Properties

Label 1560.4.a.n.1.1
Level $1560$
Weight $4$
Character 1560.1
Self dual yes
Analytic conductor $92.043$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.0429796090\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 41x^{2} - 30x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.174957\) of defining polynomial
Character \(\chi\) \(=\) 1560.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{5} -35.1436 q^{7} +9.00000 q^{9} +13.7440 q^{11} +13.0000 q^{13} +15.0000 q^{15} +87.1577 q^{17} +11.6869 q^{19} -105.431 q^{21} -108.262 q^{23} +25.0000 q^{25} +27.0000 q^{27} -150.600 q^{29} -128.191 q^{31} +41.2319 q^{33} -175.718 q^{35} -11.7756 q^{37} +39.0000 q^{39} +366.620 q^{41} +301.237 q^{43} +45.0000 q^{45} -156.862 q^{47} +892.074 q^{49} +261.473 q^{51} -229.058 q^{53} +68.7198 q^{55} +35.0607 q^{57} -377.723 q^{59} -406.355 q^{61} -316.293 q^{63} +65.0000 q^{65} -352.506 q^{67} -324.786 q^{69} -842.944 q^{71} +50.4272 q^{73} +75.0000 q^{75} -483.013 q^{77} -695.344 q^{79} +81.0000 q^{81} -675.845 q^{83} +435.789 q^{85} -451.801 q^{87} -943.389 q^{89} -456.867 q^{91} -384.572 q^{93} +58.4345 q^{95} +73.8249 q^{97} +123.696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −35.1436 −1.89758 −0.948789 0.315912i \(-0.897690\pi\)
−0.948789 + 0.315912i \(0.897690\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 13.7440 0.376724 0.188362 0.982100i \(-0.439682\pi\)
0.188362 + 0.982100i \(0.439682\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 87.1577 1.24346 0.621731 0.783231i \(-0.286430\pi\)
0.621731 + 0.783231i \(0.286430\pi\)
\(18\) 0 0
\(19\) 11.6869 0.141114 0.0705568 0.997508i \(-0.477522\pi\)
0.0705568 + 0.997508i \(0.477522\pi\)
\(20\) 0 0
\(21\) −105.431 −1.09557
\(22\) 0 0
\(23\) −108.262 −0.981485 −0.490742 0.871305i \(-0.663274\pi\)
−0.490742 + 0.871305i \(0.663274\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −150.600 −0.964337 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(30\) 0 0
\(31\) −128.191 −0.742701 −0.371350 0.928493i \(-0.621105\pi\)
−0.371350 + 0.928493i \(0.621105\pi\)
\(32\) 0 0
\(33\) 41.2319 0.217502
\(34\) 0 0
\(35\) −175.718 −0.848622
\(36\) 0 0
\(37\) −11.7756 −0.0523216 −0.0261608 0.999658i \(-0.508328\pi\)
−0.0261608 + 0.999658i \(0.508328\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 366.620 1.39650 0.698249 0.715855i \(-0.253963\pi\)
0.698249 + 0.715855i \(0.253963\pi\)
\(42\) 0 0
\(43\) 301.237 1.06833 0.534165 0.845380i \(-0.320626\pi\)
0.534165 + 0.845380i \(0.320626\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) −156.862 −0.486822 −0.243411 0.969923i \(-0.578266\pi\)
−0.243411 + 0.969923i \(0.578266\pi\)
\(48\) 0 0
\(49\) 892.074 2.60080
\(50\) 0 0
\(51\) 261.473 0.717913
\(52\) 0 0
\(53\) −229.058 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(54\) 0 0
\(55\) 68.7198 0.168476
\(56\) 0 0
\(57\) 35.0607 0.0814720
\(58\) 0 0
\(59\) −377.723 −0.833482 −0.416741 0.909025i \(-0.636828\pi\)
−0.416741 + 0.909025i \(0.636828\pi\)
\(60\) 0 0
\(61\) −406.355 −0.852925 −0.426462 0.904505i \(-0.640240\pi\)
−0.426462 + 0.904505i \(0.640240\pi\)
\(62\) 0 0
\(63\) −316.293 −0.632526
\(64\) 0 0
\(65\) 65.0000 0.124035
\(66\) 0 0
\(67\) −352.506 −0.642768 −0.321384 0.946949i \(-0.604148\pi\)
−0.321384 + 0.946949i \(0.604148\pi\)
\(68\) 0 0
\(69\) −324.786 −0.566661
\(70\) 0 0
\(71\) −842.944 −1.40900 −0.704500 0.709704i \(-0.748829\pi\)
−0.704500 + 0.709704i \(0.748829\pi\)
\(72\) 0 0
\(73\) 50.4272 0.0808502 0.0404251 0.999183i \(-0.487129\pi\)
0.0404251 + 0.999183i \(0.487129\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −483.013 −0.714862
\(78\) 0 0
\(79\) −695.344 −0.990283 −0.495141 0.868812i \(-0.664884\pi\)
−0.495141 + 0.868812i \(0.664884\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −675.845 −0.893779 −0.446889 0.894589i \(-0.647468\pi\)
−0.446889 + 0.894589i \(0.647468\pi\)
\(84\) 0 0
\(85\) 435.789 0.556093
\(86\) 0 0
\(87\) −451.801 −0.556760
\(88\) 0 0
\(89\) −943.389 −1.12358 −0.561792 0.827278i \(-0.689888\pi\)
−0.561792 + 0.827278i \(0.689888\pi\)
\(90\) 0 0
\(91\) −456.867 −0.526293
\(92\) 0 0
\(93\) −384.572 −0.428798
\(94\) 0 0
\(95\) 58.4345 0.0631080
\(96\) 0 0
\(97\) 73.8249 0.0772761 0.0386381 0.999253i \(-0.487698\pi\)
0.0386381 + 0.999253i \(0.487698\pi\)
\(98\) 0 0
\(99\) 123.696 0.125575
\(100\) 0 0
\(101\) −0.889523 −0.000876345 0 −0.000438172 1.00000i \(-0.500139\pi\)
−0.000438172 1.00000i \(0.500139\pi\)
\(102\) 0 0
\(103\) −458.408 −0.438527 −0.219263 0.975666i \(-0.570365\pi\)
−0.219263 + 0.975666i \(0.570365\pi\)
\(104\) 0 0
\(105\) −527.154 −0.489952
\(106\) 0 0
\(107\) 967.396 0.874035 0.437017 0.899453i \(-0.356035\pi\)
0.437017 + 0.899453i \(0.356035\pi\)
\(108\) 0 0
\(109\) 626.068 0.550151 0.275075 0.961423i \(-0.411297\pi\)
0.275075 + 0.961423i \(0.411297\pi\)
\(110\) 0 0
\(111\) −35.3268 −0.0302079
\(112\) 0 0
\(113\) −2241.20 −1.86579 −0.932896 0.360146i \(-0.882727\pi\)
−0.932896 + 0.360146i \(0.882727\pi\)
\(114\) 0 0
\(115\) −541.309 −0.438933
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) −3063.04 −2.35957
\(120\) 0 0
\(121\) −1142.10 −0.858079
\(122\) 0 0
\(123\) 1099.86 0.806268
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 895.939 0.625998 0.312999 0.949754i \(-0.398666\pi\)
0.312999 + 0.949754i \(0.398666\pi\)
\(128\) 0 0
\(129\) 903.711 0.616801
\(130\) 0 0
\(131\) −636.783 −0.424702 −0.212351 0.977193i \(-0.568112\pi\)
−0.212351 + 0.977193i \(0.568112\pi\)
\(132\) 0 0
\(133\) −410.720 −0.267774
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 505.892 0.315483 0.157742 0.987480i \(-0.449579\pi\)
0.157742 + 0.987480i \(0.449579\pi\)
\(138\) 0 0
\(139\) −1521.56 −0.928470 −0.464235 0.885712i \(-0.653671\pi\)
−0.464235 + 0.885712i \(0.653671\pi\)
\(140\) 0 0
\(141\) −470.585 −0.281067
\(142\) 0 0
\(143\) 178.672 0.104484
\(144\) 0 0
\(145\) −753.002 −0.431265
\(146\) 0 0
\(147\) 2676.22 1.50157
\(148\) 0 0
\(149\) −3105.92 −1.70770 −0.853849 0.520521i \(-0.825738\pi\)
−0.853849 + 0.520521i \(0.825738\pi\)
\(150\) 0 0
\(151\) −1634.44 −0.880856 −0.440428 0.897788i \(-0.645173\pi\)
−0.440428 + 0.897788i \(0.645173\pi\)
\(152\) 0 0
\(153\) 784.420 0.414487
\(154\) 0 0
\(155\) −640.953 −0.332146
\(156\) 0 0
\(157\) 3132.82 1.59253 0.796263 0.604951i \(-0.206807\pi\)
0.796263 + 0.604951i \(0.206807\pi\)
\(158\) 0 0
\(159\) −687.174 −0.342745
\(160\) 0 0
\(161\) 3804.71 1.86244
\(162\) 0 0
\(163\) 1419.27 0.681997 0.340999 0.940064i \(-0.389235\pi\)
0.340999 + 0.940064i \(0.389235\pi\)
\(164\) 0 0
\(165\) 206.159 0.0972697
\(166\) 0 0
\(167\) 3871.88 1.79410 0.897051 0.441927i \(-0.145705\pi\)
0.897051 + 0.441927i \(0.145705\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 105.182 0.0470379
\(172\) 0 0
\(173\) −1619.92 −0.711909 −0.355955 0.934503i \(-0.615844\pi\)
−0.355955 + 0.934503i \(0.615844\pi\)
\(174\) 0 0
\(175\) −878.591 −0.379515
\(176\) 0 0
\(177\) −1133.17 −0.481211
\(178\) 0 0
\(179\) −2139.73 −0.893470 −0.446735 0.894666i \(-0.647413\pi\)
−0.446735 + 0.894666i \(0.647413\pi\)
\(180\) 0 0
\(181\) −464.563 −0.190777 −0.0953887 0.995440i \(-0.530409\pi\)
−0.0953887 + 0.995440i \(0.530409\pi\)
\(182\) 0 0
\(183\) −1219.06 −0.492436
\(184\) 0 0
\(185\) −58.8780 −0.0233989
\(186\) 0 0
\(187\) 1197.89 0.468442
\(188\) 0 0
\(189\) −948.878 −0.365189
\(190\) 0 0
\(191\) 2698.18 1.02216 0.511082 0.859532i \(-0.329245\pi\)
0.511082 + 0.859532i \(0.329245\pi\)
\(192\) 0 0
\(193\) −3896.41 −1.45321 −0.726606 0.687054i \(-0.758903\pi\)
−0.726606 + 0.687054i \(0.758903\pi\)
\(194\) 0 0
\(195\) 195.000 0.0716115
\(196\) 0 0
\(197\) −1213.46 −0.438862 −0.219431 0.975628i \(-0.570420\pi\)
−0.219431 + 0.975628i \(0.570420\pi\)
\(198\) 0 0
\(199\) −4368.96 −1.55632 −0.778160 0.628066i \(-0.783847\pi\)
−0.778160 + 0.628066i \(0.783847\pi\)
\(200\) 0 0
\(201\) −1057.52 −0.371102
\(202\) 0 0
\(203\) 5292.64 1.82990
\(204\) 0 0
\(205\) 1833.10 0.624533
\(206\) 0 0
\(207\) −974.357 −0.327162
\(208\) 0 0
\(209\) 160.624 0.0531609
\(210\) 0 0
\(211\) 3107.39 1.01385 0.506923 0.861991i \(-0.330783\pi\)
0.506923 + 0.861991i \(0.330783\pi\)
\(212\) 0 0
\(213\) −2528.83 −0.813486
\(214\) 0 0
\(215\) 1506.19 0.477772
\(216\) 0 0
\(217\) 4505.08 1.40933
\(218\) 0 0
\(219\) 151.282 0.0466789
\(220\) 0 0
\(221\) 1133.05 0.344874
\(222\) 0 0
\(223\) −5594.97 −1.68012 −0.840060 0.542493i \(-0.817480\pi\)
−0.840060 + 0.542493i \(0.817480\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −2353.86 −0.688244 −0.344122 0.938925i \(-0.611823\pi\)
−0.344122 + 0.938925i \(0.611823\pi\)
\(228\) 0 0
\(229\) −1065.47 −0.307459 −0.153729 0.988113i \(-0.549128\pi\)
−0.153729 + 0.988113i \(0.549128\pi\)
\(230\) 0 0
\(231\) −1449.04 −0.412726
\(232\) 0 0
\(233\) −5318.61 −1.49542 −0.747712 0.664023i \(-0.768848\pi\)
−0.747712 + 0.664023i \(0.768848\pi\)
\(234\) 0 0
\(235\) −784.309 −0.217713
\(236\) 0 0
\(237\) −2086.03 −0.571740
\(238\) 0 0
\(239\) 623.495 0.168747 0.0843735 0.996434i \(-0.473111\pi\)
0.0843735 + 0.996434i \(0.473111\pi\)
\(240\) 0 0
\(241\) −955.412 −0.255367 −0.127684 0.991815i \(-0.540754\pi\)
−0.127684 + 0.991815i \(0.540754\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 4460.37 1.16311
\(246\) 0 0
\(247\) 151.930 0.0391379
\(248\) 0 0
\(249\) −2027.54 −0.516023
\(250\) 0 0
\(251\) 3310.75 0.832560 0.416280 0.909236i \(-0.363334\pi\)
0.416280 + 0.909236i \(0.363334\pi\)
\(252\) 0 0
\(253\) −1487.95 −0.369749
\(254\) 0 0
\(255\) 1307.37 0.321061
\(256\) 0 0
\(257\) 125.710 0.0305119 0.0152560 0.999884i \(-0.495144\pi\)
0.0152560 + 0.999884i \(0.495144\pi\)
\(258\) 0 0
\(259\) 413.838 0.0992842
\(260\) 0 0
\(261\) −1355.40 −0.321446
\(262\) 0 0
\(263\) 2980.02 0.698691 0.349346 0.936994i \(-0.386404\pi\)
0.349346 + 0.936994i \(0.386404\pi\)
\(264\) 0 0
\(265\) −1145.29 −0.265489
\(266\) 0 0
\(267\) −2830.17 −0.648702
\(268\) 0 0
\(269\) 1769.95 0.401174 0.200587 0.979676i \(-0.435715\pi\)
0.200587 + 0.979676i \(0.435715\pi\)
\(270\) 0 0
\(271\) −2492.03 −0.558597 −0.279299 0.960204i \(-0.590102\pi\)
−0.279299 + 0.960204i \(0.590102\pi\)
\(272\) 0 0
\(273\) −1370.60 −0.303856
\(274\) 0 0
\(275\) 343.599 0.0753448
\(276\) 0 0
\(277\) 8647.77 1.87579 0.937896 0.346918i \(-0.112772\pi\)
0.937896 + 0.346918i \(0.112772\pi\)
\(278\) 0 0
\(279\) −1153.72 −0.247567
\(280\) 0 0
\(281\) −4778.49 −1.01445 −0.507226 0.861813i \(-0.669329\pi\)
−0.507226 + 0.861813i \(0.669329\pi\)
\(282\) 0 0
\(283\) 5654.65 1.18775 0.593877 0.804556i \(-0.297597\pi\)
0.593877 + 0.804556i \(0.297597\pi\)
\(284\) 0 0
\(285\) 175.304 0.0364354
\(286\) 0 0
\(287\) −12884.3 −2.64996
\(288\) 0 0
\(289\) 2683.47 0.546198
\(290\) 0 0
\(291\) 221.475 0.0446154
\(292\) 0 0
\(293\) 1062.17 0.211783 0.105892 0.994378i \(-0.466230\pi\)
0.105892 + 0.994378i \(0.466230\pi\)
\(294\) 0 0
\(295\) −1888.62 −0.372744
\(296\) 0 0
\(297\) 371.087 0.0725005
\(298\) 0 0
\(299\) −1407.40 −0.272215
\(300\) 0 0
\(301\) −10586.6 −2.02724
\(302\) 0 0
\(303\) −2.66857 −0.000505958 0
\(304\) 0 0
\(305\) −2031.77 −0.381440
\(306\) 0 0
\(307\) −8643.05 −1.60679 −0.803396 0.595445i \(-0.796976\pi\)
−0.803396 + 0.595445i \(0.796976\pi\)
\(308\) 0 0
\(309\) −1375.22 −0.253184
\(310\) 0 0
\(311\) 3893.30 0.709868 0.354934 0.934891i \(-0.384503\pi\)
0.354934 + 0.934891i \(0.384503\pi\)
\(312\) 0 0
\(313\) −3637.44 −0.656871 −0.328435 0.944526i \(-0.606521\pi\)
−0.328435 + 0.944526i \(0.606521\pi\)
\(314\) 0 0
\(315\) −1581.46 −0.282874
\(316\) 0 0
\(317\) 6003.11 1.06362 0.531811 0.846863i \(-0.321512\pi\)
0.531811 + 0.846863i \(0.321512\pi\)
\(318\) 0 0
\(319\) −2069.85 −0.363289
\(320\) 0 0
\(321\) 2902.19 0.504624
\(322\) 0 0
\(323\) 1018.60 0.175470
\(324\) 0 0
\(325\) 325.000 0.0554700
\(326\) 0 0
\(327\) 1878.20 0.317630
\(328\) 0 0
\(329\) 5512.69 0.923782
\(330\) 0 0
\(331\) −6466.23 −1.07377 −0.536883 0.843657i \(-0.680398\pi\)
−0.536883 + 0.843657i \(0.680398\pi\)
\(332\) 0 0
\(333\) −105.980 −0.0174405
\(334\) 0 0
\(335\) −1762.53 −0.287455
\(336\) 0 0
\(337\) −2923.30 −0.472529 −0.236265 0.971689i \(-0.575923\pi\)
−0.236265 + 0.971689i \(0.575923\pi\)
\(338\) 0 0
\(339\) −6723.60 −1.07722
\(340\) 0 0
\(341\) −1761.85 −0.279793
\(342\) 0 0
\(343\) −19296.5 −3.03764
\(344\) 0 0
\(345\) −1623.93 −0.253418
\(346\) 0 0
\(347\) 6051.65 0.936223 0.468112 0.883669i \(-0.344934\pi\)
0.468112 + 0.883669i \(0.344934\pi\)
\(348\) 0 0
\(349\) −4539.56 −0.696267 −0.348134 0.937445i \(-0.613184\pi\)
−0.348134 + 0.937445i \(0.613184\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) 5484.84 0.826993 0.413497 0.910506i \(-0.364307\pi\)
0.413497 + 0.910506i \(0.364307\pi\)
\(354\) 0 0
\(355\) −4214.72 −0.630124
\(356\) 0 0
\(357\) −9189.12 −1.36230
\(358\) 0 0
\(359\) −8106.78 −1.19181 −0.595905 0.803055i \(-0.703206\pi\)
−0.595905 + 0.803055i \(0.703206\pi\)
\(360\) 0 0
\(361\) −6722.42 −0.980087
\(362\) 0 0
\(363\) −3426.31 −0.495412
\(364\) 0 0
\(365\) 252.136 0.0361573
\(366\) 0 0
\(367\) −6563.94 −0.933610 −0.466805 0.884360i \(-0.654595\pi\)
−0.466805 + 0.884360i \(0.654595\pi\)
\(368\) 0 0
\(369\) 3299.58 0.465499
\(370\) 0 0
\(371\) 8049.92 1.12650
\(372\) 0 0
\(373\) 14069.6 1.95307 0.976535 0.215358i \(-0.0690917\pi\)
0.976535 + 0.215358i \(0.0690917\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) −1957.80 −0.267459
\(378\) 0 0
\(379\) −5008.62 −0.678828 −0.339414 0.940637i \(-0.610229\pi\)
−0.339414 + 0.940637i \(0.610229\pi\)
\(380\) 0 0
\(381\) 2687.82 0.361420
\(382\) 0 0
\(383\) −6718.96 −0.896405 −0.448202 0.893932i \(-0.647936\pi\)
−0.448202 + 0.893932i \(0.647936\pi\)
\(384\) 0 0
\(385\) −2415.06 −0.319696
\(386\) 0 0
\(387\) 2711.13 0.356110
\(388\) 0 0
\(389\) 12362.2 1.61128 0.805640 0.592406i \(-0.201822\pi\)
0.805640 + 0.592406i \(0.201822\pi\)
\(390\) 0 0
\(391\) −9435.86 −1.22044
\(392\) 0 0
\(393\) −1910.35 −0.245202
\(394\) 0 0
\(395\) −3476.72 −0.442868
\(396\) 0 0
\(397\) 2905.75 0.367344 0.183672 0.982988i \(-0.441202\pi\)
0.183672 + 0.982988i \(0.441202\pi\)
\(398\) 0 0
\(399\) −1232.16 −0.154599
\(400\) 0 0
\(401\) 10275.9 1.27969 0.639844 0.768505i \(-0.278999\pi\)
0.639844 + 0.768505i \(0.278999\pi\)
\(402\) 0 0
\(403\) −1666.48 −0.205988
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) −161.844 −0.0197108
\(408\) 0 0
\(409\) 2615.14 0.316162 0.158081 0.987426i \(-0.449469\pi\)
0.158081 + 0.987426i \(0.449469\pi\)
\(410\) 0 0
\(411\) 1517.67 0.182144
\(412\) 0 0
\(413\) 13274.6 1.58160
\(414\) 0 0
\(415\) −3379.23 −0.399710
\(416\) 0 0
\(417\) −4564.69 −0.536052
\(418\) 0 0
\(419\) 784.644 0.0914854 0.0457427 0.998953i \(-0.485435\pi\)
0.0457427 + 0.998953i \(0.485435\pi\)
\(420\) 0 0
\(421\) 6500.38 0.752515 0.376258 0.926515i \(-0.377211\pi\)
0.376258 + 0.926515i \(0.377211\pi\)
\(422\) 0 0
\(423\) −1411.76 −0.162274
\(424\) 0 0
\(425\) 2178.94 0.248692
\(426\) 0 0
\(427\) 14280.8 1.61849
\(428\) 0 0
\(429\) 536.015 0.0603241
\(430\) 0 0
\(431\) 12519.0 1.39912 0.699558 0.714575i \(-0.253380\pi\)
0.699558 + 0.714575i \(0.253380\pi\)
\(432\) 0 0
\(433\) 13600.5 1.50947 0.754734 0.656030i \(-0.227766\pi\)
0.754734 + 0.656030i \(0.227766\pi\)
\(434\) 0 0
\(435\) −2259.01 −0.248991
\(436\) 0 0
\(437\) −1265.25 −0.138501
\(438\) 0 0
\(439\) 4044.70 0.439733 0.219867 0.975530i \(-0.429438\pi\)
0.219867 + 0.975530i \(0.429438\pi\)
\(440\) 0 0
\(441\) 8028.67 0.866933
\(442\) 0 0
\(443\) −12860.3 −1.37926 −0.689629 0.724162i \(-0.742227\pi\)
−0.689629 + 0.724162i \(0.742227\pi\)
\(444\) 0 0
\(445\) −4716.95 −0.502482
\(446\) 0 0
\(447\) −9317.76 −0.985940
\(448\) 0 0
\(449\) 16819.8 1.76788 0.883939 0.467602i \(-0.154882\pi\)
0.883939 + 0.467602i \(0.154882\pi\)
\(450\) 0 0
\(451\) 5038.81 0.526094
\(452\) 0 0
\(453\) −4903.33 −0.508562
\(454\) 0 0
\(455\) −2284.34 −0.235365
\(456\) 0 0
\(457\) −11489.0 −1.17600 −0.588000 0.808861i \(-0.700084\pi\)
−0.588000 + 0.808861i \(0.700084\pi\)
\(458\) 0 0
\(459\) 2353.26 0.239304
\(460\) 0 0
\(461\) 9109.43 0.920322 0.460161 0.887836i \(-0.347792\pi\)
0.460161 + 0.887836i \(0.347792\pi\)
\(462\) 0 0
\(463\) 12617.5 1.26649 0.633244 0.773952i \(-0.281723\pi\)
0.633244 + 0.773952i \(0.281723\pi\)
\(464\) 0 0
\(465\) −1922.86 −0.191764
\(466\) 0 0
\(467\) −13067.9 −1.29488 −0.647440 0.762116i \(-0.724160\pi\)
−0.647440 + 0.762116i \(0.724160\pi\)
\(468\) 0 0
\(469\) 12388.3 1.21970
\(470\) 0 0
\(471\) 9398.47 0.919445
\(472\) 0 0
\(473\) 4140.19 0.402466
\(474\) 0 0
\(475\) 292.173 0.0282227
\(476\) 0 0
\(477\) −2061.52 −0.197884
\(478\) 0 0
\(479\) −10980.0 −1.04737 −0.523684 0.851913i \(-0.675443\pi\)
−0.523684 + 0.851913i \(0.675443\pi\)
\(480\) 0 0
\(481\) −153.083 −0.0145114
\(482\) 0 0
\(483\) 11414.1 1.07528
\(484\) 0 0
\(485\) 369.125 0.0345589
\(486\) 0 0
\(487\) −12455.9 −1.15899 −0.579497 0.814975i \(-0.696751\pi\)
−0.579497 + 0.814975i \(0.696751\pi\)
\(488\) 0 0
\(489\) 4257.80 0.393751
\(490\) 0 0
\(491\) 2632.88 0.241996 0.120998 0.992653i \(-0.461391\pi\)
0.120998 + 0.992653i \(0.461391\pi\)
\(492\) 0 0
\(493\) −13126.0 −1.19912
\(494\) 0 0
\(495\) 618.478 0.0561587
\(496\) 0 0
\(497\) 29624.1 2.67369
\(498\) 0 0
\(499\) 4716.61 0.423136 0.211568 0.977363i \(-0.432143\pi\)
0.211568 + 0.977363i \(0.432143\pi\)
\(500\) 0 0
\(501\) 11615.6 1.03583
\(502\) 0 0
\(503\) 3448.42 0.305681 0.152841 0.988251i \(-0.451158\pi\)
0.152841 + 0.988251i \(0.451158\pi\)
\(504\) 0 0
\(505\) −4.44761 −0.000391913 0
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −16933.1 −1.47455 −0.737274 0.675593i \(-0.763888\pi\)
−0.737274 + 0.675593i \(0.763888\pi\)
\(510\) 0 0
\(511\) −1772.20 −0.153419
\(512\) 0 0
\(513\) 315.546 0.0271573
\(514\) 0 0
\(515\) −2292.04 −0.196115
\(516\) 0 0
\(517\) −2155.90 −0.183397
\(518\) 0 0
\(519\) −4859.76 −0.411021
\(520\) 0 0
\(521\) −5108.30 −0.429556 −0.214778 0.976663i \(-0.568903\pi\)
−0.214778 + 0.976663i \(0.568903\pi\)
\(522\) 0 0
\(523\) −6314.75 −0.527963 −0.263981 0.964528i \(-0.585036\pi\)
−0.263981 + 0.964528i \(0.585036\pi\)
\(524\) 0 0
\(525\) −2635.77 −0.219113
\(526\) 0 0
\(527\) −11172.8 −0.923520
\(528\) 0 0
\(529\) −446.375 −0.0366874
\(530\) 0 0
\(531\) −3399.51 −0.277827
\(532\) 0 0
\(533\) 4766.06 0.387319
\(534\) 0 0
\(535\) 4836.98 0.390880
\(536\) 0 0
\(537\) −6419.20 −0.515845
\(538\) 0 0
\(539\) 12260.6 0.979783
\(540\) 0 0
\(541\) 6811.49 0.541310 0.270655 0.962676i \(-0.412760\pi\)
0.270655 + 0.962676i \(0.412760\pi\)
\(542\) 0 0
\(543\) −1393.69 −0.110145
\(544\) 0 0
\(545\) 3130.34 0.246035
\(546\) 0 0
\(547\) 5424.93 0.424046 0.212023 0.977265i \(-0.431995\pi\)
0.212023 + 0.977265i \(0.431995\pi\)
\(548\) 0 0
\(549\) −3657.19 −0.284308
\(550\) 0 0
\(551\) −1760.05 −0.136081
\(552\) 0 0
\(553\) 24436.9 1.87914
\(554\) 0 0
\(555\) −176.634 −0.0135094
\(556\) 0 0
\(557\) −14188.1 −1.07930 −0.539649 0.841890i \(-0.681443\pi\)
−0.539649 + 0.841890i \(0.681443\pi\)
\(558\) 0 0
\(559\) 3916.08 0.296302
\(560\) 0 0
\(561\) 3593.68 0.270455
\(562\) 0 0
\(563\) 3216.80 0.240803 0.120401 0.992725i \(-0.461582\pi\)
0.120401 + 0.992725i \(0.461582\pi\)
\(564\) 0 0
\(565\) −11206.0 −0.834407
\(566\) 0 0
\(567\) −2846.63 −0.210842
\(568\) 0 0
\(569\) −5551.73 −0.409034 −0.204517 0.978863i \(-0.565562\pi\)
−0.204517 + 0.978863i \(0.565562\pi\)
\(570\) 0 0
\(571\) 381.117 0.0279322 0.0139661 0.999902i \(-0.495554\pi\)
0.0139661 + 0.999902i \(0.495554\pi\)
\(572\) 0 0
\(573\) 8094.54 0.590147
\(574\) 0 0
\(575\) −2706.55 −0.196297
\(576\) 0 0
\(577\) 3178.27 0.229312 0.114656 0.993405i \(-0.463423\pi\)
0.114656 + 0.993405i \(0.463423\pi\)
\(578\) 0 0
\(579\) −11689.2 −0.839012
\(580\) 0 0
\(581\) 23751.6 1.69601
\(582\) 0 0
\(583\) −3148.16 −0.223643
\(584\) 0 0
\(585\) 585.000 0.0413449
\(586\) 0 0
\(587\) 438.511 0.0308336 0.0154168 0.999881i \(-0.495092\pi\)
0.0154168 + 0.999881i \(0.495092\pi\)
\(588\) 0 0
\(589\) −1498.15 −0.104805
\(590\) 0 0
\(591\) −3640.39 −0.253377
\(592\) 0 0
\(593\) −7029.95 −0.486822 −0.243411 0.969923i \(-0.578266\pi\)
−0.243411 + 0.969923i \(0.578266\pi\)
\(594\) 0 0
\(595\) −15315.2 −1.05523
\(596\) 0 0
\(597\) −13106.9 −0.898542
\(598\) 0 0
\(599\) −14506.4 −0.989507 −0.494754 0.869033i \(-0.664742\pi\)
−0.494754 + 0.869033i \(0.664742\pi\)
\(600\) 0 0
\(601\) −17430.6 −1.18304 −0.591520 0.806290i \(-0.701472\pi\)
−0.591520 + 0.806290i \(0.701472\pi\)
\(602\) 0 0
\(603\) −3172.56 −0.214256
\(604\) 0 0
\(605\) −5710.52 −0.383745
\(606\) 0 0
\(607\) 13330.5 0.891383 0.445691 0.895187i \(-0.352958\pi\)
0.445691 + 0.895187i \(0.352958\pi\)
\(608\) 0 0
\(609\) 15877.9 1.05650
\(610\) 0 0
\(611\) −2039.20 −0.135020
\(612\) 0 0
\(613\) −14016.8 −0.923542 −0.461771 0.886999i \(-0.652786\pi\)
−0.461771 + 0.886999i \(0.652786\pi\)
\(614\) 0 0
\(615\) 5499.30 0.360574
\(616\) 0 0
\(617\) 14539.6 0.948688 0.474344 0.880339i \(-0.342685\pi\)
0.474344 + 0.880339i \(0.342685\pi\)
\(618\) 0 0
\(619\) −21204.4 −1.37686 −0.688429 0.725304i \(-0.741699\pi\)
−0.688429 + 0.725304i \(0.741699\pi\)
\(620\) 0 0
\(621\) −2923.07 −0.188887
\(622\) 0 0
\(623\) 33154.1 2.13209
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 481.873 0.0306924
\(628\) 0 0
\(629\) −1026.34 −0.0650599
\(630\) 0 0
\(631\) 27682.4 1.74646 0.873231 0.487307i \(-0.162021\pi\)
0.873231 + 0.487307i \(0.162021\pi\)
\(632\) 0 0
\(633\) 9322.17 0.585345
\(634\) 0 0
\(635\) 4479.69 0.279955
\(636\) 0 0
\(637\) 11597.0 0.721332
\(638\) 0 0
\(639\) −7586.49 −0.469667
\(640\) 0 0
\(641\) −25577.7 −1.57606 −0.788032 0.615634i \(-0.788900\pi\)
−0.788032 + 0.615634i \(0.788900\pi\)
\(642\) 0 0
\(643\) 13204.5 0.809852 0.404926 0.914349i \(-0.367297\pi\)
0.404926 + 0.914349i \(0.367297\pi\)
\(644\) 0 0
\(645\) 4518.56 0.275842
\(646\) 0 0
\(647\) 969.924 0.0589361 0.0294680 0.999566i \(-0.490619\pi\)
0.0294680 + 0.999566i \(0.490619\pi\)
\(648\) 0 0
\(649\) −5191.42 −0.313992
\(650\) 0 0
\(651\) 13515.3 0.813678
\(652\) 0 0
\(653\) −9040.07 −0.541754 −0.270877 0.962614i \(-0.587314\pi\)
−0.270877 + 0.962614i \(0.587314\pi\)
\(654\) 0 0
\(655\) −3183.92 −0.189933
\(656\) 0 0
\(657\) 453.845 0.0269501
\(658\) 0 0
\(659\) 27518.0 1.62663 0.813316 0.581822i \(-0.197660\pi\)
0.813316 + 0.581822i \(0.197660\pi\)
\(660\) 0 0
\(661\) 22610.3 1.33047 0.665235 0.746634i \(-0.268331\pi\)
0.665235 + 0.746634i \(0.268331\pi\)
\(662\) 0 0
\(663\) 3399.15 0.199113
\(664\) 0 0
\(665\) −2053.60 −0.119752
\(666\) 0 0
\(667\) 16304.3 0.946483
\(668\) 0 0
\(669\) −16784.9 −0.970018
\(670\) 0 0
\(671\) −5584.93 −0.321317
\(672\) 0 0
\(673\) −731.086 −0.0418741 −0.0209371 0.999781i \(-0.506665\pi\)
−0.0209371 + 0.999781i \(0.506665\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 9436.61 0.535714 0.267857 0.963459i \(-0.413685\pi\)
0.267857 + 0.963459i \(0.413685\pi\)
\(678\) 0 0
\(679\) −2594.48 −0.146637
\(680\) 0 0
\(681\) −7061.59 −0.397358
\(682\) 0 0
\(683\) 17455.9 0.977936 0.488968 0.872302i \(-0.337373\pi\)
0.488968 + 0.872302i \(0.337373\pi\)
\(684\) 0 0
\(685\) 2529.46 0.141088
\(686\) 0 0
\(687\) −3196.40 −0.177512
\(688\) 0 0
\(689\) −2977.75 −0.164649
\(690\) 0 0
\(691\) −28075.1 −1.54563 −0.772814 0.634633i \(-0.781151\pi\)
−0.772814 + 0.634633i \(0.781151\pi\)
\(692\) 0 0
\(693\) −4347.11 −0.238287
\(694\) 0 0
\(695\) −7607.81 −0.415224
\(696\) 0 0
\(697\) 31953.8 1.73649
\(698\) 0 0
\(699\) −15955.8 −0.863384
\(700\) 0 0
\(701\) −20427.0 −1.10059 −0.550297 0.834969i \(-0.685486\pi\)
−0.550297 + 0.834969i \(0.685486\pi\)
\(702\) 0 0
\(703\) −137.620 −0.00738329
\(704\) 0 0
\(705\) −2352.93 −0.125697
\(706\) 0 0
\(707\) 31.2610 0.00166293
\(708\) 0 0
\(709\) 10208.4 0.540737 0.270369 0.962757i \(-0.412854\pi\)
0.270369 + 0.962757i \(0.412854\pi\)
\(710\) 0 0
\(711\) −6258.10 −0.330094
\(712\) 0 0
\(713\) 13878.2 0.728950
\(714\) 0 0
\(715\) 893.358 0.0467268
\(716\) 0 0
\(717\) 1870.49 0.0974261
\(718\) 0 0
\(719\) 32325.4 1.67668 0.838341 0.545146i \(-0.183526\pi\)
0.838341 + 0.545146i \(0.183526\pi\)
\(720\) 0 0
\(721\) 16110.1 0.832139
\(722\) 0 0
\(723\) −2866.24 −0.147436
\(724\) 0 0
\(725\) −3765.01 −0.192867
\(726\) 0 0
\(727\) −278.152 −0.0141900 −0.00709498 0.999975i \(-0.502258\pi\)
−0.00709498 + 0.999975i \(0.502258\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 26255.1 1.32843
\(732\) 0 0
\(733\) 23210.0 1.16955 0.584775 0.811195i \(-0.301183\pi\)
0.584775 + 0.811195i \(0.301183\pi\)
\(734\) 0 0
\(735\) 13381.1 0.671524
\(736\) 0 0
\(737\) −4844.83 −0.242146
\(738\) 0 0
\(739\) −28322.4 −1.40982 −0.704910 0.709296i \(-0.749013\pi\)
−0.704910 + 0.709296i \(0.749013\pi\)
\(740\) 0 0
\(741\) 455.789 0.0225963
\(742\) 0 0
\(743\) −13502.7 −0.666710 −0.333355 0.942801i \(-0.608181\pi\)
−0.333355 + 0.942801i \(0.608181\pi\)
\(744\) 0 0
\(745\) −15529.6 −0.763706
\(746\) 0 0
\(747\) −6082.61 −0.297926
\(748\) 0 0
\(749\) −33997.8 −1.65855
\(750\) 0 0
\(751\) 3344.10 0.162487 0.0812436 0.996694i \(-0.474111\pi\)
0.0812436 + 0.996694i \(0.474111\pi\)
\(752\) 0 0
\(753\) 9932.25 0.480679
\(754\) 0 0
\(755\) −8172.22 −0.393931
\(756\) 0 0
\(757\) 16645.4 0.799190 0.399595 0.916692i \(-0.369151\pi\)
0.399595 + 0.916692i \(0.369151\pi\)
\(758\) 0 0
\(759\) −4463.84 −0.213475
\(760\) 0 0
\(761\) 21705.9 1.03395 0.516977 0.855999i \(-0.327057\pi\)
0.516977 + 0.855999i \(0.327057\pi\)
\(762\) 0 0
\(763\) −22002.3 −1.04395
\(764\) 0 0
\(765\) 3922.10 0.185364
\(766\) 0 0
\(767\) −4910.41 −0.231166
\(768\) 0 0
\(769\) −27050.4 −1.26848 −0.634240 0.773136i \(-0.718687\pi\)
−0.634240 + 0.773136i \(0.718687\pi\)
\(770\) 0 0
\(771\) 377.129 0.0176161
\(772\) 0 0
\(773\) 22869.0 1.06409 0.532045 0.846716i \(-0.321424\pi\)
0.532045 + 0.846716i \(0.321424\pi\)
\(774\) 0 0
\(775\) −3204.77 −0.148540
\(776\) 0 0
\(777\) 1241.51 0.0573218
\(778\) 0 0
\(779\) 4284.65 0.197065
\(780\) 0 0
\(781\) −11585.4 −0.530804
\(782\) 0 0
\(783\) −4066.21 −0.185587
\(784\) 0 0
\(785\) 15664.1 0.712199
\(786\) 0 0
\(787\) −35597.3 −1.61234 −0.806168 0.591687i \(-0.798462\pi\)
−0.806168 + 0.591687i \(0.798462\pi\)
\(788\) 0 0
\(789\) 8940.05 0.403390
\(790\) 0 0
\(791\) 78763.9 3.54048
\(792\) 0 0
\(793\) −5282.61 −0.236559
\(794\) 0 0
\(795\) −3435.87 −0.153280
\(796\) 0 0
\(797\) 34475.5 1.53223 0.766114 0.642705i \(-0.222188\pi\)
0.766114 + 0.642705i \(0.222188\pi\)
\(798\) 0 0
\(799\) −13671.7 −0.605345
\(800\) 0 0
\(801\) −8490.50 −0.374528
\(802\) 0 0
\(803\) 693.070 0.0304582
\(804\) 0 0
\(805\) 19023.6 0.832910
\(806\) 0 0
\(807\) 5309.85 0.231618
\(808\) 0 0
\(809\) 15321.5 0.665855 0.332928 0.942952i \(-0.391964\pi\)
0.332928 + 0.942952i \(0.391964\pi\)
\(810\) 0 0
\(811\) −38189.4 −1.65353 −0.826764 0.562548i \(-0.809821\pi\)
−0.826764 + 0.562548i \(0.809821\pi\)
\(812\) 0 0
\(813\) −7476.08 −0.322506
\(814\) 0 0
\(815\) 7096.33 0.304998
\(816\) 0 0
\(817\) 3520.53 0.150756
\(818\) 0 0
\(819\) −4111.80 −0.175431
\(820\) 0 0
\(821\) −2531.59 −0.107616 −0.0538082 0.998551i \(-0.517136\pi\)
−0.0538082 + 0.998551i \(0.517136\pi\)
\(822\) 0 0
\(823\) 41111.4 1.74126 0.870628 0.491942i \(-0.163713\pi\)
0.870628 + 0.491942i \(0.163713\pi\)
\(824\) 0 0
\(825\) 1030.80 0.0435003
\(826\) 0 0
\(827\) 12919.5 0.543233 0.271617 0.962406i \(-0.412442\pi\)
0.271617 + 0.962406i \(0.412442\pi\)
\(828\) 0 0
\(829\) 17503.1 0.733303 0.366651 0.930358i \(-0.380504\pi\)
0.366651 + 0.930358i \(0.380504\pi\)
\(830\) 0 0
\(831\) 25943.3 1.08299
\(832\) 0 0
\(833\) 77751.2 3.23400
\(834\) 0 0
\(835\) 19359.4 0.802347
\(836\) 0 0
\(837\) −3461.15 −0.142933
\(838\) 0 0
\(839\) −7315.80 −0.301036 −0.150518 0.988607i \(-0.548094\pi\)
−0.150518 + 0.988607i \(0.548094\pi\)
\(840\) 0 0
\(841\) −1708.53 −0.0700535
\(842\) 0 0
\(843\) −14335.5 −0.585694
\(844\) 0 0
\(845\) 845.000 0.0344010
\(846\) 0 0
\(847\) 40137.7 1.62827
\(848\) 0 0
\(849\) 16964.0 0.685750
\(850\) 0 0
\(851\) 1274.85 0.0513528
\(852\) 0 0
\(853\) 45161.1 1.81276 0.906382 0.422458i \(-0.138833\pi\)
0.906382 + 0.422458i \(0.138833\pi\)
\(854\) 0 0
\(855\) 525.911 0.0210360
\(856\) 0 0
\(857\) −11777.2 −0.469431 −0.234716 0.972064i \(-0.575416\pi\)
−0.234716 + 0.972064i \(0.575416\pi\)
\(858\) 0 0
\(859\) 25218.0 1.00166 0.500830 0.865545i \(-0.333028\pi\)
0.500830 + 0.865545i \(0.333028\pi\)
\(860\) 0 0
\(861\) −38653.0 −1.52996
\(862\) 0 0
\(863\) 26065.2 1.02812 0.514062 0.857753i \(-0.328140\pi\)
0.514062 + 0.857753i \(0.328140\pi\)
\(864\) 0 0
\(865\) −8099.61 −0.318376
\(866\) 0 0
\(867\) 8050.42 0.315348
\(868\) 0 0
\(869\) −9556.79 −0.373063
\(870\) 0 0
\(871\) −4582.58 −0.178272
\(872\) 0 0
\(873\) 664.424 0.0257587
\(874\) 0 0
\(875\) −4392.95 −0.169724
\(876\) 0 0
\(877\) −20173.2 −0.776740 −0.388370 0.921504i \(-0.626962\pi\)
−0.388370 + 0.921504i \(0.626962\pi\)
\(878\) 0 0
\(879\) 3186.50 0.122273
\(880\) 0 0
\(881\) 42933.8 1.64186 0.820929 0.571030i \(-0.193456\pi\)
0.820929 + 0.571030i \(0.193456\pi\)
\(882\) 0 0
\(883\) 22251.5 0.848043 0.424022 0.905652i \(-0.360618\pi\)
0.424022 + 0.905652i \(0.360618\pi\)
\(884\) 0 0
\(885\) −5665.85 −0.215204
\(886\) 0 0
\(887\) 36828.7 1.39412 0.697061 0.717012i \(-0.254491\pi\)
0.697061 + 0.717012i \(0.254491\pi\)
\(888\) 0 0
\(889\) −31486.5 −1.18788
\(890\) 0 0
\(891\) 1113.26 0.0418582
\(892\) 0 0
\(893\) −1833.23 −0.0686973
\(894\) 0 0
\(895\) −10698.7 −0.399572
\(896\) 0 0
\(897\) −4222.21 −0.157163
\(898\) 0 0
\(899\) 19305.6 0.716214
\(900\) 0 0
\(901\) −19964.2 −0.738183
\(902\) 0 0
\(903\) −31759.7 −1.17043
\(904\) 0 0
\(905\) −2322.82 −0.0853183
\(906\) 0 0
\(907\) 42790.0 1.56650 0.783252 0.621704i \(-0.213559\pi\)
0.783252 + 0.621704i \(0.213559\pi\)
\(908\) 0 0
\(909\) −8.00570 −0.000292115 0
\(910\) 0 0
\(911\) 3041.01 0.110596 0.0552981 0.998470i \(-0.482389\pi\)
0.0552981 + 0.998470i \(0.482389\pi\)
\(912\) 0 0
\(913\) −9288.79 −0.336708
\(914\) 0 0
\(915\) −6095.32 −0.220224
\(916\) 0 0
\(917\) 22378.9 0.805905
\(918\) 0 0
\(919\) −24088.9 −0.864656 −0.432328 0.901716i \(-0.642308\pi\)
−0.432328 + 0.901716i \(0.642308\pi\)
\(920\) 0 0
\(921\) −25929.2 −0.927682
\(922\) 0 0
\(923\) −10958.3 −0.390786
\(924\) 0 0
\(925\) −294.390 −0.0104643
\(926\) 0 0
\(927\) −4125.67 −0.146176
\(928\) 0 0
\(929\) 2324.68 0.0820992 0.0410496 0.999157i \(-0.486930\pi\)
0.0410496 + 0.999157i \(0.486930\pi\)
\(930\) 0 0
\(931\) 10425.6 0.367008
\(932\) 0 0
\(933\) 11679.9 0.409842
\(934\) 0 0
\(935\) 5989.47 0.209494
\(936\) 0 0
\(937\) 12489.1 0.435433 0.217717 0.976012i \(-0.430139\pi\)
0.217717 + 0.976012i \(0.430139\pi\)
\(938\) 0 0
\(939\) −10912.3 −0.379244
\(940\) 0 0
\(941\) −25862.3 −0.895949 −0.447974 0.894046i \(-0.647854\pi\)
−0.447974 + 0.894046i \(0.647854\pi\)
\(942\) 0 0
\(943\) −39690.9 −1.37064
\(944\) 0 0
\(945\) −4744.39 −0.163317
\(946\) 0 0
\(947\) 14137.1 0.485104 0.242552 0.970138i \(-0.422016\pi\)
0.242552 + 0.970138i \(0.422016\pi\)
\(948\) 0 0
\(949\) 655.554 0.0224238
\(950\) 0 0
\(951\) 18009.3 0.614083
\(952\) 0 0
\(953\) 7474.99 0.254081 0.127040 0.991898i \(-0.459452\pi\)
0.127040 + 0.991898i \(0.459452\pi\)
\(954\) 0 0
\(955\) 13490.9 0.457126
\(956\) 0 0
\(957\) −6209.54 −0.209745
\(958\) 0 0
\(959\) −17778.9 −0.598654
\(960\) 0 0
\(961\) −13358.2 −0.448396
\(962\) 0 0
\(963\) 8706.56 0.291345
\(964\) 0 0
\(965\) −19482.1 −0.649896
\(966\) 0 0
\(967\) 12889.8 0.428653 0.214327 0.976762i \(-0.431244\pi\)
0.214327 + 0.976762i \(0.431244\pi\)
\(968\) 0 0
\(969\) 3055.81 0.101307
\(970\) 0 0
\(971\) −11727.4 −0.387591 −0.193795 0.981042i \(-0.562080\pi\)
−0.193795 + 0.981042i \(0.562080\pi\)
\(972\) 0 0
\(973\) 53473.2 1.76184
\(974\) 0 0
\(975\) 975.000 0.0320256
\(976\) 0 0
\(977\) 22896.3 0.749763 0.374882 0.927073i \(-0.377683\pi\)
0.374882 + 0.927073i \(0.377683\pi\)
\(978\) 0 0
\(979\) −12965.9 −0.423281
\(980\) 0 0
\(981\) 5634.61 0.183384
\(982\) 0 0
\(983\) −36022.9 −1.16882 −0.584412 0.811457i \(-0.698675\pi\)
−0.584412 + 0.811457i \(0.698675\pi\)
\(984\) 0 0
\(985\) −6067.32 −0.196265
\(986\) 0 0
\(987\) 16538.1 0.533346
\(988\) 0 0
\(989\) −32612.5 −1.04855
\(990\) 0 0
\(991\) −868.766 −0.0278479 −0.0139240 0.999903i \(-0.504432\pi\)
−0.0139240 + 0.999903i \(0.504432\pi\)
\(992\) 0 0
\(993\) −19398.7 −0.619939
\(994\) 0 0
\(995\) −21844.8 −0.696007
\(996\) 0 0
\(997\) −42992.0 −1.36567 −0.682834 0.730574i \(-0.739253\pi\)
−0.682834 + 0.730574i \(0.739253\pi\)
\(998\) 0 0
\(999\) −317.941 −0.0100693
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 1560.4.a.n.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.4.a.n.1.1 4 1.1 even 1 trivial