Properties

Label 294.4.a.j.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -3.89949 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -3.89949 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +7.79899 q^{10} -61.3970 q^{11} -12.0000 q^{12} +53.6985 q^{13} +11.6985 q^{15} +16.0000 q^{16} +32.1005 q^{17} -18.0000 q^{18} -55.7990 q^{19} -15.5980 q^{20} +122.794 q^{22} -94.6030 q^{23} +24.0000 q^{24} -109.794 q^{25} -107.397 q^{26} -27.0000 q^{27} +138.191 q^{29} -23.3970 q^{30} +132.603 q^{31} -32.0000 q^{32} +184.191 q^{33} -64.2010 q^{34} +36.0000 q^{36} +149.206 q^{37} +111.598 q^{38} -161.095 q^{39} +31.1960 q^{40} +427.497 q^{41} +437.588 q^{43} -245.588 q^{44} -35.0955 q^{45} +189.206 q^{46} +57.0051 q^{47} -48.0000 q^{48} +219.588 q^{50} -96.3015 q^{51} +214.794 q^{52} -263.588 q^{53} +54.0000 q^{54} +239.417 q^{55} +167.397 q^{57} -276.382 q^{58} +451.799 q^{59} +46.7939 q^{60} +579.307 q^{61} -265.206 q^{62} +64.0000 q^{64} -209.397 q^{65} -368.382 q^{66} +309.588 q^{67} +128.402 q^{68} +283.809 q^{69} -1058.98 q^{71} -72.0000 q^{72} +1193.66 q^{73} -298.412 q^{74} +329.382 q^{75} -223.196 q^{76} +322.191 q^{78} +1319.56 q^{79} -62.3919 q^{80} +81.0000 q^{81} -854.995 q^{82} +1190.33 q^{83} -125.176 q^{85} -875.176 q^{86} -414.573 q^{87} +491.176 q^{88} -233.085 q^{89} +70.1909 q^{90} -378.412 q^{92} -397.809 q^{93} -114.010 q^{94} +217.588 q^{95} +96.0000 q^{96} -1609.44 q^{97} -552.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} - 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} - 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} - 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} - 16q^{8} + 18q^{9} - 24q^{10} - 4q^{11} - 24q^{12} + 48q^{13} - 36q^{15} + 32q^{16} + 84q^{17} - 36q^{18} - 72q^{19} + 48q^{20} + 8q^{22} - 308q^{23} + 48q^{24} + 18q^{25} - 96q^{26} - 54q^{27} - 80q^{29} + 72q^{30} + 384q^{31} - 64q^{32} + 12q^{33} - 168q^{34} + 72q^{36} + 536q^{37} + 144q^{38} - 144q^{39} - 96q^{40} + 756q^{41} + 400q^{43} - 16q^{44} + 108q^{45} + 616q^{46} + 312q^{47} - 96q^{48} - 36q^{50} - 252q^{51} + 192q^{52} - 52q^{53} + 108q^{54} + 1152q^{55} + 216q^{57} + 160q^{58} + 864q^{59} - 144q^{60} + 1416q^{61} - 768q^{62} + 128q^{64} - 300q^{65} - 24q^{66} + 144q^{67} + 336q^{68} + 924q^{69} - 1524q^{71} - 144q^{72} + 744q^{73} - 1072q^{74} - 54q^{75} - 288q^{76} + 288q^{78} + 976q^{79} + 192q^{80} + 162q^{81} - 1512q^{82} - 312q^{83} + 700q^{85} - 800q^{86} + 240q^{87} + 32q^{88} + 108q^{89} - 216q^{90} - 1232q^{92} - 1152q^{93} - 624q^{94} - 40q^{95} + 192q^{96} - 744q^{97} - 36q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −3.89949 −0.348781 −0.174391 0.984677i \(-0.555796\pi\)
−0.174391 + 0.984677i \(0.555796\pi\)
\(6\) 6.00000 0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 7.79899 0.246626
\(11\) −61.3970 −1.68290 −0.841449 0.540336i \(-0.818297\pi\)
−0.841449 + 0.540336i \(0.818297\pi\)
\(12\) −12.0000 −0.288675
\(13\) 53.6985 1.14564 0.572818 0.819682i \(-0.305850\pi\)
0.572818 + 0.819682i \(0.305850\pi\)
\(14\) 0 0
\(15\) 11.6985 0.201369
\(16\) 16.0000 0.250000
\(17\) 32.1005 0.457972 0.228986 0.973430i \(-0.426459\pi\)
0.228986 + 0.973430i \(0.426459\pi\)
\(18\) −18.0000 −0.235702
\(19\) −55.7990 −0.673746 −0.336873 0.941550i \(-0.609369\pi\)
−0.336873 + 0.941550i \(0.609369\pi\)
\(20\) −15.5980 −0.174391
\(21\) 0 0
\(22\) 122.794 1.18999
\(23\) −94.6030 −0.857656 −0.428828 0.903386i \(-0.641073\pi\)
−0.428828 + 0.903386i \(0.641073\pi\)
\(24\) 24.0000 0.204124
\(25\) −109.794 −0.878352
\(26\) −107.397 −0.810088
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 138.191 0.884876 0.442438 0.896799i \(-0.354114\pi\)
0.442438 + 0.896799i \(0.354114\pi\)
\(30\) −23.3970 −0.142389
\(31\) 132.603 0.768265 0.384132 0.923278i \(-0.374501\pi\)
0.384132 + 0.923278i \(0.374501\pi\)
\(32\) −32.0000 −0.176777
\(33\) 184.191 0.971622
\(34\) −64.2010 −0.323835
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 149.206 0.662955 0.331477 0.943463i \(-0.392453\pi\)
0.331477 + 0.943463i \(0.392453\pi\)
\(38\) 111.598 0.476410
\(39\) −161.095 −0.661434
\(40\) 31.1960 0.123313
\(41\) 427.497 1.62839 0.814194 0.580593i \(-0.197179\pi\)
0.814194 + 0.580593i \(0.197179\pi\)
\(42\) 0 0
\(43\) 437.588 1.55190 0.775948 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(44\) −245.588 −0.841449
\(45\) −35.0955 −0.116260
\(46\) 189.206 0.606455
\(47\) 57.0051 0.176916 0.0884579 0.996080i \(-0.471806\pi\)
0.0884579 + 0.996080i \(0.471806\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) 219.588 0.621088
\(51\) −96.3015 −0.264410
\(52\) 214.794 0.572818
\(53\) −263.588 −0.683143 −0.341572 0.939856i \(-0.610959\pi\)
−0.341572 + 0.939856i \(0.610959\pi\)
\(54\) 54.0000 0.136083
\(55\) 239.417 0.586964
\(56\) 0 0
\(57\) 167.397 0.388987
\(58\) −276.382 −0.625702
\(59\) 451.799 0.996936 0.498468 0.866908i \(-0.333896\pi\)
0.498468 + 0.866908i \(0.333896\pi\)
\(60\) 46.7939 0.100685
\(61\) 579.307 1.21594 0.607972 0.793958i \(-0.291983\pi\)
0.607972 + 0.793958i \(0.291983\pi\)
\(62\) −265.206 −0.543245
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −209.397 −0.399577
\(66\) −368.382 −0.687040
\(67\) 309.588 0.564510 0.282255 0.959339i \(-0.408918\pi\)
0.282255 + 0.959339i \(0.408918\pi\)
\(68\) 128.402 0.228986
\(69\) 283.809 0.495168
\(70\) 0 0
\(71\) −1058.98 −1.77012 −0.885059 0.465479i \(-0.845882\pi\)
−0.885059 + 0.465479i \(0.845882\pi\)
\(72\) −72.0000 −0.117851
\(73\) 1193.66 1.91380 0.956898 0.290424i \(-0.0937964\pi\)
0.956898 + 0.290424i \(0.0937964\pi\)
\(74\) −298.412 −0.468780
\(75\) 329.382 0.507116
\(76\) −223.196 −0.336873
\(77\) 0 0
\(78\) 322.191 0.467704
\(79\) 1319.56 1.87926 0.939632 0.342187i \(-0.111168\pi\)
0.939632 + 0.342187i \(0.111168\pi\)
\(80\) −62.3919 −0.0871954
\(81\) 81.0000 0.111111
\(82\) −854.995 −1.15144
\(83\) 1190.33 1.57417 0.787083 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(84\) 0 0
\(85\) −125.176 −0.159732
\(86\) −875.176 −1.09736
\(87\) −414.573 −0.510883
\(88\) 491.176 0.594994
\(89\) −233.085 −0.277607 −0.138803 0.990320i \(-0.544326\pi\)
−0.138803 + 0.990320i \(0.544326\pi\)
\(90\) 70.1909 0.0822086
\(91\) 0 0
\(92\) −378.412 −0.428828
\(93\) −397.809 −0.443558
\(94\) −114.010 −0.125098
\(95\) 217.588 0.234990
\(96\) 96.0000 0.102062
\(97\) −1609.44 −1.68468 −0.842338 0.538950i \(-0.818821\pi\)
−0.842338 + 0.538950i \(0.818821\pi\)
\(98\) 0 0
\(99\) −552.573 −0.560966
\(100\) −439.176 −0.439176
\(101\) −1479.26 −1.45734 −0.728671 0.684864i \(-0.759861\pi\)
−0.728671 + 0.684864i \(0.759861\pi\)
\(102\) 192.603 0.186966
\(103\) −1145.35 −1.09567 −0.547837 0.836585i \(-0.684548\pi\)
−0.547837 + 0.836585i \(0.684548\pi\)
\(104\) −429.588 −0.405044
\(105\) 0 0
\(106\) 527.176 0.483055
\(107\) 436.955 0.394785 0.197392 0.980325i \(-0.436753\pi\)
0.197392 + 0.980325i \(0.436753\pi\)
\(108\) −108.000 −0.0962250
\(109\) −166.352 −0.146180 −0.0730898 0.997325i \(-0.523286\pi\)
−0.0730898 + 0.997325i \(0.523286\pi\)
\(110\) −478.834 −0.415046
\(111\) −447.618 −0.382757
\(112\) 0 0
\(113\) 490.824 0.408609 0.204305 0.978907i \(-0.434507\pi\)
0.204305 + 0.978907i \(0.434507\pi\)
\(114\) −334.794 −0.275055
\(115\) 368.904 0.299135
\(116\) 552.764 0.442438
\(117\) 483.286 0.381879
\(118\) −903.598 −0.704940
\(119\) 0 0
\(120\) −93.5879 −0.0711947
\(121\) 2438.59 1.83215
\(122\) −1158.61 −0.859802
\(123\) −1282.49 −0.940150
\(124\) 530.412 0.384132
\(125\) 915.578 0.655134
\(126\) 0 0
\(127\) −2616.70 −1.82831 −0.914153 0.405369i \(-0.867143\pi\)
−0.914153 + 0.405369i \(0.867143\pi\)
\(128\) −128.000 −0.0883883
\(129\) −1312.76 −0.895988
\(130\) 418.794 0.282544
\(131\) 177.588 0.118442 0.0592211 0.998245i \(-0.481138\pi\)
0.0592211 + 0.998245i \(0.481138\pi\)
\(132\) 736.764 0.485811
\(133\) 0 0
\(134\) −619.176 −0.399169
\(135\) 105.286 0.0671230
\(136\) −256.804 −0.161917
\(137\) 27.0152 0.0168472 0.00842358 0.999965i \(-0.497319\pi\)
0.00842358 + 0.999965i \(0.497319\pi\)
\(138\) −567.618 −0.350137
\(139\) −922.754 −0.563071 −0.281536 0.959551i \(-0.590844\pi\)
−0.281536 + 0.959551i \(0.590844\pi\)
\(140\) 0 0
\(141\) −171.015 −0.102142
\(142\) 2117.97 1.25166
\(143\) −3296.92 −1.92799
\(144\) 144.000 0.0833333
\(145\) −538.875 −0.308628
\(146\) −2387.32 −1.35326
\(147\) 0 0
\(148\) 596.824 0.331477
\(149\) 746.703 0.410552 0.205276 0.978704i \(-0.434191\pi\)
0.205276 + 0.978704i \(0.434191\pi\)
\(150\) −658.764 −0.358586
\(151\) 2073.15 1.11729 0.558643 0.829408i \(-0.311322\pi\)
0.558643 + 0.829408i \(0.311322\pi\)
\(152\) 446.392 0.238205
\(153\) 288.905 0.152657
\(154\) 0 0
\(155\) −517.085 −0.267956
\(156\) −644.382 −0.330717
\(157\) 1566.22 0.796166 0.398083 0.917349i \(-0.369676\pi\)
0.398083 + 0.917349i \(0.369676\pi\)
\(158\) −2639.12 −1.32884
\(159\) 790.764 0.394413
\(160\) 124.784 0.0616564
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) 98.7333 0.0474441 0.0237221 0.999719i \(-0.492448\pi\)
0.0237221 + 0.999719i \(0.492448\pi\)
\(164\) 1709.99 0.814194
\(165\) −718.252 −0.338884
\(166\) −2380.66 −1.11310
\(167\) −2231.36 −1.03394 −0.516969 0.856004i \(-0.672940\pi\)
−0.516969 + 0.856004i \(0.672940\pi\)
\(168\) 0 0
\(169\) 686.527 0.312484
\(170\) 250.352 0.112948
\(171\) −502.191 −0.224582
\(172\) 1750.35 0.775948
\(173\) 2100.84 0.923261 0.461631 0.887072i \(-0.347265\pi\)
0.461631 + 0.887072i \(0.347265\pi\)
\(174\) 829.145 0.361249
\(175\) 0 0
\(176\) −982.352 −0.420725
\(177\) −1355.40 −0.575581
\(178\) 466.171 0.196298
\(179\) 1722.54 0.719267 0.359634 0.933094i \(-0.382902\pi\)
0.359634 + 0.933094i \(0.382902\pi\)
\(180\) −140.382 −0.0581302
\(181\) 1655.00 0.679644 0.339822 0.940490i \(-0.389633\pi\)
0.339822 + 0.940490i \(0.389633\pi\)
\(182\) 0 0
\(183\) −1737.92 −0.702026
\(184\) 756.824 0.303227
\(185\) −581.828 −0.231226
\(186\) 795.618 0.313643
\(187\) −1970.87 −0.770720
\(188\) 228.020 0.0884579
\(189\) 0 0
\(190\) −435.176 −0.166163
\(191\) −1007.69 −0.381747 −0.190874 0.981615i \(-0.561132\pi\)
−0.190874 + 0.981615i \(0.561132\pi\)
\(192\) −192.000 −0.0721688
\(193\) −7.64849 −0.00285259 −0.00142630 0.999999i \(-0.500454\pi\)
−0.00142630 + 0.999999i \(0.500454\pi\)
\(194\) 3218.87 1.19125
\(195\) 628.191 0.230696
\(196\) 0 0
\(197\) 2689.88 0.972822 0.486411 0.873730i \(-0.338306\pi\)
0.486411 + 0.873730i \(0.338306\pi\)
\(198\) 1105.15 0.396663
\(199\) 867.497 0.309021 0.154511 0.987991i \(-0.450620\pi\)
0.154511 + 0.987991i \(0.450620\pi\)
\(200\) 878.352 0.310544
\(201\) −928.764 −0.325920
\(202\) 2958.51 1.03050
\(203\) 0 0
\(204\) −385.206 −0.132205
\(205\) −1667.02 −0.567951
\(206\) 2290.69 0.774758
\(207\) −851.427 −0.285885
\(208\) 859.176 0.286409
\(209\) 3425.89 1.13385
\(210\) 0 0
\(211\) 162.030 0.0528655 0.0264328 0.999651i \(-0.491585\pi\)
0.0264328 + 0.999651i \(0.491585\pi\)
\(212\) −1054.35 −0.341572
\(213\) 3176.95 1.02198
\(214\) −873.909 −0.279155
\(215\) −1706.37 −0.541272
\(216\) 216.000 0.0680414
\(217\) 0 0
\(218\) 332.703 0.103365
\(219\) −3580.97 −1.10493
\(220\) 957.669 0.293482
\(221\) 1723.75 0.524669
\(222\) 895.236 0.270650
\(223\) −4577.85 −1.37469 −0.687344 0.726332i \(-0.741223\pi\)
−0.687344 + 0.726332i \(0.741223\pi\)
\(224\) 0 0
\(225\) −988.145 −0.292784
\(226\) −981.648 −0.288930
\(227\) −2218.19 −0.648575 −0.324287 0.945959i \(-0.605125\pi\)
−0.324287 + 0.945959i \(0.605125\pi\)
\(228\) 669.588 0.194494
\(229\) 785.217 0.226588 0.113294 0.993562i \(-0.463860\pi\)
0.113294 + 0.993562i \(0.463860\pi\)
\(230\) −737.808 −0.211520
\(231\) 0 0
\(232\) −1105.53 −0.312851
\(233\) −5369.12 −1.50963 −0.754813 0.655940i \(-0.772273\pi\)
−0.754813 + 0.655940i \(0.772273\pi\)
\(234\) −966.573 −0.270029
\(235\) −222.291 −0.0617049
\(236\) 1807.20 0.498468
\(237\) −3958.67 −1.08499
\(238\) 0 0
\(239\) 3713.28 1.00499 0.502493 0.864581i \(-0.332416\pi\)
0.502493 + 0.864581i \(0.332416\pi\)
\(240\) 187.176 0.0503423
\(241\) 6998.62 1.87063 0.935313 0.353821i \(-0.115118\pi\)
0.935313 + 0.353821i \(0.115118\pi\)
\(242\) −4877.18 −1.29552
\(243\) −243.000 −0.0641500
\(244\) 2317.23 0.607972
\(245\) 0 0
\(246\) 2564.98 0.664786
\(247\) −2996.32 −0.771868
\(248\) −1060.82 −0.271623
\(249\) −3570.99 −0.908846
\(250\) −1831.16 −0.463250
\(251\) 3722.75 0.936168 0.468084 0.883684i \(-0.344945\pi\)
0.468084 + 0.883684i \(0.344945\pi\)
\(252\) 0 0
\(253\) 5808.34 1.44335
\(254\) 5233.41 1.29281
\(255\) 375.527 0.0922213
\(256\) 256.000 0.0625000
\(257\) 1230.15 0.298578 0.149289 0.988794i \(-0.452301\pi\)
0.149289 + 0.988794i \(0.452301\pi\)
\(258\) 2625.53 0.633559
\(259\) 0 0
\(260\) −837.588 −0.199788
\(261\) 1243.72 0.294959
\(262\) −355.176 −0.0837513
\(263\) 2388.63 0.560036 0.280018 0.959995i \(-0.409660\pi\)
0.280018 + 0.959995i \(0.409660\pi\)
\(264\) −1473.53 −0.343520
\(265\) 1027.86 0.238268
\(266\) 0 0
\(267\) 699.256 0.160276
\(268\) 1238.35 0.282255
\(269\) 6702.31 1.51913 0.759567 0.650429i \(-0.225411\pi\)
0.759567 + 0.650429i \(0.225411\pi\)
\(270\) −210.573 −0.0474631
\(271\) 4950.37 1.10964 0.554822 0.831969i \(-0.312786\pi\)
0.554822 + 0.831969i \(0.312786\pi\)
\(272\) 513.608 0.114493
\(273\) 0 0
\(274\) −54.0303 −0.0119127
\(275\) 6741.02 1.47818
\(276\) 1135.24 0.247584
\(277\) −3705.18 −0.803691 −0.401846 0.915707i \(-0.631631\pi\)
−0.401846 + 0.915707i \(0.631631\pi\)
\(278\) 1845.51 0.398152
\(279\) 1193.43 0.256088
\(280\) 0 0
\(281\) 9324.74 1.97960 0.989800 0.142465i \(-0.0455029\pi\)
0.989800 + 0.142465i \(0.0455029\pi\)
\(282\) 342.030 0.0722256
\(283\) −5569.49 −1.16986 −0.584932 0.811082i \(-0.698879\pi\)
−0.584932 + 0.811082i \(0.698879\pi\)
\(284\) −4235.94 −0.885059
\(285\) −652.764 −0.135672
\(286\) 6593.85 1.36330
\(287\) 0 0
\(288\) −288.000 −0.0589256
\(289\) −3882.56 −0.790262
\(290\) 1077.75 0.218233
\(291\) 4828.31 0.972648
\(292\) 4774.63 0.956898
\(293\) −1665.31 −0.332042 −0.166021 0.986122i \(-0.553092\pi\)
−0.166021 + 0.986122i \(0.553092\pi\)
\(294\) 0 0
\(295\) −1761.79 −0.347713
\(296\) −1193.65 −0.234390
\(297\) 1657.72 0.323874
\(298\) −1493.41 −0.290304
\(299\) −5080.04 −0.982563
\(300\) 1317.53 0.253558
\(301\) 0 0
\(302\) −4146.29 −0.790040
\(303\) 4437.77 0.841396
\(304\) −892.784 −0.168436
\(305\) −2259.00 −0.424099
\(306\) −577.809 −0.107945
\(307\) −5303.32 −0.985916 −0.492958 0.870053i \(-0.664084\pi\)
−0.492958 + 0.870053i \(0.664084\pi\)
\(308\) 0 0
\(309\) 3436.04 0.632587
\(310\) 1034.17 0.189474
\(311\) 1125.99 0.205302 0.102651 0.994717i \(-0.467267\pi\)
0.102651 + 0.994717i \(0.467267\pi\)
\(312\) 1288.76 0.233852
\(313\) −8299.51 −1.49877 −0.749386 0.662133i \(-0.769651\pi\)
−0.749386 + 0.662133i \(0.769651\pi\)
\(314\) −3132.44 −0.562974
\(315\) 0 0
\(316\) 5278.23 0.939632
\(317\) −4278.76 −0.758105 −0.379053 0.925375i \(-0.623750\pi\)
−0.379053 + 0.925375i \(0.623750\pi\)
\(318\) −1581.53 −0.278892
\(319\) −8484.50 −1.48916
\(320\) −249.568 −0.0435977
\(321\) −1310.86 −0.227929
\(322\) 0 0
\(323\) −1791.18 −0.308556
\(324\) 324.000 0.0555556
\(325\) −5895.77 −1.00627
\(326\) −197.467 −0.0335481
\(327\) 499.055 0.0843969
\(328\) −3419.98 −0.575722
\(329\) 0 0
\(330\) 1436.50 0.239627
\(331\) −1707.12 −0.283479 −0.141739 0.989904i \(-0.545270\pi\)
−0.141739 + 0.989904i \(0.545270\pi\)
\(332\) 4761.33 0.787083
\(333\) 1342.85 0.220985
\(334\) 4462.71 0.731104
\(335\) −1207.24 −0.196891
\(336\) 0 0
\(337\) 1710.67 0.276517 0.138259 0.990396i \(-0.455849\pi\)
0.138259 + 0.990396i \(0.455849\pi\)
\(338\) −1373.05 −0.220960
\(339\) −1472.47 −0.235911
\(340\) −500.703 −0.0798660
\(341\) −8141.42 −1.29291
\(342\) 1004.38 0.158803
\(343\) 0 0
\(344\) −3500.70 −0.548678
\(345\) −1106.71 −0.172705
\(346\) −4201.69 −0.652844
\(347\) 8910.30 1.37847 0.689236 0.724537i \(-0.257946\pi\)
0.689236 + 0.724537i \(0.257946\pi\)
\(348\) −1658.29 −0.255442
\(349\) 5378.68 0.824969 0.412485 0.910965i \(-0.364661\pi\)
0.412485 + 0.910965i \(0.364661\pi\)
\(350\) 0 0
\(351\) −1449.86 −0.220478
\(352\) 1964.70 0.297497
\(353\) −4252.56 −0.641193 −0.320596 0.947216i \(-0.603883\pi\)
−0.320596 + 0.947216i \(0.603883\pi\)
\(354\) 2710.79 0.406997
\(355\) 4129.51 0.617384
\(356\) −932.341 −0.138803
\(357\) 0 0
\(358\) −3445.08 −0.508599
\(359\) 4903.89 0.720940 0.360470 0.932771i \(-0.382616\pi\)
0.360470 + 0.932771i \(0.382616\pi\)
\(360\) 280.764 0.0411043
\(361\) −3745.47 −0.546067
\(362\) −3310.01 −0.480581
\(363\) −7315.76 −1.05779
\(364\) 0 0
\(365\) −4654.66 −0.667497
\(366\) 3475.84 0.496407
\(367\) −4041.57 −0.574845 −0.287423 0.957804i \(-0.592798\pi\)
−0.287423 + 0.957804i \(0.592798\pi\)
\(368\) −1513.65 −0.214414
\(369\) 3847.48 0.542796
\(370\) 1163.66 0.163502
\(371\) 0 0
\(372\) −1591.24 −0.221779
\(373\) −7451.35 −1.03436 −0.517180 0.855877i \(-0.673018\pi\)
−0.517180 + 0.855877i \(0.673018\pi\)
\(374\) 3941.75 0.544981
\(375\) −2746.73 −0.378242
\(376\) −456.040 −0.0625492
\(377\) 7420.64 1.01375
\(378\) 0 0
\(379\) −12564.4 −1.70288 −0.851438 0.524456i \(-0.824269\pi\)
−0.851438 + 0.524456i \(0.824269\pi\)
\(380\) 870.352 0.117495
\(381\) 7850.11 1.05557
\(382\) 2015.38 0.269936
\(383\) −4289.93 −0.572337 −0.286169 0.958179i \(-0.592382\pi\)
−0.286169 + 0.958179i \(0.592382\pi\)
\(384\) 384.000 0.0510310
\(385\) 0 0
\(386\) 15.2970 0.00201709
\(387\) 3938.29 0.517299
\(388\) −6437.75 −0.842338
\(389\) 5662.77 0.738082 0.369041 0.929413i \(-0.379686\pi\)
0.369041 + 0.929413i \(0.379686\pi\)
\(390\) −1256.38 −0.163127
\(391\) −3036.81 −0.392782
\(392\) 0 0
\(393\) −532.764 −0.0683826
\(394\) −5379.76 −0.687889
\(395\) −5145.61 −0.655452
\(396\) −2210.29 −0.280483
\(397\) 14561.4 1.84084 0.920421 0.390928i \(-0.127846\pi\)
0.920421 + 0.390928i \(0.127846\pi\)
\(398\) −1734.99 −0.218511
\(399\) 0 0
\(400\) −1756.70 −0.219588
\(401\) −3742.19 −0.466025 −0.233013 0.972474i \(-0.574858\pi\)
−0.233013 + 0.972474i \(0.574858\pi\)
\(402\) 1857.53 0.230460
\(403\) 7120.58 0.880152
\(404\) −5917.02 −0.728671
\(405\) −315.859 −0.0387535
\(406\) 0 0
\(407\) −9160.80 −1.11569
\(408\) 770.412 0.0934830
\(409\) −3517.17 −0.425215 −0.212608 0.977138i \(-0.568196\pi\)
−0.212608 + 0.977138i \(0.568196\pi\)
\(410\) 3334.05 0.401602
\(411\) −81.0455 −0.00972671
\(412\) −4581.39 −0.547837
\(413\) 0 0
\(414\) 1702.85 0.202152
\(415\) −4641.69 −0.549040
\(416\) −1718.35 −0.202522
\(417\) 2768.26 0.325089
\(418\) −6851.78 −0.801750
\(419\) −7579.52 −0.883732 −0.441866 0.897081i \(-0.645683\pi\)
−0.441866 + 0.897081i \(0.645683\pi\)
\(420\) 0 0
\(421\) −4980.87 −0.576610 −0.288305 0.957539i \(-0.593092\pi\)
−0.288305 + 0.957539i \(0.593092\pi\)
\(422\) −324.061 −0.0373816
\(423\) 513.045 0.0589719
\(424\) 2108.70 0.241528
\(425\) −3524.44 −0.402260
\(426\) −6353.91 −0.722648
\(427\) 0 0
\(428\) 1747.82 0.197392
\(429\) 9890.77 1.11313
\(430\) 3412.74 0.382737
\(431\) 14203.3 1.58736 0.793678 0.608339i \(-0.208164\pi\)
0.793678 + 0.608339i \(0.208164\pi\)
\(432\) −432.000 −0.0481125
\(433\) 3874.82 0.430051 0.215026 0.976608i \(-0.431017\pi\)
0.215026 + 0.976608i \(0.431017\pi\)
\(434\) 0 0
\(435\) 1616.62 0.178187
\(436\) −665.406 −0.0730898
\(437\) 5278.75 0.577842
\(438\) 7161.95 0.781304
\(439\) 7763.82 0.844070 0.422035 0.906579i \(-0.361316\pi\)
0.422035 + 0.906579i \(0.361316\pi\)
\(440\) −1915.34 −0.207523
\(441\) 0 0
\(442\) −3447.50 −0.370997
\(443\) 9662.24 1.03627 0.518134 0.855299i \(-0.326627\pi\)
0.518134 + 0.855299i \(0.326627\pi\)
\(444\) −1790.47 −0.191379
\(445\) 908.915 0.0968241
\(446\) 9155.70 0.972051
\(447\) −2240.11 −0.237032
\(448\) 0 0
\(449\) −10942.6 −1.15014 −0.575069 0.818105i \(-0.695025\pi\)
−0.575069 + 0.818105i \(0.695025\pi\)
\(450\) 1976.29 0.207029
\(451\) −26247.0 −2.74041
\(452\) 1963.30 0.204305
\(453\) −6219.44 −0.645065
\(454\) 4436.38 0.458612
\(455\) 0 0
\(456\) −1339.18 −0.137528
\(457\) 13618.3 1.39396 0.696979 0.717091i \(-0.254527\pi\)
0.696979 + 0.717091i \(0.254527\pi\)
\(458\) −1570.43 −0.160222
\(459\) −866.714 −0.0881367
\(460\) 1475.62 0.149567
\(461\) 11955.8 1.20789 0.603947 0.797025i \(-0.293594\pi\)
0.603947 + 0.797025i \(0.293594\pi\)
\(462\) 0 0
\(463\) 648.503 0.0650939 0.0325470 0.999470i \(-0.489638\pi\)
0.0325470 + 0.999470i \(0.489638\pi\)
\(464\) 2211.05 0.221219
\(465\) 1551.25 0.154705
\(466\) 10738.2 1.06747
\(467\) 2784.74 0.275937 0.137969 0.990437i \(-0.455943\pi\)
0.137969 + 0.990437i \(0.455943\pi\)
\(468\) 1933.15 0.190939
\(469\) 0 0
\(470\) 444.582 0.0436320
\(471\) −4698.66 −0.459667
\(472\) −3614.39 −0.352470
\(473\) −26866.6 −2.61168
\(474\) 7917.35 0.767206
\(475\) 6126.39 0.591785
\(476\) 0 0
\(477\) −2372.29 −0.227714
\(478\) −7426.55 −0.710633
\(479\) 11113.4 1.06009 0.530046 0.847969i \(-0.322175\pi\)
0.530046 + 0.847969i \(0.322175\pi\)
\(480\) −374.352 −0.0355974
\(481\) 8012.14 0.759505
\(482\) −13997.2 −1.32273
\(483\) 0 0
\(484\) 9754.35 0.916074
\(485\) 6275.99 0.587584
\(486\) 486.000 0.0453609
\(487\) −3786.27 −0.352305 −0.176152 0.984363i \(-0.556365\pi\)
−0.176152 + 0.984363i \(0.556365\pi\)
\(488\) −4634.45 −0.429901
\(489\) −296.200 −0.0273919
\(490\) 0 0
\(491\) −9582.12 −0.880723 −0.440361 0.897821i \(-0.645150\pi\)
−0.440361 + 0.897821i \(0.645150\pi\)
\(492\) −5129.97 −0.470075
\(493\) 4436.00 0.405248
\(494\) 5992.64 0.545793
\(495\) 2154.75 0.195655
\(496\) 2121.65 0.192066
\(497\) 0 0
\(498\) 7141.99 0.642651
\(499\) 5581.55 0.500730 0.250365 0.968152i \(-0.419449\pi\)
0.250365 + 0.968152i \(0.419449\pi\)
\(500\) 3662.31 0.327567
\(501\) 6694.07 0.596944
\(502\) −7445.51 −0.661970
\(503\) −14116.3 −1.25132 −0.625661 0.780095i \(-0.715171\pi\)
−0.625661 + 0.780095i \(0.715171\pi\)
\(504\) 0 0
\(505\) 5768.35 0.508294
\(506\) −11616.7 −1.02060
\(507\) −2059.58 −0.180413
\(508\) −10466.8 −0.914153
\(509\) 16787.6 1.46188 0.730941 0.682441i \(-0.239081\pi\)
0.730941 + 0.682441i \(0.239081\pi\)
\(510\) −751.055 −0.0652103
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 1506.57 0.129662
\(514\) −2460.30 −0.211127
\(515\) 4466.27 0.382150
\(516\) −5251.05 −0.447994
\(517\) −3499.94 −0.297731
\(518\) 0 0
\(519\) −6302.53 −0.533045
\(520\) 1675.18 0.141272
\(521\) −5598.61 −0.470786 −0.235393 0.971900i \(-0.575638\pi\)
−0.235393 + 0.971900i \(0.575638\pi\)
\(522\) −2487.44 −0.208567
\(523\) −13270.7 −1.10954 −0.554769 0.832004i \(-0.687193\pi\)
−0.554769 + 0.832004i \(0.687193\pi\)
\(524\) 710.352 0.0592211
\(525\) 0 0
\(526\) −4777.27 −0.396005
\(527\) 4256.62 0.351843
\(528\) 2947.05 0.242905
\(529\) −3217.27 −0.264426
\(530\) −2055.72 −0.168481
\(531\) 4066.19 0.332312
\(532\) 0 0
\(533\) 22956.0 1.86554
\(534\) −1398.51 −0.113332
\(535\) −1703.90 −0.137694
\(536\) −2476.70 −0.199584
\(537\) −5167.63 −0.415269
\(538\) −13404.6 −1.07419
\(539\) 0 0
\(540\) 421.145 0.0335615
\(541\) −23017.5 −1.82921 −0.914603 0.404353i \(-0.867497\pi\)
−0.914603 + 0.404353i \(0.867497\pi\)
\(542\) −9900.74 −0.784637
\(543\) −4965.01 −0.392393
\(544\) −1027.22 −0.0809587
\(545\) 648.687 0.0509848
\(546\) 0 0
\(547\) 4475.84 0.349859 0.174930 0.984581i \(-0.444030\pi\)
0.174930 + 0.984581i \(0.444030\pi\)
\(548\) 108.061 0.00842358
\(549\) 5213.76 0.405315
\(550\) −13482.0 −1.04523
\(551\) −7710.91 −0.596181
\(552\) −2270.47 −0.175068
\(553\) 0 0
\(554\) 7410.35 0.568295
\(555\) 1745.48 0.133499
\(556\) −3691.01 −0.281536
\(557\) 1541.70 0.117278 0.0586390 0.998279i \(-0.481324\pi\)
0.0586390 + 0.998279i \(0.481324\pi\)
\(558\) −2386.85 −0.181082
\(559\) 23497.8 1.77791
\(560\) 0 0
\(561\) 5912.62 0.444975
\(562\) −18649.5 −1.39979
\(563\) 13079.7 0.979115 0.489557 0.871971i \(-0.337158\pi\)
0.489557 + 0.871971i \(0.337158\pi\)
\(564\) −684.061 −0.0510712
\(565\) −1913.97 −0.142515
\(566\) 11139.0 0.827219
\(567\) 0 0
\(568\) 8471.88 0.625831
\(569\) 11411.2 0.840740 0.420370 0.907353i \(-0.361900\pi\)
0.420370 + 0.907353i \(0.361900\pi\)
\(570\) 1305.53 0.0959342
\(571\) 2311.74 0.169428 0.0847139 0.996405i \(-0.473002\pi\)
0.0847139 + 0.996405i \(0.473002\pi\)
\(572\) −13187.7 −0.963995
\(573\) 3023.06 0.220402
\(574\) 0 0
\(575\) 10386.8 0.753324
\(576\) 576.000 0.0416667
\(577\) 25097.0 1.81075 0.905373 0.424617i \(-0.139591\pi\)
0.905373 + 0.424617i \(0.139591\pi\)
\(578\) 7765.12 0.558800
\(579\) 22.9455 0.00164694
\(580\) −2155.50 −0.154314
\(581\) 0 0
\(582\) −9656.62 −0.687766
\(583\) 16183.5 1.14966
\(584\) −9549.26 −0.676629
\(585\) −1884.57 −0.133192
\(586\) 3330.61 0.234789
\(587\) 19789.1 1.39145 0.695726 0.718307i \(-0.255083\pi\)
0.695726 + 0.718307i \(0.255083\pi\)
\(588\) 0 0
\(589\) −7399.12 −0.517615
\(590\) 3523.58 0.245870
\(591\) −8069.64 −0.561659
\(592\) 2387.30 0.165739
\(593\) −2774.13 −0.192108 −0.0960540 0.995376i \(-0.530622\pi\)
−0.0960540 + 0.995376i \(0.530622\pi\)
\(594\) −3315.44 −0.229013
\(595\) 0 0
\(596\) 2986.81 0.205276
\(597\) −2602.49 −0.178414
\(598\) 10160.1 0.694777
\(599\) −26190.3 −1.78649 −0.893245 0.449570i \(-0.851577\pi\)
−0.893245 + 0.449570i \(0.851577\pi\)
\(600\) −2635.05 −0.179293
\(601\) −11038.6 −0.749211 −0.374605 0.927184i \(-0.622222\pi\)
−0.374605 + 0.927184i \(0.622222\pi\)
\(602\) 0 0
\(603\) 2786.29 0.188170
\(604\) 8292.58 0.558643
\(605\) −9509.26 −0.639019
\(606\) −8875.54 −0.594957
\(607\) −1854.39 −0.123999 −0.0619996 0.998076i \(-0.519748\pi\)
−0.0619996 + 0.998076i \(0.519748\pi\)
\(608\) 1785.57 0.119103
\(609\) 0 0
\(610\) 4518.01 0.299883
\(611\) 3061.08 0.202681
\(612\) 1155.62 0.0763286
\(613\) −14856.8 −0.978894 −0.489447 0.872033i \(-0.662801\pi\)
−0.489447 + 0.872033i \(0.662801\pi\)
\(614\) 10606.6 0.697148
\(615\) 5001.07 0.327907
\(616\) 0 0
\(617\) 7600.00 0.495890 0.247945 0.968774i \(-0.420245\pi\)
0.247945 + 0.968774i \(0.420245\pi\)
\(618\) −6872.08 −0.447307
\(619\) 22685.3 1.47302 0.736509 0.676427i \(-0.236473\pi\)
0.736509 + 0.676427i \(0.236473\pi\)
\(620\) −2068.34 −0.133978
\(621\) 2554.28 0.165056
\(622\) −2251.98 −0.145171
\(623\) 0 0
\(624\) −2577.53 −0.165358
\(625\) 10154.0 0.649853
\(626\) 16599.0 1.05979
\(627\) −10277.7 −0.654626
\(628\) 6264.88 0.398083
\(629\) 4789.59 0.303614
\(630\) 0 0
\(631\) −12024.1 −0.758595 −0.379297 0.925275i \(-0.623834\pi\)
−0.379297 + 0.925275i \(0.623834\pi\)
\(632\) −10556.5 −0.664420
\(633\) −486.091 −0.0305219
\(634\) 8557.53 0.536061
\(635\) 10203.8 0.637679
\(636\) 3163.05 0.197206
\(637\) 0 0
\(638\) 16969.0 1.05299
\(639\) −9530.86 −0.590039
\(640\) 499.135 0.0308282
\(641\) 11320.6 0.697559 0.348779 0.937205i \(-0.386596\pi\)
0.348779 + 0.937205i \(0.386596\pi\)
\(642\) 2621.73 0.161170
\(643\) 16843.6 1.03304 0.516521 0.856275i \(-0.327227\pi\)
0.516521 + 0.856275i \(0.327227\pi\)
\(644\) 0 0
\(645\) 5119.12 0.312504
\(646\) 3582.35 0.218182
\(647\) 7719.32 0.469054 0.234527 0.972110i \(-0.424646\pi\)
0.234527 + 0.972110i \(0.424646\pi\)
\(648\) −648.000 −0.0392837
\(649\) −27739.1 −1.67774
\(650\) 11791.5 0.711542
\(651\) 0 0
\(652\) 394.933 0.0237221
\(653\) −30803.2 −1.84597 −0.922987 0.384832i \(-0.874259\pi\)
−0.922987 + 0.384832i \(0.874259\pi\)
\(654\) −998.109 −0.0596776
\(655\) −692.503 −0.0413104
\(656\) 6839.96 0.407097
\(657\) 10742.9 0.637932
\(658\) 0 0
\(659\) 9760.68 0.576968 0.288484 0.957485i \(-0.406849\pi\)
0.288484 + 0.957485i \(0.406849\pi\)
\(660\) −2873.01 −0.169442
\(661\) 18071.1 1.06336 0.531682 0.846944i \(-0.321560\pi\)
0.531682 + 0.846944i \(0.321560\pi\)
\(662\) 3414.23 0.200450
\(663\) −5171.25 −0.302918
\(664\) −9522.65 −0.556552
\(665\) 0 0
\(666\) −2685.71 −0.156260
\(667\) −13073.3 −0.758920
\(668\) −8925.43 −0.516969
\(669\) 13733.5 0.793676
\(670\) 2414.47 0.139223
\(671\) −35567.7 −2.04631
\(672\) 0 0
\(673\) 26591.1 1.52305 0.761524 0.648137i \(-0.224451\pi\)
0.761524 + 0.648137i \(0.224451\pi\)
\(674\) −3421.35 −0.195527
\(675\) 2964.44 0.169039
\(676\) 2746.11 0.156242
\(677\) 33080.3 1.87796 0.938979 0.343975i \(-0.111773\pi\)
0.938979 + 0.343975i \(0.111773\pi\)
\(678\) 2944.95 0.166814
\(679\) 0 0
\(680\) 1001.41 0.0564738
\(681\) 6654.57 0.374455
\(682\) 16282.8 0.914227
\(683\) −10466.1 −0.586344 −0.293172 0.956060i \(-0.594711\pi\)
−0.293172 + 0.956060i \(0.594711\pi\)
\(684\) −2008.76 −0.112291
\(685\) −105.345 −0.00587597
\(686\) 0 0
\(687\) −2355.65 −0.130820
\(688\) 7001.41 0.387974
\(689\) −14154.3 −0.782634
\(690\) 2213.42 0.122121
\(691\) 13269.9 0.730551 0.365275 0.930900i \(-0.380975\pi\)
0.365275 + 0.930900i \(0.380975\pi\)
\(692\) 8403.38 0.461631
\(693\) 0 0
\(694\) −17820.6 −0.974727
\(695\) 3598.27 0.196389
\(696\) 3316.58 0.180625
\(697\) 13722.9 0.745755
\(698\) −10757.4 −0.583341
\(699\) 16107.4 0.871583
\(700\) 0 0
\(701\) −15169.9 −0.817344 −0.408672 0.912681i \(-0.634008\pi\)
−0.408672 + 0.912681i \(0.634008\pi\)
\(702\) 2899.72 0.155901
\(703\) −8325.55 −0.446663
\(704\) −3929.41 −0.210362
\(705\) 666.873 0.0356254
\(706\) 8505.13 0.453392
\(707\) 0 0
\(708\) −5421.59 −0.287791
\(709\) 26317.7 1.39405 0.697026 0.717046i \(-0.254506\pi\)
0.697026 + 0.717046i \(0.254506\pi\)
\(710\) −8259.01 −0.436557
\(711\) 11876.0 0.626421
\(712\) 1864.68 0.0981488
\(713\) −12544.6 −0.658907
\(714\) 0 0
\(715\) 12856.3 0.672447
\(716\) 6890.17 0.359634
\(717\) −11139.8 −0.580229
\(718\) −9807.78 −0.509781
\(719\) 23013.1 1.19366 0.596831 0.802367i \(-0.296426\pi\)
0.596831 + 0.802367i \(0.296426\pi\)
\(720\) −561.527 −0.0290651
\(721\) 0 0
\(722\) 7490.95 0.386128
\(723\) −20995.9 −1.08001
\(724\) 6620.02 0.339822
\(725\) −15172.5 −0.777232
\(726\) 14631.5 0.747971
\(727\) −16265.4 −0.829780 −0.414890 0.909872i \(-0.636180\pi\)
−0.414890 + 0.909872i \(0.636180\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 9309.33 0.471991
\(731\) 14046.8 0.710724
\(732\) −6951.68 −0.351013
\(733\) −866.444 −0.0436601 −0.0218300 0.999762i \(-0.506949\pi\)
−0.0218300 + 0.999762i \(0.506949\pi\)
\(734\) 8083.14 0.406477
\(735\) 0 0
\(736\) 3027.30 0.151614
\(737\) −19007.8 −0.950013
\(738\) −7694.95 −0.383815
\(739\) 10990.1 0.547058 0.273529 0.961864i \(-0.411809\pi\)
0.273529 + 0.961864i \(0.411809\pi\)
\(740\) −2327.31 −0.115613
\(741\) 8988.96 0.445638
\(742\) 0 0
\(743\) −22416.4 −1.10683 −0.553417 0.832905i \(-0.686676\pi\)
−0.553417 + 0.832905i \(0.686676\pi\)
\(744\) 3182.47 0.156821
\(745\) −2911.76 −0.143193
\(746\) 14902.7 0.731403
\(747\) 10713.0 0.524722
\(748\) −7883.49 −0.385360
\(749\) 0 0
\(750\) 5493.47 0.267457
\(751\) −19717.9 −0.958080 −0.479040 0.877793i \(-0.659015\pi\)
−0.479040 + 0.877793i \(0.659015\pi\)
\(752\) 912.081 0.0442289
\(753\) −11168.3 −0.540497
\(754\) −14841.3 −0.716827
\(755\) −8084.22 −0.389689
\(756\) 0 0
\(757\) 839.321 0.0402981 0.0201490 0.999797i \(-0.493586\pi\)
0.0201490 + 0.999797i \(0.493586\pi\)
\(758\) 25128.8 1.20411
\(759\) −17425.0 −0.833318
\(760\) −1740.70 −0.0830815
\(761\) 18364.5 0.874786 0.437393 0.899270i \(-0.355902\pi\)
0.437393 + 0.899270i \(0.355902\pi\)
\(762\) −15700.2 −0.746403
\(763\) 0 0
\(764\) −4030.75 −0.190874
\(765\) −1126.58 −0.0532440
\(766\) 8579.86 0.404704
\(767\) 24260.9 1.14213
\(768\) −768.000 −0.0360844
\(769\) −14890.8 −0.698277 −0.349138 0.937071i \(-0.613526\pi\)
−0.349138 + 0.937071i \(0.613526\pi\)
\(770\) 0 0
\(771\) −3690.45 −0.172384
\(772\) −30.5939 −0.00142630
\(773\) −16606.2 −0.772683 −0.386341 0.922356i \(-0.626261\pi\)
−0.386341 + 0.922356i \(0.626261\pi\)
\(774\) −7876.58 −0.365785
\(775\) −14559.0 −0.674807
\(776\) 12875.5 0.595623
\(777\) 0 0
\(778\) −11325.5 −0.521903
\(779\) −23853.9 −1.09712
\(780\) 2512.76 0.115348
\(781\) 65018.5 2.97893
\(782\) 6073.61 0.277739
\(783\) −3731.15 −0.170294
\(784\) 0 0
\(785\) −6107.47 −0.277688
\(786\) 1065.53 0.0483538
\(787\) 3156.20 0.142956 0.0714781 0.997442i \(-0.477228\pi\)
0.0714781 + 0.997442i \(0.477228\pi\)
\(788\) 10759.5 0.486411
\(789\) −7165.90 −0.323337
\(790\) 10291.2 0.463475
\(791\) 0 0
\(792\) 4420.58 0.198331
\(793\) 31107.9 1.39303
\(794\) −29122.8 −1.30167
\(795\) −3083.58 −0.137564
\(796\) 3469.99 0.154511
\(797\) −13514.8 −0.600652 −0.300326 0.953837i \(-0.597096\pi\)
−0.300326 + 0.953837i \(0.597096\pi\)
\(798\) 0 0
\(799\) 1829.89 0.0810224
\(800\) 3513.41 0.155272
\(801\) −2097.77 −0.0925356
\(802\) 7484.38 0.329530
\(803\) −73287.0 −3.22072
\(804\) −3715.05 −0.162960
\(805\) 0 0
\(806\) −14241.2 −0.622362
\(807\) −20106.9 −0.877073
\(808\) 11834.0 0.515248
\(809\) −26758.1 −1.16287 −0.581437 0.813591i \(-0.697509\pi\)
−0.581437 + 0.813591i \(0.697509\pi\)
\(810\) 631.718 0.0274029
\(811\) −15920.7 −0.689338 −0.344669 0.938724i \(-0.612009\pi\)
−0.344669 + 0.938724i \(0.612009\pi\)
\(812\) 0 0
\(813\) −14851.1 −0.640653
\(814\) 18321.6 0.788909
\(815\) −385.010 −0.0165476
\(816\) −1540.82 −0.0661025
\(817\) −24417.0 −1.04558
\(818\) 7034.35 0.300673
\(819\) 0 0
\(820\) −6668.10 −0.283976
\(821\) −19306.2 −0.820696 −0.410348 0.911929i \(-0.634593\pi\)
−0.410348 + 0.911929i \(0.634593\pi\)
\(822\) 162.091 0.00687782
\(823\) 791.000 0.0335025 0.0167512 0.999860i \(-0.494668\pi\)
0.0167512 + 0.999860i \(0.494668\pi\)
\(824\) 9162.77 0.387379
\(825\) −20223.0 −0.853426
\(826\) 0 0
\(827\) 29537.6 1.24199 0.620993 0.783816i \(-0.286730\pi\)
0.620993 + 0.783816i \(0.286730\pi\)
\(828\) −3405.71 −0.142943
\(829\) −5766.52 −0.241592 −0.120796 0.992677i \(-0.538545\pi\)
−0.120796 + 0.992677i \(0.538545\pi\)
\(830\) 9283.38 0.388230
\(831\) 11115.5 0.464011
\(832\) 3436.70 0.143205
\(833\) 0 0
\(834\) −5536.52 −0.229873
\(835\) 8701.16 0.360618
\(836\) 13703.6 0.566923
\(837\) −3580.28 −0.147853
\(838\) 15159.0 0.624893
\(839\) 29726.4 1.22321 0.611603 0.791165i \(-0.290525\pi\)
0.611603 + 0.791165i \(0.290525\pi\)
\(840\) 0 0
\(841\) −5292.27 −0.216994
\(842\) 9961.75 0.407725
\(843\) −27974.2 −1.14292
\(844\) 648.121 0.0264328
\(845\) −2677.11 −0.108989
\(846\) −1026.09 −0.0416994
\(847\) 0 0
\(848\) −4217.41 −0.170786
\(849\) 16708.5 0.675421
\(850\) 7048.88 0.284441
\(851\) −14115.3 −0.568587
\(852\) 12707.8 0.510989
\(853\) −17829.6 −0.715677 −0.357838 0.933784i \(-0.616486\pi\)
−0.357838 + 0.933784i \(0.616486\pi\)
\(854\) 0 0
\(855\) 1958.29 0.0783300
\(856\) −3495.64 −0.139578
\(857\) −39682.4 −1.58171 −0.790856 0.612003i \(-0.790364\pi\)
−0.790856 + 0.612003i \(0.790364\pi\)
\(858\) −19781.5 −0.787099
\(859\) −2195.13 −0.0871909 −0.0435955 0.999049i \(-0.513881\pi\)
−0.0435955 + 0.999049i \(0.513881\pi\)
\(860\) −6825.49 −0.270636
\(861\) 0 0
\(862\) −28406.6 −1.12243
\(863\) 31917.1 1.25894 0.629472 0.777023i \(-0.283271\pi\)
0.629472 + 0.777023i \(0.283271\pi\)
\(864\) 864.000 0.0340207
\(865\) −8192.23 −0.322016
\(866\) −7749.65 −0.304092
\(867\) 11647.7 0.456258
\(868\) 0 0
\(869\) −81016.8 −3.16261
\(870\) −3233.25 −0.125997
\(871\) 16624.4 0.646724
\(872\) 1330.81 0.0516823
\(873\) −14484.9 −0.561559
\(874\) −10557.5 −0.408596
\(875\) 0 0
\(876\) −14323.9 −0.552465
\(877\) 28842.9 1.11055 0.555276 0.831666i \(-0.312613\pi\)
0.555276 + 0.831666i \(0.312613\pi\)
\(878\) −15527.6 −0.596848
\(879\) 4995.92 0.191704
\(880\) 3830.67 0.146741
\(881\) −15350.2 −0.587015 −0.293508 0.955957i \(-0.594823\pi\)
−0.293508 + 0.955957i \(0.594823\pi\)
\(882\) 0 0
\(883\) 6089.64 0.232087 0.116043 0.993244i \(-0.462979\pi\)
0.116043 + 0.993244i \(0.462979\pi\)
\(884\) 6894.99 0.262335
\(885\) 5285.36 0.200752
\(886\) −19324.5 −0.732752
\(887\) −384.441 −0.0145527 −0.00727637 0.999974i \(-0.502316\pi\)
−0.00727637 + 0.999974i \(0.502316\pi\)
\(888\) 3580.95 0.135325
\(889\) 0 0
\(890\) −1817.83 −0.0684650
\(891\) −4973.15 −0.186989
\(892\) −18311.4 −0.687344
\(893\) −3180.82 −0.119196
\(894\) 4480.22 0.167607
\(895\) −6717.05 −0.250867
\(896\) 0 0
\(897\) 15240.1 0.567283
\(898\) 21885.2 0.813271
\(899\) 18324.5 0.679819
\(900\) −3952.58 −0.146392
\(901\) −8461.30 −0.312860
\(902\) 52494.1 1.93776
\(903\) 0 0
\(904\) −3926.59 −0.144465
\(905\) −6453.68 −0.237047
\(906\) 12438.9 0.456130
\(907\) −7267.93 −0.266073 −0.133036 0.991111i \(-0.542473\pi\)
−0.133036 + 0.991111i \(0.542473\pi\)
\(908\) −8872.76 −0.324287
\(909\) −13313.3 −0.485780
\(910\) 0 0
\(911\) −8535.12 −0.310408 −0.155204 0.987882i \(-0.549603\pi\)
−0.155204 + 0.987882i \(0.549603\pi\)
\(912\) 2678.35 0.0972468
\(913\) −73082.7 −2.64916
\(914\) −27236.7 −0.985678
\(915\) 6777.01 0.244854
\(916\) 3140.87 0.113294
\(917\) 0 0
\(918\) 1733.43 0.0623220
\(919\) −10851.7 −0.389516 −0.194758 0.980851i \(-0.562392\pi\)
−0.194758 + 0.980851i \(0.562392\pi\)
\(920\) −2951.23 −0.105760
\(921\) 15909.9 0.569219
\(922\) −23911.7 −0.854110
\(923\) −56865.9 −2.02791
\(924\) 0 0
\(925\) −16381.9 −0.582307
\(926\) −1297.01 −0.0460284
\(927\) −10308.1 −0.365224
\(928\) −4422.11 −0.156425
\(929\) −1560.64 −0.0551161 −0.0275581 0.999620i \(-0.508773\pi\)
−0.0275581 + 0.999620i \(0.508773\pi\)
\(930\) −3102.51 −0.109393
\(931\) 0 0
\(932\) −21476.5 −0.754813
\(933\) −3377.97 −0.118531
\(934\) −5569.49 −0.195117
\(935\) 7685.41 0.268813
\(936\) −3866.29 −0.135015
\(937\) 11978.4 0.417627 0.208813 0.977956i \(-0.433040\pi\)
0.208813 + 0.977956i \(0.433040\pi\)
\(938\) 0 0
\(939\) 24898.5 0.865317
\(940\) −889.164 −0.0308525
\(941\) 24597.7 0.852137 0.426068 0.904691i \(-0.359898\pi\)
0.426068 + 0.904691i \(0.359898\pi\)
\(942\) 9397.32 0.325033
\(943\) −40442.6 −1.39660
\(944\) 7228.78 0.249234
\(945\) 0 0
\(946\) 53733.1 1.84674
\(947\) −10834.6 −0.371783 −0.185891 0.982570i \(-0.559517\pi\)
−0.185891 + 0.982570i \(0.559517\pi\)
\(948\) −15834.7 −0.542497
\(949\) 64097.6 2.19252
\(950\) −12252.8 −0.418456
\(951\) 12836.3 0.437692
\(952\) 0 0
\(953\) 701.418 0.0238417 0.0119209 0.999929i \(-0.496205\pi\)
0.0119209 + 0.999929i \(0.496205\pi\)
\(954\) 4744.58 0.161018
\(955\) 3929.47 0.133146
\(956\) 14853.1 0.502493
\(957\) 25453.5 0.859765
\(958\) −22226.8 −0.749599
\(959\) 0 0
\(960\) 748.703 0.0251711
\(961\) −12207.4 −0.409769
\(962\) −16024.3 −0.537051
\(963\) 3932.59 0.131595
\(964\) 27994.5 0.935313
\(965\) 29.8252 0.000994931 0
\(966\) 0 0
\(967\) 42402.5 1.41011 0.705053 0.709154i \(-0.250923\pi\)
0.705053 + 0.709154i \(0.250923\pi\)
\(968\) −19508.7 −0.647762
\(969\) 5373.53 0.178145
\(970\) −12552.0 −0.415484
\(971\) 15463.4 0.511064 0.255532 0.966801i \(-0.417749\pi\)
0.255532 + 0.966801i \(0.417749\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) 7572.55 0.249117
\(975\) 17687.3 0.580971
\(976\) 9268.91 0.303986
\(977\) −34484.1 −1.12922 −0.564609 0.825359i \(-0.690973\pi\)
−0.564609 + 0.825359i \(0.690973\pi\)
\(978\) 592.400 0.0193690
\(979\) 14310.7 0.467184
\(980\) 0 0
\(981\) −1497.16 −0.0487266
\(982\) 19164.2 0.622765
\(983\) 7335.36 0.238008 0.119004 0.992894i \(-0.462030\pi\)
0.119004 + 0.992894i \(0.462030\pi\)
\(984\) 10259.9 0.332393
\(985\) −10489.2 −0.339302
\(986\) −8872.00 −0.286554
\(987\) 0 0
\(988\) −11985.3 −0.385934
\(989\) −41397.1 −1.33099
\(990\) −4309.51 −0.138349
\(991\) −10123.0 −0.324487 −0.162243 0.986751i \(-0.551873\pi\)
−0.162243 + 0.986751i \(0.551873\pi\)
\(992\) −4243.30 −0.135811
\(993\) 5121.35 0.163667
\(994\) 0 0
\(995\) −3382.80 −0.107781
\(996\) −14284.0 −0.454423
\(997\) 56669.2 1.80013 0.900066 0.435755i \(-0.143518\pi\)
0.900066 + 0.435755i \(0.143518\pi\)
\(998\) −11163.1 −0.354070
\(999\) −4028.56 −0.127586
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 294.4.a.j.1.1 2
3.2 odd 2 882.4.a.bc.1.2 2
4.3 odd 2 2352.4.a.cd.1.1 2
7.2 even 3 294.4.e.o.67.2 4
7.3 odd 6 294.4.e.n.79.1 4
7.4 even 3 294.4.e.o.79.2 4
7.5 odd 6 294.4.e.n.67.1 4
7.6 odd 2 294.4.a.k.1.2 yes 2
21.2 odd 6 882.4.g.bd.361.1 4
21.5 even 6 882.4.g.y.361.2 4
21.11 odd 6 882.4.g.bd.667.1 4
21.17 even 6 882.4.g.y.667.2 4
21.20 even 2 882.4.a.bi.1.1 2
28.27 even 2 2352.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.1 2 1.1 even 1 trivial
294.4.a.k.1.2 yes 2 7.6 odd 2
294.4.e.n.67.1 4 7.5 odd 6
294.4.e.n.79.1 4 7.3 odd 6
294.4.e.o.67.2 4 7.2 even 3
294.4.e.o.79.2 4 7.4 even 3
882.4.a.bc.1.2 2 3.2 odd 2
882.4.a.bi.1.1 2 21.20 even 2
882.4.g.y.361.2 4 21.5 even 6
882.4.g.y.667.2 4 21.17 even 6
882.4.g.bd.361.1 4 21.2 odd 6
882.4.g.bd.667.1 4 21.11 odd 6
2352.4.a.bn.1.2 2 28.27 even 2
2352.4.a.cd.1.1 2 4.3 odd 2