Properties

Label 1849.4.a.h.1.12
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.40131 q^{2} +7.71122 q^{3} -6.03633 q^{4} +16.2919 q^{5} -10.8058 q^{6} +16.8292 q^{7} +19.6692 q^{8} +32.4629 q^{9} +O(q^{10})\) \(q-1.40131 q^{2} +7.71122 q^{3} -6.03633 q^{4} +16.2919 q^{5} -10.8058 q^{6} +16.8292 q^{7} +19.6692 q^{8} +32.4629 q^{9} -22.8300 q^{10} +16.6284 q^{11} -46.5475 q^{12} -80.3768 q^{13} -23.5829 q^{14} +125.631 q^{15} +20.7280 q^{16} -86.4572 q^{17} -45.4906 q^{18} +134.427 q^{19} -98.3436 q^{20} +129.774 q^{21} -23.3016 q^{22} +92.8088 q^{23} +151.674 q^{24} +140.427 q^{25} +112.633 q^{26} +42.1260 q^{27} -101.587 q^{28} +49.5793 q^{29} -176.048 q^{30} -59.7856 q^{31} -186.400 q^{32} +128.226 q^{33} +121.153 q^{34} +274.180 q^{35} -195.957 q^{36} -128.973 q^{37} -188.373 q^{38} -619.803 q^{39} +320.450 q^{40} +244.655 q^{41} -181.853 q^{42} -100.375 q^{44} +528.885 q^{45} -130.054 q^{46} +77.7808 q^{47} +159.838 q^{48} -59.7780 q^{49} -196.782 q^{50} -666.691 q^{51} +485.181 q^{52} +444.048 q^{53} -59.0315 q^{54} +270.910 q^{55} +331.017 q^{56} +1036.59 q^{57} -69.4759 q^{58} +539.288 q^{59} -758.350 q^{60} +627.205 q^{61} +83.7781 q^{62} +546.325 q^{63} +95.3801 q^{64} -1309.49 q^{65} -179.684 q^{66} +546.765 q^{67} +521.885 q^{68} +715.670 q^{69} -384.211 q^{70} +226.963 q^{71} +638.521 q^{72} +736.342 q^{73} +180.730 q^{74} +1082.87 q^{75} -811.444 q^{76} +279.843 q^{77} +868.536 q^{78} +440.716 q^{79} +337.700 q^{80} -551.657 q^{81} -342.838 q^{82} -85.4566 q^{83} -783.357 q^{84} -1408.56 q^{85} +382.317 q^{87} +327.069 q^{88} +882.327 q^{89} -741.130 q^{90} -1352.68 q^{91} -560.225 q^{92} -461.020 q^{93} -108.995 q^{94} +2190.07 q^{95} -1437.37 q^{96} -6.99165 q^{97} +83.7674 q^{98} +539.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 6 q^{2} + 2 q^{3} + 114 q^{4} + 27 q^{5} + 8 q^{6} + 48 q^{7} + 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} - 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} + 80 q^{18} + 254 q^{19} + 312 q^{20} - 548 q^{21} + 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} + 549 q^{26} - 10 q^{27} + 578 q^{28} + 793 q^{29} + 1560 q^{30} - 359 q^{31} + 676 q^{32} + 208 q^{33} + 1007 q^{34} - 514 q^{35} + 776 q^{36} + 510 q^{37} - 2066 q^{38} + 898 q^{39} - 1248 q^{40} - 270 q^{41} - 915 q^{42} + 3256 q^{44} + 807 q^{45} + 1960 q^{46} + 1421 q^{47} - 632 q^{48} + 386 q^{49} - 141 q^{50} + 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} + 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} + 1759 q^{61} + 395 q^{62} + 2204 q^{63} + 222 q^{64} + 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} + 1660 q^{69} + 1597 q^{70} + 727 q^{71} + 9100 q^{72} + 4623 q^{73} - 2649 q^{74} + 1027 q^{75} + 874 q^{76} + 3556 q^{77} - 4979 q^{78} + 546 q^{79} + 5809 q^{80} - 410 q^{81} - 4397 q^{82} - 492 q^{83} - 10611 q^{84} - 1723 q^{85} + 5937 q^{87} + 3974 q^{88} + 5218 q^{89} + 10492 q^{90} + 1104 q^{91} + 1060 q^{92} + 1997 q^{93} - 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} + 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40131 −0.495437 −0.247719 0.968832i \(-0.579681\pi\)
−0.247719 + 0.968832i \(0.579681\pi\)
\(3\) 7.71122 1.48403 0.742013 0.670386i \(-0.233871\pi\)
0.742013 + 0.670386i \(0.233871\pi\)
\(4\) −6.03633 −0.754542
\(5\) 16.2919 1.45720 0.728598 0.684942i \(-0.240172\pi\)
0.728598 + 0.684942i \(0.240172\pi\)
\(6\) −10.8058 −0.735242
\(7\) 16.8292 0.908691 0.454346 0.890825i \(-0.349873\pi\)
0.454346 + 0.890825i \(0.349873\pi\)
\(8\) 19.6692 0.869266
\(9\) 32.4629 1.20233
\(10\) −22.8300 −0.721949
\(11\) 16.6284 0.455787 0.227894 0.973686i \(-0.426816\pi\)
0.227894 + 0.973686i \(0.426816\pi\)
\(12\) −46.5475 −1.11976
\(13\) −80.3768 −1.71481 −0.857404 0.514643i \(-0.827924\pi\)
−0.857404 + 0.514643i \(0.827924\pi\)
\(14\) −23.5829 −0.450200
\(15\) 125.631 2.16252
\(16\) 20.7280 0.323875
\(17\) −86.4572 −1.23347 −0.616734 0.787171i \(-0.711545\pi\)
−0.616734 + 0.787171i \(0.711545\pi\)
\(18\) −45.4906 −0.595680
\(19\) 134.427 1.62314 0.811568 0.584258i \(-0.198614\pi\)
0.811568 + 0.584258i \(0.198614\pi\)
\(20\) −98.3436 −1.09952
\(21\) 129.774 1.34852
\(22\) −23.3016 −0.225814
\(23\) 92.8088 0.841391 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(24\) 151.674 1.29001
\(25\) 140.427 1.12342
\(26\) 112.633 0.849580
\(27\) 42.1260 0.300265
\(28\) −101.587 −0.685646
\(29\) 49.5793 0.317471 0.158735 0.987321i \(-0.449258\pi\)
0.158735 + 0.987321i \(0.449258\pi\)
\(30\) −176.048 −1.07139
\(31\) −59.7856 −0.346381 −0.173191 0.984888i \(-0.555408\pi\)
−0.173191 + 0.984888i \(0.555408\pi\)
\(32\) −186.400 −1.02973
\(33\) 128.226 0.676400
\(34\) 121.153 0.611106
\(35\) 274.180 1.32414
\(36\) −195.957 −0.907209
\(37\) −128.973 −0.573053 −0.286526 0.958072i \(-0.592501\pi\)
−0.286526 + 0.958072i \(0.592501\pi\)
\(38\) −188.373 −0.804162
\(39\) −619.803 −2.54482
\(40\) 320.450 1.26669
\(41\) 244.655 0.931920 0.465960 0.884806i \(-0.345709\pi\)
0.465960 + 0.884806i \(0.345709\pi\)
\(42\) −181.853 −0.668108
\(43\) 0 0
\(44\) −100.375 −0.343911
\(45\) 528.885 1.75203
\(46\) −130.054 −0.416856
\(47\) 77.7808 0.241394 0.120697 0.992689i \(-0.461487\pi\)
0.120697 + 0.992689i \(0.461487\pi\)
\(48\) 159.838 0.480639
\(49\) −59.7780 −0.174280
\(50\) −196.782 −0.556584
\(51\) −666.691 −1.83050
\(52\) 485.181 1.29390
\(53\) 444.048 1.15084 0.575422 0.817857i \(-0.304838\pi\)
0.575422 + 0.817857i \(0.304838\pi\)
\(54\) −59.0315 −0.148762
\(55\) 270.910 0.664172
\(56\) 331.017 0.789894
\(57\) 1036.59 2.40878
\(58\) −69.4759 −0.157287
\(59\) 539.288 1.18999 0.594995 0.803730i \(-0.297154\pi\)
0.594995 + 0.803730i \(0.297154\pi\)
\(60\) −758.350 −1.63171
\(61\) 627.205 1.31648 0.658241 0.752807i \(-0.271301\pi\)
0.658241 + 0.752807i \(0.271301\pi\)
\(62\) 83.7781 0.171610
\(63\) 546.325 1.09255
\(64\) 95.3801 0.186289
\(65\) −1309.49 −2.49881
\(66\) −179.684 −0.335114
\(67\) 546.765 0.996985 0.498493 0.866894i \(-0.333887\pi\)
0.498493 + 0.866894i \(0.333887\pi\)
\(68\) 521.885 0.930703
\(69\) 715.670 1.24864
\(70\) −384.211 −0.656029
\(71\) 226.963 0.379374 0.189687 0.981845i \(-0.439253\pi\)
0.189687 + 0.981845i \(0.439253\pi\)
\(72\) 638.521 1.04515
\(73\) 736.342 1.18058 0.590290 0.807191i \(-0.299013\pi\)
0.590290 + 0.807191i \(0.299013\pi\)
\(74\) 180.730 0.283912
\(75\) 1082.87 1.66718
\(76\) −811.444 −1.22472
\(77\) 279.843 0.414170
\(78\) 868.536 1.26080
\(79\) 440.716 0.627651 0.313826 0.949481i \(-0.398389\pi\)
0.313826 + 0.949481i \(0.398389\pi\)
\(80\) 337.700 0.471950
\(81\) −551.657 −0.756731
\(82\) −342.838 −0.461708
\(83\) −85.4566 −0.113013 −0.0565065 0.998402i \(-0.517996\pi\)
−0.0565065 + 0.998402i \(0.517996\pi\)
\(84\) −783.357 −1.01752
\(85\) −1408.56 −1.79740
\(86\) 0 0
\(87\) 382.317 0.471134
\(88\) 327.069 0.396200
\(89\) 882.327 1.05086 0.525429 0.850837i \(-0.323905\pi\)
0.525429 + 0.850837i \(0.323905\pi\)
\(90\) −741.130 −0.868022
\(91\) −1352.68 −1.55823
\(92\) −560.225 −0.634864
\(93\) −461.020 −0.514038
\(94\) −108.995 −0.119595
\(95\) 2190.07 2.36523
\(96\) −1437.37 −1.52814
\(97\) −6.99165 −0.00731850 −0.00365925 0.999993i \(-0.501165\pi\)
−0.00365925 + 0.999993i \(0.501165\pi\)
\(98\) 83.7674 0.0863448
\(99\) 539.808 0.548008
\(100\) −847.667 −0.847667
\(101\) 757.748 0.746522 0.373261 0.927726i \(-0.378240\pi\)
0.373261 + 0.927726i \(0.378240\pi\)
\(102\) 934.240 0.906897
\(103\) 717.838 0.686705 0.343353 0.939207i \(-0.388437\pi\)
0.343353 + 0.939207i \(0.388437\pi\)
\(104\) −1580.95 −1.49062
\(105\) 2114.27 1.96506
\(106\) −622.248 −0.570171
\(107\) 0.557731 0.000503905 0 0.000251953 1.00000i \(-0.499920\pi\)
0.000251953 1.00000i \(0.499920\pi\)
\(108\) −254.287 −0.226562
\(109\) −1205.08 −1.05895 −0.529477 0.848324i \(-0.677612\pi\)
−0.529477 + 0.848324i \(0.677612\pi\)
\(110\) −379.628 −0.329055
\(111\) −994.536 −0.850425
\(112\) 348.836 0.294303
\(113\) −1863.95 −1.55173 −0.775864 0.630900i \(-0.782686\pi\)
−0.775864 + 0.630900i \(0.782686\pi\)
\(114\) −1452.59 −1.19340
\(115\) 1512.04 1.22607
\(116\) −299.277 −0.239545
\(117\) −2609.27 −2.06177
\(118\) −755.709 −0.589565
\(119\) −1455.01 −1.12084
\(120\) 2471.06 1.87980
\(121\) −1054.50 −0.792258
\(122\) −878.908 −0.652235
\(123\) 1886.59 1.38299
\(124\) 360.886 0.261359
\(125\) 251.344 0.179847
\(126\) −765.570 −0.541289
\(127\) −2481.26 −1.73367 −0.866834 0.498597i \(-0.833849\pi\)
−0.866834 + 0.498597i \(0.833849\pi\)
\(128\) 1357.54 0.937431
\(129\) 0 0
\(130\) 1835.01 1.23801
\(131\) 962.683 0.642061 0.321031 0.947069i \(-0.395971\pi\)
0.321031 + 0.947069i \(0.395971\pi\)
\(132\) −774.012 −0.510372
\(133\) 2262.29 1.47493
\(134\) −766.187 −0.493944
\(135\) 686.314 0.437545
\(136\) −1700.55 −1.07221
\(137\) −78.1818 −0.0487556 −0.0243778 0.999703i \(-0.507760\pi\)
−0.0243778 + 0.999703i \(0.507760\pi\)
\(138\) −1002.87 −0.618625
\(139\) 2040.84 1.24534 0.622668 0.782486i \(-0.286049\pi\)
0.622668 + 0.782486i \(0.286049\pi\)
\(140\) −1655.04 −0.999120
\(141\) 599.785 0.358234
\(142\) −318.045 −0.187956
\(143\) −1336.54 −0.781588
\(144\) 672.892 0.389405
\(145\) 807.743 0.462617
\(146\) −1031.84 −0.584904
\(147\) −460.962 −0.258636
\(148\) 778.521 0.432392
\(149\) 336.571 0.185053 0.0925267 0.995710i \(-0.470506\pi\)
0.0925267 + 0.995710i \(0.470506\pi\)
\(150\) −1517.43 −0.825985
\(151\) 1352.41 0.728856 0.364428 0.931232i \(-0.381265\pi\)
0.364428 + 0.931232i \(0.381265\pi\)
\(152\) 2644.07 1.41094
\(153\) −2806.66 −1.48304
\(154\) −392.147 −0.205195
\(155\) −974.024 −0.504745
\(156\) 3741.34 1.92017
\(157\) 886.998 0.450893 0.225446 0.974256i \(-0.427616\pi\)
0.225446 + 0.974256i \(0.427616\pi\)
\(158\) −617.579 −0.310962
\(159\) 3424.15 1.70788
\(160\) −3036.82 −1.50051
\(161\) 1561.90 0.764564
\(162\) 773.041 0.374913
\(163\) −218.070 −0.104789 −0.0523944 0.998626i \(-0.516685\pi\)
−0.0523944 + 0.998626i \(0.516685\pi\)
\(164\) −1476.82 −0.703173
\(165\) 2089.04 0.985647
\(166\) 119.751 0.0559908
\(167\) −2766.96 −1.28212 −0.641058 0.767492i \(-0.721504\pi\)
−0.641058 + 0.767492i \(0.721504\pi\)
\(168\) 2552.55 1.17222
\(169\) 4263.43 1.94057
\(170\) 1973.82 0.890502
\(171\) 4363.89 1.95155
\(172\) 0 0
\(173\) −2722.10 −1.19628 −0.598142 0.801390i \(-0.704094\pi\)
−0.598142 + 0.801390i \(0.704094\pi\)
\(174\) −535.744 −0.233418
\(175\) 2363.28 1.02084
\(176\) 344.674 0.147618
\(177\) 4158.57 1.76597
\(178\) −1236.41 −0.520635
\(179\) −4376.37 −1.82741 −0.913703 0.406383i \(-0.866790\pi\)
−0.913703 + 0.406383i \(0.866790\pi\)
\(180\) −3192.52 −1.32198
\(181\) −1853.95 −0.761342 −0.380671 0.924710i \(-0.624307\pi\)
−0.380671 + 0.924710i \(0.624307\pi\)
\(182\) 1895.52 0.772006
\(183\) 4836.52 1.95369
\(184\) 1825.48 0.731392
\(185\) −2101.21 −0.835050
\(186\) 646.032 0.254674
\(187\) −1437.65 −0.562199
\(188\) −469.511 −0.182142
\(189\) 708.947 0.272848
\(190\) −3068.97 −1.17182
\(191\) −1136.38 −0.430499 −0.215249 0.976559i \(-0.569056\pi\)
−0.215249 + 0.976559i \(0.569056\pi\)
\(192\) 735.497 0.276458
\(193\) −2739.66 −1.02179 −0.510895 0.859643i \(-0.670686\pi\)
−0.510895 + 0.859643i \(0.670686\pi\)
\(194\) 9.79745 0.00362586
\(195\) −10097.8 −3.70830
\(196\) 360.840 0.131502
\(197\) 2862.90 1.03540 0.517699 0.855563i \(-0.326789\pi\)
0.517699 + 0.855563i \(0.326789\pi\)
\(198\) −756.438 −0.271503
\(199\) −15.5940 −0.00555491 −0.00277745 0.999996i \(-0.500884\pi\)
−0.00277745 + 0.999996i \(0.500884\pi\)
\(200\) 2762.10 0.976550
\(201\) 4216.23 1.47955
\(202\) −1061.84 −0.369855
\(203\) 834.380 0.288483
\(204\) 4024.37 1.38119
\(205\) 3985.91 1.35799
\(206\) −1005.91 −0.340219
\(207\) 3012.85 1.01163
\(208\) −1666.05 −0.555384
\(209\) 2235.31 0.739805
\(210\) −2962.74 −0.973564
\(211\) 4255.88 1.38856 0.694281 0.719704i \(-0.255722\pi\)
0.694281 + 0.719704i \(0.255722\pi\)
\(212\) −2680.42 −0.868359
\(213\) 1750.16 0.563000
\(214\) −0.781553 −0.000249653 0
\(215\) 0 0
\(216\) 828.586 0.261010
\(217\) −1006.14 −0.314754
\(218\) 1688.69 0.524646
\(219\) 5678.10 1.75201
\(220\) −1635.30 −0.501145
\(221\) 6949.16 2.11516
\(222\) 1393.65 0.421332
\(223\) 539.062 0.161876 0.0809378 0.996719i \(-0.474208\pi\)
0.0809378 + 0.996719i \(0.474208\pi\)
\(224\) −3136.97 −0.935703
\(225\) 4558.69 1.35072
\(226\) 2611.96 0.768784
\(227\) −5992.14 −1.75204 −0.876018 0.482278i \(-0.839810\pi\)
−0.876018 + 0.482278i \(0.839810\pi\)
\(228\) −6257.23 −1.81752
\(229\) 1856.30 0.535668 0.267834 0.963465i \(-0.413692\pi\)
0.267834 + 0.963465i \(0.413692\pi\)
\(230\) −2118.83 −0.607441
\(231\) 2157.93 0.614639
\(232\) 975.187 0.275966
\(233\) −2328.11 −0.654590 −0.327295 0.944922i \(-0.606137\pi\)
−0.327295 + 0.944922i \(0.606137\pi\)
\(234\) 3656.39 1.02148
\(235\) 1267.20 0.351758
\(236\) −3255.32 −0.897897
\(237\) 3398.46 0.931450
\(238\) 2038.91 0.555307
\(239\) −61.5635 −0.0166620 −0.00833098 0.999965i \(-0.502652\pi\)
−0.00833098 + 0.999965i \(0.502652\pi\)
\(240\) 2604.08 0.700385
\(241\) 2036.64 0.544364 0.272182 0.962246i \(-0.412255\pi\)
0.272182 + 0.962246i \(0.412255\pi\)
\(242\) 1477.67 0.392514
\(243\) −5391.35 −1.42327
\(244\) −3786.02 −0.993341
\(245\) −973.900 −0.253960
\(246\) −2643.70 −0.685187
\(247\) −10804.8 −2.78337
\(248\) −1175.94 −0.301097
\(249\) −658.975 −0.167714
\(250\) −352.210 −0.0891028
\(251\) 3260.59 0.819946 0.409973 0.912098i \(-0.365538\pi\)
0.409973 + 0.912098i \(0.365538\pi\)
\(252\) −3297.80 −0.824373
\(253\) 1543.27 0.383495
\(254\) 3477.00 0.858924
\(255\) −10861.7 −2.66739
\(256\) −2665.38 −0.650728
\(257\) −2352.03 −0.570878 −0.285439 0.958397i \(-0.592139\pi\)
−0.285439 + 0.958397i \(0.592139\pi\)
\(258\) 0 0
\(259\) −2170.50 −0.520728
\(260\) 7904.55 1.88546
\(261\) 1609.49 0.381705
\(262\) −1349.02 −0.318101
\(263\) 1854.77 0.434866 0.217433 0.976075i \(-0.430232\pi\)
0.217433 + 0.976075i \(0.430232\pi\)
\(264\) 2522.10 0.587971
\(265\) 7234.41 1.67700
\(266\) −3170.17 −0.730735
\(267\) 6803.82 1.55950
\(268\) −3300.46 −0.752267
\(269\) 7378.17 1.67232 0.836162 0.548483i \(-0.184794\pi\)
0.836162 + 0.548483i \(0.184794\pi\)
\(270\) −961.738 −0.216776
\(271\) 239.453 0.0536743 0.0268371 0.999640i \(-0.491456\pi\)
0.0268371 + 0.999640i \(0.491456\pi\)
\(272\) −1792.09 −0.399490
\(273\) −10430.8 −2.31246
\(274\) 109.557 0.0241554
\(275\) 2335.09 0.512041
\(276\) −4320.02 −0.942155
\(277\) 4600.42 0.997879 0.498940 0.866637i \(-0.333723\pi\)
0.498940 + 0.866637i \(0.333723\pi\)
\(278\) −2859.84 −0.616986
\(279\) −1940.82 −0.416465
\(280\) 5392.92 1.15103
\(281\) 3363.99 0.714160 0.357080 0.934074i \(-0.383772\pi\)
0.357080 + 0.934074i \(0.383772\pi\)
\(282\) −840.484 −0.177483
\(283\) 2786.90 0.585385 0.292693 0.956207i \(-0.405449\pi\)
0.292693 + 0.956207i \(0.405449\pi\)
\(284\) −1370.02 −0.286253
\(285\) 16888.1 3.51006
\(286\) 1872.91 0.387228
\(287\) 4117.35 0.846828
\(288\) −6051.10 −1.23807
\(289\) 2561.86 0.521444
\(290\) −1131.90 −0.229198
\(291\) −53.9141 −0.0108608
\(292\) −4444.81 −0.890797
\(293\) 1548.87 0.308826 0.154413 0.988006i \(-0.450651\pi\)
0.154413 + 0.988006i \(0.450651\pi\)
\(294\) 645.949 0.128138
\(295\) 8786.06 1.73405
\(296\) −2536.79 −0.498135
\(297\) 700.489 0.136857
\(298\) −471.640 −0.0916824
\(299\) −7459.68 −1.44282
\(300\) −6536.55 −1.25796
\(301\) 0 0
\(302\) −1895.14 −0.361102
\(303\) 5843.16 1.10786
\(304\) 2786.40 0.525694
\(305\) 10218.4 1.91837
\(306\) 3932.99 0.734752
\(307\) 702.537 0.130605 0.0653027 0.997865i \(-0.479199\pi\)
0.0653027 + 0.997865i \(0.479199\pi\)
\(308\) −1689.23 −0.312509
\(309\) 5535.41 1.01909
\(310\) 1364.91 0.250070
\(311\) −5078.98 −0.926054 −0.463027 0.886344i \(-0.653237\pi\)
−0.463027 + 0.886344i \(0.653237\pi\)
\(312\) −12191.1 −2.21212
\(313\) −5191.94 −0.937590 −0.468795 0.883307i \(-0.655312\pi\)
−0.468795 + 0.883307i \(0.655312\pi\)
\(314\) −1242.96 −0.223389
\(315\) 8900.70 1.59206
\(316\) −2660.31 −0.473589
\(317\) 369.900 0.0655384 0.0327692 0.999463i \(-0.489567\pi\)
0.0327692 + 0.999463i \(0.489567\pi\)
\(318\) −4798.29 −0.846148
\(319\) 824.426 0.144699
\(320\) 1553.93 0.271460
\(321\) 4.30078 0.000747808 0
\(322\) −2188.70 −0.378794
\(323\) −11622.2 −2.00209
\(324\) 3329.98 0.570985
\(325\) −11287.1 −1.92645
\(326\) 305.584 0.0519163
\(327\) −9292.67 −1.57152
\(328\) 4812.18 0.810086
\(329\) 1308.99 0.219352
\(330\) −2927.39 −0.488327
\(331\) −5643.44 −0.937135 −0.468567 0.883428i \(-0.655230\pi\)
−0.468567 + 0.883428i \(0.655230\pi\)
\(332\) 515.844 0.0852730
\(333\) −4186.83 −0.688999
\(334\) 3877.36 0.635208
\(335\) 8907.87 1.45280
\(336\) 2689.95 0.436752
\(337\) 11132.1 1.79942 0.899711 0.436486i \(-0.143777\pi\)
0.899711 + 0.436486i \(0.143777\pi\)
\(338\) −5974.38 −0.961431
\(339\) −14373.3 −2.30280
\(340\) 8502.52 1.35622
\(341\) −994.141 −0.157876
\(342\) −6115.15 −0.966870
\(343\) −6778.43 −1.06706
\(344\) 0 0
\(345\) 11659.6 1.81952
\(346\) 3814.50 0.592684
\(347\) −11505.4 −1.77995 −0.889973 0.456014i \(-0.849277\pi\)
−0.889973 + 0.456014i \(0.849277\pi\)
\(348\) −2307.79 −0.355491
\(349\) 2860.19 0.438689 0.219344 0.975648i \(-0.429608\pi\)
0.219344 + 0.975648i \(0.429608\pi\)
\(350\) −3311.69 −0.505763
\(351\) −3385.95 −0.514897
\(352\) −3099.54 −0.469336
\(353\) −12054.9 −1.81761 −0.908805 0.417222i \(-0.863004\pi\)
−0.908805 + 0.417222i \(0.863004\pi\)
\(354\) −5827.44 −0.874930
\(355\) 3697.67 0.552822
\(356\) −5326.02 −0.792917
\(357\) −11219.9 −1.66336
\(358\) 6132.65 0.905365
\(359\) −1496.02 −0.219935 −0.109968 0.993935i \(-0.535075\pi\)
−0.109968 + 0.993935i \(0.535075\pi\)
\(360\) 10402.8 1.52298
\(361\) 11211.5 1.63457
\(362\) 2597.95 0.377197
\(363\) −8131.45 −1.17573
\(364\) 8165.21 1.17575
\(365\) 11996.5 1.72034
\(366\) −6777.46 −0.967933
\(367\) −7457.74 −1.06074 −0.530369 0.847767i \(-0.677947\pi\)
−0.530369 + 0.847767i \(0.677947\pi\)
\(368\) 1923.74 0.272505
\(369\) 7942.23 1.12048
\(370\) 2944.45 0.413715
\(371\) 7472.97 1.04576
\(372\) 2782.87 0.387864
\(373\) 7188.19 0.997830 0.498915 0.866651i \(-0.333732\pi\)
0.498915 + 0.866651i \(0.333732\pi\)
\(374\) 2014.59 0.278535
\(375\) 1938.17 0.266897
\(376\) 1529.89 0.209835
\(377\) −3985.03 −0.544401
\(378\) −993.453 −0.135179
\(379\) 8220.25 1.11411 0.557053 0.830477i \(-0.311932\pi\)
0.557053 + 0.830477i \(0.311932\pi\)
\(380\) −13220.0 −1.78466
\(381\) −19133.5 −2.57281
\(382\) 1592.41 0.213285
\(383\) −6162.57 −0.822174 −0.411087 0.911596i \(-0.634851\pi\)
−0.411087 + 0.911596i \(0.634851\pi\)
\(384\) 10468.3 1.39117
\(385\) 4559.19 0.603527
\(386\) 3839.11 0.506233
\(387\) 0 0
\(388\) 42.2039 0.00552211
\(389\) −4726.05 −0.615990 −0.307995 0.951388i \(-0.599658\pi\)
−0.307995 + 0.951388i \(0.599658\pi\)
\(390\) 14150.1 1.83723
\(391\) −8024.00 −1.03783
\(392\) −1175.79 −0.151496
\(393\) 7423.46 0.952835
\(394\) −4011.81 −0.512975
\(395\) 7180.12 0.914611
\(396\) −3258.46 −0.413495
\(397\) 12143.2 1.53513 0.767567 0.640968i \(-0.221467\pi\)
0.767567 + 0.640968i \(0.221467\pi\)
\(398\) 21.8519 0.00275211
\(399\) 17445.0 2.18883
\(400\) 2910.78 0.363848
\(401\) −4215.81 −0.525006 −0.262503 0.964931i \(-0.584548\pi\)
−0.262503 + 0.964931i \(0.584548\pi\)
\(402\) −5908.24 −0.733025
\(403\) 4805.38 0.593978
\(404\) −4574.02 −0.563282
\(405\) −8987.56 −1.10270
\(406\) −1169.22 −0.142925
\(407\) −2144.61 −0.261190
\(408\) −13113.3 −1.59119
\(409\) 239.924 0.0290061 0.0145030 0.999895i \(-0.495383\pi\)
0.0145030 + 0.999895i \(0.495383\pi\)
\(410\) −5585.49 −0.672799
\(411\) −602.877 −0.0723546
\(412\) −4333.11 −0.518148
\(413\) 9075.79 1.08133
\(414\) −4221.93 −0.501199
\(415\) −1392.25 −0.164682
\(416\) 14982.3 1.76578
\(417\) 15737.4 1.84811
\(418\) −3132.35 −0.366527
\(419\) −31.4797 −0.00367037 −0.00183518 0.999998i \(-0.500584\pi\)
−0.00183518 + 0.999998i \(0.500584\pi\)
\(420\) −12762.4 −1.48272
\(421\) 940.763 0.108907 0.0544537 0.998516i \(-0.482658\pi\)
0.0544537 + 0.998516i \(0.482658\pi\)
\(422\) −5963.80 −0.687946
\(423\) 2524.99 0.290235
\(424\) 8734.08 1.00039
\(425\) −12141.0 −1.38570
\(426\) −2452.52 −0.278931
\(427\) 10555.4 1.19628
\(428\) −3.36665 −0.000380217 0
\(429\) −10306.4 −1.15990
\(430\) 0 0
\(431\) 2818.53 0.314997 0.157499 0.987519i \(-0.449657\pi\)
0.157499 + 0.987519i \(0.449657\pi\)
\(432\) 873.188 0.0972483
\(433\) 14883.7 1.65188 0.825942 0.563756i \(-0.190644\pi\)
0.825942 + 0.563756i \(0.190644\pi\)
\(434\) 1409.92 0.155941
\(435\) 6228.69 0.686535
\(436\) 7274.29 0.799026
\(437\) 12476.0 1.36569
\(438\) −7956.77 −0.868012
\(439\) 3638.21 0.395540 0.197770 0.980248i \(-0.436630\pi\)
0.197770 + 0.980248i \(0.436630\pi\)
\(440\) 5328.58 0.577341
\(441\) −1940.57 −0.209542
\(442\) −9737.91 −1.04793
\(443\) 5735.65 0.615144 0.307572 0.951525i \(-0.400484\pi\)
0.307572 + 0.951525i \(0.400484\pi\)
\(444\) 6003.35 0.641681
\(445\) 14374.8 1.53131
\(446\) −755.392 −0.0801992
\(447\) 2595.37 0.274624
\(448\) 1605.17 0.169279
\(449\) 13651.3 1.43484 0.717421 0.696640i \(-0.245322\pi\)
0.717421 + 0.696640i \(0.245322\pi\)
\(450\) −6388.13 −0.669199
\(451\) 4068.23 0.424758
\(452\) 11251.4 1.17084
\(453\) 10428.7 1.08164
\(454\) 8396.84 0.868025
\(455\) −22037.7 −2.27065
\(456\) 20389.0 2.09387
\(457\) −18054.8 −1.84807 −0.924037 0.382303i \(-0.875131\pi\)
−0.924037 + 0.382303i \(0.875131\pi\)
\(458\) −2601.25 −0.265390
\(459\) −3642.10 −0.370367
\(460\) −9127.16 −0.925122
\(461\) −7748.85 −0.782863 −0.391432 0.920207i \(-0.628020\pi\)
−0.391432 + 0.920207i \(0.628020\pi\)
\(462\) −3023.93 −0.304515
\(463\) −5930.29 −0.595257 −0.297629 0.954682i \(-0.596196\pi\)
−0.297629 + 0.954682i \(0.596196\pi\)
\(464\) 1027.68 0.102821
\(465\) −7510.92 −0.749055
\(466\) 3262.40 0.324308
\(467\) −3326.65 −0.329634 −0.164817 0.986324i \(-0.552703\pi\)
−0.164817 + 0.986324i \(0.552703\pi\)
\(468\) 15750.4 1.55569
\(469\) 9201.62 0.905952
\(470\) −1775.74 −0.174274
\(471\) 6839.84 0.669136
\(472\) 10607.4 1.03442
\(473\) 0 0
\(474\) −4762.29 −0.461475
\(475\) 18877.2 1.82346
\(476\) 8782.90 0.845722
\(477\) 14415.1 1.38369
\(478\) 86.2694 0.00825496
\(479\) −3999.59 −0.381516 −0.190758 0.981637i \(-0.561094\pi\)
−0.190758 + 0.981637i \(0.561094\pi\)
\(480\) −23417.6 −2.22680
\(481\) 10366.4 0.982676
\(482\) −2853.96 −0.269698
\(483\) 12044.1 1.13463
\(484\) 6365.29 0.597792
\(485\) −113.908 −0.0106645
\(486\) 7554.94 0.705142
\(487\) −17715.0 −1.64834 −0.824170 0.566342i \(-0.808358\pi\)
−0.824170 + 0.566342i \(0.808358\pi\)
\(488\) 12336.7 1.14437
\(489\) −1681.59 −0.155509
\(490\) 1364.73 0.125821
\(491\) 6351.58 0.583794 0.291897 0.956450i \(-0.405714\pi\)
0.291897 + 0.956450i \(0.405714\pi\)
\(492\) −11388.1 −1.04353
\(493\) −4286.49 −0.391590
\(494\) 15140.8 1.37898
\(495\) 8794.52 0.798554
\(496\) −1239.24 −0.112184
\(497\) 3819.60 0.344734
\(498\) 923.427 0.0830918
\(499\) 5595.78 0.502007 0.251003 0.967986i \(-0.419239\pi\)
0.251003 + 0.967986i \(0.419239\pi\)
\(500\) −1517.19 −0.135702
\(501\) −21336.6 −1.90269
\(502\) −4569.09 −0.406232
\(503\) 6219.18 0.551291 0.275646 0.961259i \(-0.411108\pi\)
0.275646 + 0.961259i \(0.411108\pi\)
\(504\) 10745.8 0.949715
\(505\) 12345.2 1.08783
\(506\) −2162.59 −0.189998
\(507\) 32876.3 2.87985
\(508\) 14977.7 1.30813
\(509\) 18178.1 1.58297 0.791485 0.611188i \(-0.209308\pi\)
0.791485 + 0.611188i \(0.209308\pi\)
\(510\) 15220.6 1.32153
\(511\) 12392.1 1.07278
\(512\) −7125.34 −0.615036
\(513\) 5662.86 0.487371
\(514\) 3295.92 0.282834
\(515\) 11695.0 1.00066
\(516\) 0 0
\(517\) 1293.37 0.110024
\(518\) 3041.55 0.257988
\(519\) −20990.7 −1.77532
\(520\) −25756.8 −2.17213
\(521\) −14404.1 −1.21124 −0.605620 0.795754i \(-0.707075\pi\)
−0.605620 + 0.795754i \(0.707075\pi\)
\(522\) −2255.39 −0.189111
\(523\) −4841.59 −0.404796 −0.202398 0.979303i \(-0.564873\pi\)
−0.202398 + 0.979303i \(0.564873\pi\)
\(524\) −5811.08 −0.484462
\(525\) 18223.8 1.51496
\(526\) −2599.10 −0.215449
\(527\) 5168.90 0.427250
\(528\) 2657.86 0.219069
\(529\) −3553.52 −0.292062
\(530\) −10137.6 −0.830850
\(531\) 17506.9 1.43076
\(532\) −13656.0 −1.11290
\(533\) −19664.6 −1.59807
\(534\) −9534.25 −0.772635
\(535\) 9.08652 0.000734288 0
\(536\) 10754.5 0.866645
\(537\) −33747.2 −2.71192
\(538\) −10339.1 −0.828532
\(539\) −994.015 −0.0794346
\(540\) −4142.82 −0.330146
\(541\) −8621.04 −0.685116 −0.342558 0.939497i \(-0.611293\pi\)
−0.342558 + 0.939497i \(0.611293\pi\)
\(542\) −335.547 −0.0265922
\(543\) −14296.2 −1.12985
\(544\) 16115.6 1.27013
\(545\) −19633.2 −1.54310
\(546\) 14616.8 1.14568
\(547\) −8074.92 −0.631186 −0.315593 0.948895i \(-0.602203\pi\)
−0.315593 + 0.948895i \(0.602203\pi\)
\(548\) 471.931 0.0367882
\(549\) 20360.9 1.58285
\(550\) −3272.18 −0.253684
\(551\) 6664.78 0.515298
\(552\) 14076.7 1.08540
\(553\) 7416.90 0.570341
\(554\) −6446.61 −0.494387
\(555\) −16202.9 −1.23924
\(556\) −12319.2 −0.939658
\(557\) 5115.39 0.389131 0.194566 0.980890i \(-0.437670\pi\)
0.194566 + 0.980890i \(0.437670\pi\)
\(558\) 2719.68 0.206332
\(559\) 0 0
\(560\) 5683.21 0.428856
\(561\) −11086.0 −0.834318
\(562\) −4713.99 −0.353822
\(563\) −4303.04 −0.322117 −0.161058 0.986945i \(-0.551491\pi\)
−0.161058 + 0.986945i \(0.551491\pi\)
\(564\) −3620.50 −0.270303
\(565\) −30367.3 −2.26117
\(566\) −3905.31 −0.290022
\(567\) −9283.94 −0.687635
\(568\) 4464.19 0.329777
\(569\) −22456.6 −1.65453 −0.827266 0.561810i \(-0.810105\pi\)
−0.827266 + 0.561810i \(0.810105\pi\)
\(570\) −23665.5 −1.73901
\(571\) 2012.73 0.147513 0.0737566 0.997276i \(-0.476501\pi\)
0.0737566 + 0.997276i \(0.476501\pi\)
\(572\) 8067.81 0.589741
\(573\) −8762.85 −0.638871
\(574\) −5769.68 −0.419550
\(575\) 13032.9 0.945235
\(576\) 3096.32 0.223981
\(577\) −6863.89 −0.495229 −0.247615 0.968859i \(-0.579647\pi\)
−0.247615 + 0.968859i \(0.579647\pi\)
\(578\) −3589.95 −0.258343
\(579\) −21126.2 −1.51636
\(580\) −4875.81 −0.349064
\(581\) −1438.17 −0.102694
\(582\) 75.5503 0.00538086
\(583\) 7383.82 0.524540
\(584\) 14483.3 1.02624
\(585\) −42510.1 −3.00440
\(586\) −2170.44 −0.153004
\(587\) −17290.7 −1.21578 −0.607891 0.794021i \(-0.707984\pi\)
−0.607891 + 0.794021i \(0.707984\pi\)
\(588\) 2782.52 0.195152
\(589\) −8036.78 −0.562224
\(590\) −12312.0 −0.859112
\(591\) 22076.5 1.53656
\(592\) −2673.34 −0.185598
\(593\) −4413.61 −0.305642 −0.152821 0.988254i \(-0.548836\pi\)
−0.152821 + 0.988254i \(0.548836\pi\)
\(594\) −981.602 −0.0678040
\(595\) −23704.9 −1.63329
\(596\) −2031.65 −0.139631
\(597\) −120.248 −0.00824362
\(598\) 10453.3 0.714829
\(599\) 14099.4 0.961743 0.480871 0.876791i \(-0.340320\pi\)
0.480871 + 0.876791i \(0.340320\pi\)
\(600\) 21299.2 1.44923
\(601\) 3049.05 0.206944 0.103472 0.994632i \(-0.467005\pi\)
0.103472 + 0.994632i \(0.467005\pi\)
\(602\) 0 0
\(603\) 17749.6 1.19871
\(604\) −8163.57 −0.549952
\(605\) −17179.8 −1.15447
\(606\) −8188.07 −0.548874
\(607\) −24259.5 −1.62218 −0.811091 0.584920i \(-0.801126\pi\)
−0.811091 + 0.584920i \(0.801126\pi\)
\(608\) −25057.2 −1.67138
\(609\) 6434.09 0.428116
\(610\) −14319.1 −0.950434
\(611\) −6251.78 −0.413944
\(612\) 16941.9 1.11901
\(613\) 17207.8 1.13380 0.566898 0.823788i \(-0.308144\pi\)
0.566898 + 0.823788i \(0.308144\pi\)
\(614\) −984.470 −0.0647068
\(615\) 30736.2 2.01529
\(616\) 5504.30 0.360024
\(617\) 18149.7 1.18425 0.592123 0.805848i \(-0.298290\pi\)
0.592123 + 0.805848i \(0.298290\pi\)
\(618\) −7756.81 −0.504894
\(619\) −1849.65 −0.120103 −0.0600514 0.998195i \(-0.519126\pi\)
−0.0600514 + 0.998195i \(0.519126\pi\)
\(620\) 5879.54 0.380851
\(621\) 3909.66 0.252640
\(622\) 7117.22 0.458802
\(623\) 14848.9 0.954906
\(624\) −12847.3 −0.824204
\(625\) −13458.6 −0.861348
\(626\) 7275.51 0.464517
\(627\) 17236.9 1.09789
\(628\) −5354.22 −0.340217
\(629\) 11150.6 0.706842
\(630\) −12472.6 −0.788764
\(631\) −13125.7 −0.828090 −0.414045 0.910256i \(-0.635884\pi\)
−0.414045 + 0.910256i \(0.635884\pi\)
\(632\) 8668.55 0.545596
\(633\) 32818.0 2.06066
\(634\) −518.345 −0.0324702
\(635\) −40424.5 −2.52629
\(636\) −20669.3 −1.28867
\(637\) 4804.77 0.298857
\(638\) −1155.28 −0.0716893
\(639\) 7367.89 0.456133
\(640\) 22117.0 1.36602
\(641\) 8940.53 0.550905 0.275452 0.961315i \(-0.411172\pi\)
0.275452 + 0.961315i \(0.411172\pi\)
\(642\) −6.02673 −0.000370492 0
\(643\) −13835.3 −0.848542 −0.424271 0.905535i \(-0.639470\pi\)
−0.424271 + 0.905535i \(0.639470\pi\)
\(644\) −9428.14 −0.576896
\(645\) 0 0
\(646\) 16286.2 0.991909
\(647\) −2206.18 −0.134055 −0.0670277 0.997751i \(-0.521352\pi\)
−0.0670277 + 0.997751i \(0.521352\pi\)
\(648\) −10850.7 −0.657800
\(649\) 8967.52 0.542382
\(650\) 15816.7 0.954435
\(651\) −7758.60 −0.467102
\(652\) 1316.35 0.0790676
\(653\) 5668.15 0.339681 0.169841 0.985472i \(-0.445675\pi\)
0.169841 + 0.985472i \(0.445675\pi\)
\(654\) 13021.9 0.778588
\(655\) 15684.0 0.935609
\(656\) 5071.22 0.301826
\(657\) 23903.8 1.41945
\(658\) −1834.30 −0.108675
\(659\) 6131.34 0.362432 0.181216 0.983443i \(-0.441997\pi\)
0.181216 + 0.983443i \(0.441997\pi\)
\(660\) −12610.2 −0.743712
\(661\) −19986.7 −1.17609 −0.588044 0.808829i \(-0.700102\pi\)
−0.588044 + 0.808829i \(0.700102\pi\)
\(662\) 7908.20 0.464292
\(663\) 53586.5 3.13896
\(664\) −1680.87 −0.0982383
\(665\) 36857.2 2.14926
\(666\) 5867.04 0.341356
\(667\) 4601.40 0.267117
\(668\) 16702.3 0.967410
\(669\) 4156.83 0.240227
\(670\) −12482.7 −0.719773
\(671\) 10429.4 0.600036
\(672\) −24189.8 −1.38861
\(673\) 8379.00 0.479921 0.239960 0.970783i \(-0.422866\pi\)
0.239960 + 0.970783i \(0.422866\pi\)
\(674\) −15599.5 −0.891501
\(675\) 5915.65 0.337323
\(676\) −25735.5 −1.46424
\(677\) −21124.8 −1.19925 −0.599626 0.800280i \(-0.704684\pi\)
−0.599626 + 0.800280i \(0.704684\pi\)
\(678\) 20141.4 1.14090
\(679\) −117.664 −0.00665025
\(680\) −27705.2 −1.56242
\(681\) −46206.7 −2.60007
\(682\) 1393.10 0.0782178
\(683\) −22857.9 −1.28058 −0.640288 0.768135i \(-0.721185\pi\)
−0.640288 + 0.768135i \(0.721185\pi\)
\(684\) −26341.9 −1.47252
\(685\) −1273.73 −0.0710465
\(686\) 9498.67 0.528660
\(687\) 14314.4 0.794945
\(688\) 0 0
\(689\) −35691.2 −1.97348
\(690\) −16338.8 −0.901458
\(691\) 28371.8 1.56196 0.780979 0.624557i \(-0.214720\pi\)
0.780979 + 0.624557i \(0.214720\pi\)
\(692\) 16431.5 0.902646
\(693\) 9084.54 0.497970
\(694\) 16122.6 0.881851
\(695\) 33249.2 1.81470
\(696\) 7519.88 0.409541
\(697\) −21152.2 −1.14949
\(698\) −4008.01 −0.217343
\(699\) −17952.6 −0.971428
\(700\) −14265.6 −0.770268
\(701\) −2096.85 −0.112977 −0.0564885 0.998403i \(-0.517990\pi\)
−0.0564885 + 0.998403i \(0.517990\pi\)
\(702\) 4744.76 0.255099
\(703\) −17337.3 −0.930143
\(704\) 1586.02 0.0849083
\(705\) 9771.67 0.522017
\(706\) 16892.6 0.900512
\(707\) 12752.3 0.678358
\(708\) −25102.5 −1.33250
\(709\) 20456.2 1.08356 0.541782 0.840519i \(-0.317750\pi\)
0.541782 + 0.840519i \(0.317750\pi\)
\(710\) −5181.57 −0.273889
\(711\) 14306.9 0.754645
\(712\) 17354.7 0.913475
\(713\) −5548.64 −0.291442
\(714\) 15722.5 0.824090
\(715\) −21774.8 −1.13893
\(716\) 26417.3 1.37885
\(717\) −474.730 −0.0247268
\(718\) 2096.38 0.108964
\(719\) −15358.6 −0.796631 −0.398316 0.917248i \(-0.630405\pi\)
−0.398316 + 0.917248i \(0.630405\pi\)
\(720\) 10962.7 0.567440
\(721\) 12080.6 0.624003
\(722\) −15710.8 −0.809828
\(723\) 15705.0 0.807850
\(724\) 11191.1 0.574465
\(725\) 6962.30 0.356653
\(726\) 11394.7 0.582501
\(727\) −23094.3 −1.17816 −0.589079 0.808075i \(-0.700509\pi\)
−0.589079 + 0.808075i \(0.700509\pi\)
\(728\) −26606.1 −1.35452
\(729\) −26679.2 −1.35544
\(730\) −16810.7 −0.852319
\(731\) 0 0
\(732\) −29194.9 −1.47414
\(733\) −24060.6 −1.21241 −0.606207 0.795307i \(-0.707310\pi\)
−0.606207 + 0.795307i \(0.707310\pi\)
\(734\) 10450.6 0.525529
\(735\) −7509.96 −0.376883
\(736\) −17299.6 −0.866401
\(737\) 9091.85 0.454413
\(738\) −11129.5 −0.555126
\(739\) 15164.2 0.754836 0.377418 0.926043i \(-0.376812\pi\)
0.377418 + 0.926043i \(0.376812\pi\)
\(740\) 12683.6 0.630080
\(741\) −83318.1 −4.13059
\(742\) −10471.9 −0.518109
\(743\) 20690.1 1.02160 0.510799 0.859700i \(-0.329349\pi\)
0.510799 + 0.859700i \(0.329349\pi\)
\(744\) −9067.92 −0.446836
\(745\) 5483.40 0.269659
\(746\) −10072.9 −0.494362
\(747\) −2774.17 −0.135879
\(748\) 8678.13 0.424203
\(749\) 9.38616 0.000457894 0
\(750\) −2715.97 −0.132231
\(751\) 9464.86 0.459890 0.229945 0.973204i \(-0.426145\pi\)
0.229945 + 0.973204i \(0.426145\pi\)
\(752\) 1612.24 0.0781814
\(753\) 25143.1 1.21682
\(754\) 5584.25 0.269717
\(755\) 22033.3 1.06209
\(756\) −4279.44 −0.205875
\(757\) −10567.2 −0.507361 −0.253680 0.967288i \(-0.581641\pi\)
−0.253680 + 0.967288i \(0.581641\pi\)
\(758\) −11519.1 −0.551969
\(759\) 11900.5 0.569117
\(760\) 43077.0 2.05601
\(761\) −11305.4 −0.538531 −0.269265 0.963066i \(-0.586781\pi\)
−0.269265 + 0.963066i \(0.586781\pi\)
\(762\) 26812.0 1.27467
\(763\) −20280.6 −0.962263
\(764\) 6859.54 0.324829
\(765\) −45725.9 −2.16108
\(766\) 8635.66 0.407335
\(767\) −43346.3 −2.04060
\(768\) −20553.3 −0.965696
\(769\) 7946.62 0.372643 0.186321 0.982489i \(-0.440343\pi\)
0.186321 + 0.982489i \(0.440343\pi\)
\(770\) −6388.83 −0.299010
\(771\) −18137.0 −0.847197
\(772\) 16537.5 0.770983
\(773\) 1530.93 0.0712340 0.0356170 0.999366i \(-0.488660\pi\)
0.0356170 + 0.999366i \(0.488660\pi\)
\(774\) 0 0
\(775\) −8395.54 −0.389131
\(776\) −137.520 −0.00636172
\(777\) −16737.2 −0.772774
\(778\) 6622.66 0.305185
\(779\) 32888.2 1.51263
\(780\) 60953.7 2.79807
\(781\) 3774.04 0.172914
\(782\) 11244.1 0.514179
\(783\) 2088.58 0.0953253
\(784\) −1239.08 −0.0564449
\(785\) 14450.9 0.657039
\(786\) −10402.6 −0.472070
\(787\) −17765.6 −0.804671 −0.402335 0.915492i \(-0.631801\pi\)
−0.402335 + 0.915492i \(0.631801\pi\)
\(788\) −17281.4 −0.781251
\(789\) 14302.5 0.645352
\(790\) −10061.6 −0.453132
\(791\) −31368.7 −1.41004
\(792\) 10617.6 0.476364
\(793\) −50412.8 −2.25752
\(794\) −17016.3 −0.760563
\(795\) 55786.1 2.48872
\(796\) 94.1304 0.00419141
\(797\) 12457.2 0.553645 0.276823 0.960921i \(-0.410719\pi\)
0.276823 + 0.960921i \(0.410719\pi\)
\(798\) −24445.9 −1.08443
\(799\) −6724.72 −0.297751
\(800\) −26175.7 −1.15681
\(801\) 28642.9 1.26348
\(802\) 5907.65 0.260108
\(803\) 12244.2 0.538094
\(804\) −25450.6 −1.11638
\(805\) 25446.4 1.11412
\(806\) −6733.82 −0.294279
\(807\) 56894.7 2.48177
\(808\) 14904.3 0.648926
\(809\) −28363.3 −1.23263 −0.616316 0.787499i \(-0.711376\pi\)
−0.616316 + 0.787499i \(0.711376\pi\)
\(810\) 12594.3 0.546321
\(811\) −5605.49 −0.242707 −0.121354 0.992609i \(-0.538723\pi\)
−0.121354 + 0.992609i \(0.538723\pi\)
\(812\) −5036.60 −0.217672
\(813\) 1846.48 0.0796540
\(814\) 3005.26 0.129403
\(815\) −3552.79 −0.152698
\(816\) −13819.2 −0.592853
\(817\) 0 0
\(818\) −336.208 −0.0143707
\(819\) −43911.9 −1.87351
\(820\) −24060.3 −1.02466
\(821\) −17256.2 −0.733549 −0.366775 0.930310i \(-0.619538\pi\)
−0.366775 + 0.930310i \(0.619538\pi\)
\(822\) 844.817 0.0358472
\(823\) −23660.3 −1.00212 −0.501060 0.865413i \(-0.667056\pi\)
−0.501060 + 0.865413i \(0.667056\pi\)
\(824\) 14119.3 0.596929
\(825\) 18006.4 0.759881
\(826\) −12718.0 −0.535733
\(827\) 30880.2 1.29844 0.649220 0.760601i \(-0.275095\pi\)
0.649220 + 0.760601i \(0.275095\pi\)
\(828\) −18186.6 −0.763317
\(829\) −25312.2 −1.06047 −0.530235 0.847851i \(-0.677896\pi\)
−0.530235 + 0.847851i \(0.677896\pi\)
\(830\) 1950.98 0.0815896
\(831\) 35474.9 1.48088
\(832\) −7666.35 −0.319451
\(833\) 5168.24 0.214969
\(834\) −22052.9 −0.915623
\(835\) −45079.1 −1.86829
\(836\) −13493.0 −0.558214
\(837\) −2518.53 −0.104006
\(838\) 44.1128 0.00181844
\(839\) 1329.25 0.0546971 0.0273485 0.999626i \(-0.491294\pi\)
0.0273485 + 0.999626i \(0.491294\pi\)
\(840\) 41586.0 1.70816
\(841\) −21930.9 −0.899212
\(842\) −1318.30 −0.0539568
\(843\) 25940.5 1.05983
\(844\) −25689.9 −1.04773
\(845\) 69459.6 2.82779
\(846\) −3538.30 −0.143793
\(847\) −17746.3 −0.719918
\(848\) 9204.23 0.372729
\(849\) 21490.4 0.868727
\(850\) 17013.2 0.686529
\(851\) −11969.8 −0.482161
\(852\) −10564.6 −0.424807
\(853\) 26615.5 1.06834 0.534171 0.845376i \(-0.320624\pi\)
0.534171 + 0.845376i \(0.320624\pi\)
\(854\) −14791.3 −0.592680
\(855\) 71096.2 2.84379
\(856\) 10.9701 0.000438027 0
\(857\) 39276.7 1.56554 0.782770 0.622311i \(-0.213806\pi\)
0.782770 + 0.622311i \(0.213806\pi\)
\(858\) 14442.4 0.574656
\(859\) 11705.3 0.464937 0.232468 0.972604i \(-0.425320\pi\)
0.232468 + 0.972604i \(0.425320\pi\)
\(860\) 0 0
\(861\) 31749.8 1.25671
\(862\) −3949.63 −0.156061
\(863\) −38432.9 −1.51596 −0.757978 0.652280i \(-0.773812\pi\)
−0.757978 + 0.652280i \(0.773812\pi\)
\(864\) −7852.29 −0.309190
\(865\) −44348.2 −1.74322
\(866\) −20856.7 −0.818405
\(867\) 19755.0 0.773836
\(868\) 6073.42 0.237495
\(869\) 7328.42 0.286076
\(870\) −8728.31 −0.340135
\(871\) −43947.3 −1.70964
\(872\) −23703.1 −0.920513
\(873\) −226.969 −0.00879926
\(874\) −17482.7 −0.676615
\(875\) 4229.91 0.163425
\(876\) −34274.9 −1.32197
\(877\) 17138.0 0.659873 0.329937 0.944003i \(-0.392973\pi\)
0.329937 + 0.944003i \(0.392973\pi\)
\(878\) −5098.25 −0.195965
\(879\) 11943.7 0.458305
\(880\) 5615.42 0.215109
\(881\) −9092.54 −0.347713 −0.173857 0.984771i \(-0.555623\pi\)
−0.173857 + 0.984771i \(0.555623\pi\)
\(882\) 2719.34 0.103815
\(883\) −29537.5 −1.12573 −0.562864 0.826550i \(-0.690300\pi\)
−0.562864 + 0.826550i \(0.690300\pi\)
\(884\) −41947.4 −1.59598
\(885\) 67751.2 2.57337
\(886\) −8037.41 −0.304765
\(887\) 42029.3 1.59099 0.795494 0.605962i \(-0.207212\pi\)
0.795494 + 0.605962i \(0.207212\pi\)
\(888\) −19561.8 −0.739245
\(889\) −41757.6 −1.57537
\(890\) −20143.6 −0.758667
\(891\) −9173.19 −0.344908
\(892\) −3253.96 −0.122142
\(893\) 10455.8 0.391815
\(894\) −3636.92 −0.136059
\(895\) −71299.6 −2.66289
\(896\) 22846.4 0.851835
\(897\) −57523.2 −2.14119
\(898\) −19129.7 −0.710875
\(899\) −2964.13 −0.109966
\(900\) −27517.8 −1.01918
\(901\) −38391.2 −1.41953
\(902\) −5700.85 −0.210441
\(903\) 0 0
\(904\) −36662.4 −1.34886
\(905\) −30204.4 −1.10942
\(906\) −14613.8 −0.535885
\(907\) −4192.38 −0.153479 −0.0767397 0.997051i \(-0.524451\pi\)
−0.0767397 + 0.997051i \(0.524451\pi\)
\(908\) 36170.6 1.32199
\(909\) 24598.7 0.897567
\(910\) 30881.7 1.12496
\(911\) −7099.74 −0.258205 −0.129103 0.991631i \(-0.541210\pi\)
−0.129103 + 0.991631i \(0.541210\pi\)
\(912\) 21486.5 0.780143
\(913\) −1421.01 −0.0515099
\(914\) 25300.4 0.915605
\(915\) 78796.3 2.84691
\(916\) −11205.3 −0.404184
\(917\) 16201.2 0.583435
\(918\) 5103.70 0.183494
\(919\) −37186.6 −1.33479 −0.667396 0.744703i \(-0.732591\pi\)
−0.667396 + 0.744703i \(0.732591\pi\)
\(920\) 29740.6 1.06578
\(921\) 5417.42 0.193822
\(922\) 10858.5 0.387860
\(923\) −18242.6 −0.650554
\(924\) −13026.0 −0.463771
\(925\) −18111.3 −0.643779
\(926\) 8310.17 0.294913
\(927\) 23303.1 0.825647
\(928\) −9241.59 −0.326908
\(929\) −24894.6 −0.879187 −0.439593 0.898197i \(-0.644877\pi\)
−0.439593 + 0.898197i \(0.644877\pi\)
\(930\) 10525.1 0.371110
\(931\) −8035.76 −0.282880
\(932\) 14053.2 0.493916
\(933\) −39165.2 −1.37429
\(934\) 4661.66 0.163313
\(935\) −23422.1 −0.819235
\(936\) −51322.3 −1.79222
\(937\) −38380.7 −1.33815 −0.669073 0.743197i \(-0.733308\pi\)
−0.669073 + 0.743197i \(0.733308\pi\)
\(938\) −12894.3 −0.448842
\(939\) −40036.2 −1.39141
\(940\) −7649.25 −0.265416
\(941\) 33578.2 1.16325 0.581625 0.813457i \(-0.302417\pi\)
0.581625 + 0.813457i \(0.302417\pi\)
\(942\) −9584.73 −0.331515
\(943\) 22706.2 0.784109
\(944\) 11178.4 0.385408
\(945\) 11550.1 0.397593
\(946\) 0 0
\(947\) −40210.2 −1.37978 −0.689892 0.723913i \(-0.742342\pi\)
−0.689892 + 0.723913i \(0.742342\pi\)
\(948\) −20514.2 −0.702818
\(949\) −59184.9 −2.02447
\(950\) −26452.8 −0.903412
\(951\) 2852.38 0.0972607
\(952\) −28618.9 −0.974309
\(953\) 18846.5 0.640607 0.320303 0.947315i \(-0.396215\pi\)
0.320303 + 0.947315i \(0.396215\pi\)
\(954\) −20200.0 −0.685534
\(955\) −18513.8 −0.627321
\(956\) 371.618 0.0125722
\(957\) 6357.33 0.214737
\(958\) 5604.66 0.189017
\(959\) −1315.74 −0.0443038
\(960\) 11982.7 0.402854
\(961\) −26216.7 −0.880020
\(962\) −14526.5 −0.486854
\(963\) 18.1056 0.000605861 0
\(964\) −12293.9 −0.410745
\(965\) −44634.5 −1.48895
\(966\) −16877.6 −0.562140
\(967\) 1189.70 0.0395637 0.0197819 0.999804i \(-0.493703\pi\)
0.0197819 + 0.999804i \(0.493703\pi\)
\(968\) −20741.1 −0.688682
\(969\) −89621.0 −2.97115
\(970\) 159.620 0.00528358
\(971\) −45974.0 −1.51944 −0.759720 0.650250i \(-0.774664\pi\)
−0.759720 + 0.650250i \(0.774664\pi\)
\(972\) 32544.0 1.07392
\(973\) 34345.7 1.13163
\(974\) 24824.1 0.816649
\(975\) −87037.4 −2.85890
\(976\) 13000.7 0.426376
\(977\) 42690.2 1.39793 0.698967 0.715154i \(-0.253644\pi\)
0.698967 + 0.715154i \(0.253644\pi\)
\(978\) 2356.42 0.0770451
\(979\) 14671.7 0.478968
\(980\) 5878.79 0.191623
\(981\) −39120.6 −1.27321
\(982\) −8900.52 −0.289233
\(983\) 19065.0 0.618596 0.309298 0.950965i \(-0.399906\pi\)
0.309298 + 0.950965i \(0.399906\pi\)
\(984\) 37107.8 1.20219
\(985\) 46642.2 1.50878
\(986\) 6006.70 0.194008
\(987\) 10093.9 0.325524
\(988\) 65221.3 2.10017
\(989\) 0 0
\(990\) −12323.8 −0.395634
\(991\) 51456.8 1.64942 0.824712 0.565553i \(-0.191337\pi\)
0.824712 + 0.565553i \(0.191337\pi\)
\(992\) 11144.1 0.356678
\(993\) −43517.8 −1.39073
\(994\) −5352.44 −0.170794
\(995\) −254.056 −0.00809459
\(996\) 3977.79 0.126547
\(997\) 2972.68 0.0944291 0.0472146 0.998885i \(-0.484966\pi\)
0.0472146 + 0.998885i \(0.484966\pi\)
\(998\) −7841.41 −0.248713
\(999\) −5433.10 −0.172068
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 1849.4.a.h.1.12 30
43.21 even 7 43.4.e.a.11.5 yes 60
43.41 even 7 43.4.e.a.4.5 60
43.42 odd 2 1849.4.a.g.1.19 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.4.5 60 43.41 even 7
43.4.e.a.11.5 yes 60 43.21 even 7
1849.4.a.g.1.19 30 43.42 odd 2
1849.4.a.h.1.12 30 1.1 even 1 trivial