Properties

Label 825.6.a.y.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.81495\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.81495 q^{2} +9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} +11.0326 q^{7} +22.8710 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-7.81495 q^{2} +9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} +11.0326 q^{7} +22.8710 q^{8} +81.0000 q^{9} +121.000 q^{11} +261.661 q^{12} +248.951 q^{13} -86.2194 q^{14} -1109.09 q^{16} -669.264 q^{17} -633.011 q^{18} +2327.66 q^{19} +99.2936 q^{21} -945.609 q^{22} +867.792 q^{23} +205.839 q^{24} -1945.54 q^{26} +729.000 q^{27} +320.756 q^{28} +2640.61 q^{29} -3381.17 q^{31} +7935.57 q^{32} +1089.00 q^{33} +5230.26 q^{34} +2354.95 q^{36} -434.439 q^{37} -18190.5 q^{38} +2240.56 q^{39} +6114.13 q^{41} -775.975 q^{42} +6682.47 q^{43} +3517.89 q^{44} -6781.75 q^{46} +20195.8 q^{47} -9981.77 q^{48} -16685.3 q^{49} -6023.37 q^{51} +7237.87 q^{52} +3335.16 q^{53} -5697.10 q^{54} +252.327 q^{56} +20948.9 q^{57} -20636.2 q^{58} -35671.0 q^{59} +16705.5 q^{61} +26423.7 q^{62} +893.642 q^{63} -26525.4 q^{64} -8510.48 q^{66} +15071.1 q^{67} -19457.8 q^{68} +7810.13 q^{69} -21017.5 q^{71} +1852.55 q^{72} +5360.83 q^{73} +3395.12 q^{74} +67673.1 q^{76} +1334.95 q^{77} -17509.9 q^{78} +11717.1 q^{79} +6561.00 q^{81} -47781.6 q^{82} +51987.1 q^{83} +2886.81 q^{84} -52223.2 q^{86} +23765.5 q^{87} +2767.39 q^{88} +89693.5 q^{89} +2746.58 q^{91} +25229.7 q^{92} -30430.5 q^{93} -157829. q^{94} +71420.2 q^{96} +19443.4 q^{97} +130395. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.81495 −1.38150 −0.690750 0.723093i \(-0.742720\pi\)
−0.690750 + 0.723093i \(0.742720\pi\)
\(3\) 9.00000 0.577350
\(4\) 29.0734 0.908545
\(5\) 0 0
\(6\) −70.3345 −0.797610
\(7\) 11.0326 0.0851008 0.0425504 0.999094i \(-0.486452\pi\)
0.0425504 + 0.999094i \(0.486452\pi\)
\(8\) 22.8710 0.126346
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 261.661 0.524549
\(13\) 248.951 0.408560 0.204280 0.978913i \(-0.434515\pi\)
0.204280 + 0.978913i \(0.434515\pi\)
\(14\) −86.2194 −0.117567
\(15\) 0 0
\(16\) −1109.09 −1.08309
\(17\) −669.264 −0.561662 −0.280831 0.959757i \(-0.590610\pi\)
−0.280831 + 0.959757i \(0.590610\pi\)
\(18\) −633.011 −0.460500
\(19\) 2327.66 1.47923 0.739615 0.673030i \(-0.235008\pi\)
0.739615 + 0.673030i \(0.235008\pi\)
\(20\) 0 0
\(21\) 99.2936 0.0491330
\(22\) −945.609 −0.416538
\(23\) 867.792 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(24\) 205.839 0.0729456
\(25\) 0 0
\(26\) −1945.54 −0.564426
\(27\) 729.000 0.192450
\(28\) 320.756 0.0773179
\(29\) 2640.61 0.583055 0.291527 0.956562i \(-0.405837\pi\)
0.291527 + 0.956562i \(0.405837\pi\)
\(30\) 0 0
\(31\) −3381.17 −0.631921 −0.315961 0.948772i \(-0.602327\pi\)
−0.315961 + 0.948772i \(0.602327\pi\)
\(32\) 7935.57 1.36995
\(33\) 1089.00 0.174078
\(34\) 5230.26 0.775937
\(35\) 0 0
\(36\) 2354.95 0.302848
\(37\) −434.439 −0.0521705 −0.0260852 0.999660i \(-0.508304\pi\)
−0.0260852 + 0.999660i \(0.508304\pi\)
\(38\) −18190.5 −2.04356
\(39\) 2240.56 0.235882
\(40\) 0 0
\(41\) 6114.13 0.568035 0.284018 0.958819i \(-0.408333\pi\)
0.284018 + 0.958819i \(0.408333\pi\)
\(42\) −775.975 −0.0678772
\(43\) 6682.47 0.551145 0.275573 0.961280i \(-0.411133\pi\)
0.275573 + 0.961280i \(0.411133\pi\)
\(44\) 3517.89 0.273937
\(45\) 0 0
\(46\) −6781.75 −0.472550
\(47\) 20195.8 1.33357 0.666787 0.745249i \(-0.267669\pi\)
0.666787 + 0.745249i \(0.267669\pi\)
\(48\) −9981.77 −0.625323
\(49\) −16685.3 −0.992758
\(50\) 0 0
\(51\) −6023.37 −0.324276
\(52\) 7237.87 0.371195
\(53\) 3335.16 0.163090 0.0815450 0.996670i \(-0.474015\pi\)
0.0815450 + 0.996670i \(0.474015\pi\)
\(54\) −5697.10 −0.265870
\(55\) 0 0
\(56\) 252.327 0.0107521
\(57\) 20948.9 0.854034
\(58\) −20636.2 −0.805491
\(59\) −35671.0 −1.33409 −0.667045 0.745017i \(-0.732441\pi\)
−0.667045 + 0.745017i \(0.732441\pi\)
\(60\) 0 0
\(61\) 16705.5 0.574825 0.287412 0.957807i \(-0.407205\pi\)
0.287412 + 0.957807i \(0.407205\pi\)
\(62\) 26423.7 0.873000
\(63\) 893.642 0.0283669
\(64\) −26525.4 −0.809490
\(65\) 0 0
\(66\) −8510.48 −0.240488
\(67\) 15071.1 0.410164 0.205082 0.978745i \(-0.434254\pi\)
0.205082 + 0.978745i \(0.434254\pi\)
\(68\) −19457.8 −0.510295
\(69\) 7810.13 0.197486
\(70\) 0 0
\(71\) −21017.5 −0.494806 −0.247403 0.968913i \(-0.579577\pi\)
−0.247403 + 0.968913i \(0.579577\pi\)
\(72\) 1852.55 0.0421152
\(73\) 5360.83 0.117740 0.0588701 0.998266i \(-0.481250\pi\)
0.0588701 + 0.998266i \(0.481250\pi\)
\(74\) 3395.12 0.0720735
\(75\) 0 0
\(76\) 67673.1 1.34395
\(77\) 1334.95 0.0256589
\(78\) −17509.9 −0.325871
\(79\) 11717.1 0.211228 0.105614 0.994407i \(-0.466319\pi\)
0.105614 + 0.994407i \(0.466319\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −47781.6 −0.784741
\(83\) 51987.1 0.828325 0.414162 0.910203i \(-0.364075\pi\)
0.414162 + 0.910203i \(0.364075\pi\)
\(84\) 2886.81 0.0446395
\(85\) 0 0
\(86\) −52223.2 −0.761408
\(87\) 23765.5 0.336627
\(88\) 2767.39 0.0380946
\(89\) 89693.5 1.20029 0.600145 0.799892i \(-0.295110\pi\)
0.600145 + 0.799892i \(0.295110\pi\)
\(90\) 0 0
\(91\) 2746.58 0.0347688
\(92\) 25229.7 0.310772
\(93\) −30430.5 −0.364840
\(94\) −157829. −1.84233
\(95\) 0 0
\(96\) 71420.2 0.790939
\(97\) 19443.4 0.209819 0.104909 0.994482i \(-0.466545\pi\)
0.104909 + 0.994482i \(0.466545\pi\)
\(98\) 130395. 1.37150
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −144534. −1.40983 −0.704917 0.709290i \(-0.749016\pi\)
−0.704917 + 0.709290i \(0.749016\pi\)
\(102\) 47072.4 0.447987
\(103\) 156184. 1.45059 0.725293 0.688441i \(-0.241704\pi\)
0.725293 + 0.688441i \(0.241704\pi\)
\(104\) 5693.76 0.0516197
\(105\) 0 0
\(106\) −26064.1 −0.225309
\(107\) −53993.7 −0.455915 −0.227957 0.973671i \(-0.573205\pi\)
−0.227957 + 0.973671i \(0.573205\pi\)
\(108\) 21194.5 0.174850
\(109\) −119467. −0.963120 −0.481560 0.876413i \(-0.659930\pi\)
−0.481560 + 0.876413i \(0.659930\pi\)
\(110\) 0 0
\(111\) −3909.95 −0.0301206
\(112\) −12236.1 −0.0921719
\(113\) 143349. 1.05609 0.528043 0.849217i \(-0.322926\pi\)
0.528043 + 0.849217i \(0.322926\pi\)
\(114\) −163715. −1.17985
\(115\) 0 0
\(116\) 76771.6 0.529731
\(117\) 20165.0 0.136187
\(118\) 278767. 1.84305
\(119\) −7383.74 −0.0477979
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −130553. −0.794121
\(123\) 55027.2 0.327955
\(124\) −98302.3 −0.574129
\(125\) 0 0
\(126\) −6983.77 −0.0391889
\(127\) −160600. −0.883562 −0.441781 0.897123i \(-0.645653\pi\)
−0.441781 + 0.897123i \(0.645653\pi\)
\(128\) −46643.9 −0.251634
\(129\) 60142.2 0.318204
\(130\) 0 0
\(131\) −6734.56 −0.0342871 −0.0171436 0.999853i \(-0.505457\pi\)
−0.0171436 + 0.999853i \(0.505457\pi\)
\(132\) 31661.0 0.158157
\(133\) 25680.2 0.125884
\(134\) −117780. −0.566642
\(135\) 0 0
\(136\) −15306.7 −0.0709635
\(137\) −117299. −0.533942 −0.266971 0.963705i \(-0.586023\pi\)
−0.266971 + 0.963705i \(0.586023\pi\)
\(138\) −61035.8 −0.272827
\(139\) −364129. −1.59852 −0.799261 0.600985i \(-0.794775\pi\)
−0.799261 + 0.600985i \(0.794775\pi\)
\(140\) 0 0
\(141\) 181762. 0.769939
\(142\) 164251. 0.683575
\(143\) 30123.1 0.123185
\(144\) −89835.9 −0.361030
\(145\) 0 0
\(146\) −41894.6 −0.162658
\(147\) −150168. −0.573169
\(148\) −12630.6 −0.0473992
\(149\) −296786. −1.09516 −0.547580 0.836753i \(-0.684451\pi\)
−0.547580 + 0.836753i \(0.684451\pi\)
\(150\) 0 0
\(151\) −464852. −1.65910 −0.829549 0.558433i \(-0.811403\pi\)
−0.829549 + 0.558433i \(0.811403\pi\)
\(152\) 53235.9 0.186894
\(153\) −54210.4 −0.187221
\(154\) −10432.5 −0.0354477
\(155\) 0 0
\(156\) 65140.8 0.214310
\(157\) −188905. −0.611639 −0.305820 0.952089i \(-0.598930\pi\)
−0.305820 + 0.952089i \(0.598930\pi\)
\(158\) −91568.4 −0.291812
\(159\) 30016.5 0.0941601
\(160\) 0 0
\(161\) 9574.02 0.0291092
\(162\) −51273.9 −0.153500
\(163\) −547363. −1.61364 −0.806820 0.590797i \(-0.798813\pi\)
−0.806820 + 0.590797i \(0.798813\pi\)
\(164\) 177759. 0.516085
\(165\) 0 0
\(166\) −406277. −1.14433
\(167\) 146681. 0.406990 0.203495 0.979076i \(-0.434770\pi\)
0.203495 + 0.979076i \(0.434770\pi\)
\(168\) 2270.94 0.00620773
\(169\) −309316. −0.833079
\(170\) 0 0
\(171\) 188540. 0.493076
\(172\) 194282. 0.500740
\(173\) 625665. 1.58937 0.794687 0.607019i \(-0.207635\pi\)
0.794687 + 0.607019i \(0.207635\pi\)
\(174\) −185726. −0.465050
\(175\) 0 0
\(176\) −134199. −0.326564
\(177\) −321039. −0.770238
\(178\) −700950. −1.65820
\(179\) 233383. 0.544422 0.272211 0.962238i \(-0.412245\pi\)
0.272211 + 0.962238i \(0.412245\pi\)
\(180\) 0 0
\(181\) 503730. 1.14288 0.571441 0.820643i \(-0.306385\pi\)
0.571441 + 0.820643i \(0.306385\pi\)
\(182\) −21464.4 −0.0480331
\(183\) 150350. 0.331875
\(184\) 19847.3 0.0432171
\(185\) 0 0
\(186\) 237813. 0.504027
\(187\) −80980.9 −0.169347
\(188\) 587162. 1.21161
\(189\) 8042.78 0.0163777
\(190\) 0 0
\(191\) 953992. 1.89217 0.946087 0.323912i \(-0.104998\pi\)
0.946087 + 0.323912i \(0.104998\pi\)
\(192\) −238728. −0.467359
\(193\) 862260. 1.66627 0.833135 0.553070i \(-0.186544\pi\)
0.833135 + 0.553070i \(0.186544\pi\)
\(194\) −151950. −0.289865
\(195\) 0 0
\(196\) −485098. −0.901965
\(197\) 497679. 0.913658 0.456829 0.889554i \(-0.348985\pi\)
0.456829 + 0.889554i \(0.348985\pi\)
\(198\) −76594.3 −0.138846
\(199\) −387099. −0.692930 −0.346465 0.938063i \(-0.612618\pi\)
−0.346465 + 0.938063i \(0.612618\pi\)
\(200\) 0 0
\(201\) 135640. 0.236808
\(202\) 1.12953e6 1.94769
\(203\) 29132.9 0.0496184
\(204\) −175120. −0.294619
\(205\) 0 0
\(206\) −1.22057e6 −2.00399
\(207\) 70291.2 0.114018
\(208\) −276108. −0.442508
\(209\) 281647. 0.446004
\(210\) 0 0
\(211\) −717688. −1.10976 −0.554880 0.831930i \(-0.687236\pi\)
−0.554880 + 0.831930i \(0.687236\pi\)
\(212\) 96964.6 0.148175
\(213\) −189158. −0.285677
\(214\) 421958. 0.629846
\(215\) 0 0
\(216\) 16672.9 0.0243152
\(217\) −37303.2 −0.0537770
\(218\) 933626. 1.33055
\(219\) 48247.5 0.0679773
\(220\) 0 0
\(221\) −166614. −0.229473
\(222\) 30556.1 0.0416117
\(223\) −113180. −0.152409 −0.0762043 0.997092i \(-0.524280\pi\)
−0.0762043 + 0.997092i \(0.524280\pi\)
\(224\) 87550.2 0.116584
\(225\) 0 0
\(226\) −1.12027e6 −1.45898
\(227\) 616694. 0.794337 0.397169 0.917746i \(-0.369993\pi\)
0.397169 + 0.917746i \(0.369993\pi\)
\(228\) 609058. 0.775928
\(229\) 502825. 0.633620 0.316810 0.948489i \(-0.397388\pi\)
0.316810 + 0.948489i \(0.397388\pi\)
\(230\) 0 0
\(231\) 12014.5 0.0148141
\(232\) 60393.4 0.0736664
\(233\) 141935. 0.171277 0.0856383 0.996326i \(-0.472707\pi\)
0.0856383 + 0.996326i \(0.472707\pi\)
\(234\) −157589. −0.188142
\(235\) 0 0
\(236\) −1.03708e6 −1.21208
\(237\) 105454. 0.121953
\(238\) 57703.5 0.0660328
\(239\) 733743. 0.830901 0.415450 0.909616i \(-0.363624\pi\)
0.415450 + 0.909616i \(0.363624\pi\)
\(240\) 0 0
\(241\) −327132. −0.362811 −0.181406 0.983408i \(-0.558065\pi\)
−0.181406 + 0.983408i \(0.558065\pi\)
\(242\) −114419. −0.125591
\(243\) 59049.0 0.0641500
\(244\) 485687. 0.522254
\(245\) 0 0
\(246\) −430035. −0.453071
\(247\) 579474. 0.604354
\(248\) −77330.7 −0.0798404
\(249\) 467884. 0.478233
\(250\) 0 0
\(251\) 36093.3 0.0361612 0.0180806 0.999837i \(-0.494244\pi\)
0.0180806 + 0.999837i \(0.494244\pi\)
\(252\) 25981.3 0.0257726
\(253\) 105003. 0.103134
\(254\) 1.25508e6 1.22064
\(255\) 0 0
\(256\) 1.21333e6 1.15712
\(257\) 1.70289e6 1.60825 0.804126 0.594459i \(-0.202634\pi\)
0.804126 + 0.594459i \(0.202634\pi\)
\(258\) −470009. −0.439599
\(259\) −4793.00 −0.00443975
\(260\) 0 0
\(261\) 213889. 0.194352
\(262\) 52630.3 0.0473677
\(263\) 411013. 0.366409 0.183205 0.983075i \(-0.441353\pi\)
0.183205 + 0.983075i \(0.441353\pi\)
\(264\) 24906.5 0.0219939
\(265\) 0 0
\(266\) −200689. −0.173908
\(267\) 807241. 0.692987
\(268\) 438169. 0.372653
\(269\) −100472. −0.0846571 −0.0423286 0.999104i \(-0.513478\pi\)
−0.0423286 + 0.999104i \(0.513478\pi\)
\(270\) 0 0
\(271\) 1.74029e6 1.43945 0.719727 0.694257i \(-0.244267\pi\)
0.719727 + 0.694257i \(0.244267\pi\)
\(272\) 742271. 0.608331
\(273\) 24719.3 0.0200738
\(274\) 916689. 0.737642
\(275\) 0 0
\(276\) 227067. 0.179425
\(277\) −1.99970e6 −1.56590 −0.782951 0.622083i \(-0.786286\pi\)
−0.782951 + 0.622083i \(0.786286\pi\)
\(278\) 2.84565e6 2.20836
\(279\) −273875. −0.210640
\(280\) 0 0
\(281\) 1.76621e6 1.33437 0.667184 0.744893i \(-0.267499\pi\)
0.667184 + 0.744893i \(0.267499\pi\)
\(282\) −1.42046e6 −1.06367
\(283\) 2.31041e6 1.71484 0.857419 0.514619i \(-0.172067\pi\)
0.857419 + 0.514619i \(0.172067\pi\)
\(284\) −611051. −0.449554
\(285\) 0 0
\(286\) −235410. −0.170181
\(287\) 67454.9 0.0483403
\(288\) 642782. 0.456649
\(289\) −971943. −0.684536
\(290\) 0 0
\(291\) 174991. 0.121139
\(292\) 155858. 0.106972
\(293\) 480796. 0.327183 0.163592 0.986528i \(-0.447692\pi\)
0.163592 + 0.986528i \(0.447692\pi\)
\(294\) 1.17355e6 0.791834
\(295\) 0 0
\(296\) −9936.05 −0.00659150
\(297\) 88209.0 0.0580259
\(298\) 2.31937e6 1.51296
\(299\) 216038. 0.139750
\(300\) 0 0
\(301\) 73725.2 0.0469029
\(302\) 3.63279e6 2.29205
\(303\) −1.30081e6 −0.813968
\(304\) −2.58157e6 −1.60214
\(305\) 0 0
\(306\) 423651. 0.258646
\(307\) −1.27658e6 −0.773043 −0.386521 0.922280i \(-0.626323\pi\)
−0.386521 + 0.922280i \(0.626323\pi\)
\(308\) 38811.5 0.0233122
\(309\) 1.40566e6 0.837496
\(310\) 0 0
\(311\) 2.50885e6 1.47087 0.735434 0.677596i \(-0.236978\pi\)
0.735434 + 0.677596i \(0.236978\pi\)
\(312\) 51243.8 0.0298027
\(313\) 1.85827e6 1.07213 0.536065 0.844177i \(-0.319910\pi\)
0.536065 + 0.844177i \(0.319910\pi\)
\(314\) 1.47629e6 0.844980
\(315\) 0 0
\(316\) 340656. 0.191910
\(317\) 2.48818e6 1.39070 0.695352 0.718670i \(-0.255249\pi\)
0.695352 + 0.718670i \(0.255249\pi\)
\(318\) −234577. −0.130082
\(319\) 319514. 0.175798
\(320\) 0 0
\(321\) −485943. −0.263222
\(322\) −74820.5 −0.0402143
\(323\) −1.55782e6 −0.830827
\(324\) 190751. 0.100949
\(325\) 0 0
\(326\) 4.27762e6 2.22924
\(327\) −1.07520e6 −0.556058
\(328\) 139836. 0.0717687
\(329\) 222813. 0.113488
\(330\) 0 0
\(331\) 2.19276e6 1.10007 0.550037 0.835141i \(-0.314614\pi\)
0.550037 + 0.835141i \(0.314614\pi\)
\(332\) 1.51144e6 0.752570
\(333\) −35189.6 −0.0173902
\(334\) −1.14631e6 −0.562257
\(335\) 0 0
\(336\) −110125. −0.0532155
\(337\) −1.08814e6 −0.521926 −0.260963 0.965349i \(-0.584040\pi\)
−0.260963 + 0.965349i \(0.584040\pi\)
\(338\) 2.41729e6 1.15090
\(339\) 1.29014e6 0.609732
\(340\) 0 0
\(341\) −409122. −0.190531
\(342\) −1.47343e6 −0.681186
\(343\) −369508. −0.169585
\(344\) 152835. 0.0696347
\(345\) 0 0
\(346\) −4.88954e6 −2.19572
\(347\) 1.48961e6 0.664123 0.332061 0.943258i \(-0.392256\pi\)
0.332061 + 0.943258i \(0.392256\pi\)
\(348\) 690944. 0.305841
\(349\) 3.61825e6 1.59014 0.795070 0.606518i \(-0.207434\pi\)
0.795070 + 0.606518i \(0.207434\pi\)
\(350\) 0 0
\(351\) 181485. 0.0786274
\(352\) 960205. 0.413054
\(353\) −346438. −0.147975 −0.0739875 0.997259i \(-0.523572\pi\)
−0.0739875 + 0.997259i \(0.523572\pi\)
\(354\) 2.50890e6 1.06408
\(355\) 0 0
\(356\) 2.60770e6 1.09052
\(357\) −66453.6 −0.0275961
\(358\) −1.82387e6 −0.752120
\(359\) −1.75327e6 −0.717981 −0.358991 0.933341i \(-0.616879\pi\)
−0.358991 + 0.933341i \(0.616879\pi\)
\(360\) 0 0
\(361\) 2.94190e6 1.18812
\(362\) −3.93663e6 −1.57889
\(363\) 131769. 0.0524864
\(364\) 79852.6 0.0315890
\(365\) 0 0
\(366\) −1.17498e6 −0.458486
\(367\) −2.68865e6 −1.04200 −0.521001 0.853556i \(-0.674441\pi\)
−0.521001 + 0.853556i \(0.674441\pi\)
\(368\) −962456. −0.370477
\(369\) 495245. 0.189345
\(370\) 0 0
\(371\) 36795.6 0.0138791
\(372\) −884720. −0.331473
\(373\) 2.73768e6 1.01885 0.509426 0.860515i \(-0.329858\pi\)
0.509426 + 0.860515i \(0.329858\pi\)
\(374\) 632862. 0.233954
\(375\) 0 0
\(376\) 461898. 0.168491
\(377\) 657383. 0.238213
\(378\) −62853.9 −0.0226257
\(379\) 1.99354e6 0.712898 0.356449 0.934315i \(-0.383987\pi\)
0.356449 + 0.934315i \(0.383987\pi\)
\(380\) 0 0
\(381\) −1.44540e6 −0.510125
\(382\) −7.45540e6 −2.61404
\(383\) −3.32117e6 −1.15690 −0.578448 0.815720i \(-0.696341\pi\)
−0.578448 + 0.815720i \(0.696341\pi\)
\(384\) −419795. −0.145281
\(385\) 0 0
\(386\) −6.73852e6 −2.30195
\(387\) 541280. 0.183715
\(388\) 565288. 0.190630
\(389\) 640065. 0.214462 0.107231 0.994234i \(-0.465802\pi\)
0.107231 + 0.994234i \(0.465802\pi\)
\(390\) 0 0
\(391\) −580782. −0.192119
\(392\) −381609. −0.125431
\(393\) −60611.1 −0.0197957
\(394\) −3.88934e6 −1.26222
\(395\) 0 0
\(396\) 284949. 0.0913122
\(397\) −806457. −0.256806 −0.128403 0.991722i \(-0.540985\pi\)
−0.128403 + 0.991722i \(0.540985\pi\)
\(398\) 3.02516e6 0.957284
\(399\) 231122. 0.0726789
\(400\) 0 0
\(401\) 2.17447e6 0.675292 0.337646 0.941273i \(-0.390369\pi\)
0.337646 + 0.941273i \(0.390369\pi\)
\(402\) −1.06002e6 −0.327151
\(403\) −841747. −0.258178
\(404\) −4.20211e6 −1.28090
\(405\) 0 0
\(406\) −227672. −0.0685479
\(407\) −52567.1 −0.0157300
\(408\) −137760. −0.0409708
\(409\) 4.98556e6 1.47369 0.736844 0.676063i \(-0.236315\pi\)
0.736844 + 0.676063i \(0.236315\pi\)
\(410\) 0 0
\(411\) −1.05569e6 −0.308272
\(412\) 4.54080e6 1.31792
\(413\) −393545. −0.113532
\(414\) −549322. −0.157517
\(415\) 0 0
\(416\) 1.97557e6 0.559705
\(417\) −3.27716e6 −0.922907
\(418\) −2.20106e6 −0.616156
\(419\) 5.22690e6 1.45448 0.727242 0.686382i \(-0.240802\pi\)
0.727242 + 0.686382i \(0.240802\pi\)
\(420\) 0 0
\(421\) 1.48521e6 0.408398 0.204199 0.978929i \(-0.434541\pi\)
0.204199 + 0.978929i \(0.434541\pi\)
\(422\) 5.60869e6 1.53314
\(423\) 1.63586e6 0.444524
\(424\) 76278.5 0.0206057
\(425\) 0 0
\(426\) 1.47826e6 0.394662
\(427\) 184306. 0.0489181
\(428\) −1.56978e6 −0.414219
\(429\) 271108. 0.0711212
\(430\) 0 0
\(431\) −301670. −0.0782238 −0.0391119 0.999235i \(-0.512453\pi\)
−0.0391119 + 0.999235i \(0.512453\pi\)
\(432\) −808523. −0.208441
\(433\) −1.78448e6 −0.457396 −0.228698 0.973497i \(-0.573447\pi\)
−0.228698 + 0.973497i \(0.573447\pi\)
\(434\) 291523. 0.0742930
\(435\) 0 0
\(436\) −3.47331e6 −0.875038
\(437\) 2.01993e6 0.505978
\(438\) −377051. −0.0939108
\(439\) −499339. −0.123661 −0.0618307 0.998087i \(-0.519694\pi\)
−0.0618307 + 0.998087i \(0.519694\pi\)
\(440\) 0 0
\(441\) −1.35151e6 −0.330919
\(442\) 1.30208e6 0.317017
\(443\) 4.62775e6 1.12037 0.560184 0.828369i \(-0.310731\pi\)
0.560184 + 0.828369i \(0.310731\pi\)
\(444\) −113676. −0.0273659
\(445\) 0 0
\(446\) 884500. 0.210553
\(447\) −2.67107e6 −0.632291
\(448\) −292645. −0.0688883
\(449\) 3.31281e6 0.775499 0.387749 0.921765i \(-0.373253\pi\)
0.387749 + 0.921765i \(0.373253\pi\)
\(450\) 0 0
\(451\) 739810. 0.171269
\(452\) 4.16766e6 0.959502
\(453\) −4.18367e6 −0.957881
\(454\) −4.81943e6 −1.09738
\(455\) 0 0
\(456\) 479123. 0.107903
\(457\) −7.09134e6 −1.58832 −0.794160 0.607709i \(-0.792089\pi\)
−0.794160 + 0.607709i \(0.792089\pi\)
\(458\) −3.92956e6 −0.875346
\(459\) −487893. −0.108092
\(460\) 0 0
\(461\) −3.68457e6 −0.807485 −0.403742 0.914873i \(-0.632291\pi\)
−0.403742 + 0.914873i \(0.632291\pi\)
\(462\) −93892.9 −0.0204658
\(463\) −5.32110e6 −1.15358 −0.576791 0.816892i \(-0.695695\pi\)
−0.576791 + 0.816892i \(0.695695\pi\)
\(464\) −2.92866e6 −0.631501
\(465\) 0 0
\(466\) −1.10921e6 −0.236619
\(467\) 9.27946e6 1.96893 0.984465 0.175581i \(-0.0561803\pi\)
0.984465 + 0.175581i \(0.0561803\pi\)
\(468\) 586267. 0.123732
\(469\) 166274. 0.0349053
\(470\) 0 0
\(471\) −1.70015e6 −0.353130
\(472\) −815831. −0.168556
\(473\) 808579. 0.166177
\(474\) −824116. −0.168478
\(475\) 0 0
\(476\) −214671. −0.0434265
\(477\) 270148. 0.0543633
\(478\) −5.73416e6 −1.14789
\(479\) −4.22915e6 −0.842199 −0.421100 0.907014i \(-0.638356\pi\)
−0.421100 + 0.907014i \(0.638356\pi\)
\(480\) 0 0
\(481\) −108154. −0.0213148
\(482\) 2.55652e6 0.501224
\(483\) 86166.2 0.0168062
\(484\) 425664. 0.0825950
\(485\) 0 0
\(486\) −461465. −0.0886233
\(487\) 8.16043e6 1.55916 0.779580 0.626302i \(-0.215432\pi\)
0.779580 + 0.626302i \(0.215432\pi\)
\(488\) 382072. 0.0726265
\(489\) −4.92627e6 −0.931635
\(490\) 0 0
\(491\) −1.72620e6 −0.323138 −0.161569 0.986861i \(-0.551655\pi\)
−0.161569 + 0.986861i \(0.551655\pi\)
\(492\) 1.59983e6 0.297962
\(493\) −1.76727e6 −0.327480
\(494\) −4.52856e6 −0.834916
\(495\) 0 0
\(496\) 3.75001e6 0.684428
\(497\) −231878. −0.0421084
\(498\) −3.65649e6 −0.660680
\(499\) 568851. 0.102270 0.0511349 0.998692i \(-0.483716\pi\)
0.0511349 + 0.998692i \(0.483716\pi\)
\(500\) 0 0
\(501\) 1.32013e6 0.234976
\(502\) −282068. −0.0499567
\(503\) 2.06495e6 0.363906 0.181953 0.983307i \(-0.441758\pi\)
0.181953 + 0.983307i \(0.441758\pi\)
\(504\) 20438.5 0.00358404
\(505\) 0 0
\(506\) −820592. −0.142479
\(507\) −2.78385e6 −0.480978
\(508\) −4.66920e6 −0.802755
\(509\) −4.03961e6 −0.691107 −0.345554 0.938399i \(-0.612309\pi\)
−0.345554 + 0.938399i \(0.612309\pi\)
\(510\) 0 0
\(511\) 59144.0 0.0100198
\(512\) −7.98952e6 −1.34693
\(513\) 1.69686e6 0.284678
\(514\) −1.33080e7 −2.22180
\(515\) 0 0
\(516\) 1.74854e6 0.289102
\(517\) 2.44369e6 0.402087
\(518\) 37457.1 0.00613352
\(519\) 5.63098e6 0.917626
\(520\) 0 0
\(521\) 2.55223e6 0.411932 0.205966 0.978559i \(-0.433966\pi\)
0.205966 + 0.978559i \(0.433966\pi\)
\(522\) −1.67154e6 −0.268497
\(523\) −1.54874e6 −0.247586 −0.123793 0.992308i \(-0.539506\pi\)
−0.123793 + 0.992308i \(0.539506\pi\)
\(524\) −195797. −0.0311514
\(525\) 0 0
\(526\) −3.21205e6 −0.506195
\(527\) 2.26290e6 0.354926
\(528\) −1.20779e6 −0.188542
\(529\) −5.68328e6 −0.882998
\(530\) 0 0
\(531\) −2.88935e6 −0.444697
\(532\) 746611. 0.114371
\(533\) 1.52212e6 0.232076
\(534\) −6.30855e6 −0.957362
\(535\) 0 0
\(536\) 344691. 0.0518224
\(537\) 2.10044e6 0.314322
\(538\) 785182. 0.116954
\(539\) −2.01892e6 −0.299328
\(540\) 0 0
\(541\) 171030. 0.0251235 0.0125617 0.999921i \(-0.496001\pi\)
0.0125617 + 0.999921i \(0.496001\pi\)
\(542\) −1.36003e7 −1.98861
\(543\) 4.53357e6 0.659843
\(544\) −5.31099e6 −0.769447
\(545\) 0 0
\(546\) −193180. −0.0277319
\(547\) 1.12874e7 1.61297 0.806487 0.591252i \(-0.201366\pi\)
0.806487 + 0.591252i \(0.201366\pi\)
\(548\) −3.41029e6 −0.485110
\(549\) 1.35315e6 0.191608
\(550\) 0 0
\(551\) 6.14644e6 0.862472
\(552\) 178625. 0.0249514
\(553\) 129270. 0.0179757
\(554\) 1.56275e7 2.16329
\(555\) 0 0
\(556\) −1.05865e7 −1.45233
\(557\) −164004. −0.0223984 −0.0111992 0.999937i \(-0.503565\pi\)
−0.0111992 + 0.999937i \(0.503565\pi\)
\(558\) 2.14032e6 0.291000
\(559\) 1.66361e6 0.225176
\(560\) 0 0
\(561\) −728828. −0.0977728
\(562\) −1.38028e7 −1.84343
\(563\) −9.26696e6 −1.23216 −0.616079 0.787685i \(-0.711280\pi\)
−0.616079 + 0.787685i \(0.711280\pi\)
\(564\) 5.28446e6 0.699524
\(565\) 0 0
\(566\) −1.80557e7 −2.36905
\(567\) 72385.0 0.00945565
\(568\) −480691. −0.0625166
\(569\) −8.70558e6 −1.12724 −0.563621 0.826034i \(-0.690592\pi\)
−0.563621 + 0.826034i \(0.690592\pi\)
\(570\) 0 0
\(571\) −3.36751e6 −0.432233 −0.216117 0.976368i \(-0.569339\pi\)
−0.216117 + 0.976368i \(0.569339\pi\)
\(572\) 875782. 0.111920
\(573\) 8.58593e6 1.09245
\(574\) −527157. −0.0667821
\(575\) 0 0
\(576\) −2.14856e6 −0.269830
\(577\) −702817. −0.0878826 −0.0439413 0.999034i \(-0.513991\pi\)
−0.0439413 + 0.999034i \(0.513991\pi\)
\(578\) 7.59568e6 0.945687
\(579\) 7.76034e6 0.962021
\(580\) 0 0
\(581\) 573554. 0.0704911
\(582\) −1.36755e6 −0.167353
\(583\) 403555. 0.0491735
\(584\) 122607. 0.0148759
\(585\) 0 0
\(586\) −3.75739e6 −0.452004
\(587\) −3.09026e6 −0.370168 −0.185084 0.982723i \(-0.559256\pi\)
−0.185084 + 0.982723i \(0.559256\pi\)
\(588\) −4.36589e6 −0.520750
\(589\) −7.87022e6 −0.934757
\(590\) 0 0
\(591\) 4.47911e6 0.527501
\(592\) 481830. 0.0565054
\(593\) −1.16268e7 −1.35777 −0.678883 0.734246i \(-0.737536\pi\)
−0.678883 + 0.734246i \(0.737536\pi\)
\(594\) −689349. −0.0801628
\(595\) 0 0
\(596\) −8.62859e6 −0.995002
\(597\) −3.48389e6 −0.400063
\(598\) −1.68833e6 −0.193065
\(599\) −2.09163e6 −0.238187 −0.119093 0.992883i \(-0.537999\pi\)
−0.119093 + 0.992883i \(0.537999\pi\)
\(600\) 0 0
\(601\) 2.26126e6 0.255367 0.127683 0.991815i \(-0.459246\pi\)
0.127683 + 0.991815i \(0.459246\pi\)
\(602\) −576159. −0.0647964
\(603\) 1.22076e6 0.136721
\(604\) −1.35148e7 −1.50737
\(605\) 0 0
\(606\) 1.01658e7 1.12450
\(607\) 1.26470e7 1.39321 0.696603 0.717457i \(-0.254694\pi\)
0.696603 + 0.717457i \(0.254694\pi\)
\(608\) 1.84713e7 2.02646
\(609\) 262196. 0.0286472
\(610\) 0 0
\(611\) 5.02777e6 0.544845
\(612\) −1.57608e6 −0.170098
\(613\) 9.20580e6 0.989488 0.494744 0.869039i \(-0.335262\pi\)
0.494744 + 0.869039i \(0.335262\pi\)
\(614\) 9.97644e6 1.06796
\(615\) 0 0
\(616\) 30531.6 0.00324188
\(617\) 1.90063e6 0.200995 0.100497 0.994937i \(-0.467957\pi\)
0.100497 + 0.994937i \(0.467957\pi\)
\(618\) −1.09851e7 −1.15700
\(619\) −3.94913e6 −0.414261 −0.207131 0.978313i \(-0.566413\pi\)
−0.207131 + 0.978313i \(0.566413\pi\)
\(620\) 0 0
\(621\) 632620. 0.0658285
\(622\) −1.96065e7 −2.03201
\(623\) 989554. 0.102146
\(624\) −2.48497e6 −0.255482
\(625\) 0 0
\(626\) −1.45223e7 −1.48115
\(627\) 2.53482e6 0.257501
\(628\) −5.49213e6 −0.555702
\(629\) 290754. 0.0293022
\(630\) 0 0
\(631\) −1.50561e7 −1.50536 −0.752680 0.658387i \(-0.771239\pi\)
−0.752680 + 0.658387i \(0.771239\pi\)
\(632\) 267981. 0.0266877
\(633\) −6.45919e6 −0.640721
\(634\) −1.94450e7 −1.92126
\(635\) 0 0
\(636\) 872682. 0.0855486
\(637\) −4.15382e6 −0.405601
\(638\) −2.49698e6 −0.242865
\(639\) −1.70242e6 −0.164935
\(640\) 0 0
\(641\) 1.51936e7 1.46055 0.730276 0.683153i \(-0.239392\pi\)
0.730276 + 0.683153i \(0.239392\pi\)
\(642\) 3.79762e6 0.363642
\(643\) −1.61149e6 −0.153709 −0.0768547 0.997042i \(-0.524488\pi\)
−0.0768547 + 0.997042i \(0.524488\pi\)
\(644\) 278350. 0.0264470
\(645\) 0 0
\(646\) 1.21743e7 1.14779
\(647\) 2.05398e7 1.92902 0.964508 0.264055i \(-0.0850600\pi\)
0.964508 + 0.264055i \(0.0850600\pi\)
\(648\) 150057. 0.0140384
\(649\) −4.31619e6 −0.402244
\(650\) 0 0
\(651\) −335729. −0.0310482
\(652\) −1.59137e7 −1.46606
\(653\) 7.96056e6 0.730568 0.365284 0.930896i \(-0.380972\pi\)
0.365284 + 0.930896i \(0.380972\pi\)
\(654\) 8.40263e6 0.768194
\(655\) 0 0
\(656\) −6.78110e6 −0.615234
\(657\) 434227. 0.0392467
\(658\) −1.74127e6 −0.156784
\(659\) 2.91266e6 0.261262 0.130631 0.991431i \(-0.458300\pi\)
0.130631 + 0.991431i \(0.458300\pi\)
\(660\) 0 0
\(661\) 1.89585e7 1.68772 0.843860 0.536564i \(-0.180278\pi\)
0.843860 + 0.536564i \(0.180278\pi\)
\(662\) −1.71363e7 −1.51975
\(663\) −1.49953e6 −0.132486
\(664\) 1.18900e6 0.104655
\(665\) 0 0
\(666\) 275005. 0.0240245
\(667\) 2.29150e6 0.199437
\(668\) 4.26453e6 0.369768
\(669\) −1.01862e6 −0.0879931
\(670\) 0 0
\(671\) 2.02137e6 0.173316
\(672\) 787952. 0.0673095
\(673\) −2.14478e7 −1.82535 −0.912674 0.408688i \(-0.865986\pi\)
−0.912674 + 0.408688i \(0.865986\pi\)
\(674\) 8.50374e6 0.721041
\(675\) 0 0
\(676\) −8.99289e6 −0.756889
\(677\) −1.84040e7 −1.54326 −0.771631 0.636071i \(-0.780559\pi\)
−0.771631 + 0.636071i \(0.780559\pi\)
\(678\) −1.00824e7 −0.842345
\(679\) 214512. 0.0178557
\(680\) 0 0
\(681\) 5.55024e6 0.458611
\(682\) 3.19727e6 0.263219
\(683\) −2.04252e7 −1.67538 −0.837692 0.546143i \(-0.816095\pi\)
−0.837692 + 0.546143i \(0.816095\pi\)
\(684\) 5.48152e6 0.447982
\(685\) 0 0
\(686\) 2.88768e6 0.234282
\(687\) 4.52543e6 0.365820
\(688\) −7.41143e6 −0.596940
\(689\) 830293. 0.0666321
\(690\) 0 0
\(691\) 1.16275e6 0.0926388 0.0463194 0.998927i \(-0.485251\pi\)
0.0463194 + 0.998927i \(0.485251\pi\)
\(692\) 1.81902e7 1.44402
\(693\) 108131. 0.00855295
\(694\) −1.16412e7 −0.917486
\(695\) 0 0
\(696\) 543540. 0.0425313
\(697\) −4.09197e6 −0.319044
\(698\) −2.82765e7 −2.19678
\(699\) 1.27741e6 0.0988866
\(700\) 0 0
\(701\) 1.80087e7 1.38416 0.692082 0.721818i \(-0.256693\pi\)
0.692082 + 0.721818i \(0.256693\pi\)
\(702\) −1.41830e6 −0.108624
\(703\) −1.01123e6 −0.0771721
\(704\) −3.20957e6 −0.244071
\(705\) 0 0
\(706\) 2.70739e6 0.204428
\(707\) −1.59459e6 −0.119978
\(708\) −9.33371e6 −0.699795
\(709\) −1.64523e7 −1.22917 −0.614583 0.788852i \(-0.710676\pi\)
−0.614583 + 0.788852i \(0.710676\pi\)
\(710\) 0 0
\(711\) 949084. 0.0704094
\(712\) 2.05138e6 0.151651
\(713\) −2.93415e6 −0.216152
\(714\) 519332. 0.0381241
\(715\) 0 0
\(716\) 6.78523e6 0.494632
\(717\) 6.60369e6 0.479721
\(718\) 1.37017e7 0.991892
\(719\) 2.09659e7 1.51249 0.756243 0.654290i \(-0.227033\pi\)
0.756243 + 0.654290i \(0.227033\pi\)
\(720\) 0 0
\(721\) 1.72312e6 0.123446
\(722\) −2.29908e7 −1.64139
\(723\) −2.94419e6 −0.209469
\(724\) 1.46452e7 1.03836
\(725\) 0 0
\(726\) −1.02977e6 −0.0725100
\(727\) −1.23702e7 −0.868040 −0.434020 0.900903i \(-0.642905\pi\)
−0.434020 + 0.900903i \(0.642905\pi\)
\(728\) 62817.1 0.00439288
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −4.47234e6 −0.309557
\(732\) 4.37118e6 0.301524
\(733\) 1.78981e7 1.23040 0.615200 0.788371i \(-0.289075\pi\)
0.615200 + 0.788371i \(0.289075\pi\)
\(734\) 2.10116e7 1.43953
\(735\) 0 0
\(736\) 6.88643e6 0.468597
\(737\) 1.82360e6 0.123669
\(738\) −3.87031e6 −0.261580
\(739\) −2.08324e6 −0.140323 −0.0701615 0.997536i \(-0.522351\pi\)
−0.0701615 + 0.997536i \(0.522351\pi\)
\(740\) 0 0
\(741\) 5.21526e6 0.348924
\(742\) −287556. −0.0191740
\(743\) −7.09623e6 −0.471580 −0.235790 0.971804i \(-0.575768\pi\)
−0.235790 + 0.971804i \(0.575768\pi\)
\(744\) −695977. −0.0460959
\(745\) 0 0
\(746\) −2.13948e7 −1.40754
\(747\) 4.21096e6 0.276108
\(748\) −2.35439e6 −0.153860
\(749\) −595692. −0.0387987
\(750\) 0 0
\(751\) −1.82817e7 −1.18282 −0.591408 0.806373i \(-0.701428\pi\)
−0.591408 + 0.806373i \(0.701428\pi\)
\(752\) −2.23989e7 −1.44438
\(753\) 324840. 0.0208777
\(754\) −5.13742e6 −0.329091
\(755\) 0 0
\(756\) 233831. 0.0148798
\(757\) −1.02358e6 −0.0649206 −0.0324603 0.999473i \(-0.510334\pi\)
−0.0324603 + 0.999473i \(0.510334\pi\)
\(758\) −1.55794e7 −0.984869
\(759\) 945026. 0.0595442
\(760\) 0 0
\(761\) 9.52820e6 0.596416 0.298208 0.954501i \(-0.403611\pi\)
0.298208 + 0.954501i \(0.403611\pi\)
\(762\) 1.12957e7 0.704738
\(763\) −1.31803e6 −0.0819623
\(764\) 2.77358e7 1.71913
\(765\) 0 0
\(766\) 2.59548e7 1.59825
\(767\) −8.88034e6 −0.545056
\(768\) 1.09200e7 0.668065
\(769\) −1.07522e6 −0.0655666 −0.0327833 0.999462i \(-0.510437\pi\)
−0.0327833 + 0.999462i \(0.510437\pi\)
\(770\) 0 0
\(771\) 1.53260e7 0.928524
\(772\) 2.50689e7 1.51388
\(773\) 6.01805e6 0.362249 0.181124 0.983460i \(-0.442026\pi\)
0.181124 + 0.983460i \(0.442026\pi\)
\(774\) −4.23008e6 −0.253803
\(775\) 0 0
\(776\) 444691. 0.0265096
\(777\) −43137.0 −0.00256329
\(778\) −5.00207e6 −0.296279
\(779\) 1.42316e7 0.840254
\(780\) 0 0
\(781\) −2.54312e6 −0.149190
\(782\) 4.53878e6 0.265413
\(783\) 1.92500e6 0.112209
\(784\) 1.85054e7 1.07525
\(785\) 0 0
\(786\) 473673. 0.0273478
\(787\) 2.82761e7 1.62736 0.813679 0.581314i \(-0.197461\pi\)
0.813679 + 0.581314i \(0.197461\pi\)
\(788\) 1.44692e7 0.830100
\(789\) 3.69912e6 0.211547
\(790\) 0 0
\(791\) 1.58152e6 0.0898738
\(792\) 224158. 0.0126982
\(793\) 4.15886e6 0.234850
\(794\) 6.30242e6 0.354777
\(795\) 0 0
\(796\) −1.12543e7 −0.629558
\(797\) 2.70959e7 1.51098 0.755490 0.655161i \(-0.227399\pi\)
0.755490 + 0.655161i \(0.227399\pi\)
\(798\) −1.80620e6 −0.100406
\(799\) −1.35163e7 −0.749017
\(800\) 0 0
\(801\) 7.26517e6 0.400096
\(802\) −1.69933e7 −0.932917
\(803\) 648660. 0.0355000
\(804\) 3.94352e6 0.215151
\(805\) 0 0
\(806\) 6.57821e6 0.356673
\(807\) −904246. −0.0488768
\(808\) −3.30564e6 −0.178126
\(809\) 1.78310e7 0.957868 0.478934 0.877851i \(-0.341023\pi\)
0.478934 + 0.877851i \(0.341023\pi\)
\(810\) 0 0
\(811\) −2.28534e7 −1.22011 −0.610055 0.792359i \(-0.708853\pi\)
−0.610055 + 0.792359i \(0.708853\pi\)
\(812\) 846992. 0.0450806
\(813\) 1.56626e7 0.831070
\(814\) 410810. 0.0217310
\(815\) 0 0
\(816\) 6.68044e6 0.351220
\(817\) 1.55545e7 0.815270
\(818\) −3.89619e7 −2.03590
\(819\) 222473. 0.0115896
\(820\) 0 0
\(821\) −3.84556e6 −0.199114 −0.0995570 0.995032i \(-0.531743\pi\)
−0.0995570 + 0.995032i \(0.531743\pi\)
\(822\) 8.25020e6 0.425878
\(823\) 2.40452e7 1.23746 0.618728 0.785605i \(-0.287648\pi\)
0.618728 + 0.785605i \(0.287648\pi\)
\(824\) 3.57208e6 0.183275
\(825\) 0 0
\(826\) 3.07553e6 0.156845
\(827\) −3.67928e7 −1.87068 −0.935339 0.353754i \(-0.884905\pi\)
−0.935339 + 0.353754i \(0.884905\pi\)
\(828\) 2.04361e6 0.103591
\(829\) 3.38058e7 1.70846 0.854231 0.519894i \(-0.174029\pi\)
0.854231 + 0.519894i \(0.174029\pi\)
\(830\) 0 0
\(831\) −1.79973e7 −0.904074
\(832\) −6.60352e6 −0.330725
\(833\) 1.11669e7 0.557594
\(834\) 2.56109e7 1.27500
\(835\) 0 0
\(836\) 8.18844e6 0.405215
\(837\) −2.46487e6 −0.121613
\(838\) −4.08479e7 −2.00937
\(839\) −2.59170e7 −1.27110 −0.635550 0.772060i \(-0.719227\pi\)
−0.635550 + 0.772060i \(0.719227\pi\)
\(840\) 0 0
\(841\) −1.35383e7 −0.660047
\(842\) −1.16069e7 −0.564202
\(843\) 1.58959e7 0.770398
\(844\) −2.08656e7 −1.00827
\(845\) 0 0
\(846\) −1.27842e7 −0.614111
\(847\) 161529. 0.00773644
\(848\) −3.69898e6 −0.176641
\(849\) 2.07937e7 0.990062
\(850\) 0 0
\(851\) −377003. −0.0178452
\(852\) −5.49946e6 −0.259550
\(853\) 1.25062e7 0.588510 0.294255 0.955727i \(-0.404929\pi\)
0.294255 + 0.955727i \(0.404929\pi\)
\(854\) −1.44034e6 −0.0675803
\(855\) 0 0
\(856\) −1.23489e6 −0.0576028
\(857\) −3.10464e7 −1.44398 −0.721988 0.691906i \(-0.756771\pi\)
−0.721988 + 0.691906i \(0.756771\pi\)
\(858\) −2.11869e6 −0.0982539
\(859\) 3.06277e7 1.41622 0.708111 0.706102i \(-0.249548\pi\)
0.708111 + 0.706102i \(0.249548\pi\)
\(860\) 0 0
\(861\) 607094. 0.0279093
\(862\) 2.35754e6 0.108066
\(863\) −5.27975e6 −0.241316 −0.120658 0.992694i \(-0.538500\pi\)
−0.120658 + 0.992694i \(0.538500\pi\)
\(864\) 5.78503e6 0.263646
\(865\) 0 0
\(866\) 1.39456e7 0.631893
\(867\) −8.74749e6 −0.395217
\(868\) −1.08453e6 −0.0488588
\(869\) 1.41777e6 0.0636877
\(870\) 0 0
\(871\) 3.75197e6 0.167577
\(872\) −2.73232e6 −0.121686
\(873\) 1.57492e6 0.0699395
\(874\) −1.57856e7 −0.699009
\(875\) 0 0
\(876\) 1.40272e6 0.0617605
\(877\) 2.47729e7 1.08762 0.543811 0.839207i \(-0.316981\pi\)
0.543811 + 0.839207i \(0.316981\pi\)
\(878\) 3.90231e6 0.170838
\(879\) 4.32716e6 0.188899
\(880\) 0 0
\(881\) 3.14613e7 1.36564 0.682822 0.730585i \(-0.260753\pi\)
0.682822 + 0.730585i \(0.260753\pi\)
\(882\) 1.05620e7 0.457165
\(883\) −5.34805e6 −0.230831 −0.115415 0.993317i \(-0.536820\pi\)
−0.115415 + 0.993317i \(0.536820\pi\)
\(884\) −4.84404e6 −0.208486
\(885\) 0 0
\(886\) −3.61656e7 −1.54779
\(887\) −5.37540e6 −0.229404 −0.114702 0.993400i \(-0.536591\pi\)
−0.114702 + 0.993400i \(0.536591\pi\)
\(888\) −89424.5 −0.00380561
\(889\) −1.77184e6 −0.0751918
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −3.29054e6 −0.138470
\(893\) 4.70090e7 1.97266
\(894\) 2.08743e7 0.873511
\(895\) 0 0
\(896\) −514604. −0.0214143
\(897\) 1.94434e6 0.0806847
\(898\) −2.58895e7 −1.07135
\(899\) −8.92836e6 −0.368445
\(900\) 0 0
\(901\) −2.23210e6 −0.0916015
\(902\) −5.78158e6 −0.236608
\(903\) 663527. 0.0270794
\(904\) 3.27854e6 0.133432
\(905\) 0 0
\(906\) 3.26951e7 1.32331
\(907\) 1.34668e7 0.543558 0.271779 0.962360i \(-0.412388\pi\)
0.271779 + 0.962360i \(0.412388\pi\)
\(908\) 1.79294e7 0.721691
\(909\) −1.17073e7 −0.469944
\(910\) 0 0
\(911\) 5.76380e6 0.230098 0.115049 0.993360i \(-0.463297\pi\)
0.115049 + 0.993360i \(0.463297\pi\)
\(912\) −2.32342e7 −0.924996
\(913\) 6.29044e6 0.249749
\(914\) 5.54185e7 2.19427
\(915\) 0 0
\(916\) 1.46189e7 0.575672
\(917\) −74299.9 −0.00291786
\(918\) 3.81286e6 0.149329
\(919\) −9.91711e6 −0.387344 −0.193672 0.981066i \(-0.562040\pi\)
−0.193672 + 0.981066i \(0.562040\pi\)
\(920\) 0 0
\(921\) −1.14893e7 −0.446316
\(922\) 2.87947e7 1.11554
\(923\) −5.23233e6 −0.202158
\(924\) 349304. 0.0134593
\(925\) 0 0
\(926\) 4.15841e7 1.59368
\(927\) 1.26509e7 0.483529
\(928\) 2.09548e7 0.798753
\(929\) −2.50662e7 −0.952905 −0.476453 0.879200i \(-0.658078\pi\)
−0.476453 + 0.879200i \(0.658078\pi\)
\(930\) 0 0
\(931\) −3.88377e7 −1.46852
\(932\) 4.12652e6 0.155613
\(933\) 2.25796e7 0.849206
\(934\) −7.25185e7 −2.72008
\(935\) 0 0
\(936\) 461194. 0.0172066
\(937\) 782497. 0.0291161 0.0145581 0.999894i \(-0.495366\pi\)
0.0145581 + 0.999894i \(0.495366\pi\)
\(938\) −1.29942e6 −0.0482217
\(939\) 1.67244e7 0.618995
\(940\) 0 0
\(941\) 1.08811e7 0.400589 0.200295 0.979736i \(-0.435810\pi\)
0.200295 + 0.979736i \(0.435810\pi\)
\(942\) 1.32866e7 0.487850
\(943\) 5.30580e6 0.194299
\(944\) 3.95622e7 1.44494
\(945\) 0 0
\(946\) −6.31900e6 −0.229573
\(947\) 3.52427e7 1.27701 0.638505 0.769617i \(-0.279553\pi\)
0.638505 + 0.769617i \(0.279553\pi\)
\(948\) 3.06590e6 0.110799
\(949\) 1.33458e6 0.0481039
\(950\) 0 0
\(951\) 2.23937e7 0.802923
\(952\) −168873. −0.00603905
\(953\) −6.73891e6 −0.240358 −0.120179 0.992752i \(-0.538347\pi\)
−0.120179 + 0.992752i \(0.538347\pi\)
\(954\) −2.11119e6 −0.0751030
\(955\) 0 0
\(956\) 2.13324e7 0.754911
\(957\) 2.87562e6 0.101497
\(958\) 3.30506e7 1.16350
\(959\) −1.29412e6 −0.0454389
\(960\) 0 0
\(961\) −1.71968e7 −0.600675
\(962\) 845219. 0.0294464
\(963\) −4.37349e6 −0.151972
\(964\) −9.51086e6 −0.329630
\(965\) 0 0
\(966\) −673385. −0.0232178
\(967\) −8.67300e6 −0.298266 −0.149133 0.988817i \(-0.547648\pi\)
−0.149133 + 0.988817i \(0.547648\pi\)
\(968\) 334854. 0.0114860
\(969\) −1.40204e7 −0.479678
\(970\) 0 0
\(971\) 1.57795e7 0.537089 0.268545 0.963267i \(-0.413457\pi\)
0.268545 + 0.963267i \(0.413457\pi\)
\(972\) 1.71676e6 0.0582832
\(973\) −4.01730e6 −0.136035
\(974\) −6.37734e7 −2.15398
\(975\) 0 0
\(976\) −1.85279e7 −0.622588
\(977\) 1.20786e6 0.0404836 0.0202418 0.999795i \(-0.493556\pi\)
0.0202418 + 0.999795i \(0.493556\pi\)
\(978\) 3.84985e7 1.28706
\(979\) 1.08529e7 0.361901
\(980\) 0 0
\(981\) −9.67680e6 −0.321040
\(982\) 1.34902e7 0.446416
\(983\) −7.42794e6 −0.245180 −0.122590 0.992457i \(-0.539120\pi\)
−0.122590 + 0.992457i \(0.539120\pi\)
\(984\) 1.25853e6 0.0414357
\(985\) 0 0
\(986\) 1.38111e7 0.452413
\(987\) 2.00532e6 0.0655224
\(988\) 1.68473e7 0.549083
\(989\) 5.79900e6 0.188522
\(990\) 0 0
\(991\) −4.26703e7 −1.38020 −0.690099 0.723715i \(-0.742433\pi\)
−0.690099 + 0.723715i \(0.742433\pi\)
\(992\) −2.68315e7 −0.865698
\(993\) 1.97349e7 0.635128
\(994\) 1.81212e6 0.0581728
\(995\) 0 0
\(996\) 1.36030e7 0.434496
\(997\) 2.10754e7 0.671486 0.335743 0.941954i \(-0.391013\pi\)
0.335743 + 0.941954i \(0.391013\pi\)
\(998\) −4.44554e6 −0.141286
\(999\) −316706. −0.0100402
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.y.1.2 13
5.2 odd 4 165.6.c.b.34.5 26
5.3 odd 4 165.6.c.b.34.22 yes 26
5.4 even 2 825.6.a.v.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.5 26 5.2 odd 4
165.6.c.b.34.22 yes 26 5.3 odd 4
825.6.a.v.1.12 13 5.4 even 2
825.6.a.y.1.2 13 1.1 even 1 trivial