Properties

Label 825.6.a.u.1.4
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 1.4
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 \(-0.601452\pi\)
−0.313354 + 0.949637i \(0.601452\pi\)
\(3\) −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 \(-0.518936\pi\)
−0.0594534 + 0.998231i \(0.518936\pi\)
\(8\) 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 \(-0.497606\pi\)
0.00752016 + 0.999972i \(0.497606\pi\)
\(14\) 54.6502 0.0745198
\(15\) 0 0
\(16\) −24.6000 −0.0240234
\(17\) 1481.22 1.24308 0.621539 0.783383i \(-0.286508\pi\)
0.621539 + 0.783383i \(0.286508\pi\)
\(18\) −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 \(-0.298090\pi\)
0.592629 + 0.805476i \(0.298090\pi\)
\(24\) −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 \(-0.538619\pi\)
−0.121027 + 0.992649i \(0.538619\pi\)
\(38\) −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 \(-0.309728\pi\)
0.562790 + 0.826600i \(0.309728\pi\)
\(44\) −2351.23 −0.183089
\(45\) 0 0
\(46\) −10660.4 −0.742809
\(47\) −6638.58 −0.438360 −0.219180 0.975684i \(-0.570338\pi\)
−0.219180 + 0.975684i \(0.570338\pi\)
\(48\) 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 \(-0.439556\pi\)
0.188751 + 0.982025i \(0.439556\pi\)
\(54\) 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 \(-0.674044\pi\)
−0.519936 + 0.854205i \(0.674044\pi\)
\(68\) −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 \(-0.682186\pi\)
−0.541613 + 0.840628i \(0.682186\pi\)
\(74\) 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 \(-0.498936\pi\)
0.00334336 + 0.999994i \(0.498936\pi\)
\(84\) −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 \(-0.494049\pi\)
0.0186935 + 0.999825i \(0.494049\pi\)
\(98\) 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 \(-0.666399\pi\)
−0.499272 + 0.866445i \(0.666399\pi\)
\(104\) 1671.03 0.0151496
\(105\) 0 0
\(106\) −27368.3 −0.236583
\(107\) −187147. −1.58024 −0.790120 0.612952i \(-0.789982\pi\)
−0.790120 + 0.612952i \(0.789982\pi\)
\(108\) 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 \(-0.367818\pi\)
0.403429 + 0.915011i \(0.367818\pi\)
\(114\) 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 \(-0.501260\pi\)
−0.00395817 + 0.999992i \(0.501260\pi\)
\(128\) 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 \(-0.723168\pi\)
−0.645062 + 0.764130i \(0.723168\pi\)
\(138\) 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 \(-0.806737\pi\)
−0.821275 + 0.570533i \(0.806737\pi\)
\(158\) −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 \(-0.298038\pi\)
0.592761 + 0.805379i \(0.298038\pi\)
\(164\) −321511. −0.933438
\(165\) 0 0
\(166\) −1487.81 −0.00419061
\(167\) −293252. −0.813672 −0.406836 0.913501i \(-0.633368\pi\)
−0.406836 + 0.913501i \(0.633368\pi\)
\(168\) 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 \(-0.471797\pi\)
0.0884875 + 0.996077i \(0.471797\pi\)
\(174\) 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 \(-0.473885\pi\)
0.0819500 + 0.996636i \(0.473885\pi\)
\(194\) −12282.6 −0.0234307
\(195\) 0 0
\(196\) 321970. 0.598653
\(197\) 15771.8 0.0289545 0.0144772 0.999895i \(-0.495392\pi\)
0.0144772 + 0.999895i \(0.495392\pi\)
\(198\) −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 \(-0.586560\pi\)
−0.268596 + 0.963253i \(0.586560\pi\)
\(224\) 88599.5 0.117981
\(225\) 0 0
\(226\) −388269. −0.505664
\(227\) −327010. −0.421208 −0.210604 0.977571i \(-0.567543\pi\)
−0.210604 + 0.977571i \(0.567543\pi\)
\(228\) 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 \(-0.212222\pi\)
0.785858 + 0.618407i \(0.212222\pi\)
\(234\) −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 \(-0.527551\pi\)
−0.0864471 + 0.996256i \(0.527551\pi\)
\(258\) 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 \(-0.204321\pi\)
0.800963 + 0.598714i \(0.204321\pi\)
\(264\) −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 \(-0.579870\pi\)
−0.248294 + 0.968685i \(0.579870\pi\)
\(278\) −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 \(-0.144181\pi\)
0.899157 + 0.437626i \(0.144181\pi\)
\(284\) 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 \(-0.418653\pi\)
0.252788 + 0.967522i \(0.418653\pi\)
\(294\) −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 \(-0.375545\pi\)
0.381102 + 0.924533i \(0.375545\pi\)
\(308\) 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 \(-0.808700\pi\)
−0.824778 + 0.565457i \(0.808700\pi\)
\(314\) 1.79849e6 1.02940
\(315\) 0 0
\(316\) −319143. −0.179791
\(317\) −1.60936e6 −0.899510 −0.449755 0.893152i \(-0.648489\pi\)
−0.449755 + 0.893152i \(0.648489\pi\)
\(318\) 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 \(-0.375036\pi\)
0.382580 + 0.923922i \(0.375036\pi\)
\(338\) 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 \(-0.731538\pi\)
−0.664930 + 0.746906i \(0.731538\pi\)
\(348\) 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 \(-0.525404\pi\)
−0.0797235 + 0.996817i \(0.525404\pi\)
\(354\) 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 \(-0.247981\pi\)
0.711578 + 0.702607i \(0.247981\pi\)
\(368\) −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 \(-0.482777\pi\)
0.0540814 + 0.998537i \(0.482777\pi\)
\(374\) −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 \(-0.595575\pi\)
−0.295766 + 0.955261i \(0.595575\pi\)
\(384\) −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 \(-0.507700\pi\)
−0.0241893 + 0.999707i \(0.507700\pi\)
\(398\) −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 \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(434\) 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 \(-0.546507\pi\)
−0.145586 + 0.989346i \(0.546507\pi\)
\(444\) −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 \(-0.481065\pi\)
0.0594504 + 0.998231i \(0.481065\pi\)
\(458\) −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 \(-0.623174\pi\)
−0.377376 + 0.926060i \(0.623174\pi\)
\(464\) −92571.8 −0.0199611
\(465\) 0 0
\(466\) −4.61746e6 −0.985006
\(467\) −6.14838e6 −1.30457 −0.652287 0.757972i \(-0.726190\pi\)
−0.652287 + 0.757972i \(0.726190\pi\)
\(468\) −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 \(-0.171004\pi\)
0.859132 + 0.511754i \(0.171004\pi\)
\(488\) −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 \(-0.399238\pi\)
0.311292 + 0.950314i \(0.399238\pi\)
\(504\) −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 \(-0.468766\pi\)
0.0979674 + 0.995190i \(0.468766\pi\)
\(524\) −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 \(-0.523836\pi\)
−0.0748118 + 0.997198i \(0.523836\pi\)
\(548\) 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 \(-0.698313\pi\)
−0.583488 + 0.812121i \(0.698313\pi\)
\(558\) −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 \(-0.568647\pi\)
−0.213992 + 0.976835i \(0.568647\pi\)
\(564\) −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 \(-0.400620\pi\)
0.307165 + 0.951656i \(0.400620\pi\)
\(578\) −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 \(-0.604124\pi\)
−0.321312 + 0.946973i \(0.604124\pi\)
\(588\) −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 \(-0.673259\pi\)
−0.517829 + 0.855484i \(0.673259\pi\)
\(594\) 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 \(-0.247379\pi\)
0.712906 + 0.701259i \(0.247379\pi\)
\(608\) −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 \(-0.441272\pi\)
0.183456 + 0.983028i \(0.441272\pi\)
\(614\) −4.46227e6 −0.477678
\(615\) 0 0
\(616\) −340100. −0.0361123
\(617\) 1.36370e7 1.44213 0.721067 0.692865i \(-0.243652\pi\)
0.721067 + 0.692865i \(0.243652\pi\)
\(618\) −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 \(-0.493594\pi\)
0.0201240 + 0.999797i \(0.493594\pi\)
\(644\) 900727. 0.0855813
\(645\) 0 0
\(646\) −5.55736e6 −0.523947
\(647\) 5.62498e6 0.528275 0.264138 0.964485i \(-0.414913\pi\)
0.264138 + 0.964485i \(0.414913\pi\)
\(648\) 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 \(-0.783067\pi\)
−0.776619 + 0.629970i \(0.783067\pi\)
\(654\) −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 \(-0.447809\pi\)
0.163229 + 0.986588i \(0.447809\pi\)
\(674\) −5.65545e6 −0.479531
\(675\) 0 0
\(676\) 7.21319e6 0.607101
\(677\) 5.74417e6 0.481676 0.240838 0.970565i \(-0.422578\pi\)
0.240838 + 0.970565i \(0.422578\pi\)
\(678\) 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 \(-0.834619\pi\)
−0.868038 + 0.496498i \(0.834619\pi\)
\(684\) −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 \(-0.440180\pi\)
0.186827 + 0.982393i \(0.440180\pi\)
\(728\) −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 \(-0.753277\pi\)
−0.714348 + 0.699790i \(0.753277\pi\)
\(734\) −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 \(-0.206709\pi\)
0.796450 + 0.604705i \(0.206709\pi\)
\(744\) −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 \(-0.295786\pi\)
0.598443 + 0.801165i \(0.295786\pi\)
\(758\) −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 \(-0.395389\pi\)
0.322762 + 0.946480i \(0.395389\pi\)
\(774\) −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 \(-0.267598\pi\)
0.666954 + 0.745099i \(0.267598\pi\)
\(788\) −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 \(-0.553037\pi\)
−0.165851 + 0.986151i \(0.553037\pi\)
\(798\) −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 \(-0.550203\pi\)
−0.157063 + 0.987589i \(0.550203\pi\)
\(824\) −1.96033e7 −1.00580
\(825\) 0 0
\(826\) 2.14007e6 0.109138
\(827\) 1.25743e7 0.639321 0.319661 0.947532i \(-0.396431\pi\)
0.319661 + 0.947532i \(0.396431\pi\)
\(828\) −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 \(-0.447097\pi\)
0.165437 + 0.986220i \(0.447097\pi\)
\(854\) −449278. −0.0210800
\(855\) 0 0
\(856\) −3.41234e7 −1.59172
\(857\) −5.35026e6 −0.248842 −0.124421 0.992230i \(-0.539707\pi\)
−0.124421 + 0.992230i \(0.539707\pi\)
\(858\) 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 \(-0.345935\pi\)
0.465331 + 0.885137i \(0.345935\pi\)
\(864\) 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 \(-0.298142\pi\)
0.592497 + 0.805573i \(0.298142\pi\)
\(878\) 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 \(-0.529603\pi\)
−0.0928661 + 0.995679i \(0.529603\pi\)
\(884\) −263782. −0.0113531
\(885\) 0 0
\(886\) 4.26381e6 0.182479
\(887\) −5.28627e6 −0.225601 −0.112800 0.993618i \(-0.535982\pi\)
−0.112800 + 0.993618i \(0.535982\pi\)
\(888\) 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 \(-0.491993\pi\)
0.0251518 + 0.999684i \(0.491993\pi\)
\(908\) 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 \(-0.298768\pi\)
0.590913 + 0.806735i \(0.298768\pi\)
\(938\) −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 \(-0.443416\pi\)
0.176830 + 0.984241i \(0.443416\pi\)
\(948\) 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 \(-0.736066\pi\)
−0.675485 + 0.737374i \(0.736066\pi\)
\(954\) −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 \(-0.588696\pi\)
−0.275056 + 0.961428i \(0.588696\pi\)
\(968\) 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 \(-0.530883\pi\)
−0.0968693 + 0.995297i \(0.530883\pi\)
\(978\) 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 \(-0.359446\pi\)
0.427355 + 0.904084i \(0.359446\pi\)
\(984\) −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 \(-0.847294\pi\)
−0.887115 + 0.461549i \(0.847294\pi\)
\(998\) 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.u.1.4 yes 10
5.4 even 2 825.6.a.t.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.7 10 5.4 even 2
825.6.a.u.1.4 yes 10 1.1 even 1 trivial