Properties

Label 5070.2.b.i.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.i.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000i q^{18} +4.00000i q^{19} +1.00000i q^{20} -4.00000i q^{21} -8.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -4.00000 q^{42} -12.0000 q^{43} -1.00000i q^{45} +8.00000i q^{46} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +10.0000 q^{53} -1.00000i q^{54} +4.00000 q^{56} +4.00000i q^{57} -2.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -2.00000 q^{68} -8.00000 q^{69} +4.00000i q^{70} -16.0000i q^{71} +1.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000i q^{83} +4.00000i q^{84} -2.00000i q^{85} +12.0000i q^{86} +2.00000 q^{87} +14.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} +4.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} - 8q^{14} + 2q^{16} + 4q^{17} - 16q^{23} - 2q^{25} + 2q^{27} + 4q^{29} - 2q^{30} - 8q^{35} - 2q^{36} + 8q^{38} + 2q^{40} - 8q^{42} - 24q^{43} + 2q^{48} - 18q^{49} + 4q^{51} + 20q^{53} + 8q^{56} - 20q^{61} - 16q^{62} - 2q^{64} - 4q^{68} - 16q^{69} - 4q^{74} - 2q^{75} - 16q^{79} + 2q^{81} - 12q^{82} + 4q^{87} - 2q^{90} + 16q^{92} + 8q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −4.00000 −1.06904
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.00000i 0.755929i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 8.00000i − 1.43684i −0.695608 0.718421i \(-0.744865\pi\)
0.695608 0.718421i \(-0.255135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 2.00000i − 0.342997i
\(35\) −4.00000 −0.676123
\(36\) −1.00000 −0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) −4.00000 −0.617213
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 8.00000i 1.17954i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 0.144338
\(49\) −9.00000 −1.28571
\(50\) 1.00000i 0.141421i
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 4.00000i 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 4.00000i 0.478091i
\(71\) − 16.0000i − 1.89885i −0.313993 0.949425i \(-0.601667\pi\)
0.313993 0.949425i \(-0.398333\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) − 4.00000i − 0.458831i
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 4.00000i 0.436436i
\(85\) − 2.00000i − 0.216930i
\(86\) 12.0000i 1.29399i
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) − 1.00000i − 0.102062i
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) − 10.0000i − 0.971286i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) − 2.00000i − 0.189832i
\(112\) − 4.00000i − 0.377964i
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000i 0.746004i
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 0 0
\(119\) − 8.00000i − 0.733359i
\(120\) 1.00000 0.0912871
\(121\) 11.0000 1.00000
\(122\) 10.0000i 0.905357i
\(123\) − 6.00000i − 0.541002i
\(124\) 8.00000i 0.718421i
\(125\) 1.00000i 0.0894427i
\(126\) −4.00000 −0.356348
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −4.00000 −0.345547
\(135\) − 1.00000i − 0.0860663i
\(136\) 2.00000i 0.171499i
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 2.00000i − 0.166091i
\(146\) 6.00000 0.496564
\(147\) −9.00000 −0.742307
\(148\) 2.00000i 0.164399i
\(149\) 10.0000i 0.819232i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 10.0000 0.793052
\(160\) −1.00000 −0.0790569
\(161\) 32.0000i 2.52195i
\(162\) − 1.00000i − 0.0785674i
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 4.00000 0.308607
\(169\) 0 0
\(170\) −2.00000 −0.153393
\(171\) 4.00000i 0.305888i
\(172\) 12.0000 0.914991
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) − 2.00000i − 0.151620i
\(175\) 4.00000i 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) − 8.00000i − 0.589768i
\(185\) −2.00000 −0.147043
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) − 4.00000i − 0.290957i
\(190\) − 4.00000i − 0.290191i
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 18.0000i − 1.29567i −0.761781 0.647834i \(-0.775675\pi\)
0.761781 0.647834i \(-0.224325\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 4.00000i − 0.282138i
\(202\) 10.0000i 0.703598i
\(203\) − 8.00000i − 0.561490i
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) − 4.00000i − 0.278693i
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 4.00000i 0.276026i
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −10.0000 −0.686803
\(213\) − 16.0000i − 1.09630i
\(214\) 12.0000i 0.820303i
\(215\) 12.0000i 0.818393i
\(216\) 1.00000i 0.0680414i
\(217\) −32.0000 −2.17230
\(218\) 14.0000 0.948200
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) 10.0000i 0.665190i
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) − 14.0000i − 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) 24.0000i 1.55243i 0.630468 + 0.776215i \(0.282863\pi\)
−0.630468 + 0.776215i \(0.717137\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 2.00000i − 0.128831i −0.997923 0.0644157i \(-0.979482\pi\)
0.997923 0.0644157i \(-0.0205183\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 9.00000i 0.574989i
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) − 4.00000i − 0.253490i
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 0 0
\(254\) − 12.0000i − 0.752947i
\(255\) − 2.00000i − 0.125245i
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 12.0000i 0.747087i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 8.00000i − 0.494242i
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) − 10.0000i − 0.614295i
\(266\) − 16.0000i − 0.981023i
\(267\) 14.0000i 0.856786i
\(268\) 4.00000i 0.244339i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 32.0000i 1.94386i 0.235267 + 0.971931i \(0.424404\pi\)
−0.235267 + 0.971931i \(0.575596\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 20.0000i 1.19952i
\(279\) − 8.00000i − 0.478947i
\(280\) − 4.00000i − 0.239046i
\(281\) − 18.0000i − 1.07379i −0.843649 0.536895i \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000i 0.949425i
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) − 1.00000i − 0.0589256i
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) − 6.00000i − 0.351726i
\(292\) − 6.00000i − 0.351123i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 9.00000i 0.524891i
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 48.0000i 2.76667i
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) 4.00000i 0.229416i
\(305\) 10.0000i 0.572598i
\(306\) − 2.00000i − 0.114332i
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 8.00000i 0.454369i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) − 18.0000i − 1.01580i
\(315\) −4.00000 −0.225374
\(316\) 8.00000 0.450035
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) −12.0000 −0.669775
\(322\) 32.0000 1.78329
\(323\) 8.00000i 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 14.0000i 0.774202i
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 4.00000i 0.219529i
\(333\) − 2.00000i − 0.109599i
\(334\) −16.0000 −0.875481
\(335\) −4.00000 −0.218543
\(336\) − 4.00000i − 0.218218i
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 2.00000i 0.108465i
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 8.00000i 0.431959i
\(344\) − 12.0000i − 0.646997i
\(345\) 8.00000i 0.430706i
\(346\) 2.00000i 0.107521i
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) − 14.0000i − 0.749403i −0.927146 0.374701i \(-0.877745\pi\)
0.927146 0.374701i \(-0.122255\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) − 14.0000i − 0.741999i
\(357\) − 8.00000i − 0.423405i
\(358\) 16.0000i 0.845626i
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 1.00000 0.0527046
\(361\) 3.00000 0.157895
\(362\) − 2.00000i − 0.105118i
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 10.0000i 0.522708i
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.00000 −0.417029
\(369\) − 6.00000i − 0.312348i
\(370\) 2.00000i 0.103975i
\(371\) − 40.0000i − 2.07670i
\(372\) 8.00000i 0.414781i
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) −4.00000 −0.205196
\(381\) 12.0000 0.614779
\(382\) − 8.00000i − 0.409316i
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −12.0000 −0.609994
\(388\) 6.00000i 0.304604i
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 9.00000i − 0.454569i
\(393\) 8.00000 0.403547
\(394\) −6.00000 −0.302276
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) − 26.0000i − 1.29838i −0.760627 0.649189i \(-0.775108\pi\)
0.760627 0.649189i \(-0.224892\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) − 1.00000i − 0.0496904i
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) 2.00000i 0.0990148i
\(409\) 10.0000i 0.494468i 0.968956 + 0.247234i \(0.0795217\pi\)
−0.968956 + 0.247234i \(0.920478\pi\)
\(410\) 6.00000i 0.296319i
\(411\) − 6.00000i − 0.295958i
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 8.00000i 0.393179i
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 4.00000 0.195180
\(421\) 30.0000i 1.46211i 0.682318 + 0.731055i \(0.260972\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) 10.0000i 0.485643i
\(425\) −2.00000 −0.0970143
\(426\) −16.0000 −0.775203
\(427\) 40.0000i 1.93574i
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 32.0000i 1.53605i
\(435\) − 2.00000i − 0.0958927i
\(436\) − 14.0000i − 0.670478i
\(437\) − 32.0000i − 1.53077i
\(438\) 6.00000 0.286691
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 2.00000i 0.0949158i
\(445\) 14.0000 0.663664
\(446\) 12.0000 0.568216
\(447\) 10.0000i 0.472984i
\(448\) 4.00000i 0.188982i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) 8.00000i 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) − 8.00000i − 0.373002i
\(461\) 2.00000i 0.0931493i 0.998915 + 0.0465746i \(0.0148305\pi\)
−0.998915 + 0.0465746i \(0.985169\pi\)
\(462\) 0 0
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 2.00000 0.0928477
\(465\) −8.00000 −0.370991
\(466\) − 26.0000i − 1.20443i
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000i 0.367452i
\(475\) − 4.00000i − 0.183533i
\(476\) 8.00000i 0.366679i
\(477\) 10.0000 0.457869
\(478\) 24.0000 1.09773
\(479\) − 8.00000i − 0.365529i −0.983157 0.182765i \(-0.941495\pi\)
0.983157 0.182765i \(-0.0585046\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 32.0000i 1.45605i
\(484\) −11.0000 −0.500000
\(485\) −6.00000 −0.272446
\(486\) − 1.00000i − 0.0453609i
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 20.0000i 0.904431i
\(490\) 9.00000 0.406579
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) − 8.00000i − 0.359211i
\(497\) −64.0000 −2.87079
\(498\) −4.00000 −0.179244
\(499\) − 44.0000i − 1.96971i −0.173379 0.984855i \(-0.555468\pi\)
0.173379 0.984855i \(-0.444532\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 16.0000i − 0.714827i
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 4.00000 0.178174
\(505\) 10.0000i 0.444994i
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) − 38.0000i − 1.68432i −0.539227 0.842160i \(-0.681284\pi\)
0.539227 0.842160i \(-0.318716\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 24.0000 1.06170
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 22.0000i 0.970378i
\(515\) − 4.00000i − 0.176261i
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −8.00000 −0.349482
\(525\) 4.00000i 0.174574i
\(526\) − 24.0000i − 1.04645i
\(527\) − 16.0000i − 0.696971i
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) 12.0000i 0.518805i
\(536\) 4.00000 0.172774
\(537\) −16.0000 −0.690451
\(538\) 14.0000i 0.603583i
\(539\) 0 0
\(540\) 1.00000i 0.0430331i
\(541\) 2.00000i 0.0859867i 0.999075 + 0.0429934i \(0.0136894\pi\)
−0.999075 + 0.0429934i \(0.986311\pi\)
\(542\) 32.0000 1.37452
\(543\) 2.00000 0.0858282
\(544\) − 2.00000i − 0.0857493i
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 8.00000i 0.340811i
\(552\) − 8.00000i − 0.340503i
\(553\) 32.0000i 1.36078i
\(554\) 18.0000i 0.764747i
\(555\) −2.00000 −0.0848953
\(556\) 20.0000 0.848189
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 10.0000i 0.420703i
\(566\) 20.0000i 0.840663i
\(567\) − 4.00000i − 0.167984i
\(568\) 16.0000 0.671345
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) − 4.00000i − 0.167542i
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 24.0000i 1.00174i
\(575\) 8.00000 0.333623
\(576\) −1.00000 −0.0416667
\(577\) − 22.0000i − 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 13.0000i 0.540729i
\(579\) − 18.0000i − 0.748054i
\(580\) 2.00000i 0.0830455i
\(581\) −16.0000 −0.663792
\(582\) −6.00000 −0.248708
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 9.00000 0.371154
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) − 6.00000i − 0.246807i
\(592\) − 2.00000i − 0.0821995i
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) − 10.0000i − 0.409616i
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 48.0000 1.95633
\(603\) − 4.00000i − 0.162893i
\(604\) − 8.00000i − 0.325515i
\(605\) − 11.0000i − 0.447214i
\(606\) 10.0000i 0.406222i
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 4.00000 0.162221
\(609\) − 8.00000i − 0.324176i
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 12.0000 0.484281
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 8.00000 0.321288
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 56.0000 2.24359
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 18.0000i − 0.719425i
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) − 4.00000i − 0.159490i
\(630\) 4.00000i 0.159364i
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) −12.0000 −0.476957
\(634\) −14.0000 −0.556011
\(635\) − 12.0000i − 0.476205i
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) − 16.0000i − 0.632950i
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) − 32.0000i − 1.26098i
\(645\) 12.0000i 0.472500i
\(646\) 8.00000 0.314756
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) − 20.0000i − 0.783260i
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 14.0000 0.547443
\(655\) − 8.00000i − 0.312586i
\(656\) − 6.00000i − 0.234261i
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 10.0000i 0.388955i 0.980907 + 0.194477i \(0.0623011\pi\)
−0.980907 + 0.194477i \(0.937699\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) − 16.0000i − 0.620453i
\(666\) −2.00000 −0.0774984
\(667\) −16.0000 −0.619522
\(668\) 16.0000i 0.619059i
\(669\) 12.0000i 0.463947i
\(670\) 4.00000i 0.154533i
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) − 22.0000i − 0.847408i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 10.0000i 0.384048i
\(679\) −24.0000 −0.921035
\(680\) 2.00000 0.0766965
\(681\) − 12.0000i − 0.459841i
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) − 4.00000i − 0.152944i
\(685\) −6.00000 −0.229248
\(686\) 8.00000 0.305441
\(687\) − 14.0000i − 0.534133i
\(688\) −12.0000 −0.457496
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) 20.0000i 0.760836i 0.924815 + 0.380418i \(0.124220\pi\)
−0.924815 + 0.380418i \(0.875780\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 4.00000i 0.151838i
\(695\) 20.0000i 0.758643i
\(696\) 2.00000i 0.0758098i
\(697\) − 12.0000i − 0.454532i
\(698\) −14.0000 −0.529908
\(699\) 26.0000 0.983410
\(700\) − 4.00000i − 0.151186i
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) 34.0000i 1.27690i 0.769665 + 0.638448i \(0.220423\pi\)
−0.769665 + 0.638448i \(0.779577\pi\)
\(710\) 16.0000i 0.600469i
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 64.0000i 2.39682i
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) 24.0000i 0.896296i
\(718\) −24.0000 −0.895672
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 16.0000i − 0.595871i
\(722\) − 3.00000i − 0.111648i
\(723\) − 2.00000i − 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) − 11.0000i − 0.408248i
\(727\) −52.0000 −1.92857 −0.964287 0.264861i \(-0.914674\pi\)
−0.964287 + 0.264861i \(0.914674\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 6.00000i − 0.222070i
\(731\) −24.0000 −0.887672
\(732\) 10.0000 0.369611
\(733\) − 22.0000i − 0.812589i −0.913742 0.406294i \(-0.866821\pi\)
0.913742 0.406294i \(-0.133179\pi\)
\(734\) − 4.00000i − 0.147643i
\(735\) 9.00000i 0.331970i
\(736\) 8.00000i 0.294884i
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) − 4.00000i − 0.147142i −0.997290 0.0735712i \(-0.976560\pi\)
0.997290 0.0735712i \(-0.0234396\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −40.0000 −1.46845
\(743\) − 40.0000i − 1.46746i −0.679442 0.733729i \(-0.737778\pi\)
0.679442 0.733729i \(-0.262222\pi\)
\(744\) 8.00000 0.293294
\(745\) 10.0000 0.366372
\(746\) 22.0000i 0.805477i
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 48.0000i 1.75388i
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 4.00000i 0.145479i
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 4.00000i 0.145095i
\(761\) − 18.0000i − 0.652499i −0.945284 0.326250i \(-0.894215\pi\)
0.945284 0.326250i \(-0.105785\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 56.0000 2.02734
\(764\) −8.00000 −0.289430
\(765\) − 2.00000i − 0.0723102i
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 2.00000i 0.0721218i 0.999350 + 0.0360609i \(0.0114810\pi\)
−0.999350 + 0.0360609i \(0.988519\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 18.0000i 0.647834i
\(773\) 10.0000i 0.359675i 0.983696 + 0.179838i \(0.0575572\pi\)
−0.983696 + 0.179838i \(0.942443\pi\)
\(774\) 12.0000i 0.431331i
\(775\) 8.00000i 0.287368i
\(776\) 6.00000 0.215387
\(777\) −8.00000 −0.286998
\(778\) − 14.0000i − 0.501924i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000i 0.572159i
\(783\) 2.00000 0.0714742
\(784\) −9.00000 −0.321429
\(785\) − 18.0000i − 0.642448i
\(786\) − 8.00000i − 0.285351i
\(787\) − 44.0000i − 1.56843i −0.620489 0.784215i \(-0.713066\pi\)
0.620489 0.784215i \(-0.286934\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 24.0000 0.854423
\(790\) 8.00000 0.284627
\(791\) 40.0000i 1.42224i
\(792\) 0 0
\(793\) 0 0
\(794\) −10.0000 −0.354887
\(795\) − 10.0000i − 0.354663i
\(796\) −8.00000 −0.283552
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 14.0000i 0.494666i
\(802\) −26.0000 −0.918092
\(803\) 0 0
\(804\) 4.00000i 0.141069i
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) − 10.0000i − 0.351799i
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 4.00000i 0.140459i 0.997531 + 0.0702295i \(0.0223732\pi\)
−0.997531 + 0.0702295i \(0.977627\pi\)
\(812\) 8.00000i 0.280745i
\(813\) 32.0000i 1.12229i
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) 2.00000 0.0700140
\(817\) − 48.0000i − 1.67931i
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 2.00000i 0.0698005i 0.999391 + 0.0349002i \(0.0111113\pi\)
−0.999391 + 0.0349002i \(0.988889\pi\)
\(822\) −6.00000 −0.209274
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 4.00000i 0.139347i
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 8.00000 0.278019
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 4.00000i 0.138842i
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 20.0000i 0.692543i
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) 24.0000i 0.829066i
\(839\) 8.00000i 0.276191i 0.990419 + 0.138095i \(0.0440980\pi\)
−0.990419 + 0.138095i \(0.955902\pi\)
\(840\) − 4.00000i − 0.138013i
\(841\) −25.0000 −0.862069
\(842\) 30.0000 1.03387
\(843\) − 18.0000i − 0.619953i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) − 44.0000i − 1.51186i
\(848\) 10.0000 0.343401
\(849\) −20.0000 −0.686398
\(850\) 2.00000i 0.0685994i
\(851\) 16.0000i 0.548473i
\(852\) 16.0000i 0.548151i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 40.0000 1.36877
\(855\) 4.00000 0.136797
\(856\) − 12.0000i − 0.410152i
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) − 12.0000i − 0.409197i
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 2.00000i 0.0680020i
\(866\) 10.0000i 0.339814i
\(867\) −13.0000 −0.441503
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) − 6.00000i − 0.203069i
\(874\) −32.0000 −1.08242
\(875\) 4.00000 0.135225
\(876\) − 6.00000i − 0.202721i
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 40.0000i 1.34993i
\(879\) 6.00000i 0.202375i
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000i 1.20944i
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 2.00000 0.0671156
\(889\) − 48.0000i − 1.60987i
\(890\) − 14.0000i − 0.469281i
\(891\) 0 0
\(892\) − 12.0000i − 0.401790i
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 16.0000i 0.534821i
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) − 16.0000i − 0.533630i
\(900\) 1.00000 0.0333333
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 48.0000i 1.59734i
\(904\) − 10.0000i − 0.332595i
\(905\) − 2.00000i − 0.0664822i
\(906\) 8.00000 0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 10.0000i 0.330590i
\(916\) 14.0000i 0.462573i
\(917\) − 32.0000i − 1.05673i
\(918\) − 2.00000i − 0.0660098i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −8.00000 −0.263752
\(921\) 12.0000i 0.395413i
\(922\) 2.00000 0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) −20.0000 −0.657241
\(927\) 4.00000 0.131377
\(928\) − 2.00000i − 0.0656532i
\(929\) 26.0000i 0.853032i 0.904480 + 0.426516i \(0.140259\pi\)
−0.904480 + 0.426516i \(0.859741\pi\)
\(930\) 8.00000i 0.262330i
\(931\) − 36.0000i − 1.17985i
\(932\) −26.0000 −0.851658
\(933\) 0 0
\(934\) − 36.0000i − 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) − 30.0000i − 0.977972i −0.872292 0.488986i \(-0.837367\pi\)
0.872292 0.488986i \(-0.162633\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 48.0000i 1.56310i
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) − 14.0000i − 0.453981i
\(952\) 8.00000 0.259281
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) − 10.0000i − 0.323762i
\(955\) − 8.00000i − 0.258874i
\(956\) − 24.0000i − 0.776215i
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) −24.0000 −0.775000
\(960\) 1.00000i 0.0322749i
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 2.00000i 0.0644157i
\(965\) −18.0000 −0.579441
\(966\) 32.0000 1.02958
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 8.00000i 0.256997i
\(970\) 6.00000i 0.192648i
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 80.0000i 2.56468i
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 38.0000i 1.21573i 0.794041 + 0.607864i \(0.207973\pi\)
−0.794041 + 0.607864i \(0.792027\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) − 9.00000i − 0.287494i
\(981\) 14.0000i 0.446986i
\(982\) 0 0
\(983\) − 56.0000i − 1.78612i −0.449935 0.893061i \(-0.648553\pi\)
0.449935 0.893061i \(-0.351447\pi\)
\(984\) 6.00000 0.191273
\(985\) −6.00000 −0.191176
\(986\) − 4.00000i − 0.127386i
\(987\) 0 0
\(988\) 0 0
\(989\) 96.0000 3.05262
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −8.00000 −0.254000
\(993\) − 20.0000i − 0.634681i
\(994\) 64.0000i 2.02996i
\(995\) − 8.00000i − 0.253617i
\(996\) 4.00000i 0.126745i
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −44.0000 −1.39280
\(999\) − 2.00000i − 0.0632772i
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 5070.2.b.i.1351.1 2
13.5 odd 4 390.2.a.c.1.1 1
13.8 odd 4 5070.2.a.u.1.1 1
13.12 even 2 inner 5070.2.b.i.1351.2 2
39.5 even 4 1170.2.a.n.1.1 1
52.31 even 4 3120.2.a.a.1.1 1
65.18 even 4 1950.2.e.e.1249.2 2
65.44 odd 4 1950.2.a.n.1.1 1
65.57 even 4 1950.2.e.e.1249.1 2
156.83 odd 4 9360.2.a.bc.1.1 1
195.44 even 4 5850.2.a.c.1.1 1
195.83 odd 4 5850.2.e.m.5149.1 2
195.122 odd 4 5850.2.e.m.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.c.1.1 1 13.5 odd 4
1170.2.a.n.1.1 1 39.5 even 4
1950.2.a.n.1.1 1 65.44 odd 4
1950.2.e.e.1249.1 2 65.57 even 4
1950.2.e.e.1249.2 2 65.18 even 4
3120.2.a.a.1.1 1 52.31 even 4
5070.2.a.u.1.1 1 13.8 odd 4
5070.2.b.i.1351.1 2 1.1 even 1 trivial
5070.2.b.i.1351.2 2 13.12 even 2 inner
5850.2.a.c.1.1 1 195.44 even 4
5850.2.e.m.5149.1 2 195.83 odd 4
5850.2.e.m.5149.2 2 195.122 odd 4
9360.2.a.bc.1.1 1 156.83 odd 4