Properties

Label 425.4.b.f.324.4
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27793984.1
Defining polynomial: \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.4
Root \(-1.93854 - 1.93854i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.f.324.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.36122i q^{2} -3.15463i q^{3} +6.14708 q^{4} +4.29415 q^{6} -7.94049i q^{7} +19.2573i q^{8} +17.0483 q^{9} +O(q^{10})\) \(q+1.36122i q^{2} -3.15463i q^{3} +6.14708 q^{4} +4.29415 q^{6} -7.94049i q^{7} +19.2573i q^{8} +17.0483 q^{9} +27.6161 q^{11} -19.3918i q^{12} -58.1117i q^{13} +10.8088 q^{14} +22.9632 q^{16} -17.0000i q^{17} +23.2065i q^{18} -89.1688 q^{19} -25.0493 q^{21} +37.5916i q^{22} +115.269i q^{23} +60.7497 q^{24} +79.1029 q^{26} -138.956i q^{27} -48.8108i q^{28} +128.558 q^{29} +273.460 q^{31} +185.316i q^{32} -87.1187i q^{33} +23.1408 q^{34} +104.797 q^{36} -132.351i q^{37} -121.379i q^{38} -183.321 q^{39} -470.559 q^{41} -34.0977i q^{42} -352.642i q^{43} +169.758 q^{44} -156.907 q^{46} +152.598i q^{47} -72.4403i q^{48} +279.949 q^{49} -53.6287 q^{51} -357.217i q^{52} -527.614i q^{53} +189.150 q^{54} +152.912 q^{56} +281.295i q^{57} +174.995i q^{58} +292.020 q^{59} -53.8962 q^{61} +372.239i q^{62} -135.372i q^{63} -68.5514 q^{64} +118.588 q^{66} +52.9572i q^{67} -104.500i q^{68} +363.632 q^{69} +788.400 q^{71} +328.304i q^{72} -295.780i q^{73} +180.159 q^{74} -548.127 q^{76} -219.285i q^{77} -249.541i q^{78} +720.325 q^{79} +21.9487 q^{81} -640.535i q^{82} +116.051i q^{83} -153.980 q^{84} +480.024 q^{86} -405.552i q^{87} +531.812i q^{88} +813.329 q^{89} -461.435 q^{91} +708.569i q^{92} -862.664i q^{93} -207.720 q^{94} +584.605 q^{96} +794.693i q^{97} +381.072i q^{98} +470.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36122i 0.481264i 0.970616 + 0.240632i \(0.0773548\pi\)
−0.970616 + 0.240632i \(0.922645\pi\)
\(3\) − 3.15463i − 0.607109i −0.952814 0.303555i \(-0.901827\pi\)
0.952814 0.303555i \(-0.0981735\pi\)
\(4\) 6.14708 0.768385
\(5\) 0 0
\(6\) 4.29415 0.292180
\(7\) − 7.94049i − 0.428746i −0.976752 0.214373i \(-0.931229\pi\)
0.976752 0.214373i \(-0.0687708\pi\)
\(8\) 19.2573i 0.851061i
\(9\) 17.0483 0.631419
\(10\) 0 0
\(11\) 27.6161 0.756961 0.378481 0.925609i \(-0.376447\pi\)
0.378481 + 0.925609i \(0.376447\pi\)
\(12\) − 19.3918i − 0.466493i
\(13\) − 58.1117i − 1.23979i −0.784684 0.619896i \(-0.787175\pi\)
0.784684 0.619896i \(-0.212825\pi\)
\(14\) 10.8088 0.206340
\(15\) 0 0
\(16\) 22.9632 0.358799
\(17\) − 17.0000i − 0.242536i
\(18\) 23.2065i 0.303879i
\(19\) −89.1688 −1.07667 −0.538335 0.842731i \(-0.680946\pi\)
−0.538335 + 0.842731i \(0.680946\pi\)
\(20\) 0 0
\(21\) −25.0493 −0.260296
\(22\) 37.5916i 0.364298i
\(23\) 115.269i 1.04501i 0.852635 + 0.522507i \(0.175003\pi\)
−0.852635 + 0.522507i \(0.824997\pi\)
\(24\) 60.7497 0.516687
\(25\) 0 0
\(26\) 79.1029 0.596668
\(27\) − 138.956i − 0.990449i
\(28\) − 48.8108i − 0.329442i
\(29\) 128.558 0.823191 0.411596 0.911367i \(-0.364972\pi\)
0.411596 + 0.911367i \(0.364972\pi\)
\(30\) 0 0
\(31\) 273.460 1.58435 0.792174 0.610295i \(-0.208949\pi\)
0.792174 + 0.610295i \(0.208949\pi\)
\(32\) 185.316i 1.02374i
\(33\) − 87.1187i − 0.459558i
\(34\) 23.1408 0.116724
\(35\) 0 0
\(36\) 104.797 0.485172
\(37\) − 132.351i − 0.588063i −0.955796 0.294031i \(-0.905003\pi\)
0.955796 0.294031i \(-0.0949970\pi\)
\(38\) − 121.379i − 0.518163i
\(39\) −183.321 −0.752689
\(40\) 0 0
\(41\) −470.559 −1.79241 −0.896207 0.443636i \(-0.853688\pi\)
−0.896207 + 0.443636i \(0.853688\pi\)
\(42\) − 34.0977i − 0.125271i
\(43\) − 352.642i − 1.25064i −0.780370 0.625318i \(-0.784969\pi\)
0.780370 0.625318i \(-0.215031\pi\)
\(44\) 169.758 0.581637
\(45\) 0 0
\(46\) −156.907 −0.502928
\(47\) 152.598i 0.473589i 0.971560 + 0.236795i \(0.0760969\pi\)
−0.971560 + 0.236795i \(0.923903\pi\)
\(48\) − 72.4403i − 0.217830i
\(49\) 279.949 0.816177
\(50\) 0 0
\(51\) −53.6287 −0.147246
\(52\) − 357.217i − 0.952637i
\(53\) − 527.614i − 1.36742i −0.729753 0.683711i \(-0.760365\pi\)
0.729753 0.683711i \(-0.239635\pi\)
\(54\) 189.150 0.476668
\(55\) 0 0
\(56\) 152.912 0.364889
\(57\) 281.295i 0.653656i
\(58\) 174.995i 0.396173i
\(59\) 292.020 0.644368 0.322184 0.946677i \(-0.395583\pi\)
0.322184 + 0.946677i \(0.395583\pi\)
\(60\) 0 0
\(61\) −53.8962 −0.113126 −0.0565632 0.998399i \(-0.518014\pi\)
−0.0565632 + 0.998399i \(0.518014\pi\)
\(62\) 372.239i 0.762490i
\(63\) − 135.372i − 0.270718i
\(64\) −68.5514 −0.133889
\(65\) 0 0
\(66\) 118.588 0.221169
\(67\) 52.9572i 0.0965635i 0.998834 + 0.0482817i \(0.0153745\pi\)
−0.998834 + 0.0482817i \(0.984625\pi\)
\(68\) − 104.500i − 0.186361i
\(69\) 363.632 0.634437
\(70\) 0 0
\(71\) 788.400 1.31783 0.658915 0.752218i \(-0.271016\pi\)
0.658915 + 0.752218i \(0.271016\pi\)
\(72\) 328.304i 0.537375i
\(73\) − 295.780i − 0.474224i −0.971482 0.237112i \(-0.923799\pi\)
0.971482 0.237112i \(-0.0762009\pi\)
\(74\) 180.159 0.283014
\(75\) 0 0
\(76\) −548.127 −0.827296
\(77\) − 219.285i − 0.324544i
\(78\) − 249.541i − 0.362242i
\(79\) 720.325 1.02586 0.512930 0.858430i \(-0.328560\pi\)
0.512930 + 0.858430i \(0.328560\pi\)
\(80\) 0 0
\(81\) 21.9487 0.0301079
\(82\) − 640.535i − 0.862625i
\(83\) 116.051i 0.153473i 0.997051 + 0.0767363i \(0.0244500\pi\)
−0.997051 + 0.0767363i \(0.975550\pi\)
\(84\) −153.980 −0.200007
\(85\) 0 0
\(86\) 480.024 0.601887
\(87\) − 405.552i − 0.499767i
\(88\) 531.812i 0.644220i
\(89\) 813.329 0.968682 0.484341 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(90\) 0 0
\(91\) −461.435 −0.531556
\(92\) 708.569i 0.802972i
\(93\) − 862.664i − 0.961872i
\(94\) −207.720 −0.227922
\(95\) 0 0
\(96\) 584.605 0.621521
\(97\) 794.693i 0.831844i 0.909400 + 0.415922i \(0.136541\pi\)
−0.909400 + 0.415922i \(0.863459\pi\)
\(98\) 381.072i 0.392797i
\(99\) 470.808 0.477959
\(100\) 0 0
\(101\) 265.513 0.261579 0.130790 0.991410i \(-0.458249\pi\)
0.130790 + 0.991410i \(0.458249\pi\)
\(102\) − 73.0006i − 0.0708641i
\(103\) − 523.107i − 0.500420i −0.968192 0.250210i \(-0.919500\pi\)
0.968192 0.250210i \(-0.0804996\pi\)
\(104\) 1119.07 1.05514
\(105\) 0 0
\(106\) 718.199 0.658091
\(107\) − 986.039i − 0.890878i −0.895312 0.445439i \(-0.853048\pi\)
0.895312 0.445439i \(-0.146952\pi\)
\(108\) − 854.174i − 0.761046i
\(109\) −1814.39 −1.59438 −0.797188 0.603732i \(-0.793680\pi\)
−0.797188 + 0.603732i \(0.793680\pi\)
\(110\) 0 0
\(111\) −417.518 −0.357018
\(112\) − 182.339i − 0.153834i
\(113\) 707.339i 0.588857i 0.955673 + 0.294429i \(0.0951293\pi\)
−0.955673 + 0.294429i \(0.904871\pi\)
\(114\) −382.904 −0.314581
\(115\) 0 0
\(116\) 790.253 0.632527
\(117\) − 990.706i − 0.782827i
\(118\) 397.503i 0.310112i
\(119\) −134.988 −0.103986
\(120\) 0 0
\(121\) −568.350 −0.427010
\(122\) − 73.3647i − 0.0544437i
\(123\) 1484.44i 1.08819i
\(124\) 1680.98 1.21739
\(125\) 0 0
\(126\) 184.271 0.130287
\(127\) 2648.18i 1.85030i 0.379600 + 0.925151i \(0.376062\pi\)
−0.379600 + 0.925151i \(0.623938\pi\)
\(128\) 1389.22i 0.959302i
\(129\) −1112.46 −0.759273
\(130\) 0 0
\(131\) −1979.08 −1.31995 −0.659974 0.751289i \(-0.729433\pi\)
−0.659974 + 0.751289i \(0.729433\pi\)
\(132\) − 535.525i − 0.353117i
\(133\) 708.044i 0.461618i
\(134\) −72.0865 −0.0464726
\(135\) 0 0
\(136\) 327.374 0.206413
\(137\) 3141.92i 1.95936i 0.200570 + 0.979679i \(0.435721\pi\)
−0.200570 + 0.979679i \(0.564279\pi\)
\(138\) 494.984i 0.305332i
\(139\) −1468.07 −0.895830 −0.447915 0.894076i \(-0.647833\pi\)
−0.447915 + 0.894076i \(0.647833\pi\)
\(140\) 0 0
\(141\) 481.390 0.287520
\(142\) 1073.19i 0.634224i
\(143\) − 1604.82i − 0.938474i
\(144\) 391.483 0.226553
\(145\) 0 0
\(146\) 402.621 0.228227
\(147\) − 883.135i − 0.495508i
\(148\) − 813.570i − 0.451858i
\(149\) 286.027 0.157263 0.0786316 0.996904i \(-0.474945\pi\)
0.0786316 + 0.996904i \(0.474945\pi\)
\(150\) 0 0
\(151\) −669.626 −0.360883 −0.180442 0.983586i \(-0.557753\pi\)
−0.180442 + 0.983586i \(0.557753\pi\)
\(152\) − 1717.15i − 0.916311i
\(153\) − 289.821i − 0.153141i
\(154\) 298.496 0.156191
\(155\) 0 0
\(156\) −1126.89 −0.578354
\(157\) 720.809i 0.366413i 0.983074 + 0.183206i \(0.0586476\pi\)
−0.983074 + 0.183206i \(0.941352\pi\)
\(158\) 980.522i 0.493710i
\(159\) −1664.43 −0.830174
\(160\) 0 0
\(161\) 915.294 0.448045
\(162\) 29.8770i 0.0144899i
\(163\) 676.599i 0.325125i 0.986698 + 0.162562i \(0.0519759\pi\)
−0.986698 + 0.162562i \(0.948024\pi\)
\(164\) −2892.56 −1.37726
\(165\) 0 0
\(166\) −157.971 −0.0738609
\(167\) − 2835.67i − 1.31396i −0.753909 0.656979i \(-0.771834\pi\)
0.753909 0.656979i \(-0.228166\pi\)
\(168\) − 482.382i − 0.221527i
\(169\) −1179.97 −0.537083
\(170\) 0 0
\(171\) −1520.18 −0.679829
\(172\) − 2167.72i − 0.960970i
\(173\) 177.314i 0.0779243i 0.999241 + 0.0389621i \(0.0124052\pi\)
−0.999241 + 0.0389621i \(0.987595\pi\)
\(174\) 552.046 0.240520
\(175\) 0 0
\(176\) 634.153 0.271597
\(177\) − 921.214i − 0.391202i
\(178\) 1107.12i 0.466192i
\(179\) 1023.76 0.427483 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(180\) 0 0
\(181\) −3450.21 −1.41686 −0.708432 0.705779i \(-0.750597\pi\)
−0.708432 + 0.705779i \(0.750597\pi\)
\(182\) − 628.116i − 0.255819i
\(183\) 170.023i 0.0686800i
\(184\) −2219.78 −0.889370
\(185\) 0 0
\(186\) 1174.28 0.462915
\(187\) − 469.474i − 0.183590i
\(188\) 938.031i 0.363899i
\(189\) −1103.38 −0.424651
\(190\) 0 0
\(191\) −490.894 −0.185968 −0.0929839 0.995668i \(-0.529640\pi\)
−0.0929839 + 0.995668i \(0.529640\pi\)
\(192\) 216.254i 0.0812855i
\(193\) 3548.80i 1.32357i 0.749696 + 0.661783i \(0.230200\pi\)
−0.749696 + 0.661783i \(0.769800\pi\)
\(194\) −1081.75 −0.400337
\(195\) 0 0
\(196\) 1720.87 0.627138
\(197\) 1363.15i 0.492996i 0.969143 + 0.246498i \(0.0792799\pi\)
−0.969143 + 0.246498i \(0.920720\pi\)
\(198\) 640.874i 0.230025i
\(199\) −3737.46 −1.33137 −0.665683 0.746235i \(-0.731860\pi\)
−0.665683 + 0.746235i \(0.731860\pi\)
\(200\) 0 0
\(201\) 167.060 0.0586246
\(202\) 361.422i 0.125889i
\(203\) − 1020.81i − 0.352940i
\(204\) −329.660 −0.113141
\(205\) 0 0
\(206\) 712.064 0.240834
\(207\) 1965.15i 0.659841i
\(208\) − 1334.43i − 0.444836i
\(209\) −2462.50 −0.814997
\(210\) 0 0
\(211\) −5266.12 −1.71817 −0.859087 0.511829i \(-0.828968\pi\)
−0.859087 + 0.511829i \(0.828968\pi\)
\(212\) − 3243.28i − 1.05071i
\(213\) − 2487.11i − 0.800066i
\(214\) 1342.22 0.428748
\(215\) 0 0
\(216\) 2675.92 0.842932
\(217\) − 2171.40i − 0.679283i
\(218\) − 2469.78i − 0.767316i
\(219\) −933.075 −0.287906
\(220\) 0 0
\(221\) −987.899 −0.300694
\(222\) − 568.334i − 0.171820i
\(223\) − 704.546i − 0.211569i −0.994389 0.105785i \(-0.966265\pi\)
0.994389 0.105785i \(-0.0337354\pi\)
\(224\) 1471.50 0.438923
\(225\) 0 0
\(226\) −962.845 −0.283396
\(227\) − 2151.26i − 0.629006i −0.949256 0.314503i \(-0.898162\pi\)
0.949256 0.314503i \(-0.101838\pi\)
\(228\) 1729.14i 0.502259i
\(229\) 3916.94 1.13030 0.565149 0.824989i \(-0.308819\pi\)
0.565149 + 0.824989i \(0.308819\pi\)
\(230\) 0 0
\(231\) −691.764 −0.197034
\(232\) 2475.67i 0.700586i
\(233\) 5192.74i 1.46003i 0.683429 + 0.730017i \(0.260488\pi\)
−0.683429 + 0.730017i \(0.739512\pi\)
\(234\) 1348.57 0.376747
\(235\) 0 0
\(236\) 1795.07 0.495123
\(237\) − 2272.36i − 0.622809i
\(238\) − 183.749i − 0.0500448i
\(239\) −334.305 −0.0904786 −0.0452393 0.998976i \(-0.514405\pi\)
−0.0452393 + 0.998976i \(0.514405\pi\)
\(240\) 0 0
\(241\) −1918.45 −0.512773 −0.256386 0.966574i \(-0.582532\pi\)
−0.256386 + 0.966574i \(0.582532\pi\)
\(242\) − 773.651i − 0.205505i
\(243\) − 3821.06i − 1.00873i
\(244\) −331.304 −0.0869245
\(245\) 0 0
\(246\) −2020.65 −0.523708
\(247\) 5181.75i 1.33485i
\(248\) 5266.09i 1.34838i
\(249\) 366.097 0.0931746
\(250\) 0 0
\(251\) 7695.71 1.93525 0.967627 0.252385i \(-0.0812148\pi\)
0.967627 + 0.252385i \(0.0812148\pi\)
\(252\) − 832.141i − 0.208016i
\(253\) 3183.29i 0.791035i
\(254\) −3604.76 −0.890484
\(255\) 0 0
\(256\) −2439.44 −0.595567
\(257\) 5335.10i 1.29492i 0.762099 + 0.647460i \(0.224169\pi\)
−0.762099 + 0.647460i \(0.775831\pi\)
\(258\) − 1514.30i − 0.365411i
\(259\) −1050.93 −0.252130
\(260\) 0 0
\(261\) 2191.69 0.519778
\(262\) − 2693.97i − 0.635244i
\(263\) − 3934.15i − 0.922396i −0.887297 0.461198i \(-0.847420\pi\)
0.887297 0.461198i \(-0.152580\pi\)
\(264\) 1677.67 0.391112
\(265\) 0 0
\(266\) −963.804 −0.222160
\(267\) − 2565.75i − 0.588095i
\(268\) 325.532i 0.0741979i
\(269\) −3424.04 −0.776088 −0.388044 0.921641i \(-0.626849\pi\)
−0.388044 + 0.921641i \(0.626849\pi\)
\(270\) 0 0
\(271\) 549.034 0.123068 0.0615340 0.998105i \(-0.480401\pi\)
0.0615340 + 0.998105i \(0.480401\pi\)
\(272\) − 390.374i − 0.0870216i
\(273\) 1455.66i 0.322712i
\(274\) −4276.85 −0.942970
\(275\) 0 0
\(276\) 2235.27 0.487492
\(277\) 5203.65i 1.12873i 0.825527 + 0.564363i \(0.190878\pi\)
−0.825527 + 0.564363i \(0.809122\pi\)
\(278\) − 1998.37i − 0.431131i
\(279\) 4662.02 1.00039
\(280\) 0 0
\(281\) −1986.73 −0.421774 −0.210887 0.977510i \(-0.567635\pi\)
−0.210887 + 0.977510i \(0.567635\pi\)
\(282\) 655.279i 0.138373i
\(283\) − 753.696i − 0.158313i −0.996862 0.0791565i \(-0.974777\pi\)
0.996862 0.0791565i \(-0.0252227\pi\)
\(284\) 4846.36 1.01260
\(285\) 0 0
\(286\) 2184.51 0.451654
\(287\) 3736.47i 0.768490i
\(288\) 3159.33i 0.646407i
\(289\) −289.000 −0.0588235
\(290\) 0 0
\(291\) 2506.96 0.505020
\(292\) − 1818.18i − 0.364387i
\(293\) 7202.22i 1.43603i 0.696025 + 0.718017i \(0.254950\pi\)
−0.696025 + 0.718017i \(0.745050\pi\)
\(294\) 1202.14 0.238471
\(295\) 0 0
\(296\) 2548.72 0.500477
\(297\) − 3837.43i − 0.749731i
\(298\) 389.345i 0.0756852i
\(299\) 6698.50 1.29560
\(300\) 0 0
\(301\) −2800.15 −0.536205
\(302\) − 911.509i − 0.173680i
\(303\) − 837.595i − 0.158807i
\(304\) −2047.60 −0.386308
\(305\) 0 0
\(306\) 394.511 0.0737016
\(307\) 2425.71i 0.450953i 0.974249 + 0.225477i \(0.0723940\pi\)
−0.974249 + 0.225477i \(0.927606\pi\)
\(308\) − 1347.96i − 0.249375i
\(309\) −1650.21 −0.303809
\(310\) 0 0
\(311\) −9544.94 −1.74033 −0.870167 0.492757i \(-0.835989\pi\)
−0.870167 + 0.492757i \(0.835989\pi\)
\(312\) − 3530.27i − 0.640584i
\(313\) − 588.379i − 0.106253i −0.998588 0.0531264i \(-0.983081\pi\)
0.998588 0.0531264i \(-0.0169186\pi\)
\(314\) −981.180 −0.176341
\(315\) 0 0
\(316\) 4427.89 0.788255
\(317\) 7653.31i 1.35600i 0.735061 + 0.678001i \(0.237154\pi\)
−0.735061 + 0.678001i \(0.762846\pi\)
\(318\) − 2265.65i − 0.399533i
\(319\) 3550.26 0.623124
\(320\) 0 0
\(321\) −3110.59 −0.540860
\(322\) 1245.92i 0.215628i
\(323\) 1515.87i 0.261131i
\(324\) 134.920 0.0231345
\(325\) 0 0
\(326\) −921.001 −0.156471
\(327\) 5723.73i 0.967960i
\(328\) − 9061.70i − 1.52545i
\(329\) 1211.70 0.203050
\(330\) 0 0
\(331\) 752.266 0.124919 0.0624597 0.998047i \(-0.480106\pi\)
0.0624597 + 0.998047i \(0.480106\pi\)
\(332\) 713.373i 0.117926i
\(333\) − 2256.36i − 0.371314i
\(334\) 3859.98 0.632361
\(335\) 0 0
\(336\) −575.211 −0.0933939
\(337\) − 1968.57i − 0.318204i −0.987262 0.159102i \(-0.949140\pi\)
0.987262 0.159102i \(-0.0508598\pi\)
\(338\) − 1606.20i − 0.258479i
\(339\) 2231.39 0.357501
\(340\) 0 0
\(341\) 7551.89 1.19929
\(342\) − 2069.30i − 0.327178i
\(343\) − 4946.52i − 0.778678i
\(344\) 6790.93 1.06437
\(345\) 0 0
\(346\) −241.363 −0.0375022
\(347\) 3983.10i 0.616207i 0.951353 + 0.308104i \(0.0996944\pi\)
−0.951353 + 0.308104i \(0.900306\pi\)
\(348\) − 2492.96i − 0.384013i
\(349\) −1495.61 −0.229393 −0.114697 0.993401i \(-0.536590\pi\)
−0.114697 + 0.993401i \(0.536590\pi\)
\(350\) 0 0
\(351\) −8074.98 −1.22795
\(352\) 5117.72i 0.774930i
\(353\) − 6482.49i − 0.977417i −0.872447 0.488708i \(-0.837468\pi\)
0.872447 0.488708i \(-0.162532\pi\)
\(354\) 1253.98 0.188272
\(355\) 0 0
\(356\) 4999.59 0.744320
\(357\) 425.838i 0.0631309i
\(358\) 1393.56i 0.205732i
\(359\) 4943.42 0.726751 0.363376 0.931643i \(-0.381624\pi\)
0.363376 + 0.931643i \(0.381624\pi\)
\(360\) 0 0
\(361\) 1092.08 0.159218
\(362\) − 4696.50i − 0.681886i
\(363\) 1792.94i 0.259242i
\(364\) −2836.48 −0.408439
\(365\) 0 0
\(366\) −231.439 −0.0330533
\(367\) − 14.8871i − 0.00211743i −0.999999 0.00105872i \(-0.999663\pi\)
0.999999 0.00105872i \(-0.000337000\pi\)
\(368\) 2646.95i 0.374950i
\(369\) −8022.23 −1.13176
\(370\) 0 0
\(371\) −4189.51 −0.586276
\(372\) − 5302.86i − 0.739088i
\(373\) − 1923.18i − 0.266966i −0.991051 0.133483i \(-0.957384\pi\)
0.991051 0.133483i \(-0.0426162\pi\)
\(374\) 639.058 0.0883554
\(375\) 0 0
\(376\) −2938.63 −0.403053
\(377\) − 7470.70i − 1.02059i
\(378\) − 1501.94i − 0.204369i
\(379\) 9592.87 1.30014 0.650069 0.759875i \(-0.274740\pi\)
0.650069 + 0.759875i \(0.274740\pi\)
\(380\) 0 0
\(381\) 8354.04 1.12333
\(382\) − 668.215i − 0.0894996i
\(383\) 9083.77i 1.21190i 0.795502 + 0.605951i \(0.207207\pi\)
−0.795502 + 0.605951i \(0.792793\pi\)
\(384\) 4382.47 0.582401
\(385\) 0 0
\(386\) −4830.70 −0.636985
\(387\) − 6011.95i − 0.789675i
\(388\) 4885.04i 0.639176i
\(389\) 1143.78 0.149079 0.0745396 0.997218i \(-0.476251\pi\)
0.0745396 + 0.997218i \(0.476251\pi\)
\(390\) 0 0
\(391\) 1959.58 0.253453
\(392\) 5391.06i 0.694616i
\(393\) 6243.27i 0.801352i
\(394\) −1855.55 −0.237262
\(395\) 0 0
\(396\) 2894.09 0.367256
\(397\) 10604.5i 1.34061i 0.742084 + 0.670307i \(0.233838\pi\)
−0.742084 + 0.670307i \(0.766162\pi\)
\(398\) − 5087.51i − 0.640739i
\(399\) 2233.62 0.280252
\(400\) 0 0
\(401\) 13785.4 1.71674 0.858368 0.513035i \(-0.171479\pi\)
0.858368 + 0.513035i \(0.171479\pi\)
\(402\) 227.406i 0.0282139i
\(403\) − 15891.2i − 1.96426i
\(404\) 1632.13 0.200993
\(405\) 0 0
\(406\) 1389.55 0.169857
\(407\) − 3655.01i − 0.445141i
\(408\) − 1032.74i − 0.125315i
\(409\) 9505.94 1.14924 0.574619 0.818421i \(-0.305150\pi\)
0.574619 + 0.818421i \(0.305150\pi\)
\(410\) 0 0
\(411\) 9911.59 1.18954
\(412\) − 3215.58i − 0.384515i
\(413\) − 2318.78i − 0.276270i
\(414\) −2675.00 −0.317558
\(415\) 0 0
\(416\) 10769.1 1.26922
\(417\) 4631.23i 0.543866i
\(418\) − 3352.00i − 0.392229i
\(419\) −9680.86 −1.12874 −0.564369 0.825523i \(-0.690880\pi\)
−0.564369 + 0.825523i \(0.690880\pi\)
\(420\) 0 0
\(421\) −12360.3 −1.43089 −0.715444 0.698671i \(-0.753775\pi\)
−0.715444 + 0.698671i \(0.753775\pi\)
\(422\) − 7168.36i − 0.826897i
\(423\) 2601.54i 0.299033i
\(424\) 10160.4 1.16376
\(425\) 0 0
\(426\) 3385.51 0.385043
\(427\) 427.962i 0.0485025i
\(428\) − 6061.25i − 0.684537i
\(429\) −5062.61 −0.569756
\(430\) 0 0
\(431\) 2970.58 0.331990 0.165995 0.986127i \(-0.446916\pi\)
0.165995 + 0.986127i \(0.446916\pi\)
\(432\) − 3190.87i − 0.355372i
\(433\) − 6131.50i − 0.680510i −0.940333 0.340255i \(-0.889487\pi\)
0.940333 0.340255i \(-0.110513\pi\)
\(434\) 2955.76 0.326915
\(435\) 0 0
\(436\) −11153.2 −1.22509
\(437\) − 10278.4i − 1.12513i
\(438\) − 1270.12i − 0.138559i
\(439\) 2544.91 0.276679 0.138339 0.990385i \(-0.455824\pi\)
0.138339 + 0.990385i \(0.455824\pi\)
\(440\) 0 0
\(441\) 4772.65 0.515349
\(442\) − 1344.75i − 0.144713i
\(443\) − 8529.82i − 0.914817i −0.889257 0.457408i \(-0.848778\pi\)
0.889257 0.457408i \(-0.151222\pi\)
\(444\) −2566.51 −0.274327
\(445\) 0 0
\(446\) 959.043 0.101821
\(447\) − 902.308i − 0.0954759i
\(448\) 544.331i 0.0574046i
\(449\) −8855.74 −0.930798 −0.465399 0.885101i \(-0.654089\pi\)
−0.465399 + 0.885101i \(0.654089\pi\)
\(450\) 0 0
\(451\) −12995.0 −1.35679
\(452\) 4348.07i 0.452469i
\(453\) 2112.42i 0.219095i
\(454\) 2928.35 0.302718
\(455\) 0 0
\(456\) −5416.98 −0.556301
\(457\) − 7154.78i − 0.732356i −0.930545 0.366178i \(-0.880666\pi\)
0.930545 0.366178i \(-0.119334\pi\)
\(458\) 5331.82i 0.543973i
\(459\) −2362.25 −0.240219
\(460\) 0 0
\(461\) −7263.06 −0.733784 −0.366892 0.930264i \(-0.619578\pi\)
−0.366892 + 0.930264i \(0.619578\pi\)
\(462\) − 941.645i − 0.0948253i
\(463\) − 352.898i − 0.0354224i −0.999843 0.0177112i \(-0.994362\pi\)
0.999843 0.0177112i \(-0.00563794\pi\)
\(464\) 2952.09 0.295360
\(465\) 0 0
\(466\) −7068.47 −0.702662
\(467\) 1483.02i 0.146951i 0.997297 + 0.0734753i \(0.0234090\pi\)
−0.997297 + 0.0734753i \(0.976591\pi\)
\(468\) − 6089.94i − 0.601512i
\(469\) 420.506 0.0414012
\(470\) 0 0
\(471\) 2273.89 0.222452
\(472\) 5623.51i 0.548396i
\(473\) − 9738.60i − 0.946683i
\(474\) 3093.19 0.299736
\(475\) 0 0
\(476\) −829.783 −0.0799014
\(477\) − 8994.92i − 0.863415i
\(478\) − 455.063i − 0.0435441i
\(479\) 9990.10 0.952942 0.476471 0.879190i \(-0.341916\pi\)
0.476471 + 0.879190i \(0.341916\pi\)
\(480\) 0 0
\(481\) −7691.13 −0.729075
\(482\) − 2611.44i − 0.246779i
\(483\) − 2887.42i − 0.272012i
\(484\) −3493.69 −0.328108
\(485\) 0 0
\(486\) 5201.30 0.485465
\(487\) − 1129.88i − 0.105133i −0.998617 0.0525663i \(-0.983260\pi\)
0.998617 0.0525663i \(-0.0167401\pi\)
\(488\) − 1037.90i − 0.0962774i
\(489\) 2134.42 0.197386
\(490\) 0 0
\(491\) 18774.9 1.72566 0.862832 0.505491i \(-0.168689\pi\)
0.862832 + 0.505491i \(0.168689\pi\)
\(492\) 9124.97i 0.836149i
\(493\) − 2185.48i − 0.199653i
\(494\) −7053.51 −0.642414
\(495\) 0 0
\(496\) 6279.49 0.568463
\(497\) − 6260.28i − 0.565014i
\(498\) 498.339i 0.0448416i
\(499\) −17329.1 −1.55462 −0.777310 0.629118i \(-0.783416\pi\)
−0.777310 + 0.629118i \(0.783416\pi\)
\(500\) 0 0
\(501\) −8945.50 −0.797716
\(502\) 10475.6i 0.931369i
\(503\) 20837.0i 1.84707i 0.383518 + 0.923533i \(0.374712\pi\)
−0.383518 + 0.923533i \(0.625288\pi\)
\(504\) 2606.90 0.230398
\(505\) 0 0
\(506\) −4333.16 −0.380697
\(507\) 3722.37i 0.326068i
\(508\) 16278.6i 1.42174i
\(509\) −11835.0 −1.03060 −0.515301 0.857009i \(-0.672320\pi\)
−0.515301 + 0.857009i \(0.672320\pi\)
\(510\) 0 0
\(511\) −2348.63 −0.203322
\(512\) 7793.12i 0.672676i
\(513\) 12390.6i 1.06639i
\(514\) −7262.26 −0.623199
\(515\) 0 0
\(516\) −6838.35 −0.583414
\(517\) 4214.16i 0.358489i
\(518\) − 1430.55i − 0.121341i
\(519\) 559.359 0.0473086
\(520\) 0 0
\(521\) 7686.37 0.646346 0.323173 0.946340i \(-0.395250\pi\)
0.323173 + 0.946340i \(0.395250\pi\)
\(522\) 2983.37i 0.250151i
\(523\) − 11476.4i − 0.959518i −0.877400 0.479759i \(-0.840724\pi\)
0.877400 0.479759i \(-0.159276\pi\)
\(524\) −12165.6 −1.01423
\(525\) 0 0
\(526\) 5355.25 0.443916
\(527\) − 4648.81i − 0.384261i
\(528\) − 2000.52i − 0.164889i
\(529\) −1120.01 −0.0920535
\(530\) 0 0
\(531\) 4978.44 0.406866
\(532\) 4352.40i 0.354700i
\(533\) 27345.0i 2.22222i
\(534\) 3492.56 0.283029
\(535\) 0 0
\(536\) −1019.81 −0.0821814
\(537\) − 3229.59i − 0.259529i
\(538\) − 4660.88i − 0.373504i
\(539\) 7731.09 0.617814
\(540\) 0 0
\(541\) −546.481 −0.0434289 −0.0217145 0.999764i \(-0.506912\pi\)
−0.0217145 + 0.999764i \(0.506912\pi\)
\(542\) 747.357i 0.0592283i
\(543\) 10884.1i 0.860191i
\(544\) 3150.38 0.248293
\(545\) 0 0
\(546\) −1981.47 −0.155310
\(547\) 8397.33i 0.656388i 0.944610 + 0.328194i \(0.106440\pi\)
−0.944610 + 0.328194i \(0.893560\pi\)
\(548\) 19313.6i 1.50554i
\(549\) −918.839 −0.0714301
\(550\) 0 0
\(551\) −11463.3 −0.886305
\(552\) 7002.58i 0.539945i
\(553\) − 5719.73i − 0.439833i
\(554\) −7083.32 −0.543215
\(555\) 0 0
\(556\) −9024.36 −0.688342
\(557\) − 4881.65i − 0.371350i −0.982611 0.185675i \(-0.940553\pi\)
0.982611 0.185675i \(-0.0594472\pi\)
\(558\) 6346.04i 0.481451i
\(559\) −20492.6 −1.55053
\(560\) 0 0
\(561\) −1481.02 −0.111459
\(562\) − 2704.38i − 0.202985i
\(563\) − 7198.57i − 0.538870i −0.963019 0.269435i \(-0.913163\pi\)
0.963019 0.269435i \(-0.0868369\pi\)
\(564\) 2959.14 0.220926
\(565\) 0 0
\(566\) 1025.95 0.0761904
\(567\) − 174.283i − 0.0129087i
\(568\) 15182.5i 1.12155i
\(569\) 23946.9 1.76433 0.882167 0.470937i \(-0.156084\pi\)
0.882167 + 0.470937i \(0.156084\pi\)
\(570\) 0 0
\(571\) 1593.15 0.116763 0.0583813 0.998294i \(-0.481406\pi\)
0.0583813 + 0.998294i \(0.481406\pi\)
\(572\) − 9864.95i − 0.721109i
\(573\) 1548.59i 0.112903i
\(574\) −5086.16 −0.369847
\(575\) 0 0
\(576\) −1168.69 −0.0845403
\(577\) 12937.4i 0.933435i 0.884406 + 0.466717i \(0.154564\pi\)
−0.884406 + 0.466717i \(0.845436\pi\)
\(578\) − 393.393i − 0.0283097i
\(579\) 11195.2 0.803549
\(580\) 0 0
\(581\) 921.499 0.0658007
\(582\) 3412.53i 0.243048i
\(583\) − 14570.6i − 1.03508i
\(584\) 5695.92 0.403594
\(585\) 0 0
\(586\) −9803.82 −0.691112
\(587\) − 12899.2i − 0.906998i −0.891257 0.453499i \(-0.850176\pi\)
0.891257 0.453499i \(-0.149824\pi\)
\(588\) − 5428.70i − 0.380741i
\(589\) −24384.1 −1.70582
\(590\) 0 0
\(591\) 4300.23 0.299302
\(592\) − 3039.19i − 0.210997i
\(593\) − 4357.13i − 0.301730i −0.988554 0.150865i \(-0.951794\pi\)
0.988554 0.150865i \(-0.0482058\pi\)
\(594\) 5223.59 0.360819
\(595\) 0 0
\(596\) 1758.23 0.120839
\(597\) 11790.3i 0.808284i
\(598\) 9118.14i 0.623526i
\(599\) −13726.8 −0.936328 −0.468164 0.883642i \(-0.655084\pi\)
−0.468164 + 0.883642i \(0.655084\pi\)
\(600\) 0 0
\(601\) 2531.41 0.171811 0.0859056 0.996303i \(-0.472622\pi\)
0.0859056 + 0.996303i \(0.472622\pi\)
\(602\) − 3811.62i − 0.258057i
\(603\) 902.830i 0.0609720i
\(604\) −4116.24 −0.277297
\(605\) 0 0
\(606\) 1140.15 0.0764283
\(607\) 185.004i 0.0123708i 0.999981 + 0.00618540i \(0.00196889\pi\)
−0.999981 + 0.00618540i \(0.998031\pi\)
\(608\) − 16524.4i − 1.10223i
\(609\) −3220.28 −0.214273
\(610\) 0 0
\(611\) 8867.73 0.587152
\(612\) − 1781.55i − 0.117672i
\(613\) 17706.9i 1.16668i 0.812228 + 0.583339i \(0.198254\pi\)
−0.812228 + 0.583339i \(0.801746\pi\)
\(614\) −3301.93 −0.217028
\(615\) 0 0
\(616\) 4222.84 0.276207
\(617\) − 6183.89i − 0.403491i −0.979438 0.201746i \(-0.935339\pi\)
0.979438 0.201746i \(-0.0646614\pi\)
\(618\) − 2246.30i − 0.146213i
\(619\) 1247.51 0.0810046 0.0405023 0.999179i \(-0.487104\pi\)
0.0405023 + 0.999179i \(0.487104\pi\)
\(620\) 0 0
\(621\) 16017.4 1.03503
\(622\) − 12992.8i − 0.837561i
\(623\) − 6458.23i − 0.415318i
\(624\) −4209.63 −0.270064
\(625\) 0 0
\(626\) 800.914 0.0511357
\(627\) 7768.27i 0.494792i
\(628\) 4430.87i 0.281546i
\(629\) −2249.96 −0.142626
\(630\) 0 0
\(631\) 24053.3 1.51750 0.758752 0.651379i \(-0.225809\pi\)
0.758752 + 0.651379i \(0.225809\pi\)
\(632\) 13871.5i 0.873069i
\(633\) 16612.7i 1.04312i
\(634\) −10417.8 −0.652596
\(635\) 0 0
\(636\) −10231.4 −0.637893
\(637\) − 16268.3i − 1.01189i
\(638\) 4832.69i 0.299887i
\(639\) 13440.9 0.832102
\(640\) 0 0
\(641\) −21286.8 −1.31167 −0.655834 0.754905i \(-0.727683\pi\)
−0.655834 + 0.754905i \(0.727683\pi\)
\(642\) − 4234.20i − 0.260297i
\(643\) 1789.41i 0.109747i 0.998493 + 0.0548736i \(0.0174756\pi\)
−0.998493 + 0.0548736i \(0.982524\pi\)
\(644\) 5626.38 0.344271
\(645\) 0 0
\(646\) −2063.43 −0.125673
\(647\) − 4378.61i − 0.266060i −0.991112 0.133030i \(-0.957529\pi\)
0.991112 0.133030i \(-0.0424707\pi\)
\(648\) 422.672i 0.0256237i
\(649\) 8064.45 0.487762
\(650\) 0 0
\(651\) −6849.97 −0.412399
\(652\) 4159.11i 0.249821i
\(653\) 7665.15i 0.459358i 0.973266 + 0.229679i \(0.0737676\pi\)
−0.973266 + 0.229679i \(0.926232\pi\)
\(654\) −7791.26 −0.465845
\(655\) 0 0
\(656\) −10805.5 −0.643117
\(657\) − 5042.54i − 0.299434i
\(658\) 1649.39i 0.0977205i
\(659\) 4710.22 0.278428 0.139214 0.990262i \(-0.455542\pi\)
0.139214 + 0.990262i \(0.455542\pi\)
\(660\) 0 0
\(661\) −31266.6 −1.83983 −0.919916 0.392116i \(-0.871743\pi\)
−0.919916 + 0.392116i \(0.871743\pi\)
\(662\) 1024.00i 0.0601192i
\(663\) 3116.46i 0.182554i
\(664\) −2234.82 −0.130614
\(665\) 0 0
\(666\) 3071.40 0.178700
\(667\) 14818.7i 0.860246i
\(668\) − 17431.1i − 1.00962i
\(669\) −2222.58 −0.128446
\(670\) 0 0
\(671\) −1488.40 −0.0856322
\(672\) − 4642.05i − 0.266474i
\(673\) − 11723.0i − 0.671454i −0.941959 0.335727i \(-0.891018\pi\)
0.941959 0.335727i \(-0.108982\pi\)
\(674\) 2679.65 0.153140
\(675\) 0 0
\(676\) −7253.37 −0.412686
\(677\) − 289.531i − 0.0164366i −0.999966 0.00821829i \(-0.997384\pi\)
0.999966 0.00821829i \(-0.00261599\pi\)
\(678\) 3037.42i 0.172052i
\(679\) 6310.25 0.356650
\(680\) 0 0
\(681\) −6786.45 −0.381875
\(682\) 10279.8i 0.577175i
\(683\) − 1720.10i − 0.0963660i −0.998839 0.0481830i \(-0.984657\pi\)
0.998839 0.0481830i \(-0.0153431\pi\)
\(684\) −9344.64 −0.522370
\(685\) 0 0
\(686\) 6733.30 0.374750
\(687\) − 12356.5i − 0.686215i
\(688\) − 8097.77i − 0.448728i
\(689\) −30660.5 −1.69532
\(690\) 0 0
\(691\) −16777.7 −0.923665 −0.461832 0.886967i \(-0.652808\pi\)
−0.461832 + 0.886967i \(0.652808\pi\)
\(692\) 1089.96i 0.0598758i
\(693\) − 3738.44i − 0.204923i
\(694\) −5421.88 −0.296559
\(695\) 0 0
\(696\) 7809.84 0.425332
\(697\) 7999.50i 0.434724i
\(698\) − 2035.86i − 0.110399i
\(699\) 16381.2 0.886400
\(700\) 0 0
\(701\) 23981.1 1.29209 0.646043 0.763301i \(-0.276423\pi\)
0.646043 + 0.763301i \(0.276423\pi\)
\(702\) − 10991.8i − 0.590969i
\(703\) 11801.6i 0.633150i
\(704\) −1893.12 −0.101349
\(705\) 0 0
\(706\) 8824.10 0.470396
\(707\) − 2108.30i − 0.112151i
\(708\) − 5662.78i − 0.300593i
\(709\) 7709.28 0.408361 0.204181 0.978933i \(-0.434547\pi\)
0.204181 + 0.978933i \(0.434547\pi\)
\(710\) 0 0
\(711\) 12280.3 0.647747
\(712\) 15662.5i 0.824407i
\(713\) 31521.5i 1.65567i
\(714\) −579.660 −0.0303827
\(715\) 0 0
\(716\) 6293.13 0.328471
\(717\) 1054.61i 0.0549304i
\(718\) 6729.09i 0.349760i
\(719\) 11976.5 0.621209 0.310605 0.950539i \(-0.399468\pi\)
0.310605 + 0.950539i \(0.399468\pi\)
\(720\) 0 0
\(721\) −4153.72 −0.214553
\(722\) 1486.56i 0.0766260i
\(723\) 6052.00i 0.311309i
\(724\) −21208.7 −1.08870
\(725\) 0 0
\(726\) −2440.58 −0.124764
\(727\) − 18597.3i − 0.948745i −0.880324 0.474372i \(-0.842675\pi\)
0.880324 0.474372i \(-0.157325\pi\)
\(728\) − 8886.00i − 0.452386i
\(729\) −11461.4 −0.582300
\(730\) 0 0
\(731\) −5994.91 −0.303324
\(732\) 1045.14i 0.0527727i
\(733\) 23569.5i 1.18767i 0.804588 + 0.593833i \(0.202386\pi\)
−0.804588 + 0.593833i \(0.797614\pi\)
\(734\) 20.2646 0.00101905
\(735\) 0 0
\(736\) −21361.3 −1.06982
\(737\) 1462.47i 0.0730948i
\(738\) − 10920.0i − 0.544678i
\(739\) −10149.1 −0.505199 −0.252599 0.967571i \(-0.581285\pi\)
−0.252599 + 0.967571i \(0.581285\pi\)
\(740\) 0 0
\(741\) 16346.5 0.810397
\(742\) − 5702.85i − 0.282154i
\(743\) − 27758.0i − 1.37058i −0.728269 0.685291i \(-0.759675\pi\)
0.728269 0.685291i \(-0.240325\pi\)
\(744\) 16612.6 0.818611
\(745\) 0 0
\(746\) 2617.87 0.128481
\(747\) 1978.47i 0.0969054i
\(748\) − 2885.89i − 0.141068i
\(749\) −7829.63 −0.381960
\(750\) 0 0
\(751\) −815.225 −0.0396112 −0.0198056 0.999804i \(-0.506305\pi\)
−0.0198056 + 0.999804i \(0.506305\pi\)
\(752\) 3504.13i 0.169924i
\(753\) − 24277.1i − 1.17491i
\(754\) 10169.3 0.491172
\(755\) 0 0
\(756\) −6782.56 −0.326295
\(757\) − 13239.4i − 0.635659i −0.948148 0.317829i \(-0.897046\pi\)
0.948148 0.317829i \(-0.102954\pi\)
\(758\) 13058.0i 0.625710i
\(759\) 10042.1 0.480244
\(760\) 0 0
\(761\) −11028.2 −0.525324 −0.262662 0.964888i \(-0.584600\pi\)
−0.262662 + 0.964888i \(0.584600\pi\)
\(762\) 11371.7i 0.540621i
\(763\) 14407.1i 0.683582i
\(764\) −3017.56 −0.142895
\(765\) 0 0
\(766\) −12365.0 −0.583246
\(767\) − 16969.8i − 0.798882i
\(768\) 7695.55i 0.361574i
\(769\) 18921.2 0.887277 0.443639 0.896206i \(-0.353687\pi\)
0.443639 + 0.896206i \(0.353687\pi\)
\(770\) 0 0
\(771\) 16830.3 0.786158
\(772\) 21814.7i 1.01701i
\(773\) − 38728.6i − 1.80203i −0.433786 0.901016i \(-0.642823\pi\)
0.433786 0.901016i \(-0.357177\pi\)
\(774\) 8183.59 0.380043
\(775\) 0 0
\(776\) −15303.6 −0.707949
\(777\) 3315.29i 0.153070i
\(778\) 1556.93i 0.0717465i
\(779\) 41959.2 1.92984
\(780\) 0 0
\(781\) 21772.5 0.997545
\(782\) 2667.42i 0.121978i
\(783\) − 17863.9i − 0.815329i
\(784\) 6428.50 0.292844
\(785\) 0 0
\(786\) −8498.47 −0.385662
\(787\) − 20587.3i − 0.932477i −0.884659 0.466239i \(-0.845609\pi\)
0.884659 0.466239i \(-0.154391\pi\)
\(788\) 8379.37i 0.378811i
\(789\) −12410.8 −0.559995
\(790\) 0 0
\(791\) 5616.62 0.252470
\(792\) 9066.49i 0.406772i
\(793\) 3132.00i 0.140253i
\(794\) −14435.0 −0.645190
\(795\) 0 0
\(796\) −22974.5 −1.02300
\(797\) − 15871.4i − 0.705385i −0.935739 0.352693i \(-0.885266\pi\)
0.935739 0.352693i \(-0.114734\pi\)
\(798\) 3040.45i 0.134876i
\(799\) 2594.17 0.114862
\(800\) 0 0
\(801\) 13865.9 0.611644
\(802\) 18765.0i 0.826204i
\(803\) − 8168.28i − 0.358969i
\(804\) 1026.93 0.0450462
\(805\) 0 0
\(806\) 21631.4 0.945329
\(807\) 10801.6i 0.471170i
\(808\) 5113.06i 0.222620i
\(809\) −39667.1 −1.72388 −0.861942 0.507007i \(-0.830752\pi\)
−0.861942 + 0.507007i \(0.830752\pi\)
\(810\) 0 0
\(811\) 8003.87 0.346552 0.173276 0.984873i \(-0.444565\pi\)
0.173276 + 0.984873i \(0.444565\pi\)
\(812\) − 6275.00i − 0.271194i
\(813\) − 1732.00i − 0.0747157i
\(814\) 4975.28 0.214230
\(815\) 0 0
\(816\) −1231.48 −0.0528316
\(817\) 31444.7i 1.34652i
\(818\) 12939.7i 0.553088i
\(819\) −7866.69 −0.335634
\(820\) 0 0
\(821\) −13279.1 −0.564489 −0.282244 0.959343i \(-0.591079\pi\)
−0.282244 + 0.959343i \(0.591079\pi\)
\(822\) 13491.9i 0.572486i
\(823\) − 28934.0i − 1.22549i −0.790281 0.612745i \(-0.790065\pi\)
0.790281 0.612745i \(-0.209935\pi\)
\(824\) 10073.6 0.425887
\(825\) 0 0
\(826\) 3156.37 0.132959
\(827\) − 13679.6i − 0.575193i −0.957752 0.287597i \(-0.907144\pi\)
0.957752 0.287597i \(-0.0928563\pi\)
\(828\) 12079.9i 0.507012i
\(829\) −16514.5 −0.691886 −0.345943 0.938256i \(-0.612441\pi\)
−0.345943 + 0.938256i \(0.612441\pi\)
\(830\) 0 0
\(831\) 16415.6 0.685260
\(832\) 3983.64i 0.165995i
\(833\) − 4759.13i − 0.197952i
\(834\) −6304.13 −0.261744
\(835\) 0 0
\(836\) −15137.1 −0.626231
\(837\) − 37998.9i − 1.56922i
\(838\) − 13177.8i − 0.543221i
\(839\) 87.9839 0.00362043 0.00181022 0.999998i \(-0.499424\pi\)
0.00181022 + 0.999998i \(0.499424\pi\)
\(840\) 0 0
\(841\) −7861.94 −0.322356
\(842\) − 16825.1i − 0.688635i
\(843\) 6267.41i 0.256063i
\(844\) −32371.3 −1.32022
\(845\) 0 0
\(846\) −3541.27 −0.143914
\(847\) 4512.98i 0.183079i
\(848\) − 12115.7i − 0.490630i
\(849\) −2377.63 −0.0961133
\(850\) 0 0
\(851\) 15256.0 0.614534
\(852\) − 15288.5i − 0.614758i
\(853\) − 8162.96i − 0.327660i −0.986489 0.163830i \(-0.947615\pi\)
0.986489 0.163830i \(-0.0523849\pi\)
\(854\) −582.551 −0.0233425
\(855\) 0 0
\(856\) 18988.4 0.758191
\(857\) 18724.9i 0.746361i 0.927759 + 0.373181i \(0.121733\pi\)
−0.927759 + 0.373181i \(0.878267\pi\)
\(858\) − 6891.34i − 0.274203i
\(859\) 46422.5 1.84391 0.921953 0.387301i \(-0.126593\pi\)
0.921953 + 0.387301i \(0.126593\pi\)
\(860\) 0 0
\(861\) 11787.2 0.466557
\(862\) 4043.61i 0.159775i
\(863\) − 29112.3i − 1.14831i −0.818746 0.574157i \(-0.805330\pi\)
0.818746 0.574157i \(-0.194670\pi\)
\(864\) 25750.8 1.01396
\(865\) 0 0
\(866\) 8346.33 0.327505
\(867\) 911.688i 0.0357123i
\(868\) − 13347.8i − 0.521950i
\(869\) 19892.6 0.776536
\(870\) 0 0
\(871\) 3077.43 0.119719
\(872\) − 34940.2i − 1.35691i
\(873\) 13548.2i 0.525241i
\(874\) 13991.2 0.541487
\(875\) 0 0
\(876\) −5735.69 −0.221222
\(877\) 39163.0i 1.50791i 0.656924 + 0.753957i \(0.271857\pi\)
−0.656924 + 0.753957i \(0.728143\pi\)
\(878\) 3464.19i 0.133156i
\(879\) 22720.3 0.871830
\(880\) 0 0
\(881\) −35073.2 −1.34125 −0.670627 0.741795i \(-0.733975\pi\)
−0.670627 + 0.741795i \(0.733975\pi\)
\(882\) 6496.63i 0.248019i
\(883\) 48775.7i 1.85893i 0.368915 + 0.929463i \(0.379729\pi\)
−0.368915 + 0.929463i \(0.620271\pi\)
\(884\) −6072.69 −0.231048
\(885\) 0 0
\(886\) 11611.0 0.440269
\(887\) 13296.0i 0.503309i 0.967817 + 0.251654i \(0.0809746\pi\)
−0.967817 + 0.251654i \(0.919025\pi\)
\(888\) − 8040.27i − 0.303844i
\(889\) 21027.9 0.793309
\(890\) 0 0
\(891\) 606.137 0.0227905
\(892\) − 4330.90i − 0.162566i
\(893\) − 13607.0i − 0.509899i
\(894\) 1228.24 0.0459492
\(895\) 0 0
\(896\) 11031.1 0.411297
\(897\) − 21131.3i − 0.786570i
\(898\) − 12054.6i − 0.447960i
\(899\) 35155.3 1.30422
\(900\) 0 0
\(901\) −8969.43 −0.331648
\(902\) − 17689.1i − 0.652974i
\(903\) 8833.44i 0.325535i
\(904\) −13621.4 −0.501153
\(905\) 0 0
\(906\) −2875.47 −0.105443
\(907\) − 11675.0i − 0.427410i −0.976898 0.213705i \(-0.931447\pi\)
0.976898 0.213705i \(-0.0685532\pi\)
\(908\) − 13224.0i − 0.483319i
\(909\) 4526.54 0.165166
\(910\) 0 0
\(911\) 18552.9 0.674738 0.337369 0.941372i \(-0.390463\pi\)
0.337369 + 0.941372i \(0.390463\pi\)
\(912\) 6459.41i 0.234531i
\(913\) 3204.87i 0.116173i
\(914\) 9739.24 0.352457
\(915\) 0 0
\(916\) 24077.7 0.868504
\(917\) 15714.9i 0.565922i
\(918\) − 3215.55i − 0.115609i
\(919\) −33956.8 −1.21886 −0.609429 0.792841i \(-0.708601\pi\)
−0.609429 + 0.792841i \(0.708601\pi\)
\(920\) 0 0
\(921\) 7652.23 0.273778
\(922\) − 9886.63i − 0.353144i
\(923\) − 45815.3i − 1.63383i
\(924\) −4252.33 −0.151398
\(925\) 0 0
\(926\) 480.372 0.0170475
\(927\) − 8918.08i − 0.315974i
\(928\) 23823.8i 0.842732i
\(929\) 23695.3 0.836832 0.418416 0.908256i \(-0.362585\pi\)
0.418416 + 0.908256i \(0.362585\pi\)
\(930\) 0 0
\(931\) −24962.7 −0.878753
\(932\) 31920.2i 1.12187i
\(933\) 30110.8i 1.05657i
\(934\) −2018.72 −0.0707221
\(935\) 0 0
\(936\) 19078.3 0.666234
\(937\) 7990.62i 0.278593i 0.990251 + 0.139297i \(0.0444842\pi\)
−0.990251 + 0.139297i \(0.955516\pi\)
\(938\) 572.402i 0.0199249i
\(939\) −1856.12 −0.0645071
\(940\) 0 0
\(941\) 24385.9 0.844799 0.422400 0.906410i \(-0.361188\pi\)
0.422400 + 0.906410i \(0.361188\pi\)
\(942\) 3095.26i 0.107058i
\(943\) − 54241.0i − 1.87310i
\(944\) 6705.69 0.231199
\(945\) 0 0
\(946\) 13256.4 0.455605
\(947\) − 1174.62i − 0.0403064i −0.999797 0.0201532i \(-0.993585\pi\)
0.999797 0.0201532i \(-0.00641539\pi\)
\(948\) − 13968.4i − 0.478557i
\(949\) −17188.3 −0.587939
\(950\) 0 0
\(951\) 24143.4 0.823241
\(952\) − 2599.51i − 0.0884985i
\(953\) 33546.9i 1.14029i 0.821546 + 0.570143i \(0.193112\pi\)
−0.821546 + 0.570143i \(0.806888\pi\)
\(954\) 12244.1 0.415531
\(955\) 0 0
\(956\) −2055.00 −0.0695224
\(957\) − 11199.8i − 0.378304i
\(958\) 13598.7i 0.458617i
\(959\) 24948.4 0.840067
\(960\) 0 0
\(961\) 44989.1 1.51016
\(962\) − 10469.3i − 0.350878i
\(963\) − 16810.3i − 0.562517i
\(964\) −11792.9 −0.394007
\(965\) 0 0
\(966\) 3930.41 0.130910
\(967\) 24766.8i 0.823625i 0.911269 + 0.411813i \(0.135104\pi\)
−0.911269 + 0.411813i \(0.864896\pi\)
\(968\) − 10944.9i − 0.363411i
\(969\) 4782.01 0.158535
\(970\) 0 0
\(971\) 42324.3 1.39882 0.699409 0.714721i \(-0.253447\pi\)
0.699409 + 0.714721i \(0.253447\pi\)
\(972\) − 23488.3i − 0.775091i
\(973\) 11657.2i 0.384083i
\(974\) 1538.01 0.0505966
\(975\) 0 0
\(976\) −1237.63 −0.0405896
\(977\) − 11320.4i − 0.370698i −0.982673 0.185349i \(-0.940658\pi\)
0.982673 0.185349i \(-0.0593416\pi\)
\(978\) 2905.42i 0.0949949i
\(979\) 22461.0 0.733254
\(980\) 0 0
\(981\) −30932.2 −1.00672
\(982\) 25556.8i 0.830500i
\(983\) 11311.9i 0.367032i 0.983017 + 0.183516i \(0.0587478\pi\)
−0.983017 + 0.183516i \(0.941252\pi\)
\(984\) −28586.3 −0.926116
\(985\) 0 0
\(986\) 2974.92 0.0960860
\(987\) − 3822.47i − 0.123273i
\(988\) 31852.6i 1.02568i
\(989\) 40648.8 1.30693
\(990\) 0 0
\(991\) −29405.5 −0.942580 −0.471290 0.881978i \(-0.656212\pi\)
−0.471290 + 0.881978i \(0.656212\pi\)
\(992\) 50676.5i 1.62196i
\(993\) − 2373.12i − 0.0758397i
\(994\) 8521.63 0.271921
\(995\) 0 0
\(996\) 2250.43 0.0715939
\(997\) − 54905.9i − 1.74412i −0.489398 0.872060i \(-0.662784\pi\)
0.489398 0.872060i \(-0.337216\pi\)
\(998\) − 23588.7i − 0.748183i
\(999\) −18390.9 −0.582446
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 425.4.b.f.324.4 6
5.2 odd 4 425.4.a.g.1.2 3
5.3 odd 4 17.4.a.b.1.2 3
5.4 even 2 inner 425.4.b.f.324.3 6
15.8 even 4 153.4.a.g.1.2 3
20.3 even 4 272.4.a.h.1.2 3
35.13 even 4 833.4.a.d.1.2 3
40.3 even 4 1088.4.a.x.1.2 3
40.13 odd 4 1088.4.a.v.1.2 3
55.43 even 4 2057.4.a.e.1.2 3
60.23 odd 4 2448.4.a.bi.1.1 3
85.13 odd 4 289.4.b.b.288.4 6
85.33 odd 4 289.4.a.b.1.2 3
85.38 odd 4 289.4.b.b.288.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.2 3 5.3 odd 4
153.4.a.g.1.2 3 15.8 even 4
272.4.a.h.1.2 3 20.3 even 4
289.4.a.b.1.2 3 85.33 odd 4
289.4.b.b.288.3 6 85.38 odd 4
289.4.b.b.288.4 6 85.13 odd 4
425.4.a.g.1.2 3 5.2 odd 4
425.4.b.f.324.3 6 5.4 even 2 inner
425.4.b.f.324.4 6 1.1 even 1 trivial
833.4.a.d.1.2 3 35.13 even 4
1088.4.a.v.1.2 3 40.13 odd 4
1088.4.a.x.1.2 3 40.3 even 4
2057.4.a.e.1.2 3 55.43 even 4
2448.4.a.bi.1.1 3 60.23 odd 4