Properties

Label 2679.2.a.n.1.23
Level $2679$
Weight $2$
Character 2679.1
Self dual yes
Analytic conductor $21.392$
Analytic rank $0$
Dimension $23$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2679,2,Mod(1,2679)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2679, 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("2679.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2679 = 3 \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2679.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: \(21.3919227015\)
Analytic rank: \(0\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.23
Character \(\chi\) \(=\) 2679.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.70026 q^{2} +1.00000 q^{3} +5.29141 q^{4} +2.20508 q^{5} +2.70026 q^{6} +0.381908 q^{7} +8.88768 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.70026 q^{2} +1.00000 q^{3} +5.29141 q^{4} +2.20508 q^{5} +2.70026 q^{6} +0.381908 q^{7} +8.88768 q^{8} +1.00000 q^{9} +5.95430 q^{10} -0.541461 q^{11} +5.29141 q^{12} -4.39015 q^{13} +1.03125 q^{14} +2.20508 q^{15} +13.4162 q^{16} -7.56447 q^{17} +2.70026 q^{18} +1.00000 q^{19} +11.6680 q^{20} +0.381908 q^{21} -1.46209 q^{22} +4.83305 q^{23} +8.88768 q^{24} -0.137605 q^{25} -11.8545 q^{26} +1.00000 q^{27} +2.02083 q^{28} -2.92584 q^{29} +5.95430 q^{30} +10.0216 q^{31} +18.4520 q^{32} -0.541461 q^{33} -20.4261 q^{34} +0.842139 q^{35} +5.29141 q^{36} -6.82713 q^{37} +2.70026 q^{38} -4.39015 q^{39} +19.5981 q^{40} -1.66489 q^{41} +1.03125 q^{42} -8.47623 q^{43} -2.86509 q^{44} +2.20508 q^{45} +13.0505 q^{46} +1.00000 q^{47} +13.4162 q^{48} -6.85415 q^{49} -0.371569 q^{50} -7.56447 q^{51} -23.2301 q^{52} +5.12841 q^{53} +2.70026 q^{54} -1.19397 q^{55} +3.39428 q^{56} +1.00000 q^{57} -7.90055 q^{58} -6.38720 q^{59} +11.6680 q^{60} +9.65918 q^{61} +27.0609 q^{62} +0.381908 q^{63} +22.9928 q^{64} -9.68064 q^{65} -1.46209 q^{66} +6.65611 q^{67} -40.0268 q^{68} +4.83305 q^{69} +2.27400 q^{70} +8.57056 q^{71} +8.88768 q^{72} -8.65829 q^{73} -18.4350 q^{74} -0.137605 q^{75} +5.29141 q^{76} -0.206788 q^{77} -11.8545 q^{78} -2.49612 q^{79} +29.5839 q^{80} +1.00000 q^{81} -4.49563 q^{82} +5.42636 q^{83} +2.02083 q^{84} -16.6803 q^{85} -22.8880 q^{86} -2.92584 q^{87} -4.81233 q^{88} -12.6622 q^{89} +5.95430 q^{90} -1.67663 q^{91} +25.5737 q^{92} +10.0216 q^{93} +2.70026 q^{94} +2.20508 q^{95} +18.4520 q^{96} -8.67832 q^{97} -18.5080 q^{98} -0.541461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23 q + 5 q^{2} + 23 q^{3} + 25 q^{4} + 12 q^{5} + 5 q^{6} - 2 q^{7} + 12 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 23 q + 5 q^{2} + 23 q^{3} + 25 q^{4} + 12 q^{5} + 5 q^{6} - 2 q^{7} + 12 q^{8} + 23 q^{9} + q^{10} + 13 q^{11} + 25 q^{12} + 5 q^{13} + 11 q^{14} + 12 q^{15} + 33 q^{16} + 22 q^{17} + 5 q^{18} + 23 q^{19} + 23 q^{20} - 2 q^{21} + q^{22} + 23 q^{23} + 12 q^{24} + 31 q^{25} + 21 q^{26} + 23 q^{27} - 22 q^{28} + 24 q^{29} + q^{30} + 20 q^{31} + 19 q^{32} + 13 q^{33} - 14 q^{34} + 15 q^{35} + 25 q^{36} + 8 q^{37} + 5 q^{38} + 5 q^{39} - 15 q^{40} + 28 q^{41} + 11 q^{42} - 6 q^{43} + q^{44} + 12 q^{45} + 29 q^{46} + 23 q^{47} + 33 q^{48} + 39 q^{49} + 27 q^{50} + 22 q^{51} + 12 q^{52} + 22 q^{53} + 5 q^{54} - 28 q^{55} + 30 q^{56} + 23 q^{57} - 5 q^{58} + 53 q^{59} + 23 q^{60} + 4 q^{61} + 38 q^{62} - 2 q^{63} + 30 q^{64} + 27 q^{65} + q^{66} - 6 q^{68} + 23 q^{69} + 30 q^{70} + 54 q^{71} + 12 q^{72} + 15 q^{73} - 12 q^{74} + 31 q^{75} + 25 q^{76} + 33 q^{77} + 21 q^{78} - 8 q^{79} + 45 q^{80} + 23 q^{81} - 69 q^{82} + 20 q^{83} - 22 q^{84} - 25 q^{85} + 50 q^{86} + 24 q^{87} + 48 q^{88} + 26 q^{89} + q^{90} + 11 q^{91} + 81 q^{92} + 20 q^{93} + 5 q^{94} + 12 q^{95} + 19 q^{96} - 4 q^{97} - 55 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70026 1.90937 0.954687 0.297612i \(-0.0961903\pi\)
0.954687 + 0.297612i \(0.0961903\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.29141 2.64571
\(5\) 2.20508 0.986144 0.493072 0.869989i \(-0.335874\pi\)
0.493072 + 0.869989i \(0.335874\pi\)
\(6\) 2.70026 1.10238
\(7\) 0.381908 0.144348 0.0721738 0.997392i \(-0.477006\pi\)
0.0721738 + 0.997392i \(0.477006\pi\)
\(8\) 8.88768 3.14227
\(9\) 1.00000 0.333333
\(10\) 5.95430 1.88292
\(11\) −0.541461 −0.163256 −0.0816282 0.996663i \(-0.526012\pi\)
−0.0816282 + 0.996663i \(0.526012\pi\)
\(12\) 5.29141 1.52750
\(13\) −4.39015 −1.21761 −0.608804 0.793321i \(-0.708350\pi\)
−0.608804 + 0.793321i \(0.708350\pi\)
\(14\) 1.03125 0.275614
\(15\) 2.20508 0.569350
\(16\) 13.4162 3.35406
\(17\) −7.56447 −1.83465 −0.917327 0.398135i \(-0.869658\pi\)
−0.917327 + 0.398135i \(0.869658\pi\)
\(18\) 2.70026 0.636458
\(19\) 1.00000 0.229416
\(20\) 11.6680 2.60905
\(21\) 0.381908 0.0833392
\(22\) −1.46209 −0.311718
\(23\) 4.83305 1.00776 0.503880 0.863774i \(-0.331905\pi\)
0.503880 + 0.863774i \(0.331905\pi\)
\(24\) 8.88768 1.81419
\(25\) −0.137605 −0.0275210
\(26\) −11.8545 −2.32487
\(27\) 1.00000 0.192450
\(28\) 2.02083 0.381902
\(29\) −2.92584 −0.543316 −0.271658 0.962394i \(-0.587572\pi\)
−0.271658 + 0.962394i \(0.587572\pi\)
\(30\) 5.95430 1.08710
\(31\) 10.0216 1.79993 0.899963 0.435965i \(-0.143593\pi\)
0.899963 + 0.435965i \(0.143593\pi\)
\(32\) 18.4520 3.26188
\(33\) −0.541461 −0.0942562
\(34\) −20.4261 −3.50304
\(35\) 0.842139 0.142348
\(36\) 5.29141 0.881902
\(37\) −6.82713 −1.12237 −0.561186 0.827690i \(-0.689655\pi\)
−0.561186 + 0.827690i \(0.689655\pi\)
\(38\) 2.70026 0.438040
\(39\) −4.39015 −0.702986
\(40\) 19.5981 3.09873
\(41\) −1.66489 −0.260012 −0.130006 0.991513i \(-0.541500\pi\)
−0.130006 + 0.991513i \(0.541500\pi\)
\(42\) 1.03125 0.159126
\(43\) −8.47623 −1.29261 −0.646306 0.763078i \(-0.723687\pi\)
−0.646306 + 0.763078i \(0.723687\pi\)
\(44\) −2.86509 −0.431929
\(45\) 2.20508 0.328715
\(46\) 13.0505 1.92419
\(47\) 1.00000 0.145865
\(48\) 13.4162 1.93647
\(49\) −6.85415 −0.979164
\(50\) −0.371569 −0.0525478
\(51\) −7.56447 −1.05924
\(52\) −23.2301 −3.22143
\(53\) 5.12841 0.704441 0.352221 0.935917i \(-0.385427\pi\)
0.352221 + 0.935917i \(0.385427\pi\)
\(54\) 2.70026 0.367459
\(55\) −1.19397 −0.160994
\(56\) 3.39428 0.453579
\(57\) 1.00000 0.132453
\(58\) −7.90055 −1.03739
\(59\) −6.38720 −0.831543 −0.415771 0.909469i \(-0.636488\pi\)
−0.415771 + 0.909469i \(0.636488\pi\)
\(60\) 11.6680 1.50633
\(61\) 9.65918 1.23673 0.618366 0.785890i \(-0.287795\pi\)
0.618366 + 0.785890i \(0.287795\pi\)
\(62\) 27.0609 3.43673
\(63\) 0.381908 0.0481159
\(64\) 22.9928 2.87410
\(65\) −9.68064 −1.20074
\(66\) −1.46209 −0.179970
\(67\) 6.65611 0.813174 0.406587 0.913612i \(-0.366719\pi\)
0.406587 + 0.913612i \(0.366719\pi\)
\(68\) −40.0268 −4.85396
\(69\) 4.83305 0.581831
\(70\) 2.27400 0.271795
\(71\) 8.57056 1.01714 0.508569 0.861021i \(-0.330175\pi\)
0.508569 + 0.861021i \(0.330175\pi\)
\(72\) 8.88768 1.04742
\(73\) −8.65829 −1.01338 −0.506688 0.862130i \(-0.669130\pi\)
−0.506688 + 0.862130i \(0.669130\pi\)
\(74\) −18.4350 −2.14303
\(75\) −0.137605 −0.0158892
\(76\) 5.29141 0.606967
\(77\) −0.206788 −0.0235657
\(78\) −11.8545 −1.34226
\(79\) −2.49612 −0.280836 −0.140418 0.990092i \(-0.544845\pi\)
−0.140418 + 0.990092i \(0.544845\pi\)
\(80\) 29.5839 3.30759
\(81\) 1.00000 0.111111
\(82\) −4.49563 −0.496459
\(83\) 5.42636 0.595621 0.297810 0.954625i \(-0.403744\pi\)
0.297810 + 0.954625i \(0.403744\pi\)
\(84\) 2.02083 0.220491
\(85\) −16.6803 −1.80923
\(86\) −22.8880 −2.46808
\(87\) −2.92584 −0.313683
\(88\) −4.81233 −0.512996
\(89\) −12.6622 −1.34219 −0.671094 0.741372i \(-0.734175\pi\)
−0.671094 + 0.741372i \(0.734175\pi\)
\(90\) 5.95430 0.627639
\(91\) −1.67663 −0.175759
\(92\) 25.5737 2.66624
\(93\) 10.0216 1.03919
\(94\) 2.70026 0.278511
\(95\) 2.20508 0.226237
\(96\) 18.4520 1.88325
\(97\) −8.67832 −0.881150 −0.440575 0.897716i \(-0.645225\pi\)
−0.440575 + 0.897716i \(0.645225\pi\)
\(98\) −18.5080 −1.86959
\(99\) −0.541461 −0.0544188
\(100\) −0.728124 −0.0728124
\(101\) 9.77259 0.972409 0.486204 0.873845i \(-0.338381\pi\)
0.486204 + 0.873845i \(0.338381\pi\)
\(102\) −20.4261 −2.02248
\(103\) 11.0004 1.08391 0.541953 0.840409i \(-0.317685\pi\)
0.541953 + 0.840409i \(0.317685\pi\)
\(104\) −39.0182 −3.82605
\(105\) 0.842139 0.0821844
\(106\) 13.8480 1.34504
\(107\) −7.79770 −0.753832 −0.376916 0.926247i \(-0.623015\pi\)
−0.376916 + 0.926247i \(0.623015\pi\)
\(108\) 5.29141 0.509167
\(109\) 16.7420 1.60359 0.801796 0.597597i \(-0.203878\pi\)
0.801796 + 0.597597i \(0.203878\pi\)
\(110\) −3.22402 −0.307398
\(111\) −6.82713 −0.648002
\(112\) 5.12377 0.484151
\(113\) −14.0364 −1.32043 −0.660216 0.751076i \(-0.729535\pi\)
−0.660216 + 0.751076i \(0.729535\pi\)
\(114\) 2.70026 0.252903
\(115\) 10.6573 0.993796
\(116\) −15.4819 −1.43745
\(117\) −4.39015 −0.405869
\(118\) −17.2471 −1.58773
\(119\) −2.88893 −0.264828
\(120\) 19.5981 1.78905
\(121\) −10.7068 −0.973347
\(122\) 26.0823 2.36138
\(123\) −1.66489 −0.150118
\(124\) 53.0283 4.76208
\(125\) −11.3288 −1.01328
\(126\) 1.03125 0.0918712
\(127\) −11.7591 −1.04345 −0.521727 0.853112i \(-0.674712\pi\)
−0.521727 + 0.853112i \(0.674712\pi\)
\(128\) 25.1825 2.22584
\(129\) −8.47623 −0.746290
\(130\) −26.1403 −2.29265
\(131\) −6.71779 −0.586936 −0.293468 0.955969i \(-0.594809\pi\)
−0.293468 + 0.955969i \(0.594809\pi\)
\(132\) −2.86509 −0.249374
\(133\) 0.381908 0.0331156
\(134\) 17.9733 1.55265
\(135\) 2.20508 0.189783
\(136\) −67.2306 −5.76498
\(137\) 12.5363 1.07104 0.535522 0.844521i \(-0.320115\pi\)
0.535522 + 0.844521i \(0.320115\pi\)
\(138\) 13.0505 1.11093
\(139\) −8.93314 −0.757699 −0.378849 0.925458i \(-0.623680\pi\)
−0.378849 + 0.925458i \(0.623680\pi\)
\(140\) 4.45611 0.376610
\(141\) 1.00000 0.0842152
\(142\) 23.1428 1.94210
\(143\) 2.37709 0.198782
\(144\) 13.4162 1.11802
\(145\) −6.45173 −0.535787
\(146\) −23.3796 −1.93491
\(147\) −6.85415 −0.565320
\(148\) −36.1252 −2.96947
\(149\) 4.20717 0.344664 0.172332 0.985039i \(-0.444870\pi\)
0.172332 + 0.985039i \(0.444870\pi\)
\(150\) −0.371569 −0.0303385
\(151\) 18.0236 1.46674 0.733369 0.679831i \(-0.237947\pi\)
0.733369 + 0.679831i \(0.237947\pi\)
\(152\) 8.88768 0.720886
\(153\) −7.56447 −0.611551
\(154\) −0.558382 −0.0449957
\(155\) 22.0984 1.77499
\(156\) −23.2301 −1.85989
\(157\) 17.2290 1.37503 0.687514 0.726171i \(-0.258702\pi\)
0.687514 + 0.726171i \(0.258702\pi\)
\(158\) −6.74019 −0.536221
\(159\) 5.12841 0.406709
\(160\) 40.6882 3.21669
\(161\) 1.84578 0.145468
\(162\) 2.70026 0.212153
\(163\) 6.95091 0.544437 0.272219 0.962235i \(-0.412243\pi\)
0.272219 + 0.962235i \(0.412243\pi\)
\(164\) −8.80960 −0.687914
\(165\) −1.19397 −0.0929501
\(166\) 14.6526 1.13726
\(167\) −9.99053 −0.773090 −0.386545 0.922270i \(-0.626332\pi\)
−0.386545 + 0.922270i \(0.626332\pi\)
\(168\) 3.39428 0.261874
\(169\) 6.27337 0.482567
\(170\) −45.0412 −3.45450
\(171\) 1.00000 0.0764719
\(172\) −44.8512 −3.41987
\(173\) −8.92035 −0.678202 −0.339101 0.940750i \(-0.610123\pi\)
−0.339101 + 0.940750i \(0.610123\pi\)
\(174\) −7.90055 −0.598939
\(175\) −0.0525524 −0.00397259
\(176\) −7.26437 −0.547572
\(177\) −6.38720 −0.480092
\(178\) −34.1912 −2.56274
\(179\) 19.7358 1.47512 0.737561 0.675281i \(-0.235978\pi\)
0.737561 + 0.675281i \(0.235978\pi\)
\(180\) 11.6680 0.869682
\(181\) −2.51847 −0.187196 −0.0935982 0.995610i \(-0.529837\pi\)
−0.0935982 + 0.995610i \(0.529837\pi\)
\(182\) −4.52735 −0.335589
\(183\) 9.65918 0.714027
\(184\) 42.9546 3.16666
\(185\) −15.0544 −1.10682
\(186\) 27.0609 1.98420
\(187\) 4.09586 0.299519
\(188\) 5.29141 0.385916
\(189\) 0.381908 0.0277797
\(190\) 5.95430 0.431971
\(191\) 18.3243 1.32590 0.662952 0.748662i \(-0.269303\pi\)
0.662952 + 0.748662i \(0.269303\pi\)
\(192\) 22.9928 1.65936
\(193\) 24.2325 1.74429 0.872146 0.489246i \(-0.162728\pi\)
0.872146 + 0.489246i \(0.162728\pi\)
\(194\) −23.4337 −1.68245
\(195\) −9.68064 −0.693245
\(196\) −36.2681 −2.59058
\(197\) 15.3067 1.09055 0.545277 0.838256i \(-0.316424\pi\)
0.545277 + 0.838256i \(0.316424\pi\)
\(198\) −1.46209 −0.103906
\(199\) 15.1884 1.07668 0.538339 0.842728i \(-0.319052\pi\)
0.538339 + 0.842728i \(0.319052\pi\)
\(200\) −1.22299 −0.0864783
\(201\) 6.65611 0.469486
\(202\) 26.3885 1.85669
\(203\) −1.11740 −0.0784264
\(204\) −40.0268 −2.80243
\(205\) −3.67121 −0.256409
\(206\) 29.7041 2.06958
\(207\) 4.83305 0.335920
\(208\) −58.8993 −4.08393
\(209\) −0.541461 −0.0374536
\(210\) 2.27400 0.156921
\(211\) −10.0965 −0.695070 −0.347535 0.937667i \(-0.612981\pi\)
−0.347535 + 0.937667i \(0.612981\pi\)
\(212\) 27.1365 1.86375
\(213\) 8.57056 0.587245
\(214\) −21.0558 −1.43935
\(215\) −18.6908 −1.27470
\(216\) 8.88768 0.604730
\(217\) 3.82732 0.259815
\(218\) 45.2078 3.06186
\(219\) −8.65829 −0.585073
\(220\) −6.31777 −0.425944
\(221\) 33.2091 2.23389
\(222\) −18.4350 −1.23728
\(223\) 19.4848 1.30480 0.652400 0.757875i \(-0.273762\pi\)
0.652400 + 0.757875i \(0.273762\pi\)
\(224\) 7.04697 0.470845
\(225\) −0.137605 −0.00917366
\(226\) −37.9019 −2.52120
\(227\) 14.1735 0.940727 0.470363 0.882473i \(-0.344123\pi\)
0.470363 + 0.882473i \(0.344123\pi\)
\(228\) 5.29141 0.350433
\(229\) −15.0866 −0.996951 −0.498475 0.866904i \(-0.666107\pi\)
−0.498475 + 0.866904i \(0.666107\pi\)
\(230\) 28.7774 1.89753
\(231\) −0.206788 −0.0136057
\(232\) −26.0040 −1.70724
\(233\) 29.5646 1.93684 0.968421 0.249321i \(-0.0802075\pi\)
0.968421 + 0.249321i \(0.0802075\pi\)
\(234\) −11.8545 −0.774956
\(235\) 2.20508 0.143844
\(236\) −33.7973 −2.20002
\(237\) −2.49612 −0.162141
\(238\) −7.80087 −0.505656
\(239\) 2.73697 0.177040 0.0885199 0.996074i \(-0.471786\pi\)
0.0885199 + 0.996074i \(0.471786\pi\)
\(240\) 29.5839 1.90964
\(241\) −13.2806 −0.855477 −0.427738 0.903903i \(-0.640690\pi\)
−0.427738 + 0.903903i \(0.640690\pi\)
\(242\) −28.9112 −1.85848
\(243\) 1.00000 0.0641500
\(244\) 51.1107 3.27203
\(245\) −15.1140 −0.965596
\(246\) −4.49563 −0.286631
\(247\) −4.39015 −0.279338
\(248\) 89.0685 5.65586
\(249\) 5.42636 0.343882
\(250\) −30.5909 −1.93474
\(251\) −13.6733 −0.863049 −0.431525 0.902101i \(-0.642024\pi\)
−0.431525 + 0.902101i \(0.642024\pi\)
\(252\) 2.02083 0.127301
\(253\) −2.61690 −0.164523
\(254\) −31.7527 −1.99234
\(255\) −16.6803 −1.04456
\(256\) 22.0138 1.37586
\(257\) −11.9308 −0.744222 −0.372111 0.928188i \(-0.621366\pi\)
−0.372111 + 0.928188i \(0.621366\pi\)
\(258\) −22.8880 −1.42495
\(259\) −2.60733 −0.162012
\(260\) −51.2243 −3.17679
\(261\) −2.92584 −0.181105
\(262\) −18.1398 −1.12068
\(263\) −20.6566 −1.27374 −0.636871 0.770970i \(-0.719772\pi\)
−0.636871 + 0.770970i \(0.719772\pi\)
\(264\) −4.81233 −0.296178
\(265\) 11.3086 0.694680
\(266\) 1.03125 0.0632301
\(267\) −12.6622 −0.774912
\(268\) 35.2203 2.15142
\(269\) −28.6700 −1.74804 −0.874020 0.485889i \(-0.838496\pi\)
−0.874020 + 0.485889i \(0.838496\pi\)
\(270\) 5.95430 0.362367
\(271\) 9.20417 0.559114 0.279557 0.960129i \(-0.409812\pi\)
0.279557 + 0.960129i \(0.409812\pi\)
\(272\) −101.487 −6.15354
\(273\) −1.67663 −0.101474
\(274\) 33.8512 2.04502
\(275\) 0.0745076 0.00449298
\(276\) 25.5737 1.53935
\(277\) −19.8425 −1.19222 −0.596111 0.802902i \(-0.703288\pi\)
−0.596111 + 0.802902i \(0.703288\pi\)
\(278\) −24.1218 −1.44673
\(279\) 10.0216 0.599976
\(280\) 7.48467 0.447294
\(281\) 32.7137 1.95153 0.975767 0.218811i \(-0.0702178\pi\)
0.975767 + 0.218811i \(0.0702178\pi\)
\(282\) 2.70026 0.160798
\(283\) −26.7479 −1.59000 −0.794999 0.606611i \(-0.792528\pi\)
−0.794999 + 0.606611i \(0.792528\pi\)
\(284\) 45.3504 2.69105
\(285\) 2.20508 0.130618
\(286\) 6.41877 0.379550
\(287\) −0.635833 −0.0375321
\(288\) 18.4520 1.08729
\(289\) 40.2212 2.36595
\(290\) −17.4214 −1.02302
\(291\) −8.67832 −0.508732
\(292\) −45.8146 −2.68110
\(293\) −5.15932 −0.301411 −0.150705 0.988579i \(-0.548154\pi\)
−0.150705 + 0.988579i \(0.548154\pi\)
\(294\) −18.5080 −1.07941
\(295\) −14.0843 −0.820021
\(296\) −60.6773 −3.52680
\(297\) −0.541461 −0.0314187
\(298\) 11.3604 0.658093
\(299\) −21.2178 −1.22706
\(300\) −0.728124 −0.0420383
\(301\) −3.23714 −0.186586
\(302\) 48.6684 2.80055
\(303\) 9.77259 0.561420
\(304\) 13.4162 0.769474
\(305\) 21.2993 1.21959
\(306\) −20.4261 −1.16768
\(307\) 5.44081 0.310523 0.155262 0.987873i \(-0.450378\pi\)
0.155262 + 0.987873i \(0.450378\pi\)
\(308\) −1.09420 −0.0623479
\(309\) 11.0004 0.625793
\(310\) 59.6715 3.38911
\(311\) 21.0326 1.19265 0.596323 0.802744i \(-0.296628\pi\)
0.596323 + 0.802744i \(0.296628\pi\)
\(312\) −39.0182 −2.20897
\(313\) −15.1762 −0.857808 −0.428904 0.903350i \(-0.641100\pi\)
−0.428904 + 0.903350i \(0.641100\pi\)
\(314\) 46.5229 2.62544
\(315\) 0.842139 0.0474492
\(316\) −13.2080 −0.743010
\(317\) −4.90155 −0.275298 −0.137649 0.990481i \(-0.543955\pi\)
−0.137649 + 0.990481i \(0.543955\pi\)
\(318\) 13.8480 0.776560
\(319\) 1.58423 0.0886998
\(320\) 50.7010 2.83427
\(321\) −7.79770 −0.435225
\(322\) 4.98409 0.277752
\(323\) −7.56447 −0.420898
\(324\) 5.29141 0.293967
\(325\) 0.604105 0.0335097
\(326\) 18.7693 1.03953
\(327\) 16.7420 0.925835
\(328\) −14.7970 −0.817027
\(329\) 0.381908 0.0210553
\(330\) −3.22402 −0.177477
\(331\) −36.1520 −1.98709 −0.993545 0.113434i \(-0.963815\pi\)
−0.993545 + 0.113434i \(0.963815\pi\)
\(332\) 28.7131 1.57584
\(333\) −6.82713 −0.374124
\(334\) −26.9770 −1.47612
\(335\) 14.6773 0.801906
\(336\) 5.12377 0.279525
\(337\) −19.5250 −1.06359 −0.531796 0.846873i \(-0.678483\pi\)
−0.531796 + 0.846873i \(0.678483\pi\)
\(338\) 16.9398 0.921401
\(339\) −14.0364 −0.762352
\(340\) −88.2623 −4.78670
\(341\) −5.42628 −0.293850
\(342\) 2.70026 0.146013
\(343\) −5.29101 −0.285688
\(344\) −75.3340 −4.06174
\(345\) 10.6573 0.573768
\(346\) −24.0873 −1.29494
\(347\) 1.91679 0.102899 0.0514493 0.998676i \(-0.483616\pi\)
0.0514493 + 0.998676i \(0.483616\pi\)
\(348\) −15.4819 −0.829915
\(349\) 17.4874 0.936079 0.468040 0.883708i \(-0.344960\pi\)
0.468040 + 0.883708i \(0.344960\pi\)
\(350\) −0.141905 −0.00758515
\(351\) −4.39015 −0.234329
\(352\) −9.99103 −0.532524
\(353\) −13.1334 −0.699021 −0.349510 0.936933i \(-0.613652\pi\)
−0.349510 + 0.936933i \(0.613652\pi\)
\(354\) −17.2471 −0.916674
\(355\) 18.8988 1.00304
\(356\) −67.0008 −3.55104
\(357\) −2.88893 −0.152899
\(358\) 53.2918 2.81656
\(359\) −16.6782 −0.880241 −0.440121 0.897939i \(-0.645064\pi\)
−0.440121 + 0.897939i \(0.645064\pi\)
\(360\) 19.5981 1.03291
\(361\) 1.00000 0.0526316
\(362\) −6.80053 −0.357428
\(363\) −10.7068 −0.561962
\(364\) −8.87175 −0.465006
\(365\) −19.0923 −0.999334
\(366\) 26.0823 1.36334
\(367\) −1.77145 −0.0924688 −0.0462344 0.998931i \(-0.514722\pi\)
−0.0462344 + 0.998931i \(0.514722\pi\)
\(368\) 64.8413 3.38009
\(369\) −1.66489 −0.0866705
\(370\) −40.6508 −2.11333
\(371\) 1.95858 0.101684
\(372\) 53.0283 2.74939
\(373\) −2.63760 −0.136570 −0.0682848 0.997666i \(-0.521753\pi\)
−0.0682848 + 0.997666i \(0.521753\pi\)
\(374\) 11.0599 0.571894
\(375\) −11.3288 −0.585019
\(376\) 8.88768 0.458347
\(377\) 12.8449 0.661545
\(378\) 1.03125 0.0530419
\(379\) 7.01437 0.360304 0.180152 0.983639i \(-0.442341\pi\)
0.180152 + 0.983639i \(0.442341\pi\)
\(380\) 11.6680 0.598556
\(381\) −11.7591 −0.602439
\(382\) 49.4805 2.53164
\(383\) −19.4874 −0.995761 −0.497880 0.867246i \(-0.665888\pi\)
−0.497880 + 0.867246i \(0.665888\pi\)
\(384\) 25.1825 1.28509
\(385\) −0.455985 −0.0232392
\(386\) 65.4340 3.33050
\(387\) −8.47623 −0.430871
\(388\) −45.9206 −2.33127
\(389\) 5.96803 0.302591 0.151296 0.988489i \(-0.451655\pi\)
0.151296 + 0.988489i \(0.451655\pi\)
\(390\) −26.1403 −1.32366
\(391\) −36.5594 −1.84889
\(392\) −60.9175 −3.07680
\(393\) −6.71779 −0.338868
\(394\) 41.3320 2.08228
\(395\) −5.50416 −0.276944
\(396\) −2.86509 −0.143976
\(397\) 6.34949 0.318672 0.159336 0.987224i \(-0.449065\pi\)
0.159336 + 0.987224i \(0.449065\pi\)
\(398\) 41.0127 2.05578
\(399\) 0.381908 0.0191193
\(400\) −1.84614 −0.0923070
\(401\) 20.4053 1.01899 0.509496 0.860473i \(-0.329832\pi\)
0.509496 + 0.860473i \(0.329832\pi\)
\(402\) 17.9733 0.896424
\(403\) −43.9961 −2.19160
\(404\) 51.7108 2.57271
\(405\) 2.20508 0.109572
\(406\) −3.01728 −0.149745
\(407\) 3.69662 0.183235
\(408\) −67.2306 −3.32841
\(409\) 12.3652 0.611421 0.305710 0.952125i \(-0.401106\pi\)
0.305710 + 0.952125i \(0.401106\pi\)
\(410\) −9.91324 −0.489580
\(411\) 12.5363 0.618368
\(412\) 58.2079 2.86770
\(413\) −2.43932 −0.120031
\(414\) 13.0505 0.641397
\(415\) 11.9656 0.587368
\(416\) −81.0070 −3.97169
\(417\) −8.93314 −0.437458
\(418\) −1.46209 −0.0715129
\(419\) −13.4073 −0.654992 −0.327496 0.944853i \(-0.606205\pi\)
−0.327496 + 0.944853i \(0.606205\pi\)
\(420\) 4.45611 0.217436
\(421\) 9.55704 0.465782 0.232891 0.972503i \(-0.425182\pi\)
0.232891 + 0.972503i \(0.425182\pi\)
\(422\) −27.2631 −1.32715
\(423\) 1.00000 0.0486217
\(424\) 45.5797 2.21354
\(425\) 1.04091 0.0504914
\(426\) 23.1428 1.12127
\(427\) 3.68892 0.178519
\(428\) −41.2609 −1.99442
\(429\) 2.37709 0.114767
\(430\) −50.4700 −2.43388
\(431\) −12.6183 −0.607804 −0.303902 0.952703i \(-0.598289\pi\)
−0.303902 + 0.952703i \(0.598289\pi\)
\(432\) 13.4162 0.645489
\(433\) 16.0281 0.770262 0.385131 0.922862i \(-0.374156\pi\)
0.385131 + 0.922862i \(0.374156\pi\)
\(434\) 10.3348 0.496084
\(435\) −6.45173 −0.309337
\(436\) 88.5889 4.24264
\(437\) 4.83305 0.231196
\(438\) −23.3796 −1.11712
\(439\) 3.06186 0.146135 0.0730674 0.997327i \(-0.476721\pi\)
0.0730674 + 0.997327i \(0.476721\pi\)
\(440\) −10.6116 −0.505888
\(441\) −6.85415 −0.326388
\(442\) 89.6733 4.26533
\(443\) 9.52124 0.452368 0.226184 0.974085i \(-0.427375\pi\)
0.226184 + 0.974085i \(0.427375\pi\)
\(444\) −36.1252 −1.71442
\(445\) −27.9212 −1.32359
\(446\) 52.6141 2.49135
\(447\) 4.20717 0.198992
\(448\) 8.78112 0.414869
\(449\) 31.2958 1.47694 0.738469 0.674287i \(-0.235549\pi\)
0.738469 + 0.674287i \(0.235549\pi\)
\(450\) −0.371569 −0.0175159
\(451\) 0.901470 0.0424486
\(452\) −74.2724 −3.49348
\(453\) 18.0236 0.846822
\(454\) 38.2721 1.79620
\(455\) −3.69711 −0.173323
\(456\) 8.88768 0.416204
\(457\) 3.14927 0.147317 0.0736583 0.997284i \(-0.476533\pi\)
0.0736583 + 0.997284i \(0.476533\pi\)
\(458\) −40.7378 −1.90355
\(459\) −7.56447 −0.353079
\(460\) 56.3921 2.62929
\(461\) 36.5366 1.70168 0.850840 0.525424i \(-0.176093\pi\)
0.850840 + 0.525424i \(0.176093\pi\)
\(462\) −0.558382 −0.0259783
\(463\) 2.42169 0.112546 0.0562728 0.998415i \(-0.482078\pi\)
0.0562728 + 0.998415i \(0.482078\pi\)
\(464\) −39.2538 −1.82231
\(465\) 22.0984 1.02479
\(466\) 79.8322 3.69815
\(467\) 9.54643 0.441756 0.220878 0.975301i \(-0.429108\pi\)
0.220878 + 0.975301i \(0.429108\pi\)
\(468\) −23.2301 −1.07381
\(469\) 2.54202 0.117380
\(470\) 5.95430 0.274652
\(471\) 17.2290 0.793873
\(472\) −56.7674 −2.61293
\(473\) 4.58954 0.211027
\(474\) −6.74019 −0.309587
\(475\) −0.137605 −0.00631374
\(476\) −15.2865 −0.700657
\(477\) 5.12841 0.234814
\(478\) 7.39053 0.338035
\(479\) −13.5642 −0.619763 −0.309881 0.950775i \(-0.600289\pi\)
−0.309881 + 0.950775i \(0.600289\pi\)
\(480\) 40.6882 1.85715
\(481\) 29.9721 1.36661
\(482\) −35.8610 −1.63342
\(483\) 1.84578 0.0839859
\(484\) −56.6542 −2.57519
\(485\) −19.1364 −0.868941
\(486\) 2.70026 0.122486
\(487\) −27.7764 −1.25867 −0.629335 0.777134i \(-0.716672\pi\)
−0.629335 + 0.777134i \(0.716672\pi\)
\(488\) 85.8478 3.88615
\(489\) 6.95091 0.314331
\(490\) −40.8117 −1.84368
\(491\) −13.7552 −0.620762 −0.310381 0.950612i \(-0.600457\pi\)
−0.310381 + 0.950612i \(0.600457\pi\)
\(492\) −8.80960 −0.397168
\(493\) 22.1325 0.996796
\(494\) −11.8545 −0.533361
\(495\) −1.19397 −0.0536648
\(496\) 134.452 6.03706
\(497\) 3.27317 0.146822
\(498\) 14.6526 0.656599
\(499\) 12.9853 0.581304 0.290652 0.956829i \(-0.406128\pi\)
0.290652 + 0.956829i \(0.406128\pi\)
\(500\) −59.9456 −2.68085
\(501\) −9.99053 −0.446344
\(502\) −36.9214 −1.64788
\(503\) 0.878196 0.0391568 0.0195784 0.999808i \(-0.493768\pi\)
0.0195784 + 0.999808i \(0.493768\pi\)
\(504\) 3.39428 0.151193
\(505\) 21.5494 0.958935
\(506\) −7.06633 −0.314137
\(507\) 6.27337 0.278610
\(508\) −62.2225 −2.76067
\(509\) 21.1285 0.936502 0.468251 0.883595i \(-0.344884\pi\)
0.468251 + 0.883595i \(0.344884\pi\)
\(510\) −45.0412 −1.99446
\(511\) −3.30667 −0.146278
\(512\) 9.07795 0.401193
\(513\) 1.00000 0.0441511
\(514\) −32.2162 −1.42100
\(515\) 24.2569 1.06889
\(516\) −44.8512 −1.97447
\(517\) −0.541461 −0.0238134
\(518\) −7.04049 −0.309341
\(519\) −8.92035 −0.391560
\(520\) −86.0384 −3.77304
\(521\) −21.4364 −0.939146 −0.469573 0.882894i \(-0.655592\pi\)
−0.469573 + 0.882894i \(0.655592\pi\)
\(522\) −7.90055 −0.345798
\(523\) 8.62069 0.376957 0.188478 0.982077i \(-0.439644\pi\)
0.188478 + 0.982077i \(0.439644\pi\)
\(524\) −35.5466 −1.55286
\(525\) −0.0525524 −0.00229357
\(526\) −55.7783 −2.43205
\(527\) −75.8079 −3.30224
\(528\) −7.26437 −0.316141
\(529\) 0.358354 0.0155806
\(530\) 30.5361 1.32640
\(531\) −6.38720 −0.277181
\(532\) 2.02083 0.0876143
\(533\) 7.30909 0.316592
\(534\) −34.1912 −1.47960
\(535\) −17.1946 −0.743387
\(536\) 59.1574 2.55521
\(537\) 19.7358 0.851662
\(538\) −77.4165 −3.33766
\(539\) 3.71125 0.159855
\(540\) 11.6680 0.502111
\(541\) 36.7599 1.58043 0.790216 0.612828i \(-0.209968\pi\)
0.790216 + 0.612828i \(0.209968\pi\)
\(542\) 24.8537 1.06756
\(543\) −2.51847 −0.108078
\(544\) −139.580 −5.98443
\(545\) 36.9175 1.58137
\(546\) −4.52735 −0.193752
\(547\) −20.6459 −0.882756 −0.441378 0.897321i \(-0.645510\pi\)
−0.441378 + 0.897321i \(0.645510\pi\)
\(548\) 66.3345 2.83367
\(549\) 9.65918 0.412244
\(550\) 0.201190 0.00857877
\(551\) −2.92584 −0.124645
\(552\) 42.9546 1.82827
\(553\) −0.953290 −0.0405380
\(554\) −53.5800 −2.27640
\(555\) −15.0544 −0.639023
\(556\) −47.2689 −2.00465
\(557\) −23.6903 −1.00379 −0.501896 0.864928i \(-0.667364\pi\)
−0.501896 + 0.864928i \(0.667364\pi\)
\(558\) 27.0609 1.14558
\(559\) 37.2119 1.57389
\(560\) 11.2983 0.477442
\(561\) 4.09586 0.172927
\(562\) 88.3355 3.72621
\(563\) 9.84108 0.414752 0.207376 0.978261i \(-0.433508\pi\)
0.207376 + 0.978261i \(0.433508\pi\)
\(564\) 5.29141 0.222809
\(565\) −30.9514 −1.30214
\(566\) −72.2263 −3.03590
\(567\) 0.381908 0.0160386
\(568\) 76.1724 3.19612
\(569\) 1.63516 0.0685495 0.0342748 0.999412i \(-0.489088\pi\)
0.0342748 + 0.999412i \(0.489088\pi\)
\(570\) 5.95430 0.249398
\(571\) −3.22029 −0.134765 −0.0673824 0.997727i \(-0.521465\pi\)
−0.0673824 + 0.997727i \(0.521465\pi\)
\(572\) 12.5782 0.525920
\(573\) 18.3243 0.765511
\(574\) −1.71692 −0.0716627
\(575\) −0.665051 −0.0277345
\(576\) 22.9928 0.958032
\(577\) 16.4921 0.686577 0.343288 0.939230i \(-0.388459\pi\)
0.343288 + 0.939230i \(0.388459\pi\)
\(578\) 108.608 4.51749
\(579\) 24.2325 1.00707
\(580\) −34.1388 −1.41754
\(581\) 2.07237 0.0859765
\(582\) −23.4337 −0.971360
\(583\) −2.77683 −0.115005
\(584\) −76.9521 −3.18430
\(585\) −9.68064 −0.400245
\(586\) −13.9315 −0.575506
\(587\) 18.4883 0.763095 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(588\) −36.2681 −1.49567
\(589\) 10.0216 0.412932
\(590\) −38.0313 −1.56573
\(591\) 15.3067 0.629632
\(592\) −91.5944 −3.76451
\(593\) −43.4614 −1.78475 −0.892373 0.451299i \(-0.850961\pi\)
−0.892373 + 0.451299i \(0.850961\pi\)
\(594\) −1.46209 −0.0599901
\(595\) −6.37034 −0.261158
\(596\) 22.2619 0.911881
\(597\) 15.1884 0.621621
\(598\) −57.2936 −2.34291
\(599\) 23.3190 0.952788 0.476394 0.879232i \(-0.341944\pi\)
0.476394 + 0.879232i \(0.341944\pi\)
\(600\) −1.22299 −0.0499283
\(601\) −34.0943 −1.39074 −0.695368 0.718654i \(-0.744759\pi\)
−0.695368 + 0.718654i \(0.744759\pi\)
\(602\) −8.74112 −0.356262
\(603\) 6.65611 0.271058
\(604\) 95.3703 3.88056
\(605\) −23.6094 −0.959860
\(606\) 26.3885 1.07196
\(607\) −11.0254 −0.447507 −0.223754 0.974646i \(-0.571831\pi\)
−0.223754 + 0.974646i \(0.571831\pi\)
\(608\) 18.4520 0.748328
\(609\) −1.11740 −0.0452795
\(610\) 57.5137 2.32866
\(611\) −4.39015 −0.177606
\(612\) −40.0268 −1.61799
\(613\) −4.46689 −0.180416 −0.0902081 0.995923i \(-0.528753\pi\)
−0.0902081 + 0.995923i \(0.528753\pi\)
\(614\) 14.6916 0.592905
\(615\) −3.67121 −0.148038
\(616\) −1.83787 −0.0740498
\(617\) 6.29671 0.253496 0.126748 0.991935i \(-0.459546\pi\)
0.126748 + 0.991935i \(0.459546\pi\)
\(618\) 29.7041 1.19487
\(619\) −36.2640 −1.45757 −0.728787 0.684741i \(-0.759915\pi\)
−0.728787 + 0.684741i \(0.759915\pi\)
\(620\) 116.932 4.69609
\(621\) 4.83305 0.193944
\(622\) 56.7934 2.27721
\(623\) −4.83579 −0.193742
\(624\) −58.8993 −2.35786
\(625\) −24.2930 −0.971722
\(626\) −40.9796 −1.63788
\(627\) −0.541461 −0.0216239
\(628\) 91.1660 3.63792
\(629\) 51.6436 2.05916
\(630\) 2.27400 0.0905982
\(631\) −10.7987 −0.429891 −0.214945 0.976626i \(-0.568957\pi\)
−0.214945 + 0.976626i \(0.568957\pi\)
\(632\) −22.1848 −0.882462
\(633\) −10.0965 −0.401299
\(634\) −13.2355 −0.525648
\(635\) −25.9299 −1.02900
\(636\) 27.1365 1.07603
\(637\) 30.0907 1.19224
\(638\) 4.27783 0.169361
\(639\) 8.57056 0.339046
\(640\) 55.5295 2.19499
\(641\) 27.5198 1.08697 0.543483 0.839420i \(-0.317105\pi\)
0.543483 + 0.839420i \(0.317105\pi\)
\(642\) −21.0558 −0.831008
\(643\) 9.75948 0.384876 0.192438 0.981309i \(-0.438361\pi\)
0.192438 + 0.981309i \(0.438361\pi\)
\(644\) 9.76679 0.384865
\(645\) −18.6908 −0.735949
\(646\) −20.4261 −0.803652
\(647\) 24.2245 0.952363 0.476181 0.879347i \(-0.342021\pi\)
0.476181 + 0.879347i \(0.342021\pi\)
\(648\) 8.88768 0.349141
\(649\) 3.45842 0.135755
\(650\) 1.63124 0.0639826
\(651\) 3.82732 0.150004
\(652\) 36.7801 1.44042
\(653\) 12.6373 0.494535 0.247268 0.968947i \(-0.420467\pi\)
0.247268 + 0.968947i \(0.420467\pi\)
\(654\) 45.2078 1.76776
\(655\) −14.8133 −0.578803
\(656\) −22.3365 −0.872094
\(657\) −8.65829 −0.337792
\(658\) 1.03125 0.0402024
\(659\) 29.8421 1.16248 0.581241 0.813731i \(-0.302567\pi\)
0.581241 + 0.813731i \(0.302567\pi\)
\(660\) −6.31777 −0.245919
\(661\) −8.01684 −0.311819 −0.155909 0.987771i \(-0.549831\pi\)
−0.155909 + 0.987771i \(0.549831\pi\)
\(662\) −97.6197 −3.79410
\(663\) 33.2091 1.28974
\(664\) 48.2278 1.87160
\(665\) 0.842139 0.0326568
\(666\) −18.4350 −0.714343
\(667\) −14.1407 −0.547532
\(668\) −52.8640 −2.04537
\(669\) 19.4848 0.753327
\(670\) 39.6325 1.53114
\(671\) −5.23007 −0.201904
\(672\) 7.04697 0.271843
\(673\) −38.4718 −1.48298 −0.741489 0.670965i \(-0.765880\pi\)
−0.741489 + 0.670965i \(0.765880\pi\)
\(674\) −52.7225 −2.03079
\(675\) −0.137605 −0.00529641
\(676\) 33.1950 1.27673
\(677\) −10.1601 −0.390486 −0.195243 0.980755i \(-0.562550\pi\)
−0.195243 + 0.980755i \(0.562550\pi\)
\(678\) −37.9019 −1.45561
\(679\) −3.31432 −0.127192
\(680\) −148.249 −5.68510
\(681\) 14.1735 0.543129
\(682\) −14.6524 −0.561069
\(683\) −26.6293 −1.01894 −0.509471 0.860488i \(-0.670159\pi\)
−0.509471 + 0.860488i \(0.670159\pi\)
\(684\) 5.29141 0.202322
\(685\) 27.6435 1.05620
\(686\) −14.2871 −0.545485
\(687\) −15.0866 −0.575590
\(688\) −113.719 −4.33550
\(689\) −22.5145 −0.857733
\(690\) 28.7774 1.09554
\(691\) −22.8073 −0.867631 −0.433816 0.901002i \(-0.642833\pi\)
−0.433816 + 0.901002i \(0.642833\pi\)
\(692\) −47.2013 −1.79432
\(693\) −0.206788 −0.00785523
\(694\) 5.17583 0.196472
\(695\) −19.6983 −0.747200
\(696\) −26.0040 −0.985678
\(697\) 12.5940 0.477031
\(698\) 47.2206 1.78732
\(699\) 29.5646 1.11824
\(700\) −0.278077 −0.0105103
\(701\) −37.4962 −1.41621 −0.708105 0.706107i \(-0.750450\pi\)
−0.708105 + 0.706107i \(0.750450\pi\)
\(702\) −11.8545 −0.447421
\(703\) −6.82713 −0.257490
\(704\) −12.4497 −0.469215
\(705\) 2.20508 0.0830483
\(706\) −35.4636 −1.33469
\(707\) 3.73223 0.140365
\(708\) −33.7973 −1.27018
\(709\) −19.6326 −0.737318 −0.368659 0.929565i \(-0.620183\pi\)
−0.368659 + 0.929565i \(0.620183\pi\)
\(710\) 51.0317 1.91519
\(711\) −2.49612 −0.0936120
\(712\) −112.537 −4.21752
\(713\) 48.4347 1.81389
\(714\) −7.80087 −0.291940
\(715\) 5.24168 0.196028
\(716\) 104.430 3.90274
\(717\) 2.73697 0.102214
\(718\) −45.0355 −1.68071
\(719\) −3.20241 −0.119430 −0.0597150 0.998215i \(-0.519019\pi\)
−0.0597150 + 0.998215i \(0.519019\pi\)
\(720\) 29.5839 1.10253
\(721\) 4.20115 0.156459
\(722\) 2.70026 0.100493
\(723\) −13.2806 −0.493910
\(724\) −13.3263 −0.495267
\(725\) 0.402610 0.0149526
\(726\) −28.9112 −1.07300
\(727\) 20.1186 0.746157 0.373078 0.927800i \(-0.378302\pi\)
0.373078 + 0.927800i \(0.378302\pi\)
\(728\) −14.9014 −0.552282
\(729\) 1.00000 0.0370370
\(730\) −51.5541 −1.90810
\(731\) 64.1182 2.37150
\(732\) 51.1107 1.88911
\(733\) −42.8614 −1.58312 −0.791560 0.611091i \(-0.790731\pi\)
−0.791560 + 0.611091i \(0.790731\pi\)
\(734\) −4.78337 −0.176557
\(735\) −15.1140 −0.557487
\(736\) 89.1794 3.28720
\(737\) −3.60402 −0.132756
\(738\) −4.49563 −0.165486
\(739\) 43.3153 1.59338 0.796689 0.604389i \(-0.206583\pi\)
0.796689 + 0.604389i \(0.206583\pi\)
\(740\) −79.6590 −2.92832
\(741\) −4.39015 −0.161276
\(742\) 5.28868 0.194154
\(743\) −15.7700 −0.578544 −0.289272 0.957247i \(-0.593413\pi\)
−0.289272 + 0.957247i \(0.593413\pi\)
\(744\) 89.0685 3.26541
\(745\) 9.27715 0.339889
\(746\) −7.12220 −0.260762
\(747\) 5.42636 0.198540
\(748\) 21.6729 0.792440
\(749\) −2.97800 −0.108814
\(750\) −30.5909 −1.11702
\(751\) 25.1146 0.916445 0.458223 0.888837i \(-0.348486\pi\)
0.458223 + 0.888837i \(0.348486\pi\)
\(752\) 13.4162 0.489240
\(753\) −13.6733 −0.498282
\(754\) 34.6845 1.26314
\(755\) 39.7435 1.44641
\(756\) 2.02083 0.0734970
\(757\) −48.6599 −1.76857 −0.884287 0.466944i \(-0.845355\pi\)
−0.884287 + 0.466944i \(0.845355\pi\)
\(758\) 18.9407 0.687955
\(759\) −2.61690 −0.0949876
\(760\) 19.5981 0.710897
\(761\) 15.5719 0.564481 0.282241 0.959344i \(-0.408922\pi\)
0.282241 + 0.959344i \(0.408922\pi\)
\(762\) −31.7527 −1.15028
\(763\) 6.39390 0.231475
\(764\) 96.9617 3.50795
\(765\) −16.6803 −0.603077
\(766\) −52.6212 −1.90128
\(767\) 28.0407 1.01249
\(768\) 22.0138 0.794353
\(769\) −24.5292 −0.884545 −0.442273 0.896881i \(-0.645828\pi\)
−0.442273 + 0.896881i \(0.645828\pi\)
\(770\) −1.23128 −0.0443722
\(771\) −11.9308 −0.429677
\(772\) 128.224 4.61488
\(773\) 24.2794 0.873271 0.436635 0.899639i \(-0.356170\pi\)
0.436635 + 0.899639i \(0.356170\pi\)
\(774\) −22.8880 −0.822693
\(775\) −1.37902 −0.0495357
\(776\) −77.1302 −2.76881
\(777\) −2.60733 −0.0935376
\(778\) 16.1152 0.577760
\(779\) −1.66489 −0.0596507
\(780\) −51.2243 −1.83412
\(781\) −4.64062 −0.166054
\(782\) −98.7201 −3.53022
\(783\) −2.92584 −0.104561
\(784\) −91.9569 −3.28417
\(785\) 37.9915 1.35597
\(786\) −18.1398 −0.647025
\(787\) −26.0922 −0.930086 −0.465043 0.885288i \(-0.653961\pi\)
−0.465043 + 0.885288i \(0.653961\pi\)
\(788\) 80.9939 2.88529
\(789\) −20.6566 −0.735395
\(790\) −14.8627 −0.528791
\(791\) −5.36061 −0.190601
\(792\) −4.81233 −0.170999
\(793\) −42.4052 −1.50585
\(794\) 17.1453 0.608463
\(795\) 11.3086 0.401074
\(796\) 80.3682 2.84858
\(797\) 12.5351 0.444017 0.222009 0.975045i \(-0.428739\pi\)
0.222009 + 0.975045i \(0.428739\pi\)
\(798\) 1.03125 0.0365059
\(799\) −7.56447 −0.267612
\(800\) −2.53909 −0.0897702
\(801\) −12.6622 −0.447396
\(802\) 55.0997 1.94564
\(803\) 4.68812 0.165440
\(804\) 35.2203 1.24212
\(805\) 4.07010 0.143452
\(806\) −118.801 −4.18459
\(807\) −28.6700 −1.00923
\(808\) 86.8556 3.05557
\(809\) 8.65520 0.304301 0.152150 0.988357i \(-0.451380\pi\)
0.152150 + 0.988357i \(0.451380\pi\)
\(810\) 5.95430 0.209213
\(811\) 32.2900 1.13385 0.566927 0.823768i \(-0.308132\pi\)
0.566927 + 0.823768i \(0.308132\pi\)
\(812\) −5.91265 −0.207493
\(813\) 9.20417 0.322804
\(814\) 9.98184 0.349863
\(815\) 15.3273 0.536893
\(816\) −101.487 −3.55275
\(817\) −8.47623 −0.296546
\(818\) 33.3893 1.16743
\(819\) −1.67663 −0.0585863
\(820\) −19.4259 −0.678382
\(821\) 47.9178 1.67234 0.836172 0.548468i \(-0.184789\pi\)
0.836172 + 0.548468i \(0.184789\pi\)
\(822\) 33.8512 1.18070
\(823\) 47.1327 1.64294 0.821471 0.570250i \(-0.193154\pi\)
0.821471 + 0.570250i \(0.193154\pi\)
\(824\) 97.7684 3.40592
\(825\) 0.0745076 0.00259402
\(826\) −6.58681 −0.229185
\(827\) 46.0328 1.60072 0.800358 0.599522i \(-0.204643\pi\)
0.800358 + 0.599522i \(0.204643\pi\)
\(828\) 25.5737 0.888746
\(829\) −18.0705 −0.627616 −0.313808 0.949486i \(-0.601605\pi\)
−0.313808 + 0.949486i \(0.601605\pi\)
\(830\) 32.3102 1.12150
\(831\) −19.8425 −0.688330
\(832\) −100.942 −3.49952
\(833\) 51.8480 1.79643
\(834\) −24.1218 −0.835270
\(835\) −22.0300 −0.762378
\(836\) −2.86509 −0.0990913
\(837\) 10.0216 0.346396
\(838\) −36.2034 −1.25062
\(839\) −8.19390 −0.282885 −0.141442 0.989946i \(-0.545174\pi\)
−0.141442 + 0.989946i \(0.545174\pi\)
\(840\) 7.48467 0.258246
\(841\) −20.4394 −0.704808
\(842\) 25.8065 0.889351
\(843\) 32.7137 1.12672
\(844\) −53.4246 −1.83895
\(845\) 13.8333 0.475881
\(846\) 2.70026 0.0928369
\(847\) −4.08902 −0.140500
\(848\) 68.8040 2.36274
\(849\) −26.7479 −0.917985
\(850\) 2.81072 0.0964070
\(851\) −32.9958 −1.13108
\(852\) 45.3504 1.55368
\(853\) 23.0917 0.790646 0.395323 0.918542i \(-0.370633\pi\)
0.395323 + 0.918542i \(0.370633\pi\)
\(854\) 9.96105 0.340860
\(855\) 2.20508 0.0754123
\(856\) −69.3035 −2.36874
\(857\) 29.1910 0.997146 0.498573 0.866848i \(-0.333858\pi\)
0.498573 + 0.866848i \(0.333858\pi\)
\(858\) 6.41877 0.219133
\(859\) −45.4744 −1.55157 −0.775783 0.630999i \(-0.782645\pi\)
−0.775783 + 0.630999i \(0.782645\pi\)
\(860\) −98.9007 −3.37249
\(861\) −0.635833 −0.0216691
\(862\) −34.0728 −1.16052
\(863\) 31.0730 1.05774 0.528869 0.848703i \(-0.322616\pi\)
0.528869 + 0.848703i \(0.322616\pi\)
\(864\) 18.4520 0.627750
\(865\) −19.6701 −0.668804
\(866\) 43.2801 1.47072
\(867\) 40.2212 1.36598
\(868\) 20.2519 0.687395
\(869\) 1.35155 0.0458483
\(870\) −17.4214 −0.590640
\(871\) −29.2213 −0.990126
\(872\) 148.798 5.03892
\(873\) −8.67832 −0.293717
\(874\) 13.0505 0.441440
\(875\) −4.32658 −0.146265
\(876\) −45.8146 −1.54793
\(877\) −4.04121 −0.136462 −0.0682311 0.997670i \(-0.521736\pi\)
−0.0682311 + 0.997670i \(0.521736\pi\)
\(878\) 8.26783 0.279026
\(879\) −5.15932 −0.174020
\(880\) −16.0185 −0.539985
\(881\) −24.3680 −0.820980 −0.410490 0.911865i \(-0.634642\pi\)
−0.410490 + 0.911865i \(0.634642\pi\)
\(882\) −18.5080 −0.623196
\(883\) −45.8443 −1.54278 −0.771391 0.636362i \(-0.780439\pi\)
−0.771391 + 0.636362i \(0.780439\pi\)
\(884\) 175.723 5.91021
\(885\) −14.0843 −0.473439
\(886\) 25.7098 0.863739
\(887\) 25.2072 0.846375 0.423187 0.906042i \(-0.360911\pi\)
0.423187 + 0.906042i \(0.360911\pi\)
\(888\) −60.6773 −2.03620
\(889\) −4.49091 −0.150620
\(890\) −75.3944 −2.52723
\(891\) −0.541461 −0.0181396
\(892\) 103.102 3.45212
\(893\) 1.00000 0.0334637
\(894\) 11.3604 0.379950
\(895\) 43.5191 1.45468
\(896\) 9.61739 0.321294
\(897\) −21.2178 −0.708441
\(898\) 84.5068 2.82003
\(899\) −29.3216 −0.977928
\(900\) −0.728124 −0.0242708
\(901\) −38.7937 −1.29241
\(902\) 2.43421 0.0810502
\(903\) −3.23714 −0.107725
\(904\) −124.751 −4.14916
\(905\) −5.55344 −0.184603
\(906\) 48.6684 1.61690
\(907\) −28.6832 −0.952409 −0.476205 0.879335i \(-0.657988\pi\)
−0.476205 + 0.879335i \(0.657988\pi\)
\(908\) 74.9978 2.48889
\(909\) 9.77259 0.324136
\(910\) −9.98318 −0.330939
\(911\) 44.0242 1.45859 0.729293 0.684201i \(-0.239849\pi\)
0.729293 + 0.684201i \(0.239849\pi\)
\(912\) 13.4162 0.444256
\(913\) −2.93816 −0.0972390
\(914\) 8.50385 0.281282
\(915\) 21.2993 0.704133
\(916\) −79.8295 −2.63764
\(917\) −2.56558 −0.0847229
\(918\) −20.4261 −0.674160
\(919\) −23.2387 −0.766573 −0.383287 0.923629i \(-0.625208\pi\)
−0.383287 + 0.923629i \(0.625208\pi\)
\(920\) 94.7185 3.12278
\(921\) 5.44081 0.179281
\(922\) 98.6585 3.24914
\(923\) −37.6260 −1.23848
\(924\) −1.09420 −0.0359966
\(925\) 0.939446 0.0308888
\(926\) 6.53920 0.214892
\(927\) 11.0004 0.361302
\(928\) −53.9877 −1.77223
\(929\) −41.4909 −1.36127 −0.680636 0.732622i \(-0.738296\pi\)
−0.680636 + 0.732622i \(0.738296\pi\)
\(930\) 59.6715 1.95670
\(931\) −6.85415 −0.224636
\(932\) 156.439 5.12432
\(933\) 21.0326 0.688575
\(934\) 25.7779 0.843477
\(935\) 9.03172 0.295369
\(936\) −39.0182 −1.27535
\(937\) −20.6869 −0.675812 −0.337906 0.941180i \(-0.609719\pi\)
−0.337906 + 0.941180i \(0.609719\pi\)
\(938\) 6.86413 0.224122
\(939\) −15.1762 −0.495256
\(940\) 11.6680 0.380569
\(941\) 42.9252 1.39932 0.699661 0.714475i \(-0.253334\pi\)
0.699661 + 0.714475i \(0.253334\pi\)
\(942\) 46.5229 1.51580
\(943\) −8.04648 −0.262029
\(944\) −85.6923 −2.78905
\(945\) 0.842139 0.0273948
\(946\) 12.3930 0.402930
\(947\) 54.2239 1.76204 0.881020 0.473079i \(-0.156857\pi\)
0.881020 + 0.473079i \(0.156857\pi\)
\(948\) −13.2080 −0.428977
\(949\) 38.0111 1.23389
\(950\) −0.371569 −0.0120553
\(951\) −4.90155 −0.158944
\(952\) −25.6759 −0.832161
\(953\) 29.3572 0.950972 0.475486 0.879723i \(-0.342272\pi\)
0.475486 + 0.879723i \(0.342272\pi\)
\(954\) 13.8480 0.448347
\(955\) 40.4067 1.30753
\(956\) 14.4824 0.468395
\(957\) 1.58423 0.0512109
\(958\) −36.6268 −1.18336
\(959\) 4.78770 0.154603
\(960\) 50.7010 1.63637
\(961\) 69.4318 2.23974
\(962\) 80.9325 2.60937
\(963\) −7.79770 −0.251277
\(964\) −70.2730 −2.26334
\(965\) 53.4346 1.72012
\(966\) 4.98409 0.160360
\(967\) −15.2193 −0.489420 −0.244710 0.969596i \(-0.578693\pi\)
−0.244710 + 0.969596i \(0.578693\pi\)
\(968\) −95.1588 −3.05852
\(969\) −7.56447 −0.243006
\(970\) −51.6734 −1.65913
\(971\) 0.850186 0.0272838 0.0136419 0.999907i \(-0.495658\pi\)
0.0136419 + 0.999907i \(0.495658\pi\)
\(972\) 5.29141 0.169722
\(973\) −3.41164 −0.109372
\(974\) −75.0036 −2.40327
\(975\) 0.604105 0.0193469
\(976\) 129.590 4.14807
\(977\) 1.29206 0.0413366 0.0206683 0.999786i \(-0.493421\pi\)
0.0206683 + 0.999786i \(0.493421\pi\)
\(978\) 18.7693 0.600175
\(979\) 6.85607 0.219121
\(980\) −79.9743 −2.55468
\(981\) 16.7420 0.534531
\(982\) −37.1426 −1.18527
\(983\) 24.2367 0.773032 0.386516 0.922283i \(-0.373678\pi\)
0.386516 + 0.922283i \(0.373678\pi\)
\(984\) −14.7970 −0.471711
\(985\) 33.7525 1.07544
\(986\) 59.7635 1.90326
\(987\) 0.381908 0.0121563
\(988\) −23.2301 −0.739047
\(989\) −40.9660 −1.30264
\(990\) −3.22402 −0.102466
\(991\) 41.8952 1.33084 0.665422 0.746467i \(-0.268251\pi\)
0.665422 + 0.746467i \(0.268251\pi\)
\(992\) 184.918 5.87115
\(993\) −36.1520 −1.14725
\(994\) 8.83841 0.280337
\(995\) 33.4917 1.06176
\(996\) 28.7131 0.909811
\(997\) −13.4242 −0.425149 −0.212574 0.977145i \(-0.568185\pi\)
−0.212574 + 0.977145i \(0.568185\pi\)
\(998\) 35.0638 1.10993
\(999\) −6.82713 −0.216001
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 2679.2.a.n.1.23 23
3.2 odd 2 8037.2.a.p.1.1 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2679.2.a.n.1.23 23 1.1 even 1 trivial
8037.2.a.p.1.1 23 3.2 odd 2