Properties

Label 503.2.a.f.1.16
Level $503$
Weight $2$
Character 503.1
Self dual yes
Analytic conductor $4.016$
Analytic rank $0$
Dimension $26$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [503,2,Mod(1,503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("503.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.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: \(4.01647522167\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 503.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.738482 q^{2} +2.27911 q^{3} -1.45464 q^{4} +3.79116 q^{5} +1.68308 q^{6} +4.03115 q^{7} -2.55119 q^{8} +2.19436 q^{9} +O(q^{10})\) \(q+0.738482 q^{2} +2.27911 q^{3} -1.45464 q^{4} +3.79116 q^{5} +1.68308 q^{6} +4.03115 q^{7} -2.55119 q^{8} +2.19436 q^{9} +2.79970 q^{10} -4.62241 q^{11} -3.31530 q^{12} -3.86363 q^{13} +2.97693 q^{14} +8.64048 q^{15} +1.02528 q^{16} +1.73504 q^{17} +1.62049 q^{18} -4.30211 q^{19} -5.51479 q^{20} +9.18744 q^{21} -3.41357 q^{22} -7.16718 q^{23} -5.81446 q^{24} +9.37288 q^{25} -2.85322 q^{26} -1.83615 q^{27} -5.86389 q^{28} -3.40299 q^{29} +6.38084 q^{30} +1.29624 q^{31} +5.85954 q^{32} -10.5350 q^{33} +1.28130 q^{34} +15.2827 q^{35} -3.19201 q^{36} +11.4689 q^{37} -3.17703 q^{38} -8.80564 q^{39} -9.67197 q^{40} -6.76299 q^{41} +6.78476 q^{42} +1.28015 q^{43} +6.72397 q^{44} +8.31915 q^{45} -5.29283 q^{46} +8.34021 q^{47} +2.33673 q^{48} +9.25015 q^{49} +6.92170 q^{50} +3.95436 q^{51} +5.62020 q^{52} +9.55893 q^{53} -1.35596 q^{54} -17.5243 q^{55} -10.2842 q^{56} -9.80499 q^{57} -2.51305 q^{58} +3.24630 q^{59} -12.5688 q^{60} -12.3562 q^{61} +0.957253 q^{62} +8.84578 q^{63} +2.27660 q^{64} -14.6476 q^{65} -7.77991 q^{66} -12.1005 q^{67} -2.52387 q^{68} -16.3348 q^{69} +11.2860 q^{70} +7.17544 q^{71} -5.59823 q^{72} +4.96822 q^{73} +8.46958 q^{74} +21.3619 q^{75} +6.25804 q^{76} -18.6336 q^{77} -6.50281 q^{78} +10.6927 q^{79} +3.88700 q^{80} -10.7679 q^{81} -4.99435 q^{82} -5.83442 q^{83} -13.3645 q^{84} +6.57783 q^{85} +0.945366 q^{86} -7.75580 q^{87} +11.7927 q^{88} +6.64931 q^{89} +6.14354 q^{90} -15.5748 q^{91} +10.4257 q^{92} +2.95429 q^{93} +6.15909 q^{94} -16.3100 q^{95} +13.3545 q^{96} -10.1113 q^{97} +6.83107 q^{98} -10.1432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q + 4 q^{2} + 4 q^{3} + 36 q^{4} + 9 q^{5} - 4 q^{6} + 11 q^{7} + 18 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q + 4 q^{2} + 4 q^{3} + 36 q^{4} + 9 q^{5} - 4 q^{6} + 11 q^{7} + 18 q^{8} + 42 q^{9} + 4 q^{10} - 17 q^{11} + 19 q^{12} + 14 q^{13} + q^{14} + 18 q^{15} + 48 q^{16} + 17 q^{17} - 10 q^{18} - 22 q^{19} - 19 q^{20} - 16 q^{21} + 38 q^{22} + 27 q^{23} - 9 q^{24} + 93 q^{25} + q^{26} + 31 q^{27} - 9 q^{28} + 13 q^{29} - 28 q^{30} + 26 q^{31} + 5 q^{32} + 6 q^{33} - 32 q^{34} - 22 q^{35} + 52 q^{36} + 55 q^{37} - 24 q^{38} - 15 q^{39} - 7 q^{40} + 24 q^{41} - 50 q^{42} + 20 q^{43} - 27 q^{44} - 8 q^{45} + 6 q^{46} - 25 q^{47} + 29 q^{48} + 65 q^{49} - 16 q^{50} + 7 q^{51} + 32 q^{52} + 30 q^{53} - 82 q^{54} + 25 q^{55} + 3 q^{56} + 9 q^{57} + 58 q^{58} - 26 q^{59} - 68 q^{60} + 15 q^{61} - 12 q^{62} - 19 q^{63} + 44 q^{64} + 20 q^{65} - 55 q^{66} - 20 q^{67} - 4 q^{68} - 27 q^{69} + 2 q^{70} - 35 q^{71} - 26 q^{72} + 38 q^{73} - 59 q^{74} + 2 q^{75} - 42 q^{76} - 6 q^{77} - 47 q^{78} + 21 q^{79} - 100 q^{80} + 70 q^{81} - 59 q^{82} - 48 q^{83} - 116 q^{84} + 6 q^{85} - 7 q^{86} - 9 q^{87} + 106 q^{88} - 5 q^{89} - 118 q^{90} - 24 q^{91} + 26 q^{92} - 8 q^{93} - 22 q^{94} + 43 q^{95} - 100 q^{96} + 142 q^{97} - 38 q^{98} - 50 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.738482 0.522186 0.261093 0.965314i \(-0.415917\pi\)
0.261093 + 0.965314i \(0.415917\pi\)
\(3\) 2.27911 1.31585 0.657923 0.753085i \(-0.271435\pi\)
0.657923 + 0.753085i \(0.271435\pi\)
\(4\) −1.45464 −0.727322
\(5\) 3.79116 1.69546 0.847729 0.530430i \(-0.177970\pi\)
0.847729 + 0.530430i \(0.177970\pi\)
\(6\) 1.68308 0.687116
\(7\) 4.03115 1.52363 0.761815 0.647794i \(-0.224308\pi\)
0.761815 + 0.647794i \(0.224308\pi\)
\(8\) −2.55119 −0.901983
\(9\) 2.19436 0.731452
\(10\) 2.79970 0.885343
\(11\) −4.62241 −1.39371 −0.696855 0.717212i \(-0.745418\pi\)
−0.696855 + 0.717212i \(0.745418\pi\)
\(12\) −3.31530 −0.957044
\(13\) −3.86363 −1.07158 −0.535788 0.844352i \(-0.679986\pi\)
−0.535788 + 0.844352i \(0.679986\pi\)
\(14\) 2.97693 0.795618
\(15\) 8.64048 2.23096
\(16\) 1.02528 0.256320
\(17\) 1.73504 0.420810 0.210405 0.977614i \(-0.432522\pi\)
0.210405 + 0.977614i \(0.432522\pi\)
\(18\) 1.62049 0.381954
\(19\) −4.30211 −0.986971 −0.493485 0.869754i \(-0.664277\pi\)
−0.493485 + 0.869754i \(0.664277\pi\)
\(20\) −5.51479 −1.23314
\(21\) 9.18744 2.00486
\(22\) −3.41357 −0.727775
\(23\) −7.16718 −1.49446 −0.747230 0.664566i \(-0.768617\pi\)
−0.747230 + 0.664566i \(0.768617\pi\)
\(24\) −5.81446 −1.18687
\(25\) 9.37288 1.87458
\(26\) −2.85322 −0.559562
\(27\) −1.83615 −0.353368
\(28\) −5.86389 −1.10817
\(29\) −3.40299 −0.631919 −0.315960 0.948773i \(-0.602326\pi\)
−0.315960 + 0.948773i \(0.602326\pi\)
\(30\) 6.38084 1.16498
\(31\) 1.29624 0.232812 0.116406 0.993202i \(-0.462863\pi\)
0.116406 + 0.993202i \(0.462863\pi\)
\(32\) 5.85954 1.03583
\(33\) −10.5350 −1.83391
\(34\) 1.28130 0.219741
\(35\) 15.2827 2.58325
\(36\) −3.19201 −0.532002
\(37\) 11.4689 1.88548 0.942738 0.333535i \(-0.108242\pi\)
0.942738 + 0.333535i \(0.108242\pi\)
\(38\) −3.17703 −0.515382
\(39\) −8.80564 −1.41003
\(40\) −9.67197 −1.52927
\(41\) −6.76299 −1.05620 −0.528101 0.849182i \(-0.677096\pi\)
−0.528101 + 0.849182i \(0.677096\pi\)
\(42\) 6.78476 1.04691
\(43\) 1.28015 0.195221 0.0976104 0.995225i \(-0.468880\pi\)
0.0976104 + 0.995225i \(0.468880\pi\)
\(44\) 6.72397 1.01368
\(45\) 8.31915 1.24015
\(46\) −5.29283 −0.780385
\(47\) 8.34021 1.21654 0.608272 0.793728i \(-0.291863\pi\)
0.608272 + 0.793728i \(0.291863\pi\)
\(48\) 2.33673 0.337278
\(49\) 9.25015 1.32145
\(50\) 6.92170 0.978876
\(51\) 3.95436 0.553722
\(52\) 5.62020 0.779382
\(53\) 9.55893 1.31302 0.656510 0.754317i \(-0.272032\pi\)
0.656510 + 0.754317i \(0.272032\pi\)
\(54\) −1.35596 −0.184523
\(55\) −17.5243 −2.36298
\(56\) −10.2842 −1.37429
\(57\) −9.80499 −1.29870
\(58\) −2.51305 −0.329979
\(59\) 3.24630 0.422632 0.211316 0.977418i \(-0.432225\pi\)
0.211316 + 0.977418i \(0.432225\pi\)
\(60\) −12.5688 −1.62263
\(61\) −12.3562 −1.58205 −0.791025 0.611784i \(-0.790452\pi\)
−0.791025 + 0.611784i \(0.790452\pi\)
\(62\) 0.957253 0.121571
\(63\) 8.84578 1.11446
\(64\) 2.27660 0.284575
\(65\) −14.6476 −1.81681
\(66\) −7.77991 −0.957641
\(67\) −12.1005 −1.47832 −0.739158 0.673532i \(-0.764777\pi\)
−0.739158 + 0.673532i \(0.764777\pi\)
\(68\) −2.52387 −0.306065
\(69\) −16.3348 −1.96648
\(70\) 11.2860 1.34894
\(71\) 7.17544 0.851568 0.425784 0.904825i \(-0.359998\pi\)
0.425784 + 0.904825i \(0.359998\pi\)
\(72\) −5.59823 −0.659757
\(73\) 4.96822 0.581486 0.290743 0.956801i \(-0.406098\pi\)
0.290743 + 0.956801i \(0.406098\pi\)
\(74\) 8.46958 0.984568
\(75\) 21.3619 2.46665
\(76\) 6.25804 0.717846
\(77\) −18.6336 −2.12350
\(78\) −6.50281 −0.736298
\(79\) 10.6927 1.20303 0.601514 0.798862i \(-0.294564\pi\)
0.601514 + 0.798862i \(0.294564\pi\)
\(80\) 3.88700 0.434579
\(81\) −10.7679 −1.19643
\(82\) −4.99435 −0.551533
\(83\) −5.83442 −0.640411 −0.320206 0.947348i \(-0.603752\pi\)
−0.320206 + 0.947348i \(0.603752\pi\)
\(84\) −13.3645 −1.45818
\(85\) 6.57783 0.713466
\(86\) 0.945366 0.101941
\(87\) −7.75580 −0.831509
\(88\) 11.7927 1.25710
\(89\) 6.64931 0.704826 0.352413 0.935845i \(-0.385361\pi\)
0.352413 + 0.935845i \(0.385361\pi\)
\(90\) 6.14354 0.647586
\(91\) −15.5748 −1.63269
\(92\) 10.4257 1.08695
\(93\) 2.95429 0.306345
\(94\) 6.15909 0.635262
\(95\) −16.3100 −1.67337
\(96\) 13.3545 1.36299
\(97\) −10.1113 −1.02665 −0.513323 0.858195i \(-0.671586\pi\)
−0.513323 + 0.858195i \(0.671586\pi\)
\(98\) 6.83107 0.690042
\(99\) −10.1432 −1.01943
\(100\) −13.6342 −1.36342
\(101\) 5.70553 0.567721 0.283861 0.958866i \(-0.408385\pi\)
0.283861 + 0.958866i \(0.408385\pi\)
\(102\) 2.92023 0.289145
\(103\) 9.24132 0.910575 0.455287 0.890345i \(-0.349537\pi\)
0.455287 + 0.890345i \(0.349537\pi\)
\(104\) 9.85685 0.966544
\(105\) 34.8310 3.39916
\(106\) 7.05910 0.685640
\(107\) −15.0186 −1.45190 −0.725951 0.687746i \(-0.758600\pi\)
−0.725951 + 0.687746i \(0.758600\pi\)
\(108\) 2.67095 0.257012
\(109\) −2.99448 −0.286819 −0.143409 0.989663i \(-0.545807\pi\)
−0.143409 + 0.989663i \(0.545807\pi\)
\(110\) −12.9414 −1.23391
\(111\) 26.1389 2.48100
\(112\) 4.13305 0.390537
\(113\) −2.05064 −0.192908 −0.0964539 0.995337i \(-0.530750\pi\)
−0.0964539 + 0.995337i \(0.530750\pi\)
\(114\) −7.24081 −0.678164
\(115\) −27.1719 −2.53379
\(116\) 4.95014 0.459609
\(117\) −8.47817 −0.783807
\(118\) 2.39733 0.220692
\(119\) 6.99422 0.641159
\(120\) −22.0435 −2.01229
\(121\) 10.3667 0.942427
\(122\) −9.12483 −0.826124
\(123\) −15.4136 −1.38980
\(124\) −1.88558 −0.169330
\(125\) 16.5783 1.48281
\(126\) 6.53245 0.581957
\(127\) −2.63530 −0.233845 −0.116923 0.993141i \(-0.537303\pi\)
−0.116923 + 0.993141i \(0.537303\pi\)
\(128\) −10.0378 −0.887228
\(129\) 2.91760 0.256881
\(130\) −10.8170 −0.948713
\(131\) −9.66184 −0.844159 −0.422080 0.906559i \(-0.638700\pi\)
−0.422080 + 0.906559i \(0.638700\pi\)
\(132\) 15.3247 1.33384
\(133\) −17.3424 −1.50378
\(134\) −8.93603 −0.771956
\(135\) −6.96114 −0.599120
\(136\) −4.42643 −0.379564
\(137\) 3.82672 0.326939 0.163469 0.986548i \(-0.447732\pi\)
0.163469 + 0.986548i \(0.447732\pi\)
\(138\) −12.0630 −1.02687
\(139\) 12.8713 1.09173 0.545863 0.837875i \(-0.316202\pi\)
0.545863 + 0.837875i \(0.316202\pi\)
\(140\) −22.2309 −1.87886
\(141\) 19.0083 1.60079
\(142\) 5.29893 0.444677
\(143\) 17.8593 1.49347
\(144\) 2.24983 0.187486
\(145\) −12.9013 −1.07139
\(146\) 3.66894 0.303644
\(147\) 21.0821 1.73883
\(148\) −16.6832 −1.37135
\(149\) 15.0108 1.22973 0.614867 0.788631i \(-0.289210\pi\)
0.614867 + 0.788631i \(0.289210\pi\)
\(150\) 15.7753 1.28805
\(151\) 7.14463 0.581422 0.290711 0.956811i \(-0.406108\pi\)
0.290711 + 0.956811i \(0.406108\pi\)
\(152\) 10.9755 0.890231
\(153\) 3.80731 0.307803
\(154\) −13.7606 −1.10886
\(155\) 4.91427 0.394724
\(156\) 12.8091 1.02555
\(157\) −3.67900 −0.293616 −0.146808 0.989165i \(-0.546900\pi\)
−0.146808 + 0.989165i \(0.546900\pi\)
\(158\) 7.89640 0.628204
\(159\) 21.7859 1.72773
\(160\) 22.2144 1.75620
\(161\) −28.8920 −2.27700
\(162\) −7.95188 −0.624758
\(163\) −6.36798 −0.498779 −0.249389 0.968403i \(-0.580230\pi\)
−0.249389 + 0.968403i \(0.580230\pi\)
\(164\) 9.83775 0.768199
\(165\) −39.9399 −3.10931
\(166\) −4.30862 −0.334413
\(167\) −1.87848 −0.145361 −0.0726804 0.997355i \(-0.523155\pi\)
−0.0726804 + 0.997355i \(0.523155\pi\)
\(168\) −23.4389 −1.80835
\(169\) 1.92760 0.148277
\(170\) 4.85761 0.372562
\(171\) −9.44036 −0.721922
\(172\) −1.86216 −0.141988
\(173\) −10.0104 −0.761080 −0.380540 0.924764i \(-0.624262\pi\)
−0.380540 + 0.924764i \(0.624262\pi\)
\(174\) −5.72752 −0.434202
\(175\) 37.7835 2.85616
\(176\) −4.73926 −0.357235
\(177\) 7.39868 0.556118
\(178\) 4.91040 0.368050
\(179\) −9.09399 −0.679717 −0.339858 0.940477i \(-0.610379\pi\)
−0.339858 + 0.940477i \(0.610379\pi\)
\(180\) −12.1014 −0.901986
\(181\) 24.7002 1.83595 0.917976 0.396635i \(-0.129822\pi\)
0.917976 + 0.396635i \(0.129822\pi\)
\(182\) −11.5017 −0.852566
\(183\) −28.1612 −2.08173
\(184\) 18.2848 1.34798
\(185\) 43.4804 3.19674
\(186\) 2.18169 0.159969
\(187\) −8.02009 −0.586487
\(188\) −12.1320 −0.884820
\(189\) −7.40180 −0.538402
\(190\) −12.0446 −0.873808
\(191\) −13.2841 −0.961204 −0.480602 0.876939i \(-0.659582\pi\)
−0.480602 + 0.876939i \(0.659582\pi\)
\(192\) 5.18864 0.374457
\(193\) 26.6666 1.91950 0.959750 0.280855i \(-0.0906180\pi\)
0.959750 + 0.280855i \(0.0906180\pi\)
\(194\) −7.46701 −0.536100
\(195\) −33.3836 −2.39065
\(196\) −13.4557 −0.961120
\(197\) 3.65462 0.260381 0.130190 0.991489i \(-0.458441\pi\)
0.130190 + 0.991489i \(0.458441\pi\)
\(198\) −7.49059 −0.532333
\(199\) 18.6263 1.32038 0.660191 0.751098i \(-0.270475\pi\)
0.660191 + 0.751098i \(0.270475\pi\)
\(200\) −23.9120 −1.69083
\(201\) −27.5785 −1.94524
\(202\) 4.21343 0.296456
\(203\) −13.7180 −0.962811
\(204\) −5.75219 −0.402734
\(205\) −25.6396 −1.79074
\(206\) 6.82455 0.475489
\(207\) −15.7273 −1.09313
\(208\) −3.96129 −0.274666
\(209\) 19.8861 1.37555
\(210\) 25.7221 1.77499
\(211\) −3.17606 −0.218649 −0.109325 0.994006i \(-0.534869\pi\)
−0.109325 + 0.994006i \(0.534869\pi\)
\(212\) −13.9048 −0.954989
\(213\) 16.3536 1.12053
\(214\) −11.0910 −0.758162
\(215\) 4.85324 0.330988
\(216\) 4.68438 0.318731
\(217\) 5.22535 0.354720
\(218\) −2.21137 −0.149773
\(219\) 11.3231 0.765146
\(220\) 25.4916 1.71864
\(221\) −6.70356 −0.450930
\(222\) 19.3031 1.29554
\(223\) −12.4850 −0.836059 −0.418030 0.908433i \(-0.637279\pi\)
−0.418030 + 0.908433i \(0.637279\pi\)
\(224\) 23.6207 1.57822
\(225\) 20.5674 1.37116
\(226\) −1.51436 −0.100734
\(227\) −16.1748 −1.07356 −0.536778 0.843723i \(-0.680359\pi\)
−0.536778 + 0.843723i \(0.680359\pi\)
\(228\) 14.2628 0.944575
\(229\) −22.6012 −1.49353 −0.746765 0.665088i \(-0.768394\pi\)
−0.746765 + 0.665088i \(0.768394\pi\)
\(230\) −20.0660 −1.32311
\(231\) −42.4682 −2.79420
\(232\) 8.68168 0.569980
\(233\) −3.70258 −0.242564 −0.121282 0.992618i \(-0.538701\pi\)
−0.121282 + 0.992618i \(0.538701\pi\)
\(234\) −6.26098 −0.409293
\(235\) 31.6191 2.06260
\(236\) −4.72221 −0.307389
\(237\) 24.3700 1.58300
\(238\) 5.16511 0.334804
\(239\) 5.99222 0.387604 0.193802 0.981041i \(-0.437918\pi\)
0.193802 + 0.981041i \(0.437918\pi\)
\(240\) 8.85890 0.571840
\(241\) 19.9256 1.28352 0.641760 0.766905i \(-0.278204\pi\)
0.641760 + 0.766905i \(0.278204\pi\)
\(242\) 7.65562 0.492122
\(243\) −19.0327 −1.22095
\(244\) 17.9739 1.15066
\(245\) 35.0688 2.24046
\(246\) −11.3827 −0.725733
\(247\) 16.6217 1.05762
\(248\) −3.30697 −0.209993
\(249\) −13.2973 −0.842683
\(250\) 12.2428 0.774300
\(251\) 5.43776 0.343228 0.171614 0.985164i \(-0.445102\pi\)
0.171614 + 0.985164i \(0.445102\pi\)
\(252\) −12.8675 −0.810574
\(253\) 33.1297 2.08284
\(254\) −1.94612 −0.122111
\(255\) 14.9916 0.938811
\(256\) −11.9660 −0.747873
\(257\) 16.9275 1.05591 0.527954 0.849273i \(-0.322959\pi\)
0.527954 + 0.849273i \(0.322959\pi\)
\(258\) 2.15460 0.134139
\(259\) 46.2328 2.87277
\(260\) 21.3071 1.32141
\(261\) −7.46737 −0.462219
\(262\) −7.13510 −0.440808
\(263\) −14.2305 −0.877489 −0.438744 0.898612i \(-0.644577\pi\)
−0.438744 + 0.898612i \(0.644577\pi\)
\(264\) 26.8768 1.65415
\(265\) 36.2394 2.22617
\(266\) −12.8071 −0.785252
\(267\) 15.1545 0.927443
\(268\) 17.6020 1.07521
\(269\) 4.78021 0.291455 0.145727 0.989325i \(-0.453448\pi\)
0.145727 + 0.989325i \(0.453448\pi\)
\(270\) −5.14068 −0.312852
\(271\) 22.8296 1.38680 0.693399 0.720554i \(-0.256112\pi\)
0.693399 + 0.720554i \(0.256112\pi\)
\(272\) 1.77891 0.107862
\(273\) −35.4968 −2.14837
\(274\) 2.82596 0.170723
\(275\) −43.3253 −2.61261
\(276\) 23.7613 1.43026
\(277\) 2.84440 0.170903 0.0854516 0.996342i \(-0.472767\pi\)
0.0854516 + 0.996342i \(0.472767\pi\)
\(278\) 9.50519 0.570083
\(279\) 2.84442 0.170291
\(280\) −38.9892 −2.33005
\(281\) −8.70598 −0.519356 −0.259678 0.965695i \(-0.583616\pi\)
−0.259678 + 0.965695i \(0.583616\pi\)
\(282\) 14.0373 0.835907
\(283\) 28.5870 1.69932 0.849659 0.527333i \(-0.176808\pi\)
0.849659 + 0.527333i \(0.176808\pi\)
\(284\) −10.4377 −0.619364
\(285\) −37.1723 −2.20189
\(286\) 13.1887 0.779867
\(287\) −27.2626 −1.60926
\(288\) 12.8579 0.757660
\(289\) −13.9896 −0.822919
\(290\) −9.52735 −0.559465
\(291\) −23.0448 −1.35091
\(292\) −7.22699 −0.422928
\(293\) −7.13843 −0.417032 −0.208516 0.978019i \(-0.566863\pi\)
−0.208516 + 0.978019i \(0.566863\pi\)
\(294\) 15.5688 0.907990
\(295\) 12.3072 0.716554
\(296\) −29.2594 −1.70067
\(297\) 8.48745 0.492492
\(298\) 11.0852 0.642149
\(299\) 27.6913 1.60143
\(300\) −31.0739 −1.79405
\(301\) 5.16047 0.297444
\(302\) 5.27618 0.303610
\(303\) 13.0035 0.747034
\(304\) −4.41086 −0.252980
\(305\) −46.8443 −2.68230
\(306\) 2.81163 0.160730
\(307\) 4.76214 0.271790 0.135895 0.990723i \(-0.456609\pi\)
0.135895 + 0.990723i \(0.456609\pi\)
\(308\) 27.1053 1.54447
\(309\) 21.0620 1.19818
\(310\) 3.62910 0.206119
\(311\) −29.6663 −1.68222 −0.841111 0.540862i \(-0.818098\pi\)
−0.841111 + 0.540862i \(0.818098\pi\)
\(312\) 22.4649 1.27182
\(313\) 3.14569 0.177805 0.0889025 0.996040i \(-0.471664\pi\)
0.0889025 + 0.996040i \(0.471664\pi\)
\(314\) −2.71687 −0.153322
\(315\) 33.5357 1.88952
\(316\) −15.5541 −0.874989
\(317\) −4.87154 −0.273613 −0.136806 0.990598i \(-0.543684\pi\)
−0.136806 + 0.990598i \(0.543684\pi\)
\(318\) 16.0885 0.902197
\(319\) 15.7300 0.880712
\(320\) 8.63096 0.482485
\(321\) −34.2291 −1.91048
\(322\) −21.3362 −1.18902
\(323\) −7.46435 −0.415327
\(324\) 15.6634 0.870190
\(325\) −36.2133 −2.00875
\(326\) −4.70264 −0.260455
\(327\) −6.82475 −0.377410
\(328\) 17.2537 0.952676
\(329\) 33.6206 1.85356
\(330\) −29.4949 −1.62364
\(331\) 19.7091 1.08331 0.541655 0.840601i \(-0.317798\pi\)
0.541655 + 0.840601i \(0.317798\pi\)
\(332\) 8.48701 0.465785
\(333\) 25.1669 1.37914
\(334\) −1.38722 −0.0759053
\(335\) −45.8751 −2.50642
\(336\) 9.41969 0.513886
\(337\) 6.18352 0.336838 0.168419 0.985716i \(-0.446134\pi\)
0.168419 + 0.985716i \(0.446134\pi\)
\(338\) 1.42350 0.0774280
\(339\) −4.67364 −0.253837
\(340\) −9.56840 −0.518919
\(341\) −5.99178 −0.324473
\(342\) −6.97153 −0.376977
\(343\) 9.07070 0.489772
\(344\) −3.26590 −0.176086
\(345\) −61.9278 −3.33408
\(346\) −7.39253 −0.397425
\(347\) −16.8542 −0.904783 −0.452392 0.891819i \(-0.649429\pi\)
−0.452392 + 0.891819i \(0.649429\pi\)
\(348\) 11.2819 0.604775
\(349\) 0.876469 0.0469163 0.0234581 0.999725i \(-0.492532\pi\)
0.0234581 + 0.999725i \(0.492532\pi\)
\(350\) 27.9024 1.49145
\(351\) 7.09420 0.378660
\(352\) −27.0852 −1.44365
\(353\) −26.5298 −1.41204 −0.706020 0.708192i \(-0.749511\pi\)
−0.706020 + 0.708192i \(0.749511\pi\)
\(354\) 5.46379 0.290397
\(355\) 27.2032 1.44380
\(356\) −9.67239 −0.512635
\(357\) 15.9406 0.843667
\(358\) −6.71575 −0.354938
\(359\) 5.36111 0.282949 0.141474 0.989942i \(-0.454816\pi\)
0.141474 + 0.989942i \(0.454816\pi\)
\(360\) −21.2238 −1.11859
\(361\) −0.491879 −0.0258884
\(362\) 18.2407 0.958708
\(363\) 23.6269 1.24009
\(364\) 22.6559 1.18749
\(365\) 18.8353 0.985885
\(366\) −20.7965 −1.08705
\(367\) 24.9627 1.30304 0.651520 0.758632i \(-0.274132\pi\)
0.651520 + 0.758632i \(0.274132\pi\)
\(368\) −7.34836 −0.383060
\(369\) −14.8404 −0.772561
\(370\) 32.1095 1.66929
\(371\) 38.5335 2.00056
\(372\) −4.29744 −0.222812
\(373\) −15.0785 −0.780734 −0.390367 0.920659i \(-0.627652\pi\)
−0.390367 + 0.920659i \(0.627652\pi\)
\(374\) −5.92269 −0.306255
\(375\) 37.7838 1.95114
\(376\) −21.2775 −1.09730
\(377\) 13.1479 0.677150
\(378\) −5.46609 −0.281146
\(379\) −14.0283 −0.720587 −0.360294 0.932839i \(-0.617324\pi\)
−0.360294 + 0.932839i \(0.617324\pi\)
\(380\) 23.7252 1.21708
\(381\) −6.00616 −0.307705
\(382\) −9.81007 −0.501927
\(383\) −13.2932 −0.679253 −0.339626 0.940560i \(-0.610301\pi\)
−0.339626 + 0.940560i \(0.610301\pi\)
\(384\) −22.8774 −1.16746
\(385\) −70.6430 −3.60030
\(386\) 19.6928 1.00234
\(387\) 2.80910 0.142795
\(388\) 14.7083 0.746703
\(389\) 24.9812 1.26660 0.633298 0.773908i \(-0.281701\pi\)
0.633298 + 0.773908i \(0.281701\pi\)
\(390\) −24.6532 −1.24836
\(391\) −12.4354 −0.628884
\(392\) −23.5989 −1.19193
\(393\) −22.0204 −1.11078
\(394\) 2.69887 0.135967
\(395\) 40.5379 2.03968
\(396\) 14.7548 0.741456
\(397\) −16.3754 −0.821860 −0.410930 0.911667i \(-0.634796\pi\)
−0.410930 + 0.911667i \(0.634796\pi\)
\(398\) 13.7552 0.689485
\(399\) −39.5254 −1.97874
\(400\) 9.60982 0.480491
\(401\) 28.3698 1.41672 0.708360 0.705851i \(-0.249435\pi\)
0.708360 + 0.705851i \(0.249435\pi\)
\(402\) −20.3662 −1.01578
\(403\) −5.00820 −0.249476
\(404\) −8.29951 −0.412916
\(405\) −40.8227 −2.02850
\(406\) −10.1305 −0.502766
\(407\) −53.0140 −2.62781
\(408\) −10.0883 −0.499447
\(409\) 25.0188 1.23710 0.618549 0.785746i \(-0.287721\pi\)
0.618549 + 0.785746i \(0.287721\pi\)
\(410\) −18.9344 −0.935101
\(411\) 8.72152 0.430201
\(412\) −13.4428 −0.662281
\(413\) 13.0863 0.643935
\(414\) −11.6144 −0.570815
\(415\) −22.1192 −1.08579
\(416\) −22.6390 −1.10997
\(417\) 29.3350 1.43654
\(418\) 14.6855 0.718293
\(419\) 16.2484 0.793788 0.396894 0.917865i \(-0.370088\pi\)
0.396894 + 0.917865i \(0.370088\pi\)
\(420\) −50.6668 −2.47229
\(421\) 27.6483 1.34750 0.673748 0.738961i \(-0.264683\pi\)
0.673748 + 0.738961i \(0.264683\pi\)
\(422\) −2.34547 −0.114175
\(423\) 18.3014 0.889844
\(424\) −24.3867 −1.18432
\(425\) 16.2624 0.788841
\(426\) 12.0769 0.585126
\(427\) −49.8097 −2.41046
\(428\) 21.8467 1.05600
\(429\) 40.7033 1.96517
\(430\) 3.58403 0.172837
\(431\) −30.8322 −1.48514 −0.742568 0.669771i \(-0.766392\pi\)
−0.742568 + 0.669771i \(0.766392\pi\)
\(432\) −1.88257 −0.0905751
\(433\) −20.9217 −1.00543 −0.502716 0.864452i \(-0.667666\pi\)
−0.502716 + 0.864452i \(0.667666\pi\)
\(434\) 3.85883 0.185230
\(435\) −29.4034 −1.40979
\(436\) 4.35590 0.208610
\(437\) 30.8340 1.47499
\(438\) 8.36193 0.399548
\(439\) −34.0600 −1.62560 −0.812798 0.582546i \(-0.802057\pi\)
−0.812798 + 0.582546i \(0.802057\pi\)
\(440\) 44.7079 2.13136
\(441\) 20.2981 0.966578
\(442\) −4.95046 −0.235469
\(443\) −28.7016 −1.36366 −0.681828 0.731513i \(-0.738815\pi\)
−0.681828 + 0.731513i \(0.738815\pi\)
\(444\) −38.0228 −1.80448
\(445\) 25.2086 1.19500
\(446\) −9.21997 −0.436578
\(447\) 34.2113 1.61814
\(448\) 9.17732 0.433588
\(449\) 35.9605 1.69708 0.848541 0.529130i \(-0.177482\pi\)
0.848541 + 0.529130i \(0.177482\pi\)
\(450\) 15.1887 0.716001
\(451\) 31.2613 1.47204
\(452\) 2.98295 0.140306
\(453\) 16.2834 0.765062
\(454\) −11.9448 −0.560596
\(455\) −59.0467 −2.76815
\(456\) 25.0144 1.17141
\(457\) 8.41951 0.393848 0.196924 0.980419i \(-0.436905\pi\)
0.196924 + 0.980419i \(0.436905\pi\)
\(458\) −16.6906 −0.779900
\(459\) −3.18581 −0.148701
\(460\) 39.5255 1.84288
\(461\) 16.3496 0.761476 0.380738 0.924683i \(-0.375670\pi\)
0.380738 + 0.924683i \(0.375670\pi\)
\(462\) −31.3620 −1.45909
\(463\) −25.5880 −1.18918 −0.594589 0.804030i \(-0.702685\pi\)
−0.594589 + 0.804030i \(0.702685\pi\)
\(464\) −3.48901 −0.161973
\(465\) 11.2002 0.519396
\(466\) −2.73429 −0.126663
\(467\) 36.2732 1.67852 0.839261 0.543729i \(-0.182988\pi\)
0.839261 + 0.543729i \(0.182988\pi\)
\(468\) 12.3327 0.570080
\(469\) −48.7791 −2.25241
\(470\) 23.3501 1.07706
\(471\) −8.38485 −0.386354
\(472\) −8.28193 −0.381207
\(473\) −5.91737 −0.272081
\(474\) 17.9968 0.826620
\(475\) −40.3231 −1.85015
\(476\) −10.1741 −0.466329
\(477\) 20.9757 0.960412
\(478\) 4.42514 0.202401
\(479\) −20.8433 −0.952357 −0.476178 0.879349i \(-0.657978\pi\)
−0.476178 + 0.879349i \(0.657978\pi\)
\(480\) 50.6292 2.31090
\(481\) −44.3115 −2.02043
\(482\) 14.7147 0.670236
\(483\) −65.8480 −2.99619
\(484\) −15.0799 −0.685448
\(485\) −38.3335 −1.74064
\(486\) −14.0553 −0.637563
\(487\) −1.61935 −0.0733798 −0.0366899 0.999327i \(-0.511681\pi\)
−0.0366899 + 0.999327i \(0.511681\pi\)
\(488\) 31.5231 1.42698
\(489\) −14.5133 −0.656316
\(490\) 25.8977 1.16994
\(491\) −18.4080 −0.830741 −0.415371 0.909652i \(-0.636348\pi\)
−0.415371 + 0.909652i \(0.636348\pi\)
\(492\) 22.4213 1.01083
\(493\) −5.90434 −0.265918
\(494\) 12.2748 0.552271
\(495\) −38.4546 −1.72840
\(496\) 1.32901 0.0596744
\(497\) 28.9253 1.29748
\(498\) −9.81982 −0.440037
\(499\) −8.70198 −0.389554 −0.194777 0.980848i \(-0.562398\pi\)
−0.194777 + 0.980848i \(0.562398\pi\)
\(500\) −24.1155 −1.07848
\(501\) −4.28126 −0.191273
\(502\) 4.01569 0.179229
\(503\) 1.00000 0.0445878
\(504\) −22.5673 −1.00523
\(505\) 21.6306 0.962547
\(506\) 24.4656 1.08763
\(507\) 4.39322 0.195110
\(508\) 3.83343 0.170081
\(509\) −6.56992 −0.291206 −0.145603 0.989343i \(-0.546512\pi\)
−0.145603 + 0.989343i \(0.546512\pi\)
\(510\) 11.0710 0.490234
\(511\) 20.0276 0.885970
\(512\) 11.2390 0.496700
\(513\) 7.89932 0.348764
\(514\) 12.5006 0.551380
\(515\) 35.0353 1.54384
\(516\) −4.24407 −0.186835
\(517\) −38.5519 −1.69551
\(518\) 34.1421 1.50012
\(519\) −22.8149 −1.00146
\(520\) 37.3689 1.63873
\(521\) −25.8155 −1.13100 −0.565499 0.824749i \(-0.691316\pi\)
−0.565499 + 0.824749i \(0.691316\pi\)
\(522\) −5.51452 −0.241364
\(523\) −0.711015 −0.0310905 −0.0155453 0.999879i \(-0.504948\pi\)
−0.0155453 + 0.999879i \(0.504948\pi\)
\(524\) 14.0545 0.613976
\(525\) 86.1128 3.75827
\(526\) −10.5090 −0.458212
\(527\) 2.24904 0.0979698
\(528\) −10.8013 −0.470067
\(529\) 28.3684 1.23341
\(530\) 26.7622 1.16247
\(531\) 7.12353 0.309135
\(532\) 25.2271 1.09373
\(533\) 26.1297 1.13180
\(534\) 11.1914 0.484297
\(535\) −56.9379 −2.46164
\(536\) 30.8708 1.33342
\(537\) −20.7262 −0.894403
\(538\) 3.53010 0.152193
\(539\) −42.7580 −1.84172
\(540\) 10.1260 0.435753
\(541\) 12.4754 0.536358 0.268179 0.963369i \(-0.413578\pi\)
0.268179 + 0.963369i \(0.413578\pi\)
\(542\) 16.8592 0.724166
\(543\) 56.2946 2.41583
\(544\) 10.1666 0.435888
\(545\) −11.3525 −0.486289
\(546\) −26.2138 −1.12185
\(547\) −2.52452 −0.107941 −0.0539704 0.998543i \(-0.517188\pi\)
−0.0539704 + 0.998543i \(0.517188\pi\)
\(548\) −5.56651 −0.237790
\(549\) −27.1139 −1.15719
\(550\) −31.9950 −1.36427
\(551\) 14.6400 0.623686
\(552\) 41.6732 1.77373
\(553\) 43.1041 1.83297
\(554\) 2.10054 0.0892432
\(555\) 99.0968 4.20642
\(556\) −18.7231 −0.794036
\(557\) −18.5546 −0.786184 −0.393092 0.919499i \(-0.628595\pi\)
−0.393092 + 0.919499i \(0.628595\pi\)
\(558\) 2.10056 0.0889236
\(559\) −4.94601 −0.209194
\(560\) 15.6691 0.662138
\(561\) −18.2787 −0.771727
\(562\) −6.42921 −0.271200
\(563\) −33.9002 −1.42872 −0.714362 0.699777i \(-0.753283\pi\)
−0.714362 + 0.699777i \(0.753283\pi\)
\(564\) −27.6503 −1.16429
\(565\) −7.77430 −0.327067
\(566\) 21.1109 0.887359
\(567\) −43.4069 −1.82292
\(568\) −18.3059 −0.768100
\(569\) −12.6955 −0.532223 −0.266111 0.963942i \(-0.585739\pi\)
−0.266111 + 0.963942i \(0.585739\pi\)
\(570\) −27.4510 −1.14980
\(571\) −12.3436 −0.516562 −0.258281 0.966070i \(-0.583156\pi\)
−0.258281 + 0.966070i \(0.583156\pi\)
\(572\) −25.9789 −1.08623
\(573\) −30.2760 −1.26480
\(574\) −20.1329 −0.840333
\(575\) −67.1771 −2.80148
\(576\) 4.99568 0.208153
\(577\) 32.8671 1.36828 0.684138 0.729353i \(-0.260179\pi\)
0.684138 + 0.729353i \(0.260179\pi\)
\(578\) −10.3311 −0.429716
\(579\) 60.7761 2.52577
\(580\) 18.7668 0.779247
\(581\) −23.5194 −0.975750
\(582\) −17.0182 −0.705426
\(583\) −44.1853 −1.82997
\(584\) −12.6749 −0.524490
\(585\) −32.1421 −1.32891
\(586\) −5.27160 −0.217768
\(587\) −29.7918 −1.22964 −0.614819 0.788668i \(-0.710771\pi\)
−0.614819 + 0.788668i \(0.710771\pi\)
\(588\) −30.6670 −1.26469
\(589\) −5.57658 −0.229779
\(590\) 9.08866 0.374174
\(591\) 8.32929 0.342621
\(592\) 11.7588 0.483285
\(593\) 45.0621 1.85048 0.925240 0.379383i \(-0.123864\pi\)
0.925240 + 0.379383i \(0.123864\pi\)
\(594\) 6.26783 0.257172
\(595\) 26.5162 1.08706
\(596\) −21.8354 −0.894412
\(597\) 42.4514 1.73742
\(598\) 20.4495 0.836243
\(599\) −41.5406 −1.69730 −0.848652 0.528952i \(-0.822585\pi\)
−0.848652 + 0.528952i \(0.822585\pi\)
\(600\) −54.4982 −2.22488
\(601\) 14.4075 0.587694 0.293847 0.955852i \(-0.405064\pi\)
0.293847 + 0.955852i \(0.405064\pi\)
\(602\) 3.81091 0.155321
\(603\) −26.5529 −1.08132
\(604\) −10.3929 −0.422881
\(605\) 39.3018 1.59785
\(606\) 9.60288 0.390090
\(607\) −18.5714 −0.753790 −0.376895 0.926256i \(-0.623008\pi\)
−0.376895 + 0.926256i \(0.623008\pi\)
\(608\) −25.2083 −1.02233
\(609\) −31.2648 −1.26691
\(610\) −34.5937 −1.40066
\(611\) −32.2234 −1.30362
\(612\) −5.53828 −0.223872
\(613\) −19.5200 −0.788404 −0.394202 0.919024i \(-0.628979\pi\)
−0.394202 + 0.919024i \(0.628979\pi\)
\(614\) 3.51675 0.141925
\(615\) −58.4355 −2.35635
\(616\) 47.5380 1.91536
\(617\) 43.6158 1.75591 0.877954 0.478745i \(-0.158908\pi\)
0.877954 + 0.478745i \(0.158908\pi\)
\(618\) 15.5539 0.625671
\(619\) −15.7737 −0.633999 −0.317000 0.948426i \(-0.602675\pi\)
−0.317000 + 0.948426i \(0.602675\pi\)
\(620\) −7.14851 −0.287091
\(621\) 13.1600 0.528094
\(622\) −21.9080 −0.878432
\(623\) 26.8044 1.07389
\(624\) −9.02824 −0.361419
\(625\) 15.9865 0.639458
\(626\) 2.32304 0.0928472
\(627\) 45.3227 1.81001
\(628\) 5.35163 0.213553
\(629\) 19.8991 0.793427
\(630\) 24.7655 0.986683
\(631\) 20.1102 0.800575 0.400288 0.916390i \(-0.368910\pi\)
0.400288 + 0.916390i \(0.368910\pi\)
\(632\) −27.2793 −1.08511
\(633\) −7.23861 −0.287709
\(634\) −3.59754 −0.142877
\(635\) −9.99085 −0.396475
\(636\) −31.6907 −1.25662
\(637\) −35.7391 −1.41604
\(638\) 11.6163 0.459895
\(639\) 15.7455 0.622882
\(640\) −38.0550 −1.50426
\(641\) −16.7023 −0.659701 −0.329850 0.944033i \(-0.606998\pi\)
−0.329850 + 0.944033i \(0.606998\pi\)
\(642\) −25.2776 −0.997625
\(643\) 11.8118 0.465811 0.232905 0.972499i \(-0.425177\pi\)
0.232905 + 0.972499i \(0.425177\pi\)
\(644\) 42.0275 1.65612
\(645\) 11.0611 0.435530
\(646\) −5.51229 −0.216878
\(647\) −5.72612 −0.225117 −0.112558 0.993645i \(-0.535905\pi\)
−0.112558 + 0.993645i \(0.535905\pi\)
\(648\) 27.4709 1.07916
\(649\) −15.0057 −0.589026
\(650\) −26.7429 −1.04894
\(651\) 11.9092 0.466757
\(652\) 9.26315 0.362773
\(653\) 25.6377 1.00328 0.501640 0.865076i \(-0.332730\pi\)
0.501640 + 0.865076i \(0.332730\pi\)
\(654\) −5.03996 −0.197078
\(655\) −36.6296 −1.43124
\(656\) −6.93395 −0.270725
\(657\) 10.9020 0.425329
\(658\) 24.8282 0.967905
\(659\) −19.6150 −0.764091 −0.382045 0.924144i \(-0.624780\pi\)
−0.382045 + 0.924144i \(0.624780\pi\)
\(660\) 58.0983 2.26147
\(661\) 6.71840 0.261316 0.130658 0.991428i \(-0.458291\pi\)
0.130658 + 0.991428i \(0.458291\pi\)
\(662\) 14.5548 0.565689
\(663\) −15.2782 −0.593355
\(664\) 14.8847 0.577640
\(665\) −65.7479 −2.54959
\(666\) 18.5853 0.720165
\(667\) 24.3898 0.944378
\(668\) 2.73251 0.105724
\(669\) −28.4548 −1.10013
\(670\) −33.8779 −1.30882
\(671\) 57.1155 2.20492
\(672\) 53.8341 2.07670
\(673\) 0.657038 0.0253270 0.0126635 0.999920i \(-0.495969\pi\)
0.0126635 + 0.999920i \(0.495969\pi\)
\(674\) 4.56642 0.175892
\(675\) −17.2100 −0.662414
\(676\) −2.80397 −0.107845
\(677\) 7.87250 0.302565 0.151282 0.988491i \(-0.451660\pi\)
0.151282 + 0.988491i \(0.451660\pi\)
\(678\) −3.45140 −0.132550
\(679\) −40.7601 −1.56423
\(680\) −16.7813 −0.643534
\(681\) −36.8641 −1.41264
\(682\) −4.42482 −0.169435
\(683\) 16.5842 0.634576 0.317288 0.948329i \(-0.397228\pi\)
0.317288 + 0.948329i \(0.397228\pi\)
\(684\) 13.7324 0.525070
\(685\) 14.5077 0.554310
\(686\) 6.69855 0.255752
\(687\) −51.5107 −1.96526
\(688\) 1.31251 0.0500389
\(689\) −36.9321 −1.40700
\(690\) −45.7326 −1.74101
\(691\) 22.9396 0.872664 0.436332 0.899786i \(-0.356277\pi\)
0.436332 + 0.899786i \(0.356277\pi\)
\(692\) 14.5616 0.553550
\(693\) −40.8888 −1.55324
\(694\) −12.4466 −0.472465
\(695\) 48.7970 1.85097
\(696\) 19.7865 0.750007
\(697\) −11.7341 −0.444460
\(698\) 0.647256 0.0244990
\(699\) −8.43859 −0.319177
\(700\) −54.9615 −2.07735
\(701\) −38.2990 −1.44653 −0.723266 0.690569i \(-0.757360\pi\)
−0.723266 + 0.690569i \(0.757360\pi\)
\(702\) 5.23894 0.197731
\(703\) −49.3404 −1.86091
\(704\) −10.5234 −0.396615
\(705\) 72.0634 2.71406
\(706\) −19.5918 −0.737347
\(707\) 22.9998 0.864998
\(708\) −10.7624 −0.404477
\(709\) 13.9055 0.522232 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(710\) 20.0891 0.753930
\(711\) 23.4637 0.879958
\(712\) −16.9637 −0.635741
\(713\) −9.29042 −0.347929
\(714\) 11.7719 0.440551
\(715\) 67.7073 2.53211
\(716\) 13.2285 0.494373
\(717\) 13.6569 0.510028
\(718\) 3.95908 0.147752
\(719\) −1.36261 −0.0508166 −0.0254083 0.999677i \(-0.508089\pi\)
−0.0254083 + 0.999677i \(0.508089\pi\)
\(720\) 8.52946 0.317874
\(721\) 37.2531 1.38738
\(722\) −0.363244 −0.0135185
\(723\) 45.4127 1.68892
\(724\) −35.9300 −1.33533
\(725\) −31.8958 −1.18458
\(726\) 17.4480 0.647557
\(727\) 23.4016 0.867918 0.433959 0.900933i \(-0.357116\pi\)
0.433959 + 0.900933i \(0.357116\pi\)
\(728\) 39.7344 1.47266
\(729\) −11.0742 −0.410154
\(730\) 13.9095 0.514815
\(731\) 2.22111 0.0821509
\(732\) 40.9645 1.51409
\(733\) −9.99331 −0.369111 −0.184556 0.982822i \(-0.559085\pi\)
−0.184556 + 0.982822i \(0.559085\pi\)
\(734\) 18.4345 0.680429
\(735\) 79.9257 2.94811
\(736\) −41.9963 −1.54801
\(737\) 55.9337 2.06034
\(738\) −10.9594 −0.403420
\(739\) 23.1704 0.852335 0.426168 0.904644i \(-0.359863\pi\)
0.426168 + 0.904644i \(0.359863\pi\)
\(740\) −63.2485 −2.32506
\(741\) 37.8828 1.39166
\(742\) 28.4563 1.04466
\(743\) 20.2086 0.741383 0.370691 0.928756i \(-0.379121\pi\)
0.370691 + 0.928756i \(0.379121\pi\)
\(744\) −7.53696 −0.276318
\(745\) 56.9083 2.08496
\(746\) −11.1352 −0.407688
\(747\) −12.8028 −0.468430
\(748\) 11.6664 0.426565
\(749\) −60.5422 −2.21216
\(750\) 27.9026 1.01886
\(751\) 11.5168 0.420253 0.210127 0.977674i \(-0.432612\pi\)
0.210127 + 0.977674i \(0.432612\pi\)
\(752\) 8.55104 0.311824
\(753\) 12.3933 0.451636
\(754\) 9.70947 0.353598
\(755\) 27.0864 0.985776
\(756\) 10.7670 0.391591
\(757\) 39.7268 1.44390 0.721948 0.691947i \(-0.243247\pi\)
0.721948 + 0.691947i \(0.243247\pi\)
\(758\) −10.3597 −0.376280
\(759\) 75.5062 2.74070
\(760\) 41.6099 1.50935
\(761\) −22.6597 −0.821415 −0.410708 0.911767i \(-0.634718\pi\)
−0.410708 + 0.911767i \(0.634718\pi\)
\(762\) −4.43544 −0.160679
\(763\) −12.0712 −0.437006
\(764\) 19.3236 0.699105
\(765\) 14.4341 0.521866
\(766\) −9.81682 −0.354696
\(767\) −12.5425 −0.452882
\(768\) −27.2718 −0.984086
\(769\) 17.6004 0.634688 0.317344 0.948311i \(-0.397209\pi\)
0.317344 + 0.948311i \(0.397209\pi\)
\(770\) −52.1686 −1.88003
\(771\) 38.5797 1.38941
\(772\) −38.7904 −1.39610
\(773\) 45.8392 1.64872 0.824361 0.566064i \(-0.191535\pi\)
0.824361 + 0.566064i \(0.191535\pi\)
\(774\) 2.07447 0.0745653
\(775\) 12.1495 0.436424
\(776\) 25.7959 0.926018
\(777\) 105.370 3.78012
\(778\) 18.4481 0.661398
\(779\) 29.0951 1.04244
\(780\) 48.5612 1.73877
\(781\) −33.1678 −1.18684
\(782\) −9.18330 −0.328394
\(783\) 6.24840 0.223300
\(784\) 9.48399 0.338714
\(785\) −13.9477 −0.497813
\(786\) −16.2617 −0.580035
\(787\) −17.4603 −0.622391 −0.311196 0.950346i \(-0.600729\pi\)
−0.311196 + 0.950346i \(0.600729\pi\)
\(788\) −5.31617 −0.189381
\(789\) −32.4329 −1.15464
\(790\) 29.9365 1.06509
\(791\) −8.26643 −0.293920
\(792\) 25.8773 0.919510
\(793\) 47.7397 1.69529
\(794\) −12.0930 −0.429163
\(795\) 82.5937 2.92930
\(796\) −27.0946 −0.960343
\(797\) 7.37081 0.261087 0.130544 0.991443i \(-0.458328\pi\)
0.130544 + 0.991443i \(0.458328\pi\)
\(798\) −29.1888 −1.03327
\(799\) 14.4706 0.511934
\(800\) 54.9207 1.94174
\(801\) 14.5910 0.515546
\(802\) 20.9506 0.739791
\(803\) −22.9652 −0.810423
\(804\) 40.1169 1.41482
\(805\) −109.534 −3.86056
\(806\) −3.69847 −0.130273
\(807\) 10.8946 0.383510
\(808\) −14.5559 −0.512075
\(809\) −7.04747 −0.247776 −0.123888 0.992296i \(-0.539536\pi\)
−0.123888 + 0.992296i \(0.539536\pi\)
\(810\) −30.1468 −1.05925
\(811\) −52.9606 −1.85970 −0.929849 0.367941i \(-0.880063\pi\)
−0.929849 + 0.367941i \(0.880063\pi\)
\(812\) 19.9547 0.700274
\(813\) 52.0312 1.82481
\(814\) −39.1499 −1.37220
\(815\) −24.1420 −0.845658
\(816\) 4.05433 0.141930
\(817\) −5.50733 −0.192677
\(818\) 18.4759 0.645995
\(819\) −34.1768 −1.19423
\(820\) 37.2964 1.30245
\(821\) −32.9925 −1.15145 −0.575724 0.817644i \(-0.695280\pi\)
−0.575724 + 0.817644i \(0.695280\pi\)
\(822\) 6.44069 0.224645
\(823\) 9.98952 0.348213 0.174106 0.984727i \(-0.444296\pi\)
0.174106 + 0.984727i \(0.444296\pi\)
\(824\) −23.5764 −0.821323
\(825\) −98.7433 −3.43780
\(826\) 9.66399 0.336253
\(827\) 18.2097 0.633213 0.316607 0.948557i \(-0.397456\pi\)
0.316607 + 0.948557i \(0.397456\pi\)
\(828\) 22.8777 0.795055
\(829\) −9.44705 −0.328110 −0.164055 0.986451i \(-0.552457\pi\)
−0.164055 + 0.986451i \(0.552457\pi\)
\(830\) −16.3346 −0.566984
\(831\) 6.48270 0.224882
\(832\) −8.79594 −0.304944
\(833\) 16.0494 0.556080
\(834\) 21.6634 0.750142
\(835\) −7.12160 −0.246453
\(836\) −28.9272 −1.00047
\(837\) −2.38010 −0.0822684
\(838\) 11.9992 0.414504
\(839\) 23.5644 0.813533 0.406766 0.913532i \(-0.366656\pi\)
0.406766 + 0.913532i \(0.366656\pi\)
\(840\) −88.8607 −3.06599
\(841\) −17.4197 −0.600678
\(842\) 20.4178 0.703643
\(843\) −19.8419 −0.683392
\(844\) 4.62004 0.159028
\(845\) 7.30783 0.251397
\(846\) 13.5153 0.464664
\(847\) 41.7897 1.43591
\(848\) 9.80057 0.336553
\(849\) 65.1529 2.23604
\(850\) 12.0095 0.411921
\(851\) −82.1996 −2.81777
\(852\) −23.7887 −0.814989
\(853\) −52.2667 −1.78958 −0.894789 0.446490i \(-0.852674\pi\)
−0.894789 + 0.446490i \(0.852674\pi\)
\(854\) −36.7836 −1.25871
\(855\) −35.7899 −1.22399
\(856\) 38.3153 1.30959
\(857\) −23.7734 −0.812083 −0.406041 0.913855i \(-0.633091\pi\)
−0.406041 + 0.913855i \(0.633091\pi\)
\(858\) 30.0587 1.02619
\(859\) 16.6060 0.566588 0.283294 0.959033i \(-0.408573\pi\)
0.283294 + 0.959033i \(0.408573\pi\)
\(860\) −7.05974 −0.240735
\(861\) −62.1346 −2.11754
\(862\) −22.7690 −0.775517
\(863\) 30.5826 1.04104 0.520521 0.853849i \(-0.325738\pi\)
0.520521 + 0.853849i \(0.325738\pi\)
\(864\) −10.7590 −0.366028
\(865\) −37.9512 −1.29038
\(866\) −15.4503 −0.525022
\(867\) −31.8839 −1.08283
\(868\) −7.60103 −0.257996
\(869\) −49.4263 −1.67667
\(870\) −21.7139 −0.736171
\(871\) 46.7520 1.58413
\(872\) 7.63949 0.258706
\(873\) −22.1878 −0.750943
\(874\) 22.7703 0.770218
\(875\) 66.8295 2.25925
\(876\) −16.4711 −0.556508
\(877\) 45.7382 1.54447 0.772235 0.635337i \(-0.219139\pi\)
0.772235 + 0.635337i \(0.219139\pi\)
\(878\) −25.1527 −0.848863
\(879\) −16.2693 −0.548750
\(880\) −17.9673 −0.605677
\(881\) −9.93887 −0.334849 −0.167425 0.985885i \(-0.553545\pi\)
−0.167425 + 0.985885i \(0.553545\pi\)
\(882\) 14.9898 0.504733
\(883\) −29.0582 −0.977886 −0.488943 0.872316i \(-0.662617\pi\)
−0.488943 + 0.872316i \(0.662617\pi\)
\(884\) 9.75130 0.327972
\(885\) 28.0495 0.942875
\(886\) −21.1956 −0.712081
\(887\) −41.7824 −1.40291 −0.701457 0.712712i \(-0.747467\pi\)
−0.701457 + 0.712712i \(0.747467\pi\)
\(888\) −66.6854 −2.23782
\(889\) −10.6233 −0.356294
\(890\) 18.6161 0.624013
\(891\) 49.7735 1.66748
\(892\) 18.1613 0.608084
\(893\) −35.8805 −1.20069
\(894\) 25.2644 0.844970
\(895\) −34.4767 −1.15243
\(896\) −40.4640 −1.35181
\(897\) 63.1116 2.10723
\(898\) 26.5562 0.886191
\(899\) −4.41111 −0.147119
\(900\) −29.9183 −0.997277
\(901\) 16.5852 0.552532
\(902\) 23.0859 0.768677
\(903\) 11.7613 0.391391
\(904\) 5.23157 0.174000
\(905\) 93.6424 3.11278
\(906\) 12.0250 0.399504
\(907\) −15.3714 −0.510397 −0.255199 0.966889i \(-0.582141\pi\)
−0.255199 + 0.966889i \(0.582141\pi\)
\(908\) 23.5285 0.780821
\(909\) 12.5200 0.415261
\(910\) −43.6049 −1.44549
\(911\) 10.3340 0.342382 0.171191 0.985238i \(-0.445238\pi\)
0.171191 + 0.985238i \(0.445238\pi\)
\(912\) −10.0529 −0.332883
\(913\) 26.9691 0.892547
\(914\) 6.21766 0.205662
\(915\) −106.764 −3.52949
\(916\) 32.8767 1.08628
\(917\) −38.9483 −1.28619
\(918\) −2.35266 −0.0776493
\(919\) 45.0421 1.48580 0.742901 0.669401i \(-0.233449\pi\)
0.742901 + 0.669401i \(0.233449\pi\)
\(920\) 69.3207 2.28544
\(921\) 10.8535 0.357634
\(922\) 12.0739 0.397632
\(923\) −27.7232 −0.912521
\(924\) 61.7761 2.03228
\(925\) 107.497 3.53447
\(926\) −18.8963 −0.620971
\(927\) 20.2788 0.666042
\(928\) −19.9399 −0.654560
\(929\) −14.7997 −0.485562 −0.242781 0.970081i \(-0.578060\pi\)
−0.242781 + 0.970081i \(0.578060\pi\)
\(930\) 8.27113 0.271221
\(931\) −39.7951 −1.30423
\(932\) 5.38593 0.176422
\(933\) −67.6129 −2.21355
\(934\) 26.7871 0.876500
\(935\) −30.4054 −0.994364
\(936\) 21.6295 0.706981
\(937\) 3.42126 0.111768 0.0558839 0.998437i \(-0.482202\pi\)
0.0558839 + 0.998437i \(0.482202\pi\)
\(938\) −36.0225 −1.17618
\(939\) 7.16939 0.233964
\(940\) −45.9945 −1.50017
\(941\) 17.9750 0.585968 0.292984 0.956117i \(-0.405352\pi\)
0.292984 + 0.956117i \(0.405352\pi\)
\(942\) −6.19206 −0.201748
\(943\) 48.4715 1.57845
\(944\) 3.32836 0.108329
\(945\) −28.0614 −0.912837
\(946\) −4.36987 −0.142077
\(947\) −9.49653 −0.308596 −0.154298 0.988024i \(-0.549312\pi\)
−0.154298 + 0.988024i \(0.549312\pi\)
\(948\) −35.4497 −1.15135
\(949\) −19.1953 −0.623107
\(950\) −29.7779 −0.966123
\(951\) −11.1028 −0.360033
\(952\) −17.8436 −0.578315
\(953\) −48.6172 −1.57487 −0.787433 0.616401i \(-0.788590\pi\)
−0.787433 + 0.616401i \(0.788590\pi\)
\(954\) 15.4902 0.501513
\(955\) −50.3621 −1.62968
\(956\) −8.71655 −0.281913
\(957\) 35.8505 1.15888
\(958\) −15.3924 −0.497307
\(959\) 15.4261 0.498134
\(960\) 19.6709 0.634877
\(961\) −29.3197 −0.945798
\(962\) −32.7233 −1.05504
\(963\) −32.9562 −1.06200
\(964\) −28.9847 −0.933533
\(965\) 101.097 3.25443
\(966\) −48.6276 −1.56457
\(967\) 36.9048 1.18678 0.593389 0.804916i \(-0.297789\pi\)
0.593389 + 0.804916i \(0.297789\pi\)
\(968\) −26.4474 −0.850053
\(969\) −17.0121 −0.546507
\(970\) −28.3086 −0.908935
\(971\) 39.9682 1.28264 0.641321 0.767273i \(-0.278387\pi\)
0.641321 + 0.767273i \(0.278387\pi\)
\(972\) 27.6859 0.888024
\(973\) 51.8859 1.66339
\(974\) −1.19586 −0.0383179
\(975\) −82.5342 −2.64321
\(976\) −12.6686 −0.405511
\(977\) 54.4292 1.74134 0.870672 0.491864i \(-0.163684\pi\)
0.870672 + 0.491864i \(0.163684\pi\)
\(978\) −10.7178 −0.342719
\(979\) −30.7359 −0.982323
\(980\) −51.0126 −1.62954
\(981\) −6.57095 −0.209794
\(982\) −13.5940 −0.433801
\(983\) −47.7647 −1.52346 −0.761728 0.647897i \(-0.775649\pi\)
−0.761728 + 0.647897i \(0.775649\pi\)
\(984\) 39.3231 1.25357
\(985\) 13.8552 0.441464
\(986\) −4.36025 −0.138859
\(987\) 76.6252 2.43901
\(988\) −24.1787 −0.769227
\(989\) −9.17505 −0.291750
\(990\) −28.3980 −0.902548
\(991\) 18.5959 0.590717 0.295359 0.955386i \(-0.404561\pi\)
0.295359 + 0.955386i \(0.404561\pi\)
\(992\) 7.59539 0.241154
\(993\) 44.9193 1.42547
\(994\) 21.3608 0.677523
\(995\) 70.6152 2.23865
\(996\) 19.3429 0.612902
\(997\) 3.78354 0.119826 0.0599129 0.998204i \(-0.480918\pi\)
0.0599129 + 0.998204i \(0.480918\pi\)
\(998\) −6.42625 −0.203420
\(999\) −21.0586 −0.666266
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 503.2.a.f.1.16 26
3.2 odd 2 4527.2.a.o.1.11 26
4.3 odd 2 8048.2.a.u.1.7 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.f.1.16 26 1.1 even 1 trivial
4527.2.a.o.1.11 26 3.2 odd 2
8048.2.a.u.1.7 26 4.3 odd 2