Properties

Label 900.4.d.d.649.1
Level $900$
Weight $4$
Character 900.649
Analytic conductor $53.102$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(53.1017190052\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 900.649
Dual form 900.4.d.d.649.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{7} +O(q^{10})\) \(q-2.00000i q^{7} -30.0000 q^{11} -4.00000i q^{13} +90.0000i q^{17} +28.0000 q^{19} -120.000i q^{23} +210.000 q^{29} -4.00000 q^{31} -200.000i q^{37} -240.000 q^{41} -136.000i q^{43} -120.000i q^{47} +339.000 q^{49} +30.0000i q^{53} -450.000 q^{59} -166.000 q^{61} -908.000i q^{67} +1020.00 q^{71} -250.000i q^{73} +60.0000i q^{77} +916.000 q^{79} +1140.00i q^{83} -420.000 q^{89} -8.00000 q^{91} -1538.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 60 q^{11} + 56 q^{19} + 420 q^{29} - 8 q^{31} - 480 q^{41} + 678 q^{49} - 900 q^{59} - 332 q^{61} + 2040 q^{71} + 1832 q^{79} - 840 q^{89} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.107990i −0.998541 0.0539949i \(-0.982805\pi\)
0.998541 0.0539949i \(-0.0171955\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −30.0000 −0.822304 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 0.0853385i −0.999089 0.0426692i \(-0.986414\pi\)
0.999089 0.0426692i \(-0.0135862\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 90.0000i 1.28401i 0.766700 + 0.642006i \(0.221898\pi\)
−0.766700 + 0.642006i \(0.778102\pi\)
\(18\) 0 0
\(19\) 28.0000 0.338086 0.169043 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 120.000i − 1.08790i −0.839117 0.543951i \(-0.816928\pi\)
0.839117 0.543951i \(-0.183072\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 210.000 1.34469 0.672345 0.740238i \(-0.265287\pi\)
0.672345 + 0.740238i \(0.265287\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.0231749 −0.0115874 0.999933i \(-0.503688\pi\)
−0.0115874 + 0.999933i \(0.503688\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 200.000i − 0.888643i −0.895867 0.444322i \(-0.853445\pi\)
0.895867 0.444322i \(-0.146555\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −240.000 −0.914188 −0.457094 0.889418i \(-0.651110\pi\)
−0.457094 + 0.889418i \(0.651110\pi\)
\(42\) 0 0
\(43\) − 136.000i − 0.482321i −0.970485 0.241161i \(-0.922472\pi\)
0.970485 0.241161i \(-0.0775280\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 120.000i − 0.372421i −0.982510 0.186211i \(-0.940379\pi\)
0.982510 0.186211i \(-0.0596207\pi\)
\(48\) 0 0
\(49\) 339.000 0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 30.0000i 0.0777513i 0.999244 + 0.0388756i \(0.0123776\pi\)
−0.999244 + 0.0388756i \(0.987622\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −450.000 −0.992966 −0.496483 0.868046i \(-0.665376\pi\)
−0.496483 + 0.868046i \(0.665376\pi\)
\(60\) 0 0
\(61\) −166.000 −0.348428 −0.174214 0.984708i \(-0.555738\pi\)
−0.174214 + 0.984708i \(0.555738\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 908.000i − 1.65567i −0.560972 0.827835i \(-0.689572\pi\)
0.560972 0.827835i \(-0.310428\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1020.00 1.70495 0.852477 0.522765i \(-0.175099\pi\)
0.852477 + 0.522765i \(0.175099\pi\)
\(72\) 0 0
\(73\) − 250.000i − 0.400826i −0.979712 0.200413i \(-0.935772\pi\)
0.979712 0.200413i \(-0.0642284\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 60.0000i 0.0888004i
\(78\) 0 0
\(79\) 916.000 1.30453 0.652266 0.757990i \(-0.273818\pi\)
0.652266 + 0.757990i \(0.273818\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1140.00i 1.50761i 0.657101 + 0.753803i \(0.271783\pi\)
−0.657101 + 0.753803i \(0.728217\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −420.000 −0.500224 −0.250112 0.968217i \(-0.580467\pi\)
−0.250112 + 0.968217i \(0.580467\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.00921569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1538.00i − 1.60990i −0.593343 0.804950i \(-0.702192\pi\)
0.593343 0.804950i \(-0.297808\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −450.000 −0.443333 −0.221667 0.975122i \(-0.571150\pi\)
−0.221667 + 0.975122i \(0.571150\pi\)
\(102\) 0 0
\(103\) − 1150.00i − 1.10012i −0.835124 0.550062i \(-0.814604\pi\)
0.835124 0.550062i \(-0.185396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1620.00i − 1.46366i −0.681489 0.731829i \(-0.738667\pi\)
0.681489 0.731829i \(-0.261333\pi\)
\(108\) 0 0
\(109\) 1702.00 1.49561 0.747807 0.663916i \(-0.231107\pi\)
0.747807 + 0.663916i \(0.231107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1350.00i − 1.12387i −0.827181 0.561935i \(-0.810057\pi\)
0.827181 0.561935i \(-0.189943\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 180.000 0.138660
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2450.00i − 1.71183i −0.517117 0.855915i \(-0.672995\pi\)
0.517117 0.855915i \(-0.327005\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −690.000 −0.460195 −0.230098 0.973168i \(-0.573905\pi\)
−0.230098 + 0.973168i \(0.573905\pi\)
\(132\) 0 0
\(133\) − 56.0000i − 0.0365099i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2070.00i 1.29089i 0.763806 + 0.645445i \(0.223328\pi\)
−0.763806 + 0.645445i \(0.776672\pi\)
\(138\) 0 0
\(139\) 1924.00 1.17404 0.587020 0.809572i \(-0.300301\pi\)
0.587020 + 0.809572i \(0.300301\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 120.000i 0.0701742i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2910.00 1.59998 0.799988 0.600016i \(-0.204839\pi\)
0.799988 + 0.600016i \(0.204839\pi\)
\(150\) 0 0
\(151\) 176.000 0.0948522 0.0474261 0.998875i \(-0.484898\pi\)
0.0474261 + 0.998875i \(0.484898\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2348.00i − 1.19357i −0.802400 0.596786i \(-0.796444\pi\)
0.802400 0.596786i \(-0.203556\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −240.000 −0.117482
\(162\) 0 0
\(163\) − 1996.00i − 0.959134i −0.877505 0.479567i \(-0.840794\pi\)
0.877505 0.479567i \(-0.159206\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3120.00i 1.44571i 0.691002 + 0.722853i \(0.257170\pi\)
−0.691002 + 0.722853i \(0.742830\pi\)
\(168\) 0 0
\(169\) 2181.00 0.992717
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1770.00i − 0.777865i −0.921266 0.388932i \(-0.872844\pi\)
0.921266 0.388932i \(-0.127156\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2130.00 −0.889406 −0.444703 0.895678i \(-0.646691\pi\)
−0.444703 + 0.895678i \(0.646691\pi\)
\(180\) 0 0
\(181\) −1654.00 −0.679231 −0.339616 0.940564i \(-0.610297\pi\)
−0.339616 + 0.940564i \(0.610297\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2700.00i − 1.05585i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1740.00 −0.659173 −0.329586 0.944125i \(-0.606909\pi\)
−0.329586 + 0.944125i \(0.606909\pi\)
\(192\) 0 0
\(193\) 86.0000i 0.0320747i 0.999871 + 0.0160373i \(0.00510506\pi\)
−0.999871 + 0.0160373i \(0.994895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2490.00i − 0.900534i −0.892894 0.450267i \(-0.851329\pi\)
0.892894 0.450267i \(-0.148671\pi\)
\(198\) 0 0
\(199\) 832.000 0.296376 0.148188 0.988959i \(-0.452656\pi\)
0.148188 + 0.988959i \(0.452656\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 420.000i − 0.145213i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −840.000 −0.278010
\(210\) 0 0
\(211\) 2084.00 0.679945 0.339973 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000i 0.00250265i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 360.000 0.109576
\(222\) 0 0
\(223\) − 1174.00i − 0.352542i −0.984342 0.176271i \(-0.943597\pi\)
0.984342 0.176271i \(-0.0564035\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3120.00i − 0.912254i −0.889915 0.456127i \(-0.849236\pi\)
0.889915 0.456127i \(-0.150764\pi\)
\(228\) 0 0
\(229\) 58.0000 0.0167369 0.00836845 0.999965i \(-0.497336\pi\)
0.00836845 + 0.999965i \(0.497336\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5910.00i 1.66170i 0.556494 + 0.830852i \(0.312146\pi\)
−0.556494 + 0.830852i \(0.687854\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3300.00 0.893135 0.446567 0.894750i \(-0.352646\pi\)
0.446567 + 0.894750i \(0.352646\pi\)
\(240\) 0 0
\(241\) −2986.00 −0.798113 −0.399056 0.916926i \(-0.630662\pi\)
−0.399056 + 0.916926i \(0.630662\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 112.000i − 0.0288518i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6630.00 1.66726 0.833629 0.552324i \(-0.186259\pi\)
0.833629 + 0.552324i \(0.186259\pi\)
\(252\) 0 0
\(253\) 3600.00i 0.894585i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1530.00i − 0.371357i −0.982611 0.185679i \(-0.940552\pi\)
0.982611 0.185679i \(-0.0594483\pi\)
\(258\) 0 0
\(259\) −400.000 −0.0959644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 2640.00i − 0.618971i −0.950904 0.309486i \(-0.899843\pi\)
0.950904 0.309486i \(-0.100157\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7350.00 −1.66594 −0.832969 0.553319i \(-0.813361\pi\)
−0.832969 + 0.553319i \(0.813361\pi\)
\(270\) 0 0
\(271\) 3512.00 0.787228 0.393614 0.919276i \(-0.371225\pi\)
0.393614 + 0.919276i \(0.371225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5368.00i 1.16437i 0.813055 + 0.582187i \(0.197803\pi\)
−0.813055 + 0.582187i \(0.802197\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3060.00 0.649624 0.324812 0.945779i \(-0.394699\pi\)
0.324812 + 0.945779i \(0.394699\pi\)
\(282\) 0 0
\(283\) − 5044.00i − 1.05949i −0.848158 0.529743i \(-0.822288\pi\)
0.848158 0.529743i \(-0.177712\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 480.000i 0.0987230i
\(288\) 0 0
\(289\) −3187.00 −0.648687
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2010.00i 0.400769i 0.979717 + 0.200385i \(0.0642192\pi\)
−0.979717 + 0.200385i \(0.935781\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −480.000 −0.0928399
\(300\) 0 0
\(301\) −272.000 −0.0520858
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2752.00i 0.511612i 0.966728 + 0.255806i \(0.0823409\pi\)
−0.966728 + 0.255806i \(0.917659\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9540.00 −1.73943 −0.869717 0.493551i \(-0.835699\pi\)
−0.869717 + 0.493551i \(0.835699\pi\)
\(312\) 0 0
\(313\) 9254.00i 1.67114i 0.549384 + 0.835570i \(0.314863\pi\)
−0.549384 + 0.835570i \(0.685137\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 150.000i 0.0265768i 0.999912 + 0.0132884i \(0.00422995\pi\)
−0.999912 + 0.0132884i \(0.995770\pi\)
\(318\) 0 0
\(319\) −6300.00 −1.10574
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2520.00i 0.434107i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −240.000 −0.0402177
\(330\) 0 0
\(331\) 1892.00 0.314180 0.157090 0.987584i \(-0.449789\pi\)
0.157090 + 0.987584i \(0.449789\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7378.00i 1.19260i 0.802763 + 0.596299i \(0.203363\pi\)
−0.802763 + 0.596299i \(0.796637\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 120.000 0.0190568
\(342\) 0 0
\(343\) − 1364.00i − 0.214720i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6720.00i 1.03962i 0.854282 + 0.519811i \(0.173997\pi\)
−0.854282 + 0.519811i \(0.826003\pi\)
\(348\) 0 0
\(349\) −5186.00 −0.795416 −0.397708 0.917512i \(-0.630194\pi\)
−0.397708 + 0.917512i \(0.630194\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 3330.00i − 0.502091i −0.967975 0.251045i \(-0.919226\pi\)
0.967975 0.251045i \(-0.0807743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9000.00 1.32312 0.661562 0.749890i \(-0.269894\pi\)
0.661562 + 0.749890i \(0.269894\pi\)
\(360\) 0 0
\(361\) −6075.00 −0.885698
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8758.00i 1.24568i 0.782350 + 0.622839i \(0.214021\pi\)
−0.782350 + 0.622839i \(0.785979\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 60.0000 0.00839635
\(372\) 0 0
\(373\) 4724.00i 0.655763i 0.944719 + 0.327881i \(0.106335\pi\)
−0.944719 + 0.327881i \(0.893665\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 840.000i − 0.114754i
\(378\) 0 0
\(379\) −7292.00 −0.988298 −0.494149 0.869377i \(-0.664520\pi\)
−0.494149 + 0.869377i \(0.664520\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 14520.0i − 1.93717i −0.248676 0.968587i \(-0.579996\pi\)
0.248676 0.968587i \(-0.420004\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7110.00 0.926713 0.463356 0.886172i \(-0.346645\pi\)
0.463356 + 0.886172i \(0.346645\pi\)
\(390\) 0 0
\(391\) 10800.0 1.39688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11488.0i 1.45231i 0.687532 + 0.726154i \(0.258694\pi\)
−0.687532 + 0.726154i \(0.741306\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −780.000 −0.0971355 −0.0485678 0.998820i \(-0.515466\pi\)
−0.0485678 + 0.998820i \(0.515466\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.00197771i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6000.00i 0.730735i
\(408\) 0 0
\(409\) −5402.00 −0.653085 −0.326542 0.945183i \(-0.605884\pi\)
−0.326542 + 0.945183i \(0.605884\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 900.000i 0.107230i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2190.00 −0.255342 −0.127671 0.991817i \(-0.540750\pi\)
−0.127671 + 0.991817i \(0.540750\pi\)
\(420\) 0 0
\(421\) −7162.00 −0.829108 −0.414554 0.910025i \(-0.636062\pi\)
−0.414554 + 0.910025i \(0.636062\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 332.000i 0.0376267i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9360.00 −1.04607 −0.523034 0.852312i \(-0.675200\pi\)
−0.523034 + 0.852312i \(0.675200\pi\)
\(432\) 0 0
\(433\) 12806.0i 1.42129i 0.703552 + 0.710643i \(0.251596\pi\)
−0.703552 + 0.710643i \(0.748404\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3360.00i − 0.367805i
\(438\) 0 0
\(439\) −11288.0 −1.22721 −0.613607 0.789612i \(-0.710282\pi\)
−0.613607 + 0.789612i \(0.710282\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8520.00i 0.913764i 0.889527 + 0.456882i \(0.151034\pi\)
−0.889527 + 0.456882i \(0.848966\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1260.00 −0.132434 −0.0662172 0.997805i \(-0.521093\pi\)
−0.0662172 + 0.997805i \(0.521093\pi\)
\(450\) 0 0
\(451\) 7200.00 0.751740
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13750.0i 1.40744i 0.710480 + 0.703718i \(0.248478\pi\)
−0.710480 + 0.703718i \(0.751522\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3210.00 0.324305 0.162152 0.986766i \(-0.448156\pi\)
0.162152 + 0.986766i \(0.448156\pi\)
\(462\) 0 0
\(463\) − 12850.0i − 1.28983i −0.764255 0.644914i \(-0.776893\pi\)
0.764255 0.644914i \(-0.223107\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8220.00i − 0.814510i −0.913314 0.407255i \(-0.866486\pi\)
0.913314 0.407255i \(-0.133514\pi\)
\(468\) 0 0
\(469\) −1816.00 −0.178795
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4080.00i 0.396614i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7020.00 −0.669628 −0.334814 0.942284i \(-0.608674\pi\)
−0.334814 + 0.942284i \(0.608674\pi\)
\(480\) 0 0
\(481\) −800.000 −0.0758355
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8122.00i 0.755735i 0.925860 + 0.377868i \(0.123343\pi\)
−0.925860 + 0.377868i \(0.876657\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13470.0 1.23807 0.619035 0.785363i \(-0.287524\pi\)
0.619035 + 0.785363i \(0.287524\pi\)
\(492\) 0 0
\(493\) 18900.0i 1.72660i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2040.00i − 0.184118i
\(498\) 0 0
\(499\) −2468.00 −0.221409 −0.110704 0.993853i \(-0.535311\pi\)
−0.110704 + 0.993853i \(0.535311\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4440.00i 0.393578i 0.980446 + 0.196789i \(0.0630514\pi\)
−0.980446 + 0.196789i \(0.936949\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11190.0 −0.974436 −0.487218 0.873280i \(-0.661988\pi\)
−0.487218 + 0.873280i \(0.661988\pi\)
\(510\) 0 0
\(511\) −500.000 −0.0432851
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3600.00i 0.306243i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4020.00 0.338041 0.169021 0.985613i \(-0.445940\pi\)
0.169021 + 0.985613i \(0.445940\pi\)
\(522\) 0 0
\(523\) − 9076.00i − 0.758826i −0.925228 0.379413i \(-0.876126\pi\)
0.925228 0.379413i \(-0.123874\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 360.000i − 0.0297568i
\(528\) 0 0
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 960.000i 0.0780154i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10170.0 −0.812714
\(540\) 0 0
\(541\) −7486.00 −0.594914 −0.297457 0.954735i \(-0.596138\pi\)
−0.297457 + 0.954735i \(0.596138\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 7400.00i − 0.578430i −0.957264 0.289215i \(-0.906606\pi\)
0.957264 0.289215i \(-0.0933942\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5880.00 0.454621
\(552\) 0 0
\(553\) − 1832.00i − 0.140876i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11490.0i 0.874052i 0.899449 + 0.437026i \(0.143968\pi\)
−0.899449 + 0.437026i \(0.856032\pi\)
\(558\) 0 0
\(559\) −544.000 −0.0411606
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19320.0i 1.44625i 0.690715 + 0.723127i \(0.257296\pi\)
−0.690715 + 0.723127i \(0.742704\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8340.00 0.614466 0.307233 0.951634i \(-0.400597\pi\)
0.307233 + 0.951634i \(0.400597\pi\)
\(570\) 0 0
\(571\) 21044.0 1.54232 0.771159 0.636642i \(-0.219677\pi\)
0.771159 + 0.636642i \(0.219677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1418.00i − 0.102309i −0.998691 0.0511543i \(-0.983710\pi\)
0.998691 0.0511543i \(-0.0162900\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2280.00 0.162806
\(582\) 0 0
\(583\) − 900.000i − 0.0639351i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22020.0i − 1.54832i −0.632991 0.774159i \(-0.718173\pi\)
0.632991 0.774159i \(-0.281827\pi\)
\(588\) 0 0
\(589\) −112.000 −0.00783511
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 25230.0i − 1.74717i −0.486671 0.873585i \(-0.661789\pi\)
0.486671 0.873585i \(-0.338211\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8280.00 0.564794 0.282397 0.959298i \(-0.408870\pi\)
0.282397 + 0.959298i \(0.408870\pi\)
\(600\) 0 0
\(601\) −18874.0 −1.28101 −0.640505 0.767954i \(-0.721275\pi\)
−0.640505 + 0.767954i \(0.721275\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 10550.0i − 0.705455i −0.935726 0.352728i \(-0.885254\pi\)
0.935726 0.352728i \(-0.114746\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −480.000 −0.0317819
\(612\) 0 0
\(613\) 11000.0i 0.724773i 0.932028 + 0.362386i \(0.118038\pi\)
−0.932028 + 0.362386i \(0.881962\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 11310.0i − 0.737963i −0.929437 0.368982i \(-0.879706\pi\)
0.929437 0.368982i \(-0.120294\pi\)
\(618\) 0 0
\(619\) 17572.0 1.14100 0.570499 0.821298i \(-0.306750\pi\)
0.570499 + 0.821298i \(0.306750\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 840.000i 0.0540191i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18000.0 1.14103
\(630\) 0 0
\(631\) 1604.00 0.101195 0.0505976 0.998719i \(-0.483887\pi\)
0.0505976 + 0.998719i \(0.483887\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1356.00i − 0.0843433i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −31320.0 −1.92990 −0.964950 0.262435i \(-0.915475\pi\)
−0.964950 + 0.262435i \(0.915475\pi\)
\(642\) 0 0
\(643\) − 31300.0i − 1.91968i −0.280555 0.959838i \(-0.590519\pi\)
0.280555 0.959838i \(-0.409481\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10920.0i 0.663539i 0.943361 + 0.331769i \(0.107646\pi\)
−0.943361 + 0.331769i \(0.892354\pi\)
\(648\) 0 0
\(649\) 13500.0 0.816520
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3210.00i 0.192369i 0.995364 + 0.0961845i \(0.0306639\pi\)
−0.995364 + 0.0961845i \(0.969336\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11910.0 −0.704018 −0.352009 0.935997i \(-0.614501\pi\)
−0.352009 + 0.935997i \(0.614501\pi\)
\(660\) 0 0
\(661\) −3382.00 −0.199008 −0.0995042 0.995037i \(-0.531726\pi\)
−0.0995042 + 0.995037i \(0.531726\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 25200.0i − 1.46289i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4980.00 0.286514
\(672\) 0 0
\(673\) 15950.0i 0.913562i 0.889579 + 0.456781i \(0.150998\pi\)
−0.889579 + 0.456781i \(0.849002\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32190.0i 1.82742i 0.406369 + 0.913709i \(0.366795\pi\)
−0.406369 + 0.913709i \(0.633205\pi\)
\(678\) 0 0
\(679\) −3076.00 −0.173853
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 22140.0i − 1.24036i −0.784461 0.620178i \(-0.787060\pi\)
0.784461 0.620178i \(-0.212940\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 120.000 0.00663518
\(690\) 0 0
\(691\) −6172.00 −0.339789 −0.169894 0.985462i \(-0.554343\pi\)
−0.169894 + 0.985462i \(0.554343\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 21600.0i − 1.17383i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −19170.0 −1.03287 −0.516434 0.856327i \(-0.672741\pi\)
−0.516434 + 0.856327i \(0.672741\pi\)
\(702\) 0 0
\(703\) − 5600.00i − 0.300438i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 900.000i 0.0478755i
\(708\) 0 0
\(709\) 21898.0 1.15994 0.579969 0.814638i \(-0.303064\pi\)
0.579969 + 0.814638i \(0.303064\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 480.000i 0.0252120i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16680.0 0.865173 0.432586 0.901593i \(-0.357601\pi\)
0.432586 + 0.901593i \(0.357601\pi\)
\(720\) 0 0
\(721\) −2300.00 −0.118802
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 6518.00i − 0.332516i −0.986082 0.166258i \(-0.946832\pi\)
0.986082 0.166258i \(-0.0531685\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12240.0 0.619306
\(732\) 0 0
\(733\) − 23200.0i − 1.16905i −0.811377 0.584524i \(-0.801281\pi\)
0.811377 0.584524i \(-0.198719\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27240.0i 1.36146i
\(738\) 0 0
\(739\) 16324.0 0.812568 0.406284 0.913747i \(-0.366824\pi\)
0.406284 + 0.913747i \(0.366824\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 120.000i 0.00592513i 0.999996 + 0.00296257i \(0.000943015\pi\)
−0.999996 + 0.00296257i \(0.999057\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3240.00 −0.158060
\(750\) 0 0
\(751\) 30548.0 1.48430 0.742152 0.670232i \(-0.233805\pi\)
0.742152 + 0.670232i \(0.233805\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 16952.0i − 0.813911i −0.913448 0.406956i \(-0.866590\pi\)
0.913448 0.406956i \(-0.133410\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20220.0 0.963173 0.481586 0.876399i \(-0.340061\pi\)
0.481586 + 0.876399i \(0.340061\pi\)
\(762\) 0 0
\(763\) − 3404.00i − 0.161511i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1800.00i 0.0847382i
\(768\) 0 0
\(769\) 20722.0 0.971722 0.485861 0.874036i \(-0.338506\pi\)
0.485861 + 0.874036i \(0.338506\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 4350.00i − 0.202404i −0.994866 0.101202i \(-0.967731\pi\)
0.994866 0.101202i \(-0.0322689\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6720.00 −0.309074
\(780\) 0 0
\(781\) −30600.0 −1.40199
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 41972.0i − 1.90107i −0.310621 0.950534i \(-0.600537\pi\)
0.310621 0.950534i \(-0.399463\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2700.00 −0.121367
\(792\) 0 0
\(793\) 664.000i 0.0297343i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 39510.0i − 1.75598i −0.478679 0.877990i \(-0.658884\pi\)
0.478679 0.877990i \(-0.341116\pi\)
\(798\) 0 0
\(799\) 10800.0 0.478193
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7500.00i 0.329601i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16680.0 0.724892 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(810\) 0 0
\(811\) −15484.0 −0.670428 −0.335214 0.942142i \(-0.608809\pi\)
−0.335214 + 0.942142i \(0.608809\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3808.00i − 0.163066i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4170.00 −0.177264 −0.0886322 0.996064i \(-0.528250\pi\)
−0.0886322 + 0.996064i \(0.528250\pi\)
\(822\) 0 0
\(823\) − 30226.0i − 1.28021i −0.768288 0.640105i \(-0.778891\pi\)
0.768288 0.640105i \(-0.221109\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 14760.0i − 0.620623i −0.950635 0.310312i \(-0.899567\pi\)
0.950635 0.310312i \(-0.100433\pi\)
\(828\) 0 0
\(829\) 9934.00 0.416191 0.208095 0.978109i \(-0.433274\pi\)
0.208095 + 0.978109i \(0.433274\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 30510.0i 1.26904i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23520.0 −0.967820 −0.483910 0.875118i \(-0.660784\pi\)
−0.483910 + 0.875118i \(0.660784\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 862.000i 0.0349689i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24000.0 −0.966756
\(852\) 0 0
\(853\) 29816.0i 1.19681i 0.801193 + 0.598406i \(0.204199\pi\)
−0.801193 + 0.598406i \(0.795801\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 35430.0i − 1.41221i −0.708106 0.706106i \(-0.750450\pi\)
0.708106 0.706106i \(-0.249550\pi\)
\(858\) 0 0
\(859\) 36196.0 1.43771 0.718854 0.695161i \(-0.244667\pi\)
0.718854 + 0.695161i \(0.244667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 480.000i − 0.0189332i −0.999955 0.00946662i \(-0.996987\pi\)
0.999955 0.00946662i \(-0.00301336\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27480.0 −1.07272
\(870\) 0 0
\(871\) −3632.00 −0.141292
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 28532.0i − 1.09858i −0.835631 0.549291i \(-0.814898\pi\)
0.835631 0.549291i \(-0.185102\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20340.0 0.777834 0.388917 0.921273i \(-0.372849\pi\)
0.388917 + 0.921273i \(0.372849\pi\)
\(882\) 0 0
\(883\) − 10756.0i − 0.409930i −0.978769 0.204965i \(-0.934292\pi\)
0.978769 0.204965i \(-0.0657081\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 600.000i 0.0227125i 0.999936 + 0.0113563i \(0.00361489\pi\)
−0.999936 + 0.0113563i \(0.996385\pi\)
\(888\) 0 0
\(889\) −4900.00 −0.184860
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 3360.00i − 0.125911i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −840.000 −0.0311630
\(900\) 0 0
\(901\) −2700.00 −0.0998336
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 25400.0i − 0.929871i −0.885345 0.464936i \(-0.846077\pi\)
0.885345 0.464936i \(-0.153923\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36240.0 1.31799 0.658993 0.752149i \(-0.270983\pi\)
0.658993 + 0.752149i \(0.270983\pi\)
\(912\) 0 0
\(913\) − 34200.0i − 1.23971i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1380.00i 0.0496964i
\(918\) 0 0
\(919\) −6572.00 −0.235898 −0.117949 0.993020i \(-0.537632\pi\)
−0.117949 + 0.993020i \(0.537632\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 4080.00i − 0.145498i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2340.00 0.0826404 0.0413202 0.999146i \(-0.486844\pi\)
0.0413202 + 0.999146i \(0.486844\pi\)
\(930\) 0 0
\(931\) 9492.00 0.334144
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2522.00i − 0.0879297i −0.999033 0.0439649i \(-0.986001\pi\)
0.999033 0.0439649i \(-0.0139990\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 52770.0 1.82811 0.914056 0.405589i \(-0.132933\pi\)
0.914056 + 0.405589i \(0.132933\pi\)
\(942\) 0 0
\(943\) 28800.0i 0.994546i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28200.0i 0.967663i 0.875161 + 0.483832i \(0.160755\pi\)
−0.875161 + 0.483832i \(0.839245\pi\)
\(948\) 0 0
\(949\) −1000.00 −0.0342059
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15570.0i 0.529236i 0.964353 + 0.264618i \(0.0852458\pi\)
−0.964353 + 0.264618i \(0.914754\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4140.00 0.139403
\(960\) 0 0
\(961\) −29775.0 −0.999463
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8350.00i 0.277681i 0.990315 + 0.138841i \(0.0443376\pi\)
−0.990315 + 0.138841i \(0.955662\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43650.0 −1.44263 −0.721316 0.692606i \(-0.756462\pi\)
−0.721316 + 0.692606i \(0.756462\pi\)
\(972\) 0 0
\(973\) − 3848.00i − 0.126784i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18810.0i 0.615952i 0.951394 + 0.307976i \(0.0996517\pi\)
−0.951394 + 0.307976i \(0.900348\pi\)
\(978\) 0 0
\(979\) 12600.0 0.411336
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25320.0i 0.821549i 0.911737 + 0.410774i \(0.134742\pi\)
−0.911737 + 0.410774i \(0.865258\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16320.0 −0.524718
\(990\) 0 0
\(991\) −6736.00 −0.215919 −0.107960 0.994155i \(-0.534432\pi\)
−0.107960 + 0.994155i \(0.534432\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 20500.0i 0.651195i 0.945508 + 0.325598i \(0.105565\pi\)
−0.945508 + 0.325598i \(0.894435\pi\)
\(998\) 0 0
\(999\) 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 900.4.d.d.649.1 2
3.2 odd 2 900.4.d.i.649.1 2
5.2 odd 4 180.4.a.b.1.1 1
5.3 odd 4 900.4.a.i.1.1 1
5.4 even 2 inner 900.4.d.d.649.2 2
15.2 even 4 180.4.a.e.1.1 yes 1
15.8 even 4 900.4.a.j.1.1 1
15.14 odd 2 900.4.d.i.649.2 2
20.7 even 4 720.4.a.h.1.1 1
45.2 even 12 1620.4.i.c.1081.1 2
45.7 odd 12 1620.4.i.i.1081.1 2
45.22 odd 12 1620.4.i.i.541.1 2
45.32 even 12 1620.4.i.c.541.1 2
60.47 odd 4 720.4.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.4.a.b.1.1 1 5.2 odd 4
180.4.a.e.1.1 yes 1 15.2 even 4
720.4.a.h.1.1 1 20.7 even 4
720.4.a.w.1.1 1 60.47 odd 4
900.4.a.i.1.1 1 5.3 odd 4
900.4.a.j.1.1 1 15.8 even 4
900.4.d.d.649.1 2 1.1 even 1 trivial
900.4.d.d.649.2 2 5.4 even 2 inner
900.4.d.i.649.1 2 3.2 odd 2
900.4.d.i.649.2 2 15.14 odd 2
1620.4.i.c.541.1 2 45.32 even 12
1620.4.i.c.1081.1 2 45.2 even 12
1620.4.i.i.541.1 2 45.22 odd 12
1620.4.i.i.1081.1 2 45.7 odd 12