Properties

Label 507.4.b.c.337.2
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{2} +3.00000 q^{3} -1.00000 q^{4} -9.00000i q^{5} +9.00000i q^{6} -2.00000i q^{7} +21.0000i q^{8} +9.00000 q^{9} +27.0000 q^{10} -30.0000i q^{11} -3.00000 q^{12} +6.00000 q^{14} -27.0000i q^{15} -71.0000 q^{16} +111.000 q^{17} +27.0000i q^{18} -46.0000i q^{19} +9.00000i q^{20} -6.00000i q^{21} +90.0000 q^{22} +6.00000 q^{23} +63.0000i q^{24} +44.0000 q^{25} +27.0000 q^{27} +2.00000i q^{28} -105.000 q^{29} +81.0000 q^{30} -100.000i q^{31} -45.0000i q^{32} -90.0000i q^{33} +333.000i q^{34} -18.0000 q^{35} -9.00000 q^{36} -17.0000i q^{37} +138.000 q^{38} +189.000 q^{40} -231.000i q^{41} +18.0000 q^{42} +514.000 q^{43} +30.0000i q^{44} -81.0000i q^{45} +18.0000i q^{46} +162.000i q^{47} -213.000 q^{48} +339.000 q^{49} +132.000i q^{50} +333.000 q^{51} +639.000 q^{53} +81.0000i q^{54} -270.000 q^{55} +42.0000 q^{56} -138.000i q^{57} -315.000i q^{58} -600.000i q^{59} +27.0000i q^{60} +233.000 q^{61} +300.000 q^{62} -18.0000i q^{63} -433.000 q^{64} +270.000 q^{66} +926.000i q^{67} -111.000 q^{68} +18.0000 q^{69} -54.0000i q^{70} -930.000i q^{71} +189.000i q^{72} +253.000i q^{73} +51.0000 q^{74} +132.000 q^{75} +46.0000i q^{76} -60.0000 q^{77} -1324.00 q^{79} +639.000i q^{80} +81.0000 q^{81} +693.000 q^{82} +810.000i q^{83} +6.00000i q^{84} -999.000i q^{85} +1542.00i q^{86} -315.000 q^{87} +630.000 q^{88} -498.000i q^{89} +243.000 q^{90} -6.00000 q^{92} -300.000i q^{93} -486.000 q^{94} -414.000 q^{95} -135.000i q^{96} +1358.00i q^{97} +1017.00i q^{98} -270.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{4} + 18 q^{9} + 54 q^{10} - 6 q^{12} + 12 q^{14} - 142 q^{16} + 222 q^{17} + 180 q^{22} + 12 q^{23} + 88 q^{25} + 54 q^{27} - 210 q^{29} + 162 q^{30} - 36 q^{35} - 18 q^{36} + 276 q^{38} + 378 q^{40} + 36 q^{42} + 1028 q^{43} - 426 q^{48} + 678 q^{49} + 666 q^{51} + 1278 q^{53} - 540 q^{55} + 84 q^{56} + 466 q^{61} + 600 q^{62} - 866 q^{64} + 540 q^{66} - 222 q^{68} + 36 q^{69} + 102 q^{74} + 264 q^{75} - 120 q^{77} - 2648 q^{79} + 162 q^{81} + 1386 q^{82} - 630 q^{87} + 1260 q^{88} + 486 q^{90} - 12 q^{92} - 972 q^{94} - 828 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000i 1.06066i 0.847791 + 0.530330i \(0.177932\pi\)
−0.847791 + 0.530330i \(0.822068\pi\)
\(3\) 3.00000 0.577350
\(4\) −1.00000 −0.125000
\(5\) − 9.00000i − 0.804984i −0.915423 0.402492i \(-0.868144\pi\)
0.915423 0.402492i \(-0.131856\pi\)
\(6\) 9.00000i 0.612372i
\(7\) − 2.00000i − 0.107990i −0.998541 0.0539949i \(-0.982805\pi\)
0.998541 0.0539949i \(-0.0171955\pi\)
\(8\) 21.0000i 0.928078i
\(9\) 9.00000 0.333333
\(10\) 27.0000 0.853815
\(11\) − 30.0000i − 0.822304i −0.911567 0.411152i \(-0.865127\pi\)
0.911567 0.411152i \(-0.134873\pi\)
\(12\) −3.00000 −0.0721688
\(13\) 0 0
\(14\) 6.00000 0.114541
\(15\) − 27.0000i − 0.464758i
\(16\) −71.0000 −1.10938
\(17\) 111.000 1.58361 0.791807 0.610771i \(-0.209140\pi\)
0.791807 + 0.610771i \(0.209140\pi\)
\(18\) 27.0000i 0.353553i
\(19\) − 46.0000i − 0.555428i −0.960664 0.277714i \(-0.910423\pi\)
0.960664 0.277714i \(-0.0895767\pi\)
\(20\) 9.00000i 0.100623i
\(21\) − 6.00000i − 0.0623480i
\(22\) 90.0000 0.872185
\(23\) 6.00000 0.0543951 0.0271975 0.999630i \(-0.491342\pi\)
0.0271975 + 0.999630i \(0.491342\pi\)
\(24\) 63.0000i 0.535826i
\(25\) 44.0000 0.352000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 2.00000i 0.0134987i
\(29\) −105.000 −0.672345 −0.336173 0.941800i \(-0.609133\pi\)
−0.336173 + 0.941800i \(0.609133\pi\)
\(30\) 81.0000 0.492950
\(31\) − 100.000i − 0.579372i −0.957122 0.289686i \(-0.906449\pi\)
0.957122 0.289686i \(-0.0935509\pi\)
\(32\) − 45.0000i − 0.248592i
\(33\) − 90.0000i − 0.474757i
\(34\) 333.000i 1.67968i
\(35\) −18.0000 −0.0869302
\(36\) −9.00000 −0.0416667
\(37\) − 17.0000i − 0.0755347i −0.999287 0.0377673i \(-0.987975\pi\)
0.999287 0.0377673i \(-0.0120246\pi\)
\(38\) 138.000 0.589120
\(39\) 0 0
\(40\) 189.000 0.747088
\(41\) − 231.000i − 0.879906i −0.898021 0.439953i \(-0.854995\pi\)
0.898021 0.439953i \(-0.145005\pi\)
\(42\) 18.0000 0.0661300
\(43\) 514.000 1.82289 0.911445 0.411422i \(-0.134968\pi\)
0.911445 + 0.411422i \(0.134968\pi\)
\(44\) 30.0000i 0.102788i
\(45\) − 81.0000i − 0.268328i
\(46\) 18.0000i 0.0576947i
\(47\) 162.000i 0.502769i 0.967887 + 0.251384i \(0.0808858\pi\)
−0.967887 + 0.251384i \(0.919114\pi\)
\(48\) −213.000 −0.640498
\(49\) 339.000 0.988338
\(50\) 132.000i 0.373352i
\(51\) 333.000 0.914301
\(52\) 0 0
\(53\) 639.000 1.65610 0.828051 0.560653i \(-0.189450\pi\)
0.828051 + 0.560653i \(0.189450\pi\)
\(54\) 81.0000i 0.204124i
\(55\) −270.000 −0.661942
\(56\) 42.0000 0.100223
\(57\) − 138.000i − 0.320676i
\(58\) − 315.000i − 0.713130i
\(59\) − 600.000i − 1.32396i −0.749524 0.661978i \(-0.769717\pi\)
0.749524 0.661978i \(-0.230283\pi\)
\(60\) 27.0000i 0.0580948i
\(61\) 233.000 0.489059 0.244529 0.969642i \(-0.421367\pi\)
0.244529 + 0.969642i \(0.421367\pi\)
\(62\) 300.000 0.614517
\(63\) − 18.0000i − 0.0359966i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) 270.000 0.503556
\(67\) 926.000i 1.68849i 0.535957 + 0.844246i \(0.319951\pi\)
−0.535957 + 0.844246i \(0.680049\pi\)
\(68\) −111.000 −0.197952
\(69\) 18.0000 0.0314050
\(70\) − 54.0000i − 0.0922033i
\(71\) − 930.000i − 1.55452i −0.629182 0.777258i \(-0.716610\pi\)
0.629182 0.777258i \(-0.283390\pi\)
\(72\) 189.000i 0.309359i
\(73\) 253.000i 0.405636i 0.979216 + 0.202818i \(0.0650099\pi\)
−0.979216 + 0.202818i \(0.934990\pi\)
\(74\) 51.0000 0.0801166
\(75\) 132.000 0.203227
\(76\) 46.0000i 0.0694284i
\(77\) −60.0000 −0.0888004
\(78\) 0 0
\(79\) −1324.00 −1.88559 −0.942795 0.333373i \(-0.891813\pi\)
−0.942795 + 0.333373i \(0.891813\pi\)
\(80\) 639.000i 0.893030i
\(81\) 81.0000 0.111111
\(82\) 693.000 0.933281
\(83\) 810.000i 1.07119i 0.844474 + 0.535597i \(0.179913\pi\)
−0.844474 + 0.535597i \(0.820087\pi\)
\(84\) 6.00000i 0.00779350i
\(85\) − 999.000i − 1.27479i
\(86\) 1542.00i 1.93347i
\(87\) −315.000 −0.388179
\(88\) 630.000 0.763162
\(89\) − 498.000i − 0.593122i −0.955014 0.296561i \(-0.904160\pi\)
0.955014 0.296561i \(-0.0958399\pi\)
\(90\) 243.000 0.284605
\(91\) 0 0
\(92\) −6.00000 −0.00679938
\(93\) − 300.000i − 0.334501i
\(94\) −486.000 −0.533267
\(95\) −414.000 −0.447111
\(96\) − 135.000i − 0.143525i
\(97\) 1358.00i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 1017.00i 1.04829i
\(99\) − 270.000i − 0.274101i
\(100\) −44.0000 −0.0440000
\(101\) 357.000 0.351711 0.175856 0.984416i \(-0.443731\pi\)
0.175856 + 0.984416i \(0.443731\pi\)
\(102\) 999.000i 0.969762i
\(103\) −1118.00 −1.06951 −0.534756 0.845006i \(-0.679597\pi\)
−0.534756 + 0.845006i \(0.679597\pi\)
\(104\) 0 0
\(105\) −54.0000 −0.0501891
\(106\) 1917.00i 1.75656i
\(107\) 714.000 0.645093 0.322547 0.946554i \(-0.395461\pi\)
0.322547 + 0.946554i \(0.395461\pi\)
\(108\) −27.0000 −0.0240563
\(109\) 2006.00i 1.76275i 0.472416 + 0.881376i \(0.343382\pi\)
−0.472416 + 0.881376i \(0.656618\pi\)
\(110\) − 810.000i − 0.702095i
\(111\) − 51.0000i − 0.0436100i
\(112\) 142.000i 0.119801i
\(113\) −1119.00 −0.931563 −0.465782 0.884900i \(-0.654227\pi\)
−0.465782 + 0.884900i \(0.654227\pi\)
\(114\) 414.000 0.340129
\(115\) − 54.0000i − 0.0437872i
\(116\) 105.000 0.0840431
\(117\) 0 0
\(118\) 1800.00 1.40427
\(119\) − 222.000i − 0.171014i
\(120\) 567.000 0.431332
\(121\) 431.000 0.323817
\(122\) 699.000i 0.518725i
\(123\) − 693.000i − 0.508014i
\(124\) 100.000i 0.0724215i
\(125\) − 1521.00i − 1.08834i
\(126\) 54.0000 0.0381802
\(127\) 604.000 0.422018 0.211009 0.977484i \(-0.432325\pi\)
0.211009 + 0.977484i \(0.432325\pi\)
\(128\) − 1659.00i − 1.14560i
\(129\) 1542.00 1.05245
\(130\) 0 0
\(131\) −1584.00 −1.05645 −0.528224 0.849105i \(-0.677142\pi\)
−0.528224 + 0.849105i \(0.677142\pi\)
\(132\) 90.0000i 0.0593447i
\(133\) −92.0000 −0.0599805
\(134\) −2778.00 −1.79092
\(135\) − 243.000i − 0.154919i
\(136\) 2331.00i 1.46972i
\(137\) − 717.000i − 0.447135i −0.974688 0.223567i \(-0.928230\pi\)
0.974688 0.223567i \(-0.0717703\pi\)
\(138\) 54.0000i 0.0333100i
\(139\) −820.000 −0.500370 −0.250185 0.968198i \(-0.580492\pi\)
−0.250185 + 0.968198i \(0.580492\pi\)
\(140\) 18.0000 0.0108663
\(141\) 486.000i 0.290274i
\(142\) 2790.00 1.64881
\(143\) 0 0
\(144\) −639.000 −0.369792
\(145\) 945.000i 0.541227i
\(146\) −759.000 −0.430242
\(147\) 1017.00 0.570617
\(148\) 17.0000i 0.00944183i
\(149\) − 1749.00i − 0.961635i −0.876821 0.480818i \(-0.840340\pi\)
0.876821 0.480818i \(-0.159660\pi\)
\(150\) 396.000i 0.215555i
\(151\) 370.000i 0.199405i 0.995017 + 0.0997026i \(0.0317891\pi\)
−0.995017 + 0.0997026i \(0.968211\pi\)
\(152\) 966.000 0.515480
\(153\) 999.000 0.527872
\(154\) − 180.000i − 0.0941871i
\(155\) −900.000 −0.466385
\(156\) 0 0
\(157\) −2611.00 −1.32726 −0.663632 0.748059i \(-0.730986\pi\)
−0.663632 + 0.748059i \(0.730986\pi\)
\(158\) − 3972.00i − 1.99997i
\(159\) 1917.00 0.956151
\(160\) −405.000 −0.200113
\(161\) − 12.0000i − 0.00587411i
\(162\) 243.000i 0.117851i
\(163\) 1636.00i 0.786144i 0.919508 + 0.393072i \(0.128588\pi\)
−0.919508 + 0.393072i \(0.871412\pi\)
\(164\) 231.000i 0.109988i
\(165\) −810.000 −0.382172
\(166\) −2430.00 −1.13617
\(167\) − 264.000i − 0.122329i −0.998128 0.0611645i \(-0.980519\pi\)
0.998128 0.0611645i \(-0.0194814\pi\)
\(168\) 126.000 0.0578638
\(169\) 0 0
\(170\) 2997.00 1.35211
\(171\) − 414.000i − 0.185143i
\(172\) −514.000 −0.227861
\(173\) −1410.00 −0.619655 −0.309827 0.950793i \(-0.600271\pi\)
−0.309827 + 0.950793i \(0.600271\pi\)
\(174\) − 945.000i − 0.411726i
\(175\) − 88.0000i − 0.0380124i
\(176\) 2130.00i 0.912243i
\(177\) − 1800.00i − 0.764386i
\(178\) 1494.00 0.629101
\(179\) 474.000 0.197924 0.0989621 0.995091i \(-0.468448\pi\)
0.0989621 + 0.995091i \(0.468448\pi\)
\(180\) 81.0000i 0.0335410i
\(181\) −2249.00 −0.923574 −0.461787 0.886991i \(-0.652792\pi\)
−0.461787 + 0.886991i \(0.652792\pi\)
\(182\) 0 0
\(183\) 699.000 0.282358
\(184\) 126.000i 0.0504828i
\(185\) −153.000 −0.0608042
\(186\) 900.000 0.354791
\(187\) − 3330.00i − 1.30221i
\(188\) − 162.000i − 0.0628461i
\(189\) − 54.0000i − 0.0207827i
\(190\) − 1242.00i − 0.474232i
\(191\) 3444.00 1.30471 0.652354 0.757915i \(-0.273782\pi\)
0.652354 + 0.757915i \(0.273782\pi\)
\(192\) −1299.00 −0.488267
\(193\) 4273.00i 1.59366i 0.604201 + 0.796832i \(0.293493\pi\)
−0.604201 + 0.796832i \(0.706507\pi\)
\(194\) −4074.00 −1.50771
\(195\) 0 0
\(196\) −339.000 −0.123542
\(197\) − 1986.00i − 0.718257i −0.933288 0.359129i \(-0.883074\pi\)
0.933288 0.359129i \(-0.116926\pi\)
\(198\) 810.000 0.290728
\(199\) 2386.00 0.849945 0.424973 0.905206i \(-0.360284\pi\)
0.424973 + 0.905206i \(0.360284\pi\)
\(200\) 924.000i 0.326683i
\(201\) 2778.00i 0.974851i
\(202\) 1071.00i 0.373046i
\(203\) 210.000i 0.0726065i
\(204\) −333.000 −0.114288
\(205\) −2079.00 −0.708311
\(206\) − 3354.00i − 1.13439i
\(207\) 54.0000 0.0181317
\(208\) 0 0
\(209\) −1380.00 −0.456730
\(210\) − 162.000i − 0.0532336i
\(211\) −1600.00 −0.522031 −0.261016 0.965335i \(-0.584057\pi\)
−0.261016 + 0.965335i \(0.584057\pi\)
\(212\) −639.000 −0.207013
\(213\) − 2790.00i − 0.897501i
\(214\) 2142.00i 0.684225i
\(215\) − 4626.00i − 1.46740i
\(216\) 567.000i 0.178609i
\(217\) −200.000 −0.0625663
\(218\) −6018.00 −1.86968
\(219\) 759.000i 0.234194i
\(220\) 270.000 0.0827427
\(221\) 0 0
\(222\) 153.000 0.0462553
\(223\) − 3832.00i − 1.15072i −0.817902 0.575358i \(-0.804863\pi\)
0.817902 0.575358i \(-0.195137\pi\)
\(224\) −90.0000 −0.0268454
\(225\) 396.000 0.117333
\(226\) − 3357.00i − 0.988072i
\(227\) − 1398.00i − 0.408760i −0.978892 0.204380i \(-0.934482\pi\)
0.978892 0.204380i \(-0.0655178\pi\)
\(228\) 138.000i 0.0400845i
\(229\) − 4466.00i − 1.28874i −0.764714 0.644370i \(-0.777120\pi\)
0.764714 0.644370i \(-0.222880\pi\)
\(230\) 162.000 0.0464433
\(231\) −180.000 −0.0512690
\(232\) − 2205.00i − 0.623989i
\(233\) 1638.00 0.460553 0.230277 0.973125i \(-0.426037\pi\)
0.230277 + 0.973125i \(0.426037\pi\)
\(234\) 0 0
\(235\) 1458.00 0.404721
\(236\) 600.000i 0.165494i
\(237\) −3972.00 −1.08865
\(238\) 666.000 0.181388
\(239\) − 594.000i − 0.160764i −0.996764 0.0803821i \(-0.974386\pi\)
0.996764 0.0803821i \(-0.0256141\pi\)
\(240\) 1917.00i 0.515591i
\(241\) − 2303.00i − 0.615557i −0.951458 0.307779i \(-0.900414\pi\)
0.951458 0.307779i \(-0.0995856\pi\)
\(242\) 1293.00i 0.343459i
\(243\) 243.000 0.0641500
\(244\) −233.000 −0.0611324
\(245\) − 3051.00i − 0.795597i
\(246\) 2079.00 0.538830
\(247\) 0 0
\(248\) 2100.00 0.537702
\(249\) 2430.00i 0.618454i
\(250\) 4563.00 1.15436
\(251\) −6324.00 −1.59031 −0.795154 0.606407i \(-0.792610\pi\)
−0.795154 + 0.606407i \(0.792610\pi\)
\(252\) 18.0000i 0.00449958i
\(253\) − 180.000i − 0.0447293i
\(254\) 1812.00i 0.447618i
\(255\) − 2997.00i − 0.735998i
\(256\) 1513.00 0.369385
\(257\) −7833.00 −1.90120 −0.950601 0.310414i \(-0.899532\pi\)
−0.950601 + 0.310414i \(0.899532\pi\)
\(258\) 4626.00i 1.11629i
\(259\) −34.0000 −0.00815698
\(260\) 0 0
\(261\) −945.000 −0.224115
\(262\) − 4752.00i − 1.12053i
\(263\) −3030.00 −0.710410 −0.355205 0.934788i \(-0.615589\pi\)
−0.355205 + 0.934788i \(0.615589\pi\)
\(264\) 1890.00 0.440612
\(265\) − 5751.00i − 1.33314i
\(266\) − 276.000i − 0.0636190i
\(267\) − 1494.00i − 0.342439i
\(268\) − 926.000i − 0.211061i
\(269\) −534.000 −0.121036 −0.0605178 0.998167i \(-0.519275\pi\)
−0.0605178 + 0.998167i \(0.519275\pi\)
\(270\) 729.000 0.164317
\(271\) 3688.00i 0.826679i 0.910577 + 0.413340i \(0.135638\pi\)
−0.910577 + 0.413340i \(0.864362\pi\)
\(272\) −7881.00 −1.75682
\(273\) 0 0
\(274\) 2151.00 0.474258
\(275\) − 1320.00i − 0.289451i
\(276\) −18.0000 −0.00392563
\(277\) −1865.00 −0.404538 −0.202269 0.979330i \(-0.564832\pi\)
−0.202269 + 0.979330i \(0.564832\pi\)
\(278\) − 2460.00i − 0.530723i
\(279\) − 900.000i − 0.193124i
\(280\) − 378.000i − 0.0806779i
\(281\) − 2997.00i − 0.636249i −0.948049 0.318125i \(-0.896947\pi\)
0.948049 0.318125i \(-0.103053\pi\)
\(282\) −1458.00 −0.307882
\(283\) 4114.00 0.864141 0.432071 0.901840i \(-0.357783\pi\)
0.432071 + 0.901840i \(0.357783\pi\)
\(284\) 930.000i 0.194315i
\(285\) −1242.00 −0.258139
\(286\) 0 0
\(287\) −462.000 −0.0950209
\(288\) − 405.000i − 0.0828641i
\(289\) 7408.00 1.50784
\(290\) −2835.00 −0.574058
\(291\) 4074.00i 0.820695i
\(292\) − 253.000i − 0.0507045i
\(293\) 4665.00i 0.930144i 0.885273 + 0.465072i \(0.153972\pi\)
−0.885273 + 0.465072i \(0.846028\pi\)
\(294\) 3051.00i 0.605231i
\(295\) −5400.00 −1.06576
\(296\) 357.000 0.0701020
\(297\) − 810.000i − 0.158252i
\(298\) 5247.00 1.01997
\(299\) 0 0
\(300\) −132.000 −0.0254034
\(301\) − 1028.00i − 0.196854i
\(302\) −1110.00 −0.211501
\(303\) 1071.00 0.203061
\(304\) 3266.00i 0.616177i
\(305\) − 2097.00i − 0.393685i
\(306\) 2997.00i 0.559892i
\(307\) − 1502.00i − 0.279230i −0.990206 0.139615i \(-0.955413\pi\)
0.990206 0.139615i \(-0.0445865\pi\)
\(308\) 60.0000 0.0111001
\(309\) −3354.00 −0.617483
\(310\) − 2700.00i − 0.494676i
\(311\) −2106.00 −0.383988 −0.191994 0.981396i \(-0.561495\pi\)
−0.191994 + 0.981396i \(0.561495\pi\)
\(312\) 0 0
\(313\) −3898.00 −0.703923 −0.351962 0.936014i \(-0.614485\pi\)
−0.351962 + 0.936014i \(0.614485\pi\)
\(314\) − 7833.00i − 1.40778i
\(315\) −162.000 −0.0289767
\(316\) 1324.00 0.235699
\(317\) 9351.00i 1.65680i 0.560140 + 0.828398i \(0.310747\pi\)
−0.560140 + 0.828398i \(0.689253\pi\)
\(318\) 5751.00i 1.01415i
\(319\) 3150.00i 0.552872i
\(320\) 3897.00i 0.680778i
\(321\) 2142.00 0.372445
\(322\) 36.0000 0.00623044
\(323\) − 5106.00i − 0.879583i
\(324\) −81.0000 −0.0138889
\(325\) 0 0
\(326\) −4908.00 −0.833831
\(327\) 6018.00i 1.01773i
\(328\) 4851.00 0.816621
\(329\) 324.000 0.0542939
\(330\) − 2430.00i − 0.405355i
\(331\) − 9172.00i − 1.52308i −0.648119 0.761539i \(-0.724444\pi\)
0.648119 0.761539i \(-0.275556\pi\)
\(332\) − 810.000i − 0.133899i
\(333\) − 153.000i − 0.0251782i
\(334\) 792.000 0.129749
\(335\) 8334.00 1.35921
\(336\) 426.000i 0.0691673i
\(337\) 11089.0 1.79245 0.896226 0.443598i \(-0.146298\pi\)
0.896226 + 0.443598i \(0.146298\pi\)
\(338\) 0 0
\(339\) −3357.00 −0.537838
\(340\) 999.000i 0.159348i
\(341\) −3000.00 −0.476420
\(342\) 1242.00 0.196373
\(343\) − 1364.00i − 0.214720i
\(344\) 10794.0i 1.69178i
\(345\) − 162.000i − 0.0252805i
\(346\) − 4230.00i − 0.657243i
\(347\) 9762.00 1.51024 0.755118 0.655589i \(-0.227580\pi\)
0.755118 + 0.655589i \(0.227580\pi\)
\(348\) 315.000 0.0485223
\(349\) 8290.00i 1.27150i 0.771895 + 0.635750i \(0.219309\pi\)
−0.771895 + 0.635750i \(0.780691\pi\)
\(350\) 264.000 0.0403183
\(351\) 0 0
\(352\) −1350.00 −0.204418
\(353\) 12405.0i 1.87040i 0.354119 + 0.935200i \(0.384781\pi\)
−0.354119 + 0.935200i \(0.615219\pi\)
\(354\) 5400.00 0.810754
\(355\) −8370.00 −1.25136
\(356\) 498.000i 0.0741403i
\(357\) − 666.000i − 0.0987352i
\(358\) 1422.00i 0.209930i
\(359\) 1098.00i 0.161421i 0.996738 + 0.0807106i \(0.0257190\pi\)
−0.996738 + 0.0807106i \(0.974281\pi\)
\(360\) 1701.00 0.249029
\(361\) 4743.00 0.691500
\(362\) − 6747.00i − 0.979598i
\(363\) 1293.00 0.186956
\(364\) 0 0
\(365\) 2277.00 0.326530
\(366\) 2097.00i 0.299486i
\(367\) −5734.00 −0.815565 −0.407783 0.913079i \(-0.633698\pi\)
−0.407783 + 0.913079i \(0.633698\pi\)
\(368\) −426.000 −0.0603445
\(369\) − 2079.00i − 0.293302i
\(370\) − 459.000i − 0.0644926i
\(371\) − 1278.00i − 0.178842i
\(372\) 300.000i 0.0418126i
\(373\) −8971.00 −1.24531 −0.622655 0.782496i \(-0.713946\pi\)
−0.622655 + 0.782496i \(0.713946\pi\)
\(374\) 9990.00 1.38120
\(375\) − 4563.00i − 0.628353i
\(376\) −3402.00 −0.466608
\(377\) 0 0
\(378\) 162.000 0.0220433
\(379\) 7244.00i 0.981792i 0.871218 + 0.490896i \(0.163331\pi\)
−0.871218 + 0.490896i \(0.836669\pi\)
\(380\) 414.000 0.0558888
\(381\) 1812.00 0.243652
\(382\) 10332.0i 1.38385i
\(383\) − 6312.00i − 0.842110i −0.907035 0.421055i \(-0.861660\pi\)
0.907035 0.421055i \(-0.138340\pi\)
\(384\) − 4977.00i − 0.661410i
\(385\) 540.000i 0.0714830i
\(386\) −12819.0 −1.69034
\(387\) 4626.00 0.607630
\(388\) − 1358.00i − 0.177686i
\(389\) −3627.00 −0.472741 −0.236370 0.971663i \(-0.575958\pi\)
−0.236370 + 0.971663i \(0.575958\pi\)
\(390\) 0 0
\(391\) 666.000 0.0861408
\(392\) 7119.00i 0.917255i
\(393\) −4752.00 −0.609941
\(394\) 5958.00 0.761827
\(395\) 11916.0i 1.51787i
\(396\) 270.000i 0.0342627i
\(397\) 3898.00i 0.492783i 0.969170 + 0.246392i \(0.0792450\pi\)
−0.969170 + 0.246392i \(0.920755\pi\)
\(398\) 7158.00i 0.901503i
\(399\) −276.000 −0.0346298
\(400\) −3124.00 −0.390500
\(401\) 5703.00i 0.710210i 0.934826 + 0.355105i \(0.115555\pi\)
−0.934826 + 0.355105i \(0.884445\pi\)
\(402\) −8334.00 −1.03399
\(403\) 0 0
\(404\) −357.000 −0.0439639
\(405\) − 729.000i − 0.0894427i
\(406\) −630.000 −0.0770108
\(407\) −510.000 −0.0621124
\(408\) 6993.00i 0.848542i
\(409\) 6311.00i 0.762980i 0.924373 + 0.381490i \(0.124589\pi\)
−0.924373 + 0.381490i \(0.875411\pi\)
\(410\) − 6237.00i − 0.751277i
\(411\) − 2151.00i − 0.258153i
\(412\) 1118.00 0.133689
\(413\) −1200.00 −0.142974
\(414\) 162.000i 0.0192316i
\(415\) 7290.00 0.862294
\(416\) 0 0
\(417\) −2460.00 −0.288889
\(418\) − 4140.00i − 0.484435i
\(419\) −2328.00 −0.271433 −0.135716 0.990748i \(-0.543334\pi\)
−0.135716 + 0.990748i \(0.543334\pi\)
\(420\) 54.0000 0.00627364
\(421\) 2045.00i 0.236739i 0.992970 + 0.118370i \(0.0377668\pi\)
−0.992970 + 0.118370i \(0.962233\pi\)
\(422\) − 4800.00i − 0.553697i
\(423\) 1458.00i 0.167590i
\(424\) 13419.0i 1.53699i
\(425\) 4884.00 0.557432
\(426\) 8370.00 0.951943
\(427\) − 466.000i − 0.0528134i
\(428\) −714.000 −0.0806367
\(429\) 0 0
\(430\) 13878.0 1.55641
\(431\) 5034.00i 0.562597i 0.959620 + 0.281298i \(0.0907651\pi\)
−0.959620 + 0.281298i \(0.909235\pi\)
\(432\) −1917.00 −0.213499
\(433\) −4283.00 −0.475353 −0.237676 0.971344i \(-0.576386\pi\)
−0.237676 + 0.971344i \(0.576386\pi\)
\(434\) − 600.000i − 0.0663616i
\(435\) 2835.00i 0.312478i
\(436\) − 2006.00i − 0.220344i
\(437\) − 276.000i − 0.0302125i
\(438\) −2277.00 −0.248400
\(439\) 1306.00 0.141986 0.0709931 0.997477i \(-0.477383\pi\)
0.0709931 + 0.997477i \(0.477383\pi\)
\(440\) − 5670.00i − 0.614333i
\(441\) 3051.00 0.329446
\(442\) 0 0
\(443\) −5796.00 −0.621617 −0.310808 0.950473i \(-0.600600\pi\)
−0.310808 + 0.950473i \(0.600600\pi\)
\(444\) 51.0000i 0.00545125i
\(445\) −4482.00 −0.477454
\(446\) 11496.0 1.22052
\(447\) − 5247.00i − 0.555200i
\(448\) 866.000i 0.0913274i
\(449\) − 2706.00i − 0.284419i −0.989837 0.142209i \(-0.954579\pi\)
0.989837 0.142209i \(-0.0454206\pi\)
\(450\) 1188.00i 0.124451i
\(451\) −6930.00 −0.723550
\(452\) 1119.00 0.116445
\(453\) 1110.00i 0.115127i
\(454\) 4194.00 0.433555
\(455\) 0 0
\(456\) 2898.00 0.297612
\(457\) − 829.000i − 0.0848555i −0.999100 0.0424278i \(-0.986491\pi\)
0.999100 0.0424278i \(-0.0135092\pi\)
\(458\) 13398.0 1.36692
\(459\) 2997.00 0.304767
\(460\) 54.0000i 0.00547340i
\(461\) − 5493.00i − 0.554956i −0.960732 0.277478i \(-0.910502\pi\)
0.960732 0.277478i \(-0.0894985\pi\)
\(462\) − 540.000i − 0.0543789i
\(463\) 15346.0i 1.54037i 0.637823 + 0.770183i \(0.279835\pi\)
−0.637823 + 0.770183i \(0.720165\pi\)
\(464\) 7455.00 0.745883
\(465\) −2700.00 −0.269268
\(466\) 4914.00i 0.488491i
\(467\) 9594.00 0.950658 0.475329 0.879808i \(-0.342329\pi\)
0.475329 + 0.879808i \(0.342329\pi\)
\(468\) 0 0
\(469\) 1852.00 0.182340
\(470\) 4374.00i 0.429271i
\(471\) −7833.00 −0.766296
\(472\) 12600.0 1.22873
\(473\) − 15420.0i − 1.49897i
\(474\) − 11916.0i − 1.15468i
\(475\) − 2024.00i − 0.195511i
\(476\) 222.000i 0.0213768i
\(477\) 5751.00 0.552034
\(478\) 1782.00 0.170516
\(479\) 12840.0i 1.22479i 0.790552 + 0.612395i \(0.209794\pi\)
−0.790552 + 0.612395i \(0.790206\pi\)
\(480\) −1215.00 −0.115535
\(481\) 0 0
\(482\) 6909.00 0.652897
\(483\) − 36.0000i − 0.00339142i
\(484\) −431.000 −0.0404771
\(485\) 12222.0 1.14427
\(486\) 729.000i 0.0680414i
\(487\) − 14086.0i − 1.31067i −0.755337 0.655336i \(-0.772527\pi\)
0.755337 0.655336i \(-0.227473\pi\)
\(488\) 4893.00i 0.453885i
\(489\) 4908.00i 0.453880i
\(490\) 9153.00 0.843858
\(491\) −11694.0 −1.07483 −0.537416 0.843317i \(-0.680600\pi\)
−0.537416 + 0.843317i \(0.680600\pi\)
\(492\) 693.000i 0.0635017i
\(493\) −11655.0 −1.06474
\(494\) 0 0
\(495\) −2430.00 −0.220647
\(496\) 7100.00i 0.642741i
\(497\) −1860.00 −0.167872
\(498\) −7290.00 −0.655969
\(499\) − 3688.00i − 0.330857i −0.986222 0.165428i \(-0.947099\pi\)
0.986222 0.165428i \(-0.0529007\pi\)
\(500\) 1521.00i 0.136042i
\(501\) − 792.000i − 0.0706266i
\(502\) − 18972.0i − 1.68678i
\(503\) −4746.00 −0.420703 −0.210352 0.977626i \(-0.567461\pi\)
−0.210352 + 0.977626i \(0.567461\pi\)
\(504\) 378.000 0.0334077
\(505\) − 3213.00i − 0.283122i
\(506\) 540.000 0.0474425
\(507\) 0 0
\(508\) −604.000 −0.0527523
\(509\) − 14505.0i − 1.26311i −0.775331 0.631555i \(-0.782417\pi\)
0.775331 0.631555i \(-0.217583\pi\)
\(510\) 8991.00 0.780643
\(511\) 506.000 0.0438045
\(512\) − 8733.00i − 0.753804i
\(513\) − 1242.00i − 0.106892i
\(514\) − 23499.0i − 2.01653i
\(515\) 10062.0i 0.860941i
\(516\) −1542.00 −0.131556
\(517\) 4860.00 0.413429
\(518\) − 102.000i − 0.00865178i
\(519\) −4230.00 −0.357758
\(520\) 0 0
\(521\) 5085.00 0.427597 0.213798 0.976878i \(-0.431416\pi\)
0.213798 + 0.976878i \(0.431416\pi\)
\(522\) − 2835.00i − 0.237710i
\(523\) −10882.0 −0.909821 −0.454911 0.890537i \(-0.650329\pi\)
−0.454911 + 0.890537i \(0.650329\pi\)
\(524\) 1584.00 0.132056
\(525\) − 264.000i − 0.0219465i
\(526\) − 9090.00i − 0.753503i
\(527\) − 11100.0i − 0.917502i
\(528\) 6390.00i 0.526684i
\(529\) −12131.0 −0.997041
\(530\) 17253.0 1.41400
\(531\) − 5400.00i − 0.441318i
\(532\) 92.0000 0.00749757
\(533\) 0 0
\(534\) 4482.00 0.363212
\(535\) − 6426.00i − 0.519290i
\(536\) −19446.0 −1.56705
\(537\) 1422.00 0.114272
\(538\) − 1602.00i − 0.128378i
\(539\) − 10170.0i − 0.812714i
\(540\) 243.000i 0.0193649i
\(541\) 4699.00i 0.373430i 0.982414 + 0.186715i \(0.0597841\pi\)
−0.982414 + 0.186715i \(0.940216\pi\)
\(542\) −11064.0 −0.876826
\(543\) −6747.00 −0.533226
\(544\) − 4995.00i − 0.393674i
\(545\) 18054.0 1.41899
\(546\) 0 0
\(547\) 8270.00 0.646434 0.323217 0.946325i \(-0.395236\pi\)
0.323217 + 0.946325i \(0.395236\pi\)
\(548\) 717.000i 0.0558918i
\(549\) 2097.00 0.163020
\(550\) 3960.00 0.307009
\(551\) 4830.00i 0.373439i
\(552\) 378.000i 0.0291463i
\(553\) 2648.00i 0.203625i
\(554\) − 5595.00i − 0.429077i
\(555\) −459.000 −0.0351053
\(556\) 820.000 0.0625463
\(557\) 22785.0i 1.73327i 0.498943 + 0.866635i \(0.333722\pi\)
−0.498943 + 0.866635i \(0.666278\pi\)
\(558\) 2700.00 0.204839
\(559\) 0 0
\(560\) 1278.00 0.0964381
\(561\) − 9990.00i − 0.751833i
\(562\) 8991.00 0.674844
\(563\) 11928.0 0.892905 0.446452 0.894807i \(-0.352687\pi\)
0.446452 + 0.894807i \(0.352687\pi\)
\(564\) − 486.000i − 0.0362842i
\(565\) 10071.0i 0.749894i
\(566\) 12342.0i 0.916560i
\(567\) − 162.000i − 0.0119989i
\(568\) 19530.0 1.44271
\(569\) 7962.00 0.586616 0.293308 0.956018i \(-0.405244\pi\)
0.293308 + 0.956018i \(0.405244\pi\)
\(570\) − 3726.00i − 0.273798i
\(571\) −20618.0 −1.51110 −0.755549 0.655093i \(-0.772630\pi\)
−0.755549 + 0.655093i \(0.772630\pi\)
\(572\) 0 0
\(573\) 10332.0 0.753273
\(574\) − 1386.00i − 0.100785i
\(575\) 264.000 0.0191471
\(576\) −3897.00 −0.281901
\(577\) − 3493.00i − 0.252020i −0.992029 0.126010i \(-0.959783\pi\)
0.992029 0.126010i \(-0.0402171\pi\)
\(578\) 22224.0i 1.59930i
\(579\) 12819.0i 0.920103i
\(580\) − 945.000i − 0.0676534i
\(581\) 1620.00 0.115678
\(582\) −12222.0 −0.870478
\(583\) − 19170.0i − 1.36182i
\(584\) −5313.00 −0.376461
\(585\) 0 0
\(586\) −13995.0 −0.986567
\(587\) 10416.0i 0.732392i 0.930538 + 0.366196i \(0.119340\pi\)
−0.930538 + 0.366196i \(0.880660\pi\)
\(588\) −1017.00 −0.0713272
\(589\) −4600.00 −0.321799
\(590\) − 16200.0i − 1.13041i
\(591\) − 5958.00i − 0.414686i
\(592\) 1207.00i 0.0837963i
\(593\) − 2061.00i − 0.142724i −0.997450 0.0713618i \(-0.977266\pi\)
0.997450 0.0713618i \(-0.0227345\pi\)
\(594\) 2430.00 0.167852
\(595\) −1998.00 −0.137664
\(596\) 1749.00i 0.120204i
\(597\) 7158.00 0.490716
\(598\) 0 0
\(599\) 12456.0 0.849647 0.424823 0.905276i \(-0.360336\pi\)
0.424823 + 0.905276i \(0.360336\pi\)
\(600\) 2772.00i 0.188611i
\(601\) −781.000 −0.0530077 −0.0265039 0.999649i \(-0.508437\pi\)
−0.0265039 + 0.999649i \(0.508437\pi\)
\(602\) 3084.00 0.208795
\(603\) 8334.00i 0.562830i
\(604\) − 370.000i − 0.0249256i
\(605\) − 3879.00i − 0.260667i
\(606\) 3213.00i 0.215378i
\(607\) 19304.0 1.29082 0.645408 0.763838i \(-0.276687\pi\)
0.645408 + 0.763838i \(0.276687\pi\)
\(608\) −2070.00 −0.138075
\(609\) 630.000i 0.0419194i
\(610\) 6291.00 0.417566
\(611\) 0 0
\(612\) −999.000 −0.0659840
\(613\) 12041.0i 0.793363i 0.917956 + 0.396681i \(0.129838\pi\)
−0.917956 + 0.396681i \(0.870162\pi\)
\(614\) 4506.00 0.296168
\(615\) −6237.00 −0.408943
\(616\) − 1260.00i − 0.0824137i
\(617\) 9717.00i 0.634022i 0.948422 + 0.317011i \(0.102679\pi\)
−0.948422 + 0.317011i \(0.897321\pi\)
\(618\) − 10062.0i − 0.654940i
\(619\) 21040.0i 1.36619i 0.730332 + 0.683093i \(0.239366\pi\)
−0.730332 + 0.683093i \(0.760634\pi\)
\(620\) 900.000 0.0582982
\(621\) 162.000 0.0104683
\(622\) − 6318.00i − 0.407281i
\(623\) −996.000 −0.0640512
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) − 11694.0i − 0.746623i
\(627\) −4140.00 −0.263693
\(628\) 2611.00 0.165908
\(629\) − 1887.00i − 0.119618i
\(630\) − 486.000i − 0.0307344i
\(631\) 5068.00i 0.319737i 0.987138 + 0.159868i \(0.0511070\pi\)
−0.987138 + 0.159868i \(0.948893\pi\)
\(632\) − 27804.0i − 1.74997i
\(633\) −4800.00 −0.301395
\(634\) −28053.0 −1.75730
\(635\) − 5436.00i − 0.339718i
\(636\) −1917.00 −0.119519
\(637\) 0 0
\(638\) −9450.00 −0.586409
\(639\) − 8370.00i − 0.518172i
\(640\) −14931.0 −0.922187
\(641\) −10185.0 −0.627587 −0.313794 0.949491i \(-0.601600\pi\)
−0.313794 + 0.949491i \(0.601600\pi\)
\(642\) 6426.00i 0.395037i
\(643\) 25928.0i 1.59020i 0.606476 + 0.795101i \(0.292582\pi\)
−0.606476 + 0.795101i \(0.707418\pi\)
\(644\) 12.0000i 0 0.000734264i
\(645\) − 13878.0i − 0.847203i
\(646\) 15318.0 0.932939
\(647\) −23160.0 −1.40729 −0.703643 0.710554i \(-0.748444\pi\)
−0.703643 + 0.710554i \(0.748444\pi\)
\(648\) 1701.00i 0.103120i
\(649\) −18000.0 −1.08869
\(650\) 0 0
\(651\) −600.000 −0.0361227
\(652\) − 1636.00i − 0.0982680i
\(653\) 16626.0 0.996364 0.498182 0.867073i \(-0.334001\pi\)
0.498182 + 0.867073i \(0.334001\pi\)
\(654\) −18054.0 −1.07946
\(655\) 14256.0i 0.850424i
\(656\) 16401.0i 0.976146i
\(657\) 2277.00i 0.135212i
\(658\) 972.000i 0.0575874i
\(659\) −14808.0 −0.875323 −0.437661 0.899140i \(-0.644193\pi\)
−0.437661 + 0.899140i \(0.644193\pi\)
\(660\) 810.000 0.0477715
\(661\) − 4853.00i − 0.285567i −0.989754 0.142784i \(-0.954395\pi\)
0.989754 0.142784i \(-0.0456053\pi\)
\(662\) 27516.0 1.61547
\(663\) 0 0
\(664\) −17010.0 −0.994151
\(665\) 828.000i 0.0482834i
\(666\) 459.000 0.0267055
\(667\) −630.000 −0.0365723
\(668\) 264.000i 0.0152911i
\(669\) − 11496.0i − 0.664366i
\(670\) 25002.0i 1.44166i
\(671\) − 6990.00i − 0.402155i
\(672\) −270.000 −0.0154992
\(673\) 16165.0 0.925877 0.462938 0.886391i \(-0.346795\pi\)
0.462938 + 0.886391i \(0.346795\pi\)
\(674\) 33267.0i 1.90118i
\(675\) 1188.00 0.0677424
\(676\) 0 0
\(677\) −25686.0 −1.45819 −0.729094 0.684414i \(-0.760058\pi\)
−0.729094 + 0.684414i \(0.760058\pi\)
\(678\) − 10071.0i − 0.570464i
\(679\) 2716.00 0.153506
\(680\) 20979.0 1.18310
\(681\) − 4194.00i − 0.235998i
\(682\) − 9000.00i − 0.505319i
\(683\) 19056.0i 1.06758i 0.845617 + 0.533790i \(0.179233\pi\)
−0.845617 + 0.533790i \(0.820767\pi\)
\(684\) 414.000i 0.0231428i
\(685\) −6453.00 −0.359936
\(686\) 4092.00 0.227745
\(687\) − 13398.0i − 0.744055i
\(688\) −36494.0 −2.02227
\(689\) 0 0
\(690\) 486.000 0.0268141
\(691\) − 16390.0i − 0.902323i −0.892442 0.451161i \(-0.851010\pi\)
0.892442 0.451161i \(-0.148990\pi\)
\(692\) 1410.00 0.0774569
\(693\) −540.000 −0.0296001
\(694\) 29286.0i 1.60185i
\(695\) 7380.00i 0.402790i
\(696\) − 6615.00i − 0.360260i
\(697\) − 25641.0i − 1.39343i
\(698\) −24870.0 −1.34863
\(699\) 4914.00 0.265901
\(700\) 88.0000i 0.00475155i
\(701\) 27846.0 1.50033 0.750163 0.661253i \(-0.229975\pi\)
0.750163 + 0.661253i \(0.229975\pi\)
\(702\) 0 0
\(703\) −782.000 −0.0419540
\(704\) 12990.0i 0.695425i
\(705\) 4374.00 0.233666
\(706\) −37215.0 −1.98386
\(707\) − 714.000i − 0.0379812i
\(708\) 1800.00i 0.0955482i
\(709\) 12283.0i 0.650632i 0.945605 + 0.325316i \(0.105471\pi\)
−0.945605 + 0.325316i \(0.894529\pi\)
\(710\) − 25110.0i − 1.32727i
\(711\) −11916.0 −0.628530
\(712\) 10458.0 0.550464
\(713\) − 600.000i − 0.0315150i
\(714\) 1998.00 0.104724
\(715\) 0 0
\(716\) −474.000 −0.0247405
\(717\) − 1782.00i − 0.0928173i
\(718\) −3294.00 −0.171213
\(719\) −25512.0 −1.32328 −0.661639 0.749822i \(-0.730139\pi\)
−0.661639 + 0.749822i \(0.730139\pi\)
\(720\) 5751.00i 0.297677i
\(721\) 2236.00i 0.115497i
\(722\) 14229.0i 0.733447i
\(723\) − 6909.00i − 0.355392i
\(724\) 2249.00 0.115447
\(725\) −4620.00 −0.236666
\(726\) 3879.00i 0.198296i
\(727\) −6110.00 −0.311702 −0.155851 0.987781i \(-0.549812\pi\)
−0.155851 + 0.987781i \(0.549812\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 6831.00i 0.346338i
\(731\) 57054.0 2.88676
\(732\) −699.000 −0.0352948
\(733\) − 27127.0i − 1.36693i −0.729984 0.683464i \(-0.760473\pi\)
0.729984 0.683464i \(-0.239527\pi\)
\(734\) − 17202.0i − 0.865037i
\(735\) − 9153.00i − 0.459338i
\(736\) − 270.000i − 0.0135222i
\(737\) 27780.0 1.38845
\(738\) 6237.00 0.311094
\(739\) 880.000i 0.0438042i 0.999760 + 0.0219021i \(0.00697222\pi\)
−0.999760 + 0.0219021i \(0.993028\pi\)
\(740\) 153.000 0.00760053
\(741\) 0 0
\(742\) 3834.00 0.189691
\(743\) − 21876.0i − 1.08015i −0.841616 0.540076i \(-0.818396\pi\)
0.841616 0.540076i \(-0.181604\pi\)
\(744\) 6300.00 0.310442
\(745\) −15741.0 −0.774102
\(746\) − 26913.0i − 1.32085i
\(747\) 7290.00i 0.357064i
\(748\) 3330.00i 0.162777i
\(749\) − 1428.00i − 0.0696635i
\(750\) 13689.0 0.666469
\(751\) −11798.0 −0.573256 −0.286628 0.958042i \(-0.592534\pi\)
−0.286628 + 0.958042i \(0.592534\pi\)
\(752\) − 11502.0i − 0.557759i
\(753\) −18972.0 −0.918165
\(754\) 0 0
\(755\) 3330.00 0.160518
\(756\) 54.0000i 0.00259783i
\(757\) −8074.00 −0.387655 −0.193827 0.981036i \(-0.562090\pi\)
−0.193827 + 0.981036i \(0.562090\pi\)
\(758\) −21732.0 −1.04135
\(759\) − 540.000i − 0.0258245i
\(760\) − 8694.00i − 0.414953i
\(761\) − 19554.0i − 0.931448i −0.884930 0.465724i \(-0.845794\pi\)
0.884930 0.465724i \(-0.154206\pi\)
\(762\) 5436.00i 0.258432i
\(763\) 4012.00 0.190359
\(764\) −3444.00 −0.163088
\(765\) − 8991.00i − 0.424928i
\(766\) 18936.0 0.893193
\(767\) 0 0
\(768\) 4539.00 0.213264
\(769\) 14030.0i 0.657913i 0.944345 + 0.328956i \(0.106697\pi\)
−0.944345 + 0.328956i \(0.893303\pi\)
\(770\) −1620.00 −0.0758192
\(771\) −23499.0 −1.09766
\(772\) − 4273.00i − 0.199208i
\(773\) 36042.0i 1.67703i 0.544882 + 0.838513i \(0.316574\pi\)
−0.544882 + 0.838513i \(0.683426\pi\)
\(774\) 13878.0i 0.644489i
\(775\) − 4400.00i − 0.203939i
\(776\) −28518.0 −1.31925
\(777\) −102.000 −0.00470943
\(778\) − 10881.0i − 0.501417i
\(779\) −10626.0 −0.488724
\(780\) 0 0
\(781\) −27900.0 −1.27828
\(782\) 1998.00i 0.0913662i
\(783\) −2835.00 −0.129393
\(784\) −24069.0 −1.09644
\(785\) 23499.0i 1.06843i
\(786\) − 14256.0i − 0.646940i
\(787\) − 28628.0i − 1.29667i −0.761356 0.648334i \(-0.775466\pi\)
0.761356 0.648334i \(-0.224534\pi\)
\(788\) 1986.00i 0.0897821i
\(789\) −9090.00 −0.410155
\(790\) −35748.0 −1.60995
\(791\) 2238.00i 0.100599i
\(792\) 5670.00 0.254387
\(793\) 0 0
\(794\) −11694.0 −0.522676
\(795\) − 17253.0i − 0.769687i
\(796\) −2386.00 −0.106243
\(797\) 37434.0 1.66371 0.831857 0.554990i \(-0.187278\pi\)
0.831857 + 0.554990i \(0.187278\pi\)
\(798\) − 828.000i − 0.0367304i
\(799\) 17982.0i 0.796192i
\(800\) − 1980.00i − 0.0875045i
\(801\) − 4482.00i − 0.197707i
\(802\) −17109.0 −0.753292
\(803\) 7590.00 0.333556
\(804\) − 2778.00i − 0.121856i
\(805\) −108.000 −0.00472857
\(806\) 0 0
\(807\) −1602.00 −0.0698799
\(808\) 7497.00i 0.326415i
\(809\) 37569.0 1.63270 0.816351 0.577556i \(-0.195994\pi\)
0.816351 + 0.577556i \(0.195994\pi\)
\(810\) 2187.00 0.0948683
\(811\) 5516.00i 0.238832i 0.992844 + 0.119416i \(0.0381023\pi\)
−0.992844 + 0.119416i \(0.961898\pi\)
\(812\) − 210.000i − 0.00907581i
\(813\) 11064.0i 0.477283i
\(814\) − 1530.00i − 0.0658802i
\(815\) 14724.0 0.632833
\(816\) −23643.0 −1.01430
\(817\) − 23644.0i − 1.01248i
\(818\) −18933.0 −0.809263
\(819\) 0 0
\(820\) 2079.00 0.0885388
\(821\) 8778.00i 0.373148i 0.982441 + 0.186574i \(0.0597384\pi\)
−0.982441 + 0.186574i \(0.940262\pi\)
\(822\) 6453.00 0.273813
\(823\) 3088.00 0.130791 0.0653955 0.997859i \(-0.479169\pi\)
0.0653955 + 0.997859i \(0.479169\pi\)
\(824\) − 23478.0i − 0.992591i
\(825\) − 3960.00i − 0.167115i
\(826\) − 3600.00i − 0.151647i
\(827\) − 13176.0i − 0.554020i −0.960867 0.277010i \(-0.910657\pi\)
0.960867 0.277010i \(-0.0893435\pi\)
\(828\) −54.0000 −0.00226646
\(829\) 2359.00 0.0988317 0.0494158 0.998778i \(-0.484264\pi\)
0.0494158 + 0.998778i \(0.484264\pi\)
\(830\) 21870.0i 0.914601i
\(831\) −5595.00 −0.233560
\(832\) 0 0
\(833\) 37629.0 1.56515
\(834\) − 7380.00i − 0.306413i
\(835\) −2376.00 −0.0984729
\(836\) 1380.00 0.0570913
\(837\) − 2700.00i − 0.111500i
\(838\) − 6984.00i − 0.287898i
\(839\) 2676.00i 0.110114i 0.998483 + 0.0550571i \(0.0175341\pi\)
−0.998483 + 0.0550571i \(0.982466\pi\)
\(840\) − 1134.00i − 0.0465794i
\(841\) −13364.0 −0.547952
\(842\) −6135.00 −0.251100
\(843\) − 8991.00i − 0.367339i
\(844\) 1600.00 0.0652539
\(845\) 0 0
\(846\) −4374.00 −0.177756
\(847\) − 862.000i − 0.0349689i
\(848\) −45369.0 −1.83724
\(849\) 12342.0 0.498912
\(850\) 14652.0i 0.591246i
\(851\) − 102.000i − 0.00410871i
\(852\) 2790.00i 0.112188i
\(853\) − 2477.00i − 0.0994266i −0.998764 0.0497133i \(-0.984169\pi\)
0.998764 0.0497133i \(-0.0158308\pi\)
\(854\) 1398.00 0.0560171
\(855\) −3726.00 −0.149037
\(856\) 14994.0i 0.598697i
\(857\) 17199.0 0.685539 0.342769 0.939420i \(-0.388635\pi\)
0.342769 + 0.939420i \(0.388635\pi\)
\(858\) 0 0
\(859\) 24338.0 0.966708 0.483354 0.875425i \(-0.339418\pi\)
0.483354 + 0.875425i \(0.339418\pi\)
\(860\) 4626.00i 0.183425i
\(861\) −1386.00 −0.0548603
\(862\) −15102.0 −0.596724
\(863\) 25146.0i 0.991865i 0.868361 + 0.495933i \(0.165174\pi\)
−0.868361 + 0.495933i \(0.834826\pi\)
\(864\) − 1215.00i − 0.0478416i
\(865\) 12690.0i 0.498813i
\(866\) − 12849.0i − 0.504188i
\(867\) 22224.0 0.870550
\(868\) 200.000 0.00782079
\(869\) 39720.0i 1.55053i
\(870\) −8505.00 −0.331433
\(871\) 0 0
\(872\) −42126.0 −1.63597
\(873\) 12222.0i 0.473828i
\(874\) 828.000 0.0320452
\(875\) −3042.00 −0.117530
\(876\) − 759.000i − 0.0292742i
\(877\) 18089.0i 0.696490i 0.937403 + 0.348245i \(0.113222\pi\)
−0.937403 + 0.348245i \(0.886778\pi\)
\(878\) 3918.00i 0.150599i
\(879\) 13995.0i 0.537019i
\(880\) 19170.0 0.734342
\(881\) 15099.0 0.577410 0.288705 0.957418i \(-0.406775\pi\)
0.288705 + 0.957418i \(0.406775\pi\)
\(882\) 9153.00i 0.349430i
\(883\) −33488.0 −1.27629 −0.638143 0.769918i \(-0.720297\pi\)
−0.638143 + 0.769918i \(0.720297\pi\)
\(884\) 0 0
\(885\) −16200.0 −0.615319
\(886\) − 17388.0i − 0.659324i
\(887\) −39768.0 −1.50539 −0.752694 0.658371i \(-0.771246\pi\)
−0.752694 + 0.658371i \(0.771246\pi\)
\(888\) 1071.00 0.0404734
\(889\) − 1208.00i − 0.0455737i
\(890\) − 13446.0i − 0.506417i
\(891\) − 2430.00i − 0.0913671i
\(892\) 3832.00i 0.143840i
\(893\) 7452.00 0.279252
\(894\) 15741.0 0.588879
\(895\) − 4266.00i − 0.159326i
\(896\) −3318.00 −0.123713
\(897\) 0 0
\(898\) 8118.00 0.301672
\(899\) 10500.0i 0.389538i
\(900\) −396.000 −0.0146667
\(901\) 70929.0 2.62263
\(902\) − 20790.0i − 0.767440i
\(903\) − 3084.00i − 0.113653i
\(904\) − 23499.0i − 0.864563i
\(905\) 20241.0i 0.743463i
\(906\) −3330.00 −0.122110
\(907\) −32156.0 −1.17720 −0.588601 0.808424i \(-0.700321\pi\)
−0.588601 + 0.808424i \(0.700321\pi\)
\(908\) 1398.00i 0.0510950i
\(909\) 3213.00 0.117237
\(910\) 0 0
\(911\) 11520.0 0.418962 0.209481 0.977813i \(-0.432823\pi\)
0.209481 + 0.977813i \(0.432823\pi\)
\(912\) 9798.00i 0.355750i
\(913\) 24300.0 0.880846
\(914\) 2487.00 0.0900029
\(915\) − 6291.00i − 0.227294i
\(916\) 4466.00i 0.161093i
\(917\) 3168.00i 0.114086i
\(918\) 8991.00i 0.323254i
\(919\) 4952.00 0.177749 0.0888745 0.996043i \(-0.471673\pi\)
0.0888745 + 0.996043i \(0.471673\pi\)
\(920\) 1134.00 0.0406379
\(921\) − 4506.00i − 0.161214i
\(922\) 16479.0 0.588619
\(923\) 0 0
\(924\) 180.000 0.00640862
\(925\) − 748.000i − 0.0265882i
\(926\) −46038.0 −1.63380
\(927\) −10062.0 −0.356504
\(928\) 4725.00i 0.167140i
\(929\) 8781.00i 0.310113i 0.987906 + 0.155057i \(0.0495560\pi\)
−0.987906 + 0.155057i \(0.950444\pi\)
\(930\) − 8100.00i − 0.285602i
\(931\) − 15594.0i − 0.548950i
\(932\) −1638.00 −0.0575692
\(933\) −6318.00 −0.221696
\(934\) 28782.0i 1.00833i
\(935\) −29970.0 −1.04826
\(936\) 0 0
\(937\) 50039.0 1.74461 0.872307 0.488959i \(-0.162623\pi\)
0.872307 + 0.488959i \(0.162623\pi\)
\(938\) 5556.00i 0.193401i
\(939\) −11694.0 −0.406410
\(940\) −1458.00 −0.0505901
\(941\) − 50670.0i − 1.75536i −0.479246 0.877681i \(-0.659090\pi\)
0.479246 0.877681i \(-0.340910\pi\)
\(942\) − 23499.0i − 0.812780i
\(943\) − 1386.00i − 0.0478625i
\(944\) 42600.0i 1.46876i
\(945\) −486.000 −0.0167297
\(946\) 46260.0 1.58990
\(947\) − 42384.0i − 1.45438i −0.686438 0.727188i \(-0.740827\pi\)
0.686438 0.727188i \(-0.259173\pi\)
\(948\) 3972.00 0.136081
\(949\) 0 0
\(950\) 6072.00 0.207370
\(951\) 28053.0i 0.956552i
\(952\) 4662.00 0.158715
\(953\) 50538.0 1.71782 0.858912 0.512123i \(-0.171141\pi\)
0.858912 + 0.512123i \(0.171141\pi\)
\(954\) 17253.0i 0.585520i
\(955\) − 30996.0i − 1.05027i
\(956\) 594.000i 0.0200955i
\(957\) 9450.00i 0.319201i
\(958\) −38520.0 −1.29909
\(959\) −1434.00 −0.0482860
\(960\) 11691.0i 0.393047i
\(961\) 19791.0 0.664328
\(962\) 0 0
\(963\) 6426.00 0.215031
\(964\) 2303.00i 0.0769446i
\(965\) 38457.0 1.28288
\(966\) 108.000 0.00359715
\(967\) − 6886.00i − 0.228996i −0.993423 0.114498i \(-0.963474\pi\)
0.993423 0.114498i \(-0.0365259\pi\)
\(968\) 9051.00i 0.300527i
\(969\) − 15318.0i − 0.507828i
\(970\) 36666.0i 1.21368i
\(971\) 9060.00 0.299433 0.149716 0.988729i \(-0.452164\pi\)
0.149716 + 0.988729i \(0.452164\pi\)
\(972\) −243.000 −0.00801875
\(973\) 1640.00i 0.0540349i
\(974\) 42258.0 1.39018
\(975\) 0 0
\(976\) −16543.0 −0.542550
\(977\) − 28311.0i − 0.927072i −0.886078 0.463536i \(-0.846581\pi\)
0.886078 0.463536i \(-0.153419\pi\)
\(978\) −14724.0 −0.481413
\(979\) −14940.0 −0.487727
\(980\) 3051.00i 0.0994496i
\(981\) 18054.0i 0.587584i
\(982\) − 35082.0i − 1.14003i
\(983\) − 4284.00i − 0.139001i −0.997582 0.0695007i \(-0.977859\pi\)
0.997582 0.0695007i \(-0.0221406\pi\)
\(984\) 14553.0 0.471476
\(985\) −17874.0 −0.578186
\(986\) − 34965.0i − 1.12932i
\(987\) 972.000 0.0313466
\(988\) 0 0
\(989\) 3084.00 0.0991562
\(990\) − 7290.00i − 0.234032i
\(991\) −2458.00 −0.0787901 −0.0393950 0.999224i \(-0.512543\pi\)
−0.0393950 + 0.999224i \(0.512543\pi\)
\(992\) −4500.00 −0.144027
\(993\) − 27516.0i − 0.879349i
\(994\) − 5580.00i − 0.178055i
\(995\) − 21474.0i − 0.684193i
\(996\) − 2430.00i − 0.0773067i
\(997\) 24101.0 0.765583 0.382792 0.923835i \(-0.374963\pi\)
0.382792 + 0.923835i \(0.374963\pi\)
\(998\) 11064.0 0.350927
\(999\) − 459.000i − 0.0145367i
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.c.337.2 2
13.2 odd 12 39.4.e.a.22.1 yes 2
13.5 odd 4 507.4.a.e.1.1 1
13.6 odd 12 39.4.e.a.16.1 2
13.8 odd 4 507.4.a.a.1.1 1
13.12 even 2 inner 507.4.b.c.337.1 2
39.2 even 12 117.4.g.b.100.1 2
39.5 even 4 1521.4.a.c.1.1 1
39.8 even 4 1521.4.a.j.1.1 1
39.32 even 12 117.4.g.b.55.1 2
52.15 even 12 624.4.q.b.529.1 2
52.19 even 12 624.4.q.b.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.a.16.1 2 13.6 odd 12
39.4.e.a.22.1 yes 2 13.2 odd 12
117.4.g.b.55.1 2 39.32 even 12
117.4.g.b.100.1 2 39.2 even 12
507.4.a.a.1.1 1 13.8 odd 4
507.4.a.e.1.1 1 13.5 odd 4
507.4.b.c.337.1 2 13.12 even 2 inner
507.4.b.c.337.2 2 1.1 even 1 trivial
624.4.q.b.289.1 2 52.19 even 12
624.4.q.b.529.1 2 52.15 even 12
1521.4.a.c.1.1 1 39.5 even 4
1521.4.a.j.1.1 1 39.8 even 4