Properties

Label 960.4.f.p.769.4
Level $960$
Weight $4$
Character 960.769
Analytic conductor $56.642$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.4
Root \(3.70156i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.p.769.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} +(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -9.00000 q^{9} -1.79063 q^{11} -58.2094i q^{13} +(-23.1047 + 24.3141i) q^{15} -18.9844i q^{17} -104.837 q^{19} -66.6281 q^{21} +49.6125i q^{23} +(6.37188 + 124.837i) q^{25} -27.0000i q^{27} -293.466 q^{29} -64.4187 q^{31} -5.37188i q^{33} +(-171.047 + 180.000i) q^{35} -19.8844i q^{37} +174.628 q^{39} -165.581 q^{41} -247.350i q^{43} +(-72.9422 - 69.3141i) q^{45} +384.544i q^{47} -150.256 q^{49} +56.9531 q^{51} -463.528i q^{53} +(-14.5125 - 13.7906i) q^{55} -314.512i q^{57} +73.7906 q^{59} +137.350 q^{61} -199.884i q^{63} +(448.303 - 471.769i) q^{65} -173.906i q^{67} -148.837 q^{69} +594.281 q^{71} +320.231i q^{73} +(-374.512 + 19.1156i) q^{75} -39.7687i q^{77} -770.469 q^{79} +81.0000 q^{81} +173.925i q^{83} +(146.209 - 153.862i) q^{85} -880.397i q^{87} -1019.02 q^{89} +1292.79 q^{91} -193.256i q^{93} +(-849.675 - 807.412i) q^{95} -384.375i q^{97} +16.1156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 36 q^{9} - 84 q^{11} - 54 q^{15} - 112 q^{19} - 36 q^{21} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 468 q^{39} - 816 q^{41} + 54 q^{45} - 140 q^{49} + 612 q^{51} + 864 q^{55} + 372 q^{59} - 680 q^{61} + 948 q^{65} - 288 q^{69} + 72 q^{71} - 576 q^{75} + 760 q^{79} + 324 q^{81} + 508 q^{85} - 2232 q^{89} + 1944 q^{91} - 2784 q^{95} + 756 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 8.10469 + 7.70156i 0.724905 + 0.688849i
\(6\) 0 0
\(7\) 22.2094i 1.19919i 0.800302 + 0.599597i \(0.204672\pi\)
−0.800302 + 0.599597i \(0.795328\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −1.79063 −0.0490813 −0.0245407 0.999699i \(-0.507812\pi\)
−0.0245407 + 0.999699i \(0.507812\pi\)
\(12\) 0 0
\(13\) 58.2094i 1.24188i −0.783860 0.620938i \(-0.786752\pi\)
0.783860 0.620938i \(-0.213248\pi\)
\(14\) 0 0
\(15\) −23.1047 + 24.3141i −0.397707 + 0.418524i
\(16\) 0 0
\(17\) 18.9844i 0.270846i −0.990788 0.135423i \(-0.956761\pi\)
0.990788 0.135423i \(-0.0432394\pi\)
\(18\) 0 0
\(19\) −104.837 −1.26586 −0.632931 0.774208i \(-0.718148\pi\)
−0.632931 + 0.774208i \(0.718148\pi\)
\(20\) 0 0
\(21\) −66.6281 −0.692355
\(22\) 0 0
\(23\) 49.6125i 0.449779i 0.974384 + 0.224890i \(0.0722021\pi\)
−0.974384 + 0.224890i \(0.927798\pi\)
\(24\) 0 0
\(25\) 6.37188 + 124.837i 0.0509751 + 0.998700i
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −293.466 −1.87914 −0.939572 0.342350i \(-0.888777\pi\)
−0.939572 + 0.342350i \(0.888777\pi\)
\(30\) 0 0
\(31\) −64.4187 −0.373224 −0.186612 0.982434i \(-0.559751\pi\)
−0.186612 + 0.982434i \(0.559751\pi\)
\(32\) 0 0
\(33\) 5.37188i 0.0283371i
\(34\) 0 0
\(35\) −171.047 + 180.000i −0.826063 + 0.869302i
\(36\) 0 0
\(37\) 19.8844i 0.0883505i −0.999024 0.0441752i \(-0.985934\pi\)
0.999024 0.0441752i \(-0.0140660\pi\)
\(38\) 0 0
\(39\) 174.628 0.716997
\(40\) 0 0
\(41\) −165.581 −0.630718 −0.315359 0.948972i \(-0.602125\pi\)
−0.315359 + 0.948972i \(0.602125\pi\)
\(42\) 0 0
\(43\) 247.350i 0.877221i −0.898677 0.438611i \(-0.855471\pi\)
0.898677 0.438611i \(-0.144529\pi\)
\(44\) 0 0
\(45\) −72.9422 69.3141i −0.241635 0.229616i
\(46\) 0 0
\(47\) 384.544i 1.19344i 0.802451 + 0.596718i \(0.203529\pi\)
−0.802451 + 0.596718i \(0.796471\pi\)
\(48\) 0 0
\(49\) −150.256 −0.438065
\(50\) 0 0
\(51\) 56.9531 0.156373
\(52\) 0 0
\(53\) 463.528i 1.20133i −0.799501 0.600665i \(-0.794903\pi\)
0.799501 0.600665i \(-0.205097\pi\)
\(54\) 0 0
\(55\) −14.5125 13.7906i −0.0355793 0.0338096i
\(56\) 0 0
\(57\) 314.512i 0.730846i
\(58\) 0 0
\(59\) 73.7906 0.162826 0.0814129 0.996680i \(-0.474057\pi\)
0.0814129 + 0.996680i \(0.474057\pi\)
\(60\) 0 0
\(61\) 137.350 0.288293 0.144146 0.989556i \(-0.453956\pi\)
0.144146 + 0.989556i \(0.453956\pi\)
\(62\) 0 0
\(63\) 199.884i 0.399731i
\(64\) 0 0
\(65\) 448.303 471.769i 0.855464 0.900242i
\(66\) 0 0
\(67\) 173.906i 0.317105i −0.987351 0.158552i \(-0.949317\pi\)
0.987351 0.158552i \(-0.0506827\pi\)
\(68\) 0 0
\(69\) −148.837 −0.259680
\(70\) 0 0
\(71\) 594.281 0.993355 0.496677 0.867935i \(-0.334553\pi\)
0.496677 + 0.867935i \(0.334553\pi\)
\(72\) 0 0
\(73\) 320.231i 0.513428i 0.966487 + 0.256714i \(0.0826398\pi\)
−0.966487 + 0.256714i \(0.917360\pi\)
\(74\) 0 0
\(75\) −374.512 + 19.1156i −0.576600 + 0.0294305i
\(76\) 0 0
\(77\) 39.7687i 0.0588580i
\(78\) 0 0
\(79\) −770.469 −1.09727 −0.548636 0.836061i \(-0.684853\pi\)
−0.548636 + 0.836061i \(0.684853\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 173.925i 0.230009i 0.993365 + 0.115004i \(0.0366882\pi\)
−0.993365 + 0.115004i \(0.963312\pi\)
\(84\) 0 0
\(85\) 146.209 153.862i 0.186572 0.196338i
\(86\) 0 0
\(87\) 880.397i 1.08492i
\(88\) 0 0
\(89\) −1019.02 −1.21367 −0.606834 0.794829i \(-0.707561\pi\)
−0.606834 + 0.794829i \(0.707561\pi\)
\(90\) 0 0
\(91\) 1292.79 1.48925
\(92\) 0 0
\(93\) 193.256i 0.215481i
\(94\) 0 0
\(95\) −849.675 807.412i −0.917630 0.871987i
\(96\) 0 0
\(97\) 384.375i 0.402344i −0.979556 0.201172i \(-0.935525\pi\)
0.979556 0.201172i \(-0.0644750\pi\)
\(98\) 0 0
\(99\) 16.1156 0.0163604
\(100\) 0 0
\(101\) −34.4906 −0.0339796 −0.0169898 0.999856i \(-0.505408\pi\)
−0.0169898 + 0.999856i \(0.505408\pi\)
\(102\) 0 0
\(103\) 1756.30i 1.68013i 0.542484 + 0.840066i \(0.317484\pi\)
−0.542484 + 0.840066i \(0.682516\pi\)
\(104\) 0 0
\(105\) −540.000 513.141i −0.501891 0.476928i
\(106\) 0 0
\(107\) 1361.74i 1.23032i 0.788403 + 0.615159i \(0.210908\pi\)
−0.788403 + 0.615159i \(0.789092\pi\)
\(108\) 0 0
\(109\) 321.119 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(110\) 0 0
\(111\) 59.6531 0.0510092
\(112\) 0 0
\(113\) 1582.25i 1.31721i −0.752487 0.658607i \(-0.771146\pi\)
0.752487 0.658607i \(-0.228854\pi\)
\(114\) 0 0
\(115\) −382.094 + 402.094i −0.309830 + 0.326047i
\(116\) 0 0
\(117\) 523.884i 0.413958i
\(118\) 0 0
\(119\) 421.631 0.324797
\(120\) 0 0
\(121\) −1327.79 −0.997591
\(122\) 0 0
\(123\) 496.744i 0.364145i
\(124\) 0 0
\(125\) −909.802 + 1060.84i −0.651001 + 0.759077i
\(126\) 0 0
\(127\) 1197.14i 0.836449i −0.908344 0.418225i \(-0.862652\pi\)
0.908344 0.418225i \(-0.137348\pi\)
\(128\) 0 0
\(129\) 742.050 0.506464
\(130\) 0 0
\(131\) −321.647 −0.214522 −0.107261 0.994231i \(-0.534208\pi\)
−0.107261 + 0.994231i \(0.534208\pi\)
\(132\) 0 0
\(133\) 2328.37i 1.51801i
\(134\) 0 0
\(135\) 207.942 218.827i 0.132569 0.139508i
\(136\) 0 0
\(137\) 354.291i 0.220942i −0.993879 0.110471i \(-0.964764\pi\)
0.993879 0.110471i \(-0.0352360\pi\)
\(138\) 0 0
\(139\) −77.2562 −0.0471424 −0.0235712 0.999722i \(-0.507504\pi\)
−0.0235712 + 0.999722i \(0.507504\pi\)
\(140\) 0 0
\(141\) −1153.63 −0.689030
\(142\) 0 0
\(143\) 104.231i 0.0609529i
\(144\) 0 0
\(145\) −2378.45 2260.14i −1.36220 1.29445i
\(146\) 0 0
\(147\) 450.769i 0.252917i
\(148\) 0 0
\(149\) −1705.38 −0.937651 −0.468826 0.883291i \(-0.655323\pi\)
−0.468826 + 0.883291i \(0.655323\pi\)
\(150\) 0 0
\(151\) −758.281 −0.408663 −0.204331 0.978902i \(-0.565502\pi\)
−0.204331 + 0.978902i \(0.565502\pi\)
\(152\) 0 0
\(153\) 170.859i 0.0902821i
\(154\) 0 0
\(155\) −522.094 496.125i −0.270552 0.257095i
\(156\) 0 0
\(157\) 1769.05i 0.899273i −0.893212 0.449636i \(-0.851554\pi\)
0.893212 0.449636i \(-0.148446\pi\)
\(158\) 0 0
\(159\) 1390.58 0.693588
\(160\) 0 0
\(161\) −1101.86 −0.539372
\(162\) 0 0
\(163\) 881.719i 0.423690i −0.977303 0.211845i \(-0.932053\pi\)
0.977303 0.211845i \(-0.0679473\pi\)
\(164\) 0 0
\(165\) 41.3719 43.5374i 0.0195200 0.0205417i
\(166\) 0 0
\(167\) 216.900i 0.100504i −0.998737 0.0502522i \(-0.983997\pi\)
0.998737 0.0502522i \(-0.0160025\pi\)
\(168\) 0 0
\(169\) −1191.33 −0.542254
\(170\) 0 0
\(171\) 943.537 0.421954
\(172\) 0 0
\(173\) 4125.91i 1.81322i −0.421970 0.906610i \(-0.638661\pi\)
0.421970 0.906610i \(-0.361339\pi\)
\(174\) 0 0
\(175\) −2772.56 + 141.515i −1.19763 + 0.0611289i
\(176\) 0 0
\(177\) 221.372i 0.0940075i
\(178\) 0 0
\(179\) 3213.14 1.34168 0.670842 0.741600i \(-0.265933\pi\)
0.670842 + 0.741600i \(0.265933\pi\)
\(180\) 0 0
\(181\) −3394.42 −1.39395 −0.696976 0.717095i \(-0.745471\pi\)
−0.696976 + 0.717095i \(0.745471\pi\)
\(182\) 0 0
\(183\) 412.050i 0.166446i
\(184\) 0 0
\(185\) 153.141 161.156i 0.0608601 0.0640457i
\(186\) 0 0
\(187\) 33.9939i 0.0132935i
\(188\) 0 0
\(189\) 599.653 0.230785
\(190\) 0 0
\(191\) 3467.49 1.31361 0.656804 0.754062i \(-0.271908\pi\)
0.656804 + 0.754062i \(0.271908\pi\)
\(192\) 0 0
\(193\) 1792.14i 0.668401i 0.942502 + 0.334200i \(0.108466\pi\)
−0.942502 + 0.334200i \(0.891534\pi\)
\(194\) 0 0
\(195\) 1415.31 + 1344.91i 0.519755 + 0.493902i
\(196\) 0 0
\(197\) 1678.19i 0.606935i 0.952842 + 0.303467i \(0.0981443\pi\)
−0.952842 + 0.303467i \(0.901856\pi\)
\(198\) 0 0
\(199\) 3108.23 1.10722 0.553610 0.832776i \(-0.313250\pi\)
0.553610 + 0.832776i \(0.313250\pi\)
\(200\) 0 0
\(201\) 521.719 0.183081
\(202\) 0 0
\(203\) 6517.69i 2.25346i
\(204\) 0 0
\(205\) −1341.98 1275.23i −0.457211 0.434469i
\(206\) 0 0
\(207\) 446.512i 0.149926i
\(208\) 0 0
\(209\) 187.725 0.0621301
\(210\) 0 0
\(211\) −4473.27 −1.45949 −0.729745 0.683719i \(-0.760361\pi\)
−0.729745 + 0.683719i \(0.760361\pi\)
\(212\) 0 0
\(213\) 1782.84i 0.573514i
\(214\) 0 0
\(215\) 1904.98 2004.69i 0.604273 0.635902i
\(216\) 0 0
\(217\) 1430.70i 0.447568i
\(218\) 0 0
\(219\) −960.694 −0.296428
\(220\) 0 0
\(221\) −1105.07 −0.336357
\(222\) 0 0
\(223\) 1753.42i 0.526535i 0.964723 + 0.263268i \(0.0848003\pi\)
−0.964723 + 0.263268i \(0.915200\pi\)
\(224\) 0 0
\(225\) −57.3469 1123.54i −0.0169917 0.332900i
\(226\) 0 0
\(227\) 936.900i 0.273939i 0.990575 + 0.136970i \(0.0437363\pi\)
−0.990575 + 0.136970i \(0.956264\pi\)
\(228\) 0 0
\(229\) −2582.06 −0.745096 −0.372548 0.928013i \(-0.621516\pi\)
−0.372548 + 0.928013i \(0.621516\pi\)
\(230\) 0 0
\(231\) 119.306 0.0339817
\(232\) 0 0
\(233\) 2295.01i 0.645284i 0.946521 + 0.322642i \(0.104571\pi\)
−0.946521 + 0.322642i \(0.895429\pi\)
\(234\) 0 0
\(235\) −2961.59 + 3116.61i −0.822096 + 0.865128i
\(236\) 0 0
\(237\) 2311.41i 0.633510i
\(238\) 0 0
\(239\) −2294.01 −0.620866 −0.310433 0.950595i \(-0.600474\pi\)
−0.310433 + 0.950595i \(0.600474\pi\)
\(240\) 0 0
\(241\) 382.287 0.102180 0.0510898 0.998694i \(-0.483731\pi\)
0.0510898 + 0.998694i \(0.483731\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −1217.78 1157.21i −0.317555 0.301760i
\(246\) 0 0
\(247\) 6102.52i 1.57204i
\(248\) 0 0
\(249\) −521.775 −0.132796
\(250\) 0 0
\(251\) −2259.98 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(252\) 0 0
\(253\) 88.8375i 0.0220758i
\(254\) 0 0
\(255\) 461.587 + 438.628i 0.113356 + 0.107717i
\(256\) 0 0
\(257\) 92.7843i 0.0225203i 0.999937 + 0.0112602i \(0.00358430\pi\)
−0.999937 + 0.0112602i \(0.996416\pi\)
\(258\) 0 0
\(259\) 441.619 0.105949
\(260\) 0 0
\(261\) 2641.19 0.626382
\(262\) 0 0
\(263\) 568.312i 0.133246i 0.997778 + 0.0666229i \(0.0212224\pi\)
−0.997778 + 0.0666229i \(0.978778\pi\)
\(264\) 0 0
\(265\) 3569.89 3756.75i 0.827534 0.870850i
\(266\) 0 0
\(267\) 3057.07i 0.700711i
\(268\) 0 0
\(269\) 7582.41 1.71862 0.859309 0.511458i \(-0.170894\pi\)
0.859309 + 0.511458i \(0.170894\pi\)
\(270\) 0 0
\(271\) −7943.69 −1.78061 −0.890304 0.455366i \(-0.849508\pi\)
−0.890304 + 0.455366i \(0.849508\pi\)
\(272\) 0 0
\(273\) 3878.38i 0.859818i
\(274\) 0 0
\(275\) −11.4097 223.537i −0.00250192 0.0490175i
\(276\) 0 0
\(277\) 6823.00i 1.47998i 0.672618 + 0.739990i \(0.265170\pi\)
−0.672618 + 0.739990i \(0.734830\pi\)
\(278\) 0 0
\(279\) 579.769 0.124408
\(280\) 0 0
\(281\) 3315.86 0.703942 0.351971 0.936011i \(-0.385512\pi\)
0.351971 + 0.936011i \(0.385512\pi\)
\(282\) 0 0
\(283\) 6602.76i 1.38690i 0.720504 + 0.693451i \(0.243910\pi\)
−0.720504 + 0.693451i \(0.756090\pi\)
\(284\) 0 0
\(285\) 2422.24 2549.02i 0.503442 0.529794i
\(286\) 0 0
\(287\) 3677.46i 0.756353i
\(288\) 0 0
\(289\) 4552.59 0.926642
\(290\) 0 0
\(291\) 1153.12 0.232293
\(292\) 0 0
\(293\) 5814.14i 1.15927i −0.814877 0.579634i \(-0.803195\pi\)
0.814877 0.579634i \(-0.196805\pi\)
\(294\) 0 0
\(295\) 598.050 + 568.303i 0.118033 + 0.112162i
\(296\) 0 0
\(297\) 48.3469i 0.00944570i
\(298\) 0 0
\(299\) 2887.91 0.558570
\(300\) 0 0
\(301\) 5493.49 1.05196
\(302\) 0 0
\(303\) 103.472i 0.0196181i
\(304\) 0 0
\(305\) 1113.18 + 1057.81i 0.208985 + 0.198590i
\(306\) 0 0
\(307\) 8124.86i 1.51046i 0.655462 + 0.755229i \(0.272474\pi\)
−0.655462 + 0.755229i \(0.727526\pi\)
\(308\) 0 0
\(309\) −5268.91 −0.970025
\(310\) 0 0
\(311\) −7336.26 −1.33762 −0.668812 0.743432i \(-0.733197\pi\)
−0.668812 + 0.743432i \(0.733197\pi\)
\(312\) 0 0
\(313\) 2202.66i 0.397768i −0.980023 0.198884i \(-0.936268\pi\)
0.980023 0.198884i \(-0.0637318\pi\)
\(314\) 0 0
\(315\) 1539.42 1620.00i 0.275354 0.289767i
\(316\) 0 0
\(317\) 10008.9i 1.77336i 0.462386 + 0.886679i \(0.346993\pi\)
−0.462386 + 0.886679i \(0.653007\pi\)
\(318\) 0 0
\(319\) 525.488 0.0922309
\(320\) 0 0
\(321\) −4085.21 −0.710325
\(322\) 0 0
\(323\) 1990.27i 0.342854i
\(324\) 0 0
\(325\) 7266.71 370.903i 1.24026 0.0633046i
\(326\) 0 0
\(327\) 963.356i 0.162917i
\(328\) 0 0
\(329\) −8540.47 −1.43116
\(330\) 0 0
\(331\) −8695.94 −1.44402 −0.722012 0.691881i \(-0.756782\pi\)
−0.722012 + 0.691881i \(0.756782\pi\)
\(332\) 0 0
\(333\) 178.959i 0.0294502i
\(334\) 0 0
\(335\) 1339.35 1409.46i 0.218437 0.229871i
\(336\) 0 0
\(337\) 7400.61i 1.19625i 0.801402 + 0.598126i \(0.204088\pi\)
−0.801402 + 0.598126i \(0.795912\pi\)
\(338\) 0 0
\(339\) 4746.74 0.760494
\(340\) 0 0
\(341\) 115.350 0.0183183
\(342\) 0 0
\(343\) 4280.72i 0.673869i
\(344\) 0 0
\(345\) −1206.28 1146.28i −0.188243 0.178880i
\(346\) 0 0
\(347\) 7841.44i 1.21311i −0.795040 0.606557i \(-0.792550\pi\)
0.795040 0.606557i \(-0.207450\pi\)
\(348\) 0 0
\(349\) −4961.26 −0.760946 −0.380473 0.924792i \(-0.624239\pi\)
−0.380473 + 0.924792i \(0.624239\pi\)
\(350\) 0 0
\(351\) −1571.65 −0.238999
\(352\) 0 0
\(353\) 12163.0i 1.83392i −0.398981 0.916959i \(-0.630636\pi\)
0.398981 0.916959i \(-0.369364\pi\)
\(354\) 0 0
\(355\) 4816.46 + 4576.89i 0.720088 + 0.684271i
\(356\) 0 0
\(357\) 1264.89i 0.187522i
\(358\) 0 0
\(359\) 5193.79 0.763559 0.381779 0.924253i \(-0.375311\pi\)
0.381779 + 0.924253i \(0.375311\pi\)
\(360\) 0 0
\(361\) 4131.90 0.602406
\(362\) 0 0
\(363\) 3983.38i 0.575959i
\(364\) 0 0
\(365\) −2466.28 + 2595.37i −0.353674 + 0.372187i
\(366\) 0 0
\(367\) 6086.09i 0.865644i −0.901479 0.432822i \(-0.857518\pi\)
0.901479 0.432822i \(-0.142482\pi\)
\(368\) 0 0
\(369\) 1490.23 0.210239
\(370\) 0 0
\(371\) 10294.7 1.44063
\(372\) 0 0
\(373\) 10581.9i 1.46893i 0.678646 + 0.734466i \(0.262567\pi\)
−0.678646 + 0.734466i \(0.737433\pi\)
\(374\) 0 0
\(375\) −3182.53 2729.40i −0.438253 0.375856i
\(376\) 0 0
\(377\) 17082.4i 2.33366i
\(378\) 0 0
\(379\) −11655.2 −1.57964 −0.789822 0.613336i \(-0.789827\pi\)
−0.789822 + 0.613336i \(0.789827\pi\)
\(380\) 0 0
\(381\) 3591.42 0.482924
\(382\) 0 0
\(383\) 6364.97i 0.849177i −0.905387 0.424588i \(-0.860419\pi\)
0.905387 0.424588i \(-0.139581\pi\)
\(384\) 0 0
\(385\) 306.281 322.313i 0.0405442 0.0426665i
\(386\) 0 0
\(387\) 2226.15i 0.292407i
\(388\) 0 0
\(389\) 6134.33 0.799545 0.399773 0.916614i \(-0.369089\pi\)
0.399773 + 0.916614i \(0.369089\pi\)
\(390\) 0 0
\(391\) 941.862 0.121821
\(392\) 0 0
\(393\) 964.941i 0.123855i
\(394\) 0 0
\(395\) −6244.41 5933.81i −0.795418 0.755854i
\(396\) 0 0
\(397\) 9746.46i 1.23214i 0.787690 + 0.616072i \(0.211277\pi\)
−0.787690 + 0.616072i \(0.788723\pi\)
\(398\) 0 0
\(399\) 6985.12 0.876425
\(400\) 0 0
\(401\) −1306.44 −0.162695 −0.0813474 0.996686i \(-0.525922\pi\)
−0.0813474 + 0.996686i \(0.525922\pi\)
\(402\) 0 0
\(403\) 3749.77i 0.463498i
\(404\) 0 0
\(405\) 656.480 + 623.827i 0.0805450 + 0.0765387i
\(406\) 0 0
\(407\) 35.6055i 0.00433636i
\(408\) 0 0
\(409\) 3876.93 0.468709 0.234354 0.972151i \(-0.424702\pi\)
0.234354 + 0.972151i \(0.424702\pi\)
\(410\) 0 0
\(411\) 1062.87 0.127561
\(412\) 0 0
\(413\) 1638.84i 0.195260i
\(414\) 0 0
\(415\) −1339.49 + 1409.61i −0.158441 + 0.166735i
\(416\) 0 0
\(417\) 231.769i 0.0272177i
\(418\) 0 0
\(419\) 16022.5 1.86814 0.934071 0.357088i \(-0.116230\pi\)
0.934071 + 0.357088i \(0.116230\pi\)
\(420\) 0 0
\(421\) 8119.73 0.939980 0.469990 0.882672i \(-0.344258\pi\)
0.469990 + 0.882672i \(0.344258\pi\)
\(422\) 0 0
\(423\) 3460.89i 0.397812i
\(424\) 0 0
\(425\) 2369.96 120.966i 0.270494 0.0138064i
\(426\) 0 0
\(427\) 3050.46i 0.345719i
\(428\) 0 0
\(429\) −312.694 −0.0351911
\(430\) 0 0
\(431\) 5713.99 0.638592 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(432\) 0 0
\(433\) 6251.34i 0.693811i 0.937900 + 0.346906i \(0.112768\pi\)
−0.937900 + 0.346906i \(0.887232\pi\)
\(434\) 0 0
\(435\) 6780.43 7135.34i 0.747349 0.786468i
\(436\) 0 0
\(437\) 5201.25i 0.569358i
\(438\) 0 0
\(439\) −4230.97 −0.459984 −0.229992 0.973192i \(-0.573870\pi\)
−0.229992 + 0.973192i \(0.573870\pi\)
\(440\) 0 0
\(441\) 1352.31 0.146022
\(442\) 0 0
\(443\) 6314.29i 0.677203i 0.940930 + 0.338601i \(0.109954\pi\)
−0.940930 + 0.338601i \(0.890046\pi\)
\(444\) 0 0
\(445\) −8258.88 7848.08i −0.879794 0.836033i
\(446\) 0 0
\(447\) 5116.13i 0.541353i
\(448\) 0 0
\(449\) 9349.71 0.982717 0.491358 0.870957i \(-0.336501\pi\)
0.491358 + 0.870957i \(0.336501\pi\)
\(450\) 0 0
\(451\) 296.494 0.0309565
\(452\) 0 0
\(453\) 2274.84i 0.235941i
\(454\) 0 0
\(455\) 10477.7 + 9956.53i 1.07956 + 1.02587i
\(456\) 0 0
\(457\) 9547.46i 0.977268i 0.872489 + 0.488634i \(0.162505\pi\)
−0.872489 + 0.488634i \(0.837495\pi\)
\(458\) 0 0
\(459\) −512.578 −0.0521244
\(460\) 0 0
\(461\) −6237.23 −0.630145 −0.315073 0.949068i \(-0.602029\pi\)
−0.315073 + 0.949068i \(0.602029\pi\)
\(462\) 0 0
\(463\) 6469.98i 0.649428i −0.945812 0.324714i \(-0.894732\pi\)
0.945812 0.324714i \(-0.105268\pi\)
\(464\) 0 0
\(465\) 1488.37 1566.28i 0.148434 0.156203i
\(466\) 0 0
\(467\) 7206.64i 0.714097i −0.934086 0.357049i \(-0.883783\pi\)
0.934086 0.357049i \(-0.116217\pi\)
\(468\) 0 0
\(469\) 3862.35 0.380270
\(470\) 0 0
\(471\) 5307.16 0.519195
\(472\) 0 0
\(473\) 442.912i 0.0430552i
\(474\) 0 0
\(475\) −668.012 13087.6i −0.0645274 1.26422i
\(476\) 0 0
\(477\) 4171.75i 0.400443i
\(478\) 0 0
\(479\) −10851.8 −1.03514 −0.517571 0.855640i \(-0.673164\pi\)
−0.517571 + 0.855640i \(0.673164\pi\)
\(480\) 0 0
\(481\) −1157.46 −0.109720
\(482\) 0 0
\(483\) 3305.59i 0.311407i
\(484\) 0 0
\(485\) 2960.29 3115.24i 0.277154 0.291661i
\(486\) 0 0
\(487\) 12757.1i 1.18702i 0.804827 + 0.593510i \(0.202258\pi\)
−0.804827 + 0.593510i \(0.797742\pi\)
\(488\) 0 0
\(489\) 2645.16 0.244618
\(490\) 0 0
\(491\) −7016.52 −0.644911 −0.322455 0.946585i \(-0.604508\pi\)
−0.322455 + 0.946585i \(0.604508\pi\)
\(492\) 0 0
\(493\) 5571.26i 0.508960i
\(494\) 0 0
\(495\) 130.612 + 124.116i 0.0118598 + 0.0112699i
\(496\) 0 0
\(497\) 13198.6i 1.19122i
\(498\) 0 0
\(499\) −11372.3 −1.02023 −0.510113 0.860107i \(-0.670396\pi\)
−0.510113 + 0.860107i \(0.670396\pi\)
\(500\) 0 0
\(501\) 650.700 0.0580262
\(502\) 0 0
\(503\) 5587.37i 0.495285i 0.968851 + 0.247643i \(0.0796559\pi\)
−0.968851 + 0.247643i \(0.920344\pi\)
\(504\) 0 0
\(505\) −279.535 265.631i −0.0246320 0.0234068i
\(506\) 0 0
\(507\) 3573.99i 0.313070i
\(508\) 0 0
\(509\) 16256.7 1.41565 0.707825 0.706388i \(-0.249676\pi\)
0.707825 + 0.706388i \(0.249676\pi\)
\(510\) 0 0
\(511\) −7112.14 −0.615699
\(512\) 0 0
\(513\) 2830.61i 0.243615i
\(514\) 0 0
\(515\) −13526.3 + 14234.3i −1.15736 + 1.21794i
\(516\) 0 0
\(517\) 688.574i 0.0585754i
\(518\) 0 0
\(519\) 12377.7 1.04686
\(520\) 0 0
\(521\) 19748.4 1.66064 0.830320 0.557286i \(-0.188157\pi\)
0.830320 + 0.557286i \(0.188157\pi\)
\(522\) 0 0
\(523\) 7843.44i 0.655774i −0.944717 0.327887i \(-0.893663\pi\)
0.944717 0.327887i \(-0.106337\pi\)
\(524\) 0 0
\(525\) −424.546 8317.69i −0.0352928 0.691455i
\(526\) 0 0
\(527\) 1222.95i 0.101086i
\(528\) 0 0
\(529\) 9705.60 0.797699
\(530\) 0 0
\(531\) −664.116 −0.0542753
\(532\) 0 0
\(533\) 9638.38i 0.783273i
\(534\) 0 0
\(535\) −10487.5 + 11036.5i −0.847504 + 0.891865i
\(536\) 0 0
\(537\) 9639.42i 0.774622i
\(538\) 0 0
\(539\) 269.053 0.0215008
\(540\) 0 0
\(541\) −7383.29 −0.586751 −0.293376 0.955997i \(-0.594779\pi\)
−0.293376 + 0.955997i \(0.594779\pi\)
\(542\) 0 0
\(543\) 10183.3i 0.804798i
\(544\) 0 0
\(545\) 2602.57 + 2473.12i 0.204554 + 0.194379i
\(546\) 0 0
\(547\) 3354.90i 0.262240i 0.991367 + 0.131120i \(0.0418573\pi\)
−0.991367 + 0.131120i \(0.958143\pi\)
\(548\) 0 0
\(549\) −1236.15 −0.0960976
\(550\) 0 0
\(551\) 30766.2 2.37874
\(552\) 0 0
\(553\) 17111.6i 1.31584i
\(554\) 0 0
\(555\) 483.469 + 459.422i 0.0369768 + 0.0351376i
\(556\) 0 0
\(557\) 20771.8i 1.58012i 0.613028 + 0.790061i \(0.289951\pi\)
−0.613028 + 0.790061i \(0.710049\pi\)
\(558\) 0 0
\(559\) −14398.1 −1.08940
\(560\) 0 0
\(561\) −101.982 −0.00767500
\(562\) 0 0
\(563\) 7194.86i 0.538592i 0.963057 + 0.269296i \(0.0867910\pi\)
−0.963057 + 0.269296i \(0.913209\pi\)
\(564\) 0 0
\(565\) 12185.8 12823.6i 0.907361 0.954856i
\(566\) 0 0
\(567\) 1798.96i 0.133244i
\(568\) 0 0
\(569\) −11549.5 −0.850931 −0.425466 0.904975i \(-0.639890\pi\)
−0.425466 + 0.904975i \(0.639890\pi\)
\(570\) 0 0
\(571\) 1482.54 0.108655 0.0543277 0.998523i \(-0.482698\pi\)
0.0543277 + 0.998523i \(0.482698\pi\)
\(572\) 0 0
\(573\) 10402.5i 0.758412i
\(574\) 0 0
\(575\) −6193.50 + 316.125i −0.449194 + 0.0229275i
\(576\) 0 0
\(577\) 15264.0i 1.10130i 0.834737 + 0.550649i \(0.185620\pi\)
−0.834737 + 0.550649i \(0.814380\pi\)
\(578\) 0 0
\(579\) −5376.43 −0.385901
\(580\) 0 0
\(581\) −3862.76 −0.275825
\(582\) 0 0
\(583\) 830.006i 0.0589628i
\(584\) 0 0
\(585\) −4034.73 + 4245.92i −0.285155 + 0.300081i
\(586\) 0 0
\(587\) 1736.89i 0.122128i −0.998134 0.0610639i \(-0.980551\pi\)
0.998134 0.0610639i \(-0.0194493\pi\)
\(588\) 0 0
\(589\) 6753.50 0.472450
\(590\) 0 0
\(591\) −5034.57 −0.350414
\(592\) 0 0
\(593\) 11764.8i 0.814707i −0.913271 0.407353i \(-0.866452\pi\)
0.913271 0.407353i \(-0.133548\pi\)
\(594\) 0 0
\(595\) 3417.19 + 3247.22i 0.235447 + 0.223736i
\(596\) 0 0
\(597\) 9324.69i 0.639253i
\(598\) 0 0
\(599\) −9451.99 −0.644737 −0.322369 0.946614i \(-0.604479\pi\)
−0.322369 + 0.946614i \(0.604479\pi\)
\(600\) 0 0
\(601\) −3131.93 −0.212569 −0.106285 0.994336i \(-0.533895\pi\)
−0.106285 + 0.994336i \(0.533895\pi\)
\(602\) 0 0
\(603\) 1565.16i 0.105702i
\(604\) 0 0
\(605\) −10761.4 10226.1i −0.723159 0.687189i
\(606\) 0 0
\(607\) 22700.8i 1.51795i 0.651120 + 0.758975i \(0.274300\pi\)
−0.651120 + 0.758975i \(0.725700\pi\)
\(608\) 0 0
\(609\) 19553.1 1.30103
\(610\) 0 0
\(611\) 22384.0 1.48210
\(612\) 0 0
\(613\) 28911.6i 1.90494i −0.304629 0.952471i \(-0.598532\pi\)
0.304629 0.952471i \(-0.401468\pi\)
\(614\) 0 0
\(615\) 3825.70 4025.95i 0.250841 0.263971i
\(616\) 0 0
\(617\) 5566.87i 0.363231i 0.983370 + 0.181616i \(0.0581326\pi\)
−0.983370 + 0.181616i \(0.941867\pi\)
\(618\) 0 0
\(619\) −4150.32 −0.269492 −0.134746 0.990880i \(-0.543022\pi\)
−0.134746 + 0.990880i \(0.543022\pi\)
\(620\) 0 0
\(621\) 1339.54 0.0865600
\(622\) 0 0
\(623\) 22631.9i 1.45542i
\(624\) 0 0
\(625\) −15543.8 + 1590.90i −0.994803 + 0.101818i
\(626\) 0 0
\(627\) 563.175i 0.0358709i
\(628\) 0 0
\(629\) −377.492 −0.0239294
\(630\) 0 0
\(631\) 4090.09 0.258041 0.129021 0.991642i \(-0.458817\pi\)
0.129021 + 0.991642i \(0.458817\pi\)
\(632\) 0 0
\(633\) 13419.8i 0.842637i
\(634\) 0 0
\(635\) 9219.85 9702.45i 0.576187 0.606346i
\(636\) 0 0
\(637\) 8746.32i 0.544022i
\(638\) 0 0
\(639\) −5348.53 −0.331118
\(640\) 0 0
\(641\) 3909.35 0.240890 0.120445 0.992720i \(-0.461568\pi\)
0.120445 + 0.992720i \(0.461568\pi\)
\(642\) 0 0
\(643\) 30539.5i 1.87303i 0.350624 + 0.936516i \(0.385969\pi\)
−0.350624 + 0.936516i \(0.614031\pi\)
\(644\) 0 0
\(645\) 6014.08 + 5714.94i 0.367138 + 0.348877i
\(646\) 0 0
\(647\) 12707.7i 0.772167i −0.922464 0.386083i \(-0.873828\pi\)
0.922464 0.386083i \(-0.126172\pi\)
\(648\) 0 0
\(649\) −132.132 −0.00799170
\(650\) 0 0
\(651\) 4292.10 0.258403
\(652\) 0 0
\(653\) 12777.6i 0.765737i −0.923803 0.382869i \(-0.874936\pi\)
0.923803 0.382869i \(-0.125064\pi\)
\(654\) 0 0
\(655\) −2606.85 2477.18i −0.155508 0.147773i
\(656\) 0 0
\(657\) 2882.08i 0.171143i
\(658\) 0 0
\(659\) −23563.5 −1.39287 −0.696435 0.717620i \(-0.745232\pi\)
−0.696435 + 0.717620i \(0.745232\pi\)
\(660\) 0 0
\(661\) 4361.31 0.256634 0.128317 0.991733i \(-0.459042\pi\)
0.128317 + 0.991733i \(0.459042\pi\)
\(662\) 0 0
\(663\) 3315.21i 0.194196i
\(664\) 0 0
\(665\) 17932.1 18870.7i 1.04568 1.10042i
\(666\) 0 0
\(667\) 14559.6i 0.845200i
\(668\) 0 0
\(669\) −5260.25 −0.303995
\(670\) 0 0
\(671\) −245.943 −0.0141498
\(672\) 0 0
\(673\) 8203.52i 0.469870i −0.972011 0.234935i \(-0.924512\pi\)
0.972011 0.234935i \(-0.0754877\pi\)
\(674\) 0 0
\(675\) 3370.61 172.041i 0.192200 0.00981015i
\(676\) 0 0
\(677\) 28057.1i 1.59279i 0.604774 + 0.796397i \(0.293263\pi\)
−0.604774 + 0.796397i \(0.706737\pi\)
\(678\) 0 0
\(679\) 8536.73 0.482488
\(680\) 0 0
\(681\) −2810.70 −0.158159
\(682\) 0 0
\(683\) 3344.62i 0.187377i −0.995602 0.0936885i \(-0.970134\pi\)
0.995602 0.0936885i \(-0.0298658\pi\)
\(684\) 0 0
\(685\) 2728.59 2871.41i 0.152196 0.160162i
\(686\) 0 0
\(687\) 7746.17i 0.430182i
\(688\) 0 0
\(689\) −26981.7 −1.49190
\(690\) 0 0
\(691\) 12964.8 0.713757 0.356879 0.934151i \(-0.383841\pi\)
0.356879 + 0.934151i \(0.383841\pi\)
\(692\) 0 0
\(693\) 357.918i 0.0196193i
\(694\) 0 0
\(695\) −626.138 594.994i −0.0341737 0.0324740i
\(696\) 0 0
\(697\) 3143.46i 0.170828i
\(698\) 0 0
\(699\) −6885.03 −0.372555
\(700\) 0 0
\(701\) 16162.1 0.870806 0.435403 0.900236i \(-0.356606\pi\)
0.435403 + 0.900236i \(0.356606\pi\)
\(702\) 0 0
\(703\) 2084.63i 0.111839i
\(704\) 0 0
\(705\) −9349.82 8884.76i −0.499482 0.474638i
\(706\) 0 0
\(707\) 766.014i 0.0407481i
\(708\) 0 0
\(709\) −14244.4 −0.754529 −0.377265 0.926105i \(-0.623135\pi\)
−0.377265 + 0.926105i \(0.623135\pi\)
\(710\) 0 0
\(711\) 6934.22 0.365757
\(712\) 0 0
\(713\) 3195.97i 0.167868i
\(714\) 0 0
\(715\) −802.744 + 844.762i −0.0419873 + 0.0441850i
\(716\) 0 0
\(717\) 6882.02i 0.358457i
\(718\) 0 0
\(719\) −27638.5 −1.43358 −0.716790 0.697289i \(-0.754389\pi\)
−0.716790 + 0.697289i \(0.754389\pi\)
\(720\) 0 0
\(721\) −39006.4 −2.01480
\(722\) 0 0
\(723\) 1146.86i 0.0589934i
\(724\) 0 0
\(725\) −1869.93 36635.5i −0.0957895 1.87670i
\(726\) 0 0
\(727\) 2525.52i 0.128840i −0.997923 0.0644199i \(-0.979480\pi\)
0.997923 0.0644199i \(-0.0205197\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −4695.79 −0.237592
\(732\) 0 0
\(733\) 8400.27i 0.423289i 0.977347 + 0.211645i \(0.0678820\pi\)
−0.977347 + 0.211645i \(0.932118\pi\)
\(734\) 0 0
\(735\) 3471.62 3653.34i 0.174221 0.183341i
\(736\) 0 0
\(737\) 311.401i 0.0155639i
\(738\) 0 0
\(739\) 19689.1 0.980074 0.490037 0.871702i \(-0.336983\pi\)
0.490037 + 0.871702i \(0.336983\pi\)
\(740\) 0 0
\(741\) −18307.6 −0.907619
\(742\) 0 0
\(743\) 22526.6i 1.11227i 0.831091 + 0.556137i \(0.187717\pi\)
−0.831091 + 0.556137i \(0.812283\pi\)
\(744\) 0 0
\(745\) −13821.6 13134.1i −0.679708 0.645900i
\(746\) 0 0
\(747\) 1565.32i 0.0766696i
\(748\) 0 0
\(749\) −30243.3 −1.47539
\(750\) 0 0
\(751\) −34691.1 −1.68562 −0.842808 0.538215i \(-0.819099\pi\)
−0.842808 + 0.538215i \(0.819099\pi\)
\(752\) 0 0
\(753\) 6779.95i 0.328121i
\(754\) 0 0
\(755\) −6145.63 5839.95i −0.296242 0.281507i
\(756\) 0 0
\(757\) 6619.98i 0.317843i 0.987291 + 0.158922i \(0.0508017\pi\)
−0.987291 + 0.158922i \(0.949198\pi\)
\(758\) 0 0
\(759\) 266.512 0.0127454
\(760\) 0 0
\(761\) −29368.7 −1.39897 −0.699483 0.714649i \(-0.746586\pi\)
−0.699483 + 0.714649i \(0.746586\pi\)
\(762\) 0 0
\(763\) 7131.84i 0.338388i
\(764\) 0 0
\(765\) −1315.88 + 1384.76i −0.0621907 + 0.0654460i
\(766\) 0 0
\(767\) 4295.31i 0.202209i
\(768\) 0 0
\(769\) −32677.4 −1.53235 −0.766174 0.642633i \(-0.777842\pi\)
−0.766174 + 0.642633i \(0.777842\pi\)
\(770\) 0 0
\(771\) −278.353 −0.0130021
\(772\) 0 0
\(773\) 28047.5i 1.30504i 0.757770 + 0.652522i \(0.226289\pi\)
−0.757770 + 0.652522i \(0.773711\pi\)
\(774\) 0 0
\(775\) −410.469 8041.87i −0.0190251 0.372739i
\(776\) 0 0
\(777\) 1324.86i 0.0611699i
\(778\) 0 0
\(779\) 17359.1 0.798402
\(780\) 0 0
\(781\) −1064.14 −0.0487552
\(782\) 0 0
\(783\) 7923.57i 0.361642i
\(784\) 0 0
\(785\) 13624.5 14337.6i 0.619463 0.651887i
\(786\) 0 0
\(787\) 22172.1i 1.00426i 0.864793 + 0.502128i \(0.167449\pi\)
−0.864793 + 0.502128i \(0.832551\pi\)
\(788\) 0 0
\(789\) −1704.94 −0.0769295
\(790\) 0 0
\(791\) 35140.7 1.57960
\(792\) 0 0
\(793\) 7995.06i 0.358024i
\(794\) 0 0
\(795\) 11270.2 + 10709.7i 0.502786 + 0.477777i
\(796\) 0 0
\(797\) 24170.3i 1.07422i