Properties

Label 5225.2.a.bb.1.13
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $22$
Inner twists $2$

Related objects

Downloads

Learn more

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 1.13
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)
 
\(f(q)\) \(=\) \(q+0.790175 q^{2} +2.57534 q^{3} -1.37562 q^{4} +2.03497 q^{6} +0.0669205 q^{7} -2.66733 q^{8} +3.63236 q^{9} +O(q^{10})\) \(q+0.790175 q^{2} +2.57534 q^{3} -1.37562 q^{4} +2.03497 q^{6} +0.0669205 q^{7} -2.66733 q^{8} +3.63236 q^{9} -1.00000 q^{11} -3.54270 q^{12} +4.14913 q^{13} +0.0528789 q^{14} +0.643589 q^{16} -6.61558 q^{17} +2.87020 q^{18} -1.00000 q^{19} +0.172343 q^{21} -0.790175 q^{22} +2.21109 q^{23} -6.86928 q^{24} +3.27854 q^{26} +1.62854 q^{27} -0.0920574 q^{28} +2.56775 q^{29} +8.77193 q^{31} +5.84321 q^{32} -2.57534 q^{33} -5.22747 q^{34} -4.99676 q^{36} +11.4277 q^{37} -0.790175 q^{38} +10.6854 q^{39} +10.9830 q^{41} +0.136181 q^{42} +3.95978 q^{43} +1.37562 q^{44} +1.74715 q^{46} -8.97459 q^{47} +1.65746 q^{48} -6.99552 q^{49} -17.0374 q^{51} -5.70765 q^{52} -3.13641 q^{53} +1.28683 q^{54} -0.178499 q^{56} -2.57534 q^{57} +2.02897 q^{58} +10.3362 q^{59} +2.76550 q^{61} +6.93135 q^{62} +0.243079 q^{63} +3.32998 q^{64} -2.03497 q^{66} +15.2686 q^{67} +9.10056 q^{68} +5.69431 q^{69} -4.11096 q^{71} -9.68871 q^{72} +7.16167 q^{73} +9.02987 q^{74} +1.37562 q^{76} -0.0669205 q^{77} +8.44334 q^{78} +8.94984 q^{79} -6.70304 q^{81} +8.67846 q^{82} +3.78408 q^{83} -0.237079 q^{84} +3.12892 q^{86} +6.61283 q^{87} +2.66733 q^{88} +0.223450 q^{89} +0.277662 q^{91} -3.04163 q^{92} +22.5907 q^{93} -7.09149 q^{94} +15.0482 q^{96} +14.1106 q^{97} -5.52768 q^{98} -3.63236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.790175 0.558738 0.279369 0.960184i \(-0.409875\pi\)
0.279369 + 0.960184i \(0.409875\pi\)
\(3\) 2.57534 1.48687 0.743436 0.668807i \(-0.233195\pi\)
0.743436 + 0.668807i \(0.233195\pi\)
\(4\) −1.37562 −0.687812
\(5\) 0 0
\(6\) 2.03497 0.830771
\(7\) 0.0669205 0.0252936 0.0126468 0.999920i \(-0.495974\pi\)
0.0126468 + 0.999920i \(0.495974\pi\)
\(8\) −2.66733 −0.943044
\(9\) 3.63236 1.21079
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −3.54270 −1.02269
\(13\) 4.14913 1.15076 0.575381 0.817885i \(-0.304854\pi\)
0.575381 + 0.817885i \(0.304854\pi\)
\(14\) 0.0528789 0.0141325
\(15\) 0 0
\(16\) 0.643589 0.160897
\(17\) −6.61558 −1.60451 −0.802257 0.596978i \(-0.796368\pi\)
−0.802257 + 0.596978i \(0.796368\pi\)
\(18\) 2.87020 0.676512
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.172343 0.0376083
\(22\) −0.790175 −0.168466
\(23\) 2.21109 0.461045 0.230522 0.973067i \(-0.425957\pi\)
0.230522 + 0.973067i \(0.425957\pi\)
\(24\) −6.86928 −1.40219
\(25\) 0 0
\(26\) 3.27854 0.642974
\(27\) 1.62854 0.313413
\(28\) −0.0920574 −0.0173972
\(29\) 2.56775 0.476820 0.238410 0.971165i \(-0.423374\pi\)
0.238410 + 0.971165i \(0.423374\pi\)
\(30\) 0 0
\(31\) 8.77193 1.57548 0.787742 0.616005i \(-0.211250\pi\)
0.787742 + 0.616005i \(0.211250\pi\)
\(32\) 5.84321 1.03294
\(33\) −2.57534 −0.448309
\(34\) −5.22747 −0.896503
\(35\) 0 0
\(36\) −4.99676 −0.832794
\(37\) 11.4277 1.87870 0.939350 0.342960i \(-0.111430\pi\)
0.939350 + 0.342960i \(0.111430\pi\)
\(38\) −0.790175 −0.128183
\(39\) 10.6854 1.71104
\(40\) 0 0
\(41\) 10.9830 1.71525 0.857626 0.514274i \(-0.171939\pi\)
0.857626 + 0.514274i \(0.171939\pi\)
\(42\) 0.136181 0.0210132
\(43\) 3.95978 0.603861 0.301931 0.953330i \(-0.402369\pi\)
0.301931 + 0.953330i \(0.402369\pi\)
\(44\) 1.37562 0.207383
\(45\) 0 0
\(46\) 1.74715 0.257603
\(47\) −8.97459 −1.30908 −0.654539 0.756028i \(-0.727137\pi\)
−0.654539 + 0.756028i \(0.727137\pi\)
\(48\) 1.65746 0.239234
\(49\) −6.99552 −0.999360
\(50\) 0 0
\(51\) −17.0374 −2.38571
\(52\) −5.70765 −0.791508
\(53\) −3.13641 −0.430819 −0.215410 0.976524i \(-0.569109\pi\)
−0.215410 + 0.976524i \(0.569109\pi\)
\(54\) 1.28683 0.175116
\(55\) 0 0
\(56\) −0.178499 −0.0238530
\(57\) −2.57534 −0.341112
\(58\) 2.02897 0.266417
\(59\) 10.3362 1.34566 0.672828 0.739799i \(-0.265079\pi\)
0.672828 + 0.739799i \(0.265079\pi\)
\(60\) 0 0
\(61\) 2.76550 0.354086 0.177043 0.984203i \(-0.443347\pi\)
0.177043 + 0.984203i \(0.443347\pi\)
\(62\) 6.93135 0.880283
\(63\) 0.243079 0.0306251
\(64\) 3.32998 0.416248
\(65\) 0 0
\(66\) −2.03497 −0.250487
\(67\) 15.2686 1.86535 0.932677 0.360713i \(-0.117467\pi\)
0.932677 + 0.360713i \(0.117467\pi\)
\(68\) 9.10056 1.10360
\(69\) 5.69431 0.685514
\(70\) 0 0
\(71\) −4.11096 −0.487881 −0.243940 0.969790i \(-0.578440\pi\)
−0.243940 + 0.969790i \(0.578440\pi\)
\(72\) −9.68871 −1.14183
\(73\) 7.16167 0.838210 0.419105 0.907938i \(-0.362344\pi\)
0.419105 + 0.907938i \(0.362344\pi\)
\(74\) 9.02987 1.04970
\(75\) 0 0
\(76\) 1.37562 0.157795
\(77\) −0.0669205 −0.00762630
\(78\) 8.44334 0.956020
\(79\) 8.94984 1.00694 0.503468 0.864014i \(-0.332057\pi\)
0.503468 + 0.864014i \(0.332057\pi\)
\(80\) 0 0
\(81\) −6.70304 −0.744782
\(82\) 8.67846 0.958376
\(83\) 3.78408 0.415357 0.207679 0.978197i \(-0.433409\pi\)
0.207679 + 0.978197i \(0.433409\pi\)
\(84\) −0.237079 −0.0258674
\(85\) 0 0
\(86\) 3.12892 0.337400
\(87\) 6.61283 0.708970
\(88\) 2.66733 0.284339
\(89\) 0.223450 0.0236856 0.0118428 0.999930i \(-0.496230\pi\)
0.0118428 + 0.999930i \(0.496230\pi\)
\(90\) 0 0
\(91\) 0.277662 0.0291069
\(92\) −3.04163 −0.317112
\(93\) 22.5907 2.34254
\(94\) −7.09149 −0.731431
\(95\) 0 0
\(96\) 15.0482 1.53585
\(97\) 14.1106 1.43272 0.716359 0.697732i \(-0.245807\pi\)
0.716359 + 0.697732i \(0.245807\pi\)
\(98\) −5.52768 −0.558380
\(99\) −3.63236 −0.365066
\(100\) 0 0
\(101\) −11.6761 −1.16182 −0.580909 0.813968i \(-0.697303\pi\)
−0.580909 + 0.813968i \(0.697303\pi\)
\(102\) −13.4625 −1.33298
\(103\) −11.6529 −1.14820 −0.574098 0.818786i \(-0.694647\pi\)
−0.574098 + 0.818786i \(0.694647\pi\)
\(104\) −11.0671 −1.08522
\(105\) 0 0
\(106\) −2.47831 −0.240715
\(107\) 2.72445 0.263382 0.131691 0.991291i \(-0.457959\pi\)
0.131691 + 0.991291i \(0.457959\pi\)
\(108\) −2.24026 −0.215569
\(109\) −3.96882 −0.380144 −0.190072 0.981770i \(-0.560872\pi\)
−0.190072 + 0.981770i \(0.560872\pi\)
\(110\) 0 0
\(111\) 29.4301 2.79339
\(112\) 0.0430693 0.00406967
\(113\) 2.81732 0.265031 0.132516 0.991181i \(-0.457695\pi\)
0.132516 + 0.991181i \(0.457695\pi\)
\(114\) −2.03497 −0.190592
\(115\) 0 0
\(116\) −3.53226 −0.327962
\(117\) 15.0711 1.39333
\(118\) 8.16739 0.751869
\(119\) −0.442718 −0.0405839
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.18523 0.197841
\(123\) 28.2848 2.55036
\(124\) −12.0669 −1.08364
\(125\) 0 0
\(126\) 0.192075 0.0171114
\(127\) −1.05561 −0.0936707 −0.0468353 0.998903i \(-0.514914\pi\)
−0.0468353 + 0.998903i \(0.514914\pi\)
\(128\) −9.05516 −0.800371
\(129\) 10.1978 0.897864
\(130\) 0 0
\(131\) −18.2156 −1.59151 −0.795753 0.605621i \(-0.792925\pi\)
−0.795753 + 0.605621i \(0.792925\pi\)
\(132\) 3.54270 0.308352
\(133\) −0.0669205 −0.00580274
\(134\) 12.0648 1.04224
\(135\) 0 0
\(136\) 17.6460 1.51313
\(137\) 7.14378 0.610334 0.305167 0.952299i \(-0.401288\pi\)
0.305167 + 0.952299i \(0.401288\pi\)
\(138\) 4.49950 0.383023
\(139\) 2.25667 0.191408 0.0957041 0.995410i \(-0.469490\pi\)
0.0957041 + 0.995410i \(0.469490\pi\)
\(140\) 0 0
\(141\) −23.1126 −1.94643
\(142\) −3.24837 −0.272598
\(143\) −4.14913 −0.346968
\(144\) 2.33775 0.194812
\(145\) 0 0
\(146\) 5.65897 0.468340
\(147\) −18.0158 −1.48592
\(148\) −15.7202 −1.29219
\(149\) 5.60288 0.459006 0.229503 0.973308i \(-0.426290\pi\)
0.229503 + 0.973308i \(0.426290\pi\)
\(150\) 0 0
\(151\) −20.8631 −1.69782 −0.848908 0.528541i \(-0.822739\pi\)
−0.848908 + 0.528541i \(0.822739\pi\)
\(152\) 2.66733 0.216349
\(153\) −24.0302 −1.94273
\(154\) −0.0528789 −0.00426110
\(155\) 0 0
\(156\) −14.6991 −1.17687
\(157\) −0.664796 −0.0530565 −0.0265282 0.999648i \(-0.508445\pi\)
−0.0265282 + 0.999648i \(0.508445\pi\)
\(158\) 7.07194 0.562613
\(159\) −8.07732 −0.640573
\(160\) 0 0
\(161\) 0.147967 0.0116615
\(162\) −5.29657 −0.416138
\(163\) 0.357030 0.0279647 0.0139824 0.999902i \(-0.495549\pi\)
0.0139824 + 0.999902i \(0.495549\pi\)
\(164\) −15.1084 −1.17977
\(165\) 0 0
\(166\) 2.99009 0.232076
\(167\) −12.2396 −0.947127 −0.473563 0.880760i \(-0.657033\pi\)
−0.473563 + 0.880760i \(0.657033\pi\)
\(168\) −0.459696 −0.0354663
\(169\) 4.21530 0.324253
\(170\) 0 0
\(171\) −3.63236 −0.277774
\(172\) −5.44717 −0.415343
\(173\) −14.4260 −1.09679 −0.548394 0.836220i \(-0.684761\pi\)
−0.548394 + 0.836220i \(0.684761\pi\)
\(174\) 5.22529 0.396128
\(175\) 0 0
\(176\) −0.643589 −0.0485124
\(177\) 26.6191 2.00082
\(178\) 0.176564 0.0132341
\(179\) 8.72275 0.651969 0.325984 0.945375i \(-0.394304\pi\)
0.325984 + 0.945375i \(0.394304\pi\)
\(180\) 0 0
\(181\) −13.9631 −1.03787 −0.518936 0.854813i \(-0.673672\pi\)
−0.518936 + 0.854813i \(0.673672\pi\)
\(182\) 0.219401 0.0162631
\(183\) 7.12209 0.526480
\(184\) −5.89772 −0.434786
\(185\) 0 0
\(186\) 17.8506 1.30887
\(187\) 6.61558 0.483779
\(188\) 12.3457 0.900400
\(189\) 0.108983 0.00792733
\(190\) 0 0
\(191\) 4.76262 0.344611 0.172306 0.985044i \(-0.444878\pi\)
0.172306 + 0.985044i \(0.444878\pi\)
\(192\) 8.57582 0.618907
\(193\) −5.69191 −0.409713 −0.204856 0.978792i \(-0.565673\pi\)
−0.204856 + 0.978792i \(0.565673\pi\)
\(194\) 11.1499 0.800514
\(195\) 0 0
\(196\) 9.62321 0.687372
\(197\) 19.1425 1.36385 0.681923 0.731424i \(-0.261144\pi\)
0.681923 + 0.731424i \(0.261144\pi\)
\(198\) −2.87020 −0.203976
\(199\) 21.1279 1.49772 0.748859 0.662730i \(-0.230602\pi\)
0.748859 + 0.662730i \(0.230602\pi\)
\(200\) 0 0
\(201\) 39.3217 2.77354
\(202\) −9.22619 −0.649152
\(203\) 0.171835 0.0120605
\(204\) 23.4370 1.64092
\(205\) 0 0
\(206\) −9.20784 −0.641541
\(207\) 8.03149 0.558227
\(208\) 2.67034 0.185155
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 21.9394 1.51037 0.755187 0.655510i \(-0.227546\pi\)
0.755187 + 0.655510i \(0.227546\pi\)
\(212\) 4.31452 0.296323
\(213\) −10.5871 −0.725416
\(214\) 2.15279 0.147162
\(215\) 0 0
\(216\) −4.34386 −0.295562
\(217\) 0.587022 0.0398496
\(218\) −3.13606 −0.212401
\(219\) 18.4437 1.24631
\(220\) 0 0
\(221\) −27.4489 −1.84641
\(222\) 23.2549 1.56077
\(223\) 23.0514 1.54363 0.771817 0.635844i \(-0.219348\pi\)
0.771817 + 0.635844i \(0.219348\pi\)
\(224\) 0.391031 0.0261268
\(225\) 0 0
\(226\) 2.22618 0.148083
\(227\) −12.1150 −0.804101 −0.402050 0.915618i \(-0.631702\pi\)
−0.402050 + 0.915618i \(0.631702\pi\)
\(228\) 3.54270 0.234621
\(229\) −17.0184 −1.12461 −0.562303 0.826931i \(-0.690085\pi\)
−0.562303 + 0.826931i \(0.690085\pi\)
\(230\) 0 0
\(231\) −0.172343 −0.0113393
\(232\) −6.84905 −0.449662
\(233\) −2.08834 −0.136812 −0.0684060 0.997658i \(-0.521791\pi\)
−0.0684060 + 0.997658i \(0.521791\pi\)
\(234\) 11.9088 0.778505
\(235\) 0 0
\(236\) −14.2187 −0.925558
\(237\) 23.0488 1.49718
\(238\) −0.349825 −0.0226758
\(239\) −16.8870 −1.09233 −0.546165 0.837677i \(-0.683913\pi\)
−0.546165 + 0.837677i \(0.683913\pi\)
\(240\) 0 0
\(241\) 19.6461 1.26551 0.632757 0.774351i \(-0.281923\pi\)
0.632757 + 0.774351i \(0.281923\pi\)
\(242\) 0.790175 0.0507944
\(243\) −22.1482 −1.42081
\(244\) −3.80429 −0.243545
\(245\) 0 0
\(246\) 22.3500 1.42498
\(247\) −4.14913 −0.264003
\(248\) −23.3976 −1.48575
\(249\) 9.74529 0.617583
\(250\) 0 0
\(251\) 21.9810 1.38743 0.693714 0.720251i \(-0.255973\pi\)
0.693714 + 0.720251i \(0.255973\pi\)
\(252\) −0.334386 −0.0210643
\(253\) −2.21109 −0.139010
\(254\) −0.834120 −0.0523374
\(255\) 0 0
\(256\) −13.8151 −0.863445
\(257\) −27.5365 −1.71768 −0.858839 0.512246i \(-0.828814\pi\)
−0.858839 + 0.512246i \(0.828814\pi\)
\(258\) 8.05802 0.501671
\(259\) 0.764746 0.0475190
\(260\) 0 0
\(261\) 9.32701 0.577327
\(262\) −14.3935 −0.889235
\(263\) −21.4214 −1.32090 −0.660449 0.750871i \(-0.729634\pi\)
−0.660449 + 0.750871i \(0.729634\pi\)
\(264\) 6.86928 0.422775
\(265\) 0 0
\(266\) −0.0528789 −0.00324221
\(267\) 0.575459 0.0352175
\(268\) −21.0038 −1.28301
\(269\) 8.06510 0.491738 0.245869 0.969303i \(-0.420927\pi\)
0.245869 + 0.969303i \(0.420927\pi\)
\(270\) 0 0
\(271\) −14.7370 −0.895211 −0.447606 0.894231i \(-0.647723\pi\)
−0.447606 + 0.894231i \(0.647723\pi\)
\(272\) −4.25772 −0.258162
\(273\) 0.715073 0.0432782
\(274\) 5.64483 0.341017
\(275\) 0 0
\(276\) −7.83323 −0.471505
\(277\) 32.5834 1.95775 0.978873 0.204470i \(-0.0655472\pi\)
0.978873 + 0.204470i \(0.0655472\pi\)
\(278\) 1.78316 0.106947
\(279\) 31.8628 1.90758
\(280\) 0 0
\(281\) −16.8550 −1.00548 −0.502741 0.864437i \(-0.667675\pi\)
−0.502741 + 0.864437i \(0.667675\pi\)
\(282\) −18.2630 −1.08754
\(283\) 6.26244 0.372263 0.186132 0.982525i \(-0.440405\pi\)
0.186132 + 0.982525i \(0.440405\pi\)
\(284\) 5.65513 0.335570
\(285\) 0 0
\(286\) −3.27854 −0.193864
\(287\) 0.734986 0.0433848
\(288\) 21.2247 1.25067
\(289\) 26.7659 1.57447
\(290\) 0 0
\(291\) 36.3396 2.13027
\(292\) −9.85177 −0.576531
\(293\) −7.67080 −0.448133 −0.224067 0.974574i \(-0.571933\pi\)
−0.224067 + 0.974574i \(0.571933\pi\)
\(294\) −14.2356 −0.830240
\(295\) 0 0
\(296\) −30.4814 −1.77170
\(297\) −1.62854 −0.0944976
\(298\) 4.42725 0.256464
\(299\) 9.17411 0.530553
\(300\) 0 0
\(301\) 0.264991 0.0152738
\(302\) −16.4855 −0.948634
\(303\) −30.0700 −1.72748
\(304\) −0.643589 −0.0369124
\(305\) 0 0
\(306\) −18.9880 −1.08547
\(307\) −0.00440478 −0.000251394 0 −0.000125697 1.00000i \(-0.500040\pi\)
−0.000125697 1.00000i \(0.500040\pi\)
\(308\) 0.0920574 0.00524546
\(309\) −30.0102 −1.70722
\(310\) 0 0
\(311\) 12.0604 0.683883 0.341942 0.939721i \(-0.388915\pi\)
0.341942 + 0.939721i \(0.388915\pi\)
\(312\) −28.5015 −1.61358
\(313\) 16.4882 0.931967 0.465983 0.884793i \(-0.345701\pi\)
0.465983 + 0.884793i \(0.345701\pi\)
\(314\) −0.525305 −0.0296447
\(315\) 0 0
\(316\) −12.3116 −0.692582
\(317\) 25.9591 1.45801 0.729005 0.684508i \(-0.239983\pi\)
0.729005 + 0.684508i \(0.239983\pi\)
\(318\) −6.38249 −0.357912
\(319\) −2.56775 −0.143767
\(320\) 0 0
\(321\) 7.01637 0.391616
\(322\) 0.116920 0.00651570
\(323\) 6.61558 0.368101
\(324\) 9.22086 0.512270
\(325\) 0 0
\(326\) 0.282116 0.0156249
\(327\) −10.2210 −0.565225
\(328\) −29.2952 −1.61756
\(329\) −0.600584 −0.0331113
\(330\) 0 0
\(331\) 25.4456 1.39862 0.699310 0.714819i \(-0.253491\pi\)
0.699310 + 0.714819i \(0.253491\pi\)
\(332\) −5.20548 −0.285688
\(333\) 41.5095 2.27470
\(334\) −9.67140 −0.529196
\(335\) 0 0
\(336\) 0.110918 0.00605107
\(337\) −28.8344 −1.57071 −0.785355 0.619046i \(-0.787520\pi\)
−0.785355 + 0.619046i \(0.787520\pi\)
\(338\) 3.33082 0.181173
\(339\) 7.25555 0.394067
\(340\) 0 0
\(341\) −8.77193 −0.475026
\(342\) −2.87020 −0.155203
\(343\) −0.936587 −0.0505710
\(344\) −10.5621 −0.569468
\(345\) 0 0
\(346\) −11.3991 −0.612817
\(347\) −20.2741 −1.08837 −0.544185 0.838965i \(-0.683161\pi\)
−0.544185 + 0.838965i \(0.683161\pi\)
\(348\) −9.09677 −0.487638
\(349\) 23.5879 1.26263 0.631316 0.775525i \(-0.282515\pi\)
0.631316 + 0.775525i \(0.282515\pi\)
\(350\) 0 0
\(351\) 6.75703 0.360664
\(352\) −5.84321 −0.311444
\(353\) −26.1715 −1.39297 −0.696484 0.717573i \(-0.745253\pi\)
−0.696484 + 0.717573i \(0.745253\pi\)
\(354\) 21.0338 1.11793
\(355\) 0 0
\(356\) −0.307383 −0.0162913
\(357\) −1.14015 −0.0603431
\(358\) 6.89249 0.364280
\(359\) 32.5322 1.71698 0.858492 0.512828i \(-0.171402\pi\)
0.858492 + 0.512828i \(0.171402\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −11.0333 −0.579898
\(363\) 2.57534 0.135170
\(364\) −0.381958 −0.0200201
\(365\) 0 0
\(366\) 5.62770 0.294164
\(367\) −9.12225 −0.476178 −0.238089 0.971243i \(-0.576521\pi\)
−0.238089 + 0.971243i \(0.576521\pi\)
\(368\) 1.42304 0.0741808
\(369\) 39.8941 2.07680
\(370\) 0 0
\(371\) −0.209890 −0.0108970
\(372\) −31.0763 −1.61123
\(373\) −32.3890 −1.67704 −0.838519 0.544872i \(-0.816578\pi\)
−0.838519 + 0.544872i \(0.816578\pi\)
\(374\) 5.22747 0.270306
\(375\) 0 0
\(376\) 23.9382 1.23452
\(377\) 10.6539 0.548706
\(378\) 0.0861155 0.00442930
\(379\) −20.8497 −1.07098 −0.535489 0.844542i \(-0.679873\pi\)
−0.535489 + 0.844542i \(0.679873\pi\)
\(380\) 0 0
\(381\) −2.71856 −0.139276
\(382\) 3.76330 0.192547
\(383\) −14.8606 −0.759342 −0.379671 0.925122i \(-0.623963\pi\)
−0.379671 + 0.925122i \(0.623963\pi\)
\(384\) −23.3201 −1.19005
\(385\) 0 0
\(386\) −4.49760 −0.228922
\(387\) 14.3834 0.731147
\(388\) −19.4109 −0.985441
\(389\) −2.79047 −0.141482 −0.0707412 0.997495i \(-0.522536\pi\)
−0.0707412 + 0.997495i \(0.522536\pi\)
\(390\) 0 0
\(391\) −14.6277 −0.739753
\(392\) 18.6594 0.942441
\(393\) −46.9114 −2.36637
\(394\) 15.1259 0.762032
\(395\) 0 0
\(396\) 4.99676 0.251097
\(397\) −9.55809 −0.479707 −0.239853 0.970809i \(-0.577099\pi\)
−0.239853 + 0.970809i \(0.577099\pi\)
\(398\) 16.6947 0.836832
\(399\) −0.172343 −0.00862793
\(400\) 0 0
\(401\) −9.84081 −0.491427 −0.245713 0.969343i \(-0.579022\pi\)
−0.245713 + 0.969343i \(0.579022\pi\)
\(402\) 31.0710 1.54968
\(403\) 36.3959 1.81301
\(404\) 16.0620 0.799113
\(405\) 0 0
\(406\) 0.135780 0.00673864
\(407\) −11.4277 −0.566449
\(408\) 45.4443 2.24983
\(409\) 5.23929 0.259066 0.129533 0.991575i \(-0.458652\pi\)
0.129533 + 0.991575i \(0.458652\pi\)
\(410\) 0 0
\(411\) 18.3976 0.907488
\(412\) 16.0300 0.789743
\(413\) 0.691702 0.0340364
\(414\) 6.34628 0.311902
\(415\) 0 0
\(416\) 24.2443 1.18867
\(417\) 5.81168 0.284599
\(418\) 0.790175 0.0386487
\(419\) −11.2351 −0.548870 −0.274435 0.961606i \(-0.588491\pi\)
−0.274435 + 0.961606i \(0.588491\pi\)
\(420\) 0 0
\(421\) 34.2914 1.67126 0.835630 0.549292i \(-0.185103\pi\)
0.835630 + 0.549292i \(0.185103\pi\)
\(422\) 17.3360 0.843903
\(423\) −32.5989 −1.58501
\(424\) 8.36586 0.406282
\(425\) 0 0
\(426\) −8.36566 −0.405318
\(427\) 0.185069 0.00895610
\(428\) −3.74782 −0.181158
\(429\) −10.6854 −0.515897
\(430\) 0 0
\(431\) 24.2325 1.16724 0.583620 0.812027i \(-0.301636\pi\)
0.583620 + 0.812027i \(0.301636\pi\)
\(432\) 1.04811 0.0504273
\(433\) −11.5535 −0.555228 −0.277614 0.960693i \(-0.589544\pi\)
−0.277614 + 0.960693i \(0.589544\pi\)
\(434\) 0.463850 0.0222655
\(435\) 0 0
\(436\) 5.45960 0.261467
\(437\) −2.21109 −0.105771
\(438\) 14.5738 0.696361
\(439\) −38.1975 −1.82307 −0.911534 0.411225i \(-0.865101\pi\)
−0.911534 + 0.411225i \(0.865101\pi\)
\(440\) 0 0
\(441\) −25.4103 −1.21001
\(442\) −21.6894 −1.03166
\(443\) 19.8322 0.942256 0.471128 0.882065i \(-0.343847\pi\)
0.471128 + 0.882065i \(0.343847\pi\)
\(444\) −40.4848 −1.92132
\(445\) 0 0
\(446\) 18.2146 0.862487
\(447\) 14.4293 0.682483
\(448\) 0.222844 0.0105284
\(449\) 19.6541 0.927532 0.463766 0.885958i \(-0.346498\pi\)
0.463766 + 0.885958i \(0.346498\pi\)
\(450\) 0 0
\(451\) −10.9830 −0.517168
\(452\) −3.87557 −0.182292
\(453\) −53.7295 −2.52443
\(454\) −9.57296 −0.449282
\(455\) 0 0
\(456\) 6.86928 0.321684
\(457\) 12.6434 0.591432 0.295716 0.955276i \(-0.404442\pi\)
0.295716 + 0.955276i \(0.404442\pi\)
\(458\) −13.4475 −0.628360
\(459\) −10.7738 −0.502876
\(460\) 0 0
\(461\) −2.24816 −0.104707 −0.0523536 0.998629i \(-0.516672\pi\)
−0.0523536 + 0.998629i \(0.516672\pi\)
\(462\) −0.136181 −0.00633571
\(463\) 16.2839 0.756778 0.378389 0.925647i \(-0.376478\pi\)
0.378389 + 0.925647i \(0.376478\pi\)
\(464\) 1.65258 0.0767190
\(465\) 0 0
\(466\) −1.65016 −0.0764420
\(467\) −2.08404 −0.0964378 −0.0482189 0.998837i \(-0.515355\pi\)
−0.0482189 + 0.998837i \(0.515355\pi\)
\(468\) −20.7322 −0.958347
\(469\) 1.02178 0.0471815
\(470\) 0 0
\(471\) −1.71207 −0.0788881
\(472\) −27.5700 −1.26901
\(473\) −3.95978 −0.182071
\(474\) 18.2126 0.836533
\(475\) 0 0
\(476\) 0.609014 0.0279141
\(477\) −11.3926 −0.521630
\(478\) −13.3437 −0.610326
\(479\) 16.3963 0.749164 0.374582 0.927194i \(-0.377786\pi\)
0.374582 + 0.927194i \(0.377786\pi\)
\(480\) 0 0
\(481\) 47.4150 2.16194
\(482\) 15.5238 0.707090
\(483\) 0.381066 0.0173391
\(484\) −1.37562 −0.0625284
\(485\) 0 0
\(486\) −17.5010 −0.793859
\(487\) −6.55183 −0.296892 −0.148446 0.988921i \(-0.547427\pi\)
−0.148446 + 0.988921i \(0.547427\pi\)
\(488\) −7.37651 −0.333919
\(489\) 0.919471 0.0415799
\(490\) 0 0
\(491\) 29.7599 1.34305 0.671524 0.740983i \(-0.265640\pi\)
0.671524 + 0.740983i \(0.265640\pi\)
\(492\) −38.9093 −1.75417
\(493\) −16.9872 −0.765064
\(494\) −3.27854 −0.147508
\(495\) 0 0
\(496\) 5.64552 0.253491
\(497\) −0.275107 −0.0123403
\(498\) 7.70048 0.345067
\(499\) −25.0420 −1.12104 −0.560518 0.828142i \(-0.689398\pi\)
−0.560518 + 0.828142i \(0.689398\pi\)
\(500\) 0 0
\(501\) −31.5210 −1.40826
\(502\) 17.3688 0.775208
\(503\) −14.9942 −0.668560 −0.334280 0.942474i \(-0.608493\pi\)
−0.334280 + 0.942474i \(0.608493\pi\)
\(504\) −0.648374 −0.0288809
\(505\) 0 0
\(506\) −1.74715 −0.0776703
\(507\) 10.8558 0.482123
\(508\) 1.45213 0.0644278
\(509\) −21.5572 −0.955506 −0.477753 0.878494i \(-0.658549\pi\)
−0.477753 + 0.878494i \(0.658549\pi\)
\(510\) 0 0
\(511\) 0.479263 0.0212013
\(512\) 7.19396 0.317931
\(513\) −1.62854 −0.0719019
\(514\) −21.7586 −0.959731
\(515\) 0 0
\(516\) −14.0283 −0.617562
\(517\) 8.97459 0.394702
\(518\) 0.604283 0.0265507
\(519\) −37.1518 −1.63078
\(520\) 0 0
\(521\) −34.7658 −1.52312 −0.761558 0.648097i \(-0.775565\pi\)
−0.761558 + 0.648097i \(0.775565\pi\)
\(522\) 7.36996 0.322575
\(523\) −26.7047 −1.16771 −0.583857 0.811856i \(-0.698457\pi\)
−0.583857 + 0.811856i \(0.698457\pi\)
\(524\) 25.0579 1.09466
\(525\) 0 0
\(526\) −16.9266 −0.738036
\(527\) −58.0314 −2.52789
\(528\) −1.65746 −0.0721316
\(529\) −18.1111 −0.787438
\(530\) 0 0
\(531\) 37.5447 1.62930
\(532\) 0.0920574 0.00399120
\(533\) 45.5698 1.97385
\(534\) 0.454713 0.0196773
\(535\) 0 0
\(536\) −40.7264 −1.75911
\(537\) 22.4640 0.969394
\(538\) 6.37284 0.274753
\(539\) 6.99552 0.301318
\(540\) 0 0
\(541\) 9.54398 0.410328 0.205164 0.978728i \(-0.434227\pi\)
0.205164 + 0.978728i \(0.434227\pi\)
\(542\) −11.6448 −0.500188
\(543\) −35.9598 −1.54318
\(544\) −38.6563 −1.65737
\(545\) 0 0
\(546\) 0.565033 0.0241812
\(547\) −27.6080 −1.18043 −0.590217 0.807245i \(-0.700958\pi\)
−0.590217 + 0.807245i \(0.700958\pi\)
\(548\) −9.82715 −0.419795
\(549\) 10.0453 0.428723
\(550\) 0 0
\(551\) −2.56775 −0.109390
\(552\) −15.1886 −0.646470
\(553\) 0.598928 0.0254690
\(554\) 25.7466 1.09387
\(555\) 0 0
\(556\) −3.10433 −0.131653
\(557\) −14.9783 −0.634651 −0.317325 0.948317i \(-0.602785\pi\)
−0.317325 + 0.948317i \(0.602785\pi\)
\(558\) 25.1772 1.06583
\(559\) 16.4297 0.694900
\(560\) 0 0
\(561\) 17.0374 0.719318
\(562\) −13.3184 −0.561801
\(563\) −0.475683 −0.0200476 −0.0100238 0.999950i \(-0.503191\pi\)
−0.0100238 + 0.999950i \(0.503191\pi\)
\(564\) 31.7942 1.33878
\(565\) 0 0
\(566\) 4.94842 0.207998
\(567\) −0.448571 −0.0188382
\(568\) 10.9653 0.460093
\(569\) −9.06417 −0.379990 −0.189995 0.981785i \(-0.560847\pi\)
−0.189995 + 0.981785i \(0.560847\pi\)
\(570\) 0 0
\(571\) 4.75692 0.199071 0.0995355 0.995034i \(-0.468264\pi\)
0.0995355 + 0.995034i \(0.468264\pi\)
\(572\) 5.70765 0.238649
\(573\) 12.2654 0.512393
\(574\) 0.580767 0.0242408
\(575\) 0 0
\(576\) 12.0957 0.503987
\(577\) −24.9499 −1.03868 −0.519338 0.854569i \(-0.673822\pi\)
−0.519338 + 0.854569i \(0.673822\pi\)
\(578\) 21.1498 0.879715
\(579\) −14.6586 −0.609190
\(580\) 0 0
\(581\) 0.253233 0.0105059
\(582\) 28.7147 1.19026
\(583\) 3.13641 0.129897
\(584\) −19.1026 −0.790470
\(585\) 0 0
\(586\) −6.06127 −0.250389
\(587\) −19.3052 −0.796810 −0.398405 0.917210i \(-0.630436\pi\)
−0.398405 + 0.917210i \(0.630436\pi\)
\(588\) 24.7830 1.02203
\(589\) −8.77193 −0.361441
\(590\) 0 0
\(591\) 49.2983 2.02786
\(592\) 7.35473 0.302278
\(593\) 6.49883 0.266875 0.133437 0.991057i \(-0.457398\pi\)
0.133437 + 0.991057i \(0.457398\pi\)
\(594\) −1.28683 −0.0527994
\(595\) 0 0
\(596\) −7.70746 −0.315710
\(597\) 54.4115 2.22691
\(598\) 7.24915 0.296440
\(599\) 8.10308 0.331083 0.165541 0.986203i \(-0.447063\pi\)
0.165541 + 0.986203i \(0.447063\pi\)
\(600\) 0 0
\(601\) −34.2650 −1.39770 −0.698848 0.715270i \(-0.746304\pi\)
−0.698848 + 0.715270i \(0.746304\pi\)
\(602\) 0.209389 0.00853405
\(603\) 55.4610 2.25855
\(604\) 28.6998 1.16778
\(605\) 0 0
\(606\) −23.7605 −0.965206
\(607\) 16.9455 0.687797 0.343898 0.939007i \(-0.388252\pi\)
0.343898 + 0.939007i \(0.388252\pi\)
\(608\) −5.84321 −0.236974
\(609\) 0.442534 0.0179324
\(610\) 0 0
\(611\) −37.2367 −1.50644
\(612\) 33.0565 1.33623
\(613\) 1.47243 0.0594709 0.0297354 0.999558i \(-0.490534\pi\)
0.0297354 + 0.999558i \(0.490534\pi\)
\(614\) −0.00348055 −0.000140464 0
\(615\) 0 0
\(616\) 0.178499 0.00719194
\(617\) 19.4675 0.783731 0.391866 0.920022i \(-0.371830\pi\)
0.391866 + 0.920022i \(0.371830\pi\)
\(618\) −23.7133 −0.953889
\(619\) 21.8377 0.877732 0.438866 0.898552i \(-0.355380\pi\)
0.438866 + 0.898552i \(0.355380\pi\)
\(620\) 0 0
\(621\) 3.60086 0.144497
\(622\) 9.52983 0.382112
\(623\) 0.0149534 0.000599094 0
\(624\) 6.87702 0.275301
\(625\) 0 0
\(626\) 13.0285 0.520725
\(627\) 2.57534 0.102849
\(628\) 0.914509 0.0364929
\(629\) −75.6008 −3.01440
\(630\) 0 0
\(631\) 0.393330 0.0156582 0.00782912 0.999969i \(-0.497508\pi\)
0.00782912 + 0.999969i \(0.497508\pi\)
\(632\) −23.8722 −0.949585
\(633\) 56.5015 2.24573
\(634\) 20.5123 0.814646
\(635\) 0 0
\(636\) 11.1114 0.440594
\(637\) −29.0253 −1.15003
\(638\) −2.02897 −0.0803278
\(639\) −14.9325 −0.590720
\(640\) 0 0
\(641\) 2.08000 0.0821549 0.0410774 0.999156i \(-0.486921\pi\)
0.0410774 + 0.999156i \(0.486921\pi\)
\(642\) 5.54416 0.218811
\(643\) −15.7787 −0.622250 −0.311125 0.950369i \(-0.600706\pi\)
−0.311125 + 0.950369i \(0.600706\pi\)
\(644\) −0.203548 −0.00802090
\(645\) 0 0
\(646\) 5.22747 0.205672
\(647\) −6.67760 −0.262524 −0.131262 0.991348i \(-0.541903\pi\)
−0.131262 + 0.991348i \(0.541903\pi\)
\(648\) 17.8792 0.702363
\(649\) −10.3362 −0.405731
\(650\) 0 0
\(651\) 1.51178 0.0592513
\(652\) −0.491138 −0.0192345
\(653\) −6.00890 −0.235147 −0.117573 0.993064i \(-0.537512\pi\)
−0.117573 + 0.993064i \(0.537512\pi\)
\(654\) −8.07641 −0.315813
\(655\) 0 0
\(656\) 7.06852 0.275979
\(657\) 26.0138 1.01489
\(658\) −0.474566 −0.0185005
\(659\) 14.3389 0.558566 0.279283 0.960209i \(-0.409903\pi\)
0.279283 + 0.960209i \(0.409903\pi\)
\(660\) 0 0
\(661\) −4.17451 −0.162370 −0.0811849 0.996699i \(-0.525870\pi\)
−0.0811849 + 0.996699i \(0.525870\pi\)
\(662\) 20.1065 0.781462
\(663\) −70.6902 −2.74538
\(664\) −10.0934 −0.391700
\(665\) 0 0
\(666\) 32.7997 1.27096
\(667\) 5.67754 0.219835
\(668\) 16.8371 0.651445
\(669\) 59.3651 2.29519
\(670\) 0 0
\(671\) −2.76550 −0.106761
\(672\) 1.00704 0.0388473
\(673\) 1.71412 0.0660745 0.0330373 0.999454i \(-0.489482\pi\)
0.0330373 + 0.999454i \(0.489482\pi\)
\(674\) −22.7842 −0.877615
\(675\) 0 0
\(676\) −5.79866 −0.223025
\(677\) −0.206434 −0.00793392 −0.00396696 0.999992i \(-0.501263\pi\)
−0.00396696 + 0.999992i \(0.501263\pi\)
\(678\) 5.73315 0.220180
\(679\) 0.944291 0.0362386
\(680\) 0 0
\(681\) −31.2002 −1.19559
\(682\) −6.93135 −0.265415
\(683\) 29.3173 1.12179 0.560897 0.827886i \(-0.310456\pi\)
0.560897 + 0.827886i \(0.310456\pi\)
\(684\) 4.99676 0.191056
\(685\) 0 0
\(686\) −0.740068 −0.0282559
\(687\) −43.8280 −1.67214
\(688\) 2.54847 0.0971596
\(689\) −13.0134 −0.495771
\(690\) 0 0
\(691\) −44.8795 −1.70730 −0.853649 0.520849i \(-0.825615\pi\)
−0.853649 + 0.520849i \(0.825615\pi\)
\(692\) 19.8448 0.754384
\(693\) −0.243079 −0.00923382
\(694\) −16.0201 −0.608114
\(695\) 0 0
\(696\) −17.6386 −0.668590
\(697\) −72.6587 −2.75215
\(698\) 18.6386 0.705481
\(699\) −5.37819 −0.203422
\(700\) 0 0
\(701\) 7.13370 0.269436 0.134718 0.990884i \(-0.456987\pi\)
0.134718 + 0.990884i \(0.456987\pi\)
\(702\) 5.33924 0.201517
\(703\) −11.4277 −0.431003
\(704\) −3.32998 −0.125503
\(705\) 0 0
\(706\) −20.6800 −0.778304
\(707\) −0.781373 −0.0293865
\(708\) −36.6179 −1.37619
\(709\) −37.5143 −1.40888 −0.704440 0.709763i \(-0.748802\pi\)
−0.704440 + 0.709763i \(0.748802\pi\)
\(710\) 0 0
\(711\) 32.5090 1.21918
\(712\) −0.596015 −0.0223366
\(713\) 19.3955 0.726369
\(714\) −0.900916 −0.0337160
\(715\) 0 0
\(716\) −11.9992 −0.448432
\(717\) −43.4898 −1.62415
\(718\) 25.7061 0.959343
\(719\) −9.92990 −0.370323 −0.185161 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(720\) 0 0
\(721\) −0.779819 −0.0290420
\(722\) 0.790175 0.0294073
\(723\) 50.5952 1.88166
\(724\) 19.2080 0.713860
\(725\) 0 0
\(726\) 2.03497 0.0755247
\(727\) −16.4750 −0.611025 −0.305513 0.952188i \(-0.598828\pi\)
−0.305513 + 0.952188i \(0.598828\pi\)
\(728\) −0.740617 −0.0274491
\(729\) −36.9300 −1.36778
\(730\) 0 0
\(731\) −26.1963 −0.968904
\(732\) −9.79732 −0.362119
\(733\) 35.6615 1.31719 0.658594 0.752498i \(-0.271151\pi\)
0.658594 + 0.752498i \(0.271151\pi\)
\(734\) −7.20817 −0.266059
\(735\) 0 0
\(736\) 12.9199 0.476233
\(737\) −15.2686 −0.562425
\(738\) 31.5233 1.16039
\(739\) 12.0150 0.441977 0.220989 0.975276i \(-0.429072\pi\)
0.220989 + 0.975276i \(0.429072\pi\)
\(740\) 0 0
\(741\) −10.6854 −0.392538
\(742\) −0.165850 −0.00608854
\(743\) 6.68005 0.245067 0.122534 0.992464i \(-0.460898\pi\)
0.122534 + 0.992464i \(0.460898\pi\)
\(744\) −60.2568 −2.20912
\(745\) 0 0
\(746\) −25.5930 −0.937025
\(747\) 13.7452 0.502909
\(748\) −9.10056 −0.332749
\(749\) 0.182321 0.00666188
\(750\) 0 0
\(751\) −6.22015 −0.226976 −0.113488 0.993539i \(-0.536202\pi\)
−0.113488 + 0.993539i \(0.536202\pi\)
\(752\) −5.77595 −0.210627
\(753\) 56.6084 2.06293
\(754\) 8.41848 0.306583
\(755\) 0 0
\(756\) −0.149919 −0.00545251
\(757\) −14.2550 −0.518105 −0.259053 0.965863i \(-0.583410\pi\)
−0.259053 + 0.965863i \(0.583410\pi\)
\(758\) −16.4749 −0.598396
\(759\) −5.69431 −0.206690
\(760\) 0 0
\(761\) −3.46765 −0.125702 −0.0628512 0.998023i \(-0.520019\pi\)
−0.0628512 + 0.998023i \(0.520019\pi\)
\(762\) −2.14814 −0.0778189
\(763\) −0.265595 −0.00961519
\(764\) −6.55158 −0.237028
\(765\) 0 0
\(766\) −11.7425 −0.424273
\(767\) 42.8862 1.54853
\(768\) −35.5786 −1.28383
\(769\) −38.6283 −1.39297 −0.696485 0.717571i \(-0.745254\pi\)
−0.696485 + 0.717571i \(0.745254\pi\)
\(770\) 0 0
\(771\) −70.9157 −2.55397
\(772\) 7.82993 0.281805
\(773\) 27.8564 1.00193 0.500963 0.865469i \(-0.332979\pi\)
0.500963 + 0.865469i \(0.332979\pi\)
\(774\) 11.3654 0.408520
\(775\) 0 0
\(776\) −37.6378 −1.35112
\(777\) 1.96948 0.0706547
\(778\) −2.20496 −0.0790515
\(779\) −10.9830 −0.393506
\(780\) 0 0
\(781\) 4.11096 0.147102
\(782\) −11.5584 −0.413328
\(783\) 4.18169 0.149441
\(784\) −4.50224 −0.160794
\(785\) 0 0
\(786\) −37.0682 −1.32218
\(787\) −9.77533 −0.348453 −0.174226 0.984706i \(-0.555742\pi\)
−0.174226 + 0.984706i \(0.555742\pi\)
\(788\) −26.3329 −0.938069
\(789\) −55.1672 −1.96401
\(790\) 0 0
\(791\) 0.188537 0.00670359
\(792\) 9.68871 0.344273
\(793\) 11.4744 0.407469
\(794\) −7.55256 −0.268030
\(795\) 0 0
\(796\) −29.0641 −1.03015
\(797\) 2.96470 0.105015 0.0525075 0.998621i \(-0.483279\pi\)
0.0525075 + 0.998621i \(0.483279\pi\)
\(798\) −0.136181 −0.00482075
\(799\) 59.3721 2.10043
\(800\) 0 0
\(801\) 0.811650 0.0286782
\(802\) −7.77596 −0.274579
\(803\) −7.16167 −0.252730
\(804\) −54.0919 −1.90767
\(805\) 0 0
\(806\) 28.7591 1.01300
\(807\) 20.7704 0.731151
\(808\) 31.1441 1.09565
\(809\) 36.2770 1.27543 0.637715 0.770272i \(-0.279880\pi\)
0.637715 + 0.770272i \(0.279880\pi\)
\(810\) 0 0
\(811\) 28.4399 0.998661 0.499330 0.866412i \(-0.333579\pi\)
0.499330 + 0.866412i \(0.333579\pi\)
\(812\) −0.236381 −0.00829534
\(813\) −37.9528 −1.33106
\(814\) −9.02987 −0.316497
\(815\) 0 0
\(816\) −10.9651 −0.383854
\(817\) −3.95978 −0.138535
\(818\) 4.13995 0.144750
\(819\) 1.00857 0.0352422
\(820\) 0 0
\(821\) 34.1352 1.19133 0.595663 0.803235i \(-0.296890\pi\)
0.595663 + 0.803235i \(0.296890\pi\)
\(822\) 14.5373 0.507048
\(823\) −21.6135 −0.753400 −0.376700 0.926335i \(-0.622941\pi\)
−0.376700 + 0.926335i \(0.622941\pi\)
\(824\) 31.0822 1.08280
\(825\) 0 0
\(826\) 0.546566 0.0190175
\(827\) 3.22960 0.112304 0.0561522 0.998422i \(-0.482117\pi\)
0.0561522 + 0.998422i \(0.482117\pi\)
\(828\) −11.0483 −0.383955
\(829\) 54.2752 1.88506 0.942528 0.334126i \(-0.108441\pi\)
0.942528 + 0.334126i \(0.108441\pi\)
\(830\) 0 0
\(831\) 83.9132 2.91092
\(832\) 13.8165 0.479002
\(833\) 46.2795 1.60349
\(834\) 4.59224 0.159016
\(835\) 0 0
\(836\) −1.37562 −0.0475769
\(837\) 14.2854 0.493777
\(838\) −8.87769 −0.306675
\(839\) 18.4633 0.637422 0.318711 0.947852i \(-0.396750\pi\)
0.318711 + 0.947852i \(0.396750\pi\)
\(840\) 0 0
\(841\) −22.4066 −0.772643
\(842\) 27.0962 0.933797
\(843\) −43.4072 −1.49502
\(844\) −30.1804 −1.03885
\(845\) 0 0
\(846\) −25.7589 −0.885608
\(847\) 0.0669205 0.00229942
\(848\) −2.01856 −0.0693177
\(849\) 16.1279 0.553508
\(850\) 0 0
\(851\) 25.2677 0.866164
\(852\) 14.5639 0.498950
\(853\) −39.7042 −1.35945 −0.679723 0.733469i \(-0.737900\pi\)
−0.679723 + 0.733469i \(0.737900\pi\)
\(854\) 0.146237 0.00500411
\(855\) 0 0
\(856\) −7.26701 −0.248381
\(857\) −52.0593 −1.77831 −0.889155 0.457606i \(-0.848707\pi\)
−0.889155 + 0.457606i \(0.848707\pi\)
\(858\) −8.44334 −0.288251
\(859\) 14.9753 0.510952 0.255476 0.966815i \(-0.417768\pi\)
0.255476 + 0.966815i \(0.417768\pi\)
\(860\) 0 0
\(861\) 1.89284 0.0645077
\(862\) 19.1479 0.652182
\(863\) 17.1586 0.584086 0.292043 0.956405i \(-0.405665\pi\)
0.292043 + 0.956405i \(0.405665\pi\)
\(864\) 9.51591 0.323738
\(865\) 0 0
\(866\) −9.12932 −0.310227
\(867\) 68.9313 2.34103
\(868\) −0.807521 −0.0274091
\(869\) −8.94984 −0.303602
\(870\) 0 0
\(871\) 63.3513 2.14658
\(872\) 10.5862 0.358493
\(873\) 51.2549 1.73472
\(874\) −1.74715 −0.0590982
\(875\) 0 0
\(876\) −25.3716 −0.857228
\(877\) 7.93413 0.267917 0.133958 0.990987i \(-0.457231\pi\)
0.133958 + 0.990987i \(0.457231\pi\)
\(878\) −30.1827 −1.01862
\(879\) −19.7549 −0.666316
\(880\) 0 0
\(881\) −15.2979 −0.515399 −0.257700 0.966225i \(-0.582964\pi\)
−0.257700 + 0.966225i \(0.582964\pi\)
\(882\) −20.0785 −0.676080
\(883\) 0.534788 0.0179971 0.00899853 0.999960i \(-0.497136\pi\)
0.00899853 + 0.999960i \(0.497136\pi\)
\(884\) 37.7594 1.26999
\(885\) 0 0
\(886\) 15.6709 0.526474
\(887\) 9.87389 0.331533 0.165766 0.986165i \(-0.446990\pi\)
0.165766 + 0.986165i \(0.446990\pi\)
\(888\) −78.5000 −2.63429
\(889\) −0.0706423 −0.00236927
\(890\) 0 0
\(891\) 6.70304 0.224560
\(892\) −31.7100 −1.06173
\(893\) 8.97459 0.300323
\(894\) 11.4017 0.381329
\(895\) 0 0
\(896\) −0.605976 −0.0202442
\(897\) 23.6264 0.788864
\(898\) 15.5301 0.518247
\(899\) 22.5241 0.751222
\(900\) 0 0
\(901\) 20.7492 0.691256
\(902\) −8.67846 −0.288961
\(903\) 0.682440 0.0227102
\(904\) −7.51473 −0.249936
\(905\) 0 0
\(906\) −42.4557 −1.41050
\(907\) −32.3531 −1.07427 −0.537133 0.843497i \(-0.680493\pi\)
−0.537133 + 0.843497i \(0.680493\pi\)
\(908\) 16.6657 0.553070
\(909\) −42.4119 −1.40671
\(910\) 0 0
\(911\) −58.8040 −1.94826 −0.974132 0.225978i \(-0.927442\pi\)
−0.974132 + 0.225978i \(0.927442\pi\)
\(912\) −1.65746 −0.0548840
\(913\) −3.78408 −0.125235
\(914\) 9.99047 0.330455
\(915\) 0 0
\(916\) 23.4109 0.773517
\(917\) −1.21900 −0.0402549
\(918\) −8.51315 −0.280976
\(919\) −22.2112 −0.732679 −0.366340 0.930481i \(-0.619389\pi\)
−0.366340 + 0.930481i \(0.619389\pi\)
\(920\) 0 0
\(921\) −0.0113438 −0.000373791 0
\(922\) −1.77644 −0.0585038
\(923\) −17.0569 −0.561435
\(924\) 0.237079 0.00779932
\(925\) 0 0
\(926\) 12.8672 0.422841
\(927\) −42.3276 −1.39022
\(928\) 15.0039 0.492528
\(929\) 8.60843 0.282433 0.141217 0.989979i \(-0.454899\pi\)
0.141217 + 0.989979i \(0.454899\pi\)
\(930\) 0 0
\(931\) 6.99552 0.229269
\(932\) 2.87278 0.0941009
\(933\) 31.0596 1.01685
\(934\) −1.64675 −0.0538835
\(935\) 0 0
\(936\) −40.1998 −1.31397
\(937\) −25.1791 −0.822566 −0.411283 0.911508i \(-0.634919\pi\)
−0.411283 + 0.911508i \(0.634919\pi\)
\(938\) 0.807385 0.0263621
\(939\) 42.4626 1.38571
\(940\) 0 0
\(941\) 25.5413 0.832622 0.416311 0.909222i \(-0.363323\pi\)
0.416311 + 0.909222i \(0.363323\pi\)
\(942\) −1.35284 −0.0440778
\(943\) 24.2844 0.790808
\(944\) 6.65225 0.216512
\(945\) 0 0
\(946\) −3.12892 −0.101730
\(947\) 21.1214 0.686353 0.343176 0.939271i \(-0.388497\pi\)
0.343176 + 0.939271i \(0.388497\pi\)
\(948\) −31.7065 −1.02978
\(949\) 29.7147 0.964581
\(950\) 0 0
\(951\) 66.8535 2.16787
\(952\) 1.18088 0.0382724
\(953\) −14.3865 −0.466025 −0.233012 0.972474i \(-0.574858\pi\)
−0.233012 + 0.972474i \(0.574858\pi\)
\(954\) −9.00213 −0.291455
\(955\) 0 0
\(956\) 23.2302 0.751318
\(957\) −6.61283 −0.213762
\(958\) 12.9559 0.418587
\(959\) 0.478065 0.0154375
\(960\) 0 0
\(961\) 45.9467 1.48215
\(962\) 37.4661 1.20796
\(963\) 9.89618 0.318900
\(964\) −27.0256 −0.870436
\(965\) 0 0
\(966\) 0.301109 0.00968801
\(967\) −41.6857 −1.34052 −0.670261 0.742125i \(-0.733818\pi\)
−0.670261 + 0.742125i \(0.733818\pi\)
\(968\) −2.66733 −0.0857313
\(969\) 17.0374 0.547319
\(970\) 0 0
\(971\) 45.0663 1.44625 0.723123 0.690719i \(-0.242706\pi\)
0.723123 + 0.690719i \(0.242706\pi\)
\(972\) 30.4676 0.977249
\(973\) 0.151017 0.00484139
\(974\) −5.17709 −0.165885
\(975\) 0 0
\(976\) 1.77985 0.0569715
\(977\) 9.51039 0.304264 0.152132 0.988360i \(-0.451386\pi\)
0.152132 + 0.988360i \(0.451386\pi\)
\(978\) 0.726543 0.0232323
\(979\) −0.223450 −0.00714149
\(980\) 0 0
\(981\) −14.4162 −0.460273
\(982\) 23.5156 0.750412
\(983\) 38.9961 1.24378 0.621891 0.783104i \(-0.286365\pi\)
0.621891 + 0.783104i \(0.286365\pi\)
\(984\) −75.4451 −2.40510
\(985\) 0 0
\(986\) −13.4228 −0.427470
\(987\) −1.54671 −0.0492322
\(988\) 5.70765 0.181584
\(989\) 8.75544 0.278407
\(990\) 0 0
\(991\) 61.4220 1.95113 0.975566 0.219705i \(-0.0705095\pi\)
0.975566 + 0.219705i \(0.0705095\pi\)
\(992\) 51.2562 1.62739
\(993\) 65.5311 2.07957
\(994\) −0.217383 −0.00689497
\(995\) 0 0
\(996\) −13.4059 −0.424781
\(997\) 8.89988 0.281862 0.140931 0.990019i \(-0.454990\pi\)
0.140931 + 0.990019i \(0.454990\pi\)
\(998\) −19.7876 −0.626365
\(999\) 18.6105 0.588809
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.1.13 22
5.2 odd 4 1045.2.b.d.419.13 yes 22
5.3 odd 4 1045.2.b.d.419.10 22
5.4 even 2 inner 5225.2.a.bb.1.10 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.d.419.10 22 5.3 odd 4
1045.2.b.d.419.13 yes 22 5.2 odd 4
5225.2.a.bb.1.10 22 5.4 even 2 inner
5225.2.a.bb.1.13 22 1.1 even 1 trivial