Properties

Label 29.6.a.a.1.1
Level $29$
Weight $6$
Character 29.1
Self dual yes
Analytic conductor $4.651$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.34807\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.21534 q^{2} +0.734546 q^{3} +52.9225 q^{4} +41.6400 q^{5} -6.76909 q^{6} -90.0205 q^{7} -192.808 q^{8} -242.460 q^{9} +O(q^{10})\) \(q-9.21534 q^{2} +0.734546 q^{3} +52.9225 q^{4} +41.6400 q^{5} -6.76909 q^{6} -90.0205 q^{7} -192.808 q^{8} -242.460 q^{9} -383.727 q^{10} -269.080 q^{11} +38.8740 q^{12} -444.579 q^{13} +829.569 q^{14} +30.5865 q^{15} +83.2700 q^{16} +485.649 q^{17} +2234.36 q^{18} -1572.80 q^{19} +2203.69 q^{20} -66.1242 q^{21} +2479.66 q^{22} -398.704 q^{23} -141.626 q^{24} -1391.11 q^{25} +4096.95 q^{26} -356.593 q^{27} -4764.11 q^{28} -841.000 q^{29} -281.865 q^{30} -8469.77 q^{31} +5402.49 q^{32} -197.652 q^{33} -4475.43 q^{34} -3748.45 q^{35} -12831.6 q^{36} +3339.33 q^{37} +14493.8 q^{38} -326.564 q^{39} -8028.52 q^{40} +18154.2 q^{41} +609.357 q^{42} +9996.71 q^{43} -14240.4 q^{44} -10096.1 q^{45} +3674.20 q^{46} +12568.0 q^{47} +61.1656 q^{48} -8703.31 q^{49} +12819.6 q^{50} +356.732 q^{51} -23528.2 q^{52} -21343.1 q^{53} +3286.12 q^{54} -11204.5 q^{55} +17356.7 q^{56} -1155.29 q^{57} +7750.10 q^{58} -30036.1 q^{59} +1618.71 q^{60} +49792.6 q^{61} +78051.8 q^{62} +21826.4 q^{63} -52450.4 q^{64} -18512.3 q^{65} +1821.43 q^{66} +47588.2 q^{67} +25701.8 q^{68} -292.867 q^{69} +34543.3 q^{70} -50164.6 q^{71} +46748.3 q^{72} -44770.3 q^{73} -30773.1 q^{74} -1021.83 q^{75} -83236.3 q^{76} +24222.7 q^{77} +3009.40 q^{78} -78464.6 q^{79} +3467.36 q^{80} +58656.0 q^{81} -167297. q^{82} +46721.8 q^{83} -3499.46 q^{84} +20222.4 q^{85} -92123.1 q^{86} -617.753 q^{87} +51880.7 q^{88} -39465.7 q^{89} +93038.6 q^{90} +40021.2 q^{91} -21100.4 q^{92} -6221.43 q^{93} -115818. q^{94} -65491.2 q^{95} +3968.38 q^{96} +48336.3 q^{97} +80204.0 q^{98} +65241.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} - 788 q^{10} - 124 q^{11} + 20 q^{12} - 460 q^{13} + 768 q^{14} + 932 q^{15} - 414 q^{16} + 184 q^{17} + 3208 q^{18} - 2392 q^{19} + 2822 q^{20} + 992 q^{21} + 5538 q^{22} - 1192 q^{23} + 6786 q^{24} + 1824 q^{25} + 4724 q^{26} + 2468 q^{27} + 44 q^{28} - 3364 q^{29} + 8186 q^{30} - 19212 q^{31} + 6552 q^{32} - 10580 q^{33} - 7612 q^{34} - 22944 q^{35} - 7468 q^{36} - 10928 q^{37} - 456 q^{38} - 8732 q^{39} - 20 q^{40} - 1120 q^{41} + 1844 q^{42} - 21420 q^{43} - 1932 q^{44} - 8344 q^{45} - 7588 q^{46} + 23772 q^{47} + 33060 q^{48} + 10452 q^{49} + 43240 q^{50} + 12744 q^{51} - 29062 q^{52} + 8860 q^{53} + 35410 q^{54} - 52652 q^{55} + 34304 q^{56} + 48944 q^{57} - 10840 q^{59} + 25200 q^{60} + 49448 q^{61} + 18518 q^{62} + 27488 q^{63} - 20734 q^{64} + 97836 q^{65} - 47744 q^{66} - 7840 q^{67} + 20724 q^{68} + 58792 q^{69} - 77496 q^{70} - 48744 q^{71} + 8088 q^{72} - 74992 q^{73} - 35920 q^{74} - 90448 q^{75} - 140792 q^{76} + 128656 q^{77} + 2982 q^{78} - 106076 q^{79} + 58638 q^{80} - 59692 q^{81} - 234132 q^{82} + 62888 q^{83} - 59832 q^{84} + 23848 q^{85} - 216014 q^{86} + 23548 q^{87} - 39426 q^{88} + 107568 q^{89} + 41552 q^{90} - 268896 q^{91} - 26268 q^{92} + 221460 q^{93} + 30542 q^{94} + 147352 q^{95} - 78606 q^{96} - 49520 q^{97} + 242304 q^{98} + 166720 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.21534 −1.62906 −0.814529 0.580123i \(-0.803004\pi\)
−0.814529 + 0.580123i \(0.803004\pi\)
\(3\) 0.734546 0.0471211 0.0235606 0.999722i \(-0.492500\pi\)
0.0235606 + 0.999722i \(0.492500\pi\)
\(4\) 52.9225 1.65383
\(5\) 41.6400 0.744879 0.372440 0.928056i \(-0.378521\pi\)
0.372440 + 0.928056i \(0.378521\pi\)
\(6\) −6.76909 −0.0767630
\(7\) −90.0205 −0.694379 −0.347189 0.937795i \(-0.612864\pi\)
−0.347189 + 0.937795i \(0.612864\pi\)
\(8\) −192.808 −1.06512
\(9\) −242.460 −0.997780
\(10\) −383.727 −1.21345
\(11\) −269.080 −0.670502 −0.335251 0.942129i \(-0.608821\pi\)
−0.335251 + 0.942129i \(0.608821\pi\)
\(12\) 38.8740 0.0779302
\(13\) −444.579 −0.729610 −0.364805 0.931084i \(-0.618864\pi\)
−0.364805 + 0.931084i \(0.618864\pi\)
\(14\) 829.569 1.13118
\(15\) 30.5865 0.0350995
\(16\) 83.2700 0.0813183
\(17\) 485.649 0.407569 0.203784 0.979016i \(-0.434676\pi\)
0.203784 + 0.979016i \(0.434676\pi\)
\(18\) 2234.36 1.62544
\(19\) −1572.80 −0.999513 −0.499756 0.866166i \(-0.666577\pi\)
−0.499756 + 0.866166i \(0.666577\pi\)
\(20\) 2203.69 1.23190
\(21\) −66.1242 −0.0327199
\(22\) 2479.66 1.09229
\(23\) −398.704 −0.157156 −0.0785781 0.996908i \(-0.525038\pi\)
−0.0785781 + 0.996908i \(0.525038\pi\)
\(24\) −141.626 −0.0501898
\(25\) −1391.11 −0.445155
\(26\) 4096.95 1.18858
\(27\) −356.593 −0.0941376
\(28\) −4764.11 −1.14838
\(29\) −841.000 −0.185695
\(30\) −281.865 −0.0571792
\(31\) −8469.77 −1.58295 −0.791475 0.611202i \(-0.790686\pi\)
−0.791475 + 0.611202i \(0.790686\pi\)
\(32\) 5402.49 0.932651
\(33\) −197.652 −0.0315948
\(34\) −4475.43 −0.663952
\(35\) −3748.45 −0.517228
\(36\) −12831.6 −1.65016
\(37\) 3339.33 0.401010 0.200505 0.979693i \(-0.435742\pi\)
0.200505 + 0.979693i \(0.435742\pi\)
\(38\) 14493.8 1.62826
\(39\) −326.564 −0.0343801
\(40\) −8028.52 −0.793388
\(41\) 18154.2 1.68662 0.843311 0.537425i \(-0.180603\pi\)
0.843311 + 0.537425i \(0.180603\pi\)
\(42\) 609.357 0.0533026
\(43\) 9996.71 0.824491 0.412246 0.911073i \(-0.364745\pi\)
0.412246 + 0.911073i \(0.364745\pi\)
\(44\) −14240.4 −1.10889
\(45\) −10096.1 −0.743225
\(46\) 3674.20 0.256016
\(47\) 12568.0 0.829890 0.414945 0.909846i \(-0.363801\pi\)
0.414945 + 0.909846i \(0.363801\pi\)
\(48\) 61.1656 0.00383181
\(49\) −8703.31 −0.517838
\(50\) 12819.6 0.725183
\(51\) 356.732 0.0192051
\(52\) −23528.2 −1.20665
\(53\) −21343.1 −1.04368 −0.521841 0.853043i \(-0.674755\pi\)
−0.521841 + 0.853043i \(0.674755\pi\)
\(54\) 3286.12 0.153356
\(55\) −11204.5 −0.499443
\(56\) 17356.7 0.739598
\(57\) −1155.29 −0.0470982
\(58\) 7750.10 0.302508
\(59\) −30036.1 −1.12334 −0.561672 0.827360i \(-0.689842\pi\)
−0.561672 + 0.827360i \(0.689842\pi\)
\(60\) 1618.71 0.0580486
\(61\) 49792.6 1.71333 0.856663 0.515877i \(-0.172534\pi\)
0.856663 + 0.515877i \(0.172534\pi\)
\(62\) 78051.8 2.57872
\(63\) 21826.4 0.692837
\(64\) −52450.4 −1.60066
\(65\) −18512.3 −0.543471
\(66\) 1821.43 0.0514697
\(67\) 47588.2 1.29513 0.647563 0.762012i \(-0.275788\pi\)
0.647563 + 0.762012i \(0.275788\pi\)
\(68\) 25701.8 0.674048
\(69\) −292.867 −0.00740538
\(70\) 34543.3 0.842594
\(71\) −50164.6 −1.18101 −0.590503 0.807036i \(-0.701070\pi\)
−0.590503 + 0.807036i \(0.701070\pi\)
\(72\) 46748.3 1.06276
\(73\) −44770.3 −0.983294 −0.491647 0.870795i \(-0.663605\pi\)
−0.491647 + 0.870795i \(0.663605\pi\)
\(74\) −30773.1 −0.653269
\(75\) −1021.83 −0.0209762
\(76\) −83236.3 −1.65302
\(77\) 24222.7 0.465582
\(78\) 3009.40 0.0560071
\(79\) −78464.6 −1.41451 −0.707255 0.706959i \(-0.750067\pi\)
−0.707255 + 0.706959i \(0.750067\pi\)
\(80\) 3467.36 0.0605723
\(81\) 58656.0 0.993344
\(82\) −167297. −2.74761
\(83\) 46721.8 0.744430 0.372215 0.928146i \(-0.378598\pi\)
0.372215 + 0.928146i \(0.378598\pi\)
\(84\) −3499.46 −0.0541131
\(85\) 20222.4 0.303589
\(86\) −92123.1 −1.34314
\(87\) −617.753 −0.00875017
\(88\) 51880.7 0.714167
\(89\) −39465.7 −0.528135 −0.264068 0.964504i \(-0.585064\pi\)
−0.264068 + 0.964504i \(0.585064\pi\)
\(90\) 93038.6 1.21076
\(91\) 40021.2 0.506626
\(92\) −21100.4 −0.259909
\(93\) −6221.43 −0.0745904
\(94\) −115818. −1.35194
\(95\) −65491.2 −0.744516
\(96\) 3968.38 0.0439476
\(97\) 48336.3 0.521608 0.260804 0.965392i \(-0.416012\pi\)
0.260804 + 0.965392i \(0.416012\pi\)
\(98\) 80204.0 0.843589
\(99\) 65241.3 0.669013
\(100\) −73621.0 −0.736210
\(101\) 111417. 1.08680 0.543399 0.839475i \(-0.317137\pi\)
0.543399 + 0.839475i \(0.317137\pi\)
\(102\) −3287.40 −0.0312862
\(103\) −1416.34 −0.0131545 −0.00657726 0.999978i \(-0.502094\pi\)
−0.00657726 + 0.999978i \(0.502094\pi\)
\(104\) 85718.4 0.777124
\(105\) −2753.41 −0.0243724
\(106\) 196684. 1.70022
\(107\) −162530. −1.37238 −0.686188 0.727424i \(-0.740717\pi\)
−0.686188 + 0.727424i \(0.740717\pi\)
\(108\) −18871.8 −0.155687
\(109\) 152425. 1.22882 0.614411 0.788986i \(-0.289394\pi\)
0.614411 + 0.788986i \(0.289394\pi\)
\(110\) 103253. 0.813621
\(111\) 2452.89 0.0188961
\(112\) −7496.00 −0.0564657
\(113\) 256365. 1.88870 0.944350 0.328942i \(-0.106692\pi\)
0.944350 + 0.328942i \(0.106692\pi\)
\(114\) 10646.4 0.0767256
\(115\) −16602.1 −0.117062
\(116\) −44507.8 −0.307108
\(117\) 107793. 0.727990
\(118\) 276792. 1.82999
\(119\) −43718.4 −0.283007
\(120\) −5897.31 −0.0373853
\(121\) −88646.9 −0.550428
\(122\) −458855. −2.79111
\(123\) 13335.1 0.0794756
\(124\) −448241. −2.61793
\(125\) −188051. −1.07647
\(126\) −201138. −1.12867
\(127\) −145336. −0.799585 −0.399792 0.916606i \(-0.630918\pi\)
−0.399792 + 0.916606i \(0.630918\pi\)
\(128\) 310469. 1.67492
\(129\) 7343.04 0.0388510
\(130\) 170597. 0.885346
\(131\) −259373. −1.32052 −0.660262 0.751035i \(-0.729555\pi\)
−0.660262 + 0.751035i \(0.729555\pi\)
\(132\) −10460.2 −0.0522523
\(133\) 141584. 0.694040
\(134\) −438541. −2.10984
\(135\) −14848.5 −0.0701212
\(136\) −93637.0 −0.434111
\(137\) 94701.8 0.431079 0.215539 0.976495i \(-0.430849\pi\)
0.215539 + 0.976495i \(0.430849\pi\)
\(138\) 2698.87 0.0120638
\(139\) 237100. 1.04086 0.520432 0.853903i \(-0.325771\pi\)
0.520432 + 0.853903i \(0.325771\pi\)
\(140\) −198377. −0.855406
\(141\) 9231.75 0.0391054
\(142\) 462284. 1.92393
\(143\) 119627. 0.489205
\(144\) −20189.7 −0.0811378
\(145\) −35019.2 −0.138321
\(146\) 412574. 1.60184
\(147\) −6392.98 −0.0244011
\(148\) 176726. 0.663202
\(149\) −81590.7 −0.301075 −0.150538 0.988604i \(-0.548100\pi\)
−0.150538 + 0.988604i \(0.548100\pi\)
\(150\) 9416.55 0.0341715
\(151\) −199008. −0.710276 −0.355138 0.934814i \(-0.615566\pi\)
−0.355138 + 0.934814i \(0.615566\pi\)
\(152\) 303247. 1.06460
\(153\) −117751. −0.406664
\(154\) −223221. −0.758460
\(155\) −352681. −1.17911
\(156\) −17282.6 −0.0568587
\(157\) −321167. −1.03988 −0.519938 0.854204i \(-0.674045\pi\)
−0.519938 + 0.854204i \(0.674045\pi\)
\(158\) 723078. 2.30432
\(159\) −15677.5 −0.0491795
\(160\) 224960. 0.694712
\(161\) 35891.6 0.109126
\(162\) −540535. −1.61821
\(163\) −621023. −1.83079 −0.915395 0.402557i \(-0.868121\pi\)
−0.915395 + 0.402557i \(0.868121\pi\)
\(164\) 960767. 2.78938
\(165\) −8230.21 −0.0235343
\(166\) −430557. −1.21272
\(167\) −59437.0 −0.164917 −0.0824585 0.996594i \(-0.526277\pi\)
−0.0824585 + 0.996594i \(0.526277\pi\)
\(168\) 12749.3 0.0348507
\(169\) −173642. −0.467669
\(170\) −186357. −0.494564
\(171\) 381341. 0.997293
\(172\) 529051. 1.36357
\(173\) 175144. 0.444918 0.222459 0.974942i \(-0.428592\pi\)
0.222459 + 0.974942i \(0.428592\pi\)
\(174\) 5692.80 0.0142545
\(175\) 125228. 0.309106
\(176\) −22406.3 −0.0545241
\(177\) −22062.9 −0.0529332
\(178\) 363690. 0.860362
\(179\) −421857. −0.984086 −0.492043 0.870571i \(-0.663750\pi\)
−0.492043 + 0.870571i \(0.663750\pi\)
\(180\) −534308. −1.22917
\(181\) 313961. 0.712326 0.356163 0.934424i \(-0.384085\pi\)
0.356163 + 0.934424i \(0.384085\pi\)
\(182\) −368809. −0.825322
\(183\) 36574.9 0.0807338
\(184\) 76873.3 0.167391
\(185\) 139050. 0.298704
\(186\) 57332.6 0.121512
\(187\) −130679. −0.273275
\(188\) 665128. 1.37250
\(189\) 32100.7 0.0653672
\(190\) 603524. 1.21286
\(191\) 85652.5 0.169886 0.0849428 0.996386i \(-0.472929\pi\)
0.0849428 + 0.996386i \(0.472929\pi\)
\(192\) −38527.2 −0.0754249
\(193\) 356203. 0.688343 0.344171 0.938907i \(-0.388160\pi\)
0.344171 + 0.938907i \(0.388160\pi\)
\(194\) −445436. −0.849729
\(195\) −13598.1 −0.0256090
\(196\) −460601. −0.856416
\(197\) 2000.17 0.00367199 0.00183599 0.999998i \(-0.499416\pi\)
0.00183599 + 0.999998i \(0.499416\pi\)
\(198\) −601220. −1.08986
\(199\) −241348. −0.432027 −0.216014 0.976390i \(-0.569306\pi\)
−0.216014 + 0.976390i \(0.569306\pi\)
\(200\) 268217. 0.474145
\(201\) 34955.7 0.0610278
\(202\) −1.02675e6 −1.77045
\(203\) 75707.2 0.128943
\(204\) 18879.1 0.0317619
\(205\) 755942. 1.25633
\(206\) 13052.1 0.0214295
\(207\) 96670.0 0.156807
\(208\) −37020.1 −0.0593307
\(209\) 423208. 0.670175
\(210\) 25373.6 0.0397040
\(211\) −611130. −0.944991 −0.472495 0.881333i \(-0.656647\pi\)
−0.472495 + 0.881333i \(0.656647\pi\)
\(212\) −1.12953e6 −1.72607
\(213\) −36848.2 −0.0556503
\(214\) 1.49777e6 2.23568
\(215\) 416263. 0.614146
\(216\) 68753.9 0.100268
\(217\) 762452. 1.09917
\(218\) −1.40465e6 −2.00182
\(219\) −32885.9 −0.0463339
\(220\) −592970. −0.825992
\(221\) −215910. −0.297366
\(222\) −22604.2 −0.0307828
\(223\) 702678. 0.946225 0.473112 0.881002i \(-0.343130\pi\)
0.473112 + 0.881002i \(0.343130\pi\)
\(224\) −486335. −0.647613
\(225\) 337289. 0.444167
\(226\) −2.36249e6 −3.07680
\(227\) 933943. 1.20297 0.601487 0.798883i \(-0.294575\pi\)
0.601487 + 0.798883i \(0.294575\pi\)
\(228\) −61140.8 −0.0778923
\(229\) −1.02337e6 −1.28956 −0.644782 0.764367i \(-0.723052\pi\)
−0.644782 + 0.764367i \(0.723052\pi\)
\(230\) 152994. 0.190701
\(231\) 17792.7 0.0219387
\(232\) 162151. 0.197788
\(233\) −994807. −1.20046 −0.600232 0.799826i \(-0.704925\pi\)
−0.600232 + 0.799826i \(0.704925\pi\)
\(234\) −993348. −1.18594
\(235\) 523330. 0.618168
\(236\) −1.58958e6 −1.85782
\(237\) −57635.8 −0.0666533
\(238\) 402880. 0.461034
\(239\) −111950. −0.126773 −0.0633867 0.997989i \(-0.520190\pi\)
−0.0633867 + 0.997989i \(0.520190\pi\)
\(240\) 2546.94 0.00285424
\(241\) 1.31520e6 1.45865 0.729323 0.684169i \(-0.239835\pi\)
0.729323 + 0.684169i \(0.239835\pi\)
\(242\) 816911. 0.896678
\(243\) 129738. 0.140945
\(244\) 2.63515e6 2.83355
\(245\) −362406. −0.385727
\(246\) −122888. −0.129470
\(247\) 699232. 0.729255
\(248\) 1.63304e6 1.68604
\(249\) 34319.3 0.0350784
\(250\) 1.73295e6 1.75362
\(251\) −440669. −0.441498 −0.220749 0.975331i \(-0.570850\pi\)
−0.220749 + 0.975331i \(0.570850\pi\)
\(252\) 1.15511e6 1.14583
\(253\) 107283. 0.105373
\(254\) 1.33932e6 1.30257
\(255\) 14854.3 0.0143055
\(256\) −1.18266e6 −1.12787
\(257\) −554939. −0.524098 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(258\) −67668.6 −0.0632904
\(259\) −300608. −0.278453
\(260\) −979716. −0.898808
\(261\) 203909. 0.185283
\(262\) 2.39021e6 2.15121
\(263\) −1.24351e6 −1.10856 −0.554279 0.832331i \(-0.687006\pi\)
−0.554279 + 0.832331i \(0.687006\pi\)
\(264\) 38108.8 0.0336523
\(265\) −888728. −0.777417
\(266\) −1.30474e6 −1.13063
\(267\) −28989.4 −0.0248863
\(268\) 2.51849e6 2.14192
\(269\) 598804. 0.504549 0.252275 0.967656i \(-0.418821\pi\)
0.252275 + 0.967656i \(0.418821\pi\)
\(270\) 136834. 0.114231
\(271\) 654368. 0.541251 0.270626 0.962685i \(-0.412769\pi\)
0.270626 + 0.962685i \(0.412769\pi\)
\(272\) 40440.0 0.0331428
\(273\) 29397.4 0.0238728
\(274\) −872709. −0.702252
\(275\) 374320. 0.298477
\(276\) −15499.2 −0.0122472
\(277\) −1.76929e6 −1.38548 −0.692739 0.721188i \(-0.743596\pi\)
−0.692739 + 0.721188i \(0.743596\pi\)
\(278\) −2.18495e6 −1.69563
\(279\) 2.05358e6 1.57943
\(280\) 722731. 0.550911
\(281\) −495309. −0.374206 −0.187103 0.982340i \(-0.559910\pi\)
−0.187103 + 0.982340i \(0.559910\pi\)
\(282\) −85073.7 −0.0637049
\(283\) −785235. −0.582819 −0.291409 0.956598i \(-0.594124\pi\)
−0.291409 + 0.956598i \(0.594124\pi\)
\(284\) −2.65484e6 −1.95318
\(285\) −48106.3 −0.0350824
\(286\) −1.10241e6 −0.796943
\(287\) −1.63425e6 −1.17115
\(288\) −1.30989e6 −0.930580
\(289\) −1.18400e6 −0.833888
\(290\) 322714. 0.225332
\(291\) 35505.2 0.0245788
\(292\) −2.36936e6 −1.62620
\(293\) 1.72437e6 1.17344 0.586719 0.809790i \(-0.300419\pi\)
0.586719 + 0.809790i \(0.300419\pi\)
\(294\) 58913.5 0.0397508
\(295\) −1.25070e6 −0.836755
\(296\) −643850. −0.427125
\(297\) 95952.0 0.0631194
\(298\) 751886. 0.490469
\(299\) 177256. 0.114663
\(300\) −54078.0 −0.0346911
\(301\) −899909. −0.572509
\(302\) 1.83392e6 1.15708
\(303\) 81841.0 0.0512111
\(304\) −130967. −0.0812787
\(305\) 2.07336e6 1.27622
\(306\) 1.08511e6 0.662478
\(307\) −3.12637e6 −1.89319 −0.946597 0.322419i \(-0.895504\pi\)
−0.946597 + 0.322419i \(0.895504\pi\)
\(308\) 1.28193e6 0.769992
\(309\) −1040.37 −0.000619856 0
\(310\) 3.25008e6 1.92083
\(311\) 2.08430e6 1.22196 0.610982 0.791644i \(-0.290775\pi\)
0.610982 + 0.791644i \(0.290775\pi\)
\(312\) 62964.1 0.0366190
\(313\) −2.27357e6 −1.31174 −0.655871 0.754873i \(-0.727698\pi\)
−0.655871 + 0.754873i \(0.727698\pi\)
\(314\) 2.95966e6 1.69402
\(315\) 908852. 0.516080
\(316\) −4.15254e6 −2.33935
\(317\) −891538. −0.498301 −0.249151 0.968465i \(-0.580151\pi\)
−0.249151 + 0.968465i \(0.580151\pi\)
\(318\) 144473. 0.0801162
\(319\) 226296. 0.124509
\(320\) −2.18404e6 −1.19230
\(321\) −119385. −0.0646679
\(322\) −330753. −0.177772
\(323\) −763827. −0.407370
\(324\) 3.10422e6 1.64282
\(325\) 618459. 0.324790
\(326\) 5.72293e6 2.98246
\(327\) 111963. 0.0579035
\(328\) −3.50028e6 −1.79646
\(329\) −1.13138e6 −0.576258
\(330\) 75844.2 0.0383387
\(331\) −768207. −0.385397 −0.192698 0.981258i \(-0.561724\pi\)
−0.192698 + 0.981258i \(0.561724\pi\)
\(332\) 2.47263e6 1.23116
\(333\) −809656. −0.400120
\(334\) 547732. 0.268659
\(335\) 1.98157e6 0.964713
\(336\) −5506.16 −0.00266073
\(337\) −740054. −0.354968 −0.177484 0.984124i \(-0.556796\pi\)
−0.177484 + 0.984124i \(0.556796\pi\)
\(338\) 1.60017e6 0.761860
\(339\) 188312. 0.0889977
\(340\) 1.07022e6 0.502084
\(341\) 2.27904e6 1.06137
\(342\) −3.51418e6 −1.62465
\(343\) 2.29645e6 1.05395
\(344\) −1.92744e6 −0.878184
\(345\) −12195.0 −0.00551611
\(346\) −1.61401e6 −0.724796
\(347\) 2.65308e6 1.18284 0.591420 0.806363i \(-0.298567\pi\)
0.591420 + 0.806363i \(0.298567\pi\)
\(348\) −32693.0 −0.0144713
\(349\) 1.02314e6 0.449646 0.224823 0.974400i \(-0.427820\pi\)
0.224823 + 0.974400i \(0.427820\pi\)
\(350\) −1.15402e6 −0.503552
\(351\) 158534. 0.0686838
\(352\) −1.45370e6 −0.625344
\(353\) −2.17906e6 −0.930747 −0.465374 0.885114i \(-0.654080\pi\)
−0.465374 + 0.885114i \(0.654080\pi\)
\(354\) 203317. 0.0862313
\(355\) −2.08886e6 −0.879706
\(356\) −2.08862e6 −0.873445
\(357\) −32113.2 −0.0133356
\(358\) 3.88756e6 1.60313
\(359\) 4.09152e6 1.67552 0.837759 0.546041i \(-0.183866\pi\)
0.837759 + 0.546041i \(0.183866\pi\)
\(360\) 1.94660e6 0.791626
\(361\) −2413.03 −0.000974529 0
\(362\) −2.89325e6 −1.16042
\(363\) −65115.2 −0.0259368
\(364\) 2.11802e6 0.837871
\(365\) −1.86424e6 −0.732435
\(366\) −337050. −0.131520
\(367\) −113276. −0.0439009 −0.0219504 0.999759i \(-0.506988\pi\)
−0.0219504 + 0.999759i \(0.506988\pi\)
\(368\) −33200.1 −0.0127797
\(369\) −4.40168e6 −1.68288
\(370\) −1.28139e6 −0.486606
\(371\) 1.92132e6 0.724711
\(372\) −329254. −0.123360
\(373\) −1.74960e6 −0.651130 −0.325565 0.945520i \(-0.605554\pi\)
−0.325565 + 0.945520i \(0.605554\pi\)
\(374\) 1.20425e6 0.445181
\(375\) −138132. −0.0507243
\(376\) −2.42320e6 −0.883935
\(377\) 373891. 0.135485
\(378\) −295819. −0.106487
\(379\) 2.62142e6 0.937429 0.468714 0.883350i \(-0.344717\pi\)
0.468714 + 0.883350i \(0.344717\pi\)
\(380\) −3.46596e6 −1.23130
\(381\) −106756. −0.0376773
\(382\) −789317. −0.276753
\(383\) 4.63046e6 1.61297 0.806486 0.591253i \(-0.201367\pi\)
0.806486 + 0.591253i \(0.201367\pi\)
\(384\) 228053. 0.0789239
\(385\) 1.00863e6 0.346802
\(386\) −3.28254e6 −1.12135
\(387\) −2.42381e6 −0.822660
\(388\) 2.55808e6 0.862650
\(389\) −3.41385e6 −1.14385 −0.571927 0.820305i \(-0.693804\pi\)
−0.571927 + 0.820305i \(0.693804\pi\)
\(390\) 125311. 0.0417185
\(391\) −193631. −0.0640519
\(392\) 1.67807e6 0.551562
\(393\) −190521. −0.0622246
\(394\) −18432.2 −0.00598188
\(395\) −3.26726e6 −1.05364
\(396\) 3.45273e6 1.10643
\(397\) −2.05992e6 −0.655954 −0.327977 0.944686i \(-0.606367\pi\)
−0.327977 + 0.944686i \(0.606367\pi\)
\(398\) 2.22410e6 0.703797
\(399\) 104000. 0.0327040
\(400\) −115838. −0.0361993
\(401\) 4.48615e6 1.39320 0.696599 0.717461i \(-0.254696\pi\)
0.696599 + 0.717461i \(0.254696\pi\)
\(402\) −322129. −0.0994178
\(403\) 3.76548e6 1.15494
\(404\) 5.89647e6 1.79738
\(405\) 2.44243e6 0.739921
\(406\) −697668. −0.210055
\(407\) −898548. −0.268878
\(408\) −68780.7 −0.0204558
\(409\) −5.12907e6 −1.51611 −0.758055 0.652191i \(-0.773850\pi\)
−0.758055 + 0.652191i \(0.773850\pi\)
\(410\) −6.96626e6 −2.04663
\(411\) 69562.8 0.0203129
\(412\) −74956.3 −0.0217553
\(413\) 2.70386e6 0.780026
\(414\) −890847. −0.255448
\(415\) 1.94549e6 0.554511
\(416\) −2.40183e6 −0.680471
\(417\) 174161. 0.0490467
\(418\) −3.90000e6 −1.09175
\(419\) −2.21170e6 −0.615446 −0.307723 0.951476i \(-0.599567\pi\)
−0.307723 + 0.951476i \(0.599567\pi\)
\(420\) −145717. −0.0403077
\(421\) 751122. 0.206540 0.103270 0.994653i \(-0.467069\pi\)
0.103270 + 0.994653i \(0.467069\pi\)
\(422\) 5.63177e6 1.53944
\(423\) −3.04724e6 −0.828047
\(424\) 4.11512e6 1.11165
\(425\) −675592. −0.181431
\(426\) 339569. 0.0906575
\(427\) −4.48235e6 −1.18970
\(428\) −8.60147e6 −2.26967
\(429\) 87871.8 0.0230519
\(430\) −3.83601e6 −1.00048
\(431\) 2.98345e6 0.773615 0.386808 0.922160i \(-0.373578\pi\)
0.386808 + 0.922160i \(0.373578\pi\)
\(432\) −29693.5 −0.00765511
\(433\) −2.17337e6 −0.557075 −0.278537 0.960425i \(-0.589850\pi\)
−0.278537 + 0.960425i \(0.589850\pi\)
\(434\) −7.02626e6 −1.79060
\(435\) −25723.2 −0.00651782
\(436\) 8.06670e6 2.03226
\(437\) 627081. 0.157080
\(438\) 303054. 0.0754806
\(439\) 24436.8 0.00605178 0.00302589 0.999995i \(-0.499037\pi\)
0.00302589 + 0.999995i \(0.499037\pi\)
\(440\) 2.16031e6 0.531968
\(441\) 2.11021e6 0.516689
\(442\) 1.98968e6 0.484426
\(443\) −6.19315e6 −1.49935 −0.749674 0.661808i \(-0.769789\pi\)
−0.749674 + 0.661808i \(0.769789\pi\)
\(444\) 129813. 0.0312508
\(445\) −1.64335e6 −0.393397
\(446\) −6.47542e6 −1.54145
\(447\) −59932.1 −0.0141870
\(448\) 4.72161e6 1.11146
\(449\) 7.56136e6 1.77004 0.885022 0.465548i \(-0.154143\pi\)
0.885022 + 0.465548i \(0.154143\pi\)
\(450\) −3.10823e6 −0.723573
\(451\) −4.88494e6 −1.13088
\(452\) 1.35675e7 3.12359
\(453\) −146180. −0.0334690
\(454\) −8.60661e6 −1.95971
\(455\) 1.66648e6 0.377375
\(456\) 222749. 0.0501653
\(457\) 5.02930e6 1.12646 0.563232 0.826299i \(-0.309558\pi\)
0.563232 + 0.826299i \(0.309558\pi\)
\(458\) 9.43067e6 2.10077
\(459\) −173179. −0.0383675
\(460\) −878622. −0.193601
\(461\) 3.58934e6 0.786615 0.393308 0.919407i \(-0.371331\pi\)
0.393308 + 0.919407i \(0.371331\pi\)
\(462\) −163966. −0.0357395
\(463\) 8.86551e6 1.92199 0.960996 0.276563i \(-0.0891955\pi\)
0.960996 + 0.276563i \(0.0891955\pi\)
\(464\) −70030.0 −0.0151004
\(465\) −259060. −0.0555608
\(466\) 9.16748e6 1.95562
\(467\) 3.24975e6 0.689537 0.344769 0.938688i \(-0.387957\pi\)
0.344769 + 0.938688i \(0.387957\pi\)
\(468\) 5.70467e6 1.20397
\(469\) −4.28391e6 −0.899308
\(470\) −4.82267e6 −1.00703
\(471\) −235912. −0.0490001
\(472\) 5.79119e6 1.19650
\(473\) −2.68992e6 −0.552823
\(474\) 531134. 0.108582
\(475\) 2.18793e6 0.444938
\(476\) −2.31369e6 −0.468045
\(477\) 5.17486e6 1.04136
\(478\) 1.03166e6 0.206521
\(479\) −2.33334e6 −0.464664 −0.232332 0.972637i \(-0.574636\pi\)
−0.232332 + 0.972637i \(0.574636\pi\)
\(480\) 165243. 0.0327356
\(481\) −1.48460e6 −0.292581
\(482\) −1.21200e7 −2.37622
\(483\) 26364.0 0.00514213
\(484\) −4.69142e6 −0.910312
\(485\) 2.01272e6 0.388535
\(486\) −1.19558e6 −0.229608
\(487\) 1.35787e6 0.259440 0.129720 0.991551i \(-0.458592\pi\)
0.129720 + 0.991551i \(0.458592\pi\)
\(488\) −9.60039e6 −1.82490
\(489\) −456170. −0.0862689
\(490\) 3.33969e6 0.628371
\(491\) 4.42097e6 0.827587 0.413794 0.910371i \(-0.364203\pi\)
0.413794 + 0.910371i \(0.364203\pi\)
\(492\) 705727. 0.131439
\(493\) −408431. −0.0756836
\(494\) −6.44366e6 −1.18800
\(495\) 2.71665e6 0.498334
\(496\) −705277. −0.128723
\(497\) 4.51585e6 0.820065
\(498\) −316264. −0.0571447
\(499\) −9.61078e6 −1.72785 −0.863927 0.503617i \(-0.832002\pi\)
−0.863927 + 0.503617i \(0.832002\pi\)
\(500\) −9.95212e6 −1.78029
\(501\) −43659.2 −0.00777108
\(502\) 4.06092e6 0.719225
\(503\) −6.10366e6 −1.07565 −0.537824 0.843057i \(-0.680754\pi\)
−0.537824 + 0.843057i \(0.680754\pi\)
\(504\) −4.20830e6 −0.737956
\(505\) 4.63941e6 0.809532
\(506\) −988653. −0.171659
\(507\) −127548. −0.0220371
\(508\) −7.69155e6 −1.32238
\(509\) −1.01562e7 −1.73755 −0.868775 0.495208i \(-0.835092\pi\)
−0.868775 + 0.495208i \(0.835092\pi\)
\(510\) −136888. −0.0233044
\(511\) 4.03025e6 0.682778
\(512\) 963628. 0.162456
\(513\) 560848. 0.0940918
\(514\) 5.11395e6 0.853785
\(515\) −58976.4 −0.00979852
\(516\) 388612. 0.0642528
\(517\) −3.38179e6 −0.556443
\(518\) 2.77021e6 0.453616
\(519\) 128651. 0.0209650
\(520\) 3.56931e6 0.578864
\(521\) 2.42580e6 0.391526 0.195763 0.980651i \(-0.437282\pi\)
0.195763 + 0.980651i \(0.437282\pi\)
\(522\) −1.87909e6 −0.301837
\(523\) −328610. −0.0525322 −0.0262661 0.999655i \(-0.508362\pi\)
−0.0262661 + 0.999655i \(0.508362\pi\)
\(524\) −1.37267e7 −2.18392
\(525\) 91986.0 0.0145654
\(526\) 1.14593e7 1.80591
\(527\) −4.11334e6 −0.645160
\(528\) −16458.4 −0.00256924
\(529\) −6.27738e6 −0.975302
\(530\) 8.18993e6 1.26646
\(531\) 7.28255e6 1.12085
\(532\) 7.49297e6 1.14782
\(533\) −8.07099e6 −1.23058
\(534\) 267147. 0.0405413
\(535\) −6.76773e6 −1.02225
\(536\) −9.17538e6 −1.37947
\(537\) −309874. −0.0463713
\(538\) −5.51818e6 −0.821940
\(539\) 2.34189e6 0.347212
\(540\) −785821. −0.115968
\(541\) 5.52172e6 0.811113 0.405557 0.914070i \(-0.367078\pi\)
0.405557 + 0.914070i \(0.367078\pi\)
\(542\) −6.03022e6 −0.881729
\(543\) 230618. 0.0335656
\(544\) 2.62372e6 0.380119
\(545\) 6.34697e6 0.915324
\(546\) −270907. −0.0388901
\(547\) 2.61929e6 0.374296 0.187148 0.982332i \(-0.440076\pi\)
0.187148 + 0.982332i \(0.440076\pi\)
\(548\) 5.01185e6 0.712930
\(549\) −1.20727e7 −1.70952
\(550\) −3.44949e6 −0.486237
\(551\) 1.32272e6 0.185605
\(552\) 56467.0 0.00788764
\(553\) 7.06342e6 0.982205
\(554\) 1.63046e7 2.25702
\(555\) 102138. 0.0140753
\(556\) 1.25479e7 1.72141
\(557\) 4.95349e6 0.676509 0.338254 0.941055i \(-0.390164\pi\)
0.338254 + 0.941055i \(0.390164\pi\)
\(558\) −1.89245e7 −2.57299
\(559\) −4.44433e6 −0.601557
\(560\) −312134. −0.0420601
\(561\) −95989.4 −0.0128770
\(562\) 4.56444e6 0.609602
\(563\) −2.28738e6 −0.304135 −0.152068 0.988370i \(-0.548593\pi\)
−0.152068 + 0.988370i \(0.548593\pi\)
\(564\) 488567. 0.0646735
\(565\) 1.06751e7 1.40685
\(566\) 7.23620e6 0.949445
\(567\) −5.28024e6 −0.689757
\(568\) 9.67214e6 1.25792
\(569\) −3.25672e6 −0.421696 −0.210848 0.977519i \(-0.567622\pi\)
−0.210848 + 0.977519i \(0.567622\pi\)
\(570\) 443316. 0.0571513
\(571\) −9.92284e6 −1.27364 −0.636819 0.771013i \(-0.719750\pi\)
−0.636819 + 0.771013i \(0.719750\pi\)
\(572\) 6.33098e6 0.809060
\(573\) 62915.7 0.00800520
\(574\) 1.50602e7 1.90788
\(575\) 554642. 0.0699589
\(576\) 1.27171e7 1.59711
\(577\) 1.23237e6 0.154100 0.0770498 0.997027i \(-0.475450\pi\)
0.0770498 + 0.997027i \(0.475450\pi\)
\(578\) 1.09110e7 1.35845
\(579\) 261648. 0.0324355
\(580\) −1.85331e6 −0.228758
\(581\) −4.20592e6 −0.516916
\(582\) −327193. −0.0400402
\(583\) 5.74301e6 0.699791
\(584\) 8.63207e6 1.04733
\(585\) 4.48850e6 0.542265
\(586\) −1.58906e7 −1.91160
\(587\) −1.02407e7 −1.22669 −0.613345 0.789815i \(-0.710177\pi\)
−0.613345 + 0.789815i \(0.710177\pi\)
\(588\) −338332. −0.0403553
\(589\) 1.33212e7 1.58218
\(590\) 1.15256e7 1.36312
\(591\) 1469.22 0.000173028 0
\(592\) 278066. 0.0326095
\(593\) −1.53106e7 −1.78795 −0.893977 0.448114i \(-0.852096\pi\)
−0.893977 + 0.448114i \(0.852096\pi\)
\(594\) −884231. −0.102825
\(595\) −1.82043e6 −0.210806
\(596\) −4.31798e6 −0.497926
\(597\) −177281. −0.0203576
\(598\) −1.63347e6 −0.186792
\(599\) −1.37662e6 −0.156764 −0.0783819 0.996923i \(-0.524975\pi\)
−0.0783819 + 0.996923i \(0.524975\pi\)
\(600\) 197018. 0.0223422
\(601\) −1.39459e7 −1.57492 −0.787462 0.616363i \(-0.788606\pi\)
−0.787462 + 0.616363i \(0.788606\pi\)
\(602\) 8.29297e6 0.932650
\(603\) −1.15383e7 −1.29225
\(604\) −1.05320e7 −1.17467
\(605\) −3.69126e6 −0.410002
\(606\) −754192. −0.0834258
\(607\) −2.10503e6 −0.231892 −0.115946 0.993255i \(-0.536990\pi\)
−0.115946 + 0.993255i \(0.536990\pi\)
\(608\) −8.49701e6 −0.932196
\(609\) 55610.4 0.00607593
\(610\) −1.91067e7 −2.07904
\(611\) −5.58746e6 −0.605496
\(612\) −6.23166e6 −0.672551
\(613\) 1.75643e7 1.88790 0.943950 0.330088i \(-0.107078\pi\)
0.943950 + 0.330088i \(0.107078\pi\)
\(614\) 2.88106e7 3.08412
\(615\) 555274. 0.0591997
\(616\) −4.67033e6 −0.495902
\(617\) −6.68190e6 −0.706622 −0.353311 0.935506i \(-0.614944\pi\)
−0.353311 + 0.935506i \(0.614944\pi\)
\(618\) 9587.34 0.00100978
\(619\) −6.69302e6 −0.702095 −0.351047 0.936358i \(-0.614174\pi\)
−0.351047 + 0.936358i \(0.614174\pi\)
\(620\) −1.86648e7 −1.95004
\(621\) 142175. 0.0147943
\(622\) −1.92075e7 −1.99065
\(623\) 3.55272e6 0.366726
\(624\) −27193.0 −0.00279573
\(625\) −3.48322e6 −0.356682
\(626\) 2.09518e7 2.13690
\(627\) 310866. 0.0315794
\(628\) −1.69969e7 −1.71977
\(629\) 1.62175e6 0.163439
\(630\) −8.37538e6 −0.840723
\(631\) 7.79549e6 0.779417 0.389709 0.920938i \(-0.372576\pi\)
0.389709 + 0.920938i \(0.372576\pi\)
\(632\) 1.51286e7 1.50663
\(633\) −448903. −0.0445290
\(634\) 8.21583e6 0.811761
\(635\) −6.05180e6 −0.595594
\(636\) −829692. −0.0813344
\(637\) 3.86931e6 0.377820
\(638\) −2.08540e6 −0.202832
\(639\) 1.21629e7 1.17838
\(640\) 1.29279e7 1.24761
\(641\) 6.27067e6 0.602794 0.301397 0.953499i \(-0.402547\pi\)
0.301397 + 0.953499i \(0.402547\pi\)
\(642\) 1.10018e6 0.105348
\(643\) −1.59445e7 −1.52084 −0.760420 0.649432i \(-0.775007\pi\)
−0.760420 + 0.649432i \(0.775007\pi\)
\(644\) 1.89947e6 0.180475
\(645\) 305764. 0.0289393
\(646\) 7.03893e6 0.663629
\(647\) 1.59290e7 1.49598 0.747992 0.663707i \(-0.231018\pi\)
0.747992 + 0.663707i \(0.231018\pi\)
\(648\) −1.13093e7 −1.05803
\(649\) 8.08210e6 0.753204
\(650\) −5.69931e6 −0.529101
\(651\) 560056. 0.0517940
\(652\) −3.28661e7 −3.02781
\(653\) 2.65262e6 0.243440 0.121720 0.992564i \(-0.461159\pi\)
0.121720 + 0.992564i \(0.461159\pi\)
\(654\) −1.03178e6 −0.0943282
\(655\) −1.08003e7 −0.983631
\(656\) 1.51170e6 0.137153
\(657\) 1.08550e7 0.981111
\(658\) 1.04260e7 0.938757
\(659\) 1.35066e7 1.21152 0.605761 0.795647i \(-0.292869\pi\)
0.605761 + 0.795647i \(0.292869\pi\)
\(660\) −435563. −0.0389217
\(661\) 1.17111e7 1.04254 0.521272 0.853391i \(-0.325458\pi\)
0.521272 + 0.853391i \(0.325458\pi\)
\(662\) 7.07928e6 0.627833
\(663\) −158596. −0.0140122
\(664\) −9.00832e6 −0.792910
\(665\) 5.89555e6 0.516976
\(666\) 7.46126e6 0.651818
\(667\) 335310. 0.0291832
\(668\) −3.14555e6 −0.272744
\(669\) 516149. 0.0445872
\(670\) −1.82609e7 −1.57157
\(671\) −1.33982e7 −1.14879
\(672\) −357235. −0.0305162
\(673\) 1.81529e7 1.54492 0.772462 0.635061i \(-0.219025\pi\)
0.772462 + 0.635061i \(0.219025\pi\)
\(674\) 6.81985e6 0.578263
\(675\) 496060. 0.0419059
\(676\) −9.18958e6 −0.773444
\(677\) −3.81675e6 −0.320053 −0.160027 0.987113i \(-0.551158\pi\)
−0.160027 + 0.987113i \(0.551158\pi\)
\(678\) −1.73536e6 −0.144982
\(679\) −4.35126e6 −0.362193
\(680\) −3.89905e6 −0.323360
\(681\) 686024. 0.0566855
\(682\) −2.10022e7 −1.72903
\(683\) −2.60743e6 −0.213876 −0.106938 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(684\) 2.01815e7 1.64935
\(685\) 3.94338e6 0.321102
\(686\) −2.11626e7 −1.71695
\(687\) −751710. −0.0607657
\(688\) 832426. 0.0670462
\(689\) 9.48871e6 0.761481
\(690\) 112381. 0.00898606
\(691\) 81457.6 0.00648988 0.00324494 0.999995i \(-0.498967\pi\)
0.00324494 + 0.999995i \(0.498967\pi\)
\(692\) 9.26905e6 0.735817
\(693\) −5.87305e6 −0.464548
\(694\) −2.44490e7 −1.92692
\(695\) 9.87283e6 0.775318
\(696\) 119108. 0.00932001
\(697\) 8.81659e6 0.687414
\(698\) −9.42856e6 −0.732499
\(699\) −730731. −0.0565672
\(700\) 6.62740e6 0.511208
\(701\) −2.32967e7 −1.79060 −0.895300 0.445464i \(-0.853039\pi\)
−0.895300 + 0.445464i \(0.853039\pi\)
\(702\) −1.46094e6 −0.111890
\(703\) −5.25209e6 −0.400815
\(704\) 1.41134e7 1.07324
\(705\) 384410. 0.0291288
\(706\) 2.00807e7 1.51624
\(707\) −1.00298e7 −0.754649
\(708\) −1.16762e6 −0.0875425
\(709\) 7.52252e6 0.562015 0.281008 0.959706i \(-0.409331\pi\)
0.281008 + 0.959706i \(0.409331\pi\)
\(710\) 1.92495e7 1.43309
\(711\) 1.90246e7 1.41137
\(712\) 7.60930e6 0.562529
\(713\) 3.37693e6 0.248770
\(714\) 295934. 0.0217245
\(715\) 4.98129e6 0.364398
\(716\) −2.23257e7 −1.62751
\(717\) −82232.2 −0.00597371
\(718\) −3.77048e7 −2.72951
\(719\) 1.30064e7 0.938289 0.469144 0.883121i \(-0.344562\pi\)
0.469144 + 0.883121i \(0.344562\pi\)
\(720\) −840698. −0.0604378
\(721\) 127500. 0.00913421
\(722\) 22236.9 0.00158756
\(723\) 966076. 0.0687331
\(724\) 1.66156e7 1.17806
\(725\) 1.16992e6 0.0826632
\(726\) 600059. 0.0422525
\(727\) 1.98038e7 1.38967 0.694835 0.719169i \(-0.255477\pi\)
0.694835 + 0.719169i \(0.255477\pi\)
\(728\) −7.71641e6 −0.539618
\(729\) −1.41581e7 −0.986702
\(730\) 1.71796e7 1.19318
\(731\) 4.85490e6 0.336037
\(732\) 1.93564e6 0.133520
\(733\) 1.81414e7 1.24713 0.623565 0.781772i \(-0.285684\pi\)
0.623565 + 0.781772i \(0.285684\pi\)
\(734\) 1.04388e6 0.0715170
\(735\) −266204. −0.0181759
\(736\) −2.15400e6 −0.146572
\(737\) −1.28050e7 −0.868385
\(738\) 4.05630e7 2.74150
\(739\) −2.22326e7 −1.49754 −0.748771 0.662829i \(-0.769355\pi\)
−0.748771 + 0.662829i \(0.769355\pi\)
\(740\) 7.35887e6 0.494005
\(741\) 513618. 0.0343633
\(742\) −1.77056e7 −1.18059
\(743\) −1.40248e6 −0.0932021 −0.0466011 0.998914i \(-0.514839\pi\)
−0.0466011 + 0.998914i \(0.514839\pi\)
\(744\) 1.19954e6 0.0794479
\(745\) −3.39744e6 −0.224264
\(746\) 1.61232e7 1.06073
\(747\) −1.13282e7 −0.742777
\(748\) −6.91584e6 −0.451950
\(749\) 1.46310e7 0.952948
\(750\) 1.27293e6 0.0826328
\(751\) 1.92135e7 1.24310 0.621552 0.783373i \(-0.286503\pi\)
0.621552 + 0.783373i \(0.286503\pi\)
\(752\) 1.04653e6 0.0674853
\(753\) −323692. −0.0208039
\(754\) −3.44553e6 −0.220713
\(755\) −8.28668e6 −0.529070
\(756\) 1.69885e6 0.108106
\(757\) −1.23078e7 −0.780621 −0.390310 0.920683i \(-0.627632\pi\)
−0.390310 + 0.920683i \(0.627632\pi\)
\(758\) −2.41573e7 −1.52713
\(759\) 78804.6 0.00496532
\(760\) 1.26272e7 0.793001
\(761\) 4.65071e6 0.291110 0.145555 0.989350i \(-0.453503\pi\)
0.145555 + 0.989350i \(0.453503\pi\)
\(762\) 983794. 0.0613786
\(763\) −1.37214e7 −0.853268
\(764\) 4.53294e6 0.280961
\(765\) −4.90314e6 −0.302915
\(766\) −4.26712e7 −2.62762
\(767\) 1.33534e7 0.819603
\(768\) −868719. −0.0531467
\(769\) 1.54937e6 0.0944800 0.0472400 0.998884i \(-0.484957\pi\)
0.0472400 + 0.998884i \(0.484957\pi\)
\(770\) −9.29491e6 −0.564961
\(771\) −407628. −0.0246961
\(772\) 1.88512e7 1.13840
\(773\) −1.71208e7 −1.03056 −0.515282 0.857021i \(-0.672313\pi\)
−0.515282 + 0.857021i \(0.672313\pi\)
\(774\) 2.23362e7 1.34016
\(775\) 1.17824e7 0.704658
\(776\) −9.31962e6 −0.555577
\(777\) −220811. −0.0131210
\(778\) 3.14598e7 1.86340
\(779\) −2.85529e7 −1.68580
\(780\) −719646. −0.0423528
\(781\) 1.34983e7 0.791866
\(782\) 1.78437e6 0.104344
\(783\) 299895. 0.0174809
\(784\) −724724. −0.0421097
\(785\) −1.33734e7 −0.774581
\(786\) 1.75572e6 0.101367
\(787\) −681275. −0.0392090 −0.0196045 0.999808i \(-0.506241\pi\)
−0.0196045 + 0.999808i \(0.506241\pi\)
\(788\) 105854. 0.00607284
\(789\) −913412. −0.0522365
\(790\) 3.01090e7 1.71644
\(791\) −2.30781e7 −1.31147
\(792\) −1.25790e7 −0.712581
\(793\) −2.21367e7 −1.25006
\(794\) 1.89828e7 1.06859
\(795\) −652811. −0.0366328
\(796\) −1.27727e7 −0.714498
\(797\) 530445. 0.0295798 0.0147899 0.999891i \(-0.495292\pi\)
0.0147899 + 0.999891i \(0.495292\pi\)
\(798\) −958394. −0.0532766
\(799\) 6.10363e6 0.338237
\(800\) −7.51546e6 −0.415174
\(801\) 9.56888e6 0.526962
\(802\) −4.13414e7 −2.26960
\(803\) 1.20468e7 0.659300
\(804\) 1.84994e6 0.100930
\(805\) 1.49452e6 0.0812856
\(806\) −3.47002e7 −1.88146
\(807\) 439849. 0.0237749
\(808\) −2.14821e7 −1.15757
\(809\) −2.65631e7 −1.42694 −0.713472 0.700684i \(-0.752878\pi\)
−0.713472 + 0.700684i \(0.752878\pi\)
\(810\) −2.25079e7 −1.20537
\(811\) 1.11660e7 0.596137 0.298069 0.954544i \(-0.403658\pi\)
0.298069 + 0.954544i \(0.403658\pi\)
\(812\) 4.00662e6 0.213249
\(813\) 480663. 0.0255044
\(814\) 8.28043e6 0.438018
\(815\) −2.58594e7 −1.36372
\(816\) 29705.0 0.00156173
\(817\) −1.57228e7 −0.824089
\(818\) 4.72661e7 2.46983
\(819\) −9.70357e6 −0.505501
\(820\) 4.00063e7 2.07775
\(821\) −3.42932e7 −1.77562 −0.887811 0.460209i \(-0.847774\pi\)
−0.887811 + 0.460209i \(0.847774\pi\)
\(822\) −641045. −0.0330909
\(823\) 1.93467e7 0.995651 0.497825 0.867277i \(-0.334132\pi\)
0.497825 + 0.867277i \(0.334132\pi\)
\(824\) 273082. 0.0140112
\(825\) 274955. 0.0140646
\(826\) −2.49170e7 −1.27071
\(827\) 2.67747e7 1.36132 0.680662 0.732598i \(-0.261692\pi\)
0.680662 + 0.732598i \(0.261692\pi\)
\(828\) 5.11602e6 0.259332
\(829\) −2.44308e7 −1.23467 −0.617335 0.786700i \(-0.711788\pi\)
−0.617335 + 0.786700i \(0.711788\pi\)
\(830\) −1.79284e7 −0.903330
\(831\) −1.29962e6 −0.0652853
\(832\) 2.33184e7 1.16786
\(833\) −4.22676e6 −0.211055
\(834\) −1.60495e6 −0.0798999
\(835\) −2.47496e6 −0.122843
\(836\) 2.23972e7 1.10835
\(837\) 3.02026e6 0.149015
\(838\) 2.03815e7 1.00260
\(839\) 9.17559e6 0.450017 0.225009 0.974357i \(-0.427759\pi\)
0.225009 + 0.974357i \(0.427759\pi\)
\(840\) 530879. 0.0259596
\(841\) 707281. 0.0344828
\(842\) −6.92184e6 −0.336466
\(843\) −363827. −0.0176330
\(844\) −3.23425e7 −1.56285
\(845\) −7.23046e6 −0.348357
\(846\) 2.80813e7 1.34894
\(847\) 7.98004e6 0.382205
\(848\) −1.77724e6 −0.0848705
\(849\) −576791. −0.0274631
\(850\) 6.22581e6 0.295562
\(851\) −1.33141e6 −0.0630212
\(852\) −1.95010e6 −0.0920360
\(853\) 1.34195e7 0.631487 0.315744 0.948845i \(-0.397746\pi\)
0.315744 + 0.948845i \(0.397746\pi\)
\(854\) 4.13064e7 1.93808
\(855\) 1.58790e7 0.742863
\(856\) 3.13370e7 1.46175
\(857\) 4.82712e6 0.224510 0.112255 0.993679i \(-0.464193\pi\)
0.112255 + 0.993679i \(0.464193\pi\)
\(858\) −809769. −0.0375528
\(859\) 2.23698e7 1.03438 0.517188 0.855872i \(-0.326979\pi\)
0.517188 + 0.855872i \(0.326979\pi\)
\(860\) 2.20297e7 1.01569
\(861\) −1.20043e6 −0.0551861
\(862\) −2.74935e7 −1.26026
\(863\) −2.28440e6 −0.104411 −0.0522054 0.998636i \(-0.516625\pi\)
−0.0522054 + 0.998636i \(0.516625\pi\)
\(864\) −1.92649e6 −0.0877975
\(865\) 7.29299e6 0.331410
\(866\) 2.00283e7 0.907507
\(867\) −869703. −0.0392937
\(868\) 4.03509e7 1.81783
\(869\) 2.11132e7 0.948431
\(870\) 237048. 0.0106179
\(871\) −2.11567e7 −0.944938
\(872\) −2.93887e7 −1.30885
\(873\) −1.17196e7 −0.520450
\(874\) −5.77876e6 −0.255892
\(875\) 1.69284e7 0.747475
\(876\) −1.74040e6 −0.0766283
\(877\) 2.19674e7 0.964452 0.482226 0.876047i \(-0.339828\pi\)
0.482226 + 0.876047i \(0.339828\pi\)
\(878\) −225194. −0.00985870
\(879\) 1.26663e6 0.0552938
\(880\) −932998. −0.0406138
\(881\) 2.81070e6 0.122004 0.0610021 0.998138i \(-0.480570\pi\)
0.0610021 + 0.998138i \(0.480570\pi\)
\(882\) −1.94463e7 −0.841715
\(883\) −3.24911e7 −1.40237 −0.701186 0.712978i \(-0.747346\pi\)
−0.701186 + 0.712978i \(0.747346\pi\)
\(884\) −1.14265e7 −0.491792
\(885\) −918697. −0.0394289
\(886\) 5.70720e7 2.44252
\(887\) 3.68463e7 1.57248 0.786239 0.617923i \(-0.212026\pi\)
0.786239 + 0.617923i \(0.212026\pi\)
\(888\) −472937. −0.0201266
\(889\) 1.30832e7 0.555215
\(890\) 1.51441e7 0.640866
\(891\) −1.57831e7 −0.666039
\(892\) 3.71875e7 1.56489
\(893\) −1.97669e7 −0.829486
\(894\) 552294. 0.0231114
\(895\) −1.75661e7 −0.733025
\(896\) −2.79485e7 −1.16303
\(897\) 130202. 0.00540304
\(898\) −6.96805e7 −2.88350
\(899\) 7.12307e6 0.293946
\(900\) 1.78502e7 0.734575
\(901\) −1.03653e7 −0.425372
\(902\) 4.50164e7 1.84227
\(903\) −661024. −0.0269773
\(904\) −4.94292e7 −2.01170
\(905\) 1.30733e7 0.530597
\(906\) 1.34710e6 0.0545230
\(907\) 6.77134e6 0.273311 0.136655 0.990619i \(-0.456365\pi\)
0.136655 + 0.990619i \(0.456365\pi\)
\(908\) 4.94266e7 1.98951
\(909\) −2.70142e7 −1.08438
\(910\) −1.53572e7 −0.614765
\(911\) 1.38061e7 0.551156 0.275578 0.961279i \(-0.411131\pi\)
0.275578 + 0.961279i \(0.411131\pi\)
\(912\) −96201.0 −0.00382994
\(913\) −1.25719e7 −0.499142
\(914\) −4.63467e7 −1.83507
\(915\) 1.52298e6 0.0601369
\(916\) −5.41591e7 −2.13272
\(917\) 2.33489e7 0.916943
\(918\) 1.59590e6 0.0625029
\(919\) −3.45509e7 −1.34949 −0.674746 0.738050i \(-0.735747\pi\)
−0.674746 + 0.738050i \(0.735747\pi\)
\(920\) 3.20101e6 0.124686
\(921\) −2.29647e6 −0.0892094
\(922\) −3.30770e7 −1.28144
\(923\) 2.23022e7 0.861673
\(924\) 941634. 0.0362829
\(925\) −4.64538e6 −0.178512
\(926\) −8.16987e7 −3.13103
\(927\) 343407. 0.0131253
\(928\) −4.54349e6 −0.173189
\(929\) −2.47472e7 −0.940776 −0.470388 0.882460i \(-0.655886\pi\)
−0.470388 + 0.882460i \(0.655886\pi\)
\(930\) 2.38733e6 0.0905118
\(931\) 1.36885e7 0.517586
\(932\) −5.26477e7 −1.98536
\(933\) 1.53101e6 0.0575803
\(934\) −2.99476e7 −1.12330
\(935\) −5.44146e6 −0.203557
\(936\) −2.07833e7 −0.775399
\(937\) 1.40114e6 0.0521352 0.0260676 0.999660i \(-0.491701\pi\)
0.0260676 + 0.999660i \(0.491701\pi\)
\(938\) 3.94777e7 1.46502
\(939\) −1.67004e6 −0.0618107
\(940\) 2.76959e7 1.02234
\(941\) −2.79452e7 −1.02881 −0.514403 0.857548i \(-0.671987\pi\)
−0.514403 + 0.857548i \(0.671987\pi\)
\(942\) 2.17400e6 0.0798240
\(943\) −7.23817e6 −0.265063
\(944\) −2.50110e6 −0.0913484
\(945\) 1.33667e6 0.0486906
\(946\) 2.47885e7 0.900580
\(947\) 3.03936e7 1.10131 0.550653 0.834734i \(-0.314379\pi\)
0.550653 + 0.834734i \(0.314379\pi\)
\(948\) −3.05023e6 −0.110233
\(949\) 1.99040e7 0.717421
\(950\) −2.01625e7 −0.724830
\(951\) −654876. −0.0234805
\(952\) 8.42925e6 0.301437
\(953\) 3.75892e7 1.34070 0.670349 0.742046i \(-0.266144\pi\)
0.670349 + 0.742046i \(0.266144\pi\)
\(954\) −4.76881e7 −1.69644
\(955\) 3.56657e6 0.126544
\(956\) −5.92466e6 −0.209661
\(957\) 166225. 0.00586701
\(958\) 2.15025e7 0.756964
\(959\) −8.52510e6 −0.299332
\(960\) −1.60427e6 −0.0561824
\(961\) 4.31078e7 1.50573
\(962\) 1.36811e7 0.476632
\(963\) 3.94070e7 1.36933
\(964\) 6.96038e7 2.41235
\(965\) 1.48323e7 0.512732
\(966\) −242953. −0.00837683
\(967\) 4.14600e7 1.42582 0.712908 0.701258i \(-0.247378\pi\)
0.712908 + 0.701258i \(0.247378\pi\)
\(968\) 1.70918e7 0.586273
\(969\) −561066. −0.0191957
\(970\) −1.85479e7 −0.632946
\(971\) −5.71685e6 −0.194585 −0.0972924 0.995256i \(-0.531018\pi\)
−0.0972924 + 0.995256i \(0.531018\pi\)
\(972\) 6.86603e6 0.233099
\(973\) −2.13438e7 −0.722754
\(974\) −1.25133e7 −0.422642
\(975\) 454286. 0.0153045
\(976\) 4.14622e6 0.139325
\(977\) 2.35225e6 0.0788400 0.0394200 0.999223i \(-0.487449\pi\)
0.0394200 + 0.999223i \(0.487449\pi\)
\(978\) 4.20376e6 0.140537
\(979\) 1.06194e7 0.354115
\(980\) −1.91794e7 −0.637926
\(981\) −3.69570e7 −1.22609
\(982\) −4.07407e7 −1.34819
\(983\) −2.67280e7 −0.882232 −0.441116 0.897450i \(-0.645417\pi\)
−0.441116 + 0.897450i \(0.645417\pi\)
\(984\) −2.57111e6 −0.0846512
\(985\) 83287.1 0.00273519
\(986\) 3.76383e6 0.123293
\(987\) −831047. −0.0271539
\(988\) 3.70051e7 1.20606
\(989\) −3.98573e6 −0.129574
\(990\) −2.50348e7 −0.811814
\(991\) −1.24011e7 −0.401122 −0.200561 0.979681i \(-0.564276\pi\)
−0.200561 + 0.979681i \(0.564276\pi\)
\(992\) −4.57578e7 −1.47634
\(993\) −564283. −0.0181603
\(994\) −4.16151e7 −1.33593
\(995\) −1.00497e7 −0.321808
\(996\) 1.81626e6 0.0580136
\(997\) 1.41176e7 0.449803 0.224902 0.974381i \(-0.427794\pi\)
0.224902 + 0.974381i \(0.427794\pi\)
\(998\) 8.85666e7 2.81477
\(999\) −1.19078e6 −0.0377502
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 29.6.a.a.1.1 4
3.2 odd 2 261.6.a.a.1.4 4
4.3 odd 2 464.6.a.i.1.2 4
5.4 even 2 725.6.a.a.1.4 4
29.28 even 2 841.6.a.a.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.1 4 1.1 even 1 trivial
261.6.a.a.1.4 4 3.2 odd 2
464.6.a.i.1.2 4 4.3 odd 2
725.6.a.a.1.4 4 5.4 even 2
841.6.a.a.1.4 4 29.28 even 2