Properties

Label 2842.2.a.bb.1.2
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2842,2,Mod(1,2842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2842, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.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: \(22.6934842544\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.401917952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 14x^{4} + 49x^{2} - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.26409\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.26409 q^{3} +1.00000 q^{4} -3.67831 q^{5} -2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.26409 q^{3} +1.00000 q^{4} -3.67831 q^{5} -2.26409 q^{6} +1.00000 q^{8} +2.12612 q^{9} -3.67831 q^{10} -2.32803 q^{11} -2.26409 q^{12} -5.27088 q^{13} +8.32803 q^{15} +1.00000 q^{16} +4.70655 q^{17} +2.12612 q^{18} -5.94240 q^{19} -3.67831 q^{20} -2.32803 q^{22} -8.65606 q^{23} -2.26409 q^{24} +8.52994 q^{25} -5.27088 q^{26} +1.97855 q^{27} -1.00000 q^{29} +8.32803 q^{30} +1.97855 q^{31} +1.00000 q^{32} +5.27088 q^{33} +4.70655 q^{34} +2.12612 q^{36} -2.00000 q^{37} -5.94240 q^{38} +11.9338 q^{39} -3.67831 q^{40} -10.3634 q^{41} -12.1860 q^{43} -2.32803 q^{44} -7.82052 q^{45} -8.65606 q^{46} +1.97855 q^{47} -2.26409 q^{48} +8.52994 q^{50} -10.6561 q^{51} -5.27088 q^{52} +6.32803 q^{53} +1.97855 q^{54} +8.56321 q^{55} +13.4542 q^{57} -1.00000 q^{58} +7.17825 q^{59} +8.32803 q^{60} +11.1129 q^{61} +1.97855 q^{62} +1.00000 q^{64} +19.3879 q^{65} +5.27088 q^{66} -2.25224 q^{67} +4.70655 q^{68} +19.5981 q^{69} +8.00000 q^{71} +2.12612 q^{72} +7.17825 q^{73} -2.00000 q^{74} -19.3126 q^{75} -5.94240 q^{76} +11.9338 q^{78} +11.1261 q^{79} -3.67831 q^{80} -10.8580 q^{81} -10.3634 q^{82} +10.3634 q^{83} -17.3121 q^{85} -12.1860 q^{86} +2.26409 q^{87} -2.32803 q^{88} -5.83521 q^{89} -7.82052 q^{90} -8.65606 q^{92} -4.47962 q^{93} +1.97855 q^{94} +21.8580 q^{95} -2.26409 q^{96} -4.13545 q^{97} -4.94967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.26409 −1.30718 −0.653588 0.756851i \(-0.726737\pi\)
−0.653588 + 0.756851i \(0.726737\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.67831 −1.64499 −0.822494 0.568773i \(-0.807418\pi\)
−0.822494 + 0.568773i \(0.807418\pi\)
\(6\) −2.26409 −0.924312
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 2.12612 0.708707
\(10\) −3.67831 −1.16318
\(11\) −2.32803 −0.701928 −0.350964 0.936389i \(-0.614146\pi\)
−0.350964 + 0.936389i \(0.614146\pi\)
\(12\) −2.26409 −0.653588
\(13\) −5.27088 −1.46188 −0.730940 0.682442i \(-0.760918\pi\)
−0.730940 + 0.682442i \(0.760918\pi\)
\(14\) 0 0
\(15\) 8.32803 2.15029
\(16\) 1.00000 0.250000
\(17\) 4.70655 1.14151 0.570753 0.821122i \(-0.306652\pi\)
0.570753 + 0.821122i \(0.306652\pi\)
\(18\) 2.12612 0.501131
\(19\) −5.94240 −1.36328 −0.681640 0.731688i \(-0.738733\pi\)
−0.681640 + 0.731688i \(0.738733\pi\)
\(20\) −3.67831 −0.822494
\(21\) 0 0
\(22\) −2.32803 −0.496338
\(23\) −8.65606 −1.80491 −0.902457 0.430780i \(-0.858238\pi\)
−0.902457 + 0.430780i \(0.858238\pi\)
\(24\) −2.26409 −0.462156
\(25\) 8.52994 1.70599
\(26\) −5.27088 −1.03370
\(27\) 1.97855 0.380772
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 8.32803 1.52048
\(31\) 1.97855 0.355358 0.177679 0.984089i \(-0.443141\pi\)
0.177679 + 0.984089i \(0.443141\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.27088 0.917543
\(34\) 4.70655 0.807166
\(35\) 0 0
\(36\) 2.12612 0.354353
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −5.94240 −0.963985
\(39\) 11.9338 1.91093
\(40\) −3.67831 −0.581591
\(41\) −10.3634 −1.61849 −0.809246 0.587470i \(-0.800124\pi\)
−0.809246 + 0.587470i \(0.800124\pi\)
\(42\) 0 0
\(43\) −12.1860 −1.85835 −0.929174 0.369642i \(-0.879480\pi\)
−0.929174 + 0.369642i \(0.879480\pi\)
\(44\) −2.32803 −0.350964
\(45\) −7.82052 −1.16581
\(46\) −8.65606 −1.27627
\(47\) 1.97855 0.288601 0.144300 0.989534i \(-0.453907\pi\)
0.144300 + 0.989534i \(0.453907\pi\)
\(48\) −2.26409 −0.326794
\(49\) 0 0
\(50\) 8.52994 1.20632
\(51\) −10.6561 −1.49215
\(52\) −5.27088 −0.730940
\(53\) 6.32803 0.869222 0.434611 0.900618i \(-0.356886\pi\)
0.434611 + 0.900618i \(0.356886\pi\)
\(54\) 1.97855 0.269246
\(55\) 8.56321 1.15466
\(56\) 0 0
\(57\) 13.4542 1.78205
\(58\) −1.00000 −0.131306
\(59\) 7.17825 0.934529 0.467265 0.884118i \(-0.345240\pi\)
0.467265 + 0.884118i \(0.345240\pi\)
\(60\) 8.32803 1.07514
\(61\) 11.1129 1.42286 0.711428 0.702759i \(-0.248049\pi\)
0.711428 + 0.702759i \(0.248049\pi\)
\(62\) 1.97855 0.251276
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 19.3879 2.40478
\(66\) 5.27088 0.648801
\(67\) −2.25224 −0.275155 −0.137577 0.990491i \(-0.543932\pi\)
−0.137577 + 0.990491i \(0.543932\pi\)
\(68\) 4.70655 0.570753
\(69\) 19.5981 2.35934
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 2.12612 0.250566
\(73\) 7.17825 0.840151 0.420076 0.907489i \(-0.362004\pi\)
0.420076 + 0.907489i \(0.362004\pi\)
\(74\) −2.00000 −0.232495
\(75\) −19.3126 −2.23003
\(76\) −5.94240 −0.681640
\(77\) 0 0
\(78\) 11.9338 1.35123
\(79\) 11.1261 1.25179 0.625893 0.779909i \(-0.284735\pi\)
0.625893 + 0.779909i \(0.284735\pi\)
\(80\) −3.67831 −0.411247
\(81\) −10.8580 −1.20644
\(82\) −10.3634 −1.14445
\(83\) 10.3634 1.13753 0.568766 0.822500i \(-0.307421\pi\)
0.568766 + 0.822500i \(0.307421\pi\)
\(84\) 0 0
\(85\) −17.3121 −1.87776
\(86\) −12.1860 −1.31405
\(87\) 2.26409 0.242736
\(88\) −2.32803 −0.248169
\(89\) −5.83521 −0.618531 −0.309266 0.950976i \(-0.600083\pi\)
−0.309266 + 0.950976i \(0.600083\pi\)
\(90\) −7.82052 −0.824355
\(91\) 0 0
\(92\) −8.65606 −0.902457
\(93\) −4.47962 −0.464515
\(94\) 1.97855 0.204072
\(95\) 21.8580 2.24258
\(96\) −2.26409 −0.231078
\(97\) −4.13545 −0.419892 −0.209946 0.977713i \(-0.567329\pi\)
−0.209946 + 0.977713i \(0.567329\pi\)
\(98\) 0 0
\(99\) −4.94967 −0.497461
\(100\) 8.52994 0.852994
\(101\) −3.75624 −0.373760 −0.186880 0.982383i \(-0.559838\pi\)
−0.186880 + 0.982383i \(0.559838\pi\)
\(102\) −10.6561 −1.05511
\(103\) −8.48528 −0.836080 −0.418040 0.908429i \(-0.637283\pi\)
−0.418040 + 0.908429i \(0.637283\pi\)
\(104\) −5.27088 −0.516852
\(105\) 0 0
\(106\) 6.32803 0.614633
\(107\) 4.65606 0.450119 0.225059 0.974345i \(-0.427742\pi\)
0.225059 + 0.974345i \(0.427742\pi\)
\(108\) 1.97855 0.190386
\(109\) −10.9841 −1.05209 −0.526043 0.850458i \(-0.676325\pi\)
−0.526043 + 0.850458i \(0.676325\pi\)
\(110\) 8.56321 0.816470
\(111\) 4.52819 0.429796
\(112\) 0 0
\(113\) 20.8076 1.95742 0.978709 0.205251i \(-0.0658010\pi\)
0.978709 + 0.205251i \(0.0658010\pi\)
\(114\) 13.4542 1.26010
\(115\) 31.8397 2.96906
\(116\) −1.00000 −0.0928477
\(117\) −11.2065 −1.03604
\(118\) 7.17825 0.660812
\(119\) 0 0
\(120\) 8.32803 0.760242
\(121\) −5.58027 −0.507297
\(122\) 11.1129 1.00611
\(123\) 23.4637 2.11565
\(124\) 1.97855 0.177679
\(125\) −12.9842 −1.16134
\(126\) 0 0
\(127\) 2.25224 0.199854 0.0999269 0.994995i \(-0.468139\pi\)
0.0999269 + 0.994995i \(0.468139\pi\)
\(128\) 1.00000 0.0883883
\(129\) 27.5903 2.42919
\(130\) 19.3879 1.70043
\(131\) −9.48427 −0.828644 −0.414322 0.910130i \(-0.635981\pi\)
−0.414322 + 0.910130i \(0.635981\pi\)
\(132\) 5.27088 0.458771
\(133\) 0 0
\(134\) −2.25224 −0.194564
\(135\) −7.27770 −0.626365
\(136\) 4.70655 0.403583
\(137\) 13.1605 1.12438 0.562190 0.827008i \(-0.309959\pi\)
0.562190 + 0.827008i \(0.309959\pi\)
\(138\) 19.5981 1.66830
\(139\) 8.66364 0.734840 0.367420 0.930055i \(-0.380241\pi\)
0.367420 + 0.930055i \(0.380241\pi\)
\(140\) 0 0
\(141\) −4.47962 −0.377252
\(142\) 8.00000 0.671345
\(143\) 12.2708 1.02613
\(144\) 2.12612 0.177177
\(145\) 3.67831 0.305467
\(146\) 7.17825 0.594077
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −5.12612 −0.419948 −0.209974 0.977707i \(-0.567338\pi\)
−0.209974 + 0.977707i \(0.567338\pi\)
\(150\) −19.3126 −1.57687
\(151\) −15.0599 −1.22556 −0.612778 0.790255i \(-0.709948\pi\)
−0.612778 + 0.790255i \(0.709948\pi\)
\(152\) −5.94240 −0.481992
\(153\) 10.0067 0.808992
\(154\) 0 0
\(155\) −7.27770 −0.584559
\(156\) 11.9338 0.955466
\(157\) 8.12856 0.648730 0.324365 0.945932i \(-0.394849\pi\)
0.324365 + 0.945932i \(0.394849\pi\)
\(158\) 11.1261 0.885147
\(159\) −14.3273 −1.13623
\(160\) −3.67831 −0.290796
\(161\) 0 0
\(162\) −10.8580 −0.853083
\(163\) 5.82355 0.456136 0.228068 0.973645i \(-0.426759\pi\)
0.228068 + 0.973645i \(0.426759\pi\)
\(164\) −10.3634 −0.809246
\(165\) −19.3879 −1.50935
\(166\) 10.3634 0.804356
\(167\) 6.22795 0.481933 0.240967 0.970533i \(-0.422536\pi\)
0.240967 + 0.970533i \(0.422536\pi\)
\(168\) 0 0
\(169\) 14.7822 1.13709
\(170\) −17.3121 −1.32778
\(171\) −12.6343 −0.966166
\(172\) −12.1860 −0.929174
\(173\) −2.36452 −0.179771 −0.0898856 0.995952i \(-0.528650\pi\)
−0.0898856 + 0.995952i \(0.528650\pi\)
\(174\) 2.26409 0.171640
\(175\) 0 0
\(176\) −2.32803 −0.175482
\(177\) −16.2522 −1.22159
\(178\) −5.83521 −0.437368
\(179\) −8.50448 −0.635655 −0.317827 0.948149i \(-0.602953\pi\)
−0.317827 + 0.948149i \(0.602953\pi\)
\(180\) −7.82052 −0.582907
\(181\) 7.99212 0.594050 0.297025 0.954870i \(-0.404006\pi\)
0.297025 + 0.954870i \(0.404006\pi\)
\(182\) 0 0
\(183\) −25.1605 −1.85992
\(184\) −8.65606 −0.638133
\(185\) 7.35661 0.540869
\(186\) −4.47962 −0.328461
\(187\) −10.9570 −0.801254
\(188\) 1.97855 0.144300
\(189\) 0 0
\(190\) 21.8580 1.58574
\(191\) 1.74776 0.126464 0.0632318 0.997999i \(-0.479859\pi\)
0.0632318 + 0.997999i \(0.479859\pi\)
\(192\) −2.26409 −0.163397
\(193\) −7.49552 −0.539539 −0.269770 0.962925i \(-0.586948\pi\)
−0.269770 + 0.962925i \(0.586948\pi\)
\(194\) −4.13545 −0.296908
\(195\) −43.8961 −3.14346
\(196\) 0 0
\(197\) 22.6561 1.61418 0.807089 0.590430i \(-0.201042\pi\)
0.807089 + 0.590430i \(0.201042\pi\)
\(198\) −4.94967 −0.351758
\(199\) 17.5417 1.24349 0.621747 0.783218i \(-0.286423\pi\)
0.621747 + 0.783218i \(0.286423\pi\)
\(200\) 8.52994 0.603158
\(201\) 5.09928 0.359675
\(202\) −3.75624 −0.264288
\(203\) 0 0
\(204\) −10.6561 −0.746074
\(205\) 38.1198 2.66240
\(206\) −8.48528 −0.591198
\(207\) −18.4038 −1.27915
\(208\) −5.27088 −0.365470
\(209\) 13.8341 0.956924
\(210\) 0 0
\(211\) −16.4796 −1.13450 −0.567252 0.823544i \(-0.691993\pi\)
−0.567252 + 0.823544i \(0.691993\pi\)
\(212\) 6.32803 0.434611
\(213\) −18.1127 −1.24107
\(214\) 4.65606 0.318282
\(215\) 44.8239 3.05696
\(216\) 1.97855 0.134623
\(217\) 0 0
\(218\) −10.9841 −0.743937
\(219\) −16.2522 −1.09822
\(220\) 8.56321 0.577332
\(221\) −24.8076 −1.66874
\(222\) 4.52819 0.303912
\(223\) 21.0835 1.41186 0.705929 0.708283i \(-0.250530\pi\)
0.705929 + 0.708283i \(0.250530\pi\)
\(224\) 0 0
\(225\) 18.1357 1.20905
\(226\) 20.8076 1.38410
\(227\) −24.3047 −1.61316 −0.806579 0.591126i \(-0.798684\pi\)
−0.806579 + 0.591126i \(0.798684\pi\)
\(228\) 13.4542 0.891023
\(229\) −12.0857 −0.798643 −0.399321 0.916811i \(-0.630754\pi\)
−0.399321 + 0.916811i \(0.630754\pi\)
\(230\) 31.8397 2.09944
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 7.63060 0.499897 0.249949 0.968259i \(-0.419586\pi\)
0.249949 + 0.968259i \(0.419586\pi\)
\(234\) −11.2065 −0.732593
\(235\) −7.27770 −0.474745
\(236\) 7.17825 0.467265
\(237\) −25.1906 −1.63630
\(238\) 0 0
\(239\) 0.504478 0.0326320 0.0163160 0.999867i \(-0.494806\pi\)
0.0163160 + 0.999867i \(0.494806\pi\)
\(240\) 8.32803 0.537572
\(241\) 21.8622 1.40827 0.704135 0.710066i \(-0.251335\pi\)
0.704135 + 0.710066i \(0.251335\pi\)
\(242\) −5.58027 −0.358713
\(243\) 18.6478 1.19626
\(244\) 11.1129 0.711428
\(245\) 0 0
\(246\) 23.4637 1.49599
\(247\) 31.3217 1.99295
\(248\) 1.97855 0.125638
\(249\) −23.4637 −1.48695
\(250\) −12.9842 −0.821194
\(251\) −9.87024 −0.623004 −0.311502 0.950245i \(-0.600832\pi\)
−0.311502 + 0.950245i \(0.600832\pi\)
\(252\) 0 0
\(253\) 20.1516 1.26692
\(254\) 2.25224 0.141318
\(255\) 39.1963 2.45457
\(256\) 1.00000 0.0625000
\(257\) −11.6772 −0.728403 −0.364201 0.931320i \(-0.618658\pi\)
−0.364201 + 0.931320i \(0.618658\pi\)
\(258\) 27.5903 1.71769
\(259\) 0 0
\(260\) 19.3879 1.20239
\(261\) −2.12612 −0.131603
\(262\) −9.48427 −0.585940
\(263\) −17.6815 −1.09029 −0.545145 0.838342i \(-0.683525\pi\)
−0.545145 + 0.838342i \(0.683525\pi\)
\(264\) 5.27088 0.324400
\(265\) −23.2764 −1.42986
\(266\) 0 0
\(267\) 13.2115 0.808529
\(268\) −2.25224 −0.137577
\(269\) 4.32733 0.263842 0.131921 0.991260i \(-0.457885\pi\)
0.131921 + 0.991260i \(0.457885\pi\)
\(270\) −7.27770 −0.442907
\(271\) −24.8690 −1.51069 −0.755343 0.655330i \(-0.772530\pi\)
−0.755343 + 0.655330i \(0.772530\pi\)
\(272\) 4.70655 0.285376
\(273\) 0 0
\(274\) 13.1605 0.795057
\(275\) −19.8580 −1.19748
\(276\) 19.5981 1.17967
\(277\) −5.64711 −0.339302 −0.169651 0.985504i \(-0.554264\pi\)
−0.169651 + 0.985504i \(0.554264\pi\)
\(278\) 8.66364 0.519611
\(279\) 4.20663 0.251844
\(280\) 0 0
\(281\) 14.2274 0.848734 0.424367 0.905490i \(-0.360497\pi\)
0.424367 + 0.905490i \(0.360497\pi\)
\(282\) −4.47962 −0.266757
\(283\) −18.2911 −1.08729 −0.543647 0.839314i \(-0.682957\pi\)
−0.543647 + 0.839314i \(0.682957\pi\)
\(284\) 8.00000 0.474713
\(285\) −49.4885 −2.93145
\(286\) 12.2708 0.725586
\(287\) 0 0
\(288\) 2.12612 0.125283
\(289\) 5.15158 0.303034
\(290\) 3.67831 0.215998
\(291\) 9.36306 0.548872
\(292\) 7.17825 0.420076
\(293\) 7.71333 0.450618 0.225309 0.974287i \(-0.427661\pi\)
0.225309 + 0.974287i \(0.427661\pi\)
\(294\) 0 0
\(295\) −26.4038 −1.53729
\(296\) −2.00000 −0.116248
\(297\) −4.60612 −0.267274
\(298\) −5.12612 −0.296948
\(299\) 45.6251 2.63857
\(300\) −19.3126 −1.11501
\(301\) 0 0
\(302\) −15.0599 −0.866599
\(303\) 8.50448 0.488569
\(304\) −5.94240 −0.340820
\(305\) −40.8765 −2.34058
\(306\) 10.0067 0.572044
\(307\) −30.5034 −1.74092 −0.870460 0.492240i \(-0.836178\pi\)
−0.870460 + 0.492240i \(0.836178\pi\)
\(308\) 0 0
\(309\) 19.2115 1.09290
\(310\) −7.27770 −0.413346
\(311\) −6.32162 −0.358466 −0.179233 0.983807i \(-0.557362\pi\)
−0.179233 + 0.983807i \(0.557362\pi\)
\(312\) 11.9338 0.675616
\(313\) −30.5970 −1.72945 −0.864724 0.502248i \(-0.832506\pi\)
−0.864724 + 0.502248i \(0.832506\pi\)
\(314\) 8.12856 0.458721
\(315\) 0 0
\(316\) 11.1261 0.625893
\(317\) 33.2115 1.86534 0.932671 0.360728i \(-0.117472\pi\)
0.932671 + 0.360728i \(0.117472\pi\)
\(318\) −14.3273 −0.803433
\(319\) 2.32803 0.130345
\(320\) −3.67831 −0.205624
\(321\) −10.5418 −0.588384
\(322\) 0 0
\(323\) −27.9682 −1.55619
\(324\) −10.8580 −0.603221
\(325\) −44.9603 −2.49395
\(326\) 5.82355 0.322537
\(327\) 24.8690 1.37526
\(328\) −10.3634 −0.572223
\(329\) 0 0
\(330\) −19.3879 −1.06727
\(331\) 11.2459 0.618130 0.309065 0.951041i \(-0.399984\pi\)
0.309065 + 0.951041i \(0.399984\pi\)
\(332\) 10.3634 0.568766
\(333\) −4.25224 −0.233021
\(334\) 6.22795 0.340778
\(335\) 8.28443 0.452627
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 14.7822 0.804045
\(339\) −47.1105 −2.55869
\(340\) −17.3121 −0.938882
\(341\) −4.60612 −0.249435
\(342\) −12.6343 −0.683182
\(343\) 0 0
\(344\) −12.1860 −0.657025
\(345\) −72.0880 −3.88109
\(346\) −2.36452 −0.127117
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 2.26409 0.121368
\(349\) 28.3622 1.51819 0.759097 0.650978i \(-0.225641\pi\)
0.759097 + 0.650978i \(0.225641\pi\)
\(350\) 0 0
\(351\) −10.4287 −0.556642
\(352\) −2.32803 −0.124084
\(353\) 5.74155 0.305592 0.152796 0.988258i \(-0.451172\pi\)
0.152796 + 0.988258i \(0.451172\pi\)
\(354\) −16.2522 −0.863797
\(355\) −29.4265 −1.56179
\(356\) −5.83521 −0.309266
\(357\) 0 0
\(358\) −8.50448 −0.449476
\(359\) 16.4382 0.867577 0.433789 0.901015i \(-0.357176\pi\)
0.433789 + 0.901015i \(0.357176\pi\)
\(360\) −7.82052 −0.412178
\(361\) 16.3121 0.858533
\(362\) 7.99212 0.420057
\(363\) 12.6343 0.663126
\(364\) 0 0
\(365\) −26.4038 −1.38204
\(366\) −25.1605 −1.31516
\(367\) −14.0485 −0.733324 −0.366662 0.930354i \(-0.619499\pi\)
−0.366662 + 0.930354i \(0.619499\pi\)
\(368\) −8.65606 −0.451228
\(369\) −22.0338 −1.14704
\(370\) 7.35661 0.382452
\(371\) 0 0
\(372\) −4.47962 −0.232257
\(373\) −17.4981 −0.906019 −0.453009 0.891506i \(-0.649650\pi\)
−0.453009 + 0.891506i \(0.649650\pi\)
\(374\) −10.9570 −0.566572
\(375\) 29.3975 1.51808
\(376\) 1.97855 0.102036
\(377\) 5.27088 0.271464
\(378\) 0 0
\(379\) −7.89935 −0.405762 −0.202881 0.979203i \(-0.565030\pi\)
−0.202881 + 0.979203i \(0.565030\pi\)
\(380\) 21.8580 1.12129
\(381\) −5.09928 −0.261244
\(382\) 1.74776 0.0894232
\(383\) 11.5146 0.588367 0.294183 0.955749i \(-0.404952\pi\)
0.294183 + 0.955749i \(0.404952\pi\)
\(384\) −2.26409 −0.115539
\(385\) 0 0
\(386\) −7.49552 −0.381512
\(387\) −25.9089 −1.31702
\(388\) −4.13545 −0.209946
\(389\) 0.756717 0.0383671 0.0191835 0.999816i \(-0.493893\pi\)
0.0191835 + 0.999816i \(0.493893\pi\)
\(390\) −43.8961 −2.22276
\(391\) −40.7402 −2.06032
\(392\) 0 0
\(393\) 21.4733 1.08318
\(394\) 22.6561 1.14140
\(395\) −40.9253 −2.05917
\(396\) −4.94967 −0.248730
\(397\) 27.4966 1.38001 0.690007 0.723803i \(-0.257607\pi\)
0.690007 + 0.723803i \(0.257607\pi\)
\(398\) 17.5417 0.879284
\(399\) 0 0
\(400\) 8.52994 0.426497
\(401\) 11.9242 0.595467 0.297733 0.954649i \(-0.403769\pi\)
0.297733 + 0.954649i \(0.403769\pi\)
\(402\) 5.09928 0.254329
\(403\) −10.4287 −0.519490
\(404\) −3.75624 −0.186880
\(405\) 39.9390 1.98458
\(406\) 0 0
\(407\) 4.65606 0.230792
\(408\) −10.6561 −0.527554
\(409\) −36.7471 −1.81703 −0.908513 0.417858i \(-0.862781\pi\)
−0.908513 + 0.417858i \(0.862781\pi\)
\(410\) 38.1198 1.88260
\(411\) −29.7967 −1.46976
\(412\) −8.48528 −0.418040
\(413\) 0 0
\(414\) −18.4038 −0.904499
\(415\) −38.1198 −1.87123
\(416\) −5.27088 −0.258426
\(417\) −19.6153 −0.960565
\(418\) 13.8341 0.676648
\(419\) 1.16468 0.0568983 0.0284492 0.999595i \(-0.490943\pi\)
0.0284492 + 0.999595i \(0.490943\pi\)
\(420\) 0 0
\(421\) −25.4128 −1.23854 −0.619272 0.785177i \(-0.712572\pi\)
−0.619272 + 0.785177i \(0.712572\pi\)
\(422\) −16.4796 −0.802215
\(423\) 4.20663 0.204533
\(424\) 6.32803 0.307316
\(425\) 40.1466 1.94740
\(426\) −18.1127 −0.877566
\(427\) 0 0
\(428\) 4.65606 0.225059
\(429\) −27.7822 −1.34134
\(430\) 44.8239 2.16160
\(431\) 30.4038 1.46450 0.732250 0.681036i \(-0.238470\pi\)
0.732250 + 0.681036i \(0.238470\pi\)
\(432\) 1.97855 0.0951929
\(433\) 14.5349 0.698501 0.349251 0.937029i \(-0.386436\pi\)
0.349251 + 0.937029i \(0.386436\pi\)
\(434\) 0 0
\(435\) −8.32803 −0.399299
\(436\) −10.9841 −0.526043
\(437\) 51.4378 2.46060
\(438\) −16.2522 −0.776562
\(439\) −1.91413 −0.0913566 −0.0456783 0.998956i \(-0.514545\pi\)
−0.0456783 + 0.998956i \(0.514545\pi\)
\(440\) 8.56321 0.408235
\(441\) 0 0
\(442\) −24.8076 −1.17998
\(443\) −37.1797 −1.76646 −0.883229 0.468941i \(-0.844636\pi\)
−0.883229 + 0.468941i \(0.844636\pi\)
\(444\) 4.52819 0.214898
\(445\) 21.4637 1.01748
\(446\) 21.0835 0.998334
\(447\) 11.6060 0.548946
\(448\) 0 0
\(449\) −36.2714 −1.71175 −0.855876 0.517182i \(-0.826981\pi\)
−0.855876 + 0.517182i \(0.826981\pi\)
\(450\) 18.1357 0.854924
\(451\) 24.1263 1.13606
\(452\) 20.8076 0.978709
\(453\) 34.0970 1.60202
\(454\) −24.3047 −1.14068
\(455\) 0 0
\(456\) 13.4542 0.630048
\(457\) −16.9592 −0.793319 −0.396660 0.917966i \(-0.629831\pi\)
−0.396660 + 0.917966i \(0.629831\pi\)
\(458\) −12.0857 −0.564726
\(459\) 9.31213 0.434653
\(460\) 31.8397 1.48453
\(461\) −24.6974 −1.15027 −0.575137 0.818057i \(-0.695051\pi\)
−0.575137 + 0.818057i \(0.695051\pi\)
\(462\) 0 0
\(463\) −0.807647 −0.0375345 −0.0187673 0.999824i \(-0.505974\pi\)
−0.0187673 + 0.999824i \(0.505974\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 16.4774 0.764121
\(466\) 7.63060 0.353481
\(467\) −27.4254 −1.26910 −0.634548 0.772883i \(-0.718814\pi\)
−0.634548 + 0.772883i \(0.718814\pi\)
\(468\) −11.2065 −0.518022
\(469\) 0 0
\(470\) −7.27770 −0.335695
\(471\) −18.4038 −0.848003
\(472\) 7.17825 0.330406
\(473\) 28.3694 1.30443
\(474\) −25.1906 −1.15704
\(475\) −50.6883 −2.32574
\(476\) 0 0
\(477\) 13.4542 0.616023
\(478\) 0.504478 0.0230743
\(479\) 18.1420 0.828929 0.414465 0.910065i \(-0.363969\pi\)
0.414465 + 0.910065i \(0.363969\pi\)
\(480\) 8.32803 0.380121
\(481\) 10.5418 0.480663
\(482\) 21.8622 0.995797
\(483\) 0 0
\(484\) −5.58027 −0.253649
\(485\) 15.2115 0.690717
\(486\) 18.6478 0.845883
\(487\) −17.8993 −0.811097 −0.405548 0.914074i \(-0.632919\pi\)
−0.405548 + 0.914074i \(0.632919\pi\)
\(488\) 11.1129 0.503055
\(489\) −13.1851 −0.596249
\(490\) 0 0
\(491\) 10.1446 0.457821 0.228910 0.973447i \(-0.426484\pi\)
0.228910 + 0.973447i \(0.426484\pi\)
\(492\) 23.4637 1.05783
\(493\) −4.70655 −0.211972
\(494\) 31.3217 1.40923
\(495\) 18.2064 0.818318
\(496\) 1.97855 0.0888394
\(497\) 0 0
\(498\) −23.4637 −1.05143
\(499\) −24.1516 −1.08117 −0.540587 0.841288i \(-0.681798\pi\)
−0.540587 + 0.841288i \(0.681798\pi\)
\(500\) −12.9842 −0.580672
\(501\) −14.1007 −0.629971
\(502\) −9.87024 −0.440530
\(503\) −33.9254 −1.51266 −0.756329 0.654191i \(-0.773009\pi\)
−0.756329 + 0.654191i \(0.773009\pi\)
\(504\) 0 0
\(505\) 13.8166 0.614831
\(506\) 20.1516 0.895847
\(507\) −33.4682 −1.48638
\(508\) 2.25224 0.0999269
\(509\) 39.1183 1.73389 0.866945 0.498404i \(-0.166080\pi\)
0.866945 + 0.498404i \(0.166080\pi\)
\(510\) 39.1963 1.73564
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −11.7573 −0.519098
\(514\) −11.6772 −0.515059
\(515\) 31.2115 1.37534
\(516\) 27.5903 1.21459
\(517\) −4.60612 −0.202577
\(518\) 0 0
\(519\) 5.35350 0.234992
\(520\) 19.3879 0.850216
\(521\) −14.7200 −0.644894 −0.322447 0.946587i \(-0.604505\pi\)
−0.322447 + 0.946587i \(0.604505\pi\)
\(522\) −2.12612 −0.0930577
\(523\) −38.8035 −1.69676 −0.848380 0.529388i \(-0.822422\pi\)
−0.848380 + 0.529388i \(0.822422\pi\)
\(524\) −9.48427 −0.414322
\(525\) 0 0
\(526\) −17.6815 −0.770951
\(527\) 9.31213 0.405643
\(528\) 5.27088 0.229386
\(529\) 51.9274 2.25771
\(530\) −23.2764 −1.01106
\(531\) 15.2618 0.662307
\(532\) 0 0
\(533\) 54.6243 2.36604
\(534\) 13.2115 0.571716
\(535\) −17.1264 −0.740440
\(536\) −2.25224 −0.0972819
\(537\) 19.2549 0.830912
\(538\) 4.32733 0.186565
\(539\) 0 0
\(540\) −7.27770 −0.313183
\(541\) 16.7758 0.721249 0.360625 0.932711i \(-0.382564\pi\)
0.360625 + 0.932711i \(0.382564\pi\)
\(542\) −24.8690 −1.06822
\(543\) −18.0949 −0.776527
\(544\) 4.70655 0.201792
\(545\) 40.4029 1.73067
\(546\) 0 0
\(547\) −19.8993 −0.850835 −0.425417 0.904997i \(-0.639873\pi\)
−0.425417 + 0.904997i \(0.639873\pi\)
\(548\) 13.1605 0.562190
\(549\) 23.6273 1.00839
\(550\) −19.8580 −0.846747
\(551\) 5.94240 0.253155
\(552\) 19.5981 0.834152
\(553\) 0 0
\(554\) −5.64711 −0.239923
\(555\) −16.6561 −0.707010
\(556\) 8.66364 0.367420
\(557\) 40.1198 1.69993 0.849965 0.526840i \(-0.176623\pi\)
0.849965 + 0.526840i \(0.176623\pi\)
\(558\) 4.20663 0.178081
\(559\) 64.2310 2.71668
\(560\) 0 0
\(561\) 24.8076 1.04738
\(562\) 14.2274 0.600146
\(563\) 39.1895 1.65164 0.825820 0.563933i \(-0.190712\pi\)
0.825820 + 0.563933i \(0.190712\pi\)
\(564\) −4.47962 −0.188626
\(565\) −76.5369 −3.21993
\(566\) −18.2911 −0.768833
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 41.2115 1.72767 0.863837 0.503771i \(-0.168054\pi\)
0.863837 + 0.503771i \(0.168054\pi\)
\(570\) −49.4885 −2.07285
\(571\) −33.4128 −1.39828 −0.699141 0.714984i \(-0.746434\pi\)
−0.699141 + 0.714984i \(0.746434\pi\)
\(572\) 12.2708 0.513067
\(573\) −3.95709 −0.165310
\(574\) 0 0
\(575\) −73.8357 −3.07916
\(576\) 2.12612 0.0885883
\(577\) 32.0765 1.33536 0.667682 0.744447i \(-0.267287\pi\)
0.667682 + 0.744447i \(0.267287\pi\)
\(578\) 5.15158 0.214278
\(579\) 16.9706 0.705273
\(580\) 3.67831 0.152733
\(581\) 0 0
\(582\) 9.36306 0.388111
\(583\) −14.7319 −0.610131
\(584\) 7.17825 0.297038
\(585\) 41.2210 1.70428
\(586\) 7.71333 0.318635
\(587\) 28.2033 1.16407 0.582037 0.813162i \(-0.302256\pi\)
0.582037 + 0.813162i \(0.302256\pi\)
\(588\) 0 0
\(589\) −11.7573 −0.484452
\(590\) −26.4038 −1.08703
\(591\) −51.2954 −2.11001
\(592\) −2.00000 −0.0821995
\(593\) 42.5890 1.74892 0.874461 0.485096i \(-0.161215\pi\)
0.874461 + 0.485096i \(0.161215\pi\)
\(594\) −4.60612 −0.188991
\(595\) 0 0
\(596\) −5.12612 −0.209974
\(597\) −39.7159 −1.62547
\(598\) 45.6251 1.86575
\(599\) −10.3185 −0.421601 −0.210801 0.977529i \(-0.567607\pi\)
−0.210801 + 0.977529i \(0.567607\pi\)
\(600\) −19.3126 −0.788433
\(601\) 34.2754 1.39812 0.699060 0.715063i \(-0.253602\pi\)
0.699060 + 0.715063i \(0.253602\pi\)
\(602\) 0 0
\(603\) −4.78853 −0.195004
\(604\) −15.0599 −0.612778
\(605\) 20.5259 0.834498
\(606\) 8.50448 0.345471
\(607\) 11.6997 0.474875 0.237438 0.971403i \(-0.423692\pi\)
0.237438 + 0.971403i \(0.423692\pi\)
\(608\) −5.94240 −0.240996
\(609\) 0 0
\(610\) −40.8765 −1.65504
\(611\) −10.4287 −0.421899
\(612\) 10.0067 0.404496
\(613\) −14.1351 −0.570910 −0.285455 0.958392i \(-0.592145\pi\)
−0.285455 + 0.958392i \(0.592145\pi\)
\(614\) −30.5034 −1.23102
\(615\) −86.3067 −3.48022
\(616\) 0 0
\(617\) −21.0599 −0.847839 −0.423920 0.905700i \(-0.639346\pi\)
−0.423920 + 0.905700i \(0.639346\pi\)
\(618\) 19.2115 0.772799
\(619\) 10.0261 0.402983 0.201492 0.979490i \(-0.435421\pi\)
0.201492 + 0.979490i \(0.435421\pi\)
\(620\) −7.27770 −0.292280
\(621\) −17.1264 −0.687260
\(622\) −6.32162 −0.253474
\(623\) 0 0
\(624\) 11.9338 0.477733
\(625\) 5.11021 0.204409
\(626\) −30.5970 −1.22290
\(627\) −31.3217 −1.25087
\(628\) 8.12856 0.324365
\(629\) −9.41309 −0.375325
\(630\) 0 0
\(631\) −14.4038 −0.573407 −0.286704 0.958019i \(-0.592559\pi\)
−0.286704 + 0.958019i \(0.592559\pi\)
\(632\) 11.1261 0.442573
\(633\) 37.3114 1.48299
\(634\) 33.2115 1.31900
\(635\) −8.28443 −0.328757
\(636\) −14.3273 −0.568113
\(637\) 0 0
\(638\) 2.32803 0.0921676
\(639\) 17.0090 0.672864
\(640\) −3.67831 −0.145398
\(641\) −40.2714 −1.59062 −0.795311 0.606201i \(-0.792693\pi\)
−0.795311 + 0.606201i \(0.792693\pi\)
\(642\) −10.5418 −0.416050
\(643\) 2.23484 0.0881335 0.0440667 0.999029i \(-0.485969\pi\)
0.0440667 + 0.999029i \(0.485969\pi\)
\(644\) 0 0
\(645\) −101.485 −3.99599
\(646\) −27.9682 −1.10039
\(647\) 11.1714 0.439192 0.219596 0.975591i \(-0.429526\pi\)
0.219596 + 0.975591i \(0.429526\pi\)
\(648\) −10.8580 −0.426542
\(649\) −16.7112 −0.655972
\(650\) −44.9603 −1.76349
\(651\) 0 0
\(652\) 5.82355 0.228068
\(653\) 26.1516 1.02339 0.511695 0.859167i \(-0.329018\pi\)
0.511695 + 0.859167i \(0.329018\pi\)
\(654\) 24.8690 0.972455
\(655\) 34.8860 1.36311
\(656\) −10.3634 −0.404623
\(657\) 15.2618 0.595421
\(658\) 0 0
\(659\) 21.9751 0.856030 0.428015 0.903772i \(-0.359213\pi\)
0.428015 + 0.903772i \(0.359213\pi\)
\(660\) −19.3879 −0.754674
\(661\) −16.1049 −0.626410 −0.313205 0.949686i \(-0.601403\pi\)
−0.313205 + 0.949686i \(0.601403\pi\)
\(662\) 11.2459 0.437084
\(663\) 56.1668 2.18134
\(664\) 10.3634 0.402178
\(665\) 0 0
\(666\) −4.25224 −0.164771
\(667\) 8.65606 0.335164
\(668\) 6.22795 0.240967
\(669\) −47.7351 −1.84554
\(670\) 8.28443 0.320055
\(671\) −25.8711 −0.998742
\(672\) 0 0
\(673\) 43.5395 1.67832 0.839162 0.543881i \(-0.183046\pi\)
0.839162 + 0.543881i \(0.183046\pi\)
\(674\) 10.0000 0.385186
\(675\) 16.8769 0.649592
\(676\) 14.7822 0.568545
\(677\) 12.0271 0.462241 0.231120 0.972925i \(-0.425761\pi\)
0.231120 + 0.972925i \(0.425761\pi\)
\(678\) −47.1105 −1.80927
\(679\) 0 0
\(680\) −17.3121 −0.663890
\(681\) 55.0281 2.10868
\(682\) −4.60612 −0.176377
\(683\) −11.2115 −0.428995 −0.214498 0.976725i \(-0.568811\pi\)
−0.214498 + 0.976725i \(0.568811\pi\)
\(684\) −12.6343 −0.483083
\(685\) −48.4085 −1.84959
\(686\) 0 0
\(687\) 27.3631 1.04397
\(688\) −12.1860 −0.464587
\(689\) −33.3543 −1.27070
\(690\) −72.0880 −2.74434
\(691\) 17.0904 0.650150 0.325075 0.945688i \(-0.394610\pi\)
0.325075 + 0.945688i \(0.394610\pi\)
\(692\) −2.36452 −0.0898856
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) −31.8675 −1.20880
\(696\) 2.26409 0.0858202
\(697\) −48.7758 −1.84752
\(698\) 28.3622 1.07353
\(699\) −17.2764 −0.653453
\(700\) 0 0
\(701\) −36.4892 −1.37818 −0.689089 0.724677i \(-0.741989\pi\)
−0.689089 + 0.724677i \(0.741989\pi\)
\(702\) −10.4287 −0.393605
\(703\) 11.8848 0.448244
\(704\) −2.32803 −0.0877410
\(705\) 16.4774 0.620575
\(706\) 5.74155 0.216086
\(707\) 0 0
\(708\) −16.2522 −0.610797
\(709\) 19.9433 0.748987 0.374494 0.927229i \(-0.377817\pi\)
0.374494 + 0.927229i \(0.377817\pi\)
\(710\) −29.4265 −1.10436
\(711\) 23.6555 0.887149
\(712\) −5.83521 −0.218684
\(713\) −17.1264 −0.641390
\(714\) 0 0
\(715\) −45.1357 −1.68798
\(716\) −8.50448 −0.317827
\(717\) −1.14219 −0.0426557
\(718\) 16.4382 0.613470
\(719\) 33.2412 1.23969 0.619844 0.784725i \(-0.287196\pi\)
0.619844 + 0.784725i \(0.287196\pi\)
\(720\) −7.82052 −0.291454
\(721\) 0 0
\(722\) 16.3121 0.607074
\(723\) −49.4981 −1.84086
\(724\) 7.99212 0.297025
\(725\) −8.52994 −0.316794
\(726\) 12.6343 0.468901
\(727\) −36.3192 −1.34700 −0.673502 0.739186i \(-0.735211\pi\)
−0.673502 + 0.739186i \(0.735211\pi\)
\(728\) 0 0
\(729\) −9.64650 −0.357278
\(730\) −26.4038 −0.977249
\(731\) −57.3540 −2.12131
\(732\) −25.1605 −0.929960
\(733\) −13.1558 −0.485921 −0.242961 0.970036i \(-0.578119\pi\)
−0.242961 + 0.970036i \(0.578119\pi\)
\(734\) −14.0485 −0.518538
\(735\) 0 0
\(736\) −8.65606 −0.319067
\(737\) 5.24328 0.193139
\(738\) −22.0338 −0.811076
\(739\) 14.7414 0.542272 0.271136 0.962541i \(-0.412601\pi\)
0.271136 + 0.962541i \(0.412601\pi\)
\(740\) 7.35661 0.270435
\(741\) −70.9152 −2.60514
\(742\) 0 0
\(743\) 50.9274 1.86835 0.934173 0.356821i \(-0.116139\pi\)
0.934173 + 0.356821i \(0.116139\pi\)
\(744\) −4.47962 −0.164231
\(745\) 18.8554 0.690810
\(746\) −17.4981 −0.640652
\(747\) 22.0338 0.806176
\(748\) −10.9570 −0.400627
\(749\) 0 0
\(750\) 29.3975 1.07344
\(751\) 10.1198 0.369276 0.184638 0.982807i \(-0.440889\pi\)
0.184638 + 0.982807i \(0.440889\pi\)
\(752\) 1.97855 0.0721502
\(753\) 22.3471 0.814375
\(754\) 5.27088 0.191954
\(755\) 55.3949 2.01603
\(756\) 0 0
\(757\) 41.3631 1.50337 0.751683 0.659525i \(-0.229242\pi\)
0.751683 + 0.659525i \(0.229242\pi\)
\(758\) −7.89935 −0.286917
\(759\) −45.6251 −1.65609
\(760\) 21.8580 0.792872
\(761\) −24.2110 −0.877649 −0.438824 0.898573i \(-0.644605\pi\)
−0.438824 + 0.898573i \(0.644605\pi\)
\(762\) −5.09928 −0.184727
\(763\) 0 0
\(764\) 1.74776 0.0632318
\(765\) −36.8076 −1.33078
\(766\) 11.5146 0.416038
\(767\) −37.8357 −1.36617
\(768\) −2.26409 −0.0816984
\(769\) −3.99311 −0.143995 −0.0719976 0.997405i \(-0.522937\pi\)
−0.0719976 + 0.997405i \(0.522937\pi\)
\(770\) 0 0
\(771\) 26.4382 0.952150
\(772\) −7.49552 −0.269770
\(773\) 2.42672 0.0872830 0.0436415 0.999047i \(-0.486104\pi\)
0.0436415 + 0.999047i \(0.486104\pi\)
\(774\) −25.9089 −0.931276
\(775\) 16.8769 0.606236
\(776\) −4.13545 −0.148454
\(777\) 0 0
\(778\) 0.756717 0.0271296
\(779\) 61.5835 2.20646
\(780\) −43.8961 −1.57173
\(781\) −18.6243 −0.666428
\(782\) −40.7402 −1.45687
\(783\) −1.97855 −0.0707075
\(784\) 0 0
\(785\) −29.8993 −1.06715
\(786\) 21.4733 0.765926
\(787\) 50.2731 1.79204 0.896021 0.444011i \(-0.146445\pi\)
0.896021 + 0.444011i \(0.146445\pi\)
\(788\) 22.6561 0.807089
\(789\) 40.0326 1.42520
\(790\) −40.9253 −1.45606
\(791\) 0 0
\(792\) −4.94967 −0.175879
\(793\) −58.5745 −2.08004
\(794\) 27.4966 0.975817
\(795\) 52.7000 1.86908
\(796\) 17.5417 0.621747
\(797\) 1.64125 0.0581361 0.0290681 0.999577i \(-0.490746\pi\)
0.0290681 + 0.999577i \(0.490746\pi\)
\(798\) 0 0
\(799\) 9.31213 0.329439
\(800\) 8.52994 0.301579
\(801\) −12.4064 −0.438357
\(802\) 11.9242 0.421058
\(803\) −16.7112 −0.589725
\(804\) 5.09928 0.179838
\(805\) 0 0
\(806\) −10.4287 −0.367335
\(807\) −9.79749 −0.344888
\(808\) −3.75624 −0.132144
\(809\) −3.34394 −0.117567 −0.0587833 0.998271i \(-0.518722\pi\)
−0.0587833 + 0.998271i \(0.518722\pi\)
\(810\) 39.9390 1.40331
\(811\) 3.00679 0.105583 0.0527913 0.998606i \(-0.483188\pi\)
0.0527913 + 0.998606i \(0.483188\pi\)
\(812\) 0 0
\(813\) 56.3058 1.97473
\(814\) 4.65606 0.163195
\(815\) −21.4208 −0.750338
\(816\) −10.6561 −0.373037
\(817\) 72.4141 2.53345
\(818\) −36.7471 −1.28483
\(819\) 0 0
\(820\) 38.1198 1.33120
\(821\) 39.9433 1.39403 0.697016 0.717056i \(-0.254511\pi\)
0.697016 + 0.717056i \(0.254511\pi\)
\(822\) −29.7967 −1.03928
\(823\) 17.0090 0.592895 0.296447 0.955049i \(-0.404198\pi\)
0.296447 + 0.955049i \(0.404198\pi\)
\(824\) −8.48528 −0.295599
\(825\) 44.9603 1.56532
\(826\) 0 0
\(827\) 33.0943 1.15080 0.575401 0.817872i \(-0.304846\pi\)
0.575401 + 0.817872i \(0.304846\pi\)
\(828\) −18.4038 −0.639577
\(829\) 7.15576 0.248530 0.124265 0.992249i \(-0.460343\pi\)
0.124265 + 0.992249i \(0.460343\pi\)
\(830\) −38.1198 −1.32316
\(831\) 12.7856 0.443527
\(832\) −5.27088 −0.182735
\(833\) 0 0
\(834\) −19.6153 −0.679222
\(835\) −22.9083 −0.792774
\(836\) 13.8341 0.478462
\(837\) 3.91465 0.135310
\(838\) 1.16468 0.0402332
\(839\) −0.684186 −0.0236207 −0.0118104 0.999930i \(-0.503759\pi\)
−0.0118104 + 0.999930i \(0.503759\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −25.4128 −0.875782
\(843\) −32.2121 −1.10944
\(844\) −16.4796 −0.567252
\(845\) −54.3734 −1.87050
\(846\) 4.20663 0.144627
\(847\) 0 0
\(848\) 6.32803 0.217305
\(849\) 41.4128 1.42128
\(850\) 40.1466 1.37702
\(851\) 17.3121 0.593452
\(852\) −18.1127 −0.620533
\(853\) 32.6701 1.11860 0.559302 0.828964i \(-0.311069\pi\)
0.559302 + 0.828964i \(0.311069\pi\)
\(854\) 0 0
\(855\) 46.4727 1.58933
\(856\) 4.65606 0.159141
\(857\) −18.0475 −0.616490 −0.308245 0.951307i \(-0.599742\pi\)
−0.308245 + 0.951307i \(0.599742\pi\)
\(858\) −27.7822 −0.948468
\(859\) −32.4490 −1.10714 −0.553572 0.832801i \(-0.686736\pi\)
−0.553572 + 0.832801i \(0.686736\pi\)
\(860\) 44.8239 1.52848
\(861\) 0 0
\(862\) 30.4038 1.03556
\(863\) 0.132467 0.00450922 0.00225461 0.999997i \(-0.499282\pi\)
0.00225461 + 0.999997i \(0.499282\pi\)
\(864\) 1.97855 0.0673115
\(865\) 8.69743 0.295722
\(866\) 14.5349 0.493915
\(867\) −11.6637 −0.396119
\(868\) 0 0
\(869\) −25.9020 −0.878664
\(870\) −8.32803 −0.282347
\(871\) 11.8713 0.402243
\(872\) −10.9841 −0.371968
\(873\) −8.79247 −0.297580
\(874\) 51.4378 1.73991
\(875\) 0 0
\(876\) −16.2522 −0.549112
\(877\) 1.56176 0.0527367 0.0263684 0.999652i \(-0.491606\pi\)
0.0263684 + 0.999652i \(0.491606\pi\)
\(878\) −1.91413 −0.0645989
\(879\) −17.4637 −0.589036
\(880\) 8.56321 0.288666
\(881\) −10.0202 −0.337589 −0.168795 0.985651i \(-0.553987\pi\)
−0.168795 + 0.985651i \(0.553987\pi\)
\(882\) 0 0
\(883\) −5.96819 −0.200846 −0.100423 0.994945i \(-0.532020\pi\)
−0.100423 + 0.994945i \(0.532020\pi\)
\(884\) −24.8076 −0.834371
\(885\) 59.7807 2.00951
\(886\) −37.1797 −1.24908
\(887\) −13.4481 −0.451544 −0.225772 0.974180i \(-0.572490\pi\)
−0.225772 + 0.974180i \(0.572490\pi\)
\(888\) 4.52819 0.151956
\(889\) 0 0
\(890\) 21.4637 0.719465
\(891\) 25.2777 0.846835
\(892\) 21.0835 0.705929
\(893\) −11.7573 −0.393444
\(894\) 11.6060 0.388163
\(895\) 31.2821 1.04564
\(896\) 0 0
\(897\) −103.299 −3.44907
\(898\) −36.2714 −1.21039
\(899\) −1.97855 −0.0659882
\(900\) 18.1357 0.604523
\(901\) 29.7832 0.992221
\(902\) 24.1263 0.803319
\(903\) 0 0
\(904\) 20.8076 0.692052
\(905\) −29.3975 −0.977205
\(906\) 34.0970 1.13280
\(907\) 33.6841 1.11846 0.559232 0.829011i \(-0.311096\pi\)
0.559232 + 0.829011i \(0.311096\pi\)
\(908\) −24.3047 −0.806579
\(909\) −7.98621 −0.264886
\(910\) 0 0
\(911\) −22.5580 −0.747380 −0.373690 0.927554i \(-0.621908\pi\)
−0.373690 + 0.927554i \(0.621908\pi\)
\(912\) 13.4542 0.445511
\(913\) −24.1263 −0.798465
\(914\) −16.9592 −0.560962
\(915\) 92.5482 3.05955
\(916\) −12.0857 −0.399321
\(917\) 0 0
\(918\) 9.31213 0.307346
\(919\) −24.0688 −0.793958 −0.396979 0.917828i \(-0.629941\pi\)
−0.396979 + 0.917828i \(0.629941\pi\)
\(920\) 31.8397 1.04972
\(921\) 69.0625 2.27569
\(922\) −24.6974 −0.813366
\(923\) −42.1670 −1.38795
\(924\) 0 0
\(925\) −17.0599 −0.560926
\(926\) −0.807647 −0.0265409
\(927\) −18.0407 −0.592535
\(928\) −1.00000 −0.0328266
\(929\) 39.9944 1.31217 0.656087 0.754685i \(-0.272210\pi\)
0.656087 + 0.754685i \(0.272210\pi\)
\(930\) 16.4774 0.540315
\(931\) 0 0
\(932\) 7.63060 0.249949
\(933\) 14.3127 0.468578
\(934\) −27.4254 −0.897387
\(935\) 40.3032 1.31805
\(936\) −11.2065 −0.366297
\(937\) 49.8812 1.62955 0.814774 0.579778i \(-0.196861\pi\)
0.814774 + 0.579778i \(0.196861\pi\)
\(938\) 0 0
\(939\) 69.2746 2.26069
\(940\) −7.27770 −0.237373
\(941\) −13.1401 −0.428354 −0.214177 0.976795i \(-0.568707\pi\)
−0.214177 + 0.976795i \(0.568707\pi\)
\(942\) −18.4038 −0.599629
\(943\) 89.7062 2.92124
\(944\) 7.17825 0.233632
\(945\) 0 0
\(946\) 28.3694 0.922369
\(947\) −27.2459 −0.885373 −0.442686 0.896677i \(-0.645974\pi\)
−0.442686 + 0.896677i \(0.645974\pi\)
\(948\) −25.1906 −0.818152
\(949\) −37.8357 −1.22820
\(950\) −50.6883 −1.64455
\(951\) −75.1939 −2.43833
\(952\) 0 0
\(953\) 40.3376 1.30666 0.653331 0.757072i \(-0.273371\pi\)
0.653331 + 0.757072i \(0.273371\pi\)
\(954\) 13.4542 0.435594
\(955\) −6.42880 −0.208031
\(956\) 0.504478 0.0163160
\(957\) −5.27088 −0.170383
\(958\) 18.1420 0.586141
\(959\) 0 0
\(960\) 8.32803 0.268786
\(961\) −27.0854 −0.873721
\(962\) 10.5418 0.339880
\(963\) 9.89935 0.319002
\(964\) 21.8622 0.704135
\(965\) 27.5708 0.887536
\(966\) 0 0
\(967\) 47.3879 1.52389 0.761946 0.647640i \(-0.224244\pi\)
0.761946 + 0.647640i \(0.224244\pi\)
\(968\) −5.58027 −0.179357
\(969\) 63.3226 2.03421
\(970\) 15.2115 0.488411
\(971\) −9.55629 −0.306676 −0.153338 0.988174i \(-0.549002\pi\)
−0.153338 + 0.988174i \(0.549002\pi\)
\(972\) 18.6478 0.598129
\(973\) 0 0
\(974\) −17.8993 −0.573532
\(975\) 101.794 3.26003
\(976\) 11.1129 0.355714
\(977\) 12.5485 0.401461 0.200730 0.979647i \(-0.435668\pi\)
0.200730 + 0.979647i \(0.435668\pi\)
\(978\) −13.1851 −0.421612
\(979\) 13.5846 0.434164
\(980\) 0 0
\(981\) −23.3535 −0.745620
\(982\) 10.1446 0.323728
\(983\) −1.51464 −0.0483095 −0.0241548 0.999708i \(-0.507689\pi\)
−0.0241548 + 0.999708i \(0.507689\pi\)
\(984\) 23.4637 0.747996
\(985\) −83.3360 −2.65530
\(986\) −4.70655 −0.149887
\(987\) 0 0
\(988\) 31.3217 0.996476
\(989\) 105.483 3.35416
\(990\) 18.2064 0.578638
\(991\) −2.53629 −0.0805679 −0.0402840 0.999188i \(-0.512826\pi\)
−0.0402840 + 0.999188i \(0.512826\pi\)
\(992\) 1.97855 0.0628189
\(993\) −25.4618 −0.808004
\(994\) 0 0
\(995\) −64.5236 −2.04554
\(996\) −23.4637 −0.743476
\(997\) 29.0112 0.918795 0.459397 0.888231i \(-0.348065\pi\)
0.459397 + 0.888231i \(0.348065\pi\)
\(998\) −24.1516 −0.764505
\(999\) −3.95709 −0.125197
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 2842.2.a.bb.1.2 6
7.6 odd 2 inner 2842.2.a.bb.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.bb.1.2 6 1.1 even 1 trivial
2842.2.a.bb.1.5 yes 6 7.6 odd 2 inner