Properties

Label 731.2.d.c.560.4
Level $731$
Weight $2$
Character 731.560
Analytic conductor $5.837$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [731,2,Mod(560,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.560");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 29 x^{18} + 358 x^{16} + 2458 x^{14} + 10298 x^{12} + 27188 x^{10} + 45053 x^{8} + 44980 x^{6} + 24400 x^{4} + 5448 x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 560.4
Root \(-1.23843i\) of defining polynomial
Character \(\chi\) \(=\) 731.560
Dual form 731.2.d.c.560.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.33801 q^{2} +3.08166i q^{3} +3.46629 q^{4} -3.53522i q^{5} -7.20495i q^{6} +0.740145i q^{7} -3.42819 q^{8} -6.49664 q^{9} +O(q^{10})\) \(q-2.33801 q^{2} +3.08166i q^{3} +3.46629 q^{4} -3.53522i q^{5} -7.20495i q^{6} +0.740145i q^{7} -3.42819 q^{8} -6.49664 q^{9} +8.26537i q^{10} +0.0284320i q^{11} +10.6819i q^{12} -6.23701 q^{13} -1.73047i q^{14} +10.8943 q^{15} +1.08257 q^{16} +(4.09808 + 0.453556i) q^{17} +15.1892 q^{18} +4.71182 q^{19} -12.2541i q^{20} -2.28088 q^{21} -0.0664743i q^{22} -0.160027i q^{23} -10.5645i q^{24} -7.49777 q^{25} +14.5822 q^{26} -10.7755i q^{27} +2.56555i q^{28} -2.33996i q^{29} -25.4711 q^{30} -5.83878i q^{31} +4.32533 q^{32} -0.0876179 q^{33} +(-9.58136 - 1.06042i) q^{34} +2.61657 q^{35} -22.5192 q^{36} +1.79732i q^{37} -11.0163 q^{38} -19.2203i q^{39} +12.1194i q^{40} -11.7697i q^{41} +5.33271 q^{42} +1.00000 q^{43} +0.0985535i q^{44} +22.9671i q^{45} +0.374145i q^{46} +6.13569 q^{47} +3.33611i q^{48} +6.45219 q^{49} +17.5298 q^{50} +(-1.39771 + 12.6289i) q^{51} -21.6193 q^{52} +4.23655 q^{53} +25.1932i q^{54} +0.100513 q^{55} -2.53736i q^{56} +14.5202i q^{57} +5.47086i q^{58} +12.6400 q^{59} +37.7629 q^{60} -3.46012i q^{61} +13.6511i q^{62} -4.80846i q^{63} -12.2778 q^{64} +22.0492i q^{65} +0.204851 q^{66} +4.99453 q^{67} +(14.2051 + 1.57215i) q^{68} +0.493150 q^{69} -6.11757 q^{70} -7.82497i q^{71} +22.2717 q^{72} -10.4028i q^{73} -4.20215i q^{74} -23.1056i q^{75} +16.3325 q^{76} -0.0210438 q^{77} +44.9374i q^{78} +3.94100i q^{79} -3.82711i q^{80} +13.7165 q^{81} +27.5177i q^{82} +8.01134 q^{83} -7.90617 q^{84} +(1.60342 - 14.4876i) q^{85} -2.33801 q^{86} +7.21098 q^{87} -0.0974704i q^{88} -5.50806 q^{89} -53.6972i q^{90} -4.61629i q^{91} -0.554701i q^{92} +17.9931 q^{93} -14.3453 q^{94} -16.6573i q^{95} +13.3292i q^{96} +13.6758i q^{97} -15.0853 q^{98} -0.184713i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 42 q^{4} + 18 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 42 q^{4} + 18 q^{8} - 18 q^{9} - 4 q^{13} + 26 q^{15} + 6 q^{16} + 16 q^{17} - 22 q^{18} - 4 q^{19} + 20 q^{21} - 2 q^{25} + 22 q^{26} - 72 q^{30} + 38 q^{32} - 12 q^{33} + 12 q^{34} - 30 q^{35} - 104 q^{36} - 22 q^{38} + 26 q^{42} + 20 q^{43} - 34 q^{47} + 22 q^{49} + 42 q^{50} + 52 q^{51} - 110 q^{52} + 14 q^{53} + 12 q^{55} + 20 q^{59} + 42 q^{60} - 22 q^{64} + 50 q^{66} - 12 q^{67} + 50 q^{68} - 82 q^{69} - 30 q^{70} - 50 q^{72} + 2 q^{76} + 78 q^{77} + 44 q^{81} + 20 q^{83} + 62 q^{84} + 76 q^{85} + 2 q^{86} + 12 q^{87} - 46 q^{89} + 58 q^{93} - 18 q^{94} - 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(562\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.33801 −1.65322 −0.826611 0.562774i \(-0.809734\pi\)
−0.826611 + 0.562774i \(0.809734\pi\)
\(3\) 3.08166i 1.77920i 0.456742 + 0.889599i \(0.349016\pi\)
−0.456742 + 0.889599i \(0.650984\pi\)
\(4\) 3.46629 1.73314
\(5\) 3.53522i 1.58100i −0.612464 0.790499i \(-0.709821\pi\)
0.612464 0.790499i \(-0.290179\pi\)
\(6\) 7.20495i 2.94141i
\(7\) 0.740145i 0.279748i 0.990169 + 0.139874i \(0.0446698\pi\)
−0.990169 + 0.139874i \(0.955330\pi\)
\(8\) −3.42819 −1.21205
\(9\) −6.49664 −2.16555
\(10\) 8.26537i 2.61374i
\(11\) 0.0284320i 0.00857258i 0.999991 + 0.00428629i \(0.00136437\pi\)
−0.999991 + 0.00428629i \(0.998636\pi\)
\(12\) 10.6819i 3.08361i
\(13\) −6.23701 −1.72983 −0.864917 0.501915i \(-0.832629\pi\)
−0.864917 + 0.501915i \(0.832629\pi\)
\(14\) 1.73047i 0.462486i
\(15\) 10.8943 2.81291
\(16\) 1.08257 0.270642
\(17\) 4.09808 + 0.453556i 0.993931 + 0.110003i
\(18\) 15.1892 3.58013
\(19\) 4.71182 1.08097 0.540483 0.841355i \(-0.318241\pi\)
0.540483 + 0.841355i \(0.318241\pi\)
\(20\) 12.2541i 2.74010i
\(21\) −2.28088 −0.497728
\(22\) 0.0664743i 0.0141724i
\(23\) 0.160027i 0.0333680i −0.999861 0.0166840i \(-0.994689\pi\)
0.999861 0.0166840i \(-0.00531093\pi\)
\(24\) 10.5645i 2.15648i
\(25\) −7.49777 −1.49955
\(26\) 14.5822 2.85980
\(27\) 10.7755i 2.07374i
\(28\) 2.56555i 0.484844i
\(29\) 2.33996i 0.434521i −0.976114 0.217260i \(-0.930288\pi\)
0.976114 0.217260i \(-0.0697120\pi\)
\(30\) −25.4711 −4.65036
\(31\) 5.83878i 1.04868i −0.851510 0.524338i \(-0.824313\pi\)
0.851510 0.524338i \(-0.175687\pi\)
\(32\) 4.32533 0.764617
\(33\) −0.0876179 −0.0152523
\(34\) −9.58136 1.06042i −1.64319 0.181860i
\(35\) 2.61657 0.442282
\(36\) −22.5192 −3.75320
\(37\) 1.79732i 0.295477i 0.989026 + 0.147739i \(0.0471995\pi\)
−0.989026 + 0.147739i \(0.952801\pi\)
\(38\) −11.0163 −1.78708
\(39\) 19.2203i 3.07772i
\(40\) 12.1194i 1.91625i
\(41\) 11.7697i 1.83812i −0.394116 0.919061i \(-0.628949\pi\)
0.394116 0.919061i \(-0.371051\pi\)
\(42\) 5.33271 0.822855
\(43\) 1.00000 0.152499
\(44\) 0.0985535i 0.0148575i
\(45\) 22.9671i 3.42373i
\(46\) 0.374145i 0.0551647i
\(47\) 6.13569 0.894983 0.447491 0.894288i \(-0.352318\pi\)
0.447491 + 0.894288i \(0.352318\pi\)
\(48\) 3.33611i 0.481526i
\(49\) 6.45219 0.921741
\(50\) 17.5298 2.47909
\(51\) −1.39771 + 12.6289i −0.195718 + 1.76840i
\(52\) −21.6193 −2.99805
\(53\) 4.23655 0.581934 0.290967 0.956733i \(-0.406023\pi\)
0.290967 + 0.956733i \(0.406023\pi\)
\(54\) 25.1932i 3.42836i
\(55\) 0.100513 0.0135532
\(56\) 2.53736i 0.339069i
\(57\) 14.5202i 1.92325i
\(58\) 5.47086i 0.718359i
\(59\) 12.6400 1.64558 0.822791 0.568344i \(-0.192416\pi\)
0.822791 + 0.568344i \(0.192416\pi\)
\(60\) 37.7629 4.87517
\(61\) 3.46012i 0.443023i −0.975158 0.221511i \(-0.928901\pi\)
0.975158 0.221511i \(-0.0710990\pi\)
\(62\) 13.6511i 1.73369i
\(63\) 4.80846i 0.605809i
\(64\) −12.2778 −1.53472
\(65\) 22.0492i 2.73486i
\(66\) 0.204851 0.0252155
\(67\) 4.99453 0.610178 0.305089 0.952324i \(-0.401314\pi\)
0.305089 + 0.952324i \(0.401314\pi\)
\(68\) 14.2051 + 1.57215i 1.72263 + 0.190652i
\(69\) 0.493150 0.0593683
\(70\) −6.11757 −0.731190
\(71\) 7.82497i 0.928653i −0.885664 0.464326i \(-0.846296\pi\)
0.885664 0.464326i \(-0.153704\pi\)
\(72\) 22.2717 2.62475
\(73\) 10.4028i 1.21756i −0.793340 0.608778i \(-0.791660\pi\)
0.793340 0.608778i \(-0.208340\pi\)
\(74\) 4.20215i 0.488490i
\(75\) 23.1056i 2.66800i
\(76\) 16.3325 1.87347
\(77\) −0.0210438 −0.00239817
\(78\) 44.9374i 5.08815i
\(79\) 3.94100i 0.443397i 0.975115 + 0.221698i \(0.0711600\pi\)
−0.975115 + 0.221698i \(0.928840\pi\)
\(80\) 3.82711i 0.427884i
\(81\) 13.7165 1.52405
\(82\) 27.5177i 3.03882i
\(83\) 8.01134 0.879359 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(84\) −7.90617 −0.862634
\(85\) 1.60342 14.4876i 0.173915 1.57140i
\(86\) −2.33801 −0.252114
\(87\) 7.21098 0.773098
\(88\) 0.0974704i 0.0103904i
\(89\) −5.50806 −0.583853 −0.291927 0.956441i \(-0.594296\pi\)
−0.291927 + 0.956441i \(0.594296\pi\)
\(90\) 53.6972i 5.66018i
\(91\) 4.61629i 0.483918i
\(92\) 0.554701i 0.0578315i
\(93\) 17.9931 1.86580
\(94\) −14.3453 −1.47961
\(95\) 16.6573i 1.70901i
\(96\) 13.3292i 1.36041i
\(97\) 13.6758i 1.38857i 0.719700 + 0.694285i \(0.244279\pi\)
−0.719700 + 0.694285i \(0.755721\pi\)
\(98\) −15.0853 −1.52384
\(99\) 0.184713i 0.0185643i
\(100\) −25.9894 −2.59894
\(101\) −6.19857 −0.616781 −0.308390 0.951260i \(-0.599790\pi\)
−0.308390 + 0.951260i \(0.599790\pi\)
\(102\) 3.26785 29.5265i 0.323565 2.92356i
\(103\) −17.1868 −1.69347 −0.846733 0.532018i \(-0.821434\pi\)
−0.846733 + 0.532018i \(0.821434\pi\)
\(104\) 21.3816 2.09664
\(105\) 8.06340i 0.786907i
\(106\) −9.90508 −0.962067
\(107\) 5.41292i 0.523287i 0.965165 + 0.261644i \(0.0842645\pi\)
−0.965165 + 0.261644i \(0.915736\pi\)
\(108\) 37.3509i 3.59409i
\(109\) 12.2793i 1.17615i 0.808807 + 0.588074i \(0.200114\pi\)
−0.808807 + 0.588074i \(0.799886\pi\)
\(110\) −0.235001 −0.0224065
\(111\) −5.53873 −0.525713
\(112\) 0.801257i 0.0757117i
\(113\) 14.1098i 1.32734i −0.748027 0.663668i \(-0.768999\pi\)
0.748027 0.663668i \(-0.231001\pi\)
\(114\) 33.9485i 3.17957i
\(115\) −0.565732 −0.0527547
\(116\) 8.11099i 0.753086i
\(117\) 40.5196 3.74604
\(118\) −29.5523 −2.72051
\(119\) −0.335697 + 3.03318i −0.0307733 + 0.278051i
\(120\) −37.3479 −3.40938
\(121\) 10.9992 0.999927
\(122\) 8.08979i 0.732415i
\(123\) 36.2703 3.27038
\(124\) 20.2389i 1.81751i
\(125\) 8.83015i 0.789793i
\(126\) 11.2422i 1.00154i
\(127\) 2.88776 0.256248 0.128124 0.991758i \(-0.459105\pi\)
0.128124 + 0.991758i \(0.459105\pi\)
\(128\) 20.0549 1.77262
\(129\) 3.08166i 0.271325i
\(130\) 51.5512i 4.52134i
\(131\) 4.11440i 0.359477i −0.983714 0.179738i \(-0.942475\pi\)
0.983714 0.179738i \(-0.0575251\pi\)
\(132\) −0.303709 −0.0264345
\(133\) 3.48743i 0.302399i
\(134\) −11.6772 −1.00876
\(135\) −38.0937 −3.27858
\(136\) −14.0490 1.55487i −1.20469 0.133329i
\(137\) 16.0702 1.37297 0.686487 0.727142i \(-0.259152\pi\)
0.686487 + 0.727142i \(0.259152\pi\)
\(138\) −1.15299 −0.0981490
\(139\) 13.4527i 1.14105i −0.821282 0.570523i \(-0.806741\pi\)
0.821282 0.570523i \(-0.193259\pi\)
\(140\) 9.06979 0.766537
\(141\) 18.9081i 1.59235i
\(142\) 18.2948i 1.53527i
\(143\) 0.177331i 0.0148291i
\(144\) −7.03306 −0.586088
\(145\) −8.27229 −0.686976
\(146\) 24.3219i 2.01289i
\(147\) 19.8835i 1.63996i
\(148\) 6.23002i 0.512105i
\(149\) −15.6285 −1.28033 −0.640167 0.768236i \(-0.721135\pi\)
−0.640167 + 0.768236i \(0.721135\pi\)
\(150\) 54.0211i 4.41080i
\(151\) −16.6485 −1.35484 −0.677419 0.735597i \(-0.736902\pi\)
−0.677419 + 0.735597i \(0.736902\pi\)
\(152\) −16.1530 −1.31018
\(153\) −26.6238 2.94659i −2.15241 0.238218i
\(154\) 0.0492006 0.00396470
\(155\) −20.6414 −1.65795
\(156\) 66.6232i 5.33413i
\(157\) 12.3538 0.985938 0.492969 0.870047i \(-0.335912\pi\)
0.492969 + 0.870047i \(0.335912\pi\)
\(158\) 9.21408i 0.733033i
\(159\) 13.0556i 1.03538i
\(160\) 15.2910i 1.20886i
\(161\) 0.118443 0.00933465
\(162\) −32.0692 −2.51959
\(163\) 10.8178i 0.847317i −0.905822 0.423659i \(-0.860746\pi\)
0.905822 0.423659i \(-0.139254\pi\)
\(164\) 40.7972i 3.18573i
\(165\) 0.309748i 0.0241139i
\(166\) −18.7306 −1.45377
\(167\) 23.8242i 1.84357i 0.387701 + 0.921785i \(0.373269\pi\)
−0.387701 + 0.921785i \(0.626731\pi\)
\(168\) 7.81928 0.603271
\(169\) 25.9003 1.99233
\(170\) −3.74881 + 33.8722i −0.287520 + 2.59788i
\(171\) −30.6110 −2.34088
\(172\) 3.46629 0.264302
\(173\) 1.83963i 0.139864i 0.997552 + 0.0699321i \(0.0222783\pi\)
−0.997552 + 0.0699321i \(0.977722\pi\)
\(174\) −16.8593 −1.27810
\(175\) 5.54943i 0.419498i
\(176\) 0.0307796i 0.00232010i
\(177\) 38.9521i 2.92782i
\(178\) 12.8779 0.965239
\(179\) −9.36447 −0.699933 −0.349966 0.936762i \(-0.613807\pi\)
−0.349966 + 0.936762i \(0.613807\pi\)
\(180\) 79.6104i 5.93381i
\(181\) 18.7895i 1.39661i −0.715799 0.698306i \(-0.753938\pi\)
0.715799 0.698306i \(-0.246062\pi\)
\(182\) 10.7929i 0.800025i
\(183\) 10.6629 0.788226
\(184\) 0.548604i 0.0404436i
\(185\) 6.35392 0.467149
\(186\) −42.0681 −3.08459
\(187\) −0.0128955 + 0.116517i −0.000943013 + 0.00852055i
\(188\) 21.2681 1.55113
\(189\) 7.97541 0.580126
\(190\) 38.9450i 2.82537i
\(191\) −9.59380 −0.694183 −0.347092 0.937831i \(-0.612831\pi\)
−0.347092 + 0.937831i \(0.612831\pi\)
\(192\) 37.8360i 2.73058i
\(193\) 3.84327i 0.276645i −0.990387 0.138322i \(-0.955829\pi\)
0.990387 0.138322i \(-0.0441710\pi\)
\(194\) 31.9742i 2.29561i
\(195\) −67.9481 −4.86587
\(196\) 22.3651 1.59751
\(197\) 9.10325i 0.648580i −0.945958 0.324290i \(-0.894875\pi\)
0.945958 0.324290i \(-0.105125\pi\)
\(198\) 0.431860i 0.0306910i
\(199\) 23.0246i 1.63217i −0.577931 0.816085i \(-0.696140\pi\)
0.577931 0.816085i \(-0.303860\pi\)
\(200\) 25.7038 1.81753
\(201\) 15.3914i 1.08563i
\(202\) 14.4923 1.01968
\(203\) 1.73191 0.121556
\(204\) −4.84485 + 43.7754i −0.339207 + 3.06489i
\(205\) −41.6085 −2.90607
\(206\) 40.1829 2.79968
\(207\) 1.03964i 0.0722600i
\(208\) −6.75198 −0.468166
\(209\) 0.133967i 0.00926667i
\(210\) 18.8523i 1.30093i
\(211\) 23.6068i 1.62516i 0.582849 + 0.812581i \(0.301938\pi\)
−0.582849 + 0.812581i \(0.698062\pi\)
\(212\) 14.6851 1.00858
\(213\) 24.1139 1.65226
\(214\) 12.6555i 0.865110i
\(215\) 3.53522i 0.241100i
\(216\) 36.9404i 2.51348i
\(217\) 4.32154 0.293365
\(218\) 28.7092i 1.94443i
\(219\) 32.0579 2.16627
\(220\) 0.348408 0.0234897
\(221\) −25.5598 2.82883i −1.71934 0.190288i
\(222\) 12.9496 0.869121
\(223\) −8.36153 −0.559930 −0.279965 0.960010i \(-0.590323\pi\)
−0.279965 + 0.960010i \(0.590323\pi\)
\(224\) 3.20137i 0.213900i
\(225\) 48.7103 3.24736
\(226\) 32.9888i 2.19438i
\(227\) 20.8368i 1.38298i −0.722384 0.691492i \(-0.756954\pi\)
0.722384 0.691492i \(-0.243046\pi\)
\(228\) 50.3313i 3.33327i
\(229\) 18.1669 1.20050 0.600252 0.799811i \(-0.295067\pi\)
0.600252 + 0.799811i \(0.295067\pi\)
\(230\) 1.32269 0.0872153
\(231\) 0.0648499i 0.00426681i
\(232\) 8.02184i 0.526660i
\(233\) 16.1982i 1.06118i −0.847628 0.530590i \(-0.821970\pi\)
0.847628 0.530590i \(-0.178030\pi\)
\(234\) −94.7352 −6.19304
\(235\) 21.6910i 1.41497i
\(236\) 43.8137 2.85203
\(237\) −12.1448 −0.788891
\(238\) 0.784862 7.09159i 0.0508751 0.459680i
\(239\) 3.56099 0.230342 0.115171 0.993346i \(-0.463258\pi\)
0.115171 + 0.993346i \(0.463258\pi\)
\(240\) 11.7939 0.761291
\(241\) 3.98278i 0.256553i 0.991738 + 0.128277i \(0.0409445\pi\)
−0.991738 + 0.128277i \(0.959055\pi\)
\(242\) −25.7162 −1.65310
\(243\) 9.94305i 0.637847i
\(244\) 11.9938i 0.767822i
\(245\) 22.8099i 1.45727i
\(246\) −84.8003 −5.40667
\(247\) −29.3877 −1.86989
\(248\) 20.0164i 1.27105i
\(249\) 24.6882i 1.56455i
\(250\) 20.6450i 1.30570i
\(251\) 24.1344 1.52335 0.761676 0.647958i \(-0.224377\pi\)
0.761676 + 0.647958i \(0.224377\pi\)
\(252\) 16.6675i 1.04995i
\(253\) 0.00454990 0.000286050
\(254\) −6.75162 −0.423634
\(255\) 44.6460 + 4.94119i 2.79584 + 0.309429i
\(256\) −22.3330 −1.39581
\(257\) −20.7034 −1.29144 −0.645720 0.763574i \(-0.723443\pi\)
−0.645720 + 0.763574i \(0.723443\pi\)
\(258\) 7.20495i 0.448561i
\(259\) −1.33028 −0.0826594
\(260\) 76.4288i 4.73991i
\(261\) 15.2019i 0.940975i
\(262\) 9.61950i 0.594295i
\(263\) 0.500679 0.0308732 0.0154366 0.999881i \(-0.495086\pi\)
0.0154366 + 0.999881i \(0.495086\pi\)
\(264\) 0.300371 0.0184865
\(265\) 14.9771i 0.920037i
\(266\) 8.15365i 0.499932i
\(267\) 16.9740i 1.03879i
\(268\) 17.3125 1.05753
\(269\) 13.4633i 0.820873i −0.911889 0.410437i \(-0.865376\pi\)
0.911889 0.410437i \(-0.134624\pi\)
\(270\) 89.0633 5.42022
\(271\) −12.7952 −0.777253 −0.388626 0.921395i \(-0.627050\pi\)
−0.388626 + 0.921395i \(0.627050\pi\)
\(272\) 4.43645 + 0.491005i 0.269000 + 0.0297715i
\(273\) 14.2258 0.860987
\(274\) −37.5724 −2.26983
\(275\) 0.213177i 0.0128550i
\(276\) 1.70940 0.102894
\(277\) 7.42987i 0.446418i 0.974771 + 0.223209i \(0.0716532\pi\)
−0.974771 + 0.223209i \(0.928347\pi\)
\(278\) 31.4526i 1.88640i
\(279\) 37.9325i 2.27096i
\(280\) −8.97011 −0.536067
\(281\) 0.280317 0.0167223 0.00836115 0.999965i \(-0.497339\pi\)
0.00836115 + 0.999965i \(0.497339\pi\)
\(282\) 44.2074i 2.63251i
\(283\) 17.5709i 1.04448i 0.852798 + 0.522240i \(0.174904\pi\)
−0.852798 + 0.522240i \(0.825096\pi\)
\(284\) 27.1236i 1.60949i
\(285\) 51.3323 3.04066
\(286\) 0.414601i 0.0245159i
\(287\) 8.71130 0.514212
\(288\) −28.1001 −1.65581
\(289\) 16.5886 + 3.71742i 0.975799 + 0.218672i
\(290\) 19.3407 1.13572
\(291\) −42.1443 −2.47054
\(292\) 36.0591i 2.11020i
\(293\) −6.68485 −0.390533 −0.195267 0.980750i \(-0.562557\pi\)
−0.195267 + 0.980750i \(0.562557\pi\)
\(294\) 46.4877i 2.71122i
\(295\) 44.6850i 2.60166i
\(296\) 6.16155i 0.358133i
\(297\) 0.306369 0.0177773
\(298\) 36.5395 2.11668
\(299\) 0.998092i 0.0577211i
\(300\) 80.0906i 4.62403i
\(301\) 0.740145i 0.0426612i
\(302\) 38.9244 2.23985
\(303\) 19.1019i 1.09738i
\(304\) 5.10087 0.292555
\(305\) −12.2323 −0.700418
\(306\) 62.2467 + 6.88915i 3.55840 + 0.393827i
\(307\) 16.1286 0.920506 0.460253 0.887788i \(-0.347759\pi\)
0.460253 + 0.887788i \(0.347759\pi\)
\(308\) −0.0729439 −0.00415636
\(309\) 52.9639i 3.01301i
\(310\) 48.2597 2.74097
\(311\) 12.0595i 0.683831i −0.939731 0.341916i \(-0.888924\pi\)
0.939731 0.341916i \(-0.111076\pi\)
\(312\) 65.8910i 3.73034i
\(313\) 28.1473i 1.59098i 0.605966 + 0.795490i \(0.292787\pi\)
−0.605966 + 0.795490i \(0.707213\pi\)
\(314\) −28.8832 −1.62997
\(315\) −16.9989 −0.957782
\(316\) 13.6606i 0.768470i
\(317\) 24.1025i 1.35373i −0.736107 0.676865i \(-0.763338\pi\)
0.736107 0.676865i \(-0.236662\pi\)
\(318\) 30.5241i 1.71171i
\(319\) 0.0665299 0.00372496
\(320\) 43.4047i 2.42639i
\(321\) −16.6808 −0.931032
\(322\) −0.276922 −0.0154322
\(323\) 19.3094 + 2.13707i 1.07441 + 0.118910i
\(324\) 47.5452 2.64140
\(325\) 46.7636 2.59398
\(326\) 25.2922i 1.40080i
\(327\) −37.8408 −2.09260
\(328\) 40.3488i 2.22789i
\(329\) 4.54130i 0.250370i
\(330\) 0.724195i 0.0398656i
\(331\) 2.10977 0.115964 0.0579818 0.998318i \(-0.481533\pi\)
0.0579818 + 0.998318i \(0.481533\pi\)
\(332\) 27.7696 1.52405
\(333\) 11.6765i 0.639871i
\(334\) 55.7011i 3.04783i
\(335\) 17.6567i 0.964691i
\(336\) −2.46920 −0.134706
\(337\) 25.7371i 1.40199i 0.713165 + 0.700996i \(0.247261\pi\)
−0.713165 + 0.700996i \(0.752739\pi\)
\(338\) −60.5550 −3.29376
\(339\) 43.4816 2.36159
\(340\) 5.55791 50.2182i 0.301420 2.72347i
\(341\) 0.166008 0.00898985
\(342\) 71.5689 3.87000
\(343\) 9.95657i 0.537604i
\(344\) −3.42819 −0.184836
\(345\) 1.74339i 0.0938612i
\(346\) 4.30106i 0.231227i
\(347\) 20.9754i 1.12602i −0.826451 0.563009i \(-0.809644\pi\)
0.826451 0.563009i \(-0.190356\pi\)
\(348\) 24.9953 1.33989
\(349\) 7.92514 0.424223 0.212112 0.977245i \(-0.431966\pi\)
0.212112 + 0.977245i \(0.431966\pi\)
\(350\) 12.9746i 0.693523i
\(351\) 67.2067i 3.58723i
\(352\) 0.122978i 0.00655474i
\(353\) −28.9441 −1.54054 −0.770269 0.637720i \(-0.779878\pi\)
−0.770269 + 0.637720i \(0.779878\pi\)
\(354\) 91.0703i 4.84033i
\(355\) −27.6630 −1.46820
\(356\) −19.0925 −1.01190
\(357\) −9.34722 1.03450i −0.494707 0.0547518i
\(358\) 21.8942 1.15714
\(359\) −7.44895 −0.393141 −0.196570 0.980490i \(-0.562980\pi\)
−0.196570 + 0.980490i \(0.562980\pi\)
\(360\) 78.7354i 4.14972i
\(361\) 3.20128 0.168488
\(362\) 43.9300i 2.30891i
\(363\) 33.8958i 1.77907i
\(364\) 16.0014i 0.838700i
\(365\) −36.7762 −1.92495
\(366\) −24.9300 −1.30311
\(367\) 33.9690i 1.77317i −0.462567 0.886584i \(-0.653072\pi\)
0.462567 0.886584i \(-0.346928\pi\)
\(368\) 0.173240i 0.00903078i
\(369\) 76.4637i 3.98054i
\(370\) −14.8555 −0.772301
\(371\) 3.13566i 0.162795i
\(372\) 62.3694 3.23370
\(373\) −21.6025 −1.11853 −0.559267 0.828988i \(-0.688917\pi\)
−0.559267 + 0.828988i \(0.688917\pi\)
\(374\) 0.0301498 0.272417i 0.00155901 0.0140864i
\(375\) −27.2115 −1.40520
\(376\) −21.0343 −1.08476
\(377\) 14.5944i 0.751649i
\(378\) −18.6466 −0.959077
\(379\) 11.1463i 0.572549i 0.958148 + 0.286275i \(0.0924170\pi\)
−0.958148 + 0.286275i \(0.907583\pi\)
\(380\) 57.7391i 2.96195i
\(381\) 8.89911i 0.455915i
\(382\) 22.4304 1.14764
\(383\) 11.5891 0.592177 0.296088 0.955161i \(-0.404318\pi\)
0.296088 + 0.955161i \(0.404318\pi\)
\(384\) 61.8025i 3.15385i
\(385\) 0.0743945i 0.00379149i
\(386\) 8.98560i 0.457355i
\(387\) −6.49664 −0.330243
\(388\) 47.4043i 2.40659i
\(389\) 3.94673 0.200107 0.100054 0.994982i \(-0.468099\pi\)
0.100054 + 0.994982i \(0.468099\pi\)
\(390\) 158.863 8.04436
\(391\) 0.0725813 0.655805i 0.00367059 0.0331655i
\(392\) −22.1193 −1.11719
\(393\) 12.6792 0.639580
\(394\) 21.2835i 1.07225i
\(395\) 13.9323 0.701009
\(396\) 0.640267i 0.0321746i
\(397\) 28.5059i 1.43067i 0.698782 + 0.715335i \(0.253726\pi\)
−0.698782 + 0.715335i \(0.746274\pi\)
\(398\) 53.8317i 2.69834i
\(399\) −10.7471 −0.538027
\(400\) −8.11684 −0.405842
\(401\) 21.5476i 1.07604i −0.842933 0.538019i \(-0.819173\pi\)
0.842933 0.538019i \(-0.180827\pi\)
\(402\) 35.9853i 1.79479i
\(403\) 36.4165i 1.81404i
\(404\) −21.4860 −1.06897
\(405\) 48.4907i 2.40952i
\(406\) −4.04923 −0.200960
\(407\) −0.0511014 −0.00253300
\(408\) 4.79160 43.2943i 0.237220 2.14339i
\(409\) 4.93591 0.244065 0.122033 0.992526i \(-0.461059\pi\)
0.122033 + 0.992526i \(0.461059\pi\)
\(410\) 97.2811 4.80437
\(411\) 49.5231i 2.44279i
\(412\) −59.5744 −2.93502
\(413\) 9.35540i 0.460349i
\(414\) 2.43069i 0.119462i
\(415\) 28.3218i 1.39026i
\(416\) −26.9771 −1.32266
\(417\) 41.4568 2.03015
\(418\) 0.313215i 0.0153199i
\(419\) 19.1063i 0.933403i 0.884415 + 0.466701i \(0.154558\pi\)
−0.884415 + 0.466701i \(0.845442\pi\)
\(420\) 27.9500i 1.36382i
\(421\) −12.5866 −0.613433 −0.306716 0.951801i \(-0.599230\pi\)
−0.306716 + 0.951801i \(0.599230\pi\)
\(422\) 55.1930i 2.68675i
\(423\) −39.8614 −1.93813
\(424\) −14.5237 −0.705333
\(425\) −30.7265 3.40065i −1.49045 0.164956i
\(426\) −56.3785 −2.73155
\(427\) 2.56099 0.123935
\(428\) 18.7627i 0.906932i
\(429\) 0.546473 0.0263840
\(430\) 8.26537i 0.398592i
\(431\) 7.68277i 0.370066i 0.982732 + 0.185033i \(0.0592392\pi\)
−0.982732 + 0.185033i \(0.940761\pi\)
\(432\) 11.6652i 0.561242i
\(433\) −9.31844 −0.447816 −0.223908 0.974610i \(-0.571881\pi\)
−0.223908 + 0.974610i \(0.571881\pi\)
\(434\) −10.1038 −0.484998
\(435\) 25.4924i 1.22227i
\(436\) 42.5637i 2.03843i
\(437\) 0.754021i 0.0360697i
\(438\) −74.9518 −3.58133
\(439\) 13.6524i 0.651592i 0.945440 + 0.325796i \(0.105632\pi\)
−0.945440 + 0.325796i \(0.894368\pi\)
\(440\) −0.344579 −0.0164272
\(441\) −41.9176 −1.99607
\(442\) 59.7590 + 6.61383i 2.84244 + 0.314588i
\(443\) −11.1088 −0.527794 −0.263897 0.964551i \(-0.585008\pi\)
−0.263897 + 0.964551i \(0.585008\pi\)
\(444\) −19.1988 −0.911136
\(445\) 19.4722i 0.923071i
\(446\) 19.5493 0.925688
\(447\) 48.1617i 2.27797i
\(448\) 9.08734i 0.429337i
\(449\) 6.26601i 0.295711i −0.989009 0.147856i \(-0.952763\pi\)
0.989009 0.147856i \(-0.0472371\pi\)
\(450\) −113.885 −5.36860
\(451\) 0.334637 0.0157574
\(452\) 48.9085i 2.30046i
\(453\) 51.3052i 2.41053i
\(454\) 48.7165i 2.28638i
\(455\) −16.3196 −0.765074
\(456\) 49.7782i 2.33108i
\(457\) −31.9074 −1.49257 −0.746283 0.665628i \(-0.768164\pi\)
−0.746283 + 0.665628i \(0.768164\pi\)
\(458\) −42.4744 −1.98470
\(459\) 4.88728 44.1588i 0.228119 2.06116i
\(460\) −1.96099 −0.0914315
\(461\) 15.2512 0.710318 0.355159 0.934806i \(-0.384427\pi\)
0.355159 + 0.934806i \(0.384427\pi\)
\(462\) 0.151620i 0.00705399i
\(463\) 41.8252 1.94378 0.971891 0.235432i \(-0.0756506\pi\)
0.971891 + 0.235432i \(0.0756506\pi\)
\(464\) 2.53317i 0.117600i
\(465\) 63.6097i 2.94983i
\(466\) 37.8716i 1.75437i
\(467\) 23.4463 1.08496 0.542482 0.840067i \(-0.317485\pi\)
0.542482 + 0.840067i \(0.317485\pi\)
\(468\) 140.453 6.49242
\(469\) 3.69667i 0.170696i
\(470\) 50.7138i 2.33925i
\(471\) 38.0701i 1.75418i
\(472\) −43.3322 −1.99453
\(473\) 0.0284320i 0.00130731i
\(474\) 28.3947 1.30421
\(475\) −35.3282 −1.62097
\(476\) −1.16362 + 10.5139i −0.0533345 + 0.481902i
\(477\) −27.5233 −1.26021
\(478\) −8.32564 −0.380806
\(479\) 10.4477i 0.477369i 0.971097 + 0.238684i \(0.0767161\pi\)
−0.971097 + 0.238684i \(0.923284\pi\)
\(480\) 47.1216 2.15080
\(481\) 11.2099i 0.511127i
\(482\) 9.31177i 0.424140i
\(483\) 0.365003i 0.0166082i
\(484\) 38.1263 1.73302
\(485\) 48.3470 2.19533
\(486\) 23.2469i 1.05450i
\(487\) 6.19164i 0.280570i 0.990111 + 0.140285i \(0.0448019\pi\)
−0.990111 + 0.140285i \(0.955198\pi\)
\(488\) 11.8619i 0.536965i
\(489\) 33.3369 1.50755
\(490\) 53.3297i 2.40919i
\(491\) 0.704438 0.0317908 0.0158954 0.999874i \(-0.494940\pi\)
0.0158954 + 0.999874i \(0.494940\pi\)
\(492\) 125.723 5.66804
\(493\) 1.06130 9.58937i 0.0477987 0.431883i
\(494\) 68.7086 3.09135
\(495\) −0.653000 −0.0293502
\(496\) 6.32088i 0.283816i
\(497\) 5.79161 0.259789
\(498\) 57.7213i 2.58655i
\(499\) 1.40530i 0.0629100i 0.999505 + 0.0314550i \(0.0100141\pi\)
−0.999505 + 0.0314550i \(0.989986\pi\)
\(500\) 30.6078i 1.36882i
\(501\) −73.4181 −3.28008
\(502\) −56.4265 −2.51844
\(503\) 32.2080i 1.43609i −0.695999 0.718043i \(-0.745038\pi\)
0.695999 0.718043i \(-0.254962\pi\)
\(504\) 16.4843i 0.734269i
\(505\) 21.9133i 0.975129i
\(506\) −0.0106377 −0.000472904
\(507\) 79.8158i 3.54475i
\(508\) 10.0098 0.444114
\(509\) −8.34894 −0.370060 −0.185030 0.982733i \(-0.559238\pi\)
−0.185030 + 0.982733i \(0.559238\pi\)
\(510\) −104.383 11.5526i −4.62214 0.511556i
\(511\) 7.69958 0.340610
\(512\) 12.1050 0.534969
\(513\) 50.7721i 2.24165i
\(514\) 48.4047 2.13504
\(515\) 60.7591i 2.67737i
\(516\) 10.6819i 0.470246i
\(517\) 0.174450i 0.00767231i
\(518\) 3.11020 0.136654
\(519\) −5.66911 −0.248846
\(520\) 75.5888i 3.31479i
\(521\) 1.54030i 0.0674818i 0.999431 + 0.0337409i \(0.0107421\pi\)
−0.999431 + 0.0337409i \(0.989258\pi\)
\(522\) 35.5422i 1.55564i
\(523\) 18.3697 0.803253 0.401626 0.915804i \(-0.368445\pi\)
0.401626 + 0.915804i \(0.368445\pi\)
\(524\) 14.2617i 0.623024i
\(525\) 17.1015 0.746370
\(526\) −1.17059 −0.0510403
\(527\) 2.64821 23.9278i 0.115358 1.04231i
\(528\) −0.0948523 −0.00412792
\(529\) 22.9744 0.998887
\(530\) 35.0166i 1.52103i
\(531\) −82.1173 −3.56359
\(532\) 12.0884i 0.524100i
\(533\) 73.4078i 3.17965i
\(534\) 39.6853i 1.71735i
\(535\) 19.1359 0.827316
\(536\) −17.1222 −0.739566
\(537\) 28.8581i 1.24532i
\(538\) 31.4774i 1.35709i
\(539\) 0.183449i 0.00790169i
\(540\) −132.044 −5.68225
\(541\) 27.4232i 1.17901i 0.807763 + 0.589507i \(0.200678\pi\)
−0.807763 + 0.589507i \(0.799322\pi\)
\(542\) 29.9153 1.28497
\(543\) 57.9029 2.48485
\(544\) 17.7256 + 1.96178i 0.759977 + 0.0841105i
\(545\) 43.4102 1.85949
\(546\) −33.2601 −1.42340
\(547\) 2.44485i 0.104534i −0.998633 0.0522672i \(-0.983355\pi\)
0.998633 0.0522672i \(-0.0166448\pi\)
\(548\) 55.7041 2.37956
\(549\) 22.4792i 0.959387i
\(550\) 0.498409i 0.0212522i
\(551\) 11.0255i 0.469702i
\(552\) −1.69061 −0.0719573
\(553\) −2.91691 −0.124040
\(554\) 17.3711i 0.738028i
\(555\) 19.5806i 0.831151i
\(556\) 46.6310i 1.97760i
\(557\) −23.3813 −0.990698 −0.495349 0.868694i \(-0.664960\pi\)
−0.495349 + 0.868694i \(0.664960\pi\)
\(558\) 88.6865i 3.75440i
\(559\) −6.23701 −0.263797
\(560\) 2.83262 0.119700
\(561\) −0.359065 0.0397396i −0.0151598 0.00167781i
\(562\) −0.655383 −0.0276457
\(563\) −18.6416 −0.785648 −0.392824 0.919614i \(-0.628502\pi\)
−0.392824 + 0.919614i \(0.628502\pi\)
\(564\) 65.5410i 2.75977i
\(565\) −49.8811 −2.09852
\(566\) 41.0809i 1.72676i
\(567\) 10.1522i 0.426351i
\(568\) 26.8255i 1.12557i
\(569\) −7.73787 −0.324389 −0.162194 0.986759i \(-0.551857\pi\)
−0.162194 + 0.986759i \(0.551857\pi\)
\(570\) −120.015 −5.02689
\(571\) 32.1225i 1.34429i 0.740421 + 0.672143i \(0.234626\pi\)
−0.740421 + 0.672143i \(0.765374\pi\)
\(572\) 0.614679i 0.0257010i
\(573\) 29.5649i 1.23509i
\(574\) −20.3671 −0.850106
\(575\) 1.19985i 0.0500371i
\(576\) 79.7644 3.32352
\(577\) −27.5133 −1.14539 −0.572696 0.819768i \(-0.694102\pi\)
−0.572696 + 0.819768i \(0.694102\pi\)
\(578\) −38.7842 8.69135i −1.61321 0.361513i
\(579\) 11.8437 0.492206
\(580\) −28.6741 −1.19063
\(581\) 5.92955i 0.245999i
\(582\) 98.5337 4.08435
\(583\) 0.120454i 0.00498868i
\(584\) 35.6628i 1.47574i
\(585\) 143.246i 5.92248i
\(586\) 15.6292 0.645638
\(587\) 13.3509 0.551051 0.275525 0.961294i \(-0.411148\pi\)
0.275525 + 0.961294i \(0.411148\pi\)
\(588\) 68.9218i 2.84229i
\(589\) 27.5113i 1.13358i
\(590\) 104.474i 4.30112i
\(591\) 28.0532 1.15395
\(592\) 1.94572i 0.0799686i
\(593\) −35.8906 −1.47385 −0.736924 0.675975i \(-0.763723\pi\)
−0.736924 + 0.675975i \(0.763723\pi\)
\(594\) −0.716293 −0.0293898
\(595\) 10.7229 + 1.18676i 0.439598 + 0.0486525i
\(596\) −54.1727 −2.21900
\(597\) 70.9540 2.90396
\(598\) 2.33355i 0.0954258i
\(599\) 18.0369 0.736969 0.368484 0.929634i \(-0.379877\pi\)
0.368484 + 0.929634i \(0.379877\pi\)
\(600\) 79.2104i 3.23375i
\(601\) 0.185843i 0.00758069i 0.999993 + 0.00379035i \(0.00120651\pi\)
−0.999993 + 0.00379035i \(0.998793\pi\)
\(602\) 1.73047i 0.0705285i
\(603\) −32.4477 −1.32137
\(604\) −57.7086 −2.34813
\(605\) 38.8845i 1.58088i
\(606\) 44.6604i 1.81421i
\(607\) 8.62574i 0.350108i −0.984559 0.175054i \(-0.943990\pi\)
0.984559 0.175054i \(-0.0560100\pi\)
\(608\) 20.3802 0.826525
\(609\) 5.33717i 0.216273i
\(610\) 28.5992 1.15795
\(611\) −38.2684 −1.54817
\(612\) −92.2857 10.2137i −3.73043 0.412865i
\(613\) 46.7758 1.88926 0.944628 0.328143i \(-0.106423\pi\)
0.944628 + 0.328143i \(0.106423\pi\)
\(614\) −37.7087 −1.52180
\(615\) 128.223i 5.17047i
\(616\) 0.0721422 0.00290669
\(617\) 22.7804i 0.917105i −0.888667 0.458552i \(-0.848368\pi\)
0.888667 0.458552i \(-0.151632\pi\)
\(618\) 123.830i 4.98118i
\(619\) 2.07620i 0.0834496i −0.999129 0.0417248i \(-0.986715\pi\)
0.999129 0.0417248i \(-0.0132853\pi\)
\(620\) −71.5489 −2.87347
\(621\) −1.72437 −0.0691966
\(622\) 28.1952i 1.13052i
\(623\) 4.07676i 0.163332i
\(624\) 20.8073i 0.832960i
\(625\) −6.27232 −0.250893
\(626\) 65.8087i 2.63024i
\(627\) −0.412840 −0.0164872
\(628\) 42.8217 1.70877
\(629\) −0.815184 + 7.36556i −0.0325035 + 0.293684i
\(630\) 39.7437 1.58343
\(631\) −0.316738 −0.0126091 −0.00630457 0.999980i \(-0.502007\pi\)
−0.00630457 + 0.999980i \(0.502007\pi\)
\(632\) 13.5105i 0.537418i
\(633\) −72.7483 −2.89149
\(634\) 56.3518i 2.23802i
\(635\) 10.2089i 0.405127i
\(636\) 45.2545i 1.79446i
\(637\) −40.2423 −1.59446
\(638\) −0.155548 −0.00615819
\(639\) 50.8360i 2.01104i
\(640\) 70.8986i 2.80251i
\(641\) 13.6477i 0.539053i −0.962993 0.269527i \(-0.913133\pi\)
0.962993 0.269527i \(-0.0868672\pi\)
\(642\) 38.9999 1.53920
\(643\) 10.1344i 0.399662i 0.979830 + 0.199831i \(0.0640394\pi\)
−0.979830 + 0.199831i \(0.935961\pi\)
\(644\) 0.410559 0.0161783
\(645\) 10.8943 0.428965
\(646\) −45.1457 4.99650i −1.77623 0.196585i
\(647\) −3.28860 −0.129288 −0.0646442 0.997908i \(-0.520591\pi\)
−0.0646442 + 0.997908i \(0.520591\pi\)
\(648\) −47.0226 −1.84722
\(649\) 0.359380i 0.0141069i
\(650\) −109.334 −4.28842
\(651\) 13.3175i 0.521955i
\(652\) 37.4977i 1.46852i
\(653\) 13.1836i 0.515915i 0.966156 + 0.257958i \(0.0830495\pi\)
−0.966156 + 0.257958i \(0.916951\pi\)
\(654\) 88.4721 3.45953
\(655\) −14.5453 −0.568332
\(656\) 12.7415i 0.497473i
\(657\) 67.5833i 2.63668i
\(658\) 10.6176i 0.413917i
\(659\) 14.3758 0.560003 0.280001 0.960000i \(-0.409665\pi\)
0.280001 + 0.960000i \(0.409665\pi\)
\(660\) 1.07368i 0.0417928i
\(661\) 43.2280 1.68138 0.840688 0.541520i \(-0.182151\pi\)
0.840688 + 0.541520i \(0.182151\pi\)
\(662\) −4.93267 −0.191714
\(663\) 8.71750 78.7666i 0.338560 3.05904i
\(664\) −27.4644 −1.06583
\(665\) 12.3288 0.478092
\(666\) 27.2999i 1.05785i
\(667\) −0.374458 −0.0144991
\(668\) 82.5814i 3.19517i
\(669\) 25.7674i 0.996226i
\(670\) 41.2816i 1.59485i
\(671\) 0.0983782 0.00379785
\(672\) −9.86554 −0.380571
\(673\) 17.3709i 0.669600i −0.942289 0.334800i \(-0.891331\pi\)
0.942289 0.334800i \(-0.108669\pi\)
\(674\) 60.1737i 2.31780i
\(675\) 80.7920i 3.10969i
\(676\) 89.7777 3.45299
\(677\) 32.2397i 1.23907i 0.784967 + 0.619537i \(0.212680\pi\)
−0.784967 + 0.619537i \(0.787320\pi\)
\(678\) −101.660 −3.90424
\(679\) −10.1221 −0.388450
\(680\) −5.49682 + 49.6663i −0.210794 + 1.90462i
\(681\) 64.2119 2.46060
\(682\) −0.388129 −0.0148622
\(683\) 7.03066i 0.269021i 0.990912 + 0.134510i \(0.0429462\pi\)
−0.990912 + 0.134510i \(0.957054\pi\)
\(684\) −106.107 −4.05709
\(685\) 56.8118i 2.17067i
\(686\) 23.2785i 0.888779i
\(687\) 55.9843i 2.13593i
\(688\) 1.08257 0.0412725
\(689\) −26.4234 −1.00665
\(690\) 4.07607i 0.155173i
\(691\) 46.9553i 1.78626i −0.449796 0.893131i \(-0.648503\pi\)
0.449796 0.893131i \(-0.351497\pi\)
\(692\) 6.37667i 0.242405i
\(693\) 0.136714 0.00519334
\(694\) 49.0406i 1.86156i
\(695\) −47.5584 −1.80399
\(696\) −24.7206 −0.937033
\(697\) 5.33822 48.2333i 0.202200 1.82697i
\(698\) −18.5291 −0.701335
\(699\) 49.9174 1.88805
\(700\) 19.2359i 0.727050i
\(701\) −4.15009 −0.156747 −0.0783734 0.996924i \(-0.524973\pi\)
−0.0783734 + 0.996924i \(0.524973\pi\)
\(702\) 157.130i 5.93049i
\(703\) 8.46865i 0.319401i
\(704\) 0.349082i 0.0131565i
\(705\) 66.8444 2.51751
\(706\) 67.6715 2.54685
\(707\) 4.58784i 0.172543i
\(708\) 135.019i 5.07433i
\(709\) 38.8785i 1.46011i −0.683388 0.730056i \(-0.739494\pi\)
0.683388 0.730056i \(-0.260506\pi\)
\(710\) 64.6763 2.42726
\(711\) 25.6032i 0.960197i
\(712\) 18.8827 0.707658
\(713\) −0.934364 −0.0349922
\(714\) 21.8539 + 2.41868i 0.817861 + 0.0905168i
\(715\) −0.626903 −0.0234448
\(716\) −32.4599 −1.21308
\(717\) 10.9738i 0.409823i
\(718\) 17.4157 0.649949
\(719\) 46.9497i 1.75093i −0.483284 0.875463i \(-0.660556\pi\)
0.483284 0.875463i \(-0.339444\pi\)
\(720\) 24.8634i 0.926604i
\(721\) 12.7207i 0.473744i
\(722\) −7.48462 −0.278549
\(723\) −12.2736 −0.456459
\(724\) 65.1298i 2.42053i
\(725\) 17.5445i 0.651587i
\(726\) 79.2487i 2.94119i
\(727\) −34.6046 −1.28341 −0.641706 0.766950i \(-0.721773\pi\)
−0.641706 + 0.766950i \(0.721773\pi\)
\(728\) 15.8255i 0.586533i
\(729\) 10.5082 0.389194
\(730\) 85.9831 3.18238
\(731\) 4.09808 + 0.453556i 0.151573 + 0.0167754i
\(732\) 36.9607 1.36611
\(733\) −36.5440 −1.34978 −0.674892 0.737917i \(-0.735810\pi\)
−0.674892 + 0.737917i \(0.735810\pi\)
\(734\) 79.4199i 2.93144i
\(735\) 70.2924 2.59277
\(736\) 0.692171i 0.0255137i
\(737\) 0.142004i 0.00523080i
\(738\) 178.773i 6.58072i
\(739\) 11.2321 0.413178 0.206589 0.978428i \(-0.433764\pi\)
0.206589 + 0.978428i \(0.433764\pi\)
\(740\) 22.0245 0.809636
\(741\) 90.5629i 3.32691i
\(742\) 7.33120i 0.269137i
\(743\) 20.9461i 0.768437i −0.923242 0.384218i \(-0.874471\pi\)
0.923242 0.384218i \(-0.125529\pi\)
\(744\) −61.6839 −2.26144
\(745\) 55.2500i 2.02420i
\(746\) 50.5068 1.84918
\(747\) −52.0468 −1.90429
\(748\) −0.0446995 + 0.403881i −0.00163438 + 0.0147673i
\(749\) −4.00635 −0.146389
\(750\) 63.6208 2.32310
\(751\) 6.18988i 0.225872i 0.993602 + 0.112936i \(0.0360255\pi\)
−0.993602 + 0.112936i \(0.963975\pi\)
\(752\) 6.64231 0.242220
\(753\) 74.3742i 2.71035i
\(754\) 34.1218i 1.24264i
\(755\) 58.8562i 2.14200i
\(756\) 27.6451 1.00544
\(757\) 5.23071 0.190113 0.0950566 0.995472i \(-0.469697\pi\)
0.0950566 + 0.995472i \(0.469697\pi\)
\(758\) 26.0603i 0.946551i
\(759\) 0.0140213i 0.000508939i
\(760\) 57.1045i 2.07140i
\(761\) 4.28808 0.155443 0.0777214 0.996975i \(-0.475236\pi\)
0.0777214 + 0.996975i \(0.475236\pi\)
\(762\) 20.8062i 0.753729i
\(763\) −9.08849 −0.329026
\(764\) −33.2549 −1.20312
\(765\) −10.4168 + 94.1209i −0.376621 + 3.40295i
\(766\) −27.0955 −0.978999
\(767\) −78.8355 −2.84658
\(768\) 68.8229i 2.48343i
\(769\) 0.199366 0.00718933 0.00359467 0.999994i \(-0.498856\pi\)
0.00359467 + 0.999994i \(0.498856\pi\)
\(770\) 0.173935i 0.00626818i
\(771\) 63.8008i 2.29773i
\(772\) 13.3219i 0.479465i
\(773\) 34.2604 1.23226 0.616131 0.787644i \(-0.288699\pi\)
0.616131 + 0.787644i \(0.288699\pi\)
\(774\) 15.1892 0.545965
\(775\) 43.7778i 1.57255i
\(776\) 46.8833i 1.68301i
\(777\) 4.09946i 0.147067i
\(778\) −9.22749 −0.330822
\(779\) 55.4568i 1.98695i
\(780\) −235.528 −8.43324
\(781\) 0.222480 0.00796095
\(782\) −0.169696 + 1.53328i −0.00606831 + 0.0548299i
\(783\) −25.2142 −0.901083
\(784\) 6.98493 0.249462
\(785\) 43.6733i 1.55877i
\(786\) −29.6440 −1.05737
\(787\) 42.4321i 1.51254i 0.654258 + 0.756271i \(0.272981\pi\)
−0.654258 + 0.756271i \(0.727019\pi\)
\(788\) 31.5545i 1.12408i
\(789\) 1.54292i 0.0549296i
\(790\) −32.5738 −1.15892
\(791\) 10.4433 0.371320
\(792\) 0.633230i 0.0225009i
\(793\) 21.5808i 0.766356i
\(794\) 66.6470i 2.36521i
\(795\) 46.1544 1.63693
\(796\) 79.8099i 2.82879i
\(797\) 8.23793 0.291802 0.145901 0.989299i \(-0.453392\pi\)
0.145901 + 0.989299i \(0.453392\pi\)
\(798\) 25.1268 0.889479
\(799\) 25.1446 + 2.78288i 0.889551 + 0.0984511i
\(800\) −32.4303 −1.14658
\(801\) 35.7839 1.26436
\(802\) 50.3786i 1.77893i
\(803\) 0.295773 0.0104376
\(804\) 53.3511i 1.88155i
\(805\) 0.418723i 0.0147581i
\(806\) 85.1421i 2.99900i
\(807\) 41.4894 1.46050
\(808\) 21.2499 0.747568
\(809\) 33.4777i 1.17701i 0.808492 + 0.588507i \(0.200284\pi\)
−0.808492 + 0.588507i \(0.799716\pi\)
\(810\) 113.372i 3.98347i
\(811\) 0.513702i 0.0180385i 0.999959 + 0.00901926i \(0.00287096\pi\)
−0.999959 + 0.00901926i \(0.997129\pi\)
\(812\) 6.00331 0.210675
\(813\) 39.4305i 1.38289i
\(814\) 0.119476 0.00418762
\(815\) −38.2434 −1.33961
\(816\) −1.51311 + 13.6717i −0.0529695 + 0.478604i
\(817\) 4.71182 0.164846
\(818\) −11.5402 −0.403494
\(819\) 29.9904i 1.04795i
\(820\) −144.227 −5.03663
\(821\) 40.8842i 1.42687i 0.700722 + 0.713435i \(0.252862\pi\)
−0.700722 + 0.713435i \(0.747138\pi\)
\(822\) 115.785i 4.03848i
\(823\) 34.3211i 1.19636i 0.801363 + 0.598179i \(0.204109\pi\)
−0.801363 + 0.598179i \(0.795891\pi\)
\(824\) 58.9196 2.05256
\(825\) 0.656939 0.0228717
\(826\) 21.8730i 0.761059i
\(827\) 34.5180i 1.20031i 0.799884 + 0.600154i \(0.204894\pi\)
−0.799884 + 0.600154i \(0.795106\pi\)
\(828\) 3.60369i 0.125237i
\(829\) −37.3709 −1.29794 −0.648972 0.760812i \(-0.724801\pi\)
−0.648972 + 0.760812i \(0.724801\pi\)
\(830\) 66.2167i 2.29841i
\(831\) −22.8964 −0.794266
\(832\) 76.5767 2.65482
\(833\) 26.4416 + 2.92642i 0.916147 + 0.101395i
\(834\) −96.9264 −3.35629
\(835\) 84.2237 2.91468
\(836\) 0.464367i 0.0160605i
\(837\) −62.9156 −2.17468
\(838\) 44.6706i 1.54312i
\(839\) 33.9006i 1.17038i 0.810897 + 0.585189i \(0.198980\pi\)
−0.810897 + 0.585189i \(0.801020\pi\)
\(840\) 27.6429i 0.953769i
\(841\) 23.5246 0.811192
\(842\) 29.4276 1.01414
\(843\) 0.863842i 0.0297523i
\(844\) 81.8280i 2.81664i
\(845\) 91.5630i 3.14986i
\(846\) 93.1964 3.20416
\(847\) 8.14099i 0.279728i
\(848\) 4.58635 0.157496
\(849\) −54.1476 −1.85834
\(850\) 71.8388 + 7.95076i 2.46405 + 0.272709i
\(851\) 0.287620 0.00985949
\(852\) 83.5857 2.86360
\(853\) 47.4275i 1.62389i −0.583737 0.811943i \(-0.698410\pi\)
0.583737 0.811943i \(-0.301590\pi\)
\(854\) −5.98762 −0.204892
\(855\) 108.217i 3.70093i
\(856\) 18.5565i 0.634249i
\(857\) 29.0994i 0.994015i 0.867746 + 0.497008i \(0.165568\pi\)
−0.867746 + 0.497008i \(0.834432\pi\)
\(858\) −1.27766 −0.0436186
\(859\) 25.0028 0.853086 0.426543 0.904467i \(-0.359731\pi\)
0.426543 + 0.904467i \(0.359731\pi\)
\(860\) 12.2541i 0.417861i
\(861\) 26.8453i 0.914885i
\(862\) 17.9624i 0.611802i
\(863\) −1.43734 −0.0489275 −0.0244638 0.999701i \(-0.507788\pi\)
−0.0244638 + 0.999701i \(0.507788\pi\)
\(864\) 46.6075i 1.58562i
\(865\) 6.50348 0.221125
\(866\) 21.7866 0.740339
\(867\) −11.4558 + 51.1204i −0.389060 + 1.73614i
\(868\) 14.9797 0.508444
\(869\) −0.112050 −0.00380105
\(870\) 59.6014i 2.02068i
\(871\) −31.1509 −1.05551
\(872\) 42.0959i 1.42555i
\(873\) 88.8470i 3.00701i
\(874\) 1.76291i 0.0596312i
\(875\) −6.53559 −0.220943
\(876\) 111.122 3.75446
\(877\) 15.8342i 0.534682i 0.963602 + 0.267341i \(0.0861450\pi\)
−0.963602 + 0.267341i \(0.913855\pi\)
\(878\) 31.9194i 1.07723i
\(879\) 20.6005i 0.694836i
\(880\) 0.108813 0.00366807
\(881\) 10.4114i 0.350771i −0.984500 0.175385i \(-0.943883\pi\)
0.984500 0.175385i \(-0.0561171\pi\)
\(882\) 98.0036 3.29995
\(883\) −50.3013 −1.69277 −0.846386 0.532569i \(-0.821227\pi\)
−0.846386 + 0.532569i \(0.821227\pi\)
\(884\) −88.5975 9.80553i −2.97986 0.329796i
\(885\) 137.704 4.62887
\(886\) 25.9724 0.872560
\(887\) 5.92453i 0.198926i 0.995041 + 0.0994631i \(0.0317125\pi\)
−0.995041 + 0.0994631i \(0.968287\pi\)
\(888\) 18.9878 0.637190
\(889\) 2.13736i 0.0716849i
\(890\) 45.5262i 1.52604i
\(891\) 0.389987i 0.0130650i
\(892\) −28.9835 −0.970438
\(893\) 28.9103 0.967446
\(894\) 112.602i 3.76599i
\(895\) 33.1054i 1.10659i
\(896\) 14.8436i 0.495888i
\(897\) −3.07578 −0.102697
\(898\) 14.6500i 0.488877i
\(899\) −13.6625 −0.455671
\(900\) 168.844 5.62813
\(901\) 17.3617 + 1.92151i 0.578403 + 0.0640148i
\(902\) −0.782384 −0.0260505
\(903\) −2.28088 −0.0759028
\(904\) 48.3710i 1.60880i
\(905\) −66.4249 −2.20804
\(906\) 119.952i 3.98514i
\(907\) 28.3773i 0.942253i −0.882066 0.471127i \(-0.843847\pi\)
0.882066 0.471127i \(-0.156153\pi\)
\(908\) 72.2262i 2.39691i
\(909\) 40.2699 1.33567
\(910\) 38.1553 1.26484
\(911\) 28.4514i 0.942638i −0.881963 0.471319i \(-0.843778\pi\)
0.881963 0.471319i \(-0.156222\pi\)
\(912\) 15.7192i 0.520513i
\(913\) 0.227779i 0.00753837i
\(914\) 74.5999 2.46754
\(915\) 37.6957i 1.24618i
\(916\) 62.9717 2.08064
\(917\) 3.04525 0.100563
\(918\) −11.4265 + 103.244i −0.377131 + 3.40755i
\(919\) −36.1952 −1.19397 −0.596984 0.802253i \(-0.703635\pi\)
−0.596984 + 0.802253i \(0.703635\pi\)
\(920\) 1.93944 0.0639413
\(921\) 49.7028i 1.63776i
\(922\) −35.6574 −1.17431
\(923\) 48.8044i 1.60642i
\(924\) 0.224788i 0.00739500i
\(925\) 13.4759i 0.443084i
\(926\) −97.7877 −3.21350
\(927\) 111.657 3.66728
\(928\) 10.1211i 0.332242i
\(929\) 33.9517i 1.11392i 0.830540 + 0.556959i \(0.188032\pi\)
−0.830540 + 0.556959i \(0.811968\pi\)
\(930\) 148.720i 4.87672i
\(931\) 30.4016 0.996371
\(932\) 56.1477i 1.83918i
\(933\) 37.1633 1.21667
\(934\) −54.8176 −1.79369
\(935\) 0.411912 + 0.0455884i 0.0134710 + 0.00149090i
\(936\) −138.909 −4.54038
\(937\) −4.22845 −0.138137 −0.0690686 0.997612i \(-0.522003\pi\)
−0.0690686 + 0.997612i \(0.522003\pi\)
\(938\) 8.64285i 0.282199i
\(939\) −86.7406 −2.83067
\(940\) 75.1873i 2.45234i
\(941\) 30.8462i 1.00556i 0.864416 + 0.502778i \(0.167689\pi\)
−0.864416 + 0.502778i \(0.832311\pi\)
\(942\) 89.0083i 2.90005i
\(943\) −1.88348 −0.0613344
\(944\) 13.6836 0.445364
\(945\) 28.1948i 0.917178i
\(946\) 0.0664743i 0.00216127i
\(947\) 7.28320i 0.236672i −0.992974 0.118336i \(-0.962244\pi\)
0.992974 0.118336i \(-0.0377560\pi\)
\(948\) −42.0974 −1.36726
\(949\) 64.8824i 2.10617i
\(950\) 82.5975 2.67982
\(951\) 74.2757 2.40856
\(952\) 1.15083 10.3983i 0.0372987 0.337011i
\(953\) 23.6911 0.767429 0.383715 0.923452i \(-0.374645\pi\)
0.383715 + 0.923452i \(0.374645\pi\)
\(954\) 64.3498 2.08340
\(955\) 33.9162i 1.09750i
\(956\) 12.3434 0.399215
\(957\) 0.205023i 0.00662745i
\(958\) 24.4269i 0.789197i
\(959\) 11.8943i 0.384087i
\(960\) −133.759 −4.31704
\(961\) −3.09135 −0.0997208
\(962\) 26.2088i 0.845007i
\(963\) 35.1658i 1.13320i
\(964\) 13.8054i 0.444644i
\(965\) −13.5868 −0.437374
\(966\) 0.853379i 0.0274570i
\(967\) 14.2289 0.457572 0.228786 0.973477i \(-0.426524\pi\)
0.228786 + 0.973477i \(0.426524\pi\)
\(968\) −37.7073 −1.21196
\(969\) −6.58574 + 59.5052i −0.211564 + 1.91158i
\(970\) −113.036 −3.62936
\(971\) 11.8113 0.379041 0.189521 0.981877i \(-0.439307\pi\)
0.189521 + 0.981877i \(0.439307\pi\)
\(972\) 34.4655i 1.10548i
\(973\) 9.95697 0.319206
\(974\) 14.4761i 0.463844i
\(975\) 144.110i 4.61520i
\(976\) 3.74581i 0.119901i
\(977\) −9.59199 −0.306875 −0.153437 0.988158i \(-0.549034\pi\)
−0.153437 + 0.988158i \(0.549034\pi\)
\(978\) −77.9419 −2.49231
\(979\) 0.156605i 0.00500513i
\(980\) 79.0656i 2.52566i
\(981\) 79.7745i 2.54700i
\(982\) −1.64698 −0.0525573
\(983\) 22.2300i 0.709026i 0.935051 + 0.354513i \(0.115353\pi\)
−0.935051 + 0.354513i \(0.884647\pi\)
\(984\) −124.342 −3.96386
\(985\) −32.1820 −1.02540
\(986\) −2.48134 + 22.4200i −0.0790219 + 0.713999i
\(987\) −13.9948 −0.445458
\(988\) −101.866 −3.24079
\(989\) 0.160027i 0.00508857i
\(990\) 1.52672 0.0485223
\(991\) 58.9752i 1.87341i −0.350120 0.936705i \(-0.613859\pi\)
0.350120 0.936705i \(-0.386141\pi\)
\(992\) 25.2546i 0.801835i
\(993\) 6.50161i 0.206322i
\(994\) −13.5408 −0.429489
\(995\) −81.3970 −2.58046
\(996\) 85.5765i 2.71160i
\(997\) 17.1366i 0.542720i 0.962478 + 0.271360i \(0.0874734\pi\)
−0.962478 + 0.271360i \(0.912527\pi\)
\(998\) 3.28561i 0.104004i
\(999\) 19.3670 0.612744
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 731.2.d.c.560.4 yes 20
17.16 even 2 inner 731.2.d.c.560.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.d.c.560.3 20 17.16 even 2 inner
731.2.d.c.560.4 yes 20 1.1 even 1 trivial