Properties

Label 1150.4.b.o.599.4
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.4
Root \(7.12571i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.o.599.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +10.1257i q^{3} -4.00000 q^{4} +20.2514 q^{6} +24.6431i q^{7} +8.00000i q^{8} -75.5301 q^{9} -17.3867 q^{11} -40.5029i q^{12} +4.00699i q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740i q^{17} +151.060i q^{18} -79.3172 q^{19} -249.529 q^{21} +34.7734i q^{22} -23.0000i q^{23} -81.0057 q^{24} +8.01398 q^{26} -491.402i q^{27} -98.5723i q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000i q^{32} -176.053i q^{33} -96.5480 q^{34} +302.120 q^{36} +422.904i q^{37} +158.634i q^{38} -40.5736 q^{39} -170.251 q^{41} +499.058i q^{42} -228.920i q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087i q^{47} +162.011i q^{48} -264.282 q^{49} +488.809 q^{51} -16.0280i q^{52} -260.354i q^{53} -982.803 q^{54} -197.145 q^{56} -803.144i q^{57} -508.535i q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392i q^{62} -1861.29i q^{63} -64.0000 q^{64} -352.106 q^{66} +820.011i q^{67} +193.096i q^{68} +232.891 q^{69} +614.845 q^{71} -604.241i q^{72} +511.586i q^{73} +845.808 q^{74} +317.269 q^{76} -428.462i q^{77} +81.1472i q^{78} -160.464 q^{79} +2936.48 q^{81} +340.502i q^{82} -32.5646i q^{83} +998.115 q^{84} -457.840 q^{86} +2574.64i q^{87} -139.094i q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000i q^{92} -2234.70i q^{93} -1160.17 q^{94} +324.023 q^{96} +249.798i q^{97} +528.563i q^{98} +1313.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 10.1257i 1.94869i 0.225050 + 0.974347i \(0.427745\pi\)
−0.225050 + 0.974347i \(0.572255\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 20.2514 1.37794
\(7\) 24.6431i 1.33060i 0.746575 + 0.665301i \(0.231697\pi\)
−0.746575 + 0.665301i \(0.768303\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −75.5301 −2.79741
\(10\) 0 0
\(11\) −17.3867 −0.476572 −0.238286 0.971195i \(-0.576586\pi\)
−0.238286 + 0.971195i \(0.576586\pi\)
\(12\) − 40.5029i − 0.974347i
\(13\) 4.00699i 0.0854876i 0.999086 + 0.0427438i \(0.0136099\pi\)
−0.999086 + 0.0427438i \(0.986390\pi\)
\(14\) 49.2862 0.940877
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 48.2740i − 0.688716i −0.938838 0.344358i \(-0.888097\pi\)
0.938838 0.344358i \(-0.111903\pi\)
\(18\) 151.060i 1.97807i
\(19\) −79.3172 −0.957717 −0.478859 0.877892i \(-0.658949\pi\)
−0.478859 + 0.877892i \(0.658949\pi\)
\(20\) 0 0
\(21\) −249.529 −2.59294
\(22\) 34.7734i 0.336987i
\(23\) − 23.0000i − 0.208514i
\(24\) −81.0057 −0.688968
\(25\) 0 0
\(26\) 8.01398 0.0604488
\(27\) − 491.402i − 3.50260i
\(28\) − 98.5723i − 0.665301i
\(29\) 254.267 1.62815 0.814074 0.580761i \(-0.197245\pi\)
0.814074 + 0.580761i \(0.197245\pi\)
\(30\) 0 0
\(31\) −220.696 −1.27865 −0.639325 0.768937i \(-0.720786\pi\)
−0.639325 + 0.768937i \(0.720786\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 176.053i − 0.928693i
\(34\) −96.5480 −0.486996
\(35\) 0 0
\(36\) 302.120 1.39870
\(37\) 422.904i 1.87905i 0.342476 + 0.939527i \(0.388735\pi\)
−0.342476 + 0.939527i \(0.611265\pi\)
\(38\) 158.634i 0.677208i
\(39\) −40.5736 −0.166589
\(40\) 0 0
\(41\) −170.251 −0.648505 −0.324252 0.945971i \(-0.605113\pi\)
−0.324252 + 0.945971i \(0.605113\pi\)
\(42\) 499.058i 1.83348i
\(43\) − 228.920i − 0.811860i −0.913904 0.405930i \(-0.866948\pi\)
0.913904 0.405930i \(-0.133052\pi\)
\(44\) 69.5468 0.238286
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) − 580.087i − 1.80031i −0.435572 0.900154i \(-0.643454\pi\)
0.435572 0.900154i \(-0.356546\pi\)
\(48\) 162.011i 0.487174i
\(49\) −264.282 −0.770500
\(50\) 0 0
\(51\) 488.809 1.34210
\(52\) − 16.0280i − 0.0427438i
\(53\) − 260.354i − 0.674762i −0.941368 0.337381i \(-0.890459\pi\)
0.941368 0.337381i \(-0.109541\pi\)
\(54\) −982.803 −2.47671
\(55\) 0 0
\(56\) −197.145 −0.470439
\(57\) − 803.144i − 1.86630i
\(58\) − 508.535i − 1.15127i
\(59\) −353.130 −0.779215 −0.389607 0.920981i \(-0.627389\pi\)
−0.389607 + 0.920981i \(0.627389\pi\)
\(60\) 0 0
\(61\) −80.6108 −0.169199 −0.0845996 0.996415i \(-0.526961\pi\)
−0.0845996 + 0.996415i \(0.526961\pi\)
\(62\) 441.392i 0.904142i
\(63\) − 1861.29i − 3.72224i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −352.106 −0.656685
\(67\) 820.011i 1.49523i 0.664133 + 0.747614i \(0.268801\pi\)
−0.664133 + 0.747614i \(0.731199\pi\)
\(68\) 193.096i 0.344358i
\(69\) 232.891 0.406331
\(70\) 0 0
\(71\) 614.845 1.02773 0.513864 0.857872i \(-0.328214\pi\)
0.513864 + 0.857872i \(0.328214\pi\)
\(72\) − 604.241i − 0.989034i
\(73\) 511.586i 0.820228i 0.912034 + 0.410114i \(0.134511\pi\)
−0.912034 + 0.410114i \(0.865489\pi\)
\(74\) 845.808 1.32869
\(75\) 0 0
\(76\) 317.269 0.478859
\(77\) − 428.462i − 0.634127i
\(78\) 81.1472i 0.117796i
\(79\) −160.464 −0.228526 −0.114263 0.993451i \(-0.536451\pi\)
−0.114263 + 0.993451i \(0.536451\pi\)
\(80\) 0 0
\(81\) 2936.48 4.02809
\(82\) 340.502i 0.458562i
\(83\) − 32.5646i − 0.0430654i −0.999768 0.0215327i \(-0.993145\pi\)
0.999768 0.0215327i \(-0.00685460\pi\)
\(84\) 998.115 1.29647
\(85\) 0 0
\(86\) −457.840 −0.574072
\(87\) 2574.64i 3.17276i
\(88\) − 139.094i − 0.168494i
\(89\) 25.0375 0.0298199 0.0149100 0.999889i \(-0.495254\pi\)
0.0149100 + 0.999889i \(0.495254\pi\)
\(90\) 0 0
\(91\) −98.7445 −0.113750
\(92\) 92.0000i 0.104257i
\(93\) − 2234.70i − 2.49170i
\(94\) −1160.17 −1.27301
\(95\) 0 0
\(96\) 324.023 0.344484
\(97\) 249.798i 0.261476i 0.991417 + 0.130738i \(0.0417347\pi\)
−0.991417 + 0.130738i \(0.958265\pi\)
\(98\) 528.563i 0.544826i
\(99\) 1313.22 1.33317
\(100\) 0 0
\(101\) 620.493 0.611300 0.305650 0.952144i \(-0.401126\pi\)
0.305650 + 0.952144i \(0.401126\pi\)
\(102\) − 977.618i − 0.949006i
\(103\) 1473.24i 1.40935i 0.709532 + 0.704673i \(0.248906\pi\)
−0.709532 + 0.704673i \(0.751094\pi\)
\(104\) −32.0559 −0.0302244
\(105\) 0 0
\(106\) −520.708 −0.477129
\(107\) 940.141i 0.849410i 0.905332 + 0.424705i \(0.139622\pi\)
−0.905332 + 0.424705i \(0.860378\pi\)
\(108\) 1965.61i 1.75130i
\(109\) 636.264 0.559111 0.279555 0.960130i \(-0.409813\pi\)
0.279555 + 0.960130i \(0.409813\pi\)
\(110\) 0 0
\(111\) −4282.20 −3.66170
\(112\) 394.289i 0.332650i
\(113\) 832.451i 0.693013i 0.938048 + 0.346506i \(0.112632\pi\)
−0.938048 + 0.346506i \(0.887368\pi\)
\(114\) −1606.29 −1.31967
\(115\) 0 0
\(116\) −1017.07 −0.814074
\(117\) − 302.648i − 0.239144i
\(118\) 706.261i 0.550988i
\(119\) 1189.62 0.916406
\(120\) 0 0
\(121\) −1028.70 −0.772879
\(122\) 161.222i 0.119642i
\(123\) − 1723.91i − 1.26374i
\(124\) 882.783 0.639325
\(125\) 0 0
\(126\) −3722.59 −2.63202
\(127\) − 1614.60i − 1.12813i −0.825729 0.564067i \(-0.809236\pi\)
0.825729 0.564067i \(-0.190764\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 2317.98 1.58207
\(130\) 0 0
\(131\) −1974.56 −1.31693 −0.658464 0.752612i \(-0.728794\pi\)
−0.658464 + 0.752612i \(0.728794\pi\)
\(132\) 704.211i 0.464346i
\(133\) − 1954.62i − 1.27434i
\(134\) 1640.02 1.05729
\(135\) 0 0
\(136\) 386.192 0.243498
\(137\) − 753.805i − 0.470087i −0.971985 0.235044i \(-0.924477\pi\)
0.971985 0.235044i \(-0.0755233\pi\)
\(138\) − 465.783i − 0.287319i
\(139\) −1014.10 −0.618810 −0.309405 0.950930i \(-0.600130\pi\)
−0.309405 + 0.950930i \(0.600130\pi\)
\(140\) 0 0
\(141\) 5873.80 3.50825
\(142\) − 1229.69i − 0.726714i
\(143\) − 69.6683i − 0.0407410i
\(144\) −1208.48 −0.699352
\(145\) 0 0
\(146\) 1023.17 0.579989
\(147\) − 2676.04i − 1.50147i
\(148\) − 1691.62i − 0.939527i
\(149\) 2771.80 1.52399 0.761995 0.647583i \(-0.224220\pi\)
0.761995 + 0.647583i \(0.224220\pi\)
\(150\) 0 0
\(151\) 3108.62 1.67534 0.837668 0.546180i \(-0.183918\pi\)
0.837668 + 0.546180i \(0.183918\pi\)
\(152\) − 634.538i − 0.338604i
\(153\) 3646.14i 1.92662i
\(154\) −856.924 −0.448396
\(155\) 0 0
\(156\) 162.294 0.0832946
\(157\) − 3712.87i − 1.88739i −0.330822 0.943693i \(-0.607326\pi\)
0.330822 0.943693i \(-0.392674\pi\)
\(158\) 320.928i 0.161593i
\(159\) 2636.27 1.31490
\(160\) 0 0
\(161\) 566.791 0.277450
\(162\) − 5872.96i − 2.84829i
\(163\) 915.791i 0.440063i 0.975493 + 0.220032i \(0.0706161\pi\)
−0.975493 + 0.220032i \(0.929384\pi\)
\(164\) 681.003 0.324252
\(165\) 0 0
\(166\) −65.1292 −0.0304518
\(167\) − 1432.32i − 0.663688i −0.943334 0.331844i \(-0.892329\pi\)
0.943334 0.331844i \(-0.107671\pi\)
\(168\) − 1996.23i − 0.916741i
\(169\) 2180.94 0.992692
\(170\) 0 0
\(171\) 5990.84 2.67913
\(172\) 915.680i 0.405930i
\(173\) − 3479.54i − 1.52916i −0.644529 0.764580i \(-0.722946\pi\)
0.644529 0.764580i \(-0.277054\pi\)
\(174\) 5149.28 2.24348
\(175\) 0 0
\(176\) −278.187 −0.119143
\(177\) − 3575.70i − 1.51845i
\(178\) − 50.0751i − 0.0210859i
\(179\) −3642.71 −1.52105 −0.760527 0.649307i \(-0.775059\pi\)
−0.760527 + 0.649307i \(0.775059\pi\)
\(180\) 0 0
\(181\) 2409.15 0.989342 0.494671 0.869080i \(-0.335289\pi\)
0.494671 + 0.869080i \(0.335289\pi\)
\(182\) 197.489i 0.0804333i
\(183\) − 816.241i − 0.329717i
\(184\) 184.000 0.0737210
\(185\) 0 0
\(186\) −4469.41 −1.76190
\(187\) 839.326i 0.328223i
\(188\) 2320.35i 0.900154i
\(189\) 12109.6 4.66057
\(190\) 0 0
\(191\) −189.608 −0.0718299 −0.0359150 0.999355i \(-0.511435\pi\)
−0.0359150 + 0.999355i \(0.511435\pi\)
\(192\) − 648.046i − 0.243587i
\(193\) − 1855.45i − 0.692012i −0.938232 0.346006i \(-0.887538\pi\)
0.938232 0.346006i \(-0.112462\pi\)
\(194\) 499.597 0.184891
\(195\) 0 0
\(196\) 1057.13 0.385250
\(197\) − 2429.85i − 0.878779i −0.898297 0.439390i \(-0.855195\pi\)
0.898297 0.439390i \(-0.144805\pi\)
\(198\) − 2626.44i − 0.942691i
\(199\) −4333.49 −1.54368 −0.771842 0.635815i \(-0.780664\pi\)
−0.771842 + 0.635815i \(0.780664\pi\)
\(200\) 0 0
\(201\) −8303.20 −2.91374
\(202\) − 1240.99i − 0.432255i
\(203\) 6265.94i 2.16642i
\(204\) −1955.24 −0.671048
\(205\) 0 0
\(206\) 2946.48 0.996558
\(207\) 1737.19i 0.583300i
\(208\) 64.1118i 0.0213719i
\(209\) 1379.07 0.456421
\(210\) 0 0
\(211\) −816.788 −0.266493 −0.133247 0.991083i \(-0.542540\pi\)
−0.133247 + 0.991083i \(0.542540\pi\)
\(212\) 1041.42i 0.337381i
\(213\) 6225.75i 2.00273i
\(214\) 1880.28 0.600624
\(215\) 0 0
\(216\) 3931.21 1.23836
\(217\) − 5438.63i − 1.70137i
\(218\) − 1272.53i − 0.395351i
\(219\) −5180.18 −1.59837
\(220\) 0 0
\(221\) 193.433 0.0588767
\(222\) 8564.41i 2.58921i
\(223\) − 4513.80i − 1.35546i −0.735313 0.677728i \(-0.762965\pi\)
0.735313 0.677728i \(-0.237035\pi\)
\(224\) 788.579 0.235219
\(225\) 0 0
\(226\) 1664.90 0.490034
\(227\) − 2792.85i − 0.816599i −0.912848 0.408300i \(-0.866122\pi\)
0.912848 0.408300i \(-0.133878\pi\)
\(228\) 3212.57i 0.933149i
\(229\) −1404.50 −0.405292 −0.202646 0.979252i \(-0.564954\pi\)
−0.202646 + 0.979252i \(0.564954\pi\)
\(230\) 0 0
\(231\) 4338.48 1.23572
\(232\) 2034.14i 0.575637i
\(233\) 1073.79i 0.301916i 0.988540 + 0.150958i \(0.0482359\pi\)
−0.988540 + 0.150958i \(0.951764\pi\)
\(234\) −605.296 −0.169100
\(235\) 0 0
\(236\) 1412.52 0.389607
\(237\) − 1624.81i − 0.445328i
\(238\) − 2379.24i − 0.647997i
\(239\) 2573.18 0.696423 0.348212 0.937416i \(-0.386789\pi\)
0.348212 + 0.937416i \(0.386789\pi\)
\(240\) 0 0
\(241\) −696.127 −0.186064 −0.0930321 0.995663i \(-0.529656\pi\)
−0.0930321 + 0.995663i \(0.529656\pi\)
\(242\) 2057.40i 0.546508i
\(243\) 16466.1i 4.34692i
\(244\) 322.443 0.0845996
\(245\) 0 0
\(246\) −3447.82 −0.893598
\(247\) − 317.823i − 0.0818729i
\(248\) − 1765.57i − 0.452071i
\(249\) 329.740 0.0839213
\(250\) 0 0
\(251\) 6467.81 1.62647 0.813236 0.581934i \(-0.197704\pi\)
0.813236 + 0.581934i \(0.197704\pi\)
\(252\) 7445.18i 1.86112i
\(253\) 399.894i 0.0993721i
\(254\) −3229.21 −0.797711
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 5332.72i − 1.29434i −0.762344 0.647172i \(-0.775952\pi\)
0.762344 0.647172i \(-0.224048\pi\)
\(258\) − 4635.96i − 1.11869i
\(259\) −10421.7 −2.50027
\(260\) 0 0
\(261\) −19204.8 −4.55460
\(262\) 3949.11i 0.931209i
\(263\) − 6872.85i − 1.61140i −0.592325 0.805699i \(-0.701790\pi\)
0.592325 0.805699i \(-0.298210\pi\)
\(264\) 1408.42 0.328342
\(265\) 0 0
\(266\) −3909.24 −0.901094
\(267\) 253.523i 0.0581099i
\(268\) − 3280.04i − 0.747614i
\(269\) 1926.13 0.436572 0.218286 0.975885i \(-0.429953\pi\)
0.218286 + 0.975885i \(0.429953\pi\)
\(270\) 0 0
\(271\) −3653.06 −0.818847 −0.409423 0.912344i \(-0.634270\pi\)
−0.409423 + 0.912344i \(0.634270\pi\)
\(272\) − 772.384i − 0.172179i
\(273\) − 999.859i − 0.221664i
\(274\) −1507.61 −0.332402
\(275\) 0 0
\(276\) −931.566 −0.203165
\(277\) 1047.37i 0.227185i 0.993527 + 0.113592i \(0.0362358\pi\)
−0.993527 + 0.113592i \(0.963764\pi\)
\(278\) 2028.19i 0.437565i
\(279\) 16669.2 3.57691
\(280\) 0 0
\(281\) −2758.90 −0.585701 −0.292851 0.956158i \(-0.594604\pi\)
−0.292851 + 0.956158i \(0.594604\pi\)
\(282\) − 11747.6i − 2.48071i
\(283\) 3355.58i 0.704837i 0.935843 + 0.352418i \(0.114641\pi\)
−0.935843 + 0.352418i \(0.885359\pi\)
\(284\) −2459.38 −0.513864
\(285\) 0 0
\(286\) −139.337 −0.0288082
\(287\) − 4195.50i − 0.862902i
\(288\) 2416.96i 0.494517i
\(289\) 2582.62 0.525670
\(290\) 0 0
\(291\) −2529.39 −0.509537
\(292\) − 2046.35i − 0.410114i
\(293\) 419.625i 0.0836681i 0.999125 + 0.0418341i \(0.0133201\pi\)
−0.999125 + 0.0418341i \(0.986680\pi\)
\(294\) −5352.08 −1.06170
\(295\) 0 0
\(296\) −3383.23 −0.664346
\(297\) 8543.85i 1.66924i
\(298\) − 5543.59i − 1.07762i
\(299\) 92.1607 0.0178254
\(300\) 0 0
\(301\) 5641.30 1.08026
\(302\) − 6217.24i − 1.18464i
\(303\) 6282.93i 1.19124i
\(304\) −1269.08 −0.239429
\(305\) 0 0
\(306\) 7292.28 1.36233
\(307\) 4133.41i 0.768425i 0.923245 + 0.384212i \(0.125527\pi\)
−0.923245 + 0.384212i \(0.874473\pi\)
\(308\) 1713.85i 0.317064i
\(309\) −14917.6 −2.74639
\(310\) 0 0
\(311\) −6991.03 −1.27468 −0.637339 0.770584i \(-0.719965\pi\)
−0.637339 + 0.770584i \(0.719965\pi\)
\(312\) − 324.589i − 0.0588982i
\(313\) 9380.53i 1.69399i 0.531600 + 0.846996i \(0.321591\pi\)
−0.531600 + 0.846996i \(0.678409\pi\)
\(314\) −7425.75 −1.33458
\(315\) 0 0
\(316\) 641.855 0.114263
\(317\) 7995.35i 1.41660i 0.705910 + 0.708302i \(0.250538\pi\)
−0.705910 + 0.708302i \(0.749462\pi\)
\(318\) − 5272.54i − 0.929778i
\(319\) −4420.87 −0.775929
\(320\) 0 0
\(321\) −9519.60 −1.65524
\(322\) − 1133.58i − 0.196186i
\(323\) 3828.96i 0.659595i
\(324\) −11745.9 −2.01405
\(325\) 0 0
\(326\) 1831.58 0.311172
\(327\) 6442.63i 1.08954i
\(328\) − 1362.01i − 0.229281i
\(329\) 14295.1 2.39549
\(330\) 0 0
\(331\) −4798.86 −0.796886 −0.398443 0.917193i \(-0.630449\pi\)
−0.398443 + 0.917193i \(0.630449\pi\)
\(332\) 130.258i 0.0215327i
\(333\) − 31942.0i − 5.25648i
\(334\) −2864.63 −0.469298
\(335\) 0 0
\(336\) −3992.46 −0.648234
\(337\) − 6895.51i − 1.11461i −0.830309 0.557303i \(-0.811836\pi\)
0.830309 0.557303i \(-0.188164\pi\)
\(338\) − 4361.89i − 0.701939i
\(339\) −8429.16 −1.35047
\(340\) 0 0
\(341\) 3837.17 0.609368
\(342\) − 11981.7i − 1.89443i
\(343\) 1939.86i 0.305372i
\(344\) 1831.36 0.287036
\(345\) 0 0
\(346\) −6959.08 −1.08128
\(347\) 8744.17i 1.35277i 0.736548 + 0.676386i \(0.236455\pi\)
−0.736548 + 0.676386i \(0.763545\pi\)
\(348\) − 10298.6i − 1.58638i
\(349\) 6387.08 0.979635 0.489818 0.871825i \(-0.337063\pi\)
0.489818 + 0.871825i \(0.337063\pi\)
\(350\) 0 0
\(351\) 1969.04 0.299429
\(352\) 556.375i 0.0842468i
\(353\) − 589.384i − 0.0888661i −0.999012 0.0444331i \(-0.985852\pi\)
0.999012 0.0444331i \(-0.0141481\pi\)
\(354\) −7151.39 −1.07371
\(355\) 0 0
\(356\) −100.150 −0.0149100
\(357\) 12045.8i 1.78580i
\(358\) 7285.41i 1.07555i
\(359\) −7214.13 −1.06058 −0.530289 0.847817i \(-0.677917\pi\)
−0.530289 + 0.847817i \(0.677917\pi\)
\(360\) 0 0
\(361\) −567.774 −0.0827780
\(362\) − 4818.30i − 0.699570i
\(363\) − 10416.3i − 1.50611i
\(364\) 394.978 0.0568749
\(365\) 0 0
\(366\) −1632.48 −0.233145
\(367\) − 12356.4i − 1.75750i −0.477286 0.878748i \(-0.658379\pi\)
0.477286 0.878748i \(-0.341621\pi\)
\(368\) − 368.000i − 0.0521286i
\(369\) 12859.1 1.81413
\(370\) 0 0
\(371\) 6415.92 0.897839
\(372\) 8938.81i 1.24585i
\(373\) 7145.98i 0.991969i 0.868331 + 0.495985i \(0.165193\pi\)
−0.868331 + 0.495985i \(0.834807\pi\)
\(374\) 1678.65 0.232088
\(375\) 0 0
\(376\) 4640.70 0.636505
\(377\) 1018.85i 0.139186i
\(378\) − 24219.3i − 3.29552i
\(379\) −2170.69 −0.294197 −0.147099 0.989122i \(-0.546993\pi\)
−0.147099 + 0.989122i \(0.546993\pi\)
\(380\) 0 0
\(381\) 16349.0 2.19839
\(382\) 379.215i 0.0507914i
\(383\) − 7967.21i − 1.06294i −0.847078 0.531469i \(-0.821640\pi\)
0.847078 0.531469i \(-0.178360\pi\)
\(384\) −1296.09 −0.172242
\(385\) 0 0
\(386\) −3710.90 −0.489326
\(387\) 17290.3i 2.27111i
\(388\) − 999.193i − 0.130738i
\(389\) 568.951 0.0741567 0.0370783 0.999312i \(-0.488195\pi\)
0.0370783 + 0.999312i \(0.488195\pi\)
\(390\) 0 0
\(391\) −1110.30 −0.143607
\(392\) − 2114.25i − 0.272413i
\(393\) − 19993.8i − 2.56629i
\(394\) −4859.70 −0.621391
\(395\) 0 0
\(396\) −5252.88 −0.666583
\(397\) − 8564.88i − 1.08277i −0.840775 0.541384i \(-0.817900\pi\)
0.840775 0.541384i \(-0.182100\pi\)
\(398\) 8666.98i 1.09155i
\(399\) 19791.9 2.48330
\(400\) 0 0
\(401\) −12455.6 −1.55113 −0.775563 0.631270i \(-0.782534\pi\)
−0.775563 + 0.631270i \(0.782534\pi\)
\(402\) 16606.4i 2.06033i
\(403\) − 884.325i − 0.109309i
\(404\) −2481.97 −0.305650
\(405\) 0 0
\(406\) 12531.9 1.53189
\(407\) − 7352.91i − 0.895504i
\(408\) 3910.47i 0.474503i
\(409\) −11838.9 −1.43129 −0.715645 0.698465i \(-0.753867\pi\)
−0.715645 + 0.698465i \(0.753867\pi\)
\(410\) 0 0
\(411\) 7632.82 0.916056
\(412\) − 5892.96i − 0.704673i
\(413\) − 8702.22i − 1.03682i
\(414\) 3474.38 0.412456
\(415\) 0 0
\(416\) 128.224 0.0151122
\(417\) − 10268.5i − 1.20587i
\(418\) − 2758.13i − 0.322738i
\(419\) 1531.77 0.178596 0.0892981 0.996005i \(-0.471538\pi\)
0.0892981 + 0.996005i \(0.471538\pi\)
\(420\) 0 0
\(421\) −9985.06 −1.15592 −0.577959 0.816066i \(-0.696151\pi\)
−0.577959 + 0.816066i \(0.696151\pi\)
\(422\) 1633.58i 0.188439i
\(423\) 43814.0i 5.03620i
\(424\) 2082.83 0.238564
\(425\) 0 0
\(426\) 12451.5 1.41614
\(427\) − 1986.50i − 0.225137i
\(428\) − 3760.56i − 0.424705i
\(429\) 705.441 0.0793917
\(430\) 0 0
\(431\) −13786.7 −1.54079 −0.770396 0.637566i \(-0.779941\pi\)
−0.770396 + 0.637566i \(0.779941\pi\)
\(432\) − 7862.42i − 0.875651i
\(433\) − 2621.92i − 0.290996i −0.989359 0.145498i \(-0.953522\pi\)
0.989359 0.145498i \(-0.0464784\pi\)
\(434\) −10877.3 −1.20305
\(435\) 0 0
\(436\) −2545.06 −0.279555
\(437\) 1824.30i 0.199698i
\(438\) 10360.4i 1.13022i
\(439\) −12062.7 −1.31143 −0.655717 0.755007i \(-0.727633\pi\)
−0.655717 + 0.755007i \(0.727633\pi\)
\(440\) 0 0
\(441\) 19961.2 2.15541
\(442\) − 386.867i − 0.0416321i
\(443\) 3659.26i 0.392453i 0.980559 + 0.196227i \(0.0628689\pi\)
−0.980559 + 0.196227i \(0.937131\pi\)
\(444\) 17128.8 1.83085
\(445\) 0 0
\(446\) −9027.60 −0.958452
\(447\) 28066.4i 2.96979i
\(448\) − 1577.16i − 0.166325i
\(449\) 10529.6 1.10674 0.553368 0.832937i \(-0.313342\pi\)
0.553368 + 0.832937i \(0.313342\pi\)
\(450\) 0 0
\(451\) 2960.10 0.309059
\(452\) − 3329.81i − 0.346506i
\(453\) 31477.0i 3.26472i
\(454\) −5585.70 −0.577423
\(455\) 0 0
\(456\) 6425.15 0.659836
\(457\) − 6443.23i − 0.659522i −0.944064 0.329761i \(-0.893032\pi\)
0.944064 0.329761i \(-0.106968\pi\)
\(458\) 2809.00i 0.286585i
\(459\) −23721.9 −2.41230
\(460\) 0 0
\(461\) −3263.86 −0.329747 −0.164873 0.986315i \(-0.552722\pi\)
−0.164873 + 0.986315i \(0.552722\pi\)
\(462\) − 8676.97i − 0.873786i
\(463\) 9518.12i 0.955388i 0.878526 + 0.477694i \(0.158527\pi\)
−0.878526 + 0.477694i \(0.841473\pi\)
\(464\) 4068.28 0.407037
\(465\) 0 0
\(466\) 2147.59 0.213487
\(467\) − 19092.6i − 1.89187i −0.324360 0.945934i \(-0.605149\pi\)
0.324360 0.945934i \(-0.394851\pi\)
\(468\) 1210.59i 0.119572i
\(469\) −20207.6 −1.98955
\(470\) 0 0
\(471\) 37595.5 3.67794
\(472\) − 2825.04i − 0.275494i
\(473\) 3980.17i 0.386910i
\(474\) −3249.62 −0.314895
\(475\) 0 0
\(476\) −4758.48 −0.458203
\(477\) 19664.6i 1.88758i
\(478\) − 5146.36i − 0.492446i
\(479\) 6324.81 0.603315 0.301658 0.953416i \(-0.402460\pi\)
0.301658 + 0.953416i \(0.402460\pi\)
\(480\) 0 0
\(481\) −1694.57 −0.160636
\(482\) 1392.25i 0.131567i
\(483\) 5739.16i 0.540664i
\(484\) 4114.81 0.386440
\(485\) 0 0
\(486\) 32932.2 3.07373
\(487\) 7873.07i 0.732573i 0.930502 + 0.366286i \(0.119371\pi\)
−0.930502 + 0.366286i \(0.880629\pi\)
\(488\) − 644.886i − 0.0598209i
\(489\) −9273.04 −0.857549
\(490\) 0 0
\(491\) −3556.82 −0.326918 −0.163459 0.986550i \(-0.552265\pi\)
−0.163459 + 0.986550i \(0.552265\pi\)
\(492\) 6895.64i 0.631869i
\(493\) − 12274.5i − 1.12133i
\(494\) −635.646 −0.0578929
\(495\) 0 0
\(496\) −3531.13 −0.319662
\(497\) 15151.7i 1.36750i
\(498\) − 659.479i − 0.0593413i
\(499\) 1933.37 0.173446 0.0867231 0.996232i \(-0.472360\pi\)
0.0867231 + 0.996232i \(0.472360\pi\)
\(500\) 0 0
\(501\) 14503.2 1.29332
\(502\) − 12935.6i − 1.15009i
\(503\) 2114.36i 0.187425i 0.995599 + 0.0937123i \(0.0298734\pi\)
−0.995599 + 0.0937123i \(0.970127\pi\)
\(504\) 14890.4 1.31601
\(505\) 0 0
\(506\) 799.789 0.0702667
\(507\) 22083.6i 1.93445i
\(508\) 6458.42i 0.564067i
\(509\) 316.452 0.0275570 0.0137785 0.999905i \(-0.495614\pi\)
0.0137785 + 0.999905i \(0.495614\pi\)
\(510\) 0 0
\(511\) −12607.1 −1.09140
\(512\) − 512.000i − 0.0441942i
\(513\) 38976.6i 3.35450i
\(514\) −10665.4 −0.915239
\(515\) 0 0
\(516\) −9271.92 −0.791034
\(517\) 10085.8i 0.857976i
\(518\) 20843.3i 1.76796i
\(519\) 35232.8 2.97987
\(520\) 0 0
\(521\) −309.041 −0.0259872 −0.0129936 0.999916i \(-0.504136\pi\)
−0.0129936 + 0.999916i \(0.504136\pi\)
\(522\) 38409.7i 3.22059i
\(523\) − 5892.46i − 0.492656i −0.969186 0.246328i \(-0.920776\pi\)
0.969186 0.246328i \(-0.0792241\pi\)
\(524\) 7898.22 0.658464
\(525\) 0 0
\(526\) −13745.7 −1.13943
\(527\) 10653.9i 0.880626i
\(528\) − 2816.84i − 0.232173i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 26672.0 2.17978
\(532\) 7818.49i 0.637170i
\(533\) − 682.193i − 0.0554391i
\(534\) 507.046 0.0410899
\(535\) 0 0
\(536\) −6560.09 −0.528643
\(537\) − 36885.0i − 2.96407i
\(538\) − 3852.25i − 0.308703i
\(539\) 4594.99 0.367199
\(540\) 0 0
\(541\) 1560.55 0.124017 0.0620085 0.998076i \(-0.480249\pi\)
0.0620085 + 0.998076i \(0.480249\pi\)
\(542\) 7306.12i 0.579012i
\(543\) 24394.4i 1.92792i
\(544\) −1544.77 −0.121749
\(545\) 0 0
\(546\) −1999.72 −0.156740
\(547\) − 17756.3i − 1.38795i −0.720001 0.693973i \(-0.755859\pi\)
0.720001 0.693973i \(-0.244141\pi\)
\(548\) 3015.22i 0.235044i
\(549\) 6088.54 0.473319
\(550\) 0 0
\(551\) −20167.8 −1.55930
\(552\) 1863.13i 0.143660i
\(553\) − 3954.32i − 0.304078i
\(554\) 2094.73 0.160644
\(555\) 0 0
\(556\) 4056.39 0.309405
\(557\) − 1212.77i − 0.0922559i −0.998936 0.0461280i \(-0.985312\pi\)
0.998936 0.0461280i \(-0.0146882\pi\)
\(558\) − 33338.3i − 2.52926i
\(559\) 917.280 0.0694040
\(560\) 0 0
\(561\) −8498.78 −0.639605
\(562\) 5517.80i 0.414153i
\(563\) 12558.2i 0.940078i 0.882646 + 0.470039i \(0.155760\pi\)
−0.882646 + 0.470039i \(0.844240\pi\)
\(564\) −23495.2 −1.75412
\(565\) 0 0
\(566\) 6711.17 0.498395
\(567\) 72363.9i 5.35978i
\(568\) 4918.76i 0.363357i
\(569\) −9776.72 −0.720319 −0.360159 0.932891i \(-0.617278\pi\)
−0.360159 + 0.932891i \(0.617278\pi\)
\(570\) 0 0
\(571\) −18733.9 −1.37301 −0.686505 0.727125i \(-0.740856\pi\)
−0.686505 + 0.727125i \(0.740856\pi\)
\(572\) 278.673i 0.0203705i
\(573\) − 1919.91i − 0.139975i
\(574\) −8391.01 −0.610164
\(575\) 0 0
\(576\) 4833.92 0.349676
\(577\) − 5113.58i − 0.368944i −0.982838 0.184472i \(-0.940942\pi\)
0.982838 0.184472i \(-0.0590576\pi\)
\(578\) − 5165.24i − 0.371705i
\(579\) 18787.8 1.34852
\(580\) 0 0
\(581\) 802.492 0.0573029
\(582\) 5058.77i 0.360297i
\(583\) 4526.70i 0.321572i
\(584\) −4092.69 −0.289995
\(585\) 0 0
\(586\) 839.250 0.0591623
\(587\) 5379.95i 0.378287i 0.981949 + 0.189143i \(0.0605711\pi\)
−0.981949 + 0.189143i \(0.939429\pi\)
\(588\) 10704.2i 0.750735i
\(589\) 17505.0 1.22458
\(590\) 0 0
\(591\) 24603.9 1.71247
\(592\) 6766.46i 0.469763i
\(593\) − 15060.3i − 1.04292i −0.853275 0.521461i \(-0.825387\pi\)
0.853275 0.521461i \(-0.174613\pi\)
\(594\) 17087.7 1.18033
\(595\) 0 0
\(596\) −11087.2 −0.761995
\(597\) − 43879.7i − 3.00817i
\(598\) − 184.321i − 0.0126045i
\(599\) 3772.04 0.257298 0.128649 0.991690i \(-0.458936\pi\)
0.128649 + 0.991690i \(0.458936\pi\)
\(600\) 0 0
\(601\) −14663.9 −0.995261 −0.497631 0.867389i \(-0.665796\pi\)
−0.497631 + 0.867389i \(0.665796\pi\)
\(602\) − 11282.6i − 0.763861i
\(603\) − 61935.5i − 4.18277i
\(604\) −12434.5 −0.837668
\(605\) 0 0
\(606\) 12565.9 0.842332
\(607\) 24514.5i 1.63923i 0.572915 + 0.819614i \(0.305812\pi\)
−0.572915 + 0.819614i \(0.694188\pi\)
\(608\) 2538.15i 0.169302i
\(609\) −63447.1 −4.22168
\(610\) 0 0
\(611\) 2324.40 0.153904
\(612\) − 14584.6i − 0.963310i
\(613\) 14451.2i 0.952167i 0.879400 + 0.476084i \(0.157944\pi\)
−0.879400 + 0.476084i \(0.842056\pi\)
\(614\) 8266.83 0.543358
\(615\) 0 0
\(616\) 3427.70 0.224198
\(617\) − 3292.19i − 0.214811i −0.994215 0.107406i \(-0.965746\pi\)
0.994215 0.107406i \(-0.0342544\pi\)
\(618\) 29835.2i 1.94199i
\(619\) 12595.5 0.817858 0.408929 0.912566i \(-0.365902\pi\)
0.408929 + 0.912566i \(0.365902\pi\)
\(620\) 0 0
\(621\) −11302.2 −0.730343
\(622\) 13982.1i 0.901333i
\(623\) 617.002i 0.0396785i
\(624\) −649.178 −0.0416473
\(625\) 0 0
\(626\) 18761.1 1.19783
\(627\) 13964.0i 0.889425i
\(628\) 14851.5i 0.943693i
\(629\) 20415.3 1.29413
\(630\) 0 0
\(631\) 19889.3 1.25480 0.627401 0.778697i \(-0.284119\pi\)
0.627401 + 0.778697i \(0.284119\pi\)
\(632\) − 1283.71i − 0.0807963i
\(633\) − 8270.56i − 0.519313i
\(634\) 15990.7 1.00169
\(635\) 0 0
\(636\) −10545.1 −0.657452
\(637\) − 1058.97i − 0.0658682i
\(638\) 8841.75i 0.548665i
\(639\) −46439.3 −2.87498
\(640\) 0 0
\(641\) −19276.0 −1.18776 −0.593880 0.804553i \(-0.702405\pi\)
−0.593880 + 0.804553i \(0.702405\pi\)
\(642\) 19039.2i 1.17043i
\(643\) − 10219.2i − 0.626758i −0.949628 0.313379i \(-0.898539\pi\)
0.949628 0.313379i \(-0.101461\pi\)
\(644\) −2267.16 −0.138725
\(645\) 0 0
\(646\) 7657.93 0.466404
\(647\) − 20818.4i − 1.26500i −0.774560 0.632500i \(-0.782029\pi\)
0.774560 0.632500i \(-0.217971\pi\)
\(648\) 23491.8i 1.42415i
\(649\) 6139.77 0.371352
\(650\) 0 0
\(651\) 55070.0 3.31546
\(652\) − 3663.17i − 0.220032i
\(653\) − 15135.3i − 0.907031i −0.891248 0.453516i \(-0.850170\pi\)
0.891248 0.453516i \(-0.149830\pi\)
\(654\) 12885.3 0.770418
\(655\) 0 0
\(656\) −2724.01 −0.162126
\(657\) − 38640.2i − 2.29451i
\(658\) − 28590.3i − 1.69387i
\(659\) 13207.9 0.780737 0.390369 0.920659i \(-0.372348\pi\)
0.390369 + 0.920659i \(0.372348\pi\)
\(660\) 0 0
\(661\) 6671.34 0.392564 0.196282 0.980547i \(-0.437113\pi\)
0.196282 + 0.980547i \(0.437113\pi\)
\(662\) 9597.72i 0.563484i
\(663\) 1958.65i 0.114733i
\(664\) 260.517 0.0152259
\(665\) 0 0
\(666\) −63883.9 −3.71689
\(667\) − 5848.15i − 0.339492i
\(668\) 5729.26i 0.331844i
\(669\) 45705.5 2.64137
\(670\) 0 0
\(671\) 1401.56 0.0806355
\(672\) 7984.92i 0.458371i
\(673\) − 9534.41i − 0.546099i −0.962000 0.273049i \(-0.911968\pi\)
0.962000 0.273049i \(-0.0880322\pi\)
\(674\) −13791.0 −0.788146
\(675\) 0 0
\(676\) −8723.78 −0.496346
\(677\) − 17748.4i − 1.00757i −0.863828 0.503787i \(-0.831940\pi\)
0.863828 0.503787i \(-0.168060\pi\)
\(678\) 16858.3i 0.954927i
\(679\) −6155.80 −0.347920
\(680\) 0 0
\(681\) 28279.6 1.59130
\(682\) − 7674.35i − 0.430888i
\(683\) 9583.57i 0.536904i 0.963293 + 0.268452i \(0.0865120\pi\)
−0.963293 + 0.268452i \(0.913488\pi\)
\(684\) −23963.3 −1.33956
\(685\) 0 0
\(686\) 3879.73 0.215931
\(687\) − 14221.5i − 0.789790i
\(688\) − 3662.72i − 0.202965i
\(689\) 1043.24 0.0576837
\(690\) 0 0
\(691\) 9410.74 0.518092 0.259046 0.965865i \(-0.416592\pi\)
0.259046 + 0.965865i \(0.416592\pi\)
\(692\) 13918.2i 0.764580i
\(693\) 32361.8i 1.77391i
\(694\) 17488.3 0.956554
\(695\) 0 0
\(696\) −20597.1 −1.12174
\(697\) 8218.69i 0.446636i
\(698\) − 12774.2i − 0.692707i
\(699\) −10872.9 −0.588343
\(700\) 0 0
\(701\) −19517.9 −1.05161 −0.525806 0.850605i \(-0.676236\pi\)
−0.525806 + 0.850605i \(0.676236\pi\)
\(702\) − 3938.08i − 0.211728i
\(703\) − 33543.6i − 1.79960i
\(704\) 1112.75 0.0595715
\(705\) 0 0
\(706\) −1178.77 −0.0628378
\(707\) 15290.9i 0.813397i
\(708\) 14302.8i 0.759225i
\(709\) 28430.7 1.50598 0.752990 0.658033i \(-0.228611\pi\)
0.752990 + 0.658033i \(0.228611\pi\)
\(710\) 0 0
\(711\) 12119.8 0.639282
\(712\) 200.300i 0.0105429i
\(713\) 5076.00i 0.266617i
\(714\) 24091.5 1.26275
\(715\) 0 0
\(716\) 14570.8 0.760527
\(717\) 26055.3i 1.35712i
\(718\) 14428.3i 0.749942i
\(719\) −18716.0 −0.970779 −0.485390 0.874298i \(-0.661322\pi\)
−0.485390 + 0.874298i \(0.661322\pi\)
\(720\) 0 0
\(721\) −36305.2 −1.87528
\(722\) 1135.55i 0.0585329i
\(723\) − 7048.78i − 0.362582i
\(724\) −9636.61 −0.494671
\(725\) 0 0
\(726\) −20832.7 −1.06498
\(727\) − 18419.5i − 0.939670i −0.882754 0.469835i \(-0.844313\pi\)
0.882754 0.469835i \(-0.155687\pi\)
\(728\) − 789.956i − 0.0402167i
\(729\) −87446.1 −4.44272
\(730\) 0 0
\(731\) −11050.9 −0.559141
\(732\) 3264.97i 0.164859i
\(733\) − 21548.4i − 1.08582i −0.839790 0.542912i \(-0.817322\pi\)
0.839790 0.542912i \(-0.182678\pi\)
\(734\) −24712.9 −1.24274
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) − 14257.3i − 0.712584i
\(738\) − 25718.1i − 1.28279i
\(739\) −12066.4 −0.600636 −0.300318 0.953839i \(-0.597093\pi\)
−0.300318 + 0.953839i \(0.597093\pi\)
\(740\) 0 0
\(741\) 3218.19 0.159545
\(742\) − 12831.8i − 0.634868i
\(743\) − 21951.0i − 1.08385i −0.840426 0.541926i \(-0.817695\pi\)
0.840426 0.541926i \(-0.182305\pi\)
\(744\) 17877.6 0.880948
\(745\) 0 0
\(746\) 14292.0 0.701428
\(747\) 2459.61i 0.120472i
\(748\) − 3357.31i − 0.164111i
\(749\) −23168.0 −1.13023
\(750\) 0 0
\(751\) −3112.43 −0.151230 −0.0756152 0.997137i \(-0.524092\pi\)
−0.0756152 + 0.997137i \(0.524092\pi\)
\(752\) − 9281.40i − 0.450077i
\(753\) 65491.2i 3.16950i
\(754\) 2037.69 0.0984197
\(755\) 0 0
\(756\) −48438.6 −2.33028
\(757\) 7684.64i 0.368960i 0.982836 + 0.184480i \(0.0590602\pi\)
−0.982836 + 0.184480i \(0.940940\pi\)
\(758\) 4341.38i 0.208029i
\(759\) −4049.21 −0.193646
\(760\) 0 0
\(761\) −36484.6 −1.73793 −0.868965 0.494874i \(-0.835214\pi\)
−0.868965 + 0.494874i \(0.835214\pi\)
\(762\) − 32698.0i − 1.55450i
\(763\) 15679.5i 0.743954i
\(764\) 758.430 0.0359150
\(765\) 0 0
\(766\) −15934.4 −0.751611
\(767\) − 1414.99i − 0.0666132i
\(768\) 2592.18i 0.121793i
\(769\) −2004.39 −0.0939925 −0.0469962 0.998895i \(-0.514965\pi\)
−0.0469962 + 0.998895i \(0.514965\pi\)
\(770\) 0 0
\(771\) 53997.6 2.52228
\(772\) 7421.81i 0.346006i
\(773\) 7716.17i 0.359031i 0.983755 + 0.179516i \(0.0574531\pi\)
−0.983755 + 0.179516i \(0.942547\pi\)
\(774\) 34580.7 1.60591
\(775\) 0 0
\(776\) −1998.39 −0.0924457
\(777\) − 105527.i − 4.87226i
\(778\) − 1137.90i − 0.0524367i
\(779\) 13503.8 0.621084
\(780\) 0 0
\(781\) −10690.1 −0.489786
\(782\) 2220.61i 0.101546i
\(783\) − 124947.i − 5.70275i
\(784\) −4228.51 −0.192625
\(785\) 0 0
\(786\) −39987.6 −1.81464
\(787\) 57.0149i 0.00258241i 0.999999 + 0.00129121i \(0.000411004\pi\)
−0.999999 + 0.00129121i \(0.999589\pi\)
\(788\) 9719.39i 0.439390i
\(789\) 69592.5 3.14012
\(790\) 0 0
\(791\) −20514.2 −0.922124
\(792\) 10505.8i 0.471346i
\(793\) − 323.006i − 0.0144644i
\(794\) −17129.8 −0.765633
\(795\) 0 0
\(796\) 17334.0 0.771842
\(797\) 15184.7i 0.674870i 0.941349 + 0.337435i \(0.109559\pi\)
−0.941349 + 0.337435i \(0.890441\pi\)
\(798\) − 39583.9i − 1.75596i
\(799\) −28003.2 −1.23990
\(800\) 0 0
\(801\) −1891.09 −0.0834186
\(802\) 24911.1i 1.09681i
\(803\) − 8894.80i − 0.390898i
\(804\) 33212.8 1.45687
\(805\) 0 0
\(806\) −1768.65 −0.0772929
\(807\) 19503.4i 0.850746i
\(808\) 4963.94i 0.216127i
\(809\) 30286.4 1.31621 0.658105 0.752926i \(-0.271358\pi\)
0.658105 + 0.752926i \(0.271358\pi\)
\(810\) 0 0
\(811\) −43936.2 −1.90235 −0.951176 0.308650i \(-0.900123\pi\)
−0.951176 + 0.308650i \(0.900123\pi\)
\(812\) − 25063.7i − 1.08321i
\(813\) − 36989.8i − 1.59568i
\(814\) −14705.8 −0.633217
\(815\) 0 0
\(816\) 7820.94 0.335524
\(817\) 18157.3i 0.777532i
\(818\) 23677.9i 1.01207i
\(819\) 7458.18 0.318205
\(820\) 0 0
\(821\) −5245.69 −0.222991 −0.111496 0.993765i \(-0.535564\pi\)
−0.111496 + 0.993765i \(0.535564\pi\)
\(822\) − 15265.6i − 0.647750i
\(823\) 10678.0i 0.452260i 0.974097 + 0.226130i \(0.0726074\pi\)
−0.974097 + 0.226130i \(0.927393\pi\)
\(824\) −11785.9 −0.498279
\(825\) 0 0
\(826\) −17404.4 −0.733145
\(827\) − 3393.69i − 0.142697i −0.997451 0.0713484i \(-0.977270\pi\)
0.997451 0.0713484i \(-0.0227302\pi\)
\(828\) − 6948.77i − 0.291650i
\(829\) −9601.74 −0.402270 −0.201135 0.979564i \(-0.564463\pi\)
−0.201135 + 0.979564i \(0.564463\pi\)
\(830\) 0 0
\(831\) −10605.3 −0.442713
\(832\) − 256.447i − 0.0106859i
\(833\) 12757.9i 0.530656i
\(834\) −20536.9 −0.852680
\(835\) 0 0
\(836\) −5516.26 −0.228210
\(837\) 108450.i 4.47860i
\(838\) − 3063.54i − 0.126287i
\(839\) −11992.8 −0.493489 −0.246744 0.969081i \(-0.579361\pi\)
−0.246744 + 0.969081i \(0.579361\pi\)
\(840\) 0 0
\(841\) 40263.0 1.65087
\(842\) 19970.1i 0.817358i
\(843\) − 27935.8i − 1.14135i
\(844\) 3267.15 0.133247
\(845\) 0 0
\(846\) 87628.1 3.56113
\(847\) − 25350.4i − 1.02839i
\(848\) − 4165.66i − 0.168690i
\(849\) −33977.7 −1.37351
\(850\) 0 0
\(851\) 9726.79 0.391810
\(852\) − 24903.0i − 1.00136i
\(853\) − 15441.5i − 0.619821i −0.950766 0.309911i \(-0.899701\pi\)
0.950766 0.309911i \(-0.100299\pi\)
\(854\) −3973.00 −0.159196
\(855\) 0 0
\(856\) −7521.13 −0.300312
\(857\) 44572.4i 1.77662i 0.459244 + 0.888310i \(0.348120\pi\)
−0.459244 + 0.888310i \(0.651880\pi\)
\(858\) − 1410.88i − 0.0561384i
\(859\) −2519.56 −0.100077 −0.0500386 0.998747i \(-0.515934\pi\)
−0.0500386 + 0.998747i \(0.515934\pi\)
\(860\) 0 0
\(861\) 42482.5 1.68153
\(862\) 27573.4i 1.08950i
\(863\) 28980.1i 1.14310i 0.820568 + 0.571548i \(0.193657\pi\)
−0.820568 + 0.571548i \(0.806343\pi\)
\(864\) −15724.8 −0.619178
\(865\) 0 0
\(866\) −5243.83 −0.205765
\(867\) 26150.9i 1.02437i
\(868\) 21754.5i 0.850687i
\(869\) 2789.94 0.108909
\(870\) 0 0
\(871\) −3285.77 −0.127823
\(872\) 5090.11i 0.197675i
\(873\) − 18867.3i − 0.731455i
\(874\) 3648.59 0.141208
\(875\) 0 0
\(876\) 20720.7 0.799187
\(877\) 7955.28i 0.306307i 0.988202 + 0.153153i \(0.0489428\pi\)
−0.988202 + 0.153153i \(0.951057\pi\)
\(878\) 24125.3i 0.927323i
\(879\) −4249.00 −0.163044
\(880\) 0 0
\(881\) 29722.0 1.13662 0.568309 0.822815i \(-0.307598\pi\)
0.568309 + 0.822815i \(0.307598\pi\)
\(882\) − 39922.4i − 1.52410i
\(883\) − 19379.7i − 0.738596i −0.929311 0.369298i \(-0.879598\pi\)
0.929311 0.369298i \(-0.120402\pi\)
\(884\) −773.734 −0.0294383
\(885\) 0 0
\(886\) 7318.53 0.277507
\(887\) − 22901.7i − 0.866925i −0.901172 0.433463i \(-0.857292\pi\)
0.901172 0.433463i \(-0.142708\pi\)
\(888\) − 34257.6i − 1.29461i
\(889\) 39788.8 1.50110
\(890\) 0 0
\(891\) −51055.7 −1.91967
\(892\) 18055.2i 0.677728i
\(893\) 46010.9i 1.72419i
\(894\) 56132.9 2.09996
\(895\) 0 0
\(896\) −3154.31 −0.117610
\(897\) 933.193i 0.0347362i
\(898\) − 21059.3i − 0.782581i
\(899\) −56115.8 −2.08183
\(900\) 0 0
\(901\) −12568.3 −0.464719
\(902\) − 5920.20i − 0.218538i
\(903\) 57122.2i 2.10510i
\(904\) −6659.61 −0.245017
\(905\) 0 0
\(906\) 62953.9 2.30850
\(907\) − 42542.2i − 1.55743i −0.627376 0.778716i \(-0.715871\pi\)
0.627376 0.778716i \(-0.284129\pi\)
\(908\) 11171.4i 0.408300i
\(909\) −46865.9 −1.71006
\(910\) 0 0
\(911\) 220.864 0.00803243 0.00401622 0.999992i \(-0.498722\pi\)
0.00401622 + 0.999992i \(0.498722\pi\)
\(912\) − 12850.3i − 0.466574i
\(913\) 566.191i 0.0205237i
\(914\) −12886.5 −0.466352
\(915\) 0 0
\(916\) 5617.99 0.202646
\(917\) − 48659.1i − 1.75231i
\(918\) 47443.9i 1.70575i
\(919\) −27835.4 −0.999135 −0.499567 0.866275i \(-0.666508\pi\)
−0.499567 + 0.866275i \(0.666508\pi\)
\(920\) 0 0
\(921\) −41853.8 −1.49743
\(922\) 6527.72i 0.233166i
\(923\) 2463.68i 0.0878580i
\(924\) −17353.9 −0.617860
\(925\) 0 0
\(926\) 19036.2 0.675562
\(927\) − 111274.i − 3.94252i
\(928\) − 8136.56i − 0.287819i
\(929\) 2172.71 0.0767323 0.0383661 0.999264i \(-0.487785\pi\)
0.0383661 + 0.999264i \(0.487785\pi\)
\(930\) 0 0
\(931\) 20962.1 0.737921
\(932\) − 4295.17i − 0.150958i
\(933\) − 70789.1i − 2.48396i
\(934\) −38185.3 −1.33775
\(935\) 0 0
\(936\) 2421.18 0.0845501
\(937\) − 54906.5i − 1.91432i −0.289560 0.957160i \(-0.593509\pi\)
0.289560 0.957160i \(-0.406491\pi\)
\(938\) 40415.2i 1.40683i
\(939\) −94984.6 −3.30107
\(940\) 0 0
\(941\) 5980.06 0.207167 0.103584 0.994621i \(-0.466969\pi\)
0.103584 + 0.994621i \(0.466969\pi\)
\(942\) − 75191.0i − 2.60070i
\(943\) 3915.77i 0.135223i
\(944\) −5650.09 −0.194804
\(945\) 0 0
\(946\) 7960.33 0.273586
\(947\) 20268.6i 0.695504i 0.937587 + 0.347752i \(0.113055\pi\)
−0.937587 + 0.347752i \(0.886945\pi\)
\(948\) 6499.24i 0.222664i
\(949\) −2049.92 −0.0701193
\(950\) 0 0
\(951\) −80958.6 −2.76053
\(952\) 9516.97i 0.323999i
\(953\) − 21797.9i − 0.740925i −0.928847 0.370463i \(-0.879199\pi\)
0.928847 0.370463i \(-0.120801\pi\)
\(954\) 39329.1 1.33472
\(955\) 0 0
\(956\) −10292.7 −0.348212
\(957\) − 44764.5i − 1.51205i
\(958\) − 12649.6i − 0.426608i
\(959\) 18576.1 0.625499
\(960\) 0 0
\(961\) 18915.6 0.634945
\(962\) 3389.14i 0.113587i
\(963\) − 71008.9i − 2.37615i
\(964\) 2784.51 0.0930321
\(965\) 0 0
\(966\) 11478.3 0.382308
\(967\) − 48950.6i − 1.62786i −0.580960 0.813932i \(-0.697323\pi\)
0.580960 0.813932i \(-0.302677\pi\)
\(968\) − 8229.62i − 0.273254i
\(969\) −38771.0 −1.28535
\(970\) 0 0
\(971\) −26426.2 −0.873386 −0.436693 0.899611i \(-0.643850\pi\)
−0.436693 + 0.899611i \(0.643850\pi\)
\(972\) − 65864.4i − 2.17346i
\(973\) − 24990.5i − 0.823389i
\(974\) 15746.1 0.518007
\(975\) 0 0
\(976\) −1289.77 −0.0422998
\(977\) − 5770.09i − 0.188947i −0.995527 0.0944736i \(-0.969883\pi\)
0.995527 0.0944736i \(-0.0301168\pi\)
\(978\) 18546.1i 0.606379i
\(979\) −435.320 −0.0142113
\(980\) 0 0
\(981\) −48057.1 −1.56406
\(982\) 7113.64i 0.231166i
\(983\) − 484.532i − 0.0157214i −0.999969 0.00786071i \(-0.997498\pi\)
0.999969 0.00786071i \(-0.00250217\pi\)
\(984\) 13791.3 0.446799
\(985\) 0 0
\(986\) −24549.0 −0.792901
\(987\) 144749.i 4.66808i
\(988\) 1271.29i 0.0409365i
\(989\) −5265.16 −0.169285
\(990\) 0 0
\(991\) −26511.5 −0.849813 −0.424907 0.905237i \(-0.639693\pi\)
−0.424907 + 0.905237i \(0.639693\pi\)
\(992\) 7062.27i 0.226035i
\(993\) − 48591.9i − 1.55289i
\(994\) 30303.4 0.966966
\(995\) 0 0
\(996\) −1318.96 −0.0419606
\(997\) 4628.20i 0.147018i 0.997295 + 0.0735088i \(0.0234197\pi\)
−0.997295 + 0.0735088i \(0.976580\pi\)
\(998\) − 3866.74i − 0.122645i
\(999\) 207816. 6.58158
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 1150.4.b.o.599.4 8
5.2 odd 4 230.4.a.j.1.4 4
5.3 odd 4 1150.4.a.n.1.1 4
5.4 even 2 inner 1150.4.b.o.599.5 8
15.2 even 4 2070.4.a.bg.1.1 4
20.7 even 4 1840.4.a.k.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.4 4 5.2 odd 4
1150.4.a.n.1.1 4 5.3 odd 4
1150.4.b.o.599.4 8 1.1 even 1 trivial
1150.4.b.o.599.5 8 5.4 even 2 inner
1840.4.a.k.1.1 4 20.7 even 4
2070.4.a.bg.1.1 4 15.2 even 4