Properties

Label 5225.2.a.bb.1.12
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.104144 q^{2} +0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} -3.21657 q^{7} -0.415445 q^{8} -2.51420 q^{9} +O(q^{10})\) \(q+0.104144 q^{2} +0.696995 q^{3} -1.98915 q^{4} +0.0725876 q^{6} -3.21657 q^{7} -0.415445 q^{8} -2.51420 q^{9} -1.00000 q^{11} -1.38643 q^{12} -4.65405 q^{13} -0.334986 q^{14} +3.93504 q^{16} +0.00602394 q^{17} -0.261838 q^{18} -1.00000 q^{19} -2.24194 q^{21} -0.104144 q^{22} +1.65858 q^{23} -0.289563 q^{24} -0.484690 q^{26} -3.84337 q^{27} +6.39826 q^{28} +3.49988 q^{29} -3.34225 q^{31} +1.24070 q^{32} -0.696995 q^{33} +0.000627356 q^{34} +5.00113 q^{36} -6.13742 q^{37} -0.104144 q^{38} -3.24385 q^{39} -9.38032 q^{41} -0.233483 q^{42} -8.61740 q^{43} +1.98915 q^{44} +0.172731 q^{46} -0.893231 q^{47} +2.74270 q^{48} +3.34635 q^{49} +0.00419866 q^{51} +9.25763 q^{52} -1.78431 q^{53} -0.400263 q^{54} +1.33631 q^{56} -0.696995 q^{57} +0.364491 q^{58} +2.57022 q^{59} -3.68861 q^{61} -0.348074 q^{62} +8.08710 q^{63} -7.74087 q^{64} -0.0725876 q^{66} -5.98644 q^{67} -0.0119826 q^{68} +1.15602 q^{69} +0.852858 q^{71} +1.04451 q^{72} -4.72786 q^{73} -0.639174 q^{74} +1.98915 q^{76} +3.21657 q^{77} -0.337827 q^{78} -4.94098 q^{79} +4.86379 q^{81} -0.976902 q^{82} +15.0339 q^{83} +4.45956 q^{84} -0.897448 q^{86} +2.43940 q^{87} +0.415445 q^{88} +5.11449 q^{89} +14.9701 q^{91} -3.29917 q^{92} -2.32953 q^{93} -0.0930244 q^{94} +0.864762 q^{96} +16.5462 q^{97} +0.348501 q^{98} +2.51420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44} + 88 q^{46} + 28 q^{49} + 8 q^{51} + 38 q^{54} + 44 q^{56} + 40 q^{59} - 6 q^{61} + 140 q^{64} + 12 q^{66} + 74 q^{69} + 62 q^{71} - 26 q^{74} - 32 q^{76} + 102 q^{79} + 94 q^{81} - 38 q^{84} + 28 q^{86} + 54 q^{89} + 88 q^{91} + 36 q^{94} + 2 q^{96} - 34 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.104144 0.0736407 0.0368204 0.999322i \(-0.488277\pi\)
0.0368204 + 0.999322i \(0.488277\pi\)
\(3\) 0.696995 0.402410 0.201205 0.979549i \(-0.435514\pi\)
0.201205 + 0.979549i \(0.435514\pi\)
\(4\) −1.98915 −0.994577
\(5\) 0 0
\(6\) 0.0725876 0.0296338
\(7\) −3.21657 −1.21575 −0.607875 0.794033i \(-0.707978\pi\)
−0.607875 + 0.794033i \(0.707978\pi\)
\(8\) −0.415445 −0.146882
\(9\) −2.51420 −0.838066
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.38643 −0.400228
\(13\) −4.65405 −1.29080 −0.645401 0.763844i \(-0.723310\pi\)
−0.645401 + 0.763844i \(0.723310\pi\)
\(14\) −0.334986 −0.0895287
\(15\) 0 0
\(16\) 3.93504 0.983761
\(17\) 0.00602394 0.00146102 0.000730510 1.00000i \(-0.499767\pi\)
0.000730510 1.00000i \(0.499767\pi\)
\(18\) −0.261838 −0.0617158
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.24194 −0.489230
\(22\) −0.104144 −0.0222035
\(23\) 1.65858 0.345838 0.172919 0.984936i \(-0.444680\pi\)
0.172919 + 0.984936i \(0.444680\pi\)
\(24\) −0.289563 −0.0591069
\(25\) 0 0
\(26\) −0.484690 −0.0950556
\(27\) −3.84337 −0.739657
\(28\) 6.39826 1.20916
\(29\) 3.49988 0.649912 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(30\) 0 0
\(31\) −3.34225 −0.600285 −0.300143 0.953894i \(-0.597034\pi\)
−0.300143 + 0.953894i \(0.597034\pi\)
\(32\) 1.24070 0.219327
\(33\) −0.696995 −0.121331
\(34\) 0.000627356 0 0.000107591 0
\(35\) 0 0
\(36\) 5.00113 0.833521
\(37\) −6.13742 −1.00899 −0.504493 0.863416i \(-0.668320\pi\)
−0.504493 + 0.863416i \(0.668320\pi\)
\(38\) −0.104144 −0.0168943
\(39\) −3.24385 −0.519432
\(40\) 0 0
\(41\) −9.38032 −1.46496 −0.732480 0.680788i \(-0.761637\pi\)
−0.732480 + 0.680788i \(0.761637\pi\)
\(42\) −0.233483 −0.0360273
\(43\) −8.61740 −1.31414 −0.657071 0.753829i \(-0.728205\pi\)
−0.657071 + 0.753829i \(0.728205\pi\)
\(44\) 1.98915 0.299876
\(45\) 0 0
\(46\) 0.172731 0.0254677
\(47\) −0.893231 −0.130291 −0.0651456 0.997876i \(-0.520751\pi\)
−0.0651456 + 0.997876i \(0.520751\pi\)
\(48\) 2.74270 0.395875
\(49\) 3.34635 0.478049
\(50\) 0 0
\(51\) 0.00419866 0.000587930 0
\(52\) 9.25763 1.28380
\(53\) −1.78431 −0.245093 −0.122547 0.992463i \(-0.539106\pi\)
−0.122547 + 0.992463i \(0.539106\pi\)
\(54\) −0.400263 −0.0544688
\(55\) 0 0
\(56\) 1.33631 0.178572
\(57\) −0.696995 −0.0923192
\(58\) 0.364491 0.0478600
\(59\) 2.57022 0.334614 0.167307 0.985905i \(-0.446493\pi\)
0.167307 + 0.985905i \(0.446493\pi\)
\(60\) 0 0
\(61\) −3.68861 −0.472278 −0.236139 0.971719i \(-0.575882\pi\)
−0.236139 + 0.971719i \(0.575882\pi\)
\(62\) −0.348074 −0.0442054
\(63\) 8.08710 1.01888
\(64\) −7.74087 −0.967609
\(65\) 0 0
\(66\) −0.0725876 −0.00893492
\(67\) −5.98644 −0.731361 −0.365680 0.930741i \(-0.619164\pi\)
−0.365680 + 0.930741i \(0.619164\pi\)
\(68\) −0.0119826 −0.00145310
\(69\) 1.15602 0.139169
\(70\) 0 0
\(71\) 0.852858 0.101216 0.0506078 0.998719i \(-0.483884\pi\)
0.0506078 + 0.998719i \(0.483884\pi\)
\(72\) 1.04451 0.123097
\(73\) −4.72786 −0.553354 −0.276677 0.960963i \(-0.589233\pi\)
−0.276677 + 0.960963i \(0.589233\pi\)
\(74\) −0.639174 −0.0743024
\(75\) 0 0
\(76\) 1.98915 0.228172
\(77\) 3.21657 0.366563
\(78\) −0.337827 −0.0382513
\(79\) −4.94098 −0.555903 −0.277952 0.960595i \(-0.589656\pi\)
−0.277952 + 0.960595i \(0.589656\pi\)
\(80\) 0 0
\(81\) 4.86379 0.540421
\(82\) −0.976902 −0.107881
\(83\) 15.0339 1.65019 0.825093 0.564997i \(-0.191123\pi\)
0.825093 + 0.564997i \(0.191123\pi\)
\(84\) 4.45956 0.486577
\(85\) 0 0
\(86\) −0.897448 −0.0967743
\(87\) 2.43940 0.261531
\(88\) 0.415445 0.0442866
\(89\) 5.11449 0.542135 0.271067 0.962560i \(-0.412623\pi\)
0.271067 + 0.962560i \(0.412623\pi\)
\(90\) 0 0
\(91\) 14.9701 1.56929
\(92\) −3.29917 −0.343962
\(93\) −2.32953 −0.241561
\(94\) −0.0930244 −0.00959474
\(95\) 0 0
\(96\) 0.864762 0.0882594
\(97\) 16.5462 1.68001 0.840005 0.542578i \(-0.182552\pi\)
0.840005 + 0.542578i \(0.182552\pi\)
\(98\) 0.348501 0.0352039
\(99\) 2.51420 0.252686
\(100\) 0 0
\(101\) 5.62840 0.560047 0.280023 0.959993i \(-0.409658\pi\)
0.280023 + 0.959993i \(0.409658\pi\)
\(102\) 0.000437264 0 4.32956e−5 0
\(103\) 7.56366 0.745270 0.372635 0.927978i \(-0.378454\pi\)
0.372635 + 0.927978i \(0.378454\pi\)
\(104\) 1.93350 0.189596
\(105\) 0 0
\(106\) −0.185824 −0.0180488
\(107\) −5.01113 −0.484444 −0.242222 0.970221i \(-0.577876\pi\)
−0.242222 + 0.970221i \(0.577876\pi\)
\(108\) 7.64505 0.735645
\(109\) 3.73155 0.357418 0.178709 0.983902i \(-0.442808\pi\)
0.178709 + 0.983902i \(0.442808\pi\)
\(110\) 0 0
\(111\) −4.27775 −0.406026
\(112\) −12.6574 −1.19601
\(113\) 16.9702 1.59642 0.798209 0.602380i \(-0.205781\pi\)
0.798209 + 0.602380i \(0.205781\pi\)
\(114\) −0.0725876 −0.00679846
\(115\) 0 0
\(116\) −6.96181 −0.646388
\(117\) 11.7012 1.08178
\(118\) 0.267672 0.0246412
\(119\) −0.0193765 −0.00177624
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.384146 −0.0347789
\(123\) −6.53804 −0.589515
\(124\) 6.64824 0.597030
\(125\) 0 0
\(126\) 0.842221 0.0750310
\(127\) −0.870581 −0.0772516 −0.0386258 0.999254i \(-0.512298\pi\)
−0.0386258 + 0.999254i \(0.512298\pi\)
\(128\) −3.28756 −0.290582
\(129\) −6.00628 −0.528824
\(130\) 0 0
\(131\) −0.332706 −0.0290687 −0.0145343 0.999894i \(-0.504627\pi\)
−0.0145343 + 0.999894i \(0.504627\pi\)
\(132\) 1.38643 0.120673
\(133\) 3.21657 0.278912
\(134\) −0.623450 −0.0538579
\(135\) 0 0
\(136\) −0.00250262 −0.000214598 0
\(137\) 13.0314 1.11335 0.556673 0.830732i \(-0.312078\pi\)
0.556673 + 0.830732i \(0.312078\pi\)
\(138\) 0.120392 0.0102485
\(139\) 12.6827 1.07573 0.537867 0.843030i \(-0.319230\pi\)
0.537867 + 0.843030i \(0.319230\pi\)
\(140\) 0 0
\(141\) −0.622578 −0.0524305
\(142\) 0.0888197 0.00745359
\(143\) 4.65405 0.389192
\(144\) −9.89348 −0.824456
\(145\) 0 0
\(146\) −0.492376 −0.0407494
\(147\) 2.33239 0.192372
\(148\) 12.2083 1.00351
\(149\) 17.2444 1.41272 0.706358 0.707855i \(-0.250337\pi\)
0.706358 + 0.707855i \(0.250337\pi\)
\(150\) 0 0
\(151\) 22.5478 1.83491 0.917456 0.397836i \(-0.130239\pi\)
0.917456 + 0.397836i \(0.130239\pi\)
\(152\) 0.415445 0.0336971
\(153\) −0.0151454 −0.00122443
\(154\) 0.334986 0.0269939
\(155\) 0 0
\(156\) 6.45252 0.516615
\(157\) −12.1662 −0.970972 −0.485486 0.874244i \(-0.661357\pi\)
−0.485486 + 0.874244i \(0.661357\pi\)
\(158\) −0.514572 −0.0409371
\(159\) −1.24365 −0.0986280
\(160\) 0 0
\(161\) −5.33494 −0.420452
\(162\) 0.506533 0.0397970
\(163\) −21.8086 −1.70818 −0.854090 0.520125i \(-0.825885\pi\)
−0.854090 + 0.520125i \(0.825885\pi\)
\(164\) 18.6589 1.45702
\(165\) 0 0
\(166\) 1.56569 0.121521
\(167\) 15.5829 1.20584 0.602919 0.797802i \(-0.294004\pi\)
0.602919 + 0.797802i \(0.294004\pi\)
\(168\) 0.931402 0.0718592
\(169\) 8.66022 0.666170
\(170\) 0 0
\(171\) 2.51420 0.192266
\(172\) 17.1413 1.30701
\(173\) −10.4213 −0.792315 −0.396157 0.918183i \(-0.629656\pi\)
−0.396157 + 0.918183i \(0.629656\pi\)
\(174\) 0.254048 0.0192594
\(175\) 0 0
\(176\) −3.93504 −0.296615
\(177\) 1.79143 0.134652
\(178\) 0.532642 0.0399232
\(179\) 20.5356 1.53490 0.767452 0.641107i \(-0.221524\pi\)
0.767452 + 0.641107i \(0.221524\pi\)
\(180\) 0 0
\(181\) 0.0667973 0.00496500 0.00248250 0.999997i \(-0.499210\pi\)
0.00248250 + 0.999997i \(0.499210\pi\)
\(182\) 1.55904 0.115564
\(183\) −2.57094 −0.190050
\(184\) −0.689049 −0.0507974
\(185\) 0 0
\(186\) −0.242606 −0.0177887
\(187\) −0.00602394 −0.000440514 0
\(188\) 1.77677 0.129585
\(189\) 12.3625 0.899238
\(190\) 0 0
\(191\) −2.09641 −0.151691 −0.0758456 0.997120i \(-0.524166\pi\)
−0.0758456 + 0.997120i \(0.524166\pi\)
\(192\) −5.39535 −0.389376
\(193\) −25.1960 −1.81365 −0.906825 0.421507i \(-0.861501\pi\)
−0.906825 + 0.421507i \(0.861501\pi\)
\(194\) 1.72318 0.123717
\(195\) 0 0
\(196\) −6.65640 −0.475457
\(197\) −5.70885 −0.406739 −0.203369 0.979102i \(-0.565189\pi\)
−0.203369 + 0.979102i \(0.565189\pi\)
\(198\) 0.261838 0.0186080
\(199\) −10.2003 −0.723081 −0.361541 0.932356i \(-0.617749\pi\)
−0.361541 + 0.932356i \(0.617749\pi\)
\(200\) 0 0
\(201\) −4.17252 −0.294307
\(202\) 0.586162 0.0412422
\(203\) −11.2576 −0.790131
\(204\) −0.00835178 −0.000584741 0
\(205\) 0 0
\(206\) 0.787708 0.0548822
\(207\) −4.17000 −0.289835
\(208\) −18.3139 −1.26984
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 1.03962 0.0715706 0.0357853 0.999360i \(-0.488607\pi\)
0.0357853 + 0.999360i \(0.488607\pi\)
\(212\) 3.54926 0.243764
\(213\) 0.594437 0.0407302
\(214\) −0.521877 −0.0356748
\(215\) 0 0
\(216\) 1.59671 0.108642
\(217\) 10.7506 0.729797
\(218\) 0.388618 0.0263205
\(219\) −3.29529 −0.222675
\(220\) 0 0
\(221\) −0.0280358 −0.00188589
\(222\) −0.445501 −0.0299001
\(223\) −6.42634 −0.430339 −0.215170 0.976577i \(-0.569030\pi\)
−0.215170 + 0.976577i \(0.569030\pi\)
\(224\) −3.99080 −0.266647
\(225\) 0 0
\(226\) 1.76734 0.117561
\(227\) −6.25815 −0.415368 −0.207684 0.978196i \(-0.566593\pi\)
−0.207684 + 0.978196i \(0.566593\pi\)
\(228\) 1.38643 0.0918186
\(229\) 26.2671 1.73578 0.867890 0.496757i \(-0.165476\pi\)
0.867890 + 0.496757i \(0.165476\pi\)
\(230\) 0 0
\(231\) 2.24194 0.147509
\(232\) −1.45401 −0.0954605
\(233\) −2.94990 −0.193255 −0.0966273 0.995321i \(-0.530805\pi\)
−0.0966273 + 0.995321i \(0.530805\pi\)
\(234\) 1.21861 0.0796629
\(235\) 0 0
\(236\) −5.11257 −0.332800
\(237\) −3.44384 −0.223701
\(238\) −0.00201794 −0.000130803 0
\(239\) 18.5942 1.20276 0.601378 0.798965i \(-0.294619\pi\)
0.601378 + 0.798965i \(0.294619\pi\)
\(240\) 0 0
\(241\) −8.68183 −0.559246 −0.279623 0.960110i \(-0.590209\pi\)
−0.279623 + 0.960110i \(0.590209\pi\)
\(242\) 0.104144 0.00669461
\(243\) 14.9201 0.957127
\(244\) 7.33722 0.469717
\(245\) 0 0
\(246\) −0.680896 −0.0434123
\(247\) 4.65405 0.296130
\(248\) 1.38852 0.0881711
\(249\) 10.4786 0.664052
\(250\) 0 0
\(251\) −27.7431 −1.75113 −0.875565 0.483100i \(-0.839511\pi\)
−0.875565 + 0.483100i \(0.839511\pi\)
\(252\) −16.0865 −1.01335
\(253\) −1.65858 −0.104274
\(254\) −0.0906655 −0.00568886
\(255\) 0 0
\(256\) 15.1394 0.946210
\(257\) 17.7003 1.10412 0.552059 0.833805i \(-0.313842\pi\)
0.552059 + 0.833805i \(0.313842\pi\)
\(258\) −0.625517 −0.0389430
\(259\) 19.7415 1.22667
\(260\) 0 0
\(261\) −8.79940 −0.544669
\(262\) −0.0346492 −0.00214064
\(263\) −22.4026 −1.38140 −0.690702 0.723139i \(-0.742698\pi\)
−0.690702 + 0.723139i \(0.742698\pi\)
\(264\) 0.289563 0.0178214
\(265\) 0 0
\(266\) 0.334986 0.0205393
\(267\) 3.56477 0.218161
\(268\) 11.9080 0.727394
\(269\) 16.1724 0.986046 0.493023 0.870016i \(-0.335892\pi\)
0.493023 + 0.870016i \(0.335892\pi\)
\(270\) 0 0
\(271\) 2.09973 0.127550 0.0637748 0.997964i \(-0.479686\pi\)
0.0637748 + 0.997964i \(0.479686\pi\)
\(272\) 0.0237045 0.00143729
\(273\) 10.4341 0.631500
\(274\) 1.35714 0.0819876
\(275\) 0 0
\(276\) −2.29950 −0.138414
\(277\) −29.9790 −1.80127 −0.900633 0.434580i \(-0.856897\pi\)
−0.900633 + 0.434580i \(0.856897\pi\)
\(278\) 1.32082 0.0792178
\(279\) 8.40307 0.503078
\(280\) 0 0
\(281\) −0.469805 −0.0280262 −0.0140131 0.999902i \(-0.504461\pi\)
−0.0140131 + 0.999902i \(0.504461\pi\)
\(282\) −0.0648376 −0.00386102
\(283\) 13.2122 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(284\) −1.69647 −0.100667
\(285\) 0 0
\(286\) 0.484690 0.0286603
\(287\) 30.1725 1.78103
\(288\) −3.11937 −0.183810
\(289\) −17.0000 −0.999998
\(290\) 0 0
\(291\) 11.5326 0.676054
\(292\) 9.40443 0.550353
\(293\) −5.47146 −0.319646 −0.159823 0.987146i \(-0.551092\pi\)
−0.159823 + 0.987146i \(0.551092\pi\)
\(294\) 0.242903 0.0141664
\(295\) 0 0
\(296\) 2.54976 0.148202
\(297\) 3.84337 0.223015
\(298\) 1.79590 0.104033
\(299\) −7.71912 −0.446408
\(300\) 0 0
\(301\) 27.7185 1.59767
\(302\) 2.34821 0.135124
\(303\) 3.92297 0.225369
\(304\) −3.93504 −0.225690
\(305\) 0 0
\(306\) −0.00157730 −9.01681e−5 0
\(307\) −30.8787 −1.76234 −0.881169 0.472801i \(-0.843243\pi\)
−0.881169 + 0.472801i \(0.843243\pi\)
\(308\) −6.39826 −0.364575
\(309\) 5.27183 0.299904
\(310\) 0 0
\(311\) 24.2528 1.37525 0.687625 0.726066i \(-0.258653\pi\)
0.687625 + 0.726066i \(0.258653\pi\)
\(312\) 1.34764 0.0762953
\(313\) 12.2922 0.694796 0.347398 0.937718i \(-0.387065\pi\)
0.347398 + 0.937718i \(0.387065\pi\)
\(314\) −1.26704 −0.0715031
\(315\) 0 0
\(316\) 9.82837 0.552889
\(317\) 20.0634 1.12687 0.563436 0.826160i \(-0.309479\pi\)
0.563436 + 0.826160i \(0.309479\pi\)
\(318\) −0.129519 −0.00726304
\(319\) −3.49988 −0.195956
\(320\) 0 0
\(321\) −3.49273 −0.194945
\(322\) −0.555601 −0.0309624
\(323\) −0.00602394 −0.000335181 0
\(324\) −9.67482 −0.537490
\(325\) 0 0
\(326\) −2.27123 −0.125792
\(327\) 2.60087 0.143829
\(328\) 3.89701 0.215176
\(329\) 2.87314 0.158402
\(330\) 0 0
\(331\) −28.3147 −1.55632 −0.778159 0.628067i \(-0.783846\pi\)
−0.778159 + 0.628067i \(0.783846\pi\)
\(332\) −29.9048 −1.64124
\(333\) 15.4307 0.845596
\(334\) 1.62286 0.0887988
\(335\) 0 0
\(336\) −8.82211 −0.481286
\(337\) −35.6067 −1.93962 −0.969810 0.243863i \(-0.921585\pi\)
−0.969810 + 0.243863i \(0.921585\pi\)
\(338\) 0.901907 0.0490573
\(339\) 11.8281 0.642415
\(340\) 0 0
\(341\) 3.34225 0.180993
\(342\) 0.261838 0.0141586
\(343\) 11.7522 0.634562
\(344\) 3.58006 0.193024
\(345\) 0 0
\(346\) −1.08531 −0.0583466
\(347\) −8.36165 −0.448877 −0.224438 0.974488i \(-0.572055\pi\)
−0.224438 + 0.974488i \(0.572055\pi\)
\(348\) −4.85235 −0.260113
\(349\) −26.3753 −1.41184 −0.705919 0.708293i \(-0.749466\pi\)
−0.705919 + 0.708293i \(0.749466\pi\)
\(350\) 0 0
\(351\) 17.8872 0.954750
\(352\) −1.24070 −0.0661296
\(353\) −13.3537 −0.710745 −0.355372 0.934725i \(-0.615646\pi\)
−0.355372 + 0.934725i \(0.615646\pi\)
\(354\) 0.186566 0.00991589
\(355\) 0 0
\(356\) −10.1735 −0.539195
\(357\) −0.0135053 −0.000714776 0
\(358\) 2.13865 0.113031
\(359\) 14.7872 0.780439 0.390220 0.920722i \(-0.372399\pi\)
0.390220 + 0.920722i \(0.372399\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0.00695651 0.000365626 0
\(363\) 0.696995 0.0365827
\(364\) −29.7778 −1.56078
\(365\) 0 0
\(366\) −0.267748 −0.0139954
\(367\) −26.2183 −1.36858 −0.684291 0.729209i \(-0.739888\pi\)
−0.684291 + 0.729209i \(0.739888\pi\)
\(368\) 6.52658 0.340221
\(369\) 23.5840 1.22773
\(370\) 0 0
\(371\) 5.73935 0.297972
\(372\) 4.63379 0.240251
\(373\) 1.30489 0.0675648 0.0337824 0.999429i \(-0.489245\pi\)
0.0337824 + 0.999429i \(0.489245\pi\)
\(374\) −0.000627356 0 −3.24398e−5 0
\(375\) 0 0
\(376\) 0.371089 0.0191374
\(377\) −16.2887 −0.838908
\(378\) 1.28747 0.0662205
\(379\) −10.0189 −0.514634 −0.257317 0.966327i \(-0.582838\pi\)
−0.257317 + 0.966327i \(0.582838\pi\)
\(380\) 0 0
\(381\) −0.606791 −0.0310868
\(382\) −0.218328 −0.0111706
\(383\) −5.44686 −0.278322 −0.139161 0.990270i \(-0.544440\pi\)
−0.139161 + 0.990270i \(0.544440\pi\)
\(384\) −2.29142 −0.116933
\(385\) 0 0
\(386\) −2.62401 −0.133559
\(387\) 21.6659 1.10134
\(388\) −32.9129 −1.67090
\(389\) −7.59394 −0.385028 −0.192514 0.981294i \(-0.561664\pi\)
−0.192514 + 0.981294i \(0.561664\pi\)
\(390\) 0 0
\(391\) 0.00999119 0.000505276 0
\(392\) −1.39022 −0.0702169
\(393\) −0.231894 −0.0116975
\(394\) −0.594541 −0.0299525
\(395\) 0 0
\(396\) −5.00113 −0.251316
\(397\) 6.27453 0.314910 0.157455 0.987526i \(-0.449671\pi\)
0.157455 + 0.987526i \(0.449671\pi\)
\(398\) −1.06230 −0.0532482
\(399\) 2.24194 0.112237
\(400\) 0 0
\(401\) 17.8048 0.889131 0.444565 0.895746i \(-0.353358\pi\)
0.444565 + 0.895746i \(0.353358\pi\)
\(402\) −0.434542 −0.0216730
\(403\) 15.5550 0.774849
\(404\) −11.1958 −0.557010
\(405\) 0 0
\(406\) −1.17241 −0.0581858
\(407\) 6.13742 0.304221
\(408\) −0.00174431 −8.63564e−5 0
\(409\) 8.47192 0.418909 0.209455 0.977818i \(-0.432831\pi\)
0.209455 + 0.977818i \(0.432831\pi\)
\(410\) 0 0
\(411\) 9.08281 0.448022
\(412\) −15.0453 −0.741228
\(413\) −8.26731 −0.406807
\(414\) −0.434279 −0.0213436
\(415\) 0 0
\(416\) −5.77429 −0.283108
\(417\) 8.83978 0.432886
\(418\) 0.104144 0.00509384
\(419\) 10.4558 0.510799 0.255399 0.966836i \(-0.417793\pi\)
0.255399 + 0.966836i \(0.417793\pi\)
\(420\) 0 0
\(421\) 32.4814 1.58305 0.791523 0.611139i \(-0.209288\pi\)
0.791523 + 0.611139i \(0.209288\pi\)
\(422\) 0.108270 0.00527051
\(423\) 2.24576 0.109193
\(424\) 0.741281 0.0359998
\(425\) 0 0
\(426\) 0.0619069 0.00299940
\(427\) 11.8647 0.574173
\(428\) 9.96790 0.481817
\(429\) 3.24385 0.156615
\(430\) 0 0
\(431\) −2.28726 −0.110174 −0.0550868 0.998482i \(-0.517544\pi\)
−0.0550868 + 0.998482i \(0.517544\pi\)
\(432\) −15.1238 −0.727645
\(433\) −36.5224 −1.75515 −0.877576 0.479437i \(-0.840841\pi\)
−0.877576 + 0.479437i \(0.840841\pi\)
\(434\) 1.11961 0.0537428
\(435\) 0 0
\(436\) −7.42264 −0.355480
\(437\) −1.65858 −0.0793406
\(438\) −0.343184 −0.0163980
\(439\) −23.0323 −1.09927 −0.549636 0.835404i \(-0.685233\pi\)
−0.549636 + 0.835404i \(0.685233\pi\)
\(440\) 0 0
\(441\) −8.41338 −0.400637
\(442\) −0.00291975 −0.000138878 0
\(443\) −3.63563 −0.172734 −0.0863671 0.996263i \(-0.527526\pi\)
−0.0863671 + 0.996263i \(0.527526\pi\)
\(444\) 8.50910 0.403824
\(445\) 0 0
\(446\) −0.669263 −0.0316905
\(447\) 12.0193 0.568492
\(448\) 24.8991 1.17637
\(449\) 25.5726 1.20685 0.603423 0.797422i \(-0.293803\pi\)
0.603423 + 0.797422i \(0.293803\pi\)
\(450\) 0 0
\(451\) 9.38032 0.441702
\(452\) −33.7563 −1.58776
\(453\) 15.7157 0.738388
\(454\) −0.651747 −0.0305880
\(455\) 0 0
\(456\) 0.289563 0.0135600
\(457\) −6.45150 −0.301789 −0.150894 0.988550i \(-0.548215\pi\)
−0.150894 + 0.988550i \(0.548215\pi\)
\(458\) 2.73555 0.127824
\(459\) −0.0231522 −0.00108065
\(460\) 0 0
\(461\) −23.1293 −1.07724 −0.538619 0.842550i \(-0.681054\pi\)
−0.538619 + 0.842550i \(0.681054\pi\)
\(462\) 0.233483 0.0108626
\(463\) −36.4434 −1.69367 −0.846834 0.531858i \(-0.821494\pi\)
−0.846834 + 0.531858i \(0.821494\pi\)
\(464\) 13.7722 0.639358
\(465\) 0 0
\(466\) −0.307214 −0.0142314
\(467\) 36.0540 1.66838 0.834191 0.551476i \(-0.185935\pi\)
0.834191 + 0.551476i \(0.185935\pi\)
\(468\) −23.2755 −1.07591
\(469\) 19.2558 0.889152
\(470\) 0 0
\(471\) −8.47981 −0.390729
\(472\) −1.06779 −0.0491488
\(473\) 8.61740 0.396228
\(474\) −0.358654 −0.0164735
\(475\) 0 0
\(476\) 0.0385428 0.00176660
\(477\) 4.48610 0.205404
\(478\) 1.93646 0.0885718
\(479\) 17.2250 0.787032 0.393516 0.919318i \(-0.371259\pi\)
0.393516 + 0.919318i \(0.371259\pi\)
\(480\) 0 0
\(481\) 28.5639 1.30240
\(482\) −0.904158 −0.0411832
\(483\) −3.71843 −0.169194
\(484\) −1.98915 −0.0904161
\(485\) 0 0
\(486\) 1.55384 0.0704835
\(487\) 31.6437 1.43391 0.716957 0.697117i \(-0.245534\pi\)
0.716957 + 0.697117i \(0.245534\pi\)
\(488\) 1.53242 0.0693692
\(489\) −15.2005 −0.687389
\(490\) 0 0
\(491\) 38.9066 1.75583 0.877916 0.478814i \(-0.158933\pi\)
0.877916 + 0.478814i \(0.158933\pi\)
\(492\) 13.0052 0.586318
\(493\) 0.0210831 0.000949536 0
\(494\) 0.484690 0.0218073
\(495\) 0 0
\(496\) −13.1519 −0.590537
\(497\) −2.74328 −0.123053
\(498\) 1.09128 0.0489013
\(499\) 16.4194 0.735033 0.367516 0.930017i \(-0.380208\pi\)
0.367516 + 0.930017i \(0.380208\pi\)
\(500\) 0 0
\(501\) 10.8612 0.485242
\(502\) −2.88927 −0.128954
\(503\) −4.73308 −0.211038 −0.105519 0.994417i \(-0.533650\pi\)
−0.105519 + 0.994417i \(0.533650\pi\)
\(504\) −3.35975 −0.149655
\(505\) 0 0
\(506\) −0.172731 −0.00767881
\(507\) 6.03613 0.268074
\(508\) 1.73172 0.0768326
\(509\) −32.8750 −1.45716 −0.728580 0.684961i \(-0.759819\pi\)
−0.728580 + 0.684961i \(0.759819\pi\)
\(510\) 0 0
\(511\) 15.2075 0.672740
\(512\) 8.15180 0.360262
\(513\) 3.84337 0.169689
\(514\) 1.84338 0.0813080
\(515\) 0 0
\(516\) 11.9474 0.525956
\(517\) 0.893231 0.0392843
\(518\) 2.05595 0.0903332
\(519\) −7.26357 −0.318835
\(520\) 0 0
\(521\) 7.50334 0.328727 0.164364 0.986400i \(-0.447443\pi\)
0.164364 + 0.986400i \(0.447443\pi\)
\(522\) −0.916402 −0.0401098
\(523\) −11.7845 −0.515300 −0.257650 0.966238i \(-0.582948\pi\)
−0.257650 + 0.966238i \(0.582948\pi\)
\(524\) 0.661803 0.0289110
\(525\) 0 0
\(526\) −2.33309 −0.101728
\(527\) −0.0201335 −0.000877029 0
\(528\) −2.74270 −0.119361
\(529\) −20.2491 −0.880396
\(530\) 0 0
\(531\) −6.46204 −0.280429
\(532\) −6.39826 −0.277400
\(533\) 43.6565 1.89097
\(534\) 0.371249 0.0160655
\(535\) 0 0
\(536\) 2.48704 0.107424
\(537\) 14.3132 0.617661
\(538\) 1.68425 0.0726131
\(539\) −3.34635 −0.144137
\(540\) 0 0
\(541\) 18.8061 0.808535 0.404268 0.914641i \(-0.367526\pi\)
0.404268 + 0.914641i \(0.367526\pi\)
\(542\) 0.218674 0.00939285
\(543\) 0.0465574 0.00199797
\(544\) 0.00747391 0.000320441 0
\(545\) 0 0
\(546\) 1.08664 0.0465041
\(547\) −11.1690 −0.477550 −0.238775 0.971075i \(-0.576746\pi\)
−0.238775 + 0.971075i \(0.576746\pi\)
\(548\) −25.9214 −1.10731
\(549\) 9.27390 0.395800
\(550\) 0 0
\(551\) −3.49988 −0.149100
\(552\) −0.480264 −0.0204414
\(553\) 15.8930 0.675840
\(554\) −3.12213 −0.132647
\(555\) 0 0
\(556\) −25.2279 −1.06990
\(557\) −26.3044 −1.11455 −0.557276 0.830327i \(-0.688154\pi\)
−0.557276 + 0.830327i \(0.688154\pi\)
\(558\) 0.875127 0.0370471
\(559\) 40.1058 1.69630
\(560\) 0 0
\(561\) −0.00419866 −0.000177267 0
\(562\) −0.0489273 −0.00206387
\(563\) −27.3566 −1.15294 −0.576472 0.817117i \(-0.695571\pi\)
−0.576472 + 0.817117i \(0.695571\pi\)
\(564\) 1.23840 0.0521462
\(565\) 0 0
\(566\) 1.37597 0.0578364
\(567\) −15.6447 −0.657017
\(568\) −0.354316 −0.0148668
\(569\) −17.7921 −0.745882 −0.372941 0.927855i \(-0.621651\pi\)
−0.372941 + 0.927855i \(0.621651\pi\)
\(570\) 0 0
\(571\) 25.9440 1.08572 0.542862 0.839822i \(-0.317341\pi\)
0.542862 + 0.839822i \(0.317341\pi\)
\(572\) −9.25763 −0.387081
\(573\) −1.46119 −0.0610421
\(574\) 3.14228 0.131156
\(575\) 0 0
\(576\) 19.4621 0.810920
\(577\) 13.7994 0.574476 0.287238 0.957859i \(-0.407263\pi\)
0.287238 + 0.957859i \(0.407263\pi\)
\(578\) −1.77044 −0.0736406
\(579\) −17.5615 −0.729831
\(580\) 0 0
\(581\) −48.3577 −2.00621
\(582\) 1.20105 0.0497851
\(583\) 1.78431 0.0738984
\(584\) 1.96417 0.0812777
\(585\) 0 0
\(586\) −0.569818 −0.0235390
\(587\) −31.3230 −1.29284 −0.646420 0.762982i \(-0.723734\pi\)
−0.646420 + 0.762982i \(0.723734\pi\)
\(588\) −4.63947 −0.191329
\(589\) 3.34225 0.137715
\(590\) 0 0
\(591\) −3.97904 −0.163676
\(592\) −24.1510 −0.992600
\(593\) 33.8912 1.39174 0.695872 0.718166i \(-0.255018\pi\)
0.695872 + 0.718166i \(0.255018\pi\)
\(594\) 0.400263 0.0164230
\(595\) 0 0
\(596\) −34.3018 −1.40506
\(597\) −7.10957 −0.290975
\(598\) −0.803897 −0.0328738
\(599\) −20.5165 −0.838280 −0.419140 0.907922i \(-0.637668\pi\)
−0.419140 + 0.907922i \(0.637668\pi\)
\(600\) 0 0
\(601\) 23.2037 0.946501 0.473250 0.880928i \(-0.343081\pi\)
0.473250 + 0.880928i \(0.343081\pi\)
\(602\) 2.88671 0.117653
\(603\) 15.0511 0.612928
\(604\) −44.8510 −1.82496
\(605\) 0 0
\(606\) 0.408552 0.0165963
\(607\) −2.28113 −0.0925881 −0.0462941 0.998928i \(-0.514741\pi\)
−0.0462941 + 0.998928i \(0.514741\pi\)
\(608\) −1.24070 −0.0503170
\(609\) −7.84652 −0.317957
\(610\) 0 0
\(611\) 4.15715 0.168180
\(612\) 0.0301265 0.00121779
\(613\) −2.06670 −0.0834732 −0.0417366 0.999129i \(-0.513289\pi\)
−0.0417366 + 0.999129i \(0.513289\pi\)
\(614\) −3.21582 −0.129780
\(615\) 0 0
\(616\) −1.33631 −0.0538415
\(617\) −39.7201 −1.59907 −0.799536 0.600618i \(-0.794921\pi\)
−0.799536 + 0.600618i \(0.794921\pi\)
\(618\) 0.549028 0.0220852
\(619\) 3.94724 0.158653 0.0793266 0.996849i \(-0.474723\pi\)
0.0793266 + 0.996849i \(0.474723\pi\)
\(620\) 0 0
\(621\) −6.37453 −0.255801
\(622\) 2.52578 0.101274
\(623\) −16.4511 −0.659101
\(624\) −12.7647 −0.510997
\(625\) 0 0
\(626\) 1.28016 0.0511653
\(627\) 0.696995 0.0278353
\(628\) 24.2005 0.965707
\(629\) −0.0369715 −0.00147415
\(630\) 0 0
\(631\) −12.3504 −0.491663 −0.245832 0.969313i \(-0.579061\pi\)
−0.245832 + 0.969313i \(0.579061\pi\)
\(632\) 2.05271 0.0816523
\(633\) 0.724612 0.0288007
\(634\) 2.08947 0.0829836
\(635\) 0 0
\(636\) 2.47382 0.0980932
\(637\) −15.5741 −0.617067
\(638\) −0.364491 −0.0144303
\(639\) −2.14425 −0.0848253
\(640\) 0 0
\(641\) 44.1646 1.74440 0.872198 0.489153i \(-0.162694\pi\)
0.872198 + 0.489153i \(0.162694\pi\)
\(642\) −0.363746 −0.0143559
\(643\) 7.27326 0.286829 0.143415 0.989663i \(-0.454192\pi\)
0.143415 + 0.989663i \(0.454192\pi\)
\(644\) 10.6120 0.418172
\(645\) 0 0
\(646\) −0.000627356 0 −2.46830e−5 0
\(647\) 11.7738 0.462878 0.231439 0.972849i \(-0.425657\pi\)
0.231439 + 0.972849i \(0.425657\pi\)
\(648\) −2.02064 −0.0793781
\(649\) −2.57022 −0.100890
\(650\) 0 0
\(651\) 7.49310 0.293678
\(652\) 43.3806 1.69892
\(653\) 31.7958 1.24427 0.622134 0.782911i \(-0.286266\pi\)
0.622134 + 0.782911i \(0.286266\pi\)
\(654\) 0.270865 0.0105916
\(655\) 0 0
\(656\) −36.9120 −1.44117
\(657\) 11.8868 0.463747
\(658\) 0.299220 0.0116648
\(659\) −23.9090 −0.931361 −0.465680 0.884953i \(-0.654190\pi\)
−0.465680 + 0.884953i \(0.654190\pi\)
\(660\) 0 0
\(661\) −39.9343 −1.55327 −0.776633 0.629953i \(-0.783074\pi\)
−0.776633 + 0.629953i \(0.783074\pi\)
\(662\) −2.94880 −0.114608
\(663\) −0.0195408 −0.000758901 0
\(664\) −6.24577 −0.242383
\(665\) 0 0
\(666\) 1.60701 0.0622703
\(667\) 5.80484 0.224764
\(668\) −30.9967 −1.19930
\(669\) −4.47913 −0.173173
\(670\) 0 0
\(671\) 3.68861 0.142397
\(672\) −2.78157 −0.107301
\(673\) 15.3378 0.591228 0.295614 0.955308i \(-0.404476\pi\)
0.295614 + 0.955308i \(0.404476\pi\)
\(674\) −3.70821 −0.142835
\(675\) 0 0
\(676\) −17.2265 −0.662558
\(677\) 20.2183 0.777054 0.388527 0.921437i \(-0.372984\pi\)
0.388527 + 0.921437i \(0.372984\pi\)
\(678\) 1.23182 0.0473079
\(679\) −53.2220 −2.04247
\(680\) 0 0
\(681\) −4.36190 −0.167148
\(682\) 0.348074 0.0133284
\(683\) 9.75882 0.373411 0.186705 0.982416i \(-0.440219\pi\)
0.186705 + 0.982416i \(0.440219\pi\)
\(684\) −5.00113 −0.191223
\(685\) 0 0
\(686\) 1.22392 0.0467296
\(687\) 18.3080 0.698495
\(688\) −33.9098 −1.29280
\(689\) 8.30425 0.316367
\(690\) 0 0
\(691\) −3.10119 −0.117975 −0.0589873 0.998259i \(-0.518787\pi\)
−0.0589873 + 0.998259i \(0.518787\pi\)
\(692\) 20.7295 0.788018
\(693\) −8.08710 −0.307204
\(694\) −0.870813 −0.0330556
\(695\) 0 0
\(696\) −1.01344 −0.0384143
\(697\) −0.0565066 −0.00214034
\(698\) −2.74682 −0.103969
\(699\) −2.05607 −0.0777676
\(700\) 0 0
\(701\) 1.49983 0.0566478 0.0283239 0.999599i \(-0.490983\pi\)
0.0283239 + 0.999599i \(0.490983\pi\)
\(702\) 1.86284 0.0703085
\(703\) 6.13742 0.231477
\(704\) 7.74087 0.291745
\(705\) 0 0
\(706\) −1.39070 −0.0523397
\(707\) −18.1042 −0.680877
\(708\) −3.56343 −0.133922
\(709\) −33.3647 −1.25304 −0.626518 0.779407i \(-0.715520\pi\)
−0.626518 + 0.779407i \(0.715520\pi\)
\(710\) 0 0
\(711\) 12.4226 0.465884
\(712\) −2.12479 −0.0796299
\(713\) −5.54338 −0.207601
\(714\) −0.00140649 −5.26366e−5 0
\(715\) 0 0
\(716\) −40.8485 −1.52658
\(717\) 12.9600 0.484001
\(718\) 1.53999 0.0574721
\(719\) −11.9782 −0.446711 −0.223355 0.974737i \(-0.571701\pi\)
−0.223355 + 0.974737i \(0.571701\pi\)
\(720\) 0 0
\(721\) −24.3291 −0.906062
\(722\) 0.104144 0.00387583
\(723\) −6.05119 −0.225046
\(724\) −0.132870 −0.00493808
\(725\) 0 0
\(726\) 0.0725876 0.00269398
\(727\) 4.56220 0.169203 0.0846014 0.996415i \(-0.473038\pi\)
0.0846014 + 0.996415i \(0.473038\pi\)
\(728\) −6.21926 −0.230501
\(729\) −4.19210 −0.155263
\(730\) 0 0
\(731\) −0.0519107 −0.00191999
\(732\) 5.11400 0.189019
\(733\) −22.2267 −0.820963 −0.410482 0.911869i \(-0.634639\pi\)
−0.410482 + 0.911869i \(0.634639\pi\)
\(734\) −2.73047 −0.100783
\(735\) 0 0
\(736\) 2.05780 0.0758515
\(737\) 5.98644 0.220513
\(738\) 2.45612 0.0904112
\(739\) 14.4390 0.531149 0.265574 0.964090i \(-0.414438\pi\)
0.265574 + 0.964090i \(0.414438\pi\)
\(740\) 0 0
\(741\) 3.24385 0.119166
\(742\) 0.597717 0.0219429
\(743\) 43.0239 1.57839 0.789197 0.614141i \(-0.210497\pi\)
0.789197 + 0.614141i \(0.210497\pi\)
\(744\) 0.967792 0.0354810
\(745\) 0 0
\(746\) 0.135896 0.00497552
\(747\) −37.7982 −1.38297
\(748\) 0.0119826 0.000438126 0
\(749\) 16.1187 0.588963
\(750\) 0 0
\(751\) 8.50342 0.310294 0.155147 0.987891i \(-0.450415\pi\)
0.155147 + 0.987891i \(0.450415\pi\)
\(752\) −3.51490 −0.128175
\(753\) −19.3368 −0.704673
\(754\) −1.69636 −0.0617778
\(755\) 0 0
\(756\) −24.5909 −0.894361
\(757\) 18.2012 0.661533 0.330766 0.943713i \(-0.392693\pi\)
0.330766 + 0.943713i \(0.392693\pi\)
\(758\) −1.04340 −0.0378980
\(759\) −1.15602 −0.0419609
\(760\) 0 0
\(761\) 45.3295 1.64319 0.821597 0.570069i \(-0.193083\pi\)
0.821597 + 0.570069i \(0.193083\pi\)
\(762\) −0.0631934 −0.00228926
\(763\) −12.0028 −0.434531
\(764\) 4.17009 0.150869
\(765\) 0 0
\(766\) −0.567256 −0.0204958
\(767\) −11.9619 −0.431921
\(768\) 10.5521 0.380765
\(769\) −27.6341 −0.996510 −0.498255 0.867031i \(-0.666026\pi\)
−0.498255 + 0.867031i \(0.666026\pi\)
\(770\) 0 0
\(771\) 12.3371 0.444308
\(772\) 50.1188 1.80382
\(773\) 27.7758 0.999027 0.499513 0.866306i \(-0.333512\pi\)
0.499513 + 0.866306i \(0.333512\pi\)
\(774\) 2.25636 0.0811033
\(775\) 0 0
\(776\) −6.87404 −0.246764
\(777\) 13.7597 0.493626
\(778\) −0.790861 −0.0283538
\(779\) 9.38032 0.336085
\(780\) 0 0
\(781\) −0.852858 −0.0305176
\(782\) 0.00104052 3.72089e−5 0
\(783\) −13.4513 −0.480712
\(784\) 13.1680 0.470286
\(785\) 0 0
\(786\) −0.0241503 −0.000861414 0
\(787\) −1.98342 −0.0707012 −0.0353506 0.999375i \(-0.511255\pi\)
−0.0353506 + 0.999375i \(0.511255\pi\)
\(788\) 11.3558 0.404533
\(789\) −15.6145 −0.555891
\(790\) 0 0
\(791\) −54.5858 −1.94085
\(792\) −1.04451 −0.0371151
\(793\) 17.1670 0.609618
\(794\) 0.653453 0.0231902
\(795\) 0 0
\(796\) 20.2900 0.719160
\(797\) −30.9790 −1.09733 −0.548666 0.836041i \(-0.684864\pi\)
−0.548666 + 0.836041i \(0.684864\pi\)
\(798\) 0.233483 0.00826523
\(799\) −0.00538078 −0.000190358 0
\(800\) 0 0
\(801\) −12.8588 −0.454345
\(802\) 1.85426 0.0654762
\(803\) 4.72786 0.166842
\(804\) 8.29979 0.292711
\(805\) 0 0
\(806\) 1.61995 0.0570605
\(807\) 11.2721 0.396795
\(808\) −2.33829 −0.0822608
\(809\) 33.4595 1.17637 0.588186 0.808725i \(-0.299842\pi\)
0.588186 + 0.808725i \(0.299842\pi\)
\(810\) 0 0
\(811\) −33.7987 −1.18683 −0.593416 0.804896i \(-0.702221\pi\)
−0.593416 + 0.804896i \(0.702221\pi\)
\(812\) 22.3932 0.785846
\(813\) 1.46350 0.0513273
\(814\) 0.639174 0.0224030
\(815\) 0 0
\(816\) 0.0165219 0.000578382 0
\(817\) 8.61740 0.301485
\(818\) 0.882297 0.0308488
\(819\) −37.6378 −1.31517
\(820\) 0 0
\(821\) −17.6687 −0.616641 −0.308320 0.951283i \(-0.599767\pi\)
−0.308320 + 0.951283i \(0.599767\pi\)
\(822\) 0.945917 0.0329927
\(823\) 46.0491 1.60517 0.802585 0.596538i \(-0.203457\pi\)
0.802585 + 0.596538i \(0.203457\pi\)
\(824\) −3.14229 −0.109467
\(825\) 0 0
\(826\) −0.860988 −0.0299576
\(827\) 23.9121 0.831503 0.415752 0.909478i \(-0.363519\pi\)
0.415752 + 0.909478i \(0.363519\pi\)
\(828\) 8.29476 0.288263
\(829\) −10.2688 −0.356649 −0.178324 0.983972i \(-0.557068\pi\)
−0.178324 + 0.983972i \(0.557068\pi\)
\(830\) 0 0
\(831\) −20.8952 −0.724848
\(832\) 36.0264 1.24899
\(833\) 0.0201582 0.000698440 0
\(834\) 0.920608 0.0318780
\(835\) 0 0
\(836\) −1.98915 −0.0687963
\(837\) 12.8455 0.444005
\(838\) 1.08890 0.0376156
\(839\) −40.5454 −1.39978 −0.699890 0.714250i \(-0.746768\pi\)
−0.699890 + 0.714250i \(0.746768\pi\)
\(840\) 0 0
\(841\) −16.7508 −0.577614
\(842\) 3.38273 0.116577
\(843\) −0.327452 −0.0112780
\(844\) −2.06797 −0.0711825
\(845\) 0 0
\(846\) 0.233882 0.00804102
\(847\) −3.21657 −0.110523
\(848\) −7.02132 −0.241113
\(849\) 9.20886 0.316047
\(850\) 0 0
\(851\) −10.1794 −0.348945
\(852\) −1.18243 −0.0405093
\(853\) 1.99694 0.0683740 0.0341870 0.999415i \(-0.489116\pi\)
0.0341870 + 0.999415i \(0.489116\pi\)
\(854\) 1.23563 0.0422825
\(855\) 0 0
\(856\) 2.08185 0.0711561
\(857\) −25.1830 −0.860235 −0.430118 0.902773i \(-0.641528\pi\)
−0.430118 + 0.902773i \(0.641528\pi\)
\(858\) 0.337827 0.0115332
\(859\) −0.915472 −0.0312355 −0.0156178 0.999878i \(-0.504971\pi\)
−0.0156178 + 0.999878i \(0.504971\pi\)
\(860\) 0 0
\(861\) 21.0301 0.716703
\(862\) −0.238204 −0.00811326
\(863\) 49.8812 1.69797 0.848987 0.528413i \(-0.177213\pi\)
0.848987 + 0.528413i \(0.177213\pi\)
\(864\) −4.76847 −0.162227
\(865\) 0 0
\(866\) −3.80357 −0.129251
\(867\) −11.8489 −0.402409
\(868\) −21.3846 −0.725839
\(869\) 4.94098 0.167611
\(870\) 0 0
\(871\) 27.8612 0.944042
\(872\) −1.55026 −0.0524983
\(873\) −41.6004 −1.40796
\(874\) −0.172731 −0.00584270
\(875\) 0 0
\(876\) 6.55484 0.221468
\(877\) −18.6769 −0.630673 −0.315337 0.948980i \(-0.602117\pi\)
−0.315337 + 0.948980i \(0.602117\pi\)
\(878\) −2.39867 −0.0809511
\(879\) −3.81358 −0.128629
\(880\) 0 0
\(881\) −46.9216 −1.58083 −0.790414 0.612573i \(-0.790134\pi\)
−0.790414 + 0.612573i \(0.790134\pi\)
\(882\) −0.876200 −0.0295032
\(883\) 44.8499 1.50932 0.754659 0.656117i \(-0.227802\pi\)
0.754659 + 0.656117i \(0.227802\pi\)
\(884\) 0.0557674 0.00187566
\(885\) 0 0
\(886\) −0.378628 −0.0127203
\(887\) −14.5451 −0.488376 −0.244188 0.969728i \(-0.578521\pi\)
−0.244188 + 0.969728i \(0.578521\pi\)
\(888\) 1.77717 0.0596380
\(889\) 2.80029 0.0939186
\(890\) 0 0
\(891\) −4.86379 −0.162943
\(892\) 12.7830 0.428006
\(893\) 0.893231 0.0298908
\(894\) 1.25173 0.0418641
\(895\) 0 0
\(896\) 10.5747 0.353276
\(897\) −5.38018 −0.179639
\(898\) 2.66322 0.0888730
\(899\) −11.6975 −0.390133
\(900\) 0 0
\(901\) −0.0107486 −0.000358086 0
\(902\) 0.976902 0.0325273
\(903\) 19.3197 0.642918
\(904\) −7.05017 −0.234485
\(905\) 0 0
\(906\) 1.63669 0.0543754
\(907\) −56.2751 −1.86858 −0.934292 0.356510i \(-0.883967\pi\)
−0.934292 + 0.356510i \(0.883967\pi\)
\(908\) 12.4484 0.413115
\(909\) −14.1509 −0.469356
\(910\) 0 0
\(911\) 11.8656 0.393123 0.196562 0.980491i \(-0.437022\pi\)
0.196562 + 0.980491i \(0.437022\pi\)
\(912\) −2.74270 −0.0908200
\(913\) −15.0339 −0.497550
\(914\) −0.671883 −0.0222239
\(915\) 0 0
\(916\) −52.2493 −1.72637
\(917\) 1.07017 0.0353402
\(918\) −0.00241116 −7.95801e−5 0
\(919\) 17.4567 0.575844 0.287922 0.957654i \(-0.407036\pi\)
0.287922 + 0.957654i \(0.407036\pi\)
\(920\) 0 0
\(921\) −21.5223 −0.709183
\(922\) −2.40877 −0.0793286
\(923\) −3.96924 −0.130649
\(924\) −4.45956 −0.146709
\(925\) 0 0
\(926\) −3.79535 −0.124723
\(927\) −19.0165 −0.624585
\(928\) 4.34231 0.142543
\(929\) 6.75452 0.221609 0.110804 0.993842i \(-0.464657\pi\)
0.110804 + 0.993842i \(0.464657\pi\)
\(930\) 0 0
\(931\) −3.34635 −0.109672
\(932\) 5.86781 0.192206
\(933\) 16.9041 0.553415
\(934\) 3.75480 0.122861
\(935\) 0 0
\(936\) −4.86121 −0.158894
\(937\) 45.1286 1.47429 0.737143 0.675736i \(-0.236174\pi\)
0.737143 + 0.675736i \(0.236174\pi\)
\(938\) 2.00537 0.0654778
\(939\) 8.56760 0.279593
\(940\) 0 0
\(941\) 26.7484 0.871972 0.435986 0.899954i \(-0.356400\pi\)
0.435986 + 0.899954i \(0.356400\pi\)
\(942\) −0.883119 −0.0287736
\(943\) −15.5580 −0.506639
\(944\) 10.1139 0.329180
\(945\) 0 0
\(946\) 0.897448 0.0291786
\(947\) 16.2900 0.529355 0.264677 0.964337i \(-0.414735\pi\)
0.264677 + 0.964337i \(0.414735\pi\)
\(948\) 6.85032 0.222488
\(949\) 22.0037 0.714270
\(950\) 0 0
\(951\) 13.9841 0.453465
\(952\) 0.00804986 0.000260897 0
\(953\) −4.10009 −0.132815 −0.0664075 0.997793i \(-0.521154\pi\)
−0.0664075 + 0.997793i \(0.521154\pi\)
\(954\) 0.467199 0.0151261
\(955\) 0 0
\(956\) −36.9866 −1.19623
\(957\) −2.43940 −0.0788547
\(958\) 1.79388 0.0579576
\(959\) −41.9164 −1.35355
\(960\) 0 0
\(961\) −19.8294 −0.639658
\(962\) 2.97475 0.0959097
\(963\) 12.5990 0.405996
\(964\) 17.2695 0.556213
\(965\) 0 0
\(966\) −0.387251 −0.0124596
\(967\) 34.0890 1.09623 0.548115 0.836403i \(-0.315346\pi\)
0.548115 + 0.836403i \(0.315346\pi\)
\(968\) −0.415445 −0.0133529
\(969\) −0.00419866 −0.000134880 0
\(970\) 0 0
\(971\) 40.8253 1.31015 0.655073 0.755565i \(-0.272638\pi\)
0.655073 + 0.755565i \(0.272638\pi\)
\(972\) −29.6785 −0.951937
\(973\) −40.7949 −1.30782
\(974\) 3.29550 0.105594
\(975\) 0 0
\(976\) −14.5148 −0.464609
\(977\) 14.7093 0.470591 0.235296 0.971924i \(-0.424394\pi\)
0.235296 + 0.971924i \(0.424394\pi\)
\(978\) −1.58303 −0.0506198
\(979\) −5.11449 −0.163460
\(980\) 0 0
\(981\) −9.38187 −0.299540
\(982\) 4.05188 0.129301
\(983\) −31.0823 −0.991372 −0.495686 0.868502i \(-0.665083\pi\)
−0.495686 + 0.868502i \(0.665083\pi\)
\(984\) 2.71620 0.0865892
\(985\) 0 0
\(986\) 0.00219567 6.99245e−5 0
\(987\) 2.00257 0.0637424
\(988\) −9.25763 −0.294524
\(989\) −14.2926 −0.454479
\(990\) 0 0
\(991\) 45.1569 1.43446 0.717228 0.696838i \(-0.245411\pi\)
0.717228 + 0.696838i \(0.245411\pi\)
\(992\) −4.14673 −0.131659
\(993\) −19.7352 −0.626279
\(994\) −0.285695 −0.00906170
\(995\) 0 0
\(996\) −20.8435 −0.660451
\(997\) 46.8604 1.48409 0.742043 0.670353i \(-0.233857\pi\)
0.742043 + 0.670353i \(0.233857\pi\)
\(998\) 1.70998 0.0541283
\(999\) 23.5884 0.746303
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 5225.2.a.bb.1.12 22
5.2 odd 4 1045.2.b.d.419.12 yes 22
5.3 odd 4 1045.2.b.d.419.11 22
5.4 even 2 inner 5225.2.a.bb.1.11 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.11 22 5.3 odd 4
1045.2.b.d.419.12 yes 22 5.2 odd 4
5225.2.a.bb.1.11 22 5.4 even 2 inner
5225.2.a.bb.1.12 22 1.1 even 1 trivial