Properties

Label 8022.2.a.z.1.11
Level $8022$
Weight $2$
Character 8022.1
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $15$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + 2130 x^{6} - 84842 x^{5} + 7822 x^{4} + 62828 x^{3} - 16144 x^{2} + \cdots + 4704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.55328\) of defining polynomial
Character \(\chi\) \(=\) 8022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.55328 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.55328 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.55328 q^{10} +2.86410 q^{11} +1.00000 q^{12} -1.19533 q^{13} -1.00000 q^{14} +1.55328 q^{15} +1.00000 q^{16} +4.13308 q^{17} -1.00000 q^{18} -4.34711 q^{19} +1.55328 q^{20} +1.00000 q^{21} -2.86410 q^{22} +5.31993 q^{23} -1.00000 q^{24} -2.58732 q^{25} +1.19533 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.80630 q^{29} -1.55328 q^{30} +8.33433 q^{31} -1.00000 q^{32} +2.86410 q^{33} -4.13308 q^{34} +1.55328 q^{35} +1.00000 q^{36} -7.50201 q^{37} +4.34711 q^{38} -1.19533 q^{39} -1.55328 q^{40} +1.58747 q^{41} -1.00000 q^{42} +5.26207 q^{43} +2.86410 q^{44} +1.55328 q^{45} -5.31993 q^{46} +4.76012 q^{47} +1.00000 q^{48} +1.00000 q^{49} +2.58732 q^{50} +4.13308 q^{51} -1.19533 q^{52} -9.50856 q^{53} -1.00000 q^{54} +4.44874 q^{55} -1.00000 q^{56} -4.34711 q^{57} -6.80630 q^{58} +0.810529 q^{59} +1.55328 q^{60} -0.00358361 q^{61} -8.33433 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.85668 q^{65} -2.86410 q^{66} +3.72543 q^{67} +4.13308 q^{68} +5.31993 q^{69} -1.55328 q^{70} -14.4977 q^{71} -1.00000 q^{72} -10.2295 q^{73} +7.50201 q^{74} -2.58732 q^{75} -4.34711 q^{76} +2.86410 q^{77} +1.19533 q^{78} +13.1530 q^{79} +1.55328 q^{80} +1.00000 q^{81} -1.58747 q^{82} +3.27581 q^{83} +1.00000 q^{84} +6.41984 q^{85} -5.26207 q^{86} +6.80630 q^{87} -2.86410 q^{88} +4.89203 q^{89} -1.55328 q^{90} -1.19533 q^{91} +5.31993 q^{92} +8.33433 q^{93} -4.76012 q^{94} -6.75228 q^{95} -1.00000 q^{96} -4.85955 q^{97} -1.00000 q^{98} +2.86410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.55328 0.694648 0.347324 0.937745i \(-0.387090\pi\)
0.347324 + 0.937745i \(0.387090\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.55328 −0.491190
\(11\) 2.86410 0.863557 0.431779 0.901980i \(-0.357886\pi\)
0.431779 + 0.901980i \(0.357886\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.19533 −0.331524 −0.165762 0.986166i \(-0.553008\pi\)
−0.165762 + 0.986166i \(0.553008\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.55328 0.401055
\(16\) 1.00000 0.250000
\(17\) 4.13308 1.00242 0.501210 0.865326i \(-0.332888\pi\)
0.501210 + 0.865326i \(0.332888\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.34711 −0.997295 −0.498648 0.866805i \(-0.666170\pi\)
−0.498648 + 0.866805i \(0.666170\pi\)
\(20\) 1.55328 0.347324
\(21\) 1.00000 0.218218
\(22\) −2.86410 −0.610627
\(23\) 5.31993 1.10928 0.554641 0.832090i \(-0.312856\pi\)
0.554641 + 0.832090i \(0.312856\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.58732 −0.517464
\(26\) 1.19533 0.234423
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.80630 1.26390 0.631949 0.775010i \(-0.282255\pi\)
0.631949 + 0.775010i \(0.282255\pi\)
\(30\) −1.55328 −0.283589
\(31\) 8.33433 1.49689 0.748445 0.663197i \(-0.230801\pi\)
0.748445 + 0.663197i \(0.230801\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.86410 0.498575
\(34\) −4.13308 −0.708818
\(35\) 1.55328 0.262552
\(36\) 1.00000 0.166667
\(37\) −7.50201 −1.23332 −0.616661 0.787229i \(-0.711515\pi\)
−0.616661 + 0.787229i \(0.711515\pi\)
\(38\) 4.34711 0.705194
\(39\) −1.19533 −0.191405
\(40\) −1.55328 −0.245595
\(41\) 1.58747 0.247921 0.123960 0.992287i \(-0.460440\pi\)
0.123960 + 0.992287i \(0.460440\pi\)
\(42\) −1.00000 −0.154303
\(43\) 5.26207 0.802458 0.401229 0.915978i \(-0.368583\pi\)
0.401229 + 0.915978i \(0.368583\pi\)
\(44\) 2.86410 0.431779
\(45\) 1.55328 0.231549
\(46\) −5.31993 −0.784381
\(47\) 4.76012 0.694336 0.347168 0.937803i \(-0.387143\pi\)
0.347168 + 0.937803i \(0.387143\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 2.58732 0.365902
\(51\) 4.13308 0.578748
\(52\) −1.19533 −0.165762
\(53\) −9.50856 −1.30610 −0.653050 0.757314i \(-0.726511\pi\)
−0.653050 + 0.757314i \(0.726511\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.44874 0.599869
\(56\) −1.00000 −0.133631
\(57\) −4.34711 −0.575789
\(58\) −6.80630 −0.893711
\(59\) 0.810529 0.105522 0.0527609 0.998607i \(-0.483198\pi\)
0.0527609 + 0.998607i \(0.483198\pi\)
\(60\) 1.55328 0.200528
\(61\) −0.00358361 −0.000458834 0 −0.000229417 1.00000i \(-0.500073\pi\)
−0.000229417 1.00000i \(0.500073\pi\)
\(62\) −8.33433 −1.05846
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −1.85668 −0.230293
\(66\) −2.86410 −0.352546
\(67\) 3.72543 0.455134 0.227567 0.973762i \(-0.426923\pi\)
0.227567 + 0.973762i \(0.426923\pi\)
\(68\) 4.13308 0.501210
\(69\) 5.31993 0.640444
\(70\) −1.55328 −0.185653
\(71\) −14.4977 −1.72056 −0.860281 0.509820i \(-0.829712\pi\)
−0.860281 + 0.509820i \(0.829712\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.2295 −1.19727 −0.598635 0.801022i \(-0.704290\pi\)
−0.598635 + 0.801022i \(0.704290\pi\)
\(74\) 7.50201 0.872090
\(75\) −2.58732 −0.298758
\(76\) −4.34711 −0.498648
\(77\) 2.86410 0.326394
\(78\) 1.19533 0.135344
\(79\) 13.1530 1.47982 0.739912 0.672704i \(-0.234867\pi\)
0.739912 + 0.672704i \(0.234867\pi\)
\(80\) 1.55328 0.173662
\(81\) 1.00000 0.111111
\(82\) −1.58747 −0.175307
\(83\) 3.27581 0.359567 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.41984 0.696329
\(86\) −5.26207 −0.567423
\(87\) 6.80630 0.729712
\(88\) −2.86410 −0.305314
\(89\) 4.89203 0.518554 0.259277 0.965803i \(-0.416516\pi\)
0.259277 + 0.965803i \(0.416516\pi\)
\(90\) −1.55328 −0.163730
\(91\) −1.19533 −0.125304
\(92\) 5.31993 0.554641
\(93\) 8.33433 0.864229
\(94\) −4.76012 −0.490969
\(95\) −6.75228 −0.692770
\(96\) −1.00000 −0.102062
\(97\) −4.85955 −0.493412 −0.246706 0.969090i \(-0.579348\pi\)
−0.246706 + 0.969090i \(0.579348\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.86410 0.287852
\(100\) −2.58732 −0.258732
\(101\) −2.55589 −0.254321 −0.127160 0.991882i \(-0.540586\pi\)
−0.127160 + 0.991882i \(0.540586\pi\)
\(102\) −4.13308 −0.409236
\(103\) −4.03327 −0.397410 −0.198705 0.980059i \(-0.563674\pi\)
−0.198705 + 0.980059i \(0.563674\pi\)
\(104\) 1.19533 0.117211
\(105\) 1.55328 0.151585
\(106\) 9.50856 0.923553
\(107\) 19.1063 1.84707 0.923536 0.383512i \(-0.125286\pi\)
0.923536 + 0.383512i \(0.125286\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.3693 1.18477 0.592383 0.805657i \(-0.298187\pi\)
0.592383 + 0.805657i \(0.298187\pi\)
\(110\) −4.44874 −0.424171
\(111\) −7.50201 −0.712059
\(112\) 1.00000 0.0944911
\(113\) 10.7853 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(114\) 4.34711 0.407144
\(115\) 8.26334 0.770561
\(116\) 6.80630 0.631949
\(117\) −1.19533 −0.110508
\(118\) −0.810529 −0.0746152
\(119\) 4.13308 0.378879
\(120\) −1.55328 −0.141794
\(121\) −2.79696 −0.254269
\(122\) 0.00358361 0.000324445 0
\(123\) 1.58747 0.143137
\(124\) 8.33433 0.748445
\(125\) −11.7852 −1.05410
\(126\) −1.00000 −0.0890871
\(127\) 0.859505 0.0762688 0.0381344 0.999273i \(-0.487859\pi\)
0.0381344 + 0.999273i \(0.487859\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.26207 0.463299
\(130\) 1.85668 0.162841
\(131\) −21.3634 −1.86652 −0.933262 0.359195i \(-0.883051\pi\)
−0.933262 + 0.359195i \(0.883051\pi\)
\(132\) 2.86410 0.249288
\(133\) −4.34711 −0.376942
\(134\) −3.72543 −0.321828
\(135\) 1.55328 0.133685
\(136\) −4.13308 −0.354409
\(137\) 1.68404 0.143878 0.0719388 0.997409i \(-0.477081\pi\)
0.0719388 + 0.997409i \(0.477081\pi\)
\(138\) −5.31993 −0.452863
\(139\) 2.96087 0.251137 0.125569 0.992085i \(-0.459924\pi\)
0.125569 + 0.992085i \(0.459924\pi\)
\(140\) 1.55328 0.131276
\(141\) 4.76012 0.400875
\(142\) 14.4977 1.21662
\(143\) −3.42353 −0.286290
\(144\) 1.00000 0.0833333
\(145\) 10.5721 0.877964
\(146\) 10.2295 0.846598
\(147\) 1.00000 0.0824786
\(148\) −7.50201 −0.616661
\(149\) −10.7462 −0.880360 −0.440180 0.897910i \(-0.645085\pi\)
−0.440180 + 0.897910i \(0.645085\pi\)
\(150\) 2.58732 0.211254
\(151\) 8.98373 0.731085 0.365543 0.930795i \(-0.380883\pi\)
0.365543 + 0.930795i \(0.380883\pi\)
\(152\) 4.34711 0.352597
\(153\) 4.13308 0.334140
\(154\) −2.86410 −0.230795
\(155\) 12.9455 1.03981
\(156\) −1.19533 −0.0957027
\(157\) 9.02424 0.720213 0.360106 0.932911i \(-0.382740\pi\)
0.360106 + 0.932911i \(0.382740\pi\)
\(158\) −13.1530 −1.04639
\(159\) −9.50856 −0.754078
\(160\) −1.55328 −0.122798
\(161\) 5.31993 0.419269
\(162\) −1.00000 −0.0785674
\(163\) 19.3344 1.51439 0.757193 0.653191i \(-0.226570\pi\)
0.757193 + 0.653191i \(0.226570\pi\)
\(164\) 1.58747 0.123960
\(165\) 4.44874 0.346334
\(166\) −3.27581 −0.254252
\(167\) −4.60141 −0.356068 −0.178034 0.984024i \(-0.556974\pi\)
−0.178034 + 0.984024i \(0.556974\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −11.5712 −0.890092
\(170\) −6.41984 −0.492379
\(171\) −4.34711 −0.332432
\(172\) 5.26207 0.401229
\(173\) 13.2466 1.00712 0.503559 0.863961i \(-0.332024\pi\)
0.503559 + 0.863961i \(0.332024\pi\)
\(174\) −6.80630 −0.515984
\(175\) −2.58732 −0.195583
\(176\) 2.86410 0.215889
\(177\) 0.810529 0.0609231
\(178\) −4.89203 −0.366673
\(179\) −23.5313 −1.75881 −0.879405 0.476075i \(-0.842059\pi\)
−0.879405 + 0.476075i \(0.842059\pi\)
\(180\) 1.55328 0.115775
\(181\) 2.50296 0.186044 0.0930220 0.995664i \(-0.470347\pi\)
0.0930220 + 0.995664i \(0.470347\pi\)
\(182\) 1.19533 0.0886035
\(183\) −0.00358361 −0.000264908 0
\(184\) −5.31993 −0.392190
\(185\) −11.6527 −0.856725
\(186\) −8.33433 −0.611102
\(187\) 11.8375 0.865647
\(188\) 4.76012 0.347168
\(189\) 1.00000 0.0727393
\(190\) 6.75228 0.489862
\(191\) 1.00000 0.0723575
\(192\) 1.00000 0.0721688
\(193\) 2.11787 0.152447 0.0762237 0.997091i \(-0.475714\pi\)
0.0762237 + 0.997091i \(0.475714\pi\)
\(194\) 4.85955 0.348895
\(195\) −1.85668 −0.132959
\(196\) 1.00000 0.0714286
\(197\) −18.5010 −1.31814 −0.659072 0.752080i \(-0.729051\pi\)
−0.659072 + 0.752080i \(0.729051\pi\)
\(198\) −2.86410 −0.203542
\(199\) 16.1068 1.14178 0.570891 0.821026i \(-0.306598\pi\)
0.570891 + 0.821026i \(0.306598\pi\)
\(200\) 2.58732 0.182951
\(201\) 3.72543 0.262772
\(202\) 2.55589 0.179832
\(203\) 6.80630 0.477708
\(204\) 4.13308 0.289374
\(205\) 2.46578 0.172218
\(206\) 4.03327 0.281012
\(207\) 5.31993 0.369761
\(208\) −1.19533 −0.0828810
\(209\) −12.4505 −0.861222
\(210\) −1.55328 −0.107187
\(211\) −12.7388 −0.876978 −0.438489 0.898736i \(-0.644486\pi\)
−0.438489 + 0.898736i \(0.644486\pi\)
\(212\) −9.50856 −0.653050
\(213\) −14.4977 −0.993367
\(214\) −19.1063 −1.30608
\(215\) 8.17347 0.557426
\(216\) −1.00000 −0.0680414
\(217\) 8.33433 0.565771
\(218\) −12.3693 −0.837755
\(219\) −10.2295 −0.691245
\(220\) 4.44874 0.299934
\(221\) −4.94039 −0.332326
\(222\) 7.50201 0.503502
\(223\) 24.3664 1.63169 0.815846 0.578269i \(-0.196272\pi\)
0.815846 + 0.578269i \(0.196272\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.58732 −0.172488
\(226\) −10.7853 −0.717429
\(227\) −12.6366 −0.838719 −0.419359 0.907820i \(-0.637745\pi\)
−0.419359 + 0.907820i \(0.637745\pi\)
\(228\) −4.34711 −0.287894
\(229\) 13.9198 0.919845 0.459922 0.887959i \(-0.347877\pi\)
0.459922 + 0.887959i \(0.347877\pi\)
\(230\) −8.26334 −0.544869
\(231\) 2.86410 0.188444
\(232\) −6.80630 −0.446855
\(233\) 5.26643 0.345015 0.172508 0.985008i \(-0.444813\pi\)
0.172508 + 0.985008i \(0.444813\pi\)
\(234\) 1.19533 0.0781410
\(235\) 7.39381 0.482319
\(236\) 0.810529 0.0527609
\(237\) 13.1530 0.854376
\(238\) −4.13308 −0.267908
\(239\) −18.1064 −1.17120 −0.585601 0.810599i \(-0.699142\pi\)
−0.585601 + 0.810599i \(0.699142\pi\)
\(240\) 1.55328 0.100264
\(241\) 29.4501 1.89705 0.948525 0.316702i \(-0.102576\pi\)
0.948525 + 0.316702i \(0.102576\pi\)
\(242\) 2.79696 0.179795
\(243\) 1.00000 0.0641500
\(244\) −0.00358361 −0.000229417 0
\(245\) 1.55328 0.0992355
\(246\) −1.58747 −0.101213
\(247\) 5.19622 0.330627
\(248\) −8.33433 −0.529230
\(249\) 3.27581 0.207596
\(250\) 11.7852 0.745364
\(251\) 16.6181 1.04893 0.524463 0.851433i \(-0.324266\pi\)
0.524463 + 0.851433i \(0.324266\pi\)
\(252\) 1.00000 0.0629941
\(253\) 15.2368 0.957929
\(254\) −0.859505 −0.0539302
\(255\) 6.41984 0.402026
\(256\) 1.00000 0.0625000
\(257\) 25.3033 1.57838 0.789189 0.614150i \(-0.210501\pi\)
0.789189 + 0.614150i \(0.210501\pi\)
\(258\) −5.26207 −0.327602
\(259\) −7.50201 −0.466152
\(260\) −1.85668 −0.115146
\(261\) 6.80630 0.421299
\(262\) 21.3634 1.31983
\(263\) −15.0413 −0.927485 −0.463743 0.885970i \(-0.653494\pi\)
−0.463743 + 0.885970i \(0.653494\pi\)
\(264\) −2.86410 −0.176273
\(265\) −14.7695 −0.907281
\(266\) 4.34711 0.266538
\(267\) 4.89203 0.299387
\(268\) 3.72543 0.227567
\(269\) 0.347738 0.0212020 0.0106010 0.999944i \(-0.496626\pi\)
0.0106010 + 0.999944i \(0.496626\pi\)
\(270\) −1.55328 −0.0945297
\(271\) −21.8338 −1.32631 −0.663155 0.748482i \(-0.730783\pi\)
−0.663155 + 0.748482i \(0.730783\pi\)
\(272\) 4.13308 0.250605
\(273\) −1.19533 −0.0723445
\(274\) −1.68404 −0.101737
\(275\) −7.41033 −0.446860
\(276\) 5.31993 0.320222
\(277\) 2.45754 0.147659 0.0738297 0.997271i \(-0.476478\pi\)
0.0738297 + 0.997271i \(0.476478\pi\)
\(278\) −2.96087 −0.177581
\(279\) 8.33433 0.498963
\(280\) −1.55328 −0.0928263
\(281\) −4.17887 −0.249291 −0.124645 0.992201i \(-0.539779\pi\)
−0.124645 + 0.992201i \(0.539779\pi\)
\(282\) −4.76012 −0.283461
\(283\) −11.7209 −0.696737 −0.348369 0.937358i \(-0.613264\pi\)
−0.348369 + 0.937358i \(0.613264\pi\)
\(284\) −14.4977 −0.860281
\(285\) −6.75228 −0.399971
\(286\) 3.42353 0.202438
\(287\) 1.58747 0.0937053
\(288\) −1.00000 −0.0589256
\(289\) 0.0823856 0.00484621
\(290\) −10.5721 −0.620814
\(291\) −4.85955 −0.284872
\(292\) −10.2295 −0.598635
\(293\) −5.34331 −0.312160 −0.156080 0.987744i \(-0.549886\pi\)
−0.156080 + 0.987744i \(0.549886\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 1.25898 0.0733006
\(296\) 7.50201 0.436045
\(297\) 2.86410 0.166192
\(298\) 10.7462 0.622508
\(299\) −6.35906 −0.367754
\(300\) −2.58732 −0.149379
\(301\) 5.26207 0.303300
\(302\) −8.98373 −0.516955
\(303\) −2.55589 −0.146832
\(304\) −4.34711 −0.249324
\(305\) −0.00556635 −0.000318728 0
\(306\) −4.13308 −0.236273
\(307\) 12.1496 0.693413 0.346707 0.937974i \(-0.387300\pi\)
0.346707 + 0.937974i \(0.387300\pi\)
\(308\) 2.86410 0.163197
\(309\) −4.03327 −0.229445
\(310\) −12.9455 −0.735258
\(311\) −16.5576 −0.938896 −0.469448 0.882960i \(-0.655547\pi\)
−0.469448 + 0.882960i \(0.655547\pi\)
\(312\) 1.19533 0.0676721
\(313\) 19.8617 1.12265 0.561324 0.827596i \(-0.310292\pi\)
0.561324 + 0.827596i \(0.310292\pi\)
\(314\) −9.02424 −0.509267
\(315\) 1.55328 0.0875175
\(316\) 13.1530 0.739912
\(317\) 17.5364 0.984942 0.492471 0.870329i \(-0.336094\pi\)
0.492471 + 0.870329i \(0.336094\pi\)
\(318\) 9.50856 0.533213
\(319\) 19.4939 1.09145
\(320\) 1.55328 0.0868310
\(321\) 19.1063 1.06641
\(322\) −5.31993 −0.296468
\(323\) −17.9670 −0.999709
\(324\) 1.00000 0.0555556
\(325\) 3.09269 0.171552
\(326\) −19.3344 −1.07083
\(327\) 12.3693 0.684024
\(328\) −1.58747 −0.0876533
\(329\) 4.76012 0.262434
\(330\) −4.44874 −0.244895
\(331\) 6.91301 0.379973 0.189987 0.981787i \(-0.439156\pi\)
0.189987 + 0.981787i \(0.439156\pi\)
\(332\) 3.27581 0.179783
\(333\) −7.50201 −0.411107
\(334\) 4.60141 0.251778
\(335\) 5.78664 0.316158
\(336\) 1.00000 0.0545545
\(337\) −23.7749 −1.29510 −0.647550 0.762023i \(-0.724206\pi\)
−0.647550 + 0.762023i \(0.724206\pi\)
\(338\) 11.5712 0.629390
\(339\) 10.7853 0.585778
\(340\) 6.41984 0.348165
\(341\) 23.8703 1.29265
\(342\) 4.34711 0.235065
\(343\) 1.00000 0.0539949
\(344\) −5.26207 −0.283712
\(345\) 8.26334 0.444884
\(346\) −13.2466 −0.712140
\(347\) 20.6915 1.11078 0.555390 0.831590i \(-0.312569\pi\)
0.555390 + 0.831590i \(0.312569\pi\)
\(348\) 6.80630 0.364856
\(349\) −13.6257 −0.729366 −0.364683 0.931132i \(-0.618823\pi\)
−0.364683 + 0.931132i \(0.618823\pi\)
\(350\) 2.58732 0.138298
\(351\) −1.19533 −0.0638018
\(352\) −2.86410 −0.152657
\(353\) 19.9014 1.05924 0.529621 0.848234i \(-0.322334\pi\)
0.529621 + 0.848234i \(0.322334\pi\)
\(354\) −0.810529 −0.0430791
\(355\) −22.5190 −1.19519
\(356\) 4.89203 0.259277
\(357\) 4.13308 0.218746
\(358\) 23.5313 1.24367
\(359\) −29.8262 −1.57416 −0.787082 0.616848i \(-0.788409\pi\)
−0.787082 + 0.616848i \(0.788409\pi\)
\(360\) −1.55328 −0.0818651
\(361\) −0.102635 −0.00540184
\(362\) −2.50296 −0.131553
\(363\) −2.79696 −0.146802
\(364\) −1.19533 −0.0626521
\(365\) −15.8893 −0.831682
\(366\) 0.00358361 0.000187318 0
\(367\) −15.6431 −0.816561 −0.408280 0.912857i \(-0.633871\pi\)
−0.408280 + 0.912857i \(0.633871\pi\)
\(368\) 5.31993 0.277321
\(369\) 1.58747 0.0826403
\(370\) 11.6527 0.605796
\(371\) −9.50856 −0.493660
\(372\) 8.33433 0.432115
\(373\) 10.4280 0.539940 0.269970 0.962869i \(-0.412986\pi\)
0.269970 + 0.962869i \(0.412986\pi\)
\(374\) −11.8375 −0.612105
\(375\) −11.7852 −0.608587
\(376\) −4.76012 −0.245485
\(377\) −8.13575 −0.419012
\(378\) −1.00000 −0.0514344
\(379\) −15.5075 −0.796567 −0.398284 0.917262i \(-0.630394\pi\)
−0.398284 + 0.917262i \(0.630394\pi\)
\(380\) −6.75228 −0.346385
\(381\) 0.859505 0.0440338
\(382\) −1.00000 −0.0511645
\(383\) −17.7275 −0.905835 −0.452918 0.891552i \(-0.649617\pi\)
−0.452918 + 0.891552i \(0.649617\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.44874 0.226729
\(386\) −2.11787 −0.107797
\(387\) 5.26207 0.267486
\(388\) −4.85955 −0.246706
\(389\) 26.9359 1.36570 0.682851 0.730558i \(-0.260740\pi\)
0.682851 + 0.730558i \(0.260740\pi\)
\(390\) 1.85668 0.0940166
\(391\) 21.9877 1.11197
\(392\) −1.00000 −0.0505076
\(393\) −21.3634 −1.07764
\(394\) 18.5010 0.932068
\(395\) 20.4302 1.02796
\(396\) 2.86410 0.143926
\(397\) −24.9829 −1.25386 −0.626928 0.779077i \(-0.715688\pi\)
−0.626928 + 0.779077i \(0.715688\pi\)
\(398\) −16.1068 −0.807361
\(399\) −4.34711 −0.217628
\(400\) −2.58732 −0.129366
\(401\) 5.32228 0.265782 0.132891 0.991131i \(-0.457574\pi\)
0.132891 + 0.991131i \(0.457574\pi\)
\(402\) −3.72543 −0.185808
\(403\) −9.96224 −0.496255
\(404\) −2.55589 −0.127160
\(405\) 1.55328 0.0771831
\(406\) −6.80630 −0.337791
\(407\) −21.4865 −1.06504
\(408\) −4.13308 −0.204618
\(409\) 14.7703 0.730345 0.365172 0.930940i \(-0.381010\pi\)
0.365172 + 0.930940i \(0.381010\pi\)
\(410\) −2.46578 −0.121776
\(411\) 1.68404 0.0830677
\(412\) −4.03327 −0.198705
\(413\) 0.810529 0.0398835
\(414\) −5.31993 −0.261460
\(415\) 5.08825 0.249772
\(416\) 1.19533 0.0586057
\(417\) 2.96087 0.144994
\(418\) 12.4505 0.608976
\(419\) 18.8559 0.921171 0.460586 0.887615i \(-0.347639\pi\)
0.460586 + 0.887615i \(0.347639\pi\)
\(420\) 1.55328 0.0757923
\(421\) 11.9762 0.583686 0.291843 0.956466i \(-0.405732\pi\)
0.291843 + 0.956466i \(0.405732\pi\)
\(422\) 12.7388 0.620117
\(423\) 4.76012 0.231445
\(424\) 9.50856 0.461776
\(425\) −10.6936 −0.518716
\(426\) 14.4977 0.702416
\(427\) −0.00358361 −0.000173423 0
\(428\) 19.1063 0.923536
\(429\) −3.42353 −0.165290
\(430\) −8.17347 −0.394160
\(431\) 33.6954 1.62305 0.811524 0.584319i \(-0.198638\pi\)
0.811524 + 0.584319i \(0.198638\pi\)
\(432\) 1.00000 0.0481125
\(433\) −25.9712 −1.24810 −0.624048 0.781386i \(-0.714513\pi\)
−0.624048 + 0.781386i \(0.714513\pi\)
\(434\) −8.33433 −0.400060
\(435\) 10.5721 0.506893
\(436\) 12.3693 0.592383
\(437\) −23.1263 −1.10628
\(438\) 10.2295 0.488784
\(439\) −10.6796 −0.509711 −0.254856 0.966979i \(-0.582028\pi\)
−0.254856 + 0.966979i \(0.582028\pi\)
\(440\) −4.44874 −0.212086
\(441\) 1.00000 0.0476190
\(442\) 4.94039 0.234990
\(443\) 28.3258 1.34580 0.672899 0.739735i \(-0.265049\pi\)
0.672899 + 0.739735i \(0.265049\pi\)
\(444\) −7.50201 −0.356029
\(445\) 7.59869 0.360213
\(446\) −24.3664 −1.15378
\(447\) −10.7462 −0.508276
\(448\) 1.00000 0.0472456
\(449\) −37.2750 −1.75912 −0.879558 0.475792i \(-0.842162\pi\)
−0.879558 + 0.475792i \(0.842162\pi\)
\(450\) 2.58732 0.121967
\(451\) 4.54666 0.214094
\(452\) 10.7853 0.507299
\(453\) 8.98373 0.422092
\(454\) 12.6366 0.593064
\(455\) −1.85668 −0.0870424
\(456\) 4.34711 0.203572
\(457\) −23.9922 −1.12231 −0.561154 0.827711i \(-0.689643\pi\)
−0.561154 + 0.827711i \(0.689643\pi\)
\(458\) −13.9198 −0.650429
\(459\) 4.13308 0.192916
\(460\) 8.26334 0.385280
\(461\) 32.3022 1.50447 0.752233 0.658898i \(-0.228977\pi\)
0.752233 + 0.658898i \(0.228977\pi\)
\(462\) −2.86410 −0.133250
\(463\) −18.2217 −0.846836 −0.423418 0.905934i \(-0.639170\pi\)
−0.423418 + 0.905934i \(0.639170\pi\)
\(464\) 6.80630 0.315974
\(465\) 12.9455 0.600335
\(466\) −5.26643 −0.243963
\(467\) −19.4182 −0.898565 −0.449283 0.893390i \(-0.648320\pi\)
−0.449283 + 0.893390i \(0.648320\pi\)
\(468\) −1.19533 −0.0552540
\(469\) 3.72543 0.172024
\(470\) −7.39381 −0.341051
\(471\) 9.02424 0.415815
\(472\) −0.810529 −0.0373076
\(473\) 15.0711 0.692968
\(474\) −13.1530 −0.604135
\(475\) 11.2474 0.516064
\(476\) 4.13308 0.189440
\(477\) −9.50856 −0.435367
\(478\) 18.1064 0.828165
\(479\) −16.8579 −0.770257 −0.385129 0.922863i \(-0.625843\pi\)
−0.385129 + 0.922863i \(0.625843\pi\)
\(480\) −1.55328 −0.0708972
\(481\) 8.96735 0.408876
\(482\) −29.4501 −1.34142
\(483\) 5.31993 0.242065
\(484\) −2.79696 −0.127134
\(485\) −7.54824 −0.342748
\(486\) −1.00000 −0.0453609
\(487\) −27.2722 −1.23582 −0.617910 0.786248i \(-0.712021\pi\)
−0.617910 + 0.786248i \(0.712021\pi\)
\(488\) 0.00358361 0.000162222 0
\(489\) 19.3344 0.874331
\(490\) −1.55328 −0.0701701
\(491\) 39.7906 1.79572 0.897862 0.440278i \(-0.145120\pi\)
0.897862 + 0.440278i \(0.145120\pi\)
\(492\) 1.58747 0.0715686
\(493\) 28.1310 1.26696
\(494\) −5.19622 −0.233789
\(495\) 4.44874 0.199956
\(496\) 8.33433 0.374222
\(497\) −14.4977 −0.650311
\(498\) −3.27581 −0.146793
\(499\) 14.3592 0.642806 0.321403 0.946943i \(-0.395846\pi\)
0.321403 + 0.946943i \(0.395846\pi\)
\(500\) −11.7852 −0.527052
\(501\) −4.60141 −0.205576
\(502\) −16.6181 −0.741703
\(503\) −8.85752 −0.394937 −0.197469 0.980309i \(-0.563272\pi\)
−0.197469 + 0.980309i \(0.563272\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −3.97002 −0.176663
\(506\) −15.2368 −0.677358
\(507\) −11.5712 −0.513895
\(508\) 0.859505 0.0381344
\(509\) −14.9787 −0.663919 −0.331959 0.943294i \(-0.607710\pi\)
−0.331959 + 0.943294i \(0.607710\pi\)
\(510\) −6.41984 −0.284275
\(511\) −10.2295 −0.452526
\(512\) −1.00000 −0.0441942
\(513\) −4.34711 −0.191930
\(514\) −25.3033 −1.11608
\(515\) −6.26481 −0.276060
\(516\) 5.26207 0.231650
\(517\) 13.6335 0.599599
\(518\) 7.50201 0.329619
\(519\) 13.2466 0.581460
\(520\) 1.85668 0.0814207
\(521\) 0.958358 0.0419864 0.0209932 0.999780i \(-0.493317\pi\)
0.0209932 + 0.999780i \(0.493317\pi\)
\(522\) −6.80630 −0.297904
\(523\) 40.3532 1.76452 0.882260 0.470762i \(-0.156021\pi\)
0.882260 + 0.470762i \(0.156021\pi\)
\(524\) −21.3634 −0.933262
\(525\) −2.58732 −0.112920
\(526\) 15.0413 0.655831
\(527\) 34.4465 1.50051
\(528\) 2.86410 0.124644
\(529\) 5.30166 0.230507
\(530\) 14.7695 0.641544
\(531\) 0.810529 0.0351739
\(532\) −4.34711 −0.188471
\(533\) −1.89754 −0.0821918
\(534\) −4.89203 −0.211699
\(535\) 29.6774 1.28307
\(536\) −3.72543 −0.160914
\(537\) −23.5313 −1.01545
\(538\) −0.347738 −0.0149920
\(539\) 2.86410 0.123365
\(540\) 1.55328 0.0668426
\(541\) 31.8764 1.37047 0.685236 0.728321i \(-0.259699\pi\)
0.685236 + 0.728321i \(0.259699\pi\)
\(542\) 21.8338 0.937843
\(543\) 2.50296 0.107413
\(544\) −4.13308 −0.177205
\(545\) 19.2130 0.822995
\(546\) 1.19533 0.0511553
\(547\) −0.0898509 −0.00384175 −0.00192087 0.999998i \(-0.500611\pi\)
−0.00192087 + 0.999998i \(0.500611\pi\)
\(548\) 1.68404 0.0719388
\(549\) −0.00358361 −0.000152945 0
\(550\) 7.41033 0.315978
\(551\) −29.5877 −1.26048
\(552\) −5.31993 −0.226431
\(553\) 13.1530 0.559321
\(554\) −2.45754 −0.104411
\(555\) −11.6527 −0.494630
\(556\) 2.96087 0.125569
\(557\) −1.06503 −0.0451266 −0.0225633 0.999745i \(-0.507183\pi\)
−0.0225633 + 0.999745i \(0.507183\pi\)
\(558\) −8.33433 −0.352820
\(559\) −6.28989 −0.266034
\(560\) 1.55328 0.0656381
\(561\) 11.8375 0.499782
\(562\) 4.17887 0.176275
\(563\) 1.32372 0.0557882 0.0278941 0.999611i \(-0.491120\pi\)
0.0278941 + 0.999611i \(0.491120\pi\)
\(564\) 4.76012 0.200437
\(565\) 16.7526 0.704788
\(566\) 11.7209 0.492667
\(567\) 1.00000 0.0419961
\(568\) 14.4977 0.608310
\(569\) −17.2551 −0.723370 −0.361685 0.932300i \(-0.617798\pi\)
−0.361685 + 0.932300i \(0.617798\pi\)
\(570\) 6.75228 0.282822
\(571\) 21.5220 0.900670 0.450335 0.892860i \(-0.351305\pi\)
0.450335 + 0.892860i \(0.351305\pi\)
\(572\) −3.42353 −0.143145
\(573\) 1.00000 0.0417756
\(574\) −1.58747 −0.0662597
\(575\) −13.7644 −0.574013
\(576\) 1.00000 0.0416667
\(577\) −24.6862 −1.02770 −0.513850 0.857880i \(-0.671781\pi\)
−0.513850 + 0.857880i \(0.671781\pi\)
\(578\) −0.0823856 −0.00342679
\(579\) 2.11787 0.0880155
\(580\) 10.5721 0.438982
\(581\) 3.27581 0.135903
\(582\) 4.85955 0.201435
\(583\) −27.2334 −1.12789
\(584\) 10.2295 0.423299
\(585\) −1.85668 −0.0767642
\(586\) 5.34331 0.220730
\(587\) 29.8961 1.23394 0.616972 0.786985i \(-0.288359\pi\)
0.616972 + 0.786985i \(0.288359\pi\)
\(588\) 1.00000 0.0412393
\(589\) −36.2302 −1.49284
\(590\) −1.25898 −0.0518313
\(591\) −18.5010 −0.761031
\(592\) −7.50201 −0.308331
\(593\) −36.7644 −1.50973 −0.754867 0.655878i \(-0.772298\pi\)
−0.754867 + 0.655878i \(0.772298\pi\)
\(594\) −2.86410 −0.117515
\(595\) 6.41984 0.263188
\(596\) −10.7462 −0.440180
\(597\) 16.1068 0.659208
\(598\) 6.35906 0.260041
\(599\) −33.8172 −1.38173 −0.690866 0.722983i \(-0.742771\pi\)
−0.690866 + 0.722983i \(0.742771\pi\)
\(600\) 2.58732 0.105627
\(601\) 16.9016 0.689429 0.344715 0.938708i \(-0.387976\pi\)
0.344715 + 0.938708i \(0.387976\pi\)
\(602\) −5.26207 −0.214466
\(603\) 3.72543 0.151711
\(604\) 8.98373 0.365543
\(605\) −4.34446 −0.176627
\(606\) 2.55589 0.103826
\(607\) −26.5600 −1.07804 −0.539019 0.842293i \(-0.681205\pi\)
−0.539019 + 0.842293i \(0.681205\pi\)
\(608\) 4.34711 0.176299
\(609\) 6.80630 0.275805
\(610\) 0.00556635 0.000225375 0
\(611\) −5.68990 −0.230189
\(612\) 4.13308 0.167070
\(613\) −43.8711 −1.77194 −0.885969 0.463745i \(-0.846505\pi\)
−0.885969 + 0.463745i \(0.846505\pi\)
\(614\) −12.1496 −0.490317
\(615\) 2.46578 0.0994300
\(616\) −2.86410 −0.115398
\(617\) 7.23786 0.291385 0.145693 0.989330i \(-0.453459\pi\)
0.145693 + 0.989330i \(0.453459\pi\)
\(618\) 4.03327 0.162242
\(619\) −8.61387 −0.346221 −0.173110 0.984902i \(-0.555382\pi\)
−0.173110 + 0.984902i \(0.555382\pi\)
\(620\) 12.9455 0.519906
\(621\) 5.31993 0.213481
\(622\) 16.5576 0.663900
\(623\) 4.89203 0.195995
\(624\) −1.19533 −0.0478514
\(625\) −5.36918 −0.214767
\(626\) −19.8617 −0.793832
\(627\) −12.4505 −0.497227
\(628\) 9.02424 0.360106
\(629\) −31.0064 −1.23631
\(630\) −1.55328 −0.0618842
\(631\) 29.7736 1.18527 0.592633 0.805472i \(-0.298088\pi\)
0.592633 + 0.805472i \(0.298088\pi\)
\(632\) −13.1530 −0.523196
\(633\) −12.7388 −0.506324
\(634\) −17.5364 −0.696459
\(635\) 1.33505 0.0529800
\(636\) −9.50856 −0.377039
\(637\) −1.19533 −0.0473606
\(638\) −19.4939 −0.771770
\(639\) −14.4977 −0.573521
\(640\) −1.55328 −0.0613988
\(641\) −38.6310 −1.52583 −0.762916 0.646498i \(-0.776233\pi\)
−0.762916 + 0.646498i \(0.776233\pi\)
\(642\) −19.1063 −0.754064
\(643\) −50.6263 −1.99651 −0.998253 0.0590920i \(-0.981179\pi\)
−0.998253 + 0.0590920i \(0.981179\pi\)
\(644\) 5.31993 0.209635
\(645\) 8.17347 0.321830
\(646\) 17.9670 0.706901
\(647\) 43.4769 1.70925 0.854626 0.519244i \(-0.173786\pi\)
0.854626 + 0.519244i \(0.173786\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 2.32143 0.0911242
\(650\) −3.09269 −0.121305
\(651\) 8.33433 0.326648
\(652\) 19.3344 0.757193
\(653\) −7.90470 −0.309335 −0.154667 0.987967i \(-0.549431\pi\)
−0.154667 + 0.987967i \(0.549431\pi\)
\(654\) −12.3693 −0.483678
\(655\) −33.1833 −1.29658
\(656\) 1.58747 0.0619802
\(657\) −10.2295 −0.399090
\(658\) −4.76012 −0.185569
\(659\) −17.9527 −0.699340 −0.349670 0.936873i \(-0.613706\pi\)
−0.349670 + 0.936873i \(0.613706\pi\)
\(660\) 4.44874 0.173167
\(661\) −7.50028 −0.291727 −0.145863 0.989305i \(-0.546596\pi\)
−0.145863 + 0.989305i \(0.546596\pi\)
\(662\) −6.91301 −0.268682
\(663\) −4.94039 −0.191869
\(664\) −3.27581 −0.127126
\(665\) −6.75228 −0.261842
\(666\) 7.50201 0.290697
\(667\) 36.2090 1.40202
\(668\) −4.60141 −0.178034
\(669\) 24.3664 0.942058
\(670\) −5.78664 −0.223557
\(671\) −0.0102638 −0.000396230 0
\(672\) −1.00000 −0.0385758
\(673\) 9.35453 0.360591 0.180295 0.983613i \(-0.442295\pi\)
0.180295 + 0.983613i \(0.442295\pi\)
\(674\) 23.7749 0.915774
\(675\) −2.58732 −0.0995860
\(676\) −11.5712 −0.445046
\(677\) 41.8837 1.60972 0.804861 0.593464i \(-0.202240\pi\)
0.804861 + 0.593464i \(0.202240\pi\)
\(678\) −10.7853 −0.414208
\(679\) −4.85955 −0.186492
\(680\) −6.41984 −0.246190
\(681\) −12.6366 −0.484235
\(682\) −23.8703 −0.914041
\(683\) −15.4568 −0.591438 −0.295719 0.955275i \(-0.595559\pi\)
−0.295719 + 0.955275i \(0.595559\pi\)
\(684\) −4.34711 −0.166216
\(685\) 2.61579 0.0999443
\(686\) −1.00000 −0.0381802
\(687\) 13.9198 0.531073
\(688\) 5.26207 0.200614
\(689\) 11.3658 0.433004
\(690\) −8.26334 −0.314580
\(691\) 27.0170 1.02777 0.513887 0.857858i \(-0.328205\pi\)
0.513887 + 0.857858i \(0.328205\pi\)
\(692\) 13.2466 0.503559
\(693\) 2.86410 0.108798
\(694\) −20.6915 −0.785440
\(695\) 4.59905 0.174452
\(696\) −6.80630 −0.257992
\(697\) 6.56114 0.248521
\(698\) 13.6257 0.515740
\(699\) 5.26643 0.199195
\(700\) −2.58732 −0.0977915
\(701\) 28.6484 1.08204 0.541018 0.841011i \(-0.318039\pi\)
0.541018 + 0.841011i \(0.318039\pi\)
\(702\) 1.19533 0.0451147
\(703\) 32.6120 1.22999
\(704\) 2.86410 0.107945
\(705\) 7.39381 0.278467
\(706\) −19.9014 −0.748997
\(707\) −2.55589 −0.0961242
\(708\) 0.810529 0.0304615
\(709\) 47.8566 1.79729 0.898647 0.438672i \(-0.144551\pi\)
0.898647 + 0.438672i \(0.144551\pi\)
\(710\) 22.5190 0.845124
\(711\) 13.1530 0.493274
\(712\) −4.89203 −0.183337
\(713\) 44.3380 1.66047
\(714\) −4.13308 −0.154677
\(715\) −5.31770 −0.198871
\(716\) −23.5313 −0.879405
\(717\) −18.1064 −0.676194
\(718\) 29.8262 1.11310
\(719\) −31.0095 −1.15646 −0.578229 0.815874i \(-0.696256\pi\)
−0.578229 + 0.815874i \(0.696256\pi\)
\(720\) 1.55328 0.0578874
\(721\) −4.03327 −0.150207
\(722\) 0.102635 0.00381968
\(723\) 29.4501 1.09526
\(724\) 2.50296 0.0930220
\(725\) −17.6101 −0.654021
\(726\) 2.79696 0.103805
\(727\) −16.8622 −0.625383 −0.312691 0.949855i \(-0.601231\pi\)
−0.312691 + 0.949855i \(0.601231\pi\)
\(728\) 1.19533 0.0443018
\(729\) 1.00000 0.0370370
\(730\) 15.8893 0.588088
\(731\) 21.7486 0.804400
\(732\) −0.00358361 −0.000132454 0
\(733\) −22.0591 −0.814770 −0.407385 0.913256i \(-0.633559\pi\)
−0.407385 + 0.913256i \(0.633559\pi\)
\(734\) 15.6431 0.577396
\(735\) 1.55328 0.0572936
\(736\) −5.31993 −0.196095
\(737\) 10.6700 0.393034
\(738\) −1.58747 −0.0584355
\(739\) −6.75125 −0.248349 −0.124174 0.992260i \(-0.539628\pi\)
−0.124174 + 0.992260i \(0.539628\pi\)
\(740\) −11.6527 −0.428363
\(741\) 5.19622 0.190888
\(742\) 9.50856 0.349070
\(743\) 46.9249 1.72151 0.860754 0.509022i \(-0.169993\pi\)
0.860754 + 0.509022i \(0.169993\pi\)
\(744\) −8.33433 −0.305551
\(745\) −16.6918 −0.611540
\(746\) −10.4280 −0.381795
\(747\) 3.27581 0.119856
\(748\) 11.8375 0.432824
\(749\) 19.1063 0.698127
\(750\) 11.7852 0.430336
\(751\) 11.2306 0.409809 0.204905 0.978782i \(-0.434312\pi\)
0.204905 + 0.978782i \(0.434312\pi\)
\(752\) 4.76012 0.173584
\(753\) 16.6181 0.605598
\(754\) 8.13575 0.296287
\(755\) 13.9542 0.507847
\(756\) 1.00000 0.0363696
\(757\) 49.8845 1.81308 0.906541 0.422117i \(-0.138713\pi\)
0.906541 + 0.422117i \(0.138713\pi\)
\(758\) 15.5075 0.563258
\(759\) 15.2368 0.553060
\(760\) 6.75228 0.244931
\(761\) 15.7307 0.570236 0.285118 0.958492i \(-0.407967\pi\)
0.285118 + 0.958492i \(0.407967\pi\)
\(762\) −0.859505 −0.0311366
\(763\) 12.3693 0.447799
\(764\) 1.00000 0.0361787
\(765\) 6.41984 0.232110
\(766\) 17.7275 0.640522
\(767\) −0.968847 −0.0349830
\(768\) 1.00000 0.0360844
\(769\) 28.3474 1.02223 0.511117 0.859511i \(-0.329232\pi\)
0.511117 + 0.859511i \(0.329232\pi\)
\(770\) −4.44874 −0.160322
\(771\) 25.3033 0.911277
\(772\) 2.11787 0.0762237
\(773\) −38.8034 −1.39566 −0.697830 0.716263i \(-0.745851\pi\)
−0.697830 + 0.716263i \(0.745851\pi\)
\(774\) −5.26207 −0.189141
\(775\) −21.5636 −0.774586
\(776\) 4.85955 0.174448
\(777\) −7.50201 −0.269133
\(778\) −26.9359 −0.965697
\(779\) −6.90090 −0.247250
\(780\) −1.85668 −0.0664797
\(781\) −41.5228 −1.48580
\(782\) −21.9877 −0.786279
\(783\) 6.80630 0.243237
\(784\) 1.00000 0.0357143
\(785\) 14.0172 0.500295
\(786\) 21.3634 0.762006
\(787\) 2.67641 0.0954039 0.0477019 0.998862i \(-0.484810\pi\)
0.0477019 + 0.998862i \(0.484810\pi\)
\(788\) −18.5010 −0.659072
\(789\) −15.0413 −0.535484
\(790\) −20.4302 −0.726875
\(791\) 10.7853 0.383482
\(792\) −2.86410 −0.101771
\(793\) 0.00428358 0.000152115 0
\(794\) 24.9829 0.886610
\(795\) −14.7695 −0.523819
\(796\) 16.1068 0.570891
\(797\) −4.33781 −0.153653 −0.0768266 0.997044i \(-0.524479\pi\)
−0.0768266 + 0.997044i \(0.524479\pi\)
\(798\) 4.34711 0.153886
\(799\) 19.6740 0.696016
\(800\) 2.58732 0.0914755
\(801\) 4.89203 0.172851
\(802\) −5.32228 −0.187936
\(803\) −29.2982 −1.03391
\(804\) 3.72543 0.131386
\(805\) 8.26334 0.291245
\(806\) 9.96224 0.350905
\(807\) 0.347738 0.0122410
\(808\) 2.55589 0.0899160
\(809\) −21.4548 −0.754310 −0.377155 0.926150i \(-0.623098\pi\)
−0.377155 + 0.926150i \(0.623098\pi\)
\(810\) −1.55328 −0.0545767
\(811\) −20.5867 −0.722898 −0.361449 0.932392i \(-0.617718\pi\)
−0.361449 + 0.932392i \(0.617718\pi\)
\(812\) 6.80630 0.238854
\(813\) −21.8338 −0.765746
\(814\) 21.4865 0.753100
\(815\) 30.0317 1.05197
\(816\) 4.13308 0.144687
\(817\) −22.8748 −0.800287
\(818\) −14.7703 −0.516432
\(819\) −1.19533 −0.0417681
\(820\) 2.46578 0.0861089
\(821\) 10.6861 0.372946 0.186473 0.982460i \(-0.440294\pi\)
0.186473 + 0.982460i \(0.440294\pi\)
\(822\) −1.68404 −0.0587377
\(823\) 55.2231 1.92496 0.962478 0.271361i \(-0.0874738\pi\)
0.962478 + 0.271361i \(0.0874738\pi\)
\(824\) 4.03327 0.140506
\(825\) −7.41033 −0.257995
\(826\) −0.810529 −0.0282019
\(827\) −18.1488 −0.631094 −0.315547 0.948910i \(-0.602188\pi\)
−0.315547 + 0.948910i \(0.602188\pi\)
\(828\) 5.31993 0.184880
\(829\) 27.9310 0.970083 0.485042 0.874491i \(-0.338804\pi\)
0.485042 + 0.874491i \(0.338804\pi\)
\(830\) −5.08825 −0.176616
\(831\) 2.45754 0.0852512
\(832\) −1.19533 −0.0414405
\(833\) 4.13308 0.143203
\(834\) −2.96087 −0.102526
\(835\) −7.14729 −0.247342
\(836\) −12.4505 −0.430611
\(837\) 8.33433 0.288076
\(838\) −18.8559 −0.651366
\(839\) −26.9531 −0.930524 −0.465262 0.885173i \(-0.654040\pi\)
−0.465262 + 0.885173i \(0.654040\pi\)
\(840\) −1.55328 −0.0535933
\(841\) 17.3257 0.597437
\(842\) −11.9762 −0.412728
\(843\) −4.17887 −0.143928
\(844\) −12.7388 −0.438489
\(845\) −17.9733 −0.618301
\(846\) −4.76012 −0.163656
\(847\) −2.79696 −0.0961045
\(848\) −9.50856 −0.326525
\(849\) −11.7209 −0.402261
\(850\) 10.6936 0.366788
\(851\) −39.9101 −1.36810
\(852\) −14.4977 −0.496683
\(853\) 35.8583 1.22776 0.613882 0.789398i \(-0.289607\pi\)
0.613882 + 0.789398i \(0.289607\pi\)
\(854\) 0.00358361 0.000122629 0
\(855\) −6.75228 −0.230923
\(856\) −19.1063 −0.653038
\(857\) 15.0412 0.513799 0.256899 0.966438i \(-0.417299\pi\)
0.256899 + 0.966438i \(0.417299\pi\)
\(858\) 3.42353 0.116877
\(859\) −5.55659 −0.189589 −0.0947943 0.995497i \(-0.530219\pi\)
−0.0947943 + 0.995497i \(0.530219\pi\)
\(860\) 8.17347 0.278713
\(861\) 1.58747 0.0541008
\(862\) −33.6954 −1.14767
\(863\) 8.47162 0.288377 0.144189 0.989550i \(-0.453943\pi\)
0.144189 + 0.989550i \(0.453943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 20.5756 0.699592
\(866\) 25.9712 0.882537
\(867\) 0.0823856 0.00279796
\(868\) 8.33433 0.282885
\(869\) 37.6713 1.27791
\(870\) −10.5721 −0.358427
\(871\) −4.45311 −0.150888
\(872\) −12.3693 −0.418878
\(873\) −4.85955 −0.164471
\(874\) 23.1263 0.782259
\(875\) −11.7852 −0.398414
\(876\) −10.2295 −0.345622
\(877\) 49.4745 1.67063 0.835317 0.549769i \(-0.185284\pi\)
0.835317 + 0.549769i \(0.185284\pi\)
\(878\) 10.6796 0.360420
\(879\) −5.34331 −0.180225
\(880\) 4.44874 0.149967
\(881\) 26.0956 0.879184 0.439592 0.898198i \(-0.355123\pi\)
0.439592 + 0.898198i \(0.355123\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 52.2438 1.75814 0.879072 0.476689i \(-0.158163\pi\)
0.879072 + 0.476689i \(0.158163\pi\)
\(884\) −4.94039 −0.166163
\(885\) 1.25898 0.0423201
\(886\) −28.3258 −0.951622
\(887\) 37.2719 1.25147 0.625733 0.780037i \(-0.284800\pi\)
0.625733 + 0.780037i \(0.284800\pi\)
\(888\) 7.50201 0.251751
\(889\) 0.859505 0.0288269
\(890\) −7.59869 −0.254709
\(891\) 2.86410 0.0959508
\(892\) 24.3664 0.815846
\(893\) −20.6928 −0.692458
\(894\) 10.7462 0.359405
\(895\) −36.5507 −1.22175
\(896\) −1.00000 −0.0334077
\(897\) −6.35906 −0.212323
\(898\) 37.2750 1.24388
\(899\) 56.7259 1.89191
\(900\) −2.58732 −0.0862440
\(901\) −39.2997 −1.30926
\(902\) −4.54666 −0.151387
\(903\) 5.26207 0.175111
\(904\) −10.7853 −0.358714
\(905\) 3.88781 0.129235
\(906\) −8.98373 −0.298464
\(907\) 18.7648 0.623075 0.311538 0.950234i \(-0.399156\pi\)
0.311538 + 0.950234i \(0.399156\pi\)
\(908\) −12.6366 −0.419359
\(909\) −2.55589 −0.0847736
\(910\) 1.85668 0.0615483
\(911\) −7.10704 −0.235467 −0.117733 0.993045i \(-0.537563\pi\)
−0.117733 + 0.993045i \(0.537563\pi\)
\(912\) −4.34711 −0.143947
\(913\) 9.38223 0.310506
\(914\) 23.9922 0.793592
\(915\) −0.00556635 −0.000184018 0
\(916\) 13.9198 0.459922
\(917\) −21.3634 −0.705480
\(918\) −4.13308 −0.136412
\(919\) −37.6799 −1.24295 −0.621473 0.783436i \(-0.713465\pi\)
−0.621473 + 0.783436i \(0.713465\pi\)
\(920\) −8.26334 −0.272434
\(921\) 12.1496 0.400342
\(922\) −32.3022 −1.06382
\(923\) 17.3295 0.570408
\(924\) 2.86410 0.0942218
\(925\) 19.4101 0.638200
\(926\) 18.2217 0.598804
\(927\) −4.03327 −0.132470
\(928\) −6.80630 −0.223428
\(929\) 40.7813 1.33799 0.668995 0.743267i \(-0.266725\pi\)
0.668995 + 0.743267i \(0.266725\pi\)
\(930\) −12.9455 −0.424501
\(931\) −4.34711 −0.142471
\(932\) 5.26643 0.172508
\(933\) −16.5576 −0.542072
\(934\) 19.4182 0.635382
\(935\) 18.3870 0.601320
\(936\) 1.19533 0.0390705
\(937\) −53.1076 −1.73495 −0.867475 0.497480i \(-0.834259\pi\)
−0.867475 + 0.497480i \(0.834259\pi\)
\(938\) −3.72543 −0.121640
\(939\) 19.8617 0.648162
\(940\) 7.39381 0.241159
\(941\) −9.22469 −0.300716 −0.150358 0.988632i \(-0.548043\pi\)
−0.150358 + 0.988632i \(0.548043\pi\)
\(942\) −9.02424 −0.294026
\(943\) 8.44522 0.275014
\(944\) 0.810529 0.0263805
\(945\) 1.55328 0.0505282
\(946\) −15.0711 −0.490002
\(947\) −21.1285 −0.686583 −0.343292 0.939229i \(-0.611542\pi\)
−0.343292 + 0.939229i \(0.611542\pi\)
\(948\) 13.1530 0.427188
\(949\) 12.2276 0.396924
\(950\) −11.2474 −0.364913
\(951\) 17.5364 0.568656
\(952\) −4.13308 −0.133954
\(953\) −0.300712 −0.00974102 −0.00487051 0.999988i \(-0.501550\pi\)
−0.00487051 + 0.999988i \(0.501550\pi\)
\(954\) 9.50856 0.307851
\(955\) 1.55328 0.0502630
\(956\) −18.1064 −0.585601
\(957\) 19.4939 0.630148
\(958\) 16.8579 0.544654
\(959\) 1.68404 0.0543806
\(960\) 1.55328 0.0501319
\(961\) 38.4610 1.24068
\(962\) −8.96735 −0.289119
\(963\) 19.1063 0.615691
\(964\) 29.4501 0.948525
\(965\) 3.28964 0.105897
\(966\) −5.31993 −0.171166
\(967\) 46.8460 1.50647 0.753233 0.657754i \(-0.228493\pi\)
0.753233 + 0.657754i \(0.228493\pi\)
\(968\) 2.79696 0.0898976
\(969\) −17.9670 −0.577182
\(970\) 7.54824 0.242359
\(971\) −31.3331 −1.00553 −0.502764 0.864424i \(-0.667683\pi\)
−0.502764 + 0.864424i \(0.667683\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.96087 0.0949210
\(974\) 27.2722 0.873857
\(975\) 3.09269 0.0990454
\(976\) −0.00358361 −0.000114709 0
\(977\) −37.1856 −1.18967 −0.594836 0.803847i \(-0.702783\pi\)
−0.594836 + 0.803847i \(0.702783\pi\)
\(978\) −19.3344 −0.618246
\(979\) 14.0112 0.447801
\(980\) 1.55328 0.0496177
\(981\) 12.3693 0.394922
\(982\) −39.7906 −1.26977
\(983\) −19.8750 −0.633915 −0.316958 0.948440i \(-0.602661\pi\)
−0.316958 + 0.948440i \(0.602661\pi\)
\(984\) −1.58747 −0.0506067
\(985\) −28.7373 −0.915646
\(986\) −28.1310 −0.895873
\(987\) 4.76012 0.151516
\(988\) 5.19622 0.165314
\(989\) 27.9938 0.890152
\(990\) −4.44874 −0.141390
\(991\) −32.2524 −1.02453 −0.512266 0.858827i \(-0.671194\pi\)
−0.512266 + 0.858827i \(0.671194\pi\)
\(992\) −8.33433 −0.264615
\(993\) 6.91301 0.219378
\(994\) 14.4977 0.459839
\(995\) 25.0184 0.793136
\(996\) 3.27581 0.103798
\(997\) −7.87706 −0.249469 −0.124734 0.992190i \(-0.539808\pi\)
−0.124734 + 0.992190i \(0.539808\pi\)
\(998\) −14.3592 −0.454532
\(999\) −7.50201 −0.237353
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 8022.2.a.z.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8022.2.a.z.1.11 15 1.1 even 1 trivial