Properties

Label 825.6.a.t.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 \(-3.54519\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.54519 q^{2} +9.00000 q^{3} -19.4316 q^{4} +31.9067 q^{6} +15.4153 q^{7} -182.335 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.54519 q^{2} +9.00000 q^{3} -19.4316 q^{4} +31.9067 q^{6} +15.4153 q^{7} -182.335 q^{8} +81.0000 q^{9} +121.000 q^{11} -174.885 q^{12} -9.16464 q^{13} +54.6502 q^{14} -24.6000 q^{16} -1481.22 q^{17} +287.160 q^{18} +1058.30 q^{19} +138.738 q^{21} +428.968 q^{22} -3006.99 q^{23} -1641.01 q^{24} -32.4904 q^{26} +729.000 q^{27} -299.544 q^{28} +3763.09 q^{29} +1375.47 q^{31} +5747.51 q^{32} +1089.00 q^{33} -5251.22 q^{34} -1573.96 q^{36} +2015.66 q^{37} +3751.87 q^{38} -82.4818 q^{39} +16545.7 q^{41} +491.852 q^{42} -13647.3 q^{43} -2351.23 q^{44} -10660.4 q^{46} +6638.58 q^{47} -221.400 q^{48} -16569.4 q^{49} -13331.0 q^{51} +178.084 q^{52} -7719.83 q^{53} +2584.44 q^{54} -2810.75 q^{56} +9524.68 q^{57} +13340.9 q^{58} +39159.4 q^{59} -8220.98 q^{61} +4876.30 q^{62} +1248.64 q^{63} +21163.2 q^{64} +3860.71 q^{66} +38209.1 q^{67} +28782.6 q^{68} -27062.9 q^{69} -48470.8 q^{71} -14769.1 q^{72} +49320.4 q^{73} +7145.91 q^{74} -20564.5 q^{76} +1865.25 q^{77} -292.414 q^{78} +16423.9 q^{79} +6561.00 q^{81} +58657.8 q^{82} -419.670 q^{83} -2695.90 q^{84} -48382.3 q^{86} +33867.8 q^{87} -22062.5 q^{88} -17081.2 q^{89} -141.276 q^{91} +58430.7 q^{92} +12379.2 q^{93} +23535.0 q^{94} +51727.5 q^{96} -3464.57 q^{97} -58741.6 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.54519 0.626707 0.313354 0.949637i \(-0.398548\pi\)
0.313354 + 0.949637i \(0.398548\pi\)
\(3\) 9.00000 0.577350
\(4\) −19.4316 −0.607238
\(5\) 0 0
\(6\) 31.9067 0.361829
\(7\) 15.4153 0.118907 0.0594534 0.998231i \(-0.481064\pi\)
0.0594534 + 0.998231i \(0.481064\pi\)
\(8\) −182.335 −1.00727
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −174.885 −0.350589
\(13\) −9.16464 −0.0150403 −0.00752016 0.999972i \(-0.502394\pi\)
−0.00752016 + 0.999972i \(0.502394\pi\)
\(14\) 54.6502 0.0745198
\(15\) 0 0
\(16\) −24.6000 −0.0240234
\(17\) −1481.22 −1.24308 −0.621539 0.783383i \(-0.713492\pi\)
−0.621539 + 0.783383i \(0.713492\pi\)
\(18\) 287.160 0.208902
\(19\) 1058.30 0.672549 0.336275 0.941764i \(-0.390833\pi\)
0.336275 + 0.941764i \(0.390833\pi\)
\(20\) 0 0
\(21\) 138.738 0.0686509
\(22\) 428.968 0.188959
\(23\) −3006.99 −1.18526 −0.592629 0.805476i \(-0.701910\pi\)
−0.592629 + 0.805476i \(0.701910\pi\)
\(24\) −1641.01 −0.581546
\(25\) 0 0
\(26\) −32.4904 −0.00942587
\(27\) 729.000 0.192450
\(28\) −299.544 −0.0722048
\(29\) 3763.09 0.830901 0.415450 0.909616i \(-0.363624\pi\)
0.415450 + 0.909616i \(0.363624\pi\)
\(30\) 0 0
\(31\) 1375.47 0.257067 0.128534 0.991705i \(-0.458973\pi\)
0.128534 + 0.991705i \(0.458973\pi\)
\(32\) 5747.51 0.992212
\(33\) 1089.00 0.174078
\(34\) −5251.22 −0.779046
\(35\) 0 0
\(36\) −1573.96 −0.202413
\(37\) 2015.66 0.242055 0.121027 0.992649i \(-0.461381\pi\)
0.121027 + 0.992649i \(0.461381\pi\)
\(38\) 3751.87 0.421491
\(39\) −82.4818 −0.00868353
\(40\) 0 0
\(41\) 16545.7 1.53719 0.768593 0.639738i \(-0.220957\pi\)
0.768593 + 0.639738i \(0.220957\pi\)
\(42\) 491.852 0.0430240
\(43\) −13647.3 −1.12558 −0.562790 0.826600i \(-0.690272\pi\)
−0.562790 + 0.826600i \(0.690272\pi\)
\(44\) −2351.23 −0.183089
\(45\) 0 0
\(46\) −10660.4 −0.742809
\(47\) 6638.58 0.438360 0.219180 0.975684i \(-0.429662\pi\)
0.219180 + 0.975684i \(0.429662\pi\)
\(48\) −221.400 −0.0138699
\(49\) −16569.4 −0.985861
\(50\) 0 0
\(51\) −13331.0 −0.717691
\(52\) 178.084 0.00913306
\(53\) −7719.83 −0.377501 −0.188751 0.982025i \(-0.560444\pi\)
−0.188751 + 0.982025i \(0.560444\pi\)
\(54\) 2584.44 0.120610
\(55\) 0 0
\(56\) −2810.75 −0.119771
\(57\) 9524.68 0.388296
\(58\) 13340.9 0.520731
\(59\) 39159.4 1.46455 0.732277 0.681007i \(-0.238458\pi\)
0.732277 + 0.681007i \(0.238458\pi\)
\(60\) 0 0
\(61\) −8220.98 −0.282878 −0.141439 0.989947i \(-0.545173\pi\)
−0.141439 + 0.989947i \(0.545173\pi\)
\(62\) 4876.30 0.161106
\(63\) 1248.64 0.0396356
\(64\) 21163.2 0.645850
\(65\) 0 0
\(66\) 3860.71 0.109096
\(67\) 38209.1 1.03987 0.519936 0.854205i \(-0.325956\pi\)
0.519936 + 0.854205i \(0.325956\pi\)
\(68\) 28782.6 0.754845
\(69\) −27062.9 −0.684308
\(70\) 0 0
\(71\) −48470.8 −1.14113 −0.570564 0.821253i \(-0.693275\pi\)
−0.570564 + 0.821253i \(0.693275\pi\)
\(72\) −14769.1 −0.335756
\(73\) 49320.4 1.08323 0.541613 0.840628i \(-0.317814\pi\)
0.541613 + 0.840628i \(0.317814\pi\)
\(74\) 7145.91 0.151698
\(75\) 0 0
\(76\) −20564.5 −0.408398
\(77\) 1865.25 0.0358518
\(78\) −292.414 −0.00544203
\(79\) 16423.9 0.296079 0.148040 0.988981i \(-0.452704\pi\)
0.148040 + 0.988981i \(0.452704\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 58657.8 0.963365
\(83\) −419.670 −0.00668671 −0.00334336 0.999994i \(-0.501064\pi\)
−0.00334336 + 0.999994i \(0.501064\pi\)
\(84\) −2695.90 −0.0416875
\(85\) 0 0
\(86\) −48382.3 −0.705409
\(87\) 33867.8 0.479721
\(88\) −22062.5 −0.303703
\(89\) −17081.2 −0.228583 −0.114291 0.993447i \(-0.536460\pi\)
−0.114291 + 0.993447i \(0.536460\pi\)
\(90\) 0 0
\(91\) −141.276 −0.00178840
\(92\) 58430.7 0.719733
\(93\) 12379.2 0.148418
\(94\) 23535.0 0.274723
\(95\) 0 0
\(96\) 51727.5 0.572854
\(97\) −3464.57 −0.0373870 −0.0186935 0.999825i \(-0.505951\pi\)
−0.0186935 + 0.999825i \(0.505951\pi\)
\(98\) −58741.6 −0.617846
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 118690. 1.15774 0.578868 0.815421i \(-0.303495\pi\)
0.578868 + 0.815421i \(0.303495\pi\)
\(102\) −47261.0 −0.449782
\(103\) 107513. 0.998544 0.499272 0.866445i \(-0.333601\pi\)
0.499272 + 0.866445i \(0.333601\pi\)
\(104\) 1671.03 0.0151496
\(105\) 0 0
\(106\) −27368.3 −0.236583
\(107\) 187147. 1.58024 0.790120 0.612952i \(-0.210018\pi\)
0.790120 + 0.612952i \(0.210018\pi\)
\(108\) −14165.7 −0.116863
\(109\) −496.203 −0.00400031 −0.00200015 0.999998i \(-0.500637\pi\)
−0.00200015 + 0.999998i \(0.500637\pi\)
\(110\) 0 0
\(111\) 18141.0 0.139750
\(112\) −379.216 −0.00285655
\(113\) −109520. −0.806859 −0.403429 0.915011i \(-0.632182\pi\)
−0.403429 + 0.915011i \(0.632182\pi\)
\(114\) 33766.8 0.243348
\(115\) 0 0
\(116\) −73122.9 −0.504555
\(117\) −742.336 −0.00501344
\(118\) 138827. 0.917847
\(119\) −22833.5 −0.147811
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −29144.9 −0.177281
\(123\) 148912. 0.887495
\(124\) −26727.6 −0.156101
\(125\) 0 0
\(126\) 4426.66 0.0248399
\(127\) 1438.91 0.00791634 0.00395817 0.999992i \(-0.498740\pi\)
0.00395817 + 0.999992i \(0.498740\pi\)
\(128\) −108893. −0.587453
\(129\) −122826. −0.649854
\(130\) 0 0
\(131\) 289785. 1.47536 0.737680 0.675151i \(-0.235921\pi\)
0.737680 + 0.675151i \(0.235921\pi\)
\(132\) −21161.0 −0.105707
\(133\) 16314.0 0.0799707
\(134\) 135459. 0.651695
\(135\) 0 0
\(136\) 270079. 1.25211
\(137\) 283421. 1.29012 0.645062 0.764130i \(-0.276832\pi\)
0.645062 + 0.764130i \(0.276832\pi\)
\(138\) −95943.2 −0.428861
\(139\) 200510. 0.880238 0.440119 0.897940i \(-0.354936\pi\)
0.440119 + 0.897940i \(0.354936\pi\)
\(140\) 0 0
\(141\) 59747.3 0.253087
\(142\) −171838. −0.715153
\(143\) −1108.92 −0.00453483
\(144\) −1992.60 −0.00800780
\(145\) 0 0
\(146\) 174850. 0.678865
\(147\) −149124. −0.569187
\(148\) −39167.6 −0.146985
\(149\) 350853. 1.29467 0.647336 0.762205i \(-0.275883\pi\)
0.647336 + 0.762205i \(0.275883\pi\)
\(150\) 0 0
\(151\) −66420.6 −0.237061 −0.118531 0.992950i \(-0.537818\pi\)
−0.118531 + 0.992950i \(0.537818\pi\)
\(152\) −192965. −0.677437
\(153\) −119979. −0.414359
\(154\) 6612.67 0.0224686
\(155\) 0 0
\(156\) 1602.75 0.00527297
\(157\) 507303. 1.64255 0.821275 0.570533i \(-0.193263\pi\)
0.821275 + 0.570533i \(0.193263\pi\)
\(158\) 58225.8 0.185555
\(159\) −69478.5 −0.217950
\(160\) 0 0
\(161\) −46353.7 −0.140935
\(162\) 23260.0 0.0696341
\(163\) −402141. −1.18552 −0.592761 0.805379i \(-0.701962\pi\)
−0.592761 + 0.805379i \(0.701962\pi\)
\(164\) −321511. −0.933438
\(165\) 0 0
\(166\) −1487.81 −0.00419061
\(167\) 293252. 0.813672 0.406836 0.913501i \(-0.366632\pi\)
0.406836 + 0.913501i \(0.366632\pi\)
\(168\) −25296.7 −0.0691498
\(169\) −371209. −0.999774
\(170\) 0 0
\(171\) 85722.2 0.224183
\(172\) 265190. 0.683495
\(173\) −69667.0 −0.176975 −0.0884875 0.996077i \(-0.528203\pi\)
−0.0884875 + 0.996077i \(0.528203\pi\)
\(174\) 120068. 0.300644
\(175\) 0 0
\(176\) −2976.60 −0.00724333
\(177\) 352434. 0.845561
\(178\) −60556.1 −0.143254
\(179\) 137994. 0.321904 0.160952 0.986962i \(-0.448544\pi\)
0.160952 + 0.986962i \(0.448544\pi\)
\(180\) 0 0
\(181\) 365502. 0.829264 0.414632 0.909989i \(-0.363910\pi\)
0.414632 + 0.909989i \(0.363910\pi\)
\(182\) −500.849 −0.00112080
\(183\) −73988.8 −0.163320
\(184\) 548279. 1.19387
\(185\) 0 0
\(186\) 43886.7 0.0930145
\(187\) −179228. −0.374802
\(188\) −128998. −0.266189
\(189\) 11237.8 0.0228836
\(190\) 0 0
\(191\) 339173. 0.672725 0.336363 0.941733i \(-0.390803\pi\)
0.336363 + 0.941733i \(0.390803\pi\)
\(192\) 190469. 0.372881
\(193\) −84815.0 −0.163900 −0.0819500 0.996636i \(-0.526115\pi\)
−0.0819500 + 0.996636i \(0.526115\pi\)
\(194\) −12282.6 −0.0234307
\(195\) 0 0
\(196\) 321970. 0.598653
\(197\) −15771.8 −0.0289545 −0.0144772 0.999895i \(-0.504608\pi\)
−0.0144772 + 0.999895i \(0.504608\pi\)
\(198\) 34746.4 0.0629864
\(199\) 937964. 1.67901 0.839505 0.543351i \(-0.182845\pi\)
0.839505 + 0.543351i \(0.182845\pi\)
\(200\) 0 0
\(201\) 343882. 0.600371
\(202\) 420777. 0.725561
\(203\) 58009.1 0.0987998
\(204\) 259043. 0.435810
\(205\) 0 0
\(206\) 381154. 0.625795
\(207\) −243566. −0.395086
\(208\) 225.450 0.000361320 0
\(209\) 128054. 0.202781
\(210\) 0 0
\(211\) −680076. −1.05160 −0.525801 0.850608i \(-0.676234\pi\)
−0.525801 + 0.850608i \(0.676234\pi\)
\(212\) 150009. 0.229233
\(213\) −436237. −0.658831
\(214\) 663471. 0.990348
\(215\) 0 0
\(216\) −132922. −0.193849
\(217\) 21203.3 0.0305671
\(218\) −1759.14 −0.00250702
\(219\) 443883. 0.625401
\(220\) 0 0
\(221\) 13574.9 0.0186963
\(222\) 64313.2 0.0875826
\(223\) 398926. 0.537193 0.268596 0.963253i \(-0.413440\pi\)
0.268596 + 0.963253i \(0.413440\pi\)
\(224\) 88599.5 0.117981
\(225\) 0 0
\(226\) −388269. −0.505664
\(227\) 327010. 0.421208 0.210604 0.977571i \(-0.432457\pi\)
0.210604 + 0.977571i \(0.432457\pi\)
\(228\) −185080. −0.235788
\(229\) 855131. 1.07757 0.538783 0.842445i \(-0.318884\pi\)
0.538783 + 0.842445i \(0.318884\pi\)
\(230\) 0 0
\(231\) 16787.3 0.0206990
\(232\) −686142. −0.836940
\(233\) −1.30246e6 −1.57172 −0.785858 0.618407i \(-0.787778\pi\)
−0.785858 + 0.618407i \(0.787778\pi\)
\(234\) −2631.72 −0.00314196
\(235\) 0 0
\(236\) −760930. −0.889333
\(237\) 147815. 0.170941
\(238\) −80949.2 −0.0926339
\(239\) −887999. −1.00558 −0.502791 0.864408i \(-0.667694\pi\)
−0.502791 + 0.864408i \(0.667694\pi\)
\(240\) 0 0
\(241\) 787905. 0.873838 0.436919 0.899501i \(-0.356070\pi\)
0.436919 + 0.899501i \(0.356070\pi\)
\(242\) 51905.1 0.0569734
\(243\) 59049.0 0.0641500
\(244\) 159747. 0.171774
\(245\) 0 0
\(246\) 527920. 0.556199
\(247\) −9698.92 −0.0101154
\(248\) −250796. −0.258936
\(249\) −3777.03 −0.00386057
\(250\) 0 0
\(251\) −1.07431e6 −1.07633 −0.538167 0.842838i \(-0.680883\pi\)
−0.538167 + 0.842838i \(0.680883\pi\)
\(252\) −24263.1 −0.0240683
\(253\) −363846. −0.357368
\(254\) 5101.21 0.00496123
\(255\) 0 0
\(256\) −1.06327e6 −1.01401
\(257\) 183068. 0.172894 0.0864471 0.996256i \(-0.472449\pi\)
0.0864471 + 0.996256i \(0.472449\pi\)
\(258\) −435441. −0.407268
\(259\) 31072.1 0.0287820
\(260\) 0 0
\(261\) 304810. 0.276967
\(262\) 1.02734e6 0.924618
\(263\) −1.79693e6 −1.60193 −0.800963 0.598714i \(-0.795679\pi\)
−0.800963 + 0.598714i \(0.795679\pi\)
\(264\) −198563. −0.175343
\(265\) 0 0
\(266\) 57836.2 0.0501182
\(267\) −153731. −0.131972
\(268\) −742465. −0.631450
\(269\) −198575. −0.167319 −0.0836593 0.996494i \(-0.526661\pi\)
−0.0836593 + 0.996494i \(0.526661\pi\)
\(270\) 0 0
\(271\) 2.09316e6 1.73133 0.865663 0.500627i \(-0.166897\pi\)
0.865663 + 0.500627i \(0.166897\pi\)
\(272\) 36438.1 0.0298630
\(273\) −1271.48 −0.00103253
\(274\) 1.00478e6 0.808530
\(275\) 0 0
\(276\) 525876. 0.415538
\(277\) 634155. 0.496588 0.248294 0.968685i \(-0.420130\pi\)
0.248294 + 0.968685i \(0.420130\pi\)
\(278\) 710848. 0.551651
\(279\) 111413. 0.0856891
\(280\) 0 0
\(281\) 724201. 0.547133 0.273567 0.961853i \(-0.411797\pi\)
0.273567 + 0.961853i \(0.411797\pi\)
\(282\) 211815. 0.158612
\(283\) −2.42288e6 −1.79831 −0.899157 0.437626i \(-0.855819\pi\)
−0.899157 + 0.437626i \(0.855819\pi\)
\(284\) 941867. 0.692937
\(285\) 0 0
\(286\) −3931.34 −0.00284201
\(287\) 255058. 0.182782
\(288\) 465548. 0.330737
\(289\) 774167. 0.545243
\(290\) 0 0
\(291\) −31181.2 −0.0215854
\(292\) −958375. −0.657776
\(293\) −742942. −0.505575 −0.252788 0.967522i \(-0.581347\pi\)
−0.252788 + 0.967522i \(0.581347\pi\)
\(294\) −528674. −0.356714
\(295\) 0 0
\(296\) −367526. −0.243814
\(297\) 88209.0 0.0580259
\(298\) 1.24384e6 0.811379
\(299\) 27558.0 0.0178266
\(300\) 0 0
\(301\) −210378. −0.133839
\(302\) −235474. −0.148568
\(303\) 1.06821e6 0.668419
\(304\) −26034.1 −0.0161569
\(305\) 0 0
\(306\) −425349. −0.259682
\(307\) −1.25868e6 −0.762203 −0.381102 0.924533i \(-0.624455\pi\)
−0.381102 + 0.924533i \(0.624455\pi\)
\(308\) −36244.9 −0.0217706
\(309\) 967616. 0.576510
\(310\) 0 0
\(311\) −1.71512e6 −1.00553 −0.502763 0.864424i \(-0.667683\pi\)
−0.502763 + 0.864424i \(0.667683\pi\)
\(312\) 15039.3 0.00874664
\(313\) 2.85909e6 1.64956 0.824778 0.565457i \(-0.191300\pi\)
0.824778 + 0.565457i \(0.191300\pi\)
\(314\) 1.79849e6 1.02940
\(315\) 0 0
\(316\) −319143. −0.179791
\(317\) 1.60936e6 0.899510 0.449755 0.893152i \(-0.351511\pi\)
0.449755 + 0.893152i \(0.351511\pi\)
\(318\) −246315. −0.136591
\(319\) 455334. 0.250526
\(320\) 0 0
\(321\) 1.68432e6 0.912352
\(322\) −164333. −0.0883251
\(323\) −1.56758e6 −0.836031
\(324\) −127491. −0.0674709
\(325\) 0 0
\(326\) −1.42567e6 −0.742975
\(327\) −4465.83 −0.00230958
\(328\) −3.01687e6 −1.54836
\(329\) 102336. 0.0521240
\(330\) 0 0
\(331\) −1.21843e6 −0.611264 −0.305632 0.952150i \(-0.598868\pi\)
−0.305632 + 0.952150i \(0.598868\pi\)
\(332\) 8154.87 0.00406043
\(333\) 163269. 0.0806850
\(334\) 1.03963e6 0.509934
\(335\) 0 0
\(336\) −3412.94 −0.00164923
\(337\) −1.59524e6 −0.765160 −0.382580 0.923922i \(-0.624964\pi\)
−0.382580 + 0.923922i \(0.624964\pi\)
\(338\) −1.31601e6 −0.626565
\(339\) −985681. −0.465840
\(340\) 0 0
\(341\) 166432. 0.0775087
\(342\) 303901. 0.140497
\(343\) −514507. −0.236133
\(344\) 2.48838e6 1.13376
\(345\) 0 0
\(346\) −246983. −0.110911
\(347\) 2.98284e6 1.32986 0.664930 0.746906i \(-0.268462\pi\)
0.664930 + 0.746906i \(0.268462\pi\)
\(348\) −658106. −0.291305
\(349\) 1.45121e6 0.637775 0.318887 0.947793i \(-0.396691\pi\)
0.318887 + 0.947793i \(0.396691\pi\)
\(350\) 0 0
\(351\) −6681.02 −0.00289451
\(352\) 695448. 0.299163
\(353\) 373296. 0.159447 0.0797235 0.996817i \(-0.474596\pi\)
0.0797235 + 0.996817i \(0.474596\pi\)
\(354\) 1.24945e6 0.529919
\(355\) 0 0
\(356\) 331915. 0.138804
\(357\) −205502. −0.0853384
\(358\) 489213. 0.201739
\(359\) 1.59973e6 0.655103 0.327551 0.944833i \(-0.393777\pi\)
0.327551 + 0.944833i \(0.393777\pi\)
\(360\) 0 0
\(361\) −1.35610e6 −0.547678
\(362\) 1.29577e6 0.519706
\(363\) 131769. 0.0524864
\(364\) 2745.22 0.00108598
\(365\) 0 0
\(366\) −262304. −0.102354
\(367\) −3.67213e6 −1.42316 −0.711578 0.702607i \(-0.752019\pi\)
−0.711578 + 0.702607i \(0.752019\pi\)
\(368\) 73971.9 0.0284739
\(369\) 1.34020e6 0.512395
\(370\) 0 0
\(371\) −119004. −0.0448875
\(372\) −240549. −0.0901250
\(373\) −290636. −0.108163 −0.0540814 0.998537i \(-0.517223\pi\)
−0.0540814 + 0.998537i \(0.517223\pi\)
\(374\) −635398. −0.234891
\(375\) 0 0
\(376\) −1.21045e6 −0.441546
\(377\) −34487.3 −0.0124970
\(378\) 39840.0 0.0143413
\(379\) 4.72607e6 1.69006 0.845031 0.534718i \(-0.179582\pi\)
0.845031 + 0.534718i \(0.179582\pi\)
\(380\) 0 0
\(381\) 12950.2 0.00457050
\(382\) 1.20243e6 0.421602
\(383\) 1.69814e6 0.591531 0.295766 0.955261i \(-0.404425\pi\)
0.295766 + 0.955261i \(0.404425\pi\)
\(384\) −980033. −0.339166
\(385\) 0 0
\(386\) −300685. −0.102717
\(387\) −1.10543e6 −0.375193
\(388\) 67322.3 0.0227028
\(389\) −614215. −0.205800 −0.102900 0.994692i \(-0.532812\pi\)
−0.102900 + 0.994692i \(0.532812\pi\)
\(390\) 0 0
\(391\) 4.45403e6 1.47337
\(392\) 3.02117e6 0.993026
\(393\) 2.60807e6 0.851799
\(394\) −55914.0 −0.0181460
\(395\) 0 0
\(396\) −190449. −0.0610297
\(397\) 151925. 0.0483786 0.0241893 0.999707i \(-0.492300\pi\)
0.0241893 + 0.999707i \(0.492300\pi\)
\(398\) 3.32526e6 1.05225
\(399\) 146826. 0.0461711
\(400\) 0 0
\(401\) −411747. −0.127870 −0.0639350 0.997954i \(-0.520365\pi\)
−0.0639350 + 0.997954i \(0.520365\pi\)
\(402\) 1.21913e6 0.376256
\(403\) −12605.7 −0.00386637
\(404\) −2.30633e6 −0.703021
\(405\) 0 0
\(406\) 205653. 0.0619186
\(407\) 243895. 0.0729823
\(408\) 2.43071e6 0.722907
\(409\) −522118. −0.154334 −0.0771668 0.997018i \(-0.524587\pi\)
−0.0771668 + 0.997018i \(0.524587\pi\)
\(410\) 0 0
\(411\) 2.55079e6 0.744853
\(412\) −2.08915e6 −0.606354
\(413\) 603653. 0.174146
\(414\) −863489. −0.247603
\(415\) 0 0
\(416\) −52673.8 −0.0149232
\(417\) 1.80459e6 0.508205
\(418\) 453976. 0.127084
\(419\) 4.42421e6 1.23112 0.615561 0.788090i \(-0.288930\pi\)
0.615561 + 0.788090i \(0.288930\pi\)
\(420\) 0 0
\(421\) 41986.3 0.0115452 0.00577261 0.999983i \(-0.498163\pi\)
0.00577261 + 0.999983i \(0.498163\pi\)
\(422\) −2.41100e6 −0.659047
\(423\) 537725. 0.146120
\(424\) 1.40760e6 0.380245
\(425\) 0 0
\(426\) −1.54654e6 −0.412894
\(427\) −126729. −0.0336361
\(428\) −3.63657e6 −0.959582
\(429\) −9980.29 −0.00261818
\(430\) 0 0
\(431\) 2.90650e6 0.753664 0.376832 0.926282i \(-0.377013\pi\)
0.376832 + 0.926282i \(0.377013\pi\)
\(432\) −17933.4 −0.00462331
\(433\) −1.49232e6 −0.382508 −0.191254 0.981541i \(-0.561255\pi\)
−0.191254 + 0.981541i \(0.561255\pi\)
\(434\) 75169.7 0.0191566
\(435\) 0 0
\(436\) 9642.04 0.00242914
\(437\) −3.18229e6 −0.797144
\(438\) 1.57365e6 0.391943
\(439\) −6.08351e6 −1.50658 −0.753291 0.657687i \(-0.771535\pi\)
−0.753291 + 0.657687i \(0.771535\pi\)
\(440\) 0 0
\(441\) −1.34212e6 −0.328620
\(442\) 48125.5 0.0117171
\(443\) 1.20270e6 0.291171 0.145586 0.989346i \(-0.453493\pi\)
0.145586 + 0.989346i \(0.453493\pi\)
\(444\) −352509. −0.0848618
\(445\) 0 0
\(446\) 1.41427e6 0.336662
\(447\) 3.15768e6 0.747479
\(448\) 326237. 0.0767960
\(449\) −935321. −0.218950 −0.109475 0.993990i \(-0.534917\pi\)
−0.109475 + 0.993990i \(0.534917\pi\)
\(450\) 0 0
\(451\) 2.00203e6 0.463479
\(452\) 2.12815e6 0.489956
\(453\) −597786. −0.136867
\(454\) 1.15931e6 0.263974
\(455\) 0 0
\(456\) −1.73668e6 −0.391118
\(457\) −530855. −0.118901 −0.0594504 0.998231i \(-0.518935\pi\)
−0.0594504 + 0.998231i \(0.518935\pi\)
\(458\) 3.03160e6 0.675318
\(459\) −1.07981e6 −0.239230
\(460\) 0 0
\(461\) 2.26064e6 0.495426 0.247713 0.968833i \(-0.420321\pi\)
0.247713 + 0.968833i \(0.420321\pi\)
\(462\) 59514.1 0.0129722
\(463\) 3.48143e6 0.754753 0.377376 0.926060i \(-0.376826\pi\)
0.377376 + 0.926060i \(0.376826\pi\)
\(464\) −92571.8 −0.0199611
\(465\) 0 0
\(466\) −4.61746e6 −0.985006
\(467\) 6.14838e6 1.30457 0.652287 0.757972i \(-0.273810\pi\)
0.652287 + 0.757972i \(0.273810\pi\)
\(468\) 14424.8 0.00304435
\(469\) 589005. 0.123648
\(470\) 0 0
\(471\) 4.56573e6 0.948326
\(472\) −7.14012e6 −1.47520
\(473\) −1.65133e6 −0.339375
\(474\) 524032. 0.107130
\(475\) 0 0
\(476\) 443692. 0.0897562
\(477\) −625307. −0.125834
\(478\) −3.14813e6 −0.630206
\(479\) −5.92932e6 −1.18077 −0.590386 0.807121i \(-0.701024\pi\)
−0.590386 + 0.807121i \(0.701024\pi\)
\(480\) 0 0
\(481\) −18472.8 −0.00364058
\(482\) 2.79327e6 0.547641
\(483\) −417183. −0.0813690
\(484\) −284498. −0.0552035
\(485\) 0 0
\(486\) 209340. 0.0402033
\(487\) −8.99316e6 −1.71826 −0.859132 0.511754i \(-0.828996\pi\)
−0.859132 + 0.511754i \(0.828996\pi\)
\(488\) 1.49897e6 0.284934
\(489\) −3.61927e6 −0.684461
\(490\) 0 0
\(491\) −2.74306e6 −0.513489 −0.256745 0.966479i \(-0.582650\pi\)
−0.256745 + 0.966479i \(0.582650\pi\)
\(492\) −2.89359e6 −0.538921
\(493\) −5.57397e6 −1.03287
\(494\) −34384.5 −0.00633936
\(495\) 0 0
\(496\) −33836.5 −0.00617563
\(497\) −747192. −0.135688
\(498\) −13390.3 −0.00241945
\(499\) −6.85332e6 −1.23211 −0.616055 0.787703i \(-0.711270\pi\)
−0.616055 + 0.787703i \(0.711270\pi\)
\(500\) 0 0
\(501\) 2.63927e6 0.469774
\(502\) −3.80865e6 −0.674546
\(503\) −3.53279e6 −0.622583 −0.311292 0.950314i \(-0.600762\pi\)
−0.311292 + 0.950314i \(0.600762\pi\)
\(504\) −227671. −0.0399237
\(505\) 0 0
\(506\) −1.28990e6 −0.223965
\(507\) −3.34088e6 −0.577220
\(508\) −27960.4 −0.00480711
\(509\) −8.42114e6 −1.44071 −0.720354 0.693606i \(-0.756021\pi\)
−0.720354 + 0.693606i \(0.756021\pi\)
\(510\) 0 0
\(511\) 760288. 0.128803
\(512\) −284922. −0.0480343
\(513\) 771499. 0.129432
\(514\) 649012. 0.108354
\(515\) 0 0
\(516\) 2.38671e6 0.394616
\(517\) 803269. 0.132170
\(518\) 110156. 0.0180379
\(519\) −627003. −0.102177
\(520\) 0 0
\(521\) 1.11375e6 0.179761 0.0898804 0.995953i \(-0.471352\pi\)
0.0898804 + 0.995953i \(0.471352\pi\)
\(522\) 1.08061e6 0.173577
\(523\) −1.22565e6 −0.195935 −0.0979674 0.995190i \(-0.531234\pi\)
−0.0979674 + 0.995190i \(0.531234\pi\)
\(524\) −5.63100e6 −0.895895
\(525\) 0 0
\(526\) −6.37047e6 −1.00394
\(527\) −2.03738e6 −0.319555
\(528\) −26789.4 −0.00418194
\(529\) 2.60565e6 0.404834
\(530\) 0 0
\(531\) 3.17191e6 0.488185
\(532\) −317007. −0.0485613
\(533\) −151636. −0.0231198
\(534\) −545005. −0.0827080
\(535\) 0 0
\(536\) −6.96686e6 −1.04743
\(537\) 1.24194e6 0.185851
\(538\) −703987. −0.104860
\(539\) −2.00489e6 −0.297248
\(540\) 0 0
\(541\) −47884.8 −0.00703403 −0.00351701 0.999994i \(-0.501120\pi\)
−0.00351701 + 0.999994i \(0.501120\pi\)
\(542\) 7.42065e6 1.08503
\(543\) 3.28951e6 0.478776
\(544\) −8.51334e6 −1.23340
\(545\) 0 0
\(546\) −4507.64 −0.000647095 0
\(547\) 1.04705e6 0.149624 0.0748118 0.997198i \(-0.476164\pi\)
0.0748118 + 0.997198i \(0.476164\pi\)
\(548\) −5.50734e6 −0.783412
\(549\) −665899. −0.0942926
\(550\) 0 0
\(551\) 3.98247e6 0.558822
\(552\) 4.93451e6 0.689282
\(553\) 253179. 0.0352059
\(554\) 2.24820e6 0.311215
\(555\) 0 0
\(556\) −3.89624e6 −0.534514
\(557\) 8.54477e6 1.16698 0.583488 0.812121i \(-0.301687\pi\)
0.583488 + 0.812121i \(0.301687\pi\)
\(558\) 394980. 0.0537020
\(559\) 125073. 0.0169291
\(560\) 0 0
\(561\) −1.61305e6 −0.216392
\(562\) 2.56743e6 0.342892
\(563\) 3.21883e6 0.427984 0.213992 0.976835i \(-0.431353\pi\)
0.213992 + 0.976835i \(0.431353\pi\)
\(564\) −1.16099e6 −0.153684
\(565\) 0 0
\(566\) −8.58957e6 −1.12702
\(567\) 101140. 0.0132119
\(568\) 8.83792e6 1.14942
\(569\) −9.31971e6 −1.20676 −0.603381 0.797453i \(-0.706180\pi\)
−0.603381 + 0.797453i \(0.706180\pi\)
\(570\) 0 0
\(571\) 1.38799e7 1.78154 0.890769 0.454457i \(-0.150167\pi\)
0.890769 + 0.454457i \(0.150167\pi\)
\(572\) 21548.1 0.00275372
\(573\) 3.05256e6 0.388398
\(574\) 904228. 0.114551
\(575\) 0 0
\(576\) 1.71422e6 0.215283
\(577\) −4.91293e6 −0.614330 −0.307165 0.951656i \(-0.599380\pi\)
−0.307165 + 0.951656i \(0.599380\pi\)
\(578\) 2.74457e6 0.341708
\(579\) −763335. −0.0946278
\(580\) 0 0
\(581\) −6469.34 −0.000795096 0
\(582\) −110543. −0.0135277
\(583\) −934100. −0.113821
\(584\) −8.99282e6 −1.09110
\(585\) 0 0
\(586\) −2.63387e6 −0.316848
\(587\) 5.36479e6 0.642624 0.321312 0.946973i \(-0.395876\pi\)
0.321312 + 0.946973i \(0.395876\pi\)
\(588\) 2.89773e6 0.345632
\(589\) 1.45566e6 0.172890
\(590\) 0 0
\(591\) −141946. −0.0167169
\(592\) −49585.3 −0.00581498
\(593\) 8.86855e6 1.03566 0.517829 0.855484i \(-0.326741\pi\)
0.517829 + 0.855484i \(0.326741\pi\)
\(594\) 312718. 0.0363652
\(595\) 0 0
\(596\) −6.81764e6 −0.786174
\(597\) 8.44168e6 0.969377
\(598\) 97698.3 0.0111721
\(599\) −1.10799e7 −1.26174 −0.630869 0.775889i \(-0.717302\pi\)
−0.630869 + 0.775889i \(0.717302\pi\)
\(600\) 0 0
\(601\) −5.50666e6 −0.621873 −0.310937 0.950431i \(-0.600643\pi\)
−0.310937 + 0.950431i \(0.600643\pi\)
\(602\) −745828. −0.0838779
\(603\) 3.09494e6 0.346624
\(604\) 1.29066e6 0.143953
\(605\) 0 0
\(606\) 3.78700e6 0.418903
\(607\) −1.29430e7 −1.42581 −0.712906 0.701259i \(-0.752621\pi\)
−0.712906 + 0.701259i \(0.752621\pi\)
\(608\) 6.08257e6 0.667311
\(609\) 522082. 0.0570421
\(610\) 0 0
\(611\) −60840.2 −0.00659307
\(612\) 2.33139e6 0.251615
\(613\) −3.41360e6 −0.366911 −0.183456 0.983028i \(-0.558728\pi\)
−0.183456 + 0.983028i \(0.558728\pi\)
\(614\) −4.46227e6 −0.477678
\(615\) 0 0
\(616\) −340100. −0.0361123
\(617\) −1.36370e7 −1.44213 −0.721067 0.692865i \(-0.756348\pi\)
−0.721067 + 0.692865i \(0.756348\pi\)
\(618\) 3.43038e6 0.361303
\(619\) −6.21254e6 −0.651692 −0.325846 0.945423i \(-0.605649\pi\)
−0.325846 + 0.945423i \(0.605649\pi\)
\(620\) 0 0
\(621\) −2.19210e6 −0.228103
\(622\) −6.08042e6 −0.630170
\(623\) −263312. −0.0271801
\(624\) 2029.05 0.000208608 0
\(625\) 0 0
\(626\) 1.01360e7 1.03379
\(627\) 1.15249e6 0.117076
\(628\) −9.85772e6 −0.997419
\(629\) −2.98565e6 −0.300893
\(630\) 0 0
\(631\) −3.32349e6 −0.332293 −0.166146 0.986101i \(-0.553132\pi\)
−0.166146 + 0.986101i \(0.553132\pi\)
\(632\) −2.99465e6 −0.298231
\(633\) −6.12069e6 −0.607143
\(634\) 5.70550e6 0.563729
\(635\) 0 0
\(636\) 1.35008e6 0.132348
\(637\) 151852. 0.0148277
\(638\) 1.61424e6 0.157006
\(639\) −3.92614e6 −0.380376
\(640\) 0 0
\(641\) −1.81780e6 −0.174744 −0.0873720 0.996176i \(-0.527847\pi\)
−0.0873720 + 0.996176i \(0.527847\pi\)
\(642\) 5.97124e6 0.571777
\(643\) −421961. −0.0402481 −0.0201240 0.999797i \(-0.506406\pi\)
−0.0201240 + 0.999797i \(0.506406\pi\)
\(644\) 900727. 0.0855813
\(645\) 0 0
\(646\) −5.55736e6 −0.523947
\(647\) −5.62498e6 −0.528275 −0.264138 0.964485i \(-0.585087\pi\)
−0.264138 + 0.964485i \(0.585087\pi\)
\(648\) −1.19630e6 −0.111919
\(649\) 4.73828e6 0.441580
\(650\) 0 0
\(651\) 190830. 0.0176479
\(652\) 7.81426e6 0.719894
\(653\) 1.69247e7 1.55324 0.776619 0.629970i \(-0.216933\pi\)
0.776619 + 0.629970i \(0.216933\pi\)
\(654\) −15832.2 −0.00144743
\(655\) 0 0
\(656\) −407025. −0.0369285
\(657\) 3.99495e6 0.361075
\(658\) 362800. 0.0326665
\(659\) −480657. −0.0431144 −0.0215572 0.999768i \(-0.506862\pi\)
−0.0215572 + 0.999768i \(0.506862\pi\)
\(660\) 0 0
\(661\) −1.53510e7 −1.36657 −0.683285 0.730151i \(-0.739449\pi\)
−0.683285 + 0.730151i \(0.739449\pi\)
\(662\) −4.31955e6 −0.383083
\(663\) 122174. 0.0107943
\(664\) 76520.5 0.00673531
\(665\) 0 0
\(666\) 578819. 0.0505658
\(667\) −1.13156e7 −0.984831
\(668\) −5.69836e6 −0.494093
\(669\) 3.59033e6 0.310148
\(670\) 0 0
\(671\) −994738. −0.0852909
\(672\) 797396. 0.0681163
\(673\) −3.83589e6 −0.326459 −0.163229 0.986588i \(-0.552191\pi\)
−0.163229 + 0.986588i \(0.552191\pi\)
\(674\) −5.65545e6 −0.479531
\(675\) 0 0
\(676\) 7.21319e6 0.607101
\(677\) −5.74417e6 −0.481676 −0.240838 0.970565i \(-0.577422\pi\)
−0.240838 + 0.970565i \(0.577422\pi\)
\(678\) −3.49443e6 −0.291945
\(679\) −53407.5 −0.00444557
\(680\) 0 0
\(681\) 2.94309e6 0.243185
\(682\) 590033. 0.0485753
\(683\) 2.11651e7 1.73608 0.868038 0.496498i \(-0.165381\pi\)
0.868038 + 0.496498i \(0.165381\pi\)
\(684\) −1.66572e6 −0.136133
\(685\) 0 0
\(686\) −1.82402e6 −0.147986
\(687\) 7.69618e6 0.622133
\(688\) 335724. 0.0270403
\(689\) 70749.5 0.00567774
\(690\) 0 0
\(691\) −2.20475e7 −1.75657 −0.878283 0.478142i \(-0.841311\pi\)
−0.878283 + 0.478142i \(0.841311\pi\)
\(692\) 1.35374e6 0.107466
\(693\) 151085. 0.0119506
\(694\) 1.05747e7 0.833432
\(695\) 0 0
\(696\) −6.17528e6 −0.483207
\(697\) −2.45079e7 −1.91084
\(698\) 5.14482e6 0.399698
\(699\) −1.17221e7 −0.907431
\(700\) 0 0
\(701\) 1.26680e7 0.973676 0.486838 0.873492i \(-0.338150\pi\)
0.486838 + 0.873492i \(0.338150\pi\)
\(702\) −23685.5 −0.00181401
\(703\) 2.13317e6 0.162794
\(704\) 2.56075e6 0.194731
\(705\) 0 0
\(706\) 1.32341e6 0.0999266
\(707\) 1.82964e6 0.137663
\(708\) −6.84837e6 −0.513457
\(709\) −6.24919e6 −0.466883 −0.233441 0.972371i \(-0.574999\pi\)
−0.233441 + 0.972371i \(0.574999\pi\)
\(710\) 0 0
\(711\) 1.33033e6 0.0986931
\(712\) 3.11450e6 0.230244
\(713\) −4.13603e6 −0.304691
\(714\) −728542. −0.0534822
\(715\) 0 0
\(716\) −2.68144e6 −0.195472
\(717\) −7.99199e6 −0.580573
\(718\) 5.67133e6 0.410558
\(719\) 2.29008e6 0.165207 0.0826036 0.996582i \(-0.473676\pi\)
0.0826036 + 0.996582i \(0.473676\pi\)
\(720\) 0 0
\(721\) 1.65734e6 0.118734
\(722\) −4.80765e6 −0.343233
\(723\) 7.09114e6 0.504511
\(724\) −7.10229e6 −0.503561
\(725\) 0 0
\(726\) 467146. 0.0328936
\(727\) −5.32482e6 −0.373653 −0.186827 0.982393i \(-0.559820\pi\)
−0.186827 + 0.982393i \(0.559820\pi\)
\(728\) 25759.5 0.00180139
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.02147e7 1.39918
\(732\) 1.43772e6 0.0991739
\(733\) 2.07826e7 1.42870 0.714348 0.699790i \(-0.246723\pi\)
0.714348 + 0.699790i \(0.246723\pi\)
\(734\) −1.30184e7 −0.891902
\(735\) 0 0
\(736\) −1.72827e7 −1.17603
\(737\) 4.62330e6 0.313533
\(738\) 4.75128e6 0.321122
\(739\) 1.24923e7 0.841454 0.420727 0.907187i \(-0.361775\pi\)
0.420727 + 0.907187i \(0.361775\pi\)
\(740\) 0 0
\(741\) −87290.3 −0.00584010
\(742\) −421890. −0.0281313
\(743\) −2.39696e7 −1.59290 −0.796450 0.604705i \(-0.793291\pi\)
−0.796450 + 0.604705i \(0.793291\pi\)
\(744\) −2.25717e6 −0.149497
\(745\) 0 0
\(746\) −1.03036e6 −0.0677864
\(747\) −33993.3 −0.00222890
\(748\) 3.48269e6 0.227594
\(749\) 2.88493e6 0.187901
\(750\) 0 0
\(751\) 1.76984e6 0.114508 0.0572539 0.998360i \(-0.481766\pi\)
0.0572539 + 0.998360i \(0.481766\pi\)
\(752\) −163309. −0.0105309
\(753\) −9.66883e6 −0.621422
\(754\) −122264. −0.00783197
\(755\) 0 0
\(756\) −218368. −0.0138958
\(757\) −1.88709e7 −1.19689 −0.598443 0.801165i \(-0.704214\pi\)
−0.598443 + 0.801165i \(0.704214\pi\)
\(758\) 1.67548e7 1.05917
\(759\) −3.27461e6 −0.206327
\(760\) 0 0
\(761\) 9.56502e6 0.598721 0.299360 0.954140i \(-0.403227\pi\)
0.299360 + 0.954140i \(0.403227\pi\)
\(762\) 45910.9 0.00286437
\(763\) −7649.12 −0.000475664 0
\(764\) −6.59068e6 −0.408504
\(765\) 0 0
\(766\) 6.02025e6 0.370717
\(767\) −358881. −0.0220274
\(768\) −9.56941e6 −0.585439
\(769\) 2.63227e7 1.60514 0.802572 0.596556i \(-0.203464\pi\)
0.802572 + 0.596556i \(0.203464\pi\)
\(770\) 0 0
\(771\) 1.64761e6 0.0998205
\(772\) 1.64809e6 0.0995264
\(773\) −1.07241e7 −0.645525 −0.322762 0.946480i \(-0.604611\pi\)
−0.322762 + 0.946480i \(0.604611\pi\)
\(774\) −3.91897e6 −0.235136
\(775\) 0 0
\(776\) 631713. 0.0376587
\(777\) 279649. 0.0166173
\(778\) −2.17751e6 −0.128977
\(779\) 1.75103e7 1.03383
\(780\) 0 0
\(781\) −5.86497e6 −0.344063
\(782\) 1.57904e7 0.923369
\(783\) 2.74329e6 0.159907
\(784\) 407606. 0.0236837
\(785\) 0 0
\(786\) 9.24609e6 0.533829
\(787\) −2.31773e7 −1.33391 −0.666954 0.745099i \(-0.732402\pi\)
−0.666954 + 0.745099i \(0.732402\pi\)
\(788\) 306472. 0.0175823
\(789\) −1.61724e7 −0.924873
\(790\) 0 0
\(791\) −1.68828e6 −0.0959411
\(792\) −1.78706e6 −0.101234
\(793\) 75342.3 0.00425457
\(794\) 538604. 0.0303192
\(795\) 0 0
\(796\) −1.82262e7 −1.01956
\(797\) 5.94831e6 0.331702 0.165851 0.986151i \(-0.446963\pi\)
0.165851 + 0.986151i \(0.446963\pi\)
\(798\) 520526. 0.0289358
\(799\) −9.83323e6 −0.544916
\(800\) 0 0
\(801\) −1.38358e6 −0.0761942
\(802\) −1.45972e6 −0.0801371
\(803\) 5.96776e6 0.326605
\(804\) −6.68219e6 −0.364568
\(805\) 0 0
\(806\) −44689.6 −0.00242308
\(807\) −1.78718e6 −0.0966014
\(808\) −2.16413e7 −1.16615
\(809\) 2.46026e6 0.132163 0.0660815 0.997814i \(-0.478950\pi\)
0.0660815 + 0.997814i \(0.478950\pi\)
\(810\) 0 0
\(811\) −1.67473e7 −0.894112 −0.447056 0.894506i \(-0.647528\pi\)
−0.447056 + 0.894506i \(0.647528\pi\)
\(812\) −1.12721e6 −0.0599950
\(813\) 1.88384e7 0.999582
\(814\) 864655. 0.0457385
\(815\) 0 0
\(816\) 327943. 0.0172414
\(817\) −1.44429e7 −0.757008
\(818\) −1.85101e6 −0.0967220
\(819\) −11443.3 −0.000596132 0
\(820\) 0 0
\(821\) 2.04537e7 1.05904 0.529521 0.848297i \(-0.322372\pi\)
0.529521 + 0.848297i \(0.322372\pi\)
\(822\) 9.04305e6 0.466805
\(823\) 6.10386e6 0.314127 0.157063 0.987589i \(-0.449797\pi\)
0.157063 + 0.987589i \(0.449797\pi\)
\(824\) −1.96033e7 −1.00580
\(825\) 0 0
\(826\) 2.14007e6 0.109138
\(827\) −1.25743e7 −0.639321 −0.319661 0.947532i \(-0.603569\pi\)
−0.319661 + 0.947532i \(0.603569\pi\)
\(828\) 4.73289e6 0.239911
\(829\) −3.64808e6 −0.184365 −0.0921824 0.995742i \(-0.529384\pi\)
−0.0921824 + 0.995742i \(0.529384\pi\)
\(830\) 0 0
\(831\) 5.70740e6 0.286705
\(832\) −193953. −0.00971378
\(833\) 2.45429e7 1.22550
\(834\) 6.39763e6 0.318496
\(835\) 0 0
\(836\) −2.48830e6 −0.123137
\(837\) 1.00272e6 0.0494726
\(838\) 1.56847e7 0.771552
\(839\) 3.98638e7 1.95512 0.977560 0.210657i \(-0.0675602\pi\)
0.977560 + 0.210657i \(0.0675602\pi\)
\(840\) 0 0
\(841\) −6.35033e6 −0.309604
\(842\) 148849. 0.00723547
\(843\) 6.51781e6 0.315888
\(844\) 1.32150e7 0.638573
\(845\) 0 0
\(846\) 1.90634e6 0.0915744
\(847\) 225695. 0.0108097
\(848\) 189908. 0.00906886
\(849\) −2.18059e7 −1.03826
\(850\) 0 0
\(851\) −6.06108e6 −0.286897
\(852\) 8.47680e6 0.400067
\(853\) −7.03128e6 −0.330873 −0.165437 0.986220i \(-0.552903\pi\)
−0.165437 + 0.986220i \(0.552903\pi\)
\(854\) −449278. −0.0210800
\(855\) 0 0
\(856\) −3.41234e7 −1.59172
\(857\) 5.35026e6 0.248842 0.124421 0.992230i \(-0.460293\pi\)
0.124421 + 0.992230i \(0.460293\pi\)
\(858\) −35382.0 −0.00164083
\(859\) 3.74127e7 1.72996 0.864981 0.501805i \(-0.167331\pi\)
0.864981 + 0.501805i \(0.167331\pi\)
\(860\) 0 0
\(861\) 2.29552e6 0.105529
\(862\) 1.03041e7 0.472326
\(863\) −2.03619e7 −0.930661 −0.465331 0.885137i \(-0.654065\pi\)
−0.465331 + 0.885137i \(0.654065\pi\)
\(864\) 4.18993e6 0.190951
\(865\) 0 0
\(866\) −5.29054e6 −0.239721
\(867\) 6.96750e6 0.314796
\(868\) −412014. −0.0185615
\(869\) 1.98729e6 0.0892713
\(870\) 0 0
\(871\) −350173. −0.0156400
\(872\) 90475.2 0.00402938
\(873\) −280630. −0.0124623
\(874\) −1.12818e7 −0.499576
\(875\) 0 0
\(876\) −8.62537e6 −0.379767
\(877\) −2.69907e7 −1.18499 −0.592497 0.805573i \(-0.701858\pi\)
−0.592497 + 0.805573i \(0.701858\pi\)
\(878\) −2.15672e7 −0.944186
\(879\) −6.68648e6 −0.291894
\(880\) 0 0
\(881\) −5.82848e6 −0.252997 −0.126499 0.991967i \(-0.540374\pi\)
−0.126499 + 0.991967i \(0.540374\pi\)
\(882\) −4.75807e6 −0.205949
\(883\) 4.30317e6 0.185732 0.0928661 0.995679i \(-0.470397\pi\)
0.0928661 + 0.995679i \(0.470397\pi\)
\(884\) −263782. −0.0113531
\(885\) 0 0
\(886\) 4.26381e6 0.182479
\(887\) 5.28627e6 0.225601 0.112800 0.993618i \(-0.464018\pi\)
0.112800 + 0.993618i \(0.464018\pi\)
\(888\) −3.30773e6 −0.140766
\(889\) 22181.2 0.000941308 0
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −7.75178e6 −0.326204
\(893\) 7.02560e6 0.294819
\(894\) 1.11946e7 0.468450
\(895\) 0 0
\(896\) −1.67861e6 −0.0698522
\(897\) 248022. 0.0102922
\(898\) −3.31589e6 −0.137217
\(899\) 5.17601e6 0.213597
\(900\) 0 0
\(901\) 1.14348e7 0.469263
\(902\) 7.09759e6 0.290466
\(903\) −1.89340e6 −0.0772721
\(904\) 1.99693e7 0.812723
\(905\) 0 0
\(906\) −2.11926e6 −0.0857758
\(907\) −1.24628e6 −0.0503035 −0.0251518 0.999684i \(-0.508007\pi\)
−0.0251518 + 0.999684i \(0.508007\pi\)
\(908\) −6.35434e6 −0.255774
\(909\) 9.61386e6 0.385912
\(910\) 0 0
\(911\) −7.11472e6 −0.284028 −0.142014 0.989865i \(-0.545358\pi\)
−0.142014 + 0.989865i \(0.545358\pi\)
\(912\) −234307. −0.00932820
\(913\) −50780.1 −0.00201612
\(914\) −1.88198e6 −0.0745160
\(915\) 0 0
\(916\) −1.66166e7 −0.654339
\(917\) 4.46713e6 0.175430
\(918\) −3.82814e6 −0.149927
\(919\) 8.70813e6 0.340123 0.170061 0.985433i \(-0.445603\pi\)
0.170061 + 0.985433i \(0.445603\pi\)
\(920\) 0 0
\(921\) −1.13282e7 −0.440058
\(922\) 8.01440e6 0.310487
\(923\) 444218. 0.0171629
\(924\) −326204. −0.0125692
\(925\) 0 0
\(926\) 1.23423e7 0.473009
\(927\) 8.70854e6 0.332848
\(928\) 2.16284e7 0.824430
\(929\) −1.65935e6 −0.0630809 −0.0315405 0.999502i \(-0.510041\pi\)
−0.0315405 + 0.999502i \(0.510041\pi\)
\(930\) 0 0
\(931\) −1.75353e7 −0.663040
\(932\) 2.53089e7 0.954406
\(933\) −1.54361e7 −0.580541
\(934\) 2.17972e7 0.817585
\(935\) 0 0
\(936\) 135354. 0.00504987
\(937\) −3.17616e7 −1.18183 −0.590913 0.806735i \(-0.701232\pi\)
−0.590913 + 0.806735i \(0.701232\pi\)
\(938\) 2.08814e6 0.0774911
\(939\) 2.57318e7 0.952371
\(940\) 0 0
\(941\) 4.87851e7 1.79603 0.898014 0.439966i \(-0.145010\pi\)
0.898014 + 0.439966i \(0.145010\pi\)
\(942\) 1.61864e7 0.594323
\(943\) −4.97529e7 −1.82196
\(944\) −963319. −0.0351836
\(945\) 0 0
\(946\) −5.85426e6 −0.212689
\(947\) −9.76025e6 −0.353660 −0.176830 0.984241i \(-0.556584\pi\)
−0.176830 + 0.984241i \(0.556584\pi\)
\(948\) −2.87228e6 −0.103802
\(949\) −452003. −0.0162921
\(950\) 0 0
\(951\) 1.44843e7 0.519332
\(952\) 4.16335e6 0.148885
\(953\) 3.78772e7 1.35097 0.675485 0.737374i \(-0.263934\pi\)
0.675485 + 0.737374i \(0.263934\pi\)
\(954\) −2.21683e6 −0.0788609
\(955\) 0 0
\(956\) 1.72553e7 0.610628
\(957\) 4.09800e6 0.144641
\(958\) −2.10206e7 −0.739998
\(959\) 4.36903e6 0.153405
\(960\) 0 0
\(961\) −2.67372e7 −0.933916
\(962\) −65489.7 −0.00228158
\(963\) 1.51589e7 0.526747
\(964\) −1.53103e7 −0.530628
\(965\) 0 0
\(966\) −1.47899e6 −0.0509945
\(967\) 1.59962e7 0.550113 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(968\) −2.66957e6 −0.0915698
\(969\) −1.41082e7 −0.482683
\(970\) 0 0
\(971\) 5.23637e7 1.78231 0.891153 0.453703i \(-0.149897\pi\)
0.891153 + 0.453703i \(0.149897\pi\)
\(972\) −1.14742e6 −0.0389544
\(973\) 3.09093e6 0.104666
\(974\) −3.18825e7 −1.07685
\(975\) 0 0
\(976\) 202236. 0.00679569
\(977\) 5.78033e6 0.193739 0.0968693 0.995297i \(-0.469117\pi\)
0.0968693 + 0.995297i \(0.469117\pi\)
\(978\) −1.28310e7 −0.428957
\(979\) −2.06683e6 −0.0689203
\(980\) 0 0
\(981\) −40192.5 −0.00133344
\(982\) −9.72467e6 −0.321807
\(983\) −2.58942e7 −0.854709 −0.427355 0.904084i \(-0.640554\pi\)
−0.427355 + 0.904084i \(0.640554\pi\)
\(984\) −2.71518e7 −0.893945
\(985\) 0 0
\(986\) −1.97608e7 −0.647310
\(987\) 921022. 0.0300938
\(988\) 188466. 0.00614243
\(989\) 4.10374e7 1.33410
\(990\) 0 0
\(991\) 3.10539e7 1.00446 0.502230 0.864734i \(-0.332513\pi\)
0.502230 + 0.864734i \(0.332513\pi\)
\(992\) 7.90552e6 0.255065
\(993\) −1.09658e7 −0.352913
\(994\) −2.64894e6 −0.0850366
\(995\) 0 0
\(996\) 73393.8 0.00234429
\(997\) 5.56862e7 1.77423 0.887115 0.461549i \(-0.152706\pi\)
0.887115 + 0.461549i \(0.152706\pi\)
\(998\) −2.42963e7 −0.772172
\(999\) 1.46942e6 0.0465835
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.t.1.7 10
5.4 even 2 825.6.a.u.1.4 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.7 10 1.1 even 1 trivial
825.6.a.u.1.4 yes 10 5.4 even 2