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Properties

Label 4650.2.a.l.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
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 \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +3.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} +6.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} +2.00000 q^{38} +3.00000 q^{39} +7.00000 q^{41} +3.00000 q^{42} -11.0000 q^{43} -1.00000 q^{44} +4.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{51} -3.00000 q^{52} -5.00000 q^{53} +1.00000 q^{54} -3.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +1.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -14.0000 q^{67} +4.00000 q^{68} +4.00000 q^{69} -9.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +7.00000 q^{74} -2.00000 q^{76} -3.00000 q^{77} -3.00000 q^{78} +1.00000 q^{81} -7.00000 q^{82} -1.00000 q^{83} -3.00000 q^{84} +11.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} -9.00000 q^{91} -4.00000 q^{92} +1.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} -6.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.00000 −0.832050 −0.416025 − 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 3.00000 0.588348
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 2.00000 0.324443
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 − 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 3.00000 0.462910
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −3.00000 −0.416025
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 1.00000 0.127000
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 4.00000 0.485071
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −3.00000 −0.341882
\(78\) −3.00000 −0.339683
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.00000 −0.773021
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 4.00000 0.396059
\(103\) 11.0000 1.08386 0.541931 − 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 3.00000 0.283473
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −3.00000 −0.277350
\(118\) −6.00000 −0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) −7.00000 −0.631169
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 − 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 1.00000 0.0870388
\(133\) −6.00000 −0.520266
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 4.00000 0.341743 0.170872 − 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −4.00000 −0.340503
\(139\) 5.00000 0.424094 0.212047 − 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 9.00000 0.755263
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −2.00000 −0.164957
\(148\) −7.00000 −0.575396
\(149\) 8.00000 0.655386 0.327693 − 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) 4.00000 0.323381
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) −1.00000 −0.0785674
\(163\) 10.0000 0.783260 0.391630 − 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 3.00000 0.231455
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −11.0000 −0.838742
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −6.00000 −0.450988
\(178\) −10.0000 −0.749532
\(179\) 13.0000 0.971666 0.485833 − 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 9.00000 0.667124
\(183\) −1.00000 −0.0739221
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −4.00000 −0.292509
\(188\) −3.00000 −0.218797
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 7.00000 0.498729 0.249365 − 0.968410i \(-0.419778\pi\)
0.249365 + 0.968410i \(0.419778\pi\)
\(198\) 1.00000 0.0710669
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) 18.0000 1.26335
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) −4.00000 −0.278019
\(208\) −3.00000 −0.208013
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −5.00000 −0.343401
\(213\) 9.00000 0.616670
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −3.00000 −0.203653
\(218\) 18.0000 1.21911
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −7.00000 −0.469809
\(223\) −14.0000 −0.937509 −0.468755 − 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −28.0000 −1.85843 −0.929213 − 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 2.00000 0.132453
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) −6.00000 −0.393919
\(233\) −7.00000 −0.458585 −0.229293 − 0.973358i \(-0.573641\pi\)
−0.229293 + 0.973358i \(0.573641\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) −30.0000 −1.94054 −0.970269 − 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 7.00000 0.446304
\(247\) 6.00000 0.381771
\(248\) 1.00000 0.0635001
\(249\) 1.00000 0.0633724
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 3.00000 0.188982
\(253\) 4.00000 0.251478
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.0000 −0.686161 −0.343081 − 0.939306i \(-0.611470\pi\)
−0.343081 + 0.939306i \(0.611470\pi\)
\(258\) −11.0000 −0.684830
\(259\) −21.0000 −1.30488
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 10.0000 0.617802
\(263\) 2.00000 0.123325 0.0616626 − 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) −10.0000 −0.611990
\(268\) −14.0000 −0.855186
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 9.00000 0.544705
\(274\) −4.00000 −0.241649
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −5.00000 −0.299880
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 − 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) −3.00000 −0.178647
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) 21.0000 1.23959
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) −2.00000 −0.117041
\(293\) −4.00000 −0.233682 −0.116841 − 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 1.00000 0.0580259
\(298\) −8.00000 −0.463428
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −33.0000 −1.90209
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −3.00000 −0.170941
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) −33.0000 −1.87126 −0.935629 − 0.352985i \(-0.885167\pi\)
−0.935629 + 0.352985i \(0.885167\pi\)
\(312\) −3.00000 −0.169842
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 − 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −5.00000 −0.280386
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 12.0000 0.668734
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 18.0000 0.995402
\(328\) −7.00000 −0.386510
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 − 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −1.00000 −0.0548821
\(333\) −7.00000 −0.383598
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 4.00000 0.217571
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 1.00000 0.0541530
\(342\) 2.00000 0.108148
\(343\) −15.0000 −0.809924
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 11.0000 0.590511 0.295255 − 0.955418i \(-0.404595\pi\)
0.295255 + 0.955418i \(0.404595\pi\)
\(348\) −6.00000 −0.321634
\(349\) −18.0000 −0.963518 −0.481759 − 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 1.00000 0.0533002
\(353\) 8.00000 0.425797 0.212899 − 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −12.0000 −0.635107
\(358\) −13.0000 −0.687071
\(359\) −28.0000 −1.47778 −0.738892 − 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −5.00000 −0.262794
\(363\) 10.0000 0.524864
\(364\) −9.00000 −0.471728
\(365\) 0 0
\(366\) 1.00000 0.0522708
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −4.00000 −0.208514
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) −15.0000 −0.778761
\(372\) 1.00000 0.0518476
\(373\) −8.00000 −0.414224 −0.207112 − 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −18.0000 −0.927047
\(378\) 3.00000 0.154303
\(379\) 6.00000 0.308199 0.154100 − 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 12.0000 0.613973
\(383\) −4.00000 −0.204390 −0.102195 − 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) −11.0000 −0.559161
\(388\) −6.00000 −0.304604
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −2.00000 −0.101015
\(393\) 10.0000 0.504433
\(394\) −7.00000 −0.352655
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −10.0000 −0.501255
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 32.0000 1.59800 0.799002 − 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) −14.0000 −0.698257
\(403\) 3.00000 0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) 7.00000 0.346977
\(408\) 4.00000 0.198030
\(409\) −38.0000 −1.87898 −0.939490 − 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 11.0000 0.541931
\(413\) 18.0000 0.885722
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 3.00000 0.147087
\(417\) −5.00000 −0.244851
\(418\) −2.00000 −0.0978232
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 20.0000 0.973585
\(423\) −3.00000 −0.145865
\(424\) 5.00000 0.242821
\(425\) 0 0
\(426\) −9.00000 −0.436051
\(427\) 3.00000 0.145180
\(428\) 4.00000 0.193347
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 − 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.00000 −0.384455 −0.192228 − 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 8.00000 0.382692
\(438\) −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
\(441\) 2.00000 0.0952381
\(442\) 12.0000 0.570782
\(443\) −34.0000 −1.61539 −0.807694 − 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 7.00000 0.332205
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) −8.00000 −0.378387
\(448\) 3.00000 0.141737
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −7.00000 −0.329617
\(452\) 2.00000 0.0940721
\(453\) −8.00000 −0.375873
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −26.0000 −1.21490
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 − 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) −3.00000 −0.139573
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 7.00000 0.324269
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −3.00000 −0.138675
\(469\) −42.0000 −1.93938
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −6.00000 −0.276172
\(473\) 11.0000 0.505781
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) −5.00000 −0.228934
\(478\) 30.0000 1.37217
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 21.0000 0.957518
\(482\) 18.0000 0.819878
\(483\) 12.0000 0.546019
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −7.00000 −0.315584
\(493\) 24.0000 1.08091
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −27.0000 −1.21112
\(498\) −1.00000 −0.0448111
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 9.00000 0.401690
\(503\) −27.0000 −1.20387 −0.601935 − 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 4.00000 0.177646
\(508\) 12.0000 0.532414
\(509\) 35.0000 1.55135 0.775674 − 0.631134i \(-0.217410\pi\)
0.775674 + 0.631134i \(0.217410\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 11.0000 0.485189
\(515\) 0 0
\(516\) 11.0000 0.484248
\(517\) 3.00000 0.131940
\(518\) 21.0000 0.922687
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 − 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) −4.00000 −0.174243
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −6.00000 −0.260133
\(533\) −21.0000 −0.909611
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) −13.0000 −0.560991
\(538\) −26.0000 −1.12094
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −20.0000 −0.859074
\(543\) −5.00000 −0.214571
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −9.00000 −0.385164
\(547\) 32.0000 1.36822 0.684111 − 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 4.00000 0.170872
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 1.00000 0.0423334
\(559\) 33.0000 1.39575
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −7.00000 −0.295277
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 3.00000 0.125988
\(568\) 9.00000 0.377632
\(569\) 28.0000 1.17382 0.586911 − 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −5.00000 −0.209243 −0.104622 − 0.994512i \(-0.533363\pi\)
−0.104622 + 0.994512i \(0.533363\pi\)
\(572\) 3.00000 0.125436
\(573\) 12.0000 0.501307
\(574\) −21.0000 −0.876523
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 1.00000 0.0415945
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) −3.00000 −0.124461
\(582\) −6.00000 −0.248708
\(583\) 5.00000 0.207079
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) −3.00000 −0.123823 −0.0619116 − 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −7.00000 −0.287942
\(592\) −7.00000 −0.287698
\(593\) 15.0000 0.615976 0.307988 − 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 8.00000 0.327693
\(597\) −10.0000 −0.409273
\(598\) −12.0000 −0.490716
\(599\) 25.0000 1.02147 0.510736 − 0.859738i \(-0.329373\pi\)
0.510736 + 0.859738i \(0.329373\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 33.0000 1.34498
\(603\) −14.0000 −0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 19.0000 0.771186 0.385593 − 0.922669i \(-0.373997\pi\)
0.385593 + 0.922669i \(0.373997\pi\)
\(608\) 2.00000 0.0811107
\(609\) −18.0000 −0.729397
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 4.00000 0.161690
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −17.0000 −0.684394 −0.342197 − 0.939628i \(-0.611171\pi\)
−0.342197 + 0.939628i \(0.611171\pi\)
\(618\) 11.0000 0.442485
\(619\) −27.0000 −1.08522 −0.542611 − 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 33.0000 1.32318
\(623\) 30.0000 1.20192
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) −2.00000 −0.0798723
\(628\) −14.0000 −0.558661
\(629\) −28.0000 −1.11643
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 5.00000 0.198263
\(637\) −6.00000 −0.237729
\(638\) 6.00000 0.237542
\(639\) −9.00000 −0.356034
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 − 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 4.00000 0.157867
\(643\) 7.00000 0.276053 0.138027 − 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 10.0000 0.391630
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) 7.00000 0.273304
\(657\) −2.00000 −0.0780274
\(658\) 9.00000 0.350857
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 − 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 17.0000 0.660724
\(663\) 12.0000 0.466041
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) −24.0000 −0.929284
\(668\) 12.0000 0.464294
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) 3.00000 0.115728
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −25.0000 −0.960828 −0.480414 − 0.877042i \(-0.659514\pi\)
−0.480414 + 0.877042i \(0.659514\pi\)
\(678\) 2.00000 0.0768095
\(679\) −18.0000 −0.690777
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) −1.00000 −0.0382920
\(683\) 16.0000 0.612223 0.306111 − 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) −26.0000 −0.991962
\(688\) −11.0000 −0.419371
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −14.0000 −0.532200
\(693\) −3.00000 −0.113961
\(694\) −11.0000 −0.417554
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 28.0000 1.06058
\(698\) 18.0000 0.681310
\(699\) 7.00000 0.264764
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 − 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) −3.00000 −0.113228
\(703\) 14.0000 0.528020
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −8.00000 −0.301084
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 4.00000 0.149801
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 13.0000 0.485833
\(717\) 30.0000 1.12037
\(718\) 28.0000 1.04495
\(719\) −8.00000 −0.298350 −0.149175 − 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 33.0000 1.22898
\(722\) 15.0000 0.558242
\(723\) 18.0000 0.669427
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) −3.00000 −0.111264 −0.0556319 − 0.998451i \(-0.517717\pi\)
−0.0556319 + 0.998451i \(0.517717\pi\)
\(728\) 9.00000 0.333562
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.0000 −1.62740
\(732\) −1.00000 −0.0369611
\(733\) −52.0000 −1.92066 −0.960332 − 0.278859i \(-0.910044\pi\)
−0.960332 + 0.278859i \(0.910044\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 14.0000 0.515697
\(738\) −7.00000 −0.257674
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 15.0000 0.550667
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) −1.00000 −0.0365881
\(748\) −4.00000 −0.146254
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 − 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) −3.00000 −0.109399
\(753\) 9.00000 0.327978
\(754\) 18.0000 0.655521
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) −29.0000 −1.05402 −0.527011 − 0.849858i \(-0.676688\pi\)
−0.527011 + 0.849858i \(0.676688\pi\)
\(758\) −6.00000 −0.217930
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 − 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 12.0000 0.434714
\(763\) −54.0000 −1.95493
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) −18.0000 −0.649942
\(768\) −1.00000 −0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 − 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 11.0000 0.396155
\(772\) 5.00000 0.179954
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 21.0000 0.753371
\(778\) 6.00000 0.215110
\(779\) −14.0000 −0.501602
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 16.0000 0.572159
\(783\) −6.00000 −0.214423
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) −45.0000 −1.60408 −0.802038 − 0.597272i \(-0.796251\pi\)
−0.802038 + 0.597272i \(0.796251\pi\)
\(788\) 7.00000 0.249365
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 1.00000 0.0355335
\(793\) −3.00000 −0.106533
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −6.00000 −0.212398
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −32.0000 −1.12996
\(803\) 2.00000 0.0705785
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −3.00000 −0.105670
\(807\) −26.0000 −0.915243
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 18.0000 0.631676
\(813\) −20.0000 −0.701431
\(814\) −7.00000 −0.245350
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 22.0000 0.769683
\(818\) 38.0000 1.32864
\(819\) −9.00000 −0.314485
\(820\) 0 0
\(821\) 21.0000 0.732905 0.366453 − 0.930437i \(-0.380572\pi\)
0.366453 + 0.930437i \(0.380572\pi\)
\(822\) 4.00000 0.139516
\(823\) −56.0000 −1.95204 −0.976019 − 0.217687i \(-0.930149\pi\)
−0.976019 + 0.217687i \(0.930149\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) −4.00000 −0.139010
\(829\) −23.0000 −0.798823 −0.399412 − 0.916772i \(-0.630786\pi\)
−0.399412 + 0.916772i \(0.630786\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) −3.00000 −0.104006
\(833\) 8.00000 0.277184
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 2.00000 0.0691714
\(837\) 1.00000 0.0345651
\(838\) −4.00000 −0.138178
\(839\) 3.00000 0.103572 0.0517858 − 0.998658i \(-0.483509\pi\)
0.0517858 + 0.998658i \(0.483509\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) −7.00000 −0.241093
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −30.0000 −1.03081
\(848\) −5.00000 −0.171701
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 28.0000 0.959828
\(852\) 9.00000 0.308335
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −21.0000 −0.717346 −0.358673 − 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 3.00000 0.102418
\(859\) −31.0000 −1.05771 −0.528853 − 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) −21.0000 −0.715678
\(862\) 33.0000 1.12398
\(863\) −44.0000 −1.49778 −0.748889 − 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 8.00000 0.271851
\(867\) 1.00000 0.0339618
\(868\) −3.00000 −0.101827
\(869\) 0 0
\(870\) 0 0
\(871\) 42.0000 1.42312
\(872\) 18.0000 0.609557
\(873\) −6.00000 −0.203069
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 20.0000 0.675352 0.337676 − 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 8.00000 0.269987
\(879\) 4.00000 0.134917
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 − 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −1.00000 −0.0336527 −0.0168263 − 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) −27.0000 −0.906571 −0.453286 − 0.891365i \(-0.649748\pi\)
−0.453286 + 0.891365i \(0.649748\pi\)
\(888\) −7.00000 −0.234905
\(889\) 36.0000 1.20740
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −14.0000 −0.468755
\(893\) 6.00000 0.200782
\(894\) 8.00000 0.267560
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) −12.0000 −0.400668
\(898\) −6.00000 −0.200223
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 7.00000 0.233075
\(903\) 33.0000 1.09817
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 24.0000 0.796907 0.398453 − 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −28.0000 −0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 − 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 2.00000 0.0662266
\(913\) 1.00000 0.0330952
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) −30.0000 −0.990687
\(918\) 4.00000 0.132020
\(919\) −11.0000 −0.362857 −0.181428 − 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) −3.00000 −0.0987997
\(923\) 27.0000 0.888716
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 11.0000 0.361287
\(928\) −6.00000 −0.196960
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −7.00000 −0.229293
\(933\) 33.0000 1.08037
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 42.0000 1.37135
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) −14.0000 −0.456145
\(943\) −28.0000 −0.911805
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −11.0000 −0.357641
\(947\) 21.0000 0.682408 0.341204 − 0.939989i \(-0.389165\pi\)
0.341204 + 0.939989i \(0.389165\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) −12.0000 −0.388922
\(953\) 48.0000 1.55487 0.777436 − 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 5.00000 0.161881
\(955\) 0 0
\(956\) −30.0000 −0.970269
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −21.0000 −0.677067
\(963\) 4.00000 0.128898
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 26.0000 0.836104 0.418052 − 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 10.0000 0.321412
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 − 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 15.0000 0.480878
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) 45.0000 1.43968 0.719839 − 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) 10.0000 0.319765
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 20.0000 0.638226
\(983\) −30.0000 −0.956851 −0.478426 − 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 7.00000 0.223152
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 9.00000 0.286473
\(988\) 6.00000 0.190885
\(989\) 44.0000 1.39912
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 − 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 1.00000 0.0317500
\(993\) 17.0000 0.539479
\(994\) 27.0000 0.856388
\(995\) 0 0
\(996\) 1.00000 0.0316862
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 32.0000 1.01294
\(999\) 7.00000 0.221470
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 4650.2.a.l.1.1 ✓ 1
5.2 odd 4 4650.2.d.t.3349.1 2
5.3 odd 4 4650.2.d.t.3349.2 2
5.4 even 2 4650.2.a.bj.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4650.2.a.l.1.1 ✓ 1 1.1 even 1 trivial
4650.2.a.bj.1.1 yes 1 5.4 even 2
4650.2.d.t.3349.1 2 5.2 odd 4
4650.2.d.t.3349.2 2 5.3 odd 4