Properties

Label 1875.4.a.g.1.13
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 1.13
Root \(4.65416\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.65416 q^{2} +3.00000 q^{3} +13.6612 q^{4} +13.9625 q^{6} -26.0445 q^{7} +26.3483 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.65416 q^{2} +3.00000 q^{3} +13.6612 q^{4} +13.9625 q^{6} -26.0445 q^{7} +26.3483 q^{8} +9.00000 q^{9} +2.51593 q^{11} +40.9837 q^{12} +39.6201 q^{13} -121.215 q^{14} +13.3393 q^{16} -84.9913 q^{17} +41.8875 q^{18} -142.827 q^{19} -78.1335 q^{21} +11.7096 q^{22} +157.199 q^{23} +79.0448 q^{24} +184.399 q^{26} +27.0000 q^{27} -355.800 q^{28} -100.233 q^{29} -173.361 q^{31} -148.703 q^{32} +7.54780 q^{33} -395.563 q^{34} +122.951 q^{36} -59.3956 q^{37} -664.741 q^{38} +118.860 q^{39} -355.427 q^{41} -363.646 q^{42} +109.742 q^{43} +34.3707 q^{44} +731.630 q^{46} -58.1116 q^{47} +40.0178 q^{48} +335.317 q^{49} -254.974 q^{51} +541.260 q^{52} -390.876 q^{53} +125.662 q^{54} -686.228 q^{56} -428.481 q^{57} -466.501 q^{58} +412.549 q^{59} +358.635 q^{61} -806.851 q^{62} -234.401 q^{63} -798.802 q^{64} +35.1287 q^{66} +351.010 q^{67} -1161.09 q^{68} +471.597 q^{69} +170.828 q^{71} +237.134 q^{72} -932.891 q^{73} -276.437 q^{74} -1951.19 q^{76} -65.5263 q^{77} +553.196 q^{78} -1313.26 q^{79} +81.0000 q^{81} -1654.21 q^{82} -653.900 q^{83} -1067.40 q^{84} +510.757 q^{86} -300.699 q^{87} +66.2905 q^{88} +1059.65 q^{89} -1031.89 q^{91} +2147.53 q^{92} -520.083 q^{93} -270.461 q^{94} -446.109 q^{96} +641.500 q^{97} +1560.62 q^{98} +22.6434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.65416 1.64549 0.822747 0.568407i \(-0.192440\pi\)
0.822747 + 0.568407i \(0.192440\pi\)
\(3\) 3.00000 0.577350
\(4\) 13.6612 1.70765
\(5\) 0 0
\(6\) 13.9625 0.950027
\(7\) −26.0445 −1.40627 −0.703136 0.711056i \(-0.748217\pi\)
−0.703136 + 0.711056i \(0.748217\pi\)
\(8\) 26.3483 1.16444
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 2.51593 0.0689621 0.0344810 0.999405i \(-0.489022\pi\)
0.0344810 + 0.999405i \(0.489022\pi\)
\(12\) 40.9837 0.985914
\(13\) 39.6201 0.845281 0.422640 0.906298i \(-0.361103\pi\)
0.422640 + 0.906298i \(0.361103\pi\)
\(14\) −121.215 −2.31401
\(15\) 0 0
\(16\) 13.3393 0.208426
\(17\) −84.9913 −1.21255 −0.606277 0.795253i \(-0.707338\pi\)
−0.606277 + 0.795253i \(0.707338\pi\)
\(18\) 41.8875 0.548498
\(19\) −142.827 −1.72457 −0.862284 0.506425i \(-0.830967\pi\)
−0.862284 + 0.506425i \(0.830967\pi\)
\(20\) 0 0
\(21\) −78.1335 −0.811911
\(22\) 11.7096 0.113477
\(23\) 157.199 1.42514 0.712571 0.701600i \(-0.247531\pi\)
0.712571 + 0.701600i \(0.247531\pi\)
\(24\) 79.0448 0.672289
\(25\) 0 0
\(26\) 184.399 1.39090
\(27\) 27.0000 0.192450
\(28\) −355.800 −2.40142
\(29\) −100.233 −0.641821 −0.320911 0.947109i \(-0.603989\pi\)
−0.320911 + 0.947109i \(0.603989\pi\)
\(30\) 0 0
\(31\) −173.361 −1.00441 −0.502203 0.864750i \(-0.667477\pi\)
−0.502203 + 0.864750i \(0.667477\pi\)
\(32\) −148.703 −0.821476
\(33\) 7.54780 0.0398153
\(34\) −395.563 −1.99525
\(35\) 0 0
\(36\) 122.951 0.569218
\(37\) −59.3956 −0.263907 −0.131954 0.991256i \(-0.542125\pi\)
−0.131954 + 0.991256i \(0.542125\pi\)
\(38\) −664.741 −2.83777
\(39\) 118.860 0.488023
\(40\) 0 0
\(41\) −355.427 −1.35386 −0.676930 0.736047i \(-0.736690\pi\)
−0.676930 + 0.736047i \(0.736690\pi\)
\(42\) −363.646 −1.33600
\(43\) 109.742 0.389198 0.194599 0.980883i \(-0.437659\pi\)
0.194599 + 0.980883i \(0.437659\pi\)
\(44\) 34.3707 0.117763
\(45\) 0 0
\(46\) 731.630 2.34506
\(47\) −58.1116 −0.180350 −0.0901750 0.995926i \(-0.528743\pi\)
−0.0901750 + 0.995926i \(0.528743\pi\)
\(48\) 40.0178 0.120335
\(49\) 335.317 0.977600
\(50\) 0 0
\(51\) −254.974 −0.700069
\(52\) 541.260 1.44345
\(53\) −390.876 −1.01304 −0.506518 0.862229i \(-0.669068\pi\)
−0.506518 + 0.862229i \(0.669068\pi\)
\(54\) 125.662 0.316676
\(55\) 0 0
\(56\) −686.228 −1.63752
\(57\) −428.481 −0.995680
\(58\) −466.501 −1.05611
\(59\) 412.549 0.910327 0.455163 0.890408i \(-0.349581\pi\)
0.455163 + 0.890408i \(0.349581\pi\)
\(60\) 0 0
\(61\) 358.635 0.752763 0.376381 0.926465i \(-0.377168\pi\)
0.376381 + 0.926465i \(0.377168\pi\)
\(62\) −806.851 −1.65274
\(63\) −234.401 −0.468757
\(64\) −798.802 −1.56016
\(65\) 0 0
\(66\) 35.1287 0.0655158
\(67\) 351.010 0.640041 0.320020 0.947411i \(-0.396310\pi\)
0.320020 + 0.947411i \(0.396310\pi\)
\(68\) −1161.09 −2.07062
\(69\) 471.597 0.822806
\(70\) 0 0
\(71\) 170.828 0.285543 0.142771 0.989756i \(-0.454399\pi\)
0.142771 + 0.989756i \(0.454399\pi\)
\(72\) 237.134 0.388146
\(73\) −932.891 −1.49571 −0.747853 0.663864i \(-0.768915\pi\)
−0.747853 + 0.663864i \(0.768915\pi\)
\(74\) −276.437 −0.434258
\(75\) 0 0
\(76\) −1951.19 −2.94496
\(77\) −65.5263 −0.0969794
\(78\) 553.196 0.803039
\(79\) −1313.26 −1.87029 −0.935147 0.354261i \(-0.884733\pi\)
−0.935147 + 0.354261i \(0.884733\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −1654.21 −2.22777
\(83\) −653.900 −0.864757 −0.432378 0.901692i \(-0.642326\pi\)
−0.432378 + 0.901692i \(0.642326\pi\)
\(84\) −1067.40 −1.38646
\(85\) 0 0
\(86\) 510.757 0.640423
\(87\) −300.699 −0.370556
\(88\) 66.2905 0.0803021
\(89\) 1059.65 1.26205 0.631025 0.775763i \(-0.282635\pi\)
0.631025 + 0.775763i \(0.282635\pi\)
\(90\) 0 0
\(91\) −1031.89 −1.18869
\(92\) 2147.53 2.43365
\(93\) −520.083 −0.579894
\(94\) −270.461 −0.296765
\(95\) 0 0
\(96\) −446.109 −0.474279
\(97\) 641.500 0.671489 0.335745 0.941953i \(-0.391012\pi\)
0.335745 + 0.941953i \(0.391012\pi\)
\(98\) 1560.62 1.60864
\(99\) 22.6434 0.0229874
\(100\) 0 0
\(101\) 864.030 0.851229 0.425615 0.904904i \(-0.360058\pi\)
0.425615 + 0.904904i \(0.360058\pi\)
\(102\) −1186.69 −1.15196
\(103\) 1832.11 1.75265 0.876324 0.481722i \(-0.159988\pi\)
0.876324 + 0.481722i \(0.159988\pi\)
\(104\) 1043.92 0.984278
\(105\) 0 0
\(106\) −1819.20 −1.66695
\(107\) −1037.99 −0.937816 −0.468908 0.883247i \(-0.655352\pi\)
−0.468908 + 0.883247i \(0.655352\pi\)
\(108\) 368.853 0.328638
\(109\) 1557.03 1.36823 0.684113 0.729376i \(-0.260189\pi\)
0.684113 + 0.729376i \(0.260189\pi\)
\(110\) 0 0
\(111\) −178.187 −0.152367
\(112\) −347.415 −0.293104
\(113\) −975.199 −0.811850 −0.405925 0.913906i \(-0.633051\pi\)
−0.405925 + 0.913906i \(0.633051\pi\)
\(114\) −1994.22 −1.63839
\(115\) 0 0
\(116\) −1369.31 −1.09601
\(117\) 356.581 0.281760
\(118\) 1920.07 1.49794
\(119\) 2213.56 1.70518
\(120\) 0 0
\(121\) −1324.67 −0.995244
\(122\) 1669.15 1.23867
\(123\) −1066.28 −0.781652
\(124\) −2368.33 −1.71518
\(125\) 0 0
\(126\) −1090.94 −0.771338
\(127\) −1028.89 −0.718894 −0.359447 0.933166i \(-0.617035\pi\)
−0.359447 + 0.933166i \(0.617035\pi\)
\(128\) −2528.13 −1.74576
\(129\) 329.226 0.224704
\(130\) 0 0
\(131\) −2669.90 −1.78069 −0.890346 0.455285i \(-0.849537\pi\)
−0.890346 + 0.455285i \(0.849537\pi\)
\(132\) 103.112 0.0679907
\(133\) 3719.86 2.42521
\(134\) 1633.66 1.05318
\(135\) 0 0
\(136\) −2239.37 −1.41195
\(137\) −1219.42 −0.760454 −0.380227 0.924893i \(-0.624154\pi\)
−0.380227 + 0.924893i \(0.624154\pi\)
\(138\) 2194.89 1.35392
\(139\) 875.292 0.534110 0.267055 0.963681i \(-0.413949\pi\)
0.267055 + 0.963681i \(0.413949\pi\)
\(140\) 0 0
\(141\) −174.335 −0.104125
\(142\) 795.061 0.469859
\(143\) 99.6816 0.0582923
\(144\) 120.053 0.0694753
\(145\) 0 0
\(146\) −4341.82 −2.46118
\(147\) 1005.95 0.564417
\(148\) −811.416 −0.450662
\(149\) −819.633 −0.450650 −0.225325 0.974284i \(-0.572344\pi\)
−0.225325 + 0.974284i \(0.572344\pi\)
\(150\) 0 0
\(151\) 567.457 0.305821 0.152911 0.988240i \(-0.451135\pi\)
0.152911 + 0.988240i \(0.451135\pi\)
\(152\) −3763.25 −2.00815
\(153\) −764.922 −0.404185
\(154\) −304.970 −0.159579
\(155\) 0 0
\(156\) 1623.78 0.833374
\(157\) −1643.63 −0.835516 −0.417758 0.908558i \(-0.637184\pi\)
−0.417758 + 0.908558i \(0.637184\pi\)
\(158\) −6112.12 −3.07756
\(159\) −1172.63 −0.584877
\(160\) 0 0
\(161\) −4094.17 −2.00414
\(162\) 376.987 0.182833
\(163\) −1280.29 −0.615217 −0.307608 0.951513i \(-0.599529\pi\)
−0.307608 + 0.951513i \(0.599529\pi\)
\(164\) −4855.56 −2.31192
\(165\) 0 0
\(166\) −3043.36 −1.42295
\(167\) −1576.61 −0.730547 −0.365274 0.930900i \(-0.619025\pi\)
−0.365274 + 0.930900i \(0.619025\pi\)
\(168\) −2058.68 −0.945422
\(169\) −627.245 −0.285501
\(170\) 0 0
\(171\) −1285.44 −0.574856
\(172\) 1499.21 0.664615
\(173\) 2221.62 0.976337 0.488169 0.872749i \(-0.337665\pi\)
0.488169 + 0.872749i \(0.337665\pi\)
\(174\) −1399.50 −0.609747
\(175\) 0 0
\(176\) 33.5607 0.0143735
\(177\) 1237.65 0.525577
\(178\) 4931.77 2.07670
\(179\) 3015.07 1.25898 0.629489 0.777010i \(-0.283264\pi\)
0.629489 + 0.777010i \(0.283264\pi\)
\(180\) 0 0
\(181\) 2472.66 1.01542 0.507710 0.861528i \(-0.330492\pi\)
0.507710 + 0.861528i \(0.330492\pi\)
\(182\) −4802.57 −1.95599
\(183\) 1075.91 0.434608
\(184\) 4141.92 1.65949
\(185\) 0 0
\(186\) −2420.55 −0.954212
\(187\) −213.833 −0.0836203
\(188\) −793.876 −0.307975
\(189\) −703.202 −0.270637
\(190\) 0 0
\(191\) −640.945 −0.242812 −0.121406 0.992603i \(-0.538740\pi\)
−0.121406 + 0.992603i \(0.538740\pi\)
\(192\) −2396.41 −0.900759
\(193\) −658.953 −0.245764 −0.122882 0.992421i \(-0.539214\pi\)
−0.122882 + 0.992421i \(0.539214\pi\)
\(194\) 2985.65 1.10493
\(195\) 0 0
\(196\) 4580.84 1.66940
\(197\) 2802.27 1.01347 0.506734 0.862102i \(-0.330853\pi\)
0.506734 + 0.862102i \(0.330853\pi\)
\(198\) 105.386 0.0378256
\(199\) −2715.89 −0.967459 −0.483729 0.875218i \(-0.660718\pi\)
−0.483729 + 0.875218i \(0.660718\pi\)
\(200\) 0 0
\(201\) 1053.03 0.369528
\(202\) 4021.33 1.40069
\(203\) 2610.52 0.902575
\(204\) −3483.26 −1.19547
\(205\) 0 0
\(206\) 8526.92 2.88397
\(207\) 1414.79 0.475047
\(208\) 528.503 0.176178
\(209\) −359.344 −0.118930
\(210\) 0 0
\(211\) −1733.25 −0.565507 −0.282753 0.959193i \(-0.591248\pi\)
−0.282753 + 0.959193i \(0.591248\pi\)
\(212\) −5339.84 −1.72991
\(213\) 512.484 0.164858
\(214\) −4830.97 −1.54317
\(215\) 0 0
\(216\) 711.403 0.224096
\(217\) 4515.11 1.41247
\(218\) 7246.68 2.25141
\(219\) −2798.67 −0.863547
\(220\) 0 0
\(221\) −3367.37 −1.02495
\(222\) −829.310 −0.250719
\(223\) 4377.14 1.31442 0.657208 0.753709i \(-0.271737\pi\)
0.657208 + 0.753709i \(0.271737\pi\)
\(224\) 3872.90 1.15522
\(225\) 0 0
\(226\) −4538.74 −1.33589
\(227\) −2248.73 −0.657503 −0.328752 0.944416i \(-0.606628\pi\)
−0.328752 + 0.944416i \(0.606628\pi\)
\(228\) −5853.58 −1.70028
\(229\) 2245.32 0.647926 0.323963 0.946070i \(-0.394985\pi\)
0.323963 + 0.946070i \(0.394985\pi\)
\(230\) 0 0
\(231\) −196.579 −0.0559911
\(232\) −2640.97 −0.747362
\(233\) 708.378 0.199173 0.0995866 0.995029i \(-0.468248\pi\)
0.0995866 + 0.995029i \(0.468248\pi\)
\(234\) 1659.59 0.463635
\(235\) 0 0
\(236\) 5635.92 1.55452
\(237\) −3939.78 −1.07981
\(238\) 10302.3 2.80587
\(239\) 5452.75 1.47577 0.737885 0.674927i \(-0.235825\pi\)
0.737885 + 0.674927i \(0.235825\pi\)
\(240\) 0 0
\(241\) 5720.73 1.52906 0.764532 0.644585i \(-0.222970\pi\)
0.764532 + 0.644585i \(0.222970\pi\)
\(242\) −6165.23 −1.63767
\(243\) 243.000 0.0641500
\(244\) 4899.39 1.28546
\(245\) 0 0
\(246\) −4962.64 −1.28620
\(247\) −5658.83 −1.45774
\(248\) −4567.76 −1.16957
\(249\) −1961.70 −0.499268
\(250\) 0 0
\(251\) 2045.23 0.514319 0.257159 0.966369i \(-0.417213\pi\)
0.257159 + 0.966369i \(0.417213\pi\)
\(252\) −3202.20 −0.800475
\(253\) 395.502 0.0982807
\(254\) −4788.64 −1.18294
\(255\) 0 0
\(256\) −5375.91 −1.31248
\(257\) 407.547 0.0989186 0.0494593 0.998776i \(-0.484250\pi\)
0.0494593 + 0.998776i \(0.484250\pi\)
\(258\) 1532.27 0.369748
\(259\) 1546.93 0.371125
\(260\) 0 0
\(261\) −902.098 −0.213940
\(262\) −12426.2 −2.93012
\(263\) −887.485 −0.208079 −0.104039 0.994573i \(-0.533177\pi\)
−0.104039 + 0.994573i \(0.533177\pi\)
\(264\) 198.871 0.0463625
\(265\) 0 0
\(266\) 17312.8 3.99067
\(267\) 3178.94 0.728645
\(268\) 4795.23 1.09297
\(269\) 6033.48 1.36754 0.683769 0.729699i \(-0.260340\pi\)
0.683769 + 0.729699i \(0.260340\pi\)
\(270\) 0 0
\(271\) 6534.92 1.46483 0.732413 0.680860i \(-0.238394\pi\)
0.732413 + 0.680860i \(0.238394\pi\)
\(272\) −1133.72 −0.252728
\(273\) −3095.66 −0.686293
\(274\) −5675.39 −1.25132
\(275\) 0 0
\(276\) 6442.59 1.40507
\(277\) −5506.53 −1.19442 −0.597212 0.802084i \(-0.703725\pi\)
−0.597212 + 0.802084i \(0.703725\pi\)
\(278\) 4073.75 0.878875
\(279\) −1560.25 −0.334802
\(280\) 0 0
\(281\) −7519.12 −1.59627 −0.798137 0.602476i \(-0.794181\pi\)
−0.798137 + 0.602476i \(0.794181\pi\)
\(282\) −811.382 −0.171337
\(283\) 4480.16 0.941052 0.470526 0.882386i \(-0.344064\pi\)
0.470526 + 0.882386i \(0.344064\pi\)
\(284\) 2333.72 0.487608
\(285\) 0 0
\(286\) 463.934 0.0959197
\(287\) 9256.91 1.90390
\(288\) −1338.33 −0.273825
\(289\) 2310.53 0.470289
\(290\) 0 0
\(291\) 1924.50 0.387685
\(292\) −12744.4 −2.55415
\(293\) −6672.17 −1.33035 −0.665174 0.746688i \(-0.731643\pi\)
−0.665174 + 0.746688i \(0.731643\pi\)
\(294\) 4681.85 0.928746
\(295\) 0 0
\(296\) −1564.97 −0.307304
\(297\) 67.9302 0.0132718
\(298\) −3814.70 −0.741543
\(299\) 6228.25 1.20464
\(300\) 0 0
\(301\) −2858.18 −0.547318
\(302\) 2641.04 0.503227
\(303\) 2592.09 0.491458
\(304\) −1905.21 −0.359445
\(305\) 0 0
\(306\) −3560.07 −0.665084
\(307\) 5433.28 1.01008 0.505039 0.863097i \(-0.331478\pi\)
0.505039 + 0.863097i \(0.331478\pi\)
\(308\) −895.169 −0.165607
\(309\) 5496.32 1.01189
\(310\) 0 0
\(311\) 513.012 0.0935377 0.0467689 0.998906i \(-0.485108\pi\)
0.0467689 + 0.998906i \(0.485108\pi\)
\(312\) 3131.76 0.568273
\(313\) −996.467 −0.179948 −0.0899739 0.995944i \(-0.528678\pi\)
−0.0899739 + 0.995944i \(0.528678\pi\)
\(314\) −7649.72 −1.37484
\(315\) 0 0
\(316\) −17940.7 −3.19381
\(317\) 4280.57 0.758425 0.379213 0.925310i \(-0.376195\pi\)
0.379213 + 0.925310i \(0.376195\pi\)
\(318\) −5457.60 −0.962411
\(319\) −252.180 −0.0442613
\(320\) 0 0
\(321\) −3113.97 −0.541448
\(322\) −19054.9 −3.29780
\(323\) 12139.1 2.09113
\(324\) 1106.56 0.189739
\(325\) 0 0
\(326\) −5958.70 −1.01234
\(327\) 4671.10 0.789946
\(328\) −9364.87 −1.57649
\(329\) 1513.49 0.253621
\(330\) 0 0
\(331\) −2972.75 −0.493647 −0.246823 0.969060i \(-0.579387\pi\)
−0.246823 + 0.969060i \(0.579387\pi\)
\(332\) −8933.07 −1.47670
\(333\) −534.560 −0.0879691
\(334\) −7337.78 −1.20211
\(335\) 0 0
\(336\) −1042.24 −0.169223
\(337\) 1542.78 0.249379 0.124690 0.992196i \(-0.460206\pi\)
0.124690 + 0.992196i \(0.460206\pi\)
\(338\) −2919.30 −0.469790
\(339\) −2925.60 −0.468722
\(340\) 0 0
\(341\) −436.165 −0.0692659
\(342\) −5982.67 −0.945922
\(343\) 200.108 0.0315009
\(344\) 2891.51 0.453197
\(345\) 0 0
\(346\) 10339.8 1.60656
\(347\) −5624.32 −0.870114 −0.435057 0.900403i \(-0.643272\pi\)
−0.435057 + 0.900403i \(0.643272\pi\)
\(348\) −4107.92 −0.632780
\(349\) 3546.44 0.543945 0.271972 0.962305i \(-0.412324\pi\)
0.271972 + 0.962305i \(0.412324\pi\)
\(350\) 0 0
\(351\) 1069.74 0.162674
\(352\) −374.127 −0.0566506
\(353\) −1916.05 −0.288898 −0.144449 0.989512i \(-0.546141\pi\)
−0.144449 + 0.989512i \(0.546141\pi\)
\(354\) 5760.21 0.864835
\(355\) 0 0
\(356\) 14476.1 2.15514
\(357\) 6640.67 0.984487
\(358\) 14032.6 2.07164
\(359\) 2633.09 0.387101 0.193550 0.981090i \(-0.438000\pi\)
0.193550 + 0.981090i \(0.438000\pi\)
\(360\) 0 0
\(361\) 13540.6 1.97413
\(362\) 11508.1 1.67087
\(363\) −3974.01 −0.574605
\(364\) −14096.8 −2.02988
\(365\) 0 0
\(366\) 5007.44 0.715145
\(367\) −6802.59 −0.967554 −0.483777 0.875191i \(-0.660735\pi\)
−0.483777 + 0.875191i \(0.660735\pi\)
\(368\) 2096.92 0.297037
\(369\) −3198.84 −0.451287
\(370\) 0 0
\(371\) 10180.2 1.42460
\(372\) −7104.98 −0.990258
\(373\) −6705.12 −0.930772 −0.465386 0.885108i \(-0.654084\pi\)
−0.465386 + 0.885108i \(0.654084\pi\)
\(374\) −995.212 −0.137597
\(375\) 0 0
\(376\) −1531.14 −0.210007
\(377\) −3971.25 −0.542519
\(378\) −3272.82 −0.445332
\(379\) 3890.69 0.527312 0.263656 0.964617i \(-0.415072\pi\)
0.263656 + 0.964617i \(0.415072\pi\)
\(380\) 0 0
\(381\) −3086.68 −0.415054
\(382\) −2983.06 −0.399547
\(383\) −5642.19 −0.752748 −0.376374 0.926468i \(-0.622829\pi\)
−0.376374 + 0.926468i \(0.622829\pi\)
\(384\) −7584.39 −1.00791
\(385\) 0 0
\(386\) −3066.87 −0.404403
\(387\) 987.679 0.129733
\(388\) 8763.68 1.14667
\(389\) −4678.95 −0.609852 −0.304926 0.952376i \(-0.598632\pi\)
−0.304926 + 0.952376i \(0.598632\pi\)
\(390\) 0 0
\(391\) −13360.6 −1.72806
\(392\) 8835.01 1.13836
\(393\) −8009.71 −1.02808
\(394\) 13042.2 1.66766
\(395\) 0 0
\(396\) 309.337 0.0392544
\(397\) 8379.25 1.05930 0.529650 0.848216i \(-0.322323\pi\)
0.529650 + 0.848216i \(0.322323\pi\)
\(398\) −12640.2 −1.59195
\(399\) 11159.6 1.40020
\(400\) 0 0
\(401\) 6042.30 0.752464 0.376232 0.926525i \(-0.377220\pi\)
0.376232 + 0.926525i \(0.377220\pi\)
\(402\) 4900.98 0.608056
\(403\) −6868.59 −0.849005
\(404\) 11803.7 1.45360
\(405\) 0 0
\(406\) 12149.8 1.48518
\(407\) −149.435 −0.0181996
\(408\) −6718.12 −0.815188
\(409\) 6539.69 0.790628 0.395314 0.918546i \(-0.370636\pi\)
0.395314 + 0.918546i \(0.370636\pi\)
\(410\) 0 0
\(411\) −3658.27 −0.439049
\(412\) 25028.8 2.99292
\(413\) −10744.6 −1.28017
\(414\) 6584.67 0.781688
\(415\) 0 0
\(416\) −5891.63 −0.694377
\(417\) 2625.88 0.308368
\(418\) −1672.44 −0.195698
\(419\) 13109.1 1.52846 0.764228 0.644946i \(-0.223120\pi\)
0.764228 + 0.644946i \(0.223120\pi\)
\(420\) 0 0
\(421\) 3413.16 0.395124 0.197562 0.980290i \(-0.436698\pi\)
0.197562 + 0.980290i \(0.436698\pi\)
\(422\) −8066.83 −0.930538
\(423\) −523.004 −0.0601167
\(424\) −10298.9 −1.17962
\(425\) 0 0
\(426\) 2385.18 0.271273
\(427\) −9340.48 −1.05859
\(428\) −14180.2 −1.60146
\(429\) 299.045 0.0336551
\(430\) 0 0
\(431\) −1511.82 −0.168960 −0.0844801 0.996425i \(-0.526923\pi\)
−0.0844801 + 0.996425i \(0.526923\pi\)
\(432\) 360.160 0.0401116
\(433\) −8624.67 −0.957218 −0.478609 0.878028i \(-0.658859\pi\)
−0.478609 + 0.878028i \(0.658859\pi\)
\(434\) 21014.0 2.32421
\(435\) 0 0
\(436\) 21271.0 2.33646
\(437\) −22452.3 −2.45775
\(438\) −13025.5 −1.42096
\(439\) −3528.80 −0.383645 −0.191823 0.981430i \(-0.561440\pi\)
−0.191823 + 0.981430i \(0.561440\pi\)
\(440\) 0 0
\(441\) 3017.85 0.325867
\(442\) −15672.3 −1.68655
\(443\) −9589.40 −1.02846 −0.514228 0.857653i \(-0.671922\pi\)
−0.514228 + 0.857653i \(0.671922\pi\)
\(444\) −2434.25 −0.260190
\(445\) 0 0
\(446\) 20371.9 2.16286
\(447\) −2458.90 −0.260183
\(448\) 20804.4 2.19401
\(449\) −9820.81 −1.03223 −0.516117 0.856518i \(-0.672623\pi\)
−0.516117 + 0.856518i \(0.672623\pi\)
\(450\) 0 0
\(451\) −894.230 −0.0933650
\(452\) −13322.4 −1.38636
\(453\) 1702.37 0.176566
\(454\) −10465.9 −1.08192
\(455\) 0 0
\(456\) −11289.7 −1.15941
\(457\) 1597.23 0.163491 0.0817455 0.996653i \(-0.473951\pi\)
0.0817455 + 0.996653i \(0.473951\pi\)
\(458\) 10450.1 1.06616
\(459\) −2294.77 −0.233356
\(460\) 0 0
\(461\) −3903.07 −0.394325 −0.197163 0.980371i \(-0.563173\pi\)
−0.197163 + 0.980371i \(0.563173\pi\)
\(462\) −914.910 −0.0921330
\(463\) −15661.9 −1.57208 −0.786039 0.618177i \(-0.787871\pi\)
−0.786039 + 0.618177i \(0.787871\pi\)
\(464\) −1337.04 −0.133772
\(465\) 0 0
\(466\) 3296.90 0.327739
\(467\) −6397.32 −0.633903 −0.316951 0.948442i \(-0.602659\pi\)
−0.316951 + 0.948442i \(0.602659\pi\)
\(468\) 4871.34 0.481149
\(469\) −9141.89 −0.900071
\(470\) 0 0
\(471\) −4930.89 −0.482385
\(472\) 10869.9 1.06002
\(473\) 276.104 0.0268399
\(474\) −18336.4 −1.77683
\(475\) 0 0
\(476\) 30239.9 2.91186
\(477\) −3517.88 −0.337679
\(478\) 25378.0 2.42837
\(479\) −10285.8 −0.981153 −0.490576 0.871398i \(-0.663214\pi\)
−0.490576 + 0.871398i \(0.663214\pi\)
\(480\) 0 0
\(481\) −2353.26 −0.223076
\(482\) 26625.2 2.51607
\(483\) −12282.5 −1.15709
\(484\) −18096.6 −1.69953
\(485\) 0 0
\(486\) 1130.96 0.105559
\(487\) 2083.46 0.193861 0.0969306 0.995291i \(-0.469098\pi\)
0.0969306 + 0.995291i \(0.469098\pi\)
\(488\) 9449.41 0.876546
\(489\) −3840.88 −0.355196
\(490\) 0 0
\(491\) 17704.5 1.62727 0.813636 0.581374i \(-0.197485\pi\)
0.813636 + 0.581374i \(0.197485\pi\)
\(492\) −14566.7 −1.33479
\(493\) 8518.94 0.778243
\(494\) −26337.1 −2.39871
\(495\) 0 0
\(496\) −2312.51 −0.209344
\(497\) −4449.13 −0.401551
\(498\) −9130.07 −0.821542
\(499\) 5018.92 0.450256 0.225128 0.974329i \(-0.427720\pi\)
0.225128 + 0.974329i \(0.427720\pi\)
\(500\) 0 0
\(501\) −4729.82 −0.421782
\(502\) 9518.85 0.846309
\(503\) −10061.9 −0.891924 −0.445962 0.895052i \(-0.647138\pi\)
−0.445962 + 0.895052i \(0.647138\pi\)
\(504\) −6176.05 −0.545839
\(505\) 0 0
\(506\) 1840.73 0.161720
\(507\) −1881.73 −0.164834
\(508\) −14055.9 −1.22762
\(509\) −5317.01 −0.463011 −0.231505 0.972834i \(-0.574365\pi\)
−0.231505 + 0.972834i \(0.574365\pi\)
\(510\) 0 0
\(511\) 24296.7 2.10337
\(512\) −4795.32 −0.413916
\(513\) −3856.33 −0.331893
\(514\) 1896.79 0.162770
\(515\) 0 0
\(516\) 4497.63 0.383716
\(517\) −146.205 −0.0124373
\(518\) 7199.66 0.610685
\(519\) 6664.85 0.563689
\(520\) 0 0
\(521\) −6274.02 −0.527582 −0.263791 0.964580i \(-0.584973\pi\)
−0.263791 + 0.964580i \(0.584973\pi\)
\(522\) −4198.51 −0.352038
\(523\) 14059.6 1.17549 0.587746 0.809045i \(-0.300015\pi\)
0.587746 + 0.809045i \(0.300015\pi\)
\(524\) −36474.1 −3.04080
\(525\) 0 0
\(526\) −4130.50 −0.342392
\(527\) 14734.2 1.21790
\(528\) 100.682 0.00829853
\(529\) 12544.5 1.03103
\(530\) 0 0
\(531\) 3712.94 0.303442
\(532\) 50817.9 4.14142
\(533\) −14082.0 −1.14439
\(534\) 14795.3 1.19898
\(535\) 0 0
\(536\) 9248.51 0.745289
\(537\) 9045.21 0.726871
\(538\) 28080.8 2.25028
\(539\) 843.635 0.0674173
\(540\) 0 0
\(541\) −20130.6 −1.59978 −0.799891 0.600145i \(-0.795110\pi\)
−0.799891 + 0.600145i \(0.795110\pi\)
\(542\) 30414.6 2.41036
\(543\) 7417.97 0.586253
\(544\) 12638.5 0.996084
\(545\) 0 0
\(546\) −14407.7 −1.12929
\(547\) −13220.8 −1.03342 −0.516711 0.856160i \(-0.672844\pi\)
−0.516711 + 0.856160i \(0.672844\pi\)
\(548\) −16658.8 −1.29859
\(549\) 3227.72 0.250921
\(550\) 0 0
\(551\) 14316.0 1.10686
\(552\) 12425.8 0.958108
\(553\) 34203.2 2.63014
\(554\) −25628.3 −1.96542
\(555\) 0 0
\(556\) 11957.6 0.912074
\(557\) −14165.4 −1.07757 −0.538786 0.842442i \(-0.681117\pi\)
−0.538786 + 0.842442i \(0.681117\pi\)
\(558\) −7261.66 −0.550915
\(559\) 4348.00 0.328981
\(560\) 0 0
\(561\) −641.498 −0.0482782
\(562\) −34995.2 −2.62666
\(563\) 11266.9 0.843413 0.421707 0.906732i \(-0.361431\pi\)
0.421707 + 0.906732i \(0.361431\pi\)
\(564\) −2381.63 −0.177810
\(565\) 0 0
\(566\) 20851.4 1.54850
\(567\) −2109.61 −0.156252
\(568\) 4501.02 0.332497
\(569\) −7401.52 −0.545322 −0.272661 0.962110i \(-0.587904\pi\)
−0.272661 + 0.962110i \(0.587904\pi\)
\(570\) 0 0
\(571\) −13070.2 −0.957919 −0.478959 0.877837i \(-0.658986\pi\)
−0.478959 + 0.877837i \(0.658986\pi\)
\(572\) 1361.77 0.0995430
\(573\) −1922.84 −0.140188
\(574\) 43083.2 3.13285
\(575\) 0 0
\(576\) −7189.22 −0.520053
\(577\) 8216.81 0.592843 0.296421 0.955057i \(-0.404207\pi\)
0.296421 + 0.955057i \(0.404207\pi\)
\(578\) 10753.6 0.773857
\(579\) −1976.86 −0.141892
\(580\) 0 0
\(581\) 17030.5 1.21608
\(582\) 8956.94 0.637933
\(583\) −983.417 −0.0698610
\(584\) −24580.0 −1.74166
\(585\) 0 0
\(586\) −31053.3 −2.18908
\(587\) 19965.0 1.40382 0.701911 0.712265i \(-0.252331\pi\)
0.701911 + 0.712265i \(0.252331\pi\)
\(588\) 13742.5 0.963829
\(589\) 24760.7 1.73217
\(590\) 0 0
\(591\) 8406.80 0.585126
\(592\) −792.293 −0.0550051
\(593\) −14429.9 −0.999267 −0.499634 0.866237i \(-0.666532\pi\)
−0.499634 + 0.866237i \(0.666532\pi\)
\(594\) 316.158 0.0218386
\(595\) 0 0
\(596\) −11197.2 −0.769555
\(597\) −8147.67 −0.558563
\(598\) 28987.3 1.98224
\(599\) −7838.61 −0.534686 −0.267343 0.963601i \(-0.586146\pi\)
−0.267343 + 0.963601i \(0.586146\pi\)
\(600\) 0 0
\(601\) −25163.3 −1.70787 −0.853937 0.520376i \(-0.825792\pi\)
−0.853937 + 0.520376i \(0.825792\pi\)
\(602\) −13302.4 −0.900609
\(603\) 3159.09 0.213347
\(604\) 7752.16 0.522237
\(605\) 0 0
\(606\) 12064.0 0.808691
\(607\) −563.698 −0.0376932 −0.0188466 0.999822i \(-0.505999\pi\)
−0.0188466 + 0.999822i \(0.505999\pi\)
\(608\) 21238.8 1.41669
\(609\) 7831.56 0.521102
\(610\) 0 0
\(611\) −2302.39 −0.152446
\(612\) −10449.8 −0.690208
\(613\) −6736.20 −0.443838 −0.221919 0.975065i \(-0.571232\pi\)
−0.221919 + 0.975065i \(0.571232\pi\)
\(614\) 25287.4 1.66208
\(615\) 0 0
\(616\) −1726.50 −0.112927
\(617\) −10198.6 −0.665448 −0.332724 0.943024i \(-0.607968\pi\)
−0.332724 + 0.943024i \(0.607968\pi\)
\(618\) 25580.8 1.66506
\(619\) 7579.25 0.492142 0.246071 0.969252i \(-0.420860\pi\)
0.246071 + 0.969252i \(0.420860\pi\)
\(620\) 0 0
\(621\) 4244.37 0.274269
\(622\) 2387.64 0.153916
\(623\) −27598.0 −1.77478
\(624\) 1585.51 0.101717
\(625\) 0 0
\(626\) −4637.72 −0.296103
\(627\) −1078.03 −0.0686641
\(628\) −22454.0 −1.42677
\(629\) 5048.11 0.320002
\(630\) 0 0
\(631\) 4575.59 0.288671 0.144335 0.989529i \(-0.453896\pi\)
0.144335 + 0.989529i \(0.453896\pi\)
\(632\) −34602.1 −2.17784
\(633\) −5199.75 −0.326495
\(634\) 19922.5 1.24799
\(635\) 0 0
\(636\) −16019.5 −0.998766
\(637\) 13285.3 0.826346
\(638\) −1173.69 −0.0728317
\(639\) 1537.45 0.0951809
\(640\) 0 0
\(641\) −17713.6 −1.09149 −0.545745 0.837951i \(-0.683753\pi\)
−0.545745 + 0.837951i \(0.683753\pi\)
\(642\) −14492.9 −0.890950
\(643\) −13212.2 −0.810323 −0.405162 0.914245i \(-0.632785\pi\)
−0.405162 + 0.914245i \(0.632785\pi\)
\(644\) −55931.4 −3.42237
\(645\) 0 0
\(646\) 56497.2 3.44095
\(647\) 4215.72 0.256163 0.128081 0.991764i \(-0.459118\pi\)
0.128081 + 0.991764i \(0.459118\pi\)
\(648\) 2134.21 0.129382
\(649\) 1037.95 0.0627780
\(650\) 0 0
\(651\) 13545.3 0.815488
\(652\) −17490.4 −1.05058
\(653\) 26191.1 1.56958 0.784792 0.619759i \(-0.212770\pi\)
0.784792 + 0.619759i \(0.212770\pi\)
\(654\) 21740.1 1.29985
\(655\) 0 0
\(656\) −4741.13 −0.282180
\(657\) −8396.02 −0.498569
\(658\) 7044.02 0.417332
\(659\) −12868.3 −0.760665 −0.380332 0.924850i \(-0.624190\pi\)
−0.380332 + 0.924850i \(0.624190\pi\)
\(660\) 0 0
\(661\) −21599.8 −1.27101 −0.635503 0.772099i \(-0.719207\pi\)
−0.635503 + 0.772099i \(0.719207\pi\)
\(662\) −13835.7 −0.812293
\(663\) −10102.1 −0.591755
\(664\) −17229.1 −1.00696
\(665\) 0 0
\(666\) −2487.93 −0.144753
\(667\) −15756.5 −0.914686
\(668\) −21538.4 −1.24752
\(669\) 13131.4 0.758878
\(670\) 0 0
\(671\) 902.302 0.0519121
\(672\) 11618.7 0.666965
\(673\) 10572.3 0.605546 0.302773 0.953063i \(-0.402088\pi\)
0.302773 + 0.953063i \(0.402088\pi\)
\(674\) 7180.37 0.410352
\(675\) 0 0
\(676\) −8568.93 −0.487536
\(677\) −15661.0 −0.889070 −0.444535 0.895762i \(-0.646631\pi\)
−0.444535 + 0.895762i \(0.646631\pi\)
\(678\) −13616.2 −0.771279
\(679\) −16707.6 −0.944296
\(680\) 0 0
\(681\) −6746.18 −0.379610
\(682\) −2029.98 −0.113977
\(683\) 23697.9 1.32763 0.663817 0.747895i \(-0.268935\pi\)
0.663817 + 0.747895i \(0.268935\pi\)
\(684\) −17560.7 −0.981655
\(685\) 0 0
\(686\) 931.335 0.0518346
\(687\) 6735.97 0.374080
\(688\) 1463.88 0.0811189
\(689\) −15486.5 −0.856300
\(690\) 0 0
\(691\) 4426.30 0.243682 0.121841 0.992550i \(-0.461120\pi\)
0.121841 + 0.992550i \(0.461120\pi\)
\(692\) 30350.0 1.66725
\(693\) −589.736 −0.0323265
\(694\) −26176.5 −1.43177
\(695\) 0 0
\(696\) −7922.90 −0.431490
\(697\) 30208.2 1.64163
\(698\) 16505.7 0.895059
\(699\) 2125.13 0.114993
\(700\) 0 0
\(701\) 5192.49 0.279768 0.139884 0.990168i \(-0.455327\pi\)
0.139884 + 0.990168i \(0.455327\pi\)
\(702\) 4978.76 0.267680
\(703\) 8483.30 0.455126
\(704\) −2009.73 −0.107592
\(705\) 0 0
\(706\) −8917.59 −0.475379
\(707\) −22503.2 −1.19706
\(708\) 16907.8 0.897504
\(709\) −1120.17 −0.0593356 −0.0296678 0.999560i \(-0.509445\pi\)
−0.0296678 + 0.999560i \(0.509445\pi\)
\(710\) 0 0
\(711\) −11819.3 −0.623431
\(712\) 27919.9 1.46958
\(713\) −27252.2 −1.43142
\(714\) 30906.8 1.61997
\(715\) 0 0
\(716\) 41189.5 2.14990
\(717\) 16358.2 0.852036
\(718\) 12254.8 0.636973
\(719\) 22785.4 1.18185 0.590927 0.806725i \(-0.298762\pi\)
0.590927 + 0.806725i \(0.298762\pi\)
\(720\) 0 0
\(721\) −47716.3 −2.46470
\(722\) 63020.1 3.24843
\(723\) 17162.2 0.882806
\(724\) 33779.5 1.73399
\(725\) 0 0
\(726\) −18495.7 −0.945509
\(727\) 32028.6 1.63394 0.816970 0.576680i \(-0.195652\pi\)
0.816970 + 0.576680i \(0.195652\pi\)
\(728\) −27188.4 −1.38416
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −9327.13 −0.471924
\(732\) 14698.2 0.742159
\(733\) 21723.1 1.09463 0.547313 0.836928i \(-0.315651\pi\)
0.547313 + 0.836928i \(0.315651\pi\)
\(734\) −31660.4 −1.59211
\(735\) 0 0
\(736\) −23376.0 −1.17072
\(737\) 883.119 0.0441385
\(738\) −14887.9 −0.742590
\(739\) 30000.0 1.49332 0.746662 0.665203i \(-0.231655\pi\)
0.746662 + 0.665203i \(0.231655\pi\)
\(740\) 0 0
\(741\) −16976.5 −0.841629
\(742\) 47380.2 2.34418
\(743\) −5551.03 −0.274088 −0.137044 0.990565i \(-0.543760\pi\)
−0.137044 + 0.990565i \(0.543760\pi\)
\(744\) −13703.3 −0.675251
\(745\) 0 0
\(746\) −31206.7 −1.53158
\(747\) −5885.10 −0.288252
\(748\) −2921.21 −0.142794
\(749\) 27033.9 1.31882
\(750\) 0 0
\(751\) 5337.87 0.259363 0.129682 0.991556i \(-0.458604\pi\)
0.129682 + 0.991556i \(0.458604\pi\)
\(752\) −775.166 −0.0375896
\(753\) 6135.70 0.296942
\(754\) −18482.8 −0.892712
\(755\) 0 0
\(756\) −9606.60 −0.462154
\(757\) −17462.2 −0.838409 −0.419205 0.907892i \(-0.637691\pi\)
−0.419205 + 0.907892i \(0.637691\pi\)
\(758\) 18107.9 0.867690
\(759\) 1186.51 0.0567424
\(760\) 0 0
\(761\) 13611.2 0.648367 0.324183 0.945994i \(-0.394911\pi\)
0.324183 + 0.945994i \(0.394911\pi\)
\(762\) −14365.9 −0.682969
\(763\) −40552.2 −1.92410
\(764\) −8756.10 −0.414639
\(765\) 0 0
\(766\) −26259.7 −1.23864
\(767\) 16345.2 0.769481
\(768\) −16127.7 −0.757759
\(769\) −23956.9 −1.12342 −0.561709 0.827335i \(-0.689856\pi\)
−0.561709 + 0.827335i \(0.689856\pi\)
\(770\) 0 0
\(771\) 1222.64 0.0571107
\(772\) −9002.10 −0.419680
\(773\) −35279.7 −1.64156 −0.820779 0.571246i \(-0.806460\pi\)
−0.820779 + 0.571246i \(0.806460\pi\)
\(774\) 4596.82 0.213474
\(775\) 0 0
\(776\) 16902.4 0.781909
\(777\) 4640.79 0.214269
\(778\) −21776.6 −1.00351
\(779\) 50764.5 2.33482
\(780\) 0 0
\(781\) 429.792 0.0196916
\(782\) −62182.2 −2.84352
\(783\) −2706.29 −0.123519
\(784\) 4472.88 0.203757
\(785\) 0 0
\(786\) −37278.5 −1.69170
\(787\) −3904.96 −0.176870 −0.0884351 0.996082i \(-0.528187\pi\)
−0.0884351 + 0.996082i \(0.528187\pi\)
\(788\) 38282.4 1.73065
\(789\) −2662.46 −0.120134
\(790\) 0 0
\(791\) 25398.6 1.14168
\(792\) 596.614 0.0267674
\(793\) 14209.2 0.636296
\(794\) 38998.4 1.74307
\(795\) 0 0
\(796\) −37102.4 −1.65208
\(797\) 6743.01 0.299686 0.149843 0.988710i \(-0.452123\pi\)
0.149843 + 0.988710i \(0.452123\pi\)
\(798\) 51938.5 2.30402
\(799\) 4938.98 0.218684
\(800\) 0 0
\(801\) 9536.83 0.420683
\(802\) 28121.8 1.23818
\(803\) −2347.09 −0.103147
\(804\) 14385.7 0.631025
\(805\) 0 0
\(806\) −31967.5 −1.39703
\(807\) 18100.4 0.789548
\(808\) 22765.7 0.991205
\(809\) 13083.1 0.568577 0.284288 0.958739i \(-0.408243\pi\)
0.284288 + 0.958739i \(0.408243\pi\)
\(810\) 0 0
\(811\) 7501.06 0.324782 0.162391 0.986727i \(-0.448079\pi\)
0.162391 + 0.986727i \(0.448079\pi\)
\(812\) 35662.9 1.54128
\(813\) 19604.8 0.845718
\(814\) −695.496 −0.0299473
\(815\) 0 0
\(816\) −3401.17 −0.145913
\(817\) −15674.1 −0.671198
\(818\) 30436.8 1.30097
\(819\) −9286.98 −0.396231
\(820\) 0 0
\(821\) −23440.2 −0.996431 −0.498215 0.867053i \(-0.666011\pi\)
−0.498215 + 0.867053i \(0.666011\pi\)
\(822\) −17026.2 −0.722452
\(823\) −6213.94 −0.263189 −0.131594 0.991304i \(-0.542010\pi\)
−0.131594 + 0.991304i \(0.542010\pi\)
\(824\) 48272.8 2.04085
\(825\) 0 0
\(826\) −50007.3 −2.10651
\(827\) −589.434 −0.0247843 −0.0123922 0.999923i \(-0.503945\pi\)
−0.0123922 + 0.999923i \(0.503945\pi\)
\(828\) 19327.8 0.811216
\(829\) 39666.0 1.66183 0.830916 0.556398i \(-0.187817\pi\)
0.830916 + 0.556398i \(0.187817\pi\)
\(830\) 0 0
\(831\) −16519.6 −0.689601
\(832\) −31648.6 −1.31877
\(833\) −28499.0 −1.18539
\(834\) 12221.2 0.507419
\(835\) 0 0
\(836\) −4909.07 −0.203091
\(837\) −4680.75 −0.193298
\(838\) 61012.0 2.51507
\(839\) −6048.43 −0.248885 −0.124443 0.992227i \(-0.539714\pi\)
−0.124443 + 0.992227i \(0.539714\pi\)
\(840\) 0 0
\(841\) −14342.3 −0.588066
\(842\) 15885.4 0.650174
\(843\) −22557.4 −0.921610
\(844\) −23678.3 −0.965689
\(845\) 0 0
\(846\) −2434.15 −0.0989216
\(847\) 34500.4 1.39958
\(848\) −5213.99 −0.211143
\(849\) 13440.5 0.543317
\(850\) 0 0
\(851\) −9336.93 −0.376105
\(852\) 7001.16 0.281521
\(853\) −31051.4 −1.24640 −0.623200 0.782063i \(-0.714168\pi\)
−0.623200 + 0.782063i \(0.714168\pi\)
\(854\) −43472.1 −1.74190
\(855\) 0 0
\(856\) −27349.2 −1.09203
\(857\) 10170.1 0.405371 0.202685 0.979244i \(-0.435033\pi\)
0.202685 + 0.979244i \(0.435033\pi\)
\(858\) 1391.80 0.0553792
\(859\) −41323.5 −1.64137 −0.820687 0.571378i \(-0.806409\pi\)
−0.820687 + 0.571378i \(0.806409\pi\)
\(860\) 0 0
\(861\) 27770.7 1.09921
\(862\) −7036.26 −0.278023
\(863\) 30910.9 1.21926 0.609628 0.792688i \(-0.291319\pi\)
0.609628 + 0.792688i \(0.291319\pi\)
\(864\) −4014.98 −0.158093
\(865\) 0 0
\(866\) −40140.6 −1.57510
\(867\) 6931.58 0.271521
\(868\) 61681.9 2.41200
\(869\) −3304.07 −0.128979
\(870\) 0 0
\(871\) 13907.1 0.541014
\(872\) 41025.1 1.59322
\(873\) 5773.50 0.223830
\(874\) −104497. −4.04422
\(875\) 0 0
\(876\) −38233.3 −1.47464
\(877\) −10750.3 −0.413923 −0.206961 0.978349i \(-0.566357\pi\)
−0.206961 + 0.978349i \(0.566357\pi\)
\(878\) −16423.6 −0.631286
\(879\) −20016.5 −0.768077
\(880\) 0 0
\(881\) −20982.9 −0.802419 −0.401209 0.915986i \(-0.631410\pi\)
−0.401209 + 0.915986i \(0.631410\pi\)
\(882\) 14045.6 0.536212
\(883\) −30286.5 −1.15427 −0.577136 0.816648i \(-0.695830\pi\)
−0.577136 + 0.816648i \(0.695830\pi\)
\(884\) −46002.4 −1.75026
\(885\) 0 0
\(886\) −44630.6 −1.69232
\(887\) 18397.5 0.696423 0.348212 0.937416i \(-0.386789\pi\)
0.348212 + 0.937416i \(0.386789\pi\)
\(888\) −4694.91 −0.177422
\(889\) 26797.0 1.01096
\(890\) 0 0
\(891\) 203.791 0.00766245
\(892\) 59797.0 2.24457
\(893\) 8299.91 0.311026
\(894\) −11444.1 −0.428130
\(895\) 0 0
\(896\) 65843.9 2.45501
\(897\) 18684.7 0.695502
\(898\) −45707.6 −1.69853
\(899\) 17376.5 0.644649
\(900\) 0 0
\(901\) 33221.1 1.22836
\(902\) −4161.89 −0.153632
\(903\) −8574.54 −0.315994
\(904\) −25694.8 −0.945350
\(905\) 0 0
\(906\) 7923.12 0.290539
\(907\) −22143.3 −0.810648 −0.405324 0.914173i \(-0.632841\pi\)
−0.405324 + 0.914173i \(0.632841\pi\)
\(908\) −30720.4 −1.12279
\(909\) 7776.27 0.283743
\(910\) 0 0
\(911\) 42399.1 1.54198 0.770991 0.636846i \(-0.219761\pi\)
0.770991 + 0.636846i \(0.219761\pi\)
\(912\) −5715.63 −0.207526
\(913\) −1645.17 −0.0596354
\(914\) 7433.78 0.269024
\(915\) 0 0
\(916\) 30673.9 1.10643
\(917\) 69536.3 2.50414
\(918\) −10680.2 −0.383986
\(919\) 11557.8 0.414859 0.207429 0.978250i \(-0.433490\pi\)
0.207429 + 0.978250i \(0.433490\pi\)
\(920\) 0 0
\(921\) 16299.9 0.583169
\(922\) −18165.5 −0.648860
\(923\) 6768.22 0.241364
\(924\) −2685.51 −0.0956133
\(925\) 0 0
\(926\) −72893.2 −2.58685
\(927\) 16489.0 0.584216
\(928\) 14905.0 0.527240
\(929\) −16273.9 −0.574735 −0.287367 0.957820i \(-0.592780\pi\)
−0.287367 + 0.957820i \(0.592780\pi\)
\(930\) 0 0
\(931\) −47892.3 −1.68594
\(932\) 9677.31 0.340119
\(933\) 1539.04 0.0540040
\(934\) −29774.2 −1.04308
\(935\) 0 0
\(936\) 9395.29 0.328093
\(937\) −39685.9 −1.38365 −0.691825 0.722065i \(-0.743193\pi\)
−0.691825 + 0.722065i \(0.743193\pi\)
\(938\) −42547.8 −1.48106
\(939\) −2989.40 −0.103893
\(940\) 0 0
\(941\) −21618.8 −0.748940 −0.374470 0.927239i \(-0.622175\pi\)
−0.374470 + 0.927239i \(0.622175\pi\)
\(942\) −22949.2 −0.793762
\(943\) −55872.7 −1.92944
\(944\) 5503.10 0.189736
\(945\) 0 0
\(946\) 1285.03 0.0441649
\(947\) 3083.65 0.105813 0.0529066 0.998599i \(-0.483151\pi\)
0.0529066 + 0.998599i \(0.483151\pi\)
\(948\) −53822.2 −1.84395
\(949\) −36961.2 −1.26429
\(950\) 0 0
\(951\) 12841.7 0.437877
\(952\) 58323.4 1.98558
\(953\) 35749.5 1.21515 0.607576 0.794261i \(-0.292142\pi\)
0.607576 + 0.794261i \(0.292142\pi\)
\(954\) −16372.8 −0.555649
\(955\) 0 0
\(956\) 74491.2 2.52010
\(957\) −756.539 −0.0255543
\(958\) −47872.0 −1.61448
\(959\) 31759.3 1.06941
\(960\) 0 0
\(961\) 263.073 0.00883061
\(962\) −10952.5 −0.367070
\(963\) −9341.91 −0.312605
\(964\) 78152.2 2.61111
\(965\) 0 0
\(966\) −57164.8 −1.90398
\(967\) −23629.2 −0.785796 −0.392898 0.919582i \(-0.628527\pi\)
−0.392898 + 0.919582i \(0.628527\pi\)
\(968\) −34902.7 −1.15890
\(969\) 36417.2 1.20732
\(970\) 0 0
\(971\) 15734.9 0.520039 0.260019 0.965603i \(-0.416271\pi\)
0.260019 + 0.965603i \(0.416271\pi\)
\(972\) 3319.68 0.109546
\(973\) −22796.5 −0.751104
\(974\) 9696.74 0.318997
\(975\) 0 0
\(976\) 4783.93 0.156895
\(977\) −4163.71 −0.136345 −0.0681724 0.997674i \(-0.521717\pi\)
−0.0681724 + 0.997674i \(0.521717\pi\)
\(978\) −17876.1 −0.584473
\(979\) 2666.00 0.0870335
\(980\) 0 0
\(981\) 14013.3 0.456076
\(982\) 82399.4 2.67767
\(983\) −40241.6 −1.30571 −0.652853 0.757485i \(-0.726428\pi\)
−0.652853 + 0.757485i \(0.726428\pi\)
\(984\) −28094.6 −0.910186
\(985\) 0 0
\(986\) 39648.5 1.28060
\(987\) 4540.47 0.146428
\(988\) −77306.5 −2.48932
\(989\) 17251.3 0.554662
\(990\) 0 0
\(991\) 52220.5 1.67390 0.836952 0.547276i \(-0.184335\pi\)
0.836952 + 0.547276i \(0.184335\pi\)
\(992\) 25779.3 0.825095
\(993\) −8918.25 −0.285007
\(994\) −20707.0 −0.660750
\(995\) 0 0
\(996\) −26799.2 −0.852576
\(997\) −57238.6 −1.81822 −0.909109 0.416558i \(-0.863236\pi\)
−0.909109 + 0.416558i \(0.863236\pi\)
\(998\) 23358.9 0.740894
\(999\) −1603.68 −0.0507890
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.1.13 14
5.4 even 2 1875.4.a.f.1.2 14
25.6 even 5 75.4.g.b.61.1 yes 28
25.21 even 5 75.4.g.b.16.1 28
75.56 odd 10 225.4.h.a.136.7 28
75.71 odd 10 225.4.h.a.91.7 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.1 28 25.21 even 5
75.4.g.b.61.1 yes 28 25.6 even 5
225.4.h.a.91.7 28 75.71 odd 10
225.4.h.a.136.7 28 75.56 odd 10
1875.4.a.f.1.2 14 5.4 even 2
1875.4.a.g.1.13 14 1.1 even 1 trivial