Properties

Label 29.6.a.a.1.2
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 1.2
Root \(6.17343\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.863638 q^{2} +7.07413 q^{3} -31.2541 q^{4} -44.4758 q^{5} -6.10949 q^{6} -36.7447 q^{7} +54.6287 q^{8} -192.957 q^{9} +O(q^{10})\) \(q-0.863638 q^{2} +7.07413 q^{3} -31.2541 q^{4} -44.4758 q^{5} -6.10949 q^{6} -36.7447 q^{7} +54.6287 q^{8} -192.957 q^{9} +38.4110 q^{10} -302.283 q^{11} -221.096 q^{12} -373.472 q^{13} +31.7341 q^{14} -314.627 q^{15} +952.953 q^{16} +280.365 q^{17} +166.645 q^{18} +1371.41 q^{19} +1390.05 q^{20} -259.937 q^{21} +261.063 q^{22} +1861.10 q^{23} +386.450 q^{24} -1146.91 q^{25} +322.545 q^{26} -3084.01 q^{27} +1148.42 q^{28} -841.000 q^{29} +271.724 q^{30} +1472.03 q^{31} -2571.12 q^{32} -2138.39 q^{33} -242.134 q^{34} +1634.25 q^{35} +6030.69 q^{36} -11730.4 q^{37} -1184.40 q^{38} -2641.99 q^{39} -2429.65 q^{40} -2177.39 q^{41} +224.491 q^{42} -9679.03 q^{43} +9447.58 q^{44} +8581.90 q^{45} -1607.32 q^{46} -15909.1 q^{47} +6741.31 q^{48} -15456.8 q^{49} +990.512 q^{50} +1983.34 q^{51} +11672.6 q^{52} +24359.3 q^{53} +2663.47 q^{54} +13444.3 q^{55} -2007.32 q^{56} +9701.53 q^{57} +726.320 q^{58} +36304.7 q^{59} +9833.40 q^{60} -22316.1 q^{61} -1271.30 q^{62} +7090.14 q^{63} -28274.0 q^{64} +16610.5 q^{65} +1846.79 q^{66} -54808.6 q^{67} -8762.58 q^{68} +13165.7 q^{69} -1411.40 q^{70} +27790.4 q^{71} -10541.0 q^{72} +31685.5 q^{73} +10130.8 q^{74} -8113.36 q^{75} -42862.2 q^{76} +11107.3 q^{77} +2281.73 q^{78} -55328.4 q^{79} -42383.3 q^{80} +25071.8 q^{81} +1880.47 q^{82} -46888.8 q^{83} +8124.10 q^{84} -12469.5 q^{85} +8359.18 q^{86} -5949.34 q^{87} -16513.3 q^{88} +2564.30 q^{89} -7411.65 q^{90} +13723.1 q^{91} -58167.2 q^{92} +10413.3 q^{93} +13739.7 q^{94} -60994.5 q^{95} -18188.5 q^{96} +34940.3 q^{97} +13349.1 q^{98} +58327.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.863638 −0.152671 −0.0763356 0.997082i \(-0.524322\pi\)
−0.0763356 + 0.997082i \(0.524322\pi\)
\(3\) 7.07413 0.453806 0.226903 0.973917i \(-0.427140\pi\)
0.226903 + 0.973917i \(0.427140\pi\)
\(4\) −31.2541 −0.976692
\(5\) −44.4758 −0.795607 −0.397803 0.917471i \(-0.630227\pi\)
−0.397803 + 0.917471i \(0.630227\pi\)
\(6\) −6.10949 −0.0692830
\(7\) −36.7447 −0.283433 −0.141716 0.989907i \(-0.545262\pi\)
−0.141716 + 0.989907i \(0.545262\pi\)
\(8\) 54.6287 0.301784
\(9\) −192.957 −0.794060
\(10\) 38.4110 0.121466
\(11\) −302.283 −0.753237 −0.376619 0.926368i \(-0.622913\pi\)
−0.376619 + 0.926368i \(0.622913\pi\)
\(12\) −221.096 −0.443228
\(13\) −373.472 −0.612915 −0.306457 0.951884i \(-0.599144\pi\)
−0.306457 + 0.951884i \(0.599144\pi\)
\(14\) 31.7341 0.0432720
\(15\) −314.627 −0.361051
\(16\) 952.953 0.930618
\(17\) 280.365 0.235289 0.117645 0.993056i \(-0.462466\pi\)
0.117645 + 0.993056i \(0.462466\pi\)
\(18\) 166.645 0.121230
\(19\) 1371.41 0.871532 0.435766 0.900060i \(-0.356478\pi\)
0.435766 + 0.900060i \(0.356478\pi\)
\(20\) 1390.05 0.777062
\(21\) −259.937 −0.128623
\(22\) 261.063 0.114998
\(23\) 1861.10 0.733586 0.366793 0.930303i \(-0.380456\pi\)
0.366793 + 0.930303i \(0.380456\pi\)
\(24\) 386.450 0.136951
\(25\) −1146.91 −0.367010
\(26\) 322.545 0.0935744
\(27\) −3084.01 −0.814155
\(28\) 1148.42 0.276826
\(29\) −841.000 −0.185695
\(30\) 271.724 0.0551220
\(31\) 1472.03 0.275114 0.137557 0.990494i \(-0.456075\pi\)
0.137557 + 0.990494i \(0.456075\pi\)
\(32\) −2571.12 −0.443862
\(33\) −2138.39 −0.341823
\(34\) −242.134 −0.0359219
\(35\) 1634.25 0.225501
\(36\) 6030.69 0.775552
\(37\) −11730.4 −1.40867 −0.704335 0.709868i \(-0.748755\pi\)
−0.704335 + 0.709868i \(0.748755\pi\)
\(38\) −1184.40 −0.133058
\(39\) −2641.99 −0.278144
\(40\) −2429.65 −0.240101
\(41\) −2177.39 −0.202291 −0.101145 0.994872i \(-0.532251\pi\)
−0.101145 + 0.994872i \(0.532251\pi\)
\(42\) 224.491 0.0196371
\(43\) −9679.03 −0.798290 −0.399145 0.916888i \(-0.630693\pi\)
−0.399145 + 0.916888i \(0.630693\pi\)
\(44\) 9447.58 0.735680
\(45\) 8581.90 0.631760
\(46\) −1607.32 −0.111997
\(47\) −15909.1 −1.05051 −0.525255 0.850945i \(-0.676030\pi\)
−0.525255 + 0.850945i \(0.676030\pi\)
\(48\) 6741.31 0.422320
\(49\) −15456.8 −0.919666
\(50\) 990.512 0.0560318
\(51\) 1983.34 0.106776
\(52\) 11672.6 0.598629
\(53\) 24359.3 1.19117 0.595586 0.803291i \(-0.296920\pi\)
0.595586 + 0.803291i \(0.296920\pi\)
\(54\) 2663.47 0.124298
\(55\) 13444.3 0.599280
\(56\) −2007.32 −0.0855353
\(57\) 9701.53 0.395506
\(58\) 726.320 0.0283503
\(59\) 36304.7 1.35779 0.678894 0.734236i \(-0.262459\pi\)
0.678894 + 0.734236i \(0.262459\pi\)
\(60\) 9833.40 0.352635
\(61\) −22316.1 −0.767880 −0.383940 0.923358i \(-0.625433\pi\)
−0.383940 + 0.923358i \(0.625433\pi\)
\(62\) −1271.30 −0.0420019
\(63\) 7090.14 0.225063
\(64\) −28274.0 −0.862853
\(65\) 16610.5 0.487639
\(66\) 1846.79 0.0521865
\(67\) −54808.6 −1.49163 −0.745817 0.666151i \(-0.767940\pi\)
−0.745817 + 0.666151i \(0.767940\pi\)
\(68\) −8762.58 −0.229805
\(69\) 13165.7 0.332905
\(70\) −1411.40 −0.0344275
\(71\) 27790.4 0.654258 0.327129 0.944980i \(-0.393919\pi\)
0.327129 + 0.944980i \(0.393919\pi\)
\(72\) −10541.0 −0.239635
\(73\) 31685.5 0.695910 0.347955 0.937511i \(-0.386876\pi\)
0.347955 + 0.937511i \(0.386876\pi\)
\(74\) 10130.8 0.215063
\(75\) −8113.36 −0.166551
\(76\) −42862.2 −0.851218
\(77\) 11107.3 0.213492
\(78\) 2281.73 0.0424646
\(79\) −55328.4 −0.997426 −0.498713 0.866767i \(-0.666194\pi\)
−0.498713 + 0.866767i \(0.666194\pi\)
\(80\) −42383.3 −0.740406
\(81\) 25071.8 0.424592
\(82\) 1880.47 0.0308839
\(83\) −46888.8 −0.747092 −0.373546 0.927612i \(-0.621858\pi\)
−0.373546 + 0.927612i \(0.621858\pi\)
\(84\) 8124.10 0.125625
\(85\) −12469.5 −0.187198
\(86\) 8359.18 0.121876
\(87\) −5949.34 −0.0842696
\(88\) −16513.3 −0.227315
\(89\) 2564.30 0.0343157 0.0171579 0.999853i \(-0.494538\pi\)
0.0171579 + 0.999853i \(0.494538\pi\)
\(90\) −7411.65 −0.0964515
\(91\) 13723.1 0.173720
\(92\) −58167.2 −0.716487
\(93\) 10413.3 0.124848
\(94\) 13739.7 0.160382
\(95\) −60994.5 −0.693396
\(96\) −18188.5 −0.201427
\(97\) 34940.3 0.377048 0.188524 0.982069i \(-0.439630\pi\)
0.188524 + 0.982069i \(0.439630\pi\)
\(98\) 13349.1 0.140406
\(99\) 58327.5 0.598116
\(100\) 35845.6 0.358456
\(101\) 170224. 1.66042 0.830210 0.557451i \(-0.188221\pi\)
0.830210 + 0.557451i \(0.188221\pi\)
\(102\) −1712.89 −0.0163015
\(103\) 83962.9 0.779820 0.389910 0.920853i \(-0.372506\pi\)
0.389910 + 0.920853i \(0.372506\pi\)
\(104\) −20402.3 −0.184968
\(105\) 11560.9 0.102334
\(106\) −21037.6 −0.181858
\(107\) 16842.4 0.142215 0.0711074 0.997469i \(-0.477347\pi\)
0.0711074 + 0.997469i \(0.477347\pi\)
\(108\) 96388.2 0.795178
\(109\) −3855.63 −0.0310834 −0.0155417 0.999879i \(-0.504947\pi\)
−0.0155417 + 0.999879i \(0.504947\pi\)
\(110\) −11611.0 −0.0914928
\(111\) −82982.5 −0.639262
\(112\) −35016.0 −0.263767
\(113\) 107319. 0.790644 0.395322 0.918543i \(-0.370633\pi\)
0.395322 + 0.918543i \(0.370633\pi\)
\(114\) −8378.61 −0.0603823
\(115\) −82774.0 −0.583646
\(116\) 26284.7 0.181367
\(117\) 72064.0 0.486691
\(118\) −31354.1 −0.207295
\(119\) −10301.9 −0.0666886
\(120\) −17187.7 −0.108959
\(121\) −69676.1 −0.432634
\(122\) 19273.0 0.117233
\(123\) −15403.1 −0.0918006
\(124\) −46007.0 −0.268701
\(125\) 189996. 1.08760
\(126\) −6123.31 −0.0343606
\(127\) −14823.9 −0.0815554 −0.0407777 0.999168i \(-0.512984\pi\)
−0.0407777 + 0.999168i \(0.512984\pi\)
\(128\) 106694. 0.575595
\(129\) −68470.7 −0.362269
\(130\) −14345.4 −0.0744484
\(131\) −313171. −1.59442 −0.797211 0.603701i \(-0.793692\pi\)
−0.797211 + 0.603701i \(0.793692\pi\)
\(132\) 66833.4 0.333856
\(133\) −50392.1 −0.247020
\(134\) 47334.8 0.227729
\(135\) 137164. 0.647747
\(136\) 15316.0 0.0710065
\(137\) −254312. −1.15762 −0.578810 0.815463i \(-0.696483\pi\)
−0.578810 + 0.815463i \(0.696483\pi\)
\(138\) −11370.4 −0.0508250
\(139\) −390931. −1.71618 −0.858091 0.513498i \(-0.828349\pi\)
−0.858091 + 0.513498i \(0.828349\pi\)
\(140\) −51077.0 −0.220245
\(141\) −112543. −0.476727
\(142\) −24000.9 −0.0998863
\(143\) 112894. 0.461670
\(144\) −183879. −0.738967
\(145\) 37404.1 0.147740
\(146\) −27364.8 −0.106245
\(147\) −109344. −0.417350
\(148\) 366624. 1.37584
\(149\) 72345.7 0.266961 0.133480 0.991051i \(-0.457385\pi\)
0.133480 + 0.991051i \(0.457385\pi\)
\(150\) 7007.01 0.0254276
\(151\) −227419. −0.811679 −0.405840 0.913944i \(-0.633021\pi\)
−0.405840 + 0.913944i \(0.633021\pi\)
\(152\) 74918.3 0.263014
\(153\) −54098.4 −0.186834
\(154\) −9592.68 −0.0325940
\(155\) −65469.6 −0.218882
\(156\) 82573.2 0.271661
\(157\) 539281. 1.74609 0.873044 0.487641i \(-0.162143\pi\)
0.873044 + 0.487641i \(0.162143\pi\)
\(158\) 47783.7 0.152278
\(159\) 172321. 0.540561
\(160\) 114353. 0.353140
\(161\) −68385.7 −0.207922
\(162\) −21652.9 −0.0648230
\(163\) 87996.5 0.259416 0.129708 0.991552i \(-0.458596\pi\)
0.129708 + 0.991552i \(0.458596\pi\)
\(164\) 68052.3 0.197576
\(165\) 95106.4 0.271957
\(166\) 40495.0 0.114059
\(167\) 199897. 0.554644 0.277322 0.960777i \(-0.410553\pi\)
0.277322 + 0.960777i \(0.410553\pi\)
\(168\) −14200.0 −0.0388164
\(169\) −231811. −0.624335
\(170\) 10769.1 0.0285797
\(171\) −264623. −0.692049
\(172\) 302510. 0.779683
\(173\) −313797. −0.797139 −0.398569 0.917138i \(-0.630493\pi\)
−0.398569 + 0.917138i \(0.630493\pi\)
\(174\) 5138.08 0.0128655
\(175\) 42142.7 0.104023
\(176\) −288061. −0.700976
\(177\) 256824. 0.616172
\(178\) −2214.63 −0.00523902
\(179\) −59814.7 −0.139533 −0.0697663 0.997563i \(-0.522225\pi\)
−0.0697663 + 0.997563i \(0.522225\pi\)
\(180\) −268220. −0.617034
\(181\) 402231. 0.912597 0.456298 0.889827i \(-0.349175\pi\)
0.456298 + 0.889827i \(0.349175\pi\)
\(182\) −11851.8 −0.0265220
\(183\) −157867. −0.348468
\(184\) 101670. 0.221384
\(185\) 521719. 1.12075
\(186\) −8993.35 −0.0190607
\(187\) −84749.6 −0.177229
\(188\) 497224. 1.02602
\(189\) 113321. 0.230758
\(190\) 52677.2 0.105862
\(191\) −903529. −1.79209 −0.896043 0.443967i \(-0.853571\pi\)
−0.896043 + 0.443967i \(0.853571\pi\)
\(192\) −200014. −0.391568
\(193\) 1.02870e6 1.98791 0.993956 0.109780i \(-0.0350148\pi\)
0.993956 + 0.109780i \(0.0350148\pi\)
\(194\) −30175.7 −0.0575644
\(195\) 117505. 0.221293
\(196\) 483090. 0.898230
\(197\) −866099. −1.59002 −0.795009 0.606598i \(-0.792534\pi\)
−0.795009 + 0.606598i \(0.792534\pi\)
\(198\) −50373.8 −0.0913150
\(199\) 285836. 0.511663 0.255832 0.966721i \(-0.417651\pi\)
0.255832 + 0.966721i \(0.417651\pi\)
\(200\) −62654.0 −0.110758
\(201\) −387723. −0.676911
\(202\) −147012. −0.253498
\(203\) 30902.3 0.0526321
\(204\) −61987.6 −0.104287
\(205\) 96840.9 0.160944
\(206\) −72513.6 −0.119056
\(207\) −359112. −0.582512
\(208\) −355901. −0.570389
\(209\) −414554. −0.656470
\(210\) −9984.43 −0.0156234
\(211\) −918930. −1.42094 −0.710471 0.703727i \(-0.751518\pi\)
−0.710471 + 0.703727i \(0.751518\pi\)
\(212\) −761328. −1.16341
\(213\) 196593. 0.296906
\(214\) −14545.8 −0.0217121
\(215\) 430482. 0.635125
\(216\) −168476. −0.245699
\(217\) −54089.3 −0.0779762
\(218\) 3329.87 0.00474554
\(219\) 224147. 0.315808
\(220\) −420189. −0.585312
\(221\) −104709. −0.144212
\(222\) 71666.8 0.0975969
\(223\) 486134. 0.654626 0.327313 0.944916i \(-0.393857\pi\)
0.327313 + 0.944916i \(0.393857\pi\)
\(224\) 94475.2 0.125805
\(225\) 221303. 0.291428
\(226\) −92684.9 −0.120708
\(227\) −557639. −0.718271 −0.359136 0.933285i \(-0.616928\pi\)
−0.359136 + 0.933285i \(0.616928\pi\)
\(228\) −303213. −0.386287
\(229\) 440289. 0.554816 0.277408 0.960752i \(-0.410525\pi\)
0.277408 + 0.960752i \(0.410525\pi\)
\(230\) 71486.8 0.0891059
\(231\) 78574.4 0.0968838
\(232\) −45942.7 −0.0560398
\(233\) 1.46183e6 1.76403 0.882015 0.471221i \(-0.156187\pi\)
0.882015 + 0.471221i \(0.156187\pi\)
\(234\) −62237.2 −0.0743037
\(235\) 707568. 0.835792
\(236\) −1.13467e6 −1.32614
\(237\) −391400. −0.452637
\(238\) 8897.16 0.0101814
\(239\) 1.51292e6 1.71325 0.856627 0.515936i \(-0.172556\pi\)
0.856627 + 0.515936i \(0.172556\pi\)
\(240\) −299825. −0.336000
\(241\) −800679. −0.888006 −0.444003 0.896025i \(-0.646442\pi\)
−0.444003 + 0.896025i \(0.646442\pi\)
\(242\) 60175.0 0.0660507
\(243\) 926776. 1.00684
\(244\) 697470. 0.749982
\(245\) 687454. 0.731692
\(246\) 13302.7 0.0140153
\(247\) −512184. −0.534175
\(248\) 80415.0 0.0830249
\(249\) −331698. −0.339035
\(250\) −164088. −0.166045
\(251\) 1.66023e6 1.66335 0.831676 0.555261i \(-0.187382\pi\)
0.831676 + 0.555261i \(0.187382\pi\)
\(252\) −221596. −0.219817
\(253\) −562580. −0.552564
\(254\) 12802.5 0.0124512
\(255\) −88210.6 −0.0849514
\(256\) 812621. 0.774976
\(257\) −1.00723e6 −0.951250 −0.475625 0.879648i \(-0.657778\pi\)
−0.475625 + 0.879648i \(0.657778\pi\)
\(258\) 59133.9 0.0553079
\(259\) 431031. 0.399263
\(260\) −519146. −0.476273
\(261\) 162277. 0.147453
\(262\) 270466. 0.243422
\(263\) 1.01326e6 0.903298 0.451649 0.892196i \(-0.350836\pi\)
0.451649 + 0.892196i \(0.350836\pi\)
\(264\) −116817. −0.103157
\(265\) −1.08340e6 −0.947705
\(266\) 43520.5 0.0377129
\(267\) 18140.2 0.0155727
\(268\) 1.71300e6 1.45687
\(269\) −1.45518e6 −1.22613 −0.613064 0.790034i \(-0.710063\pi\)
−0.613064 + 0.790034i \(0.710063\pi\)
\(270\) −118460. −0.0988922
\(271\) −1.61146e6 −1.33290 −0.666450 0.745550i \(-0.732187\pi\)
−0.666450 + 0.745550i \(0.732187\pi\)
\(272\) 267175. 0.218964
\(273\) 97079.2 0.0788351
\(274\) 219634. 0.176735
\(275\) 346690. 0.276446
\(276\) −411482. −0.325146
\(277\) 2.06358e6 1.61593 0.807966 0.589230i \(-0.200569\pi\)
0.807966 + 0.589230i \(0.200569\pi\)
\(278\) 337623. 0.262011
\(279\) −284038. −0.218457
\(280\) 89276.9 0.0680525
\(281\) 21664.7 0.0163676 0.00818382 0.999967i \(-0.497395\pi\)
0.00818382 + 0.999967i \(0.497395\pi\)
\(282\) 97196.2 0.0727824
\(283\) −552726. −0.410245 −0.205123 0.978736i \(-0.565759\pi\)
−0.205123 + 0.978736i \(0.565759\pi\)
\(284\) −868565. −0.639008
\(285\) −431483. −0.314667
\(286\) −97499.8 −0.0704837
\(287\) 80007.4 0.0573358
\(288\) 496116. 0.352453
\(289\) −1.34125e6 −0.944639
\(290\) −32303.6 −0.0225557
\(291\) 247172. 0.171107
\(292\) −990301. −0.679689
\(293\) −1.53503e6 −1.04459 −0.522297 0.852764i \(-0.674925\pi\)
−0.522297 + 0.852764i \(0.674925\pi\)
\(294\) 94433.3 0.0637172
\(295\) −1.61468e6 −1.08027
\(296\) −640817. −0.425114
\(297\) 932244. 0.613251
\(298\) −62480.5 −0.0407572
\(299\) −695071. −0.449626
\(300\) 253576. 0.162669
\(301\) 355653. 0.226261
\(302\) 196408. 0.123920
\(303\) 1.20419e6 0.753508
\(304\) 1.30689e6 0.811063
\(305\) 992525. 0.610930
\(306\) 46721.4 0.0285241
\(307\) −2.69140e6 −1.62979 −0.814895 0.579609i \(-0.803205\pi\)
−0.814895 + 0.579609i \(0.803205\pi\)
\(308\) −347149. −0.208516
\(309\) 593964. 0.353887
\(310\) 56542.1 0.0334170
\(311\) −1.88553e6 −1.10543 −0.552717 0.833369i \(-0.686409\pi\)
−0.552717 + 0.833369i \(0.686409\pi\)
\(312\) −144329. −0.0839394
\(313\) −2.48142e6 −1.43166 −0.715828 0.698276i \(-0.753951\pi\)
−0.715828 + 0.698276i \(0.753951\pi\)
\(314\) −465744. −0.266577
\(315\) −315339. −0.179061
\(316\) 1.72924e6 0.974177
\(317\) 204476. 0.114286 0.0571432 0.998366i \(-0.481801\pi\)
0.0571432 + 0.998366i \(0.481801\pi\)
\(318\) −148823. −0.0825280
\(319\) 254220. 0.139873
\(320\) 1.25751e6 0.686492
\(321\) 119145. 0.0645379
\(322\) 59060.5 0.0317437
\(323\) 384496. 0.205062
\(324\) −783596. −0.414696
\(325\) 428338. 0.224946
\(326\) −75997.1 −0.0396053
\(327\) −27275.2 −0.0141058
\(328\) −118948. −0.0610480
\(329\) 584574. 0.297749
\(330\) −82137.5 −0.0415199
\(331\) 1.58800e6 0.796672 0.398336 0.917240i \(-0.369588\pi\)
0.398336 + 0.917240i \(0.369588\pi\)
\(332\) 1.46547e6 0.729678
\(333\) 2.26346e6 1.11857
\(334\) −172638. −0.0846782
\(335\) 2.43766e6 1.18675
\(336\) −247708. −0.119699
\(337\) −813616. −0.390252 −0.195126 0.980778i \(-0.562512\pi\)
−0.195126 + 0.980778i \(0.562512\pi\)
\(338\) 200201. 0.0953180
\(339\) 759189. 0.358799
\(340\) 389722. 0.182834
\(341\) −444969. −0.207226
\(342\) 228538. 0.105656
\(343\) 1.18553e6 0.544096
\(344\) −528753. −0.240911
\(345\) −585554. −0.264862
\(346\) 271007. 0.121700
\(347\) 516038. 0.230069 0.115034 0.993362i \(-0.463302\pi\)
0.115034 + 0.993362i \(0.463302\pi\)
\(348\) 185942. 0.0823054
\(349\) −2.65164e6 −1.16533 −0.582667 0.812711i \(-0.697991\pi\)
−0.582667 + 0.812711i \(0.697991\pi\)
\(350\) −36396.1 −0.0158812
\(351\) 1.15179e6 0.499007
\(352\) 777207. 0.334333
\(353\) 391323. 0.167147 0.0835734 0.996502i \(-0.473367\pi\)
0.0835734 + 0.996502i \(0.473367\pi\)
\(354\) −221803. −0.0940717
\(355\) −1.23600e6 −0.520532
\(356\) −80144.9 −0.0335159
\(357\) −72877.3 −0.0302637
\(358\) 51658.3 0.0213026
\(359\) 3.03988e6 1.24486 0.622430 0.782675i \(-0.286146\pi\)
0.622430 + 0.782675i \(0.286146\pi\)
\(360\) 468818. 0.190655
\(361\) −595334. −0.240432
\(362\) −347382. −0.139327
\(363\) −492898. −0.196332
\(364\) −428905. −0.169671
\(365\) −1.40924e6 −0.553670
\(366\) 136340. 0.0532010
\(367\) 1.02897e6 0.398785 0.199392 0.979920i \(-0.436103\pi\)
0.199392 + 0.979920i \(0.436103\pi\)
\(368\) 1.77354e6 0.682688
\(369\) 420141. 0.160631
\(370\) −450577. −0.171106
\(371\) −895074. −0.337617
\(372\) −325460. −0.121938
\(373\) −4.42358e6 −1.64627 −0.823136 0.567844i \(-0.807778\pi\)
−0.823136 + 0.567844i \(0.807778\pi\)
\(374\) 73193.0 0.0270577
\(375\) 1.34406e6 0.493560
\(376\) −869091. −0.317027
\(377\) 314090. 0.113815
\(378\) −97868.5 −0.0352301
\(379\) −4.23499e6 −1.51445 −0.757223 0.653156i \(-0.773445\pi\)
−0.757223 + 0.653156i \(0.773445\pi\)
\(380\) 1.90633e6 0.677234
\(381\) −104866. −0.0370103
\(382\) 780323. 0.273600
\(383\) −894484. −0.311584 −0.155792 0.987790i \(-0.549793\pi\)
−0.155792 + 0.987790i \(0.549793\pi\)
\(384\) 754770. 0.261208
\(385\) −494005. −0.169856
\(386\) −888428. −0.303497
\(387\) 1.86763e6 0.633891
\(388\) −1.09203e6 −0.368260
\(389\) 3.60641e6 1.20837 0.604186 0.796843i \(-0.293498\pi\)
0.604186 + 0.796843i \(0.293498\pi\)
\(390\) −101481. −0.0337851
\(391\) 521789. 0.172605
\(392\) −844386. −0.277540
\(393\) −2.21541e6 −0.723557
\(394\) 747996. 0.242750
\(395\) 2.46077e6 0.793559
\(396\) −1.82297e6 −0.584175
\(397\) −1.96195e6 −0.624757 −0.312379 0.949958i \(-0.601126\pi\)
−0.312379 + 0.949958i \(0.601126\pi\)
\(398\) −246859. −0.0781162
\(399\) −356480. −0.112099
\(400\) −1.09295e6 −0.341546
\(401\) −721116. −0.223947 −0.111973 0.993711i \(-0.535717\pi\)
−0.111973 + 0.993711i \(0.535717\pi\)
\(402\) 334853. 0.103345
\(403\) −549762. −0.168621
\(404\) −5.32021e6 −1.62172
\(405\) −1.11509e6 −0.337809
\(406\) −26688.4 −0.00803540
\(407\) 3.54590e6 1.06106
\(408\) 108347. 0.0322231
\(409\) 1.33383e6 0.394269 0.197135 0.980376i \(-0.436836\pi\)
0.197135 + 0.980376i \(0.436836\pi\)
\(410\) −83635.5 −0.0245715
\(411\) −1.79904e6 −0.525334
\(412\) −2.62419e6 −0.761644
\(413\) −1.33400e6 −0.384842
\(414\) 310143. 0.0889327
\(415\) 2.08542e6 0.594391
\(416\) 960244. 0.272050
\(417\) −2.76550e6 −0.778813
\(418\) 358024. 0.100224
\(419\) −4.51294e6 −1.25581 −0.627905 0.778290i \(-0.716087\pi\)
−0.627905 + 0.778290i \(0.716087\pi\)
\(420\) −361326. −0.0999483
\(421\) 982181. 0.270076 0.135038 0.990840i \(-0.456884\pi\)
0.135038 + 0.990840i \(0.456884\pi\)
\(422\) 793623. 0.216937
\(423\) 3.06976e6 0.834168
\(424\) 1.33071e6 0.359476
\(425\) −321553. −0.0863535
\(426\) −169785. −0.0453290
\(427\) 819998. 0.217642
\(428\) −526395. −0.138900
\(429\) 798629. 0.209508
\(430\) −371781. −0.0969652
\(431\) 5.31784e6 1.37893 0.689465 0.724319i \(-0.257846\pi\)
0.689465 + 0.724319i \(0.257846\pi\)
\(432\) −2.93892e6 −0.757667
\(433\) 679278. 0.174112 0.0870558 0.996203i \(-0.472254\pi\)
0.0870558 + 0.996203i \(0.472254\pi\)
\(434\) 46713.6 0.0119047
\(435\) 264602. 0.0670454
\(436\) 120504. 0.0303589
\(437\) 2.55234e6 0.639344
\(438\) −193582. −0.0482147
\(439\) 3.53619e6 0.875739 0.437870 0.899039i \(-0.355733\pi\)
0.437870 + 0.899039i \(0.355733\pi\)
\(440\) 734442. 0.180853
\(441\) 2.98250e6 0.730270
\(442\) 90430.5 0.0220170
\(443\) −1.35298e6 −0.327553 −0.163776 0.986497i \(-0.552368\pi\)
−0.163776 + 0.986497i \(0.552368\pi\)
\(444\) 2.59355e6 0.624362
\(445\) −114049. −0.0273018
\(446\) −419844. −0.0999425
\(447\) 511783. 0.121148
\(448\) 1.03892e6 0.244561
\(449\) −2.04554e6 −0.478843 −0.239421 0.970916i \(-0.576958\pi\)
−0.239421 + 0.970916i \(0.576958\pi\)
\(450\) −191126. −0.0444927
\(451\) 658186. 0.152373
\(452\) −3.35416e6 −0.772215
\(453\) −1.60879e6 −0.368345
\(454\) 481598. 0.109659
\(455\) −610347. −0.138213
\(456\) 529982. 0.119357
\(457\) 8.48379e6 1.90020 0.950100 0.311944i \(-0.100980\pi\)
0.950100 + 0.311944i \(0.100980\pi\)
\(458\) −380250. −0.0847044
\(459\) −864651. −0.191562
\(460\) 2.58703e6 0.570042
\(461\) −2.06795e6 −0.453198 −0.226599 0.973988i \(-0.572761\pi\)
−0.226599 + 0.973988i \(0.572761\pi\)
\(462\) −67859.9 −0.0147914
\(463\) −368290. −0.0798430 −0.0399215 0.999203i \(-0.512711\pi\)
−0.0399215 + 0.999203i \(0.512711\pi\)
\(464\) −801433. −0.172811
\(465\) −463141. −0.0993300
\(466\) −1.26249e6 −0.269317
\(467\) −2.78365e6 −0.590640 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(468\) −2.25230e6 −0.475347
\(469\) 2.01393e6 0.422777
\(470\) −611082. −0.127601
\(471\) 3.81494e6 0.792385
\(472\) 1.98328e6 0.409759
\(473\) 2.92580e6 0.601302
\(474\) 338028. 0.0691047
\(475\) −1.57288e6 −0.319861
\(476\) 321978. 0.0651342
\(477\) −4.70028e6 −0.945863
\(478\) −1.30662e6 −0.261564
\(479\) 1.49820e6 0.298352 0.149176 0.988811i \(-0.452338\pi\)
0.149176 + 0.988811i \(0.452338\pi\)
\(480\) 808946. 0.160257
\(481\) 4.38099e6 0.863394
\(482\) 691497. 0.135573
\(483\) −483770. −0.0943563
\(484\) 2.17767e6 0.422550
\(485\) −1.55399e6 −0.299982
\(486\) −800399. −0.153715
\(487\) 7.95675e6 1.52024 0.760122 0.649781i \(-0.225139\pi\)
0.760122 + 0.649781i \(0.225139\pi\)
\(488\) −1.21910e6 −0.231734
\(489\) 622498. 0.117724
\(490\) −593712. −0.111708
\(491\) −1.87342e6 −0.350697 −0.175348 0.984506i \(-0.556105\pi\)
−0.175348 + 0.984506i \(0.556105\pi\)
\(492\) 481411. 0.0896609
\(493\) −235787. −0.0436921
\(494\) 442341. 0.0815530
\(495\) −2.59416e6 −0.475865
\(496\) 1.40277e6 0.256026
\(497\) −1.02115e6 −0.185438
\(498\) 286467. 0.0517608
\(499\) −352286. −0.0633350 −0.0316675 0.999498i \(-0.510082\pi\)
−0.0316675 + 0.999498i \(0.510082\pi\)
\(500\) −5.93817e6 −1.06225
\(501\) 1.41410e6 0.251701
\(502\) −1.43384e6 −0.253946
\(503\) −3.57266e6 −0.629611 −0.314805 0.949156i \(-0.601939\pi\)
−0.314805 + 0.949156i \(0.601939\pi\)
\(504\) 387325. 0.0679202
\(505\) −7.57085e6 −1.32104
\(506\) 485865. 0.0843606
\(507\) −1.63986e6 −0.283327
\(508\) 463308. 0.0796545
\(509\) −1.05719e7 −1.80867 −0.904337 0.426819i \(-0.859634\pi\)
−0.904337 + 0.426819i \(0.859634\pi\)
\(510\) 76182.1 0.0129696
\(511\) −1.16427e6 −0.197243
\(512\) −4.11603e6 −0.693911
\(513\) −4.22945e6 −0.709562
\(514\) 869880. 0.145228
\(515\) −3.73431e6 −0.620430
\(516\) 2.13999e6 0.353825
\(517\) 4.80903e6 0.791282
\(518\) −372255. −0.0609559
\(519\) −2.21984e6 −0.361746
\(520\) 907408. 0.147162
\(521\) 7.42701e6 1.19872 0.599362 0.800478i \(-0.295421\pi\)
0.599362 + 0.800478i \(0.295421\pi\)
\(522\) −140148. −0.0225119
\(523\) −1.03292e7 −1.65125 −0.825623 0.564222i \(-0.809176\pi\)
−0.825623 + 0.564222i \(0.809176\pi\)
\(524\) 9.78788e6 1.55726
\(525\) 298123. 0.0472060
\(526\) −875089. −0.137908
\(527\) 412706. 0.0647313
\(528\) −2.03778e6 −0.318107
\(529\) −2.97263e6 −0.461851
\(530\) 935663. 0.144687
\(531\) −7.00523e6 −1.07817
\(532\) 1.57496e6 0.241263
\(533\) 813193. 0.123987
\(534\) −15666.5 −0.00237750
\(535\) −749079. −0.113147
\(536\) −2.99412e6 −0.450151
\(537\) −423137. −0.0633207
\(538\) 1.25675e6 0.187194
\(539\) 4.67233e6 0.692726
\(540\) −4.28694e6 −0.632649
\(541\) 4.60833e6 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(542\) 1.39172e6 0.203495
\(543\) 2.84543e6 0.414142
\(544\) −720854. −0.104436
\(545\) 171482. 0.0247302
\(546\) −83841.3 −0.0120358
\(547\) −1.79146e6 −0.255999 −0.127999 0.991774i \(-0.540856\pi\)
−0.127999 + 0.991774i \(0.540856\pi\)
\(548\) 7.94831e6 1.13064
\(549\) 4.30604e6 0.609743
\(550\) −299415. −0.0422052
\(551\) −1.15336e6 −0.161839
\(552\) 719224. 0.100465
\(553\) 2.03303e6 0.282703
\(554\) −1.78219e6 −0.246706
\(555\) 3.69071e6 0.508601
\(556\) 1.22182e7 1.67618
\(557\) 1.07840e7 1.47280 0.736399 0.676548i \(-0.236525\pi\)
0.736399 + 0.676548i \(0.236525\pi\)
\(558\) 245306. 0.0333521
\(559\) 3.61485e6 0.489284
\(560\) 1.55736e6 0.209855
\(561\) −599530. −0.0804273
\(562\) −18710.4 −0.00249887
\(563\) −1.40598e7 −1.86942 −0.934710 0.355411i \(-0.884341\pi\)
−0.934710 + 0.355411i \(0.884341\pi\)
\(564\) 3.51743e6 0.465615
\(565\) −4.77310e6 −0.629041
\(566\) 477355. 0.0626326
\(567\) −921255. −0.120343
\(568\) 1.51815e6 0.197444
\(569\) 4.09890e6 0.530746 0.265373 0.964146i \(-0.414505\pi\)
0.265373 + 0.964146i \(0.414505\pi\)
\(570\) 372645. 0.0480406
\(571\) −2.46148e6 −0.315941 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(572\) −3.52841e6 −0.450909
\(573\) −6.39168e6 −0.813259
\(574\) −69097.5 −0.00875351
\(575\) −2.13451e6 −0.269233
\(576\) 5.45565e6 0.685157
\(577\) 7.09785e6 0.887539 0.443770 0.896141i \(-0.353641\pi\)
0.443770 + 0.896141i \(0.353641\pi\)
\(578\) 1.15836e6 0.144219
\(579\) 7.27718e6 0.902125
\(580\) −1.16903e6 −0.144297
\(581\) 1.72292e6 0.211750
\(582\) −213467. −0.0261230
\(583\) −7.36339e6 −0.897235
\(584\) 1.73094e6 0.210014
\(585\) −3.20510e6 −0.387215
\(586\) 1.32571e6 0.159479
\(587\) −1.43549e7 −1.71951 −0.859753 0.510710i \(-0.829383\pi\)
−0.859753 + 0.510710i \(0.829383\pi\)
\(588\) 3.41744e6 0.407622
\(589\) 2.01876e6 0.239770
\(590\) 1.39450e6 0.164925
\(591\) −6.12690e6 −0.721559
\(592\) −1.11785e7 −1.31093
\(593\) 1.01492e7 1.18521 0.592606 0.805493i \(-0.298099\pi\)
0.592606 + 0.805493i \(0.298099\pi\)
\(594\) −805122. −0.0936258
\(595\) 458187. 0.0530579
\(596\) −2.26110e6 −0.260738
\(597\) 2.02204e6 0.232196
\(598\) 600290. 0.0686449
\(599\) −8.76777e6 −0.998440 −0.499220 0.866475i \(-0.666380\pi\)
−0.499220 + 0.866475i \(0.666380\pi\)
\(600\) −443222. −0.0502624
\(601\) 1.49619e7 1.68966 0.844831 0.535034i \(-0.179701\pi\)
0.844831 + 0.535034i \(0.179701\pi\)
\(602\) −307156. −0.0345436
\(603\) 1.05757e7 1.18445
\(604\) 7.10779e6 0.792760
\(605\) 3.09890e6 0.344206
\(606\) −1.03998e6 −0.115039
\(607\) 7.03290e6 0.774752 0.387376 0.921922i \(-0.373382\pi\)
0.387376 + 0.921922i \(0.373382\pi\)
\(608\) −3.52606e6 −0.386840
\(609\) 218607. 0.0238847
\(610\) −857182. −0.0932714
\(611\) 5.94159e6 0.643873
\(612\) 1.69080e6 0.182479
\(613\) −806845. −0.0867239 −0.0433620 0.999059i \(-0.513807\pi\)
−0.0433620 + 0.999059i \(0.513807\pi\)
\(614\) 2.32439e6 0.248822
\(615\) 685065. 0.0730372
\(616\) 606777. 0.0644284
\(617\) 7.36483e6 0.778843 0.389421 0.921060i \(-0.372675\pi\)
0.389421 + 0.921060i \(0.372675\pi\)
\(618\) −512970. −0.0540283
\(619\) −1.30528e7 −1.36923 −0.684617 0.728903i \(-0.740031\pi\)
−0.684617 + 0.728903i \(0.740031\pi\)
\(620\) 2.04620e6 0.213781
\(621\) −5.73967e6 −0.597253
\(622\) 1.62842e6 0.168768
\(623\) −94224.4 −0.00972620
\(624\) −2.51769e6 −0.258846
\(625\) −4.86615e6 −0.498294
\(626\) 2.14305e6 0.218573
\(627\) −2.93261e6 −0.297910
\(628\) −1.68548e7 −1.70539
\(629\) −3.28880e6 −0.331445
\(630\) 272339. 0.0273375
\(631\) 2.89105e6 0.289056 0.144528 0.989501i \(-0.453834\pi\)
0.144528 + 0.989501i \(0.453834\pi\)
\(632\) −3.02252e6 −0.301007
\(633\) −6.50063e6 −0.644831
\(634\) −176594. −0.0174482
\(635\) 659304. 0.0648860
\(636\) −5.38573e6 −0.527961
\(637\) 5.77270e6 0.563677
\(638\) −219554. −0.0213545
\(639\) −5.36234e6 −0.519520
\(640\) −4.74532e6 −0.457947
\(641\) −1.31397e7 −1.26311 −0.631555 0.775331i \(-0.717583\pi\)
−0.631555 + 0.775331i \(0.717583\pi\)
\(642\) −102899. −0.00985307
\(643\) −1.17266e7 −1.11852 −0.559259 0.828993i \(-0.688914\pi\)
−0.559259 + 0.828993i \(0.688914\pi\)
\(644\) 2.13734e6 0.203076
\(645\) 3.04529e6 0.288223
\(646\) −332065. −0.0313071
\(647\) 1.10662e7 1.03929 0.519646 0.854382i \(-0.326064\pi\)
0.519646 + 0.854382i \(0.326064\pi\)
\(648\) 1.36964e6 0.128135
\(649\) −1.09743e7 −1.02274
\(650\) −369929. −0.0343427
\(651\) −382635. −0.0353860
\(652\) −2.75025e6 −0.253369
\(653\) −3.44597e6 −0.316249 −0.158124 0.987419i \(-0.550545\pi\)
−0.158124 + 0.987419i \(0.550545\pi\)
\(654\) 23555.9 0.00215355
\(655\) 1.39285e7 1.26853
\(656\) −2.07495e6 −0.188255
\(657\) −6.11392e6 −0.552594
\(658\) −504860. −0.0454576
\(659\) −9.12181e6 −0.818215 −0.409108 0.912486i \(-0.634160\pi\)
−0.409108 + 0.912486i \(0.634160\pi\)
\(660\) −2.97247e6 −0.265618
\(661\) 1.68113e7 1.49657 0.748287 0.663375i \(-0.230877\pi\)
0.748287 + 0.663375i \(0.230877\pi\)
\(662\) −1.37145e6 −0.121629
\(663\) −740723. −0.0654443
\(664\) −2.56147e6 −0.225460
\(665\) 2.24123e6 0.196531
\(666\) −1.95481e6 −0.170773
\(667\) −1.56519e6 −0.136224
\(668\) −6.24760e6 −0.541716
\(669\) 3.43897e6 0.297073
\(670\) −2.10525e6 −0.181183
\(671\) 6.74577e6 0.578396
\(672\) 668330. 0.0570910
\(673\) 8.53511e6 0.726393 0.363196 0.931713i \(-0.381685\pi\)
0.363196 + 0.931713i \(0.381685\pi\)
\(674\) 702670. 0.0595802
\(675\) 3.53708e6 0.298803
\(676\) 7.24506e6 0.609783
\(677\) −912690. −0.0765335 −0.0382667 0.999268i \(-0.512184\pi\)
−0.0382667 + 0.999268i \(0.512184\pi\)
\(678\) −655665. −0.0547782
\(679\) −1.28387e6 −0.106868
\(680\) −681191. −0.0564932
\(681\) −3.94481e6 −0.325955
\(682\) 384292. 0.0316374
\(683\) −4.20567e6 −0.344972 −0.172486 0.985012i \(-0.555180\pi\)
−0.172486 + 0.985012i \(0.555180\pi\)
\(684\) 8.27055e6 0.675918
\(685\) 1.13107e7 0.921010
\(686\) −1.02386e6 −0.0830677
\(687\) 3.11466e6 0.251779
\(688\) −9.22366e6 −0.742903
\(689\) −9.09751e6 −0.730087
\(690\) 505707. 0.0404367
\(691\) 6.23030e6 0.496379 0.248190 0.968711i \(-0.420164\pi\)
0.248190 + 0.968711i \(0.420164\pi\)
\(692\) 9.80746e6 0.778559
\(693\) −2.14323e6 −0.169525
\(694\) −445670. −0.0351249
\(695\) 1.73870e7 1.36541
\(696\) −325005. −0.0254312
\(697\) −610464. −0.0475968
\(698\) 2.29005e6 0.177913
\(699\) 1.03412e7 0.800527
\(700\) −1.31713e6 −0.101598
\(701\) 1.69453e7 1.30243 0.651215 0.758893i \(-0.274259\pi\)
0.651215 + 0.758893i \(0.274259\pi\)
\(702\) −994733. −0.0761840
\(703\) −1.60872e7 −1.22770
\(704\) 8.54673e6 0.649933
\(705\) 5.00542e6 0.379287
\(706\) −337961. −0.0255185
\(707\) −6.25484e6 −0.470617
\(708\) −8.02680e6 −0.601810
\(709\) −1.87326e7 −1.39953 −0.699766 0.714372i \(-0.746712\pi\)
−0.699766 + 0.714372i \(0.746712\pi\)
\(710\) 1.06746e6 0.0794702
\(711\) 1.06760e7 0.792016
\(712\) 140084. 0.0103559
\(713\) 2.73960e6 0.201820
\(714\) 62939.6 0.00462039
\(715\) −5.02106e6 −0.367308
\(716\) 1.86946e6 0.136280
\(717\) 1.07026e7 0.777485
\(718\) −2.62536e6 −0.190054
\(719\) 2.15611e6 0.155542 0.0777711 0.996971i \(-0.475220\pi\)
0.0777711 + 0.996971i \(0.475220\pi\)
\(720\) 8.17814e6 0.587927
\(721\) −3.08519e6 −0.221026
\(722\) 514154. 0.0367071
\(723\) −5.66410e6 −0.402982
\(724\) −1.25714e7 −0.891326
\(725\) 964548. 0.0681520
\(726\) 425686. 0.0299742
\(727\) 4.87466e6 0.342065 0.171032 0.985265i \(-0.445290\pi\)
0.171032 + 0.985265i \(0.445290\pi\)
\(728\) 749677. 0.0524259
\(729\) 463697. 0.0323159
\(730\) 1.21707e6 0.0845295
\(731\) −2.71367e6 −0.187829
\(732\) 4.93399e6 0.340346
\(733\) 5.15920e6 0.354669 0.177334 0.984151i \(-0.443253\pi\)
0.177334 + 0.984151i \(0.443253\pi\)
\(734\) −888660. −0.0608829
\(735\) 4.86314e6 0.332046
\(736\) −4.78513e6 −0.325611
\(737\) 1.65677e7 1.12355
\(738\) −362850. −0.0245237
\(739\) −4.60821e6 −0.310400 −0.155200 0.987883i \(-0.549602\pi\)
−0.155200 + 0.987883i \(0.549602\pi\)
\(740\) −1.63059e7 −1.09462
\(741\) −3.62325e6 −0.242411
\(742\) 773020. 0.0515444
\(743\) 2.50473e7 1.66452 0.832261 0.554384i \(-0.187046\pi\)
0.832261 + 0.554384i \(0.187046\pi\)
\(744\) 568866. 0.0376771
\(745\) −3.21763e6 −0.212396
\(746\) 3.82037e6 0.251338
\(747\) 9.04751e6 0.593236
\(748\) 2.64878e6 0.173098
\(749\) −618870. −0.0403083
\(750\) −1.16078e6 −0.0753524
\(751\) −1.87279e7 −1.21168 −0.605841 0.795586i \(-0.707163\pi\)
−0.605841 + 0.795586i \(0.707163\pi\)
\(752\) −1.51606e7 −0.977623
\(753\) 1.17447e7 0.754839
\(754\) −271260. −0.0173763
\(755\) 1.01146e7 0.645778
\(756\) −3.54176e6 −0.225379
\(757\) −5.50814e6 −0.349354 −0.174677 0.984626i \(-0.555888\pi\)
−0.174677 + 0.984626i \(0.555888\pi\)
\(758\) 3.65750e6 0.231212
\(759\) −3.97976e6 −0.250757
\(760\) −3.33205e6 −0.209256
\(761\) −2.67826e7 −1.67645 −0.838226 0.545323i \(-0.816407\pi\)
−0.838226 + 0.545323i \(0.816407\pi\)
\(762\) 90566.4 0.00565040
\(763\) 141674. 0.00881005
\(764\) 2.82390e7 1.75032
\(765\) 2.40607e6 0.148646
\(766\) 772511. 0.0475699
\(767\) −1.35588e7 −0.832209
\(768\) 5.74859e6 0.351689
\(769\) 1.76374e7 1.07552 0.537761 0.843097i \(-0.319270\pi\)
0.537761 + 0.843097i \(0.319270\pi\)
\(770\) 426642. 0.0259320
\(771\) −7.12526e6 −0.431683
\(772\) −3.21512e7 −1.94158
\(773\) −6.59079e6 −0.396724 −0.198362 0.980129i \(-0.563562\pi\)
−0.198362 + 0.980129i \(0.563562\pi\)
\(774\) −1.61296e6 −0.0967768
\(775\) −1.68828e6 −0.100970
\(776\) 1.90874e6 0.113787
\(777\) 3.04917e6 0.181188
\(778\) −3.11463e6 −0.184484
\(779\) −2.98609e6 −0.176303
\(780\) −3.67250e6 −0.216135
\(781\) −8.40056e6 −0.492811
\(782\) −450637. −0.0263518
\(783\) 2.59366e6 0.151185
\(784\) −1.47296e7 −0.855858
\(785\) −2.39849e7 −1.38920
\(786\) 1.91331e6 0.110466
\(787\) 1.83730e7 1.05741 0.528704 0.848806i \(-0.322678\pi\)
0.528704 + 0.848806i \(0.322678\pi\)
\(788\) 2.70692e7 1.55296
\(789\) 7.16792e6 0.409922
\(790\) −2.12522e6 −0.121153
\(791\) −3.94341e6 −0.224094
\(792\) 3.18635e6 0.180502
\(793\) 8.33444e6 0.470645
\(794\) 1.69441e6 0.0953824
\(795\) −7.66409e6 −0.430074
\(796\) −8.93356e6 −0.499737
\(797\) −3.06769e7 −1.71067 −0.855334 0.518078i \(-0.826648\pi\)
−0.855334 + 0.518078i \(0.826648\pi\)
\(798\) 307870. 0.0171143
\(799\) −4.46035e6 −0.247174
\(800\) 2.94884e6 0.162902
\(801\) −494798. −0.0272488
\(802\) 622784. 0.0341902
\(803\) −9.57797e6 −0.524185
\(804\) 1.21180e7 0.661134
\(805\) 3.04151e6 0.165424
\(806\) 474796. 0.0257436
\(807\) −1.02941e7 −0.556423
\(808\) 9.29912e6 0.501088
\(809\) −2.81720e6 −0.151338 −0.0756688 0.997133i \(-0.524109\pi\)
−0.0756688 + 0.997133i \(0.524109\pi\)
\(810\) 963031. 0.0515736
\(811\) −1.57819e7 −0.842574 −0.421287 0.906927i \(-0.638421\pi\)
−0.421287 + 0.906927i \(0.638421\pi\)
\(812\) −965825. −0.0514053
\(813\) −1.13997e7 −0.604877
\(814\) −3.06238e6 −0.161994
\(815\) −3.91371e6 −0.206393
\(816\) 1.89003e6 0.0993673
\(817\) −1.32739e7 −0.695735
\(818\) −1.15195e6 −0.0601935
\(819\) −2.64797e6 −0.137944
\(820\) −3.02668e6 −0.157192
\(821\) −2.40563e6 −0.124558 −0.0622789 0.998059i \(-0.519837\pi\)
−0.0622789 + 0.998059i \(0.519837\pi\)
\(822\) 1.55372e6 0.0802034
\(823\) 4.47397e6 0.230247 0.115123 0.993351i \(-0.463274\pi\)
0.115123 + 0.993351i \(0.463274\pi\)
\(824\) 4.58678e6 0.235337
\(825\) 2.45253e6 0.125453
\(826\) 1.15210e6 0.0587542
\(827\) 2.36054e7 1.20018 0.600091 0.799932i \(-0.295131\pi\)
0.600091 + 0.799932i \(0.295131\pi\)
\(828\) 1.12237e7 0.568934
\(829\) −7.52668e6 −0.380380 −0.190190 0.981747i \(-0.560910\pi\)
−0.190190 + 0.981747i \(0.560910\pi\)
\(830\) −1.80105e6 −0.0907464
\(831\) 1.45981e7 0.733319
\(832\) 1.05595e7 0.528855
\(833\) −4.33356e6 −0.216388
\(834\) 2.38839e6 0.118902
\(835\) −8.89056e6 −0.441279
\(836\) 1.29565e7 0.641169
\(837\) −4.53976e6 −0.223985
\(838\) 3.89754e6 0.191726
\(839\) −7.28878e6 −0.357478 −0.178739 0.983896i \(-0.557202\pi\)
−0.178739 + 0.983896i \(0.557202\pi\)
\(840\) 631556. 0.0308826
\(841\) 707281. 0.0344828
\(842\) −848249. −0.0412328
\(843\) 153259. 0.00742773
\(844\) 2.87204e7 1.38782
\(845\) 1.03100e7 0.496725
\(846\) −2.65116e6 −0.127353
\(847\) 2.56023e6 0.122623
\(848\) 2.32132e7 1.10853
\(849\) −3.91005e6 −0.186172
\(850\) 277705. 0.0131837
\(851\) −2.18315e7 −1.03338
\(852\) −6.14434e6 −0.289985
\(853\) 2.18767e7 1.02946 0.514729 0.857353i \(-0.327893\pi\)
0.514729 + 0.857353i \(0.327893\pi\)
\(854\) −708182. −0.0332277
\(855\) 1.17693e7 0.550599
\(856\) 920079. 0.0429181
\(857\) −1.62738e7 −0.756897 −0.378448 0.925622i \(-0.623542\pi\)
−0.378448 + 0.925622i \(0.623542\pi\)
\(858\) −689726. −0.0319859
\(859\) 1.50220e7 0.694615 0.347308 0.937751i \(-0.387096\pi\)
0.347308 + 0.937751i \(0.387096\pi\)
\(860\) −1.34544e7 −0.620321
\(861\) 565983. 0.0260193
\(862\) −4.59269e6 −0.210523
\(863\) 2.24278e7 1.02508 0.512542 0.858662i \(-0.328704\pi\)
0.512542 + 0.858662i \(0.328704\pi\)
\(864\) 7.92938e6 0.361372
\(865\) 1.39564e7 0.634209
\(866\) −586650. −0.0265818
\(867\) −9.48819e6 −0.428682
\(868\) 1.69051e6 0.0761587
\(869\) 1.67248e7 0.751298
\(870\) −228520. −0.0102359
\(871\) 2.04695e7 0.914244
\(872\) −210628. −0.00938047
\(873\) −6.74196e6 −0.299399
\(874\) −2.20430e6 −0.0976093
\(875\) −6.98136e6 −0.308262
\(876\) −7.00552e6 −0.308447
\(877\) 1.25450e7 0.550771 0.275385 0.961334i \(-0.411195\pi\)
0.275385 + 0.961334i \(0.411195\pi\)
\(878\) −3.05399e6 −0.133700
\(879\) −1.08590e7 −0.474043
\(880\) 1.28117e7 0.557701
\(881\) 7.60341e6 0.330042 0.165021 0.986290i \(-0.447231\pi\)
0.165021 + 0.986290i \(0.447231\pi\)
\(882\) −2.57580e6 −0.111491
\(883\) 2.15220e7 0.928923 0.464462 0.885593i \(-0.346248\pi\)
0.464462 + 0.885593i \(0.346248\pi\)
\(884\) 3.27258e6 0.140851
\(885\) −1.14224e7 −0.490231
\(886\) 1.16848e6 0.0500079
\(887\) −333279. −0.0142232 −0.00711162 0.999975i \(-0.502264\pi\)
−0.00711162 + 0.999975i \(0.502264\pi\)
\(888\) −4.53322e6 −0.192919
\(889\) 544700. 0.0231155
\(890\) 98497.2 0.00416820
\(891\) −7.57876e6 −0.319819
\(892\) −1.51937e7 −0.639368
\(893\) −2.18178e7 −0.915552
\(894\) −441995. −0.0184958
\(895\) 2.66031e6 0.111013
\(896\) −3.92046e6 −0.163142
\(897\) −4.91702e6 −0.204043
\(898\) 1.76661e6 0.0731055
\(899\) −1.23798e6 −0.0510873
\(900\) −6.91664e6 −0.284635
\(901\) 6.82950e6 0.280270
\(902\) −568435. −0.0232629
\(903\) 2.51594e6 0.102679
\(904\) 5.86270e6 0.238603
\(905\) −1.78895e7 −0.726068
\(906\) 1.38941e6 0.0562356
\(907\) 1.65687e7 0.668759 0.334380 0.942439i \(-0.391473\pi\)
0.334380 + 0.942439i \(0.391473\pi\)
\(908\) 1.74285e7 0.701529
\(909\) −3.28459e7 −1.31847
\(910\) 527119. 0.0211011
\(911\) 1.30648e7 0.521561 0.260781 0.965398i \(-0.416020\pi\)
0.260781 + 0.965398i \(0.416020\pi\)
\(912\) 9.24510e6 0.368065
\(913\) 1.41737e7 0.562737
\(914\) −7.32693e6 −0.290106
\(915\) 7.02125e6 0.277244
\(916\) −1.37608e7 −0.541884
\(917\) 1.15074e7 0.451911
\(918\) 746746. 0.0292460
\(919\) −7.30805e6 −0.285439 −0.142719 0.989763i \(-0.545585\pi\)
−0.142719 + 0.989763i \(0.545585\pi\)
\(920\) −4.52184e6 −0.176135
\(921\) −1.90393e7 −0.739608
\(922\) 1.78596e6 0.0691902
\(923\) −1.03789e7 −0.401004
\(924\) −2.45578e6 −0.0946256
\(925\) 1.34537e7 0.516996
\(926\) 318069. 0.0121897
\(927\) −1.62012e7 −0.619224
\(928\) 2.16232e6 0.0824231
\(929\) 2.69866e7 1.02591 0.512955 0.858416i \(-0.328551\pi\)
0.512955 + 0.858416i \(0.328551\pi\)
\(930\) 399986. 0.0151648
\(931\) −2.11976e7 −0.801518
\(932\) −4.56881e7 −1.72291
\(933\) −1.33385e7 −0.501652
\(934\) 2.40407e6 0.0901737
\(935\) 3.76930e6 0.141004
\(936\) 3.93676e6 0.146876
\(937\) −2.15536e7 −0.801993 −0.400997 0.916080i \(-0.631336\pi\)
−0.400997 + 0.916080i \(0.631336\pi\)
\(938\) −1.73930e6 −0.0645459
\(939\) −1.75539e7 −0.649694
\(940\) −2.21144e7 −0.816311
\(941\) −1.22668e7 −0.451602 −0.225801 0.974173i \(-0.572500\pi\)
−0.225801 + 0.974173i \(0.572500\pi\)
\(942\) −3.29473e6 −0.120974
\(943\) −4.05234e6 −0.148398
\(944\) 3.45966e7 1.26358
\(945\) −5.04005e6 −0.183593
\(946\) −2.52684e6 −0.0918014
\(947\) 5.13280e7 1.85986 0.929928 0.367741i \(-0.119869\pi\)
0.929928 + 0.367741i \(0.119869\pi\)
\(948\) 1.22329e7 0.442087
\(949\) −1.18336e7 −0.426533
\(950\) 1.35840e6 0.0488335
\(951\) 1.44649e6 0.0518638
\(952\) −562782. −0.0201255
\(953\) −3.30301e7 −1.17809 −0.589044 0.808101i \(-0.700496\pi\)
−0.589044 + 0.808101i \(0.700496\pi\)
\(954\) 4.05934e6 0.144406
\(955\) 4.01852e7 1.42580
\(956\) −4.72851e7 −1.67332
\(957\) 1.79838e6 0.0634750
\(958\) −1.29390e6 −0.0455498
\(959\) 9.34463e6 0.328107
\(960\) 8.89576e6 0.311534
\(961\) −2.64623e7 −0.924312
\(962\) −3.78359e6 −0.131815
\(963\) −3.24986e6 −0.112927
\(964\) 2.50245e7 0.867307
\(965\) −4.57524e7 −1.58160
\(966\) 417802. 0.0144055
\(967\) −7.77281e6 −0.267308 −0.133654 0.991028i \(-0.542671\pi\)
−0.133654 + 0.991028i \(0.542671\pi\)
\(968\) −3.80632e6 −0.130562
\(969\) 2.71997e6 0.0930583
\(970\) 1.34209e6 0.0457986
\(971\) 2.31111e7 0.786635 0.393317 0.919403i \(-0.371327\pi\)
0.393317 + 0.919403i \(0.371327\pi\)
\(972\) −2.89656e7 −0.983369
\(973\) 1.43647e7 0.486422
\(974\) −6.87175e6 −0.232097
\(975\) 3.03012e6 0.102082
\(976\) −2.12662e7 −0.714603
\(977\) 1.04792e7 0.351231 0.175616 0.984459i \(-0.443808\pi\)
0.175616 + 0.984459i \(0.443808\pi\)
\(978\) −537613. −0.0179731
\(979\) −775143. −0.0258479
\(980\) −2.14858e7 −0.714638
\(981\) 743969. 0.0246821
\(982\) 1.61796e6 0.0535413
\(983\) 3.94189e7 1.30113 0.650565 0.759450i \(-0.274532\pi\)
0.650565 + 0.759450i \(0.274532\pi\)
\(984\) −841452. −0.0277039
\(985\) 3.85204e7 1.26503
\(986\) 203635. 0.00667052
\(987\) 4.13535e6 0.135120
\(988\) 1.60079e7 0.521724
\(989\) −1.80137e7 −0.585615
\(990\) 2.24042e6 0.0726508
\(991\) −4.53294e7 −1.46621 −0.733104 0.680117i \(-0.761929\pi\)
−0.733104 + 0.680117i \(0.761929\pi\)
\(992\) −3.78477e6 −0.122113
\(993\) 1.12337e7 0.361534
\(994\) 881905. 0.0283110
\(995\) −1.27128e7 −0.407083
\(996\) 1.03669e7 0.331132
\(997\) 4.74723e7 1.51253 0.756263 0.654268i \(-0.227023\pi\)
0.756263 + 0.654268i \(0.227023\pi\)
\(998\) 304247. 0.00966942
\(999\) 3.61768e7 1.14687
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.1.2 4
3.2 odd 2 261.6.a.a.1.3 4
4.3 odd 2 464.6.a.i.1.1 4
5.4 even 2 725.6.a.a.1.3 4
29.28 even 2 841.6.a.a.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.2 4 1.1 even 1 trivial
261.6.a.a.1.3 4 3.2 odd 2
464.6.a.i.1.1 4 4.3 odd 2
725.6.a.a.1.3 4 5.4 even 2
841.6.a.a.1.3 4 29.28 even 2