Properties

Label 825.6.a.u.1.7
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.16483\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.16483 q^{2} -9.00000 q^{3} -21.9838 q^{4} -28.4835 q^{6} +223.585 q^{7} -170.850 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.16483 q^{2} -9.00000 q^{3} -21.9838 q^{4} -28.4835 q^{6} +223.585 q^{7} -170.850 q^{8} +81.0000 q^{9} +121.000 q^{11} +197.855 q^{12} +999.193 q^{13} +707.609 q^{14} +162.772 q^{16} +2010.59 q^{17} +256.351 q^{18} -856.966 q^{19} -2012.27 q^{21} +382.945 q^{22} +2374.78 q^{23} +1537.65 q^{24} +3162.28 q^{26} -729.000 q^{27} -4915.26 q^{28} +4716.76 q^{29} -9118.77 q^{31} +5982.34 q^{32} -1089.00 q^{33} +6363.19 q^{34} -1780.69 q^{36} -3138.40 q^{37} -2712.15 q^{38} -8992.73 q^{39} -11989.9 q^{41} -6368.48 q^{42} -8608.69 q^{43} -2660.04 q^{44} +7515.77 q^{46} +18196.2 q^{47} -1464.95 q^{48} +33183.3 q^{49} -18095.3 q^{51} -21966.1 q^{52} +10919.2 q^{53} -2307.16 q^{54} -38199.5 q^{56} +7712.69 q^{57} +14927.7 q^{58} -38947.0 q^{59} -36320.3 q^{61} -28859.4 q^{62} +18110.4 q^{63} +13724.4 q^{64} -3446.50 q^{66} +53109.7 q^{67} -44200.5 q^{68} -21373.0 q^{69} +52703.0 q^{71} -13838.8 q^{72} +55220.6 q^{73} -9932.51 q^{74} +18839.4 q^{76} +27053.8 q^{77} -28460.5 q^{78} +71829.6 q^{79} +6561.00 q^{81} -37945.9 q^{82} +17356.7 q^{83} +44237.3 q^{84} -27245.0 q^{86} -42450.8 q^{87} -20672.8 q^{88} -44666.7 q^{89} +223405. q^{91} -52206.7 q^{92} +82068.9 q^{93} +57587.9 q^{94} -53841.0 q^{96} -136962. q^{97} +105019. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.16483 0.559469 0.279734 0.960077i \(-0.409754\pi\)
0.279734 + 0.960077i \(0.409754\pi\)
\(3\) −9.00000 −0.577350
\(4\) −21.9838 −0.686995
\(5\) 0 0
\(6\) −28.4835 −0.323009
\(7\) 223.585 1.72464 0.862318 0.506366i \(-0.169012\pi\)
0.862318 + 0.506366i \(0.169012\pi\)
\(8\) −170.850 −0.943821
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 197.855 0.396637
\(13\) 999.193 1.63980 0.819900 0.572507i \(-0.194029\pi\)
0.819900 + 0.572507i \(0.194029\pi\)
\(14\) 707.609 0.964880
\(15\) 0 0
\(16\) 162.772 0.158957
\(17\) 2010.59 1.68734 0.843668 0.536865i \(-0.180392\pi\)
0.843668 + 0.536865i \(0.180392\pi\)
\(18\) 256.351 0.186490
\(19\) −856.966 −0.544602 −0.272301 0.962212i \(-0.587785\pi\)
−0.272301 + 0.962212i \(0.587785\pi\)
\(20\) 0 0
\(21\) −2012.27 −0.995720
\(22\) 382.945 0.168686
\(23\) 2374.78 0.936059 0.468030 0.883713i \(-0.344964\pi\)
0.468030 + 0.883713i \(0.344964\pi\)
\(24\) 1537.65 0.544915
\(25\) 0 0
\(26\) 3162.28 0.917416
\(27\) −729.000 −0.192450
\(28\) −4915.26 −1.18482
\(29\) 4716.76 1.04147 0.520737 0.853717i \(-0.325657\pi\)
0.520737 + 0.853717i \(0.325657\pi\)
\(30\) 0 0
\(31\) −9118.77 −1.70425 −0.852123 0.523342i \(-0.824685\pi\)
−0.852123 + 0.523342i \(0.824685\pi\)
\(32\) 5982.34 1.03275
\(33\) −1089.00 −0.174078
\(34\) 6363.19 0.944012
\(35\) 0 0
\(36\) −1780.69 −0.228998
\(37\) −3138.40 −0.376881 −0.188440 0.982085i \(-0.560343\pi\)
−0.188440 + 0.982085i \(0.560343\pi\)
\(38\) −2712.15 −0.304688
\(39\) −8992.73 −0.946739
\(40\) 0 0
\(41\) −11989.9 −1.11392 −0.556961 0.830539i \(-0.688033\pi\)
−0.556961 + 0.830539i \(0.688033\pi\)
\(42\) −6368.48 −0.557074
\(43\) −8608.69 −0.710012 −0.355006 0.934864i \(-0.615521\pi\)
−0.355006 + 0.934864i \(0.615521\pi\)
\(44\) −2660.04 −0.207137
\(45\) 0 0
\(46\) 7515.77 0.523696
\(47\) 18196.2 1.20153 0.600767 0.799424i \(-0.294862\pi\)
0.600767 + 0.799424i \(0.294862\pi\)
\(48\) −1464.95 −0.0917738
\(49\) 33183.3 1.97437
\(50\) 0 0
\(51\) −18095.3 −0.974184
\(52\) −21966.1 −1.12653
\(53\) 10919.2 0.533952 0.266976 0.963703i \(-0.413976\pi\)
0.266976 + 0.963703i \(0.413976\pi\)
\(54\) −2307.16 −0.107670
\(55\) 0 0
\(56\) −38199.5 −1.62775
\(57\) 7712.69 0.314426
\(58\) 14927.7 0.582672
\(59\) −38947.0 −1.45661 −0.728305 0.685253i \(-0.759692\pi\)
−0.728305 + 0.685253i \(0.759692\pi\)
\(60\) 0 0
\(61\) −36320.3 −1.24976 −0.624878 0.780722i \(-0.714851\pi\)
−0.624878 + 0.780722i \(0.714851\pi\)
\(62\) −28859.4 −0.953472
\(63\) 18110.4 0.574879
\(64\) 13724.4 0.418835
\(65\) 0 0
\(66\) −3446.50 −0.0973910
\(67\) 53109.7 1.44540 0.722698 0.691164i \(-0.242902\pi\)
0.722698 + 0.691164i \(0.242902\pi\)
\(68\) −44200.5 −1.15919
\(69\) −21373.0 −0.540434
\(70\) 0 0
\(71\) 52703.0 1.24077 0.620383 0.784299i \(-0.286977\pi\)
0.620383 + 0.784299i \(0.286977\pi\)
\(72\) −13838.8 −0.314607
\(73\) 55220.6 1.21281 0.606406 0.795155i \(-0.292610\pi\)
0.606406 + 0.795155i \(0.292610\pi\)
\(74\) −9932.51 −0.210853
\(75\) 0 0
\(76\) 18839.4 0.374139
\(77\) 27053.8 0.519998
\(78\) −28460.5 −0.529671
\(79\) 71829.6 1.29490 0.647450 0.762108i \(-0.275836\pi\)
0.647450 + 0.762108i \(0.275836\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −37945.9 −0.623204
\(83\) 17356.7 0.276548 0.138274 0.990394i \(-0.455845\pi\)
0.138274 + 0.990394i \(0.455845\pi\)
\(84\) 44237.3 0.684054
\(85\) 0 0
\(86\) −27245.0 −0.397229
\(87\) −42450.8 −0.601295
\(88\) −20672.8 −0.284573
\(89\) −44666.7 −0.597736 −0.298868 0.954294i \(-0.596609\pi\)
−0.298868 + 0.954294i \(0.596609\pi\)
\(90\) 0 0
\(91\) 223405. 2.82806
\(92\) −52206.7 −0.643068
\(93\) 82068.9 0.983946
\(94\) 57587.9 0.672221
\(95\) 0 0
\(96\) −53841.0 −0.596260
\(97\) −136962. −1.47799 −0.738993 0.673713i \(-0.764698\pi\)
−0.738993 + 0.673713i \(0.764698\pi\)
\(98\) 105019. 1.10460
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −159090. −1.55182 −0.775909 0.630845i \(-0.782708\pi\)
−0.775909 + 0.630845i \(0.782708\pi\)
\(102\) −57268.7 −0.545025
\(103\) −153555. −1.42616 −0.713082 0.701080i \(-0.752701\pi\)
−0.713082 + 0.701080i \(0.752701\pi\)
\(104\) −170712. −1.54768
\(105\) 0 0
\(106\) 34557.5 0.298729
\(107\) 52408.5 0.442529 0.221265 0.975214i \(-0.428982\pi\)
0.221265 + 0.975214i \(0.428982\pi\)
\(108\) 16026.2 0.132212
\(109\) −90098.5 −0.726359 −0.363180 0.931719i \(-0.618309\pi\)
−0.363180 + 0.931719i \(0.618309\pi\)
\(110\) 0 0
\(111\) 28245.6 0.217592
\(112\) 36393.3 0.274143
\(113\) 50729.4 0.373735 0.186867 0.982385i \(-0.440167\pi\)
0.186867 + 0.982385i \(0.440167\pi\)
\(114\) 24409.4 0.175912
\(115\) 0 0
\(116\) −103692. −0.715487
\(117\) 80934.6 0.546600
\(118\) −123261. −0.814928
\(119\) 449538. 2.91004
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −114948. −0.699199
\(123\) 107909. 0.643123
\(124\) 200466. 1.17081
\(125\) 0 0
\(126\) 57316.3 0.321627
\(127\) 338912. 1.86456 0.932282 0.361733i \(-0.117815\pi\)
0.932282 + 0.361733i \(0.117815\pi\)
\(128\) −147999. −0.798427
\(129\) 77478.2 0.409926
\(130\) 0 0
\(131\) −214136. −1.09021 −0.545106 0.838367i \(-0.683511\pi\)
−0.545106 + 0.838367i \(0.683511\pi\)
\(132\) 23940.4 0.119590
\(133\) −191605. −0.939241
\(134\) 168083. 0.808653
\(135\) 0 0
\(136\) −343509. −1.59254
\(137\) −61579.4 −0.280307 −0.140153 0.990130i \(-0.544760\pi\)
−0.140153 + 0.990130i \(0.544760\pi\)
\(138\) −67641.9 −0.302356
\(139\) 241459. 1.06000 0.530000 0.847998i \(-0.322192\pi\)
0.530000 + 0.847998i \(0.322192\pi\)
\(140\) 0 0
\(141\) −163766. −0.693706
\(142\) 166796. 0.694169
\(143\) 120902. 0.494418
\(144\) 13184.5 0.0529856
\(145\) 0 0
\(146\) 174764. 0.678531
\(147\) −298649. −1.13990
\(148\) 68994.1 0.258915
\(149\) 209027. 0.771325 0.385662 0.922640i \(-0.373973\pi\)
0.385662 + 0.922640i \(0.373973\pi\)
\(150\) 0 0
\(151\) 128993. 0.460389 0.230195 0.973145i \(-0.426064\pi\)
0.230195 + 0.973145i \(0.426064\pi\)
\(152\) 146412. 0.514007
\(153\) 162858. 0.562445
\(154\) 85620.7 0.290922
\(155\) 0 0
\(156\) 197695. 0.650405
\(157\) 192991. 0.624866 0.312433 0.949940i \(-0.398856\pi\)
0.312433 + 0.949940i \(0.398856\pi\)
\(158\) 227329. 0.724456
\(159\) −98273.1 −0.308277
\(160\) 0 0
\(161\) 530965. 1.61436
\(162\) 20764.5 0.0621632
\(163\) 98844.2 0.291395 0.145698 0.989329i \(-0.453457\pi\)
0.145698 + 0.989329i \(0.453457\pi\)
\(164\) 263583. 0.765258
\(165\) 0 0
\(166\) 54930.9 0.154720
\(167\) −12212.4 −0.0338851 −0.0169425 0.999856i \(-0.505393\pi\)
−0.0169425 + 0.999856i \(0.505393\pi\)
\(168\) 343795. 0.939781
\(169\) 627093. 1.68894
\(170\) 0 0
\(171\) −69414.2 −0.181534
\(172\) 189252. 0.487775
\(173\) 209579. 0.532393 0.266197 0.963919i \(-0.414233\pi\)
0.266197 + 0.963919i \(0.414233\pi\)
\(174\) −134350. −0.336406
\(175\) 0 0
\(176\) 19695.4 0.0479273
\(177\) 350523. 0.840975
\(178\) −141363. −0.334414
\(179\) −245093. −0.571740 −0.285870 0.958268i \(-0.592283\pi\)
−0.285870 + 0.958268i \(0.592283\pi\)
\(180\) 0 0
\(181\) 420637. 0.954356 0.477178 0.878807i \(-0.341660\pi\)
0.477178 + 0.878807i \(0.341660\pi\)
\(182\) 707038. 1.58221
\(183\) 326883. 0.721547
\(184\) −405730. −0.883472
\(185\) 0 0
\(186\) 259734. 0.550487
\(187\) 243282. 0.508751
\(188\) −400022. −0.825448
\(189\) −162993. −0.331907
\(190\) 0 0
\(191\) 838901. 1.66390 0.831950 0.554851i \(-0.187225\pi\)
0.831950 + 0.554851i \(0.187225\pi\)
\(192\) −123520. −0.241815
\(193\) 847703. 1.63814 0.819069 0.573696i \(-0.194491\pi\)
0.819069 + 0.573696i \(0.194491\pi\)
\(194\) −433461. −0.826886
\(195\) 0 0
\(196\) −729496. −1.35638
\(197\) −877903. −1.61169 −0.805844 0.592128i \(-0.798288\pi\)
−0.805844 + 0.592128i \(0.798288\pi\)
\(198\) 31018.5 0.0562287
\(199\) 132070. 0.236413 0.118207 0.992989i \(-0.462286\pi\)
0.118207 + 0.992989i \(0.462286\pi\)
\(200\) 0 0
\(201\) −477987. −0.834499
\(202\) −503495. −0.868193
\(203\) 1.05460e6 1.79616
\(204\) 397805. 0.669260
\(205\) 0 0
\(206\) −485975. −0.797894
\(207\) 192357. 0.312020
\(208\) 162640. 0.260657
\(209\) −103693. −0.164204
\(210\) 0 0
\(211\) −638957. −0.988020 −0.494010 0.869456i \(-0.664469\pi\)
−0.494010 + 0.869456i \(0.664469\pi\)
\(212\) −240047. −0.366822
\(213\) −474327. −0.716356
\(214\) 165864. 0.247581
\(215\) 0 0
\(216\) 124549. 0.181638
\(217\) −2.03882e6 −2.93920
\(218\) −285147. −0.406375
\(219\) −496985. −0.700218
\(220\) 0 0
\(221\) 2.00897e6 2.76689
\(222\) 89392.6 0.121736
\(223\) 75080.5 0.101103 0.0505516 0.998721i \(-0.483902\pi\)
0.0505516 + 0.998721i \(0.483902\pi\)
\(224\) 1.33756e6 1.78112
\(225\) 0 0
\(226\) 160550. 0.209093
\(227\) 480604. 0.619046 0.309523 0.950892i \(-0.399831\pi\)
0.309523 + 0.950892i \(0.399831\pi\)
\(228\) −169555. −0.216009
\(229\) 331611. 0.417870 0.208935 0.977930i \(-0.433000\pi\)
0.208935 + 0.977930i \(0.433000\pi\)
\(230\) 0 0
\(231\) −243484. −0.300221
\(232\) −805857. −0.982965
\(233\) −69038.0 −0.0833102 −0.0416551 0.999132i \(-0.513263\pi\)
−0.0416551 + 0.999132i \(0.513263\pi\)
\(234\) 256144. 0.305805
\(235\) 0 0
\(236\) 856203. 1.00068
\(237\) −646467. −0.747610
\(238\) 1.42271e6 1.62808
\(239\) 735334. 0.832703 0.416351 0.909204i \(-0.363309\pi\)
0.416351 + 0.909204i \(0.363309\pi\)
\(240\) 0 0
\(241\) −70515.7 −0.0782065 −0.0391033 0.999235i \(-0.512450\pi\)
−0.0391033 + 0.999235i \(0.512450\pi\)
\(242\) 46336.3 0.0508608
\(243\) −59049.0 −0.0641500
\(244\) 798460. 0.858576
\(245\) 0 0
\(246\) 341513. 0.359807
\(247\) −856274. −0.893038
\(248\) 1.55794e6 1.60850
\(249\) −156210. −0.159665
\(250\) 0 0
\(251\) 451414. 0.452262 0.226131 0.974097i \(-0.427392\pi\)
0.226131 + 0.974097i \(0.427392\pi\)
\(252\) −398136. −0.394939
\(253\) 287348. 0.282232
\(254\) 1.07260e6 1.04316
\(255\) 0 0
\(256\) −907574. −0.865530
\(257\) −794147. −0.750012 −0.375006 0.927022i \(-0.622359\pi\)
−0.375006 + 0.927022i \(0.622359\pi\)
\(258\) 245205. 0.229341
\(259\) −701699. −0.649982
\(260\) 0 0
\(261\) 382057. 0.347158
\(262\) −677703. −0.609939
\(263\) −889131. −0.792641 −0.396320 0.918112i \(-0.629713\pi\)
−0.396320 + 0.918112i \(0.629713\pi\)
\(264\) 186055. 0.164298
\(265\) 0 0
\(266\) −606397. −0.525476
\(267\) 402001. 0.345103
\(268\) −1.16755e6 −0.992979
\(269\) −1.74480e6 −1.47016 −0.735082 0.677978i \(-0.762856\pi\)
−0.735082 + 0.677978i \(0.762856\pi\)
\(270\) 0 0
\(271\) 806384. 0.666989 0.333495 0.942752i \(-0.391772\pi\)
0.333495 + 0.942752i \(0.391772\pi\)
\(272\) 327268. 0.268214
\(273\) −2.01064e6 −1.63278
\(274\) −194888. −0.156823
\(275\) 0 0
\(276\) 469860. 0.371275
\(277\) 1.76740e6 1.38400 0.692000 0.721897i \(-0.256730\pi\)
0.692000 + 0.721897i \(0.256730\pi\)
\(278\) 764176. 0.593037
\(279\) −738620. −0.568082
\(280\) 0 0
\(281\) 79693.3 0.0602082 0.0301041 0.999547i \(-0.490416\pi\)
0.0301041 + 0.999547i \(0.490416\pi\)
\(282\) −518291. −0.388107
\(283\) −763339. −0.566567 −0.283283 0.959036i \(-0.591424\pi\)
−0.283283 + 0.959036i \(0.591424\pi\)
\(284\) −1.15861e6 −0.852400
\(285\) 0 0
\(286\) 382636. 0.276611
\(287\) −2.68075e6 −1.92111
\(288\) 484569. 0.344251
\(289\) 2.62262e6 1.84710
\(290\) 0 0
\(291\) 1.23266e6 0.853315
\(292\) −1.21396e6 −0.833196
\(293\) 402977. 0.274228 0.137114 0.990555i \(-0.456217\pi\)
0.137114 + 0.990555i \(0.456217\pi\)
\(294\) −945175. −0.637741
\(295\) 0 0
\(296\) 536195. 0.355708
\(297\) −88209.0 −0.0580259
\(298\) 661536. 0.431532
\(299\) 2.37286e6 1.53495
\(300\) 0 0
\(301\) −1.92477e6 −1.22451
\(302\) 408243. 0.257573
\(303\) 1.43181e6 0.895943
\(304\) −139490. −0.0865682
\(305\) 0 0
\(306\) 515418. 0.314671
\(307\) −2.24468e6 −1.35928 −0.679639 0.733546i \(-0.737864\pi\)
−0.679639 + 0.733546i \(0.737864\pi\)
\(308\) −594746. −0.357236
\(309\) 1.38199e6 0.823397
\(310\) 0 0
\(311\) −378005. −0.221614 −0.110807 0.993842i \(-0.535344\pi\)
−0.110807 + 0.993842i \(0.535344\pi\)
\(312\) 1.53641e6 0.893552
\(313\) 2.25734e6 1.30237 0.651187 0.758917i \(-0.274271\pi\)
0.651187 + 0.758917i \(0.274271\pi\)
\(314\) 610783. 0.349593
\(315\) 0 0
\(316\) −1.57909e6 −0.889589
\(317\) −3.00613e6 −1.68019 −0.840097 0.542436i \(-0.817502\pi\)
−0.840097 + 0.542436i \(0.817502\pi\)
\(318\) −311018. −0.172472
\(319\) 570727. 0.314016
\(320\) 0 0
\(321\) −471676. −0.255494
\(322\) 1.68041e6 0.903185
\(323\) −1.72301e6 −0.918927
\(324\) −144236. −0.0763328
\(325\) 0 0
\(326\) 312825. 0.163026
\(327\) 810887. 0.419364
\(328\) 2.04847e6 1.05134
\(329\) 4.06840e6 2.07221
\(330\) 0 0
\(331\) −289324. −0.145149 −0.0725746 0.997363i \(-0.523122\pi\)
−0.0725746 + 0.997363i \(0.523122\pi\)
\(332\) −381566. −0.189987
\(333\) −254210. −0.125627
\(334\) −38650.1 −0.0189576
\(335\) 0 0
\(336\) −327540. −0.158276
\(337\) −290901. −0.139531 −0.0697655 0.997563i \(-0.522225\pi\)
−0.0697655 + 0.997563i \(0.522225\pi\)
\(338\) 1.98464e6 0.944911
\(339\) −456564. −0.215776
\(340\) 0 0
\(341\) −1.10337e6 −0.513849
\(342\) −219684. −0.101563
\(343\) 3.66149e6 1.68044
\(344\) 1.47079e6 0.670124
\(345\) 0 0
\(346\) 663282. 0.297857
\(347\) 3.62678e6 1.61695 0.808476 0.588529i \(-0.200293\pi\)
0.808476 + 0.588529i \(0.200293\pi\)
\(348\) 933231. 0.413087
\(349\) 3.99131e6 1.75409 0.877046 0.480407i \(-0.159511\pi\)
0.877046 + 0.480407i \(0.159511\pi\)
\(350\) 0 0
\(351\) −728411. −0.315580
\(352\) 723863. 0.311386
\(353\) −3.83884e6 −1.63969 −0.819847 0.572582i \(-0.805942\pi\)
−0.819847 + 0.572582i \(0.805942\pi\)
\(354\) 1.10935e6 0.470499
\(355\) 0 0
\(356\) 981946. 0.410641
\(357\) −4.04584e6 −1.68011
\(358\) −775678. −0.319870
\(359\) −3.98280e6 −1.63099 −0.815496 0.578762i \(-0.803536\pi\)
−0.815496 + 0.578762i \(0.803536\pi\)
\(360\) 0 0
\(361\) −1.74171e6 −0.703408
\(362\) 1.33124e6 0.533932
\(363\) −131769. −0.0524864
\(364\) −4.91129e6 −1.94286
\(365\) 0 0
\(366\) 1.03453e6 0.403683
\(367\) 1.13099e6 0.438323 0.219162 0.975689i \(-0.429668\pi\)
0.219162 + 0.975689i \(0.429668\pi\)
\(368\) 386547. 0.148793
\(369\) −971179. −0.371307
\(370\) 0 0
\(371\) 2.44138e6 0.920874
\(372\) −1.80419e6 −0.675966
\(373\) −2.59821e6 −0.966945 −0.483472 0.875360i \(-0.660625\pi\)
−0.483472 + 0.875360i \(0.660625\pi\)
\(374\) 769946. 0.284630
\(375\) 0 0
\(376\) −3.10882e6 −1.13403
\(377\) 4.71295e6 1.70781
\(378\) −515847. −0.185691
\(379\) 2.07353e6 0.741500 0.370750 0.928733i \(-0.379101\pi\)
0.370750 + 0.928733i \(0.379101\pi\)
\(380\) 0 0
\(381\) −3.05020e6 −1.07651
\(382\) 2.65498e6 0.930899
\(383\) −2.26397e6 −0.788632 −0.394316 0.918975i \(-0.629018\pi\)
−0.394316 + 0.918975i \(0.629018\pi\)
\(384\) 1.33199e6 0.460972
\(385\) 0 0
\(386\) 2.68284e6 0.916486
\(387\) −697304. −0.236671
\(388\) 3.01095e6 1.01537
\(389\) 989122. 0.331418 0.165709 0.986175i \(-0.447009\pi\)
0.165709 + 0.986175i \(0.447009\pi\)
\(390\) 0 0
\(391\) 4.77471e6 1.57945
\(392\) −5.66935e6 −1.86345
\(393\) 1.92722e6 0.629434
\(394\) −2.77842e6 −0.901689
\(395\) 0 0
\(396\) −215464. −0.0690456
\(397\) −1.93874e6 −0.617368 −0.308684 0.951165i \(-0.599889\pi\)
−0.308684 + 0.951165i \(0.599889\pi\)
\(398\) 417979. 0.132266
\(399\) 1.72444e6 0.542271
\(400\) 0 0
\(401\) −3.89960e6 −1.21104 −0.605520 0.795830i \(-0.707035\pi\)
−0.605520 + 0.795830i \(0.707035\pi\)
\(402\) −1.51275e6 −0.466876
\(403\) −9.11141e6 −2.79462
\(404\) 3.49742e6 1.06609
\(405\) 0 0
\(406\) 3.33762e6 1.00490
\(407\) −379746. −0.113634
\(408\) 3.09158e6 0.919455
\(409\) 1.63513e6 0.483331 0.241665 0.970360i \(-0.422306\pi\)
0.241665 + 0.970360i \(0.422306\pi\)
\(410\) 0 0
\(411\) 554214. 0.161835
\(412\) 3.37572e6 0.979768
\(413\) −8.70796e6 −2.51212
\(414\) 608777. 0.174565
\(415\) 0 0
\(416\) 5.97751e6 1.69351
\(417\) −2.17313e6 −0.611991
\(418\) −328170. −0.0918668
\(419\) −3.50652e6 −0.975755 −0.487878 0.872912i \(-0.662229\pi\)
−0.487878 + 0.872912i \(0.662229\pi\)
\(420\) 0 0
\(421\) 233311. 0.0641549 0.0320774 0.999485i \(-0.489788\pi\)
0.0320774 + 0.999485i \(0.489788\pi\)
\(422\) −2.02219e6 −0.552766
\(423\) 1.47389e6 0.400511
\(424\) −1.86555e6 −0.503955
\(425\) 0 0
\(426\) −1.50117e6 −0.400779
\(427\) −8.12068e6 −2.15538
\(428\) −1.15214e6 −0.304015
\(429\) −1.08812e6 −0.285452
\(430\) 0 0
\(431\) 5.88958e6 1.52718 0.763591 0.645700i \(-0.223434\pi\)
0.763591 + 0.645700i \(0.223434\pi\)
\(432\) −118661. −0.0305913
\(433\) −1.07678e6 −0.276000 −0.138000 0.990432i \(-0.544067\pi\)
−0.138000 + 0.990432i \(0.544067\pi\)
\(434\) −6.45253e6 −1.64439
\(435\) 0 0
\(436\) 1.98071e6 0.499005
\(437\) −2.03510e6 −0.509780
\(438\) −1.57287e6 −0.391750
\(439\) −3.45853e6 −0.856505 −0.428253 0.903659i \(-0.640871\pi\)
−0.428253 + 0.903659i \(0.640871\pi\)
\(440\) 0 0
\(441\) 2.68784e6 0.658124
\(442\) 6.35805e6 1.54799
\(443\) 3.46996e6 0.840070 0.420035 0.907508i \(-0.362018\pi\)
0.420035 + 0.907508i \(0.362018\pi\)
\(444\) −620947. −0.149485
\(445\) 0 0
\(446\) 237617. 0.0565640
\(447\) −1.88125e6 −0.445325
\(448\) 3.06857e6 0.722339
\(449\) −2.98659e6 −0.699133 −0.349566 0.936912i \(-0.613671\pi\)
−0.349566 + 0.936912i \(0.613671\pi\)
\(450\) 0 0
\(451\) −1.45077e6 −0.335860
\(452\) −1.11523e6 −0.256754
\(453\) −1.16094e6 −0.265806
\(454\) 1.52103e6 0.346337
\(455\) 0 0
\(456\) −1.31771e6 −0.296762
\(457\) 8.27627e6 1.85372 0.926860 0.375408i \(-0.122498\pi\)
0.926860 + 0.375408i \(0.122498\pi\)
\(458\) 1.04949e6 0.233785
\(459\) −1.46572e6 −0.324728
\(460\) 0 0
\(461\) 5.25876e6 1.15247 0.576237 0.817283i \(-0.304521\pi\)
0.576237 + 0.817283i \(0.304521\pi\)
\(462\) −770586. −0.167964
\(463\) 2.91066e6 0.631013 0.315507 0.948923i \(-0.397826\pi\)
0.315507 + 0.948923i \(0.397826\pi\)
\(464\) 767755. 0.165549
\(465\) 0 0
\(466\) −218494. −0.0466095
\(467\) 905276. 0.192083 0.0960415 0.995377i \(-0.469382\pi\)
0.0960415 + 0.995377i \(0.469382\pi\)
\(468\) −1.77925e6 −0.375511
\(469\) 1.18745e7 2.49278
\(470\) 0 0
\(471\) −1.73692e6 −0.360767
\(472\) 6.65408e6 1.37478
\(473\) −1.04165e6 −0.214077
\(474\) −2.04596e6 −0.418265
\(475\) 0 0
\(476\) −9.88258e6 −1.99918
\(477\) 884458. 0.177984
\(478\) 2.32721e6 0.465871
\(479\) −1.83868e6 −0.366158 −0.183079 0.983098i \(-0.558606\pi\)
−0.183079 + 0.983098i \(0.558606\pi\)
\(480\) 0 0
\(481\) −3.13587e6 −0.618009
\(482\) −223170. −0.0437541
\(483\) −4.77868e6 −0.932052
\(484\) −321865. −0.0624541
\(485\) 0 0
\(486\) −186880. −0.0358899
\(487\) 2.81488e6 0.537821 0.268911 0.963165i \(-0.413336\pi\)
0.268911 + 0.963165i \(0.413336\pi\)
\(488\) 6.20532e6 1.17955
\(489\) −889598. −0.168237
\(490\) 0 0
\(491\) −6.61534e6 −1.23836 −0.619182 0.785247i \(-0.712536\pi\)
−0.619182 + 0.785247i \(0.712536\pi\)
\(492\) −2.37225e6 −0.441822
\(493\) 9.48347e6 1.75732
\(494\) −2.70996e6 −0.499627
\(495\) 0 0
\(496\) −1.48428e6 −0.270901
\(497\) 1.17836e7 2.13987
\(498\) −494378. −0.0893276
\(499\) 2.83864e6 0.510339 0.255170 0.966896i \(-0.417869\pi\)
0.255170 + 0.966896i \(0.417869\pi\)
\(500\) 0 0
\(501\) 109911. 0.0195636
\(502\) 1.42865e6 0.253026
\(503\) −906183. −0.159697 −0.0798483 0.996807i \(-0.525444\pi\)
−0.0798483 + 0.996807i \(0.525444\pi\)
\(504\) −3.09416e6 −0.542583
\(505\) 0 0
\(506\) 909408. 0.157900
\(507\) −5.64383e6 −0.975112
\(508\) −7.45058e6 −1.28095
\(509\) −3.13716e6 −0.536714 −0.268357 0.963320i \(-0.586481\pi\)
−0.268357 + 0.963320i \(0.586481\pi\)
\(510\) 0 0
\(511\) 1.23465e7 2.09166
\(512\) 1.86366e6 0.314190
\(513\) 624728. 0.104809
\(514\) −2.51334e6 −0.419608
\(515\) 0 0
\(516\) −1.70327e6 −0.281617
\(517\) 2.20174e6 0.362276
\(518\) −2.22076e6 −0.363645
\(519\) −1.88621e6 −0.307377
\(520\) 0 0
\(521\) −1.29715e6 −0.209362 −0.104681 0.994506i \(-0.533382\pi\)
−0.104681 + 0.994506i \(0.533382\pi\)
\(522\) 1.20915e6 0.194224
\(523\) 7.45988e6 1.19255 0.596276 0.802779i \(-0.296646\pi\)
0.596276 + 0.802779i \(0.296646\pi\)
\(524\) 4.70752e6 0.748970
\(525\) 0 0
\(526\) −2.81395e6 −0.443458
\(527\) −1.83341e7 −2.87564
\(528\) −177259. −0.0276708
\(529\) −796777. −0.123793
\(530\) 0 0
\(531\) −3.15470e6 −0.485537
\(532\) 4.21221e6 0.645254
\(533\) −1.19802e7 −1.82661
\(534\) 1.27226e6 0.193074
\(535\) 0 0
\(536\) −9.07378e6 −1.36419
\(537\) 2.20584e6 0.330094
\(538\) −5.52201e6 −0.822511
\(539\) 4.01518e6 0.595296
\(540\) 0 0
\(541\) −1.01922e7 −1.49718 −0.748591 0.663032i \(-0.769270\pi\)
−0.748591 + 0.663032i \(0.769270\pi\)
\(542\) 2.55207e6 0.373159
\(543\) −3.78573e6 −0.550998
\(544\) 1.20280e7 1.74260
\(545\) 0 0
\(546\) −6.36334e6 −0.913489
\(547\) 8.45753e6 1.20858 0.604289 0.796765i \(-0.293457\pi\)
0.604289 + 0.796765i \(0.293457\pi\)
\(548\) 1.35375e6 0.192569
\(549\) −2.94195e6 −0.416585
\(550\) 0 0
\(551\) −4.04210e6 −0.567189
\(552\) 3.65157e6 0.510073
\(553\) 1.60600e7 2.23323
\(554\) 5.59354e6 0.774305
\(555\) 0 0
\(556\) −5.30819e6 −0.728215
\(557\) 1.27511e7 1.74144 0.870720 0.491778i \(-0.163653\pi\)
0.870720 + 0.491778i \(0.163653\pi\)
\(558\) −2.33761e6 −0.317824
\(559\) −8.60173e6 −1.16428
\(560\) 0 0
\(561\) −2.18953e6 −0.293728
\(562\) 252216. 0.0336846
\(563\) 6.44710e6 0.857223 0.428611 0.903489i \(-0.359003\pi\)
0.428611 + 0.903489i \(0.359003\pi\)
\(564\) 3.60020e6 0.476573
\(565\) 0 0
\(566\) −2.41584e6 −0.316976
\(567\) 1.46694e6 0.191626
\(568\) −9.00430e6 −1.17106
\(569\) −7.05536e6 −0.913563 −0.456782 0.889579i \(-0.650998\pi\)
−0.456782 + 0.889579i \(0.650998\pi\)
\(570\) 0 0
\(571\) 393546. 0.0505132 0.0252566 0.999681i \(-0.491960\pi\)
0.0252566 + 0.999681i \(0.491960\pi\)
\(572\) −2.65790e6 −0.339663
\(573\) −7.55010e6 −0.960653
\(574\) −8.48414e6 −1.07480
\(575\) 0 0
\(576\) 1.11168e6 0.139612
\(577\) 2.27712e6 0.284738 0.142369 0.989814i \(-0.454528\pi\)
0.142369 + 0.989814i \(0.454528\pi\)
\(578\) 8.30016e6 1.03340
\(579\) −7.62932e6 −0.945779
\(580\) 0 0
\(581\) 3.88069e6 0.476945
\(582\) 3.90115e6 0.477403
\(583\) 1.32123e6 0.160993
\(584\) −9.43442e6 −1.14468
\(585\) 0 0
\(586\) 1.27536e6 0.153422
\(587\) 1.39306e6 0.166868 0.0834341 0.996513i \(-0.473411\pi\)
0.0834341 + 0.996513i \(0.473411\pi\)
\(588\) 6.56546e6 0.783108
\(589\) 7.81447e6 0.928136
\(590\) 0 0
\(591\) 7.90113e6 0.930508
\(592\) −510843. −0.0599078
\(593\) 4.53743e6 0.529874 0.264937 0.964266i \(-0.414649\pi\)
0.264937 + 0.964266i \(0.414649\pi\)
\(594\) −279167. −0.0324637
\(595\) 0 0
\(596\) −4.59522e6 −0.529896
\(597\) −1.18863e6 −0.136493
\(598\) 7.50970e6 0.858756
\(599\) −4.08022e6 −0.464640 −0.232320 0.972639i \(-0.574632\pi\)
−0.232320 + 0.972639i \(0.574632\pi\)
\(600\) 0 0
\(601\) −1.33295e7 −1.50532 −0.752660 0.658409i \(-0.771230\pi\)
−0.752660 + 0.658409i \(0.771230\pi\)
\(602\) −6.09158e6 −0.685076
\(603\) 4.30188e6 0.481798
\(604\) −2.83577e6 −0.316285
\(605\) 0 0
\(606\) 4.53145e6 0.501252
\(607\) 5.04375e6 0.555625 0.277812 0.960635i \(-0.410391\pi\)
0.277812 + 0.960635i \(0.410391\pi\)
\(608\) −5.12666e6 −0.562439
\(609\) −9.49136e6 −1.03702
\(610\) 0 0
\(611\) 1.81815e7 1.97028
\(612\) −3.58024e6 −0.386397
\(613\) −4.29244e6 −0.461374 −0.230687 0.973028i \(-0.574097\pi\)
−0.230687 + 0.973028i \(0.574097\pi\)
\(614\) −7.10404e6 −0.760474
\(615\) 0 0
\(616\) −4.62213e6 −0.490784
\(617\) 6.86553e6 0.726041 0.363020 0.931781i \(-0.381746\pi\)
0.363020 + 0.931781i \(0.381746\pi\)
\(618\) 4.37377e6 0.460665
\(619\) 1.81177e6 0.190054 0.0950269 0.995475i \(-0.469706\pi\)
0.0950269 + 0.995475i \(0.469706\pi\)
\(620\) 0 0
\(621\) −1.73121e6 −0.180145
\(622\) −1.19632e6 −0.123986
\(623\) −9.98682e6 −1.03088
\(624\) −1.46376e6 −0.150491
\(625\) 0 0
\(626\) 7.14410e6 0.728638
\(627\) 933235. 0.0948031
\(628\) −4.24267e6 −0.429280
\(629\) −6.31004e6 −0.635925
\(630\) 0 0
\(631\) −8.04264e6 −0.804129 −0.402064 0.915611i \(-0.631707\pi\)
−0.402064 + 0.915611i \(0.631707\pi\)
\(632\) −1.22721e7 −1.22215
\(633\) 5.75062e6 0.570434
\(634\) −9.51389e6 −0.940016
\(635\) 0 0
\(636\) 2.16042e6 0.211785
\(637\) 3.31565e7 3.23757
\(638\) 1.80626e6 0.175682
\(639\) 4.26895e6 0.413589
\(640\) 0 0
\(641\) 3.75102e6 0.360583 0.180291 0.983613i \(-0.442296\pi\)
0.180291 + 0.983613i \(0.442296\pi\)
\(642\) −1.49278e6 −0.142941
\(643\) −1.92270e7 −1.83393 −0.916966 0.398964i \(-0.869370\pi\)
−0.916966 + 0.398964i \(0.869370\pi\)
\(644\) −1.16726e7 −1.10906
\(645\) 0 0
\(646\) −5.45303e6 −0.514111
\(647\) 1.65974e7 1.55876 0.779382 0.626549i \(-0.215533\pi\)
0.779382 + 0.626549i \(0.215533\pi\)
\(648\) −1.12095e6 −0.104869
\(649\) −4.71258e6 −0.439185
\(650\) 0 0
\(651\) 1.83494e7 1.69695
\(652\) −2.17297e6 −0.200187
\(653\) 5.30043e6 0.486439 0.243219 0.969971i \(-0.421797\pi\)
0.243219 + 0.969971i \(0.421797\pi\)
\(654\) 2.56632e6 0.234621
\(655\) 0 0
\(656\) −1.95161e6 −0.177065
\(657\) 4.47287e6 0.404271
\(658\) 1.28758e7 1.15934
\(659\) 1.22198e7 1.09610 0.548051 0.836445i \(-0.315370\pi\)
0.548051 + 0.836445i \(0.315370\pi\)
\(660\) 0 0
\(661\) 5.92583e6 0.527528 0.263764 0.964587i \(-0.415036\pi\)
0.263764 + 0.964587i \(0.415036\pi\)
\(662\) −915662. −0.0812064
\(663\) −1.80807e7 −1.59747
\(664\) −2.96538e6 −0.261012
\(665\) 0 0
\(666\) −804533. −0.0702843
\(667\) 1.12012e7 0.974881
\(668\) 268474. 0.0232789
\(669\) −675724. −0.0583719
\(670\) 0 0
\(671\) −4.39476e6 −0.376816
\(672\) −1.20381e7 −1.02833
\(673\) 1.44387e7 1.22882 0.614412 0.788986i \(-0.289393\pi\)
0.614412 + 0.788986i \(0.289393\pi\)
\(674\) −920653. −0.0780632
\(675\) 0 0
\(676\) −1.37859e7 −1.16030
\(677\) 2.08078e7 1.74483 0.872417 0.488763i \(-0.162552\pi\)
0.872417 + 0.488763i \(0.162552\pi\)
\(678\) −1.44495e6 −0.120720
\(679\) −3.06226e7 −2.54899
\(680\) 0 0
\(681\) −4.32543e6 −0.357406
\(682\) −3.49199e6 −0.287483
\(683\) −1.34086e7 −1.09985 −0.549923 0.835215i \(-0.685343\pi\)
−0.549923 + 0.835215i \(0.685343\pi\)
\(684\) 1.52599e6 0.124713
\(685\) 0 0
\(686\) 1.15880e7 0.940152
\(687\) −2.98450e6 −0.241257
\(688\) −1.40125e6 −0.112861
\(689\) 1.09104e7 0.875575
\(690\) 0 0
\(691\) −1.76242e7 −1.40415 −0.702076 0.712102i \(-0.747743\pi\)
−0.702076 + 0.712102i \(0.747743\pi\)
\(692\) −4.60735e6 −0.365751
\(693\) 2.19136e6 0.173333
\(694\) 1.14781e7 0.904634
\(695\) 0 0
\(696\) 7.25271e6 0.567515
\(697\) −2.41067e7 −1.87956
\(698\) 1.26318e7 0.981359
\(699\) 621342. 0.0480992
\(700\) 0 0
\(701\) 2.59651e6 0.199570 0.0997850 0.995009i \(-0.468184\pi\)
0.0997850 + 0.995009i \(0.468184\pi\)
\(702\) −2.30530e6 −0.176557
\(703\) 2.68950e6 0.205250
\(704\) 1.66065e6 0.126284
\(705\) 0 0
\(706\) −1.21493e7 −0.917358
\(707\) −3.55703e7 −2.67632
\(708\) −7.70583e6 −0.577745
\(709\) −1.04437e7 −0.780259 −0.390129 0.920760i \(-0.627570\pi\)
−0.390129 + 0.920760i \(0.627570\pi\)
\(710\) 0 0
\(711\) 5.81820e6 0.431633
\(712\) 7.63130e6 0.564155
\(713\) −2.16550e7 −1.59527
\(714\) −1.28044e7 −0.939971
\(715\) 0 0
\(716\) 5.38809e6 0.392782
\(717\) −6.61801e6 −0.480761
\(718\) −1.26049e7 −0.912489
\(719\) −1.16530e7 −0.840650 −0.420325 0.907374i \(-0.638084\pi\)
−0.420325 + 0.907374i \(0.638084\pi\)
\(720\) 0 0
\(721\) −3.43325e7 −2.45962
\(722\) −5.51222e6 −0.393535
\(723\) 634641. 0.0451526
\(724\) −9.24721e6 −0.655638
\(725\) 0 0
\(726\) −417027. −0.0293645
\(727\) 9.42203e6 0.661163 0.330581 0.943777i \(-0.392755\pi\)
0.330581 + 0.943777i \(0.392755\pi\)
\(728\) −3.81686e7 −2.66918
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.73086e7 −1.19803
\(732\) −7.18614e6 −0.495699
\(733\) 6.06936e6 0.417237 0.208619 0.977997i \(-0.433103\pi\)
0.208619 + 0.977997i \(0.433103\pi\)
\(734\) 3.57940e6 0.245228
\(735\) 0 0
\(736\) 1.42067e7 0.966717
\(737\) 6.42627e6 0.435803
\(738\) −3.07362e6 −0.207735
\(739\) 1.54739e7 1.04229 0.521144 0.853469i \(-0.325505\pi\)
0.521144 + 0.853469i \(0.325505\pi\)
\(740\) 0 0
\(741\) 7.70646e6 0.515596
\(742\) 7.72655e6 0.515200
\(743\) −1.84749e7 −1.22775 −0.613876 0.789402i \(-0.710391\pi\)
−0.613876 + 0.789402i \(0.710391\pi\)
\(744\) −1.40215e7 −0.928669
\(745\) 0 0
\(746\) −8.22289e6 −0.540975
\(747\) 1.40589e6 0.0921827
\(748\) −5.34826e6 −0.349509
\(749\) 1.17177e7 0.763202
\(750\) 0 0
\(751\) 2.07376e7 1.34171 0.670856 0.741588i \(-0.265927\pi\)
0.670856 + 0.741588i \(0.265927\pi\)
\(752\) 2.96183e6 0.190992
\(753\) −4.06272e6 −0.261114
\(754\) 1.49157e7 0.955465
\(755\) 0 0
\(756\) 3.58322e6 0.228018
\(757\) −1.55685e6 −0.0987432 −0.0493716 0.998780i \(-0.515722\pi\)
−0.0493716 + 0.998780i \(0.515722\pi\)
\(758\) 6.56236e6 0.414846
\(759\) −2.58613e6 −0.162947
\(760\) 0 0
\(761\) 2.49641e7 1.56262 0.781312 0.624140i \(-0.214551\pi\)
0.781312 + 0.624140i \(0.214551\pi\)
\(762\) −9.65339e6 −0.602271
\(763\) −2.01447e7 −1.25271
\(764\) −1.84423e7 −1.14309
\(765\) 0 0
\(766\) −7.16510e6 −0.441215
\(767\) −3.89155e7 −2.38855
\(768\) 8.16817e6 0.499714
\(769\) 3.71595e6 0.226597 0.113298 0.993561i \(-0.463858\pi\)
0.113298 + 0.993561i \(0.463858\pi\)
\(770\) 0 0
\(771\) 7.14733e6 0.433020
\(772\) −1.86358e7 −1.12539
\(773\) −3.04530e7 −1.83308 −0.916539 0.399946i \(-0.869029\pi\)
−0.916539 + 0.399946i \(0.869029\pi\)
\(774\) −2.20685e6 −0.132410
\(775\) 0 0
\(776\) 2.33999e7 1.39495
\(777\) 6.31529e6 0.375267
\(778\) 3.13041e6 0.185418
\(779\) 1.02749e7 0.606644
\(780\) 0 0
\(781\) 6.37707e6 0.374105
\(782\) 1.51112e7 0.883651
\(783\) −3.43851e6 −0.200432
\(784\) 5.40130e6 0.313840
\(785\) 0 0
\(786\) 6.09933e6 0.352149
\(787\) 2.23713e7 1.28752 0.643760 0.765228i \(-0.277374\pi\)
0.643760 + 0.765228i \(0.277374\pi\)
\(788\) 1.92997e7 1.10722
\(789\) 8.00218e6 0.457631
\(790\) 0 0
\(791\) 1.13423e7 0.644557
\(792\) −1.67450e6 −0.0948575
\(793\) −3.62910e7 −2.04935
\(794\) −6.13580e6 −0.345398
\(795\) 0 0
\(796\) −2.90341e6 −0.162415
\(797\) 4.96966e6 0.277128 0.138564 0.990353i \(-0.455751\pi\)
0.138564 + 0.990353i \(0.455751\pi\)
\(798\) 5.45757e6 0.303384
\(799\) 3.65851e7 2.02739
\(800\) 0 0
\(801\) −3.61801e6 −0.199245
\(802\) −1.23416e7 −0.677539
\(803\) 6.68169e6 0.365677
\(804\) 1.05080e7 0.573297
\(805\) 0 0
\(806\) −2.88361e7 −1.56350
\(807\) 1.57032e7 0.848800
\(808\) 2.71806e7 1.46464
\(809\) −2.69703e7 −1.44882 −0.724410 0.689369i \(-0.757888\pi\)
−0.724410 + 0.689369i \(0.757888\pi\)
\(810\) 0 0
\(811\) −1.73142e7 −0.924378 −0.462189 0.886782i \(-0.652936\pi\)
−0.462189 + 0.886782i \(0.652936\pi\)
\(812\) −2.31841e7 −1.23396
\(813\) −7.25746e6 −0.385086
\(814\) −1.20183e6 −0.0635746
\(815\) 0 0
\(816\) −2.94541e6 −0.154853
\(817\) 7.37735e6 0.386674
\(818\) 5.17492e6 0.270408
\(819\) 1.80958e7 0.942686
\(820\) 0 0
\(821\) 1.77928e7 0.921269 0.460634 0.887590i \(-0.347622\pi\)
0.460634 + 0.887590i \(0.347622\pi\)
\(822\) 1.75399e6 0.0905417
\(823\) −1.50458e7 −0.774314 −0.387157 0.922014i \(-0.626543\pi\)
−0.387157 + 0.922014i \(0.626543\pi\)
\(824\) 2.62348e7 1.34604
\(825\) 0 0
\(826\) −2.75592e7 −1.40545
\(827\) −1.07676e7 −0.547461 −0.273731 0.961806i \(-0.588258\pi\)
−0.273731 + 0.961806i \(0.588258\pi\)
\(828\) −4.22874e6 −0.214356
\(829\) −2.87834e6 −0.145464 −0.0727322 0.997352i \(-0.523172\pi\)
−0.0727322 + 0.997352i \(0.523172\pi\)
\(830\) 0 0
\(831\) −1.59066e7 −0.799053
\(832\) 1.37133e7 0.686806
\(833\) 6.67180e7 3.33143
\(834\) −6.87759e6 −0.342390
\(835\) 0 0
\(836\) 2.27957e6 0.112807
\(837\) 6.64758e6 0.327982
\(838\) −1.10975e7 −0.545905
\(839\) −3.26365e7 −1.60066 −0.800330 0.599560i \(-0.795342\pi\)
−0.800330 + 0.599560i \(0.795342\pi\)
\(840\) 0 0
\(841\) 1.73663e6 0.0846678
\(842\) 738390. 0.0358926
\(843\) −717239. −0.0347612
\(844\) 1.40467e7 0.678765
\(845\) 0 0
\(846\) 4.66462e6 0.224074
\(847\) 3.27351e6 0.156785
\(848\) 1.77734e6 0.0848754
\(849\) 6.87005e6 0.327108
\(850\) 0 0
\(851\) −7.45300e6 −0.352783
\(852\) 1.04275e7 0.492133
\(853\) 1.64332e7 0.773305 0.386652 0.922226i \(-0.373631\pi\)
0.386652 + 0.922226i \(0.373631\pi\)
\(854\) −2.57006e7 −1.20586
\(855\) 0 0
\(856\) −8.95398e6 −0.417668
\(857\) 1.27204e7 0.591628 0.295814 0.955246i \(-0.404409\pi\)
0.295814 + 0.955246i \(0.404409\pi\)
\(858\) −3.44372e6 −0.159702
\(859\) 1.60847e7 0.743755 0.371878 0.928282i \(-0.378714\pi\)
0.371878 + 0.928282i \(0.378714\pi\)
\(860\) 0 0
\(861\) 2.41268e7 1.10915
\(862\) 1.86395e7 0.854410
\(863\) 1.66292e7 0.760056 0.380028 0.924975i \(-0.375914\pi\)
0.380028 + 0.924975i \(0.375914\pi\)
\(864\) −4.36113e6 −0.198753
\(865\) 0 0
\(866\) −3.40784e6 −0.154413
\(867\) −2.36036e7 −1.06643
\(868\) 4.48211e7 2.01922
\(869\) 8.69139e6 0.390427
\(870\) 0 0
\(871\) 5.30668e7 2.37016
\(872\) 1.53933e7 0.685553
\(873\) −1.10939e7 −0.492662
\(874\) −6.44076e6 −0.285206
\(875\) 0 0
\(876\) 1.09256e7 0.481046
\(877\) 2.13465e7 0.937191 0.468595 0.883413i \(-0.344760\pi\)
0.468595 + 0.883413i \(0.344760\pi\)
\(878\) −1.09457e7 −0.479188
\(879\) −3.62679e6 −0.158325
\(880\) 0 0
\(881\) −3.40726e7 −1.47899 −0.739495 0.673162i \(-0.764935\pi\)
−0.739495 + 0.673162i \(0.764935\pi\)
\(882\) 8.50658e6 0.368200
\(883\) −7.87284e6 −0.339805 −0.169902 0.985461i \(-0.554345\pi\)
−0.169902 + 0.985461i \(0.554345\pi\)
\(884\) −4.41648e7 −1.90084
\(885\) 0 0
\(886\) 1.09819e7 0.469993
\(887\) 2.57330e7 1.09820 0.549100 0.835757i \(-0.314971\pi\)
0.549100 + 0.835757i \(0.314971\pi\)
\(888\) −4.82575e6 −0.205368
\(889\) 7.57756e7 3.21569
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −1.65056e6 −0.0694574
\(893\) −1.55935e7 −0.654358
\(894\) −5.95383e6 −0.249145
\(895\) 0 0
\(896\) −3.30905e7 −1.37700
\(897\) −2.13557e7 −0.886203
\(898\) −9.45205e6 −0.391143
\(899\) −4.30110e7 −1.77493
\(900\) 0 0
\(901\) 2.19541e7 0.900957
\(902\) −4.59145e6 −0.187903
\(903\) 1.73230e7 0.706973
\(904\) −8.66710e6 −0.352738
\(905\) 0 0
\(906\) −3.67418e6 −0.148710
\(907\) 2.93343e7 1.18402 0.592008 0.805932i \(-0.298335\pi\)
0.592008 + 0.805932i \(0.298335\pi\)
\(908\) −1.05655e7 −0.425281
\(909\) −1.28863e7 −0.517273
\(910\) 0 0
\(911\) 6.38249e6 0.254797 0.127399 0.991852i \(-0.459337\pi\)
0.127399 + 0.991852i \(0.459337\pi\)
\(912\) 1.25541e6 0.0499802
\(913\) 2.10015e6 0.0833824
\(914\) 2.61930e7 1.03710
\(915\) 0 0
\(916\) −7.29009e6 −0.287074
\(917\) −4.78775e7 −1.88022
\(918\) −4.63876e6 −0.181675
\(919\) −4.36179e6 −0.170363 −0.0851817 0.996365i \(-0.527147\pi\)
−0.0851817 + 0.996365i \(0.527147\pi\)
\(920\) 0 0
\(921\) 2.02021e7 0.784780
\(922\) 1.66431e7 0.644772
\(923\) 5.26605e7 2.03461
\(924\) 5.35271e6 0.206250
\(925\) 0 0
\(926\) 9.21174e6 0.353032
\(927\) −1.24379e7 −0.475388
\(928\) 2.82172e7 1.07558
\(929\) 1.47406e7 0.560371 0.280186 0.959946i \(-0.409604\pi\)
0.280186 + 0.959946i \(0.409604\pi\)
\(930\) 0 0
\(931\) −2.84369e7 −1.07525
\(932\) 1.51772e6 0.0572337
\(933\) 3.40205e6 0.127949
\(934\) 2.86505e6 0.107464
\(935\) 0 0
\(936\) −1.38277e7 −0.515892
\(937\) −2.20101e7 −0.818978 −0.409489 0.912315i \(-0.634293\pi\)
−0.409489 + 0.912315i \(0.634293\pi\)
\(938\) 3.75809e7 1.39463
\(939\) −2.03161e7 −0.751927
\(940\) 0 0
\(941\) 1.04925e7 0.386282 0.193141 0.981171i \(-0.438132\pi\)
0.193141 + 0.981171i \(0.438132\pi\)
\(942\) −5.49705e6 −0.201838
\(943\) −2.84732e7 −1.04270
\(944\) −6.33947e6 −0.231538
\(945\) 0 0
\(946\) −3.29665e6 −0.119769
\(947\) −2.19095e7 −0.793885 −0.396943 0.917843i \(-0.629929\pi\)
−0.396943 + 0.917843i \(0.629929\pi\)
\(948\) 1.42118e7 0.513605
\(949\) 5.51760e7 1.98877
\(950\) 0 0
\(951\) 2.70552e7 0.970060
\(952\) −7.68035e7 −2.74656
\(953\) −2.07187e7 −0.738974 −0.369487 0.929236i \(-0.620467\pi\)
−0.369487 + 0.929236i \(0.620467\pi\)
\(954\) 2.79916e6 0.0995765
\(955\) 0 0
\(956\) −1.61655e7 −0.572063
\(957\) −5.13655e6 −0.181297
\(958\) −5.81912e6 −0.204854
\(959\) −1.37682e7 −0.483428
\(960\) 0 0
\(961\) 5.45228e7 1.90445
\(962\) −9.92449e6 −0.345757
\(963\) 4.24509e6 0.147510
\(964\) 1.55021e6 0.0537275
\(965\) 0 0
\(966\) −1.51237e7 −0.521454
\(967\) 4.84470e7 1.66610 0.833050 0.553197i \(-0.186593\pi\)
0.833050 + 0.553197i \(0.186593\pi\)
\(968\) −2.50141e6 −0.0858019
\(969\) 1.55071e7 0.530543
\(970\) 0 0
\(971\) −2.80579e7 −0.955007 −0.477503 0.878630i \(-0.658458\pi\)
−0.477503 + 0.878630i \(0.658458\pi\)
\(972\) 1.29812e6 0.0440707
\(973\) 5.39866e7 1.82811
\(974\) 8.90864e6 0.300894
\(975\) 0 0
\(976\) −5.91193e6 −0.198657
\(977\) −1.41314e7 −0.473640 −0.236820 0.971554i \(-0.576105\pi\)
−0.236820 + 0.971554i \(0.576105\pi\)
\(978\) −2.81543e6 −0.0941233
\(979\) −5.40468e6 −0.180224
\(980\) 0 0
\(981\) −7.29798e6 −0.242120
\(982\) −2.09364e7 −0.692826
\(983\) −3.18529e7 −1.05139 −0.525697 0.850672i \(-0.676195\pi\)
−0.525697 + 0.850672i \(0.676195\pi\)
\(984\) −1.84362e7 −0.606993
\(985\) 0 0
\(986\) 3.00136e7 0.983164
\(987\) −3.66156e7 −1.19639
\(988\) 1.88242e7 0.613513
\(989\) −2.04437e7 −0.664613
\(990\) 0 0
\(991\) 9.56789e6 0.309480 0.154740 0.987955i \(-0.450546\pi\)
0.154740 + 0.987955i \(0.450546\pi\)
\(992\) −5.45516e7 −1.76006
\(993\) 2.60392e6 0.0838019
\(994\) 3.72932e7 1.19719
\(995\) 0 0
\(996\) 3.43409e6 0.109689
\(997\) −954420. −0.0304090 −0.0152045 0.999884i \(-0.504840\pi\)
−0.0152045 + 0.999884i \(0.504840\pi\)
\(998\) 8.98382e6 0.285519
\(999\) 2.28789e6 0.0725307
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 825.6.a.u.1.7 yes 10
5.4 even 2 825.6.a.t.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.4 10 5.4 even 2
825.6.a.u.1.7 yes 10 1.1 even 1 trivial