Properties

Label 4007.2.a.a.1.14
Level $4007$
Weight $2$
Character 4007.1
Self dual yes
Analytic conductor $31.996$
Analytic rank $1$
Dimension $139$
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] = [4007,2,Mod(1,4007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4007, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4007.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: \(31.9960560899\)
Analytic rank: \(1\)
Dimension: \(139\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 4007.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.39846 q^{2} -2.50987 q^{3} +3.75261 q^{4} +0.204658 q^{5} +6.01982 q^{6} -4.29057 q^{7} -4.20358 q^{8} +3.29943 q^{9} +O(q^{10})\) \(q-2.39846 q^{2} -2.50987 q^{3} +3.75261 q^{4} +0.204658 q^{5} +6.01982 q^{6} -4.29057 q^{7} -4.20358 q^{8} +3.29943 q^{9} -0.490864 q^{10} +5.44379 q^{11} -9.41856 q^{12} +4.38906 q^{13} +10.2908 q^{14} -0.513664 q^{15} +2.57688 q^{16} -2.66989 q^{17} -7.91356 q^{18} +5.03946 q^{19} +0.768001 q^{20} +10.7688 q^{21} -13.0567 q^{22} -4.73681 q^{23} +10.5504 q^{24} -4.95812 q^{25} -10.5270 q^{26} -0.751541 q^{27} -16.1008 q^{28} -5.83511 q^{29} +1.23200 q^{30} -2.59154 q^{31} +2.22660 q^{32} -13.6632 q^{33} +6.40362 q^{34} -0.878098 q^{35} +12.3815 q^{36} -0.316382 q^{37} -12.0869 q^{38} -11.0160 q^{39} -0.860294 q^{40} +4.20345 q^{41} -25.8284 q^{42} -5.47822 q^{43} +20.4285 q^{44} +0.675255 q^{45} +11.3610 q^{46} +0.230350 q^{47} -6.46763 q^{48} +11.4090 q^{49} +11.8918 q^{50} +6.70107 q^{51} +16.4704 q^{52} +1.38245 q^{53} +1.80254 q^{54} +1.11411 q^{55} +18.0357 q^{56} -12.6484 q^{57} +13.9953 q^{58} -6.50513 q^{59} -1.92758 q^{60} -3.83527 q^{61} +6.21571 q^{62} -14.1564 q^{63} -10.4942 q^{64} +0.898255 q^{65} +32.7707 q^{66} +7.74657 q^{67} -10.0191 q^{68} +11.8888 q^{69} +2.10608 q^{70} +7.88074 q^{71} -13.8694 q^{72} -1.60794 q^{73} +0.758829 q^{74} +12.4442 q^{75} +18.9111 q^{76} -23.3570 q^{77} +26.4213 q^{78} +14.5083 q^{79} +0.527379 q^{80} -8.01203 q^{81} -10.0818 q^{82} -1.36997 q^{83} +40.4110 q^{84} -0.546413 q^{85} +13.1393 q^{86} +14.6453 q^{87} -22.8834 q^{88} -2.34048 q^{89} -1.61957 q^{90} -18.8316 q^{91} -17.7754 q^{92} +6.50443 q^{93} -0.552484 q^{94} +1.03136 q^{95} -5.58847 q^{96} -17.4363 q^{97} -27.3640 q^{98} +17.9614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 139 q - 13 q^{2} - 22 q^{3} + 113 q^{4} - 16 q^{5} - 15 q^{6} - 44 q^{7} - 36 q^{8} + 87 q^{9} - 40 q^{10} - 17 q^{11} - 59 q^{12} - 89 q^{13} - 15 q^{14} - 29 q^{15} + 73 q^{16} - 58 q^{17} - 51 q^{18} - 37 q^{19} - 24 q^{20} - 37 q^{21} - 99 q^{22} - 42 q^{23} - 27 q^{24} - 11 q^{25} + 2 q^{26} - 73 q^{27} - 113 q^{28} - 57 q^{29} - 29 q^{30} - 51 q^{31} - 80 q^{32} - 78 q^{33} - 28 q^{34} - 34 q^{35} + 28 q^{36} - 117 q^{37} - 31 q^{38} - 36 q^{39} - 107 q^{40} - 60 q^{41} - 41 q^{42} - 109 q^{43} - 21 q^{44} - 62 q^{45} - 92 q^{46} - 26 q^{47} - 90 q^{48} - 7 q^{49} - 22 q^{50} - 47 q^{51} - 182 q^{52} - 83 q^{53} - 19 q^{54} - 53 q^{55} - 23 q^{56} - 201 q^{57} - 112 q^{58} + 14 q^{59} - 64 q^{60} - 73 q^{61} - 21 q^{62} - 94 q^{63} + 14 q^{64} - 123 q^{65} - 10 q^{66} - 135 q^{67} - 84 q^{68} - 50 q^{69} - 35 q^{70} - 29 q^{71} - 143 q^{72} - 266 q^{73} - 53 q^{74} - 32 q^{75} - 66 q^{76} - 69 q^{77} - 59 q^{78} - 124 q^{79} - 20 q^{80} - 33 q^{81} - 93 q^{82} - 28 q^{83} - 4 q^{84} - 179 q^{85} + 6 q^{86} - 40 q^{87} - 259 q^{88} - 41 q^{89} + 2 q^{90} - 50 q^{91} - 77 q^{92} - 60 q^{93} - 48 q^{94} - 37 q^{95} + 3 q^{96} - 220 q^{97} - 9 q^{98} - 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39846 −1.69597 −0.847984 0.530022i \(-0.822184\pi\)
−0.847984 + 0.530022i \(0.822184\pi\)
\(3\) −2.50987 −1.44907 −0.724536 0.689237i \(-0.757946\pi\)
−0.724536 + 0.689237i \(0.757946\pi\)
\(4\) 3.75261 1.87631
\(5\) 0.204658 0.0915257 0.0457629 0.998952i \(-0.485428\pi\)
0.0457629 + 0.998952i \(0.485428\pi\)
\(6\) 6.01982 2.45758
\(7\) −4.29057 −1.62168 −0.810841 0.585266i \(-0.800990\pi\)
−0.810841 + 0.585266i \(0.800990\pi\)
\(8\) −4.20358 −1.48619
\(9\) 3.29943 1.09981
\(10\) −0.490864 −0.155225
\(11\) 5.44379 1.64137 0.820683 0.571384i \(-0.193593\pi\)
0.820683 + 0.571384i \(0.193593\pi\)
\(12\) −9.41856 −2.71891
\(13\) 4.38906 1.21731 0.608653 0.793437i \(-0.291710\pi\)
0.608653 + 0.793437i \(0.291710\pi\)
\(14\) 10.2908 2.75032
\(15\) −0.513664 −0.132627
\(16\) 2.57688 0.644221
\(17\) −2.66989 −0.647543 −0.323772 0.946135i \(-0.604951\pi\)
−0.323772 + 0.946135i \(0.604951\pi\)
\(18\) −7.91356 −1.86525
\(19\) 5.03946 1.15613 0.578065 0.815990i \(-0.303808\pi\)
0.578065 + 0.815990i \(0.303808\pi\)
\(20\) 0.768001 0.171730
\(21\) 10.7688 2.34994
\(22\) −13.0567 −2.78370
\(23\) −4.73681 −0.987692 −0.493846 0.869549i \(-0.664409\pi\)
−0.493846 + 0.869549i \(0.664409\pi\)
\(24\) 10.5504 2.15359
\(25\) −4.95812 −0.991623
\(26\) −10.5270 −2.06451
\(27\) −0.751541 −0.144634
\(28\) −16.1008 −3.04277
\(29\) −5.83511 −1.08355 −0.541776 0.840523i \(-0.682248\pi\)
−0.541776 + 0.840523i \(0.682248\pi\)
\(30\) 1.23200 0.224932
\(31\) −2.59154 −0.465455 −0.232727 0.972542i \(-0.574765\pi\)
−0.232727 + 0.972542i \(0.574765\pi\)
\(32\) 2.22660 0.393611
\(33\) −13.6632 −2.37846
\(34\) 6.40362 1.09821
\(35\) −0.878098 −0.148426
\(36\) 12.3815 2.06358
\(37\) −0.316382 −0.0520128 −0.0260064 0.999662i \(-0.508279\pi\)
−0.0260064 + 0.999662i \(0.508279\pi\)
\(38\) −12.0869 −1.96076
\(39\) −11.0160 −1.76396
\(40\) −0.860294 −0.136024
\(41\) 4.20345 0.656469 0.328234 0.944596i \(-0.393546\pi\)
0.328234 + 0.944596i \(0.393546\pi\)
\(42\) −25.8284 −3.98541
\(43\) −5.47822 −0.835420 −0.417710 0.908580i \(-0.637167\pi\)
−0.417710 + 0.908580i \(0.637167\pi\)
\(44\) 20.4285 3.07971
\(45\) 0.675255 0.100661
\(46\) 11.3610 1.67509
\(47\) 0.230350 0.0335999 0.0168000 0.999859i \(-0.494652\pi\)
0.0168000 + 0.999859i \(0.494652\pi\)
\(48\) −6.46763 −0.933523
\(49\) 11.4090 1.62985
\(50\) 11.8918 1.68176
\(51\) 6.70107 0.938337
\(52\) 16.4704 2.28404
\(53\) 1.38245 0.189895 0.0949474 0.995482i \(-0.469732\pi\)
0.0949474 + 0.995482i \(0.469732\pi\)
\(54\) 1.80254 0.245295
\(55\) 1.11411 0.150227
\(56\) 18.0357 2.41013
\(57\) −12.6484 −1.67532
\(58\) 13.9953 1.83767
\(59\) −6.50513 −0.846896 −0.423448 0.905920i \(-0.639180\pi\)
−0.423448 + 0.905920i \(0.639180\pi\)
\(60\) −1.92758 −0.248850
\(61\) −3.83527 −0.491056 −0.245528 0.969389i \(-0.578961\pi\)
−0.245528 + 0.969389i \(0.578961\pi\)
\(62\) 6.21571 0.789396
\(63\) −14.1564 −1.78354
\(64\) −10.4942 −1.31177
\(65\) 0.898255 0.111415
\(66\) 32.7707 4.03379
\(67\) 7.74657 0.946394 0.473197 0.880957i \(-0.343100\pi\)
0.473197 + 0.880957i \(0.343100\pi\)
\(68\) −10.0191 −1.21499
\(69\) 11.8888 1.43124
\(70\) 2.10608 0.251725
\(71\) 7.88074 0.935271 0.467636 0.883921i \(-0.345106\pi\)
0.467636 + 0.883921i \(0.345106\pi\)
\(72\) −13.8694 −1.63453
\(73\) −1.60794 −0.188195 −0.0940977 0.995563i \(-0.529997\pi\)
−0.0940977 + 0.995563i \(0.529997\pi\)
\(74\) 0.758829 0.0882121
\(75\) 12.4442 1.43693
\(76\) 18.9111 2.16926
\(77\) −23.3570 −2.66177
\(78\) 26.4213 2.99163
\(79\) 14.5083 1.63231 0.816153 0.577836i \(-0.196102\pi\)
0.816153 + 0.577836i \(0.196102\pi\)
\(80\) 0.527379 0.0589628
\(81\) −8.01203 −0.890226
\(82\) −10.0818 −1.11335
\(83\) −1.36997 −0.150373 −0.0751866 0.997169i \(-0.523955\pi\)
−0.0751866 + 0.997169i \(0.523955\pi\)
\(84\) 40.4110 4.40920
\(85\) −0.546413 −0.0592668
\(86\) 13.1393 1.41685
\(87\) 14.6453 1.57015
\(88\) −22.8834 −2.43938
\(89\) −2.34048 −0.248091 −0.124045 0.992277i \(-0.539587\pi\)
−0.124045 + 0.992277i \(0.539587\pi\)
\(90\) −1.61957 −0.170718
\(91\) −18.8316 −1.97408
\(92\) −17.7754 −1.85321
\(93\) 6.50443 0.674478
\(94\) −0.552484 −0.0569844
\(95\) 1.03136 0.105816
\(96\) −5.58847 −0.570371
\(97\) −17.4363 −1.77038 −0.885192 0.465225i \(-0.845973\pi\)
−0.885192 + 0.465225i \(0.845973\pi\)
\(98\) −27.3640 −2.76418
\(99\) 17.9614 1.80519
\(100\) −18.6059 −1.86059
\(101\) 1.92809 0.191852 0.0959260 0.995388i \(-0.469419\pi\)
0.0959260 + 0.995388i \(0.469419\pi\)
\(102\) −16.0722 −1.59139
\(103\) 1.85189 0.182473 0.0912363 0.995829i \(-0.470918\pi\)
0.0912363 + 0.995829i \(0.470918\pi\)
\(104\) −18.4497 −1.80915
\(105\) 2.20391 0.215080
\(106\) −3.31576 −0.322055
\(107\) 8.95957 0.866154 0.433077 0.901357i \(-0.357428\pi\)
0.433077 + 0.901357i \(0.357428\pi\)
\(108\) −2.82024 −0.271378
\(109\) −7.30336 −0.699535 −0.349768 0.936836i \(-0.613739\pi\)
−0.349768 + 0.936836i \(0.613739\pi\)
\(110\) −2.67216 −0.254780
\(111\) 0.794076 0.0753703
\(112\) −11.0563 −1.04472
\(113\) −9.67194 −0.909860 −0.454930 0.890527i \(-0.650336\pi\)
−0.454930 + 0.890527i \(0.650336\pi\)
\(114\) 30.3366 2.84129
\(115\) −0.969424 −0.0903993
\(116\) −21.8969 −2.03308
\(117\) 14.4814 1.33881
\(118\) 15.6023 1.43631
\(119\) 11.4553 1.05011
\(120\) 2.15922 0.197109
\(121\) 18.6349 1.69408
\(122\) 9.19875 0.832815
\(123\) −10.5501 −0.951271
\(124\) −9.72506 −0.873336
\(125\) −2.03801 −0.182285
\(126\) 33.9537 3.02483
\(127\) 10.7486 0.953784 0.476892 0.878962i \(-0.341763\pi\)
0.476892 + 0.878962i \(0.341763\pi\)
\(128\) 20.7167 1.83111
\(129\) 13.7496 1.21058
\(130\) −2.15443 −0.188956
\(131\) 7.27828 0.635906 0.317953 0.948106i \(-0.397005\pi\)
0.317953 + 0.948106i \(0.397005\pi\)
\(132\) −51.2727 −4.46272
\(133\) −21.6221 −1.87488
\(134\) −18.5798 −1.60505
\(135\) −0.153809 −0.0132377
\(136\) 11.2231 0.962371
\(137\) 17.6360 1.50675 0.753373 0.657594i \(-0.228426\pi\)
0.753373 + 0.657594i \(0.228426\pi\)
\(138\) −28.5147 −2.42733
\(139\) 14.9558 1.26853 0.634265 0.773115i \(-0.281303\pi\)
0.634265 + 0.773115i \(0.281303\pi\)
\(140\) −3.29516 −0.278492
\(141\) −0.578147 −0.0486887
\(142\) −18.9016 −1.58619
\(143\) 23.8931 1.99804
\(144\) 8.50226 0.708521
\(145\) −1.19420 −0.0991729
\(146\) 3.85659 0.319173
\(147\) −28.6350 −2.36178
\(148\) −1.18726 −0.0975920
\(149\) −14.5422 −1.19135 −0.595673 0.803227i \(-0.703114\pi\)
−0.595673 + 0.803227i \(0.703114\pi\)
\(150\) −29.8470 −2.43699
\(151\) 5.05088 0.411035 0.205517 0.978653i \(-0.434112\pi\)
0.205517 + 0.978653i \(0.434112\pi\)
\(152\) −21.1837 −1.71823
\(153\) −8.80912 −0.712175
\(154\) 56.0208 4.51428
\(155\) −0.530379 −0.0426011
\(156\) −41.3386 −3.30974
\(157\) 5.06912 0.404560 0.202280 0.979328i \(-0.435165\pi\)
0.202280 + 0.979328i \(0.435165\pi\)
\(158\) −34.7975 −2.76834
\(159\) −3.46978 −0.275171
\(160\) 0.455691 0.0360255
\(161\) 20.3236 1.60172
\(162\) 19.2166 1.50979
\(163\) −4.69674 −0.367877 −0.183939 0.982938i \(-0.558885\pi\)
−0.183939 + 0.982938i \(0.558885\pi\)
\(164\) 15.7739 1.23174
\(165\) −2.79628 −0.217690
\(166\) 3.28581 0.255028
\(167\) −12.2300 −0.946383 −0.473191 0.880960i \(-0.656898\pi\)
−0.473191 + 0.880960i \(0.656898\pi\)
\(168\) −45.2673 −3.49245
\(169\) 6.26383 0.481833
\(170\) 1.31055 0.100515
\(171\) 16.6274 1.27153
\(172\) −20.5576 −1.56750
\(173\) 20.7435 1.57710 0.788550 0.614970i \(-0.210832\pi\)
0.788550 + 0.614970i \(0.210832\pi\)
\(174\) −35.1263 −2.66292
\(175\) 21.2731 1.60810
\(176\) 14.0280 1.05740
\(177\) 16.3270 1.22721
\(178\) 5.61356 0.420754
\(179\) 24.7729 1.85161 0.925807 0.377997i \(-0.123387\pi\)
0.925807 + 0.377997i \(0.123387\pi\)
\(180\) 2.53397 0.188871
\(181\) −20.0145 −1.48766 −0.743832 0.668367i \(-0.766994\pi\)
−0.743832 + 0.668367i \(0.766994\pi\)
\(182\) 45.1667 3.34798
\(183\) 9.62602 0.711576
\(184\) 19.9115 1.46790
\(185\) −0.0647499 −0.00476051
\(186\) −15.6006 −1.14389
\(187\) −14.5343 −1.06285
\(188\) 0.864413 0.0630438
\(189\) 3.22454 0.234551
\(190\) −2.47369 −0.179460
\(191\) 19.2432 1.39239 0.696194 0.717854i \(-0.254875\pi\)
0.696194 + 0.717854i \(0.254875\pi\)
\(192\) 26.3390 1.90085
\(193\) −16.7280 −1.20411 −0.602054 0.798455i \(-0.705651\pi\)
−0.602054 + 0.798455i \(0.705651\pi\)
\(194\) 41.8202 3.00252
\(195\) −2.25450 −0.161448
\(196\) 42.8135 3.05810
\(197\) 13.3001 0.947591 0.473796 0.880635i \(-0.342884\pi\)
0.473796 + 0.880635i \(0.342884\pi\)
\(198\) −43.0798 −3.06155
\(199\) −2.39195 −0.169561 −0.0847804 0.996400i \(-0.527019\pi\)
−0.0847804 + 0.996400i \(0.527019\pi\)
\(200\) 20.8418 1.47374
\(201\) −19.4429 −1.37139
\(202\) −4.62445 −0.325375
\(203\) 25.0359 1.75718
\(204\) 25.1465 1.76061
\(205\) 0.860269 0.0600838
\(206\) −4.44170 −0.309468
\(207\) −15.6288 −1.08628
\(208\) 11.3101 0.784214
\(209\) 27.4338 1.89763
\(210\) −5.28599 −0.364768
\(211\) −22.5413 −1.55180 −0.775902 0.630853i \(-0.782705\pi\)
−0.775902 + 0.630853i \(0.782705\pi\)
\(212\) 5.18782 0.356301
\(213\) −19.7796 −1.35528
\(214\) −21.4892 −1.46897
\(215\) −1.12116 −0.0764624
\(216\) 3.15916 0.214954
\(217\) 11.1192 0.754820
\(218\) 17.5168 1.18639
\(219\) 4.03572 0.272709
\(220\) 4.18084 0.281872
\(221\) −11.7183 −0.788258
\(222\) −1.90456 −0.127826
\(223\) 13.5262 0.905780 0.452890 0.891566i \(-0.350393\pi\)
0.452890 + 0.891566i \(0.350393\pi\)
\(224\) −9.55337 −0.638311
\(225\) −16.3590 −1.09060
\(226\) 23.1978 1.54309
\(227\) −11.9248 −0.791475 −0.395737 0.918364i \(-0.629511\pi\)
−0.395737 + 0.918364i \(0.629511\pi\)
\(228\) −47.4645 −3.14341
\(229\) −23.8963 −1.57911 −0.789557 0.613677i \(-0.789690\pi\)
−0.789557 + 0.613677i \(0.789690\pi\)
\(230\) 2.32513 0.153314
\(231\) 58.6229 3.85710
\(232\) 24.5283 1.61036
\(233\) 0.363211 0.0237947 0.0118974 0.999929i \(-0.496213\pi\)
0.0118974 + 0.999929i \(0.496213\pi\)
\(234\) −34.7331 −2.27057
\(235\) 0.0471428 0.00307526
\(236\) −24.4113 −1.58904
\(237\) −36.4138 −2.36533
\(238\) −27.4752 −1.78095
\(239\) −8.23364 −0.532590 −0.266295 0.963892i \(-0.585799\pi\)
−0.266295 + 0.963892i \(0.585799\pi\)
\(240\) −1.32365 −0.0854413
\(241\) 24.0996 1.55239 0.776195 0.630492i \(-0.217147\pi\)
0.776195 + 0.630492i \(0.217147\pi\)
\(242\) −44.6951 −2.87311
\(243\) 22.3638 1.43464
\(244\) −14.3923 −0.921372
\(245\) 2.33493 0.149173
\(246\) 25.3040 1.61332
\(247\) 22.1185 1.40736
\(248\) 10.8937 0.691754
\(249\) 3.43843 0.217902
\(250\) 4.88808 0.309149
\(251\) −24.6273 −1.55446 −0.777231 0.629215i \(-0.783377\pi\)
−0.777231 + 0.629215i \(0.783377\pi\)
\(252\) −53.1237 −3.34648
\(253\) −25.7862 −1.62116
\(254\) −25.7801 −1.61759
\(255\) 1.37143 0.0858820
\(256\) −28.6998 −1.79374
\(257\) 5.82941 0.363629 0.181814 0.983333i \(-0.441803\pi\)
0.181814 + 0.983333i \(0.441803\pi\)
\(258\) −32.9779 −2.05311
\(259\) 1.35746 0.0843483
\(260\) 3.37080 0.209048
\(261\) −19.2526 −1.19170
\(262\) −17.4567 −1.07848
\(263\) 29.9351 1.84587 0.922937 0.384950i \(-0.125781\pi\)
0.922937 + 0.384950i \(0.125781\pi\)
\(264\) 57.4343 3.53484
\(265\) 0.282930 0.0173803
\(266\) 51.8598 3.17973
\(267\) 5.87430 0.359501
\(268\) 29.0699 1.77573
\(269\) 2.02825 0.123664 0.0618322 0.998087i \(-0.480306\pi\)
0.0618322 + 0.998087i \(0.480306\pi\)
\(270\) 0.368904 0.0224508
\(271\) −4.78683 −0.290779 −0.145390 0.989374i \(-0.546444\pi\)
−0.145390 + 0.989374i \(0.546444\pi\)
\(272\) −6.87999 −0.417161
\(273\) 47.2647 2.86059
\(274\) −42.2993 −2.55539
\(275\) −26.9910 −1.62762
\(276\) 44.6139 2.68544
\(277\) −10.4445 −0.627547 −0.313774 0.949498i \(-0.601593\pi\)
−0.313774 + 0.949498i \(0.601593\pi\)
\(278\) −35.8708 −2.15139
\(279\) −8.55063 −0.511913
\(280\) 3.69115 0.220588
\(281\) −16.5320 −0.986215 −0.493107 0.869968i \(-0.664139\pi\)
−0.493107 + 0.869968i \(0.664139\pi\)
\(282\) 1.38666 0.0825745
\(283\) −14.1500 −0.841128 −0.420564 0.907263i \(-0.638168\pi\)
−0.420564 + 0.907263i \(0.638168\pi\)
\(284\) 29.5734 1.75486
\(285\) −2.58859 −0.153335
\(286\) −57.3067 −3.38862
\(287\) −18.0352 −1.06458
\(288\) 7.34652 0.432898
\(289\) −9.87170 −0.580688
\(290\) 2.86424 0.168194
\(291\) 43.7627 2.56542
\(292\) −6.03399 −0.353112
\(293\) 33.5494 1.95998 0.979988 0.199056i \(-0.0637874\pi\)
0.979988 + 0.199056i \(0.0637874\pi\)
\(294\) 68.6799 4.00549
\(295\) −1.33133 −0.0775128
\(296\) 1.32993 0.0773008
\(297\) −4.09124 −0.237398
\(298\) 34.8790 2.02048
\(299\) −20.7901 −1.20232
\(300\) 46.6983 2.69613
\(301\) 23.5047 1.35479
\(302\) −12.1143 −0.697102
\(303\) −4.83925 −0.278008
\(304\) 12.9861 0.744804
\(305\) −0.784918 −0.0449443
\(306\) 21.1283 1.20783
\(307\) −22.2473 −1.26972 −0.634860 0.772628i \(-0.718942\pi\)
−0.634860 + 0.772628i \(0.718942\pi\)
\(308\) −87.6497 −4.99430
\(309\) −4.64801 −0.264416
\(310\) 1.27209 0.0722501
\(311\) 22.8842 1.29765 0.648823 0.760939i \(-0.275262\pi\)
0.648823 + 0.760939i \(0.275262\pi\)
\(312\) 46.3064 2.62158
\(313\) −14.2580 −0.805912 −0.402956 0.915219i \(-0.632017\pi\)
−0.402956 + 0.915219i \(0.632017\pi\)
\(314\) −12.1581 −0.686120
\(315\) −2.89723 −0.163240
\(316\) 54.4439 3.06271
\(317\) −14.1865 −0.796794 −0.398397 0.917213i \(-0.630433\pi\)
−0.398397 + 0.917213i \(0.630433\pi\)
\(318\) 8.32213 0.466682
\(319\) −31.7651 −1.77851
\(320\) −2.14771 −0.120061
\(321\) −22.4873 −1.25512
\(322\) −48.7453 −2.71647
\(323\) −13.4548 −0.748645
\(324\) −30.0661 −1.67034
\(325\) −21.7615 −1.20711
\(326\) 11.2649 0.623908
\(327\) 18.3305 1.01368
\(328\) −17.6695 −0.975636
\(329\) −0.988330 −0.0544884
\(330\) 6.70677 0.369195
\(331\) 5.32710 0.292804 0.146402 0.989225i \(-0.453231\pi\)
0.146402 + 0.989225i \(0.453231\pi\)
\(332\) −5.14095 −0.282146
\(333\) −1.04388 −0.0572043
\(334\) 29.3331 1.60503
\(335\) 1.58540 0.0866194
\(336\) 27.7498 1.51388
\(337\) −21.1201 −1.15048 −0.575241 0.817984i \(-0.695092\pi\)
−0.575241 + 0.817984i \(0.695092\pi\)
\(338\) −15.0236 −0.817174
\(339\) 24.2753 1.31845
\(340\) −2.05048 −0.111203
\(341\) −14.1078 −0.763982
\(342\) −39.8801 −2.15647
\(343\) −18.9170 −1.02142
\(344\) 23.0281 1.24159
\(345\) 2.43313 0.130995
\(346\) −49.7525 −2.67471
\(347\) −10.9753 −0.589183 −0.294592 0.955623i \(-0.595184\pi\)
−0.294592 + 0.955623i \(0.595184\pi\)
\(348\) 54.9583 2.94608
\(349\) 10.0533 0.538141 0.269071 0.963120i \(-0.413284\pi\)
0.269071 + 0.963120i \(0.413284\pi\)
\(350\) −51.0228 −2.72728
\(351\) −3.29856 −0.176064
\(352\) 12.1211 0.646059
\(353\) 20.4812 1.09011 0.545053 0.838402i \(-0.316510\pi\)
0.545053 + 0.838402i \(0.316510\pi\)
\(354\) −39.1597 −2.08132
\(355\) 1.61285 0.0856014
\(356\) −8.78293 −0.465494
\(357\) −28.7514 −1.52168
\(358\) −59.4168 −3.14028
\(359\) −19.6592 −1.03757 −0.518787 0.854903i \(-0.673616\pi\)
−0.518787 + 0.854903i \(0.673616\pi\)
\(360\) −2.83848 −0.149601
\(361\) 6.39614 0.336639
\(362\) 48.0039 2.52303
\(363\) −46.7711 −2.45485
\(364\) −70.6675 −3.70399
\(365\) −0.329078 −0.0172247
\(366\) −23.0876 −1.20681
\(367\) −32.0867 −1.67491 −0.837456 0.546505i \(-0.815958\pi\)
−0.837456 + 0.546505i \(0.815958\pi\)
\(368\) −12.2062 −0.636292
\(369\) 13.8690 0.721992
\(370\) 0.155300 0.00807367
\(371\) −5.93152 −0.307949
\(372\) 24.4086 1.26553
\(373\) 29.2769 1.51590 0.757951 0.652312i \(-0.226201\pi\)
0.757951 + 0.652312i \(0.226201\pi\)
\(374\) 34.8600 1.80257
\(375\) 5.11512 0.264144
\(376\) −0.968292 −0.0499358
\(377\) −25.6106 −1.31901
\(378\) −7.73393 −0.397790
\(379\) −17.2225 −0.884660 −0.442330 0.896852i \(-0.645848\pi\)
−0.442330 + 0.896852i \(0.645848\pi\)
\(380\) 3.87031 0.198543
\(381\) −26.9776 −1.38210
\(382\) −46.1540 −2.36144
\(383\) 31.5618 1.61273 0.806367 0.591415i \(-0.201431\pi\)
0.806367 + 0.591415i \(0.201431\pi\)
\(384\) −51.9961 −2.65341
\(385\) −4.78018 −0.243621
\(386\) 40.1215 2.04213
\(387\) −18.0750 −0.918805
\(388\) −65.4316 −3.32178
\(389\) 23.6722 1.20023 0.600115 0.799914i \(-0.295122\pi\)
0.600115 + 0.799914i \(0.295122\pi\)
\(390\) 5.40733 0.273811
\(391\) 12.6467 0.639573
\(392\) −47.9585 −2.42227
\(393\) −18.2675 −0.921474
\(394\) −31.8997 −1.60708
\(395\) 2.96923 0.149398
\(396\) 67.4024 3.38710
\(397\) −30.8537 −1.54850 −0.774251 0.632879i \(-0.781873\pi\)
−0.774251 + 0.632879i \(0.781873\pi\)
\(398\) 5.73700 0.287570
\(399\) 54.2687 2.71683
\(400\) −12.7765 −0.638824
\(401\) 7.66836 0.382940 0.191470 0.981498i \(-0.438675\pi\)
0.191470 + 0.981498i \(0.438675\pi\)
\(402\) 46.6329 2.32584
\(403\) −11.3744 −0.566601
\(404\) 7.23537 0.359973
\(405\) −1.63972 −0.0814786
\(406\) −60.0477 −2.98012
\(407\) −1.72232 −0.0853720
\(408\) −28.1684 −1.39455
\(409\) 23.1618 1.14528 0.572638 0.819808i \(-0.305920\pi\)
0.572638 + 0.819808i \(0.305920\pi\)
\(410\) −2.06332 −0.101900
\(411\) −44.2640 −2.18338
\(412\) 6.94944 0.342375
\(413\) 27.9107 1.37340
\(414\) 37.4850 1.84229
\(415\) −0.280374 −0.0137630
\(416\) 9.77267 0.479145
\(417\) −37.5370 −1.83819
\(418\) −65.7988 −3.21833
\(419\) −16.7624 −0.818895 −0.409447 0.912334i \(-0.634278\pi\)
−0.409447 + 0.912334i \(0.634278\pi\)
\(420\) 8.27042 0.403555
\(421\) −7.00713 −0.341507 −0.170753 0.985314i \(-0.554620\pi\)
−0.170753 + 0.985314i \(0.554620\pi\)
\(422\) 54.0643 2.63181
\(423\) 0.760023 0.0369536
\(424\) −5.81125 −0.282219
\(425\) 13.2376 0.642119
\(426\) 47.4406 2.29850
\(427\) 16.4555 0.796337
\(428\) 33.6218 1.62517
\(429\) −59.9686 −2.89531
\(430\) 2.68906 0.129678
\(431\) −30.4756 −1.46796 −0.733980 0.679171i \(-0.762340\pi\)
−0.733980 + 0.679171i \(0.762340\pi\)
\(432\) −1.93663 −0.0931763
\(433\) 0.765557 0.0367903 0.0183952 0.999831i \(-0.494144\pi\)
0.0183952 + 0.999831i \(0.494144\pi\)
\(434\) −26.6689 −1.28015
\(435\) 2.99728 0.143709
\(436\) −27.4067 −1.31254
\(437\) −23.8709 −1.14190
\(438\) −9.67952 −0.462505
\(439\) 6.80740 0.324900 0.162450 0.986717i \(-0.448060\pi\)
0.162450 + 0.986717i \(0.448060\pi\)
\(440\) −4.68326 −0.223266
\(441\) 37.6432 1.79253
\(442\) 28.1059 1.33686
\(443\) −24.5249 −1.16521 −0.582605 0.812755i \(-0.697967\pi\)
−0.582605 + 0.812755i \(0.697967\pi\)
\(444\) 2.97986 0.141418
\(445\) −0.478998 −0.0227067
\(446\) −32.4420 −1.53617
\(447\) 36.4991 1.72635
\(448\) 45.0260 2.12728
\(449\) 11.4903 0.542261 0.271130 0.962543i \(-0.412603\pi\)
0.271130 + 0.962543i \(0.412603\pi\)
\(450\) 39.2364 1.84962
\(451\) 22.8827 1.07751
\(452\) −36.2951 −1.70718
\(453\) −12.6770 −0.595619
\(454\) 28.6011 1.34232
\(455\) −3.85402 −0.180679
\(456\) 53.1684 2.48984
\(457\) −15.6501 −0.732079 −0.366039 0.930599i \(-0.619286\pi\)
−0.366039 + 0.930599i \(0.619286\pi\)
\(458\) 57.3144 2.67813
\(459\) 2.00653 0.0936569
\(460\) −3.63787 −0.169617
\(461\) −40.3046 −1.87717 −0.938586 0.345044i \(-0.887864\pi\)
−0.938586 + 0.345044i \(0.887864\pi\)
\(462\) −140.605 −6.54152
\(463\) 2.65746 0.123503 0.0617513 0.998092i \(-0.480331\pi\)
0.0617513 + 0.998092i \(0.480331\pi\)
\(464\) −15.0364 −0.698047
\(465\) 1.33118 0.0617321
\(466\) −0.871147 −0.0403551
\(467\) −38.0290 −1.75977 −0.879887 0.475183i \(-0.842382\pi\)
−0.879887 + 0.475183i \(0.842382\pi\)
\(468\) 54.3431 2.51201
\(469\) −33.2372 −1.53475
\(470\) −0.113070 −0.00521554
\(471\) −12.7228 −0.586236
\(472\) 27.3448 1.25865
\(473\) −29.8223 −1.37123
\(474\) 87.3371 4.01152
\(475\) −24.9862 −1.14645
\(476\) 42.9875 1.97033
\(477\) 4.56132 0.208848
\(478\) 19.7481 0.903255
\(479\) −18.8091 −0.859410 −0.429705 0.902969i \(-0.641382\pi\)
−0.429705 + 0.902969i \(0.641382\pi\)
\(480\) −1.14372 −0.0522036
\(481\) −1.38862 −0.0633155
\(482\) −57.8019 −2.63281
\(483\) −51.0095 −2.32101
\(484\) 69.9296 3.17862
\(485\) −3.56847 −0.162036
\(486\) −53.6386 −2.43310
\(487\) −39.5206 −1.79085 −0.895424 0.445214i \(-0.853128\pi\)
−0.895424 + 0.445214i \(0.853128\pi\)
\(488\) 16.1218 0.729802
\(489\) 11.7882 0.533081
\(490\) −5.60025 −0.252993
\(491\) −1.77764 −0.0802238 −0.0401119 0.999195i \(-0.512771\pi\)
−0.0401119 + 0.999195i \(0.512771\pi\)
\(492\) −39.5905 −1.78488
\(493\) 15.5791 0.701647
\(494\) −53.0503 −2.38685
\(495\) 3.67595 0.165222
\(496\) −6.67810 −0.299856
\(497\) −33.8128 −1.51671
\(498\) −8.24694 −0.369554
\(499\) 15.9181 0.712591 0.356296 0.934373i \(-0.384040\pi\)
0.356296 + 0.934373i \(0.384040\pi\)
\(500\) −7.64785 −0.342022
\(501\) 30.6956 1.37138
\(502\) 59.0677 2.63632
\(503\) −3.09482 −0.137991 −0.0689956 0.997617i \(-0.521979\pi\)
−0.0689956 + 0.997617i \(0.521979\pi\)
\(504\) 59.5077 2.65068
\(505\) 0.394598 0.0175594
\(506\) 61.8472 2.74944
\(507\) −15.7214 −0.698211
\(508\) 40.3354 1.78959
\(509\) −3.76643 −0.166944 −0.0834721 0.996510i \(-0.526601\pi\)
−0.0834721 + 0.996510i \(0.526601\pi\)
\(510\) −3.28931 −0.145653
\(511\) 6.89898 0.305193
\(512\) 27.4019 1.21101
\(513\) −3.78736 −0.167216
\(514\) −13.9816 −0.616702
\(515\) 0.379004 0.0167009
\(516\) 51.5969 2.27143
\(517\) 1.25398 0.0551498
\(518\) −3.25581 −0.143052
\(519\) −52.0635 −2.28533
\(520\) −3.77588 −0.165583
\(521\) 37.4208 1.63944 0.819718 0.572767i \(-0.194130\pi\)
0.819718 + 0.572767i \(0.194130\pi\)
\(522\) 46.1765 2.02109
\(523\) −25.7081 −1.12414 −0.562069 0.827090i \(-0.689995\pi\)
−0.562069 + 0.827090i \(0.689995\pi\)
\(524\) 27.3126 1.19316
\(525\) −53.3927 −2.33025
\(526\) −71.7981 −3.13054
\(527\) 6.91913 0.301402
\(528\) −35.2085 −1.53225
\(529\) −0.562667 −0.0244638
\(530\) −0.678597 −0.0294764
\(531\) −21.4633 −0.931426
\(532\) −81.1395 −3.51784
\(533\) 18.4492 0.799123
\(534\) −14.0893 −0.609703
\(535\) 1.83365 0.0792754
\(536\) −32.5633 −1.40652
\(537\) −62.1767 −2.68312
\(538\) −4.86467 −0.209731
\(539\) 62.1081 2.67518
\(540\) −0.577185 −0.0248381
\(541\) −9.97394 −0.428813 −0.214407 0.976744i \(-0.568782\pi\)
−0.214407 + 0.976744i \(0.568782\pi\)
\(542\) 11.4810 0.493152
\(543\) 50.2337 2.15573
\(544\) −5.94477 −0.254880
\(545\) −1.49469 −0.0640255
\(546\) −113.363 −4.85147
\(547\) 28.9440 1.23756 0.618778 0.785566i \(-0.287628\pi\)
0.618778 + 0.785566i \(0.287628\pi\)
\(548\) 66.1811 2.82712
\(549\) −12.6542 −0.540069
\(550\) 64.7368 2.76038
\(551\) −29.4058 −1.25273
\(552\) −49.9753 −2.12709
\(553\) −62.2487 −2.64708
\(554\) 25.0506 1.06430
\(555\) 0.162514 0.00689833
\(556\) 56.1232 2.38015
\(557\) −26.4802 −1.12200 −0.561001 0.827815i \(-0.689584\pi\)
−0.561001 + 0.827815i \(0.689584\pi\)
\(558\) 20.5083 0.868187
\(559\) −24.0442 −1.01696
\(560\) −2.26276 −0.0956189
\(561\) 36.4792 1.54015
\(562\) 39.6513 1.67259
\(563\) 24.5064 1.03282 0.516412 0.856340i \(-0.327267\pi\)
0.516412 + 0.856340i \(0.327267\pi\)
\(564\) −2.16956 −0.0913550
\(565\) −1.97944 −0.0832756
\(566\) 33.9381 1.42653
\(567\) 34.3762 1.44366
\(568\) −33.1273 −1.38999
\(569\) −19.7815 −0.829283 −0.414641 0.909985i \(-0.636093\pi\)
−0.414641 + 0.909985i \(0.636093\pi\)
\(570\) 6.20862 0.260051
\(571\) 1.18458 0.0495731 0.0247866 0.999693i \(-0.492109\pi\)
0.0247866 + 0.999693i \(0.492109\pi\)
\(572\) 89.6617 3.74894
\(573\) −48.2978 −2.01767
\(574\) 43.2567 1.80550
\(575\) 23.4856 0.979419
\(576\) −34.6248 −1.44270
\(577\) −39.6255 −1.64963 −0.824815 0.565403i \(-0.808721\pi\)
−0.824815 + 0.565403i \(0.808721\pi\)
\(578\) 23.6769 0.984828
\(579\) 41.9851 1.74484
\(580\) −4.48137 −0.186079
\(581\) 5.87793 0.243858
\(582\) −104.963 −4.35086
\(583\) 7.52580 0.311687
\(584\) 6.75911 0.279694
\(585\) 2.96373 0.122535
\(586\) −80.4669 −3.32406
\(587\) 34.2360 1.41307 0.706536 0.707677i \(-0.250257\pi\)
0.706536 + 0.707677i \(0.250257\pi\)
\(588\) −107.456 −4.43142
\(589\) −13.0600 −0.538127
\(590\) 3.19313 0.131459
\(591\) −33.3814 −1.37313
\(592\) −0.815278 −0.0335077
\(593\) −3.10027 −0.127313 −0.0636565 0.997972i \(-0.520276\pi\)
−0.0636565 + 0.997972i \(0.520276\pi\)
\(594\) 9.81267 0.402619
\(595\) 2.34442 0.0961120
\(596\) −54.5713 −2.23533
\(597\) 6.00348 0.245706
\(598\) 49.8643 2.03910
\(599\) −1.47930 −0.0604424 −0.0302212 0.999543i \(-0.509621\pi\)
−0.0302212 + 0.999543i \(0.509621\pi\)
\(600\) −52.3102 −2.13555
\(601\) −17.0845 −0.696892 −0.348446 0.937329i \(-0.613291\pi\)
−0.348446 + 0.937329i \(0.613291\pi\)
\(602\) −56.3750 −2.29767
\(603\) 25.5593 1.04086
\(604\) 18.9540 0.771227
\(605\) 3.81377 0.155052
\(606\) 11.6067 0.471492
\(607\) −24.5135 −0.994972 −0.497486 0.867472i \(-0.665743\pi\)
−0.497486 + 0.867472i \(0.665743\pi\)
\(608\) 11.2208 0.455066
\(609\) −62.8368 −2.54628
\(610\) 1.88259 0.0762240
\(611\) 1.01102 0.0409014
\(612\) −33.0572 −1.33626
\(613\) 0.0222735 0.000899619 0 0.000449810 1.00000i \(-0.499857\pi\)
0.000449810 1.00000i \(0.499857\pi\)
\(614\) 53.3592 2.15340
\(615\) −2.15916 −0.0870658
\(616\) 98.1828 3.95590
\(617\) −20.6034 −0.829461 −0.414730 0.909944i \(-0.636124\pi\)
−0.414730 + 0.909944i \(0.636124\pi\)
\(618\) 11.1481 0.448441
\(619\) 31.5214 1.26695 0.633476 0.773762i \(-0.281628\pi\)
0.633476 + 0.773762i \(0.281628\pi\)
\(620\) −1.99031 −0.0799327
\(621\) 3.55991 0.142854
\(622\) −54.8869 −2.20077
\(623\) 10.0420 0.402324
\(624\) −28.3868 −1.13638
\(625\) 24.3735 0.974939
\(626\) 34.1973 1.36680
\(627\) −68.8551 −2.74981
\(628\) 19.0224 0.759078
\(629\) 0.844704 0.0336805
\(630\) 6.94888 0.276850
\(631\) 9.21836 0.366977 0.183489 0.983022i \(-0.441261\pi\)
0.183489 + 0.983022i \(0.441261\pi\)
\(632\) −60.9866 −2.42591
\(633\) 56.5756 2.24868
\(634\) 34.0258 1.35134
\(635\) 2.19979 0.0872958
\(636\) −13.0207 −0.516306
\(637\) 50.0746 1.98403
\(638\) 76.1874 3.01629
\(639\) 26.0020 1.02862
\(640\) 4.23983 0.167594
\(641\) −16.2032 −0.639988 −0.319994 0.947420i \(-0.603681\pi\)
−0.319994 + 0.947420i \(0.603681\pi\)
\(642\) 53.9350 2.12864
\(643\) 21.3454 0.841778 0.420889 0.907112i \(-0.361718\pi\)
0.420889 + 0.907112i \(0.361718\pi\)
\(644\) 76.2666 3.00532
\(645\) 2.81396 0.110800
\(646\) 32.2708 1.26968
\(647\) 28.3383 1.11409 0.557047 0.830481i \(-0.311934\pi\)
0.557047 + 0.830481i \(0.311934\pi\)
\(648\) 33.6792 1.32304
\(649\) −35.4126 −1.39007
\(650\) 52.1940 2.04722
\(651\) −27.9077 −1.09379
\(652\) −17.6251 −0.690250
\(653\) 33.5963 1.31473 0.657363 0.753574i \(-0.271672\pi\)
0.657363 + 0.753574i \(0.271672\pi\)
\(654\) −43.9649 −1.71916
\(655\) 1.48956 0.0582018
\(656\) 10.8318 0.422911
\(657\) −5.30530 −0.206979
\(658\) 2.37047 0.0924106
\(659\) −36.9324 −1.43868 −0.719341 0.694657i \(-0.755556\pi\)
−0.719341 + 0.694657i \(0.755556\pi\)
\(660\) −10.4934 −0.408453
\(661\) −15.9484 −0.620321 −0.310160 0.950684i \(-0.600383\pi\)
−0.310160 + 0.950684i \(0.600383\pi\)
\(662\) −12.7768 −0.496586
\(663\) 29.4114 1.14224
\(664\) 5.75875 0.223483
\(665\) −4.42514 −0.171599
\(666\) 2.50371 0.0970166
\(667\) 27.6398 1.07022
\(668\) −45.8943 −1.77570
\(669\) −33.9489 −1.31254
\(670\) −3.80251 −0.146904
\(671\) −20.8784 −0.806003
\(672\) 23.9777 0.924960
\(673\) −42.2541 −1.62878 −0.814388 0.580320i \(-0.802927\pi\)
−0.814388 + 0.580320i \(0.802927\pi\)
\(674\) 50.6556 1.95118
\(675\) 3.72623 0.143423
\(676\) 23.5057 0.904067
\(677\) 35.6139 1.36875 0.684377 0.729129i \(-0.260074\pi\)
0.684377 + 0.729129i \(0.260074\pi\)
\(678\) −58.2233 −2.23605
\(679\) 74.8115 2.87100
\(680\) 2.29689 0.0880817
\(681\) 29.9296 1.14690
\(682\) 33.8371 1.29569
\(683\) −10.5247 −0.402718 −0.201359 0.979518i \(-0.564536\pi\)
−0.201359 + 0.979518i \(0.564536\pi\)
\(684\) 62.3961 2.38577
\(685\) 3.60935 0.137906
\(686\) 45.3716 1.73230
\(687\) 59.9766 2.28825
\(688\) −14.1167 −0.538195
\(689\) 6.06767 0.231160
\(690\) −5.83576 −0.222163
\(691\) 4.48104 0.170467 0.0852333 0.996361i \(-0.472836\pi\)
0.0852333 + 0.996361i \(0.472836\pi\)
\(692\) 77.8424 2.95912
\(693\) −77.0648 −2.92745
\(694\) 26.3237 0.999236
\(695\) 3.06081 0.116103
\(696\) −61.5628 −2.33353
\(697\) −11.2227 −0.425092
\(698\) −24.1125 −0.912670
\(699\) −0.911611 −0.0344803
\(700\) 79.8298 3.01728
\(701\) −3.99773 −0.150992 −0.0754961 0.997146i \(-0.524054\pi\)
−0.0754961 + 0.997146i \(0.524054\pi\)
\(702\) 7.91146 0.298599
\(703\) −1.59439 −0.0601336
\(704\) −57.1281 −2.15310
\(705\) −0.118322 −0.00445627
\(706\) −49.1234 −1.84878
\(707\) −8.27260 −0.311123
\(708\) 61.2690 2.30263
\(709\) 2.98427 0.112077 0.0560383 0.998429i \(-0.482153\pi\)
0.0560383 + 0.998429i \(0.482153\pi\)
\(710\) −3.86837 −0.145177
\(711\) 47.8690 1.79523
\(712\) 9.83840 0.368710
\(713\) 12.2756 0.459726
\(714\) 68.9591 2.58073
\(715\) 4.88991 0.182872
\(716\) 92.9632 3.47420
\(717\) 20.6653 0.771761
\(718\) 47.1519 1.75969
\(719\) 37.6332 1.40348 0.701741 0.712432i \(-0.252406\pi\)
0.701741 + 0.712432i \(0.252406\pi\)
\(720\) 1.74005 0.0648479
\(721\) −7.94568 −0.295912
\(722\) −15.3409 −0.570929
\(723\) −60.4868 −2.24953
\(724\) −75.1066 −2.79131
\(725\) 28.9311 1.07448
\(726\) 112.179 4.16334
\(727\) 21.0326 0.780056 0.390028 0.920803i \(-0.372465\pi\)
0.390028 + 0.920803i \(0.372465\pi\)
\(728\) 79.1598 2.93386
\(729\) −32.0940 −1.18867
\(730\) 0.789280 0.0292126
\(731\) 14.6262 0.540971
\(732\) 36.1227 1.33513
\(733\) 15.2159 0.562011 0.281006 0.959706i \(-0.409332\pi\)
0.281006 + 0.959706i \(0.409332\pi\)
\(734\) 76.9587 2.84060
\(735\) −5.86037 −0.216163
\(736\) −10.5470 −0.388766
\(737\) 42.1707 1.55338
\(738\) −33.2643 −1.22447
\(739\) 9.62533 0.354073 0.177037 0.984204i \(-0.443349\pi\)
0.177037 + 0.984204i \(0.443349\pi\)
\(740\) −0.242982 −0.00893218
\(741\) −55.5144 −2.03937
\(742\) 14.2265 0.522272
\(743\) 3.47732 0.127570 0.0637852 0.997964i \(-0.479683\pi\)
0.0637852 + 0.997964i \(0.479683\pi\)
\(744\) −27.3419 −1.00240
\(745\) −2.97618 −0.109039
\(746\) −70.2196 −2.57092
\(747\) −4.52011 −0.165382
\(748\) −54.5417 −1.99424
\(749\) −38.4416 −1.40463
\(750\) −12.2684 −0.447979
\(751\) 4.63809 0.169246 0.0846231 0.996413i \(-0.473031\pi\)
0.0846231 + 0.996413i \(0.473031\pi\)
\(752\) 0.593584 0.0216458
\(753\) 61.8113 2.25253
\(754\) 61.4261 2.23701
\(755\) 1.03370 0.0376203
\(756\) 12.1004 0.440089
\(757\) −1.31767 −0.0478916 −0.0239458 0.999713i \(-0.507623\pi\)
−0.0239458 + 0.999713i \(0.507623\pi\)
\(758\) 41.3074 1.50035
\(759\) 64.7199 2.34918
\(760\) −4.33542 −0.157262
\(761\) 25.8210 0.936010 0.468005 0.883726i \(-0.344973\pi\)
0.468005 + 0.883726i \(0.344973\pi\)
\(762\) 64.7046 2.34400
\(763\) 31.3356 1.13442
\(764\) 72.2122 2.61255
\(765\) −1.80286 −0.0651824
\(766\) −75.6998 −2.73515
\(767\) −28.5514 −1.03093
\(768\) 72.0326 2.59925
\(769\) −37.1369 −1.33919 −0.669596 0.742726i \(-0.733533\pi\)
−0.669596 + 0.742726i \(0.733533\pi\)
\(770\) 11.4651 0.413173
\(771\) −14.6310 −0.526924
\(772\) −62.7737 −2.25928
\(773\) −31.2596 −1.12433 −0.562165 0.827025i \(-0.690031\pi\)
−0.562165 + 0.827025i \(0.690031\pi\)
\(774\) 43.3522 1.55826
\(775\) 12.8492 0.461556
\(776\) 73.2947 2.63112
\(777\) −3.40704 −0.122227
\(778\) −56.7769 −2.03555
\(779\) 21.1831 0.758964
\(780\) −8.46027 −0.302926
\(781\) 42.9011 1.53512
\(782\) −30.3327 −1.08470
\(783\) 4.38532 0.156719
\(784\) 29.3996 1.04998
\(785\) 1.03743 0.0370276
\(786\) 43.8139 1.56279
\(787\) 37.3036 1.32973 0.664865 0.746964i \(-0.268489\pi\)
0.664865 + 0.746964i \(0.268489\pi\)
\(788\) 49.9101 1.77797
\(789\) −75.1330 −2.67481
\(790\) −7.12157 −0.253374
\(791\) 41.4981 1.47550
\(792\) −75.5023 −2.68286
\(793\) −16.8332 −0.597765
\(794\) 74.0013 2.62621
\(795\) −0.710117 −0.0251853
\(796\) −8.97606 −0.318148
\(797\) 12.9795 0.459756 0.229878 0.973219i \(-0.426167\pi\)
0.229878 + 0.973219i \(0.426167\pi\)
\(798\) −130.161 −4.60766
\(799\) −0.615007 −0.0217574
\(800\) −11.0397 −0.390313
\(801\) −7.72227 −0.272853
\(802\) −18.3923 −0.649454
\(803\) −8.75331 −0.308897
\(804\) −72.9616 −2.57316
\(805\) 4.15938 0.146599
\(806\) 27.2811 0.960937
\(807\) −5.09063 −0.179199
\(808\) −8.10487 −0.285128
\(809\) 27.3774 0.962539 0.481269 0.876573i \(-0.340176\pi\)
0.481269 + 0.876573i \(0.340176\pi\)
\(810\) 3.93282 0.138185
\(811\) −46.5830 −1.63575 −0.817874 0.575397i \(-0.804848\pi\)
−0.817874 + 0.575397i \(0.804848\pi\)
\(812\) 93.9502 3.29700
\(813\) 12.0143 0.421360
\(814\) 4.13091 0.144788
\(815\) −0.961224 −0.0336702
\(816\) 17.2679 0.604496
\(817\) −27.6072 −0.965855
\(818\) −55.5526 −1.94235
\(819\) −62.1335 −2.17112
\(820\) 3.22826 0.112736
\(821\) −54.4996 −1.90205 −0.951024 0.309116i \(-0.899967\pi\)
−0.951024 + 0.309116i \(0.899967\pi\)
\(822\) 106.166 3.70295
\(823\) 22.6504 0.789542 0.394771 0.918779i \(-0.370824\pi\)
0.394771 + 0.918779i \(0.370824\pi\)
\(824\) −7.78458 −0.271189
\(825\) 67.7437 2.35853
\(826\) −66.9428 −2.32924
\(827\) −22.0027 −0.765110 −0.382555 0.923933i \(-0.624956\pi\)
−0.382555 + 0.923933i \(0.624956\pi\)
\(828\) −58.6488 −2.03819
\(829\) −22.3821 −0.777363 −0.388681 0.921372i \(-0.627069\pi\)
−0.388681 + 0.921372i \(0.627069\pi\)
\(830\) 0.672466 0.0233416
\(831\) 26.2142 0.909362
\(832\) −46.0595 −1.59683
\(833\) −30.4607 −1.05540
\(834\) 90.0309 3.11752
\(835\) −2.50296 −0.0866184
\(836\) 102.948 3.56054
\(837\) 1.94765 0.0673207
\(838\) 40.2039 1.38882
\(839\) 7.51523 0.259454 0.129727 0.991550i \(-0.458590\pi\)
0.129727 + 0.991550i \(0.458590\pi\)
\(840\) −9.26430 −0.319649
\(841\) 5.04848 0.174086
\(842\) 16.8063 0.579184
\(843\) 41.4931 1.42910
\(844\) −84.5887 −2.91166
\(845\) 1.28194 0.0441001
\(846\) −1.82289 −0.0626721
\(847\) −79.9543 −2.74726
\(848\) 3.56242 0.122334
\(849\) 35.5145 1.21886
\(850\) −31.7499 −1.08901
\(851\) 1.49864 0.0513727
\(852\) −74.2252 −2.54291
\(853\) 38.3776 1.31402 0.657012 0.753880i \(-0.271820\pi\)
0.657012 + 0.753880i \(0.271820\pi\)
\(854\) −39.4678 −1.35056
\(855\) 3.40292 0.116377
\(856\) −37.6622 −1.28727
\(857\) −50.0740 −1.71050 −0.855248 0.518219i \(-0.826595\pi\)
−0.855248 + 0.518219i \(0.826595\pi\)
\(858\) 143.832 4.91035
\(859\) −30.5936 −1.04384 −0.521920 0.852994i \(-0.674784\pi\)
−0.521920 + 0.852994i \(0.674784\pi\)
\(860\) −4.20728 −0.143467
\(861\) 45.2659 1.54266
\(862\) 73.0946 2.48961
\(863\) −42.4567 −1.44524 −0.722622 0.691243i \(-0.757063\pi\)
−0.722622 + 0.691243i \(0.757063\pi\)
\(864\) −1.67338 −0.0569296
\(865\) 4.24532 0.144345
\(866\) −1.83616 −0.0623952
\(867\) 24.7766 0.841459
\(868\) 41.7260 1.41627
\(869\) 78.9800 2.67921
\(870\) −7.18887 −0.243725
\(871\) 34.0001 1.15205
\(872\) 30.7002 1.03964
\(873\) −57.5298 −1.94709
\(874\) 57.2535 1.93663
\(875\) 8.74420 0.295608
\(876\) 15.1445 0.511685
\(877\) 6.07266 0.205059 0.102530 0.994730i \(-0.467306\pi\)
0.102530 + 0.994730i \(0.467306\pi\)
\(878\) −16.3273 −0.551019
\(879\) −84.2045 −2.84015
\(880\) 2.87094 0.0967795
\(881\) −28.1383 −0.948004 −0.474002 0.880524i \(-0.657191\pi\)
−0.474002 + 0.880524i \(0.657191\pi\)
\(882\) −90.2856 −3.04007
\(883\) 8.93733 0.300765 0.150383 0.988628i \(-0.451949\pi\)
0.150383 + 0.988628i \(0.451949\pi\)
\(884\) −43.9742 −1.47901
\(885\) 3.34145 0.112322
\(886\) 58.8219 1.97616
\(887\) 9.64244 0.323761 0.161881 0.986810i \(-0.448244\pi\)
0.161881 + 0.986810i \(0.448244\pi\)
\(888\) −3.33796 −0.112015
\(889\) −46.1176 −1.54674
\(890\) 1.14886 0.0385098
\(891\) −43.6159 −1.46119
\(892\) 50.7585 1.69952
\(893\) 1.16084 0.0388459
\(894\) −87.5415 −2.92783
\(895\) 5.06997 0.169470
\(896\) −88.8863 −2.96948
\(897\) 52.1804 1.74225
\(898\) −27.5590 −0.919657
\(899\) 15.1219 0.504345
\(900\) −61.3889 −2.04630
\(901\) −3.69100 −0.122965
\(902\) −54.8833 −1.82741
\(903\) −58.9936 −1.96318
\(904\) 40.6567 1.35222
\(905\) −4.09612 −0.136160
\(906\) 30.4054 1.01015
\(907\) −31.4379 −1.04388 −0.521940 0.852982i \(-0.674791\pi\)
−0.521940 + 0.852982i \(0.674791\pi\)
\(908\) −44.7490 −1.48505
\(909\) 6.36160 0.211001
\(910\) 9.24372 0.306426
\(911\) −37.9061 −1.25588 −0.627942 0.778260i \(-0.716103\pi\)
−0.627942 + 0.778260i \(0.716103\pi\)
\(912\) −32.5934 −1.07927
\(913\) −7.45781 −0.246817
\(914\) 37.5360 1.24158
\(915\) 1.97004 0.0651275
\(916\) −89.6737 −2.96290
\(917\) −31.2279 −1.03124
\(918\) −4.81259 −0.158839
\(919\) −12.4739 −0.411476 −0.205738 0.978607i \(-0.565959\pi\)
−0.205738 + 0.978607i \(0.565959\pi\)
\(920\) 4.07505 0.134350
\(921\) 55.8377 1.83992
\(922\) 96.6690 3.18362
\(923\) 34.5890 1.13851
\(924\) 219.989 7.23711
\(925\) 1.56866 0.0515771
\(926\) −6.37381 −0.209456
\(927\) 6.11020 0.200685
\(928\) −12.9924 −0.426498
\(929\) −23.1401 −0.759201 −0.379601 0.925150i \(-0.623939\pi\)
−0.379601 + 0.925150i \(0.623939\pi\)
\(930\) −3.19279 −0.104696
\(931\) 57.4950 1.88432
\(932\) 1.36299 0.0446462
\(933\) −57.4364 −1.88038
\(934\) 91.2112 2.98452
\(935\) −2.97456 −0.0972786
\(936\) −60.8737 −1.98972
\(937\) −7.59503 −0.248119 −0.124059 0.992275i \(-0.539591\pi\)
−0.124059 + 0.992275i \(0.539591\pi\)
\(938\) 79.7181 2.60289
\(939\) 35.7858 1.16782
\(940\) 0.176909 0.00577013
\(941\) 29.3897 0.958078 0.479039 0.877794i \(-0.340985\pi\)
0.479039 + 0.877794i \(0.340985\pi\)
\(942\) 30.5152 0.994238
\(943\) −19.9109 −0.648389
\(944\) −16.7630 −0.545588
\(945\) 0.659927 0.0214674
\(946\) 71.5276 2.32556
\(947\) 13.0235 0.423208 0.211604 0.977355i \(-0.432131\pi\)
0.211604 + 0.977355i \(0.432131\pi\)
\(948\) −136.647 −4.43809
\(949\) −7.05735 −0.229091
\(950\) 59.9285 1.94434
\(951\) 35.6062 1.15461
\(952\) −48.1534 −1.56066
\(953\) −7.02814 −0.227664 −0.113832 0.993500i \(-0.536313\pi\)
−0.113832 + 0.993500i \(0.536313\pi\)
\(954\) −10.9401 −0.354200
\(955\) 3.93827 0.127439
\(956\) −30.8977 −0.999302
\(957\) 79.7262 2.57718
\(958\) 45.1129 1.45753
\(959\) −75.6685 −2.44346
\(960\) 5.39048 0.173977
\(961\) −24.2839 −0.783352
\(962\) 3.33054 0.107381
\(963\) 29.5615 0.952607
\(964\) 90.4364 2.91276
\(965\) −3.42352 −0.110207
\(966\) 122.344 3.93636
\(967\) 57.8670 1.86088 0.930438 0.366449i \(-0.119427\pi\)
0.930438 + 0.366449i \(0.119427\pi\)
\(968\) −78.3332 −2.51772
\(969\) 33.7697 1.08484
\(970\) 8.55883 0.274807
\(971\) −8.61825 −0.276573 −0.138286 0.990392i \(-0.544159\pi\)
−0.138286 + 0.990392i \(0.544159\pi\)
\(972\) 83.9226 2.69182
\(973\) −64.1687 −2.05715
\(974\) 94.7886 3.03722
\(975\) 54.6184 1.74919
\(976\) −9.88304 −0.316349
\(977\) 13.9995 0.447884 0.223942 0.974602i \(-0.428107\pi\)
0.223942 + 0.974602i \(0.428107\pi\)
\(978\) −28.2735 −0.904088
\(979\) −12.7411 −0.407208
\(980\) 8.76211 0.279895
\(981\) −24.0970 −0.769357
\(982\) 4.26360 0.136057
\(983\) 45.9125 1.46438 0.732191 0.681100i \(-0.238498\pi\)
0.732191 + 0.681100i \(0.238498\pi\)
\(984\) 44.3482 1.41377
\(985\) 2.72196 0.0867290
\(986\) −37.3658 −1.18997
\(987\) 2.48058 0.0789577
\(988\) 83.0021 2.64065
\(989\) 25.9492 0.825138
\(990\) −8.81662 −0.280210
\(991\) −33.9616 −1.07883 −0.539413 0.842041i \(-0.681354\pi\)
−0.539413 + 0.842041i \(0.681354\pi\)
\(992\) −5.77033 −0.183208
\(993\) −13.3703 −0.424294
\(994\) 81.0987 2.57230
\(995\) −0.489531 −0.0155192
\(996\) 12.9031 0.408851
\(997\) −52.6533 −1.66755 −0.833774 0.552106i \(-0.813824\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(998\) −38.1789 −1.20853
\(999\) 0.237774 0.00752283
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 4007.2.a.a.1.14 139
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4007.2.a.a.1.14 139 1.1 even 1 trivial