Properties

Label 75.4.a.c.1.2
Level $75$
Weight $4$
Character 75.1
Self dual yes
Analytic conductor $4.425$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.42514325043\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.70156 q^{2} -3.00000 q^{3} -5.10469 q^{4} -5.10469 q^{6} -22.2094 q^{7} -22.2984 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.70156 q^{2} -3.00000 q^{3} -5.10469 q^{4} -5.10469 q^{6} -22.2094 q^{7} -22.2984 q^{8} +9.00000 q^{9} -1.79063 q^{11} +15.3141 q^{12} -58.2094 q^{13} -37.7906 q^{14} +2.89531 q^{16} -18.9844 q^{17} +15.3141 q^{18} +104.837 q^{19} +66.6281 q^{21} -3.04686 q^{22} +49.6125 q^{23} +66.8953 q^{24} -99.0469 q^{26} -27.0000 q^{27} +113.372 q^{28} -293.466 q^{29} +64.4187 q^{31} +183.314 q^{32} +5.37188 q^{33} -32.3031 q^{34} -45.9422 q^{36} +19.8844 q^{37} +178.388 q^{38} +174.628 q^{39} -165.581 q^{41} +113.372 q^{42} +247.350 q^{43} +9.14059 q^{44} +84.4187 q^{46} -384.544 q^{47} -8.68594 q^{48} +150.256 q^{49} +56.9531 q^{51} +297.141 q^{52} -463.528 q^{53} -45.9422 q^{54} +495.234 q^{56} -314.512 q^{57} -499.350 q^{58} -73.7906 q^{59} -137.350 q^{61} +109.612 q^{62} -199.884 q^{63} +288.758 q^{64} +9.14059 q^{66} -173.906 q^{67} +96.9093 q^{68} -148.837 q^{69} -594.281 q^{71} -200.686 q^{72} -320.231 q^{73} +33.8345 q^{74} -535.163 q^{76} +39.7687 q^{77} +297.141 q^{78} -770.469 q^{79} +81.0000 q^{81} -281.747 q^{82} -173.925 q^{83} -340.116 q^{84} +420.881 q^{86} +880.397 q^{87} +39.9282 q^{88} +1019.02 q^{89} +1292.79 q^{91} -253.256 q^{92} -193.256 q^{93} -654.325 q^{94} -549.942 q^{96} -384.375 q^{97} +255.670 q^{98} -16.1156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9} - 42 q^{11} - 27 q^{12} - 78 q^{13} - 114 q^{14} + 25 q^{16} - 102 q^{17} - 27 q^{18} + 56 q^{19} + 18 q^{21} + 186 q^{22} + 48 q^{23} + 153 q^{24} - 6 q^{26} - 54 q^{27} + 342 q^{28} - 318 q^{29} + 52 q^{31} + 309 q^{32} + 126 q^{33} + 358 q^{34} + 81 q^{36} - 306 q^{37} + 408 q^{38} + 234 q^{39} - 408 q^{41} + 342 q^{42} - 120 q^{43} - 558 q^{44} + 92 q^{46} - 180 q^{47} - 75 q^{48} + 70 q^{49} + 306 q^{51} + 18 q^{52} - 402 q^{53} + 81 q^{54} + 30 q^{56} - 168 q^{57} - 384 q^{58} - 186 q^{59} + 340 q^{61} + 168 q^{62} - 54 q^{63} - 479 q^{64} - 558 q^{66} - 732 q^{67} - 1074 q^{68} - 144 q^{69} - 36 q^{71} - 459 q^{72} - 1332 q^{73} + 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 18 q^{78} + 380 q^{79} + 162 q^{81} + 858 q^{82} + 984 q^{83} - 1026 q^{84} + 2148 q^{86} + 954 q^{87} + 1194 q^{88} + 1116 q^{89} + 972 q^{91} - 276 q^{92} - 156 q^{93} - 1616 q^{94} - 927 q^{96} + 768 q^{97} + 633 q^{98} - 378 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.70156 0.601593 0.300797 0.953688i \(-0.402747\pi\)
0.300797 + 0.953688i \(0.402747\pi\)
\(3\) −3.00000 −0.577350
\(4\) −5.10469 −0.638086
\(5\) 0 0
\(6\) −5.10469 −0.347330
\(7\) −22.2094 −1.19919 −0.599597 0.800302i \(-0.704672\pi\)
−0.599597 + 0.800302i \(0.704672\pi\)
\(8\) −22.2984 −0.985461
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −1.79063 −0.0490813 −0.0245407 0.999699i \(-0.507812\pi\)
−0.0245407 + 0.999699i \(0.507812\pi\)
\(12\) 15.3141 0.368399
\(13\) −58.2094 −1.24188 −0.620938 0.783860i \(-0.713248\pi\)
−0.620938 + 0.783860i \(0.713248\pi\)
\(14\) −37.7906 −0.721426
\(15\) 0 0
\(16\) 2.89531 0.0452393
\(17\) −18.9844 −0.270846 −0.135423 0.990788i \(-0.543239\pi\)
−0.135423 + 0.990788i \(0.543239\pi\)
\(18\) 15.3141 0.200531
\(19\) 104.837 1.26586 0.632931 0.774208i \(-0.281852\pi\)
0.632931 + 0.774208i \(0.281852\pi\)
\(20\) 0 0
\(21\) 66.6281 0.692355
\(22\) −3.04686 −0.0295270
\(23\) 49.6125 0.449779 0.224890 0.974384i \(-0.427798\pi\)
0.224890 + 0.974384i \(0.427798\pi\)
\(24\) 66.8953 0.568956
\(25\) 0 0
\(26\) −99.0469 −0.747103
\(27\) −27.0000 −0.192450
\(28\) 113.372 0.765188
\(29\) −293.466 −1.87914 −0.939572 0.342350i \(-0.888777\pi\)
−0.939572 + 0.342350i \(0.888777\pi\)
\(30\) 0 0
\(31\) 64.4187 0.373224 0.186612 0.982434i \(-0.440249\pi\)
0.186612 + 0.982434i \(0.440249\pi\)
\(32\) 183.314 1.01268
\(33\) 5.37188 0.0283371
\(34\) −32.3031 −0.162939
\(35\) 0 0
\(36\) −45.9422 −0.212695
\(37\) 19.8844 0.0883505 0.0441752 0.999024i \(-0.485934\pi\)
0.0441752 + 0.999024i \(0.485934\pi\)
\(38\) 178.388 0.761534
\(39\) 174.628 0.716997
\(40\) 0 0
\(41\) −165.581 −0.630718 −0.315359 0.948972i \(-0.602125\pi\)
−0.315359 + 0.948972i \(0.602125\pi\)
\(42\) 113.372 0.416516
\(43\) 247.350 0.877221 0.438611 0.898677i \(-0.355471\pi\)
0.438611 + 0.898677i \(0.355471\pi\)
\(44\) 9.14059 0.0313181
\(45\) 0 0
\(46\) 84.4187 0.270584
\(47\) −384.544 −1.19344 −0.596718 0.802451i \(-0.703529\pi\)
−0.596718 + 0.802451i \(0.703529\pi\)
\(48\) −8.68594 −0.0261189
\(49\) 150.256 0.438065
\(50\) 0 0
\(51\) 56.9531 0.156373
\(52\) 297.141 0.792423
\(53\) −463.528 −1.20133 −0.600665 0.799501i \(-0.705097\pi\)
−0.600665 + 0.799501i \(0.705097\pi\)
\(54\) −45.9422 −0.115777
\(55\) 0 0
\(56\) 495.234 1.18176
\(57\) −314.512 −0.730846
\(58\) −499.350 −1.13048
\(59\) −73.7906 −0.162826 −0.0814129 0.996680i \(-0.525943\pi\)
−0.0814129 + 0.996680i \(0.525943\pi\)
\(60\) 0 0
\(61\) −137.350 −0.288293 −0.144146 0.989556i \(-0.546044\pi\)
−0.144146 + 0.989556i \(0.546044\pi\)
\(62\) 109.612 0.224529
\(63\) −199.884 −0.399731
\(64\) 288.758 0.563980
\(65\) 0 0
\(66\) 9.14059 0.0170474
\(67\) −173.906 −0.317105 −0.158552 0.987351i \(-0.550683\pi\)
−0.158552 + 0.987351i \(0.550683\pi\)
\(68\) 96.9093 0.172823
\(69\) −148.837 −0.259680
\(70\) 0 0
\(71\) −594.281 −0.993355 −0.496677 0.867935i \(-0.665447\pi\)
−0.496677 + 0.867935i \(0.665447\pi\)
\(72\) −200.686 −0.328487
\(73\) −320.231 −0.513428 −0.256714 0.966487i \(-0.582640\pi\)
−0.256714 + 0.966487i \(0.582640\pi\)
\(74\) 33.8345 0.0531510
\(75\) 0 0
\(76\) −535.163 −0.807728
\(77\) 39.7687 0.0588580
\(78\) 297.141 0.431340
\(79\) −770.469 −1.09727 −0.548636 0.836061i \(-0.684853\pi\)
−0.548636 + 0.836061i \(0.684853\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −281.747 −0.379436
\(83\) −173.925 −0.230009 −0.115004 0.993365i \(-0.536688\pi\)
−0.115004 + 0.993365i \(0.536688\pi\)
\(84\) −340.116 −0.441782
\(85\) 0 0
\(86\) 420.881 0.527730
\(87\) 880.397 1.08492
\(88\) 39.9282 0.0483677
\(89\) 1019.02 1.21367 0.606834 0.794829i \(-0.292439\pi\)
0.606834 + 0.794829i \(0.292439\pi\)
\(90\) 0 0
\(91\) 1292.79 1.48925
\(92\) −253.256 −0.286998
\(93\) −193.256 −0.215481
\(94\) −654.325 −0.717962
\(95\) 0 0
\(96\) −549.942 −0.584669
\(97\) −384.375 −0.402344 −0.201172 0.979556i \(-0.564475\pi\)
−0.201172 + 0.979556i \(0.564475\pi\)
\(98\) 255.670 0.263537
\(99\) −16.1156 −0.0163604
\(100\) 0 0
\(101\) 34.4906 0.0339796 0.0169898 0.999856i \(-0.494592\pi\)
0.0169898 + 0.999856i \(0.494592\pi\)
\(102\) 96.9093 0.0940730
\(103\) 1756.30 1.68013 0.840066 0.542484i \(-0.182516\pi\)
0.840066 + 0.542484i \(0.182516\pi\)
\(104\) 1297.98 1.22382
\(105\) 0 0
\(106\) −788.722 −0.722712
\(107\) 1361.74 1.23032 0.615159 0.788403i \(-0.289092\pi\)
0.615159 + 0.788403i \(0.289092\pi\)
\(108\) 137.827 0.122800
\(109\) 321.119 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(110\) 0 0
\(111\) −59.6531 −0.0510092
\(112\) −64.3031 −0.0542506
\(113\) 1582.25 1.31721 0.658607 0.752487i \(-0.271146\pi\)
0.658607 + 0.752487i \(0.271146\pi\)
\(114\) −535.163 −0.439672
\(115\) 0 0
\(116\) 1498.05 1.19906
\(117\) −523.884 −0.413958
\(118\) −125.559 −0.0979549
\(119\) 421.631 0.324797
\(120\) 0 0
\(121\) −1327.79 −0.997591
\(122\) −233.709 −0.173435
\(123\) 496.744 0.364145
\(124\) −328.837 −0.238149
\(125\) 0 0
\(126\) −340.116 −0.240475
\(127\) 1197.14 0.836449 0.418225 0.908344i \(-0.362652\pi\)
0.418225 + 0.908344i \(0.362652\pi\)
\(128\) −975.173 −0.673390
\(129\) −742.050 −0.506464
\(130\) 0 0
\(131\) −321.647 −0.214522 −0.107261 0.994231i \(-0.534208\pi\)
−0.107261 + 0.994231i \(0.534208\pi\)
\(132\) −27.4218 −0.0180815
\(133\) −2328.37 −1.51801
\(134\) −295.912 −0.190768
\(135\) 0 0
\(136\) 423.322 0.266909
\(137\) −354.291 −0.220942 −0.110471 0.993879i \(-0.535236\pi\)
−0.110471 + 0.993879i \(0.535236\pi\)
\(138\) −253.256 −0.156222
\(139\) 77.2562 0.0471424 0.0235712 0.999722i \(-0.492496\pi\)
0.0235712 + 0.999722i \(0.492496\pi\)
\(140\) 0 0
\(141\) 1153.63 0.689030
\(142\) −1011.21 −0.597595
\(143\) 104.231 0.0609529
\(144\) 26.0578 0.0150798
\(145\) 0 0
\(146\) −544.893 −0.308875
\(147\) −450.769 −0.252917
\(148\) −101.503 −0.0563752
\(149\) −1705.38 −0.937651 −0.468826 0.883291i \(-0.655323\pi\)
−0.468826 + 0.883291i \(0.655323\pi\)
\(150\) 0 0
\(151\) 758.281 0.408663 0.204331 0.978902i \(-0.434498\pi\)
0.204331 + 0.978902i \(0.434498\pi\)
\(152\) −2337.71 −1.24746
\(153\) −170.859 −0.0902821
\(154\) 67.6689 0.0354086
\(155\) 0 0
\(156\) −891.422 −0.457506
\(157\) 1769.05 0.899273 0.449636 0.893212i \(-0.351554\pi\)
0.449636 + 0.893212i \(0.351554\pi\)
\(158\) −1311.00 −0.660111
\(159\) 1390.58 0.693588
\(160\) 0 0
\(161\) −1101.86 −0.539372
\(162\) 137.827 0.0668437
\(163\) 881.719 0.423690 0.211845 0.977303i \(-0.432053\pi\)
0.211845 + 0.977303i \(0.432053\pi\)
\(164\) 845.240 0.402452
\(165\) 0 0
\(166\) −295.944 −0.138372
\(167\) 216.900 0.100504 0.0502522 0.998737i \(-0.483997\pi\)
0.0502522 + 0.998737i \(0.483997\pi\)
\(168\) −1485.70 −0.682289
\(169\) 1191.33 0.542254
\(170\) 0 0
\(171\) 943.537 0.421954
\(172\) −1262.64 −0.559742
\(173\) −4125.91 −1.81322 −0.906610 0.421970i \(-0.861339\pi\)
−0.906610 + 0.421970i \(0.861339\pi\)
\(174\) 1498.05 0.652683
\(175\) 0 0
\(176\) −5.18443 −0.00222040
\(177\) 221.372 0.0940075
\(178\) 1733.93 0.730134
\(179\) −3213.14 −1.34168 −0.670842 0.741600i \(-0.734067\pi\)
−0.670842 + 0.741600i \(0.734067\pi\)
\(180\) 0 0
\(181\) 3394.42 1.39395 0.696976 0.717095i \(-0.254529\pi\)
0.696976 + 0.717095i \(0.254529\pi\)
\(182\) 2199.77 0.895921
\(183\) 412.050 0.166446
\(184\) −1106.28 −0.443240
\(185\) 0 0
\(186\) −328.837 −0.129632
\(187\) 33.9939 0.0132935
\(188\) 1962.98 0.761514
\(189\) 599.653 0.230785
\(190\) 0 0
\(191\) −3467.49 −1.31361 −0.656804 0.754062i \(-0.728092\pi\)
−0.656804 + 0.754062i \(0.728092\pi\)
\(192\) −866.273 −0.325614
\(193\) −1792.14 −0.668401 −0.334200 0.942502i \(-0.608466\pi\)
−0.334200 + 0.942502i \(0.608466\pi\)
\(194\) −654.038 −0.242047
\(195\) 0 0
\(196\) −767.011 −0.279523
\(197\) −1678.19 −0.606935 −0.303467 0.952842i \(-0.598144\pi\)
−0.303467 + 0.952842i \(0.598144\pi\)
\(198\) −27.4218 −0.00984233
\(199\) 3108.23 1.10722 0.553610 0.832776i \(-0.313250\pi\)
0.553610 + 0.832776i \(0.313250\pi\)
\(200\) 0 0
\(201\) 521.719 0.183081
\(202\) 58.6878 0.0204419
\(203\) 6517.69 2.25346
\(204\) −290.728 −0.0997795
\(205\) 0 0
\(206\) 2988.46 1.01076
\(207\) 446.512 0.149926
\(208\) −168.534 −0.0561815
\(209\) −187.725 −0.0621301
\(210\) 0 0
\(211\) −4473.27 −1.45949 −0.729745 0.683719i \(-0.760361\pi\)
−0.729745 + 0.683719i \(0.760361\pi\)
\(212\) 2366.17 0.766551
\(213\) 1782.84 0.573514
\(214\) 2317.08 0.740151
\(215\) 0 0
\(216\) 602.058 0.189652
\(217\) −1430.70 −0.447568
\(218\) 546.403 0.169757
\(219\) 960.694 0.296428
\(220\) 0 0
\(221\) 1105.07 0.336357
\(222\) −101.503 −0.0306868
\(223\) 1753.42 0.526535 0.263268 0.964723i \(-0.415200\pi\)
0.263268 + 0.964723i \(0.415200\pi\)
\(224\) −4071.29 −1.21440
\(225\) 0 0
\(226\) 2692.29 0.792427
\(227\) 936.900 0.273939 0.136970 0.990575i \(-0.456264\pi\)
0.136970 + 0.990575i \(0.456264\pi\)
\(228\) 1605.49 0.466342
\(229\) −2582.06 −0.745096 −0.372548 0.928013i \(-0.621516\pi\)
−0.372548 + 0.928013i \(0.621516\pi\)
\(230\) 0 0
\(231\) −119.306 −0.0339817
\(232\) 6543.82 1.85182
\(233\) −2295.01 −0.645284 −0.322642 0.946521i \(-0.604571\pi\)
−0.322642 + 0.946521i \(0.604571\pi\)
\(234\) −891.422 −0.249034
\(235\) 0 0
\(236\) 376.678 0.103897
\(237\) 2311.41 0.633510
\(238\) 717.432 0.195396
\(239\) −2294.01 −0.620866 −0.310433 0.950595i \(-0.600474\pi\)
−0.310433 + 0.950595i \(0.600474\pi\)
\(240\) 0 0
\(241\) 382.287 0.102180 0.0510898 0.998694i \(-0.483731\pi\)
0.0510898 + 0.998694i \(0.483731\pi\)
\(242\) −2259.32 −0.600144
\(243\) −243.000 −0.0641500
\(244\) 701.128 0.183956
\(245\) 0 0
\(246\) 845.240 0.219067
\(247\) −6102.52 −1.57204
\(248\) −1436.44 −0.367798
\(249\) 521.775 0.132796
\(250\) 0 0
\(251\) −2259.98 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(252\) 1020.35 0.255063
\(253\) −88.8375 −0.0220758
\(254\) 2037.01 0.503202
\(255\) 0 0
\(256\) −3969.38 −0.969087
\(257\) 92.7843 0.0225203 0.0112602 0.999937i \(-0.496416\pi\)
0.0112602 + 0.999937i \(0.496416\pi\)
\(258\) −1262.64 −0.304685
\(259\) −441.619 −0.105949
\(260\) 0 0
\(261\) −2641.19 −0.626382
\(262\) −547.302 −0.129055
\(263\) 568.312 0.133246 0.0666229 0.997778i \(-0.478778\pi\)
0.0666229 + 0.997778i \(0.478778\pi\)
\(264\) −119.785 −0.0279251
\(265\) 0 0
\(266\) −3961.87 −0.913226
\(267\) −3057.07 −0.700711
\(268\) 887.737 0.202340
\(269\) 7582.41 1.71862 0.859309 0.511458i \(-0.170894\pi\)
0.859309 + 0.511458i \(0.170894\pi\)
\(270\) 0 0
\(271\) 7943.69 1.78061 0.890304 0.455366i \(-0.150492\pi\)
0.890304 + 0.455366i \(0.150492\pi\)
\(272\) −54.9657 −0.0122529
\(273\) −3878.38 −0.859818
\(274\) −602.847 −0.132917
\(275\) 0 0
\(276\) 759.769 0.165698
\(277\) −6823.00 −1.47998 −0.739990 0.672618i \(-0.765170\pi\)
−0.739990 + 0.672618i \(0.765170\pi\)
\(278\) 131.456 0.0283605
\(279\) 579.769 0.124408
\(280\) 0 0
\(281\) 3315.86 0.703942 0.351971 0.936011i \(-0.385512\pi\)
0.351971 + 0.936011i \(0.385512\pi\)
\(282\) 1962.98 0.414516
\(283\) −6602.76 −1.38690 −0.693451 0.720504i \(-0.743910\pi\)
−0.693451 + 0.720504i \(0.743910\pi\)
\(284\) 3033.62 0.633846
\(285\) 0 0
\(286\) 177.356 0.0366688
\(287\) 3677.46 0.756353
\(288\) 1649.83 0.337559
\(289\) −4552.59 −0.926642
\(290\) 0 0
\(291\) 1153.12 0.232293
\(292\) 1634.68 0.327611
\(293\) −5814.14 −1.15927 −0.579634 0.814877i \(-0.696805\pi\)
−0.579634 + 0.814877i \(0.696805\pi\)
\(294\) −767.011 −0.152153
\(295\) 0 0
\(296\) −443.390 −0.0870660
\(297\) 48.3469 0.00944570
\(298\) −2901.81 −0.564084
\(299\) −2887.91 −0.558570
\(300\) 0 0
\(301\) −5493.49 −1.05196
\(302\) 1290.26 0.245849
\(303\) −103.472 −0.0196181
\(304\) 303.537 0.0572667
\(305\) 0 0
\(306\) −290.728 −0.0543131
\(307\) 8124.86 1.51046 0.755229 0.655462i \(-0.227526\pi\)
0.755229 + 0.655462i \(0.227526\pi\)
\(308\) −203.007 −0.0375564
\(309\) −5268.91 −0.970025
\(310\) 0 0
\(311\) 7336.26 1.33762 0.668812 0.743432i \(-0.266803\pi\)
0.668812 + 0.743432i \(0.266803\pi\)
\(312\) −3893.93 −0.706572
\(313\) 2202.66 0.397768 0.198884 0.980023i \(-0.436268\pi\)
0.198884 + 0.980023i \(0.436268\pi\)
\(314\) 3010.15 0.540996
\(315\) 0 0
\(316\) 3933.00 0.700154
\(317\) −10008.9 −1.77336 −0.886679 0.462386i \(-0.846993\pi\)
−0.886679 + 0.462386i \(0.846993\pi\)
\(318\) 2366.17 0.417258
\(319\) 525.488 0.0922309
\(320\) 0 0
\(321\) −4085.21 −0.710325
\(322\) −1874.89 −0.324483
\(323\) −1990.27 −0.342854
\(324\) −413.480 −0.0708984
\(325\) 0 0
\(326\) 1500.30 0.254889
\(327\) −963.356 −0.162917
\(328\) 3692.20 0.621548
\(329\) 8540.47 1.43116
\(330\) 0 0
\(331\) −8695.94 −1.44402 −0.722012 0.691881i \(-0.756782\pi\)
−0.722012 + 0.691881i \(0.756782\pi\)
\(332\) 887.832 0.146765
\(333\) 178.959 0.0294502
\(334\) 369.069 0.0604627
\(335\) 0 0
\(336\) 192.909 0.0313216
\(337\) 7400.61 1.19625 0.598126 0.801402i \(-0.295912\pi\)
0.598126 + 0.801402i \(0.295912\pi\)
\(338\) 2027.12 0.326216
\(339\) −4746.74 −0.760494
\(340\) 0 0
\(341\) −115.350 −0.0183183
\(342\) 1605.49 0.253845
\(343\) 4280.72 0.673869
\(344\) −5515.52 −0.864467
\(345\) 0 0
\(346\) −7020.49 −1.09082
\(347\) −7841.44 −1.21311 −0.606557 0.795040i \(-0.707450\pi\)
−0.606557 + 0.795040i \(0.707450\pi\)
\(348\) −4494.15 −0.692275
\(349\) −4961.26 −0.760946 −0.380473 0.924792i \(-0.624239\pi\)
−0.380473 + 0.924792i \(0.624239\pi\)
\(350\) 0 0
\(351\) 1571.65 0.238999
\(352\) −328.247 −0.0497035
\(353\) 12163.0 1.83392 0.916959 0.398981i \(-0.130636\pi\)
0.916959 + 0.398981i \(0.130636\pi\)
\(354\) 376.678 0.0565543
\(355\) 0 0
\(356\) −5201.80 −0.774424
\(357\) −1264.89 −0.187522
\(358\) −5467.36 −0.807148
\(359\) 5193.79 0.763559 0.381779 0.924253i \(-0.375311\pi\)
0.381779 + 0.924253i \(0.375311\pi\)
\(360\) 0 0
\(361\) 4131.90 0.602406
\(362\) 5775.81 0.838591
\(363\) 3983.38 0.575959
\(364\) −6599.31 −0.950268
\(365\) 0 0
\(366\) 701.128 0.100133
\(367\) 6086.09 0.865644 0.432822 0.901479i \(-0.357518\pi\)
0.432822 + 0.901479i \(0.357518\pi\)
\(368\) 143.644 0.0203477
\(369\) −1490.23 −0.210239
\(370\) 0 0
\(371\) 10294.7 1.44063
\(372\) 986.512 0.137495
\(373\) 10581.9 1.46893 0.734466 0.678646i \(-0.237433\pi\)
0.734466 + 0.678646i \(0.237433\pi\)
\(374\) 57.8428 0.00799727
\(375\) 0 0
\(376\) 8574.72 1.17608
\(377\) 17082.4 2.33366
\(378\) 1020.35 0.138839
\(379\) 11655.2 1.57964 0.789822 0.613336i \(-0.210173\pi\)
0.789822 + 0.613336i \(0.210173\pi\)
\(380\) 0 0
\(381\) −3591.42 −0.482924
\(382\) −5900.16 −0.790257
\(383\) −6364.97 −0.849177 −0.424588 0.905387i \(-0.639581\pi\)
−0.424588 + 0.905387i \(0.639581\pi\)
\(384\) 2925.52 0.388782
\(385\) 0 0
\(386\) −3049.44 −0.402105
\(387\) 2226.15 0.292407
\(388\) 1962.11 0.256730
\(389\) 6134.33 0.799545 0.399773 0.916614i \(-0.369089\pi\)
0.399773 + 0.916614i \(0.369089\pi\)
\(390\) 0 0
\(391\) −941.862 −0.121821
\(392\) −3350.48 −0.431696
\(393\) 964.941 0.123855
\(394\) −2855.55 −0.365128
\(395\) 0 0
\(396\) 82.2653 0.0104394
\(397\) −9746.46 −1.23214 −0.616072 0.787690i \(-0.711277\pi\)
−0.616072 + 0.787690i \(0.711277\pi\)
\(398\) 5288.85 0.666096
\(399\) 6985.12 0.876425
\(400\) 0 0
\(401\) −1306.44 −0.162695 −0.0813474 0.996686i \(-0.525922\pi\)
−0.0813474 + 0.996686i \(0.525922\pi\)
\(402\) 887.737 0.110140
\(403\) −3749.77 −0.463498
\(404\) −176.063 −0.0216819
\(405\) 0 0
\(406\) 11090.2 1.35566
\(407\) −35.6055 −0.00433636
\(408\) −1269.97 −0.154100
\(409\) −3876.93 −0.468709 −0.234354 0.972151i \(-0.575298\pi\)
−0.234354 + 0.972151i \(0.575298\pi\)
\(410\) 0 0
\(411\) 1062.87 0.127561
\(412\) −8965.38 −1.07207
\(413\) 1638.84 0.195260
\(414\) 759.769 0.0901947
\(415\) 0 0
\(416\) −10670.6 −1.25762
\(417\) −231.769 −0.0272177
\(418\) −319.426 −0.0373771
\(419\) −16022.5 −1.86814 −0.934071 0.357088i \(-0.883770\pi\)
−0.934071 + 0.357088i \(0.883770\pi\)
\(420\) 0 0
\(421\) −8119.73 −0.939980 −0.469990 0.882672i \(-0.655742\pi\)
−0.469990 + 0.882672i \(0.655742\pi\)
\(422\) −7611.54 −0.878019
\(423\) −3460.89 −0.397812
\(424\) 10336.0 1.18386
\(425\) 0 0
\(426\) 3033.62 0.345022
\(427\) 3050.46 0.345719
\(428\) −6951.24 −0.785049
\(429\) −312.694 −0.0351911
\(430\) 0 0
\(431\) −5713.99 −0.638592 −0.319296 0.947655i \(-0.603446\pi\)
−0.319296 + 0.947655i \(0.603446\pi\)
\(432\) −78.1735 −0.00870630
\(433\) −6251.34 −0.693811 −0.346906 0.937900i \(-0.612768\pi\)
−0.346906 + 0.937900i \(0.612768\pi\)
\(434\) −2434.42 −0.269254
\(435\) 0 0
\(436\) −1639.21 −0.180055
\(437\) 5201.25 0.569358
\(438\) 1634.68 0.178329
\(439\) −4230.97 −0.459984 −0.229992 0.973192i \(-0.573870\pi\)
−0.229992 + 0.973192i \(0.573870\pi\)
\(440\) 0 0
\(441\) 1352.31 0.146022
\(442\) 1880.34 0.202350
\(443\) −6314.29 −0.677203 −0.338601 0.940930i \(-0.609954\pi\)
−0.338601 + 0.940930i \(0.609954\pi\)
\(444\) 304.510 0.0325482
\(445\) 0 0
\(446\) 2983.55 0.316760
\(447\) 5116.13 0.541353
\(448\) −6413.13 −0.676321
\(449\) −9349.71 −0.982717 −0.491358 0.870957i \(-0.663499\pi\)
−0.491358 + 0.870957i \(0.663499\pi\)
\(450\) 0 0
\(451\) 296.494 0.0309565
\(452\) −8076.87 −0.840496
\(453\) −2274.84 −0.235941
\(454\) 1594.19 0.164800
\(455\) 0 0
\(456\) 7013.14 0.720220
\(457\) 9547.46 0.977268 0.488634 0.872489i \(-0.337495\pi\)
0.488634 + 0.872489i \(0.337495\pi\)
\(458\) −4393.53 −0.448245
\(459\) 512.578 0.0521244
\(460\) 0 0
\(461\) 6237.23 0.630145 0.315073 0.949068i \(-0.397971\pi\)
0.315073 + 0.949068i \(0.397971\pi\)
\(462\) −203.007 −0.0204431
\(463\) −6469.98 −0.649428 −0.324714 0.945812i \(-0.605268\pi\)
−0.324714 + 0.945812i \(0.605268\pi\)
\(464\) −849.675 −0.0850111
\(465\) 0 0
\(466\) −3905.10 −0.388198
\(467\) −7206.64 −0.714097 −0.357049 0.934086i \(-0.616217\pi\)
−0.357049 + 0.934086i \(0.616217\pi\)
\(468\) 2674.27 0.264141
\(469\) 3862.35 0.380270
\(470\) 0 0
\(471\) −5307.16 −0.519195
\(472\) 1645.42 0.160458
\(473\) −442.912 −0.0430552
\(474\) 3933.00 0.381115
\(475\) 0 0
\(476\) −2152.29 −0.207248
\(477\) −4171.75 −0.400443
\(478\) −3903.39 −0.373509
\(479\) −10851.8 −1.03514 −0.517571 0.855640i \(-0.673164\pi\)
−0.517571 + 0.855640i \(0.673164\pi\)
\(480\) 0 0
\(481\) −1157.46 −0.109720
\(482\) 650.485 0.0614705
\(483\) 3305.59 0.311407
\(484\) 6777.97 0.636549
\(485\) 0 0
\(486\) −413.480 −0.0385922
\(487\) −12757.1 −1.18702 −0.593510 0.804827i \(-0.702258\pi\)
−0.593510 + 0.804827i \(0.702258\pi\)
\(488\) 3062.69 0.284101
\(489\) −2645.16 −0.244618
\(490\) 0 0
\(491\) −7016.52 −0.644911 −0.322455 0.946585i \(-0.604508\pi\)
−0.322455 + 0.946585i \(0.604508\pi\)
\(492\) −2535.72 −0.232356
\(493\) 5571.26 0.508960
\(494\) −10383.8 −0.945729
\(495\) 0 0
\(496\) 186.512 0.0168844
\(497\) 13198.6 1.19122
\(498\) 887.832 0.0798890
\(499\) 11372.3 1.02023 0.510113 0.860107i \(-0.329604\pi\)
0.510113 + 0.860107i \(0.329604\pi\)
\(500\) 0 0
\(501\) −650.700 −0.0580262
\(502\) −3845.50 −0.341899
\(503\) 5587.37 0.495285 0.247643 0.968851i \(-0.420344\pi\)
0.247643 + 0.968851i \(0.420344\pi\)
\(504\) 4457.11 0.393919
\(505\) 0 0
\(506\) −151.163 −0.0132806
\(507\) −3573.99 −0.313070
\(508\) −6111.03 −0.533726
\(509\) 16256.7 1.41565 0.707825 0.706388i \(-0.249676\pi\)
0.707825 + 0.706388i \(0.249676\pi\)
\(510\) 0 0
\(511\) 7112.14 0.615699
\(512\) 1047.24 0.0903943
\(513\) −2830.61 −0.243615
\(514\) 157.878 0.0135481
\(515\) 0 0
\(516\) 3787.93 0.323167
\(517\) 688.574 0.0585754
\(518\) −751.442 −0.0637384
\(519\) 12377.7 1.04686
\(520\) 0 0
\(521\) 19748.4 1.66064 0.830320 0.557286i \(-0.188157\pi\)
0.830320 + 0.557286i \(0.188157\pi\)
\(522\) −4494.15 −0.376827
\(523\) 7843.44 0.655774 0.327887 0.944717i \(-0.393663\pi\)
0.327887 + 0.944717i \(0.393663\pi\)
\(524\) 1641.91 0.136884
\(525\) 0 0
\(526\) 967.019 0.0801597
\(527\) −1222.95 −0.101086
\(528\) 15.5533 0.00128195
\(529\) −9705.60 −0.797699
\(530\) 0 0
\(531\) −664.116 −0.0542753
\(532\) 11885.6 0.968622
\(533\) 9638.38 0.783273
\(534\) −5201.80 −0.421543
\(535\) 0 0
\(536\) 3877.84 0.312495
\(537\) 9639.42 0.774622
\(538\) 12902.0 1.03391
\(539\) −269.053 −0.0215008
\(540\) 0 0
\(541\) 7383.29 0.586751 0.293376 0.955997i \(-0.405221\pi\)
0.293376 + 0.955997i \(0.405221\pi\)
\(542\) 13516.7 1.07120
\(543\) −10183.3 −0.804798
\(544\) −3480.10 −0.274280
\(545\) 0 0
\(546\) −6599.31 −0.517260
\(547\) 3354.90 0.262240 0.131120 0.991367i \(-0.458143\pi\)
0.131120 + 0.991367i \(0.458143\pi\)
\(548\) 1808.54 0.140980
\(549\) −1236.15 −0.0960976
\(550\) 0 0
\(551\) −30766.2 −2.37874
\(552\) 3318.84 0.255905
\(553\) 17111.6 1.31584
\(554\) −11609.8 −0.890346
\(555\) 0 0
\(556\) −394.369 −0.0300809
\(557\) −20771.8 −1.58012 −0.790061 0.613028i \(-0.789951\pi\)
−0.790061 + 0.613028i \(0.789951\pi\)
\(558\) 986.512 0.0748430
\(559\) −14398.1 −1.08940
\(560\) 0 0
\(561\) −101.982 −0.00767500
\(562\) 5642.15 0.423487
\(563\) −7194.86 −0.538592 −0.269296 0.963057i \(-0.586791\pi\)
−0.269296 + 0.963057i \(0.586791\pi\)
\(564\) −5888.93 −0.439660
\(565\) 0 0
\(566\) −11235.0 −0.834350
\(567\) −1798.96 −0.133244
\(568\) 13251.5 0.978913
\(569\) 11549.5 0.850931 0.425466 0.904975i \(-0.360110\pi\)
0.425466 + 0.904975i \(0.360110\pi\)
\(570\) 0 0
\(571\) 1482.54 0.108655 0.0543277 0.998523i \(-0.482698\pi\)
0.0543277 + 0.998523i \(0.482698\pi\)
\(572\) −532.068 −0.0388932
\(573\) 10402.5 0.758412
\(574\) 6257.42 0.455017
\(575\) 0 0
\(576\) 2598.82 0.187993
\(577\) 15264.0 1.10130 0.550649 0.834737i \(-0.314380\pi\)
0.550649 + 0.834737i \(0.314380\pi\)
\(578\) −7746.52 −0.557462
\(579\) 5376.43 0.385901
\(580\) 0 0
\(581\) 3862.76 0.275825
\(582\) 1962.11 0.139746
\(583\) 830.006 0.0589628
\(584\) 7140.66 0.505963
\(585\) 0 0
\(586\) −9893.12 −0.697408
\(587\) −1736.89 −0.122128 −0.0610639 0.998134i \(-0.519449\pi\)
−0.0610639 + 0.998134i \(0.519449\pi\)
\(588\) 2301.03 0.161383
\(589\) 6753.50 0.472450
\(590\) 0 0
\(591\) 5034.57 0.350414
\(592\) 57.5714 0.00399691
\(593\) 11764.8 0.814707 0.407353 0.913271i \(-0.366452\pi\)
0.407353 + 0.913271i \(0.366452\pi\)
\(594\) 82.2653 0.00568247
\(595\) 0 0
\(596\) 8705.42 0.598302
\(597\) −9324.69 −0.639253
\(598\) −4913.96 −0.336032
\(599\) −9451.99 −0.644737 −0.322369 0.946614i \(-0.604479\pi\)
−0.322369 + 0.946614i \(0.604479\pi\)
\(600\) 0 0
\(601\) −3131.93 −0.212569 −0.106285 0.994336i \(-0.533895\pi\)
−0.106285 + 0.994336i \(0.533895\pi\)
\(602\) −9347.51 −0.632851
\(603\) −1565.16 −0.105702
\(604\) −3870.79 −0.260762
\(605\) 0 0
\(606\) −176.063 −0.0118021
\(607\) −22700.8 −1.51795 −0.758975 0.651120i \(-0.774300\pi\)
−0.758975 + 0.651120i \(0.774300\pi\)
\(608\) 19218.2 1.28191
\(609\) −19553.1 −1.30103
\(610\) 0 0
\(611\) 22384.0 1.48210
\(612\) 872.184 0.0576077
\(613\) −28911.6 −1.90494 −0.952471 0.304629i \(-0.901468\pi\)
−0.952471 + 0.304629i \(0.901468\pi\)
\(614\) 13825.0 0.908681
\(615\) 0 0
\(616\) −886.780 −0.0580023
\(617\) 5566.87 0.363231 0.181616 0.983370i \(-0.441867\pi\)
0.181616 + 0.983370i \(0.441867\pi\)
\(618\) −8965.38 −0.583560
\(619\) 4150.32 0.269492 0.134746 0.990880i \(-0.456978\pi\)
0.134746 + 0.990880i \(0.456978\pi\)
\(620\) 0 0
\(621\) −1339.54 −0.0865600
\(622\) 12483.1 0.804705
\(623\) −22631.9 −1.45542
\(624\) 505.603 0.0324364
\(625\) 0 0
\(626\) 3747.96 0.239295
\(627\) 563.175 0.0358709
\(628\) −9030.46 −0.573813
\(629\) −377.492 −0.0239294
\(630\) 0 0
\(631\) −4090.09 −0.258041 −0.129021 0.991642i \(-0.541183\pi\)
−0.129021 + 0.991642i \(0.541183\pi\)
\(632\) 17180.2 1.08132
\(633\) 13419.8 0.842637
\(634\) −17030.7 −1.06684
\(635\) 0 0
\(636\) −7098.50 −0.442569
\(637\) −8746.32 −0.544022
\(638\) 894.150 0.0554855
\(639\) −5348.53 −0.331118
\(640\) 0 0
\(641\) 3909.35 0.240890 0.120445 0.992720i \(-0.461568\pi\)
0.120445 + 0.992720i \(0.461568\pi\)
\(642\) −6951.24 −0.427327
\(643\) −30539.5 −1.87303 −0.936516 0.350624i \(-0.885969\pi\)
−0.936516 + 0.350624i \(0.885969\pi\)
\(644\) 5624.66 0.344166
\(645\) 0 0
\(646\) −3386.58 −0.206259
\(647\) 12707.7 0.772167 0.386083 0.922464i \(-0.373828\pi\)
0.386083 + 0.922464i \(0.373828\pi\)
\(648\) −1806.17 −0.109496
\(649\) 132.132 0.00799170
\(650\) 0 0
\(651\) 4292.10 0.258403
\(652\) −4500.90 −0.270351
\(653\) −12777.6 −0.765737 −0.382869 0.923803i \(-0.625064\pi\)
−0.382869 + 0.923803i \(0.625064\pi\)
\(654\) −1639.21 −0.0980095
\(655\) 0 0
\(656\) −479.410 −0.0285332
\(657\) −2882.08 −0.171143
\(658\) 14532.1 0.860976
\(659\) 23563.5 1.39287 0.696435 0.717620i \(-0.254768\pi\)
0.696435 + 0.717620i \(0.254768\pi\)
\(660\) 0 0
\(661\) −4361.31 −0.256634 −0.128317 0.991733i \(-0.540958\pi\)
−0.128317 + 0.991733i \(0.540958\pi\)
\(662\) −14796.7 −0.868715
\(663\) −3315.21 −0.194196
\(664\) 3878.25 0.226665
\(665\) 0 0
\(666\) 304.510 0.0177170
\(667\) −14559.6 −0.845200
\(668\) −1107.21 −0.0641304
\(669\) −5260.25 −0.303995
\(670\) 0 0
\(671\) 245.943 0.0141498
\(672\) 12213.9 0.701131
\(673\) 8203.52 0.469870 0.234935 0.972011i \(-0.424512\pi\)
0.234935 + 0.972011i \(0.424512\pi\)
\(674\) 12592.6 0.719657
\(675\) 0 0
\(676\) −6081.37 −0.346004
\(677\) −28057.1 −1.59279 −0.796397 0.604774i \(-0.793263\pi\)
−0.796397 + 0.604774i \(0.793263\pi\)
\(678\) −8076.87 −0.457508
\(679\) 8536.73 0.482488
\(680\) 0 0
\(681\) −2810.70 −0.158159
\(682\) −196.275 −0.0110202
\(683\) 3344.62 0.187377 0.0936885 0.995602i \(-0.470134\pi\)
0.0936885 + 0.995602i \(0.470134\pi\)
\(684\) −4816.46 −0.269243
\(685\) 0 0
\(686\) 7283.91 0.405395
\(687\) 7746.17 0.430182
\(688\) 716.156 0.0396849
\(689\) 26981.7 1.49190
\(690\) 0 0
\(691\) 12964.8 0.713757 0.356879 0.934151i \(-0.383841\pi\)
0.356879 + 0.934151i \(0.383841\pi\)
\(692\) 21061.5 1.15699
\(693\) 357.918 0.0196193
\(694\) −13342.7 −0.729801
\(695\) 0 0
\(696\) −19631.5 −1.06915
\(697\) 3143.46 0.170828
\(698\) −8441.90 −0.457780
\(699\) 6885.03 0.372555
\(700\) 0 0
\(701\) −16162.1 −0.870806 −0.435403 0.900236i \(-0.643394\pi\)
−0.435403 + 0.900236i \(0.643394\pi\)
\(702\) 2674.27 0.143780
\(703\) 2084.63 0.111839
\(704\) −517.058 −0.0276809
\(705\) 0 0
\(706\) 20696.2 1.10327
\(707\) −766.014 −0.0407481
\(708\) −1130.03 −0.0599849
\(709\) −14244.4 −0.754529 −0.377265 0.926105i \(-0.623135\pi\)
−0.377265 + 0.926105i \(0.623135\pi\)
\(710\) 0 0
\(711\) −6934.22 −0.365757
\(712\) −22722.7 −1.19602
\(713\) 3195.97 0.167868
\(714\) −2152.29 −0.112812
\(715\) 0 0
\(716\) 16402.1 0.856109
\(717\) 6882.02 0.358457
\(718\) 8837.55 0.459352
\(719\) −27638.5 −1.43358 −0.716790 0.697289i \(-0.754389\pi\)
−0.716790 + 0.697289i \(0.754389\pi\)
\(720\) 0 0
\(721\) −39006.4 −2.01480
\(722\) 7030.68 0.362403
\(723\) −1146.86 −0.0589934
\(724\) −17327.4 −0.889460
\(725\) 0 0
\(726\) 6777.97 0.346493
\(727\) 2525.52 0.128840 0.0644199 0.997923i \(-0.479480\pi\)
0.0644199 + 0.997923i \(0.479480\pi\)
\(728\) −28827.3 −1.46760
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4695.79 −0.237592
\(732\) −2103.39 −0.106207
\(733\) 8400.27 0.423289 0.211645 0.977347i \(-0.432118\pi\)
0.211645 + 0.977347i \(0.432118\pi\)
\(734\) 10355.9 0.520765
\(735\) 0 0
\(736\) 9094.67 0.455481
\(737\) 311.401 0.0155639
\(738\) −2535.72 −0.126479
\(739\) −19689.1 −0.980074 −0.490037 0.871702i \(-0.663017\pi\)
−0.490037 + 0.871702i \(0.663017\pi\)
\(740\) 0 0
\(741\) 18307.6 0.907619
\(742\) 17517.0 0.866671
\(743\) 22526.6 1.11227 0.556137 0.831091i \(-0.312283\pi\)
0.556137 + 0.831091i \(0.312283\pi\)
\(744\) 4309.31 0.212348
\(745\) 0 0
\(746\) 18005.8 0.883699
\(747\) −1565.32 −0.0766696
\(748\) −173.528 −0.00848239
\(749\) −30243.3 −1.47539
\(750\) 0 0
\(751\) 34691.1 1.68562 0.842808 0.538215i \(-0.180901\pi\)
0.842808 + 0.538215i \(0.180901\pi\)
\(752\) −1113.37 −0.0539902
\(753\) 6779.95 0.328121
\(754\) 29066.8 1.40392
\(755\) 0 0
\(756\) −3061.04 −0.147261
\(757\) −6619.98 −0.317843 −0.158922 0.987291i \(-0.550802\pi\)
−0.158922 + 0.987291i \(0.550802\pi\)
\(758\) 19832.0 0.950303
\(759\) 266.512 0.0127454
\(760\) 0 0
\(761\) −29368.7 −1.39897 −0.699483 0.714649i \(-0.746586\pi\)
−0.699483 + 0.714649i \(0.746586\pi\)
\(762\) −6111.03 −0.290524
\(763\) −7131.84 −0.338388
\(764\) 17700.5 0.838194
\(765\) 0 0
\(766\) −10830.4 −0.510859
\(767\) 4295.31 0.202209
\(768\) 11908.1 0.559503
\(769\) 32677.4 1.53235 0.766174 0.642633i \(-0.222158\pi\)
0.766174 + 0.642633i \(0.222158\pi\)
\(770\) 0 0
\(771\) −278.353 −0.0130021
\(772\) 9148.33 0.426497
\(773\) 28047.5 1.30504 0.652522 0.757770i \(-0.273711\pi\)
0.652522 + 0.757770i \(0.273711\pi\)
\(774\) 3787.93 0.175910
\(775\) 0 0
\(776\) 8570.96 0.396494
\(777\) 1324.86 0.0611699
\(778\) 10438.0 0.481001
\(779\) −17359.1 −0.798402
\(780\) 0 0
\(781\) 1064.14 0.0487552
\(782\) −1602.64 −0.0732867
\(783\) 7923.57 0.361642
\(784\) 435.039 0.0198177
\(785\) 0 0
\(786\) 1641.91 0.0745100
\(787\) 22172.1 1.00426 0.502128 0.864793i \(-0.332551\pi\)
0.502128 + 0.864793i \(0.332551\pi\)
\(788\) 8566.64 0.387276
\(789\) −1704.94 −0.0769295
\(790\) 0 0
\(791\) −35140.7 −1.57960
\(792\) 359.354 0.0161226
\(793\) 7995.06 0.358024
\(794\) −16584.2 −0.741249
\(795\) 0 0
\(796\) −15866.5 −0.706501
\(797\) 24170.3 1.07422 0.537112 0.843511i \(-0.319515\pi\)
0.537112 + 0.843511i \(0.319515\pi\)
\(798\) 11885.6 0.527251
\(799\) 7300.32 0.323238
\(800\) 0 0
\(801\) 9171.22 0.404556
\(802\) −2222.99 −0.0978761
\(803\) 573.415 0.0251997
\(804\) −2663.21 −0.116821
\(805\) 0 0
\(806\) −6380.47 −0.278837
\(807\) −22747.2 −0.992244
\(808\) −769.085 −0.0334856
\(809\) 15304.2 0.665102 0.332551 0.943085i \(-0.392091\pi\)
0.332551 + 0.943085i \(0.392091\pi\)
\(810\) 0 0
\(811\) −27002.2 −1.16914 −0.584572 0.811342i \(-0.698738\pi\)
−0.584572 + 0.811342i \(0.698738\pi\)
\(812\) −33270.7 −1.43790
\(813\) −23831.1 −1.02803
\(814\) −60.5849 −0.00260872
\(815\) 0 0
\(816\) 164.897 0.00707421
\(817\) 25931.5 1.11044
\(818\) −6596.84 −0.281972
\(819\) 11635.1 0.496416
\(820\) 0 0
\(821\) 25061.4 1.06535 0.532673 0.846321i \(-0.321188\pi\)
0.532673 + 0.846321i \(0.321188\pi\)
\(822\) 1808.54 0.0767398
\(823\) 24896.4 1.05448 0.527238 0.849718i \(-0.323228\pi\)
0.527238 + 0.849718i \(0.323228\pi\)
\(824\) −39162.8 −1.65571
\(825\) 0 0
\(826\) 2788.59 0.117467
\(827\) 20063.2 0.843612 0.421806 0.906686i \(-0.361396\pi\)
0.421806 + 0.906686i \(0.361396\pi\)
\(828\) −2279.31 −0.0956659
\(829\) −13884.2 −0.581687 −0.290844 0.956771i \(-0.593936\pi\)
−0.290844 + 0.956771i \(0.593936\pi\)
\(830\) 0 0
\(831\) 20469.0 0.854467
\(832\) −16808.4 −0.700393
\(833\) −2852.52 −0.118648
\(834\) −394.369 −0.0163740
\(835\) 0 0
\(836\) 958.277 0.0396444
\(837\) −1739.31 −0.0718270
\(838\) −27263.3 −1.12386
\(839\) −13678.1 −0.562838 −0.281419 0.959585i \(-0.590805\pi\)
−0.281419 + 0.959585i \(0.590805\pi\)
\(840\) 0 0
\(841\) 61733.1 2.53118
\(842\) −13816.2 −0.565485
\(843\) −9947.59 −0.406421
\(844\) 22834.6 0.931280
\(845\) 0 0
\(846\) −5888.93 −0.239321
\(847\) 29489.5 1.19630
\(848\) −1342.06 −0.0543473
\(849\) 19808.3 0.800728
\(850\) 0 0
\(851\) 986.512 0.0397382
\(852\) −9100.86 −0.365951
\(853\) −29802.9 −1.19629 −0.598143 0.801390i \(-0.704094\pi\)
−0.598143 + 0.801390i \(0.704094\pi\)
\(854\) 5190.54 0.207982
\(855\) 0 0
\(856\) −30364.6 −1.21243
\(857\) −22045.2 −0.878706 −0.439353 0.898314i \(-0.644792\pi\)
−0.439353 + 0.898314i \(0.644792\pi\)
\(858\) −532.068 −0.0211708
\(859\) 33609.5 1.33497 0.667487 0.744622i \(-0.267370\pi\)
0.667487 + 0.744622i \(0.267370\pi\)
\(860\) 0 0
\(861\) −11032.4 −0.436681
\(862\) −9722.70 −0.384172
\(863\) −33775.6 −1.33226 −0.666128 0.745838i \(-0.732049\pi\)
−0.666128 + 0.745838i \(0.732049\pi\)
\(864\) −4949.48 −0.194890
\(865\) 0 0
\(866\) −10637.0 −0.417392
\(867\) 13657.8 0.534997
\(868\) 7303.27 0.285587
\(869\) 1379.62 0.0538556
\(870\) 0 0
\(871\) 10123.0 0.393805
\(872\) −7160.44 −0.278077
\(873\) −3459.37 −0.134115
\(874\) 8850.25 0.342522
\(875\) 0 0
\(876\) −4904.04 −0.189146
\(877\) −12637.0 −0.486570 −0.243285 0.969955i \(-0.578225\pi\)
−0.243285 + 0.969955i \(0.578225\pi\)
\(878\) −7199.25 −0.276723
\(879\) 17442.4 0.669304
\(880\) 0 0
\(881\) −6579.45 −0.251609 −0.125804 0.992055i \(-0.540151\pi\)
−0.125804 + 0.992055i \(0.540151\pi\)
\(882\) 2301.03 0.0878456
\(883\) 50442.1 1.92244 0.961219 0.275786i \(-0.0889382\pi\)
0.961219 + 0.275786i \(0.0889382\pi\)
\(884\) −5641.03 −0.214625
\(885\) 0 0
\(886\) −10744.2 −0.407401
\(887\) −984.823 −0.0372797 −0.0186399 0.999826i \(-0.505934\pi\)
−0.0186399 + 0.999826i \(0.505934\pi\)
\(888\) 1330.17 0.0502676
\(889\) −26587.7 −1.00306
\(890\) 0 0
\(891\) −145.041 −0.00545348
\(892\) −8950.64 −0.335975
\(893\) −40314.6 −1.51072
\(894\) 8705.42 0.325674
\(895\) 0 0
\(896\) 21658.0 0.807525
\(897\) 8663.74 0.322490
\(898\) −15909.1 −0.591196
\(899\) −18904.7 −0.701342
\(900\) 0 0
\(901\) 8799.79 0.325376
\(902\) 504.503 0.0186232
\(903\) 16480.5 0.607348
\(904\) −35281.6 −1.29806
\(905\) 0 0
\(906\) −3870.79 −0.141941
\(907\) −43679.9 −1.59908 −0.799541 0.600612i \(-0.794924\pi\)
−0.799541 + 0.600612i \(0.794924\pi\)
\(908\) −4782.58 −0.174797
\(909\) 310.415 0.0113265
\(910\) 0 0
\(911\) −10364.3 −0.376930 −0.188465 0.982080i \(-0.560351\pi\)
−0.188465 + 0.982080i \(0.560351\pi\)
\(912\) −910.612 −0.0330629
\(913\) 311.435 0.0112891
\(914\) 16245.6 0.587918
\(915\) 0 0
\(916\) 13180.6 0.475435
\(917\) 7143.58 0.257254
\(918\) 872.184 0.0313577
\(919\) 11451.9 0.411059 0.205530 0.978651i \(-0.434108\pi\)
0.205530 + 0.978651i \(0.434108\pi\)
\(920\) 0 0
\(921\) −24374.6 −0.872063
\(922\) 10613.0 0.379091
\(923\) 34592.7 1.23362
\(924\) 609.020 0.0216832
\(925\) 0 0
\(926\) −11009.1 −0.390692
\(927\) 15806.7 0.560044
\(928\) −53796.4 −1.90297
\(929\) −27701.8 −0.978326 −0.489163 0.872192i \(-0.662698\pi\)
−0.489163 + 0.872192i \(0.662698\pi\)
\(930\) 0 0
\(931\) 15752.5 0.554529
\(932\) 11715.3 0.411746
\(933\) −22008.8 −0.772277
\(934\) −12262.5 −0.429596
\(935\) 0 0
\(936\) 11681.8 0.407940
\(937\) 5878.01 0.204937 0.102469 0.994736i \(-0.467326\pi\)
0.102469 + 0.994736i \(0.467326\pi\)
\(938\) 6572.03 0.228768
\(939\) −6607.97 −0.229652
\(940\) 0 0
\(941\) −28786.0 −0.997234 −0.498617 0.866823i \(-0.666159\pi\)
−0.498617 + 0.866823i \(0.666159\pi\)
\(942\) −9030.46 −0.312344
\(943\) −8214.90 −0.283684
\(944\) −213.647 −0.00736612
\(945\) 0 0
\(946\) −753.642 −0.0259017
\(947\) 1695.04 0.0581641 0.0290821 0.999577i \(-0.490742\pi\)
0.0290821 + 0.999577i \(0.490742\pi\)
\(948\) −11799.0 −0.404234
\(949\) 18640.5 0.637613
\(950\) 0 0
\(951\) 30026.6 1.02385
\(952\) −9401.72 −0.320075
\(953\) −31929.4 −1.08530 −0.542651 0.839958i \(-0.682580\pi\)
−0.542651 + 0.839958i \(0.682580\pi\)
\(954\) −7098.50 −0.240904
\(955\) 0 0
\(956\) 11710.2 0.396166
\(957\) −1576.46 −0.0532495
\(958\) −18465.1 −0.622735
\(959\) 7868.57 0.264952
\(960\) 0 0
\(961\) −25641.2 −0.860704
\(962\) −1969.48 −0.0660069
\(963\) 12255.6 0.410106
\(964\) −1951.46 −0.0651994
\(965\) 0 0
\(966\) 5624.66 0.187340
\(967\) 10897.1 0.362385 0.181193 0.983448i \(-0.442004\pi\)
0.181193 + 0.983448i \(0.442004\pi\)
\(968\) 29607.7 0.983087
\(969\) 5970.82 0.197947
\(970\) 0 0
\(971\) 7041.97 0.232737 0.116368 0.993206i \(-0.462875\pi\)
0.116368 + 0.993206i \(0.462875\pi\)
\(972\) 1240.44 0.0409332
\(973\) −1715.81 −0.0565328
\(974\) −21707.0 −0.714103
\(975\) 0 0
\(976\) −397.671 −0.0130422
\(977\) 37607.6 1.23150 0.615749 0.787943i \(-0.288854\pi\)
0.615749 + 0.787943i \(0.288854\pi\)
\(978\) −4500.90 −0.147160
\(979\) −1824.69 −0.0595684
\(980\) 0 0
\(981\) 2890.07 0.0940599
\(982\) −11939.0 −0.387974
\(983\) 25297.7 0.820826 0.410413 0.911900i \(-0.365385\pi\)
0.410413 + 0.911900i \(0.365385\pi\)
\(984\) −11076.6 −0.358851
\(985\) 0 0
\(986\) 9479.85 0.306187
\(987\) −25621.4 −0.826281
\(988\) 31151.5 1.00310
\(989\) 12271.6 0.394556
\(990\) 0 0
\(991\) −41686.5 −1.33624 −0.668120 0.744053i \(-0.732901\pi\)
−0.668120 + 0.744053i \(0.732901\pi\)
\(992\) 11808.9 0.377955
\(993\) 26087.8 0.833708
\(994\) 22458.3 0.716633
\(995\) 0 0
\(996\) −2663.50 −0.0847351
\(997\) −25465.9 −0.808939 −0.404470 0.914551i \(-0.632544\pi\)
−0.404470 + 0.914551i \(0.632544\pi\)
\(998\) 19350.6 0.613761
\(999\) −536.878 −0.0170031
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 75.4.a.c.1.2 2
3.2 odd 2 225.4.a.o.1.1 2
4.3 odd 2 1200.4.a.bt.1.2 2
5.2 odd 4 15.4.b.a.4.3 yes 4
5.3 odd 4 15.4.b.a.4.2 4
5.4 even 2 75.4.a.f.1.1 2
15.2 even 4 45.4.b.b.19.2 4
15.8 even 4 45.4.b.b.19.3 4
15.14 odd 2 225.4.a.i.1.2 2
20.3 even 4 240.4.f.f.49.3 4
20.7 even 4 240.4.f.f.49.1 4
20.19 odd 2 1200.4.a.bn.1.1 2
40.3 even 4 960.4.f.p.769.2 4
40.13 odd 4 960.4.f.q.769.4 4
40.27 even 4 960.4.f.p.769.4 4
40.37 odd 4 960.4.f.q.769.2 4
60.23 odd 4 720.4.f.j.289.3 4
60.47 odd 4 720.4.f.j.289.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.2 4 5.3 odd 4
15.4.b.a.4.3 yes 4 5.2 odd 4
45.4.b.b.19.2 4 15.2 even 4
45.4.b.b.19.3 4 15.8 even 4
75.4.a.c.1.2 2 1.1 even 1 trivial
75.4.a.f.1.1 2 5.4 even 2
225.4.a.i.1.2 2 15.14 odd 2
225.4.a.o.1.1 2 3.2 odd 2
240.4.f.f.49.1 4 20.7 even 4
240.4.f.f.49.3 4 20.3 even 4
720.4.f.j.289.3 4 60.23 odd 4
720.4.f.j.289.4 4 60.47 odd 4
960.4.f.p.769.2 4 40.3 even 4
960.4.f.p.769.4 4 40.27 even 4
960.4.f.q.769.2 4 40.37 odd 4
960.4.f.q.769.4 4 40.13 odd 4
1200.4.a.bn.1.1 2 20.19 odd 2
1200.4.a.bt.1.2 2 4.3 odd 2