Properties

Label 975.2.c.g.274.2
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.g.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} -5.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} -3.00000 q^{21} -1.00000i q^{22} +3.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} +5.00000i q^{32} +1.00000i q^{33} +5.00000 q^{34} -1.00000 q^{36} -8.00000i q^{37} +8.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} -3.00000i q^{42} -8.00000i q^{43} -1.00000 q^{44} -11.0000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -5.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} +9.00000 q^{56} -8.00000i q^{57} -1.00000i q^{58} -5.00000 q^{59} +1.00000 q^{61} +3.00000i q^{62} +3.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} +3.00000i q^{67} -5.00000i q^{68} +16.0000 q^{71} -3.00000i q^{72} +4.00000i q^{73} +8.00000 q^{74} +8.00000 q^{76} +3.00000i q^{77} -1.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +3.00000i q^{83} -3.00000 q^{84} +8.00000 q^{86} +1.00000i q^{87} -3.00000i q^{88} -3.00000 q^{91} -3.00000i q^{93} +11.0000 q^{94} +5.00000 q^{96} -2.00000i q^{97} -2.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 6 q^{21} + 6 q^{24} + 2 q^{26} - 2 q^{29} + 6 q^{31} + 10 q^{34} - 2 q^{36} - 2 q^{39} - 4 q^{41} - 2 q^{44} - 4 q^{49} - 10 q^{51} - 2 q^{54} + 18 q^{56} - 10 q^{59} + 2 q^{61} - 14 q^{64} - 2 q^{66} + 32 q^{71} + 16 q^{74} + 16 q^{76} - 24 q^{79} + 2 q^{81} - 6 q^{84} + 16 q^{86} - 6 q^{91} + 22 q^{94} + 10 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) − 1.00000i − 0.213201i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000i 0.192450i
\(28\) − 3.00000i − 0.566947i
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 1.00000i 0.174078i
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 8.00000i 1.29777i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) − 3.00000i − 0.462910i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.0000i − 1.60451i −0.596978 0.802257i \(-0.703632\pi\)
0.596978 0.802257i \(-0.296368\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) − 1.00000i − 0.138675i
\(53\) 11.0000i 1.51097i 0.655168 + 0.755483i \(0.272598\pi\)
−0.655168 + 0.755483i \(0.727402\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 9.00000 1.20268
\(57\) − 8.00000i − 1.05963i
\(58\) − 1.00000i − 0.131306i
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 3.00000i 0.381000i
\(63\) 3.00000i 0.377964i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) − 5.00000i − 0.606339i
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 3.00000i 0.341882i
\(78\) − 1.00000i − 0.113228i
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 3.00000i 0.329293i 0.986353 + 0.164646i \(0.0526483\pi\)
−0.986353 + 0.164646i \(0.947352\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 1.00000i 0.107211i
\(88\) − 3.00000i − 0.319801i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) − 3.00000i − 0.311086i
\(94\) 11.0000 1.13456
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) − 5.00000i − 0.495074i
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 3.00000i 0.283473i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 1.00000i 0.0924500i
\(118\) − 5.00000i − 0.460287i
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 1.00000i 0.0905357i
\(123\) 2.00000i 0.180334i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 3.00000i 0.265165i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) − 24.0000i − 2.08106i
\(134\) −3.00000 −0.259161
\(135\) 0 0
\(136\) 15.0000 1.28624
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −11.0000 −0.926367
\(142\) 16.0000i 1.34269i
\(143\) 1.00000i 0.0836242i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 2.00000i 0.164957i
\(148\) − 8.00000i − 0.657596i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 24.0000i 1.94666i
\(153\) 5.00000i 0.404226i
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 11.0000i − 0.877896i −0.898513 0.438948i \(-0.855351\pi\)
0.898513 0.438948i \(-0.144649\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) − 9.00000i − 0.694365i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) − 8.00000i − 0.609994i
\(173\) 21.0000i 1.59660i 0.602260 + 0.798300i \(0.294267\pi\)
−0.602260 + 0.798300i \(0.705733\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 5.00000i 0.375823i
\(178\) 0 0
\(179\) −26.0000 −1.94333 −0.971666 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) − 3.00000i − 0.222375i
\(183\) − 1.00000i − 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 5.00000i 0.365636i
\(188\) − 11.0000i − 0.802257i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 7.00000i 0.505181i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 9.00000i 0.633238i
\(203\) 3.00000i 0.210559i
\(204\) −5.00000 −0.350070
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 1.00000i 0.0693375i
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 11.0000i 0.755483i
\(213\) − 16.0000i − 1.09630i
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) − 9.00000i − 0.610960i
\(218\) 10.0000i 0.677285i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) − 8.00000i − 0.536925i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 15.0000 1.00223
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 23.0000i 1.52656i 0.646066 + 0.763282i \(0.276413\pi\)
−0.646066 + 0.763282i \(0.723587\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) − 3.00000i − 0.196960i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −5.00000 −0.325472
\(237\) 12.0000i 0.779484i
\(238\) − 15.0000i − 0.972306i
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 10.0000i − 0.642824i
\(243\) − 1.00000i − 0.0641500i
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 8.00000i − 0.509028i
\(248\) 9.00000i 0.571501i
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 3.00000i − 0.187135i −0.995613 0.0935674i \(-0.970173\pi\)
0.995613 0.0935674i \(-0.0298271\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 6.00000i 0.370681i
\(263\) − 14.0000i − 0.863277i −0.902047 0.431638i \(-0.857936\pi\)
0.902047 0.431638i \(-0.142064\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) 3.00000i 0.183254i
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) 5.00000i 0.303170i
\(273\) 3.00000i 0.181568i
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 6.00000i 0.360505i 0.983620 + 0.180253i \(0.0576915\pi\)
−0.983620 + 0.180253i \(0.942309\pi\)
\(278\) 10.0000i 0.599760i
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) − 11.0000i − 0.655040i
\(283\) 18.0000i 1.06999i 0.844856 + 0.534994i \(0.179686\pi\)
−0.844856 + 0.534994i \(0.820314\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 6.00000i 0.354169i
\(288\) − 5.00000i − 0.294628i
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 4.00000i 0.234082i
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 24.0000 1.39497
\(297\) − 1.00000i − 0.0580259i
\(298\) − 10.0000i − 0.579284i
\(299\) 0 0
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) − 17.0000i − 0.978240i
\(303\) − 9.00000i − 0.517036i
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −5.00000 −0.285831
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 3.00000i 0.170941i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) − 3.00000i − 0.169842i
\(313\) 29.0000i 1.63918i 0.572953 + 0.819588i \(0.305798\pi\)
−0.572953 + 0.819588i \(0.694202\pi\)
\(314\) 11.0000 0.620766
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) − 16.0000i − 0.898650i −0.893368 0.449325i \(-0.851665\pi\)
0.893368 0.449325i \(-0.148335\pi\)
\(318\) 11.0000i 0.616849i
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) − 40.0000i − 2.22566i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) − 10.0000i − 0.553001i
\(328\) − 6.00000i − 0.331295i
\(329\) −33.0000 −1.81935
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 3.00000i 0.164646i
\(333\) 8.00000i 0.438397i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 5.00000i 0.272367i 0.990684 + 0.136184i \(0.0434837\pi\)
−0.990684 + 0.136184i \(0.956516\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) − 8.00000i − 0.432590i
\(343\) − 15.0000i − 0.809924i
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 1.00000i 0.0536056i
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) − 5.00000i − 0.266501i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) −5.00000 −0.265747
\(355\) 0 0
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) − 26.0000i − 1.37414i
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 5.00000i 0.262794i
\(363\) 10.0000i 0.524864i
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) 1.00000 0.0522708
\(367\) − 12.0000i − 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 33.0000 1.71327
\(372\) − 3.00000i − 0.155543i
\(373\) 31.0000i 1.60512i 0.596572 + 0.802560i \(0.296529\pi\)
−0.596572 + 0.802560i \(0.703471\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) 33.0000 1.70185
\(377\) 1.00000i 0.0515026i
\(378\) 3.00000i 0.154303i
\(379\) 35.0000 1.79783 0.898915 0.438124i \(-0.144357\pi\)
0.898915 + 0.438124i \(0.144357\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 10.0000i 0.511645i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 8.00000i 0.406663i
\(388\) − 2.00000i − 0.101535i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 6.00000i − 0.303046i
\(393\) − 6.00000i − 0.302660i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 10.0000i − 0.501255i
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 3.00000i 0.149626i
\(403\) − 3.00000i − 0.149441i
\(404\) 9.00000 0.447767
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 8.00000i 0.396545i
\(408\) − 15.0000i − 0.742611i
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 14.0000i 0.689730i
\(413\) 15.0000i 0.738102i
\(414\) 0 0
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) − 10.0000i − 0.489702i
\(418\) − 8.00000i − 0.391293i
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 11.0000i 0.534838i
\(424\) −33.0000 −1.60262
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) − 3.00000i − 0.145180i
\(428\) 18.0000i 0.870063i
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 9.00000 0.432014
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 4.00000i 0.191127i
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) − 5.00000i − 0.237826i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 10.0000i 0.472984i
\(448\) 21.0000i 0.992157i
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 2.00000i 0.0940721i
\(453\) 17.0000i 0.798730i
\(454\) −23.0000 −1.07944
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) − 34.0000i − 1.59045i −0.606313 0.795226i \(-0.707352\pi\)
0.606313 0.795226i \(-0.292648\pi\)
\(458\) 18.0000i 0.841085i
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 3.00000i 0.139573i
\(463\) 9.00000i 0.418265i 0.977887 + 0.209133i \(0.0670641\pi\)
−0.977887 + 0.209133i \(0.932936\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 32.0000i 1.48078i 0.672176 + 0.740392i \(0.265360\pi\)
−0.672176 + 0.740392i \(0.734640\pi\)
\(468\) 1.00000i 0.0462250i
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) − 15.0000i − 0.690431i
\(473\) 8.00000i 0.367840i
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) −15.0000 −0.687524
\(477\) − 11.0000i − 0.503655i
\(478\) 19.0000i 0.869040i
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 37.0000i 1.67663i 0.545186 + 0.838315i \(0.316459\pi\)
−0.545186 + 0.838315i \(0.683541\pi\)
\(488\) 3.00000i 0.135804i
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 5.00000i 0.225189i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) − 48.0000i − 2.15309i
\(498\) 3.00000i 0.134433i
\(499\) 41.0000 1.83541 0.917706 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) − 30.0000i − 1.33897i
\(503\) − 26.0000i − 1.15928i −0.814872 0.579641i \(-0.803193\pi\)
0.814872 0.579641i \(-0.196807\pi\)
\(504\) −9.00000 −0.400892
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 8.00000i 0.354943i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) − 11.0000i − 0.486136i
\(513\) 8.00000i 0.353209i
\(514\) 3.00000 0.132324
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 11.0000i 0.483779i
\(518\) − 24.0000i − 1.05450i
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 1.00000i 0.0437688i
\(523\) − 26.0000i − 1.13690i −0.822718 0.568450i \(-0.807543\pi\)
0.822718 0.568450i \(-0.192457\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) − 15.0000i − 0.653410i
\(528\) − 1.00000i − 0.0435194i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) − 24.0000i − 1.04053i
\(533\) 2.00000i 0.0866296i
\(534\) 0 0
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) 26.0000i 1.12198i
\(538\) − 17.0000i − 0.732922i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) − 5.00000i − 0.214768i
\(543\) − 5.00000i − 0.214571i
\(544\) 25.0000 1.07187
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) − 22.0000i − 0.940652i −0.882493 0.470326i \(-0.844136\pi\)
0.882493 0.470326i \(-0.155864\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 36.0000i 1.53088i
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) − 3.00000i − 0.127000i
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) − 18.0000i − 0.759284i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) −11.0000 −0.463184
\(565\) 0 0
\(566\) −18.0000 −0.756596
\(567\) − 3.00000i − 0.125988i
\(568\) 48.0000i 2.01404i
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 1.00000i 0.0418121i
\(573\) − 10.0000i − 0.417756i
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) − 2.00000i − 0.0829027i
\(583\) − 11.0000i − 0.455573i
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) − 45.0000i − 1.85735i −0.370896 0.928674i \(-0.620949\pi\)
0.370896 0.928674i \(-0.379051\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000i 0.328798i
\(593\) − 26.0000i − 1.06769i −0.845582 0.533846i \(-0.820746\pi\)
0.845582 0.533846i \(-0.179254\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 10.0000i 0.409273i
\(598\) 0 0
\(599\) −46.0000 −1.87951 −0.939755 0.341850i \(-0.888947\pi\)
−0.939755 + 0.341850i \(0.888947\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) − 24.0000i − 0.978167i
\(603\) − 3.00000i − 0.122169i
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) 38.0000i 1.54237i 0.636610 + 0.771186i \(0.280336\pi\)
−0.636610 + 0.771186i \(0.719664\pi\)
\(608\) 40.0000i 1.62221i
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) −11.0000 −0.445012
\(612\) 5.00000i 0.202113i
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 14.0000i 0.563163i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.00000i 0.240578i
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −29.0000 −1.15907
\(627\) 8.00000i 0.319489i
\(628\) − 11.0000i − 0.438948i
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) − 36.0000i − 1.43200i
\(633\) 16.0000i 0.635943i
\(634\) 16.0000 0.635441
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) 2.00000i 0.0792429i
\(638\) 1.00000i 0.0395904i
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 18.0000i 0.710403i
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) 16.0000i 0.629025i 0.949253 + 0.314512i \(0.101841\pi\)
−0.949253 + 0.314512i \(0.898159\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 20.0000i 0.783260i
\(653\) − 35.0000i − 1.36966i −0.728705 0.684828i \(-0.759877\pi\)
0.728705 0.684828i \(-0.240123\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) − 4.00000i − 0.156055i
\(658\) − 33.0000i − 1.28647i
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) 5.00000i 0.194184i
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) − 15.0000i − 0.578638i
\(673\) − 45.0000i − 1.73462i −0.497766 0.867311i \(-0.665846\pi\)
0.497766 0.867311i \(-0.334154\pi\)
\(674\) −5.00000 −0.192593
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 2.00000i 0.0768095i
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 23.0000 0.881362
\(682\) − 3.00000i − 0.114876i
\(683\) − 9.00000i − 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) − 18.0000i − 0.686743i
\(688\) 8.00000i 0.304997i
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) −29.0000 −1.10321 −0.551606 0.834105i \(-0.685985\pi\)
−0.551606 + 0.834105i \(0.685985\pi\)
\(692\) 21.0000i 0.798300i
\(693\) − 3.00000i − 0.113961i
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 10.0000i 0.378777i
\(698\) − 16.0000i − 0.605609i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 31.0000 1.17085 0.585427 0.810725i \(-0.300927\pi\)
0.585427 + 0.810725i \(0.300927\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) − 64.0000i − 2.41381i
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) − 27.0000i − 1.01544i
\(708\) 5.00000i 0.187912i
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) −15.0000 −0.561361
\(715\) 0 0
\(716\) −26.0000 −0.971666
\(717\) − 19.0000i − 0.709568i
\(718\) − 15.0000i − 0.559795i
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 45.0000i 1.67473i
\(723\) − 10.0000i − 0.371904i
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) − 2.00000i − 0.0741759i −0.999312 0.0370879i \(-0.988192\pi\)
0.999312 0.0370879i \(-0.0118082\pi\)
\(728\) − 9.00000i − 0.333562i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) − 1.00000i − 0.0369611i
\(733\) 20.0000i 0.738717i 0.929287 + 0.369358i \(0.120423\pi\)
−0.929287 + 0.369358i \(0.879577\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) − 3.00000i − 0.110506i
\(738\) 2.00000i 0.0736210i
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 33.0000i 1.21147i
\(743\) − 13.0000i − 0.476924i −0.971152 0.238462i \(-0.923357\pi\)
0.971152 0.238462i \(-0.0766432\pi\)
\(744\) 9.00000 0.329956
\(745\) 0 0
\(746\) −31.0000 −1.13499
\(747\) − 3.00000i − 0.109764i
\(748\) 5.00000i 0.182818i
\(749\) 54.0000 1.97312
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 11.0000i 0.401129i
\(753\) 30.0000i 1.09326i
\(754\) −1.00000 −0.0364179
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) 23.0000i 0.835949i 0.908459 + 0.417975i \(0.137260\pi\)
−0.908459 + 0.417975i \(0.862740\pi\)
\(758\) 35.0000i 1.27126i
\(759\) 0 0
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 8.00000i 0.289809i
\(763\) − 30.0000i − 1.08607i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 5.00000i 0.180540i
\(768\) 17.0000i 0.613435i
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) − 10.0000i − 0.359908i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 24.0000i 0.860995i
\(778\) 30.0000i 1.07555i
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) − 1.00000i − 0.0357371i
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 3.00000i 0.106600i
\(793\) − 1.00000i − 0.0355110i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) − 43.0000i − 1.52314i −0.648084 0.761569i \(-0.724429\pi\)
0.648084 0.761569i \(-0.275571\pi\)
\(798\) − 24.0000i − 0.849591i
\(799\) −55.0000 −1.94576
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 4.00000i − 0.141157i
\(804\) 3.00000 0.105802
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) 17.0000i 0.598428i
\(808\) 27.0000i 0.949857i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 3.00000i 0.105279i
\(813\) 5.00000i 0.175358i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 5.00000 0.175035
\(817\) − 64.0000i − 2.23908i
\(818\) − 2.00000i − 0.0699284i
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) −42.0000 −1.46314
\(825\) 0 0
\(826\) −15.0000 −0.521917
\(827\) 1.00000i 0.0347734i 0.999849 + 0.0173867i \(0.00553464\pi\)
−0.999849 + 0.0173867i \(0.994465\pi\)
\(828\) 0 0
\(829\) −51.0000 −1.77130 −0.885652 0.464350i \(-0.846288\pi\)
−0.885652 + 0.464350i \(0.846288\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) 7.00000i 0.242681i
\(833\) 10.0000i 0.346479i
\(834\) 10.0000 0.346272
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 3.00000i 0.103695i
\(838\) 26.0000i 0.898155i
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 28.0000i 0.964944i
\(843\) 18.0000i 0.619953i
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) −11.0000 −0.378188
\(847\) 30.0000i 1.03081i
\(848\) − 11.0000i − 0.377742i
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) 0 0
\(852\) − 16.0000i − 0.548151i
\(853\) − 24.0000i − 0.821744i −0.911693 0.410872i \(-0.865224\pi\)
0.911693 0.410872i \(-0.134776\pi\)
\(854\) 3.00000 0.102658
\(855\) 0 0
\(856\) −54.0000 −1.84568
\(857\) − 38.0000i − 1.29806i −0.760765 0.649028i \(-0.775176\pi\)
0.760765 0.649028i \(-0.224824\pi\)
\(858\) 1.00000i 0.0341394i
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) − 12.0000i − 0.408722i
\(863\) 7.00000i 0.238283i 0.992877 + 0.119141i \(0.0380142\pi\)
−0.992877 + 0.119141i \(0.961986\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 8.00000i 0.271694i
\(868\) − 9.00000i − 0.305480i
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 30.0000i 1.01593i
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 24.0000i 0.810422i 0.914223 + 0.405211i \(0.132802\pi\)
−0.914223 + 0.405211i \(0.867198\pi\)
\(878\) − 4.00000i − 0.134993i
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −25.0000 −0.842271 −0.421136 0.906998i \(-0.638368\pi\)
−0.421136 + 0.906998i \(0.638368\pi\)
\(882\) 2.00000i 0.0673435i
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) −5.00000 −0.168168
\(885\) 0 0
\(886\) 0 0
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) − 24.0000i − 0.805387i
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 8.00000i 0.267860i
\(893\) − 88.0000i − 2.94481i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 9.00000 0.300669
\(897\) 0 0
\(898\) 28.0000i 0.934372i
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) 55.0000 1.83232
\(902\) 2.00000i 0.0665927i
\(903\) 24.0000i 0.798670i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −17.0000 −0.564787
\(907\) 24.0000i 0.796907i 0.917189 + 0.398453i \(0.130453\pi\)
−0.917189 + 0.398453i \(0.869547\pi\)
\(908\) 23.0000i 0.763282i
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 3.00000i − 0.0992855i
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) 18.0000 0.594737
\(917\) − 18.0000i − 0.594412i
\(918\) 5.00000i 0.165025i
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 42.0000i 1.38320i
\(923\) − 16.0000i − 0.526646i
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −9.00000 −0.295758
\(927\) − 14.0000i − 0.459820i
\(928\) − 5.00000i − 0.164133i
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) 6.00000i 0.196537i
\(933\) − 6.00000i − 0.196431i
\(934\) −32.0000 −1.04707
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 45.0000i 1.47009i 0.678021 + 0.735043i \(0.262838\pi\)
−0.678021 + 0.735043i \(0.737162\pi\)
\(938\) 9.00000i 0.293860i
\(939\) 29.0000 0.946379
\(940\) 0 0
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) − 11.0000i − 0.358399i
\(943\) 0 0
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 21.0000i 0.682408i 0.939989 + 0.341204i \(0.110835\pi\)
−0.939989 + 0.341204i \(0.889165\pi\)
\(948\) 12.0000i 0.389742i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −16.0000 −0.518836
\(952\) − 45.0000i − 1.45846i
\(953\) − 37.0000i − 1.19855i −0.800544 0.599274i \(-0.795456\pi\)
0.800544 0.599274i \(-0.204544\pi\)
\(954\) 11.0000 0.356138
\(955\) 0 0
\(956\) 19.0000 0.614504
\(957\) − 1.00000i − 0.0323254i
\(958\) 15.0000i 0.484628i
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 8.00000i − 0.257930i
\(963\) − 18.0000i − 0.580042i
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0000i 1.18984i 0.803785 + 0.594920i \(0.202816\pi\)
−0.803785 + 0.594920i \(0.797184\pi\)
\(968\) − 30.0000i − 0.964237i
\(969\) −40.0000 −1.28499
\(970\) 0 0
\(971\) 54.0000 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 30.0000i − 0.961756i
\(974\) −37.0000 −1.18556
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 28.0000i 0.895799i 0.894084 + 0.447900i \(0.147828\pi\)
−0.894084 + 0.447900i \(0.852172\pi\)
\(978\) 20.0000i 0.639529i
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 26.0000i − 0.829693i
\(983\) 21.0000i 0.669796i 0.942254 + 0.334898i \(0.108702\pi\)
−0.942254 + 0.334898i \(0.891298\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −5.00000 −0.159232
\(987\) 33.0000i 1.05040i
\(988\) − 8.00000i − 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 15.0000i 0.476250i
\(993\) 12.0000i 0.380808i
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 3.00000 0.0950586
\(997\) 3.00000i 0.0950110i 0.998871 + 0.0475055i \(0.0151272\pi\)
−0.998871 + 0.0475055i \(0.984873\pi\)
\(998\) 41.0000i 1.29783i
\(999\) 8.00000 0.253109
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 975.2.c.g.274.2 2
3.2 odd 2 2925.2.c.i.2224.1 2
5.2 odd 4 975.2.a.d.1.1 1
5.3 odd 4 975.2.a.k.1.1 yes 1
5.4 even 2 inner 975.2.c.g.274.1 2
15.2 even 4 2925.2.a.o.1.1 1
15.8 even 4 2925.2.a.b.1.1 1
15.14 odd 2 2925.2.c.i.2224.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
975.2.a.d.1.1 1 5.2 odd 4
975.2.a.k.1.1 yes 1 5.3 odd 4
975.2.c.g.274.1 2 5.4 even 2 inner
975.2.c.g.274.2 2 1.1 even 1 trivial
2925.2.a.b.1.1 1 15.8 even 4
2925.2.a.o.1.1 1 15.2 even 4
2925.2.c.i.2224.1 2 3.2 odd 2
2925.2.c.i.2224.2 2 15.14 odd 2