Properties

Label 1575.4.a.x.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +O(q^{10})\) \(q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +5.19224 q^{11} -54.5464 q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619 q^{17} +87.4470 q^{19} +18.4924 q^{22} +176.477 q^{23} -194.270 q^{26} +32.7926 q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793 q^{32} +57.5616 q^{34} -17.3305 q^{37} +311.447 q^{38} -210.270 q^{41} -521.570 q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417 q^{47} +49.0000 q^{49} -255.531 q^{52} -108.978 q^{53} -82.6543 q^{56} -506.348 q^{58} -210.365 q^{59} -674.304 q^{61} -335.855 q^{62} -36.1449 q^{64} -324.929 q^{67} +75.7131 q^{68} -793.965 q^{71} -315.417 q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457 q^{77} -425.840 q^{79} -748.887 q^{82} -283.029 q^{83} -1857.60 q^{86} -61.3087 q^{88} +843.131 q^{89} -381.825 q^{91} +826.736 q^{92} -375.447 q^{94} -1537.33 q^{97} +174.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.56155 1.25920 0.629600 0.776920i \(-0.283219\pi\)
0.629600 + 0.776920i \(0.283219\pi\)
\(3\) 0 0
\(4\) 4.68466 0.585582
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −11.8078 −0.521834
\(9\) 0 0
\(10\) 0 0
\(11\) 5.19224 0.142320 0.0711599 0.997465i \(-0.477330\pi\)
0.0711599 + 0.997465i \(0.477330\pi\)
\(12\) 0 0
\(13\) −54.5464 −1.16373 −0.581863 0.813287i \(-0.697676\pi\)
−0.581863 + 0.813287i \(0.697676\pi\)
\(14\) 24.9309 0.475933
\(15\) 0 0
\(16\) −79.5312 −1.24268
\(17\) 16.1619 0.230579 0.115289 0.993332i \(-0.463220\pi\)
0.115289 + 0.993332i \(0.463220\pi\)
\(18\) 0 0
\(19\) 87.4470 1.05588 0.527940 0.849282i \(-0.322965\pi\)
0.527940 + 0.849282i \(0.322965\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 18.4924 0.179209
\(23\) 176.477 1.59992 0.799958 0.600056i \(-0.204855\pi\)
0.799958 + 0.600056i \(0.204855\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −194.270 −1.46536
\(27\) 0 0
\(28\) 32.7926 0.221329
\(29\) −142.170 −0.910358 −0.455179 0.890400i \(-0.650425\pi\)
−0.455179 + 0.890400i \(0.650425\pi\)
\(30\) 0 0
\(31\) −94.3002 −0.546349 −0.273174 0.961965i \(-0.588074\pi\)
−0.273174 + 0.961965i \(0.588074\pi\)
\(32\) −188.793 −1.04294
\(33\) 0 0
\(34\) 57.5616 0.290345
\(35\) 0 0
\(36\) 0 0
\(37\) −17.3305 −0.0770031 −0.0385016 0.999259i \(-0.512258\pi\)
−0.0385016 + 0.999259i \(0.512258\pi\)
\(38\) 311.447 1.32956
\(39\) 0 0
\(40\) 0 0
\(41\) −210.270 −0.800942 −0.400471 0.916309i \(-0.631154\pi\)
−0.400471 + 0.916309i \(0.631154\pi\)
\(42\) 0 0
\(43\) −521.570 −1.84974 −0.924868 0.380287i \(-0.875825\pi\)
−0.924868 + 0.380287i \(0.875825\pi\)
\(44\) 24.3239 0.0833400
\(45\) 0 0
\(46\) 628.533 2.01461
\(47\) −105.417 −0.327162 −0.163581 0.986530i \(-0.552304\pi\)
−0.163581 + 0.986530i \(0.552304\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −255.531 −0.681458
\(53\) −108.978 −0.282440 −0.141220 0.989978i \(-0.545102\pi\)
−0.141220 + 0.989978i \(0.545102\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −82.6543 −0.197235
\(57\) 0 0
\(58\) −506.348 −1.14632
\(59\) −210.365 −0.464189 −0.232094 0.972693i \(-0.574558\pi\)
−0.232094 + 0.972693i \(0.574558\pi\)
\(60\) 0 0
\(61\) −674.304 −1.41534 −0.707670 0.706543i \(-0.750254\pi\)
−0.707670 + 0.706543i \(0.750254\pi\)
\(62\) −335.855 −0.687962
\(63\) 0 0
\(64\) −36.1449 −0.0705955
\(65\) 0 0
\(66\) 0 0
\(67\) −324.929 −0.592484 −0.296242 0.955113i \(-0.595733\pi\)
−0.296242 + 0.955113i \(0.595733\pi\)
\(68\) 75.7131 0.135023
\(69\) 0 0
\(70\) 0 0
\(71\) −793.965 −1.32713 −0.663565 0.748118i \(-0.730958\pi\)
−0.663565 + 0.748118i \(0.730958\pi\)
\(72\) 0 0
\(73\) −315.417 −0.505709 −0.252854 0.967504i \(-0.581369\pi\)
−0.252854 + 0.967504i \(0.581369\pi\)
\(74\) −61.7235 −0.0969623
\(75\) 0 0
\(76\) 409.659 0.618304
\(77\) 36.3457 0.0537918
\(78\) 0 0
\(79\) −425.840 −0.606465 −0.303233 0.952917i \(-0.598066\pi\)
−0.303233 + 0.952917i \(0.598066\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −748.887 −1.00855
\(83\) −283.029 −0.374295 −0.187148 0.982332i \(-0.559924\pi\)
−0.187148 + 0.982332i \(0.559924\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1857.60 −2.32919
\(87\) 0 0
\(88\) −61.3087 −0.0742674
\(89\) 843.131 1.00418 0.502088 0.864817i \(-0.332565\pi\)
0.502088 + 0.864817i \(0.332565\pi\)
\(90\) 0 0
\(91\) −381.825 −0.439847
\(92\) 826.736 0.936882
\(93\) 0 0
\(94\) −375.447 −0.411962
\(95\) 0 0
\(96\) 0 0
\(97\) −1537.33 −1.60920 −0.804601 0.593816i \(-0.797621\pi\)
−0.804601 + 0.593816i \(0.797621\pi\)
\(98\) 174.516 0.179886
\(99\) 0 0
\(100\) 0 0
\(101\) 1589.99 1.56644 0.783219 0.621745i \(-0.213576\pi\)
0.783219 + 0.621745i \(0.213576\pi\)
\(102\) 0 0
\(103\) 164.793 0.157646 0.0788228 0.996889i \(-0.474884\pi\)
0.0788228 + 0.996889i \(0.474884\pi\)
\(104\) 644.071 0.607273
\(105\) 0 0
\(106\) −388.132 −0.355648
\(107\) 1184.08 1.06981 0.534904 0.844913i \(-0.320348\pi\)
0.534904 + 0.844913i \(0.320348\pi\)
\(108\) 0 0
\(109\) 333.247 0.292837 0.146419 0.989223i \(-0.453225\pi\)
0.146419 + 0.989223i \(0.453225\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −556.719 −0.469687
\(113\) 1881.49 1.56634 0.783168 0.621810i \(-0.213602\pi\)
0.783168 + 0.621810i \(0.213602\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −666.020 −0.533090
\(117\) 0 0
\(118\) −749.224 −0.584506
\(119\) 113.133 0.0871507
\(120\) 0 0
\(121\) −1304.04 −0.979745
\(122\) −2401.57 −1.78220
\(123\) 0 0
\(124\) −441.764 −0.319932
\(125\) 0 0
\(126\) 0 0
\(127\) −1638.79 −1.14503 −0.572516 0.819893i \(-0.694033\pi\)
−0.572516 + 0.819893i \(0.694033\pi\)
\(128\) 1381.61 0.954048
\(129\) 0 0
\(130\) 0 0
\(131\) 598.142 0.398931 0.199465 0.979905i \(-0.436079\pi\)
0.199465 + 0.979905i \(0.436079\pi\)
\(132\) 0 0
\(133\) 612.129 0.399085
\(134\) −1157.25 −0.746055
\(135\) 0 0
\(136\) −190.836 −0.120324
\(137\) −1005.25 −0.626894 −0.313447 0.949606i \(-0.601484\pi\)
−0.313447 + 0.949606i \(0.601484\pi\)
\(138\) 0 0
\(139\) −1875.01 −1.14414 −0.572072 0.820204i \(-0.693860\pi\)
−0.572072 + 0.820204i \(0.693860\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2827.75 −1.67112
\(143\) −283.218 −0.165621
\(144\) 0 0
\(145\) 0 0
\(146\) −1123.37 −0.636788
\(147\) 0 0
\(148\) −81.1875 −0.0450917
\(149\) 1051.80 0.578299 0.289150 0.957284i \(-0.406627\pi\)
0.289150 + 0.957284i \(0.406627\pi\)
\(150\) 0 0
\(151\) −750.383 −0.404406 −0.202203 0.979344i \(-0.564810\pi\)
−0.202203 + 0.979344i \(0.564810\pi\)
\(152\) −1032.55 −0.550994
\(153\) 0 0
\(154\) 129.447 0.0677346
\(155\) 0 0
\(156\) 0 0
\(157\) 1453.90 0.739067 0.369533 0.929217i \(-0.379518\pi\)
0.369533 + 0.929217i \(0.379518\pi\)
\(158\) −1516.65 −0.763660
\(159\) 0 0
\(160\) 0 0
\(161\) 1235.34 0.604711
\(162\) 0 0
\(163\) −1300.49 −0.624921 −0.312461 0.949931i \(-0.601153\pi\)
−0.312461 + 0.949931i \(0.601153\pi\)
\(164\) −985.043 −0.469018
\(165\) 0 0
\(166\) −1008.02 −0.471312
\(167\) −2111.46 −0.978381 −0.489191 0.872177i \(-0.662708\pi\)
−0.489191 + 0.872177i \(0.662708\pi\)
\(168\) 0 0
\(169\) 778.310 0.354260
\(170\) 0 0
\(171\) 0 0
\(172\) −2443.38 −1.08317
\(173\) 335.292 0.147351 0.0736756 0.997282i \(-0.476527\pi\)
0.0736756 + 0.997282i \(0.476527\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −412.945 −0.176857
\(177\) 0 0
\(178\) 3002.85 1.26446
\(179\) −2322.23 −0.969672 −0.484836 0.874605i \(-0.661121\pi\)
−0.484836 + 0.874605i \(0.661121\pi\)
\(180\) 0 0
\(181\) −1525.59 −0.626500 −0.313250 0.949671i \(-0.601418\pi\)
−0.313250 + 0.949671i \(0.601418\pi\)
\(182\) −1359.89 −0.553855
\(183\) 0 0
\(184\) −2083.80 −0.834891
\(185\) 0 0
\(186\) 0 0
\(187\) 83.9165 0.0328160
\(188\) −493.841 −0.191580
\(189\) 0 0
\(190\) 0 0
\(191\) −293.912 −0.111344 −0.0556721 0.998449i \(-0.517730\pi\)
−0.0556721 + 0.998449i \(0.517730\pi\)
\(192\) 0 0
\(193\) 3664.91 1.36687 0.683435 0.730012i \(-0.260485\pi\)
0.683435 + 0.730012i \(0.260485\pi\)
\(194\) −5475.29 −2.02630
\(195\) 0 0
\(196\) 229.548 0.0836546
\(197\) −5101.89 −1.84515 −0.922576 0.385816i \(-0.873920\pi\)
−0.922576 + 0.385816i \(0.873920\pi\)
\(198\) 0 0
\(199\) 5025.86 1.79032 0.895161 0.445743i \(-0.147061\pi\)
0.895161 + 0.445743i \(0.147061\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 5662.85 1.97246
\(203\) −995.193 −0.344083
\(204\) 0 0
\(205\) 0 0
\(206\) 586.918 0.198507
\(207\) 0 0
\(208\) 4338.14 1.44614
\(209\) 454.045 0.150273
\(210\) 0 0
\(211\) −3267.98 −1.06624 −0.533122 0.846039i \(-0.678981\pi\)
−0.533122 + 0.846039i \(0.678981\pi\)
\(212\) −510.526 −0.165392
\(213\) 0 0
\(214\) 4217.17 1.34710
\(215\) 0 0
\(216\) 0 0
\(217\) −660.101 −0.206500
\(218\) 1186.88 0.368741
\(219\) 0 0
\(220\) 0 0
\(221\) −881.575 −0.268331
\(222\) 0 0
\(223\) −5457.65 −1.63888 −0.819442 0.573162i \(-0.805717\pi\)
−0.819442 + 0.573162i \(0.805717\pi\)
\(224\) −1321.55 −0.394195
\(225\) 0 0
\(226\) 6701.04 1.97233
\(227\) 281.023 0.0821682 0.0410841 0.999156i \(-0.486919\pi\)
0.0410841 + 0.999156i \(0.486919\pi\)
\(228\) 0 0
\(229\) 2776.64 0.801248 0.400624 0.916243i \(-0.368793\pi\)
0.400624 + 0.916243i \(0.368793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1678.71 0.475056
\(233\) 5781.09 1.62546 0.812729 0.582642i \(-0.197981\pi\)
0.812729 + 0.582642i \(0.197981\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −985.486 −0.271821
\(237\) 0 0
\(238\) 402.931 0.109740
\(239\) −1588.17 −0.429833 −0.214916 0.976632i \(-0.568948\pi\)
−0.214916 + 0.976632i \(0.568948\pi\)
\(240\) 0 0
\(241\) −4330.01 −1.15735 −0.578673 0.815560i \(-0.696429\pi\)
−0.578673 + 0.815560i \(0.696429\pi\)
\(242\) −4644.41 −1.23369
\(243\) 0 0
\(244\) −3158.88 −0.828798
\(245\) 0 0
\(246\) 0 0
\(247\) −4769.92 −1.22876
\(248\) 1113.47 0.285104
\(249\) 0 0
\(250\) 0 0
\(251\) −1400.53 −0.352195 −0.176097 0.984373i \(-0.556347\pi\)
−0.176097 + 0.984373i \(0.556347\pi\)
\(252\) 0 0
\(253\) 916.312 0.227700
\(254\) −5836.64 −1.44182
\(255\) 0 0
\(256\) 5209.83 1.27193
\(257\) 4304.86 1.04486 0.522431 0.852681i \(-0.325025\pi\)
0.522431 + 0.852681i \(0.325025\pi\)
\(258\) 0 0
\(259\) −121.313 −0.0291045
\(260\) 0 0
\(261\) 0 0
\(262\) 2130.31 0.502333
\(263\) −1724.69 −0.404369 −0.202184 0.979347i \(-0.564804\pi\)
−0.202184 + 0.979347i \(0.564804\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2180.13 0.502527
\(267\) 0 0
\(268\) −1522.18 −0.346948
\(269\) −8004.82 −1.81436 −0.907180 0.420744i \(-0.861769\pi\)
−0.907180 + 0.420744i \(0.861769\pi\)
\(270\) 0 0
\(271\) −1963.65 −0.440160 −0.220080 0.975482i \(-0.570632\pi\)
−0.220080 + 0.975482i \(0.570632\pi\)
\(272\) −1285.38 −0.286535
\(273\) 0 0
\(274\) −3580.26 −0.789384
\(275\) 0 0
\(276\) 0 0
\(277\) 3278.33 0.711104 0.355552 0.934656i \(-0.384293\pi\)
0.355552 + 0.934656i \(0.384293\pi\)
\(278\) −6677.93 −1.44070
\(279\) 0 0
\(280\) 0 0
\(281\) −2859.04 −0.606961 −0.303480 0.952838i \(-0.598149\pi\)
−0.303480 + 0.952838i \(0.598149\pi\)
\(282\) 0 0
\(283\) 5433.66 1.14134 0.570668 0.821181i \(-0.306685\pi\)
0.570668 + 0.821181i \(0.306685\pi\)
\(284\) −3719.45 −0.777144
\(285\) 0 0
\(286\) −1008.70 −0.208550
\(287\) −1471.89 −0.302728
\(288\) 0 0
\(289\) −4651.79 −0.946833
\(290\) 0 0
\(291\) 0 0
\(292\) −1477.62 −0.296134
\(293\) 8583.43 1.71143 0.855715 0.517447i \(-0.173117\pi\)
0.855715 + 0.517447i \(0.173117\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 204.634 0.0401829
\(297\) 0 0
\(298\) 3746.03 0.728194
\(299\) −9626.20 −1.86186
\(300\) 0 0
\(301\) −3650.99 −0.699135
\(302\) −2672.53 −0.509228
\(303\) 0 0
\(304\) −6954.77 −1.31212
\(305\) 0 0
\(306\) 0 0
\(307\) 5269.83 0.979691 0.489846 0.871809i \(-0.337053\pi\)
0.489846 + 0.871809i \(0.337053\pi\)
\(308\) 170.267 0.0314995
\(309\) 0 0
\(310\) 0 0
\(311\) 4761.43 0.868154 0.434077 0.900876i \(-0.357075\pi\)
0.434077 + 0.900876i \(0.357075\pi\)
\(312\) 0 0
\(313\) 7602.95 1.37298 0.686492 0.727137i \(-0.259150\pi\)
0.686492 + 0.727137i \(0.259150\pi\)
\(314\) 5178.13 0.930632
\(315\) 0 0
\(316\) −1994.91 −0.355135
\(317\) 8064.55 1.42886 0.714432 0.699704i \(-0.246685\pi\)
0.714432 + 0.699704i \(0.246685\pi\)
\(318\) 0 0
\(319\) −738.182 −0.129562
\(320\) 0 0
\(321\) 0 0
\(322\) 4399.73 0.761452
\(323\) 1413.31 0.243464
\(324\) 0 0
\(325\) 0 0
\(326\) −4631.76 −0.786901
\(327\) 0 0
\(328\) 2482.82 0.417959
\(329\) −737.917 −0.123655
\(330\) 0 0
\(331\) 6960.79 1.15589 0.577945 0.816076i \(-0.303855\pi\)
0.577945 + 0.816076i \(0.303855\pi\)
\(332\) −1325.90 −0.219181
\(333\) 0 0
\(334\) −7520.08 −1.23198
\(335\) 0 0
\(336\) 0 0
\(337\) −4731.61 −0.764828 −0.382414 0.923991i \(-0.624907\pi\)
−0.382414 + 0.923991i \(0.624907\pi\)
\(338\) 2771.99 0.446084
\(339\) 0 0
\(340\) 0 0
\(341\) −489.629 −0.0777563
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 6158.58 0.965256
\(345\) 0 0
\(346\) 1194.16 0.185544
\(347\) 9796.67 1.51560 0.757800 0.652487i \(-0.226274\pi\)
0.757800 + 0.652487i \(0.226274\pi\)
\(348\) 0 0
\(349\) 12702.4 1.94827 0.974134 0.225971i \(-0.0725554\pi\)
0.974134 + 0.225971i \(0.0725554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −980.256 −0.148431
\(353\) −9970.21 −1.50329 −0.751644 0.659569i \(-0.770739\pi\)
−0.751644 + 0.659569i \(0.770739\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3949.78 0.588028
\(357\) 0 0
\(358\) −8270.73 −1.22101
\(359\) −4388.21 −0.645128 −0.322564 0.946548i \(-0.604545\pi\)
−0.322564 + 0.946548i \(0.604545\pi\)
\(360\) 0 0
\(361\) 787.970 0.114881
\(362\) −5433.49 −0.788889
\(363\) 0 0
\(364\) −1788.72 −0.257567
\(365\) 0 0
\(366\) 0 0
\(367\) 9441.30 1.34287 0.671433 0.741065i \(-0.265679\pi\)
0.671433 + 0.741065i \(0.265679\pi\)
\(368\) −14035.5 −1.98818
\(369\) 0 0
\(370\) 0 0
\(371\) −762.847 −0.106752
\(372\) 0 0
\(373\) 3219.40 0.446901 0.223451 0.974715i \(-0.428268\pi\)
0.223451 + 0.974715i \(0.428268\pi\)
\(374\) 298.873 0.0413218
\(375\) 0 0
\(376\) 1244.73 0.170724
\(377\) 7754.89 1.05941
\(378\) 0 0
\(379\) −14011.4 −1.89899 −0.949495 0.313783i \(-0.898403\pi\)
−0.949495 + 0.313783i \(0.898403\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1046.78 −0.140204
\(383\) 5322.87 0.710147 0.355073 0.934838i \(-0.384456\pi\)
0.355073 + 0.934838i \(0.384456\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 13052.8 1.72116
\(387\) 0 0
\(388\) −7201.88 −0.942320
\(389\) 3844.51 0.501091 0.250545 0.968105i \(-0.419390\pi\)
0.250545 + 0.968105i \(0.419390\pi\)
\(390\) 0 0
\(391\) 2852.21 0.368907
\(392\) −578.580 −0.0745478
\(393\) 0 0
\(394\) −18170.7 −2.32341
\(395\) 0 0
\(396\) 0 0
\(397\) 8046.40 1.01722 0.508611 0.860996i \(-0.330159\pi\)
0.508611 + 0.860996i \(0.330159\pi\)
\(398\) 17899.9 2.25437
\(399\) 0 0
\(400\) 0 0
\(401\) −7741.38 −0.964055 −0.482027 0.876156i \(-0.660099\pi\)
−0.482027 + 0.876156i \(0.660099\pi\)
\(402\) 0 0
\(403\) 5143.74 0.635801
\(404\) 7448.58 0.917279
\(405\) 0 0
\(406\) −3544.43 −0.433269
\(407\) −89.9840 −0.0109591
\(408\) 0 0
\(409\) −8966.94 −1.08407 −0.542037 0.840354i \(-0.682347\pi\)
−0.542037 + 0.840354i \(0.682347\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 771.997 0.0923145
\(413\) −1472.55 −0.175447
\(414\) 0 0
\(415\) 0 0
\(416\) 10298.0 1.21370
\(417\) 0 0
\(418\) 1617.11 0.189223
\(419\) 12413.6 1.44736 0.723681 0.690135i \(-0.242449\pi\)
0.723681 + 0.690135i \(0.242449\pi\)
\(420\) 0 0
\(421\) −1672.14 −0.193575 −0.0967875 0.995305i \(-0.530857\pi\)
−0.0967875 + 0.995305i \(0.530857\pi\)
\(422\) −11639.1 −1.34261
\(423\) 0 0
\(424\) 1286.79 0.147387
\(425\) 0 0
\(426\) 0 0
\(427\) −4720.13 −0.534948
\(428\) 5547.02 0.626461
\(429\) 0 0
\(430\) 0 0
\(431\) −16021.8 −1.79059 −0.895296 0.445472i \(-0.853036\pi\)
−0.895296 + 0.445472i \(0.853036\pi\)
\(432\) 0 0
\(433\) −10882.7 −1.20782 −0.603912 0.797051i \(-0.706392\pi\)
−0.603912 + 0.797051i \(0.706392\pi\)
\(434\) −2350.99 −0.260025
\(435\) 0 0
\(436\) 1561.15 0.171480
\(437\) 15432.4 1.68932
\(438\) 0 0
\(439\) 7738.40 0.841307 0.420653 0.907221i \(-0.361801\pi\)
0.420653 + 0.907221i \(0.361801\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3139.78 −0.337882
\(443\) 8766.56 0.940207 0.470103 0.882611i \(-0.344217\pi\)
0.470103 + 0.882611i \(0.344217\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −19437.7 −2.06368
\(447\) 0 0
\(448\) −253.014 −0.0266826
\(449\) −3099.58 −0.325786 −0.162893 0.986644i \(-0.552083\pi\)
−0.162893 + 0.986644i \(0.552083\pi\)
\(450\) 0 0
\(451\) −1091.77 −0.113990
\(452\) 8814.15 0.917219
\(453\) 0 0
\(454\) 1000.88 0.103466
\(455\) 0 0
\(456\) 0 0
\(457\) 6122.94 0.626737 0.313369 0.949632i \(-0.398542\pi\)
0.313369 + 0.949632i \(0.398542\pi\)
\(458\) 9889.16 1.00893
\(459\) 0 0
\(460\) 0 0
\(461\) 10412.2 1.05194 0.525970 0.850503i \(-0.323703\pi\)
0.525970 + 0.850503i \(0.323703\pi\)
\(462\) 0 0
\(463\) 11278.5 1.13209 0.566043 0.824376i \(-0.308474\pi\)
0.566043 + 0.824376i \(0.308474\pi\)
\(464\) 11307.0 1.13128
\(465\) 0 0
\(466\) 20589.6 2.04677
\(467\) 14923.2 1.47872 0.739359 0.673311i \(-0.235128\pi\)
0.739359 + 0.673311i \(0.235128\pi\)
\(468\) 0 0
\(469\) −2274.50 −0.223938
\(470\) 0 0
\(471\) 0 0
\(472\) 2483.93 0.242230
\(473\) −2708.11 −0.263254
\(474\) 0 0
\(475\) 0 0
\(476\) 529.992 0.0510339
\(477\) 0 0
\(478\) −5656.34 −0.541245
\(479\) 4674.21 0.445867 0.222933 0.974834i \(-0.428437\pi\)
0.222933 + 0.974834i \(0.428437\pi\)
\(480\) 0 0
\(481\) 945.316 0.0896106
\(482\) −15421.6 −1.45733
\(483\) 0 0
\(484\) −6108.99 −0.573721
\(485\) 0 0
\(486\) 0 0
\(487\) −17081.7 −1.58941 −0.794706 0.606994i \(-0.792375\pi\)
−0.794706 + 0.606994i \(0.792375\pi\)
\(488\) 7962.02 0.738573
\(489\) 0 0
\(490\) 0 0
\(491\) −18203.9 −1.67318 −0.836588 0.547832i \(-0.815453\pi\)
−0.836588 + 0.547832i \(0.815453\pi\)
\(492\) 0 0
\(493\) −2297.75 −0.209909
\(494\) −16988.3 −1.54725
\(495\) 0 0
\(496\) 7499.81 0.678934
\(497\) −5557.75 −0.501608
\(498\) 0 0
\(499\) 7109.47 0.637803 0.318901 0.947788i \(-0.396686\pi\)
0.318901 + 0.947788i \(0.396686\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4988.07 −0.443483
\(503\) −15402.0 −1.36529 −0.682647 0.730748i \(-0.739171\pi\)
−0.682647 + 0.730748i \(0.739171\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3263.49 0.286719
\(507\) 0 0
\(508\) −7677.17 −0.670511
\(509\) −6404.72 −0.557730 −0.278865 0.960330i \(-0.589958\pi\)
−0.278865 + 0.960330i \(0.589958\pi\)
\(510\) 0 0
\(511\) −2207.92 −0.191140
\(512\) 7502.22 0.647567
\(513\) 0 0
\(514\) 15332.0 1.31569
\(515\) 0 0
\(516\) 0 0
\(517\) −547.348 −0.0465616
\(518\) −432.064 −0.0366483
\(519\) 0 0
\(520\) 0 0
\(521\) 8916.72 0.749806 0.374903 0.927064i \(-0.377676\pi\)
0.374903 + 0.927064i \(0.377676\pi\)
\(522\) 0 0
\(523\) 6929.40 0.579353 0.289677 0.957125i \(-0.406452\pi\)
0.289677 + 0.957125i \(0.406452\pi\)
\(524\) 2802.09 0.233607
\(525\) 0 0
\(526\) −6142.58 −0.509181
\(527\) −1524.07 −0.125977
\(528\) 0 0
\(529\) 18977.2 1.55973
\(530\) 0 0
\(531\) 0 0
\(532\) 2867.61 0.233697
\(533\) 11469.5 0.932078
\(534\) 0 0
\(535\) 0 0
\(536\) 3836.69 0.309178
\(537\) 0 0
\(538\) −28509.6 −2.28464
\(539\) 254.420 0.0203314
\(540\) 0 0
\(541\) −6929.23 −0.550667 −0.275334 0.961349i \(-0.588788\pi\)
−0.275334 + 0.961349i \(0.588788\pi\)
\(542\) −6993.65 −0.554249
\(543\) 0 0
\(544\) −3051.25 −0.240480
\(545\) 0 0
\(546\) 0 0
\(547\) 8509.95 0.665190 0.332595 0.943070i \(-0.392076\pi\)
0.332595 + 0.943070i \(0.392076\pi\)
\(548\) −4709.26 −0.367098
\(549\) 0 0
\(550\) 0 0
\(551\) −12432.4 −0.961228
\(552\) 0 0
\(553\) −2980.88 −0.229222
\(554\) 11676.0 0.895422
\(555\) 0 0
\(556\) −8783.76 −0.669990
\(557\) −4043.94 −0.307625 −0.153813 0.988100i \(-0.549155\pi\)
−0.153813 + 0.988100i \(0.549155\pi\)
\(558\) 0 0
\(559\) 28449.8 2.15259
\(560\) 0 0
\(561\) 0 0
\(562\) −10182.6 −0.764285
\(563\) 15878.9 1.18866 0.594331 0.804221i \(-0.297417\pi\)
0.594331 + 0.804221i \(0.297417\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 19352.3 1.43717
\(567\) 0 0
\(568\) 9374.95 0.692543
\(569\) −11611.6 −0.855510 −0.427755 0.903895i \(-0.640695\pi\)
−0.427755 + 0.903895i \(0.640695\pi\)
\(570\) 0 0
\(571\) 17395.7 1.27493 0.637466 0.770478i \(-0.279982\pi\)
0.637466 + 0.770478i \(0.279982\pi\)
\(572\) −1326.78 −0.0969850
\(573\) 0 0
\(574\) −5242.21 −0.381195
\(575\) 0 0
\(576\) 0 0
\(577\) 11474.0 0.827848 0.413924 0.910311i \(-0.364158\pi\)
0.413924 + 0.910311i \(0.364158\pi\)
\(578\) −16567.6 −1.19225
\(579\) 0 0
\(580\) 0 0
\(581\) −1981.20 −0.141470
\(582\) 0 0
\(583\) −565.841 −0.0401968
\(584\) 3724.37 0.263896
\(585\) 0 0
\(586\) 30570.3 2.15503
\(587\) 11870.4 0.834659 0.417330 0.908755i \(-0.362966\pi\)
0.417330 + 0.908755i \(0.362966\pi\)
\(588\) 0 0
\(589\) −8246.26 −0.576878
\(590\) 0 0
\(591\) 0 0
\(592\) 1378.32 0.0956899
\(593\) −5760.65 −0.398923 −0.199462 0.979906i \(-0.563919\pi\)
−0.199462 + 0.979906i \(0.563919\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4927.31 0.338642
\(597\) 0 0
\(598\) −34284.2 −2.34446
\(599\) 21696.5 1.47996 0.739978 0.672631i \(-0.234836\pi\)
0.739978 + 0.672631i \(0.234836\pi\)
\(600\) 0 0
\(601\) −12403.0 −0.841812 −0.420906 0.907104i \(-0.638288\pi\)
−0.420906 + 0.907104i \(0.638288\pi\)
\(602\) −13003.2 −0.880350
\(603\) 0 0
\(604\) −3515.29 −0.236813
\(605\) 0 0
\(606\) 0 0
\(607\) −17066.5 −1.14120 −0.570600 0.821228i \(-0.693289\pi\)
−0.570600 + 0.821228i \(0.693289\pi\)
\(608\) −16509.3 −1.10122
\(609\) 0 0
\(610\) 0 0
\(611\) 5750.10 0.380727
\(612\) 0 0
\(613\) −2707.50 −0.178393 −0.0891965 0.996014i \(-0.528430\pi\)
−0.0891965 + 0.996014i \(0.528430\pi\)
\(614\) 18768.8 1.23363
\(615\) 0 0
\(616\) −429.161 −0.0280704
\(617\) −23226.3 −1.51549 −0.757743 0.652553i \(-0.773698\pi\)
−0.757743 + 0.652553i \(0.773698\pi\)
\(618\) 0 0
\(619\) −2298.43 −0.149243 −0.0746216 0.997212i \(-0.523775\pi\)
−0.0746216 + 0.997212i \(0.523775\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16958.1 1.09318
\(623\) 5901.91 0.379543
\(624\) 0 0
\(625\) 0 0
\(626\) 27078.3 1.72886
\(627\) 0 0
\(628\) 6811.01 0.432785
\(629\) −280.094 −0.0177553
\(630\) 0 0
\(631\) −663.913 −0.0418858 −0.0209429 0.999781i \(-0.506667\pi\)
−0.0209429 + 0.999781i \(0.506667\pi\)
\(632\) 5028.22 0.316474
\(633\) 0 0
\(634\) 28722.3 1.79923
\(635\) 0 0
\(636\) 0 0
\(637\) −2672.77 −0.166247
\(638\) −2629.08 −0.163144
\(639\) 0 0
\(640\) 0 0
\(641\) −15215.6 −0.937566 −0.468783 0.883313i \(-0.655307\pi\)
−0.468783 + 0.883313i \(0.655307\pi\)
\(642\) 0 0
\(643\) −12904.0 −0.791420 −0.395710 0.918375i \(-0.629502\pi\)
−0.395710 + 0.918375i \(0.629502\pi\)
\(644\) 5787.15 0.354108
\(645\) 0 0
\(646\) 5033.58 0.306569
\(647\) −9425.54 −0.572730 −0.286365 0.958121i \(-0.592447\pi\)
−0.286365 + 0.958121i \(0.592447\pi\)
\(648\) 0 0
\(649\) −1092.26 −0.0660632
\(650\) 0 0
\(651\) 0 0
\(652\) −6092.35 −0.365943
\(653\) −29894.7 −1.79153 −0.895765 0.444528i \(-0.853372\pi\)
−0.895765 + 0.444528i \(0.853372\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 16723.0 0.995312
\(657\) 0 0
\(658\) −2628.13 −0.155707
\(659\) 11593.6 0.685313 0.342656 0.939461i \(-0.388673\pi\)
0.342656 + 0.939461i \(0.388673\pi\)
\(660\) 0 0
\(661\) −17149.2 −1.00911 −0.504557 0.863378i \(-0.668344\pi\)
−0.504557 + 0.863378i \(0.668344\pi\)
\(662\) 24791.2 1.45550
\(663\) 0 0
\(664\) 3341.94 0.195320
\(665\) 0 0
\(666\) 0 0
\(667\) −25089.9 −1.45650
\(668\) −9891.47 −0.572923
\(669\) 0 0
\(670\) 0 0
\(671\) −3501.15 −0.201431
\(672\) 0 0
\(673\) 16475.0 0.943633 0.471817 0.881697i \(-0.343598\pi\)
0.471817 + 0.881697i \(0.343598\pi\)
\(674\) −16851.9 −0.963071
\(675\) 0 0
\(676\) 3646.11 0.207448
\(677\) −4559.89 −0.258864 −0.129432 0.991588i \(-0.541315\pi\)
−0.129432 + 0.991588i \(0.541315\pi\)
\(678\) 0 0
\(679\) −10761.3 −0.608221
\(680\) 0 0
\(681\) 0 0
\(682\) −1743.84 −0.0979106
\(683\) −27895.9 −1.56282 −0.781411 0.624017i \(-0.785500\pi\)
−0.781411 + 0.624017i \(0.785500\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1221.61 0.0679904
\(687\) 0 0
\(688\) 41481.1 2.29862
\(689\) 5944.37 0.328683
\(690\) 0 0
\(691\) −28178.5 −1.55132 −0.775659 0.631152i \(-0.782582\pi\)
−0.775659 + 0.631152i \(0.782582\pi\)
\(692\) 1570.73 0.0862862
\(693\) 0 0
\(694\) 34891.4 1.90844
\(695\) 0 0
\(696\) 0 0
\(697\) −3398.37 −0.184680
\(698\) 45240.4 2.45326
\(699\) 0 0
\(700\) 0 0
\(701\) −3912.96 −0.210828 −0.105414 0.994428i \(-0.533617\pi\)
−0.105414 + 0.994428i \(0.533617\pi\)
\(702\) 0 0
\(703\) −1515.50 −0.0813060
\(704\) −187.673 −0.0100471
\(705\) 0 0
\(706\) −35509.4 −1.89294
\(707\) 11130.0 0.592058
\(708\) 0 0
\(709\) −7782.72 −0.412251 −0.206126 0.978526i \(-0.566086\pi\)
−0.206126 + 0.978526i \(0.566086\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −9955.49 −0.524014
\(713\) −16641.8 −0.874112
\(714\) 0 0
\(715\) 0 0
\(716\) −10878.8 −0.567823
\(717\) 0 0
\(718\) −15628.9 −0.812345
\(719\) 27868.8 1.44552 0.722762 0.691097i \(-0.242872\pi\)
0.722762 + 0.691097i \(0.242872\pi\)
\(720\) 0 0
\(721\) 1153.55 0.0595844
\(722\) 2806.40 0.144658
\(723\) 0 0
\(724\) −7146.89 −0.366868
\(725\) 0 0
\(726\) 0 0
\(727\) 34202.4 1.74484 0.872419 0.488759i \(-0.162550\pi\)
0.872419 + 0.488759i \(0.162550\pi\)
\(728\) 4508.50 0.229527
\(729\) 0 0
\(730\) 0 0
\(731\) −8429.58 −0.426510
\(732\) 0 0
\(733\) 12544.2 0.632101 0.316051 0.948742i \(-0.397643\pi\)
0.316051 + 0.948742i \(0.397643\pi\)
\(734\) 33625.7 1.69094
\(735\) 0 0
\(736\) −33317.6 −1.66862
\(737\) −1687.11 −0.0843221
\(738\) 0 0
\(739\) 4563.19 0.227144 0.113572 0.993530i \(-0.463771\pi\)
0.113572 + 0.993530i \(0.463771\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2716.92 −0.134422
\(743\) 10369.7 0.512017 0.256009 0.966674i \(-0.417592\pi\)
0.256009 + 0.966674i \(0.417592\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 11466.1 0.562738
\(747\) 0 0
\(748\) 393.120 0.0192164
\(749\) 8288.57 0.404349
\(750\) 0 0
\(751\) −36808.0 −1.78847 −0.894237 0.447595i \(-0.852281\pi\)
−0.894237 + 0.447595i \(0.852281\pi\)
\(752\) 8383.92 0.406556
\(753\) 0 0
\(754\) 27619.4 1.33401
\(755\) 0 0
\(756\) 0 0
\(757\) 12516.6 0.600955 0.300477 0.953789i \(-0.402854\pi\)
0.300477 + 0.953789i \(0.402854\pi\)
\(758\) −49902.3 −2.39121
\(759\) 0 0
\(760\) 0 0
\(761\) −11745.1 −0.559473 −0.279736 0.960077i \(-0.590247\pi\)
−0.279736 + 0.960077i \(0.590247\pi\)
\(762\) 0 0
\(763\) 2332.73 0.110682
\(764\) −1376.88 −0.0652011
\(765\) 0 0
\(766\) 18957.7 0.894216
\(767\) 11474.6 0.540189
\(768\) 0 0
\(769\) 36497.1 1.71147 0.855735 0.517414i \(-0.173105\pi\)
0.855735 + 0.517414i \(0.173105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17168.8 0.800414
\(773\) −29858.1 −1.38929 −0.694644 0.719353i \(-0.744438\pi\)
−0.694644 + 0.719353i \(0.744438\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 18152.5 0.839737
\(777\) 0 0
\(778\) 13692.4 0.630973
\(779\) −18387.5 −0.845699
\(780\) 0 0
\(781\) −4122.45 −0.188877
\(782\) 10158.3 0.464527
\(783\) 0 0
\(784\) −3897.03 −0.177525
\(785\) 0 0
\(786\) 0 0
\(787\) −3168.00 −0.143491 −0.0717453 0.997423i \(-0.522857\pi\)
−0.0717453 + 0.997423i \(0.522857\pi\)
\(788\) −23900.6 −1.08049
\(789\) 0 0
\(790\) 0 0
\(791\) 13170.5 0.592019
\(792\) 0 0
\(793\) 36780.8 1.64707
\(794\) 28657.7 1.28089
\(795\) 0 0
\(796\) 23544.5 1.04838
\(797\) 29317.0 1.30296 0.651481 0.758665i \(-0.274148\pi\)
0.651481 + 0.758665i \(0.274148\pi\)
\(798\) 0 0
\(799\) −1703.74 −0.0754366
\(800\) 0 0
\(801\) 0 0
\(802\) −27571.3 −1.21394
\(803\) −1637.72 −0.0719724
\(804\) 0 0
\(805\) 0 0
\(806\) 18319.7 0.800600
\(807\) 0 0
\(808\) −18774.3 −0.817422
\(809\) −16657.3 −0.723904 −0.361952 0.932197i \(-0.617890\pi\)
−0.361952 + 0.932197i \(0.617890\pi\)
\(810\) 0 0
\(811\) 5144.55 0.222749 0.111375 0.993779i \(-0.464475\pi\)
0.111375 + 0.993779i \(0.464475\pi\)
\(812\) −4662.14 −0.201489
\(813\) 0 0
\(814\) −320.483 −0.0137997
\(815\) 0 0
\(816\) 0 0
\(817\) −45609.7 −1.95310
\(818\) −31936.2 −1.36507
\(819\) 0 0
\(820\) 0 0
\(821\) 5217.18 0.221779 0.110890 0.993833i \(-0.464630\pi\)
0.110890 + 0.993833i \(0.464630\pi\)
\(822\) 0 0
\(823\) −42326.5 −1.79272 −0.896360 0.443327i \(-0.853798\pi\)
−0.896360 + 0.443327i \(0.853798\pi\)
\(824\) −1945.83 −0.0822649
\(825\) 0 0
\(826\) −5244.57 −0.220922
\(827\) −31675.8 −1.33189 −0.665946 0.746000i \(-0.731972\pi\)
−0.665946 + 0.746000i \(0.731972\pi\)
\(828\) 0 0
\(829\) −3471.22 −0.145429 −0.0727144 0.997353i \(-0.523166\pi\)
−0.0727144 + 0.997353i \(0.523166\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1971.57 0.0821539
\(833\) 791.934 0.0329399
\(834\) 0 0
\(835\) 0 0
\(836\) 2127.05 0.0879970
\(837\) 0 0
\(838\) 44211.7 1.82252
\(839\) 20964.9 0.862682 0.431341 0.902189i \(-0.358041\pi\)
0.431341 + 0.902189i \(0.358041\pi\)
\(840\) 0 0
\(841\) −4176.57 −0.171248
\(842\) −5955.41 −0.243749
\(843\) 0 0
\(844\) −15309.4 −0.624373
\(845\) 0 0
\(846\) 0 0
\(847\) −9128.28 −0.370309
\(848\) 8667.17 0.350981
\(849\) 0 0
\(850\) 0 0
\(851\) −3058.44 −0.123199
\(852\) 0 0
\(853\) −1084.77 −0.0435426 −0.0217713 0.999763i \(-0.506931\pi\)
−0.0217713 + 0.999763i \(0.506931\pi\)
\(854\) −16811.0 −0.673607
\(855\) 0 0
\(856\) −13981.4 −0.558263
\(857\) −12661.6 −0.504679 −0.252340 0.967639i \(-0.581200\pi\)
−0.252340 + 0.967639i \(0.581200\pi\)
\(858\) 0 0
\(859\) −39678.7 −1.57604 −0.788021 0.615648i \(-0.788894\pi\)
−0.788021 + 0.615648i \(0.788894\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −57062.6 −2.25471
\(863\) −41614.0 −1.64143 −0.820717 0.571334i \(-0.806426\pi\)
−0.820717 + 0.571334i \(0.806426\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −38759.2 −1.52089
\(867\) 0 0
\(868\) −3092.35 −0.120923
\(869\) −2211.06 −0.0863120
\(870\) 0 0
\(871\) 17723.7 0.689489
\(872\) −3934.90 −0.152813
\(873\) 0 0
\(874\) 54963.3 2.12719
\(875\) 0 0
\(876\) 0 0
\(877\) 37061.0 1.42698 0.713490 0.700665i \(-0.247113\pi\)
0.713490 + 0.700665i \(0.247113\pi\)
\(878\) 27560.7 1.05937
\(879\) 0 0
\(880\) 0 0
\(881\) 25468.7 0.973962 0.486981 0.873412i \(-0.338098\pi\)
0.486981 + 0.873412i \(0.338098\pi\)
\(882\) 0 0
\(883\) −34428.3 −1.31212 −0.656062 0.754707i \(-0.727779\pi\)
−0.656062 + 0.754707i \(0.727779\pi\)
\(884\) −4129.88 −0.157130
\(885\) 0 0
\(886\) 31222.6 1.18391
\(887\) 41295.4 1.56321 0.781603 0.623777i \(-0.214403\pi\)
0.781603 + 0.623777i \(0.214403\pi\)
\(888\) 0 0
\(889\) −11471.5 −0.432782
\(890\) 0 0
\(891\) 0 0
\(892\) −25567.2 −0.959702
\(893\) −9218.37 −0.345443
\(894\) 0 0
\(895\) 0 0
\(896\) 9671.26 0.360596
\(897\) 0 0
\(898\) −11039.3 −0.410230
\(899\) 13406.7 0.497373
\(900\) 0 0
\(901\) −1761.30 −0.0651247
\(902\) −3888.40 −0.143536
\(903\) 0 0
\(904\) −22216.2 −0.817368
\(905\) 0 0
\(906\) 0 0
\(907\) 53733.8 1.96715 0.983573 0.180509i \(-0.0577746\pi\)
0.983573 + 0.180509i \(0.0577746\pi\)
\(908\) 1316.50 0.0481162
\(909\) 0 0
\(910\) 0 0
\(911\) −24296.8 −0.883634 −0.441817 0.897105i \(-0.645666\pi\)
−0.441817 + 0.897105i \(0.645666\pi\)
\(912\) 0 0
\(913\) −1469.55 −0.0532696
\(914\) 21807.2 0.789187
\(915\) 0 0
\(916\) 13007.6 0.469197
\(917\) 4186.99 0.150782
\(918\) 0 0
\(919\) 4280.46 0.153645 0.0768223 0.997045i \(-0.475523\pi\)
0.0768223 + 0.997045i \(0.475523\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 37083.6 1.32460
\(923\) 43307.9 1.54442
\(924\) 0 0
\(925\) 0 0
\(926\) 40168.9 1.42552
\(927\) 0 0
\(928\) 26840.7 0.949450
\(929\) 31884.5 1.12604 0.563022 0.826442i \(-0.309639\pi\)
0.563022 + 0.826442i \(0.309639\pi\)
\(930\) 0 0
\(931\) 4284.90 0.150840
\(932\) 27082.4 0.951839
\(933\) 0 0
\(934\) 53149.6 1.86200
\(935\) 0 0
\(936\) 0 0
\(937\) 44523.1 1.55230 0.776151 0.630548i \(-0.217170\pi\)
0.776151 + 0.630548i \(0.217170\pi\)
\(938\) −8100.76 −0.281982
\(939\) 0 0
\(940\) 0 0
\(941\) −46374.9 −1.60657 −0.803283 0.595598i \(-0.796915\pi\)
−0.803283 + 0.595598i \(0.796915\pi\)
\(942\) 0 0
\(943\) −37107.9 −1.28144
\(944\) 16730.6 0.576836
\(945\) 0 0
\(946\) −9645.09 −0.331489
\(947\) −20348.2 −0.698234 −0.349117 0.937079i \(-0.613518\pi\)
−0.349117 + 0.937079i \(0.613518\pi\)
\(948\) 0 0
\(949\) 17204.8 0.588507
\(950\) 0 0
\(951\) 0 0
\(952\) −1335.85 −0.0454782
\(953\) −45012.9 −1.53002 −0.765010 0.644018i \(-0.777266\pi\)
−0.765010 + 0.644018i \(0.777266\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −7440.02 −0.251702
\(957\) 0 0
\(958\) 16647.4 0.561435
\(959\) −7036.76 −0.236944
\(960\) 0 0
\(961\) −20898.5 −0.701503
\(962\) 3366.79 0.112838
\(963\) 0 0
\(964\) −20284.6 −0.677721
\(965\) 0 0
\(966\) 0 0
\(967\) 40305.8 1.34038 0.670190 0.742190i \(-0.266213\pi\)
0.670190 + 0.742190i \(0.266213\pi\)
\(968\) 15397.8 0.511265
\(969\) 0 0
\(970\) 0 0
\(971\) −33991.8 −1.12343 −0.561713 0.827332i \(-0.689858\pi\)
−0.561713 + 0.827332i \(0.689858\pi\)
\(972\) 0 0
\(973\) −13125.0 −0.432445
\(974\) −60837.2 −2.00139
\(975\) 0 0
\(976\) 53628.2 1.75881
\(977\) −18219.0 −0.596600 −0.298300 0.954472i \(-0.596420\pi\)
−0.298300 + 0.954472i \(0.596420\pi\)
\(978\) 0 0
\(979\) 4377.73 0.142914
\(980\) 0 0
\(981\) 0 0
\(982\) −64834.1 −2.10686
\(983\) 7676.89 0.249089 0.124545 0.992214i \(-0.460253\pi\)
0.124545 + 0.992214i \(0.460253\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8183.55 −0.264318
\(987\) 0 0
\(988\) −22345.4 −0.719537
\(989\) −92045.3 −2.95942
\(990\) 0 0
\(991\) 50585.6 1.62150 0.810748 0.585395i \(-0.199061\pi\)
0.810748 + 0.585395i \(0.199061\pi\)
\(992\) 17803.2 0.569810
\(993\) 0 0
\(994\) −19794.2 −0.631625
\(995\) 0 0
\(996\) 0 0
\(997\) 53060.5 1.68550 0.842750 0.538305i \(-0.180935\pi\)
0.842750 + 0.538305i \(0.180935\pi\)
\(998\) 25320.8 0.803121
\(999\) 0 0
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 1575.4.a.x.1.2 2
3.2 odd 2 525.4.a.j.1.1 2
5.4 even 2 1575.4.a.o.1.1 2
15.2 even 4 525.4.d.m.274.1 4
15.8 even 4 525.4.d.m.274.4 4
15.14 odd 2 525.4.a.m.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 3.2 odd 2
525.4.a.m.1.2 yes 2 15.14 odd 2
525.4.d.m.274.1 4 15.2 even 4
525.4.d.m.274.4 4 15.8 even 4
1575.4.a.o.1.1 2 5.4 even 2
1575.4.a.x.1.2 2 1.1 even 1 trivial