Properties

Label 4023.2.a.e.1.3
Level $4023$
Weight $2$
Character 4023.1
Self dual yes
Analytic conductor $32.124$
Analytic rank $1$
Dimension $25$
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] = [4023,2,Mod(1,4023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4023, 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("4023.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4023 = 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4023.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: \(32.1238167332\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 4023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.51871 q^{2} +4.34389 q^{4} +3.54254 q^{5} +3.81300 q^{7} -5.90358 q^{8} +O(q^{10})\) \(q-2.51871 q^{2} +4.34389 q^{4} +3.54254 q^{5} +3.81300 q^{7} -5.90358 q^{8} -8.92261 q^{10} -2.43666 q^{11} -0.859287 q^{13} -9.60384 q^{14} +6.18161 q^{16} -5.04387 q^{17} -6.23507 q^{19} +15.3884 q^{20} +6.13723 q^{22} -6.49720 q^{23} +7.54956 q^{25} +2.16429 q^{26} +16.5633 q^{28} +1.60049 q^{29} +2.29467 q^{31} -3.76251 q^{32} +12.7040 q^{34} +13.5077 q^{35} -0.904111 q^{37} +15.7043 q^{38} -20.9136 q^{40} -7.92721 q^{41} -4.50374 q^{43} -10.5846 q^{44} +16.3646 q^{46} -2.73389 q^{47} +7.53897 q^{49} -19.0151 q^{50} -3.73265 q^{52} -13.3347 q^{53} -8.63194 q^{55} -22.5103 q^{56} -4.03118 q^{58} -5.37203 q^{59} +1.31743 q^{61} -5.77959 q^{62} -2.88654 q^{64} -3.04406 q^{65} +1.24840 q^{67} -21.9100 q^{68} -34.0219 q^{70} +2.94792 q^{71} +2.46161 q^{73} +2.27719 q^{74} -27.0845 q^{76} -9.29097 q^{77} -10.5158 q^{79} +21.8986 q^{80} +19.9663 q^{82} +7.75670 q^{83} -17.8681 q^{85} +11.3436 q^{86} +14.3850 q^{88} -0.331629 q^{89} -3.27646 q^{91} -28.2231 q^{92} +6.88587 q^{94} -22.0880 q^{95} -18.4028 q^{97} -18.9885 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.51871 −1.78100 −0.890498 0.454987i \(-0.849644\pi\)
−0.890498 + 0.454987i \(0.849644\pi\)
\(3\) 0 0
\(4\) 4.34389 2.17195
\(5\) 3.54254 1.58427 0.792135 0.610346i \(-0.208970\pi\)
0.792135 + 0.610346i \(0.208970\pi\)
\(6\) 0 0
\(7\) 3.81300 1.44118 0.720589 0.693362i \(-0.243871\pi\)
0.720589 + 0.693362i \(0.243871\pi\)
\(8\) −5.90358 −2.08723
\(9\) 0 0
\(10\) −8.92261 −2.82158
\(11\) −2.43666 −0.734680 −0.367340 0.930087i \(-0.619731\pi\)
−0.367340 + 0.930087i \(0.619731\pi\)
\(12\) 0 0
\(13\) −0.859287 −0.238323 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(14\) −9.60384 −2.56673
\(15\) 0 0
\(16\) 6.18161 1.54540
\(17\) −5.04387 −1.22332 −0.611659 0.791121i \(-0.709498\pi\)
−0.611659 + 0.791121i \(0.709498\pi\)
\(18\) 0 0
\(19\) −6.23507 −1.43042 −0.715212 0.698908i \(-0.753670\pi\)
−0.715212 + 0.698908i \(0.753670\pi\)
\(20\) 15.3884 3.44095
\(21\) 0 0
\(22\) 6.13723 1.30846
\(23\) −6.49720 −1.35476 −0.677380 0.735633i \(-0.736885\pi\)
−0.677380 + 0.735633i \(0.736885\pi\)
\(24\) 0 0
\(25\) 7.54956 1.50991
\(26\) 2.16429 0.424453
\(27\) 0 0
\(28\) 16.5633 3.13016
\(29\) 1.60049 0.297204 0.148602 0.988897i \(-0.452523\pi\)
0.148602 + 0.988897i \(0.452523\pi\)
\(30\) 0 0
\(31\) 2.29467 0.412134 0.206067 0.978538i \(-0.433934\pi\)
0.206067 + 0.978538i \(0.433934\pi\)
\(32\) −3.76251 −0.665125
\(33\) 0 0
\(34\) 12.7040 2.17873
\(35\) 13.5077 2.28322
\(36\) 0 0
\(37\) −0.904111 −0.148635 −0.0743175 0.997235i \(-0.523678\pi\)
−0.0743175 + 0.997235i \(0.523678\pi\)
\(38\) 15.7043 2.54758
\(39\) 0 0
\(40\) −20.9136 −3.30674
\(41\) −7.92721 −1.23802 −0.619011 0.785382i \(-0.712466\pi\)
−0.619011 + 0.785382i \(0.712466\pi\)
\(42\) 0 0
\(43\) −4.50374 −0.686814 −0.343407 0.939187i \(-0.611581\pi\)
−0.343407 + 0.939187i \(0.611581\pi\)
\(44\) −10.5846 −1.59568
\(45\) 0 0
\(46\) 16.3646 2.41282
\(47\) −2.73389 −0.398779 −0.199389 0.979920i \(-0.563896\pi\)
−0.199389 + 0.979920i \(0.563896\pi\)
\(48\) 0 0
\(49\) 7.53897 1.07700
\(50\) −19.0151 −2.68915
\(51\) 0 0
\(52\) −3.73265 −0.517626
\(53\) −13.3347 −1.83166 −0.915832 0.401562i \(-0.868467\pi\)
−0.915832 + 0.401562i \(0.868467\pi\)
\(54\) 0 0
\(55\) −8.63194 −1.16393
\(56\) −22.5103 −3.00807
\(57\) 0 0
\(58\) −4.03118 −0.529319
\(59\) −5.37203 −0.699379 −0.349689 0.936866i \(-0.613713\pi\)
−0.349689 + 0.936866i \(0.613713\pi\)
\(60\) 0 0
\(61\) 1.31743 0.168680 0.0843400 0.996437i \(-0.473122\pi\)
0.0843400 + 0.996437i \(0.473122\pi\)
\(62\) −5.77959 −0.734009
\(63\) 0 0
\(64\) −2.88654 −0.360818
\(65\) −3.04406 −0.377569
\(66\) 0 0
\(67\) 1.24840 0.152516 0.0762581 0.997088i \(-0.475703\pi\)
0.0762581 + 0.997088i \(0.475703\pi\)
\(68\) −21.9100 −2.65698
\(69\) 0 0
\(70\) −34.0219 −4.06640
\(71\) 2.94792 0.349853 0.174927 0.984581i \(-0.444031\pi\)
0.174927 + 0.984581i \(0.444031\pi\)
\(72\) 0 0
\(73\) 2.46161 0.288109 0.144055 0.989570i \(-0.453986\pi\)
0.144055 + 0.989570i \(0.453986\pi\)
\(74\) 2.27719 0.264718
\(75\) 0 0
\(76\) −27.0845 −3.10680
\(77\) −9.29097 −1.05880
\(78\) 0 0
\(79\) −10.5158 −1.18312 −0.591562 0.806259i \(-0.701489\pi\)
−0.591562 + 0.806259i \(0.701489\pi\)
\(80\) 21.8986 2.44833
\(81\) 0 0
\(82\) 19.9663 2.20491
\(83\) 7.75670 0.851409 0.425704 0.904862i \(-0.360026\pi\)
0.425704 + 0.904862i \(0.360026\pi\)
\(84\) 0 0
\(85\) −17.8681 −1.93807
\(86\) 11.3436 1.22321
\(87\) 0 0
\(88\) 14.3850 1.53345
\(89\) −0.331629 −0.0351526 −0.0175763 0.999846i \(-0.505595\pi\)
−0.0175763 + 0.999846i \(0.505595\pi\)
\(90\) 0 0
\(91\) −3.27646 −0.343467
\(92\) −28.2231 −2.94247
\(93\) 0 0
\(94\) 6.88587 0.710223
\(95\) −22.0880 −2.26618
\(96\) 0 0
\(97\) −18.4028 −1.86852 −0.934260 0.356591i \(-0.883939\pi\)
−0.934260 + 0.356591i \(0.883939\pi\)
\(98\) −18.9885 −1.91813
\(99\) 0 0
\(100\) 32.7945 3.27945
\(101\) −10.4299 −1.03781 −0.518906 0.854832i \(-0.673660\pi\)
−0.518906 + 0.854832i \(0.673660\pi\)
\(102\) 0 0
\(103\) 9.94376 0.979787 0.489894 0.871782i \(-0.337036\pi\)
0.489894 + 0.871782i \(0.337036\pi\)
\(104\) 5.07287 0.497436
\(105\) 0 0
\(106\) 33.5863 3.26219
\(107\) 5.63676 0.544926 0.272463 0.962166i \(-0.412162\pi\)
0.272463 + 0.962166i \(0.412162\pi\)
\(108\) 0 0
\(109\) 11.1052 1.06368 0.531841 0.846844i \(-0.321500\pi\)
0.531841 + 0.846844i \(0.321500\pi\)
\(110\) 21.7414 2.07296
\(111\) 0 0
\(112\) 23.5705 2.22720
\(113\) −19.5556 −1.83963 −0.919816 0.392349i \(-0.871662\pi\)
−0.919816 + 0.392349i \(0.871662\pi\)
\(114\) 0 0
\(115\) −23.0166 −2.14631
\(116\) 6.95237 0.645511
\(117\) 0 0
\(118\) 13.5306 1.24559
\(119\) −19.2323 −1.76302
\(120\) 0 0
\(121\) −5.06270 −0.460246
\(122\) −3.31823 −0.300418
\(123\) 0 0
\(124\) 9.96778 0.895133
\(125\) 9.03190 0.807838
\(126\) 0 0
\(127\) 20.0860 1.78235 0.891174 0.453662i \(-0.149883\pi\)
0.891174 + 0.453662i \(0.149883\pi\)
\(128\) 14.7954 1.30774
\(129\) 0 0
\(130\) 7.66709 0.672448
\(131\) 6.31752 0.551965 0.275982 0.961163i \(-0.410997\pi\)
0.275982 + 0.961163i \(0.410997\pi\)
\(132\) 0 0
\(133\) −23.7743 −2.06150
\(134\) −3.14435 −0.271631
\(135\) 0 0
\(136\) 29.7769 2.55335
\(137\) −6.45572 −0.551549 −0.275775 0.961222i \(-0.588934\pi\)
−0.275775 + 0.961222i \(0.588934\pi\)
\(138\) 0 0
\(139\) −15.4023 −1.30640 −0.653202 0.757184i \(-0.726575\pi\)
−0.653202 + 0.757184i \(0.726575\pi\)
\(140\) 58.6759 4.95902
\(141\) 0 0
\(142\) −7.42494 −0.623087
\(143\) 2.09379 0.175091
\(144\) 0 0
\(145\) 5.66981 0.470852
\(146\) −6.20007 −0.513121
\(147\) 0 0
\(148\) −3.92736 −0.322827
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) 19.0141 1.54735 0.773674 0.633584i \(-0.218417\pi\)
0.773674 + 0.633584i \(0.218417\pi\)
\(152\) 36.8092 2.98562
\(153\) 0 0
\(154\) 23.4013 1.88573
\(155\) 8.12893 0.652932
\(156\) 0 0
\(157\) −17.4113 −1.38958 −0.694788 0.719215i \(-0.744502\pi\)
−0.694788 + 0.719215i \(0.744502\pi\)
\(158\) 26.4863 2.10714
\(159\) 0 0
\(160\) −13.3288 −1.05374
\(161\) −24.7738 −1.95245
\(162\) 0 0
\(163\) −20.4329 −1.60043 −0.800215 0.599713i \(-0.795281\pi\)
−0.800215 + 0.599713i \(0.795281\pi\)
\(164\) −34.4349 −2.68892
\(165\) 0 0
\(166\) −19.5369 −1.51636
\(167\) 10.3545 0.801255 0.400628 0.916241i \(-0.368792\pi\)
0.400628 + 0.916241i \(0.368792\pi\)
\(168\) 0 0
\(169\) −12.2616 −0.943202
\(170\) 45.0045 3.45169
\(171\) 0 0
\(172\) −19.5637 −1.49172
\(173\) 20.7152 1.57494 0.787472 0.616350i \(-0.211389\pi\)
0.787472 + 0.616350i \(0.211389\pi\)
\(174\) 0 0
\(175\) 28.7865 2.17605
\(176\) −15.0625 −1.13538
\(177\) 0 0
\(178\) 0.835277 0.0626067
\(179\) −15.2082 −1.13671 −0.568357 0.822782i \(-0.692421\pi\)
−0.568357 + 0.822782i \(0.692421\pi\)
\(180\) 0 0
\(181\) 1.03982 0.0772892 0.0386446 0.999253i \(-0.487696\pi\)
0.0386446 + 0.999253i \(0.487696\pi\)
\(182\) 8.25246 0.611713
\(183\) 0 0
\(184\) 38.3567 2.82770
\(185\) −3.20285 −0.235478
\(186\) 0 0
\(187\) 12.2902 0.898748
\(188\) −11.8757 −0.866126
\(189\) 0 0
\(190\) 55.6331 4.03605
\(191\) 13.0784 0.946322 0.473161 0.880976i \(-0.343113\pi\)
0.473161 + 0.880976i \(0.343113\pi\)
\(192\) 0 0
\(193\) 15.1240 1.08865 0.544325 0.838874i \(-0.316786\pi\)
0.544325 + 0.838874i \(0.316786\pi\)
\(194\) 46.3513 3.32783
\(195\) 0 0
\(196\) 32.7485 2.33918
\(197\) 19.2160 1.36909 0.684543 0.728972i \(-0.260002\pi\)
0.684543 + 0.728972i \(0.260002\pi\)
\(198\) 0 0
\(199\) −14.3009 −1.01377 −0.506883 0.862015i \(-0.669202\pi\)
−0.506883 + 0.862015i \(0.669202\pi\)
\(200\) −44.5694 −3.15153
\(201\) 0 0
\(202\) 26.2698 1.84834
\(203\) 6.10268 0.428324
\(204\) 0 0
\(205\) −28.0824 −1.96136
\(206\) −25.0454 −1.74500
\(207\) 0 0
\(208\) −5.31178 −0.368306
\(209\) 15.1927 1.05090
\(210\) 0 0
\(211\) 8.92394 0.614349 0.307175 0.951653i \(-0.400616\pi\)
0.307175 + 0.951653i \(0.400616\pi\)
\(212\) −57.9245 −3.97827
\(213\) 0 0
\(214\) −14.1974 −0.970511
\(215\) −15.9547 −1.08810
\(216\) 0 0
\(217\) 8.74956 0.593959
\(218\) −27.9707 −1.89441
\(219\) 0 0
\(220\) −37.4962 −2.52800
\(221\) 4.33414 0.291546
\(222\) 0 0
\(223\) −1.01768 −0.0681491 −0.0340746 0.999419i \(-0.510848\pi\)
−0.0340746 + 0.999419i \(0.510848\pi\)
\(224\) −14.3465 −0.958564
\(225\) 0 0
\(226\) 49.2548 3.27638
\(227\) 0.986066 0.0654475 0.0327238 0.999464i \(-0.489582\pi\)
0.0327238 + 0.999464i \(0.489582\pi\)
\(228\) 0 0
\(229\) 16.2376 1.07301 0.536505 0.843897i \(-0.319744\pi\)
0.536505 + 0.843897i \(0.319744\pi\)
\(230\) 57.9720 3.82256
\(231\) 0 0
\(232\) −9.44864 −0.620334
\(233\) −5.54175 −0.363052 −0.181526 0.983386i \(-0.558104\pi\)
−0.181526 + 0.983386i \(0.558104\pi\)
\(234\) 0 0
\(235\) −9.68490 −0.631773
\(236\) −23.3355 −1.51901
\(237\) 0 0
\(238\) 48.4405 3.13993
\(239\) 4.34193 0.280856 0.140428 0.990091i \(-0.455152\pi\)
0.140428 + 0.990091i \(0.455152\pi\)
\(240\) 0 0
\(241\) −11.1086 −0.715566 −0.357783 0.933805i \(-0.616467\pi\)
−0.357783 + 0.933805i \(0.616467\pi\)
\(242\) 12.7515 0.819696
\(243\) 0 0
\(244\) 5.72279 0.366364
\(245\) 26.7071 1.70625
\(246\) 0 0
\(247\) 5.35772 0.340903
\(248\) −13.5467 −0.860219
\(249\) 0 0
\(250\) −22.7487 −1.43876
\(251\) 28.3565 1.78984 0.894922 0.446222i \(-0.147231\pi\)
0.894922 + 0.446222i \(0.147231\pi\)
\(252\) 0 0
\(253\) 15.8315 0.995315
\(254\) −50.5909 −3.17435
\(255\) 0 0
\(256\) −31.4922 −1.96826
\(257\) 24.5994 1.53447 0.767234 0.641368i \(-0.221633\pi\)
0.767234 + 0.641368i \(0.221633\pi\)
\(258\) 0 0
\(259\) −3.44738 −0.214210
\(260\) −13.2230 −0.820059
\(261\) 0 0
\(262\) −15.9120 −0.983046
\(263\) −4.51251 −0.278253 −0.139126 0.990275i \(-0.544429\pi\)
−0.139126 + 0.990275i \(0.544429\pi\)
\(264\) 0 0
\(265\) −47.2387 −2.90185
\(266\) 59.8806 3.67151
\(267\) 0 0
\(268\) 5.42291 0.331257
\(269\) 27.6112 1.68348 0.841741 0.539881i \(-0.181531\pi\)
0.841741 + 0.539881i \(0.181531\pi\)
\(270\) 0 0
\(271\) −2.25459 −0.136957 −0.0684784 0.997653i \(-0.521814\pi\)
−0.0684784 + 0.997653i \(0.521814\pi\)
\(272\) −31.1793 −1.89052
\(273\) 0 0
\(274\) 16.2601 0.982307
\(275\) −18.3957 −1.10930
\(276\) 0 0
\(277\) 8.57741 0.515366 0.257683 0.966229i \(-0.417041\pi\)
0.257683 + 0.966229i \(0.417041\pi\)
\(278\) 38.7938 2.32670
\(279\) 0 0
\(280\) −79.7437 −4.76560
\(281\) 29.5805 1.76462 0.882312 0.470666i \(-0.155986\pi\)
0.882312 + 0.470666i \(0.155986\pi\)
\(282\) 0 0
\(283\) 19.8967 1.18274 0.591368 0.806402i \(-0.298588\pi\)
0.591368 + 0.806402i \(0.298588\pi\)
\(284\) 12.8054 0.759862
\(285\) 0 0
\(286\) −5.27364 −0.311837
\(287\) −30.2265 −1.78421
\(288\) 0 0
\(289\) 8.44065 0.496509
\(290\) −14.2806 −0.838585
\(291\) 0 0
\(292\) 10.6929 0.625757
\(293\) −27.2289 −1.59073 −0.795365 0.606131i \(-0.792721\pi\)
−0.795365 + 0.606131i \(0.792721\pi\)
\(294\) 0 0
\(295\) −19.0306 −1.10801
\(296\) 5.33749 0.310235
\(297\) 0 0
\(298\) 2.51871 0.145905
\(299\) 5.58296 0.322871
\(300\) 0 0
\(301\) −17.1728 −0.989821
\(302\) −47.8910 −2.75582
\(303\) 0 0
\(304\) −38.5428 −2.21058
\(305\) 4.66705 0.267235
\(306\) 0 0
\(307\) −3.04427 −0.173746 −0.0868728 0.996219i \(-0.527687\pi\)
−0.0868728 + 0.996219i \(0.527687\pi\)
\(308\) −40.3590 −2.29967
\(309\) 0 0
\(310\) −20.4744 −1.16287
\(311\) −16.1735 −0.917115 −0.458558 0.888665i \(-0.651634\pi\)
−0.458558 + 0.888665i \(0.651634\pi\)
\(312\) 0 0
\(313\) −19.7255 −1.11495 −0.557476 0.830193i \(-0.688230\pi\)
−0.557476 + 0.830193i \(0.688230\pi\)
\(314\) 43.8541 2.47483
\(315\) 0 0
\(316\) −45.6797 −2.56968
\(317\) −12.7244 −0.714673 −0.357337 0.933976i \(-0.616315\pi\)
−0.357337 + 0.933976i \(0.616315\pi\)
\(318\) 0 0
\(319\) −3.89985 −0.218350
\(320\) −10.2257 −0.571633
\(321\) 0 0
\(322\) 62.3981 3.47731
\(323\) 31.4489 1.74986
\(324\) 0 0
\(325\) −6.48724 −0.359847
\(326\) 51.4646 2.85036
\(327\) 0 0
\(328\) 46.7989 2.58404
\(329\) −10.4243 −0.574711
\(330\) 0 0
\(331\) 31.1086 1.70988 0.854942 0.518723i \(-0.173593\pi\)
0.854942 + 0.518723i \(0.173593\pi\)
\(332\) 33.6943 1.84921
\(333\) 0 0
\(334\) −26.0800 −1.42703
\(335\) 4.42250 0.241627
\(336\) 0 0
\(337\) −22.6804 −1.23548 −0.617741 0.786382i \(-0.711952\pi\)
−0.617741 + 0.786382i \(0.711952\pi\)
\(338\) 30.8835 1.67984
\(339\) 0 0
\(340\) −77.6171 −4.20938
\(341\) −5.59131 −0.302787
\(342\) 0 0
\(343\) 2.05511 0.110965
\(344\) 26.5882 1.43354
\(345\) 0 0
\(346\) −52.1754 −2.80497
\(347\) −24.2199 −1.30019 −0.650095 0.759853i \(-0.725271\pi\)
−0.650095 + 0.759853i \(0.725271\pi\)
\(348\) 0 0
\(349\) −31.2438 −1.67244 −0.836220 0.548394i \(-0.815239\pi\)
−0.836220 + 0.548394i \(0.815239\pi\)
\(350\) −72.5047 −3.87554
\(351\) 0 0
\(352\) 9.16796 0.488654
\(353\) −16.8284 −0.895688 −0.447844 0.894112i \(-0.647808\pi\)
−0.447844 + 0.894112i \(0.647808\pi\)
\(354\) 0 0
\(355\) 10.4431 0.554262
\(356\) −1.44056 −0.0763496
\(357\) 0 0
\(358\) 38.3050 2.02448
\(359\) 8.82572 0.465804 0.232902 0.972500i \(-0.425178\pi\)
0.232902 + 0.972500i \(0.425178\pi\)
\(360\) 0 0
\(361\) 19.8761 1.04611
\(362\) −2.61900 −0.137652
\(363\) 0 0
\(364\) −14.2326 −0.745991
\(365\) 8.72032 0.456443
\(366\) 0 0
\(367\) −25.3302 −1.32223 −0.661113 0.750286i \(-0.729916\pi\)
−0.661113 + 0.750286i \(0.729916\pi\)
\(368\) −40.1632 −2.09365
\(369\) 0 0
\(370\) 8.06704 0.419385
\(371\) −50.8453 −2.63975
\(372\) 0 0
\(373\) −14.2292 −0.736760 −0.368380 0.929675i \(-0.620088\pi\)
−0.368380 + 0.929675i \(0.620088\pi\)
\(374\) −30.9554 −1.60067
\(375\) 0 0
\(376\) 16.1397 0.832343
\(377\) −1.37528 −0.0708307
\(378\) 0 0
\(379\) 7.53201 0.386893 0.193447 0.981111i \(-0.438033\pi\)
0.193447 + 0.981111i \(0.438033\pi\)
\(380\) −95.9477 −4.92201
\(381\) 0 0
\(382\) −32.9407 −1.68539
\(383\) 36.0995 1.84460 0.922298 0.386479i \(-0.126309\pi\)
0.922298 + 0.386479i \(0.126309\pi\)
\(384\) 0 0
\(385\) −32.9136 −1.67743
\(386\) −38.0930 −1.93888
\(387\) 0 0
\(388\) −79.9397 −4.05833
\(389\) 16.6283 0.843089 0.421544 0.906808i \(-0.361488\pi\)
0.421544 + 0.906808i \(0.361488\pi\)
\(390\) 0 0
\(391\) 32.7711 1.65730
\(392\) −44.5069 −2.24794
\(393\) 0 0
\(394\) −48.3996 −2.43834
\(395\) −37.2528 −1.87439
\(396\) 0 0
\(397\) −37.3483 −1.87446 −0.937229 0.348715i \(-0.886618\pi\)
−0.937229 + 0.348715i \(0.886618\pi\)
\(398\) 36.0199 1.80551
\(399\) 0 0
\(400\) 46.6684 2.33342
\(401\) −12.7934 −0.638873 −0.319436 0.947608i \(-0.603494\pi\)
−0.319436 + 0.947608i \(0.603494\pi\)
\(402\) 0 0
\(403\) −1.97178 −0.0982212
\(404\) −45.3062 −2.25407
\(405\) 0 0
\(406\) −15.3709 −0.762844
\(407\) 2.20301 0.109199
\(408\) 0 0
\(409\) 27.9878 1.38391 0.691954 0.721942i \(-0.256750\pi\)
0.691954 + 0.721942i \(0.256750\pi\)
\(410\) 70.7314 3.49318
\(411\) 0 0
\(412\) 43.1946 2.12805
\(413\) −20.4836 −1.00793
\(414\) 0 0
\(415\) 27.4784 1.34886
\(416\) 3.23308 0.158515
\(417\) 0 0
\(418\) −38.2660 −1.87165
\(419\) −26.3038 −1.28503 −0.642513 0.766275i \(-0.722108\pi\)
−0.642513 + 0.766275i \(0.722108\pi\)
\(420\) 0 0
\(421\) 5.74782 0.280131 0.140066 0.990142i \(-0.455269\pi\)
0.140066 + 0.990142i \(0.455269\pi\)
\(422\) −22.4768 −1.09415
\(423\) 0 0
\(424\) 78.7225 3.82310
\(425\) −38.0790 −1.84710
\(426\) 0 0
\(427\) 5.02337 0.243098
\(428\) 24.4855 1.18355
\(429\) 0 0
\(430\) 40.1851 1.93790
\(431\) 8.62945 0.415666 0.207833 0.978164i \(-0.433359\pi\)
0.207833 + 0.978164i \(0.433359\pi\)
\(432\) 0 0
\(433\) 18.5910 0.893425 0.446712 0.894678i \(-0.352595\pi\)
0.446712 + 0.894678i \(0.352595\pi\)
\(434\) −22.0376 −1.05784
\(435\) 0 0
\(436\) 48.2397 2.31026
\(437\) 40.5105 1.93788
\(438\) 0 0
\(439\) 2.84186 0.135635 0.0678173 0.997698i \(-0.478397\pi\)
0.0678173 + 0.997698i \(0.478397\pi\)
\(440\) 50.9594 2.42939
\(441\) 0 0
\(442\) −10.9164 −0.519241
\(443\) 29.6998 1.41108 0.705541 0.708670i \(-0.250704\pi\)
0.705541 + 0.708670i \(0.250704\pi\)
\(444\) 0 0
\(445\) −1.17481 −0.0556913
\(446\) 2.56325 0.121373
\(447\) 0 0
\(448\) −11.0064 −0.520003
\(449\) −7.67404 −0.362161 −0.181080 0.983468i \(-0.557959\pi\)
−0.181080 + 0.983468i \(0.557959\pi\)
\(450\) 0 0
\(451\) 19.3159 0.909550
\(452\) −84.9472 −3.99558
\(453\) 0 0
\(454\) −2.48361 −0.116562
\(455\) −11.6070 −0.544144
\(456\) 0 0
\(457\) −7.83854 −0.366672 −0.183336 0.983050i \(-0.558690\pi\)
−0.183336 + 0.983050i \(0.558690\pi\)
\(458\) −40.8977 −1.91103
\(459\) 0 0
\(460\) −99.9815 −4.66166
\(461\) 22.5521 1.05035 0.525177 0.850993i \(-0.323999\pi\)
0.525177 + 0.850993i \(0.323999\pi\)
\(462\) 0 0
\(463\) −3.11066 −0.144565 −0.0722823 0.997384i \(-0.523028\pi\)
−0.0722823 + 0.997384i \(0.523028\pi\)
\(464\) 9.89363 0.459300
\(465\) 0 0
\(466\) 13.9581 0.646595
\(467\) 9.43684 0.436685 0.218342 0.975872i \(-0.429935\pi\)
0.218342 + 0.975872i \(0.429935\pi\)
\(468\) 0 0
\(469\) 4.76015 0.219803
\(470\) 24.3934 1.12519
\(471\) 0 0
\(472\) 31.7142 1.45976
\(473\) 10.9741 0.504588
\(474\) 0 0
\(475\) −47.0720 −2.15981
\(476\) −83.5430 −3.82919
\(477\) 0 0
\(478\) −10.9361 −0.500204
\(479\) 21.7130 0.992094 0.496047 0.868296i \(-0.334784\pi\)
0.496047 + 0.868296i \(0.334784\pi\)
\(480\) 0 0
\(481\) 0.776891 0.0354232
\(482\) 27.9792 1.27442
\(483\) 0 0
\(484\) −21.9918 −0.999629
\(485\) −65.1926 −2.96024
\(486\) 0 0
\(487\) −30.9987 −1.40468 −0.702342 0.711840i \(-0.747862\pi\)
−0.702342 + 0.711840i \(0.747862\pi\)
\(488\) −7.77757 −0.352074
\(489\) 0 0
\(490\) −67.2674 −3.03883
\(491\) −14.9510 −0.674731 −0.337366 0.941374i \(-0.609536\pi\)
−0.337366 + 0.941374i \(0.609536\pi\)
\(492\) 0 0
\(493\) −8.07269 −0.363576
\(494\) −13.4945 −0.607147
\(495\) 0 0
\(496\) 14.1847 0.636913
\(497\) 11.2404 0.504201
\(498\) 0 0
\(499\) −14.3098 −0.640596 −0.320298 0.947317i \(-0.603783\pi\)
−0.320298 + 0.947317i \(0.603783\pi\)
\(500\) 39.2336 1.75458
\(501\) 0 0
\(502\) −71.4217 −3.18771
\(503\) 38.7190 1.72640 0.863198 0.504866i \(-0.168458\pi\)
0.863198 + 0.504866i \(0.168458\pi\)
\(504\) 0 0
\(505\) −36.9482 −1.64417
\(506\) −39.8748 −1.77265
\(507\) 0 0
\(508\) 87.2515 3.87116
\(509\) 26.9274 1.19354 0.596768 0.802414i \(-0.296451\pi\)
0.596768 + 0.802414i \(0.296451\pi\)
\(510\) 0 0
\(511\) 9.38610 0.415217
\(512\) 49.7288 2.19773
\(513\) 0 0
\(514\) −61.9587 −2.73288
\(515\) 35.2261 1.55225
\(516\) 0 0
\(517\) 6.66155 0.292975
\(518\) 8.68294 0.381506
\(519\) 0 0
\(520\) 17.9708 0.788073
\(521\) 37.0750 1.62429 0.812143 0.583458i \(-0.198301\pi\)
0.812143 + 0.583458i \(0.198301\pi\)
\(522\) 0 0
\(523\) −16.3357 −0.714310 −0.357155 0.934045i \(-0.616253\pi\)
−0.357155 + 0.934045i \(0.616253\pi\)
\(524\) 27.4426 1.19884
\(525\) 0 0
\(526\) 11.3657 0.495567
\(527\) −11.5740 −0.504171
\(528\) 0 0
\(529\) 19.2136 0.835375
\(530\) 118.981 5.16818
\(531\) 0 0
\(532\) −103.273 −4.47746
\(533\) 6.81175 0.295050
\(534\) 0 0
\(535\) 19.9684 0.863310
\(536\) −7.37002 −0.318336
\(537\) 0 0
\(538\) −69.5445 −2.99827
\(539\) −18.3699 −0.791247
\(540\) 0 0
\(541\) 19.3725 0.832889 0.416445 0.909161i \(-0.363276\pi\)
0.416445 + 0.909161i \(0.363276\pi\)
\(542\) 5.67866 0.243919
\(543\) 0 0
\(544\) 18.9776 0.813660
\(545\) 39.3405 1.68516
\(546\) 0 0
\(547\) −15.0673 −0.644232 −0.322116 0.946700i \(-0.604394\pi\)
−0.322116 + 0.946700i \(0.604394\pi\)
\(548\) −28.0429 −1.19794
\(549\) 0 0
\(550\) 46.3334 1.97566
\(551\) −9.97919 −0.425128
\(552\) 0 0
\(553\) −40.0969 −1.70509
\(554\) −21.6040 −0.917865
\(555\) 0 0
\(556\) −66.9058 −2.83744
\(557\) 17.9877 0.762164 0.381082 0.924541i \(-0.375552\pi\)
0.381082 + 0.924541i \(0.375552\pi\)
\(558\) 0 0
\(559\) 3.87000 0.163684
\(560\) 83.4993 3.52849
\(561\) 0 0
\(562\) −74.5046 −3.14279
\(563\) −31.6510 −1.33393 −0.666965 0.745089i \(-0.732407\pi\)
−0.666965 + 0.745089i \(0.732407\pi\)
\(564\) 0 0
\(565\) −69.2763 −2.91447
\(566\) −50.1140 −2.10645
\(567\) 0 0
\(568\) −17.4033 −0.730224
\(569\) 28.5709 1.19776 0.598878 0.800840i \(-0.295613\pi\)
0.598878 + 0.800840i \(0.295613\pi\)
\(570\) 0 0
\(571\) 26.2318 1.09777 0.548883 0.835899i \(-0.315053\pi\)
0.548883 + 0.835899i \(0.315053\pi\)
\(572\) 9.09519 0.380289
\(573\) 0 0
\(574\) 76.1316 3.17767
\(575\) −49.0510 −2.04557
\(576\) 0 0
\(577\) 28.7950 1.19875 0.599376 0.800467i \(-0.295415\pi\)
0.599376 + 0.800467i \(0.295415\pi\)
\(578\) −21.2595 −0.884280
\(579\) 0 0
\(580\) 24.6290 1.02266
\(581\) 29.5763 1.22703
\(582\) 0 0
\(583\) 32.4921 1.34569
\(584\) −14.5323 −0.601350
\(585\) 0 0
\(586\) 68.5817 2.83308
\(587\) 15.5798 0.643048 0.321524 0.946901i \(-0.395805\pi\)
0.321524 + 0.946901i \(0.395805\pi\)
\(588\) 0 0
\(589\) −14.3074 −0.589526
\(590\) 47.9326 1.97335
\(591\) 0 0
\(592\) −5.58886 −0.229701
\(593\) −6.82097 −0.280104 −0.140052 0.990144i \(-0.544727\pi\)
−0.140052 + 0.990144i \(0.544727\pi\)
\(594\) 0 0
\(595\) −68.1311 −2.79310
\(596\) −4.34389 −0.177933
\(597\) 0 0
\(598\) −14.0619 −0.575032
\(599\) 7.40614 0.302607 0.151303 0.988487i \(-0.451653\pi\)
0.151303 + 0.988487i \(0.451653\pi\)
\(600\) 0 0
\(601\) 29.0323 1.18425 0.592127 0.805845i \(-0.298289\pi\)
0.592127 + 0.805845i \(0.298289\pi\)
\(602\) 43.2532 1.76287
\(603\) 0 0
\(604\) 82.5953 3.36075
\(605\) −17.9348 −0.729153
\(606\) 0 0
\(607\) −38.5152 −1.56328 −0.781641 0.623728i \(-0.785617\pi\)
−0.781641 + 0.623728i \(0.785617\pi\)
\(608\) 23.4595 0.951410
\(609\) 0 0
\(610\) −11.7549 −0.475944
\(611\) 2.34920 0.0950383
\(612\) 0 0
\(613\) −15.0901 −0.609485 −0.304742 0.952435i \(-0.598570\pi\)
−0.304742 + 0.952435i \(0.598570\pi\)
\(614\) 7.66763 0.309440
\(615\) 0 0
\(616\) 54.8500 2.20997
\(617\) −22.5652 −0.908442 −0.454221 0.890889i \(-0.650082\pi\)
−0.454221 + 0.890889i \(0.650082\pi\)
\(618\) 0 0
\(619\) 0.705183 0.0283437 0.0141719 0.999900i \(-0.495489\pi\)
0.0141719 + 0.999900i \(0.495489\pi\)
\(620\) 35.3112 1.41813
\(621\) 0 0
\(622\) 40.7363 1.63338
\(623\) −1.26450 −0.0506612
\(624\) 0 0
\(625\) −5.75195 −0.230078
\(626\) 49.6828 1.98572
\(627\) 0 0
\(628\) −75.6330 −3.01808
\(629\) 4.56022 0.181828
\(630\) 0 0
\(631\) 31.0301 1.23529 0.617645 0.786457i \(-0.288087\pi\)
0.617645 + 0.786457i \(0.288087\pi\)
\(632\) 62.0811 2.46945
\(633\) 0 0
\(634\) 32.0490 1.27283
\(635\) 71.1555 2.82372
\(636\) 0 0
\(637\) −6.47814 −0.256673
\(638\) 9.82260 0.388880
\(639\) 0 0
\(640\) 52.4132 2.07181
\(641\) −47.6989 −1.88399 −0.941996 0.335624i \(-0.891053\pi\)
−0.941996 + 0.335624i \(0.891053\pi\)
\(642\) 0 0
\(643\) 7.51878 0.296512 0.148256 0.988949i \(-0.452634\pi\)
0.148256 + 0.988949i \(0.452634\pi\)
\(644\) −107.615 −4.24062
\(645\) 0 0
\(646\) −79.2106 −3.11650
\(647\) −44.3497 −1.74357 −0.871783 0.489892i \(-0.837036\pi\)
−0.871783 + 0.489892i \(0.837036\pi\)
\(648\) 0 0
\(649\) 13.0898 0.513819
\(650\) 16.3395 0.640887
\(651\) 0 0
\(652\) −88.7584 −3.47605
\(653\) −22.8452 −0.894000 −0.447000 0.894534i \(-0.647508\pi\)
−0.447000 + 0.894534i \(0.647508\pi\)
\(654\) 0 0
\(655\) 22.3800 0.874461
\(656\) −49.0029 −1.91324
\(657\) 0 0
\(658\) 26.2558 1.02356
\(659\) −0.270041 −0.0105193 −0.00525964 0.999986i \(-0.501674\pi\)
−0.00525964 + 0.999986i \(0.501674\pi\)
\(660\) 0 0
\(661\) −0.272894 −0.0106144 −0.00530718 0.999986i \(-0.501689\pi\)
−0.00530718 + 0.999986i \(0.501689\pi\)
\(662\) −78.3535 −3.04530
\(663\) 0 0
\(664\) −45.7923 −1.77709
\(665\) −84.2214 −3.26597
\(666\) 0 0
\(667\) −10.3987 −0.402640
\(668\) 44.9788 1.74028
\(669\) 0 0
\(670\) −11.1390 −0.430336
\(671\) −3.21013 −0.123926
\(672\) 0 0
\(673\) 19.7981 0.763160 0.381580 0.924336i \(-0.375380\pi\)
0.381580 + 0.924336i \(0.375380\pi\)
\(674\) 57.1254 2.20039
\(675\) 0 0
\(676\) −53.2632 −2.04858
\(677\) −23.6048 −0.907205 −0.453603 0.891204i \(-0.649861\pi\)
−0.453603 + 0.891204i \(0.649861\pi\)
\(678\) 0 0
\(679\) −70.1699 −2.69287
\(680\) 105.486 4.04519
\(681\) 0 0
\(682\) 14.0829 0.539261
\(683\) −11.2704 −0.431248 −0.215624 0.976476i \(-0.569179\pi\)
−0.215624 + 0.976476i \(0.569179\pi\)
\(684\) 0 0
\(685\) −22.8696 −0.873803
\(686\) −5.17621 −0.197629
\(687\) 0 0
\(688\) −27.8403 −1.06140
\(689\) 11.4584 0.436528
\(690\) 0 0
\(691\) 32.3062 1.22899 0.614493 0.788922i \(-0.289361\pi\)
0.614493 + 0.788922i \(0.289361\pi\)
\(692\) 89.9844 3.42069
\(693\) 0 0
\(694\) 61.0027 2.31563
\(695\) −54.5631 −2.06970
\(696\) 0 0
\(697\) 39.9838 1.51450
\(698\) 78.6939 2.97861
\(699\) 0 0
\(700\) 125.045 4.72627
\(701\) −8.35579 −0.315594 −0.157797 0.987472i \(-0.550439\pi\)
−0.157797 + 0.987472i \(0.550439\pi\)
\(702\) 0 0
\(703\) 5.63720 0.212611
\(704\) 7.03352 0.265086
\(705\) 0 0
\(706\) 42.3859 1.59522
\(707\) −39.7691 −1.49567
\(708\) 0 0
\(709\) 51.8270 1.94640 0.973201 0.229955i \(-0.0738579\pi\)
0.973201 + 0.229955i \(0.0738579\pi\)
\(710\) −26.3031 −0.987138
\(711\) 0 0
\(712\) 1.95780 0.0733716
\(713\) −14.9089 −0.558343
\(714\) 0 0
\(715\) 7.41732 0.277392
\(716\) −66.0627 −2.46888
\(717\) 0 0
\(718\) −22.2294 −0.829594
\(719\) −16.7450 −0.624484 −0.312242 0.950003i \(-0.601080\pi\)
−0.312242 + 0.950003i \(0.601080\pi\)
\(720\) 0 0
\(721\) 37.9156 1.41205
\(722\) −50.0621 −1.86312
\(723\) 0 0
\(724\) 4.51687 0.167868
\(725\) 12.0830 0.448752
\(726\) 0 0
\(727\) −5.13040 −0.190276 −0.0951379 0.995464i \(-0.530329\pi\)
−0.0951379 + 0.995464i \(0.530329\pi\)
\(728\) 19.3429 0.716894
\(729\) 0 0
\(730\) −21.9640 −0.812922
\(731\) 22.7163 0.840192
\(732\) 0 0
\(733\) −13.8555 −0.511763 −0.255882 0.966708i \(-0.582366\pi\)
−0.255882 + 0.966708i \(0.582366\pi\)
\(734\) 63.7994 2.35488
\(735\) 0 0
\(736\) 24.4458 0.901085
\(737\) −3.04192 −0.112051
\(738\) 0 0
\(739\) 12.2565 0.450864 0.225432 0.974259i \(-0.427621\pi\)
0.225432 + 0.974259i \(0.427621\pi\)
\(740\) −13.9128 −0.511445
\(741\) 0 0
\(742\) 128.064 4.70139
\(743\) −6.73262 −0.246996 −0.123498 0.992345i \(-0.539411\pi\)
−0.123498 + 0.992345i \(0.539411\pi\)
\(744\) 0 0
\(745\) −3.54254 −0.129788
\(746\) 35.8392 1.31217
\(747\) 0 0
\(748\) 53.3872 1.95203
\(749\) 21.4930 0.785336
\(750\) 0 0
\(751\) −27.9110 −1.01849 −0.509244 0.860622i \(-0.670075\pi\)
−0.509244 + 0.860622i \(0.670075\pi\)
\(752\) −16.8998 −0.616273
\(753\) 0 0
\(754\) 3.46394 0.126149
\(755\) 67.3582 2.45142
\(756\) 0 0
\(757\) 6.28185 0.228318 0.114159 0.993463i \(-0.463583\pi\)
0.114159 + 0.993463i \(0.463583\pi\)
\(758\) −18.9709 −0.689056
\(759\) 0 0
\(760\) 130.398 4.73003
\(761\) 8.56141 0.310351 0.155175 0.987887i \(-0.450406\pi\)
0.155175 + 0.987887i \(0.450406\pi\)
\(762\) 0 0
\(763\) 42.3440 1.53296
\(764\) 56.8113 2.05536
\(765\) 0 0
\(766\) −90.9240 −3.28522
\(767\) 4.61612 0.166678
\(768\) 0 0
\(769\) 13.9571 0.503304 0.251652 0.967818i \(-0.419026\pi\)
0.251652 + 0.967818i \(0.419026\pi\)
\(770\) 82.8998 2.98750
\(771\) 0 0
\(772\) 65.6971 2.36449
\(773\) −28.6210 −1.02943 −0.514713 0.857362i \(-0.672102\pi\)
−0.514713 + 0.857362i \(0.672102\pi\)
\(774\) 0 0
\(775\) 17.3237 0.622286
\(776\) 108.642 3.90003
\(777\) 0 0
\(778\) −41.8819 −1.50154
\(779\) 49.4267 1.77090
\(780\) 0 0
\(781\) −7.18306 −0.257030
\(782\) −82.5407 −2.95165
\(783\) 0 0
\(784\) 46.6030 1.66439
\(785\) −61.6803 −2.20146
\(786\) 0 0
\(787\) 25.4572 0.907451 0.453725 0.891142i \(-0.350095\pi\)
0.453725 + 0.891142i \(0.350095\pi\)
\(788\) 83.4724 2.97358
\(789\) 0 0
\(790\) 93.8288 3.33828
\(791\) −74.5654 −2.65124
\(792\) 0 0
\(793\) −1.13205 −0.0402004
\(794\) 94.0695 3.33840
\(795\) 0 0
\(796\) −62.1217 −2.20184
\(797\) 11.8383 0.419333 0.209666 0.977773i \(-0.432762\pi\)
0.209666 + 0.977773i \(0.432762\pi\)
\(798\) 0 0
\(799\) 13.7894 0.487833
\(800\) −28.4053 −1.00428
\(801\) 0 0
\(802\) 32.2229 1.13783
\(803\) −5.99809 −0.211668
\(804\) 0 0
\(805\) −87.7622 −3.09321
\(806\) 4.96633 0.174932
\(807\) 0 0
\(808\) 61.5736 2.16615
\(809\) 38.7147 1.36114 0.680569 0.732684i \(-0.261733\pi\)
0.680569 + 0.732684i \(0.261733\pi\)
\(810\) 0 0
\(811\) 52.7892 1.85368 0.926840 0.375455i \(-0.122514\pi\)
0.926840 + 0.375455i \(0.122514\pi\)
\(812\) 26.5094 0.930297
\(813\) 0 0
\(814\) −5.54874 −0.194483
\(815\) −72.3844 −2.53551
\(816\) 0 0
\(817\) 28.0811 0.982434
\(818\) −70.4931 −2.46473
\(819\) 0 0
\(820\) −121.987 −4.25997
\(821\) −27.1384 −0.947136 −0.473568 0.880757i \(-0.657034\pi\)
−0.473568 + 0.880757i \(0.657034\pi\)
\(822\) 0 0
\(823\) −19.3029 −0.672856 −0.336428 0.941709i \(-0.609219\pi\)
−0.336428 + 0.941709i \(0.609219\pi\)
\(824\) −58.7038 −2.04504
\(825\) 0 0
\(826\) 51.5921 1.79512
\(827\) 28.5457 0.992633 0.496316 0.868142i \(-0.334686\pi\)
0.496316 + 0.868142i \(0.334686\pi\)
\(828\) 0 0
\(829\) −51.5908 −1.79182 −0.895912 0.444232i \(-0.853477\pi\)
−0.895912 + 0.444232i \(0.853477\pi\)
\(830\) −69.2101 −2.40232
\(831\) 0 0
\(832\) 2.48037 0.0859914
\(833\) −38.0256 −1.31751
\(834\) 0 0
\(835\) 36.6812 1.26941
\(836\) 65.9956 2.28250
\(837\) 0 0
\(838\) 66.2517 2.28863
\(839\) −43.8462 −1.51374 −0.756870 0.653565i \(-0.773273\pi\)
−0.756870 + 0.653565i \(0.773273\pi\)
\(840\) 0 0
\(841\) −26.4384 −0.911670
\(842\) −14.4771 −0.498913
\(843\) 0 0
\(844\) 38.7646 1.33433
\(845\) −43.4372 −1.49429
\(846\) 0 0
\(847\) −19.3041 −0.663296
\(848\) −82.4300 −2.83066
\(849\) 0 0
\(850\) 95.9099 3.28968
\(851\) 5.87419 0.201365
\(852\) 0 0
\(853\) −4.31118 −0.147612 −0.0738061 0.997273i \(-0.523515\pi\)
−0.0738061 + 0.997273i \(0.523515\pi\)
\(854\) −12.6524 −0.432957
\(855\) 0 0
\(856\) −33.2771 −1.13739
\(857\) 14.3215 0.489212 0.244606 0.969623i \(-0.421341\pi\)
0.244606 + 0.969623i \(0.421341\pi\)
\(858\) 0 0
\(859\) 11.8911 0.405720 0.202860 0.979208i \(-0.434976\pi\)
0.202860 + 0.979208i \(0.434976\pi\)
\(860\) −69.3053 −2.36329
\(861\) 0 0
\(862\) −21.7351 −0.740299
\(863\) −18.9929 −0.646526 −0.323263 0.946309i \(-0.604780\pi\)
−0.323263 + 0.946309i \(0.604780\pi\)
\(864\) 0 0
\(865\) 73.3842 2.49514
\(866\) −46.8252 −1.59119
\(867\) 0 0
\(868\) 38.0071 1.29005
\(869\) 25.6235 0.869218
\(870\) 0 0
\(871\) −1.07273 −0.0363482
\(872\) −65.5603 −2.22015
\(873\) 0 0
\(874\) −102.034 −3.45136
\(875\) 34.4387 1.16424
\(876\) 0 0
\(877\) −53.0314 −1.79074 −0.895372 0.445319i \(-0.853090\pi\)
−0.895372 + 0.445319i \(0.853090\pi\)
\(878\) −7.15781 −0.241565
\(879\) 0 0
\(880\) −53.3593 −1.79874
\(881\) 2.62841 0.0885533 0.0442766 0.999019i \(-0.485902\pi\)
0.0442766 + 0.999019i \(0.485902\pi\)
\(882\) 0 0
\(883\) −11.1139 −0.374011 −0.187006 0.982359i \(-0.559878\pi\)
−0.187006 + 0.982359i \(0.559878\pi\)
\(884\) 18.8270 0.633221
\(885\) 0 0
\(886\) −74.8052 −2.51313
\(887\) −20.8917 −0.701473 −0.350737 0.936474i \(-0.614069\pi\)
−0.350737 + 0.936474i \(0.614069\pi\)
\(888\) 0 0
\(889\) 76.5881 2.56868
\(890\) 2.95900 0.0991859
\(891\) 0 0
\(892\) −4.42071 −0.148016
\(893\) 17.0460 0.570422
\(894\) 0 0
\(895\) −53.8755 −1.80086
\(896\) 56.4148 1.88469
\(897\) 0 0
\(898\) 19.3287 0.645006
\(899\) 3.67260 0.122488
\(900\) 0 0
\(901\) 67.2586 2.24071
\(902\) −48.6511 −1.61990
\(903\) 0 0
\(904\) 115.448 3.83974
\(905\) 3.68360 0.122447
\(906\) 0 0
\(907\) −3.86295 −0.128267 −0.0641335 0.997941i \(-0.520428\pi\)
−0.0641335 + 0.997941i \(0.520428\pi\)
\(908\) 4.28336 0.142148
\(909\) 0 0
\(910\) 29.2346 0.969118
\(911\) −33.9317 −1.12421 −0.562103 0.827067i \(-0.690008\pi\)
−0.562103 + 0.827067i \(0.690008\pi\)
\(912\) 0 0
\(913\) −18.9004 −0.625513
\(914\) 19.7430 0.653040
\(915\) 0 0
\(916\) 70.5343 2.33052
\(917\) 24.0887 0.795480
\(918\) 0 0
\(919\) 53.1931 1.75468 0.877339 0.479871i \(-0.159316\pi\)
0.877339 + 0.479871i \(0.159316\pi\)
\(920\) 135.880 4.47984
\(921\) 0 0
\(922\) −56.8021 −1.87068
\(923\) −2.53311 −0.0833782
\(924\) 0 0
\(925\) −6.82564 −0.224426
\(926\) 7.83485 0.257469
\(927\) 0 0
\(928\) −6.02188 −0.197678
\(929\) −26.3041 −0.863010 −0.431505 0.902111i \(-0.642017\pi\)
−0.431505 + 0.902111i \(0.642017\pi\)
\(930\) 0 0
\(931\) −47.0060 −1.54056
\(932\) −24.0728 −0.788530
\(933\) 0 0
\(934\) −23.7686 −0.777734
\(935\) 43.5384 1.42386
\(936\) 0 0
\(937\) −16.5656 −0.541174 −0.270587 0.962695i \(-0.587218\pi\)
−0.270587 + 0.962695i \(0.587218\pi\)
\(938\) −11.9894 −0.391468
\(939\) 0 0
\(940\) −42.0701 −1.37218
\(941\) −52.1923 −1.70142 −0.850711 0.525634i \(-0.823828\pi\)
−0.850711 + 0.525634i \(0.823828\pi\)
\(942\) 0 0
\(943\) 51.5047 1.67722
\(944\) −33.2078 −1.08082
\(945\) 0 0
\(946\) −27.6405 −0.898669
\(947\) 21.4194 0.696037 0.348018 0.937488i \(-0.386855\pi\)
0.348018 + 0.937488i \(0.386855\pi\)
\(948\) 0 0
\(949\) −2.11523 −0.0686632
\(950\) 118.561 3.84662
\(951\) 0 0
\(952\) 113.539 3.67983
\(953\) −28.4814 −0.922604 −0.461302 0.887243i \(-0.652617\pi\)
−0.461302 + 0.887243i \(0.652617\pi\)
\(954\) 0 0
\(955\) 46.3308 1.49923
\(956\) 18.8609 0.610005
\(957\) 0 0
\(958\) −54.6888 −1.76692
\(959\) −24.6157 −0.794881
\(960\) 0 0
\(961\) −25.7345 −0.830146
\(962\) −1.95676 −0.0630886
\(963\) 0 0
\(964\) −48.2544 −1.55417
\(965\) 53.5774 1.72472
\(966\) 0 0
\(967\) −46.0595 −1.48117 −0.740587 0.671961i \(-0.765452\pi\)
−0.740587 + 0.671961i \(0.765452\pi\)
\(968\) 29.8881 0.960639
\(969\) 0 0
\(970\) 164.201 5.27218
\(971\) 11.9120 0.382273 0.191137 0.981563i \(-0.438783\pi\)
0.191137 + 0.981563i \(0.438783\pi\)
\(972\) 0 0
\(973\) −58.7289 −1.88276
\(974\) 78.0766 2.50174
\(975\) 0 0
\(976\) 8.14386 0.260679
\(977\) 5.22896 0.167289 0.0836447 0.996496i \(-0.473344\pi\)
0.0836447 + 0.996496i \(0.473344\pi\)
\(978\) 0 0
\(979\) 0.808067 0.0258259
\(980\) 116.013 3.70589
\(981\) 0 0
\(982\) 37.6573 1.20169
\(983\) 31.5001 1.00470 0.502349 0.864665i \(-0.332469\pi\)
0.502349 + 0.864665i \(0.332469\pi\)
\(984\) 0 0
\(985\) 68.0735 2.16900
\(986\) 20.3327 0.647526
\(987\) 0 0
\(988\) 23.2733 0.740424
\(989\) 29.2617 0.930468
\(990\) 0 0
\(991\) 31.3670 0.996407 0.498203 0.867060i \(-0.333993\pi\)
0.498203 + 0.867060i \(0.333993\pi\)
\(992\) −8.63371 −0.274121
\(993\) 0 0
\(994\) −28.3113 −0.897980
\(995\) −50.6615 −1.60608
\(996\) 0 0
\(997\) 46.0663 1.45894 0.729468 0.684015i \(-0.239768\pi\)
0.729468 + 0.684015i \(0.239768\pi\)
\(998\) 36.0423 1.14090
\(999\) 0 0
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 4023.2.a.e.1.3 25
3.2 odd 2 4023.2.a.f.1.23 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4023.2.a.e.1.3 25 1.1 even 1 trivial
4023.2.a.f.1.23 yes 25 3.2 odd 2