Properties

Label 175.2.b.a.99.1
Level $175$
Weight $2$
Character 175.99
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.2.b.a.99.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} +2.00000 q^{9} -3.00000 q^{11} -2.00000i q^{12} -5.00000i q^{13} +4.00000 q^{16} +3.00000i q^{17} -2.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} -5.00000i q^{27} +2.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} +3.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -5.00000 q^{39} -12.0000 q^{41} +10.0000i q^{43} -6.00000 q^{44} +9.00000i q^{47} -4.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} -10.0000i q^{52} -12.0000i q^{53} +2.00000i q^{57} +8.00000 q^{61} +2.00000i q^{63} +8.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{69} -2.00000i q^{73} -4.00000 q^{76} -3.00000i q^{77} +1.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +2.00000 q^{84} +3.00000i q^{87} +12.0000 q^{89} +5.00000 q^{91} +12.0000i q^{92} +4.00000i q^{93} -1.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{4} + 4q^{9} - 6q^{11} + 8q^{16} - 4q^{19} + 2q^{21} - 6q^{29} - 8q^{31} + 8q^{36} - 10q^{39} - 24q^{41} - 12q^{44} - 2q^{49} + 6q^{51} + 16q^{61} + 16q^{64} + 12q^{69} - 8q^{76} + 2q^{79} + 2q^{81} + 4q^{84} + 24q^{89} + 10q^{91} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 2.00000i 0.377964i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 4.00000 0.666667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) − 10.0000i − 1.38675i
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 3.00000i − 0.341882i
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 0 0
\(87\) 3.00000i 0.321634i
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 12.0000i 1.25109i
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.00000i − 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 5.00000i − 0.492665i −0.969185 0.246332i \(-0.920775\pi\)
0.969185 0.246332i \(-0.0792255\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) − 10.0000i − 0.962250i
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000i 0.377964i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) − 10.0000i − 0.924500i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 6.00000i 0.522233i
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 15.0000i 1.25436i
\(144\) 8.00000 0.666667
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 4.00000i 0.328798i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −10.0000 −0.800641
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) − 2.00000i − 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) −24.0000 −1.87409
\(165\) 0 0
\(166\) 0 0
\(167\) − 3.00000i − 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 20.0000i 1.52499i
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) − 8.00000i − 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.00000i − 0.658145i
\(188\) 18.0000i 1.31278i
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) − 3.00000i − 0.210559i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000i 0.834058i
\(208\) − 20.0000i − 1.38675i
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) − 24.0000i − 1.64833i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) 19.0000i 1.27233i 0.771551 + 0.636167i \(0.219481\pi\)
−0.771551 + 0.636167i \(0.780519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.00000i − 0.199117i −0.995032 0.0995585i \(-0.968257\pi\)
0.995032 0.0995585i \(-0.0317430\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.00000i − 0.0649570i
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 16.0000 1.02430
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000i 0.636285i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 4.00000i 0.251976i
\(253\) − 18.0000i − 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 30.0000i 1.87135i 0.352865 + 0.935674i \(0.385208\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) − 8.00000i − 0.488678i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 12.0000i 0.727607i
\(273\) − 5.00000i − 0.302614i
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 13.0000i 0.772770i 0.922338 + 0.386385i \(0.126276\pi\)
−0.922338 + 0.386385i \(0.873724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 12.0000i − 0.708338i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) − 4.00000i − 0.234082i
\(293\) 21.0000i 1.22683i 0.789760 + 0.613417i \(0.210205\pi\)
−0.789760 + 0.613417i \(0.789795\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000i 0.870388i
\(298\) 0 0
\(299\) 30.0000 1.73494
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) 11.0000i 0.627803i 0.949456 + 0.313902i \(0.101636\pi\)
−0.949456 + 0.313902i \(0.898364\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) −5.00000 −0.284440
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 19.0000i 1.07394i 0.843600 + 0.536972i \(0.180432\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) − 6.00000i − 0.333849i
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.00000i − 0.387101i
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 24.0000i − 1.31717i
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.0000i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 6.00000i 0.321634i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −25.0000 −1.33440
\(352\) 0 0
\(353\) − 15.0000i − 0.798369i −0.916871 0.399185i \(-0.869293\pi\)
0.916871 0.399185i \(-0.130707\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) 3.00000i 0.158777i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000i 0.104973i
\(364\) 10.0000 0.524142
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 24.0000i 1.25109i
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000i 0.414781i
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000i 0.772539i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.0000i 1.01666i
\(388\) − 2.00000i − 0.101535i
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) − 25.0000i − 1.25471i −0.778732 0.627357i \(-0.784137\pi\)
0.778732 0.627357i \(-0.215863\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 20.0000i 0.996271i
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.00000i − 0.297409i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) − 10.0000i − 0.492665i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) 18.0000i 0.875190i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 12.0000i 0.580042i
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) − 20.0000i − 0.962250i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) − 12.0000i − 0.574038i
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 18.0000i 0.855206i 0.903967 + 0.427603i \(0.140642\pi\)
−0.903967 + 0.427603i \(0.859358\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 0 0
\(447\) − 6.00000i − 0.283790i
\(448\) 8.00000i 0.377964i
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) − 12.0000i − 0.564433i
\(453\) 1.00000i 0.0469841i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) −12.0000 −0.557086
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0000i 0.694117i 0.937843 + 0.347059i \(0.112820\pi\)
−0.937843 + 0.347059i \(0.887180\pi\)
\(468\) − 20.0000i − 0.924500i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) − 30.0000i − 1.37940i
\(474\) 0 0
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) − 24.0000i − 1.09888i
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) −4.00000 −0.181818
\(485\) 0 0
\(486\) 0 0
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 24.0000i 1.08200i
\(493\) − 9.00000i − 0.405340i
\(494\) 0 0
\(495\) 0 0
\(496\) −16.0000 −0.718421
\(497\) 0 0
\(498\) 0 0
\(499\) 31.0000 1.38775 0.693875 0.720095i \(-0.255902\pi\)
0.693875 + 0.720095i \(0.255902\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) 0 0
\(503\) − 27.0000i − 1.20387i −0.798545 0.601935i \(-0.794397\pi\)
0.798545 0.601935i \(-0.205603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000i 0.532939i
\(508\) − 32.0000i − 1.41977i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 10.0000i 0.441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 20.0000 0.880451
\(517\) − 27.0000i − 1.18746i
\(518\) 0 0
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) − 12.0000i − 0.522728i
\(528\) 12.0000i 0.522233i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 4.00000i − 0.173422i
\(533\) 60.0000i 2.59889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) − 20.0000i − 0.858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) − 24.0000i − 1.02523i
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 1.00000i 0.0425243i
\(554\) 0 0
\(555\) 0 0
\(556\) −28.0000 −1.18746
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) 50.0000 2.11477
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 18.0000 0.757937
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 30.0000i 1.25436i
\(573\) − 9.00000i − 0.375980i
\(574\) 0 0
\(575\) 0 0
\(576\) 16.0000 0.666667
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 24.0000i − 0.990586i −0.868726 0.495293i \(-0.835061\pi\)
0.868726 0.495293i \(-0.164939\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000i 0.328798i
\(593\) 39.0000i 1.60154i 0.598973 + 0.800769i \(0.295576\pi\)
−0.598973 + 0.800769i \(0.704424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) − 16.0000i − 0.654836i
\(598\) 0 0
\(599\) −45.0000 −1.83865 −0.919325 0.393499i \(-0.871265\pi\)
−0.919325 + 0.393499i \(0.871265\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) − 13.0000i − 0.527654i −0.964570 0.263827i \(-0.915015\pi\)
0.964570 0.263827i \(-0.0849848\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 45.0000 1.82051
\(612\) 12.0000i 0.485071i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) −20.0000 −0.800641
\(625\) 0 0
\(626\) 0 0
\(627\) − 6.00000i − 0.239617i
\(628\) 28.0000i 1.11732i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 13.0000i 0.516704i
\(634\) 0 0
\(635\) 0 0
\(636\) −24.0000 −0.951662
\(637\) 5.00000i 0.198107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) − 41.0000i − 1.61688i −0.588577 0.808441i \(-0.700312\pi\)
0.588577 0.808441i \(-0.299688\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) − 4.00000i − 0.156652i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −48.0000 −1.87409
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) − 15.0000i − 0.582552i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 18.0000i − 0.696963i
\(668\) − 6.00000i − 0.232147i
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 28.0000i 1.07932i 0.841883 + 0.539660i \(0.181447\pi\)
−0.841883 + 0.539660i \(0.818553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 45.0000i 1.72949i 0.502211 + 0.864745i \(0.332520\pi\)
−0.502211 + 0.864745i \(0.667480\pi\)
\(678\) 0 0
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.00000i − 0.152610i
\(688\) 40.0000i 1.52499i
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000i 0.684257i
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −9.00000 −0.339925 −0.169963 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) −24.0000 −0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) − 24.0000i − 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) − 21.0000i − 0.784259i
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) 0 0
\(723\) 10.0000i 0.371904i
\(724\) 40.0000 1.48659
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) − 16.0000i − 0.591377i
\(733\) 31.0000i 1.14501i 0.819901 + 0.572506i \(0.194029\pi\)
−0.819901 + 0.572506i \(0.805971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000i 0.442026i
\(738\) 0 0
\(739\) 43.0000 1.58178 0.790890 0.611958i \(-0.209618\pi\)
0.790890 + 0.611958i \(0.209618\pi\)
\(740\) 0 0
\(741\) 10.0000 0.367359
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 24.0000i − 0.878114i
\(748\) − 18.0000i − 0.658145i
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 36.0000i 1.31278i
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 10.0000 0.363696
\(757\) − 16.0000i − 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) 0 0
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) − 16.0000i − 0.577350i
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 8.00000i 0.287926i
\(773\) − 21.0000i − 0.755318i −0.925945 0.377659i \(-0.876729\pi\)
0.925945 0.377659i \(-0.123271\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.00000i 0.0717496i
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 15.0000i 0.536056i
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 0 0
\(787\) 5.00000i 0.178231i 0.996021 + 0.0891154i \(0.0284040\pi\)
−0.996021 + 0.0891154i \(0.971596\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) − 40.0000i − 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 32.0000 1.13421
\(797\) 15.0000i 0.531327i 0.964066 + 0.265664i \(0.0855911\pi\)
−0.964066 + 0.265664i \(0.914409\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 6.00000i 0.211735i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) − 20.0000i − 0.699711i
\(818\) 0 0
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 54.0000i 1.87776i 0.344239 + 0.938882i \(0.388137\pi\)
−0.344239 + 0.938882i \(0.611863\pi\)
\(828\) 24.0000i 0.834058i
\(829\) 52.0000 1.80603 0.903017 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) − 40.0000i − 1.38675i
\(833\) − 3.00000i − 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 20.0000i 0.691301i
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 3.00000i − 0.103325i
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) − 48.0000i − 1.64833i
\(849\) 13.0000 0.446159
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 30.0000i − 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) − 8.00000i − 0.271538i
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) 0 0
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) 0 0
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 38.0000i 1.27233i
\(893\) − 18.0000i − 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 30.0000i − 1.00167i
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 10.0000i 0.332779i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 26.0000i 0.863316i 0.902037 + 0.431658i \(0.142071\pi\)
−0.902037 + 0.431658i \(0.857929\pi\)
\(908\) − 6.00000i − 0.199117i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 8.00000i 0.264906i
\(913\) 36.0000i 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) − 6.00000i − 0.198137i
\(918\) 0 0
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 0 0
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 0 0
\(927\) − 10.0000i − 0.328443i
\(928\) 0 0
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) − 48.0000i − 1.57229i
\(933\) − 18.0000i − 0.589294i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47.0000i 1.53542i 0.640796 + 0.767712i \(0.278605\pi\)
−0.640796 + 0.767712i \(0.721395\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) − 72.0000i − 2.34464i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) − 2.00000i − 0.0649570i
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 42.0000 1.35838
\(957\) − 9.00000i − 0.290929i
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) 0 0
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) − 32.0000i − 1.02640i
\(973\) − 14.0000i − 0.448819i
\(974\) 0 0
\(975\) 0 0
\(976\) 32.0000 1.02430
\(977\) 54.0000i 1.72761i 0.503824 + 0.863807i \(0.331926\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 21.0000i 0.669796i 0.942254 + 0.334898i \(0.108702\pi\)
−0.942254 + 0.334898i \(0.891298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.00000i 0.286473i
\(988\) 20.0000i 0.636285i
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) − 37.0000i − 1.17180i −0.810383 0.585901i \(-0.800741\pi\)
0.810383 0.585901i \(-0.199259\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 175.2.b.a.99.1 2
3.2 odd 2 1575.2.d.c.1324.2 2
4.3 odd 2 2800.2.g.l.449.2 2
5.2 odd 4 175.2.a.b.1.1 1
5.3 odd 4 35.2.a.a.1.1 1
5.4 even 2 inner 175.2.b.a.99.2 2
7.6 odd 2 1225.2.b.d.99.2 2
15.2 even 4 1575.2.a.f.1.1 1
15.8 even 4 315.2.a.b.1.1 1
15.14 odd 2 1575.2.d.c.1324.1 2
20.3 even 4 560.2.a.b.1.1 1
20.7 even 4 2800.2.a.z.1.1 1
20.19 odd 2 2800.2.g.l.449.1 2
35.3 even 12 245.2.e.b.226.1 2
35.13 even 4 245.2.a.c.1.1 1
35.18 odd 12 245.2.e.a.226.1 2
35.23 odd 12 245.2.e.a.116.1 2
35.27 even 4 1225.2.a.e.1.1 1
35.33 even 12 245.2.e.b.116.1 2
35.34 odd 2 1225.2.b.d.99.1 2
40.3 even 4 2240.2.a.u.1.1 1
40.13 odd 4 2240.2.a.k.1.1 1
55.43 even 4 4235.2.a.c.1.1 1
60.23 odd 4 5040.2.a.v.1.1 1
65.38 odd 4 5915.2.a.f.1.1 1
105.83 odd 4 2205.2.a.e.1.1 1
140.83 odd 4 3920.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 1 5.3 odd 4
175.2.a.b.1.1 1 5.2 odd 4
175.2.b.a.99.1 2 1.1 even 1 trivial
175.2.b.a.99.2 2 5.4 even 2 inner
245.2.a.c.1.1 1 35.13 even 4
245.2.e.a.116.1 2 35.23 odd 12
245.2.e.a.226.1 2 35.18 odd 12
245.2.e.b.116.1 2 35.33 even 12
245.2.e.b.226.1 2 35.3 even 12
315.2.a.b.1.1 1 15.8 even 4
560.2.a.b.1.1 1 20.3 even 4
1225.2.a.e.1.1 1 35.27 even 4
1225.2.b.d.99.1 2 35.34 odd 2
1225.2.b.d.99.2 2 7.6 odd 2
1575.2.a.f.1.1 1 15.2 even 4
1575.2.d.c.1324.1 2 15.14 odd 2
1575.2.d.c.1324.2 2 3.2 odd 2
2205.2.a.e.1.1 1 105.83 odd 4
2240.2.a.k.1.1 1 40.13 odd 4
2240.2.a.u.1.1 1 40.3 even 4
2800.2.a.z.1.1 1 20.7 even 4
2800.2.g.l.449.1 2 20.19 odd 2
2800.2.g.l.449.2 2 4.3 odd 2
3920.2.a.ba.1.1 1 140.83 odd 4
4235.2.a.c.1.1 1 55.43 even 4
5040.2.a.v.1.1 1 60.23 odd 4
5915.2.a.f.1.1 1 65.38 odd 4