Properties

Label 2352.4.a.cf.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +19.8995 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +19.8995 q^{5} +9.00000 q^{9} -23.9411 q^{11} +87.3553 q^{13} +59.6985 q^{15} -5.63961 q^{17} -64.8873 q^{19} +25.5980 q^{23} +270.990 q^{25} +27.0000 q^{27} +60.3188 q^{29} +122.711 q^{31} -71.8234 q^{33} -56.1177 q^{37} +262.066 q^{39} -299.713 q^{41} +501.421 q^{43} +179.095 q^{45} -305.553 q^{47} -16.9188 q^{51} -375.117 q^{53} -476.416 q^{55} -194.662 q^{57} +627.612 q^{59} +3.75736 q^{61} +1738.33 q^{65} +813.048 q^{67} +76.7939 q^{69} -165.902 q^{71} +619.100 q^{73} +812.970 q^{75} +138.246 q^{79} +81.0000 q^{81} -621.137 q^{83} -112.225 q^{85} +180.956 q^{87} +285.418 q^{89} +368.132 q^{93} -1291.22 q^{95} -603.114 q^{97} -215.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 20q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 20q^{5} + 18q^{9} + 20q^{11} + 104q^{13} + 60q^{15} + 116q^{17} - 192q^{19} - 28q^{23} + 146q^{25} + 54q^{27} + 296q^{29} + 104q^{31} + 60q^{33} - 248q^{37} + 312q^{39} + 20q^{41} + 720q^{43} + 180q^{45} + 96q^{47} + 348q^{51} + 268q^{53} - 472q^{55} - 576q^{57} + 616q^{59} + 16q^{61} + 1740q^{65} + 144q^{67} - 84q^{69} - 988q^{71} + 104q^{73} + 438q^{75} + 944q^{79} + 162q^{81} - 1016q^{83} - 100q^{85} + 888q^{87} - 388q^{89} + 312q^{93} - 1304q^{95} + 488q^{97} + 180q^{99} + O(q^{100}) \)

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
\(5\) 19.8995 1.77986 0.889932 0.456092i \(-0.150751\pi\)
0.889932 + 0.456092i \(0.150751\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −23.9411 −0.656229 −0.328115 0.944638i \(-0.606413\pi\)
−0.328115 + 0.944638i \(0.606413\pi\)
\(12\) 0 0
\(13\) 87.3553 1.86369 0.931847 0.362852i \(-0.118197\pi\)
0.931847 + 0.362852i \(0.118197\pi\)
\(14\) 0 0
\(15\) 59.6985 1.02761
\(16\) 0 0
\(17\) −5.63961 −0.0804592 −0.0402296 0.999190i \(-0.512809\pi\)
−0.0402296 + 0.999190i \(0.512809\pi\)
\(18\) 0 0
\(19\) −64.8873 −0.783483 −0.391741 0.920075i \(-0.628127\pi\)
−0.391741 + 0.920075i \(0.628127\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 25.5980 0.232067 0.116034 0.993245i \(-0.462982\pi\)
0.116034 + 0.993245i \(0.462982\pi\)
\(24\) 0 0
\(25\) 270.990 2.16792
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 60.3188 0.386238 0.193119 0.981175i \(-0.438140\pi\)
0.193119 + 0.981175i \(0.438140\pi\)
\(30\) 0 0
\(31\) 122.711 0.710951 0.355476 0.934686i \(-0.384319\pi\)
0.355476 + 0.934686i \(0.384319\pi\)
\(32\) 0 0
\(33\) −71.8234 −0.378874
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −56.1177 −0.249343 −0.124672 0.992198i \(-0.539788\pi\)
−0.124672 + 0.992198i \(0.539788\pi\)
\(38\) 0 0
\(39\) 262.066 1.07600
\(40\) 0 0
\(41\) −299.713 −1.14164 −0.570820 0.821075i \(-0.693375\pi\)
−0.570820 + 0.821075i \(0.693375\pi\)
\(42\) 0 0
\(43\) 501.421 1.77828 0.889140 0.457635i \(-0.151303\pi\)
0.889140 + 0.457635i \(0.151303\pi\)
\(44\) 0 0
\(45\) 179.095 0.593288
\(46\) 0 0
\(47\) −305.553 −0.948288 −0.474144 0.880447i \(-0.657242\pi\)
−0.474144 + 0.880447i \(0.657242\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −16.9188 −0.0464531
\(52\) 0 0
\(53\) −375.117 −0.972194 −0.486097 0.873905i \(-0.661580\pi\)
−0.486097 + 0.873905i \(0.661580\pi\)
\(54\) 0 0
\(55\) −476.416 −1.16800
\(56\) 0 0
\(57\) −194.662 −0.452344
\(58\) 0 0
\(59\) 627.612 1.38488 0.692442 0.721474i \(-0.256535\pi\)
0.692442 + 0.721474i \(0.256535\pi\)
\(60\) 0 0
\(61\) 3.75736 0.00788657 0.00394328 0.999992i \(-0.498745\pi\)
0.00394328 + 0.999992i \(0.498745\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1738.33 3.31712
\(66\) 0 0
\(67\) 813.048 1.48253 0.741266 0.671212i \(-0.234226\pi\)
0.741266 + 0.671212i \(0.234226\pi\)
\(68\) 0 0
\(69\) 76.7939 0.133984
\(70\) 0 0
\(71\) −165.902 −0.277310 −0.138655 0.990341i \(-0.544278\pi\)
−0.138655 + 0.990341i \(0.544278\pi\)
\(72\) 0 0
\(73\) 619.100 0.992605 0.496302 0.868150i \(-0.334691\pi\)
0.496302 + 0.868150i \(0.334691\pi\)
\(74\) 0 0
\(75\) 812.970 1.25165
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 138.246 0.196884 0.0984421 0.995143i \(-0.468614\pi\)
0.0984421 + 0.995143i \(0.468614\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −621.137 −0.821430 −0.410715 0.911764i \(-0.634721\pi\)
−0.410715 + 0.911764i \(0.634721\pi\)
\(84\) 0 0
\(85\) −112.225 −0.143207
\(86\) 0 0
\(87\) 180.956 0.222995
\(88\) 0 0
\(89\) 285.418 0.339936 0.169968 0.985450i \(-0.445634\pi\)
0.169968 + 0.985450i \(0.445634\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 368.132 0.410468
\(94\) 0 0
\(95\) −1291.22 −1.39449
\(96\) 0 0
\(97\) −603.114 −0.631309 −0.315654 0.948874i \(-0.602224\pi\)
−0.315654 + 0.948874i \(0.602224\pi\)
\(98\) 0 0
\(99\) −215.470 −0.218743
\(100\) 0 0
\(101\) 457.209 0.450436 0.225218 0.974308i \(-0.427691\pi\)
0.225218 + 0.974308i \(0.427691\pi\)
\(102\) 0 0
\(103\) −786.045 −0.751954 −0.375977 0.926629i \(-0.622693\pi\)
−0.375977 + 0.926629i \(0.622693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 196.461 0.177501 0.0887504 0.996054i \(-0.471713\pi\)
0.0887504 + 0.996054i \(0.471713\pi\)
\(108\) 0 0
\(109\) −306.343 −0.269196 −0.134598 0.990900i \(-0.542974\pi\)
−0.134598 + 0.990900i \(0.542974\pi\)
\(110\) 0 0
\(111\) −168.353 −0.143958
\(112\) 0 0
\(113\) 1997.63 1.66302 0.831508 0.555512i \(-0.187478\pi\)
0.831508 + 0.555512i \(0.187478\pi\)
\(114\) 0 0
\(115\) 509.387 0.413048
\(116\) 0 0
\(117\) 786.198 0.621231
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −757.823 −0.569363
\(122\) 0 0
\(123\) −899.138 −0.659127
\(124\) 0 0
\(125\) 2905.13 2.07874
\(126\) 0 0
\(127\) 2311.40 1.61499 0.807494 0.589875i \(-0.200823\pi\)
0.807494 + 0.589875i \(0.200823\pi\)
\(128\) 0 0
\(129\) 1504.26 1.02669
\(130\) 0 0
\(131\) 155.018 0.103389 0.0516945 0.998663i \(-0.483538\pi\)
0.0516945 + 0.998663i \(0.483538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 537.286 0.342535
\(136\) 0 0
\(137\) 516.936 0.322371 0.161186 0.986924i \(-0.448468\pi\)
0.161186 + 0.986924i \(0.448468\pi\)
\(138\) 0 0
\(139\) −958.067 −0.584620 −0.292310 0.956324i \(-0.594424\pi\)
−0.292310 + 0.956324i \(0.594424\pi\)
\(140\) 0 0
\(141\) −916.660 −0.547494
\(142\) 0 0
\(143\) −2091.39 −1.22301
\(144\) 0 0
\(145\) 1200.31 0.687452
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1770.63 −0.973526 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(150\) 0 0
\(151\) 2540.24 1.36902 0.684508 0.729005i \(-0.260017\pi\)
0.684508 + 0.729005i \(0.260017\pi\)
\(152\) 0 0
\(153\) −50.7565 −0.0268197
\(154\) 0 0
\(155\) 2441.88 1.26540
\(156\) 0 0
\(157\) −1083.34 −0.550702 −0.275351 0.961344i \(-0.588794\pi\)
−0.275351 + 0.961344i \(0.588794\pi\)
\(158\) 0 0
\(159\) −1125.35 −0.561296
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2968.72 −1.42655 −0.713277 0.700882i \(-0.752790\pi\)
−0.713277 + 0.700882i \(0.752790\pi\)
\(164\) 0 0
\(165\) −1429.25 −0.674345
\(166\) 0 0
\(167\) 2091.53 0.969149 0.484574 0.874750i \(-0.338975\pi\)
0.484574 + 0.874750i \(0.338975\pi\)
\(168\) 0 0
\(169\) 5433.96 2.47335
\(170\) 0 0
\(171\) −583.986 −0.261161
\(172\) 0 0
\(173\) −470.148 −0.206617 −0.103308 0.994649i \(-0.532943\pi\)
−0.103308 + 0.994649i \(0.532943\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1882.84 0.799563
\(178\) 0 0
\(179\) −1056.46 −0.441137 −0.220569 0.975371i \(-0.570791\pi\)
−0.220569 + 0.975371i \(0.570791\pi\)
\(180\) 0 0
\(181\) −406.470 −0.166921 −0.0834605 0.996511i \(-0.526597\pi\)
−0.0834605 + 0.996511i \(0.526597\pi\)
\(182\) 0 0
\(183\) 11.2721 0.00455331
\(184\) 0 0
\(185\) −1116.71 −0.443797
\(186\) 0 0
\(187\) 135.019 0.0527997
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −179.410 −0.0679666 −0.0339833 0.999422i \(-0.510819\pi\)
−0.0339833 + 0.999422i \(0.510819\pi\)
\(192\) 0 0
\(193\) −2388.37 −0.890769 −0.445385 0.895339i \(-0.646933\pi\)
−0.445385 + 0.895339i \(0.646933\pi\)
\(194\) 0 0
\(195\) 5214.98 1.91514
\(196\) 0 0
\(197\) −2665.35 −0.963952 −0.481976 0.876184i \(-0.660081\pi\)
−0.481976 + 0.876184i \(0.660081\pi\)
\(198\) 0 0
\(199\) 1342.31 0.478159 0.239079 0.971000i \(-0.423154\pi\)
0.239079 + 0.971000i \(0.423154\pi\)
\(200\) 0 0
\(201\) 2439.14 0.855940
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5964.13 −2.03197
\(206\) 0 0
\(207\) 230.382 0.0773558
\(208\) 0 0
\(209\) 1553.48 0.514144
\(210\) 0 0
\(211\) −628.442 −0.205042 −0.102521 0.994731i \(-0.532691\pi\)
−0.102521 + 0.994731i \(0.532691\pi\)
\(212\) 0 0
\(213\) −497.707 −0.160105
\(214\) 0 0
\(215\) 9978.03 3.16510
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1857.30 0.573081
\(220\) 0 0
\(221\) −492.650 −0.149951
\(222\) 0 0
\(223\) 969.970 0.291273 0.145637 0.989338i \(-0.453477\pi\)
0.145637 + 0.989338i \(0.453477\pi\)
\(224\) 0 0
\(225\) 2438.91 0.722640
\(226\) 0 0
\(227\) 4748.64 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(228\) 0 0
\(229\) −4801.99 −1.38570 −0.692848 0.721083i \(-0.743644\pi\)
−0.692848 + 0.721083i \(0.743644\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3155.29 −0.887166 −0.443583 0.896233i \(-0.646293\pi\)
−0.443583 + 0.896233i \(0.646293\pi\)
\(234\) 0 0
\(235\) −6080.36 −1.68782
\(236\) 0 0
\(237\) 414.737 0.113671
\(238\) 0 0
\(239\) −4241.93 −1.14806 −0.574032 0.818833i \(-0.694622\pi\)
−0.574032 + 0.818833i \(0.694622\pi\)
\(240\) 0 0
\(241\) −4342.99 −1.16081 −0.580407 0.814326i \(-0.697107\pi\)
−0.580407 + 0.814326i \(0.697107\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5668.25 −1.46017
\(248\) 0 0
\(249\) −1863.41 −0.474253
\(250\) 0 0
\(251\) −3003.01 −0.755172 −0.377586 0.925974i \(-0.623246\pi\)
−0.377586 + 0.925974i \(0.623246\pi\)
\(252\) 0 0
\(253\) −612.844 −0.152289
\(254\) 0 0
\(255\) −336.676 −0.0826803
\(256\) 0 0
\(257\) 4468.84 1.08466 0.542332 0.840164i \(-0.317541\pi\)
0.542332 + 0.840164i \(0.317541\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 542.869 0.128746
\(262\) 0 0
\(263\) −6160.13 −1.44430 −0.722148 0.691738i \(-0.756845\pi\)
−0.722148 + 0.691738i \(0.756845\pi\)
\(264\) 0 0
\(265\) −7464.64 −1.73037
\(266\) 0 0
\(267\) 856.255 0.196262
\(268\) 0 0
\(269\) 4988.90 1.13078 0.565388 0.824825i \(-0.308726\pi\)
0.565388 + 0.824825i \(0.308726\pi\)
\(270\) 0 0
\(271\) −4433.73 −0.993837 −0.496918 0.867797i \(-0.665535\pi\)
−0.496918 + 0.867797i \(0.665535\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6487.80 −1.42265
\(276\) 0 0
\(277\) 1112.37 0.241284 0.120642 0.992696i \(-0.461505\pi\)
0.120642 + 0.992696i \(0.461505\pi\)
\(278\) 0 0
\(279\) 1104.40 0.236984
\(280\) 0 0
\(281\) 2813.22 0.597233 0.298616 0.954373i \(-0.403475\pi\)
0.298616 + 0.954373i \(0.403475\pi\)
\(282\) 0 0
\(283\) −3147.54 −0.661137 −0.330569 0.943782i \(-0.607241\pi\)
−0.330569 + 0.943782i \(0.607241\pi\)
\(284\) 0 0
\(285\) −3873.67 −0.805111
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4881.19 −0.993526
\(290\) 0 0
\(291\) −1809.34 −0.364486
\(292\) 0 0
\(293\) 9143.04 1.82301 0.911505 0.411289i \(-0.134921\pi\)
0.911505 + 0.411289i \(0.134921\pi\)
\(294\) 0 0
\(295\) 12489.2 2.46491
\(296\) 0 0
\(297\) −646.410 −0.126291
\(298\) 0 0
\(299\) 2236.12 0.432502
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1371.63 0.260059
\(304\) 0 0
\(305\) 74.7696 0.0140370
\(306\) 0 0
\(307\) 4648.90 0.864257 0.432129 0.901812i \(-0.357763\pi\)
0.432129 + 0.901812i \(0.357763\pi\)
\(308\) 0 0
\(309\) −2358.13 −0.434141
\(310\) 0 0
\(311\) 6417.18 1.17005 0.585024 0.811016i \(-0.301085\pi\)
0.585024 + 0.811016i \(0.301085\pi\)
\(312\) 0 0
\(313\) 5868.22 1.05972 0.529858 0.848086i \(-0.322245\pi\)
0.529858 + 0.848086i \(0.322245\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3974.52 −0.704200 −0.352100 0.935962i \(-0.614532\pi\)
−0.352100 + 0.935962i \(0.614532\pi\)
\(318\) 0 0
\(319\) −1444.10 −0.253461
\(320\) 0 0
\(321\) 589.383 0.102480
\(322\) 0 0
\(323\) 365.939 0.0630384
\(324\) 0 0
\(325\) 23672.4 4.04034
\(326\) 0 0
\(327\) −919.029 −0.155420
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8912.82 1.48004 0.740020 0.672585i \(-0.234816\pi\)
0.740020 + 0.672585i \(0.234816\pi\)
\(332\) 0 0
\(333\) −505.060 −0.0831144
\(334\) 0 0
\(335\) 16179.2 2.63871
\(336\) 0 0
\(337\) 3977.06 0.642862 0.321431 0.946933i \(-0.395836\pi\)
0.321431 + 0.946933i \(0.395836\pi\)
\(338\) 0 0
\(339\) 5992.88 0.960143
\(340\) 0 0
\(341\) −2937.83 −0.466547
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1528.16 0.238474
\(346\) 0 0
\(347\) −6826.43 −1.05609 −0.528043 0.849218i \(-0.677074\pi\)
−0.528043 + 0.849218i \(0.677074\pi\)
\(348\) 0 0
\(349\) −807.342 −0.123828 −0.0619141 0.998081i \(-0.519720\pi\)
−0.0619141 + 0.998081i \(0.519720\pi\)
\(350\) 0 0
\(351\) 2358.59 0.358668
\(352\) 0 0
\(353\) −7919.20 −1.19404 −0.597020 0.802226i \(-0.703649\pi\)
−0.597020 + 0.802226i \(0.703649\pi\)
\(354\) 0 0
\(355\) −3301.38 −0.493574
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8819.21 1.29655 0.648273 0.761408i \(-0.275491\pi\)
0.648273 + 0.761408i \(0.275491\pi\)
\(360\) 0 0
\(361\) −2648.64 −0.386155
\(362\) 0 0
\(363\) −2273.47 −0.328722
\(364\) 0 0
\(365\) 12319.8 1.76670
\(366\) 0 0
\(367\) 11161.8 1.58758 0.793788 0.608194i \(-0.208106\pi\)
0.793788 + 0.608194i \(0.208106\pi\)
\(368\) 0 0
\(369\) −2697.41 −0.380547
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2727.86 0.378668 0.189334 0.981913i \(-0.439367\pi\)
0.189334 + 0.981913i \(0.439367\pi\)
\(374\) 0 0
\(375\) 8715.38 1.20016
\(376\) 0 0
\(377\) 5269.17 0.719830
\(378\) 0 0
\(379\) −4086.49 −0.553849 −0.276924 0.960892i \(-0.589315\pi\)
−0.276924 + 0.960892i \(0.589315\pi\)
\(380\) 0 0
\(381\) 6934.20 0.932414
\(382\) 0 0
\(383\) −13032.9 −1.73878 −0.869389 0.494129i \(-0.835487\pi\)
−0.869389 + 0.494129i \(0.835487\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4512.79 0.592760
\(388\) 0 0
\(389\) −194.991 −0.0254151 −0.0127075 0.999919i \(-0.504045\pi\)
−0.0127075 + 0.999919i \(0.504045\pi\)
\(390\) 0 0
\(391\) −144.363 −0.0186719
\(392\) 0 0
\(393\) 465.053 0.0596916
\(394\) 0 0
\(395\) 2751.02 0.350427
\(396\) 0 0
\(397\) −14183.0 −1.79300 −0.896501 0.443042i \(-0.853899\pi\)
−0.896501 + 0.443042i \(0.853899\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10005.0 1.24596 0.622978 0.782239i \(-0.285923\pi\)
0.622978 + 0.782239i \(0.285923\pi\)
\(402\) 0 0
\(403\) 10719.4 1.32500
\(404\) 0 0
\(405\) 1611.86 0.197763
\(406\) 0 0
\(407\) 1343.52 0.163626
\(408\) 0 0
\(409\) 4634.93 0.560349 0.280174 0.959949i \(-0.409608\pi\)
0.280174 + 0.959949i \(0.409608\pi\)
\(410\) 0 0
\(411\) 1550.81 0.186121
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12360.3 −1.46203
\(416\) 0 0
\(417\) −2874.20 −0.337531
\(418\) 0 0
\(419\) −4998.31 −0.582777 −0.291388 0.956605i \(-0.594117\pi\)
−0.291388 + 0.956605i \(0.594117\pi\)
\(420\) 0 0
\(421\) −704.160 −0.0815170 −0.0407585 0.999169i \(-0.512977\pi\)
−0.0407585 + 0.999169i \(0.512977\pi\)
\(422\) 0 0
\(423\) −2749.98 −0.316096
\(424\) 0 0
\(425\) −1528.28 −0.174429
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6274.16 −0.706105
\(430\) 0 0
\(431\) 10332.8 1.15479 0.577393 0.816466i \(-0.304070\pi\)
0.577393 + 0.816466i \(0.304070\pi\)
\(432\) 0 0
\(433\) 11106.8 1.23270 0.616348 0.787474i \(-0.288611\pi\)
0.616348 + 0.787474i \(0.288611\pi\)
\(434\) 0 0
\(435\) 3600.94 0.396901
\(436\) 0 0
\(437\) −1660.98 −0.181821
\(438\) 0 0
\(439\) 7299.28 0.793566 0.396783 0.917912i \(-0.370126\pi\)
0.396783 + 0.917912i \(0.370126\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16089.7 −1.72560 −0.862802 0.505542i \(-0.831293\pi\)
−0.862802 + 0.505542i \(0.831293\pi\)
\(444\) 0 0
\(445\) 5679.68 0.605040
\(446\) 0 0
\(447\) −5311.88 −0.562065
\(448\) 0 0
\(449\) 13561.7 1.42543 0.712715 0.701454i \(-0.247466\pi\)
0.712715 + 0.701454i \(0.247466\pi\)
\(450\) 0 0
\(451\) 7175.46 0.749178
\(452\) 0 0
\(453\) 7620.71 0.790402
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10848.6 1.11045 0.555224 0.831701i \(-0.312633\pi\)
0.555224 + 0.831701i \(0.312633\pi\)
\(458\) 0 0
\(459\) −152.269 −0.0154844
\(460\) 0 0
\(461\) −1758.69 −0.177679 −0.0888397 0.996046i \(-0.528316\pi\)
−0.0888397 + 0.996046i \(0.528316\pi\)
\(462\) 0 0
\(463\) 5411.95 0.543228 0.271614 0.962406i \(-0.412443\pi\)
0.271614 + 0.962406i \(0.412443\pi\)
\(464\) 0 0
\(465\) 7325.64 0.730577
\(466\) 0 0
\(467\) −8111.34 −0.803744 −0.401872 0.915696i \(-0.631640\pi\)
−0.401872 + 0.915696i \(0.631640\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3250.03 −0.317948
\(472\) 0 0
\(473\) −12004.6 −1.16696
\(474\) 0 0
\(475\) −17583.8 −1.69853
\(476\) 0 0
\(477\) −3376.05 −0.324065
\(478\) 0 0
\(479\) −2095.76 −0.199912 −0.0999559 0.994992i \(-0.531870\pi\)
−0.0999559 + 0.994992i \(0.531870\pi\)
\(480\) 0 0
\(481\) −4902.18 −0.464699
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12001.7 −1.12364
\(486\) 0 0
\(487\) −9610.08 −0.894197 −0.447099 0.894485i \(-0.647543\pi\)
−0.447099 + 0.894485i \(0.647543\pi\)
\(488\) 0 0
\(489\) −8906.17 −0.823622
\(490\) 0 0
\(491\) 11717.3 1.07698 0.538488 0.842633i \(-0.318996\pi\)
0.538488 + 0.842633i \(0.318996\pi\)
\(492\) 0 0
\(493\) −340.174 −0.0310764
\(494\) 0 0
\(495\) −4287.75 −0.389333
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9195.19 0.824916 0.412458 0.910977i \(-0.364670\pi\)
0.412458 + 0.910977i \(0.364670\pi\)
\(500\) 0 0
\(501\) 6274.60 0.559538
\(502\) 0 0
\(503\) −16118.8 −1.42883 −0.714414 0.699724i \(-0.753307\pi\)
−0.714414 + 0.699724i \(0.753307\pi\)
\(504\) 0 0
\(505\) 9098.23 0.801715
\(506\) 0 0
\(507\) 16301.9 1.42799
\(508\) 0 0
\(509\) 4918.78 0.428333 0.214166 0.976797i \(-0.431297\pi\)
0.214166 + 0.976797i \(0.431297\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1751.96 −0.150781
\(514\) 0 0
\(515\) −15641.9 −1.33838
\(516\) 0 0
\(517\) 7315.29 0.622294
\(518\) 0 0
\(519\) −1410.44 −0.119290
\(520\) 0 0
\(521\) 13963.4 1.17418 0.587089 0.809522i \(-0.300274\pi\)
0.587089 + 0.809522i \(0.300274\pi\)
\(522\) 0 0
\(523\) 13755.3 1.15005 0.575024 0.818136i \(-0.304993\pi\)
0.575024 + 0.818136i \(0.304993\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −692.040 −0.0572026
\(528\) 0 0
\(529\) −11511.7 −0.946145
\(530\) 0 0
\(531\) 5648.51 0.461628
\(532\) 0 0
\(533\) −26181.5 −2.12767
\(534\) 0 0
\(535\) 3909.47 0.315928
\(536\) 0 0
\(537\) −3169.38 −0.254691
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14462.7 1.14935 0.574676 0.818381i \(-0.305128\pi\)
0.574676 + 0.818381i \(0.305128\pi\)
\(542\) 0 0
\(543\) −1219.41 −0.0963719
\(544\) 0 0
\(545\) −6096.07 −0.479132
\(546\) 0 0
\(547\) −13682.5 −1.06951 −0.534755 0.845007i \(-0.679596\pi\)
−0.534755 + 0.845007i \(0.679596\pi\)
\(548\) 0 0
\(549\) 33.8162 0.00262886
\(550\) 0 0
\(551\) −3913.92 −0.302611
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3350.14 −0.256227
\(556\) 0 0
\(557\) −7663.13 −0.582939 −0.291470 0.956580i \(-0.594144\pi\)
−0.291470 + 0.956580i \(0.594144\pi\)
\(558\) 0 0
\(559\) 43801.8 3.31417
\(560\) 0 0
\(561\) 405.056 0.0304839
\(562\) 0 0
\(563\) −17470.5 −1.30780 −0.653902 0.756580i \(-0.726869\pi\)
−0.653902 + 0.756580i \(0.726869\pi\)
\(564\) 0 0
\(565\) 39751.8 2.95995
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13873.9 −1.02219 −0.511094 0.859525i \(-0.670760\pi\)
−0.511094 + 0.859525i \(0.670760\pi\)
\(570\) 0 0
\(571\) 3777.52 0.276855 0.138428 0.990373i \(-0.455795\pi\)
0.138428 + 0.990373i \(0.455795\pi\)
\(572\) 0 0
\(573\) −538.229 −0.0392405
\(574\) 0 0
\(575\) 6936.79 0.503103
\(576\) 0 0
\(577\) 13880.5 1.00148 0.500738 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(578\) 0 0
\(579\) −7165.10 −0.514286
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8980.72 0.637982
\(584\) 0 0
\(585\) 15644.9 1.10571
\(586\) 0 0
\(587\) 2395.61 0.168445 0.0842227 0.996447i \(-0.473159\pi\)
0.0842227 + 0.996447i \(0.473159\pi\)
\(588\) 0 0
\(589\) −7962.36 −0.557018
\(590\) 0 0
\(591\) −7996.06 −0.556538
\(592\) 0 0
\(593\) 6603.50 0.457290 0.228645 0.973510i \(-0.426570\pi\)
0.228645 + 0.973510i \(0.426570\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4026.92 0.276065
\(598\) 0 0
\(599\) −17252.1 −1.17680 −0.588399 0.808571i \(-0.700241\pi\)
−0.588399 + 0.808571i \(0.700241\pi\)
\(600\) 0 0
\(601\) 12833.1 0.871005 0.435503 0.900187i \(-0.356571\pi\)
0.435503 + 0.900187i \(0.356571\pi\)
\(602\) 0 0
\(603\) 7317.43 0.494177
\(604\) 0 0
\(605\) −15080.3 −1.01339
\(606\) 0 0
\(607\) −8620.16 −0.576411 −0.288206 0.957569i \(-0.593059\pi\)
−0.288206 + 0.957569i \(0.593059\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −26691.7 −1.76732
\(612\) 0 0
\(613\) 5136.73 0.338451 0.169226 0.985577i \(-0.445873\pi\)
0.169226 + 0.985577i \(0.445873\pi\)
\(614\) 0 0
\(615\) −17892.4 −1.17316
\(616\) 0 0
\(617\) −1759.82 −0.114826 −0.0574131 0.998351i \(-0.518285\pi\)
−0.0574131 + 0.998351i \(0.518285\pi\)
\(618\) 0 0
\(619\) 3560.24 0.231176 0.115588 0.993297i \(-0.463125\pi\)
0.115588 + 0.993297i \(0.463125\pi\)
\(620\) 0 0
\(621\) 691.145 0.0446614
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 23936.8 1.53195
\(626\) 0 0
\(627\) 4660.43 0.296841
\(628\) 0 0
\(629\) 316.482 0.0200620
\(630\) 0 0
\(631\) 27321.4 1.72369 0.861845 0.507172i \(-0.169309\pi\)
0.861845 + 0.507172i \(0.169309\pi\)
\(632\) 0 0
\(633\) −1885.33 −0.118381
\(634\) 0 0
\(635\) 45995.7 2.87446
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1493.12 −0.0924366
\(640\) 0 0
\(641\) −21927.0 −1.35112 −0.675558 0.737307i \(-0.736097\pi\)
−0.675558 + 0.737307i \(0.736097\pi\)
\(642\) 0 0
\(643\) 5826.04 0.357320 0.178660 0.983911i \(-0.442824\pi\)
0.178660 + 0.983911i \(0.442824\pi\)
\(644\) 0 0
\(645\) 29934.1 1.82737
\(646\) 0 0
\(647\) −24210.7 −1.47113 −0.735565 0.677454i \(-0.763083\pi\)
−0.735565 + 0.677454i \(0.763083\pi\)
\(648\) 0 0
\(649\) −15025.7 −0.908801
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25623.3 −1.53556 −0.767778 0.640716i \(-0.778637\pi\)
−0.767778 + 0.640716i \(0.778637\pi\)
\(654\) 0 0
\(655\) 3084.77 0.184018
\(656\) 0 0
\(657\) 5571.90 0.330868
\(658\) 0 0
\(659\) 23273.7 1.37574 0.687871 0.725833i \(-0.258545\pi\)
0.687871 + 0.725833i \(0.258545\pi\)
\(660\) 0 0
\(661\) −20036.5 −1.17902 −0.589508 0.807763i \(-0.700678\pi\)
−0.589508 + 0.807763i \(0.700678\pi\)
\(662\) 0 0
\(663\) −1477.95 −0.0865744
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1544.04 0.0896333
\(668\) 0 0
\(669\) 2909.91 0.168167
\(670\) 0 0
\(671\) −89.9554 −0.00517540
\(672\) 0 0
\(673\) −18127.8 −1.03830 −0.519149 0.854684i \(-0.673751\pi\)
−0.519149 + 0.854684i \(0.673751\pi\)
\(674\) 0 0
\(675\) 7316.73 0.417216
\(676\) 0 0
\(677\) 13815.5 0.784301 0.392150 0.919901i \(-0.371731\pi\)
0.392150 + 0.919901i \(0.371731\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14245.9 0.801622
\(682\) 0 0
\(683\) 4604.23 0.257944 0.128972 0.991648i \(-0.458832\pi\)
0.128972 + 0.991648i \(0.458832\pi\)
\(684\) 0 0
\(685\) 10286.8 0.573777
\(686\) 0 0
\(687\) −14406.0 −0.800032
\(688\) 0 0
\(689\) −32768.5 −1.81187
\(690\) 0 0
\(691\) −17913.7 −0.986205 −0.493103 0.869971i \(-0.664137\pi\)
−0.493103 + 0.869971i \(0.664137\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −19065.1 −1.04054
\(696\) 0 0
\(697\) 1690.26 0.0918555
\(698\) 0 0
\(699\) −9465.86 −0.512206
\(700\) 0 0
\(701\) 11303.7 0.609035 0.304518 0.952507i \(-0.401505\pi\)
0.304518 + 0.952507i \(0.401505\pi\)
\(702\) 0 0
\(703\) 3641.33 0.195356
\(704\) 0 0
\(705\) −18241.1 −0.974466
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −16046.3 −0.849973 −0.424987 0.905200i \(-0.639721\pi\)
−0.424987 + 0.905200i \(0.639721\pi\)
\(710\) 0 0
\(711\) 1244.21 0.0656280
\(712\) 0 0
\(713\) 3141.15 0.164989
\(714\) 0 0
\(715\) −41617.5 −2.17679
\(716\) 0 0
\(717\) −12725.8 −0.662836
\(718\) 0 0
\(719\) 25190.5 1.30660 0.653300 0.757099i \(-0.273384\pi\)
0.653300 + 0.757099i \(0.273384\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −13029.0 −0.670197
\(724\) 0 0
\(725\) 16345.8 0.837334
\(726\) 0 0
\(727\) −11277.2 −0.575307 −0.287653 0.957735i \(-0.592875\pi\)
−0.287653 + 0.957735i \(0.592875\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2827.82 −0.143079
\(732\) 0 0
\(733\) 23720.0 1.19525 0.597626 0.801775i \(-0.296111\pi\)
0.597626 + 0.801775i \(0.296111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19465.3 −0.972880
\(738\) 0 0
\(739\) −8124.72 −0.404428 −0.202214 0.979341i \(-0.564814\pi\)
−0.202214 + 0.979341i \(0.564814\pi\)
\(740\) 0 0
\(741\) −17004.8 −0.843030
\(742\) 0 0
\(743\) −20955.3 −1.03469 −0.517346 0.855777i \(-0.673080\pi\)
−0.517346 + 0.855777i \(0.673080\pi\)
\(744\) 0 0
\(745\) −35234.6 −1.73274
\(746\) 0 0
\(747\) −5590.23 −0.273810
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 38208.1 1.85650 0.928251 0.371954i \(-0.121312\pi\)
0.928251 + 0.371954i \(0.121312\pi\)
\(752\) 0 0
\(753\) −9009.02 −0.435999
\(754\) 0 0
\(755\) 50549.4 2.43666
\(756\) 0 0
\(757\) 30958.1 1.48638 0.743191 0.669079i \(-0.233311\pi\)
0.743191 + 0.669079i \(0.233311\pi\)
\(758\) 0 0
\(759\) −1838.53 −0.0879243
\(760\) 0 0
\(761\) −40049.4 −1.90774 −0.953871 0.300218i \(-0.902941\pi\)
−0.953871 + 0.300218i \(0.902941\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1010.03 −0.0477355
\(766\) 0 0
\(767\) 54825.3 2.58100
\(768\) 0 0
\(769\) −8002.01 −0.375240 −0.187620 0.982242i \(-0.560077\pi\)
−0.187620 + 0.982242i \(0.560077\pi\)
\(770\) 0 0
\(771\) 13406.5 0.626231
\(772\) 0 0
\(773\) −13933.4 −0.648316 −0.324158 0.946003i \(-0.605081\pi\)
−0.324158 + 0.946003i \(0.605081\pi\)
\(774\) 0 0
\(775\) 33253.4 1.54128
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19447.6 0.894456
\(780\) 0 0
\(781\) 3971.89 0.181979
\(782\) 0 0
\(783\) 1628.61 0.0743316
\(784\) 0 0
\(785\) −21558.0 −0.980175
\(786\) 0 0
\(787\) −37581.7 −1.70222 −0.851108 0.524991i \(-0.824069\pi\)
−0.851108 + 0.524991i \(0.824069\pi\)
\(788\) 0 0
\(789\) −18480.4 −0.833865
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 328.225 0.0146981
\(794\) 0 0
\(795\) −22393.9 −0.999032
\(796\) 0 0
\(797\) −9458.78 −0.420385 −0.210193 0.977660i \(-0.567409\pi\)
−0.210193 + 0.977660i \(0.567409\pi\)
\(798\) 0 0
\(799\) 1723.20 0.0762985
\(800\) 0 0
\(801\) 2568.77 0.113312
\(802\) 0 0
\(803\) −14821.9 −0.651376
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14966.7 0.652854
\(808\) 0 0
\(809\) −1909.94 −0.0830034 −0.0415017 0.999138i \(-0.513214\pi\)
−0.0415017 + 0.999138i \(0.513214\pi\)
\(810\) 0 0
\(811\) −43110.6 −1.86661 −0.933303 0.359091i \(-0.883087\pi\)
−0.933303 + 0.359091i \(0.883087\pi\)
\(812\) 0 0
\(813\) −13301.2 −0.573792
\(814\) 0 0
\(815\) −59076.1 −2.53907
\(816\) 0 0
\(817\) −32535.9 −1.39325
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4026.56 0.171167 0.0855834 0.996331i \(-0.472725\pi\)
0.0855834 + 0.996331i \(0.472725\pi\)
\(822\) 0 0
\(823\) −39668.1 −1.68012 −0.840062 0.542491i \(-0.817481\pi\)
−0.840062 + 0.542491i \(0.817481\pi\)
\(824\) 0 0
\(825\) −19463.4 −0.821368
\(826\) 0 0
\(827\) −30137.7 −1.26722 −0.633611 0.773652i \(-0.718428\pi\)
−0.633611 + 0.773652i \(0.718428\pi\)
\(828\) 0 0
\(829\) 23278.0 0.975245 0.487622 0.873055i \(-0.337864\pi\)
0.487622 + 0.873055i \(0.337864\pi\)
\(830\) 0 0
\(831\) 3337.10 0.139306
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 41620.5 1.72495
\(836\) 0 0
\(837\) 3313.19 0.136823
\(838\) 0 0
\(839\) 9494.43 0.390684 0.195342 0.980735i \(-0.437418\pi\)
0.195342 + 0.980735i \(0.437418\pi\)
\(840\) 0 0
\(841\) −20750.6 −0.850820
\(842\) 0 0
\(843\) 8439.65 0.344813
\(844\) 0 0
\(845\) 108133. 4.40223
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9442.62 −0.381708
\(850\) 0 0
\(851\) −1436.50 −0.0578644
\(852\) 0 0
\(853\) −12692.4 −0.509471 −0.254736 0.967011i \(-0.581988\pi\)
−0.254736 + 0.967011i \(0.581988\pi\)
\(854\) 0 0
\(855\) −11621.0 −0.464831
\(856\) 0 0
\(857\) 22206.0 0.885114 0.442557 0.896740i \(-0.354071\pi\)
0.442557 + 0.896740i \(0.354071\pi\)
\(858\) 0 0
\(859\) −19820.5 −0.787271 −0.393636 0.919267i \(-0.628783\pi\)
−0.393636 + 0.919267i \(0.628783\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36413.8 −1.43631 −0.718157 0.695881i \(-0.755014\pi\)
−0.718157 + 0.695881i \(0.755014\pi\)
\(864\) 0 0
\(865\) −9355.70 −0.367749
\(866\) 0 0
\(867\) −14643.6 −0.573613
\(868\) 0 0
\(869\) −3309.76 −0.129201
\(870\) 0 0
\(871\) 71024.1 2.76298
\(872\) 0 0
\(873\) −5428.03 −0.210436
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19442.6 0.748609 0.374305 0.927306i \(-0.377881\pi\)
0.374305 + 0.927306i \(0.377881\pi\)
\(878\) 0 0
\(879\) 27429.1 1.05252
\(880\) 0 0
\(881\) −25184.2 −0.963082 −0.481541 0.876423i \(-0.659923\pi\)
−0.481541 + 0.876423i \(0.659923\pi\)
\(882\) 0 0
\(883\) 4050.03 0.154354 0.0771769 0.997017i \(-0.475409\pi\)
0.0771769 + 0.997017i \(0.475409\pi\)
\(884\) 0 0
\(885\) 37467.5 1.42311
\(886\) 0 0
\(887\) −41604.2 −1.57489 −0.787447 0.616383i \(-0.788598\pi\)
−0.787447 + 0.616383i \(0.788598\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1939.23 −0.0729144
\(892\) 0 0
\(893\) 19826.5 0.742967
\(894\) 0 0
\(895\) −21023.0 −0.785165
\(896\) 0 0
\(897\) 6708.36 0.249705
\(898\) 0 0
\(899\) 7401.76 0.274597
\(900\) 0 0
\(901\) 2115.51 0.0782219
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8088.56 −0.297097
\(906\) 0 0
\(907\) 3572.60 0.130790 0.0653949 0.997859i \(-0.479169\pi\)
0.0653949 + 0.997859i \(0.479169\pi\)
\(908\) 0 0
\(909\) 4114.88 0.150145
\(910\) 0 0
\(911\) −29457.7 −1.07133 −0.535663 0.844432i \(-0.679938\pi\)
−0.535663 + 0.844432i \(0.679938\pi\)
\(912\) 0 0
\(913\) 14870.7 0.539046
\(914\) 0 0
\(915\) 224.309 0.00810428
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3310.65 0.118834 0.0594168 0.998233i \(-0.481076\pi\)
0.0594168 + 0.998233i \(0.481076\pi\)
\(920\) 0 0
\(921\) 13946.7 0.498979
\(922\) 0 0
\(923\) −14492.5 −0.516820
\(924\) 0 0
\(925\) −15207.3 −0.540556
\(926\) 0 0
\(927\) −7074.40 −0.250651
\(928\) 0 0
\(929\) −31467.5 −1.11132 −0.555660 0.831410i \(-0.687534\pi\)
−0.555660 + 0.831410i \(0.687534\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 19251.5 0.675527
\(934\) 0 0
\(935\) 2686.80 0.0939763
\(936\) 0 0
\(937\) −17363.4 −0.605375 −0.302688 0.953090i \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(938\) 0 0
\(939\) 17604.6 0.611828
\(940\) 0 0
\(941\) −5547.77 −0.192192 −0.0960958 0.995372i \(-0.530636\pi\)
−0.0960958 + 0.995372i \(0.530636\pi\)
\(942\) 0 0
\(943\) −7672.04 −0.264937
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37960.3 1.30258 0.651290 0.758829i \(-0.274228\pi\)
0.651290 + 0.758829i \(0.274228\pi\)
\(948\) 0 0
\(949\) 54081.7 1.84991
\(950\) 0 0
\(951\) −11923.6 −0.406570
\(952\) 0 0
\(953\) −10019.3 −0.340563 −0.170282 0.985395i \(-0.554468\pi\)
−0.170282 + 0.985395i \(0.554468\pi\)
\(954\) 0 0
\(955\) −3570.16 −0.120971
\(956\) 0 0
\(957\) −4332.30 −0.146336
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14733.1 −0.494548
\(962\) 0 0
\(963\) 1768.15 0.0591670
\(964\) 0 0
\(965\) −47527.3 −1.58545
\(966\) 0 0
\(967\) −27834.4 −0.925641 −0.462820 0.886452i \(-0.653163\pi\)
−0.462820 + 0.886452i \(0.653163\pi\)
\(968\) 0 0
\(969\) 1097.82 0.0363952
\(970\) 0 0
\(971\) 18275.3 0.604000 0.302000 0.953308i \(-0.402346\pi\)
0.302000 + 0.953308i \(0.402346\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 71017.2 2.33269
\(976\) 0 0
\(977\) −43028.9 −1.40903 −0.704513 0.709691i \(-0.748834\pi\)
−0.704513 + 0.709691i \(0.748834\pi\)
\(978\) 0 0
\(979\) −6833.24 −0.223076
\(980\) 0 0
\(981\) −2757.09 −0.0897320
\(982\) 0 0
\(983\) −30559.2 −0.991544 −0.495772 0.868453i \(-0.665115\pi\)
−0.495772 + 0.868453i \(0.665115\pi\)
\(984\) 0 0
\(985\) −53039.2 −1.71571
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12835.4 0.412681
\(990\) 0 0
\(991\) 44945.9 1.44072 0.720360 0.693600i \(-0.243976\pi\)
0.720360 + 0.693600i \(0.243976\pi\)
\(992\) 0 0
\(993\) 26738.5 0.854501
\(994\) 0 0
\(995\) 26711.2 0.851058
\(996\) 0 0
\(997\) 29006.1 0.921397 0.460698 0.887557i \(-0.347599\pi\)
0.460698 + 0.887557i \(0.347599\pi\)
\(998\) 0 0
\(999\) −1515.18 −0.0479861
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.cf.1.2 2
4.3 odd 2 147.4.a.j.1.2 2
7.6 odd 2 2352.4.a.bl.1.1 2
12.11 even 2 441.4.a.n.1.1 2
28.3 even 6 147.4.e.j.79.1 4
28.11 odd 6 147.4.e.k.79.1 4
28.19 even 6 147.4.e.j.67.1 4
28.23 odd 6 147.4.e.k.67.1 4
28.27 even 2 147.4.a.k.1.2 yes 2
84.11 even 6 441.4.e.v.226.2 4
84.23 even 6 441.4.e.v.361.2 4
84.47 odd 6 441.4.e.u.361.2 4
84.59 odd 6 441.4.e.u.226.2 4
84.83 odd 2 441.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.2 2 4.3 odd 2
147.4.a.k.1.2 yes 2 28.27 even 2
147.4.e.j.67.1 4 28.19 even 6
147.4.e.j.79.1 4 28.3 even 6
147.4.e.k.67.1 4 28.23 odd 6
147.4.e.k.79.1 4 28.11 odd 6
441.4.a.n.1.1 2 12.11 even 2
441.4.a.o.1.1 2 84.83 odd 2
441.4.e.u.226.2 4 84.59 odd 6
441.4.e.u.361.2 4 84.47 odd 6
441.4.e.v.226.2 4 84.11 even 6
441.4.e.v.361.2 4 84.23 even 6
2352.4.a.bl.1.1 2 7.6 odd 2
2352.4.a.cf.1.2 2 1.1 even 1 trivial