↑

Properties

Label 3850.2.a.a.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3850,2,Mod(1,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +8.00000 q^{19} +1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +11.0000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} +7.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} +7.00000 q^{43} -1.00000 q^{44} -9.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} -3.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} -8.00000 q^{57} +6.00000 q^{58} -3.00000 q^{59} -10.0000 q^{61} -11.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} +6.00000 q^{71} +2.00000 q^{72} -5.00000 q^{73} -7.00000 q^{74} +8.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} +1.00000 q^{84} -7.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +2.00000 q^{91} -11.0000 q^{93} +9.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 2.00000 0.471405
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 5.00000 0.962250
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 11.0000 1.97566 0.987829 − 0.155543i \(-0.0497126\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 7.00000 1.06749 0.533745 − 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −8.00000 −1.05963
\(58\) 6.00000 0.787839
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −11.0000 −1.39700
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 2.00000 0.235702
\(73\) −5.00000 −0.585206 −0.292603 − 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −11.0000 −1.14065
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 − 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) −3.00000 −0.297044
\(103\) −5.00000 −0.492665 −0.246332 − 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 5.00000 0.481125
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −1.00000 −0.0944911
\(113\) −21.0000 −1.97551 −0.987757 − 0.156001i \(-0.950140\pi\)
−0.987757 + 0.156001i \(0.950140\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 4.00000 0.369800
\(118\) 3.00000 0.276172
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 11.0000 0.987829
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −17.0000 −1.50851 −0.754253 − 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.00000 −0.616316
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 1.00000 0.0870388
\(133\) −8.00000 −0.693688
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 − 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −6.00000 −0.503509
\(143\) 2.00000 0.167248
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 5.00000 0.413803
\(147\) −1.00000 −0.0824786
\(148\) 7.00000 0.575396
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −8.00000 −0.648886
\(153\) 6.00000 0.485071
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 1.00000 0.0795557
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 10.0000 0.783260 0.391630 − 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −16.0000 −1.22355
\(172\) 7.00000 0.533745
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 11.0000 0.806559
\(187\) 3.00000 0.219382
\(188\) −9.00000 −0.656392
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −2.00000 −0.142134
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −15.0000 −1.05540
\(203\) 6.00000 0.421117
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 5.00000 0.348367
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −3.00000 −0.206041
\(213\) −6.00000 −0.411113
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) −5.00000 −0.340207
\(217\) −11.0000 −0.746729
\(218\) 16.0000 1.08366
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 7.00000 0.469809
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 21.0000 1.39690
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −8.00000 −0.529813
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 6.00000 0.393919
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 1.00000 0.0649570
\(238\) −3.00000 −0.194461
\(239\) −27.0000 −1.74648 −0.873242 − 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 − 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −16.0000 −1.02640
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −16.0000 −1.01806
\(248\) −11.0000 −0.698501
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 − 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 17.0000 1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 − 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 7.00000 0.435801
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) −21.0000 −1.29492 −0.647458 − 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −3.00000 −0.181902
\(273\) −2.00000 −0.121046
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 − 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 22.0000 1.31947
\(279\) −22.0000 −1.31711
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −9.00000 −0.535942
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) 2.00000 0.117851
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −5.00000 −0.292603
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) −5.00000 −0.290129
\(298\) −12.0000 −0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) −8.00000 −0.460348
\(303\) −15.0000 −0.861727
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 1.00000 0.0569803
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) −33.0000 −1.87126 −0.935629 − 0.352985i \(-0.885167\pi\)
−0.935629 + 0.352985i \(0.885167\pi\)
\(312\) −2.00000 −0.113228
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 27.0000 1.51647 0.758236 − 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) −3.00000 −0.168232
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 16.0000 0.884802
\(328\) −6.00000 −0.331295
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −6.00000 −0.329293
\(333\) −14.0000 −0.767195
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −32.0000 −1.74315 −0.871576 − 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 9.00000 0.489535
\(339\) 21.0000 1.14056
\(340\) 0 0
\(341\) −11.0000 −0.595683
\(342\) 16.0000 0.865181
\(343\) −1.00000 −0.0539949
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −9.00000 −0.483145 −0.241573 − 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) 6.00000 0.321634
\(349\) −7.00000 −0.374701 −0.187351 − 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 1.00000 0.0533002
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −3.00000 −0.159448
\(355\) 0 0
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 12.0000 0.634220
\(359\) −36.0000 −1.90001 −0.950004 − 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) −1.00000 −0.0524864
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −11.0000 −0.570323
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) 12.0000 0.618031
\(378\) 5.00000 0.257172
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 17.0000 0.870936
\(382\) −24.0000 −1.22795
\(383\) −15.0000 −0.766464 −0.383232 − 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −14.0000 −0.711660
\(388\) 10.0000 0.507673
\(389\) −9.00000 −0.456318 −0.228159 − 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 16.0000 0.802008
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 − 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) −8.00000 −0.399004
\(403\) −22.0000 −1.09590
\(404\) 15.0000 0.746278
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −7.00000 −0.346977
\(408\) −3.00000 −0.148522
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −5.00000 −0.246332
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 22.0000 1.07734
\(418\) 8.00000 0.391293
\(419\) −3.00000 −0.146560 −0.0732798 − 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 13.0000 0.632830
\(423\) 18.0000 0.875190
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 10.0000 0.483934
\(428\) −3.00000 −0.145010
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 − 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 5.00000 0.240563
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 11.0000 0.528017
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) −5.00000 −0.238909
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −6.00000 −0.285391
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −7.00000 −0.332205
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −12.0000 −0.567581
\(448\) −1.00000 −0.0472456
\(449\) 15.0000 0.707894 0.353947 − 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −21.0000 −0.987757
\(453\) −8.00000 −0.375873
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 4.00000 0.186908
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) −27.0000 −1.25752 −0.628758 − 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 1.00000 0.0465242
\(463\) 22.0000 1.02243 0.511213 − 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −21.0000 −0.971764 −0.485882 − 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 4.00000 0.184900
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 3.00000 0.138086
\(473\) −7.00000 −0.321860
\(474\) −1.00000 −0.0459315
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 6.00000 0.274721
\(478\) 27.0000 1.23495
\(479\) 18.0000 0.822441 0.411220 − 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 25.0000 1.13872
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 16.0000 0.725775
\(487\) 34.0000 1.54069 0.770344 − 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 10.0000 0.452679
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −27.0000 −1.21849 −0.609246 − 0.792981i \(-0.708528\pi\)
−0.609246 + 0.792981i \(0.708528\pi\)
\(492\) −6.00000 −0.270501
\(493\) 18.0000 0.810679
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 11.0000 0.493915
\(497\) −6.00000 −0.269137
\(498\) −6.00000 −0.268866
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 15.0000 0.669483
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −17.0000 −0.754253
\(509\) 12.0000 0.531891 0.265945 − 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 5.00000 0.221187
\(512\) −1.00000 −0.0441942
\(513\) 40.0000 1.76604
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) −7.00000 −0.308158
\(517\) 9.00000 0.395820
\(518\) 7.00000 0.307562
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 − 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) −12.0000 −0.525226
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) −33.0000 −1.43750
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −8.00000 −0.346844
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −2.00000 −0.0858282
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) −35.0000 −1.49649 −0.748246 − 0.663421i \(-0.769104\pi\)
−0.748246 + 0.663421i \(0.769104\pi\)
\(548\) −6.00000 −0.256307
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −22.0000 −0.933008
\(557\) −36.0000 −1.52537 −0.762684 − 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 22.0000 0.931334
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) −12.0000 −0.506189
\(563\) −42.0000 −1.77009 −0.885044 − 0.465506i \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) −1.00000 −0.0419961
\(568\) −6.00000 −0.251754
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 2.00000 0.0836242
\(573\) −24.0000 −1.00261
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 8.00000 0.332756
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 10.0000 0.414513
\(583\) 3.00000 0.124247
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 88.0000 3.62598
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 7.00000 0.287698
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 − 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 − 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 7.00000 0.285299
\(603\) 16.0000 0.651570
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 15.0000 0.609333
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −8.00000 −0.324443
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 18.0000 0.728202
\(612\) 6.00000 0.242536
\(613\) 28.0000 1.13091 0.565455 − 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 33.0000 1.32853 0.664265 − 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) −5.00000 −0.201129
\(619\) −13.0000 −0.522514 −0.261257 − 0.965269i \(-0.584137\pi\)
−0.261257 + 0.965269i \(0.584137\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 33.0000 1.32318
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 8.00000 0.319489
\(628\) 10.0000 0.399043
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 1.00000 0.0397779
\(633\) 13.0000 0.516704
\(634\) −27.0000 −1.07231
\(635\) 0 0
\(636\) 3.00000 0.118958
\(637\) −2.00000 −0.0792429
\(638\) −6.00000 −0.237542
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) −3.00000 −0.118401
\(643\) 25.0000 0.985904 0.492952 − 0.870057i \(-0.335918\pi\)
0.492952 + 0.870057i \(0.335918\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 39.0000 1.53325 0.766624 − 0.642096i \(-0.221935\pi\)
0.766624 + 0.642096i \(0.221935\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 11.0000 0.431124
\(652\) 10.0000 0.391630
\(653\) −21.0000 −0.821794 −0.410897 − 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(654\) −16.0000 −0.625650
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 10.0000 0.390137
\(658\) −9.00000 −0.350857
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) −6.00000 −0.233021
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 14.0000 0.542489
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) −1.00000 −0.0385758
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −15.0000 −0.576497 −0.288248 − 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) −21.0000 −0.806500
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 11.0000 0.421212
\(683\) −18.0000 −0.688751 −0.344375 − 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) −16.0000 −0.611775
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 4.00000 0.152610
\(688\) 7.00000 0.266872
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 − 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) 6.00000 0.228086
\(693\) −2.00000 −0.0759737
\(694\) 9.00000 0.341635
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −18.0000 −0.681799
\(698\) 7.00000 0.264954
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 10.0000 0.377426
\(703\) 56.0000 2.11208
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 0 0
\(707\) −15.0000 −0.564133
\(708\) 3.00000 0.112747
\(709\) 17.0000 0.638448 0.319224 − 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) 0 0
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 27.0000 1.00833
\(718\) 36.0000 1.34351
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) −45.0000 −1.67473
\(723\) 25.0000 0.929760
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 1.00000 0.0371135
\(727\) 19.0000 0.704671 0.352335 − 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −21.0000 −0.776713
\(732\) 10.0000 0.369611
\(733\) −5.00000 −0.184679 −0.0923396 − 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 12.0000 0.441726
\(739\) 5.00000 0.183928 0.0919640 − 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) −3.00000 −0.110133
\(743\) 39.0000 1.43077 0.715386 − 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 11.0000 0.403280
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 12.0000 0.439057
\(748\) 3.00000 0.109691
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −46.0000 −1.67856 −0.839282 − 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) −9.00000 −0.328196
\(753\) 15.0000 0.546630
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −5.00000 −0.181848
\(757\) −11.0000 −0.399802 −0.199901 − 0.979816i \(-0.564062\pi\)
−0.199901 + 0.979816i \(0.564062\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 − 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) −17.0000 −0.615845
\(763\) 16.0000 0.579239
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 15.0000 0.541972
\(767\) 6.00000 0.216647
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 − 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 10.0000 0.359908
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 14.0000 0.503220
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 7.00000 0.251124
\(778\) 9.00000 0.322666
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) −30.0000 −1.07211
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) 21.0000 0.746674
\(792\) −2.00000 −0.0710669
\(793\) 20.0000 0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −8.00000 −0.283197
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 0 0
\(802\) −3.00000 −0.105934
\(803\) 5.00000 0.176446
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 22.0000 0.774917
\(807\) 6.00000 0.211210
\(808\) −15.0000 −0.527698
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 6.00000 0.210559
\(813\) −2.00000 −0.0701431
\(814\) 7.00000 0.245350
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 56.0000 1.95919
\(818\) −5.00000 −0.174821
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −6.00000 −0.209274
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 − 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) −2.00000 −0.0693375
\(833\) −3.00000 −0.103944
\(834\) −22.0000 −0.761798
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 55.0000 1.90108
\(838\) 3.00000 0.103633
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −12.0000 −0.413302
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) −1.00000 −0.0343604
\(848\) −3.00000 −0.103020
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 0 0
\(852\) −6.00000 −0.205557
\(853\) −35.0000 −1.19838 −0.599189 − 0.800608i \(-0.704510\pi\)
−0.599189 + 0.800608i \(0.704510\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 2.00000 0.0682789
\(859\) −1.00000 −0.0341196 −0.0170598 − 0.999854i \(-0.505431\pi\)
−0.0170598 + 0.999854i \(0.505431\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 21.0000 0.715263
\(863\) 42.0000 1.42970 0.714848 − 0.699280i \(-0.246496\pi\)
0.714848 + 0.699280i \(0.246496\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 8.00000 0.271694
\(868\) −11.0000 −0.373364
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 16.0000 0.541828
\(873\) −20.0000 −0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 5.00000 0.168934
\(877\) −32.0000 −1.08056 −0.540282 − 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) −26.0000 −0.877457
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 − 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 2.00000 0.0673435
\(883\) −2.00000 −0.0673054 −0.0336527 − 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 7.00000 0.234905
\(889\) 17.0000 0.570162
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) −72.0000 −2.40939
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −15.0000 −0.500556
\(899\) −66.0000 −2.20122
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 6.00000 0.199778
\(903\) 7.00000 0.232945
\(904\) 21.0000 0.698450
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −26.0000 −0.863316 −0.431658 − 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 18.0000 0.597351
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 − 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) −8.00000 −0.264906
\(913\) 6.00000 0.198571
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 15.0000 0.495074
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 27.0000 0.889198
\(923\) −12.0000 −0.394985
\(924\) −1.00000 −0.0328976
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) 10.0000 0.328443
\(928\) 6.00000 0.196960
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 24.0000 0.786146
\(933\) 33.0000 1.08037
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −29.0000 −0.947389 −0.473694 − 0.880689i \(-0.657080\pi\)
−0.473694 + 0.880689i \(0.657080\pi\)
\(938\) −8.00000 −0.261209
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 9.00000 0.293392 0.146696 − 0.989182i \(-0.453136\pi\)
0.146696 + 0.989182i \(0.453136\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) 7.00000 0.227590
\(947\) 48.0000 1.55979 0.779895 − 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 1.00000 0.0324785
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) −3.00000 −0.0972306
\(953\) −12.0000 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −27.0000 −0.873242
\(957\) −6.00000 −0.193952
\(958\) −18.0000 −0.581554
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 90.0000 2.90323
\(962\) 14.0000 0.451378
\(963\) 6.00000 0.193347
\(964\) −25.0000 −0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 − 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −16.0000 −0.513200
\(973\) 22.0000 0.705288
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 10.0000 0.319765
\(979\) 0 0
\(980\) 0 0
\(981\) 32.0000 1.02168
\(982\) 27.0000 0.861605
\(983\) 33.0000 1.05254 0.526268 − 0.850319i \(-0.323591\pi\)
0.526268 + 0.850319i \(0.323591\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) −9.00000 −0.286473
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 − 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) −11.0000 −0.349250
\(993\) −20.0000 −0.634681
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −47.0000 −1.48850 −0.744252 − 0.667898i \(-0.767194\pi\)
−0.744252 + 0.667898i \(0.767194\pi\)
\(998\) −32.0000 −1.01294
\(999\) 35.0000 1.10735
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 3850.2.a.a.1.1 ✓ 1
5.2 odd 4 3850.2.c.n.1849.1 2
5.3 odd 4 3850.2.c.n.1849.2 2
5.4 even 2 3850.2.a.z.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3850.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
3850.2.a.z.1.1 yes 1 5.4 even 2
3850.2.c.n.1849.1 2 5.2 odd 4
3850.2.c.n.1849.2 2 5.3 odd 4