Properties

Label 825.6.a.t.1.8
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 1.8
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 \(-0.263314\pi\)
0.676921 + 0.736055i \(0.263314\pi\)
\(3\) 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 \(-0.357554\pi\)
0.432720 + 0.901528i \(0.357554\pi\)
\(8\) −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 \(-0.254621\pi\)
0.696767 + 0.717298i \(0.254621\pi\)
\(14\) 859.262 1.17167
\(15\) 0 0
\(16\) −1166.52 −1.13918
\(17\) 608.382 0.510568 0.255284 0.966866i \(-0.417831\pi\)
0.255284 + 0.966866i \(0.417831\pi\)
\(18\) 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 \(-0.418347\pi\)
0.253717 + 0.967279i \(0.418347\pi\)
\(24\) −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 \(-0.243456\pi\)
0.721493 + 0.692422i \(0.243456\pi\)
\(38\) −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 \(-0.314427\pi\)
0.550526 + 0.834818i \(0.314427\pi\)
\(44\) 3224.95 0.251126
\(45\) 0 0
\(46\) 9859.21 0.686986
\(47\) 6897.13 0.455433 0.227716 0.973728i \(-0.426874\pi\)
0.227716 + 0.973728i \(0.426874\pi\)
\(48\) −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 \(-0.515435\pi\)
−0.0484719 + 0.998825i \(0.515435\pi\)
\(54\) 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 \(-0.661331\pi\)
−0.485415 + 0.874284i \(0.661331\pi\)
\(68\) 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 \(-0.381267\pi\)
0.364420 + 0.931235i \(0.381267\pi\)
\(74\) 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 \(-0.241838\pi\)
0.725004 + 0.688745i \(0.241838\pi\)
\(84\) 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 \(-0.201736\pi\)
0.805800 + 0.592188i \(0.201736\pi\)
\(98\) −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 \(-0.561436\pi\)
−0.191811 + 0.981432i \(0.561436\pi\)
\(104\) −34775.3 −0.315274
\(105\) 0 0
\(106\) −15182.8 −0.131247
\(107\) 133537. 1.12756 0.563782 0.825924i \(-0.309346\pi\)
0.563782 + 0.825924i \(0.309346\pi\)
\(108\) 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 \(-0.320793\pi\)
0.533722 + 0.845660i \(0.320793\pi\)
\(114\) −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 \(-0.241617\pi\)
0.725482 + 0.688241i \(0.241617\pi\)
\(128\) 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 \(-0.371301\pi\)
0.393393 + 0.919370i \(0.371301\pi\)
\(138\) 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 \(-0.393845\pi\)
0.327348 + 0.944904i \(0.393845\pi\)
\(158\) −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 \(-0.655153\pi\)
−0.468354 + 0.883541i \(0.655153\pi\)
\(164\) −16482.1 −0.0478524
\(165\) 0 0
\(166\) 696962. 1.96308
\(167\) 590220. 1.63766 0.818828 0.574038i \(-0.194624\pi\)
0.818828 + 0.574038i \(0.194624\pi\)
\(168\) −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 \(-0.699517\pi\)
−0.586557 + 0.809908i \(0.699517\pi\)
\(174\) 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 \(-0.908237\pi\)
−0.958734 + 0.284304i \(0.908237\pi\)
\(194\) 1.14375e6 2.18185
\(195\) 0 0
\(196\) −112441. −0.209066
\(197\) 854973. 1.56959 0.784796 0.619754i \(-0.212767\pi\)
0.784796 + 0.619754i \(0.212767\pi\)
\(198\) 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 \(-0.439882\pi\)
0.187745 + 0.982218i \(0.439882\pi\)
\(224\) −855313. −1.13895
\(225\) 0 0
\(226\) 1.10965e6 1.44515
\(227\) −770180. −0.992036 −0.496018 0.868312i \(-0.665205\pi\)
−0.496018 + 0.868312i \(0.665205\pi\)
\(228\) −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 \(-0.790446\pi\)
−0.791014 + 0.611799i \(0.790446\pi\)
\(234\) 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 \(-0.612699\pi\)
−0.346703 + 0.937975i \(0.612699\pi\)
\(258\) 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 \(-0.512681\pi\)
−0.0398273 + 0.999207i \(0.512681\pi\)
\(264\) −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 \(-0.783907\pi\)
−0.778278 + 0.627920i \(0.783907\pi\)
\(278\) −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 \(-0.200495\pi\)
0.808102 + 0.589043i \(0.200495\pi\)
\(284\) 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 \(-0.562514\pi\)
−0.195134 + 0.980777i \(0.562514\pi\)
\(294\) −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 \(-0.576291\pi\)
−0.237388 + 0.971415i \(0.576291\pi\)
\(308\) 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 \(-0.639368\pi\)
−0.423981 + 0.905671i \(0.639368\pi\)
\(314\) 1.54858e6 0.886356
\(315\) 0 0
\(316\) −849735. −0.478703
\(317\) 943173. 0.527161 0.263580 0.964637i \(-0.415097\pi\)
0.263580 + 0.964637i \(0.415097\pi\)
\(318\) −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 \(-0.369066\pi\)
0.399839 + 0.916585i \(0.369066\pi\)
\(338\) 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 \(-0.267042\pi\)
0.668254 + 0.743933i \(0.267042\pi\)
\(348\) 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 \(-0.618808\pi\)
−0.364640 + 0.931149i \(0.618808\pi\)
\(354\) −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 \(-0.631097\pi\)
−0.400308 + 0.916381i \(0.631097\pi\)
\(368\) −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 \(-0.689263\pi\)
−0.560166 + 0.828380i \(0.689263\pi\)
\(374\) 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 \(-0.688807\pi\)
−0.558979 + 0.829182i \(0.688807\pi\)
\(384\) 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 \(-0.622335\pi\)
−0.374935 + 0.927051i \(0.622335\pi\)
\(398\) 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 \(-0.191564\pi\)
0.824309 + 0.566140i \(0.191564\pi\)
\(434\) 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 \(-0.677557\pi\)
−0.529331 + 0.848415i \(0.677557\pi\)
\(444\) 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 \(-0.610162\pi\)
−0.339216 + 0.940709i \(0.610162\pi\)
\(458\) −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 \(-0.546156\pi\)
−0.144497 + 0.989505i \(0.546156\pi\)
\(464\) −3.12556e6 −0.673959
\(465\) 0 0
\(466\) −1.00403e7 −2.14182
\(467\) −2.53362e6 −0.537587 −0.268793 0.963198i \(-0.586625\pi\)
−0.268793 + 0.963198i \(0.586625\pi\)
\(468\) 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 \(-0.639045\pi\)
−0.423061 + 0.906101i \(0.639045\pi\)
\(488\) −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 \(-0.360158\pi\)
0.425330 + 0.905038i \(0.360158\pi\)
\(504\) −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 \(-0.413800\pi\)
0.267508 + 0.963556i \(0.413800\pi\)
\(524\) −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 \(-0.749068\pi\)
−0.705034 + 0.709173i \(0.749068\pi\)
\(548\) 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 \(-0.482611\pi\)
0.0546008 + 0.998508i \(0.482611\pi\)
\(558\) 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 \(-0.541448\pi\)
−0.129847 + 0.991534i \(0.541448\pi\)
\(564\) 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 \(-0.655290\pi\)
−0.468736 + 0.883338i \(0.655290\pi\)
\(578\) −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 \(-0.669219\pi\)
−0.506928 + 0.861988i \(0.669219\pi\)
\(588\) −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 \(-0.615899\pi\)
−0.356114 + 0.934442i \(0.615899\pi\)
\(594\) 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 \(-0.254912\pi\)
0.696112 + 0.717933i \(0.254912\pi\)
\(608\) 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 \(-0.289990\pi\)
0.612932 + 0.790136i \(0.289990\pi\)
\(614\) −6.00452e6 −0.642772
\(615\) 0 0
\(616\) −555985. −0.0590352
\(617\) 1.57880e7 1.66961 0.834804 0.550547i \(-0.185581\pi\)
0.834804 + 0.550547i \(0.185581\pi\)
\(618\) −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 \(-0.301576\pi\)
0.583772 + 0.811917i \(0.301576\pi\)
\(644\) 3.84963e6 0.365767
\(645\) 0 0
\(646\) −1.04616e7 −0.986318
\(647\) −1.57395e7 −1.47819 −0.739095 0.673601i \(-0.764746\pi\)
−0.739095 + 0.673601i \(0.764746\pi\)
\(648\) −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 \(-0.420641\pi\)
0.246738 + 0.969082i \(0.420641\pi\)
\(654\) −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 \(-0.566849\pi\)
−0.208473 + 0.978028i \(0.566849\pi\)
\(674\) 1.27683e7 1.08264
\(675\) 0 0
\(676\) 9.32127e6 0.784528
\(677\) 2.13572e7 1.79091 0.895453 0.445156i \(-0.146852\pi\)
0.895453 + 0.445156i \(0.146852\pi\)
\(678\) 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 \(-0.694255\pi\)
−0.573089 + 0.819493i \(0.694255\pi\)
\(684\) −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 \(-0.845312\pi\)
−0.884223 + 0.467065i \(0.845312\pi\)
\(728\) −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 \(-0.282482\pi\)
0.631397 + 0.775460i \(0.282482\pi\)
\(734\) −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 \(-0.566492\pi\)
−0.207375 + 0.978261i \(0.566492\pi\)
\(744\) −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 \(-0.673756\pi\)
−0.519164 + 0.854675i \(0.673756\pi\)
\(758\) 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 \(-0.245363\pi\)
0.717333 + 0.696730i \(0.245363\pi\)
\(774\) 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 \(-0.563181\pi\)
−0.197188 + 0.980366i \(0.563181\pi\)
\(788\) 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 \(-0.544183\pi\)
−0.138358 + 0.990382i \(0.544183\pi\)
\(798\) −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 \(-0.592352\pi\)
−0.286078 + 0.958206i \(0.592352\pi\)
\(824\) 1.69158e6 0.0867908
\(825\) 0 0
\(826\) −3.27786e7 −1.67163
\(827\) −1.84798e7 −0.939580 −0.469790 0.882778i \(-0.655670\pi\)
−0.469790 + 0.882778i \(0.655670\pi\)
\(828\) 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 \(-0.610958\pi\)
−0.341569 + 0.939857i \(0.610958\pi\)
\(854\) 4.81570e6 0.225951
\(855\) 0 0
\(856\) −5.46885e6 −0.255101
\(857\) 2.63178e7 1.22405 0.612024 0.790840i \(-0.290356\pi\)
0.612024 + 0.790840i \(0.290356\pi\)
\(858\) 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 \(-0.486897\pi\)
0.0411542 + 0.999153i \(0.486897\pi\)
\(864\) −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 \(-0.878539\pi\)
−0.928077 + 0.372388i \(0.878539\pi\)
\(878\) −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 \(-0.773312\pi\)
−0.756951 + 0.653472i \(0.773312\pi\)
\(884\) 1.37686e7 0.592596
\(885\) 0 0
\(886\) −3.34896e7 −1.43326
\(887\) 3.66149e7 1.56260 0.781302 0.624153i \(-0.214556\pi\)
0.781302 + 0.624153i \(0.214556\pi\)
\(888\) −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 \(-0.111415\pi\)
0.939366 + 0.342916i \(0.111415\pi\)
\(908\) −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 \(-0.681019\pi\)
−0.538528 + 0.842608i \(0.681019\pi\)
\(938\) −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 \(-0.683645\pi\)
−0.545460 + 0.838137i \(0.683645\pi\)
\(948\) −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 \(-0.270789\pi\)
0.659450 + 0.751748i \(0.270789\pi\)
\(954\) −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 \(-0.528676\pi\)
−0.0899666 + 0.995945i \(0.528676\pi\)
\(968\) −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 \(-0.717017\pi\)
−0.630176 + 0.776452i \(0.717017\pi\)
\(978\) −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 \(-0.521948\pi\)
−0.0688957 + 0.997624i \(0.521948\pi\)
\(984\) 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 \(-0.451893\pi\)
0.150558 + 0.988601i \(0.451893\pi\)
\(998\) −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.t.1.8 10
5.4 even 2 825.6.a.u.1.3 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.8 10 1.1 even 1 trivial
825.6.a.u.1.3 yes 10 5.4 even 2