Properties

Label 207.2.a.d.1.1
Level $207$
Weight $2$
Character 207.1
Self dual yes
Analytic conductor $1.653$
Analytic rank $0$
Dimension $2$
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] = [207,2,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, 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("207.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} +0.763932 q^{10} +5.23607 q^{11} +3.00000 q^{13} -2.00000 q^{14} +1.85410 q^{16} -0.763932 q^{17} -2.00000 q^{19} +2.00000 q^{20} -3.23607 q^{22} -1.00000 q^{23} -3.47214 q^{25} -1.85410 q^{26} -5.23607 q^{28} +3.00000 q^{29} +6.70820 q^{31} -5.61803 q^{32} +0.472136 q^{34} -4.00000 q^{35} -1.23607 q^{37} +1.23607 q^{38} -2.76393 q^{40} +3.47214 q^{41} -8.47214 q^{44} +0.618034 q^{46} +2.23607 q^{47} +3.47214 q^{49} +2.14590 q^{50} -4.85410 q^{52} -0.472136 q^{53} -6.47214 q^{55} +7.23607 q^{56} -1.85410 q^{58} -6.47214 q^{59} -6.94427 q^{61} -4.14590 q^{62} -0.236068 q^{64} -3.70820 q^{65} -2.76393 q^{67} +1.23607 q^{68} +2.47214 q^{70} -12.2361 q^{71} +6.52786 q^{73} +0.763932 q^{74} +3.23607 q^{76} +16.9443 q^{77} -10.9443 q^{79} -2.29180 q^{80} -2.14590 q^{82} +8.76393 q^{83} +0.944272 q^{85} +11.7082 q^{88} +10.4721 q^{89} +9.70820 q^{91} +1.61803 q^{92} -1.38197 q^{94} +2.47214 q^{95} +17.7082 q^{97} -2.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 4 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} - 6 q^{28} + 6 q^{29} - 9 q^{32} - 8 q^{34} - 8 q^{35} + 2 q^{37} - 2 q^{38} - 10 q^{40} - 2 q^{41} - 8 q^{44} - q^{46} - 2 q^{49} + 11 q^{50} - 3 q^{52} + 8 q^{53} - 4 q^{55} + 10 q^{56} + 3 q^{58} - 4 q^{59} + 4 q^{61} - 15 q^{62} + 4 q^{64} + 6 q^{65} - 10 q^{67} - 2 q^{68} - 4 q^{70} - 20 q^{71} + 22 q^{73} + 6 q^{74} + 2 q^{76} + 16 q^{77} - 4 q^{79} - 18 q^{80} - 11 q^{82} + 22 q^{83} - 16 q^{85} + 10 q^{88} + 12 q^{89} + 6 q^{91} + q^{92} - 5 q^{94} - 4 q^{95} + 22 q^{97} - 11 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0.763932 0.241577
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −3.23607 −0.689932
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) −1.85410 −0.363619
\(27\) 0 0
\(28\) −5.23607 −0.989524
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) 0.472136 0.0809706
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −1.23607 −0.203208 −0.101604 0.994825i \(-0.532398\pi\)
−0.101604 + 0.994825i \(0.532398\pi\)
\(38\) 1.23607 0.200517
\(39\) 0 0
\(40\) −2.76393 −0.437016
\(41\) 3.47214 0.542257 0.271128 0.962543i \(-0.412603\pi\)
0.271128 + 0.962543i \(0.412603\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −8.47214 −1.27722
\(45\) 0 0
\(46\) 0.618034 0.0911241
\(47\) 2.23607 0.326164 0.163082 0.986613i \(-0.447856\pi\)
0.163082 + 0.986613i \(0.447856\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 2.14590 0.303476
\(51\) 0 0
\(52\) −4.85410 −0.673143
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 7.23607 0.966960
\(57\) 0 0
\(58\) −1.85410 −0.243456
\(59\) −6.47214 −0.842600 −0.421300 0.906921i \(-0.638426\pi\)
−0.421300 + 0.906921i \(0.638426\pi\)
\(60\) 0 0
\(61\) −6.94427 −0.889123 −0.444561 0.895748i \(-0.646640\pi\)
−0.444561 + 0.895748i \(0.646640\pi\)
\(62\) −4.14590 −0.526530
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −3.70820 −0.459946
\(66\) 0 0
\(67\) −2.76393 −0.337668 −0.168834 0.985644i \(-0.554000\pi\)
−0.168834 + 0.985644i \(0.554000\pi\)
\(68\) 1.23607 0.149895
\(69\) 0 0
\(70\) 2.47214 0.295477
\(71\) −12.2361 −1.45215 −0.726077 0.687613i \(-0.758658\pi\)
−0.726077 + 0.687613i \(0.758658\pi\)
\(72\) 0 0
\(73\) 6.52786 0.764029 0.382014 0.924156i \(-0.375230\pi\)
0.382014 + 0.924156i \(0.375230\pi\)
\(74\) 0.763932 0.0888053
\(75\) 0 0
\(76\) 3.23607 0.371202
\(77\) 16.9443 1.93098
\(78\) 0 0
\(79\) −10.9443 −1.23133 −0.615663 0.788009i \(-0.711112\pi\)
−0.615663 + 0.788009i \(0.711112\pi\)
\(80\) −2.29180 −0.256231
\(81\) 0 0
\(82\) −2.14590 −0.236975
\(83\) 8.76393 0.961967 0.480983 0.876730i \(-0.340280\pi\)
0.480983 + 0.876730i \(0.340280\pi\)
\(84\) 0 0
\(85\) 0.944272 0.102421
\(86\) 0 0
\(87\) 0 0
\(88\) 11.7082 1.24810
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 9.70820 1.01770
\(92\) 1.61803 0.168692
\(93\) 0 0
\(94\) −1.38197 −0.142539
\(95\) 2.47214 0.253636
\(96\) 0 0
\(97\) 17.7082 1.79800 0.898998 0.437953i \(-0.144296\pi\)
0.898998 + 0.437953i \(0.144296\pi\)
\(98\) −2.14590 −0.216768
\(99\) 0 0
\(100\) 5.61803 0.561803
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) −4.18034 −0.411901 −0.205951 0.978562i \(-0.566029\pi\)
−0.205951 + 0.978562i \(0.566029\pi\)
\(104\) 6.70820 0.657794
\(105\) 0 0
\(106\) 0.291796 0.0283417
\(107\) −13.4164 −1.29701 −0.648507 0.761209i \(-0.724606\pi\)
−0.648507 + 0.761209i \(0.724606\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 6.00000 0.566947
\(113\) −8.76393 −0.824441 −0.412221 0.911084i \(-0.635247\pi\)
−0.412221 + 0.911084i \(0.635247\pi\)
\(114\) 0 0
\(115\) 1.23607 0.115264
\(116\) −4.85410 −0.450692
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −2.47214 −0.226620
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 4.29180 0.388561
\(123\) 0 0
\(124\) −10.8541 −0.974727
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −7.29180 −0.647042 −0.323521 0.946221i \(-0.604867\pi\)
−0.323521 + 0.946221i \(0.604867\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) 2.29180 0.201004
\(131\) −18.7082 −1.63454 −0.817272 0.576253i \(-0.804514\pi\)
−0.817272 + 0.576253i \(0.804514\pi\)
\(132\) 0 0
\(133\) −6.47214 −0.561205
\(134\) 1.70820 0.147566
\(135\) 0 0
\(136\) −1.70820 −0.146477
\(137\) 21.8885 1.87006 0.935032 0.354563i \(-0.115370\pi\)
0.935032 + 0.354563i \(0.115370\pi\)
\(138\) 0 0
\(139\) −10.7082 −0.908258 −0.454129 0.890936i \(-0.650049\pi\)
−0.454129 + 0.890936i \(0.650049\pi\)
\(140\) 6.47214 0.546995
\(141\) 0 0
\(142\) 7.56231 0.634615
\(143\) 15.7082 1.31359
\(144\) 0 0
\(145\) −3.70820 −0.307950
\(146\) −4.03444 −0.333893
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −23.8885 −1.95703 −0.978513 0.206186i \(-0.933895\pi\)
−0.978513 + 0.206186i \(0.933895\pi\)
\(150\) 0 0
\(151\) 4.23607 0.344726 0.172363 0.985033i \(-0.444860\pi\)
0.172363 + 0.985033i \(0.444860\pi\)
\(152\) −4.47214 −0.362738
\(153\) 0 0
\(154\) −10.4721 −0.843869
\(155\) −8.29180 −0.666013
\(156\) 0 0
\(157\) −11.4164 −0.911129 −0.455564 0.890203i \(-0.650562\pi\)
−0.455564 + 0.890203i \(0.650562\pi\)
\(158\) 6.76393 0.538110
\(159\) 0 0
\(160\) 6.94427 0.548993
\(161\) −3.23607 −0.255038
\(162\) 0 0
\(163\) −5.76393 −0.451466 −0.225733 0.974189i \(-0.572478\pi\)
−0.225733 + 0.974189i \(0.572478\pi\)
\(164\) −5.61803 −0.438695
\(165\) 0 0
\(166\) −5.41641 −0.420395
\(167\) −1.52786 −0.118230 −0.0591148 0.998251i \(-0.518828\pi\)
−0.0591148 + 0.998251i \(0.518828\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −0.583592 −0.0447595
\(171\) 0 0
\(172\) 0 0
\(173\) −22.9443 −1.74442 −0.872210 0.489131i \(-0.837314\pi\)
−0.872210 + 0.489131i \(0.837314\pi\)
\(174\) 0 0
\(175\) −11.2361 −0.849367
\(176\) 9.70820 0.731783
\(177\) 0 0
\(178\) −6.47214 −0.485107
\(179\) −0.708204 −0.0529336 −0.0264668 0.999650i \(-0.508426\pi\)
−0.0264668 + 0.999650i \(0.508426\pi\)
\(180\) 0 0
\(181\) 16.6525 1.23777 0.618884 0.785482i \(-0.287585\pi\)
0.618884 + 0.785482i \(0.287585\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −2.23607 −0.164845
\(185\) 1.52786 0.112331
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) −3.61803 −0.263872
\(189\) 0 0
\(190\) −1.52786 −0.110843
\(191\) 26.1803 1.89434 0.947171 0.320728i \(-0.103927\pi\)
0.947171 + 0.320728i \(0.103927\pi\)
\(192\) 0 0
\(193\) 9.94427 0.715804 0.357902 0.933759i \(-0.383492\pi\)
0.357902 + 0.933759i \(0.383492\pi\)
\(194\) −10.9443 −0.785753
\(195\) 0 0
\(196\) −5.61803 −0.401288
\(197\) 1.47214 0.104885 0.0524427 0.998624i \(-0.483299\pi\)
0.0524427 + 0.998624i \(0.483299\pi\)
\(198\) 0 0
\(199\) −12.2918 −0.871342 −0.435671 0.900106i \(-0.643489\pi\)
−0.435671 + 0.900106i \(0.643489\pi\)
\(200\) −7.76393 −0.548993
\(201\) 0 0
\(202\) 2.76393 0.194470
\(203\) 9.70820 0.681382
\(204\) 0 0
\(205\) −4.29180 −0.299752
\(206\) 2.58359 0.180007
\(207\) 0 0
\(208\) 5.56231 0.385677
\(209\) −10.4721 −0.724373
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) 0.763932 0.0524671
\(213\) 0 0
\(214\) 8.29180 0.566816
\(215\) 0 0
\(216\) 0 0
\(217\) 21.7082 1.47365
\(218\) 0 0
\(219\) 0 0
\(220\) 10.4721 0.706031
\(221\) −2.29180 −0.154163
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −18.1803 −1.21473
\(225\) 0 0
\(226\) 5.41641 0.360294
\(227\) 12.1803 0.808438 0.404219 0.914662i \(-0.367543\pi\)
0.404219 + 0.914662i \(0.367543\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −0.763932 −0.0503722
\(231\) 0 0
\(232\) 6.70820 0.440415
\(233\) 6.52786 0.427655 0.213827 0.976871i \(-0.431407\pi\)
0.213827 + 0.976871i \(0.431407\pi\)
\(234\) 0 0
\(235\) −2.76393 −0.180299
\(236\) 10.4721 0.681678
\(237\) 0 0
\(238\) 1.52786 0.0990367
\(239\) −13.7639 −0.890315 −0.445157 0.895452i \(-0.646852\pi\)
−0.445157 + 0.895452i \(0.646852\pi\)
\(240\) 0 0
\(241\) −23.1246 −1.48959 −0.744794 0.667295i \(-0.767452\pi\)
−0.744794 + 0.667295i \(0.767452\pi\)
\(242\) −10.1459 −0.652203
\(243\) 0 0
\(244\) 11.2361 0.719316
\(245\) −4.29180 −0.274193
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 15.0000 0.952501
\(249\) 0 0
\(250\) −6.47214 −0.409334
\(251\) −2.29180 −0.144657 −0.0723284 0.997381i \(-0.523043\pi\)
−0.0723284 + 0.997381i \(0.523043\pi\)
\(252\) 0 0
\(253\) −5.23607 −0.329189
\(254\) 4.50658 0.282768
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 7.47214 0.466099 0.233050 0.972465i \(-0.425130\pi\)
0.233050 + 0.972465i \(0.425130\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) 11.5623 0.714322
\(263\) −2.94427 −0.181552 −0.0907758 0.995871i \(-0.528935\pi\)
−0.0907758 + 0.995871i \(0.528935\pi\)
\(264\) 0 0
\(265\) 0.583592 0.0358498
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 4.47214 0.273179
\(269\) 7.94427 0.484371 0.242185 0.970230i \(-0.422136\pi\)
0.242185 + 0.970230i \(0.422136\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −1.41641 −0.0858823
\(273\) 0 0
\(274\) −13.5279 −0.817248
\(275\) −18.1803 −1.09632
\(276\) 0 0
\(277\) 15.4721 0.929631 0.464815 0.885408i \(-0.346121\pi\)
0.464815 + 0.885408i \(0.346121\pi\)
\(278\) 6.61803 0.396923
\(279\) 0 0
\(280\) −8.94427 −0.534522
\(281\) 8.76393 0.522812 0.261406 0.965229i \(-0.415814\pi\)
0.261406 + 0.965229i \(0.415814\pi\)
\(282\) 0 0
\(283\) 27.7082 1.64708 0.823541 0.567257i \(-0.191995\pi\)
0.823541 + 0.567257i \(0.191995\pi\)
\(284\) 19.7984 1.17482
\(285\) 0 0
\(286\) −9.70820 −0.574058
\(287\) 11.2361 0.663244
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 2.29180 0.134579
\(291\) 0 0
\(292\) −10.5623 −0.618112
\(293\) 1.52786 0.0892588 0.0446294 0.999004i \(-0.485789\pi\)
0.0446294 + 0.999004i \(0.485789\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −2.76393 −0.160650
\(297\) 0 0
\(298\) 14.7639 0.855252
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 0 0
\(302\) −2.61803 −0.150651
\(303\) 0 0
\(304\) −3.70820 −0.212680
\(305\) 8.58359 0.491495
\(306\) 0 0
\(307\) 9.52786 0.543784 0.271892 0.962328i \(-0.412351\pi\)
0.271892 + 0.962328i \(0.412351\pi\)
\(308\) −27.4164 −1.56219
\(309\) 0 0
\(310\) 5.12461 0.291058
\(311\) −13.1803 −0.747389 −0.373694 0.927552i \(-0.621909\pi\)
−0.373694 + 0.927552i \(0.621909\pi\)
\(312\) 0 0
\(313\) 24.3607 1.37695 0.688474 0.725261i \(-0.258281\pi\)
0.688474 + 0.725261i \(0.258281\pi\)
\(314\) 7.05573 0.398178
\(315\) 0 0
\(316\) 17.7082 0.996164
\(317\) −25.4164 −1.42753 −0.713764 0.700386i \(-0.753011\pi\)
−0.713764 + 0.700386i \(0.753011\pi\)
\(318\) 0 0
\(319\) 15.7082 0.879491
\(320\) 0.291796 0.0163119
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) 1.52786 0.0850126
\(324\) 0 0
\(325\) −10.4164 −0.577798
\(326\) 3.56231 0.197298
\(327\) 0 0
\(328\) 7.76393 0.428691
\(329\) 7.23607 0.398937
\(330\) 0 0
\(331\) −19.6525 −1.08020 −0.540099 0.841602i \(-0.681613\pi\)
−0.540099 + 0.841602i \(0.681613\pi\)
\(332\) −14.1803 −0.778247
\(333\) 0 0
\(334\) 0.944272 0.0516683
\(335\) 3.41641 0.186658
\(336\) 0 0
\(337\) 23.4164 1.27557 0.637787 0.770213i \(-0.279850\pi\)
0.637787 + 0.770213i \(0.279850\pi\)
\(338\) 2.47214 0.134466
\(339\) 0 0
\(340\) −1.52786 −0.0828601
\(341\) 35.1246 1.90210
\(342\) 0 0
\(343\) −11.4164 −0.616428
\(344\) 0 0
\(345\) 0 0
\(346\) 14.1803 0.762340
\(347\) 9.88854 0.530845 0.265422 0.964132i \(-0.414489\pi\)
0.265422 + 0.964132i \(0.414489\pi\)
\(348\) 0 0
\(349\) 24.4164 1.30698 0.653490 0.756935i \(-0.273304\pi\)
0.653490 + 0.756935i \(0.273304\pi\)
\(350\) 6.94427 0.371187
\(351\) 0 0
\(352\) −29.4164 −1.56790
\(353\) −9.36068 −0.498219 −0.249109 0.968475i \(-0.580138\pi\)
−0.249109 + 0.968475i \(0.580138\pi\)
\(354\) 0 0
\(355\) 15.1246 0.802731
\(356\) −16.9443 −0.898045
\(357\) 0 0
\(358\) 0.437694 0.0231329
\(359\) 19.8885 1.04968 0.524839 0.851202i \(-0.324126\pi\)
0.524839 + 0.851202i \(0.324126\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −10.2918 −0.540925
\(363\) 0 0
\(364\) −15.7082 −0.823334
\(365\) −8.06888 −0.422345
\(366\) 0 0
\(367\) −4.18034 −0.218212 −0.109106 0.994030i \(-0.534799\pi\)
−0.109106 + 0.994030i \(0.534799\pi\)
\(368\) −1.85410 −0.0966517
\(369\) 0 0
\(370\) −0.944272 −0.0490904
\(371\) −1.52786 −0.0793227
\(372\) 0 0
\(373\) 7.70820 0.399116 0.199558 0.979886i \(-0.436049\pi\)
0.199558 + 0.979886i \(0.436049\pi\)
\(374\) 2.47214 0.127831
\(375\) 0 0
\(376\) 5.00000 0.257855
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) 24.3607 1.25132 0.625662 0.780094i \(-0.284829\pi\)
0.625662 + 0.780094i \(0.284829\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −16.1803 −0.827858
\(383\) −7.05573 −0.360531 −0.180265 0.983618i \(-0.557696\pi\)
−0.180265 + 0.983618i \(0.557696\pi\)
\(384\) 0 0
\(385\) −20.9443 −1.06742
\(386\) −6.14590 −0.312818
\(387\) 0 0
\(388\) −28.6525 −1.45461
\(389\) −25.5279 −1.29431 −0.647157 0.762357i \(-0.724042\pi\)
−0.647157 + 0.762357i \(0.724042\pi\)
\(390\) 0 0
\(391\) 0.763932 0.0386337
\(392\) 7.76393 0.392138
\(393\) 0 0
\(394\) −0.909830 −0.0458366
\(395\) 13.5279 0.680661
\(396\) 0 0
\(397\) −24.4164 −1.22542 −0.612712 0.790306i \(-0.709922\pi\)
−0.612712 + 0.790306i \(0.709922\pi\)
\(398\) 7.59675 0.380791
\(399\) 0 0
\(400\) −6.43769 −0.321885
\(401\) 14.1803 0.708132 0.354066 0.935220i \(-0.384799\pi\)
0.354066 + 0.935220i \(0.384799\pi\)
\(402\) 0 0
\(403\) 20.1246 1.00248
\(404\) 7.23607 0.360008
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −6.47214 −0.320812
\(408\) 0 0
\(409\) 21.3607 1.05622 0.528109 0.849177i \(-0.322901\pi\)
0.528109 + 0.849177i \(0.322901\pi\)
\(410\) 2.65248 0.130996
\(411\) 0 0
\(412\) 6.76393 0.333235
\(413\) −20.9443 −1.03060
\(414\) 0 0
\(415\) −10.8328 −0.531762
\(416\) −16.8541 −0.826340
\(417\) 0 0
\(418\) 6.47214 0.316563
\(419\) 4.58359 0.223923 0.111962 0.993713i \(-0.464287\pi\)
0.111962 + 0.993713i \(0.464287\pi\)
\(420\) 0 0
\(421\) −10.2918 −0.501591 −0.250796 0.968040i \(-0.580692\pi\)
−0.250796 + 0.968040i \(0.580692\pi\)
\(422\) 14.4721 0.704493
\(423\) 0 0
\(424\) −1.05573 −0.0512707
\(425\) 2.65248 0.128664
\(426\) 0 0
\(427\) −22.4721 −1.08750
\(428\) 21.7082 1.04931
\(429\) 0 0
\(430\) 0 0
\(431\) 17.5279 0.844288 0.422144 0.906529i \(-0.361278\pi\)
0.422144 + 0.906529i \(0.361278\pi\)
\(432\) 0 0
\(433\) 17.8197 0.856358 0.428179 0.903694i \(-0.359155\pi\)
0.428179 + 0.903694i \(0.359155\pi\)
\(434\) −13.4164 −0.644008
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −18.7082 −0.892894 −0.446447 0.894810i \(-0.647311\pi\)
−0.446447 + 0.894810i \(0.647311\pi\)
\(440\) −14.4721 −0.689932
\(441\) 0 0
\(442\) 1.41641 0.0673717
\(443\) −38.1246 −1.81135 −0.905677 0.423967i \(-0.860637\pi\)
−0.905677 + 0.423967i \(0.860637\pi\)
\(444\) 0 0
\(445\) −12.9443 −0.613617
\(446\) −2.47214 −0.117059
\(447\) 0 0
\(448\) −0.763932 −0.0360924
\(449\) 14.9443 0.705264 0.352632 0.935762i \(-0.385287\pi\)
0.352632 + 0.935762i \(0.385287\pi\)
\(450\) 0 0
\(451\) 18.1803 0.856079
\(452\) 14.1803 0.666987
\(453\) 0 0
\(454\) −7.52786 −0.353300
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −5.12461 −0.239719 −0.119860 0.992791i \(-0.538244\pi\)
−0.119860 + 0.992791i \(0.538244\pi\)
\(458\) 7.41641 0.346546
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 1.47214 0.0685642 0.0342821 0.999412i \(-0.489086\pi\)
0.0342821 + 0.999412i \(0.489086\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 5.56231 0.258224
\(465\) 0 0
\(466\) −4.03444 −0.186892
\(467\) 13.0557 0.604147 0.302074 0.953285i \(-0.402321\pi\)
0.302074 + 0.953285i \(0.402321\pi\)
\(468\) 0 0
\(469\) −8.94427 −0.413008
\(470\) 1.70820 0.0787936
\(471\) 0 0
\(472\) −14.4721 −0.666134
\(473\) 0 0
\(474\) 0 0
\(475\) 6.94427 0.318625
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 8.50658 0.389082
\(479\) −31.5967 −1.44369 −0.721846 0.692054i \(-0.756706\pi\)
−0.721846 + 0.692054i \(0.756706\pi\)
\(480\) 0 0
\(481\) −3.70820 −0.169080
\(482\) 14.2918 0.650973
\(483\) 0 0
\(484\) −26.5623 −1.20738
\(485\) −21.8885 −0.993908
\(486\) 0 0
\(487\) −14.7082 −0.666492 −0.333246 0.942840i \(-0.608144\pi\)
−0.333246 + 0.942840i \(0.608144\pi\)
\(488\) −15.5279 −0.702913
\(489\) 0 0
\(490\) 2.65248 0.119827
\(491\) −8.34752 −0.376718 −0.188359 0.982100i \(-0.560317\pi\)
−0.188359 + 0.982100i \(0.560317\pi\)
\(492\) 0 0
\(493\) −2.29180 −0.103217
\(494\) 3.70820 0.166840
\(495\) 0 0
\(496\) 12.4377 0.558469
\(497\) −39.5967 −1.77616
\(498\) 0 0
\(499\) 19.2918 0.863619 0.431810 0.901965i \(-0.357875\pi\)
0.431810 + 0.901965i \(0.357875\pi\)
\(500\) −16.9443 −0.757771
\(501\) 0 0
\(502\) 1.41641 0.0632174
\(503\) 26.9443 1.20139 0.600693 0.799480i \(-0.294891\pi\)
0.600693 + 0.799480i \(0.294891\pi\)
\(504\) 0 0
\(505\) 5.52786 0.245987
\(506\) 3.23607 0.143861
\(507\) 0 0
\(508\) 11.7984 0.523468
\(509\) 28.3050 1.25459 0.627297 0.778780i \(-0.284161\pi\)
0.627297 + 0.778780i \(0.284161\pi\)
\(510\) 0 0
\(511\) 21.1246 0.934498
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) −4.61803 −0.203693
\(515\) 5.16718 0.227693
\(516\) 0 0
\(517\) 11.7082 0.514926
\(518\) 2.47214 0.108619
\(519\) 0 0
\(520\) −8.29180 −0.363619
\(521\) −31.4164 −1.37638 −0.688189 0.725532i \(-0.741594\pi\)
−0.688189 + 0.725532i \(0.741594\pi\)
\(522\) 0 0
\(523\) 41.1246 1.79825 0.899127 0.437688i \(-0.144203\pi\)
0.899127 + 0.437688i \(0.144203\pi\)
\(524\) 30.2705 1.32237
\(525\) 0 0
\(526\) 1.81966 0.0793410
\(527\) −5.12461 −0.223232
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −0.360680 −0.0156669
\(531\) 0 0
\(532\) 10.4721 0.454025
\(533\) 10.4164 0.451185
\(534\) 0 0
\(535\) 16.5836 0.716971
\(536\) −6.18034 −0.266950
\(537\) 0 0
\(538\) −4.90983 −0.211678
\(539\) 18.1803 0.783083
\(540\) 0 0
\(541\) −34.4164 −1.47968 −0.739838 0.672785i \(-0.765098\pi\)
−0.739838 + 0.672785i \(0.765098\pi\)
\(542\) −4.94427 −0.212375
\(543\) 0 0
\(544\) 4.29180 0.184009
\(545\) 0 0
\(546\) 0 0
\(547\) −29.5410 −1.26308 −0.631541 0.775342i \(-0.717577\pi\)
−0.631541 + 0.775342i \(0.717577\pi\)
\(548\) −35.4164 −1.51291
\(549\) 0 0
\(550\) 11.2361 0.479108
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −35.4164 −1.50606
\(554\) −9.56231 −0.406263
\(555\) 0 0
\(556\) 17.3262 0.734796
\(557\) 7.41641 0.314243 0.157122 0.987579i \(-0.449779\pi\)
0.157122 + 0.987579i \(0.449779\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7.41641 −0.313400
\(561\) 0 0
\(562\) −5.41641 −0.228477
\(563\) 32.9443 1.38844 0.694218 0.719765i \(-0.255750\pi\)
0.694218 + 0.719765i \(0.255750\pi\)
\(564\) 0 0
\(565\) 10.8328 0.455740
\(566\) −17.1246 −0.719801
\(567\) 0 0
\(568\) −27.3607 −1.14803
\(569\) 22.1803 0.929848 0.464924 0.885351i \(-0.346082\pi\)
0.464924 + 0.885351i \(0.346082\pi\)
\(570\) 0 0
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) −25.4164 −1.06271
\(573\) 0 0
\(574\) −6.94427 −0.289848
\(575\) 3.47214 0.144798
\(576\) 0 0
\(577\) 22.8885 0.952863 0.476431 0.879212i \(-0.341930\pi\)
0.476431 + 0.879212i \(0.341930\pi\)
\(578\) 10.1459 0.422014
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 28.3607 1.17660
\(582\) 0 0
\(583\) −2.47214 −0.102385
\(584\) 14.5967 0.604018
\(585\) 0 0
\(586\) −0.944272 −0.0390075
\(587\) 24.7082 1.01982 0.509908 0.860229i \(-0.329679\pi\)
0.509908 + 0.860229i \(0.329679\pi\)
\(588\) 0 0
\(589\) −13.4164 −0.552813
\(590\) −4.94427 −0.203552
\(591\) 0 0
\(592\) −2.29180 −0.0941922
\(593\) 2.94427 0.120907 0.0604534 0.998171i \(-0.480745\pi\)
0.0604534 + 0.998171i \(0.480745\pi\)
\(594\) 0 0
\(595\) 3.05573 0.125273
\(596\) 38.6525 1.58327
\(597\) 0 0
\(598\) 1.85410 0.0758199
\(599\) −33.8885 −1.38465 −0.692324 0.721587i \(-0.743413\pi\)
−0.692324 + 0.721587i \(0.743413\pi\)
\(600\) 0 0
\(601\) 46.8885 1.91262 0.956312 0.292349i \(-0.0944368\pi\)
0.956312 + 0.292349i \(0.0944368\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −6.85410 −0.278889
\(605\) −20.2918 −0.824979
\(606\) 0 0
\(607\) 26.4721 1.07447 0.537235 0.843432i \(-0.319469\pi\)
0.537235 + 0.843432i \(0.319469\pi\)
\(608\) 11.2361 0.455683
\(609\) 0 0
\(610\) −5.30495 −0.214791
\(611\) 6.70820 0.271385
\(612\) 0 0
\(613\) 5.70820 0.230552 0.115276 0.993333i \(-0.463225\pi\)
0.115276 + 0.993333i \(0.463225\pi\)
\(614\) −5.88854 −0.237642
\(615\) 0 0
\(616\) 37.8885 1.52657
\(617\) 7.52786 0.303060 0.151530 0.988453i \(-0.451580\pi\)
0.151530 + 0.988453i \(0.451580\pi\)
\(618\) 0 0
\(619\) 19.4164 0.780411 0.390206 0.920728i \(-0.372404\pi\)
0.390206 + 0.920728i \(0.372404\pi\)
\(620\) 13.4164 0.538816
\(621\) 0 0
\(622\) 8.14590 0.326621
\(623\) 33.8885 1.35772
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) −15.0557 −0.601748
\(627\) 0 0
\(628\) 18.4721 0.737118
\(629\) 0.944272 0.0376506
\(630\) 0 0
\(631\) 12.3607 0.492071 0.246035 0.969261i \(-0.420872\pi\)
0.246035 + 0.969261i \(0.420872\pi\)
\(632\) −24.4721 −0.973449
\(633\) 0 0
\(634\) 15.7082 0.623852
\(635\) 9.01316 0.357676
\(636\) 0 0
\(637\) 10.4164 0.412713
\(638\) −9.70820 −0.384351
\(639\) 0 0
\(640\) −14.0689 −0.556121
\(641\) 17.3050 0.683504 0.341752 0.939790i \(-0.388980\pi\)
0.341752 + 0.939790i \(0.388980\pi\)
\(642\) 0 0
\(643\) −29.5967 −1.16718 −0.583591 0.812048i \(-0.698353\pi\)
−0.583591 + 0.812048i \(0.698353\pi\)
\(644\) 5.23607 0.206330
\(645\) 0 0
\(646\) −0.944272 −0.0371519
\(647\) −6.70820 −0.263727 −0.131863 0.991268i \(-0.542096\pi\)
−0.131863 + 0.991268i \(0.542096\pi\)
\(648\) 0 0
\(649\) −33.8885 −1.33024
\(650\) 6.43769 0.252507
\(651\) 0 0
\(652\) 9.32624 0.365244
\(653\) 38.3050 1.49899 0.749494 0.662011i \(-0.230297\pi\)
0.749494 + 0.662011i \(0.230297\pi\)
\(654\) 0 0
\(655\) 23.1246 0.903553
\(656\) 6.43769 0.251350
\(657\) 0 0
\(658\) −4.47214 −0.174342
\(659\) 10.6525 0.414962 0.207481 0.978239i \(-0.433474\pi\)
0.207481 + 0.978239i \(0.433474\pi\)
\(660\) 0 0
\(661\) −22.9443 −0.892429 −0.446214 0.894926i \(-0.647228\pi\)
−0.446214 + 0.894926i \(0.647228\pi\)
\(662\) 12.1459 0.472064
\(663\) 0 0
\(664\) 19.5967 0.760501
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 2.47214 0.0956498
\(669\) 0 0
\(670\) −2.11146 −0.0815727
\(671\) −36.3607 −1.40369
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) −14.4721 −0.557446
\(675\) 0 0
\(676\) 6.47214 0.248928
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 57.3050 2.19916
\(680\) 2.11146 0.0809706
\(681\) 0 0
\(682\) −21.7082 −0.831250
\(683\) −26.5967 −1.01770 −0.508848 0.860856i \(-0.669929\pi\)
−0.508848 + 0.860856i \(0.669929\pi\)
\(684\) 0 0
\(685\) −27.0557 −1.03375
\(686\) 7.05573 0.269389
\(687\) 0 0
\(688\) 0 0
\(689\) −1.41641 −0.0539608
\(690\) 0 0
\(691\) 7.05573 0.268413 0.134206 0.990953i \(-0.457152\pi\)
0.134206 + 0.990953i \(0.457152\pi\)
\(692\) 37.1246 1.41127
\(693\) 0 0
\(694\) −6.11146 −0.231988
\(695\) 13.2361 0.502073
\(696\) 0 0
\(697\) −2.65248 −0.100470
\(698\) −15.0902 −0.571171
\(699\) 0 0
\(700\) 18.1803 0.687152
\(701\) 3.81966 0.144267 0.0721333 0.997395i \(-0.477019\pi\)
0.0721333 + 0.997395i \(0.477019\pi\)
\(702\) 0 0
\(703\) 2.47214 0.0932384
\(704\) −1.23607 −0.0465861
\(705\) 0 0
\(706\) 5.78522 0.217730
\(707\) −14.4721 −0.544281
\(708\) 0 0
\(709\) −42.0689 −1.57993 −0.789965 0.613152i \(-0.789901\pi\)
−0.789965 + 0.613152i \(0.789901\pi\)
\(710\) −9.34752 −0.350806
\(711\) 0 0
\(712\) 23.4164 0.877567
\(713\) −6.70820 −0.251224
\(714\) 0 0
\(715\) −19.4164 −0.726132
\(716\) 1.14590 0.0428242
\(717\) 0 0
\(718\) −12.2918 −0.458726
\(719\) 3.05573 0.113959 0.0569797 0.998375i \(-0.481853\pi\)
0.0569797 + 0.998375i \(0.481853\pi\)
\(720\) 0 0
\(721\) −13.5279 −0.503804
\(722\) 9.27051 0.345013
\(723\) 0 0
\(724\) −26.9443 −1.00138
\(725\) −10.4164 −0.386856
\(726\) 0 0
\(727\) −27.7082 −1.02764 −0.513820 0.857898i \(-0.671770\pi\)
−0.513820 + 0.857898i \(0.671770\pi\)
\(728\) 21.7082 0.804560
\(729\) 0 0
\(730\) 4.98684 0.184571
\(731\) 0 0
\(732\) 0 0
\(733\) −31.2361 −1.15373 −0.576865 0.816839i \(-0.695724\pi\)
−0.576865 + 0.816839i \(0.695724\pi\)
\(734\) 2.58359 0.0953621
\(735\) 0 0
\(736\) 5.61803 0.207083
\(737\) −14.4721 −0.533088
\(738\) 0 0
\(739\) 26.8197 0.986577 0.493289 0.869866i \(-0.335795\pi\)
0.493289 + 0.869866i \(0.335795\pi\)
\(740\) −2.47214 −0.0908775
\(741\) 0 0
\(742\) 0.944272 0.0346653
\(743\) −41.1246 −1.50872 −0.754358 0.656463i \(-0.772052\pi\)
−0.754358 + 0.656463i \(0.772052\pi\)
\(744\) 0 0
\(745\) 29.5279 1.08182
\(746\) −4.76393 −0.174420
\(747\) 0 0
\(748\) 6.47214 0.236645
\(749\) −43.4164 −1.58640
\(750\) 0 0
\(751\) 0.360680 0.0131614 0.00658070 0.999978i \(-0.497905\pi\)
0.00658070 + 0.999978i \(0.497905\pi\)
\(752\) 4.14590 0.151185
\(753\) 0 0
\(754\) −5.56231 −0.202567
\(755\) −5.23607 −0.190560
\(756\) 0 0
\(757\) 1.59675 0.0580348 0.0290174 0.999579i \(-0.490762\pi\)
0.0290174 + 0.999579i \(0.490762\pi\)
\(758\) −15.0557 −0.546849
\(759\) 0 0
\(760\) 5.52786 0.200517
\(761\) −46.3050 −1.67855 −0.839277 0.543705i \(-0.817021\pi\)
−0.839277 + 0.543705i \(0.817021\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −42.3607 −1.53256
\(765\) 0 0
\(766\) 4.36068 0.157558
\(767\) −19.4164 −0.701086
\(768\) 0 0
\(769\) −23.1246 −0.833895 −0.416947 0.908931i \(-0.636900\pi\)
−0.416947 + 0.908931i \(0.636900\pi\)
\(770\) 12.9443 0.466479
\(771\) 0 0
\(772\) −16.0902 −0.579098
\(773\) 5.52786 0.198823 0.0994117 0.995046i \(-0.468304\pi\)
0.0994117 + 0.995046i \(0.468304\pi\)
\(774\) 0 0
\(775\) −23.2918 −0.836666
\(776\) 39.5967 1.42144
\(777\) 0 0
\(778\) 15.7771 0.565636
\(779\) −6.94427 −0.248804
\(780\) 0 0
\(781\) −64.0689 −2.29256
\(782\) −0.472136 −0.0168835
\(783\) 0 0
\(784\) 6.43769 0.229918
\(785\) 14.1115 0.503659
\(786\) 0 0
\(787\) 24.5836 0.876310 0.438155 0.898899i \(-0.355632\pi\)
0.438155 + 0.898899i \(0.355632\pi\)
\(788\) −2.38197 −0.0848540
\(789\) 0 0
\(790\) −8.36068 −0.297460
\(791\) −28.3607 −1.00839
\(792\) 0 0
\(793\) −20.8328 −0.739795
\(794\) 15.0902 0.535530
\(795\) 0 0
\(796\) 19.8885 0.704931
\(797\) 34.3607 1.21712 0.608559 0.793509i \(-0.291748\pi\)
0.608559 + 0.793509i \(0.291748\pi\)
\(798\) 0 0
\(799\) −1.70820 −0.0604319
\(800\) 19.5066 0.689662
\(801\) 0 0
\(802\) −8.76393 −0.309465
\(803\) 34.1803 1.20620
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −12.4377 −0.438099
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −12.1115 −0.425816 −0.212908 0.977072i \(-0.568293\pi\)
−0.212908 + 0.977072i \(0.568293\pi\)
\(810\) 0 0
\(811\) −24.3475 −0.854957 −0.427479 0.904025i \(-0.640598\pi\)
−0.427479 + 0.904025i \(0.640598\pi\)
\(812\) −15.7082 −0.551250
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 7.12461 0.249564
\(816\) 0 0
\(817\) 0 0
\(818\) −13.2016 −0.461584
\(819\) 0 0
\(820\) 6.94427 0.242504
\(821\) 38.9443 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(822\) 0 0
\(823\) −39.5410 −1.37831 −0.689157 0.724612i \(-0.742019\pi\)
−0.689157 + 0.724612i \(0.742019\pi\)
\(824\) −9.34752 −0.325636
\(825\) 0 0
\(826\) 12.9443 0.450389
\(827\) −1.52786 −0.0531290 −0.0265645 0.999647i \(-0.508457\pi\)
−0.0265645 + 0.999647i \(0.508457\pi\)
\(828\) 0 0
\(829\) 40.2492 1.39791 0.698957 0.715164i \(-0.253648\pi\)
0.698957 + 0.715164i \(0.253648\pi\)
\(830\) 6.69505 0.232389
\(831\) 0 0
\(832\) −0.708204 −0.0245526
\(833\) −2.65248 −0.0919028
\(834\) 0 0
\(835\) 1.88854 0.0653558
\(836\) 16.9443 0.586030
\(837\) 0 0
\(838\) −2.83282 −0.0978580
\(839\) 41.1246 1.41978 0.709890 0.704313i \(-0.248745\pi\)
0.709890 + 0.704313i \(0.248745\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 6.36068 0.219204
\(843\) 0 0
\(844\) 37.8885 1.30418
\(845\) 4.94427 0.170088
\(846\) 0 0
\(847\) 53.1246 1.82538
\(848\) −0.875388 −0.0300610
\(849\) 0 0
\(850\) −1.63932 −0.0562282
\(851\) 1.23607 0.0423719
\(852\) 0 0
\(853\) −10.5836 −0.362375 −0.181188 0.983449i \(-0.557994\pi\)
−0.181188 + 0.983449i \(0.557994\pi\)
\(854\) 13.8885 0.475256
\(855\) 0 0
\(856\) −30.0000 −1.02538
\(857\) −1.47214 −0.0502872 −0.0251436 0.999684i \(-0.508004\pi\)
−0.0251436 + 0.999684i \(0.508004\pi\)
\(858\) 0 0
\(859\) −16.7082 −0.570077 −0.285038 0.958516i \(-0.592006\pi\)
−0.285038 + 0.958516i \(0.592006\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −10.8328 −0.368967
\(863\) 21.5410 0.733265 0.366632 0.930366i \(-0.380511\pi\)
0.366632 + 0.930366i \(0.380511\pi\)
\(864\) 0 0
\(865\) 28.3607 0.964292
\(866\) −11.0132 −0.374242
\(867\) 0 0
\(868\) −35.1246 −1.19221
\(869\) −57.3050 −1.94394
\(870\) 0 0
\(871\) −8.29180 −0.280957
\(872\) 0 0
\(873\) 0 0
\(874\) −1.23607 −0.0418106
\(875\) 33.8885 1.14564
\(876\) 0 0
\(877\) −36.4721 −1.23158 −0.615788 0.787912i \(-0.711162\pi\)
−0.615788 + 0.787912i \(0.711162\pi\)
\(878\) 11.5623 0.390209
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) −44.1803 −1.48847 −0.744237 0.667916i \(-0.767187\pi\)
−0.744237 + 0.667916i \(0.767187\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 3.70820 0.124720
\(885\) 0 0
\(886\) 23.5623 0.791591
\(887\) −23.0689 −0.774577 −0.387289 0.921959i \(-0.626588\pi\)
−0.387289 + 0.921959i \(0.626588\pi\)
\(888\) 0 0
\(889\) −23.5967 −0.791410
\(890\) 8.00000 0.268161
\(891\) 0 0
\(892\) −6.47214 −0.216703
\(893\) −4.47214 −0.149654
\(894\) 0 0
\(895\) 0.875388 0.0292610
\(896\) 36.8328 1.23050
\(897\) 0 0
\(898\) −9.23607 −0.308212
\(899\) 20.1246 0.671193
\(900\) 0 0
\(901\) 0.360680 0.0120160
\(902\) −11.2361 −0.374120
\(903\) 0 0
\(904\) −19.5967 −0.651778
\(905\) −20.5836 −0.684222
\(906\) 0 0
\(907\) −40.2492 −1.33645 −0.668227 0.743958i \(-0.732946\pi\)
−0.668227 + 0.743958i \(0.732946\pi\)
\(908\) −19.7082 −0.654040
\(909\) 0 0
\(910\) 7.41641 0.245852
\(911\) −31.3050 −1.03718 −0.518590 0.855023i \(-0.673543\pi\)
−0.518590 + 0.855023i \(0.673543\pi\)
\(912\) 0 0
\(913\) 45.8885 1.51869
\(914\) 3.16718 0.104761
\(915\) 0 0
\(916\) 19.4164 0.641536
\(917\) −60.5410 −1.99924
\(918\) 0 0
\(919\) 41.1246 1.35658 0.678288 0.734796i \(-0.262722\pi\)
0.678288 + 0.734796i \(0.262722\pi\)
\(920\) 2.76393 0.0911241
\(921\) 0 0
\(922\) −0.909830 −0.0299637
\(923\) −36.7082 −1.20827
\(924\) 0 0
\(925\) 4.29180 0.141113
\(926\) 12.3607 0.406197
\(927\) 0 0
\(928\) −16.8541 −0.553263
\(929\) 24.0557 0.789243 0.394621 0.918844i \(-0.370876\pi\)
0.394621 + 0.918844i \(0.370876\pi\)
\(930\) 0 0
\(931\) −6.94427 −0.227589
\(932\) −10.5623 −0.345980
\(933\) 0 0
\(934\) −8.06888 −0.264022
\(935\) 4.94427 0.161695
\(936\) 0 0
\(937\) 34.1803 1.11662 0.558312 0.829631i \(-0.311449\pi\)
0.558312 + 0.829631i \(0.311449\pi\)
\(938\) 5.52786 0.180491
\(939\) 0 0
\(940\) 4.47214 0.145865
\(941\) −6.65248 −0.216865 −0.108432 0.994104i \(-0.534583\pi\)
−0.108432 + 0.994104i \(0.534583\pi\)
\(942\) 0 0
\(943\) −3.47214 −0.113068
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 10.8197 0.351592 0.175796 0.984427i \(-0.443750\pi\)
0.175796 + 0.984427i \(0.443750\pi\)
\(948\) 0 0
\(949\) 19.5836 0.635710
\(950\) −4.29180 −0.139244
\(951\) 0 0
\(952\) −5.52786 −0.179159
\(953\) −20.4721 −0.663158 −0.331579 0.943428i \(-0.607581\pi\)
−0.331579 + 0.943428i \(0.607581\pi\)
\(954\) 0 0
\(955\) −32.3607 −1.04717
\(956\) 22.2705 0.720280
\(957\) 0 0
\(958\) 19.5279 0.630917
\(959\) 70.8328 2.28731
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 2.29180 0.0738905
\(963\) 0 0
\(964\) 37.4164 1.20510
\(965\) −12.2918 −0.395687
\(966\) 0 0
\(967\) 27.5410 0.885659 0.442830 0.896606i \(-0.353975\pi\)
0.442830 + 0.896606i \(0.353975\pi\)
\(968\) 36.7082 1.17985
\(969\) 0 0
\(970\) 13.5279 0.434354
\(971\) −16.4721 −0.528616 −0.264308 0.964438i \(-0.585144\pi\)
−0.264308 + 0.964438i \(0.585144\pi\)
\(972\) 0 0
\(973\) −34.6525 −1.11091
\(974\) 9.09017 0.291268
\(975\) 0 0
\(976\) −12.8754 −0.412131
\(977\) 23.3475 0.746953 0.373477 0.927640i \(-0.378166\pi\)
0.373477 + 0.927640i \(0.378166\pi\)
\(978\) 0 0
\(979\) 54.8328 1.75246
\(980\) 6.94427 0.221827
\(981\) 0 0
\(982\) 5.15905 0.164632
\(983\) 40.4721 1.29086 0.645430 0.763819i \(-0.276678\pi\)
0.645430 + 0.763819i \(0.276678\pi\)
\(984\) 0 0
\(985\) −1.81966 −0.0579792
\(986\) 1.41641 0.0451076
\(987\) 0 0
\(988\) 9.70820 0.308859
\(989\) 0 0
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) −37.6869 −1.19656
\(993\) 0 0
\(994\) 24.4721 0.776209
\(995\) 15.1935 0.481666
\(996\) 0 0
\(997\) 16.8328 0.533101 0.266550 0.963821i \(-0.414116\pi\)
0.266550 + 0.963821i \(0.414116\pi\)
\(998\) −11.9230 −0.377416
\(999\) 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 207.2.a.d.1.1 2
3.2 odd 2 23.2.a.a.1.2 2
4.3 odd 2 3312.2.a.ba.1.1 2
5.4 even 2 5175.2.a.be.1.2 2
12.11 even 2 368.2.a.h.1.2 2
15.2 even 4 575.2.b.d.24.3 4
15.8 even 4 575.2.b.d.24.2 4
15.14 odd 2 575.2.a.f.1.1 2
21.20 even 2 1127.2.a.c.1.2 2
23.22 odd 2 4761.2.a.w.1.1 2
24.5 odd 2 1472.2.a.t.1.2 2
24.11 even 2 1472.2.a.s.1.1 2
33.32 even 2 2783.2.a.c.1.1 2
39.38 odd 2 3887.2.a.i.1.1 2
51.50 odd 2 6647.2.a.b.1.2 2
57.56 even 2 8303.2.a.e.1.1 2
60.59 even 2 9200.2.a.bt.1.1 2
69.2 odd 22 529.2.c.o.487.2 20
69.5 even 22 529.2.c.n.255.2 20
69.8 odd 22 529.2.c.o.501.1 20
69.11 even 22 529.2.c.n.466.2 20
69.14 even 22 529.2.c.n.334.2 20
69.17 even 22 529.2.c.n.266.1 20
69.20 even 22 529.2.c.n.170.1 20
69.26 odd 22 529.2.c.o.170.1 20
69.29 odd 22 529.2.c.o.266.1 20
69.32 odd 22 529.2.c.o.334.2 20
69.35 odd 22 529.2.c.o.466.2 20
69.38 even 22 529.2.c.n.501.1 20
69.41 odd 22 529.2.c.o.255.2 20
69.44 even 22 529.2.c.n.487.2 20
69.50 odd 22 529.2.c.o.177.1 20
69.53 even 22 529.2.c.n.118.1 20
69.56 even 22 529.2.c.n.399.1 20
69.59 odd 22 529.2.c.o.399.1 20
69.62 odd 22 529.2.c.o.118.1 20
69.65 even 22 529.2.c.n.177.1 20
69.68 even 2 529.2.a.a.1.2 2
276.275 odd 2 8464.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.2.a.a.1.2 2 3.2 odd 2
207.2.a.d.1.1 2 1.1 even 1 trivial
368.2.a.h.1.2 2 12.11 even 2
529.2.a.a.1.2 2 69.68 even 2
529.2.c.n.118.1 20 69.53 even 22
529.2.c.n.170.1 20 69.20 even 22
529.2.c.n.177.1 20 69.65 even 22
529.2.c.n.255.2 20 69.5 even 22
529.2.c.n.266.1 20 69.17 even 22
529.2.c.n.334.2 20 69.14 even 22
529.2.c.n.399.1 20 69.56 even 22
529.2.c.n.466.2 20 69.11 even 22
529.2.c.n.487.2 20 69.44 even 22
529.2.c.n.501.1 20 69.38 even 22
529.2.c.o.118.1 20 69.62 odd 22
529.2.c.o.170.1 20 69.26 odd 22
529.2.c.o.177.1 20 69.50 odd 22
529.2.c.o.255.2 20 69.41 odd 22
529.2.c.o.266.1 20 69.29 odd 22
529.2.c.o.334.2 20 69.32 odd 22
529.2.c.o.399.1 20 69.59 odd 22
529.2.c.o.466.2 20 69.35 odd 22
529.2.c.o.487.2 20 69.2 odd 22
529.2.c.o.501.1 20 69.8 odd 22
575.2.a.f.1.1 2 15.14 odd 2
575.2.b.d.24.2 4 15.8 even 4
575.2.b.d.24.3 4 15.2 even 4
1127.2.a.c.1.2 2 21.20 even 2
1472.2.a.s.1.1 2 24.11 even 2
1472.2.a.t.1.2 2 24.5 odd 2
2783.2.a.c.1.1 2 33.32 even 2
3312.2.a.ba.1.1 2 4.3 odd 2
3887.2.a.i.1.1 2 39.38 odd 2
4761.2.a.w.1.1 2 23.22 odd 2
5175.2.a.be.1.2 2 5.4 even 2
6647.2.a.b.1.2 2 51.50 odd 2
8303.2.a.e.1.1 2 57.56 even 2
8464.2.a.bb.1.2 2 276.275 odd 2
9200.2.a.bt.1.1 2 60.59 even 2