Properties

Label 960.4.f.q.769.2
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 769.2
Root \(-3.70156i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.q.769.4

$q$-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 \(-0.795328\pi\)
0.800302 0.599597i \(-0.204672\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 1.79063 0.0490813 0.0245407 0.999699i \(-0.492188\pi\)
0.0245407 + 0.999699i \(0.492188\pi\)
\(12\) 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 \(-0.281852\pi\)
0.632931 + 0.774208i \(0.281852\pi\)
\(20\) 0 0
\(21\) −66.6281 −0.692355
\(22\) 0 0
\(23\) 49.6125i 0.449779i −0.974384 0.224890i \(-0.927798\pi\)
0.974384 0.224890i \(-0.0722021\pi\)
\(24\) 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 \(-0.440249\pi\)
0.186612 + 0.982434i \(0.440249\pi\)
\(32\) 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 \(0.144529\pi\)
−0.898677 + 0.438611i \(0.855471\pi\)
\(44\) 0 0
\(45\) −72.9422 69.3141i −0.241635 0.229616i
\(46\) 0 0
\(47\) 384.544i 1.19344i −0.802451 0.596718i \(-0.796471\pi\)
0.802451 0.596718i \(-0.203529\pi\)
\(48\) 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 \(-0.525943\pi\)
−0.0814129 + 0.996680i \(0.525943\pi\)
\(60\) 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 \(0.0506827\pi\)
−0.987351 + 0.158552i \(0.949317\pi\)
\(68\) 0 0
\(69\) −148.837 −0.259680
\(70\) 0 0
\(71\) −594.281 −0.993355 −0.496677 0.867935i \(-0.665447\pi\)
−0.496677 + 0.867935i \(0.665447\pi\)
\(72\) 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 \(-0.315147\pi\)
0.548636 + 0.836061i \(0.315147\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 173.925i 0.230009i −0.993365 0.115004i \(-0.963312\pi\)
0.993365 0.115004i \(-0.0366882\pi\)
\(84\) 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 \(-0.682516\pi\)
0.542484 0.840066i \(-0.317484\pi\)
\(104\) 0 0
\(105\) −540.000 513.141i −0.501891 0.476928i
\(106\) 0 0
\(107\) 1361.74i 1.23032i −0.788403 0.615159i \(-0.789092\pi\)
0.788403 0.615159i \(-0.210908\pi\)
\(108\) 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 \(0.137348\pi\)
−0.908344 + 0.418225i \(0.862652\pi\)
\(128\) 0 0
\(129\) 742.050 0.506464
\(130\) 0 0
\(131\) 321.647 0.214522 0.107261 0.994231i \(-0.465792\pi\)
0.107261 + 0.994231i \(0.465792\pi\)
\(132\) 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 \(-0.492496\pi\)
0.0235712 + 0.999722i \(0.492496\pi\)
\(140\) 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 \(-0.434498\pi\)
0.204331 + 0.978902i \(0.434498\pi\)
\(152\) 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 \(0.0679473\pi\)
−0.977303 + 0.211845i \(0.932053\pi\)
\(164\) 0 0
\(165\) 41.3719 43.5374i 0.0195200 0.0205417i
\(166\) 0 0
\(167\) 216.900i 0.100504i 0.998737 + 0.0502522i \(0.0160025\pi\)
−0.998737 + 0.0502522i \(0.983997\pi\)
\(168\) 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 \(-0.734067\pi\)
−0.670842 + 0.741600i \(0.734067\pi\)
\(180\) 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 \(-0.728092\pi\)
−0.656804 + 0.754062i \(0.728092\pi\)
\(192\) 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 \(-0.686750\pi\)
−0.553610 + 0.832776i \(0.686750\pi\)
\(200\) 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 \(-0.239639\pi\)
0.729745 + 0.683719i \(0.239639\pi\)
\(212\) 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 \(-0.915200\pi\)
0.964723 0.263268i \(-0.0848003\pi\)
\(224\) 0 0
\(225\) −57.3469 1123.54i −0.0169917 0.332900i
\(226\) 0 0
\(227\) 936.900i 0.273939i −0.990575 0.136970i \(-0.956264\pi\)
0.990575 0.136970i \(-0.0437363\pi\)
\(228\) 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 \(-0.399526\pi\)
0.310433 + 0.950595i \(0.399526\pi\)
\(240\) 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 \(-0.408285\pi\)
0.284161 + 0.958777i \(0.408285\pi\)
\(252\) 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 \(-0.978778\pi\)
0.997778 0.0666229i \(-0.0212224\pi\)
\(264\) 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 \(-0.150492\pi\)
0.890304 + 0.455366i \(0.150492\pi\)
\(272\) 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 \(-0.756090\pi\)
0.720504 0.693451i \(-0.243910\pi\)
\(284\) 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 \(-0.727526\pi\)
0.655462 0.755229i \(-0.272474\pi\)
\(308\) 0 0
\(309\) −5268.91 −0.970025
\(310\) 0 0
\(311\) 7336.26 1.33762 0.668812 0.743432i \(-0.266803\pi\)
0.668812 + 0.743432i \(0.266803\pi\)
\(312\) 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 \(-0.243218\pi\)
0.722012 + 0.691881i \(0.243218\pi\)
\(332\) 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 \(0.207450\pi\)
−0.795040 + 0.606557i \(0.792550\pi\)
\(348\) 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 \(-0.624689\pi\)
−0.381779 + 0.924253i \(0.624689\pi\)
\(360\) 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 \(0.142482\pi\)
−0.901479 + 0.432822i \(0.857518\pi\)
\(368\) 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 \(-0.210173\pi\)
0.789822 + 0.613336i \(0.210173\pi\)
\(380\) 0 0
\(381\) 3591.42 0.482924
\(382\) 0 0
\(383\) 6364.97i 0.849177i 0.905387 + 0.424588i \(0.139581\pi\)
−0.905387 + 0.424588i \(0.860419\pi\)
\(384\) 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 \(-0.883770\pi\)
−0.934071 + 0.357088i \(0.883770\pi\)
\(420\) 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 \(-0.603446\pi\)
−0.319296 + 0.947655i \(0.603446\pi\)
\(432\) 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 \(-0.426130\pi\)
0.229992 + 0.973192i \(0.426130\pi\)
\(440\) 0 0
\(441\) 1352.31 0.146022
\(442\) 0 0
\(443\) 6314.29i 0.677203i −0.940930 0.338601i \(-0.890046\pi\)
0.940930 0.338601i \(-0.109954\pi\)
\(444\) 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 \(0.105268\pi\)
−0.945812 + 0.324714i \(0.894732\pi\)
\(464\) 0 0
\(465\) 1488.37 1566.28i 0.148434 0.156203i
\(466\) 0 0
\(467\) 7206.64i 0.714097i 0.934086 + 0.357049i \(0.116217\pi\)
−0.934086 + 0.357049i \(0.883783\pi\)
\(468\) 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 \(-0.326836\pi\)
0.517571 + 0.855640i \(0.326836\pi\)
\(480\) 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 \(-0.797742\pi\)
0.804827 0.593510i \(-0.202258\pi\)
\(488\) 0 0
\(489\) 2645.16 0.244618
\(490\) 0 0
\(491\) 7016.52 0.644911 0.322455 0.946585i \(-0.395492\pi\)
0.322455 + 0.946585i \(0.395492\pi\)
\(492\) 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 \(-0.329604\pi\)
0.510113 + 0.860107i \(0.329604\pi\)
\(500\) 0 0
\(501\) 650.700 0.0580262
\(502\) 0 0
\(503\) 5587.37i 0.495285i −0.968851 0.247643i \(-0.920344\pi\)
0.968851 0.247643i \(-0.0796559\pi\)
\(504\) 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 \(0.106337\pi\)
−0.944717 + 0.327887i \(0.893663\pi\)
\(524\) 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 \(-0.958143\pi\)
0.991367 0.131120i \(-0.0418573\pi\)
\(548\) 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 \(-0.913209\pi\)
0.963057 0.269296i \(-0.0867910\pi\)
\(564\) 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 \(-0.517302\pi\)
−0.0543277 + 0.998523i \(0.517302\pi\)
\(572\) 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 \(0.0194493\pi\)
−0.998134 + 0.0610639i \(0.980551\pi\)
\(588\) 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 \(-0.395521\pi\)
0.322369 + 0.946614i \(0.395521\pi\)
\(600\) 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 \(-0.725700\pi\)
0.651120 0.758975i \(-0.274300\pi\)
\(608\) 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 \(-0.456978\pi\)
0.134746 + 0.990880i \(0.456978\pi\)
\(620\) 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 \(-0.541183\pi\)
−0.129021 + 0.991642i \(0.541183\pi\)
\(632\) 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 \(-0.614031\pi\)
0.350624 0.936516i \(-0.385969\pi\)
\(644\) 0 0
\(645\) 6014.08 + 5714.94i 0.367138 + 0.348877i
\(646\) 0 0
\(647\) 12707.7i 0.772167i 0.922464 + 0.386083i \(0.126172\pi\)
−0.922464 + 0.386083i \(0.873828\pi\)
\(648\) 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 \(-0.254768\pi\)
0.696435 + 0.717620i \(0.254768\pi\)
\(660\) 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 \(0.0298658\pi\)
−0.995602 + 0.0936885i \(0.970134\pi\)
\(684\) 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 \(-0.616159\pi\)
−0.356879 + 0.934151i \(0.616159\pi\)
\(692\) 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 \(-0.245611\pi\)
0.716790 + 0.697289i \(0.245611\pi\)
\(720\) 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 \(0.0205197\pi\)
−0.997923 + 0.0644199i \(0.979480\pi\)
\(728\) 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 \(-0.663017\pi\)
−0.490037 + 0.871702i \(0.663017\pi\)
\(740\) 0 0
\(741\) −18307.6 −0.907619
\(742\) 0 0
\(743\) 22526.6i 1.11227i −0.831091 0.556137i \(-0.812283\pi\)
0.831091 0.556137i \(-0.187717\pi\)
\(744\) 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 \(-0.180901\pi\)
0.842808 + 0.538215i \(0.180901\pi\)
\(752\) 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 \(-0.832551\pi\)
0.864793 0.502128i \(-0.167449\pi\)
\(788\) 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