Properties

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

$q$-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 \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 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 \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) − 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 \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) 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 \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) −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 \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) − 4.00000i − 0.478091i
\(71\) 16.0000i 1.89885i 0.313993 + 0.949425i \(0.398333\pi\)
−0.313993 + 0.949425i \(0.601667\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −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 \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) − 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 \(-0.733878\pi\)
0.670402 0.741999i \(-0.266122\pi\)
\(90\) −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 \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) − 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 \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 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 \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 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 \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) − 8.00000i − 0.651031i −0.945537 0.325515i \(-0.894462\pi\)
0.945537 0.325515i \(-0.105538\pi\)
\(152\) −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 \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 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 \(0.224325\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 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 \(-0.868388\pi\)
0.915732 0.401790i \(-0.131612\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) − 10.0000i − 0.665190i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 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 \(-0.717137\pi\)
0.630468 0.776215i \(-0.282863\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 2.00000i 0.128831i 0.997923 + 0.0644157i \(0.0205183\pi\)
−0.997923 + 0.0644157i \(0.979482\pi\)
\(242\) 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 \(-0.575596\pi\)
0.235267 0.971931i \(-0.424404\pi\)
\(272\) 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 \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(282\) 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 \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) − 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 \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 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 \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 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 \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) − 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 \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 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 \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 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 \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) −4.00000 −0.205196
\(381\) 12.0000 0.614779
\(382\) 8.00000i 0.409316i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) − 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 \(0.0807406\pi\)
−0.968002 + 0.250943i \(0.919259\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) 26.0000i 1.29838i 0.760627 + 0.649189i \(0.224892\pi\)
−0.760627 + 0.649189i \(0.775108\pi\)
\(402\) −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 \(-0.920478\pi\)
0.968956 0.247234i \(-0.0795217\pi\)
\(410\) − 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 \(-0.739028\pi\)
0.682318 0.731055i \(-0.260972\pi\)
\(422\) − 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 \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) − 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 \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 8.00000i 0.373002i
\(461\) − 2.00000i − 0.0931493i −0.998915 0.0465746i \(-0.985169\pi\)
0.998915 0.0465746i \(-0.0148305\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 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 \(0.0585046\pi\)
−0.983157 + 0.182765i \(0.941495\pi\)
\(480\) −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 \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 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 \(0.444532\pi\)
−0.173379 + 0.984855i \(0.555468\pi\)
\(500\) 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 \(0.318716\pi\)
−0.539227 + 0.842160i \(0.681284\pi\)
\(510\) −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 \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) 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 \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) −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 \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) − 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 \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 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 \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 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 \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) 12.0000 0.484281
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 4.00000i 0.160904i
\(619\) − 28.0000i − 1.12542i −0.826656 0.562708i \(-0.809760\pi\)
0.826656 0.562708i \(-0.190240\pi\)
\(620\) 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 \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 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 \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) −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 \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 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 \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 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 \(-0.779577\pi\)
0.769665 0.638448i \(-0.220423\pi\)
\(710\) − 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 \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 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 \(0.0234396\pi\)
−0.997290 + 0.0735712i \(0.976560\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −40.0000 −1.46845
\(743\) 40.0000i 1.46746i 0.679442 + 0.733729i \(0.262222\pi\)
−0.679442 + 0.733729i \(0.737778\pi\)
\(744\) 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 \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) 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 \(-0.988519\pi\)
0.999350 0.0360609i \(-0.0114810\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) − 18.0000i − 0.647834i
\(773\) − 10.0000i − 0.359675i −0.983696 0.179838i \(-0.942443\pi\)
0.983696 0.179838i \(-0.0575572\pi\)
\(774\) − 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 \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 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 \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) − 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 \(-0.988889\pi\)
0.999391 0.0349002i \(-0.0111113\pi\)
\(822\) −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 \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 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 \(-0.955902\pi\)
0.990419 0.138095i \(-0.0440980\pi\)
\(840\) 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 \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 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 \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 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 \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) − 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 \(-0.859741\pi\)
0.904480 0.426516i \(-0.140259\pi\)
\(930\) − 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 \(0.162633\pi\)
−0.872292 + 0.488986i \(0.837367\pi\)
\(942\) 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 \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 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 \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) − 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 \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 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 \(0.351447\pi\)
−0.449935 + 0.893061i \(0.648553\pi\)
\(984\) 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.1351.2 2
13.5 odd 4 5070.2.a.u.1.1 1
13.8 odd 4 390.2.a.c.1.1 1
13.12 even 2 inner 5070.2.b.i.1351.1 2
39.8 even 4 1170.2.a.n.1.1 1
52.47 even 4 3120.2.a.a.1.1 1
65.8 even 4 1950.2.e.e.1249.2 2
65.34 odd 4 1950.2.a.n.1.1 1
65.47 even 4 1950.2.e.e.1249.1 2
156.47 odd 4 9360.2.a.bc.1.1 1
195.8 odd 4 5850.2.e.m.5149.1 2
195.47 odd 4 5850.2.e.m.5149.2 2
195.164 even 4 5850.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.c.1.1 1 13.8 odd 4
1170.2.a.n.1.1 1 39.8 even 4
1950.2.a.n.1.1 1 65.34 odd 4
1950.2.e.e.1249.1 2 65.47 even 4
1950.2.e.e.1249.2 2 65.8 even 4
3120.2.a.a.1.1 1 52.47 even 4
5070.2.a.u.1.1 1 13.5 odd 4
5070.2.b.i.1351.1 2 13.12 even 2 inner
5070.2.b.i.1351.2 2 1.1 even 1 trivial
5850.2.a.c.1.1 1 195.164 even 4
5850.2.e.m.5149.1 2 195.8 odd 4
5850.2.e.m.5149.2 2 195.47 odd 4
9360.2.a.bc.1.1 1 156.47 odd 4