Properties

Label 825.6.a.v.1.12
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
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: \(1\)
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 1.12
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 \(-0.257280\pi\)
0.690750 + 0.723093i \(0.257280\pi\)
\(3\) −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 \(-0.513548\pi\)
−0.0425504 + 0.999094i \(0.513548\pi\)
\(8\) −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 \(-0.565485\pi\)
−0.204280 + 0.978913i \(0.565485\pi\)
\(14\) −86.2194 −0.117567
\(15\) 0 0
\(16\) −1109.09 −1.08309
\(17\) 669.264 0.561662 0.280831 0.959757i \(-0.409390\pi\)
0.280831 + 0.959757i \(0.409390\pi\)
\(18\) 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 \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(24\) 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 \(-0.491696\pi\)
0.0260852 + 0.999660i \(0.491696\pi\)
\(38\) 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 \(-0.588867\pi\)
−0.275573 + 0.961280i \(0.588867\pi\)
\(44\) 3517.89 0.273937
\(45\) 0 0
\(46\) −6781.75 −0.472550
\(47\) −20195.8 −1.33357 −0.666787 0.745249i \(-0.732331\pi\)
−0.666787 + 0.745249i \(0.732331\pi\)
\(48\) 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 \(-0.525985\pi\)
−0.0815450 + 0.996670i \(0.525985\pi\)
\(54\) −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 \(-0.565746\pi\)
−0.205082 + 0.978745i \(0.565746\pi\)
\(68\) 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 \(-0.518750\pi\)
−0.0588701 + 0.998266i \(0.518750\pi\)
\(74\) 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 \(-0.635925\pi\)
−0.414162 + 0.910203i \(0.635925\pi\)
\(84\) 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 \(-0.533455\pi\)
−0.104909 + 0.994482i \(0.533455\pi\)
\(98\) −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 \(-0.758296\pi\)
−0.725293 + 0.688441i \(0.758296\pi\)
\(104\) 5693.76 0.0516197
\(105\) 0 0
\(106\) −26064.1 −0.225309
\(107\) 53993.7 0.455915 0.227957 0.973671i \(-0.426795\pi\)
0.227957 + 0.973671i \(0.426795\pi\)
\(108\) −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 \(-0.677074\pi\)
−0.528043 + 0.849217i \(0.677074\pi\)
\(114\) −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 \(-0.354347\pi\)
0.441781 + 0.897123i \(0.354347\pi\)
\(128\) 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 \(-0.413977\pi\)
0.266971 + 0.963705i \(0.413977\pi\)
\(138\) 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 \(-0.401070\pi\)
0.305820 + 0.952089i \(0.401070\pi\)
\(158\) 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 \(-0.201187\pi\)
0.806820 + 0.590797i \(0.201187\pi\)
\(164\) 177759. 0.516085
\(165\) 0 0
\(166\) −406277. −1.14433
\(167\) −146681. −0.406990 −0.203495 0.979076i \(-0.565230\pi\)
−0.203495 + 0.979076i \(0.565230\pi\)
\(168\) −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 \(-0.792365\pi\)
−0.794687 + 0.607019i \(0.792365\pi\)
\(174\) −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 \(-0.813456\pi\)
−0.833135 + 0.553070i \(0.813456\pi\)
\(194\) −151950. −0.289865
\(195\) 0 0
\(196\) −485098. −0.901965
\(197\) −497679. −0.913658 −0.456829 0.889554i \(-0.651015\pi\)
−0.456829 + 0.889554i \(0.651015\pi\)
\(198\) 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 \(-0.475720\pi\)
0.0762043 + 0.997092i \(0.475720\pi\)
\(224\) 87550.2 0.116584
\(225\) 0 0
\(226\) −1.12027e6 −1.45898
\(227\) −616694. −0.794337 −0.397169 0.917746i \(-0.630007\pi\)
−0.397169 + 0.917746i \(0.630007\pi\)
\(228\) −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 \(-0.527293\pi\)
−0.0856383 + 0.996326i \(0.527293\pi\)
\(234\) −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 \(-0.797366\pi\)
−0.804126 + 0.594459i \(0.797366\pi\)
\(258\) 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 \(-0.558647\pi\)
−0.183205 + 0.983075i \(0.558647\pi\)
\(264\) 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 \(-0.213714\pi\)
0.782951 + 0.622083i \(0.213714\pi\)
\(278\) −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 \(-0.827933\pi\)
−0.857419 + 0.514619i \(0.827933\pi\)
\(284\) −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 \(-0.552308\pi\)
−0.163592 + 0.986528i \(0.552308\pi\)
\(294\) 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 \(-0.373677\pi\)
0.386521 + 0.922280i \(0.373677\pi\)
\(308\) −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 \(-0.680090\pi\)
−0.536065 + 0.844177i \(0.680090\pi\)
\(314\) 1.47629e6 0.844980
\(315\) 0 0
\(316\) 340656. 0.191910
\(317\) −2.48818e6 −1.39070 −0.695352 0.718670i \(-0.744751\pi\)
−0.695352 + 0.718670i \(0.744751\pi\)
\(318\) 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 \(-0.415960\pi\)
0.260963 + 0.965349i \(0.415960\pi\)
\(338\) −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 \(-0.607744\pi\)
−0.332061 + 0.943258i \(0.607744\pi\)
\(348\) −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 \(-0.476428\pi\)
0.0739875 + 0.997259i \(0.476428\pi\)
\(354\) 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 \(-0.325559\pi\)
0.521001 + 0.853556i \(0.325559\pi\)
\(368\) 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 \(-0.670142\pi\)
−0.509426 + 0.860515i \(0.670142\pi\)
\(374\) 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 \(-0.303659\pi\)
0.578448 + 0.815720i \(0.303659\pi\)
\(384\) −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 \(-0.459015\pi\)
0.128403 + 0.991722i \(0.459015\pi\)
\(398\) −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 \(-0.426553\pi\)
0.228698 + 0.973497i \(0.426553\pi\)
\(434\) 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 \(-0.689269\pi\)
−0.560184 + 0.828369i \(0.689269\pi\)
\(444\) −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 \(-0.207911\pi\)
0.794160 + 0.607709i \(0.207911\pi\)
\(458\) 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 \(-0.304305\pi\)
0.576791 + 0.816892i \(0.304305\pi\)
\(464\) −2.92866e6 −0.631501
\(465\) 0 0
\(466\) −1.10921e6 −0.236619
\(467\) −9.27946e6 −1.96893 −0.984465 0.175581i \(-0.943820\pi\)
−0.984465 + 0.175581i \(0.943820\pi\)
\(468\) −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 \(-0.784568\pi\)
−0.779580 + 0.626302i \(0.784568\pi\)
\(488\) −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 \(-0.558242\pi\)
−0.181953 + 0.983307i \(0.558242\pi\)
\(504\) 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 \(-0.460494\pi\)
0.123793 + 0.992308i \(0.460494\pi\)
\(524\) −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 \(-0.798634\pi\)
−0.806487 + 0.591252i \(0.798634\pi\)
\(548\) 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 \(-0.496435\pi\)
0.0111992 + 0.999937i \(0.496435\pi\)
\(558\) −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 \(-0.288720\pi\)
0.616079 + 0.787685i \(0.288720\pi\)
\(564\) 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 \(-0.486009\pi\)
0.0439413 + 0.999034i \(0.486009\pi\)
\(578\) −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 \(-0.440744\pi\)
0.185084 + 0.982723i \(0.440744\pi\)
\(588\) 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 \(-0.262464\pi\)
0.678883 + 0.734246i \(0.262464\pi\)
\(594\) −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 \(-0.745306\pi\)
−0.696603 + 0.717457i \(0.745306\pi\)
\(608\) −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 \(-0.664738\pi\)
−0.494744 + 0.869039i \(0.664738\pi\)
\(614\) 9.97644e6 1.06796
\(615\) 0 0
\(616\) 30531.6 0.00324188
\(617\) −1.90063e6 −0.200995 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(618\) 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 \(-0.475512\pi\)
0.0768547 + 0.997042i \(0.475512\pi\)
\(644\) 278350. 0.0264470
\(645\) 0 0
\(646\) 1.21743e7 1.14779
\(647\) −2.05398e7 −1.92902 −0.964508 0.264055i \(-0.914940\pi\)
−0.964508 + 0.264055i \(0.914940\pi\)
\(648\) −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 \(-0.619028\pi\)
−0.365284 + 0.930896i \(0.619028\pi\)
\(654\) 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 \(-0.134014\pi\)
0.912674 + 0.408688i \(0.134014\pi\)
\(674\) 8.50374e6 0.721041
\(675\) 0 0
\(676\) −8.99289e6 −0.756889
\(677\) 1.84040e7 1.54326 0.771631 0.636071i \(-0.219441\pi\)
0.771631 + 0.636071i \(0.219441\pi\)
\(678\) 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 \(-0.183905\pi\)
0.837692 + 0.546143i \(0.183905\pi\)
\(684\) 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 \(-0.357095\pi\)
0.434020 + 0.900903i \(0.357095\pi\)
\(728\) −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 \(-0.710925\pi\)
−0.615200 + 0.788371i \(0.710925\pi\)
\(734\) 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 \(-0.424232\pi\)
0.235790 + 0.971804i \(0.424232\pi\)
\(744\) −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 \(-0.489666\pi\)
0.0324603 + 0.999473i \(0.489666\pi\)
\(758\) 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 \(-0.557974\pi\)
−0.181124 + 0.983460i \(0.557974\pi\)
\(774\) −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 \(-0.802539\pi\)
−0.813679 + 0.581314i \(0.802539\pi\)
\(788\) −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 \(-0.772601\pi\)
−0.755490 + 0.655161i \(0.772601\pi\)
\(798\) 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 \(-0.712352\pi\)
−0.618728 + 0.785605i \(0.712352\pi\)
\(824\) 3.57208e6 0.183275
\(825\) 0 0
\(826\) 3.07553e6 0.156845
\(827\) 3.67928e7 1.87068 0.935339 0.353754i \(-0.115095\pi\)
0.935339 + 0.353754i \(0.115095\pi\)
\(828\) −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 \(-0.595071\pi\)
−0.294255 + 0.955727i \(0.595071\pi\)
\(854\) −1.44034e6 −0.0675803
\(855\) 0 0
\(856\) −1.23489e6 −0.0576028
\(857\) 3.10464e7 1.44398 0.721988 0.691906i \(-0.243229\pi\)
0.721988 + 0.691906i \(0.243229\pi\)
\(858\) 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 \(-0.461500\pi\)
0.120658 + 0.992694i \(0.461500\pi\)
\(864\) 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 \(-0.683019\pi\)
−0.543811 + 0.839207i \(0.683019\pi\)
\(878\) −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 \(-0.463180\pi\)
0.115415 + 0.993317i \(0.463180\pi\)
\(884\) −4.84404e6 −0.208486
\(885\) 0 0
\(886\) −3.61656e7 −1.54779
\(887\) 5.37540e6 0.229404 0.114702 0.993400i \(-0.463409\pi\)
0.114702 + 0.993400i \(0.463409\pi\)
\(888\) 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 \(-0.587612\pi\)
−0.271779 + 0.962360i \(0.587612\pi\)
\(908\) −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 \(-0.504634\pi\)
−0.0145581 + 0.999894i \(0.504634\pi\)
\(938\) 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 \(-0.720447\pi\)
−0.638505 + 0.769617i \(0.720447\pi\)
\(948\) −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 \(-0.461653\pi\)
0.120179 + 0.992752i \(0.461653\pi\)
\(954\) −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 \(-0.452352\pi\)
0.149133 + 0.988817i \(0.452352\pi\)
\(968\) −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 \(-0.506444\pi\)
−0.0202418 + 0.999795i \(0.506444\pi\)
\(978\) −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 \(-0.460880\pi\)
0.122590 + 0.992457i \(0.460880\pi\)
\(984\) 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 \(-0.608987\pi\)
−0.335743 + 0.941954i \(0.608987\pi\)
\(998\) 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.v.1.12 13
5.2 odd 4 165.6.c.b.34.22 yes 26
5.3 odd 4 165.6.c.b.34.5 26
5.4 even 2 825.6.a.y.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.5 26 5.3 odd 4
165.6.c.b.34.22 yes 26 5.2 odd 4
825.6.a.v.1.12 13 1.1 even 1 trivial
825.6.a.y.1.2 13 5.4 even 2