Properties

Label 507.4.b.d.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.d.337.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +7.00000 q^{10} +22.0000i q^{11} +21.0000 q^{12} +10.0000 q^{14} +21.0000i q^{15} +41.0000 q^{16} -37.0000 q^{17} -9.00000i q^{18} +30.0000i q^{19} +49.0000i q^{20} +30.0000i q^{21} +22.0000 q^{22} +162.000 q^{23} -45.0000i q^{24} +76.0000 q^{25} +27.0000 q^{27} +70.0000i q^{28} -113.000 q^{29} +21.0000 q^{30} +196.000i q^{31} -161.000i q^{32} +66.0000i q^{33} +37.0000i q^{34} -70.0000 q^{35} +63.0000 q^{36} -13.0000i q^{37} +30.0000 q^{38} +105.000 q^{40} +285.000i q^{41} +30.0000 q^{42} +246.000 q^{43} +154.000i q^{44} +63.0000i q^{45} -162.000i q^{46} +462.000i q^{47} +123.000 q^{48} +243.000 q^{49} -76.0000i q^{50} -111.000 q^{51} -537.000 q^{53} -27.0000i q^{54} -154.000 q^{55} +150.000 q^{56} +90.0000i q^{57} +113.000i q^{58} -576.000i q^{59} +147.000i q^{60} -635.000 q^{61} +196.000 q^{62} +90.0000i q^{63} +167.000 q^{64} +66.0000 q^{66} +202.000i q^{67} -259.000 q^{68} +486.000 q^{69} +70.0000i q^{70} -1086.00i q^{71} -135.000i q^{72} +805.000i q^{73} -13.0000 q^{74} +228.000 q^{75} +210.000i q^{76} -220.000 q^{77} +884.000 q^{79} +287.000i q^{80} +81.0000 q^{81} +285.000 q^{82} +518.000i q^{83} +210.000i q^{84} -259.000i q^{85} -246.000i q^{86} -339.000 q^{87} +330.000 q^{88} -194.000i q^{89} +63.0000 q^{90} +1134.00 q^{92} +588.000i q^{93} +462.000 q^{94} -210.000 q^{95} -483.000i q^{96} -1202.00i q^{97} -243.000i q^{98} +198.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.353553i −0.984251 0.176777i \(-0.943433\pi\)
0.984251 0.176777i \(-0.0565670\pi\)
\(3\) 3.00000 0.577350
\(4\) 7.00000 0.875000
\(5\) 7.00000i 0.626099i 0.949737 + 0.313050i \(0.101351\pi\)
−0.949737 + 0.313050i \(0.898649\pi\)
\(6\) − 3.00000i − 0.204124i
\(7\) 10.0000i 0.539949i 0.962867 + 0.269975i \(0.0870153\pi\)
−0.962867 + 0.269975i \(0.912985\pi\)
\(8\) − 15.0000i − 0.662913i
\(9\) 9.00000 0.333333
\(10\) 7.00000 0.221359
\(11\) 22.0000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 21.0000 0.505181
\(13\) 0 0
\(14\) 10.0000 0.190901
\(15\) 21.0000i 0.361478i
\(16\) 41.0000 0.640625
\(17\) −37.0000 −0.527872 −0.263936 0.964540i \(-0.585021\pi\)
−0.263936 + 0.964540i \(0.585021\pi\)
\(18\) − 9.00000i − 0.117851i
\(19\) 30.0000i 0.362235i 0.983461 + 0.181118i \(0.0579715\pi\)
−0.983461 + 0.181118i \(0.942029\pi\)
\(20\) 49.0000i 0.547837i
\(21\) 30.0000i 0.311740i
\(22\) 22.0000 0.213201
\(23\) 162.000 1.46867 0.734333 0.678789i \(-0.237495\pi\)
0.734333 + 0.678789i \(0.237495\pi\)
\(24\) − 45.0000i − 0.382733i
\(25\) 76.0000 0.608000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 70.0000i 0.472456i
\(29\) −113.000 −0.723571 −0.361786 0.932261i \(-0.617833\pi\)
−0.361786 + 0.932261i \(0.617833\pi\)
\(30\) 21.0000 0.127802
\(31\) 196.000i 1.13557i 0.823177 + 0.567785i \(0.192199\pi\)
−0.823177 + 0.567785i \(0.807801\pi\)
\(32\) − 161.000i − 0.889408i
\(33\) 66.0000i 0.348155i
\(34\) 37.0000i 0.186631i
\(35\) −70.0000 −0.338062
\(36\) 63.0000 0.291667
\(37\) − 13.0000i − 0.0577618i −0.999583 0.0288809i \(-0.990806\pi\)
0.999583 0.0288809i \(-0.00919436\pi\)
\(38\) 30.0000 0.128070
\(39\) 0 0
\(40\) 105.000 0.415049
\(41\) 285.000i 1.08560i 0.839863 + 0.542799i \(0.182635\pi\)
−0.839863 + 0.542799i \(0.817365\pi\)
\(42\) 30.0000 0.110217
\(43\) 246.000 0.872434 0.436217 0.899842i \(-0.356318\pi\)
0.436217 + 0.899842i \(0.356318\pi\)
\(44\) 154.000i 0.527645i
\(45\) 63.0000i 0.208700i
\(46\) − 162.000i − 0.519252i
\(47\) 462.000i 1.43382i 0.697165 + 0.716911i \(0.254445\pi\)
−0.697165 + 0.716911i \(0.745555\pi\)
\(48\) 123.000 0.369865
\(49\) 243.000 0.708455
\(50\) − 76.0000i − 0.214960i
\(51\) −111.000 −0.304767
\(52\) 0 0
\(53\) −537.000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(54\) − 27.0000i − 0.0680414i
\(55\) −154.000 −0.377552
\(56\) 150.000 0.357939
\(57\) 90.0000i 0.209137i
\(58\) 113.000i 0.255821i
\(59\) − 576.000i − 1.27100i −0.772102 0.635498i \(-0.780795\pi\)
0.772102 0.635498i \(-0.219205\pi\)
\(60\) 147.000i 0.316294i
\(61\) −635.000 −1.33284 −0.666421 0.745575i \(-0.732175\pi\)
−0.666421 + 0.745575i \(0.732175\pi\)
\(62\) 196.000 0.401484
\(63\) 90.0000i 0.179983i
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) 66.0000 0.123091
\(67\) 202.000i 0.368332i 0.982895 + 0.184166i \(0.0589584\pi\)
−0.982895 + 0.184166i \(0.941042\pi\)
\(68\) −259.000 −0.461888
\(69\) 486.000 0.847935
\(70\) 70.0000i 0.119523i
\(71\) − 1086.00i − 1.81527i −0.419755 0.907637i \(-0.637884\pi\)
0.419755 0.907637i \(-0.362116\pi\)
\(72\) − 135.000i − 0.220971i
\(73\) 805.000i 1.29066i 0.763904 + 0.645330i \(0.223280\pi\)
−0.763904 + 0.645330i \(0.776720\pi\)
\(74\) −13.0000 −0.0204219
\(75\) 228.000 0.351029
\(76\) 210.000i 0.316956i
\(77\) −220.000 −0.325602
\(78\) 0 0
\(79\) 884.000 1.25896 0.629480 0.777017i \(-0.283268\pi\)
0.629480 + 0.777017i \(0.283268\pi\)
\(80\) 287.000i 0.401095i
\(81\) 81.0000 0.111111
\(82\) 285.000 0.383817
\(83\) 518.000i 0.685035i 0.939511 + 0.342517i \(0.111280\pi\)
−0.939511 + 0.342517i \(0.888720\pi\)
\(84\) 210.000i 0.272772i
\(85\) − 259.000i − 0.330500i
\(86\) − 246.000i − 0.308452i
\(87\) −339.000 −0.417754
\(88\) 330.000 0.399751
\(89\) − 194.000i − 0.231056i −0.993304 0.115528i \(-0.963144\pi\)
0.993304 0.115528i \(-0.0368560\pi\)
\(90\) 63.0000 0.0737865
\(91\) 0 0
\(92\) 1134.00 1.28508
\(93\) 588.000i 0.655621i
\(94\) 462.000 0.506933
\(95\) −210.000 −0.226795
\(96\) − 483.000i − 0.513500i
\(97\) − 1202.00i − 1.25819i −0.777328 0.629096i \(-0.783425\pi\)
0.777328 0.629096i \(-0.216575\pi\)
\(98\) − 243.000i − 0.250477i
\(99\) 198.000i 0.201008i
\(100\) 532.000 0.532000
\(101\) 429.000 0.422645 0.211322 0.977416i \(-0.432223\pi\)
0.211322 + 0.977416i \(0.432223\pi\)
\(102\) 111.000i 0.107751i
\(103\) 1302.00 1.24553 0.622766 0.782408i \(-0.286009\pi\)
0.622766 + 0.782408i \(0.286009\pi\)
\(104\) 0 0
\(105\) −210.000 −0.195180
\(106\) 537.000i 0.492057i
\(107\) −1338.00 −1.20887 −0.604436 0.796654i \(-0.706602\pi\)
−0.604436 + 0.796654i \(0.706602\pi\)
\(108\) 189.000 0.168394
\(109\) − 1034.00i − 0.908617i −0.890844 0.454308i \(-0.849886\pi\)
0.890844 0.454308i \(-0.150114\pi\)
\(110\) 154.000i 0.133485i
\(111\) − 39.0000i − 0.0333488i
\(112\) 410.000i 0.345905i
\(113\) 1077.00 0.896599 0.448299 0.893884i \(-0.352030\pi\)
0.448299 + 0.893884i \(0.352030\pi\)
\(114\) 90.0000 0.0739410
\(115\) 1134.00i 0.919531i
\(116\) −791.000 −0.633125
\(117\) 0 0
\(118\) −576.000 −0.449365
\(119\) − 370.000i − 0.285024i
\(120\) 315.000 0.239629
\(121\) 847.000 0.636364
\(122\) 635.000i 0.471231i
\(123\) 855.000i 0.626770i
\(124\) 1372.00i 0.993623i
\(125\) 1407.00i 1.00677i
\(126\) 90.0000 0.0636336
\(127\) 988.000 0.690321 0.345161 0.938544i \(-0.387824\pi\)
0.345161 + 0.938544i \(0.387824\pi\)
\(128\) − 1455.00i − 1.00473i
\(129\) 738.000 0.503700
\(130\) 0 0
\(131\) 560.000 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(132\) 462.000i 0.304636i
\(133\) −300.000 −0.195589
\(134\) 202.000 0.130225
\(135\) 189.000i 0.120493i
\(136\) 555.000i 0.349933i
\(137\) 519.000i 0.323658i 0.986819 + 0.161829i \(0.0517393\pi\)
−0.986819 + 0.161829i \(0.948261\pi\)
\(138\) − 486.000i − 0.299790i
\(139\) −348.000 −0.212352 −0.106176 0.994347i \(-0.533861\pi\)
−0.106176 + 0.994347i \(0.533861\pi\)
\(140\) −490.000 −0.295804
\(141\) 1386.00i 0.827817i
\(142\) −1086.00 −0.641796
\(143\) 0 0
\(144\) 369.000 0.213542
\(145\) − 791.000i − 0.453027i
\(146\) 805.000 0.456317
\(147\) 729.000 0.409027
\(148\) − 91.0000i − 0.0505416i
\(149\) − 645.000i − 0.354634i −0.984154 0.177317i \(-0.943258\pi\)
0.984154 0.177317i \(-0.0567418\pi\)
\(150\) − 228.000i − 0.124107i
\(151\) − 2914.00i − 1.57045i −0.619211 0.785225i \(-0.712547\pi\)
0.619211 0.785225i \(-0.287453\pi\)
\(152\) 450.000 0.240130
\(153\) −333.000 −0.175957
\(154\) 220.000i 0.115118i
\(155\) −1372.00 −0.710979
\(156\) 0 0
\(157\) −2079.00 −1.05683 −0.528415 0.848986i \(-0.677213\pi\)
−0.528415 + 0.848986i \(0.677213\pi\)
\(158\) − 884.000i − 0.445109i
\(159\) −1611.00 −0.803526
\(160\) 1127.00 0.556857
\(161\) 1620.00i 0.793006i
\(162\) − 81.0000i − 0.0392837i
\(163\) − 1700.00i − 0.816897i −0.912781 0.408449i \(-0.866070\pi\)
0.912781 0.408449i \(-0.133930\pi\)
\(164\) 1995.00i 0.949898i
\(165\) −462.000 −0.217980
\(166\) 518.000 0.242196
\(167\) − 3680.00i − 1.70519i −0.522571 0.852596i \(-0.675027\pi\)
0.522571 0.852596i \(-0.324973\pi\)
\(168\) 450.000 0.206656
\(169\) 0 0
\(170\) −259.000 −0.116849
\(171\) 270.000i 0.120745i
\(172\) 1722.00 0.763379
\(173\) −4146.00 −1.82205 −0.911025 0.412352i \(-0.864707\pi\)
−0.911025 + 0.412352i \(0.864707\pi\)
\(174\) 339.000i 0.147698i
\(175\) 760.000i 0.328289i
\(176\) 902.000i 0.386311i
\(177\) − 1728.00i − 0.733810i
\(178\) −194.000 −0.0816905
\(179\) −3674.00 −1.53412 −0.767060 0.641575i \(-0.778281\pi\)
−0.767060 + 0.641575i \(0.778281\pi\)
\(180\) 441.000i 0.182612i
\(181\) 3283.00 1.34820 0.674098 0.738642i \(-0.264533\pi\)
0.674098 + 0.738642i \(0.264533\pi\)
\(182\) 0 0
\(183\) −1905.00 −0.769517
\(184\) − 2430.00i − 0.973598i
\(185\) 91.0000 0.0361646
\(186\) 588.000 0.231797
\(187\) − 814.000i − 0.318319i
\(188\) 3234.00i 1.25459i
\(189\) 270.000i 0.103913i
\(190\) 210.000i 0.0801842i
\(191\) −596.000 −0.225786 −0.112893 0.993607i \(-0.536012\pi\)
−0.112893 + 0.993607i \(0.536012\pi\)
\(192\) 501.000 0.188315
\(193\) 393.000i 0.146574i 0.997311 + 0.0732869i \(0.0233489\pi\)
−0.997311 + 0.0732869i \(0.976651\pi\)
\(194\) −1202.00 −0.444838
\(195\) 0 0
\(196\) 1701.00 0.619898
\(197\) − 3522.00i − 1.27377i −0.770960 0.636884i \(-0.780223\pi\)
0.770960 0.636884i \(-0.219777\pi\)
\(198\) 198.000 0.0710669
\(199\) −2018.00 −0.718855 −0.359428 0.933173i \(-0.617028\pi\)
−0.359428 + 0.933173i \(0.617028\pi\)
\(200\) − 1140.00i − 0.403051i
\(201\) 606.000i 0.212656i
\(202\) − 429.000i − 0.149427i
\(203\) − 1130.00i − 0.390692i
\(204\) −777.000 −0.266671
\(205\) −1995.00 −0.679692
\(206\) − 1302.00i − 0.440362i
\(207\) 1458.00 0.489556
\(208\) 0 0
\(209\) −660.000 −0.218436
\(210\) 210.000i 0.0690066i
\(211\) 160.000 0.0522031 0.0261016 0.999659i \(-0.491691\pi\)
0.0261016 + 0.999659i \(0.491691\pi\)
\(212\) −3759.00 −1.21778
\(213\) − 3258.00i − 1.04805i
\(214\) 1338.00i 0.427401i
\(215\) 1722.00i 0.546230i
\(216\) − 405.000i − 0.127578i
\(217\) −1960.00 −0.613150
\(218\) −1034.00 −0.321245
\(219\) 2415.00i 0.745162i
\(220\) −1078.00 −0.330358
\(221\) 0 0
\(222\) −39.0000 −0.0117906
\(223\) 4072.00i 1.22279i 0.791327 + 0.611393i \(0.209391\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(224\) 1610.00 0.480235
\(225\) 684.000 0.202667
\(226\) − 1077.00i − 0.316995i
\(227\) − 5794.00i − 1.69410i −0.531511 0.847051i \(-0.678376\pi\)
0.531511 0.847051i \(-0.321624\pi\)
\(228\) 630.000i 0.182995i
\(229\) − 6482.00i − 1.87049i −0.353999 0.935246i \(-0.615178\pi\)
0.353999 0.935246i \(-0.384822\pi\)
\(230\) 1134.00 0.325103
\(231\) −660.000 −0.187986
\(232\) 1695.00i 0.479665i
\(233\) −6890.00 −1.93725 −0.968624 0.248530i \(-0.920053\pi\)
−0.968624 + 0.248530i \(0.920053\pi\)
\(234\) 0 0
\(235\) −3234.00 −0.897714
\(236\) − 4032.00i − 1.11212i
\(237\) 2652.00 0.726860
\(238\) −370.000 −0.100771
\(239\) 2466.00i 0.667415i 0.942677 + 0.333708i \(0.108300\pi\)
−0.942677 + 0.333708i \(0.891700\pi\)
\(240\) 861.000i 0.231572i
\(241\) 3617.00i 0.966770i 0.875408 + 0.483385i \(0.160593\pi\)
−0.875408 + 0.483385i \(0.839407\pi\)
\(242\) − 847.000i − 0.224989i
\(243\) 243.000 0.0641500
\(244\) −4445.00 −1.16624
\(245\) 1701.00i 0.443563i
\(246\) 855.000 0.221597
\(247\) 0 0
\(248\) 2940.00 0.752783
\(249\) 1554.00i 0.395505i
\(250\) 1407.00 0.355946
\(251\) −4860.00 −1.22215 −0.611077 0.791571i \(-0.709263\pi\)
−0.611077 + 0.791571i \(0.709263\pi\)
\(252\) 630.000i 0.157485i
\(253\) 3564.00i 0.885639i
\(254\) − 988.000i − 0.244065i
\(255\) − 777.000i − 0.190814i
\(256\) −119.000 −0.0290527
\(257\) −565.000 −0.137135 −0.0685676 0.997646i \(-0.521843\pi\)
−0.0685676 + 0.997646i \(0.521843\pi\)
\(258\) − 738.000i − 0.178085i
\(259\) 130.000 0.0311884
\(260\) 0 0
\(261\) −1017.00 −0.241190
\(262\) − 560.000i − 0.132049i
\(263\) −498.000 −0.116760 −0.0583802 0.998294i \(-0.518594\pi\)
−0.0583802 + 0.998294i \(0.518594\pi\)
\(264\) 990.000 0.230797
\(265\) − 3759.00i − 0.871372i
\(266\) 300.000i 0.0691511i
\(267\) − 582.000i − 0.133400i
\(268\) 1414.00i 0.322290i
\(269\) 5546.00 1.25705 0.628523 0.777791i \(-0.283660\pi\)
0.628523 + 0.777791i \(0.283660\pi\)
\(270\) 189.000 0.0426006
\(271\) 2256.00i 0.505691i 0.967507 + 0.252845i \(0.0813664\pi\)
−0.967507 + 0.252845i \(0.918634\pi\)
\(272\) −1517.00 −0.338168
\(273\) 0 0
\(274\) 519.000 0.114430
\(275\) 1672.00i 0.366638i
\(276\) 3402.00 0.741943
\(277\) −2309.00 −0.500846 −0.250423 0.968137i \(-0.580570\pi\)
−0.250423 + 0.968137i \(0.580570\pi\)
\(278\) 348.000i 0.0750779i
\(279\) 1764.00i 0.378523i
\(280\) 1050.00i 0.224105i
\(281\) − 5833.00i − 1.23832i −0.785265 0.619159i \(-0.787473\pi\)
0.785265 0.619159i \(-0.212527\pi\)
\(282\) 1386.00 0.292678
\(283\) −1650.00 −0.346581 −0.173290 0.984871i \(-0.555440\pi\)
−0.173290 + 0.984871i \(0.555440\pi\)
\(284\) − 7602.00i − 1.58837i
\(285\) −630.000 −0.130940
\(286\) 0 0
\(287\) −2850.00 −0.586168
\(288\) − 1449.00i − 0.296469i
\(289\) −3544.00 −0.721352
\(290\) −791.000 −0.160169
\(291\) − 3606.00i − 0.726417i
\(292\) 5635.00i 1.12933i
\(293\) − 2991.00i − 0.596369i −0.954508 0.298184i \(-0.903619\pi\)
0.954508 0.298184i \(-0.0963811\pi\)
\(294\) − 729.000i − 0.144613i
\(295\) 4032.00 0.795770
\(296\) −195.000 −0.0382910
\(297\) 594.000i 0.116052i
\(298\) −645.000 −0.125382
\(299\) 0 0
\(300\) 1596.00 0.307150
\(301\) 2460.00i 0.471070i
\(302\) −2914.00 −0.555238
\(303\) 1287.00 0.244014
\(304\) 1230.00i 0.232057i
\(305\) − 4445.00i − 0.834492i
\(306\) 333.000i 0.0622103i
\(307\) 2422.00i 0.450263i 0.974328 + 0.225132i \(0.0722812\pi\)
−0.974328 + 0.225132i \(0.927719\pi\)
\(308\) −1540.00 −0.284901
\(309\) 3906.00 0.719109
\(310\) 1372.00i 0.251369i
\(311\) 3402.00 0.620288 0.310144 0.950690i \(-0.399623\pi\)
0.310144 + 0.950690i \(0.399623\pi\)
\(312\) 0 0
\(313\) 2310.00 0.417153 0.208577 0.978006i \(-0.433117\pi\)
0.208577 + 0.978006i \(0.433117\pi\)
\(314\) 2079.00i 0.373646i
\(315\) −630.000 −0.112687
\(316\) 6188.00 1.10159
\(317\) − 257.000i − 0.0455349i −0.999741 0.0227674i \(-0.992752\pi\)
0.999741 0.0227674i \(-0.00724773\pi\)
\(318\) 1611.00i 0.284089i
\(319\) − 2486.00i − 0.436330i
\(320\) 1169.00i 0.204216i
\(321\) −4014.00 −0.697943
\(322\) 1620.00 0.280370
\(323\) − 1110.00i − 0.191214i
\(324\) 567.000 0.0972222
\(325\) 0 0
\(326\) −1700.00 −0.288817
\(327\) − 3102.00i − 0.524590i
\(328\) 4275.00 0.719657
\(329\) −4620.00 −0.774191
\(330\) 462.000i 0.0770675i
\(331\) 1028.00i 0.170707i 0.996351 + 0.0853535i \(0.0272019\pi\)
−0.996351 + 0.0853535i \(0.972798\pi\)
\(332\) 3626.00i 0.599405i
\(333\) − 117.000i − 0.0192539i
\(334\) −3680.00 −0.602876
\(335\) −1414.00 −0.230612
\(336\) 1230.00i 0.199708i
\(337\) −2487.00 −0.402005 −0.201002 0.979591i \(-0.564420\pi\)
−0.201002 + 0.979591i \(0.564420\pi\)
\(338\) 0 0
\(339\) 3231.00 0.517651
\(340\) − 1813.00i − 0.289187i
\(341\) −4312.00 −0.684774
\(342\) 270.000 0.0426898
\(343\) 5860.00i 0.922479i
\(344\) − 3690.00i − 0.578347i
\(345\) 3402.00i 0.530891i
\(346\) 4146.00i 0.644192i
\(347\) −2850.00 −0.440911 −0.220455 0.975397i \(-0.570754\pi\)
−0.220455 + 0.975397i \(0.570754\pi\)
\(348\) −2373.00 −0.365535
\(349\) 2018.00i 0.309516i 0.987952 + 0.154758i \(0.0494598\pi\)
−0.987952 + 0.154758i \(0.950540\pi\)
\(350\) 760.000 0.116068
\(351\) 0 0
\(352\) 3542.00 0.536333
\(353\) − 5287.00i − 0.797163i −0.917133 0.398582i \(-0.869503\pi\)
0.917133 0.398582i \(-0.130497\pi\)
\(354\) −1728.00 −0.259441
\(355\) 7602.00 1.13654
\(356\) − 1358.00i − 0.202174i
\(357\) − 1110.00i − 0.164559i
\(358\) 3674.00i 0.542394i
\(359\) 7278.00i 1.06997i 0.844863 + 0.534983i \(0.179682\pi\)
−0.844863 + 0.534983i \(0.820318\pi\)
\(360\) 945.000 0.138350
\(361\) 5959.00 0.868786
\(362\) − 3283.00i − 0.476659i
\(363\) 2541.00 0.367405
\(364\) 0 0
\(365\) −5635.00 −0.808080
\(366\) 1905.00i 0.272065i
\(367\) −4202.00 −0.597664 −0.298832 0.954306i \(-0.596597\pi\)
−0.298832 + 0.954306i \(0.596597\pi\)
\(368\) 6642.00 0.940865
\(369\) 2565.00i 0.361866i
\(370\) − 91.0000i − 0.0127861i
\(371\) − 5370.00i − 0.751473i
\(372\) 4116.00i 0.573668i
\(373\) −1583.00 −0.219744 −0.109872 0.993946i \(-0.535044\pi\)
−0.109872 + 0.993946i \(0.535044\pi\)
\(374\) −814.000 −0.112543
\(375\) 4221.00i 0.581257i
\(376\) 6930.00 0.950499
\(377\) 0 0
\(378\) 270.000 0.0367389
\(379\) − 2052.00i − 0.278111i −0.990285 0.139056i \(-0.955593\pi\)
0.990285 0.139056i \(-0.0444067\pi\)
\(380\) −1470.00 −0.198446
\(381\) 2964.00 0.398557
\(382\) 596.000i 0.0798273i
\(383\) − 6872.00i − 0.916822i −0.888740 0.458411i \(-0.848419\pi\)
0.888740 0.458411i \(-0.151581\pi\)
\(384\) − 4365.00i − 0.580079i
\(385\) − 1540.00i − 0.203859i
\(386\) 393.000 0.0518217
\(387\) 2214.00 0.290811
\(388\) − 8414.00i − 1.10092i
\(389\) 11653.0 1.51884 0.759422 0.650598i \(-0.225482\pi\)
0.759422 + 0.650598i \(0.225482\pi\)
\(390\) 0 0
\(391\) −5994.00 −0.775268
\(392\) − 3645.00i − 0.469644i
\(393\) 1680.00 0.215636
\(394\) −3522.00 −0.450345
\(395\) 6188.00i 0.788233i
\(396\) 1386.00i 0.175882i
\(397\) − 6134.00i − 0.775458i −0.921774 0.387729i \(-0.873260\pi\)
0.921774 0.387729i \(-0.126740\pi\)
\(398\) 2018.00i 0.254154i
\(399\) −900.000 −0.112923
\(400\) 3116.00 0.389500
\(401\) 10795.0i 1.34433i 0.740401 + 0.672165i \(0.234636\pi\)
−0.740401 + 0.672165i \(0.765364\pi\)
\(402\) 606.000 0.0751854
\(403\) 0 0
\(404\) 3003.00 0.369814
\(405\) 567.000i 0.0695666i
\(406\) −1130.00 −0.138130
\(407\) 286.000 0.0348317
\(408\) 1665.00i 0.202034i
\(409\) − 8489.00i − 1.02629i −0.858301 0.513147i \(-0.828480\pi\)
0.858301 0.513147i \(-0.171520\pi\)
\(410\) 1995.00i 0.240307i
\(411\) 1557.00i 0.186864i
\(412\) 9114.00 1.08984
\(413\) 5760.00 0.686274
\(414\) − 1458.00i − 0.173084i
\(415\) −3626.00 −0.428900
\(416\) 0 0
\(417\) −1044.00 −0.122602
\(418\) 660.000i 0.0772288i
\(419\) 1496.00 0.174426 0.0872129 0.996190i \(-0.472204\pi\)
0.0872129 + 0.996190i \(0.472204\pi\)
\(420\) −1470.00 −0.170783
\(421\) − 11695.0i − 1.35387i −0.736043 0.676935i \(-0.763308\pi\)
0.736043 0.676935i \(-0.236692\pi\)
\(422\) − 160.000i − 0.0184566i
\(423\) 4158.00i 0.477941i
\(424\) 8055.00i 0.922607i
\(425\) −2812.00 −0.320946
\(426\) −3258.00 −0.370541
\(427\) − 6350.00i − 0.719668i
\(428\) −9366.00 −1.05776
\(429\) 0 0
\(430\) 1722.00 0.193121
\(431\) 10590.0i 1.18353i 0.806110 + 0.591766i \(0.201569\pi\)
−0.806110 + 0.591766i \(0.798431\pi\)
\(432\) 1107.00 0.123288
\(433\) 13949.0 1.54814 0.774072 0.633098i \(-0.218217\pi\)
0.774072 + 0.633098i \(0.218217\pi\)
\(434\) 1960.00i 0.216781i
\(435\) − 2373.00i − 0.261555i
\(436\) − 7238.00i − 0.795040i
\(437\) 4860.00i 0.532003i
\(438\) 2415.00 0.263455
\(439\) 10726.0 1.16611 0.583057 0.812431i \(-0.301856\pi\)
0.583057 + 0.812431i \(0.301856\pi\)
\(440\) 2310.00i 0.250284i
\(441\) 2187.00 0.236152
\(442\) 0 0
\(443\) 16228.0 1.74044 0.870221 0.492662i \(-0.163976\pi\)
0.870221 + 0.492662i \(0.163976\pi\)
\(444\) − 273.000i − 0.0291802i
\(445\) 1358.00 0.144664
\(446\) 4072.00 0.432320
\(447\) − 1935.00i − 0.204748i
\(448\) 1670.00i 0.176116i
\(449\) − 7538.00i − 0.792294i −0.918187 0.396147i \(-0.870347\pi\)
0.918187 0.396147i \(-0.129653\pi\)
\(450\) − 684.000i − 0.0716535i
\(451\) −6270.00 −0.654640
\(452\) 7539.00 0.784524
\(453\) − 8742.00i − 0.906700i
\(454\) −5794.00 −0.598956
\(455\) 0 0
\(456\) 1350.00 0.138639
\(457\) 15539.0i 1.59056i 0.606245 + 0.795278i \(0.292675\pi\)
−0.606245 + 0.795278i \(0.707325\pi\)
\(458\) −6482.00 −0.661319
\(459\) −999.000 −0.101589
\(460\) 7938.00i 0.804589i
\(461\) 4811.00i 0.486053i 0.970020 + 0.243027i \(0.0781403\pi\)
−0.970020 + 0.243027i \(0.921860\pi\)
\(462\) 660.000i 0.0664632i
\(463\) − 562.000i − 0.0564111i −0.999602 0.0282056i \(-0.991021\pi\)
0.999602 0.0282056i \(-0.00897930\pi\)
\(464\) −4633.00 −0.463538
\(465\) −4116.00 −0.410484
\(466\) 6890.00i 0.684921i
\(467\) −4914.00 −0.486922 −0.243461 0.969911i \(-0.578283\pi\)
−0.243461 + 0.969911i \(0.578283\pi\)
\(468\) 0 0
\(469\) −2020.00 −0.198880
\(470\) 3234.00i 0.317390i
\(471\) −6237.00 −0.610161
\(472\) −8640.00 −0.842560
\(473\) 5412.00i 0.526097i
\(474\) − 2652.00i − 0.256984i
\(475\) 2280.00i 0.220239i
\(476\) − 2590.00i − 0.249396i
\(477\) −4833.00 −0.463916
\(478\) 2466.00 0.235967
\(479\) 3600.00i 0.343399i 0.985149 + 0.171700i \(0.0549258\pi\)
−0.985149 + 0.171700i \(0.945074\pi\)
\(480\) 3381.00 0.321502
\(481\) 0 0
\(482\) 3617.00 0.341805
\(483\) 4860.00i 0.457842i
\(484\) 5929.00 0.556818
\(485\) 8414.00 0.787753
\(486\) − 243.000i − 0.0226805i
\(487\) − 17130.0i − 1.59391i −0.604038 0.796955i \(-0.706443\pi\)
0.604038 0.796955i \(-0.293557\pi\)
\(488\) 9525.00i 0.883558i
\(489\) − 5100.00i − 0.471636i
\(490\) 1701.00 0.156823
\(491\) 11838.0 1.08807 0.544034 0.839063i \(-0.316896\pi\)
0.544034 + 0.839063i \(0.316896\pi\)
\(492\) 5985.00i 0.548424i
\(493\) 4181.00 0.381953
\(494\) 0 0
\(495\) −1386.00 −0.125851
\(496\) 8036.00i 0.727474i
\(497\) 10860.0 0.980156
\(498\) 1554.00 0.139832
\(499\) 8976.00i 0.805252i 0.915364 + 0.402626i \(0.131903\pi\)
−0.915364 + 0.402626i \(0.868097\pi\)
\(500\) 9849.00i 0.880921i
\(501\) − 11040.0i − 0.984493i
\(502\) 4860.00i 0.432096i
\(503\) 1682.00 0.149099 0.0745494 0.997217i \(-0.476248\pi\)
0.0745494 + 0.997217i \(0.476248\pi\)
\(504\) 1350.00 0.119313
\(505\) 3003.00i 0.264617i
\(506\) 3564.00 0.313121
\(507\) 0 0
\(508\) 6916.00 0.604031
\(509\) 15167.0i 1.32076i 0.750933 + 0.660379i \(0.229604\pi\)
−0.750933 + 0.660379i \(0.770396\pi\)
\(510\) −777.000 −0.0674630
\(511\) −8050.00 −0.696890
\(512\) − 11521.0i − 0.994455i
\(513\) 810.000i 0.0697122i
\(514\) 565.000i 0.0484846i
\(515\) 9114.00i 0.779827i
\(516\) 5166.00 0.440737
\(517\) −10164.0 −0.864627
\(518\) − 130.000i − 0.0110268i
\(519\) −12438.0 −1.05196
\(520\) 0 0
\(521\) −6783.00 −0.570381 −0.285191 0.958471i \(-0.592057\pi\)
−0.285191 + 0.958471i \(0.592057\pi\)
\(522\) 1017.00i 0.0852737i
\(523\) −13918.0 −1.16366 −0.581828 0.813312i \(-0.697662\pi\)
−0.581828 + 0.813312i \(0.697662\pi\)
\(524\) 3920.00 0.326805
\(525\) 2280.00i 0.189538i
\(526\) 498.000i 0.0412810i
\(527\) − 7252.00i − 0.599435i
\(528\) 2706.00i 0.223037i
\(529\) 14077.0 1.15698
\(530\) −3759.00 −0.308076
\(531\) − 5184.00i − 0.423666i
\(532\) −2100.00 −0.171140
\(533\) 0 0
\(534\) −582.000 −0.0471641
\(535\) − 9366.00i − 0.756874i
\(536\) 3030.00 0.244172
\(537\) −11022.0 −0.885725
\(538\) − 5546.00i − 0.444433i
\(539\) 5346.00i 0.427214i
\(540\) 1323.00i 0.105431i
\(541\) 1335.00i 0.106093i 0.998592 + 0.0530463i \(0.0168931\pi\)
−0.998592 + 0.0530463i \(0.983107\pi\)
\(542\) 2256.00 0.178789
\(543\) 9849.00 0.778381
\(544\) 5957.00i 0.469493i
\(545\) 7238.00 0.568884
\(546\) 0 0
\(547\) −3806.00 −0.297501 −0.148750 0.988875i \(-0.547525\pi\)
−0.148750 + 0.988875i \(0.547525\pi\)
\(548\) 3633.00i 0.283201i
\(549\) −5715.00 −0.444281
\(550\) 1672.00 0.129626
\(551\) − 3390.00i − 0.262103i
\(552\) − 7290.00i − 0.562107i
\(553\) 8840.00i 0.679774i
\(554\) 2309.00i 0.177076i
\(555\) 273.000 0.0208796
\(556\) −2436.00 −0.185808
\(557\) 1905.00i 0.144915i 0.997372 + 0.0724573i \(0.0230841\pi\)
−0.997372 + 0.0724573i \(0.976916\pi\)
\(558\) 1764.00 0.133828
\(559\) 0 0
\(560\) −2870.00 −0.216571
\(561\) − 2442.00i − 0.183781i
\(562\) −5833.00 −0.437812
\(563\) 4800.00 0.359318 0.179659 0.983729i \(-0.442501\pi\)
0.179659 + 0.983729i \(0.442501\pi\)
\(564\) 9702.00i 0.724340i
\(565\) 7539.00i 0.561359i
\(566\) 1650.00i 0.122535i
\(567\) 810.000i 0.0599944i
\(568\) −16290.0 −1.20337
\(569\) −14678.0 −1.08143 −0.540715 0.841206i \(-0.681846\pi\)
−0.540715 + 0.841206i \(0.681846\pi\)
\(570\) 630.000i 0.0462944i
\(571\) 586.000 0.0429481 0.0214740 0.999769i \(-0.493164\pi\)
0.0214740 + 0.999769i \(0.493164\pi\)
\(572\) 0 0
\(573\) −1788.00 −0.130357
\(574\) 2850.00i 0.207242i
\(575\) 12312.0 0.892949
\(576\) 1503.00 0.108724
\(577\) 8939.00i 0.644949i 0.946578 + 0.322474i \(0.104515\pi\)
−0.946578 + 0.322474i \(0.895485\pi\)
\(578\) 3544.00i 0.255036i
\(579\) 1179.00i 0.0846245i
\(580\) − 5537.00i − 0.396399i
\(581\) −5180.00 −0.369884
\(582\) −3606.00 −0.256827
\(583\) − 11814.0i − 0.839255i
\(584\) 12075.0 0.855594
\(585\) 0 0
\(586\) −2991.00 −0.210848
\(587\) − 13792.0i − 0.969773i −0.874577 0.484887i \(-0.838861\pi\)
0.874577 0.484887i \(-0.161139\pi\)
\(588\) 5103.00 0.357898
\(589\) −5880.00 −0.411343
\(590\) − 4032.00i − 0.281347i
\(591\) − 10566.0i − 0.735410i
\(592\) − 533.000i − 0.0370037i
\(593\) − 9569.00i − 0.662650i −0.943517 0.331325i \(-0.892504\pi\)
0.943517 0.331325i \(-0.107496\pi\)
\(594\) 594.000 0.0410305
\(595\) 2590.00 0.178453
\(596\) − 4515.00i − 0.310305i
\(597\) −6054.00 −0.415031
\(598\) 0 0
\(599\) −5192.00 −0.354156 −0.177078 0.984197i \(-0.556664\pi\)
−0.177078 + 0.984197i \(0.556664\pi\)
\(600\) − 3420.00i − 0.232702i
\(601\) −3677.00 −0.249564 −0.124782 0.992184i \(-0.539823\pi\)
−0.124782 + 0.992184i \(0.539823\pi\)
\(602\) 2460.00 0.166548
\(603\) 1818.00i 0.122777i
\(604\) − 20398.0i − 1.37414i
\(605\) 5929.00i 0.398427i
\(606\) − 1287.00i − 0.0862719i
\(607\) −10960.0 −0.732871 −0.366435 0.930443i \(-0.619422\pi\)
−0.366435 + 0.930443i \(0.619422\pi\)
\(608\) 4830.00 0.322175
\(609\) − 3390.00i − 0.225566i
\(610\) −4445.00 −0.295037
\(611\) 0 0
\(612\) −2331.00 −0.153963
\(613\) − 26027.0i − 1.71488i −0.514585 0.857439i \(-0.672054\pi\)
0.514585 0.857439i \(-0.327946\pi\)
\(614\) 2422.00 0.159192
\(615\) −5985.00 −0.392420
\(616\) 3300.00i 0.215845i
\(617\) 17681.0i 1.15366i 0.816863 + 0.576832i \(0.195711\pi\)
−0.816863 + 0.576832i \(0.804289\pi\)
\(618\) − 3906.00i − 0.254243i
\(619\) − 3192.00i − 0.207265i −0.994616 0.103633i \(-0.966953\pi\)
0.994616 0.103633i \(-0.0330467\pi\)
\(620\) −9604.00 −0.622106
\(621\) 4374.00 0.282645
\(622\) − 3402.00i − 0.219305i
\(623\) 1940.00 0.124758
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) − 2310.00i − 0.147486i
\(627\) −1980.00 −0.126114
\(628\) −14553.0 −0.924726
\(629\) 481.000i 0.0304908i
\(630\) 630.000i 0.0398410i
\(631\) − 7580.00i − 0.478217i −0.970993 0.239109i \(-0.923145\pi\)
0.970993 0.239109i \(-0.0768552\pi\)
\(632\) − 13260.0i − 0.834580i
\(633\) 480.000 0.0301395
\(634\) −257.000 −0.0160990
\(635\) 6916.00i 0.432210i
\(636\) −11277.0 −0.703085
\(637\) 0 0
\(638\) −2486.00 −0.154266
\(639\) − 9774.00i − 0.605091i
\(640\) 10185.0 0.629059
\(641\) 27707.0 1.70727 0.853635 0.520871i \(-0.174393\pi\)
0.853635 + 0.520871i \(0.174393\pi\)
\(642\) 4014.00i 0.246760i
\(643\) 11216.0i 0.687894i 0.938989 + 0.343947i \(0.111764\pi\)
−0.938989 + 0.343947i \(0.888236\pi\)
\(644\) 11340.0i 0.693880i
\(645\) 5166.00i 0.315366i
\(646\) −1110.00 −0.0676043
\(647\) 2536.00 0.154097 0.0770483 0.997027i \(-0.475450\pi\)
0.0770483 + 0.997027i \(0.475450\pi\)
\(648\) − 1215.00i − 0.0736570i
\(649\) 12672.0 0.766440
\(650\) 0 0
\(651\) −5880.00 −0.354002
\(652\) − 11900.0i − 0.714785i
\(653\) 17730.0 1.06252 0.531262 0.847207i \(-0.321718\pi\)
0.531262 + 0.847207i \(0.321718\pi\)
\(654\) −3102.00 −0.185471
\(655\) 3920.00i 0.233843i
\(656\) 11685.0i 0.695461i
\(657\) 7245.00i 0.430220i
\(658\) 4620.00i 0.273718i
\(659\) 18920.0 1.11839 0.559195 0.829036i \(-0.311110\pi\)
0.559195 + 0.829036i \(0.311110\pi\)
\(660\) −3234.00 −0.190732
\(661\) − 5241.00i − 0.308398i −0.988040 0.154199i \(-0.950720\pi\)
0.988040 0.154199i \(-0.0492797\pi\)
\(662\) 1028.00 0.0603540
\(663\) 0 0
\(664\) 7770.00 0.454118
\(665\) − 2100.00i − 0.122458i
\(666\) −117.000 −0.00680729
\(667\) −18306.0 −1.06269
\(668\) − 25760.0i − 1.49204i
\(669\) 12216.0i 0.705976i
\(670\) 1414.00i 0.0815337i
\(671\) − 13970.0i − 0.803735i
\(672\) 4830.00 0.277264
\(673\) −20467.0 −1.17228 −0.586140 0.810210i \(-0.699353\pi\)
−0.586140 + 0.810210i \(0.699353\pi\)
\(674\) 2487.00i 0.142130i
\(675\) 2052.00 0.117010
\(676\) 0 0
\(677\) −70.0000 −0.00397388 −0.00198694 0.999998i \(-0.500632\pi\)
−0.00198694 + 0.999998i \(0.500632\pi\)
\(678\) − 3231.00i − 0.183017i
\(679\) 12020.0 0.679360
\(680\) −3885.00 −0.219093
\(681\) − 17382.0i − 0.978091i
\(682\) 4312.00i 0.242104i
\(683\) − 6432.00i − 0.360342i −0.983635 0.180171i \(-0.942335\pi\)
0.983635 0.180171i \(-0.0576651\pi\)
\(684\) 1890.00i 0.105652i
\(685\) −3633.00 −0.202642
\(686\) 5860.00 0.326146
\(687\) − 19446.0i − 1.07993i
\(688\) 10086.0 0.558903
\(689\) 0 0
\(690\) 3402.00 0.187698
\(691\) − 6666.00i − 0.366985i −0.983021 0.183492i \(-0.941260\pi\)
0.983021 0.183492i \(-0.0587403\pi\)
\(692\) −29022.0 −1.59429
\(693\) −1980.00 −0.108534
\(694\) 2850.00i 0.155885i
\(695\) − 2436.00i − 0.132954i
\(696\) 5085.00i 0.276935i
\(697\) − 10545.0i − 0.573056i
\(698\) 2018.00 0.109430
\(699\) −20670.0 −1.11847
\(700\) 5320.00i 0.287253i
\(701\) 14054.0 0.757221 0.378611 0.925556i \(-0.376402\pi\)
0.378611 + 0.925556i \(0.376402\pi\)
\(702\) 0 0
\(703\) 390.000 0.0209234
\(704\) 3674.00i 0.196689i
\(705\) −9702.00 −0.518296
\(706\) −5287.00 −0.281840
\(707\) 4290.00i 0.228207i
\(708\) − 12096.0i − 0.642084i
\(709\) 71.0000i 0.00376088i 0.999998 + 0.00188044i \(0.000598562\pi\)
−0.999998 + 0.00188044i \(0.999401\pi\)
\(710\) − 7602.00i − 0.401828i
\(711\) 7956.00 0.419653
\(712\) −2910.00 −0.153170
\(713\) 31752.0i 1.66777i
\(714\) −1110.00 −0.0581803
\(715\) 0 0
\(716\) −25718.0 −1.34236
\(717\) 7398.00i 0.385332i
\(718\) 7278.00 0.378290
\(719\) −3936.00 −0.204156 −0.102078 0.994776i \(-0.532549\pi\)
−0.102078 + 0.994776i \(0.532549\pi\)
\(720\) 2583.00i 0.133698i
\(721\) 13020.0i 0.672524i
\(722\) − 5959.00i − 0.307162i
\(723\) 10851.0i 0.558165i
\(724\) 22981.0 1.17967
\(725\) −8588.00 −0.439931
\(726\) − 2541.00i − 0.129897i
\(727\) −34202.0 −1.74482 −0.872409 0.488777i \(-0.837443\pi\)
−0.872409 + 0.488777i \(0.837443\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 5635.00i 0.285700i
\(731\) −9102.00 −0.460533
\(732\) −13335.0 −0.673328
\(733\) − 27363.0i − 1.37882i −0.724371 0.689410i \(-0.757870\pi\)
0.724371 0.689410i \(-0.242130\pi\)
\(734\) 4202.00i 0.211306i
\(735\) 5103.00i 0.256091i
\(736\) − 26082.0i − 1.30624i
\(737\) −4444.00 −0.222112
\(738\) 2565.00 0.127939
\(739\) − 21776.0i − 1.08396i −0.840393 0.541978i \(-0.817676\pi\)
0.840393 0.541978i \(-0.182324\pi\)
\(740\) 637.000 0.0316440
\(741\) 0 0
\(742\) −5370.00 −0.265686
\(743\) 2484.00i 0.122650i 0.998118 + 0.0613251i \(0.0195326\pi\)
−0.998118 + 0.0613251i \(0.980467\pi\)
\(744\) 8820.00 0.434619
\(745\) 4515.00 0.222036
\(746\) 1583.00i 0.0776914i
\(747\) 4662.00i 0.228345i
\(748\) − 5698.00i − 0.278529i
\(749\) − 13380.0i − 0.652730i
\(750\) 4221.00 0.205506
\(751\) −32906.0 −1.59888 −0.799439 0.600748i \(-0.794870\pi\)
−0.799439 + 0.600748i \(0.794870\pi\)
\(752\) 18942.0i 0.918542i
\(753\) −14580.0 −0.705611
\(754\) 0 0
\(755\) 20398.0 0.983257
\(756\) 1890.00i 0.0909241i
\(757\) −3914.00 −0.187922 −0.0939609 0.995576i \(-0.529953\pi\)
−0.0939609 + 0.995576i \(0.529953\pi\)
\(758\) −2052.00 −0.0983272
\(759\) 10692.0i 0.511324i
\(760\) 3150.00i 0.150345i
\(761\) 33038.0i 1.57375i 0.617110 + 0.786877i \(0.288303\pi\)
−0.617110 + 0.786877i \(0.711697\pi\)
\(762\) − 2964.00i − 0.140911i
\(763\) 10340.0 0.490607
\(764\) −4172.00 −0.197562
\(765\) − 2331.00i − 0.110167i
\(766\) −6872.00 −0.324145
\(767\) 0 0
\(768\) −357.000 −0.0167736
\(769\) − 17586.0i − 0.824665i −0.911033 0.412332i \(-0.864714\pi\)
0.911033 0.412332i \(-0.135286\pi\)
\(770\) −1540.00 −0.0720750
\(771\) −1695.00 −0.0791750
\(772\) 2751.00i 0.128252i
\(773\) 18314.0i 0.852146i 0.904689 + 0.426073i \(0.140103\pi\)
−0.904689 + 0.426073i \(0.859897\pi\)
\(774\) − 2214.00i − 0.102817i
\(775\) 14896.0i 0.690426i
\(776\) −18030.0 −0.834071
\(777\) 390.000 0.0180067
\(778\) − 11653.0i − 0.536993i
\(779\) −8550.00 −0.393242
\(780\) 0 0
\(781\) 23892.0 1.09465
\(782\) 5994.00i 0.274098i
\(783\) −3051.00 −0.139251
\(784\) 9963.00 0.453854
\(785\) − 14553.0i − 0.661680i
\(786\) − 1680.00i − 0.0762387i
\(787\) 42068.0i 1.90542i 0.303888 + 0.952708i \(0.401715\pi\)
−0.303888 + 0.952708i \(0.598285\pi\)
\(788\) − 24654.0i − 1.11455i
\(789\) −1494.00 −0.0674117
\(790\) 6188.00 0.278682
\(791\) 10770.0i 0.484118i
\(792\) 2970.00 0.133250
\(793\) 0 0
\(794\) −6134.00 −0.274166
\(795\) − 11277.0i − 0.503087i
\(796\) −14126.0 −0.628998
\(797\) 4282.00 0.190309 0.0951545 0.995463i \(-0.469665\pi\)
0.0951545 + 0.995463i \(0.469665\pi\)
\(798\) 900.000i 0.0399244i
\(799\) − 17094.0i − 0.756874i
\(800\) − 12236.0i − 0.540760i
\(801\) − 1746.00i − 0.0770186i
\(802\) 10795.0 0.475293
\(803\) −17710.0 −0.778297
\(804\) 4242.00i 0.186074i
\(805\) −11340.0 −0.496500
\(806\) 0 0
\(807\) 16638.0 0.725756
\(808\) − 6435.00i − 0.280176i
\(809\) 40221.0 1.74795 0.873977 0.485967i \(-0.161532\pi\)
0.873977 + 0.485967i \(0.161532\pi\)
\(810\) 567.000 0.0245955
\(811\) − 7084.00i − 0.306724i −0.988170 0.153362i \(-0.950990\pi\)
0.988170 0.153362i \(-0.0490100\pi\)
\(812\) − 7910.00i − 0.341855i
\(813\) 6768.00i 0.291961i
\(814\) − 286.000i − 0.0123149i
\(815\) 11900.0 0.511459
\(816\) −4551.00 −0.195241
\(817\) 7380.00i 0.316026i
\(818\) −8489.00 −0.362850
\(819\) 0 0
\(820\) −13965.0 −0.594730
\(821\) 17338.0i 0.737028i 0.929622 + 0.368514i \(0.120133\pi\)
−0.929622 + 0.368514i \(0.879867\pi\)
\(822\) 1557.00 0.0660664
\(823\) −35496.0 −1.50342 −0.751709 0.659495i \(-0.770770\pi\)
−0.751709 + 0.659495i \(0.770770\pi\)
\(824\) − 19530.0i − 0.825679i
\(825\) 5016.00i 0.211678i
\(826\) − 5760.00i − 0.242634i
\(827\) 14992.0i 0.630378i 0.949029 + 0.315189i \(0.102068\pi\)
−0.949029 + 0.315189i \(0.897932\pi\)
\(828\) 10206.0 0.428361
\(829\) 20659.0 0.865521 0.432760 0.901509i \(-0.357540\pi\)
0.432760 + 0.901509i \(0.357540\pi\)
\(830\) 3626.00i 0.151639i
\(831\) −6927.00 −0.289164
\(832\) 0 0
\(833\) −8991.00 −0.373973
\(834\) 1044.00i 0.0433462i
\(835\) 25760.0 1.06762
\(836\) −4620.00 −0.191132
\(837\) 5292.00i 0.218540i
\(838\) − 1496.00i − 0.0616688i
\(839\) − 28716.0i − 1.18163i −0.806808 0.590814i \(-0.798807\pi\)
0.806808 0.590814i \(-0.201193\pi\)
\(840\) 3150.00i 0.129387i
\(841\) −11620.0 −0.476444
\(842\) −11695.0 −0.478665
\(843\) − 17499.0i − 0.714944i
\(844\) 1120.00 0.0456777
\(845\) 0 0
\(846\) 4158.00 0.168978
\(847\) 8470.00i 0.343604i
\(848\) −22017.0 −0.891588
\(849\) −4950.00 −0.200098
\(850\) 2812.00i 0.113472i
\(851\) − 2106.00i − 0.0848328i
\(852\) − 22806.0i − 0.917043i
\(853\) − 13377.0i − 0.536952i −0.963286 0.268476i \(-0.913480\pi\)
0.963286 0.268476i \(-0.0865199\pi\)
\(854\) −6350.00 −0.254441
\(855\) −1890.00 −0.0755984
\(856\) 20070.0i 0.801377i
\(857\) 27419.0 1.09290 0.546450 0.837492i \(-0.315979\pi\)
0.546450 + 0.837492i \(0.315979\pi\)
\(858\) 0 0
\(859\) 2422.00 0.0962021 0.0481010 0.998842i \(-0.484683\pi\)
0.0481010 + 0.998842i \(0.484683\pi\)
\(860\) 12054.0i 0.477951i
\(861\) −8550.00 −0.338424
\(862\) 10590.0 0.418442
\(863\) − 34522.0i − 1.36169i −0.732425 0.680847i \(-0.761612\pi\)
0.732425 0.680847i \(-0.238388\pi\)
\(864\) − 4347.00i − 0.171167i
\(865\) − 29022.0i − 1.14078i
\(866\) − 13949.0i − 0.547351i
\(867\) −10632.0 −0.416472
\(868\) −13720.0 −0.536506
\(869\) 19448.0i 0.759181i
\(870\) −2373.00 −0.0924738
\(871\) 0 0
\(872\) −15510.0 −0.602334
\(873\) − 10818.0i − 0.419397i
\(874\) 4860.00 0.188091
\(875\) −14070.0 −0.543603
\(876\) 16905.0i 0.652017i
\(877\) 13733.0i 0.528769i 0.964417 + 0.264385i \(0.0851688\pi\)
−0.964417 + 0.264385i \(0.914831\pi\)
\(878\) − 10726.0i − 0.412284i
\(879\) − 8973.00i − 0.344314i
\(880\) −6314.00 −0.241869
\(881\) 22759.0 0.870341 0.435170 0.900348i \(-0.356688\pi\)
0.435170 + 0.900348i \(0.356688\pi\)
\(882\) − 2187.00i − 0.0834922i
\(883\) 2168.00 0.0826263 0.0413131 0.999146i \(-0.486846\pi\)
0.0413131 + 0.999146i \(0.486846\pi\)
\(884\) 0 0
\(885\) 12096.0 0.459438
\(886\) − 16228.0i − 0.615339i
\(887\) 15888.0 0.601428 0.300714 0.953714i \(-0.402775\pi\)
0.300714 + 0.953714i \(0.402775\pi\)
\(888\) −585.000 −0.0221073
\(889\) 9880.00i 0.372739i
\(890\) − 1358.00i − 0.0511464i
\(891\) 1782.00i 0.0670025i
\(892\) 28504.0i 1.06994i
\(893\) −13860.0 −0.519381
\(894\) −1935.00 −0.0723894
\(895\) − 25718.0i − 0.960512i
\(896\) 14550.0 0.542502
\(897\) 0 0
\(898\) −7538.00 −0.280118
\(899\) − 22148.0i − 0.821665i
\(900\) 4788.00 0.177333
\(901\) 19869.0 0.734664
\(902\) 6270.00i 0.231450i
\(903\) 7380.00i 0.271972i
\(904\) − 16155.0i − 0.594366i
\(905\) 22981.0i 0.844104i
\(906\) −8742.00 −0.320567
\(907\) 11628.0 0.425691 0.212845 0.977086i \(-0.431727\pi\)
0.212845 + 0.977086i \(0.431727\pi\)
\(908\) − 40558.0i − 1.48234i
\(909\) 3861.00 0.140882
\(910\) 0 0
\(911\) −12584.0 −0.457658 −0.228829 0.973467i \(-0.573490\pi\)
−0.228829 + 0.973467i \(0.573490\pi\)
\(912\) 3690.00i 0.133978i
\(913\) −11396.0 −0.413092
\(914\) 15539.0 0.562346
\(915\) − 13335.0i − 0.481794i
\(916\) − 45374.0i − 1.63668i
\(917\) 5600.00i 0.201667i
\(918\) 999.000i 0.0359171i
\(919\) 17184.0 0.616809 0.308405 0.951255i \(-0.400205\pi\)
0.308405 + 0.951255i \(0.400205\pi\)
\(920\) 17010.0 0.609569
\(921\) 7266.00i 0.259960i
\(922\) 4811.00 0.171846
\(923\) 0 0
\(924\) −4620.00 −0.164488
\(925\) − 988.000i − 0.0351192i
\(926\) −562.000 −0.0199443
\(927\) 11718.0 0.415178
\(928\) 18193.0i 0.643550i
\(929\) 12777.0i 0.451238i 0.974216 + 0.225619i \(0.0724404\pi\)
−0.974216 + 0.225619i \(0.927560\pi\)
\(930\) 4116.00i 0.145128i
\(931\) 7290.00i 0.256627i
\(932\) −48230.0 −1.69509
\(933\) 10206.0 0.358124
\(934\) 4914.00i 0.172153i
\(935\) 5698.00 0.199299
\(936\) 0 0
\(937\) 9191.00 0.320445 0.160222 0.987081i \(-0.448779\pi\)
0.160222 + 0.987081i \(0.448779\pi\)
\(938\) 2020.00i 0.0703149i
\(939\) 6930.00 0.240843
\(940\) −22638.0 −0.785500
\(941\) 50498.0i 1.74940i 0.484662 + 0.874701i \(0.338942\pi\)
−0.484662 + 0.874701i \(0.661058\pi\)
\(942\) 6237.00i 0.215724i
\(943\) 46170.0i 1.59438i
\(944\) − 23616.0i − 0.814232i
\(945\) −1890.00 −0.0650600
\(946\) 5412.00 0.186003
\(947\) − 1560.00i − 0.0535303i −0.999642 0.0267651i \(-0.991479\pi\)
0.999642 0.0267651i \(-0.00852063\pi\)
\(948\) 18564.0 0.636003
\(949\) 0 0
\(950\) 2280.00 0.0778663
\(951\) − 771.000i − 0.0262896i
\(952\) −5550.00 −0.188946
\(953\) 21498.0 0.730733 0.365366 0.930864i \(-0.380944\pi\)
0.365366 + 0.930864i \(0.380944\pi\)
\(954\) 4833.00i 0.164019i
\(955\) − 4172.00i − 0.141364i
\(956\) 17262.0i 0.583988i
\(957\) − 7458.00i − 0.251915i
\(958\) 3600.00 0.121410
\(959\) −5190.00 −0.174759
\(960\) 3507.00i 0.117904i
\(961\) −8625.00 −0.289517
\(962\) 0 0
\(963\) −12042.0 −0.402957
\(964\) 25319.0i 0.845923i
\(965\) −2751.00 −0.0917698
\(966\) 4860.00 0.161872
\(967\) − 418.000i − 0.0139007i −0.999976 0.00695035i \(-0.997788\pi\)
0.999976 0.00695035i \(-0.00221238\pi\)
\(968\) − 12705.0i − 0.421853i
\(969\) − 3330.00i − 0.110397i
\(970\) − 8414.00i − 0.278513i
\(971\) 18132.0 0.599262 0.299631 0.954055i \(-0.403136\pi\)
0.299631 + 0.954055i \(0.403136\pi\)
\(972\) 1701.00 0.0561313
\(973\) − 3480.00i − 0.114659i
\(974\) −17130.0 −0.563532
\(975\) 0 0
\(976\) −26035.0 −0.853853
\(977\) 12501.0i 0.409358i 0.978829 + 0.204679i \(0.0656150\pi\)
−0.978829 + 0.204679i \(0.934385\pi\)
\(978\) −5100.00 −0.166748
\(979\) 4268.00 0.139332
\(980\) 11907.0i 0.388118i
\(981\) − 9306.00i − 0.302872i
\(982\) − 11838.0i − 0.384690i
\(983\) 43708.0i 1.41818i 0.705119 + 0.709089i \(0.250894\pi\)
−0.705119 + 0.709089i \(0.749106\pi\)
\(984\) 12825.0 0.415494
\(985\) 24654.0 0.797504
\(986\) − 4181.00i − 0.135041i
\(987\) −13860.0 −0.446979
\(988\) 0 0
\(989\) 39852.0 1.28131
\(990\) 1386.00i 0.0444949i
\(991\) −39614.0 −1.26981 −0.634904 0.772591i \(-0.718960\pi\)
−0.634904 + 0.772591i \(0.718960\pi\)
\(992\) 31556.0 1.00998
\(993\) 3084.00i 0.0985577i
\(994\) − 10860.0i − 0.346538i
\(995\) − 14126.0i − 0.450075i
\(996\) 10878.0i 0.346067i
\(997\) −36503.0 −1.15954 −0.579770 0.814780i \(-0.696858\pi\)
−0.579770 + 0.814780i \(0.696858\pi\)
\(998\) 8976.00 0.284700
\(999\) − 351.000i − 0.0111163i
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 507.4.b.d.337.1 2
13.2 odd 12 39.4.e.b.22.1 yes 2
13.5 odd 4 507.4.a.b.1.1 1
13.6 odd 12 39.4.e.b.16.1 2
13.8 odd 4 507.4.a.d.1.1 1
13.12 even 2 inner 507.4.b.d.337.2 2
39.2 even 12 117.4.g.a.100.1 2
39.5 even 4 1521.4.a.h.1.1 1
39.8 even 4 1521.4.a.e.1.1 1
39.32 even 12 117.4.g.a.55.1 2
52.15 even 12 624.4.q.c.529.1 2
52.19 even 12 624.4.q.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 13.6 odd 12
39.4.e.b.22.1 yes 2 13.2 odd 12
117.4.g.a.55.1 2 39.32 even 12
117.4.g.a.100.1 2 39.2 even 12
507.4.a.b.1.1 1 13.5 odd 4
507.4.a.d.1.1 1 13.8 odd 4
507.4.b.d.337.1 2 1.1 even 1 trivial
507.4.b.d.337.2 2 13.12 even 2 inner
624.4.q.c.289.1 2 52.19 even 12
624.4.q.c.529.1 2 52.15 even 12
1521.4.a.e.1.1 1 39.8 even 4
1521.4.a.h.1.1 1 39.5 even 4