Properties

Label 1045.4.a.c.1.8
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.01989\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.01989 q^{2} +7.09682 q^{3} -3.92003 q^{4} -5.00000 q^{5} -14.3348 q^{6} -8.46588 q^{7} +24.0772 q^{8} +23.3649 q^{9} +O(q^{10})\) \(q-2.01989 q^{2} +7.09682 q^{3} -3.92003 q^{4} -5.00000 q^{5} -14.3348 q^{6} -8.46588 q^{7} +24.0772 q^{8} +23.3649 q^{9} +10.0995 q^{10} -11.0000 q^{11} -27.8198 q^{12} -33.6704 q^{13} +17.1002 q^{14} -35.4841 q^{15} -17.2731 q^{16} +98.1464 q^{17} -47.1945 q^{18} +19.0000 q^{19} +19.6002 q^{20} -60.0808 q^{21} +22.2188 q^{22} +90.8592 q^{23} +170.871 q^{24} +25.0000 q^{25} +68.0106 q^{26} -25.7979 q^{27} +33.1865 q^{28} -142.201 q^{29} +71.6741 q^{30} +38.3284 q^{31} -157.728 q^{32} -78.0650 q^{33} -198.245 q^{34} +42.3294 q^{35} -91.5910 q^{36} -66.4096 q^{37} -38.3780 q^{38} -238.953 q^{39} -120.386 q^{40} +501.764 q^{41} +121.357 q^{42} -249.116 q^{43} +43.1204 q^{44} -116.824 q^{45} -183.526 q^{46} +269.945 q^{47} -122.584 q^{48} -271.329 q^{49} -50.4973 q^{50} +696.527 q^{51} +131.989 q^{52} -252.766 q^{53} +52.1091 q^{54} +55.0000 q^{55} -203.835 q^{56} +134.840 q^{57} +287.231 q^{58} -669.048 q^{59} +139.099 q^{60} -546.193 q^{61} -77.4193 q^{62} -197.804 q^{63} +456.778 q^{64} +168.352 q^{65} +157.683 q^{66} +648.266 q^{67} -384.737 q^{68} +644.812 q^{69} -85.5008 q^{70} +773.047 q^{71} +562.560 q^{72} -685.480 q^{73} +134.140 q^{74} +177.421 q^{75} -74.4806 q^{76} +93.1247 q^{77} +482.659 q^{78} -1258.69 q^{79} +86.3654 q^{80} -813.935 q^{81} -1013.51 q^{82} -1290.53 q^{83} +235.519 q^{84} -490.732 q^{85} +503.187 q^{86} -1009.18 q^{87} -264.849 q^{88} -734.991 q^{89} +235.973 q^{90} +285.050 q^{91} -356.171 q^{92} +272.010 q^{93} -545.260 q^{94} -95.0000 q^{95} -1119.37 q^{96} -1006.26 q^{97} +548.055 q^{98} -257.013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.01989 −0.714140 −0.357070 0.934078i \(-0.616224\pi\)
−0.357070 + 0.934078i \(0.616224\pi\)
\(3\) 7.09682 1.36578 0.682892 0.730519i \(-0.260722\pi\)
0.682892 + 0.730519i \(0.260722\pi\)
\(4\) −3.92003 −0.490004
\(5\) −5.00000 −0.447214
\(6\) −14.3348 −0.975361
\(7\) −8.46588 −0.457114 −0.228557 0.973530i \(-0.573401\pi\)
−0.228557 + 0.973530i \(0.573401\pi\)
\(8\) 24.0772 1.06407
\(9\) 23.3649 0.865365
\(10\) 10.0995 0.319373
\(11\) −11.0000 −0.301511
\(12\) −27.8198 −0.669240
\(13\) −33.6704 −0.718346 −0.359173 0.933271i \(-0.616941\pi\)
−0.359173 + 0.933271i \(0.616941\pi\)
\(14\) 17.1002 0.326444
\(15\) −35.4841 −0.610797
\(16\) −17.2731 −0.269892
\(17\) 98.1464 1.40023 0.700117 0.714028i \(-0.253131\pi\)
0.700117 + 0.714028i \(0.253131\pi\)
\(18\) −47.1945 −0.617992
\(19\) 19.0000 0.229416
\(20\) 19.6002 0.219137
\(21\) −60.0808 −0.624319
\(22\) 22.2188 0.215321
\(23\) 90.8592 0.823716 0.411858 0.911248i \(-0.364880\pi\)
0.411858 + 0.911248i \(0.364880\pi\)
\(24\) 170.871 1.45329
\(25\) 25.0000 0.200000
\(26\) 68.0106 0.512999
\(27\) −25.7979 −0.183882
\(28\) 33.1865 0.223988
\(29\) −142.201 −0.910554 −0.455277 0.890350i \(-0.650460\pi\)
−0.455277 + 0.890350i \(0.650460\pi\)
\(30\) 71.6741 0.436195
\(31\) 38.3284 0.222064 0.111032 0.993817i \(-0.464584\pi\)
0.111032 + 0.993817i \(0.464584\pi\)
\(32\) −157.728 −0.871331
\(33\) −78.0650 −0.411799
\(34\) −198.245 −0.999964
\(35\) 42.3294 0.204428
\(36\) −91.5910 −0.424033
\(37\) −66.4096 −0.295072 −0.147536 0.989057i \(-0.547134\pi\)
−0.147536 + 0.989057i \(0.547134\pi\)
\(38\) −38.3780 −0.163835
\(39\) −238.953 −0.981105
\(40\) −120.386 −0.475867
\(41\) 501.764 1.91128 0.955639 0.294540i \(-0.0951663\pi\)
0.955639 + 0.294540i \(0.0951663\pi\)
\(42\) 121.357 0.445851
\(43\) −249.116 −0.883483 −0.441741 0.897142i \(-0.645639\pi\)
−0.441741 + 0.897142i \(0.645639\pi\)
\(44\) 43.1204 0.147742
\(45\) −116.824 −0.387003
\(46\) −183.526 −0.588248
\(47\) 269.945 0.837776 0.418888 0.908038i \(-0.362420\pi\)
0.418888 + 0.908038i \(0.362420\pi\)
\(48\) −122.584 −0.368614
\(49\) −271.329 −0.791046
\(50\) −50.4973 −0.142828
\(51\) 696.527 1.91242
\(52\) 131.989 0.351992
\(53\) −252.766 −0.655097 −0.327548 0.944834i \(-0.606222\pi\)
−0.327548 + 0.944834i \(0.606222\pi\)
\(54\) 52.1091 0.131318
\(55\) 55.0000 0.134840
\(56\) −203.835 −0.486402
\(57\) 134.840 0.313332
\(58\) 287.231 0.650263
\(59\) −669.048 −1.47632 −0.738158 0.674628i \(-0.764304\pi\)
−0.738158 + 0.674628i \(0.764304\pi\)
\(60\) 139.099 0.299293
\(61\) −546.193 −1.14644 −0.573220 0.819402i \(-0.694306\pi\)
−0.573220 + 0.819402i \(0.694306\pi\)
\(62\) −77.4193 −0.158585
\(63\) −197.804 −0.395571
\(64\) 456.778 0.892144
\(65\) 168.352 0.321254
\(66\) 157.683 0.294082
\(67\) 648.266 1.18206 0.591032 0.806648i \(-0.298721\pi\)
0.591032 + 0.806648i \(0.298721\pi\)
\(68\) −384.737 −0.686121
\(69\) 644.812 1.12502
\(70\) −85.5008 −0.145990
\(71\) 773.047 1.29217 0.646083 0.763267i \(-0.276406\pi\)
0.646083 + 0.763267i \(0.276406\pi\)
\(72\) 562.560 0.920810
\(73\) −685.480 −1.09903 −0.549516 0.835483i \(-0.685188\pi\)
−0.549516 + 0.835483i \(0.685188\pi\)
\(74\) 134.140 0.210723
\(75\) 177.421 0.273157
\(76\) −74.4806 −0.112415
\(77\) 93.1247 0.137825
\(78\) 482.659 0.700646
\(79\) −1258.69 −1.79258 −0.896290 0.443468i \(-0.853748\pi\)
−0.896290 + 0.443468i \(0.853748\pi\)
\(80\) 86.3654 0.120699
\(81\) −813.935 −1.11651
\(82\) −1013.51 −1.36492
\(83\) −1290.53 −1.70668 −0.853338 0.521358i \(-0.825426\pi\)
−0.853338 + 0.521358i \(0.825426\pi\)
\(84\) 235.519 0.305919
\(85\) −490.732 −0.626204
\(86\) 503.187 0.630930
\(87\) −1009.18 −1.24362
\(88\) −264.849 −0.320830
\(89\) −734.991 −0.875381 −0.437690 0.899126i \(-0.644203\pi\)
−0.437690 + 0.899126i \(0.644203\pi\)
\(90\) 235.973 0.276374
\(91\) 285.050 0.328366
\(92\) −356.171 −0.403624
\(93\) 272.010 0.303292
\(94\) −545.260 −0.598290
\(95\) −95.0000 −0.102598
\(96\) −1119.37 −1.19005
\(97\) −1006.26 −1.05330 −0.526650 0.850082i \(-0.676552\pi\)
−0.526650 + 0.850082i \(0.676552\pi\)
\(98\) 548.055 0.564918
\(99\) −257.013 −0.260917
\(100\) −98.0008 −0.0980008
\(101\) 1207.14 1.18925 0.594626 0.804002i \(-0.297300\pi\)
0.594626 + 0.804002i \(0.297300\pi\)
\(102\) −1406.91 −1.36573
\(103\) −696.236 −0.666041 −0.333020 0.942920i \(-0.608068\pi\)
−0.333020 + 0.942920i \(0.608068\pi\)
\(104\) −810.689 −0.764371
\(105\) 300.404 0.279204
\(106\) 510.561 0.467831
\(107\) 611.211 0.552224 0.276112 0.961125i \(-0.410954\pi\)
0.276112 + 0.961125i \(0.410954\pi\)
\(108\) 101.129 0.0901030
\(109\) 1109.55 0.975006 0.487503 0.873121i \(-0.337908\pi\)
0.487503 + 0.873121i \(0.337908\pi\)
\(110\) −111.094 −0.0962946
\(111\) −471.297 −0.403005
\(112\) 146.232 0.123371
\(113\) −801.022 −0.666848 −0.333424 0.942777i \(-0.608204\pi\)
−0.333424 + 0.942777i \(0.608204\pi\)
\(114\) −272.362 −0.223763
\(115\) −454.296 −0.368377
\(116\) 557.433 0.446175
\(117\) −786.705 −0.621631
\(118\) 1351.40 1.05430
\(119\) −830.895 −0.640068
\(120\) −854.357 −0.649932
\(121\) 121.000 0.0909091
\(122\) 1103.25 0.818719
\(123\) 3560.93 2.61039
\(124\) −150.249 −0.108812
\(125\) −125.000 −0.0894427
\(126\) 399.543 0.282493
\(127\) −1926.14 −1.34581 −0.672904 0.739729i \(-0.734953\pi\)
−0.672904 + 0.739729i \(0.734953\pi\)
\(128\) 339.180 0.234215
\(129\) −1767.93 −1.20665
\(130\) −340.053 −0.229420
\(131\) −2340.12 −1.56074 −0.780370 0.625318i \(-0.784969\pi\)
−0.780370 + 0.625318i \(0.784969\pi\)
\(132\) 306.017 0.201783
\(133\) −160.852 −0.104869
\(134\) −1309.43 −0.844159
\(135\) 128.990 0.0822346
\(136\) 2363.09 1.48995
\(137\) −351.320 −0.219090 −0.109545 0.993982i \(-0.534939\pi\)
−0.109545 + 0.993982i \(0.534939\pi\)
\(138\) −1302.45 −0.803420
\(139\) 544.833 0.332461 0.166231 0.986087i \(-0.446840\pi\)
0.166231 + 0.986087i \(0.446840\pi\)
\(140\) −165.933 −0.100170
\(141\) 1915.75 1.14422
\(142\) −1561.47 −0.922788
\(143\) 370.375 0.216589
\(144\) −403.583 −0.233555
\(145\) 711.005 0.407212
\(146\) 1384.60 0.784863
\(147\) −1925.57 −1.08040
\(148\) 260.328 0.144587
\(149\) 2517.27 1.38405 0.692024 0.721875i \(-0.256719\pi\)
0.692024 + 0.721875i \(0.256719\pi\)
\(150\) −358.370 −0.195072
\(151\) 2695.70 1.45280 0.726400 0.687272i \(-0.241192\pi\)
0.726400 + 0.687272i \(0.241192\pi\)
\(152\) 457.467 0.244115
\(153\) 2293.18 1.21171
\(154\) −188.102 −0.0984265
\(155\) −191.642 −0.0993101
\(156\) 936.703 0.480745
\(157\) −2298.33 −1.16832 −0.584162 0.811637i \(-0.698577\pi\)
−0.584162 + 0.811637i \(0.698577\pi\)
\(158\) 2542.42 1.28015
\(159\) −1793.84 −0.894721
\(160\) 788.639 0.389671
\(161\) −769.203 −0.376532
\(162\) 1644.06 0.797343
\(163\) −3635.99 −1.74720 −0.873598 0.486649i \(-0.838219\pi\)
−0.873598 + 0.486649i \(0.838219\pi\)
\(164\) −1966.93 −0.936534
\(165\) 390.325 0.184162
\(166\) 2606.73 1.21881
\(167\) 4033.97 1.86921 0.934604 0.355689i \(-0.115754\pi\)
0.934604 + 0.355689i \(0.115754\pi\)
\(168\) −1446.58 −0.664320
\(169\) −1063.30 −0.483979
\(170\) 991.226 0.447197
\(171\) 443.932 0.198528
\(172\) 976.541 0.432910
\(173\) −985.837 −0.433247 −0.216624 0.976255i \(-0.569504\pi\)
−0.216624 + 0.976255i \(0.569504\pi\)
\(174\) 2038.43 0.888119
\(175\) −211.647 −0.0914229
\(176\) 190.004 0.0813754
\(177\) −4748.11 −2.01633
\(178\) 1484.60 0.625144
\(179\) −1643.02 −0.686063 −0.343032 0.939324i \(-0.611454\pi\)
−0.343032 + 0.939324i \(0.611454\pi\)
\(180\) 457.955 0.189633
\(181\) 896.230 0.368046 0.184023 0.982922i \(-0.441088\pi\)
0.184023 + 0.982922i \(0.441088\pi\)
\(182\) −575.770 −0.234499
\(183\) −3876.23 −1.56579
\(184\) 2187.64 0.876492
\(185\) 332.048 0.131960
\(186\) −549.431 −0.216593
\(187\) −1079.61 −0.422187
\(188\) −1058.19 −0.410514
\(189\) 218.402 0.0840552
\(190\) 191.890 0.0732692
\(191\) 2241.19 0.849041 0.424521 0.905418i \(-0.360443\pi\)
0.424521 + 0.905418i \(0.360443\pi\)
\(192\) 3241.67 1.21848
\(193\) 3670.91 1.36911 0.684553 0.728963i \(-0.259997\pi\)
0.684553 + 0.728963i \(0.259997\pi\)
\(194\) 2032.53 0.752203
\(195\) 1194.76 0.438763
\(196\) 1063.62 0.387616
\(197\) −3468.41 −1.25439 −0.627193 0.778864i \(-0.715796\pi\)
−0.627193 + 0.778864i \(0.715796\pi\)
\(198\) 519.140 0.186332
\(199\) 999.241 0.355951 0.177976 0.984035i \(-0.443045\pi\)
0.177976 + 0.984035i \(0.443045\pi\)
\(200\) 601.930 0.212814
\(201\) 4600.63 1.61444
\(202\) −2438.29 −0.849293
\(203\) 1203.86 0.416227
\(204\) −2730.41 −0.937093
\(205\) −2508.82 −0.854750
\(206\) 1406.32 0.475646
\(207\) 2122.91 0.712815
\(208\) 581.592 0.193876
\(209\) −209.000 −0.0691714
\(210\) −606.784 −0.199391
\(211\) −5828.82 −1.90176 −0.950882 0.309553i \(-0.899821\pi\)
−0.950882 + 0.309553i \(0.899821\pi\)
\(212\) 990.853 0.321000
\(213\) 5486.18 1.76482
\(214\) −1234.58 −0.394365
\(215\) 1245.58 0.395106
\(216\) −621.142 −0.195664
\(217\) −324.484 −0.101509
\(218\) −2241.17 −0.696291
\(219\) −4864.73 −1.50104
\(220\) −215.602 −0.0660721
\(221\) −3304.63 −1.00585
\(222\) 951.970 0.287802
\(223\) −4403.88 −1.32245 −0.661223 0.750190i \(-0.729962\pi\)
−0.661223 + 0.750190i \(0.729962\pi\)
\(224\) 1335.30 0.398298
\(225\) 584.121 0.173073
\(226\) 1617.98 0.476223
\(227\) −5588.56 −1.63403 −0.817017 0.576614i \(-0.804374\pi\)
−0.817017 + 0.576614i \(0.804374\pi\)
\(228\) −528.576 −0.153534
\(229\) −5112.12 −1.47519 −0.737595 0.675243i \(-0.764039\pi\)
−0.737595 + 0.675243i \(0.764039\pi\)
\(230\) 917.630 0.263073
\(231\) 660.889 0.188239
\(232\) −3423.80 −0.968895
\(233\) −3928.08 −1.10445 −0.552225 0.833695i \(-0.686221\pi\)
−0.552225 + 0.833695i \(0.686221\pi\)
\(234\) 1589.06 0.443932
\(235\) −1349.72 −0.374665
\(236\) 2622.69 0.723401
\(237\) −8932.71 −2.44828
\(238\) 1678.32 0.457098
\(239\) 4642.18 1.25639 0.628196 0.778055i \(-0.283794\pi\)
0.628196 + 0.778055i \(0.283794\pi\)
\(240\) 612.919 0.164849
\(241\) −3635.15 −0.971620 −0.485810 0.874064i \(-0.661475\pi\)
−0.485810 + 0.874064i \(0.661475\pi\)
\(242\) −244.407 −0.0649218
\(243\) −5079.80 −1.34103
\(244\) 2141.09 0.561760
\(245\) 1356.64 0.353767
\(246\) −7192.70 −1.86419
\(247\) −639.738 −0.164800
\(248\) 922.841 0.236292
\(249\) −9158.66 −2.33095
\(250\) 252.487 0.0638746
\(251\) −1114.30 −0.280214 −0.140107 0.990136i \(-0.544745\pi\)
−0.140107 + 0.990136i \(0.544745\pi\)
\(252\) 775.398 0.193831
\(253\) −999.452 −0.248360
\(254\) 3890.61 0.961096
\(255\) −3482.64 −0.855259
\(256\) −4339.33 −1.05941
\(257\) −1864.36 −0.452512 −0.226256 0.974068i \(-0.572649\pi\)
−0.226256 + 0.974068i \(0.572649\pi\)
\(258\) 3571.02 0.861714
\(259\) 562.216 0.134882
\(260\) −659.946 −0.157416
\(261\) −3322.51 −0.787962
\(262\) 4726.78 1.11459
\(263\) −2985.71 −0.700025 −0.350012 0.936745i \(-0.613823\pi\)
−0.350012 + 0.936745i \(0.613823\pi\)
\(264\) −1879.59 −0.438184
\(265\) 1263.83 0.292968
\(266\) 324.903 0.0748913
\(267\) −5216.10 −1.19558
\(268\) −2541.22 −0.579216
\(269\) 2454.88 0.556419 0.278210 0.960520i \(-0.410259\pi\)
0.278210 + 0.960520i \(0.410259\pi\)
\(270\) −260.545 −0.0587270
\(271\) 4937.48 1.10676 0.553378 0.832930i \(-0.313339\pi\)
0.553378 + 0.832930i \(0.313339\pi\)
\(272\) −1695.29 −0.377912
\(273\) 2022.95 0.448477
\(274\) 709.629 0.156461
\(275\) −275.000 −0.0603023
\(276\) −2527.68 −0.551263
\(277\) 8417.50 1.82584 0.912921 0.408135i \(-0.133821\pi\)
0.912921 + 0.408135i \(0.133821\pi\)
\(278\) −1100.50 −0.237424
\(279\) 895.538 0.192167
\(280\) 1019.17 0.217526
\(281\) −3428.72 −0.727902 −0.363951 0.931418i \(-0.618572\pi\)
−0.363951 + 0.931418i \(0.618572\pi\)
\(282\) −3869.61 −0.817134
\(283\) 3319.00 0.697153 0.348576 0.937280i \(-0.386665\pi\)
0.348576 + 0.937280i \(0.386665\pi\)
\(284\) −3030.37 −0.633167
\(285\) −674.198 −0.140126
\(286\) −748.117 −0.154675
\(287\) −4247.88 −0.873673
\(288\) −3685.29 −0.754019
\(289\) 4719.71 0.960658
\(290\) −1436.15 −0.290806
\(291\) −7141.24 −1.43858
\(292\) 2687.11 0.538531
\(293\) −3387.73 −0.675473 −0.337736 0.941241i \(-0.609661\pi\)
−0.337736 + 0.941241i \(0.609661\pi\)
\(294\) 3889.45 0.771556
\(295\) 3345.24 0.660228
\(296\) −1598.96 −0.313978
\(297\) 283.777 0.0554425
\(298\) −5084.62 −0.988404
\(299\) −3059.27 −0.591713
\(300\) −695.494 −0.133848
\(301\) 2108.98 0.403853
\(302\) −5445.02 −1.03750
\(303\) 8566.83 1.62426
\(304\) −328.188 −0.0619174
\(305\) 2730.96 0.512704
\(306\) −4631.97 −0.865334
\(307\) −282.993 −0.0526100 −0.0263050 0.999654i \(-0.508374\pi\)
−0.0263050 + 0.999654i \(0.508374\pi\)
\(308\) −365.052 −0.0675349
\(309\) −4941.06 −0.909667
\(310\) 387.097 0.0709213
\(311\) −1137.96 −0.207484 −0.103742 0.994604i \(-0.533082\pi\)
−0.103742 + 0.994604i \(0.533082\pi\)
\(312\) −5753.31 −1.04397
\(313\) 6864.16 1.23957 0.619784 0.784772i \(-0.287220\pi\)
0.619784 + 0.784772i \(0.287220\pi\)
\(314\) 4642.39 0.834347
\(315\) 989.020 0.176905
\(316\) 4934.11 0.878372
\(317\) −7369.52 −1.30572 −0.652861 0.757478i \(-0.726431\pi\)
−0.652861 + 0.757478i \(0.726431\pi\)
\(318\) 3623.36 0.638956
\(319\) 1564.21 0.274542
\(320\) −2283.89 −0.398979
\(321\) 4337.66 0.754219
\(322\) 1553.71 0.268897
\(323\) 1864.78 0.321236
\(324\) 3190.65 0.547094
\(325\) −841.761 −0.143669
\(326\) 7344.31 1.24774
\(327\) 7874.28 1.33165
\(328\) 12081.1 2.03374
\(329\) −2285.32 −0.382960
\(330\) −788.415 −0.131518
\(331\) −3486.74 −0.578998 −0.289499 0.957178i \(-0.593489\pi\)
−0.289499 + 0.957178i \(0.593489\pi\)
\(332\) 5058.92 0.836278
\(333\) −1551.65 −0.255345
\(334\) −8148.18 −1.33488
\(335\) −3241.33 −0.528635
\(336\) 1037.78 0.168499
\(337\) 8742.45 1.41315 0.706575 0.707638i \(-0.250239\pi\)
0.706575 + 0.707638i \(0.250239\pi\)
\(338\) 2147.76 0.345629
\(339\) −5684.71 −0.910771
\(340\) 1923.69 0.306843
\(341\) −421.613 −0.0669549
\(342\) −896.696 −0.141777
\(343\) 5200.83 0.818713
\(344\) −5998.00 −0.940089
\(345\) −3224.06 −0.503123
\(346\) 1991.28 0.309399
\(347\) −2887.00 −0.446635 −0.223318 0.974746i \(-0.571689\pi\)
−0.223318 + 0.974746i \(0.571689\pi\)
\(348\) 3956.00 0.609379
\(349\) −3735.62 −0.572961 −0.286480 0.958086i \(-0.592485\pi\)
−0.286480 + 0.958086i \(0.592485\pi\)
\(350\) 427.504 0.0652887
\(351\) 868.628 0.132091
\(352\) 1735.01 0.262716
\(353\) 1311.14 0.197690 0.0988452 0.995103i \(-0.468485\pi\)
0.0988452 + 0.995103i \(0.468485\pi\)
\(354\) 9590.68 1.43994
\(355\) −3865.24 −0.577874
\(356\) 2881.19 0.428940
\(357\) −5896.71 −0.874194
\(358\) 3318.73 0.489945
\(359\) 567.020 0.0833598 0.0416799 0.999131i \(-0.486729\pi\)
0.0416799 + 0.999131i \(0.486729\pi\)
\(360\) −2812.80 −0.411799
\(361\) 361.000 0.0526316
\(362\) −1810.29 −0.262836
\(363\) 858.715 0.124162
\(364\) −1117.40 −0.160901
\(365\) 3427.40 0.491502
\(366\) 7829.58 1.11819
\(367\) −8364.22 −1.18967 −0.594835 0.803848i \(-0.702782\pi\)
−0.594835 + 0.803848i \(0.702782\pi\)
\(368\) −1569.42 −0.222314
\(369\) 11723.7 1.65395
\(370\) −670.702 −0.0942381
\(371\) 2139.89 0.299454
\(372\) −1066.29 −0.148614
\(373\) 11870.5 1.64780 0.823899 0.566736i \(-0.191794\pi\)
0.823899 + 0.566736i \(0.191794\pi\)
\(374\) 2180.70 0.301500
\(375\) −887.103 −0.122159
\(376\) 6499.51 0.891454
\(377\) 4787.97 0.654093
\(378\) −441.149 −0.0600271
\(379\) −9903.32 −1.34222 −0.671108 0.741360i \(-0.734181\pi\)
−0.671108 + 0.741360i \(0.734181\pi\)
\(380\) 372.403 0.0502734
\(381\) −13669.5 −1.83808
\(382\) −4526.96 −0.606334
\(383\) 8875.98 1.18418 0.592090 0.805872i \(-0.298303\pi\)
0.592090 + 0.805872i \(0.298303\pi\)
\(384\) 2407.10 0.319888
\(385\) −465.623 −0.0616373
\(386\) −7414.84 −0.977734
\(387\) −5820.55 −0.764535
\(388\) 3944.57 0.516121
\(389\) 3628.59 0.472948 0.236474 0.971638i \(-0.424008\pi\)
0.236474 + 0.971638i \(0.424008\pi\)
\(390\) −2413.30 −0.313339
\(391\) 8917.51 1.15340
\(392\) −6532.84 −0.841730
\(393\) −16607.4 −2.13163
\(394\) 7005.82 0.895807
\(395\) 6293.46 0.801666
\(396\) 1007.50 0.127851
\(397\) 8027.83 1.01487 0.507437 0.861689i \(-0.330593\pi\)
0.507437 + 0.861689i \(0.330593\pi\)
\(398\) −2018.36 −0.254199
\(399\) −1141.54 −0.143229
\(400\) −431.827 −0.0539783
\(401\) 1041.43 0.129692 0.0648462 0.997895i \(-0.479344\pi\)
0.0648462 + 0.997895i \(0.479344\pi\)
\(402\) −9292.77 −1.15294
\(403\) −1290.53 −0.159519
\(404\) −4732.01 −0.582739
\(405\) 4069.67 0.499318
\(406\) −2431.66 −0.297245
\(407\) 730.506 0.0889676
\(408\) 16770.4 2.03495
\(409\) −14008.7 −1.69361 −0.846805 0.531903i \(-0.821477\pi\)
−0.846805 + 0.531903i \(0.821477\pi\)
\(410\) 5067.55 0.610411
\(411\) −2493.26 −0.299229
\(412\) 2729.27 0.326363
\(413\) 5664.08 0.674845
\(414\) −4288.06 −0.509050
\(415\) 6452.65 0.763249
\(416\) 5310.76 0.625917
\(417\) 3866.58 0.454070
\(418\) 422.158 0.0493981
\(419\) −10197.5 −1.18897 −0.594487 0.804105i \(-0.702645\pi\)
−0.594487 + 0.804105i \(0.702645\pi\)
\(420\) −1177.59 −0.136811
\(421\) −2664.97 −0.308510 −0.154255 0.988031i \(-0.549298\pi\)
−0.154255 + 0.988031i \(0.549298\pi\)
\(422\) 11773.6 1.35813
\(423\) 6307.22 0.724983
\(424\) −6085.90 −0.697070
\(425\) 2453.66 0.280047
\(426\) −11081.5 −1.26033
\(427\) 4624.00 0.524054
\(428\) −2395.97 −0.270592
\(429\) 2628.48 0.295814
\(430\) −2515.93 −0.282161
\(431\) 5565.87 0.622038 0.311019 0.950404i \(-0.399330\pi\)
0.311019 + 0.950404i \(0.399330\pi\)
\(432\) 445.610 0.0496283
\(433\) 10951.1 1.21542 0.607710 0.794159i \(-0.292088\pi\)
0.607710 + 0.794159i \(0.292088\pi\)
\(434\) 655.422 0.0724914
\(435\) 5045.88 0.556164
\(436\) −4349.48 −0.477757
\(437\) 1726.33 0.188973
\(438\) 9826.23 1.07195
\(439\) −3597.17 −0.391079 −0.195540 0.980696i \(-0.562646\pi\)
−0.195540 + 0.980696i \(0.562646\pi\)
\(440\) 1324.25 0.143479
\(441\) −6339.56 −0.684544
\(442\) 6675.00 0.718320
\(443\) 7027.57 0.753702 0.376851 0.926274i \(-0.377007\pi\)
0.376851 + 0.926274i \(0.377007\pi\)
\(444\) 1847.50 0.197474
\(445\) 3674.96 0.391482
\(446\) 8895.36 0.944411
\(447\) 17864.6 1.89031
\(448\) −3867.02 −0.407812
\(449\) −12874.1 −1.35316 −0.676578 0.736371i \(-0.736538\pi\)
−0.676578 + 0.736371i \(0.736538\pi\)
\(450\) −1179.86 −0.123598
\(451\) −5519.41 −0.576272
\(452\) 3140.03 0.326758
\(453\) 19130.9 1.98421
\(454\) 11288.3 1.16693
\(455\) −1425.25 −0.146850
\(456\) 3246.56 0.333408
\(457\) 12730.1 1.30304 0.651522 0.758630i \(-0.274131\pi\)
0.651522 + 0.758630i \(0.274131\pi\)
\(458\) 10325.9 1.05349
\(459\) −2531.97 −0.257478
\(460\) 1780.86 0.180506
\(461\) −5008.81 −0.506038 −0.253019 0.967461i \(-0.581424\pi\)
−0.253019 + 0.967461i \(0.581424\pi\)
\(462\) −1334.92 −0.134429
\(463\) −8923.88 −0.895741 −0.447870 0.894098i \(-0.647817\pi\)
−0.447870 + 0.894098i \(0.647817\pi\)
\(464\) 2456.25 0.245751
\(465\) −1360.05 −0.135636
\(466\) 7934.29 0.788732
\(467\) 10392.4 1.02977 0.514886 0.857259i \(-0.327834\pi\)
0.514886 + 0.857259i \(0.327834\pi\)
\(468\) 3083.91 0.304602
\(469\) −5488.14 −0.540339
\(470\) 2726.30 0.267563
\(471\) −16310.9 −1.59568
\(472\) −16108.8 −1.57091
\(473\) 2740.27 0.266380
\(474\) 18043.1 1.74841
\(475\) 475.000 0.0458831
\(476\) 3257.14 0.313636
\(477\) −5905.85 −0.566898
\(478\) −9376.71 −0.897240
\(479\) 8337.70 0.795322 0.397661 0.917532i \(-0.369822\pi\)
0.397661 + 0.917532i \(0.369822\pi\)
\(480\) 5596.83 0.532206
\(481\) 2236.04 0.211964
\(482\) 7342.61 0.693873
\(483\) −5458.90 −0.514262
\(484\) −474.324 −0.0445458
\(485\) 5031.29 0.471050
\(486\) 10260.7 0.957681
\(487\) 2788.06 0.259423 0.129712 0.991552i \(-0.458595\pi\)
0.129712 + 0.991552i \(0.458595\pi\)
\(488\) −13150.8 −1.21989
\(489\) −25804.0 −2.38629
\(490\) −2740.28 −0.252639
\(491\) −9107.39 −0.837089 −0.418545 0.908196i \(-0.637460\pi\)
−0.418545 + 0.908196i \(0.637460\pi\)
\(492\) −13959.0 −1.27910
\(493\) −13956.5 −1.27499
\(494\) 1292.20 0.117690
\(495\) 1285.07 0.116686
\(496\) −662.050 −0.0599333
\(497\) −6544.52 −0.590668
\(498\) 18499.5 1.66462
\(499\) 4366.94 0.391765 0.195883 0.980627i \(-0.437243\pi\)
0.195883 + 0.980627i \(0.437243\pi\)
\(500\) 490.004 0.0438273
\(501\) 28628.4 2.55294
\(502\) 2250.76 0.200112
\(503\) 11008.0 0.975793 0.487896 0.872902i \(-0.337764\pi\)
0.487896 + 0.872902i \(0.337764\pi\)
\(504\) −4762.57 −0.420916
\(505\) −6035.68 −0.531850
\(506\) 2018.79 0.177364
\(507\) −7546.07 −0.661011
\(508\) 7550.55 0.659452
\(509\) −8378.75 −0.729629 −0.364815 0.931080i \(-0.618868\pi\)
−0.364815 + 0.931080i \(0.618868\pi\)
\(510\) 7034.55 0.610775
\(511\) 5803.19 0.502384
\(512\) 6051.54 0.522349
\(513\) −490.161 −0.0421854
\(514\) 3765.81 0.323157
\(515\) 3481.18 0.297862
\(516\) 6930.34 0.591262
\(517\) −2969.39 −0.252599
\(518\) −1135.62 −0.0963245
\(519\) −6996.30 −0.591722
\(520\) 4053.45 0.341837
\(521\) 14913.0 1.25403 0.627016 0.779006i \(-0.284276\pi\)
0.627016 + 0.779006i \(0.284276\pi\)
\(522\) 6711.11 0.562715
\(523\) 12149.1 1.01576 0.507882 0.861426i \(-0.330428\pi\)
0.507882 + 0.861426i \(0.330428\pi\)
\(524\) 9173.33 0.764769
\(525\) −1502.02 −0.124864
\(526\) 6030.81 0.499916
\(527\) 3761.80 0.310942
\(528\) 1348.42 0.111141
\(529\) −3911.60 −0.321492
\(530\) −2552.80 −0.209220
\(531\) −15632.2 −1.27755
\(532\) 630.544 0.0513864
\(533\) −16894.6 −1.37296
\(534\) 10536.0 0.853812
\(535\) −3056.06 −0.246962
\(536\) 15608.4 1.25780
\(537\) −11660.2 −0.937014
\(538\) −4958.60 −0.397361
\(539\) 2984.62 0.238509
\(540\) −505.644 −0.0402953
\(541\) −2442.51 −0.194107 −0.0970535 0.995279i \(-0.530942\pi\)
−0.0970535 + 0.995279i \(0.530942\pi\)
\(542\) −9973.19 −0.790378
\(543\) 6360.38 0.502671
\(544\) −15480.4 −1.22007
\(545\) −5547.75 −0.436036
\(546\) −4086.13 −0.320275
\(547\) 20864.1 1.63087 0.815433 0.578851i \(-0.196499\pi\)
0.815433 + 0.578851i \(0.196499\pi\)
\(548\) 1377.19 0.107355
\(549\) −12761.7 −0.992089
\(550\) 555.471 0.0430643
\(551\) −2701.82 −0.208895
\(552\) 15525.3 1.19710
\(553\) 10655.9 0.819415
\(554\) −17002.4 −1.30391
\(555\) 2356.49 0.180229
\(556\) −2135.76 −0.162907
\(557\) 172.091 0.0130911 0.00654553 0.999979i \(-0.497916\pi\)
0.00654553 + 0.999979i \(0.497916\pi\)
\(558\) −1808.89 −0.137234
\(559\) 8387.82 0.634646
\(560\) −731.159 −0.0551734
\(561\) −7661.80 −0.576616
\(562\) 6925.65 0.519824
\(563\) 18482.5 1.38356 0.691779 0.722109i \(-0.256827\pi\)
0.691779 + 0.722109i \(0.256827\pi\)
\(564\) −7509.80 −0.560673
\(565\) 4005.11 0.298224
\(566\) −6704.03 −0.497865
\(567\) 6890.67 0.510372
\(568\) 18612.8 1.37496
\(569\) 14030.7 1.03374 0.516868 0.856065i \(-0.327098\pi\)
0.516868 + 0.856065i \(0.327098\pi\)
\(570\) 1361.81 0.100070
\(571\) 185.650 0.0136063 0.00680317 0.999977i \(-0.497834\pi\)
0.00680317 + 0.999977i \(0.497834\pi\)
\(572\) −1451.88 −0.106130
\(573\) 15905.3 1.15961
\(574\) 8580.25 0.623925
\(575\) 2271.48 0.164743
\(576\) 10672.5 0.772030
\(577\) −16102.5 −1.16180 −0.580898 0.813976i \(-0.697299\pi\)
−0.580898 + 0.813976i \(0.697299\pi\)
\(578\) −9533.31 −0.686044
\(579\) 26051.8 1.86990
\(580\) −2787.16 −0.199536
\(581\) 10925.5 0.780146
\(582\) 14424.5 1.02735
\(583\) 2780.43 0.197519
\(584\) −16504.4 −1.16945
\(585\) 3933.52 0.278002
\(586\) 6842.86 0.482382
\(587\) −10494.4 −0.737902 −0.368951 0.929449i \(-0.620283\pi\)
−0.368951 + 0.929449i \(0.620283\pi\)
\(588\) 7548.31 0.529400
\(589\) 728.240 0.0509450
\(590\) −6757.02 −0.471495
\(591\) −24614.7 −1.71322
\(592\) 1147.10 0.0796376
\(593\) −22833.3 −1.58120 −0.790600 0.612333i \(-0.790231\pi\)
−0.790600 + 0.612333i \(0.790231\pi\)
\(594\) −573.200 −0.0395937
\(595\) 4154.48 0.286247
\(596\) −9867.80 −0.678189
\(597\) 7091.44 0.486153
\(598\) 6179.40 0.422566
\(599\) 23137.1 1.57822 0.789112 0.614250i \(-0.210541\pi\)
0.789112 + 0.614250i \(0.210541\pi\)
\(600\) 4271.79 0.290658
\(601\) 24923.2 1.69158 0.845790 0.533516i \(-0.179130\pi\)
0.845790 + 0.533516i \(0.179130\pi\)
\(602\) −4259.92 −0.288407
\(603\) 15146.6 1.02292
\(604\) −10567.2 −0.711878
\(605\) −605.000 −0.0406558
\(606\) −17304.1 −1.15995
\(607\) 25216.6 1.68618 0.843088 0.537776i \(-0.180735\pi\)
0.843088 + 0.537776i \(0.180735\pi\)
\(608\) −2996.83 −0.199897
\(609\) 8543.55 0.568477
\(610\) −5516.26 −0.366142
\(611\) −9089.15 −0.601813
\(612\) −8989.33 −0.593745
\(613\) −12880.9 −0.848701 −0.424351 0.905498i \(-0.639498\pi\)
−0.424351 + 0.905498i \(0.639498\pi\)
\(614\) 571.616 0.0375709
\(615\) −17804.7 −1.16740
\(616\) 2242.18 0.146656
\(617\) −3004.83 −0.196062 −0.0980308 0.995183i \(-0.531254\pi\)
−0.0980308 + 0.995183i \(0.531254\pi\)
\(618\) 9980.42 0.649630
\(619\) 18673.8 1.21254 0.606271 0.795258i \(-0.292665\pi\)
0.606271 + 0.795258i \(0.292665\pi\)
\(620\) 751.244 0.0486624
\(621\) −2343.98 −0.151467
\(622\) 2298.55 0.148173
\(623\) 6222.34 0.400149
\(624\) 4127.45 0.264792
\(625\) 625.000 0.0400000
\(626\) −13864.9 −0.885226
\(627\) −1483.24 −0.0944732
\(628\) 9009.55 0.572484
\(629\) −6517.86 −0.413170
\(630\) −1997.71 −0.126335
\(631\) 7662.68 0.483433 0.241717 0.970347i \(-0.422290\pi\)
0.241717 + 0.970347i \(0.422290\pi\)
\(632\) −30305.8 −1.90743
\(633\) −41366.1 −2.59740
\(634\) 14885.6 0.932468
\(635\) 9630.72 0.601864
\(636\) 7031.90 0.438417
\(637\) 9135.76 0.568245
\(638\) −3159.54 −0.196062
\(639\) 18062.1 1.11820
\(640\) −1695.90 −0.104744
\(641\) −23554.0 −1.45137 −0.725685 0.688027i \(-0.758477\pi\)
−0.725685 + 0.688027i \(0.758477\pi\)
\(642\) −8761.60 −0.538618
\(643\) 31637.7 1.94039 0.970194 0.242331i \(-0.0779120\pi\)
0.970194 + 0.242331i \(0.0779120\pi\)
\(644\) 3015.30 0.184502
\(645\) 8839.64 0.539629
\(646\) −3766.66 −0.229407
\(647\) 4828.07 0.293371 0.146686 0.989183i \(-0.453139\pi\)
0.146686 + 0.989183i \(0.453139\pi\)
\(648\) −19597.3 −1.18804
\(649\) 7359.53 0.445126
\(650\) 1700.27 0.102600
\(651\) −2302.80 −0.138639
\(652\) 14253.2 0.856133
\(653\) 22266.6 1.33439 0.667197 0.744881i \(-0.267494\pi\)
0.667197 + 0.744881i \(0.267494\pi\)
\(654\) −15905.2 −0.950983
\(655\) 11700.6 0.697984
\(656\) −8667.01 −0.515838
\(657\) −16016.1 −0.951065
\(658\) 4616.10 0.273487
\(659\) 23347.4 1.38010 0.690050 0.723761i \(-0.257588\pi\)
0.690050 + 0.723761i \(0.257588\pi\)
\(660\) −1530.09 −0.0902403
\(661\) −690.780 −0.0406478 −0.0203239 0.999793i \(-0.506470\pi\)
−0.0203239 + 0.999793i \(0.506470\pi\)
\(662\) 7042.83 0.413485
\(663\) −23452.4 −1.37378
\(664\) −31072.3 −1.81603
\(665\) 804.258 0.0468990
\(666\) 3134.17 0.182352
\(667\) −12920.3 −0.750038
\(668\) −15813.3 −0.915920
\(669\) −31253.5 −1.80617
\(670\) 6547.14 0.377519
\(671\) 6008.12 0.345665
\(672\) 9476.41 0.543989
\(673\) 20393.0 1.16804 0.584021 0.811739i \(-0.301479\pi\)
0.584021 + 0.811739i \(0.301479\pi\)
\(674\) −17658.8 −1.00919
\(675\) −644.949 −0.0367764
\(676\) 4168.18 0.237152
\(677\) −17385.9 −0.986995 −0.493497 0.869747i \(-0.664282\pi\)
−0.493497 + 0.869747i \(0.664282\pi\)
\(678\) 11482.5 0.650418
\(679\) 8518.86 0.481478
\(680\) −11815.4 −0.666326
\(681\) −39661.0 −2.23174
\(682\) 851.612 0.0478151
\(683\) −24654.3 −1.38122 −0.690609 0.723228i \(-0.742657\pi\)
−0.690609 + 0.723228i \(0.742657\pi\)
\(684\) −1740.23 −0.0972797
\(685\) 1756.60 0.0979799
\(686\) −10505.1 −0.584676
\(687\) −36279.8 −2.01479
\(688\) 4302.99 0.238445
\(689\) 8510.75 0.470586
\(690\) 6512.25 0.359300
\(691\) −24883.5 −1.36992 −0.684959 0.728581i \(-0.740180\pi\)
−0.684959 + 0.728581i \(0.740180\pi\)
\(692\) 3864.51 0.212293
\(693\) 2175.84 0.119269
\(694\) 5831.44 0.318960
\(695\) −2724.17 −0.148681
\(696\) −24298.1 −1.32330
\(697\) 49246.3 2.67624
\(698\) 7545.56 0.409174
\(699\) −27876.9 −1.50844
\(700\) 829.663 0.0447976
\(701\) 26129.9 1.40786 0.703932 0.710267i \(-0.251426\pi\)
0.703932 + 0.710267i \(0.251426\pi\)
\(702\) −1754.53 −0.0943314
\(703\) −1261.78 −0.0676942
\(704\) −5024.55 −0.268992
\(705\) −9578.75 −0.511711
\(706\) −2648.35 −0.141179
\(707\) −10219.5 −0.543625
\(708\) 18612.8 0.988009
\(709\) −32752.9 −1.73492 −0.867462 0.497503i \(-0.834250\pi\)
−0.867462 + 0.497503i \(0.834250\pi\)
\(710\) 7807.36 0.412683
\(711\) −29409.2 −1.55124
\(712\) −17696.5 −0.931468
\(713\) 3482.49 0.182918
\(714\) 11910.7 0.624297
\(715\) −1851.87 −0.0968617
\(716\) 6440.70 0.336174
\(717\) 32944.7 1.71596
\(718\) −1145.32 −0.0595306
\(719\) 1736.28 0.0900587 0.0450294 0.998986i \(-0.485662\pi\)
0.0450294 + 0.998986i \(0.485662\pi\)
\(720\) 2017.91 0.104449
\(721\) 5894.25 0.304457
\(722\) −729.181 −0.0375863
\(723\) −25798.0 −1.32702
\(724\) −3513.25 −0.180344
\(725\) −3555.03 −0.182111
\(726\) −1734.51 −0.0886692
\(727\) 10714.3 0.546589 0.273294 0.961930i \(-0.411887\pi\)
0.273294 + 0.961930i \(0.411887\pi\)
\(728\) 6863.19 0.349405
\(729\) −14074.2 −0.715044
\(730\) −6922.98 −0.351001
\(731\) −24449.8 −1.23708
\(732\) 15195.0 0.767243
\(733\) −17240.1 −0.868729 −0.434365 0.900737i \(-0.643027\pi\)
−0.434365 + 0.900737i \(0.643027\pi\)
\(734\) 16894.8 0.849590
\(735\) 9627.86 0.483169
\(736\) −14331.0 −0.717729
\(737\) −7130.93 −0.356406
\(738\) −23680.5 −1.18115
\(739\) 25523.9 1.27052 0.635260 0.772299i \(-0.280893\pi\)
0.635260 + 0.772299i \(0.280893\pi\)
\(740\) −1301.64 −0.0646611
\(741\) −4540.11 −0.225081
\(742\) −4322.35 −0.213852
\(743\) −5396.65 −0.266466 −0.133233 0.991085i \(-0.542536\pi\)
−0.133233 + 0.991085i \(0.542536\pi\)
\(744\) 6549.23 0.322724
\(745\) −12586.4 −0.618965
\(746\) −23977.0 −1.17676
\(747\) −30153.1 −1.47690
\(748\) 4232.11 0.206873
\(749\) −5174.44 −0.252430
\(750\) 1791.85 0.0872389
\(751\) 16662.9 0.809635 0.404818 0.914397i \(-0.367335\pi\)
0.404818 + 0.914397i \(0.367335\pi\)
\(752\) −4662.78 −0.226109
\(753\) −7907.96 −0.382712
\(754\) −9671.18 −0.467114
\(755\) −13478.5 −0.649712
\(756\) −856.144 −0.0411874
\(757\) −35292.9 −1.69451 −0.847253 0.531189i \(-0.821745\pi\)
−0.847253 + 0.531189i \(0.821745\pi\)
\(758\) 20003.7 0.958530
\(759\) −7092.93 −0.339206
\(760\) −2287.33 −0.109171
\(761\) −23776.0 −1.13256 −0.566281 0.824212i \(-0.691618\pi\)
−0.566281 + 0.824212i \(0.691618\pi\)
\(762\) 27610.9 1.31265
\(763\) −9393.32 −0.445689
\(764\) −8785.54 −0.416034
\(765\) −11465.9 −0.541895
\(766\) −17928.5 −0.845671
\(767\) 22527.1 1.06050
\(768\) −30795.4 −1.44692
\(769\) 6885.24 0.322872 0.161436 0.986883i \(-0.448388\pi\)
0.161436 + 0.986883i \(0.448388\pi\)
\(770\) 940.509 0.0440177
\(771\) −13231.0 −0.618034
\(772\) −14390.1 −0.670868
\(773\) 6932.66 0.322575 0.161288 0.986907i \(-0.448435\pi\)
0.161288 + 0.986907i \(0.448435\pi\)
\(774\) 11756.9 0.545985
\(775\) 958.211 0.0444128
\(776\) −24227.9 −1.12079
\(777\) 3989.94 0.184219
\(778\) −7329.36 −0.337751
\(779\) 9533.52 0.438477
\(780\) −4683.52 −0.214996
\(781\) −8503.52 −0.389603
\(782\) −18012.4 −0.823686
\(783\) 3668.49 0.167435
\(784\) 4686.68 0.213497
\(785\) 11491.7 0.522491
\(786\) 33545.1 1.52228
\(787\) 35502.7 1.60805 0.804023 0.594598i \(-0.202689\pi\)
0.804023 + 0.594598i \(0.202689\pi\)
\(788\) 13596.3 0.614655
\(789\) −21189.0 −0.956083
\(790\) −12712.1 −0.572502
\(791\) 6781.36 0.304826
\(792\) −6188.16 −0.277635
\(793\) 18390.5 0.823540
\(794\) −16215.4 −0.724763
\(795\) 8969.19 0.400131
\(796\) −3917.06 −0.174418
\(797\) −25023.3 −1.11213 −0.556067 0.831137i \(-0.687690\pi\)
−0.556067 + 0.831137i \(0.687690\pi\)
\(798\) 2305.78 0.102285
\(799\) 26494.1 1.17308
\(800\) −3943.19 −0.174266
\(801\) −17173.0 −0.757524
\(802\) −2103.58 −0.0926185
\(803\) 7540.28 0.331371
\(804\) −18034.6 −0.791084
\(805\) 3846.02 0.168390
\(806\) 2606.74 0.113919
\(807\) 17421.8 0.759948
\(808\) 29064.4 1.26545
\(809\) 5913.07 0.256975 0.128487 0.991711i \(-0.458988\pi\)
0.128487 + 0.991711i \(0.458988\pi\)
\(810\) −8220.30 −0.356583
\(811\) −17566.5 −0.760594 −0.380297 0.924864i \(-0.624178\pi\)
−0.380297 + 0.924864i \(0.624178\pi\)
\(812\) −4719.16 −0.203953
\(813\) 35040.4 1.51159
\(814\) −1475.54 −0.0635353
\(815\) 18180.0 0.781369
\(816\) −12031.2 −0.516146
\(817\) −4733.19 −0.202685
\(818\) 28296.1 1.20947
\(819\) 6660.15 0.284157
\(820\) 9834.66 0.418831
\(821\) 8776.96 0.373103 0.186552 0.982445i \(-0.440269\pi\)
0.186552 + 0.982445i \(0.440269\pi\)
\(822\) 5036.11 0.213692
\(823\) −23197.1 −0.982505 −0.491253 0.871017i \(-0.663461\pi\)
−0.491253 + 0.871017i \(0.663461\pi\)
\(824\) −16763.4 −0.708715
\(825\) −1951.63 −0.0823599
\(826\) −11440.8 −0.481934
\(827\) −29552.0 −1.24259 −0.621296 0.783576i \(-0.713393\pi\)
−0.621296 + 0.783576i \(0.713393\pi\)
\(828\) −8321.89 −0.349282
\(829\) −19467.4 −0.815597 −0.407799 0.913072i \(-0.633703\pi\)
−0.407799 + 0.913072i \(0.633703\pi\)
\(830\) −13033.7 −0.545066
\(831\) 59737.5 2.49371
\(832\) −15379.9 −0.640868
\(833\) −26629.9 −1.10765
\(834\) −7810.08 −0.324270
\(835\) −20169.8 −0.835936
\(836\) 819.287 0.0338943
\(837\) −988.794 −0.0408336
\(838\) 20597.8 0.849094
\(839\) −38868.5 −1.59939 −0.799696 0.600405i \(-0.795006\pi\)
−0.799696 + 0.600405i \(0.795006\pi\)
\(840\) 7232.89 0.297093
\(841\) −4167.87 −0.170891
\(842\) 5382.96 0.220319
\(843\) −24333.0 −0.994156
\(844\) 22849.2 0.931872
\(845\) 5316.51 0.216442
\(846\) −12739.9 −0.517739
\(847\) −1024.37 −0.0415559
\(848\) 4366.05 0.176805
\(849\) 23554.4 0.952160
\(850\) −4956.13 −0.199993
\(851\) −6033.93 −0.243056
\(852\) −21506.0 −0.864769
\(853\) −5235.53 −0.210154 −0.105077 0.994464i \(-0.533509\pi\)
−0.105077 + 0.994464i \(0.533509\pi\)
\(854\) −9339.99 −0.374248
\(855\) −2219.66 −0.0887846
\(856\) 14716.2 0.587606
\(857\) −42559.1 −1.69637 −0.848186 0.529698i \(-0.822305\pi\)
−0.848186 + 0.529698i \(0.822305\pi\)
\(858\) −5309.25 −0.211253
\(859\) 9352.08 0.371465 0.185733 0.982600i \(-0.440534\pi\)
0.185733 + 0.982600i \(0.440534\pi\)
\(860\) −4882.71 −0.193603
\(861\) −30146.4 −1.19325
\(862\) −11242.5 −0.444222
\(863\) −21160.7 −0.834666 −0.417333 0.908754i \(-0.637035\pi\)
−0.417333 + 0.908754i \(0.637035\pi\)
\(864\) 4069.05 0.160222
\(865\) 4929.18 0.193754
\(866\) −22120.1 −0.867980
\(867\) 33494.9 1.31205
\(868\) 1271.99 0.0497397
\(869\) 13845.6 0.540483
\(870\) −10192.1 −0.397179
\(871\) −21827.4 −0.849131
\(872\) 26714.9 1.03748
\(873\) −23511.1 −0.911489
\(874\) −3486.99 −0.134953
\(875\) 1058.23 0.0408856
\(876\) 19069.9 0.735516
\(877\) −10461.9 −0.402820 −0.201410 0.979507i \(-0.564552\pi\)
−0.201410 + 0.979507i \(0.564552\pi\)
\(878\) 7265.91 0.279285
\(879\) −24042.1 −0.922550
\(880\) −950.019 −0.0363922
\(881\) 3846.51 0.147097 0.0735484 0.997292i \(-0.476568\pi\)
0.0735484 + 0.997292i \(0.476568\pi\)
\(882\) 12805.2 0.488860
\(883\) −14907.1 −0.568135 −0.284068 0.958804i \(-0.591684\pi\)
−0.284068 + 0.958804i \(0.591684\pi\)
\(884\) 12954.3 0.492872
\(885\) 23740.6 0.901729
\(886\) −14194.9 −0.538249
\(887\) 33284.1 1.25995 0.629973 0.776617i \(-0.283066\pi\)
0.629973 + 0.776617i \(0.283066\pi\)
\(888\) −11347.5 −0.428826
\(889\) 16306.5 0.615189
\(890\) −7423.02 −0.279573
\(891\) 8953.28 0.336640
\(892\) 17263.3 0.648004
\(893\) 5128.95 0.192199
\(894\) −36084.7 −1.34995
\(895\) 8215.11 0.306817
\(896\) −2871.46 −0.107063
\(897\) −21711.1 −0.808152
\(898\) 26004.3 0.966343
\(899\) −5450.34 −0.202201
\(900\) −2289.78 −0.0848065
\(901\) −24808.1 −0.917289
\(902\) 11148.6 0.411539
\(903\) 14967.1 0.551575
\(904\) −19286.4 −0.709574
\(905\) −4481.15 −0.164595
\(906\) −38642.3 −1.41700
\(907\) −22644.1 −0.828979 −0.414489 0.910054i \(-0.636040\pi\)
−0.414489 + 0.910054i \(0.636040\pi\)
\(908\) 21907.3 0.800683
\(909\) 28204.6 1.02914
\(910\) 2878.85 0.104871
\(911\) −53413.8 −1.94256 −0.971282 0.237929i \(-0.923531\pi\)
−0.971282 + 0.237929i \(0.923531\pi\)
\(912\) −2329.09 −0.0845658
\(913\) 14195.8 0.514582
\(914\) −25713.5 −0.930555
\(915\) 19381.2 0.700242
\(916\) 20039.7 0.722849
\(917\) 19811.1 0.713437
\(918\) 5114.32 0.183875
\(919\) −45492.3 −1.63292 −0.816459 0.577403i \(-0.804066\pi\)
−0.816459 + 0.577403i \(0.804066\pi\)
\(920\) −10938.2 −0.391979
\(921\) −2008.35 −0.0718539
\(922\) 10117.3 0.361382
\(923\) −26028.8 −0.928222
\(924\) −2590.71 −0.0922381
\(925\) −1660.24 −0.0590145
\(926\) 18025.3 0.639684
\(927\) −16267.5 −0.576368
\(928\) 22429.0 0.793394
\(929\) 46757.6 1.65131 0.825655 0.564175i \(-0.190806\pi\)
0.825655 + 0.564175i \(0.190806\pi\)
\(930\) 2747.15 0.0968632
\(931\) −5155.25 −0.181478
\(932\) 15398.2 0.541185
\(933\) −8075.87 −0.283378
\(934\) −20991.5 −0.735401
\(935\) 5398.05 0.188808
\(936\) −18941.6 −0.661460
\(937\) 32214.0 1.12314 0.561571 0.827428i \(-0.310197\pi\)
0.561571 + 0.827428i \(0.310197\pi\)
\(938\) 11085.5 0.385877
\(939\) 48713.7 1.69298
\(940\) 5290.96 0.183587
\(941\) −5294.98 −0.183434 −0.0917170 0.995785i \(-0.529236\pi\)
−0.0917170 + 0.995785i \(0.529236\pi\)
\(942\) 32946.2 1.13954
\(943\) 45589.9 1.57435
\(944\) 11556.5 0.398445
\(945\) −1092.01 −0.0375906
\(946\) −5535.05 −0.190233
\(947\) 24960.7 0.856508 0.428254 0.903658i \(-0.359129\pi\)
0.428254 + 0.903658i \(0.359129\pi\)
\(948\) 35016.5 1.19967
\(949\) 23080.4 0.789486
\(950\) −959.449 −0.0327670
\(951\) −52300.2 −1.78333
\(952\) −20005.6 −0.681078
\(953\) −11933.9 −0.405642 −0.202821 0.979216i \(-0.565011\pi\)
−0.202821 + 0.979216i \(0.565011\pi\)
\(954\) 11929.2 0.404844
\(955\) −11206.0 −0.379703
\(956\) −18197.5 −0.615637
\(957\) 11100.9 0.374966
\(958\) −16841.3 −0.567971
\(959\) 2974.23 0.100149
\(960\) −16208.3 −0.544919
\(961\) −28321.9 −0.950688
\(962\) −4516.56 −0.151372
\(963\) 14280.9 0.477876
\(964\) 14249.9 0.476098
\(965\) −18354.5 −0.612283
\(966\) 11026.4 0.367255
\(967\) 1573.62 0.0523312 0.0261656 0.999658i \(-0.491670\pi\)
0.0261656 + 0.999658i \(0.491670\pi\)
\(968\) 2913.34 0.0967338
\(969\) 13234.0 0.438739
\(970\) −10162.7 −0.336396
\(971\) −42094.8 −1.39123 −0.695616 0.718414i \(-0.744868\pi\)
−0.695616 + 0.718414i \(0.744868\pi\)
\(972\) 19913.0 0.657109
\(973\) −4612.49 −0.151973
\(974\) −5631.59 −0.185265
\(975\) −5973.82 −0.196221
\(976\) 9434.43 0.309415
\(977\) 23184.2 0.759188 0.379594 0.925153i \(-0.376064\pi\)
0.379594 + 0.925153i \(0.376064\pi\)
\(978\) 52121.3 1.70415
\(979\) 8084.90 0.263937
\(980\) −5318.09 −0.173347
\(981\) 25924.5 0.843736
\(982\) 18395.9 0.597799
\(983\) −56868.7 −1.84520 −0.922599 0.385760i \(-0.873939\pi\)
−0.922599 + 0.385760i \(0.873939\pi\)
\(984\) 85737.2 2.77764
\(985\) 17342.1 0.560979
\(986\) 28190.7 0.910521
\(987\) −16218.5 −0.523040
\(988\) 2507.79 0.0807526
\(989\) −22634.4 −0.727739
\(990\) −2595.70 −0.0833300
\(991\) 32533.9 1.04286 0.521430 0.853294i \(-0.325399\pi\)
0.521430 + 0.853294i \(0.325399\pi\)
\(992\) −6045.46 −0.193491
\(993\) −24744.7 −0.790786
\(994\) 13219.2 0.421820
\(995\) −4996.21 −0.159186
\(996\) 35902.3 1.14218
\(997\) 60795.3 1.93120 0.965600 0.260033i \(-0.0837335\pi\)
0.965600 + 0.260033i \(0.0837335\pi\)
\(998\) −8820.74 −0.279775
\(999\) 1713.23 0.0542585
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 1045.4.a.c.1.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.8 20 1.1 even 1 trivial