Properties

Label 1875.2.a.k.1.3
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
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] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.44400625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.246759\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.246759 q^{2} +1.00000 q^{3} -1.93911 q^{4} -0.246759 q^{6} +1.24676 q^{7} +0.972011 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.246759 q^{2} +1.00000 q^{3} -1.93911 q^{4} -0.246759 q^{6} +1.24676 q^{7} +0.972011 q^{8} +1.00000 q^{9} +2.56971 q^{11} -1.93911 q^{12} +4.68126 q^{13} -0.307649 q^{14} +3.63837 q^{16} -5.83914 q^{17} -0.246759 q^{18} +4.16857 q^{19} +1.24676 q^{21} -0.634099 q^{22} -1.60547 q^{23} +0.972011 q^{24} -1.15514 q^{26} +1.00000 q^{27} -2.41760 q^{28} -3.21573 q^{29} -9.19312 q^{31} -2.84182 q^{32} +2.56971 q^{33} +1.44086 q^{34} -1.93911 q^{36} +1.27778 q^{37} -1.02863 q^{38} +4.68126 q^{39} +8.69592 q^{41} -0.307649 q^{42} +3.88086 q^{43} -4.98295 q^{44} +0.396163 q^{46} +3.20952 q^{47} +3.63837 q^{48} -5.44559 q^{49} -5.83914 q^{51} -9.07748 q^{52} +13.3343 q^{53} -0.246759 q^{54} +1.21186 q^{56} +4.16857 q^{57} +0.793511 q^{58} -6.39279 q^{59} +3.82692 q^{61} +2.26848 q^{62} +1.24676 q^{63} -6.57549 q^{64} -0.634099 q^{66} +10.1238 q^{67} +11.3227 q^{68} -1.60547 q^{69} +13.8791 q^{71} +0.972011 q^{72} +11.3814 q^{73} -0.315305 q^{74} -8.08332 q^{76} +3.20381 q^{77} -1.15514 q^{78} +11.4832 q^{79} +1.00000 q^{81} -2.14580 q^{82} +2.83229 q^{83} -2.41760 q^{84} -0.957636 q^{86} -3.21573 q^{87} +2.49779 q^{88} -1.38077 q^{89} +5.83641 q^{91} +3.11318 q^{92} -9.19312 q^{93} -0.791979 q^{94} -2.84182 q^{96} -0.0305081 q^{97} +1.34375 q^{98} +2.56971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 10 q^{4} + 6 q^{7} + 3 q^{8} + 6 q^{9} + 3 q^{11} + 10 q^{12} + 6 q^{13} - 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} + 6 q^{21} + 16 q^{22} + 13 q^{23} + 3 q^{24} - 28 q^{26} + 6 q^{27} + 7 q^{28} - 3 q^{29} - 11 q^{31} + 16 q^{32} + 3 q^{33} + 15 q^{34} + 10 q^{36} + 21 q^{37} - 9 q^{38} + 6 q^{39} - q^{41} - 22 q^{42} + 2 q^{43} + 9 q^{44} + 19 q^{46} + 14 q^{47} + 18 q^{48} - 14 q^{49} + 13 q^{51} + 13 q^{52} + 23 q^{53} - 35 q^{56} + 11 q^{57} + 22 q^{58} + 9 q^{59} + 11 q^{61} - 23 q^{62} + 6 q^{63} - 23 q^{64} + 16 q^{66} + 8 q^{67} + 50 q^{68} + 13 q^{69} - 8 q^{71} + 3 q^{72} + 13 q^{73} - 22 q^{74} - 26 q^{76} - 13 q^{77} - 28 q^{78} - 5 q^{79} + 6 q^{81} - 13 q^{82} - 20 q^{83} + 7 q^{84} - 37 q^{86} - 3 q^{87} + 28 q^{88} - 4 q^{89} + 34 q^{91} + 61 q^{92} - 11 q^{93} + 41 q^{94} + 16 q^{96} - 7 q^{97} - 41 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.246759 −0.174485 −0.0872425 0.996187i \(-0.527805\pi\)
−0.0872425 + 0.996187i \(0.527805\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.93911 −0.969555
\(5\) 0 0
\(6\) −0.246759 −0.100739
\(7\) 1.24676 0.471231 0.235615 0.971846i \(-0.424289\pi\)
0.235615 + 0.971846i \(0.424289\pi\)
\(8\) 0.972011 0.343658
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.56971 0.774797 0.387398 0.921912i \(-0.373374\pi\)
0.387398 + 0.921912i \(0.373374\pi\)
\(12\) −1.93911 −0.559773
\(13\) 4.68126 1.29835 0.649174 0.760640i \(-0.275115\pi\)
0.649174 + 0.760640i \(0.275115\pi\)
\(14\) −0.307649 −0.0822226
\(15\) 0 0
\(16\) 3.63837 0.909592
\(17\) −5.83914 −1.41620 −0.708100 0.706112i \(-0.750447\pi\)
−0.708100 + 0.706112i \(0.750447\pi\)
\(18\) −0.246759 −0.0581616
\(19\) 4.16857 0.956336 0.478168 0.878268i \(-0.341301\pi\)
0.478168 + 0.878268i \(0.341301\pi\)
\(20\) 0 0
\(21\) 1.24676 0.272065
\(22\) −0.634099 −0.135190
\(23\) −1.60547 −0.334763 −0.167382 0.985892i \(-0.553531\pi\)
−0.167382 + 0.985892i \(0.553531\pi\)
\(24\) 0.972011 0.198411
\(25\) 0 0
\(26\) −1.15514 −0.226542
\(27\) 1.00000 0.192450
\(28\) −2.41760 −0.456884
\(29\) −3.21573 −0.597147 −0.298573 0.954387i \(-0.596511\pi\)
−0.298573 + 0.954387i \(0.596511\pi\)
\(30\) 0 0
\(31\) −9.19312 −1.65113 −0.825566 0.564305i \(-0.809144\pi\)
−0.825566 + 0.564305i \(0.809144\pi\)
\(32\) −2.84182 −0.502368
\(33\) 2.56971 0.447329
\(34\) 1.44086 0.247106
\(35\) 0 0
\(36\) −1.93911 −0.323185
\(37\) 1.27778 0.210066 0.105033 0.994469i \(-0.466505\pi\)
0.105033 + 0.994469i \(0.466505\pi\)
\(38\) −1.02863 −0.166866
\(39\) 4.68126 0.749602
\(40\) 0 0
\(41\) 8.69592 1.35808 0.679038 0.734104i \(-0.262397\pi\)
0.679038 + 0.734104i \(0.262397\pi\)
\(42\) −0.307649 −0.0474713
\(43\) 3.88086 0.591825 0.295913 0.955215i \(-0.404376\pi\)
0.295913 + 0.955215i \(0.404376\pi\)
\(44\) −4.98295 −0.751208
\(45\) 0 0
\(46\) 0.396163 0.0584111
\(47\) 3.20952 0.468157 0.234079 0.972218i \(-0.424793\pi\)
0.234079 + 0.972218i \(0.424793\pi\)
\(48\) 3.63837 0.525153
\(49\) −5.44559 −0.777942
\(50\) 0 0
\(51\) −5.83914 −0.817643
\(52\) −9.07748 −1.25882
\(53\) 13.3343 1.83161 0.915803 0.401628i \(-0.131556\pi\)
0.915803 + 0.401628i \(0.131556\pi\)
\(54\) −0.246759 −0.0335796
\(55\) 0 0
\(56\) 1.21186 0.161942
\(57\) 4.16857 0.552141
\(58\) 0.793511 0.104193
\(59\) −6.39279 −0.832270 −0.416135 0.909303i \(-0.636616\pi\)
−0.416135 + 0.909303i \(0.636616\pi\)
\(60\) 0 0
\(61\) 3.82692 0.489986 0.244993 0.969525i \(-0.421214\pi\)
0.244993 + 0.969525i \(0.421214\pi\)
\(62\) 2.26848 0.288098
\(63\) 1.24676 0.157077
\(64\) −6.57549 −0.821936
\(65\) 0 0
\(66\) −0.634099 −0.0780522
\(67\) 10.1238 1.23682 0.618411 0.785855i \(-0.287777\pi\)
0.618411 + 0.785855i \(0.287777\pi\)
\(68\) 11.3227 1.37308
\(69\) −1.60547 −0.193276
\(70\) 0 0
\(71\) 13.8791 1.64715 0.823574 0.567209i \(-0.191977\pi\)
0.823574 + 0.567209i \(0.191977\pi\)
\(72\) 0.972011 0.114553
\(73\) 11.3814 1.33209 0.666045 0.745912i \(-0.267986\pi\)
0.666045 + 0.745912i \(0.267986\pi\)
\(74\) −0.315305 −0.0366534
\(75\) 0 0
\(76\) −8.08332 −0.927220
\(77\) 3.20381 0.365108
\(78\) −1.15514 −0.130794
\(79\) 11.4832 1.29196 0.645979 0.763355i \(-0.276449\pi\)
0.645979 + 0.763355i \(0.276449\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.14580 −0.236964
\(83\) 2.83229 0.310884 0.155442 0.987845i \(-0.450320\pi\)
0.155442 + 0.987845i \(0.450320\pi\)
\(84\) −2.41760 −0.263782
\(85\) 0 0
\(86\) −0.957636 −0.103265
\(87\) −3.21573 −0.344763
\(88\) 2.49779 0.266265
\(89\) −1.38077 −0.146362 −0.0731808 0.997319i \(-0.523315\pi\)
−0.0731808 + 0.997319i \(0.523315\pi\)
\(90\) 0 0
\(91\) 5.83641 0.611822
\(92\) 3.11318 0.324571
\(93\) −9.19312 −0.953282
\(94\) −0.791979 −0.0816864
\(95\) 0 0
\(96\) −2.84182 −0.290042
\(97\) −0.0305081 −0.00309763 −0.00154881 0.999999i \(-0.500493\pi\)
−0.00154881 + 0.999999i \(0.500493\pi\)
\(98\) 1.34375 0.135739
\(99\) 2.56971 0.258266
\(100\) 0 0
\(101\) −11.2308 −1.11750 −0.558751 0.829336i \(-0.688719\pi\)
−0.558751 + 0.829336i \(0.688719\pi\)
\(102\) 1.44086 0.142666
\(103\) 0.720983 0.0710406 0.0355203 0.999369i \(-0.488691\pi\)
0.0355203 + 0.999369i \(0.488691\pi\)
\(104\) 4.55024 0.446187
\(105\) 0 0
\(106\) −3.29036 −0.319588
\(107\) −2.11668 −0.204627 −0.102313 0.994752i \(-0.532624\pi\)
−0.102313 + 0.994752i \(0.532624\pi\)
\(108\) −1.93911 −0.186591
\(109\) 2.23483 0.214058 0.107029 0.994256i \(-0.465866\pi\)
0.107029 + 0.994256i \(0.465866\pi\)
\(110\) 0 0
\(111\) 1.27778 0.121282
\(112\) 4.53617 0.428628
\(113\) 3.14940 0.296270 0.148135 0.988967i \(-0.452673\pi\)
0.148135 + 0.988967i \(0.452673\pi\)
\(114\) −1.02863 −0.0963402
\(115\) 0 0
\(116\) 6.23566 0.578967
\(117\) 4.68126 0.432783
\(118\) 1.57748 0.145219
\(119\) −7.28000 −0.667357
\(120\) 0 0
\(121\) −4.39659 −0.399690
\(122\) −0.944326 −0.0854952
\(123\) 8.69592 0.784085
\(124\) 17.8265 1.60086
\(125\) 0 0
\(126\) −0.307649 −0.0274075
\(127\) 15.6603 1.38962 0.694812 0.719191i \(-0.255487\pi\)
0.694812 + 0.719191i \(0.255487\pi\)
\(128\) 7.30620 0.645783
\(129\) 3.88086 0.341691
\(130\) 0 0
\(131\) −14.9564 −1.30675 −0.653374 0.757036i \(-0.726647\pi\)
−0.653374 + 0.757036i \(0.726647\pi\)
\(132\) −4.98295 −0.433710
\(133\) 5.19720 0.450655
\(134\) −2.49814 −0.215807
\(135\) 0 0
\(136\) −5.67571 −0.486688
\(137\) 10.4946 0.896611 0.448305 0.893881i \(-0.352028\pi\)
0.448305 + 0.893881i \(0.352028\pi\)
\(138\) 0.396163 0.0337237
\(139\) 12.4508 1.05606 0.528030 0.849226i \(-0.322931\pi\)
0.528030 + 0.849226i \(0.322931\pi\)
\(140\) 0 0
\(141\) 3.20952 0.270291
\(142\) −3.42480 −0.287402
\(143\) 12.0295 1.00596
\(144\) 3.63837 0.303197
\(145\) 0 0
\(146\) −2.80846 −0.232430
\(147\) −5.44559 −0.449145
\(148\) −2.47776 −0.203671
\(149\) −17.0680 −1.39826 −0.699131 0.714994i \(-0.746430\pi\)
−0.699131 + 0.714994i \(0.746430\pi\)
\(150\) 0 0
\(151\) −1.57516 −0.128185 −0.0640923 0.997944i \(-0.520415\pi\)
−0.0640923 + 0.997944i \(0.520415\pi\)
\(152\) 4.05190 0.328652
\(153\) −5.83914 −0.472067
\(154\) −0.790569 −0.0637059
\(155\) 0 0
\(156\) −9.07748 −0.726780
\(157\) 9.20058 0.734286 0.367143 0.930164i \(-0.380336\pi\)
0.367143 + 0.930164i \(0.380336\pi\)
\(158\) −2.83358 −0.225427
\(159\) 13.3343 1.05748
\(160\) 0 0
\(161\) −2.00163 −0.157751
\(162\) −0.246759 −0.0193872
\(163\) 4.58076 0.358793 0.179396 0.983777i \(-0.442586\pi\)
0.179396 + 0.983777i \(0.442586\pi\)
\(164\) −16.8624 −1.31673
\(165\) 0 0
\(166\) −0.698893 −0.0542446
\(167\) −0.142531 −0.0110294 −0.00551469 0.999985i \(-0.501755\pi\)
−0.00551469 + 0.999985i \(0.501755\pi\)
\(168\) 1.21186 0.0934973
\(169\) 8.91422 0.685709
\(170\) 0 0
\(171\) 4.16857 0.318779
\(172\) −7.52541 −0.573807
\(173\) −24.6228 −1.87203 −0.936017 0.351954i \(-0.885517\pi\)
−0.936017 + 0.351954i \(0.885517\pi\)
\(174\) 0.793511 0.0601559
\(175\) 0 0
\(176\) 9.34955 0.704749
\(177\) −6.39279 −0.480511
\(178\) 0.340718 0.0255379
\(179\) −24.1553 −1.80545 −0.902726 0.430216i \(-0.858437\pi\)
−0.902726 + 0.430216i \(0.858437\pi\)
\(180\) 0 0
\(181\) 1.96805 0.146284 0.0731419 0.997322i \(-0.476697\pi\)
0.0731419 + 0.997322i \(0.476697\pi\)
\(182\) −1.44019 −0.106754
\(183\) 3.82692 0.282894
\(184\) −1.56053 −0.115044
\(185\) 0 0
\(186\) 2.26848 0.166333
\(187\) −15.0049 −1.09727
\(188\) −6.22362 −0.453904
\(189\) 1.24676 0.0906884
\(190\) 0 0
\(191\) −7.76854 −0.562112 −0.281056 0.959691i \(-0.590685\pi\)
−0.281056 + 0.959691i \(0.590685\pi\)
\(192\) −6.57549 −0.474545
\(193\) −14.3458 −1.03264 −0.516318 0.856397i \(-0.672698\pi\)
−0.516318 + 0.856397i \(0.672698\pi\)
\(194\) 0.00752814 0.000540489 0
\(195\) 0 0
\(196\) 10.5596 0.754257
\(197\) −0.163540 −0.0116517 −0.00582587 0.999983i \(-0.501854\pi\)
−0.00582587 + 0.999983i \(0.501854\pi\)
\(198\) −0.634099 −0.0450635
\(199\) 4.96974 0.352295 0.176148 0.984364i \(-0.443636\pi\)
0.176148 + 0.984364i \(0.443636\pi\)
\(200\) 0 0
\(201\) 10.1238 0.714079
\(202\) 2.77129 0.194987
\(203\) −4.00925 −0.281394
\(204\) 11.3227 0.792750
\(205\) 0 0
\(206\) −0.177909 −0.0123955
\(207\) −1.60547 −0.111588
\(208\) 17.0322 1.18097
\(209\) 10.7120 0.740966
\(210\) 0 0
\(211\) 3.29381 0.226755 0.113378 0.993552i \(-0.463833\pi\)
0.113378 + 0.993552i \(0.463833\pi\)
\(212\) −25.8567 −1.77584
\(213\) 13.8791 0.950981
\(214\) 0.522309 0.0357043
\(215\) 0 0
\(216\) 0.972011 0.0661369
\(217\) −11.4616 −0.778064
\(218\) −0.551465 −0.0373500
\(219\) 11.3814 0.769082
\(220\) 0 0
\(221\) −27.3346 −1.83872
\(222\) −0.315305 −0.0211619
\(223\) 15.7732 1.05625 0.528127 0.849166i \(-0.322895\pi\)
0.528127 + 0.849166i \(0.322895\pi\)
\(224\) −3.54307 −0.236731
\(225\) 0 0
\(226\) −0.777142 −0.0516947
\(227\) −13.5406 −0.898721 −0.449360 0.893351i \(-0.648348\pi\)
−0.449360 + 0.893351i \(0.648348\pi\)
\(228\) −8.08332 −0.535331
\(229\) −14.7829 −0.976879 −0.488440 0.872598i \(-0.662434\pi\)
−0.488440 + 0.872598i \(0.662434\pi\)
\(230\) 0 0
\(231\) 3.20381 0.210795
\(232\) −3.12573 −0.205214
\(233\) −0.518980 −0.0339995 −0.0169998 0.999855i \(-0.505411\pi\)
−0.0169998 + 0.999855i \(0.505411\pi\)
\(234\) −1.15514 −0.0755141
\(235\) 0 0
\(236\) 12.3963 0.806932
\(237\) 11.4832 0.745912
\(238\) 1.79641 0.116444
\(239\) 8.23032 0.532375 0.266188 0.963921i \(-0.414236\pi\)
0.266188 + 0.963921i \(0.414236\pi\)
\(240\) 0 0
\(241\) −1.28118 −0.0825280 −0.0412640 0.999148i \(-0.513138\pi\)
−0.0412640 + 0.999148i \(0.513138\pi\)
\(242\) 1.08490 0.0697398
\(243\) 1.00000 0.0641500
\(244\) −7.42081 −0.475069
\(245\) 0 0
\(246\) −2.14580 −0.136811
\(247\) 19.5142 1.24166
\(248\) −8.93581 −0.567424
\(249\) 2.83229 0.179489
\(250\) 0 0
\(251\) 4.63494 0.292555 0.146277 0.989244i \(-0.453271\pi\)
0.146277 + 0.989244i \(0.453271\pi\)
\(252\) −2.41760 −0.152295
\(253\) −4.12559 −0.259373
\(254\) −3.86431 −0.242469
\(255\) 0 0
\(256\) 11.3481 0.709257
\(257\) 18.3001 1.14153 0.570766 0.821113i \(-0.306647\pi\)
0.570766 + 0.821113i \(0.306647\pi\)
\(258\) −0.957636 −0.0596198
\(259\) 1.59309 0.0989897
\(260\) 0 0
\(261\) −3.21573 −0.199049
\(262\) 3.69063 0.228008
\(263\) −16.6666 −1.02771 −0.513854 0.857877i \(-0.671783\pi\)
−0.513854 + 0.857877i \(0.671783\pi\)
\(264\) 2.49779 0.153728
\(265\) 0 0
\(266\) −1.28246 −0.0786324
\(267\) −1.38077 −0.0845020
\(268\) −19.6312 −1.19917
\(269\) 2.63211 0.160483 0.0802414 0.996775i \(-0.474431\pi\)
0.0802414 + 0.996775i \(0.474431\pi\)
\(270\) 0 0
\(271\) 4.56346 0.277210 0.138605 0.990348i \(-0.455738\pi\)
0.138605 + 0.990348i \(0.455738\pi\)
\(272\) −21.2449 −1.28816
\(273\) 5.83641 0.353235
\(274\) −2.58963 −0.156445
\(275\) 0 0
\(276\) 3.11318 0.187391
\(277\) −1.25926 −0.0756614 −0.0378307 0.999284i \(-0.512045\pi\)
−0.0378307 + 0.999284i \(0.512045\pi\)
\(278\) −3.07234 −0.184266
\(279\) −9.19312 −0.550378
\(280\) 0 0
\(281\) 13.4167 0.800370 0.400185 0.916434i \(-0.368946\pi\)
0.400185 + 0.916434i \(0.368946\pi\)
\(282\) −0.791979 −0.0471617
\(283\) −24.0890 −1.43194 −0.715971 0.698130i \(-0.754016\pi\)
−0.715971 + 0.698130i \(0.754016\pi\)
\(284\) −26.9131 −1.59700
\(285\) 0 0
\(286\) −2.96838 −0.175524
\(287\) 10.8417 0.639966
\(288\) −2.84182 −0.167456
\(289\) 17.0956 1.00562
\(290\) 0 0
\(291\) −0.0305081 −0.00178841
\(292\) −22.0697 −1.29153
\(293\) 32.1727 1.87955 0.939776 0.341792i \(-0.111034\pi\)
0.939776 + 0.341792i \(0.111034\pi\)
\(294\) 1.34375 0.0783690
\(295\) 0 0
\(296\) 1.24202 0.0721909
\(297\) 2.56971 0.149110
\(298\) 4.21167 0.243976
\(299\) −7.51561 −0.434639
\(300\) 0 0
\(301\) 4.83849 0.278886
\(302\) 0.388685 0.0223663
\(303\) −11.2308 −0.645190
\(304\) 15.1668 0.869875
\(305\) 0 0
\(306\) 1.44086 0.0823685
\(307\) −20.2962 −1.15837 −0.579183 0.815198i \(-0.696628\pi\)
−0.579183 + 0.815198i \(0.696628\pi\)
\(308\) −6.21254 −0.353992
\(309\) 0.720983 0.0410153
\(310\) 0 0
\(311\) −18.7224 −1.06165 −0.530825 0.847481i \(-0.678118\pi\)
−0.530825 + 0.847481i \(0.678118\pi\)
\(312\) 4.55024 0.257606
\(313\) −24.8839 −1.40652 −0.703261 0.710932i \(-0.748274\pi\)
−0.703261 + 0.710932i \(0.748274\pi\)
\(314\) −2.27033 −0.128122
\(315\) 0 0
\(316\) −22.2671 −1.25262
\(317\) 25.1880 1.41470 0.707350 0.706863i \(-0.249890\pi\)
0.707350 + 0.706863i \(0.249890\pi\)
\(318\) −3.29036 −0.184514
\(319\) −8.26351 −0.462668
\(320\) 0 0
\(321\) −2.11668 −0.118141
\(322\) 0.493920 0.0275251
\(323\) −24.3409 −1.35436
\(324\) −1.93911 −0.107728
\(325\) 0 0
\(326\) −1.13034 −0.0626039
\(327\) 2.23483 0.123587
\(328\) 8.45253 0.466713
\(329\) 4.00150 0.220610
\(330\) 0 0
\(331\) 30.4193 1.67199 0.835997 0.548735i \(-0.184890\pi\)
0.835997 + 0.548735i \(0.184890\pi\)
\(332\) −5.49212 −0.301419
\(333\) 1.27778 0.0700221
\(334\) 0.0351708 0.00192446
\(335\) 0 0
\(336\) 4.53617 0.247468
\(337\) 4.57455 0.249192 0.124596 0.992208i \(-0.460237\pi\)
0.124596 + 0.992208i \(0.460237\pi\)
\(338\) −2.19966 −0.119646
\(339\) 3.14940 0.171052
\(340\) 0 0
\(341\) −23.6237 −1.27929
\(342\) −1.02863 −0.0556221
\(343\) −15.5167 −0.837821
\(344\) 3.77224 0.203385
\(345\) 0 0
\(346\) 6.07589 0.326642
\(347\) 16.8153 0.902691 0.451346 0.892349i \(-0.350944\pi\)
0.451346 + 0.892349i \(0.350944\pi\)
\(348\) 6.23566 0.334267
\(349\) 0.373581 0.0199973 0.00999866 0.999950i \(-0.496817\pi\)
0.00999866 + 0.999950i \(0.496817\pi\)
\(350\) 0 0
\(351\) 4.68126 0.249867
\(352\) −7.30266 −0.389233
\(353\) 23.9685 1.27572 0.637858 0.770154i \(-0.279821\pi\)
0.637858 + 0.770154i \(0.279821\pi\)
\(354\) 1.57748 0.0838420
\(355\) 0 0
\(356\) 2.67747 0.141906
\(357\) −7.28000 −0.385299
\(358\) 5.96054 0.315024
\(359\) −4.75020 −0.250706 −0.125353 0.992112i \(-0.540006\pi\)
−0.125353 + 0.992112i \(0.540006\pi\)
\(360\) 0 0
\(361\) −1.62302 −0.0854221
\(362\) −0.485633 −0.0255243
\(363\) −4.39659 −0.230761
\(364\) −11.3174 −0.593195
\(365\) 0 0
\(366\) −0.944326 −0.0493607
\(367\) 17.3700 0.906706 0.453353 0.891331i \(-0.350228\pi\)
0.453353 + 0.891331i \(0.350228\pi\)
\(368\) −5.84128 −0.304498
\(369\) 8.69592 0.452692
\(370\) 0 0
\(371\) 16.6246 0.863109
\(372\) 17.8265 0.924259
\(373\) −2.10877 −0.109188 −0.0545939 0.998509i \(-0.517386\pi\)
−0.0545939 + 0.998509i \(0.517386\pi\)
\(374\) 3.70260 0.191457
\(375\) 0 0
\(376\) 3.11969 0.160886
\(377\) −15.0537 −0.775305
\(378\) −0.307649 −0.0158238
\(379\) 12.4090 0.637405 0.318703 0.947855i \(-0.396753\pi\)
0.318703 + 0.947855i \(0.396753\pi\)
\(380\) 0 0
\(381\) 15.6603 0.802300
\(382\) 1.91696 0.0980801
\(383\) −33.3056 −1.70184 −0.850919 0.525297i \(-0.823954\pi\)
−0.850919 + 0.525297i \(0.823954\pi\)
\(384\) 7.30620 0.372843
\(385\) 0 0
\(386\) 3.53996 0.180179
\(387\) 3.88086 0.197275
\(388\) 0.0591585 0.00300332
\(389\) −16.9777 −0.860802 −0.430401 0.902638i \(-0.641628\pi\)
−0.430401 + 0.902638i \(0.641628\pi\)
\(390\) 0 0
\(391\) 9.37455 0.474091
\(392\) −5.29317 −0.267346
\(393\) −14.9564 −0.754451
\(394\) 0.0403549 0.00203305
\(395\) 0 0
\(396\) −4.98295 −0.250403
\(397\) 0.866570 0.0434919 0.0217460 0.999764i \(-0.493078\pi\)
0.0217460 + 0.999764i \(0.493078\pi\)
\(398\) −1.22633 −0.0614702
\(399\) 5.19720 0.260186
\(400\) 0 0
\(401\) 9.68680 0.483736 0.241868 0.970309i \(-0.422240\pi\)
0.241868 + 0.970309i \(0.422240\pi\)
\(402\) −2.49814 −0.124596
\(403\) −43.0354 −2.14375
\(404\) 21.7777 1.08348
\(405\) 0 0
\(406\) 0.989317 0.0490990
\(407\) 3.28353 0.162759
\(408\) −5.67571 −0.280989
\(409\) 1.51785 0.0750531 0.0375265 0.999296i \(-0.488052\pi\)
0.0375265 + 0.999296i \(0.488052\pi\)
\(410\) 0 0
\(411\) 10.4946 0.517658
\(412\) −1.39807 −0.0688778
\(413\) −7.97027 −0.392191
\(414\) 0.396163 0.0194704
\(415\) 0 0
\(416\) −13.3033 −0.652249
\(417\) 12.4508 0.609716
\(418\) −2.64329 −0.129287
\(419\) −30.5620 −1.49305 −0.746526 0.665356i \(-0.768280\pi\)
−0.746526 + 0.665356i \(0.768280\pi\)
\(420\) 0 0
\(421\) −6.97162 −0.339776 −0.169888 0.985463i \(-0.554341\pi\)
−0.169888 + 0.985463i \(0.554341\pi\)
\(422\) −0.812777 −0.0395654
\(423\) 3.20952 0.156052
\(424\) 12.9611 0.629445
\(425\) 0 0
\(426\) −3.42480 −0.165932
\(427\) 4.77124 0.230897
\(428\) 4.10447 0.198397
\(429\) 12.0295 0.580789
\(430\) 0 0
\(431\) 10.9472 0.527310 0.263655 0.964617i \(-0.415072\pi\)
0.263655 + 0.964617i \(0.415072\pi\)
\(432\) 3.63837 0.175051
\(433\) −10.4393 −0.501683 −0.250841 0.968028i \(-0.580707\pi\)
−0.250841 + 0.968028i \(0.580707\pi\)
\(434\) 2.82825 0.135761
\(435\) 0 0
\(436\) −4.33359 −0.207541
\(437\) −6.69250 −0.320146
\(438\) −2.80846 −0.134193
\(439\) 15.4334 0.736594 0.368297 0.929708i \(-0.379941\pi\)
0.368297 + 0.929708i \(0.379941\pi\)
\(440\) 0 0
\(441\) −5.44559 −0.259314
\(442\) 6.74505 0.320829
\(443\) −18.9105 −0.898463 −0.449231 0.893415i \(-0.648302\pi\)
−0.449231 + 0.893415i \(0.648302\pi\)
\(444\) −2.47776 −0.117589
\(445\) 0 0
\(446\) −3.89218 −0.184300
\(447\) −17.0680 −0.807287
\(448\) −8.19805 −0.387322
\(449\) −11.4152 −0.538714 −0.269357 0.963040i \(-0.586811\pi\)
−0.269357 + 0.963040i \(0.586811\pi\)
\(450\) 0 0
\(451\) 22.3460 1.05223
\(452\) −6.10703 −0.287250
\(453\) −1.57516 −0.0740074
\(454\) 3.34126 0.156813
\(455\) 0 0
\(456\) 4.05190 0.189747
\(457\) 33.9739 1.58923 0.794616 0.607112i \(-0.207672\pi\)
0.794616 + 0.607112i \(0.207672\pi\)
\(458\) 3.64780 0.170451
\(459\) −5.83914 −0.272548
\(460\) 0 0
\(461\) −24.6907 −1.14996 −0.574981 0.818167i \(-0.694991\pi\)
−0.574981 + 0.818167i \(0.694991\pi\)
\(462\) −0.790569 −0.0367806
\(463\) 13.6315 0.633510 0.316755 0.948507i \(-0.397407\pi\)
0.316755 + 0.948507i \(0.397407\pi\)
\(464\) −11.7000 −0.543160
\(465\) 0 0
\(466\) 0.128063 0.00593241
\(467\) −9.47115 −0.438272 −0.219136 0.975694i \(-0.570324\pi\)
−0.219136 + 0.975694i \(0.570324\pi\)
\(468\) −9.07748 −0.419607
\(469\) 12.6220 0.582828
\(470\) 0 0
\(471\) 9.20058 0.423940
\(472\) −6.21386 −0.286016
\(473\) 9.97268 0.458544
\(474\) −2.83358 −0.130150
\(475\) 0 0
\(476\) 14.1167 0.647039
\(477\) 13.3343 0.610535
\(478\) −2.03091 −0.0928914
\(479\) −25.7286 −1.17557 −0.587784 0.809018i \(-0.700000\pi\)
−0.587784 + 0.809018i \(0.700000\pi\)
\(480\) 0 0
\(481\) 5.98164 0.272739
\(482\) 0.316142 0.0143999
\(483\) −2.00163 −0.0910773
\(484\) 8.52546 0.387521
\(485\) 0 0
\(486\) −0.246759 −0.0111932
\(487\) 28.4046 1.28714 0.643568 0.765389i \(-0.277453\pi\)
0.643568 + 0.765389i \(0.277453\pi\)
\(488\) 3.71980 0.168388
\(489\) 4.58076 0.207149
\(490\) 0 0
\(491\) −6.10299 −0.275424 −0.137712 0.990472i \(-0.543975\pi\)
−0.137712 + 0.990472i \(0.543975\pi\)
\(492\) −16.8624 −0.760214
\(493\) 18.7771 0.845679
\(494\) −4.81530 −0.216650
\(495\) 0 0
\(496\) −33.4479 −1.50186
\(497\) 17.3039 0.776186
\(498\) −0.698893 −0.0313181
\(499\) −20.3163 −0.909481 −0.454740 0.890624i \(-0.650268\pi\)
−0.454740 + 0.890624i \(0.650268\pi\)
\(500\) 0 0
\(501\) −0.142531 −0.00636782
\(502\) −1.14371 −0.0510464
\(503\) −14.6267 −0.652171 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(504\) 1.21186 0.0539807
\(505\) 0 0
\(506\) 1.01803 0.0452567
\(507\) 8.91422 0.395894
\(508\) −30.3670 −1.34732
\(509\) −4.30584 −0.190853 −0.0954265 0.995436i \(-0.530422\pi\)
−0.0954265 + 0.995436i \(0.530422\pi\)
\(510\) 0 0
\(511\) 14.1898 0.627721
\(512\) −17.4127 −0.769538
\(513\) 4.16857 0.184047
\(514\) −4.51572 −0.199180
\(515\) 0 0
\(516\) −7.52541 −0.331288
\(517\) 8.24755 0.362727
\(518\) −0.393109 −0.0172722
\(519\) −24.6228 −1.08082
\(520\) 0 0
\(521\) −39.6413 −1.73672 −0.868358 0.495938i \(-0.834824\pi\)
−0.868358 + 0.495938i \(0.834824\pi\)
\(522\) 0.793511 0.0347310
\(523\) −36.2874 −1.58674 −0.793369 0.608741i \(-0.791675\pi\)
−0.793369 + 0.608741i \(0.791675\pi\)
\(524\) 29.0021 1.26696
\(525\) 0 0
\(526\) 4.11264 0.179320
\(527\) 53.6799 2.33833
\(528\) 9.34955 0.406887
\(529\) −20.4225 −0.887934
\(530\) 0 0
\(531\) −6.39279 −0.277423
\(532\) −10.0779 −0.436934
\(533\) 40.7079 1.76325
\(534\) 0.340718 0.0147443
\(535\) 0 0
\(536\) 9.84046 0.425043
\(537\) −24.1553 −1.04238
\(538\) −0.649497 −0.0280018
\(539\) −13.9936 −0.602747
\(540\) 0 0
\(541\) −38.6044 −1.65973 −0.829866 0.557962i \(-0.811583\pi\)
−0.829866 + 0.557962i \(0.811583\pi\)
\(542\) −1.12607 −0.0483690
\(543\) 1.96805 0.0844570
\(544\) 16.5938 0.711453
\(545\) 0 0
\(546\) −1.44019 −0.0616343
\(547\) −10.3620 −0.443047 −0.221524 0.975155i \(-0.571103\pi\)
−0.221524 + 0.975155i \(0.571103\pi\)
\(548\) −20.3501 −0.869313
\(549\) 3.82692 0.163329
\(550\) 0 0
\(551\) −13.4050 −0.571073
\(552\) −1.56053 −0.0664206
\(553\) 14.3167 0.608810
\(554\) 0.310733 0.0132018
\(555\) 0 0
\(556\) −24.1434 −1.02391
\(557\) −12.6830 −0.537398 −0.268699 0.963224i \(-0.586594\pi\)
−0.268699 + 0.963224i \(0.586594\pi\)
\(558\) 2.26848 0.0960326
\(559\) 18.1673 0.768396
\(560\) 0 0
\(561\) −15.0049 −0.633508
\(562\) −3.31068 −0.139653
\(563\) −13.0892 −0.551645 −0.275823 0.961209i \(-0.588950\pi\)
−0.275823 + 0.961209i \(0.588950\pi\)
\(564\) −6.22362 −0.262062
\(565\) 0 0
\(566\) 5.94417 0.249852
\(567\) 1.24676 0.0523590
\(568\) 13.4906 0.566055
\(569\) −2.58803 −0.108496 −0.0542478 0.998528i \(-0.517276\pi\)
−0.0542478 + 0.998528i \(0.517276\pi\)
\(570\) 0 0
\(571\) −10.5754 −0.442568 −0.221284 0.975209i \(-0.571025\pi\)
−0.221284 + 0.975209i \(0.571025\pi\)
\(572\) −23.3265 −0.975330
\(573\) −7.76854 −0.324536
\(574\) −2.67529 −0.111665
\(575\) 0 0
\(576\) −6.57549 −0.273979
\(577\) 0.0142904 0.000594917 0 0.000297458 1.00000i \(-0.499905\pi\)
0.000297458 1.00000i \(0.499905\pi\)
\(578\) −4.21849 −0.175466
\(579\) −14.3458 −0.596193
\(580\) 0 0
\(581\) 3.53118 0.146498
\(582\) 0.00752814 0.000312051 0
\(583\) 34.2653 1.41912
\(584\) 11.0628 0.457783
\(585\) 0 0
\(586\) −7.93891 −0.327953
\(587\) 22.6389 0.934408 0.467204 0.884149i \(-0.345261\pi\)
0.467204 + 0.884149i \(0.345261\pi\)
\(588\) 10.5596 0.435471
\(589\) −38.3222 −1.57904
\(590\) 0 0
\(591\) −0.163540 −0.00672713
\(592\) 4.64905 0.191075
\(593\) −12.7423 −0.523264 −0.261632 0.965168i \(-0.584261\pi\)
−0.261632 + 0.965168i \(0.584261\pi\)
\(594\) −0.634099 −0.0260174
\(595\) 0 0
\(596\) 33.0967 1.35569
\(597\) 4.96974 0.203398
\(598\) 1.85454 0.0758380
\(599\) −6.83599 −0.279311 −0.139656 0.990200i \(-0.544600\pi\)
−0.139656 + 0.990200i \(0.544600\pi\)
\(600\) 0 0
\(601\) 31.8422 1.29887 0.649435 0.760417i \(-0.275005\pi\)
0.649435 + 0.760417i \(0.275005\pi\)
\(602\) −1.19394 −0.0486614
\(603\) 10.1238 0.412274
\(604\) 3.05441 0.124282
\(605\) 0 0
\(606\) 2.77129 0.112576
\(607\) −27.7763 −1.12741 −0.563703 0.825977i \(-0.690624\pi\)
−0.563703 + 0.825977i \(0.690624\pi\)
\(608\) −11.8463 −0.480432
\(609\) −4.00925 −0.162463
\(610\) 0 0
\(611\) 15.0246 0.607831
\(612\) 11.3227 0.457695
\(613\) 3.61694 0.146087 0.0730433 0.997329i \(-0.476729\pi\)
0.0730433 + 0.997329i \(0.476729\pi\)
\(614\) 5.00827 0.202117
\(615\) 0 0
\(616\) 3.11414 0.125472
\(617\) −44.6425 −1.79724 −0.898621 0.438727i \(-0.855430\pi\)
−0.898621 + 0.438727i \(0.855430\pi\)
\(618\) −0.177909 −0.00715655
\(619\) 20.5085 0.824307 0.412153 0.911115i \(-0.364777\pi\)
0.412153 + 0.911115i \(0.364777\pi\)
\(620\) 0 0
\(621\) −1.60547 −0.0644252
\(622\) 4.61992 0.185242
\(623\) −1.72149 −0.0689701
\(624\) 17.0322 0.681832
\(625\) 0 0
\(626\) 6.14033 0.245417
\(627\) 10.7120 0.427797
\(628\) −17.8409 −0.711931
\(629\) −7.46116 −0.297496
\(630\) 0 0
\(631\) −37.7357 −1.50223 −0.751117 0.660169i \(-0.770485\pi\)
−0.751117 + 0.660169i \(0.770485\pi\)
\(632\) 11.1618 0.443991
\(633\) 3.29381 0.130917
\(634\) −6.21537 −0.246844
\(635\) 0 0
\(636\) −25.8567 −1.02528
\(637\) −25.4922 −1.01004
\(638\) 2.03909 0.0807285
\(639\) 13.8791 0.549049
\(640\) 0 0
\(641\) −9.18606 −0.362828 −0.181414 0.983407i \(-0.558067\pi\)
−0.181414 + 0.983407i \(0.558067\pi\)
\(642\) 0.522309 0.0206139
\(643\) −3.42111 −0.134915 −0.0674577 0.997722i \(-0.521489\pi\)
−0.0674577 + 0.997722i \(0.521489\pi\)
\(644\) 3.88138 0.152948
\(645\) 0 0
\(646\) 6.00633 0.236316
\(647\) −3.26735 −0.128453 −0.0642264 0.997935i \(-0.520458\pi\)
−0.0642264 + 0.997935i \(0.520458\pi\)
\(648\) 0.972011 0.0381842
\(649\) −16.4276 −0.644840
\(650\) 0 0
\(651\) −11.4616 −0.449216
\(652\) −8.88259 −0.347869
\(653\) −12.6362 −0.494494 −0.247247 0.968952i \(-0.579526\pi\)
−0.247247 + 0.968952i \(0.579526\pi\)
\(654\) −0.551465 −0.0215640
\(655\) 0 0
\(656\) 31.6390 1.23529
\(657\) 11.3814 0.444030
\(658\) −0.987407 −0.0384931
\(659\) −9.14896 −0.356393 −0.178197 0.983995i \(-0.557026\pi\)
−0.178197 + 0.983995i \(0.557026\pi\)
\(660\) 0 0
\(661\) 6.39083 0.248574 0.124287 0.992246i \(-0.460336\pi\)
0.124287 + 0.992246i \(0.460336\pi\)
\(662\) −7.50622 −0.291738
\(663\) −27.3346 −1.06159
\(664\) 2.75302 0.106838
\(665\) 0 0
\(666\) −0.315305 −0.0122178
\(667\) 5.16276 0.199903
\(668\) 0.276383 0.0106936
\(669\) 15.7732 0.609828
\(670\) 0 0
\(671\) 9.83407 0.379640
\(672\) −3.54307 −0.136677
\(673\) 33.7063 1.29928 0.649641 0.760241i \(-0.274919\pi\)
0.649641 + 0.760241i \(0.274919\pi\)
\(674\) −1.12881 −0.0434802
\(675\) 0 0
\(676\) −17.2857 −0.664833
\(677\) −9.20340 −0.353715 −0.176858 0.984236i \(-0.556593\pi\)
−0.176858 + 0.984236i \(0.556593\pi\)
\(678\) −0.777142 −0.0298459
\(679\) −0.0380362 −0.00145970
\(680\) 0 0
\(681\) −13.5406 −0.518877
\(682\) 5.82935 0.223217
\(683\) −27.2216 −1.04161 −0.520803 0.853677i \(-0.674367\pi\)
−0.520803 + 0.853677i \(0.674367\pi\)
\(684\) −8.08332 −0.309073
\(685\) 0 0
\(686\) 3.82887 0.146187
\(687\) −14.7829 −0.564001
\(688\) 14.1200 0.538320
\(689\) 62.4213 2.37806
\(690\) 0 0
\(691\) 29.8238 1.13455 0.567276 0.823528i \(-0.307997\pi\)
0.567276 + 0.823528i \(0.307997\pi\)
\(692\) 47.7463 1.81504
\(693\) 3.20381 0.121703
\(694\) −4.14932 −0.157506
\(695\) 0 0
\(696\) −3.12573 −0.118480
\(697\) −50.7767 −1.92331
\(698\) −0.0921844 −0.00348923
\(699\) −0.518980 −0.0196296
\(700\) 0 0
\(701\) −19.1081 −0.721704 −0.360852 0.932623i \(-0.617514\pi\)
−0.360852 + 0.932623i \(0.617514\pi\)
\(702\) −1.15514 −0.0435981
\(703\) 5.32653 0.200894
\(704\) −16.8971 −0.636834
\(705\) 0 0
\(706\) −5.91445 −0.222593
\(707\) −14.0020 −0.526601
\(708\) 12.3963 0.465882
\(709\) 14.5146 0.545107 0.272553 0.962141i \(-0.412132\pi\)
0.272553 + 0.962141i \(0.412132\pi\)
\(710\) 0 0
\(711\) 11.4832 0.430653
\(712\) −1.34213 −0.0502983
\(713\) 14.7592 0.552738
\(714\) 1.79641 0.0672288
\(715\) 0 0
\(716\) 46.8398 1.75048
\(717\) 8.23032 0.307367
\(718\) 1.17215 0.0437444
\(719\) 34.7507 1.29598 0.647992 0.761647i \(-0.275609\pi\)
0.647992 + 0.761647i \(0.275609\pi\)
\(720\) 0 0
\(721\) 0.898892 0.0334765
\(722\) 0.400494 0.0149049
\(723\) −1.28118 −0.0476476
\(724\) −3.81626 −0.141830
\(725\) 0 0
\(726\) 1.08490 0.0402643
\(727\) −42.6279 −1.58098 −0.790490 0.612475i \(-0.790174\pi\)
−0.790490 + 0.612475i \(0.790174\pi\)
\(728\) 5.67305 0.210257
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −22.6609 −0.838143
\(732\) −7.42081 −0.274281
\(733\) −1.96230 −0.0724793 −0.0362397 0.999343i \(-0.511538\pi\)
−0.0362397 + 0.999343i \(0.511538\pi\)
\(734\) −4.28620 −0.158207
\(735\) 0 0
\(736\) 4.56245 0.168174
\(737\) 26.0153 0.958286
\(738\) −2.14580 −0.0789879
\(739\) 12.3601 0.454672 0.227336 0.973816i \(-0.426998\pi\)
0.227336 + 0.973816i \(0.426998\pi\)
\(740\) 0 0
\(741\) 19.5142 0.716871
\(742\) −4.10228 −0.150599
\(743\) −8.63742 −0.316876 −0.158438 0.987369i \(-0.550646\pi\)
−0.158438 + 0.987369i \(0.550646\pi\)
\(744\) −8.93581 −0.327603
\(745\) 0 0
\(746\) 0.520357 0.0190516
\(747\) 2.83229 0.103628
\(748\) 29.0962 1.06386
\(749\) −2.63898 −0.0964264
\(750\) 0 0
\(751\) −24.3654 −0.889105 −0.444552 0.895753i \(-0.646637\pi\)
−0.444552 + 0.895753i \(0.646637\pi\)
\(752\) 11.6774 0.425832
\(753\) 4.63494 0.168906
\(754\) 3.71463 0.135279
\(755\) 0 0
\(756\) −2.41760 −0.0879274
\(757\) 5.18352 0.188398 0.0941991 0.995553i \(-0.469971\pi\)
0.0941991 + 0.995553i \(0.469971\pi\)
\(758\) −3.06202 −0.111218
\(759\) −4.12559 −0.149749
\(760\) 0 0
\(761\) 31.2636 1.13331 0.566653 0.823957i \(-0.308238\pi\)
0.566653 + 0.823957i \(0.308238\pi\)
\(762\) −3.86431 −0.139989
\(763\) 2.78630 0.100871
\(764\) 15.0641 0.544999
\(765\) 0 0
\(766\) 8.21846 0.296945
\(767\) −29.9263 −1.08058
\(768\) 11.3481 0.409490
\(769\) −44.1891 −1.59350 −0.796750 0.604309i \(-0.793449\pi\)
−0.796750 + 0.604309i \(0.793449\pi\)
\(770\) 0 0
\(771\) 18.3001 0.659063
\(772\) 27.8182 1.00120
\(773\) 21.5881 0.776471 0.388236 0.921560i \(-0.373085\pi\)
0.388236 + 0.921560i \(0.373085\pi\)
\(774\) −0.957636 −0.0344215
\(775\) 0 0
\(776\) −0.0296542 −0.00106452
\(777\) 1.59309 0.0571517
\(778\) 4.18939 0.150197
\(779\) 36.2496 1.29878
\(780\) 0 0
\(781\) 35.6653 1.27621
\(782\) −2.31325 −0.0827218
\(783\) −3.21573 −0.114921
\(784\) −19.8131 −0.707610
\(785\) 0 0
\(786\) 3.69063 0.131640
\(787\) −14.4212 −0.514059 −0.257029 0.966404i \(-0.582744\pi\)
−0.257029 + 0.966404i \(0.582744\pi\)
\(788\) 0.317122 0.0112970
\(789\) −16.6666 −0.593348
\(790\) 0 0
\(791\) 3.92654 0.139612
\(792\) 2.49779 0.0887550
\(793\) 17.9148 0.636173
\(794\) −0.213834 −0.00758868
\(795\) 0 0
\(796\) −9.63687 −0.341570
\(797\) 39.4567 1.39763 0.698814 0.715304i \(-0.253712\pi\)
0.698814 + 0.715304i \(0.253712\pi\)
\(798\) −1.28246 −0.0453985
\(799\) −18.7409 −0.663004
\(800\) 0 0
\(801\) −1.38077 −0.0487872
\(802\) −2.39031 −0.0844046
\(803\) 29.2469 1.03210
\(804\) −19.6312 −0.692339
\(805\) 0 0
\(806\) 10.6194 0.374051
\(807\) 2.63211 0.0926547
\(808\) −10.9164 −0.384038
\(809\) −5.96204 −0.209614 −0.104807 0.994493i \(-0.533423\pi\)
−0.104807 + 0.994493i \(0.533423\pi\)
\(810\) 0 0
\(811\) −39.1764 −1.37567 −0.687835 0.725867i \(-0.741439\pi\)
−0.687835 + 0.725867i \(0.741439\pi\)
\(812\) 7.77437 0.272827
\(813\) 4.56346 0.160048
\(814\) −0.810242 −0.0283990
\(815\) 0 0
\(816\) −21.2449 −0.743722
\(817\) 16.1776 0.565984
\(818\) −0.374544 −0.0130956
\(819\) 5.83641 0.203941
\(820\) 0 0
\(821\) 20.6329 0.720092 0.360046 0.932934i \(-0.382761\pi\)
0.360046 + 0.932934i \(0.382761\pi\)
\(822\) −2.58963 −0.0903236
\(823\) −39.0162 −1.36002 −0.680009 0.733203i \(-0.738024\pi\)
−0.680009 + 0.733203i \(0.738024\pi\)
\(824\) 0.700803 0.0244136
\(825\) 0 0
\(826\) 1.96673 0.0684314
\(827\) −19.5976 −0.681474 −0.340737 0.940159i \(-0.610677\pi\)
−0.340737 + 0.940159i \(0.610677\pi\)
\(828\) 3.11318 0.108190
\(829\) −8.67511 −0.301299 −0.150650 0.988587i \(-0.548136\pi\)
−0.150650 + 0.988587i \(0.548136\pi\)
\(830\) 0 0
\(831\) −1.25926 −0.0436831
\(832\) −30.7816 −1.06716
\(833\) 31.7976 1.10172
\(834\) −3.07234 −0.106386
\(835\) 0 0
\(836\) −20.7718 −0.718407
\(837\) −9.19312 −0.317761
\(838\) 7.54145 0.260515
\(839\) −3.85921 −0.133235 −0.0666173 0.997779i \(-0.521221\pi\)
−0.0666173 + 0.997779i \(0.521221\pi\)
\(840\) 0 0
\(841\) −18.6591 −0.643416
\(842\) 1.72031 0.0592858
\(843\) 13.4167 0.462094
\(844\) −6.38706 −0.219852
\(845\) 0 0
\(846\) −0.791979 −0.0272288
\(847\) −5.48148 −0.188346
\(848\) 48.5151 1.66601
\(849\) −24.0890 −0.826732
\(850\) 0 0
\(851\) −2.05144 −0.0703224
\(852\) −26.9131 −0.922029
\(853\) 29.7898 1.01998 0.509991 0.860180i \(-0.329649\pi\)
0.509991 + 0.860180i \(0.329649\pi\)
\(854\) −1.17735 −0.0402880
\(855\) 0 0
\(856\) −2.05743 −0.0703216
\(857\) −36.2041 −1.23671 −0.618354 0.785899i \(-0.712200\pi\)
−0.618354 + 0.785899i \(0.712200\pi\)
\(858\) −2.96838 −0.101339
\(859\) −37.6533 −1.28471 −0.642357 0.766406i \(-0.722043\pi\)
−0.642357 + 0.766406i \(0.722043\pi\)
\(860\) 0 0
\(861\) 10.8417 0.369485
\(862\) −2.70133 −0.0920077
\(863\) 2.49913 0.0850715 0.0425357 0.999095i \(-0.486456\pi\)
0.0425357 + 0.999095i \(0.486456\pi\)
\(864\) −2.84182 −0.0966807
\(865\) 0 0
\(866\) 2.57600 0.0875361
\(867\) 17.0956 0.580596
\(868\) 22.2253 0.754376
\(869\) 29.5084 1.00100
\(870\) 0 0
\(871\) 47.3923 1.60583
\(872\) 2.17228 0.0735628
\(873\) −0.0305081 −0.00103254
\(874\) 1.65143 0.0558606
\(875\) 0 0
\(876\) −22.0697 −0.745668
\(877\) 32.4491 1.09573 0.547864 0.836567i \(-0.315441\pi\)
0.547864 + 0.836567i \(0.315441\pi\)
\(878\) −3.80832 −0.128525
\(879\) 32.1727 1.08516
\(880\) 0 0
\(881\) −33.2625 −1.12064 −0.560321 0.828275i \(-0.689322\pi\)
−0.560321 + 0.828275i \(0.689322\pi\)
\(882\) 1.34375 0.0452464
\(883\) 10.9730 0.369269 0.184635 0.982807i \(-0.440890\pi\)
0.184635 + 0.982807i \(0.440890\pi\)
\(884\) 53.0047 1.78274
\(885\) 0 0
\(886\) 4.66632 0.156768
\(887\) 26.0052 0.873168 0.436584 0.899664i \(-0.356188\pi\)
0.436584 + 0.899664i \(0.356188\pi\)
\(888\) 1.24202 0.0416794
\(889\) 19.5246 0.654834
\(890\) 0 0
\(891\) 2.56971 0.0860886
\(892\) −30.5860 −1.02410
\(893\) 13.3791 0.447715
\(894\) 4.21167 0.140859
\(895\) 0 0
\(896\) 9.10907 0.304313
\(897\) −7.51561 −0.250939
\(898\) 2.81679 0.0939975
\(899\) 29.5626 0.985969
\(900\) 0 0
\(901\) −77.8608 −2.59392
\(902\) −5.51408 −0.183599
\(903\) 4.83849 0.161015
\(904\) 3.06125 0.101816
\(905\) 0 0
\(906\) 0.388685 0.0129132
\(907\) 55.0108 1.82660 0.913302 0.407283i \(-0.133524\pi\)
0.913302 + 0.407283i \(0.133524\pi\)
\(908\) 26.2567 0.871359
\(909\) −11.2308 −0.372501
\(910\) 0 0
\(911\) −12.1075 −0.401139 −0.200569 0.979679i \(-0.564279\pi\)
−0.200569 + 0.979679i \(0.564279\pi\)
\(912\) 15.1668 0.502223
\(913\) 7.27817 0.240872
\(914\) −8.38337 −0.277297
\(915\) 0 0
\(916\) 28.6656 0.947138
\(917\) −18.6470 −0.615779
\(918\) 1.44086 0.0475555
\(919\) 5.97191 0.196995 0.0984975 0.995137i \(-0.468596\pi\)
0.0984975 + 0.995137i \(0.468596\pi\)
\(920\) 0 0
\(921\) −20.2962 −0.668783
\(922\) 6.09266 0.200651
\(923\) 64.9718 2.13857
\(924\) −6.21254 −0.204378
\(925\) 0 0
\(926\) −3.36370 −0.110538
\(927\) 0.720983 0.0236802
\(928\) 9.13854 0.299987
\(929\) 46.8756 1.53794 0.768970 0.639285i \(-0.220770\pi\)
0.768970 + 0.639285i \(0.220770\pi\)
\(930\) 0 0
\(931\) −22.7003 −0.743973
\(932\) 1.00636 0.0329644
\(933\) −18.7224 −0.612944
\(934\) 2.33709 0.0764719
\(935\) 0 0
\(936\) 4.55024 0.148729
\(937\) 19.4335 0.634864 0.317432 0.948281i \(-0.397179\pi\)
0.317432 + 0.948281i \(0.397179\pi\)
\(938\) −3.11458 −0.101695
\(939\) −24.8839 −0.812056
\(940\) 0 0
\(941\) 21.7097 0.707717 0.353858 0.935299i \(-0.384870\pi\)
0.353858 + 0.935299i \(0.384870\pi\)
\(942\) −2.27033 −0.0739712
\(943\) −13.9610 −0.454633
\(944\) −23.2593 −0.757026
\(945\) 0 0
\(946\) −2.46085 −0.0800091
\(947\) 6.31560 0.205229 0.102615 0.994721i \(-0.467279\pi\)
0.102615 + 0.994721i \(0.467279\pi\)
\(948\) −22.2671 −0.723203
\(949\) 53.2792 1.72952
\(950\) 0 0
\(951\) 25.1880 0.816778
\(952\) −7.07624 −0.229342
\(953\) −21.8017 −0.706227 −0.353114 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(954\) −3.29036 −0.106529
\(955\) 0 0
\(956\) −15.9595 −0.516167
\(957\) −8.26351 −0.267121
\(958\) 6.34876 0.205119
\(959\) 13.0842 0.422510
\(960\) 0 0
\(961\) 53.5134 1.72624
\(962\) −1.47602 −0.0475889
\(963\) −2.11668 −0.0682089
\(964\) 2.48435 0.0800154
\(965\) 0 0
\(966\) 0.493920 0.0158916
\(967\) 7.01268 0.225513 0.112756 0.993623i \(-0.464032\pi\)
0.112756 + 0.993623i \(0.464032\pi\)
\(968\) −4.27353 −0.137356
\(969\) −24.3409 −0.781942
\(970\) 0 0
\(971\) 7.41028 0.237807 0.118904 0.992906i \(-0.462062\pi\)
0.118904 + 0.992906i \(0.462062\pi\)
\(972\) −1.93911 −0.0621970
\(973\) 15.5231 0.497647
\(974\) −7.00909 −0.224586
\(975\) 0 0
\(976\) 13.9237 0.445688
\(977\) −39.2585 −1.25599 −0.627995 0.778217i \(-0.716124\pi\)
−0.627995 + 0.778217i \(0.716124\pi\)
\(978\) −1.13034 −0.0361444
\(979\) −3.54819 −0.113401
\(980\) 0 0
\(981\) 2.23483 0.0713528
\(982\) 1.50597 0.0480573
\(983\) 18.3827 0.586317 0.293159 0.956064i \(-0.405294\pi\)
0.293159 + 0.956064i \(0.405294\pi\)
\(984\) 8.45253 0.269457
\(985\) 0 0
\(986\) −4.63343 −0.147558
\(987\) 4.00150 0.127369
\(988\) −37.8401 −1.20385
\(989\) −6.23059 −0.198121
\(990\) 0 0
\(991\) 52.4658 1.66663 0.833316 0.552797i \(-0.186440\pi\)
0.833316 + 0.552797i \(0.186440\pi\)
\(992\) 26.1252 0.829476
\(993\) 30.4193 0.965326
\(994\) −4.26989 −0.135433
\(995\) 0 0
\(996\) −5.49212 −0.174025
\(997\) 36.3720 1.15191 0.575956 0.817480i \(-0.304630\pi\)
0.575956 + 0.817480i \(0.304630\pi\)
\(998\) 5.01322 0.158691
\(999\) 1.27778 0.0404273
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 1875.2.a.k.1.3 6
3.2 odd 2 5625.2.a.q.1.4 6
5.2 odd 4 1875.2.b.f.1249.5 12
5.3 odd 4 1875.2.b.f.1249.8 12
5.4 even 2 1875.2.a.j.1.4 6
15.14 odd 2 5625.2.a.p.1.3 6
25.2 odd 20 375.2.i.d.274.3 24
25.9 even 10 75.2.g.c.31.2 12
25.11 even 5 375.2.g.c.226.2 12
25.12 odd 20 375.2.i.d.349.4 24
25.13 odd 20 375.2.i.d.349.3 24
25.14 even 10 75.2.g.c.46.2 yes 12
25.16 even 5 375.2.g.c.151.2 12
25.23 odd 20 375.2.i.d.274.4 24
75.14 odd 10 225.2.h.d.46.2 12
75.59 odd 10 225.2.h.d.181.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.31.2 12 25.9 even 10
75.2.g.c.46.2 yes 12 25.14 even 10
225.2.h.d.46.2 12 75.14 odd 10
225.2.h.d.181.2 12 75.59 odd 10
375.2.g.c.151.2 12 25.16 even 5
375.2.g.c.226.2 12 25.11 even 5
375.2.i.d.274.3 24 25.2 odd 20
375.2.i.d.274.4 24 25.23 odd 20
375.2.i.d.349.3 24 25.13 odd 20
375.2.i.d.349.4 24 25.12 odd 20
1875.2.a.j.1.4 6 5.4 even 2
1875.2.a.k.1.3 6 1.1 even 1 trivial
1875.2.b.f.1249.5 12 5.2 odd 4
1875.2.b.f.1249.8 12 5.3 odd 4
5625.2.a.p.1.3 6 15.14 odd 2
5625.2.a.q.1.4 6 3.2 odd 2