Properties

Label 8450.2.a.bl.1.2
Level $8450$
Weight $2$
Character 8450.1
Self dual yes
Analytic conductor $67.474$
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] = [8450,2,Mod(1,8450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8450, 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("8450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.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: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 650)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 8450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.30278 q^{3} +1.00000 q^{4} +2.30278 q^{6} +0.697224 q^{7} +1.00000 q^{8} +2.30278 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.30278 q^{3} +1.00000 q^{4} +2.30278 q^{6} +0.697224 q^{7} +1.00000 q^{8} +2.30278 q^{9} +0.697224 q^{11} +2.30278 q^{12} +0.697224 q^{14} +1.00000 q^{16} -2.90833 q^{17} +2.30278 q^{18} +0.394449 q^{19} +1.60555 q^{21} +0.697224 q^{22} +5.60555 q^{23} +2.30278 q^{24} -1.60555 q^{27} +0.697224 q^{28} +3.30278 q^{29} +8.60555 q^{31} +1.00000 q^{32} +1.60555 q^{33} -2.90833 q^{34} +2.30278 q^{36} -3.69722 q^{37} +0.394449 q^{38} -1.00000 q^{41} +1.60555 q^{42} -8.51388 q^{43} +0.697224 q^{44} +5.60555 q^{46} +10.3028 q^{47} +2.30278 q^{48} -6.51388 q^{49} -6.69722 q^{51} +12.2111 q^{53} -1.60555 q^{54} +0.697224 q^{56} +0.908327 q^{57} +3.30278 q^{58} +10.2111 q^{59} +4.21110 q^{61} +8.60555 q^{62} +1.60555 q^{63} +1.00000 q^{64} +1.60555 q^{66} +5.60555 q^{67} -2.90833 q^{68} +12.9083 q^{69} +10.6056 q^{71} +2.30278 q^{72} -8.00000 q^{73} -3.69722 q^{74} +0.394449 q^{76} +0.486122 q^{77} +9.30278 q^{79} -10.6056 q^{81} -1.00000 q^{82} +16.8167 q^{83} +1.60555 q^{84} -8.51388 q^{86} +7.60555 q^{87} +0.697224 q^{88} -0.0916731 q^{89} +5.60555 q^{92} +19.8167 q^{93} +10.3028 q^{94} +2.30278 q^{96} -11.6972 q^{97} -6.51388 q^{98} +1.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 5 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + 5 q^{7} + 2 q^{8} + q^{9} + 5 q^{11} + q^{12} + 5 q^{14} + 2 q^{16} + 5 q^{17} + q^{18} + 8 q^{19} - 4 q^{21} + 5 q^{22} + 4 q^{23} + q^{24} + 4 q^{27} + 5 q^{28} + 3 q^{29} + 10 q^{31} + 2 q^{32} - 4 q^{33} + 5 q^{34} + q^{36} - 11 q^{37} + 8 q^{38} - 2 q^{41} - 4 q^{42} + q^{43} + 5 q^{44} + 4 q^{46} + 17 q^{47} + q^{48} + 5 q^{49} - 17 q^{51} + 10 q^{53} + 4 q^{54} + 5 q^{56} - 9 q^{57} + 3 q^{58} + 6 q^{59} - 6 q^{61} + 10 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{66} + 4 q^{67} + 5 q^{68} + 15 q^{69} + 14 q^{71} + q^{72} - 16 q^{73} - 11 q^{74} + 8 q^{76} + 19 q^{77} + 15 q^{79} - 14 q^{81} - 2 q^{82} + 12 q^{83} - 4 q^{84} + q^{86} + 8 q^{87} + 5 q^{88} - 11 q^{89} + 4 q^{92} + 18 q^{93} + 17 q^{94} + q^{96} - 27 q^{97} + 5 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.30278 1.32951 0.664754 0.747062i \(-0.268536\pi\)
0.664754 + 0.747062i \(0.268536\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.30278 0.940104
\(7\) 0.697224 0.263526 0.131763 0.991281i \(-0.457936\pi\)
0.131763 + 0.991281i \(0.457936\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 0.697224 0.210221 0.105111 0.994461i \(-0.466480\pi\)
0.105111 + 0.994461i \(0.466480\pi\)
\(12\) 2.30278 0.664754
\(13\) 0 0
\(14\) 0.697224 0.186341
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.90833 −0.705373 −0.352686 0.935742i \(-0.614732\pi\)
−0.352686 + 0.935742i \(0.614732\pi\)
\(18\) 2.30278 0.542769
\(19\) 0.394449 0.0904927 0.0452464 0.998976i \(-0.485593\pi\)
0.0452464 + 0.998976i \(0.485593\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 0.697224 0.148649
\(23\) 5.60555 1.16884 0.584419 0.811452i \(-0.301322\pi\)
0.584419 + 0.811452i \(0.301322\pi\)
\(24\) 2.30278 0.470052
\(25\) 0 0
\(26\) 0 0
\(27\) −1.60555 −0.308988
\(28\) 0.697224 0.131763
\(29\) 3.30278 0.613310 0.306655 0.951821i \(-0.400790\pi\)
0.306655 + 0.951821i \(0.400790\pi\)
\(30\) 0 0
\(31\) 8.60555 1.54560 0.772801 0.634648i \(-0.218855\pi\)
0.772801 + 0.634648i \(0.218855\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.60555 0.279491
\(34\) −2.90833 −0.498774
\(35\) 0 0
\(36\) 2.30278 0.383796
\(37\) −3.69722 −0.607820 −0.303910 0.952701i \(-0.598292\pi\)
−0.303910 + 0.952701i \(0.598292\pi\)
\(38\) 0.394449 0.0639880
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 1.60555 0.247742
\(43\) −8.51388 −1.29835 −0.649177 0.760637i \(-0.724887\pi\)
−0.649177 + 0.760637i \(0.724887\pi\)
\(44\) 0.697224 0.105111
\(45\) 0 0
\(46\) 5.60555 0.826493
\(47\) 10.3028 1.50281 0.751407 0.659839i \(-0.229375\pi\)
0.751407 + 0.659839i \(0.229375\pi\)
\(48\) 2.30278 0.332377
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) −6.69722 −0.937799
\(52\) 0 0
\(53\) 12.2111 1.67732 0.838662 0.544652i \(-0.183338\pi\)
0.838662 + 0.544652i \(0.183338\pi\)
\(54\) −1.60555 −0.218488
\(55\) 0 0
\(56\) 0.697224 0.0931705
\(57\) 0.908327 0.120311
\(58\) 3.30278 0.433676
\(59\) 10.2111 1.32937 0.664686 0.747123i \(-0.268565\pi\)
0.664686 + 0.747123i \(0.268565\pi\)
\(60\) 0 0
\(61\) 4.21110 0.539176 0.269588 0.962976i \(-0.413112\pi\)
0.269588 + 0.962976i \(0.413112\pi\)
\(62\) 8.60555 1.09291
\(63\) 1.60555 0.202280
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.60555 0.197630
\(67\) 5.60555 0.684827 0.342414 0.939549i \(-0.388756\pi\)
0.342414 + 0.939549i \(0.388756\pi\)
\(68\) −2.90833 −0.352686
\(69\) 12.9083 1.55398
\(70\) 0 0
\(71\) 10.6056 1.25865 0.629324 0.777143i \(-0.283332\pi\)
0.629324 + 0.777143i \(0.283332\pi\)
\(72\) 2.30278 0.271385
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −3.69722 −0.429794
\(75\) 0 0
\(76\) 0.394449 0.0452464
\(77\) 0.486122 0.0553987
\(78\) 0 0
\(79\) 9.30278 1.04664 0.523322 0.852135i \(-0.324692\pi\)
0.523322 + 0.852135i \(0.324692\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) −1.00000 −0.110432
\(83\) 16.8167 1.84587 0.922934 0.384959i \(-0.125784\pi\)
0.922934 + 0.384959i \(0.125784\pi\)
\(84\) 1.60555 0.175180
\(85\) 0 0
\(86\) −8.51388 −0.918075
\(87\) 7.60555 0.815401
\(88\) 0.697224 0.0743244
\(89\) −0.0916731 −0.00971733 −0.00485866 0.999988i \(-0.501547\pi\)
−0.00485866 + 0.999988i \(0.501547\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.60555 0.584419
\(93\) 19.8167 2.05489
\(94\) 10.3028 1.06265
\(95\) 0 0
\(96\) 2.30278 0.235026
\(97\) −11.6972 −1.18767 −0.593837 0.804586i \(-0.702387\pi\)
−0.593837 + 0.804586i \(0.702387\pi\)
\(98\) −6.51388 −0.658001
\(99\) 1.60555 0.161364
\(100\) 0 0
\(101\) −6.21110 −0.618028 −0.309014 0.951058i \(-0.599999\pi\)
−0.309014 + 0.951058i \(0.599999\pi\)
\(102\) −6.69722 −0.663124
\(103\) −6.39445 −0.630064 −0.315032 0.949081i \(-0.602015\pi\)
−0.315032 + 0.949081i \(0.602015\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.2111 1.18605
\(107\) −13.8167 −1.33571 −0.667853 0.744293i \(-0.732787\pi\)
−0.667853 + 0.744293i \(0.732787\pi\)
\(108\) −1.60555 −0.154494
\(109\) −11.6056 −1.11161 −0.555805 0.831313i \(-0.687590\pi\)
−0.555805 + 0.831313i \(0.687590\pi\)
\(110\) 0 0
\(111\) −8.51388 −0.808102
\(112\) 0.697224 0.0658815
\(113\) 10.5139 0.989062 0.494531 0.869160i \(-0.335340\pi\)
0.494531 + 0.869160i \(0.335340\pi\)
\(114\) 0.908327 0.0850726
\(115\) 0 0
\(116\) 3.30278 0.306655
\(117\) 0 0
\(118\) 10.2111 0.940008
\(119\) −2.02776 −0.185884
\(120\) 0 0
\(121\) −10.5139 −0.955807
\(122\) 4.21110 0.381255
\(123\) −2.30278 −0.207634
\(124\) 8.60555 0.772801
\(125\) 0 0
\(126\) 1.60555 0.143034
\(127\) −4.90833 −0.435544 −0.217772 0.976000i \(-0.569879\pi\)
−0.217772 + 0.976000i \(0.569879\pi\)
\(128\) 1.00000 0.0883883
\(129\) −19.6056 −1.72617
\(130\) 0 0
\(131\) 8.81665 0.770315 0.385157 0.922851i \(-0.374147\pi\)
0.385157 + 0.922851i \(0.374147\pi\)
\(132\) 1.60555 0.139745
\(133\) 0.275019 0.0238472
\(134\) 5.60555 0.484246
\(135\) 0 0
\(136\) −2.90833 −0.249387
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 12.9083 1.09883
\(139\) −9.81665 −0.832638 −0.416319 0.909219i \(-0.636680\pi\)
−0.416319 + 0.909219i \(0.636680\pi\)
\(140\) 0 0
\(141\) 23.7250 1.99800
\(142\) 10.6056 0.889998
\(143\) 0 0
\(144\) 2.30278 0.191898
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) −15.0000 −1.23718
\(148\) −3.69722 −0.303910
\(149\) 8.30278 0.680190 0.340095 0.940391i \(-0.389541\pi\)
0.340095 + 0.940391i \(0.389541\pi\)
\(150\) 0 0
\(151\) 12.1194 0.986265 0.493133 0.869954i \(-0.335852\pi\)
0.493133 + 0.869954i \(0.335852\pi\)
\(152\) 0.394449 0.0319940
\(153\) −6.69722 −0.541438
\(154\) 0.486122 0.0391728
\(155\) 0 0
\(156\) 0 0
\(157\) 24.4222 1.94910 0.974552 0.224161i \(-0.0719642\pi\)
0.974552 + 0.224161i \(0.0719642\pi\)
\(158\) 9.30278 0.740089
\(159\) 28.1194 2.23002
\(160\) 0 0
\(161\) 3.90833 0.308019
\(162\) −10.6056 −0.833251
\(163\) −1.90833 −0.149472 −0.0747358 0.997203i \(-0.523811\pi\)
−0.0747358 + 0.997203i \(0.523811\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) 16.8167 1.30523
\(167\) −3.60555 −0.279006 −0.139503 0.990222i \(-0.544550\pi\)
−0.139503 + 0.990222i \(0.544550\pi\)
\(168\) 1.60555 0.123871
\(169\) 0 0
\(170\) 0 0
\(171\) 0.908327 0.0694615
\(172\) −8.51388 −0.649177
\(173\) −6.48612 −0.493131 −0.246565 0.969126i \(-0.579302\pi\)
−0.246565 + 0.969126i \(0.579302\pi\)
\(174\) 7.60555 0.576575
\(175\) 0 0
\(176\) 0.697224 0.0525553
\(177\) 23.5139 1.76741
\(178\) −0.0916731 −0.00687119
\(179\) 7.21110 0.538983 0.269492 0.963003i \(-0.413144\pi\)
0.269492 + 0.963003i \(0.413144\pi\)
\(180\) 0 0
\(181\) −17.9083 −1.33112 −0.665558 0.746346i \(-0.731806\pi\)
−0.665558 + 0.746346i \(0.731806\pi\)
\(182\) 0 0
\(183\) 9.69722 0.716839
\(184\) 5.60555 0.413247
\(185\) 0 0
\(186\) 19.8167 1.45303
\(187\) −2.02776 −0.148284
\(188\) 10.3028 0.751407
\(189\) −1.11943 −0.0814265
\(190\) 0 0
\(191\) −10.8167 −0.782666 −0.391333 0.920249i \(-0.627986\pi\)
−0.391333 + 0.920249i \(0.627986\pi\)
\(192\) 2.30278 0.166189
\(193\) 17.5139 1.26068 0.630338 0.776321i \(-0.282916\pi\)
0.630338 + 0.776321i \(0.282916\pi\)
\(194\) −11.6972 −0.839812
\(195\) 0 0
\(196\) −6.51388 −0.465277
\(197\) −22.3028 −1.58901 −0.794503 0.607260i \(-0.792269\pi\)
−0.794503 + 0.607260i \(0.792269\pi\)
\(198\) 1.60555 0.114102
\(199\) 3.90833 0.277054 0.138527 0.990359i \(-0.455763\pi\)
0.138527 + 0.990359i \(0.455763\pi\)
\(200\) 0 0
\(201\) 12.9083 0.910483
\(202\) −6.21110 −0.437012
\(203\) 2.30278 0.161623
\(204\) −6.69722 −0.468899
\(205\) 0 0
\(206\) −6.39445 −0.445522
\(207\) 12.9083 0.897191
\(208\) 0 0
\(209\) 0.275019 0.0190235
\(210\) 0 0
\(211\) −6.81665 −0.469278 −0.234639 0.972083i \(-0.575391\pi\)
−0.234639 + 0.972083i \(0.575391\pi\)
\(212\) 12.2111 0.838662
\(213\) 24.4222 1.67338
\(214\) −13.8167 −0.944487
\(215\) 0 0
\(216\) −1.60555 −0.109244
\(217\) 6.00000 0.407307
\(218\) −11.6056 −0.786027
\(219\) −18.4222 −1.24486
\(220\) 0 0
\(221\) 0 0
\(222\) −8.51388 −0.571414
\(223\) 26.8167 1.79578 0.897888 0.440224i \(-0.145101\pi\)
0.897888 + 0.440224i \(0.145101\pi\)
\(224\) 0.697224 0.0465853
\(225\) 0 0
\(226\) 10.5139 0.699373
\(227\) −19.0000 −1.26107 −0.630537 0.776159i \(-0.717165\pi\)
−0.630537 + 0.776159i \(0.717165\pi\)
\(228\) 0.908327 0.0601554
\(229\) −7.51388 −0.496531 −0.248266 0.968692i \(-0.579861\pi\)
−0.248266 + 0.968692i \(0.579861\pi\)
\(230\) 0 0
\(231\) 1.11943 0.0736531
\(232\) 3.30278 0.216838
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.2111 0.664686
\(237\) 21.4222 1.39152
\(238\) −2.02776 −0.131440
\(239\) 0.577795 0.0373744 0.0186872 0.999825i \(-0.494051\pi\)
0.0186872 + 0.999825i \(0.494051\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −10.5139 −0.675858
\(243\) −19.6056 −1.25770
\(244\) 4.21110 0.269588
\(245\) 0 0
\(246\) −2.30278 −0.146820
\(247\) 0 0
\(248\) 8.60555 0.546453
\(249\) 38.7250 2.45410
\(250\) 0 0
\(251\) −17.3305 −1.09389 −0.546947 0.837167i \(-0.684210\pi\)
−0.546947 + 0.837167i \(0.684210\pi\)
\(252\) 1.60555 0.101140
\(253\) 3.90833 0.245714
\(254\) −4.90833 −0.307976
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.60555 0.349665 0.174832 0.984598i \(-0.444062\pi\)
0.174832 + 0.984598i \(0.444062\pi\)
\(258\) −19.6056 −1.22059
\(259\) −2.57779 −0.160176
\(260\) 0 0
\(261\) 7.60555 0.470772
\(262\) 8.81665 0.544695
\(263\) 18.7250 1.15463 0.577316 0.816521i \(-0.304100\pi\)
0.577316 + 0.816521i \(0.304100\pi\)
\(264\) 1.60555 0.0988149
\(265\) 0 0
\(266\) 0.275019 0.0168625
\(267\) −0.211103 −0.0129193
\(268\) 5.60555 0.342414
\(269\) −28.8167 −1.75698 −0.878491 0.477759i \(-0.841449\pi\)
−0.878491 + 0.477759i \(0.841449\pi\)
\(270\) 0 0
\(271\) −2.42221 −0.147138 −0.0735692 0.997290i \(-0.523439\pi\)
−0.0735692 + 0.997290i \(0.523439\pi\)
\(272\) −2.90833 −0.176343
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 12.9083 0.776990
\(277\) −32.1194 −1.92987 −0.964935 0.262490i \(-0.915456\pi\)
−0.964935 + 0.262490i \(0.915456\pi\)
\(278\) −9.81665 −0.588764
\(279\) 19.8167 1.18639
\(280\) 0 0
\(281\) −13.3028 −0.793577 −0.396789 0.917910i \(-0.629875\pi\)
−0.396789 + 0.917910i \(0.629875\pi\)
\(282\) 23.7250 1.41280
\(283\) 22.5139 1.33831 0.669156 0.743122i \(-0.266656\pi\)
0.669156 + 0.743122i \(0.266656\pi\)
\(284\) 10.6056 0.629324
\(285\) 0 0
\(286\) 0 0
\(287\) −0.697224 −0.0411559
\(288\) 2.30278 0.135692
\(289\) −8.54163 −0.502449
\(290\) 0 0
\(291\) −26.9361 −1.57902
\(292\) −8.00000 −0.468165
\(293\) −2.51388 −0.146862 −0.0734312 0.997300i \(-0.523395\pi\)
−0.0734312 + 0.997300i \(0.523395\pi\)
\(294\) −15.0000 −0.874818
\(295\) 0 0
\(296\) −3.69722 −0.214897
\(297\) −1.11943 −0.0649559
\(298\) 8.30278 0.480967
\(299\) 0 0
\(300\) 0 0
\(301\) −5.93608 −0.342150
\(302\) 12.1194 0.697395
\(303\) −14.3028 −0.821673
\(304\) 0.394449 0.0226232
\(305\) 0 0
\(306\) −6.69722 −0.382855
\(307\) −11.2111 −0.639851 −0.319926 0.947443i \(-0.603658\pi\)
−0.319926 + 0.947443i \(0.603658\pi\)
\(308\) 0.486122 0.0276994
\(309\) −14.7250 −0.837675
\(310\) 0 0
\(311\) −9.90833 −0.561850 −0.280925 0.959730i \(-0.590641\pi\)
−0.280925 + 0.959730i \(0.590641\pi\)
\(312\) 0 0
\(313\) −15.9083 −0.899192 −0.449596 0.893232i \(-0.648432\pi\)
−0.449596 + 0.893232i \(0.648432\pi\)
\(314\) 24.4222 1.37822
\(315\) 0 0
\(316\) 9.30278 0.523322
\(317\) −11.4861 −0.645125 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(318\) 28.1194 1.57686
\(319\) 2.30278 0.128931
\(320\) 0 0
\(321\) −31.8167 −1.77583
\(322\) 3.90833 0.217803
\(323\) −1.14719 −0.0638311
\(324\) −10.6056 −0.589197
\(325\) 0 0
\(326\) −1.90833 −0.105692
\(327\) −26.7250 −1.47789
\(328\) −1.00000 −0.0552158
\(329\) 7.18335 0.396031
\(330\) 0 0
\(331\) 20.9083 1.14923 0.574613 0.818425i \(-0.305153\pi\)
0.574613 + 0.818425i \(0.305153\pi\)
\(332\) 16.8167 0.922934
\(333\) −8.51388 −0.466558
\(334\) −3.60555 −0.197287
\(335\) 0 0
\(336\) 1.60555 0.0875900
\(337\) 14.0278 0.764141 0.382070 0.924133i \(-0.375211\pi\)
0.382070 + 0.924133i \(0.375211\pi\)
\(338\) 0 0
\(339\) 24.2111 1.31497
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0.908327 0.0491167
\(343\) −9.42221 −0.508751
\(344\) −8.51388 −0.459038
\(345\) 0 0
\(346\) −6.48612 −0.348696
\(347\) −2.21110 −0.118698 −0.0593491 0.998237i \(-0.518903\pi\)
−0.0593491 + 0.998237i \(0.518903\pi\)
\(348\) 7.60555 0.407700
\(349\) 33.7250 1.80526 0.902628 0.430421i \(-0.141635\pi\)
0.902628 + 0.430421i \(0.141635\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.697224 0.0371622
\(353\) −20.5139 −1.09184 −0.545922 0.837836i \(-0.683820\pi\)
−0.545922 + 0.837836i \(0.683820\pi\)
\(354\) 23.5139 1.24975
\(355\) 0 0
\(356\) −0.0916731 −0.00485866
\(357\) −4.66947 −0.247134
\(358\) 7.21110 0.381119
\(359\) 22.9361 1.21052 0.605260 0.796028i \(-0.293069\pi\)
0.605260 + 0.796028i \(0.293069\pi\)
\(360\) 0 0
\(361\) −18.8444 −0.991811
\(362\) −17.9083 −0.941241
\(363\) −24.2111 −1.27075
\(364\) 0 0
\(365\) 0 0
\(366\) 9.69722 0.506882
\(367\) −25.8167 −1.34762 −0.673809 0.738905i \(-0.735343\pi\)
−0.673809 + 0.738905i \(0.735343\pi\)
\(368\) 5.60555 0.292210
\(369\) −2.30278 −0.119878
\(370\) 0 0
\(371\) 8.51388 0.442019
\(372\) 19.8167 1.02745
\(373\) −15.2111 −0.787601 −0.393801 0.919196i \(-0.628840\pi\)
−0.393801 + 0.919196i \(0.628840\pi\)
\(374\) −2.02776 −0.104853
\(375\) 0 0
\(376\) 10.3028 0.531325
\(377\) 0 0
\(378\) −1.11943 −0.0575772
\(379\) 34.8167 1.78841 0.894206 0.447656i \(-0.147741\pi\)
0.894206 + 0.447656i \(0.147741\pi\)
\(380\) 0 0
\(381\) −11.3028 −0.579059
\(382\) −10.8167 −0.553428
\(383\) −11.6056 −0.593016 −0.296508 0.955030i \(-0.595822\pi\)
−0.296508 + 0.955030i \(0.595822\pi\)
\(384\) 2.30278 0.117513
\(385\) 0 0
\(386\) 17.5139 0.891433
\(387\) −19.6056 −0.996606
\(388\) −11.6972 −0.593837
\(389\) −9.21110 −0.467021 −0.233511 0.972354i \(-0.575021\pi\)
−0.233511 + 0.972354i \(0.575021\pi\)
\(390\) 0 0
\(391\) −16.3028 −0.824467
\(392\) −6.51388 −0.329001
\(393\) 20.3028 1.02414
\(394\) −22.3028 −1.12360
\(395\) 0 0
\(396\) 1.60555 0.0806820
\(397\) 10.3305 0.518475 0.259237 0.965814i \(-0.416529\pi\)
0.259237 + 0.965814i \(0.416529\pi\)
\(398\) 3.90833 0.195907
\(399\) 0.633308 0.0317050
\(400\) 0 0
\(401\) −14.7250 −0.735330 −0.367665 0.929958i \(-0.619843\pi\)
−0.367665 + 0.929958i \(0.619843\pi\)
\(402\) 12.9083 0.643809
\(403\) 0 0
\(404\) −6.21110 −0.309014
\(405\) 0 0
\(406\) 2.30278 0.114285
\(407\) −2.57779 −0.127777
\(408\) −6.69722 −0.331562
\(409\) 34.2111 1.69163 0.845815 0.533476i \(-0.179115\pi\)
0.845815 + 0.533476i \(0.179115\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.39445 −0.315032
\(413\) 7.11943 0.350324
\(414\) 12.9083 0.634410
\(415\) 0 0
\(416\) 0 0
\(417\) −22.6056 −1.10700
\(418\) 0.275019 0.0134516
\(419\) 17.3028 0.845296 0.422648 0.906294i \(-0.361101\pi\)
0.422648 + 0.906294i \(0.361101\pi\)
\(420\) 0 0
\(421\) 29.9083 1.45764 0.728821 0.684704i \(-0.240068\pi\)
0.728821 + 0.684704i \(0.240068\pi\)
\(422\) −6.81665 −0.331830
\(423\) 23.7250 1.15355
\(424\) 12.2111 0.593024
\(425\) 0 0
\(426\) 24.4222 1.18326
\(427\) 2.93608 0.142087
\(428\) −13.8167 −0.667853
\(429\) 0 0
\(430\) 0 0
\(431\) −0.908327 −0.0437526 −0.0218763 0.999761i \(-0.506964\pi\)
−0.0218763 + 0.999761i \(0.506964\pi\)
\(432\) −1.60555 −0.0772471
\(433\) 36.8167 1.76930 0.884648 0.466260i \(-0.154399\pi\)
0.884648 + 0.466260i \(0.154399\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) −11.6056 −0.555805
\(437\) 2.21110 0.105771
\(438\) −18.4222 −0.880247
\(439\) 11.0278 0.526326 0.263163 0.964751i \(-0.415234\pi\)
0.263163 + 0.964751i \(0.415234\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) 7.88057 0.374417 0.187209 0.982320i \(-0.440056\pi\)
0.187209 + 0.982320i \(0.440056\pi\)
\(444\) −8.51388 −0.404051
\(445\) 0 0
\(446\) 26.8167 1.26981
\(447\) 19.1194 0.904318
\(448\) 0.697224 0.0329408
\(449\) −36.9083 −1.74181 −0.870906 0.491450i \(-0.836467\pi\)
−0.870906 + 0.491450i \(0.836467\pi\)
\(450\) 0 0
\(451\) −0.697224 −0.0328310
\(452\) 10.5139 0.494531
\(453\) 27.9083 1.31125
\(454\) −19.0000 −0.891714
\(455\) 0 0
\(456\) 0.908327 0.0425363
\(457\) −10.5139 −0.491818 −0.245909 0.969293i \(-0.579087\pi\)
−0.245909 + 0.969293i \(0.579087\pi\)
\(458\) −7.51388 −0.351100
\(459\) 4.66947 0.217952
\(460\) 0 0
\(461\) −8.48612 −0.395238 −0.197619 0.980279i \(-0.563321\pi\)
−0.197619 + 0.980279i \(0.563321\pi\)
\(462\) 1.11943 0.0520806
\(463\) −10.4861 −0.487331 −0.243666 0.969859i \(-0.578350\pi\)
−0.243666 + 0.969859i \(0.578350\pi\)
\(464\) 3.30278 0.153328
\(465\) 0 0
\(466\) −21.0000 −0.972806
\(467\) 16.4861 0.762887 0.381443 0.924392i \(-0.375427\pi\)
0.381443 + 0.924392i \(0.375427\pi\)
\(468\) 0 0
\(469\) 3.90833 0.180470
\(470\) 0 0
\(471\) 56.2389 2.59135
\(472\) 10.2111 0.470004
\(473\) −5.93608 −0.272941
\(474\) 21.4222 0.983954
\(475\) 0 0
\(476\) −2.02776 −0.0929421
\(477\) 28.1194 1.28750
\(478\) 0.577795 0.0264277
\(479\) −31.7250 −1.44955 −0.724776 0.688985i \(-0.758057\pi\)
−0.724776 + 0.688985i \(0.758057\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −5.00000 −0.227744
\(483\) 9.00000 0.409514
\(484\) −10.5139 −0.477904
\(485\) 0 0
\(486\) −19.6056 −0.889326
\(487\) 0.302776 0.0137201 0.00686004 0.999976i \(-0.497816\pi\)
0.00686004 + 0.999976i \(0.497816\pi\)
\(488\) 4.21110 0.190628
\(489\) −4.39445 −0.198724
\(490\) 0 0
\(491\) −37.4222 −1.68884 −0.844420 0.535681i \(-0.820055\pi\)
−0.844420 + 0.535681i \(0.820055\pi\)
\(492\) −2.30278 −0.103817
\(493\) −9.60555 −0.432612
\(494\) 0 0
\(495\) 0 0
\(496\) 8.60555 0.386401
\(497\) 7.39445 0.331686
\(498\) 38.7250 1.73531
\(499\) 28.6333 1.28180 0.640901 0.767623i \(-0.278561\pi\)
0.640901 + 0.767623i \(0.278561\pi\)
\(500\) 0 0
\(501\) −8.30278 −0.370941
\(502\) −17.3305 −0.773499
\(503\) −24.6056 −1.09711 −0.548554 0.836115i \(-0.684821\pi\)
−0.548554 + 0.836115i \(0.684821\pi\)
\(504\) 1.60555 0.0715169
\(505\) 0 0
\(506\) 3.90833 0.173746
\(507\) 0 0
\(508\) −4.90833 −0.217772
\(509\) 4.18335 0.185424 0.0927118 0.995693i \(-0.470446\pi\)
0.0927118 + 0.995693i \(0.470446\pi\)
\(510\) 0 0
\(511\) −5.57779 −0.246747
\(512\) 1.00000 0.0441942
\(513\) −0.633308 −0.0279612
\(514\) 5.60555 0.247250
\(515\) 0 0
\(516\) −19.6056 −0.863086
\(517\) 7.18335 0.315923
\(518\) −2.57779 −0.113262
\(519\) −14.9361 −0.655621
\(520\) 0 0
\(521\) −22.8444 −1.00083 −0.500416 0.865785i \(-0.666820\pi\)
−0.500416 + 0.865785i \(0.666820\pi\)
\(522\) 7.60555 0.332886
\(523\) 23.2111 1.01495 0.507475 0.861666i \(-0.330579\pi\)
0.507475 + 0.861666i \(0.330579\pi\)
\(524\) 8.81665 0.385157
\(525\) 0 0
\(526\) 18.7250 0.816448
\(527\) −25.0278 −1.09023
\(528\) 1.60555 0.0698727
\(529\) 8.42221 0.366183
\(530\) 0 0
\(531\) 23.5139 1.02042
\(532\) 0.275019 0.0119236
\(533\) 0 0
\(534\) −0.211103 −0.00913530
\(535\) 0 0
\(536\) 5.60555 0.242123
\(537\) 16.6056 0.716582
\(538\) −28.8167 −1.24237
\(539\) −4.54163 −0.195622
\(540\) 0 0
\(541\) −17.8167 −0.765998 −0.382999 0.923749i \(-0.625109\pi\)
−0.382999 + 0.923749i \(0.625109\pi\)
\(542\) −2.42221 −0.104043
\(543\) −41.2389 −1.76973
\(544\) −2.90833 −0.124693
\(545\) 0 0
\(546\) 0 0
\(547\) −21.6056 −0.923787 −0.461893 0.886935i \(-0.652830\pi\)
−0.461893 + 0.886935i \(0.652830\pi\)
\(548\) 0 0
\(549\) 9.69722 0.413867
\(550\) 0 0
\(551\) 1.30278 0.0555001
\(552\) 12.9083 0.549415
\(553\) 6.48612 0.275818
\(554\) −32.1194 −1.36462
\(555\) 0 0
\(556\) −9.81665 −0.416319
\(557\) 9.81665 0.415945 0.207972 0.978135i \(-0.433314\pi\)
0.207972 + 0.978135i \(0.433314\pi\)
\(558\) 19.8167 0.838906
\(559\) 0 0
\(560\) 0 0
\(561\) −4.66947 −0.197145
\(562\) −13.3028 −0.561144
\(563\) 20.7250 0.873454 0.436727 0.899594i \(-0.356138\pi\)
0.436727 + 0.899594i \(0.356138\pi\)
\(564\) 23.7250 0.999002
\(565\) 0 0
\(566\) 22.5139 0.946329
\(567\) −7.39445 −0.310538
\(568\) 10.6056 0.444999
\(569\) 28.1472 1.17999 0.589996 0.807406i \(-0.299129\pi\)
0.589996 + 0.807406i \(0.299129\pi\)
\(570\) 0 0
\(571\) 13.5139 0.565538 0.282769 0.959188i \(-0.408747\pi\)
0.282769 + 0.959188i \(0.408747\pi\)
\(572\) 0 0
\(573\) −24.9083 −1.04056
\(574\) −0.697224 −0.0291016
\(575\) 0 0
\(576\) 2.30278 0.0959490
\(577\) −23.6056 −0.982712 −0.491356 0.870959i \(-0.663499\pi\)
−0.491356 + 0.870959i \(0.663499\pi\)
\(578\) −8.54163 −0.355285
\(579\) 40.3305 1.67608
\(580\) 0 0
\(581\) 11.7250 0.486434
\(582\) −26.9361 −1.11654
\(583\) 8.51388 0.352609
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) −2.51388 −0.103847
\(587\) −14.4861 −0.597906 −0.298953 0.954268i \(-0.596637\pi\)
−0.298953 + 0.954268i \(0.596637\pi\)
\(588\) −15.0000 −0.618590
\(589\) 3.39445 0.139866
\(590\) 0 0
\(591\) −51.3583 −2.11260
\(592\) −3.69722 −0.151955
\(593\) −42.6611 −1.75188 −0.875940 0.482420i \(-0.839758\pi\)
−0.875940 + 0.482420i \(0.839758\pi\)
\(594\) −1.11943 −0.0459307
\(595\) 0 0
\(596\) 8.30278 0.340095
\(597\) 9.00000 0.368345
\(598\) 0 0
\(599\) 37.6611 1.53879 0.769395 0.638774i \(-0.220558\pi\)
0.769395 + 0.638774i \(0.220558\pi\)
\(600\) 0 0
\(601\) −8.39445 −0.342417 −0.171208 0.985235i \(-0.554767\pi\)
−0.171208 + 0.985235i \(0.554767\pi\)
\(602\) −5.93608 −0.241937
\(603\) 12.9083 0.525668
\(604\) 12.1194 0.493133
\(605\) 0 0
\(606\) −14.3028 −0.581011
\(607\) −31.0555 −1.26050 −0.630252 0.776390i \(-0.717049\pi\)
−0.630252 + 0.776390i \(0.717049\pi\)
\(608\) 0.394449 0.0159970
\(609\) 5.30278 0.214879
\(610\) 0 0
\(611\) 0 0
\(612\) −6.69722 −0.270719
\(613\) −21.4500 −0.866356 −0.433178 0.901308i \(-0.642608\pi\)
−0.433178 + 0.901308i \(0.642608\pi\)
\(614\) −11.2111 −0.452443
\(615\) 0 0
\(616\) 0.486122 0.0195864
\(617\) −23.0555 −0.928180 −0.464090 0.885788i \(-0.653619\pi\)
−0.464090 + 0.885788i \(0.653619\pi\)
\(618\) −14.7250 −0.592326
\(619\) −7.33053 −0.294639 −0.147319 0.989089i \(-0.547065\pi\)
−0.147319 + 0.989089i \(0.547065\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) −9.90833 −0.397288
\(623\) −0.0639167 −0.00256077
\(624\) 0 0
\(625\) 0 0
\(626\) −15.9083 −0.635825
\(627\) 0.633308 0.0252919
\(628\) 24.4222 0.974552
\(629\) 10.7527 0.428740
\(630\) 0 0
\(631\) 5.09167 0.202696 0.101348 0.994851i \(-0.467684\pi\)
0.101348 + 0.994851i \(0.467684\pi\)
\(632\) 9.30278 0.370045
\(633\) −15.6972 −0.623909
\(634\) −11.4861 −0.456172
\(635\) 0 0
\(636\) 28.1194 1.11501
\(637\) 0 0
\(638\) 2.30278 0.0911678
\(639\) 24.4222 0.966128
\(640\) 0 0
\(641\) 47.1749 1.86330 0.931649 0.363359i \(-0.118370\pi\)
0.931649 + 0.363359i \(0.118370\pi\)
\(642\) −31.8167 −1.25570
\(643\) −35.4222 −1.39692 −0.698458 0.715651i \(-0.746130\pi\)
−0.698458 + 0.715651i \(0.746130\pi\)
\(644\) 3.90833 0.154010
\(645\) 0 0
\(646\) −1.14719 −0.0451354
\(647\) −16.3028 −0.640928 −0.320464 0.947261i \(-0.603839\pi\)
−0.320464 + 0.947261i \(0.603839\pi\)
\(648\) −10.6056 −0.416625
\(649\) 7.11943 0.279462
\(650\) 0 0
\(651\) 13.8167 0.541517
\(652\) −1.90833 −0.0747358
\(653\) 8.21110 0.321325 0.160663 0.987009i \(-0.448637\pi\)
0.160663 + 0.987009i \(0.448637\pi\)
\(654\) −26.7250 −1.04503
\(655\) 0 0
\(656\) −1.00000 −0.0390434
\(657\) −18.4222 −0.718719
\(658\) 7.18335 0.280036
\(659\) −9.21110 −0.358814 −0.179407 0.983775i \(-0.557418\pi\)
−0.179407 + 0.983775i \(0.557418\pi\)
\(660\) 0 0
\(661\) 28.3028 1.10085 0.550425 0.834884i \(-0.314466\pi\)
0.550425 + 0.834884i \(0.314466\pi\)
\(662\) 20.9083 0.812625
\(663\) 0 0
\(664\) 16.8167 0.652613
\(665\) 0 0
\(666\) −8.51388 −0.329906
\(667\) 18.5139 0.716860
\(668\) −3.60555 −0.139503
\(669\) 61.7527 2.38750
\(670\) 0 0
\(671\) 2.93608 0.113346
\(672\) 1.60555 0.0619355
\(673\) −15.0278 −0.579277 −0.289639 0.957136i \(-0.593535\pi\)
−0.289639 + 0.957136i \(0.593535\pi\)
\(674\) 14.0278 0.540329
\(675\) 0 0
\(676\) 0 0
\(677\) −7.42221 −0.285259 −0.142629 0.989776i \(-0.545556\pi\)
−0.142629 + 0.989776i \(0.545556\pi\)
\(678\) 24.2111 0.929822
\(679\) −8.15559 −0.312983
\(680\) 0 0
\(681\) −43.7527 −1.67661
\(682\) 6.00000 0.229752
\(683\) 16.6333 0.636456 0.318228 0.948014i \(-0.396912\pi\)
0.318228 + 0.948014i \(0.396912\pi\)
\(684\) 0.908327 0.0347307
\(685\) 0 0
\(686\) −9.42221 −0.359741
\(687\) −17.3028 −0.660142
\(688\) −8.51388 −0.324589
\(689\) 0 0
\(690\) 0 0
\(691\) 23.3944 0.889967 0.444983 0.895539i \(-0.353210\pi\)
0.444983 + 0.895539i \(0.353210\pi\)
\(692\) −6.48612 −0.246565
\(693\) 1.11943 0.0425236
\(694\) −2.21110 −0.0839323
\(695\) 0 0
\(696\) 7.60555 0.288288
\(697\) 2.90833 0.110161
\(698\) 33.7250 1.27651
\(699\) −48.3583 −1.82908
\(700\) 0 0
\(701\) −3.09167 −0.116771 −0.0583854 0.998294i \(-0.518595\pi\)
−0.0583854 + 0.998294i \(0.518595\pi\)
\(702\) 0 0
\(703\) −1.45837 −0.0550033
\(704\) 0.697224 0.0262776
\(705\) 0 0
\(706\) −20.5139 −0.772050
\(707\) −4.33053 −0.162866
\(708\) 23.5139 0.883706
\(709\) −6.54163 −0.245676 −0.122838 0.992427i \(-0.539200\pi\)
−0.122838 + 0.992427i \(0.539200\pi\)
\(710\) 0 0
\(711\) 21.4222 0.803395
\(712\) −0.0916731 −0.00343559
\(713\) 48.2389 1.80656
\(714\) −4.66947 −0.174750
\(715\) 0 0
\(716\) 7.21110 0.269492
\(717\) 1.33053 0.0496896
\(718\) 22.9361 0.855967
\(719\) −7.93608 −0.295966 −0.147983 0.988990i \(-0.547278\pi\)
−0.147983 + 0.988990i \(0.547278\pi\)
\(720\) 0 0
\(721\) −4.45837 −0.166038
\(722\) −18.8444 −0.701316
\(723\) −11.5139 −0.428206
\(724\) −17.9083 −0.665558
\(725\) 0 0
\(726\) −24.2111 −0.898558
\(727\) −42.0555 −1.55975 −0.779876 0.625934i \(-0.784718\pi\)
−0.779876 + 0.625934i \(0.784718\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) 24.7611 0.915824
\(732\) 9.69722 0.358420
\(733\) −41.5416 −1.53438 −0.767188 0.641423i \(-0.778344\pi\)
−0.767188 + 0.641423i \(0.778344\pi\)
\(734\) −25.8167 −0.952910
\(735\) 0 0
\(736\) 5.60555 0.206623
\(737\) 3.90833 0.143965
\(738\) −2.30278 −0.0847663
\(739\) 14.7250 0.541667 0.270834 0.962626i \(-0.412701\pi\)
0.270834 + 0.962626i \(0.412701\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.51388 0.312554
\(743\) −35.2111 −1.29177 −0.645885 0.763435i \(-0.723511\pi\)
−0.645885 + 0.763435i \(0.723511\pi\)
\(744\) 19.8167 0.726514
\(745\) 0 0
\(746\) −15.2111 −0.556918
\(747\) 38.7250 1.41687
\(748\) −2.02776 −0.0741421
\(749\) −9.63331 −0.351993
\(750\) 0 0
\(751\) 32.6333 1.19081 0.595403 0.803427i \(-0.296992\pi\)
0.595403 + 0.803427i \(0.296992\pi\)
\(752\) 10.3028 0.375704
\(753\) −39.9083 −1.45434
\(754\) 0 0
\(755\) 0 0
\(756\) −1.11943 −0.0407133
\(757\) −9.78890 −0.355784 −0.177892 0.984050i \(-0.556928\pi\)
−0.177892 + 0.984050i \(0.556928\pi\)
\(758\) 34.8167 1.26460
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) 19.6056 0.710701 0.355350 0.934733i \(-0.384362\pi\)
0.355350 + 0.934733i \(0.384362\pi\)
\(762\) −11.3028 −0.409456
\(763\) −8.09167 −0.292938
\(764\) −10.8167 −0.391333
\(765\) 0 0
\(766\) −11.6056 −0.419326
\(767\) 0 0
\(768\) 2.30278 0.0830943
\(769\) −21.4500 −0.773505 −0.386753 0.922183i \(-0.626403\pi\)
−0.386753 + 0.922183i \(0.626403\pi\)
\(770\) 0 0
\(771\) 12.9083 0.464882
\(772\) 17.5139 0.630338
\(773\) 40.8444 1.46907 0.734536 0.678570i \(-0.237400\pi\)
0.734536 + 0.678570i \(0.237400\pi\)
\(774\) −19.6056 −0.704707
\(775\) 0 0
\(776\) −11.6972 −0.419906
\(777\) −5.93608 −0.212956
\(778\) −9.21110 −0.330234
\(779\) −0.394449 −0.0141326
\(780\) 0 0
\(781\) 7.39445 0.264594
\(782\) −16.3028 −0.582986
\(783\) −5.30278 −0.189506
\(784\) −6.51388 −0.232639
\(785\) 0 0
\(786\) 20.3028 0.724176
\(787\) −18.3028 −0.652423 −0.326212 0.945297i \(-0.605772\pi\)
−0.326212 + 0.945297i \(0.605772\pi\)
\(788\) −22.3028 −0.794503
\(789\) 43.1194 1.53509
\(790\) 0 0
\(791\) 7.33053 0.260644
\(792\) 1.60555 0.0570508
\(793\) 0 0
\(794\) 10.3305 0.366617
\(795\) 0 0
\(796\) 3.90833 0.138527
\(797\) 1.00000 0.0354218 0.0177109 0.999843i \(-0.494362\pi\)
0.0177109 + 0.999843i \(0.494362\pi\)
\(798\) 0.633308 0.0224188
\(799\) −29.9638 −1.06004
\(800\) 0 0
\(801\) −0.211103 −0.00745894
\(802\) −14.7250 −0.519957
\(803\) −5.57779 −0.196836
\(804\) 12.9083 0.455242
\(805\) 0 0
\(806\) 0 0
\(807\) −66.3583 −2.33592
\(808\) −6.21110 −0.218506
\(809\) −13.6333 −0.479322 −0.239661 0.970857i \(-0.577036\pi\)
−0.239661 + 0.970857i \(0.577036\pi\)
\(810\) 0 0
\(811\) 21.6972 0.761893 0.380946 0.924597i \(-0.375598\pi\)
0.380946 + 0.924597i \(0.375598\pi\)
\(812\) 2.30278 0.0808116
\(813\) −5.57779 −0.195622
\(814\) −2.57779 −0.0903517
\(815\) 0 0
\(816\) −6.69722 −0.234450
\(817\) −3.35829 −0.117492
\(818\) 34.2111 1.19616
\(819\) 0 0
\(820\) 0 0
\(821\) −39.4222 −1.37584 −0.687922 0.725784i \(-0.741477\pi\)
−0.687922 + 0.725784i \(0.741477\pi\)
\(822\) 0 0
\(823\) −12.1833 −0.424685 −0.212342 0.977195i \(-0.568109\pi\)
−0.212342 + 0.977195i \(0.568109\pi\)
\(824\) −6.39445 −0.222761
\(825\) 0 0
\(826\) 7.11943 0.247717
\(827\) −50.4500 −1.75432 −0.877159 0.480201i \(-0.840564\pi\)
−0.877159 + 0.480201i \(0.840564\pi\)
\(828\) 12.9083 0.448595
\(829\) 38.9083 1.35134 0.675671 0.737203i \(-0.263854\pi\)
0.675671 + 0.737203i \(0.263854\pi\)
\(830\) 0 0
\(831\) −73.9638 −2.56578
\(832\) 0 0
\(833\) 18.9445 0.656388
\(834\) −22.6056 −0.782766
\(835\) 0 0
\(836\) 0.275019 0.00951174
\(837\) −13.8167 −0.477573
\(838\) 17.3028 0.597714
\(839\) −12.9361 −0.446603 −0.223302 0.974749i \(-0.571683\pi\)
−0.223302 + 0.974749i \(0.571683\pi\)
\(840\) 0 0
\(841\) −18.0917 −0.623851
\(842\) 29.9083 1.03071
\(843\) −30.6333 −1.05507
\(844\) −6.81665 −0.234639
\(845\) 0 0
\(846\) 23.7250 0.815682
\(847\) −7.33053 −0.251880
\(848\) 12.2111 0.419331
\(849\) 51.8444 1.77930
\(850\) 0 0
\(851\) −20.7250 −0.710443
\(852\) 24.4222 0.836691
\(853\) −8.27502 −0.283331 −0.141666 0.989915i \(-0.545246\pi\)
−0.141666 + 0.989915i \(0.545246\pi\)
\(854\) 2.93608 0.100471
\(855\) 0 0
\(856\) −13.8167 −0.472244
\(857\) −20.5778 −0.702924 −0.351462 0.936202i \(-0.614315\pi\)
−0.351462 + 0.936202i \(0.614315\pi\)
\(858\) 0 0
\(859\) 15.1194 0.515868 0.257934 0.966163i \(-0.416958\pi\)
0.257934 + 0.966163i \(0.416958\pi\)
\(860\) 0 0
\(861\) −1.60555 −0.0547170
\(862\) −0.908327 −0.0309377
\(863\) −4.02776 −0.137106 −0.0685532 0.997647i \(-0.521838\pi\)
−0.0685532 + 0.997647i \(0.521838\pi\)
\(864\) −1.60555 −0.0546220
\(865\) 0 0
\(866\) 36.8167 1.25108
\(867\) −19.6695 −0.668010
\(868\) 6.00000 0.203653
\(869\) 6.48612 0.220027
\(870\) 0 0
\(871\) 0 0
\(872\) −11.6056 −0.393014
\(873\) −26.9361 −0.911648
\(874\) 2.21110 0.0747917
\(875\) 0 0
\(876\) −18.4222 −0.622429
\(877\) −47.4777 −1.60321 −0.801604 0.597855i \(-0.796020\pi\)
−0.801604 + 0.597855i \(0.796020\pi\)
\(878\) 11.0278 0.372169
\(879\) −5.78890 −0.195255
\(880\) 0 0
\(881\) 32.5694 1.09729 0.548645 0.836055i \(-0.315144\pi\)
0.548645 + 0.836055i \(0.315144\pi\)
\(882\) −15.0000 −0.505076
\(883\) −2.11943 −0.0713245 −0.0356622 0.999364i \(-0.511354\pi\)
−0.0356622 + 0.999364i \(0.511354\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 7.88057 0.264753
\(887\) −38.4222 −1.29009 −0.645046 0.764144i \(-0.723162\pi\)
−0.645046 + 0.764144i \(0.723162\pi\)
\(888\) −8.51388 −0.285707
\(889\) −3.42221 −0.114777
\(890\) 0 0
\(891\) −7.39445 −0.247723
\(892\) 26.8167 0.897888
\(893\) 4.06392 0.135994
\(894\) 19.1194 0.639449
\(895\) 0 0
\(896\) 0.697224 0.0232926
\(897\) 0 0
\(898\) −36.9083 −1.23165
\(899\) 28.4222 0.947934
\(900\) 0 0
\(901\) −35.5139 −1.18314
\(902\) −0.697224 −0.0232150
\(903\) −13.6695 −0.454891
\(904\) 10.5139 0.349686
\(905\) 0 0
\(906\) 27.9083 0.927192
\(907\) 45.6611 1.51615 0.758075 0.652167i \(-0.226140\pi\)
0.758075 + 0.652167i \(0.226140\pi\)
\(908\) −19.0000 −0.630537
\(909\) −14.3028 −0.474393
\(910\) 0 0
\(911\) 8.57779 0.284195 0.142098 0.989853i \(-0.454615\pi\)
0.142098 + 0.989853i \(0.454615\pi\)
\(912\) 0.908327 0.0300777
\(913\) 11.7250 0.388040
\(914\) −10.5139 −0.347768
\(915\) 0 0
\(916\) −7.51388 −0.248266
\(917\) 6.14719 0.202998
\(918\) 4.66947 0.154115
\(919\) 28.2389 0.931514 0.465757 0.884913i \(-0.345782\pi\)
0.465757 + 0.884913i \(0.345782\pi\)
\(920\) 0 0
\(921\) −25.8167 −0.850688
\(922\) −8.48612 −0.279476
\(923\) 0 0
\(924\) 1.11943 0.0368265
\(925\) 0 0
\(926\) −10.4861 −0.344595
\(927\) −14.7250 −0.483632
\(928\) 3.30278 0.108419
\(929\) 40.6611 1.33405 0.667023 0.745037i \(-0.267568\pi\)
0.667023 + 0.745037i \(0.267568\pi\)
\(930\) 0 0
\(931\) −2.56939 −0.0842084
\(932\) −21.0000 −0.687878
\(933\) −22.8167 −0.746984
\(934\) 16.4861 0.539442
\(935\) 0 0
\(936\) 0 0
\(937\) 38.3305 1.25220 0.626102 0.779741i \(-0.284649\pi\)
0.626102 + 0.779741i \(0.284649\pi\)
\(938\) 3.90833 0.127611
\(939\) −36.6333 −1.19548
\(940\) 0 0
\(941\) 42.5694 1.38772 0.693861 0.720109i \(-0.255908\pi\)
0.693861 + 0.720109i \(0.255908\pi\)
\(942\) 56.2389 1.83236
\(943\) −5.60555 −0.182542
\(944\) 10.2111 0.332343
\(945\) 0 0
\(946\) −5.93608 −0.192999
\(947\) 51.3305 1.66802 0.834009 0.551751i \(-0.186040\pi\)
0.834009 + 0.551751i \(0.186040\pi\)
\(948\) 21.4222 0.695761
\(949\) 0 0
\(950\) 0 0
\(951\) −26.4500 −0.857699
\(952\) −2.02776 −0.0657200
\(953\) −57.4777 −1.86189 −0.930943 0.365165i \(-0.881013\pi\)
−0.930943 + 0.365165i \(0.881013\pi\)
\(954\) 28.1194 0.910400
\(955\) 0 0
\(956\) 0.577795 0.0186872
\(957\) 5.30278 0.171414
\(958\) −31.7250 −1.02499
\(959\) 0 0
\(960\) 0 0
\(961\) 43.0555 1.38889
\(962\) 0 0
\(963\) −31.8167 −1.02528
\(964\) −5.00000 −0.161039
\(965\) 0 0
\(966\) 9.00000 0.289570
\(967\) 41.0278 1.31936 0.659682 0.751545i \(-0.270691\pi\)
0.659682 + 0.751545i \(0.270691\pi\)
\(968\) −10.5139 −0.337929
\(969\) −2.64171 −0.0848640
\(970\) 0 0
\(971\) −18.4500 −0.592087 −0.296044 0.955174i \(-0.595667\pi\)
−0.296044 + 0.955174i \(0.595667\pi\)
\(972\) −19.6056 −0.628848
\(973\) −6.84441 −0.219422
\(974\) 0.302776 0.00970156
\(975\) 0 0
\(976\) 4.21110 0.134794
\(977\) 16.8167 0.538012 0.269006 0.963138i \(-0.413305\pi\)
0.269006 + 0.963138i \(0.413305\pi\)
\(978\) −4.39445 −0.140519
\(979\) −0.0639167 −0.00204279
\(980\) 0 0
\(981\) −26.7250 −0.853263
\(982\) −37.4222 −1.19419
\(983\) 28.9638 0.923803 0.461902 0.886931i \(-0.347167\pi\)
0.461902 + 0.886931i \(0.347167\pi\)
\(984\) −2.30278 −0.0734098
\(985\) 0 0
\(986\) −9.60555 −0.305903
\(987\) 16.5416 0.526526
\(988\) 0 0
\(989\) −47.7250 −1.51757
\(990\) 0 0
\(991\) 2.84441 0.0903557 0.0451778 0.998979i \(-0.485615\pi\)
0.0451778 + 0.998979i \(0.485615\pi\)
\(992\) 8.60555 0.273227
\(993\) 48.1472 1.52790
\(994\) 7.39445 0.234538
\(995\) 0 0
\(996\) 38.7250 1.22705
\(997\) 1.09167 0.0345736 0.0172868 0.999851i \(-0.494497\pi\)
0.0172868 + 0.999851i \(0.494497\pi\)
\(998\) 28.6333 0.906372
\(999\) 5.93608 0.187809
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 8450.2.a.bl.1.2 2
5.4 even 2 8450.2.a.ba.1.1 2
13.4 even 6 650.2.e.g.601.1 yes 4
13.10 even 6 650.2.e.g.451.1 yes 4
13.12 even 2 8450.2.a.bd.1.2 2
65.4 even 6 650.2.e.e.601.2 yes 4
65.17 odd 12 650.2.o.h.549.1 8
65.23 odd 12 650.2.o.h.399.1 8
65.43 odd 12 650.2.o.h.549.4 8
65.49 even 6 650.2.e.e.451.2 4
65.62 odd 12 650.2.o.h.399.4 8
65.64 even 2 8450.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
650.2.e.e.451.2 4 65.49 even 6
650.2.e.e.601.2 yes 4 65.4 even 6
650.2.e.g.451.1 yes 4 13.10 even 6
650.2.e.g.601.1 yes 4 13.4 even 6
650.2.o.h.399.1 8 65.23 odd 12
650.2.o.h.399.4 8 65.62 odd 12
650.2.o.h.549.1 8 65.17 odd 12
650.2.o.h.549.4 8 65.43 odd 12
8450.2.a.ba.1.1 2 5.4 even 2
8450.2.a.bd.1.2 2 13.12 even 2
8450.2.a.bi.1.1 2 65.64 even 2
8450.2.a.bl.1.2 2 1.1 even 1 trivial