Properties

Label 23.6.a.a.1.2
Level $23$
Weight $6$
Character 23.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7925.1
Defining polynomial: \( x^{3} - x^{2} - 13x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.917748\) of defining polynomial
Character \(\chi\) \(=\) 23.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.164504 q^{2} -1.43100 q^{3} -31.9729 q^{4} -37.9865 q^{5} +0.235406 q^{6} -43.8366 q^{7} +10.5238 q^{8} -240.952 q^{9} +O(q^{10})\) \(q-0.164504 q^{2} -1.43100 q^{3} -31.9729 q^{4} -37.9865 q^{5} +0.235406 q^{6} -43.8366 q^{7} +10.5238 q^{8} -240.952 q^{9} +6.24894 q^{10} -163.213 q^{11} +45.7533 q^{12} +430.356 q^{13} +7.21131 q^{14} +54.3587 q^{15} +1021.40 q^{16} +740.415 q^{17} +39.6377 q^{18} -916.578 q^{19} +1214.54 q^{20} +62.7302 q^{21} +26.8493 q^{22} -529.000 q^{23} -15.0596 q^{24} -1682.03 q^{25} -70.7955 q^{26} +692.536 q^{27} +1401.59 q^{28} -5112.96 q^{29} -8.94223 q^{30} -5702.60 q^{31} -504.787 q^{32} +233.558 q^{33} -121.802 q^{34} +1665.20 q^{35} +7703.95 q^{36} -10913.1 q^{37} +150.781 q^{38} -615.840 q^{39} -399.763 q^{40} +11092.6 q^{41} -10.3194 q^{42} +5528.76 q^{43} +5218.41 q^{44} +9152.92 q^{45} +87.0228 q^{46} +14435.2 q^{47} -1461.63 q^{48} -14885.4 q^{49} +276.701 q^{50} -1059.53 q^{51} -13759.8 q^{52} +6852.79 q^{53} -113.925 q^{54} +6199.90 q^{55} -461.329 q^{56} +1311.62 q^{57} +841.104 q^{58} +4155.17 q^{59} -1738.01 q^{60} -21911.4 q^{61} +938.102 q^{62} +10562.5 q^{63} -32601.9 q^{64} -16347.7 q^{65} -38.4213 q^{66} -15164.2 q^{67} -23673.3 q^{68} +756.999 q^{69} -273.932 q^{70} +14031.4 q^{71} -2535.74 q^{72} -23310.1 q^{73} +1795.25 q^{74} +2406.98 q^{75} +29305.7 q^{76} +7154.72 q^{77} +101.308 q^{78} +64276.8 q^{79} -38799.5 q^{80} +57560.4 q^{81} -1824.78 q^{82} +114349. q^{83} -2005.67 q^{84} -28125.8 q^{85} -909.504 q^{86} +7316.65 q^{87} -1717.63 q^{88} -63420.7 q^{89} -1505.70 q^{90} -18865.4 q^{91} +16913.7 q^{92} +8160.42 q^{93} -2374.66 q^{94} +34817.6 q^{95} +722.351 q^{96} -91987.6 q^{97} +2448.70 q^{98} +39326.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 20 q^{3} + 16 q^{4} - 58 q^{5} - 230 q^{6} - 282 q^{7} - 360 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} - 20 q^{3} + 16 q^{4} - 58 q^{5} - 230 q^{6} - 282 q^{7} - 360 q^{8} + 121 q^{9} - 156 q^{10} + 136 q^{11} + 620 q^{12} - 1116 q^{13} + 2016 q^{14} + 750 q^{15} + 128 q^{16} - 896 q^{17} + 4282 q^{18} + 1654 q^{19} + 1504 q^{20} - 1670 q^{21} + 1352 q^{22} - 1587 q^{23} + 3600 q^{24} - 7347 q^{25} + 1998 q^{26} - 10700 q^{27} - 10264 q^{28} - 844 q^{29} + 3180 q^{30} - 3020 q^{31} + 6656 q^{32} - 7370 q^{33} - 11212 q^{34} + 1072 q^{35} - 2548 q^{36} + 8938 q^{37} + 10728 q^{38} + 16020 q^{39} + 3440 q^{40} - 12792 q^{41} + 20560 q^{42} - 16730 q^{43} + 5112 q^{44} - 3936 q^{45} + 2116 q^{46} + 22500 q^{47} + 23120 q^{48} + 2887 q^{49} + 15156 q^{50} + 50290 q^{51} - 47412 q^{52} + 17108 q^{53} - 7610 q^{54} - 436 q^{55} + 42640 q^{56} - 61960 q^{57} - 55678 q^{58} + 54176 q^{59} - 2400 q^{60} - 71324 q^{61} - 72710 q^{62} + 40696 q^{63} - 49984 q^{64} + 846 q^{65} - 42860 q^{66} - 62960 q^{67} - 8352 q^{68} + 10580 q^{69} + 9224 q^{70} + 98400 q^{71} - 72840 q^{72} - 81772 q^{73} - 59044 q^{74} + 44800 q^{75} + 31488 q^{76} - 304 q^{77} + 182070 q^{78} + 58224 q^{79} - 17568 q^{80} + 149947 q^{81} + 61926 q^{82} + 9892 q^{83} - 109720 q^{84} + 15536 q^{85} + 191140 q^{86} + 90500 q^{87} - 58400 q^{88} + 27542 q^{89} - 62112 q^{90} + 151974 q^{91} - 8464 q^{92} + 157330 q^{93} - 146990 q^{94} - 20644 q^{95} + 16160 q^{96} - 273672 q^{97} - 401276 q^{98} + 183082 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.164504 −0.0290805 −0.0145403 0.999894i \(-0.504628\pi\)
−0.0145403 + 0.999894i \(0.504628\pi\)
\(3\) −1.43100 −0.0917987 −0.0458994 0.998946i \(-0.514615\pi\)
−0.0458994 + 0.998946i \(0.514615\pi\)
\(4\) −31.9729 −0.999154
\(5\) −37.9865 −0.679523 −0.339761 0.940512i \(-0.610346\pi\)
−0.339761 + 0.940512i \(0.610346\pi\)
\(6\) 0.235406 0.00266955
\(7\) −43.8366 −0.338136 −0.169068 0.985604i \(-0.554076\pi\)
−0.169068 + 0.985604i \(0.554076\pi\)
\(8\) 10.5238 0.0581365
\(9\) −240.952 −0.991573
\(10\) 6.24894 0.0197609
\(11\) −163.213 −0.406700 −0.203350 0.979106i \(-0.565183\pi\)
−0.203350 + 0.979106i \(0.565183\pi\)
\(12\) 45.7533 0.0917211
\(13\) 430.356 0.706269 0.353134 0.935573i \(-0.385116\pi\)
0.353134 + 0.935573i \(0.385116\pi\)
\(14\) 7.21131 0.00983318
\(15\) 54.3587 0.0623793
\(16\) 1021.40 0.997464
\(17\) 740.415 0.621374 0.310687 0.950512i \(-0.399441\pi\)
0.310687 + 0.950512i \(0.399441\pi\)
\(18\) 39.6377 0.0288355
\(19\) −916.578 −0.582486 −0.291243 0.956649i \(-0.594069\pi\)
−0.291243 + 0.956649i \(0.594069\pi\)
\(20\) 1214.54 0.678948
\(21\) 62.7302 0.0310405
\(22\) 26.8493 0.0118270
\(23\) −529.000 −0.208514
\(24\) −15.0596 −0.00533685
\(25\) −1682.03 −0.538249
\(26\) −70.7955 −0.0205387
\(27\) 692.536 0.182824
\(28\) 1401.59 0.337850
\(29\) −5112.96 −1.12896 −0.564478 0.825448i \(-0.690923\pi\)
−0.564478 + 0.825448i \(0.690923\pi\)
\(30\) −8.94223 −0.00181402
\(31\) −5702.60 −1.06578 −0.532891 0.846184i \(-0.678895\pi\)
−0.532891 + 0.846184i \(0.678895\pi\)
\(32\) −504.787 −0.0871432
\(33\) 233.558 0.0373345
\(34\) −121.802 −0.0180699
\(35\) 1665.20 0.229771
\(36\) 7703.95 0.990734
\(37\) −10913.1 −1.31052 −0.655261 0.755402i \(-0.727441\pi\)
−0.655261 + 0.755402i \(0.727441\pi\)
\(38\) 150.781 0.0169390
\(39\) −615.840 −0.0648346
\(40\) −399.763 −0.0395050
\(41\) 11092.6 1.03056 0.515282 0.857021i \(-0.327687\pi\)
0.515282 + 0.857021i \(0.327687\pi\)
\(42\) −10.3194 −0.000902673 0
\(43\) 5528.76 0.455991 0.227996 0.973662i \(-0.426783\pi\)
0.227996 + 0.973662i \(0.426783\pi\)
\(44\) 5218.41 0.406356
\(45\) 9152.92 0.673796
\(46\) 87.0228 0.00606371
\(47\) 14435.2 0.953189 0.476594 0.879123i \(-0.341871\pi\)
0.476594 + 0.879123i \(0.341871\pi\)
\(48\) −1461.63 −0.0915659
\(49\) −14885.4 −0.885664
\(50\) 276.701 0.0156526
\(51\) −1059.53 −0.0570413
\(52\) −13759.8 −0.705671
\(53\) 6852.79 0.335102 0.167551 0.985863i \(-0.446414\pi\)
0.167551 + 0.985863i \(0.446414\pi\)
\(54\) −113.925 −0.00531661
\(55\) 6199.90 0.276362
\(56\) −461.329 −0.0196580
\(57\) 1311.62 0.0534715
\(58\) 841.104 0.0328307
\(59\) 4155.17 0.155403 0.0777014 0.996977i \(-0.475242\pi\)
0.0777014 + 0.996977i \(0.475242\pi\)
\(60\) −1738.01 −0.0623266
\(61\) −21911.4 −0.753955 −0.376978 0.926222i \(-0.623037\pi\)
−0.376978 + 0.926222i \(0.623037\pi\)
\(62\) 938.102 0.0309935
\(63\) 10562.5 0.335287
\(64\) −32601.9 −0.994930
\(65\) −16347.7 −0.479925
\(66\) −38.4213 −0.00108571
\(67\) −15164.2 −0.412699 −0.206349 0.978478i \(-0.566158\pi\)
−0.206349 + 0.978478i \(0.566158\pi\)
\(68\) −23673.3 −0.620849
\(69\) 756.999 0.0191414
\(70\) −273.932 −0.00668187
\(71\) 14031.4 0.330334 0.165167 0.986266i \(-0.447184\pi\)
0.165167 + 0.986266i \(0.447184\pi\)
\(72\) −2535.74 −0.0576465
\(73\) −23310.1 −0.511962 −0.255981 0.966682i \(-0.582398\pi\)
−0.255981 + 0.966682i \(0.582398\pi\)
\(74\) 1795.25 0.0381107
\(75\) 2406.98 0.0494106
\(76\) 29305.7 0.581994
\(77\) 7154.72 0.137520
\(78\) 101.308 0.00188542
\(79\) 64276.8 1.15874 0.579371 0.815064i \(-0.303298\pi\)
0.579371 + 0.815064i \(0.303298\pi\)
\(80\) −38799.5 −0.677799
\(81\) 57560.4 0.974790
\(82\) −1824.78 −0.0299693
\(83\) 114349. 1.82195 0.910974 0.412465i \(-0.135332\pi\)
0.910974 + 0.412465i \(0.135332\pi\)
\(84\) −2005.67 −0.0310142
\(85\) −28125.8 −0.422238
\(86\) −909.504 −0.0132605
\(87\) 7316.65 0.103637
\(88\) −1717.63 −0.0236441
\(89\) −63420.7 −0.848703 −0.424351 0.905498i \(-0.639498\pi\)
−0.424351 + 0.905498i \(0.639498\pi\)
\(90\) −1505.70 −0.0195943
\(91\) −18865.4 −0.238815
\(92\) 16913.7 0.208338
\(93\) 8160.42 0.0978375
\(94\) −2374.66 −0.0277192
\(95\) 34817.6 0.395813
\(96\) 722.351 0.00799964
\(97\) −91987.6 −0.992658 −0.496329 0.868134i \(-0.665319\pi\)
−0.496329 + 0.868134i \(0.665319\pi\)
\(98\) 2448.70 0.0257556
\(99\) 39326.6 0.403272
\(100\) 53779.4 0.537794
\(101\) 122133. 1.19132 0.595660 0.803236i \(-0.296890\pi\)
0.595660 + 0.803236i \(0.296890\pi\)
\(102\) 174.298 0.00165879
\(103\) −145517. −1.35151 −0.675756 0.737125i \(-0.736183\pi\)
−0.675756 + 0.737125i \(0.736183\pi\)
\(104\) 4528.99 0.0410600
\(105\) −2382.90 −0.0210927
\(106\) −1127.31 −0.00974495
\(107\) −129578. −1.09414 −0.547068 0.837088i \(-0.684256\pi\)
−0.547068 + 0.837088i \(0.684256\pi\)
\(108\) −22142.4 −0.182669
\(109\) −10434.1 −0.0841178 −0.0420589 0.999115i \(-0.513392\pi\)
−0.0420589 + 0.999115i \(0.513392\pi\)
\(110\) −1019.91 −0.00803674
\(111\) 15616.7 0.120304
\(112\) −44774.8 −0.337279
\(113\) −253268. −1.86589 −0.932943 0.360025i \(-0.882768\pi\)
−0.932943 + 0.360025i \(0.882768\pi\)
\(114\) −215.768 −0.00155498
\(115\) 20094.8 0.141690
\(116\) 163476. 1.12800
\(117\) −103695. −0.700317
\(118\) −683.543 −0.00451919
\(119\) −32457.3 −0.210109
\(120\) 572.061 0.00362651
\(121\) −134412. −0.834595
\(122\) 3604.52 0.0219254
\(123\) −15873.6 −0.0946044
\(124\) 182329. 1.06488
\(125\) 182602. 1.04527
\(126\) −1737.58 −0.00975032
\(127\) 235260. 1.29431 0.647156 0.762358i \(-0.275958\pi\)
0.647156 + 0.762358i \(0.275958\pi\)
\(128\) 21516.3 0.116076
\(129\) −7911.65 −0.0418594
\(130\) 2689.27 0.0139565
\(131\) 62921.5 0.320347 0.160174 0.987089i \(-0.448795\pi\)
0.160174 + 0.987089i \(0.448795\pi\)
\(132\) −7467.54 −0.0373029
\(133\) 40179.7 0.196960
\(134\) 2494.58 0.0120015
\(135\) −26307.0 −0.124233
\(136\) 7792.00 0.0361245
\(137\) −134562. −0.612521 −0.306260 0.951948i \(-0.599078\pi\)
−0.306260 + 0.951948i \(0.599078\pi\)
\(138\) −124.530 −0.000556641 0
\(139\) 287261. 1.26107 0.630535 0.776161i \(-0.282835\pi\)
0.630535 + 0.776161i \(0.282835\pi\)
\(140\) −53241.3 −0.229577
\(141\) −20656.8 −0.0875015
\(142\) −2308.22 −0.00960629
\(143\) −70239.9 −0.287239
\(144\) −246109. −0.989058
\(145\) 194223. 0.767152
\(146\) 3834.61 0.0148881
\(147\) 21300.9 0.0813028
\(148\) 348924. 1.30941
\(149\) −418105. −1.54284 −0.771418 0.636329i \(-0.780452\pi\)
−0.771418 + 0.636329i \(0.780452\pi\)
\(150\) −395.959 −0.00143689
\(151\) 486983. 1.73809 0.869043 0.494736i \(-0.164735\pi\)
0.869043 + 0.494736i \(0.164735\pi\)
\(152\) −9645.91 −0.0338637
\(153\) −178405. −0.616138
\(154\) −1176.98 −0.00399915
\(155\) 216622. 0.724223
\(156\) 19690.2 0.0647797
\(157\) −439288. −1.42233 −0.711165 0.703026i \(-0.751832\pi\)
−0.711165 + 0.703026i \(0.751832\pi\)
\(158\) −10573.8 −0.0336968
\(159\) −9806.34 −0.0307620
\(160\) 19175.1 0.0592158
\(161\) 23189.6 0.0705063
\(162\) −9468.93 −0.0283474
\(163\) −125475. −0.369905 −0.184952 0.982747i \(-0.559213\pi\)
−0.184952 + 0.982747i \(0.559213\pi\)
\(164\) −354664. −1.02969
\(165\) −8872.05 −0.0253696
\(166\) −18810.8 −0.0529832
\(167\) −626996. −1.73970 −0.869848 0.493319i \(-0.835783\pi\)
−0.869848 + 0.493319i \(0.835783\pi\)
\(168\) 660.162 0.00180458
\(169\) −186086. −0.501185
\(170\) 4626.81 0.0122789
\(171\) 220852. 0.577578
\(172\) −176771. −0.455605
\(173\) 502364. 1.27615 0.638077 0.769972i \(-0.279730\pi\)
0.638077 + 0.769972i \(0.279730\pi\)
\(174\) −1203.62 −0.00301381
\(175\) 73734.4 0.182002
\(176\) −166706. −0.405668
\(177\) −5946.05 −0.0142658
\(178\) 10433.0 0.0246807
\(179\) −131681. −0.307178 −0.153589 0.988135i \(-0.549083\pi\)
−0.153589 + 0.988135i \(0.549083\pi\)
\(180\) −292646. −0.673226
\(181\) 343209. 0.778686 0.389343 0.921093i \(-0.372702\pi\)
0.389343 + 0.921093i \(0.372702\pi\)
\(182\) 3103.43 0.00694487
\(183\) 31355.2 0.0692121
\(184\) −5567.10 −0.0121223
\(185\) 414551. 0.890529
\(186\) −1342.42 −0.00284516
\(187\) −120846. −0.252713
\(188\) −461536. −0.952383
\(189\) −30358.4 −0.0618194
\(190\) −5727.64 −0.0115104
\(191\) 831093. 1.64841 0.824207 0.566289i \(-0.191621\pi\)
0.824207 + 0.566289i \(0.191621\pi\)
\(192\) 46653.3 0.0913333
\(193\) 498764. 0.963832 0.481916 0.876217i \(-0.339941\pi\)
0.481916 + 0.876217i \(0.339941\pi\)
\(194\) 15132.3 0.0288670
\(195\) 23393.6 0.0440565
\(196\) 475928. 0.884915
\(197\) 415236. 0.762306 0.381153 0.924512i \(-0.375527\pi\)
0.381153 + 0.924512i \(0.375527\pi\)
\(198\) −6469.39 −0.0117274
\(199\) −969194. −1.73491 −0.867457 0.497512i \(-0.834247\pi\)
−0.867457 + 0.497512i \(0.834247\pi\)
\(200\) −17701.4 −0.0312919
\(201\) 21700.0 0.0378852
\(202\) −20091.4 −0.0346442
\(203\) 224135. 0.381741
\(204\) 33876.4 0.0569931
\(205\) −421370. −0.700291
\(206\) 23938.1 0.0393027
\(207\) 127464. 0.206757
\(208\) 439567. 0.704477
\(209\) 149598. 0.236897
\(210\) 391.997 0.000613387 0
\(211\) 372669. 0.576259 0.288129 0.957592i \(-0.406967\pi\)
0.288129 + 0.957592i \(0.406967\pi\)
\(212\) −219104. −0.334819
\(213\) −20078.9 −0.0303243
\(214\) 21316.1 0.0318181
\(215\) −210018. −0.309856
\(216\) 7288.12 0.0106287
\(217\) 249983. 0.360380
\(218\) 1716.45 0.00244619
\(219\) 33356.8 0.0469974
\(220\) −198229. −0.276128
\(221\) 318643. 0.438857
\(222\) −2569.01 −0.00349851
\(223\) −668196. −0.899791 −0.449895 0.893081i \(-0.648539\pi\)
−0.449895 + 0.893081i \(0.648539\pi\)
\(224\) 22128.2 0.0294663
\(225\) 405288. 0.533713
\(226\) 41663.7 0.0542609
\(227\) −1.01295e6 −1.30473 −0.652367 0.757903i \(-0.726224\pi\)
−0.652367 + 0.757903i \(0.726224\pi\)
\(228\) −41936.5 −0.0534263
\(229\) −342003. −0.430964 −0.215482 0.976508i \(-0.569132\pi\)
−0.215482 + 0.976508i \(0.569132\pi\)
\(230\) −3305.69 −0.00412043
\(231\) −10238.4 −0.0126242
\(232\) −53807.9 −0.0656335
\(233\) −792950. −0.956877 −0.478438 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(234\) 17058.3 0.0203656
\(235\) −548343. −0.647713
\(236\) −132853. −0.155271
\(237\) −91980.2 −0.106371
\(238\) 5339.36 0.00611008
\(239\) 520763. 0.589719 0.294859 0.955541i \(-0.404727\pi\)
0.294859 + 0.955541i \(0.404727\pi\)
\(240\) 55522.1 0.0622211
\(241\) −867128. −0.961702 −0.480851 0.876802i \(-0.659672\pi\)
−0.480851 + 0.876802i \(0.659672\pi\)
\(242\) 22111.4 0.0242705
\(243\) −250655. −0.272308
\(244\) 700572. 0.753317
\(245\) 565442. 0.601829
\(246\) 2611.27 0.00275115
\(247\) −394455. −0.411392
\(248\) −60013.1 −0.0619608
\(249\) −163633. −0.167252
\(250\) −30038.8 −0.0303971
\(251\) −590873. −0.591983 −0.295992 0.955191i \(-0.595650\pi\)
−0.295992 + 0.955191i \(0.595650\pi\)
\(252\) −337715. −0.335003
\(253\) 86339.8 0.0848027
\(254\) −38701.3 −0.0376393
\(255\) 40248.0 0.0387609
\(256\) 1.03972e6 0.991554
\(257\) −448650. −0.423716 −0.211858 0.977300i \(-0.567951\pi\)
−0.211858 + 0.977300i \(0.567951\pi\)
\(258\) 1301.50 0.00121729
\(259\) 478394. 0.443135
\(260\) 522685. 0.479520
\(261\) 1.23198e6 1.11944
\(262\) −10350.9 −0.00931586
\(263\) 434704. 0.387529 0.193765 0.981048i \(-0.437930\pi\)
0.193765 + 0.981048i \(0.437930\pi\)
\(264\) 2457.93 0.00217050
\(265\) −260313. −0.227710
\(266\) −6609.73 −0.00572769
\(267\) 90755.0 0.0779098
\(268\) 484845. 0.412350
\(269\) 873326. 0.735861 0.367930 0.929853i \(-0.380067\pi\)
0.367930 + 0.929853i \(0.380067\pi\)
\(270\) 4327.61 0.00361276
\(271\) −1.16572e6 −0.964208 −0.482104 0.876114i \(-0.660127\pi\)
−0.482104 + 0.876114i \(0.660127\pi\)
\(272\) 756262. 0.619798
\(273\) 26996.3 0.0219229
\(274\) 22136.0 0.0178124
\(275\) 274529. 0.218906
\(276\) −24203.5 −0.0191252
\(277\) −604025. −0.472994 −0.236497 0.971632i \(-0.575999\pi\)
−0.236497 + 0.971632i \(0.575999\pi\)
\(278\) −47255.6 −0.0366726
\(279\) 1.37405e6 1.05680
\(280\) 17524.2 0.0133581
\(281\) 1.72154e6 1.30062 0.650310 0.759669i \(-0.274639\pi\)
0.650310 + 0.759669i \(0.274639\pi\)
\(282\) 3398.13 0.00254459
\(283\) −2.01557e6 −1.49600 −0.748000 0.663699i \(-0.768986\pi\)
−0.748000 + 0.663699i \(0.768986\pi\)
\(284\) −448624. −0.330055
\(285\) −49824.0 −0.0363351
\(286\) 11554.8 0.00835306
\(287\) −486263. −0.348471
\(288\) 121630. 0.0864089
\(289\) −871642. −0.613894
\(290\) −31950.6 −0.0223092
\(291\) 131634. 0.0911248
\(292\) 745293. 0.511529
\(293\) 1.45451e6 0.989798 0.494899 0.868951i \(-0.335205\pi\)
0.494899 + 0.868951i \(0.335205\pi\)
\(294\) −3504.10 −0.00236433
\(295\) −157840. −0.105600
\(296\) −114848. −0.0761891
\(297\) −113031. −0.0743544
\(298\) 68780.1 0.0448665
\(299\) −227659. −0.147267
\(300\) −76958.3 −0.0493688
\(301\) −242362. −0.154187
\(302\) −80110.8 −0.0505445
\(303\) −174772. −0.109362
\(304\) −936196. −0.581009
\(305\) 832337. 0.512329
\(306\) 29348.3 0.0179176
\(307\) 272463. 0.164991 0.0824957 0.996591i \(-0.473711\pi\)
0.0824957 + 0.996591i \(0.473711\pi\)
\(308\) −228757. −0.137404
\(309\) 208235. 0.124067
\(310\) −35635.2 −0.0210608
\(311\) −2.33041e6 −1.36625 −0.683126 0.730301i \(-0.739380\pi\)
−0.683126 + 0.730301i \(0.739380\pi\)
\(312\) −6480.99 −0.00376925
\(313\) −1.67227e6 −0.964818 −0.482409 0.875946i \(-0.660238\pi\)
−0.482409 + 0.875946i \(0.660238\pi\)
\(314\) 72264.7 0.0413621
\(315\) −401233. −0.227835
\(316\) −2.05512e6 −1.15776
\(317\) 1.18782e6 0.663901 0.331950 0.943297i \(-0.392293\pi\)
0.331950 + 0.943297i \(0.392293\pi\)
\(318\) 1613.19 0.000894574 0
\(319\) 834503. 0.459146
\(320\) 1.23843e6 0.676077
\(321\) 185426. 0.100440
\(322\) −3814.78 −0.00205036
\(323\) −678649. −0.361942
\(324\) −1.84037e6 −0.973966
\(325\) −723872. −0.380148
\(326\) 20641.3 0.0107570
\(327\) 14931.2 0.00772191
\(328\) 116737. 0.0599133
\(329\) −632791. −0.322308
\(330\) 1459.49 0.000737762 0
\(331\) −2.09261e6 −1.04983 −0.524915 0.851155i \(-0.675903\pi\)
−0.524915 + 0.851155i \(0.675903\pi\)
\(332\) −3.65606e6 −1.82041
\(333\) 2.62954e6 1.29948
\(334\) 103144. 0.0505913
\(335\) 576035. 0.280438
\(336\) 64072.8 0.0309618
\(337\) 2.13994e6 1.02642 0.513211 0.858262i \(-0.328456\pi\)
0.513211 + 0.858262i \(0.328456\pi\)
\(338\) 30612.0 0.0145747
\(339\) 362427. 0.171286
\(340\) 899263. 0.421881
\(341\) 930740. 0.433453
\(342\) −36331.0 −0.0167963
\(343\) 1.38929e6 0.637611
\(344\) 58183.6 0.0265097
\(345\) −28755.7 −0.0130070
\(346\) −82641.0 −0.0371112
\(347\) 3.78048e6 1.68548 0.842739 0.538322i \(-0.180941\pi\)
0.842739 + 0.538322i \(0.180941\pi\)
\(348\) −233935. −0.103549
\(349\) 3.46473e6 1.52267 0.761334 0.648360i \(-0.224545\pi\)
0.761334 + 0.648360i \(0.224545\pi\)
\(350\) −12129.6 −0.00529270
\(351\) 298037. 0.129123
\(352\) 82388.0 0.0354411
\(353\) −2.39834e6 −1.02441 −0.512206 0.858863i \(-0.671171\pi\)
−0.512206 + 0.858863i \(0.671171\pi\)
\(354\) 978.151 0.000414856 0
\(355\) −533002. −0.224470
\(356\) 2.02774e6 0.847985
\(357\) 46446.4 0.0192878
\(358\) 21662.0 0.00893289
\(359\) −2.04171e6 −0.836098 −0.418049 0.908424i \(-0.637286\pi\)
−0.418049 + 0.908424i \(0.637286\pi\)
\(360\) 96323.7 0.0391721
\(361\) −1.63598e6 −0.660710
\(362\) −56459.3 −0.0226446
\(363\) 192344. 0.0766148
\(364\) 603181. 0.238613
\(365\) 885469. 0.347889
\(366\) −5158.07 −0.00201272
\(367\) 2.41363e6 0.935417 0.467709 0.883883i \(-0.345080\pi\)
0.467709 + 0.883883i \(0.345080\pi\)
\(368\) −540322. −0.207986
\(369\) −2.67279e6 −1.02188
\(370\) −68195.4 −0.0258971
\(371\) −300403. −0.113310
\(372\) −260913. −0.0977547
\(373\) 1.83073e6 0.681321 0.340660 0.940186i \(-0.389349\pi\)
0.340660 + 0.940186i \(0.389349\pi\)
\(374\) 19879.6 0.00734901
\(375\) −261304. −0.0959549
\(376\) 151914. 0.0554150
\(377\) −2.20039e6 −0.797347
\(378\) 4994.09 0.00179774
\(379\) 5.24532e6 1.87575 0.937874 0.346977i \(-0.112792\pi\)
0.937874 + 0.346977i \(0.112792\pi\)
\(380\) −1.11322e6 −0.395478
\(381\) −336657. −0.118816
\(382\) −136718. −0.0479367
\(383\) −754501. −0.262823 −0.131411 0.991328i \(-0.541951\pi\)
−0.131411 + 0.991328i \(0.541951\pi\)
\(384\) −30789.9 −0.0106557
\(385\) −271782. −0.0934479
\(386\) −82048.7 −0.0280287
\(387\) −1.33217e6 −0.452148
\(388\) 2.94111e6 0.991819
\(389\) 2.05221e6 0.687618 0.343809 0.939040i \(-0.388283\pi\)
0.343809 + 0.939040i \(0.388283\pi\)
\(390\) −3848.35 −0.00128119
\(391\) −391680. −0.129565
\(392\) −156651. −0.0514894
\(393\) −90040.7 −0.0294074
\(394\) −68308.0 −0.0221682
\(395\) −2.44165e6 −0.787391
\(396\) −1.25739e6 −0.402931
\(397\) 3.99917e6 1.27348 0.636742 0.771077i \(-0.280281\pi\)
0.636742 + 0.771077i \(0.280281\pi\)
\(398\) 159437. 0.0504522
\(399\) −57497.2 −0.0180807
\(400\) −1.71803e6 −0.536884
\(401\) −5.10734e6 −1.58611 −0.793056 0.609149i \(-0.791511\pi\)
−0.793056 + 0.609149i \(0.791511\pi\)
\(402\) −3569.74 −0.00110172
\(403\) −2.45415e6 −0.752729
\(404\) −3.90494e6 −1.19031
\(405\) −2.18652e6 −0.662392
\(406\) −36871.1 −0.0111012
\(407\) 1.78116e6 0.532989
\(408\) −11150.4 −0.00331618
\(409\) −3.06823e6 −0.906942 −0.453471 0.891271i \(-0.649814\pi\)
−0.453471 + 0.891271i \(0.649814\pi\)
\(410\) 69317.1 0.0203648
\(411\) 192558. 0.0562286
\(412\) 4.65260e6 1.35037
\(413\) −182149. −0.0525473
\(414\) −20968.3 −0.00601261
\(415\) −4.34370e6 −1.23805
\(416\) −217239. −0.0615465
\(417\) −411070. −0.115765
\(418\) −24609.5 −0.00688908
\(419\) 3.33601e6 0.928308 0.464154 0.885754i \(-0.346358\pi\)
0.464154 + 0.885754i \(0.346358\pi\)
\(420\) 76188.3 0.0210749
\(421\) −2.75080e6 −0.756404 −0.378202 0.925723i \(-0.623458\pi\)
−0.378202 + 0.925723i \(0.623458\pi\)
\(422\) −61305.7 −0.0167579
\(423\) −3.47820e6 −0.945156
\(424\) 72117.5 0.0194817
\(425\) −1.24540e6 −0.334454
\(426\) 3303.06 0.000881846 0
\(427\) 960521. 0.254940
\(428\) 4.14299e6 1.09321
\(429\) 100513. 0.0263682
\(430\) 34548.8 0.00901078
\(431\) 926380. 0.240213 0.120106 0.992761i \(-0.461676\pi\)
0.120106 + 0.992761i \(0.461676\pi\)
\(432\) 707358. 0.182360
\(433\) 680820. 0.174507 0.0872535 0.996186i \(-0.472191\pi\)
0.0872535 + 0.996186i \(0.472191\pi\)
\(434\) −41123.2 −0.0104800
\(435\) −277934. −0.0704236
\(436\) 333608. 0.0840467
\(437\) 484870. 0.121457
\(438\) −5487.33 −0.00136671
\(439\) 2.17406e6 0.538406 0.269203 0.963083i \(-0.413240\pi\)
0.269203 + 0.963083i \(0.413240\pi\)
\(440\) 65246.6 0.0160667
\(441\) 3.58666e6 0.878200
\(442\) −52418.1 −0.0127622
\(443\) 4.17517e6 1.01080 0.505400 0.862885i \(-0.331345\pi\)
0.505400 + 0.862885i \(0.331345\pi\)
\(444\) −499311. −0.120203
\(445\) 2.40913e6 0.576713
\(446\) 109921. 0.0261664
\(447\) 598308. 0.141630
\(448\) 1.42915e6 0.336422
\(449\) −4.68946e6 −1.09776 −0.548880 0.835901i \(-0.684945\pi\)
−0.548880 + 0.835901i \(0.684945\pi\)
\(450\) −66671.7 −0.0155207
\(451\) −1.81046e6 −0.419130
\(452\) 8.09774e6 1.86431
\(453\) −696873. −0.159554
\(454\) 166634. 0.0379424
\(455\) 716629. 0.162280
\(456\) 13803.3 0.00310864
\(457\) −1.68983e6 −0.378487 −0.189244 0.981930i \(-0.560604\pi\)
−0.189244 + 0.981930i \(0.560604\pi\)
\(458\) 56260.9 0.0125326
\(459\) 512764. 0.113602
\(460\) −642491. −0.141570
\(461\) −4.28765e6 −0.939652 −0.469826 0.882759i \(-0.655683\pi\)
−0.469826 + 0.882759i \(0.655683\pi\)
\(462\) 1684.26 0.000367117 0
\(463\) 5.26738e6 1.14194 0.570969 0.820972i \(-0.306568\pi\)
0.570969 + 0.820972i \(0.306568\pi\)
\(464\) −5.22239e6 −1.12609
\(465\) −309986. −0.0664828
\(466\) 130444. 0.0278265
\(467\) −3.66305e6 −0.777231 −0.388616 0.921400i \(-0.627047\pi\)
−0.388616 + 0.921400i \(0.627047\pi\)
\(468\) 3.31544e6 0.699725
\(469\) 664748. 0.139548
\(470\) 90204.8 0.0188358
\(471\) 628621. 0.130568
\(472\) 43728.3 0.00903457
\(473\) −902366. −0.185451
\(474\) 15131.1 0.00309332
\(475\) 1.54171e6 0.313523
\(476\) 1.03776e6 0.209931
\(477\) −1.65119e6 −0.332279
\(478\) −85667.7 −0.0171493
\(479\) 7.03353e6 1.40067 0.700333 0.713816i \(-0.253035\pi\)
0.700333 + 0.713816i \(0.253035\pi\)
\(480\) −27439.6 −0.00543593
\(481\) −4.69653e6 −0.925581
\(482\) 142646. 0.0279668
\(483\) −33184.3 −0.00647239
\(484\) 4.29756e6 0.833890
\(485\) 3.49428e6 0.674534
\(486\) 41233.8 0.00791887
\(487\) 5.05170e6 0.965195 0.482598 0.875842i \(-0.339693\pi\)
0.482598 + 0.875842i \(0.339693\pi\)
\(488\) −230592. −0.0438323
\(489\) 179555. 0.0339568
\(490\) −93017.6 −0.0175015
\(491\) −3.90292e6 −0.730610 −0.365305 0.930888i \(-0.619035\pi\)
−0.365305 + 0.930888i \(0.619035\pi\)
\(492\) 507524. 0.0945244
\(493\) −3.78571e6 −0.701505
\(494\) 64889.6 0.0119635
\(495\) −1.49388e6 −0.274033
\(496\) −5.82465e6 −1.06308
\(497\) −615087. −0.111698
\(498\) 26918.3 0.00486379
\(499\) 1.13818e6 0.204625 0.102312 0.994752i \(-0.467376\pi\)
0.102312 + 0.994752i \(0.467376\pi\)
\(500\) −5.83832e6 −1.04439
\(501\) 897231. 0.159702
\(502\) 97201.1 0.0172152
\(503\) −7.48030e6 −1.31825 −0.659127 0.752032i \(-0.729074\pi\)
−0.659127 + 0.752032i \(0.729074\pi\)
\(504\) 111158. 0.0194924
\(505\) −4.63939e6 −0.809529
\(506\) −14203.3 −0.00246611
\(507\) 266290. 0.0460081
\(508\) −7.52195e6 −1.29322
\(509\) 3.03026e6 0.518424 0.259212 0.965821i \(-0.416537\pi\)
0.259212 + 0.965821i \(0.416537\pi\)
\(510\) −6620.97 −0.00112719
\(511\) 1.02184e6 0.173113
\(512\) −859561. −0.144911
\(513\) −634763. −0.106492
\(514\) 73804.8 0.0123219
\(515\) 5.52767e6 0.918383
\(516\) 252959. 0.0418240
\(517\) −2.35602e6 −0.387661
\(518\) −78697.8 −0.0128866
\(519\) −718883. −0.117149
\(520\) −172040. −0.0279012
\(521\) −1.90771e6 −0.307905 −0.153953 0.988078i \(-0.549200\pi\)
−0.153953 + 0.988078i \(0.549200\pi\)
\(522\) −202666. −0.0325540
\(523\) 2.40656e6 0.384718 0.192359 0.981325i \(-0.438386\pi\)
0.192359 + 0.981325i \(0.438386\pi\)
\(524\) −2.01178e6 −0.320076
\(525\) −105514. −0.0167075
\(526\) −71510.7 −0.0112696
\(527\) −4.22229e6 −0.662250
\(528\) 238557. 0.0372398
\(529\) 279841. 0.0434783
\(530\) 42822.6 0.00662192
\(531\) −1.00120e6 −0.154093
\(532\) −1.28466e6 −0.196793
\(533\) 4.77378e6 0.727855
\(534\) −14929.6 −0.00226566
\(535\) 4.92221e6 0.743490
\(536\) −159586. −0.0239928
\(537\) 188435. 0.0281985
\(538\) −143666. −0.0213992
\(539\) 2.42949e6 0.360199
\(540\) 841112. 0.124128
\(541\) −711055. −0.104450 −0.0522252 0.998635i \(-0.516631\pi\)
−0.0522252 + 0.998635i \(0.516631\pi\)
\(542\) 191766. 0.0280397
\(543\) −491132. −0.0714823
\(544\) −373752. −0.0541485
\(545\) 396354. 0.0571600
\(546\) −4441.01 −0.000637530 0
\(547\) 7.21528e6 1.03106 0.515531 0.856871i \(-0.327595\pi\)
0.515531 + 0.856871i \(0.327595\pi\)
\(548\) 4.30234e6 0.612003
\(549\) 5.27960e6 0.747601
\(550\) −45161.2 −0.00636589
\(551\) 4.68643e6 0.657602
\(552\) 7966.53 0.00111281
\(553\) −2.81768e6 −0.391813
\(554\) 99364.7 0.0137549
\(555\) −593222. −0.0817495
\(556\) −9.18457e6 −1.26000
\(557\) −920675. −0.125739 −0.0628693 0.998022i \(-0.520025\pi\)
−0.0628693 + 0.998022i \(0.520025\pi\)
\(558\) −226038. −0.0307323
\(559\) 2.37934e6 0.322052
\(560\) 1.70084e6 0.229189
\(561\) 172930. 0.0231987
\(562\) −283200. −0.0378227
\(563\) −7.13949e6 −0.949284 −0.474642 0.880179i \(-0.657422\pi\)
−0.474642 + 0.880179i \(0.657422\pi\)
\(564\) 660459. 0.0874275
\(565\) 9.62077e6 1.26791
\(566\) 331570. 0.0435045
\(567\) −2.52325e6 −0.329612
\(568\) 147663. 0.0192045
\(569\) 1.32562e7 1.71648 0.858239 0.513250i \(-0.171559\pi\)
0.858239 + 0.513250i \(0.171559\pi\)
\(570\) 8196.26 0.00105664
\(571\) −8.74927e6 −1.12300 −0.561502 0.827475i \(-0.689776\pi\)
−0.561502 + 0.827475i \(0.689776\pi\)
\(572\) 2.24578e6 0.286996
\(573\) −1.18929e6 −0.151322
\(574\) 79992.4 0.0101337
\(575\) 889793. 0.112233
\(576\) 7.85549e6 0.986545
\(577\) 2.28815e6 0.286118 0.143059 0.989714i \(-0.454306\pi\)
0.143059 + 0.989714i \(0.454306\pi\)
\(578\) 143389. 0.0178524
\(579\) −713731. −0.0884786
\(580\) −6.20989e6 −0.766503
\(581\) −5.01266e6 −0.616067
\(582\) −21654.4 −0.00264996
\(583\) −1.11847e6 −0.136286
\(584\) −245312. −0.0297636
\(585\) 3.93902e6 0.475881
\(586\) −239272. −0.0287838
\(587\) 1.48384e7 1.77743 0.888714 0.458463i \(-0.151600\pi\)
0.888714 + 0.458463i \(0.151600\pi\)
\(588\) −681054. −0.0812341
\(589\) 5.22688e6 0.620804
\(590\) 25965.4 0.00307090
\(591\) −594202. −0.0699787
\(592\) −1.11467e7 −1.30720
\(593\) −7.67836e6 −0.896668 −0.448334 0.893866i \(-0.647982\pi\)
−0.448334 + 0.893866i \(0.647982\pi\)
\(594\) 18594.1 0.00216226
\(595\) 1.23294e6 0.142774
\(596\) 1.33680e7 1.54153
\(597\) 1.38692e6 0.159263
\(598\) 37450.8 0.00428261
\(599\) −599539. −0.0682733 −0.0341366 0.999417i \(-0.510868\pi\)
−0.0341366 + 0.999417i \(0.510868\pi\)
\(600\) 25330.7 0.00287256
\(601\) −6.92017e6 −0.781503 −0.390751 0.920496i \(-0.627785\pi\)
−0.390751 + 0.920496i \(0.627785\pi\)
\(602\) 39869.6 0.00448384
\(603\) 3.65385e6 0.409221
\(604\) −1.55703e7 −1.73662
\(605\) 5.10585e6 0.567126
\(606\) 28750.7 0.00318030
\(607\) −6.65094e6 −0.732675 −0.366337 0.930482i \(-0.619388\pi\)
−0.366337 + 0.930482i \(0.619388\pi\)
\(608\) 462677. 0.0507597
\(609\) −320737. −0.0350434
\(610\) −136923. −0.0148988
\(611\) 6.21229e6 0.673207
\(612\) 5.70412e6 0.615617
\(613\) −1.07456e7 −1.15499 −0.577497 0.816393i \(-0.695970\pi\)
−0.577497 + 0.816393i \(0.695970\pi\)
\(614\) −44821.3 −0.00479804
\(615\) 602980. 0.0642859
\(616\) 75295.0 0.00799492
\(617\) −1.66965e7 −1.76568 −0.882841 0.469673i \(-0.844372\pi\)
−0.882841 + 0.469673i \(0.844372\pi\)
\(618\) −34255.5 −0.00360794
\(619\) −1.06977e7 −1.12218 −0.561090 0.827755i \(-0.689618\pi\)
−0.561090 + 0.827755i \(0.689618\pi\)
\(620\) −6.92603e6 −0.723611
\(621\) −366351. −0.0381214
\(622\) 383362. 0.0397313
\(623\) 2.78015e6 0.286977
\(624\) −629021. −0.0646701
\(625\) −1.68007e6 −0.172039
\(626\) 275095. 0.0280574
\(627\) −214074. −0.0217468
\(628\) 1.40453e7 1.42113
\(629\) −8.08024e6 −0.814325
\(630\) 66004.6 0.00662556
\(631\) 1.31871e7 1.31849 0.659243 0.751930i \(-0.270877\pi\)
0.659243 + 0.751930i \(0.270877\pi\)
\(632\) 676438. 0.0673651
\(633\) −533290. −0.0528998
\(634\) −195402. −0.0193066
\(635\) −8.93670e6 −0.879514
\(636\) 313538. 0.0307360
\(637\) −6.40601e6 −0.625517
\(638\) −137279. −0.0133522
\(639\) −3.38089e6 −0.327551
\(640\) −817330. −0.0788765
\(641\) −849150. −0.0816280 −0.0408140 0.999167i \(-0.512995\pi\)
−0.0408140 + 0.999167i \(0.512995\pi\)
\(642\) −30503.4 −0.00292086
\(643\) −2.31164e6 −0.220491 −0.110246 0.993904i \(-0.535164\pi\)
−0.110246 + 0.993904i \(0.535164\pi\)
\(644\) −741439. −0.0704467
\(645\) 300536. 0.0284444
\(646\) 111641. 0.0105255
\(647\) −1.51827e7 −1.42590 −0.712949 0.701216i \(-0.752641\pi\)
−0.712949 + 0.701216i \(0.752641\pi\)
\(648\) 605755. 0.0566708
\(649\) −678179. −0.0632023
\(650\) 119080. 0.0110549
\(651\) −357725. −0.0330824
\(652\) 4.01182e6 0.369592
\(653\) 1.57201e7 1.44269 0.721343 0.692578i \(-0.243525\pi\)
0.721343 + 0.692578i \(0.243525\pi\)
\(654\) −2456.24 −0.000224557 0
\(655\) −2.39016e6 −0.217683
\(656\) 1.13300e7 1.02795
\(657\) 5.61663e6 0.507647
\(658\) 104097. 0.00937288
\(659\) −1.71369e7 −1.53716 −0.768580 0.639754i \(-0.779036\pi\)
−0.768580 + 0.639754i \(0.779036\pi\)
\(660\) 283666. 0.0253482
\(661\) 1.52598e7 1.35845 0.679227 0.733928i \(-0.262315\pi\)
0.679227 + 0.733928i \(0.262315\pi\)
\(662\) 344244. 0.0305296
\(663\) −455978. −0.0402865
\(664\) 1.20338e6 0.105922
\(665\) −1.52628e6 −0.133839
\(666\) −432570. −0.0377895
\(667\) 2.70476e6 0.235404
\(668\) 2.00469e7 1.73823
\(669\) 956188. 0.0825996
\(670\) −94760.3 −0.00815529
\(671\) 3.57623e6 0.306633
\(672\) −31665.4 −0.00270497
\(673\) −1.35466e7 −1.15290 −0.576451 0.817132i \(-0.695563\pi\)
−0.576451 + 0.817132i \(0.695563\pi\)
\(674\) −352029. −0.0298489
\(675\) −1.16486e6 −0.0984048
\(676\) 5.94973e6 0.500761
\(677\) 1.13711e7 0.953520 0.476760 0.879034i \(-0.341811\pi\)
0.476760 + 0.879034i \(0.341811\pi\)
\(678\) −59620.8 −0.00498108
\(679\) 4.03242e6 0.335654
\(680\) −295991. −0.0245474
\(681\) 1.44953e6 0.119773
\(682\) −153111. −0.0126050
\(683\) −498823. −0.0409161 −0.0204581 0.999791i \(-0.506512\pi\)
−0.0204581 + 0.999791i \(0.506512\pi\)
\(684\) −7.06127e6 −0.577089
\(685\) 5.11153e6 0.416222
\(686\) −228543. −0.0185421
\(687\) 489406. 0.0395619
\(688\) 5.64709e6 0.454834
\(689\) 2.94914e6 0.236672
\(690\) 4730.44 0.000378250 0
\(691\) 1.03020e7 0.820783 0.410392 0.911909i \(-0.365392\pi\)
0.410392 + 0.911909i \(0.365392\pi\)
\(692\) −1.60621e7 −1.27508
\(693\) −1.72394e6 −0.136361
\(694\) −621905. −0.0490146
\(695\) −1.09120e7 −0.856926
\(696\) 76999.1 0.00602508
\(697\) 8.21315e6 0.640366
\(698\) −569962. −0.0442800
\(699\) 1.13471e6 0.0878401
\(700\) −2.35751e6 −0.181848
\(701\) −3.29623e6 −0.253351 −0.126675 0.991944i \(-0.540431\pi\)
−0.126675 + 0.991944i \(0.540431\pi\)
\(702\) −49028.4 −0.00375496
\(703\) 1.00027e7 0.763361
\(704\) 5.32105e6 0.404637
\(705\) 784679. 0.0594592
\(706\) 394537. 0.0297904
\(707\) −5.35388e6 −0.402829
\(708\) 190113. 0.0142537
\(709\) 1.35819e7 1.01472 0.507359 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(710\) 87681.0 0.00652769
\(711\) −1.54876e7 −1.14898
\(712\) −667428. −0.0493406
\(713\) 3.01667e6 0.222231
\(714\) −7640.63 −0.000560898 0
\(715\) 2.66816e6 0.195185
\(716\) 4.21022e6 0.306918
\(717\) −745212. −0.0541354
\(718\) 335870. 0.0243142
\(719\) −8.30096e6 −0.598834 −0.299417 0.954122i \(-0.596792\pi\)
−0.299417 + 0.954122i \(0.596792\pi\)
\(720\) 9.34882e6 0.672087
\(721\) 6.37896e6 0.456995
\(722\) 269126. 0.0192138
\(723\) 1.24086e6 0.0882831
\(724\) −1.09734e7 −0.778027
\(725\) 8.60014e6 0.607660
\(726\) −31641.5 −0.00222800
\(727\) −5.01347e6 −0.351806 −0.175903 0.984408i \(-0.556284\pi\)
−0.175903 + 0.984408i \(0.556284\pi\)
\(728\) −198536. −0.0138839
\(729\) −1.36285e7 −0.949792
\(730\) −145663. −0.0101168
\(731\) 4.09358e6 0.283341
\(732\) −1.00252e6 −0.0691536
\(733\) −1.48950e7 −1.02395 −0.511977 0.858999i \(-0.671087\pi\)
−0.511977 + 0.858999i \(0.671087\pi\)
\(734\) −397052. −0.0272024
\(735\) −809148. −0.0552471
\(736\) 267033. 0.0181706
\(737\) 2.47500e6 0.167844
\(738\) 439686. 0.0297168
\(739\) 8.39705e6 0.565608 0.282804 0.959178i \(-0.408735\pi\)
0.282804 + 0.959178i \(0.408735\pi\)
\(740\) −1.32544e7 −0.889776
\(741\) 564466. 0.0377652
\(742\) 49417.6 0.00329512
\(743\) −1.54362e7 −1.02582 −0.512908 0.858444i \(-0.671432\pi\)
−0.512908 + 0.858444i \(0.671432\pi\)
\(744\) 85878.8 0.00568792
\(745\) 1.58823e7 1.04839
\(746\) −301163. −0.0198132
\(747\) −2.75526e7 −1.80659
\(748\) 3.86379e6 0.252499
\(749\) 5.68025e6 0.369967
\(750\) 42985.6 0.00279042
\(751\) −5.26274e6 −0.340496 −0.170248 0.985401i \(-0.554457\pi\)
−0.170248 + 0.985401i \(0.554457\pi\)
\(752\) 1.47442e7 0.950771
\(753\) 845539. 0.0543433
\(754\) 361974. 0.0231873
\(755\) −1.84988e7 −1.18107
\(756\) 970648. 0.0617671
\(757\) −2.58188e7 −1.63756 −0.818779 0.574109i \(-0.805348\pi\)
−0.818779 + 0.574109i \(0.805348\pi\)
\(758\) −862878. −0.0545477
\(759\) −123552. −0.00778478
\(760\) 366414. 0.0230111
\(761\) 4.74356e6 0.296922 0.148461 0.988918i \(-0.452568\pi\)
0.148461 + 0.988918i \(0.452568\pi\)
\(762\) 55381.5 0.00345524
\(763\) 457395. 0.0284433
\(764\) −2.65725e7 −1.64702
\(765\) 6.77697e6 0.418680
\(766\) 124119. 0.00764302
\(767\) 1.78820e6 0.109756
\(768\) −1.48784e6 −0.0910234
\(769\) 2.16055e7 1.31749 0.658746 0.752365i \(-0.271087\pi\)
0.658746 + 0.752365i \(0.271087\pi\)
\(770\) 44709.4 0.00271751
\(771\) 642018. 0.0388966
\(772\) −1.59469e7 −0.963017
\(773\) 4.60087e6 0.276944 0.138472 0.990366i \(-0.455781\pi\)
0.138472 + 0.990366i \(0.455781\pi\)
\(774\) 219147. 0.0131487
\(775\) 9.59193e6 0.573656
\(776\) −968061. −0.0577096
\(777\) −684582. −0.0406792
\(778\) −337597. −0.0199963
\(779\) −1.01673e7 −0.600289
\(780\) −747962. −0.0440193
\(781\) −2.29010e6 −0.134347
\(782\) 64433.0 0.00376783
\(783\) −3.54091e6 −0.206400
\(784\) −1.52039e7 −0.883417
\(785\) 1.66870e7 0.966505
\(786\) 14812.1 0.000855184 0
\(787\) 3.20780e7 1.84616 0.923082 0.384603i \(-0.125662\pi\)
0.923082 + 0.384603i \(0.125662\pi\)
\(788\) −1.32763e7 −0.761661
\(789\) −622062. −0.0355747
\(790\) 401662. 0.0228977
\(791\) 1.11024e7 0.630924
\(792\) 413866. 0.0234448
\(793\) −9.42971e6 −0.532495
\(794\) −657881. −0.0370336
\(795\) 372508. 0.0209035
\(796\) 3.09880e7 1.73345
\(797\) 950693. 0.0530145 0.0265072 0.999649i \(-0.491561\pi\)
0.0265072 + 0.999649i \(0.491561\pi\)
\(798\) 9458.53 0.000525795 0
\(799\) 1.06881e7 0.592287
\(800\) 849067. 0.0469047
\(801\) 1.52813e7 0.841551
\(802\) 840179. 0.0461250
\(803\) 3.80452e6 0.208215
\(804\) −693813. −0.0378532
\(805\) −880890. −0.0479106
\(806\) 403718. 0.0218897
\(807\) −1.24973e6 −0.0675511
\(808\) 1.28530e6 0.0692592
\(809\) 1.89226e7 1.01650 0.508252 0.861208i \(-0.330292\pi\)
0.508252 + 0.861208i \(0.330292\pi\)
\(810\) 359691. 0.0192627
\(811\) 4.78517e6 0.255473 0.127737 0.991808i \(-0.459229\pi\)
0.127737 + 0.991808i \(0.459229\pi\)
\(812\) −7.16625e6 −0.381419
\(813\) 1.66814e6 0.0885130
\(814\) −293009. −0.0154996
\(815\) 4.76637e6 0.251359
\(816\) −1.08221e6 −0.0568967
\(817\) −5.06754e6 −0.265608
\(818\) 504737. 0.0263744
\(819\) 4.54565e6 0.236803
\(820\) 1.34724e7 0.699699
\(821\) 5.51231e6 0.285414 0.142707 0.989765i \(-0.454419\pi\)
0.142707 + 0.989765i \(0.454419\pi\)
\(822\) −31676.6 −0.00163516
\(823\) −1.44617e7 −0.744249 −0.372125 0.928183i \(-0.621371\pi\)
−0.372125 + 0.928183i \(0.621371\pi\)
\(824\) −1.53139e6 −0.0785721
\(825\) −392852. −0.0200953
\(826\) 29964.2 0.00152810
\(827\) −2.78469e7 −1.41584 −0.707919 0.706294i \(-0.750366\pi\)
−0.707919 + 0.706294i \(0.750366\pi\)
\(828\) −4.07539e6 −0.206582
\(829\) 1.89279e6 0.0956569 0.0478285 0.998856i \(-0.484770\pi\)
0.0478285 + 0.998856i \(0.484770\pi\)
\(830\) 714557. 0.0360033
\(831\) 864360. 0.0434202
\(832\) −1.40304e7 −0.702687
\(833\) −1.10213e7 −0.550329
\(834\) 67622.8 0.00336650
\(835\) 2.38174e7 1.18216
\(836\) −4.78308e6 −0.236697
\(837\) −3.94925e6 −0.194850
\(838\) −548788. −0.0269957
\(839\) 8.81372e6 0.432269 0.216135 0.976364i \(-0.430655\pi\)
0.216135 + 0.976364i \(0.430655\pi\)
\(840\) −25077.2 −0.00122626
\(841\) 5.63121e6 0.274544
\(842\) 452518. 0.0219966
\(843\) −2.46352e6 −0.119395
\(844\) −1.19153e7 −0.575771
\(845\) 7.06876e6 0.340566
\(846\) 572179. 0.0274856
\(847\) 5.89219e6 0.282207
\(848\) 6.99946e6 0.334253
\(849\) 2.88428e6 0.137331
\(850\) 204874. 0.00972610
\(851\) 5.77304e6 0.273263
\(852\) 641981. 0.0302986
\(853\) 4.21641e6 0.198413 0.0992064 0.995067i \(-0.468370\pi\)
0.0992064 + 0.995067i \(0.468370\pi\)
\(854\) −158010. −0.00741378
\(855\) −8.38937e6 −0.392477
\(856\) −1.36365e6 −0.0636092
\(857\) 1.36923e7 0.636829 0.318415 0.947952i \(-0.396850\pi\)
0.318415 + 0.947952i \(0.396850\pi\)
\(858\) −16534.9 −0.000766801 0
\(859\) 1.31118e7 0.606291 0.303145 0.952944i \(-0.401963\pi\)
0.303145 + 0.952944i \(0.401963\pi\)
\(860\) 6.71489e6 0.309594
\(861\) 695843. 0.0319892
\(862\) −152393. −0.00698551
\(863\) 5.67126e6 0.259210 0.129605 0.991566i \(-0.458629\pi\)
0.129605 + 0.991566i \(0.458629\pi\)
\(864\) −349583. −0.0159319
\(865\) −1.90830e7 −0.867176
\(866\) −111998. −0.00507475
\(867\) 1.24732e6 0.0563547
\(868\) −7.99268e6 −0.360075
\(869\) −1.04908e7 −0.471260
\(870\) 45721.3 0.00204795
\(871\) −6.52602e6 −0.291476
\(872\) −109806. −0.00489031
\(873\) 2.21646e7 0.984293
\(874\) −79763.2 −0.00353203
\(875\) −8.00465e6 −0.353445
\(876\) −1.06651e6 −0.0469577
\(877\) −2.51970e7 −1.10624 −0.553120 0.833101i \(-0.686563\pi\)
−0.553120 + 0.833101i \(0.686563\pi\)
\(878\) −357642. −0.0156571
\(879\) −2.08140e6 −0.0908622
\(880\) 6.33259e6 0.275661
\(881\) −2.99532e7 −1.30018 −0.650090 0.759857i \(-0.725269\pi\)
−0.650090 + 0.759857i \(0.725269\pi\)
\(882\) −590021. −0.0255385
\(883\) −2.98802e7 −1.28968 −0.644840 0.764318i \(-0.723076\pi\)
−0.644840 + 0.764318i \(0.723076\pi\)
\(884\) −1.01879e7 −0.438486
\(885\) 225869. 0.00969392
\(886\) −686834. −0.0293946
\(887\) −7.65547e6 −0.326710 −0.163355 0.986567i \(-0.552232\pi\)
−0.163355 + 0.986567i \(0.552232\pi\)
\(888\) 164347. 0.00699406
\(889\) −1.03130e7 −0.437654
\(890\) −396312. −0.0167711
\(891\) −9.39462e6 −0.396447
\(892\) 2.13642e7 0.899030
\(893\) −1.32310e7 −0.555219
\(894\) −98424.3 −0.00411868
\(895\) 5.00208e6 0.208734
\(896\) −943203. −0.0392496
\(897\) 325779. 0.0135189
\(898\) 771436. 0.0319234
\(899\) 2.91572e7 1.20322
\(900\) −1.29583e7 −0.533262
\(901\) 5.07391e6 0.208224
\(902\) 297829. 0.0121885
\(903\) 346820. 0.0141542
\(904\) −2.66535e6 −0.108476
\(905\) −1.30373e7 −0.529134
\(906\) 114639. 0.00463992
\(907\) −3.50928e7 −1.41644 −0.708222 0.705990i \(-0.750502\pi\)
−0.708222 + 0.705990i \(0.750502\pi\)
\(908\) 3.23869e7 1.30363
\(909\) −2.94282e7 −1.18128
\(910\) −117888. −0.00471919
\(911\) −3.03942e7 −1.21337 −0.606687 0.794941i \(-0.707502\pi\)
−0.606687 + 0.794941i \(0.707502\pi\)
\(912\) 1.33970e6 0.0533359
\(913\) −1.86632e7 −0.740985
\(914\) 277983. 0.0110066
\(915\) −1.19107e6 −0.0470312
\(916\) 1.09348e7 0.430599
\(917\) −2.75826e6 −0.108321
\(918\) −84351.9 −0.00330361
\(919\) 3.59276e7 1.40326 0.701632 0.712540i \(-0.252455\pi\)
0.701632 + 0.712540i \(0.252455\pi\)
\(920\) 211475. 0.00823737
\(921\) −389894. −0.0151460
\(922\) 705337. 0.0273256
\(923\) 6.03848e6 0.233305
\(924\) 327352. 0.0126135
\(925\) 1.83562e7 0.705387
\(926\) −866507. −0.0332081
\(927\) 3.50626e7 1.34012
\(928\) 2.58096e6 0.0983809
\(929\) −9.41890e6 −0.358064 −0.179032 0.983843i \(-0.557297\pi\)
−0.179032 + 0.983843i \(0.557297\pi\)
\(930\) 50994.0 0.00193335
\(931\) 1.36436e7 0.515887
\(932\) 2.53529e7 0.956068
\(933\) 3.33481e6 0.125420
\(934\) 602587. 0.0226023
\(935\) 4.59050e6 0.171724
\(936\) −1.09127e6 −0.0407139
\(937\) −2.13837e7 −0.795673 −0.397836 0.917456i \(-0.630239\pi\)
−0.397836 + 0.917456i \(0.630239\pi\)
\(938\) −109354. −0.00405814
\(939\) 2.39302e6 0.0885690
\(940\) 1.75321e7 0.647165
\(941\) 4.31706e7 1.58933 0.794666 0.607047i \(-0.207646\pi\)
0.794666 + 0.607047i \(0.207646\pi\)
\(942\) −103411. −0.00379699
\(943\) −5.86800e6 −0.214887
\(944\) 4.24410e6 0.155009
\(945\) 1.15321e6 0.0420077
\(946\) 148443. 0.00539302
\(947\) 1.75109e7 0.634504 0.317252 0.948341i \(-0.397240\pi\)
0.317252 + 0.948341i \(0.397240\pi\)
\(948\) 2.94088e6 0.106281
\(949\) −1.00317e7 −0.361582
\(950\) −253618. −0.00911740
\(951\) −1.69977e6 −0.0609452
\(952\) −341575. −0.0122150
\(953\) −4.05106e6 −0.144489 −0.0722447 0.997387i \(-0.523016\pi\)
−0.0722447 + 0.997387i \(0.523016\pi\)
\(954\) 271629. 0.00966283
\(955\) −3.15703e7 −1.12013
\(956\) −1.66503e7 −0.589220
\(957\) −1.19417e6 −0.0421490
\(958\) −1.15705e6 −0.0407321
\(959\) 5.89874e6 0.207116
\(960\) −1.77219e6 −0.0620630
\(961\) 3.89048e6 0.135892
\(962\) 772599. 0.0269164
\(963\) 3.12221e7 1.08492
\(964\) 2.77246e7 0.960889
\(965\) −1.89463e7 −0.654946
\(966\) 5458.96 0.000188220 0
\(967\) −2.54319e7 −0.874608 −0.437304 0.899314i \(-0.644067\pi\)
−0.437304 + 0.899314i \(0.644067\pi\)
\(968\) −1.41453e6 −0.0485204
\(969\) 971147. 0.0332258
\(970\) −574824. −0.0196158
\(971\) −4.42829e6 −0.150726 −0.0753630 0.997156i \(-0.524012\pi\)
−0.0753630 + 0.997156i \(0.524012\pi\)
\(972\) 8.01418e6 0.272078
\(973\) −1.25925e7 −0.426414
\(974\) −831026. −0.0280684
\(975\) 1.03586e6 0.0348971
\(976\) −2.23804e7 −0.752043
\(977\) 5.60569e7 1.87885 0.939427 0.342750i \(-0.111358\pi\)
0.939427 + 0.342750i \(0.111358\pi\)
\(978\) −29537.6 −0.000987481 0
\(979\) 1.03511e7 0.345167
\(980\) −1.80788e7 −0.601320
\(981\) 2.51412e6 0.0834090
\(982\) 642046. 0.0212465
\(983\) 3.41997e7 1.12886 0.564428 0.825482i \(-0.309097\pi\)
0.564428 + 0.825482i \(0.309097\pi\)
\(984\) −167050. −0.00549997
\(985\) −1.57733e7 −0.518004
\(986\) 622766. 0.0204001
\(987\) 905525. 0.0295874
\(988\) 1.26119e7 0.411044
\(989\) −2.92471e6 −0.0950807
\(990\) 245749. 0.00796901
\(991\) −904509. −0.0292569 −0.0146285 0.999893i \(-0.504657\pi\)
−0.0146285 + 0.999893i \(0.504657\pi\)
\(992\) 2.87860e6 0.0928757
\(993\) 2.99453e6 0.0963730
\(994\) 101184. 0.00324824
\(995\) 3.68163e7 1.17891
\(996\) 5.23183e6 0.167111
\(997\) −4.00447e7 −1.27587 −0.637936 0.770089i \(-0.720212\pi\)
−0.637936 + 0.770089i \(0.720212\pi\)
\(998\) −187235. −0.00595060
\(999\) −7.55772e6 −0.239595
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 23.6.a.a.1.2 3
3.2 odd 2 207.6.a.b.1.2 3
4.3 odd 2 368.6.a.e.1.2 3
5.4 even 2 575.6.a.b.1.2 3
23.22 odd 2 529.6.a.a.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.6.a.a.1.2 3 1.1 even 1 trivial
207.6.a.b.1.2 3 3.2 odd 2
368.6.a.e.1.2 3 4.3 odd 2
529.6.a.a.1.2 3 23.22 odd 2
575.6.a.b.1.2 3 5.4 even 2