Properties

Label 5225.2.a.h.1.5
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $5$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.245526\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.18524 q^{2} -2.15766 q^{3} +2.77529 q^{4} -4.71500 q^{6} -3.93972 q^{7} +1.69419 q^{8} +1.65548 q^{9} +O(q^{10})\) \(q+2.18524 q^{2} -2.15766 q^{3} +2.77529 q^{4} -4.71500 q^{6} -3.93972 q^{7} +1.69419 q^{8} +1.65548 q^{9} +1.00000 q^{11} -5.98812 q^{12} -3.31182 q^{13} -8.60924 q^{14} -1.84836 q^{16} -2.80637 q^{17} +3.61763 q^{18} -1.00000 q^{19} +8.50056 q^{21} +2.18524 q^{22} -6.88998 q^{23} -3.65548 q^{24} -7.23713 q^{26} +2.90101 q^{27} -10.9338 q^{28} +5.67979 q^{29} +2.51864 q^{31} -7.42749 q^{32} -2.15766 q^{33} -6.13259 q^{34} +4.59444 q^{36} +6.39893 q^{37} -2.18524 q^{38} +7.14577 q^{39} +0.560629 q^{41} +18.5758 q^{42} +9.40080 q^{43} +2.77529 q^{44} -15.0563 q^{46} +12.1742 q^{47} +3.98812 q^{48} +8.52137 q^{49} +6.05517 q^{51} -9.19126 q^{52} -5.68316 q^{53} +6.33941 q^{54} -6.67463 q^{56} +2.15766 q^{57} +12.4117 q^{58} +4.35730 q^{59} -3.56412 q^{61} +5.50384 q^{62} -6.52213 q^{63} -12.5342 q^{64} -4.71500 q^{66} +9.95563 q^{67} -7.78847 q^{68} +14.8662 q^{69} -11.4671 q^{71} +2.80470 q^{72} +8.95834 q^{73} +13.9832 q^{74} -2.77529 q^{76} -3.93972 q^{77} +15.6153 q^{78} +8.49105 q^{79} -11.2258 q^{81} +1.22511 q^{82} +5.21960 q^{83} +23.5915 q^{84} +20.5430 q^{86} -12.2550 q^{87} +1.69419 q^{88} -7.28423 q^{89} +13.0476 q^{91} -19.1217 q^{92} -5.43436 q^{93} +26.6036 q^{94} +16.0260 q^{96} -10.6574 q^{97} +18.6213 q^{98} +1.65548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24} - 6 q^{26} + 11 q^{27} + 10 q^{28} + 10 q^{29} + 11 q^{31} - 14 q^{32} - q^{33} - 4 q^{34} - 26 q^{36} - q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{41} + 16 q^{42} - 20 q^{43} + 6 q^{44} - 4 q^{46} + 20 q^{47} - 4 q^{48} + 3 q^{49} + 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 38 q^{56} + q^{57} + 6 q^{58} + 3 q^{59} - 10 q^{61} + 6 q^{62} - 24 q^{63} - 2 q^{66} - 9 q^{67} - 24 q^{68} - 5 q^{69} + 23 q^{71} + 12 q^{72} + 8 q^{74} - 6 q^{76} - 6 q^{77} + 22 q^{78} + 44 q^{79} + q^{81} + 30 q^{82} + 14 q^{83} + 14 q^{84} + 52 q^{86} - 28 q^{87} - 6 q^{88} - 27 q^{89} + 24 q^{91} - 58 q^{92} + 27 q^{93} - 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.18524 1.54520 0.772600 0.634893i \(-0.218956\pi\)
0.772600 + 0.634893i \(0.218956\pi\)
\(3\) −2.15766 −1.24572 −0.622862 0.782332i \(-0.714030\pi\)
−0.622862 + 0.782332i \(0.714030\pi\)
\(4\) 2.77529 1.38764
\(5\) 0 0
\(6\) −4.71500 −1.92489
\(7\) −3.93972 −1.48907 −0.744537 0.667582i \(-0.767329\pi\)
−0.744537 + 0.667582i \(0.767329\pi\)
\(8\) 1.69419 0.598987
\(9\) 1.65548 0.551827
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −5.98812 −1.72862
\(13\) −3.31182 −0.918534 −0.459267 0.888298i \(-0.651888\pi\)
−0.459267 + 0.888298i \(0.651888\pi\)
\(14\) −8.60924 −2.30092
\(15\) 0 0
\(16\) −1.84836 −0.462089
\(17\) −2.80637 −0.680644 −0.340322 0.940309i \(-0.610536\pi\)
−0.340322 + 0.940309i \(0.610536\pi\)
\(18\) 3.61763 0.852684
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 8.50056 1.85497
\(22\) 2.18524 0.465895
\(23\) −6.88998 −1.43666 −0.718330 0.695702i \(-0.755093\pi\)
−0.718330 + 0.695702i \(0.755093\pi\)
\(24\) −3.65548 −0.746172
\(25\) 0 0
\(26\) −7.23713 −1.41932
\(27\) 2.90101 0.558299
\(28\) −10.9338 −2.06630
\(29\) 5.67979 1.05471 0.527355 0.849645i \(-0.323184\pi\)
0.527355 + 0.849645i \(0.323184\pi\)
\(30\) 0 0
\(31\) 2.51864 0.452361 0.226180 0.974085i \(-0.427376\pi\)
0.226180 + 0.974085i \(0.427376\pi\)
\(32\) −7.42749 −1.31301
\(33\) −2.15766 −0.375600
\(34\) −6.13259 −1.05173
\(35\) 0 0
\(36\) 4.59444 0.765740
\(37\) 6.39893 1.05198 0.525989 0.850491i \(-0.323695\pi\)
0.525989 + 0.850491i \(0.323695\pi\)
\(38\) −2.18524 −0.354493
\(39\) 7.14577 1.14424
\(40\) 0 0
\(41\) 0.560629 0.0875555 0.0437778 0.999041i \(-0.486061\pi\)
0.0437778 + 0.999041i \(0.486061\pi\)
\(42\) 18.5758 2.86631
\(43\) 9.40080 1.43361 0.716805 0.697274i \(-0.245604\pi\)
0.716805 + 0.697274i \(0.245604\pi\)
\(44\) 2.77529 0.418390
\(45\) 0 0
\(46\) −15.0563 −2.21993
\(47\) 12.1742 1.77579 0.887896 0.460044i \(-0.152166\pi\)
0.887896 + 0.460044i \(0.152166\pi\)
\(48\) 3.98812 0.575635
\(49\) 8.52137 1.21734
\(50\) 0 0
\(51\) 6.05517 0.847894
\(52\) −9.19126 −1.27460
\(53\) −5.68316 −0.780643 −0.390321 0.920679i \(-0.627636\pi\)
−0.390321 + 0.920679i \(0.627636\pi\)
\(54\) 6.33941 0.862684
\(55\) 0 0
\(56\) −6.67463 −0.891935
\(57\) 2.15766 0.285789
\(58\) 12.4117 1.62974
\(59\) 4.35730 0.567273 0.283636 0.958932i \(-0.408459\pi\)
0.283636 + 0.958932i \(0.408459\pi\)
\(60\) 0 0
\(61\) −3.56412 −0.456339 −0.228169 0.973621i \(-0.573274\pi\)
−0.228169 + 0.973621i \(0.573274\pi\)
\(62\) 5.50384 0.698988
\(63\) −6.52213 −0.821711
\(64\) −12.5342 −1.56677
\(65\) 0 0
\(66\) −4.71500 −0.580377
\(67\) 9.95563 1.21627 0.608137 0.793832i \(-0.291917\pi\)
0.608137 + 0.793832i \(0.291917\pi\)
\(68\) −7.78847 −0.944491
\(69\) 14.8662 1.78968
\(70\) 0 0
\(71\) −11.4671 −1.36089 −0.680447 0.732797i \(-0.738214\pi\)
−0.680447 + 0.732797i \(0.738214\pi\)
\(72\) 2.80470 0.330537
\(73\) 8.95834 1.04849 0.524247 0.851566i \(-0.324347\pi\)
0.524247 + 0.851566i \(0.324347\pi\)
\(74\) 13.9832 1.62552
\(75\) 0 0
\(76\) −2.77529 −0.318347
\(77\) −3.93972 −0.448972
\(78\) 15.6153 1.76808
\(79\) 8.49105 0.955318 0.477659 0.878545i \(-0.341485\pi\)
0.477659 + 0.878545i \(0.341485\pi\)
\(80\) 0 0
\(81\) −11.2258 −1.24731
\(82\) 1.22511 0.135291
\(83\) 5.21960 0.572926 0.286463 0.958091i \(-0.407520\pi\)
0.286463 + 0.958091i \(0.407520\pi\)
\(84\) 23.5915 2.57404
\(85\) 0 0
\(86\) 20.5430 2.21521
\(87\) −12.2550 −1.31388
\(88\) 1.69419 0.180601
\(89\) −7.28423 −0.772127 −0.386064 0.922472i \(-0.626166\pi\)
−0.386064 + 0.922472i \(0.626166\pi\)
\(90\) 0 0
\(91\) 13.0476 1.36776
\(92\) −19.1217 −1.99357
\(93\) −5.43436 −0.563517
\(94\) 26.6036 2.74395
\(95\) 0 0
\(96\) 16.0260 1.63564
\(97\) −10.6574 −1.08209 −0.541045 0.840993i \(-0.681971\pi\)
−0.541045 + 0.840993i \(0.681971\pi\)
\(98\) 18.6213 1.88103
\(99\) 1.65548 0.166382
\(100\) 0 0
\(101\) −11.4716 −1.14147 −0.570735 0.821134i \(-0.693342\pi\)
−0.570735 + 0.821134i \(0.693342\pi\)
\(102\) 13.2320 1.31017
\(103\) −18.3034 −1.80349 −0.901745 0.432268i \(-0.857714\pi\)
−0.901745 + 0.432268i \(0.857714\pi\)
\(104\) −5.61086 −0.550190
\(105\) 0 0
\(106\) −12.4191 −1.20625
\(107\) −1.38838 −0.134220 −0.0671100 0.997746i \(-0.521378\pi\)
−0.0671100 + 0.997746i \(0.521378\pi\)
\(108\) 8.05113 0.774720
\(109\) −0.412113 −0.0394732 −0.0197366 0.999805i \(-0.506283\pi\)
−0.0197366 + 0.999805i \(0.506283\pi\)
\(110\) 0 0
\(111\) −13.8067 −1.31047
\(112\) 7.28200 0.688084
\(113\) 6.54003 0.615234 0.307617 0.951510i \(-0.400468\pi\)
0.307617 + 0.951510i \(0.400468\pi\)
\(114\) 4.71500 0.441601
\(115\) 0 0
\(116\) 15.7630 1.46356
\(117\) −5.48266 −0.506872
\(118\) 9.52177 0.876550
\(119\) 11.0563 1.01353
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −7.78847 −0.705135
\(123\) −1.20964 −0.109070
\(124\) 6.98995 0.627716
\(125\) 0 0
\(126\) −14.2524 −1.26971
\(127\) −9.08005 −0.805724 −0.402862 0.915261i \(-0.631985\pi\)
−0.402862 + 0.915261i \(0.631985\pi\)
\(128\) −12.5352 −1.10797
\(129\) −20.2837 −1.78588
\(130\) 0 0
\(131\) −10.3876 −0.907571 −0.453785 0.891111i \(-0.649927\pi\)
−0.453785 + 0.891111i \(0.649927\pi\)
\(132\) −5.98812 −0.521199
\(133\) 3.93972 0.341617
\(134\) 21.7555 1.87939
\(135\) 0 0
\(136\) −4.75452 −0.407697
\(137\) 0.798293 0.0682028 0.0341014 0.999418i \(-0.489143\pi\)
0.0341014 + 0.999418i \(0.489143\pi\)
\(138\) 32.4863 2.76542
\(139\) 5.03184 0.426795 0.213398 0.976965i \(-0.431547\pi\)
0.213398 + 0.976965i \(0.431547\pi\)
\(140\) 0 0
\(141\) −26.2678 −2.21215
\(142\) −25.0584 −2.10285
\(143\) −3.31182 −0.276948
\(144\) −3.05992 −0.254993
\(145\) 0 0
\(146\) 19.5761 1.62013
\(147\) −18.3862 −1.51647
\(148\) 17.7589 1.45977
\(149\) 19.8351 1.62496 0.812479 0.582991i \(-0.198118\pi\)
0.812479 + 0.582991i \(0.198118\pi\)
\(150\) 0 0
\(151\) 22.5447 1.83466 0.917331 0.398125i \(-0.130339\pi\)
0.917331 + 0.398125i \(0.130339\pi\)
\(152\) −1.69419 −0.137417
\(153\) −4.64589 −0.375598
\(154\) −8.60924 −0.693752
\(155\) 0 0
\(156\) 19.8316 1.58780
\(157\) 11.8013 0.941843 0.470921 0.882175i \(-0.343922\pi\)
0.470921 + 0.882175i \(0.343922\pi\)
\(158\) 18.5550 1.47616
\(159\) 12.2623 0.972465
\(160\) 0 0
\(161\) 27.1446 2.13929
\(162\) −24.5312 −1.92735
\(163\) 24.8395 1.94558 0.972789 0.231691i \(-0.0744259\pi\)
0.972789 + 0.231691i \(0.0744259\pi\)
\(164\) 1.55591 0.121496
\(165\) 0 0
\(166\) 11.4061 0.885285
\(167\) 2.79938 0.216623 0.108311 0.994117i \(-0.465456\pi\)
0.108311 + 0.994117i \(0.465456\pi\)
\(168\) 14.4016 1.11110
\(169\) −2.03184 −0.156295
\(170\) 0 0
\(171\) −1.65548 −0.126598
\(172\) 26.0899 1.98934
\(173\) 6.43926 0.489568 0.244784 0.969578i \(-0.421283\pi\)
0.244784 + 0.969578i \(0.421283\pi\)
\(174\) −26.7802 −2.03020
\(175\) 0 0
\(176\) −1.84836 −0.139325
\(177\) −9.40156 −0.706665
\(178\) −15.9178 −1.19309
\(179\) 12.5241 0.936095 0.468048 0.883703i \(-0.344958\pi\)
0.468048 + 0.883703i \(0.344958\pi\)
\(180\) 0 0
\(181\) 13.7515 1.02214 0.511071 0.859538i \(-0.329249\pi\)
0.511071 + 0.859538i \(0.329249\pi\)
\(182\) 28.5123 2.11347
\(183\) 7.69015 0.568472
\(184\) −11.6729 −0.860541
\(185\) 0 0
\(186\) −11.8754 −0.870746
\(187\) −2.80637 −0.205222
\(188\) 33.7869 2.46417
\(189\) −11.4292 −0.831348
\(190\) 0 0
\(191\) 3.10678 0.224799 0.112399 0.993663i \(-0.464146\pi\)
0.112399 + 0.993663i \(0.464146\pi\)
\(192\) 27.0444 1.95176
\(193\) 0.747815 0.0538289 0.0269144 0.999638i \(-0.491432\pi\)
0.0269144 + 0.999638i \(0.491432\pi\)
\(194\) −23.2889 −1.67205
\(195\) 0 0
\(196\) 23.6492 1.68923
\(197\) 3.41798 0.243521 0.121761 0.992559i \(-0.461146\pi\)
0.121761 + 0.992559i \(0.461146\pi\)
\(198\) 3.61763 0.257094
\(199\) 5.36785 0.380517 0.190258 0.981734i \(-0.439067\pi\)
0.190258 + 0.981734i \(0.439067\pi\)
\(200\) 0 0
\(201\) −21.4808 −1.51514
\(202\) −25.0683 −1.76380
\(203\) −22.3768 −1.57054
\(204\) 16.8048 1.17657
\(205\) 0 0
\(206\) −39.9974 −2.78675
\(207\) −11.4062 −0.792789
\(208\) 6.12142 0.424444
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.55492 0.175888 0.0879441 0.996125i \(-0.471970\pi\)
0.0879441 + 0.996125i \(0.471970\pi\)
\(212\) −15.7724 −1.08325
\(213\) 24.7421 1.69530
\(214\) −3.03395 −0.207397
\(215\) 0 0
\(216\) 4.91486 0.334414
\(217\) −9.92272 −0.673598
\(218\) −0.900566 −0.0609940
\(219\) −19.3290 −1.30613
\(220\) 0 0
\(221\) 9.29418 0.625194
\(222\) −30.1710 −2.02494
\(223\) 24.9404 1.67013 0.835066 0.550149i \(-0.185429\pi\)
0.835066 + 0.550149i \(0.185429\pi\)
\(224\) 29.2622 1.95516
\(225\) 0 0
\(226\) 14.2915 0.950660
\(227\) 22.8254 1.51497 0.757487 0.652851i \(-0.226427\pi\)
0.757487 + 0.652851i \(0.226427\pi\)
\(228\) 5.98812 0.396573
\(229\) −0.603546 −0.0398834 −0.0199417 0.999801i \(-0.506348\pi\)
−0.0199417 + 0.999801i \(0.506348\pi\)
\(230\) 0 0
\(231\) 8.50056 0.559296
\(232\) 9.62264 0.631757
\(233\) −17.3705 −1.13798 −0.568988 0.822346i \(-0.692665\pi\)
−0.568988 + 0.822346i \(0.692665\pi\)
\(234\) −11.9809 −0.783219
\(235\) 0 0
\(236\) 12.0928 0.787172
\(237\) −18.3208 −1.19006
\(238\) 24.1607 1.56610
\(239\) 7.23486 0.467984 0.233992 0.972238i \(-0.424821\pi\)
0.233992 + 0.972238i \(0.424821\pi\)
\(240\) 0 0
\(241\) −12.2034 −0.786090 −0.393045 0.919519i \(-0.628578\pi\)
−0.393045 + 0.919519i \(0.628578\pi\)
\(242\) 2.18524 0.140473
\(243\) 15.5185 0.995509
\(244\) −9.89146 −0.633236
\(245\) 0 0
\(246\) −2.64337 −0.168535
\(247\) 3.31182 0.210726
\(248\) 4.26705 0.270958
\(249\) −11.2621 −0.713707
\(250\) 0 0
\(251\) −14.0923 −0.889499 −0.444750 0.895655i \(-0.646707\pi\)
−0.444750 + 0.895655i \(0.646707\pi\)
\(252\) −18.1008 −1.14024
\(253\) −6.88998 −0.433169
\(254\) −19.8421 −1.24501
\(255\) 0 0
\(256\) −2.32415 −0.145259
\(257\) −0.440920 −0.0275038 −0.0137519 0.999905i \(-0.504378\pi\)
−0.0137519 + 0.999905i \(0.504378\pi\)
\(258\) −44.3248 −2.75954
\(259\) −25.2100 −1.56647
\(260\) 0 0
\(261\) 9.40279 0.582018
\(262\) −22.6995 −1.40238
\(263\) 15.0661 0.929016 0.464508 0.885569i \(-0.346231\pi\)
0.464508 + 0.885569i \(0.346231\pi\)
\(264\) −3.65548 −0.224979
\(265\) 0 0
\(266\) 8.60924 0.527866
\(267\) 15.7169 0.961857
\(268\) 27.6297 1.68775
\(269\) −7.25751 −0.442498 −0.221249 0.975217i \(-0.571013\pi\)
−0.221249 + 0.975217i \(0.571013\pi\)
\(270\) 0 0
\(271\) −16.8878 −1.02586 −0.512931 0.858430i \(-0.671440\pi\)
−0.512931 + 0.858430i \(0.671440\pi\)
\(272\) 5.18716 0.314518
\(273\) −28.1523 −1.70386
\(274\) 1.74446 0.105387
\(275\) 0 0
\(276\) 41.2580 2.48344
\(277\) 15.3818 0.924204 0.462102 0.886827i \(-0.347095\pi\)
0.462102 + 0.886827i \(0.347095\pi\)
\(278\) 10.9958 0.659484
\(279\) 4.16956 0.249625
\(280\) 0 0
\(281\) −25.8974 −1.54491 −0.772456 0.635069i \(-0.780972\pi\)
−0.772456 + 0.635069i \(0.780972\pi\)
\(282\) −57.4015 −3.41821
\(283\) −18.6882 −1.11090 −0.555450 0.831550i \(-0.687454\pi\)
−0.555450 + 0.831550i \(0.687454\pi\)
\(284\) −31.8245 −1.88844
\(285\) 0 0
\(286\) −7.23713 −0.427941
\(287\) −2.20872 −0.130377
\(288\) −12.2961 −0.724553
\(289\) −9.12431 −0.536724
\(290\) 0 0
\(291\) 22.9949 1.34799
\(292\) 24.8620 1.45494
\(293\) −26.7471 −1.56258 −0.781291 0.624167i \(-0.785439\pi\)
−0.781291 + 0.624167i \(0.785439\pi\)
\(294\) −40.1783 −2.34325
\(295\) 0 0
\(296\) 10.8410 0.630121
\(297\) 2.90101 0.168334
\(298\) 43.3446 2.51088
\(299\) 22.8184 1.31962
\(300\) 0 0
\(301\) −37.0365 −2.13475
\(302\) 49.2657 2.83492
\(303\) 24.7518 1.42196
\(304\) 1.84836 0.106010
\(305\) 0 0
\(306\) −10.1524 −0.580374
\(307\) −22.6415 −1.29222 −0.646109 0.763245i \(-0.723605\pi\)
−0.646109 + 0.763245i \(0.723605\pi\)
\(308\) −10.9338 −0.623014
\(309\) 39.4925 2.24665
\(310\) 0 0
\(311\) −1.38723 −0.0786628 −0.0393314 0.999226i \(-0.512523\pi\)
−0.0393314 + 0.999226i \(0.512523\pi\)
\(312\) 12.1063 0.685384
\(313\) −12.9018 −0.729255 −0.364627 0.931153i \(-0.618804\pi\)
−0.364627 + 0.931153i \(0.618804\pi\)
\(314\) 25.7886 1.45534
\(315\) 0 0
\(316\) 23.5651 1.32564
\(317\) 19.0712 1.07114 0.535572 0.844489i \(-0.320096\pi\)
0.535572 + 0.844489i \(0.320096\pi\)
\(318\) 26.7961 1.50265
\(319\) 5.67979 0.318007
\(320\) 0 0
\(321\) 2.99565 0.167201
\(322\) 59.3175 3.30564
\(323\) 2.80637 0.156150
\(324\) −31.1549 −1.73083
\(325\) 0 0
\(326\) 54.2803 3.00631
\(327\) 0.889197 0.0491727
\(328\) 0.949812 0.0524446
\(329\) −47.9630 −2.64428
\(330\) 0 0
\(331\) −12.4616 −0.684952 −0.342476 0.939527i \(-0.611265\pi\)
−0.342476 + 0.939527i \(0.611265\pi\)
\(332\) 14.4859 0.795017
\(333\) 10.5933 0.580510
\(334\) 6.11733 0.334725
\(335\) 0 0
\(336\) −15.7121 −0.857163
\(337\) 0.401035 0.0218458 0.0109229 0.999940i \(-0.496523\pi\)
0.0109229 + 0.999940i \(0.496523\pi\)
\(338\) −4.44006 −0.241508
\(339\) −14.1111 −0.766411
\(340\) 0 0
\(341\) 2.51864 0.136392
\(342\) −3.61763 −0.195619
\(343\) −5.99377 −0.323633
\(344\) 15.9268 0.858713
\(345\) 0 0
\(346\) 14.0713 0.756480
\(347\) −1.06608 −0.0572304 −0.0286152 0.999591i \(-0.509110\pi\)
−0.0286152 + 0.999591i \(0.509110\pi\)
\(348\) −34.0112 −1.82319
\(349\) 22.2695 1.19206 0.596029 0.802963i \(-0.296744\pi\)
0.596029 + 0.802963i \(0.296744\pi\)
\(350\) 0 0
\(351\) −9.60762 −0.512817
\(352\) −7.42749 −0.395886
\(353\) −12.3631 −0.658023 −0.329012 0.944326i \(-0.606716\pi\)
−0.329012 + 0.944326i \(0.606716\pi\)
\(354\) −20.5447 −1.09194
\(355\) 0 0
\(356\) −20.2158 −1.07144
\(357\) −23.8557 −1.26258
\(358\) 27.3682 1.44645
\(359\) 32.1914 1.69899 0.849497 0.527593i \(-0.176905\pi\)
0.849497 + 0.527593i \(0.176905\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 30.0504 1.57941
\(363\) −2.15766 −0.113248
\(364\) 36.2109 1.89797
\(365\) 0 0
\(366\) 16.8048 0.878403
\(367\) 18.9777 0.990628 0.495314 0.868714i \(-0.335053\pi\)
0.495314 + 0.868714i \(0.335053\pi\)
\(368\) 12.7351 0.663865
\(369\) 0.928111 0.0483155
\(370\) 0 0
\(371\) 22.3901 1.16243
\(372\) −15.0819 −0.781960
\(373\) −1.15238 −0.0596678 −0.0298339 0.999555i \(-0.509498\pi\)
−0.0298339 + 0.999555i \(0.509498\pi\)
\(374\) −6.13259 −0.317109
\(375\) 0 0
\(376\) 20.6254 1.06368
\(377\) −18.8104 −0.968787
\(378\) −24.9755 −1.28460
\(379\) 13.5365 0.695325 0.347663 0.937620i \(-0.386975\pi\)
0.347663 + 0.937620i \(0.386975\pi\)
\(380\) 0 0
\(381\) 19.5916 1.00371
\(382\) 6.78907 0.347359
\(383\) −13.1671 −0.672809 −0.336404 0.941718i \(-0.609211\pi\)
−0.336404 + 0.941718i \(0.609211\pi\)
\(384\) 27.0467 1.38022
\(385\) 0 0
\(386\) 1.63416 0.0831764
\(387\) 15.5629 0.791105
\(388\) −29.5772 −1.50156
\(389\) 0.223588 0.0113364 0.00566819 0.999984i \(-0.498196\pi\)
0.00566819 + 0.999984i \(0.498196\pi\)
\(390\) 0 0
\(391\) 19.3358 0.977854
\(392\) 14.4368 0.729170
\(393\) 22.4129 1.13058
\(394\) 7.46913 0.376289
\(395\) 0 0
\(396\) 4.59444 0.230879
\(397\) 21.6504 1.08660 0.543301 0.839538i \(-0.317174\pi\)
0.543301 + 0.839538i \(0.317174\pi\)
\(398\) 11.7301 0.587975
\(399\) −8.50056 −0.425560
\(400\) 0 0
\(401\) 1.30180 0.0650089 0.0325045 0.999472i \(-0.489652\pi\)
0.0325045 + 0.999472i \(0.489652\pi\)
\(402\) −46.9408 −2.34120
\(403\) −8.34128 −0.415509
\(404\) −31.8371 −1.58395
\(405\) 0 0
\(406\) −48.8986 −2.42680
\(407\) 6.39893 0.317183
\(408\) 10.2586 0.507877
\(409\) −2.15631 −0.106622 −0.0533112 0.998578i \(-0.516978\pi\)
−0.0533112 + 0.998578i \(0.516978\pi\)
\(410\) 0 0
\(411\) −1.72244 −0.0849618
\(412\) −50.7973 −2.50260
\(413\) −17.1665 −0.844710
\(414\) −24.9254 −1.22502
\(415\) 0 0
\(416\) 24.5985 1.20604
\(417\) −10.8570 −0.531669
\(418\) −2.18524 −0.106884
\(419\) −29.6335 −1.44769 −0.723844 0.689963i \(-0.757627\pi\)
−0.723844 + 0.689963i \(0.757627\pi\)
\(420\) 0 0
\(421\) −5.25385 −0.256057 −0.128028 0.991770i \(-0.540865\pi\)
−0.128028 + 0.991770i \(0.540865\pi\)
\(422\) 5.58313 0.271782
\(423\) 20.1542 0.979931
\(424\) −9.62837 −0.467595
\(425\) 0 0
\(426\) 54.0674 2.61957
\(427\) 14.0416 0.679522
\(428\) −3.85316 −0.186249
\(429\) 7.14577 0.345001
\(430\) 0 0
\(431\) 17.8489 0.859752 0.429876 0.902888i \(-0.358557\pi\)
0.429876 + 0.902888i \(0.358557\pi\)
\(432\) −5.36209 −0.257984
\(433\) 0.696383 0.0334660 0.0167330 0.999860i \(-0.494673\pi\)
0.0167330 + 0.999860i \(0.494673\pi\)
\(434\) −21.6836 −1.04084
\(435\) 0 0
\(436\) −1.14373 −0.0547748
\(437\) 6.88998 0.329593
\(438\) −42.2386 −2.01824
\(439\) −24.8210 −1.18464 −0.592321 0.805702i \(-0.701788\pi\)
−0.592321 + 0.805702i \(0.701788\pi\)
\(440\) 0 0
\(441\) 14.1070 0.671761
\(442\) 20.3100 0.966050
\(443\) 3.93424 0.186922 0.0934608 0.995623i \(-0.470207\pi\)
0.0934608 + 0.995623i \(0.470207\pi\)
\(444\) −38.3175 −1.81847
\(445\) 0 0
\(446\) 54.5008 2.58069
\(447\) −42.7974 −2.02425
\(448\) 49.3810 2.33303
\(449\) 6.13394 0.289479 0.144739 0.989470i \(-0.453766\pi\)
0.144739 + 0.989470i \(0.453766\pi\)
\(450\) 0 0
\(451\) 0.560629 0.0263990
\(452\) 18.1505 0.853725
\(453\) −48.6437 −2.28548
\(454\) 49.8790 2.34094
\(455\) 0 0
\(456\) 3.65548 0.171184
\(457\) −8.40189 −0.393024 −0.196512 0.980501i \(-0.562961\pi\)
−0.196512 + 0.980501i \(0.562961\pi\)
\(458\) −1.31889 −0.0616279
\(459\) −8.14129 −0.380003
\(460\) 0 0
\(461\) 25.9903 1.21049 0.605246 0.796039i \(-0.293075\pi\)
0.605246 + 0.796039i \(0.293075\pi\)
\(462\) 18.5758 0.864224
\(463\) −19.3724 −0.900311 −0.450155 0.892950i \(-0.648631\pi\)
−0.450155 + 0.892950i \(0.648631\pi\)
\(464\) −10.4983 −0.487370
\(465\) 0 0
\(466\) −37.9587 −1.75840
\(467\) 34.6720 1.60443 0.802214 0.597037i \(-0.203655\pi\)
0.802214 + 0.597037i \(0.203655\pi\)
\(468\) −15.2160 −0.703358
\(469\) −39.2224 −1.81112
\(470\) 0 0
\(471\) −25.4631 −1.17328
\(472\) 7.38210 0.339789
\(473\) 9.40080 0.432249
\(474\) −40.0353 −1.83888
\(475\) 0 0
\(476\) 30.6844 1.40642
\(477\) −9.40838 −0.430780
\(478\) 15.8099 0.723130
\(479\) 23.2352 1.06164 0.530821 0.847484i \(-0.321883\pi\)
0.530821 + 0.847484i \(0.321883\pi\)
\(480\) 0 0
\(481\) −21.1921 −0.966277
\(482\) −26.6674 −1.21467
\(483\) −58.5687 −2.66497
\(484\) 2.77529 0.126149
\(485\) 0 0
\(486\) 33.9116 1.53826
\(487\) −14.2680 −0.646545 −0.323272 0.946306i \(-0.604783\pi\)
−0.323272 + 0.946306i \(0.604783\pi\)
\(488\) −6.03830 −0.273341
\(489\) −53.5951 −2.42365
\(490\) 0 0
\(491\) 41.6168 1.87814 0.939071 0.343724i \(-0.111689\pi\)
0.939071 + 0.343724i \(0.111689\pi\)
\(492\) −3.35711 −0.151350
\(493\) −15.9396 −0.717882
\(494\) 7.23713 0.325614
\(495\) 0 0
\(496\) −4.65534 −0.209031
\(497\) 45.1771 2.02647
\(498\) −24.6104 −1.10282
\(499\) 19.9664 0.893820 0.446910 0.894579i \(-0.352524\pi\)
0.446910 + 0.894579i \(0.352524\pi\)
\(500\) 0 0
\(501\) −6.04010 −0.269852
\(502\) −30.7951 −1.37445
\(503\) −2.31362 −0.103159 −0.0515797 0.998669i \(-0.516426\pi\)
−0.0515797 + 0.998669i \(0.516426\pi\)
\(504\) −11.0497 −0.492194
\(505\) 0 0
\(506\) −15.0563 −0.669334
\(507\) 4.38401 0.194701
\(508\) −25.1998 −1.11806
\(509\) −13.2009 −0.585120 −0.292560 0.956247i \(-0.594507\pi\)
−0.292560 + 0.956247i \(0.594507\pi\)
\(510\) 0 0
\(511\) −35.2933 −1.56128
\(512\) 19.9916 0.883511
\(513\) −2.90101 −0.128083
\(514\) −0.963517 −0.0424989
\(515\) 0 0
\(516\) −56.2931 −2.47817
\(517\) 12.1742 0.535421
\(518\) −55.0899 −2.42051
\(519\) −13.8937 −0.609866
\(520\) 0 0
\(521\) −15.6498 −0.685629 −0.342814 0.939403i \(-0.611380\pi\)
−0.342814 + 0.939403i \(0.611380\pi\)
\(522\) 20.5474 0.899334
\(523\) 17.7350 0.775498 0.387749 0.921765i \(-0.373253\pi\)
0.387749 + 0.921765i \(0.373253\pi\)
\(524\) −28.8286 −1.25938
\(525\) 0 0
\(526\) 32.9231 1.43552
\(527\) −7.06822 −0.307897
\(528\) 3.98812 0.173560
\(529\) 24.4719 1.06399
\(530\) 0 0
\(531\) 7.21344 0.313037
\(532\) 10.9338 0.474042
\(533\) −1.85670 −0.0804227
\(534\) 34.3452 1.48626
\(535\) 0 0
\(536\) 16.8667 0.728532
\(537\) −27.0227 −1.16612
\(538\) −15.8594 −0.683748
\(539\) 8.52137 0.367041
\(540\) 0 0
\(541\) −34.5196 −1.48412 −0.742058 0.670336i \(-0.766150\pi\)
−0.742058 + 0.670336i \(0.766150\pi\)
\(542\) −36.9040 −1.58516
\(543\) −29.6711 −1.27331
\(544\) 20.8442 0.893690
\(545\) 0 0
\(546\) −61.5197 −2.63280
\(547\) −1.99561 −0.0853263 −0.0426632 0.999090i \(-0.513584\pi\)
−0.0426632 + 0.999090i \(0.513584\pi\)
\(548\) 2.21549 0.0946411
\(549\) −5.90034 −0.251820
\(550\) 0 0
\(551\) −5.67979 −0.241967
\(552\) 25.1862 1.07200
\(553\) −33.4523 −1.42254
\(554\) 33.6130 1.42808
\(555\) 0 0
\(556\) 13.9648 0.592239
\(557\) 23.5272 0.996881 0.498440 0.866924i \(-0.333906\pi\)
0.498440 + 0.866924i \(0.333906\pi\)
\(558\) 9.11150 0.385721
\(559\) −31.1338 −1.31682
\(560\) 0 0
\(561\) 6.05517 0.255650
\(562\) −56.5922 −2.38720
\(563\) −14.2122 −0.598972 −0.299486 0.954101i \(-0.596815\pi\)
−0.299486 + 0.954101i \(0.596815\pi\)
\(564\) −72.9006 −3.06967
\(565\) 0 0
\(566\) −40.8383 −1.71656
\(567\) 44.2266 1.85734
\(568\) −19.4275 −0.815158
\(569\) 22.1906 0.930277 0.465138 0.885238i \(-0.346005\pi\)
0.465138 + 0.885238i \(0.346005\pi\)
\(570\) 0 0
\(571\) 8.67954 0.363227 0.181614 0.983370i \(-0.441868\pi\)
0.181614 + 0.983370i \(0.441868\pi\)
\(572\) −9.19126 −0.384306
\(573\) −6.70337 −0.280037
\(574\) −4.82659 −0.201458
\(575\) 0 0
\(576\) −20.7501 −0.864586
\(577\) −40.9150 −1.70331 −0.851657 0.524100i \(-0.824402\pi\)
−0.851657 + 0.524100i \(0.824402\pi\)
\(578\) −19.9388 −0.829346
\(579\) −1.61353 −0.0670559
\(580\) 0 0
\(581\) −20.5638 −0.853128
\(582\) 50.2495 2.08291
\(583\) −5.68316 −0.235373
\(584\) 15.1771 0.628034
\(585\) 0 0
\(586\) −58.4489 −2.41450
\(587\) 21.2684 0.877840 0.438920 0.898526i \(-0.355361\pi\)
0.438920 + 0.898526i \(0.355361\pi\)
\(588\) −51.0270 −2.10432
\(589\) −2.51864 −0.103779
\(590\) 0 0
\(591\) −7.37484 −0.303360
\(592\) −11.8275 −0.486107
\(593\) 23.0212 0.945368 0.472684 0.881232i \(-0.343285\pi\)
0.472684 + 0.881232i \(0.343285\pi\)
\(594\) 6.33941 0.260109
\(595\) 0 0
\(596\) 55.0482 2.25486
\(597\) −11.5820 −0.474019
\(598\) 49.8637 2.03908
\(599\) −41.6249 −1.70075 −0.850374 0.526179i \(-0.823624\pi\)
−0.850374 + 0.526179i \(0.823624\pi\)
\(600\) 0 0
\(601\) −20.8629 −0.851016 −0.425508 0.904955i \(-0.639905\pi\)
−0.425508 + 0.904955i \(0.639905\pi\)
\(602\) −80.9338 −3.29861
\(603\) 16.4814 0.671173
\(604\) 62.5680 2.54586
\(605\) 0 0
\(606\) 54.0888 2.19721
\(607\) 0.146703 0.00595450 0.00297725 0.999996i \(-0.499052\pi\)
0.00297725 + 0.999996i \(0.499052\pi\)
\(608\) 7.42749 0.301224
\(609\) 48.2813 1.95646
\(610\) 0 0
\(611\) −40.3188 −1.63113
\(612\) −12.8937 −0.521196
\(613\) 18.3393 0.740716 0.370358 0.928889i \(-0.379235\pi\)
0.370358 + 0.928889i \(0.379235\pi\)
\(614\) −49.4772 −1.99674
\(615\) 0 0
\(616\) −6.67463 −0.268929
\(617\) −4.12050 −0.165885 −0.0829425 0.996554i \(-0.526432\pi\)
−0.0829425 + 0.996554i \(0.526432\pi\)
\(618\) 86.3007 3.47153
\(619\) 28.5699 1.14832 0.574161 0.818743i \(-0.305328\pi\)
0.574161 + 0.818743i \(0.305328\pi\)
\(620\) 0 0
\(621\) −19.9879 −0.802087
\(622\) −3.03144 −0.121550
\(623\) 28.6978 1.14975
\(624\) −13.2079 −0.528740
\(625\) 0 0
\(626\) −28.1937 −1.12684
\(627\) 2.15766 0.0861685
\(628\) 32.7519 1.30694
\(629\) −17.9577 −0.716022
\(630\) 0 0
\(631\) 33.9323 1.35082 0.675412 0.737441i \(-0.263966\pi\)
0.675412 + 0.737441i \(0.263966\pi\)
\(632\) 14.3855 0.572223
\(633\) −5.51265 −0.219108
\(634\) 41.6752 1.65513
\(635\) 0 0
\(636\) 34.0315 1.34943
\(637\) −28.2213 −1.11817
\(638\) 12.4117 0.491385
\(639\) −18.9836 −0.750979
\(640\) 0 0
\(641\) 19.7694 0.780844 0.390422 0.920636i \(-0.372329\pi\)
0.390422 + 0.920636i \(0.372329\pi\)
\(642\) 6.54623 0.258359
\(643\) −36.7642 −1.44984 −0.724920 0.688833i \(-0.758123\pi\)
−0.724920 + 0.688833i \(0.758123\pi\)
\(644\) 75.3340 2.96858
\(645\) 0 0
\(646\) 6.13259 0.241284
\(647\) −24.9430 −0.980610 −0.490305 0.871551i \(-0.663115\pi\)
−0.490305 + 0.871551i \(0.663115\pi\)
\(648\) −19.0187 −0.747125
\(649\) 4.35730 0.171039
\(650\) 0 0
\(651\) 21.4098 0.839117
\(652\) 68.9367 2.69977
\(653\) −8.60802 −0.336858 −0.168429 0.985714i \(-0.553869\pi\)
−0.168429 + 0.985714i \(0.553869\pi\)
\(654\) 1.94311 0.0759817
\(655\) 0 0
\(656\) −1.03624 −0.0404584
\(657\) 14.8304 0.578588
\(658\) −104.811 −4.08595
\(659\) 27.7805 1.08217 0.541087 0.840967i \(-0.318013\pi\)
0.541087 + 0.840967i \(0.318013\pi\)
\(660\) 0 0
\(661\) 30.8037 1.19812 0.599062 0.800702i \(-0.295540\pi\)
0.599062 + 0.800702i \(0.295540\pi\)
\(662\) −27.2316 −1.05839
\(663\) −20.0537 −0.778819
\(664\) 8.84300 0.343175
\(665\) 0 0
\(666\) 23.1490 0.897004
\(667\) −39.1336 −1.51526
\(668\) 7.76909 0.300595
\(669\) −53.8128 −2.08052
\(670\) 0 0
\(671\) −3.56412 −0.137591
\(672\) −63.1378 −2.43559
\(673\) −15.2015 −0.585974 −0.292987 0.956116i \(-0.594649\pi\)
−0.292987 + 0.956116i \(0.594649\pi\)
\(674\) 0.876359 0.0337561
\(675\) 0 0
\(676\) −5.63894 −0.216882
\(677\) 9.49027 0.364741 0.182370 0.983230i \(-0.441623\pi\)
0.182370 + 0.983230i \(0.441623\pi\)
\(678\) −30.8362 −1.18426
\(679\) 41.9870 1.61131
\(680\) 0 0
\(681\) −49.2493 −1.88724
\(682\) 5.50384 0.210753
\(683\) −15.0960 −0.577631 −0.288816 0.957385i \(-0.593261\pi\)
−0.288816 + 0.957385i \(0.593261\pi\)
\(684\) −4.59444 −0.175673
\(685\) 0 0
\(686\) −13.0978 −0.500078
\(687\) 1.30224 0.0496837
\(688\) −17.3760 −0.662455
\(689\) 18.8216 0.717047
\(690\) 0 0
\(691\) 3.46673 0.131881 0.0659403 0.997824i \(-0.478995\pi\)
0.0659403 + 0.997824i \(0.478995\pi\)
\(692\) 17.8708 0.679345
\(693\) −6.52213 −0.247755
\(694\) −2.32965 −0.0884325
\(695\) 0 0
\(696\) −20.7624 −0.786995
\(697\) −1.57333 −0.0595941
\(698\) 48.6643 1.84197
\(699\) 37.4795 1.41760
\(700\) 0 0
\(701\) −26.2612 −0.991872 −0.495936 0.868359i \(-0.665175\pi\)
−0.495936 + 0.868359i \(0.665175\pi\)
\(702\) −20.9950 −0.792405
\(703\) −6.39893 −0.241340
\(704\) −12.5342 −0.472399
\(705\) 0 0
\(706\) −27.0165 −1.01678
\(707\) 45.1950 1.69973
\(708\) −26.0920 −0.980599
\(709\) −20.1488 −0.756704 −0.378352 0.925662i \(-0.623509\pi\)
−0.378352 + 0.925662i \(0.623509\pi\)
\(710\) 0 0
\(711\) 14.0568 0.527171
\(712\) −12.3409 −0.462494
\(713\) −17.3534 −0.649889
\(714\) −52.1304 −1.95093
\(715\) 0 0
\(716\) 34.7580 1.29897
\(717\) −15.6104 −0.582979
\(718\) 70.3459 2.62529
\(719\) −43.4738 −1.62130 −0.810650 0.585532i \(-0.800886\pi\)
−0.810650 + 0.585532i \(0.800886\pi\)
\(720\) 0 0
\(721\) 72.1103 2.68553
\(722\) 2.18524 0.0813263
\(723\) 26.3307 0.979250
\(724\) 38.1644 1.41837
\(725\) 0 0
\(726\) −4.71500 −0.174990
\(727\) −9.55640 −0.354427 −0.177214 0.984172i \(-0.556708\pi\)
−0.177214 + 0.984172i \(0.556708\pi\)
\(728\) 22.1052 0.819273
\(729\) 0.193986 0.00718467
\(730\) 0 0
\(731\) −26.3821 −0.975777
\(732\) 21.3424 0.788837
\(733\) 20.3072 0.750065 0.375032 0.927012i \(-0.377632\pi\)
0.375032 + 0.927012i \(0.377632\pi\)
\(734\) 41.4709 1.53072
\(735\) 0 0
\(736\) 51.1753 1.88635
\(737\) 9.95563 0.366720
\(738\) 2.02815 0.0746572
\(739\) −6.60855 −0.243099 −0.121550 0.992585i \(-0.538786\pi\)
−0.121550 + 0.992585i \(0.538786\pi\)
\(740\) 0 0
\(741\) −7.14577 −0.262507
\(742\) 48.9277 1.79619
\(743\) 39.1431 1.43602 0.718010 0.696033i \(-0.245053\pi\)
0.718010 + 0.696033i \(0.245053\pi\)
\(744\) −9.20684 −0.337539
\(745\) 0 0
\(746\) −2.51822 −0.0921988
\(747\) 8.64096 0.316156
\(748\) −7.78847 −0.284775
\(749\) 5.46983 0.199863
\(750\) 0 0
\(751\) 43.2173 1.57702 0.788511 0.615020i \(-0.210852\pi\)
0.788511 + 0.615020i \(0.210852\pi\)
\(752\) −22.5023 −0.820574
\(753\) 30.4064 1.10807
\(754\) −41.1054 −1.49697
\(755\) 0 0
\(756\) −31.7192 −1.15362
\(757\) −9.93714 −0.361171 −0.180586 0.983559i \(-0.557799\pi\)
−0.180586 + 0.983559i \(0.557799\pi\)
\(758\) 29.5806 1.07442
\(759\) 14.8662 0.539609
\(760\) 0 0
\(761\) 14.5829 0.528630 0.264315 0.964436i \(-0.414854\pi\)
0.264315 + 0.964436i \(0.414854\pi\)
\(762\) 42.8125 1.55093
\(763\) 1.62361 0.0587785
\(764\) 8.62221 0.311941
\(765\) 0 0
\(766\) −28.7734 −1.03962
\(767\) −14.4306 −0.521059
\(768\) 5.01471 0.180953
\(769\) −11.8924 −0.428853 −0.214426 0.976740i \(-0.568788\pi\)
−0.214426 + 0.976740i \(0.568788\pi\)
\(770\) 0 0
\(771\) 0.951354 0.0342622
\(772\) 2.07540 0.0746953
\(773\) −15.0935 −0.542876 −0.271438 0.962456i \(-0.587499\pi\)
−0.271438 + 0.962456i \(0.587499\pi\)
\(774\) 34.0086 1.22242
\(775\) 0 0
\(776\) −18.0556 −0.648158
\(777\) 54.3945 1.95139
\(778\) 0.488595 0.0175170
\(779\) −0.560629 −0.0200866
\(780\) 0 0
\(781\) −11.4671 −0.410325
\(782\) 42.2534 1.51098
\(783\) 16.4771 0.588844
\(784\) −15.7505 −0.562519
\(785\) 0 0
\(786\) 48.9777 1.74698
\(787\) 5.98214 0.213240 0.106620 0.994300i \(-0.465997\pi\)
0.106620 + 0.994300i \(0.465997\pi\)
\(788\) 9.48589 0.337921
\(789\) −32.5075 −1.15730
\(790\) 0 0
\(791\) −25.7659 −0.916128
\(792\) 2.80470 0.0996608
\(793\) 11.8037 0.419163
\(794\) 47.3114 1.67902
\(795\) 0 0
\(796\) 14.8973 0.528022
\(797\) −24.2718 −0.859751 −0.429876 0.902888i \(-0.641443\pi\)
−0.429876 + 0.902888i \(0.641443\pi\)
\(798\) −18.5758 −0.657576
\(799\) −34.1653 −1.20868
\(800\) 0 0
\(801\) −12.0589 −0.426081
\(802\) 2.84476 0.100452
\(803\) 8.95834 0.316133
\(804\) −59.6155 −2.10248
\(805\) 0 0
\(806\) −18.2277 −0.642044
\(807\) 15.6592 0.551230
\(808\) −19.4351 −0.683726
\(809\) −17.9557 −0.631289 −0.315644 0.948878i \(-0.602221\pi\)
−0.315644 + 0.948878i \(0.602221\pi\)
\(810\) 0 0
\(811\) 5.48300 0.192534 0.0962671 0.995356i \(-0.469310\pi\)
0.0962671 + 0.995356i \(0.469310\pi\)
\(812\) −62.1019 −2.17935
\(813\) 36.4381 1.27794
\(814\) 13.9832 0.490111
\(815\) 0 0
\(816\) −11.1921 −0.391802
\(817\) −9.40080 −0.328892
\(818\) −4.71205 −0.164753
\(819\) 21.6001 0.754770
\(820\) 0 0
\(821\) 52.6324 1.83688 0.918441 0.395558i \(-0.129449\pi\)
0.918441 + 0.395558i \(0.129449\pi\)
\(822\) −3.76395 −0.131283
\(823\) 20.4784 0.713831 0.356916 0.934137i \(-0.383828\pi\)
0.356916 + 0.934137i \(0.383828\pi\)
\(824\) −31.0095 −1.08027
\(825\) 0 0
\(826\) −37.5131 −1.30525
\(827\) 0.359755 0.0125099 0.00625496 0.999980i \(-0.498009\pi\)
0.00625496 + 0.999980i \(0.498009\pi\)
\(828\) −31.6556 −1.10011
\(829\) 22.7422 0.789870 0.394935 0.918709i \(-0.370767\pi\)
0.394935 + 0.918709i \(0.370767\pi\)
\(830\) 0 0
\(831\) −33.1887 −1.15130
\(832\) 41.5109 1.43913
\(833\) −23.9141 −0.828574
\(834\) −23.7251 −0.821535
\(835\) 0 0
\(836\) −2.77529 −0.0959853
\(837\) 7.30659 0.252553
\(838\) −64.7563 −2.23697
\(839\) 13.1616 0.454388 0.227194 0.973849i \(-0.427045\pi\)
0.227194 + 0.973849i \(0.427045\pi\)
\(840\) 0 0
\(841\) 3.25998 0.112413
\(842\) −11.4809 −0.395659
\(843\) 55.8778 1.92453
\(844\) 7.09065 0.244070
\(845\) 0 0
\(846\) 44.0418 1.51419
\(847\) −3.93972 −0.135370
\(848\) 10.5045 0.360726
\(849\) 40.3228 1.38387
\(850\) 0 0
\(851\) −44.0885 −1.51133
\(852\) 68.6663 2.35247
\(853\) 36.9102 1.26378 0.631891 0.775057i \(-0.282279\pi\)
0.631891 + 0.775057i \(0.282279\pi\)
\(854\) 30.6844 1.05000
\(855\) 0 0
\(856\) −2.35218 −0.0803960
\(857\) −24.3615 −0.832174 −0.416087 0.909325i \(-0.636599\pi\)
−0.416087 + 0.909325i \(0.636599\pi\)
\(858\) 15.6153 0.533096
\(859\) −7.69750 −0.262635 −0.131318 0.991340i \(-0.541921\pi\)
−0.131318 + 0.991340i \(0.541921\pi\)
\(860\) 0 0
\(861\) 4.76566 0.162413
\(862\) 39.0042 1.32849
\(863\) 29.0225 0.987937 0.493969 0.869480i \(-0.335546\pi\)
0.493969 + 0.869480i \(0.335546\pi\)
\(864\) −21.5472 −0.733051
\(865\) 0 0
\(866\) 1.52177 0.0517117
\(867\) 19.6871 0.668610
\(868\) −27.5384 −0.934714
\(869\) 8.49105 0.288039
\(870\) 0 0
\(871\) −32.9713 −1.11719
\(872\) −0.698197 −0.0236439
\(873\) −17.6431 −0.597127
\(874\) 15.0563 0.509286
\(875\) 0 0
\(876\) −53.6436 −1.81245
\(877\) 36.0010 1.21567 0.607834 0.794064i \(-0.292039\pi\)
0.607834 + 0.794064i \(0.292039\pi\)
\(878\) −54.2399 −1.83051
\(879\) 57.7111 1.94655
\(880\) 0 0
\(881\) −15.3587 −0.517448 −0.258724 0.965951i \(-0.583302\pi\)
−0.258724 + 0.965951i \(0.583302\pi\)
\(882\) 30.8272 1.03800
\(883\) −40.2870 −1.35577 −0.677883 0.735170i \(-0.737102\pi\)
−0.677883 + 0.735170i \(0.737102\pi\)
\(884\) 25.7940 0.867547
\(885\) 0 0
\(886\) 8.59728 0.288831
\(887\) 7.09855 0.238346 0.119173 0.992874i \(-0.461976\pi\)
0.119173 + 0.992874i \(0.461976\pi\)
\(888\) −23.3912 −0.784956
\(889\) 35.7728 1.19978
\(890\) 0 0
\(891\) −11.2258 −0.376079
\(892\) 69.2168 2.31755
\(893\) −12.1742 −0.407395
\(894\) −93.5228 −3.12787
\(895\) 0 0
\(896\) 49.3851 1.64984
\(897\) −49.2342 −1.64388
\(898\) 13.4042 0.447302
\(899\) 14.3053 0.477110
\(900\) 0 0
\(901\) 15.9490 0.531339
\(902\) 1.22511 0.0407917
\(903\) 79.9121 2.65931
\(904\) 11.0801 0.368517
\(905\) 0 0
\(906\) −106.298 −3.53153
\(907\) −28.4435 −0.944452 −0.472226 0.881478i \(-0.656549\pi\)
−0.472226 + 0.881478i \(0.656549\pi\)
\(908\) 63.3470 2.10224
\(909\) −18.9911 −0.629895
\(910\) 0 0
\(911\) 49.5740 1.64246 0.821230 0.570598i \(-0.193288\pi\)
0.821230 + 0.570598i \(0.193288\pi\)
\(912\) −3.98812 −0.132060
\(913\) 5.21960 0.172744
\(914\) −18.3602 −0.607301
\(915\) 0 0
\(916\) −1.67501 −0.0553440
\(917\) 40.9243 1.35144
\(918\) −17.7907 −0.587180
\(919\) 17.8421 0.588556 0.294278 0.955720i \(-0.404921\pi\)
0.294278 + 0.955720i \(0.404921\pi\)
\(920\) 0 0
\(921\) 48.8526 1.60975
\(922\) 56.7952 1.87045
\(923\) 37.9770 1.25003
\(924\) 23.5915 0.776103
\(925\) 0 0
\(926\) −42.3334 −1.39116
\(927\) −30.3010 −0.995215
\(928\) −42.1865 −1.38484
\(929\) 36.0244 1.18192 0.590961 0.806700i \(-0.298749\pi\)
0.590961 + 0.806700i \(0.298749\pi\)
\(930\) 0 0
\(931\) −8.52137 −0.279277
\(932\) −48.2080 −1.57911
\(933\) 2.99317 0.0979921
\(934\) 75.7667 2.47916
\(935\) 0 0
\(936\) −9.28867 −0.303610
\(937\) 3.50371 0.114461 0.0572307 0.998361i \(-0.481773\pi\)
0.0572307 + 0.998361i \(0.481773\pi\)
\(938\) −85.7104 −2.79854
\(939\) 27.8377 0.908450
\(940\) 0 0
\(941\) 44.2074 1.44112 0.720560 0.693392i \(-0.243885\pi\)
0.720560 + 0.693392i \(0.243885\pi\)
\(942\) −55.6430 −1.81295
\(943\) −3.86272 −0.125788
\(944\) −8.05385 −0.262130
\(945\) 0 0
\(946\) 20.5430 0.667912
\(947\) −37.9515 −1.23326 −0.616629 0.787254i \(-0.711502\pi\)
−0.616629 + 0.787254i \(0.711502\pi\)
\(948\) −50.8454 −1.65138
\(949\) −29.6684 −0.963077
\(950\) 0 0
\(951\) −41.1491 −1.33435
\(952\) 18.7315 0.607090
\(953\) −13.0303 −0.422094 −0.211047 0.977476i \(-0.567687\pi\)
−0.211047 + 0.977476i \(0.567687\pi\)
\(954\) −20.5596 −0.665641
\(955\) 0 0
\(956\) 20.0788 0.649396
\(957\) −12.2550 −0.396149
\(958\) 50.7745 1.64045
\(959\) −3.14505 −0.101559
\(960\) 0 0
\(961\) −24.6565 −0.795370
\(962\) −46.3099 −1.49309
\(963\) −2.29844 −0.0740662
\(964\) −33.8679 −1.09081
\(965\) 0 0
\(966\) −127.987 −4.11791
\(967\) −24.0504 −0.773409 −0.386705 0.922204i \(-0.626387\pi\)
−0.386705 + 0.922204i \(0.626387\pi\)
\(968\) 1.69419 0.0544534
\(969\) −6.05517 −0.194520
\(970\) 0 0
\(971\) 54.1335 1.73723 0.868614 0.495490i \(-0.165012\pi\)
0.868614 + 0.495490i \(0.165012\pi\)
\(972\) 43.0682 1.38141
\(973\) −19.8240 −0.635529
\(974\) −31.1791 −0.999041
\(975\) 0 0
\(976\) 6.58776 0.210869
\(977\) 50.2507 1.60766 0.803832 0.594857i \(-0.202791\pi\)
0.803832 + 0.594857i \(0.202791\pi\)
\(978\) −117.118 −3.74503
\(979\) −7.28423 −0.232805
\(980\) 0 0
\(981\) −0.682245 −0.0217824
\(982\) 90.9429 2.90210
\(983\) −6.64633 −0.211985 −0.105993 0.994367i \(-0.533802\pi\)
−0.105993 + 0.994367i \(0.533802\pi\)
\(984\) −2.04937 −0.0653315
\(985\) 0 0
\(986\) −34.8318 −1.10927
\(987\) 103.488 3.29405
\(988\) 9.19126 0.292413
\(989\) −64.7714 −2.05961
\(990\) 0 0
\(991\) 51.7993 1.64546 0.822729 0.568434i \(-0.192451\pi\)
0.822729 + 0.568434i \(0.192451\pi\)
\(992\) −18.7072 −0.593953
\(993\) 26.8879 0.853260
\(994\) 98.7230 3.13130
\(995\) 0 0
\(996\) −31.2556 −0.990371
\(997\) 14.2042 0.449851 0.224925 0.974376i \(-0.427786\pi\)
0.224925 + 0.974376i \(0.427786\pi\)
\(998\) 43.6315 1.38113
\(999\) 18.5633 0.587318
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 5225.2.a.h.1.5 5
5.4 even 2 209.2.a.c.1.1 5
15.14 odd 2 1881.2.a.k.1.5 5
20.19 odd 2 3344.2.a.t.1.2 5
55.54 odd 2 2299.2.a.n.1.5 5
95.94 odd 2 3971.2.a.h.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.1 5 5.4 even 2
1881.2.a.k.1.5 5 15.14 odd 2
2299.2.a.n.1.5 5 55.54 odd 2
3344.2.a.t.1.2 5 20.19 odd 2
3971.2.a.h.1.5 5 95.94 odd 2
5225.2.a.h.1.5 5 1.1 even 1 trivial