Properties

Label 825.6.a.u.1.3
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 1.3
Root \(-7.65849\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-7.65849 q^{2} -9.00000 q^{3} +26.6525 q^{4} +68.9264 q^{6} -112.197 q^{7} +40.9539 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.872 q^{12} -849.133 q^{13} +859.262 q^{14} -1166.52 q^{16} -608.382 q^{17} -620.338 q^{18} -2245.32 q^{19} +1009.78 q^{21} -926.677 q^{22} -1287.36 q^{23} -368.585 q^{24} +6503.08 q^{26} -729.000 q^{27} -2990.34 q^{28} +2679.38 q^{29} +3838.89 q^{31} +7623.29 q^{32} -1089.00 q^{33} +4659.28 q^{34} +2158.85 q^{36} -12016.2 q^{37} +17195.8 q^{38} +7642.20 q^{39} -618.409 q^{41} -7733.36 q^{42} -13349.9 q^{43} +3224.95 q^{44} +9859.21 q^{46} -6897.13 q^{47} +10498.7 q^{48} -4218.77 q^{49} +5475.43 q^{51} -22631.5 q^{52} +1982.49 q^{53} +5583.04 q^{54} -4594.92 q^{56} +20207.9 q^{57} -20520.0 q^{58} -38147.4 q^{59} +5604.46 q^{61} -29400.1 q^{62} -9087.98 q^{63} -21054.1 q^{64} +8340.10 q^{66} +35672.2 q^{67} -16214.9 q^{68} +11586.2 q^{69} +36035.5 q^{71} +3317.27 q^{72} -33184.8 q^{73} +92025.8 q^{74} -59843.4 q^{76} -13575.9 q^{77} -58527.7 q^{78} -31882.0 q^{79} +6561.00 q^{81} +4736.08 q^{82} -91005.1 q^{83} +26913.0 q^{84} +102240. q^{86} -24114.4 q^{87} +4955.42 q^{88} +50403.1 q^{89} +95270.4 q^{91} -34311.3 q^{92} -34550.0 q^{93} +52821.6 q^{94} -68609.6 q^{96} -149344. q^{97} +32309.4 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.65849 −1.35384 −0.676921 0.736055i \(-0.736686\pi\)
−0.676921 + 0.736055i \(0.736686\pi\)
\(3\) −9.00000 −0.577350
\(4\) 26.6525 0.832890
\(5\) 0 0
\(6\) 68.9264 0.781641
\(7\) −112.197 −0.865440 −0.432720 0.901528i \(-0.642446\pi\)
−0.432720 + 0.901528i \(0.642446\pi\)
\(8\) 40.9539 0.226241
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −239.872 −0.480869
\(13\) −849.133 −1.39353 −0.696767 0.717298i \(-0.745379\pi\)
−0.696767 + 0.717298i \(0.745379\pi\)
\(14\) 859.262 1.17167
\(15\) 0 0
\(16\) −1166.52 −1.13918
\(17\) −608.382 −0.510568 −0.255284 0.966866i \(-0.582169\pi\)
−0.255284 + 0.966866i \(0.582169\pi\)
\(18\) −620.338 −0.451281
\(19\) −2245.32 −1.42690 −0.713452 0.700704i \(-0.752869\pi\)
−0.713452 + 0.700704i \(0.752869\pi\)
\(20\) 0 0
\(21\) 1009.78 0.499662
\(22\) −926.677 −0.408199
\(23\) −1287.36 −0.507434 −0.253717 0.967279i \(-0.581653\pi\)
−0.253717 + 0.967279i \(0.581653\pi\)
\(24\) −368.585 −0.130620
\(25\) 0 0
\(26\) 6503.08 1.88662
\(27\) −729.000 −0.192450
\(28\) −2990.34 −0.720817
\(29\) 2679.38 0.591616 0.295808 0.955247i \(-0.404411\pi\)
0.295808 + 0.955247i \(0.404411\pi\)
\(30\) 0 0
\(31\) 3838.89 0.717467 0.358733 0.933440i \(-0.383209\pi\)
0.358733 + 0.933440i \(0.383209\pi\)
\(32\) 7623.29 1.31604
\(33\) −1089.00 −0.174078
\(34\) 4659.28 0.691229
\(35\) 0 0
\(36\) 2158.85 0.277630
\(37\) −12016.2 −1.44299 −0.721493 0.692422i \(-0.756544\pi\)
−0.721493 + 0.692422i \(0.756544\pi\)
\(38\) 17195.8 1.93180
\(39\) 7642.20 0.804557
\(40\) 0 0
\(41\) −618.409 −0.0574534 −0.0287267 0.999587i \(-0.509145\pi\)
−0.0287267 + 0.999587i \(0.509145\pi\)
\(42\) −7733.36 −0.676464
\(43\) −13349.9 −1.10105 −0.550526 0.834818i \(-0.685573\pi\)
−0.550526 + 0.834818i \(0.685573\pi\)
\(44\) 3224.95 0.251126
\(45\) 0 0
\(46\) 9859.21 0.686986
\(47\) −6897.13 −0.455433 −0.227716 0.973728i \(-0.573126\pi\)
−0.227716 + 0.973728i \(0.573126\pi\)
\(48\) 10498.7 0.657708
\(49\) −4218.77 −0.251013
\(50\) 0 0
\(51\) 5475.43 0.294777
\(52\) −22631.5 −1.16066
\(53\) 1982.49 0.0969438 0.0484719 0.998825i \(-0.484565\pi\)
0.0484719 + 0.998825i \(0.484565\pi\)
\(54\) 5583.04 0.260547
\(55\) 0 0
\(56\) −4594.92 −0.195798
\(57\) 20207.9 0.823824
\(58\) −20520.0 −0.800954
\(59\) −38147.4 −1.42671 −0.713353 0.700805i \(-0.752824\pi\)
−0.713353 + 0.700805i \(0.752824\pi\)
\(60\) 0 0
\(61\) 5604.46 0.192845 0.0964226 0.995340i \(-0.469260\pi\)
0.0964226 + 0.995340i \(0.469260\pi\)
\(62\) −29400.1 −0.971337
\(63\) −9087.98 −0.288480
\(64\) −21054.1 −0.642521
\(65\) 0 0
\(66\) 8340.10 0.235674
\(67\) 35672.2 0.970830 0.485415 0.874284i \(-0.338669\pi\)
0.485415 + 0.874284i \(0.338669\pi\)
\(68\) −16214.9 −0.425247
\(69\) 11586.2 0.292967
\(70\) 0 0
\(71\) 36035.5 0.848368 0.424184 0.905576i \(-0.360561\pi\)
0.424184 + 0.905576i \(0.360561\pi\)
\(72\) 3317.27 0.0754135
\(73\) −33184.8 −0.728841 −0.364420 0.931235i \(-0.618733\pi\)
−0.364420 + 0.931235i \(0.618733\pi\)
\(74\) 92025.8 1.95358
\(75\) 0 0
\(76\) −59843.4 −1.18845
\(77\) −13575.9 −0.260940
\(78\) −58527.7 −1.08924
\(79\) −31882.0 −0.574749 −0.287374 0.957818i \(-0.592782\pi\)
−0.287374 + 0.957818i \(0.592782\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 4736.08 0.0777829
\(83\) −91005.1 −1.45001 −0.725004 0.688745i \(-0.758162\pi\)
−0.725004 + 0.688745i \(0.758162\pi\)
\(84\) 26913.0 0.416164
\(85\) 0 0
\(86\) 102240. 1.49065
\(87\) −24114.4 −0.341569
\(88\) 4955.42 0.0682141
\(89\) 50403.1 0.674501 0.337250 0.941415i \(-0.390503\pi\)
0.337250 + 0.941415i \(0.390503\pi\)
\(90\) 0 0
\(91\) 95270.4 1.20602
\(92\) −34311.3 −0.422637
\(93\) −34550.0 −0.414230
\(94\) 52821.6 0.616584
\(95\) 0 0
\(96\) −68609.6 −0.759814
\(97\) −149344. −1.61160 −0.805800 0.592188i \(-0.798264\pi\)
−0.805800 + 0.592188i \(0.798264\pi\)
\(98\) 32309.4 0.339832
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −55801.3 −0.544304 −0.272152 0.962254i \(-0.587735\pi\)
−0.272152 + 0.962254i \(0.587735\pi\)
\(102\) −41933.6 −0.399081
\(103\) 41304.4 0.383622 0.191811 0.981432i \(-0.438564\pi\)
0.191811 + 0.981432i \(0.438564\pi\)
\(104\) −34775.3 −0.315274
\(105\) 0 0
\(106\) −15182.8 −0.131247
\(107\) −133537. −1.12756 −0.563782 0.825924i \(-0.690654\pi\)
−0.563782 + 0.825924i \(0.690654\pi\)
\(108\) −19429.7 −0.160290
\(109\) −217266. −1.75156 −0.875782 0.482706i \(-0.839654\pi\)
−0.875782 + 0.482706i \(0.839654\pi\)
\(110\) 0 0
\(111\) 108146. 0.833108
\(112\) 130881. 0.985896
\(113\) −144891. −1.06744 −0.533722 0.845660i \(-0.679207\pi\)
−0.533722 + 0.845660i \(0.679207\pi\)
\(114\) −154762. −1.11533
\(115\) 0 0
\(116\) 71412.2 0.492751
\(117\) −68779.8 −0.464511
\(118\) 292151. 1.93154
\(119\) 68258.7 0.441866
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −42921.7 −0.261082
\(123\) 5565.68 0.0331707
\(124\) 102316. 0.597571
\(125\) 0 0
\(126\) 69600.2 0.390557
\(127\) −263734. −1.45096 −0.725482 0.688241i \(-0.758383\pi\)
−0.725482 + 0.688241i \(0.758383\pi\)
\(128\) −82702.5 −0.446163
\(129\) 120149. 0.635692
\(130\) 0 0
\(131\) −343441. −1.74853 −0.874266 0.485447i \(-0.838657\pi\)
−0.874266 + 0.485447i \(0.838657\pi\)
\(132\) −29024.6 −0.144988
\(133\) 251919. 1.23490
\(134\) −273195. −1.31435
\(135\) 0 0
\(136\) −24915.6 −0.115511
\(137\) −172845. −0.786786 −0.393393 0.919370i \(-0.628699\pi\)
−0.393393 + 0.919370i \(0.628699\pi\)
\(138\) −88732.9 −0.396631
\(139\) −79230.1 −0.347819 −0.173909 0.984762i \(-0.555640\pi\)
−0.173909 + 0.984762i \(0.555640\pi\)
\(140\) 0 0
\(141\) 62074.2 0.262944
\(142\) −275977. −1.14856
\(143\) −102745. −0.420166
\(144\) −94488.5 −0.379728
\(145\) 0 0
\(146\) 254146. 0.986736
\(147\) 37969.0 0.144922
\(148\) −320261. −1.20185
\(149\) −112367. −0.414641 −0.207321 0.978273i \(-0.566474\pi\)
−0.207321 + 0.978273i \(0.566474\pi\)
\(150\) 0 0
\(151\) 351417. 1.25424 0.627119 0.778923i \(-0.284234\pi\)
0.627119 + 0.778923i \(0.284234\pi\)
\(152\) −91954.8 −0.322824
\(153\) −49278.9 −0.170189
\(154\) 103971. 0.353272
\(155\) 0 0
\(156\) 203684. 0.670107
\(157\) −202204. −0.654696 −0.327348 0.944904i \(-0.606155\pi\)
−0.327348 + 0.944904i \(0.606155\pi\)
\(158\) 244168. 0.778120
\(159\) −17842.4 −0.0559706
\(160\) 0 0
\(161\) 144438. 0.439154
\(162\) −50247.4 −0.150427
\(163\) 317741. 0.936708 0.468354 0.883541i \(-0.344847\pi\)
0.468354 + 0.883541i \(0.344847\pi\)
\(164\) −16482.1 −0.0478524
\(165\) 0 0
\(166\) 696962. 1.96308
\(167\) −590220. −1.63766 −0.818828 0.574038i \(-0.805376\pi\)
−0.818828 + 0.574038i \(0.805376\pi\)
\(168\) 41354.2 0.113044
\(169\) 349734. 0.941934
\(170\) 0 0
\(171\) −181871. −0.475635
\(172\) −355809. −0.917055
\(173\) 461802. 1.17311 0.586557 0.809908i \(-0.300483\pi\)
0.586557 + 0.809908i \(0.300483\pi\)
\(174\) 184680. 0.462431
\(175\) 0 0
\(176\) −141149. −0.343477
\(177\) 343326. 0.823709
\(178\) −386012. −0.913168
\(179\) 60459.9 0.141038 0.0705188 0.997510i \(-0.477535\pi\)
0.0705188 + 0.997510i \(0.477535\pi\)
\(180\) 0 0
\(181\) −472232. −1.07142 −0.535709 0.844403i \(-0.679955\pi\)
−0.535709 + 0.844403i \(0.679955\pi\)
\(182\) −729627. −1.63276
\(183\) −50440.1 −0.111339
\(184\) −52722.3 −0.114802
\(185\) 0 0
\(186\) 264601. 0.560802
\(187\) −73614.2 −0.153942
\(188\) −183826. −0.379325
\(189\) 81791.8 0.166554
\(190\) 0 0
\(191\) −85661.8 −0.169904 −0.0849521 0.996385i \(-0.527074\pi\)
−0.0849521 + 0.996385i \(0.527074\pi\)
\(192\) 189487. 0.370960
\(193\) 992251. 1.91747 0.958734 0.284304i \(-0.0917625\pi\)
0.958734 + 0.284304i \(0.0917625\pi\)
\(194\) 1.14375e6 2.18185
\(195\) 0 0
\(196\) −112441. −0.209066
\(197\) −854973. −1.56959 −0.784796 0.619754i \(-0.787233\pi\)
−0.784796 + 0.619754i \(0.787233\pi\)
\(198\) −75060.9 −0.136066
\(199\) 77645.6 0.138990 0.0694951 0.997582i \(-0.477861\pi\)
0.0694951 + 0.997582i \(0.477861\pi\)
\(200\) 0 0
\(201\) −321050. −0.560509
\(202\) 427354. 0.736902
\(203\) −300619. −0.512008
\(204\) 145934. 0.245517
\(205\) 0 0
\(206\) −316330. −0.519364
\(207\) −104276. −0.169145
\(208\) 990534. 1.58749
\(209\) −271684. −0.430228
\(210\) 0 0
\(211\) 741653. 1.14682 0.573409 0.819269i \(-0.305621\pi\)
0.573409 + 0.819269i \(0.305621\pi\)
\(212\) 52838.1 0.0807436
\(213\) −324319. −0.489806
\(214\) 1.02269e6 1.52654
\(215\) 0 0
\(216\) −29855.4 −0.0435400
\(217\) −430713. −0.620925
\(218\) 1.66393e6 2.37134
\(219\) 298664. 0.420796
\(220\) 0 0
\(221\) 516597. 0.711494
\(222\) −828232. −1.12790
\(223\) −278843. −0.375490 −0.187745 0.982218i \(-0.560118\pi\)
−0.187745 + 0.982218i \(0.560118\pi\)
\(224\) −855313. −1.13895
\(225\) 0 0
\(226\) 1.10965e6 1.44515
\(227\) 770180. 0.992036 0.496018 0.868312i \(-0.334795\pi\)
0.496018 + 0.868312i \(0.334795\pi\)
\(228\) 538591. 0.686155
\(229\) −1.03700e6 −1.30674 −0.653370 0.757039i \(-0.726645\pi\)
−0.653370 + 0.757039i \(0.726645\pi\)
\(230\) 0 0
\(231\) 122183. 0.150654
\(232\) 109731. 0.133847
\(233\) 1.31100e6 1.58203 0.791014 0.611799i \(-0.209554\pi\)
0.791014 + 0.611799i \(0.209554\pi\)
\(234\) 526749. 0.628875
\(235\) 0 0
\(236\) −1.01672e6 −1.18829
\(237\) 286938. 0.331831
\(238\) −522759. −0.598217
\(239\) 173528. 0.196506 0.0982529 0.995161i \(-0.468675\pi\)
0.0982529 + 0.995161i \(0.468675\pi\)
\(240\) 0 0
\(241\) −496038. −0.550138 −0.275069 0.961424i \(-0.588701\pi\)
−0.275069 + 0.961424i \(0.588701\pi\)
\(242\) −112128. −0.123077
\(243\) −59049.0 −0.0641500
\(244\) 149373. 0.160619
\(245\) 0 0
\(246\) −42624.7 −0.0449080
\(247\) 1.90658e6 1.98844
\(248\) 157218. 0.162320
\(249\) 819046. 0.837162
\(250\) 0 0
\(251\) 831223. 0.832785 0.416393 0.909185i \(-0.363294\pi\)
0.416393 + 0.909185i \(0.363294\pi\)
\(252\) −242217. −0.240272
\(253\) −155770. −0.152997
\(254\) 2.01980e6 1.96438
\(255\) 0 0
\(256\) 1.30711e6 1.24656
\(257\) 734210. 0.693406 0.346703 0.937975i \(-0.387301\pi\)
0.346703 + 0.937975i \(0.387301\pi\)
\(258\) −920162. −0.860627
\(259\) 1.34818e6 1.24882
\(260\) 0 0
\(261\) 217030. 0.197205
\(262\) 2.63024e6 2.36724
\(263\) 89351.2 0.0796547 0.0398273 0.999207i \(-0.487319\pi\)
0.0398273 + 0.999207i \(0.487319\pi\)
\(264\) −44598.8 −0.0393834
\(265\) 0 0
\(266\) −1.92932e6 −1.67186
\(267\) −453628. −0.389423
\(268\) 950753. 0.808594
\(269\) −927745. −0.781714 −0.390857 0.920451i \(-0.627821\pi\)
−0.390857 + 0.920451i \(0.627821\pi\)
\(270\) 0 0
\(271\) 494541. 0.409052 0.204526 0.978861i \(-0.434435\pi\)
0.204526 + 0.978861i \(0.434435\pi\)
\(272\) 709692. 0.581631
\(273\) −857434. −0.696296
\(274\) 1.32373e6 1.06518
\(275\) 0 0
\(276\) 308801. 0.244009
\(277\) 1.98776e6 1.55656 0.778278 0.627920i \(-0.216093\pi\)
0.778278 + 0.627920i \(0.216093\pi\)
\(278\) 606783. 0.470892
\(279\) 310950. 0.239156
\(280\) 0 0
\(281\) −760827. −0.574804 −0.287402 0.957810i \(-0.592792\pi\)
−0.287402 + 0.957810i \(0.592792\pi\)
\(282\) −475395. −0.355985
\(283\) −2.17752e6 −1.61620 −0.808102 0.589043i \(-0.799505\pi\)
−0.808102 + 0.589043i \(0.799505\pi\)
\(284\) 960435. 0.706598
\(285\) 0 0
\(286\) 786872. 0.568839
\(287\) 69383.8 0.0497225
\(288\) 617487. 0.438679
\(289\) −1.04973e6 −0.739320
\(290\) 0 0
\(291\) 1.34409e6 0.930458
\(292\) −884458. −0.607044
\(293\) 573498. 0.390268 0.195134 0.980777i \(-0.437486\pi\)
0.195134 + 0.980777i \(0.437486\pi\)
\(294\) −290785. −0.196202
\(295\) 0 0
\(296\) −492109. −0.326462
\(297\) −88209.0 −0.0580259
\(298\) 860560. 0.561359
\(299\) 1.09314e6 0.707126
\(300\) 0 0
\(301\) 1.49782e6 0.952894
\(302\) −2.69132e6 −1.69804
\(303\) 502212. 0.314254
\(304\) 2.61922e6 1.62551
\(305\) 0 0
\(306\) 377402. 0.230410
\(307\) 784034. 0.474776 0.237388 0.971415i \(-0.423709\pi\)
0.237388 + 0.971415i \(0.423709\pi\)
\(308\) −361831. −0.217334
\(309\) −371740. −0.221484
\(310\) 0 0
\(311\) −1.20531e6 −0.706639 −0.353320 0.935503i \(-0.614947\pi\)
−0.353320 + 0.935503i \(0.614947\pi\)
\(312\) 312978. 0.182023
\(313\) 1.46973e6 0.847962 0.423981 0.905671i \(-0.360632\pi\)
0.423981 + 0.905671i \(0.360632\pi\)
\(314\) 1.54858e6 0.886356
\(315\) 0 0
\(316\) −849735. −0.478703
\(317\) −943173. −0.527161 −0.263580 0.964637i \(-0.584903\pi\)
−0.263580 + 0.964637i \(0.584903\pi\)
\(318\) 136646. 0.0757753
\(319\) 324205. 0.178379
\(320\) 0 0
\(321\) 1.20183e6 0.651000
\(322\) −1.10618e6 −0.594545
\(323\) 1.36601e6 0.728532
\(324\) 174867. 0.0925433
\(325\) 0 0
\(326\) −2.43342e6 −1.26816
\(327\) 1.95540e6 1.01127
\(328\) −25326.2 −0.0129983
\(329\) 773840. 0.394150
\(330\) 0 0
\(331\) −2.82790e6 −1.41871 −0.709357 0.704850i \(-0.751014\pi\)
−0.709357 + 0.704850i \(0.751014\pi\)
\(332\) −2.42551e6 −1.20770
\(333\) −973310. −0.480995
\(334\) 4.52020e6 2.21713
\(335\) 0 0
\(336\) −1.17793e6 −0.569207
\(337\) −1.66721e6 −0.799678 −0.399839 0.916585i \(-0.630934\pi\)
−0.399839 + 0.916585i \(0.630934\pi\)
\(338\) −2.67843e6 −1.27523
\(339\) 1.30402e6 0.616289
\(340\) 0 0
\(341\) 464506. 0.216324
\(342\) 1.39286e6 0.643935
\(343\) 2.35903e6 1.08268
\(344\) −546732. −0.249102
\(345\) 0 0
\(346\) −3.53671e6 −1.58821
\(347\) −2.99775e6 −1.33651 −0.668254 0.743933i \(-0.732958\pi\)
−0.668254 + 0.743933i \(0.732958\pi\)
\(348\) −642710. −0.284490
\(349\) −46625.4 −0.0204908 −0.0102454 0.999948i \(-0.503261\pi\)
−0.0102454 + 0.999948i \(0.503261\pi\)
\(350\) 0 0
\(351\) 619018. 0.268186
\(352\) 922418. 0.396800
\(353\) 1.70738e6 0.729280 0.364640 0.931149i \(-0.381192\pi\)
0.364640 + 0.931149i \(0.381192\pi\)
\(354\) −2.62936e6 −1.11517
\(355\) 0 0
\(356\) 1.34337e6 0.561785
\(357\) −614329. −0.255112
\(358\) −463032. −0.190943
\(359\) −2.51698e6 −1.03073 −0.515364 0.856971i \(-0.672343\pi\)
−0.515364 + 0.856971i \(0.672343\pi\)
\(360\) 0 0
\(361\) 2.56538e6 1.03606
\(362\) 3.61658e6 1.45053
\(363\) −131769. −0.0524864
\(364\) 2.53919e6 1.00448
\(365\) 0 0
\(366\) 386295. 0.150736
\(367\) 2.06581e6 0.800616 0.400308 0.916381i \(-0.368903\pi\)
0.400308 + 0.916381i \(0.368903\pi\)
\(368\) 1.50173e6 0.578061
\(369\) −50091.1 −0.0191511
\(370\) 0 0
\(371\) −222429. −0.0838991
\(372\) −920844. −0.345008
\(373\) 3.01036e6 1.12033 0.560166 0.828380i \(-0.310737\pi\)
0.560166 + 0.828380i \(0.310737\pi\)
\(374\) 563773. 0.208413
\(375\) 0 0
\(376\) −282465. −0.103037
\(377\) −2.27515e6 −0.824436
\(378\) −626402. −0.225488
\(379\) 510365. 0.182508 0.0912541 0.995828i \(-0.470912\pi\)
0.0912541 + 0.995828i \(0.470912\pi\)
\(380\) 0 0
\(381\) 2.37361e6 0.837715
\(382\) 656040. 0.230024
\(383\) 3.20939e6 1.11796 0.558979 0.829182i \(-0.311193\pi\)
0.558979 + 0.829182i \(0.311193\pi\)
\(384\) 744322. 0.257592
\(385\) 0 0
\(386\) −7.59914e6 −2.59595
\(387\) −1.08134e6 −0.367017
\(388\) −3.98038e6 −1.34229
\(389\) −4.52943e6 −1.51764 −0.758822 0.651298i \(-0.774225\pi\)
−0.758822 + 0.651298i \(0.774225\pi\)
\(390\) 0 0
\(391\) 783204. 0.259080
\(392\) −172775. −0.0567893
\(393\) 3.09097e6 1.00952
\(394\) 6.54781e6 2.12498
\(395\) 0 0
\(396\) 261221. 0.0837086
\(397\) 2.35485e6 0.749870 0.374935 0.927051i \(-0.377665\pi\)
0.374935 + 0.927051i \(0.377665\pi\)
\(398\) −594648. −0.188171
\(399\) −2.26727e6 −0.712970
\(400\) 0 0
\(401\) 3.24320e6 1.00719 0.503597 0.863939i \(-0.332010\pi\)
0.503597 + 0.863939i \(0.332010\pi\)
\(402\) 2.45876e6 0.758841
\(403\) −3.25973e6 −0.999814
\(404\) −1.48724e6 −0.453345
\(405\) 0 0
\(406\) 2.30229e6 0.693178
\(407\) −1.45396e6 −0.435076
\(408\) 224240. 0.0666904
\(409\) 2.99435e6 0.885103 0.442551 0.896743i \(-0.354073\pi\)
0.442551 + 0.896743i \(0.354073\pi\)
\(410\) 0 0
\(411\) 1.55561e6 0.454251
\(412\) 1.10087e6 0.319515
\(413\) 4.28003e6 1.23473
\(414\) 798596. 0.228995
\(415\) 0 0
\(416\) −6.47319e6 −1.83394
\(417\) 713071. 0.200813
\(418\) 2.08069e6 0.582461
\(419\) 1.09825e6 0.305608 0.152804 0.988257i \(-0.451170\pi\)
0.152804 + 0.988257i \(0.451170\pi\)
\(420\) 0 0
\(421\) 52153.9 0.0143411 0.00717053 0.999974i \(-0.497718\pi\)
0.00717053 + 0.999974i \(0.497718\pi\)
\(422\) −5.67994e6 −1.55261
\(423\) −558668. −0.151811
\(424\) 81190.5 0.0219326
\(425\) 0 0
\(426\) 2.48380e6 0.663120
\(427\) −628805. −0.166896
\(428\) −3.55909e6 −0.939137
\(429\) 924706. 0.242583
\(430\) 0 0
\(431\) 380799. 0.0987423 0.0493711 0.998781i \(-0.484278\pi\)
0.0493711 + 0.998781i \(0.484278\pi\)
\(432\) 850396. 0.219236
\(433\) −6.43191e6 −1.64862 −0.824309 0.566140i \(-0.808436\pi\)
−0.824309 + 0.566140i \(0.808436\pi\)
\(434\) 3.29861e6 0.840634
\(435\) 0 0
\(436\) −5.79069e6 −1.45886
\(437\) 2.89053e6 0.724060
\(438\) −2.28731e6 −0.569692
\(439\) −2.43398e6 −0.602776 −0.301388 0.953502i \(-0.597450\pi\)
−0.301388 + 0.953502i \(0.597450\pi\)
\(440\) 0 0
\(441\) −341721. −0.0836710
\(442\) −3.95635e6 −0.963250
\(443\) 4.37287e6 1.05866 0.529331 0.848415i \(-0.322443\pi\)
0.529331 + 0.848415i \(0.322443\pi\)
\(444\) 2.88235e6 0.693887
\(445\) 0 0
\(446\) 2.13552e6 0.508354
\(447\) 1.01130e6 0.239393
\(448\) 2.36222e6 0.556064
\(449\) 6.51066e6 1.52409 0.762043 0.647527i \(-0.224197\pi\)
0.762043 + 0.647527i \(0.224197\pi\)
\(450\) 0 0
\(451\) −74827.4 −0.0173229
\(452\) −3.86171e6 −0.889064
\(453\) −3.16275e6 −0.724135
\(454\) −5.89841e6 −1.34306
\(455\) 0 0
\(456\) 827593. 0.186382
\(457\) 3.02898e6 0.678431 0.339216 0.940709i \(-0.389838\pi\)
0.339216 + 0.940709i \(0.389838\pi\)
\(458\) 7.94184e6 1.76912
\(459\) 443510. 0.0982589
\(460\) 0 0
\(461\) −6.31089e6 −1.38305 −0.691525 0.722352i \(-0.743061\pi\)
−0.691525 + 0.722352i \(0.743061\pi\)
\(462\) −935736. −0.203962
\(463\) 1.33303e6 0.288993 0.144497 0.989505i \(-0.453844\pi\)
0.144497 + 0.989505i \(0.453844\pi\)
\(464\) −3.12556e6 −0.673959
\(465\) 0 0
\(466\) −1.00403e7 −2.14182
\(467\) 2.53362e6 0.537587 0.268793 0.963198i \(-0.413375\pi\)
0.268793 + 0.963198i \(0.413375\pi\)
\(468\) −1.83315e6 −0.386887
\(469\) −4.00232e6 −0.840195
\(470\) 0 0
\(471\) 1.81983e6 0.377989
\(472\) −1.56228e6 −0.322779
\(473\) −1.61534e6 −0.331979
\(474\) −2.19751e6 −0.449248
\(475\) 0 0
\(476\) 1.81927e6 0.368026
\(477\) 160581. 0.0323146
\(478\) −1.32896e6 −0.266038
\(479\) −5.07626e6 −1.01089 −0.505447 0.862858i \(-0.668672\pi\)
−0.505447 + 0.862858i \(0.668672\pi\)
\(480\) 0 0
\(481\) 1.02033e7 2.01085
\(482\) 3.79890e6 0.744801
\(483\) −1.29994e6 −0.253546
\(484\) 390219. 0.0757173
\(485\) 0 0
\(486\) 452226. 0.0868491
\(487\) 4.42849e6 0.846123 0.423061 0.906101i \(-0.360955\pi\)
0.423061 + 0.906101i \(0.360955\pi\)
\(488\) 229524. 0.0436294
\(489\) −2.85967e6 −0.540809
\(490\) 0 0
\(491\) 3.27064e6 0.612250 0.306125 0.951991i \(-0.400967\pi\)
0.306125 + 0.951991i \(0.400967\pi\)
\(492\) 148339. 0.0276276
\(493\) −1.63009e6 −0.302060
\(494\) −1.46015e7 −2.69203
\(495\) 0 0
\(496\) −4.47816e6 −0.817327
\(497\) −4.04308e6 −0.734212
\(498\) −6.27265e6 −1.13339
\(499\) −160223. −0.0288053 −0.0144027 0.999896i \(-0.504585\pi\)
−0.0144027 + 0.999896i \(0.504585\pi\)
\(500\) 0 0
\(501\) 5.31198e6 0.945502
\(502\) −6.36591e6 −1.12746
\(503\) −4.82699e6 −0.850661 −0.425330 0.905038i \(-0.639842\pi\)
−0.425330 + 0.905038i \(0.639842\pi\)
\(504\) −372188. −0.0652659
\(505\) 0 0
\(506\) 1.19296e6 0.207134
\(507\) −3.14760e6 −0.543826
\(508\) −7.02917e6 −1.20849
\(509\) −3.12997e6 −0.535483 −0.267741 0.963491i \(-0.586277\pi\)
−0.267741 + 0.963491i \(0.586277\pi\)
\(510\) 0 0
\(511\) 3.72325e6 0.630768
\(512\) −7.36400e6 −1.24148
\(513\) 1.63684e6 0.274608
\(514\) −5.62294e6 −0.938763
\(515\) 0 0
\(516\) 3.20228e6 0.529462
\(517\) −834553. −0.137318
\(518\) −1.03250e7 −1.69070
\(519\) −4.15622e6 −0.677298
\(520\) 0 0
\(521\) 7.30204e6 1.17856 0.589278 0.807930i \(-0.299412\pi\)
0.589278 + 0.807930i \(0.299412\pi\)
\(522\) −1.66212e6 −0.266985
\(523\) −3.34673e6 −0.535016 −0.267508 0.963556i \(-0.586200\pi\)
−0.267508 + 0.963556i \(0.586200\pi\)
\(524\) −9.15355e6 −1.45634
\(525\) 0 0
\(526\) −684296. −0.107840
\(527\) −2.33551e6 −0.366316
\(528\) 1.27035e6 0.198307
\(529\) −4.77905e6 −0.742511
\(530\) 0 0
\(531\) −3.08994e6 −0.475569
\(532\) 6.71427e6 1.02854
\(533\) 525111. 0.0800632
\(534\) 3.47411e6 0.527218
\(535\) 0 0
\(536\) 1.46092e6 0.219641
\(537\) −544139. −0.0814281
\(538\) 7.10513e6 1.05832
\(539\) −510472. −0.0756832
\(540\) 0 0
\(541\) −4.74690e6 −0.697296 −0.348648 0.937254i \(-0.613359\pi\)
−0.348648 + 0.937254i \(0.613359\pi\)
\(542\) −3.78744e6 −0.553793
\(543\) 4.25008e6 0.618583
\(544\) −4.63787e6 −0.671926
\(545\) 0 0
\(546\) 6.56665e6 0.942675
\(547\) 9.86753e6 1.41007 0.705034 0.709173i \(-0.250932\pi\)
0.705034 + 0.709173i \(0.250932\pi\)
\(548\) −4.60676e6 −0.655306
\(549\) 453961. 0.0642818
\(550\) 0 0
\(551\) −6.01608e6 −0.844179
\(552\) 474501. 0.0662810
\(553\) 3.57708e6 0.497411
\(554\) −1.52232e7 −2.10733
\(555\) 0 0
\(556\) −2.11168e6 −0.289695
\(557\) −799590. −0.109202 −0.0546008 0.998508i \(-0.517389\pi\)
−0.0546008 + 0.998508i \(0.517389\pi\)
\(558\) −2.38141e6 −0.323779
\(559\) 1.13359e7 1.53435
\(560\) 0 0
\(561\) 662527. 0.0888785
\(562\) 5.82678e6 0.778194
\(563\) 1.95313e6 0.259693 0.129847 0.991534i \(-0.458552\pi\)
0.129847 + 0.991534i \(0.458552\pi\)
\(564\) 1.65443e6 0.219004
\(565\) 0 0
\(566\) 1.66765e7 2.18809
\(567\) −736126. −0.0961600
\(568\) 1.47579e6 0.191935
\(569\) −1.55722e6 −0.201637 −0.100819 0.994905i \(-0.532146\pi\)
−0.100819 + 0.994905i \(0.532146\pi\)
\(570\) 0 0
\(571\) 1.77525e6 0.227861 0.113930 0.993489i \(-0.463656\pi\)
0.113930 + 0.993489i \(0.463656\pi\)
\(572\) −2.73841e6 −0.349952
\(573\) 770957. 0.0980942
\(574\) −531375. −0.0673165
\(575\) 0 0
\(576\) −1.70538e6 −0.214174
\(577\) 7.49718e6 0.937472 0.468736 0.883338i \(-0.344710\pi\)
0.468736 + 0.883338i \(0.344710\pi\)
\(578\) 8.03934e6 1.00092
\(579\) −8.93026e6 −1.10705
\(580\) 0 0
\(581\) 1.02105e7 1.25490
\(582\) −1.02937e7 −1.25969
\(583\) 239881. 0.0292297
\(584\) −1.35905e6 −0.164893
\(585\) 0 0
\(586\) −4.39213e6 −0.528361
\(587\) 8.46392e6 1.01386 0.506928 0.861988i \(-0.330781\pi\)
0.506928 + 0.861988i \(0.330781\pi\)
\(588\) 1.01197e6 0.120704
\(589\) −8.61956e6 −1.02376
\(590\) 0 0
\(591\) 7.69476e6 0.906205
\(592\) 1.40172e7 1.64383
\(593\) 6.09896e6 0.712228 0.356114 0.934442i \(-0.384101\pi\)
0.356114 + 0.934442i \(0.384101\pi\)
\(594\) 675548. 0.0785579
\(595\) 0 0
\(596\) −2.99486e6 −0.345351
\(597\) −698811. −0.0802461
\(598\) −8.37178e6 −0.957337
\(599\) −4.91865e6 −0.560118 −0.280059 0.959983i \(-0.590354\pi\)
−0.280059 + 0.959983i \(0.590354\pi\)
\(600\) 0 0
\(601\) 1.00491e7 1.13486 0.567431 0.823421i \(-0.307937\pi\)
0.567431 + 0.823421i \(0.307937\pi\)
\(602\) −1.14711e7 −1.29007
\(603\) 2.88945e6 0.323610
\(604\) 9.36613e6 1.04464
\(605\) 0 0
\(606\) −3.84619e6 −0.425450
\(607\) −1.26381e7 −1.39222 −0.696112 0.717933i \(-0.745088\pi\)
−0.696112 + 0.717933i \(0.745088\pi\)
\(608\) −1.71168e7 −1.87786
\(609\) 2.70557e6 0.295608
\(610\) 0 0
\(611\) 5.85658e6 0.634660
\(612\) −1.31341e6 −0.141749
\(613\) −1.14049e7 −1.22586 −0.612932 0.790136i \(-0.710010\pi\)
−0.612932 + 0.790136i \(0.710010\pi\)
\(614\) −6.00452e6 −0.642772
\(615\) 0 0
\(616\) −555985. −0.0590352
\(617\) −1.57880e7 −1.66961 −0.834804 0.550547i \(-0.814419\pi\)
−0.834804 + 0.550547i \(0.814419\pi\)
\(618\) 2.84697e6 0.299855
\(619\) −3.22994e6 −0.338819 −0.169409 0.985546i \(-0.554186\pi\)
−0.169409 + 0.985546i \(0.554186\pi\)
\(620\) 0 0
\(621\) 938483. 0.0976557
\(622\) 9.23086e6 0.956679
\(623\) −5.65509e6 −0.583740
\(624\) −8.91481e6 −0.916538
\(625\) 0 0
\(626\) −1.12559e7 −1.14801
\(627\) 2.44516e6 0.248392
\(628\) −5.38923e6 −0.545290
\(629\) 7.31042e6 0.736742
\(630\) 0 0
\(631\) −1.46803e7 −1.46778 −0.733890 0.679268i \(-0.762297\pi\)
−0.733890 + 0.679268i \(0.762297\pi\)
\(632\) −1.30569e6 −0.130031
\(633\) −6.67488e6 −0.662116
\(634\) 7.22328e6 0.713693
\(635\) 0 0
\(636\) −475543. −0.0466173
\(637\) 3.58230e6 0.349795
\(638\) −2.48292e6 −0.241497
\(639\) 2.91887e6 0.282789
\(640\) 0 0
\(641\) −3.89836e6 −0.374746 −0.187373 0.982289i \(-0.559997\pi\)
−0.187373 + 0.982289i \(0.559997\pi\)
\(642\) −9.20421e6 −0.881351
\(643\) −1.22406e7 −1.16754 −0.583772 0.811917i \(-0.698424\pi\)
−0.583772 + 0.811917i \(0.698424\pi\)
\(644\) 3.84963e6 0.365767
\(645\) 0 0
\(646\) −1.04616e7 −0.986318
\(647\) 1.57395e7 1.47819 0.739095 0.673601i \(-0.235254\pi\)
0.739095 + 0.673601i \(0.235254\pi\)
\(648\) 268699. 0.0251378
\(649\) −4.61583e6 −0.430168
\(650\) 0 0
\(651\) 3.87642e6 0.358491
\(652\) 8.46859e6 0.780175
\(653\) −5.37711e6 −0.493476 −0.246738 0.969082i \(-0.579359\pi\)
−0.246738 + 0.969082i \(0.579359\pi\)
\(654\) −1.49754e7 −1.36910
\(655\) 0 0
\(656\) 721389. 0.0654500
\(657\) −2.68797e6 −0.242947
\(658\) −5.92644e6 −0.533617
\(659\) −5.58169e6 −0.500671 −0.250335 0.968159i \(-0.580541\pi\)
−0.250335 + 0.968159i \(0.580541\pi\)
\(660\) 0 0
\(661\) 2.97463e6 0.264807 0.132403 0.991196i \(-0.457731\pi\)
0.132403 + 0.991196i \(0.457731\pi\)
\(662\) 2.16575e7 1.92071
\(663\) −4.64937e6 −0.410781
\(664\) −3.72701e6 −0.328051
\(665\) 0 0
\(666\) 7.45409e6 0.651192
\(667\) −3.44932e6 −0.300206
\(668\) −1.57308e7 −1.36399
\(669\) 2.50959e6 0.216789
\(670\) 0 0
\(671\) 678139. 0.0581450
\(672\) 7.69781e6 0.657573
\(673\) 4.89911e6 0.416946 0.208473 0.978028i \(-0.433151\pi\)
0.208473 + 0.978028i \(0.433151\pi\)
\(674\) 1.27683e7 1.08264
\(675\) 0 0
\(676\) 9.32127e6 0.784528
\(677\) −2.13572e7 −1.79091 −0.895453 0.445156i \(-0.853148\pi\)
−0.895453 + 0.445156i \(0.853148\pi\)
\(678\) −9.98682e6 −0.834359
\(679\) 1.67559e7 1.39474
\(680\) 0 0
\(681\) −6.93162e6 −0.572752
\(682\) −3.55741e6 −0.292869
\(683\) 1.39735e7 1.14618 0.573089 0.819493i \(-0.305745\pi\)
0.573089 + 0.819493i \(0.305745\pi\)
\(684\) −4.84732e6 −0.396152
\(685\) 0 0
\(686\) −1.80666e7 −1.46577
\(687\) 9.33298e6 0.754446
\(688\) 1.55730e7 1.25430
\(689\) −1.68339e6 −0.135094
\(690\) 0 0
\(691\) 1.97099e7 1.57033 0.785164 0.619288i \(-0.212579\pi\)
0.785164 + 0.619288i \(0.212579\pi\)
\(692\) 1.23082e7 0.977076
\(693\) −1.09965e6 −0.0869800
\(694\) 2.29582e7 1.80942
\(695\) 0 0
\(696\) −987580. −0.0772768
\(697\) 376228. 0.0293339
\(698\) 357080. 0.0277413
\(699\) −1.17990e7 −0.913384
\(700\) 0 0
\(701\) −1.31790e7 −1.01295 −0.506473 0.862256i \(-0.669051\pi\)
−0.506473 + 0.862256i \(0.669051\pi\)
\(702\) −4.74074e6 −0.363081
\(703\) 2.69802e7 2.05900
\(704\) −2.54755e6 −0.193727
\(705\) 0 0
\(706\) −1.30760e7 −0.987330
\(707\) 6.26076e6 0.471062
\(708\) 9.15050e6 0.686059
\(709\) −2.28382e7 −1.70626 −0.853131 0.521697i \(-0.825299\pi\)
−0.853131 + 0.521697i \(0.825299\pi\)
\(710\) 0 0
\(711\) −2.58244e6 −0.191583
\(712\) 2.06420e6 0.152599
\(713\) −4.94203e6 −0.364067
\(714\) 4.70483e6 0.345381
\(715\) 0 0
\(716\) 1.61141e6 0.117469
\(717\) −1.56175e6 −0.113453
\(718\) 1.92763e7 1.39544
\(719\) −3.78947e6 −0.273373 −0.136687 0.990614i \(-0.543645\pi\)
−0.136687 + 0.990614i \(0.543645\pi\)
\(720\) 0 0
\(721\) −4.63424e6 −0.332002
\(722\) −1.96469e7 −1.40266
\(723\) 4.46434e6 0.317623
\(724\) −1.25861e7 −0.892373
\(725\) 0 0
\(726\) 1.00915e6 0.0710583
\(727\) 2.52016e7 1.76845 0.884223 0.467065i \(-0.154688\pi\)
0.884223 + 0.467065i \(0.154688\pi\)
\(728\) 3.90169e6 0.272851
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 8.12185e6 0.562162
\(732\) −1.34435e6 −0.0927334
\(733\) −1.83693e7 −1.26279 −0.631397 0.775460i \(-0.717518\pi\)
−0.631397 + 0.775460i \(0.717518\pi\)
\(734\) −1.58210e7 −1.08391
\(735\) 0 0
\(736\) −9.81390e6 −0.667801
\(737\) 4.31634e6 0.292716
\(738\) 383622. 0.0259276
\(739\) 4.22050e6 0.284284 0.142142 0.989846i \(-0.454601\pi\)
0.142142 + 0.989846i \(0.454601\pi\)
\(740\) 0 0
\(741\) −1.71592e7 −1.14803
\(742\) 1.70347e6 0.113586
\(743\) 6.24107e6 0.414750 0.207375 0.978261i \(-0.433508\pi\)
0.207375 + 0.978261i \(0.433508\pi\)
\(744\) −1.41496e6 −0.0937155
\(745\) 0 0
\(746\) −2.30548e7 −1.51675
\(747\) −7.37141e6 −0.483336
\(748\) −1.96200e6 −0.128217
\(749\) 1.49825e7 0.975840
\(750\) 0 0
\(751\) −542004. −0.0350673 −0.0175337 0.999846i \(-0.505581\pi\)
−0.0175337 + 0.999846i \(0.505581\pi\)
\(752\) 8.04568e6 0.518822
\(753\) −7.48100e6 −0.480809
\(754\) 1.74242e7 1.11616
\(755\) 0 0
\(756\) 2.17995e6 0.138721
\(757\) 1.63709e7 1.03833 0.519164 0.854675i \(-0.326244\pi\)
0.519164 + 0.854675i \(0.326244\pi\)
\(758\) −3.90862e6 −0.247087
\(759\) 1.40193e6 0.0883329
\(760\) 0 0
\(761\) −1.17890e7 −0.737931 −0.368966 0.929443i \(-0.620288\pi\)
−0.368966 + 0.929443i \(0.620288\pi\)
\(762\) −1.81782e7 −1.13413
\(763\) 2.43767e7 1.51587
\(764\) −2.28310e6 −0.141511
\(765\) 0 0
\(766\) −2.45791e7 −1.51354
\(767\) 3.23922e7 1.98816
\(768\) −1.17640e7 −0.719699
\(769\) −2.22979e7 −1.35972 −0.679858 0.733344i \(-0.737958\pi\)
−0.679858 + 0.733344i \(0.737958\pi\)
\(770\) 0 0
\(771\) −6.60789e6 −0.400338
\(772\) 2.64459e7 1.59704
\(773\) −2.38341e7 −1.43467 −0.717333 0.696730i \(-0.754637\pi\)
−0.717333 + 0.696730i \(0.754637\pi\)
\(774\) 8.28146e6 0.496883
\(775\) 0 0
\(776\) −6.11620e6 −0.364609
\(777\) −1.21336e7 −0.721005
\(778\) 3.46886e7 2.05465
\(779\) 1.38853e6 0.0819805
\(780\) 0 0
\(781\) 4.36029e6 0.255793
\(782\) −5.99816e6 −0.350753
\(783\) −1.95327e6 −0.113856
\(784\) 4.92130e6 0.285950
\(785\) 0 0
\(786\) −2.36721e7 −1.36673
\(787\) 6.85248e6 0.394376 0.197188 0.980366i \(-0.436819\pi\)
0.197188 + 0.980366i \(0.436819\pi\)
\(788\) −2.27872e7 −1.30730
\(789\) −804161. −0.0459886
\(790\) 0 0
\(791\) 1.62564e7 0.923810
\(792\) 401389. 0.0227380
\(793\) −4.75893e6 −0.268736
\(794\) −1.80346e7 −1.01521
\(795\) 0 0
\(796\) 2.06945e6 0.115764
\(797\) 4.96228e6 0.276717 0.138358 0.990382i \(-0.455817\pi\)
0.138358 + 0.990382i \(0.455817\pi\)
\(798\) 1.73639e7 0.965250
\(799\) 4.19609e6 0.232529
\(800\) 0 0
\(801\) 4.08265e6 0.224834
\(802\) −2.48380e7 −1.36358
\(803\) −4.01537e6 −0.219754
\(804\) −8.55678e6 −0.466842
\(805\) 0 0
\(806\) 2.49646e7 1.35359
\(807\) 8.34971e6 0.451323
\(808\) −2.28528e6 −0.123144
\(809\) −1.48303e6 −0.0796669 −0.0398335 0.999206i \(-0.512683\pi\)
−0.0398335 + 0.999206i \(0.512683\pi\)
\(810\) 0 0
\(811\) −8.26617e6 −0.441319 −0.220659 0.975351i \(-0.570821\pi\)
−0.220659 + 0.975351i \(0.570821\pi\)
\(812\) −8.01225e6 −0.426446
\(813\) −4.45087e6 −0.236167
\(814\) 1.11351e7 0.589025
\(815\) 0 0
\(816\) −6.38723e6 −0.335805
\(817\) 2.99749e7 1.57110
\(818\) −2.29322e7 −1.19829
\(819\) 7.71690e6 0.402007
\(820\) 0 0
\(821\) 2.39516e7 1.24016 0.620078 0.784540i \(-0.287101\pi\)
0.620078 + 0.784540i \(0.287101\pi\)
\(822\) −1.19136e7 −0.614984
\(823\) 1.11177e7 0.572156 0.286078 0.958206i \(-0.407648\pi\)
0.286078 + 0.958206i \(0.407648\pi\)
\(824\) 1.69158e6 0.0867908
\(825\) 0 0
\(826\) −3.27786e7 −1.67163
\(827\) 1.84798e7 0.939580 0.469790 0.882778i \(-0.344330\pi\)
0.469790 + 0.882778i \(0.344330\pi\)
\(828\) −2.77921e6 −0.140879
\(829\) 1.97075e7 0.995969 0.497984 0.867186i \(-0.334074\pi\)
0.497984 + 0.867186i \(0.334074\pi\)
\(830\) 0 0
\(831\) −1.78898e7 −0.898678
\(832\) 1.78778e7 0.895374
\(833\) 2.56662e6 0.128159
\(834\) −5.46105e6 −0.271870
\(835\) 0 0
\(836\) −7.24106e6 −0.358333
\(837\) −2.79855e6 −0.138077
\(838\) −8.41090e6 −0.413745
\(839\) 3.06964e7 1.50550 0.752752 0.658304i \(-0.228726\pi\)
0.752752 + 0.658304i \(0.228726\pi\)
\(840\) 0 0
\(841\) −1.33321e7 −0.649991
\(842\) −399420. −0.0194155
\(843\) 6.84744e6 0.331863
\(844\) 1.97669e7 0.955173
\(845\) 0 0
\(846\) 4.27855e6 0.205528
\(847\) −1.64268e6 −0.0786764
\(848\) −2.31262e6 −0.110437
\(849\) 1.95977e7 0.933116
\(850\) 0 0
\(851\) 1.54691e7 0.732220
\(852\) −8.64392e6 −0.407954
\(853\) 1.45171e7 0.683137 0.341569 0.939857i \(-0.389042\pi\)
0.341569 + 0.939857i \(0.389042\pi\)
\(854\) 4.81570e6 0.225951
\(855\) 0 0
\(856\) −5.46885e6 −0.255101
\(857\) −2.63178e7 −1.22405 −0.612024 0.790840i \(-0.709644\pi\)
−0.612024 + 0.790840i \(0.709644\pi\)
\(858\) −7.08185e6 −0.328419
\(859\) −7.00715e6 −0.324010 −0.162005 0.986790i \(-0.551796\pi\)
−0.162005 + 0.986790i \(0.551796\pi\)
\(860\) 0 0
\(861\) −624454. −0.0287073
\(862\) −2.91635e6 −0.133681
\(863\) −1.80082e6 −0.0823084 −0.0411542 0.999153i \(-0.513103\pi\)
−0.0411542 + 0.999153i \(0.513103\pi\)
\(864\) −5.55738e6 −0.253271
\(865\) 0 0
\(866\) 4.92587e7 2.23197
\(867\) 9.44756e6 0.426847
\(868\) −1.14796e7 −0.517162
\(869\) −3.85772e6 −0.173293
\(870\) 0 0
\(871\) −3.02904e7 −1.35288
\(872\) −8.89790e6 −0.396275
\(873\) −1.20968e7 −0.537200
\(874\) −2.21371e7 −0.980263
\(875\) 0 0
\(876\) 7.96013e6 0.350477
\(877\) 4.22779e7 1.85615 0.928077 0.372388i \(-0.121461\pi\)
0.928077 + 0.372388i \(0.121461\pi\)
\(878\) 1.86406e7 0.816064
\(879\) −5.16148e6 −0.225321
\(880\) 0 0
\(881\) 9.29766e6 0.403584 0.201792 0.979428i \(-0.435323\pi\)
0.201792 + 0.979428i \(0.435323\pi\)
\(882\) 2.61706e6 0.113277
\(883\) 3.50752e7 1.51390 0.756951 0.653472i \(-0.226688\pi\)
0.756951 + 0.653472i \(0.226688\pi\)
\(884\) 1.37686e7 0.592596
\(885\) 0 0
\(886\) −3.34896e7 −1.43326
\(887\) −3.66149e7 −1.56260 −0.781302 0.624153i \(-0.785444\pi\)
−0.781302 + 0.624153i \(0.785444\pi\)
\(888\) 4.42898e6 0.188483
\(889\) 2.95902e7 1.25572
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −7.43186e6 −0.312742
\(893\) 1.54863e7 0.649859
\(894\) −7.74504e6 −0.324101
\(895\) 0 0
\(896\) 9.27899e6 0.386128
\(897\) −9.83823e6 −0.408259
\(898\) −4.98618e7 −2.06337
\(899\) 1.02859e7 0.424464
\(900\) 0 0
\(901\) −1.20611e6 −0.0494964
\(902\) 573065. 0.0234524
\(903\) −1.34804e7 −0.550154
\(904\) −5.93385e6 −0.241499
\(905\) 0 0
\(906\) 2.42219e7 0.980365
\(907\) −4.65461e7 −1.87873 −0.939366 0.342916i \(-0.888585\pi\)
−0.939366 + 0.342916i \(0.888585\pi\)
\(908\) 2.05272e7 0.826257
\(909\) −4.51991e6 −0.181435
\(910\) 0 0
\(911\) 3.05339e7 1.21895 0.609476 0.792805i \(-0.291380\pi\)
0.609476 + 0.792805i \(0.291380\pi\)
\(912\) −2.35730e7 −0.938487
\(913\) −1.10116e7 −0.437194
\(914\) −2.31974e7 −0.918490
\(915\) 0 0
\(916\) −2.76386e7 −1.08837
\(917\) 3.85331e7 1.51325
\(918\) −3.39662e6 −0.133027
\(919\) −2.39764e7 −0.936472 −0.468236 0.883603i \(-0.655110\pi\)
−0.468236 + 0.883603i \(0.655110\pi\)
\(920\) 0 0
\(921\) −7.05630e6 −0.274112
\(922\) 4.83319e7 1.87243
\(923\) −3.05989e7 −1.18223
\(924\) 3.25648e6 0.125478
\(925\) 0 0
\(926\) −1.02090e7 −0.391251
\(927\) 3.34566e6 0.127874
\(928\) 2.04257e7 0.778587
\(929\) 3.60560e7 1.37069 0.685343 0.728221i \(-0.259652\pi\)
0.685343 + 0.728221i \(0.259652\pi\)
\(930\) 0 0
\(931\) 9.47251e6 0.358171
\(932\) 3.49415e7 1.31765
\(933\) 1.08478e7 0.407978
\(934\) −1.94037e7 −0.727808
\(935\) 0 0
\(936\) −2.81680e6 −0.105091
\(937\) 2.89459e7 1.07706 0.538528 0.842608i \(-0.318981\pi\)
0.538528 + 0.842608i \(0.318981\pi\)
\(938\) 3.06518e7 1.13749
\(939\) −1.32276e7 −0.489571
\(940\) 0 0
\(941\) −2.26496e7 −0.833849 −0.416924 0.908941i \(-0.636892\pi\)
−0.416924 + 0.908941i \(0.636892\pi\)
\(942\) −1.39372e7 −0.511738
\(943\) 796113. 0.0291538
\(944\) 4.44998e7 1.62528
\(945\) 0 0
\(946\) 1.23711e7 0.449448
\(947\) 3.01070e7 1.09092 0.545460 0.838137i \(-0.316355\pi\)
0.545460 + 0.838137i \(0.316355\pi\)
\(948\) 7.64762e6 0.276379
\(949\) 2.81783e7 1.01566
\(950\) 0 0
\(951\) 8.48855e6 0.304356
\(952\) 2.79546e6 0.0999681
\(953\) −3.69781e7 −1.31890 −0.659450 0.751748i \(-0.729211\pi\)
−0.659450 + 0.751748i \(0.729211\pi\)
\(954\) −1.22981e6 −0.0437489
\(955\) 0 0
\(956\) 4.62496e6 0.163668
\(957\) −2.91785e6 −0.102987
\(958\) 3.88765e7 1.36859
\(959\) 1.93928e7 0.680916
\(960\) 0 0
\(961\) −1.38921e7 −0.485242
\(962\) −7.81421e7 −2.72237
\(963\) −1.08165e7 −0.375855
\(964\) −1.32206e7 −0.458205
\(965\) 0 0
\(966\) 9.95559e6 0.343261
\(967\) 5.23212e6 0.179933 0.0899666 0.995945i \(-0.471324\pi\)
0.0899666 + 0.995945i \(0.471324\pi\)
\(968\) 599606. 0.0205673
\(969\) −1.22941e7 −0.420618
\(970\) 0 0
\(971\) 5.23574e7 1.78209 0.891046 0.453913i \(-0.149972\pi\)
0.891046 + 0.453913i \(0.149972\pi\)
\(972\) −1.57380e6 −0.0534299
\(973\) 8.88940e6 0.301017
\(974\) −3.39156e7 −1.14552
\(975\) 0 0
\(976\) −6.53774e6 −0.219686
\(977\) 3.76035e7 1.26035 0.630176 0.776452i \(-0.282983\pi\)
0.630176 + 0.776452i \(0.282983\pi\)
\(978\) 2.19007e7 0.732170
\(979\) 6.09878e6 0.203370
\(980\) 0 0
\(981\) −1.75986e7 −0.583855
\(982\) −2.50482e7 −0.828890
\(983\) 4.17452e6 0.137791 0.0688957 0.997624i \(-0.478052\pi\)
0.0688957 + 0.997624i \(0.478052\pi\)
\(984\) 227936. 0.00750457
\(985\) 0 0
\(986\) 1.24840e7 0.408942
\(987\) −6.96456e6 −0.227562
\(988\) 5.08150e7 1.65615
\(989\) 1.71861e7 0.558711
\(990\) 0 0
\(991\) −7.89607e6 −0.255404 −0.127702 0.991813i \(-0.540760\pi\)
−0.127702 + 0.991813i \(0.540760\pi\)
\(992\) 2.92650e7 0.944212
\(993\) 2.54511e7 0.819095
\(994\) 3.09639e7 0.994008
\(995\) 0 0
\(996\) 2.18296e7 0.697264
\(997\) −9.45089e6 −0.301117 −0.150558 0.988601i \(-0.548107\pi\)
−0.150558 + 0.988601i \(0.548107\pi\)
\(998\) 1.22706e6 0.0389979
\(999\) 8.75979e6 0.277703
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.1.3 yes 10
5.4 even 2 825.6.a.t.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.8 10 5.4 even 2
825.6.a.u.1.3 yes 10 1.1 even 1 trivial