Properties

Label 5070.2.b.k.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $1$
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: \(1\)
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 30)
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.k.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +4.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -1.00000i q^{20} -4.00000i q^{21} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} -6.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} +1.00000i q^{32} -6.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +4.00000 q^{42} +4.00000 q^{43} +1.00000i q^{45} +1.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} -4.00000 q^{56} +4.00000i q^{57} -6.00000i q^{58} -1.00000i q^{60} -10.0000 q^{61} +8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000 q^{68} +4.00000i q^{70} -1.00000i q^{72} +2.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} +8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000i q^{83} +4.00000i q^{84} -6.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} +18.0000i q^{89} -1.00000 q^{90} -8.00000i q^{93} -4.00000 q^{95} +1.00000i q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} + 8q^{14} + 2q^{16} - 12q^{17} - 2q^{25} + 2q^{27} - 12q^{29} - 2q^{30} + 8q^{35} - 2q^{36} - 8q^{38} + 2q^{40} + 8q^{42} + 8q^{43} + 2q^{48} - 18q^{49} - 12q^{51} - 12q^{53} - 8q^{56} - 20q^{61} + 16q^{62} - 2q^{64} + 12q^{68} - 4q^{74} - 2q^{75} + 16q^{79} + 2q^{81} - 12q^{82} - 12q^{87} - 2q^{90} - 8q^{95} + O(q^{100}) \)

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
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.00000i 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −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
\(34\) − 6.00000i − 1.02899i
\(35\) 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
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 0.144338
\(49\) −9.00000 −1.28571
\(50\) − 1.00000i − 0.141421i
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 4.00000i 0.529813i
\(58\) − 6.00000i − 0.787839i
\(59\) 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 \(0.921429\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 4.00000i 0.478091i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −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 \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 4.00000i 0.436436i
\(85\) − 6.00000i − 0.650791i
\(86\) 4.00000i 0.431331i
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 18.0000i 1.90800i 0.299813 + 0.953998i \(0.403076\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 1.00000i 0.102062i
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 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
\(106\) − 6.00000i − 0.582772i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) 2.00000i 0.189832i
\(112\) − 4.00000i − 0.377964i
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 24.0000i 2.20008i
\(120\) 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
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −4.00000 −0.345547
\(135\) 1.00000i 0.0860663i
\(136\) 6.00000i 0.514496i
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 6.00000i − 0.498273i
\(146\) −2.00000 −0.165521
\(147\) −9.00000 −0.742307
\(148\) − 2.00000i − 0.164399i
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) − 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
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) 6.00000 0.460179
\(171\) 4.00000i 0.305888i
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) − 6.00000i − 0.454859i
\(175\) 4.00000i 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −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
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 22.0000i − 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) 2.00000 0.143592
\(195\) 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 \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 4.00000i 0.282138i
\(202\) − 18.0000i − 1.26648i
\(203\) 24.0000i 1.68447i
\(204\) 6.00000 0.420084
\(205\) −6.00000 −0.419058
\(206\) 4.00000i 0.278693i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 4.00000i 0.276026i
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) − 12.0000i − 0.820303i
\(215\) 4.00000i 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) −32.0000 −2.17230
\(218\) −10.0000 −0.677285
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) − 20.0000i − 1.33930i −0.742677 0.669650i \(-0.766444\pi\)
0.742677 0.669650i \(-0.233556\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) − 18.0000i − 1.19734i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) − 10.0000i − 0.660819i −0.943838 0.330409i \(-0.892813\pi\)
0.943838 0.330409i \(-0.107187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) −24.0000 −1.55569
\(239\) − 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
\(249\) − 12.0000i − 0.760469i
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 0 0
\(254\) − 20.0000i − 1.25491i
\(255\) − 6.00000i − 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) − 6.00000i − 0.368577i
\(266\) 16.0000i 0.981023i
\(267\) 18.0000i 1.10158i
\(268\) − 4.00000i − 0.244339i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 16.0000i − 0.971931i −0.873978 0.485965i \(-0.838468\pi\)
0.873978 0.485965i \(-0.161532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) − 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
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 1.00000i 0.0589256i
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) − 2.00000i − 0.117242i
\(292\) − 2.00000i − 0.117041i
\(293\) − 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
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) − 16.0000i − 0.922225i
\(302\) −8.00000 −0.460348
\(303\) −18.0000 −1.03407
\(304\) 4.00000i 0.229416i
\(305\) − 10.0000i − 0.572598i
\(306\) − 6.00000i − 0.342997i
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 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
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 2.00000i 0.112867i
\(315\) 4.00000 0.225374
\(316\) −8.00000 −0.450035
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) − 1.00000i − 0.0559017i
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 10.0000i 0.553001i
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) − 4.00000i − 0.218218i
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 6.00000i 0.325396i
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 8.00000i 0.431959i
\(344\) − 4.00000i − 0.215666i
\(345\) 0 0
\(346\) − 18.0000i − 0.967686i
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) − 10.0000i − 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 18.0000i − 0.953998i
\(357\) 24.0000i 1.27021i
\(358\) − 24.0000i − 1.26844i
\(359\) − 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
\(362\) − 14.0000i − 0.735824i
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) − 10.0000i − 0.522708i
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 6.00000i 0.312348i
\(370\) − 2.00000i − 0.103975i
\(371\) 24.0000i 1.24602i
\(372\) 8.00000i 0.414781i
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) − 1.00000i − 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 4.00000 0.205196
\(381\) −20.0000 −1.02463
\(382\) − 24.0000i − 1.22795i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 4.00000 0.203331
\(388\) 2.00000i 0.101535i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000i 0.454569i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) − 6.00000i − 0.299626i −0.988714 0.149813i \(-0.952133\pi\)
0.988714 0.149813i \(-0.0478671\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 1.00000i 0.0496904i
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) 6.00000i 0.297044i
\(409\) − 26.0000i − 1.28562i −0.766027 0.642809i \(-0.777769\pi\)
0.766027 0.642809i \(-0.222231\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) 6.00000i 0.295958i
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −4.00000 −0.195180
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 40.0000i 1.93574i
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) − 32.0000i − 1.53605i
\(435\) − 6.00000i − 0.287678i
\(436\) − 10.0000i − 0.478913i
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) − 2.00000i − 0.0949158i
\(445\) −18.0000 −0.853282
\(446\) 20.0000 0.947027
\(447\) 6.00000i 0.283790i
\(448\) 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
\(452\) 18.0000 0.846649
\(453\) 8.00000i 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) −6.00000 −0.278543
\(465\) 8.00000 0.370991
\(466\) 18.0000i 0.833834i
\(467\) 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
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000i 0.367452i
\(475\) − 4.00000i − 0.183533i
\(476\) − 24.0000i − 1.10004i
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 2.00000 0.0908153
\(486\) 1.00000i 0.0453609i
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 10.0000i 0.452679i
\(489\) − 4.00000i − 0.180886i
\(490\) 9.00000 0.406579
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) − 8.00000i − 0.359211i
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 0 0
\(502\) 24.0000i 1.07117i
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −4.00000 −0.178174
\(505\) − 18.0000i − 0.800989i
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 6.00000 0.265684
\(511\) 8.00000 0.353899
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 18.0000i 0.793946i
\(515\) 4.00000i 0.176261i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) − 12.0000i − 0.518805i
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) − 6.00000i − 0.258678i
\(539\) 0 0
\(540\) − 1.00000i − 0.0430331i
\(541\) − 10.0000i − 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) 16.0000 0.687259
\(543\) −14.0000 −0.600798
\(544\) − 6.00000i − 0.257248i
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) − 24.0000i − 1.02243i
\(552\) 0 0
\(553\) − 32.0000i − 1.36078i
\(554\) − 2.00000i − 0.0849719i
\(555\) −2.00000 −0.0848953
\(556\) 4.00000 0.169638
\(557\) 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
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) − 18.0000i − 0.757266i
\(566\) 28.0000i 1.17693i
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) − 4.00000i − 0.167542i
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 24.0000i 1.00174i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 19.0000i 0.790296i
\(579\) − 22.0000i − 0.914289i
\(580\) 6.00000i 0.249136i
\(581\) −48.0000 −1.99138
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 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
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) −24.0000 −0.983904
\(596\) − 6.00000i − 0.245770i
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 16.0000 0.652111
\(603\) 4.00000i 0.162893i
\(604\) − 8.00000i − 0.325515i
\(605\) 11.0000i 0.447214i
\(606\) − 18.0000i − 0.731200i
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −4.00000 −0.162221
\(609\) 24.0000i 0.972529i
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) −20.0000 −0.807134
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 44.0000i 1.76851i 0.467005 + 0.884255i \(0.345333\pi\)
−0.467005 + 0.884255i \(0.654667\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 72.0000 2.88462
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.00000i 0.0799361i
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) − 12.0000i − 0.478471i
\(630\) 4.00000i 0.159364i
\(631\) 32.0000i 1.27390i 0.770905 + 0.636950i \(0.219804\pi\)
−0.770905 + 0.636950i \(0.780196\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 20.0000 0.794929
\(634\) 18.0000 0.714871
\(635\) − 20.0000i − 0.793676i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 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
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 4.00000i 0.157500i
\(646\) 24.0000 0.944267
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 4.00000i 0.156652i
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 6.00000i 0.234261i
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 14.0000i 0.544537i 0.962221 + 0.272268i \(0.0877739\pi\)
−0.962221 + 0.272268i \(0.912226\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 16.0000i 0.620453i
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) − 20.0000i − 0.773245i
\(670\) − 4.00000i − 0.154533i
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) − 26.0000i − 1.00148i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) − 18.0000i − 0.691286i
\(679\) −8.00000 −0.307012
\(680\) −6.00000 −0.230089
\(681\) 12.0000i 0.459841i
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) − 4.00000i − 0.152944i
\(685\) −6.00000 −0.229248
\(686\) −8.00000 −0.305441
\(687\) − 10.0000i − 0.381524i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) − 44.0000i − 1.67384i −0.547326 0.836919i \(-0.684354\pi\)
0.547326 0.836919i \(-0.315646\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 12.0000i 0.455514i
\(695\) − 4.00000i − 0.151729i
\(696\) 6.00000i 0.227429i
\(697\) − 36.0000i − 1.36360i
\(698\) 10.0000 0.378506
\(699\) 18.0000 0.680823
\(700\) − 4.00000i − 0.151186i
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 72.0000i 2.70784i
\(708\) 0 0
\(709\) 38.0000i 1.42712i 0.700594 + 0.713560i \(0.252918\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) − 24.0000i − 0.896296i
\(718\) 24.0000 0.895672
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 16.0000i − 0.595871i
\(722\) 3.00000i 0.111648i
\(723\) 2.00000i 0.0743808i
\(724\) 14.0000 0.520306
\(725\) 6.00000 0.222834
\(726\) 11.0000i 0.408248i
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 2.00000i − 0.0740233i
\(731\) −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 \(0.866821\pi\)
\(734\) − 28.0000i − 1.03350i
\(735\) − 9.00000i − 0.331970i
\(736\) 0 0
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) − 52.0000i − 1.91285i −0.291977 0.956425i \(-0.594313\pi\)
0.291977 0.956425i \(-0.405687\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −8.00000 −0.293294
\(745\) −6.00000 −0.219823
\(746\) 26.0000i 0.951928i
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 48.0000i 1.75388i
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 4.00000i 0.145479i
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 4.00000i 0.145095i
\(761\) 18.0000i 0.652499i 0.945284 + 0.326250i \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) − 20.0000i − 0.724524i
\(763\) 40.0000 1.44810
\(764\) 24.0000 0.868290
\(765\) − 6.00000i − 0.216930i
\(766\) 0 0
\(767\) 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
\(771\) 18.0000 0.648254
\(772\) 22.0000i 0.791797i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 8.00000i 0.287368i
\(776\) −2.00000 −0.0717958
\(777\) 8.00000 0.286998
\(778\) 6.00000i 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −9.00000 −0.321429
\(785\) 2.00000i 0.0713831i
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 72.0000i 2.56003i
\(792\) 0 0
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) − 6.00000i − 0.212798i
\(796\) 8.00000 0.283552
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 16.0000i 0.566394i
\(799\) 0 0
\(800\) − 1.00000i − 0.0353553i
\(801\) 18.0000i 0.635999i
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) − 4.00000i − 0.141069i
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 18.0000i 0.633238i
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 4.00000i 0.140459i 0.997531 + 0.0702295i \(0.0223732\pi\)
−0.997531 + 0.0702295i \(0.977627\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) − 16.0000i − 0.561144i
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) −6.00000 −0.210042
\(817\) 16.0000i 0.559769i
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) − 18.0000i − 0.628204i −0.949389 0.314102i \(-0.898297\pi\)
0.949389 0.314102i \(-0.101703\pi\)
\(822\) −6.00000 −0.209274
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) − 4.00000i − 0.139347i
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 12.0000i 0.416526i
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) − 4.00000i − 0.138509i
\(835\) 0 0
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) 0 0
\(839\) 24.0000i 0.828572i 0.910147 + 0.414286i \(0.135969\pi\)
−0.910147 + 0.414286i \(0.864031\pi\)
\(840\) − 4.00000i − 0.138013i
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 18.0000i 0.619953i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 44.0000i − 1.51186i
\(848\) −6.00000 −0.206041
\(849\) 28.0000 0.960958
\(850\) 6.00000i 0.205798i
\(851\) 0 0
\(852\) 0 0
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) −40.0000 −1.36877
\(855\) −4.00000 −0.136797
\(856\) 12.0000i 0.410152i
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) − 4.00000i − 0.136399i
\(861\) 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
\(865\) − 18.0000i − 0.612018i
\(866\) − 26.0000i − 0.883516i
\(867\) 19.0000 0.645274
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) 10.0000 0.338643
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) − 2.00000i − 0.0675737i
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) − 6.00000i − 0.202375i
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000i 0.403148i
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 2.00000 0.0671156
\(889\) 80.0000i 2.68311i
\(890\) − 18.0000i − 0.603361i
\(891\) 0 0
\(892\) 20.0000i 0.669650i
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) − 24.0000i − 0.802232i
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 48.0000i 1.60089i
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 18.0000i 0.598671i
\(905\) − 14.0000i − 0.465376i
\(906\) −8.00000 −0.265782
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) − 10.0000i − 0.330590i
\(916\) 10.0000i 0.330409i
\(917\) 0 0
\(918\) − 6.00000i − 0.198030i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 20.0000i 0.659022i
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) − 2.00000i − 0.0657596i
\(926\) 4.00000 0.131448
\(927\) 4.00000 0.131377
\(928\) − 6.00000i − 0.196960i
\(929\) 6.00000i 0.196854i 0.995144 + 0.0984268i \(0.0313810\pi\)
−0.995144 + 0.0984268i \(0.968619\pi\)
\(930\) 8.00000i 0.262330i
\(931\) − 36.0000i − 1.17985i
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 36.0000i 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 2.00000 0.0652675
\(940\) 0 0
\(941\) − 18.0000i − 0.586783i −0.955992 0.293392i \(-0.905216\pi\)
0.955992 0.293392i \(-0.0947840\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) − 18.0000i − 0.583690i
\(952\) 24.0000 0.777844
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) − 24.0000i − 0.776622i
\(956\) 24.0000i 0.776215i
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 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
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) 4.00000i 0.128631i 0.997930 + 0.0643157i \(0.0204865\pi\)
−0.997930 + 0.0643157i \(0.979514\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) − 24.0000i − 0.770991i
\(970\) 2.00000i 0.0642161i
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000i 0.512936i
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) 9.00000i 0.287494i
\(981\) 10.0000i 0.319275i
\(982\) − 24.0000i − 0.765871i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 6.00000 0.191273
\(985\) −6.00000 −0.191176
\(986\) 36.0000i 1.14647i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) − 8.00000i − 0.253617i
\(996\) 12.0000i 0.380235i
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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.k.1351.2 2
13.5 odd 4 5070.2.a.w.1.1 1
13.8 odd 4 30.2.a.a.1.1 1
13.12 even 2 inner 5070.2.b.k.1351.1 2
39.8 even 4 90.2.a.c.1.1 1
52.47 even 4 240.2.a.b.1.1 1
65.8 even 4 150.2.c.a.49.2 2
65.34 odd 4 150.2.a.b.1.1 1
65.47 even 4 150.2.c.a.49.1 2
91.34 even 4 1470.2.a.d.1.1 1
91.47 even 12 1470.2.i.q.361.1 2
91.60 odd 12 1470.2.i.o.961.1 2
91.73 even 12 1470.2.i.q.961.1 2
91.86 odd 12 1470.2.i.o.361.1 2
104.21 odd 4 960.2.a.e.1.1 1
104.99 even 4 960.2.a.p.1.1 1
117.34 odd 12 810.2.e.l.271.1 2
117.47 even 12 810.2.e.b.271.1 2
117.86 even 12 810.2.e.b.541.1 2
117.112 odd 12 810.2.e.l.541.1 2
143.21 even 4 3630.2.a.w.1.1 1
156.47 odd 4 720.2.a.j.1.1 1
195.8 odd 4 450.2.c.b.199.1 2
195.47 odd 4 450.2.c.b.199.2 2
195.164 even 4 450.2.a.d.1.1 1
208.21 odd 4 3840.2.k.y.1921.1 2
208.99 even 4 3840.2.k.f.1921.1 2
208.125 odd 4 3840.2.k.y.1921.2 2
208.203 even 4 3840.2.k.f.1921.2 2
221.203 odd 4 8670.2.a.g.1.1 1
260.47 odd 4 1200.2.f.e.49.2 2
260.99 even 4 1200.2.a.k.1.1 1
260.203 odd 4 1200.2.f.e.49.1 2
273.125 odd 4 4410.2.a.z.1.1 1
312.125 even 4 2880.2.a.a.1.1 1
312.203 odd 4 2880.2.a.q.1.1 1
455.34 even 4 7350.2.a.ct.1.1 1
520.99 even 4 4800.2.a.d.1.1 1
520.203 odd 4 4800.2.f.w.3649.2 2
520.229 odd 4 4800.2.a.cq.1.1 1
520.307 odd 4 4800.2.f.w.3649.1 2
520.333 even 4 4800.2.f.p.3649.1 2
520.437 even 4 4800.2.f.p.3649.2 2
780.47 even 4 3600.2.f.i.2449.2 2
780.203 even 4 3600.2.f.i.2449.1 2
780.359 odd 4 3600.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 13.8 odd 4
90.2.a.c.1.1 1 39.8 even 4
150.2.a.b.1.1 1 65.34 odd 4
150.2.c.a.49.1 2 65.47 even 4
150.2.c.a.49.2 2 65.8 even 4
240.2.a.b.1.1 1 52.47 even 4
450.2.a.d.1.1 1 195.164 even 4
450.2.c.b.199.1 2 195.8 odd 4
450.2.c.b.199.2 2 195.47 odd 4
720.2.a.j.1.1 1 156.47 odd 4
810.2.e.b.271.1 2 117.47 even 12
810.2.e.b.541.1 2 117.86 even 12
810.2.e.l.271.1 2 117.34 odd 12
810.2.e.l.541.1 2 117.112 odd 12
960.2.a.e.1.1 1 104.21 odd 4
960.2.a.p.1.1 1 104.99 even 4
1200.2.a.k.1.1 1 260.99 even 4
1200.2.f.e.49.1 2 260.203 odd 4
1200.2.f.e.49.2 2 260.47 odd 4
1470.2.a.d.1.1 1 91.34 even 4
1470.2.i.o.361.1 2 91.86 odd 12
1470.2.i.o.961.1 2 91.60 odd 12
1470.2.i.q.361.1 2 91.47 even 12
1470.2.i.q.961.1 2 91.73 even 12
2880.2.a.a.1.1 1 312.125 even 4
2880.2.a.q.1.1 1 312.203 odd 4
3600.2.a.f.1.1 1 780.359 odd 4
3600.2.f.i.2449.1 2 780.203 even 4
3600.2.f.i.2449.2 2 780.47 even 4
3630.2.a.w.1.1 1 143.21 even 4
3840.2.k.f.1921.1 2 208.99 even 4
3840.2.k.f.1921.2 2 208.203 even 4
3840.2.k.y.1921.1 2 208.21 odd 4
3840.2.k.y.1921.2 2 208.125 odd 4
4410.2.a.z.1.1 1 273.125 odd 4
4800.2.a.d.1.1 1 520.99 even 4
4800.2.a.cq.1.1 1 520.229 odd 4
4800.2.f.p.3649.1 2 520.333 even 4
4800.2.f.p.3649.2 2 520.437 even 4
4800.2.f.w.3649.1 2 520.307 odd 4
4800.2.f.w.3649.2 2 520.203 odd 4
5070.2.a.w.1.1 1 13.5 odd 4
5070.2.b.k.1351.1 2 13.12 even 2 inner
5070.2.b.k.1351.2 2 1.1 even 1 trivial
7350.2.a.ct.1.1 1 455.34 even 4
8670.2.a.g.1.1 1 221.203 odd 4