Properties

Label 5225.2.a.v.1.9
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.814490\) 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)
 
\(f(q)\) \(=\) \(q+0.814490 q^{2} -2.50015 q^{3} -1.33661 q^{4} -2.03634 q^{6} -4.83914 q^{7} -2.71763 q^{8} +3.25073 q^{9} +O(q^{10})\) \(q+0.814490 q^{2} -2.50015 q^{3} -1.33661 q^{4} -2.03634 q^{6} -4.83914 q^{7} -2.71763 q^{8} +3.25073 q^{9} -1.00000 q^{11} +3.34171 q^{12} -4.12188 q^{13} -3.94143 q^{14} +0.459728 q^{16} +2.02683 q^{17} +2.64769 q^{18} +1.00000 q^{19} +12.0985 q^{21} -0.814490 q^{22} +6.38724 q^{23} +6.79448 q^{24} -3.35723 q^{26} -0.626858 q^{27} +6.46802 q^{28} +7.42993 q^{29} -4.40903 q^{31} +5.80971 q^{32} +2.50015 q^{33} +1.65083 q^{34} -4.34494 q^{36} +1.77785 q^{37} +0.814490 q^{38} +10.3053 q^{39} +9.92201 q^{41} +9.85414 q^{42} -9.89030 q^{43} +1.33661 q^{44} +5.20234 q^{46} +0.145235 q^{47} -1.14939 q^{48} +16.4172 q^{49} -5.06736 q^{51} +5.50933 q^{52} -12.9926 q^{53} -0.510570 q^{54} +13.1510 q^{56} -2.50015 q^{57} +6.05160 q^{58} +1.17115 q^{59} +2.03091 q^{61} -3.59111 q^{62} -15.7307 q^{63} +3.81249 q^{64} +2.03634 q^{66} +7.84101 q^{67} -2.70907 q^{68} -15.9690 q^{69} -11.1719 q^{71} -8.83428 q^{72} -3.27201 q^{73} +1.44804 q^{74} -1.33661 q^{76} +4.83914 q^{77} +8.39357 q^{78} +11.3472 q^{79} -8.18495 q^{81} +8.08137 q^{82} +2.37099 q^{83} -16.1710 q^{84} -8.05555 q^{86} -18.5759 q^{87} +2.71763 q^{88} -9.26921 q^{89} +19.9463 q^{91} -8.53722 q^{92} +11.0232 q^{93} +0.118292 q^{94} -14.5251 q^{96} +8.29514 q^{97} +13.3717 q^{98} -3.25073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 17 q^{7} - 3 q^{8} + 15 q^{9} - 15 q^{11} - 9 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} - q^{17} + 10 q^{18} + 15 q^{19} + 16 q^{21} - q^{22} - 18 q^{23} - 27 q^{24} + 15 q^{26} - 31 q^{27} - 34 q^{28} + 11 q^{29} - 2 q^{31} - 10 q^{32} + 4 q^{33} - 13 q^{34} + 8 q^{36} - 27 q^{37} + q^{38} + 4 q^{39} + 6 q^{41} - 2 q^{42} - 38 q^{43} - 13 q^{44} - 9 q^{46} - 14 q^{47} - 32 q^{48} + 28 q^{49} - 32 q^{51} - 16 q^{52} - 11 q^{53} - 11 q^{54} + 2 q^{56} - 4 q^{57} + 6 q^{58} + 13 q^{59} + 8 q^{61} - 7 q^{62} - 49 q^{63} + 9 q^{64} + q^{66} - 31 q^{67} + 26 q^{68} + 39 q^{69} - 3 q^{71} + 18 q^{72} - 18 q^{73} + 7 q^{74} + 13 q^{76} + 17 q^{77} - 18 q^{78} + 10 q^{79} + 31 q^{81} - 58 q^{82} - 16 q^{83} + 112 q^{84} - 63 q^{86} - 67 q^{87} + 3 q^{88} - 7 q^{89} - 50 q^{91} - 98 q^{92} - 26 q^{93} + 22 q^{94} - 37 q^{96} - 24 q^{97} + 46 q^{98} - 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.814490 0.575931 0.287966 0.957641i \(-0.407021\pi\)
0.287966 + 0.957641i \(0.407021\pi\)
\(3\) −2.50015 −1.44346 −0.721730 0.692175i \(-0.756653\pi\)
−0.721730 + 0.692175i \(0.756653\pi\)
\(4\) −1.33661 −0.668303
\(5\) 0 0
\(6\) −2.03634 −0.831334
\(7\) −4.83914 −1.82902 −0.914511 0.404562i \(-0.867424\pi\)
−0.914511 + 0.404562i \(0.867424\pi\)
\(8\) −2.71763 −0.960828
\(9\) 3.25073 1.08358
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.34171 0.964669
\(13\) −4.12188 −1.14320 −0.571602 0.820531i \(-0.693678\pi\)
−0.571602 + 0.820531i \(0.693678\pi\)
\(14\) −3.94143 −1.05339
\(15\) 0 0
\(16\) 0.459728 0.114932
\(17\) 2.02683 0.491578 0.245789 0.969323i \(-0.420953\pi\)
0.245789 + 0.969323i \(0.420953\pi\)
\(18\) 2.64769 0.624065
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 12.0985 2.64012
\(22\) −0.814490 −0.173650
\(23\) 6.38724 1.33183 0.665915 0.746027i \(-0.268041\pi\)
0.665915 + 0.746027i \(0.268041\pi\)
\(24\) 6.79448 1.38692
\(25\) 0 0
\(26\) −3.35723 −0.658407
\(27\) −0.626858 −0.120639
\(28\) 6.46802 1.22234
\(29\) 7.42993 1.37970 0.689851 0.723951i \(-0.257676\pi\)
0.689851 + 0.723951i \(0.257676\pi\)
\(30\) 0 0
\(31\) −4.40903 −0.791885 −0.395943 0.918275i \(-0.629582\pi\)
−0.395943 + 0.918275i \(0.629582\pi\)
\(32\) 5.80971 1.02702
\(33\) 2.50015 0.435220
\(34\) 1.65083 0.283115
\(35\) 0 0
\(36\) −4.34494 −0.724157
\(37\) 1.77785 0.292277 0.146139 0.989264i \(-0.453315\pi\)
0.146139 + 0.989264i \(0.453315\pi\)
\(38\) 0.814490 0.132128
\(39\) 10.3053 1.65017
\(40\) 0 0
\(41\) 9.92201 1.54956 0.774778 0.632233i \(-0.217861\pi\)
0.774778 + 0.632233i \(0.217861\pi\)
\(42\) 9.85414 1.52053
\(43\) −9.89030 −1.50826 −0.754129 0.656727i \(-0.771941\pi\)
−0.754129 + 0.656727i \(0.771941\pi\)
\(44\) 1.33661 0.201501
\(45\) 0 0
\(46\) 5.20234 0.767043
\(47\) 0.145235 0.0211847 0.0105923 0.999944i \(-0.496628\pi\)
0.0105923 + 0.999944i \(0.496628\pi\)
\(48\) −1.14939 −0.165900
\(49\) 16.4172 2.34532
\(50\) 0 0
\(51\) −5.06736 −0.709573
\(52\) 5.50933 0.764007
\(53\) −12.9926 −1.78467 −0.892333 0.451378i \(-0.850933\pi\)
−0.892333 + 0.451378i \(0.850933\pi\)
\(54\) −0.510570 −0.0694797
\(55\) 0 0
\(56\) 13.1510 1.75737
\(57\) −2.50015 −0.331152
\(58\) 6.05160 0.794614
\(59\) 1.17115 0.152471 0.0762354 0.997090i \(-0.475710\pi\)
0.0762354 + 0.997090i \(0.475710\pi\)
\(60\) 0 0
\(61\) 2.03091 0.260032 0.130016 0.991512i \(-0.458497\pi\)
0.130016 + 0.991512i \(0.458497\pi\)
\(62\) −3.59111 −0.456071
\(63\) −15.7307 −1.98188
\(64\) 3.81249 0.476561
\(65\) 0 0
\(66\) 2.03634 0.250657
\(67\) 7.84101 0.957932 0.478966 0.877833i \(-0.341012\pi\)
0.478966 + 0.877833i \(0.341012\pi\)
\(68\) −2.70907 −0.328523
\(69\) −15.9690 −1.92244
\(70\) 0 0
\(71\) −11.1719 −1.32586 −0.662928 0.748683i \(-0.730686\pi\)
−0.662928 + 0.748683i \(0.730686\pi\)
\(72\) −8.83428 −1.04113
\(73\) −3.27201 −0.382959 −0.191480 0.981497i \(-0.561329\pi\)
−0.191480 + 0.981497i \(0.561329\pi\)
\(74\) 1.44804 0.168332
\(75\) 0 0
\(76\) −1.33661 −0.153319
\(77\) 4.83914 0.551471
\(78\) 8.39357 0.950385
\(79\) 11.3472 1.27666 0.638331 0.769762i \(-0.279625\pi\)
0.638331 + 0.769762i \(0.279625\pi\)
\(80\) 0 0
\(81\) −8.18495 −0.909439
\(82\) 8.08137 0.892438
\(83\) 2.37099 0.260250 0.130125 0.991498i \(-0.458462\pi\)
0.130125 + 0.991498i \(0.458462\pi\)
\(84\) −16.1710 −1.76440
\(85\) 0 0
\(86\) −8.05555 −0.868653
\(87\) −18.5759 −1.99155
\(88\) 2.71763 0.289701
\(89\) −9.26921 −0.982534 −0.491267 0.871009i \(-0.663466\pi\)
−0.491267 + 0.871009i \(0.663466\pi\)
\(90\) 0 0
\(91\) 19.9463 2.09095
\(92\) −8.53722 −0.890067
\(93\) 11.0232 1.14305
\(94\) 0.118292 0.0122009
\(95\) 0 0
\(96\) −14.5251 −1.48246
\(97\) 8.29514 0.842244 0.421122 0.907004i \(-0.361636\pi\)
0.421122 + 0.907004i \(0.361636\pi\)
\(98\) 13.3717 1.35074
\(99\) −3.25073 −0.326711
\(100\) 0 0
\(101\) 4.99870 0.497389 0.248695 0.968582i \(-0.419998\pi\)
0.248695 + 0.968582i \(0.419998\pi\)
\(102\) −4.12732 −0.408665
\(103\) −17.6185 −1.73600 −0.867999 0.496565i \(-0.834594\pi\)
−0.867999 + 0.496565i \(0.834594\pi\)
\(104\) 11.2018 1.09842
\(105\) 0 0
\(106\) −10.5823 −1.02784
\(107\) −4.68661 −0.453072 −0.226536 0.974003i \(-0.572740\pi\)
−0.226536 + 0.974003i \(0.572740\pi\)
\(108\) 0.837862 0.0806233
\(109\) −5.02273 −0.481090 −0.240545 0.970638i \(-0.577326\pi\)
−0.240545 + 0.970638i \(0.577326\pi\)
\(110\) 0 0
\(111\) −4.44489 −0.421890
\(112\) −2.22469 −0.210213
\(113\) 17.8589 1.68002 0.840011 0.542569i \(-0.182548\pi\)
0.840011 + 0.542569i \(0.182548\pi\)
\(114\) −2.03634 −0.190721
\(115\) 0 0
\(116\) −9.93089 −0.922060
\(117\) −13.3991 −1.23875
\(118\) 0.953891 0.0878128
\(119\) −9.80809 −0.899106
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.65416 0.149760
\(123\) −24.8065 −2.23672
\(124\) 5.89314 0.529219
\(125\) 0 0
\(126\) −12.8125 −1.14143
\(127\) 5.52662 0.490409 0.245204 0.969471i \(-0.421145\pi\)
0.245204 + 0.969471i \(0.421145\pi\)
\(128\) −8.51418 −0.752554
\(129\) 24.7272 2.17711
\(130\) 0 0
\(131\) −4.31319 −0.376846 −0.188423 0.982088i \(-0.560338\pi\)
−0.188423 + 0.982088i \(0.560338\pi\)
\(132\) −3.34171 −0.290859
\(133\) −4.83914 −0.419606
\(134\) 6.38643 0.551703
\(135\) 0 0
\(136\) −5.50817 −0.472322
\(137\) 10.9200 0.932958 0.466479 0.884532i \(-0.345522\pi\)
0.466479 + 0.884532i \(0.345522\pi\)
\(138\) −13.0066 −1.10720
\(139\) 16.1839 1.37270 0.686352 0.727270i \(-0.259211\pi\)
0.686352 + 0.727270i \(0.259211\pi\)
\(140\) 0 0
\(141\) −0.363108 −0.0305792
\(142\) −9.09936 −0.763602
\(143\) 4.12188 0.344689
\(144\) 1.49445 0.124538
\(145\) 0 0
\(146\) −2.66502 −0.220558
\(147\) −41.0455 −3.38537
\(148\) −2.37629 −0.195330
\(149\) −14.5778 −1.19426 −0.597129 0.802145i \(-0.703692\pi\)
−0.597129 + 0.802145i \(0.703692\pi\)
\(150\) 0 0
\(151\) −22.0082 −1.79100 −0.895501 0.445060i \(-0.853182\pi\)
−0.895501 + 0.445060i \(0.853182\pi\)
\(152\) −2.71763 −0.220429
\(153\) 6.58867 0.532662
\(154\) 3.94143 0.317609
\(155\) 0 0
\(156\) −13.7741 −1.10281
\(157\) 18.5410 1.47974 0.739868 0.672752i \(-0.234888\pi\)
0.739868 + 0.672752i \(0.234888\pi\)
\(158\) 9.24220 0.735270
\(159\) 32.4833 2.57609
\(160\) 0 0
\(161\) −30.9087 −2.43595
\(162\) −6.66656 −0.523774
\(163\) 3.79670 0.297381 0.148690 0.988884i \(-0.452494\pi\)
0.148690 + 0.988884i \(0.452494\pi\)
\(164\) −13.2618 −1.03557
\(165\) 0 0
\(166\) 1.93114 0.149886
\(167\) −18.4689 −1.42917 −0.714583 0.699550i \(-0.753384\pi\)
−0.714583 + 0.699550i \(0.753384\pi\)
\(168\) −32.8794 −2.53670
\(169\) 3.98992 0.306917
\(170\) 0 0
\(171\) 3.25073 0.248589
\(172\) 13.2194 1.00797
\(173\) 3.71022 0.282082 0.141041 0.990004i \(-0.454955\pi\)
0.141041 + 0.990004i \(0.454955\pi\)
\(174\) −15.1299 −1.14699
\(175\) 0 0
\(176\) −0.459728 −0.0346533
\(177\) −2.92805 −0.220086
\(178\) −7.54968 −0.565872
\(179\) 18.6382 1.39308 0.696542 0.717516i \(-0.254721\pi\)
0.696542 + 0.717516i \(0.254721\pi\)
\(180\) 0 0
\(181\) 8.27637 0.615178 0.307589 0.951519i \(-0.400478\pi\)
0.307589 + 0.951519i \(0.400478\pi\)
\(182\) 16.2461 1.20424
\(183\) −5.07758 −0.375346
\(184\) −17.3582 −1.27966
\(185\) 0 0
\(186\) 8.97830 0.658321
\(187\) −2.02683 −0.148216
\(188\) −0.194122 −0.0141578
\(189\) 3.03345 0.220651
\(190\) 0 0
\(191\) −20.9577 −1.51645 −0.758223 0.651996i \(-0.773932\pi\)
−0.758223 + 0.651996i \(0.773932\pi\)
\(192\) −9.53179 −0.687897
\(193\) 1.14465 0.0823936 0.0411968 0.999151i \(-0.486883\pi\)
0.0411968 + 0.999151i \(0.486883\pi\)
\(194\) 6.75631 0.485075
\(195\) 0 0
\(196\) −21.9434 −1.56738
\(197\) 0.139556 0.00994294 0.00497147 0.999988i \(-0.498418\pi\)
0.00497147 + 0.999988i \(0.498418\pi\)
\(198\) −2.64769 −0.188163
\(199\) 10.3017 0.730266 0.365133 0.930955i \(-0.381023\pi\)
0.365133 + 0.930955i \(0.381023\pi\)
\(200\) 0 0
\(201\) −19.6037 −1.38274
\(202\) 4.07139 0.286462
\(203\) −35.9544 −2.52351
\(204\) 6.77307 0.474210
\(205\) 0 0
\(206\) −14.3501 −0.999816
\(207\) 20.7632 1.44314
\(208\) −1.89495 −0.131391
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 27.3988 1.88621 0.943107 0.332490i \(-0.107889\pi\)
0.943107 + 0.332490i \(0.107889\pi\)
\(212\) 17.3659 1.19270
\(213\) 27.9313 1.91382
\(214\) −3.81720 −0.260938
\(215\) 0 0
\(216\) 1.70357 0.115913
\(217\) 21.3359 1.44837
\(218\) −4.09096 −0.277075
\(219\) 8.18050 0.552787
\(220\) 0 0
\(221\) −8.35435 −0.561974
\(222\) −3.62032 −0.242980
\(223\) −13.3605 −0.894688 −0.447344 0.894362i \(-0.647630\pi\)
−0.447344 + 0.894362i \(0.647630\pi\)
\(224\) −28.1140 −1.87844
\(225\) 0 0
\(226\) 14.5459 0.967577
\(227\) 22.2383 1.47601 0.738005 0.674795i \(-0.235768\pi\)
0.738005 + 0.674795i \(0.235768\pi\)
\(228\) 3.34171 0.221310
\(229\) 14.3077 0.945481 0.472740 0.881202i \(-0.343265\pi\)
0.472740 + 0.881202i \(0.343265\pi\)
\(230\) 0 0
\(231\) −12.0985 −0.796026
\(232\) −20.1918 −1.32566
\(233\) 0.655778 0.0429615 0.0214807 0.999769i \(-0.493162\pi\)
0.0214807 + 0.999769i \(0.493162\pi\)
\(234\) −10.9135 −0.713435
\(235\) 0 0
\(236\) −1.56537 −0.101897
\(237\) −28.3697 −1.84281
\(238\) −7.98859 −0.517824
\(239\) −1.24483 −0.0805216 −0.0402608 0.999189i \(-0.512819\pi\)
−0.0402608 + 0.999189i \(0.512819\pi\)
\(240\) 0 0
\(241\) 26.2028 1.68787 0.843935 0.536446i \(-0.180233\pi\)
0.843935 + 0.536446i \(0.180233\pi\)
\(242\) 0.814490 0.0523574
\(243\) 22.3441 1.43338
\(244\) −2.71453 −0.173780
\(245\) 0 0
\(246\) −20.2046 −1.28820
\(247\) −4.12188 −0.262269
\(248\) 11.9821 0.760865
\(249\) −5.92781 −0.375660
\(250\) 0 0
\(251\) 6.42911 0.405802 0.202901 0.979199i \(-0.434963\pi\)
0.202901 + 0.979199i \(0.434963\pi\)
\(252\) 21.0258 1.32450
\(253\) −6.38724 −0.401562
\(254\) 4.50138 0.282442
\(255\) 0 0
\(256\) −14.5597 −0.909981
\(257\) −20.6502 −1.28813 −0.644064 0.764972i \(-0.722753\pi\)
−0.644064 + 0.764972i \(0.722753\pi\)
\(258\) 20.1401 1.25387
\(259\) −8.60327 −0.534581
\(260\) 0 0
\(261\) 24.1527 1.49501
\(262\) −3.51305 −0.217037
\(263\) −6.43749 −0.396953 −0.198476 0.980106i \(-0.563599\pi\)
−0.198476 + 0.980106i \(0.563599\pi\)
\(264\) −6.79448 −0.418171
\(265\) 0 0
\(266\) −3.94143 −0.241664
\(267\) 23.1744 1.41825
\(268\) −10.4803 −0.640189
\(269\) −3.76698 −0.229677 −0.114838 0.993384i \(-0.536635\pi\)
−0.114838 + 0.993384i \(0.536635\pi\)
\(270\) 0 0
\(271\) −7.83339 −0.475845 −0.237922 0.971284i \(-0.576466\pi\)
−0.237922 + 0.971284i \(0.576466\pi\)
\(272\) 0.931790 0.0564981
\(273\) −49.8688 −3.01820
\(274\) 8.89422 0.537320
\(275\) 0 0
\(276\) 21.3443 1.28478
\(277\) −30.7008 −1.84463 −0.922315 0.386439i \(-0.873705\pi\)
−0.922315 + 0.386439i \(0.873705\pi\)
\(278\) 13.1817 0.790583
\(279\) −14.3326 −0.858068
\(280\) 0 0
\(281\) 25.8170 1.54011 0.770057 0.637975i \(-0.220228\pi\)
0.770057 + 0.637975i \(0.220228\pi\)
\(282\) −0.295748 −0.0176115
\(283\) −8.25782 −0.490876 −0.245438 0.969412i \(-0.578932\pi\)
−0.245438 + 0.969412i \(0.578932\pi\)
\(284\) 14.9324 0.886073
\(285\) 0 0
\(286\) 3.35723 0.198517
\(287\) −48.0139 −2.83417
\(288\) 18.8858 1.11286
\(289\) −12.8920 −0.758351
\(290\) 0 0
\(291\) −20.7391 −1.21575
\(292\) 4.37339 0.255933
\(293\) 12.2486 0.715574 0.357787 0.933803i \(-0.383531\pi\)
0.357787 + 0.933803i \(0.383531\pi\)
\(294\) −33.4311 −1.94974
\(295\) 0 0
\(296\) −4.83155 −0.280828
\(297\) 0.626858 0.0363740
\(298\) −11.8735 −0.687811
\(299\) −26.3274 −1.52256
\(300\) 0 0
\(301\) 47.8605 2.75863
\(302\) −17.9255 −1.03149
\(303\) −12.4975 −0.717961
\(304\) 0.459728 0.0263672
\(305\) 0 0
\(306\) 5.36640 0.306777
\(307\) 1.13694 0.0648885 0.0324443 0.999474i \(-0.489671\pi\)
0.0324443 + 0.999474i \(0.489671\pi\)
\(308\) −6.46802 −0.368550
\(309\) 44.0487 2.50584
\(310\) 0 0
\(311\) −5.71535 −0.324088 −0.162044 0.986784i \(-0.551809\pi\)
−0.162044 + 0.986784i \(0.551809\pi\)
\(312\) −28.0060 −1.58553
\(313\) −9.16694 −0.518146 −0.259073 0.965858i \(-0.583417\pi\)
−0.259073 + 0.965858i \(0.583417\pi\)
\(314\) 15.1015 0.852226
\(315\) 0 0
\(316\) −15.1668 −0.853197
\(317\) −8.67039 −0.486977 −0.243489 0.969904i \(-0.578292\pi\)
−0.243489 + 0.969904i \(0.578292\pi\)
\(318\) 26.4573 1.48365
\(319\) −7.42993 −0.415996
\(320\) 0 0
\(321\) 11.7172 0.653991
\(322\) −25.1748 −1.40294
\(323\) 2.02683 0.112776
\(324\) 10.9401 0.607781
\(325\) 0 0
\(326\) 3.09238 0.171271
\(327\) 12.5575 0.694434
\(328\) −26.9644 −1.48886
\(329\) −0.702810 −0.0387472
\(330\) 0 0
\(331\) −17.6479 −0.970019 −0.485009 0.874509i \(-0.661184\pi\)
−0.485009 + 0.874509i \(0.661184\pi\)
\(332\) −3.16908 −0.173926
\(333\) 5.77932 0.316705
\(334\) −15.0427 −0.823102
\(335\) 0 0
\(336\) 5.56204 0.303434
\(337\) −23.9468 −1.30447 −0.652233 0.758019i \(-0.726168\pi\)
−0.652233 + 0.758019i \(0.726168\pi\)
\(338\) 3.24975 0.176763
\(339\) −44.6498 −2.42504
\(340\) 0 0
\(341\) 4.40903 0.238762
\(342\) 2.64769 0.143170
\(343\) −45.5712 −2.46062
\(344\) 26.8782 1.44918
\(345\) 0 0
\(346\) 3.02193 0.162460
\(347\) −4.75858 −0.255454 −0.127727 0.991809i \(-0.540768\pi\)
−0.127727 + 0.991809i \(0.540768\pi\)
\(348\) 24.8287 1.33096
\(349\) −21.1755 −1.13350 −0.566748 0.823891i \(-0.691799\pi\)
−0.566748 + 0.823891i \(0.691799\pi\)
\(350\) 0 0
\(351\) 2.58384 0.137915
\(352\) −5.80971 −0.309658
\(353\) 25.7740 1.37181 0.685905 0.727691i \(-0.259406\pi\)
0.685905 + 0.727691i \(0.259406\pi\)
\(354\) −2.38487 −0.126754
\(355\) 0 0
\(356\) 12.3893 0.656631
\(357\) 24.5217 1.29782
\(358\) 15.1806 0.802321
\(359\) −14.4460 −0.762432 −0.381216 0.924486i \(-0.624495\pi\)
−0.381216 + 0.924486i \(0.624495\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.74102 0.354300
\(363\) −2.50015 −0.131224
\(364\) −26.6604 −1.39739
\(365\) 0 0
\(366\) −4.13564 −0.216173
\(367\) −19.6353 −1.02496 −0.512478 0.858700i \(-0.671272\pi\)
−0.512478 + 0.858700i \(0.671272\pi\)
\(368\) 2.93639 0.153070
\(369\) 32.2537 1.67906
\(370\) 0 0
\(371\) 62.8728 3.26419
\(372\) −14.7337 −0.763907
\(373\) −2.25454 −0.116735 −0.0583677 0.998295i \(-0.518590\pi\)
−0.0583677 + 0.998295i \(0.518590\pi\)
\(374\) −1.65083 −0.0853624
\(375\) 0 0
\(376\) −0.394694 −0.0203548
\(377\) −30.6253 −1.57728
\(378\) 2.47071 0.127080
\(379\) 17.2791 0.887570 0.443785 0.896133i \(-0.353635\pi\)
0.443785 + 0.896133i \(0.353635\pi\)
\(380\) 0 0
\(381\) −13.8174 −0.707885
\(382\) −17.0698 −0.873369
\(383\) −16.9020 −0.863649 −0.431825 0.901958i \(-0.642130\pi\)
−0.431825 + 0.901958i \(0.642130\pi\)
\(384\) 21.2867 1.08628
\(385\) 0 0
\(386\) 0.932304 0.0474530
\(387\) −32.1507 −1.63431
\(388\) −11.0873 −0.562874
\(389\) −6.06683 −0.307600 −0.153800 0.988102i \(-0.549151\pi\)
−0.153800 + 0.988102i \(0.549151\pi\)
\(390\) 0 0
\(391\) 12.9458 0.654699
\(392\) −44.6160 −2.25345
\(393\) 10.7836 0.543961
\(394\) 0.113667 0.00572645
\(395\) 0 0
\(396\) 4.34494 0.218342
\(397\) 23.1083 1.15977 0.579887 0.814697i \(-0.303097\pi\)
0.579887 + 0.814697i \(0.303097\pi\)
\(398\) 8.39061 0.420583
\(399\) 12.0985 0.605685
\(400\) 0 0
\(401\) −27.5387 −1.37522 −0.687608 0.726082i \(-0.741339\pi\)
−0.687608 + 0.726082i \(0.741339\pi\)
\(402\) −15.9670 −0.796361
\(403\) 18.1735 0.905287
\(404\) −6.68129 −0.332407
\(405\) 0 0
\(406\) −29.2845 −1.45337
\(407\) −1.77785 −0.0881249
\(408\) 13.7712 0.681778
\(409\) 22.0368 1.08965 0.544824 0.838551i \(-0.316597\pi\)
0.544824 + 0.838551i \(0.316597\pi\)
\(410\) 0 0
\(411\) −27.3016 −1.34669
\(412\) 23.5489 1.16017
\(413\) −5.66736 −0.278872
\(414\) 16.9114 0.831150
\(415\) 0 0
\(416\) −23.9469 −1.17410
\(417\) −40.4622 −1.98144
\(418\) −0.814490 −0.0398380
\(419\) −6.24131 −0.304908 −0.152454 0.988311i \(-0.548718\pi\)
−0.152454 + 0.988311i \(0.548718\pi\)
\(420\) 0 0
\(421\) −14.9972 −0.730918 −0.365459 0.930827i \(-0.619088\pi\)
−0.365459 + 0.930827i \(0.619088\pi\)
\(422\) 22.3161 1.08633
\(423\) 0.472119 0.0229552
\(424\) 35.3090 1.71476
\(425\) 0 0
\(426\) 22.7497 1.10223
\(427\) −9.82787 −0.475604
\(428\) 6.26416 0.302789
\(429\) −10.3053 −0.497545
\(430\) 0 0
\(431\) 21.4445 1.03295 0.516473 0.856303i \(-0.327245\pi\)
0.516473 + 0.856303i \(0.327245\pi\)
\(432\) −0.288184 −0.0138653
\(433\) 10.6928 0.513861 0.256931 0.966430i \(-0.417289\pi\)
0.256931 + 0.966430i \(0.417289\pi\)
\(434\) 17.3779 0.834164
\(435\) 0 0
\(436\) 6.71341 0.321514
\(437\) 6.38724 0.305543
\(438\) 6.66293 0.318367
\(439\) −36.6563 −1.74951 −0.874755 0.484565i \(-0.838978\pi\)
−0.874755 + 0.484565i \(0.838978\pi\)
\(440\) 0 0
\(441\) 53.3680 2.54133
\(442\) −6.80453 −0.323659
\(443\) −21.8919 −1.04012 −0.520058 0.854131i \(-0.674090\pi\)
−0.520058 + 0.854131i \(0.674090\pi\)
\(444\) 5.94107 0.281951
\(445\) 0 0
\(446\) −10.8820 −0.515279
\(447\) 36.4466 1.72386
\(448\) −18.4492 −0.871641
\(449\) −26.2989 −1.24112 −0.620561 0.784158i \(-0.713095\pi\)
−0.620561 + 0.784158i \(0.713095\pi\)
\(450\) 0 0
\(451\) −9.92201 −0.467209
\(452\) −23.8703 −1.12276
\(453\) 55.0237 2.58524
\(454\) 18.1129 0.850081
\(455\) 0 0
\(456\) 6.79448 0.318180
\(457\) 33.5722 1.57044 0.785221 0.619215i \(-0.212549\pi\)
0.785221 + 0.619215i \(0.212549\pi\)
\(458\) 11.6535 0.544532
\(459\) −1.27053 −0.0593034
\(460\) 0 0
\(461\) −3.64687 −0.169852 −0.0849258 0.996387i \(-0.527065\pi\)
−0.0849258 + 0.996387i \(0.527065\pi\)
\(462\) −9.85414 −0.458456
\(463\) 14.8684 0.690991 0.345495 0.938420i \(-0.387711\pi\)
0.345495 + 0.938420i \(0.387711\pi\)
\(464\) 3.41575 0.158572
\(465\) 0 0
\(466\) 0.534125 0.0247429
\(467\) −27.8778 −1.29003 −0.645015 0.764170i \(-0.723149\pi\)
−0.645015 + 0.764170i \(0.723149\pi\)
\(468\) 17.9094 0.827860
\(469\) −37.9437 −1.75208
\(470\) 0 0
\(471\) −46.3553 −2.13594
\(472\) −3.18276 −0.146498
\(473\) 9.89030 0.454757
\(474\) −23.1068 −1.06133
\(475\) 0 0
\(476\) 13.1096 0.600876
\(477\) −42.2353 −1.93382
\(478\) −1.01390 −0.0463749
\(479\) −2.90602 −0.132779 −0.0663897 0.997794i \(-0.521148\pi\)
−0.0663897 + 0.997794i \(0.521148\pi\)
\(480\) 0 0
\(481\) −7.32810 −0.334133
\(482\) 21.3419 0.972097
\(483\) 77.2762 3.51619
\(484\) −1.33661 −0.0607548
\(485\) 0 0
\(486\) 18.1991 0.825527
\(487\) 17.6547 0.800010 0.400005 0.916513i \(-0.369008\pi\)
0.400005 + 0.916513i \(0.369008\pi\)
\(488\) −5.51928 −0.249846
\(489\) −9.49231 −0.429257
\(490\) 0 0
\(491\) 27.3096 1.23247 0.616233 0.787564i \(-0.288658\pi\)
0.616233 + 0.787564i \(0.288658\pi\)
\(492\) 33.1565 1.49481
\(493\) 15.0592 0.678231
\(494\) −3.35723 −0.151049
\(495\) 0 0
\(496\) −2.02696 −0.0910130
\(497\) 54.0621 2.42502
\(498\) −4.82814 −0.216354
\(499\) −19.0909 −0.854627 −0.427314 0.904104i \(-0.640540\pi\)
−0.427314 + 0.904104i \(0.640540\pi\)
\(500\) 0 0
\(501\) 46.1749 2.06294
\(502\) 5.23644 0.233714
\(503\) 39.8092 1.77500 0.887501 0.460805i \(-0.152439\pi\)
0.887501 + 0.460805i \(0.152439\pi\)
\(504\) 42.7503 1.90425
\(505\) 0 0
\(506\) −5.20234 −0.231272
\(507\) −9.97538 −0.443022
\(508\) −7.38692 −0.327742
\(509\) −9.96761 −0.441806 −0.220903 0.975296i \(-0.570900\pi\)
−0.220903 + 0.975296i \(0.570900\pi\)
\(510\) 0 0
\(511\) 15.8337 0.700441
\(512\) 5.16963 0.228468
\(513\) −0.626858 −0.0276765
\(514\) −16.8194 −0.741873
\(515\) 0 0
\(516\) −33.0505 −1.45497
\(517\) −0.145235 −0.00638742
\(518\) −7.00728 −0.307882
\(519\) −9.27608 −0.407175
\(520\) 0 0
\(521\) 24.1830 1.05948 0.529738 0.848162i \(-0.322290\pi\)
0.529738 + 0.848162i \(0.322290\pi\)
\(522\) 19.6721 0.861025
\(523\) 12.1479 0.531189 0.265594 0.964085i \(-0.414432\pi\)
0.265594 + 0.964085i \(0.414432\pi\)
\(524\) 5.76504 0.251847
\(525\) 0 0
\(526\) −5.24327 −0.228618
\(527\) −8.93634 −0.389273
\(528\) 1.14939 0.0500207
\(529\) 17.7968 0.773773
\(530\) 0 0
\(531\) 3.80709 0.165214
\(532\) 6.46802 0.280424
\(533\) −40.8973 −1.77146
\(534\) 18.8753 0.816814
\(535\) 0 0
\(536\) −21.3090 −0.920408
\(537\) −46.5982 −2.01086
\(538\) −3.06817 −0.132278
\(539\) −16.4172 −0.707140
\(540\) 0 0
\(541\) −17.1167 −0.735905 −0.367952 0.929845i \(-0.619941\pi\)
−0.367952 + 0.929845i \(0.619941\pi\)
\(542\) −6.38022 −0.274054
\(543\) −20.6921 −0.887984
\(544\) 11.7753 0.504861
\(545\) 0 0
\(546\) −40.6176 −1.73827
\(547\) 27.0558 1.15682 0.578411 0.815746i \(-0.303673\pi\)
0.578411 + 0.815746i \(0.303673\pi\)
\(548\) −14.5957 −0.623499
\(549\) 6.60195 0.281764
\(550\) 0 0
\(551\) 7.42993 0.316526
\(552\) 43.3979 1.84714
\(553\) −54.9107 −2.33504
\(554\) −25.0055 −1.06238
\(555\) 0 0
\(556\) −21.6316 −0.917382
\(557\) 32.5288 1.37829 0.689144 0.724624i \(-0.257987\pi\)
0.689144 + 0.724624i \(0.257987\pi\)
\(558\) −11.6737 −0.494188
\(559\) 40.7667 1.72425
\(560\) 0 0
\(561\) 5.06736 0.213944
\(562\) 21.0277 0.887000
\(563\) −3.59287 −0.151421 −0.0757107 0.997130i \(-0.524123\pi\)
−0.0757107 + 0.997130i \(0.524123\pi\)
\(564\) 0.485332 0.0204362
\(565\) 0 0
\(566\) −6.72591 −0.282711
\(567\) 39.6081 1.66338
\(568\) 30.3610 1.27392
\(569\) −32.6119 −1.36716 −0.683582 0.729874i \(-0.739579\pi\)
−0.683582 + 0.729874i \(0.739579\pi\)
\(570\) 0 0
\(571\) 35.9109 1.50283 0.751413 0.659832i \(-0.229373\pi\)
0.751413 + 0.659832i \(0.229373\pi\)
\(572\) −5.50933 −0.230357
\(573\) 52.3973 2.18893
\(574\) −39.1069 −1.63229
\(575\) 0 0
\(576\) 12.3934 0.516391
\(577\) 34.5334 1.43764 0.718822 0.695194i \(-0.244682\pi\)
0.718822 + 0.695194i \(0.244682\pi\)
\(578\) −10.5004 −0.436758
\(579\) −2.86179 −0.118932
\(580\) 0 0
\(581\) −11.4735 −0.476002
\(582\) −16.8918 −0.700186
\(583\) 12.9926 0.538097
\(584\) 8.89211 0.367958
\(585\) 0 0
\(586\) 9.97640 0.412121
\(587\) −37.3612 −1.54206 −0.771031 0.636798i \(-0.780259\pi\)
−0.771031 + 0.636798i \(0.780259\pi\)
\(588\) 54.8616 2.26246
\(589\) −4.40903 −0.181671
\(590\) 0 0
\(591\) −0.348910 −0.0143522
\(592\) 0.817330 0.0335920
\(593\) −8.14523 −0.334484 −0.167242 0.985916i \(-0.553486\pi\)
−0.167242 + 0.985916i \(0.553486\pi\)
\(594\) 0.510570 0.0209489
\(595\) 0 0
\(596\) 19.4847 0.798126
\(597\) −25.7557 −1.05411
\(598\) −21.4434 −0.876887
\(599\) −27.7885 −1.13541 −0.567704 0.823233i \(-0.692168\pi\)
−0.567704 + 0.823233i \(0.692168\pi\)
\(600\) 0 0
\(601\) 3.66613 0.149544 0.0747722 0.997201i \(-0.476177\pi\)
0.0747722 + 0.997201i \(0.476177\pi\)
\(602\) 38.9819 1.58878
\(603\) 25.4890 1.03799
\(604\) 29.4163 1.19693
\(605\) 0 0
\(606\) −10.1791 −0.413496
\(607\) 20.5553 0.834315 0.417158 0.908834i \(-0.363026\pi\)
0.417158 + 0.908834i \(0.363026\pi\)
\(608\) 5.80971 0.235615
\(609\) 89.8913 3.64258
\(610\) 0 0
\(611\) −0.598640 −0.0242184
\(612\) −8.80645 −0.355980
\(613\) 18.0377 0.728534 0.364267 0.931295i \(-0.381320\pi\)
0.364267 + 0.931295i \(0.381320\pi\)
\(614\) 0.926025 0.0373713
\(615\) 0 0
\(616\) −13.1510 −0.529868
\(617\) 1.78298 0.0717801 0.0358901 0.999356i \(-0.488573\pi\)
0.0358901 + 0.999356i \(0.488573\pi\)
\(618\) 35.8772 1.44319
\(619\) 23.1501 0.930479 0.465240 0.885185i \(-0.345968\pi\)
0.465240 + 0.885185i \(0.345968\pi\)
\(620\) 0 0
\(621\) −4.00389 −0.160671
\(622\) −4.65510 −0.186652
\(623\) 44.8550 1.79708
\(624\) 4.73764 0.189658
\(625\) 0 0
\(626\) −7.46638 −0.298417
\(627\) 2.50015 0.0998462
\(628\) −24.7821 −0.988912
\(629\) 3.60340 0.143677
\(630\) 0 0
\(631\) 16.4227 0.653778 0.326889 0.945063i \(-0.394000\pi\)
0.326889 + 0.945063i \(0.394000\pi\)
\(632\) −30.8376 −1.22665
\(633\) −68.5011 −2.72267
\(634\) −7.06194 −0.280466
\(635\) 0 0
\(636\) −43.4174 −1.72161
\(637\) −67.6699 −2.68118
\(638\) −6.05160 −0.239585
\(639\) −36.3167 −1.43667
\(640\) 0 0
\(641\) −16.3288 −0.644947 −0.322474 0.946578i \(-0.604514\pi\)
−0.322474 + 0.946578i \(0.604514\pi\)
\(642\) 9.54356 0.376654
\(643\) −25.4635 −1.00418 −0.502091 0.864815i \(-0.667436\pi\)
−0.502091 + 0.864815i \(0.667436\pi\)
\(644\) 41.3128 1.62795
\(645\) 0 0
\(646\) 1.65083 0.0649511
\(647\) 4.22559 0.166125 0.0830625 0.996544i \(-0.473530\pi\)
0.0830625 + 0.996544i \(0.473530\pi\)
\(648\) 22.2437 0.873814
\(649\) −1.17115 −0.0459717
\(650\) 0 0
\(651\) −53.3428 −2.09067
\(652\) −5.07470 −0.198741
\(653\) −6.29198 −0.246224 −0.123112 0.992393i \(-0.539287\pi\)
−0.123112 + 0.992393i \(0.539287\pi\)
\(654\) 10.2280 0.399946
\(655\) 0 0
\(656\) 4.56143 0.178094
\(657\) −10.6364 −0.414966
\(658\) −0.572432 −0.0223157
\(659\) −32.2776 −1.25736 −0.628678 0.777666i \(-0.716404\pi\)
−0.628678 + 0.777666i \(0.716404\pi\)
\(660\) 0 0
\(661\) 41.5732 1.61701 0.808505 0.588489i \(-0.200277\pi\)
0.808505 + 0.588489i \(0.200277\pi\)
\(662\) −14.3741 −0.558664
\(663\) 20.8871 0.811187
\(664\) −6.44347 −0.250055
\(665\) 0 0
\(666\) 4.70720 0.182400
\(667\) 47.4567 1.83753
\(668\) 24.6856 0.955116
\(669\) 33.4033 1.29145
\(670\) 0 0
\(671\) −2.03091 −0.0784026
\(672\) 70.2890 2.71146
\(673\) −16.4972 −0.635921 −0.317960 0.948104i \(-0.602998\pi\)
−0.317960 + 0.948104i \(0.602998\pi\)
\(674\) −19.5044 −0.751283
\(675\) 0 0
\(676\) −5.33295 −0.205113
\(677\) −34.3832 −1.32145 −0.660726 0.750627i \(-0.729752\pi\)
−0.660726 + 0.750627i \(0.729752\pi\)
\(678\) −36.3668 −1.39666
\(679\) −40.1413 −1.54048
\(680\) 0 0
\(681\) −55.5991 −2.13056
\(682\) 3.59111 0.137511
\(683\) −26.1835 −1.00188 −0.500941 0.865481i \(-0.667013\pi\)
−0.500941 + 0.865481i \(0.667013\pi\)
\(684\) −4.34494 −0.166133
\(685\) 0 0
\(686\) −37.1173 −1.41715
\(687\) −35.7714 −1.36476
\(688\) −4.54685 −0.173347
\(689\) 53.5538 2.04024
\(690\) 0 0
\(691\) −45.2927 −1.72301 −0.861507 0.507745i \(-0.830479\pi\)
−0.861507 + 0.507745i \(0.830479\pi\)
\(692\) −4.95910 −0.188517
\(693\) 15.7307 0.597560
\(694\) −3.87582 −0.147124
\(695\) 0 0
\(696\) 50.4825 1.91353
\(697\) 20.1102 0.761728
\(698\) −17.2472 −0.652816
\(699\) −1.63954 −0.0620132
\(700\) 0 0
\(701\) −12.7781 −0.482624 −0.241312 0.970448i \(-0.577578\pi\)
−0.241312 + 0.970448i \(0.577578\pi\)
\(702\) 2.10451 0.0794295
\(703\) 1.77785 0.0670530
\(704\) −3.81249 −0.143689
\(705\) 0 0
\(706\) 20.9926 0.790069
\(707\) −24.1894 −0.909736
\(708\) 3.91365 0.147084
\(709\) −16.4621 −0.618247 −0.309124 0.951022i \(-0.600036\pi\)
−0.309124 + 0.951022i \(0.600036\pi\)
\(710\) 0 0
\(711\) 36.8867 1.38336
\(712\) 25.1903 0.944047
\(713\) −28.1615 −1.05466
\(714\) 19.9726 0.747458
\(715\) 0 0
\(716\) −24.9119 −0.931003
\(717\) 3.11227 0.116230
\(718\) −11.7661 −0.439109
\(719\) 6.34240 0.236532 0.118266 0.992982i \(-0.462266\pi\)
0.118266 + 0.992982i \(0.462266\pi\)
\(720\) 0 0
\(721\) 85.2581 3.17518
\(722\) 0.814490 0.0303122
\(723\) −65.5108 −2.43637
\(724\) −11.0622 −0.411125
\(725\) 0 0
\(726\) −2.03634 −0.0755758
\(727\) 23.3638 0.866517 0.433258 0.901270i \(-0.357364\pi\)
0.433258 + 0.901270i \(0.357364\pi\)
\(728\) −54.2068 −2.00904
\(729\) −31.3088 −1.15958
\(730\) 0 0
\(731\) −20.0459 −0.741426
\(732\) 6.78672 0.250845
\(733\) 1.38798 0.0512661 0.0256331 0.999671i \(-0.491840\pi\)
0.0256331 + 0.999671i \(0.491840\pi\)
\(734\) −15.9928 −0.590304
\(735\) 0 0
\(736\) 37.1080 1.36782
\(737\) −7.84101 −0.288827
\(738\) 26.2704 0.967025
\(739\) 3.75342 0.138072 0.0690360 0.997614i \(-0.478008\pi\)
0.0690360 + 0.997614i \(0.478008\pi\)
\(740\) 0 0
\(741\) 10.3053 0.378575
\(742\) 51.2092 1.87995
\(743\) 6.00085 0.220150 0.110075 0.993923i \(-0.464891\pi\)
0.110075 + 0.993923i \(0.464891\pi\)
\(744\) −29.9570 −1.09828
\(745\) 0 0
\(746\) −1.83630 −0.0672316
\(747\) 7.70743 0.282000
\(748\) 2.70907 0.0990534
\(749\) 22.6792 0.828678
\(750\) 0 0
\(751\) −29.4168 −1.07343 −0.536717 0.843763i \(-0.680336\pi\)
−0.536717 + 0.843763i \(0.680336\pi\)
\(752\) 0.0667685 0.00243480
\(753\) −16.0737 −0.585758
\(754\) −24.9440 −0.908407
\(755\) 0 0
\(756\) −4.05453 −0.147462
\(757\) −11.1487 −0.405207 −0.202604 0.979261i \(-0.564940\pi\)
−0.202604 + 0.979261i \(0.564940\pi\)
\(758\) 14.0737 0.511179
\(759\) 15.9690 0.579639
\(760\) 0 0
\(761\) −19.0044 −0.688909 −0.344454 0.938803i \(-0.611936\pi\)
−0.344454 + 0.938803i \(0.611936\pi\)
\(762\) −11.2541 −0.407693
\(763\) 24.3057 0.879924
\(764\) 28.0122 1.01345
\(765\) 0 0
\(766\) −13.7665 −0.497403
\(767\) −4.82735 −0.174305
\(768\) 36.4014 1.31352
\(769\) −12.9925 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(770\) 0 0
\(771\) 51.6286 1.85936
\(772\) −1.52994 −0.0550639
\(773\) 8.76741 0.315342 0.157671 0.987492i \(-0.449602\pi\)
0.157671 + 0.987492i \(0.449602\pi\)
\(774\) −26.1864 −0.941251
\(775\) 0 0
\(776\) −22.5431 −0.809252
\(777\) 21.5094 0.771646
\(778\) −4.94137 −0.177157
\(779\) 9.92201 0.355493
\(780\) 0 0
\(781\) 11.1719 0.399760
\(782\) 10.5442 0.377061
\(783\) −4.65751 −0.166446
\(784\) 7.54747 0.269552
\(785\) 0 0
\(786\) 8.78314 0.313284
\(787\) −42.7634 −1.52435 −0.762175 0.647371i \(-0.775868\pi\)
−0.762175 + 0.647371i \(0.775868\pi\)
\(788\) −0.186531 −0.00664489
\(789\) 16.0947 0.572986
\(790\) 0 0
\(791\) −86.4215 −3.07280
\(792\) 8.83428 0.313913
\(793\) −8.37119 −0.297270
\(794\) 18.8215 0.667950
\(795\) 0 0
\(796\) −13.7693 −0.488039
\(797\) 19.2140 0.680593 0.340297 0.940318i \(-0.389473\pi\)
0.340297 + 0.940318i \(0.389473\pi\)
\(798\) 9.85414 0.348833
\(799\) 0.294366 0.0104139
\(800\) 0 0
\(801\) −30.1317 −1.06465
\(802\) −22.4300 −0.792030
\(803\) 3.27201 0.115467
\(804\) 26.2024 0.924087
\(805\) 0 0
\(806\) 14.8021 0.521383
\(807\) 9.41800 0.331529
\(808\) −13.5846 −0.477906
\(809\) −51.4141 −1.80762 −0.903812 0.427930i \(-0.859243\pi\)
−0.903812 + 0.427930i \(0.859243\pi\)
\(810\) 0 0
\(811\) −38.7105 −1.35931 −0.679654 0.733533i \(-0.737870\pi\)
−0.679654 + 0.733533i \(0.737870\pi\)
\(812\) 48.0569 1.68647
\(813\) 19.5846 0.686863
\(814\) −1.44804 −0.0507539
\(815\) 0 0
\(816\) −2.32961 −0.0815527
\(817\) −9.89030 −0.346018
\(818\) 17.9487 0.627562
\(819\) 64.8402 2.26570
\(820\) 0 0
\(821\) 27.2774 0.951986 0.475993 0.879449i \(-0.342089\pi\)
0.475993 + 0.879449i \(0.342089\pi\)
\(822\) −22.2369 −0.775599
\(823\) 29.7389 1.03663 0.518316 0.855189i \(-0.326559\pi\)
0.518316 + 0.855189i \(0.326559\pi\)
\(824\) 47.8805 1.66800
\(825\) 0 0
\(826\) −4.61601 −0.160611
\(827\) −7.43328 −0.258481 −0.129240 0.991613i \(-0.541254\pi\)
−0.129240 + 0.991613i \(0.541254\pi\)
\(828\) −27.7522 −0.964455
\(829\) 19.8954 0.690996 0.345498 0.938419i \(-0.387710\pi\)
0.345498 + 0.938419i \(0.387710\pi\)
\(830\) 0 0
\(831\) 76.7564 2.66265
\(832\) −15.7146 −0.544807
\(833\) 33.2749 1.15291
\(834\) −32.9561 −1.14117
\(835\) 0 0
\(836\) 1.33661 0.0462275
\(837\) 2.76384 0.0955321
\(838\) −5.08349 −0.175606
\(839\) 33.5693 1.15894 0.579471 0.814993i \(-0.303259\pi\)
0.579471 + 0.814993i \(0.303259\pi\)
\(840\) 0 0
\(841\) 26.2038 0.903580
\(842\) −12.2151 −0.420959
\(843\) −64.5463 −2.22309
\(844\) −36.6214 −1.26056
\(845\) 0 0
\(846\) 0.384536 0.0132206
\(847\) −4.83914 −0.166275
\(848\) −5.97305 −0.205115
\(849\) 20.6457 0.708560
\(850\) 0 0
\(851\) 11.3556 0.389264
\(852\) −37.3331 −1.27901
\(853\) −2.86820 −0.0982052 −0.0491026 0.998794i \(-0.515636\pi\)
−0.0491026 + 0.998794i \(0.515636\pi\)
\(854\) −8.00470 −0.273915
\(855\) 0 0
\(856\) 12.7365 0.435324
\(857\) 38.0209 1.29877 0.649385 0.760460i \(-0.275026\pi\)
0.649385 + 0.760460i \(0.275026\pi\)
\(858\) −8.39357 −0.286552
\(859\) −14.8502 −0.506681 −0.253341 0.967377i \(-0.581529\pi\)
−0.253341 + 0.967377i \(0.581529\pi\)
\(860\) 0 0
\(861\) 120.042 4.09101
\(862\) 17.4664 0.594906
\(863\) 15.6023 0.531109 0.265554 0.964096i \(-0.414445\pi\)
0.265554 + 0.964096i \(0.414445\pi\)
\(864\) −3.64186 −0.123899
\(865\) 0 0
\(866\) 8.70914 0.295949
\(867\) 32.2318 1.09465
\(868\) −28.5177 −0.967953
\(869\) −11.3472 −0.384928
\(870\) 0 0
\(871\) −32.3197 −1.09511
\(872\) 13.6499 0.462245
\(873\) 26.9653 0.912635
\(874\) 5.20234 0.175972
\(875\) 0 0
\(876\) −10.9341 −0.369429
\(877\) 37.4675 1.26519 0.632594 0.774484i \(-0.281990\pi\)
0.632594 + 0.774484i \(0.281990\pi\)
\(878\) −29.8562 −1.00760
\(879\) −30.6234 −1.03290
\(880\) 0 0
\(881\) −43.7824 −1.47507 −0.737533 0.675311i \(-0.764009\pi\)
−0.737533 + 0.675311i \(0.764009\pi\)
\(882\) 43.4677 1.46363
\(883\) −56.9866 −1.91775 −0.958876 0.283825i \(-0.908397\pi\)
−0.958876 + 0.283825i \(0.908397\pi\)
\(884\) 11.1665 0.375569
\(885\) 0 0
\(886\) −17.8307 −0.599036
\(887\) −41.5001 −1.39344 −0.696718 0.717345i \(-0.745357\pi\)
−0.696718 + 0.717345i \(0.745357\pi\)
\(888\) 12.0796 0.405364
\(889\) −26.7441 −0.896968
\(890\) 0 0
\(891\) 8.18495 0.274206
\(892\) 17.8578 0.597923
\(893\) 0.145235 0.00486009
\(894\) 29.6854 0.992827
\(895\) 0 0
\(896\) 41.2013 1.37644
\(897\) 65.8224 2.19775
\(898\) −21.4202 −0.714801
\(899\) −32.7588 −1.09257
\(900\) 0 0
\(901\) −26.3337 −0.877302
\(902\) −8.08137 −0.269080
\(903\) −119.658 −3.98198
\(904\) −48.5339 −1.61421
\(905\) 0 0
\(906\) 44.8163 1.48892
\(907\) −11.8437 −0.393263 −0.196632 0.980477i \(-0.563000\pi\)
−0.196632 + 0.980477i \(0.563000\pi\)
\(908\) −29.7239 −0.986423
\(909\) 16.2494 0.538959
\(910\) 0 0
\(911\) −23.4751 −0.777765 −0.388882 0.921287i \(-0.627139\pi\)
−0.388882 + 0.921287i \(0.627139\pi\)
\(912\) −1.14939 −0.0380600
\(913\) −2.37099 −0.0784682
\(914\) 27.3442 0.904467
\(915\) 0 0
\(916\) −19.1238 −0.631868
\(917\) 20.8721 0.689258
\(918\) −1.03484 −0.0341547
\(919\) 16.6026 0.547668 0.273834 0.961777i \(-0.411708\pi\)
0.273834 + 0.961777i \(0.411708\pi\)
\(920\) 0 0
\(921\) −2.84251 −0.0936640
\(922\) −2.97034 −0.0978229
\(923\) 46.0491 1.51572
\(924\) 16.1710 0.531986
\(925\) 0 0
\(926\) 12.1101 0.397963
\(927\) −57.2728 −1.88109
\(928\) 43.1657 1.41698
\(929\) 24.2704 0.796287 0.398144 0.917323i \(-0.369655\pi\)
0.398144 + 0.917323i \(0.369655\pi\)
\(930\) 0 0
\(931\) 16.4172 0.538053
\(932\) −0.876518 −0.0287113
\(933\) 14.2892 0.467808
\(934\) −22.7062 −0.742969
\(935\) 0 0
\(936\) 36.4139 1.19023
\(937\) −4.73959 −0.154836 −0.0774179 0.996999i \(-0.524668\pi\)
−0.0774179 + 0.996999i \(0.524668\pi\)
\(938\) −30.9048 −1.00908
\(939\) 22.9187 0.747923
\(940\) 0 0
\(941\) −17.3884 −0.566846 −0.283423 0.958995i \(-0.591470\pi\)
−0.283423 + 0.958995i \(0.591470\pi\)
\(942\) −37.7559 −1.23015
\(943\) 63.3742 2.06375
\(944\) 0.538412 0.0175238
\(945\) 0 0
\(946\) 8.05555 0.261909
\(947\) 36.0870 1.17267 0.586335 0.810069i \(-0.300570\pi\)
0.586335 + 0.810069i \(0.300570\pi\)
\(948\) 37.9191 1.23156
\(949\) 13.4868 0.437801
\(950\) 0 0
\(951\) 21.6772 0.702932
\(952\) 26.6548 0.863887
\(953\) −2.22478 −0.0720676 −0.0360338 0.999351i \(-0.511472\pi\)
−0.0360338 + 0.999351i \(0.511472\pi\)
\(954\) −34.4002 −1.11375
\(955\) 0 0
\(956\) 1.66385 0.0538128
\(957\) 18.5759 0.600474
\(958\) −2.36692 −0.0764718
\(959\) −52.8433 −1.70640
\(960\) 0 0
\(961\) −11.5605 −0.372918
\(962\) −5.96866 −0.192437
\(963\) −15.2349 −0.490938
\(964\) −35.0228 −1.12801
\(965\) 0 0
\(966\) 62.9407 2.02508
\(967\) −11.2409 −0.361484 −0.180742 0.983531i \(-0.557850\pi\)
−0.180742 + 0.983531i \(0.557850\pi\)
\(968\) −2.71763 −0.0873480
\(969\) −5.06736 −0.162787
\(970\) 0 0
\(971\) −41.8105 −1.34176 −0.670881 0.741565i \(-0.734084\pi\)
−0.670881 + 0.741565i \(0.734084\pi\)
\(972\) −29.8653 −0.957930
\(973\) −78.3163 −2.51070
\(974\) 14.3796 0.460751
\(975\) 0 0
\(976\) 0.933669 0.0298860
\(977\) 6.49730 0.207867 0.103934 0.994584i \(-0.466857\pi\)
0.103934 + 0.994584i \(0.466857\pi\)
\(978\) −7.73139 −0.247223
\(979\) 9.26921 0.296245
\(980\) 0 0
\(981\) −16.3275 −0.521298
\(982\) 22.2434 0.709816
\(983\) 49.7817 1.58779 0.793895 0.608055i \(-0.208050\pi\)
0.793895 + 0.608055i \(0.208050\pi\)
\(984\) 67.4148 2.14911
\(985\) 0 0
\(986\) 12.2656 0.390615
\(987\) 1.75713 0.0559300
\(988\) 5.50933 0.175275
\(989\) −63.1717 −2.00874
\(990\) 0 0
\(991\) 3.05414 0.0970179 0.0485090 0.998823i \(-0.484553\pi\)
0.0485090 + 0.998823i \(0.484553\pi\)
\(992\) −25.6152 −0.813283
\(993\) 44.1224 1.40018
\(994\) 44.0331 1.39664
\(995\) 0 0
\(996\) 7.92315 0.251055
\(997\) −14.2020 −0.449783 −0.224892 0.974384i \(-0.572203\pi\)
−0.224892 + 0.974384i \(0.572203\pi\)
\(998\) −15.5494 −0.492207
\(999\) −1.11446 −0.0352600
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.v.1.9 yes 15
5.4 even 2 5225.2.a.u.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.u.1.7 15 5.4 even 2
5225.2.a.v.1.9 yes 15 1.1 even 1 trivial