Properties

Label 23.6.a.a.1.3
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 1.3
Root \(3.65736\) of defining polynomial
Character \(\chi\) \(=\) 23.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.31473 q^{2} -27.6631 q^{3} -3.75366 q^{4} -23.8768 q^{5} -147.022 q^{6} -11.7843 q^{7} -190.021 q^{8} +522.249 q^{9} +O(q^{10})\) \(q+5.31473 q^{2} -27.6631 q^{3} -3.75366 q^{4} -23.8768 q^{5} -147.022 q^{6} -11.7843 q^{7} -190.021 q^{8} +522.249 q^{9} -126.899 q^{10} +280.887 q^{11} +103.838 q^{12} -835.171 q^{13} -62.6305 q^{14} +660.508 q^{15} -889.793 q^{16} -1801.86 q^{17} +2775.61 q^{18} +2357.32 q^{19} +89.6256 q^{20} +325.991 q^{21} +1492.84 q^{22} -529.000 q^{23} +5256.58 q^{24} -2554.90 q^{25} -4438.71 q^{26} -7724.90 q^{27} +44.2344 q^{28} -1206.86 q^{29} +3510.42 q^{30} -3394.53 q^{31} +1351.66 q^{32} -7770.22 q^{33} -9576.38 q^{34} +281.372 q^{35} -1960.35 q^{36} +8351.42 q^{37} +12528.5 q^{38} +23103.5 q^{39} +4537.10 q^{40} -10701.7 q^{41} +1732.56 q^{42} -803.535 q^{43} -1054.36 q^{44} -12469.7 q^{45} -2811.49 q^{46} -4896.03 q^{47} +24614.5 q^{48} -16668.1 q^{49} -13578.6 q^{50} +49845.0 q^{51} +3134.95 q^{52} +39358.6 q^{53} -41055.8 q^{54} -6706.69 q^{55} +2239.27 q^{56} -65211.0 q^{57} -6414.13 q^{58} +39514.6 q^{59} -2479.32 q^{60} -15309.3 q^{61} -18041.0 q^{62} -6154.35 q^{63} +35657.1 q^{64} +19941.2 q^{65} -41296.6 q^{66} -60783.9 q^{67} +6763.56 q^{68} +14633.8 q^{69} +1495.42 q^{70} +39125.6 q^{71} -99238.3 q^{72} -5872.67 q^{73} +44385.5 q^{74} +70676.5 q^{75} -8848.60 q^{76} -3310.07 q^{77} +122789. q^{78} -63656.1 q^{79} +21245.4 q^{80} +86788.5 q^{81} -56876.4 q^{82} -63639.8 q^{83} -1223.66 q^{84} +43022.6 q^{85} -4270.57 q^{86} +33385.5 q^{87} -53374.5 q^{88} +36262.8 q^{89} -66272.8 q^{90} +9841.93 q^{91} +1985.69 q^{92} +93903.4 q^{93} -26021.1 q^{94} -56285.4 q^{95} -37391.3 q^{96} -64187.5 q^{97} -88586.6 q^{98} +146693. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 20 q^{3} + 16 q^{4} - 58 q^{5} - 230 q^{6} - 282 q^{7} - 360 q^{8} + 121 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.31473 0.939520 0.469760 0.882794i \(-0.344340\pi\)
0.469760 + 0.882794i \(0.344340\pi\)
\(3\) −27.6631 −1.77459 −0.887295 0.461202i \(-0.847419\pi\)
−0.887295 + 0.461202i \(0.847419\pi\)
\(4\) −3.75366 −0.117302
\(5\) −23.8768 −0.427122 −0.213561 0.976930i \(-0.568506\pi\)
−0.213561 + 0.976930i \(0.568506\pi\)
\(6\) −147.022 −1.66726
\(7\) −11.7843 −0.0908991 −0.0454496 0.998967i \(-0.514472\pi\)
−0.0454496 + 0.998967i \(0.514472\pi\)
\(8\) −190.021 −1.04973
\(9\) 522.249 2.14917
\(10\) −126.899 −0.401289
\(11\) 280.887 0.699923 0.349961 0.936764i \(-0.386195\pi\)
0.349961 + 0.936764i \(0.386195\pi\)
\(12\) 103.838 0.208163
\(13\) −835.171 −1.37062 −0.685310 0.728251i \(-0.740333\pi\)
−0.685310 + 0.728251i \(0.740333\pi\)
\(14\) −62.6305 −0.0854015
\(15\) 660.508 0.757966
\(16\) −889.793 −0.868938
\(17\) −1801.86 −1.51216 −0.756080 0.654479i \(-0.772888\pi\)
−0.756080 + 0.654479i \(0.772888\pi\)
\(18\) 2775.61 2.01919
\(19\) 2357.32 1.49808 0.749041 0.662524i \(-0.230515\pi\)
0.749041 + 0.662524i \(0.230515\pi\)
\(20\) 89.6256 0.0501022
\(21\) 325.991 0.161309
\(22\) 1492.84 0.657592
\(23\) −529.000 −0.208514
\(24\) 5256.58 1.86284
\(25\) −2554.90 −0.817567
\(26\) −4438.71 −1.28773
\(27\) −7724.90 −2.03931
\(28\) 44.2344 0.0106626
\(29\) −1206.86 −0.266478 −0.133239 0.991084i \(-0.542538\pi\)
−0.133239 + 0.991084i \(0.542538\pi\)
\(30\) 3510.42 0.712125
\(31\) −3394.53 −0.634418 −0.317209 0.948356i \(-0.602746\pi\)
−0.317209 + 0.948356i \(0.602746\pi\)
\(32\) 1351.66 0.233343
\(33\) −7770.22 −1.24208
\(34\) −9576.38 −1.42071
\(35\) 281.372 0.0388250
\(36\) −1960.35 −0.252102
\(37\) 8351.42 1.00290 0.501448 0.865188i \(-0.332801\pi\)
0.501448 + 0.865188i \(0.332801\pi\)
\(38\) 12528.5 1.40748
\(39\) 23103.5 2.43229
\(40\) 4537.10 0.448362
\(41\) −10701.7 −0.994240 −0.497120 0.867682i \(-0.665609\pi\)
−0.497120 + 0.867682i \(0.665609\pi\)
\(42\) 1732.56 0.151553
\(43\) −803.535 −0.0662726 −0.0331363 0.999451i \(-0.510550\pi\)
−0.0331363 + 0.999451i \(0.510550\pi\)
\(44\) −1054.36 −0.0821023
\(45\) −12469.7 −0.917958
\(46\) −2811.49 −0.195903
\(47\) −4896.03 −0.323295 −0.161648 0.986849i \(-0.551681\pi\)
−0.161648 + 0.986849i \(0.551681\pi\)
\(48\) 24614.5 1.54201
\(49\) −16668.1 −0.991737
\(50\) −13578.6 −0.768121
\(51\) 49845.0 2.68347
\(52\) 3134.95 0.160776
\(53\) 39358.6 1.92464 0.962322 0.271913i \(-0.0876562\pi\)
0.962322 + 0.271913i \(0.0876562\pi\)
\(54\) −41055.8 −1.91597
\(55\) −6706.69 −0.298952
\(56\) 2239.27 0.0954193
\(57\) −65211.0 −2.65848
\(58\) −6414.13 −0.250362
\(59\) 39514.6 1.47784 0.738921 0.673792i \(-0.235336\pi\)
0.738921 + 0.673792i \(0.235336\pi\)
\(60\) −2479.32 −0.0889109
\(61\) −15309.3 −0.526783 −0.263392 0.964689i \(-0.584841\pi\)
−0.263392 + 0.964689i \(0.584841\pi\)
\(62\) −18041.0 −0.596048
\(63\) −6154.35 −0.195358
\(64\) 35657.1 1.08817
\(65\) 19941.2 0.585422
\(66\) −41296.6 −1.16696
\(67\) −60783.9 −1.65425 −0.827126 0.562016i \(-0.810026\pi\)
−0.827126 + 0.562016i \(0.810026\pi\)
\(68\) 6763.56 0.177379
\(69\) 14633.8 0.370028
\(70\) 1495.42 0.0364769
\(71\) 39125.6 0.921118 0.460559 0.887629i \(-0.347649\pi\)
0.460559 + 0.887629i \(0.347649\pi\)
\(72\) −99238.3 −2.25605
\(73\) −5872.67 −0.128982 −0.0644909 0.997918i \(-0.520542\pi\)
−0.0644909 + 0.997918i \(0.520542\pi\)
\(74\) 44385.5 0.942241
\(75\) 70676.5 1.45085
\(76\) −8848.60 −0.175728
\(77\) −3310.07 −0.0636224
\(78\) 122789. 2.28519
\(79\) −63656.1 −1.14755 −0.573776 0.819012i \(-0.694522\pi\)
−0.573776 + 0.819012i \(0.694522\pi\)
\(80\) 21245.4 0.371142
\(81\) 86788.5 1.46977
\(82\) −56876.4 −0.934109
\(83\) −63639.8 −1.01399 −0.506995 0.861949i \(-0.669244\pi\)
−0.506995 + 0.861949i \(0.669244\pi\)
\(84\) −1223.66 −0.0189218
\(85\) 43022.6 0.645877
\(86\) −4270.57 −0.0622644
\(87\) 33385.5 0.472890
\(88\) −53374.5 −0.734728
\(89\) 36262.8 0.485273 0.242637 0.970117i \(-0.421988\pi\)
0.242637 + 0.970117i \(0.421988\pi\)
\(90\) −66272.8 −0.862440
\(91\) 9841.93 0.124588
\(92\) 1985.69 0.0244591
\(93\) 93903.4 1.12583
\(94\) −26021.1 −0.303742
\(95\) −56285.4 −0.639863
\(96\) −37391.3 −0.414088
\(97\) −64187.5 −0.692662 −0.346331 0.938112i \(-0.612573\pi\)
−0.346331 + 0.938112i \(0.612573\pi\)
\(98\) −88586.6 −0.931757
\(99\) 146693. 1.50425
\(100\) 9590.22 0.0959022
\(101\) −108356. −1.05693 −0.528467 0.848954i \(-0.677233\pi\)
−0.528467 + 0.848954i \(0.677233\pi\)
\(102\) 264913. 2.52117
\(103\) −88102.3 −0.818265 −0.409133 0.912475i \(-0.634169\pi\)
−0.409133 + 0.912475i \(0.634169\pi\)
\(104\) 158700. 1.43878
\(105\) −7783.64 −0.0688985
\(106\) 209180. 1.80824
\(107\) −16813.2 −0.141968 −0.0709842 0.997477i \(-0.522614\pi\)
−0.0709842 + 0.997477i \(0.522614\pi\)
\(108\) 28996.7 0.239215
\(109\) −192169. −1.54923 −0.774616 0.632432i \(-0.782057\pi\)
−0.774616 + 0.632432i \(0.782057\pi\)
\(110\) −35644.3 −0.280872
\(111\) −231026. −1.77973
\(112\) 10485.6 0.0789857
\(113\) 24975.1 0.183997 0.0919985 0.995759i \(-0.470674\pi\)
0.0919985 + 0.995759i \(0.470674\pi\)
\(114\) −346579. −2.49770
\(115\) 12630.8 0.0890610
\(116\) 4530.14 0.0312584
\(117\) −436167. −2.94570
\(118\) 210010. 1.38846
\(119\) 21233.7 0.137454
\(120\) −125510. −0.795658
\(121\) −82153.4 −0.510108
\(122\) −81365.0 −0.494923
\(123\) 296041. 1.76437
\(124\) 12741.9 0.0744185
\(125\) 135618. 0.776322
\(126\) −32708.7 −0.183543
\(127\) 215290. 1.18445 0.592223 0.805774i \(-0.298251\pi\)
0.592223 + 0.805774i \(0.298251\pi\)
\(128\) 146255. 0.789013
\(129\) 22228.3 0.117607
\(130\) 105982. 0.550015
\(131\) 34626.7 0.176292 0.0881461 0.996108i \(-0.471906\pi\)
0.0881461 + 0.996108i \(0.471906\pi\)
\(132\) 29166.8 0.145698
\(133\) −27779.5 −0.136174
\(134\) −323050. −1.55420
\(135\) 184446. 0.871034
\(136\) 342391. 1.58736
\(137\) 172235. 0.784007 0.392003 0.919964i \(-0.371782\pi\)
0.392003 + 0.919964i \(0.371782\pi\)
\(138\) 77774.7 0.347649
\(139\) 153367. 0.673278 0.336639 0.941634i \(-0.390710\pi\)
0.336639 + 0.941634i \(0.390710\pi\)
\(140\) −1056.18 −0.00455425
\(141\) 135440. 0.573717
\(142\) 207942. 0.865409
\(143\) −234589. −0.959328
\(144\) −464693. −1.86750
\(145\) 28816.0 0.113819
\(146\) −31211.7 −0.121181
\(147\) 461093. 1.75993
\(148\) −31348.4 −0.117642
\(149\) −28131.3 −0.103807 −0.0519033 0.998652i \(-0.516529\pi\)
−0.0519033 + 0.998652i \(0.516529\pi\)
\(150\) 375626. 1.36310
\(151\) 160593. 0.573172 0.286586 0.958055i \(-0.407480\pi\)
0.286586 + 0.958055i \(0.407480\pi\)
\(152\) −447941. −1.57258
\(153\) −941018. −3.24989
\(154\) −17592.1 −0.0597745
\(155\) 81050.6 0.270974
\(156\) −86722.6 −0.285312
\(157\) 242015. 0.783599 0.391800 0.920051i \(-0.371853\pi\)
0.391800 + 0.920051i \(0.371853\pi\)
\(158\) −338315. −1.07815
\(159\) −1.08878e6 −3.41546
\(160\) −32273.5 −0.0996657
\(161\) 6233.91 0.0189538
\(162\) 461257. 1.38088
\(163\) −136177. −0.401453 −0.200727 0.979647i \(-0.564330\pi\)
−0.200727 + 0.979647i \(0.564330\pi\)
\(164\) 40170.4 0.116626
\(165\) 185528. 0.530518
\(166\) −338228. −0.952664
\(167\) −9384.75 −0.0260394 −0.0130197 0.999915i \(-0.504144\pi\)
−0.0130197 + 0.999915i \(0.504144\pi\)
\(168\) −61945.2 −0.169330
\(169\) 326218. 0.878600
\(170\) 228654. 0.606814
\(171\) 1.23111e6 3.21964
\(172\) 3016.20 0.00777390
\(173\) −398962. −1.01348 −0.506741 0.862098i \(-0.669150\pi\)
−0.506741 + 0.862098i \(0.669150\pi\)
\(174\) 177435. 0.444290
\(175\) 30107.7 0.0743161
\(176\) −249931. −0.608190
\(177\) −1.09310e6 −2.62256
\(178\) 192727. 0.455924
\(179\) 456085. 1.06393 0.531965 0.846766i \(-0.321454\pi\)
0.531965 + 0.846766i \(0.321454\pi\)
\(180\) 46806.9 0.107678
\(181\) −120816. −0.274111 −0.137056 0.990563i \(-0.543764\pi\)
−0.137056 + 0.990563i \(0.543764\pi\)
\(182\) 52307.2 0.117053
\(183\) 423504. 0.934825
\(184\) 100521. 0.218883
\(185\) −199405. −0.428359
\(186\) 499071. 1.05774
\(187\) −506118. −1.05840
\(188\) 18378.0 0.0379232
\(189\) 91032.8 0.185372
\(190\) −299142. −0.601164
\(191\) 416377. 0.825854 0.412927 0.910764i \(-0.364506\pi\)
0.412927 + 0.910764i \(0.364506\pi\)
\(192\) −986387. −1.93105
\(193\) −381178. −0.736605 −0.368302 0.929706i \(-0.620061\pi\)
−0.368302 + 0.929706i \(0.620061\pi\)
\(194\) −341139. −0.650770
\(195\) −551637. −1.03888
\(196\) 62566.5 0.116333
\(197\) 617955. 1.13447 0.567233 0.823558i \(-0.308014\pi\)
0.567233 + 0.823558i \(0.308014\pi\)
\(198\) 779634. 1.41328
\(199\) 455121. 0.814693 0.407346 0.913274i \(-0.366454\pi\)
0.407346 + 0.913274i \(0.366454\pi\)
\(200\) 485484. 0.858223
\(201\) 1.68147e6 2.93562
\(202\) −575881. −0.993011
\(203\) 14222.0 0.0242226
\(204\) −187101. −0.314776
\(205\) 255522. 0.424662
\(206\) −468240. −0.768776
\(207\) −276270. −0.448133
\(208\) 743129. 1.19098
\(209\) 662142. 1.04854
\(210\) −41367.9 −0.0647315
\(211\) −870625. −1.34625 −0.673124 0.739530i \(-0.735048\pi\)
−0.673124 + 0.739530i \(0.735048\pi\)
\(212\) −147739. −0.225764
\(213\) −1.08234e6 −1.63461
\(214\) −89357.8 −0.133382
\(215\) 19185.9 0.0283065
\(216\) 1.46789e6 2.14072
\(217\) 40002.3 0.0576680
\(218\) −1.02132e6 −1.45553
\(219\) 162457. 0.228890
\(220\) 25174.7 0.0350677
\(221\) 1.50486e6 2.07260
\(222\) −1.22784e6 −1.67209
\(223\) −135558. −0.182541 −0.0912707 0.995826i \(-0.529093\pi\)
−0.0912707 + 0.995826i \(0.529093\pi\)
\(224\) −15928.5 −0.0212106
\(225\) −1.33429e6 −1.75709
\(226\) 132736. 0.172869
\(227\) −443206. −0.570875 −0.285438 0.958397i \(-0.592139\pi\)
−0.285438 + 0.958397i \(0.592139\pi\)
\(228\) 244780. 0.311845
\(229\) 689970. 0.869444 0.434722 0.900565i \(-0.356847\pi\)
0.434722 + 0.900565i \(0.356847\pi\)
\(230\) 67129.5 0.0836746
\(231\) 91566.8 0.112904
\(232\) 229329. 0.279730
\(233\) 487389. 0.588147 0.294074 0.955783i \(-0.404989\pi\)
0.294074 + 0.955783i \(0.404989\pi\)
\(234\) −2.31811e6 −2.76754
\(235\) 116902. 0.138086
\(236\) −148325. −0.173354
\(237\) 1.76093e6 2.03644
\(238\) 112851. 0.129141
\(239\) −1.12530e6 −1.27430 −0.637150 0.770740i \(-0.719887\pi\)
−0.637150 + 0.770740i \(0.719887\pi\)
\(240\) −587715. −0.658626
\(241\) −1.48985e6 −1.65235 −0.826173 0.563416i \(-0.809487\pi\)
−0.826173 + 0.563416i \(0.809487\pi\)
\(242\) −436623. −0.479257
\(243\) −523690. −0.568930
\(244\) 57466.1 0.0617927
\(245\) 397982. 0.423593
\(246\) 1.57338e6 1.65766
\(247\) −1.96877e6 −2.05330
\(248\) 645032. 0.665966
\(249\) 1.76048e6 1.79942
\(250\) 720773. 0.729371
\(251\) 63456.5 0.0635758 0.0317879 0.999495i \(-0.489880\pi\)
0.0317879 + 0.999495i \(0.489880\pi\)
\(252\) 23101.4 0.0229159
\(253\) −148589. −0.145944
\(254\) 1.14421e6 1.11281
\(255\) −1.19014e6 −1.14617
\(256\) −363724. −0.346874
\(257\) 1.08874e6 1.02823 0.514116 0.857720i \(-0.328120\pi\)
0.514116 + 0.857720i \(0.328120\pi\)
\(258\) 118137. 0.110494
\(259\) −98415.8 −0.0911624
\(260\) −74852.7 −0.0686711
\(261\) −630281. −0.572708
\(262\) 184032. 0.165630
\(263\) −542790. −0.483885 −0.241943 0.970291i \(-0.577785\pi\)
−0.241943 + 0.970291i \(0.577785\pi\)
\(264\) 1.47650e6 1.30384
\(265\) −939760. −0.822057
\(266\) −147640. −0.127938
\(267\) −1.00314e6 −0.861162
\(268\) 228162. 0.194047
\(269\) −1.66169e6 −1.40013 −0.700065 0.714079i \(-0.746846\pi\)
−0.700065 + 0.714079i \(0.746846\pi\)
\(270\) 980281. 0.818354
\(271\) 621380. 0.513966 0.256983 0.966416i \(-0.417272\pi\)
0.256983 + 0.966416i \(0.417272\pi\)
\(272\) 1.60328e6 1.31397
\(273\) −272259. −0.221093
\(274\) 915382. 0.736590
\(275\) −717638. −0.572234
\(276\) −54930.3 −0.0434050
\(277\) 266442. 0.208643 0.104321 0.994544i \(-0.466733\pi\)
0.104321 + 0.994544i \(0.466733\pi\)
\(278\) 815103. 0.632558
\(279\) −1.77279e6 −1.36347
\(280\) −53466.7 −0.0407557
\(281\) 472574. 0.357029 0.178515 0.983937i \(-0.442871\pi\)
0.178515 + 0.983937i \(0.442871\pi\)
\(282\) 719824. 0.539018
\(283\) 914756. 0.678952 0.339476 0.940615i \(-0.389750\pi\)
0.339476 + 0.940615i \(0.389750\pi\)
\(284\) −146864. −0.108049
\(285\) 1.55703e6 1.13550
\(286\) −1.24678e6 −0.901308
\(287\) 126112. 0.0903755
\(288\) 705906. 0.501494
\(289\) 1.82683e6 1.28663
\(290\) 153149. 0.106935
\(291\) 1.77563e6 1.22919
\(292\) 22044.0 0.0151298
\(293\) −906414. −0.616819 −0.308409 0.951254i \(-0.599797\pi\)
−0.308409 + 0.951254i \(0.599797\pi\)
\(294\) 2.45058e6 1.65349
\(295\) −943484. −0.631218
\(296\) −1.58695e6 −1.05277
\(297\) −2.16983e6 −1.42736
\(298\) −149510. −0.0975283
\(299\) 441806. 0.285794
\(300\) −265296. −0.170187
\(301\) 9469.12 0.00602412
\(302\) 853509. 0.538506
\(303\) 2.99746e6 1.87563
\(304\) −2.09753e6 −1.30174
\(305\) 365539. 0.225001
\(306\) −5.00125e6 −3.05334
\(307\) 102614. 0.0621387 0.0310694 0.999517i \(-0.490109\pi\)
0.0310694 + 0.999517i \(0.490109\pi\)
\(308\) 12424.9 0.00746303
\(309\) 2.43718e6 1.45209
\(310\) 430762. 0.254585
\(311\) 1.21303e6 0.711168 0.355584 0.934644i \(-0.384282\pi\)
0.355584 + 0.934644i \(0.384282\pi\)
\(312\) −4.39014e6 −2.55324
\(313\) −3.04190e6 −1.75503 −0.877514 0.479551i \(-0.840800\pi\)
−0.877514 + 0.479551i \(0.840800\pi\)
\(314\) 1.28625e6 0.736207
\(315\) 146946. 0.0834416
\(316\) 238944. 0.134610
\(317\) −2.10095e6 −1.17427 −0.587133 0.809490i \(-0.699744\pi\)
−0.587133 + 0.809490i \(0.699744\pi\)
\(318\) −5.78659e6 −3.20889
\(319\) −338991. −0.186514
\(320\) −851379. −0.464780
\(321\) 465107. 0.251936
\(322\) 33131.5 0.0178075
\(323\) −4.24756e6 −2.26534
\(324\) −325775. −0.172407
\(325\) 2.13378e6 1.12057
\(326\) −723744. −0.377173
\(327\) 5.31599e6 2.74925
\(328\) 2.03354e6 1.04368
\(329\) 57696.4 0.0293873
\(330\) 986032. 0.498432
\(331\) −2.91623e6 −1.46303 −0.731513 0.681827i \(-0.761186\pi\)
−0.731513 + 0.681827i \(0.761186\pi\)
\(332\) 238882. 0.118943
\(333\) 4.36152e6 2.15540
\(334\) −49877.4 −0.0244646
\(335\) 1.45133e6 0.706567
\(336\) −290065. −0.140167
\(337\) −2.93330e6 −1.40696 −0.703480 0.710715i \(-0.748372\pi\)
−0.703480 + 0.710715i \(0.748372\pi\)
\(338\) 1.73376e6 0.825462
\(339\) −690889. −0.326520
\(340\) −161492. −0.0757626
\(341\) −953480. −0.444044
\(342\) 6.54302e6 3.02491
\(343\) 394482. 0.181047
\(344\) 152689. 0.0695682
\(345\) −349409. −0.158047
\(346\) −2.12038e6 −0.952188
\(347\) −1.48233e6 −0.660877 −0.330438 0.943828i \(-0.607197\pi\)
−0.330438 + 0.943828i \(0.607197\pi\)
\(348\) −125318. −0.0554709
\(349\) 3.14342e6 1.38146 0.690731 0.723112i \(-0.257289\pi\)
0.690731 + 0.723112i \(0.257289\pi\)
\(350\) 160014. 0.0698215
\(351\) 6.45162e6 2.79512
\(352\) 379665. 0.163322
\(353\) 2.99556e6 1.27950 0.639751 0.768582i \(-0.279037\pi\)
0.639751 + 0.768582i \(0.279037\pi\)
\(354\) −5.80952e6 −2.46395
\(355\) −934196. −0.393430
\(356\) −136118. −0.0569235
\(357\) −587390. −0.243925
\(358\) 2.42397e6 0.999584
\(359\) −2.23362e6 −0.914687 −0.457344 0.889290i \(-0.651199\pi\)
−0.457344 + 0.889290i \(0.651199\pi\)
\(360\) 2.36950e6 0.963606
\(361\) 3.08088e6 1.24425
\(362\) −642102. −0.257533
\(363\) 2.27262e6 0.905233
\(364\) −36943.3 −0.0146144
\(365\) 140221. 0.0550910
\(366\) 2.25081e6 0.878287
\(367\) −923000. −0.357715 −0.178857 0.983875i \(-0.557240\pi\)
−0.178857 + 0.983875i \(0.557240\pi\)
\(368\) 470700. 0.181186
\(369\) −5.58893e6 −2.13679
\(370\) −1.05979e6 −0.402452
\(371\) −463815. −0.174948
\(372\) −352481. −0.132062
\(373\) −2.31000e6 −0.859688 −0.429844 0.902903i \(-0.641431\pi\)
−0.429844 + 0.902903i \(0.641431\pi\)
\(374\) −2.68988e6 −0.994384
\(375\) −3.75162e6 −1.37765
\(376\) 930348. 0.339372
\(377\) 1.00793e6 0.365240
\(378\) 483814. 0.174160
\(379\) −1.89297e6 −0.676934 −0.338467 0.940978i \(-0.609908\pi\)
−0.338467 + 0.940978i \(0.609908\pi\)
\(380\) 211277. 0.0750572
\(381\) −5.95560e6 −2.10191
\(382\) 2.21293e6 0.775907
\(383\) 2.56252e6 0.892629 0.446315 0.894876i \(-0.352736\pi\)
0.446315 + 0.894876i \(0.352736\pi\)
\(384\) −4.04586e6 −1.40018
\(385\) 79033.9 0.0271745
\(386\) −2.02586e6 −0.692055
\(387\) −419645. −0.142431
\(388\) 240938. 0.0812506
\(389\) 5.33851e6 1.78873 0.894367 0.447334i \(-0.147627\pi\)
0.894367 + 0.447334i \(0.147627\pi\)
\(390\) −2.93180e6 −0.976052
\(391\) 953182. 0.315307
\(392\) 3.16729e6 1.04105
\(393\) −957883. −0.312846
\(394\) 3.28426e6 1.06585
\(395\) 1.51991e6 0.490145
\(396\) −550636. −0.176452
\(397\) 2.26002e6 0.719676 0.359838 0.933015i \(-0.382832\pi\)
0.359838 + 0.933015i \(0.382832\pi\)
\(398\) 2.41884e6 0.765420
\(399\) 768467. 0.241654
\(400\) 2.27333e6 0.710415
\(401\) −5.78253e6 −1.79580 −0.897899 0.440202i \(-0.854907\pi\)
−0.897899 + 0.440202i \(0.854907\pi\)
\(402\) 8.93658e6 2.75808
\(403\) 2.83501e6 0.869546
\(404\) 406730. 0.123980
\(405\) −2.07223e6 −0.627771
\(406\) 75586.2 0.0227577
\(407\) 2.34581e6 0.701950
\(408\) −9.47160e6 −2.81691
\(409\) −1.80868e6 −0.534629 −0.267315 0.963609i \(-0.586136\pi\)
−0.267315 + 0.963609i \(0.586136\pi\)
\(410\) 1.35803e6 0.398978
\(411\) −4.76456e6 −1.39129
\(412\) 330706. 0.0959841
\(413\) −465653. −0.134335
\(414\) −1.46830e6 −0.421030
\(415\) 1.51952e6 0.433097
\(416\) −1.12887e6 −0.319824
\(417\) −4.24261e6 −1.19479
\(418\) 3.51911e6 0.985126
\(419\) −2.44304e6 −0.679822 −0.339911 0.940458i \(-0.610397\pi\)
−0.339911 + 0.940458i \(0.610397\pi\)
\(420\) 29217.2 0.00808192
\(421\) −66516.0 −0.0182903 −0.00914515 0.999958i \(-0.502911\pi\)
−0.00914515 + 0.999958i \(0.502911\pi\)
\(422\) −4.62713e6 −1.26483
\(423\) −2.55695e6 −0.694817
\(424\) −7.47897e6 −2.02035
\(425\) 4.60356e6 1.23629
\(426\) −5.75233e6 −1.53575
\(427\) 180410. 0.0478841
\(428\) 63111.2 0.0166532
\(429\) 6.48946e6 1.70242
\(430\) 101968. 0.0265945
\(431\) −583815. −0.151385 −0.0756924 0.997131i \(-0.524117\pi\)
−0.0756924 + 0.997131i \(0.524117\pi\)
\(432\) 6.87356e6 1.77204
\(433\) 2.28085e6 0.584624 0.292312 0.956323i \(-0.405575\pi\)
0.292312 + 0.956323i \(0.405575\pi\)
\(434\) 212601. 0.0541803
\(435\) −797140. −0.201982
\(436\) 721337. 0.181728
\(437\) −1.24702e6 −0.312372
\(438\) 863412. 0.215047
\(439\) 6.19515e6 1.53423 0.767116 0.641509i \(-0.221691\pi\)
0.767116 + 0.641509i \(0.221691\pi\)
\(440\) 1.27441e6 0.313818
\(441\) −8.70491e6 −2.13141
\(442\) 7.99791e6 1.94725
\(443\) −1.01675e6 −0.246153 −0.123076 0.992397i \(-0.539276\pi\)
−0.123076 + 0.992397i \(0.539276\pi\)
\(444\) 867195. 0.208766
\(445\) −865841. −0.207271
\(446\) −720451. −0.171501
\(447\) 778201. 0.184214
\(448\) −420195. −0.0989135
\(449\) −3.04664e6 −0.713191 −0.356595 0.934259i \(-0.616062\pi\)
−0.356595 + 0.934259i \(0.616062\pi\)
\(450\) −7.09140e6 −1.65082
\(451\) −3.00596e6 −0.695891
\(452\) −93748.0 −0.0215832
\(453\) −4.44251e6 −1.01715
\(454\) −2.35552e6 −0.536349
\(455\) −234994. −0.0532143
\(456\) 1.23915e7 2.79068
\(457\) 4.24293e6 0.950333 0.475166 0.879896i \(-0.342388\pi\)
0.475166 + 0.879896i \(0.342388\pi\)
\(458\) 3.66700e6 0.816860
\(459\) 1.39192e7 3.08377
\(460\) −47411.9 −0.0104470
\(461\) 1.08445e6 0.237662 0.118831 0.992915i \(-0.462085\pi\)
0.118831 + 0.992915i \(0.462085\pi\)
\(462\) 486653. 0.106075
\(463\) 932916. 0.202251 0.101125 0.994874i \(-0.467756\pi\)
0.101125 + 0.994874i \(0.467756\pi\)
\(464\) 1.07386e6 0.231553
\(465\) −2.24211e6 −0.480867
\(466\) 2.59034e6 0.552576
\(467\) −252067. −0.0534839 −0.0267420 0.999642i \(-0.508513\pi\)
−0.0267420 + 0.999642i \(0.508513\pi\)
\(468\) 1.63722e6 0.345536
\(469\) 716298. 0.150370
\(470\) 621301. 0.129735
\(471\) −6.69491e6 −1.39057
\(472\) −7.50861e6 −1.55133
\(473\) −225703. −0.0463857
\(474\) 9.35885e6 1.91327
\(475\) −6.02272e6 −1.22478
\(476\) −79704.0 −0.0161236
\(477\) 2.05550e7 4.13639
\(478\) −5.98064e6 −1.19723
\(479\) −638652. −0.127182 −0.0635909 0.997976i \(-0.520255\pi\)
−0.0635909 + 0.997976i \(0.520255\pi\)
\(480\) 892786. 0.176866
\(481\) −6.97487e6 −1.37459
\(482\) −7.91817e6 −1.55241
\(483\) −172449. −0.0336352
\(484\) 308376. 0.0598367
\(485\) 1.53259e6 0.295851
\(486\) −2.78327e6 −0.534521
\(487\) −6.40830e6 −1.22439 −0.612196 0.790706i \(-0.709714\pi\)
−0.612196 + 0.790706i \(0.709714\pi\)
\(488\) 2.90910e6 0.552979
\(489\) 3.76708e6 0.712415
\(490\) 2.11517e6 0.397974
\(491\) −8.07458e6 −1.51153 −0.755764 0.654844i \(-0.772734\pi\)
−0.755764 + 0.654844i \(0.772734\pi\)
\(492\) −1.11124e6 −0.206964
\(493\) 2.17459e6 0.402958
\(494\) −1.04635e7 −1.92912
\(495\) −3.50256e6 −0.642500
\(496\) 3.02043e6 0.551270
\(497\) −461069. −0.0837288
\(498\) 9.35646e6 1.69059
\(499\) 7.25360e6 1.30407 0.652037 0.758187i \(-0.273915\pi\)
0.652037 + 0.758187i \(0.273915\pi\)
\(500\) −509064. −0.0910641
\(501\) 259612. 0.0462093
\(502\) 337254. 0.0597308
\(503\) −4.02697e6 −0.709673 −0.354836 0.934928i \(-0.615463\pi\)
−0.354836 + 0.934928i \(0.615463\pi\)
\(504\) 1.16946e6 0.205073
\(505\) 2.58719e6 0.451440
\(506\) −789712. −0.137117
\(507\) −9.02421e6 −1.55916
\(508\) −808127. −0.138938
\(509\) −98493.6 −0.0168505 −0.00842526 0.999965i \(-0.502682\pi\)
−0.00842526 + 0.999965i \(0.502682\pi\)
\(510\) −6.32527e6 −1.07685
\(511\) 69205.5 0.0117243
\(512\) −6.61324e6 −1.11491
\(513\) −1.82101e7 −3.05505
\(514\) 5.78636e6 0.966045
\(515\) 2.10360e6 0.349499
\(516\) −83437.6 −0.0137955
\(517\) −1.37523e6 −0.226282
\(518\) −523053. −0.0856489
\(519\) 1.10365e7 1.79852
\(520\) −3.78925e6 −0.614533
\(521\) −7.76729e6 −1.25365 −0.626824 0.779161i \(-0.715645\pi\)
−0.626824 + 0.779161i \(0.715645\pi\)
\(522\) −3.34977e6 −0.538070
\(523\) 1.05753e7 1.69058 0.845292 0.534305i \(-0.179427\pi\)
0.845292 + 0.534305i \(0.179427\pi\)
\(524\) −129977. −0.0206794
\(525\) −832874. −0.131881
\(526\) −2.88478e6 −0.454620
\(527\) 6.11646e6 0.959342
\(528\) 6.91388e6 1.07929
\(529\) 279841. 0.0434783
\(530\) −4.99457e6 −0.772339
\(531\) 2.06365e7 3.17614
\(532\) 104275. 0.0159735
\(533\) 8.93771e6 1.36273
\(534\) −5.33143e6 −0.809079
\(535\) 401447. 0.0606378
\(536\) 1.15502e7 1.73651
\(537\) −1.26167e7 −1.88804
\(538\) −8.83141e6 −1.31545
\(539\) −4.68186e6 −0.694140
\(540\) −692349. −0.102174
\(541\) −2.72554e6 −0.400369 −0.200184 0.979758i \(-0.564154\pi\)
−0.200184 + 0.979758i \(0.564154\pi\)
\(542\) 3.30247e6 0.482881
\(543\) 3.34214e6 0.486435
\(544\) −2.43551e6 −0.352852
\(545\) 4.58838e6 0.661711
\(546\) −1.44698e6 −0.207721
\(547\) −1.12681e7 −1.61020 −0.805101 0.593137i \(-0.797889\pi\)
−0.805101 + 0.593137i \(0.797889\pi\)
\(548\) −646512. −0.0919655
\(549\) −7.99529e6 −1.13215
\(550\) −3.81405e6 −0.537625
\(551\) −2.84496e6 −0.399206
\(552\) −2.78073e6 −0.388428
\(553\) 750145. 0.104311
\(554\) 1.41607e6 0.196024
\(555\) 5.51618e6 0.760162
\(556\) −575687. −0.0789768
\(557\) 1.42068e7 1.94025 0.970125 0.242607i \(-0.0780024\pi\)
0.970125 + 0.242607i \(0.0780024\pi\)
\(558\) −9.42190e6 −1.28101
\(559\) 671090. 0.0908345
\(560\) −250363. −0.0337365
\(561\) 1.40008e7 1.87822
\(562\) 2.51160e6 0.335436
\(563\) 9.93624e6 1.32115 0.660574 0.750761i \(-0.270313\pi\)
0.660574 + 0.750761i \(0.270313\pi\)
\(564\) −508394. −0.0672981
\(565\) −596326. −0.0785892
\(566\) 4.86168e6 0.637889
\(567\) −1.02274e6 −0.133601
\(568\) −7.43469e6 −0.966923
\(569\) −2.43903e6 −0.315818 −0.157909 0.987454i \(-0.550475\pi\)
−0.157909 + 0.987454i \(0.550475\pi\)
\(570\) 8.27520e6 1.06682
\(571\) 1.54691e7 1.98552 0.992759 0.120124i \(-0.0383292\pi\)
0.992759 + 0.120124i \(0.0383292\pi\)
\(572\) 880567. 0.112531
\(573\) −1.15183e7 −1.46555
\(574\) 670250. 0.0849096
\(575\) 1.35154e6 0.170475
\(576\) 1.86219e7 2.33866
\(577\) −1.30892e7 −1.63672 −0.818360 0.574706i \(-0.805116\pi\)
−0.818360 + 0.574706i \(0.805116\pi\)
\(578\) 9.70910e6 1.20881
\(579\) 1.05446e7 1.30717
\(580\) −108165. −0.0133511
\(581\) 749952. 0.0921708
\(582\) 9.43698e6 1.15485
\(583\) 1.10553e7 1.34710
\(584\) 1.11593e6 0.135396
\(585\) 1.04143e7 1.25817
\(586\) −4.81734e6 −0.579514
\(587\) 2.37887e6 0.284954 0.142477 0.989798i \(-0.454493\pi\)
0.142477 + 0.989798i \(0.454493\pi\)
\(588\) −1.73079e6 −0.206443
\(589\) −8.00201e6 −0.950410
\(590\) −5.01436e6 −0.593042
\(591\) −1.70946e7 −2.01321
\(592\) −7.43103e6 −0.871455
\(593\) −3.47325e6 −0.405601 −0.202800 0.979220i \(-0.565004\pi\)
−0.202800 + 0.979220i \(0.565004\pi\)
\(594\) −1.15320e7 −1.34103
\(595\) −506993. −0.0587096
\(596\) 105596. 0.0121767
\(597\) −1.25901e7 −1.44575
\(598\) 2.34808e6 0.268509
\(599\) 5.31390e6 0.605127 0.302564 0.953129i \(-0.402158\pi\)
0.302564 + 0.953129i \(0.402158\pi\)
\(600\) −1.34300e7 −1.52299
\(601\) −1.35246e6 −0.152734 −0.0763672 0.997080i \(-0.524332\pi\)
−0.0763672 + 0.997080i \(0.524332\pi\)
\(602\) 50325.8 0.00565978
\(603\) −3.17443e7 −3.55527
\(604\) −602813. −0.0672341
\(605\) 1.96156e6 0.217878
\(606\) 1.59307e7 1.76219
\(607\) −384207. −0.0423247 −0.0211623 0.999776i \(-0.506737\pi\)
−0.0211623 + 0.999776i \(0.506737\pi\)
\(608\) 3.18631e6 0.349566
\(609\) −393426. −0.0429853
\(610\) 1.94274e6 0.211393
\(611\) 4.08902e6 0.443115
\(612\) 3.53226e6 0.381219
\(613\) −1.67749e7 −1.80306 −0.901530 0.432718i \(-0.857555\pi\)
−0.901530 + 0.432718i \(0.857555\pi\)
\(614\) 545368. 0.0583806
\(615\) −7.06853e6 −0.753601
\(616\) 628982. 0.0667862
\(617\) 1.27567e6 0.134905 0.0674523 0.997723i \(-0.478513\pi\)
0.0674523 + 0.997723i \(0.478513\pi\)
\(618\) 1.29530e7 1.36426
\(619\) 1.46795e7 1.53988 0.769938 0.638118i \(-0.220287\pi\)
0.769938 + 0.638118i \(0.220287\pi\)
\(620\) −304237. −0.0317857
\(621\) 4.08647e6 0.425226
\(622\) 6.44695e6 0.668157
\(623\) −427333. −0.0441109
\(624\) −2.05573e7 −2.11351
\(625\) 4.74593e6 0.485983
\(626\) −1.61669e7 −1.64888
\(627\) −1.83169e7 −1.86073
\(628\) −908444. −0.0919177
\(629\) −1.50481e7 −1.51654
\(630\) 780980. 0.0783951
\(631\) −1.65066e7 −1.65038 −0.825190 0.564856i \(-0.808932\pi\)
−0.825190 + 0.564856i \(0.808932\pi\)
\(632\) 1.20960e7 1.20462
\(633\) 2.40842e7 2.38904
\(634\) −1.11660e7 −1.10325
\(635\) −5.14045e6 −0.505902
\(636\) 4.08692e6 0.400640
\(637\) 1.39207e7 1.35930
\(638\) −1.80165e6 −0.175234
\(639\) 2.04333e7 1.97964
\(640\) −3.49209e6 −0.337005
\(641\) 1.87653e6 0.180389 0.0901947 0.995924i \(-0.471251\pi\)
0.0901947 + 0.995924i \(0.471251\pi\)
\(642\) 2.47192e6 0.236699
\(643\) 1.72531e6 0.164566 0.0822830 0.996609i \(-0.473779\pi\)
0.0822830 + 0.996609i \(0.473779\pi\)
\(644\) −23400.0 −0.00222331
\(645\) −530741. −0.0502324
\(646\) −2.25746e7 −2.12833
\(647\) 1.18002e7 1.10822 0.554112 0.832442i \(-0.313058\pi\)
0.554112 + 0.832442i \(0.313058\pi\)
\(648\) −1.64916e7 −1.54286
\(649\) 1.10992e7 1.03438
\(650\) 1.13404e7 1.05280
\(651\) −1.10659e6 −0.102337
\(652\) 511163. 0.0470912
\(653\) 1.00372e7 0.921147 0.460573 0.887622i \(-0.347644\pi\)
0.460573 + 0.887622i \(0.347644\pi\)
\(654\) 2.82530e7 2.58298
\(655\) −826776. −0.0752982
\(656\) 9.52226e6 0.863933
\(657\) −3.06700e6 −0.277204
\(658\) 306641. 0.0276099
\(659\) 1.02223e7 0.916929 0.458464 0.888713i \(-0.348400\pi\)
0.458464 + 0.888713i \(0.348400\pi\)
\(660\) −696410. −0.0622308
\(661\) −1.87007e6 −0.166477 −0.0832386 0.996530i \(-0.526526\pi\)
−0.0832386 + 0.996530i \(0.526526\pi\)
\(662\) −1.54990e7 −1.37454
\(663\) −4.16291e7 −3.67801
\(664\) 1.20929e7 1.06441
\(665\) 663286. 0.0581630
\(666\) 2.31803e7 2.02504
\(667\) 638429. 0.0555646
\(668\) 35227.2 0.00305448
\(669\) 3.74995e6 0.323936
\(670\) 7.71341e6 0.663834
\(671\) −4.30020e6 −0.368708
\(672\) 440631. 0.0376402
\(673\) −2.14057e6 −0.182176 −0.0910881 0.995843i \(-0.529034\pi\)
−0.0910881 + 0.995843i \(0.529034\pi\)
\(674\) −1.55897e7 −1.32187
\(675\) 1.97363e7 1.66727
\(676\) −1.22451e6 −0.103061
\(677\) 1.42893e7 1.19822 0.599112 0.800665i \(-0.295520\pi\)
0.599112 + 0.800665i \(0.295520\pi\)
\(678\) −3.67189e6 −0.306772
\(679\) 756407. 0.0629624
\(680\) −8.17520e6 −0.677995
\(681\) 1.22605e7 1.01307
\(682\) −5.06749e6 −0.417188
\(683\) −7.29980e6 −0.598769 −0.299384 0.954133i \(-0.596781\pi\)
−0.299384 + 0.954133i \(0.596781\pi\)
\(684\) −4.62117e6 −0.377670
\(685\) −4.11242e6 −0.334866
\(686\) 2.09656e6 0.170097
\(687\) −1.90867e7 −1.54291
\(688\) 714980. 0.0575868
\(689\) −3.28712e7 −2.63796
\(690\) −1.85701e6 −0.148488
\(691\) 3.77617e6 0.300855 0.150427 0.988621i \(-0.451935\pi\)
0.150427 + 0.988621i \(0.451935\pi\)
\(692\) 1.49757e6 0.118884
\(693\) −1.72868e6 −0.136735
\(694\) −7.87817e6 −0.620907
\(695\) −3.66191e6 −0.287572
\(696\) −6.34395e6 −0.496406
\(697\) 1.92828e7 1.50345
\(698\) 1.67064e7 1.29791
\(699\) −1.34827e7 −1.04372
\(700\) −113014. −0.00871743
\(701\) −1.27732e7 −0.981761 −0.490880 0.871227i \(-0.663325\pi\)
−0.490880 + 0.871227i \(0.663325\pi\)
\(702\) 3.42886e7 2.62607
\(703\) 1.96870e7 1.50242
\(704\) 1.00156e7 0.761634
\(705\) −3.23387e6 −0.245047
\(706\) 1.59206e7 1.20212
\(707\) 1.27690e6 0.0960744
\(708\) 4.10312e6 0.307632
\(709\) −2.21287e7 −1.65326 −0.826628 0.562748i \(-0.809744\pi\)
−0.826628 + 0.562748i \(0.809744\pi\)
\(710\) −4.96500e6 −0.369635
\(711\) −3.32443e7 −2.46629
\(712\) −6.89069e6 −0.509405
\(713\) 1.79571e6 0.132285
\(714\) −3.12182e6 −0.229172
\(715\) 5.60124e6 0.409750
\(716\) −1.71199e6 −0.124801
\(717\) 3.11292e7 2.26136
\(718\) −1.18711e7 −0.859367
\(719\) −6.86509e6 −0.495249 −0.247625 0.968856i \(-0.579650\pi\)
−0.247625 + 0.968856i \(0.579650\pi\)
\(720\) 1.10954e7 0.797649
\(721\) 1.03823e6 0.0743796
\(722\) 1.63740e7 1.16900
\(723\) 4.12140e7 2.93224
\(724\) 453501. 0.0321538
\(725\) 3.08340e6 0.217864
\(726\) 1.20784e7 0.850485
\(727\) 4.75733e6 0.333831 0.166916 0.985971i \(-0.446619\pi\)
0.166916 + 0.985971i \(0.446619\pi\)
\(728\) −1.87017e6 −0.130784
\(729\) −6.60269e6 −0.460153
\(730\) 745236. 0.0517591
\(731\) 1.44786e6 0.100215
\(732\) −1.58969e6 −0.109657
\(733\) 2.41232e7 1.65835 0.829173 0.558993i \(-0.188812\pi\)
0.829173 + 0.558993i \(0.188812\pi\)
\(734\) −4.90550e6 −0.336080
\(735\) −1.10094e7 −0.751704
\(736\) −715031. −0.0486553
\(737\) −1.70734e7 −1.15785
\(738\) −2.97036e7 −2.00756
\(739\) −1.01851e7 −0.686050 −0.343025 0.939326i \(-0.611452\pi\)
−0.343025 + 0.939326i \(0.611452\pi\)
\(740\) 748501. 0.0502473
\(741\) 5.44623e7 3.64377
\(742\) −2.46505e6 −0.164368
\(743\) −1.55515e7 −1.03348 −0.516738 0.856144i \(-0.672854\pi\)
−0.516738 + 0.856144i \(0.672854\pi\)
\(744\) −1.78436e7 −1.18182
\(745\) 671687. 0.0443380
\(746\) −1.22770e7 −0.807694
\(747\) −3.32358e7 −2.17924
\(748\) 1.89980e6 0.124152
\(749\) 198133. 0.0129048
\(750\) −1.99388e7 −1.29433
\(751\) −1.03320e7 −0.668473 −0.334237 0.942489i \(-0.608478\pi\)
−0.334237 + 0.942489i \(0.608478\pi\)
\(752\) 4.35645e6 0.280924
\(753\) −1.75541e6 −0.112821
\(754\) 5.35690e6 0.343151
\(755\) −3.83446e6 −0.244814
\(756\) −341706. −0.0217444
\(757\) −1.61676e7 −1.02543 −0.512714 0.858559i \(-0.671360\pi\)
−0.512714 + 0.858559i \(0.671360\pi\)
\(758\) −1.00606e7 −0.635993
\(759\) 4.11045e6 0.258991
\(760\) 1.06954e7 0.671682
\(761\) 8.40313e6 0.525993 0.262996 0.964797i \(-0.415289\pi\)
0.262996 + 0.964797i \(0.415289\pi\)
\(762\) −3.16524e7 −1.97478
\(763\) 2.26458e6 0.140824
\(764\) −1.56294e6 −0.0968743
\(765\) 2.24685e7 1.38810
\(766\) 1.36191e7 0.838643
\(767\) −3.30015e7 −2.02556
\(768\) 1.00618e7 0.615560
\(769\) −1.74997e7 −1.06713 −0.533563 0.845760i \(-0.679147\pi\)
−0.533563 + 0.845760i \(0.679147\pi\)
\(770\) 420044. 0.0255310
\(771\) −3.01180e7 −1.82469
\(772\) 1.43081e6 0.0864052
\(773\) −2.11605e7 −1.27373 −0.636863 0.770977i \(-0.719768\pi\)
−0.636863 + 0.770977i \(0.719768\pi\)
\(774\) −2.23030e6 −0.133817
\(775\) 8.67268e6 0.518679
\(776\) 1.21970e7 0.727106
\(777\) 2.72249e6 0.161776
\(778\) 2.83727e7 1.68055
\(779\) −2.52273e7 −1.48945
\(780\) 2.07066e6 0.121863
\(781\) 1.09899e7 0.644712
\(782\) 5.06590e6 0.296237
\(783\) 9.32287e6 0.543432
\(784\) 1.48312e7 0.861759
\(785\) −5.77856e6 −0.334692
\(786\) −5.09089e6 −0.293926
\(787\) −1.19600e7 −0.688325 −0.344163 0.938910i \(-0.611837\pi\)
−0.344163 + 0.938910i \(0.611837\pi\)
\(788\) −2.31959e6 −0.133075
\(789\) 1.50153e7 0.858698
\(790\) 8.07789e6 0.460501
\(791\) −294315. −0.0167252
\(792\) −2.78748e7 −1.57906
\(793\) 1.27859e7 0.722020
\(794\) 1.20114e7 0.676150
\(795\) 2.59967e7 1.45882
\(796\) −1.70837e6 −0.0955650
\(797\) 1.32264e7 0.737559 0.368779 0.929517i \(-0.379776\pi\)
0.368779 + 0.929517i \(0.379776\pi\)
\(798\) 4.08420e6 0.227038
\(799\) 8.82194e6 0.488874
\(800\) −3.45336e6 −0.190773
\(801\) 1.89382e7 1.04294
\(802\) −3.07326e7 −1.68719
\(803\) −1.64956e6 −0.0902773
\(804\) −6.31169e6 −0.344354
\(805\) −148846. −0.00809557
\(806\) 1.50673e7 0.816956
\(807\) 4.59674e7 2.48466
\(808\) 2.05898e7 1.10949
\(809\) 2.81536e6 0.151239 0.0756194 0.997137i \(-0.475907\pi\)
0.0756194 + 0.997137i \(0.475907\pi\)
\(810\) −1.10134e7 −0.589803
\(811\) 2.03162e7 1.08465 0.542325 0.840169i \(-0.317544\pi\)
0.542325 + 0.840169i \(0.317544\pi\)
\(812\) −53384.7 −0.00284136
\(813\) −1.71893e7 −0.912079
\(814\) 1.24673e7 0.659496
\(815\) 3.25148e6 0.171469
\(816\) −4.43517e7 −2.33177
\(817\) −1.89419e6 −0.0992817
\(818\) −9.61263e6 −0.502295
\(819\) 5.13994e6 0.267761
\(820\) −959142. −0.0498136
\(821\) 2.06237e7 1.06785 0.533924 0.845532i \(-0.320717\pi\)
0.533924 + 0.845532i \(0.320717\pi\)
\(822\) −2.53223e7 −1.30715
\(823\) 3.05683e7 1.57316 0.786578 0.617491i \(-0.211851\pi\)
0.786578 + 0.617491i \(0.211851\pi\)
\(824\) 1.67413e7 0.858955
\(825\) 1.98521e7 1.01548
\(826\) −2.47482e6 −0.126210
\(827\) 3.23053e7 1.64252 0.821258 0.570556i \(-0.193272\pi\)
0.821258 + 0.570556i \(0.193272\pi\)
\(828\) 1.03702e6 0.0525669
\(829\) 1.30561e7 0.659823 0.329912 0.944012i \(-0.392981\pi\)
0.329912 + 0.944012i \(0.392981\pi\)
\(830\) 8.07582e6 0.406904
\(831\) −7.37063e6 −0.370256
\(832\) −2.97798e7 −1.49147
\(833\) 3.00336e7 1.49967
\(834\) −2.25483e7 −1.12253
\(835\) 224078. 0.0111220
\(836\) −2.48546e6 −0.122996
\(837\) 2.62224e7 1.29378
\(838\) −1.29841e7 −0.638707
\(839\) −6.50241e6 −0.318911 −0.159455 0.987205i \(-0.550974\pi\)
−0.159455 + 0.987205i \(0.550974\pi\)
\(840\) 1.47906e6 0.0723246
\(841\) −1.90546e7 −0.928989
\(842\) −353514. −0.0171841
\(843\) −1.30729e7 −0.633581
\(844\) 3.26803e6 0.157917
\(845\) −7.78905e6 −0.375269
\(846\) −1.35895e7 −0.652795
\(847\) 968123. 0.0463684
\(848\) −3.50210e7 −1.67240
\(849\) −2.53050e7 −1.20486
\(850\) 2.44667e7 1.16152
\(851\) −4.41790e6 −0.209118
\(852\) 4.06273e6 0.191743
\(853\) −3.18228e7 −1.49750 −0.748749 0.662854i \(-0.769345\pi\)
−0.748749 + 0.662854i \(0.769345\pi\)
\(854\) 958832. 0.0449881
\(855\) −2.93950e7 −1.37518
\(856\) 3.19487e6 0.149028
\(857\) −1.82430e7 −0.848487 −0.424243 0.905548i \(-0.639460\pi\)
−0.424243 + 0.905548i \(0.639460\pi\)
\(858\) 3.44897e7 1.59945
\(859\) 2.66125e7 1.23056 0.615281 0.788308i \(-0.289043\pi\)
0.615281 + 0.788308i \(0.289043\pi\)
\(860\) −72017.3 −0.00332040
\(861\) −3.48865e6 −0.160380
\(862\) −3.10282e6 −0.142229
\(863\) 4.88600e6 0.223320 0.111660 0.993746i \(-0.464383\pi\)
0.111660 + 0.993746i \(0.464383\pi\)
\(864\) −1.04415e7 −0.475858
\(865\) 9.52595e6 0.432881
\(866\) 1.21221e7 0.549266
\(867\) −5.05358e7 −2.28324
\(868\) −150155. −0.00676457
\(869\) −1.78802e7 −0.803198
\(870\) −4.23658e6 −0.189766
\(871\) 5.07650e7 2.26735
\(872\) 3.65161e7 1.62627
\(873\) −3.35219e7 −1.48865
\(874\) −6.62760e6 −0.293479
\(875\) −1.59817e6 −0.0705670
\(876\) −609807. −0.0268492
\(877\) −2.36006e7 −1.03615 −0.518077 0.855334i \(-0.673352\pi\)
−0.518077 + 0.855334i \(0.673352\pi\)
\(878\) 3.29256e7 1.44144
\(879\) 2.50743e7 1.09460
\(880\) 5.96757e6 0.259771
\(881\) 7.11920e6 0.309023 0.154512 0.987991i \(-0.450620\pi\)
0.154512 + 0.987991i \(0.450620\pi\)
\(882\) −4.62643e7 −2.00251
\(883\) −3.69545e7 −1.59502 −0.797509 0.603307i \(-0.793849\pi\)
−0.797509 + 0.603307i \(0.793849\pi\)
\(884\) −5.64873e6 −0.243120
\(885\) 2.60997e7 1.12015
\(886\) −5.40374e6 −0.231265
\(887\) 2.40116e6 0.102474 0.0512368 0.998687i \(-0.483684\pi\)
0.0512368 + 0.998687i \(0.483684\pi\)
\(888\) 4.38999e7 1.86823
\(889\) −2.53705e6 −0.107665
\(890\) −4.60171e6 −0.194735
\(891\) 2.43778e7 1.02873
\(892\) 508837. 0.0214125
\(893\) −1.15415e7 −0.484323
\(894\) 4.13593e6 0.173073
\(895\) −1.08899e7 −0.454428
\(896\) −1.72351e6 −0.0717206
\(897\) −1.22217e7 −0.507168
\(898\) −1.61921e7 −0.670057
\(899\) 4.09672e6 0.169059
\(900\) 5.00848e6 0.206110
\(901\) −7.09186e7 −2.91037
\(902\) −1.59758e7 −0.653804
\(903\) −261946. −0.0106903
\(904\) −4.74579e6 −0.193147
\(905\) 2.88469e6 0.117079
\(906\) −2.36107e7 −0.955628
\(907\) 1.34643e7 0.543458 0.271729 0.962374i \(-0.412405\pi\)
0.271729 + 0.962374i \(0.412405\pi\)
\(908\) 1.66365e6 0.0669648
\(909\) −5.65886e7 −2.27153
\(910\) −1.24893e6 −0.0499959
\(911\) 3.07348e7 1.22697 0.613485 0.789707i \(-0.289767\pi\)
0.613485 + 0.789707i \(0.289767\pi\)
\(912\) 5.80243e7 2.31006
\(913\) −1.78756e7 −0.709715
\(914\) 2.25500e7 0.892857
\(915\) −1.01119e7 −0.399284
\(916\) −2.58991e6 −0.101987
\(917\) −408052. −0.0160248
\(918\) 7.39766e7 2.89726
\(919\) 2.52416e7 0.985888 0.492944 0.870061i \(-0.335921\pi\)
0.492944 + 0.870061i \(0.335921\pi\)
\(920\) −2.40013e6 −0.0934898
\(921\) −2.83864e6 −0.110271
\(922\) 5.76358e6 0.223288
\(923\) −3.26766e7 −1.26250
\(924\) −343711. −0.0132438
\(925\) −2.13370e7 −0.819935
\(926\) 4.95820e6 0.190019
\(927\) −4.60113e7 −1.75859
\(928\) −1.63127e6 −0.0621807
\(929\) 1.43072e7 0.543897 0.271948 0.962312i \(-0.412332\pi\)
0.271948 + 0.962312i \(0.412332\pi\)
\(930\) −1.19162e7 −0.451785
\(931\) −3.92922e7 −1.48570
\(932\) −1.82949e6 −0.0689908
\(933\) −3.35563e7 −1.26203
\(934\) −1.33967e6 −0.0502492
\(935\) 1.20845e7 0.452064
\(936\) 8.28810e7 3.09218
\(937\) −1.37068e7 −0.510020 −0.255010 0.966938i \(-0.582079\pi\)
−0.255010 + 0.966938i \(0.582079\pi\)
\(938\) 3.80693e6 0.141276
\(939\) 8.41485e7 3.11446
\(940\) −438809. −0.0161978
\(941\) 7.51582e6 0.276696 0.138348 0.990384i \(-0.455821\pi\)
0.138348 + 0.990384i \(0.455821\pi\)
\(942\) −3.55816e7 −1.30647
\(943\) 5.66118e6 0.207313
\(944\) −3.51598e7 −1.28415
\(945\) −2.17357e6 −0.0791762
\(946\) −1.19955e6 −0.0435803
\(947\) 1.59343e7 0.577375 0.288688 0.957423i \(-0.406781\pi\)
0.288688 + 0.957423i \(0.406781\pi\)
\(948\) −6.60993e6 −0.238878
\(949\) 4.90469e6 0.176785
\(950\) −3.20091e7 −1.15071
\(951\) 5.81187e7 2.08384
\(952\) −4.03484e6 −0.144289
\(953\) 2.00561e7 0.715343 0.357672 0.933847i \(-0.383571\pi\)
0.357672 + 0.933847i \(0.383571\pi\)
\(954\) 1.09244e8 3.88622
\(955\) −9.94177e6 −0.352740
\(956\) 4.22398e6 0.149478
\(957\) 9.37756e6 0.330986
\(958\) −3.39426e6 −0.119490
\(959\) −2.02967e6 −0.0712655
\(960\) 2.35518e7 0.824795
\(961\) −1.71063e7 −0.597514
\(962\) −3.70695e7 −1.29145
\(963\) −8.78070e6 −0.305115
\(964\) 5.59241e6 0.193823
\(965\) 9.10132e6 0.314620
\(966\) −916522. −0.0316009
\(967\) 1.66228e7 0.571661 0.285831 0.958280i \(-0.407731\pi\)
0.285831 + 0.958280i \(0.407731\pi\)
\(968\) 1.56109e7 0.535475
\(969\) 1.17501e8 4.02005
\(970\) 8.14533e6 0.277958
\(971\) −5.74792e7 −1.95642 −0.978212 0.207609i \(-0.933432\pi\)
−0.978212 + 0.207609i \(0.933432\pi\)
\(972\) 1.96576e6 0.0667366
\(973\) −1.80732e6 −0.0612004
\(974\) −3.40584e7 −1.15034
\(975\) −5.90269e7 −1.98856
\(976\) 1.36221e7 0.457742
\(977\) 2.65537e7 0.889996 0.444998 0.895532i \(-0.353204\pi\)
0.444998 + 0.895532i \(0.353204\pi\)
\(978\) 2.00210e7 0.669328
\(979\) 1.01858e7 0.339654
\(980\) −1.49389e6 −0.0496882
\(981\) −1.00360e8 −3.32957
\(982\) −4.29142e7 −1.42011
\(983\) 2.70907e6 0.0894205 0.0447102 0.999000i \(-0.485764\pi\)
0.0447102 + 0.999000i \(0.485764\pi\)
\(984\) −5.62541e7 −1.85211
\(985\) −1.47548e7 −0.484555
\(986\) 1.15573e7 0.378587
\(987\) −1.59606e6 −0.0521503
\(988\) 7.39010e6 0.240856
\(989\) 425070. 0.0138188
\(990\) −1.86152e7 −0.603642
\(991\) 2.13175e7 0.689530 0.344765 0.938689i \(-0.387959\pi\)
0.344765 + 0.938689i \(0.387959\pi\)
\(992\) −4.58827e6 −0.148037
\(993\) 8.06721e7 2.59627
\(994\) −2.45046e6 −0.0786649
\(995\) −1.08668e7 −0.347973
\(996\) −6.60824e6 −0.211075
\(997\) 4.42548e7 1.41001 0.705005 0.709202i \(-0.250945\pi\)
0.705005 + 0.709202i \(0.250945\pi\)
\(998\) 3.85509e7 1.22520
\(999\) −6.45139e7 −2.04522
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.1.3 3
3.2 odd 2 207.6.a.b.1.1 3
4.3 odd 2 368.6.a.e.1.3 3
5.4 even 2 575.6.a.b.1.1 3
23.22 odd 2 529.6.a.a.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.6.a.a.1.3 3 1.1 even 1 trivial
207.6.a.b.1.1 3 3.2 odd 2
368.6.a.e.1.3 3 4.3 odd 2
529.6.a.a.1.3 3 23.22 odd 2
575.6.a.b.1.1 3 5.4 even 2