Properties

Label 495.4.a.l.1.1
Level $495$
Weight $4$
Character 495.1
Self dual yes
Analytic conductor $29.206$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.59056\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} +O(q^{10})\) \(q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} -17.9528 q^{10} -11.0000 q^{11} -54.1214 q^{13} +57.9749 q^{14} -79.2041 q^{16} +107.010 q^{17} +48.7496 q^{19} +24.4606 q^{20} +39.4962 q^{22} -11.9498 q^{23} +25.0000 q^{25} +194.326 q^{26} -78.9905 q^{28} -239.733 q^{29} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} -21.7573 q^{37} -175.038 q^{38} +55.7952 q^{40} +124.835 q^{41} +224.459 q^{43} -53.8133 q^{44} +42.9064 q^{46} +186.832 q^{47} -82.2913 q^{49} -89.7640 q^{50} -264.768 q^{52} -233.997 q^{53} -55.0000 q^{55} -180.179 q^{56} +860.774 q^{58} -232.936 q^{59} +163.849 q^{61} +294.731 q^{62} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} +289.874 q^{70} +733.141 q^{71} +1161.97 q^{73} +78.1208 q^{74} +238.489 q^{76} +177.611 q^{77} -588.831 q^{79} -396.021 q^{80} -448.226 q^{82} +1161.06 q^{83} +535.048 q^{85} -805.933 q^{86} -122.749 q^{88} +1042.16 q^{89} +873.869 q^{91} -58.4597 q^{92} -670.831 q^{94} +243.748 q^{95} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8} + O(q^{10}) \) \( 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8} + 20 q^{10} - 33 q^{11} + 56 q^{14} + 50 q^{16} + 218 q^{17} + 146 q^{19} + 110 q^{20} - 44 q^{22} + 200 q^{23} + 75 q^{25} + 508 q^{26} - 340 q^{28} - 68 q^{29} - 68 q^{31} + 688 q^{32} - 176 q^{34} - 20 q^{35} - 390 q^{37} + 316 q^{38} + 240 q^{40} + 196 q^{41} - 524 q^{43} - 242 q^{44} + 1160 q^{46} + 60 q^{47} - 157 q^{49} + 100 q^{50} + 1020 q^{52} + 158 q^{53} - 165 q^{55} - 1368 q^{56} + 1092 q^{58} + 1044 q^{59} + 642 q^{61} - 88 q^{62} + 1166 q^{64} - 236 q^{67} - 144 q^{68} + 280 q^{70} + 544 q^{71} + 900 q^{73} - 1536 q^{74} + 1996 q^{76} + 44 q^{77} - 1586 q^{79} + 250 q^{80} + 380 q^{82} + 1582 q^{83} + 1090 q^{85} - 3568 q^{86} - 528 q^{88} + 2122 q^{89} - 8 q^{91} + 4128 q^{92} - 2152 q^{94} + 730 q^{95} + 618 q^{97} - 572 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.59056 −1.26945 −0.634727 0.772736i \(-0.718888\pi\)
−0.634727 + 0.772736i \(0.718888\pi\)
\(3\) 0 0
\(4\) 4.89212 0.611515
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −16.1465 −0.871828 −0.435914 0.899988i \(-0.643575\pi\)
−0.435914 + 0.899988i \(0.643575\pi\)
\(8\) 11.1590 0.493164
\(9\) 0 0
\(10\) −17.9528 −0.567717
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −54.1214 −1.15466 −0.577329 0.816511i \(-0.695905\pi\)
−0.577329 + 0.816511i \(0.695905\pi\)
\(14\) 57.9749 1.10675
\(15\) 0 0
\(16\) −79.2041 −1.23756
\(17\) 107.010 1.52668 0.763342 0.645995i \(-0.223557\pi\)
0.763342 + 0.645995i \(0.223557\pi\)
\(18\) 0 0
\(19\) 48.7496 0.588628 0.294314 0.955709i \(-0.404909\pi\)
0.294314 + 0.955709i \(0.404909\pi\)
\(20\) 24.4606 0.273478
\(21\) 0 0
\(22\) 39.4962 0.382755
\(23\) −11.9498 −0.108335 −0.0541674 0.998532i \(-0.517250\pi\)
−0.0541674 + 0.998532i \(0.517250\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 194.326 1.46579
\(27\) 0 0
\(28\) −78.9905 −0.533136
\(29\) −239.733 −1.53508 −0.767538 0.641003i \(-0.778519\pi\)
−0.767538 + 0.641003i \(0.778519\pi\)
\(30\) 0 0
\(31\) −82.0851 −0.475578 −0.237789 0.971317i \(-0.576423\pi\)
−0.237789 + 0.971317i \(0.576423\pi\)
\(32\) 195.115 1.07787
\(33\) 0 0
\(34\) −384.224 −1.93806
\(35\) −80.7324 −0.389893
\(36\) 0 0
\(37\) −21.7573 −0.0966723 −0.0483361 0.998831i \(-0.515392\pi\)
−0.0483361 + 0.998831i \(0.515392\pi\)
\(38\) −175.038 −0.747236
\(39\) 0 0
\(40\) 55.7952 0.220550
\(41\) 124.835 0.475510 0.237755 0.971325i \(-0.423588\pi\)
0.237755 + 0.971325i \(0.423588\pi\)
\(42\) 0 0
\(43\) 224.459 0.796039 0.398019 0.917377i \(-0.369698\pi\)
0.398019 + 0.917377i \(0.369698\pi\)
\(44\) −53.8133 −0.184379
\(45\) 0 0
\(46\) 42.9064 0.137526
\(47\) 186.832 0.579835 0.289917 0.957052i \(-0.406372\pi\)
0.289917 + 0.957052i \(0.406372\pi\)
\(48\) 0 0
\(49\) −82.2913 −0.239916
\(50\) −89.7640 −0.253891
\(51\) 0 0
\(52\) −264.768 −0.706091
\(53\) −233.997 −0.606453 −0.303226 0.952919i \(-0.598064\pi\)
−0.303226 + 0.952919i \(0.598064\pi\)
\(54\) 0 0
\(55\) −55.0000 −0.134840
\(56\) −180.179 −0.429954
\(57\) 0 0
\(58\) 860.774 1.94871
\(59\) −232.936 −0.513996 −0.256998 0.966412i \(-0.582733\pi\)
−0.256998 + 0.966412i \(0.582733\pi\)
\(60\) 0 0
\(61\) 163.849 0.343913 0.171957 0.985105i \(-0.444991\pi\)
0.171957 + 0.985105i \(0.444991\pi\)
\(62\) 294.731 0.603725
\(63\) 0 0
\(64\) −66.9386 −0.130739
\(65\) −270.607 −0.516379
\(66\) 0 0
\(67\) −876.918 −1.59899 −0.799497 0.600670i \(-0.794900\pi\)
−0.799497 + 0.600670i \(0.794900\pi\)
\(68\) 523.503 0.933590
\(69\) 0 0
\(70\) 289.874 0.494952
\(71\) 733.141 1.22546 0.612731 0.790291i \(-0.290071\pi\)
0.612731 + 0.790291i \(0.290071\pi\)
\(72\) 0 0
\(73\) 1161.97 1.86299 0.931496 0.363750i \(-0.118504\pi\)
0.931496 + 0.363750i \(0.118504\pi\)
\(74\) 78.1208 0.122721
\(75\) 0 0
\(76\) 238.489 0.359954
\(77\) 177.611 0.262866
\(78\) 0 0
\(79\) −588.831 −0.838591 −0.419296 0.907850i \(-0.637723\pi\)
−0.419296 + 0.907850i \(0.637723\pi\)
\(80\) −396.021 −0.553456
\(81\) 0 0
\(82\) −448.226 −0.603638
\(83\) 1161.06 1.53546 0.767731 0.640772i \(-0.221386\pi\)
0.767731 + 0.640772i \(0.221386\pi\)
\(84\) 0 0
\(85\) 535.048 0.682754
\(86\) −805.933 −1.01053
\(87\) 0 0
\(88\) −122.749 −0.148695
\(89\) 1042.16 1.24122 0.620610 0.784120i \(-0.286885\pi\)
0.620610 + 0.784120i \(0.286885\pi\)
\(90\) 0 0
\(91\) 873.869 1.00666
\(92\) −58.4597 −0.0662483
\(93\) 0 0
\(94\) −670.831 −0.736074
\(95\) 243.748 0.263242
\(96\) 0 0
\(97\) 1546.63 1.61893 0.809464 0.587169i \(-0.199758\pi\)
0.809464 + 0.587169i \(0.199758\pi\)
\(98\) 295.472 0.304563
\(99\) 0 0
\(100\) 122.303 0.122303
\(101\) 662.282 0.652470 0.326235 0.945289i \(-0.394220\pi\)
0.326235 + 0.945289i \(0.394220\pi\)
\(102\) 0 0
\(103\) 399.592 0.382262 0.191131 0.981565i \(-0.438784\pi\)
0.191131 + 0.981565i \(0.438784\pi\)
\(104\) −603.942 −0.569436
\(105\) 0 0
\(106\) 840.181 0.769864
\(107\) 1591.22 1.43765 0.718827 0.695189i \(-0.244679\pi\)
0.718827 + 0.695189i \(0.244679\pi\)
\(108\) 0 0
\(109\) 755.128 0.663561 0.331780 0.943357i \(-0.392351\pi\)
0.331780 + 0.943357i \(0.392351\pi\)
\(110\) 197.481 0.171173
\(111\) 0 0
\(112\) 1278.87 1.07894
\(113\) 1145.65 0.953753 0.476876 0.878970i \(-0.341769\pi\)
0.476876 + 0.878970i \(0.341769\pi\)
\(114\) 0 0
\(115\) −59.7488 −0.0484488
\(116\) −1172.80 −0.938722
\(117\) 0 0
\(118\) 836.372 0.652494
\(119\) −1727.83 −1.33101
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −588.309 −0.436582
\(123\) 0 0
\(124\) −401.570 −0.290823
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1461.29 1.02101 0.510505 0.859875i \(-0.329458\pi\)
0.510505 + 0.859875i \(0.329458\pi\)
\(128\) −1320.57 −0.911900
\(129\) 0 0
\(130\) 971.630 0.655520
\(131\) 1524.94 1.01706 0.508528 0.861045i \(-0.330190\pi\)
0.508528 + 0.861045i \(0.330190\pi\)
\(132\) 0 0
\(133\) −787.134 −0.513182
\(134\) 3148.63 2.02985
\(135\) 0 0
\(136\) 1194.12 0.752906
\(137\) 2125.68 1.32561 0.662805 0.748792i \(-0.269366\pi\)
0.662805 + 0.748792i \(0.269366\pi\)
\(138\) 0 0
\(139\) −1774.28 −1.08268 −0.541339 0.840805i \(-0.682082\pi\)
−0.541339 + 0.840805i \(0.682082\pi\)
\(140\) −394.952 −0.238425
\(141\) 0 0
\(142\) −2632.39 −1.55567
\(143\) 595.335 0.348143
\(144\) 0 0
\(145\) −1198.66 −0.686507
\(146\) −4172.13 −2.36498
\(147\) 0 0
\(148\) −106.439 −0.0591165
\(149\) −1575.78 −0.866393 −0.433197 0.901299i \(-0.642614\pi\)
−0.433197 + 0.901299i \(0.642614\pi\)
\(150\) 0 0
\(151\) 420.978 0.226879 0.113439 0.993545i \(-0.463813\pi\)
0.113439 + 0.993545i \(0.463813\pi\)
\(152\) 543.998 0.290290
\(153\) 0 0
\(154\) −637.724 −0.333696
\(155\) −410.425 −0.212685
\(156\) 0 0
\(157\) −2224.30 −1.13069 −0.565345 0.824854i \(-0.691257\pi\)
−0.565345 + 0.824854i \(0.691257\pi\)
\(158\) 2114.23 1.06455
\(159\) 0 0
\(160\) 975.574 0.482037
\(161\) 192.947 0.0944492
\(162\) 0 0
\(163\) −3093.37 −1.48645 −0.743226 0.669040i \(-0.766705\pi\)
−0.743226 + 0.669040i \(0.766705\pi\)
\(164\) 610.706 0.290781
\(165\) 0 0
\(166\) −4168.87 −1.94920
\(167\) 2416.43 1.11970 0.559848 0.828595i \(-0.310860\pi\)
0.559848 + 0.828595i \(0.310860\pi\)
\(168\) 0 0
\(169\) 732.122 0.333237
\(170\) −1921.12 −0.866725
\(171\) 0 0
\(172\) 1098.08 0.486789
\(173\) 3758.02 1.65154 0.825771 0.564005i \(-0.190740\pi\)
0.825771 + 0.564005i \(0.190740\pi\)
\(174\) 0 0
\(175\) −403.662 −0.174366
\(176\) 871.245 0.373140
\(177\) 0 0
\(178\) −3741.93 −1.57567
\(179\) −2533.99 −1.05810 −0.529049 0.848591i \(-0.677451\pi\)
−0.529049 + 0.848591i \(0.677451\pi\)
\(180\) 0 0
\(181\) −13.8995 −0.00570798 −0.00285399 0.999996i \(-0.500908\pi\)
−0.00285399 + 0.999996i \(0.500908\pi\)
\(182\) −3137.68 −1.27791
\(183\) 0 0
\(184\) −133.348 −0.0534268
\(185\) −108.786 −0.0432332
\(186\) 0 0
\(187\) −1177.10 −0.460312
\(188\) 914.004 0.354577
\(189\) 0 0
\(190\) −875.192 −0.334174
\(191\) −3495.39 −1.32417 −0.662087 0.749427i \(-0.730329\pi\)
−0.662087 + 0.749427i \(0.730329\pi\)
\(192\) 0 0
\(193\) 3469.33 1.29393 0.646963 0.762522i \(-0.276039\pi\)
0.646963 + 0.762522i \(0.276039\pi\)
\(194\) −5553.25 −2.05516
\(195\) 0 0
\(196\) −402.579 −0.146712
\(197\) −3638.39 −1.31586 −0.657930 0.753079i \(-0.728568\pi\)
−0.657930 + 0.753079i \(0.728568\pi\)
\(198\) 0 0
\(199\) 51.6049 0.0183828 0.00919140 0.999958i \(-0.497074\pi\)
0.00919140 + 0.999958i \(0.497074\pi\)
\(200\) 278.976 0.0986329
\(201\) 0 0
\(202\) −2377.96 −0.828281
\(203\) 3870.84 1.33832
\(204\) 0 0
\(205\) 624.173 0.212654
\(206\) −1434.76 −0.485265
\(207\) 0 0
\(208\) 4286.63 1.42896
\(209\) −536.246 −0.177478
\(210\) 0 0
\(211\) 2084.57 0.680131 0.340065 0.940402i \(-0.389551\pi\)
0.340065 + 0.940402i \(0.389551\pi\)
\(212\) −1144.74 −0.370855
\(213\) 0 0
\(214\) −5713.37 −1.82504
\(215\) 1122.29 0.355999
\(216\) 0 0
\(217\) 1325.38 0.414622
\(218\) −2711.33 −0.842361
\(219\) 0 0
\(220\) −269.067 −0.0824566
\(221\) −5791.50 −1.76280
\(222\) 0 0
\(223\) 1887.36 0.566757 0.283378 0.959008i \(-0.408545\pi\)
0.283378 + 0.959008i \(0.408545\pi\)
\(224\) −3150.42 −0.939715
\(225\) 0 0
\(226\) −4113.54 −1.21075
\(227\) 1150.73 0.336462 0.168231 0.985748i \(-0.446194\pi\)
0.168231 + 0.985748i \(0.446194\pi\)
\(228\) 0 0
\(229\) 4106.79 1.18508 0.592542 0.805540i \(-0.298125\pi\)
0.592542 + 0.805540i \(0.298125\pi\)
\(230\) 214.532 0.0615035
\(231\) 0 0
\(232\) −2675.18 −0.757045
\(233\) 5733.58 1.61210 0.806050 0.591848i \(-0.201601\pi\)
0.806050 + 0.591848i \(0.201601\pi\)
\(234\) 0 0
\(235\) 934.159 0.259310
\(236\) −1139.55 −0.314316
\(237\) 0 0
\(238\) 6203.86 1.68965
\(239\) −6036.18 −1.63367 −0.816837 0.576868i \(-0.804275\pi\)
−0.816837 + 0.576868i \(0.804275\pi\)
\(240\) 0 0
\(241\) 3720.90 0.994540 0.497270 0.867596i \(-0.334336\pi\)
0.497270 + 0.867596i \(0.334336\pi\)
\(242\) −434.458 −0.115405
\(243\) 0 0
\(244\) 801.568 0.210308
\(245\) −411.457 −0.107294
\(246\) 0 0
\(247\) −2638.39 −0.679664
\(248\) −915.990 −0.234538
\(249\) 0 0
\(250\) −448.820 −0.113543
\(251\) 3809.88 0.958077 0.479038 0.877794i \(-0.340985\pi\)
0.479038 + 0.877794i \(0.340985\pi\)
\(252\) 0 0
\(253\) 131.447 0.0326641
\(254\) −5246.84 −1.29613
\(255\) 0 0
\(256\) 5277.10 1.28835
\(257\) −1225.95 −0.297559 −0.148780 0.988870i \(-0.547535\pi\)
−0.148780 + 0.988870i \(0.547535\pi\)
\(258\) 0 0
\(259\) 351.303 0.0842816
\(260\) −1323.84 −0.315773
\(261\) 0 0
\(262\) −5475.38 −1.29111
\(263\) −7397.68 −1.73445 −0.867225 0.497916i \(-0.834099\pi\)
−0.867225 + 0.497916i \(0.834099\pi\)
\(264\) 0 0
\(265\) −1169.99 −0.271214
\(266\) 2826.25 0.651461
\(267\) 0 0
\(268\) −4289.99 −0.977808
\(269\) 3214.48 0.728589 0.364295 0.931284i \(-0.381310\pi\)
0.364295 + 0.931284i \(0.381310\pi\)
\(270\) 0 0
\(271\) −7377.37 −1.65367 −0.826833 0.562448i \(-0.809860\pi\)
−0.826833 + 0.562448i \(0.809860\pi\)
\(272\) −8475.60 −1.88937
\(273\) 0 0
\(274\) −7632.36 −1.68280
\(275\) −275.000 −0.0603023
\(276\) 0 0
\(277\) −810.606 −0.175829 −0.0879144 0.996128i \(-0.528020\pi\)
−0.0879144 + 0.996128i \(0.528020\pi\)
\(278\) 6370.65 1.37441
\(279\) 0 0
\(280\) −900.895 −0.192281
\(281\) −1114.72 −0.236651 −0.118325 0.992975i \(-0.537753\pi\)
−0.118325 + 0.992975i \(0.537753\pi\)
\(282\) 0 0
\(283\) 2265.40 0.475844 0.237922 0.971284i \(-0.423534\pi\)
0.237922 + 0.971284i \(0.423534\pi\)
\(284\) 3586.61 0.749388
\(285\) 0 0
\(286\) −2137.59 −0.441951
\(287\) −2015.64 −0.414563
\(288\) 0 0
\(289\) 6538.04 1.33076
\(290\) 4303.87 0.871490
\(291\) 0 0
\(292\) 5684.50 1.13925
\(293\) −3802.06 −0.758084 −0.379042 0.925380i \(-0.623746\pi\)
−0.379042 + 0.925380i \(0.623746\pi\)
\(294\) 0 0
\(295\) −1164.68 −0.229866
\(296\) −242.790 −0.0476753
\(297\) 0 0
\(298\) 5657.92 1.09985
\(299\) 646.738 0.125090
\(300\) 0 0
\(301\) −3624.22 −0.694009
\(302\) −1511.55 −0.288013
\(303\) 0 0
\(304\) −3861.17 −0.728465
\(305\) 819.244 0.153803
\(306\) 0 0
\(307\) −1356.35 −0.252153 −0.126077 0.992021i \(-0.540239\pi\)
−0.126077 + 0.992021i \(0.540239\pi\)
\(308\) 868.895 0.160746
\(309\) 0 0
\(310\) 1473.66 0.269994
\(311\) 8078.07 1.47288 0.736440 0.676503i \(-0.236505\pi\)
0.736440 + 0.676503i \(0.236505\pi\)
\(312\) 0 0
\(313\) 5761.54 1.04045 0.520226 0.854029i \(-0.325848\pi\)
0.520226 + 0.854029i \(0.325848\pi\)
\(314\) 7986.48 1.43536
\(315\) 0 0
\(316\) −2880.63 −0.512811
\(317\) −5107.00 −0.904851 −0.452426 0.891802i \(-0.649441\pi\)
−0.452426 + 0.891802i \(0.649441\pi\)
\(318\) 0 0
\(319\) 2637.06 0.462843
\(320\) −334.693 −0.0584684
\(321\) 0 0
\(322\) −692.786 −0.119899
\(323\) 5216.67 0.898648
\(324\) 0 0
\(325\) −1353.03 −0.230932
\(326\) 11106.9 1.88698
\(327\) 0 0
\(328\) 1393.03 0.234504
\(329\) −3016.68 −0.505516
\(330\) 0 0
\(331\) −2780.94 −0.461796 −0.230898 0.972978i \(-0.574166\pi\)
−0.230898 + 0.972978i \(0.574166\pi\)
\(332\) 5680.06 0.938958
\(333\) 0 0
\(334\) −8676.35 −1.42140
\(335\) −4384.59 −0.715092
\(336\) 0 0
\(337\) 4939.28 0.798397 0.399198 0.916865i \(-0.369288\pi\)
0.399198 + 0.916865i \(0.369288\pi\)
\(338\) −2628.73 −0.423029
\(339\) 0 0
\(340\) 2617.52 0.417514
\(341\) 902.936 0.143392
\(342\) 0 0
\(343\) 6866.96 1.08099
\(344\) 2504.74 0.392578
\(345\) 0 0
\(346\) −13493.4 −2.09656
\(347\) 2711.58 0.419496 0.209748 0.977755i \(-0.432736\pi\)
0.209748 + 0.977755i \(0.432736\pi\)
\(348\) 0 0
\(349\) 5496.03 0.842967 0.421484 0.906836i \(-0.361510\pi\)
0.421484 + 0.906836i \(0.361510\pi\)
\(350\) 1449.37 0.221349
\(351\) 0 0
\(352\) −2146.26 −0.324989
\(353\) −6372.23 −0.960792 −0.480396 0.877052i \(-0.659507\pi\)
−0.480396 + 0.877052i \(0.659507\pi\)
\(354\) 0 0
\(355\) 3665.71 0.548043
\(356\) 5098.36 0.759024
\(357\) 0 0
\(358\) 9098.46 1.34321
\(359\) −630.622 −0.0927101 −0.0463551 0.998925i \(-0.514761\pi\)
−0.0463551 + 0.998925i \(0.514761\pi\)
\(360\) 0 0
\(361\) −4482.48 −0.653518
\(362\) 49.9071 0.00724602
\(363\) 0 0
\(364\) 4275.07 0.615590
\(365\) 5809.86 0.833156
\(366\) 0 0
\(367\) −5374.49 −0.764431 −0.382216 0.924073i \(-0.624839\pi\)
−0.382216 + 0.924073i \(0.624839\pi\)
\(368\) 946.471 0.134071
\(369\) 0 0
\(370\) 390.604 0.0548825
\(371\) 3778.23 0.528722
\(372\) 0 0
\(373\) 7520.12 1.04391 0.521953 0.852974i \(-0.325204\pi\)
0.521953 + 0.852974i \(0.325204\pi\)
\(374\) 4226.46 0.584346
\(375\) 0 0
\(376\) 2084.86 0.285954
\(377\) 12974.7 1.77249
\(378\) 0 0
\(379\) −12509.1 −1.69538 −0.847690 0.530492i \(-0.822007\pi\)
−0.847690 + 0.530492i \(0.822007\pi\)
\(380\) 1192.44 0.160977
\(381\) 0 0
\(382\) 12550.4 1.68098
\(383\) 11149.9 1.48755 0.743775 0.668430i \(-0.233033\pi\)
0.743775 + 0.668430i \(0.233033\pi\)
\(384\) 0 0
\(385\) 888.056 0.117557
\(386\) −12456.8 −1.64258
\(387\) 0 0
\(388\) 7566.28 0.989999
\(389\) −3194.22 −0.416333 −0.208166 0.978093i \(-0.566750\pi\)
−0.208166 + 0.978093i \(0.566750\pi\)
\(390\) 0 0
\(391\) −1278.74 −0.165393
\(392\) −918.292 −0.118318
\(393\) 0 0
\(394\) 13063.8 1.67042
\(395\) −2944.16 −0.375029
\(396\) 0 0
\(397\) −584.410 −0.0738809 −0.0369404 0.999317i \(-0.511761\pi\)
−0.0369404 + 0.999317i \(0.511761\pi\)
\(398\) −185.291 −0.0233361
\(399\) 0 0
\(400\) −1980.10 −0.247513
\(401\) 6951.24 0.865657 0.432829 0.901476i \(-0.357516\pi\)
0.432829 + 0.901476i \(0.357516\pi\)
\(402\) 0 0
\(403\) 4442.56 0.549130
\(404\) 3239.96 0.398995
\(405\) 0 0
\(406\) −13898.5 −1.69894
\(407\) 239.330 0.0291478
\(408\) 0 0
\(409\) 11754.1 1.42104 0.710519 0.703678i \(-0.248460\pi\)
0.710519 + 0.703678i \(0.248460\pi\)
\(410\) −2241.13 −0.269955
\(411\) 0 0
\(412\) 1954.85 0.233759
\(413\) 3761.10 0.448116
\(414\) 0 0
\(415\) 5805.32 0.686680
\(416\) −10559.9 −1.24457
\(417\) 0 0
\(418\) 1925.42 0.225300
\(419\) −11829.8 −1.37929 −0.689646 0.724146i \(-0.742234\pi\)
−0.689646 + 0.724146i \(0.742234\pi\)
\(420\) 0 0
\(421\) −2000.07 −0.231538 −0.115769 0.993276i \(-0.536933\pi\)
−0.115769 + 0.993276i \(0.536933\pi\)
\(422\) −7484.76 −0.863395
\(423\) 0 0
\(424\) −2611.18 −0.299081
\(425\) 2675.24 0.305337
\(426\) 0 0
\(427\) −2645.58 −0.299833
\(428\) 7784.43 0.879147
\(429\) 0 0
\(430\) −4029.67 −0.451925
\(431\) 8064.11 0.901240 0.450620 0.892716i \(-0.351203\pi\)
0.450620 + 0.892716i \(0.351203\pi\)
\(432\) 0 0
\(433\) −10710.3 −1.18869 −0.594345 0.804210i \(-0.702589\pi\)
−0.594345 + 0.804210i \(0.702589\pi\)
\(434\) −4758.87 −0.526344
\(435\) 0 0
\(436\) 3694.18 0.405777
\(437\) −582.546 −0.0637688
\(438\) 0 0
\(439\) 4658.08 0.506419 0.253210 0.967411i \(-0.418514\pi\)
0.253210 + 0.967411i \(0.418514\pi\)
\(440\) −613.747 −0.0664983
\(441\) 0 0
\(442\) 20794.7 2.23779
\(443\) −2094.73 −0.224658 −0.112329 0.993671i \(-0.535831\pi\)
−0.112329 + 0.993671i \(0.535831\pi\)
\(444\) 0 0
\(445\) 5210.79 0.555090
\(446\) −6776.67 −0.719472
\(447\) 0 0
\(448\) 1080.82 0.113982
\(449\) 1402.21 0.147382 0.0736909 0.997281i \(-0.476522\pi\)
0.0736909 + 0.997281i \(0.476522\pi\)
\(450\) 0 0
\(451\) −1373.18 −0.143372
\(452\) 5604.67 0.583234
\(453\) 0 0
\(454\) −4131.78 −0.427124
\(455\) 4369.35 0.450194
\(456\) 0 0
\(457\) −4914.19 −0.503011 −0.251506 0.967856i \(-0.580926\pi\)
−0.251506 + 0.967856i \(0.580926\pi\)
\(458\) −14745.7 −1.50441
\(459\) 0 0
\(460\) −292.298 −0.0296271
\(461\) 2214.08 0.223688 0.111844 0.993726i \(-0.464324\pi\)
0.111844 + 0.993726i \(0.464324\pi\)
\(462\) 0 0
\(463\) −5567.02 −0.558793 −0.279396 0.960176i \(-0.590134\pi\)
−0.279396 + 0.960176i \(0.590134\pi\)
\(464\) 18987.8 1.89976
\(465\) 0 0
\(466\) −20586.8 −2.04649
\(467\) −497.054 −0.0492525 −0.0246263 0.999697i \(-0.507840\pi\)
−0.0246263 + 0.999697i \(0.507840\pi\)
\(468\) 0 0
\(469\) 14159.1 1.39405
\(470\) −3354.15 −0.329182
\(471\) 0 0
\(472\) −2599.35 −0.253484
\(473\) −2469.05 −0.240015
\(474\) 0 0
\(475\) 1218.74 0.117726
\(476\) −8452.73 −0.813929
\(477\) 0 0
\(478\) 21673.3 2.07388
\(479\) 9349.28 0.891815 0.445908 0.895079i \(-0.352881\pi\)
0.445908 + 0.895079i \(0.352881\pi\)
\(480\) 0 0
\(481\) 1177.53 0.111624
\(482\) −13360.1 −1.26252
\(483\) 0 0
\(484\) 591.946 0.0555923
\(485\) 7733.13 0.724007
\(486\) 0 0
\(487\) 197.750 0.0184003 0.00920013 0.999958i \(-0.497071\pi\)
0.00920013 + 0.999958i \(0.497071\pi\)
\(488\) 1828.40 0.169606
\(489\) 0 0
\(490\) 1477.36 0.136205
\(491\) 9997.05 0.918861 0.459430 0.888214i \(-0.348054\pi\)
0.459430 + 0.888214i \(0.348054\pi\)
\(492\) 0 0
\(493\) −25653.7 −2.34358
\(494\) 9473.31 0.862803
\(495\) 0 0
\(496\) 6501.48 0.588558
\(497\) −11837.6 −1.06839
\(498\) 0 0
\(499\) −8714.73 −0.781813 −0.390907 0.920430i \(-0.627838\pi\)
−0.390907 + 0.920430i \(0.627838\pi\)
\(500\) 611.515 0.0546955
\(501\) 0 0
\(502\) −13679.6 −1.21623
\(503\) −5978.53 −0.529959 −0.264979 0.964254i \(-0.585365\pi\)
−0.264979 + 0.964254i \(0.585365\pi\)
\(504\) 0 0
\(505\) 3311.41 0.291794
\(506\) −471.970 −0.0414657
\(507\) 0 0
\(508\) 7148.79 0.624363
\(509\) 8205.79 0.714569 0.357284 0.933996i \(-0.383703\pi\)
0.357284 + 0.933996i \(0.383703\pi\)
\(510\) 0 0
\(511\) −18761.7 −1.62421
\(512\) −8383.17 −0.723608
\(513\) 0 0
\(514\) 4401.85 0.377738
\(515\) 1997.96 0.170953
\(516\) 0 0
\(517\) −2055.15 −0.174827
\(518\) −1261.38 −0.106992
\(519\) 0 0
\(520\) −3019.71 −0.254660
\(521\) 5266.06 0.442822 0.221411 0.975181i \(-0.428934\pi\)
0.221411 + 0.975181i \(0.428934\pi\)
\(522\) 0 0
\(523\) −22398.6 −1.87270 −0.936350 0.351068i \(-0.885819\pi\)
−0.936350 + 0.351068i \(0.885819\pi\)
\(524\) 7460.18 0.621945
\(525\) 0 0
\(526\) 26561.8 2.20181
\(527\) −8783.89 −0.726057
\(528\) 0 0
\(529\) −12024.2 −0.988264
\(530\) 4200.90 0.344294
\(531\) 0 0
\(532\) −3850.75 −0.313818
\(533\) −6756.22 −0.549052
\(534\) 0 0
\(535\) 7956.10 0.642938
\(536\) −9785.56 −0.788566
\(537\) 0 0
\(538\) −11541.8 −0.924911
\(539\) 905.205 0.0723375
\(540\) 0 0
\(541\) −13030.2 −1.03551 −0.517757 0.855528i \(-0.673233\pi\)
−0.517757 + 0.855528i \(0.673233\pi\)
\(542\) 26488.9 2.09925
\(543\) 0 0
\(544\) 20879.1 1.64556
\(545\) 3775.64 0.296753
\(546\) 0 0
\(547\) 10448.9 0.816747 0.408374 0.912815i \(-0.366096\pi\)
0.408374 + 0.912815i \(0.366096\pi\)
\(548\) 10399.1 0.810631
\(549\) 0 0
\(550\) 987.404 0.0765510
\(551\) −11686.9 −0.903588
\(552\) 0 0
\(553\) 9507.55 0.731107
\(554\) 2910.53 0.223207
\(555\) 0 0
\(556\) −8679.97 −0.662073
\(557\) 11448.7 0.870911 0.435456 0.900210i \(-0.356587\pi\)
0.435456 + 0.900210i \(0.356587\pi\)
\(558\) 0 0
\(559\) −12148.0 −0.919153
\(560\) 6394.34 0.482518
\(561\) 0 0
\(562\) 4002.49 0.300418
\(563\) 26035.8 1.94898 0.974492 0.224422i \(-0.0720494\pi\)
0.974492 + 0.224422i \(0.0720494\pi\)
\(564\) 0 0
\(565\) 5728.27 0.426531
\(566\) −8134.05 −0.604063
\(567\) 0 0
\(568\) 8181.15 0.604354
\(569\) −25075.0 −1.84745 −0.923725 0.383057i \(-0.874871\pi\)
−0.923725 + 0.383057i \(0.874871\pi\)
\(570\) 0 0
\(571\) 19056.6 1.39666 0.698330 0.715776i \(-0.253927\pi\)
0.698330 + 0.715776i \(0.253927\pi\)
\(572\) 2912.45 0.212894
\(573\) 0 0
\(574\) 7237.28 0.526268
\(575\) −298.744 −0.0216669
\(576\) 0 0
\(577\) −6482.55 −0.467716 −0.233858 0.972271i \(-0.575135\pi\)
−0.233858 + 0.972271i \(0.575135\pi\)
\(578\) −23475.2 −1.68934
\(579\) 0 0
\(580\) −5864.00 −0.419809
\(581\) −18747.1 −1.33866
\(582\) 0 0
\(583\) 2573.97 0.182852
\(584\) 12966.5 0.918762
\(585\) 0 0
\(586\) 13651.5 0.962353
\(587\) −24650.3 −1.73327 −0.866633 0.498946i \(-0.833720\pi\)
−0.866633 + 0.498946i \(0.833720\pi\)
\(588\) 0 0
\(589\) −4001.61 −0.279938
\(590\) 4181.86 0.291804
\(591\) 0 0
\(592\) 1723.27 0.119638
\(593\) −23163.2 −1.60405 −0.802024 0.597292i \(-0.796243\pi\)
−0.802024 + 0.597292i \(0.796243\pi\)
\(594\) 0 0
\(595\) −8639.13 −0.595244
\(596\) −7708.88 −0.529812
\(597\) 0 0
\(598\) −2322.15 −0.158796
\(599\) −5355.44 −0.365304 −0.182652 0.983178i \(-0.558468\pi\)
−0.182652 + 0.983178i \(0.558468\pi\)
\(600\) 0 0
\(601\) −20417.6 −1.38578 −0.692889 0.721045i \(-0.743662\pi\)
−0.692889 + 0.721045i \(0.743662\pi\)
\(602\) 13013.0 0.881012
\(603\) 0 0
\(604\) 2059.48 0.138740
\(605\) 605.000 0.0406558
\(606\) 0 0
\(607\) −6749.81 −0.451345 −0.225673 0.974203i \(-0.572458\pi\)
−0.225673 + 0.974203i \(0.572458\pi\)
\(608\) 9511.77 0.634463
\(609\) 0 0
\(610\) −2941.55 −0.195245
\(611\) −10111.6 −0.669511
\(612\) 0 0
\(613\) −30321.0 −1.99780 −0.998901 0.0468613i \(-0.985078\pi\)
−0.998901 + 0.0468613i \(0.985078\pi\)
\(614\) 4870.06 0.320097
\(615\) 0 0
\(616\) 1981.97 0.129636
\(617\) −15236.8 −0.994182 −0.497091 0.867699i \(-0.665598\pi\)
−0.497091 + 0.867699i \(0.665598\pi\)
\(618\) 0 0
\(619\) −20875.4 −1.35550 −0.677749 0.735293i \(-0.737044\pi\)
−0.677749 + 0.735293i \(0.737044\pi\)
\(620\) −2007.85 −0.130060
\(621\) 0 0
\(622\) −29004.8 −1.86975
\(623\) −16827.2 −1.08213
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −20687.2 −1.32081
\(627\) 0 0
\(628\) −10881.5 −0.691434
\(629\) −2328.24 −0.147588
\(630\) 0 0
\(631\) 27966.1 1.76437 0.882183 0.470907i \(-0.156073\pi\)
0.882183 + 0.470907i \(0.156073\pi\)
\(632\) −6570.79 −0.413563
\(633\) 0 0
\(634\) 18337.0 1.14867
\(635\) 7306.44 0.456610
\(636\) 0 0
\(637\) 4453.72 0.277022
\(638\) −9468.52 −0.587558
\(639\) 0 0
\(640\) −6602.86 −0.407814
\(641\) 17992.0 1.10865 0.554323 0.832301i \(-0.312977\pi\)
0.554323 + 0.832301i \(0.312977\pi\)
\(642\) 0 0
\(643\) −9448.64 −0.579499 −0.289750 0.957102i \(-0.593572\pi\)
−0.289750 + 0.957102i \(0.593572\pi\)
\(644\) 943.918 0.0577571
\(645\) 0 0
\(646\) −18730.8 −1.14079
\(647\) 7429.22 0.451426 0.225713 0.974194i \(-0.427529\pi\)
0.225713 + 0.974194i \(0.427529\pi\)
\(648\) 0 0
\(649\) 2562.30 0.154976
\(650\) 4858.15 0.293157
\(651\) 0 0
\(652\) −15133.2 −0.908988
\(653\) 4488.63 0.268995 0.134497 0.990914i \(-0.457058\pi\)
0.134497 + 0.990914i \(0.457058\pi\)
\(654\) 0 0
\(655\) 7624.69 0.454842
\(656\) −9887.42 −0.588474
\(657\) 0 0
\(658\) 10831.6 0.641730
\(659\) 25326.7 1.49710 0.748550 0.663078i \(-0.230750\pi\)
0.748550 + 0.663078i \(0.230750\pi\)
\(660\) 0 0
\(661\) 15192.3 0.893969 0.446984 0.894542i \(-0.352498\pi\)
0.446984 + 0.894542i \(0.352498\pi\)
\(662\) 9985.15 0.586229
\(663\) 0 0
\(664\) 12956.4 0.757235
\(665\) −3935.67 −0.229502
\(666\) 0 0
\(667\) 2864.75 0.166302
\(668\) 11821.5 0.684711
\(669\) 0 0
\(670\) 15743.1 0.907776
\(671\) −1802.34 −0.103694
\(672\) 0 0
\(673\) 11718.2 0.671180 0.335590 0.942008i \(-0.391064\pi\)
0.335590 + 0.942008i \(0.391064\pi\)
\(674\) −17734.8 −1.01353
\(675\) 0 0
\(676\) 3581.63 0.203779
\(677\) 15649.1 0.888397 0.444198 0.895928i \(-0.353489\pi\)
0.444198 + 0.895928i \(0.353489\pi\)
\(678\) 0 0
\(679\) −24972.6 −1.41143
\(680\) 5970.61 0.336710
\(681\) 0 0
\(682\) −3242.05 −0.182030
\(683\) 18162.8 1.01754 0.508770 0.860903i \(-0.330101\pi\)
0.508770 + 0.860903i \(0.330101\pi\)
\(684\) 0 0
\(685\) 10628.4 0.592831
\(686\) −24656.2 −1.37227
\(687\) 0 0
\(688\) −17778.1 −0.985149
\(689\) 12664.2 0.700246
\(690\) 0 0
\(691\) 29606.7 1.62995 0.814973 0.579500i \(-0.196752\pi\)
0.814973 + 0.579500i \(0.196752\pi\)
\(692\) 18384.7 1.00994
\(693\) 0 0
\(694\) −9736.08 −0.532531
\(695\) −8871.38 −0.484188
\(696\) 0 0
\(697\) 13358.5 0.725953
\(698\) −19733.8 −1.07011
\(699\) 0 0
\(700\) −1974.76 −0.106627
\(701\) 26164.7 1.40974 0.704870 0.709336i \(-0.251005\pi\)
0.704870 + 0.709336i \(0.251005\pi\)
\(702\) 0 0
\(703\) −1060.66 −0.0569040
\(704\) 736.324 0.0394194
\(705\) 0 0
\(706\) 22879.9 1.21968
\(707\) −10693.5 −0.568842
\(708\) 0 0
\(709\) −14508.9 −0.768535 −0.384268 0.923222i \(-0.625546\pi\)
−0.384268 + 0.923222i \(0.625546\pi\)
\(710\) −13161.9 −0.695716
\(711\) 0 0
\(712\) 11629.5 0.612125
\(713\) 980.898 0.0515216
\(714\) 0 0
\(715\) 2976.67 0.155694
\(716\) −12396.6 −0.647043
\(717\) 0 0
\(718\) 2264.28 0.117691
\(719\) 2545.80 0.132048 0.0660239 0.997818i \(-0.478969\pi\)
0.0660239 + 0.997818i \(0.478969\pi\)
\(720\) 0 0
\(721\) −6452.01 −0.333267
\(722\) 16094.6 0.829611
\(723\) 0 0
\(724\) −67.9982 −0.00349051
\(725\) −5993.31 −0.307015
\(726\) 0 0
\(727\) −18984.8 −0.968511 −0.484255 0.874927i \(-0.660909\pi\)
−0.484255 + 0.874927i \(0.660909\pi\)
\(728\) 9751.54 0.496451
\(729\) 0 0
\(730\) −20860.6 −1.05765
\(731\) 24019.2 1.21530
\(732\) 0 0
\(733\) 36740.4 1.85134 0.925672 0.378326i \(-0.123500\pi\)
0.925672 + 0.378326i \(0.123500\pi\)
\(734\) 19297.4 0.970411
\(735\) 0 0
\(736\) −2331.58 −0.116770
\(737\) 9646.10 0.482115
\(738\) 0 0
\(739\) 1755.17 0.0873682 0.0436841 0.999045i \(-0.486090\pi\)
0.0436841 + 0.999045i \(0.486090\pi\)
\(740\) −532.196 −0.0264377
\(741\) 0 0
\(742\) −13566.0 −0.671189
\(743\) −17972.6 −0.887419 −0.443709 0.896171i \(-0.646338\pi\)
−0.443709 + 0.896171i \(0.646338\pi\)
\(744\) 0 0
\(745\) −7878.88 −0.387463
\(746\) −27001.4 −1.32519
\(747\) 0 0
\(748\) −5758.54 −0.281488
\(749\) −25692.6 −1.25339
\(750\) 0 0
\(751\) 22704.9 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(752\) −14797.9 −0.717583
\(753\) 0 0
\(754\) −46586.3 −2.25010
\(755\) 2104.89 0.101463
\(756\) 0 0
\(757\) 39860.3 1.91380 0.956901 0.290414i \(-0.0937929\pi\)
0.956901 + 0.290414i \(0.0937929\pi\)
\(758\) 44914.6 2.15221
\(759\) 0 0
\(760\) 2719.99 0.129822
\(761\) −19631.4 −0.935135 −0.467568 0.883957i \(-0.654870\pi\)
−0.467568 + 0.883957i \(0.654870\pi\)
\(762\) 0 0
\(763\) −12192.7 −0.578511
\(764\) −17099.8 −0.809752
\(765\) 0 0
\(766\) −40034.3 −1.88838
\(767\) 12606.8 0.593490
\(768\) 0 0
\(769\) 35029.9 1.64267 0.821333 0.570449i \(-0.193231\pi\)
0.821333 + 0.570449i \(0.193231\pi\)
\(770\) −3188.62 −0.149234
\(771\) 0 0
\(772\) 16972.4 0.791255
\(773\) 26736.9 1.24406 0.622032 0.782992i \(-0.286307\pi\)
0.622032 + 0.782992i \(0.286307\pi\)
\(774\) 0 0
\(775\) −2052.13 −0.0951156
\(776\) 17258.8 0.798398
\(777\) 0 0
\(778\) 11469.0 0.528515
\(779\) 6085.64 0.279898
\(780\) 0 0
\(781\) −8064.55 −0.369491
\(782\) 4591.39 0.209959
\(783\) 0 0
\(784\) 6517.81 0.296912
\(785\) −11121.5 −0.505660
\(786\) 0 0
\(787\) 4799.45 0.217385 0.108693 0.994075i \(-0.465334\pi\)
0.108693 + 0.994075i \(0.465334\pi\)
\(788\) −17799.4 −0.804668
\(789\) 0 0
\(790\) 10571.2 0.476083
\(791\) −18498.3 −0.831508
\(792\) 0 0
\(793\) −8867.72 −0.397102
\(794\) 2098.36 0.0937884
\(795\) 0 0
\(796\) 252.457 0.0112413
\(797\) −38438.9 −1.70838 −0.854189 0.519963i \(-0.825946\pi\)
−0.854189 + 0.519963i \(0.825946\pi\)
\(798\) 0 0
\(799\) 19992.8 0.885224
\(800\) 4877.87 0.215574
\(801\) 0 0
\(802\) −24958.9 −1.09891
\(803\) −12781.7 −0.561713
\(804\) 0 0
\(805\) 964.733 0.0422390
\(806\) −15951.3 −0.697096
\(807\) 0 0
\(808\) 7390.42 0.321775
\(809\) −9960.91 −0.432889 −0.216444 0.976295i \(-0.569446\pi\)
−0.216444 + 0.976295i \(0.569446\pi\)
\(810\) 0 0
\(811\) −34199.9 −1.48079 −0.740396 0.672171i \(-0.765362\pi\)
−0.740396 + 0.672171i \(0.765362\pi\)
\(812\) 18936.6 0.818404
\(813\) 0 0
\(814\) −859.329 −0.0370018
\(815\) −15466.9 −0.664762
\(816\) 0 0
\(817\) 10942.3 0.468570
\(818\) −42203.9 −1.80394
\(819\) 0 0
\(820\) 3053.53 0.130041
\(821\) 31796.4 1.35165 0.675823 0.737064i \(-0.263788\pi\)
0.675823 + 0.737064i \(0.263788\pi\)
\(822\) 0 0
\(823\) −10783.6 −0.456733 −0.228366 0.973575i \(-0.573338\pi\)
−0.228366 + 0.973575i \(0.573338\pi\)
\(824\) 4459.07 0.188518
\(825\) 0 0
\(826\) −13504.5 −0.568862
\(827\) −44197.5 −1.85840 −0.929201 0.369576i \(-0.879503\pi\)
−0.929201 + 0.369576i \(0.879503\pi\)
\(828\) 0 0
\(829\) 22487.7 0.942137 0.471069 0.882097i \(-0.343868\pi\)
0.471069 + 0.882097i \(0.343868\pi\)
\(830\) −20844.4 −0.871709
\(831\) 0 0
\(832\) 3622.81 0.150959
\(833\) −8805.96 −0.366276
\(834\) 0 0
\(835\) 12082.2 0.500743
\(836\) −2623.38 −0.108530
\(837\) 0 0
\(838\) 42475.6 1.75095
\(839\) −22491.0 −0.925477 −0.462738 0.886495i \(-0.653133\pi\)
−0.462738 + 0.886495i \(0.653133\pi\)
\(840\) 0 0
\(841\) 33082.7 1.35646
\(842\) 7181.37 0.293927
\(843\) 0 0
\(844\) 10198.0 0.415910
\(845\) 3660.61 0.149028
\(846\) 0 0
\(847\) −1953.72 −0.0792571
\(848\) 18533.5 0.750524
\(849\) 0 0
\(850\) −9605.60 −0.387611
\(851\) 259.994 0.0104730
\(852\) 0 0
\(853\) 22327.9 0.896241 0.448120 0.893973i \(-0.352094\pi\)
0.448120 + 0.893973i \(0.352094\pi\)
\(854\) 9499.12 0.380624
\(855\) 0 0
\(856\) 17756.5 0.708999
\(857\) 14505.9 0.578193 0.289096 0.957300i \(-0.406645\pi\)
0.289096 + 0.957300i \(0.406645\pi\)
\(858\) 0 0
\(859\) −8411.45 −0.334104 −0.167052 0.985948i \(-0.553425\pi\)
−0.167052 + 0.985948i \(0.553425\pi\)
\(860\) 5490.40 0.217699
\(861\) 0 0
\(862\) −28954.7 −1.14408
\(863\) −32499.6 −1.28192 −0.640961 0.767573i \(-0.721464\pi\)
−0.640961 + 0.767573i \(0.721464\pi\)
\(864\) 0 0
\(865\) 18790.1 0.738592
\(866\) 38455.9 1.50899
\(867\) 0 0
\(868\) 6483.94 0.253548
\(869\) 6477.15 0.252845
\(870\) 0 0
\(871\) 47460.0 1.84629
\(872\) 8426.50 0.327245
\(873\) 0 0
\(874\) 2091.67 0.0809516
\(875\) −2018.31 −0.0779786
\(876\) 0 0
\(877\) −32183.1 −1.23916 −0.619581 0.784933i \(-0.712697\pi\)
−0.619581 + 0.784933i \(0.712697\pi\)
\(878\) −16725.1 −0.642876
\(879\) 0 0
\(880\) 4356.23 0.166873
\(881\) 6246.34 0.238870 0.119435 0.992842i \(-0.461892\pi\)
0.119435 + 0.992842i \(0.461892\pi\)
\(882\) 0 0
\(883\) −11801.1 −0.449762 −0.224881 0.974386i \(-0.572199\pi\)
−0.224881 + 0.974386i \(0.572199\pi\)
\(884\) −28332.7 −1.07798
\(885\) 0 0
\(886\) 7521.24 0.285193
\(887\) −32375.1 −1.22553 −0.612767 0.790264i \(-0.709944\pi\)
−0.612767 + 0.790264i \(0.709944\pi\)
\(888\) 0 0
\(889\) −23594.6 −0.890145
\(890\) −18709.7 −0.704662
\(891\) 0 0
\(892\) 9233.17 0.346580
\(893\) 9107.98 0.341307
\(894\) 0 0
\(895\) −12670.0 −0.473196
\(896\) 21322.6 0.795020
\(897\) 0 0
\(898\) −5034.72 −0.187095
\(899\) 19678.5 0.730049
\(900\) 0 0
\(901\) −25039.9 −0.925861
\(902\) 4930.49 0.182004
\(903\) 0 0
\(904\) 12784.4 0.470357
\(905\) −69.4977 −0.00255269
\(906\) 0 0
\(907\) −19592.8 −0.717277 −0.358638 0.933477i \(-0.616759\pi\)
−0.358638 + 0.933477i \(0.616759\pi\)
\(908\) 5629.53 0.205752
\(909\) 0 0
\(910\) −15688.4 −0.571500
\(911\) −18673.8 −0.679133 −0.339567 0.940582i \(-0.610280\pi\)
−0.339567 + 0.940582i \(0.610280\pi\)
\(912\) 0 0
\(913\) −12771.7 −0.462959
\(914\) 17644.7 0.638550
\(915\) 0 0
\(916\) 20090.9 0.724696
\(917\) −24622.4 −0.886698
\(918\) 0 0
\(919\) −4572.90 −0.164142 −0.0820708 0.996627i \(-0.526153\pi\)
−0.0820708 + 0.996627i \(0.526153\pi\)
\(920\) −666.739 −0.0238932
\(921\) 0 0
\(922\) −7949.78 −0.283961
\(923\) −39678.6 −1.41499
\(924\) 0 0
\(925\) −543.932 −0.0193345
\(926\) 19988.7 0.709362
\(927\) 0 0
\(928\) −46775.4 −1.65461
\(929\) 44222.1 1.56176 0.780882 0.624679i \(-0.214770\pi\)
0.780882 + 0.624679i \(0.214770\pi\)
\(930\) 0 0
\(931\) −4011.67 −0.141221
\(932\) 28049.3 0.985823
\(933\) 0 0
\(934\) 1784.70 0.0625238
\(935\) −5885.52 −0.205858
\(936\) 0 0
\(937\) 9218.62 0.321408 0.160704 0.987003i \(-0.448624\pi\)
0.160704 + 0.987003i \(0.448624\pi\)
\(938\) −50839.2 −1.76968
\(939\) 0 0
\(940\) 4570.02 0.158572
\(941\) 26484.9 0.917516 0.458758 0.888561i \(-0.348294\pi\)
0.458758 + 0.888561i \(0.348294\pi\)
\(942\) 0 0
\(943\) −1491.75 −0.0515142
\(944\) 18449.5 0.636103
\(945\) 0 0
\(946\) 8865.26 0.304688
\(947\) 44972.4 1.54320 0.771599 0.636110i \(-0.219457\pi\)
0.771599 + 0.636110i \(0.219457\pi\)
\(948\) 0 0
\(949\) −62887.5 −2.15112
\(950\) −4375.96 −0.149447
\(951\) 0 0
\(952\) −19280.9 −0.656404
\(953\) 2052.50 0.0697659 0.0348829 0.999391i \(-0.488894\pi\)
0.0348829 + 0.999391i \(0.488894\pi\)
\(954\) 0 0
\(955\) −17476.9 −0.592189
\(956\) −29529.7 −0.999016
\(957\) 0 0
\(958\) −33569.2 −1.13212
\(959\) −34322.2 −1.15570
\(960\) 0 0
\(961\) −23053.0 −0.773826
\(962\) −4228.00 −0.141701
\(963\) 0 0
\(964\) 18203.1 0.608176
\(965\) 17346.6 0.578661
\(966\) 0 0
\(967\) 14950.9 0.497195 0.248598 0.968607i \(-0.420030\pi\)
0.248598 + 0.968607i \(0.420030\pi\)
\(968\) 1350.24 0.0448331
\(969\) 0 0
\(970\) −27766.3 −0.919094
\(971\) −26751.9 −0.884151 −0.442076 0.896978i \(-0.645758\pi\)
−0.442076 + 0.896978i \(0.645758\pi\)
\(972\) 0 0
\(973\) 28648.3 0.943908
\(974\) −710.034 −0.0233583
\(975\) 0 0
\(976\) −12977.5 −0.425615
\(977\) 56893.7 1.86304 0.931520 0.363690i \(-0.118483\pi\)
0.931520 + 0.363690i \(0.118483\pi\)
\(978\) 0 0
\(979\) −11463.7 −0.374242
\(980\) −2012.89 −0.0656118
\(981\) 0 0
\(982\) −35895.0 −1.16645
\(983\) 34633.1 1.12373 0.561863 0.827230i \(-0.310085\pi\)
0.561863 + 0.827230i \(0.310085\pi\)
\(984\) 0 0
\(985\) −18191.9 −0.588470
\(986\) 92111.0 2.97506
\(987\) 0 0
\(988\) −12907.3 −0.415625
\(989\) −2682.23 −0.0862386
\(990\) 0 0
\(991\) 1961.79 0.0628843 0.0314422 0.999506i \(-0.489990\pi\)
0.0314422 + 0.999506i \(0.489990\pi\)
\(992\) −16016.0 −0.512610
\(993\) 0 0
\(994\) 42503.8 1.35628
\(995\) 258.025 0.00822103
\(996\) 0 0
\(997\) 13096.5 0.416017 0.208009 0.978127i \(-0.433302\pi\)
0.208009 + 0.978127i \(0.433302\pi\)
\(998\) 31290.8 0.992476
\(999\) 0 0
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 495.4.a.l.1.1 3
3.2 odd 2 165.4.a.d.1.3 3
5.4 even 2 2475.4.a.s.1.3 3
15.2 even 4 825.4.c.l.199.5 6
15.8 even 4 825.4.c.l.199.2 6
15.14 odd 2 825.4.a.s.1.1 3
33.32 even 2 1815.4.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 3.2 odd 2
495.4.a.l.1.1 3 1.1 even 1 trivial
825.4.a.s.1.1 3 15.14 odd 2
825.4.c.l.199.2 6 15.8 even 4
825.4.c.l.199.5 6 15.2 even 4
1815.4.a.s.1.1 3 33.32 even 2
2475.4.a.s.1.3 3 5.4 even 2