Properties

Label 5070.2.b.k.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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

$q$-expansion

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

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 \(0.272814\pi\)
−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
\(17\) −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 \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 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 \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) − 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 \(-0.947432\pi\)
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
\(43\) 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 \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 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 \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −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 \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) − 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 \(-0.596924\pi\)
0.299813 0.953998i \(-0.403076\pi\)
\(90\) −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 \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 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 \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 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 \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 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 \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 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
\(153\) −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 \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 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 \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 2.00000 0.143592
\(195\) 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 \(-0.591513\pi\)
−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 \(0.233556\pi\)
−0.742677 + 0.669650i \(0.766444\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) 18.0000i 1.19734i
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 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 \(0.282863\pi\)
−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
\(249\) 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 \(0.161532\pi\)
−0.873978 + 0.485965i \(0.838468\pi\)
\(272\) −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 \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 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 \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 9.00000i 0.524891i
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −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 \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 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 \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 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 \(-0.720501\pi\)
0.638635 0.769510i \(-0.279499\pi\)
\(332\) − 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 \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 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 \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 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 \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 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 \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) 6.00000i 0.299626i 0.988714 + 0.149813i \(0.0478671\pi\)
−0.988714 + 0.149813i \(0.952133\pi\)
\(402\) −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 \(0.222231\pi\)
−0.766027 + 0.642809i \(0.777769\pi\)
\(410\) 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 \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) − 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 \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 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 \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −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 \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) −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 \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) − 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 \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) − 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 \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 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 \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 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 \(-0.875462\pi\)
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
\(563\) 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 \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 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 \(-0.920343\pi\)
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
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 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 \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −20.0000 −0.807134
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) − 44.0000i − 1.76851i −0.467005 0.884255i \(-0.654667\pi\)
0.467005 0.884255i \(-0.345333\pi\)
\(620\) −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 \(-0.780196\pi\)
0.770905 0.636950i \(-0.219804\pi\)
\(632\) 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 \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 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 \(-0.912226\pi\)
0.962221 0.272268i \(-0.0877739\pi\)
\(662\) −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 \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 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 \(0.315646\pi\)
−0.547326 + 0.836919i \(0.684354\pi\)
\(692\) 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 \(-0.747082\pi\)
0.700594 0.713560i \(-0.252918\pi\)
\(710\) 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 \(-0.866821\pi\)
0.913742 0.406294i \(-0.133179\pi\)
\(734\) 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 \(0.405687\pi\)
−0.291977 + 0.956425i \(0.594313\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) −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 \(-0.894215\pi\)
0.945284 0.326250i \(-0.105785\pi\)
\(762\) 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 \(0.0114810\pi\)
−0.999350 + 0.0360609i \(0.988519\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 22.0000i − 0.791797i
\(773\) 42.0000i 1.51064i 0.655359 + 0.755318i \(0.272517\pi\)
−0.655359 + 0.755318i \(0.727483\pi\)
\(774\) − 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 \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 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 \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 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 \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) −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 \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 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 \(-0.864031\pi\)
0.910147 0.414286i \(-0.135969\pi\)
\(840\) 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 \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) −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 \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) − 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 \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 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 \(-0.968619\pi\)
0.995144 0.0984268i \(-0.0313810\pi\)
\(930\) − 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 \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) − 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 \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) −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 \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) 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 \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 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 \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 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.1351.1 2
13.5 odd 4 30.2.a.a.1.1 1
13.8 odd 4 5070.2.a.w.1.1 1
13.12 even 2 inner 5070.2.b.k.1351.2 2
39.5 even 4 90.2.a.c.1.1 1
52.31 even 4 240.2.a.b.1.1 1
65.18 even 4 150.2.c.a.49.2 2
65.44 odd 4 150.2.a.b.1.1 1
65.57 even 4 150.2.c.a.49.1 2
91.5 even 12 1470.2.i.q.361.1 2
91.18 odd 12 1470.2.i.o.961.1 2
91.31 even 12 1470.2.i.q.961.1 2
91.44 odd 12 1470.2.i.o.361.1 2
91.83 even 4 1470.2.a.d.1.1 1
104.5 odd 4 960.2.a.e.1.1 1
104.83 even 4 960.2.a.p.1.1 1
117.5 even 12 810.2.e.b.541.1 2
117.31 odd 12 810.2.e.l.541.1 2
117.70 odd 12 810.2.e.l.271.1 2
117.83 even 12 810.2.e.b.271.1 2
143.109 even 4 3630.2.a.w.1.1 1
156.83 odd 4 720.2.a.j.1.1 1
195.44 even 4 450.2.a.d.1.1 1
195.83 odd 4 450.2.c.b.199.1 2
195.122 odd 4 450.2.c.b.199.2 2
208.5 odd 4 3840.2.k.y.1921.1 2
208.83 even 4 3840.2.k.f.1921.1 2
208.109 odd 4 3840.2.k.y.1921.2 2
208.187 even 4 3840.2.k.f.1921.2 2
221.135 odd 4 8670.2.a.g.1.1 1
260.83 odd 4 1200.2.f.e.49.1 2
260.187 odd 4 1200.2.f.e.49.2 2
260.239 even 4 1200.2.a.k.1.1 1
273.83 odd 4 4410.2.a.z.1.1 1
312.5 even 4 2880.2.a.a.1.1 1
312.83 odd 4 2880.2.a.q.1.1 1
455.174 even 4 7350.2.a.ct.1.1 1
520.83 odd 4 4800.2.f.w.3649.2 2
520.109 odd 4 4800.2.a.cq.1.1 1
520.187 odd 4 4800.2.f.w.3649.1 2
520.213 even 4 4800.2.f.p.3649.1 2
520.317 even 4 4800.2.f.p.3649.2 2
520.499 even 4 4800.2.a.d.1.1 1
780.83 even 4 3600.2.f.i.2449.1 2
780.239 odd 4 3600.2.a.f.1.1 1
780.707 even 4 3600.2.f.i.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 13.5 odd 4
90.2.a.c.1.1 1 39.5 even 4
150.2.a.b.1.1 1 65.44 odd 4
150.2.c.a.49.1 2 65.57 even 4
150.2.c.a.49.2 2 65.18 even 4
240.2.a.b.1.1 1 52.31 even 4
450.2.a.d.1.1 1 195.44 even 4
450.2.c.b.199.1 2 195.83 odd 4
450.2.c.b.199.2 2 195.122 odd 4
720.2.a.j.1.1 1 156.83 odd 4
810.2.e.b.271.1 2 117.83 even 12
810.2.e.b.541.1 2 117.5 even 12
810.2.e.l.271.1 2 117.70 odd 12
810.2.e.l.541.1 2 117.31 odd 12
960.2.a.e.1.1 1 104.5 odd 4
960.2.a.p.1.1 1 104.83 even 4
1200.2.a.k.1.1 1 260.239 even 4
1200.2.f.e.49.1 2 260.83 odd 4
1200.2.f.e.49.2 2 260.187 odd 4
1470.2.a.d.1.1 1 91.83 even 4
1470.2.i.o.361.1 2 91.44 odd 12
1470.2.i.o.961.1 2 91.18 odd 12
1470.2.i.q.361.1 2 91.5 even 12
1470.2.i.q.961.1 2 91.31 even 12
2880.2.a.a.1.1 1 312.5 even 4
2880.2.a.q.1.1 1 312.83 odd 4
3600.2.a.f.1.1 1 780.239 odd 4
3600.2.f.i.2449.1 2 780.83 even 4
3600.2.f.i.2449.2 2 780.707 even 4
3630.2.a.w.1.1 1 143.109 even 4
3840.2.k.f.1921.1 2 208.83 even 4
3840.2.k.f.1921.2 2 208.187 even 4
3840.2.k.y.1921.1 2 208.5 odd 4
3840.2.k.y.1921.2 2 208.109 odd 4
4410.2.a.z.1.1 1 273.83 odd 4
4800.2.a.d.1.1 1 520.499 even 4
4800.2.a.cq.1.1 1 520.109 odd 4
4800.2.f.p.3649.1 2 520.213 even 4
4800.2.f.p.3649.2 2 520.317 even 4
4800.2.f.w.3649.1 2 520.187 odd 4
4800.2.f.w.3649.2 2 520.83 odd 4
5070.2.a.w.1.1 1 13.8 odd 4
5070.2.b.k.1351.1 2 1.1 even 1 trivial
5070.2.b.k.1351.2 2 13.12 even 2 inner
7350.2.a.ct.1.1 1 455.174 even 4
8670.2.a.g.1.1 1 221.135 odd 4