Properties

Label 1350.4.c.p.649.2
Level $1350$
Weight $4$
Character 1350.649
Analytic conductor $79.653$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.6525785077\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.4.c.p.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} -23.0000i q^{7} -8.00000i q^{8} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} -23.0000i q^{7} -8.00000i q^{8} +30.0000 q^{11} -34.0000i q^{13} +46.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +139.000 q^{19} +60.0000i q^{22} +192.000i q^{23} +68.0000 q^{26} +92.0000i q^{28} -234.000 q^{29} -55.0000 q^{31} +32.0000i q^{32} -84.0000 q^{34} -191.000i q^{37} +278.000i q^{38} +138.000 q^{41} +53.0000i q^{43} -120.000 q^{44} -384.000 q^{46} -366.000i q^{47} -186.000 q^{49} +136.000i q^{52} -330.000i q^{53} -184.000 q^{56} -468.000i q^{58} +396.000 q^{59} +23.0000 q^{61} -110.000i q^{62} -64.0000 q^{64} -452.000i q^{67} -168.000i q^{68} +204.000 q^{71} -691.000i q^{73} +382.000 q^{74} -556.000 q^{76} -690.000i q^{77} +709.000 q^{79} +276.000i q^{82} +1098.00i q^{83} -106.000 q^{86} -240.000i q^{88} +816.000 q^{89} -782.000 q^{91} -768.000i q^{92} +732.000 q^{94} -905.000i q^{97} -372.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 60 q^{11} + 92 q^{14} + 32 q^{16} + 278 q^{19} + 136 q^{26} - 468 q^{29} - 110 q^{31} - 168 q^{34} + 276 q^{41} - 240 q^{44} - 768 q^{46} - 372 q^{49} - 368 q^{56} + 792 q^{59} + 46 q^{61} - 128 q^{64} + 408 q^{71} + 764 q^{74} - 1112 q^{76} + 1418 q^{79} - 212 q^{86} + 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 23.0000i − 1.24188i −0.783857 0.620942i \(-0.786750\pi\)
0.783857 0.620942i \(-0.213250\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) − 34.0000i − 0.725377i −0.931910 0.362689i \(-0.881859\pi\)
0.931910 0.362689i \(-0.118141\pi\)
\(14\) 46.0000 0.878144
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 42.0000i 0.599206i 0.954064 + 0.299603i \(0.0968542\pi\)
−0.954064 + 0.299603i \(0.903146\pi\)
\(18\) 0 0
\(19\) 139.000 1.67836 0.839179 0.543856i \(-0.183036\pi\)
0.839179 + 0.543856i \(0.183036\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 60.0000i 0.581456i
\(23\) 192.000i 1.74064i 0.492485 + 0.870321i \(0.336089\pi\)
−0.492485 + 0.870321i \(0.663911\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 68.0000 0.512919
\(27\) 0 0
\(28\) 92.0000i 0.620942i
\(29\) −234.000 −1.49837 −0.749185 0.662361i \(-0.769554\pi\)
−0.749185 + 0.662361i \(0.769554\pi\)
\(30\) 0 0
\(31\) −55.0000 −0.318655 −0.159327 0.987226i \(-0.550933\pi\)
−0.159327 + 0.987226i \(0.550933\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) −84.0000 −0.423702
\(35\) 0 0
\(36\) 0 0
\(37\) − 191.000i − 0.848654i −0.905509 0.424327i \(-0.860511\pi\)
0.905509 0.424327i \(-0.139489\pi\)
\(38\) 278.000i 1.18678i
\(39\) 0 0
\(40\) 0 0
\(41\) 138.000 0.525658 0.262829 0.964842i \(-0.415344\pi\)
0.262829 + 0.964842i \(0.415344\pi\)
\(42\) 0 0
\(43\) 53.0000i 0.187963i 0.995574 + 0.0939817i \(0.0299595\pi\)
−0.995574 + 0.0939817i \(0.970040\pi\)
\(44\) −120.000 −0.411152
\(45\) 0 0
\(46\) −384.000 −1.23082
\(47\) − 366.000i − 1.13588i −0.823068 0.567942i \(-0.807740\pi\)
0.823068 0.567942i \(-0.192260\pi\)
\(48\) 0 0
\(49\) −186.000 −0.542274
\(50\) 0 0
\(51\) 0 0
\(52\) 136.000i 0.362689i
\(53\) − 330.000i − 0.855264i −0.903953 0.427632i \(-0.859348\pi\)
0.903953 0.427632i \(-0.140652\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −184.000 −0.439072
\(57\) 0 0
\(58\) − 468.000i − 1.05951i
\(59\) 396.000 0.873810 0.436905 0.899508i \(-0.356075\pi\)
0.436905 + 0.899508i \(0.356075\pi\)
\(60\) 0 0
\(61\) 23.0000 0.0482762 0.0241381 0.999709i \(-0.492316\pi\)
0.0241381 + 0.999709i \(0.492316\pi\)
\(62\) − 110.000i − 0.225323i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 452.000i − 0.824188i −0.911141 0.412094i \(-0.864798\pi\)
0.911141 0.412094i \(-0.135202\pi\)
\(68\) − 168.000i − 0.299603i
\(69\) 0 0
\(70\) 0 0
\(71\) 204.000 0.340991 0.170495 0.985358i \(-0.445463\pi\)
0.170495 + 0.985358i \(0.445463\pi\)
\(72\) 0 0
\(73\) − 691.000i − 1.10788i −0.832556 0.553941i \(-0.813123\pi\)
0.832556 0.553941i \(-0.186877\pi\)
\(74\) 382.000 0.600089
\(75\) 0 0
\(76\) −556.000 −0.839179
\(77\) − 690.000i − 1.02121i
\(78\) 0 0
\(79\) 709.000 1.00973 0.504865 0.863198i \(-0.331542\pi\)
0.504865 + 0.863198i \(0.331542\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 276.000i 0.371696i
\(83\) 1098.00i 1.45206i 0.687662 + 0.726031i \(0.258637\pi\)
−0.687662 + 0.726031i \(0.741363\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −106.000 −0.132910
\(87\) 0 0
\(88\) − 240.000i − 0.290728i
\(89\) 816.000 0.971863 0.485932 0.873997i \(-0.338480\pi\)
0.485932 + 0.873997i \(0.338480\pi\)
\(90\) 0 0
\(91\) −782.000 −0.900834
\(92\) − 768.000i − 0.870321i
\(93\) 0 0
\(94\) 732.000 0.803192
\(95\) 0 0
\(96\) 0 0
\(97\) − 905.000i − 0.947308i −0.880711 0.473654i \(-0.842935\pi\)
0.880711 0.473654i \(-0.157065\pi\)
\(98\) − 372.000i − 0.383446i
\(99\) 0 0
\(100\) 0 0
\(101\) −1278.00 −1.25907 −0.629533 0.776973i \(-0.716754\pi\)
−0.629533 + 0.776973i \(0.716754\pi\)
\(102\) 0 0
\(103\) 605.000i 0.578761i 0.957214 + 0.289381i \(0.0934493\pi\)
−0.957214 + 0.289381i \(0.906551\pi\)
\(104\) −272.000 −0.256460
\(105\) 0 0
\(106\) 660.000 0.604763
\(107\) − 1488.00i − 1.34440i −0.740371 0.672198i \(-0.765350\pi\)
0.740371 0.672198i \(-0.234650\pi\)
\(108\) 0 0
\(109\) −593.000 −0.521093 −0.260546 0.965461i \(-0.583903\pi\)
−0.260546 + 0.965461i \(0.583903\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 368.000i − 0.310471i
\(113\) 324.000i 0.269729i 0.990864 + 0.134864i \(0.0430599\pi\)
−0.990864 + 0.134864i \(0.956940\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 936.000 0.749185
\(117\) 0 0
\(118\) 792.000i 0.617877i
\(119\) 966.000 0.744143
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 46.0000i 0.0341364i
\(123\) 0 0
\(124\) 220.000 0.159327
\(125\) 0 0
\(126\) 0 0
\(127\) − 1928.00i − 1.34711i −0.739139 0.673553i \(-0.764768\pi\)
0.739139 0.673553i \(-0.235232\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 2742.00 1.82878 0.914388 0.404839i \(-0.132672\pi\)
0.914388 + 0.404839i \(0.132672\pi\)
\(132\) 0 0
\(133\) − 3197.00i − 2.08432i
\(134\) 904.000 0.582789
\(135\) 0 0
\(136\) 336.000 0.211851
\(137\) − 1326.00i − 0.826918i −0.910523 0.413459i \(-0.864320\pi\)
0.910523 0.413459i \(-0.135680\pi\)
\(138\) 0 0
\(139\) −893.000 −0.544916 −0.272458 0.962168i \(-0.587837\pi\)
−0.272458 + 0.962168i \(0.587837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 408.000i 0.241117i
\(143\) − 1020.00i − 0.596480i
\(144\) 0 0
\(145\) 0 0
\(146\) 1382.00 0.783391
\(147\) 0 0
\(148\) 764.000i 0.424327i
\(149\) 2502.00 1.37565 0.687825 0.725877i \(-0.258566\pi\)
0.687825 + 0.725877i \(0.258566\pi\)
\(150\) 0 0
\(151\) −2767.00 −1.49123 −0.745613 0.666379i \(-0.767843\pi\)
−0.745613 + 0.666379i \(0.767843\pi\)
\(152\) − 1112.00i − 0.593389i
\(153\) 0 0
\(154\) 1380.00 0.722101
\(155\) 0 0
\(156\) 0 0
\(157\) 2701.00i 1.37301i 0.727123 + 0.686507i \(0.240857\pi\)
−0.727123 + 0.686507i \(0.759143\pi\)
\(158\) 1418.00i 0.713987i
\(159\) 0 0
\(160\) 0 0
\(161\) 4416.00 2.16167
\(162\) 0 0
\(163\) 1748.00i 0.839963i 0.907533 + 0.419981i \(0.137963\pi\)
−0.907533 + 0.419981i \(0.862037\pi\)
\(164\) −552.000 −0.262829
\(165\) 0 0
\(166\) −2196.00 −1.02676
\(167\) 534.000i 0.247438i 0.992317 + 0.123719i \(0.0394822\pi\)
−0.992317 + 0.123719i \(0.960518\pi\)
\(168\) 0 0
\(169\) 1041.00 0.473828
\(170\) 0 0
\(171\) 0 0
\(172\) − 212.000i − 0.0939817i
\(173\) − 192.000i − 0.0843786i −0.999110 0.0421893i \(-0.986567\pi\)
0.999110 0.0421893i \(-0.0134333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 480.000 0.205576
\(177\) 0 0
\(178\) 1632.00i 0.687211i
\(179\) −1140.00 −0.476020 −0.238010 0.971263i \(-0.576495\pi\)
−0.238010 + 0.971263i \(0.576495\pi\)
\(180\) 0 0
\(181\) 398.000 0.163443 0.0817213 0.996655i \(-0.473958\pi\)
0.0817213 + 0.996655i \(0.473958\pi\)
\(182\) − 1564.00i − 0.636986i
\(183\) 0 0
\(184\) 1536.00 0.615410
\(185\) 0 0
\(186\) 0 0
\(187\) 1260.00i 0.492729i
\(188\) 1464.00i 0.567942i
\(189\) 0 0
\(190\) 0 0
\(191\) 3474.00 1.31607 0.658036 0.752986i \(-0.271387\pi\)
0.658036 + 0.752986i \(0.271387\pi\)
\(192\) 0 0
\(193\) − 2713.00i − 1.01184i −0.862579 0.505922i \(-0.831152\pi\)
0.862579 0.505922i \(-0.168848\pi\)
\(194\) 1810.00 0.669848
\(195\) 0 0
\(196\) 744.000 0.271137
\(197\) − 4734.00i − 1.71210i −0.516894 0.856050i \(-0.672912\pi\)
0.516894 0.856050i \(-0.327088\pi\)
\(198\) 0 0
\(199\) −5132.00 −1.82813 −0.914065 0.405568i \(-0.867074\pi\)
−0.914065 + 0.405568i \(0.867074\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 2556.00i − 0.890295i
\(203\) 5382.00i 1.86080i
\(204\) 0 0
\(205\) 0 0
\(206\) −1210.00 −0.409246
\(207\) 0 0
\(208\) − 544.000i − 0.181344i
\(209\) 4170.00 1.38012
\(210\) 0 0
\(211\) 5240.00 1.70965 0.854826 0.518915i \(-0.173664\pi\)
0.854826 + 0.518915i \(0.173664\pi\)
\(212\) 1320.00i 0.427632i
\(213\) 0 0
\(214\) 2976.00 0.950632
\(215\) 0 0
\(216\) 0 0
\(217\) 1265.00i 0.395732i
\(218\) − 1186.00i − 0.368468i
\(219\) 0 0
\(220\) 0 0
\(221\) 1428.00 0.434650
\(222\) 0 0
\(223\) − 4519.00i − 1.35702i −0.734593 0.678508i \(-0.762627\pi\)
0.734593 0.678508i \(-0.237373\pi\)
\(224\) 736.000 0.219536
\(225\) 0 0
\(226\) −648.000 −0.190727
\(227\) − 5064.00i − 1.48066i −0.672244 0.740329i \(-0.734670\pi\)
0.672244 0.740329i \(-0.265330\pi\)
\(228\) 0 0
\(229\) −2573.00 −0.742483 −0.371242 0.928536i \(-0.621068\pi\)
−0.371242 + 0.928536i \(0.621068\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1872.00i 0.529754i
\(233\) 4098.00i 1.15223i 0.817370 + 0.576114i \(0.195431\pi\)
−0.817370 + 0.576114i \(0.804569\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1584.00 −0.436905
\(237\) 0 0
\(238\) 1932.00i 0.526189i
\(239\) 306.000 0.0828180 0.0414090 0.999142i \(-0.486815\pi\)
0.0414090 + 0.999142i \(0.486815\pi\)
\(240\) 0 0
\(241\) 6482.00 1.73254 0.866270 0.499575i \(-0.166511\pi\)
0.866270 + 0.499575i \(0.166511\pi\)
\(242\) − 862.000i − 0.228973i
\(243\) 0 0
\(244\) −92.0000 −0.0241381
\(245\) 0 0
\(246\) 0 0
\(247\) − 4726.00i − 1.21744i
\(248\) 440.000i 0.112661i
\(249\) 0 0
\(250\) 0 0
\(251\) −5850.00 −1.47111 −0.735555 0.677465i \(-0.763079\pi\)
−0.735555 + 0.677465i \(0.763079\pi\)
\(252\) 0 0
\(253\) 5760.00i 1.43134i
\(254\) 3856.00 0.952547
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 5598.00i − 1.35873i −0.733800 0.679365i \(-0.762255\pi\)
0.733800 0.679365i \(-0.237745\pi\)
\(258\) 0 0
\(259\) −4393.00 −1.05393
\(260\) 0 0
\(261\) 0 0
\(262\) 5484.00i 1.29314i
\(263\) − 8286.00i − 1.94272i −0.237603 0.971362i \(-0.576362\pi\)
0.237603 0.971362i \(-0.423638\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6394.00 1.47384
\(267\) 0 0
\(268\) 1808.00i 0.412094i
\(269\) 504.000 0.114236 0.0571179 0.998367i \(-0.481809\pi\)
0.0571179 + 0.998367i \(0.481809\pi\)
\(270\) 0 0
\(271\) −4489.00 −1.00623 −0.503113 0.864221i \(-0.667812\pi\)
−0.503113 + 0.864221i \(0.667812\pi\)
\(272\) 672.000i 0.149801i
\(273\) 0 0
\(274\) 2652.00 0.584720
\(275\) 0 0
\(276\) 0 0
\(277\) − 2213.00i − 0.480023i −0.970770 0.240011i \(-0.922849\pi\)
0.970770 0.240011i \(-0.0771512\pi\)
\(278\) − 1786.00i − 0.385314i
\(279\) 0 0
\(280\) 0 0
\(281\) −2718.00 −0.577019 −0.288509 0.957477i \(-0.593160\pi\)
−0.288509 + 0.957477i \(0.593160\pi\)
\(282\) 0 0
\(283\) 5615.00i 1.17942i 0.807613 + 0.589712i \(0.200759\pi\)
−0.807613 + 0.589712i \(0.799241\pi\)
\(284\) −816.000 −0.170495
\(285\) 0 0
\(286\) 2040.00 0.421775
\(287\) − 3174.00i − 0.652806i
\(288\) 0 0
\(289\) 3149.00 0.640953
\(290\) 0 0
\(291\) 0 0
\(292\) 2764.00i 0.553941i
\(293\) 1488.00i 0.296689i 0.988936 + 0.148345i \(0.0473945\pi\)
−0.988936 + 0.148345i \(0.952606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1528.00 −0.300045
\(297\) 0 0
\(298\) 5004.00i 0.972731i
\(299\) 6528.00 1.26262
\(300\) 0 0
\(301\) 1219.00 0.233429
\(302\) − 5534.00i − 1.05446i
\(303\) 0 0
\(304\) 2224.00 0.419589
\(305\) 0 0
\(306\) 0 0
\(307\) − 7487.00i − 1.39188i −0.718102 0.695938i \(-0.754989\pi\)
0.718102 0.695938i \(-0.245011\pi\)
\(308\) 2760.00i 0.510603i
\(309\) 0 0
\(310\) 0 0
\(311\) 8118.00 1.48016 0.740080 0.672519i \(-0.234788\pi\)
0.740080 + 0.672519i \(0.234788\pi\)
\(312\) 0 0
\(313\) − 5002.00i − 0.903290i −0.892198 0.451645i \(-0.850837\pi\)
0.892198 0.451645i \(-0.149163\pi\)
\(314\) −5402.00 −0.970868
\(315\) 0 0
\(316\) −2836.00 −0.504865
\(317\) − 5082.00i − 0.900421i −0.892922 0.450211i \(-0.851349\pi\)
0.892922 0.450211i \(-0.148651\pi\)
\(318\) 0 0
\(319\) −7020.00 −1.23211
\(320\) 0 0
\(321\) 0 0
\(322\) 8832.00i 1.52853i
\(323\) 5838.00i 1.00568i
\(324\) 0 0
\(325\) 0 0
\(326\) −3496.00 −0.593943
\(327\) 0 0
\(328\) − 1104.00i − 0.185848i
\(329\) −8418.00 −1.41064
\(330\) 0 0
\(331\) 7625.00 1.26619 0.633094 0.774075i \(-0.281785\pi\)
0.633094 + 0.774075i \(0.281785\pi\)
\(332\) − 4392.00i − 0.726031i
\(333\) 0 0
\(334\) −1068.00 −0.174965
\(335\) 0 0
\(336\) 0 0
\(337\) 3778.00i 0.610685i 0.952243 + 0.305342i \(0.0987709\pi\)
−0.952243 + 0.305342i \(0.901229\pi\)
\(338\) 2082.00i 0.335047i
\(339\) 0 0
\(340\) 0 0
\(341\) −1650.00 −0.262031
\(342\) 0 0
\(343\) − 3611.00i − 0.568442i
\(344\) 424.000 0.0664551
\(345\) 0 0
\(346\) 384.000 0.0596646
\(347\) 8268.00i 1.27911i 0.768747 + 0.639553i \(0.220880\pi\)
−0.768747 + 0.639553i \(0.779120\pi\)
\(348\) 0 0
\(349\) −1379.00 −0.211508 −0.105754 0.994392i \(-0.533726\pi\)
−0.105754 + 0.994392i \(0.533726\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 960.000i 0.145364i
\(353\) 3072.00i 0.463190i 0.972812 + 0.231595i \(0.0743944\pi\)
−0.972812 + 0.231595i \(0.925606\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3264.00 −0.485932
\(357\) 0 0
\(358\) − 2280.00i − 0.336597i
\(359\) 7446.00 1.09467 0.547333 0.836915i \(-0.315643\pi\)
0.547333 + 0.836915i \(0.315643\pi\)
\(360\) 0 0
\(361\) 12462.0 1.81688
\(362\) 796.000i 0.115571i
\(363\) 0 0
\(364\) 3128.00 0.450417
\(365\) 0 0
\(366\) 0 0
\(367\) 6496.00i 0.923947i 0.886894 + 0.461973i \(0.152858\pi\)
−0.886894 + 0.461973i \(0.847142\pi\)
\(368\) 3072.00i 0.435161i
\(369\) 0 0
\(370\) 0 0
\(371\) −7590.00 −1.06214
\(372\) 0 0
\(373\) − 1633.00i − 0.226685i −0.993556 0.113343i \(-0.963844\pi\)
0.993556 0.113343i \(-0.0361557\pi\)
\(374\) −2520.00 −0.348412
\(375\) 0 0
\(376\) −2928.00 −0.401596
\(377\) 7956.00i 1.08688i
\(378\) 0 0
\(379\) −6788.00 −0.919990 −0.459995 0.887922i \(-0.652149\pi\)
−0.459995 + 0.887922i \(0.652149\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6948.00i 0.930604i
\(383\) 6582.00i 0.878132i 0.898455 + 0.439066i \(0.144691\pi\)
−0.898455 + 0.439066i \(0.855309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5426.00 0.715482
\(387\) 0 0
\(388\) 3620.00i 0.473654i
\(389\) −2850.00 −0.371467 −0.185734 0.982600i \(-0.559466\pi\)
−0.185734 + 0.982600i \(0.559466\pi\)
\(390\) 0 0
\(391\) −8064.00 −1.04300
\(392\) 1488.00i 0.191723i
\(393\) 0 0
\(394\) 9468.00 1.21064
\(395\) 0 0
\(396\) 0 0
\(397\) − 7451.00i − 0.941952i −0.882146 0.470976i \(-0.843902\pi\)
0.882146 0.470976i \(-0.156098\pi\)
\(398\) − 10264.0i − 1.29268i
\(399\) 0 0
\(400\) 0 0
\(401\) 14124.0 1.75890 0.879450 0.475991i \(-0.157911\pi\)
0.879450 + 0.475991i \(0.157911\pi\)
\(402\) 0 0
\(403\) 1870.00i 0.231145i
\(404\) 5112.00 0.629533
\(405\) 0 0
\(406\) −10764.0 −1.31578
\(407\) − 5730.00i − 0.697851i
\(408\) 0 0
\(409\) −6374.00 −0.770597 −0.385298 0.922792i \(-0.625901\pi\)
−0.385298 + 0.922792i \(0.625901\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 2420.00i − 0.289381i
\(413\) − 9108.00i − 1.08517i
\(414\) 0 0
\(415\) 0 0
\(416\) 1088.00 0.128230
\(417\) 0 0
\(418\) 8340.00i 0.975892i
\(419\) −3948.00 −0.460316 −0.230158 0.973153i \(-0.573924\pi\)
−0.230158 + 0.973153i \(0.573924\pi\)
\(420\) 0 0
\(421\) 12629.0 1.46199 0.730997 0.682380i \(-0.239055\pi\)
0.730997 + 0.682380i \(0.239055\pi\)
\(422\) 10480.0i 1.20891i
\(423\) 0 0
\(424\) −2640.00 −0.302381
\(425\) 0 0
\(426\) 0 0
\(427\) − 529.000i − 0.0599534i
\(428\) 5952.00i 0.672198i
\(429\) 0 0
\(430\) 0 0
\(431\) 4842.00 0.541139 0.270570 0.962700i \(-0.412788\pi\)
0.270570 + 0.962700i \(0.412788\pi\)
\(432\) 0 0
\(433\) 3851.00i 0.427407i 0.976899 + 0.213704i \(0.0685526\pi\)
−0.976899 + 0.213704i \(0.931447\pi\)
\(434\) −2530.00 −0.279825
\(435\) 0 0
\(436\) 2372.00 0.260546
\(437\) 26688.0i 2.92142i
\(438\) 0 0
\(439\) 7435.00 0.808322 0.404161 0.914688i \(-0.367564\pi\)
0.404161 + 0.914688i \(0.367564\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2856.00i 0.307344i
\(443\) − 5760.00i − 0.617756i −0.951102 0.308878i \(-0.900047\pi\)
0.951102 0.308878i \(-0.0999535\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9038.00 0.959555
\(447\) 0 0
\(448\) 1472.00i 0.155235i
\(449\) 2190.00 0.230184 0.115092 0.993355i \(-0.463284\pi\)
0.115092 + 0.993355i \(0.463284\pi\)
\(450\) 0 0
\(451\) 4140.00 0.432251
\(452\) − 1296.00i − 0.134864i
\(453\) 0 0
\(454\) 10128.0 1.04698
\(455\) 0 0
\(456\) 0 0
\(457\) − 7202.00i − 0.737189i −0.929590 0.368594i \(-0.879839\pi\)
0.929590 0.368594i \(-0.120161\pi\)
\(458\) − 5146.00i − 0.525015i
\(459\) 0 0
\(460\) 0 0
\(461\) −13476.0 −1.36147 −0.680737 0.732528i \(-0.738341\pi\)
−0.680737 + 0.732528i \(0.738341\pi\)
\(462\) 0 0
\(463\) − 10843.0i − 1.08837i −0.838964 0.544187i \(-0.816838\pi\)
0.838964 0.544187i \(-0.183162\pi\)
\(464\) −3744.00 −0.374592
\(465\) 0 0
\(466\) −8196.00 −0.814748
\(467\) − 6108.00i − 0.605235i −0.953112 0.302617i \(-0.902140\pi\)
0.953112 0.302617i \(-0.0978604\pi\)
\(468\) 0 0
\(469\) −10396.0 −1.02355
\(470\) 0 0
\(471\) 0 0
\(472\) − 3168.00i − 0.308939i
\(473\) 1590.00i 0.154563i
\(474\) 0 0
\(475\) 0 0
\(476\) −3864.00 −0.372072
\(477\) 0 0
\(478\) 612.000i 0.0585611i
\(479\) −9852.00 −0.939769 −0.469885 0.882728i \(-0.655704\pi\)
−0.469885 + 0.882728i \(0.655704\pi\)
\(480\) 0 0
\(481\) −6494.00 −0.615594
\(482\) 12964.0i 1.22509i
\(483\) 0 0
\(484\) 1724.00 0.161908
\(485\) 0 0
\(486\) 0 0
\(487\) − 7796.00i − 0.725401i −0.931906 0.362701i \(-0.881855\pi\)
0.931906 0.362701i \(-0.118145\pi\)
\(488\) − 184.000i − 0.0170682i
\(489\) 0 0
\(490\) 0 0
\(491\) 2454.00 0.225555 0.112777 0.993620i \(-0.464025\pi\)
0.112777 + 0.993620i \(0.464025\pi\)
\(492\) 0 0
\(493\) − 9828.00i − 0.897831i
\(494\) 9452.00 0.860862
\(495\) 0 0
\(496\) −880.000 −0.0796636
\(497\) − 4692.00i − 0.423471i
\(498\) 0 0
\(499\) 2953.00 0.264919 0.132459 0.991188i \(-0.457713\pi\)
0.132459 + 0.991188i \(0.457713\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 11700.0i − 1.04023i
\(503\) − 11322.0i − 1.00362i −0.864977 0.501812i \(-0.832667\pi\)
0.864977 0.501812i \(-0.167333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −11520.0 −1.01211
\(507\) 0 0
\(508\) 7712.00i 0.673553i
\(509\) −9696.00 −0.844337 −0.422169 0.906517i \(-0.638731\pi\)
−0.422169 + 0.906517i \(0.638731\pi\)
\(510\) 0 0
\(511\) −15893.0 −1.37586
\(512\) 512.000i 0.0441942i
\(513\) 0 0
\(514\) 11196.0 0.960767
\(515\) 0 0
\(516\) 0 0
\(517\) − 10980.0i − 0.934042i
\(518\) − 8786.00i − 0.745241i
\(519\) 0 0
\(520\) 0 0
\(521\) −12192.0 −1.02522 −0.512612 0.858621i \(-0.671322\pi\)
−0.512612 + 0.858621i \(0.671322\pi\)
\(522\) 0 0
\(523\) − 8491.00i − 0.709915i −0.934882 0.354957i \(-0.884495\pi\)
0.934882 0.354957i \(-0.115505\pi\)
\(524\) −10968.0 −0.914388
\(525\) 0 0
\(526\) 16572.0 1.37371
\(527\) − 2310.00i − 0.190940i
\(528\) 0 0
\(529\) −24697.0 −2.02983
\(530\) 0 0
\(531\) 0 0
\(532\) 12788.0i 1.04216i
\(533\) − 4692.00i − 0.381300i
\(534\) 0 0
\(535\) 0 0
\(536\) −3616.00 −0.291394
\(537\) 0 0
\(538\) 1008.00i 0.0807769i
\(539\) −5580.00 −0.445914
\(540\) 0 0
\(541\) −9355.00 −0.743443 −0.371722 0.928344i \(-0.621232\pi\)
−0.371722 + 0.928344i \(0.621232\pi\)
\(542\) − 8978.00i − 0.711509i
\(543\) 0 0
\(544\) −1344.00 −0.105926
\(545\) 0 0
\(546\) 0 0
\(547\) 295.000i 0.0230590i 0.999934 + 0.0115295i \(0.00367004\pi\)
−0.999934 + 0.0115295i \(0.996330\pi\)
\(548\) 5304.00i 0.413459i
\(549\) 0 0
\(550\) 0 0
\(551\) −32526.0 −2.51480
\(552\) 0 0
\(553\) − 16307.0i − 1.25397i
\(554\) 4426.00 0.339427
\(555\) 0 0
\(556\) 3572.00 0.272458
\(557\) 16914.0i 1.28666i 0.765589 + 0.643330i \(0.222448\pi\)
−0.765589 + 0.643330i \(0.777552\pi\)
\(558\) 0 0
\(559\) 1802.00 0.136344
\(560\) 0 0
\(561\) 0 0
\(562\) − 5436.00i − 0.408014i
\(563\) − 12108.0i − 0.906379i −0.891414 0.453189i \(-0.850286\pi\)
0.891414 0.453189i \(-0.149714\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −11230.0 −0.833979
\(567\) 0 0
\(568\) − 1632.00i − 0.120558i
\(569\) 6960.00 0.512792 0.256396 0.966572i \(-0.417465\pi\)
0.256396 + 0.966572i \(0.417465\pi\)
\(570\) 0 0
\(571\) −10687.0 −0.783252 −0.391626 0.920124i \(-0.628087\pi\)
−0.391626 + 0.920124i \(0.628087\pi\)
\(572\) 4080.00i 0.298240i
\(573\) 0 0
\(574\) 6348.00 0.461603
\(575\) 0 0
\(576\) 0 0
\(577\) 8329.00i 0.600937i 0.953792 + 0.300469i \(0.0971431\pi\)
−0.953792 + 0.300469i \(0.902857\pi\)
\(578\) 6298.00i 0.453222i
\(579\) 0 0
\(580\) 0 0
\(581\) 25254.0 1.80329
\(582\) 0 0
\(583\) − 9900.00i − 0.703287i
\(584\) −5528.00 −0.391696
\(585\) 0 0
\(586\) −2976.00 −0.209791
\(587\) 25302.0i 1.77909i 0.456848 + 0.889545i \(0.348978\pi\)
−0.456848 + 0.889545i \(0.651022\pi\)
\(588\) 0 0
\(589\) −7645.00 −0.534816
\(590\) 0 0
\(591\) 0 0
\(592\) − 3056.00i − 0.212164i
\(593\) − 22248.0i − 1.54067i −0.637641 0.770334i \(-0.720090\pi\)
0.637641 0.770334i \(-0.279910\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10008.0 −0.687825
\(597\) 0 0
\(598\) 13056.0i 0.892809i
\(599\) −24252.0 −1.65427 −0.827137 0.562001i \(-0.810032\pi\)
−0.827137 + 0.562001i \(0.810032\pi\)
\(600\) 0 0
\(601\) −9829.00 −0.667110 −0.333555 0.942731i \(-0.608248\pi\)
−0.333555 + 0.942731i \(0.608248\pi\)
\(602\) 2438.00i 0.165059i
\(603\) 0 0
\(604\) 11068.0 0.745613
\(605\) 0 0
\(606\) 0 0
\(607\) 14155.0i 0.946514i 0.880925 + 0.473257i \(0.156922\pi\)
−0.880925 + 0.473257i \(0.843078\pi\)
\(608\) 4448.00i 0.296694i
\(609\) 0 0
\(610\) 0 0
\(611\) −12444.0 −0.823945
\(612\) 0 0
\(613\) 23051.0i 1.51879i 0.650627 + 0.759397i \(0.274506\pi\)
−0.650627 + 0.759397i \(0.725494\pi\)
\(614\) 14974.0 0.984204
\(615\) 0 0
\(616\) −5520.00 −0.361051
\(617\) 8352.00i 0.544958i 0.962162 + 0.272479i \(0.0878435\pi\)
−0.962162 + 0.272479i \(0.912157\pi\)
\(618\) 0 0
\(619\) 24331.0 1.57988 0.789940 0.613184i \(-0.210112\pi\)
0.789940 + 0.613184i \(0.210112\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16236.0i 1.04663i
\(623\) − 18768.0i − 1.20694i
\(624\) 0 0
\(625\) 0 0
\(626\) 10004.0 0.638722
\(627\) 0 0
\(628\) − 10804.0i − 0.686507i
\(629\) 8022.00 0.508518
\(630\) 0 0
\(631\) 2216.00 0.139806 0.0699030 0.997554i \(-0.477731\pi\)
0.0699030 + 0.997554i \(0.477731\pi\)
\(632\) − 5672.00i − 0.356994i
\(633\) 0 0
\(634\) 10164.0 0.636694
\(635\) 0 0
\(636\) 0 0
\(637\) 6324.00i 0.393353i
\(638\) − 14040.0i − 0.871237i
\(639\) 0 0
\(640\) 0 0
\(641\) −28500.0 −1.75613 −0.878067 0.478537i \(-0.841167\pi\)
−0.878067 + 0.478537i \(0.841167\pi\)
\(642\) 0 0
\(643\) − 25252.0i − 1.54874i −0.632731 0.774371i \(-0.718066\pi\)
0.632731 0.774371i \(-0.281934\pi\)
\(644\) −17664.0 −1.08084
\(645\) 0 0
\(646\) −11676.0 −0.711124
\(647\) 28920.0i 1.75728i 0.477481 + 0.878642i \(0.341550\pi\)
−0.477481 + 0.878642i \(0.658450\pi\)
\(648\) 0 0
\(649\) 11880.0 0.718537
\(650\) 0 0
\(651\) 0 0
\(652\) − 6992.00i − 0.419981i
\(653\) 32268.0i 1.93376i 0.255234 + 0.966879i \(0.417848\pi\)
−0.255234 + 0.966879i \(0.582152\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2208.00 0.131415
\(657\) 0 0
\(658\) − 16836.0i − 0.997471i
\(659\) 10050.0 0.594070 0.297035 0.954867i \(-0.404002\pi\)
0.297035 + 0.954867i \(0.404002\pi\)
\(660\) 0 0
\(661\) −4561.00 −0.268385 −0.134192 0.990955i \(-0.542844\pi\)
−0.134192 + 0.990955i \(0.542844\pi\)
\(662\) 15250.0i 0.895329i
\(663\) 0 0
\(664\) 8784.00 0.513381
\(665\) 0 0
\(666\) 0 0
\(667\) − 44928.0i − 2.60812i
\(668\) − 2136.00i − 0.123719i
\(669\) 0 0
\(670\) 0 0
\(671\) 690.000 0.0396977
\(672\) 0 0
\(673\) 21359.0i 1.22337i 0.791101 + 0.611686i \(0.209508\pi\)
−0.791101 + 0.611686i \(0.790492\pi\)
\(674\) −7556.00 −0.431819
\(675\) 0 0
\(676\) −4164.00 −0.236914
\(677\) − 15042.0i − 0.853931i −0.904268 0.426965i \(-0.859583\pi\)
0.904268 0.426965i \(-0.140417\pi\)
\(678\) 0 0
\(679\) −20815.0 −1.17645
\(680\) 0 0
\(681\) 0 0
\(682\) − 3300.00i − 0.185284i
\(683\) 27462.0i 1.53851i 0.638940 + 0.769256i \(0.279373\pi\)
−0.638940 + 0.769256i \(0.720627\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7222.00 0.401949
\(687\) 0 0
\(688\) 848.000i 0.0469908i
\(689\) −11220.0 −0.620389
\(690\) 0 0
\(691\) 6212.00 0.341991 0.170995 0.985272i \(-0.445302\pi\)
0.170995 + 0.985272i \(0.445302\pi\)
\(692\) 768.000i 0.0421893i
\(693\) 0 0
\(694\) −16536.0 −0.904464
\(695\) 0 0
\(696\) 0 0
\(697\) 5796.00i 0.314977i
\(698\) − 2758.00i − 0.149559i
\(699\) 0 0
\(700\) 0 0
\(701\) −13224.0 −0.712502 −0.356251 0.934390i \(-0.615945\pi\)
−0.356251 + 0.934390i \(0.615945\pi\)
\(702\) 0 0
\(703\) − 26549.0i − 1.42434i
\(704\) −1920.00 −0.102788
\(705\) 0 0
\(706\) −6144.00 −0.327525
\(707\) 29394.0i 1.56361i
\(708\) 0 0
\(709\) −34709.0 −1.83854 −0.919269 0.393629i \(-0.871219\pi\)
−0.919269 + 0.393629i \(0.871219\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6528.00i − 0.343606i
\(713\) − 10560.0i − 0.554664i
\(714\) 0 0
\(715\) 0 0
\(716\) 4560.00 0.238010
\(717\) 0 0
\(718\) 14892.0i 0.774045i
\(719\) −32070.0 −1.66343 −0.831717 0.555200i \(-0.812642\pi\)
−0.831717 + 0.555200i \(0.812642\pi\)
\(720\) 0 0
\(721\) 13915.0 0.718754
\(722\) 24924.0i 1.28473i
\(723\) 0 0
\(724\) −1592.00 −0.0817213
\(725\) 0 0
\(726\) 0 0
\(727\) − 125.000i − 0.00637688i −0.999995 0.00318844i \(-0.998985\pi\)
0.999995 0.00318844i \(-0.00101491\pi\)
\(728\) 6256.00i 0.318493i
\(729\) 0 0
\(730\) 0 0
\(731\) −2226.00 −0.112629
\(732\) 0 0
\(733\) 32222.0i 1.62367i 0.583890 + 0.811833i \(0.301530\pi\)
−0.583890 + 0.811833i \(0.698470\pi\)
\(734\) −12992.0 −0.653329
\(735\) 0 0
\(736\) −6144.00 −0.307705
\(737\) − 13560.0i − 0.677733i
\(738\) 0 0
\(739\) 19240.0 0.957720 0.478860 0.877891i \(-0.341050\pi\)
0.478860 + 0.877891i \(0.341050\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 15180.0i − 0.751045i
\(743\) 30252.0i 1.49373i 0.664978 + 0.746863i \(0.268441\pi\)
−0.664978 + 0.746863i \(0.731559\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3266.00 0.160291
\(747\) 0 0
\(748\) − 5040.00i − 0.246365i
\(749\) −34224.0 −1.66958
\(750\) 0 0
\(751\) −12517.0 −0.608192 −0.304096 0.952641i \(-0.598354\pi\)
−0.304096 + 0.952641i \(0.598354\pi\)
\(752\) − 5856.00i − 0.283971i
\(753\) 0 0
\(754\) −15912.0 −0.768542
\(755\) 0 0
\(756\) 0 0
\(757\) − 2201.00i − 0.105676i −0.998603 0.0528380i \(-0.983173\pi\)
0.998603 0.0528380i \(-0.0168267\pi\)
\(758\) − 13576.0i − 0.650531i
\(759\) 0 0
\(760\) 0 0
\(761\) 8742.00 0.416422 0.208211 0.978084i \(-0.433236\pi\)
0.208211 + 0.978084i \(0.433236\pi\)
\(762\) 0 0
\(763\) 13639.0i 0.647136i
\(764\) −13896.0 −0.658036
\(765\) 0 0
\(766\) −13164.0 −0.620933
\(767\) − 13464.0i − 0.633842i
\(768\) 0 0
\(769\) 28618.0 1.34199 0.670996 0.741461i \(-0.265867\pi\)
0.670996 + 0.741461i \(0.265867\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10852.0i 0.505922i
\(773\) 6594.00i 0.306817i 0.988163 + 0.153409i \(0.0490251\pi\)
−0.988163 + 0.153409i \(0.950975\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7240.00 −0.334924
\(777\) 0 0
\(778\) − 5700.00i − 0.262667i
\(779\) 19182.0 0.882242
\(780\) 0 0
\(781\) 6120.00 0.280398
\(782\) − 16128.0i − 0.737514i
\(783\) 0 0
\(784\) −2976.00 −0.135569
\(785\) 0 0
\(786\) 0 0
\(787\) − 15881.0i − 0.719309i −0.933085 0.359655i \(-0.882894\pi\)
0.933085 0.359655i \(-0.117106\pi\)
\(788\) 18936.0i 0.856050i
\(789\) 0 0
\(790\) 0 0
\(791\) 7452.00 0.334972
\(792\) 0 0
\(793\) − 782.000i − 0.0350185i
\(794\) 14902.0 0.666061
\(795\) 0 0
\(796\) 20528.0 0.914065
\(797\) 26052.0i 1.15785i 0.815380 + 0.578927i \(0.196528\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(798\) 0 0
\(799\) 15372.0 0.680629
\(800\) 0 0
\(801\) 0 0
\(802\) 28248.0i 1.24373i
\(803\) − 20730.0i − 0.911016i
\(804\) 0 0
\(805\) 0 0
\(806\) −3740.00 −0.163444
\(807\) 0 0
\(808\) 10224.0i 0.445147i
\(809\) −12648.0 −0.549666 −0.274833 0.961492i \(-0.588623\pi\)
−0.274833 + 0.961492i \(0.588623\pi\)
\(810\) 0 0
\(811\) 27179.0 1.17680 0.588399 0.808570i \(-0.299758\pi\)
0.588399 + 0.808570i \(0.299758\pi\)
\(812\) − 21528.0i − 0.930400i
\(813\) 0 0
\(814\) 11460.0 0.493456
\(815\) 0 0
\(816\) 0 0
\(817\) 7367.00i 0.315470i
\(818\) − 12748.0i − 0.544894i
\(819\) 0 0
\(820\) 0 0
\(821\) 11874.0 0.504757 0.252378 0.967629i \(-0.418787\pi\)
0.252378 + 0.967629i \(0.418787\pi\)
\(822\) 0 0
\(823\) − 18448.0i − 0.781357i −0.920527 0.390679i \(-0.872240\pi\)
0.920527 0.390679i \(-0.127760\pi\)
\(824\) 4840.00 0.204623
\(825\) 0 0
\(826\) 18216.0 0.767331
\(827\) 3234.00i 0.135982i 0.997686 + 0.0679911i \(0.0216589\pi\)
−0.997686 + 0.0679911i \(0.978341\pi\)
\(828\) 0 0
\(829\) 32155.0 1.34715 0.673576 0.739118i \(-0.264757\pi\)
0.673576 + 0.739118i \(0.264757\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2176.00i 0.0906721i
\(833\) − 7812.00i − 0.324934i
\(834\) 0 0
\(835\) 0 0
\(836\) −16680.0 −0.690060
\(837\) 0 0
\(838\) − 7896.00i − 0.325493i
\(839\) 21996.0 0.905109 0.452554 0.891737i \(-0.350513\pi\)
0.452554 + 0.891737i \(0.350513\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 25258.0i 1.03379i
\(843\) 0 0
\(844\) −20960.0 −0.854826
\(845\) 0 0
\(846\) 0 0
\(847\) 9913.00i 0.402143i
\(848\) − 5280.00i − 0.213816i
\(849\) 0 0
\(850\) 0 0
\(851\) 36672.0 1.47720
\(852\) 0 0
\(853\) 3278.00i 0.131579i 0.997834 + 0.0657893i \(0.0209565\pi\)
−0.997834 + 0.0657893i \(0.979043\pi\)
\(854\) 1058.00 0.0423935
\(855\) 0 0
\(856\) −11904.0 −0.475316
\(857\) 21228.0i 0.846131i 0.906099 + 0.423066i \(0.139046\pi\)
−0.906099 + 0.423066i \(0.860954\pi\)
\(858\) 0 0
\(859\) 5767.00 0.229066 0.114533 0.993419i \(-0.463463\pi\)
0.114533 + 0.993419i \(0.463463\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 9684.00i 0.382643i
\(863\) − 5322.00i − 0.209922i −0.994476 0.104961i \(-0.966528\pi\)
0.994476 0.104961i \(-0.0334718\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −7702.00 −0.302222
\(867\) 0 0
\(868\) − 5060.00i − 0.197866i
\(869\) 21270.0 0.830305
\(870\) 0 0
\(871\) −15368.0 −0.597847
\(872\) 4744.00i 0.184234i
\(873\) 0 0
\(874\) −53376.0 −2.06576
\(875\) 0 0
\(876\) 0 0
\(877\) 42097.0i 1.62088i 0.585819 + 0.810442i \(0.300773\pi\)
−0.585819 + 0.810442i \(0.699227\pi\)
\(878\) 14870.0i 0.571570i
\(879\) 0 0
\(880\) 0 0
\(881\) −17658.0 −0.675270 −0.337635 0.941277i \(-0.609627\pi\)
−0.337635 + 0.941277i \(0.609627\pi\)
\(882\) 0 0
\(883\) − 22297.0i − 0.849778i −0.905246 0.424889i \(-0.860313\pi\)
0.905246 0.424889i \(-0.139687\pi\)
\(884\) −5712.00 −0.217325
\(885\) 0 0
\(886\) 11520.0 0.436819
\(887\) 10542.0i 0.399059i 0.979892 + 0.199530i \(0.0639414\pi\)
−0.979892 + 0.199530i \(0.936059\pi\)
\(888\) 0 0
\(889\) −44344.0 −1.67295
\(890\) 0 0
\(891\) 0 0
\(892\) 18076.0i 0.678508i
\(893\) − 50874.0i − 1.90642i
\(894\) 0 0
\(895\) 0 0
\(896\) −2944.00 −0.109768
\(897\) 0 0
\(898\) 4380.00i 0.162764i
\(899\) 12870.0 0.477462
\(900\) 0 0
\(901\) 13860.0 0.512479
\(902\) 8280.00i 0.305647i
\(903\) 0 0
\(904\) 2592.00 0.0953635
\(905\) 0 0
\(906\) 0 0
\(907\) − 47639.0i − 1.74402i −0.489487 0.872010i \(-0.662816\pi\)
0.489487 0.872010i \(-0.337184\pi\)
\(908\) 20256.0i 0.740329i
\(909\) 0 0
\(910\) 0 0
\(911\) −10326.0 −0.375539 −0.187769 0.982213i \(-0.560126\pi\)
−0.187769 + 0.982213i \(0.560126\pi\)
\(912\) 0 0
\(913\) 32940.0i 1.19404i
\(914\) 14404.0 0.521271
\(915\) 0 0
\(916\) 10292.0 0.371242
\(917\) − 63066.0i − 2.27113i
\(918\) 0 0
\(919\) −5147.00 −0.184748 −0.0923742 0.995724i \(-0.529446\pi\)
−0.0923742 + 0.995724i \(0.529446\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 26952.0i − 0.962708i
\(923\) − 6936.00i − 0.247347i
\(924\) 0 0
\(925\) 0 0
\(926\) 21686.0 0.769596
\(927\) 0 0
\(928\) − 7488.00i − 0.264877i
\(929\) 47064.0 1.66213 0.831066 0.556175i \(-0.187731\pi\)
0.831066 + 0.556175i \(0.187731\pi\)
\(930\) 0 0
\(931\) −25854.0 −0.910130
\(932\) − 16392.0i − 0.576114i
\(933\) 0 0
\(934\) 12216.0 0.427965
\(935\) 0 0
\(936\) 0 0
\(937\) 15385.0i 0.536399i 0.963363 + 0.268200i \(0.0864287\pi\)
−0.963363 + 0.268200i \(0.913571\pi\)
\(938\) − 20792.0i − 0.723756i
\(939\) 0 0
\(940\) 0 0
\(941\) −34914.0 −1.20953 −0.604763 0.796406i \(-0.706732\pi\)
−0.604763 + 0.796406i \(0.706732\pi\)
\(942\) 0 0
\(943\) 26496.0i 0.914982i
\(944\) 6336.00 0.218453
\(945\) 0 0
\(946\) −3180.00 −0.109293
\(947\) − 37434.0i − 1.28452i −0.766486 0.642261i \(-0.777997\pi\)
0.766486 0.642261i \(-0.222003\pi\)
\(948\) 0 0
\(949\) −23494.0 −0.803633
\(950\) 0 0
\(951\) 0 0
\(952\) − 7728.00i − 0.263094i
\(953\) 8778.00i 0.298371i 0.988809 + 0.149185i \(0.0476651\pi\)
−0.988809 + 0.149185i \(0.952335\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1224.00 −0.0414090
\(957\) 0 0
\(958\) − 19704.0i − 0.664517i
\(959\) −30498.0 −1.02694
\(960\) 0 0
\(961\) −26766.0 −0.898459
\(962\) − 12988.0i − 0.435291i
\(963\) 0 0
\(964\) −25928.0 −0.866270
\(965\) 0 0
\(966\) 0 0
\(967\) 54061.0i 1.79781i 0.438141 + 0.898906i \(0.355637\pi\)
−0.438141 + 0.898906i \(0.644363\pi\)
\(968\) 3448.00i 0.114486i
\(969\) 0 0
\(970\) 0 0
\(971\) −3084.00 −0.101926 −0.0509631 0.998701i \(-0.516229\pi\)
−0.0509631 + 0.998701i \(0.516229\pi\)
\(972\) 0 0
\(973\) 20539.0i 0.676722i
\(974\) 15592.0 0.512936
\(975\) 0 0
\(976\) 368.000 0.0120691
\(977\) − 12048.0i − 0.394524i −0.980351 0.197262i \(-0.936795\pi\)
0.980351 0.197262i \(-0.0632049\pi\)
\(978\) 0 0
\(979\) 24480.0 0.799167
\(980\) 0 0
\(981\) 0 0
\(982\) 4908.00i 0.159491i
\(983\) 33618.0i 1.09079i 0.838179 + 0.545396i \(0.183621\pi\)
−0.838179 + 0.545396i \(0.816379\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19656.0 0.634863
\(987\) 0 0
\(988\) 18904.0i 0.608721i
\(989\) −10176.0 −0.327177
\(990\) 0 0
\(991\) 25043.0 0.802742 0.401371 0.915916i \(-0.368534\pi\)
0.401371 + 0.915916i \(0.368534\pi\)
\(992\) − 1760.00i − 0.0563307i
\(993\) 0 0
\(994\) 9384.00 0.299439
\(995\) 0 0
\(996\) 0 0
\(997\) 17710.0i 0.562569i 0.959624 + 0.281285i \(0.0907605\pi\)
−0.959624 + 0.281285i \(0.909240\pi\)
\(998\) 5906.00i 0.187326i
\(999\) 0 0
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 1350.4.c.p.649.2 2
3.2 odd 2 1350.4.c.e.649.1 2
5.2 odd 4 1350.4.a.m.1.1 yes 1
5.3 odd 4 1350.4.a.p.1.1 yes 1
5.4 even 2 inner 1350.4.c.p.649.1 2
15.2 even 4 1350.4.a.ba.1.1 yes 1
15.8 even 4 1350.4.a.b.1.1 1
15.14 odd 2 1350.4.c.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1350.4.a.b.1.1 1 15.8 even 4
1350.4.a.m.1.1 yes 1 5.2 odd 4
1350.4.a.p.1.1 yes 1 5.3 odd 4
1350.4.a.ba.1.1 yes 1 15.2 even 4
1350.4.c.e.649.1 2 3.2 odd 2
1350.4.c.e.649.2 2 15.14 odd 2
1350.4.c.p.649.1 2 5.4 even 2 inner
1350.4.c.p.649.2 2 1.1 even 1 trivial