Properties

Label 5225.2.a.ba.1.9
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $20$
Inner twists $2$

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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: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.386431\) 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)
 
\(f(q)\) \(=\) \(q-0.386431 q^{2} -0.261531 q^{3} -1.85067 q^{4} +0.101064 q^{6} -4.13791 q^{7} +1.48802 q^{8} -2.93160 q^{9} +O(q^{10})\) \(q-0.386431 q^{2} -0.261531 q^{3} -1.85067 q^{4} +0.101064 q^{6} -4.13791 q^{7} +1.48802 q^{8} -2.93160 q^{9} +1.00000 q^{11} +0.484009 q^{12} -0.244900 q^{13} +1.59902 q^{14} +3.12633 q^{16} +4.03954 q^{17} +1.13286 q^{18} -1.00000 q^{19} +1.08219 q^{21} -0.386431 q^{22} -0.483419 q^{23} -0.389163 q^{24} +0.0946368 q^{26} +1.55130 q^{27} +7.65791 q^{28} +5.24273 q^{29} -9.76294 q^{31} -4.18414 q^{32} -0.261531 q^{33} -1.56100 q^{34} +5.42543 q^{36} +1.52969 q^{37} +0.386431 q^{38} +0.0640490 q^{39} +6.06547 q^{41} -0.418193 q^{42} +7.60200 q^{43} -1.85067 q^{44} +0.186808 q^{46} +9.01914 q^{47} -0.817633 q^{48} +10.1223 q^{49} -1.05647 q^{51} +0.453229 q^{52} -5.86595 q^{53} -0.599470 q^{54} -6.15728 q^{56} +0.261531 q^{57} -2.02595 q^{58} -12.7234 q^{59} +7.58454 q^{61} +3.77270 q^{62} +12.1307 q^{63} -4.63577 q^{64} +0.101064 q^{66} -2.72263 q^{67} -7.47585 q^{68} +0.126429 q^{69} +2.89186 q^{71} -4.36227 q^{72} +7.24330 q^{73} -0.591117 q^{74} +1.85067 q^{76} -4.13791 q^{77} -0.0247505 q^{78} +5.91489 q^{79} +8.38909 q^{81} -2.34388 q^{82} +4.94874 q^{83} -2.00278 q^{84} -2.93765 q^{86} -1.37114 q^{87} +1.48802 q^{88} -3.41496 q^{89} +1.01337 q^{91} +0.894649 q^{92} +2.55331 q^{93} -3.48527 q^{94} +1.09428 q^{96} +12.5567 q^{97} -3.91157 q^{98} -2.93160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.386431 −0.273248 −0.136624 0.990623i \(-0.543625\pi\)
−0.136624 + 0.990623i \(0.543625\pi\)
\(3\) −0.261531 −0.150995 −0.0754976 0.997146i \(-0.524055\pi\)
−0.0754976 + 0.997146i \(0.524055\pi\)
\(4\) −1.85067 −0.925336
\(5\) 0 0
\(6\) 0.101064 0.0412591
\(7\) −4.13791 −1.56398 −0.781992 0.623289i \(-0.785796\pi\)
−0.781992 + 0.623289i \(0.785796\pi\)
\(8\) 1.48802 0.526094
\(9\) −2.93160 −0.977200
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0.484009 0.139721
\(13\) −0.244900 −0.0679230 −0.0339615 0.999423i \(-0.510812\pi\)
−0.0339615 + 0.999423i \(0.510812\pi\)
\(14\) 1.59902 0.427355
\(15\) 0 0
\(16\) 3.12633 0.781582
\(17\) 4.03954 0.979732 0.489866 0.871798i \(-0.337046\pi\)
0.489866 + 0.871798i \(0.337046\pi\)
\(18\) 1.13286 0.267018
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.08219 0.236154
\(22\) −0.386431 −0.0823873
\(23\) −0.483419 −0.100800 −0.0503999 0.998729i \(-0.516050\pi\)
−0.0503999 + 0.998729i \(0.516050\pi\)
\(24\) −0.389163 −0.0794376
\(25\) 0 0
\(26\) 0.0946368 0.0185598
\(27\) 1.55130 0.298548
\(28\) 7.65791 1.44721
\(29\) 5.24273 0.973550 0.486775 0.873527i \(-0.338173\pi\)
0.486775 + 0.873527i \(0.338173\pi\)
\(30\) 0 0
\(31\) −9.76294 −1.75348 −0.876738 0.480969i \(-0.840285\pi\)
−0.876738 + 0.480969i \(0.840285\pi\)
\(32\) −4.18414 −0.739659
\(33\) −0.261531 −0.0455268
\(34\) −1.56100 −0.267709
\(35\) 0 0
\(36\) 5.42543 0.904238
\(37\) 1.52969 0.251479 0.125739 0.992063i \(-0.459870\pi\)
0.125739 + 0.992063i \(0.459870\pi\)
\(38\) 0.386431 0.0626873
\(39\) 0.0640490 0.0102560
\(40\) 0 0
\(41\) 6.06547 0.947268 0.473634 0.880722i \(-0.342942\pi\)
0.473634 + 0.880722i \(0.342942\pi\)
\(42\) −0.418193 −0.0645285
\(43\) 7.60200 1.15929 0.579647 0.814868i \(-0.303190\pi\)
0.579647 + 0.814868i \(0.303190\pi\)
\(44\) −1.85067 −0.278999
\(45\) 0 0
\(46\) 0.186808 0.0275433
\(47\) 9.01914 1.31558 0.657788 0.753203i \(-0.271492\pi\)
0.657788 + 0.753203i \(0.271492\pi\)
\(48\) −0.817633 −0.118015
\(49\) 10.1223 1.44604
\(50\) 0 0
\(51\) −1.05647 −0.147935
\(52\) 0.453229 0.0628516
\(53\) −5.86595 −0.805751 −0.402875 0.915255i \(-0.631989\pi\)
−0.402875 + 0.915255i \(0.631989\pi\)
\(54\) −0.599470 −0.0815775
\(55\) 0 0
\(56\) −6.15728 −0.822802
\(57\) 0.261531 0.0346407
\(58\) −2.02595 −0.266020
\(59\) −12.7234 −1.65644 −0.828220 0.560403i \(-0.810646\pi\)
−0.828220 + 0.560403i \(0.810646\pi\)
\(60\) 0 0
\(61\) 7.58454 0.971101 0.485551 0.874209i \(-0.338619\pi\)
0.485551 + 0.874209i \(0.338619\pi\)
\(62\) 3.77270 0.479133
\(63\) 12.1307 1.52833
\(64\) −4.63577 −0.579472
\(65\) 0 0
\(66\) 0.101064 0.0124401
\(67\) −2.72263 −0.332623 −0.166311 0.986073i \(-0.553186\pi\)
−0.166311 + 0.986073i \(0.553186\pi\)
\(68\) −7.47585 −0.906581
\(69\) 0.126429 0.0152203
\(70\) 0 0
\(71\) 2.89186 0.343201 0.171600 0.985167i \(-0.445106\pi\)
0.171600 + 0.985167i \(0.445106\pi\)
\(72\) −4.36227 −0.514099
\(73\) 7.24330 0.847764 0.423882 0.905717i \(-0.360667\pi\)
0.423882 + 0.905717i \(0.360667\pi\)
\(74\) −0.591117 −0.0687160
\(75\) 0 0
\(76\) 1.85067 0.212287
\(77\) −4.13791 −0.471559
\(78\) −0.0247505 −0.00280244
\(79\) 5.91489 0.665477 0.332739 0.943019i \(-0.392027\pi\)
0.332739 + 0.943019i \(0.392027\pi\)
\(80\) 0 0
\(81\) 8.38909 0.932121
\(82\) −2.34388 −0.258839
\(83\) 4.94874 0.543195 0.271598 0.962411i \(-0.412448\pi\)
0.271598 + 0.962411i \(0.412448\pi\)
\(84\) −2.00278 −0.218522
\(85\) 0 0
\(86\) −2.93765 −0.316774
\(87\) −1.37114 −0.147001
\(88\) 1.48802 0.158623
\(89\) −3.41496 −0.361985 −0.180993 0.983484i \(-0.557931\pi\)
−0.180993 + 0.983484i \(0.557931\pi\)
\(90\) 0 0
\(91\) 1.01337 0.106230
\(92\) 0.894649 0.0932736
\(93\) 2.55331 0.264766
\(94\) −3.48527 −0.359478
\(95\) 0 0
\(96\) 1.09428 0.111685
\(97\) 12.5567 1.27494 0.637468 0.770477i \(-0.279982\pi\)
0.637468 + 0.770477i \(0.279982\pi\)
\(98\) −3.91157 −0.395128
\(99\) −2.93160 −0.294637
\(100\) 0 0
\(101\) −11.5009 −1.14438 −0.572192 0.820120i \(-0.693907\pi\)
−0.572192 + 0.820120i \(0.693907\pi\)
\(102\) 0.408251 0.0404228
\(103\) −13.0487 −1.28573 −0.642864 0.765980i \(-0.722254\pi\)
−0.642864 + 0.765980i \(0.722254\pi\)
\(104\) −0.364415 −0.0357339
\(105\) 0 0
\(106\) 2.26678 0.220169
\(107\) 15.0538 1.45531 0.727654 0.685945i \(-0.240611\pi\)
0.727654 + 0.685945i \(0.240611\pi\)
\(108\) −2.87095 −0.276257
\(109\) −3.97702 −0.380929 −0.190465 0.981694i \(-0.560999\pi\)
−0.190465 + 0.981694i \(0.560999\pi\)
\(110\) 0 0
\(111\) −0.400061 −0.0379721
\(112\) −12.9365 −1.22238
\(113\) −0.880484 −0.0828290 −0.0414145 0.999142i \(-0.513186\pi\)
−0.0414145 + 0.999142i \(0.513186\pi\)
\(114\) −0.101064 −0.00946548
\(115\) 0 0
\(116\) −9.70257 −0.900861
\(117\) 0.717949 0.0663744
\(118\) 4.91670 0.452619
\(119\) −16.7152 −1.53228
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.93090 −0.265351
\(123\) −1.58631 −0.143033
\(124\) 18.0680 1.62255
\(125\) 0 0
\(126\) −4.68768 −0.417611
\(127\) −14.6685 −1.30162 −0.650809 0.759241i \(-0.725570\pi\)
−0.650809 + 0.759241i \(0.725570\pi\)
\(128\) 10.1597 0.897998
\(129\) −1.98816 −0.175048
\(130\) 0 0
\(131\) 1.37384 0.120033 0.0600165 0.998197i \(-0.480885\pi\)
0.0600165 + 0.998197i \(0.480885\pi\)
\(132\) 0.484009 0.0421275
\(133\) 4.13791 0.358802
\(134\) 1.05211 0.0908884
\(135\) 0 0
\(136\) 6.01090 0.515430
\(137\) −3.57696 −0.305600 −0.152800 0.988257i \(-0.548829\pi\)
−0.152800 + 0.988257i \(0.548829\pi\)
\(138\) −0.0488561 −0.00415891
\(139\) 7.87917 0.668303 0.334151 0.942519i \(-0.391550\pi\)
0.334151 + 0.942519i \(0.391550\pi\)
\(140\) 0 0
\(141\) −2.35879 −0.198646
\(142\) −1.11750 −0.0937788
\(143\) −0.244900 −0.0204796
\(144\) −9.16515 −0.763762
\(145\) 0 0
\(146\) −2.79903 −0.231650
\(147\) −2.64730 −0.218346
\(148\) −2.83094 −0.232702
\(149\) 0.608700 0.0498666 0.0249333 0.999689i \(-0.492063\pi\)
0.0249333 + 0.999689i \(0.492063\pi\)
\(150\) 0 0
\(151\) −22.8471 −1.85927 −0.929636 0.368479i \(-0.879879\pi\)
−0.929636 + 0.368479i \(0.879879\pi\)
\(152\) −1.48802 −0.120694
\(153\) −11.8423 −0.957394
\(154\) 1.59902 0.128852
\(155\) 0 0
\(156\) −0.118534 −0.00949029
\(157\) 9.47273 0.756006 0.378003 0.925804i \(-0.376611\pi\)
0.378003 + 0.925804i \(0.376611\pi\)
\(158\) −2.28569 −0.181840
\(159\) 1.53413 0.121664
\(160\) 0 0
\(161\) 2.00034 0.157649
\(162\) −3.24180 −0.254700
\(163\) 23.5045 1.84101 0.920507 0.390727i \(-0.127776\pi\)
0.920507 + 0.390727i \(0.127776\pi\)
\(164\) −11.2252 −0.876540
\(165\) 0 0
\(166\) −1.91235 −0.148427
\(167\) −18.0231 −1.39467 −0.697334 0.716746i \(-0.745631\pi\)
−0.697334 + 0.716746i \(0.745631\pi\)
\(168\) 1.61032 0.124239
\(169\) −12.9400 −0.995386
\(170\) 0 0
\(171\) 2.93160 0.224185
\(172\) −14.0688 −1.07274
\(173\) −18.1676 −1.38126 −0.690628 0.723210i \(-0.742666\pi\)
−0.690628 + 0.723210i \(0.742666\pi\)
\(174\) 0.529850 0.0401678
\(175\) 0 0
\(176\) 3.12633 0.235656
\(177\) 3.32756 0.250115
\(178\) 1.31965 0.0989117
\(179\) −21.8821 −1.63555 −0.817774 0.575540i \(-0.804792\pi\)
−0.817774 + 0.575540i \(0.804792\pi\)
\(180\) 0 0
\(181\) −17.5731 −1.30620 −0.653100 0.757271i \(-0.726532\pi\)
−0.653100 + 0.757271i \(0.726532\pi\)
\(182\) −0.391599 −0.0290272
\(183\) −1.98360 −0.146632
\(184\) −0.719335 −0.0530301
\(185\) 0 0
\(186\) −0.986679 −0.0723468
\(187\) 4.03954 0.295400
\(188\) −16.6915 −1.21735
\(189\) −6.41914 −0.466924
\(190\) 0 0
\(191\) −20.5263 −1.48523 −0.742615 0.669719i \(-0.766415\pi\)
−0.742615 + 0.669719i \(0.766415\pi\)
\(192\) 1.21240 0.0874974
\(193\) 14.2453 1.02540 0.512699 0.858569i \(-0.328646\pi\)
0.512699 + 0.858569i \(0.328646\pi\)
\(194\) −4.85228 −0.348373
\(195\) 0 0
\(196\) −18.7331 −1.33808
\(197\) −3.26945 −0.232939 −0.116469 0.993194i \(-0.537158\pi\)
−0.116469 + 0.993194i \(0.537158\pi\)
\(198\) 1.13286 0.0805089
\(199\) −14.9387 −1.05897 −0.529487 0.848318i \(-0.677616\pi\)
−0.529487 + 0.848318i \(0.677616\pi\)
\(200\) 0 0
\(201\) 0.712054 0.0502244
\(202\) 4.44431 0.312700
\(203\) −21.6940 −1.52262
\(204\) 1.95517 0.136889
\(205\) 0 0
\(206\) 5.04243 0.351322
\(207\) 1.41719 0.0985016
\(208\) −0.765637 −0.0530874
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 7.07216 0.486867 0.243434 0.969918i \(-0.421726\pi\)
0.243434 + 0.969918i \(0.421726\pi\)
\(212\) 10.8560 0.745590
\(213\) −0.756312 −0.0518217
\(214\) −5.81725 −0.397659
\(215\) 0 0
\(216\) 2.30836 0.157064
\(217\) 40.3982 2.74241
\(218\) 1.53684 0.104088
\(219\) −1.89435 −0.128008
\(220\) 0 0
\(221\) −0.989282 −0.0665463
\(222\) 0.154596 0.0103758
\(223\) −6.07334 −0.406701 −0.203351 0.979106i \(-0.565183\pi\)
−0.203351 + 0.979106i \(0.565183\pi\)
\(224\) 17.3136 1.15681
\(225\) 0 0
\(226\) 0.340246 0.0226328
\(227\) −1.60412 −0.106469 −0.0532345 0.998582i \(-0.516953\pi\)
−0.0532345 + 0.998582i \(0.516953\pi\)
\(228\) −0.484009 −0.0320542
\(229\) 5.47550 0.361831 0.180916 0.983499i \(-0.442094\pi\)
0.180916 + 0.983499i \(0.442094\pi\)
\(230\) 0 0
\(231\) 1.08219 0.0712031
\(232\) 7.80127 0.512179
\(233\) −15.8903 −1.04101 −0.520505 0.853859i \(-0.674256\pi\)
−0.520505 + 0.853859i \(0.674256\pi\)
\(234\) −0.277437 −0.0181366
\(235\) 0 0
\(236\) 23.5468 1.53276
\(237\) −1.54693 −0.100484
\(238\) 6.45928 0.418693
\(239\) −29.9052 −1.93440 −0.967202 0.254008i \(-0.918251\pi\)
−0.967202 + 0.254008i \(0.918251\pi\)
\(240\) 0 0
\(241\) −7.02083 −0.452251 −0.226126 0.974098i \(-0.572606\pi\)
−0.226126 + 0.974098i \(0.572606\pi\)
\(242\) −0.386431 −0.0248407
\(243\) −6.84791 −0.439294
\(244\) −14.0365 −0.898595
\(245\) 0 0
\(246\) 0.612999 0.0390834
\(247\) 0.244900 0.0155826
\(248\) −14.5274 −0.922492
\(249\) −1.29425 −0.0820198
\(250\) 0 0
\(251\) 11.8694 0.749191 0.374596 0.927188i \(-0.377782\pi\)
0.374596 + 0.927188i \(0.377782\pi\)
\(252\) −22.4500 −1.41421
\(253\) −0.483419 −0.0303923
\(254\) 5.66835 0.355664
\(255\) 0 0
\(256\) 5.34553 0.334096
\(257\) 24.0606 1.50086 0.750430 0.660950i \(-0.229846\pi\)
0.750430 + 0.660950i \(0.229846\pi\)
\(258\) 0.768286 0.0478314
\(259\) −6.32970 −0.393309
\(260\) 0 0
\(261\) −15.3696 −0.951354
\(262\) −0.530894 −0.0327987
\(263\) 8.07318 0.497814 0.248907 0.968527i \(-0.419929\pi\)
0.248907 + 0.968527i \(0.419929\pi\)
\(264\) −0.389163 −0.0239513
\(265\) 0 0
\(266\) −1.59902 −0.0980419
\(267\) 0.893120 0.0546581
\(268\) 5.03870 0.307788
\(269\) 9.12119 0.556129 0.278064 0.960562i \(-0.410307\pi\)
0.278064 + 0.960562i \(0.410307\pi\)
\(270\) 0 0
\(271\) 17.0227 1.03406 0.517028 0.855969i \(-0.327038\pi\)
0.517028 + 0.855969i \(0.327038\pi\)
\(272\) 12.6289 0.765740
\(273\) −0.265029 −0.0160403
\(274\) 1.38225 0.0835045
\(275\) 0 0
\(276\) −0.233979 −0.0140839
\(277\) 26.3075 1.58066 0.790331 0.612680i \(-0.209908\pi\)
0.790331 + 0.612680i \(0.209908\pi\)
\(278\) −3.04475 −0.182612
\(279\) 28.6210 1.71350
\(280\) 0 0
\(281\) −0.467398 −0.0278826 −0.0139413 0.999903i \(-0.504438\pi\)
−0.0139413 + 0.999903i \(0.504438\pi\)
\(282\) 0.911508 0.0542795
\(283\) −20.3573 −1.21011 −0.605057 0.796182i \(-0.706850\pi\)
−0.605057 + 0.796182i \(0.706850\pi\)
\(284\) −5.35188 −0.317576
\(285\) 0 0
\(286\) 0.0946368 0.00559599
\(287\) −25.0984 −1.48151
\(288\) 12.2662 0.722795
\(289\) −0.682144 −0.0401261
\(290\) 0 0
\(291\) −3.28396 −0.192509
\(292\) −13.4050 −0.784467
\(293\) −29.4685 −1.72157 −0.860784 0.508970i \(-0.830027\pi\)
−0.860784 + 0.508970i \(0.830027\pi\)
\(294\) 1.02300 0.0596625
\(295\) 0 0
\(296\) 2.27620 0.132301
\(297\) 1.55130 0.0900155
\(298\) −0.235220 −0.0136259
\(299\) 0.118389 0.00684662
\(300\) 0 0
\(301\) −31.4564 −1.81312
\(302\) 8.82883 0.508042
\(303\) 3.00785 0.172797
\(304\) −3.12633 −0.179307
\(305\) 0 0
\(306\) 4.57623 0.261606
\(307\) 13.9125 0.794028 0.397014 0.917813i \(-0.370046\pi\)
0.397014 + 0.917813i \(0.370046\pi\)
\(308\) 7.65791 0.436350
\(309\) 3.41265 0.194139
\(310\) 0 0
\(311\) 8.17133 0.463354 0.231677 0.972793i \(-0.425579\pi\)
0.231677 + 0.972793i \(0.425579\pi\)
\(312\) 0.0953060 0.00539564
\(313\) 24.9259 1.40889 0.704447 0.709757i \(-0.251195\pi\)
0.704447 + 0.709757i \(0.251195\pi\)
\(314\) −3.66055 −0.206577
\(315\) 0 0
\(316\) −10.9465 −0.615790
\(317\) 22.6159 1.27023 0.635117 0.772416i \(-0.280952\pi\)
0.635117 + 0.772416i \(0.280952\pi\)
\(318\) −0.592835 −0.0332445
\(319\) 5.24273 0.293537
\(320\) 0 0
\(321\) −3.93704 −0.219744
\(322\) −0.772994 −0.0430773
\(323\) −4.03954 −0.224766
\(324\) −15.5255 −0.862525
\(325\) 0 0
\(326\) −9.08285 −0.503053
\(327\) 1.04012 0.0575185
\(328\) 9.02553 0.498351
\(329\) −37.3204 −2.05754
\(330\) 0 0
\(331\) −19.8439 −1.09072 −0.545358 0.838203i \(-0.683606\pi\)
−0.545358 + 0.838203i \(0.683606\pi\)
\(332\) −9.15850 −0.502638
\(333\) −4.48443 −0.245745
\(334\) 6.96467 0.381090
\(335\) 0 0
\(336\) 3.38329 0.184574
\(337\) −17.4056 −0.948144 −0.474072 0.880486i \(-0.657216\pi\)
−0.474072 + 0.880486i \(0.657216\pi\)
\(338\) 5.00042 0.271987
\(339\) 0.230274 0.0125068
\(340\) 0 0
\(341\) −9.76294 −0.528693
\(342\) −1.13286 −0.0612581
\(343\) −12.9199 −0.697607
\(344\) 11.3119 0.609897
\(345\) 0 0
\(346\) 7.02051 0.377425
\(347\) −3.85633 −0.207019 −0.103509 0.994628i \(-0.533007\pi\)
−0.103509 + 0.994628i \(0.533007\pi\)
\(348\) 2.53753 0.136026
\(349\) −5.00844 −0.268096 −0.134048 0.990975i \(-0.542798\pi\)
−0.134048 + 0.990975i \(0.542798\pi\)
\(350\) 0 0
\(351\) −0.379913 −0.0202783
\(352\) −4.18414 −0.223016
\(353\) −16.8399 −0.896296 −0.448148 0.893959i \(-0.647916\pi\)
−0.448148 + 0.893959i \(0.647916\pi\)
\(354\) −1.28587 −0.0683432
\(355\) 0 0
\(356\) 6.31998 0.334958
\(357\) 4.37156 0.231368
\(358\) 8.45593 0.446910
\(359\) −1.05288 −0.0555691 −0.0277846 0.999614i \(-0.508845\pi\)
−0.0277846 + 0.999614i \(0.508845\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.79080 0.356916
\(363\) −0.261531 −0.0137268
\(364\) −1.87542 −0.0982988
\(365\) 0 0
\(366\) 0.766522 0.0400667
\(367\) −5.43166 −0.283531 −0.141765 0.989900i \(-0.545278\pi\)
−0.141765 + 0.989900i \(0.545278\pi\)
\(368\) −1.51133 −0.0787833
\(369\) −17.7815 −0.925670
\(370\) 0 0
\(371\) 24.2728 1.26018
\(372\) −4.72535 −0.244998
\(373\) 23.1593 1.19914 0.599572 0.800321i \(-0.295338\pi\)
0.599572 + 0.800321i \(0.295338\pi\)
\(374\) −1.56100 −0.0807174
\(375\) 0 0
\(376\) 13.4206 0.692116
\(377\) −1.28394 −0.0661265
\(378\) 2.48055 0.127586
\(379\) −20.8260 −1.06976 −0.534879 0.844929i \(-0.679643\pi\)
−0.534879 + 0.844929i \(0.679643\pi\)
\(380\) 0 0
\(381\) 3.83627 0.196538
\(382\) 7.93198 0.405835
\(383\) 23.5549 1.20360 0.601800 0.798647i \(-0.294451\pi\)
0.601800 + 0.798647i \(0.294451\pi\)
\(384\) −2.65708 −0.135593
\(385\) 0 0
\(386\) −5.50481 −0.280187
\(387\) −22.2860 −1.13286
\(388\) −23.2382 −1.17974
\(389\) 16.9626 0.860037 0.430019 0.902820i \(-0.358507\pi\)
0.430019 + 0.902820i \(0.358507\pi\)
\(390\) 0 0
\(391\) −1.95279 −0.0987567
\(392\) 15.0622 0.760755
\(393\) −0.359302 −0.0181244
\(394\) 1.26342 0.0636500
\(395\) 0 0
\(396\) 5.42543 0.272638
\(397\) −11.2020 −0.562210 −0.281105 0.959677i \(-0.590701\pi\)
−0.281105 + 0.959677i \(0.590701\pi\)
\(398\) 5.77276 0.289362
\(399\) −1.08219 −0.0541774
\(400\) 0 0
\(401\) −7.62060 −0.380555 −0.190277 0.981730i \(-0.560939\pi\)
−0.190277 + 0.981730i \(0.560939\pi\)
\(402\) −0.275159 −0.0137237
\(403\) 2.39094 0.119101
\(404\) 21.2844 1.05894
\(405\) 0 0
\(406\) 8.38321 0.416052
\(407\) 1.52969 0.0758237
\(408\) −1.57204 −0.0778275
\(409\) −30.7305 −1.51953 −0.759763 0.650200i \(-0.774685\pi\)
−0.759763 + 0.650200i \(0.774685\pi\)
\(410\) 0 0
\(411\) 0.935486 0.0461441
\(412\) 24.1489 1.18973
\(413\) 52.6482 2.59065
\(414\) −0.547646 −0.0269153
\(415\) 0 0
\(416\) 1.02470 0.0502399
\(417\) −2.06065 −0.100910
\(418\) 0.386431 0.0189009
\(419\) −23.9749 −1.17125 −0.585625 0.810582i \(-0.699151\pi\)
−0.585625 + 0.810582i \(0.699151\pi\)
\(420\) 0 0
\(421\) −9.89352 −0.482181 −0.241090 0.970503i \(-0.577505\pi\)
−0.241090 + 0.970503i \(0.577505\pi\)
\(422\) −2.73290 −0.133035
\(423\) −26.4405 −1.28558
\(424\) −8.72864 −0.423900
\(425\) 0 0
\(426\) 0.292262 0.0141601
\(427\) −31.3842 −1.51879
\(428\) −27.8597 −1.34665
\(429\) 0.0640490 0.00309231
\(430\) 0 0
\(431\) 20.9302 1.00817 0.504086 0.863653i \(-0.331829\pi\)
0.504086 + 0.863653i \(0.331829\pi\)
\(432\) 4.84987 0.233340
\(433\) 4.10135 0.197098 0.0985490 0.995132i \(-0.468580\pi\)
0.0985490 + 0.995132i \(0.468580\pi\)
\(434\) −15.6111 −0.749356
\(435\) 0 0
\(436\) 7.36016 0.352488
\(437\) 0.483419 0.0231251
\(438\) 0.732035 0.0349780
\(439\) 29.1785 1.39262 0.696308 0.717743i \(-0.254825\pi\)
0.696308 + 0.717743i \(0.254825\pi\)
\(440\) 0 0
\(441\) −29.6746 −1.41308
\(442\) 0.382289 0.0181836
\(443\) −19.1722 −0.910899 −0.455450 0.890262i \(-0.650522\pi\)
−0.455450 + 0.890262i \(0.650522\pi\)
\(444\) 0.740381 0.0351369
\(445\) 0 0
\(446\) 2.34693 0.111130
\(447\) −0.159194 −0.00752962
\(448\) 19.1824 0.906284
\(449\) 3.01676 0.142370 0.0711849 0.997463i \(-0.477322\pi\)
0.0711849 + 0.997463i \(0.477322\pi\)
\(450\) 0 0
\(451\) 6.06547 0.285612
\(452\) 1.62949 0.0766446
\(453\) 5.97524 0.280741
\(454\) 0.619880 0.0290924
\(455\) 0 0
\(456\) 0.389163 0.0182242
\(457\) −6.32717 −0.295973 −0.147986 0.988989i \(-0.547279\pi\)
−0.147986 + 0.988989i \(0.547279\pi\)
\(458\) −2.11590 −0.0988695
\(459\) 6.26653 0.292497
\(460\) 0 0
\(461\) −26.9740 −1.25631 −0.628153 0.778090i \(-0.716189\pi\)
−0.628153 + 0.778090i \(0.716189\pi\)
\(462\) −0.418193 −0.0194561
\(463\) 5.17767 0.240627 0.120313 0.992736i \(-0.461610\pi\)
0.120313 + 0.992736i \(0.461610\pi\)
\(464\) 16.3905 0.760909
\(465\) 0 0
\(466\) 6.14051 0.284454
\(467\) 12.5932 0.582744 0.291372 0.956610i \(-0.405888\pi\)
0.291372 + 0.956610i \(0.405888\pi\)
\(468\) −1.32869 −0.0614186
\(469\) 11.2660 0.520216
\(470\) 0 0
\(471\) −2.47742 −0.114153
\(472\) −18.9326 −0.871443
\(473\) 7.60200 0.349540
\(474\) 0.597781 0.0274570
\(475\) 0 0
\(476\) 30.9344 1.41788
\(477\) 17.1966 0.787380
\(478\) 11.5563 0.528571
\(479\) 36.1790 1.65306 0.826530 0.562893i \(-0.190312\pi\)
0.826530 + 0.562893i \(0.190312\pi\)
\(480\) 0 0
\(481\) −0.374620 −0.0170812
\(482\) 2.71306 0.123577
\(483\) −0.523153 −0.0238043
\(484\) −1.85067 −0.0841214
\(485\) 0 0
\(486\) 2.64624 0.120036
\(487\) 13.3983 0.607133 0.303566 0.952810i \(-0.401823\pi\)
0.303566 + 0.952810i \(0.401823\pi\)
\(488\) 11.2859 0.510890
\(489\) −6.14716 −0.277984
\(490\) 0 0
\(491\) −26.7013 −1.20501 −0.602506 0.798115i \(-0.705831\pi\)
−0.602506 + 0.798115i \(0.705831\pi\)
\(492\) 2.93574 0.132353
\(493\) 21.1782 0.953818
\(494\) −0.0946368 −0.00425791
\(495\) 0 0
\(496\) −30.5221 −1.37048
\(497\) −11.9663 −0.536760
\(498\) 0.500138 0.0224117
\(499\) 4.18714 0.187442 0.0937211 0.995598i \(-0.470124\pi\)
0.0937211 + 0.995598i \(0.470124\pi\)
\(500\) 0 0
\(501\) 4.71360 0.210588
\(502\) −4.58671 −0.204715
\(503\) 3.27450 0.146003 0.0730014 0.997332i \(-0.476742\pi\)
0.0730014 + 0.997332i \(0.476742\pi\)
\(504\) 18.0507 0.804042
\(505\) 0 0
\(506\) 0.186808 0.00830462
\(507\) 3.38422 0.150299
\(508\) 27.1466 1.20443
\(509\) 4.77423 0.211614 0.105807 0.994387i \(-0.466257\pi\)
0.105807 + 0.994387i \(0.466257\pi\)
\(510\) 0 0
\(511\) −29.9721 −1.32589
\(512\) −22.3851 −0.989289
\(513\) −1.55130 −0.0684915
\(514\) −9.29776 −0.410107
\(515\) 0 0
\(516\) 3.67943 0.161978
\(517\) 9.01914 0.396661
\(518\) 2.44599 0.107471
\(519\) 4.75139 0.208563
\(520\) 0 0
\(521\) −40.1413 −1.75862 −0.879311 0.476247i \(-0.841997\pi\)
−0.879311 + 0.476247i \(0.841997\pi\)
\(522\) 5.93928 0.259955
\(523\) −25.7012 −1.12384 −0.561918 0.827193i \(-0.689936\pi\)
−0.561918 + 0.827193i \(0.689936\pi\)
\(524\) −2.54253 −0.111071
\(525\) 0 0
\(526\) −3.11972 −0.136026
\(527\) −39.4377 −1.71794
\(528\) −0.817633 −0.0355829
\(529\) −22.7663 −0.989839
\(530\) 0 0
\(531\) 37.2998 1.61867
\(532\) −7.65791 −0.332013
\(533\) −1.48543 −0.0643413
\(534\) −0.345129 −0.0149352
\(535\) 0 0
\(536\) −4.05133 −0.174991
\(537\) 5.72286 0.246960
\(538\) −3.52471 −0.151961
\(539\) 10.1223 0.435999
\(540\) 0 0
\(541\) 13.5359 0.581952 0.290976 0.956730i \(-0.406020\pi\)
0.290976 + 0.956730i \(0.406020\pi\)
\(542\) −6.57809 −0.282553
\(543\) 4.59592 0.197230
\(544\) −16.9020 −0.724667
\(545\) 0 0
\(546\) 0.102415 0.00438297
\(547\) −16.5470 −0.707498 −0.353749 0.935340i \(-0.615093\pi\)
−0.353749 + 0.935340i \(0.615093\pi\)
\(548\) 6.61977 0.282783
\(549\) −22.2349 −0.948961
\(550\) 0 0
\(551\) −5.24273 −0.223348
\(552\) 0.188129 0.00800729
\(553\) −24.4753 −1.04080
\(554\) −10.1660 −0.431912
\(555\) 0 0
\(556\) −14.5818 −0.618404
\(557\) −20.7288 −0.878307 −0.439153 0.898412i \(-0.644721\pi\)
−0.439153 + 0.898412i \(0.644721\pi\)
\(558\) −11.0600 −0.468209
\(559\) −1.86173 −0.0787427
\(560\) 0 0
\(561\) −1.05647 −0.0446040
\(562\) 0.180617 0.00761886
\(563\) −4.80258 −0.202405 −0.101202 0.994866i \(-0.532269\pi\)
−0.101202 + 0.994866i \(0.532269\pi\)
\(564\) 4.36534 0.183814
\(565\) 0 0
\(566\) 7.86667 0.330661
\(567\) −34.7133 −1.45782
\(568\) 4.30314 0.180556
\(569\) 18.7387 0.785565 0.392783 0.919631i \(-0.371512\pi\)
0.392783 + 0.919631i \(0.371512\pi\)
\(570\) 0 0
\(571\) −34.1368 −1.42858 −0.714291 0.699849i \(-0.753251\pi\)
−0.714291 + 0.699849i \(0.753251\pi\)
\(572\) 0.453229 0.0189505
\(573\) 5.36826 0.224262
\(574\) 9.69878 0.404819
\(575\) 0 0
\(576\) 13.5902 0.566260
\(577\) 43.9533 1.82980 0.914901 0.403679i \(-0.132269\pi\)
0.914901 + 0.403679i \(0.132269\pi\)
\(578\) 0.263601 0.0109644
\(579\) −3.72559 −0.154830
\(580\) 0 0
\(581\) −20.4775 −0.849548
\(582\) 1.26902 0.0526027
\(583\) −5.86595 −0.242943
\(584\) 10.7782 0.446003
\(585\) 0 0
\(586\) 11.3875 0.470415
\(587\) −15.3671 −0.634267 −0.317134 0.948381i \(-0.602720\pi\)
−0.317134 + 0.948381i \(0.602720\pi\)
\(588\) 4.89929 0.202043
\(589\) 9.76294 0.402275
\(590\) 0 0
\(591\) 0.855064 0.0351726
\(592\) 4.78230 0.196551
\(593\) −16.1352 −0.662595 −0.331297 0.943526i \(-0.607486\pi\)
−0.331297 + 0.943526i \(0.607486\pi\)
\(594\) −0.599470 −0.0245965
\(595\) 0 0
\(596\) −1.12650 −0.0461434
\(597\) 3.90693 0.159900
\(598\) −0.0457492 −0.00187082
\(599\) −9.65248 −0.394390 −0.197195 0.980364i \(-0.563183\pi\)
−0.197195 + 0.980364i \(0.563183\pi\)
\(600\) 0 0
\(601\) 29.8944 1.21942 0.609708 0.792626i \(-0.291287\pi\)
0.609708 + 0.792626i \(0.291287\pi\)
\(602\) 12.1557 0.495430
\(603\) 7.98168 0.325039
\(604\) 42.2825 1.72045
\(605\) 0 0
\(606\) −1.16233 −0.0472162
\(607\) −15.3867 −0.624528 −0.312264 0.949995i \(-0.601087\pi\)
−0.312264 + 0.949995i \(0.601087\pi\)
\(608\) 4.18414 0.169689
\(609\) 5.67365 0.229908
\(610\) 0 0
\(611\) −2.20879 −0.0893579
\(612\) 21.9162 0.885911
\(613\) −43.3990 −1.75287 −0.876435 0.481521i \(-0.840085\pi\)
−0.876435 + 0.481521i \(0.840085\pi\)
\(614\) −5.37621 −0.216966
\(615\) 0 0
\(616\) −6.15728 −0.248084
\(617\) −32.5803 −1.31164 −0.655818 0.754919i \(-0.727676\pi\)
−0.655818 + 0.754919i \(0.727676\pi\)
\(618\) −1.31875 −0.0530480
\(619\) −4.04214 −0.162467 −0.0812336 0.996695i \(-0.525886\pi\)
−0.0812336 + 0.996695i \(0.525886\pi\)
\(620\) 0 0
\(621\) −0.749927 −0.0300935
\(622\) −3.15765 −0.126610
\(623\) 14.1308 0.566139
\(624\) 0.200238 0.00801594
\(625\) 0 0
\(626\) −9.63212 −0.384977
\(627\) 0.261531 0.0104446
\(628\) −17.5309 −0.699560
\(629\) 6.17922 0.246382
\(630\) 0 0
\(631\) 39.0445 1.55434 0.777169 0.629292i \(-0.216655\pi\)
0.777169 + 0.629292i \(0.216655\pi\)
\(632\) 8.80146 0.350103
\(633\) −1.84959 −0.0735146
\(634\) −8.73946 −0.347088
\(635\) 0 0
\(636\) −2.83917 −0.112580
\(637\) −2.47895 −0.0982197
\(638\) −2.02595 −0.0802082
\(639\) −8.47778 −0.335376
\(640\) 0 0
\(641\) 30.5292 1.20583 0.602916 0.797805i \(-0.294006\pi\)
0.602916 + 0.797805i \(0.294006\pi\)
\(642\) 1.52139 0.0600446
\(643\) 27.4261 1.08158 0.540790 0.841158i \(-0.318125\pi\)
0.540790 + 0.841158i \(0.318125\pi\)
\(644\) −3.70198 −0.145878
\(645\) 0 0
\(646\) 1.56100 0.0614167
\(647\) −42.2102 −1.65945 −0.829726 0.558171i \(-0.811503\pi\)
−0.829726 + 0.558171i \(0.811503\pi\)
\(648\) 12.4831 0.490383
\(649\) −12.7234 −0.499436
\(650\) 0 0
\(651\) −10.5654 −0.414090
\(652\) −43.4991 −1.70356
\(653\) −28.8516 −1.12905 −0.564526 0.825415i \(-0.690941\pi\)
−0.564526 + 0.825415i \(0.690941\pi\)
\(654\) −0.401932 −0.0157168
\(655\) 0 0
\(656\) 18.9627 0.740367
\(657\) −21.2345 −0.828436
\(658\) 14.4217 0.562218
\(659\) −11.5297 −0.449135 −0.224567 0.974459i \(-0.572097\pi\)
−0.224567 + 0.974459i \(0.572097\pi\)
\(660\) 0 0
\(661\) 16.8814 0.656610 0.328305 0.944572i \(-0.393523\pi\)
0.328305 + 0.944572i \(0.393523\pi\)
\(662\) 7.66827 0.298036
\(663\) 0.258728 0.0100482
\(664\) 7.36381 0.285771
\(665\) 0 0
\(666\) 1.73292 0.0671493
\(667\) −2.53443 −0.0981337
\(668\) 33.3548 1.29054
\(669\) 1.58837 0.0614099
\(670\) 0 0
\(671\) 7.58454 0.292798
\(672\) −4.52805 −0.174673
\(673\) 7.27270 0.280342 0.140171 0.990127i \(-0.455235\pi\)
0.140171 + 0.990127i \(0.455235\pi\)
\(674\) 6.72606 0.259078
\(675\) 0 0
\(676\) 23.9477 0.921067
\(677\) −0.515123 −0.0197978 −0.00989888 0.999951i \(-0.503151\pi\)
−0.00989888 + 0.999951i \(0.503151\pi\)
\(678\) −0.0889850 −0.00341745
\(679\) −51.9583 −1.99398
\(680\) 0 0
\(681\) 0.419527 0.0160763
\(682\) 3.77270 0.144464
\(683\) 29.0661 1.11218 0.556091 0.831121i \(-0.312300\pi\)
0.556091 + 0.831121i \(0.312300\pi\)
\(684\) −5.42543 −0.207447
\(685\) 0 0
\(686\) 4.99263 0.190619
\(687\) −1.43201 −0.0546348
\(688\) 23.7663 0.906083
\(689\) 1.43657 0.0547290
\(690\) 0 0
\(691\) −1.60956 −0.0612306 −0.0306153 0.999531i \(-0.509747\pi\)
−0.0306153 + 0.999531i \(0.509747\pi\)
\(692\) 33.6222 1.27813
\(693\) 12.1307 0.460807
\(694\) 1.49020 0.0565673
\(695\) 0 0
\(696\) −2.04028 −0.0773365
\(697\) 24.5017 0.928068
\(698\) 1.93541 0.0732565
\(699\) 4.15582 0.157187
\(700\) 0 0
\(701\) −26.6908 −1.00810 −0.504048 0.863675i \(-0.668157\pi\)
−0.504048 + 0.863675i \(0.668157\pi\)
\(702\) 0.146810 0.00554099
\(703\) −1.52969 −0.0576932
\(704\) −4.63577 −0.174717
\(705\) 0 0
\(706\) 6.50744 0.244911
\(707\) 47.5898 1.78980
\(708\) −6.15822 −0.231440
\(709\) 34.7167 1.30381 0.651907 0.758299i \(-0.273969\pi\)
0.651907 + 0.758299i \(0.273969\pi\)
\(710\) 0 0
\(711\) −17.3401 −0.650305
\(712\) −5.08153 −0.190438
\(713\) 4.71959 0.176750
\(714\) −1.68930 −0.0632206
\(715\) 0 0
\(716\) 40.4966 1.51343
\(717\) 7.82114 0.292086
\(718\) 0.406867 0.0151841
\(719\) 0.454145 0.0169368 0.00846838 0.999964i \(-0.497304\pi\)
0.00846838 + 0.999964i \(0.497304\pi\)
\(720\) 0 0
\(721\) 53.9945 2.01086
\(722\) −0.386431 −0.0143815
\(723\) 1.83617 0.0682878
\(724\) 32.5221 1.20867
\(725\) 0 0
\(726\) 0.101064 0.00375083
\(727\) −34.4667 −1.27830 −0.639150 0.769082i \(-0.720714\pi\)
−0.639150 + 0.769082i \(0.720714\pi\)
\(728\) 1.50792 0.0558872
\(729\) −23.3763 −0.865790
\(730\) 0 0
\(731\) 30.7086 1.13580
\(732\) 3.67098 0.135683
\(733\) −33.3530 −1.23192 −0.615960 0.787777i \(-0.711232\pi\)
−0.615960 + 0.787777i \(0.711232\pi\)
\(734\) 2.09896 0.0774741
\(735\) 0 0
\(736\) 2.02269 0.0745574
\(737\) −2.72263 −0.100290
\(738\) 6.87133 0.252937
\(739\) 14.6031 0.537185 0.268592 0.963254i \(-0.413442\pi\)
0.268592 + 0.963254i \(0.413442\pi\)
\(740\) 0 0
\(741\) −0.0640490 −0.00235290
\(742\) −9.37975 −0.344341
\(743\) 29.2470 1.07297 0.536484 0.843910i \(-0.319752\pi\)
0.536484 + 0.843910i \(0.319752\pi\)
\(744\) 3.79938 0.139292
\(745\) 0 0
\(746\) −8.94946 −0.327663
\(747\) −14.5077 −0.530810
\(748\) −7.47585 −0.273344
\(749\) −62.2914 −2.27608
\(750\) 0 0
\(751\) −26.1796 −0.955307 −0.477654 0.878548i \(-0.658513\pi\)
−0.477654 + 0.878548i \(0.658513\pi\)
\(752\) 28.1968 1.02823
\(753\) −3.10422 −0.113124
\(754\) 0.496155 0.0180689
\(755\) 0 0
\(756\) 11.8797 0.432061
\(757\) −35.4067 −1.28688 −0.643439 0.765497i \(-0.722493\pi\)
−0.643439 + 0.765497i \(0.722493\pi\)
\(758\) 8.04778 0.292309
\(759\) 0.126429 0.00458909
\(760\) 0 0
\(761\) 3.04755 0.110474 0.0552368 0.998473i \(-0.482409\pi\)
0.0552368 + 0.998473i \(0.482409\pi\)
\(762\) −1.48245 −0.0537036
\(763\) 16.4566 0.595767
\(764\) 37.9874 1.37434
\(765\) 0 0
\(766\) −9.10233 −0.328881
\(767\) 3.11595 0.112510
\(768\) −1.39802 −0.0504469
\(769\) 48.6008 1.75259 0.876295 0.481776i \(-0.160008\pi\)
0.876295 + 0.481776i \(0.160008\pi\)
\(770\) 0 0
\(771\) −6.29261 −0.226623
\(772\) −26.3633 −0.948837
\(773\) −51.0426 −1.83587 −0.917937 0.396727i \(-0.870146\pi\)
−0.917937 + 0.396727i \(0.870146\pi\)
\(774\) 8.61200 0.309552
\(775\) 0 0
\(776\) 18.6845 0.670735
\(777\) 1.65542 0.0593877
\(778\) −6.55486 −0.235003
\(779\) −6.06547 −0.217318
\(780\) 0 0
\(781\) 2.89186 0.103479
\(782\) 0.754617 0.0269850
\(783\) 8.13304 0.290651
\(784\) 31.6457 1.13020
\(785\) 0 0
\(786\) 0.138845 0.00495245
\(787\) 3.01361 0.107424 0.0537118 0.998556i \(-0.482895\pi\)
0.0537118 + 0.998556i \(0.482895\pi\)
\(788\) 6.05068 0.215547
\(789\) −2.11139 −0.0751675
\(790\) 0 0
\(791\) 3.64337 0.129543
\(792\) −4.36227 −0.155007
\(793\) −1.85745 −0.0659601
\(794\) 4.32878 0.153623
\(795\) 0 0
\(796\) 27.6466 0.979907
\(797\) 45.9063 1.62608 0.813042 0.582206i \(-0.197810\pi\)
0.813042 + 0.582206i \(0.197810\pi\)
\(798\) 0.418193 0.0148039
\(799\) 36.4331 1.28891
\(800\) 0 0
\(801\) 10.0113 0.353732
\(802\) 2.94483 0.103986
\(803\) 7.24330 0.255611
\(804\) −1.31778 −0.0464744
\(805\) 0 0
\(806\) −0.923933 −0.0325442
\(807\) −2.38548 −0.0839728
\(808\) −17.1136 −0.602053
\(809\) 41.8470 1.47126 0.735631 0.677383i \(-0.236886\pi\)
0.735631 + 0.677383i \(0.236886\pi\)
\(810\) 0 0
\(811\) 28.5357 1.00202 0.501011 0.865441i \(-0.332962\pi\)
0.501011 + 0.865441i \(0.332962\pi\)
\(812\) 40.1484 1.40893
\(813\) −4.45197 −0.156137
\(814\) −0.591117 −0.0207186
\(815\) 0 0
\(816\) −3.30286 −0.115623
\(817\) −7.60200 −0.265960
\(818\) 11.8752 0.415207
\(819\) −2.97081 −0.103808
\(820\) 0 0
\(821\) −18.4483 −0.643849 −0.321924 0.946765i \(-0.604330\pi\)
−0.321924 + 0.946765i \(0.604330\pi\)
\(822\) −0.361500 −0.0126088
\(823\) 8.59026 0.299438 0.149719 0.988729i \(-0.452163\pi\)
0.149719 + 0.988729i \(0.452163\pi\)
\(824\) −19.4167 −0.676414
\(825\) 0 0
\(826\) −20.3449 −0.707888
\(827\) −22.3427 −0.776931 −0.388466 0.921463i \(-0.626995\pi\)
−0.388466 + 0.921463i \(0.626995\pi\)
\(828\) −2.62275 −0.0911470
\(829\) −15.8776 −0.551450 −0.275725 0.961237i \(-0.588918\pi\)
−0.275725 + 0.961237i \(0.588918\pi\)
\(830\) 0 0
\(831\) −6.88023 −0.238672
\(832\) 1.13530 0.0393595
\(833\) 40.8895 1.41674
\(834\) 0.796298 0.0275736
\(835\) 0 0
\(836\) 1.85067 0.0640068
\(837\) −15.1452 −0.523496
\(838\) 9.26463 0.320042
\(839\) 50.1265 1.73056 0.865280 0.501289i \(-0.167141\pi\)
0.865280 + 0.501289i \(0.167141\pi\)
\(840\) 0 0
\(841\) −1.51378 −0.0521994
\(842\) 3.82316 0.131755
\(843\) 0.122239 0.00421014
\(844\) −13.0882 −0.450516
\(845\) 0 0
\(846\) 10.2174 0.351282
\(847\) −4.13791 −0.142180
\(848\) −18.3389 −0.629760
\(849\) 5.32406 0.182721
\(850\) 0 0
\(851\) −0.739478 −0.0253490
\(852\) 1.39969 0.0479524
\(853\) −31.6970 −1.08528 −0.542642 0.839964i \(-0.682576\pi\)
−0.542642 + 0.839964i \(0.682576\pi\)
\(854\) 12.1278 0.415005
\(855\) 0 0
\(856\) 22.4003 0.765628
\(857\) −20.4944 −0.700075 −0.350037 0.936736i \(-0.613831\pi\)
−0.350037 + 0.936736i \(0.613831\pi\)
\(858\) −0.0247505 −0.000844968 0
\(859\) 8.63383 0.294583 0.147291 0.989093i \(-0.452945\pi\)
0.147291 + 0.989093i \(0.452945\pi\)
\(860\) 0 0
\(861\) 6.56401 0.223701
\(862\) −8.08807 −0.275481
\(863\) −49.4127 −1.68203 −0.841014 0.541014i \(-0.818041\pi\)
−0.841014 + 0.541014i \(0.818041\pi\)
\(864\) −6.49086 −0.220824
\(865\) 0 0
\(866\) −1.58489 −0.0538566
\(867\) 0.178402 0.00605885
\(868\) −74.7638 −2.53765
\(869\) 5.91489 0.200649
\(870\) 0 0
\(871\) 0.666773 0.0225927
\(872\) −5.91787 −0.200404
\(873\) −36.8111 −1.24587
\(874\) −0.186808 −0.00631887
\(875\) 0 0
\(876\) 3.50582 0.118451
\(877\) −12.9391 −0.436922 −0.218461 0.975846i \(-0.570104\pi\)
−0.218461 + 0.975846i \(0.570104\pi\)
\(878\) −11.2755 −0.380529
\(879\) 7.70694 0.259949
\(880\) 0 0
\(881\) −18.7876 −0.632970 −0.316485 0.948598i \(-0.602503\pi\)
−0.316485 + 0.948598i \(0.602503\pi\)
\(882\) 11.4672 0.386120
\(883\) 49.3212 1.65979 0.829895 0.557919i \(-0.188400\pi\)
0.829895 + 0.557919i \(0.188400\pi\)
\(884\) 1.83084 0.0615777
\(885\) 0 0
\(886\) 7.40873 0.248901
\(887\) −43.4343 −1.45838 −0.729191 0.684310i \(-0.760103\pi\)
−0.729191 + 0.684310i \(0.760103\pi\)
\(888\) −0.595297 −0.0199769
\(889\) 60.6969 2.03571
\(890\) 0 0
\(891\) 8.38909 0.281045
\(892\) 11.2398 0.376335
\(893\) −9.01914 −0.301814
\(894\) 0.0615175 0.00205745
\(895\) 0 0
\(896\) −42.0399 −1.40445
\(897\) −0.0309625 −0.00103381
\(898\) −1.16577 −0.0389022
\(899\) −51.1844 −1.70710
\(900\) 0 0
\(901\) −23.6957 −0.789419
\(902\) −2.34388 −0.0780428
\(903\) 8.22684 0.273772
\(904\) −1.31018 −0.0435758
\(905\) 0 0
\(906\) −2.30901 −0.0767119
\(907\) −39.3136 −1.30539 −0.652694 0.757622i \(-0.726361\pi\)
−0.652694 + 0.757622i \(0.726361\pi\)
\(908\) 2.96869 0.0985196
\(909\) 33.7161 1.11829
\(910\) 0 0
\(911\) 2.30597 0.0764001 0.0382001 0.999270i \(-0.487838\pi\)
0.0382001 + 0.999270i \(0.487838\pi\)
\(912\) 0.817633 0.0270745
\(913\) 4.94874 0.163779
\(914\) 2.44501 0.0808738
\(915\) 0 0
\(916\) −10.1334 −0.334815
\(917\) −5.68483 −0.187730
\(918\) −2.42158 −0.0799240
\(919\) 35.1934 1.16092 0.580462 0.814287i \(-0.302872\pi\)
0.580462 + 0.814287i \(0.302872\pi\)
\(920\) 0 0
\(921\) −3.63855 −0.119894
\(922\) 10.4236 0.343283
\(923\) −0.708216 −0.0233112
\(924\) −2.00278 −0.0658868
\(925\) 0 0
\(926\) −2.00081 −0.0657507
\(927\) 38.2537 1.25641
\(928\) −21.9363 −0.720095
\(929\) 26.4097 0.866474 0.433237 0.901280i \(-0.357371\pi\)
0.433237 + 0.901280i \(0.357371\pi\)
\(930\) 0 0
\(931\) −10.1223 −0.331745
\(932\) 29.4078 0.963284
\(933\) −2.13706 −0.0699642
\(934\) −4.86640 −0.159233
\(935\) 0 0
\(936\) 1.06832 0.0349191
\(937\) −25.8042 −0.842987 −0.421493 0.906831i \(-0.638494\pi\)
−0.421493 + 0.906831i \(0.638494\pi\)
\(938\) −4.35353 −0.142148
\(939\) −6.51890 −0.212736
\(940\) 0 0
\(941\) 57.3538 1.86968 0.934841 0.355067i \(-0.115542\pi\)
0.934841 + 0.355067i \(0.115542\pi\)
\(942\) 0.957349 0.0311921
\(943\) −2.93216 −0.0954843
\(944\) −39.7774 −1.29464
\(945\) 0 0
\(946\) −2.93765 −0.0955111
\(947\) 59.3155 1.92749 0.963747 0.266818i \(-0.0859723\pi\)
0.963747 + 0.266818i \(0.0859723\pi\)
\(948\) 2.86286 0.0929813
\(949\) −1.77388 −0.0575827
\(950\) 0 0
\(951\) −5.91476 −0.191799
\(952\) −24.8726 −0.806125
\(953\) 1.58112 0.0512174 0.0256087 0.999672i \(-0.491848\pi\)
0.0256087 + 0.999672i \(0.491848\pi\)
\(954\) −6.64531 −0.215150
\(955\) 0 0
\(956\) 55.3446 1.78997
\(957\) −1.37114 −0.0443226
\(958\) −13.9807 −0.451695
\(959\) 14.8011 0.477953
\(960\) 0 0
\(961\) 64.3150 2.07468
\(962\) 0.144765 0.00466740
\(963\) −44.1318 −1.42213
\(964\) 12.9932 0.418484
\(965\) 0 0
\(966\) 0.202162 0.00650446
\(967\) 38.7297 1.24546 0.622731 0.782436i \(-0.286023\pi\)
0.622731 + 0.782436i \(0.286023\pi\)
\(968\) 1.48802 0.0478267
\(969\) 1.05647 0.0339386
\(970\) 0 0
\(971\) −44.2299 −1.41941 −0.709703 0.704501i \(-0.751171\pi\)
−0.709703 + 0.704501i \(0.751171\pi\)
\(972\) 12.6732 0.406494
\(973\) −32.6033 −1.04521
\(974\) −5.17750 −0.165898
\(975\) 0 0
\(976\) 23.7118 0.758995
\(977\) 0.637601 0.0203987 0.0101993 0.999948i \(-0.496753\pi\)
0.0101993 + 0.999948i \(0.496753\pi\)
\(978\) 2.37545 0.0759585
\(979\) −3.41496 −0.109143
\(980\) 0 0
\(981\) 11.6590 0.372244
\(982\) 10.3182 0.329267
\(983\) 27.4345 0.875025 0.437512 0.899212i \(-0.355860\pi\)
0.437512 + 0.899212i \(0.355860\pi\)
\(984\) −2.36046 −0.0752486
\(985\) 0 0
\(986\) −8.18390 −0.260629
\(987\) 9.76046 0.310679
\(988\) −0.453229 −0.0144191
\(989\) −3.67495 −0.116857
\(990\) 0 0
\(991\) −49.4562 −1.57103 −0.785514 0.618844i \(-0.787601\pi\)
−0.785514 + 0.618844i \(0.787601\pi\)
\(992\) 40.8495 1.29697
\(993\) 5.18979 0.164693
\(994\) 4.62413 0.146669
\(995\) 0 0
\(996\) 2.39523 0.0758959
\(997\) 23.2499 0.736331 0.368165 0.929760i \(-0.379986\pi\)
0.368165 + 0.929760i \(0.379986\pi\)
\(998\) −1.61804 −0.0512182
\(999\) 2.37300 0.0750784
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.ba.1.9 20
5.2 odd 4 1045.2.b.c.419.9 20
5.3 odd 4 1045.2.b.c.419.12 yes 20
5.4 even 2 inner 5225.2.a.ba.1.12 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.c.419.9 20 5.2 odd 4
1045.2.b.c.419.12 yes 20 5.3 odd 4
5225.2.a.ba.1.9 20 1.1 even 1 trivial
5225.2.a.ba.1.12 20 5.4 even 2 inner