Properties

Label 29.6.a.a.1.3
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 1.3
Root \(-1.10057\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.16235 q^{2} -17.5258 q^{3} -14.6748 q^{4} +32.5670 q^{5} -72.9487 q^{6} -220.793 q^{7} -194.277 q^{8} +64.1549 q^{9} +O(q^{10})\) \(q+4.16235 q^{2} -17.5258 q^{3} -14.6748 q^{4} +32.5670 q^{5} -72.9487 q^{6} -220.793 q^{7} -194.277 q^{8} +64.1549 q^{9} +135.555 q^{10} -85.7296 q^{11} +257.189 q^{12} +1034.02 q^{13} -919.018 q^{14} -570.765 q^{15} -339.054 q^{16} -313.020 q^{17} +267.035 q^{18} +458.534 q^{19} -477.916 q^{20} +3869.58 q^{21} -356.836 q^{22} -3448.84 q^{23} +3404.87 q^{24} -2064.39 q^{25} +4303.94 q^{26} +3134.41 q^{27} +3240.10 q^{28} -841.000 q^{29} -2375.72 q^{30} -7983.23 q^{31} +4805.60 q^{32} +1502.48 q^{33} -1302.90 q^{34} -7190.58 q^{35} -941.463 q^{36} +152.624 q^{37} +1908.58 q^{38} -18122.0 q^{39} -6327.03 q^{40} -18492.2 q^{41} +16106.6 q^{42} +2072.84 q^{43} +1258.07 q^{44} +2089.33 q^{45} -14355.3 q^{46} +15845.6 q^{47} +5942.21 q^{48} +31942.6 q^{49} -8592.70 q^{50} +5485.94 q^{51} -15174.0 q^{52} +9240.52 q^{53} +13046.5 q^{54} -2791.96 q^{55} +42895.0 q^{56} -8036.19 q^{57} -3500.54 q^{58} -14323.2 q^{59} +8375.88 q^{60} -19580.2 q^{61} -33229.0 q^{62} -14165.0 q^{63} +30852.3 q^{64} +33674.9 q^{65} +6253.86 q^{66} -9193.70 q^{67} +4593.53 q^{68} +60443.8 q^{69} -29929.7 q^{70} -19374.7 q^{71} -12463.8 q^{72} -56912.4 q^{73} +635.276 q^{74} +36180.1 q^{75} -6728.91 q^{76} +18928.5 q^{77} -75430.1 q^{78} +51573.6 q^{79} -11042.0 q^{80} -70522.8 q^{81} -76971.0 q^{82} +19978.1 q^{83} -56785.5 q^{84} -10194.2 q^{85} +8627.87 q^{86} +14739.2 q^{87} +16655.3 q^{88} +130663. q^{89} +8696.54 q^{90} -228304. q^{91} +50611.1 q^{92} +139913. q^{93} +65954.9 q^{94} +14933.1 q^{95} -84222.2 q^{96} +43603.5 q^{97} +132956. q^{98} -5499.97 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.16235 0.735806 0.367903 0.929864i \(-0.380076\pi\)
0.367903 + 0.929864i \(0.380076\pi\)
\(3\) −17.5258 −1.12428 −0.562141 0.827041i \(-0.690022\pi\)
−0.562141 + 0.827041i \(0.690022\pi\)
\(4\) −14.6748 −0.458589
\(5\) 32.5670 0.582577 0.291289 0.956635i \(-0.405916\pi\)
0.291289 + 0.956635i \(0.405916\pi\)
\(6\) −72.9487 −0.827255
\(7\) −220.793 −1.70310 −0.851551 0.524272i \(-0.824337\pi\)
−0.851551 + 0.524272i \(0.824337\pi\)
\(8\) −194.277 −1.07324
\(9\) 64.1549 0.264012
\(10\) 135.555 0.428664
\(11\) −85.7296 −0.213623 −0.106812 0.994279i \(-0.534064\pi\)
−0.106812 + 0.994279i \(0.534064\pi\)
\(12\) 257.189 0.515584
\(13\) 1034.02 1.69695 0.848475 0.529235i \(-0.177521\pi\)
0.848475 + 0.529235i \(0.177521\pi\)
\(14\) −919.018 −1.25315
\(15\) −570.765 −0.654981
\(16\) −339.054 −0.331108
\(17\) −313.020 −0.262694 −0.131347 0.991336i \(-0.541930\pi\)
−0.131347 + 0.991336i \(0.541930\pi\)
\(18\) 267.035 0.194262
\(19\) 458.534 0.291398 0.145699 0.989329i \(-0.453457\pi\)
0.145699 + 0.989329i \(0.453457\pi\)
\(20\) −477.916 −0.267163
\(21\) 3869.58 1.91477
\(22\) −356.836 −0.157186
\(23\) −3448.84 −1.35942 −0.679709 0.733482i \(-0.737894\pi\)
−0.679709 + 0.733482i \(0.737894\pi\)
\(24\) 3404.87 1.20662
\(25\) −2064.39 −0.660604
\(26\) 4303.94 1.24863
\(27\) 3134.41 0.827459
\(28\) 3240.10 0.781023
\(29\) −841.000 −0.185695
\(30\) −2375.72 −0.481940
\(31\) −7983.23 −1.49202 −0.746009 0.665936i \(-0.768033\pi\)
−0.746009 + 0.665936i \(0.768033\pi\)
\(32\) 4805.60 0.829608
\(33\) 1502.48 0.240173
\(34\) −1302.90 −0.193292
\(35\) −7190.58 −0.992188
\(36\) −941.463 −0.121073
\(37\) 152.624 0.0183282 0.00916409 0.999958i \(-0.497083\pi\)
0.00916409 + 0.999958i \(0.497083\pi\)
\(38\) 1908.58 0.214413
\(39\) −18122.0 −1.90785
\(40\) −6327.03 −0.625244
\(41\) −18492.2 −1.71802 −0.859011 0.511957i \(-0.828921\pi\)
−0.859011 + 0.511957i \(0.828921\pi\)
\(42\) 16106.6 1.40890
\(43\) 2072.84 0.170960 0.0854799 0.996340i \(-0.472758\pi\)
0.0854799 + 0.996340i \(0.472758\pi\)
\(44\) 1258.07 0.0979653
\(45\) 2089.33 0.153807
\(46\) −14355.3 −1.00027
\(47\) 15845.6 1.04632 0.523159 0.852235i \(-0.324753\pi\)
0.523159 + 0.852235i \(0.324753\pi\)
\(48\) 5942.21 0.372258
\(49\) 31942.6 1.90055
\(50\) −8592.70 −0.486077
\(51\) 5485.94 0.295343
\(52\) −15174.0 −0.778203
\(53\) 9240.52 0.451863 0.225931 0.974143i \(-0.427457\pi\)
0.225931 + 0.974143i \(0.427457\pi\)
\(54\) 13046.5 0.608850
\(55\) −2791.96 −0.124452
\(56\) 42895.0 1.82783
\(57\) −8036.19 −0.327614
\(58\) −3500.54 −0.136636
\(59\) −14323.2 −0.535686 −0.267843 0.963463i \(-0.586311\pi\)
−0.267843 + 0.963463i \(0.586311\pi\)
\(60\) 8375.88 0.300367
\(61\) −19580.2 −0.673739 −0.336870 0.941551i \(-0.609368\pi\)
−0.336870 + 0.941551i \(0.609368\pi\)
\(62\) −33229.0 −1.09784
\(63\) −14165.0 −0.449639
\(64\) 30852.3 0.941539
\(65\) 33674.9 0.988604
\(66\) 6253.86 0.176721
\(67\) −9193.70 −0.250209 −0.125105 0.992144i \(-0.539927\pi\)
−0.125105 + 0.992144i \(0.539927\pi\)
\(68\) 4593.53 0.120469
\(69\) 60443.8 1.52837
\(70\) −29929.7 −0.730058
\(71\) −19374.7 −0.456131 −0.228066 0.973646i \(-0.573240\pi\)
−0.228066 + 0.973646i \(0.573240\pi\)
\(72\) −12463.8 −0.283348
\(73\) −56912.4 −1.24997 −0.624985 0.780637i \(-0.714895\pi\)
−0.624985 + 0.780637i \(0.714895\pi\)
\(74\) 635.276 0.0134860
\(75\) 36180.1 0.742706
\(76\) −6728.91 −0.133632
\(77\) 18928.5 0.363822
\(78\) −75430.1 −1.40381
\(79\) 51573.6 0.929736 0.464868 0.885380i \(-0.346102\pi\)
0.464868 + 0.885380i \(0.346102\pi\)
\(80\) −11042.0 −0.192896
\(81\) −70522.8 −1.19431
\(82\) −76971.0 −1.26413
\(83\) 19978.1 0.318317 0.159158 0.987253i \(-0.449122\pi\)
0.159158 + 0.987253i \(0.449122\pi\)
\(84\) −56785.5 −0.878091
\(85\) −10194.2 −0.153040
\(86\) 8627.87 0.125793
\(87\) 14739.2 0.208774
\(88\) 16655.3 0.229269
\(89\) 130663. 1.74855 0.874274 0.485432i \(-0.161338\pi\)
0.874274 + 0.485432i \(0.161338\pi\)
\(90\) 8696.54 0.113172
\(91\) −228304. −2.89008
\(92\) 50611.1 0.623414
\(93\) 139913. 1.67745
\(94\) 65954.9 0.769888
\(95\) 14933.1 0.169762
\(96\) −84222.2 −0.932714
\(97\) 43603.5 0.470535 0.235268 0.971931i \(-0.424403\pi\)
0.235268 + 0.971931i \(0.424403\pi\)
\(98\) 132956. 1.39844
\(99\) −5499.97 −0.0563991
\(100\) 30294.6 0.302946
\(101\) −56686.0 −0.552933 −0.276467 0.961024i \(-0.589164\pi\)
−0.276467 + 0.961024i \(0.589164\pi\)
\(102\) 22834.4 0.217315
\(103\) −111450. −1.03511 −0.517556 0.855649i \(-0.673158\pi\)
−0.517556 + 0.855649i \(0.673158\pi\)
\(104\) −200886. −1.82123
\(105\) 126021. 1.11550
\(106\) 38462.3 0.332484
\(107\) 189434. 1.59955 0.799775 0.600300i \(-0.204952\pi\)
0.799775 + 0.600300i \(0.204952\pi\)
\(108\) −45997.0 −0.379463
\(109\) 100232. 0.808054 0.404027 0.914747i \(-0.367610\pi\)
0.404027 + 0.914747i \(0.367610\pi\)
\(110\) −11621.1 −0.0915727
\(111\) −2674.87 −0.0206061
\(112\) 74860.8 0.563910
\(113\) −103444. −0.762092 −0.381046 0.924556i \(-0.624436\pi\)
−0.381046 + 0.924556i \(0.624436\pi\)
\(114\) −33449.4 −0.241061
\(115\) −112318. −0.791966
\(116\) 12341.5 0.0851578
\(117\) 66337.2 0.448015
\(118\) −59618.2 −0.394161
\(119\) 69112.8 0.447395
\(120\) 110886. 0.702952
\(121\) −153701. −0.954365
\(122\) −81499.5 −0.495742
\(123\) 324091. 1.93154
\(124\) 117153. 0.684223
\(125\) −169003. −0.967430
\(126\) −58959.5 −0.330847
\(127\) −247538. −1.36186 −0.680930 0.732349i \(-0.738424\pi\)
−0.680930 + 0.732349i \(0.738424\pi\)
\(128\) −25361.1 −0.136818
\(129\) −36328.2 −0.192207
\(130\) 140167. 0.727422
\(131\) 198458. 1.01040 0.505198 0.863004i \(-0.331420\pi\)
0.505198 + 0.863004i \(0.331420\pi\)
\(132\) −22048.7 −0.110141
\(133\) −101241. −0.496281
\(134\) −38267.4 −0.184106
\(135\) 102078. 0.482059
\(136\) 60812.7 0.281934
\(137\) 141442. 0.643839 0.321920 0.946767i \(-0.395672\pi\)
0.321920 + 0.946767i \(0.395672\pi\)
\(138\) 251588. 1.12459
\(139\) −325330. −1.42819 −0.714097 0.700047i \(-0.753162\pi\)
−0.714097 + 0.700047i \(0.753162\pi\)
\(140\) 105521. 0.455006
\(141\) −277707. −1.17636
\(142\) −80644.4 −0.335624
\(143\) −88645.8 −0.362508
\(144\) −21752.0 −0.0874163
\(145\) −27388.9 −0.108182
\(146\) −236889. −0.919736
\(147\) −559821. −2.13676
\(148\) −2239.74 −0.00840510
\(149\) −391462. −1.44452 −0.722261 0.691621i \(-0.756897\pi\)
−0.722261 + 0.691621i \(0.756897\pi\)
\(150\) 150594. 0.546488
\(151\) −216585. −0.773013 −0.386507 0.922287i \(-0.626318\pi\)
−0.386507 + 0.922287i \(0.626318\pi\)
\(152\) −89082.6 −0.312740
\(153\) −20081.8 −0.0693544
\(154\) 78787.0 0.267703
\(155\) −259990. −0.869216
\(156\) 265938. 0.874920
\(157\) 564396. 1.82740 0.913702 0.406385i \(-0.133211\pi\)
0.913702 + 0.406385i \(0.133211\pi\)
\(158\) 214667. 0.684106
\(159\) −161948. −0.508022
\(160\) 156504. 0.483311
\(161\) 761480. 2.31523
\(162\) −293541. −0.878781
\(163\) 476986. 1.40617 0.703083 0.711108i \(-0.251806\pi\)
0.703083 + 0.711108i \(0.251806\pi\)
\(164\) 271370. 0.787866
\(165\) 48931.4 0.139919
\(166\) 83155.9 0.234219
\(167\) −203397. −0.564357 −0.282178 0.959362i \(-0.591057\pi\)
−0.282178 + 0.959362i \(0.591057\pi\)
\(168\) −751771. −2.05500
\(169\) 697898. 1.87964
\(170\) −42431.6 −0.112608
\(171\) 29417.2 0.0769327
\(172\) −30418.6 −0.0784003
\(173\) −95409.0 −0.242367 −0.121184 0.992630i \(-0.538669\pi\)
−0.121184 + 0.992630i \(0.538669\pi\)
\(174\) 61349.8 0.153617
\(175\) 455803. 1.12508
\(176\) 29067.0 0.0707323
\(177\) 251026. 0.602263
\(178\) 543865. 1.28659
\(179\) −327011. −0.762833 −0.381416 0.924403i \(-0.624564\pi\)
−0.381416 + 0.924403i \(0.624564\pi\)
\(180\) −30660.7 −0.0705343
\(181\) 108581. 0.246353 0.123177 0.992385i \(-0.460692\pi\)
0.123177 + 0.992385i \(0.460692\pi\)
\(182\) −950280. −2.12654
\(183\) 343159. 0.757473
\(184\) 670030. 1.45898
\(185\) 4970.52 0.0106776
\(186\) 582366. 1.23428
\(187\) 26835.1 0.0561176
\(188\) −232532. −0.479830
\(189\) −692056. −1.40925
\(190\) 62156.7 0.124912
\(191\) −315738. −0.626244 −0.313122 0.949713i \(-0.601375\pi\)
−0.313122 + 0.949713i \(0.601375\pi\)
\(192\) −540713. −1.05856
\(193\) 41432.1 0.0800651 0.0400326 0.999198i \(-0.487254\pi\)
0.0400326 + 0.999198i \(0.487254\pi\)
\(194\) 181493. 0.346223
\(195\) −590180. −1.11147
\(196\) −468753. −0.871573
\(197\) 354739. 0.651243 0.325622 0.945500i \(-0.394426\pi\)
0.325622 + 0.945500i \(0.394426\pi\)
\(198\) −22892.8 −0.0414988
\(199\) −949129. −1.69900 −0.849498 0.527591i \(-0.823095\pi\)
−0.849498 + 0.527591i \(0.823095\pi\)
\(200\) 401063. 0.708986
\(201\) 161127. 0.281306
\(202\) −235947. −0.406852
\(203\) 185687. 0.316258
\(204\) −80505.4 −0.135441
\(205\) −602236. −1.00088
\(206\) −463894. −0.761642
\(207\) −221260. −0.358903
\(208\) −350588. −0.561873
\(209\) −39309.9 −0.0622495
\(210\) 524543. 0.820792
\(211\) −380574. −0.588481 −0.294241 0.955731i \(-0.595067\pi\)
−0.294241 + 0.955731i \(0.595067\pi\)
\(212\) −135603. −0.207219
\(213\) 339558. 0.512820
\(214\) 788489. 1.17696
\(215\) 67506.2 0.0995973
\(216\) −608944. −0.888061
\(217\) 1.76264e6 2.54106
\(218\) 417201. 0.594571
\(219\) 997437. 1.40532
\(220\) 40971.5 0.0570723
\(221\) −323668. −0.445779
\(222\) −11133.7 −0.0151621
\(223\) −68362.6 −0.0920569 −0.0460284 0.998940i \(-0.514656\pi\)
−0.0460284 + 0.998940i \(0.514656\pi\)
\(224\) −1.06104e6 −1.41291
\(225\) −132441. −0.174407
\(226\) −430569. −0.560753
\(227\) −1.07914e6 −1.39000 −0.694998 0.719012i \(-0.744595\pi\)
−0.694998 + 0.719012i \(0.744595\pi\)
\(228\) 117930. 0.150240
\(229\) −724221. −0.912604 −0.456302 0.889825i \(-0.650826\pi\)
−0.456302 + 0.889825i \(0.650826\pi\)
\(230\) −467509. −0.582734
\(231\) −331738. −0.409039
\(232\) 163387. 0.199296
\(233\) 1.37311e6 1.65697 0.828486 0.560009i \(-0.189203\pi\)
0.828486 + 0.560009i \(0.189203\pi\)
\(234\) 276119. 0.329652
\(235\) 516044. 0.609561
\(236\) 210191. 0.245660
\(237\) −903870. −1.04529
\(238\) 287672. 0.329196
\(239\) 1.11446e6 1.26204 0.631018 0.775769i \(-0.282638\pi\)
0.631018 + 0.775769i \(0.282638\pi\)
\(240\) 193520. 0.216869
\(241\) −1.47058e6 −1.63097 −0.815486 0.578776i \(-0.803530\pi\)
−0.815486 + 0.578776i \(0.803530\pi\)
\(242\) −639759. −0.702228
\(243\) 474309. 0.515283
\(244\) 287336. 0.308969
\(245\) 1.04028e6 1.10722
\(246\) 1.34898e6 1.42124
\(247\) 474132. 0.494489
\(248\) 1.55096e6 1.60129
\(249\) −350133. −0.357878
\(250\) −703450. −0.711841
\(251\) 375105. 0.375810 0.187905 0.982187i \(-0.439830\pi\)
0.187905 + 0.982187i \(0.439830\pi\)
\(252\) 207868. 0.206199
\(253\) 295667. 0.290404
\(254\) −1.03034e6 −1.00207
\(255\) 178661. 0.172060
\(256\) −1.09284e6 −1.04221
\(257\) −471074. −0.444894 −0.222447 0.974945i \(-0.571404\pi\)
−0.222447 + 0.974945i \(0.571404\pi\)
\(258\) −151211. −0.141427
\(259\) −33698.4 −0.0312147
\(260\) −494173. −0.453363
\(261\) −53954.3 −0.0490258
\(262\) 826053. 0.743455
\(263\) 1.11373e6 0.992865 0.496433 0.868075i \(-0.334643\pi\)
0.496433 + 0.868075i \(0.334643\pi\)
\(264\) −291898. −0.257763
\(265\) 300936. 0.263245
\(266\) −421401. −0.365167
\(267\) −2.28998e6 −1.96586
\(268\) 134916. 0.114743
\(269\) −381568. −0.321508 −0.160754 0.986995i \(-0.551393\pi\)
−0.160754 + 0.986995i \(0.551393\pi\)
\(270\) 424886. 0.354702
\(271\) 1.08834e6 0.900208 0.450104 0.892976i \(-0.351387\pi\)
0.450104 + 0.892976i \(0.351387\pi\)
\(272\) 106131. 0.0869800
\(273\) 4.00121e6 3.24927
\(274\) 588732. 0.473741
\(275\) 176979. 0.141120
\(276\) −887003. −0.700894
\(277\) −2.18958e6 −1.71459 −0.857295 0.514825i \(-0.827857\pi\)
−0.857295 + 0.514825i \(0.827857\pi\)
\(278\) −1.35414e6 −1.05087
\(279\) −512163. −0.393911
\(280\) 1.39696e6 1.06485
\(281\) −1.02712e6 −0.775992 −0.387996 0.921661i \(-0.626833\pi\)
−0.387996 + 0.921661i \(0.626833\pi\)
\(282\) −1.15592e6 −0.865572
\(283\) 889156. 0.659952 0.329976 0.943989i \(-0.392959\pi\)
0.329976 + 0.943989i \(0.392959\pi\)
\(284\) 284321. 0.209177
\(285\) −261715. −0.190861
\(286\) −368975. −0.266736
\(287\) 4.08295e6 2.92597
\(288\) 308303. 0.219026
\(289\) −1.32188e6 −0.930992
\(290\) −114002. −0.0796009
\(291\) −764188. −0.529015
\(292\) 835180. 0.573222
\(293\) −482000. −0.328003 −0.164001 0.986460i \(-0.552440\pi\)
−0.164001 + 0.986460i \(0.552440\pi\)
\(294\) −2.33017e6 −1.57224
\(295\) −466465. −0.312079
\(296\) −29651.4 −0.0196705
\(297\) −268712. −0.176765
\(298\) −1.62940e6 −1.06289
\(299\) −3.56616e6 −2.30687
\(300\) −530937. −0.340597
\(301\) −457668. −0.291162
\(302\) −901505. −0.568788
\(303\) 993470. 0.621653
\(304\) −155468. −0.0964842
\(305\) −637668. −0.392505
\(306\) −83587.4 −0.0510314
\(307\) 229158. 0.138768 0.0693839 0.997590i \(-0.477897\pi\)
0.0693839 + 0.997590i \(0.477897\pi\)
\(308\) −277773. −0.166845
\(309\) 1.95326e6 1.16376
\(310\) −1.08217e6 −0.639575
\(311\) 2.49699e6 1.46392 0.731958 0.681350i \(-0.238607\pi\)
0.731958 + 0.681350i \(0.238607\pi\)
\(312\) 3.52069e6 2.04758
\(313\) −2.78111e6 −1.60457 −0.802283 0.596944i \(-0.796381\pi\)
−0.802283 + 0.596944i \(0.796381\pi\)
\(314\) 2.34921e6 1.34462
\(315\) −461311. −0.261949
\(316\) −756834. −0.426366
\(317\) 1.94375e6 1.08641 0.543203 0.839602i \(-0.317212\pi\)
0.543203 + 0.839602i \(0.317212\pi\)
\(318\) −674083. −0.373806
\(319\) 72098.6 0.0396689
\(320\) 1.00477e6 0.548519
\(321\) −3.31998e6 −1.79835
\(322\) 3.16955e6 1.70356
\(323\) −143530. −0.0765487
\(324\) 1.03491e6 0.547697
\(325\) −2.13461e6 −1.12101
\(326\) 1.98538e6 1.03467
\(327\) −1.75665e6 −0.908481
\(328\) 3.59261e6 1.84385
\(329\) −3.49860e6 −1.78199
\(330\) 203670. 0.102954
\(331\) −60445.1 −0.0303243 −0.0151622 0.999885i \(-0.504826\pi\)
−0.0151622 + 0.999885i \(0.504826\pi\)
\(332\) −293176. −0.145976
\(333\) 9791.59 0.00483886
\(334\) −846611. −0.415258
\(335\) −299412. −0.145766
\(336\) −1.31200e6 −0.633994
\(337\) −1.72999e6 −0.829791 −0.414895 0.909869i \(-0.636182\pi\)
−0.414895 + 0.909869i \(0.636182\pi\)
\(338\) 2.90489e6 1.38305
\(339\) 1.81294e6 0.856807
\(340\) 149598. 0.0701822
\(341\) 684398. 0.318730
\(342\) 122445. 0.0566075
\(343\) −3.34184e6 −1.53373
\(344\) −402705. −0.183481
\(345\) 1.96847e6 0.890394
\(346\) −397126. −0.178335
\(347\) 1.17667e6 0.524605 0.262303 0.964986i \(-0.415518\pi\)
0.262303 + 0.964986i \(0.415518\pi\)
\(348\) −216296. −0.0957415
\(349\) 1.20893e6 0.531297 0.265648 0.964070i \(-0.414414\pi\)
0.265648 + 0.964070i \(0.414414\pi\)
\(350\) 1.89721e6 0.827838
\(351\) 3.24103e6 1.40416
\(352\) −411982. −0.177224
\(353\) 1.00723e6 0.430223 0.215111 0.976589i \(-0.430989\pi\)
0.215111 + 0.976589i \(0.430989\pi\)
\(354\) 1.04486e6 0.443149
\(355\) −630978. −0.265732
\(356\) −1.91746e6 −0.801865
\(357\) −1.21126e6 −0.502998
\(358\) −1.36113e6 −0.561297
\(359\) 1.10145e6 0.451055 0.225527 0.974237i \(-0.427589\pi\)
0.225527 + 0.974237i \(0.427589\pi\)
\(360\) −405910. −0.165072
\(361\) −2.26585e6 −0.915087
\(362\) 451953. 0.181268
\(363\) 2.69375e6 1.07298
\(364\) 3.35032e6 1.32536
\(365\) −1.85347e6 −0.728204
\(366\) 1.42835e6 0.557354
\(367\) 2.99288e6 1.15991 0.579955 0.814648i \(-0.303070\pi\)
0.579955 + 0.814648i \(0.303070\pi\)
\(368\) 1.16934e6 0.450114
\(369\) −1.18636e6 −0.453578
\(370\) 20689.0 0.00785663
\(371\) −2.04024e6 −0.769568
\(372\) −2.05320e6 −0.769260
\(373\) 4.23178e6 1.57489 0.787446 0.616384i \(-0.211403\pi\)
0.787446 + 0.616384i \(0.211403\pi\)
\(374\) 111697. 0.0412917
\(375\) 2.96192e6 1.08766
\(376\) −3.07844e6 −1.12295
\(377\) −869608. −0.315116
\(378\) −2.88058e6 −1.03693
\(379\) 1.06268e6 0.380018 0.190009 0.981782i \(-0.439148\pi\)
0.190009 + 0.981782i \(0.439148\pi\)
\(380\) −219141. −0.0778510
\(381\) 4.33831e6 1.53112
\(382\) −1.31421e6 −0.460794
\(383\) −3.75736e6 −1.30884 −0.654418 0.756133i \(-0.727087\pi\)
−0.654418 + 0.756133i \(0.727087\pi\)
\(384\) 444474. 0.153822
\(385\) 616445. 0.211955
\(386\) 172455. 0.0589125
\(387\) 132983. 0.0451354
\(388\) −639875. −0.215782
\(389\) −3.88561e6 −1.30192 −0.650961 0.759112i \(-0.725634\pi\)
−0.650961 + 0.759112i \(0.725634\pi\)
\(390\) −2.45654e6 −0.817828
\(391\) 1.07956e6 0.357111
\(392\) −6.20571e6 −2.03975
\(393\) −3.47815e6 −1.13597
\(394\) 1.47655e6 0.479189
\(395\) 1.67960e6 0.541643
\(396\) 80711.2 0.0258640
\(397\) 719537. 0.229127 0.114564 0.993416i \(-0.463453\pi\)
0.114564 + 0.993416i \(0.463453\pi\)
\(398\) −3.95061e6 −1.25013
\(399\) 1.77433e6 0.557960
\(400\) 699939. 0.218731
\(401\) −1.57519e6 −0.489185 −0.244592 0.969626i \(-0.578654\pi\)
−0.244592 + 0.969626i \(0.578654\pi\)
\(402\) 670668. 0.206987
\(403\) −8.25479e6 −2.53188
\(404\) 831859. 0.253569
\(405\) −2.29672e6 −0.695777
\(406\) 772894. 0.232705
\(407\) −13084.4 −0.00391533
\(408\) −1.06579e6 −0.316973
\(409\) −3.35355e6 −0.991279 −0.495639 0.868528i \(-0.665066\pi\)
−0.495639 + 0.868528i \(0.665066\pi\)
\(410\) −2.50672e6 −0.736454
\(411\) −2.47889e6 −0.723857
\(412\) 1.63551e6 0.474691
\(413\) 3.16247e6 0.912328
\(414\) −920961. −0.264083
\(415\) 650628. 0.185444
\(416\) 4.96907e6 1.40780
\(417\) 5.70168e6 1.60569
\(418\) −163622. −0.0458036
\(419\) 6.79853e6 1.89182 0.945911 0.324427i \(-0.105171\pi\)
0.945911 + 0.324427i \(0.105171\pi\)
\(420\) −1.84934e6 −0.511556
\(421\) −1.77259e6 −0.487420 −0.243710 0.969848i \(-0.578365\pi\)
−0.243710 + 0.969848i \(0.578365\pi\)
\(422\) −1.58408e6 −0.433008
\(423\) 1.01657e6 0.276241
\(424\) −1.79522e6 −0.484957
\(425\) 646196. 0.173537
\(426\) 1.41336e6 0.377337
\(427\) 4.32317e6 1.14745
\(428\) −2.77991e6 −0.733536
\(429\) 1.55359e6 0.407562
\(430\) 280984. 0.0732843
\(431\) 1.30221e6 0.337666 0.168833 0.985645i \(-0.446000\pi\)
0.168833 + 0.985645i \(0.446000\pi\)
\(432\) −1.06273e6 −0.273978
\(433\) 2.15386e6 0.552074 0.276037 0.961147i \(-0.410979\pi\)
0.276037 + 0.961147i \(0.410979\pi\)
\(434\) 7.33673e6 1.86973
\(435\) 480013. 0.121627
\(436\) −1.47089e6 −0.370564
\(437\) −1.58141e6 −0.396133
\(438\) 4.15168e6 1.03404
\(439\) −5.81916e6 −1.44112 −0.720558 0.693394i \(-0.756114\pi\)
−0.720558 + 0.693394i \(0.756114\pi\)
\(440\) 542413. 0.133567
\(441\) 2.04927e6 0.501769
\(442\) −1.34722e6 −0.328007
\(443\) 4.34332e6 1.05151 0.525754 0.850637i \(-0.323783\pi\)
0.525754 + 0.850637i \(0.323783\pi\)
\(444\) 39253.3 0.00944970
\(445\) 4.25531e6 1.01866
\(446\) −284549. −0.0677361
\(447\) 6.86070e6 1.62405
\(448\) −6.81198e6 −1.60354
\(449\) 2.68108e6 0.627615 0.313807 0.949487i \(-0.398395\pi\)
0.313807 + 0.949487i \(0.398395\pi\)
\(450\) −551264. −0.128330
\(451\) 1.58533e6 0.367010
\(452\) 1.51802e6 0.349487
\(453\) 3.79584e6 0.869086
\(454\) −4.49176e6 −1.02277
\(455\) −7.43518e6 −1.68369
\(456\) 1.56125e6 0.351609
\(457\) −4.29834e6 −0.962743 −0.481371 0.876517i \(-0.659861\pi\)
−0.481371 + 0.876517i \(0.659861\pi\)
\(458\) −3.01446e6 −0.671500
\(459\) −981134. −0.217369
\(460\) 1.64826e6 0.363187
\(461\) 2.28441e6 0.500636 0.250318 0.968164i \(-0.419465\pi\)
0.250318 + 0.968164i \(0.419465\pi\)
\(462\) −1.38081e6 −0.300974
\(463\) 1.51960e6 0.329440 0.164720 0.986340i \(-0.447328\pi\)
0.164720 + 0.986340i \(0.447328\pi\)
\(464\) 285144. 0.0614851
\(465\) 4.55654e6 0.977244
\(466\) 5.71536e6 1.21921
\(467\) −4.71846e6 −1.00117 −0.500585 0.865687i \(-0.666882\pi\)
−0.500585 + 0.865687i \(0.666882\pi\)
\(468\) −973488. −0.205455
\(469\) 2.02991e6 0.426132
\(470\) 2.14796e6 0.448519
\(471\) −9.89151e6 −2.05452
\(472\) 2.78267e6 0.574919
\(473\) −177703. −0.0365210
\(474\) −3.76222e6 −0.769128
\(475\) −946591. −0.192499
\(476\) −1.01422e6 −0.205170
\(477\) 592824. 0.119297
\(478\) 4.63879e6 0.928614
\(479\) 5.85157e6 1.16529 0.582644 0.812727i \(-0.302018\pi\)
0.582644 + 0.812727i \(0.302018\pi\)
\(480\) −2.74287e6 −0.543378
\(481\) 157816. 0.0311020
\(482\) −6.12108e6 −1.20008
\(483\) −1.33456e7 −2.60297
\(484\) 2.25554e6 0.437661
\(485\) 1.42004e6 0.274123
\(486\) 1.97424e6 0.379149
\(487\) 1.83575e6 0.350745 0.175372 0.984502i \(-0.443887\pi\)
0.175372 + 0.984502i \(0.443887\pi\)
\(488\) 3.80398e6 0.723083
\(489\) −8.35958e6 −1.58093
\(490\) 4.32999e6 0.814699
\(491\) −5.75635e6 −1.07757 −0.538783 0.842445i \(-0.681116\pi\)
−0.538783 + 0.842445i \(0.681116\pi\)
\(492\) −4.75599e6 −0.885784
\(493\) 263250. 0.0487811
\(494\) 1.97350e6 0.363848
\(495\) −179118. −0.0328568
\(496\) 2.70675e6 0.494019
\(497\) 4.27781e6 0.776838
\(498\) −1.45738e6 −0.263329
\(499\) 4.10166e6 0.737409 0.368705 0.929547i \(-0.379801\pi\)
0.368705 + 0.929547i \(0.379801\pi\)
\(500\) 2.48009e6 0.443652
\(501\) 3.56471e6 0.634497
\(502\) 1.56132e6 0.276523
\(503\) −1.12157e6 −0.197654 −0.0988270 0.995105i \(-0.531509\pi\)
−0.0988270 + 0.995105i \(0.531509\pi\)
\(504\) 2.75193e6 0.482570
\(505\) −1.84610e6 −0.322126
\(506\) 1.23067e6 0.213681
\(507\) −1.22312e7 −2.11325
\(508\) 3.63258e6 0.624534
\(509\) −2.68902e6 −0.460045 −0.230023 0.973185i \(-0.573880\pi\)
−0.230023 + 0.973185i \(0.573880\pi\)
\(510\) 743650. 0.126603
\(511\) 1.25659e7 2.12883
\(512\) −3.73721e6 −0.630047
\(513\) 1.43723e6 0.241120
\(514\) −1.96078e6 −0.327356
\(515\) −3.62960e6 −0.603033
\(516\) 533111. 0.0881441
\(517\) −1.35844e6 −0.223518
\(518\) −140264. −0.0229680
\(519\) 1.67212e6 0.272489
\(520\) −6.54225e6 −1.06101
\(521\) 2.88911e6 0.466305 0.233152 0.972440i \(-0.425096\pi\)
0.233152 + 0.972440i \(0.425096\pi\)
\(522\) −224576. −0.0360735
\(523\) −1.40417e6 −0.224474 −0.112237 0.993681i \(-0.535802\pi\)
−0.112237 + 0.993681i \(0.535802\pi\)
\(524\) −2.91235e6 −0.463356
\(525\) −7.98832e6 −1.26490
\(526\) 4.63573e6 0.730557
\(527\) 2.49891e6 0.391945
\(528\) −509423. −0.0795231
\(529\) 5.45814e6 0.848019
\(530\) 1.25260e6 0.193697
\(531\) −918904. −0.141428
\(532\) 1.48570e6 0.227589
\(533\) −1.91212e7 −2.91540
\(534\) −9.53169e6 −1.44650
\(535\) 6.16930e6 0.931861
\(536\) 1.78612e6 0.268534
\(537\) 5.73113e6 0.857640
\(538\) −1.58822e6 −0.236568
\(539\) −2.73843e6 −0.406003
\(540\) −1.49799e6 −0.221067
\(541\) −211154. −0.0310174 −0.0155087 0.999880i \(-0.504937\pi\)
−0.0155087 + 0.999880i \(0.504937\pi\)
\(542\) 4.53007e6 0.662379
\(543\) −1.90298e6 −0.276971
\(544\) −1.50425e6 −0.217933
\(545\) 3.26426e6 0.470754
\(546\) 1.66545e7 2.39083
\(547\) −4.61571e6 −0.659584 −0.329792 0.944054i \(-0.606979\pi\)
−0.329792 + 0.944054i \(0.606979\pi\)
\(548\) −2.07564e6 −0.295257
\(549\) −1.25616e6 −0.177875
\(550\) 736649. 0.103837
\(551\) −385627. −0.0541113
\(552\) −1.17428e7 −1.64031
\(553\) −1.13871e7 −1.58343
\(554\) −9.11378e6 −1.26161
\(555\) −87112.5 −0.0120046
\(556\) 4.77416e6 0.654953
\(557\) 1.01209e7 1.38223 0.691114 0.722746i \(-0.257120\pi\)
0.691114 + 0.722746i \(0.257120\pi\)
\(558\) −2.13180e6 −0.289842
\(559\) 2.14335e6 0.290110
\(560\) 2.43800e6 0.328521
\(561\) −470308. −0.0630921
\(562\) −4.27525e6 −0.570980
\(563\) 3.84774e6 0.511604 0.255802 0.966729i \(-0.417660\pi\)
0.255802 + 0.966729i \(0.417660\pi\)
\(564\) 4.07531e6 0.539465
\(565\) −3.36885e6 −0.443978
\(566\) 3.70098e6 0.485597
\(567\) 1.55709e7 2.03403
\(568\) 3.76406e6 0.489538
\(569\) 4.38443e6 0.567718 0.283859 0.958866i \(-0.408385\pi\)
0.283859 + 0.958866i \(0.408385\pi\)
\(570\) −1.08935e6 −0.140436
\(571\) 2.61241e6 0.335313 0.167656 0.985845i \(-0.446380\pi\)
0.167656 + 0.985845i \(0.446380\pi\)
\(572\) 1.30086e6 0.166242
\(573\) 5.53357e6 0.704075
\(574\) 1.69947e7 2.15295
\(575\) 7.11974e6 0.898037
\(576\) 1.97933e6 0.248577
\(577\) −7.06124e6 −0.882961 −0.441480 0.897271i \(-0.645547\pi\)
−0.441480 + 0.897271i \(0.645547\pi\)
\(578\) −5.50211e6 −0.685030
\(579\) −726132. −0.0900159
\(580\) 401928. 0.0496110
\(581\) −4.41103e6 −0.542125
\(582\) −3.18082e6 −0.389252
\(583\) −792186. −0.0965285
\(584\) 1.10568e7 1.34152
\(585\) 2.16041e6 0.261003
\(586\) −2.00625e6 −0.241347
\(587\) −8.18957e6 −0.980993 −0.490496 0.871443i \(-0.663185\pi\)
−0.490496 + 0.871443i \(0.663185\pi\)
\(588\) 8.21528e6 0.979894
\(589\) −3.66058e6 −0.434772
\(590\) −1.94159e6 −0.229629
\(591\) −6.21710e6 −0.732182
\(592\) −51747.9 −0.00606860
\(593\) −9.78828e6 −1.14306 −0.571531 0.820580i \(-0.693650\pi\)
−0.571531 + 0.820580i \(0.693650\pi\)
\(594\) −1.11847e6 −0.130065
\(595\) 2.25080e6 0.260642
\(596\) 5.74465e6 0.662442
\(597\) 1.66343e7 1.91015
\(598\) −1.48436e7 −1.69741
\(599\) 4.44247e6 0.505892 0.252946 0.967480i \(-0.418600\pi\)
0.252946 + 0.967480i \(0.418600\pi\)
\(600\) −7.02897e6 −0.797101
\(601\) 248973. 0.0281168 0.0140584 0.999901i \(-0.495525\pi\)
0.0140584 + 0.999901i \(0.495525\pi\)
\(602\) −1.90498e6 −0.214239
\(603\) −589821. −0.0660582
\(604\) 3.17836e6 0.354495
\(605\) −5.00560e6 −0.555991
\(606\) 4.13517e6 0.457417
\(607\) −5.61487e6 −0.618540 −0.309270 0.950974i \(-0.600085\pi\)
−0.309270 + 0.950974i \(0.600085\pi\)
\(608\) 2.20353e6 0.241747
\(609\) −3.25432e6 −0.355563
\(610\) −2.65420e6 −0.288808
\(611\) 1.63846e7 1.77555
\(612\) 294697. 0.0318051
\(613\) 7.11213e6 0.764449 0.382225 0.924069i \(-0.375158\pi\)
0.382225 + 0.924069i \(0.375158\pi\)
\(614\) 953834. 0.102106
\(615\) 1.05547e7 1.12527
\(616\) −3.67737e6 −0.390468
\(617\) 872669. 0.0922862 0.0461431 0.998935i \(-0.485307\pi\)
0.0461431 + 0.998935i \(0.485307\pi\)
\(618\) 8.13014e6 0.856301
\(619\) −9.40244e6 −0.986312 −0.493156 0.869941i \(-0.664157\pi\)
−0.493156 + 0.869941i \(0.664157\pi\)
\(620\) 3.81531e6 0.398613
\(621\) −1.08101e7 −1.12486
\(622\) 1.03934e7 1.07716
\(623\) −2.88495e7 −2.97796
\(624\) 6.14434e6 0.631704
\(625\) 947282. 0.0970017
\(626\) −1.15760e7 −1.18065
\(627\) 688939. 0.0699861
\(628\) −8.28242e6 −0.838027
\(629\) −47774.5 −0.00481470
\(630\) −1.92014e6 −0.192744
\(631\) −1.90990e7 −1.90957 −0.954786 0.297293i \(-0.903916\pi\)
−0.954786 + 0.297293i \(0.903916\pi\)
\(632\) −1.00196e7 −0.997829
\(633\) 6.66987e6 0.661619
\(634\) 8.09056e6 0.799384
\(635\) −8.06158e6 −0.793388
\(636\) 2.37656e6 0.232973
\(637\) 3.30292e7 3.22515
\(638\) 300099. 0.0291886
\(639\) −1.24298e6 −0.120424
\(640\) −825936. −0.0797070
\(641\) −4.06757e6 −0.391012 −0.195506 0.980703i \(-0.562635\pi\)
−0.195506 + 0.980703i \(0.562635\pi\)
\(642\) −1.38189e7 −1.32324
\(643\) −1.00899e6 −0.0962410 −0.0481205 0.998842i \(-0.515323\pi\)
−0.0481205 + 0.998842i \(0.515323\pi\)
\(644\) −1.11746e7 −1.06174
\(645\) −1.18310e6 −0.111975
\(646\) −597424. −0.0563250
\(647\) 1.99986e7 1.87819 0.939095 0.343657i \(-0.111666\pi\)
0.939095 + 0.343657i \(0.111666\pi\)
\(648\) 1.37010e7 1.28178
\(649\) 1.22792e6 0.114435
\(650\) −8.88500e6 −0.824848
\(651\) −3.08918e7 −2.85687
\(652\) −6.99970e6 −0.644852
\(653\) −1.16599e7 −1.07007 −0.535035 0.844830i \(-0.679702\pi\)
−0.535035 + 0.844830i \(0.679702\pi\)
\(654\) −7.31179e6 −0.668466
\(655\) 6.46320e6 0.588633
\(656\) 6.26986e6 0.568850
\(657\) −3.65121e6 −0.330007
\(658\) −1.45624e7 −1.31120
\(659\) 1.40204e7 1.25761 0.628806 0.777562i \(-0.283544\pi\)
0.628806 + 0.777562i \(0.283544\pi\)
\(660\) −718061. −0.0641654
\(661\) 5.39932e6 0.480657 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(662\) −251594. −0.0223128
\(663\) 5.67256e6 0.501182
\(664\) −3.88129e6 −0.341630
\(665\) −3.29712e6 −0.289122
\(666\) 40756.0 0.00356046
\(667\) 2.90047e6 0.252438
\(668\) 2.98482e6 0.258808
\(669\) 1.19811e6 0.103498
\(670\) −1.24626e6 −0.107256
\(671\) 1.67860e6 0.143926
\(672\) 1.85957e7 1.58851
\(673\) 2.37236e6 0.201903 0.100951 0.994891i \(-0.467811\pi\)
0.100951 + 0.994891i \(0.467811\pi\)
\(674\) −7.20082e6 −0.610565
\(675\) −6.47064e6 −0.546623
\(676\) −1.02415e7 −0.861983
\(677\) 1.48279e7 1.24339 0.621695 0.783260i \(-0.286444\pi\)
0.621695 + 0.783260i \(0.286444\pi\)
\(678\) 7.54607e6 0.630444
\(679\) −9.62736e6 −0.801369
\(680\) 1.98049e6 0.164248
\(681\) 1.89128e7 1.56275
\(682\) 2.84871e6 0.234524
\(683\) −1.45710e7 −1.19519 −0.597594 0.801799i \(-0.703877\pi\)
−0.597594 + 0.801799i \(0.703877\pi\)
\(684\) −431692. −0.0352805
\(685\) 4.60635e6 0.375086
\(686\) −1.39099e7 −1.12853
\(687\) 1.26926e7 1.02603
\(688\) −702804. −0.0566061
\(689\) 9.55485e6 0.766789
\(690\) 8.19348e6 0.655158
\(691\) −2.39101e6 −0.190496 −0.0952480 0.995454i \(-0.530364\pi\)
−0.0952480 + 0.995454i \(0.530364\pi\)
\(692\) 1.40011e6 0.111147
\(693\) 1.21436e6 0.0960534
\(694\) 4.89773e6 0.386008
\(695\) −1.05950e7 −0.832033
\(696\) −2.86349e6 −0.224065
\(697\) 5.78844e6 0.451315
\(698\) 5.03198e6 0.390932
\(699\) −2.40649e7 −1.86291
\(700\) −6.68883e6 −0.515947
\(701\) −2.67685e6 −0.205745 −0.102872 0.994695i \(-0.532803\pi\)
−0.102872 + 0.994695i \(0.532803\pi\)
\(702\) 1.34903e7 1.03319
\(703\) 69983.4 0.00534080
\(704\) −2.64496e6 −0.201135
\(705\) −9.04411e6 −0.685319
\(706\) 4.19246e6 0.316561
\(707\) 1.25159e7 0.941701
\(708\) −3.68377e6 −0.276191
\(709\) 1.61177e7 1.20417 0.602085 0.798432i \(-0.294337\pi\)
0.602085 + 0.798432i \(0.294337\pi\)
\(710\) −2.62635e6 −0.195527
\(711\) 3.30870e6 0.245461
\(712\) −2.53848e7 −1.87661
\(713\) 2.75329e7 2.02828
\(714\) −5.04168e6 −0.370109
\(715\) −2.88693e6 −0.211189
\(716\) 4.79883e6 0.349826
\(717\) −1.95319e7 −1.41888
\(718\) 4.58463e6 0.331889
\(719\) 2.74012e7 1.97673 0.988363 0.152111i \(-0.0486070\pi\)
0.988363 + 0.152111i \(0.0486070\pi\)
\(720\) −708398. −0.0509267
\(721\) 2.46074e7 1.76290
\(722\) −9.43124e6 −0.673327
\(723\) 2.57732e7 1.83367
\(724\) −1.59341e6 −0.112975
\(725\) 1.73615e6 0.122671
\(726\) 1.12123e7 0.789503
\(727\) −1.44905e7 −1.01683 −0.508413 0.861113i \(-0.669768\pi\)
−0.508413 + 0.861113i \(0.669768\pi\)
\(728\) 4.43542e7 3.10175
\(729\) 8.82438e6 0.614986
\(730\) −7.71478e6 −0.535817
\(731\) −648840. −0.0449101
\(732\) −5.03580e6 −0.347369
\(733\) 9.24243e6 0.635370 0.317685 0.948196i \(-0.397095\pi\)
0.317685 + 0.948196i \(0.397095\pi\)
\(734\) 1.24574e7 0.853470
\(735\) −1.82317e7 −1.24483
\(736\) −1.65737e7 −1.12778
\(737\) 788172. 0.0534505
\(738\) −4.93807e6 −0.333746
\(739\) −270924. −0.0182489 −0.00912445 0.999958i \(-0.502904\pi\)
−0.00912445 + 0.999958i \(0.502904\pi\)
\(740\) −72941.6 −0.00489662
\(741\) −8.30955e6 −0.555945
\(742\) −8.49221e6 −0.566253
\(743\) 1.56128e7 1.03755 0.518776 0.854910i \(-0.326388\pi\)
0.518776 + 0.854910i \(0.326388\pi\)
\(744\) −2.71818e7 −1.80031
\(745\) −1.27488e7 −0.841545
\(746\) 1.76141e7 1.15882
\(747\) 1.28169e6 0.0840394
\(748\) −393801. −0.0257349
\(749\) −4.18257e7 −2.72420
\(750\) 1.23285e7 0.800311
\(751\) 2.81796e6 0.182320 0.0911600 0.995836i \(-0.470943\pi\)
0.0911600 + 0.995836i \(0.470943\pi\)
\(752\) −5.37251e6 −0.346444
\(753\) −6.57403e6 −0.422517
\(754\) −3.61961e6 −0.231864
\(755\) −7.05355e6 −0.450340
\(756\) 1.01558e7 0.646264
\(757\) 1.47622e6 0.0936293 0.0468146 0.998904i \(-0.485093\pi\)
0.0468146 + 0.998904i \(0.485093\pi\)
\(758\) 4.42325e6 0.279620
\(759\) −5.18182e6 −0.326496
\(760\) −2.90116e6 −0.182195
\(761\) −1.59710e7 −0.999700 −0.499850 0.866112i \(-0.666612\pi\)
−0.499850 + 0.866112i \(0.666612\pi\)
\(762\) 1.80576e7 1.12660
\(763\) −2.21305e7 −1.37620
\(764\) 4.63340e6 0.287188
\(765\) −654005. −0.0404043
\(766\) −1.56394e7 −0.963051
\(767\) −1.48104e7 −0.909033
\(768\) 1.91529e7 1.17174
\(769\) −2.69038e7 −1.64058 −0.820291 0.571947i \(-0.806188\pi\)
−0.820291 + 0.571947i \(0.806188\pi\)
\(770\) 2.56586e6 0.155958
\(771\) 8.25597e6 0.500187
\(772\) −608009. −0.0367170
\(773\) 2.10506e7 1.26711 0.633556 0.773697i \(-0.281595\pi\)
0.633556 + 0.773697i \(0.281595\pi\)
\(774\) 553520. 0.0332109
\(775\) 1.64805e7 0.985633
\(776\) −8.47116e6 −0.504997
\(777\) 590592. 0.0350942
\(778\) −1.61732e7 −0.957962
\(779\) −8.47930e6 −0.500629
\(780\) 8.66080e6 0.509708
\(781\) 1.66099e6 0.0974403
\(782\) 4.49349e6 0.262765
\(783\) −2.63604e6 −0.153655
\(784\) −1.08303e7 −0.629288
\(785\) 1.83807e7 1.06460
\(786\) −1.44773e7 −0.835854
\(787\) −1.46428e7 −0.842728 −0.421364 0.906892i \(-0.638448\pi\)
−0.421364 + 0.906892i \(0.638448\pi\)
\(788\) −5.20574e6 −0.298653
\(789\) −1.95190e7 −1.11626
\(790\) 6.99108e6 0.398544
\(791\) 2.28396e7 1.29792
\(792\) 1.06852e6 0.0605297
\(793\) −2.02462e7 −1.14330
\(794\) 2.99496e6 0.168593
\(795\) −5.27416e6 −0.295962
\(796\) 1.39283e7 0.779141
\(797\) −1.53622e7 −0.856660 −0.428330 0.903622i \(-0.640898\pi\)
−0.428330 + 0.903622i \(0.640898\pi\)
\(798\) 7.38540e6 0.410551
\(799\) −4.96000e6 −0.274862
\(800\) −9.92063e6 −0.548042
\(801\) 8.38267e6 0.461638
\(802\) −6.55651e6 −0.359945
\(803\) 4.87907e6 0.267023
\(804\) −2.36452e6 −0.129004
\(805\) 2.47991e7 1.34880
\(806\) −3.43593e7 −1.86298
\(807\) 6.68730e6 0.361466
\(808\) 1.10128e7 0.593430
\(809\) −7.26215e6 −0.390116 −0.195058 0.980792i \(-0.562490\pi\)
−0.195058 + 0.980792i \(0.562490\pi\)
\(810\) −9.55975e6 −0.511957
\(811\) 4.17728e6 0.223019 0.111510 0.993763i \(-0.464431\pi\)
0.111510 + 0.993763i \(0.464431\pi\)
\(812\) −2.72493e6 −0.145032
\(813\) −1.90741e7 −1.01209
\(814\) −54461.9 −0.00288092
\(815\) 1.55340e7 0.819200
\(816\) −1.86003e6 −0.0977901
\(817\) 950466. 0.0498174
\(818\) −1.39586e7 −0.729390
\(819\) −1.46468e7 −0.763015
\(820\) 8.83772e6 0.458993
\(821\) 2.70285e7 1.39947 0.699736 0.714402i \(-0.253301\pi\)
0.699736 + 0.714402i \(0.253301\pi\)
\(822\) −1.03180e7 −0.532619
\(823\) −6.95032e6 −0.357689 −0.178844 0.983877i \(-0.557236\pi\)
−0.178844 + 0.983877i \(0.557236\pi\)
\(824\) 2.16522e7 1.11092
\(825\) −3.10171e6 −0.158659
\(826\) 1.31633e7 0.671297
\(827\) −370060. −0.0188152 −0.00940759 0.999956i \(-0.502995\pi\)
−0.00940759 + 0.999956i \(0.502995\pi\)
\(828\) 3.24695e6 0.164589
\(829\) 1.38451e7 0.699698 0.349849 0.936806i \(-0.386233\pi\)
0.349849 + 0.936806i \(0.386233\pi\)
\(830\) 2.70814e6 0.136451
\(831\) 3.83741e7 1.92768
\(832\) 3.19018e7 1.59774
\(833\) −9.99869e6 −0.499264
\(834\) 2.37324e7 1.18148
\(835\) −6.62405e6 −0.328781
\(836\) 576867. 0.0285469
\(837\) −2.50227e7 −1.23458
\(838\) 2.82979e7 1.39201
\(839\) 1.86174e7 0.913089 0.456544 0.889701i \(-0.349087\pi\)
0.456544 + 0.889701i \(0.349087\pi\)
\(840\) −2.44830e7 −1.19720
\(841\) 707281. 0.0344828
\(842\) −7.37815e6 −0.358647
\(843\) 1.80012e7 0.872435
\(844\) 5.58486e6 0.269871
\(845\) 2.27285e7 1.09504
\(846\) 4.23133e6 0.203260
\(847\) 3.39362e7 1.62538
\(848\) −3.13304e6 −0.149615
\(849\) −1.55832e7 −0.741972
\(850\) 2.68969e6 0.127690
\(851\) −526376. −0.0249157
\(852\) −4.98296e6 −0.235174
\(853\) 2.15389e6 0.101356 0.0506782 0.998715i \(-0.483862\pi\)
0.0506782 + 0.998715i \(0.483862\pi\)
\(854\) 1.79945e7 0.844298
\(855\) 958031. 0.0448192
\(856\) −3.68026e7 −1.71670
\(857\) −3.33513e7 −1.55118 −0.775588 0.631239i \(-0.782547\pi\)
−0.775588 + 0.631239i \(0.782547\pi\)
\(858\) 6.46659e6 0.299887
\(859\) −3.11061e7 −1.43834 −0.719171 0.694833i \(-0.755478\pi\)
−0.719171 + 0.694833i \(0.755478\pi\)
\(860\) −990643. −0.0456742
\(861\) −7.15571e7 −3.28961
\(862\) 5.42025e6 0.248457
\(863\) −1.59387e7 −0.728494 −0.364247 0.931302i \(-0.618674\pi\)
−0.364247 + 0.931302i \(0.618674\pi\)
\(864\) 1.50627e7 0.686467
\(865\) −3.10719e6 −0.141198
\(866\) 8.96511e6 0.406220
\(867\) 2.31670e7 1.04670
\(868\) −2.58665e7 −1.16530
\(869\) −4.42138e6 −0.198613
\(870\) 1.99798e6 0.0894939
\(871\) −9.50644e6 −0.424593
\(872\) −1.94728e7 −0.867235
\(873\) 2.79738e6 0.124227
\(874\) −6.58238e6 −0.291477
\(875\) 3.73147e7 1.64763
\(876\) −1.46372e7 −0.644464
\(877\) 666578. 0.0292652 0.0146326 0.999893i \(-0.495342\pi\)
0.0146326 + 0.999893i \(0.495342\pi\)
\(878\) −2.42214e7 −1.06038
\(879\) 8.44745e6 0.368768
\(880\) 946625. 0.0412070
\(881\) −2.30910e7 −1.00231 −0.501155 0.865358i \(-0.667091\pi\)
−0.501155 + 0.865358i \(0.667091\pi\)
\(882\) 8.52980e6 0.369205
\(883\) 3.85060e6 0.166198 0.0830992 0.996541i \(-0.473518\pi\)
0.0830992 + 0.996541i \(0.473518\pi\)
\(884\) 4.74978e6 0.204429
\(885\) 8.17519e6 0.350864
\(886\) 1.80784e7 0.773706
\(887\) −4.14426e7 −1.76863 −0.884316 0.466889i \(-0.845375\pi\)
−0.884316 + 0.466889i \(0.845375\pi\)
\(888\) 519665. 0.0221152
\(889\) 5.46547e7 2.31939
\(890\) 1.77121e7 0.749540
\(891\) 6.04589e6 0.255133
\(892\) 1.00321e6 0.0422163
\(893\) 7.26574e6 0.304896
\(894\) 2.85566e7 1.19499
\(895\) −1.06498e7 −0.444409
\(896\) 5.59955e6 0.233015
\(897\) 6.24999e7 2.59357
\(898\) 1.11596e7 0.461803
\(899\) 6.71389e6 0.277061
\(900\) 1.94354e6 0.0799812
\(901\) −2.89247e6 −0.118702
\(902\) 6.59869e6 0.270048
\(903\) 8.02102e6 0.327348
\(904\) 2.00967e7 0.817907
\(905\) 3.53617e6 0.143520
\(906\) 1.57996e7 0.639479
\(907\) −4.63163e7 −1.86946 −0.934729 0.355363i \(-0.884357\pi\)
−0.934729 + 0.355363i \(0.884357\pi\)
\(908\) 1.58362e7 0.637436
\(909\) −3.63669e6 −0.145981
\(910\) −3.09478e7 −1.23887
\(911\) 2.69515e6 0.107594 0.0537969 0.998552i \(-0.482868\pi\)
0.0537969 + 0.998552i \(0.482868\pi\)
\(912\) 2.72470e6 0.108476
\(913\) −1.71272e6 −0.0679999
\(914\) −1.78912e7 −0.708392
\(915\) 1.11757e7 0.441286
\(916\) 1.06278e7 0.418510
\(917\) −4.38183e7 −1.72081
\(918\) −4.08383e6 −0.159941
\(919\) 1.39792e7 0.546000 0.273000 0.962014i \(-0.411984\pi\)
0.273000 + 0.962014i \(0.411984\pi\)
\(920\) 2.18209e7 0.849969
\(921\) −4.01618e6 −0.156014
\(922\) 9.50852e6 0.368371
\(923\) −2.00338e7 −0.774032
\(924\) 4.86820e6 0.187581
\(925\) −315076. −0.0121077
\(926\) 6.32510e6 0.242404
\(927\) −7.15007e6 −0.273282
\(928\) −4.04151e6 −0.154054
\(929\) −7.25580e6 −0.275833 −0.137916 0.990444i \(-0.544041\pi\)
−0.137916 + 0.990444i \(0.544041\pi\)
\(930\) 1.89659e7 0.719063
\(931\) 1.46468e7 0.553819
\(932\) −2.01502e7 −0.759869
\(933\) −4.37619e7 −1.64586
\(934\) −1.96399e7 −0.736667
\(935\) 873940. 0.0326928
\(936\) −1.28878e7 −0.480827
\(937\) 3.95850e6 0.147293 0.0736464 0.997284i \(-0.476536\pi\)
0.0736464 + 0.997284i \(0.476536\pi\)
\(938\) 8.44918e6 0.313550
\(939\) 4.87413e7 1.80399
\(940\) −7.57287e6 −0.279538
\(941\) −4.07672e7 −1.50085 −0.750425 0.660956i \(-0.770151\pi\)
−0.750425 + 0.660956i \(0.770151\pi\)
\(942\) −4.11719e7 −1.51173
\(943\) 6.37766e7 2.33551
\(944\) 4.85635e6 0.177370
\(945\) −2.25382e7 −0.820994
\(946\) −739664. −0.0268724
\(947\) −2.67878e7 −0.970650 −0.485325 0.874334i \(-0.661299\pi\)
−0.485325 + 0.874334i \(0.661299\pi\)
\(948\) 1.32641e7 0.479356
\(949\) −5.88484e7 −2.12114
\(950\) −3.94004e6 −0.141642
\(951\) −3.40658e7 −1.22143
\(952\) −1.34270e7 −0.480162
\(953\) −1.81430e7 −0.647107 −0.323553 0.946210i \(-0.604877\pi\)
−0.323553 + 0.946210i \(0.604877\pi\)
\(954\) 2.46754e6 0.0877796
\(955\) −1.02827e7 −0.364835
\(956\) −1.63546e7 −0.578755
\(957\) −1.26359e6 −0.0445990
\(958\) 2.43563e7 0.857427
\(959\) −3.12294e7 −1.09652
\(960\) −1.76094e7 −0.616690
\(961\) 3.51028e7 1.22612
\(962\) 656886. 0.0228851
\(963\) 1.21531e7 0.422300
\(964\) 2.15806e7 0.747946
\(965\) 1.34932e6 0.0466441
\(966\) −5.55489e7 −1.91528
\(967\) −1.40529e7 −0.483282 −0.241641 0.970366i \(-0.577686\pi\)
−0.241641 + 0.970366i \(0.577686\pi\)
\(968\) 2.98607e7 1.02426
\(969\) 2.51549e6 0.0860624
\(970\) 5.91069e6 0.201702
\(971\) 1.77622e7 0.604573 0.302287 0.953217i \(-0.402250\pi\)
0.302287 + 0.953217i \(0.402250\pi\)
\(972\) −6.96041e6 −0.236303
\(973\) 7.18306e7 2.43236
\(974\) 7.64103e6 0.258080
\(975\) 3.74108e7 1.26034
\(976\) 6.63873e6 0.223080
\(977\) −1.66625e7 −0.558474 −0.279237 0.960222i \(-0.590082\pi\)
−0.279237 + 0.960222i \(0.590082\pi\)
\(978\) −3.47955e7 −1.16326
\(979\) −1.12017e7 −0.373531
\(980\) −1.52659e7 −0.507758
\(981\) 6.43037e6 0.213336
\(982\) −2.39600e7 −0.792880
\(983\) −2.28571e7 −0.754461 −0.377231 0.926119i \(-0.623124\pi\)
−0.377231 + 0.926119i \(0.623124\pi\)
\(984\) −6.29635e7 −2.07301
\(985\) 1.15528e7 0.379399
\(986\) 1.09574e6 0.0358934
\(987\) 6.13159e7 2.00346
\(988\) −6.95781e6 −0.226767
\(989\) −7.14888e6 −0.232406
\(990\) −745551. −0.0241763
\(991\) −2.52864e7 −0.817904 −0.408952 0.912556i \(-0.634106\pi\)
−0.408952 + 0.912556i \(0.634106\pi\)
\(992\) −3.83642e7 −1.23779
\(993\) 1.05935e6 0.0340931
\(994\) 1.78057e7 0.571602
\(995\) −3.09103e7 −0.989796
\(996\) 5.13815e6 0.164119
\(997\) 4.44014e7 1.41468 0.707340 0.706873i \(-0.249895\pi\)
0.707340 + 0.706873i \(0.249895\pi\)
\(998\) 1.70726e7 0.542590
\(999\) 478387. 0.0151658
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.1.3 4
3.2 odd 2 261.6.a.a.1.2 4
4.3 odd 2 464.6.a.i.1.3 4
5.4 even 2 725.6.a.a.1.2 4
29.28 even 2 841.6.a.a.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.3 4 1.1 even 1 trivial
261.6.a.a.1.2 4 3.2 odd 2
464.6.a.i.1.3 4 4.3 odd 2
725.6.a.a.1.2 4 5.4 even 2
841.6.a.a.1.2 4 29.28 even 2