Properties

Label 507.4.b.k.337.10
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.10
Root \(-0.588238i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.9

$q$-expansion

\(f(q)\) \(=\) \(q+0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} +15.3391i q^{5} +0.641100i q^{6} +32.3928i q^{7} +3.40944i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} +15.3391i q^{5} +0.641100i q^{6} +32.3928i q^{7} +3.40944i q^{8} +9.00000 q^{9} -3.27797 q^{10} -29.5925i q^{11} +23.8630 q^{12} -6.92233 q^{14} +46.0174i q^{15} +62.9061 q^{16} +78.1958 q^{17} +1.92330i q^{18} +10.6600i q^{19} +122.013i q^{20} +97.1783i q^{21} +6.32392 q^{22} +26.8789 q^{23} +10.2283i q^{24} -110.289 q^{25} +27.0000 q^{27} +257.663i q^{28} -190.785 q^{29} -9.83392 q^{30} -128.108i q^{31} +40.7185i q^{32} -88.7776i q^{33} +16.7104i q^{34} -496.877 q^{35} +71.5890 q^{36} +379.934i q^{37} -2.27805 q^{38} -52.2979 q^{40} -464.631i q^{41} -20.7670 q^{42} -322.758 q^{43} -235.389i q^{44} +138.052i q^{45} +5.74401i q^{46} +248.529i q^{47} +188.718 q^{48} -706.292 q^{49} -23.5688i q^{50} +234.587 q^{51} +740.167 q^{53} +5.76990i q^{54} +453.924 q^{55} -110.441 q^{56} +31.9801i q^{57} -40.7706i q^{58} +340.673i q^{59} +366.038i q^{60} -590.834 q^{61} +27.3767 q^{62} +291.535i q^{63} +494.547 q^{64} +18.9718 q^{66} -340.777i q^{67} +621.995 q^{68} +80.6366 q^{69} -106.183i q^{70} -36.2243i q^{71} +30.6850i q^{72} +164.572i q^{73} -81.1919 q^{74} -330.868 q^{75} +84.7935i q^{76} +958.584 q^{77} -327.242 q^{79} +964.925i q^{80} +81.0000 q^{81} +99.2916 q^{82} -1404.48i q^{83} +772.989i q^{84} +1199.46i q^{85} -68.9734i q^{86} -572.354 q^{87} +100.894 q^{88} -736.986i q^{89} -29.5018 q^{90} +213.803 q^{92} -384.324i q^{93} -53.1107 q^{94} -163.516 q^{95} +122.156i q^{96} +1494.87i q^{97} -150.935i q^{98} -266.333i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.213700i 0.0755543i 0.999286 + 0.0377772i \(0.0120277\pi\)
−0.999286 + 0.0377772i \(0.987972\pi\)
\(3\) 3.00000 0.577350
\(4\) 7.95433 0.994292
\(5\) 15.3391i 1.37197i 0.727614 + 0.685987i \(0.240629\pi\)
−0.727614 + 0.685987i \(0.759371\pi\)
\(6\) 0.641100i 0.0436213i
\(7\) 32.3928i 1.74905i 0.484984 + 0.874523i \(0.338825\pi\)
−0.484984 + 0.874523i \(0.661175\pi\)
\(8\) 3.40944i 0.150677i
\(9\) 9.00000 0.333333
\(10\) −3.27797 −0.103659
\(11\) − 29.5925i − 0.811135i −0.914065 0.405567i \(-0.867074\pi\)
0.914065 0.405567i \(-0.132926\pi\)
\(12\) 23.8630 0.574054
\(13\) 0 0
\(14\) −6.92233 −0.132148
\(15\) 46.0174i 0.792110i
\(16\) 62.9061 0.982907
\(17\) 78.1958 1.11560 0.557802 0.829974i \(-0.311645\pi\)
0.557802 + 0.829974i \(0.311645\pi\)
\(18\) 1.92330i 0.0251848i
\(19\) 10.6600i 0.128715i 0.997927 + 0.0643574i \(0.0204998\pi\)
−0.997927 + 0.0643574i \(0.979500\pi\)
\(20\) 122.013i 1.36414i
\(21\) 97.1783i 1.00981i
\(22\) 6.32392 0.0612847
\(23\) 26.8789 0.243680 0.121840 0.992550i \(-0.461121\pi\)
0.121840 + 0.992550i \(0.461121\pi\)
\(24\) 10.2283i 0.0869936i
\(25\) −110.289 −0.882314
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 257.663i 1.73906i
\(29\) −190.785 −1.22165 −0.610824 0.791766i \(-0.709162\pi\)
−0.610824 + 0.791766i \(0.709162\pi\)
\(30\) −9.83392 −0.0598473
\(31\) − 128.108i − 0.742222i −0.928589 0.371111i \(-0.878977\pi\)
0.928589 0.371111i \(-0.121023\pi\)
\(32\) 40.7185i 0.224940i
\(33\) − 88.7776i − 0.468309i
\(34\) 16.7104i 0.0842887i
\(35\) −496.877 −2.39965
\(36\) 71.5890 0.331431
\(37\) 379.934i 1.68813i 0.536241 + 0.844065i \(0.319844\pi\)
−0.536241 + 0.844065i \(0.680156\pi\)
\(38\) −2.27805 −0.00972496
\(39\) 0 0
\(40\) −52.2979 −0.206725
\(41\) − 464.631i − 1.76983i −0.465749 0.884917i \(-0.654215\pi\)
0.465749 0.884917i \(-0.345785\pi\)
\(42\) −20.7670 −0.0762957
\(43\) −322.758 −1.14466 −0.572328 0.820025i \(-0.693959\pi\)
−0.572328 + 0.820025i \(0.693959\pi\)
\(44\) − 235.389i − 0.806504i
\(45\) 138.052i 0.457325i
\(46\) 5.74401i 0.0184110i
\(47\) 248.529i 0.771314i 0.922642 + 0.385657i \(0.126025\pi\)
−0.922642 + 0.385657i \(0.873975\pi\)
\(48\) 188.718 0.567482
\(49\) −706.292 −2.05916
\(50\) − 23.5688i − 0.0666626i
\(51\) 234.587 0.644094
\(52\) 0 0
\(53\) 740.167 1.91830 0.959148 0.282904i \(-0.0912976\pi\)
0.959148 + 0.282904i \(0.0912976\pi\)
\(54\) 5.76990i 0.0145404i
\(55\) 453.924 1.11286
\(56\) −110.441 −0.263542
\(57\) 31.9801i 0.0743135i
\(58\) − 40.7706i − 0.0923008i
\(59\) 340.673i 0.751727i 0.926675 + 0.375864i \(0.122654\pi\)
−0.926675 + 0.375864i \(0.877346\pi\)
\(60\) 366.038i 0.787588i
\(61\) −590.834 −1.24014 −0.620070 0.784547i \(-0.712896\pi\)
−0.620070 + 0.784547i \(0.712896\pi\)
\(62\) 27.3767 0.0560781
\(63\) 291.535i 0.583015i
\(64\) 494.547 0.965912
\(65\) 0 0
\(66\) 18.9718 0.0353828
\(67\) − 340.777i − 0.621382i −0.950511 0.310691i \(-0.899440\pi\)
0.950511 0.310691i \(-0.100560\pi\)
\(68\) 621.995 1.10924
\(69\) 80.6366 0.140688
\(70\) − 106.183i − 0.181304i
\(71\) − 36.2243i − 0.0605498i −0.999542 0.0302749i \(-0.990362\pi\)
0.999542 0.0302749i \(-0.00963827\pi\)
\(72\) 30.6850i 0.0502258i
\(73\) 164.572i 0.263859i 0.991259 + 0.131929i \(0.0421172\pi\)
−0.991259 + 0.131929i \(0.957883\pi\)
\(74\) −81.1919 −0.127546
\(75\) −330.868 −0.509404
\(76\) 84.7935i 0.127980i
\(77\) 958.584 1.41871
\(78\) 0 0
\(79\) −327.242 −0.466046 −0.233023 0.972471i \(-0.574862\pi\)
−0.233023 + 0.972471i \(0.574862\pi\)
\(80\) 964.925i 1.34852i
\(81\) 81.0000 0.111111
\(82\) 99.2916 0.133719
\(83\) − 1404.48i − 1.85737i −0.370871 0.928684i \(-0.620941\pi\)
0.370871 0.928684i \(-0.379059\pi\)
\(84\) 772.989i 1.00405i
\(85\) 1199.46i 1.53058i
\(86\) − 68.9734i − 0.0864837i
\(87\) −572.354 −0.705319
\(88\) 100.894 0.122220
\(89\) − 736.986i − 0.877756i −0.898547 0.438878i \(-0.855376\pi\)
0.898547 0.438878i \(-0.144624\pi\)
\(90\) −29.5018 −0.0345529
\(91\) 0 0
\(92\) 213.803 0.242289
\(93\) − 384.324i − 0.428522i
\(94\) −53.1107 −0.0582761
\(95\) −163.516 −0.176593
\(96\) 122.156i 0.129869i
\(97\) 1494.87i 1.56476i 0.622804 + 0.782378i \(0.285993\pi\)
−0.622804 + 0.782378i \(0.714007\pi\)
\(98\) − 150.935i − 0.155578i
\(99\) − 266.333i − 0.270378i
\(100\) −877.277 −0.877277
\(101\) 484.010 0.476839 0.238420 0.971162i \(-0.423371\pi\)
0.238420 + 0.971162i \(0.423371\pi\)
\(102\) 50.1313i 0.0486641i
\(103\) −214.049 −0.204766 −0.102383 0.994745i \(-0.532647\pi\)
−0.102383 + 0.994745i \(0.532647\pi\)
\(104\) 0 0
\(105\) −1490.63 −1.38544
\(106\) 158.174i 0.144936i
\(107\) 148.288 0.133977 0.0669884 0.997754i \(-0.478661\pi\)
0.0669884 + 0.997754i \(0.478661\pi\)
\(108\) 214.767 0.191351
\(109\) − 504.881i − 0.443659i −0.975086 0.221829i \(-0.928797\pi\)
0.975086 0.221829i \(-0.0712028\pi\)
\(110\) 97.0035i 0.0840811i
\(111\) 1139.80i 0.974642i
\(112\) 2037.70i 1.71915i
\(113\) −1136.49 −0.946120 −0.473060 0.881030i \(-0.656851\pi\)
−0.473060 + 0.881030i \(0.656851\pi\)
\(114\) −6.83415 −0.00561471
\(115\) 412.299i 0.334322i
\(116\) −1517.56 −1.21467
\(117\) 0 0
\(118\) −72.8019 −0.0567962
\(119\) 2532.98i 1.95124i
\(120\) −156.894 −0.119353
\(121\) 455.282 0.342060
\(122\) − 126.261i − 0.0936979i
\(123\) − 1393.89i − 1.02181i
\(124\) − 1019.01i − 0.737985i
\(125\) 225.651i 0.161463i
\(126\) −62.3010 −0.0440493
\(127\) 1286.84 0.899122 0.449561 0.893250i \(-0.351580\pi\)
0.449561 + 0.893250i \(0.351580\pi\)
\(128\) 431.433i 0.297919i
\(129\) −968.275 −0.660867
\(130\) 0 0
\(131\) 2229.88 1.48722 0.743610 0.668614i \(-0.233112\pi\)
0.743610 + 0.668614i \(0.233112\pi\)
\(132\) − 706.166i − 0.465636i
\(133\) −345.308 −0.225128
\(134\) 72.8241 0.0469481
\(135\) 414.157i 0.264037i
\(136\) 266.604i 0.168096i
\(137\) − 474.745i − 0.296060i −0.988983 0.148030i \(-0.952707\pi\)
0.988983 0.148030i \(-0.0472932\pi\)
\(138\) 17.2320i 0.0106296i
\(139\) 1927.02 1.17588 0.587941 0.808904i \(-0.299939\pi\)
0.587941 + 0.808904i \(0.299939\pi\)
\(140\) −3952.33 −2.38595
\(141\) 745.588i 0.445318i
\(142\) 7.74113 0.00457480
\(143\) 0 0
\(144\) 566.155 0.327636
\(145\) − 2926.47i − 1.67607i
\(146\) −35.1690 −0.0199357
\(147\) −2118.88 −1.18886
\(148\) 3022.12i 1.67849i
\(149\) − 1065.73i − 0.585959i −0.956119 0.292979i \(-0.905353\pi\)
0.956119 0.292979i \(-0.0946467\pi\)
\(150\) − 70.7064i − 0.0384877i
\(151\) 877.888i 0.473123i 0.971617 + 0.236561i \(0.0760204\pi\)
−0.971617 + 0.236561i \(0.923980\pi\)
\(152\) −36.3448 −0.0193944
\(153\) 703.762 0.371868
\(154\) 204.849i 0.107190i
\(155\) 1965.07 1.01831
\(156\) 0 0
\(157\) 2314.94 1.17676 0.588382 0.808583i \(-0.299765\pi\)
0.588382 + 0.808583i \(0.299765\pi\)
\(158\) − 69.9317i − 0.0352118i
\(159\) 2220.50 1.10753
\(160\) −624.587 −0.308612
\(161\) 870.681i 0.426207i
\(162\) 17.3097i 0.00839492i
\(163\) 331.399i 0.159246i 0.996825 + 0.0796232i \(0.0253717\pi\)
−0.996825 + 0.0796232i \(0.974628\pi\)
\(164\) − 3695.83i − 1.75973i
\(165\) 1361.77 0.642508
\(166\) 300.137 0.140332
\(167\) − 2171.62i − 1.00626i −0.864212 0.503129i \(-0.832182\pi\)
0.864212 0.503129i \(-0.167818\pi\)
\(168\) −331.324 −0.152156
\(169\) 0 0
\(170\) −256.324 −0.115642
\(171\) 95.9404i 0.0429049i
\(172\) −2567.33 −1.13812
\(173\) 2936.69 1.29059 0.645295 0.763933i \(-0.276734\pi\)
0.645295 + 0.763933i \(0.276734\pi\)
\(174\) − 122.312i − 0.0532899i
\(175\) − 3572.57i − 1.54321i
\(176\) − 1861.55i − 0.797270i
\(177\) 1022.02i 0.434010i
\(178\) 157.494 0.0663183
\(179\) −1615.57 −0.674599 −0.337300 0.941397i \(-0.609514\pi\)
−0.337300 + 0.941397i \(0.609514\pi\)
\(180\) 1098.11i 0.454714i
\(181\) −725.019 −0.297736 −0.148868 0.988857i \(-0.547563\pi\)
−0.148868 + 0.988857i \(0.547563\pi\)
\(182\) 0 0
\(183\) −1772.50 −0.715995
\(184\) 91.6419i 0.0367170i
\(185\) −5827.87 −2.31607
\(186\) 82.1300 0.0323767
\(187\) − 2314.01i − 0.904905i
\(188\) 1976.89i 0.766911i
\(189\) 874.605i 0.336604i
\(190\) − 34.9433i − 0.0133424i
\(191\) 1717.08 0.650490 0.325245 0.945630i \(-0.394553\pi\)
0.325245 + 0.945630i \(0.394553\pi\)
\(192\) 1483.64 0.557670
\(193\) 435.830i 0.162548i 0.996692 + 0.0812740i \(0.0258989\pi\)
−0.996692 + 0.0812740i \(0.974101\pi\)
\(194\) −319.454 −0.118224
\(195\) 0 0
\(196\) −5618.08 −2.04741
\(197\) − 694.556i − 0.251193i −0.992081 0.125597i \(-0.959915\pi\)
0.992081 0.125597i \(-0.0400845\pi\)
\(198\) 56.9153 0.0204282
\(199\) −2899.24 −1.03277 −0.516386 0.856356i \(-0.672723\pi\)
−0.516386 + 0.856356i \(0.672723\pi\)
\(200\) − 376.024i − 0.132945i
\(201\) − 1022.33i − 0.358755i
\(202\) 103.433i 0.0360273i
\(203\) − 6180.04i − 2.13672i
\(204\) 1865.99 0.640417
\(205\) 7127.04 2.42817
\(206\) − 45.7423i − 0.0154709i
\(207\) 241.910 0.0812265
\(208\) 0 0
\(209\) 315.457 0.104405
\(210\) − 318.548i − 0.104676i
\(211\) 5250.30 1.71301 0.856507 0.516136i \(-0.172630\pi\)
0.856507 + 0.516136i \(0.172630\pi\)
\(212\) 5887.53 1.90735
\(213\) − 108.673i − 0.0349584i
\(214\) 31.6891i 0.0101225i
\(215\) − 4950.84i − 1.57044i
\(216\) 92.0549i 0.0289979i
\(217\) 4149.77 1.29818
\(218\) 107.893 0.0335203
\(219\) 493.716i 0.152339i
\(220\) 3610.66 1.10650
\(221\) 0 0
\(222\) −243.576 −0.0736384
\(223\) 1382.46i 0.415141i 0.978220 + 0.207570i \(0.0665556\pi\)
−0.978220 + 0.207570i \(0.933444\pi\)
\(224\) −1318.99 −0.393431
\(225\) −992.603 −0.294105
\(226\) − 242.867i − 0.0714835i
\(227\) − 1223.57i − 0.357760i −0.983871 0.178880i \(-0.942753\pi\)
0.983871 0.178880i \(-0.0572474\pi\)
\(228\) 254.380i 0.0738893i
\(229\) − 4019.61i − 1.15993i −0.814642 0.579964i \(-0.803067\pi\)
0.814642 0.579964i \(-0.196933\pi\)
\(230\) −88.1082 −0.0252595
\(231\) 2875.75 0.819094
\(232\) − 650.468i − 0.184075i
\(233\) −1658.40 −0.466290 −0.233145 0.972442i \(-0.574902\pi\)
−0.233145 + 0.972442i \(0.574902\pi\)
\(234\) 0 0
\(235\) −3812.23 −1.05822
\(236\) 2709.83i 0.747436i
\(237\) −981.727 −0.269072
\(238\) −541.297 −0.147425
\(239\) 618.554i 0.167410i 0.996491 + 0.0837049i \(0.0266753\pi\)
−0.996491 + 0.0837049i \(0.973325\pi\)
\(240\) 2894.77i 0.778570i
\(241\) − 1135.25i − 0.303434i −0.988424 0.151717i \(-0.951520\pi\)
0.988424 0.151717i \(-0.0484803\pi\)
\(242\) 97.2938i 0.0258441i
\(243\) 243.000 0.0641500
\(244\) −4699.69 −1.23306
\(245\) − 10833.9i − 2.82512i
\(246\) 297.875 0.0772025
\(247\) 0 0
\(248\) 436.777 0.111836
\(249\) − 4213.44i − 1.07235i
\(250\) −48.2215 −0.0121992
\(251\) −2287.83 −0.575326 −0.287663 0.957732i \(-0.592878\pi\)
−0.287663 + 0.957732i \(0.592878\pi\)
\(252\) 2318.97i 0.579687i
\(253\) − 795.414i − 0.197657i
\(254\) 274.997i 0.0679326i
\(255\) 3598.37i 0.883681i
\(256\) 3864.18 0.943403
\(257\) 5113.49 1.24113 0.620566 0.784154i \(-0.286903\pi\)
0.620566 + 0.784154i \(0.286903\pi\)
\(258\) − 206.920i − 0.0499314i
\(259\) −12307.1 −2.95262
\(260\) 0 0
\(261\) −1717.06 −0.407216
\(262\) 476.526i 0.112366i
\(263\) 2760.17 0.647145 0.323573 0.946203i \(-0.395116\pi\)
0.323573 + 0.946203i \(0.395116\pi\)
\(264\) 302.682 0.0705635
\(265\) 11353.5i 2.63185i
\(266\) − 73.7923i − 0.0170094i
\(267\) − 2210.96i − 0.506773i
\(268\) − 2710.66i − 0.617834i
\(269\) −3310.32 −0.750312 −0.375156 0.926962i \(-0.622411\pi\)
−0.375156 + 0.926962i \(0.622411\pi\)
\(270\) −88.5053 −0.0199491
\(271\) 2522.26i 0.565374i 0.959212 + 0.282687i \(0.0912258\pi\)
−0.959212 + 0.282687i \(0.908774\pi\)
\(272\) 4918.99 1.09654
\(273\) 0 0
\(274\) 101.453 0.0223686
\(275\) 3263.74i 0.715675i
\(276\) 641.410 0.139885
\(277\) 3455.48 0.749530 0.374765 0.927120i \(-0.377723\pi\)
0.374765 + 0.927120i \(0.377723\pi\)
\(278\) 411.804i 0.0888430i
\(279\) − 1152.97i − 0.247407i
\(280\) − 1694.07i − 0.361572i
\(281\) − 981.649i − 0.208400i −0.994556 0.104200i \(-0.966772\pi\)
0.994556 0.104200i \(-0.0332281\pi\)
\(282\) −159.332 −0.0336457
\(283\) −5331.10 −1.11979 −0.559896 0.828563i \(-0.689159\pi\)
−0.559896 + 0.828563i \(0.689159\pi\)
\(284\) − 288.140i − 0.0602041i
\(285\) −490.547 −0.101956
\(286\) 0 0
\(287\) 15050.7 3.09552
\(288\) 366.467i 0.0749801i
\(289\) 1201.58 0.244572
\(290\) 625.387 0.126634
\(291\) 4484.62i 0.903412i
\(292\) 1309.06i 0.262353i
\(293\) − 2420.39i − 0.482597i −0.970451 0.241299i \(-0.922427\pi\)
0.970451 0.241299i \(-0.0775733\pi\)
\(294\) − 452.804i − 0.0898233i
\(295\) −5225.64 −1.03135
\(296\) −1295.36 −0.254363
\(297\) − 798.998i − 0.156103i
\(298\) 227.746 0.0442717
\(299\) 0 0
\(300\) −2631.83 −0.506496
\(301\) − 10455.0i − 2.00205i
\(302\) −187.605 −0.0357465
\(303\) 1452.03 0.275303
\(304\) 670.581i 0.126515i
\(305\) − 9062.88i − 1.70144i
\(306\) 150.394i 0.0280962i
\(307\) − 875.509i − 0.162762i −0.996683 0.0813810i \(-0.974067\pi\)
0.996683 0.0813810i \(-0.0259330\pi\)
\(308\) 7624.90 1.41061
\(309\) −642.147 −0.118222
\(310\) 419.935i 0.0769377i
\(311\) −6419.10 −1.17040 −0.585199 0.810890i \(-0.698984\pi\)
−0.585199 + 0.810890i \(0.698984\pi\)
\(312\) 0 0
\(313\) −5015.58 −0.905742 −0.452871 0.891576i \(-0.649600\pi\)
−0.452871 + 0.891576i \(0.649600\pi\)
\(314\) 494.701i 0.0889096i
\(315\) −4471.90 −0.799882
\(316\) −2602.99 −0.463386
\(317\) − 5228.81i − 0.926433i −0.886245 0.463217i \(-0.846695\pi\)
0.886245 0.463217i \(-0.153305\pi\)
\(318\) 474.521i 0.0836786i
\(319\) 5645.80i 0.990922i
\(320\) 7585.92i 1.32521i
\(321\) 444.863 0.0773515
\(322\) −186.064 −0.0322018
\(323\) 833.570i 0.143595i
\(324\) 644.301 0.110477
\(325\) 0 0
\(326\) −70.8199 −0.0120318
\(327\) − 1514.64i − 0.256146i
\(328\) 1584.13 0.266674
\(329\) −8050.56 −1.34906
\(330\) 291.010i 0.0485442i
\(331\) 3186.81i 0.529193i 0.964359 + 0.264596i \(0.0852387\pi\)
−0.964359 + 0.264596i \(0.914761\pi\)
\(332\) − 11171.7i − 1.84677i
\(333\) 3419.41i 0.562710i
\(334\) 464.075 0.0760271
\(335\) 5227.23 0.852520
\(336\) 6113.11i 0.992551i
\(337\) 5600.69 0.905309 0.452655 0.891686i \(-0.350477\pi\)
0.452655 + 0.891686i \(0.350477\pi\)
\(338\) 0 0
\(339\) −3409.46 −0.546243
\(340\) 9540.87i 1.52184i
\(341\) −3791.04 −0.602042
\(342\) −20.5024 −0.00324165
\(343\) − 11768.0i − 1.85252i
\(344\) − 1100.42i − 0.172474i
\(345\) 1236.90i 0.193021i
\(346\) 627.570i 0.0975097i
\(347\) −6982.33 −1.08020 −0.540102 0.841599i \(-0.681614\pi\)
−0.540102 + 0.841599i \(0.681614\pi\)
\(348\) −4552.69 −0.701293
\(349\) 2872.20i 0.440531i 0.975440 + 0.220265i \(0.0706924\pi\)
−0.975440 + 0.220265i \(0.929308\pi\)
\(350\) 763.459 0.116596
\(351\) 0 0
\(352\) 1204.96 0.182457
\(353\) 5104.55i 0.769655i 0.922989 + 0.384827i \(0.125739\pi\)
−0.922989 + 0.384827i \(0.874261\pi\)
\(354\) −218.406 −0.0327913
\(355\) 555.650 0.0830728
\(356\) − 5862.23i − 0.872746i
\(357\) 7598.94i 1.12655i
\(358\) − 345.247i − 0.0509689i
\(359\) − 10771.0i − 1.58348i −0.610857 0.791741i \(-0.709175\pi\)
0.610857 0.791741i \(-0.290825\pi\)
\(360\) −470.681 −0.0689085
\(361\) 6745.36 0.983433
\(362\) − 154.936i − 0.0224952i
\(363\) 1365.85 0.197489
\(364\) 0 0
\(365\) −2524.39 −0.362008
\(366\) − 378.783i − 0.0540965i
\(367\) −11380.5 −1.61868 −0.809340 0.587340i \(-0.800175\pi\)
−0.809340 + 0.587340i \(0.800175\pi\)
\(368\) 1690.84 0.239514
\(369\) − 4181.68i − 0.589945i
\(370\) − 1245.41i − 0.174989i
\(371\) 23976.1i 3.35519i
\(372\) − 3057.04i − 0.426076i
\(373\) 3196.75 0.443758 0.221879 0.975074i \(-0.428781\pi\)
0.221879 + 0.975074i \(0.428781\pi\)
\(374\) 494.504 0.0683695
\(375\) 676.952i 0.0932204i
\(376\) −847.346 −0.116220
\(377\) 0 0
\(378\) −186.903 −0.0254319
\(379\) − 2050.75i − 0.277942i −0.990296 0.138971i \(-0.955621\pi\)
0.990296 0.138971i \(-0.0443795\pi\)
\(380\) −1300.66 −0.175585
\(381\) 3860.52 0.519108
\(382\) 366.940i 0.0491473i
\(383\) 3507.54i 0.467955i 0.972242 + 0.233978i \(0.0751742\pi\)
−0.972242 + 0.233978i \(0.924826\pi\)
\(384\) 1294.30i 0.172004i
\(385\) 14703.9i 1.94644i
\(386\) −93.1369 −0.0122812
\(387\) −2904.83 −0.381552
\(388\) 11890.7i 1.55582i
\(389\) 6572.66 0.856676 0.428338 0.903618i \(-0.359099\pi\)
0.428338 + 0.903618i \(0.359099\pi\)
\(390\) 0 0
\(391\) 2101.81 0.271850
\(392\) − 2408.06i − 0.310269i
\(393\) 6689.65 0.858646
\(394\) 148.427 0.0189787
\(395\) − 5019.62i − 0.639403i
\(396\) − 2118.50i − 0.268835i
\(397\) 5285.98i 0.668251i 0.942529 + 0.334126i \(0.108441\pi\)
−0.942529 + 0.334126i \(0.891559\pi\)
\(398\) − 619.567i − 0.0780304i
\(399\) −1035.92 −0.129978
\(400\) −6937.86 −0.867233
\(401\) 919.541i 0.114513i 0.998360 + 0.0572565i \(0.0182353\pi\)
−0.998360 + 0.0572565i \(0.981765\pi\)
\(402\) 218.472 0.0271055
\(403\) 0 0
\(404\) 3849.97 0.474117
\(405\) 1242.47i 0.152442i
\(406\) 1320.67 0.161438
\(407\) 11243.2 1.36930
\(408\) 799.811i 0.0970504i
\(409\) − 1539.17i − 0.186081i −0.995662 0.0930403i \(-0.970341\pi\)
0.995662 0.0930403i \(-0.0296585\pi\)
\(410\) 1523.05i 0.183458i
\(411\) − 1424.23i − 0.170930i
\(412\) −1702.62 −0.203597
\(413\) −11035.4 −1.31481
\(414\) 51.6961i 0.00613702i
\(415\) 21543.5 2.54826
\(416\) 0 0
\(417\) 5781.06 0.678896
\(418\) 67.4132i 0.00788825i
\(419\) 10579.9 1.23357 0.616783 0.787133i \(-0.288436\pi\)
0.616783 + 0.787133i \(0.288436\pi\)
\(420\) −11857.0 −1.37753
\(421\) 74.9351i 0.00867485i 0.999991 + 0.00433743i \(0.00138065\pi\)
−0.999991 + 0.00433743i \(0.998619\pi\)
\(422\) 1121.99i 0.129426i
\(423\) 2236.77i 0.257105i
\(424\) 2523.55i 0.289044i
\(425\) −8624.15 −0.984313
\(426\) 23.2234 0.00264126
\(427\) − 19138.8i − 2.16906i
\(428\) 1179.53 0.133212
\(429\) 0 0
\(430\) 1057.99 0.118653
\(431\) − 12165.6i − 1.35962i −0.733388 0.679810i \(-0.762062\pi\)
0.733388 0.679810i \(-0.237938\pi\)
\(432\) 1698.46 0.189161
\(433\) 2869.23 0.318445 0.159222 0.987243i \(-0.449101\pi\)
0.159222 + 0.987243i \(0.449101\pi\)
\(434\) 886.806i 0.0980831i
\(435\) − 8779.41i − 0.967680i
\(436\) − 4015.99i − 0.441126i
\(437\) 286.530i 0.0313652i
\(438\) −105.507 −0.0115099
\(439\) 3845.12 0.418035 0.209018 0.977912i \(-0.432973\pi\)
0.209018 + 0.977912i \(0.432973\pi\)
\(440\) 1547.63i 0.167682i
\(441\) −6356.63 −0.686387
\(442\) 0 0
\(443\) −3858.30 −0.413799 −0.206900 0.978362i \(-0.566337\pi\)
−0.206900 + 0.978362i \(0.566337\pi\)
\(444\) 9066.37i 0.969079i
\(445\) 11304.7 1.20426
\(446\) −295.432 −0.0313657
\(447\) − 3197.18i − 0.338303i
\(448\) 16019.7i 1.68942i
\(449\) 5550.51i 0.583395i 0.956511 + 0.291698i \(0.0942201\pi\)
−0.956511 + 0.291698i \(0.905780\pi\)
\(450\) − 212.119i − 0.0222209i
\(451\) −13749.6 −1.43557
\(452\) −9039.99 −0.940719
\(453\) 2633.66i 0.273157i
\(454\) 261.478 0.0270303
\(455\) 0 0
\(456\) −109.034 −0.0111974
\(457\) 8102.42i 0.829355i 0.909968 + 0.414677i \(0.136106\pi\)
−0.909968 + 0.414677i \(0.863894\pi\)
\(458\) 858.991 0.0876376
\(459\) 2111.29 0.214698
\(460\) 3279.56i 0.332414i
\(461\) 9230.86i 0.932590i 0.884629 + 0.466295i \(0.154412\pi\)
−0.884629 + 0.466295i \(0.845588\pi\)
\(462\) 614.548i 0.0618861i
\(463\) 13934.7i 1.39870i 0.714777 + 0.699352i \(0.246528\pi\)
−0.714777 + 0.699352i \(0.753472\pi\)
\(464\) −12001.5 −1.20077
\(465\) 5895.20 0.587921
\(466\) − 354.400i − 0.0352302i
\(467\) −10918.0 −1.08185 −0.540926 0.841070i \(-0.681926\pi\)
−0.540926 + 0.841070i \(0.681926\pi\)
\(468\) 0 0
\(469\) 11038.7 1.08682
\(470\) − 814.673i − 0.0799533i
\(471\) 6944.81 0.679405
\(472\) −1161.51 −0.113268
\(473\) 9551.23i 0.928470i
\(474\) − 209.795i − 0.0203295i
\(475\) − 1175.69i − 0.113567i
\(476\) 20148.2i 1.94010i
\(477\) 6661.50 0.639432
\(478\) −132.185 −0.0126485
\(479\) 11028.4i 1.05198i 0.850490 + 0.525992i \(0.176306\pi\)
−0.850490 + 0.525992i \(0.823694\pi\)
\(480\) −1873.76 −0.178177
\(481\) 0 0
\(482\) 242.602 0.0229258
\(483\) 2612.04i 0.246071i
\(484\) 3621.47 0.340108
\(485\) −22930.0 −2.14680
\(486\) 51.9291i 0.00484681i
\(487\) − 1078.10i − 0.100315i −0.998741 0.0501576i \(-0.984028\pi\)
0.998741 0.0501576i \(-0.0159724\pi\)
\(488\) − 2014.41i − 0.186861i
\(489\) 994.197i 0.0919410i
\(490\) 2315.21 0.213450
\(491\) −6572.88 −0.604134 −0.302067 0.953287i \(-0.597677\pi\)
−0.302067 + 0.953287i \(0.597677\pi\)
\(492\) − 11087.5i − 1.01598i
\(493\) −14918.6 −1.36288
\(494\) 0 0
\(495\) 4085.32 0.370952
\(496\) − 8058.77i − 0.729535i
\(497\) 1173.41 0.105904
\(498\) 900.411 0.0810208
\(499\) − 9956.79i − 0.893241i −0.894724 0.446620i \(-0.852627\pi\)
0.894724 0.446620i \(-0.147373\pi\)
\(500\) 1794.90i 0.160541i
\(501\) − 6514.86i − 0.580963i
\(502\) − 488.910i − 0.0434683i
\(503\) −11965.0 −1.06062 −0.530310 0.847804i \(-0.677924\pi\)
−0.530310 + 0.847804i \(0.677924\pi\)
\(504\) −993.971 −0.0878472
\(505\) 7424.29i 0.654211i
\(506\) 169.980 0.0149338
\(507\) 0 0
\(508\) 10235.9 0.893990
\(509\) 7160.44i 0.623538i 0.950158 + 0.311769i \(0.100922\pi\)
−0.950158 + 0.311769i \(0.899078\pi\)
\(510\) −768.971 −0.0667659
\(511\) −5330.94 −0.461501
\(512\) 4277.24i 0.369197i
\(513\) 287.821i 0.0247712i
\(514\) 1092.75i 0.0937729i
\(515\) − 3283.33i − 0.280933i
\(516\) −7701.98 −0.657095
\(517\) 7354.61 0.625639
\(518\) − 2630.03i − 0.223083i
\(519\) 8810.06 0.745123
\(520\) 0 0
\(521\) −17213.1 −1.44745 −0.723725 0.690089i \(-0.757572\pi\)
−0.723725 + 0.690089i \(0.757572\pi\)
\(522\) − 366.936i − 0.0307669i
\(523\) −10110.1 −0.845281 −0.422640 0.906297i \(-0.638897\pi\)
−0.422640 + 0.906297i \(0.638897\pi\)
\(524\) 17737.2 1.47873
\(525\) − 10717.7i − 0.890971i
\(526\) 589.847i 0.0488946i
\(527\) − 10017.5i − 0.828026i
\(528\) − 5584.65i − 0.460304i
\(529\) −11444.5 −0.940620
\(530\) −2426.25 −0.198848
\(531\) 3066.06i 0.250576i
\(532\) −2746.70 −0.223843
\(533\) 0 0
\(534\) 472.481 0.0382889
\(535\) 2274.61i 0.183813i
\(536\) 1161.86 0.0936281
\(537\) −4846.71 −0.389480
\(538\) − 707.416i − 0.0566893i
\(539\) 20901.0i 1.67026i
\(540\) 3294.34i 0.262529i
\(541\) − 5951.54i − 0.472970i −0.971635 0.236485i \(-0.924005\pi\)
0.971635 0.236485i \(-0.0759954\pi\)
\(542\) −539.007 −0.0427165
\(543\) −2175.06 −0.171898
\(544\) 3184.02i 0.250944i
\(545\) 7744.43 0.608688
\(546\) 0 0
\(547\) 5157.62 0.403152 0.201576 0.979473i \(-0.435394\pi\)
0.201576 + 0.979473i \(0.435394\pi\)
\(548\) − 3776.28i − 0.294370i
\(549\) −5317.51 −0.413380
\(550\) −697.460 −0.0540724
\(551\) − 2033.77i − 0.157244i
\(552\) 274.926i 0.0211986i
\(553\) − 10600.3i − 0.815136i
\(554\) 738.436i 0.0566302i
\(555\) −17483.6 −1.33718
\(556\) 15328.2 1.16917
\(557\) 23314.4i 1.77354i 0.462206 + 0.886772i \(0.347058\pi\)
−0.462206 + 0.886772i \(0.652942\pi\)
\(558\) 246.390 0.0186927
\(559\) 0 0
\(560\) −31256.6 −2.35863
\(561\) − 6942.03i − 0.522447i
\(562\) 209.778 0.0157455
\(563\) −2329.12 −0.174353 −0.0871764 0.996193i \(-0.527784\pi\)
−0.0871764 + 0.996193i \(0.527784\pi\)
\(564\) 5930.66i 0.442776i
\(565\) − 17432.7i − 1.29805i
\(566\) − 1139.26i − 0.0846051i
\(567\) 2623.81i 0.194338i
\(568\) 123.505 0.00912348
\(569\) 17446.6 1.28541 0.642706 0.766113i \(-0.277812\pi\)
0.642706 + 0.766113i \(0.277812\pi\)
\(570\) − 104.830i − 0.00770323i
\(571\) 413.526 0.0303074 0.0151537 0.999885i \(-0.495176\pi\)
0.0151537 + 0.999885i \(0.495176\pi\)
\(572\) 0 0
\(573\) 5151.24 0.375560
\(574\) 3216.33i 0.233880i
\(575\) −2964.45 −0.215002
\(576\) 4450.92 0.321971
\(577\) − 21258.8i − 1.53382i −0.641754 0.766911i \(-0.721793\pi\)
0.641754 0.766911i \(-0.278207\pi\)
\(578\) 256.778i 0.0184785i
\(579\) 1307.49i 0.0938471i
\(580\) − 23278.1i − 1.66650i
\(581\) 45495.0 3.24862
\(582\) −958.362 −0.0682567
\(583\) − 21903.4i − 1.55600i
\(584\) −561.098 −0.0397575
\(585\) 0 0
\(586\) 517.238 0.0364623
\(587\) 2965.69i 0.208530i 0.994550 + 0.104265i \(0.0332490\pi\)
−0.994550 + 0.104265i \(0.966751\pi\)
\(588\) −16854.2 −1.18207
\(589\) 1365.64 0.0955349
\(590\) − 1116.72i − 0.0779230i
\(591\) − 2083.67i − 0.145027i
\(592\) 23900.2i 1.65928i
\(593\) 17786.8i 1.23173i 0.787852 + 0.615865i \(0.211193\pi\)
−0.787852 + 0.615865i \(0.788807\pi\)
\(594\) 170.746 0.0117943
\(595\) −38853.7 −2.67705
\(596\) − 8477.16i − 0.582614i
\(597\) −8697.72 −0.596271
\(598\) 0 0
\(599\) 21796.4 1.48677 0.743387 0.668861i \(-0.233218\pi\)
0.743387 + 0.668861i \(0.233218\pi\)
\(600\) − 1128.07i − 0.0767557i
\(601\) 20468.7 1.38924 0.694622 0.719375i \(-0.255572\pi\)
0.694622 + 0.719375i \(0.255572\pi\)
\(602\) 2234.24 0.151264
\(603\) − 3066.99i − 0.207127i
\(604\) 6983.01i 0.470422i
\(605\) 6983.64i 0.469298i
\(606\) 310.298i 0.0208004i
\(607\) 7852.53 0.525081 0.262541 0.964921i \(-0.415440\pi\)
0.262541 + 0.964921i \(0.415440\pi\)
\(608\) −434.061 −0.0289531
\(609\) − 18540.1i − 1.23364i
\(610\) 1936.74 0.128551
\(611\) 0 0
\(612\) 5597.96 0.369745
\(613\) − 2460.21i − 0.162099i −0.996710 0.0810497i \(-0.974173\pi\)
0.996710 0.0810497i \(-0.0258273\pi\)
\(614\) 187.096 0.0122974
\(615\) 21381.1 1.40190
\(616\) 3268.23i 0.213768i
\(617\) − 17829.3i − 1.16334i −0.813425 0.581670i \(-0.802400\pi\)
0.813425 0.581670i \(-0.197600\pi\)
\(618\) − 137.227i − 0.00893215i
\(619\) − 16901.7i − 1.09748i −0.835994 0.548739i \(-0.815108\pi\)
0.835994 0.548739i \(-0.184892\pi\)
\(620\) 15630.8 1.01250
\(621\) 725.729 0.0468962
\(622\) − 1371.76i − 0.0884286i
\(623\) 23873.0 1.53524
\(624\) 0 0
\(625\) −17247.4 −1.10384
\(626\) − 1071.83i − 0.0684327i
\(627\) 946.372 0.0602783
\(628\) 18413.8 1.17005
\(629\) 29709.3i 1.88328i
\(630\) − 955.644i − 0.0604345i
\(631\) 1833.82i 0.115695i 0.998325 + 0.0578473i \(0.0184237\pi\)
−0.998325 + 0.0578473i \(0.981576\pi\)
\(632\) − 1115.71i − 0.0702226i
\(633\) 15750.9 0.989009
\(634\) 1117.40 0.0699960
\(635\) 19739.0i 1.23357i
\(636\) 17662.6 1.10121
\(637\) 0 0
\(638\) −1206.51 −0.0748684
\(639\) − 326.019i − 0.0201833i
\(640\) −6617.81 −0.408737
\(641\) −29172.1 −1.79755 −0.898773 0.438414i \(-0.855540\pi\)
−0.898773 + 0.438414i \(0.855540\pi\)
\(642\) 95.0672i 0.00584424i
\(643\) − 25103.8i − 1.53965i −0.638254 0.769826i \(-0.720343\pi\)
0.638254 0.769826i \(-0.279657\pi\)
\(644\) 6925.69i 0.423774i
\(645\) − 14852.5i − 0.906693i
\(646\) −178.134 −0.0108492
\(647\) 24471.6 1.48698 0.743492 0.668745i \(-0.233168\pi\)
0.743492 + 0.668745i \(0.233168\pi\)
\(648\) 276.165i 0.0167419i
\(649\) 10081.4 0.609752
\(650\) 0 0
\(651\) 12449.3 0.749505
\(652\) 2636.06i 0.158337i
\(653\) 6632.68 0.397484 0.198742 0.980052i \(-0.436314\pi\)
0.198742 + 0.980052i \(0.436314\pi\)
\(654\) 323.679 0.0193530
\(655\) 34204.5i 2.04043i
\(656\) − 29228.1i − 1.73958i
\(657\) 1481.15i 0.0879529i
\(658\) − 1720.40i − 0.101928i
\(659\) 26436.9 1.56272 0.781361 0.624080i \(-0.214526\pi\)
0.781361 + 0.624080i \(0.214526\pi\)
\(660\) 10832.0 0.638840
\(661\) 27596.7i 1.62389i 0.583737 + 0.811943i \(0.301590\pi\)
−0.583737 + 0.811943i \(0.698410\pi\)
\(662\) −681.020 −0.0399828
\(663\) 0 0
\(664\) 4788.49 0.279863
\(665\) − 5296.73i − 0.308870i
\(666\) −730.727 −0.0425152
\(667\) −5128.07 −0.297691
\(668\) − 17273.8i − 1.00051i
\(669\) 4147.38i 0.239682i
\(670\) 1117.06i 0.0644115i
\(671\) 17484.3i 1.00592i
\(672\) −3956.96 −0.227147
\(673\) −18860.1 −1.08024 −0.540122 0.841587i \(-0.681622\pi\)
−0.540122 + 0.841587i \(0.681622\pi\)
\(674\) 1196.87i 0.0684000i
\(675\) −2977.81 −0.169801
\(676\) 0 0
\(677\) −14705.5 −0.834825 −0.417412 0.908717i \(-0.637063\pi\)
−0.417412 + 0.908717i \(0.637063\pi\)
\(678\) − 728.601i − 0.0412710i
\(679\) −48423.1 −2.73683
\(680\) −4089.47 −0.230624
\(681\) − 3670.72i − 0.206553i
\(682\) − 810.145i − 0.0454869i
\(683\) − 17950.2i − 1.00563i −0.864395 0.502814i \(-0.832298\pi\)
0.864395 0.502814i \(-0.167702\pi\)
\(684\) 763.141i 0.0426600i
\(685\) 7282.18 0.406186
\(686\) 2514.83 0.139966
\(687\) − 12058.8i − 0.669685i
\(688\) −20303.5 −1.12509
\(689\) 0 0
\(690\) −264.325 −0.0145836
\(691\) 8692.38i 0.478544i 0.970953 + 0.239272i \(0.0769087\pi\)
−0.970953 + 0.239272i \(0.923091\pi\)
\(692\) 23359.4 1.28322
\(693\) 8627.26 0.472904
\(694\) − 1492.12i − 0.0816141i
\(695\) 29558.8i 1.61328i
\(696\) − 1951.41i − 0.106276i
\(697\) − 36332.2i − 1.97443i
\(698\) −613.789 −0.0332840
\(699\) −4975.20 −0.269212
\(700\) − 28417.4i − 1.53440i
\(701\) −23275.9 −1.25409 −0.627047 0.778982i \(-0.715736\pi\)
−0.627047 + 0.778982i \(0.715736\pi\)
\(702\) 0 0
\(703\) −4050.11 −0.217287
\(704\) − 14634.9i − 0.783485i
\(705\) −11436.7 −0.610965
\(706\) −1090.84 −0.0581507
\(707\) 15678.4i 0.834014i
\(708\) 8129.49i 0.431532i
\(709\) − 34164.7i − 1.80971i −0.425722 0.904854i \(-0.639980\pi\)
0.425722 0.904854i \(-0.360020\pi\)
\(710\) 118.742i 0.00627651i
\(711\) −2945.18 −0.155349
\(712\) 2512.71 0.132258
\(713\) − 3443.40i − 0.180864i
\(714\) −1623.89 −0.0851157
\(715\) 0 0
\(716\) −12850.8 −0.670748
\(717\) 1855.66i 0.0966541i
\(718\) 2301.75 0.119639
\(719\) 8391.74 0.435270 0.217635 0.976030i \(-0.430166\pi\)
0.217635 + 0.976030i \(0.430166\pi\)
\(720\) 8684.32i 0.449508i
\(721\) − 6933.64i − 0.358145i
\(722\) 1441.48i 0.0743026i
\(723\) − 3405.74i − 0.175188i
\(724\) −5767.04 −0.296036
\(725\) 21041.5 1.07788
\(726\) 291.881i 0.0149211i
\(727\) −23500.4 −1.19887 −0.599436 0.800423i \(-0.704608\pi\)
−0.599436 + 0.800423i \(0.704608\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 539.462i − 0.0273512i
\(731\) −25238.3 −1.27698
\(732\) −14099.1 −0.711908
\(733\) − 1289.07i − 0.0649563i −0.999472 0.0324781i \(-0.989660\pi\)
0.999472 0.0324781i \(-0.0103399\pi\)
\(734\) − 2432.01i − 0.122298i
\(735\) − 32501.7i − 1.63108i
\(736\) 1094.47i 0.0548133i
\(737\) −10084.5 −0.504024
\(738\) 893.624 0.0445729
\(739\) − 18197.8i − 0.905844i −0.891550 0.452922i \(-0.850382\pi\)
0.891550 0.452922i \(-0.149618\pi\)
\(740\) −46356.8 −2.30285
\(741\) 0 0
\(742\) −5123.68 −0.253499
\(743\) − 21277.2i − 1.05059i −0.850921 0.525294i \(-0.823956\pi\)
0.850921 0.525294i \(-0.176044\pi\)
\(744\) 1310.33 0.0645686
\(745\) 16347.4 0.803920
\(746\) 683.146i 0.0335278i
\(747\) − 12640.3i − 0.619123i
\(748\) − 18406.4i − 0.899739i
\(749\) 4803.45i 0.234331i
\(750\) −144.665 −0.00704321
\(751\) 25653.4 1.24648 0.623240 0.782030i \(-0.285816\pi\)
0.623240 + 0.782030i \(0.285816\pi\)
\(752\) 15634.0i 0.758130i
\(753\) −6863.50 −0.332164
\(754\) 0 0
\(755\) −13466.0 −0.649112
\(756\) 6956.90i 0.334682i
\(757\) 26521.2 1.27336 0.636678 0.771130i \(-0.280308\pi\)
0.636678 + 0.771130i \(0.280308\pi\)
\(758\) 438.245 0.0209997
\(759\) − 2386.24i − 0.114117i
\(760\) − 557.497i − 0.0266086i
\(761\) 21607.2i 1.02925i 0.857415 + 0.514625i \(0.172069\pi\)
−0.857415 + 0.514625i \(0.827931\pi\)
\(762\) 824.992i 0.0392209i
\(763\) 16354.5 0.775979
\(764\) 13658.2 0.646776
\(765\) 10795.1i 0.510193i
\(766\) −749.560 −0.0353560
\(767\) 0 0
\(768\) 11592.5 0.544674
\(769\) 17179.2i 0.805591i 0.915290 + 0.402795i \(0.131961\pi\)
−0.915290 + 0.402795i \(0.868039\pi\)
\(770\) −3142.21 −0.147062
\(771\) 15340.5 0.716568
\(772\) 3466.74i 0.161620i
\(773\) 12822.1i 0.596609i 0.954471 + 0.298304i \(0.0964211\pi\)
−0.954471 + 0.298304i \(0.903579\pi\)
\(774\) − 620.761i − 0.0288279i
\(775\) 14128.9i 0.654873i
\(776\) −5096.67 −0.235773
\(777\) −36921.4 −1.70469
\(778\) 1404.58i 0.0647256i
\(779\) 4952.98 0.227804
\(780\) 0 0
\(781\) −1071.97 −0.0491140
\(782\) 449.157i 0.0205394i
\(783\) −5151.18 −0.235106
\(784\) −44430.0 −2.02396
\(785\) 35509.1i 1.61449i
\(786\) 1429.58i 0.0648744i
\(787\) − 35213.4i − 1.59495i −0.603354 0.797474i \(-0.706169\pi\)
0.603354 0.797474i \(-0.293831\pi\)
\(788\) − 5524.73i − 0.249759i
\(789\) 8280.50 0.373629
\(790\) 1072.69 0.0483097
\(791\) − 36813.9i − 1.65481i
\(792\) 908.045 0.0407399
\(793\) 0 0
\(794\) −1129.61 −0.0504893
\(795\) 34060.6i 1.51950i
\(796\) −23061.5 −1.02688
\(797\) −7500.15 −0.333336 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(798\) − 221.377i − 0.00982038i
\(799\) 19434.0i 0.860481i
\(800\) − 4490.82i − 0.198468i
\(801\) − 6632.87i − 0.292585i
\(802\) −196.506 −0.00865195
\(803\) 4870.10 0.214025
\(804\) − 8131.97i − 0.356707i
\(805\) −13355.5 −0.584745
\(806\) 0 0
\(807\) −9930.97 −0.433193
\(808\) 1650.20i 0.0718489i
\(809\) −22020.5 −0.956982 −0.478491 0.878092i \(-0.658816\pi\)
−0.478491 + 0.878092i \(0.658816\pi\)
\(810\) −265.516 −0.0115176
\(811\) − 20444.6i − 0.885210i −0.896717 0.442605i \(-0.854054\pi\)
0.896717 0.442605i \(-0.145946\pi\)
\(812\) − 49158.1i − 2.12452i
\(813\) 7566.78i 0.326419i
\(814\) 2402.67i 0.103457i
\(815\) −5083.37 −0.218482
\(816\) 14757.0 0.633085
\(817\) − 3440.62i − 0.147334i
\(818\) 328.920 0.0140592
\(819\) 0 0
\(820\) 56690.8 2.41431
\(821\) − 40353.0i − 1.71538i −0.514164 0.857692i \(-0.671898\pi\)
0.514164 0.857692i \(-0.328102\pi\)
\(822\) 304.359 0.0129145
\(823\) 33110.7 1.40239 0.701194 0.712971i \(-0.252651\pi\)
0.701194 + 0.712971i \(0.252651\pi\)
\(824\) − 729.787i − 0.0308536i
\(825\) 9791.21i 0.413195i
\(826\) − 2358.26i − 0.0993392i
\(827\) − 43410.4i − 1.82531i −0.408735 0.912653i \(-0.634030\pi\)
0.408735 0.912653i \(-0.365970\pi\)
\(828\) 1924.23 0.0807629
\(829\) 5502.87 0.230546 0.115273 0.993334i \(-0.463226\pi\)
0.115273 + 0.993334i \(0.463226\pi\)
\(830\) 4603.84i 0.192532i
\(831\) 10366.4 0.432741
\(832\) 0 0
\(833\) −55229.1 −2.29721
\(834\) 1235.41i 0.0512935i
\(835\) 33310.8 1.38056
\(836\) 2509.25 0.103809
\(837\) − 3458.92i − 0.142841i
\(838\) 2260.93i 0.0932013i
\(839\) − 1698.27i − 0.0698819i −0.999389 0.0349409i \(-0.988876\pi\)
0.999389 0.0349409i \(-0.0111243\pi\)
\(840\) − 5082.22i − 0.208754i
\(841\) 12009.8 0.492425
\(842\) −16.0136 −0.000655423 0
\(843\) − 2944.95i − 0.120320i
\(844\) 41762.7 1.70323
\(845\) 0 0
\(846\) −477.996 −0.0194254
\(847\) 14747.9i 0.598279i
\(848\) 46561.0 1.88551
\(849\) −15993.3 −0.646512
\(850\) − 1842.98i − 0.0743691i
\(851\) 10212.2i 0.411363i
\(852\) − 864.421i − 0.0347589i
\(853\) 18686.8i 0.750087i 0.927007 + 0.375044i \(0.122372\pi\)
−0.927007 + 0.375044i \(0.877628\pi\)
\(854\) 4089.95 0.163882
\(855\) −1471.64 −0.0588645
\(856\) 505.578i 0.0201873i
\(857\) 9659.16 0.385007 0.192503 0.981296i \(-0.438339\pi\)
0.192503 + 0.981296i \(0.438339\pi\)
\(858\) 0 0
\(859\) −1559.90 −0.0619595 −0.0309798 0.999520i \(-0.509863\pi\)
−0.0309798 + 0.999520i \(0.509863\pi\)
\(860\) − 39380.6i − 1.56147i
\(861\) 45152.1 1.78720
\(862\) 2599.79 0.102725
\(863\) 11358.7i 0.448034i 0.974585 + 0.224017i \(0.0719171\pi\)
−0.974585 + 0.224017i \(0.928083\pi\)
\(864\) 1099.40i 0.0432898i
\(865\) 45046.3i 1.77066i
\(866\) 613.155i 0.0240599i
\(867\) 3604.74 0.141204
\(868\) 33008.7 1.29077
\(869\) 9683.93i 0.378026i
\(870\) 1876.16 0.0731124
\(871\) 0 0
\(872\) 1721.36 0.0668493
\(873\) 13453.8i 0.521585i
\(874\) −61.2314 −0.00236977
\(875\) −7309.45 −0.282405
\(876\) 3927.18i 0.151469i
\(877\) 33301.6i 1.28223i 0.767444 + 0.641115i \(0.221528\pi\)
−0.767444 + 0.641115i \(0.778472\pi\)
\(878\) 821.701i 0.0315844i
\(879\) − 7261.18i − 0.278628i
\(880\) 28554.6 1.09383
\(881\) −28912.3 −1.10565 −0.552826 0.833296i \(-0.686451\pi\)
−0.552826 + 0.833296i \(0.686451\pi\)
\(882\) − 1358.41i − 0.0518595i
\(883\) 49100.9 1.87132 0.935661 0.352901i \(-0.114805\pi\)
0.935661 + 0.352901i \(0.114805\pi\)
\(884\) 0 0
\(885\) −15676.9 −0.595451
\(886\) − 824.517i − 0.0312643i
\(887\) 4566.49 0.172861 0.0864304 0.996258i \(-0.472454\pi\)
0.0864304 + 0.996258i \(0.472454\pi\)
\(888\) −3886.09 −0.146857
\(889\) 41684.3i 1.57261i
\(890\) 2415.82i 0.0909870i
\(891\) − 2396.99i − 0.0901261i
\(892\) 10996.5i 0.412771i
\(893\) −2649.33 −0.0992795
\(894\) 683.238 0.0255603
\(895\) − 24781.4i − 0.925533i
\(896\) −13975.3 −0.521074
\(897\) 0 0
\(898\) −1186.14 −0.0440780
\(899\) 24441.0i 0.906734i
\(900\) −7895.49 −0.292426
\(901\) 57877.9 2.14006
\(902\) − 2938.29i − 0.108464i
\(903\) − 31365.1i − 1.15589i
\(904\) − 3874.78i − 0.142559i
\(905\) − 11121.2i − 0.408486i
\(906\) −562.814 −0.0206382
\(907\) −48873.3 −1.78921 −0.894604 0.446860i \(-0.852542\pi\)
−0.894604 + 0.446860i \(0.852542\pi\)
\(908\) − 9732.72i − 0.355718i
\(909\) 4356.09 0.158946
\(910\) 0 0
\(911\) 31550.1 1.14742 0.573710 0.819058i \(-0.305503\pi\)
0.573710 + 0.819058i \(0.305503\pi\)
\(912\) 2011.74i 0.0730433i
\(913\) −41562.1 −1.50658
\(914\) −1731.49 −0.0626613
\(915\) − 27188.7i − 0.982327i
\(916\) − 31973.3i − 1.15331i
\(917\) 72232.1i 2.60121i
\(918\) 451.182i 0.0162214i
\(919\) −11720.3 −0.420694 −0.210347 0.977627i \(-0.567459\pi\)
−0.210347 + 0.977627i \(0.567459\pi\)
\(920\) −1405.71 −0.0503748
\(921\) − 2626.53i − 0.0939707i
\(922\) −1972.63 −0.0704612
\(923\) 0 0
\(924\) 22874.7 0.814418
\(925\) − 41902.7i − 1.48946i
\(926\) −2977.84 −0.105678
\(927\) −1926.44 −0.0682553
\(928\) − 7768.47i − 0.274798i
\(929\) 6911.23i 0.244080i 0.992525 + 0.122040i \(0.0389436\pi\)
−0.992525 + 0.122040i \(0.961056\pi\)
\(930\) 1259.80i 0.0444200i
\(931\) − 7529.10i − 0.265044i
\(932\) −13191.5 −0.463628
\(933\) −19257.3 −0.675730
\(934\) − 2333.18i − 0.0817386i
\(935\) 35494.9 1.24151
\(936\) 0 0
\(937\) −28673.9 −0.999717 −0.499858 0.866107i \(-0.666615\pi\)
−0.499858 + 0.866107i \(0.666615\pi\)
\(938\) 2358.97i 0.0821143i
\(939\) −15046.7 −0.522930
\(940\) −30323.7 −1.05218
\(941\) − 22463.6i − 0.778207i −0.921194 0.389103i \(-0.872785\pi\)
0.921194 0.389103i \(-0.127215\pi\)
\(942\) 1484.10i 0.0513320i
\(943\) − 12488.8i − 0.431272i
\(944\) 21430.4i 0.738878i
\(945\) −13415.7 −0.461812
\(946\) −2041.10 −0.0701499
\(947\) 36806.6i 1.26299i 0.775379 + 0.631497i \(0.217559\pi\)
−0.775379 + 0.631497i \(0.782441\pi\)
\(948\) −7808.98 −0.267536
\(949\) 0 0
\(950\) 251.244 0.00858046
\(951\) − 15686.4i − 0.534876i
\(952\) −8636.04 −0.294008
\(953\) 1781.07 0.0605400 0.0302700 0.999542i \(-0.490363\pi\)
0.0302700 + 0.999542i \(0.490363\pi\)
\(954\) 1423.56i 0.0483119i
\(955\) 26338.5i 0.892455i
\(956\) 4920.19i 0.166454i
\(957\) 16937.4i 0.572109i
\(958\) −2356.77 −0.0794819
\(959\) 15378.3 0.517822
\(960\) 22757.8i 0.765108i
\(961\) 13379.3 0.449107
\(962\) 0 0
\(963\) 1334.59 0.0446589
\(964\) − 9030.12i − 0.301702i
\(965\) −6685.26 −0.223012
\(966\) −558.193 −0.0185917
\(967\) − 54777.2i − 1.82163i −0.412815 0.910815i \(-0.635455\pi\)
0.412815 0.910815i \(-0.364545\pi\)
\(968\) 1552.26i 0.0515408i
\(969\) 2500.71i 0.0829044i
\(970\) − 4900.15i − 0.162200i
\(971\) −2391.54 −0.0790403 −0.0395201 0.999219i \(-0.512583\pi\)
−0.0395201 + 0.999219i \(0.512583\pi\)
\(972\) 1932.90 0.0637838
\(973\) 62421.5i 2.05667i
\(974\) 230.391 0.00757925
\(975\) 0 0
\(976\) −37167.0 −1.21894
\(977\) 43738.5i 1.43226i 0.697967 + 0.716130i \(0.254088\pi\)
−0.697967 + 0.716130i \(0.745912\pi\)
\(978\) −212.460 −0.00694654
\(979\) −21809.3 −0.711979
\(980\) − 86176.5i − 2.80899i
\(981\) − 4543.92i − 0.147886i
\(982\) − 1404.62i − 0.0456449i
\(983\) 27712.4i 0.899174i 0.893236 + 0.449587i \(0.148429\pi\)
−0.893236 + 0.449587i \(0.851571\pi\)
\(984\) 4752.39 0.153964
\(985\) 10653.9 0.344631
\(986\) − 3188.09i − 0.102971i
\(987\) −24151.7 −0.778882
\(988\) 0 0
\(989\) −8675.38 −0.278929
\(990\) 873.031i 0.0280270i
\(991\) 3550.23 0.113801 0.0569005 0.998380i \(-0.481878\pi\)
0.0569005 + 0.998380i \(0.481878\pi\)
\(992\) 5216.37 0.166956
\(993\) 9560.42i 0.305529i
\(994\) 250.757i 0.00800153i
\(995\) − 44471.8i − 1.41694i
\(996\) − 33515.1i − 1.06623i
\(997\) −28338.4 −0.900188 −0.450094 0.892981i \(-0.648610\pi\)
−0.450094 + 0.892981i \(0.648610\pi\)
\(998\) 2127.76 0.0674882
\(999\) 10258.2i 0.324881i
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 507.4.b.k.337.10 18
13.5 odd 4 507.4.a.o.1.6 9
13.8 odd 4 507.4.a.p.1.4 yes 9
13.12 even 2 inner 507.4.b.k.337.9 18
39.5 even 4 1521.4.a.bi.1.4 9
39.8 even 4 1521.4.a.bf.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.6 9 13.5 odd 4
507.4.a.p.1.4 yes 9 13.8 odd 4
507.4.b.k.337.9 18 13.12 even 2 inner
507.4.b.k.337.10 18 1.1 even 1 trivial
1521.4.a.bf.1.6 9 39.8 even 4
1521.4.a.bi.1.4 9 39.5 even 4