Properties

Label 294.4.a.l.1.2
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.58579 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.58579 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} -9.17157 q^{10} -6.48528 q^{11} -12.0000 q^{12} -45.2132 q^{13} +13.7574 q^{15} +16.0000 q^{16} -81.5563 q^{17} +18.0000 q^{18} +5.05382 q^{19} -18.3431 q^{20} -12.9706 q^{22} +106.250 q^{23} -24.0000 q^{24} -103.971 q^{25} -90.4264 q^{26} -27.0000 q^{27} -268.132 q^{29} +27.5147 q^{30} -292.368 q^{31} +32.0000 q^{32} +19.4558 q^{33} -163.113 q^{34} +36.0000 q^{36} +114.558 q^{37} +10.1076 q^{38} +135.640 q^{39} -36.6863 q^{40} +161.605 q^{41} -471.294 q^{43} -25.9411 q^{44} -41.2721 q^{45} +212.500 q^{46} -346.004 q^{47} -48.0000 q^{48} -207.941 q^{50} +244.669 q^{51} -180.853 q^{52} +405.529 q^{53} -54.0000 q^{54} +29.7401 q^{55} -15.1615 q^{57} -536.264 q^{58} +253.436 q^{59} +55.0294 q^{60} +751.217 q^{61} -584.735 q^{62} +64.0000 q^{64} +207.338 q^{65} +38.9117 q^{66} +11.6468 q^{67} -326.225 q^{68} -318.749 q^{69} -681.661 q^{71} +72.0000 q^{72} -685.457 q^{73} +229.117 q^{74} +311.912 q^{75} +20.2153 q^{76} +271.279 q^{78} +0.264069 q^{79} -73.3726 q^{80} +81.0000 q^{81} +323.210 q^{82} +437.137 q^{83} +374.000 q^{85} -942.587 q^{86} +804.396 q^{87} -51.8823 q^{88} +58.5126 q^{89} -82.5442 q^{90} +424.999 q^{92} +877.103 q^{93} -692.008 q^{94} -23.1758 q^{95} -96.0000 q^{96} +1280.09 q^{97} -58.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} + 16q^{8} + 18q^{9} - 24q^{10} + 4q^{11} - 24q^{12} - 48q^{13} + 36q^{15} + 32q^{16} - 132q^{17} + 36q^{18} - 120q^{19} - 48q^{20} + 8q^{22} - 76q^{23} - 48q^{24} - 174q^{25} - 96q^{26} - 54q^{27} - 112q^{29} + 72q^{30} - 432q^{31} + 64q^{32} - 12q^{33} - 264q^{34} + 72q^{36} - 280q^{37} - 240q^{38} + 144q^{39} - 96q^{40} - 36q^{41} - 128q^{43} + 16q^{44} - 108q^{45} - 152q^{46} + 264q^{47} - 96q^{48} - 348q^{50} + 396q^{51} - 192q^{52} + 268q^{53} - 108q^{54} - 48q^{55} + 360q^{57} - 224q^{58} - 336q^{59} + 144q^{60} + 504q^{61} - 864q^{62} + 128q^{64} + 228q^{65} - 24q^{66} - 384q^{67} - 528q^{68} + 228q^{69} - 396q^{71} + 144q^{72} + 312q^{73} - 560q^{74} + 522q^{75} - 480q^{76} + 288q^{78} - 848q^{79} - 192q^{80} + 162q^{81} - 72q^{82} + 648q^{83} + 748q^{85} - 256q^{86} + 336q^{87} + 32q^{88} + 612q^{89} - 216q^{90} - 304q^{92} + 1296q^{93} + 528q^{94} + 904q^{95} - 192q^{96} + 2184q^{97} + 36q^{99} + O(q^{100}) \)

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
\(5\) −4.58579 −0.410165 −0.205083 0.978745i \(-0.565746\pi\)
−0.205083 + 0.978745i \(0.565746\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −9.17157 −0.290031
\(11\) −6.48528 −0.177762 −0.0888812 0.996042i \(-0.528329\pi\)
−0.0888812 + 0.996042i \(0.528329\pi\)
\(12\) −12.0000 −0.288675
\(13\) −45.2132 −0.964607 −0.482303 0.876004i \(-0.660200\pi\)
−0.482303 + 0.876004i \(0.660200\pi\)
\(14\) 0 0
\(15\) 13.7574 0.236809
\(16\) 16.0000 0.250000
\(17\) −81.5563 −1.16355 −0.581774 0.813350i \(-0.697641\pi\)
−0.581774 + 0.813350i \(0.697641\pi\)
\(18\) 18.0000 0.235702
\(19\) 5.05382 0.0610225 0.0305112 0.999534i \(-0.490286\pi\)
0.0305112 + 0.999534i \(0.490286\pi\)
\(20\) −18.3431 −0.205083
\(21\) 0 0
\(22\) −12.9706 −0.125697
\(23\) 106.250 0.963244 0.481622 0.876379i \(-0.340048\pi\)
0.481622 + 0.876379i \(0.340048\pi\)
\(24\) −24.0000 −0.204124
\(25\) −103.971 −0.831765
\(26\) −90.4264 −0.682080
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −268.132 −1.71693 −0.858463 0.512875i \(-0.828580\pi\)
−0.858463 + 0.512875i \(0.828580\pi\)
\(30\) 27.5147 0.167449
\(31\) −292.368 −1.69390 −0.846948 0.531676i \(-0.821562\pi\)
−0.846948 + 0.531676i \(0.821562\pi\)
\(32\) 32.0000 0.176777
\(33\) 19.4558 0.102631
\(34\) −163.113 −0.822753
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 114.558 0.509008 0.254504 0.967072i \(-0.418088\pi\)
0.254504 + 0.967072i \(0.418088\pi\)
\(38\) 10.1076 0.0431494
\(39\) 135.640 0.556916
\(40\) −36.6863 −0.145015
\(41\) 161.605 0.615573 0.307786 0.951456i \(-0.400412\pi\)
0.307786 + 0.951456i \(0.400412\pi\)
\(42\) 0 0
\(43\) −471.294 −1.67143 −0.835716 0.549162i \(-0.814947\pi\)
−0.835716 + 0.549162i \(0.814947\pi\)
\(44\) −25.9411 −0.0888812
\(45\) −41.2721 −0.136722
\(46\) 212.500 0.681116
\(47\) −346.004 −1.07383 −0.536914 0.843637i \(-0.680410\pi\)
−0.536914 + 0.843637i \(0.680410\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) −207.941 −0.588146
\(51\) 244.669 0.671775
\(52\) −180.853 −0.482303
\(53\) 405.529 1.05101 0.525507 0.850790i \(-0.323876\pi\)
0.525507 + 0.850790i \(0.323876\pi\)
\(54\) −54.0000 −0.136083
\(55\) 29.7401 0.0729119
\(56\) 0 0
\(57\) −15.1615 −0.0352313
\(58\) −536.264 −1.21405
\(59\) 253.436 0.559229 0.279614 0.960112i \(-0.409793\pi\)
0.279614 + 0.960112i \(0.409793\pi\)
\(60\) 55.0294 0.118404
\(61\) 751.217 1.57678 0.788390 0.615176i \(-0.210915\pi\)
0.788390 + 0.615176i \(0.210915\pi\)
\(62\) −584.735 −1.19776
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 207.338 0.395648
\(66\) 38.9117 0.0725712
\(67\) 11.6468 0.0212370 0.0106185 0.999944i \(-0.496620\pi\)
0.0106185 + 0.999944i \(0.496620\pi\)
\(68\) −326.225 −0.581774
\(69\) −318.749 −0.556129
\(70\) 0 0
\(71\) −681.661 −1.13941 −0.569706 0.821848i \(-0.692943\pi\)
−0.569706 + 0.821848i \(0.692943\pi\)
\(72\) 72.0000 0.117851
\(73\) −685.457 −1.09900 −0.549498 0.835495i \(-0.685181\pi\)
−0.549498 + 0.835495i \(0.685181\pi\)
\(74\) 229.117 0.359923
\(75\) 311.912 0.480219
\(76\) 20.2153 0.0305112
\(77\) 0 0
\(78\) 271.279 0.393799
\(79\) 0.264069 0.000376077 0 0.000188038 1.00000i \(-0.499940\pi\)
0.000188038 1.00000i \(0.499940\pi\)
\(80\) −73.3726 −0.102541
\(81\) 81.0000 0.111111
\(82\) 323.210 0.435276
\(83\) 437.137 0.578097 0.289048 0.957314i \(-0.406661\pi\)
0.289048 + 0.957314i \(0.406661\pi\)
\(84\) 0 0
\(85\) 374.000 0.477247
\(86\) −942.587 −1.18188
\(87\) 804.396 0.991268
\(88\) −51.8823 −0.0628485
\(89\) 58.5126 0.0696891 0.0348445 0.999393i \(-0.488906\pi\)
0.0348445 + 0.999393i \(0.488906\pi\)
\(90\) −82.5442 −0.0966769
\(91\) 0 0
\(92\) 424.999 0.481622
\(93\) 877.103 0.977971
\(94\) −692.008 −0.759311
\(95\) −23.1758 −0.0250293
\(96\) −96.0000 −0.102062
\(97\) 1280.09 1.33993 0.669966 0.742391i \(-0.266308\pi\)
0.669966 + 0.742391i \(0.266308\pi\)
\(98\) 0 0
\(99\) −58.3675 −0.0592541
\(100\) −415.882 −0.415882
\(101\) 1306.39 1.28704 0.643518 0.765431i \(-0.277474\pi\)
0.643518 + 0.765431i \(0.277474\pi\)
\(102\) 489.338 0.475017
\(103\) 758.975 0.726058 0.363029 0.931778i \(-0.381743\pi\)
0.363029 + 0.931778i \(0.381743\pi\)
\(104\) −361.706 −0.341040
\(105\) 0 0
\(106\) 811.058 0.743178
\(107\) 1262.51 1.14067 0.570336 0.821412i \(-0.306813\pi\)
0.570336 + 0.821412i \(0.306813\pi\)
\(108\) −108.000 −0.0962250
\(109\) −2105.53 −1.85021 −0.925105 0.379711i \(-0.876023\pi\)
−0.925105 + 0.379711i \(0.876023\pi\)
\(110\) 59.4802 0.0515565
\(111\) −343.675 −0.293876
\(112\) 0 0
\(113\) 1535.76 1.27852 0.639258 0.768992i \(-0.279241\pi\)
0.639258 + 0.768992i \(0.279241\pi\)
\(114\) −30.3229 −0.0249123
\(115\) −487.239 −0.395089
\(116\) −1072.53 −0.858463
\(117\) −406.919 −0.321536
\(118\) 506.871 0.395435
\(119\) 0 0
\(120\) 110.059 0.0837246
\(121\) −1288.94 −0.968401
\(122\) 1502.43 1.11495
\(123\) −484.815 −0.355401
\(124\) −1169.47 −0.846948
\(125\) 1050.01 0.751326
\(126\) 0 0
\(127\) 24.1749 0.0168911 0.00844557 0.999964i \(-0.497312\pi\)
0.00844557 + 0.999964i \(0.497312\pi\)
\(128\) 128.000 0.0883883
\(129\) 1413.88 0.965002
\(130\) 414.676 0.279765
\(131\) −1581.53 −1.05480 −0.527400 0.849617i \(-0.676833\pi\)
−0.527400 + 0.849617i \(0.676833\pi\)
\(132\) 77.8234 0.0513156
\(133\) 0 0
\(134\) 23.2935 0.0150168
\(135\) 123.816 0.0789363
\(136\) −652.451 −0.411376
\(137\) −745.188 −0.464713 −0.232357 0.972631i \(-0.574644\pi\)
−0.232357 + 0.972631i \(0.574644\pi\)
\(138\) −637.499 −0.393243
\(139\) 1373.60 0.838179 0.419090 0.907945i \(-0.362349\pi\)
0.419090 + 0.907945i \(0.362349\pi\)
\(140\) 0 0
\(141\) 1038.01 0.619975
\(142\) −1363.32 −0.805686
\(143\) 293.220 0.171471
\(144\) 144.000 0.0833333
\(145\) 1229.60 0.704224
\(146\) −1370.91 −0.777107
\(147\) 0 0
\(148\) 458.234 0.254504
\(149\) 620.530 0.341180 0.170590 0.985342i \(-0.445433\pi\)
0.170590 + 0.985342i \(0.445433\pi\)
\(150\) 623.823 0.339566
\(151\) −1939.26 −1.04513 −0.522567 0.852598i \(-0.675025\pi\)
−0.522567 + 0.852598i \(0.675025\pi\)
\(152\) 40.4306 0.0215747
\(153\) −734.007 −0.387849
\(154\) 0 0
\(155\) 1340.74 0.694777
\(156\) 542.558 0.278458
\(157\) 412.843 0.209863 0.104931 0.994479i \(-0.466538\pi\)
0.104931 + 0.994479i \(0.466538\pi\)
\(158\) 0.528137 0.000265926 0
\(159\) −1216.59 −0.606803
\(160\) −146.745 −0.0725077
\(161\) 0 0
\(162\) 162.000 0.0785674
\(163\) −3907.44 −1.87763 −0.938817 0.344417i \(-0.888077\pi\)
−0.938817 + 0.344417i \(0.888077\pi\)
\(164\) 646.420 0.307786
\(165\) −89.2203 −0.0420957
\(166\) 874.274 0.408776
\(167\) 1286.41 0.596082 0.298041 0.954553i \(-0.403667\pi\)
0.298041 + 0.954553i \(0.403667\pi\)
\(168\) 0 0
\(169\) −152.766 −0.0695340
\(170\) 748.000 0.337465
\(171\) 45.4844 0.0203408
\(172\) −1885.17 −0.835716
\(173\) 1251.26 0.549892 0.274946 0.961460i \(-0.411340\pi\)
0.274946 + 0.961460i \(0.411340\pi\)
\(174\) 1608.79 0.700932
\(175\) 0 0
\(176\) −103.765 −0.0444406
\(177\) −760.307 −0.322871
\(178\) 117.025 0.0492776
\(179\) 3623.51 1.51304 0.756520 0.653971i \(-0.226898\pi\)
0.756520 + 0.653971i \(0.226898\pi\)
\(180\) −165.088 −0.0683609
\(181\) −181.727 −0.0746280 −0.0373140 0.999304i \(-0.511880\pi\)
−0.0373140 + 0.999304i \(0.511880\pi\)
\(182\) 0 0
\(183\) −2253.65 −0.910354
\(184\) 849.998 0.340558
\(185\) −525.341 −0.208777
\(186\) 1754.21 0.691530
\(187\) 528.916 0.206835
\(188\) −1384.02 −0.536914
\(189\) 0 0
\(190\) −46.3515 −0.0176984
\(191\) 1481.28 0.561160 0.280580 0.959831i \(-0.409473\pi\)
0.280580 + 0.959831i \(0.409473\pi\)
\(192\) −192.000 −0.0721688
\(193\) −356.708 −0.133038 −0.0665192 0.997785i \(-0.521189\pi\)
−0.0665192 + 0.997785i \(0.521189\pi\)
\(194\) 2560.18 0.947476
\(195\) −622.014 −0.228428
\(196\) 0 0
\(197\) 4890.53 1.76871 0.884355 0.466816i \(-0.154599\pi\)
0.884355 + 0.466816i \(0.154599\pi\)
\(198\) −116.735 −0.0418990
\(199\) −3542.85 −1.26204 −0.631020 0.775766i \(-0.717364\pi\)
−0.631020 + 0.775766i \(0.717364\pi\)
\(200\) −831.765 −0.294073
\(201\) −34.9403 −0.0122612
\(202\) 2612.78 0.910071
\(203\) 0 0
\(204\) 978.676 0.335887
\(205\) −741.087 −0.252486
\(206\) 1517.95 0.513401
\(207\) 956.248 0.321081
\(208\) −723.411 −0.241152
\(209\) −32.7755 −0.0108475
\(210\) 0 0
\(211\) −4289.50 −1.39953 −0.699765 0.714373i \(-0.746712\pi\)
−0.699765 + 0.714373i \(0.746712\pi\)
\(212\) 1622.12 0.525507
\(213\) 2044.98 0.657840
\(214\) 2525.03 0.806576
\(215\) 2161.25 0.685563
\(216\) −216.000 −0.0680414
\(217\) 0 0
\(218\) −4211.05 −1.30830
\(219\) 2056.37 0.634505
\(220\) 118.960 0.0364560
\(221\) 3687.42 1.12237
\(222\) −687.351 −0.207802
\(223\) −5795.73 −1.74041 −0.870204 0.492692i \(-0.836013\pi\)
−0.870204 + 0.492692i \(0.836013\pi\)
\(224\) 0 0
\(225\) −935.735 −0.277255
\(226\) 3071.53 0.904048
\(227\) −4104.04 −1.19998 −0.599989 0.800008i \(-0.704828\pi\)
−0.599989 + 0.800008i \(0.704828\pi\)
\(228\) −60.6459 −0.0176157
\(229\) 1296.83 0.374223 0.187111 0.982339i \(-0.440087\pi\)
0.187111 + 0.982339i \(0.440087\pi\)
\(230\) −974.478 −0.279370
\(231\) 0 0
\(232\) −2145.06 −0.607025
\(233\) −1478.33 −0.415660 −0.207830 0.978165i \(-0.566640\pi\)
−0.207830 + 0.978165i \(0.566640\pi\)
\(234\) −813.838 −0.227360
\(235\) 1586.70 0.440447
\(236\) 1013.74 0.279614
\(237\) −0.792206 −0.000217128 0
\(238\) 0 0
\(239\) −3776.92 −1.02221 −0.511106 0.859518i \(-0.670764\pi\)
−0.511106 + 0.859518i \(0.670764\pi\)
\(240\) 220.118 0.0592022
\(241\) 3996.38 1.06817 0.534086 0.845430i \(-0.320656\pi\)
0.534086 + 0.845430i \(0.320656\pi\)
\(242\) −2577.88 −0.684763
\(243\) −243.000 −0.0641500
\(244\) 3004.87 0.788390
\(245\) 0 0
\(246\) −969.631 −0.251306
\(247\) −228.500 −0.0588627
\(248\) −2338.94 −0.598882
\(249\) −1311.41 −0.333764
\(250\) 2100.02 0.531268
\(251\) −5423.58 −1.36388 −0.681939 0.731409i \(-0.738863\pi\)
−0.681939 + 0.731409i \(0.738863\pi\)
\(252\) 0 0
\(253\) −689.060 −0.171229
\(254\) 48.3498 0.0119438
\(255\) −1122.00 −0.275539
\(256\) 256.000 0.0625000
\(257\) −5964.22 −1.44762 −0.723809 0.690000i \(-0.757611\pi\)
−0.723809 + 0.690000i \(0.757611\pi\)
\(258\) 2827.76 0.682359
\(259\) 0 0
\(260\) 829.352 0.197824
\(261\) −2413.19 −0.572309
\(262\) −3163.05 −0.745856
\(263\) 5166.01 1.21122 0.605608 0.795763i \(-0.292930\pi\)
0.605608 + 0.795763i \(0.292930\pi\)
\(264\) 155.647 0.0362856
\(265\) −1859.67 −0.431089
\(266\) 0 0
\(267\) −175.538 −0.0402350
\(268\) 46.5870 0.0106185
\(269\) −3883.29 −0.880180 −0.440090 0.897954i \(-0.645054\pi\)
−0.440090 + 0.897954i \(0.645054\pi\)
\(270\) 247.632 0.0558164
\(271\) −5527.65 −1.23904 −0.619522 0.784979i \(-0.712674\pi\)
−0.619522 + 0.784979i \(0.712674\pi\)
\(272\) −1304.90 −0.290887
\(273\) 0 0
\(274\) −1490.38 −0.328602
\(275\) 674.278 0.147856
\(276\) −1275.00 −0.278065
\(277\) 2268.12 0.491979 0.245989 0.969273i \(-0.420887\pi\)
0.245989 + 0.969273i \(0.420887\pi\)
\(278\) 2747.19 0.592682
\(279\) −2631.31 −0.564632
\(280\) 0 0
\(281\) 725.656 0.154053 0.0770267 0.997029i \(-0.475457\pi\)
0.0770267 + 0.997029i \(0.475457\pi\)
\(282\) 2076.03 0.438388
\(283\) −4237.00 −0.889976 −0.444988 0.895536i \(-0.646792\pi\)
−0.444988 + 0.895536i \(0.646792\pi\)
\(284\) −2726.64 −0.569706
\(285\) 69.5273 0.0144507
\(286\) 586.441 0.121248
\(287\) 0 0
\(288\) 288.000 0.0589256
\(289\) 1738.44 0.353845
\(290\) 2459.19 0.497961
\(291\) −3840.27 −0.773611
\(292\) −2741.83 −0.549498
\(293\) 4373.78 0.872079 0.436039 0.899928i \(-0.356381\pi\)
0.436039 + 0.899928i \(0.356381\pi\)
\(294\) 0 0
\(295\) −1162.20 −0.229376
\(296\) 916.468 0.179961
\(297\) 175.103 0.0342104
\(298\) 1241.06 0.241251
\(299\) −4803.89 −0.929152
\(300\) 1247.65 0.240110
\(301\) 0 0
\(302\) −3878.53 −0.739021
\(303\) −3919.17 −0.743070
\(304\) 80.8612 0.0152556
\(305\) −3444.92 −0.646740
\(306\) −1468.01 −0.274251
\(307\) −4133.47 −0.768435 −0.384217 0.923243i \(-0.625529\pi\)
−0.384217 + 0.923243i \(0.625529\pi\)
\(308\) 0 0
\(309\) −2276.92 −0.419190
\(310\) 2681.47 0.491282
\(311\) −5063.23 −0.923182 −0.461591 0.887093i \(-0.652721\pi\)
−0.461591 + 0.887093i \(0.652721\pi\)
\(312\) 1085.12 0.196900
\(313\) −7411.56 −1.33842 −0.669211 0.743073i \(-0.733368\pi\)
−0.669211 + 0.743073i \(0.733368\pi\)
\(314\) 825.686 0.148395
\(315\) 0 0
\(316\) 1.05627 0.000188038 0
\(317\) −6737.05 −1.19366 −0.596831 0.802367i \(-0.703574\pi\)
−0.596831 + 0.802367i \(0.703574\pi\)
\(318\) −2433.17 −0.429074
\(319\) 1738.91 0.305205
\(320\) −293.490 −0.0512707
\(321\) −3787.54 −0.658567
\(322\) 0 0
\(323\) −412.171 −0.0710026
\(324\) 324.000 0.0555556
\(325\) 4700.84 0.802326
\(326\) −7814.88 −1.32769
\(327\) 6316.58 1.06822
\(328\) 1292.84 0.217638
\(329\) 0 0
\(330\) −178.441 −0.0297662
\(331\) 11175.9 1.85585 0.927923 0.372771i \(-0.121592\pi\)
0.927923 + 0.372771i \(0.121592\pi\)
\(332\) 1748.55 0.289048
\(333\) 1031.03 0.169669
\(334\) 2572.83 0.421494
\(335\) −53.4095 −0.00871067
\(336\) 0 0
\(337\) 9379.78 1.51617 0.758085 0.652156i \(-0.226135\pi\)
0.758085 + 0.652156i \(0.226135\pi\)
\(338\) −305.532 −0.0491680
\(339\) −4607.29 −0.738152
\(340\) 1496.00 0.238624
\(341\) 1896.09 0.301111
\(342\) 90.9688 0.0143831
\(343\) 0 0
\(344\) −3770.35 −0.590941
\(345\) 1461.72 0.228105
\(346\) 2502.51 0.388832
\(347\) 5681.46 0.878953 0.439476 0.898254i \(-0.355164\pi\)
0.439476 + 0.898254i \(0.355164\pi\)
\(348\) 3217.58 0.495634
\(349\) 704.250 0.108016 0.0540080 0.998541i \(-0.482800\pi\)
0.0540080 + 0.998541i \(0.482800\pi\)
\(350\) 0 0
\(351\) 1220.76 0.185639
\(352\) −207.529 −0.0314242
\(353\) 4284.96 0.646078 0.323039 0.946386i \(-0.395295\pi\)
0.323039 + 0.946386i \(0.395295\pi\)
\(354\) −1520.61 −0.228304
\(355\) 3125.95 0.467347
\(356\) 234.051 0.0348445
\(357\) 0 0
\(358\) 7247.03 1.06988
\(359\) 4661.27 0.685272 0.342636 0.939468i \(-0.388680\pi\)
0.342636 + 0.939468i \(0.388680\pi\)
\(360\) −330.177 −0.0483384
\(361\) −6833.46 −0.996276
\(362\) −363.454 −0.0527700
\(363\) 3866.82 0.559106
\(364\) 0 0
\(365\) 3143.36 0.450770
\(366\) −4507.30 −0.643717
\(367\) 6935.30 0.986430 0.493215 0.869907i \(-0.335822\pi\)
0.493215 + 0.869907i \(0.335822\pi\)
\(368\) 1700.00 0.240811
\(369\) 1454.45 0.205191
\(370\) −1050.68 −0.147628
\(371\) 0 0
\(372\) 3508.41 0.488985
\(373\) −3081.10 −0.427704 −0.213852 0.976866i \(-0.568601\pi\)
−0.213852 + 0.976866i \(0.568601\pi\)
\(374\) 1057.83 0.146254
\(375\) −3150.03 −0.433778
\(376\) −2768.03 −0.379655
\(377\) 12123.1 1.65616
\(378\) 0 0
\(379\) 941.827 0.127647 0.0638237 0.997961i \(-0.479670\pi\)
0.0638237 + 0.997961i \(0.479670\pi\)
\(380\) −92.7030 −0.0125146
\(381\) −72.5247 −0.00975210
\(382\) 2962.56 0.396800
\(383\) 677.526 0.0903915 0.0451958 0.998978i \(-0.485609\pi\)
0.0451958 + 0.998978i \(0.485609\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −713.416 −0.0940724
\(387\) −4241.64 −0.557144
\(388\) 5120.36 0.669966
\(389\) −11865.4 −1.54653 −0.773266 0.634082i \(-0.781378\pi\)
−0.773266 + 0.634082i \(0.781378\pi\)
\(390\) −1244.03 −0.161523
\(391\) −8665.34 −1.12078
\(392\) 0 0
\(393\) 4744.58 0.608989
\(394\) 9781.06 1.25067
\(395\) −1.21096 −0.000154254 0
\(396\) −233.470 −0.0296271
\(397\) 5140.76 0.649893 0.324947 0.945732i \(-0.394654\pi\)
0.324947 + 0.945732i \(0.394654\pi\)
\(398\) −7085.70 −0.892397
\(399\) 0 0
\(400\) −1663.53 −0.207941
\(401\) −12382.0 −1.54196 −0.770981 0.636858i \(-0.780234\pi\)
−0.770981 + 0.636858i \(0.780234\pi\)
\(402\) −69.8805 −0.00866996
\(403\) 13218.9 1.63394
\(404\) 5225.56 0.643518
\(405\) −371.449 −0.0455739
\(406\) 0 0
\(407\) −742.944 −0.0904824
\(408\) 1957.35 0.237508
\(409\) 15875.6 1.91931 0.959657 0.281173i \(-0.0907235\pi\)
0.959657 + 0.281173i \(0.0907235\pi\)
\(410\) −1482.17 −0.178535
\(411\) 2235.56 0.268302
\(412\) 3035.90 0.363029
\(413\) 0 0
\(414\) 1912.50 0.227039
\(415\) −2004.62 −0.237115
\(416\) −1446.82 −0.170520
\(417\) −4120.79 −0.483923
\(418\) −65.5509 −0.00767034
\(419\) 16111.9 1.87857 0.939283 0.343145i \(-0.111492\pi\)
0.939283 + 0.343145i \(0.111492\pi\)
\(420\) 0 0
\(421\) 8691.58 1.00618 0.503090 0.864234i \(-0.332197\pi\)
0.503090 + 0.864234i \(0.332197\pi\)
\(422\) −8578.99 −0.989618
\(423\) −3114.04 −0.357943
\(424\) 3244.23 0.371589
\(425\) 8479.46 0.967798
\(426\) 4089.97 0.465163
\(427\) 0 0
\(428\) 5050.06 0.570336
\(429\) −879.661 −0.0989987
\(430\) 4322.50 0.484766
\(431\) −4195.10 −0.468842 −0.234421 0.972135i \(-0.575319\pi\)
−0.234421 + 0.972135i \(0.575319\pi\)
\(432\) −432.000 −0.0481125
\(433\) 5426.54 0.602270 0.301135 0.953582i \(-0.402635\pi\)
0.301135 + 0.953582i \(0.402635\pi\)
\(434\) 0 0
\(435\) −3688.79 −0.406584
\(436\) −8422.11 −0.925105
\(437\) 536.968 0.0587795
\(438\) 4112.74 0.448663
\(439\) 3771.24 0.410003 0.205002 0.978762i \(-0.434280\pi\)
0.205002 + 0.978762i \(0.434280\pi\)
\(440\) 237.921 0.0257783
\(441\) 0 0
\(442\) 7374.85 0.793633
\(443\) −5930.30 −0.636020 −0.318010 0.948087i \(-0.603015\pi\)
−0.318010 + 0.948087i \(0.603015\pi\)
\(444\) −1374.70 −0.146938
\(445\) −268.326 −0.0285840
\(446\) −11591.5 −1.23065
\(447\) −1861.59 −0.196980
\(448\) 0 0
\(449\) −529.065 −0.0556083 −0.0278041 0.999613i \(-0.508851\pi\)
−0.0278041 + 0.999613i \(0.508851\pi\)
\(450\) −1871.47 −0.196049
\(451\) −1048.05 −0.109426
\(452\) 6143.05 0.639258
\(453\) 5817.79 0.603408
\(454\) −8208.09 −0.848512
\(455\) 0 0
\(456\) −121.292 −0.0124562
\(457\) 10057.2 1.02944 0.514721 0.857358i \(-0.327896\pi\)
0.514721 + 0.857358i \(0.327896\pi\)
\(458\) 2593.66 0.264616
\(459\) 2202.02 0.223925
\(460\) −1948.96 −0.197545
\(461\) 5010.31 0.506190 0.253095 0.967441i \(-0.418552\pi\)
0.253095 + 0.967441i \(0.418552\pi\)
\(462\) 0 0
\(463\) −7124.38 −0.715114 −0.357557 0.933891i \(-0.616390\pi\)
−0.357557 + 0.933891i \(0.616390\pi\)
\(464\) −4290.11 −0.429232
\(465\) −4022.21 −0.401130
\(466\) −2956.67 −0.293916
\(467\) −7501.44 −0.743309 −0.371654 0.928371i \(-0.621209\pi\)
−0.371654 + 0.928371i \(0.621209\pi\)
\(468\) −1627.68 −0.160768
\(469\) 0 0
\(470\) 3173.40 0.311443
\(471\) −1238.53 −0.121164
\(472\) 2027.49 0.197717
\(473\) 3056.47 0.297118
\(474\) −1.58441 −0.000153533 0
\(475\) −525.449 −0.0507563
\(476\) 0 0
\(477\) 3649.76 0.350338
\(478\) −7553.84 −0.722813
\(479\) 8173.80 0.779688 0.389844 0.920881i \(-0.372529\pi\)
0.389844 + 0.920881i \(0.372529\pi\)
\(480\) 440.235 0.0418623
\(481\) −5179.55 −0.490992
\(482\) 7992.76 0.755312
\(483\) 0 0
\(484\) −5155.76 −0.484200
\(485\) −5870.22 −0.549594
\(486\) −486.000 −0.0453609
\(487\) −11968.8 −1.11367 −0.556835 0.830623i \(-0.687985\pi\)
−0.556835 + 0.830623i \(0.687985\pi\)
\(488\) 6009.74 0.557476
\(489\) 11722.3 1.08405
\(490\) 0 0
\(491\) 2079.96 0.191176 0.0955878 0.995421i \(-0.469527\pi\)
0.0955878 + 0.995421i \(0.469527\pi\)
\(492\) −1939.26 −0.177701
\(493\) 21867.9 1.99773
\(494\) −456.999 −0.0416222
\(495\) 267.661 0.0243040
\(496\) −4677.88 −0.423474
\(497\) 0 0
\(498\) −2622.82 −0.236007
\(499\) 12834.4 1.15140 0.575699 0.817662i \(-0.304730\pi\)
0.575699 + 0.817662i \(0.304730\pi\)
\(500\) 4200.04 0.375663
\(501\) −3859.24 −0.344148
\(502\) −10847.2 −0.964408
\(503\) −16808.8 −1.48999 −0.744997 0.667068i \(-0.767549\pi\)
−0.744997 + 0.667068i \(0.767549\pi\)
\(504\) 0 0
\(505\) −5990.82 −0.527897
\(506\) −1378.12 −0.121077
\(507\) 458.299 0.0401455
\(508\) 96.6996 0.00844557
\(509\) 4270.26 0.371859 0.185929 0.982563i \(-0.440470\pi\)
0.185929 + 0.982563i \(0.440470\pi\)
\(510\) −2244.00 −0.194835
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) −136.453 −0.0117438
\(514\) −11928.4 −1.02362
\(515\) −3480.50 −0.297804
\(516\) 5655.52 0.482501
\(517\) 2243.93 0.190886
\(518\) 0 0
\(519\) −3753.77 −0.317480
\(520\) 1658.70 0.139883
\(521\) −15283.3 −1.28517 −0.642584 0.766215i \(-0.722138\pi\)
−0.642584 + 0.766215i \(0.722138\pi\)
\(522\) −4826.38 −0.404683
\(523\) 4499.33 0.376180 0.188090 0.982152i \(-0.439770\pi\)
0.188090 + 0.982152i \(0.439770\pi\)
\(524\) −6326.11 −0.527400
\(525\) 0 0
\(526\) 10332.0 0.856459
\(527\) 23844.4 1.97093
\(528\) 311.294 0.0256578
\(529\) −877.984 −0.0721611
\(530\) −3719.34 −0.304826
\(531\) 2280.92 0.186410
\(532\) 0 0
\(533\) −7306.69 −0.593785
\(534\) −351.076 −0.0284504
\(535\) −5789.62 −0.467864
\(536\) 93.1740 0.00750840
\(537\) −10870.5 −0.873554
\(538\) −7766.59 −0.622382
\(539\) 0 0
\(540\) 495.265 0.0394682
\(541\) 3970.82 0.315561 0.157781 0.987474i \(-0.449566\pi\)
0.157781 + 0.987474i \(0.449566\pi\)
\(542\) −11055.3 −0.876137
\(543\) 545.181 0.0430865
\(544\) −2609.80 −0.205688
\(545\) 9655.50 0.758892
\(546\) 0 0
\(547\) −2703.90 −0.211353 −0.105677 0.994401i \(-0.533701\pi\)
−0.105677 + 0.994401i \(0.533701\pi\)
\(548\) −2980.75 −0.232357
\(549\) 6760.96 0.525593
\(550\) 1348.56 0.104550
\(551\) −1355.09 −0.104771
\(552\) −2549.99 −0.196621
\(553\) 0 0
\(554\) 4536.24 0.347881
\(555\) 1576.02 0.120538
\(556\) 5494.38 0.419090
\(557\) 790.824 0.0601585 0.0300793 0.999548i \(-0.490424\pi\)
0.0300793 + 0.999548i \(0.490424\pi\)
\(558\) −5262.62 −0.399255
\(559\) 21308.7 1.61227
\(560\) 0 0
\(561\) −1586.75 −0.119416
\(562\) 1451.31 0.108932
\(563\) −7517.15 −0.562718 −0.281359 0.959603i \(-0.590785\pi\)
−0.281359 + 0.959603i \(0.590785\pi\)
\(564\) 4152.05 0.309987
\(565\) −7042.68 −0.524403
\(566\) −8473.99 −0.629308
\(567\) 0 0
\(568\) −5453.29 −0.402843
\(569\) 13945.4 1.02746 0.513728 0.857953i \(-0.328264\pi\)
0.513728 + 0.857953i \(0.328264\pi\)
\(570\) 139.055 0.0102182
\(571\) −2118.49 −0.155265 −0.0776323 0.996982i \(-0.524736\pi\)
−0.0776323 + 0.996982i \(0.524736\pi\)
\(572\) 1172.88 0.0857354
\(573\) −4443.84 −0.323986
\(574\) 0 0
\(575\) −11046.8 −0.801192
\(576\) 576.000 0.0416667
\(577\) −22857.7 −1.64918 −0.824592 0.565728i \(-0.808595\pi\)
−0.824592 + 0.565728i \(0.808595\pi\)
\(578\) 3476.88 0.250206
\(579\) 1070.12 0.0768098
\(580\) 4918.38 0.352112
\(581\) 0 0
\(582\) −7680.54 −0.547025
\(583\) −2629.97 −0.186831
\(584\) −5483.66 −0.388554
\(585\) 1866.04 0.131883
\(586\) 8747.56 0.616653
\(587\) −23955.3 −1.68440 −0.842199 0.539167i \(-0.818739\pi\)
−0.842199 + 0.539167i \(0.818739\pi\)
\(588\) 0 0
\(589\) −1477.57 −0.103366
\(590\) −2324.40 −0.162194
\(591\) −14671.6 −1.02116
\(592\) 1832.94 0.127252
\(593\) 10778.0 0.746373 0.373186 0.927756i \(-0.378265\pi\)
0.373186 + 0.927756i \(0.378265\pi\)
\(594\) 350.205 0.0241904
\(595\) 0 0
\(596\) 2482.12 0.170590
\(597\) 10628.6 0.728639
\(598\) −9607.79 −0.657009
\(599\) 7597.58 0.518245 0.259122 0.965845i \(-0.416567\pi\)
0.259122 + 0.965845i \(0.416567\pi\)
\(600\) 2495.29 0.169783
\(601\) −19956.1 −1.35445 −0.677225 0.735776i \(-0.736818\pi\)
−0.677225 + 0.735776i \(0.736818\pi\)
\(602\) 0 0
\(603\) 104.821 0.00707899
\(604\) −7757.06 −0.522567
\(605\) 5910.81 0.397204
\(606\) −7838.33 −0.525430
\(607\) 236.311 0.0158016 0.00790079 0.999969i \(-0.497485\pi\)
0.00790079 + 0.999969i \(0.497485\pi\)
\(608\) 161.722 0.0107873
\(609\) 0 0
\(610\) −6889.85 −0.457314
\(611\) 15644.0 1.03582
\(612\) −2936.03 −0.193925
\(613\) 26414.9 1.74044 0.870219 0.492664i \(-0.163977\pi\)
0.870219 + 0.492664i \(0.163977\pi\)
\(614\) −8266.94 −0.543365
\(615\) 2223.26 0.145773
\(616\) 0 0
\(617\) 18473.6 1.20538 0.602689 0.797976i \(-0.294096\pi\)
0.602689 + 0.797976i \(0.294096\pi\)
\(618\) −4553.85 −0.296412
\(619\) −16047.9 −1.04204 −0.521018 0.853546i \(-0.674448\pi\)
−0.521018 + 0.853546i \(0.674448\pi\)
\(620\) 5362.94 0.347388
\(621\) −2868.74 −0.185376
\(622\) −10126.5 −0.652788
\(623\) 0 0
\(624\) 2170.23 0.139229
\(625\) 8181.20 0.523597
\(626\) −14823.1 −0.946407
\(627\) 98.3264 0.00626280
\(628\) 1651.37 0.104931
\(629\) −9342.97 −0.592255
\(630\) 0 0
\(631\) −15065.7 −0.950487 −0.475243 0.879854i \(-0.657640\pi\)
−0.475243 + 0.879854i \(0.657640\pi\)
\(632\) 2.11255 0.000132963 0
\(633\) 12868.5 0.808020
\(634\) −13474.1 −0.844046
\(635\) −110.861 −0.00692816
\(636\) −4866.35 −0.303401
\(637\) 0 0
\(638\) 3477.82 0.215812
\(639\) −6134.95 −0.379804
\(640\) −586.981 −0.0362538
\(641\) −31198.4 −1.92241 −0.961203 0.275842i \(-0.911043\pi\)
−0.961203 + 0.275842i \(0.911043\pi\)
\(642\) −7575.08 −0.465677
\(643\) 12497.9 0.766517 0.383259 0.923641i \(-0.374802\pi\)
0.383259 + 0.923641i \(0.374802\pi\)
\(644\) 0 0
\(645\) −6483.75 −0.395810
\(646\) −824.343 −0.0502064
\(647\) 9929.72 0.603366 0.301683 0.953408i \(-0.402452\pi\)
0.301683 + 0.953408i \(0.402452\pi\)
\(648\) 648.000 0.0392837
\(649\) −1643.60 −0.0994099
\(650\) 9401.68 0.567330
\(651\) 0 0
\(652\) −15629.8 −0.938817
\(653\) −8145.58 −0.488149 −0.244075 0.969756i \(-0.578484\pi\)
−0.244075 + 0.969756i \(0.578484\pi\)
\(654\) 12633.2 0.755345
\(655\) 7252.55 0.432642
\(656\) 2585.68 0.153893
\(657\) −6169.11 −0.366332
\(658\) 0 0
\(659\) −16975.8 −1.00347 −0.501733 0.865022i \(-0.667304\pi\)
−0.501733 + 0.865022i \(0.667304\pi\)
\(660\) −356.881 −0.0210479
\(661\) −20637.8 −1.21440 −0.607199 0.794550i \(-0.707707\pi\)
−0.607199 + 0.794550i \(0.707707\pi\)
\(662\) 22351.9 1.31228
\(663\) −11062.3 −0.647999
\(664\) 3497.10 0.204388
\(665\) 0 0
\(666\) 2062.05 0.119974
\(667\) −28489.0 −1.65382
\(668\) 5145.66 0.298041
\(669\) 17387.2 1.00482
\(670\) −106.819 −0.00615937
\(671\) −4871.86 −0.280292
\(672\) 0 0
\(673\) −2150.29 −0.123161 −0.0615807 0.998102i \(-0.519614\pi\)
−0.0615807 + 0.998102i \(0.519614\pi\)
\(674\) 18759.6 1.07209
\(675\) 2807.21 0.160073
\(676\) −611.065 −0.0347670
\(677\) 27783.4 1.57726 0.788628 0.614871i \(-0.210792\pi\)
0.788628 + 0.614871i \(0.210792\pi\)
\(678\) −9214.58 −0.521952
\(679\) 0 0
\(680\) 2992.00 0.168732
\(681\) 12312.1 0.692807
\(682\) 3792.17 0.212918
\(683\) 18181.8 1.01860 0.509302 0.860588i \(-0.329904\pi\)
0.509302 + 0.860588i \(0.329904\pi\)
\(684\) 181.938 0.0101704
\(685\) 3417.27 0.190609
\(686\) 0 0
\(687\) −3890.49 −0.216058
\(688\) −7540.70 −0.417858
\(689\) −18335.3 −1.01381
\(690\) 2923.43 0.161294
\(691\) −23935.1 −1.31771 −0.658853 0.752272i \(-0.728958\pi\)
−0.658853 + 0.752272i \(0.728958\pi\)
\(692\) 5005.02 0.274946
\(693\) 0 0
\(694\) 11362.9 0.621513
\(695\) −6299.02 −0.343792
\(696\) 6435.17 0.350466
\(697\) −13179.9 −0.716249
\(698\) 1408.50 0.0763789
\(699\) 4435.00 0.239982
\(700\) 0 0
\(701\) −20627.2 −1.11138 −0.555691 0.831389i \(-0.687546\pi\)
−0.555691 + 0.831389i \(0.687546\pi\)
\(702\) 2441.51 0.131266
\(703\) 578.958 0.0310609
\(704\) −415.058 −0.0222203
\(705\) −4760.10 −0.254292
\(706\) 8569.93 0.456846
\(707\) 0 0
\(708\) −3041.23 −0.161436
\(709\) −6584.64 −0.348789 −0.174394 0.984676i \(-0.555797\pi\)
−0.174394 + 0.984676i \(0.555797\pi\)
\(710\) 6251.90 0.330464
\(711\) 2.37662 0.000125359 0
\(712\) 468.101 0.0246388
\(713\) −31064.0 −1.63163
\(714\) 0 0
\(715\) −1344.65 −0.0703313
\(716\) 14494.1 0.756520
\(717\) 11330.8 0.590174
\(718\) 9322.54 0.484560
\(719\) −170.886 −0.00886366 −0.00443183 0.999990i \(-0.501411\pi\)
−0.00443183 + 0.999990i \(0.501411\pi\)
\(720\) −660.353 −0.0341804
\(721\) 0 0
\(722\) −13666.9 −0.704474
\(723\) −11989.1 −0.616710
\(724\) −726.908 −0.0373140
\(725\) 27877.8 1.42808
\(726\) 7733.65 0.395348
\(727\) 11127.5 0.567671 0.283836 0.958873i \(-0.408393\pi\)
0.283836 + 0.958873i \(0.408393\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 6286.72 0.318742
\(731\) 38437.0 1.94479
\(732\) −9014.61 −0.455177
\(733\) 22575.9 1.13760 0.568800 0.822476i \(-0.307408\pi\)
0.568800 + 0.822476i \(0.307408\pi\)
\(734\) 13870.6 0.697511
\(735\) 0 0
\(736\) 3399.99 0.170279
\(737\) −75.5325 −0.00377513
\(738\) 2908.89 0.145092
\(739\) −22936.4 −1.14172 −0.570860 0.821048i \(-0.693390\pi\)
−0.570860 + 0.821048i \(0.693390\pi\)
\(740\) −2101.36 −0.104389
\(741\) 685.499 0.0339844
\(742\) 0 0
\(743\) 16973.4 0.838081 0.419041 0.907967i \(-0.362366\pi\)
0.419041 + 0.907967i \(0.362366\pi\)
\(744\) 7016.82 0.345765
\(745\) −2845.62 −0.139940
\(746\) −6162.21 −0.302432
\(747\) 3934.23 0.192699
\(748\) 2115.66 0.103418
\(749\) 0 0
\(750\) −6300.06 −0.306728
\(751\) −21197.9 −1.02999 −0.514994 0.857194i \(-0.672206\pi\)
−0.514994 + 0.857194i \(0.672206\pi\)
\(752\) −5536.07 −0.268457
\(753\) 16270.8 0.787436
\(754\) 24246.2 1.17108
\(755\) 8893.05 0.428677
\(756\) 0 0
\(757\) −7962.24 −0.382289 −0.191144 0.981562i \(-0.561220\pi\)
−0.191144 + 0.981562i \(0.561220\pi\)
\(758\) 1883.65 0.0902604
\(759\) 2067.18 0.0988588
\(760\) −185.406 −0.00884919
\(761\) −26856.2 −1.27928 −0.639642 0.768673i \(-0.720917\pi\)
−0.639642 + 0.768673i \(0.720917\pi\)
\(762\) −145.049 −0.00689578
\(763\) 0 0
\(764\) 5925.12 0.280580
\(765\) 3366.00 0.159082
\(766\) 1355.05 0.0639165
\(767\) −11458.6 −0.539436
\(768\) −768.000 −0.0360844
\(769\) 12183.6 0.571331 0.285666 0.958329i \(-0.407785\pi\)
0.285666 + 0.958329i \(0.407785\pi\)
\(770\) 0 0
\(771\) 17892.7 0.835783
\(772\) −1426.83 −0.0665192
\(773\) 17455.2 0.812187 0.406093 0.913832i \(-0.366891\pi\)
0.406093 + 0.913832i \(0.366891\pi\)
\(774\) −8483.28 −0.393960
\(775\) 30397.6 1.40892
\(776\) 10240.7 0.473738
\(777\) 0 0
\(778\) −23730.8 −1.09356
\(779\) 816.724 0.0375638
\(780\) −2488.06 −0.114214
\(781\) 4420.76 0.202545
\(782\) −17330.7 −0.792512
\(783\) 7239.56 0.330423
\(784\) 0 0
\(785\) −1893.21 −0.0860785
\(786\) 9489.16 0.430620
\(787\) −30981.0 −1.40324 −0.701622 0.712549i \(-0.747540\pi\)
−0.701622 + 0.712549i \(0.747540\pi\)
\(788\) 19562.1 0.884355
\(789\) −15498.0 −0.699296
\(790\) −2.42193 −0.000109074 0
\(791\) 0 0
\(792\) −466.940 −0.0209495
\(793\) −33964.9 −1.52097
\(794\) 10281.5 0.459544
\(795\) 5579.01 0.248889
\(796\) −14171.4 −0.631020
\(797\) 10517.6 0.467446 0.233723 0.972303i \(-0.424909\pi\)
0.233723 + 0.972303i \(0.424909\pi\)
\(798\) 0 0
\(799\) 28218.8 1.24945
\(800\) −3327.06 −0.147037
\(801\) 526.614 0.0232297
\(802\) −24764.0 −1.09033
\(803\) 4445.38 0.195360
\(804\) −139.761 −0.00613059
\(805\) 0 0
\(806\) 26437.7 1.15537
\(807\) 11649.9 0.508172
\(808\) 10451.1 0.455036
\(809\) −41778.4 −1.81564 −0.907819 0.419361i \(-0.862254\pi\)
−0.907819 + 0.419361i \(0.862254\pi\)
\(810\) −742.897 −0.0322256
\(811\) 12935.4 0.560079 0.280039 0.959988i \(-0.409652\pi\)
0.280039 + 0.959988i \(0.409652\pi\)
\(812\) 0 0
\(813\) 16583.0 0.715363
\(814\) −1485.89 −0.0639807
\(815\) 17918.7 0.770140
\(816\) 3914.70 0.167944
\(817\) −2381.83 −0.101995
\(818\) 31751.3 1.35716
\(819\) 0 0
\(820\) −2964.35 −0.126243
\(821\) −14499.2 −0.616354 −0.308177 0.951329i \(-0.599719\pi\)
−0.308177 + 0.951329i \(0.599719\pi\)
\(822\) 4471.13 0.189718
\(823\) −11977.1 −0.507287 −0.253643 0.967298i \(-0.581629\pi\)
−0.253643 + 0.967298i \(0.581629\pi\)
\(824\) 6071.80 0.256700
\(825\) −2022.84 −0.0853649
\(826\) 0 0
\(827\) 27613.5 1.16108 0.580541 0.814231i \(-0.302841\pi\)
0.580541 + 0.814231i \(0.302841\pi\)
\(828\) 3824.99 0.160541
\(829\) 677.836 0.0283983 0.0141992 0.999899i \(-0.495480\pi\)
0.0141992 + 0.999899i \(0.495480\pi\)
\(830\) −4009.23 −0.167666
\(831\) −6804.36 −0.284044
\(832\) −2893.65 −0.120576
\(833\) 0 0
\(834\) −8241.58 −0.342185
\(835\) −5899.22 −0.244492
\(836\) −131.102 −0.00542375
\(837\) 7893.92 0.325990
\(838\) 32223.8 1.32835
\(839\) −42209.6 −1.73687 −0.868436 0.495801i \(-0.834874\pi\)
−0.868436 + 0.495801i \(0.834874\pi\)
\(840\) 0 0
\(841\) 47505.8 1.94784
\(842\) 17383.2 0.711476
\(843\) −2176.97 −0.0889428
\(844\) −17158.0 −0.699765
\(845\) 700.553 0.0285204
\(846\) −6228.08 −0.253104
\(847\) 0 0
\(848\) 6488.46 0.262753
\(849\) 12711.0 0.513828
\(850\) 16958.9 0.684337
\(851\) 12171.8 0.490299
\(852\) 8179.93 0.328920
\(853\) 2796.45 0.112249 0.0561247 0.998424i \(-0.482126\pi\)
0.0561247 + 0.998424i \(0.482126\pi\)
\(854\) 0 0
\(855\) −208.582 −0.00834310
\(856\) 10100.1 0.403288
\(857\) −23181.1 −0.923979 −0.461989 0.886886i \(-0.652864\pi\)
−0.461989 + 0.886886i \(0.652864\pi\)
\(858\) −1759.32 −0.0700026
\(859\) 11897.5 0.472569 0.236285 0.971684i \(-0.424070\pi\)
0.236285 + 0.971684i \(0.424070\pi\)
\(860\) 8645.01 0.342782
\(861\) 0 0
\(862\) −8390.21 −0.331521
\(863\) −29815.9 −1.17607 −0.588033 0.808837i \(-0.700098\pi\)
−0.588033 + 0.808837i \(0.700098\pi\)
\(864\) −864.000 −0.0340207
\(865\) −5737.99 −0.225546
\(866\) 10853.1 0.425869
\(867\) −5215.31 −0.204292
\(868\) 0 0
\(869\) −1.71256 −6.68523e−5 0
\(870\) −7377.58 −0.287498
\(871\) −526.587 −0.0204853
\(872\) −16844.2 −0.654148
\(873\) 11520.8 0.446644
\(874\) 1073.94 0.0415634
\(875\) 0 0
\(876\) 8225.48 0.317253
\(877\) −10338.8 −0.398082 −0.199041 0.979991i \(-0.563783\pi\)
−0.199041 + 0.979991i \(0.563783\pi\)
\(878\) 7542.48 0.289916
\(879\) −13121.3 −0.503495
\(880\) 475.842 0.0182280
\(881\) −24140.2 −0.923160 −0.461580 0.887099i \(-0.652717\pi\)
−0.461580 + 0.887099i \(0.652717\pi\)
\(882\) 0 0
\(883\) 12997.6 0.495361 0.247681 0.968842i \(-0.420332\pi\)
0.247681 + 0.968842i \(0.420332\pi\)
\(884\) 14749.7 0.561183
\(885\) 3486.61 0.132430
\(886\) −11860.6 −0.449734
\(887\) −45266.6 −1.71353 −0.856766 0.515705i \(-0.827530\pi\)
−0.856766 + 0.515705i \(0.827530\pi\)
\(888\) −2749.40 −0.103901
\(889\) 0 0
\(890\) −536.653 −0.0202120
\(891\) −525.308 −0.0197514
\(892\) −23182.9 −0.870204
\(893\) −1748.64 −0.0655276
\(894\) −3723.18 −0.139286
\(895\) −16616.7 −0.620596
\(896\) 0 0
\(897\) 14411.7 0.536446
\(898\) −1058.13 −0.0393210
\(899\) 78393.1 2.90829
\(900\) −3742.94 −0.138627
\(901\) −33073.5 −1.22290
\(902\) −2096.11 −0.0773756
\(903\) 0 0
\(904\) 12286.1 0.452024
\(905\) 833.361 0.0306098
\(906\) 11635.6 0.426674
\(907\) −27766.5 −1.01651 −0.508253 0.861208i \(-0.669708\pi\)
−0.508253 + 0.861208i \(0.669708\pi\)
\(908\) −16416.2 −0.599989
\(909\) 11757.5 0.429012
\(910\) 0 0
\(911\) 18531.2 0.673948 0.336974 0.941514i \(-0.390597\pi\)
0.336974 + 0.941514i \(0.390597\pi\)
\(912\) −242.584 −0.00880783
\(913\) −2834.96 −0.102764
\(914\) 20114.4 0.727925
\(915\) 10334.8 0.373395
\(916\) 5187.33 0.187111
\(917\) 0 0
\(918\) 4404.04 0.158339
\(919\) −18093.4 −0.649452 −0.324726 0.945808i \(-0.605272\pi\)
−0.324726 + 0.945808i \(0.605272\pi\)
\(920\) −3897.91 −0.139685
\(921\) 12400.4 0.443656
\(922\) 10020.6 0.357930
\(923\) 30820.1 1.09908
\(924\) 0 0
\(925\) −11910.7 −0.423375
\(926\) −14248.8 −0.505662
\(927\) 6830.77 0.242019
\(928\) −8580.23 −0.303513
\(929\) −10622.5 −0.375149 −0.187574 0.982250i \(-0.560063\pi\)
−0.187574 + 0.982250i \(0.560063\pi\)
\(930\) −8044.41 −0.283642
\(931\) 0 0
\(932\) −5913.34 −0.207830
\(933\) 15189.7 0.532999
\(934\) −15002.9 −0.525599
\(935\) −2425.50 −0.0848366
\(936\) −3255.35 −0.113680
\(937\) −16057.6 −0.559851 −0.279925 0.960022i \(-0.590310\pi\)
−0.279925 + 0.960022i \(0.590310\pi\)
\(938\) 0 0
\(939\) 22234.7 0.772738
\(940\) 6346.81 0.220223
\(941\) −55153.3 −1.91068 −0.955338 0.295516i \(-0.904508\pi\)
−0.955338 + 0.295516i \(0.904508\pi\)
\(942\) −2477.06 −0.0856762
\(943\) 17170.5 0.592947
\(944\) 4054.97 0.139807
\(945\) 0 0
\(946\) 6112.94 0.210094
\(947\) 18785.7 0.644619 0.322309 0.946634i \(-0.395541\pi\)
0.322309 + 0.946634i \(0.395541\pi\)
\(948\) −3.16882 −0.000108564 0
\(949\) 30991.7 1.06010
\(950\) −1050.90 −0.0358901
\(951\) 20211.2 0.689161
\(952\) 0 0
\(953\) −36499.4 −1.24064 −0.620321 0.784348i \(-0.712998\pi\)
−0.620321 + 0.784348i \(0.712998\pi\)
\(954\) 7299.52 0.247726
\(955\) −6792.83 −0.230168
\(956\) −15107.7 −0.511106
\(957\) −5216.74 −0.176210
\(958\) 16347.6 0.551323
\(959\) 0 0
\(960\) 880.471 0.0296011
\(961\) 55687.8 1.86928
\(962\) −10359.1 −0.347184
\(963\) 11362.6 0.380224
\(964\) 15985.5 0.534086
\(965\) 1635.79 0.0545677
\(966\) 0 0
\(967\) 26059.9 0.866627 0.433314 0.901243i \(-0.357344\pi\)
0.433314 + 0.901243i \(0.357344\pi\)
\(968\) −10311.5 −0.342381
\(969\) 1236.51 0.0409934
\(970\) −11740.4 −0.388622
\(971\) −21689.0 −0.716820 −0.358410 0.933564i \(-0.616681\pi\)
−0.358410 + 0.933564i \(0.616681\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) −23937.6 −0.787484
\(975\) −14102.5 −0.463223
\(976\) 12019.5 0.394195
\(977\) 2119.19 0.0693950 0.0346975 0.999398i \(-0.488953\pi\)
0.0346975 + 0.999398i \(0.488953\pi\)
\(978\) 23444.6 0.766541
\(979\) −379.471 −0.0123881
\(980\) 0 0
\(981\) −18949.7 −0.616737
\(982\) 4159.92 0.135182
\(983\) 48504.1 1.57380 0.786898 0.617083i \(-0.211686\pi\)
0.786898 + 0.617083i \(0.211686\pi\)
\(984\) −3878.52 −0.125653
\(985\) −22426.9 −0.725463
\(986\) 43735.7 1.41261
\(987\) 0 0
\(988\) −913.998 −0.0294313
\(989\) −50074.8 −1.61000
\(990\) 535.322 0.0171855
\(991\) 2043.84 0.0655144 0.0327572 0.999463i \(-0.489571\pi\)
0.0327572 + 0.999463i \(0.489571\pi\)
\(992\) −9355.76 −0.299441
\(993\) −33527.8 −1.07147
\(994\) 0 0
\(995\) 16246.8 0.517645
\(996\) −5245.65 −0.166882
\(997\) 26738.9 0.849378 0.424689 0.905339i \(-0.360383\pi\)
0.424689 + 0.905339i \(0.360383\pi\)
\(998\) 25668.8 0.814161
\(999\) −3093.08 −0.0979586
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.l.1.2 2
3.2 odd 2 882.4.a.bb.1.1 2
4.3 odd 2 2352.4.a.bw.1.2 2
7.2 even 3 294.4.e.m.67.1 4
7.3 odd 6 294.4.e.k.79.2 4
7.4 even 3 294.4.e.m.79.1 4
7.5 odd 6 294.4.e.k.67.2 4
7.6 odd 2 294.4.a.o.1.1 yes 2
21.2 odd 6 882.4.g.be.361.2 4
21.5 even 6 882.4.g.bk.361.1 4
21.11 odd 6 882.4.g.be.667.2 4
21.17 even 6 882.4.g.bk.667.1 4
21.20 even 2 882.4.a.t.1.2 2
28.27 even 2 2352.4.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.l.1.2 2 1.1 even 1 trivial
294.4.a.o.1.1 yes 2 7.6 odd 2
294.4.e.k.67.2 4 7.5 odd 6
294.4.e.k.79.2 4 7.3 odd 6
294.4.e.m.67.1 4 7.2 even 3
294.4.e.m.79.1 4 7.4 even 3
882.4.a.t.1.2 2 21.20 even 2
882.4.a.bb.1.1 2 3.2 odd 2
882.4.g.be.361.2 4 21.2 odd 6
882.4.g.be.667.2 4 21.11 odd 6
882.4.g.bk.361.1 4 21.5 even 6
882.4.g.bk.667.1 4 21.17 even 6
2352.4.a.bu.1.1 2 28.27 even 2
2352.4.a.bw.1.2 2 4.3 odd 2