Properties

Label 525.4.d.c.274.1
Level 525
Weight 4
Character 525.274
Analytic conductor 30.976
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.c.274.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} +7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} +7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} -36.0000 q^{11} -3.00000i q^{12} +34.0000i q^{13} +21.0000 q^{14} -71.0000 q^{16} +42.0000i q^{17} +27.0000i q^{18} +124.000 q^{19} -21.0000 q^{21} +108.000i q^{22} +63.0000 q^{24} +102.000 q^{26} -27.0000i q^{27} -7.00000i q^{28} -102.000 q^{29} -160.000 q^{31} +45.0000i q^{32} -108.000i q^{33} +126.000 q^{34} +9.00000 q^{36} +398.000i q^{37} -372.000i q^{38} -102.000 q^{39} -318.000 q^{41} +63.0000i q^{42} +268.000i q^{43} +36.0000 q^{44} +240.000i q^{47} -213.000i q^{48} -49.0000 q^{49} -126.000 q^{51} -34.0000i q^{52} +498.000i q^{53} -81.0000 q^{54} +147.000 q^{56} +372.000i q^{57} +306.000i q^{58} +132.000 q^{59} +398.000 q^{61} +480.000i q^{62} -63.0000i q^{63} -433.000 q^{64} -324.000 q^{66} +92.0000i q^{67} -42.0000i q^{68} -720.000 q^{71} +189.000i q^{72} +502.000i q^{73} +1194.00 q^{74} -124.000 q^{76} -252.000i q^{77} +306.000i q^{78} +1024.00 q^{79} +81.0000 q^{81} +954.000i q^{82} +204.000i q^{83} +21.0000 q^{84} +804.000 q^{86} -306.000i q^{87} +756.000i q^{88} -354.000 q^{89} -238.000 q^{91} -480.000i q^{93} +720.000 q^{94} -135.000 q^{96} -286.000i q^{97} +147.000i q^{98} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} - 72q^{11} + 42q^{14} - 142q^{16} + 248q^{19} - 42q^{21} + 126q^{24} + 204q^{26} - 204q^{29} - 320q^{31} + 252q^{34} + 18q^{36} - 204q^{39} - 636q^{41} + 72q^{44} - 98q^{49} - 252q^{51} - 162q^{54} + 294q^{56} + 264q^{59} + 796q^{61} - 866q^{64} - 648q^{66} - 1440q^{71} + 2388q^{74} - 248q^{76} + 2048q^{79} + 162q^{81} + 42q^{84} + 1608q^{86} - 708q^{89} - 476q^{91} + 1440q^{94} - 270q^{96} + 648q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 9.00000 0.612372
\(7\) 7.00000i 0.377964i
\(8\) − 21.0000i − 0.928078i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) − 3.00000i − 0.0721688i
\(13\) 34.0000i 0.725377i 0.931910 + 0.362689i \(0.118141\pi\)
−0.931910 + 0.362689i \(0.881859\pi\)
\(14\) 21.0000 0.400892
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 42.0000i 0.599206i 0.954064 + 0.299603i \(0.0968542\pi\)
−0.954064 + 0.299603i \(0.903146\pi\)
\(18\) 27.0000i 0.353553i
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 108.000i 1.04662i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 63.0000 0.535826
\(25\) 0 0
\(26\) 102.000 0.769379
\(27\) − 27.0000i − 0.192450i
\(28\) − 7.00000i − 0.0472456i
\(29\) −102.000 −0.653135 −0.326568 0.945174i \(-0.605892\pi\)
−0.326568 + 0.945174i \(0.605892\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 45.0000i 0.248592i
\(33\) − 108.000i − 0.569709i
\(34\) 126.000 0.635554
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) 398.000i 1.76840i 0.467109 + 0.884200i \(0.345296\pi\)
−0.467109 + 0.884200i \(0.654704\pi\)
\(38\) − 372.000i − 1.58806i
\(39\) −102.000 −0.418797
\(40\) 0 0
\(41\) −318.000 −1.21130 −0.605649 0.795732i \(-0.707087\pi\)
−0.605649 + 0.795732i \(0.707087\pi\)
\(42\) 63.0000i 0.231455i
\(43\) 268.000i 0.950456i 0.879863 + 0.475228i \(0.157634\pi\)
−0.879863 + 0.475228i \(0.842366\pi\)
\(44\) 36.0000 0.123346
\(45\) 0 0
\(46\) 0 0
\(47\) 240.000i 0.744843i 0.928064 + 0.372421i \(0.121472\pi\)
−0.928064 + 0.372421i \(0.878528\pi\)
\(48\) − 213.000i − 0.640498i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −126.000 −0.345952
\(52\) − 34.0000i − 0.0906721i
\(53\) 498.000i 1.29067i 0.763899 + 0.645335i \(0.223282\pi\)
−0.763899 + 0.645335i \(0.776718\pi\)
\(54\) −81.0000 −0.204124
\(55\) 0 0
\(56\) 147.000 0.350780
\(57\) 372.000i 0.864432i
\(58\) 306.000i 0.692755i
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) 0 0
\(61\) 398.000 0.835388 0.417694 0.908588i \(-0.362838\pi\)
0.417694 + 0.908588i \(0.362838\pi\)
\(62\) 480.000i 0.983227i
\(63\) − 63.0000i − 0.125988i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −324.000 −0.604267
\(67\) 92.0000i 0.167755i 0.996476 + 0.0838775i \(0.0267305\pi\)
−0.996476 + 0.0838775i \(0.973270\pi\)
\(68\) − 42.0000i − 0.0749007i
\(69\) 0 0
\(70\) 0 0
\(71\) −720.000 −1.20350 −0.601748 0.798686i \(-0.705529\pi\)
−0.601748 + 0.798686i \(0.705529\pi\)
\(72\) 189.000i 0.309359i
\(73\) 502.000i 0.804858i 0.915451 + 0.402429i \(0.131834\pi\)
−0.915451 + 0.402429i \(0.868166\pi\)
\(74\) 1194.00 1.87567
\(75\) 0 0
\(76\) −124.000 −0.187155
\(77\) − 252.000i − 0.372962i
\(78\) 306.000i 0.444201i
\(79\) 1024.00 1.45834 0.729171 0.684332i \(-0.239906\pi\)
0.729171 + 0.684332i \(0.239906\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 954.000i 1.28478i
\(83\) 204.000i 0.269782i 0.990860 + 0.134891i \(0.0430684\pi\)
−0.990860 + 0.134891i \(0.956932\pi\)
\(84\) 21.0000 0.0272772
\(85\) 0 0
\(86\) 804.000 1.00811
\(87\) − 306.000i − 0.377088i
\(88\) 756.000i 0.915794i
\(89\) −354.000 −0.421617 −0.210809 0.977527i \(-0.567610\pi\)
−0.210809 + 0.977527i \(0.567610\pi\)
\(90\) 0 0
\(91\) −238.000 −0.274167
\(92\) 0 0
\(93\) − 480.000i − 0.535201i
\(94\) 720.000 0.790025
\(95\) 0 0
\(96\) −135.000 −0.143525
\(97\) − 286.000i − 0.299370i −0.988734 0.149685i \(-0.952174\pi\)
0.988734 0.149685i \(-0.0478260\pi\)
\(98\) 147.000i 0.151523i
\(99\) 324.000 0.328921
\(100\) 0 0
\(101\) 414.000 0.407867 0.203933 0.978985i \(-0.434627\pi\)
0.203933 + 0.978985i \(0.434627\pi\)
\(102\) 378.000i 0.366937i
\(103\) − 56.0000i − 0.0535713i −0.999641 0.0267857i \(-0.991473\pi\)
0.999641 0.0267857i \(-0.00852716\pi\)
\(104\) 714.000 0.673206
\(105\) 0 0
\(106\) 1494.00 1.36896
\(107\) 12.0000i 0.0108419i 0.999985 + 0.00542095i \(0.00172555\pi\)
−0.999985 + 0.00542095i \(0.998274\pi\)
\(108\) 27.0000i 0.0240563i
\(109\) −1478.00 −1.29878 −0.649389 0.760457i \(-0.724975\pi\)
−0.649389 + 0.760457i \(0.724975\pi\)
\(110\) 0 0
\(111\) −1194.00 −1.02099
\(112\) − 497.000i − 0.419304i
\(113\) − 402.000i − 0.334664i −0.985901 0.167332i \(-0.946485\pi\)
0.985901 0.167332i \(-0.0535151\pi\)
\(114\) 1116.00 0.916868
\(115\) 0 0
\(116\) 102.000 0.0816419
\(117\) − 306.000i − 0.241792i
\(118\) − 396.000i − 0.308939i
\(119\) −294.000 −0.226478
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) − 1194.00i − 0.886063i
\(123\) − 954.000i − 0.699344i
\(124\) 160.000 0.115874
\(125\) 0 0
\(126\) −189.000 −0.133631
\(127\) 1280.00i 0.894344i 0.894448 + 0.447172i \(0.147569\pi\)
−0.894448 + 0.447172i \(0.852431\pi\)
\(128\) 1659.00i 1.14560i
\(129\) −804.000 −0.548746
\(130\) 0 0
\(131\) 1764.00 1.17650 0.588250 0.808679i \(-0.299817\pi\)
0.588250 + 0.808679i \(0.299817\pi\)
\(132\) 108.000i 0.0712136i
\(133\) 868.000i 0.565903i
\(134\) 276.000 0.177931
\(135\) 0 0
\(136\) 882.000 0.556109
\(137\) − 2358.00i − 1.47049i −0.677800 0.735246i \(-0.737066\pi\)
0.677800 0.735246i \(-0.262934\pi\)
\(138\) 0 0
\(139\) 52.0000 0.0317308 0.0158654 0.999874i \(-0.494950\pi\)
0.0158654 + 0.999874i \(0.494950\pi\)
\(140\) 0 0
\(141\) −720.000 −0.430035
\(142\) 2160.00i 1.27650i
\(143\) − 1224.00i − 0.715776i
\(144\) 639.000 0.369792
\(145\) 0 0
\(146\) 1506.00 0.853681
\(147\) − 147.000i − 0.0824786i
\(148\) − 398.000i − 0.221050i
\(149\) 1746.00 0.959986 0.479993 0.877272i \(-0.340639\pi\)
0.479993 + 0.877272i \(0.340639\pi\)
\(150\) 0 0
\(151\) −232.000 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(152\) − 2604.00i − 1.38955i
\(153\) − 378.000i − 0.199735i
\(154\) −756.000 −0.395586
\(155\) 0 0
\(156\) 102.000 0.0523496
\(157\) 1694.00i 0.861120i 0.902562 + 0.430560i \(0.141684\pi\)
−0.902562 + 0.430560i \(0.858316\pi\)
\(158\) − 3072.00i − 1.54681i
\(159\) −1494.00 −0.745169
\(160\) 0 0
\(161\) 0 0
\(162\) − 243.000i − 0.117851i
\(163\) 2932.00i 1.40891i 0.709750 + 0.704454i \(0.248808\pi\)
−0.709750 + 0.704454i \(0.751192\pi\)
\(164\) 318.000 0.151412
\(165\) 0 0
\(166\) 612.000 0.286147
\(167\) 1176.00i 0.544920i 0.962167 + 0.272460i \(0.0878372\pi\)
−0.962167 + 0.272460i \(0.912163\pi\)
\(168\) 441.000i 0.202523i
\(169\) 1041.00 0.473828
\(170\) 0 0
\(171\) −1116.00 −0.499080
\(172\) − 268.000i − 0.118807i
\(173\) − 870.000i − 0.382340i −0.981557 0.191170i \(-0.938772\pi\)
0.981557 0.191170i \(-0.0612282\pi\)
\(174\) −918.000 −0.399962
\(175\) 0 0
\(176\) 2556.00 1.09469
\(177\) 396.000i 0.168165i
\(178\) 1062.00i 0.447193i
\(179\) 2316.00 0.967072 0.483536 0.875324i \(-0.339352\pi\)
0.483536 + 0.875324i \(0.339352\pi\)
\(180\) 0 0
\(181\) −106.000 −0.0435299 −0.0217650 0.999763i \(-0.506929\pi\)
−0.0217650 + 0.999763i \(0.506929\pi\)
\(182\) 714.000i 0.290798i
\(183\) 1194.00i 0.482312i
\(184\) 0 0
\(185\) 0 0
\(186\) −1440.00 −0.567666
\(187\) − 1512.00i − 0.591275i
\(188\) − 240.000i − 0.0931053i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −1128.00 −0.427326 −0.213663 0.976907i \(-0.568539\pi\)
−0.213663 + 0.976907i \(0.568539\pi\)
\(192\) − 1299.00i − 0.488267i
\(193\) − 4034.00i − 1.50453i −0.658862 0.752263i \(-0.728962\pi\)
0.658862 0.752263i \(-0.271038\pi\)
\(194\) −858.000 −0.317530
\(195\) 0 0
\(196\) 49.0000 0.0178571
\(197\) − 1314.00i − 0.475221i −0.971360 0.237611i \(-0.923636\pi\)
0.971360 0.237611i \(-0.0763643\pi\)
\(198\) − 972.000i − 0.348874i
\(199\) −5096.00 −1.81531 −0.907653 0.419722i \(-0.862128\pi\)
−0.907653 + 0.419722i \(0.862128\pi\)
\(200\) 0 0
\(201\) −276.000 −0.0968534
\(202\) − 1242.00i − 0.432608i
\(203\) − 714.000i − 0.246862i
\(204\) 126.000 0.0432439
\(205\) 0 0
\(206\) −168.000 −0.0568209
\(207\) 0 0
\(208\) − 2414.00i − 0.804715i
\(209\) −4464.00 −1.47742
\(210\) 0 0
\(211\) −3076.00 −1.00360 −0.501802 0.864982i \(-0.667330\pi\)
−0.501802 + 0.864982i \(0.667330\pi\)
\(212\) − 498.000i − 0.161334i
\(213\) − 2160.00i − 0.694839i
\(214\) 36.0000 0.0114996
\(215\) 0 0
\(216\) −567.000 −0.178609
\(217\) − 1120.00i − 0.350371i
\(218\) 4434.00i 1.37756i
\(219\) −1506.00 −0.464685
\(220\) 0 0
\(221\) −1428.00 −0.434650
\(222\) 3582.00i 1.08292i
\(223\) 1888.00i 0.566950i 0.958980 + 0.283475i \(0.0914873\pi\)
−0.958980 + 0.283475i \(0.908513\pi\)
\(224\) −315.000 −0.0939590
\(225\) 0 0
\(226\) −1206.00 −0.354964
\(227\) − 4716.00i − 1.37891i −0.724330 0.689454i \(-0.757851\pi\)
0.724330 0.689454i \(-0.242149\pi\)
\(228\) − 372.000i − 0.108054i
\(229\) 1690.00 0.487678 0.243839 0.969816i \(-0.421593\pi\)
0.243839 + 0.969816i \(0.421593\pi\)
\(230\) 0 0
\(231\) 756.000 0.215330
\(232\) 2142.00i 0.606160i
\(233\) − 138.000i − 0.0388012i −0.999812 0.0194006i \(-0.993824\pi\)
0.999812 0.0194006i \(-0.00617579\pi\)
\(234\) −918.000 −0.256460
\(235\) 0 0
\(236\) −132.000 −0.0364088
\(237\) 3072.00i 0.841974i
\(238\) 882.000i 0.240217i
\(239\) −1896.00 −0.513147 −0.256573 0.966525i \(-0.582594\pi\)
−0.256573 + 0.966525i \(0.582594\pi\)
\(240\) 0 0
\(241\) −3598.00 −0.961691 −0.480846 0.876805i \(-0.659670\pi\)
−0.480846 + 0.876805i \(0.659670\pi\)
\(242\) 105.000i 0.0278911i
\(243\) 243.000i 0.0641500i
\(244\) −398.000 −0.104424
\(245\) 0 0
\(246\) −2862.00 −0.741766
\(247\) 4216.00i 1.08606i
\(248\) 3360.00i 0.860323i
\(249\) −612.000 −0.155759
\(250\) 0 0
\(251\) −3060.00 −0.769504 −0.384752 0.923020i \(-0.625713\pi\)
−0.384752 + 0.923020i \(0.625713\pi\)
\(252\) 63.0000i 0.0157485i
\(253\) 0 0
\(254\) 3840.00 0.948595
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) − 6822.00i − 1.65582i −0.560864 0.827908i \(-0.689531\pi\)
0.560864 0.827908i \(-0.310469\pi\)
\(258\) 2412.00i 0.582033i
\(259\) −2786.00 −0.668392
\(260\) 0 0
\(261\) 918.000 0.217712
\(262\) − 5292.00i − 1.24787i
\(263\) − 2592.00i − 0.607717i −0.952717 0.303858i \(-0.901725\pi\)
0.952717 0.303858i \(-0.0982750\pi\)
\(264\) −2268.00 −0.528734
\(265\) 0 0
\(266\) 2604.00 0.600231
\(267\) − 1062.00i − 0.243421i
\(268\) − 92.0000i − 0.0209694i
\(269\) −8214.00 −1.86177 −0.930886 0.365311i \(-0.880963\pi\)
−0.930886 + 0.365311i \(0.880963\pi\)
\(270\) 0 0
\(271\) −5344.00 −1.19788 −0.598939 0.800795i \(-0.704411\pi\)
−0.598939 + 0.800795i \(0.704411\pi\)
\(272\) − 2982.00i − 0.664744i
\(273\) − 714.000i − 0.158290i
\(274\) −7074.00 −1.55969
\(275\) 0 0
\(276\) 0 0
\(277\) − 6514.00i − 1.41295i −0.707736 0.706477i \(-0.750283\pi\)
0.707736 0.706477i \(-0.249717\pi\)
\(278\) − 156.000i − 0.0336556i
\(279\) 1440.00 0.308998
\(280\) 0 0
\(281\) 6618.00 1.40497 0.702485 0.711698i \(-0.252074\pi\)
0.702485 + 0.711698i \(0.252074\pi\)
\(282\) 2160.00i 0.456121i
\(283\) − 3260.00i − 0.684759i −0.939562 0.342380i \(-0.888767\pi\)
0.939562 0.342380i \(-0.111233\pi\)
\(284\) 720.000 0.150437
\(285\) 0 0
\(286\) −3672.00 −0.759195
\(287\) − 2226.00i − 0.457828i
\(288\) − 405.000i − 0.0828641i
\(289\) 3149.00 0.640953
\(290\) 0 0
\(291\) 858.000 0.172841
\(292\) − 502.000i − 0.100607i
\(293\) − 5118.00i − 1.02047i −0.860036 0.510233i \(-0.829559\pi\)
0.860036 0.510233i \(-0.170441\pi\)
\(294\) −441.000 −0.0874818
\(295\) 0 0
\(296\) 8358.00 1.64121
\(297\) 972.000i 0.189903i
\(298\) − 5238.00i − 1.01822i
\(299\) 0 0
\(300\) 0 0
\(301\) −1876.00 −0.359239
\(302\) 696.000i 0.132617i
\(303\) 1242.00i 0.235482i
\(304\) −8804.00 −1.66100
\(305\) 0 0
\(306\) −1134.00 −0.211851
\(307\) 452.000i 0.0840293i 0.999117 + 0.0420147i \(0.0133776\pi\)
−0.999117 + 0.0420147i \(0.986622\pi\)
\(308\) 252.000i 0.0466202i
\(309\) 168.000 0.0309294
\(310\) 0 0
\(311\) 5016.00 0.914570 0.457285 0.889320i \(-0.348822\pi\)
0.457285 + 0.889320i \(0.348822\pi\)
\(312\) 2142.00i 0.388676i
\(313\) − 5402.00i − 0.975524i −0.872977 0.487762i \(-0.837813\pi\)
0.872977 0.487762i \(-0.162187\pi\)
\(314\) 5082.00 0.913356
\(315\) 0 0
\(316\) −1024.00 −0.182293
\(317\) 10086.0i 1.78702i 0.449041 + 0.893511i \(0.351766\pi\)
−0.449041 + 0.893511i \(0.648234\pi\)
\(318\) 4482.00i 0.790371i
\(319\) 3672.00 0.644491
\(320\) 0 0
\(321\) −36.0000 −0.00625958
\(322\) 0 0
\(323\) 5208.00i 0.897154i
\(324\) −81.0000 −0.0138889
\(325\) 0 0
\(326\) 8796.00 1.49437
\(327\) − 4434.00i − 0.749849i
\(328\) 6678.00i 1.12418i
\(329\) −1680.00 −0.281524
\(330\) 0 0
\(331\) −8044.00 −1.33577 −0.667883 0.744267i \(-0.732799\pi\)
−0.667883 + 0.744267i \(0.732799\pi\)
\(332\) − 204.000i − 0.0337228i
\(333\) − 3582.00i − 0.589467i
\(334\) 3528.00 0.577975
\(335\) 0 0
\(336\) 1491.00 0.242085
\(337\) 4178.00i 0.675342i 0.941264 + 0.337671i \(0.109639\pi\)
−0.941264 + 0.337671i \(0.890361\pi\)
\(338\) − 3123.00i − 0.502570i
\(339\) 1206.00 0.193218
\(340\) 0 0
\(341\) 5760.00 0.914726
\(342\) 3348.00i 0.529354i
\(343\) − 343.000i − 0.0539949i
\(344\) 5628.00 0.882097
\(345\) 0 0
\(346\) −2610.00 −0.405533
\(347\) 156.000i 0.0241341i 0.999927 + 0.0120670i \(0.00384115\pi\)
−0.999927 + 0.0120670i \(0.996159\pi\)
\(348\) 306.000i 0.0471360i
\(349\) 12418.0 1.90464 0.952321 0.305097i \(-0.0986888\pi\)
0.952321 + 0.305097i \(0.0986888\pi\)
\(350\) 0 0
\(351\) 918.000 0.139599
\(352\) − 1620.00i − 0.245302i
\(353\) 7830.00i 1.18059i 0.807187 + 0.590296i \(0.200989\pi\)
−0.807187 + 0.590296i \(0.799011\pi\)
\(354\) 1188.00 0.178366
\(355\) 0 0
\(356\) 354.000 0.0527021
\(357\) − 882.000i − 0.130757i
\(358\) − 6948.00i − 1.02574i
\(359\) 9312.00 1.36899 0.684497 0.729016i \(-0.260022\pi\)
0.684497 + 0.729016i \(0.260022\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 318.000i 0.0461705i
\(363\) − 105.000i − 0.0151820i
\(364\) 238.000 0.0342709
\(365\) 0 0
\(366\) 3582.00 0.511569
\(367\) − 3760.00i − 0.534797i −0.963586 0.267398i \(-0.913836\pi\)
0.963586 0.267398i \(-0.0861640\pi\)
\(368\) 0 0
\(369\) 2862.00 0.403766
\(370\) 0 0
\(371\) −3486.00 −0.487828
\(372\) 480.000i 0.0669001i
\(373\) − 5870.00i − 0.814845i −0.913240 0.407422i \(-0.866428\pi\)
0.913240 0.407422i \(-0.133572\pi\)
\(374\) −4536.00 −0.627142
\(375\) 0 0
\(376\) 5040.00 0.691272
\(377\) − 3468.00i − 0.473769i
\(378\) − 567.000i − 0.0771517i
\(379\) 1852.00 0.251005 0.125502 0.992093i \(-0.459946\pi\)
0.125502 + 0.992093i \(0.459946\pi\)
\(380\) 0 0
\(381\) −3840.00 −0.516350
\(382\) 3384.00i 0.453247i
\(383\) − 2160.00i − 0.288175i −0.989565 0.144087i \(-0.953975\pi\)
0.989565 0.144087i \(-0.0460246\pi\)
\(384\) −4977.00 −0.661410
\(385\) 0 0
\(386\) −12102.0 −1.59579
\(387\) − 2412.00i − 0.316819i
\(388\) 286.000i 0.0374213i
\(389\) 6786.00 0.884483 0.442241 0.896896i \(-0.354183\pi\)
0.442241 + 0.896896i \(0.354183\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1029.00i 0.132583i
\(393\) 5292.00i 0.679252i
\(394\) −3942.00 −0.504048
\(395\) 0 0
\(396\) −324.000 −0.0411152
\(397\) − 6514.00i − 0.823497i −0.911298 0.411748i \(-0.864918\pi\)
0.911298 0.411748i \(-0.135082\pi\)
\(398\) 15288.0i 1.92542i
\(399\) −2604.00 −0.326724
\(400\) 0 0
\(401\) 3330.00 0.414694 0.207347 0.978267i \(-0.433517\pi\)
0.207347 + 0.978267i \(0.433517\pi\)
\(402\) 828.000i 0.102729i
\(403\) − 5440.00i − 0.672421i
\(404\) −414.000 −0.0509833
\(405\) 0 0
\(406\) −2142.00 −0.261837
\(407\) − 14328.0i − 1.74499i
\(408\) 2646.00i 0.321070i
\(409\) 5398.00 0.652601 0.326301 0.945266i \(-0.394198\pi\)
0.326301 + 0.945266i \(0.394198\pi\)
\(410\) 0 0
\(411\) 7074.00 0.848990
\(412\) 56.0000i 0.00669641i
\(413\) 924.000i 0.110090i
\(414\) 0 0
\(415\) 0 0
\(416\) −1530.00 −0.180323
\(417\) 156.000i 0.0183198i
\(418\) 13392.0i 1.56704i
\(419\) −13092.0 −1.52646 −0.763229 0.646128i \(-0.776387\pi\)
−0.763229 + 0.646128i \(0.776387\pi\)
\(420\) 0 0
\(421\) −322.000 −0.0372763 −0.0186381 0.999826i \(-0.505933\pi\)
−0.0186381 + 0.999826i \(0.505933\pi\)
\(422\) 9228.00i 1.06448i
\(423\) − 2160.00i − 0.248281i
\(424\) 10458.0 1.19784
\(425\) 0 0
\(426\) −6480.00 −0.736988
\(427\) 2786.00i 0.315747i
\(428\) − 12.0000i − 0.00135524i
\(429\) 3672.00 0.413254
\(430\) 0 0
\(431\) 2616.00 0.292363 0.146181 0.989258i \(-0.453302\pi\)
0.146181 + 0.989258i \(0.453302\pi\)
\(432\) 1917.00i 0.213499i
\(433\) − 4322.00i − 0.479681i −0.970812 0.239841i \(-0.922905\pi\)
0.970812 0.239841i \(-0.0770952\pi\)
\(434\) −3360.00 −0.371625
\(435\) 0 0
\(436\) 1478.00 0.162347
\(437\) 0 0
\(438\) 4518.00i 0.492873i
\(439\) 9016.00 0.980205 0.490103 0.871665i \(-0.336959\pi\)
0.490103 + 0.871665i \(0.336959\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 4284.00i 0.461016i
\(443\) 5268.00i 0.564989i 0.959269 + 0.282495i \(0.0911619\pi\)
−0.959269 + 0.282495i \(0.908838\pi\)
\(444\) 1194.00 0.127623
\(445\) 0 0
\(446\) 5664.00 0.601341
\(447\) 5238.00i 0.554248i
\(448\) − 3031.00i − 0.319646i
\(449\) 5310.00 0.558117 0.279058 0.960274i \(-0.409978\pi\)
0.279058 + 0.960274i \(0.409978\pi\)
\(450\) 0 0
\(451\) 11448.0 1.19527
\(452\) 402.000i 0.0418329i
\(453\) − 696.000i − 0.0721875i
\(454\) −14148.0 −1.46255
\(455\) 0 0
\(456\) 7812.00 0.802260
\(457\) 15770.0i 1.61420i 0.590415 + 0.807100i \(0.298964\pi\)
−0.590415 + 0.807100i \(0.701036\pi\)
\(458\) − 5070.00i − 0.517261i
\(459\) 1134.00 0.115317
\(460\) 0 0
\(461\) −5370.00 −0.542529 −0.271264 0.962505i \(-0.587442\pi\)
−0.271264 + 0.962505i \(0.587442\pi\)
\(462\) − 2268.00i − 0.228392i
\(463\) 3328.00i 0.334050i 0.985953 + 0.167025i \(0.0534161\pi\)
−0.985953 + 0.167025i \(0.946584\pi\)
\(464\) 7242.00 0.724572
\(465\) 0 0
\(466\) −414.000 −0.0411549
\(467\) 4548.00i 0.450656i 0.974283 + 0.225328i \(0.0723454\pi\)
−0.974283 + 0.225328i \(0.927655\pi\)
\(468\) 306.000i 0.0302240i
\(469\) −644.000 −0.0634055
\(470\) 0 0
\(471\) −5082.00 −0.497168
\(472\) − 2772.00i − 0.270321i
\(473\) − 9648.00i − 0.937876i
\(474\) 9216.00 0.893048
\(475\) 0 0
\(476\) 294.000 0.0283098
\(477\) − 4482.00i − 0.430224i
\(478\) 5688.00i 0.544274i
\(479\) 8064.00 0.769214 0.384607 0.923080i \(-0.374337\pi\)
0.384607 + 0.923080i \(0.374337\pi\)
\(480\) 0 0
\(481\) −13532.0 −1.28276
\(482\) 10794.0i 1.02003i
\(483\) 0 0
\(484\) 35.0000 0.00328700
\(485\) 0 0
\(486\) 729.000 0.0680414
\(487\) 16616.0i 1.54608i 0.634355 + 0.773042i \(0.281266\pi\)
−0.634355 + 0.773042i \(0.718734\pi\)
\(488\) − 8358.00i − 0.775305i
\(489\) −8796.00 −0.813433
\(490\) 0 0
\(491\) −7140.00 −0.656260 −0.328130 0.944633i \(-0.606418\pi\)
−0.328130 + 0.944633i \(0.606418\pi\)
\(492\) 954.000i 0.0874180i
\(493\) − 4284.00i − 0.391362i
\(494\) 12648.0 1.15194
\(495\) 0 0
\(496\) 11360.0 1.02839
\(497\) − 5040.00i − 0.454879i
\(498\) 1836.00i 0.165207i
\(499\) 9124.00 0.818530 0.409265 0.912416i \(-0.365785\pi\)
0.409265 + 0.912416i \(0.365785\pi\)
\(500\) 0 0
\(501\) −3528.00 −0.314610
\(502\) 9180.00i 0.816182i
\(503\) 6552.00i 0.580794i 0.956906 + 0.290397i \(0.0937873\pi\)
−0.956906 + 0.290397i \(0.906213\pi\)
\(504\) −1323.00 −0.116927
\(505\) 0 0
\(506\) 0 0
\(507\) 3123.00i 0.273565i
\(508\) − 1280.00i − 0.111793i
\(509\) −2790.00 −0.242956 −0.121478 0.992594i \(-0.538763\pi\)
−0.121478 + 0.992594i \(0.538763\pi\)
\(510\) 0 0
\(511\) −3514.00 −0.304208
\(512\) 8733.00i 0.753804i
\(513\) − 3348.00i − 0.288144i
\(514\) −20466.0 −1.75626
\(515\) 0 0
\(516\) 804.000 0.0685933
\(517\) − 8640.00i − 0.734984i
\(518\) 8358.00i 0.708937i
\(519\) 2610.00 0.220744
\(520\) 0 0
\(521\) −14862.0 −1.24974 −0.624871 0.780728i \(-0.714849\pi\)
−0.624871 + 0.780728i \(0.714849\pi\)
\(522\) − 2754.00i − 0.230918i
\(523\) − 17660.0i − 1.47652i −0.674518 0.738258i \(-0.735649\pi\)
0.674518 0.738258i \(-0.264351\pi\)
\(524\) −1764.00 −0.147062
\(525\) 0 0
\(526\) −7776.00 −0.644581
\(527\) − 6720.00i − 0.555461i
\(528\) 7668.00i 0.632021i
\(529\) 12167.0 1.00000
\(530\) 0 0
\(531\) −1188.00 −0.0970900
\(532\) − 868.000i − 0.0707379i
\(533\) − 10812.0i − 0.878649i
\(534\) −3186.00 −0.258187
\(535\) 0 0
\(536\) 1932.00 0.155690
\(537\) 6948.00i 0.558340i
\(538\) 24642.0i 1.97471i
\(539\) 1764.00 0.140966
\(540\) 0 0
\(541\) −19834.0 −1.57621 −0.788106 0.615540i \(-0.788938\pi\)
−0.788106 + 0.615540i \(0.788938\pi\)
\(542\) 16032.0i 1.27054i
\(543\) − 318.000i − 0.0251320i
\(544\) −1890.00 −0.148958
\(545\) 0 0
\(546\) −2142.00 −0.167892
\(547\) 20972.0i 1.63930i 0.572863 + 0.819651i \(0.305833\pi\)
−0.572863 + 0.819651i \(0.694167\pi\)
\(548\) 2358.00i 0.183812i
\(549\) −3582.00 −0.278463
\(550\) 0 0
\(551\) −12648.0 −0.977900
\(552\) 0 0
\(553\) 7168.00i 0.551201i
\(554\) −19542.0 −1.49866
\(555\) 0 0
\(556\) −52.0000 −0.00396635
\(557\) 21174.0i 1.61072i 0.592786 + 0.805360i \(0.298028\pi\)
−0.592786 + 0.805360i \(0.701972\pi\)
\(558\) − 4320.00i − 0.327742i
\(559\) −9112.00 −0.689439
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) − 19854.0i − 1.49020i
\(563\) 17772.0i 1.33037i 0.746677 + 0.665187i \(0.231648\pi\)
−0.746677 + 0.665187i \(0.768352\pi\)
\(564\) 720.000 0.0537544
\(565\) 0 0
\(566\) −9780.00 −0.726297
\(567\) 567.000i 0.0419961i
\(568\) 15120.0i 1.11694i
\(569\) −8250.00 −0.607835 −0.303917 0.952698i \(-0.598295\pi\)
−0.303917 + 0.952698i \(0.598295\pi\)
\(570\) 0 0
\(571\) 20756.0 1.52121 0.760606 0.649214i \(-0.224902\pi\)
0.760606 + 0.649214i \(0.224902\pi\)
\(572\) 1224.00i 0.0894720i
\(573\) − 3384.00i − 0.246717i
\(574\) −6678.00 −0.485600
\(575\) 0 0
\(576\) 3897.00 0.281901
\(577\) 2.00000i 0 0.000144300i 1.00000 7.21500e-5i \(2.29661e-5\pi\)
−1.00000 7.21500e-5i \(0.999977\pi\)
\(578\) − 9447.00i − 0.679833i
\(579\) 12102.0 0.868639
\(580\) 0 0
\(581\) −1428.00 −0.101968
\(582\) − 2574.00i − 0.183326i
\(583\) − 17928.0i − 1.27359i
\(584\) 10542.0 0.746971
\(585\) 0 0
\(586\) −15354.0 −1.08237
\(587\) 26364.0i 1.85376i 0.375354 + 0.926881i \(0.377521\pi\)
−0.375354 + 0.926881i \(0.622479\pi\)
\(588\) 147.000i 0.0103098i
\(589\) −19840.0 −1.38793
\(590\) 0 0
\(591\) 3942.00 0.274369
\(592\) − 28258.0i − 1.96182i
\(593\) − 2298.00i − 0.159136i −0.996829 0.0795679i \(-0.974646\pi\)
0.996829 0.0795679i \(-0.0253541\pi\)
\(594\) 2916.00 0.201422
\(595\) 0 0
\(596\) −1746.00 −0.119998
\(597\) − 15288.0i − 1.04807i
\(598\) 0 0
\(599\) −3072.00 −0.209547 −0.104773 0.994496i \(-0.533412\pi\)
−0.104773 + 0.994496i \(0.533412\pi\)
\(600\) 0 0
\(601\) 24554.0 1.66652 0.833260 0.552881i \(-0.186472\pi\)
0.833260 + 0.552881i \(0.186472\pi\)
\(602\) 5628.00i 0.381030i
\(603\) − 828.000i − 0.0559184i
\(604\) 232.000 0.0156290
\(605\) 0 0
\(606\) 3726.00 0.249766
\(607\) 16832.0i 1.12552i 0.826621 + 0.562759i \(0.190260\pi\)
−0.826621 + 0.562759i \(0.809740\pi\)
\(608\) 5580.00i 0.372202i
\(609\) 2142.00 0.142526
\(610\) 0 0
\(611\) −8160.00 −0.540292
\(612\) 378.000i 0.0249669i
\(613\) 2482.00i 0.163535i 0.996651 + 0.0817676i \(0.0260565\pi\)
−0.996651 + 0.0817676i \(0.973943\pi\)
\(614\) 1356.00 0.0891266
\(615\) 0 0
\(616\) −5292.00 −0.346138
\(617\) − 15798.0i − 1.03080i −0.856950 0.515400i \(-0.827643\pi\)
0.856950 0.515400i \(-0.172357\pi\)
\(618\) − 504.000i − 0.0328056i
\(619\) 15460.0 1.00386 0.501930 0.864908i \(-0.332623\pi\)
0.501930 + 0.864908i \(0.332623\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 15048.0i − 0.970048i
\(623\) − 2478.00i − 0.159356i
\(624\) 7242.00 0.464603
\(625\) 0 0
\(626\) −16206.0 −1.03470
\(627\) − 13392.0i − 0.852990i
\(628\) − 1694.00i − 0.107640i
\(629\) −16716.0 −1.05964
\(630\) 0 0
\(631\) −7720.00 −0.487050 −0.243525 0.969895i \(-0.578304\pi\)
−0.243525 + 0.969895i \(0.578304\pi\)
\(632\) − 21504.0i − 1.35345i
\(633\) − 9228.00i − 0.579431i
\(634\) 30258.0 1.89542
\(635\) 0 0
\(636\) 1494.00 0.0931462
\(637\) − 1666.00i − 0.103625i
\(638\) − 11016.0i − 0.683586i
\(639\) 6480.00 0.401166
\(640\) 0 0
\(641\) −17262.0 −1.06366 −0.531832 0.846850i \(-0.678496\pi\)
−0.531832 + 0.846850i \(0.678496\pi\)
\(642\) 108.000i 0.00663928i
\(643\) 12220.0i 0.749471i 0.927132 + 0.374735i \(0.122266\pi\)
−0.927132 + 0.374735i \(0.877734\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 15624.0 0.951576
\(647\) 13560.0i 0.823955i 0.911194 + 0.411977i \(0.135162\pi\)
−0.911194 + 0.411977i \(0.864838\pi\)
\(648\) − 1701.00i − 0.103120i
\(649\) −4752.00 −0.287415
\(650\) 0 0
\(651\) 3360.00 0.202287
\(652\) − 2932.00i − 0.176113i
\(653\) − 23094.0i − 1.38398i −0.721908 0.691989i \(-0.756735\pi\)
0.721908 0.691989i \(-0.243265\pi\)
\(654\) −13302.0 −0.795335
\(655\) 0 0
\(656\) 22578.0 1.34378
\(657\) − 4518.00i − 0.268286i
\(658\) 5040.00i 0.298601i
\(659\) −22548.0 −1.33285 −0.666423 0.745574i \(-0.732175\pi\)
−0.666423 + 0.745574i \(0.732175\pi\)
\(660\) 0 0
\(661\) 17462.0 1.02752 0.513762 0.857933i \(-0.328252\pi\)
0.513762 + 0.857933i \(0.328252\pi\)
\(662\) 24132.0i 1.41679i
\(663\) − 4284.00i − 0.250945i
\(664\) 4284.00 0.250379
\(665\) 0 0
\(666\) −10746.0 −0.625224
\(667\) 0 0
\(668\) − 1176.00i − 0.0681150i
\(669\) −5664.00 −0.327329
\(670\) 0 0
\(671\) −14328.0 −0.824331
\(672\) − 945.000i − 0.0542473i
\(673\) 22462.0i 1.28655i 0.765636 + 0.643274i \(0.222424\pi\)
−0.765636 + 0.643274i \(0.777576\pi\)
\(674\) 12534.0 0.716308
\(675\) 0 0
\(676\) −1041.00 −0.0592285
\(677\) − 25554.0i − 1.45069i −0.688383 0.725347i \(-0.741679\pi\)
0.688383 0.725347i \(-0.258321\pi\)
\(678\) − 3618.00i − 0.204939i
\(679\) 2002.00 0.113151
\(680\) 0 0
\(681\) 14148.0 0.796112
\(682\) − 17280.0i − 0.970213i
\(683\) − 9276.00i − 0.519672i −0.965653 0.259836i \(-0.916331\pi\)
0.965653 0.259836i \(-0.0836686\pi\)
\(684\) 1116.00 0.0623850
\(685\) 0 0
\(686\) −1029.00 −0.0572703
\(687\) 5070.00i 0.281561i
\(688\) − 19028.0i − 1.05441i
\(689\) −16932.0 −0.936223
\(690\) 0 0
\(691\) 27380.0 1.50736 0.753679 0.657243i \(-0.228277\pi\)
0.753679 + 0.657243i \(0.228277\pi\)
\(692\) 870.000i 0.0477925i
\(693\) 2268.00i 0.124321i
\(694\) 468.000 0.0255980
\(695\) 0 0
\(696\) −6426.00 −0.349967
\(697\) − 13356.0i − 0.725817i
\(698\) − 37254.0i − 2.02018i
\(699\) 414.000 0.0224019
\(700\) 0 0
\(701\) 25830.0 1.39171 0.695853 0.718184i \(-0.255027\pi\)
0.695853 + 0.718184i \(0.255027\pi\)
\(702\) − 2754.00i − 0.148067i
\(703\) 49352.0i 2.64772i
\(704\) 15588.0 0.834510
\(705\) 0 0
\(706\) 23490.0 1.25221
\(707\) 2898.00i 0.154159i
\(708\) − 396.000i − 0.0210206i
\(709\) 6226.00 0.329792 0.164896 0.986311i \(-0.447271\pi\)
0.164896 + 0.986311i \(0.447271\pi\)
\(710\) 0 0
\(711\) −9216.00 −0.486114
\(712\) 7434.00i 0.391293i
\(713\) 0 0
\(714\) −2646.00 −0.138689
\(715\) 0 0
\(716\) −2316.00 −0.120884
\(717\) − 5688.00i − 0.296265i
\(718\) − 27936.0i − 1.45204i
\(719\) 15072.0 0.781767 0.390884 0.920440i \(-0.372169\pi\)
0.390884 + 0.920440i \(0.372169\pi\)
\(720\) 0 0
\(721\) 392.000 0.0202480
\(722\) − 25551.0i − 1.31705i
\(723\) − 10794.0i − 0.555233i
\(724\) 106.000 0.00544124
\(725\) 0 0
\(726\) −315.000 −0.0161030
\(727\) − 32920.0i − 1.67942i −0.543038 0.839708i \(-0.682726\pi\)
0.543038 0.839708i \(-0.317274\pi\)
\(728\) 4998.00i 0.254448i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −11256.0 −0.569519
\(732\) − 1194.00i − 0.0602889i
\(733\) 6946.00i 0.350009i 0.984568 + 0.175004i \(0.0559939\pi\)
−0.984568 + 0.175004i \(0.944006\pi\)
\(734\) −11280.0 −0.567238
\(735\) 0 0
\(736\) 0 0
\(737\) − 3312.00i − 0.165535i
\(738\) − 8586.00i − 0.428259i
\(739\) 2356.00 0.117276 0.0586379 0.998279i \(-0.481324\pi\)
0.0586379 + 0.998279i \(0.481324\pi\)
\(740\) 0 0
\(741\) −12648.0 −0.627039
\(742\) 10458.0i 0.517419i
\(743\) 23520.0i 1.16133i 0.814144 + 0.580663i \(0.197207\pi\)
−0.814144 + 0.580663i \(0.802793\pi\)
\(744\) −10080.0 −0.496708
\(745\) 0 0
\(746\) −17610.0 −0.864273
\(747\) − 1836.00i − 0.0899273i
\(748\) 1512.00i 0.0739094i
\(749\) −84.0000 −0.00409785
\(750\) 0 0
\(751\) 3008.00 0.146156 0.0730782 0.997326i \(-0.476718\pi\)
0.0730782 + 0.997326i \(0.476718\pi\)
\(752\) − 17040.0i − 0.826310i
\(753\) − 9180.00i − 0.444273i
\(754\) −10404.0 −0.502508
\(755\) 0 0
\(756\) −189.000 −0.00909241
\(757\) − 20770.0i − 0.997224i −0.866825 0.498612i \(-0.833843\pi\)
0.866825 0.498612i \(-0.166157\pi\)
\(758\) − 5556.00i − 0.266231i
\(759\) 0 0
\(760\) 0 0
\(761\) 11538.0 0.549609 0.274804 0.961500i \(-0.411387\pi\)
0.274804 + 0.961500i \(0.411387\pi\)
\(762\) 11520.0i 0.547671i
\(763\) − 10346.0i − 0.490892i
\(764\) 1128.00 0.0534157
\(765\) 0 0
\(766\) −6480.00 −0.305655
\(767\) 4488.00i 0.211281i
\(768\) 4539.00i 0.213264i
\(769\) −8498.00 −0.398499 −0.199249 0.979949i \(-0.563850\pi\)
−0.199249 + 0.979949i \(0.563850\pi\)
\(770\) 0 0
\(771\) 20466.0 0.955986
\(772\) 4034.00i 0.188066i
\(773\) 32322.0i 1.50393i 0.659200 + 0.751967i \(0.270895\pi\)
−0.659200 + 0.751967i \(0.729105\pi\)
\(774\) −7236.00 −0.336037
\(775\) 0 0
\(776\) −6006.00 −0.277839
\(777\) − 8358.00i − 0.385896i
\(778\) − 20358.0i − 0.938136i
\(779\) −39432.0 −1.81360
\(780\) 0 0
\(781\) 25920.0 1.18757
\(782\) 0 0
\(783\) 2754.00i 0.125696i
\(784\) 3479.00 0.158482
\(785\) 0 0
\(786\) 15876.0 0.720456
\(787\) 26228.0i 1.18796i 0.804479 + 0.593982i \(0.202445\pi\)
−0.804479 + 0.593982i \(0.797555\pi\)
\(788\) 1314.00i 0.0594027i
\(789\) 7776.00 0.350866
\(790\) 0 0
\(791\) 2814.00 0.126491
\(792\) − 6804.00i − 0.305265i
\(793\) 13532.0i 0.605972i
\(794\) −19542.0 −0.873450
\(795\) 0 0
\(796\) 5096.00 0.226913
\(797\) − 43338.0i − 1.92611i −0.269302 0.963056i \(-0.586793\pi\)
0.269302 0.963056i \(-0.413207\pi\)
\(798\) 7812.00i 0.346544i
\(799\) −10080.0 −0.446314
\(800\) 0 0
\(801\) 3186.00 0.140539
\(802\) − 9990.00i − 0.439849i
\(803\) − 18072.0i − 0.794206i
\(804\) 276.000 0.0121067
\(805\) 0 0
\(806\) −16320.0 −0.713210
\(807\) − 24642.0i − 1.07489i
\(808\) − 8694.00i − 0.378532i
\(809\) 28902.0 1.25604 0.628022 0.778195i \(-0.283865\pi\)
0.628022 + 0.778195i \(0.283865\pi\)
\(810\) 0 0
\(811\) 27164.0 1.17615 0.588075 0.808807i \(-0.299886\pi\)
0.588075 + 0.808807i \(0.299886\pi\)
\(812\) 714.000i 0.0308577i
\(813\) − 16032.0i − 0.691595i
\(814\) −42984.0 −1.85085
\(815\) 0 0
\(816\) 8946.00 0.383790
\(817\) 33232.0i 1.42306i
\(818\) − 16194.0i − 0.692188i
\(819\) 2142.00 0.0913889
\(820\) 0 0
\(821\) −17202.0 −0.731247 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(822\) − 21222.0i − 0.900489i
\(823\) 5992.00i 0.253789i 0.991916 + 0.126894i \(0.0405009\pi\)
−0.991916 + 0.126894i \(0.959499\pi\)
\(824\) −1176.00 −0.0497183
\(825\) 0 0
\(826\) 2772.00 0.116768
\(827\) 25884.0i 1.08836i 0.838968 + 0.544181i \(0.183159\pi\)
−0.838968 + 0.544181i \(0.816841\pi\)
\(828\) 0 0
\(829\) 1474.00 0.0617541 0.0308770 0.999523i \(-0.490170\pi\)
0.0308770 + 0.999523i \(0.490170\pi\)
\(830\) 0 0
\(831\) 19542.0 0.815770
\(832\) − 14722.0i − 0.613454i
\(833\) − 2058.00i − 0.0856008i
\(834\) 468.000 0.0194311
\(835\) 0 0
\(836\) 4464.00 0.184678
\(837\) 4320.00i 0.178400i
\(838\) 39276.0i 1.61905i
\(839\) −33528.0 −1.37964 −0.689818 0.723983i \(-0.742310\pi\)
−0.689818 + 0.723983i \(0.742310\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 966.000i 0.0395375i
\(843\) 19854.0i 0.811160i
\(844\) 3076.00 0.125451
\(845\) 0 0
\(846\) −6480.00 −0.263342
\(847\) − 245.000i − 0.00993896i
\(848\) − 35358.0i − 1.43184i
\(849\) 9780.00 0.395346
\(850\) 0 0
\(851\) 0 0
\(852\) 2160.00i 0.0868549i
\(853\) − 1190.00i − 0.0477665i −0.999715 0.0238832i \(-0.992397\pi\)
0.999715 0.0238832i \(-0.00760300\pi\)
\(854\) 8358.00 0.334900
\(855\) 0 0
\(856\) 252.000 0.0100621
\(857\) 34578.0i 1.37825i 0.724642 + 0.689126i \(0.242005\pi\)
−0.724642 + 0.689126i \(0.757995\pi\)
\(858\) − 11016.0i − 0.438322i
\(859\) 44404.0 1.76373 0.881865 0.471501i \(-0.156288\pi\)
0.881865 + 0.471501i \(0.156288\pi\)
\(860\) 0 0
\(861\) 6678.00 0.264327
\(862\) − 7848.00i − 0.310097i
\(863\) 38328.0i 1.51182i 0.654676 + 0.755910i \(0.272805\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(864\) 1215.00 0.0478416
\(865\) 0 0
\(866\) −12966.0 −0.508779
\(867\) 9447.00i 0.370054i
\(868\) 1120.00i 0.0437964i
\(869\) −36864.0 −1.43904
\(870\) 0 0
\(871\) −3128.00 −0.121686
\(872\) 31038.0i 1.20537i
\(873\) 2574.00i 0.0997900i
\(874\) 0 0
\(875\) 0 0
\(876\) 1506.00 0.0580856
\(877\) − 38842.0i − 1.49555i −0.663950 0.747777i \(-0.731121\pi\)
0.663950 0.747777i \(-0.268879\pi\)
\(878\) − 27048.0i − 1.03966i
\(879\) 15354.0 0.589167
\(880\) 0 0
\(881\) −35046.0 −1.34022 −0.670108 0.742264i \(-0.733752\pi\)
−0.670108 + 0.742264i \(0.733752\pi\)
\(882\) − 1323.00i − 0.0505076i
\(883\) − 14204.0i − 0.541339i −0.962672 0.270670i \(-0.912755\pi\)
0.962672 0.270670i \(-0.0872451\pi\)
\(884\) 1428.00 0.0543313
\(885\) 0 0
\(886\) 15804.0 0.599262
\(887\) − 26136.0i − 0.989359i −0.869076 0.494679i \(-0.835286\pi\)
0.869076 0.494679i \(-0.164714\pi\)
\(888\) 25074.0i 0.947554i
\(889\) −8960.00 −0.338030
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) − 1888.00i − 0.0708687i
\(893\) 29760.0i 1.11521i
\(894\) 15714.0 0.587869
\(895\) 0 0
\(896\) −11613.0 −0.432995
\(897\) 0 0
\(898\) − 15930.0i − 0.591972i
\(899\) 16320.0 0.605453
\(900\) 0 0
\(901\) −20916.0 −0.773377
\(902\) − 34344.0i − 1.26777i
\(903\) − 5628.00i − 0.207407i
\(904\) −8442.00 −0.310594
\(905\) 0 0
\(906\) −2088.00 −0.0765664
\(907\) − 9052.00i − 0.331386i −0.986177 0.165693i \(-0.947014\pi\)
0.986177 0.165693i \(-0.0529860\pi\)
\(908\) 4716.00i 0.172363i
\(909\) −3726.00 −0.135956
\(910\) 0 0
\(911\) 5016.00 0.182423 0.0912116 0.995832i \(-0.470926\pi\)
0.0912116 + 0.995832i \(0.470926\pi\)
\(912\) − 26412.0i − 0.958979i
\(913\) − 7344.00i − 0.266211i
\(914\) 47310.0 1.71212
\(915\) 0 0
\(916\) −1690.00 −0.0609598
\(917\) 12348.0i 0.444675i
\(918\) − 3402.00i − 0.122312i
\(919\) −44552.0 −1.59917 −0.799584 0.600555i \(-0.794946\pi\)
−0.799584 + 0.600555i \(0.794946\pi\)
\(920\) 0 0
\(921\) −1356.00 −0.0485144
\(922\) 16110.0i 0.575439i
\(923\) − 24480.0i − 0.872989i
\(924\) −756.000 −0.0269162
\(925\) 0 0
\(926\) 9984.00 0.354314
\(927\) 504.000i 0.0178571i
\(928\) − 4590.00i − 0.162364i
\(929\) −24234.0 −0.855858 −0.427929 0.903812i \(-0.640757\pi\)
−0.427929 + 0.903812i \(0.640757\pi\)
\(930\) 0 0
\(931\) −6076.00 −0.213891
\(932\) 138.000i 0.00485015i
\(933\) 15048.0i 0.528027i
\(934\) 13644.0 0.477993
\(935\) 0 0
\(936\) −6426.00 −0.224402
\(937\) − 13894.0i − 0.484415i −0.970224 0.242208i \(-0.922128\pi\)
0.970224 0.242208i \(-0.0778715\pi\)
\(938\) 1932.00i 0.0672516i
\(939\) 16206.0 0.563219
\(940\) 0 0
\(941\) 46758.0 1.61984 0.809919 0.586542i \(-0.199511\pi\)
0.809919 + 0.586542i \(0.199511\pi\)
\(942\) 15246.0i 0.527326i
\(943\) 0 0
\(944\) −9372.00 −0.323128
\(945\) 0 0
\(946\) −28944.0 −0.994768
\(947\) 13812.0i 0.473949i 0.971516 + 0.236974i \(0.0761558\pi\)
−0.971516 + 0.236974i \(0.923844\pi\)
\(948\) − 3072.00i − 0.105247i
\(949\) −17068.0 −0.583826
\(950\) 0 0
\(951\) −30258.0 −1.03174
\(952\) 6174.00i 0.210190i
\(953\) 58518.0i 1.98907i 0.104402 + 0.994535i \(0.466707\pi\)
−0.104402 + 0.994535i \(0.533293\pi\)
\(954\) −13446.0 −0.456321
\(955\) 0 0
\(956\) 1896.00 0.0641433
\(957\) 11016.0i 0.372097i
\(958\) − 24192.0i − 0.815875i
\(959\) 16506.0 0.555794
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 40596.0i 1.36057i
\(963\) − 108.000i − 0.00361397i
\(964\) 3598.00 0.120211
\(965\) 0 0
\(966\) 0 0
\(967\) 19640.0i 0.653133i 0.945174 + 0.326567i \(0.105892\pi\)
−0.945174 + 0.326567i \(0.894108\pi\)
\(968\) 735.000i 0.0244047i
\(969\) −15624.0 −0.517972
\(970\) 0 0
\(971\) −58308.0 −1.92708 −0.963539 0.267568i \(-0.913780\pi\)
−0.963539 + 0.267568i \(0.913780\pi\)
\(972\) − 243.000i − 0.00801875i
\(973\) 364.000i 0.0119931i
\(974\) 49848.0 1.63987
\(975\) 0 0
\(976\) −28258.0 −0.926759
\(977\) − 23550.0i − 0.771168i −0.922673 0.385584i \(-0.874000\pi\)
0.922673 0.385584i \(-0.126000\pi\)
\(978\) 26388.0i 0.862776i
\(979\) 12744.0 0.416037
\(980\) 0 0
\(981\) 13302.0 0.432926
\(982\) 21420.0i 0.696069i
\(983\) − 15768.0i − 0.511619i −0.966727 0.255809i \(-0.917658\pi\)
0.966727 0.255809i \(-0.0823419\pi\)
\(984\) −20034.0 −0.649045
\(985\) 0 0
\(986\) −12852.0 −0.415102
\(987\) − 5040.00i − 0.162538i
\(988\) − 4216.00i − 0.135758i
\(989\) 0 0
\(990\) 0 0
\(991\) 35264.0 1.13037 0.565186 0.824964i \(-0.308805\pi\)
0.565186 + 0.824964i \(0.308805\pi\)
\(992\) − 7200.00i − 0.230444i
\(993\) − 24132.0i − 0.771204i
\(994\) −15120.0 −0.482472
\(995\) 0 0
\(996\) 612.000 0.0194698
\(997\) − 29338.0i − 0.931940i −0.884801 0.465970i \(-0.845706\pi\)
0.884801 0.465970i \(-0.154294\pi\)
\(998\) − 27372.0i − 0.868182i
\(999\) 10746.0 0.340329
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 525.4.d.c.274.1 2
5.2 odd 4 525.4.a.g.1.1 1
5.3 odd 4 21.4.a.a.1.1 1
5.4 even 2 inner 525.4.d.c.274.2 2
15.2 even 4 1575.4.a.b.1.1 1
15.8 even 4 63.4.a.c.1.1 1
20.3 even 4 336.4.a.f.1.1 1
35.3 even 12 147.4.e.g.79.1 2
35.13 even 4 147.4.a.c.1.1 1
35.18 odd 12 147.4.e.i.79.1 2
35.23 odd 12 147.4.e.i.67.1 2
35.33 even 12 147.4.e.g.67.1 2
40.3 even 4 1344.4.a.n.1.1 1
40.13 odd 4 1344.4.a.ba.1.1 1
60.23 odd 4 1008.4.a.v.1.1 1
105.23 even 12 441.4.e.b.361.1 2
105.38 odd 12 441.4.e.d.226.1 2
105.53 even 12 441.4.e.b.226.1 2
105.68 odd 12 441.4.e.d.361.1 2
105.83 odd 4 441.4.a.j.1.1 1
140.83 odd 4 2352.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 5.3 odd 4
63.4.a.c.1.1 1 15.8 even 4
147.4.a.c.1.1 1 35.13 even 4
147.4.e.g.67.1 2 35.33 even 12
147.4.e.g.79.1 2 35.3 even 12
147.4.e.i.67.1 2 35.23 odd 12
147.4.e.i.79.1 2 35.18 odd 12
336.4.a.f.1.1 1 20.3 even 4
441.4.a.j.1.1 1 105.83 odd 4
441.4.e.b.226.1 2 105.53 even 12
441.4.e.b.361.1 2 105.23 even 12
441.4.e.d.226.1 2 105.38 odd 12
441.4.e.d.361.1 2 105.68 odd 12
525.4.a.g.1.1 1 5.2 odd 4
525.4.d.c.274.1 2 1.1 even 1 trivial
525.4.d.c.274.2 2 5.4 even 2 inner
1008.4.a.v.1.1 1 60.23 odd 4
1344.4.a.n.1.1 1 40.3 even 4
1344.4.a.ba.1.1 1 40.13 odd 4
1575.4.a.b.1.1 1 15.2 even 4
2352.4.a.r.1.1 1 140.83 odd 4