Properties

Label 575.2.a.h.1.1
Level $575$
Weight $2$
Character 575.1
Self dual yes
Analytic conductor $4.591$
Analytic rank $0$
Dimension $4$
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] = [575,2,Mod(1,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("575.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.69353\) of defining polynomial
Character \(\chi\) \(=\) 575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.69353 q^{2} +2.56155 q^{3} +5.25508 q^{4} -6.89961 q^{6} +2.74252 q^{7} -8.76763 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q-2.69353 q^{2} +2.56155 q^{3} +5.25508 q^{4} -6.89961 q^{6} +2.74252 q^{7} -8.76763 q^{8} +3.56155 q^{9} -3.38705 q^{11} +13.4612 q^{12} +2.46356 q^{13} -7.38705 q^{14} +13.1057 q^{16} +2.64453 q^{17} -9.59313 q^{18} +3.38705 q^{19} +7.02511 q^{21} +9.12311 q^{22} +1.00000 q^{23} -22.4588 q^{24} -6.63566 q^{26} +1.43845 q^{27} +14.4122 q^{28} +9.20608 q^{29} -5.10809 q^{31} -17.7652 q^{32} -8.67611 q^{33} -7.12311 q^{34} +18.7162 q^{36} -5.50369 q^{37} -9.12311 q^{38} +6.31054 q^{39} -1.20608 q^{41} -18.9223 q^{42} -3.02511 q^{43} -17.7992 q^{44} -2.69353 q^{46} -8.21255 q^{47} +33.5709 q^{48} +0.521423 q^{49} +6.77410 q^{51} +12.9462 q^{52} +10.5417 q^{53} -3.87449 q^{54} -24.0454 q^{56} +8.67611 q^{57} -24.7968 q^{58} +8.28259 q^{59} +0.263945 q^{61} +13.7588 q^{62} +9.76763 q^{63} +21.6397 q^{64} +23.3693 q^{66} +7.66964 q^{67} +13.8972 q^{68} +2.56155 q^{69} -0.0150161 q^{71} -31.2264 q^{72} +5.53644 q^{73} +14.8243 q^{74} +17.7992 q^{76} -9.28906 q^{77} -16.9976 q^{78} -8.67611 q^{79} -7.00000 q^{81} +3.24861 q^{82} +3.52142 q^{83} +36.9175 q^{84} +8.14822 q^{86} +23.5819 q^{87} +29.6964 q^{88} -4.66318 q^{89} +6.75637 q^{91} +5.25508 q^{92} -13.0846 q^{93} +22.1207 q^{94} -45.5066 q^{96} +4.09799 q^{97} -1.40447 q^{98} -12.0632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 19 q^{12} - 12 q^{14} + 8 q^{16} + q^{17} - 3 q^{18} - 4 q^{19} + 10 q^{21} + 20 q^{22} + 4 q^{23} - 30 q^{24} - q^{26} + 14 q^{27} + 22 q^{28} + 19 q^{29} - q^{31} - 20 q^{32} + 2 q^{33} - 12 q^{34} + 23 q^{36} + 3 q^{37} - 20 q^{38} + 13 q^{41} - 6 q^{42} + 6 q^{43} - 18 q^{44} - 2 q^{46} - 6 q^{47} + 21 q^{48} + 9 q^{49} - 8 q^{51} + q^{52} - 19 q^{53} - 7 q^{54} - 10 q^{56} - 2 q^{57} - 21 q^{58} + 23 q^{59} + 13 q^{62} + 13 q^{63} + 27 q^{64} + 44 q^{66} + 3 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{71} - 39 q^{72} + 32 q^{73} - 12 q^{74} + 18 q^{76} - 18 q^{77} - 43 q^{78} + 2 q^{79} - 28 q^{81} + 5 q^{82} + 21 q^{83} + 28 q^{84} - 2 q^{86} + 18 q^{87} + 14 q^{88} - 40 q^{91} + 4 q^{92} + 8 q^{93} + 47 q^{94} - 61 q^{96} + 18 q^{97} - 16 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.69353 −1.90461 −0.952305 0.305148i \(-0.901294\pi\)
−0.952305 + 0.305148i \(0.901294\pi\)
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 5.25508 2.62754
\(5\) 0 0
\(6\) −6.89961 −2.81675
\(7\) 2.74252 1.03658 0.518288 0.855206i \(-0.326570\pi\)
0.518288 + 0.855206i \(0.326570\pi\)
\(8\) −8.76763 −3.09983
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −3.38705 −1.02123 −0.510617 0.859808i \(-0.670583\pi\)
−0.510617 + 0.859808i \(0.670583\pi\)
\(12\) 13.4612 3.88590
\(13\) 2.46356 0.683269 0.341634 0.939833i \(-0.389020\pi\)
0.341634 + 0.939833i \(0.389020\pi\)
\(14\) −7.38705 −1.97427
\(15\) 0 0
\(16\) 13.1057 3.27642
\(17\) 2.64453 0.641393 0.320696 0.947182i \(-0.396083\pi\)
0.320696 + 0.947182i \(0.396083\pi\)
\(18\) −9.59313 −2.26112
\(19\) 3.38705 0.777043 0.388521 0.921440i \(-0.372986\pi\)
0.388521 + 0.921440i \(0.372986\pi\)
\(20\) 0 0
\(21\) 7.02511 1.53301
\(22\) 9.12311 1.94505
\(23\) 1.00000 0.208514
\(24\) −22.4588 −4.58438
\(25\) 0 0
\(26\) −6.63566 −1.30136
\(27\) 1.43845 0.276829
\(28\) 14.4122 2.72364
\(29\) 9.20608 1.70953 0.854763 0.519018i \(-0.173702\pi\)
0.854763 + 0.519018i \(0.173702\pi\)
\(30\) 0 0
\(31\) −5.10809 −0.917440 −0.458720 0.888581i \(-0.651692\pi\)
−0.458720 + 0.888581i \(0.651692\pi\)
\(32\) −17.7652 −3.14048
\(33\) −8.67611 −1.51032
\(34\) −7.12311 −1.22160
\(35\) 0 0
\(36\) 18.7162 3.11937
\(37\) −5.50369 −0.904801 −0.452401 0.891815i \(-0.649432\pi\)
−0.452401 + 0.891815i \(0.649432\pi\)
\(38\) −9.12311 −1.47996
\(39\) 6.31054 1.01050
\(40\) 0 0
\(41\) −1.20608 −0.188358 −0.0941792 0.995555i \(-0.530023\pi\)
−0.0941792 + 0.995555i \(0.530023\pi\)
\(42\) −18.9223 −2.91978
\(43\) −3.02511 −0.461325 −0.230663 0.973034i \(-0.574089\pi\)
−0.230663 + 0.973034i \(0.574089\pi\)
\(44\) −17.7992 −2.68333
\(45\) 0 0
\(46\) −2.69353 −0.397139
\(47\) −8.21255 −1.19792 −0.598962 0.800778i \(-0.704420\pi\)
−0.598962 + 0.800778i \(0.704420\pi\)
\(48\) 33.5709 4.84554
\(49\) 0.521423 0.0744891
\(50\) 0 0
\(51\) 6.77410 0.948564
\(52\) 12.9462 1.79532
\(53\) 10.5417 1.44802 0.724009 0.689790i \(-0.242297\pi\)
0.724009 + 0.689790i \(0.242297\pi\)
\(54\) −3.87449 −0.527252
\(55\) 0 0
\(56\) −24.0454 −3.21321
\(57\) 8.67611 1.14918
\(58\) −24.7968 −3.25598
\(59\) 8.28259 1.07830 0.539151 0.842209i \(-0.318745\pi\)
0.539151 + 0.842209i \(0.318745\pi\)
\(60\) 0 0
\(61\) 0.263945 0.0337947 0.0168973 0.999857i \(-0.494621\pi\)
0.0168973 + 0.999857i \(0.494621\pi\)
\(62\) 13.7588 1.74737
\(63\) 9.76763 1.23061
\(64\) 21.6397 2.70497
\(65\) 0 0
\(66\) 23.3693 2.87656
\(67\) 7.66964 0.936996 0.468498 0.883465i \(-0.344795\pi\)
0.468498 + 0.883465i \(0.344795\pi\)
\(68\) 13.8972 1.68528
\(69\) 2.56155 0.308375
\(70\) 0 0
\(71\) −0.0150161 −0.00178208 −0.000891041 1.00000i \(-0.500284\pi\)
−0.000891041 1.00000i \(0.500284\pi\)
\(72\) −31.2264 −3.68007
\(73\) 5.53644 0.647991 0.323996 0.946059i \(-0.394974\pi\)
0.323996 + 0.946059i \(0.394974\pi\)
\(74\) 14.8243 1.72329
\(75\) 0 0
\(76\) 17.7992 2.04171
\(77\) −9.28906 −1.05859
\(78\) −16.9976 −1.92460
\(79\) −8.67611 −0.976138 −0.488069 0.872805i \(-0.662299\pi\)
−0.488069 + 0.872805i \(0.662299\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 3.24861 0.358749
\(83\) 3.52142 0.386526 0.193263 0.981147i \(-0.438093\pi\)
0.193263 + 0.981147i \(0.438093\pi\)
\(84\) 36.9175 4.02803
\(85\) 0 0
\(86\) 8.14822 0.878645
\(87\) 23.5819 2.52824
\(88\) 29.6964 3.16565
\(89\) −4.66318 −0.494296 −0.247148 0.968978i \(-0.579493\pi\)
−0.247148 + 0.968978i \(0.579493\pi\)
\(90\) 0 0
\(91\) 6.75637 0.708260
\(92\) 5.25508 0.547880
\(93\) −13.0846 −1.35681
\(94\) 22.1207 2.28158
\(95\) 0 0
\(96\) −45.5066 −4.64450
\(97\) 4.09799 0.416088 0.208044 0.978119i \(-0.433290\pi\)
0.208044 + 0.978119i \(0.433290\pi\)
\(98\) −1.40447 −0.141873
\(99\) −12.0632 −1.21239
\(100\) 0 0
\(101\) 12.2955 1.22345 0.611725 0.791070i \(-0.290476\pi\)
0.611725 + 0.791070i \(0.290476\pi\)
\(102\) −18.2462 −1.80664
\(103\) −6.05023 −0.596147 −0.298073 0.954543i \(-0.596344\pi\)
−0.298073 + 0.954543i \(0.596344\pi\)
\(104\) −21.5996 −2.11801
\(105\) 0 0
\(106\) −28.3944 −2.75791
\(107\) −13.1547 −1.27171 −0.635856 0.771808i \(-0.719353\pi\)
−0.635856 + 0.771808i \(0.719353\pi\)
\(108\) 7.55915 0.727380
\(109\) −17.1183 −1.63964 −0.819818 0.572624i \(-0.805925\pi\)
−0.819818 + 0.572624i \(0.805925\pi\)
\(110\) 0 0
\(111\) −14.0980 −1.33812
\(112\) 35.9426 3.39626
\(113\) −4.38058 −0.412091 −0.206045 0.978542i \(-0.566059\pi\)
−0.206045 + 0.978542i \(0.566059\pi\)
\(114\) −23.3693 −2.18874
\(115\) 0 0
\(116\) 48.3787 4.49185
\(117\) 8.77410 0.811166
\(118\) −22.3094 −2.05374
\(119\) 7.25268 0.664852
\(120\) 0 0
\(121\) 0.472110 0.0429191
\(122\) −0.710942 −0.0643657
\(123\) −3.08944 −0.278566
\(124\) −26.8434 −2.41061
\(125\) 0 0
\(126\) −26.3094 −2.34382
\(127\) 18.4588 1.63795 0.818975 0.573829i \(-0.194543\pi\)
0.818975 + 0.573829i \(0.194543\pi\)
\(128\) −22.7567 −2.01143
\(129\) −7.74899 −0.682260
\(130\) 0 0
\(131\) −3.86834 −0.337979 −0.168989 0.985618i \(-0.554050\pi\)
−0.168989 + 0.985618i \(0.554050\pi\)
\(132\) −45.5936 −3.96842
\(133\) 9.28906 0.805463
\(134\) −20.6584 −1.78461
\(135\) 0 0
\(136\) −23.1863 −1.98821
\(137\) −20.6713 −1.76607 −0.883034 0.469308i \(-0.844503\pi\)
−0.883034 + 0.469308i \(0.844503\pi\)
\(138\) −6.89961 −0.587334
\(139\) 12.5802 1.06704 0.533519 0.845788i \(-0.320869\pi\)
0.533519 + 0.845788i \(0.320869\pi\)
\(140\) 0 0
\(141\) −21.0369 −1.77162
\(142\) 0.0404462 0.00339417
\(143\) −8.34420 −0.697777
\(144\) 46.6766 3.88972
\(145\) 0 0
\(146\) −14.9125 −1.23417
\(147\) 1.33565 0.110163
\(148\) −28.9223 −2.37740
\(149\) 11.4850 0.940891 0.470446 0.882429i \(-0.344093\pi\)
0.470446 + 0.882429i \(0.344093\pi\)
\(150\) 0 0
\(151\) −5.58667 −0.454636 −0.227318 0.973821i \(-0.572996\pi\)
−0.227318 + 0.973821i \(0.572996\pi\)
\(152\) −29.6964 −2.40870
\(153\) 9.41863 0.761451
\(154\) 25.0203 2.01619
\(155\) 0 0
\(156\) 33.1624 2.65512
\(157\) −5.86563 −0.468128 −0.234064 0.972221i \(-0.575203\pi\)
−0.234064 + 0.972221i \(0.575203\pi\)
\(158\) 23.3693 1.85916
\(159\) 27.0032 2.14149
\(160\) 0 0
\(161\) 2.74252 0.216141
\(162\) 18.8547 1.48136
\(163\) −0.0465955 −0.00364964 −0.00182482 0.999998i \(-0.500581\pi\)
−0.00182482 + 0.999998i \(0.500581\pi\)
\(164\) −6.33805 −0.494919
\(165\) 0 0
\(166\) −9.48504 −0.736182
\(167\) 2.24621 0.173817 0.0869085 0.996216i \(-0.472301\pi\)
0.0869085 + 0.996216i \(0.472301\pi\)
\(168\) −61.5936 −4.75205
\(169\) −6.93087 −0.533144
\(170\) 0 0
\(171\) 12.0632 0.922493
\(172\) −15.8972 −1.21215
\(173\) −8.01293 −0.609212 −0.304606 0.952478i \(-0.598525\pi\)
−0.304606 + 0.952478i \(0.598525\pi\)
\(174\) −63.5183 −4.81531
\(175\) 0 0
\(176\) −44.3896 −3.34599
\(177\) 21.2163 1.59471
\(178\) 12.5604 0.941440
\(179\) −11.9615 −0.894047 −0.447024 0.894522i \(-0.647516\pi\)
−0.447024 + 0.894522i \(0.647516\pi\)
\(180\) 0 0
\(181\) −17.4802 −1.29930 −0.649648 0.760235i \(-0.725084\pi\)
−0.649648 + 0.760235i \(0.725084\pi\)
\(182\) −18.1984 −1.34896
\(183\) 0.676108 0.0499794
\(184\) −8.76763 −0.646359
\(185\) 0 0
\(186\) 35.2438 2.58420
\(187\) −8.95715 −0.655012
\(188\) −43.1576 −3.14759
\(189\) 3.94497 0.286954
\(190\) 0 0
\(191\) −13.1231 −0.949555 −0.474777 0.880106i \(-0.657471\pi\)
−0.474777 + 0.880106i \(0.657471\pi\)
\(192\) 55.4313 4.00041
\(193\) 12.5438 0.902924 0.451462 0.892290i \(-0.350903\pi\)
0.451462 + 0.892290i \(0.350903\pi\)
\(194\) −11.0380 −0.792485
\(195\) 0 0
\(196\) 2.74012 0.195723
\(197\) −3.88544 −0.276826 −0.138413 0.990375i \(-0.544200\pi\)
−0.138413 + 0.990375i \(0.544200\pi\)
\(198\) 32.4924 2.30914
\(199\) 22.1612 1.57096 0.785481 0.618886i \(-0.212416\pi\)
0.785481 + 0.618886i \(0.212416\pi\)
\(200\) 0 0
\(201\) 19.6462 1.38574
\(202\) −33.1183 −2.33020
\(203\) 25.2479 1.77205
\(204\) 35.5984 2.49239
\(205\) 0 0
\(206\) 16.2964 1.13543
\(207\) 3.56155 0.247545
\(208\) 32.2867 2.23868
\(209\) −11.4721 −0.793542
\(210\) 0 0
\(211\) 4.81048 0.331167 0.165584 0.986196i \(-0.447049\pi\)
0.165584 + 0.986196i \(0.447049\pi\)
\(212\) 55.3976 3.80473
\(213\) −0.0384645 −0.00263554
\(214\) 35.4325 2.42211
\(215\) 0 0
\(216\) −12.6118 −0.858123
\(217\) −14.0090 −0.950996
\(218\) 46.1086 3.12287
\(219\) 14.1819 0.958323
\(220\) 0 0
\(221\) 6.51496 0.438243
\(222\) 37.9733 2.54860
\(223\) −13.7815 −0.922876 −0.461438 0.887172i \(-0.652666\pi\)
−0.461438 + 0.887172i \(0.652666\pi\)
\(224\) −48.7215 −3.25534
\(225\) 0 0
\(226\) 11.7992 0.784872
\(227\) −29.1012 −1.93151 −0.965757 0.259447i \(-0.916460\pi\)
−0.965757 + 0.259447i \(0.916460\pi\)
\(228\) 45.5936 3.01951
\(229\) −9.38705 −0.620314 −0.310157 0.950685i \(-0.600382\pi\)
−0.310157 + 0.950685i \(0.600382\pi\)
\(230\) 0 0
\(231\) −23.7944 −1.56556
\(232\) −80.7156 −5.29924
\(233\) 12.6725 0.830202 0.415101 0.909775i \(-0.363746\pi\)
0.415101 + 0.909775i \(0.363746\pi\)
\(234\) −23.6333 −1.54495
\(235\) 0 0
\(236\) 43.5257 2.83328
\(237\) −22.2243 −1.44362
\(238\) −19.5353 −1.26628
\(239\) 19.1883 1.24119 0.620596 0.784131i \(-0.286891\pi\)
0.620596 + 0.784131i \(0.286891\pi\)
\(240\) 0 0
\(241\) 16.7741 1.08051 0.540257 0.841500i \(-0.318327\pi\)
0.540257 + 0.841500i \(0.318327\pi\)
\(242\) −1.27164 −0.0817442
\(243\) −22.2462 −1.42710
\(244\) 1.38705 0.0887968
\(245\) 0 0
\(246\) 8.32149 0.530559
\(247\) 8.34420 0.530929
\(248\) 44.7859 2.84391
\(249\) 9.02031 0.571639
\(250\) 0 0
\(251\) −22.8114 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(252\) 51.3297 3.23347
\(253\) −3.38705 −0.212942
\(254\) −49.7191 −3.11966
\(255\) 0 0
\(256\) 18.0162 1.12602
\(257\) 23.7526 1.48165 0.740824 0.671699i \(-0.234435\pi\)
0.740824 + 0.671699i \(0.234435\pi\)
\(258\) 20.8721 1.29944
\(259\) −15.0940 −0.937895
\(260\) 0 0
\(261\) 32.7879 2.02952
\(262\) 10.4195 0.643718
\(263\) −16.5895 −1.02295 −0.511476 0.859297i \(-0.670901\pi\)
−0.511476 + 0.859297i \(0.670901\pi\)
\(264\) 76.0689 4.68172
\(265\) 0 0
\(266\) −25.0203 −1.53409
\(267\) −11.9450 −0.731020
\(268\) 40.3046 2.46199
\(269\) −8.47495 −0.516727 −0.258363 0.966048i \(-0.583183\pi\)
−0.258363 + 0.966048i \(0.583183\pi\)
\(270\) 0 0
\(271\) −21.3029 −1.29406 −0.647030 0.762465i \(-0.723989\pi\)
−0.647030 + 0.762465i \(0.723989\pi\)
\(272\) 34.6584 2.10147
\(273\) 17.3068 1.04745
\(274\) 55.6787 3.36367
\(275\) 0 0
\(276\) 13.4612 0.810267
\(277\) −21.9615 −1.31954 −0.659770 0.751467i \(-0.729346\pi\)
−0.659770 + 0.751467i \(0.729346\pi\)
\(278\) −33.8851 −2.03229
\(279\) −18.1927 −1.08917
\(280\) 0 0
\(281\) 19.9020 1.18725 0.593627 0.804740i \(-0.297695\pi\)
0.593627 + 0.804740i \(0.297695\pi\)
\(282\) 56.6634 3.37425
\(283\) 8.87781 0.527731 0.263865 0.964559i \(-0.415003\pi\)
0.263865 + 0.964559i \(0.415003\pi\)
\(284\) −0.0789107 −0.00468249
\(285\) 0 0
\(286\) 22.4753 1.32899
\(287\) −3.30771 −0.195248
\(288\) −63.2718 −3.72833
\(289\) −10.0065 −0.588616
\(290\) 0 0
\(291\) 10.4972 0.615358
\(292\) 29.0944 1.70262
\(293\) −23.9709 −1.40039 −0.700197 0.713950i \(-0.746904\pi\)
−0.700197 + 0.713950i \(0.746904\pi\)
\(294\) −3.59762 −0.209817
\(295\) 0 0
\(296\) 48.2543 2.80473
\(297\) −4.87209 −0.282708
\(298\) −30.9353 −1.79203
\(299\) 2.46356 0.142471
\(300\) 0 0
\(301\) −8.29644 −0.478199
\(302\) 15.0478 0.865905
\(303\) 31.4956 1.80938
\(304\) 44.3896 2.54592
\(305\) 0 0
\(306\) −25.3693 −1.45027
\(307\) −12.0632 −0.688481 −0.344240 0.938882i \(-0.611864\pi\)
−0.344240 + 0.938882i \(0.611864\pi\)
\(308\) −48.8147 −2.78148
\(309\) −15.4980 −0.881649
\(310\) 0 0
\(311\) −6.81257 −0.386305 −0.193153 0.981169i \(-0.561871\pi\)
−0.193153 + 0.981169i \(0.561871\pi\)
\(312\) −55.3285 −3.13236
\(313\) 6.38539 0.360923 0.180462 0.983582i \(-0.442241\pi\)
0.180462 + 0.983582i \(0.442241\pi\)
\(314\) 15.7992 0.891601
\(315\) 0 0
\(316\) −45.5936 −2.56484
\(317\) −16.8114 −0.944222 −0.472111 0.881539i \(-0.656508\pi\)
−0.472111 + 0.881539i \(0.656508\pi\)
\(318\) −72.7338 −4.07871
\(319\) −31.1815 −1.74583
\(320\) 0 0
\(321\) −33.6964 −1.88075
\(322\) −7.38705 −0.411664
\(323\) 8.95715 0.498389
\(324\) −36.7855 −2.04364
\(325\) 0 0
\(326\) 0.125506 0.00695115
\(327\) −43.8494 −2.42488
\(328\) 10.5745 0.583878
\(329\) −22.5231 −1.24174
\(330\) 0 0
\(331\) 3.29114 0.180898 0.0904488 0.995901i \(-0.471170\pi\)
0.0904488 + 0.995901i \(0.471170\pi\)
\(332\) 18.5054 1.01561
\(333\) −19.6017 −1.07417
\(334\) −6.05023 −0.331054
\(335\) 0 0
\(336\) 92.0689 5.02277
\(337\) 7.90201 0.430450 0.215225 0.976565i \(-0.430952\pi\)
0.215225 + 0.976565i \(0.430952\pi\)
\(338\) 18.6685 1.01543
\(339\) −11.2211 −0.609446
\(340\) 0 0
\(341\) 17.3014 0.936921
\(342\) −32.4924 −1.75699
\(343\) −17.7676 −0.959362
\(344\) 26.5231 1.43003
\(345\) 0 0
\(346\) 21.5830 1.16031
\(347\) 19.4024 1.04158 0.520789 0.853686i \(-0.325638\pi\)
0.520789 + 0.853686i \(0.325638\pi\)
\(348\) 123.925 6.64305
\(349\) −33.1664 −1.77536 −0.887680 0.460462i \(-0.847684\pi\)
−0.887680 + 0.460462i \(0.847684\pi\)
\(350\) 0 0
\(351\) 3.54370 0.189149
\(352\) 60.1717 3.20716
\(353\) 29.8960 1.59121 0.795603 0.605819i \(-0.207154\pi\)
0.795603 + 0.605819i \(0.207154\pi\)
\(354\) −57.1466 −3.03731
\(355\) 0 0
\(356\) −24.5054 −1.29878
\(357\) 18.5781 0.983258
\(358\) 32.2187 1.70281
\(359\) −24.9725 −1.31800 −0.659000 0.752143i \(-0.729020\pi\)
−0.659000 + 0.752143i \(0.729020\pi\)
\(360\) 0 0
\(361\) −7.52789 −0.396205
\(362\) 47.0835 2.47465
\(363\) 1.20934 0.0634737
\(364\) 35.5052 1.86098
\(365\) 0 0
\(366\) −1.82112 −0.0951912
\(367\) −15.8179 −0.825686 −0.412843 0.910802i \(-0.635464\pi\)
−0.412843 + 0.910802i \(0.635464\pi\)
\(368\) 13.1057 0.683181
\(369\) −4.29552 −0.223616
\(370\) 0 0
\(371\) 28.9109 1.50098
\(372\) −68.7608 −3.56508
\(373\) −22.6283 −1.17165 −0.585826 0.810437i \(-0.699230\pi\)
−0.585826 + 0.810437i \(0.699230\pi\)
\(374\) 24.1263 1.24754
\(375\) 0 0
\(376\) 72.0046 3.71335
\(377\) 22.6797 1.16807
\(378\) −10.6259 −0.546536
\(379\) 10.2721 0.527641 0.263821 0.964572i \(-0.415017\pi\)
0.263821 + 0.964572i \(0.415017\pi\)
\(380\) 0 0
\(381\) 47.2831 2.42239
\(382\) 35.3474 1.80853
\(383\) −6.21463 −0.317553 −0.158776 0.987315i \(-0.550755\pi\)
−0.158776 + 0.987315i \(0.550755\pi\)
\(384\) −58.2924 −2.97472
\(385\) 0 0
\(386\) −33.7871 −1.71972
\(387\) −10.7741 −0.547678
\(388\) 21.5353 1.09329
\(389\) 27.6414 1.40147 0.700737 0.713420i \(-0.252855\pi\)
0.700737 + 0.713420i \(0.252855\pi\)
\(390\) 0 0
\(391\) 2.64453 0.133740
\(392\) −4.57165 −0.230903
\(393\) −9.90897 −0.499841
\(394\) 10.4655 0.527246
\(395\) 0 0
\(396\) −63.3928 −3.18561
\(397\) −1.07171 −0.0537875 −0.0268938 0.999638i \(-0.508562\pi\)
−0.0268938 + 0.999638i \(0.508562\pi\)
\(398\) −59.6916 −2.99207
\(399\) 23.7944 1.19121
\(400\) 0 0
\(401\) 1.49798 0.0748053 0.0374027 0.999300i \(-0.488092\pi\)
0.0374027 + 0.999300i \(0.488092\pi\)
\(402\) −52.9175 −2.63929
\(403\) −12.5841 −0.626858
\(404\) 64.6139 3.21466
\(405\) 0 0
\(406\) −68.0058 −3.37507
\(407\) 18.6413 0.924014
\(408\) −59.3928 −2.94038
\(409\) 20.9373 1.03528 0.517642 0.855597i \(-0.326810\pi\)
0.517642 + 0.855597i \(0.326810\pi\)
\(410\) 0 0
\(411\) −52.9506 −2.61186
\(412\) −31.7944 −1.56640
\(413\) 22.7152 1.11774
\(414\) −9.59313 −0.471477
\(415\) 0 0
\(416\) −43.7657 −2.14579
\(417\) 32.2248 1.57806
\(418\) 30.9004 1.51139
\(419\) −18.7564 −0.916309 −0.458154 0.888873i \(-0.651489\pi\)
−0.458154 + 0.888873i \(0.651489\pi\)
\(420\) 0 0
\(421\) 20.6163 1.00478 0.502388 0.864642i \(-0.332455\pi\)
0.502388 + 0.864642i \(0.332455\pi\)
\(422\) −12.9572 −0.630744
\(423\) −29.2494 −1.42216
\(424\) −92.4261 −4.48861
\(425\) 0 0
\(426\) 0.103605 0.00501968
\(427\) 0.723874 0.0350307
\(428\) −69.1289 −3.34147
\(429\) −21.3741 −1.03195
\(430\) 0 0
\(431\) 27.0155 1.30129 0.650646 0.759381i \(-0.274498\pi\)
0.650646 + 0.759381i \(0.274498\pi\)
\(432\) 18.8518 0.907010
\(433\) 25.2900 1.21536 0.607679 0.794183i \(-0.292101\pi\)
0.607679 + 0.794183i \(0.292101\pi\)
\(434\) 37.7337 1.81128
\(435\) 0 0
\(436\) −89.9580 −4.30821
\(437\) 3.38705 0.162025
\(438\) −38.1993 −1.82523
\(439\) 34.4110 1.64235 0.821174 0.570679i \(-0.193320\pi\)
0.821174 + 0.570679i \(0.193320\pi\)
\(440\) 0 0
\(441\) 1.85708 0.0884322
\(442\) −17.5482 −0.834683
\(443\) 29.8281 1.41717 0.708587 0.705623i \(-0.249333\pi\)
0.708587 + 0.705623i \(0.249333\pi\)
\(444\) −74.0860 −3.51597
\(445\) 0 0
\(446\) 37.1208 1.75772
\(447\) 29.4195 1.39150
\(448\) 59.3474 2.80390
\(449\) −2.67456 −0.126220 −0.0631102 0.998007i \(-0.520102\pi\)
−0.0631102 + 0.998007i \(0.520102\pi\)
\(450\) 0 0
\(451\) 4.08506 0.192358
\(452\) −23.0203 −1.08278
\(453\) −14.3105 −0.672368
\(454\) 78.3848 3.67878
\(455\) 0 0
\(456\) −76.0689 −3.56225
\(457\) 30.9037 1.44561 0.722806 0.691051i \(-0.242852\pi\)
0.722806 + 0.691051i \(0.242852\pi\)
\(458\) 25.2843 1.18146
\(459\) 3.80402 0.177556
\(460\) 0 0
\(461\) −26.6022 −1.23899 −0.619493 0.785002i \(-0.712662\pi\)
−0.619493 + 0.785002i \(0.712662\pi\)
\(462\) 64.0909 2.98178
\(463\) −26.7013 −1.24092 −0.620458 0.784239i \(-0.713053\pi\)
−0.620458 + 0.784239i \(0.713053\pi\)
\(464\) 120.652 5.60113
\(465\) 0 0
\(466\) −34.1336 −1.58121
\(467\) −35.2503 −1.63119 −0.815596 0.578622i \(-0.803590\pi\)
−0.815596 + 0.578622i \(0.803590\pi\)
\(468\) 46.1086 2.13137
\(469\) 21.0342 0.971267
\(470\) 0 0
\(471\) −15.0251 −0.692321
\(472\) −72.6187 −3.34255
\(473\) 10.2462 0.471121
\(474\) 59.8617 2.74954
\(475\) 0 0
\(476\) 38.1134 1.74692
\(477\) 37.5449 1.71907
\(478\) −51.6843 −2.36398
\(479\) −39.0705 −1.78518 −0.892589 0.450871i \(-0.851114\pi\)
−0.892589 + 0.450871i \(0.851114\pi\)
\(480\) 0 0
\(481\) −13.5587 −0.618222
\(482\) −45.1815 −2.05796
\(483\) 7.02511 0.319654
\(484\) 2.48098 0.112772
\(485\) 0 0
\(486\) 59.9207 2.71806
\(487\) −24.0839 −1.09135 −0.545673 0.837998i \(-0.683726\pi\)
−0.545673 + 0.837998i \(0.683726\pi\)
\(488\) −2.31417 −0.104758
\(489\) −0.119357 −0.00539751
\(490\) 0 0
\(491\) 5.60051 0.252748 0.126374 0.991983i \(-0.459666\pi\)
0.126374 + 0.991983i \(0.459666\pi\)
\(492\) −16.2353 −0.731942
\(493\) 24.3458 1.09648
\(494\) −22.4753 −1.01121
\(495\) 0 0
\(496\) −66.9450 −3.00592
\(497\) −0.0411819 −0.00184726
\(498\) −24.2964 −1.08875
\(499\) 9.53319 0.426764 0.213382 0.976969i \(-0.431552\pi\)
0.213382 + 0.976969i \(0.431552\pi\)
\(500\) 0 0
\(501\) 5.75379 0.257060
\(502\) 61.4431 2.74234
\(503\) −14.0568 −0.626762 −0.313381 0.949627i \(-0.601462\pi\)
−0.313381 + 0.949627i \(0.601462\pi\)
\(504\) −85.6391 −3.81467
\(505\) 0 0
\(506\) 9.12311 0.405572
\(507\) −17.7538 −0.788473
\(508\) 97.0022 4.30378
\(509\) 21.7105 0.962302 0.481151 0.876638i \(-0.340219\pi\)
0.481151 + 0.876638i \(0.340219\pi\)
\(510\) 0 0
\(511\) 15.1838 0.671692
\(512\) −3.01385 −0.133194
\(513\) 4.87209 0.215108
\(514\) −63.9783 −2.82196
\(515\) 0 0
\(516\) −40.7215 −1.79267
\(517\) 27.8163 1.22336
\(518\) 40.6560 1.78632
\(519\) −20.5255 −0.900972
\(520\) 0 0
\(521\) −8.87689 −0.388904 −0.194452 0.980912i \(-0.562293\pi\)
−0.194452 + 0.980912i \(0.562293\pi\)
\(522\) −88.3152 −3.86545
\(523\) 0.0550279 0.00240620 0.00120310 0.999999i \(-0.499617\pi\)
0.00120310 + 0.999999i \(0.499617\pi\)
\(524\) −20.3285 −0.888053
\(525\) 0 0
\(526\) 44.6842 1.94833
\(527\) −13.5085 −0.588439
\(528\) −113.706 −4.94843
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 29.4989 1.28014
\(532\) 48.8147 2.11639
\(533\) −2.97126 −0.128699
\(534\) 32.1741 1.39231
\(535\) 0 0
\(536\) −67.2446 −2.90453
\(537\) −30.6401 −1.32222
\(538\) 22.8275 0.984163
\(539\) −1.76609 −0.0760708
\(540\) 0 0
\(541\) 5.99070 0.257560 0.128780 0.991673i \(-0.458894\pi\)
0.128780 + 0.991673i \(0.458894\pi\)
\(542\) 57.3799 2.46468
\(543\) −44.7766 −1.92155
\(544\) −46.9807 −2.01428
\(545\) 0 0
\(546\) −46.6163 −1.99499
\(547\) 0.408533 0.0174676 0.00873380 0.999962i \(-0.497220\pi\)
0.00873380 + 0.999962i \(0.497220\pi\)
\(548\) −108.629 −4.64042
\(549\) 0.940053 0.0401205
\(550\) 0 0
\(551\) 31.1815 1.32837
\(552\) −22.4588 −0.955908
\(553\) −23.7944 −1.01184
\(554\) 59.1539 2.51321
\(555\) 0 0
\(556\) 66.1099 2.80369
\(557\) 5.46406 0.231519 0.115760 0.993277i \(-0.463070\pi\)
0.115760 + 0.993277i \(0.463070\pi\)
\(558\) 49.0026 2.07444
\(559\) −7.45255 −0.315209
\(560\) 0 0
\(561\) −22.9442 −0.968706
\(562\) −53.6066 −2.26126
\(563\) 41.8212 1.76255 0.881277 0.472601i \(-0.156685\pi\)
0.881277 + 0.472601i \(0.156685\pi\)
\(564\) −110.550 −4.65501
\(565\) 0 0
\(566\) −23.9126 −1.00512
\(567\) −19.1976 −0.806225
\(568\) 0.131656 0.00552415
\(569\) 39.5127 1.65646 0.828230 0.560388i \(-0.189348\pi\)
0.828230 + 0.560388i \(0.189348\pi\)
\(570\) 0 0
\(571\) −24.4924 −1.02498 −0.512488 0.858694i \(-0.671276\pi\)
−0.512488 + 0.858694i \(0.671276\pi\)
\(572\) −43.8494 −1.83344
\(573\) −33.6155 −1.40431
\(574\) 8.90939 0.371871
\(575\) 0 0
\(576\) 77.0710 3.21129
\(577\) 36.3382 1.51278 0.756390 0.654121i \(-0.226961\pi\)
0.756390 + 0.654121i \(0.226961\pi\)
\(578\) 26.9527 1.12108
\(579\) 32.1317 1.33535
\(580\) 0 0
\(581\) 9.65758 0.400664
\(582\) −28.2745 −1.17202
\(583\) −35.7054 −1.47877
\(584\) −48.5415 −2.00866
\(585\) 0 0
\(586\) 64.5662 2.66720
\(587\) −5.89358 −0.243254 −0.121627 0.992576i \(-0.538811\pi\)
−0.121627 + 0.992576i \(0.538811\pi\)
\(588\) 7.01896 0.289457
\(589\) −17.3014 −0.712890
\(590\) 0 0
\(591\) −9.95277 −0.409402
\(592\) −72.1296 −2.96451
\(593\) 11.0931 0.455538 0.227769 0.973715i \(-0.426857\pi\)
0.227769 + 0.973715i \(0.426857\pi\)
\(594\) 13.1231 0.538448
\(595\) 0 0
\(596\) 60.3548 2.47223
\(597\) 56.7670 2.32332
\(598\) −6.63566 −0.271352
\(599\) 40.1864 1.64197 0.820986 0.570949i \(-0.193425\pi\)
0.820986 + 0.570949i \(0.193425\pi\)
\(600\) 0 0
\(601\) 41.1965 1.68044 0.840220 0.542246i \(-0.182426\pi\)
0.840220 + 0.542246i \(0.182426\pi\)
\(602\) 22.3467 0.910782
\(603\) 27.3158 1.11239
\(604\) −29.3584 −1.19457
\(605\) 0 0
\(606\) −84.8343 −3.44616
\(607\) 16.6283 0.674924 0.337462 0.941339i \(-0.390432\pi\)
0.337462 + 0.941339i \(0.390432\pi\)
\(608\) −60.1717 −2.44029
\(609\) 64.6738 2.62071
\(610\) 0 0
\(611\) −20.2321 −0.818504
\(612\) 49.4956 2.00074
\(613\) −25.5052 −1.03015 −0.515073 0.857146i \(-0.672235\pi\)
−0.515073 + 0.857146i \(0.672235\pi\)
\(614\) 32.4924 1.31129
\(615\) 0 0
\(616\) 81.4431 3.28143
\(617\) 3.14742 0.126710 0.0633552 0.997991i \(-0.479820\pi\)
0.0633552 + 0.997991i \(0.479820\pi\)
\(618\) 41.7442 1.67920
\(619\) −39.2665 −1.57825 −0.789127 0.614230i \(-0.789467\pi\)
−0.789127 + 0.614230i \(0.789467\pi\)
\(620\) 0 0
\(621\) 1.43845 0.0577229
\(622\) 18.3498 0.735761
\(623\) −12.7889 −0.512375
\(624\) 82.7040 3.31081
\(625\) 0 0
\(626\) −17.1992 −0.687418
\(627\) −29.3864 −1.17358
\(628\) −30.8243 −1.23002
\(629\) −14.5547 −0.580333
\(630\) 0 0
\(631\) 18.2916 0.728179 0.364089 0.931364i \(-0.381380\pi\)
0.364089 + 0.931364i \(0.381380\pi\)
\(632\) 76.0689 3.02586
\(633\) 12.3223 0.489768
\(634\) 45.2819 1.79837
\(635\) 0 0
\(636\) 141.904 5.62686
\(637\) 1.28456 0.0508960
\(638\) 83.9881 3.32512
\(639\) −0.0534806 −0.00211566
\(640\) 0 0
\(641\) 28.5304 1.12688 0.563441 0.826157i \(-0.309477\pi\)
0.563441 + 0.826157i \(0.309477\pi\)
\(642\) 90.7622 3.58210
\(643\) 12.2430 0.482815 0.241408 0.970424i \(-0.422391\pi\)
0.241408 + 0.970424i \(0.422391\pi\)
\(644\) 14.4122 0.567919
\(645\) 0 0
\(646\) −24.1263 −0.949237
\(647\) −3.11947 −0.122639 −0.0613196 0.998118i \(-0.519531\pi\)
−0.0613196 + 0.998118i \(0.519531\pi\)
\(648\) 61.3734 2.41098
\(649\) −28.0536 −1.10120
\(650\) 0 0
\(651\) −35.8849 −1.40644
\(652\) −0.244863 −0.00958958
\(653\) 21.7301 0.850364 0.425182 0.905108i \(-0.360210\pi\)
0.425182 + 0.905108i \(0.360210\pi\)
\(654\) 118.110 4.61845
\(655\) 0 0
\(656\) −15.8065 −0.617141
\(657\) 19.7183 0.769285
\(658\) 60.6665 2.36503
\(659\) −12.2333 −0.476541 −0.238270 0.971199i \(-0.576580\pi\)
−0.238270 + 0.971199i \(0.576580\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −8.86477 −0.344539
\(663\) 16.6884 0.648124
\(664\) −30.8746 −1.19817
\(665\) 0 0
\(666\) 52.7976 2.04587
\(667\) 9.20608 0.356461
\(668\) 11.8040 0.456711
\(669\) −35.3020 −1.36485
\(670\) 0 0
\(671\) −0.893994 −0.0345123
\(672\) −124.803 −4.81437
\(673\) −10.7900 −0.415925 −0.207963 0.978137i \(-0.566683\pi\)
−0.207963 + 0.978137i \(0.566683\pi\)
\(674\) −21.2843 −0.819839
\(675\) 0 0
\(676\) −36.4223 −1.40086
\(677\) −23.6721 −0.909793 −0.454896 0.890544i \(-0.650324\pi\)
−0.454896 + 0.890544i \(0.650324\pi\)
\(678\) 30.2243 1.16076
\(679\) 11.2388 0.431307
\(680\) 0 0
\(681\) −74.5443 −2.85654
\(682\) −46.6016 −1.78447
\(683\) 10.3583 0.396350 0.198175 0.980167i \(-0.436499\pi\)
0.198175 + 0.980167i \(0.436499\pi\)
\(684\) 63.3928 2.42389
\(685\) 0 0
\(686\) 47.8576 1.82721
\(687\) −24.0454 −0.917390
\(688\) −39.6462 −1.51150
\(689\) 25.9702 0.989386
\(690\) 0 0
\(691\) 13.5578 0.515763 0.257882 0.966177i \(-0.416976\pi\)
0.257882 + 0.966177i \(0.416976\pi\)
\(692\) −42.1086 −1.60073
\(693\) −33.0835 −1.25674
\(694\) −52.2610 −1.98380
\(695\) 0 0
\(696\) −206.757 −7.83711
\(697\) −3.18952 −0.120812
\(698\) 89.3347 3.38137
\(699\) 32.4612 1.22780
\(700\) 0 0
\(701\) −21.3774 −0.807415 −0.403708 0.914888i \(-0.632279\pi\)
−0.403708 + 0.914888i \(0.632279\pi\)
\(702\) −9.54505 −0.360255
\(703\) −18.6413 −0.703069
\(704\) −73.2948 −2.76240
\(705\) 0 0
\(706\) −80.5257 −3.03063
\(707\) 33.7207 1.26820
\(708\) 111.493 4.19017
\(709\) 30.3719 1.14064 0.570320 0.821422i \(-0.306819\pi\)
0.570320 + 0.821422i \(0.306819\pi\)
\(710\) 0 0
\(711\) −30.9004 −1.15886
\(712\) 40.8850 1.53223
\(713\) −5.10809 −0.191299
\(714\) −50.0406 −1.87272
\(715\) 0 0
\(716\) −62.8588 −2.34914
\(717\) 49.1520 1.83561
\(718\) 67.2642 2.51028
\(719\) 8.41851 0.313958 0.156979 0.987602i \(-0.449825\pi\)
0.156979 + 0.987602i \(0.449825\pi\)
\(720\) 0 0
\(721\) −16.5929 −0.617951
\(722\) 20.2766 0.754615
\(723\) 42.9677 1.59799
\(724\) −91.8600 −3.41395
\(725\) 0 0
\(726\) −3.25738 −0.120893
\(727\) −19.7505 −0.732507 −0.366253 0.930515i \(-0.619360\pi\)
−0.366253 + 0.930515i \(0.619360\pi\)
\(728\) −59.2374 −2.19548
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 3.55300 0.131323
\(733\) −19.7280 −0.728670 −0.364335 0.931268i \(-0.618704\pi\)
−0.364335 + 0.931268i \(0.618704\pi\)
\(734\) 42.6058 1.57261
\(735\) 0 0
\(736\) −17.7652 −0.654835
\(737\) −25.9775 −0.956892
\(738\) 11.5701 0.425901
\(739\) −0.180969 −0.00665704 −0.00332852 0.999994i \(-0.501060\pi\)
−0.00332852 + 0.999994i \(0.501060\pi\)
\(740\) 0 0
\(741\) 21.3741 0.785198
\(742\) −77.8723 −2.85878
\(743\) −6.05023 −0.221961 −0.110981 0.993823i \(-0.535399\pi\)
−0.110981 + 0.993823i \(0.535399\pi\)
\(744\) 114.721 4.20589
\(745\) 0 0
\(746\) 60.9500 2.23154
\(747\) 12.5417 0.458878
\(748\) −47.0705 −1.72107
\(749\) −36.0770 −1.31823
\(750\) 0 0
\(751\) −21.5659 −0.786952 −0.393476 0.919335i \(-0.628728\pi\)
−0.393476 + 0.919335i \(0.628728\pi\)
\(752\) −107.631 −3.92490
\(753\) −58.4326 −2.12940
\(754\) −61.0885 −2.22471
\(755\) 0 0
\(756\) 20.7311 0.753984
\(757\) 24.7798 0.900638 0.450319 0.892868i \(-0.351310\pi\)
0.450319 + 0.892868i \(0.351310\pi\)
\(758\) −27.6681 −1.00495
\(759\) −8.67611 −0.314923
\(760\) 0 0
\(761\) 2.29916 0.0833443 0.0416722 0.999131i \(-0.486731\pi\)
0.0416722 + 0.999131i \(0.486731\pi\)
\(762\) −127.358 −4.61370
\(763\) −46.9473 −1.69961
\(764\) −68.9629 −2.49499
\(765\) 0 0
\(766\) 16.7393 0.604814
\(767\) 20.4047 0.736770
\(768\) 46.1496 1.66528
\(769\) −10.3692 −0.373923 −0.186961 0.982367i \(-0.559864\pi\)
−0.186961 + 0.982367i \(0.559864\pi\)
\(770\) 0 0
\(771\) 60.8436 2.19123
\(772\) 65.9187 2.37247
\(773\) −12.7864 −0.459895 −0.229947 0.973203i \(-0.573855\pi\)
−0.229947 + 0.973203i \(0.573855\pi\)
\(774\) 29.0203 1.04311
\(775\) 0 0
\(776\) −35.9297 −1.28980
\(777\) −38.6640 −1.38706
\(778\) −74.4528 −2.66926
\(779\) −4.08506 −0.146362
\(780\) 0 0
\(781\) 0.0508602 0.00181992
\(782\) −7.12311 −0.254722
\(783\) 13.2425 0.473247
\(784\) 6.83361 0.244058
\(785\) 0 0
\(786\) 26.6901 0.952003
\(787\) 34.9239 1.24490 0.622451 0.782659i \(-0.286137\pi\)
0.622451 + 0.782659i \(0.286137\pi\)
\(788\) −20.4183 −0.727372
\(789\) −42.4949 −1.51286
\(790\) 0 0
\(791\) −12.0138 −0.427163
\(792\) 105.765 3.75821
\(793\) 0.650244 0.0230908
\(794\) 2.88667 0.102444
\(795\) 0 0
\(796\) 116.459 4.12776
\(797\) −4.87844 −0.172803 −0.0864016 0.996260i \(-0.527537\pi\)
−0.0864016 + 0.996260i \(0.527537\pi\)
\(798\) −64.0909 −2.26879
\(799\) −21.7183 −0.768339
\(800\) 0 0
\(801\) −16.6081 −0.586820
\(802\) −4.03483 −0.142475
\(803\) −18.7522 −0.661751
\(804\) 103.242 3.64107
\(805\) 0 0
\(806\) 33.8956 1.19392
\(807\) −21.7090 −0.764194
\(808\) −107.803 −3.79248
\(809\) 18.1498 0.638112 0.319056 0.947736i \(-0.396634\pi\)
0.319056 + 0.947736i \(0.396634\pi\)
\(810\) 0 0
\(811\) 17.8019 0.625110 0.312555 0.949900i \(-0.398815\pi\)
0.312555 + 0.949900i \(0.398815\pi\)
\(812\) 132.680 4.65614
\(813\) −54.5685 −1.91380
\(814\) −50.2107 −1.75989
\(815\) 0 0
\(816\) 88.7793 3.10790
\(817\) −10.2462 −0.358470
\(818\) −56.3952 −1.97181
\(819\) 24.0632 0.840835
\(820\) 0 0
\(821\) 32.9701 1.15066 0.575332 0.817920i \(-0.304873\pi\)
0.575332 + 0.817920i \(0.304873\pi\)
\(822\) 142.624 4.97458
\(823\) 39.5819 1.37974 0.689869 0.723935i \(-0.257668\pi\)
0.689869 + 0.723935i \(0.257668\pi\)
\(824\) 53.0462 1.84795
\(825\) 0 0
\(826\) −61.1839 −2.12886
\(827\) 1.50849 0.0524554 0.0262277 0.999656i \(-0.491651\pi\)
0.0262277 + 0.999656i \(0.491651\pi\)
\(828\) 18.7162 0.650434
\(829\) −7.66718 −0.266292 −0.133146 0.991096i \(-0.542508\pi\)
−0.133146 + 0.991096i \(0.542508\pi\)
\(830\) 0 0
\(831\) −56.2556 −1.95149
\(832\) 53.3108 1.84822
\(833\) 1.37892 0.0477767
\(834\) −86.7984 −3.00558
\(835\) 0 0
\(836\) −60.2868 −2.08506
\(837\) −7.34772 −0.253974
\(838\) 50.5207 1.74521
\(839\) −8.53602 −0.294696 −0.147348 0.989085i \(-0.547074\pi\)
−0.147348 + 0.989085i \(0.547074\pi\)
\(840\) 0 0
\(841\) 55.7519 1.92248
\(842\) −55.5305 −1.91371
\(843\) 50.9800 1.75585
\(844\) 25.2795 0.870155
\(845\) 0 0
\(846\) 78.7841 2.70865
\(847\) 1.29477 0.0444889
\(848\) 138.157 4.74432
\(849\) 22.7410 0.780468
\(850\) 0 0
\(851\) −5.50369 −0.188664
\(852\) −0.202134 −0.00692500
\(853\) 48.0665 1.64577 0.822883 0.568211i \(-0.192364\pi\)
0.822883 + 0.568211i \(0.192364\pi\)
\(854\) −1.94977 −0.0667199
\(855\) 0 0
\(856\) 115.335 3.94209
\(857\) −29.4313 −1.00535 −0.502677 0.864474i \(-0.667652\pi\)
−0.502677 + 0.864474i \(0.667652\pi\)
\(858\) 57.5717 1.96547
\(859\) −33.5308 −1.14406 −0.572029 0.820234i \(-0.693843\pi\)
−0.572029 + 0.820234i \(0.693843\pi\)
\(860\) 0 0
\(861\) −8.47286 −0.288754
\(862\) −72.7670 −2.47845
\(863\) −6.79483 −0.231299 −0.115649 0.993290i \(-0.536895\pi\)
−0.115649 + 0.993290i \(0.536895\pi\)
\(864\) −25.5544 −0.869377
\(865\) 0 0
\(866\) −68.1192 −2.31478
\(867\) −25.6321 −0.870511
\(868\) −73.6186 −2.49878
\(869\) 29.3864 0.996866
\(870\) 0 0
\(871\) 18.8946 0.640220
\(872\) 150.087 5.08259
\(873\) 14.5952 0.493973
\(874\) −9.12311 −0.308594
\(875\) 0 0
\(876\) 74.5269 2.51803
\(877\) 21.9904 0.742563 0.371281 0.928520i \(-0.378918\pi\)
0.371281 + 0.928520i \(0.378918\pi\)
\(878\) −92.6869 −3.12803
\(879\) −61.4027 −2.07106
\(880\) 0 0
\(881\) 3.66242 0.123390 0.0616951 0.998095i \(-0.480349\pi\)
0.0616951 + 0.998095i \(0.480349\pi\)
\(882\) −5.00208 −0.168429
\(883\) −10.6640 −0.358874 −0.179437 0.983769i \(-0.557428\pi\)
−0.179437 + 0.983769i \(0.557428\pi\)
\(884\) 34.2366 1.15150
\(885\) 0 0
\(886\) −80.3427 −2.69916
\(887\) −0.481294 −0.0161603 −0.00808014 0.999967i \(-0.502572\pi\)
−0.00808014 + 0.999967i \(0.502572\pi\)
\(888\) 123.606 4.14795
\(889\) 50.6235 1.69786
\(890\) 0 0
\(891\) 23.7094 0.794293
\(892\) −72.4228 −2.42489
\(893\) −27.8163 −0.930837
\(894\) −79.2423 −2.65026
\(895\) 0 0
\(896\) −62.4107 −2.08499
\(897\) 6.31054 0.210703
\(898\) 7.20400 0.240401
\(899\) −47.0255 −1.56839
\(900\) 0 0
\(901\) 27.8779 0.928748
\(902\) −11.0032 −0.366367
\(903\) −21.2518 −0.707214
\(904\) 38.4074 1.27741
\(905\) 0 0
\(906\) 38.5458 1.28060
\(907\) 50.3078 1.67044 0.835222 0.549913i \(-0.185339\pi\)
0.835222 + 0.549913i \(0.185339\pi\)
\(908\) −152.929 −5.07513
\(909\) 43.7912 1.45246
\(910\) 0 0
\(911\) −39.5103 −1.30903 −0.654517 0.756047i \(-0.727128\pi\)
−0.654517 + 0.756047i \(0.727128\pi\)
\(912\) 113.706 3.76519
\(913\) −11.9272 −0.394734
\(914\) −83.2398 −2.75333
\(915\) 0 0
\(916\) −49.3297 −1.62990
\(917\) −10.6090 −0.350341
\(918\) −10.2462 −0.338175
\(919\) −10.0307 −0.330881 −0.165441 0.986220i \(-0.552905\pi\)
−0.165441 + 0.986220i \(0.552905\pi\)
\(920\) 0 0
\(921\) −30.9004 −1.01820
\(922\) 71.6536 2.35979
\(923\) −0.0369930 −0.00121764
\(924\) −125.041 −4.11356
\(925\) 0 0
\(926\) 71.9207 2.36346
\(927\) −21.5482 −0.707736
\(928\) −163.548 −5.36873
\(929\) 5.83676 0.191498 0.0957490 0.995406i \(-0.469475\pi\)
0.0957490 + 0.995406i \(0.469475\pi\)
\(930\) 0 0
\(931\) 1.76609 0.0578812
\(932\) 66.5949 2.18139
\(933\) −17.4507 −0.571312
\(934\) 94.9477 3.10678
\(935\) 0 0
\(936\) −76.9281 −2.51447
\(937\) −37.5175 −1.22564 −0.612822 0.790221i \(-0.709966\pi\)
−0.612822 + 0.790221i \(0.709966\pi\)
\(938\) −56.6560 −1.84989
\(939\) 16.3565 0.533774
\(940\) 0 0
\(941\) 45.6189 1.48713 0.743566 0.668662i \(-0.233133\pi\)
0.743566 + 0.668662i \(0.233133\pi\)
\(942\) 40.4705 1.31860
\(943\) −1.20608 −0.0392754
\(944\) 108.549 3.53297
\(945\) 0 0
\(946\) −27.5984 −0.897302
\(947\) 17.3357 0.563333 0.281667 0.959512i \(-0.409113\pi\)
0.281667 + 0.959512i \(0.409113\pi\)
\(948\) −116.790 −3.79318
\(949\) 13.6394 0.442752
\(950\) 0 0
\(951\) −43.0633 −1.39642
\(952\) −63.5888 −2.06093
\(953\) −14.2706 −0.462269 −0.231135 0.972922i \(-0.574244\pi\)
−0.231135 + 0.972922i \(0.574244\pi\)
\(954\) −101.128 −3.27415
\(955\) 0 0
\(956\) 100.836 3.26128
\(957\) −79.8730 −2.58193
\(958\) 105.237 3.40007
\(959\) −56.6915 −1.83066
\(960\) 0 0
\(961\) −4.90742 −0.158304
\(962\) 36.5206 1.17747
\(963\) −46.8511 −1.50976
\(964\) 88.1492 2.83909
\(965\) 0 0
\(966\) −18.9223 −0.608816
\(967\) 4.07663 0.131096 0.0655478 0.997849i \(-0.479121\pi\)
0.0655478 + 0.997849i \(0.479121\pi\)
\(968\) −4.13929 −0.133042
\(969\) 22.9442 0.737075
\(970\) 0 0
\(971\) −20.2988 −0.651419 −0.325709 0.945470i \(-0.605603\pi\)
−0.325709 + 0.945470i \(0.605603\pi\)
\(972\) −116.906 −3.74975
\(973\) 34.5015 1.10607
\(974\) 64.8706 2.07859
\(975\) 0 0
\(976\) 3.45918 0.110726
\(977\) 40.3782 1.29181 0.645907 0.763416i \(-0.276479\pi\)
0.645907 + 0.763416i \(0.276479\pi\)
\(978\) 0.321491 0.0102801
\(979\) 15.7944 0.504792
\(980\) 0 0
\(981\) −60.9677 −1.94655
\(982\) −15.0851 −0.481386
\(983\) 53.1628 1.69563 0.847815 0.530292i \(-0.177918\pi\)
0.847815 + 0.530292i \(0.177918\pi\)
\(984\) 27.0871 0.863505
\(985\) 0 0
\(986\) −65.5759 −2.08836
\(987\) −57.6941 −1.83642
\(988\) 43.8494 1.39504
\(989\) −3.02511 −0.0961930
\(990\) 0 0
\(991\) 36.6746 1.16501 0.582503 0.812829i \(-0.302073\pi\)
0.582503 + 0.812829i \(0.302073\pi\)
\(992\) 90.7464 2.88120
\(993\) 8.43043 0.267532
\(994\) 0.110925 0.00351831
\(995\) 0 0
\(996\) 47.4024 1.50200
\(997\) 11.1189 0.352140 0.176070 0.984378i \(-0.443661\pi\)
0.176070 + 0.984378i \(0.443661\pi\)
\(998\) −25.6779 −0.812819
\(999\) −7.91677 −0.250475
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 575.2.a.h.1.1 4
3.2 odd 2 5175.2.a.bx.1.4 4
4.3 odd 2 9200.2.a.cl.1.1 4
5.2 odd 4 575.2.b.e.24.1 8
5.3 odd 4 575.2.b.e.24.8 8
5.4 even 2 115.2.a.c.1.4 4
15.14 odd 2 1035.2.a.o.1.1 4
20.19 odd 2 1840.2.a.u.1.4 4
35.34 odd 2 5635.2.a.v.1.4 4
40.19 odd 2 7360.2.a.cg.1.2 4
40.29 even 2 7360.2.a.cj.1.3 4
115.114 odd 2 2645.2.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.a.c.1.4 4 5.4 even 2
575.2.a.h.1.1 4 1.1 even 1 trivial
575.2.b.e.24.1 8 5.2 odd 4
575.2.b.e.24.8 8 5.3 odd 4
1035.2.a.o.1.1 4 15.14 odd 2
1840.2.a.u.1.4 4 20.19 odd 2
2645.2.a.m.1.4 4 115.114 odd 2
5175.2.a.bx.1.4 4 3.2 odd 2
5635.2.a.v.1.4 4 35.34 odd 2
7360.2.a.cg.1.2 4 40.19 odd 2
7360.2.a.cj.1.3 4 40.29 even 2
9200.2.a.cl.1.1 4 4.3 odd 2