Properties

Label 1045.4.a.c.1.7
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
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] = [1045,4,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.10393\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.10393 q^{2} -1.35130 q^{3} -3.57349 q^{4} -5.00000 q^{5} +2.84303 q^{6} -13.9401 q^{7} +24.3498 q^{8} -25.1740 q^{9} +O(q^{10})\) \(q-2.10393 q^{2} -1.35130 q^{3} -3.57349 q^{4} -5.00000 q^{5} +2.84303 q^{6} -13.9401 q^{7} +24.3498 q^{8} -25.1740 q^{9} +10.5196 q^{10} -11.0000 q^{11} +4.82885 q^{12} +44.0744 q^{13} +29.3289 q^{14} +6.75648 q^{15} -22.6422 q^{16} -43.5563 q^{17} +52.9643 q^{18} +19.0000 q^{19} +17.8675 q^{20} +18.8372 q^{21} +23.1432 q^{22} -78.4251 q^{23} -32.9038 q^{24} +25.0000 q^{25} -92.7293 q^{26} +70.5025 q^{27} +49.8148 q^{28} +20.2635 q^{29} -14.2151 q^{30} +277.768 q^{31} -147.161 q^{32} +14.8643 q^{33} +91.6394 q^{34} +69.7004 q^{35} +89.9591 q^{36} +118.093 q^{37} -39.9746 q^{38} -59.5576 q^{39} -121.749 q^{40} +293.707 q^{41} -39.6321 q^{42} +171.485 q^{43} +39.3084 q^{44} +125.870 q^{45} +165.001 q^{46} +265.721 q^{47} +30.5964 q^{48} -148.674 q^{49} -52.5982 q^{50} +58.8575 q^{51} -157.499 q^{52} -384.897 q^{53} -148.332 q^{54} +55.0000 q^{55} -339.438 q^{56} -25.6746 q^{57} -42.6329 q^{58} -472.620 q^{59} -24.1442 q^{60} +617.674 q^{61} -584.404 q^{62} +350.928 q^{63} +490.753 q^{64} -220.372 q^{65} -31.2733 q^{66} -24.8664 q^{67} +155.648 q^{68} +105.976 q^{69} -146.645 q^{70} +828.984 q^{71} -612.981 q^{72} -350.335 q^{73} -248.459 q^{74} -33.7824 q^{75} -67.8963 q^{76} +153.341 q^{77} +125.305 q^{78} -475.784 q^{79} +113.211 q^{80} +584.428 q^{81} -617.937 q^{82} +757.088 q^{83} -67.3145 q^{84} +217.782 q^{85} -360.792 q^{86} -27.3820 q^{87} -267.848 q^{88} -182.170 q^{89} -264.821 q^{90} -614.400 q^{91} +280.251 q^{92} -375.347 q^{93} -559.058 q^{94} -95.0000 q^{95} +198.858 q^{96} +161.592 q^{97} +312.800 q^{98} +276.914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.10393 −0.743850 −0.371925 0.928263i \(-0.621302\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(3\) −1.35130 −0.260057 −0.130029 0.991510i \(-0.541507\pi\)
−0.130029 + 0.991510i \(0.541507\pi\)
\(4\) −3.57349 −0.446686
\(5\) −5.00000 −0.447214
\(6\) 2.84303 0.193444
\(7\) −13.9401 −0.752694 −0.376347 0.926479i \(-0.622820\pi\)
−0.376347 + 0.926479i \(0.622820\pi\)
\(8\) 24.3498 1.07612
\(9\) −25.1740 −0.932370
\(10\) 10.5196 0.332660
\(11\) −11.0000 −0.301511
\(12\) 4.82885 0.116164
\(13\) 44.0744 0.940310 0.470155 0.882584i \(-0.344198\pi\)
0.470155 + 0.882584i \(0.344198\pi\)
\(14\) 29.3289 0.559892
\(15\) 6.75648 0.116301
\(16\) −22.6422 −0.353785
\(17\) −43.5563 −0.621410 −0.310705 0.950506i \(-0.600565\pi\)
−0.310705 + 0.950506i \(0.600565\pi\)
\(18\) 52.9643 0.693544
\(19\) 19.0000 0.229416
\(20\) 17.8675 0.199764
\(21\) 18.8372 0.195743
\(22\) 23.1432 0.224279
\(23\) −78.4251 −0.710989 −0.355495 0.934678i \(-0.615688\pi\)
−0.355495 + 0.934678i \(0.615688\pi\)
\(24\) −32.9038 −0.279852
\(25\) 25.0000 0.200000
\(26\) −92.7293 −0.699450
\(27\) 70.5025 0.502527
\(28\) 49.8148 0.336218
\(29\) 20.2635 0.129753 0.0648764 0.997893i \(-0.479335\pi\)
0.0648764 + 0.997893i \(0.479335\pi\)
\(30\) −14.2151 −0.0865106
\(31\) 277.768 1.60931 0.804655 0.593743i \(-0.202350\pi\)
0.804655 + 0.593743i \(0.202350\pi\)
\(32\) −147.161 −0.812955
\(33\) 14.8643 0.0784102
\(34\) 91.6394 0.462236
\(35\) 69.7004 0.336615
\(36\) 89.9591 0.416477
\(37\) 118.093 0.524713 0.262357 0.964971i \(-0.415500\pi\)
0.262357 + 0.964971i \(0.415500\pi\)
\(38\) −39.9746 −0.170651
\(39\) −59.5576 −0.244534
\(40\) −121.749 −0.481255
\(41\) 293.707 1.11876 0.559381 0.828911i \(-0.311039\pi\)
0.559381 + 0.828911i \(0.311039\pi\)
\(42\) −39.6321 −0.145604
\(43\) 171.485 0.608168 0.304084 0.952645i \(-0.401650\pi\)
0.304084 + 0.952645i \(0.401650\pi\)
\(44\) 39.3084 0.134681
\(45\) 125.870 0.416969
\(46\) 165.001 0.528870
\(47\) 265.721 0.824668 0.412334 0.911033i \(-0.364714\pi\)
0.412334 + 0.911033i \(0.364714\pi\)
\(48\) 30.5964 0.0920042
\(49\) −148.674 −0.433452
\(50\) −52.5982 −0.148770
\(51\) 58.8575 0.161602
\(52\) −157.499 −0.420024
\(53\) −384.897 −0.997542 −0.498771 0.866734i \(-0.666215\pi\)
−0.498771 + 0.866734i \(0.666215\pi\)
\(54\) −148.332 −0.373805
\(55\) 55.0000 0.134840
\(56\) −339.438 −0.809987
\(57\) −25.6746 −0.0596612
\(58\) −42.6329 −0.0965168
\(59\) −472.620 −1.04288 −0.521440 0.853288i \(-0.674605\pi\)
−0.521440 + 0.853288i \(0.674605\pi\)
\(60\) −24.1442 −0.0519501
\(61\) 617.674 1.29648 0.648238 0.761438i \(-0.275506\pi\)
0.648238 + 0.761438i \(0.275506\pi\)
\(62\) −584.404 −1.19709
\(63\) 350.928 0.701789
\(64\) 490.753 0.958502
\(65\) −220.372 −0.420520
\(66\) −31.2733 −0.0583254
\(67\) −24.8664 −0.0453420 −0.0226710 0.999743i \(-0.507217\pi\)
−0.0226710 + 0.999743i \(0.507217\pi\)
\(68\) 155.648 0.277575
\(69\) 105.976 0.184898
\(70\) −146.645 −0.250391
\(71\) 828.984 1.38567 0.692833 0.721098i \(-0.256362\pi\)
0.692833 + 0.721098i \(0.256362\pi\)
\(72\) −612.981 −1.00334
\(73\) −350.335 −0.561693 −0.280847 0.959753i \(-0.590615\pi\)
−0.280847 + 0.959753i \(0.590615\pi\)
\(74\) −248.459 −0.390308
\(75\) −33.7824 −0.0520114
\(76\) −67.8963 −0.102477
\(77\) 153.341 0.226946
\(78\) 125.305 0.181897
\(79\) −475.784 −0.677593 −0.338797 0.940860i \(-0.610020\pi\)
−0.338797 + 0.940860i \(0.610020\pi\)
\(80\) 113.211 0.158217
\(81\) 584.428 0.801685
\(82\) −617.937 −0.832192
\(83\) 757.088 1.00122 0.500609 0.865673i \(-0.333109\pi\)
0.500609 + 0.865673i \(0.333109\pi\)
\(84\) −67.3145 −0.0874359
\(85\) 217.782 0.277903
\(86\) −360.792 −0.452386
\(87\) −27.3820 −0.0337432
\(88\) −267.848 −0.324462
\(89\) −182.170 −0.216966 −0.108483 0.994098i \(-0.534599\pi\)
−0.108483 + 0.994098i \(0.534599\pi\)
\(90\) −264.821 −0.310162
\(91\) −614.400 −0.707766
\(92\) 280.251 0.317589
\(93\) −375.347 −0.418513
\(94\) −559.058 −0.613430
\(95\) −95.0000 −0.102598
\(96\) 198.858 0.211415
\(97\) 161.592 0.169146 0.0845730 0.996417i \(-0.473047\pi\)
0.0845730 + 0.996417i \(0.473047\pi\)
\(98\) 312.800 0.322424
\(99\) 276.914 0.281120
\(100\) −89.3373 −0.0893373
\(101\) −409.325 −0.403261 −0.201630 0.979462i \(-0.564624\pi\)
−0.201630 + 0.979462i \(0.564624\pi\)
\(102\) −123.832 −0.120208
\(103\) −151.866 −0.145279 −0.0726397 0.997358i \(-0.523142\pi\)
−0.0726397 + 0.997358i \(0.523142\pi\)
\(104\) 1073.20 1.01189
\(105\) −94.1859 −0.0875391
\(106\) 809.796 0.742022
\(107\) −1472.78 −1.33064 −0.665322 0.746556i \(-0.731706\pi\)
−0.665322 + 0.746556i \(0.731706\pi\)
\(108\) −251.940 −0.224472
\(109\) −737.377 −0.647962 −0.323981 0.946064i \(-0.605021\pi\)
−0.323981 + 0.946064i \(0.605021\pi\)
\(110\) −115.716 −0.100301
\(111\) −159.579 −0.136455
\(112\) 315.634 0.266292
\(113\) 1092.53 0.909525 0.454763 0.890613i \(-0.349724\pi\)
0.454763 + 0.890613i \(0.349724\pi\)
\(114\) 54.0175 0.0443790
\(115\) 392.125 0.317964
\(116\) −72.4114 −0.0579589
\(117\) −1109.53 −0.876717
\(118\) 994.358 0.775746
\(119\) 607.179 0.467731
\(120\) 164.519 0.125154
\(121\) 121.000 0.0909091
\(122\) −1299.54 −0.964384
\(123\) −396.885 −0.290942
\(124\) −992.602 −0.718857
\(125\) −125.000 −0.0894427
\(126\) −738.326 −0.522026
\(127\) −215.621 −0.150656 −0.0753278 0.997159i \(-0.524000\pi\)
−0.0753278 + 0.997159i \(0.524000\pi\)
\(128\) 144.777 0.0999732
\(129\) −231.727 −0.158158
\(130\) 463.646 0.312804
\(131\) −683.511 −0.455868 −0.227934 0.973677i \(-0.573197\pi\)
−0.227934 + 0.973677i \(0.573197\pi\)
\(132\) −53.1173 −0.0350248
\(133\) −264.862 −0.172680
\(134\) 52.3171 0.0337277
\(135\) −352.513 −0.224737
\(136\) −1060.59 −0.668710
\(137\) 353.862 0.220675 0.110337 0.993894i \(-0.464807\pi\)
0.110337 + 0.993894i \(0.464807\pi\)
\(138\) −222.965 −0.137536
\(139\) −456.332 −0.278457 −0.139229 0.990260i \(-0.544462\pi\)
−0.139229 + 0.990260i \(0.544462\pi\)
\(140\) −249.074 −0.150361
\(141\) −359.068 −0.214461
\(142\) −1744.12 −1.03073
\(143\) −484.818 −0.283514
\(144\) 569.995 0.329858
\(145\) −101.317 −0.0580273
\(146\) 737.079 0.417816
\(147\) 200.903 0.112722
\(148\) −422.005 −0.234382
\(149\) −2470.77 −1.35848 −0.679238 0.733918i \(-0.737690\pi\)
−0.679238 + 0.733918i \(0.737690\pi\)
\(150\) 71.0757 0.0386887
\(151\) −2923.75 −1.57570 −0.787852 0.615865i \(-0.788807\pi\)
−0.787852 + 0.615865i \(0.788807\pi\)
\(152\) 462.646 0.246878
\(153\) 1096.49 0.579384
\(154\) −322.618 −0.168814
\(155\) −1388.84 −0.719705
\(156\) 212.828 0.109230
\(157\) 693.959 0.352764 0.176382 0.984322i \(-0.443561\pi\)
0.176382 + 0.984322i \(0.443561\pi\)
\(158\) 1001.01 0.504028
\(159\) 520.111 0.259418
\(160\) 735.803 0.363565
\(161\) 1093.25 0.535157
\(162\) −1229.59 −0.596334
\(163\) 3123.87 1.50110 0.750552 0.660811i \(-0.229788\pi\)
0.750552 + 0.660811i \(0.229788\pi\)
\(164\) −1049.56 −0.499736
\(165\) −74.3213 −0.0350661
\(166\) −1592.86 −0.744757
\(167\) −715.299 −0.331446 −0.165723 0.986172i \(-0.552996\pi\)
−0.165723 + 0.986172i \(0.552996\pi\)
\(168\) 458.681 0.210643
\(169\) −254.449 −0.115817
\(170\) −458.197 −0.206718
\(171\) −478.306 −0.213900
\(172\) −612.801 −0.271661
\(173\) 1071.76 0.471008 0.235504 0.971873i \(-0.424326\pi\)
0.235504 + 0.971873i \(0.424326\pi\)
\(174\) 57.6097 0.0250999
\(175\) −348.502 −0.150539
\(176\) 249.064 0.106670
\(177\) 638.650 0.271208
\(178\) 383.272 0.161390
\(179\) 1302.68 0.543948 0.271974 0.962305i \(-0.412324\pi\)
0.271974 + 0.962305i \(0.412324\pi\)
\(180\) −449.795 −0.186254
\(181\) −1133.87 −0.465634 −0.232817 0.972521i \(-0.574794\pi\)
−0.232817 + 0.972521i \(0.574794\pi\)
\(182\) 1292.65 0.526472
\(183\) −834.660 −0.337158
\(184\) −1909.63 −0.765109
\(185\) −590.465 −0.234659
\(186\) 789.703 0.311311
\(187\) 479.120 0.187362
\(188\) −949.552 −0.368368
\(189\) −982.811 −0.378249
\(190\) 199.873 0.0763174
\(191\) −3282.67 −1.24359 −0.621794 0.783181i \(-0.713596\pi\)
−0.621794 + 0.783181i \(0.713596\pi\)
\(192\) −663.153 −0.249265
\(193\) −2704.78 −1.00878 −0.504390 0.863476i \(-0.668283\pi\)
−0.504390 + 0.863476i \(0.668283\pi\)
\(194\) −339.977 −0.125819
\(195\) 297.788 0.109359
\(196\) 531.286 0.193617
\(197\) −3177.48 −1.14917 −0.574584 0.818446i \(-0.694836\pi\)
−0.574584 + 0.818446i \(0.694836\pi\)
\(198\) −582.607 −0.209111
\(199\) −2469.85 −0.879814 −0.439907 0.898043i \(-0.644989\pi\)
−0.439907 + 0.898043i \(0.644989\pi\)
\(200\) 608.745 0.215224
\(201\) 33.6019 0.0117915
\(202\) 861.190 0.299966
\(203\) −282.475 −0.0976642
\(204\) −210.327 −0.0721854
\(205\) −1468.53 −0.500326
\(206\) 319.514 0.108066
\(207\) 1974.27 0.662905
\(208\) −997.942 −0.332667
\(209\) −209.000 −0.0691714
\(210\) 198.160 0.0651160
\(211\) −3087.64 −1.00740 −0.503701 0.863878i \(-0.668029\pi\)
−0.503701 + 0.863878i \(0.668029\pi\)
\(212\) 1375.43 0.445589
\(213\) −1120.20 −0.360352
\(214\) 3098.62 0.989801
\(215\) −857.426 −0.271981
\(216\) 1716.72 0.540778
\(217\) −3872.11 −1.21132
\(218\) 1551.39 0.481987
\(219\) 473.406 0.146072
\(220\) −196.542 −0.0602312
\(221\) −1919.72 −0.584318
\(222\) 335.742 0.101502
\(223\) −2151.13 −0.645965 −0.322983 0.946405i \(-0.604686\pi\)
−0.322983 + 0.946405i \(0.604686\pi\)
\(224\) 2051.43 0.611906
\(225\) −629.350 −0.186474
\(226\) −2298.60 −0.676551
\(227\) 4431.47 1.29571 0.647856 0.761763i \(-0.275666\pi\)
0.647856 + 0.761763i \(0.275666\pi\)
\(228\) 91.7481 0.0266498
\(229\) 5246.32 1.51392 0.756958 0.653463i \(-0.226685\pi\)
0.756958 + 0.653463i \(0.226685\pi\)
\(230\) −825.003 −0.236518
\(231\) −207.209 −0.0590188
\(232\) 493.411 0.139629
\(233\) 5077.70 1.42769 0.713844 0.700305i \(-0.246953\pi\)
0.713844 + 0.700305i \(0.246953\pi\)
\(234\) 2334.37 0.652147
\(235\) −1328.61 −0.368803
\(236\) 1688.90 0.465840
\(237\) 642.925 0.176213
\(238\) −1277.46 −0.347922
\(239\) −3494.23 −0.945702 −0.472851 0.881142i \(-0.656775\pi\)
−0.472851 + 0.881142i \(0.656775\pi\)
\(240\) −152.982 −0.0411455
\(241\) −131.158 −0.0350566 −0.0175283 0.999846i \(-0.505580\pi\)
−0.0175283 + 0.999846i \(0.505580\pi\)
\(242\) −254.575 −0.0676228
\(243\) −2693.30 −0.711010
\(244\) −2207.25 −0.579118
\(245\) 743.371 0.193846
\(246\) 835.016 0.216417
\(247\) 837.413 0.215722
\(248\) 6763.59 1.73181
\(249\) −1023.05 −0.260374
\(250\) 262.991 0.0665320
\(251\) −1093.75 −0.275048 −0.137524 0.990498i \(-0.543914\pi\)
−0.137524 + 0.990498i \(0.543914\pi\)
\(252\) −1254.04 −0.313480
\(253\) 862.676 0.214371
\(254\) 453.651 0.112065
\(255\) −294.288 −0.0722706
\(256\) −4230.62 −1.03287
\(257\) −1440.11 −0.349540 −0.174770 0.984609i \(-0.555918\pi\)
−0.174770 + 0.984609i \(0.555918\pi\)
\(258\) 487.537 0.117646
\(259\) −1646.23 −0.394948
\(260\) 787.497 0.187840
\(261\) −510.113 −0.120978
\(262\) 1438.06 0.339097
\(263\) −5960.11 −1.39740 −0.698699 0.715415i \(-0.746237\pi\)
−0.698699 + 0.715415i \(0.746237\pi\)
\(264\) 361.941 0.0843786
\(265\) 1924.49 0.446114
\(266\) 557.249 0.128448
\(267\) 246.166 0.0564236
\(268\) 88.8599 0.0202537
\(269\) −2603.80 −0.590173 −0.295087 0.955471i \(-0.595349\pi\)
−0.295087 + 0.955471i \(0.595349\pi\)
\(270\) 741.661 0.167171
\(271\) 5159.24 1.15646 0.578231 0.815873i \(-0.303743\pi\)
0.578231 + 0.815873i \(0.303743\pi\)
\(272\) 986.213 0.219845
\(273\) 830.237 0.184059
\(274\) −744.500 −0.164149
\(275\) −275.000 −0.0603023
\(276\) −378.703 −0.0825914
\(277\) −3966.30 −0.860331 −0.430165 0.902750i \(-0.641545\pi\)
−0.430165 + 0.902750i \(0.641545\pi\)
\(278\) 960.090 0.207131
\(279\) −6992.53 −1.50047
\(280\) 1697.19 0.362237
\(281\) −6393.99 −1.35742 −0.678708 0.734409i \(-0.737460\pi\)
−0.678708 + 0.734409i \(0.737460\pi\)
\(282\) 755.453 0.159527
\(283\) 3112.65 0.653808 0.326904 0.945058i \(-0.393995\pi\)
0.326904 + 0.945058i \(0.393995\pi\)
\(284\) −2962.37 −0.618958
\(285\) 128.373 0.0266813
\(286\) 1020.02 0.210892
\(287\) −4094.29 −0.842085
\(288\) 3704.62 0.757976
\(289\) −3015.84 −0.613850
\(290\) 213.164 0.0431636
\(291\) −218.358 −0.0439876
\(292\) 1251.92 0.250901
\(293\) 4021.36 0.801810 0.400905 0.916120i \(-0.368696\pi\)
0.400905 + 0.916120i \(0.368696\pi\)
\(294\) −422.685 −0.0838486
\(295\) 2363.10 0.466390
\(296\) 2875.54 0.564653
\(297\) −775.528 −0.151517
\(298\) 5198.31 1.01050
\(299\) −3456.54 −0.668551
\(300\) 120.721 0.0232328
\(301\) −2390.52 −0.457764
\(302\) 6151.35 1.17209
\(303\) 553.119 0.104871
\(304\) −430.202 −0.0811638
\(305\) −3088.37 −0.579802
\(306\) −2306.93 −0.430975
\(307\) 2712.90 0.504343 0.252172 0.967683i \(-0.418855\pi\)
0.252172 + 0.967683i \(0.418855\pi\)
\(308\) −547.962 −0.101374
\(309\) 205.216 0.0377809
\(310\) 2922.02 0.535353
\(311\) −8728.35 −1.59144 −0.795722 0.605662i \(-0.792908\pi\)
−0.795722 + 0.605662i \(0.792908\pi\)
\(312\) −1450.21 −0.263148
\(313\) −10402.1 −1.87847 −0.939233 0.343279i \(-0.888462\pi\)
−0.939233 + 0.343279i \(0.888462\pi\)
\(314\) −1460.04 −0.262404
\(315\) −1754.64 −0.313850
\(316\) 1700.21 0.302672
\(317\) 277.488 0.0491650 0.0245825 0.999698i \(-0.492174\pi\)
0.0245825 + 0.999698i \(0.492174\pi\)
\(318\) −1094.27 −0.192968
\(319\) −222.898 −0.0391220
\(320\) −2453.77 −0.428655
\(321\) 1990.16 0.346044
\(322\) −2300.12 −0.398077
\(323\) −827.571 −0.142561
\(324\) −2088.45 −0.358102
\(325\) 1101.86 0.188062
\(326\) −6572.38 −1.11660
\(327\) 996.415 0.168507
\(328\) 7151.69 1.20392
\(329\) −3704.17 −0.620723
\(330\) 156.367 0.0260839
\(331\) 6416.27 1.06547 0.532734 0.846283i \(-0.321165\pi\)
0.532734 + 0.846283i \(0.321165\pi\)
\(332\) −2705.45 −0.447231
\(333\) −2972.88 −0.489227
\(334\) 1504.94 0.246546
\(335\) 124.332 0.0202776
\(336\) −426.516 −0.0692510
\(337\) 10858.3 1.75517 0.877584 0.479423i \(-0.159154\pi\)
0.877584 + 0.479423i \(0.159154\pi\)
\(338\) 535.342 0.0861502
\(339\) −1476.33 −0.236528
\(340\) −778.241 −0.124135
\(341\) −3055.45 −0.485225
\(342\) 1006.32 0.159110
\(343\) 6853.98 1.07895
\(344\) 4175.62 0.654461
\(345\) −529.878 −0.0826888
\(346\) −2254.90 −0.350359
\(347\) −4827.37 −0.746821 −0.373411 0.927666i \(-0.621812\pi\)
−0.373411 + 0.927666i \(0.621812\pi\)
\(348\) 97.8492 0.0150726
\(349\) 12166.2 1.86603 0.933013 0.359843i \(-0.117170\pi\)
0.933013 + 0.359843i \(0.117170\pi\)
\(350\) 733.223 0.111978
\(351\) 3107.36 0.472531
\(352\) 1618.77 0.245115
\(353\) 3040.19 0.458394 0.229197 0.973380i \(-0.426390\pi\)
0.229197 + 0.973380i \(0.426390\pi\)
\(354\) −1343.67 −0.201738
\(355\) −4144.92 −0.619689
\(356\) 650.983 0.0969158
\(357\) −820.479 −0.121637
\(358\) −2740.74 −0.404616
\(359\) 176.300 0.0259186 0.0129593 0.999916i \(-0.495875\pi\)
0.0129593 + 0.999916i \(0.495875\pi\)
\(360\) 3064.91 0.448708
\(361\) 361.000 0.0526316
\(362\) 2385.58 0.346362
\(363\) −163.507 −0.0236416
\(364\) 2195.55 0.316149
\(365\) 1751.67 0.251197
\(366\) 1756.06 0.250795
\(367\) −860.001 −0.122321 −0.0611604 0.998128i \(-0.519480\pi\)
−0.0611604 + 0.998128i \(0.519480\pi\)
\(368\) 1775.72 0.251537
\(369\) −7393.77 −1.04310
\(370\) 1242.30 0.174551
\(371\) 5365.50 0.750844
\(372\) 1341.30 0.186944
\(373\) −9198.13 −1.27684 −0.638420 0.769688i \(-0.720412\pi\)
−0.638420 + 0.769688i \(0.720412\pi\)
\(374\) −1008.03 −0.139369
\(375\) 168.912 0.0232602
\(376\) 6470.25 0.887441
\(377\) 893.100 0.122008
\(378\) 2067.76 0.281360
\(379\) −7433.64 −1.00749 −0.503747 0.863851i \(-0.668046\pi\)
−0.503747 + 0.863851i \(0.668046\pi\)
\(380\) 339.482 0.0458291
\(381\) 291.368 0.0391791
\(382\) 6906.49 0.925044
\(383\) 9654.84 1.28809 0.644046 0.764987i \(-0.277255\pi\)
0.644046 + 0.764987i \(0.277255\pi\)
\(384\) −195.636 −0.0259987
\(385\) −766.704 −0.101493
\(386\) 5690.67 0.750382
\(387\) −4316.97 −0.567038
\(388\) −577.447 −0.0755552
\(389\) 11358.3 1.48044 0.740218 0.672367i \(-0.234722\pi\)
0.740218 + 0.672367i \(0.234722\pi\)
\(390\) −626.524 −0.0813468
\(391\) 3415.91 0.441816
\(392\) −3620.18 −0.466446
\(393\) 923.626 0.118552
\(394\) 6685.18 0.854809
\(395\) 2378.92 0.303029
\(396\) −989.550 −0.125573
\(397\) 4939.10 0.624399 0.312199 0.950017i \(-0.398934\pi\)
0.312199 + 0.950017i \(0.398934\pi\)
\(398\) 5196.38 0.654450
\(399\) 357.906 0.0449066
\(400\) −566.056 −0.0707570
\(401\) −1507.40 −0.187721 −0.0938604 0.995585i \(-0.529921\pi\)
−0.0938604 + 0.995585i \(0.529921\pi\)
\(402\) −70.6959 −0.00877112
\(403\) 12242.5 1.51325
\(404\) 1462.72 0.180131
\(405\) −2922.14 −0.358524
\(406\) 594.306 0.0726475
\(407\) −1299.02 −0.158207
\(408\) 1433.17 0.173903
\(409\) 11953.9 1.44519 0.722597 0.691270i \(-0.242948\pi\)
0.722597 + 0.691270i \(0.242948\pi\)
\(410\) 3089.69 0.372168
\(411\) −478.172 −0.0573881
\(412\) 542.691 0.0648943
\(413\) 6588.36 0.784969
\(414\) −4153.72 −0.493102
\(415\) −3785.44 −0.447759
\(416\) −6486.01 −0.764430
\(417\) 616.640 0.0724148
\(418\) 439.721 0.0514532
\(419\) 13232.6 1.54285 0.771424 0.636321i \(-0.219545\pi\)
0.771424 + 0.636321i \(0.219545\pi\)
\(420\) 336.573 0.0391025
\(421\) −840.471 −0.0972970 −0.0486485 0.998816i \(-0.515491\pi\)
−0.0486485 + 0.998816i \(0.515491\pi\)
\(422\) 6496.17 0.749357
\(423\) −6689.26 −0.768896
\(424\) −9372.17 −1.07347
\(425\) −1088.91 −0.124282
\(426\) 2356.83 0.268048
\(427\) −8610.42 −0.975849
\(428\) 5262.97 0.594381
\(429\) 655.133 0.0737299
\(430\) 1803.96 0.202313
\(431\) −9747.18 −1.08934 −0.544669 0.838651i \(-0.683345\pi\)
−0.544669 + 0.838651i \(0.683345\pi\)
\(432\) −1596.33 −0.177786
\(433\) 7573.69 0.840574 0.420287 0.907391i \(-0.361929\pi\)
0.420287 + 0.907391i \(0.361929\pi\)
\(434\) 8146.63 0.901039
\(435\) 136.910 0.0150904
\(436\) 2635.01 0.289436
\(437\) −1490.08 −0.163112
\(438\) −996.012 −0.108656
\(439\) 652.502 0.0709390 0.0354695 0.999371i \(-0.488707\pi\)
0.0354695 + 0.999371i \(0.488707\pi\)
\(440\) 1339.24 0.145104
\(441\) 3742.72 0.404138
\(442\) 4038.95 0.434645
\(443\) −6013.35 −0.644927 −0.322464 0.946582i \(-0.604511\pi\)
−0.322464 + 0.946582i \(0.604511\pi\)
\(444\) 570.253 0.0609528
\(445\) 910.850 0.0970302
\(446\) 4525.82 0.480501
\(447\) 3338.74 0.353281
\(448\) −6841.14 −0.721458
\(449\) −4456.99 −0.468459 −0.234230 0.972181i \(-0.575257\pi\)
−0.234230 + 0.972181i \(0.575257\pi\)
\(450\) 1324.11 0.138709
\(451\) −3230.77 −0.337320
\(452\) −3904.14 −0.406273
\(453\) 3950.85 0.409773
\(454\) −9323.48 −0.963816
\(455\) 3072.00 0.316522
\(456\) −625.172 −0.0642025
\(457\) −2578.27 −0.263909 −0.131954 0.991256i \(-0.542125\pi\)
−0.131954 + 0.991256i \(0.542125\pi\)
\(458\) −11037.9 −1.12613
\(459\) −3070.83 −0.312275
\(460\) −1401.26 −0.142030
\(461\) −4415.71 −0.446117 −0.223059 0.974805i \(-0.571604\pi\)
−0.223059 + 0.974805i \(0.571604\pi\)
\(462\) 435.953 0.0439012
\(463\) 7177.27 0.720423 0.360211 0.932871i \(-0.382705\pi\)
0.360211 + 0.932871i \(0.382705\pi\)
\(464\) −458.810 −0.0459046
\(465\) 1876.73 0.187165
\(466\) −10683.1 −1.06199
\(467\) 9415.10 0.932931 0.466466 0.884539i \(-0.345527\pi\)
0.466466 + 0.884539i \(0.345527\pi\)
\(468\) 3964.89 0.391618
\(469\) 346.640 0.0341286
\(470\) 2795.29 0.274334
\(471\) −937.744 −0.0917388
\(472\) −11508.2 −1.12226
\(473\) −1886.34 −0.183370
\(474\) −1352.67 −0.131076
\(475\) 475.000 0.0458831
\(476\) −2169.75 −0.208929
\(477\) 9689.41 0.930079
\(478\) 7351.60 0.703461
\(479\) 14544.7 1.38740 0.693701 0.720263i \(-0.255979\pi\)
0.693701 + 0.720263i \(0.255979\pi\)
\(480\) −994.288 −0.0945476
\(481\) 5204.88 0.493393
\(482\) 275.947 0.0260768
\(483\) −1477.31 −0.139171
\(484\) −432.392 −0.0406079
\(485\) −807.959 −0.0756444
\(486\) 5666.51 0.528885
\(487\) −925.751 −0.0861392 −0.0430696 0.999072i \(-0.513714\pi\)
−0.0430696 + 0.999072i \(0.513714\pi\)
\(488\) 15040.2 1.39516
\(489\) −4221.27 −0.390373
\(490\) −1564.00 −0.144192
\(491\) 3372.03 0.309934 0.154967 0.987920i \(-0.450473\pi\)
0.154967 + 0.987920i \(0.450473\pi\)
\(492\) 1418.26 0.129960
\(493\) −882.603 −0.0806297
\(494\) −1761.86 −0.160465
\(495\) −1384.57 −0.125721
\(496\) −6289.29 −0.569349
\(497\) −11556.1 −1.04298
\(498\) 2152.42 0.193679
\(499\) 2491.89 0.223552 0.111776 0.993733i \(-0.464346\pi\)
0.111776 + 0.993733i \(0.464346\pi\)
\(500\) 446.686 0.0399529
\(501\) 966.580 0.0861949
\(502\) 2301.18 0.204595
\(503\) −6378.28 −0.565395 −0.282697 0.959209i \(-0.591229\pi\)
−0.282697 + 0.959209i \(0.591229\pi\)
\(504\) 8545.01 0.755208
\(505\) 2046.62 0.180344
\(506\) −1815.01 −0.159460
\(507\) 343.836 0.0301189
\(508\) 770.520 0.0672958
\(509\) −6553.75 −0.570707 −0.285353 0.958422i \(-0.592111\pi\)
−0.285353 + 0.958422i \(0.592111\pi\)
\(510\) 619.160 0.0537585
\(511\) 4883.70 0.422783
\(512\) 7742.71 0.668326
\(513\) 1339.55 0.115288
\(514\) 3029.89 0.260005
\(515\) 759.329 0.0649709
\(516\) 828.075 0.0706472
\(517\) −2922.93 −0.248647
\(518\) 3463.54 0.293782
\(519\) −1448.26 −0.122489
\(520\) −5366.01 −0.452529
\(521\) −5965.32 −0.501623 −0.250811 0.968036i \(-0.580697\pi\)
−0.250811 + 0.968036i \(0.580697\pi\)
\(522\) 1073.24 0.0899894
\(523\) 17544.9 1.46689 0.733446 0.679748i \(-0.237911\pi\)
0.733446 + 0.679748i \(0.237911\pi\)
\(524\) 2442.52 0.203630
\(525\) 470.930 0.0391487
\(526\) 12539.6 1.03946
\(527\) −12098.6 −1.00004
\(528\) −336.560 −0.0277403
\(529\) −6016.51 −0.494494
\(530\) −4048.98 −0.331842
\(531\) 11897.7 0.972350
\(532\) 946.480 0.0771337
\(533\) 12944.9 1.05198
\(534\) −517.914 −0.0419707
\(535\) 7363.90 0.595082
\(536\) −605.491 −0.0487934
\(537\) −1760.30 −0.141458
\(538\) 5478.21 0.439001
\(539\) 1635.42 0.130691
\(540\) 1259.70 0.100387
\(541\) 14642.0 1.16361 0.581803 0.813330i \(-0.302347\pi\)
0.581803 + 0.813330i \(0.302347\pi\)
\(542\) −10854.7 −0.860235
\(543\) 1532.19 0.121091
\(544\) 6409.78 0.505178
\(545\) 3686.88 0.289778
\(546\) −1746.76 −0.136913
\(547\) −10032.1 −0.784171 −0.392086 0.919929i \(-0.628246\pi\)
−0.392086 + 0.919929i \(0.628246\pi\)
\(548\) −1264.52 −0.0985725
\(549\) −15549.3 −1.20880
\(550\) 578.580 0.0448559
\(551\) 385.006 0.0297674
\(552\) 2580.48 0.198972
\(553\) 6632.47 0.510020
\(554\) 8344.80 0.639957
\(555\) 797.894 0.0610247
\(556\) 1630.70 0.124383
\(557\) 1967.94 0.149702 0.0748512 0.997195i \(-0.476152\pi\)
0.0748512 + 0.997195i \(0.476152\pi\)
\(558\) 14711.8 1.11613
\(559\) 7558.10 0.571867
\(560\) −1578.17 −0.119089
\(561\) −647.433 −0.0487248
\(562\) 13452.5 1.00971
\(563\) −10183.1 −0.762288 −0.381144 0.924516i \(-0.624470\pi\)
−0.381144 + 0.924516i \(0.624470\pi\)
\(564\) 1283.13 0.0957968
\(565\) −5462.64 −0.406752
\(566\) −6548.78 −0.486335
\(567\) −8146.98 −0.603423
\(568\) 20185.6 1.49114
\(569\) −23440.2 −1.72700 −0.863502 0.504345i \(-0.831734\pi\)
−0.863502 + 0.504345i \(0.831734\pi\)
\(570\) −270.088 −0.0198469
\(571\) 3958.88 0.290147 0.145074 0.989421i \(-0.453658\pi\)
0.145074 + 0.989421i \(0.453658\pi\)
\(572\) 1732.49 0.126642
\(573\) 4435.85 0.323404
\(574\) 8614.09 0.626386
\(575\) −1960.63 −0.142198
\(576\) −12354.2 −0.893679
\(577\) 24216.8 1.74724 0.873620 0.486609i \(-0.161766\pi\)
0.873620 + 0.486609i \(0.161766\pi\)
\(578\) 6345.12 0.456613
\(579\) 3654.96 0.262340
\(580\) 362.057 0.0259200
\(581\) −10553.9 −0.753611
\(582\) 459.410 0.0327202
\(583\) 4233.87 0.300770
\(584\) −8530.58 −0.604448
\(585\) 5547.64 0.392080
\(586\) −8460.64 −0.596427
\(587\) −16503.0 −1.16040 −0.580199 0.814475i \(-0.697025\pi\)
−0.580199 + 0.814475i \(0.697025\pi\)
\(588\) −717.925 −0.0503515
\(589\) 5277.59 0.369201
\(590\) −4971.79 −0.346924
\(591\) 4293.71 0.298849
\(592\) −2673.89 −0.185636
\(593\) 4271.85 0.295824 0.147912 0.989001i \(-0.452745\pi\)
0.147912 + 0.989001i \(0.452745\pi\)
\(594\) 1631.65 0.112706
\(595\) −3035.89 −0.209176
\(596\) 8829.26 0.606813
\(597\) 3337.50 0.228802
\(598\) 7272.30 0.497302
\(599\) −18163.3 −1.23895 −0.619476 0.785016i \(-0.712655\pi\)
−0.619476 + 0.785016i \(0.712655\pi\)
\(600\) −822.594 −0.0559705
\(601\) −25285.8 −1.71619 −0.858094 0.513492i \(-0.828352\pi\)
−0.858094 + 0.513492i \(0.828352\pi\)
\(602\) 5029.47 0.340508
\(603\) 625.987 0.0422755
\(604\) 10448.0 0.703845
\(605\) −605.000 −0.0406558
\(606\) −1163.72 −0.0780082
\(607\) −599.666 −0.0400983 −0.0200492 0.999799i \(-0.506382\pi\)
−0.0200492 + 0.999799i \(0.506382\pi\)
\(608\) −2796.05 −0.186505
\(609\) 381.707 0.0253983
\(610\) 6497.70 0.431286
\(611\) 11711.5 0.775444
\(612\) −3918.29 −0.258803
\(613\) 17067.6 1.12456 0.562280 0.826947i \(-0.309924\pi\)
0.562280 + 0.826947i \(0.309924\pi\)
\(614\) −5707.74 −0.375156
\(615\) 1984.42 0.130113
\(616\) 3733.82 0.244220
\(617\) 10691.4 0.697600 0.348800 0.937197i \(-0.386589\pi\)
0.348800 + 0.937197i \(0.386589\pi\)
\(618\) −431.759 −0.0281034
\(619\) −19853.6 −1.28915 −0.644573 0.764543i \(-0.722965\pi\)
−0.644573 + 0.764543i \(0.722965\pi\)
\(620\) 4963.01 0.321483
\(621\) −5529.17 −0.357291
\(622\) 18363.8 1.18380
\(623\) 2539.46 0.163309
\(624\) 1348.52 0.0865125
\(625\) 625.000 0.0400000
\(626\) 21885.2 1.39730
\(627\) 282.421 0.0179885
\(628\) −2479.86 −0.157575
\(629\) −5143.70 −0.326062
\(630\) 3691.63 0.233457
\(631\) 5829.93 0.367806 0.183903 0.982944i \(-0.441127\pi\)
0.183903 + 0.982944i \(0.441127\pi\)
\(632\) −11585.2 −0.729171
\(633\) 4172.32 0.261982
\(634\) −583.815 −0.0365714
\(635\) 1078.10 0.0673752
\(636\) −1858.61 −0.115878
\(637\) −6552.72 −0.407580
\(638\) 468.962 0.0291009
\(639\) −20868.8 −1.29195
\(640\) −723.883 −0.0447094
\(641\) −14587.1 −0.898840 −0.449420 0.893320i \(-0.648369\pi\)
−0.449420 + 0.893320i \(0.648369\pi\)
\(642\) −4187.16 −0.257405
\(643\) 3107.91 0.190613 0.0953065 0.995448i \(-0.469617\pi\)
0.0953065 + 0.995448i \(0.469617\pi\)
\(644\) −3906.73 −0.239047
\(645\) 1158.64 0.0707306
\(646\) 1741.15 0.106044
\(647\) −30018.7 −1.82405 −0.912023 0.410139i \(-0.865480\pi\)
−0.912023 + 0.410139i \(0.865480\pi\)
\(648\) 14230.7 0.862708
\(649\) 5198.82 0.314440
\(650\) −2318.23 −0.139890
\(651\) 5232.37 0.315012
\(652\) −11163.1 −0.670523
\(653\) −15017.8 −0.899987 −0.449994 0.893032i \(-0.648574\pi\)
−0.449994 + 0.893032i \(0.648574\pi\)
\(654\) −2096.38 −0.125344
\(655\) 3417.56 0.203870
\(656\) −6650.17 −0.395801
\(657\) 8819.33 0.523706
\(658\) 7793.31 0.461725
\(659\) −12515.7 −0.739822 −0.369911 0.929067i \(-0.620612\pi\)
−0.369911 + 0.929067i \(0.620612\pi\)
\(660\) 265.587 0.0156635
\(661\) −8228.73 −0.484207 −0.242103 0.970250i \(-0.577837\pi\)
−0.242103 + 0.970250i \(0.577837\pi\)
\(662\) −13499.4 −0.792549
\(663\) 2594.11 0.151956
\(664\) 18434.9 1.07743
\(665\) 1324.31 0.0772247
\(666\) 6254.71 0.363912
\(667\) −1589.16 −0.0922529
\(668\) 2556.11 0.148052
\(669\) 2906.81 0.167988
\(670\) −261.585 −0.0150835
\(671\) −6794.41 −0.390902
\(672\) −2772.09 −0.159131
\(673\) 10839.5 0.620848 0.310424 0.950598i \(-0.399529\pi\)
0.310424 + 0.950598i \(0.399529\pi\)
\(674\) −22845.2 −1.30558
\(675\) 1762.56 0.100505
\(676\) 909.272 0.0517337
\(677\) 26166.0 1.48544 0.742719 0.669604i \(-0.233536\pi\)
0.742719 + 0.669604i \(0.233536\pi\)
\(678\) 3106.09 0.175942
\(679\) −2252.60 −0.127315
\(680\) 5302.94 0.299056
\(681\) −5988.22 −0.336959
\(682\) 6428.44 0.360935
\(683\) 21585.3 1.20928 0.604641 0.796498i \(-0.293317\pi\)
0.604641 + 0.796498i \(0.293317\pi\)
\(684\) 1709.22 0.0955464
\(685\) −1769.31 −0.0986888
\(686\) −14420.3 −0.802578
\(687\) −7089.34 −0.393705
\(688\) −3882.80 −0.215161
\(689\) −16964.1 −0.937999
\(690\) 1114.82 0.0615081
\(691\) 12782.1 0.703696 0.351848 0.936057i \(-0.385553\pi\)
0.351848 + 0.936057i \(0.385553\pi\)
\(692\) −3829.92 −0.210393
\(693\) −3860.20 −0.211597
\(694\) 10156.4 0.555523
\(695\) 2281.66 0.124530
\(696\) −666.745 −0.0363116
\(697\) −12792.8 −0.695210
\(698\) −25596.8 −1.38804
\(699\) −6861.48 −0.371280
\(700\) 1245.37 0.0672436
\(701\) −17148.3 −0.923943 −0.461971 0.886895i \(-0.652858\pi\)
−0.461971 + 0.886895i \(0.652858\pi\)
\(702\) −6537.65 −0.351492
\(703\) 2243.77 0.120377
\(704\) −5398.28 −0.288999
\(705\) 1795.34 0.0959098
\(706\) −6396.34 −0.340977
\(707\) 5706.02 0.303532
\(708\) −2282.21 −0.121145
\(709\) −20910.8 −1.10765 −0.553824 0.832634i \(-0.686832\pi\)
−0.553824 + 0.832634i \(0.686832\pi\)
\(710\) 8720.61 0.460956
\(711\) 11977.4 0.631768
\(712\) −4435.80 −0.233481
\(713\) −21784.0 −1.14420
\(714\) 1726.23 0.0904796
\(715\) 2424.09 0.126791
\(716\) −4655.11 −0.242974
\(717\) 4721.74 0.245937
\(718\) −370.922 −0.0192795
\(719\) −25881.4 −1.34244 −0.671218 0.741260i \(-0.734229\pi\)
−0.671218 + 0.741260i \(0.734229\pi\)
\(720\) −2849.98 −0.147517
\(721\) 2117.02 0.109351
\(722\) −759.518 −0.0391500
\(723\) 177.233 0.00911671
\(724\) 4051.87 0.207992
\(725\) 506.587 0.0259506
\(726\) 344.006 0.0175858
\(727\) 5329.67 0.271893 0.135947 0.990716i \(-0.456592\pi\)
0.135947 + 0.990716i \(0.456592\pi\)
\(728\) −14960.5 −0.761640
\(729\) −12140.1 −0.616781
\(730\) −3685.40 −0.186853
\(731\) −7469.26 −0.377922
\(732\) 2982.65 0.150604
\(733\) 15533.6 0.782738 0.391369 0.920234i \(-0.372002\pi\)
0.391369 + 0.920234i \(0.372002\pi\)
\(734\) 1809.38 0.0909883
\(735\) −1004.51 −0.0504110
\(736\) 11541.1 0.578003
\(737\) 273.530 0.0136711
\(738\) 15555.9 0.775911
\(739\) −18629.2 −0.927315 −0.463658 0.886014i \(-0.653463\pi\)
−0.463658 + 0.886014i \(0.653463\pi\)
\(740\) 2110.02 0.104819
\(741\) −1131.59 −0.0561000
\(742\) −11288.6 −0.558515
\(743\) −12179.0 −0.601350 −0.300675 0.953727i \(-0.597212\pi\)
−0.300675 + 0.953727i \(0.597212\pi\)
\(744\) −9139.62 −0.450369
\(745\) 12353.8 0.607529
\(746\) 19352.2 0.949778
\(747\) −19058.9 −0.933507
\(748\) −1712.13 −0.0836921
\(749\) 20530.7 1.00157
\(750\) −355.379 −0.0173021
\(751\) −17345.0 −0.842779 −0.421389 0.906880i \(-0.638457\pi\)
−0.421389 + 0.906880i \(0.638457\pi\)
\(752\) −6016.52 −0.291755
\(753\) 1477.98 0.0715282
\(754\) −1879.02 −0.0907557
\(755\) 14618.7 0.704676
\(756\) 3512.07 0.168959
\(757\) −8740.90 −0.419674 −0.209837 0.977736i \(-0.567293\pi\)
−0.209837 + 0.977736i \(0.567293\pi\)
\(758\) 15639.8 0.749426
\(759\) −1165.73 −0.0557488
\(760\) −2313.23 −0.110407
\(761\) 12237.7 0.582940 0.291470 0.956580i \(-0.405856\pi\)
0.291470 + 0.956580i \(0.405856\pi\)
\(762\) −613.017 −0.0291434
\(763\) 10279.1 0.487717
\(764\) 11730.6 0.555494
\(765\) −5482.44 −0.259108
\(766\) −20313.1 −0.958148
\(767\) −20830.4 −0.980630
\(768\) 5716.83 0.268604
\(769\) −30994.0 −1.45341 −0.726704 0.686950i \(-0.758949\pi\)
−0.726704 + 0.686950i \(0.758949\pi\)
\(770\) 1613.09 0.0754958
\(771\) 1946.02 0.0909003
\(772\) 9665.52 0.450608
\(773\) 3179.76 0.147953 0.0739767 0.997260i \(-0.476431\pi\)
0.0739767 + 0.997260i \(0.476431\pi\)
\(774\) 9082.58 0.421791
\(775\) 6944.20 0.321862
\(776\) 3934.72 0.182021
\(777\) 2224.54 0.102709
\(778\) −23897.1 −1.10122
\(779\) 5580.42 0.256662
\(780\) −1064.14 −0.0488492
\(781\) −9118.83 −0.417794
\(782\) −7186.82 −0.328645
\(783\) 1428.63 0.0652043
\(784\) 3366.31 0.153349
\(785\) −3469.80 −0.157761
\(786\) −1943.24 −0.0881847
\(787\) −4618.33 −0.209181 −0.104591 0.994515i \(-0.533353\pi\)
−0.104591 + 0.994515i \(0.533353\pi\)
\(788\) 11354.7 0.513317
\(789\) 8053.87 0.363403
\(790\) −5005.07 −0.225408
\(791\) −15229.9 −0.684594
\(792\) 6742.79 0.302519
\(793\) 27223.6 1.21909
\(794\) −10391.5 −0.464459
\(795\) −2600.55 −0.116015
\(796\) 8825.99 0.393001
\(797\) −11910.6 −0.529356 −0.264678 0.964337i \(-0.585266\pi\)
−0.264678 + 0.964337i \(0.585266\pi\)
\(798\) −753.009 −0.0334038
\(799\) −11573.8 −0.512457
\(800\) −3679.02 −0.162591
\(801\) 4585.95 0.202293
\(802\) 3171.46 0.139636
\(803\) 3853.68 0.169357
\(804\) −120.076 −0.00526711
\(805\) −5466.26 −0.239330
\(806\) −25757.2 −1.12563
\(807\) 3518.51 0.153479
\(808\) −9966.97 −0.433956
\(809\) 14183.9 0.616417 0.308208 0.951319i \(-0.400271\pi\)
0.308208 + 0.951319i \(0.400271\pi\)
\(810\) 6147.97 0.266688
\(811\) −9815.70 −0.425001 −0.212501 0.977161i \(-0.568161\pi\)
−0.212501 + 0.977161i \(0.568161\pi\)
\(812\) 1009.42 0.0436253
\(813\) −6971.66 −0.300746
\(814\) 2733.05 0.117682
\(815\) −15619.3 −0.671314
\(816\) −1332.67 −0.0571723
\(817\) 3258.22 0.139523
\(818\) −25150.2 −1.07501
\(819\) 15466.9 0.659900
\(820\) 5247.79 0.223489
\(821\) −28522.4 −1.21247 −0.606236 0.795285i \(-0.707321\pi\)
−0.606236 + 0.795285i \(0.707321\pi\)
\(822\) 1006.04 0.0426882
\(823\) −36441.1 −1.54345 −0.771724 0.635957i \(-0.780605\pi\)
−0.771724 + 0.635957i \(0.780605\pi\)
\(824\) −3697.90 −0.156338
\(825\) 371.607 0.0156820
\(826\) −13861.4 −0.583899
\(827\) 17468.3 0.734502 0.367251 0.930122i \(-0.380299\pi\)
0.367251 + 0.930122i \(0.380299\pi\)
\(828\) −7055.05 −0.296111
\(829\) 23483.0 0.983834 0.491917 0.870642i \(-0.336296\pi\)
0.491917 + 0.870642i \(0.336296\pi\)
\(830\) 7964.29 0.333066
\(831\) 5359.64 0.223735
\(832\) 21629.6 0.901289
\(833\) 6475.70 0.269351
\(834\) −1297.37 −0.0538658
\(835\) 3576.49 0.148227
\(836\) 746.860 0.0308979
\(837\) 19583.4 0.808721
\(838\) −27840.4 −1.14765
\(839\) −3723.45 −0.153216 −0.0766078 0.997061i \(-0.524409\pi\)
−0.0766078 + 0.997061i \(0.524409\pi\)
\(840\) −2293.41 −0.0942024
\(841\) −23978.4 −0.983164
\(842\) 1768.29 0.0723744
\(843\) 8640.18 0.353005
\(844\) 11033.7 0.449993
\(845\) 1272.25 0.0517948
\(846\) 14073.7 0.571944
\(847\) −1686.75 −0.0684267
\(848\) 8714.94 0.352915
\(849\) −4206.11 −0.170027
\(850\) 2290.98 0.0924472
\(851\) −9261.46 −0.373065
\(852\) 4003.04 0.160965
\(853\) 15522.6 0.623076 0.311538 0.950234i \(-0.399156\pi\)
0.311538 + 0.950234i \(0.399156\pi\)
\(854\) 18115.7 0.725886
\(855\) 2391.53 0.0956592
\(856\) −35861.9 −1.43193
\(857\) −41729.3 −1.66330 −0.831649 0.555302i \(-0.812603\pi\)
−0.831649 + 0.555302i \(0.812603\pi\)
\(858\) −1378.35 −0.0548440
\(859\) −10863.6 −0.431502 −0.215751 0.976448i \(-0.569220\pi\)
−0.215751 + 0.976448i \(0.569220\pi\)
\(860\) 3064.00 0.121490
\(861\) 5532.60 0.218990
\(862\) 20507.3 0.810305
\(863\) −29609.2 −1.16791 −0.583956 0.811785i \(-0.698496\pi\)
−0.583956 + 0.811785i \(0.698496\pi\)
\(864\) −10375.2 −0.408532
\(865\) −5358.80 −0.210641
\(866\) −15934.5 −0.625261
\(867\) 4075.30 0.159636
\(868\) 13836.9 0.541079
\(869\) 5233.62 0.204302
\(870\) −288.048 −0.0112250
\(871\) −1095.97 −0.0426355
\(872\) −17955.0 −0.697284
\(873\) −4067.91 −0.157707
\(874\) 3135.01 0.121331
\(875\) 1742.51 0.0673230
\(876\) −1691.71 −0.0652485
\(877\) −20633.9 −0.794477 −0.397239 0.917715i \(-0.630031\pi\)
−0.397239 + 0.917715i \(0.630031\pi\)
\(878\) −1372.82 −0.0527680
\(879\) −5434.05 −0.208516
\(880\) −1245.32 −0.0477043
\(881\) −26074.6 −0.997134 −0.498567 0.866851i \(-0.666140\pi\)
−0.498567 + 0.866851i \(0.666140\pi\)
\(882\) −7874.41 −0.300618
\(883\) −8802.66 −0.335485 −0.167742 0.985831i \(-0.553648\pi\)
−0.167742 + 0.985831i \(0.553648\pi\)
\(884\) 6860.10 0.261007
\(885\) −3193.25 −0.121288
\(886\) 12651.6 0.479729
\(887\) 6551.53 0.248003 0.124002 0.992282i \(-0.460427\pi\)
0.124002 + 0.992282i \(0.460427\pi\)
\(888\) −3885.71 −0.146842
\(889\) 3005.77 0.113398
\(890\) −1916.36 −0.0721759
\(891\) −6428.71 −0.241717
\(892\) 7687.04 0.288544
\(893\) 5048.70 0.189192
\(894\) −7024.46 −0.262789
\(895\) −6513.39 −0.243261
\(896\) −2018.20 −0.0752492
\(897\) 4670.80 0.173861
\(898\) 9377.17 0.348464
\(899\) 5628.55 0.208813
\(900\) 2248.98 0.0832954
\(901\) 16764.7 0.619882
\(902\) 6797.31 0.250915
\(903\) 3230.30 0.119045
\(904\) 26602.8 0.978757
\(905\) 5669.34 0.208238
\(906\) −8312.30 −0.304810
\(907\) −27373.0 −1.00210 −0.501050 0.865418i \(-0.667053\pi\)
−0.501050 + 0.865418i \(0.667053\pi\)
\(908\) −15835.8 −0.578777
\(909\) 10304.3 0.375988
\(910\) −6463.27 −0.235445
\(911\) −27648.6 −1.00553 −0.502766 0.864422i \(-0.667684\pi\)
−0.502766 + 0.864422i \(0.667684\pi\)
\(912\) 581.331 0.0211072
\(913\) −8327.96 −0.301879
\(914\) 5424.49 0.196309
\(915\) 4173.30 0.150782
\(916\) −18747.7 −0.676246
\(917\) 9528.20 0.343129
\(918\) 6460.81 0.232286
\(919\) 13285.7 0.476883 0.238442 0.971157i \(-0.423363\pi\)
0.238442 + 0.971157i \(0.423363\pi\)
\(920\) 9548.17 0.342167
\(921\) −3665.93 −0.131158
\(922\) 9290.33 0.331844
\(923\) 36537.0 1.30296
\(924\) 740.460 0.0263629
\(925\) 2952.33 0.104943
\(926\) −15100.4 −0.535887
\(927\) 3823.07 0.135454
\(928\) −2981.99 −0.105483
\(929\) −26022.5 −0.919021 −0.459510 0.888172i \(-0.651975\pi\)
−0.459510 + 0.888172i \(0.651975\pi\)
\(930\) −3948.51 −0.139222
\(931\) −2824.81 −0.0994408
\(932\) −18145.1 −0.637729
\(933\) 11794.6 0.413866
\(934\) −19808.7 −0.693961
\(935\) −2395.60 −0.0837909
\(936\) −27016.8 −0.943452
\(937\) 46641.3 1.62615 0.813075 0.582158i \(-0.197792\pi\)
0.813075 + 0.582158i \(0.197792\pi\)
\(938\) −729.304 −0.0253866
\(939\) 14056.3 0.488509
\(940\) 4747.76 0.164739
\(941\) −27230.8 −0.943357 −0.471679 0.881771i \(-0.656352\pi\)
−0.471679 + 0.881771i \(0.656352\pi\)
\(942\) 1972.95 0.0682400
\(943\) −23034.0 −0.795428
\(944\) 10701.2 0.368955
\(945\) 4914.06 0.169158
\(946\) 3968.71 0.136400
\(947\) −37535.1 −1.28799 −0.643996 0.765029i \(-0.722725\pi\)
−0.643996 + 0.765029i \(0.722725\pi\)
\(948\) −2297.49 −0.0787120
\(949\) −15440.8 −0.528166
\(950\) −999.365 −0.0341302
\(951\) −374.969 −0.0127857
\(952\) 14784.7 0.503334
\(953\) 2383.13 0.0810044 0.0405022 0.999179i \(-0.487104\pi\)
0.0405022 + 0.999179i \(0.487104\pi\)
\(954\) −20385.8 −0.691839
\(955\) 16413.3 0.556150
\(956\) 12486.6 0.422432
\(957\) 301.202 0.0101739
\(958\) −30601.1 −1.03202
\(959\) −4932.86 −0.166101
\(960\) 3315.76 0.111475
\(961\) 47364.1 1.58988
\(962\) −10950.7 −0.367011
\(963\) 37075.8 1.24065
\(964\) 468.692 0.0156593
\(965\) 13523.9 0.451140
\(966\) 3108.15 0.103523
\(967\) 21589.3 0.717957 0.358979 0.933346i \(-0.383125\pi\)
0.358979 + 0.933346i \(0.383125\pi\)
\(968\) 2946.32 0.0978289
\(969\) 1118.29 0.0370740
\(970\) 1699.89 0.0562681
\(971\) −36213.7 −1.19686 −0.598431 0.801174i \(-0.704209\pi\)
−0.598431 + 0.801174i \(0.704209\pi\)
\(972\) 9624.50 0.317599
\(973\) 6361.31 0.209593
\(974\) 1947.71 0.0640747
\(975\) −1488.94 −0.0489069
\(976\) −13985.5 −0.458673
\(977\) 29395.9 0.962599 0.481300 0.876556i \(-0.340165\pi\)
0.481300 + 0.876556i \(0.340165\pi\)
\(978\) 8881.24 0.290379
\(979\) 2003.87 0.0654177
\(980\) −2656.43 −0.0865883
\(981\) 18562.7 0.604141
\(982\) −7094.50 −0.230544
\(983\) 33250.4 1.07886 0.539432 0.842029i \(-0.318639\pi\)
0.539432 + 0.842029i \(0.318639\pi\)
\(984\) −9664.05 −0.313088
\(985\) 15887.4 0.513923
\(986\) 1856.93 0.0599764
\(987\) 5005.44 0.161423
\(988\) −2992.49 −0.0963601
\(989\) −13448.7 −0.432401
\(990\) 2913.03 0.0935175
\(991\) −33387.9 −1.07024 −0.535118 0.844778i \(-0.679733\pi\)
−0.535118 + 0.844778i \(0.679733\pi\)
\(992\) −40876.5 −1.30830
\(993\) −8670.28 −0.277083
\(994\) 24313.2 0.775823
\(995\) 12349.3 0.393465
\(996\) 3655.86 0.116306
\(997\) −22983.7 −0.730092 −0.365046 0.930990i \(-0.618947\pi\)
−0.365046 + 0.930990i \(0.618947\pi\)
\(998\) −5242.76 −0.166289
\(999\) 8325.86 0.263682
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 1045.4.a.c.1.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.7 20 1.1 even 1 trivial