Properties

Label 717.2.a.d.1.5
Level $717$
Weight $2$
Character 717.1
Self dual yes
Analytic conductor $5.725$
Analytic rank $1$
Dimension $6$
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] = [717,2,Mod(1,717)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(717, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("717.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 717 = 3 \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 717.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: \(5.72527382493\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 7x^{4} - x^{3} + 11x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.30642\) of defining polynomial
Character \(\chi\) \(=\) 717.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.05161 q^{2} -1.00000 q^{3} -0.894124 q^{4} +2.87285 q^{5} -1.05161 q^{6} -4.11383 q^{7} -3.04348 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.05161 q^{2} -1.00000 q^{3} -0.894124 q^{4} +2.87285 q^{5} -1.05161 q^{6} -4.11383 q^{7} -3.04348 q^{8} +1.00000 q^{9} +3.02110 q^{10} -1.17944 q^{11} +0.894124 q^{12} +0.346441 q^{13} -4.32613 q^{14} -2.87285 q^{15} -1.41230 q^{16} -3.26277 q^{17} +1.05161 q^{18} -1.31974 q^{19} -2.56868 q^{20} +4.11383 q^{21} -1.24031 q^{22} -9.10667 q^{23} +3.04348 q^{24} +3.25325 q^{25} +0.364319 q^{26} -1.00000 q^{27} +3.67827 q^{28} -0.639311 q^{29} -3.02110 q^{30} -1.52951 q^{31} +4.60178 q^{32} +1.17944 q^{33} -3.43115 q^{34} -11.8184 q^{35} -0.894124 q^{36} +1.40670 q^{37} -1.38785 q^{38} -0.346441 q^{39} -8.74345 q^{40} -2.17524 q^{41} +4.32613 q^{42} -1.97537 q^{43} +1.05456 q^{44} +2.87285 q^{45} -9.57663 q^{46} -1.69565 q^{47} +1.41230 q^{48} +9.92360 q^{49} +3.42114 q^{50} +3.26277 q^{51} -0.309761 q^{52} -0.0493538 q^{53} -1.05161 q^{54} -3.38835 q^{55} +12.5204 q^{56} +1.31974 q^{57} -0.672304 q^{58} +0.806525 q^{59} +2.56868 q^{60} -1.25891 q^{61} -1.60844 q^{62} -4.11383 q^{63} +7.66385 q^{64} +0.995271 q^{65} +1.24031 q^{66} +4.01099 q^{67} +2.91732 q^{68} +9.10667 q^{69} -12.4283 q^{70} +0.361936 q^{71} -3.04348 q^{72} -11.9650 q^{73} +1.47930 q^{74} -3.25325 q^{75} +1.18001 q^{76} +4.85201 q^{77} -0.364319 q^{78} +4.27659 q^{79} -4.05731 q^{80} +1.00000 q^{81} -2.28749 q^{82} +14.0559 q^{83} -3.67827 q^{84} -9.37343 q^{85} -2.07731 q^{86} +0.639311 q^{87} +3.58960 q^{88} -10.6550 q^{89} +3.02110 q^{90} -1.42520 q^{91} +8.14249 q^{92} +1.52951 q^{93} -1.78316 q^{94} -3.79142 q^{95} -4.60178 q^{96} -10.2134 q^{97} +10.4357 q^{98} -1.17944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 6 q^{3} + 4 q^{4} + 5 q^{5} + 2 q^{6} - 9 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 6 q^{3} + 4 q^{4} + 5 q^{5} + 2 q^{6} - 9 q^{7} - 3 q^{8} + 6 q^{9} - 11 q^{10} - 13 q^{11} - 4 q^{12} - q^{13} - 5 q^{15} - 4 q^{16} + 11 q^{17} - 2 q^{18} - 22 q^{19} - q^{20} + 9 q^{21} - 2 q^{22} - 12 q^{23} + 3 q^{24} - q^{25} + 12 q^{26} - 6 q^{27} - 16 q^{28} + 11 q^{30} - 18 q^{31} + 7 q^{32} + 13 q^{33} - 3 q^{34} - 9 q^{35} + 4 q^{36} - 8 q^{37} - 5 q^{38} + q^{39} - 11 q^{40} + 10 q^{41} - 14 q^{43} - 4 q^{44} + 5 q^{45} - 18 q^{46} - 9 q^{47} + 4 q^{48} + 5 q^{49} + 4 q^{50} - 11 q^{51} - 16 q^{52} - 8 q^{53} + 2 q^{54} - 20 q^{55} + 11 q^{56} + 22 q^{57} - 15 q^{58} - 10 q^{59} + q^{60} - 12 q^{61} - 13 q^{62} - 9 q^{63} - 31 q^{64} - 11 q^{65} + 2 q^{66} - 36 q^{67} + 22 q^{68} + 12 q^{69} + q^{70} - 3 q^{71} - 3 q^{72} - 32 q^{73} + 9 q^{74} + q^{75} - 4 q^{76} + 6 q^{77} - 12 q^{78} - q^{79} - 7 q^{80} + 6 q^{81} + 7 q^{82} - 7 q^{83} + 16 q^{84} - 14 q^{85} + 45 q^{86} - 15 q^{88} + 17 q^{89} - 11 q^{90} - 23 q^{91} - 12 q^{92} + 18 q^{93} + 50 q^{94} - 7 q^{96} - 28 q^{97} + 13 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05161 0.743598 0.371799 0.928313i \(-0.378741\pi\)
0.371799 + 0.928313i \(0.378741\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.894124 −0.447062
\(5\) 2.87285 1.28478 0.642388 0.766380i \(-0.277944\pi\)
0.642388 + 0.766380i \(0.277944\pi\)
\(6\) −1.05161 −0.429317
\(7\) −4.11383 −1.55488 −0.777441 0.628956i \(-0.783483\pi\)
−0.777441 + 0.628956i \(0.783483\pi\)
\(8\) −3.04348 −1.07603
\(9\) 1.00000 0.333333
\(10\) 3.02110 0.955357
\(11\) −1.17944 −0.355614 −0.177807 0.984065i \(-0.556900\pi\)
−0.177807 + 0.984065i \(0.556900\pi\)
\(12\) 0.894124 0.258111
\(13\) 0.346441 0.0960854 0.0480427 0.998845i \(-0.484702\pi\)
0.0480427 + 0.998845i \(0.484702\pi\)
\(14\) −4.32613 −1.15621
\(15\) −2.87285 −0.741766
\(16\) −1.41230 −0.353074
\(17\) −3.26277 −0.791337 −0.395669 0.918393i \(-0.629487\pi\)
−0.395669 + 0.918393i \(0.629487\pi\)
\(18\) 1.05161 0.247866
\(19\) −1.31974 −0.302770 −0.151385 0.988475i \(-0.548373\pi\)
−0.151385 + 0.988475i \(0.548373\pi\)
\(20\) −2.56868 −0.574374
\(21\) 4.11383 0.897711
\(22\) −1.24031 −0.264434
\(23\) −9.10667 −1.89887 −0.949436 0.313960i \(-0.898344\pi\)
−0.949436 + 0.313960i \(0.898344\pi\)
\(24\) 3.04348 0.621248
\(25\) 3.25325 0.650649
\(26\) 0.364319 0.0714489
\(27\) −1.00000 −0.192450
\(28\) 3.67827 0.695128
\(29\) −0.639311 −0.118717 −0.0593585 0.998237i \(-0.518906\pi\)
−0.0593585 + 0.998237i \(0.518906\pi\)
\(30\) −3.02110 −0.551576
\(31\) −1.52951 −0.274708 −0.137354 0.990522i \(-0.543860\pi\)
−0.137354 + 0.990522i \(0.543860\pi\)
\(32\) 4.60178 0.813487
\(33\) 1.17944 0.205314
\(34\) −3.43115 −0.588437
\(35\) −11.8184 −1.99767
\(36\) −0.894124 −0.149021
\(37\) 1.40670 0.231260 0.115630 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(38\) −1.38785 −0.225139
\(39\) −0.346441 −0.0554749
\(40\) −8.74345 −1.38246
\(41\) −2.17524 −0.339715 −0.169857 0.985469i \(-0.554331\pi\)
−0.169857 + 0.985469i \(0.554331\pi\)
\(42\) 4.32613 0.667537
\(43\) −1.97537 −0.301241 −0.150621 0.988592i \(-0.548127\pi\)
−0.150621 + 0.988592i \(0.548127\pi\)
\(44\) 1.05456 0.158982
\(45\) 2.87285 0.428259
\(46\) −9.57663 −1.41200
\(47\) −1.69565 −0.247336 −0.123668 0.992324i \(-0.539466\pi\)
−0.123668 + 0.992324i \(0.539466\pi\)
\(48\) 1.41230 0.203847
\(49\) 9.92360 1.41766
\(50\) 3.42114 0.483822
\(51\) 3.26277 0.456879
\(52\) −0.309761 −0.0429561
\(53\) −0.0493538 −0.00677926 −0.00338963 0.999994i \(-0.501079\pi\)
−0.00338963 + 0.999994i \(0.501079\pi\)
\(54\) −1.05161 −0.143106
\(55\) −3.38835 −0.456884
\(56\) 12.5204 1.67310
\(57\) 1.31974 0.174804
\(58\) −0.672304 −0.0882778
\(59\) 0.806525 0.105001 0.0525003 0.998621i \(-0.483281\pi\)
0.0525003 + 0.998621i \(0.483281\pi\)
\(60\) 2.56868 0.331615
\(61\) −1.25891 −0.161187 −0.0805935 0.996747i \(-0.525682\pi\)
−0.0805935 + 0.996747i \(0.525682\pi\)
\(62\) −1.60844 −0.204272
\(63\) −4.11383 −0.518294
\(64\) 7.66385 0.957982
\(65\) 0.995271 0.123448
\(66\) 1.24031 0.152671
\(67\) 4.01099 0.490021 0.245010 0.969520i \(-0.421209\pi\)
0.245010 + 0.969520i \(0.421209\pi\)
\(68\) 2.91732 0.353777
\(69\) 9.10667 1.09631
\(70\) −12.4283 −1.48547
\(71\) 0.361936 0.0429539 0.0214770 0.999769i \(-0.493163\pi\)
0.0214770 + 0.999769i \(0.493163\pi\)
\(72\) −3.04348 −0.358677
\(73\) −11.9650 −1.40040 −0.700198 0.713949i \(-0.746905\pi\)
−0.700198 + 0.713949i \(0.746905\pi\)
\(74\) 1.47930 0.171965
\(75\) −3.25325 −0.375653
\(76\) 1.18001 0.135357
\(77\) 4.85201 0.552938
\(78\) −0.364319 −0.0412511
\(79\) 4.27659 0.481154 0.240577 0.970630i \(-0.422663\pi\)
0.240577 + 0.970630i \(0.422663\pi\)
\(80\) −4.05731 −0.453621
\(81\) 1.00000 0.111111
\(82\) −2.28749 −0.252611
\(83\) 14.0559 1.54284 0.771418 0.636329i \(-0.219548\pi\)
0.771418 + 0.636329i \(0.219548\pi\)
\(84\) −3.67827 −0.401333
\(85\) −9.37343 −1.01669
\(86\) −2.07731 −0.224003
\(87\) 0.639311 0.0685413
\(88\) 3.58960 0.382652
\(89\) −10.6550 −1.12943 −0.564714 0.825287i \(-0.691013\pi\)
−0.564714 + 0.825287i \(0.691013\pi\)
\(90\) 3.02110 0.318452
\(91\) −1.42520 −0.149401
\(92\) 8.14249 0.848913
\(93\) 1.52951 0.158603
\(94\) −1.78316 −0.183919
\(95\) −3.79142 −0.388991
\(96\) −4.60178 −0.469667
\(97\) −10.2134 −1.03701 −0.518505 0.855075i \(-0.673511\pi\)
−0.518505 + 0.855075i \(0.673511\pi\)
\(98\) 10.4357 1.05417
\(99\) −1.17944 −0.118538
\(100\) −2.90881 −0.290881
\(101\) 13.3232 1.32571 0.662855 0.748748i \(-0.269345\pi\)
0.662855 + 0.748748i \(0.269345\pi\)
\(102\) 3.43115 0.339734
\(103\) −16.3775 −1.61372 −0.806860 0.590743i \(-0.798835\pi\)
−0.806860 + 0.590743i \(0.798835\pi\)
\(104\) −1.05439 −0.103391
\(105\) 11.8184 1.15336
\(106\) −0.0519008 −0.00504105
\(107\) −10.8242 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(108\) 0.894124 0.0860371
\(109\) 17.1432 1.64202 0.821012 0.570910i \(-0.193410\pi\)
0.821012 + 0.570910i \(0.193410\pi\)
\(110\) −3.56321 −0.339738
\(111\) −1.40670 −0.133518
\(112\) 5.80994 0.548988
\(113\) 13.1922 1.24102 0.620511 0.784198i \(-0.286925\pi\)
0.620511 + 0.784198i \(0.286925\pi\)
\(114\) 1.38785 0.129984
\(115\) −26.1621 −2.43963
\(116\) 0.571623 0.0530739
\(117\) 0.346441 0.0320285
\(118\) 0.848147 0.0780782
\(119\) 13.4225 1.23044
\(120\) 8.74345 0.798164
\(121\) −9.60892 −0.873539
\(122\) −1.32388 −0.119858
\(123\) 2.17524 0.196134
\(124\) 1.36757 0.122812
\(125\) −5.01815 −0.448837
\(126\) −4.32613 −0.385402
\(127\) 13.0405 1.15716 0.578580 0.815625i \(-0.303607\pi\)
0.578580 + 0.815625i \(0.303607\pi\)
\(128\) −1.14420 −0.101134
\(129\) 1.97537 0.173922
\(130\) 1.04663 0.0917959
\(131\) 20.5593 1.79628 0.898138 0.439714i \(-0.144920\pi\)
0.898138 + 0.439714i \(0.144920\pi\)
\(132\) −1.05456 −0.0917880
\(133\) 5.42920 0.470771
\(134\) 4.21799 0.364378
\(135\) −2.87285 −0.247255
\(136\) 9.93016 0.851504
\(137\) 20.2062 1.72633 0.863165 0.504923i \(-0.168479\pi\)
0.863165 + 0.504923i \(0.168479\pi\)
\(138\) 9.57663 0.815217
\(139\) 14.9894 1.27139 0.635694 0.771941i \(-0.280714\pi\)
0.635694 + 0.771941i \(0.280714\pi\)
\(140\) 10.5671 0.893084
\(141\) 1.69565 0.142800
\(142\) 0.380614 0.0319404
\(143\) −0.408606 −0.0341693
\(144\) −1.41230 −0.117691
\(145\) −1.83664 −0.152525
\(146\) −12.5825 −1.04133
\(147\) −9.92360 −0.818485
\(148\) −1.25776 −0.103388
\(149\) 5.24611 0.429778 0.214889 0.976638i \(-0.431061\pi\)
0.214889 + 0.976638i \(0.431061\pi\)
\(150\) −3.42114 −0.279335
\(151\) 5.59004 0.454911 0.227456 0.973788i \(-0.426959\pi\)
0.227456 + 0.973788i \(0.426959\pi\)
\(152\) 4.01661 0.325790
\(153\) −3.26277 −0.263779
\(154\) 5.10241 0.411164
\(155\) −4.39405 −0.352938
\(156\) 0.309761 0.0248007
\(157\) 17.6356 1.40747 0.703736 0.710461i \(-0.251514\pi\)
0.703736 + 0.710461i \(0.251514\pi\)
\(158\) 4.49729 0.357785
\(159\) 0.0493538 0.00391401
\(160\) 13.2202 1.04515
\(161\) 37.4633 2.95252
\(162\) 1.05161 0.0826220
\(163\) −12.8967 −1.01015 −0.505073 0.863077i \(-0.668534\pi\)
−0.505073 + 0.863077i \(0.668534\pi\)
\(164\) 1.94493 0.151874
\(165\) 3.38835 0.263782
\(166\) 14.7813 1.14725
\(167\) −4.91384 −0.380244 −0.190122 0.981760i \(-0.560888\pi\)
−0.190122 + 0.981760i \(0.560888\pi\)
\(168\) −12.5204 −0.965967
\(169\) −12.8800 −0.990768
\(170\) −9.85716 −0.756010
\(171\) −1.31974 −0.100923
\(172\) 1.76623 0.134674
\(173\) −15.5398 −1.18147 −0.590736 0.806865i \(-0.701163\pi\)
−0.590736 + 0.806865i \(0.701163\pi\)
\(174\) 0.672304 0.0509672
\(175\) −13.3833 −1.01168
\(176\) 1.66572 0.125558
\(177\) −0.806525 −0.0606221
\(178\) −11.2049 −0.839840
\(179\) −16.9641 −1.26796 −0.633979 0.773350i \(-0.718580\pi\)
−0.633979 + 0.773350i \(0.718580\pi\)
\(180\) −2.56868 −0.191458
\(181\) 3.30222 0.245452 0.122726 0.992441i \(-0.460836\pi\)
0.122726 + 0.992441i \(0.460836\pi\)
\(182\) −1.49875 −0.111095
\(183\) 1.25891 0.0930613
\(184\) 27.7160 2.04325
\(185\) 4.04124 0.297118
\(186\) 1.60844 0.117937
\(187\) 3.84823 0.281411
\(188\) 1.51612 0.110575
\(189\) 4.11383 0.299237
\(190\) −3.98708 −0.289253
\(191\) −12.2407 −0.885704 −0.442852 0.896595i \(-0.646033\pi\)
−0.442852 + 0.896595i \(0.646033\pi\)
\(192\) −7.66385 −0.553091
\(193\) −15.3502 −1.10493 −0.552465 0.833536i \(-0.686313\pi\)
−0.552465 + 0.833536i \(0.686313\pi\)
\(194\) −10.7404 −0.771119
\(195\) −0.995271 −0.0712729
\(196\) −8.87293 −0.633781
\(197\) −20.6755 −1.47307 −0.736533 0.676402i \(-0.763538\pi\)
−0.736533 + 0.676402i \(0.763538\pi\)
\(198\) −1.24031 −0.0881447
\(199\) −24.9536 −1.76892 −0.884459 0.466619i \(-0.845472\pi\)
−0.884459 + 0.466619i \(0.845472\pi\)
\(200\) −9.90119 −0.700120
\(201\) −4.01099 −0.282914
\(202\) 14.0108 0.985795
\(203\) 2.63002 0.184591
\(204\) −2.91732 −0.204253
\(205\) −6.24912 −0.436457
\(206\) −17.2226 −1.19996
\(207\) −9.10667 −0.632957
\(208\) −0.489277 −0.0339252
\(209\) 1.55655 0.107669
\(210\) 12.4283 0.857635
\(211\) −3.86876 −0.266336 −0.133168 0.991093i \(-0.542515\pi\)
−0.133168 + 0.991093i \(0.542515\pi\)
\(212\) 0.0441284 0.00303075
\(213\) −0.361936 −0.0247994
\(214\) −11.3828 −0.778109
\(215\) −5.67494 −0.387028
\(216\) 3.04348 0.207083
\(217\) 6.29214 0.427139
\(218\) 18.0279 1.22101
\(219\) 11.9650 0.808518
\(220\) 3.02960 0.204256
\(221\) −1.13036 −0.0760360
\(222\) −1.47930 −0.0992838
\(223\) 9.50369 0.636414 0.318207 0.948021i \(-0.396919\pi\)
0.318207 + 0.948021i \(0.396919\pi\)
\(224\) −18.9309 −1.26488
\(225\) 3.25325 0.216883
\(226\) 13.8730 0.922821
\(227\) 7.95147 0.527758 0.263879 0.964556i \(-0.414998\pi\)
0.263879 + 0.964556i \(0.414998\pi\)
\(228\) −1.18001 −0.0781482
\(229\) −11.1993 −0.740070 −0.370035 0.929018i \(-0.620654\pi\)
−0.370035 + 0.929018i \(0.620654\pi\)
\(230\) −27.5122 −1.81410
\(231\) −4.85201 −0.319239
\(232\) 1.94573 0.127743
\(233\) −15.6891 −1.02783 −0.513914 0.857842i \(-0.671805\pi\)
−0.513914 + 0.857842i \(0.671805\pi\)
\(234\) 0.364319 0.0238163
\(235\) −4.87135 −0.317772
\(236\) −0.721133 −0.0469418
\(237\) −4.27659 −0.277794
\(238\) 14.1152 0.914950
\(239\) −1.00000 −0.0646846
\(240\) 4.05731 0.261898
\(241\) −25.5643 −1.64674 −0.823371 0.567503i \(-0.807910\pi\)
−0.823371 + 0.567503i \(0.807910\pi\)
\(242\) −10.1048 −0.649562
\(243\) −1.00000 −0.0641500
\(244\) 1.12562 0.0720605
\(245\) 28.5090 1.82137
\(246\) 2.28749 0.145845
\(247\) −0.457213 −0.0290917
\(248\) 4.65503 0.295595
\(249\) −14.0559 −0.890757
\(250\) −5.27712 −0.333754
\(251\) 16.7833 1.05935 0.529676 0.848200i \(-0.322314\pi\)
0.529676 + 0.848200i \(0.322314\pi\)
\(252\) 3.67827 0.231709
\(253\) 10.7408 0.675266
\(254\) 13.7135 0.860462
\(255\) 9.37343 0.586987
\(256\) −16.5310 −1.03318
\(257\) −6.98956 −0.435996 −0.217998 0.975949i \(-0.569953\pi\)
−0.217998 + 0.975949i \(0.569953\pi\)
\(258\) 2.07731 0.129328
\(259\) −5.78693 −0.359582
\(260\) −0.889896 −0.0551890
\(261\) −0.639311 −0.0395724
\(262\) 21.6203 1.33571
\(263\) −13.7752 −0.849413 −0.424707 0.905331i \(-0.639623\pi\)
−0.424707 + 0.905331i \(0.639623\pi\)
\(264\) −3.58960 −0.220924
\(265\) −0.141786 −0.00870984
\(266\) 5.70938 0.350064
\(267\) 10.6550 0.652075
\(268\) −3.58632 −0.219070
\(269\) 11.2058 0.683228 0.341614 0.939840i \(-0.389026\pi\)
0.341614 + 0.939840i \(0.389026\pi\)
\(270\) −3.02110 −0.183859
\(271\) −20.5528 −1.24850 −0.624248 0.781226i \(-0.714595\pi\)
−0.624248 + 0.781226i \(0.714595\pi\)
\(272\) 4.60799 0.279400
\(273\) 1.42520 0.0862570
\(274\) 21.2489 1.28370
\(275\) −3.83701 −0.231380
\(276\) −8.14249 −0.490120
\(277\) −17.2928 −1.03902 −0.519511 0.854463i \(-0.673886\pi\)
−0.519511 + 0.854463i \(0.673886\pi\)
\(278\) 15.7630 0.945401
\(279\) −1.52951 −0.0915694
\(280\) 35.9691 2.14956
\(281\) 24.1530 1.44085 0.720425 0.693533i \(-0.243947\pi\)
0.720425 + 0.693533i \(0.243947\pi\)
\(282\) 1.78316 0.106186
\(283\) −12.4777 −0.741722 −0.370861 0.928688i \(-0.620937\pi\)
−0.370861 + 0.928688i \(0.620937\pi\)
\(284\) −0.323616 −0.0192031
\(285\) 3.79142 0.224584
\(286\) −0.429692 −0.0254082
\(287\) 8.94855 0.528216
\(288\) 4.60178 0.271162
\(289\) −6.35435 −0.373785
\(290\) −1.93143 −0.113417
\(291\) 10.2134 0.598718
\(292\) 10.6982 0.626063
\(293\) −24.0095 −1.40265 −0.701325 0.712842i \(-0.747408\pi\)
−0.701325 + 0.712842i \(0.747408\pi\)
\(294\) −10.4357 −0.608624
\(295\) 2.31702 0.134902
\(296\) −4.28126 −0.248843
\(297\) 1.17944 0.0684380
\(298\) 5.51685 0.319582
\(299\) −3.15492 −0.182454
\(300\) 2.90881 0.167940
\(301\) 8.12635 0.468395
\(302\) 5.87853 0.338271
\(303\) −13.3232 −0.765399
\(304\) 1.86387 0.106900
\(305\) −3.61666 −0.207089
\(306\) −3.43115 −0.196146
\(307\) 7.54396 0.430557 0.215278 0.976553i \(-0.430934\pi\)
0.215278 + 0.976553i \(0.430934\pi\)
\(308\) −4.33830 −0.247197
\(309\) 16.3775 0.931681
\(310\) −4.62081 −0.262444
\(311\) −7.22920 −0.409930 −0.204965 0.978769i \(-0.565708\pi\)
−0.204965 + 0.978769i \(0.565708\pi\)
\(312\) 1.05439 0.0596928
\(313\) −21.2712 −1.20232 −0.601161 0.799128i \(-0.705295\pi\)
−0.601161 + 0.799128i \(0.705295\pi\)
\(314\) 18.5457 1.04659
\(315\) −11.8184 −0.665892
\(316\) −3.82380 −0.215106
\(317\) 31.0998 1.74674 0.873369 0.487059i \(-0.161930\pi\)
0.873369 + 0.487059i \(0.161930\pi\)
\(318\) 0.0519008 0.00291045
\(319\) 0.754028 0.0422175
\(320\) 22.0171 1.23079
\(321\) 10.8242 0.604146
\(322\) 39.3967 2.19549
\(323\) 4.30601 0.239593
\(324\) −0.894124 −0.0496735
\(325\) 1.12706 0.0625179
\(326\) −13.5622 −0.751142
\(327\) −17.1432 −0.948023
\(328\) 6.62029 0.365544
\(329\) 6.97562 0.384579
\(330\) 3.56321 0.196148
\(331\) −4.80463 −0.264086 −0.132043 0.991244i \(-0.542154\pi\)
−0.132043 + 0.991244i \(0.542154\pi\)
\(332\) −12.5677 −0.689743
\(333\) 1.40670 0.0770867
\(334\) −5.16743 −0.282749
\(335\) 11.5230 0.629567
\(336\) −5.80994 −0.316958
\(337\) −1.20117 −0.0654317 −0.0327159 0.999465i \(-0.510416\pi\)
−0.0327159 + 0.999465i \(0.510416\pi\)
\(338\) −13.5447 −0.736733
\(339\) −13.1922 −0.716504
\(340\) 8.38100 0.454524
\(341\) 1.80396 0.0976901
\(342\) −1.38785 −0.0750463
\(343\) −12.0272 −0.649408
\(344\) 6.01200 0.324146
\(345\) 26.1621 1.40852
\(346\) −16.3418 −0.878541
\(347\) −1.61746 −0.0868296 −0.0434148 0.999057i \(-0.513824\pi\)
−0.0434148 + 0.999057i \(0.513824\pi\)
\(348\) −0.571623 −0.0306422
\(349\) 31.2253 1.67145 0.835727 0.549145i \(-0.185047\pi\)
0.835727 + 0.549145i \(0.185047\pi\)
\(350\) −14.0740 −0.752286
\(351\) −0.346441 −0.0184916
\(352\) −5.42752 −0.289288
\(353\) −14.6883 −0.781779 −0.390889 0.920438i \(-0.627832\pi\)
−0.390889 + 0.920438i \(0.627832\pi\)
\(354\) −0.848147 −0.0450785
\(355\) 1.03979 0.0551861
\(356\) 9.52689 0.504924
\(357\) −13.4225 −0.710392
\(358\) −17.8396 −0.942851
\(359\) 6.81390 0.359624 0.179812 0.983701i \(-0.442451\pi\)
0.179812 + 0.983701i \(0.442451\pi\)
\(360\) −8.74345 −0.460820
\(361\) −17.2583 −0.908331
\(362\) 3.47264 0.182518
\(363\) 9.60892 0.504338
\(364\) 1.27430 0.0667917
\(365\) −34.3736 −1.79919
\(366\) 1.32388 0.0692002
\(367\) 9.51165 0.496504 0.248252 0.968695i \(-0.420144\pi\)
0.248252 + 0.968695i \(0.420144\pi\)
\(368\) 12.8613 0.670442
\(369\) −2.17524 −0.113238
\(370\) 4.24979 0.220936
\(371\) 0.203033 0.0105410
\(372\) −1.36757 −0.0709053
\(373\) −12.7503 −0.660186 −0.330093 0.943949i \(-0.607080\pi\)
−0.330093 + 0.943949i \(0.607080\pi\)
\(374\) 4.04683 0.209256
\(375\) 5.01815 0.259136
\(376\) 5.16068 0.266142
\(377\) −0.221483 −0.0114070
\(378\) 4.32613 0.222512
\(379\) −10.2372 −0.525852 −0.262926 0.964816i \(-0.584688\pi\)
−0.262926 + 0.964816i \(0.584688\pi\)
\(380\) 3.39000 0.173903
\(381\) −13.0405 −0.668087
\(382\) −12.8724 −0.658608
\(383\) −24.8455 −1.26954 −0.634772 0.772699i \(-0.718906\pi\)
−0.634772 + 0.772699i \(0.718906\pi\)
\(384\) 1.14420 0.0583898
\(385\) 13.9391 0.710401
\(386\) −16.1423 −0.821624
\(387\) −1.97537 −0.100414
\(388\) 9.13201 0.463608
\(389\) 3.01669 0.152952 0.0764760 0.997071i \(-0.475633\pi\)
0.0764760 + 0.997071i \(0.475633\pi\)
\(390\) −1.04663 −0.0529984
\(391\) 29.7129 1.50265
\(392\) −30.2023 −1.52545
\(393\) −20.5593 −1.03708
\(394\) −21.7424 −1.09537
\(395\) 12.2860 0.618175
\(396\) 1.05456 0.0529938
\(397\) 0.428312 0.0214964 0.0107482 0.999942i \(-0.496579\pi\)
0.0107482 + 0.999942i \(0.496579\pi\)
\(398\) −26.2414 −1.31536
\(399\) −5.42920 −0.271800
\(400\) −4.59455 −0.229727
\(401\) −14.0036 −0.699305 −0.349653 0.936879i \(-0.613700\pi\)
−0.349653 + 0.936879i \(0.613700\pi\)
\(402\) −4.21799 −0.210374
\(403\) −0.529885 −0.0263954
\(404\) −11.9126 −0.592674
\(405\) 2.87285 0.142753
\(406\) 2.76574 0.137262
\(407\) −1.65912 −0.0822394
\(408\) −9.93016 −0.491616
\(409\) −39.9092 −1.97338 −0.986692 0.162600i \(-0.948012\pi\)
−0.986692 + 0.162600i \(0.948012\pi\)
\(410\) −6.57162 −0.324549
\(411\) −20.2062 −0.996697
\(412\) 14.6435 0.721432
\(413\) −3.31791 −0.163264
\(414\) −9.57663 −0.470666
\(415\) 40.3805 1.98220
\(416\) 1.59424 0.0781643
\(417\) −14.9894 −0.734036
\(418\) 1.63688 0.0800626
\(419\) 23.6201 1.15392 0.576959 0.816773i \(-0.304239\pi\)
0.576959 + 0.816773i \(0.304239\pi\)
\(420\) −10.5671 −0.515622
\(421\) −16.4261 −0.800559 −0.400279 0.916393i \(-0.631087\pi\)
−0.400279 + 0.916393i \(0.631087\pi\)
\(422\) −4.06841 −0.198047
\(423\) −1.69565 −0.0824454
\(424\) 0.150207 0.00729471
\(425\) −10.6146 −0.514883
\(426\) −0.380614 −0.0184408
\(427\) 5.17894 0.250627
\(428\) 9.67814 0.467810
\(429\) 0.408606 0.0197277
\(430\) −5.96780 −0.287793
\(431\) 24.1771 1.16457 0.582285 0.812985i \(-0.302159\pi\)
0.582285 + 0.812985i \(0.302159\pi\)
\(432\) 1.41230 0.0679491
\(433\) −7.20673 −0.346333 −0.173167 0.984893i \(-0.555400\pi\)
−0.173167 + 0.984893i \(0.555400\pi\)
\(434\) 6.61686 0.317619
\(435\) 1.83664 0.0880603
\(436\) −15.3282 −0.734087
\(437\) 12.0185 0.574921
\(438\) 12.5825 0.601213
\(439\) 21.1400 1.00896 0.504479 0.863424i \(-0.331684\pi\)
0.504479 + 0.863424i \(0.331684\pi\)
\(440\) 10.3124 0.491623
\(441\) 9.92360 0.472552
\(442\) −1.18869 −0.0565402
\(443\) −11.1509 −0.529797 −0.264899 0.964276i \(-0.585338\pi\)
−0.264899 + 0.964276i \(0.585338\pi\)
\(444\) 1.25776 0.0596909
\(445\) −30.6102 −1.45106
\(446\) 9.99415 0.473237
\(447\) −5.24611 −0.248133
\(448\) −31.5278 −1.48955
\(449\) −2.12495 −0.100283 −0.0501414 0.998742i \(-0.515967\pi\)
−0.0501414 + 0.998742i \(0.515967\pi\)
\(450\) 3.42114 0.161274
\(451\) 2.56556 0.120807
\(452\) −11.7955 −0.554814
\(453\) −5.59004 −0.262643
\(454\) 8.36182 0.392440
\(455\) −4.09438 −0.191947
\(456\) −4.01661 −0.188095
\(457\) 37.1322 1.73697 0.868485 0.495715i \(-0.165094\pi\)
0.868485 + 0.495715i \(0.165094\pi\)
\(458\) −11.7772 −0.550315
\(459\) 3.26277 0.152293
\(460\) 23.3921 1.09066
\(461\) 29.8834 1.39181 0.695905 0.718134i \(-0.255004\pi\)
0.695905 + 0.718134i \(0.255004\pi\)
\(462\) −5.10241 −0.237385
\(463\) 20.4052 0.948311 0.474156 0.880441i \(-0.342753\pi\)
0.474156 + 0.880441i \(0.342753\pi\)
\(464\) 0.902896 0.0419159
\(465\) 4.39405 0.203769
\(466\) −16.4988 −0.764291
\(467\) 4.72528 0.218660 0.109330 0.994006i \(-0.465129\pi\)
0.109330 + 0.994006i \(0.465129\pi\)
\(468\) −0.309761 −0.0143187
\(469\) −16.5005 −0.761924
\(470\) −5.12274 −0.236294
\(471\) −17.6356 −0.812604
\(472\) −2.45464 −0.112984
\(473\) 2.32983 0.107126
\(474\) −4.49729 −0.206567
\(475\) −4.29345 −0.196997
\(476\) −12.0013 −0.550081
\(477\) −0.0493538 −0.00225975
\(478\) −1.05161 −0.0480994
\(479\) 39.4927 1.80447 0.902234 0.431247i \(-0.141926\pi\)
0.902234 + 0.431247i \(0.141926\pi\)
\(480\) −13.2202 −0.603417
\(481\) 0.487339 0.0222207
\(482\) −26.8836 −1.22451
\(483\) −37.4633 −1.70464
\(484\) 8.59157 0.390526
\(485\) −29.3414 −1.33233
\(486\) −1.05161 −0.0477018
\(487\) −26.1172 −1.18349 −0.591743 0.806127i \(-0.701560\pi\)
−0.591743 + 0.806127i \(0.701560\pi\)
\(488\) 3.83147 0.173442
\(489\) 12.8967 0.583208
\(490\) 29.9802 1.35437
\(491\) −20.1522 −0.909454 −0.454727 0.890631i \(-0.650263\pi\)
−0.454727 + 0.890631i \(0.650263\pi\)
\(492\) −1.94493 −0.0876842
\(493\) 2.08592 0.0939453
\(494\) −0.480808 −0.0216326
\(495\) −3.38835 −0.152295
\(496\) 2.16012 0.0969922
\(497\) −1.48894 −0.0667882
\(498\) −14.7813 −0.662365
\(499\) 16.6298 0.744452 0.372226 0.928142i \(-0.378595\pi\)
0.372226 + 0.928142i \(0.378595\pi\)
\(500\) 4.48685 0.200658
\(501\) 4.91384 0.219534
\(502\) 17.6494 0.787732
\(503\) −13.0732 −0.582906 −0.291453 0.956585i \(-0.594139\pi\)
−0.291453 + 0.956585i \(0.594139\pi\)
\(504\) 12.5204 0.557701
\(505\) 38.2755 1.70324
\(506\) 11.2951 0.502126
\(507\) 12.8800 0.572020
\(508\) −11.6599 −0.517322
\(509\) −12.4973 −0.553934 −0.276967 0.960879i \(-0.589329\pi\)
−0.276967 + 0.960879i \(0.589329\pi\)
\(510\) 9.85716 0.436482
\(511\) 49.2219 2.17745
\(512\) −15.0957 −0.667140
\(513\) 1.31974 0.0582680
\(514\) −7.35026 −0.324206
\(515\) −47.0499 −2.07327
\(516\) −1.76623 −0.0777538
\(517\) 1.99992 0.0879563
\(518\) −6.08557 −0.267385
\(519\) 15.5398 0.682123
\(520\) −3.02909 −0.132834
\(521\) 3.31677 0.145310 0.0726552 0.997357i \(-0.476853\pi\)
0.0726552 + 0.997357i \(0.476853\pi\)
\(522\) −0.672304 −0.0294259
\(523\) 32.7979 1.43415 0.717075 0.696996i \(-0.245480\pi\)
0.717075 + 0.696996i \(0.245480\pi\)
\(524\) −18.3826 −0.803047
\(525\) 13.3833 0.584095
\(526\) −14.4861 −0.631622
\(527\) 4.99043 0.217387
\(528\) −1.66572 −0.0724910
\(529\) 59.9315 2.60572
\(530\) −0.149103 −0.00647662
\(531\) 0.806525 0.0350002
\(532\) −4.85437 −0.210464
\(533\) −0.753591 −0.0326416
\(534\) 11.2049 0.484882
\(535\) −31.0961 −1.34440
\(536\) −12.2074 −0.527278
\(537\) 16.9641 0.732056
\(538\) 11.7841 0.508047
\(539\) −11.7043 −0.504139
\(540\) 2.56868 0.110538
\(541\) −5.52107 −0.237369 −0.118685 0.992932i \(-0.537868\pi\)
−0.118685 + 0.992932i \(0.537868\pi\)
\(542\) −21.6135 −0.928380
\(543\) −3.30222 −0.141712
\(544\) −15.0145 −0.643743
\(545\) 49.2499 2.10963
\(546\) 1.49875 0.0641405
\(547\) 12.7057 0.543258 0.271629 0.962402i \(-0.412438\pi\)
0.271629 + 0.962402i \(0.412438\pi\)
\(548\) −18.0668 −0.771776
\(549\) −1.25891 −0.0537290
\(550\) −4.03502 −0.172054
\(551\) 0.843726 0.0359439
\(552\) −27.7160 −1.17967
\(553\) −17.5932 −0.748138
\(554\) −18.1852 −0.772615
\(555\) −4.04124 −0.171541
\(556\) −13.4024 −0.568389
\(557\) 42.9858 1.82137 0.910683 0.413105i \(-0.135556\pi\)
0.910683 + 0.413105i \(0.135556\pi\)
\(558\) −1.60844 −0.0680908
\(559\) −0.684350 −0.0289449
\(560\) 16.6911 0.705327
\(561\) −3.84823 −0.162473
\(562\) 25.3995 1.07141
\(563\) 20.3703 0.858504 0.429252 0.903185i \(-0.358777\pi\)
0.429252 + 0.903185i \(0.358777\pi\)
\(564\) −1.51612 −0.0638403
\(565\) 37.8993 1.59444
\(566\) −13.1216 −0.551543
\(567\) −4.11383 −0.172765
\(568\) −1.10154 −0.0462198
\(569\) −31.4900 −1.32013 −0.660064 0.751210i \(-0.729471\pi\)
−0.660064 + 0.751210i \(0.729471\pi\)
\(570\) 3.98708 0.167000
\(571\) −31.3410 −1.31158 −0.655789 0.754944i \(-0.727664\pi\)
−0.655789 + 0.754944i \(0.727664\pi\)
\(572\) 0.365344 0.0152758
\(573\) 12.2407 0.511362
\(574\) 9.41036 0.392781
\(575\) −29.6263 −1.23550
\(576\) 7.66385 0.319327
\(577\) 23.8525 0.992992 0.496496 0.868039i \(-0.334620\pi\)
0.496496 + 0.868039i \(0.334620\pi\)
\(578\) −6.68228 −0.277946
\(579\) 15.3502 0.637932
\(580\) 1.64219 0.0681881
\(581\) −57.8236 −2.39893
\(582\) 10.7404 0.445206
\(583\) 0.0582098 0.00241080
\(584\) 36.4152 1.50687
\(585\) 0.995271 0.0411494
\(586\) −25.2485 −1.04301
\(587\) 8.14382 0.336131 0.168066 0.985776i \(-0.446248\pi\)
0.168066 + 0.985776i \(0.446248\pi\)
\(588\) 8.87293 0.365913
\(589\) 2.01856 0.0831733
\(590\) 2.43660 0.100313
\(591\) 20.6755 0.850475
\(592\) −1.98668 −0.0816519
\(593\) 1.77048 0.0727049 0.0363524 0.999339i \(-0.488426\pi\)
0.0363524 + 0.999339i \(0.488426\pi\)
\(594\) 1.24031 0.0508903
\(595\) 38.5607 1.58083
\(596\) −4.69067 −0.192138
\(597\) 24.9536 1.02128
\(598\) −3.31774 −0.135672
\(599\) −24.1665 −0.987415 −0.493707 0.869628i \(-0.664359\pi\)
−0.493707 + 0.869628i \(0.664359\pi\)
\(600\) 9.90119 0.404214
\(601\) 10.2259 0.417123 0.208561 0.978009i \(-0.433122\pi\)
0.208561 + 0.978009i \(0.433122\pi\)
\(602\) 8.54572 0.348297
\(603\) 4.01099 0.163340
\(604\) −4.99819 −0.203373
\(605\) −27.6050 −1.12230
\(606\) −14.0108 −0.569149
\(607\) −21.0640 −0.854960 −0.427480 0.904025i \(-0.640599\pi\)
−0.427480 + 0.904025i \(0.640599\pi\)
\(608\) −6.07316 −0.246299
\(609\) −2.63002 −0.106574
\(610\) −3.80330 −0.153991
\(611\) −0.587443 −0.0237654
\(612\) 2.91732 0.117926
\(613\) −6.72545 −0.271638 −0.135819 0.990734i \(-0.543367\pi\)
−0.135819 + 0.990734i \(0.543367\pi\)
\(614\) 7.93328 0.320161
\(615\) 6.24912 0.251989
\(616\) −14.7670 −0.594979
\(617\) −0.401847 −0.0161778 −0.00808888 0.999967i \(-0.502575\pi\)
−0.00808888 + 0.999967i \(0.502575\pi\)
\(618\) 17.2226 0.692796
\(619\) −20.6279 −0.829104 −0.414552 0.910026i \(-0.636062\pi\)
−0.414552 + 0.910026i \(0.636062\pi\)
\(620\) 3.92882 0.157785
\(621\) 9.10667 0.365438
\(622\) −7.60227 −0.304823
\(623\) 43.8329 1.75613
\(624\) 0.489277 0.0195867
\(625\) −30.6826 −1.22730
\(626\) −22.3690 −0.894044
\(627\) −1.55655 −0.0621628
\(628\) −15.7684 −0.629227
\(629\) −4.58974 −0.183005
\(630\) −12.4283 −0.495156
\(631\) −29.6371 −1.17984 −0.589918 0.807463i \(-0.700840\pi\)
−0.589918 + 0.807463i \(0.700840\pi\)
\(632\) −13.0157 −0.517737
\(633\) 3.86876 0.153769
\(634\) 32.7047 1.29887
\(635\) 37.4635 1.48669
\(636\) −0.0441284 −0.00174980
\(637\) 3.43794 0.136216
\(638\) 0.792941 0.0313928
\(639\) 0.361936 0.0143180
\(640\) −3.28712 −0.129935
\(641\) 11.3866 0.449745 0.224872 0.974388i \(-0.427803\pi\)
0.224872 + 0.974388i \(0.427803\pi\)
\(642\) 11.3828 0.449242
\(643\) −33.4917 −1.32079 −0.660393 0.750921i \(-0.729610\pi\)
−0.660393 + 0.750921i \(0.729610\pi\)
\(644\) −33.4968 −1.31996
\(645\) 5.67494 0.223451
\(646\) 4.52823 0.178161
\(647\) 48.9050 1.92265 0.961326 0.275412i \(-0.0888143\pi\)
0.961326 + 0.275412i \(0.0888143\pi\)
\(648\) −3.04348 −0.119559
\(649\) −0.951247 −0.0373397
\(650\) 1.18522 0.0464882
\(651\) −6.29214 −0.246609
\(652\) 11.5312 0.451598
\(653\) 5.23254 0.204765 0.102383 0.994745i \(-0.467353\pi\)
0.102383 + 0.994745i \(0.467353\pi\)
\(654\) −18.0279 −0.704948
\(655\) 59.0638 2.30781
\(656\) 3.07208 0.119944
\(657\) −11.9650 −0.466798
\(658\) 7.33561 0.285972
\(659\) −3.59290 −0.139960 −0.0699798 0.997548i \(-0.522293\pi\)
−0.0699798 + 0.997548i \(0.522293\pi\)
\(660\) −3.02960 −0.117927
\(661\) −15.9161 −0.619066 −0.309533 0.950889i \(-0.600173\pi\)
−0.309533 + 0.950889i \(0.600173\pi\)
\(662\) −5.05258 −0.196374
\(663\) 1.13036 0.0438994
\(664\) −42.7789 −1.66014
\(665\) 15.5972 0.604835
\(666\) 1.47930 0.0573215
\(667\) 5.82200 0.225429
\(668\) 4.39358 0.169993
\(669\) −9.50369 −0.367434
\(670\) 12.1176 0.468145
\(671\) 1.48481 0.0573204
\(672\) 18.9309 0.730277
\(673\) 34.3829 1.32536 0.662682 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(674\) −1.26315 −0.0486549
\(675\) −3.25325 −0.125218
\(676\) 11.5163 0.442934
\(677\) −27.8827 −1.07162 −0.535809 0.844339i \(-0.679993\pi\)
−0.535809 + 0.844339i \(0.679993\pi\)
\(678\) −13.8730 −0.532791
\(679\) 42.0161 1.61243
\(680\) 28.5278 1.09399
\(681\) −7.95147 −0.304701
\(682\) 1.89706 0.0726422
\(683\) 1.84211 0.0704864 0.0352432 0.999379i \(-0.488779\pi\)
0.0352432 + 0.999379i \(0.488779\pi\)
\(684\) 1.18001 0.0451189
\(685\) 58.0492 2.21795
\(686\) −12.6479 −0.482898
\(687\) 11.1993 0.427280
\(688\) 2.78981 0.106360
\(689\) −0.0170982 −0.000651388 0
\(690\) 27.5122 1.04737
\(691\) 18.0372 0.686166 0.343083 0.939305i \(-0.388529\pi\)
0.343083 + 0.939305i \(0.388529\pi\)
\(692\) 13.8945 0.528191
\(693\) 4.85201 0.184313
\(694\) −1.70093 −0.0645663
\(695\) 43.0623 1.63345
\(696\) −1.94573 −0.0737527
\(697\) 7.09729 0.268829
\(698\) 32.8368 1.24289
\(699\) 15.6891 0.593417
\(700\) 11.9663 0.452285
\(701\) −6.10724 −0.230667 −0.115334 0.993327i \(-0.536794\pi\)
−0.115334 + 0.993327i \(0.536794\pi\)
\(702\) −0.364319 −0.0137504
\(703\) −1.85648 −0.0700186
\(704\) −9.03904 −0.340672
\(705\) 4.87135 0.183466
\(706\) −15.4463 −0.581329
\(707\) −54.8094 −2.06132
\(708\) 0.721133 0.0271018
\(709\) −29.3897 −1.10375 −0.551877 0.833926i \(-0.686088\pi\)
−0.551877 + 0.833926i \(0.686088\pi\)
\(710\) 1.09345 0.0410363
\(711\) 4.27659 0.160385
\(712\) 32.4283 1.21530
\(713\) 13.9287 0.521636
\(714\) −14.1152 −0.528246
\(715\) −1.17386 −0.0438999
\(716\) 15.1680 0.566856
\(717\) 1.00000 0.0373457
\(718\) 7.16554 0.267416
\(719\) −38.2319 −1.42581 −0.712904 0.701262i \(-0.752621\pi\)
−0.712904 + 0.701262i \(0.752621\pi\)
\(720\) −4.05731 −0.151207
\(721\) 67.3741 2.50914
\(722\) −18.1489 −0.675433
\(723\) 25.5643 0.950747
\(724\) −2.95260 −0.109732
\(725\) −2.07984 −0.0772432
\(726\) 10.1048 0.375025
\(727\) 16.8660 0.625526 0.312763 0.949831i \(-0.398745\pi\)
0.312763 + 0.949831i \(0.398745\pi\)
\(728\) 4.33756 0.160761
\(729\) 1.00000 0.0370370
\(730\) −36.1475 −1.33788
\(731\) 6.44518 0.238384
\(732\) −1.12562 −0.0416042
\(733\) −9.13669 −0.337471 −0.168736 0.985661i \(-0.553968\pi\)
−0.168736 + 0.985661i \(0.553968\pi\)
\(734\) 10.0025 0.369200
\(735\) −28.5090 −1.05157
\(736\) −41.9069 −1.54471
\(737\) −4.73072 −0.174258
\(738\) −2.28749 −0.0842038
\(739\) −28.4826 −1.04775 −0.523875 0.851795i \(-0.675514\pi\)
−0.523875 + 0.851795i \(0.675514\pi\)
\(740\) −3.61336 −0.132830
\(741\) 0.457213 0.0167961
\(742\) 0.213511 0.00783823
\(743\) −37.7287 −1.38413 −0.692066 0.721834i \(-0.743299\pi\)
−0.692066 + 0.721834i \(0.743299\pi\)
\(744\) −4.65503 −0.170662
\(745\) 15.0713 0.552169
\(746\) −13.4083 −0.490913
\(747\) 14.0559 0.514279
\(748\) −3.44080 −0.125808
\(749\) 44.5288 1.62705
\(750\) 5.27712 0.192693
\(751\) −32.0210 −1.16846 −0.584231 0.811588i \(-0.698604\pi\)
−0.584231 + 0.811588i \(0.698604\pi\)
\(752\) 2.39476 0.0873280
\(753\) −16.7833 −0.611617
\(754\) −0.232913 −0.00848221
\(755\) 16.0593 0.584459
\(756\) −3.67827 −0.133778
\(757\) −13.7562 −0.499976 −0.249988 0.968249i \(-0.580427\pi\)
−0.249988 + 0.968249i \(0.580427\pi\)
\(758\) −10.7656 −0.391023
\(759\) −10.7408 −0.389865
\(760\) 11.5391 0.418567
\(761\) 45.3489 1.64390 0.821948 0.569563i \(-0.192888\pi\)
0.821948 + 0.569563i \(0.192888\pi\)
\(762\) −13.7135 −0.496788
\(763\) −70.5244 −2.55315
\(764\) 10.9447 0.395965
\(765\) −9.37343 −0.338897
\(766\) −26.1277 −0.944031
\(767\) 0.279413 0.0100890
\(768\) 16.5310 0.596509
\(769\) −15.6445 −0.564153 −0.282077 0.959392i \(-0.591023\pi\)
−0.282077 + 0.959392i \(0.591023\pi\)
\(770\) 14.6584 0.528253
\(771\) 6.98956 0.251723
\(772\) 13.7250 0.493972
\(773\) −22.9648 −0.825988 −0.412994 0.910734i \(-0.635517\pi\)
−0.412994 + 0.910734i \(0.635517\pi\)
\(774\) −2.07731 −0.0746675
\(775\) −4.97587 −0.178739
\(776\) 31.0842 1.11586
\(777\) 5.78693 0.207605
\(778\) 3.17237 0.113735
\(779\) 2.87075 0.102855
\(780\) 0.889896 0.0318634
\(781\) −0.426881 −0.0152750
\(782\) 31.2463 1.11737
\(783\) 0.639311 0.0228471
\(784\) −14.0151 −0.500538
\(785\) 50.6643 1.80829
\(786\) −21.6203 −0.771171
\(787\) 11.3151 0.403341 0.201671 0.979453i \(-0.435363\pi\)
0.201671 + 0.979453i \(0.435363\pi\)
\(788\) 18.4864 0.658551
\(789\) 13.7752 0.490409
\(790\) 12.9200 0.459674
\(791\) −54.2707 −1.92964
\(792\) 3.58960 0.127551
\(793\) −0.436138 −0.0154877
\(794\) 0.450416 0.0159847
\(795\) 0.141786 0.00502863
\(796\) 22.3116 0.790815
\(797\) 13.4591 0.476745 0.238372 0.971174i \(-0.423386\pi\)
0.238372 + 0.971174i \(0.423386\pi\)
\(798\) −5.70938 −0.202110
\(799\) 5.53252 0.195726
\(800\) 14.9707 0.529295
\(801\) −10.6550 −0.376476
\(802\) −14.7263 −0.520002
\(803\) 14.1120 0.498000
\(804\) 3.58632 0.126480
\(805\) 107.626 3.79333
\(806\) −0.557230 −0.0196276
\(807\) −11.2058 −0.394462
\(808\) −40.5489 −1.42651
\(809\) 29.8582 1.04976 0.524878 0.851177i \(-0.324111\pi\)
0.524878 + 0.851177i \(0.324111\pi\)
\(810\) 3.02110 0.106151
\(811\) 23.1835 0.814083 0.407042 0.913410i \(-0.366560\pi\)
0.407042 + 0.913410i \(0.366560\pi\)
\(812\) −2.35156 −0.0825236
\(813\) 20.5528 0.720820
\(814\) −1.74474 −0.0611531
\(815\) −37.0502 −1.29781
\(816\) −4.60799 −0.161312
\(817\) 2.60698 0.0912067
\(818\) −41.9688 −1.46740
\(819\) −1.42520 −0.0498005
\(820\) 5.58749 0.195123
\(821\) 16.7615 0.584980 0.292490 0.956269i \(-0.405516\pi\)
0.292490 + 0.956269i \(0.405516\pi\)
\(822\) −21.2489 −0.741142
\(823\) −9.77979 −0.340902 −0.170451 0.985366i \(-0.554522\pi\)
−0.170451 + 0.985366i \(0.554522\pi\)
\(824\) 49.8445 1.73641
\(825\) 3.83701 0.133587
\(826\) −3.48913 −0.121402
\(827\) −38.6934 −1.34550 −0.672751 0.739869i \(-0.734887\pi\)
−0.672751 + 0.739869i \(0.734887\pi\)
\(828\) 8.14249 0.282971
\(829\) 7.90150 0.274430 0.137215 0.990541i \(-0.456185\pi\)
0.137215 + 0.990541i \(0.456185\pi\)
\(830\) 42.4644 1.47396
\(831\) 17.2928 0.599880
\(832\) 2.65507 0.0920480
\(833\) −32.3784 −1.12184
\(834\) −15.7630 −0.545828
\(835\) −14.1167 −0.488529
\(836\) −1.39175 −0.0481348
\(837\) 1.52951 0.0528676
\(838\) 24.8391 0.858051
\(839\) −1.29253 −0.0446230 −0.0223115 0.999751i \(-0.507103\pi\)
−0.0223115 + 0.999751i \(0.507103\pi\)
\(840\) −35.9691 −1.24105
\(841\) −28.5913 −0.985906
\(842\) −17.2738 −0.595294
\(843\) −24.1530 −0.831875
\(844\) 3.45915 0.119069
\(845\) −37.0022 −1.27291
\(846\) −1.78316 −0.0613063
\(847\) 39.5295 1.35825
\(848\) 0.0697021 0.00239358
\(849\) 12.4777 0.428234
\(850\) −11.1624 −0.382866
\(851\) −12.8104 −0.439134
\(852\) 0.323616 0.0110869
\(853\) 47.2923 1.61926 0.809629 0.586942i \(-0.199668\pi\)
0.809629 + 0.586942i \(0.199668\pi\)
\(854\) 5.44621 0.186366
\(855\) −3.79142 −0.129664
\(856\) 32.9431 1.12597
\(857\) 41.5916 1.42074 0.710371 0.703827i \(-0.248527\pi\)
0.710371 + 0.703827i \(0.248527\pi\)
\(858\) 0.429692 0.0146695
\(859\) 39.2094 1.33781 0.668904 0.743349i \(-0.266764\pi\)
0.668904 + 0.743349i \(0.266764\pi\)
\(860\) 5.07410 0.173025
\(861\) −8.94855 −0.304966
\(862\) 25.4248 0.865972
\(863\) 37.3913 1.27281 0.636407 0.771354i \(-0.280420\pi\)
0.636407 + 0.771354i \(0.280420\pi\)
\(864\) −4.60178 −0.156556
\(865\) −44.6436 −1.51793
\(866\) −7.57864 −0.257533
\(867\) 6.35435 0.215805
\(868\) −5.62596 −0.190957
\(869\) −5.04398 −0.171105
\(870\) 1.93143 0.0654815
\(871\) 1.38957 0.0470838
\(872\) −52.1751 −1.76687
\(873\) −10.2134 −0.345670
\(874\) 12.6387 0.427510
\(875\) 20.6438 0.697889
\(876\) −10.6982 −0.361458
\(877\) −42.5083 −1.43540 −0.717701 0.696351i \(-0.754806\pi\)
−0.717701 + 0.696351i \(0.754806\pi\)
\(878\) 22.2310 0.750260
\(879\) 24.0095 0.809820
\(880\) 4.78535 0.161314
\(881\) 30.1193 1.01475 0.507373 0.861726i \(-0.330617\pi\)
0.507373 + 0.861726i \(0.330617\pi\)
\(882\) 10.4357 0.351389
\(883\) −34.4505 −1.15935 −0.579676 0.814847i \(-0.696821\pi\)
−0.579676 + 0.814847i \(0.696821\pi\)
\(884\) 1.01068 0.0339928
\(885\) −2.31702 −0.0778858
\(886\) −11.7264 −0.393956
\(887\) 47.0599 1.58012 0.790059 0.613031i \(-0.210050\pi\)
0.790059 + 0.613031i \(0.210050\pi\)
\(888\) 4.28126 0.143670
\(889\) −53.6466 −1.79925
\(890\) −32.1899 −1.07901
\(891\) −1.17944 −0.0395127
\(892\) −8.49748 −0.284517
\(893\) 2.23782 0.0748859
\(894\) −5.51685 −0.184511
\(895\) −48.7353 −1.62904
\(896\) 4.70705 0.157252
\(897\) 3.15492 0.105340
\(898\) −2.23462 −0.0745701
\(899\) 0.977833 0.0326125
\(900\) −2.90881 −0.0969602
\(901\) 0.161030 0.00536468
\(902\) 2.69796 0.0898321
\(903\) −8.12635 −0.270428
\(904\) −40.1503 −1.33538
\(905\) 9.48678 0.315351
\(906\) −5.87853 −0.195301
\(907\) 53.8831 1.78916 0.894579 0.446910i \(-0.147476\pi\)
0.894579 + 0.446910i \(0.147476\pi\)
\(908\) −7.10960 −0.235940
\(909\) 13.3232 0.441903
\(910\) −4.30567 −0.142732
\(911\) 2.77566 0.0919616 0.0459808 0.998942i \(-0.485359\pi\)
0.0459808 + 0.998942i \(0.485359\pi\)
\(912\) −1.86387 −0.0617188
\(913\) −16.5781 −0.548654
\(914\) 39.0484 1.29161
\(915\) 3.61666 0.119563
\(916\) 10.0136 0.330857
\(917\) −84.5776 −2.79300
\(918\) 3.43115 0.113245
\(919\) 19.6148 0.647031 0.323516 0.946223i \(-0.395135\pi\)
0.323516 + 0.946223i \(0.395135\pi\)
\(920\) 79.6237 2.62512
\(921\) −7.54396 −0.248582
\(922\) 31.4256 1.03495
\(923\) 0.125389 0.00412724
\(924\) 4.33830 0.142720
\(925\) 4.57635 0.150469
\(926\) 21.4583 0.705163
\(927\) −16.3775 −0.537906
\(928\) −2.94197 −0.0965749
\(929\) 53.7834 1.76458 0.882289 0.470709i \(-0.156002\pi\)
0.882289 + 0.470709i \(0.156002\pi\)
\(930\) 4.62081 0.151522
\(931\) −13.0966 −0.429224
\(932\) 14.0280 0.459503
\(933\) 7.22920 0.236673
\(934\) 4.96914 0.162595
\(935\) 11.0554 0.361550
\(936\) −1.05439 −0.0344637
\(937\) 31.1352 1.01714 0.508572 0.861020i \(-0.330174\pi\)
0.508572 + 0.861020i \(0.330174\pi\)
\(938\) −17.3521 −0.566565
\(939\) 21.2712 0.694161
\(940\) 4.35559 0.142064
\(941\) 29.4881 0.961286 0.480643 0.876916i \(-0.340403\pi\)
0.480643 + 0.876916i \(0.340403\pi\)
\(942\) −18.5457 −0.604251
\(943\) 19.8092 0.645075
\(944\) −1.13905 −0.0370730
\(945\) 11.8184 0.384453
\(946\) 2.45006 0.0796585
\(947\) −43.4174 −1.41087 −0.705437 0.708772i \(-0.749249\pi\)
−0.705437 + 0.708772i \(0.749249\pi\)
\(948\) 3.82380 0.124191
\(949\) −4.14516 −0.134558
\(950\) −4.51502 −0.146487
\(951\) −31.0998 −1.00848
\(952\) −40.8510 −1.32399
\(953\) 14.4497 0.468071 0.234035 0.972228i \(-0.424807\pi\)
0.234035 + 0.972228i \(0.424807\pi\)
\(954\) −0.0519008 −0.00168035
\(955\) −35.1656 −1.13793
\(956\) 0.894124 0.0289180
\(957\) −0.754028 −0.0243743
\(958\) 41.5308 1.34180
\(959\) −83.1247 −2.68424
\(960\) −22.0171 −0.710598
\(961\) −28.6606 −0.924535
\(962\) 0.512488 0.0165233
\(963\) −10.8242 −0.348804
\(964\) 22.8577 0.736196
\(965\) −44.0987 −1.41959
\(966\) −39.3967 −1.26757
\(967\) 6.88995 0.221566 0.110783 0.993845i \(-0.464664\pi\)
0.110783 + 0.993845i \(0.464664\pi\)
\(968\) 29.2446 0.939956
\(969\) −4.30601 −0.138329
\(970\) −30.8556 −0.990715
\(971\) −1.06821 −0.0342806 −0.0171403 0.999853i \(-0.505456\pi\)
−0.0171403 + 0.999853i \(0.505456\pi\)
\(972\) 0.894124 0.0286790
\(973\) −61.6640 −1.97686
\(974\) −27.4651 −0.880037
\(975\) −1.12706 −0.0360947
\(976\) 1.77795 0.0569109
\(977\) −36.3679 −1.16351 −0.581755 0.813364i \(-0.697634\pi\)
−0.581755 + 0.813364i \(0.697634\pi\)
\(978\) 13.5622 0.433672
\(979\) 12.5669 0.401640
\(980\) −25.4906 −0.814266
\(981\) 17.1432 0.547342
\(982\) −21.1921 −0.676268
\(983\) 5.70415 0.181934 0.0909671 0.995854i \(-0.471004\pi\)
0.0909671 + 0.995854i \(0.471004\pi\)
\(984\) −6.62029 −0.211047
\(985\) −59.3974 −1.89256
\(986\) 2.19357 0.0698575
\(987\) −6.97562 −0.222037
\(988\) 0.408805 0.0130058
\(989\) 17.9891 0.572019
\(990\) −3.56321 −0.113246
\(991\) −6.98693 −0.221947 −0.110974 0.993823i \(-0.535397\pi\)
−0.110974 + 0.993823i \(0.535397\pi\)
\(992\) −7.03847 −0.223472
\(993\) 4.80463 0.152470
\(994\) −1.56578 −0.0496636
\(995\) −71.6880 −2.27266
\(996\) 12.5677 0.398223
\(997\) −28.4466 −0.900914 −0.450457 0.892798i \(-0.648739\pi\)
−0.450457 + 0.892798i \(0.648739\pi\)
\(998\) 17.4880 0.553573
\(999\) −1.40670 −0.0445060
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 717.2.a.d.1.5 6
3.2 odd 2 2151.2.a.e.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.5 6 1.1 even 1 trivial
2151.2.a.e.1.2 6 3.2 odd 2