Properties

Label 6015.2.a.e.1.3
Level $6015$
Weight $2$
Character 6015.1
Self dual yes
Analytic conductor $48.030$
Analytic rank $0$
Dimension $31$
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] = [6015,2,Mod(1,6015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6015, 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("6015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6015.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: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 6015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.16689 q^{2} -1.00000 q^{3} +2.69543 q^{4} -1.00000 q^{5} +2.16689 q^{6} +0.00504296 q^{7} -1.50693 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.16689 q^{2} -1.00000 q^{3} +2.69543 q^{4} -1.00000 q^{5} +2.16689 q^{6} +0.00504296 q^{7} -1.50693 q^{8} +1.00000 q^{9} +2.16689 q^{10} +1.76259 q^{11} -2.69543 q^{12} +3.82887 q^{13} -0.0109276 q^{14} +1.00000 q^{15} -2.12551 q^{16} +4.15472 q^{17} -2.16689 q^{18} +0.970934 q^{19} -2.69543 q^{20} -0.00504296 q^{21} -3.81934 q^{22} -2.02549 q^{23} +1.50693 q^{24} +1.00000 q^{25} -8.29675 q^{26} -1.00000 q^{27} +0.0135929 q^{28} +7.57134 q^{29} -2.16689 q^{30} -2.56990 q^{31} +7.61961 q^{32} -1.76259 q^{33} -9.00284 q^{34} -0.00504296 q^{35} +2.69543 q^{36} -2.34651 q^{37} -2.10391 q^{38} -3.82887 q^{39} +1.50693 q^{40} +8.18663 q^{41} +0.0109276 q^{42} +10.0295 q^{43} +4.75093 q^{44} -1.00000 q^{45} +4.38903 q^{46} +3.69280 q^{47} +2.12551 q^{48} -6.99997 q^{49} -2.16689 q^{50} -4.15472 q^{51} +10.3205 q^{52} +11.0767 q^{53} +2.16689 q^{54} -1.76259 q^{55} -0.00759937 q^{56} -0.970934 q^{57} -16.4063 q^{58} +3.40656 q^{59} +2.69543 q^{60} +2.14095 q^{61} +5.56870 q^{62} +0.00504296 q^{63} -12.2599 q^{64} -3.82887 q^{65} +3.81934 q^{66} +15.4956 q^{67} +11.1988 q^{68} +2.02549 q^{69} +0.0109276 q^{70} +6.55114 q^{71} -1.50693 q^{72} -2.90544 q^{73} +5.08465 q^{74} -1.00000 q^{75} +2.61709 q^{76} +0.00888865 q^{77} +8.29675 q^{78} -16.7124 q^{79} +2.12551 q^{80} +1.00000 q^{81} -17.7396 q^{82} +10.7751 q^{83} -0.0135929 q^{84} -4.15472 q^{85} -21.7328 q^{86} -7.57134 q^{87} -2.65609 q^{88} -10.0083 q^{89} +2.16689 q^{90} +0.0193088 q^{91} -5.45957 q^{92} +2.56990 q^{93} -8.00190 q^{94} -0.970934 q^{95} -7.61961 q^{96} +8.39573 q^{97} +15.1682 q^{98} +1.76259 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.16689 −1.53223 −0.766113 0.642706i \(-0.777812\pi\)
−0.766113 + 0.642706i \(0.777812\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.69543 1.34772
\(5\) −1.00000 −0.447214
\(6\) 2.16689 0.884631
\(7\) 0.00504296 0.00190606 0.000953030 1.00000i \(-0.499697\pi\)
0.000953030 1.00000i \(0.499697\pi\)
\(8\) −1.50693 −0.532779
\(9\) 1.00000 0.333333
\(10\) 2.16689 0.685232
\(11\) 1.76259 0.531440 0.265720 0.964050i \(-0.414390\pi\)
0.265720 + 0.964050i \(0.414390\pi\)
\(12\) −2.69543 −0.778104
\(13\) 3.82887 1.06194 0.530969 0.847392i \(-0.321828\pi\)
0.530969 + 0.847392i \(0.321828\pi\)
\(14\) −0.0109276 −0.00292051
\(15\) 1.00000 0.258199
\(16\) −2.12551 −0.531378
\(17\) 4.15472 1.00767 0.503834 0.863801i \(-0.331922\pi\)
0.503834 + 0.863801i \(0.331922\pi\)
\(18\) −2.16689 −0.510742
\(19\) 0.970934 0.222747 0.111374 0.993779i \(-0.464475\pi\)
0.111374 + 0.993779i \(0.464475\pi\)
\(20\) −2.69543 −0.602717
\(21\) −0.00504296 −0.00110046
\(22\) −3.81934 −0.814286
\(23\) −2.02549 −0.422344 −0.211172 0.977449i \(-0.567728\pi\)
−0.211172 + 0.977449i \(0.567728\pi\)
\(24\) 1.50693 0.307600
\(25\) 1.00000 0.200000
\(26\) −8.29675 −1.62713
\(27\) −1.00000 −0.192450
\(28\) 0.0135929 0.00256883
\(29\) 7.57134 1.40596 0.702981 0.711209i \(-0.251852\pi\)
0.702981 + 0.711209i \(0.251852\pi\)
\(30\) −2.16689 −0.395619
\(31\) −2.56990 −0.461568 −0.230784 0.973005i \(-0.574129\pi\)
−0.230784 + 0.973005i \(0.574129\pi\)
\(32\) 7.61961 1.34697
\(33\) −1.76259 −0.306827
\(34\) −9.00284 −1.54397
\(35\) −0.00504296 −0.000852416 0
\(36\) 2.69543 0.449239
\(37\) −2.34651 −0.385764 −0.192882 0.981222i \(-0.561784\pi\)
−0.192882 + 0.981222i \(0.561784\pi\)
\(38\) −2.10391 −0.341299
\(39\) −3.82887 −0.613110
\(40\) 1.50693 0.238266
\(41\) 8.18663 1.27854 0.639269 0.768984i \(-0.279237\pi\)
0.639269 + 0.768984i \(0.279237\pi\)
\(42\) 0.0109276 0.00168616
\(43\) 10.0295 1.52948 0.764739 0.644340i \(-0.222868\pi\)
0.764739 + 0.644340i \(0.222868\pi\)
\(44\) 4.75093 0.716230
\(45\) −1.00000 −0.149071
\(46\) 4.38903 0.647126
\(47\) 3.69280 0.538650 0.269325 0.963049i \(-0.413199\pi\)
0.269325 + 0.963049i \(0.413199\pi\)
\(48\) 2.12551 0.306791
\(49\) −6.99997 −0.999996
\(50\) −2.16689 −0.306445
\(51\) −4.15472 −0.581777
\(52\) 10.3205 1.43119
\(53\) 11.0767 1.52150 0.760750 0.649045i \(-0.224831\pi\)
0.760750 + 0.649045i \(0.224831\pi\)
\(54\) 2.16689 0.294877
\(55\) −1.76259 −0.237667
\(56\) −0.00759937 −0.00101551
\(57\) −0.970934 −0.128603
\(58\) −16.4063 −2.15425
\(59\) 3.40656 0.443497 0.221748 0.975104i \(-0.428824\pi\)
0.221748 + 0.975104i \(0.428824\pi\)
\(60\) 2.69543 0.347979
\(61\) 2.14095 0.274120 0.137060 0.990563i \(-0.456235\pi\)
0.137060 + 0.990563i \(0.456235\pi\)
\(62\) 5.56870 0.707226
\(63\) 0.00504296 0.000635353 0
\(64\) −12.2599 −1.53248
\(65\) −3.82887 −0.474913
\(66\) 3.81934 0.470128
\(67\) 15.4956 1.89309 0.946545 0.322570i \(-0.104547\pi\)
0.946545 + 0.322570i \(0.104547\pi\)
\(68\) 11.1988 1.35805
\(69\) 2.02549 0.243840
\(70\) 0.0109276 0.00130609
\(71\) 6.55114 0.777477 0.388738 0.921348i \(-0.372911\pi\)
0.388738 + 0.921348i \(0.372911\pi\)
\(72\) −1.50693 −0.177593
\(73\) −2.90544 −0.340056 −0.170028 0.985439i \(-0.554386\pi\)
−0.170028 + 0.985439i \(0.554386\pi\)
\(74\) 5.08465 0.591078
\(75\) −1.00000 −0.115470
\(76\) 2.61709 0.300200
\(77\) 0.00888865 0.00101296
\(78\) 8.29675 0.939422
\(79\) −16.7124 −1.88029 −0.940145 0.340776i \(-0.889310\pi\)
−0.940145 + 0.340776i \(0.889310\pi\)
\(80\) 2.12551 0.237639
\(81\) 1.00000 0.111111
\(82\) −17.7396 −1.95901
\(83\) 10.7751 1.18272 0.591361 0.806407i \(-0.298591\pi\)
0.591361 + 0.806407i \(0.298591\pi\)
\(84\) −0.0135929 −0.00148311
\(85\) −4.15472 −0.450643
\(86\) −21.7328 −2.34351
\(87\) −7.57134 −0.811733
\(88\) −2.65609 −0.283140
\(89\) −10.0083 −1.06087 −0.530437 0.847725i \(-0.677972\pi\)
−0.530437 + 0.847725i \(0.677972\pi\)
\(90\) 2.16689 0.228411
\(91\) 0.0193088 0.00202411
\(92\) −5.45957 −0.569200
\(93\) 2.56990 0.266486
\(94\) −8.00190 −0.825333
\(95\) −0.970934 −0.0996157
\(96\) −7.61961 −0.777674
\(97\) 8.39573 0.852457 0.426229 0.904615i \(-0.359842\pi\)
0.426229 + 0.904615i \(0.359842\pi\)
\(98\) 15.1682 1.53222
\(99\) 1.76259 0.177147
\(100\) 2.69543 0.269543
\(101\) 0.662610 0.0659321 0.0329661 0.999456i \(-0.489505\pi\)
0.0329661 + 0.999456i \(0.489505\pi\)
\(102\) 9.00284 0.891414
\(103\) −8.12507 −0.800587 −0.400293 0.916387i \(-0.631092\pi\)
−0.400293 + 0.916387i \(0.631092\pi\)
\(104\) −5.76983 −0.565778
\(105\) 0.00504296 0.000492142 0
\(106\) −24.0020 −2.33128
\(107\) −11.8467 −1.14527 −0.572633 0.819812i \(-0.694078\pi\)
−0.572633 + 0.819812i \(0.694078\pi\)
\(108\) −2.69543 −0.259368
\(109\) 10.1499 0.972188 0.486094 0.873906i \(-0.338421\pi\)
0.486094 + 0.873906i \(0.338421\pi\)
\(110\) 3.81934 0.364160
\(111\) 2.34651 0.222721
\(112\) −0.0107189 −0.00101284
\(113\) 1.13141 0.106434 0.0532172 0.998583i \(-0.483052\pi\)
0.0532172 + 0.998583i \(0.483052\pi\)
\(114\) 2.10391 0.197049
\(115\) 2.02549 0.188878
\(116\) 20.4080 1.89484
\(117\) 3.82887 0.353979
\(118\) −7.38166 −0.679537
\(119\) 0.0209521 0.00192067
\(120\) −1.50693 −0.137563
\(121\) −7.89329 −0.717571
\(122\) −4.63921 −0.420014
\(123\) −8.18663 −0.738164
\(124\) −6.92699 −0.622062
\(125\) −1.00000 −0.0894427
\(126\) −0.0109276 −0.000973504 0
\(127\) −6.34172 −0.562737 −0.281369 0.959600i \(-0.590788\pi\)
−0.281369 + 0.959600i \(0.590788\pi\)
\(128\) 11.3266 1.00114
\(129\) −10.0295 −0.883045
\(130\) 8.29675 0.727673
\(131\) 6.08939 0.532032 0.266016 0.963969i \(-0.414293\pi\)
0.266016 + 0.963969i \(0.414293\pi\)
\(132\) −4.75093 −0.413516
\(133\) 0.00489638 0.000424570 0
\(134\) −33.5774 −2.90064
\(135\) 1.00000 0.0860663
\(136\) −6.26086 −0.536864
\(137\) 2.67382 0.228440 0.114220 0.993455i \(-0.463563\pi\)
0.114220 + 0.993455i \(0.463563\pi\)
\(138\) −4.38903 −0.373619
\(139\) 2.73592 0.232058 0.116029 0.993246i \(-0.462983\pi\)
0.116029 + 0.993246i \(0.462983\pi\)
\(140\) −0.0135929 −0.00114881
\(141\) −3.69280 −0.310990
\(142\) −14.1956 −1.19127
\(143\) 6.74871 0.564356
\(144\) −2.12551 −0.177126
\(145\) −7.57134 −0.628765
\(146\) 6.29579 0.521043
\(147\) 6.99997 0.577348
\(148\) −6.32487 −0.519901
\(149\) 5.25042 0.430131 0.215066 0.976600i \(-0.431003\pi\)
0.215066 + 0.976600i \(0.431003\pi\)
\(150\) 2.16689 0.176926
\(151\) −12.0736 −0.982535 −0.491268 0.871009i \(-0.663466\pi\)
−0.491268 + 0.871009i \(0.663466\pi\)
\(152\) −1.46313 −0.118675
\(153\) 4.15472 0.335889
\(154\) −0.0192608 −0.00155208
\(155\) 2.56990 0.206419
\(156\) −10.3205 −0.826298
\(157\) 12.0714 0.963401 0.481701 0.876336i \(-0.340019\pi\)
0.481701 + 0.876336i \(0.340019\pi\)
\(158\) 36.2140 2.88103
\(159\) −11.0767 −0.878439
\(160\) −7.61961 −0.602383
\(161\) −0.0102145 −0.000805013 0
\(162\) −2.16689 −0.170247
\(163\) 1.17314 0.0918878 0.0459439 0.998944i \(-0.485370\pi\)
0.0459439 + 0.998944i \(0.485370\pi\)
\(164\) 22.0665 1.72310
\(165\) 1.76259 0.137217
\(166\) −23.3485 −1.81220
\(167\) −14.6844 −1.13631 −0.568156 0.822921i \(-0.692343\pi\)
−0.568156 + 0.822921i \(0.692343\pi\)
\(168\) 0.00759937 0.000586304 0
\(169\) 1.66023 0.127710
\(170\) 9.00284 0.690486
\(171\) 0.970934 0.0742491
\(172\) 27.0337 2.06130
\(173\) 0.0371186 0.00282208 0.00141104 0.999999i \(-0.499551\pi\)
0.00141104 + 0.999999i \(0.499551\pi\)
\(174\) 16.4063 1.24376
\(175\) 0.00504296 0.000381212 0
\(176\) −3.74640 −0.282395
\(177\) −3.40656 −0.256053
\(178\) 21.6868 1.62550
\(179\) −6.30136 −0.470986 −0.235493 0.971876i \(-0.575670\pi\)
−0.235493 + 0.971876i \(0.575670\pi\)
\(180\) −2.69543 −0.200906
\(181\) −3.22464 −0.239686 −0.119843 0.992793i \(-0.538239\pi\)
−0.119843 + 0.992793i \(0.538239\pi\)
\(182\) −0.0418402 −0.00310140
\(183\) −2.14095 −0.158264
\(184\) 3.05227 0.225016
\(185\) 2.34651 0.172519
\(186\) −5.56870 −0.408317
\(187\) 7.32306 0.535515
\(188\) 9.95368 0.725947
\(189\) −0.00504296 −0.000366821 0
\(190\) 2.10391 0.152634
\(191\) −6.82580 −0.493898 −0.246949 0.969029i \(-0.579428\pi\)
−0.246949 + 0.969029i \(0.579428\pi\)
\(192\) 12.2599 0.884780
\(193\) 9.33805 0.672168 0.336084 0.941832i \(-0.390897\pi\)
0.336084 + 0.941832i \(0.390897\pi\)
\(194\) −18.1927 −1.30616
\(195\) 3.82887 0.274191
\(196\) −18.8680 −1.34771
\(197\) −12.3305 −0.878514 −0.439257 0.898362i \(-0.644758\pi\)
−0.439257 + 0.898362i \(0.644758\pi\)
\(198\) −3.81934 −0.271429
\(199\) −6.80836 −0.482632 −0.241316 0.970447i \(-0.577579\pi\)
−0.241316 + 0.970447i \(0.577579\pi\)
\(200\) −1.50693 −0.106556
\(201\) −15.4956 −1.09298
\(202\) −1.43580 −0.101023
\(203\) 0.0381819 0.00267985
\(204\) −11.1988 −0.784070
\(205\) −8.18663 −0.571779
\(206\) 17.6062 1.22668
\(207\) −2.02549 −0.140781
\(208\) −8.13830 −0.564290
\(209\) 1.71136 0.118377
\(210\) −0.0109276 −0.000754073 0
\(211\) −14.4508 −0.994835 −0.497418 0.867511i \(-0.665718\pi\)
−0.497418 + 0.867511i \(0.665718\pi\)
\(212\) 29.8565 2.05055
\(213\) −6.55114 −0.448877
\(214\) 25.6706 1.75481
\(215\) −10.0295 −0.684004
\(216\) 1.50693 0.102533
\(217\) −0.0129599 −0.000879775 0
\(218\) −21.9939 −1.48961
\(219\) 2.90544 0.196332
\(220\) −4.75093 −0.320308
\(221\) 15.9079 1.07008
\(222\) −5.08465 −0.341259
\(223\) −4.76381 −0.319008 −0.159504 0.987197i \(-0.550990\pi\)
−0.159504 + 0.987197i \(0.550990\pi\)
\(224\) 0.0384254 0.00256740
\(225\) 1.00000 0.0666667
\(226\) −2.45165 −0.163082
\(227\) −4.42996 −0.294027 −0.147014 0.989134i \(-0.546966\pi\)
−0.147014 + 0.989134i \(0.546966\pi\)
\(228\) −2.61709 −0.173321
\(229\) −5.80113 −0.383349 −0.191675 0.981459i \(-0.561392\pi\)
−0.191675 + 0.981459i \(0.561392\pi\)
\(230\) −4.38903 −0.289404
\(231\) −0.00888865 −0.000584830 0
\(232\) −11.4095 −0.749067
\(233\) 18.4705 1.21004 0.605020 0.796210i \(-0.293165\pi\)
0.605020 + 0.796210i \(0.293165\pi\)
\(234\) −8.29675 −0.542376
\(235\) −3.69280 −0.240891
\(236\) 9.18216 0.597707
\(237\) 16.7124 1.08559
\(238\) −0.0454009 −0.00294291
\(239\) 9.36356 0.605678 0.302839 0.953042i \(-0.402066\pi\)
0.302839 + 0.953042i \(0.402066\pi\)
\(240\) −2.12551 −0.137201
\(241\) −5.55603 −0.357895 −0.178948 0.983859i \(-0.557269\pi\)
−0.178948 + 0.983859i \(0.557269\pi\)
\(242\) 17.1039 1.09948
\(243\) −1.00000 −0.0641500
\(244\) 5.77078 0.369436
\(245\) 6.99997 0.447212
\(246\) 17.7396 1.13103
\(247\) 3.71758 0.236544
\(248\) 3.87265 0.245914
\(249\) −10.7751 −0.682845
\(250\) 2.16689 0.137046
\(251\) 2.48181 0.156650 0.0783252 0.996928i \(-0.475043\pi\)
0.0783252 + 0.996928i \(0.475043\pi\)
\(252\) 0.0135929 0.000856275 0
\(253\) −3.57010 −0.224451
\(254\) 13.7418 0.862240
\(255\) 4.15472 0.260179
\(256\) −0.0238591 −0.00149119
\(257\) 17.5860 1.09698 0.548491 0.836157i \(-0.315203\pi\)
0.548491 + 0.836157i \(0.315203\pi\)
\(258\) 21.7328 1.35302
\(259\) −0.0118334 −0.000735290 0
\(260\) −10.3205 −0.640047
\(261\) 7.57134 0.468654
\(262\) −13.1951 −0.815194
\(263\) −13.2922 −0.819633 −0.409816 0.912168i \(-0.634407\pi\)
−0.409816 + 0.912168i \(0.634407\pi\)
\(264\) 2.65609 0.163471
\(265\) −11.0767 −0.680436
\(266\) −0.0106099 −0.000650537 0
\(267\) 10.0083 0.612495
\(268\) 41.7674 2.55135
\(269\) 15.1862 0.925921 0.462960 0.886379i \(-0.346787\pi\)
0.462960 + 0.886379i \(0.346787\pi\)
\(270\) −2.16689 −0.131873
\(271\) 0.0501761 0.00304798 0.00152399 0.999999i \(-0.499515\pi\)
0.00152399 + 0.999999i \(0.499515\pi\)
\(272\) −8.83091 −0.535452
\(273\) −0.0193088 −0.00116862
\(274\) −5.79388 −0.350021
\(275\) 1.76259 0.106288
\(276\) 5.45957 0.328628
\(277\) −6.76742 −0.406615 −0.203307 0.979115i \(-0.565169\pi\)
−0.203307 + 0.979115i \(0.565169\pi\)
\(278\) −5.92846 −0.355565
\(279\) −2.56990 −0.153856
\(280\) 0.00759937 0.000454149 0
\(281\) 0.611028 0.0364509 0.0182254 0.999834i \(-0.494198\pi\)
0.0182254 + 0.999834i \(0.494198\pi\)
\(282\) 8.00190 0.476506
\(283\) 0.373208 0.0221849 0.0110925 0.999938i \(-0.496469\pi\)
0.0110925 + 0.999938i \(0.496469\pi\)
\(284\) 17.6581 1.04782
\(285\) 0.970934 0.0575131
\(286\) −14.6238 −0.864721
\(287\) 0.0412848 0.00243697
\(288\) 7.61961 0.448990
\(289\) 0.261700 0.0153941
\(290\) 16.4063 0.963411
\(291\) −8.39573 −0.492166
\(292\) −7.83142 −0.458299
\(293\) 4.10295 0.239697 0.119849 0.992792i \(-0.461759\pi\)
0.119849 + 0.992792i \(0.461759\pi\)
\(294\) −15.1682 −0.884628
\(295\) −3.40656 −0.198338
\(296\) 3.53602 0.205527
\(297\) −1.76259 −0.102276
\(298\) −11.3771 −0.659058
\(299\) −7.75534 −0.448503
\(300\) −2.69543 −0.155621
\(301\) 0.0505782 0.00291528
\(302\) 26.1622 1.50547
\(303\) −0.662610 −0.0380659
\(304\) −2.06373 −0.118363
\(305\) −2.14095 −0.122590
\(306\) −9.00284 −0.514658
\(307\) −14.6558 −0.836452 −0.418226 0.908343i \(-0.637348\pi\)
−0.418226 + 0.908343i \(0.637348\pi\)
\(308\) 0.0239588 0.00136518
\(309\) 8.12507 0.462219
\(310\) −5.56870 −0.316281
\(311\) −25.1912 −1.42846 −0.714232 0.699909i \(-0.753224\pi\)
−0.714232 + 0.699909i \(0.753224\pi\)
\(312\) 5.76983 0.326652
\(313\) 7.25873 0.410288 0.205144 0.978732i \(-0.434234\pi\)
0.205144 + 0.978732i \(0.434234\pi\)
\(314\) −26.1574 −1.47615
\(315\) −0.00504296 −0.000284139 0
\(316\) −45.0471 −2.53410
\(317\) −18.5700 −1.04299 −0.521497 0.853253i \(-0.674626\pi\)
−0.521497 + 0.853253i \(0.674626\pi\)
\(318\) 24.0020 1.34597
\(319\) 13.3451 0.747185
\(320\) 12.2599 0.685348
\(321\) 11.8467 0.661219
\(322\) 0.0221337 0.00123346
\(323\) 4.03396 0.224455
\(324\) 2.69543 0.149746
\(325\) 3.82887 0.212387
\(326\) −2.54208 −0.140793
\(327\) −10.1499 −0.561293
\(328\) −12.3367 −0.681178
\(329\) 0.0186226 0.00102670
\(330\) −3.81934 −0.210248
\(331\) −19.8133 −1.08904 −0.544519 0.838749i \(-0.683288\pi\)
−0.544519 + 0.838749i \(0.683288\pi\)
\(332\) 29.0436 1.59397
\(333\) −2.34651 −0.128588
\(334\) 31.8195 1.74109
\(335\) −15.4956 −0.846616
\(336\) 0.0107189 0.000584762 0
\(337\) 6.66980 0.363327 0.181664 0.983361i \(-0.441852\pi\)
0.181664 + 0.983361i \(0.441852\pi\)
\(338\) −3.59755 −0.195681
\(339\) −1.13141 −0.0614499
\(340\) −11.1988 −0.607338
\(341\) −4.52967 −0.245296
\(342\) −2.10391 −0.113766
\(343\) −0.0706013 −0.00381211
\(344\) −15.1137 −0.814874
\(345\) −2.02549 −0.109049
\(346\) −0.0804321 −0.00432406
\(347\) 26.4112 1.41783 0.708915 0.705294i \(-0.249185\pi\)
0.708915 + 0.705294i \(0.249185\pi\)
\(348\) −20.4080 −1.09398
\(349\) −16.5407 −0.885405 −0.442702 0.896669i \(-0.645980\pi\)
−0.442702 + 0.896669i \(0.645980\pi\)
\(350\) −0.0109276 −0.000584103 0
\(351\) −3.82887 −0.204370
\(352\) 13.4302 0.715834
\(353\) −12.4234 −0.661231 −0.330615 0.943766i \(-0.607256\pi\)
−0.330615 + 0.943766i \(0.607256\pi\)
\(354\) 7.38166 0.392331
\(355\) −6.55114 −0.347698
\(356\) −26.9766 −1.42976
\(357\) −0.0209521 −0.00110890
\(358\) 13.6544 0.721657
\(359\) 22.1737 1.17028 0.585141 0.810932i \(-0.301039\pi\)
0.585141 + 0.810932i \(0.301039\pi\)
\(360\) 1.50693 0.0794220
\(361\) −18.0573 −0.950384
\(362\) 6.98745 0.367252
\(363\) 7.89329 0.414290
\(364\) 0.0520456 0.00272793
\(365\) 2.90544 0.152078
\(366\) 4.63921 0.242495
\(367\) 5.21685 0.272317 0.136159 0.990687i \(-0.456524\pi\)
0.136159 + 0.990687i \(0.456524\pi\)
\(368\) 4.30520 0.224424
\(369\) 8.18663 0.426179
\(370\) −5.08465 −0.264338
\(371\) 0.0558593 0.00290007
\(372\) 6.92699 0.359148
\(373\) 21.8700 1.13238 0.566192 0.824274i \(-0.308416\pi\)
0.566192 + 0.824274i \(0.308416\pi\)
\(374\) −15.8683 −0.820530
\(375\) 1.00000 0.0516398
\(376\) −5.56478 −0.286981
\(377\) 28.9897 1.49304
\(378\) 0.0109276 0.000562053 0
\(379\) 9.16916 0.470988 0.235494 0.971876i \(-0.424329\pi\)
0.235494 + 0.971876i \(0.424329\pi\)
\(380\) −2.61709 −0.134254
\(381\) 6.34172 0.324896
\(382\) 14.7908 0.756763
\(383\) −0.0369414 −0.00188762 −0.000943809 1.00000i \(-0.500300\pi\)
−0.000943809 1.00000i \(0.500300\pi\)
\(384\) −11.3266 −0.578010
\(385\) −0.00888865 −0.000453008 0
\(386\) −20.2346 −1.02991
\(387\) 10.0295 0.509826
\(388\) 22.6301 1.14887
\(389\) −22.3190 −1.13162 −0.565809 0.824536i \(-0.691436\pi\)
−0.565809 + 0.824536i \(0.691436\pi\)
\(390\) −8.29675 −0.420122
\(391\) −8.41535 −0.425582
\(392\) 10.5485 0.532777
\(393\) −6.08939 −0.307169
\(394\) 26.7190 1.34608
\(395\) 16.7124 0.840891
\(396\) 4.75093 0.238743
\(397\) 7.32213 0.367487 0.183743 0.982974i \(-0.441178\pi\)
0.183743 + 0.982974i \(0.441178\pi\)
\(398\) 14.7530 0.739501
\(399\) −0.00489638 −0.000245125 0
\(400\) −2.12551 −0.106276
\(401\) 1.00000 0.0499376
\(402\) 33.5774 1.67469
\(403\) −9.83981 −0.490156
\(404\) 1.78602 0.0888577
\(405\) −1.00000 −0.0496904
\(406\) −0.0827362 −0.00410613
\(407\) −4.13593 −0.205011
\(408\) 6.26086 0.309959
\(409\) −15.2133 −0.752250 −0.376125 0.926569i \(-0.622744\pi\)
−0.376125 + 0.926569i \(0.622744\pi\)
\(410\) 17.7396 0.876095
\(411\) −2.67382 −0.131890
\(412\) −21.9006 −1.07896
\(413\) 0.0171792 0.000845331 0
\(414\) 4.38903 0.215709
\(415\) −10.7751 −0.528929
\(416\) 29.1745 1.43040
\(417\) −2.73592 −0.133979
\(418\) −3.70833 −0.181380
\(419\) −18.7772 −0.917326 −0.458663 0.888610i \(-0.651671\pi\)
−0.458663 + 0.888610i \(0.651671\pi\)
\(420\) 0.0135929 0.000663268 0
\(421\) 20.4730 0.997791 0.498895 0.866662i \(-0.333739\pi\)
0.498895 + 0.866662i \(0.333739\pi\)
\(422\) 31.3134 1.52431
\(423\) 3.69280 0.179550
\(424\) −16.6918 −0.810624
\(425\) 4.15472 0.201534
\(426\) 14.1956 0.687780
\(427\) 0.0107967 0.000522490 0
\(428\) −31.9320 −1.54349
\(429\) −6.74871 −0.325831
\(430\) 21.7328 1.04805
\(431\) −8.80104 −0.423931 −0.211966 0.977277i \(-0.567987\pi\)
−0.211966 + 0.977277i \(0.567987\pi\)
\(432\) 2.12551 0.102264
\(433\) 21.6902 1.04236 0.521182 0.853446i \(-0.325491\pi\)
0.521182 + 0.853446i \(0.325491\pi\)
\(434\) 0.0280827 0.00134801
\(435\) 7.57134 0.363018
\(436\) 27.3585 1.31023
\(437\) −1.96662 −0.0940761
\(438\) −6.29579 −0.300824
\(439\) −8.09050 −0.386139 −0.193069 0.981185i \(-0.561844\pi\)
−0.193069 + 0.981185i \(0.561844\pi\)
\(440\) 2.65609 0.126624
\(441\) −6.99997 −0.333332
\(442\) −34.4707 −1.63960
\(443\) 6.48024 0.307886 0.153943 0.988080i \(-0.450803\pi\)
0.153943 + 0.988080i \(0.450803\pi\)
\(444\) 6.32487 0.300165
\(445\) 10.0083 0.474437
\(446\) 10.3227 0.488793
\(447\) −5.25042 −0.248336
\(448\) −0.0618260 −0.00292101
\(449\) 40.0075 1.88807 0.944035 0.329844i \(-0.106996\pi\)
0.944035 + 0.329844i \(0.106996\pi\)
\(450\) −2.16689 −0.102148
\(451\) 14.4297 0.679466
\(452\) 3.04965 0.143443
\(453\) 12.0736 0.567267
\(454\) 9.59927 0.450516
\(455\) −0.0193088 −0.000905212 0
\(456\) 1.46313 0.0685172
\(457\) −36.3259 −1.69925 −0.849626 0.527386i \(-0.823172\pi\)
−0.849626 + 0.527386i \(0.823172\pi\)
\(458\) 12.5704 0.587378
\(459\) −4.15472 −0.193926
\(460\) 5.45957 0.254554
\(461\) 1.80384 0.0840132 0.0420066 0.999117i \(-0.486625\pi\)
0.0420066 + 0.999117i \(0.486625\pi\)
\(462\) 0.0192608 0.000896092 0
\(463\) 23.7101 1.10190 0.550950 0.834538i \(-0.314265\pi\)
0.550950 + 0.834538i \(0.314265\pi\)
\(464\) −16.0930 −0.747097
\(465\) −2.56990 −0.119176
\(466\) −40.0236 −1.85406
\(467\) 3.87922 0.179509 0.0897545 0.995964i \(-0.471392\pi\)
0.0897545 + 0.995964i \(0.471392\pi\)
\(468\) 10.3205 0.477063
\(469\) 0.0781438 0.00360834
\(470\) 8.00190 0.369100
\(471\) −12.0714 −0.556220
\(472\) −5.13344 −0.236286
\(473\) 17.6778 0.812826
\(474\) −36.2140 −1.66336
\(475\) 0.970934 0.0445495
\(476\) 0.0564749 0.00258852
\(477\) 11.0767 0.507167
\(478\) −20.2898 −0.928036
\(479\) 8.29971 0.379224 0.189612 0.981859i \(-0.439277\pi\)
0.189612 + 0.981859i \(0.439277\pi\)
\(480\) 7.61961 0.347786
\(481\) −8.98449 −0.409657
\(482\) 12.0393 0.548377
\(483\) 0.0102145 0.000464774 0
\(484\) −21.2758 −0.967082
\(485\) −8.39573 −0.381231
\(486\) 2.16689 0.0982923
\(487\) −13.7270 −0.622028 −0.311014 0.950405i \(-0.600669\pi\)
−0.311014 + 0.950405i \(0.600669\pi\)
\(488\) −3.22625 −0.146046
\(489\) −1.17314 −0.0530514
\(490\) −15.1682 −0.685230
\(491\) −4.29842 −0.193985 −0.0969925 0.995285i \(-0.530922\pi\)
−0.0969925 + 0.995285i \(0.530922\pi\)
\(492\) −22.0665 −0.994835
\(493\) 31.4568 1.41674
\(494\) −8.05560 −0.362438
\(495\) −1.76259 −0.0792224
\(496\) 5.46235 0.245267
\(497\) 0.0330371 0.00148192
\(498\) 23.3485 1.04627
\(499\) 12.9791 0.581024 0.290512 0.956871i \(-0.406174\pi\)
0.290512 + 0.956871i \(0.406174\pi\)
\(500\) −2.69543 −0.120543
\(501\) 14.6844 0.656050
\(502\) −5.37782 −0.240024
\(503\) 40.9671 1.82663 0.913316 0.407251i \(-0.133513\pi\)
0.913316 + 0.407251i \(0.133513\pi\)
\(504\) −0.00759937 −0.000338503 0
\(505\) −0.662610 −0.0294857
\(506\) 7.73604 0.343909
\(507\) −1.66023 −0.0737336
\(508\) −17.0937 −0.758410
\(509\) 30.8686 1.36823 0.684113 0.729376i \(-0.260189\pi\)
0.684113 + 0.729376i \(0.260189\pi\)
\(510\) −9.00284 −0.398652
\(511\) −0.0146520 −0.000648168 0
\(512\) −22.6016 −0.998857
\(513\) −0.970934 −0.0428678
\(514\) −38.1069 −1.68082
\(515\) 8.12507 0.358033
\(516\) −27.0337 −1.19009
\(517\) 6.50888 0.286260
\(518\) 0.0256417 0.00112663
\(519\) −0.0371186 −0.00162933
\(520\) 5.76983 0.253024
\(521\) −7.91428 −0.346731 −0.173365 0.984858i \(-0.555464\pi\)
−0.173365 + 0.984858i \(0.555464\pi\)
\(522\) −16.4063 −0.718084
\(523\) 0.783187 0.0342464 0.0171232 0.999853i \(-0.494549\pi\)
0.0171232 + 0.999853i \(0.494549\pi\)
\(524\) 16.4135 0.717029
\(525\) −0.00504296 −0.000220093 0
\(526\) 28.8028 1.25586
\(527\) −10.6772 −0.465107
\(528\) 3.74640 0.163041
\(529\) −18.8974 −0.821625
\(530\) 24.0020 1.04258
\(531\) 3.40656 0.147832
\(532\) 0.0131979 0.000572199 0
\(533\) 31.3455 1.35773
\(534\) −21.6868 −0.938481
\(535\) 11.8467 0.512178
\(536\) −23.3508 −1.00860
\(537\) 6.30136 0.271924
\(538\) −32.9070 −1.41872
\(539\) −12.3381 −0.531438
\(540\) 2.69543 0.115993
\(541\) −5.28262 −0.227117 −0.113559 0.993531i \(-0.536225\pi\)
−0.113559 + 0.993531i \(0.536225\pi\)
\(542\) −0.108726 −0.00467019
\(543\) 3.22464 0.138383
\(544\) 31.6574 1.35730
\(545\) −10.1499 −0.434776
\(546\) 0.0418402 0.00179059
\(547\) −10.6008 −0.453260 −0.226630 0.973981i \(-0.572771\pi\)
−0.226630 + 0.973981i \(0.572771\pi\)
\(548\) 7.20710 0.307872
\(549\) 2.14095 0.0913735
\(550\) −3.81934 −0.162857
\(551\) 7.35127 0.313174
\(552\) −3.05227 −0.129913
\(553\) −0.0842798 −0.00358394
\(554\) 14.6643 0.623025
\(555\) −2.34651 −0.0996039
\(556\) 7.37450 0.312748
\(557\) −24.4387 −1.03550 −0.517750 0.855532i \(-0.673230\pi\)
−0.517750 + 0.855532i \(0.673230\pi\)
\(558\) 5.56870 0.235742
\(559\) 38.4015 1.62421
\(560\) 0.0107189 0.000452955 0
\(561\) −7.32306 −0.309180
\(562\) −1.32403 −0.0558510
\(563\) −24.3687 −1.02702 −0.513508 0.858085i \(-0.671654\pi\)
−0.513508 + 0.858085i \(0.671654\pi\)
\(564\) −9.95368 −0.419126
\(565\) −1.13141 −0.0475989
\(566\) −0.808703 −0.0339923
\(567\) 0.00504296 0.000211784 0
\(568\) −9.87209 −0.414224
\(569\) 20.1924 0.846509 0.423254 0.906011i \(-0.360888\pi\)
0.423254 + 0.906011i \(0.360888\pi\)
\(570\) −2.10391 −0.0881231
\(571\) 38.5531 1.61340 0.806698 0.590963i \(-0.201252\pi\)
0.806698 + 0.590963i \(0.201252\pi\)
\(572\) 18.1907 0.760591
\(573\) 6.82580 0.285152
\(574\) −0.0894599 −0.00373398
\(575\) −2.02549 −0.0844688
\(576\) −12.2599 −0.510828
\(577\) 16.2255 0.675477 0.337738 0.941240i \(-0.390338\pi\)
0.337738 + 0.941240i \(0.390338\pi\)
\(578\) −0.567077 −0.0235873
\(579\) −9.33805 −0.388076
\(580\) −20.4080 −0.847397
\(581\) 0.0543384 0.00225434
\(582\) 18.1927 0.754110
\(583\) 19.5236 0.808586
\(584\) 4.37829 0.181175
\(585\) −3.82887 −0.158304
\(586\) −8.89066 −0.367270
\(587\) 3.97374 0.164014 0.0820069 0.996632i \(-0.473867\pi\)
0.0820069 + 0.996632i \(0.473867\pi\)
\(588\) 18.8680 0.778101
\(589\) −2.49520 −0.102813
\(590\) 7.38166 0.303898
\(591\) 12.3305 0.507210
\(592\) 4.98754 0.204987
\(593\) 33.5065 1.37595 0.687974 0.725736i \(-0.258500\pi\)
0.687974 + 0.725736i \(0.258500\pi\)
\(594\) 3.81934 0.156709
\(595\) −0.0209521 −0.000858952 0
\(596\) 14.1522 0.579695
\(597\) 6.80836 0.278648
\(598\) 16.8050 0.687208
\(599\) −17.3766 −0.709989 −0.354994 0.934868i \(-0.615517\pi\)
−0.354994 + 0.934868i \(0.615517\pi\)
\(600\) 1.50693 0.0615200
\(601\) −15.9888 −0.652198 −0.326099 0.945336i \(-0.605734\pi\)
−0.326099 + 0.945336i \(0.605734\pi\)
\(602\) −0.109598 −0.00446686
\(603\) 15.4956 0.631030
\(604\) −32.5436 −1.32418
\(605\) 7.89329 0.320908
\(606\) 1.43580 0.0583256
\(607\) 8.10826 0.329104 0.164552 0.986368i \(-0.447382\pi\)
0.164552 + 0.986368i \(0.447382\pi\)
\(608\) 7.39814 0.300034
\(609\) −0.0381819 −0.00154721
\(610\) 4.63921 0.187836
\(611\) 14.1392 0.572012
\(612\) 11.1988 0.452683
\(613\) 7.67715 0.310077 0.155039 0.987908i \(-0.450450\pi\)
0.155039 + 0.987908i \(0.450450\pi\)
\(614\) 31.7576 1.28163
\(615\) 8.18663 0.330117
\(616\) −0.0133946 −0.000539682 0
\(617\) 31.4420 1.26581 0.632904 0.774230i \(-0.281863\pi\)
0.632904 + 0.774230i \(0.281863\pi\)
\(618\) −17.6062 −0.708224
\(619\) −22.2006 −0.892318 −0.446159 0.894954i \(-0.647209\pi\)
−0.446159 + 0.894954i \(0.647209\pi\)
\(620\) 6.92699 0.278195
\(621\) 2.02549 0.0812802
\(622\) 54.5868 2.18873
\(623\) −0.0504712 −0.00202209
\(624\) 8.13830 0.325793
\(625\) 1.00000 0.0400000
\(626\) −15.7289 −0.628653
\(627\) −1.71136 −0.0683449
\(628\) 32.5376 1.29839
\(629\) −9.74911 −0.388722
\(630\) 0.0109276 0.000435364 0
\(631\) 37.4356 1.49029 0.745143 0.666905i \(-0.232381\pi\)
0.745143 + 0.666905i \(0.232381\pi\)
\(632\) 25.1843 1.00178
\(633\) 14.4508 0.574368
\(634\) 40.2392 1.59810
\(635\) 6.34172 0.251664
\(636\) −29.8565 −1.18389
\(637\) −26.8020 −1.06193
\(638\) −28.9175 −1.14486
\(639\) 6.55114 0.259159
\(640\) −11.3266 −0.447724
\(641\) 20.5587 0.812019 0.406010 0.913869i \(-0.366920\pi\)
0.406010 + 0.913869i \(0.366920\pi\)
\(642\) −25.6706 −1.01314
\(643\) 35.6074 1.40422 0.702110 0.712068i \(-0.252241\pi\)
0.702110 + 0.712068i \(0.252241\pi\)
\(644\) −0.0275324 −0.00108493
\(645\) 10.0295 0.394910
\(646\) −8.74116 −0.343916
\(647\) 19.6236 0.771482 0.385741 0.922607i \(-0.373946\pi\)
0.385741 + 0.922607i \(0.373946\pi\)
\(648\) −1.50693 −0.0591977
\(649\) 6.00436 0.235692
\(650\) −8.29675 −0.325425
\(651\) 0.0129599 0.000507938 0
\(652\) 3.16213 0.123839
\(653\) 14.0953 0.551593 0.275797 0.961216i \(-0.411058\pi\)
0.275797 + 0.961216i \(0.411058\pi\)
\(654\) 21.9939 0.860028
\(655\) −6.08939 −0.237932
\(656\) −17.4008 −0.679386
\(657\) −2.90544 −0.113352
\(658\) −0.0403532 −0.00157313
\(659\) 6.89872 0.268736 0.134368 0.990932i \(-0.457100\pi\)
0.134368 + 0.990932i \(0.457100\pi\)
\(660\) 4.75093 0.184930
\(661\) −4.74671 −0.184626 −0.0923129 0.995730i \(-0.529426\pi\)
−0.0923129 + 0.995730i \(0.529426\pi\)
\(662\) 42.9333 1.66865
\(663\) −15.9079 −0.617811
\(664\) −16.2373 −0.630130
\(665\) −0.00489638 −0.000189873 0
\(666\) 5.08465 0.197026
\(667\) −15.3357 −0.593800
\(668\) −39.5807 −1.53142
\(669\) 4.76381 0.184179
\(670\) 33.5774 1.29721
\(671\) 3.77361 0.145679
\(672\) −0.0384254 −0.00148229
\(673\) 16.5598 0.638334 0.319167 0.947699i \(-0.396597\pi\)
0.319167 + 0.947699i \(0.396597\pi\)
\(674\) −14.4528 −0.556699
\(675\) −1.00000 −0.0384900
\(676\) 4.47505 0.172117
\(677\) −19.9707 −0.767535 −0.383767 0.923430i \(-0.625374\pi\)
−0.383767 + 0.923430i \(0.625374\pi\)
\(678\) 2.45165 0.0941552
\(679\) 0.0423393 0.00162483
\(680\) 6.26086 0.240093
\(681\) 4.42996 0.169757
\(682\) 9.81532 0.375848
\(683\) 38.4201 1.47011 0.735053 0.678010i \(-0.237157\pi\)
0.735053 + 0.678010i \(0.237157\pi\)
\(684\) 2.61709 0.100067
\(685\) −2.67382 −0.102161
\(686\) 0.152986 0.00584101
\(687\) 5.80113 0.221327
\(688\) −21.3177 −0.812731
\(689\) 42.4112 1.61574
\(690\) 4.38903 0.167087
\(691\) 13.0693 0.497179 0.248590 0.968609i \(-0.420033\pi\)
0.248590 + 0.968609i \(0.420033\pi\)
\(692\) 0.100051 0.00380336
\(693\) 0.00888865 0.000337652 0
\(694\) −57.2304 −2.17243
\(695\) −2.73592 −0.103780
\(696\) 11.4095 0.432474
\(697\) 34.0132 1.28834
\(698\) 35.8420 1.35664
\(699\) −18.4705 −0.698617
\(700\) 0.0135929 0.000513765 0
\(701\) −33.9526 −1.28237 −0.641186 0.767386i \(-0.721557\pi\)
−0.641186 + 0.767386i \(0.721557\pi\)
\(702\) 8.29675 0.313141
\(703\) −2.27831 −0.0859280
\(704\) −21.6091 −0.814424
\(705\) 3.69280 0.139079
\(706\) 26.9202 1.01316
\(707\) 0.00334151 0.000125670 0
\(708\) −9.18216 −0.345087
\(709\) 3.44067 0.129217 0.0646085 0.997911i \(-0.479420\pi\)
0.0646085 + 0.997911i \(0.479420\pi\)
\(710\) 14.1956 0.532752
\(711\) −16.7124 −0.626763
\(712\) 15.0817 0.565211
\(713\) 5.20531 0.194940
\(714\) 0.0454009 0.00169909
\(715\) −6.74871 −0.252388
\(716\) −16.9849 −0.634755
\(717\) −9.36356 −0.349688
\(718\) −48.0480 −1.79314
\(719\) 26.7896 0.999084 0.499542 0.866290i \(-0.333502\pi\)
0.499542 + 0.866290i \(0.333502\pi\)
\(720\) 2.12551 0.0792131
\(721\) −0.0409744 −0.00152597
\(722\) 39.1282 1.45620
\(723\) 5.55603 0.206631
\(724\) −8.69179 −0.323028
\(725\) 7.57134 0.281192
\(726\) −17.1039 −0.634786
\(727\) 4.36497 0.161888 0.0809439 0.996719i \(-0.474207\pi\)
0.0809439 + 0.996719i \(0.474207\pi\)
\(728\) −0.0290970 −0.00107841
\(729\) 1.00000 0.0370370
\(730\) −6.29579 −0.233018
\(731\) 41.6696 1.54121
\(732\) −5.77078 −0.213294
\(733\) 10.4218 0.384939 0.192470 0.981303i \(-0.438350\pi\)
0.192470 + 0.981303i \(0.438350\pi\)
\(734\) −11.3044 −0.417252
\(735\) −6.99997 −0.258198
\(736\) −15.4335 −0.568885
\(737\) 27.3124 1.00606
\(738\) −17.7396 −0.653002
\(739\) 11.3457 0.417359 0.208680 0.977984i \(-0.433083\pi\)
0.208680 + 0.977984i \(0.433083\pi\)
\(740\) 6.32487 0.232507
\(741\) −3.71758 −0.136569
\(742\) −0.121041 −0.00444356
\(743\) −39.3912 −1.44512 −0.722561 0.691307i \(-0.757035\pi\)
−0.722561 + 0.691307i \(0.757035\pi\)
\(744\) −3.87265 −0.141978
\(745\) −5.25042 −0.192361
\(746\) −47.3899 −1.73507
\(747\) 10.7751 0.394241
\(748\) 19.7388 0.721722
\(749\) −0.0597425 −0.00218294
\(750\) −2.16689 −0.0791238
\(751\) 14.4578 0.527574 0.263787 0.964581i \(-0.415028\pi\)
0.263787 + 0.964581i \(0.415028\pi\)
\(752\) −7.84908 −0.286227
\(753\) −2.48181 −0.0904421
\(754\) −62.8175 −2.28768
\(755\) 12.0736 0.439403
\(756\) −0.0135929 −0.000494371 0
\(757\) 16.8295 0.611677 0.305839 0.952083i \(-0.401063\pi\)
0.305839 + 0.952083i \(0.401063\pi\)
\(758\) −19.8686 −0.721660
\(759\) 3.57010 0.129587
\(760\) 1.46313 0.0530732
\(761\) −30.2009 −1.09478 −0.547390 0.836878i \(-0.684379\pi\)
−0.547390 + 0.836878i \(0.684379\pi\)
\(762\) −13.7418 −0.497815
\(763\) 0.0511857 0.00185305
\(764\) −18.3985 −0.665634
\(765\) −4.15472 −0.150214
\(766\) 0.0800481 0.00289226
\(767\) 13.0433 0.470966
\(768\) 0.0238591 0.000860940 0
\(769\) −7.53603 −0.271756 −0.135878 0.990726i \(-0.543386\pi\)
−0.135878 + 0.990726i \(0.543386\pi\)
\(770\) 0.0192608 0.000694110 0
\(771\) −17.5860 −0.633343
\(772\) 25.1701 0.905891
\(773\) −15.0483 −0.541249 −0.270625 0.962685i \(-0.587230\pi\)
−0.270625 + 0.962685i \(0.587230\pi\)
\(774\) −21.7328 −0.781169
\(775\) −2.56990 −0.0923135
\(776\) −12.6518 −0.454172
\(777\) 0.0118334 0.000424520 0
\(778\) 48.3629 1.73390
\(779\) 7.94868 0.284791
\(780\) 10.3205 0.369532
\(781\) 11.5470 0.413182
\(782\) 18.2352 0.652088
\(783\) −7.57134 −0.270578
\(784\) 14.8785 0.531376
\(785\) −12.0714 −0.430846
\(786\) 13.1951 0.470652
\(787\) −26.7369 −0.953066 −0.476533 0.879157i \(-0.658107\pi\)
−0.476533 + 0.879157i \(0.658107\pi\)
\(788\) −33.2361 −1.18399
\(789\) 13.2922 0.473215
\(790\) −36.2140 −1.28843
\(791\) 0.00570567 0.000202870 0
\(792\) −2.65609 −0.0943801
\(793\) 8.19741 0.291099
\(794\) −15.8663 −0.563073
\(795\) 11.0767 0.392850
\(796\) −18.3515 −0.650451
\(797\) −42.0180 −1.48835 −0.744177 0.667983i \(-0.767158\pi\)
−0.744177 + 0.667983i \(0.767158\pi\)
\(798\) 0.0106099 0.000375588 0
\(799\) 15.3425 0.542780
\(800\) 7.61961 0.269394
\(801\) −10.0083 −0.353624
\(802\) −2.16689 −0.0765157
\(803\) −5.12110 −0.180720
\(804\) −41.7674 −1.47302
\(805\) 0.0102145 0.000360013 0
\(806\) 21.3218 0.751029
\(807\) −15.1862 −0.534581
\(808\) −0.998504 −0.0351273
\(809\) 31.7848 1.11749 0.558747 0.829338i \(-0.311282\pi\)
0.558747 + 0.829338i \(0.311282\pi\)
\(810\) 2.16689 0.0761369
\(811\) 49.1303 1.72520 0.862599 0.505888i \(-0.168835\pi\)
0.862599 + 0.505888i \(0.168835\pi\)
\(812\) 0.102917 0.00361167
\(813\) −0.0501761 −0.00175975
\(814\) 8.96213 0.314123
\(815\) −1.17314 −0.0410935
\(816\) 8.83091 0.309144
\(817\) 9.73794 0.340687
\(818\) 32.9656 1.15262
\(819\) 0.0193088 0.000674705 0
\(820\) −22.0665 −0.770596
\(821\) −26.6244 −0.929197 −0.464599 0.885521i \(-0.653801\pi\)
−0.464599 + 0.885521i \(0.653801\pi\)
\(822\) 5.79388 0.202085
\(823\) 40.0432 1.39582 0.697908 0.716187i \(-0.254114\pi\)
0.697908 + 0.716187i \(0.254114\pi\)
\(824\) 12.2439 0.426536
\(825\) −1.76259 −0.0613654
\(826\) −0.0372254 −0.00129524
\(827\) −38.0400 −1.32278 −0.661391 0.750042i \(-0.730033\pi\)
−0.661391 + 0.750042i \(0.730033\pi\)
\(828\) −5.45957 −0.189733
\(829\) −20.5302 −0.713044 −0.356522 0.934287i \(-0.616038\pi\)
−0.356522 + 0.934287i \(0.616038\pi\)
\(830\) 23.3485 0.810439
\(831\) 6.76742 0.234759
\(832\) −46.9414 −1.62740
\(833\) −29.0829 −1.00766
\(834\) 5.92846 0.205286
\(835\) 14.6844 0.508174
\(836\) 4.61284 0.159538
\(837\) 2.56990 0.0888287
\(838\) 40.6882 1.40555
\(839\) 20.2718 0.699860 0.349930 0.936776i \(-0.386205\pi\)
0.349930 + 0.936776i \(0.386205\pi\)
\(840\) −0.00759937 −0.000262203 0
\(841\) 28.3252 0.976730
\(842\) −44.3627 −1.52884
\(843\) −0.611028 −0.0210449
\(844\) −38.9512 −1.34076
\(845\) −1.66023 −0.0571138
\(846\) −8.00190 −0.275111
\(847\) −0.0398055 −0.00136773
\(848\) −23.5436 −0.808492
\(849\) −0.373208 −0.0128085
\(850\) −9.00284 −0.308795
\(851\) 4.75284 0.162925
\(852\) −17.6581 −0.604958
\(853\) −9.91138 −0.339359 −0.169680 0.985499i \(-0.554273\pi\)
−0.169680 + 0.985499i \(0.554273\pi\)
\(854\) −0.0233953 −0.000800572 0
\(855\) −0.970934 −0.0332052
\(856\) 17.8521 0.610174
\(857\) 19.2380 0.657156 0.328578 0.944477i \(-0.393431\pi\)
0.328578 + 0.944477i \(0.393431\pi\)
\(858\) 14.6238 0.499247
\(859\) 31.2756 1.06711 0.533554 0.845766i \(-0.320856\pi\)
0.533554 + 0.845766i \(0.320856\pi\)
\(860\) −27.0337 −0.921842
\(861\) −0.0412848 −0.00140698
\(862\) 19.0709 0.649559
\(863\) −9.59780 −0.326713 −0.163357 0.986567i \(-0.552232\pi\)
−0.163357 + 0.986567i \(0.552232\pi\)
\(864\) −7.61961 −0.259225
\(865\) −0.0371186 −0.00126207
\(866\) −47.0003 −1.59714
\(867\) −0.261700 −0.00888781
\(868\) −0.0349325 −0.00118569
\(869\) −29.4570 −0.999261
\(870\) −16.4063 −0.556225
\(871\) 59.3307 2.01034
\(872\) −15.2952 −0.517962
\(873\) 8.39573 0.284152
\(874\) 4.26145 0.144146
\(875\) −0.00504296 −0.000170483 0
\(876\) 7.83142 0.264599
\(877\) −23.1994 −0.783390 −0.391695 0.920095i \(-0.628111\pi\)
−0.391695 + 0.920095i \(0.628111\pi\)
\(878\) 17.5313 0.591652
\(879\) −4.10295 −0.138389
\(880\) 3.74640 0.126291
\(881\) 23.6639 0.797258 0.398629 0.917112i \(-0.369486\pi\)
0.398629 + 0.917112i \(0.369486\pi\)
\(882\) 15.1682 0.510740
\(883\) 31.3497 1.05500 0.527500 0.849555i \(-0.323129\pi\)
0.527500 + 0.849555i \(0.323129\pi\)
\(884\) 42.8786 1.44216
\(885\) 3.40656 0.114510
\(886\) −14.0420 −0.471750
\(887\) 48.8031 1.63865 0.819323 0.573332i \(-0.194349\pi\)
0.819323 + 0.573332i \(0.194349\pi\)
\(888\) −3.53602 −0.118661
\(889\) −0.0319811 −0.00107261
\(890\) −21.6868 −0.726944
\(891\) 1.76259 0.0590489
\(892\) −12.8405 −0.429932
\(893\) 3.58546 0.119983
\(894\) 11.3771 0.380508
\(895\) 6.30136 0.210631
\(896\) 0.0571197 0.00190824
\(897\) 7.75534 0.258943
\(898\) −86.6920 −2.89295
\(899\) −19.4576 −0.648947
\(900\) 2.69543 0.0898477
\(901\) 46.0205 1.53317
\(902\) −31.2675 −1.04109
\(903\) −0.0505782 −0.00168314
\(904\) −1.70496 −0.0567060
\(905\) 3.22464 0.107191
\(906\) −26.1622 −0.869181
\(907\) 15.3571 0.509925 0.254962 0.966951i \(-0.417937\pi\)
0.254962 + 0.966951i \(0.417937\pi\)
\(908\) −11.9407 −0.396265
\(909\) 0.662610 0.0219774
\(910\) 0.0418402 0.00138699
\(911\) 10.8585 0.359759 0.179879 0.983689i \(-0.442429\pi\)
0.179879 + 0.983689i \(0.442429\pi\)
\(912\) 2.06373 0.0683369
\(913\) 18.9921 0.628546
\(914\) 78.7143 2.60364
\(915\) 2.14095 0.0707776
\(916\) −15.6365 −0.516646
\(917\) 0.0307085 0.00101409
\(918\) 9.00284 0.297138
\(919\) −10.5753 −0.348847 −0.174424 0.984671i \(-0.555806\pi\)
−0.174424 + 0.984671i \(0.555806\pi\)
\(920\) −3.05227 −0.100630
\(921\) 14.6558 0.482926
\(922\) −3.90873 −0.128727
\(923\) 25.0834 0.825632
\(924\) −0.0239588 −0.000788185 0
\(925\) −2.34651 −0.0771529
\(926\) −51.3772 −1.68836
\(927\) −8.12507 −0.266862
\(928\) 57.6907 1.89379
\(929\) 21.1469 0.693807 0.346903 0.937901i \(-0.387233\pi\)
0.346903 + 0.937901i \(0.387233\pi\)
\(930\) 5.56870 0.182605
\(931\) −6.79651 −0.222747
\(932\) 49.7859 1.63079
\(933\) 25.1912 0.824724
\(934\) −8.40587 −0.275048
\(935\) −7.32306 −0.239490
\(936\) −5.76983 −0.188593
\(937\) −39.3882 −1.28676 −0.643379 0.765548i \(-0.722468\pi\)
−0.643379 + 0.765548i \(0.722468\pi\)
\(938\) −0.169329 −0.00552880
\(939\) −7.25873 −0.236880
\(940\) −9.95368 −0.324653
\(941\) 55.5735 1.81164 0.905821 0.423660i \(-0.139255\pi\)
0.905821 + 0.423660i \(0.139255\pi\)
\(942\) 26.1574 0.852254
\(943\) −16.5819 −0.539983
\(944\) −7.24069 −0.235664
\(945\) 0.00504296 0.000164047 0
\(946\) −38.3059 −1.24543
\(947\) −45.6057 −1.48199 −0.740993 0.671513i \(-0.765645\pi\)
−0.740993 + 0.671513i \(0.765645\pi\)
\(948\) 45.0471 1.46306
\(949\) −11.1246 −0.361118
\(950\) −2.10391 −0.0682599
\(951\) 18.5700 0.602173
\(952\) −0.0315733 −0.00102330
\(953\) −13.1788 −0.426904 −0.213452 0.976954i \(-0.568471\pi\)
−0.213452 + 0.976954i \(0.568471\pi\)
\(954\) −24.0020 −0.777094
\(955\) 6.82580 0.220878
\(956\) 25.2388 0.816282
\(957\) −13.3451 −0.431387
\(958\) −17.9846 −0.581056
\(959\) 0.0134840 0.000435420 0
\(960\) −12.2599 −0.395686
\(961\) −24.3956 −0.786955
\(962\) 19.4684 0.627688
\(963\) −11.8467 −0.381755
\(964\) −14.9759 −0.482341
\(965\) −9.33805 −0.300603
\(966\) −0.0221337 −0.000712139 0
\(967\) −21.3582 −0.686834 −0.343417 0.939183i \(-0.611584\pi\)
−0.343417 + 0.939183i \(0.611584\pi\)
\(968\) 11.8946 0.382307
\(969\) −4.03396 −0.129589
\(970\) 18.1927 0.584131
\(971\) −25.9007 −0.831191 −0.415596 0.909549i \(-0.636427\pi\)
−0.415596 + 0.909549i \(0.636427\pi\)
\(972\) −2.69543 −0.0864560
\(973\) 0.0137972 0.000442316 0
\(974\) 29.7449 0.953087
\(975\) −3.82887 −0.122622
\(976\) −4.55061 −0.145662
\(977\) −47.7440 −1.52747 −0.763733 0.645533i \(-0.776635\pi\)
−0.763733 + 0.645533i \(0.776635\pi\)
\(978\) 2.54208 0.0812868
\(979\) −17.6404 −0.563790
\(980\) 18.8680 0.602715
\(981\) 10.1499 0.324063
\(982\) 9.31422 0.297229
\(983\) 13.0238 0.415395 0.207698 0.978193i \(-0.433403\pi\)
0.207698 + 0.978193i \(0.433403\pi\)
\(984\) 12.3367 0.393278
\(985\) 12.3305 0.392883
\(986\) −68.1635 −2.17077
\(987\) −0.0186226 −0.000592764 0
\(988\) 10.0205 0.318794
\(989\) −20.3146 −0.645966
\(990\) 3.81934 0.121387
\(991\) −13.6345 −0.433113 −0.216557 0.976270i \(-0.569483\pi\)
−0.216557 + 0.976270i \(0.569483\pi\)
\(992\) −19.5816 −0.621718
\(993\) 19.8133 0.628756
\(994\) −0.0715879 −0.00227063
\(995\) 6.80836 0.215840
\(996\) −29.0436 −0.920281
\(997\) 17.0307 0.539367 0.269684 0.962949i \(-0.413081\pi\)
0.269684 + 0.962949i \(0.413081\pi\)
\(998\) −28.1243 −0.890260
\(999\) 2.34651 0.0742404
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 6015.2.a.e.1.3 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6015.2.a.e.1.3 31 1.1 even 1 trivial