Properties

Label 1875.4.a.g.1.8
Level $1875$
Weight $4$
Character 1875.1
Self dual yes
Analytic conductor $110.629$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.628581261\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.134967\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.134967 q^{2} +3.00000 q^{3} -7.98178 q^{4} -0.404902 q^{6} -17.3099 q^{7} +2.15702 q^{8} +9.00000 q^{9} +42.1544 q^{11} -23.9454 q^{12} -84.5240 q^{13} +2.33627 q^{14} +63.5631 q^{16} -45.1024 q^{17} -1.21471 q^{18} +83.6634 q^{19} -51.9297 q^{21} -5.68947 q^{22} +20.9307 q^{23} +6.47105 q^{24} +11.4080 q^{26} +27.0000 q^{27} +138.164 q^{28} +220.065 q^{29} +160.220 q^{31} -25.8351 q^{32} +126.463 q^{33} +6.08735 q^{34} -71.8361 q^{36} -263.227 q^{37} -11.2918 q^{38} -253.572 q^{39} +410.142 q^{41} +7.00881 q^{42} +290.699 q^{43} -336.468 q^{44} -2.82495 q^{46} -460.997 q^{47} +190.689 q^{48} -43.3675 q^{49} -135.307 q^{51} +674.652 q^{52} -142.443 q^{53} -3.64412 q^{54} -37.3377 q^{56} +250.990 q^{57} -29.7016 q^{58} -472.475 q^{59} +54.0384 q^{61} -21.6244 q^{62} -155.789 q^{63} -505.018 q^{64} -17.0684 q^{66} +449.961 q^{67} +359.998 q^{68} +62.7920 q^{69} -626.058 q^{71} +19.4132 q^{72} +48.7415 q^{73} +35.5271 q^{74} -667.783 q^{76} -729.689 q^{77} +34.2239 q^{78} -647.571 q^{79} +81.0000 q^{81} -55.3557 q^{82} +669.851 q^{83} +414.492 q^{84} -39.2349 q^{86} +660.196 q^{87} +90.9278 q^{88} -866.036 q^{89} +1463.10 q^{91} -167.064 q^{92} +480.659 q^{93} +62.2195 q^{94} -77.5052 q^{96} -177.047 q^{97} +5.85319 q^{98} +379.390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.134967 −0.0477181 −0.0238591 0.999715i \(-0.507595\pi\)
−0.0238591 + 0.999715i \(0.507595\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.98178 −0.997723
\(5\) 0 0
\(6\) −0.404902 −0.0275501
\(7\) −17.3099 −0.934647 −0.467323 0.884087i \(-0.654782\pi\)
−0.467323 + 0.884087i \(0.654782\pi\)
\(8\) 2.15702 0.0953276
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 42.1544 1.15546 0.577729 0.816229i \(-0.303939\pi\)
0.577729 + 0.816229i \(0.303939\pi\)
\(12\) −23.9454 −0.576036
\(13\) −84.5240 −1.80329 −0.901643 0.432480i \(-0.857639\pi\)
−0.901643 + 0.432480i \(0.857639\pi\)
\(14\) 2.33627 0.0445996
\(15\) 0 0
\(16\) 63.5631 0.993174
\(17\) −45.1024 −0.643467 −0.321734 0.946830i \(-0.604266\pi\)
−0.321734 + 0.946830i \(0.604266\pi\)
\(18\) −1.21471 −0.0159060
\(19\) 83.6634 1.01019 0.505097 0.863062i \(-0.331457\pi\)
0.505097 + 0.863062i \(0.331457\pi\)
\(20\) 0 0
\(21\) −51.9297 −0.539618
\(22\) −5.68947 −0.0551363
\(23\) 20.9307 0.189754 0.0948771 0.995489i \(-0.469754\pi\)
0.0948771 + 0.995489i \(0.469754\pi\)
\(24\) 6.47105 0.0550374
\(25\) 0 0
\(26\) 11.4080 0.0860495
\(27\) 27.0000 0.192450
\(28\) 138.164 0.932518
\(29\) 220.065 1.40914 0.704571 0.709634i \(-0.251139\pi\)
0.704571 + 0.709634i \(0.251139\pi\)
\(30\) 0 0
\(31\) 160.220 0.928267 0.464134 0.885765i \(-0.346366\pi\)
0.464134 + 0.885765i \(0.346366\pi\)
\(32\) −25.8351 −0.142720
\(33\) 126.463 0.667104
\(34\) 6.08735 0.0307050
\(35\) 0 0
\(36\) −71.8361 −0.332574
\(37\) −263.227 −1.16958 −0.584788 0.811186i \(-0.698822\pi\)
−0.584788 + 0.811186i \(0.698822\pi\)
\(38\) −11.2918 −0.0482046
\(39\) −253.572 −1.04113
\(40\) 0 0
\(41\) 410.142 1.56228 0.781139 0.624357i \(-0.214639\pi\)
0.781139 + 0.624357i \(0.214639\pi\)
\(42\) 7.00881 0.0257496
\(43\) 290.699 1.03096 0.515479 0.856902i \(-0.327614\pi\)
0.515479 + 0.856902i \(0.327614\pi\)
\(44\) −336.468 −1.15283
\(45\) 0 0
\(46\) −2.82495 −0.00905471
\(47\) −460.997 −1.43071 −0.715355 0.698761i \(-0.753735\pi\)
−0.715355 + 0.698761i \(0.753735\pi\)
\(48\) 190.689 0.573409
\(49\) −43.3675 −0.126436
\(50\) 0 0
\(51\) −135.307 −0.371506
\(52\) 674.652 1.79918
\(53\) −142.443 −0.369170 −0.184585 0.982817i \(-0.559094\pi\)
−0.184585 + 0.982817i \(0.559094\pi\)
\(54\) −3.64412 −0.00918336
\(55\) 0 0
\(56\) −37.3377 −0.0890976
\(57\) 250.990 0.583236
\(58\) −29.7016 −0.0672416
\(59\) −472.475 −1.04256 −0.521280 0.853386i \(-0.674545\pi\)
−0.521280 + 0.853386i \(0.674545\pi\)
\(60\) 0 0
\(61\) 54.0384 0.113425 0.0567124 0.998391i \(-0.481938\pi\)
0.0567124 + 0.998391i \(0.481938\pi\)
\(62\) −21.6244 −0.0442952
\(63\) −155.789 −0.311549
\(64\) −505.018 −0.986364
\(65\) 0 0
\(66\) −17.0684 −0.0318330
\(67\) 449.961 0.820470 0.410235 0.911980i \(-0.365447\pi\)
0.410235 + 0.911980i \(0.365447\pi\)
\(68\) 359.998 0.642002
\(69\) 62.7920 0.109555
\(70\) 0 0
\(71\) −626.058 −1.04647 −0.523235 0.852188i \(-0.675275\pi\)
−0.523235 + 0.852188i \(0.675275\pi\)
\(72\) 19.4132 0.0317759
\(73\) 48.7415 0.0781474 0.0390737 0.999236i \(-0.487559\pi\)
0.0390737 + 0.999236i \(0.487559\pi\)
\(74\) 35.5271 0.0558099
\(75\) 0 0
\(76\) −667.783 −1.00789
\(77\) −729.689 −1.07995
\(78\) 34.2239 0.0496807
\(79\) −647.571 −0.922246 −0.461123 0.887336i \(-0.652553\pi\)
−0.461123 + 0.887336i \(0.652553\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −55.3557 −0.0745490
\(83\) 669.851 0.885852 0.442926 0.896558i \(-0.353940\pi\)
0.442926 + 0.896558i \(0.353940\pi\)
\(84\) 414.492 0.538390
\(85\) 0 0
\(86\) −39.2349 −0.0491954
\(87\) 660.196 0.813569
\(88\) 90.9278 0.110147
\(89\) −866.036 −1.03146 −0.515728 0.856752i \(-0.672479\pi\)
−0.515728 + 0.856752i \(0.672479\pi\)
\(90\) 0 0
\(91\) 1463.10 1.68544
\(92\) −167.064 −0.189322
\(93\) 480.659 0.535935
\(94\) 62.2195 0.0682708
\(95\) 0 0
\(96\) −77.5052 −0.0823994
\(97\) −177.047 −0.185324 −0.0926621 0.995698i \(-0.529538\pi\)
−0.0926621 + 0.995698i \(0.529538\pi\)
\(98\) 5.85319 0.00603328
\(99\) 379.390 0.385153
\(100\) 0 0
\(101\) 810.637 0.798628 0.399314 0.916814i \(-0.369248\pi\)
0.399314 + 0.916814i \(0.369248\pi\)
\(102\) 18.2620 0.0177276
\(103\) 1364.33 1.30516 0.652580 0.757720i \(-0.273687\pi\)
0.652580 + 0.757720i \(0.273687\pi\)
\(104\) −182.320 −0.171903
\(105\) 0 0
\(106\) 19.2251 0.0176161
\(107\) −1109.53 −1.00245 −0.501227 0.865316i \(-0.667118\pi\)
−0.501227 + 0.865316i \(0.667118\pi\)
\(108\) −215.508 −0.192012
\(109\) 1263.96 1.11069 0.555345 0.831620i \(-0.312586\pi\)
0.555345 + 0.831620i \(0.312586\pi\)
\(110\) 0 0
\(111\) −789.682 −0.675255
\(112\) −1100.27 −0.928267
\(113\) −1144.53 −0.952816 −0.476408 0.879224i \(-0.658062\pi\)
−0.476408 + 0.879224i \(0.658062\pi\)
\(114\) −33.8755 −0.0278309
\(115\) 0 0
\(116\) −1756.51 −1.40593
\(117\) −760.716 −0.601096
\(118\) 63.7686 0.0497490
\(119\) 780.718 0.601414
\(120\) 0 0
\(121\) 445.997 0.335084
\(122\) −7.29341 −0.00541242
\(123\) 1230.43 0.901982
\(124\) −1278.84 −0.926153
\(125\) 0 0
\(126\) 21.0264 0.0148665
\(127\) −2283.93 −1.59579 −0.797897 0.602793i \(-0.794054\pi\)
−0.797897 + 0.602793i \(0.794054\pi\)
\(128\) 274.842 0.189787
\(129\) 872.097 0.595224
\(130\) 0 0
\(131\) −203.205 −0.135527 −0.0677637 0.997701i \(-0.521586\pi\)
−0.0677637 + 0.997701i \(0.521586\pi\)
\(132\) −1009.40 −0.665585
\(133\) −1448.21 −0.944175
\(134\) −60.7300 −0.0391513
\(135\) 0 0
\(136\) −97.2867 −0.0613402
\(137\) 63.6065 0.0396662 0.0198331 0.999803i \(-0.493687\pi\)
0.0198331 + 0.999803i \(0.493687\pi\)
\(138\) −8.47486 −0.00522774
\(139\) 1603.91 0.978722 0.489361 0.872081i \(-0.337230\pi\)
0.489361 + 0.872081i \(0.337230\pi\)
\(140\) 0 0
\(141\) −1382.99 −0.826021
\(142\) 84.4973 0.0499356
\(143\) −3563.06 −2.08362
\(144\) 572.068 0.331058
\(145\) 0 0
\(146\) −6.57851 −0.00372905
\(147\) −130.102 −0.0729977
\(148\) 2101.02 1.16691
\(149\) −2731.00 −1.50156 −0.750780 0.660552i \(-0.770322\pi\)
−0.750780 + 0.660552i \(0.770322\pi\)
\(150\) 0 0
\(151\) −879.795 −0.474150 −0.237075 0.971491i \(-0.576189\pi\)
−0.237075 + 0.971491i \(0.576189\pi\)
\(152\) 180.463 0.0962994
\(153\) −405.922 −0.214489
\(154\) 98.4841 0.0515330
\(155\) 0 0
\(156\) 2023.96 1.03876
\(157\) 366.317 0.186212 0.0931059 0.995656i \(-0.470320\pi\)
0.0931059 + 0.995656i \(0.470320\pi\)
\(158\) 87.4009 0.0440078
\(159\) −427.328 −0.213140
\(160\) 0 0
\(161\) −362.308 −0.177353
\(162\) −10.9323 −0.00530201
\(163\) 843.200 0.405181 0.202591 0.979264i \(-0.435064\pi\)
0.202591 + 0.979264i \(0.435064\pi\)
\(164\) −3273.66 −1.55872
\(165\) 0 0
\(166\) −90.4079 −0.0422712
\(167\) −2230.09 −1.03335 −0.516675 0.856182i \(-0.672830\pi\)
−0.516675 + 0.856182i \(0.672830\pi\)
\(168\) −112.013 −0.0514405
\(169\) 4947.30 2.25184
\(170\) 0 0
\(171\) 752.971 0.336732
\(172\) −2320.30 −1.02861
\(173\) −2893.55 −1.27163 −0.635816 0.771841i \(-0.719336\pi\)
−0.635816 + 0.771841i \(0.719336\pi\)
\(174\) −89.1048 −0.0388220
\(175\) 0 0
\(176\) 2679.47 1.14757
\(177\) −1417.42 −0.601922
\(178\) 116.886 0.0492192
\(179\) −2705.86 −1.12986 −0.564931 0.825138i \(-0.691097\pi\)
−0.564931 + 0.825138i \(0.691097\pi\)
\(180\) 0 0
\(181\) −3554.78 −1.45980 −0.729902 0.683552i \(-0.760434\pi\)
−0.729902 + 0.683552i \(0.760434\pi\)
\(182\) −197.471 −0.0804258
\(183\) 162.115 0.0654858
\(184\) 45.1478 0.0180888
\(185\) 0 0
\(186\) −64.8732 −0.0255738
\(187\) −1901.27 −0.743500
\(188\) 3679.58 1.42745
\(189\) −467.367 −0.179873
\(190\) 0 0
\(191\) 172.435 0.0653244 0.0326622 0.999466i \(-0.489601\pi\)
0.0326622 + 0.999466i \(0.489601\pi\)
\(192\) −1515.05 −0.569477
\(193\) 1203.86 0.448993 0.224496 0.974475i \(-0.427926\pi\)
0.224496 + 0.974475i \(0.427926\pi\)
\(194\) 23.8956 0.00884332
\(195\) 0 0
\(196\) 346.150 0.126148
\(197\) 4195.96 1.51751 0.758756 0.651375i \(-0.225808\pi\)
0.758756 + 0.651375i \(0.225808\pi\)
\(198\) −51.2052 −0.0183788
\(199\) −2413.09 −0.859594 −0.429797 0.902926i \(-0.641415\pi\)
−0.429797 + 0.902926i \(0.641415\pi\)
\(200\) 0 0
\(201\) 1349.88 0.473699
\(202\) −109.409 −0.0381090
\(203\) −3809.31 −1.31705
\(204\) 1079.99 0.370660
\(205\) 0 0
\(206\) −184.140 −0.0622798
\(207\) 188.376 0.0632514
\(208\) −5372.61 −1.79098
\(209\) 3526.78 1.16724
\(210\) 0 0
\(211\) 675.120 0.220271 0.110136 0.993917i \(-0.464872\pi\)
0.110136 + 0.993917i \(0.464872\pi\)
\(212\) 1136.95 0.368329
\(213\) −1878.17 −0.604180
\(214\) 149.750 0.0478352
\(215\) 0 0
\(216\) 58.2395 0.0183458
\(217\) −2773.38 −0.867602
\(218\) −170.593 −0.0530000
\(219\) 146.225 0.0451184
\(220\) 0 0
\(221\) 3812.23 1.16036
\(222\) 106.581 0.0322219
\(223\) −5739.58 −1.72355 −0.861773 0.507294i \(-0.830646\pi\)
−0.861773 + 0.507294i \(0.830646\pi\)
\(224\) 447.203 0.133393
\(225\) 0 0
\(226\) 154.474 0.0454666
\(227\) −2649.74 −0.774756 −0.387378 0.921921i \(-0.626619\pi\)
−0.387378 + 0.921921i \(0.626619\pi\)
\(228\) −2003.35 −0.581908
\(229\) −3685.69 −1.06357 −0.531785 0.846879i \(-0.678479\pi\)
−0.531785 + 0.846879i \(0.678479\pi\)
\(230\) 0 0
\(231\) −2189.07 −0.623507
\(232\) 474.685 0.134330
\(233\) 2551.40 0.717373 0.358687 0.933458i \(-0.383225\pi\)
0.358687 + 0.933458i \(0.383225\pi\)
\(234\) 102.672 0.0286832
\(235\) 0 0
\(236\) 3771.19 1.04019
\(237\) −1942.71 −0.532459
\(238\) −105.371 −0.0286984
\(239\) −6229.73 −1.68606 −0.843028 0.537869i \(-0.819230\pi\)
−0.843028 + 0.537869i \(0.819230\pi\)
\(240\) 0 0
\(241\) −4084.06 −1.09161 −0.545804 0.837913i \(-0.683776\pi\)
−0.545804 + 0.837913i \(0.683776\pi\)
\(242\) −60.1949 −0.0159896
\(243\) 243.000 0.0641500
\(244\) −431.323 −0.113166
\(245\) 0 0
\(246\) −166.067 −0.0430409
\(247\) −7071.56 −1.82167
\(248\) 345.596 0.0884895
\(249\) 2009.55 0.511447
\(250\) 0 0
\(251\) −1158.16 −0.291245 −0.145622 0.989340i \(-0.546518\pi\)
−0.145622 + 0.989340i \(0.546518\pi\)
\(252\) 1243.47 0.310839
\(253\) 882.321 0.219253
\(254\) 308.256 0.0761483
\(255\) 0 0
\(256\) 4003.05 0.977308
\(257\) −4275.75 −1.03780 −0.518899 0.854836i \(-0.673658\pi\)
−0.518899 + 0.854836i \(0.673658\pi\)
\(258\) −117.705 −0.0284030
\(259\) 4556.44 1.09314
\(260\) 0 0
\(261\) 1980.59 0.469714
\(262\) 27.4260 0.00646711
\(263\) −267.835 −0.0627964 −0.0313982 0.999507i \(-0.509996\pi\)
−0.0313982 + 0.999507i \(0.509996\pi\)
\(264\) 272.784 0.0635934
\(265\) 0 0
\(266\) 195.460 0.0450543
\(267\) −2598.11 −0.595512
\(268\) −3591.49 −0.818602
\(269\) 6221.73 1.41021 0.705103 0.709105i \(-0.250901\pi\)
0.705103 + 0.709105i \(0.250901\pi\)
\(270\) 0 0
\(271\) −1485.86 −0.333061 −0.166531 0.986036i \(-0.553256\pi\)
−0.166531 + 0.986036i \(0.553256\pi\)
\(272\) −2866.85 −0.639075
\(273\) 4389.30 0.973087
\(274\) −8.58479 −0.00189280
\(275\) 0 0
\(276\) −501.192 −0.109305
\(277\) 1125.01 0.244025 0.122013 0.992529i \(-0.461065\pi\)
0.122013 + 0.992529i \(0.461065\pi\)
\(278\) −216.476 −0.0467028
\(279\) 1441.98 0.309422
\(280\) 0 0
\(281\) 3631.61 0.770974 0.385487 0.922713i \(-0.374033\pi\)
0.385487 + 0.922713i \(0.374033\pi\)
\(282\) 186.659 0.0394162
\(283\) 5262.72 1.10543 0.552715 0.833371i \(-0.313592\pi\)
0.552715 + 0.833371i \(0.313592\pi\)
\(284\) 4997.06 1.04409
\(285\) 0 0
\(286\) 480.896 0.0994266
\(287\) −7099.51 −1.46018
\(288\) −232.516 −0.0475733
\(289\) −2878.77 −0.585950
\(290\) 0 0
\(291\) −531.142 −0.106997
\(292\) −389.044 −0.0779695
\(293\) 3619.55 0.721693 0.360847 0.932625i \(-0.382488\pi\)
0.360847 + 0.932625i \(0.382488\pi\)
\(294\) 17.5596 0.00348331
\(295\) 0 0
\(296\) −567.786 −0.111493
\(297\) 1138.17 0.222368
\(298\) 368.596 0.0716516
\(299\) −1769.14 −0.342181
\(300\) 0 0
\(301\) −5031.97 −0.963581
\(302\) 118.743 0.0226256
\(303\) 2431.91 0.461088
\(304\) 5317.91 1.00330
\(305\) 0 0
\(306\) 54.7861 0.0102350
\(307\) −1819.08 −0.338176 −0.169088 0.985601i \(-0.554082\pi\)
−0.169088 + 0.985601i \(0.554082\pi\)
\(308\) 5824.22 1.07749
\(309\) 4092.99 0.753534
\(310\) 0 0
\(311\) 341.860 0.0623315 0.0311657 0.999514i \(-0.490078\pi\)
0.0311657 + 0.999514i \(0.490078\pi\)
\(312\) −546.959 −0.0992482
\(313\) −3913.30 −0.706686 −0.353343 0.935494i \(-0.614955\pi\)
−0.353343 + 0.935494i \(0.614955\pi\)
\(314\) −49.4408 −0.00888568
\(315\) 0 0
\(316\) 5168.77 0.920146
\(317\) 6577.26 1.16535 0.582675 0.812705i \(-0.302006\pi\)
0.582675 + 0.812705i \(0.302006\pi\)
\(318\) 57.6752 0.0101706
\(319\) 9276.73 1.62820
\(320\) 0 0
\(321\) −3328.60 −0.578767
\(322\) 48.8997 0.00846296
\(323\) −3773.42 −0.650027
\(324\) −646.524 −0.110858
\(325\) 0 0
\(326\) −113.804 −0.0193345
\(327\) 3791.87 0.641257
\(328\) 884.683 0.148928
\(329\) 7979.81 1.33721
\(330\) 0 0
\(331\) −9580.26 −1.59087 −0.795436 0.606038i \(-0.792758\pi\)
−0.795436 + 0.606038i \(0.792758\pi\)
\(332\) −5346.61 −0.883835
\(333\) −2369.05 −0.389859
\(334\) 300.989 0.0493095
\(335\) 0 0
\(336\) −3300.81 −0.535935
\(337\) −1790.88 −0.289482 −0.144741 0.989470i \(-0.546235\pi\)
−0.144741 + 0.989470i \(0.546235\pi\)
\(338\) −667.723 −0.107454
\(339\) −3433.59 −0.550109
\(340\) 0 0
\(341\) 6753.97 1.07257
\(342\) −101.626 −0.0160682
\(343\) 6687.98 1.05282
\(344\) 627.043 0.0982787
\(345\) 0 0
\(346\) 390.534 0.0606799
\(347\) −9080.61 −1.40482 −0.702411 0.711772i \(-0.747893\pi\)
−0.702411 + 0.711772i \(0.747893\pi\)
\(348\) −5269.54 −0.811716
\(349\) 1036.94 0.159043 0.0795216 0.996833i \(-0.474661\pi\)
0.0795216 + 0.996833i \(0.474661\pi\)
\(350\) 0 0
\(351\) −2282.15 −0.347043
\(352\) −1089.06 −0.164907
\(353\) 3307.28 0.498665 0.249332 0.968418i \(-0.419789\pi\)
0.249332 + 0.968418i \(0.419789\pi\)
\(354\) 191.306 0.0287226
\(355\) 0 0
\(356\) 6912.51 1.02911
\(357\) 2342.15 0.347227
\(358\) 365.202 0.0539149
\(359\) −12468.5 −1.83304 −0.916521 0.399987i \(-0.869015\pi\)
−0.916521 + 0.399987i \(0.869015\pi\)
\(360\) 0 0
\(361\) 140.567 0.0204939
\(362\) 479.779 0.0696591
\(363\) 1337.99 0.193461
\(364\) −11678.2 −1.68160
\(365\) 0 0
\(366\) −21.8802 −0.00312486
\(367\) −4956.10 −0.704923 −0.352461 0.935826i \(-0.614655\pi\)
−0.352461 + 0.935826i \(0.614655\pi\)
\(368\) 1330.42 0.188459
\(369\) 3691.28 0.520759
\(370\) 0 0
\(371\) 2465.67 0.345043
\(372\) −3836.51 −0.534715
\(373\) 2903.60 0.403063 0.201532 0.979482i \(-0.435408\pi\)
0.201532 + 0.979482i \(0.435408\pi\)
\(374\) 256.609 0.0354784
\(375\) 0 0
\(376\) −994.379 −0.136386
\(377\) −18600.8 −2.54109
\(378\) 63.0793 0.00858319
\(379\) 4784.12 0.648400 0.324200 0.945989i \(-0.394905\pi\)
0.324200 + 0.945989i \(0.394905\pi\)
\(380\) 0 0
\(381\) −6851.79 −0.921333
\(382\) −23.2731 −0.00311716
\(383\) 1537.67 0.205146 0.102573 0.994725i \(-0.467292\pi\)
0.102573 + 0.994725i \(0.467292\pi\)
\(384\) 824.525 0.109574
\(385\) 0 0
\(386\) −162.481 −0.0214251
\(387\) 2616.29 0.343653
\(388\) 1413.15 0.184902
\(389\) −3208.85 −0.418239 −0.209119 0.977890i \(-0.567060\pi\)
−0.209119 + 0.977890i \(0.567060\pi\)
\(390\) 0 0
\(391\) −944.024 −0.122101
\(392\) −93.5444 −0.0120528
\(393\) −609.614 −0.0782468
\(394\) −566.317 −0.0724128
\(395\) 0 0
\(396\) −3028.21 −0.384276
\(397\) 6227.13 0.787231 0.393616 0.919275i \(-0.371224\pi\)
0.393616 + 0.919275i \(0.371224\pi\)
\(398\) 325.687 0.0410182
\(399\) −4344.62 −0.545120
\(400\) 0 0
\(401\) −3174.14 −0.395284 −0.197642 0.980274i \(-0.563328\pi\)
−0.197642 + 0.980274i \(0.563328\pi\)
\(402\) −182.190 −0.0226040
\(403\) −13542.4 −1.67393
\(404\) −6470.33 −0.796810
\(405\) 0 0
\(406\) 514.132 0.0628471
\(407\) −11096.2 −1.35140
\(408\) −291.860 −0.0354148
\(409\) 10096.0 1.22057 0.610286 0.792182i \(-0.291055\pi\)
0.610286 + 0.792182i \(0.291055\pi\)
\(410\) 0 0
\(411\) 190.819 0.0229013
\(412\) −10889.8 −1.30219
\(413\) 8178.49 0.974424
\(414\) −25.4246 −0.00301824
\(415\) 0 0
\(416\) 2183.68 0.257365
\(417\) 4811.74 0.565065
\(418\) −476.000 −0.0556984
\(419\) −4337.34 −0.505711 −0.252855 0.967504i \(-0.581370\pi\)
−0.252855 + 0.967504i \(0.581370\pi\)
\(420\) 0 0
\(421\) −16427.6 −1.90173 −0.950867 0.309598i \(-0.899805\pi\)
−0.950867 + 0.309598i \(0.899805\pi\)
\(422\) −91.1191 −0.0105109
\(423\) −4148.98 −0.476903
\(424\) −307.251 −0.0351920
\(425\) 0 0
\(426\) 253.492 0.0288303
\(427\) −935.399 −0.106012
\(428\) 8856.04 1.00017
\(429\) −10689.2 −1.20298
\(430\) 0 0
\(431\) 1395.49 0.155959 0.0779797 0.996955i \(-0.475153\pi\)
0.0779797 + 0.996955i \(0.475153\pi\)
\(432\) 1716.20 0.191136
\(433\) −1905.84 −0.211522 −0.105761 0.994392i \(-0.533728\pi\)
−0.105761 + 0.994392i \(0.533728\pi\)
\(434\) 374.316 0.0414003
\(435\) 0 0
\(436\) −10088.6 −1.10816
\(437\) 1751.13 0.191689
\(438\) −19.7355 −0.00215297
\(439\) −7973.28 −0.866843 −0.433421 0.901191i \(-0.642694\pi\)
−0.433421 + 0.901191i \(0.642694\pi\)
\(440\) 0 0
\(441\) −390.307 −0.0421453
\(442\) −514.527 −0.0553700
\(443\) 2866.93 0.307476 0.153738 0.988112i \(-0.450869\pi\)
0.153738 + 0.988112i \(0.450869\pi\)
\(444\) 6303.07 0.673717
\(445\) 0 0
\(446\) 774.655 0.0822444
\(447\) −8193.01 −0.866926
\(448\) 8741.81 0.921902
\(449\) −10711.2 −1.12582 −0.562908 0.826520i \(-0.690317\pi\)
−0.562908 + 0.826520i \(0.690317\pi\)
\(450\) 0 0
\(451\) 17289.3 1.80515
\(452\) 9135.38 0.950647
\(453\) −2639.38 −0.273751
\(454\) 357.628 0.0369699
\(455\) 0 0
\(456\) 541.390 0.0555985
\(457\) −5306.67 −0.543185 −0.271593 0.962412i \(-0.587550\pi\)
−0.271593 + 0.962412i \(0.587550\pi\)
\(458\) 497.448 0.0507515
\(459\) −1217.77 −0.123835
\(460\) 0 0
\(461\) −8288.61 −0.837395 −0.418697 0.908126i \(-0.637513\pi\)
−0.418697 + 0.908126i \(0.637513\pi\)
\(462\) 295.452 0.0297526
\(463\) 1162.48 0.116685 0.0583424 0.998297i \(-0.481418\pi\)
0.0583424 + 0.998297i \(0.481418\pi\)
\(464\) 13988.0 1.39952
\(465\) 0 0
\(466\) −344.356 −0.0342317
\(467\) 7585.71 0.751659 0.375829 0.926689i \(-0.377358\pi\)
0.375829 + 0.926689i \(0.377358\pi\)
\(468\) 6071.87 0.599727
\(469\) −7788.78 −0.766849
\(470\) 0 0
\(471\) 1098.95 0.107509
\(472\) −1019.14 −0.0993846
\(473\) 12254.3 1.19123
\(474\) 262.203 0.0254079
\(475\) 0 0
\(476\) −6231.52 −0.600045
\(477\) −1281.98 −0.123057
\(478\) 840.809 0.0804555
\(479\) −7436.68 −0.709375 −0.354687 0.934985i \(-0.615413\pi\)
−0.354687 + 0.934985i \(0.615413\pi\)
\(480\) 0 0
\(481\) 22249.0 2.10908
\(482\) 551.214 0.0520895
\(483\) −1086.92 −0.102395
\(484\) −3559.85 −0.334321
\(485\) 0 0
\(486\) −32.7970 −0.00306112
\(487\) 21171.7 1.96998 0.984991 0.172604i \(-0.0552182\pi\)
0.984991 + 0.172604i \(0.0552182\pi\)
\(488\) 116.562 0.0108125
\(489\) 2529.60 0.233931
\(490\) 0 0
\(491\) 9419.29 0.865756 0.432878 0.901452i \(-0.357498\pi\)
0.432878 + 0.901452i \(0.357498\pi\)
\(492\) −9820.99 −0.899928
\(493\) −9925.48 −0.906737
\(494\) 954.429 0.0869267
\(495\) 0 0
\(496\) 10184.1 0.921931
\(497\) 10837.0 0.978080
\(498\) −271.224 −0.0244053
\(499\) 3295.44 0.295640 0.147820 0.989014i \(-0.452774\pi\)
0.147820 + 0.989014i \(0.452774\pi\)
\(500\) 0 0
\(501\) −6690.27 −0.596605
\(502\) 156.314 0.0138976
\(503\) 5611.35 0.497411 0.248706 0.968579i \(-0.419995\pi\)
0.248706 + 0.968579i \(0.419995\pi\)
\(504\) −336.040 −0.0296992
\(505\) 0 0
\(506\) −119.084 −0.0104623
\(507\) 14841.9 1.30010
\(508\) 18229.8 1.59216
\(509\) 20552.6 1.78974 0.894872 0.446323i \(-0.147267\pi\)
0.894872 + 0.446323i \(0.147267\pi\)
\(510\) 0 0
\(511\) −843.711 −0.0730402
\(512\) −2739.01 −0.236423
\(513\) 2258.91 0.194412
\(514\) 577.086 0.0495218
\(515\) 0 0
\(516\) −6960.89 −0.593869
\(517\) −19433.1 −1.65313
\(518\) −614.970 −0.0521626
\(519\) −8680.64 −0.734177
\(520\) 0 0
\(521\) 541.011 0.0454936 0.0227468 0.999741i \(-0.492759\pi\)
0.0227468 + 0.999741i \(0.492759\pi\)
\(522\) −267.315 −0.0224139
\(523\) 2248.74 0.188012 0.0940061 0.995572i \(-0.470033\pi\)
0.0940061 + 0.995572i \(0.470033\pi\)
\(524\) 1621.94 0.135219
\(525\) 0 0
\(526\) 36.1490 0.00299652
\(527\) −7226.29 −0.597309
\(528\) 8038.41 0.662551
\(529\) −11728.9 −0.963993
\(530\) 0 0
\(531\) −4252.27 −0.347520
\(532\) 11559.3 0.942025
\(533\) −34666.8 −2.81724
\(534\) 350.659 0.0284167
\(535\) 0 0
\(536\) 970.574 0.0782134
\(537\) −8117.57 −0.652326
\(538\) −839.730 −0.0672924
\(539\) −1828.13 −0.146091
\(540\) 0 0
\(541\) −18663.3 −1.48318 −0.741590 0.670854i \(-0.765928\pi\)
−0.741590 + 0.670854i \(0.765928\pi\)
\(542\) 200.543 0.0158931
\(543\) −10664.3 −0.842819
\(544\) 1165.22 0.0918356
\(545\) 0 0
\(546\) −592.412 −0.0464339
\(547\) 10867.7 0.849484 0.424742 0.905314i \(-0.360365\pi\)
0.424742 + 0.905314i \(0.360365\pi\)
\(548\) −507.693 −0.0395759
\(549\) 486.346 0.0378082
\(550\) 0 0
\(551\) 18411.4 1.42351
\(552\) 135.443 0.0104436
\(553\) 11209.4 0.861974
\(554\) −151.839 −0.0116444
\(555\) 0 0
\(556\) −12802.1 −0.976493
\(557\) −2740.22 −0.208450 −0.104225 0.994554i \(-0.533236\pi\)
−0.104225 + 0.994554i \(0.533236\pi\)
\(558\) −194.619 −0.0147651
\(559\) −24571.0 −1.85911
\(560\) 0 0
\(561\) −5703.80 −0.429260
\(562\) −490.148 −0.0367894
\(563\) −19190.9 −1.43659 −0.718294 0.695740i \(-0.755077\pi\)
−0.718294 + 0.695740i \(0.755077\pi\)
\(564\) 11038.7 0.824140
\(565\) 0 0
\(566\) −710.295 −0.0527490
\(567\) −1402.10 −0.103850
\(568\) −1350.42 −0.0997575
\(569\) 20217.4 1.48956 0.744779 0.667311i \(-0.232555\pi\)
0.744779 + 0.667311i \(0.232555\pi\)
\(570\) 0 0
\(571\) 5129.67 0.375954 0.187977 0.982173i \(-0.439807\pi\)
0.187977 + 0.982173i \(0.439807\pi\)
\(572\) 28439.6 2.07888
\(573\) 517.305 0.0377151
\(574\) 958.202 0.0696769
\(575\) 0 0
\(576\) −4545.16 −0.328788
\(577\) 14063.5 1.01468 0.507342 0.861745i \(-0.330628\pi\)
0.507342 + 0.861745i \(0.330628\pi\)
\(578\) 388.540 0.0279604
\(579\) 3611.58 0.259226
\(580\) 0 0
\(581\) −11595.1 −0.827958
\(582\) 71.6868 0.00510569
\(583\) −6004.58 −0.426560
\(584\) 105.136 0.00744961
\(585\) 0 0
\(586\) −488.520 −0.0344379
\(587\) −3057.11 −0.214958 −0.107479 0.994207i \(-0.534278\pi\)
−0.107479 + 0.994207i \(0.534278\pi\)
\(588\) 1038.45 0.0728315
\(589\) 13404.5 0.937731
\(590\) 0 0
\(591\) 12587.9 0.876136
\(592\) −16731.6 −1.16159
\(593\) −13287.5 −0.920153 −0.460077 0.887879i \(-0.652178\pi\)
−0.460077 + 0.887879i \(0.652178\pi\)
\(594\) −153.616 −0.0106110
\(595\) 0 0
\(596\) 21798.3 1.49814
\(597\) −7239.26 −0.496287
\(598\) 238.776 0.0163282
\(599\) 12749.7 0.869682 0.434841 0.900507i \(-0.356805\pi\)
0.434841 + 0.900507i \(0.356805\pi\)
\(600\) 0 0
\(601\) 21184.3 1.43782 0.718908 0.695106i \(-0.244642\pi\)
0.718908 + 0.695106i \(0.244642\pi\)
\(602\) 679.151 0.0459803
\(603\) 4049.65 0.273490
\(604\) 7022.33 0.473071
\(605\) 0 0
\(606\) −328.228 −0.0220023
\(607\) 2387.17 0.159625 0.0798124 0.996810i \(-0.474568\pi\)
0.0798124 + 0.996810i \(0.474568\pi\)
\(608\) −2161.45 −0.144175
\(609\) −11427.9 −0.760399
\(610\) 0 0
\(611\) 38965.3 2.57998
\(612\) 3239.98 0.214001
\(613\) −9470.85 −0.624020 −0.312010 0.950079i \(-0.601002\pi\)
−0.312010 + 0.950079i \(0.601002\pi\)
\(614\) 245.516 0.0161371
\(615\) 0 0
\(616\) −1573.95 −0.102949
\(617\) 27465.6 1.79210 0.896049 0.443955i \(-0.146425\pi\)
0.896049 + 0.443955i \(0.146425\pi\)
\(618\) −552.420 −0.0359572
\(619\) −19231.1 −1.24873 −0.624365 0.781133i \(-0.714642\pi\)
−0.624365 + 0.781133i \(0.714642\pi\)
\(620\) 0 0
\(621\) 565.128 0.0365182
\(622\) −46.1399 −0.00297434
\(623\) 14991.0 0.964047
\(624\) −16117.8 −1.03402
\(625\) 0 0
\(626\) 528.167 0.0337217
\(627\) 10580.4 0.673905
\(628\) −2923.86 −0.185788
\(629\) 11872.2 0.752583
\(630\) 0 0
\(631\) −894.541 −0.0564360 −0.0282180 0.999602i \(-0.508983\pi\)
−0.0282180 + 0.999602i \(0.508983\pi\)
\(632\) −1396.82 −0.0879155
\(633\) 2025.36 0.127174
\(634\) −887.715 −0.0556083
\(635\) 0 0
\(636\) 3410.84 0.212655
\(637\) 3665.59 0.228000
\(638\) −1252.05 −0.0776949
\(639\) −5634.52 −0.348823
\(640\) 0 0
\(641\) −8943.89 −0.551111 −0.275556 0.961285i \(-0.588862\pi\)
−0.275556 + 0.961285i \(0.588862\pi\)
\(642\) 449.251 0.0276177
\(643\) −4002.17 −0.245459 −0.122730 0.992440i \(-0.539165\pi\)
−0.122730 + 0.992440i \(0.539165\pi\)
\(644\) 2891.86 0.176949
\(645\) 0 0
\(646\) 509.288 0.0310181
\(647\) −13315.6 −0.809104 −0.404552 0.914515i \(-0.632573\pi\)
−0.404552 + 0.914515i \(0.632573\pi\)
\(648\) 174.718 0.0105920
\(649\) −19916.9 −1.20463
\(650\) 0 0
\(651\) −8320.15 −0.500910
\(652\) −6730.24 −0.404258
\(653\) −22994.6 −1.37802 −0.689011 0.724750i \(-0.741955\pi\)
−0.689011 + 0.724750i \(0.741955\pi\)
\(654\) −511.779 −0.0305996
\(655\) 0 0
\(656\) 26069.9 1.55161
\(657\) 438.674 0.0260491
\(658\) −1077.01 −0.0638090
\(659\) −4002.57 −0.236598 −0.118299 0.992978i \(-0.537744\pi\)
−0.118299 + 0.992978i \(0.537744\pi\)
\(660\) 0 0
\(661\) −13629.6 −0.802009 −0.401005 0.916076i \(-0.631339\pi\)
−0.401005 + 0.916076i \(0.631339\pi\)
\(662\) 1293.02 0.0759134
\(663\) 11436.7 0.669932
\(664\) 1444.88 0.0844461
\(665\) 0 0
\(666\) 319.743 0.0186033
\(667\) 4606.12 0.267391
\(668\) 17800.1 1.03100
\(669\) −17218.7 −0.995090
\(670\) 0 0
\(671\) 2277.96 0.131058
\(672\) 1341.61 0.0770143
\(673\) −25216.1 −1.44429 −0.722146 0.691741i \(-0.756844\pi\)
−0.722146 + 0.691741i \(0.756844\pi\)
\(674\) 241.710 0.0138135
\(675\) 0 0
\(676\) −39488.3 −2.24672
\(677\) 529.842 0.0300790 0.0150395 0.999887i \(-0.495213\pi\)
0.0150395 + 0.999887i \(0.495213\pi\)
\(678\) 463.422 0.0262502
\(679\) 3064.67 0.173213
\(680\) 0 0
\(681\) −7949.22 −0.447305
\(682\) −911.564 −0.0511812
\(683\) −2509.91 −0.140614 −0.0703068 0.997525i \(-0.522398\pi\)
−0.0703068 + 0.997525i \(0.522398\pi\)
\(684\) −6010.05 −0.335965
\(685\) 0 0
\(686\) −902.658 −0.0502386
\(687\) −11057.1 −0.614052
\(688\) 18477.7 1.02392
\(689\) 12039.8 0.665719
\(690\) 0 0
\(691\) −25676.8 −1.41359 −0.706796 0.707417i \(-0.749860\pi\)
−0.706796 + 0.707417i \(0.749860\pi\)
\(692\) 23095.7 1.26874
\(693\) −6567.20 −0.359982
\(694\) 1225.59 0.0670354
\(695\) 0 0
\(696\) 1424.05 0.0775555
\(697\) −18498.4 −1.00527
\(698\) −139.953 −0.00758924
\(699\) 7654.21 0.414176
\(700\) 0 0
\(701\) 34424.0 1.85475 0.927373 0.374139i \(-0.122062\pi\)
0.927373 + 0.374139i \(0.122062\pi\)
\(702\) 308.015 0.0165602
\(703\) −22022.5 −1.18150
\(704\) −21288.8 −1.13970
\(705\) 0 0
\(706\) −446.374 −0.0237953
\(707\) −14032.0 −0.746435
\(708\) 11313.6 0.600551
\(709\) −6915.53 −0.366316 −0.183158 0.983083i \(-0.558632\pi\)
−0.183158 + 0.983083i \(0.558632\pi\)
\(710\) 0 0
\(711\) −5828.14 −0.307415
\(712\) −1868.05 −0.0983263
\(713\) 3353.50 0.176143
\(714\) −316.114 −0.0165690
\(715\) 0 0
\(716\) 21597.6 1.12729
\(717\) −18689.2 −0.973445
\(718\) 1682.84 0.0874693
\(719\) −29938.5 −1.55288 −0.776438 0.630193i \(-0.782976\pi\)
−0.776438 + 0.630193i \(0.782976\pi\)
\(720\) 0 0
\(721\) −23616.4 −1.21986
\(722\) −18.9720 −0.000977929 0
\(723\) −12252.2 −0.630240
\(724\) 28373.5 1.45648
\(725\) 0 0
\(726\) −180.585 −0.00923158
\(727\) 17305.3 0.882832 0.441416 0.897303i \(-0.354476\pi\)
0.441416 + 0.897303i \(0.354476\pi\)
\(728\) 3155.93 0.160669
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −13111.2 −0.663388
\(732\) −1293.97 −0.0653367
\(733\) 4581.82 0.230878 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(734\) 668.912 0.0336376
\(735\) 0 0
\(736\) −540.746 −0.0270817
\(737\) 18967.9 0.948019
\(738\) −498.201 −0.0248497
\(739\) −1715.91 −0.0854139 −0.0427069 0.999088i \(-0.513598\pi\)
−0.0427069 + 0.999088i \(0.513598\pi\)
\(740\) 0 0
\(741\) −21214.7 −1.05174
\(742\) −332.784 −0.0164648
\(743\) 12824.6 0.633228 0.316614 0.948554i \(-0.397454\pi\)
0.316614 + 0.948554i \(0.397454\pi\)
\(744\) 1036.79 0.0510894
\(745\) 0 0
\(746\) −391.890 −0.0192334
\(747\) 6028.66 0.295284
\(748\) 15175.5 0.741807
\(749\) 19205.9 0.936939
\(750\) 0 0
\(751\) −38918.8 −1.89103 −0.945516 0.325574i \(-0.894442\pi\)
−0.945516 + 0.325574i \(0.894442\pi\)
\(752\) −29302.4 −1.42094
\(753\) −3474.48 −0.168150
\(754\) 2510.50 0.121256
\(755\) 0 0
\(756\) 3730.42 0.179463
\(757\) 22740.4 1.09183 0.545915 0.837841i \(-0.316182\pi\)
0.545915 + 0.837841i \(0.316182\pi\)
\(758\) −645.699 −0.0309404
\(759\) 2646.96 0.126586
\(760\) 0 0
\(761\) 40270.0 1.91825 0.959123 0.282988i \(-0.0913257\pi\)
0.959123 + 0.282988i \(0.0913257\pi\)
\(762\) 924.767 0.0439643
\(763\) −21879.0 −1.03810
\(764\) −1376.34 −0.0651757
\(765\) 0 0
\(766\) −207.534 −0.00978919
\(767\) 39935.5 1.88003
\(768\) 12009.2 0.564249
\(769\) −30661.9 −1.43784 −0.718919 0.695094i \(-0.755363\pi\)
−0.718919 + 0.695094i \(0.755363\pi\)
\(770\) 0 0
\(771\) −12827.3 −0.599173
\(772\) −9608.94 −0.447971
\(773\) −30438.6 −1.41630 −0.708151 0.706061i \(-0.750471\pi\)
−0.708151 + 0.706061i \(0.750471\pi\)
\(774\) −353.114 −0.0163985
\(775\) 0 0
\(776\) −381.894 −0.0176665
\(777\) 13669.3 0.631125
\(778\) 433.089 0.0199576
\(779\) 34313.9 1.57821
\(780\) 0 0
\(781\) −26391.1 −1.20915
\(782\) 127.412 0.00582641
\(783\) 5941.77 0.271190
\(784\) −2756.57 −0.125573
\(785\) 0 0
\(786\) 82.2780 0.00373379
\(787\) 24386.3 1.10455 0.552274 0.833663i \(-0.313760\pi\)
0.552274 + 0.833663i \(0.313760\pi\)
\(788\) −33491.2 −1.51406
\(789\) −803.506 −0.0362555
\(790\) 0 0
\(791\) 19811.7 0.890547
\(792\) 818.351 0.0367157
\(793\) −4567.54 −0.204537
\(794\) −840.459 −0.0375652
\(795\) 0 0
\(796\) 19260.7 0.857636
\(797\) 41143.2 1.82857 0.914283 0.405076i \(-0.132755\pi\)
0.914283 + 0.405076i \(0.132755\pi\)
\(798\) 586.381 0.0260121
\(799\) 20792.1 0.920615
\(800\) 0 0
\(801\) −7794.32 −0.343819
\(802\) 428.404 0.0188622
\(803\) 2054.67 0.0902961
\(804\) −10774.5 −0.472620
\(805\) 0 0
\(806\) 1827.78 0.0798769
\(807\) 18665.2 0.814183
\(808\) 1748.56 0.0761313
\(809\) −16879.6 −0.733566 −0.366783 0.930307i \(-0.619541\pi\)
−0.366783 + 0.930307i \(0.619541\pi\)
\(810\) 0 0
\(811\) 11423.5 0.494616 0.247308 0.968937i \(-0.420454\pi\)
0.247308 + 0.968937i \(0.420454\pi\)
\(812\) 30405.1 1.31405
\(813\) −4457.58 −0.192293
\(814\) 1497.62 0.0644861
\(815\) 0 0
\(816\) −8600.55 −0.368970
\(817\) 24320.9 1.04147
\(818\) −1362.62 −0.0582434
\(819\) 13167.9 0.561812
\(820\) 0 0
\(821\) −3507.29 −0.149093 −0.0745466 0.997218i \(-0.523751\pi\)
−0.0745466 + 0.997218i \(0.523751\pi\)
\(822\) −25.7544 −0.00109281
\(823\) −10475.2 −0.443674 −0.221837 0.975084i \(-0.571205\pi\)
−0.221837 + 0.975084i \(0.571205\pi\)
\(824\) 2942.88 0.124418
\(825\) 0 0
\(826\) −1103.83 −0.0464977
\(827\) 26560.2 1.11679 0.558397 0.829574i \(-0.311417\pi\)
0.558397 + 0.829574i \(0.311417\pi\)
\(828\) −1503.58 −0.0631074
\(829\) 7296.73 0.305701 0.152850 0.988249i \(-0.451155\pi\)
0.152850 + 0.988249i \(0.451155\pi\)
\(830\) 0 0
\(831\) 3375.02 0.140888
\(832\) 42686.1 1.77870
\(833\) 1955.98 0.0813573
\(834\) −649.428 −0.0269638
\(835\) 0 0
\(836\) −28150.0 −1.16458
\(837\) 4325.93 0.178645
\(838\) 585.398 0.0241316
\(839\) 46997.4 1.93389 0.966944 0.254989i \(-0.0820718\pi\)
0.966944 + 0.254989i \(0.0820718\pi\)
\(840\) 0 0
\(841\) 24039.8 0.985681
\(842\) 2217.18 0.0907472
\(843\) 10894.8 0.445122
\(844\) −5388.66 −0.219770
\(845\) 0 0
\(846\) 559.976 0.0227569
\(847\) −7720.16 −0.313185
\(848\) −9054.10 −0.366650
\(849\) 15788.2 0.638220
\(850\) 0 0
\(851\) −5509.52 −0.221932
\(852\) 14991.2 0.602804
\(853\) 30588.9 1.22784 0.613918 0.789370i \(-0.289592\pi\)
0.613918 + 0.789370i \(0.289592\pi\)
\(854\) 126.248 0.00505870
\(855\) 0 0
\(856\) −2393.28 −0.0955614
\(857\) −39552.7 −1.57654 −0.788271 0.615329i \(-0.789023\pi\)
−0.788271 + 0.615329i \(0.789023\pi\)
\(858\) 1442.69 0.0574040
\(859\) 14881.4 0.591089 0.295545 0.955329i \(-0.404499\pi\)
0.295545 + 0.955329i \(0.404499\pi\)
\(860\) 0 0
\(861\) −21298.5 −0.843034
\(862\) −188.346 −0.00744209
\(863\) 6318.56 0.249231 0.124616 0.992205i \(-0.460230\pi\)
0.124616 + 0.992205i \(0.460230\pi\)
\(864\) −697.547 −0.0274665
\(865\) 0 0
\(866\) 257.226 0.0100934
\(867\) −8636.32 −0.338298
\(868\) 22136.6 0.865626
\(869\) −27298.0 −1.06562
\(870\) 0 0
\(871\) −38032.5 −1.47954
\(872\) 2726.38 0.105879
\(873\) −1593.43 −0.0617747
\(874\) −236.345 −0.00914703
\(875\) 0 0
\(876\) −1167.13 −0.0450157
\(877\) 25188.2 0.969835 0.484918 0.874560i \(-0.338850\pi\)
0.484918 + 0.874560i \(0.338850\pi\)
\(878\) 1076.13 0.0413641
\(879\) 10858.6 0.416670
\(880\) 0 0
\(881\) −1209.46 −0.0462518 −0.0231259 0.999733i \(-0.507362\pi\)
−0.0231259 + 0.999733i \(0.507362\pi\)
\(882\) 52.6787 0.00201109
\(883\) 6253.63 0.238337 0.119168 0.992874i \(-0.461977\pi\)
0.119168 + 0.992874i \(0.461977\pi\)
\(884\) −30428.4 −1.15771
\(885\) 0 0
\(886\) −386.941 −0.0146722
\(887\) −20271.5 −0.767360 −0.383680 0.923466i \(-0.625344\pi\)
−0.383680 + 0.923466i \(0.625344\pi\)
\(888\) −1703.36 −0.0643704
\(889\) 39534.6 1.49150
\(890\) 0 0
\(891\) 3414.51 0.128384
\(892\) 45812.1 1.71962
\(893\) −38568.6 −1.44530
\(894\) 1105.79 0.0413681
\(895\) 0 0
\(896\) −4757.48 −0.177384
\(897\) −5307.43 −0.197558
\(898\) 1445.66 0.0537218
\(899\) 35258.8 1.30806
\(900\) 0 0
\(901\) 6424.50 0.237548
\(902\) −2333.49 −0.0861382
\(903\) −15095.9 −0.556324
\(904\) −2468.77 −0.0908297
\(905\) 0 0
\(906\) 356.230 0.0130629
\(907\) 20161.0 0.738077 0.369039 0.929414i \(-0.379687\pi\)
0.369039 + 0.929414i \(0.379687\pi\)
\(908\) 21149.7 0.772991
\(909\) 7295.74 0.266209
\(910\) 0 0
\(911\) 12771.2 0.464468 0.232234 0.972660i \(-0.425397\pi\)
0.232234 + 0.972660i \(0.425397\pi\)
\(912\) 15953.7 0.579255
\(913\) 28237.2 1.02356
\(914\) 716.227 0.0259198
\(915\) 0 0
\(916\) 29418.4 1.06115
\(917\) 3517.45 0.126670
\(918\) 164.358 0.00590919
\(919\) 22979.7 0.824844 0.412422 0.910993i \(-0.364683\pi\)
0.412422 + 0.910993i \(0.364683\pi\)
\(920\) 0 0
\(921\) −5457.23 −0.195246
\(922\) 1118.69 0.0399589
\(923\) 52916.9 1.88709
\(924\) 17472.7 0.622087
\(925\) 0 0
\(926\) −156.897 −0.00556798
\(927\) 12279.0 0.435053
\(928\) −5685.41 −0.201113
\(929\) 31009.9 1.09516 0.547579 0.836754i \(-0.315549\pi\)
0.547579 + 0.836754i \(0.315549\pi\)
\(930\) 0 0
\(931\) −3628.27 −0.127725
\(932\) −20364.7 −0.715740
\(933\) 1025.58 0.0359871
\(934\) −1023.82 −0.0358677
\(935\) 0 0
\(936\) −1640.88 −0.0573010
\(937\) −37193.1 −1.29674 −0.648371 0.761325i \(-0.724549\pi\)
−0.648371 + 0.761325i \(0.724549\pi\)
\(938\) 1051.23 0.0365926
\(939\) −11739.9 −0.408005
\(940\) 0 0
\(941\) −2159.48 −0.0748109 −0.0374055 0.999300i \(-0.511909\pi\)
−0.0374055 + 0.999300i \(0.511909\pi\)
\(942\) −148.322 −0.00513015
\(943\) 8584.55 0.296449
\(944\) −30032.0 −1.03544
\(945\) 0 0
\(946\) −1653.92 −0.0568432
\(947\) 9462.47 0.324698 0.162349 0.986733i \(-0.448093\pi\)
0.162349 + 0.986733i \(0.448093\pi\)
\(948\) 15506.3 0.531247
\(949\) −4119.83 −0.140922
\(950\) 0 0
\(951\) 19731.8 0.672815
\(952\) 1684.02 0.0573314
\(953\) −11718.0 −0.398304 −0.199152 0.979969i \(-0.563819\pi\)
−0.199152 + 0.979969i \(0.563819\pi\)
\(954\) 173.026 0.00587203
\(955\) 0 0
\(956\) 49724.3 1.68222
\(957\) 27830.2 0.940044
\(958\) 1003.71 0.0338500
\(959\) −1101.02 −0.0370739
\(960\) 0 0
\(961\) −4120.70 −0.138320
\(962\) −3002.89 −0.100641
\(963\) −9985.79 −0.334151
\(964\) 32598.1 1.08912
\(965\) 0 0
\(966\) 146.699 0.00488609
\(967\) −40593.1 −1.34993 −0.674967 0.737848i \(-0.735842\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(968\) 962.022 0.0319427
\(969\) −11320.3 −0.375293
\(970\) 0 0
\(971\) 51359.1 1.69742 0.848708 0.528862i \(-0.177381\pi\)
0.848708 + 0.528862i \(0.177381\pi\)
\(972\) −1939.57 −0.0640040
\(973\) −27763.6 −0.914759
\(974\) −2857.49 −0.0940039
\(975\) 0 0
\(976\) 3434.85 0.112651
\(977\) −45722.4 −1.49723 −0.748613 0.663008i \(-0.769280\pi\)
−0.748613 + 0.663008i \(0.769280\pi\)
\(978\) −341.413 −0.0111628
\(979\) −36507.3 −1.19181
\(980\) 0 0
\(981\) 11375.6 0.370230
\(982\) −1271.29 −0.0413123
\(983\) 12933.8 0.419657 0.209828 0.977738i \(-0.432709\pi\)
0.209828 + 0.977738i \(0.432709\pi\)
\(984\) 2654.05 0.0859837
\(985\) 0 0
\(986\) 1339.61 0.0432678
\(987\) 23939.4 0.772037
\(988\) 56443.7 1.81752
\(989\) 6084.53 0.195629
\(990\) 0 0
\(991\) 13580.2 0.435308 0.217654 0.976026i \(-0.430159\pi\)
0.217654 + 0.976026i \(0.430159\pi\)
\(992\) −4139.28 −0.132482
\(993\) −28740.8 −0.918490
\(994\) −1462.64 −0.0466721
\(995\) 0 0
\(996\) −16039.8 −0.510282
\(997\) 19249.8 0.611480 0.305740 0.952115i \(-0.401096\pi\)
0.305740 + 0.952115i \(0.401096\pi\)
\(998\) −444.777 −0.0141074
\(999\) −7107.14 −0.225085
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 1875.4.a.g.1.8 14
5.4 even 2 1875.4.a.f.1.7 14
25.6 even 5 75.4.g.b.61.4 yes 28
25.21 even 5 75.4.g.b.16.4 28
75.56 odd 10 225.4.h.a.136.4 28
75.71 odd 10 225.4.h.a.91.4 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.4 28 25.21 even 5
75.4.g.b.61.4 yes 28 25.6 even 5
225.4.h.a.91.4 28 75.71 odd 10
225.4.h.a.136.4 28 75.56 odd 10
1875.4.a.f.1.7 14 5.4 even 2
1875.4.a.g.1.8 14 1.1 even 1 trivial