Properties

Label 245.4.a.o.1.4
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.05886\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} +5.00000 q^{5} +2.69906 q^{6} -10.0465 q^{8} -9.47008 q^{9} +O(q^{10})\) \(q+0.644648 q^{2} +4.18687 q^{3} -7.58443 q^{4} +5.00000 q^{5} +2.69906 q^{6} -10.0465 q^{8} -9.47008 q^{9} +3.22324 q^{10} -47.7013 q^{11} -31.7550 q^{12} -57.2256 q^{13} +20.9344 q^{15} +54.1990 q^{16} +36.9686 q^{17} -6.10487 q^{18} -30.7659 q^{19} -37.9221 q^{20} -30.7506 q^{22} +53.1282 q^{23} -42.0633 q^{24} +25.0000 q^{25} -36.8904 q^{26} -152.696 q^{27} -195.663 q^{29} +13.4953 q^{30} -257.870 q^{31} +115.311 q^{32} -199.719 q^{33} +23.8317 q^{34} +71.8252 q^{36} +346.423 q^{37} -19.8332 q^{38} -239.596 q^{39} -50.2324 q^{40} -267.050 q^{41} -176.859 q^{43} +361.787 q^{44} -47.3504 q^{45} +34.2490 q^{46} +311.598 q^{47} +226.924 q^{48} +16.1162 q^{50} +154.783 q^{51} +434.024 q^{52} -492.270 q^{53} -98.4349 q^{54} -238.507 q^{55} -128.813 q^{57} -126.134 q^{58} -98.7653 q^{59} -158.775 q^{60} -82.1682 q^{61} -166.235 q^{62} -359.257 q^{64} -286.128 q^{65} -128.749 q^{66} +654.668 q^{67} -280.386 q^{68} +222.441 q^{69} +779.658 q^{71} +95.1409 q^{72} +829.673 q^{73} +223.321 q^{74} +104.672 q^{75} +233.342 q^{76} -154.455 q^{78} -769.426 q^{79} +270.995 q^{80} -383.625 q^{81} -172.153 q^{82} -613.203 q^{83} +184.843 q^{85} -114.012 q^{86} -819.215 q^{87} +479.230 q^{88} +457.666 q^{89} -30.5244 q^{90} -402.947 q^{92} -1079.67 q^{93} +200.871 q^{94} -153.830 q^{95} +482.793 q^{96} +1412.11 q^{97} +451.736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 16 q^{3} + 14 q^{4} + 30 q^{5} - 24 q^{6} - 66 q^{8} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 16 q^{3} + 14 q^{4} + 30 q^{5} - 24 q^{6} - 66 q^{8} + 70 q^{9} - 10 q^{10} - 16 q^{11} - 160 q^{12} - 168 q^{13} - 80 q^{15} + 298 q^{16} + 4 q^{17} + 354 q^{18} - 308 q^{19} + 70 q^{20} - 236 q^{22} - 336 q^{23} + 92 q^{24} + 150 q^{25} - 56 q^{26} - 964 q^{27} + 176 q^{29} - 120 q^{30} - 392 q^{31} - 770 q^{32} - 188 q^{33} - 812 q^{34} + 230 q^{36} - 140 q^{37} - 20 q^{38} + 140 q^{39} - 330 q^{40} - 656 q^{41} - 388 q^{43} - 160 q^{44} + 350 q^{45} - 388 q^{46} - 628 q^{47} - 1396 q^{48} - 50 q^{50} + 744 q^{51} - 1520 q^{52} - 676 q^{53} - 2284 q^{54} - 80 q^{55} + 1468 q^{57} - 2012 q^{58} - 996 q^{59} - 800 q^{60} - 740 q^{61} + 364 q^{62} + 1426 q^{64} - 840 q^{65} + 3620 q^{66} + 1768 q^{67} + 2940 q^{68} + 1048 q^{69} - 224 q^{71} + 2858 q^{72} - 2640 q^{73} + 928 q^{74} - 400 q^{75} + 1340 q^{76} + 8 q^{78} + 1636 q^{79} + 1490 q^{80} + 4442 q^{81} + 1756 q^{82} - 140 q^{83} + 20 q^{85} + 1180 q^{86} - 1940 q^{87} - 5652 q^{88} + 1904 q^{89} + 1770 q^{90} - 1952 q^{92} - 1592 q^{93} + 3332 q^{94} - 1540 q^{95} + 6460 q^{96} - 516 q^{97} - 2804 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.644648 0.227917 0.113959 0.993485i \(-0.463647\pi\)
0.113959 + 0.993485i \(0.463647\pi\)
\(3\) 4.18687 0.805764 0.402882 0.915252i \(-0.368008\pi\)
0.402882 + 0.915252i \(0.368008\pi\)
\(4\) −7.58443 −0.948054
\(5\) 5.00000 0.447214
\(6\) 2.69906 0.183648
\(7\) 0 0
\(8\) −10.0465 −0.443995
\(9\) −9.47008 −0.350744
\(10\) 3.22324 0.101928
\(11\) −47.7013 −1.30750 −0.653750 0.756711i \(-0.726805\pi\)
−0.653750 + 0.756711i \(0.726805\pi\)
\(12\) −31.7550 −0.763908
\(13\) −57.2256 −1.22089 −0.610443 0.792060i \(-0.709009\pi\)
−0.610443 + 0.792060i \(0.709009\pi\)
\(14\) 0 0
\(15\) 20.9344 0.360349
\(16\) 54.1990 0.846859
\(17\) 36.9686 0.527423 0.263712 0.964602i \(-0.415053\pi\)
0.263712 + 0.964602i \(0.415053\pi\)
\(18\) −6.10487 −0.0799407
\(19\) −30.7659 −0.371484 −0.185742 0.982599i \(-0.559469\pi\)
−0.185742 + 0.982599i \(0.559469\pi\)
\(20\) −37.9221 −0.423982
\(21\) 0 0
\(22\) −30.7506 −0.298002
\(23\) 53.1282 0.481652 0.240826 0.970568i \(-0.422582\pi\)
0.240826 + 0.970568i \(0.422582\pi\)
\(24\) −42.0633 −0.357756
\(25\) 25.0000 0.200000
\(26\) −36.8904 −0.278261
\(27\) −152.696 −1.08838
\(28\) 0 0
\(29\) −195.663 −1.25288 −0.626442 0.779468i \(-0.715490\pi\)
−0.626442 + 0.779468i \(0.715490\pi\)
\(30\) 13.4953 0.0821298
\(31\) −257.870 −1.49403 −0.747014 0.664809i \(-0.768513\pi\)
−0.747014 + 0.664809i \(0.768513\pi\)
\(32\) 115.311 0.637010
\(33\) −199.719 −1.05354
\(34\) 23.8317 0.120209
\(35\) 0 0
\(36\) 71.8252 0.332524
\(37\) 346.423 1.53923 0.769616 0.638507i \(-0.220447\pi\)
0.769616 + 0.638507i \(0.220447\pi\)
\(38\) −19.8332 −0.0846676
\(39\) −239.596 −0.983747
\(40\) −50.2324 −0.198561
\(41\) −267.050 −1.01722 −0.508611 0.860996i \(-0.669841\pi\)
−0.508611 + 0.860996i \(0.669841\pi\)
\(42\) 0 0
\(43\) −176.859 −0.627227 −0.313614 0.949551i \(-0.601540\pi\)
−0.313614 + 0.949551i \(0.601540\pi\)
\(44\) 361.787 1.23958
\(45\) −47.3504 −0.156857
\(46\) 34.2490 0.109777
\(47\) 311.598 0.967049 0.483524 0.875331i \(-0.339356\pi\)
0.483524 + 0.875331i \(0.339356\pi\)
\(48\) 226.924 0.682369
\(49\) 0 0
\(50\) 16.1162 0.0455835
\(51\) 154.783 0.424979
\(52\) 434.024 1.15747
\(53\) −492.270 −1.27582 −0.637910 0.770111i \(-0.720201\pi\)
−0.637910 + 0.770111i \(0.720201\pi\)
\(54\) −98.4349 −0.248061
\(55\) −238.507 −0.584731
\(56\) 0 0
\(57\) −128.813 −0.299328
\(58\) −126.134 −0.285554
\(59\) −98.7653 −0.217935 −0.108967 0.994045i \(-0.534754\pi\)
−0.108967 + 0.994045i \(0.534754\pi\)
\(60\) −158.775 −0.341630
\(61\) −82.1682 −0.172468 −0.0862340 0.996275i \(-0.527483\pi\)
−0.0862340 + 0.996275i \(0.527483\pi\)
\(62\) −166.235 −0.340515
\(63\) 0 0
\(64\) −359.257 −0.701674
\(65\) −286.128 −0.545997
\(66\) −128.749 −0.240119
\(67\) 654.668 1.19374 0.596869 0.802339i \(-0.296411\pi\)
0.596869 + 0.802339i \(0.296411\pi\)
\(68\) −280.386 −0.500026
\(69\) 222.441 0.388098
\(70\) 0 0
\(71\) 779.658 1.30322 0.651608 0.758556i \(-0.274095\pi\)
0.651608 + 0.758556i \(0.274095\pi\)
\(72\) 95.1409 0.155729
\(73\) 829.673 1.33022 0.665109 0.746747i \(-0.268385\pi\)
0.665109 + 0.746747i \(0.268385\pi\)
\(74\) 223.321 0.350818
\(75\) 104.672 0.161153
\(76\) 233.342 0.352186
\(77\) 0 0
\(78\) −154.455 −0.224213
\(79\) −769.426 −1.09579 −0.547894 0.836548i \(-0.684570\pi\)
−0.547894 + 0.836548i \(0.684570\pi\)
\(80\) 270.995 0.378727
\(81\) −383.625 −0.526235
\(82\) −172.153 −0.231843
\(83\) −613.203 −0.810937 −0.405469 0.914109i \(-0.632892\pi\)
−0.405469 + 0.914109i \(0.632892\pi\)
\(84\) 0 0
\(85\) 184.843 0.235871
\(86\) −114.012 −0.142956
\(87\) −819.215 −1.00953
\(88\) 479.230 0.580524
\(89\) 457.666 0.545084 0.272542 0.962144i \(-0.412136\pi\)
0.272542 + 0.962144i \(0.412136\pi\)
\(90\) −30.5244 −0.0357506
\(91\) 0 0
\(92\) −402.947 −0.456632
\(93\) −1079.67 −1.20383
\(94\) 200.871 0.220407
\(95\) −153.830 −0.166132
\(96\) 482.793 0.513280
\(97\) 1412.11 1.47813 0.739063 0.673636i \(-0.235268\pi\)
0.739063 + 0.673636i \(0.235268\pi\)
\(98\) 0 0
\(99\) 451.736 0.458597
\(100\) −189.611 −0.189611
\(101\) −1823.79 −1.79677 −0.898386 0.439208i \(-0.855259\pi\)
−0.898386 + 0.439208i \(0.855259\pi\)
\(102\) 99.7804 0.0968601
\(103\) −407.775 −0.390090 −0.195045 0.980794i \(-0.562485\pi\)
−0.195045 + 0.980794i \(0.562485\pi\)
\(104\) 574.915 0.542068
\(105\) 0 0
\(106\) −317.341 −0.290782
\(107\) −370.109 −0.334390 −0.167195 0.985924i \(-0.553471\pi\)
−0.167195 + 0.985924i \(0.553471\pi\)
\(108\) 1158.11 1.03184
\(109\) 975.570 0.857272 0.428636 0.903477i \(-0.358994\pi\)
0.428636 + 0.903477i \(0.358994\pi\)
\(110\) −153.753 −0.133270
\(111\) 1450.43 1.24026
\(112\) 0 0
\(113\) 1978.85 1.64739 0.823693 0.567036i \(-0.191910\pi\)
0.823693 + 0.567036i \(0.191910\pi\)
\(114\) −83.0391 −0.0682221
\(115\) 265.641 0.215401
\(116\) 1483.99 1.18780
\(117\) 541.931 0.428219
\(118\) −63.6689 −0.0496711
\(119\) 0 0
\(120\) −210.317 −0.159993
\(121\) 944.416 0.709554
\(122\) −52.9695 −0.0393085
\(123\) −1118.10 −0.819642
\(124\) 1955.80 1.41642
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1392.38 0.972867 0.486433 0.873718i \(-0.338298\pi\)
0.486433 + 0.873718i \(0.338298\pi\)
\(128\) −1154.08 −0.796933
\(129\) −740.487 −0.505397
\(130\) −184.452 −0.124442
\(131\) 1777.02 1.18518 0.592591 0.805504i \(-0.298105\pi\)
0.592591 + 0.805504i \(0.298105\pi\)
\(132\) 1514.76 0.998809
\(133\) 0 0
\(134\) 422.031 0.272074
\(135\) −763.478 −0.486739
\(136\) −371.404 −0.234174
\(137\) −1980.48 −1.23506 −0.617532 0.786545i \(-0.711868\pi\)
−0.617532 + 0.786545i \(0.711868\pi\)
\(138\) 143.396 0.0884543
\(139\) −2182.09 −1.33153 −0.665763 0.746163i \(-0.731894\pi\)
−0.665763 + 0.746163i \(0.731894\pi\)
\(140\) 0 0
\(141\) 1304.62 0.779213
\(142\) 502.605 0.297026
\(143\) 2729.74 1.59631
\(144\) −513.269 −0.297031
\(145\) −978.313 −0.560307
\(146\) 534.847 0.303180
\(147\) 0 0
\(148\) −2627.42 −1.45928
\(149\) 670.154 0.368464 0.184232 0.982883i \(-0.441020\pi\)
0.184232 + 0.982883i \(0.441020\pi\)
\(150\) 67.4765 0.0367296
\(151\) 3348.84 1.80480 0.902401 0.430898i \(-0.141803\pi\)
0.902401 + 0.430898i \(0.141803\pi\)
\(152\) 309.089 0.164937
\(153\) −350.096 −0.184990
\(154\) 0 0
\(155\) −1289.35 −0.668149
\(156\) 1817.20 0.932645
\(157\) −2408.41 −1.22428 −0.612140 0.790749i \(-0.709691\pi\)
−0.612140 + 0.790749i \(0.709691\pi\)
\(158\) −496.009 −0.249749
\(159\) −2061.07 −1.02801
\(160\) 576.555 0.284879
\(161\) 0 0
\(162\) −247.303 −0.119938
\(163\) −3811.03 −1.83131 −0.915654 0.401968i \(-0.868326\pi\)
−0.915654 + 0.401968i \(0.868326\pi\)
\(164\) 2025.42 0.964382
\(165\) −998.597 −0.471156
\(166\) −395.300 −0.184827
\(167\) −1207.15 −0.559354 −0.279677 0.960094i \(-0.590227\pi\)
−0.279677 + 0.960094i \(0.590227\pi\)
\(168\) 0 0
\(169\) 1077.77 0.490564
\(170\) 119.159 0.0537591
\(171\) 291.356 0.130296
\(172\) 1341.38 0.594645
\(173\) −3243.28 −1.42533 −0.712665 0.701505i \(-0.752512\pi\)
−0.712665 + 0.701505i \(0.752512\pi\)
\(174\) −528.105 −0.230089
\(175\) 0 0
\(176\) −2585.36 −1.10727
\(177\) −413.518 −0.175604
\(178\) 295.034 0.124234
\(179\) 859.728 0.358989 0.179495 0.983759i \(-0.442554\pi\)
0.179495 + 0.983759i \(0.442554\pi\)
\(180\) 359.126 0.148709
\(181\) 290.504 0.119298 0.0596491 0.998219i \(-0.481002\pi\)
0.0596491 + 0.998219i \(0.481002\pi\)
\(182\) 0 0
\(183\) −344.028 −0.138969
\(184\) −533.751 −0.213851
\(185\) 1732.12 0.688366
\(186\) −696.007 −0.274375
\(187\) −1763.45 −0.689605
\(188\) −2363.30 −0.916814
\(189\) 0 0
\(190\) −99.1660 −0.0378645
\(191\) 4895.61 1.85463 0.927315 0.374282i \(-0.122111\pi\)
0.927315 + 0.374282i \(0.122111\pi\)
\(192\) −1504.16 −0.565384
\(193\) −3548.73 −1.32354 −0.661770 0.749707i \(-0.730194\pi\)
−0.661770 + 0.749707i \(0.730194\pi\)
\(194\) 910.315 0.336891
\(195\) −1197.98 −0.439945
\(196\) 0 0
\(197\) −650.107 −0.235118 −0.117559 0.993066i \(-0.537507\pi\)
−0.117559 + 0.993066i \(0.537507\pi\)
\(198\) 291.210 0.104522
\(199\) −4055.09 −1.44451 −0.722256 0.691626i \(-0.756895\pi\)
−0.722256 + 0.691626i \(0.756895\pi\)
\(200\) −251.162 −0.0887991
\(201\) 2741.01 0.961871
\(202\) −1175.70 −0.409516
\(203\) 0 0
\(204\) −1173.94 −0.402903
\(205\) −1335.25 −0.454916
\(206\) −262.871 −0.0889083
\(207\) −503.128 −0.168936
\(208\) −3101.57 −1.03392
\(209\) 1467.58 0.485714
\(210\) 0 0
\(211\) −1569.67 −0.512134 −0.256067 0.966659i \(-0.582427\pi\)
−0.256067 + 0.966659i \(0.582427\pi\)
\(212\) 3733.59 1.20955
\(213\) 3264.33 1.05008
\(214\) −238.590 −0.0762134
\(215\) −884.296 −0.280504
\(216\) 1534.05 0.483236
\(217\) 0 0
\(218\) 628.900 0.195387
\(219\) 3473.74 1.07184
\(220\) 1808.94 0.554357
\(221\) −2115.55 −0.643924
\(222\) 935.017 0.282677
\(223\) −4723.86 −1.41853 −0.709267 0.704940i \(-0.750974\pi\)
−0.709267 + 0.704940i \(0.750974\pi\)
\(224\) 0 0
\(225\) −236.752 −0.0701488
\(226\) 1275.66 0.375468
\(227\) −3685.02 −1.07746 −0.538730 0.842479i \(-0.681096\pi\)
−0.538730 + 0.842479i \(0.681096\pi\)
\(228\) 976.973 0.283779
\(229\) −3356.78 −0.968656 −0.484328 0.874887i \(-0.660936\pi\)
−0.484328 + 0.874887i \(0.660936\pi\)
\(230\) 171.245 0.0490937
\(231\) 0 0
\(232\) 1965.72 0.556275
\(233\) 2314.01 0.650627 0.325314 0.945606i \(-0.394530\pi\)
0.325314 + 0.945606i \(0.394530\pi\)
\(234\) 349.355 0.0975985
\(235\) 1557.99 0.432477
\(236\) 749.079 0.206614
\(237\) −3221.49 −0.882946
\(238\) 0 0
\(239\) 941.179 0.254727 0.127364 0.991856i \(-0.459348\pi\)
0.127364 + 0.991856i \(0.459348\pi\)
\(240\) 1134.62 0.305165
\(241\) −5638.24 −1.50702 −0.753509 0.657438i \(-0.771640\pi\)
−0.753509 + 0.657438i \(0.771640\pi\)
\(242\) 608.816 0.161720
\(243\) 2516.59 0.664360
\(244\) 623.199 0.163509
\(245\) 0 0
\(246\) −720.783 −0.186811
\(247\) 1760.60 0.453539
\(248\) 2590.69 0.663341
\(249\) −2567.40 −0.653424
\(250\) 80.5810 0.0203856
\(251\) −365.822 −0.0919940 −0.0459970 0.998942i \(-0.514646\pi\)
−0.0459970 + 0.998942i \(0.514646\pi\)
\(252\) 0 0
\(253\) −2534.28 −0.629759
\(254\) 897.598 0.221733
\(255\) 773.914 0.190056
\(256\) 2130.08 0.520039
\(257\) 6276.46 1.52340 0.761702 0.647928i \(-0.224364\pi\)
0.761702 + 0.647928i \(0.224364\pi\)
\(258\) −477.353 −0.115189
\(259\) 0 0
\(260\) 2170.12 0.517635
\(261\) 1852.94 0.439442
\(262\) 1145.55 0.270124
\(263\) −4225.97 −0.990817 −0.495408 0.868660i \(-0.664982\pi\)
−0.495408 + 0.868660i \(0.664982\pi\)
\(264\) 2006.48 0.467765
\(265\) −2461.35 −0.570564
\(266\) 0 0
\(267\) 1916.19 0.439210
\(268\) −4965.28 −1.13173
\(269\) −981.591 −0.222486 −0.111243 0.993793i \(-0.535483\pi\)
−0.111243 + 0.993793i \(0.535483\pi\)
\(270\) −492.175 −0.110936
\(271\) −3884.42 −0.870708 −0.435354 0.900259i \(-0.643377\pi\)
−0.435354 + 0.900259i \(0.643377\pi\)
\(272\) 2003.66 0.446653
\(273\) 0 0
\(274\) −1276.71 −0.281493
\(275\) −1192.53 −0.261500
\(276\) −1687.09 −0.367938
\(277\) 3614.93 0.784116 0.392058 0.919941i \(-0.371763\pi\)
0.392058 + 0.919941i \(0.371763\pi\)
\(278\) −1406.68 −0.303478
\(279\) 2442.05 0.524021
\(280\) 0 0
\(281\) 72.6835 0.0154304 0.00771518 0.999970i \(-0.497544\pi\)
0.00771518 + 0.999970i \(0.497544\pi\)
\(282\) 841.023 0.177596
\(283\) −1743.04 −0.366125 −0.183062 0.983101i \(-0.558601\pi\)
−0.183062 + 0.983101i \(0.558601\pi\)
\(284\) −5913.26 −1.23552
\(285\) −644.065 −0.133864
\(286\) 1759.72 0.363827
\(287\) 0 0
\(288\) −1092.01 −0.223427
\(289\) −3546.32 −0.721825
\(290\) −630.668 −0.127704
\(291\) 5912.34 1.19102
\(292\) −6292.59 −1.26112
\(293\) 4989.29 0.994804 0.497402 0.867520i \(-0.334287\pi\)
0.497402 + 0.867520i \(0.334287\pi\)
\(294\) 0 0
\(295\) −493.827 −0.0974634
\(296\) −3480.33 −0.683412
\(297\) 7283.78 1.42306
\(298\) 432.013 0.0839794
\(299\) −3040.29 −0.588042
\(300\) −793.876 −0.152782
\(301\) 0 0
\(302\) 2158.82 0.411346
\(303\) −7635.98 −1.44777
\(304\) −1667.48 −0.314594
\(305\) −410.841 −0.0771301
\(306\) −225.688 −0.0421626
\(307\) −1664.61 −0.309461 −0.154731 0.987957i \(-0.549451\pi\)
−0.154731 + 0.987957i \(0.549451\pi\)
\(308\) 0 0
\(309\) −1707.30 −0.314320
\(310\) −831.177 −0.152283
\(311\) −545.623 −0.0994838 −0.0497419 0.998762i \(-0.515840\pi\)
−0.0497419 + 0.998762i \(0.515840\pi\)
\(312\) 2407.10 0.436779
\(313\) −213.564 −0.0385667 −0.0192833 0.999814i \(-0.506138\pi\)
−0.0192833 + 0.999814i \(0.506138\pi\)
\(314\) −1552.58 −0.279035
\(315\) 0 0
\(316\) 5835.66 1.03887
\(317\) −2683.93 −0.475535 −0.237767 0.971322i \(-0.576416\pi\)
−0.237767 + 0.971322i \(0.576416\pi\)
\(318\) −1328.67 −0.234302
\(319\) 9333.37 1.63815
\(320\) −1796.28 −0.313798
\(321\) −1549.60 −0.269440
\(322\) 0 0
\(323\) −1137.37 −0.195929
\(324\) 2909.58 0.498899
\(325\) −1430.64 −0.244177
\(326\) −2456.77 −0.417387
\(327\) 4084.59 0.690760
\(328\) 2682.91 0.451642
\(329\) 0 0
\(330\) −643.744 −0.107385
\(331\) −5067.52 −0.841500 −0.420750 0.907177i \(-0.638233\pi\)
−0.420750 + 0.907177i \(0.638233\pi\)
\(332\) 4650.80 0.768812
\(333\) −3280.66 −0.539876
\(334\) −778.187 −0.127487
\(335\) 3273.34 0.533856
\(336\) 0 0
\(337\) −9353.21 −1.51187 −0.755937 0.654644i \(-0.772818\pi\)
−0.755937 + 0.654644i \(0.772818\pi\)
\(338\) 694.782 0.111808
\(339\) 8285.20 1.32741
\(340\) −1401.93 −0.223618
\(341\) 12300.7 1.95344
\(342\) 187.822 0.0296966
\(343\) 0 0
\(344\) 1776.81 0.278486
\(345\) 1112.21 0.173563
\(346\) −2090.77 −0.324858
\(347\) 2349.34 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(348\) 6213.28 0.957088
\(349\) 10472.6 1.60626 0.803128 0.595806i \(-0.203167\pi\)
0.803128 + 0.595806i \(0.203167\pi\)
\(350\) 0 0
\(351\) 8738.10 1.32879
\(352\) −5500.49 −0.832889
\(353\) 7174.63 1.08178 0.540888 0.841095i \(-0.318088\pi\)
0.540888 + 0.841095i \(0.318088\pi\)
\(354\) −266.574 −0.0400232
\(355\) 3898.29 0.582816
\(356\) −3471.14 −0.516769
\(357\) 0 0
\(358\) 554.222 0.0818199
\(359\) −7534.13 −1.10762 −0.553811 0.832643i \(-0.686827\pi\)
−0.553811 + 0.832643i \(0.686827\pi\)
\(360\) 475.705 0.0696440
\(361\) −5912.46 −0.862000
\(362\) 187.273 0.0271902
\(363\) 3954.15 0.571733
\(364\) 0 0
\(365\) 4148.36 0.594891
\(366\) −221.777 −0.0316734
\(367\) −5453.06 −0.775606 −0.387803 0.921742i \(-0.626766\pi\)
−0.387803 + 0.921742i \(0.626766\pi\)
\(368\) 2879.49 0.407891
\(369\) 2528.98 0.356785
\(370\) 1116.60 0.156891
\(371\) 0 0
\(372\) 8188.68 1.14130
\(373\) 8231.29 1.14263 0.571314 0.820732i \(-0.306434\pi\)
0.571314 + 0.820732i \(0.306434\pi\)
\(374\) −1136.80 −0.157173
\(375\) 523.359 0.0720698
\(376\) −3130.46 −0.429365
\(377\) 11196.9 1.52963
\(378\) 0 0
\(379\) 1670.06 0.226346 0.113173 0.993575i \(-0.463898\pi\)
0.113173 + 0.993575i \(0.463898\pi\)
\(380\) 1166.71 0.157502
\(381\) 5829.74 0.783901
\(382\) 3155.95 0.422703
\(383\) −3220.14 −0.429612 −0.214806 0.976657i \(-0.568912\pi\)
−0.214806 + 0.976657i \(0.568912\pi\)
\(384\) −4832.00 −0.642140
\(385\) 0 0
\(386\) −2287.68 −0.301658
\(387\) 1674.87 0.219996
\(388\) −10710.1 −1.40134
\(389\) 3522.23 0.459085 0.229543 0.973299i \(-0.426277\pi\)
0.229543 + 0.973299i \(0.426277\pi\)
\(390\) −772.277 −0.100271
\(391\) 1964.07 0.254034
\(392\) 0 0
\(393\) 7440.15 0.954977
\(394\) −419.090 −0.0535875
\(395\) −3847.13 −0.490051
\(396\) −3426.16 −0.434775
\(397\) 5455.16 0.689639 0.344819 0.938669i \(-0.387940\pi\)
0.344819 + 0.938669i \(0.387940\pi\)
\(398\) −2614.11 −0.329229
\(399\) 0 0
\(400\) 1354.97 0.169372
\(401\) −1161.80 −0.144682 −0.0723409 0.997380i \(-0.523047\pi\)
−0.0723409 + 0.997380i \(0.523047\pi\)
\(402\) 1766.99 0.219227
\(403\) 14756.8 1.82404
\(404\) 13832.4 1.70344
\(405\) −1918.13 −0.235339
\(406\) 0 0
\(407\) −16524.8 −2.01255
\(408\) −1555.02 −0.188689
\(409\) 7398.65 0.894473 0.447237 0.894416i \(-0.352408\pi\)
0.447237 + 0.894416i \(0.352408\pi\)
\(410\) −860.765 −0.103683
\(411\) −8292.02 −0.995171
\(412\) 3092.74 0.369826
\(413\) 0 0
\(414\) −324.341 −0.0385036
\(415\) −3066.02 −0.362662
\(416\) −6598.74 −0.777716
\(417\) −9136.12 −1.07290
\(418\) 946.069 0.110703
\(419\) −2134.46 −0.248867 −0.124433 0.992228i \(-0.539711\pi\)
−0.124433 + 0.992228i \(0.539711\pi\)
\(420\) 0 0
\(421\) −3902.36 −0.451756 −0.225878 0.974156i \(-0.572525\pi\)
−0.225878 + 0.974156i \(0.572525\pi\)
\(422\) −1011.88 −0.116724
\(423\) −2950.86 −0.339186
\(424\) 4945.58 0.566459
\(425\) 924.214 0.105485
\(426\) 2104.34 0.239333
\(427\) 0 0
\(428\) 2807.06 0.317020
\(429\) 11429.1 1.28625
\(430\) −570.059 −0.0639319
\(431\) −3618.13 −0.404360 −0.202180 0.979348i \(-0.564803\pi\)
−0.202180 + 0.979348i \(0.564803\pi\)
\(432\) −8275.95 −0.921706
\(433\) −4222.37 −0.468624 −0.234312 0.972161i \(-0.575284\pi\)
−0.234312 + 0.972161i \(0.575284\pi\)
\(434\) 0 0
\(435\) −4096.08 −0.451475
\(436\) −7399.14 −0.812740
\(437\) −1634.54 −0.178926
\(438\) 2239.34 0.244291
\(439\) −13518.0 −1.46965 −0.734826 0.678256i \(-0.762736\pi\)
−0.734826 + 0.678256i \(0.762736\pi\)
\(440\) 2396.15 0.259618
\(441\) 0 0
\(442\) −1363.78 −0.146762
\(443\) 16591.2 1.77939 0.889695 0.456555i \(-0.150917\pi\)
0.889695 + 0.456555i \(0.150917\pi\)
\(444\) −11000.7 −1.17583
\(445\) 2288.33 0.243769
\(446\) −3045.23 −0.323309
\(447\) 2805.85 0.296895
\(448\) 0 0
\(449\) 8354.32 0.878095 0.439048 0.898464i \(-0.355316\pi\)
0.439048 + 0.898464i \(0.355316\pi\)
\(450\) −152.622 −0.0159881
\(451\) 12738.6 1.33002
\(452\) −15008.5 −1.56181
\(453\) 14021.2 1.45424
\(454\) −2375.54 −0.245572
\(455\) 0 0
\(456\) 1294.12 0.132900
\(457\) −1280.34 −0.131054 −0.0655269 0.997851i \(-0.520873\pi\)
−0.0655269 + 0.997851i \(0.520873\pi\)
\(458\) −2163.94 −0.220774
\(459\) −5644.94 −0.574038
\(460\) −2014.73 −0.204212
\(461\) −6986.72 −0.705865 −0.352932 0.935649i \(-0.614815\pi\)
−0.352932 + 0.935649i \(0.614815\pi\)
\(462\) 0 0
\(463\) −5587.32 −0.560831 −0.280416 0.959879i \(-0.590472\pi\)
−0.280416 + 0.959879i \(0.590472\pi\)
\(464\) −10604.7 −1.06102
\(465\) −5398.35 −0.538371
\(466\) 1491.73 0.148289
\(467\) −9933.83 −0.984332 −0.492166 0.870501i \(-0.663795\pi\)
−0.492166 + 0.870501i \(0.663795\pi\)
\(468\) −4110.24 −0.405974
\(469\) 0 0
\(470\) 1004.36 0.0985691
\(471\) −10083.7 −0.986481
\(472\) 992.243 0.0967620
\(473\) 8436.41 0.820099
\(474\) −2076.73 −0.201239
\(475\) −769.148 −0.0742967
\(476\) 0 0
\(477\) 4661.84 0.447486
\(478\) 606.729 0.0580568
\(479\) 5068.62 0.483489 0.241744 0.970340i \(-0.422280\pi\)
0.241744 + 0.970340i \(0.422280\pi\)
\(480\) 2413.96 0.229546
\(481\) −19824.3 −1.87923
\(482\) −3634.68 −0.343476
\(483\) 0 0
\(484\) −7162.86 −0.672695
\(485\) 7060.56 0.661038
\(486\) 1622.32 0.151419
\(487\) 264.353 0.0245975 0.0122988 0.999924i \(-0.496085\pi\)
0.0122988 + 0.999924i \(0.496085\pi\)
\(488\) 825.500 0.0765751
\(489\) −15956.3 −1.47560
\(490\) 0 0
\(491\) −7459.47 −0.685623 −0.342812 0.939404i \(-0.611379\pi\)
−0.342812 + 0.939404i \(0.611379\pi\)
\(492\) 8480.17 0.777064
\(493\) −7233.37 −0.660800
\(494\) 1134.97 0.103370
\(495\) 2258.68 0.205091
\(496\) −13976.3 −1.26523
\(497\) 0 0
\(498\) −1655.07 −0.148927
\(499\) 7206.67 0.646523 0.323261 0.946310i \(-0.395221\pi\)
0.323261 + 0.946310i \(0.395221\pi\)
\(500\) −948.054 −0.0847965
\(501\) −5054.19 −0.450707
\(502\) −235.827 −0.0209670
\(503\) 10886.7 0.965037 0.482519 0.875886i \(-0.339722\pi\)
0.482519 + 0.875886i \(0.339722\pi\)
\(504\) 0 0
\(505\) −9118.95 −0.803541
\(506\) −1633.72 −0.143533
\(507\) 4512.49 0.395279
\(508\) −10560.4 −0.922330
\(509\) 7504.97 0.653540 0.326770 0.945104i \(-0.394040\pi\)
0.326770 + 0.945104i \(0.394040\pi\)
\(510\) 498.902 0.0433172
\(511\) 0 0
\(512\) 10605.8 0.915459
\(513\) 4697.82 0.404316
\(514\) 4046.11 0.347210
\(515\) −2038.87 −0.174453
\(516\) 5616.17 0.479144
\(517\) −14863.7 −1.26442
\(518\) 0 0
\(519\) −13579.2 −1.14848
\(520\) 2874.58 0.242420
\(521\) 22264.8 1.87224 0.936120 0.351681i \(-0.114390\pi\)
0.936120 + 0.351681i \(0.114390\pi\)
\(522\) 1194.50 0.100156
\(523\) −11286.7 −0.943655 −0.471828 0.881691i \(-0.656405\pi\)
−0.471828 + 0.881691i \(0.656405\pi\)
\(524\) −13477.7 −1.12362
\(525\) 0 0
\(526\) −2724.27 −0.225824
\(527\) −9533.09 −0.787985
\(528\) −10824.6 −0.892197
\(529\) −9344.40 −0.768012
\(530\) −1586.70 −0.130042
\(531\) 935.316 0.0764393
\(532\) 0 0
\(533\) 15282.1 1.24191
\(534\) 1235.27 0.100104
\(535\) −1850.54 −0.149544
\(536\) −6577.10 −0.530014
\(537\) 3599.57 0.289261
\(538\) −632.781 −0.0507084
\(539\) 0 0
\(540\) 5790.55 0.461455
\(541\) 16406.8 1.30385 0.651924 0.758284i \(-0.273962\pi\)
0.651924 + 0.758284i \(0.273962\pi\)
\(542\) −2504.08 −0.198450
\(543\) 1216.30 0.0961263
\(544\) 4262.88 0.335974
\(545\) 4877.85 0.383384
\(546\) 0 0
\(547\) −8692.48 −0.679458 −0.339729 0.940523i \(-0.610335\pi\)
−0.339729 + 0.940523i \(0.610335\pi\)
\(548\) 15020.8 1.17091
\(549\) 778.139 0.0604921
\(550\) −768.764 −0.0596004
\(551\) 6019.74 0.465426
\(552\) −2234.75 −0.172314
\(553\) 0 0
\(554\) 2330.36 0.178714
\(555\) 7252.15 0.554661
\(556\) 16549.9 1.26236
\(557\) −13935.7 −1.06010 −0.530049 0.847967i \(-0.677826\pi\)
−0.530049 + 0.847967i \(0.677826\pi\)
\(558\) 1574.26 0.119434
\(559\) 10120.9 0.765773
\(560\) 0 0
\(561\) −7383.34 −0.555659
\(562\) 46.8553 0.00351685
\(563\) −22263.6 −1.66661 −0.833304 0.552815i \(-0.813553\pi\)
−0.833304 + 0.552815i \(0.813553\pi\)
\(564\) −9894.82 −0.738736
\(565\) 9894.26 0.736734
\(566\) −1123.65 −0.0834462
\(567\) 0 0
\(568\) −7832.81 −0.578622
\(569\) 9525.66 0.701822 0.350911 0.936409i \(-0.385872\pi\)
0.350911 + 0.936409i \(0.385872\pi\)
\(570\) −415.195 −0.0305099
\(571\) 7669.99 0.562135 0.281068 0.959688i \(-0.409311\pi\)
0.281068 + 0.959688i \(0.409311\pi\)
\(572\) −20703.5 −1.51339
\(573\) 20497.3 1.49439
\(574\) 0 0
\(575\) 1328.20 0.0963304
\(576\) 3402.19 0.246108
\(577\) 3907.15 0.281901 0.140950 0.990017i \(-0.454984\pi\)
0.140950 + 0.990017i \(0.454984\pi\)
\(578\) −2286.13 −0.164516
\(579\) −14858.1 −1.06646
\(580\) 7419.95 0.531201
\(581\) 0 0
\(582\) 3811.38 0.271455
\(583\) 23481.9 1.66813
\(584\) −8335.28 −0.590610
\(585\) 2709.66 0.191505
\(586\) 3216.34 0.226733
\(587\) −19891.6 −1.39866 −0.699331 0.714798i \(-0.746519\pi\)
−0.699331 + 0.714798i \(0.746519\pi\)
\(588\) 0 0
\(589\) 7933.61 0.555007
\(590\) −318.344 −0.0222136
\(591\) −2721.92 −0.189450
\(592\) 18775.8 1.30351
\(593\) 4780.36 0.331038 0.165519 0.986207i \(-0.447070\pi\)
0.165519 + 0.986207i \(0.447070\pi\)
\(594\) 4695.48 0.324340
\(595\) 0 0
\(596\) −5082.74 −0.349324
\(597\) −16978.2 −1.16394
\(598\) −1959.92 −0.134025
\(599\) 11085.6 0.756170 0.378085 0.925771i \(-0.376583\pi\)
0.378085 + 0.925771i \(0.376583\pi\)
\(600\) −1051.58 −0.0715511
\(601\) −1573.44 −0.106792 −0.0533958 0.998573i \(-0.517005\pi\)
−0.0533958 + 0.998573i \(0.517005\pi\)
\(602\) 0 0
\(603\) −6199.76 −0.418696
\(604\) −25399.1 −1.71105
\(605\) 4722.08 0.317322
\(606\) −4922.52 −0.329973
\(607\) 8542.73 0.571233 0.285617 0.958344i \(-0.407802\pi\)
0.285617 + 0.958344i \(0.407802\pi\)
\(608\) −3547.65 −0.236639
\(609\) 0 0
\(610\) −264.848 −0.0175793
\(611\) −17831.4 −1.18066
\(612\) 2655.27 0.175381
\(613\) 15068.2 0.992817 0.496409 0.868089i \(-0.334652\pi\)
0.496409 + 0.868089i \(0.334652\pi\)
\(614\) −1073.09 −0.0705316
\(615\) −5590.52 −0.366555
\(616\) 0 0
\(617\) 2524.58 0.164725 0.0823627 0.996602i \(-0.473753\pi\)
0.0823627 + 0.996602i \(0.473753\pi\)
\(618\) −1100.61 −0.0716391
\(619\) −21238.9 −1.37910 −0.689551 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(620\) 9778.99 0.633441
\(621\) −8112.44 −0.524221
\(622\) −351.735 −0.0226741
\(623\) 0 0
\(624\) −12985.9 −0.833095
\(625\) 625.000 0.0400000
\(626\) −137.674 −0.00879002
\(627\) 6144.55 0.391371
\(628\) 18266.4 1.16068
\(629\) 12806.8 0.811827
\(630\) 0 0
\(631\) −8885.83 −0.560601 −0.280300 0.959912i \(-0.590434\pi\)
−0.280300 + 0.959912i \(0.590434\pi\)
\(632\) 7730.02 0.486525
\(633\) −6571.99 −0.412659
\(634\) −1730.19 −0.108383
\(635\) 6961.92 0.435079
\(636\) 15632.1 0.974609
\(637\) 0 0
\(638\) 6016.74 0.373362
\(639\) −7383.42 −0.457095
\(640\) −5770.41 −0.356399
\(641\) 3655.63 0.225255 0.112628 0.993637i \(-0.464073\pi\)
0.112628 + 0.993637i \(0.464073\pi\)
\(642\) −998.946 −0.0614100
\(643\) 4221.22 0.258894 0.129447 0.991586i \(-0.458680\pi\)
0.129447 + 0.991586i \(0.458680\pi\)
\(644\) 0 0
\(645\) −3702.43 −0.226020
\(646\) −733.205 −0.0446557
\(647\) 103.679 0.00629989 0.00314995 0.999995i \(-0.498997\pi\)
0.00314995 + 0.999995i \(0.498997\pi\)
\(648\) 3854.08 0.233646
\(649\) 4711.24 0.284949
\(650\) −922.259 −0.0556523
\(651\) 0 0
\(652\) 28904.5 1.73618
\(653\) 4476.11 0.268245 0.134122 0.990965i \(-0.457179\pi\)
0.134122 + 0.990965i \(0.457179\pi\)
\(654\) 2633.12 0.157436
\(655\) 8885.09 0.530029
\(656\) −14473.8 −0.861445
\(657\) −7857.07 −0.466566
\(658\) 0 0
\(659\) 12022.0 0.710641 0.355321 0.934745i \(-0.384372\pi\)
0.355321 + 0.934745i \(0.384372\pi\)
\(660\) 7573.79 0.446681
\(661\) 13451.0 0.791504 0.395752 0.918357i \(-0.370484\pi\)
0.395752 + 0.918357i \(0.370484\pi\)
\(662\) −3266.77 −0.191792
\(663\) −8857.54 −0.518851
\(664\) 6160.53 0.360053
\(665\) 0 0
\(666\) −2114.87 −0.123047
\(667\) −10395.2 −0.603454
\(668\) 9155.55 0.530297
\(669\) −19778.2 −1.14300
\(670\) 2110.15 0.121675
\(671\) 3919.53 0.225502
\(672\) 0 0
\(673\) 9774.83 0.559869 0.279935 0.960019i \(-0.409687\pi\)
0.279935 + 0.960019i \(0.409687\pi\)
\(674\) −6029.53 −0.344583
\(675\) −3817.39 −0.217676
\(676\) −8174.27 −0.465081
\(677\) 23857.4 1.35438 0.677190 0.735808i \(-0.263198\pi\)
0.677190 + 0.735808i \(0.263198\pi\)
\(678\) 5341.04 0.302539
\(679\) 0 0
\(680\) −1857.02 −0.104726
\(681\) −15428.7 −0.868179
\(682\) 7929.65 0.445223
\(683\) 11919.4 0.667763 0.333881 0.942615i \(-0.391642\pi\)
0.333881 + 0.942615i \(0.391642\pi\)
\(684\) −2209.77 −0.123527
\(685\) −9902.40 −0.552338
\(686\) 0 0
\(687\) −14054.4 −0.780508
\(688\) −9585.59 −0.531173
\(689\) 28170.5 1.55763
\(690\) 716.981 0.0395580
\(691\) −8203.58 −0.451634 −0.225817 0.974170i \(-0.572505\pi\)
−0.225817 + 0.974170i \(0.572505\pi\)
\(692\) 24598.4 1.35129
\(693\) 0 0
\(694\) 1514.50 0.0828380
\(695\) −10910.4 −0.595477
\(696\) 8230.22 0.448227
\(697\) −9872.44 −0.536507
\(698\) 6751.12 0.366094
\(699\) 9688.49 0.524252
\(700\) 0 0
\(701\) 449.084 0.0241964 0.0120982 0.999927i \(-0.496149\pi\)
0.0120982 + 0.999927i \(0.496149\pi\)
\(702\) 5633.00 0.302855
\(703\) −10658.0 −0.571800
\(704\) 17137.0 0.917438
\(705\) 6523.11 0.348475
\(706\) 4625.11 0.246556
\(707\) 0 0
\(708\) 3136.30 0.166482
\(709\) 1897.64 0.100518 0.0502590 0.998736i \(-0.483995\pi\)
0.0502590 + 0.998736i \(0.483995\pi\)
\(710\) 2513.02 0.132834
\(711\) 7286.53 0.384341
\(712\) −4597.93 −0.242015
\(713\) −13700.2 −0.719601
\(714\) 0 0
\(715\) 13648.7 0.713891
\(716\) −6520.54 −0.340341
\(717\) 3940.60 0.205250
\(718\) −4856.86 −0.252446
\(719\) 6494.70 0.336873 0.168436 0.985713i \(-0.446128\pi\)
0.168436 + 0.985713i \(0.446128\pi\)
\(720\) −2566.35 −0.132836
\(721\) 0 0
\(722\) −3811.45 −0.196465
\(723\) −23606.6 −1.21430
\(724\) −2203.31 −0.113101
\(725\) −4891.57 −0.250577
\(726\) 2549.04 0.130308
\(727\) −18311.2 −0.934148 −0.467074 0.884218i \(-0.654692\pi\)
−0.467074 + 0.884218i \(0.654692\pi\)
\(728\) 0 0
\(729\) 20894.5 1.06155
\(730\) 2674.23 0.135586
\(731\) −6538.23 −0.330814
\(732\) 2609.25 0.131750
\(733\) 21233.3 1.06994 0.534972 0.844870i \(-0.320322\pi\)
0.534972 + 0.844870i \(0.320322\pi\)
\(734\) −3515.30 −0.176774
\(735\) 0 0
\(736\) 6126.27 0.306817
\(737\) −31228.5 −1.56081
\(738\) 1630.30 0.0813175
\(739\) −22023.5 −1.09628 −0.548138 0.836388i \(-0.684663\pi\)
−0.548138 + 0.836388i \(0.684663\pi\)
\(740\) −13137.1 −0.652608
\(741\) 7371.40 0.365446
\(742\) 0 0
\(743\) 9436.77 0.465951 0.232975 0.972483i \(-0.425154\pi\)
0.232975 + 0.972483i \(0.425154\pi\)
\(744\) 10846.9 0.534497
\(745\) 3350.77 0.164782
\(746\) 5306.29 0.260425
\(747\) 5807.09 0.284431
\(748\) 13374.8 0.653783
\(749\) 0 0
\(750\) 337.383 0.0164260
\(751\) −39161.8 −1.90284 −0.951421 0.307893i \(-0.900376\pi\)
−0.951421 + 0.307893i \(0.900376\pi\)
\(752\) 16888.3 0.818954
\(753\) −1531.65 −0.0741255
\(754\) 7218.07 0.348629
\(755\) 16744.2 0.807132
\(756\) 0 0
\(757\) 20340.6 0.976607 0.488303 0.872674i \(-0.337616\pi\)
0.488303 + 0.872674i \(0.337616\pi\)
\(758\) 1076.60 0.0515883
\(759\) −10610.7 −0.507438
\(760\) 1545.44 0.0737621
\(761\) 3307.90 0.157570 0.0787852 0.996892i \(-0.474896\pi\)
0.0787852 + 0.996892i \(0.474896\pi\)
\(762\) 3758.13 0.178665
\(763\) 0 0
\(764\) −37130.4 −1.75829
\(765\) −1750.48 −0.0827303
\(766\) −2075.85 −0.0979160
\(767\) 5651.91 0.266074
\(768\) 8918.37 0.419029
\(769\) 17234.0 0.808159 0.404079 0.914724i \(-0.367592\pi\)
0.404079 + 0.914724i \(0.367592\pi\)
\(770\) 0 0
\(771\) 26278.7 1.22750
\(772\) 26915.1 1.25479
\(773\) −8726.33 −0.406034 −0.203017 0.979175i \(-0.565075\pi\)
−0.203017 + 0.979175i \(0.565075\pi\)
\(774\) 1079.70 0.0501409
\(775\) −6446.75 −0.298806
\(776\) −14186.7 −0.656281
\(777\) 0 0
\(778\) 2270.60 0.104634
\(779\) 8216.03 0.377882
\(780\) 9086.01 0.417091
\(781\) −37190.7 −1.70395
\(782\) 1266.14 0.0578989
\(783\) 29876.8 1.36362
\(784\) 0 0
\(785\) −12042.0 −0.547515
\(786\) 4796.28 0.217656
\(787\) −10642.1 −0.482018 −0.241009 0.970523i \(-0.577478\pi\)
−0.241009 + 0.970523i \(0.577478\pi\)
\(788\) 4930.69 0.222904
\(789\) −17693.6 −0.798365
\(790\) −2480.05 −0.111691
\(791\) 0 0
\(792\) −4538.35 −0.203615
\(793\) 4702.12 0.210564
\(794\) 3516.66 0.157181
\(795\) −10305.4 −0.459740
\(796\) 30755.5 1.36947
\(797\) 29234.1 1.29928 0.649640 0.760242i \(-0.274920\pi\)
0.649640 + 0.760242i \(0.274920\pi\)
\(798\) 0 0
\(799\) 11519.3 0.510044
\(800\) 2882.78 0.127402
\(801\) −4334.14 −0.191185
\(802\) −748.950 −0.0329755
\(803\) −39576.5 −1.73926
\(804\) −20789.0 −0.911906
\(805\) 0 0
\(806\) 9512.93 0.415730
\(807\) −4109.80 −0.179271
\(808\) 18322.7 0.797758
\(809\) 36211.2 1.57369 0.786846 0.617150i \(-0.211713\pi\)
0.786846 + 0.617150i \(0.211713\pi\)
\(810\) −1236.52 −0.0536380
\(811\) −27995.2 −1.21214 −0.606069 0.795412i \(-0.707254\pi\)
−0.606069 + 0.795412i \(0.707254\pi\)
\(812\) 0 0
\(813\) −16263.6 −0.701585
\(814\) −10652.7 −0.458694
\(815\) −19055.2 −0.818985
\(816\) 8389.07 0.359897
\(817\) 5441.23 0.233005
\(818\) 4769.52 0.203866
\(819\) 0 0
\(820\) 10127.1 0.431285
\(821\) 44805.7 1.90467 0.952333 0.305060i \(-0.0986766\pi\)
0.952333 + 0.305060i \(0.0986766\pi\)
\(822\) −5345.44 −0.226817
\(823\) −15850.9 −0.671360 −0.335680 0.941976i \(-0.608966\pi\)
−0.335680 + 0.941976i \(0.608966\pi\)
\(824\) 4096.70 0.173198
\(825\) −4992.99 −0.210707
\(826\) 0 0
\(827\) −45013.9 −1.89273 −0.946363 0.323104i \(-0.895274\pi\)
−0.946363 + 0.323104i \(0.895274\pi\)
\(828\) 3815.94 0.160161
\(829\) 5655.83 0.236954 0.118477 0.992957i \(-0.462199\pi\)
0.118477 + 0.992957i \(0.462199\pi\)
\(830\) −1976.50 −0.0826571
\(831\) 15135.3 0.631812
\(832\) 20558.7 0.856664
\(833\) 0 0
\(834\) −5889.58 −0.244532
\(835\) −6035.75 −0.250151
\(836\) −11130.7 −0.460483
\(837\) 39375.7 1.62607
\(838\) −1375.97 −0.0567211
\(839\) −22037.0 −0.906797 −0.453398 0.891308i \(-0.649789\pi\)
−0.453398 + 0.891308i \(0.649789\pi\)
\(840\) 0 0
\(841\) 13894.9 0.569719
\(842\) −2515.65 −0.102963
\(843\) 304.317 0.0124332
\(844\) 11905.0 0.485530
\(845\) 5388.85 0.219387
\(846\) −1902.27 −0.0773065
\(847\) 0 0
\(848\) −26680.5 −1.08044
\(849\) −7297.91 −0.295010
\(850\) 595.793 0.0240418
\(851\) 18404.8 0.741374
\(852\) −24758.1 −0.995537
\(853\) −44486.7 −1.78569 −0.892846 0.450362i \(-0.851295\pi\)
−0.892846 + 0.450362i \(0.851295\pi\)
\(854\) 0 0
\(855\) 1456.78 0.0582699
\(856\) 3718.29 0.148468
\(857\) −38559.6 −1.53696 −0.768478 0.639877i \(-0.778985\pi\)
−0.768478 + 0.639877i \(0.778985\pi\)
\(858\) 7367.72 0.293158
\(859\) 8426.01 0.334682 0.167341 0.985899i \(-0.446482\pi\)
0.167341 + 0.985899i \(0.446482\pi\)
\(860\) 6706.88 0.265933
\(861\) 0 0
\(862\) −2332.42 −0.0921608
\(863\) −20054.3 −0.791029 −0.395514 0.918460i \(-0.629434\pi\)
−0.395514 + 0.918460i \(0.629434\pi\)
\(864\) −17607.5 −0.693309
\(865\) −16216.4 −0.637427
\(866\) −2721.94 −0.106808
\(867\) −14848.0 −0.581621
\(868\) 0 0
\(869\) 36702.6 1.43274
\(870\) −2640.53 −0.102899
\(871\) −37463.8 −1.45742
\(872\) −9801.04 −0.380625
\(873\) −13372.8 −0.518444
\(874\) −1053.70 −0.0407803
\(875\) 0 0
\(876\) −26346.3 −1.01616
\(877\) 47082.1 1.81283 0.906414 0.422391i \(-0.138809\pi\)
0.906414 + 0.422391i \(0.138809\pi\)
\(878\) −8714.32 −0.334959
\(879\) 20889.5 0.801577
\(880\) −12926.8 −0.495185
\(881\) −9467.24 −0.362042 −0.181021 0.983479i \(-0.557940\pi\)
−0.181021 + 0.983479i \(0.557940\pi\)
\(882\) 0 0
\(883\) −3049.49 −0.116221 −0.0581106 0.998310i \(-0.518508\pi\)
−0.0581106 + 0.998310i \(0.518508\pi\)
\(884\) 16045.2 0.610475
\(885\) −2067.59 −0.0785325
\(886\) 10695.5 0.405554
\(887\) −19468.6 −0.736968 −0.368484 0.929634i \(-0.620123\pi\)
−0.368484 + 0.929634i \(0.620123\pi\)
\(888\) −14571.7 −0.550669
\(889\) 0 0
\(890\) 1475.17 0.0555592
\(891\) 18299.4 0.688052
\(892\) 35827.8 1.34485
\(893\) −9586.61 −0.359243
\(894\) 1808.79 0.0676676
\(895\) 4298.64 0.160545
\(896\) 0 0
\(897\) −12729.3 −0.473823
\(898\) 5385.60 0.200133
\(899\) 50455.6 1.87184
\(900\) 1795.63 0.0665048
\(901\) −18198.5 −0.672898
\(902\) 8211.92 0.303134
\(903\) 0 0
\(904\) −19880.5 −0.731432
\(905\) 1452.52 0.0533518
\(906\) 9038.73 0.331448
\(907\) −32014.2 −1.17201 −0.586006 0.810307i \(-0.699301\pi\)
−0.586006 + 0.810307i \(0.699301\pi\)
\(908\) 27948.8 1.02149
\(909\) 17271.4 0.630206
\(910\) 0 0
\(911\) 20921.1 0.760866 0.380433 0.924809i \(-0.375775\pi\)
0.380433 + 0.924809i \(0.375775\pi\)
\(912\) −6981.54 −0.253489
\(913\) 29250.6 1.06030
\(914\) −825.366 −0.0298694
\(915\) −1720.14 −0.0621487
\(916\) 25459.2 0.918337
\(917\) 0 0
\(918\) −3639.00 −0.130833
\(919\) 43467.4 1.56024 0.780118 0.625632i \(-0.215159\pi\)
0.780118 + 0.625632i \(0.215159\pi\)
\(920\) −2668.75 −0.0956372
\(921\) −6969.53 −0.249353
\(922\) −4503.97 −0.160879
\(923\) −44616.4 −1.59108
\(924\) 0 0
\(925\) 8660.58 0.307847
\(926\) −3601.85 −0.127823
\(927\) 3861.66 0.136822
\(928\) −22562.1 −0.798099
\(929\) −18542.1 −0.654840 −0.327420 0.944879i \(-0.606179\pi\)
−0.327420 + 0.944879i \(0.606179\pi\)
\(930\) −3480.04 −0.122704
\(931\) 0 0
\(932\) −17550.5 −0.616830
\(933\) −2284.46 −0.0801605
\(934\) −6403.82 −0.224346
\(935\) −8817.25 −0.308401
\(936\) −5444.50 −0.190127
\(937\) 13842.5 0.482619 0.241309 0.970448i \(-0.422423\pi\)
0.241309 + 0.970448i \(0.422423\pi\)
\(938\) 0 0
\(939\) −894.167 −0.0310757
\(940\) −11816.5 −0.410012
\(941\) −53393.2 −1.84970 −0.924850 0.380333i \(-0.875809\pi\)
−0.924850 + 0.380333i \(0.875809\pi\)
\(942\) −6500.44 −0.224836
\(943\) −14187.9 −0.489947
\(944\) −5352.98 −0.184560
\(945\) 0 0
\(946\) 5438.52 0.186915
\(947\) 22344.2 0.766727 0.383364 0.923598i \(-0.374766\pi\)
0.383364 + 0.923598i \(0.374766\pi\)
\(948\) 24433.2 0.837081
\(949\) −47478.5 −1.62404
\(950\) −495.830 −0.0169335
\(951\) −11237.3 −0.383169
\(952\) 0 0
\(953\) −8902.63 −0.302607 −0.151304 0.988487i \(-0.548347\pi\)
−0.151304 + 0.988487i \(0.548347\pi\)
\(954\) 3005.25 0.101990
\(955\) 24478.1 0.829416
\(956\) −7138.31 −0.241495
\(957\) 39077.6 1.31996
\(958\) 3267.48 0.110196
\(959\) 0 0
\(960\) −7520.82 −0.252847
\(961\) 36706.0 1.23212
\(962\) −12779.7 −0.428309
\(963\) 3504.96 0.117285
\(964\) 42762.8 1.42873
\(965\) −17743.6 −0.591905
\(966\) 0 0
\(967\) −2225.57 −0.0740119 −0.0370059 0.999315i \(-0.511782\pi\)
−0.0370059 + 0.999315i \(0.511782\pi\)
\(968\) −9488.05 −0.315039
\(969\) −4762.03 −0.157873
\(970\) 4551.58 0.150662
\(971\) −11324.8 −0.374285 −0.187143 0.982333i \(-0.559923\pi\)
−0.187143 + 0.982333i \(0.559923\pi\)
\(972\) −19086.9 −0.629849
\(973\) 0 0
\(974\) 170.415 0.00560621
\(975\) −5989.91 −0.196749
\(976\) −4453.43 −0.146056
\(977\) −57695.9 −1.88931 −0.944654 0.328067i \(-0.893603\pi\)
−0.944654 + 0.328067i \(0.893603\pi\)
\(978\) −10286.2 −0.336315
\(979\) −21831.3 −0.712697
\(980\) 0 0
\(981\) −9238.73 −0.300683
\(982\) −4808.73 −0.156266
\(983\) 59657.3 1.93568 0.967839 0.251570i \(-0.0809469\pi\)
0.967839 + 0.251570i \(0.0809469\pi\)
\(984\) 11233.0 0.363917
\(985\) −3250.54 −0.105148
\(986\) −4662.98 −0.150608
\(987\) 0 0
\(988\) −13353.1 −0.429980
\(989\) −9396.20 −0.302105
\(990\) 1456.05 0.0467438
\(991\) 2890.60 0.0926568 0.0463284 0.998926i \(-0.485248\pi\)
0.0463284 + 0.998926i \(0.485248\pi\)
\(992\) −29735.3 −0.951710
\(993\) −21217.1 −0.678050
\(994\) 0 0
\(995\) −20275.5 −0.646005
\(996\) 19472.3 0.619481
\(997\) −10619.6 −0.337337 −0.168669 0.985673i \(-0.553947\pi\)
−0.168669 + 0.985673i \(0.553947\pi\)
\(998\) 4645.77 0.147354
\(999\) −52897.3 −1.67527
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 245.4.a.o.1.4 6
3.2 odd 2 2205.4.a.bz.1.3 6
5.4 even 2 1225.4.a.bj.1.3 6
7.2 even 3 245.4.e.q.116.3 12
7.3 odd 6 245.4.e.p.226.3 12
7.4 even 3 245.4.e.q.226.3 12
7.5 odd 6 245.4.e.p.116.3 12
7.6 odd 2 245.4.a.p.1.4 yes 6
21.20 even 2 2205.4.a.ca.1.3 6
35.34 odd 2 1225.4.a.bi.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.4 6 1.1 even 1 trivial
245.4.a.p.1.4 yes 6 7.6 odd 2
245.4.e.p.116.3 12 7.5 odd 6
245.4.e.p.226.3 12 7.3 odd 6
245.4.e.q.116.3 12 7.2 even 3
245.4.e.q.226.3 12 7.4 even 3
1225.4.a.bi.1.3 6 35.34 odd 2
1225.4.a.bj.1.3 6 5.4 even 2
2205.4.a.bz.1.3 6 3.2 odd 2
2205.4.a.ca.1.3 6 21.20 even 2