Properties

Label 1875.4.a.g.1.2
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 1.2
Root \(-4.79301\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.79301 q^{2} +3.00000 q^{3} +14.9730 q^{4} -14.3790 q^{6} +0.140520 q^{7} -33.4216 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.79301 q^{2} +3.00000 q^{3} +14.9730 q^{4} -14.3790 q^{6} +0.140520 q^{7} -33.4216 q^{8} +9.00000 q^{9} +47.3504 q^{11} +44.9190 q^{12} +34.3424 q^{13} -0.673515 q^{14} +40.4065 q^{16} +16.5164 q^{17} -43.1371 q^{18} +91.0020 q^{19} +0.421560 q^{21} -226.951 q^{22} -161.582 q^{23} -100.265 q^{24} -164.604 q^{26} +27.0000 q^{27} +2.10400 q^{28} -12.3327 q^{29} -333.038 q^{31} +73.7042 q^{32} +142.051 q^{33} -79.1633 q^{34} +134.757 q^{36} -281.402 q^{37} -436.174 q^{38} +103.027 q^{39} +125.610 q^{41} -2.02054 q^{42} -529.821 q^{43} +708.977 q^{44} +774.463 q^{46} +75.0713 q^{47} +121.220 q^{48} -342.980 q^{49} +49.5492 q^{51} +514.208 q^{52} -249.426 q^{53} -129.411 q^{54} -4.69641 q^{56} +273.006 q^{57} +59.1108 q^{58} -268.564 q^{59} +369.848 q^{61} +1596.26 q^{62} +1.26468 q^{63} -676.517 q^{64} -680.854 q^{66} -176.506 q^{67} +247.300 q^{68} -484.745 q^{69} +722.104 q^{71} -300.795 q^{72} -2.44271 q^{73} +1348.77 q^{74} +1362.57 q^{76} +6.65368 q^{77} -493.811 q^{78} +607.662 q^{79} +81.0000 q^{81} -602.049 q^{82} -1268.55 q^{83} +6.31201 q^{84} +2539.44 q^{86} -36.9981 q^{87} -1582.53 q^{88} -1337.53 q^{89} +4.82579 q^{91} -2419.36 q^{92} -999.114 q^{93} -359.818 q^{94} +221.113 q^{96} -1431.21 q^{97} +1643.91 q^{98} +426.154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.79301 −1.69459 −0.847293 0.531125i \(-0.821769\pi\)
−0.847293 + 0.531125i \(0.821769\pi\)
\(3\) 3.00000 0.577350
\(4\) 14.9730 1.87162
\(5\) 0 0
\(6\) −14.3790 −0.978370
\(7\) 0.140520 0.00758737 0.00379368 0.999993i \(-0.498792\pi\)
0.00379368 + 0.999993i \(0.498792\pi\)
\(8\) −33.4216 −1.47704
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 47.3504 1.29788 0.648940 0.760839i \(-0.275212\pi\)
0.648940 + 0.760839i \(0.275212\pi\)
\(12\) 44.9190 1.08058
\(13\) 34.3424 0.732682 0.366341 0.930481i \(-0.380610\pi\)
0.366341 + 0.930481i \(0.380610\pi\)
\(14\) −0.673515 −0.0128575
\(15\) 0 0
\(16\) 40.4065 0.631352
\(17\) 16.5164 0.235636 0.117818 0.993035i \(-0.462410\pi\)
0.117818 + 0.993035i \(0.462410\pi\)
\(18\) −43.1371 −0.564862
\(19\) 91.0020 1.09881 0.549403 0.835558i \(-0.314855\pi\)
0.549403 + 0.835558i \(0.314855\pi\)
\(20\) 0 0
\(21\) 0.421560 0.00438057
\(22\) −226.951 −2.19937
\(23\) −161.582 −1.46487 −0.732436 0.680835i \(-0.761617\pi\)
−0.732436 + 0.680835i \(0.761617\pi\)
\(24\) −100.265 −0.852771
\(25\) 0 0
\(26\) −164.604 −1.24159
\(27\) 27.0000 0.192450
\(28\) 2.10400 0.0142007
\(29\) −12.3327 −0.0789698 −0.0394849 0.999220i \(-0.512572\pi\)
−0.0394849 + 0.999220i \(0.512572\pi\)
\(30\) 0 0
\(31\) −333.038 −1.92953 −0.964765 0.263115i \(-0.915250\pi\)
−0.964765 + 0.263115i \(0.915250\pi\)
\(32\) 73.7042 0.407162
\(33\) 142.051 0.749332
\(34\) −79.1633 −0.399306
\(35\) 0 0
\(36\) 134.757 0.623875
\(37\) −281.402 −1.25033 −0.625166 0.780492i \(-0.714969\pi\)
−0.625166 + 0.780492i \(0.714969\pi\)
\(38\) −436.174 −1.86202
\(39\) 103.027 0.423014
\(40\) 0 0
\(41\) 125.610 0.478462 0.239231 0.970963i \(-0.423105\pi\)
0.239231 + 0.970963i \(0.423105\pi\)
\(42\) −2.02054 −0.00742325
\(43\) −529.821 −1.87900 −0.939500 0.342549i \(-0.888710\pi\)
−0.939500 + 0.342549i \(0.888710\pi\)
\(44\) 708.977 2.42914
\(45\) 0 0
\(46\) 774.463 2.48235
\(47\) 75.0713 0.232985 0.116492 0.993192i \(-0.462835\pi\)
0.116492 + 0.993192i \(0.462835\pi\)
\(48\) 121.220 0.364511
\(49\) −342.980 −0.999942
\(50\) 0 0
\(51\) 49.5492 0.136045
\(52\) 514.208 1.37130
\(53\) −249.426 −0.646438 −0.323219 0.946324i \(-0.604765\pi\)
−0.323219 + 0.946324i \(0.604765\pi\)
\(54\) −129.411 −0.326123
\(55\) 0 0
\(56\) −4.69641 −0.0112069
\(57\) 273.006 0.634395
\(58\) 59.1108 0.133821
\(59\) −268.564 −0.592611 −0.296305 0.955093i \(-0.595755\pi\)
−0.296305 + 0.955093i \(0.595755\pi\)
\(60\) 0 0
\(61\) 369.848 0.776297 0.388149 0.921597i \(-0.373115\pi\)
0.388149 + 0.921597i \(0.373115\pi\)
\(62\) 1596.26 3.26975
\(63\) 1.26468 0.00252912
\(64\) −676.517 −1.32132
\(65\) 0 0
\(66\) −680.854 −1.26981
\(67\) −176.506 −0.321846 −0.160923 0.986967i \(-0.551447\pi\)
−0.160923 + 0.986967i \(0.551447\pi\)
\(68\) 247.300 0.441022
\(69\) −484.745 −0.845745
\(70\) 0 0
\(71\) 722.104 1.20701 0.603507 0.797358i \(-0.293770\pi\)
0.603507 + 0.797358i \(0.293770\pi\)
\(72\) −300.795 −0.492347
\(73\) −2.44271 −0.00391641 −0.00195820 0.999998i \(-0.500623\pi\)
−0.00195820 + 0.999998i \(0.500623\pi\)
\(74\) 1348.77 2.11879
\(75\) 0 0
\(76\) 1362.57 2.05655
\(77\) 6.65368 0.00984750
\(78\) −493.811 −0.716834
\(79\) 607.662 0.865409 0.432705 0.901536i \(-0.357559\pi\)
0.432705 + 0.901536i \(0.357559\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −602.049 −0.810795
\(83\) −1268.55 −1.67761 −0.838807 0.544429i \(-0.816747\pi\)
−0.838807 + 0.544429i \(0.816747\pi\)
\(84\) 6.31201 0.00819878
\(85\) 0 0
\(86\) 2539.44 3.18413
\(87\) −36.9981 −0.0455933
\(88\) −1582.53 −1.91702
\(89\) −1337.53 −1.59301 −0.796507 0.604629i \(-0.793321\pi\)
−0.796507 + 0.604629i \(0.793321\pi\)
\(90\) 0 0
\(91\) 4.82579 0.00555913
\(92\) −2419.36 −2.74169
\(93\) −999.114 −1.11401
\(94\) −359.818 −0.394812
\(95\) 0 0
\(96\) 221.113 0.235075
\(97\) −1431.21 −1.49812 −0.749059 0.662503i \(-0.769494\pi\)
−0.749059 + 0.662503i \(0.769494\pi\)
\(98\) 1643.91 1.69449
\(99\) 426.154 0.432627
\(100\) 0 0
\(101\) −374.102 −0.368559 −0.184280 0.982874i \(-0.558995\pi\)
−0.184280 + 0.982874i \(0.558995\pi\)
\(102\) −237.490 −0.230539
\(103\) −2017.87 −1.93035 −0.965177 0.261597i \(-0.915751\pi\)
−0.965177 + 0.261597i \(0.915751\pi\)
\(104\) −1147.78 −1.08220
\(105\) 0 0
\(106\) 1195.50 1.09545
\(107\) 1679.71 1.51761 0.758804 0.651319i \(-0.225784\pi\)
0.758804 + 0.651319i \(0.225784\pi\)
\(108\) 404.271 0.360194
\(109\) 20.2283 0.0177754 0.00888772 0.999961i \(-0.497171\pi\)
0.00888772 + 0.999961i \(0.497171\pi\)
\(110\) 0 0
\(111\) −844.207 −0.721879
\(112\) 5.67792 0.00479030
\(113\) −642.520 −0.534896 −0.267448 0.963572i \(-0.586180\pi\)
−0.267448 + 0.963572i \(0.586180\pi\)
\(114\) −1308.52 −1.07504
\(115\) 0 0
\(116\) −184.657 −0.147802
\(117\) 309.081 0.244227
\(118\) 1287.23 1.00423
\(119\) 2.32088 0.00178786
\(120\) 0 0
\(121\) 911.063 0.684495
\(122\) −1772.69 −1.31550
\(123\) 376.829 0.276240
\(124\) −4986.58 −3.61135
\(125\) 0 0
\(126\) −6.06163 −0.00428582
\(127\) −1185.91 −0.828603 −0.414302 0.910140i \(-0.635974\pi\)
−0.414302 + 0.910140i \(0.635974\pi\)
\(128\) 2652.92 1.83193
\(129\) −1589.46 −1.08484
\(130\) 0 0
\(131\) −223.094 −0.148793 −0.0743963 0.997229i \(-0.523703\pi\)
−0.0743963 + 0.997229i \(0.523703\pi\)
\(132\) 2126.93 1.40247
\(133\) 12.7876 0.00833704
\(134\) 845.997 0.545396
\(135\) 0 0
\(136\) −552.005 −0.348045
\(137\) −1579.55 −0.985035 −0.492517 0.870303i \(-0.663923\pi\)
−0.492517 + 0.870303i \(0.663923\pi\)
\(138\) 2323.39 1.43319
\(139\) 1565.25 0.955128 0.477564 0.878597i \(-0.341520\pi\)
0.477564 + 0.878597i \(0.341520\pi\)
\(140\) 0 0
\(141\) 225.214 0.134514
\(142\) −3461.05 −2.04539
\(143\) 1626.13 0.950934
\(144\) 363.659 0.210451
\(145\) 0 0
\(146\) 11.7080 0.00663669
\(147\) −1028.94 −0.577317
\(148\) −4213.43 −2.34015
\(149\) 192.635 0.105915 0.0529574 0.998597i \(-0.483135\pi\)
0.0529574 + 0.998597i \(0.483135\pi\)
\(150\) 0 0
\(151\) 1052.83 0.567406 0.283703 0.958912i \(-0.408437\pi\)
0.283703 + 0.958912i \(0.408437\pi\)
\(152\) −3041.44 −1.62298
\(153\) 148.648 0.0785454
\(154\) −31.8912 −0.0166874
\(155\) 0 0
\(156\) 1542.62 0.791723
\(157\) 2045.26 1.03968 0.519838 0.854265i \(-0.325992\pi\)
0.519838 + 0.854265i \(0.325992\pi\)
\(158\) −2912.53 −1.46651
\(159\) −748.277 −0.373221
\(160\) 0 0
\(161\) −22.7054 −0.0111145
\(162\) −388.234 −0.188287
\(163\) 848.013 0.407494 0.203747 0.979024i \(-0.434688\pi\)
0.203747 + 0.979024i \(0.434688\pi\)
\(164\) 1880.75 0.895500
\(165\) 0 0
\(166\) 6080.20 2.84286
\(167\) 1972.45 0.913968 0.456984 0.889475i \(-0.348930\pi\)
0.456984 + 0.889475i \(0.348930\pi\)
\(168\) −14.0892 −0.00647028
\(169\) −1017.60 −0.463177
\(170\) 0 0
\(171\) 819.018 0.366268
\(172\) −7933.01 −3.51678
\(173\) 2707.65 1.18993 0.594967 0.803750i \(-0.297165\pi\)
0.594967 + 0.803750i \(0.297165\pi\)
\(174\) 177.332 0.0772617
\(175\) 0 0
\(176\) 1913.26 0.819419
\(177\) −805.692 −0.342144
\(178\) 6410.82 2.69950
\(179\) 3132.25 1.30791 0.653953 0.756535i \(-0.273109\pi\)
0.653953 + 0.756535i \(0.273109\pi\)
\(180\) 0 0
\(181\) 317.051 0.130200 0.0651000 0.997879i \(-0.479263\pi\)
0.0651000 + 0.997879i \(0.479263\pi\)
\(182\) −23.1301 −0.00942042
\(183\) 1109.54 0.448195
\(184\) 5400.32 2.16368
\(185\) 0 0
\(186\) 4788.77 1.88779
\(187\) 782.059 0.305828
\(188\) 1124.04 0.436059
\(189\) 3.79404 0.00146019
\(190\) 0 0
\(191\) 1679.46 0.636237 0.318119 0.948051i \(-0.396949\pi\)
0.318119 + 0.948051i \(0.396949\pi\)
\(192\) −2029.55 −0.762866
\(193\) −1449.62 −0.540653 −0.270326 0.962769i \(-0.587132\pi\)
−0.270326 + 0.962769i \(0.587132\pi\)
\(194\) 6859.82 2.53869
\(195\) 0 0
\(196\) −5135.44 −1.87152
\(197\) −1060.62 −0.383586 −0.191793 0.981435i \(-0.561430\pi\)
−0.191793 + 0.981435i \(0.561430\pi\)
\(198\) −2042.56 −0.733124
\(199\) −183.077 −0.0652159 −0.0326080 0.999468i \(-0.510381\pi\)
−0.0326080 + 0.999468i \(0.510381\pi\)
\(200\) 0 0
\(201\) −529.519 −0.185818
\(202\) 1793.07 0.624556
\(203\) −1.73299 −0.000599173 0
\(204\) 741.900 0.254624
\(205\) 0 0
\(206\) 9671.67 3.27115
\(207\) −1454.23 −0.488291
\(208\) 1387.66 0.462580
\(209\) 4308.98 1.42612
\(210\) 0 0
\(211\) −4904.73 −1.60026 −0.800131 0.599825i \(-0.795237\pi\)
−0.800131 + 0.599825i \(0.795237\pi\)
\(212\) −3734.65 −1.20989
\(213\) 2166.31 0.696870
\(214\) −8050.90 −2.57172
\(215\) 0 0
\(216\) −902.384 −0.284257
\(217\) −46.7985 −0.0146400
\(218\) −96.9547 −0.0301220
\(219\) −7.32813 −0.00226114
\(220\) 0 0
\(221\) 567.212 0.172646
\(222\) 4046.30 1.22329
\(223\) −4837.42 −1.45263 −0.726317 0.687360i \(-0.758770\pi\)
−0.726317 + 0.687360i \(0.758770\pi\)
\(224\) 10.3569 0.00308929
\(225\) 0 0
\(226\) 3079.61 0.906428
\(227\) −44.8176 −0.0131042 −0.00655209 0.999979i \(-0.502086\pi\)
−0.00655209 + 0.999979i \(0.502086\pi\)
\(228\) 4087.72 1.18735
\(229\) −4159.78 −1.20038 −0.600188 0.799859i \(-0.704908\pi\)
−0.600188 + 0.799859i \(0.704908\pi\)
\(230\) 0 0
\(231\) 19.9610 0.00568546
\(232\) 412.179 0.116642
\(233\) 2945.40 0.828152 0.414076 0.910242i \(-0.364105\pi\)
0.414076 + 0.910242i \(0.364105\pi\)
\(234\) −1481.43 −0.413864
\(235\) 0 0
\(236\) −4021.21 −1.10914
\(237\) 1822.99 0.499644
\(238\) −11.1240 −0.00302968
\(239\) 2174.54 0.588533 0.294266 0.955723i \(-0.404925\pi\)
0.294266 + 0.955723i \(0.404925\pi\)
\(240\) 0 0
\(241\) 2300.69 0.614939 0.307470 0.951558i \(-0.400518\pi\)
0.307470 + 0.951558i \(0.400518\pi\)
\(242\) −4366.74 −1.15994
\(243\) 243.000 0.0641500
\(244\) 5537.72 1.45294
\(245\) 0 0
\(246\) −1806.15 −0.468112
\(247\) 3125.23 0.805074
\(248\) 11130.7 2.85000
\(249\) −3805.66 −0.968571
\(250\) 0 0
\(251\) 3368.55 0.847096 0.423548 0.905874i \(-0.360785\pi\)
0.423548 + 0.905874i \(0.360785\pi\)
\(252\) 18.9360 0.00473357
\(253\) −7650.95 −1.90123
\(254\) 5684.09 1.40414
\(255\) 0 0
\(256\) −7303.36 −1.78305
\(257\) 404.905 0.0982772 0.0491386 0.998792i \(-0.484352\pi\)
0.0491386 + 0.998792i \(0.484352\pi\)
\(258\) 7618.33 1.83836
\(259\) −39.5427 −0.00948672
\(260\) 0 0
\(261\) −110.994 −0.0263233
\(262\) 1069.29 0.252142
\(263\) −1514.70 −0.355135 −0.177567 0.984109i \(-0.556823\pi\)
−0.177567 + 0.984109i \(0.556823\pi\)
\(264\) −4747.59 −1.10679
\(265\) 0 0
\(266\) −61.2912 −0.0141278
\(267\) −4012.60 −0.919727
\(268\) −2642.83 −0.602374
\(269\) −1891.88 −0.428811 −0.214405 0.976745i \(-0.568781\pi\)
−0.214405 + 0.976745i \(0.568781\pi\)
\(270\) 0 0
\(271\) −2094.31 −0.469448 −0.234724 0.972062i \(-0.575419\pi\)
−0.234724 + 0.972062i \(0.575419\pi\)
\(272\) 667.370 0.148769
\(273\) 14.4774 0.00320956
\(274\) 7570.79 1.66923
\(275\) 0 0
\(276\) −7258.08 −1.58292
\(277\) −1492.73 −0.323789 −0.161894 0.986808i \(-0.551760\pi\)
−0.161894 + 0.986808i \(0.551760\pi\)
\(278\) −7502.27 −1.61855
\(279\) −2997.34 −0.643176
\(280\) 0 0
\(281\) 2069.49 0.439344 0.219672 0.975574i \(-0.429501\pi\)
0.219672 + 0.975574i \(0.429501\pi\)
\(282\) −1079.45 −0.227945
\(283\) 3398.90 0.713935 0.356967 0.934117i \(-0.383811\pi\)
0.356967 + 0.934117i \(0.383811\pi\)
\(284\) 10812.1 2.25908
\(285\) 0 0
\(286\) −7794.05 −1.61144
\(287\) 17.6507 0.00363026
\(288\) 663.338 0.135721
\(289\) −4640.21 −0.944476
\(290\) 0 0
\(291\) −4293.64 −0.864939
\(292\) −36.5747 −0.00733004
\(293\) 7677.60 1.53082 0.765410 0.643543i \(-0.222536\pi\)
0.765410 + 0.643543i \(0.222536\pi\)
\(294\) 4931.73 0.978314
\(295\) 0 0
\(296\) 9404.93 1.84679
\(297\) 1278.46 0.249777
\(298\) −923.304 −0.179482
\(299\) −5549.09 −1.07329
\(300\) 0 0
\(301\) −74.4505 −0.0142567
\(302\) −5046.24 −0.961518
\(303\) −1122.31 −0.212788
\(304\) 3677.07 0.693732
\(305\) 0 0
\(306\) −712.470 −0.133102
\(307\) −4507.90 −0.838043 −0.419022 0.907976i \(-0.637627\pi\)
−0.419022 + 0.907976i \(0.637627\pi\)
\(308\) 99.6255 0.0184308
\(309\) −6053.61 −1.11449
\(310\) 0 0
\(311\) 7065.19 1.28820 0.644100 0.764941i \(-0.277232\pi\)
0.644100 + 0.764941i \(0.277232\pi\)
\(312\) −3443.34 −0.624809
\(313\) 2063.81 0.372694 0.186347 0.982484i \(-0.440335\pi\)
0.186347 + 0.982484i \(0.440335\pi\)
\(314\) −9802.94 −1.76182
\(315\) 0 0
\(316\) 9098.52 1.61972
\(317\) 8596.49 1.52311 0.761557 0.648098i \(-0.224435\pi\)
0.761557 + 0.648098i \(0.224435\pi\)
\(318\) 3586.50 0.632456
\(319\) −583.959 −0.102493
\(320\) 0 0
\(321\) 5039.14 0.876192
\(322\) 108.827 0.0188345
\(323\) 1503.03 0.258918
\(324\) 1212.81 0.207958
\(325\) 0 0
\(326\) −4064.54 −0.690533
\(327\) 60.6850 0.0102627
\(328\) −4198.08 −0.706708
\(329\) 10.5490 0.00176774
\(330\) 0 0
\(331\) 259.628 0.0431131 0.0215566 0.999768i \(-0.493138\pi\)
0.0215566 + 0.999768i \(0.493138\pi\)
\(332\) −18994.1 −3.13986
\(333\) −2532.62 −0.416777
\(334\) −9453.97 −1.54880
\(335\) 0 0
\(336\) 17.0338 0.00276568
\(337\) 6721.19 1.08643 0.543215 0.839594i \(-0.317207\pi\)
0.543215 + 0.839594i \(0.317207\pi\)
\(338\) 4877.38 0.784894
\(339\) −1927.56 −0.308822
\(340\) 0 0
\(341\) −15769.5 −2.50430
\(342\) −3925.57 −0.620673
\(343\) −96.3940 −0.0151743
\(344\) 17707.5 2.77536
\(345\) 0 0
\(346\) −12977.8 −2.01645
\(347\) −7139.56 −1.10453 −0.552264 0.833669i \(-0.686236\pi\)
−0.552264 + 0.833669i \(0.686236\pi\)
\(348\) −553.972 −0.0853334
\(349\) −11473.4 −1.75976 −0.879879 0.475198i \(-0.842376\pi\)
−0.879879 + 0.475198i \(0.842376\pi\)
\(350\) 0 0
\(351\) 927.244 0.141005
\(352\) 3489.92 0.528448
\(353\) 59.6020 0.00898667 0.00449333 0.999990i \(-0.498570\pi\)
0.00449333 + 0.999990i \(0.498570\pi\)
\(354\) 3861.69 0.579793
\(355\) 0 0
\(356\) −20026.9 −2.98152
\(357\) 6.96265 0.00103222
\(358\) −15012.9 −2.21636
\(359\) −7801.88 −1.14698 −0.573492 0.819211i \(-0.694412\pi\)
−0.573492 + 0.819211i \(0.694412\pi\)
\(360\) 0 0
\(361\) 1422.37 0.207373
\(362\) −1519.63 −0.220635
\(363\) 2733.19 0.395193
\(364\) 72.2565 0.0104046
\(365\) 0 0
\(366\) −5318.06 −0.759506
\(367\) −1583.29 −0.225197 −0.112599 0.993641i \(-0.535917\pi\)
−0.112599 + 0.993641i \(0.535917\pi\)
\(368\) −6528.94 −0.924850
\(369\) 1130.49 0.159487
\(370\) 0 0
\(371\) −35.0493 −0.00490477
\(372\) −14959.7 −2.08502
\(373\) −11181.2 −1.55212 −0.776059 0.630661i \(-0.782784\pi\)
−0.776059 + 0.630661i \(0.782784\pi\)
\(374\) −3748.42 −0.518252
\(375\) 0 0
\(376\) −2509.01 −0.344128
\(377\) −423.534 −0.0578598
\(378\) −18.1849 −0.00247442
\(379\) −11723.0 −1.58885 −0.794423 0.607365i \(-0.792227\pi\)
−0.794423 + 0.607365i \(0.792227\pi\)
\(380\) 0 0
\(381\) −3557.73 −0.478394
\(382\) −8049.67 −1.07816
\(383\) −9272.02 −1.23702 −0.618509 0.785778i \(-0.712263\pi\)
−0.618509 + 0.785778i \(0.712263\pi\)
\(384\) 7958.77 1.05767
\(385\) 0 0
\(386\) 6948.05 0.916183
\(387\) −4768.39 −0.626333
\(388\) −21429.5 −2.80391
\(389\) −1341.15 −0.174805 −0.0874023 0.996173i \(-0.527857\pi\)
−0.0874023 + 0.996173i \(0.527857\pi\)
\(390\) 0 0
\(391\) −2668.74 −0.345177
\(392\) 11463.0 1.47696
\(393\) −669.283 −0.0859054
\(394\) 5083.59 0.650019
\(395\) 0 0
\(396\) 6380.80 0.809715
\(397\) 11366.3 1.43692 0.718459 0.695569i \(-0.244848\pi\)
0.718459 + 0.695569i \(0.244848\pi\)
\(398\) 877.490 0.110514
\(399\) 38.3628 0.00481339
\(400\) 0 0
\(401\) −12421.7 −1.54690 −0.773451 0.633856i \(-0.781471\pi\)
−0.773451 + 0.633856i \(0.781471\pi\)
\(402\) 2537.99 0.314884
\(403\) −11437.3 −1.41373
\(404\) −5601.42 −0.689805
\(405\) 0 0
\(406\) 8.30625 0.00101535
\(407\) −13324.5 −1.62278
\(408\) −1656.02 −0.200944
\(409\) 12917.8 1.56172 0.780860 0.624706i \(-0.214781\pi\)
0.780860 + 0.624706i \(0.214781\pi\)
\(410\) 0 0
\(411\) −4738.64 −0.568710
\(412\) −30213.5 −3.61290
\(413\) −37.7386 −0.00449636
\(414\) 6970.16 0.827451
\(415\) 0 0
\(416\) 2531.18 0.298320
\(417\) 4695.75 0.551443
\(418\) −20653.0 −2.41668
\(419\) −4162.80 −0.485361 −0.242680 0.970106i \(-0.578027\pi\)
−0.242680 + 0.970106i \(0.578027\pi\)
\(420\) 0 0
\(421\) −14947.3 −1.73037 −0.865186 0.501451i \(-0.832800\pi\)
−0.865186 + 0.501451i \(0.832800\pi\)
\(422\) 23508.4 2.71178
\(423\) 675.642 0.0776615
\(424\) 8336.21 0.954817
\(425\) 0 0
\(426\) −10383.2 −1.18091
\(427\) 51.9710 0.00589005
\(428\) 25150.4 2.84039
\(429\) 4878.38 0.549022
\(430\) 0 0
\(431\) −12762.9 −1.42638 −0.713188 0.700973i \(-0.752749\pi\)
−0.713188 + 0.700973i \(0.752749\pi\)
\(432\) 1090.98 0.121504
\(433\) 777.497 0.0862913 0.0431456 0.999069i \(-0.486262\pi\)
0.0431456 + 0.999069i \(0.486262\pi\)
\(434\) 224.306 0.0248088
\(435\) 0 0
\(436\) 302.879 0.0332689
\(437\) −14704.2 −1.60961
\(438\) 35.1239 0.00383170
\(439\) 9936.45 1.08028 0.540138 0.841577i \(-0.318372\pi\)
0.540138 + 0.841577i \(0.318372\pi\)
\(440\) 0 0
\(441\) −3086.82 −0.333314
\(442\) −2718.66 −0.292564
\(443\) 7139.86 0.765745 0.382873 0.923801i \(-0.374935\pi\)
0.382873 + 0.923801i \(0.374935\pi\)
\(444\) −12640.3 −1.35109
\(445\) 0 0
\(446\) 23185.8 2.46162
\(447\) 577.906 0.0611499
\(448\) −95.0642 −0.0100254
\(449\) 10438.2 1.09712 0.548560 0.836111i \(-0.315176\pi\)
0.548560 + 0.836111i \(0.315176\pi\)
\(450\) 0 0
\(451\) 5947.67 0.620986
\(452\) −9620.45 −1.00112
\(453\) 3158.50 0.327592
\(454\) 214.811 0.0222062
\(455\) 0 0
\(456\) −9124.31 −0.937029
\(457\) 5039.76 0.515864 0.257932 0.966163i \(-0.416959\pi\)
0.257932 + 0.966163i \(0.416959\pi\)
\(458\) 19937.9 2.03414
\(459\) 445.943 0.0453482
\(460\) 0 0
\(461\) 1832.68 0.185155 0.0925775 0.995705i \(-0.470489\pi\)
0.0925775 + 0.995705i \(0.470489\pi\)
\(462\) −95.6736 −0.00963450
\(463\) 1887.49 0.189458 0.0947290 0.995503i \(-0.469802\pi\)
0.0947290 + 0.995503i \(0.469802\pi\)
\(464\) −498.321 −0.0498577
\(465\) 0 0
\(466\) −14117.3 −1.40337
\(467\) −3944.31 −0.390837 −0.195419 0.980720i \(-0.562607\pi\)
−0.195419 + 0.980720i \(0.562607\pi\)
\(468\) 4627.87 0.457101
\(469\) −24.8027 −0.00244196
\(470\) 0 0
\(471\) 6135.77 0.600257
\(472\) 8975.85 0.875311
\(473\) −25087.3 −2.43872
\(474\) −8737.60 −0.846690
\(475\) 0 0
\(476\) 34.7506 0.00334620
\(477\) −2244.83 −0.215479
\(478\) −10422.6 −0.997320
\(479\) −10759.1 −1.02630 −0.513150 0.858299i \(-0.671522\pi\)
−0.513150 + 0.858299i \(0.671522\pi\)
\(480\) 0 0
\(481\) −9664.02 −0.916095
\(482\) −11027.2 −1.04207
\(483\) −68.1163 −0.00641698
\(484\) 13641.3 1.28112
\(485\) 0 0
\(486\) −1164.70 −0.108708
\(487\) 3191.34 0.296948 0.148474 0.988916i \(-0.452564\pi\)
0.148474 + 0.988916i \(0.452564\pi\)
\(488\) −12360.9 −1.14662
\(489\) 2544.04 0.235267
\(490\) 0 0
\(491\) 933.900 0.0858377 0.0429189 0.999079i \(-0.486334\pi\)
0.0429189 + 0.999079i \(0.486334\pi\)
\(492\) 5642.25 0.517017
\(493\) −203.692 −0.0186082
\(494\) −14979.3 −1.36427
\(495\) 0 0
\(496\) −13456.9 −1.21821
\(497\) 101.470 0.00915806
\(498\) 18240.6 1.64133
\(499\) 5499.10 0.493333 0.246667 0.969100i \(-0.420665\pi\)
0.246667 + 0.969100i \(0.420665\pi\)
\(500\) 0 0
\(501\) 5917.34 0.527679
\(502\) −16145.5 −1.43548
\(503\) −11172.5 −0.990374 −0.495187 0.868787i \(-0.664900\pi\)
−0.495187 + 0.868787i \(0.664900\pi\)
\(504\) −42.2677 −0.00373562
\(505\) 0 0
\(506\) 36671.1 3.22180
\(507\) −3052.80 −0.267416
\(508\) −17756.6 −1.55083
\(509\) −8576.81 −0.746877 −0.373439 0.927655i \(-0.621821\pi\)
−0.373439 + 0.927655i \(0.621821\pi\)
\(510\) 0 0
\(511\) −0.343250 −2.97152e−5 0
\(512\) 13781.7 1.18960
\(513\) 2457.05 0.211465
\(514\) −1940.71 −0.166539
\(515\) 0 0
\(516\) −23799.0 −2.03041
\(517\) 3554.66 0.302386
\(518\) 189.529 0.0160761
\(519\) 8122.94 0.687008
\(520\) 0 0
\(521\) −19732.4 −1.65929 −0.829647 0.558289i \(-0.811458\pi\)
−0.829647 + 0.558289i \(0.811458\pi\)
\(522\) 531.997 0.0446071
\(523\) 16353.7 1.36730 0.683651 0.729809i \(-0.260391\pi\)
0.683651 + 0.729809i \(0.260391\pi\)
\(524\) −3340.39 −0.278484
\(525\) 0 0
\(526\) 7259.98 0.601807
\(527\) −5500.59 −0.454667
\(528\) 5739.79 0.473092
\(529\) 13941.6 1.14585
\(530\) 0 0
\(531\) −2417.08 −0.197537
\(532\) 191.469 0.0156038
\(533\) 4313.73 0.350560
\(534\) 19232.5 1.55856
\(535\) 0 0
\(536\) 5899.13 0.475380
\(537\) 9396.75 0.755120
\(538\) 9067.82 0.726657
\(539\) −16240.3 −1.29781
\(540\) 0 0
\(541\) 14238.7 1.13155 0.565776 0.824559i \(-0.308577\pi\)
0.565776 + 0.824559i \(0.308577\pi\)
\(542\) 10038.1 0.795521
\(543\) 951.153 0.0751710
\(544\) 1217.33 0.0959421
\(545\) 0 0
\(546\) −69.3903 −0.00543888
\(547\) 7018.76 0.548630 0.274315 0.961640i \(-0.411549\pi\)
0.274315 + 0.961640i \(0.411549\pi\)
\(548\) −23650.5 −1.84361
\(549\) 3328.63 0.258766
\(550\) 0 0
\(551\) −1122.30 −0.0867725
\(552\) 16201.0 1.24920
\(553\) 85.3887 0.00656618
\(554\) 7154.68 0.548688
\(555\) 0 0
\(556\) 23436.5 1.78764
\(557\) 6144.57 0.467421 0.233711 0.972306i \(-0.424913\pi\)
0.233711 + 0.972306i \(0.424913\pi\)
\(558\) 14366.3 1.08992
\(559\) −18195.3 −1.37671
\(560\) 0 0
\(561\) 2346.18 0.176570
\(562\) −9919.12 −0.744507
\(563\) 17169.5 1.28527 0.642635 0.766172i \(-0.277841\pi\)
0.642635 + 0.766172i \(0.277841\pi\)
\(564\) 3372.12 0.251759
\(565\) 0 0
\(566\) −16291.0 −1.20982
\(567\) 11.3821 0.000843041 0
\(568\) −24133.9 −1.78281
\(569\) −15575.2 −1.14754 −0.573768 0.819018i \(-0.694519\pi\)
−0.573768 + 0.819018i \(0.694519\pi\)
\(570\) 0 0
\(571\) −21207.6 −1.55431 −0.777156 0.629308i \(-0.783339\pi\)
−0.777156 + 0.629308i \(0.783339\pi\)
\(572\) 24348.0 1.77979
\(573\) 5038.37 0.367332
\(574\) −84.5999 −0.00615180
\(575\) 0 0
\(576\) −6088.65 −0.440441
\(577\) −2071.25 −0.149441 −0.0747205 0.997205i \(-0.523806\pi\)
−0.0747205 + 0.997205i \(0.523806\pi\)
\(578\) 22240.6 1.60050
\(579\) −4348.86 −0.312146
\(580\) 0 0
\(581\) −178.257 −0.0127287
\(582\) 20579.5 1.46571
\(583\) −11810.4 −0.839000
\(584\) 81.6394 0.00578470
\(585\) 0 0
\(586\) −36798.9 −2.59411
\(587\) 20730.6 1.45766 0.728828 0.684697i \(-0.240065\pi\)
0.728828 + 0.684697i \(0.240065\pi\)
\(588\) −15406.3 −1.08052
\(589\) −30307.1 −2.12018
\(590\) 0 0
\(591\) −3181.87 −0.221463
\(592\) −11370.5 −0.789399
\(593\) −1924.00 −0.133236 −0.0666182 0.997779i \(-0.521221\pi\)
−0.0666182 + 0.997779i \(0.521221\pi\)
\(594\) −6127.68 −0.423269
\(595\) 0 0
\(596\) 2884.33 0.198233
\(597\) −549.230 −0.0376524
\(598\) 26596.9 1.81878
\(599\) −8088.34 −0.551720 −0.275860 0.961198i \(-0.588963\pi\)
−0.275860 + 0.961198i \(0.588963\pi\)
\(600\) 0 0
\(601\) 19953.8 1.35430 0.677149 0.735846i \(-0.263215\pi\)
0.677149 + 0.735846i \(0.263215\pi\)
\(602\) 356.842 0.0241592
\(603\) −1588.56 −0.107282
\(604\) 15764.0 1.06197
\(605\) 0 0
\(606\) 5379.22 0.360588
\(607\) −3633.68 −0.242976 −0.121488 0.992593i \(-0.538767\pi\)
−0.121488 + 0.992593i \(0.538767\pi\)
\(608\) 6707.23 0.447392
\(609\) −5.19897 −0.000345933 0
\(610\) 0 0
\(611\) 2578.13 0.170704
\(612\) 2225.70 0.147007
\(613\) −1581.91 −0.104230 −0.0521148 0.998641i \(-0.516596\pi\)
−0.0521148 + 0.998641i \(0.516596\pi\)
\(614\) 21606.4 1.42014
\(615\) 0 0
\(616\) −222.377 −0.0145452
\(617\) 4454.03 0.290620 0.145310 0.989386i \(-0.453582\pi\)
0.145310 + 0.989386i \(0.453582\pi\)
\(618\) 29015.0 1.88860
\(619\) −2630.26 −0.170790 −0.0853951 0.996347i \(-0.527215\pi\)
−0.0853951 + 0.996347i \(0.527215\pi\)
\(620\) 0 0
\(621\) −4362.70 −0.281915
\(622\) −33863.6 −2.18297
\(623\) −187.950 −0.0120868
\(624\) 4162.97 0.267071
\(625\) 0 0
\(626\) −9891.85 −0.631562
\(627\) 12927.0 0.823370
\(628\) 30623.6 1.94588
\(629\) −4647.75 −0.294623
\(630\) 0 0
\(631\) −16896.4 −1.06599 −0.532993 0.846120i \(-0.678933\pi\)
−0.532993 + 0.846120i \(0.678933\pi\)
\(632\) −20309.1 −1.27825
\(633\) −14714.2 −0.923912
\(634\) −41203.1 −2.58105
\(635\) 0 0
\(636\) −11203.9 −0.698530
\(637\) −11778.8 −0.732640
\(638\) 2798.92 0.173684
\(639\) 6498.93 0.402338
\(640\) 0 0
\(641\) 28804.5 1.77490 0.887448 0.460908i \(-0.152476\pi\)
0.887448 + 0.460908i \(0.152476\pi\)
\(642\) −24152.7 −1.48478
\(643\) 10219.9 0.626801 0.313400 0.949621i \(-0.398532\pi\)
0.313400 + 0.949621i \(0.398532\pi\)
\(644\) −339.968 −0.0208022
\(645\) 0 0
\(646\) −7204.03 −0.438759
\(647\) −1792.38 −0.108912 −0.0544558 0.998516i \(-0.517342\pi\)
−0.0544558 + 0.998516i \(0.517342\pi\)
\(648\) −2707.15 −0.164116
\(649\) −12716.6 −0.769138
\(650\) 0 0
\(651\) −140.396 −0.00845243
\(652\) 12697.3 0.762675
\(653\) −13405.4 −0.803359 −0.401679 0.915780i \(-0.631573\pi\)
−0.401679 + 0.915780i \(0.631573\pi\)
\(654\) −290.864 −0.0173910
\(655\) 0 0
\(656\) 5075.44 0.302077
\(657\) −21.9844 −0.00130547
\(658\) −50.5616 −0.00299559
\(659\) −15441.6 −0.912778 −0.456389 0.889780i \(-0.650858\pi\)
−0.456389 + 0.889780i \(0.650858\pi\)
\(660\) 0 0
\(661\) 4356.59 0.256356 0.128178 0.991751i \(-0.459087\pi\)
0.128178 + 0.991751i \(0.459087\pi\)
\(662\) −1244.40 −0.0730589
\(663\) 1701.64 0.0996774
\(664\) 42397.2 2.47791
\(665\) 0 0
\(666\) 12138.9 0.706265
\(667\) 1992.74 0.115681
\(668\) 29533.4 1.71060
\(669\) −14512.3 −0.838679
\(670\) 0 0
\(671\) 17512.4 1.00754
\(672\) 31.0707 0.00178360
\(673\) −17617.2 −1.00905 −0.504527 0.863396i \(-0.668333\pi\)
−0.504527 + 0.863396i \(0.668333\pi\)
\(674\) −32214.8 −1.84105
\(675\) 0 0
\(676\) −15236.5 −0.866894
\(677\) 3294.63 0.187036 0.0935178 0.995618i \(-0.470189\pi\)
0.0935178 + 0.995618i \(0.470189\pi\)
\(678\) 9238.83 0.523326
\(679\) −201.114 −0.0113668
\(680\) 0 0
\(681\) −134.453 −0.00756570
\(682\) 75583.4 4.24375
\(683\) 4829.50 0.270565 0.135282 0.990807i \(-0.456806\pi\)
0.135282 + 0.990807i \(0.456806\pi\)
\(684\) 12263.2 0.685517
\(685\) 0 0
\(686\) 462.018 0.0257142
\(687\) −12479.3 −0.693037
\(688\) −21408.2 −1.18631
\(689\) −8565.87 −0.473634
\(690\) 0 0
\(691\) 23330.5 1.28442 0.642211 0.766528i \(-0.278017\pi\)
0.642211 + 0.766528i \(0.278017\pi\)
\(692\) 40541.5 2.22711
\(693\) 59.8831 0.00328250
\(694\) 34220.0 1.87172
\(695\) 0 0
\(696\) 1236.54 0.0673432
\(697\) 2074.62 0.112743
\(698\) 54992.0 2.98206
\(699\) 8836.19 0.478134
\(700\) 0 0
\(701\) 27619.7 1.48813 0.744066 0.668106i \(-0.232895\pi\)
0.744066 + 0.668106i \(0.232895\pi\)
\(702\) −4444.30 −0.238945
\(703\) −25608.2 −1.37387
\(704\) −32033.4 −1.71492
\(705\) 0 0
\(706\) −285.673 −0.0152287
\(707\) −52.5688 −0.00279640
\(708\) −12063.6 −0.640365
\(709\) 25819.3 1.36765 0.683826 0.729645i \(-0.260315\pi\)
0.683826 + 0.729645i \(0.260315\pi\)
\(710\) 0 0
\(711\) 5468.96 0.288470
\(712\) 44702.6 2.35295
\(713\) 53812.8 2.82651
\(714\) −33.3721 −0.00174919
\(715\) 0 0
\(716\) 46899.1 2.44791
\(717\) 6523.62 0.339790
\(718\) 37394.5 1.94366
\(719\) −16339.0 −0.847484 −0.423742 0.905783i \(-0.639284\pi\)
−0.423742 + 0.905783i \(0.639284\pi\)
\(720\) 0 0
\(721\) −283.551 −0.0146463
\(722\) −6817.44 −0.351411
\(723\) 6902.06 0.355035
\(724\) 4747.20 0.243685
\(725\) 0 0
\(726\) −13100.2 −0.669689
\(727\) −7287.52 −0.371773 −0.185887 0.982571i \(-0.559516\pi\)
−0.185887 + 0.982571i \(0.559516\pi\)
\(728\) −161.286 −0.00821106
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −8750.74 −0.442760
\(732\) 16613.2 0.838853
\(733\) −12030.1 −0.606197 −0.303098 0.952959i \(-0.598021\pi\)
−0.303098 + 0.952959i \(0.598021\pi\)
\(734\) 7588.76 0.381616
\(735\) 0 0
\(736\) −11909.2 −0.596440
\(737\) −8357.64 −0.417718
\(738\) −5418.44 −0.270265
\(739\) −15163.0 −0.754778 −0.377389 0.926055i \(-0.623178\pi\)
−0.377389 + 0.926055i \(0.623178\pi\)
\(740\) 0 0
\(741\) 9375.68 0.464810
\(742\) 167.992 0.00831155
\(743\) −22772.8 −1.12443 −0.562216 0.826991i \(-0.690051\pi\)
−0.562216 + 0.826991i \(0.690051\pi\)
\(744\) 33392.0 1.64545
\(745\) 0 0
\(746\) 53591.6 2.63020
\(747\) −11417.0 −0.559205
\(748\) 11709.8 0.572394
\(749\) 236.034 0.0115147
\(750\) 0 0
\(751\) −27855.5 −1.35348 −0.676739 0.736223i \(-0.736608\pi\)
−0.676739 + 0.736223i \(0.736608\pi\)
\(752\) 3033.37 0.147095
\(753\) 10105.6 0.489071
\(754\) 2030.01 0.0980484
\(755\) 0 0
\(756\) 56.8081 0.00273293
\(757\) 302.235 0.0145111 0.00725557 0.999974i \(-0.497690\pi\)
0.00725557 + 0.999974i \(0.497690\pi\)
\(758\) 56188.7 2.69244
\(759\) −22952.9 −1.09768
\(760\) 0 0
\(761\) −4497.19 −0.214222 −0.107111 0.994247i \(-0.534160\pi\)
−0.107111 + 0.994247i \(0.534160\pi\)
\(762\) 17052.3 0.810680
\(763\) 2.84249 0.000134869 0
\(764\) 25146.5 1.19080
\(765\) 0 0
\(766\) 44440.9 2.09623
\(767\) −9223.13 −0.434195
\(768\) −21910.1 −1.02944
\(769\) 2438.76 0.114362 0.0571808 0.998364i \(-0.481789\pi\)
0.0571808 + 0.998364i \(0.481789\pi\)
\(770\) 0 0
\(771\) 1214.71 0.0567404
\(772\) −21705.1 −1.01190
\(773\) −37464.5 −1.74321 −0.871607 0.490205i \(-0.836922\pi\)
−0.871607 + 0.490205i \(0.836922\pi\)
\(774\) 22855.0 1.06138
\(775\) 0 0
\(776\) 47833.4 2.21278
\(777\) −118.628 −0.00547716
\(778\) 6428.15 0.296222
\(779\) 11430.7 0.525736
\(780\) 0 0
\(781\) 34191.9 1.56656
\(782\) 12791.3 0.584932
\(783\) −332.983 −0.0151978
\(784\) −13858.6 −0.631315
\(785\) 0 0
\(786\) 3207.88 0.145574
\(787\) 30317.8 1.37321 0.686603 0.727033i \(-0.259101\pi\)
0.686603 + 0.727033i \(0.259101\pi\)
\(788\) −15880.7 −0.717928
\(789\) −4544.10 −0.205037
\(790\) 0 0
\(791\) −90.2870 −0.00405845
\(792\) −14242.8 −0.639008
\(793\) 12701.4 0.568779
\(794\) −54478.7 −2.43498
\(795\) 0 0
\(796\) −2741.21 −0.122060
\(797\) −23627.1 −1.05008 −0.525041 0.851077i \(-0.675950\pi\)
−0.525041 + 0.851077i \(0.675950\pi\)
\(798\) −183.874 −0.00815671
\(799\) 1239.91 0.0548996
\(800\) 0 0
\(801\) −12037.8 −0.531005
\(802\) 59537.2 2.62136
\(803\) −115.663 −0.00508303
\(804\) −7928.48 −0.347781
\(805\) 0 0
\(806\) 54819.2 2.39569
\(807\) −5675.65 −0.247574
\(808\) 12503.1 0.544378
\(809\) −30566.8 −1.32839 −0.664197 0.747558i \(-0.731226\pi\)
−0.664197 + 0.747558i \(0.731226\pi\)
\(810\) 0 0
\(811\) 36976.4 1.60101 0.800504 0.599328i \(-0.204565\pi\)
0.800504 + 0.599328i \(0.204565\pi\)
\(812\) −25.9481 −0.00112143
\(813\) −6282.94 −0.271036
\(814\) 63864.6 2.74994
\(815\) 0 0
\(816\) 2002.11 0.0858920
\(817\) −48214.8 −2.06466
\(818\) −61915.1 −2.64647
\(819\) 43.4321 0.00185304
\(820\) 0 0
\(821\) −9499.03 −0.403798 −0.201899 0.979406i \(-0.564711\pi\)
−0.201899 + 0.979406i \(0.564711\pi\)
\(822\) 22712.4 0.963728
\(823\) −4753.25 −0.201322 −0.100661 0.994921i \(-0.532096\pi\)
−0.100661 + 0.994921i \(0.532096\pi\)
\(824\) 67440.5 2.85121
\(825\) 0 0
\(826\) 180.882 0.00761947
\(827\) 13224.0 0.556039 0.278020 0.960575i \(-0.410322\pi\)
0.278020 + 0.960575i \(0.410322\pi\)
\(828\) −21774.2 −0.913897
\(829\) −10210.7 −0.427782 −0.213891 0.976858i \(-0.568614\pi\)
−0.213891 + 0.976858i \(0.568614\pi\)
\(830\) 0 0
\(831\) −4478.19 −0.186939
\(832\) −23233.2 −0.968109
\(833\) −5664.80 −0.235623
\(834\) −22506.8 −0.934469
\(835\) 0 0
\(836\) 64518.4 2.66916
\(837\) −8992.03 −0.371338
\(838\) 19952.4 0.822485
\(839\) −6722.41 −0.276619 −0.138310 0.990389i \(-0.544167\pi\)
−0.138310 + 0.990389i \(0.544167\pi\)
\(840\) 0 0
\(841\) −24236.9 −0.993764
\(842\) 71642.6 2.93227
\(843\) 6208.48 0.253655
\(844\) −73438.4 −2.99509
\(845\) 0 0
\(846\) −3238.36 −0.131604
\(847\) 128.023 0.00519351
\(848\) −10078.4 −0.408130
\(849\) 10196.7 0.412191
\(850\) 0 0
\(851\) 45469.4 1.83158
\(852\) 32436.2 1.30428
\(853\) −3785.52 −0.151950 −0.0759752 0.997110i \(-0.524207\pi\)
−0.0759752 + 0.997110i \(0.524207\pi\)
\(854\) −249.098 −0.00998121
\(855\) 0 0
\(856\) −56138.8 −2.24157
\(857\) 13421.7 0.534979 0.267489 0.963561i \(-0.413806\pi\)
0.267489 + 0.963561i \(0.413806\pi\)
\(858\) −23382.1 −0.930365
\(859\) 491.965 0.0195409 0.00977045 0.999952i \(-0.496890\pi\)
0.00977045 + 0.999952i \(0.496890\pi\)
\(860\) 0 0
\(861\) 52.9520 0.00209593
\(862\) 61172.8 2.41712
\(863\) −8088.10 −0.319029 −0.159515 0.987196i \(-0.550993\pi\)
−0.159515 + 0.987196i \(0.550993\pi\)
\(864\) 1990.01 0.0783583
\(865\) 0 0
\(866\) −3726.55 −0.146228
\(867\) −13920.6 −0.545293
\(868\) −700.714 −0.0274007
\(869\) 28773.1 1.12320
\(870\) 0 0
\(871\) −6061.64 −0.235811
\(872\) −676.064 −0.0262551
\(873\) −12880.9 −0.499373
\(874\) 70477.7 2.72762
\(875\) 0 0
\(876\) −109.724 −0.00423200
\(877\) 12000.4 0.462058 0.231029 0.972947i \(-0.425791\pi\)
0.231029 + 0.972947i \(0.425791\pi\)
\(878\) −47625.6 −1.83062
\(879\) 23032.8 0.883820
\(880\) 0 0
\(881\) −2285.38 −0.0873964 −0.0436982 0.999045i \(-0.513914\pi\)
−0.0436982 + 0.999045i \(0.513914\pi\)
\(882\) 14795.2 0.564830
\(883\) 19226.6 0.732759 0.366379 0.930466i \(-0.380597\pi\)
0.366379 + 0.930466i \(0.380597\pi\)
\(884\) 8492.87 0.323129
\(885\) 0 0
\(886\) −34221.5 −1.29762
\(887\) 14952.8 0.566028 0.283014 0.959116i \(-0.408666\pi\)
0.283014 + 0.959116i \(0.408666\pi\)
\(888\) 28214.8 1.06625
\(889\) −166.644 −0.00628692
\(890\) 0 0
\(891\) 3835.38 0.144209
\(892\) −72430.6 −2.71879
\(893\) 6831.64 0.256005
\(894\) −2769.91 −0.103624
\(895\) 0 0
\(896\) 372.789 0.0138996
\(897\) −16647.3 −0.619662
\(898\) −50030.2 −1.85916
\(899\) 4107.26 0.152375
\(900\) 0 0
\(901\) −4119.61 −0.152324
\(902\) −28507.3 −1.05231
\(903\) −223.352 −0.00823109
\(904\) 21474.1 0.790064
\(905\) 0 0
\(906\) −15138.7 −0.555133
\(907\) 48744.8 1.78450 0.892251 0.451540i \(-0.149125\pi\)
0.892251 + 0.451540i \(0.149125\pi\)
\(908\) −671.053 −0.0245261
\(909\) −3366.92 −0.122853
\(910\) 0 0
\(911\) −35064.7 −1.27524 −0.637621 0.770350i \(-0.720081\pi\)
−0.637621 + 0.770350i \(0.720081\pi\)
\(912\) 11031.2 0.400527
\(913\) −60066.6 −2.17734
\(914\) −24155.6 −0.874177
\(915\) 0 0
\(916\) −62284.4 −2.24665
\(917\) −31.3492 −0.00112894
\(918\) −2137.41 −0.0768465
\(919\) −20159.6 −0.723617 −0.361808 0.932253i \(-0.617841\pi\)
−0.361808 + 0.932253i \(0.617841\pi\)
\(920\) 0 0
\(921\) −13523.7 −0.483845
\(922\) −8784.07 −0.313761
\(923\) 24798.8 0.884357
\(924\) 298.877 0.0106410
\(925\) 0 0
\(926\) −9046.76 −0.321053
\(927\) −18160.8 −0.643451
\(928\) −908.972 −0.0321535
\(929\) 22221.8 0.784793 0.392397 0.919796i \(-0.371646\pi\)
0.392397 + 0.919796i \(0.371646\pi\)
\(930\) 0 0
\(931\) −31211.9 −1.09874
\(932\) 44101.4 1.54999
\(933\) 21195.6 0.743743
\(934\) 18905.2 0.662308
\(935\) 0 0
\(936\) −10330.0 −0.360734
\(937\) 8123.54 0.283228 0.141614 0.989922i \(-0.454771\pi\)
0.141614 + 0.989922i \(0.454771\pi\)
\(938\) 118.880 0.00413812
\(939\) 6191.42 0.215175
\(940\) 0 0
\(941\) −41826.3 −1.44899 −0.724494 0.689281i \(-0.757927\pi\)
−0.724494 + 0.689281i \(0.757927\pi\)
\(942\) −29408.8 −1.01719
\(943\) −20296.2 −0.700885
\(944\) −10851.7 −0.374146
\(945\) 0 0
\(946\) 120244. 4.13262
\(947\) 32843.5 1.12700 0.563501 0.826115i \(-0.309454\pi\)
0.563501 + 0.826115i \(0.309454\pi\)
\(948\) 27295.6 0.935146
\(949\) −83.8885 −0.00286948
\(950\) 0 0
\(951\) 25789.5 0.879370
\(952\) −77.5678 −0.00264074
\(953\) −12579.4 −0.427583 −0.213791 0.976879i \(-0.568581\pi\)
−0.213791 + 0.976879i \(0.568581\pi\)
\(954\) 10759.5 0.365149
\(955\) 0 0
\(956\) 32559.4 1.10151
\(957\) −1751.88 −0.0591746
\(958\) 51568.8 1.73916
\(959\) −221.958 −0.00747382
\(960\) 0 0
\(961\) 81123.3 2.72308
\(962\) 46319.8 1.55240
\(963\) 15117.4 0.505870
\(964\) 34448.2 1.15093
\(965\) 0 0
\(966\) 326.482 0.0108741
\(967\) 41883.1 1.39283 0.696417 0.717637i \(-0.254777\pi\)
0.696417 + 0.717637i \(0.254777\pi\)
\(968\) −30449.2 −1.01103
\(969\) 4509.08 0.149487
\(970\) 0 0
\(971\) 55390.3 1.83065 0.915323 0.402720i \(-0.131935\pi\)
0.915323 + 0.402720i \(0.131935\pi\)
\(972\) 3638.44 0.120065
\(973\) 219.949 0.00724691
\(974\) −15296.2 −0.503204
\(975\) 0 0
\(976\) 14944.2 0.490117
\(977\) −35463.4 −1.16128 −0.580642 0.814159i \(-0.697198\pi\)
−0.580642 + 0.814159i \(0.697198\pi\)
\(978\) −12193.6 −0.398680
\(979\) −63332.8 −2.06754
\(980\) 0 0
\(981\) 182.055 0.00592515
\(982\) −4476.20 −0.145459
\(983\) −34670.9 −1.12495 −0.562477 0.826813i \(-0.690151\pi\)
−0.562477 + 0.826813i \(0.690151\pi\)
\(984\) −12594.2 −0.408018
\(985\) 0 0
\(986\) 976.298 0.0315331
\(987\) 31.6471 0.00102060
\(988\) 46794.0 1.50680
\(989\) 85609.3 2.75250
\(990\) 0 0
\(991\) −37280.3 −1.19500 −0.597501 0.801868i \(-0.703840\pi\)
−0.597501 + 0.801868i \(0.703840\pi\)
\(992\) −24546.3 −0.785631
\(993\) 778.884 0.0248914
\(994\) −486.347 −0.0155191
\(995\) 0 0
\(996\) −56982.2 −1.81280
\(997\) −34383.2 −1.09220 −0.546101 0.837719i \(-0.683889\pi\)
−0.546101 + 0.837719i \(0.683889\pi\)
\(998\) −26357.3 −0.835996
\(999\) −7597.86 −0.240626
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.1.2 14
5.4 even 2 1875.4.a.f.1.13 14
25.6 even 5 75.4.g.b.61.7 yes 28
25.21 even 5 75.4.g.b.16.7 28
75.56 odd 10 225.4.h.a.136.1 28
75.71 odd 10 225.4.h.a.91.1 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.7 28 25.21 even 5
75.4.g.b.61.7 yes 28 25.6 even 5
225.4.h.a.91.1 28 75.71 odd 10
225.4.h.a.136.1 28 75.56 odd 10
1875.4.a.f.1.13 14 5.4 even 2
1875.4.a.g.1.2 14 1.1 even 1 trivial