500\pi\)
−0.710431 + 0.703767i \(0.751500\pi\)
\(12\) 0 0
\(13\) 38.8371 0.828575 0.414288 0.910146i \(-0.364031\pi\)
0.414288 + 0.910146i \(0.364031\pi\)
\(14\) 0 0
\(15\) 47.5114 0.817825
\(16\) 0 0
\(17\) 27.3485 0.390175 0.195088 0.980786i \(-0.437501\pi\)
0.195088 + 0.980786i \(0.437501\pi\)
\(18\) 0 0
\(19\) −76.5114 −0.923837 −0.461919 0.886922i \(-0.652839\pi\)
−0.461919 + 0.886922i \(0.652839\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −147.348 −1.33584 −0.667919 0.744234i \(-0.732815\pi\)
−0.667919 + 0.744234i \(0.732815\pi\)
\(24\) 0 0
\(25\) 125.814 1.00652
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 240.208 1.53812 0.769061 0.639175i \(-0.220724\pi\)
0.769061 + 0.639175i \(0.220724\pi\)
\(30\) 0 0
\(31\) 296.674 1.71885 0.859424 0.511264i \(-0.170823\pi\)
0.859424 + 0.511264i \(0.170823\pi\)
\(32\) 0 0
\(33\) 155.511 0.820335
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −161.534 −0.717731 −0.358865 0.933389i \(-0.616836\pi\)
−0.358865 + 0.933389i \(0.616836\pi\)
\(38\) 0 0
\(39\) −116.511 −0.478378
\(40\) 0 0
\(41\) −102.977 −0.392252 −0.196126 0.980579i \(-0.562836\pi\)
−0.196126 + 0.980579i \(0.562836\pi\)
\(42\) 0 0
\(43\) 328.557 1.16522 0.582610 0.812752i \(-0.302032\pi\)
0.582610 + 0.812752i \(0.302032\pi\)
\(44\) 0 0
\(45\) −142.534 −0.472172
\(46\) 0 0
\(47\) −67.9546 −0.210898 −0.105449 0.994425i \(-0.533628\pi\)
−0.105449 + 0.994425i \(0.533628\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −82.0454 −0.225268
\(52\) 0 0
\(53\) −66.4886 −0.172319 −0.0861596 0.996281i \(-0.527459\pi\)
−0.0861596 + 0.996281i \(0.527459\pi\)
\(54\) 0 0
\(55\) 820.951 2.01267
\(56\) 0 0
\(57\) 229.534 0.533378
\(58\) 0 0
\(59\) 461.928 1.01929 0.509643 0.860386i \(-0.329777\pi\)
0.509643 + 0.860386i \(0.329777\pi\)
\(60\) 0 0
\(61\) 185.348 0.389040 0.194520 0.980899i \(-0.437685\pi\)
0.194520 + 0.980899i \(0.437685\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −615.068 −1.17369
\(66\) 0 0
\(67\) −545.208 −0.994146 −0.497073 0.867709i \(-0.665592\pi\)
−0.497073 + 0.867709i \(0.665592\pi\)
\(68\) 0 0
\(69\) 442.045 0.771247
\(70\) 0 0
\(71\) 130.742 0.218539 0.109270 0.994012i \(-0.465149\pi\)
0.109270 + 0.994012i \(0.465149\pi\)
\(72\) 0 0
\(73\) 181.299 0.290678 0.145339 0.989382i \(-0.453573\pi\)
0.145339 + 0.989382i \(0.453573\pi\)
\(74\) 0 0
\(75\) −377.443 −0.581112
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 409.697 0.583475 0.291737 0.956498i \(-0.405767\pi\)
0.291737 + 0.956498i \(0.405767\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −347.928 −0.460121 −0.230061 0.973176i \(-0.573892\pi\)
−0.230061 + 0.973176i \(0.573892\pi\)
\(84\) 0 0
\(85\) −433.121 −0.552689
\(86\) 0 0
\(87\) −720.625 −0.888036
\(88\) 0 0
\(89\) 1157.16 1.37819 0.689093 0.724673i \(-0.258009\pi\)
0.689093 + 0.724673i \(0.258009\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −890.023 −0.992377
\(94\) 0 0
\(95\) 1211.72 1.30863
\(96\) 0 0
\(97\) 1618.30 1.69395 0.846976 0.531631i \(-0.178421\pi\)
0.846976 + 0.531631i \(0.178421\pi\)
\(98\) 0 0
\(99\) −466.534 −0.473621
\(100\) 0 0
\(101\) 718.742 0.708094 0.354047 0.935228i \(-0.384805\pi\)
0.354047 + 0.935228i \(0.384805\pi\)
\(102\) 0 0
\(103\) 1611.58 1.54169 0.770843 0.637025i \(-0.219835\pi\)
0.770843 + 0.637025i \(0.219835\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −934.670 −0.844467 −0.422234 0.906487i \(-0.638754\pi\)
−0.422234 + 0.906487i \(0.638754\pi\)
\(108\) 0 0
\(109\) −1197.02 −1.05187 −0.525934 0.850525i \(-0.676284\pi\)
−0.525934 + 0.850525i \(0.676284\pi\)
\(110\) 0 0
\(111\) 484.602 0.414382
\(112\) 0 0
\(113\) −2384.64 −1.98521 −0.992604 0.121400i \(-0.961262\pi\)
−0.992604 + 0.121400i \(0.961262\pi\)
\(114\) 0 0
\(115\) 2333.58 1.89224
\(116\) 0 0
\(117\) 349.534 0.276192
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1356.09 1.01885
\(122\) 0 0
\(123\) 308.932 0.226467
\(124\) 0 0
\(125\) −12.8977 −0.00922883
\(126\) 0 0
\(127\) 2673.92 1.86829 0.934143 0.356898i \(-0.116166\pi\)
0.934143 + 0.356898i \(0.116166\pi\)
\(128\) 0 0
\(129\) −985.670 −0.672740
\(130\) 0 0
\(131\) 38.8598 0.0259176 0.0129588 0.999916i \(-0.495875\pi\)
0.0129588 + 0.999916i \(0.495875\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 427.602 0.272608
\(136\) 0 0
\(137\) −768.144 −0.479029 −0.239514 0.970893i \(-0.576988\pi\)
−0.239514 + 0.970893i \(0.576988\pi\)
\(138\) 0 0
\(139\) −1052.55 −0.642274 −0.321137 0.947033i \(-0.604065\pi\)
−0.321137 + 0.947033i \(0.604065\pi\)
\(140\) 0 0
\(141\) 203.864 0.121762
\(142\) 0 0
\(143\) −2013.20 −1.17729
\(144\) 0 0
\(145\) −3804.21 −2.17877
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 360.977 0.198473 0.0992363 0.995064i \(-0.468360\pi\)
0.0992363 + 0.995064i \(0.468360\pi\)
\(150\) 0 0
\(151\) −1548.39 −0.834478 −0.417239 0.908797i \(-0.637002\pi\)
−0.417239 + 0.908797i \(0.637002\pi\)
\(152\) 0 0
\(153\) 246.136 0.130058
\(154\) 0 0
\(155\) −4698.47 −2.43477
\(156\) 0 0
\(157\) −967.068 −0.491595 −0.245798 0.969321i \(-0.579050\pi\)
−0.245798 + 0.969321i \(0.579050\pi\)
\(158\) 0 0
\(159\) 199.466 0.0994885
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1326.50 0.637420 0.318710 0.947852i \(-0.396750\pi\)
0.318710 + 0.947852i \(0.396750\pi\)
\(164\) 0 0
\(165\) −2462.85 −1.16202
\(166\) 0 0
\(167\) −1416.70 −0.656451 −0.328225 0.944599i \(-0.606451\pi\)
−0.328225 + 0.944599i \(0.606451\pi\)
\(168\) 0 0
\(169\) −688.678 −0.313463
\(170\) 0 0
\(171\) −688.602 −0.307946
\(172\) 0 0
\(173\) −1036.60 −0.455556 −0.227778 0.973713i \(-0.573146\pi\)
−0.227778 + 0.973713i \(0.573146\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1385.78 −0.588485
\(178\) 0 0
\(179\) 767.432 0.320450 0.160225 0.987081i \(-0.448778\pi\)
0.160225 + 0.987081i \(0.448778\pi\)
\(180\) 0 0
\(181\) −3957.71 −1.62527 −0.812636 0.582772i \(-0.801968\pi\)
−0.812636 + 0.582772i \(0.801968\pi\)
\(182\) 0 0
\(183\) −556.045 −0.224612
\(184\) 0 0
\(185\) 2558.23 1.01668
\(186\) 0 0
\(187\) −1417.67 −0.554385
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1805.30 0.683909 0.341954 0.939717i \(-0.388911\pi\)
0.341954 + 0.939717i \(0.388911\pi\)
\(192\) 0 0
\(193\) 3370.84 1.25719 0.628597 0.777731i \(-0.283630\pi\)
0.628597 + 0.777731i \(0.283630\pi\)
\(194\) 0 0
\(195\) 1845.20 0.677630
\(196\) 0 0
\(197\) −4612.31 −1.66809 −0.834044 0.551697i \(-0.813980\pi\)
−0.834044 + 0.551697i \(0.813980\pi\)
\(198\) 0 0
\(199\) 2229.86 0.794326 0.397163 0.917748i \(-0.369995\pi\)
0.397163 + 0.917748i \(0.369995\pi\)
\(200\) 0 0
\(201\) 1635.62 0.573971
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1630.86 0.555631
\(206\) 0 0
\(207\) −1326.14 −0.445279
\(208\) 0 0
\(209\) 3966.13 1.31265
\(210\) 0 0
\(211\) −912.614 −0.297758 −0.148879 0.988855i \(-0.547566\pi\)
−0.148879 + 0.988855i \(0.547566\pi\)
\(212\) 0 0
\(213\) −392.227 −0.126174
\(214\) 0 0
\(215\) −5203.39 −1.65055
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −543.898 −0.167823
\(220\) 0 0
\(221\) 1062.14 0.323290
\(222\) 0 0
\(223\) 4319.47 1.29710 0.648549 0.761173i \(-0.275376\pi\)
0.648549 + 0.761173i \(0.275376\pi\)
\(224\) 0 0
\(225\) 1132.33 0.335505
\(226\) 0 0
\(227\) −2061.28 −0.602697 −0.301349 0.953514i \(-0.597437\pi\)
−0.301349 + 0.953514i \(0.597437\pi\)
\(228\) 0 0
\(229\) −3474.63 −1.00266 −0.501332 0.865255i \(-0.667157\pi\)
−0.501332 + 0.865255i \(0.667157\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 776.099 0.218214 0.109107 0.994030i \(-0.465201\pi\)
0.109107 + 0.994030i \(0.465201\pi\)
\(234\) 0 0
\(235\) 1076.20 0.298740
\(236\) 0 0
\(237\) −1229.09 −0.336869
\(238\) 0 0
\(239\) −2006.80 −0.543133 −0.271567 0.962420i \(-0.587542\pi\)
−0.271567 + 0.962420i \(0.587542\pi\)
\(240\) 0 0
\(241\) 805.648 0.215337 0.107669 0.994187i \(-0.465661\pi\)
0.107669 + 0.994187i \(0.465661\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2971.48 −0.765469
\(248\) 0 0
\(249\) 1043.78 0.265651
\(250\) 0 0
\(251\) −1421.78 −0.357539 −0.178769 0.983891i \(-0.557212\pi\)
−0.178769 + 0.983891i \(0.557212\pi\)
\(252\) 0 0
\(253\) 7638.12 1.89804
\(254\) 0 0
\(255\) 1299.36 0.319095
\(256\) 0 0
\(257\) −1465.82 −0.355779 −0.177890 0.984050i \(-0.556927\pi\)
−0.177890 + 0.984050i \(0.556927\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2161.87 0.512708
\(262\) 0 0
\(263\) −6991.39 −1.63919 −0.819596 0.572942i \(-0.805802\pi\)
−0.819596 + 0.572942i \(0.805802\pi\)
\(264\) 0 0
\(265\) 1052.99 0.244093
\(266\) 0 0
\(267\) −3471.48 −0.795696
\(268\) 0 0
\(269\) 808.958 0.183357 0.0916786 0.995789i \(-0.470777\pi\)
0.0916786 + 0.995789i \(0.470777\pi\)
\(270\) 0 0
\(271\) −6661.77 −1.49326 −0.746630 0.665239i \(-0.768330\pi\)
−0.746630 + 0.665239i \(0.768330\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6521.86 −1.43012
\(276\) 0 0
\(277\) −7531.47 −1.63365 −0.816827 0.576883i \(-0.804269\pi\)
−0.816827 + 0.576883i \(0.804269\pi\)
\(278\) 0 0
\(279\) 2670.07 0.572949
\(280\) 0 0
\(281\) 1690.19 0.358819 0.179410 0.983774i \(-0.442581\pi\)
0.179410 + 0.983774i \(0.442581\pi\)
\(282\) 0 0
\(283\) −3178.23 −0.667584 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(284\) 0 0
\(285\) −3635.16 −0.755538
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4165.06 −0.847763
\(290\) 0 0
\(291\) −4854.90 −0.978004
\(292\) 0 0
\(293\) −2176.53 −0.433974 −0.216987 0.976174i \(-0.569623\pi\)
−0.216987 + 0.976174i \(0.569623\pi\)
\(294\) 0 0
\(295\) −7315.61 −1.44383
\(296\) 0 0
\(297\) 1399.60 0.273445
\(298\) 0 0
\(299\) −5722.59 −1.10684
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2156.23 −0.408819
\(304\) 0 0
\(305\) −2935.39 −0.551081
\(306\) 0 0
\(307\) −623.504 −0.115913 −0.0579564 0.998319i \(-0.518458\pi\)
−0.0579564 + 0.998319i \(0.518458\pi\)
\(308\) 0 0
\(309\) −4834.74 −0.890093
\(310\) 0 0
\(311\) −467.992 −0.0853293 −0.0426647 0.999089i \(-0.513585\pi\)
−0.0426647 + 0.999089i \(0.513585\pi\)
\(312\) 0 0
\(313\) 3612.81 0.652422 0.326211 0.945297i \(-0.394228\pi\)
0.326211 + 0.945297i \(0.394228\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4531.74 0.802927 0.401463 0.915875i \(-0.368502\pi\)
0.401463 + 0.915875i \(0.368502\pi\)
\(318\) 0 0
\(319\) −12451.7 −2.18546
\(320\) 0 0
\(321\) 2804.01 0.487553
\(322\) 0 0
\(323\) −2092.47 −0.360459
\(324\) 0 0
\(325\) 4886.27 0.833974
\(326\) 0 0
\(327\) 3591.06 0.607296
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1237.06 0.205422 0.102711 0.994711i \(-0.467248\pi\)
0.102711 + 0.994711i \(0.467248\pi\)
\(332\) 0 0
\(333\) −1453.81 −0.239244
\(334\) 0 0
\(335\) 8634.53 1.40822
\(336\) 0 0
\(337\) −1867.83 −0.301921 −0.150960 0.988540i \(-0.548237\pi\)
−0.150960 + 0.988540i \(0.548237\pi\)
\(338\) 0 0
\(339\) 7153.93 1.14616
\(340\) 0 0
\(341\) −15378.7 −2.44224
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −7000.73 −1.09248
\(346\) 0 0
\(347\) 63.3637 0.00980271 0.00490136 0.999988i \(-0.498440\pi\)
0.00490136 + 0.999988i \(0.498440\pi\)
\(348\) 0 0
\(349\) 1223.79 0.187702 0.0938508 0.995586i \(-0.470082\pi\)
0.0938508 + 0.995586i \(0.470082\pi\)
\(350\) 0 0
\(351\) −1048.60 −0.159459
\(352\) 0 0
\(353\) −4515.61 −0.680855 −0.340428 0.940271i \(-0.610572\pi\)
−0.340428 + 0.940271i \(0.610572\pi\)
\(354\) 0 0
\(355\) −2070.58 −0.309564
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2228.49 −0.327619 −0.163810 0.986492i \(-0.552378\pi\)
−0.163810 + 0.986492i \(0.552378\pi\)
\(360\) 0 0
\(361\) −1005.01 −0.146524
\(362\) 0 0
\(363\) −4068.26 −0.588232
\(364\) 0 0
\(365\) −2871.26 −0.411749
\(366\) 0 0
\(367\) 1437.34 0.204438 0.102219 0.994762i \(-0.467406\pi\)
0.102219 + 0.994762i \(0.467406\pi\)
\(368\) 0 0
\(369\) −926.795 −0.130751
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12237.4 −1.69874 −0.849370 0.527798i \(-0.823018\pi\)
−0.849370 + 0.527798i \(0.823018\pi\)
\(374\) 0 0
\(375\) 38.6931 0.00532827
\(376\) 0 0
\(377\) 9329.00 1.27445
\(378\) 0 0
\(379\) 10647.0 1.44301 0.721503 0.692411i \(-0.243451\pi\)
0.721503 + 0.692411i \(0.243451\pi\)
\(380\) 0 0
\(381\) −8021.77 −1.07866
\(382\) 0 0
\(383\) 6714.81 0.895851 0.447925 0.894071i \(-0.352163\pi\)
0.447925 + 0.894071i \(0.352163\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2957.01 0.388407
\(388\) 0 0
\(389\) −10653.1 −1.38852 −0.694258 0.719727i \(-0.744267\pi\)
−0.694258 + 0.719727i \(0.744267\pi\)
\(390\) 0 0
\(391\) −4029.76 −0.521211
\(392\) 0 0
\(393\) −116.580 −0.0149635
\(394\) 0 0
\(395\) −6488.42 −0.826501
\(396\) 0 0
\(397\) 3221.04 0.407203 0.203601 0.979054i \(-0.434735\pi\)
0.203601 + 0.979054i \(0.434735\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12485.0 1.55479 0.777395 0.629012i \(-0.216540\pi\)
0.777395 + 0.629012i \(0.216540\pi\)
\(402\) 0 0
\(403\) 11522.0 1.42419
\(404\) 0 0
\(405\) −1282.81 −0.157391
\(406\) 0 0
\(407\) 8373.46 1.01980
\(408\) 0 0
\(409\) 7037.39 0.850798 0.425399 0.905006i \(-0.360134\pi\)
0.425399 + 0.905006i \(0.360134\pi\)
\(410\) 0 0
\(411\) 2304.43 0.276567
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 5510.18 0.651769
\(416\) 0 0
\(417\) 3157.65 0.370817
\(418\) 0 0
\(419\) 1549.66 0.180682 0.0903410 0.995911i \(-0.471204\pi\)
0.0903410 + 0.995911i \(0.471204\pi\)
\(420\) 0 0
\(421\) 5531.63 0.640369 0.320184 0.947355i \(-0.396255\pi\)
0.320184 + 0.947355i \(0.396255\pi\)
\(422\) 0 0
\(423\) −611.591 −0.0702992
\(424\) 0 0
\(425\) 3440.83 0.392717
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6039.61 0.679709
\(430\) 0 0
\(431\) 2029.93 0.226864 0.113432 0.993546i \(-0.463816\pi\)
0.113432 + 0.993546i \(0.463816\pi\)
\(432\) 0 0
\(433\) −327.739 −0.0363744 −0.0181872 0.999835i \(-0.505789\pi\)
−0.0181872 + 0.999835i \(0.505789\pi\)
\(434\) 0 0
\(435\) 11412.6 1.25792
\(436\) 0 0
\(437\) 11273.8 1.23410
\(438\) 0 0
\(439\) −7908.68 −0.859819 −0.429910 0.902872i \(-0.641455\pi\)
−0.429910 + 0.902872i \(0.641455\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2920.82 0.313256 0.156628 0.987658i \(-0.449938\pi\)
0.156628 + 0.987658i \(0.449938\pi\)
\(444\) 0 0
\(445\) −18326.1 −1.95222
\(446\) 0 0
\(447\) −1082.93 −0.114588
\(448\) 0 0
\(449\) −10240.2 −1.07631 −0.538156 0.842845i \(-0.680879\pi\)
−0.538156 + 0.842845i \(0.680879\pi\)
\(450\) 0 0
\(451\) 5338.05 0.557336
\(452\) 0 0
\(453\) 4645.17 0.481786
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5892.63 0.603163 0.301582 0.953440i \(-0.402485\pi\)
0.301582 + 0.953440i \(0.402485\pi\)
\(458\) 0 0
\(459\) −738.409 −0.0750893
\(460\) 0 0
\(461\) −12643.4 −1.27735 −0.638677 0.769475i \(-0.720518\pi\)
−0.638677 + 0.769475i \(0.720518\pi\)
\(462\) 0 0
\(463\) −15093.2 −1.51499 −0.757494 0.652842i \(-0.773577\pi\)
−0.757494 + 0.652842i \(0.773577\pi\)
\(464\) 0 0
\(465\) 14095.4 1.40572
\(466\) 0 0
\(467\) 2820.23 0.279454 0.139727 0.990190i \(-0.455378\pi\)
0.139727 + 0.990190i \(0.455378\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2901.20 0.283823
\(472\) 0 0
\(473\) −17031.4 −1.65562
\(474\) 0 0
\(475\) −9626.23 −0.929856
\(476\) 0 0
\(477\) −598.398 −0.0574397
\(478\) 0 0
\(479\) 16448.5 1.56900 0.784500 0.620129i \(-0.212920\pi\)
0.784500 + 0.620129i \(0.212920\pi\)
\(480\) 0 0
\(481\) −6273.52 −0.594694
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25629.2 −2.39951
\(486\) 0 0
\(487\) −6331.07 −0.589093 −0.294546 0.955637i \(-0.595169\pi\)
−0.294546 + 0.955637i \(0.595169\pi\)
\(488\) 0 0
\(489\) −3979.50 −0.368015
\(490\) 0 0
\(491\) 9286.90 0.853588 0.426794 0.904349i \(-0.359643\pi\)
0.426794 + 0.904349i \(0.359643\pi\)
\(492\) 0 0
\(493\) 6569.33 0.600138
\(494\) 0 0
\(495\) 7388.56 0.670891
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 243.451 0.0218404 0.0109202 0.999940i \(-0.496524\pi\)
0.0109202 + 0.999940i \(0.496524\pi\)
\(500\) 0 0
\(501\) 4250.09 0.379002
\(502\) 0 0
\(503\) −8499.30 −0.753409 −0.376705 0.926333i \(-0.622943\pi\)
−0.376705 + 0.926333i \(0.622943\pi\)
\(504\) 0 0
\(505\) −11382.8 −1.00303
\(506\) 0 0
\(507\) 2066.03 0.180978
\(508\) 0 0
\(509\) −7683.10 −0.669052 −0.334526 0.942387i \(-0.608576\pi\)
−0.334526 + 0.942387i \(0.608576\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2065.81 0.177793
\(514\) 0 0
\(515\) −25522.8 −2.18382
\(516\) 0 0
\(517\) 3522.57 0.299656
\(518\) 0 0
\(519\) 3109.80 0.263015
\(520\) 0 0
\(521\) −21530.6 −1.81051 −0.905253 0.424874i \(-0.860319\pi\)
−0.905253 + 0.424874i \(0.860319\pi\)
\(522\) 0 0
\(523\) 16847.1 1.40855 0.704274 0.709929i \(-0.251273\pi\)
0.704274 + 0.709929i \(0.251273\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8113.59 0.670652
\(528\) 0 0
\(529\) 9544.58 0.784464
\(530\) 0 0
\(531\) 4157.35 0.339762
\(532\) 0 0
\(533\) −3999.34 −0.325011
\(534\) 0 0
\(535\) 14802.5 1.19620
\(536\) 0 0
\(537\) −2302.30 −0.185012
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17440.0 1.38596 0.692981 0.720955i \(-0.256297\pi\)
0.692981 + 0.720955i \(0.256297\pi\)
\(542\) 0 0
\(543\) 11873.1 0.938351
\(544\) 0 0
\(545\) 18957.3 1.48999
\(546\) 0 0
\(547\) −11520.7 −0.900530 −0.450265 0.892895i \(-0.648671\pi\)
−0.450265 + 0.892895i \(0.648671\pi\)
\(548\) 0 0
\(549\) 1668.14 0.129680
\(550\) 0 0
\(551\) −18378.7 −1.42098
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −7674.70 −0.586978
\(556\) 0 0
\(557\) 11493.0 0.874282 0.437141 0.899393i \(-0.355991\pi\)
0.437141 + 0.899393i \(0.355991\pi\)
\(558\) 0 0
\(559\) 12760.2 0.965472
\(560\) 0 0
\(561\) 4253.00 0.320075
\(562\) 0 0
\(563\) −18111.3 −1.35577 −0.677886 0.735167i \(-0.737104\pi\)
−0.677886 + 0.735167i \(0.737104\pi\)
\(564\) 0 0
\(565\) 37765.9 2.81208
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4417.61 0.325476 0.162738 0.986669i \(-0.447967\pi\)
0.162738 + 0.986669i \(0.447967\pi\)
\(570\) 0 0
\(571\) −13219.7 −0.968878 −0.484439 0.874825i \(-0.660976\pi\)
−0.484439 + 0.874825i \(0.660976\pi\)
\(572\) 0 0
\(573\) −5415.89 −0.394855
\(574\) 0 0
\(575\) −18538.6 −1.34454
\(576\) 0 0
\(577\) −17496.4 −1.26236 −0.631182 0.775634i \(-0.717430\pi\)
−0.631182 + 0.775634i \(0.717430\pi\)
\(578\) 0 0
\(579\) −10112.5 −0.725841
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3446.58 0.244842
\(584\) 0 0
\(585\) −5535.61 −0.391230
\(586\) 0 0
\(587\) 4280.53 0.300982 0.150491 0.988611i \(-0.451915\pi\)
0.150491 + 0.988611i \(0.451915\pi\)
\(588\) 0 0
\(589\) −22699.0 −1.58794
\(590\) 0 0
\(591\) 13836.9 0.963072
\(592\) 0 0
\(593\) −1590.93 −0.110172 −0.0550858 0.998482i \(-0.517543\pi\)
−0.0550858 + 0.998482i \(0.517543\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6689.59 −0.458604
\(598\) 0 0
\(599\) 13922.8 0.949703 0.474851 0.880066i \(-0.342502\pi\)
0.474851 + 0.880066i \(0.342502\pi\)
\(600\) 0 0
\(601\) 12559.7 0.852446 0.426223 0.904618i \(-0.359844\pi\)
0.426223 + 0.904618i \(0.359844\pi\)
\(602\) 0 0
\(603\) −4906.87 −0.331382
\(604\) 0 0
\(605\) −21476.5 −1.44321
\(606\) 0 0
\(607\) −7678.37 −0.513436 −0.256718 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2639.16 −0.174745
\(612\) 0 0
\(613\) 6158.37 0.405766 0.202883 0.979203i \(-0.434969\pi\)
0.202883 + 0.979203i \(0.434969\pi\)
\(614\) 0 0
\(615\) −4892.59 −0.320794
\(616\) 0 0
\(617\) 8813.12 0.575045 0.287523 0.957774i \(-0.407168\pi\)
0.287523 + 0.957774i \(0.407168\pi\)
\(618\) 0 0
\(619\) −23189.9 −1.50579 −0.752894 0.658142i \(-0.771342\pi\)
−0.752894 + 0.658142i \(0.771342\pi\)
\(620\) 0 0
\(621\) 3978.41 0.257082
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −15522.5 −0.993442
\(626\) 0 0
\(627\) −11898.4 −0.757856
\(628\) 0 0
\(629\) −4417.71 −0.280041
\(630\) 0 0
\(631\) −7936.94 −0.500736 −0.250368 0.968151i \(-0.580552\pi\)
−0.250368 + 0.968151i \(0.580552\pi\)
\(632\) 0 0
\(633\) 2737.84 0.171911
\(634\) 0 0
\(635\) −42347.3 −2.64646
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1176.68 0.0728463
\(640\) 0 0
\(641\) 32114.6 1.97886 0.989432 0.144996i \(-0.0463169\pi\)
0.989432 + 0.144996i \(0.0463169\pi\)
\(642\) 0 0
\(643\) 24786.7 1.52021 0.760104 0.649802i \(-0.225148\pi\)
0.760104 + 0.649802i \(0.225148\pi\)
\(644\) 0 0
\(645\) 15610.2 0.952946
\(646\) 0 0
\(647\) −7545.59 −0.458497 −0.229249 0.973368i \(-0.573627\pi\)
−0.229249 + 0.973368i \(0.573627\pi\)
\(648\) 0 0
\(649\) −23945.0 −1.44827
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4888.99 −0.292988 −0.146494 0.989212i \(-0.546799\pi\)
−0.146494 + 0.989212i \(0.546799\pi\)
\(654\) 0 0
\(655\) −615.428 −0.0367126
\(656\) 0 0
\(657\) 1631.69 0.0968926
\(658\) 0 0
\(659\) −25895.9 −1.53075 −0.765374 0.643586i \(-0.777446\pi\)
−0.765374 + 0.643586i \(0.777446\pi\)
\(660\) 0 0
\(661\) 8183.37 0.481537 0.240769 0.970583i \(-0.422601\pi\)
0.240769 + 0.970583i \(0.422601\pi\)
\(662\) 0 0
\(663\) −3186.41 −0.186651
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −35394.3 −2.05468
\(668\) 0 0
\(669\) −12958.4 −0.748880
\(670\) 0 0
\(671\) −9607.93 −0.552772
\(672\) 0 0
\(673\) −4635.02 −0.265478 −0.132739 0.991151i \(-0.542377\pi\)
−0.132739 + 0.991151i \(0.542377\pi\)
\(674\) 0 0
\(675\) −3396.99 −0.193704
\(676\) 0 0
\(677\) −24385.8 −1.38437 −0.692187 0.721718i \(-0.743353\pi\)
−0.692187 + 0.721718i \(0.743353\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6183.85 0.347967
\(682\) 0 0
\(683\) 18393.4 1.03046 0.515230 0.857052i \(-0.327706\pi\)
0.515230 + 0.857052i \(0.327706\pi\)
\(684\) 0 0
\(685\) 12165.2 0.678552
\(686\) 0 0
\(687\) 10423.9 0.578889
\(688\) 0 0
\(689\) −2582.23 −0.142779
\(690\) 0 0
\(691\) 14898.9 0.820231 0.410116 0.912034i \(-0.365488\pi\)
0.410116 + 0.912034i \(0.365488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16669.3 0.909791
\(696\) 0 0
\(697\) −2816.27 −0.153047
\(698\) 0 0
\(699\) −2328.30 −0.125986
\(700\) 0 0
\(701\) −5725.70 −0.308497 −0.154249 0.988032i \(-0.549296\pi\)
−0.154249 + 0.988032i \(0.549296\pi\)
\(702\) 0 0
\(703\) 12359.2 0.663067
\(704\) 0 0
\(705\) −3228.61 −0.172477
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −23456.8 −1.24251 −0.621255 0.783609i \(-0.713377\pi\)
−0.621255 + 0.783609i \(0.713377\pi\)
\(710\) 0 0
\(711\) 3687.27 0.194492
\(712\) 0 0
\(713\) −43714.5 −2.29610
\(714\) 0 0
\(715\) 31883.4 1.66765
\(716\) 0 0
\(717\) 6020.39 0.313578
\(718\) 0 0
\(719\) 8459.00 0.438759 0.219379 0.975640i \(-0.429597\pi\)
0.219379 + 0.975640i \(0.429597\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2416.94 −0.124325
\(724\) 0 0
\(725\) 30221.7 1.54814
\(726\) 0 0
\(727\) 11822.2 0.603111 0.301555 0.953449i \(-0.402494\pi\)
0.301555 + 0.953449i \(0.402494\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8985.53 0.454640
\(732\) 0 0
\(733\) 5028.95 0.253409 0.126704 0.991941i \(-0.459560\pi\)
0.126704 + 0.991941i \(0.459560\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28262.0 1.41254
\(738\) 0 0
\(739\) 17743.9 0.883247 0.441624 0.897200i \(-0.354403\pi\)
0.441624 + 0.897200i \(0.354403\pi\)
\(740\) 0 0
\(741\) 8914.44 0.441944
\(742\) 0 0
\(743\) 13202.3 0.651877 0.325938 0.945391i \(-0.394320\pi\)
0.325938 + 0.945391i \(0.394320\pi\)
\(744\) 0 0
\(745\) −5716.84 −0.281139
\(746\) 0 0
\(747\) −3131.35 −0.153374
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15601.0 −0.758040 −0.379020 0.925388i \(-0.623739\pi\)
−0.379020 + 0.925388i \(0.623739\pi\)
\(752\) 0 0
\(753\) 4265.35 0.206425
\(754\) 0 0
\(755\) 24522.0 1.18205
\(756\) 0 0
\(757\) 2948.08 0.141545 0.0707725 0.997492i \(-0.477454\pi\)
0.0707725 + 0.997492i \(0.477454\pi\)
\(758\) 0 0
\(759\) −22914.4 −1.09583
\(760\) 0 0
\(761\) 1697.03 0.0808375 0.0404187 0.999183i \(-0.487131\pi\)
0.0404187 + 0.999183i \(0.487131\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3898.09 −0.184230
\(766\) 0 0
\(767\) 17940.0 0.844556
\(768\) 0 0
\(769\) −96.7799 −0.00453833 −0.00226916 0.999997i \(-0.500722\pi\)
−0.00226916 + 0.999997i \(0.500722\pi\)
\(770\) 0 0
\(771\) 4397.45 0.205409
\(772\) 0 0
\(773\) 36326.8 1.69028 0.845138 0.534548i \(-0.179518\pi\)
0.845138 + 0.534548i \(0.179518\pi\)
\(774\) 0 0
\(775\) 37325.9 1.73005
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7878.93 0.362377
\(780\) 0 0
\(781\) −6777.31 −0.310514
\(782\) 0 0
\(783\) −6485.62 −0.296012
\(784\) 0 0
\(785\) 15315.6 0.696352
\(786\) 0 0
\(787\) −7096.46 −0.321425 −0.160713 0.987001i \(-0.551379\pi\)
−0.160713 + 0.987001i \(0.551379\pi\)
\(788\) 0 0
\(789\) 20974.2 0.946388
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7198.40 0.322349
\(794\) 0 0
\(795\) −3158.97 −0.140927
\(796\) 0 0
\(797\) 40289.6 1.79063 0.895314 0.445436i \(-0.146951\pi\)
0.895314 + 0.445436i \(0.146951\pi\)
\(798\) 0 0
\(799\) −1858.45 −0.0822871
\(800\) 0 0
\(801\) 10414.4 0.459396
\(802\) 0 0
\(803\) −9398.03 −0.413013
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2426.88 −0.105861
\(808\) 0 0
\(809\) −11566.0 −0.502642 −0.251321 0.967904i \(-0.580865\pi\)
−0.251321 + 0.967904i \(0.580865\pi\)
\(810\) 0 0
\(811\) −18014.2 −0.779981 −0.389991 0.920819i \(-0.627522\pi\)
−0.389991 + 0.920819i \(0.627522\pi\)
\(812\) 0 0
\(813\) 19985.3 0.862134
\(814\) 0 0
\(815\) −21007.9 −0.902915
\(816\) 0 0
\(817\) −25138.3 −1.07647
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16792.4 −0.713837 −0.356918 0.934136i \(-0.616173\pi\)
−0.356918 + 0.934136i \(0.616173\pi\)
\(822\) 0 0
\(823\) 8819.03 0.373526 0.186763 0.982405i \(-0.440200\pi\)
0.186763 + 0.982405i \(0.440200\pi\)
\(824\) 0 0
\(825\) 19565.6 0.825680
\(826\) 0 0
\(827\) −8250.13 −0.346898 −0.173449 0.984843i \(-0.555491\pi\)
−0.173449 + 0.984843i \(0.555491\pi\)
\(828\) 0 0
\(829\) −21550.5 −0.902871 −0.451435 0.892304i \(-0.649088\pi\)
−0.451435 + 0.892304i \(0.649088\pi\)
\(830\) 0 0
\(831\) 22594.4 0.943190
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 22436.4 0.929873
\(836\) 0 0
\(837\) −8010.20 −0.330792
\(838\) 0 0
\(839\) −30130.4 −1.23983 −0.619916 0.784669i \(-0.712833\pi\)
−0.619916 + 0.784669i \(0.712833\pi\)
\(840\) 0 0
\(841\) 33311.0 1.36582
\(842\) 0 0
\(843\) −5070.57 −0.207164
\(844\) 0 0
\(845\) 10906.7 0.444025
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9534.69 0.385430
\(850\) 0 0
\(851\) 23801.8 0.958772
\(852\) 0 0
\(853\) −40738.6 −1.63525 −0.817623 0.575754i \(-0.804709\pi\)
−0.817623 + 0.575754i \(0.804709\pi\)
\(854\) 0 0
\(855\) 10905.5 0.436210
\(856\) 0 0
\(857\) −36508.2 −1.45519 −0.727594 0.686008i \(-0.759362\pi\)
−0.727594 + 0.686008i \(0.759362\pi\)
\(858\) 0 0
\(859\) 21980.8 0.873081 0.436541 0.899685i \(-0.356204\pi\)
0.436541 + 0.899685i \(0.356204\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23426.1 0.924024 0.462012 0.886874i \(-0.347128\pi\)
0.462012 + 0.886874i \(0.347128\pi\)
\(864\) 0 0
\(865\) 16416.7 0.645301
\(866\) 0 0
\(867\) 12495.2 0.489456
\(868\) 0 0
\(869\) −21237.5 −0.829037
\(870\) 0 0
\(871\) −21174.3 −0.823725
\(872\) 0 0
\(873\) 14564.7 0.564651
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 307.401 0.0118360 0.00591802 0.999982i \(-0.498116\pi\)
0.00591802 + 0.999982i \(0.498116\pi\)
\(878\) 0 0
\(879\) 6529.60 0.250555
\(880\) 0 0
\(881\) −19941.7 −0.762605 −0.381302 0.924450i \(-0.624524\pi\)
−0.381302 + 0.924450i \(0.624524\pi\)
\(882\) 0 0
\(883\) 37524.1 1.43011 0.715056 0.699068i \(-0.246401\pi\)
0.715056 + 0.699068i \(0.246401\pi\)
\(884\) 0 0
\(885\) 21946.8 0.833598
\(886\) 0 0
\(887\) 2880.20 0.109028 0.0545140 0.998513i \(-0.482639\pi\)
0.0545140 + 0.998513i \(0.482639\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4198.81 −0.157874
\(892\) 0 0
\(893\) 5199.30 0.194835
\(894\) 0 0
\(895\) −12153.9 −0.453922
\(896\) 0 0
\(897\) 17167.8 0.639036
\(898\) 0 0
\(899\) 71263.6 2.64380
\(900\) 0 0
\(901\) −1818.36 −0.0672347
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 62678.7 2.30222
\(906\) 0 0
\(907\) −18319.2 −0.670651 −0.335326 0.942102i \(-0.608846\pi\)
−0.335326 + 0.942102i \(0.608846\pi\)
\(908\) 0 0
\(909\) 6468.68 0.236031
\(910\) 0 0
\(911\) −46150.7 −1.67842 −0.839210 0.543807i \(-0.816982\pi\)
−0.839210 + 0.543807i \(0.816982\pi\)
\(912\) 0 0
\(913\) 18035.6 0.653769
\(914\) 0 0
\(915\) 8806.16 0.318167
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −47264.3 −1.69652 −0.848261 0.529578i \(-0.822350\pi\)
−0.848261 + 0.529578i \(0.822350\pi\)
\(920\) 0 0
\(921\) 1870.51 0.0669223
\(922\) 0 0
\(923\) 5077.66 0.181076
\(924\) 0 0
\(925\) −20323.3 −0.722407
\(926\) 0 0
\(927\) 14504.2 0.513895
\(928\) 0 0
\(929\) −53271.5 −1.88136 −0.940678 0.339301i \(-0.889810\pi\)
−0.940678 + 0.339301i \(0.889810\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1403.98 0.0492649
\(934\) 0 0
\(935\) 22451.8 0.785295
\(936\) 0 0
\(937\) 17197.8 0.599602 0.299801 0.954002i \(-0.403080\pi\)
0.299801 + 0.954002i \(0.403080\pi\)
\(938\) 0 0
\(939\) −10838.4 −0.376676
\(940\) 0 0
\(941\) 28834.9 0.998926 0.499463 0.866335i \(-0.333531\pi\)
0.499463 + 0.866335i \(0.333531\pi\)
\(942\) 0 0
\(943\) 15173.5 0.523986
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51841.2 1.77890 0.889448 0.457037i \(-0.151089\pi\)
0.889448 + 0.457037i \(0.151089\pi\)
\(948\) 0 0
\(949\) 7041.14 0.240848
\(950\) 0 0
\(951\) −13595.2 −0.463570
\(952\) 0 0
\(953\) −5887.31 −0.200114 −0.100057 0.994982i \(-0.531903\pi\)
−0.100057 + 0.994982i \(0.531903\pi\)
\(954\) 0 0
\(955\) −28590.7 −0.968767
\(956\) 0 0
\(957\) 37355.1 1.26178
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 58224.6 1.95444
\(962\) 0 0
\(963\) −8412.03 −0.281489
\(964\) 0 0
\(965\) −53384.4 −1.78083
\(966\) 0 0
\(967\) 36620.0 1.21781 0.608904 0.793244i \(-0.291609\pi\)
0.608904 + 0.793244i \(0.291609\pi\)
\(968\) 0 0
\(969\) 6277.41 0.208111
\(970\) 0 0
\(971\) 42364.9 1.40016 0.700079 0.714065i \(-0.253148\pi\)
0.700079 + 0.714065i \(0.253148\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −14658.8 −0.481495
\(976\) 0 0
\(977\) 5022.32 0.164461 0.0822304 0.996613i \(-0.473796\pi\)
0.0822304 + 0.996613i \(0.473796\pi\)
\(978\) 0 0
\(979\) −59983.8 −1.95821
\(980\) 0 0
\(981\) −10773.2 −0.350623
\(982\) 0 0
\(983\) −28292.8 −0.918005 −0.459003 0.888435i \(-0.651793\pi\)
−0.459003 + 0.888435i \(0.651793\pi\)
\(984\) 0 0
\(985\) 73045.7 2.36287
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48412.3 −1.55655
\(990\) 0 0
\(991\) −36401.1 −1.16682 −0.583410 0.812178i \(-0.698282\pi\)
−0.583410 + 0.812178i \(0.698282\pi\)
\(992\) 0 0
\(993\) −3711.17 −0.118601
\(994\) 0 0
\(995\) −35314.6 −1.12517
\(996\) 0 0
\(997\) −2357.19 −0.0748777 −0.0374389 0.999299i \(-0.511920\pi\)
−0.0374389 + 0.999299i \(0.511920\pi\)
\(998\) 0 0
\(999\) 4361.42 0.138127
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 2352.4.a.bq.1.1 2
4.3 odd 2 294.4.a.n.1.1 2
7.2 even 3 336.4.q.j.193.2 4
7.4 even 3 336.4.q.j.289.2 4
7.6 odd 2 2352.4.a.ca.1.2 2
12.11 even 2 882.4.a.v.1.2 2
28.3 even 6 294.4.e.l.79.1 4
28.11 odd 6 42.4.e.c.37.2 yes 4
28.19 even 6 294.4.e.l.67.1 4
28.23 odd 6 42.4.e.c.25.2 4
28.27 even 2 294.4.a.m.1.2 2
84.11 even 6 126.4.g.g.37.1 4
84.23 even 6 126.4.g.g.109.1 4
84.47 odd 6 882.4.g.bf.361.2 4
84.59 odd 6 882.4.g.bf.667.2 4
84.83 odd 2 882.4.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.c.25.2 4 28.23 odd 6
42.4.e.c.37.2 yes 4 28.11 odd 6
126.4.g.g.37.1 4 84.11 even 6
126.4.g.g.109.1 4 84.23 even 6
294.4.a.m.1.2 2 28.27 even 2
294.4.a.n.1.1 2 4.3 odd 2
294.4.e.l.67.1 4 28.19 even 6
294.4.e.l.79.1 4 28.3 even 6
336.4.q.j.193.2 4 7.2 even 3
336.4.q.j.289.2 4 7.4 even 3
882.4.a.v.1.2 2 12.11 even 2
882.4.a.z.1.1 2 84.83 odd 2
882.4.g.bf.361.2 4 84.47 odd 6
882.4.g.bf.667.2 4 84.59 odd 6
2352.4.a.bq.1.1 2 1.1 even 1 trivial
2352.4.a.ca.1.2 2 7.6 odd 2