Properties

Label 650.4.b.i.599.3
Level $650$
Weight $4$
Character 650.599
Analytic conductor $38.351$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,4,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-16,0,-28,0,0,-46,0,-102] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.3512415037\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{105})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 53x^{2} + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-4.62348i\) of defining polynomial
Character \(\chi\) \(=\) 650.599
Dual form 650.4.b.i.599.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -1.62348i q^{3} -4.00000 q^{4} +3.24695 q^{6} +2.62348i q^{7} -8.00000i q^{8} +24.3643 q^{9} -51.1174 q^{11} +6.49390i q^{12} -13.0000i q^{13} -5.24695 q^{14} +16.0000 q^{16} +4.36433i q^{17} +48.7287i q^{18} +47.4817 q^{19} +4.25915 q^{21} -102.235i q^{22} +91.5991i q^{23} -12.9878 q^{24} +26.0000 q^{26} -83.3887i q^{27} -10.4939i q^{28} +139.988 q^{29} +31.1418 q^{31} +32.0000i q^{32} +82.9878i q^{33} -8.72866 q^{34} -97.4573 q^{36} -377.482i q^{37} +94.9634i q^{38} -21.1052 q^{39} -7.71646 q^{41} +8.51830i q^{42} -75.5991i q^{43} +204.470 q^{44} -183.198 q^{46} +186.785i q^{47} -25.9756i q^{48} +336.117 q^{49} +7.08538 q^{51} +52.0000i q^{52} -236.457i q^{53} +166.777 q^{54} +20.9878 q^{56} -77.0854i q^{57} +279.976i q^{58} +4.36433 q^{59} +380.360 q^{61} +62.2835i q^{62} +63.9192i q^{63} -64.0000 q^{64} -165.976 q^{66} +98.2424i q^{67} -17.4573i q^{68} +148.709 q^{69} +1168.46 q^{71} -194.915i q^{72} +404.357i q^{73} +754.963 q^{74} -189.927 q^{76} -134.105i q^{77} -42.2104i q^{78} +856.748 q^{79} +522.457 q^{81} -15.4329i q^{82} +920.898i q^{83} -17.0366 q^{84} +151.198 q^{86} -227.267i q^{87} +408.939i q^{88} +1526.04 q^{89} +34.1052 q^{91} -366.396i q^{92} -50.5579i q^{93} -373.570 q^{94} +51.9512 q^{96} -960.979i q^{97} +672.235i q^{98} -1245.44 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} - 28 q^{6} - 46 q^{9} - 102 q^{11} + 20 q^{14} + 64 q^{16} - 56 q^{19} + 140 q^{21} + 112 q^{24} + 104 q^{26} + 396 q^{29} + 350 q^{31} + 252 q^{34} + 184 q^{36} + 182 q^{39} + 420 q^{41}+ \cdots - 2502 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) − 1.62348i − 0.312438i −0.987722 0.156219i \(-0.950069\pi\)
0.987722 0.156219i \(-0.0499306\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 3.24695 0.220927
\(7\) 2.62348i 0.141654i 0.997489 + 0.0708272i \(0.0225639\pi\)
−0.997489 + 0.0708272i \(0.977436\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 24.3643 0.902383
\(10\) 0 0
\(11\) −51.1174 −1.40113 −0.700567 0.713587i \(-0.747069\pi\)
−0.700567 + 0.713587i \(0.747069\pi\)
\(12\) 6.49390i 0.156219i
\(13\) − 13.0000i − 0.277350i
\(14\) −5.24695 −0.100165
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 4.36433i 0.0622650i 0.999515 + 0.0311325i \(0.00991138\pi\)
−0.999515 + 0.0311325i \(0.990089\pi\)
\(18\) 48.7287i 0.638081i
\(19\) 47.4817 0.573318 0.286659 0.958033i \(-0.407455\pi\)
0.286659 + 0.958033i \(0.407455\pi\)
\(20\) 0 0
\(21\) 4.25915 0.0442582
\(22\) − 102.235i − 0.990751i
\(23\) 91.5991i 0.830423i 0.909725 + 0.415211i \(0.136292\pi\)
−0.909725 + 0.415211i \(0.863708\pi\)
\(24\) −12.9878 −0.110464
\(25\) 0 0
\(26\) 26.0000 0.196116
\(27\) − 83.3887i − 0.594377i
\(28\) − 10.4939i − 0.0708272i
\(29\) 139.988 0.896382 0.448191 0.893938i \(-0.352068\pi\)
0.448191 + 0.893938i \(0.352068\pi\)
\(30\) 0 0
\(31\) 31.1418 0.180427 0.0902133 0.995922i \(-0.471245\pi\)
0.0902133 + 0.995922i \(0.471245\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 82.9878i 0.437767i
\(34\) −8.72866 −0.0440280
\(35\) 0 0
\(36\) −97.4573 −0.451191
\(37\) − 377.482i − 1.67723i −0.544723 0.838616i \(-0.683365\pi\)
0.544723 0.838616i \(-0.316635\pi\)
\(38\) 94.9634i 0.405397i
\(39\) −21.1052 −0.0866547
\(40\) 0 0
\(41\) −7.71646 −0.0293929 −0.0146964 0.999892i \(-0.504678\pi\)
−0.0146964 + 0.999892i \(0.504678\pi\)
\(42\) 8.51830i 0.0312953i
\(43\) − 75.5991i − 0.268111i −0.990974 0.134055i \(-0.957200\pi\)
0.990974 0.134055i \(-0.0428000\pi\)
\(44\) 204.470 0.700567
\(45\) 0 0
\(46\) −183.198 −0.587198
\(47\) 186.785i 0.579689i 0.957074 + 0.289845i \(0.0936037\pi\)
−0.957074 + 0.289845i \(0.906396\pi\)
\(48\) − 25.9756i − 0.0781095i
\(49\) 336.117 0.979934
\(50\) 0 0
\(51\) 7.08538 0.0194539
\(52\) 52.0000i 0.138675i
\(53\) − 236.457i − 0.612828i −0.951898 0.306414i \(-0.900871\pi\)
0.951898 0.306414i \(-0.0991293\pi\)
\(54\) 166.777 0.420288
\(55\) 0 0
\(56\) 20.9878 0.0500824
\(57\) − 77.0854i − 0.179126i
\(58\) 279.976i 0.633838i
\(59\) 4.36433 0.00963029 0.00481514 0.999988i \(-0.498467\pi\)
0.00481514 + 0.999988i \(0.498467\pi\)
\(60\) 0 0
\(61\) 380.360 0.798362 0.399181 0.916872i \(-0.369295\pi\)
0.399181 + 0.916872i \(0.369295\pi\)
\(62\) 62.2835i 0.127581i
\(63\) 63.9192i 0.127826i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −165.976 −0.309548
\(67\) 98.2424i 0.179138i 0.995981 + 0.0895688i \(0.0285489\pi\)
−0.995981 + 0.0895688i \(0.971451\pi\)
\(68\) − 17.4573i − 0.0311325i
\(69\) 148.709 0.259456
\(70\) 0 0
\(71\) 1168.46 1.95311 0.976554 0.215271i \(-0.0690634\pi\)
0.976554 + 0.215271i \(0.0690634\pi\)
\(72\) − 194.915i − 0.319040i
\(73\) 404.357i 0.648307i 0.946005 + 0.324153i \(0.105079\pi\)
−0.946005 + 0.324153i \(0.894921\pi\)
\(74\) 754.963 1.18598
\(75\) 0 0
\(76\) −189.927 −0.286659
\(77\) − 134.105i − 0.198477i
\(78\) − 42.2104i − 0.0612741i
\(79\) 856.748 1.22015 0.610074 0.792344i \(-0.291140\pi\)
0.610074 + 0.792344i \(0.291140\pi\)
\(80\) 0 0
\(81\) 522.457 0.716677
\(82\) − 15.4329i − 0.0207839i
\(83\) 920.898i 1.21785i 0.793227 + 0.608926i \(0.208399\pi\)
−0.793227 + 0.608926i \(0.791601\pi\)
\(84\) −17.0366 −0.0221291
\(85\) 0 0
\(86\) 151.198 0.189583
\(87\) − 227.267i − 0.280064i
\(88\) 408.939i 0.495376i
\(89\) 1526.04 1.81753 0.908763 0.417312i \(-0.137028\pi\)
0.908763 + 0.417312i \(0.137028\pi\)
\(90\) 0 0
\(91\) 34.1052 0.0392878
\(92\) − 366.396i − 0.415211i
\(93\) − 50.5579i − 0.0563721i
\(94\) −373.570 −0.409902
\(95\) 0 0
\(96\) 51.9512 0.0552318
\(97\) − 960.979i − 1.00590i −0.864315 0.502952i \(-0.832247\pi\)
0.864315 0.502952i \(-0.167753\pi\)
\(98\) 672.235i 0.692918i
\(99\) −1245.44 −1.26436
\(100\) 0 0
\(101\) 848.854 0.836278 0.418139 0.908383i \(-0.362682\pi\)
0.418139 + 0.908383i \(0.362682\pi\)
\(102\) 14.1708i 0.0137560i
\(103\) 1819.89i 1.74096i 0.492203 + 0.870480i \(0.336192\pi\)
−0.492203 + 0.870480i \(0.663808\pi\)
\(104\) −104.000 −0.0980581
\(105\) 0 0
\(106\) 472.915 0.433335
\(107\) − 165.008i − 0.149083i −0.997218 0.0745415i \(-0.976251\pi\)
0.997218 0.0745415i \(-0.0237493\pi\)
\(108\) 333.555i 0.297188i
\(109\) −920.719 −0.809073 −0.404536 0.914522i \(-0.632567\pi\)
−0.404536 + 0.914522i \(0.632567\pi\)
\(110\) 0 0
\(111\) −612.832 −0.524031
\(112\) 41.9756i 0.0354136i
\(113\) − 965.986i − 0.804180i −0.915600 0.402090i \(-0.868284\pi\)
0.915600 0.402090i \(-0.131716\pi\)
\(114\) 154.171 0.126662
\(115\) 0 0
\(116\) −559.951 −0.448191
\(117\) − 316.736i − 0.250276i
\(118\) 8.72866i 0.00680964i
\(119\) −11.4497 −0.00882011
\(120\) 0 0
\(121\) 1281.99 0.963175
\(122\) 760.719i 0.564527i
\(123\) 12.5275i 0.00918345i
\(124\) −124.567 −0.0902133
\(125\) 0 0
\(126\) −127.838 −0.0903869
\(127\) − 1105.46i − 0.772393i −0.922417 0.386196i \(-0.873789\pi\)
0.922417 0.386196i \(-0.126211\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −122.733 −0.0837679
\(130\) 0 0
\(131\) 700.020 0.466878 0.233439 0.972371i \(-0.425002\pi\)
0.233439 + 0.972371i \(0.425002\pi\)
\(132\) − 331.951i − 0.218884i
\(133\) 124.567i 0.0812131i
\(134\) −196.485 −0.126669
\(135\) 0 0
\(136\) 34.9146 0.0220140
\(137\) 1230.03i 0.767070i 0.923526 + 0.383535i \(0.125293\pi\)
−0.923526 + 0.383535i \(0.874707\pi\)
\(138\) 297.418i 0.183463i
\(139\) 1004.05 0.612679 0.306340 0.951922i \(-0.400896\pi\)
0.306340 + 0.951922i \(0.400896\pi\)
\(140\) 0 0
\(141\) 303.241 0.181117
\(142\) 2336.92i 1.38106i
\(143\) 664.526i 0.388605i
\(144\) 389.829 0.225596
\(145\) 0 0
\(146\) −808.713 −0.458422
\(147\) − 545.678i − 0.306169i
\(148\) 1509.93i 0.838616i
\(149\) −2310.81 −1.27053 −0.635264 0.772295i \(-0.719109\pi\)
−0.635264 + 0.772295i \(0.719109\pi\)
\(150\) 0 0
\(151\) 2268.83 1.22275 0.611374 0.791341i \(-0.290617\pi\)
0.611374 + 0.791341i \(0.290617\pi\)
\(152\) − 379.854i − 0.202699i
\(153\) 106.334i 0.0561868i
\(154\) 268.210 0.140344
\(155\) 0 0
\(156\) 84.4207 0.0433274
\(157\) − 1943.62i − 0.988011i −0.869459 0.494006i \(-0.835532\pi\)
0.869459 0.494006i \(-0.164468\pi\)
\(158\) 1713.50i 0.862775i
\(159\) −383.883 −0.191471
\(160\) 0 0
\(161\) −240.308 −0.117633
\(162\) 1044.91i 0.506767i
\(163\) − 311.814i − 0.149835i −0.997190 0.0749177i \(-0.976131\pi\)
0.997190 0.0749177i \(-0.0238694\pi\)
\(164\) 30.8658 0.0146964
\(165\) 0 0
\(166\) −1841.80 −0.861151
\(167\) 331.570i 0.153639i 0.997045 + 0.0768194i \(0.0244765\pi\)
−0.997045 + 0.0768194i \(0.975524\pi\)
\(168\) − 34.0732i − 0.0156476i
\(169\) −169.000 −0.0769231
\(170\) 0 0
\(171\) 1156.86 0.517353
\(172\) 302.396i 0.134055i
\(173\) 234.096i 0.102879i 0.998676 + 0.0514393i \(0.0163809\pi\)
−0.998676 + 0.0514393i \(0.983619\pi\)
\(174\) 454.534 0.198035
\(175\) 0 0
\(176\) −817.878 −0.350283
\(177\) − 7.08538i − 0.00300887i
\(178\) 3052.08i 1.28519i
\(179\) −21.5807 −0.00901127 −0.00450564 0.999990i \(-0.501434\pi\)
−0.00450564 + 0.999990i \(0.501434\pi\)
\(180\) 0 0
\(181\) 1132.38 0.465024 0.232512 0.972594i \(-0.425306\pi\)
0.232512 + 0.972594i \(0.425306\pi\)
\(182\) 68.2104i 0.0277807i
\(183\) − 617.505i − 0.249439i
\(184\) 732.793 0.293599
\(185\) 0 0
\(186\) 101.116 0.0398611
\(187\) − 223.093i − 0.0872416i
\(188\) − 747.140i − 0.289845i
\(189\) 218.768 0.0841960
\(190\) 0 0
\(191\) −1547.72 −0.586331 −0.293165 0.956062i \(-0.594709\pi\)
−0.293165 + 0.956062i \(0.594709\pi\)
\(192\) 103.902i 0.0390547i
\(193\) 1721.73i 0.642140i 0.947056 + 0.321070i \(0.104042\pi\)
−0.947056 + 0.321070i \(0.895958\pi\)
\(194\) 1921.96 0.711281
\(195\) 0 0
\(196\) −1344.47 −0.489967
\(197\) − 44.9786i − 0.0162670i −0.999967 0.00813349i \(-0.997411\pi\)
0.999967 0.00813349i \(-0.00258900\pi\)
\(198\) − 2490.88i − 0.894036i
\(199\) −2577.95 −0.918323 −0.459161 0.888353i \(-0.651850\pi\)
−0.459161 + 0.888353i \(0.651850\pi\)
\(200\) 0 0
\(201\) 159.494 0.0559694
\(202\) 1697.71i 0.591338i
\(203\) 367.255i 0.126976i
\(204\) −28.3415 −0.00972697
\(205\) 0 0
\(206\) −3639.78 −1.23105
\(207\) 2231.75i 0.749359i
\(208\) − 208.000i − 0.0693375i
\(209\) −2427.14 −0.803296
\(210\) 0 0
\(211\) 1434.93 0.468174 0.234087 0.972216i \(-0.424790\pi\)
0.234087 + 0.972216i \(0.424790\pi\)
\(212\) 945.829i 0.306414i
\(213\) − 1896.97i − 0.610225i
\(214\) 330.015 0.105418
\(215\) 0 0
\(216\) −667.110 −0.210144
\(217\) 81.6997i 0.0255582i
\(218\) − 1841.44i − 0.572101i
\(219\) 656.463 0.202556
\(220\) 0 0
\(221\) 56.7363 0.0172692
\(222\) − 1225.66i − 0.370546i
\(223\) − 5222.63i − 1.56831i −0.620565 0.784155i \(-0.713097\pi\)
0.620565 0.784155i \(-0.286903\pi\)
\(224\) −83.9512 −0.0250412
\(225\) 0 0
\(226\) 1931.97 0.568641
\(227\) − 2349.65i − 0.687011i −0.939151 0.343506i \(-0.888386\pi\)
0.939151 0.343506i \(-0.111614\pi\)
\(228\) 308.342i 0.0895632i
\(229\) −317.860 −0.0917238 −0.0458619 0.998948i \(-0.514603\pi\)
−0.0458619 + 0.998948i \(0.514603\pi\)
\(230\) 0 0
\(231\) −217.716 −0.0620117
\(232\) − 1119.90i − 0.316919i
\(233\) − 1295.68i − 0.364303i −0.983271 0.182151i \(-0.941694\pi\)
0.983271 0.182151i \(-0.0583061\pi\)
\(234\) 633.473 0.176972
\(235\) 0 0
\(236\) −17.4573 −0.00481514
\(237\) − 1390.91i − 0.381221i
\(238\) − 22.8994i − 0.00623676i
\(239\) −2149.67 −0.581801 −0.290900 0.956753i \(-0.593955\pi\)
−0.290900 + 0.956753i \(0.593955\pi\)
\(240\) 0 0
\(241\) −6687.39 −1.78744 −0.893719 0.448628i \(-0.851913\pi\)
−0.893719 + 0.448628i \(0.851913\pi\)
\(242\) 2563.97i 0.681068i
\(243\) − 3099.69i − 0.818294i
\(244\) −1521.44 −0.399181
\(245\) 0 0
\(246\) −25.0550 −0.00649368
\(247\) − 617.262i − 0.159010i
\(248\) − 249.134i − 0.0637905i
\(249\) 1495.05 0.380503
\(250\) 0 0
\(251\) 577.116 0.145128 0.0725642 0.997364i \(-0.476882\pi\)
0.0725642 + 0.997364i \(0.476882\pi\)
\(252\) − 255.677i − 0.0639132i
\(253\) − 4682.30i − 1.16353i
\(254\) 2210.92 0.546164
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 4004.15i 0.971876i 0.873993 + 0.485938i \(0.161522\pi\)
−0.873993 + 0.485938i \(0.838478\pi\)
\(258\) − 245.466i − 0.0592329i
\(259\) 990.314 0.237587
\(260\) 0 0
\(261\) 3410.71 0.808880
\(262\) 1400.04i 0.330133i
\(263\) 290.393i 0.0680852i 0.999420 + 0.0340426i \(0.0108382\pi\)
−0.999420 + 0.0340426i \(0.989162\pi\)
\(264\) 663.902 0.154774
\(265\) 0 0
\(266\) −249.134 −0.0574263
\(267\) − 2477.49i − 0.567864i
\(268\) − 392.969i − 0.0895688i
\(269\) −1004.21 −0.227612 −0.113806 0.993503i \(-0.536304\pi\)
−0.113806 + 0.993503i \(0.536304\pi\)
\(270\) 0 0
\(271\) 5506.55 1.23431 0.617157 0.786840i \(-0.288284\pi\)
0.617157 + 0.786840i \(0.288284\pi\)
\(272\) 69.8292i 0.0155662i
\(273\) − 55.3689i − 0.0122750i
\(274\) −2460.06 −0.542400
\(275\) 0 0
\(276\) −594.835 −0.129728
\(277\) 4208.88i 0.912950i 0.889736 + 0.456475i \(0.150888\pi\)
−0.889736 + 0.456475i \(0.849112\pi\)
\(278\) 2008.10i 0.433230i
\(279\) 758.748 0.162814
\(280\) 0 0
\(281\) −6804.56 −1.44458 −0.722288 0.691592i \(-0.756910\pi\)
−0.722288 + 0.691592i \(0.756910\pi\)
\(282\) 606.482i 0.128069i
\(283\) − 3351.59i − 0.703997i −0.936001 0.351999i \(-0.885502\pi\)
0.936001 0.351999i \(-0.114498\pi\)
\(284\) −4673.84 −0.976554
\(285\) 0 0
\(286\) −1329.05 −0.274785
\(287\) − 20.2439i − 0.00416363i
\(288\) 779.658i 0.159520i
\(289\) 4893.95 0.996123
\(290\) 0 0
\(291\) −1560.13 −0.314282
\(292\) − 1617.43i − 0.324153i
\(293\) − 3486.87i − 0.695239i −0.937636 0.347620i \(-0.886990\pi\)
0.937636 0.347620i \(-0.113010\pi\)
\(294\) 1091.36 0.216494
\(295\) 0 0
\(296\) −3019.85 −0.592991
\(297\) 4262.61i 0.832801i
\(298\) − 4621.62i − 0.898399i
\(299\) 1190.79 0.230318
\(300\) 0 0
\(301\) 198.332 0.0379790
\(302\) 4537.67i 0.864614i
\(303\) − 1378.09i − 0.261285i
\(304\) 759.707 0.143330
\(305\) 0 0
\(306\) −212.668 −0.0397301
\(307\) − 8145.25i − 1.51425i −0.653271 0.757124i \(-0.726604\pi\)
0.653271 0.757124i \(-0.273396\pi\)
\(308\) 536.421i 0.0992383i
\(309\) 2954.54 0.543942
\(310\) 0 0
\(311\) −511.038 −0.0931779 −0.0465889 0.998914i \(-0.514835\pi\)
−0.0465889 + 0.998914i \(0.514835\pi\)
\(312\) 168.841i 0.0306371i
\(313\) − 3285.96i − 0.593397i −0.954971 0.296699i \(-0.904114\pi\)
0.954971 0.296699i \(-0.0958857\pi\)
\(314\) 3887.24 0.698630
\(315\) 0 0
\(316\) −3426.99 −0.610074
\(317\) − 8375.89i − 1.48403i −0.670385 0.742014i \(-0.733871\pi\)
0.670385 0.742014i \(-0.266129\pi\)
\(318\) − 767.765i − 0.135390i
\(319\) −7155.81 −1.25595
\(320\) 0 0
\(321\) −267.886 −0.0465792
\(322\) − 480.616i − 0.0831791i
\(323\) 207.226i 0.0356977i
\(324\) −2089.83 −0.358338
\(325\) 0 0
\(326\) 623.628 0.105950
\(327\) 1494.77i 0.252785i
\(328\) 61.7317i 0.0103920i
\(329\) −490.026 −0.0821155
\(330\) 0 0
\(331\) −1475.80 −0.245068 −0.122534 0.992464i \(-0.539102\pi\)
−0.122534 + 0.992464i \(0.539102\pi\)
\(332\) − 3683.59i − 0.608926i
\(333\) − 9197.09i − 1.51351i
\(334\) −663.140 −0.108639
\(335\) 0 0
\(336\) 68.1464 0.0110646
\(337\) − 9267.32i − 1.49799i −0.662575 0.748996i \(-0.730536\pi\)
0.662575 0.748996i \(-0.269464\pi\)
\(338\) − 338.000i − 0.0543928i
\(339\) −1568.25 −0.251256
\(340\) 0 0
\(341\) −1591.89 −0.252802
\(342\) 2313.72i 0.365823i
\(343\) 1781.65i 0.280466i
\(344\) −604.793 −0.0947914
\(345\) 0 0
\(346\) −468.192 −0.0727461
\(347\) 12251.7i 1.89541i 0.319152 + 0.947703i \(0.396602\pi\)
−0.319152 + 0.947703i \(0.603398\pi\)
\(348\) 909.067i 0.140032i
\(349\) −9704.59 −1.48847 −0.744233 0.667920i \(-0.767185\pi\)
−0.744233 + 0.667920i \(0.767185\pi\)
\(350\) 0 0
\(351\) −1084.05 −0.164850
\(352\) − 1635.76i − 0.247688i
\(353\) − 10183.3i − 1.53542i −0.640797 0.767711i \(-0.721396\pi\)
0.640797 0.767711i \(-0.278604\pi\)
\(354\) 14.1708 0.00212759
\(355\) 0 0
\(356\) −6104.16 −0.908763
\(357\) 18.5883i 0.00275574i
\(358\) − 43.1614i − 0.00637193i
\(359\) −8393.64 −1.23398 −0.616991 0.786970i \(-0.711649\pi\)
−0.616991 + 0.786970i \(0.711649\pi\)
\(360\) 0 0
\(361\) −4604.49 −0.671306
\(362\) 2264.77i 0.328822i
\(363\) − 2081.27i − 0.300933i
\(364\) −136.421 −0.0196439
\(365\) 0 0
\(366\) 1235.01 0.176380
\(367\) 1694.93i 0.241075i 0.992709 + 0.120538i \(0.0384618\pi\)
−0.992709 + 0.120538i \(0.961538\pi\)
\(368\) 1465.59i 0.207606i
\(369\) −188.006 −0.0265236
\(370\) 0 0
\(371\) 620.340 0.0868098
\(372\) 202.232i 0.0281861i
\(373\) 4925.86i 0.683784i 0.939739 + 0.341892i \(0.111068\pi\)
−0.939739 + 0.341892i \(0.888932\pi\)
\(374\) 446.186 0.0616891
\(375\) 0 0
\(376\) 1494.28 0.204951
\(377\) − 1819.84i − 0.248612i
\(378\) 437.537i 0.0595356i
\(379\) −4563.89 −0.618552 −0.309276 0.950972i \(-0.600087\pi\)
−0.309276 + 0.950972i \(0.600087\pi\)
\(380\) 0 0
\(381\) −1794.69 −0.241325
\(382\) − 3095.44i − 0.414598i
\(383\) 6385.74i 0.851948i 0.904735 + 0.425974i \(0.140068\pi\)
−0.904735 + 0.425974i \(0.859932\pi\)
\(384\) −207.805 −0.0276159
\(385\) 0 0
\(386\) −3443.46 −0.454061
\(387\) − 1841.92i − 0.241938i
\(388\) 3843.91i 0.502952i
\(389\) 12304.3 1.60374 0.801870 0.597498i \(-0.203838\pi\)
0.801870 + 0.597498i \(0.203838\pi\)
\(390\) 0 0
\(391\) −399.768 −0.0517063
\(392\) − 2688.94i − 0.346459i
\(393\) − 1136.46i − 0.145870i
\(394\) 89.9572 0.0115025
\(395\) 0 0
\(396\) 4981.76 0.632179
\(397\) − 3593.57i − 0.454298i −0.973860 0.227149i \(-0.927060\pi\)
0.973860 0.227149i \(-0.0729404\pi\)
\(398\) − 5155.91i − 0.649352i
\(399\) 202.232 0.0253740
\(400\) 0 0
\(401\) 5943.15 0.740116 0.370058 0.929009i \(-0.379338\pi\)
0.370058 + 0.929009i \(0.379338\pi\)
\(402\) 318.988i 0.0395763i
\(403\) − 404.843i − 0.0500414i
\(404\) −3395.41 −0.418139
\(405\) 0 0
\(406\) −734.509 −0.0897859
\(407\) 19295.9i 2.35003i
\(408\) − 56.6830i − 0.00687801i
\(409\) 3673.76 0.444146 0.222073 0.975030i \(-0.428718\pi\)
0.222073 + 0.975030i \(0.428718\pi\)
\(410\) 0 0
\(411\) 1996.92 0.239662
\(412\) − 7279.55i − 0.870480i
\(413\) 11.4497i 0.00136417i
\(414\) −4463.50 −0.529877
\(415\) 0 0
\(416\) 416.000 0.0490290
\(417\) − 1630.05i − 0.191424i
\(418\) − 4854.28i − 0.568016i
\(419\) 6367.72 0.742443 0.371221 0.928544i \(-0.378939\pi\)
0.371221 + 0.928544i \(0.378939\pi\)
\(420\) 0 0
\(421\) −5184.48 −0.600180 −0.300090 0.953911i \(-0.597017\pi\)
−0.300090 + 0.953911i \(0.597017\pi\)
\(422\) 2869.86i 0.331049i
\(423\) 4550.89i 0.523102i
\(424\) −1891.66 −0.216668
\(425\) 0 0
\(426\) 3793.93 0.431495
\(427\) 997.864i 0.113091i
\(428\) 660.030i 0.0745415i
\(429\) 1078.84 0.121415
\(430\) 0 0
\(431\) 6252.06 0.698727 0.349363 0.936987i \(-0.386398\pi\)
0.349363 + 0.936987i \(0.386398\pi\)
\(432\) − 1334.22i − 0.148594i
\(433\) 13563.3i 1.50533i 0.658402 + 0.752667i \(0.271233\pi\)
−0.658402 + 0.752667i \(0.728767\pi\)
\(434\) −163.399 −0.0180724
\(435\) 0 0
\(436\) 3682.88 0.404536
\(437\) 4349.28i 0.476097i
\(438\) 1312.93i 0.143228i
\(439\) −4955.82 −0.538789 −0.269394 0.963030i \(-0.586824\pi\)
−0.269394 + 0.963030i \(0.586824\pi\)
\(440\) 0 0
\(441\) 8189.27 0.884275
\(442\) 113.473i 0.0122112i
\(443\) 10721.6i 1.14988i 0.818194 + 0.574942i \(0.194975\pi\)
−0.818194 + 0.574942i \(0.805025\pi\)
\(444\) 2451.33 0.262016
\(445\) 0 0
\(446\) 10445.3 1.10896
\(447\) 3751.54i 0.396961i
\(448\) − 167.902i − 0.0177068i
\(449\) 3704.31 0.389348 0.194674 0.980868i \(-0.437635\pi\)
0.194674 + 0.980868i \(0.437635\pi\)
\(450\) 0 0
\(451\) 394.445 0.0411834
\(452\) 3863.94i 0.402090i
\(453\) − 3683.40i − 0.382033i
\(454\) 4699.30 0.485790
\(455\) 0 0
\(456\) −616.683 −0.0633308
\(457\) 6710.74i 0.686904i 0.939170 + 0.343452i \(0.111596\pi\)
−0.939170 + 0.343452i \(0.888404\pi\)
\(458\) − 635.719i − 0.0648585i
\(459\) 363.936 0.0370089
\(460\) 0 0
\(461\) −977.491 −0.0987555 −0.0493777 0.998780i \(-0.515724\pi\)
−0.0493777 + 0.998780i \(0.515724\pi\)
\(462\) − 435.433i − 0.0438489i
\(463\) 14489.1i 1.45435i 0.686452 + 0.727176i \(0.259167\pi\)
−0.686452 + 0.727176i \(0.740833\pi\)
\(464\) 2239.80 0.224096
\(465\) 0 0
\(466\) 2591.35 0.257601
\(467\) 14849.7i 1.47144i 0.677285 + 0.735721i \(0.263156\pi\)
−0.677285 + 0.735721i \(0.736844\pi\)
\(468\) 1266.95i 0.125138i
\(469\) −257.736 −0.0253756
\(470\) 0 0
\(471\) −3155.42 −0.308692
\(472\) − 34.9146i − 0.00340482i
\(473\) 3864.43i 0.375659i
\(474\) 2781.82 0.269564
\(475\) 0 0
\(476\) 45.7988 0.00441005
\(477\) − 5761.12i − 0.553006i
\(478\) − 4299.33i − 0.411395i
\(479\) −8109.85 −0.773588 −0.386794 0.922166i \(-0.626418\pi\)
−0.386794 + 0.922166i \(0.626418\pi\)
\(480\) 0 0
\(481\) −4907.26 −0.465181
\(482\) − 13374.8i − 1.26391i
\(483\) 390.134i 0.0367530i
\(484\) −5127.94 −0.481588
\(485\) 0 0
\(486\) 6199.38 0.578621
\(487\) 1215.92i 0.113138i 0.998399 + 0.0565692i \(0.0180162\pi\)
−0.998399 + 0.0565692i \(0.981984\pi\)
\(488\) − 3042.88i − 0.282264i
\(489\) −506.222 −0.0468143
\(490\) 0 0
\(491\) −12639.4 −1.16173 −0.580866 0.814000i \(-0.697286\pi\)
−0.580866 + 0.814000i \(0.697286\pi\)
\(492\) − 50.1099i − 0.00459173i
\(493\) 610.953i 0.0558132i
\(494\) 1234.52 0.112437
\(495\) 0 0
\(496\) 498.268 0.0451067
\(497\) 3065.43i 0.276666i
\(498\) 2990.11i 0.269056i
\(499\) −2658.21 −0.238473 −0.119236 0.992866i \(-0.538045\pi\)
−0.119236 + 0.992866i \(0.538045\pi\)
\(500\) 0 0
\(501\) 538.296 0.0480026
\(502\) 1154.23i 0.102621i
\(503\) 18698.8i 1.65753i 0.559599 + 0.828764i \(0.310955\pi\)
−0.559599 + 0.828764i \(0.689045\pi\)
\(504\) 511.354 0.0451935
\(505\) 0 0
\(506\) 9364.61 0.822742
\(507\) 274.367i 0.0240337i
\(508\) 4421.85i 0.386196i
\(509\) 20005.8 1.74212 0.871062 0.491173i \(-0.163432\pi\)
0.871062 + 0.491173i \(0.163432\pi\)
\(510\) 0 0
\(511\) −1060.82 −0.0918354
\(512\) 512.000i 0.0441942i
\(513\) − 3959.44i − 0.340767i
\(514\) −8008.30 −0.687220
\(515\) 0 0
\(516\) 490.933 0.0418840
\(517\) − 9547.96i − 0.812222i
\(518\) 1980.63i 0.168000i
\(519\) 380.049 0.0321432
\(520\) 0 0
\(521\) −9981.25 −0.839321 −0.419661 0.907681i \(-0.637851\pi\)
−0.419661 + 0.907681i \(0.637851\pi\)
\(522\) 6821.42i 0.571964i
\(523\) 398.152i 0.0332887i 0.999861 + 0.0166443i \(0.00529831\pi\)
−0.999861 + 0.0166443i \(0.994702\pi\)
\(524\) −2800.08 −0.233439
\(525\) 0 0
\(526\) −580.786 −0.0481435
\(527\) 135.913i 0.0112343i
\(528\) 1327.80i 0.109442i
\(529\) 3776.61 0.310398
\(530\) 0 0
\(531\) 106.334 0.00869020
\(532\) − 498.268i − 0.0406065i
\(533\) 100.314i 0.00815212i
\(534\) 4954.98 0.401541
\(535\) 0 0
\(536\) 785.939 0.0633347
\(537\) 35.0358i 0.00281546i
\(538\) − 2008.41i − 0.160946i
\(539\) −17181.4 −1.37302
\(540\) 0 0
\(541\) 7263.34 0.577219 0.288609 0.957447i \(-0.406807\pi\)
0.288609 + 0.957447i \(0.406807\pi\)
\(542\) 11013.1i 0.872792i
\(543\) − 1838.40i − 0.145291i
\(544\) −139.658 −0.0110070
\(545\) 0 0
\(546\) 110.738 0.00867975
\(547\) − 16741.6i − 1.30863i −0.756222 0.654315i \(-0.772957\pi\)
0.756222 0.654315i \(-0.227043\pi\)
\(548\) − 4920.12i − 0.383535i
\(549\) 9267.21 0.720428
\(550\) 0 0
\(551\) 6646.86 0.513912
\(552\) − 1189.67i − 0.0917314i
\(553\) 2247.66i 0.172839i
\(554\) −8417.76 −0.645553
\(555\) 0 0
\(556\) −4016.20 −0.306340
\(557\) 6000.76i 0.456482i 0.973605 + 0.228241i \(0.0732974\pi\)
−0.973605 + 0.228241i \(0.926703\pi\)
\(558\) 1517.50i 0.115127i
\(559\) −982.788 −0.0743605
\(560\) 0 0
\(561\) −362.186 −0.0272576
\(562\) − 13609.1i − 1.02147i
\(563\) − 147.660i − 0.0110535i −0.999985 0.00552674i \(-0.998241\pi\)
0.999985 0.00552674i \(-0.00175923\pi\)
\(564\) −1212.96 −0.0905585
\(565\) 0 0
\(566\) 6703.18 0.497801
\(567\) 1370.65i 0.101520i
\(568\) − 9347.68i − 0.690528i
\(569\) 19161.2 1.41174 0.705868 0.708343i \(-0.250557\pi\)
0.705868 + 0.708343i \(0.250557\pi\)
\(570\) 0 0
\(571\) 20439.8 1.49804 0.749019 0.662549i \(-0.230525\pi\)
0.749019 + 0.662549i \(0.230525\pi\)
\(572\) − 2658.10i − 0.194302i
\(573\) 2512.69i 0.183192i
\(574\) 40.4879 0.00294413
\(575\) 0 0
\(576\) −1559.32 −0.112798
\(577\) − 22889.0i − 1.65144i −0.564081 0.825720i \(-0.690769\pi\)
0.564081 0.825720i \(-0.309231\pi\)
\(578\) 9787.91i 0.704365i
\(579\) 2795.19 0.200629
\(580\) 0 0
\(581\) −2415.95 −0.172514
\(582\) − 3120.25i − 0.222231i
\(583\) 12087.1i 0.858655i
\(584\) 3234.85 0.229211
\(585\) 0 0
\(586\) 6973.74 0.491608
\(587\) 1492.73i 0.104960i 0.998622 + 0.0524799i \(0.0167126\pi\)
−0.998622 + 0.0524799i \(0.983287\pi\)
\(588\) 2182.71i 0.153084i
\(589\) 1478.66 0.103442
\(590\) 0 0
\(591\) −73.0217 −0.00508242
\(592\) − 6039.71i − 0.419308i
\(593\) 23920.6i 1.65650i 0.560360 + 0.828249i \(0.310663\pi\)
−0.560360 + 0.828249i \(0.689337\pi\)
\(594\) −8525.23 −0.588879
\(595\) 0 0
\(596\) 9243.23 0.635264
\(597\) 4185.24i 0.286919i
\(598\) 2381.58i 0.162859i
\(599\) 23867.0 1.62801 0.814005 0.580857i \(-0.197283\pi\)
0.814005 + 0.580857i \(0.197283\pi\)
\(600\) 0 0
\(601\) −6618.71 −0.449222 −0.224611 0.974448i \(-0.572111\pi\)
−0.224611 + 0.974448i \(0.572111\pi\)
\(602\) 396.665i 0.0268552i
\(603\) 2393.61i 0.161651i
\(604\) −9075.34 −0.611374
\(605\) 0 0
\(606\) 2756.19 0.184756
\(607\) − 13982.7i − 0.934994i −0.883995 0.467497i \(-0.845156\pi\)
0.883995 0.467497i \(-0.154844\pi\)
\(608\) 1519.41i 0.101349i
\(609\) 596.229 0.0396723
\(610\) 0 0
\(611\) 2428.21 0.160777
\(612\) − 425.336i − 0.0280934i
\(613\) − 4043.90i − 0.266446i −0.991086 0.133223i \(-0.957467\pi\)
0.991086 0.133223i \(-0.0425327\pi\)
\(614\) 16290.5 1.07074
\(615\) 0 0
\(616\) −1072.84 −0.0701721
\(617\) 2871.83i 0.187384i 0.995601 + 0.0936918i \(0.0298668\pi\)
−0.995601 + 0.0936918i \(0.970133\pi\)
\(618\) 5909.09i 0.384625i
\(619\) −26324.3 −1.70931 −0.854656 0.519195i \(-0.826232\pi\)
−0.854656 + 0.519195i \(0.826232\pi\)
\(620\) 0 0
\(621\) 7638.33 0.493584
\(622\) − 1022.08i − 0.0658867i
\(623\) 4003.53i 0.257461i
\(624\) −337.683 −0.0216637
\(625\) 0 0
\(626\) 6571.91 0.419595
\(627\) 3940.40i 0.250980i
\(628\) 7774.48i 0.494006i
\(629\) 1647.45 0.104433
\(630\) 0 0
\(631\) −17428.2 −1.09953 −0.549766 0.835319i \(-0.685283\pi\)
−0.549766 + 0.835319i \(0.685283\pi\)
\(632\) − 6853.99i − 0.431388i
\(633\) − 2329.57i − 0.146275i
\(634\) 16751.8 1.04937
\(635\) 0 0
\(636\) 1535.53 0.0957354
\(637\) − 4369.53i − 0.271785i
\(638\) − 14311.6i − 0.888092i
\(639\) 28468.8 1.76245
\(640\) 0 0
\(641\) 22541.7 1.38899 0.694496 0.719496i \(-0.255627\pi\)
0.694496 + 0.719496i \(0.255627\pi\)
\(642\) − 535.772i − 0.0329365i
\(643\) 21498.3i 1.31852i 0.751913 + 0.659262i \(0.229131\pi\)
−0.751913 + 0.659262i \(0.770869\pi\)
\(644\) 961.232 0.0588165
\(645\) 0 0
\(646\) −414.451 −0.0252421
\(647\) − 8290.26i − 0.503746i −0.967760 0.251873i \(-0.918953\pi\)
0.967760 0.251873i \(-0.0810466\pi\)
\(648\) − 4179.66i − 0.253383i
\(649\) −223.093 −0.0134933
\(650\) 0 0
\(651\) 132.637 0.00798536
\(652\) 1247.26i 0.0749177i
\(653\) 8650.10i 0.518383i 0.965826 + 0.259192i \(0.0834562\pi\)
−0.965826 + 0.259192i \(0.916544\pi\)
\(654\) −2989.53 −0.178746
\(655\) 0 0
\(656\) −123.463 −0.00734822
\(657\) 9851.88i 0.585020i
\(658\) − 980.052i − 0.0580644i
\(659\) 29128.3 1.72182 0.860908 0.508761i \(-0.169896\pi\)
0.860908 + 0.508761i \(0.169896\pi\)
\(660\) 0 0
\(661\) −14221.5 −0.836839 −0.418420 0.908254i \(-0.637416\pi\)
−0.418420 + 0.908254i \(0.637416\pi\)
\(662\) − 2951.61i − 0.173289i
\(663\) − 92.1099i − 0.00539555i
\(664\) 7367.18 0.430576
\(665\) 0 0
\(666\) 18394.2 1.07021
\(667\) 12822.8i 0.744376i
\(668\) − 1326.28i − 0.0768194i
\(669\) −8478.81 −0.490000
\(670\) 0 0
\(671\) −19443.0 −1.11861
\(672\) 136.293i 0.00782382i
\(673\) − 33065.1i − 1.89386i −0.321444 0.946929i \(-0.604168\pi\)
0.321444 0.946929i \(-0.395832\pi\)
\(674\) 18534.6 1.05924
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) − 24785.0i − 1.40704i −0.710675 0.703520i \(-0.751610\pi\)
0.710675 0.703520i \(-0.248390\pi\)
\(678\) − 3136.51i − 0.177665i
\(679\) 2521.10 0.142491
\(680\) 0 0
\(681\) −3814.60 −0.214648
\(682\) − 3183.77i − 0.178758i
\(683\) 8376.31i 0.469269i 0.972084 + 0.234634i \(0.0753893\pi\)
−0.972084 + 0.234634i \(0.924611\pi\)
\(684\) −4627.44 −0.258676
\(685\) 0 0
\(686\) −3563.30 −0.198320
\(687\) 516.037i 0.0286580i
\(688\) − 1209.59i − 0.0670276i
\(689\) −3073.95 −0.169968
\(690\) 0 0
\(691\) −1387.54 −0.0763883 −0.0381942 0.999270i \(-0.512161\pi\)
−0.0381942 + 0.999270i \(0.512161\pi\)
\(692\) − 936.384i − 0.0514393i
\(693\) − 3267.38i − 0.179102i
\(694\) −24503.4 −1.34026
\(695\) 0 0
\(696\) −1818.13 −0.0990175
\(697\) − 33.6772i − 0.00183015i
\(698\) − 19409.2i − 1.05250i
\(699\) −2103.50 −0.113822
\(700\) 0 0
\(701\) −23975.7 −1.29180 −0.645898 0.763424i \(-0.723517\pi\)
−0.645898 + 0.763424i \(0.723517\pi\)
\(702\) − 2168.11i − 0.116567i
\(703\) − 17923.5i − 0.961588i
\(704\) 3271.51 0.175142
\(705\) 0 0
\(706\) 20366.7 1.08571
\(707\) 2226.95i 0.118462i
\(708\) 28.3415i 0.00150443i
\(709\) 6875.84 0.364214 0.182107 0.983279i \(-0.441708\pi\)
0.182107 + 0.983279i \(0.441708\pi\)
\(710\) 0 0
\(711\) 20874.1 1.10104
\(712\) − 12208.3i − 0.642593i
\(713\) 2852.56i 0.149830i
\(714\) −37.1766 −0.00194860
\(715\) 0 0
\(716\) 86.3228 0.00450564
\(717\) 3489.93i 0.181777i
\(718\) − 16787.3i − 0.872557i
\(719\) −35132.8 −1.82230 −0.911149 0.412078i \(-0.864803\pi\)
−0.911149 + 0.412078i \(0.864803\pi\)
\(720\) 0 0
\(721\) −4774.43 −0.246615
\(722\) − 9208.98i − 0.474685i
\(723\) 10856.8i 0.558463i
\(724\) −4529.53 −0.232512
\(725\) 0 0
\(726\) 4162.55 0.212791
\(727\) 21820.6i 1.11318i 0.830788 + 0.556589i \(0.187890\pi\)
−0.830788 + 0.556589i \(0.812110\pi\)
\(728\) − 272.841i − 0.0138904i
\(729\) 9074.07 0.461011
\(730\) 0 0
\(731\) 329.939 0.0166939
\(732\) 2470.02i 0.124719i
\(733\) 39336.5i 1.98216i 0.133258 + 0.991081i \(0.457456\pi\)
−0.133258 + 0.991081i \(0.542544\pi\)
\(734\) −3389.86 −0.170466
\(735\) 0 0
\(736\) −2931.17 −0.146799
\(737\) − 5021.89i − 0.250996i
\(738\) − 376.013i − 0.0187550i
\(739\) −16015.2 −0.797198 −0.398599 0.917125i \(-0.630503\pi\)
−0.398599 + 0.917125i \(0.630503\pi\)
\(740\) 0 0
\(741\) −1002.11 −0.0496807
\(742\) 1240.68i 0.0613838i
\(743\) − 7885.18i − 0.389340i −0.980869 0.194670i \(-0.937636\pi\)
0.980869 0.194670i \(-0.0623635\pi\)
\(744\) −404.463 −0.0199306
\(745\) 0 0
\(746\) −9851.72 −0.483508
\(747\) 22437.1i 1.09897i
\(748\) 892.372i 0.0436208i
\(749\) 432.893 0.0211183
\(750\) 0 0
\(751\) −22509.9 −1.09374 −0.546870 0.837218i \(-0.684181\pi\)
−0.546870 + 0.837218i \(0.684181\pi\)
\(752\) 2988.56i 0.144922i
\(753\) − 936.933i − 0.0453436i
\(754\) 3639.68 0.175795
\(755\) 0 0
\(756\) −875.073 −0.0420980
\(757\) − 11376.9i − 0.546234i −0.961981 0.273117i \(-0.911945\pi\)
0.961981 0.273117i \(-0.0880546\pi\)
\(758\) − 9127.78i − 0.437382i
\(759\) −7601.61 −0.363532
\(760\) 0 0
\(761\) −1967.75 −0.0937329 −0.0468665 0.998901i \(-0.514924\pi\)
−0.0468665 + 0.998901i \(0.514924\pi\)
\(762\) − 3589.38i − 0.170642i
\(763\) − 2415.48i − 0.114609i
\(764\) 6190.88 0.293165
\(765\) 0 0
\(766\) −12771.5 −0.602418
\(767\) − 56.7363i − 0.00267096i
\(768\) − 415.610i − 0.0195274i
\(769\) 29055.2 1.36250 0.681248 0.732053i \(-0.261438\pi\)
0.681248 + 0.732053i \(0.261438\pi\)
\(770\) 0 0
\(771\) 6500.64 0.303651
\(772\) − 6886.93i − 0.321070i
\(773\) − 17339.6i − 0.806809i −0.915022 0.403404i \(-0.867827\pi\)
0.915022 0.403404i \(-0.132173\pi\)
\(774\) 3683.84 0.171076
\(775\) 0 0
\(776\) −7687.83 −0.355640
\(777\) − 1607.75i − 0.0742313i
\(778\) 24608.7i 1.13402i
\(779\) −366.391 −0.0168515
\(780\) 0 0
\(781\) −59728.6 −2.73657
\(782\) − 799.537i − 0.0365619i
\(783\) − 11673.4i − 0.532789i
\(784\) 5377.88 0.244984
\(785\) 0 0
\(786\) 2272.93 0.103146
\(787\) − 20499.1i − 0.928482i −0.885709 0.464241i \(-0.846327\pi\)
0.885709 0.464241i \(-0.153673\pi\)
\(788\) 179.914i 0.00813349i
\(789\) 471.446 0.0212724
\(790\) 0 0
\(791\) 2534.24 0.113916
\(792\) 9963.52i 0.447018i
\(793\) − 4944.68i − 0.221426i
\(794\) 7187.14 0.321237
\(795\) 0 0
\(796\) 10311.8 0.459161
\(797\) 20639.2i 0.917289i 0.888620 + 0.458644i \(0.151665\pi\)
−0.888620 + 0.458644i \(0.848335\pi\)
\(798\) 404.463i 0.0179422i
\(799\) −815.191 −0.0360944
\(800\) 0 0
\(801\) 37180.9 1.64010
\(802\) 11886.3i 0.523341i
\(803\) − 20669.7i − 0.908364i
\(804\) −637.976 −0.0279847
\(805\) 0 0
\(806\) 809.686 0.0353846
\(807\) 1630.31i 0.0711146i
\(808\) − 6790.83i − 0.295669i
\(809\) 24472.6 1.06355 0.531773 0.846887i \(-0.321526\pi\)
0.531773 + 0.846887i \(0.321526\pi\)
\(810\) 0 0
\(811\) −928.998 −0.0402238 −0.0201119 0.999798i \(-0.506402\pi\)
−0.0201119 + 0.999798i \(0.506402\pi\)
\(812\) − 1469.02i − 0.0634882i
\(813\) − 8939.75i − 0.385647i
\(814\) −38591.7 −1.66172
\(815\) 0 0
\(816\) 113.366 0.00486349
\(817\) − 3589.57i − 0.153713i
\(818\) 7347.52i 0.314059i
\(819\) 830.950 0.0354527
\(820\) 0 0
\(821\) 3761.43 0.159896 0.0799481 0.996799i \(-0.474525\pi\)
0.0799481 + 0.996799i \(0.474525\pi\)
\(822\) 3993.85i 0.169466i
\(823\) 5718.32i 0.242197i 0.992640 + 0.121098i \(0.0386417\pi\)
−0.992640 + 0.121098i \(0.961358\pi\)
\(824\) 14559.1 0.615523
\(825\) 0 0
\(826\) −22.8994 −0.000964616 0
\(827\) 30541.0i 1.28418i 0.766630 + 0.642089i \(0.221932\pi\)
−0.766630 + 0.642089i \(0.778068\pi\)
\(828\) − 8927.00i − 0.374680i
\(829\) 41273.1 1.72916 0.864581 0.502493i \(-0.167584\pi\)
0.864581 + 0.502493i \(0.167584\pi\)
\(830\) 0 0
\(831\) 6833.02 0.285240
\(832\) 832.000i 0.0346688i
\(833\) 1466.93i 0.0610156i
\(834\) 3260.10 0.135357
\(835\) 0 0
\(836\) 9708.56 0.401648
\(837\) − 2596.87i − 0.107241i
\(838\) 12735.4i 0.524986i
\(839\) −38391.2 −1.57975 −0.789875 0.613268i \(-0.789855\pi\)
−0.789875 + 0.613268i \(0.789855\pi\)
\(840\) 0 0
\(841\) −4792.41 −0.196499
\(842\) − 10369.0i − 0.424392i
\(843\) 11047.0i 0.451341i
\(844\) −5739.72 −0.234087
\(845\) 0 0
\(846\) −9101.78 −0.369889
\(847\) 3363.26i 0.136438i
\(848\) − 3783.32i − 0.153207i
\(849\) −5441.22 −0.219955
\(850\) 0 0
\(851\) 34577.0 1.39281
\(852\) 7587.87i 0.305113i
\(853\) − 10176.9i − 0.408498i −0.978919 0.204249i \(-0.934525\pi\)
0.978919 0.204249i \(-0.0654752\pi\)
\(854\) −1995.73 −0.0799677
\(855\) 0 0
\(856\) −1320.06 −0.0527088
\(857\) 20605.8i 0.821331i 0.911786 + 0.410666i \(0.134704\pi\)
−0.911786 + 0.410666i \(0.865296\pi\)
\(858\) 2157.68i 0.0858532i
\(859\) −23472.9 −0.932345 −0.466173 0.884694i \(-0.654367\pi\)
−0.466173 + 0.884694i \(0.654367\pi\)
\(860\) 0 0
\(861\) −32.8655 −0.00130088
\(862\) 12504.1i 0.494074i
\(863\) − 22922.3i − 0.904153i −0.891979 0.452076i \(-0.850683\pi\)
0.891979 0.452076i \(-0.149317\pi\)
\(864\) 2668.44 0.105072
\(865\) 0 0
\(866\) −27126.6 −1.06443
\(867\) − 7945.21i − 0.311227i
\(868\) − 326.799i − 0.0127791i
\(869\) −43794.7 −1.70959
\(870\) 0 0
\(871\) 1277.15 0.0496838
\(872\) 7365.76i 0.286050i
\(873\) − 23413.6i − 0.907709i
\(874\) −8698.56 −0.336651
\(875\) 0 0
\(876\) −2625.85 −0.101278
\(877\) 12732.4i 0.490243i 0.969492 + 0.245122i \(0.0788279\pi\)
−0.969492 + 0.245122i \(0.921172\pi\)
\(878\) − 9911.63i − 0.380981i
\(879\) −5660.85 −0.217219
\(880\) 0 0
\(881\) 32910.4 1.25855 0.629273 0.777184i \(-0.283353\pi\)
0.629273 + 0.777184i \(0.283353\pi\)
\(882\) 16378.5i 0.625277i
\(883\) − 17053.9i − 0.649952i −0.945722 0.324976i \(-0.894644\pi\)
0.945722 0.324976i \(-0.105356\pi\)
\(884\) −226.945 −0.00863460
\(885\) 0 0
\(886\) −21443.2 −0.813090
\(887\) 7601.23i 0.287739i 0.989597 + 0.143869i \(0.0459545\pi\)
−0.989597 + 0.143869i \(0.954046\pi\)
\(888\) 4902.66i 0.185273i
\(889\) 2900.15 0.109413
\(890\) 0 0
\(891\) −26706.6 −1.00416
\(892\) 20890.5i 0.784155i
\(893\) 8868.87i 0.332347i
\(894\) −7503.08 −0.280694
\(895\) 0 0
\(896\) 335.805 0.0125206
\(897\) − 1933.22i − 0.0719601i
\(898\) 7408.63i 0.275311i
\(899\) 4359.47 0.161731
\(900\) 0 0
\(901\) 1031.98 0.0381578
\(902\) 788.890i 0.0291210i
\(903\) − 321.988i − 0.0118661i
\(904\) −7727.89 −0.284321
\(905\) 0 0
\(906\) 7366.79 0.270138
\(907\) − 49076.2i − 1.79664i −0.439347 0.898318i \(-0.644790\pi\)
0.439347 0.898318i \(-0.355210\pi\)
\(908\) 9398.59i 0.343506i
\(909\) 20681.7 0.754643
\(910\) 0 0
\(911\) −22427.3 −0.815642 −0.407821 0.913062i \(-0.633711\pi\)
−0.407821 + 0.913062i \(0.633711\pi\)
\(912\) − 1233.37i − 0.0447816i
\(913\) − 47073.9i − 1.70637i
\(914\) −13421.5 −0.485715
\(915\) 0 0
\(916\) 1271.44 0.0458619
\(917\) 1836.48i 0.0661353i
\(918\) 727.871i 0.0261692i
\(919\) −47062.8 −1.68929 −0.844645 0.535327i \(-0.820188\pi\)
−0.844645 + 0.535327i \(0.820188\pi\)
\(920\) 0 0
\(921\) −13223.6 −0.473109
\(922\) − 1954.98i − 0.0698307i
\(923\) − 15190.0i − 0.541695i
\(924\) 870.866 0.0310058
\(925\) 0 0
\(926\) −28978.2 −1.02838
\(927\) 44340.4i 1.57101i
\(928\) 4479.61i 0.158459i
\(929\) −18074.8 −0.638338 −0.319169 0.947698i \(-0.603404\pi\)
−0.319169 + 0.947698i \(0.603404\pi\)
\(930\) 0 0
\(931\) 15959.4 0.561814
\(932\) 5182.70i 0.182151i
\(933\) 829.658i 0.0291123i
\(934\) −29699.4 −1.04047
\(935\) 0 0
\(936\) −2533.89 −0.0884859
\(937\) − 39939.1i − 1.39248i −0.717809 0.696241i \(-0.754855\pi\)
0.717809 0.696241i \(-0.245145\pi\)
\(938\) − 515.473i − 0.0179433i
\(939\) −5334.67 −0.185400
\(940\) 0 0
\(941\) −12975.7 −0.449518 −0.224759 0.974414i \(-0.572159\pi\)
−0.224759 + 0.974414i \(0.572159\pi\)
\(942\) − 6310.84i − 0.218278i
\(943\) − 706.821i − 0.0244085i
\(944\) 69.8292 0.00240757
\(945\) 0 0
\(946\) −7728.85 −0.265631
\(947\) 43713.9i 1.50001i 0.661432 + 0.750005i \(0.269949\pi\)
−0.661432 + 0.750005i \(0.730051\pi\)
\(948\) 5563.64i 0.190610i
\(949\) 5256.64 0.179808
\(950\) 0 0
\(951\) −13598.0 −0.463667
\(952\) 91.5976i 0.00311838i
\(953\) − 13116.8i − 0.445850i −0.974836 0.222925i \(-0.928439\pi\)
0.974836 0.222925i \(-0.0715606\pi\)
\(954\) 11522.2 0.391034
\(955\) 0 0
\(956\) 8598.66 0.290900
\(957\) 11617.3i 0.392407i
\(958\) − 16219.7i − 0.547009i
\(959\) −3226.95 −0.108659
\(960\) 0 0
\(961\) −28821.2 −0.967446
\(962\) − 9814.52i − 0.328932i
\(963\) − 4020.30i − 0.134530i
\(964\) 26749.5 0.893719
\(965\) 0 0
\(966\) −780.268 −0.0259883
\(967\) 20837.0i 0.692941i 0.938061 + 0.346471i \(0.112620\pi\)
−0.938061 + 0.346471i \(0.887380\pi\)
\(968\) − 10255.9i − 0.340534i
\(969\) 336.426 0.0111533
\(970\) 0 0
\(971\) 36214.0 1.19687 0.598436 0.801170i \(-0.295789\pi\)
0.598436 + 0.801170i \(0.295789\pi\)
\(972\) 12398.8i 0.409147i
\(973\) 2634.10i 0.0867887i
\(974\) −2431.83 −0.0800010
\(975\) 0 0
\(976\) 6085.76 0.199590
\(977\) − 4171.16i − 0.136589i −0.997665 0.0682943i \(-0.978244\pi\)
0.997665 0.0682943i \(-0.0217557\pi\)
\(978\) − 1012.44i − 0.0331027i
\(979\) −78007.1 −2.54660
\(980\) 0 0
\(981\) −22432.7 −0.730093
\(982\) − 25278.9i − 0.821468i
\(983\) 45275.4i 1.46903i 0.678590 + 0.734517i \(0.262591\pi\)
−0.678590 + 0.734517i \(0.737409\pi\)
\(984\) 100.220 0.00324684
\(985\) 0 0
\(986\) −1221.91 −0.0394659
\(987\) 795.545i 0.0256560i
\(988\) 2469.05i 0.0795050i
\(989\) 6924.81 0.222645
\(990\) 0 0
\(991\) −2463.81 −0.0789764 −0.0394882 0.999220i \(-0.512573\pi\)
−0.0394882 + 0.999220i \(0.512573\pi\)
\(992\) 996.537i 0.0318952i
\(993\) 2395.93i 0.0765686i
\(994\) −6130.85 −0.195633
\(995\) 0 0
\(996\) −5980.22 −0.190252
\(997\) − 29662.7i − 0.942253i −0.882066 0.471127i \(-0.843848\pi\)
0.882066 0.471127i \(-0.156152\pi\)
\(998\) − 5316.42i − 0.168626i
\(999\) −31477.7 −0.996908
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 650.4.b.i.599.3 4
5.2 odd 4 650.4.a.m.1.1 2
5.3 odd 4 650.4.a.p.1.2 yes 2
5.4 even 2 inner 650.4.b.i.599.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
650.4.a.m.1.1 2 5.2 odd 4
650.4.a.p.1.2 yes 2 5.3 odd 4
650.4.b.i.599.2 4 5.4 even 2 inner
650.4.b.i.599.3 4 1.1 even 1 trivial