Properties

Label 3971.2.a.h.1.5
Level $3971$
Weight $2$
Character 3971.1
Self dual yes
Analytic conductor $31.709$
Analytic rank $1$
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] = [3971,2,Mod(1,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3971.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.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: \(31.7085946427\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 3971.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.18524 q^{2} -2.15766 q^{3} +2.77529 q^{4} -3.43077 q^{5} -4.71500 q^{6} +3.93972 q^{7} +1.69419 q^{8} +1.65548 q^{9} +O(q^{10})\) \(q+2.18524 q^{2} -2.15766 q^{3} +2.77529 q^{4} -3.43077 q^{5} -4.71500 q^{6} +3.93972 q^{7} +1.69419 q^{8} +1.65548 q^{9} -7.49706 q^{10} +1.00000 q^{11} -5.98812 q^{12} -3.31182 q^{13} +8.60924 q^{14} +7.40242 q^{15} -1.84836 q^{16} +2.80637 q^{17} +3.61763 q^{18} -9.52137 q^{20} -8.50056 q^{21} +2.18524 q^{22} +6.88998 q^{23} -3.65548 q^{24} +6.77018 q^{25} -7.23713 q^{26} +2.90101 q^{27} +10.9338 q^{28} -5.67979 q^{29} +16.1761 q^{30} -2.51864 q^{31} -7.42749 q^{32} -2.15766 q^{33} +6.13259 q^{34} -13.5163 q^{35} +4.59444 q^{36} +6.39893 q^{37} +7.14577 q^{39} -5.81238 q^{40} -0.560629 q^{41} -18.5758 q^{42} -9.40080 q^{43} +2.77529 q^{44} -5.67958 q^{45} +15.0563 q^{46} -12.1742 q^{47} +3.98812 q^{48} +8.52137 q^{49} +14.7945 q^{50} -6.05517 q^{51} -9.19126 q^{52} -5.68316 q^{53} +6.33941 q^{54} -3.43077 q^{55} +6.67463 q^{56} -12.4117 q^{58} -4.35730 q^{59} +20.5438 q^{60} -3.56412 q^{61} -5.50384 q^{62} +6.52213 q^{63} -12.5342 q^{64} +11.3621 q^{65} -4.71500 q^{66} +9.95563 q^{67} +7.78847 q^{68} -14.8662 q^{69} -29.5363 q^{70} +11.4671 q^{71} +2.80470 q^{72} -8.95834 q^{73} +13.9832 q^{74} -14.6077 q^{75} +3.93972 q^{77} +15.6153 q^{78} -8.49105 q^{79} +6.34128 q^{80} -11.2258 q^{81} -1.22511 q^{82} -5.21960 q^{83} -23.5915 q^{84} -9.62799 q^{85} -20.5430 q^{86} +12.2550 q^{87} +1.69419 q^{88} +7.28423 q^{89} -12.4113 q^{90} -13.0476 q^{91} +19.1217 q^{92} +5.43436 q^{93} -26.6036 q^{94} +16.0260 q^{96} -10.6574 q^{97} +18.6213 q^{98} +1.65548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 5 q^{11} - 6 q^{12} - 4 q^{13} + 14 q^{14} - 3 q^{15} + 8 q^{16} - 4 q^{17} + 20 q^{18} - 8 q^{20} - 10 q^{21} - 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} + 11 q^{27} - 10 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 14 q^{32} - q^{33} + 4 q^{34} - 8 q^{35} - 26 q^{36} - q^{37} + 2 q^{39} + 16 q^{40} - 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} + 4 q^{46} - 20 q^{47} - 4 q^{48} + 3 q^{49} + 32 q^{50} - 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} + 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} - 9 q^{67} + 24 q^{68} + 5 q^{69} - 50 q^{70} - 23 q^{71} + 12 q^{72} + 8 q^{74} + 18 q^{75} + 6 q^{77} + 22 q^{78} - 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} - 14 q^{84} - 12 q^{85} - 52 q^{86} + 28 q^{87} - 6 q^{88} + 27 q^{89} - 26 q^{90} - 24 q^{91} + 58 q^{92} - 27 q^{93} + 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(5\) −3.43077 −1.53429 −0.767143 0.641476i \(-0.778322\pi\)
−0.767143 + 0.641476i \(0.778322\pi\)
\(6\) −4.71500 −1.92489
\(7\) 3.93972 1.48907 0.744537 0.667582i \(-0.232671\pi\)
0.744537 + 0.667582i \(0.232671\pi\)
\(8\) 1.69419 0.598987
\(9\) 1.65548 0.551827
\(10\) −7.49706 −2.37078
\(11\) 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
\(15\) 7.40242 1.91130
\(16\) −1.84836 −0.462089
\(17\) 2.80637 0.680644 0.340322 0.940309i \(-0.389464\pi\)
0.340322 + 0.940309i \(0.389464\pi\)
\(18\) 3.61763 0.852684
\(19\) 0 0
\(20\) −9.52137 −2.12904
\(21\) −8.50056 −1.85497
\(22\) 2.18524 0.465895
\(23\) 6.88998 1.43666 0.718330 0.695702i \(-0.244907\pi\)
0.718330 + 0.695702i \(0.244907\pi\)
\(24\) −3.65548 −0.746172
\(25\) 6.77018 1.35404
\(26\) −7.23713 −1.41932
\(27\) 2.90101 0.558299
\(28\) 10.9338 2.06630
\(29\) −5.67979 −1.05471 −0.527355 0.849645i \(-0.676816\pi\)
−0.527355 + 0.849645i \(0.676816\pi\)
\(30\) 16.1761 2.95334
\(31\) −2.51864 −0.452361 −0.226180 0.974085i \(-0.572624\pi\)
−0.226180 + 0.974085i \(0.572624\pi\)
\(32\) −7.42749 −1.31301
\(33\) −2.15766 −0.375600
\(34\) 6.13259 1.05173
\(35\) −13.5163 −2.28466
\(36\) 4.59444 0.765740
\(37\) 6.39893 1.05198 0.525989 0.850491i \(-0.323695\pi\)
0.525989 + 0.850491i \(0.323695\pi\)
\(38\) 0 0
\(39\) 7.14577 1.14424
\(40\) −5.81238 −0.919018
\(41\) −0.560629 −0.0875555 −0.0437778 0.999041i \(-0.513939\pi\)
−0.0437778 + 0.999041i \(0.513939\pi\)
\(42\) −18.5758 −2.86631
\(43\) −9.40080 −1.43361 −0.716805 0.697274i \(-0.754396\pi\)
−0.716805 + 0.697274i \(0.754396\pi\)
\(44\) 2.77529 0.418390
\(45\) −5.67958 −0.846661
\(46\) 15.0563 2.21993
\(47\) −12.1742 −1.77579 −0.887896 0.460044i \(-0.847834\pi\)
−0.887896 + 0.460044i \(0.847834\pi\)
\(48\) 3.98812 0.575635
\(49\) 8.52137 1.21734
\(50\) 14.7945 2.09226
\(51\) −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
\(55\) −3.43077 −0.462605
\(56\) 6.67463 0.891935
\(57\) 0 0
\(58\) −12.4117 −1.62974
\(59\) −4.35730 −0.567273 −0.283636 0.958932i \(-0.591541\pi\)
−0.283636 + 0.958932i \(0.591541\pi\)
\(60\) 20.5438 2.65220
\(61\) −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
\(65\) 11.3621 1.40929
\(66\) −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
\(70\) −29.5363 −3.53026
\(71\) 11.4671 1.36089 0.680447 0.732797i \(-0.261786\pi\)
0.680447 + 0.732797i \(0.261786\pi\)
\(72\) 2.80470 0.330537
\(73\) −8.95834 −1.04849 −0.524247 0.851566i \(-0.675653\pi\)
−0.524247 + 0.851566i \(0.675653\pi\)
\(74\) 13.9832 1.62552
\(75\) −14.6077 −1.68675
\(76\) 0 0
\(77\) 3.93972 0.448972
\(78\) 15.6153 1.76808
\(79\) −8.49105 −0.955318 −0.477659 0.878545i \(-0.658515\pi\)
−0.477659 + 0.878545i \(0.658515\pi\)
\(80\) 6.34128 0.708977
\(81\) −11.2258 −1.24731
\(82\) −1.22511 −0.135291
\(83\) −5.21960 −0.572926 −0.286463 0.958091i \(-0.592480\pi\)
−0.286463 + 0.958091i \(0.592480\pi\)
\(84\) −23.5915 −2.57404
\(85\) −9.62799 −1.04430
\(86\) −20.5430 −2.21521
\(87\) 12.2550 1.31388
\(88\) 1.69419 0.180601
\(89\) 7.28423 0.772127 0.386064 0.922472i \(-0.373834\pi\)
0.386064 + 0.922472i \(0.373834\pi\)
\(90\) −12.4113 −1.30826
\(91\) −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
\(100\) 18.7892 1.87892
\(101\) −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
\(105\) 29.1634 2.84606
\(106\) −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 \(-0.493717\pi\)
0.0197366 + 0.999805i \(0.493717\pi\)
\(110\) −7.49706 −0.714817
\(111\) −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 \(0.400468\pi\)
\(114\) 0 0
\(115\) −23.6379 −2.20425
\(116\) −15.7630 −1.46356
\(117\) −5.48266 −0.506872
\(118\) −9.52177 −0.876550
\(119\) 11.0563 1.01353
\(120\) 12.5411 1.14484
\(121\) 1.00000 0.0909091
\(122\) −7.78847 −0.705135
\(123\) 1.20964 0.109070
\(124\) −6.98995 −0.627716
\(125\) −6.07307 −0.543192
\(126\) 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
\(130\) 24.8289 2.17764
\(131\) −10.3876 −0.907571 −0.453785 0.891111i \(-0.649927\pi\)
−0.453785 + 0.891111i \(0.649927\pi\)
\(132\) −5.98812 −0.521199
\(133\) 0 0
\(134\) 21.7555 1.87939
\(135\) −9.95269 −0.856591
\(136\) 4.75452 0.407697
\(137\) −0.798293 −0.0682028 −0.0341014 0.999418i \(-0.510857\pi\)
−0.0341014 + 0.999418i \(0.510857\pi\)
\(138\) −32.4863 −2.76542
\(139\) 5.03184 0.426795 0.213398 0.976965i \(-0.431547\pi\)
0.213398 + 0.976965i \(0.431547\pi\)
\(140\) −37.5115 −3.17030
\(141\) 26.2678 2.21215
\(142\) 25.0584 2.10285
\(143\) −3.31182 −0.276948
\(144\) −3.05992 −0.254993
\(145\) 19.4860 1.61823
\(146\) −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 \(0.198118\pi\)
\(150\) −31.9214 −2.60637
\(151\) −22.5447 −1.83466 −0.917331 0.398125i \(-0.869661\pi\)
−0.917331 + 0.398125i \(0.869661\pi\)
\(152\) 0 0
\(153\) 4.64589 0.375598
\(154\) 8.60924 0.693752
\(155\) 8.64087 0.694051
\(156\) 19.8316 1.58780
\(157\) −11.8013 −0.941843 −0.470921 0.882175i \(-0.656078\pi\)
−0.470921 + 0.882175i \(0.656078\pi\)
\(158\) −18.5550 −1.47616
\(159\) 12.2623 0.972465
\(160\) 25.4820 2.01453
\(161\) 27.1446 2.13929
\(162\) −24.5312 −1.92735
\(163\) −24.8395 −1.94558 −0.972789 0.231691i \(-0.925574\pi\)
−0.972789 + 0.231691i \(0.925574\pi\)
\(164\) −1.55591 −0.121496
\(165\) 7.40242 0.576278
\(166\) −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
\(170\) −21.0395 −1.61366
\(171\) 0 0
\(172\) −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
\(175\) 26.6726 2.01626
\(176\) −1.84836 −0.139325
\(177\) 9.40156 0.706665
\(178\) 15.9178 1.19309
\(179\) −12.5241 −0.936095 −0.468048 0.883703i \(-0.655042\pi\)
−0.468048 + 0.883703i \(0.655042\pi\)
\(180\) −15.7625 −1.17486
\(181\) −13.7515 −1.02214 −0.511071 0.859538i \(-0.670751\pi\)
−0.511071 + 0.859538i \(0.670751\pi\)
\(182\) −28.5123 −2.11347
\(183\) 7.69015 0.568472
\(184\) 11.6729 0.860541
\(185\) −21.9533 −1.61404
\(186\) 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
\(195\) −24.5155 −1.75559
\(196\) 23.6492 1.68923
\(197\) −3.41798 −0.243521 −0.121761 0.992559i \(-0.538854\pi\)
−0.121761 + 0.992559i \(0.538854\pi\)
\(198\) 3.61763 0.257094
\(199\) 5.36785 0.380517 0.190258 0.981734i \(-0.439067\pi\)
0.190258 + 0.981734i \(0.439067\pi\)
\(200\) 11.4700 0.811049
\(201\) −21.4808 −1.51514
\(202\) −25.0683 −1.76380
\(203\) −22.3768 −1.57054
\(204\) −16.8048 −1.17657
\(205\) 1.92339 0.134335
\(206\) −39.9974 −2.78675
\(207\) 11.4062 0.792789
\(208\) 6.12142 0.424444
\(209\) 0 0
\(210\) 63.7292 4.39773
\(211\) −2.55492 −0.175888 −0.0879441 0.996125i \(-0.528030\pi\)
−0.0879441 + 0.996125i \(0.528030\pi\)
\(212\) −15.7724 −1.08325
\(213\) −24.7421 −1.69530
\(214\) −3.03395 −0.207397
\(215\) 32.2520 2.19957
\(216\) 4.91486 0.334414
\(217\) −9.92272 −0.673598
\(218\) 0.900566 0.0609940
\(219\) 19.3290 1.30613
\(220\) −9.52137 −0.641931
\(221\) −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
\(225\) 11.2079 0.747194
\(226\) 14.2915 0.950660
\(227\) 22.8254 1.51497 0.757487 0.652851i \(-0.226427\pi\)
0.757487 + 0.652851i \(0.226427\pi\)
\(228\) 0 0
\(229\) −0.603546 −0.0398834 −0.0199417 0.999801i \(-0.506348\pi\)
−0.0199417 + 0.999801i \(0.506348\pi\)
\(230\) −51.6546 −3.40601
\(231\) −8.50056 −0.559296
\(232\) −9.62264 −0.631757
\(233\) 17.3705 1.13798 0.568988 0.822346i \(-0.307335\pi\)
0.568988 + 0.822346i \(0.307335\pi\)
\(234\) −11.9809 −0.783219
\(235\) 41.7669 2.72457
\(236\) −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 \(0.424821\pi\)
\(240\) −13.6823 −0.883189
\(241\) 12.2034 0.786090 0.393045 0.919519i \(-0.371422\pi\)
0.393045 + 0.919519i \(0.371422\pi\)
\(242\) 2.18524 0.140473
\(243\) 15.5185 0.995509
\(244\) −9.89146 −0.633236
\(245\) −29.2349 −1.86775
\(246\) 2.64337 0.168535
\(247\) 0 0
\(248\) −4.26705 −0.270958
\(249\) 11.2621 0.713707
\(250\) −13.2711 −0.839340
\(251\) −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
\(255\) 20.7739 1.30091
\(256\) −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
\(260\) 31.5331 1.95560
\(261\) −9.40279 −0.582018
\(262\) −22.6995 −1.40238
\(263\) −15.0661 −0.929016 −0.464508 0.885569i \(-0.653769\pi\)
−0.464508 + 0.885569i \(0.653769\pi\)
\(264\) −3.65548 −0.224979
\(265\) 19.4976 1.19773
\(266\) 0 0
\(267\) −15.7169 −0.961857
\(268\) 27.6297 1.68775
\(269\) 7.25751 0.442498 0.221249 0.975217i \(-0.428987\pi\)
0.221249 + 0.975217i \(0.428987\pi\)
\(270\) −21.7490 −1.32360
\(271\) −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
\(275\) 6.77018 0.408257
\(276\) −41.2580 −2.48344
\(277\) −15.3818 −0.924204 −0.462102 0.886827i \(-0.652905\pi\)
−0.462102 + 0.886827i \(0.652905\pi\)
\(278\) 10.9958 0.659484
\(279\) −4.16956 −0.249625
\(280\) −22.8991 −1.36848
\(281\) 25.8974 1.54491 0.772456 0.635069i \(-0.219028\pi\)
0.772456 + 0.635069i \(0.219028\pi\)
\(282\) 57.4015 3.41821
\(283\) 18.6882 1.11090 0.555450 0.831550i \(-0.312546\pi\)
0.555450 + 0.831550i \(0.312546\pi\)
\(284\) 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
\(290\) 42.5817 2.50049
\(291\) 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
\(295\) 14.9489 0.870359
\(296\) 10.8410 0.630121
\(297\) 2.90101 0.168334
\(298\) 43.3446 2.51088
\(299\) −22.8184 −1.31962
\(300\) −40.5406 −2.34061
\(301\) −37.0365 −2.13475
\(302\) −49.2657 −2.83492
\(303\) 24.7518 1.42196
\(304\) 0 0
\(305\) 12.2277 0.700155
\(306\) 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
\(310\) 18.8824 1.07245
\(311\) −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 \(-0.381196\pi\)
0.364627 + 0.931153i \(0.381196\pi\)
\(314\) −25.7886 −1.45534
\(315\) −22.3759 −1.26074
\(316\) −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
\(320\) 43.0018 2.40387
\(321\) 2.99565 0.167201
\(322\) 59.3175 3.30564
\(323\) 0 0
\(324\) −31.1549 −1.73083
\(325\) −22.4216 −1.24373
\(326\) −54.2803 −3.00631
\(327\) −0.889197 −0.0491727
\(328\) −0.949812 −0.0524446
\(329\) −47.9630 −2.64428
\(330\) 16.1761 0.890464
\(331\) 12.4616 0.684952 0.342476 0.939527i \(-0.388735\pi\)
0.342476 + 0.939527i \(0.388735\pi\)
\(332\) −14.4859 −0.795017
\(333\) 10.5933 0.580510
\(334\) 6.11733 0.334725
\(335\) −34.1555 −1.86611
\(336\) 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
\(340\) −26.7204 −1.44912
\(341\) −2.51864 −0.136392
\(342\) 0 0
\(343\) 5.99377 0.323633
\(344\) −15.9268 −0.858713
\(345\) 51.0026 2.74589
\(346\) 14.0713 0.756480
\(347\) 1.06608 0.0572304 0.0286152 0.999591i \(-0.490890\pi\)
0.0286152 + 0.999591i \(0.490890\pi\)
\(348\) 34.0112 1.82319
\(349\) 22.2695 1.19206 0.596029 0.802963i \(-0.296744\pi\)
0.596029 + 0.802963i \(0.296744\pi\)
\(350\) 58.2861 3.11552
\(351\) −9.60762 −0.512817
\(352\) −7.42749 −0.395886
\(353\) 12.3631 0.658023 0.329012 0.944326i \(-0.393284\pi\)
0.329012 + 0.944326i \(0.393284\pi\)
\(354\) 20.5447 1.09194
\(355\) −39.3410 −2.08800
\(356\) 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 \(0.176905\pi\)
\(360\) −9.62229 −0.507139
\(361\) 0 0
\(362\) −30.0504 −1.57941
\(363\) −2.15766 −0.113248
\(364\) −36.2109 −1.89797
\(365\) 30.7340 1.60869
\(366\) 16.8048 0.878403
\(367\) −18.9777 −0.990628 −0.495314 0.868714i \(-0.664947\pi\)
−0.495314 + 0.868714i \(0.664947\pi\)
\(368\) −12.7351 −0.663865
\(369\) −0.928111 −0.0483155
\(370\) −47.9732 −2.49401
\(371\) −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
\(375\) 13.1036 0.676667
\(376\) −20.6254 −1.06368
\(377\) 18.8104 0.968787
\(378\) 24.9755 1.28460
\(379\) −13.5365 −0.695325 −0.347663 0.937620i \(-0.613025\pi\)
−0.347663 + 0.937620i \(0.613025\pi\)
\(380\) 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
\(385\) −13.5163 −0.688852
\(386\) 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 \(0.498196\pi\)
\(390\) −53.5723 −2.71274
\(391\) 19.3358 0.977854
\(392\) 14.4368 0.729170
\(393\) 22.4129 1.13058
\(394\) −7.46913 −0.376289
\(395\) 29.1308 1.46573
\(396\) 4.59444 0.230879
\(397\) −21.6504 −1.08660 −0.543301 0.839538i \(-0.682826\pi\)
−0.543301 + 0.839538i \(0.682826\pi\)
\(398\) 11.7301 0.587975
\(399\) 0 0
\(400\) −12.5137 −0.625685
\(401\) −1.30180 −0.0650089 −0.0325045 0.999472i \(-0.510348\pi\)
−0.0325045 + 0.999472i \(0.510348\pi\)
\(402\) −46.9408 −2.34120
\(403\) 8.34128 0.415509
\(404\) −31.8371 −1.58395
\(405\) 38.5132 1.91374
\(406\) −48.8986 −2.42680
\(407\) 6.39893 0.317183
\(408\) −10.2586 −0.507877
\(409\) 2.15631 0.106622 0.0533112 0.998578i \(-0.483022\pi\)
0.0533112 + 0.998578i \(0.483022\pi\)
\(410\) 4.20307 0.207575
\(411\) 1.72244 0.0849618
\(412\) −50.7973 −2.50260
\(413\) −17.1665 −0.844710
\(414\) 24.9254 1.22502
\(415\) 17.9073 0.879032
\(416\) 24.5985 1.20604
\(417\) −10.8570 −0.531669
\(418\) 0 0
\(419\) −29.6335 −1.44769 −0.723844 0.689963i \(-0.757627\pi\)
−0.723844 + 0.689963i \(0.757627\pi\)
\(420\) 80.9369 3.94932
\(421\) 5.25385 0.256057 0.128028 0.991770i \(-0.459135\pi\)
0.128028 + 0.991770i \(0.459135\pi\)
\(422\) −5.58313 −0.271782
\(423\) −20.1542 −0.979931
\(424\) −9.62837 −0.467595
\(425\) 18.9996 0.921616
\(426\) −54.0674 −2.61957
\(427\) −14.0416 −0.679522
\(428\) −3.85316 −0.186249
\(429\) 7.14577 0.345001
\(430\) 70.4784 3.39877
\(431\) −17.8489 −0.859752 −0.429876 0.902888i \(-0.641443\pi\)
−0.429876 + 0.902888i \(0.641443\pi\)
\(432\) −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
\(435\) −42.0442 −2.01586
\(436\) 1.14373 0.0547748
\(437\) 0 0
\(438\) 42.2386 2.01824
\(439\) 24.8210 1.18464 0.592321 0.805702i \(-0.298212\pi\)
0.592321 + 0.805702i \(0.298212\pi\)
\(440\) −5.81238 −0.277094
\(441\) 14.1070 0.671761
\(442\) −20.3100 −0.966050
\(443\) −3.93424 −0.186922 −0.0934608 0.995623i \(-0.529793\pi\)
−0.0934608 + 0.995623i \(0.529793\pi\)
\(444\) −38.3175 −1.81847
\(445\) −24.9905 −1.18466
\(446\) 54.5008 2.58069
\(447\) −42.7974 −2.02425
\(448\) −49.3810 −2.33303
\(449\) −6.13394 −0.289479 −0.144739 0.989470i \(-0.546234\pi\)
−0.144739 + 0.989470i \(0.546234\pi\)
\(450\) 24.4920 1.15456
\(451\) −0.560629 −0.0263990
\(452\) 18.1505 0.853725
\(453\) 48.6437 2.28548
\(454\) 49.8790 2.34094
\(455\) 44.7634 2.09854
\(456\) 0 0
\(457\) 8.40189 0.393024 0.196512 0.980501i \(-0.437039\pi\)
0.196512 + 0.980501i \(0.437039\pi\)
\(458\) −1.31889 −0.0616279
\(459\) 8.14129 0.380003
\(460\) −65.6021 −3.05871
\(461\) 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 \(-0.351369\pi\)
0.450155 + 0.892950i \(0.351369\pi\)
\(464\) 10.4983 0.487370
\(465\) −18.6440 −0.864596
\(466\) 37.9587 1.75840
\(467\) −34.6720 −1.60443 −0.802214 0.597037i \(-0.796345\pi\)
−0.802214 + 0.597037i \(0.796345\pi\)
\(468\) −15.2160 −0.703358
\(469\) 39.2224 1.81112
\(470\) 91.2709 4.21001
\(471\) 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 \(0.321883\pi\)
\(480\) −54.9814 −2.50955
\(481\) −21.1921 −0.966277
\(482\) 26.6674 1.21467
\(483\) −58.5687 −2.66497
\(484\) 2.77529 0.126149
\(485\) 36.5629 1.66024
\(486\) 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
\(490\) −63.8853 −2.88604
\(491\) 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
\(494\) 0 0
\(495\) −5.67958 −0.255278
\(496\) 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 \(0.352524\pi\)
\(500\) −16.8545 −0.753757
\(501\) −6.04010 −0.269852
\(502\) −30.7951 −1.37445
\(503\) 2.31362 0.103159 0.0515797 0.998669i \(-0.483574\pi\)
0.0515797 + 0.998669i \(0.483574\pi\)
\(504\) 11.0497 0.492194
\(505\) 39.3565 1.75134
\(506\) 15.0563 0.669334
\(507\) 4.38401 0.194701
\(508\) −25.1998 −1.11806
\(509\) 13.2009 0.585120 0.292560 0.956247i \(-0.405493\pi\)
0.292560 + 0.956247i \(0.405493\pi\)
\(510\) 45.3960 2.01017
\(511\) −35.2933 −1.56128
\(512\) 19.9916 0.883511
\(513\) 0 0
\(514\) −0.963517 −0.0424989
\(515\) 62.7948 2.76707
\(516\) 56.2931 2.47817
\(517\) −12.1742 −0.535421
\(518\) 55.0899 2.42051
\(519\) −13.8937 −0.609866
\(520\) 19.2496 0.844149
\(521\) 15.6498 0.685629 0.342814 0.939403i \(-0.388620\pi\)
0.342814 + 0.939403i \(0.388620\pi\)
\(522\) −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
\(525\) −57.5503 −2.51170
\(526\) −32.9231 −1.43552
\(527\) −7.06822 −0.307897
\(528\) 3.98812 0.173560
\(529\) 24.4719 1.06399
\(530\) 42.6071 1.85073
\(531\) −7.21344 −0.313037
\(532\) 0 0
\(533\) 1.85670 0.0804227
\(534\) −34.3452 −1.48626
\(535\) 4.76322 0.205932
\(536\) 16.8667 0.728532
\(537\) 27.0227 1.16612
\(538\) 15.8594 0.683748
\(539\) 8.52137 0.367041
\(540\) −27.6216 −1.18864
\(541\) −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
\(545\) −1.41386 −0.0605632
\(546\) 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
\(550\) 14.7945 0.630839
\(551\) 0 0
\(552\) −25.1862 −1.07200
\(553\) −33.4523 −1.42254
\(554\) −33.6130 −1.42808
\(555\) 47.3676 2.01064
\(556\) 13.9648 0.592239
\(557\) −23.5272 −0.996881 −0.498440 0.866924i \(-0.666094\pi\)
−0.498440 + 0.866924i \(0.666094\pi\)
\(558\) −9.11150 −0.385721
\(559\) 31.1338 1.31682
\(560\) 24.9829 1.05572
\(561\) −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
\(565\) −22.4373 −0.943945
\(566\) 40.8383 1.71656
\(567\) −44.2266 −1.85734
\(568\) 19.4275 0.815158
\(569\) −22.1906 −0.930277 −0.465138 0.885238i \(-0.653995\pi\)
−0.465138 + 0.885238i \(0.653995\pi\)
\(570\) 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
\(575\) 46.6464 1.94529
\(576\) −20.7501 −0.864586
\(577\) 40.9150 1.70331 0.851657 0.524100i \(-0.175598\pi\)
0.851657 + 0.524100i \(0.175598\pi\)
\(578\) −19.9388 −0.829346
\(579\) −1.61353 −0.0670559
\(580\) 54.0794 2.24552
\(581\) −20.5638 −0.853128
\(582\) 50.2495 2.08291
\(583\) −5.68316 −0.235373
\(584\) −15.1771 −0.628034
\(585\) 18.8097 0.777687
\(586\) −58.4489 −2.41450
\(587\) −21.2684 −0.877840 −0.438920 0.898526i \(-0.644639\pi\)
−0.438920 + 0.898526i \(0.644639\pi\)
\(588\) −51.0270 −2.10432
\(589\) 0 0
\(590\) 32.6670 1.34488
\(591\) 7.37484 0.303360
\(592\) −11.8275 −0.486107
\(593\) −23.0212 −0.945368 −0.472684 0.881232i \(-0.656715\pi\)
−0.472684 + 0.881232i \(0.656715\pi\)
\(594\) 6.33941 0.260109
\(595\) −37.9316 −1.55504
\(596\) 55.0482 2.25486
\(597\) −11.5820 −0.474019
\(598\) −49.8637 −2.03908
\(599\) 41.6249 1.70075 0.850374 0.526179i \(-0.176376\pi\)
0.850374 + 0.526179i \(0.176376\pi\)
\(600\) −24.7483 −1.01034
\(601\) 20.8629 0.851016 0.425508 0.904955i \(-0.360095\pi\)
0.425508 + 0.904955i \(0.360095\pi\)
\(602\) −80.9338 −3.29861
\(603\) 16.4814 0.671173
\(604\) −62.5680 −2.54586
\(605\) −3.43077 −0.139481
\(606\) 54.0888 2.19721
\(607\) 0.146703 0.00595450 0.00297725 0.999996i \(-0.499052\pi\)
0.00297725 + 0.999996i \(0.499052\pi\)
\(608\) 0 0
\(609\) 48.2813 1.95646
\(610\) 26.7204 1.08188
\(611\) 40.3188 1.63113
\(612\) 12.8937 0.521196
\(613\) −18.3393 −0.740716 −0.370358 0.928889i \(-0.620765\pi\)
−0.370358 + 0.928889i \(0.620765\pi\)
\(614\) −49.4772 −1.99674
\(615\) −4.15001 −0.167345
\(616\) 6.67463 0.268929
\(617\) 4.12050 0.165885 0.0829425 0.996554i \(-0.473568\pi\)
0.0829425 + 0.996554i \(0.473568\pi\)
\(618\) 86.3007 3.47153
\(619\) 28.5699 1.14832 0.574161 0.818743i \(-0.305328\pi\)
0.574161 + 0.818743i \(0.305328\pi\)
\(620\) 23.9809 0.963096
\(621\) 19.9879 0.802087
\(622\) −3.03144 −0.121550
\(623\) 28.6978 1.14975
\(624\) −13.2079 −0.528740
\(625\) −13.0156 −0.520624
\(626\) 28.1937 1.12684
\(627\) 0 0
\(628\) −32.7519 −1.30694
\(629\) 17.9577 0.716022
\(630\) −48.8968 −1.94810
\(631\) 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
\(635\) 31.1516 1.23621
\(636\) 34.0315 1.34943
\(637\) −28.2213 −1.11817
\(638\) −12.4117 −0.491385
\(639\) 18.9836 0.750979
\(640\) 43.0054 1.69994
\(641\) −19.7694 −0.780844 −0.390422 0.920636i \(-0.627671\pi\)
−0.390422 + 0.920636i \(0.627671\pi\)
\(642\) 6.54623 0.258359
\(643\) 36.7642 1.44984 0.724920 0.688833i \(-0.241877\pi\)
0.724920 + 0.688833i \(0.241877\pi\)
\(644\) 75.3340 2.96858
\(645\) −69.5887 −2.74005
\(646\) 0 0
\(647\) 24.9430 0.980610 0.490305 0.871551i \(-0.336885\pi\)
0.490305 + 0.871551i \(0.336885\pi\)
\(648\) −19.0187 −0.747125
\(649\) −4.35730 −0.171039
\(650\) −48.9967 −1.92181
\(651\) 21.4098 0.839117
\(652\) −68.9367 −2.69977
\(653\) 8.60802 0.336858 0.168429 0.985714i \(-0.446131\pi\)
0.168429 + 0.985714i \(0.446131\pi\)
\(654\) −1.94311 −0.0759817
\(655\) 35.6375 1.39247
\(656\) 1.03624 0.0404584
\(657\) −14.8304 −0.578588
\(658\) −104.811 −4.08595
\(659\) −27.7805 −1.08217 −0.541087 0.840967i \(-0.681987\pi\)
−0.541087 + 0.840967i \(0.681987\pi\)
\(660\) 20.5438 0.799668
\(661\) −30.8037 −1.19812 −0.599062 0.800702i \(-0.704460\pi\)
−0.599062 + 0.800702i \(0.704460\pi\)
\(662\) 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
\(670\) −74.6380 −2.88352
\(671\) −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
\(675\) 19.6403 0.755957
\(676\) −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
\(680\) −16.3117 −0.625523
\(681\) −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 \(0.593261\pi\)
\(684\) 0 0
\(685\) 2.73876 0.104643
\(686\) 13.0978 0.500078
\(687\) 1.30224 0.0496837
\(688\) 17.3760 0.662455
\(689\) 18.8216 0.717047
\(690\) 111.453 4.24294
\(691\) 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
\(695\) −17.2631 −0.654826
\(696\) 20.7624 0.786995
\(697\) −1.57333 −0.0595941
\(698\) 48.6643 1.84197
\(699\) −37.4795 −1.41760
\(700\) 74.0241 2.79785
\(701\) −26.2612 −0.991872 −0.495936 0.868359i \(-0.665175\pi\)
−0.495936 + 0.868359i \(0.665175\pi\)
\(702\) −20.9950 −0.792405
\(703\) 0 0
\(704\) −12.5342 −0.472399
\(705\) −90.1187 −3.39407
\(706\) 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 \(0.623509\pi\)
\(710\) −85.9696 −3.22638
\(711\) −14.0568 −0.527171
\(712\) 12.3409 0.462494
\(713\) −17.3534 −0.649889
\(714\) −52.1304 −1.95093
\(715\) 11.3621 0.424918
\(716\) −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 \(0.800886\pi\)
\(720\) 10.4979 0.391233
\(721\) −72.1103 −2.68553
\(722\) 0 0
\(723\) −26.3307 −0.979250
\(724\) −38.1644 −1.41837
\(725\) −38.4532 −1.42811
\(726\) −4.71500 −0.174990
\(727\) 9.55640 0.354427 0.177214 0.984172i \(-0.443292\pi\)
0.177214 + 0.984172i \(0.443292\pi\)
\(728\) −22.1052 −0.819273
\(729\) 0.193986 0.00718467
\(730\) 67.1612 2.48575
\(731\) −26.3821 −0.975777
\(732\) 21.3424 0.788837
\(733\) −20.3072 −0.750065 −0.375032 0.927012i \(-0.622368\pi\)
−0.375032 + 0.927012i \(0.622368\pi\)
\(734\) −41.4709 −1.53072
\(735\) 63.0788 2.32670
\(736\) −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 \(0.538786\pi\)
\(740\) −60.9266 −2.23971
\(741\) 0 0
\(742\) −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
\(745\) −68.0498 −2.49315
\(746\) −2.51822 −0.0921988
\(747\) −8.64096 −0.316156
\(748\) 7.78847 0.284775
\(749\) −5.46983 −0.199863
\(750\) 28.6345 1.04559
\(751\) −43.2173 −1.57702 −0.788511 0.615020i \(-0.789148\pi\)
−0.788511 + 0.615020i \(0.789148\pi\)
\(752\) 22.5023 0.820574
\(753\) 30.4064 1.10807
\(754\) 41.1054 1.49697
\(755\) 77.3457 2.81490
\(756\) 31.7192 1.15362
\(757\) 9.93714 0.361171 0.180586 0.983559i \(-0.442201\pi\)
0.180586 + 0.983559i \(0.442201\pi\)
\(758\) −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
\(765\) −15.9390 −0.576275
\(766\) −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 \(0.568788\pi\)
\(770\) −29.5363 −1.06441
\(771\) 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
\(775\) −17.0516 −0.612513
\(776\) −18.0556 −0.648158
\(777\) −54.3945 −1.95139
\(778\) 0.488595 0.0175170
\(779\) 0 0
\(780\) −68.0375 −2.43613
\(781\) 11.4671 0.410325
\(782\) 42.2534 1.51098
\(783\) −16.4771 −0.588844
\(784\) −15.7505 −0.562519
\(785\) 40.4874 1.44506
\(786\) 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
\(790\) 63.6580 2.26485
\(791\) 25.7659 0.916128
\(792\) 2.80470 0.0996608
\(793\) 11.8037 0.419163
\(794\) −47.3114 −1.67902
\(795\) −42.0692 −1.49204
\(796\) 14.8973 0.528022
\(797\) −24.2718 −0.859751 −0.429876 0.902888i \(-0.641443\pi\)
−0.429876 + 0.902888i \(0.641443\pi\)
\(798\) 0 0
\(799\) −34.1653 −1.20868
\(800\) −50.2854 −1.77786
\(801\) 12.0589 0.426081
\(802\) −2.84476 −0.100452
\(803\) −8.95834 −0.316133
\(804\) −59.6155 −2.10248
\(805\) −93.1268 −3.28229
\(806\) 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 \(0.602221\pi\)
\(810\) 84.1607 2.95711
\(811\) −5.48300 −0.192534 −0.0962671 0.995356i \(-0.530690\pi\)
−0.0962671 + 0.995356i \(0.530690\pi\)
\(812\) −62.1019 −2.17935
\(813\) 36.4381 1.27794
\(814\) 13.9832 0.490111
\(815\) 85.2185 2.98508
\(816\) 11.1921 0.391802
\(817\) 0 0
\(818\) 4.71205 0.164753
\(819\) −21.6001 −0.754770
\(820\) 5.33795 0.186409
\(821\) 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 \(-0.616172\pi\)
−0.356916 + 0.934137i \(0.616172\pi\)
\(824\) −31.0095 −1.08027
\(825\) −14.6077 −0.508575
\(826\) −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 \(-0.629233\pi\)
−0.394935 + 0.918709i \(0.629233\pi\)
\(830\) 39.1317 1.35828
\(831\) 33.1887 1.15130
\(832\) 41.5109 1.43913
\(833\) 23.9141 0.828574
\(834\) −23.7251 −0.821535
\(835\) −9.60403 −0.332361
\(836\) 0 0
\(837\) −7.30659 −0.252553
\(838\) −64.7563 −2.23697
\(839\) −13.1616 −0.454388 −0.227194 0.973849i \(-0.572955\pi\)
−0.227194 + 0.973849i \(0.572955\pi\)
\(840\) 49.4084 1.70475
\(841\) 3.25998 0.112413
\(842\) 11.4809 0.395659
\(843\) −55.8778 −1.92453
\(844\) −7.09065 −0.244070
\(845\) 6.97077 0.239802
\(846\) −44.0418 −1.51419
\(847\) 3.93972 0.135370
\(848\) 10.5045 0.360726
\(849\) −40.3228 −1.38387
\(850\) 41.5187 1.42408
\(851\) 44.0885 1.51133
\(852\) −68.6663 −2.35247
\(853\) −36.9102 −1.26378 −0.631891 0.775057i \(-0.717721\pi\)
−0.631891 + 0.775057i \(0.717721\pi\)
\(854\) −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 \(0.541921\pi\)
\(860\) 89.5085 3.05222
\(861\) 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
\(865\) −22.0916 −0.751137
\(866\) 1.52177 0.0517117
\(867\) 19.6871 0.668610
\(868\) −27.5384 −0.934714
\(869\) −8.49105 −0.288039
\(870\) −91.8768 −3.11491
\(871\) −32.9713 −1.11719
\(872\) 0.698197 0.0236439
\(873\) −17.6431 −0.597127
\(874\) 0 0
\(875\) −23.9262 −0.808852
\(876\) 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
\(880\) 6.34128 0.213765
\(881\) −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 \(-0.262898\pi\)
0.677883 + 0.735170i \(0.262898\pi\)
\(884\) −25.7940 −0.867547
\(885\) −32.2546 −1.08423
\(886\) −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
\(890\) −54.6104 −1.83054
\(891\) −11.2258 −0.376079
\(892\) 69.2168 2.31755
\(893\) 0 0
\(894\) −93.5228 −3.12787
\(895\) 42.9673 1.43624
\(896\) −49.3851 −1.64984
\(897\) 49.2342 1.64388
\(898\) −13.4042 −0.447302
\(899\) 14.3053 0.477110
\(900\) 31.1052 1.03684
\(901\) −15.9490 −0.531339
\(902\) −1.22511 −0.0407917
\(903\) 79.9121 2.65931
\(904\) 11.0801 0.368517
\(905\) 47.1783 1.56826
\(906\) 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
\(910\) 97.8190 3.24267
\(911\) −49.5740 −1.64246 −0.821230 0.570598i \(-0.806712\pi\)
−0.821230 + 0.570598i \(0.806712\pi\)
\(912\) 0 0
\(913\) −5.21960 −0.172744
\(914\) 18.3602 0.607301
\(915\) −26.3831 −0.872199
\(916\) −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 \(0.404921\pi\)
\(920\) −40.0472 −1.32032
\(921\) 48.8526 1.60975
\(922\) 56.7952 1.87045
\(923\) −37.9770 −1.25003
\(924\) −23.5915 −0.776103
\(925\) 43.3219 1.42441
\(926\) 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 \(0.298749\pi\)
\(930\) −40.7417 −1.33597
\(931\) 0 0
\(932\) 48.2080 1.57911
\(933\) 2.99317 0.0979921
\(934\) −75.7667 −2.47916
\(935\) −9.62799 −0.314869
\(936\) −9.28867 −0.303610
\(937\) −3.50371 −0.114461 −0.0572307 0.998361i \(-0.518227\pi\)
−0.0572307 + 0.998361i \(0.518227\pi\)
\(938\) 85.7104 2.79854
\(939\) −27.8377 −0.908450
\(940\) 115.915 3.78074
\(941\) −44.2074 −1.44112 −0.720560 0.693392i \(-0.756115\pi\)
−0.720560 + 0.693392i \(0.756115\pi\)
\(942\) 55.6430 1.81295
\(943\) −3.86272 −0.125788
\(944\) 8.05385 0.262130
\(945\) −39.2108 −1.27553
\(946\) −20.5430 −0.667912
\(947\) 37.9515 1.23326 0.616629 0.787254i \(-0.288498\pi\)
0.616629 + 0.787254i \(0.288498\pi\)
\(948\) 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
\(955\) −10.6586 −0.344906
\(956\) 20.0788 0.649396
\(957\) 12.2550 0.396149
\(958\) 50.7745 1.64045
\(959\) −3.14505 −0.101559
\(960\) −92.7831 −2.99456
\(961\) −24.6565 −0.795370
\(962\) −46.3099 −1.49309
\(963\) −2.29844 −0.0740662
\(964\) 33.8679 1.09081
\(965\) −2.56558 −0.0825889
\(966\) −127.987 −4.11791
\(967\) 24.0504 0.773409 0.386705 0.922204i \(-0.373613\pi\)
0.386705 + 0.922204i \(0.373613\pi\)
\(968\) 1.69419 0.0544534
\(969\) 0 0
\(970\) 79.8989 2.56540
\(971\) −54.1335 −1.73723 −0.868614 0.495490i \(-0.834988\pi\)
−0.868614 + 0.495490i \(0.834988\pi\)
\(972\) 43.0682 1.38141
\(973\) 19.8240 0.635529
\(974\) −31.1791 −0.999041
\(975\) 48.3781 1.54934
\(976\) 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
\(980\) −81.1351 −2.59177
\(981\) 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
\(985\) 11.7263 0.373631
\(986\) −34.8318 −1.10927
\(987\) 103.488 3.29405
\(988\) 0 0
\(989\) −64.7714 −2.05961
\(990\) −12.4113 −0.394456
\(991\) −51.7993 −1.64546 −0.822729 0.568434i \(-0.807549\pi\)
−0.822729 + 0.568434i \(0.807549\pi\)
\(992\) 18.7072 0.593953
\(993\) −26.8879 −0.853260
\(994\) 98.7230 3.13130
\(995\) −18.4159 −0.583822
\(996\) 31.2556 0.990371
\(997\) −14.2042 −0.449851 −0.224925 0.974376i \(-0.572214\pi\)
−0.224925 + 0.974376i \(0.572214\pi\)
\(998\) 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 3971.2.a.h.1.5 5
19.18 odd 2 209.2.a.c.1.1 5
57.56 even 2 1881.2.a.k.1.5 5
76.75 even 2 3344.2.a.t.1.2 5
95.94 odd 2 5225.2.a.h.1.5 5
209.208 even 2 2299.2.a.n.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.1 5 19.18 odd 2
1881.2.a.k.1.5 5 57.56 even 2
2299.2.a.n.1.5 5 209.208 even 2
3344.2.a.t.1.2 5 76.75 even 2
3971.2.a.h.1.5 5 1.1 even 1 trivial
5225.2.a.h.1.5 5 95.94 odd 2