Properties

Label 900.3.f.d.199.5
Level $900$
Weight $3$
Character 900.199
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(199,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18084870400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 40x^{4} + 17x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.5
Root \(-2.53999 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 900.199
Dual form 900.3.f.d.199.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.758030 - 1.85078i) q^{2} +(-2.85078 - 2.80590i) q^{4} +4.09573 q^{7} +(-7.35408 + 3.14922i) q^{8} +O(q^{10})\) \(q+(0.758030 - 1.85078i) q^{2} +(-2.85078 - 2.80590i) q^{4} +4.09573 q^{7} +(-7.35408 + 3.14922i) q^{8} -0.984255i q^{11} -14.8062i q^{13} +(3.10469 - 7.58030i) q^{14} +(0.253905 + 15.9980i) q^{16} +9.80625i q^{17} -27.3684i q^{19} +(-1.82164 - 0.746095i) q^{22} -18.3514 q^{23} +(-27.4031 - 11.2236i) q^{26} +(-11.6760 - 11.4922i) q^{28} -2.80625 q^{29} -1.96851i q^{31} +(29.8012 + 11.6570i) q^{32} +(18.1492 + 7.43343i) q^{34} -44.8062i q^{37} +(-50.6530 - 20.7461i) q^{38} -17.0000 q^{41} -54.8956 q^{43} +(-2.76172 + 2.80590i) q^{44} +(-13.9109 + 33.9645i) q^{46} -58.8326 q^{47} -32.2250 q^{49} +(-41.5448 + 42.2094i) q^{52} -89.2250i q^{53} +(-30.1203 + 12.8984i) q^{56} +(-2.12722 + 5.19375i) q^{58} +67.3415i q^{59} +36.0312 q^{61} +(-3.64328 - 1.49219i) q^{62} +(44.1648 - 46.3192i) q^{64} +25.5586 q^{67} +(27.5153 - 27.9555i) q^{68} +120.110i q^{71} -69.8062i q^{73} +(-82.9266 - 33.9645i) q^{74} +(-76.7930 + 78.0214i) q^{76} -4.03124i q^{77} +32.2897i q^{79} +(-12.8865 + 31.4633i) q^{82} -68.1670 q^{83} +(-41.6125 + 101.600i) q^{86} +(3.09963 + 7.23828i) q^{88} -67.8062 q^{89} -60.6424i q^{91} +(52.3159 + 51.4922i) q^{92} +(-44.5969 + 108.886i) q^{94} +1.16251i q^{97} +(-24.4275 + 59.6414i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} - 52 q^{14} - 62 q^{16} - 168 q^{26} + 80 q^{29} + 158 q^{34} - 136 q^{41} + 170 q^{44} + 68 q^{46} + 152 q^{49} + 92 q^{56} - 224 q^{61} + 110 q^{64} - 228 q^{74} + 90 q^{76} - 128 q^{86} - 440 q^{89} - 408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.758030 1.85078i 0.379015 0.925391i
\(3\) 0 0
\(4\) −2.85078 2.80590i −0.712695 0.701474i
\(5\) 0 0
\(6\) 0 0
\(7\) 4.09573 0.585104 0.292552 0.956250i \(-0.405495\pi\)
0.292552 + 0.956250i \(0.405495\pi\)
\(8\) −7.35408 + 3.14922i −0.919259 + 0.393652i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.984255i 0.0894777i −0.998999 0.0447389i \(-0.985754\pi\)
0.998999 0.0447389i \(-0.0142456\pi\)
\(12\) 0 0
\(13\) 14.8062i 1.13894i −0.822011 0.569471i \(-0.807148\pi\)
0.822011 0.569471i \(-0.192852\pi\)
\(14\) 3.10469 7.58030i 0.221763 0.541450i
\(15\) 0 0
\(16\) 0.253905 + 15.9980i 0.0158691 + 0.999874i
\(17\) 9.80625i 0.576838i 0.957504 + 0.288419i \(0.0931296\pi\)
−0.957504 + 0.288419i \(0.906870\pi\)
\(18\) 0 0
\(19\) 27.3684i 1.44044i −0.693744 0.720222i \(-0.744040\pi\)
0.693744 0.720222i \(-0.255960\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.82164 0.746095i −0.0828018 0.0339134i
\(23\) −18.3514 −0.797888 −0.398944 0.916975i \(-0.630623\pi\)
−0.398944 + 0.916975i \(0.630623\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −27.4031 11.2236i −1.05397 0.431676i
\(27\) 0 0
\(28\) −11.6760 11.4922i −0.417001 0.410435i
\(29\) −2.80625 −0.0967672 −0.0483836 0.998829i \(-0.515407\pi\)
−0.0483836 + 0.998829i \(0.515407\pi\)
\(30\) 0 0
\(31\) 1.96851i 0.0635003i −0.999496 0.0317502i \(-0.989892\pi\)
0.999496 0.0317502i \(-0.0101081\pi\)
\(32\) 29.8012 + 11.6570i 0.931289 + 0.364282i
\(33\) 0 0
\(34\) 18.1492 + 7.43343i 0.533801 + 0.218630i
\(35\) 0 0
\(36\) 0 0
\(37\) 44.8062i 1.21098i −0.795853 0.605490i \(-0.792977\pi\)
0.795853 0.605490i \(-0.207023\pi\)
\(38\) −50.6530 20.7461i −1.33297 0.545950i
\(39\) 0 0
\(40\) 0 0
\(41\) −17.0000 −0.414634 −0.207317 0.978274i \(-0.566473\pi\)
−0.207317 + 0.978274i \(0.566473\pi\)
\(42\) 0 0
\(43\) −54.8956 −1.27664 −0.638321 0.769771i \(-0.720371\pi\)
−0.638321 + 0.769771i \(0.720371\pi\)
\(44\) −2.76172 + 2.80590i −0.0627663 + 0.0637703i
\(45\) 0 0
\(46\) −13.9109 + 33.9645i −0.302412 + 0.738358i
\(47\) −58.8326 −1.25176 −0.625879 0.779920i \(-0.715260\pi\)
−0.625879 + 0.779920i \(0.715260\pi\)
\(48\) 0 0
\(49\) −32.2250 −0.657653
\(50\) 0 0
\(51\) 0 0
\(52\) −41.5448 + 42.2094i −0.798938 + 0.811719i
\(53\) 89.2250i 1.68349i −0.539875 0.841745i \(-0.681528\pi\)
0.539875 0.841745i \(-0.318472\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −30.1203 + 12.8984i −0.537863 + 0.230328i
\(57\) 0 0
\(58\) −2.12722 + 5.19375i −0.0366762 + 0.0895474i
\(59\) 67.3415i 1.14138i 0.821165 + 0.570691i \(0.193324\pi\)
−0.821165 + 0.570691i \(0.806676\pi\)
\(60\) 0 0
\(61\) 36.0312 0.590676 0.295338 0.955393i \(-0.404568\pi\)
0.295338 + 0.955393i \(0.404568\pi\)
\(62\) −3.64328 1.49219i −0.0587626 0.0240676i
\(63\) 0 0
\(64\) 44.1648 46.3192i 0.690076 0.723737i
\(65\) 0 0
\(66\) 0 0
\(67\) 25.5586 0.381472 0.190736 0.981641i \(-0.438913\pi\)
0.190736 + 0.981641i \(0.438913\pi\)
\(68\) 27.5153 27.9555i 0.404637 0.411110i
\(69\) 0 0
\(70\) 0 0
\(71\) 120.110i 1.69169i 0.533430 + 0.845844i \(0.320903\pi\)
−0.533430 + 0.845844i \(0.679097\pi\)
\(72\) 0 0
\(73\) 69.8062i 0.956250i −0.878292 0.478125i \(-0.841317\pi\)
0.878292 0.478125i \(-0.158683\pi\)
\(74\) −82.9266 33.9645i −1.12063 0.458979i
\(75\) 0 0
\(76\) −76.7930 + 78.0214i −1.01043 + 1.02660i
\(77\) 4.03124i 0.0523538i
\(78\) 0 0
\(79\) 32.2897i 0.408730i 0.978895 + 0.204365i \(0.0655130\pi\)
−0.978895 + 0.204365i \(0.934487\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.8865 + 31.4633i −0.157153 + 0.383699i
\(83\) −68.1670 −0.821289 −0.410645 0.911795i \(-0.634696\pi\)
−0.410645 + 0.911795i \(0.634696\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −41.6125 + 101.600i −0.483866 + 1.18139i
\(87\) 0 0
\(88\) 3.09963 + 7.23828i 0.0352231 + 0.0822532i
\(89\) −67.8062 −0.761868 −0.380934 0.924602i \(-0.624397\pi\)
−0.380934 + 0.924602i \(0.624397\pi\)
\(90\) 0 0
\(91\) 60.6424i 0.666400i
\(92\) 52.3159 + 51.4922i 0.568651 + 0.559698i
\(93\) 0 0
\(94\) −44.5969 + 108.886i −0.474435 + 1.15836i
\(95\) 0 0
\(96\) 0 0
\(97\) 1.16251i 0.0119846i 0.999982 + 0.00599232i \(0.00190742\pi\)
−0.999982 + 0.00599232i \(0.998093\pi\)
\(98\) −24.4275 + 59.6414i −0.249260 + 0.608586i
\(99\) 0 0
\(100\) 0 0
\(101\) 132.031 1.30724 0.653620 0.756823i \(-0.273249\pi\)
0.653620 + 0.756823i \(0.273249\pi\)
\(102\) 0 0
\(103\) −111.601 −1.08350 −0.541752 0.840538i \(-0.682239\pi\)
−0.541752 + 0.840538i \(0.682239\pi\)
\(104\) 46.6281 + 108.886i 0.448347 + 1.04698i
\(105\) 0 0
\(106\) −165.136 67.6352i −1.55789 0.638068i
\(107\) 62.2615 0.581883 0.290942 0.956741i \(-0.406031\pi\)
0.290942 + 0.956741i \(0.406031\pi\)
\(108\) 0 0
\(109\) −87.1938 −0.799943 −0.399971 0.916528i \(-0.630980\pi\)
−0.399971 + 0.916528i \(0.630980\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.03993 + 65.5234i 0.00928507 + 0.585031i
\(113\) 84.2250i 0.745354i −0.927961 0.372677i \(-0.878440\pi\)
0.927961 0.372677i \(-0.121560\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.00000 + 7.87404i 0.0689655 + 0.0678796i
\(117\) 0 0
\(118\) 124.634 + 51.0469i 1.05622 + 0.432601i
\(119\) 40.1637i 0.337510i
\(120\) 0 0
\(121\) 120.031 0.991994
\(122\) 27.3128 66.6859i 0.223875 0.546606i
\(123\) 0 0
\(124\) −5.52343 + 5.61179i −0.0445438 + 0.0452564i
\(125\) 0 0
\(126\) 0 0
\(127\) −75.0568 −0.590999 −0.295499 0.955343i \(-0.595486\pi\)
−0.295499 + 0.955343i \(0.595486\pi\)
\(128\) −52.2484 116.851i −0.408191 0.912897i
\(129\) 0 0
\(130\) 0 0
\(131\) 139.795i 1.06714i −0.845757 0.533568i \(-0.820851\pi\)
0.845757 0.533568i \(-0.179149\pi\)
\(132\) 0 0
\(133\) 112.094i 0.842810i
\(134\) 19.3742 47.3034i 0.144584 0.353011i
\(135\) 0 0
\(136\) −30.8820 72.1159i −0.227074 0.530264i
\(137\) 56.1625i 0.409945i −0.978768 0.204973i \(-0.934289\pi\)
0.978768 0.204973i \(-0.0657105\pi\)
\(138\) 0 0
\(139\) 194.341i 1.39814i 0.715054 + 0.699069i \(0.246402\pi\)
−0.715054 + 0.699069i \(0.753598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 222.297 + 91.0469i 1.56547 + 0.641175i
\(143\) −14.5731 −0.101910
\(144\) 0 0
\(145\) 0 0
\(146\) −129.196 52.9152i −0.884905 0.362433i
\(147\) 0 0
\(148\) −125.722 + 127.733i −0.849470 + 0.863059i
\(149\) −110.869 −0.744085 −0.372043 0.928216i \(-0.621343\pi\)
−0.372043 + 0.928216i \(0.621343\pi\)
\(150\) 0 0
\(151\) 141.763i 0.938831i −0.882977 0.469415i \(-0.844465\pi\)
0.882977 0.469415i \(-0.155535\pi\)
\(152\) 86.1892 + 201.270i 0.567034 + 1.32414i
\(153\) 0 0
\(154\) −7.46095 3.05580i −0.0484477 0.0198429i
\(155\) 0 0
\(156\) 0 0
\(157\) 178.837i 1.13909i −0.821959 0.569546i \(-0.807119\pi\)
0.821959 0.569546i \(-0.192881\pi\)
\(158\) 59.7612 + 24.4766i 0.378235 + 0.154915i
\(159\) 0 0
\(160\) 0 0
\(161\) −75.1625 −0.466848
\(162\) 0 0
\(163\) 222.535 1.36525 0.682623 0.730771i \(-0.260839\pi\)
0.682623 + 0.730771i \(0.260839\pi\)
\(164\) 48.4633 + 47.7002i 0.295508 + 0.290855i
\(165\) 0 0
\(166\) −51.6727 + 126.162i −0.311281 + 0.760014i
\(167\) 212.883 1.27475 0.637375 0.770554i \(-0.280020\pi\)
0.637375 + 0.770554i \(0.280020\pi\)
\(168\) 0 0
\(169\) −50.2250 −0.297189
\(170\) 0 0
\(171\) 0 0
\(172\) 156.495 + 154.031i 0.909856 + 0.895530i
\(173\) 20.7750i 0.120087i 0.998196 + 0.0600434i \(0.0191239\pi\)
−0.998196 + 0.0600434i \(0.980876\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.7461 0.249908i 0.0894664 0.00141993i
\(177\) 0 0
\(178\) −51.3992 + 125.495i −0.288759 + 0.705025i
\(179\) 37.2110i 0.207883i −0.994583 0.103941i \(-0.966855\pi\)
0.994583 0.103941i \(-0.0331454\pi\)
\(180\) 0 0
\(181\) 318.125 1.75760 0.878798 0.477193i \(-0.158346\pi\)
0.878798 + 0.477193i \(0.158346\pi\)
\(182\) −112.236 45.9688i −0.616680 0.252576i
\(183\) 0 0
\(184\) 134.958 57.7927i 0.733466 0.314091i
\(185\) 0 0
\(186\) 0 0
\(187\) 9.65185 0.0516142
\(188\) 167.719 + 165.078i 0.892122 + 0.878075i
\(189\) 0 0
\(190\) 0 0
\(191\) 257.936i 1.35045i −0.737611 0.675226i \(-0.764046\pi\)
0.737611 0.675226i \(-0.235954\pi\)
\(192\) 0 0
\(193\) 127.869i 0.662532i −0.943537 0.331266i \(-0.892524\pi\)
0.943537 0.331266i \(-0.107476\pi\)
\(194\) 2.15155 + 0.881217i 0.0110905 + 0.00454235i
\(195\) 0 0
\(196\) 91.8664 + 90.4200i 0.468706 + 0.461326i
\(197\) 97.2875i 0.493845i −0.969035 0.246923i \(-0.920581\pi\)
0.969035 0.246923i \(-0.0794193\pi\)
\(198\) 0 0
\(199\) 19.6851i 0.0989201i 0.998776 + 0.0494600i \(0.0157500\pi\)
−0.998776 + 0.0494600i \(0.984250\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 100.084 244.361i 0.495464 1.20971i
\(203\) −11.4936 −0.0566189
\(204\) 0 0
\(205\) 0 0
\(206\) −84.5969 + 206.549i −0.410664 + 1.00266i
\(207\) 0 0
\(208\) 236.870 3.75938i 1.13880 0.0180740i
\(209\) −26.9375 −0.128888
\(210\) 0 0
\(211\) 235.298i 1.11516i 0.830124 + 0.557579i \(0.188270\pi\)
−0.830124 + 0.557579i \(0.811730\pi\)
\(212\) −250.356 + 254.361i −1.18092 + 1.19982i
\(213\) 0 0
\(214\) 47.1961 115.232i 0.220542 0.538469i
\(215\) 0 0
\(216\) 0 0
\(217\) 8.06248i 0.0371543i
\(218\) −66.0955 + 161.377i −0.303190 + 0.740259i
\(219\) 0 0
\(220\) 0 0
\(221\) 145.194 0.656985
\(222\) 0 0
\(223\) −32.4484 −0.145509 −0.0727543 0.997350i \(-0.523179\pi\)
−0.0727543 + 0.997350i \(0.523179\pi\)
\(224\) 122.058 + 47.7440i 0.544901 + 0.213143i
\(225\) 0 0
\(226\) −155.882 63.8451i −0.689743 0.282500i
\(227\) −289.591 −1.27573 −0.637866 0.770147i \(-0.720183\pi\)
−0.637866 + 0.770147i \(0.720183\pi\)
\(228\) 0 0
\(229\) −99.2875 −0.433570 −0.216785 0.976219i \(-0.569557\pi\)
−0.216785 + 0.976219i \(0.569557\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 20.6374 8.83749i 0.0889541 0.0380926i
\(233\) 50.7750i 0.217918i 0.994046 + 0.108959i \(0.0347518\pi\)
−0.994046 + 0.108959i \(0.965248\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 188.953 191.976i 0.800649 0.813457i
\(237\) 0 0
\(238\) 74.3343 + 30.4453i 0.312329 + 0.127922i
\(239\) 311.498i 1.30334i −0.758503 0.651670i \(-0.774069\pi\)
0.758503 0.651670i \(-0.225931\pi\)
\(240\) 0 0
\(241\) 319.094 1.32404 0.662020 0.749486i \(-0.269699\pi\)
0.662020 + 0.749486i \(0.269699\pi\)
\(242\) 90.9873 222.152i 0.375981 0.917982i
\(243\) 0 0
\(244\) −102.717 101.100i −0.420972 0.414344i
\(245\) 0 0
\(246\) 0 0
\(247\) −405.224 −1.64058
\(248\) 6.19927 + 14.4766i 0.0249970 + 0.0583733i
\(249\) 0 0
\(250\) 0 0
\(251\) 87.6294i 0.349121i −0.984646 0.174561i \(-0.944149\pi\)
0.984646 0.174561i \(-0.0558505\pi\)
\(252\) 0 0
\(253\) 18.0625i 0.0713932i
\(254\) −56.8953 + 138.914i −0.223997 + 0.546904i
\(255\) 0 0
\(256\) −255.871 + 8.12395i −0.999496 + 0.0317342i
\(257\) 117.287i 0.456372i −0.973618 0.228186i \(-0.926721\pi\)
0.973618 0.228186i \(-0.0732794\pi\)
\(258\) 0 0
\(259\) 183.514i 0.708549i
\(260\) 0 0
\(261\) 0 0
\(262\) −258.730 105.969i −0.987518 0.404461i
\(263\) 47.8151 0.181806 0.0909032 0.995860i \(-0.471025\pi\)
0.0909032 + 0.995860i \(0.471025\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −207.461 84.9704i −0.779928 0.319438i
\(267\) 0 0
\(268\) −72.8621 71.7149i −0.271873 0.267593i
\(269\) 479.287 1.78174 0.890869 0.454260i \(-0.150096\pi\)
0.890869 + 0.454260i \(0.150096\pi\)
\(270\) 0 0
\(271\) 242.188i 0.893683i −0.894613 0.446842i \(-0.852549\pi\)
0.894613 0.446842i \(-0.147451\pi\)
\(272\) −156.880 + 2.48986i −0.576766 + 0.00915389i
\(273\) 0 0
\(274\) −103.945 42.5729i −0.379360 0.155375i
\(275\) 0 0
\(276\) 0 0
\(277\) 33.2562i 0.120059i 0.998197 + 0.0600293i \(0.0191194\pi\)
−0.998197 + 0.0600293i \(0.980881\pi\)
\(278\) 359.683 + 147.316i 1.29382 + 0.529915i
\(279\) 0 0
\(280\) 0 0
\(281\) 246.062 0.875667 0.437834 0.899056i \(-0.355746\pi\)
0.437834 + 0.899056i \(0.355746\pi\)
\(282\) 0 0
\(283\) −1.52437 −0.00538646 −0.00269323 0.999996i \(-0.500857\pi\)
−0.00269323 + 0.999996i \(0.500857\pi\)
\(284\) 337.016 342.407i 1.18667 1.20566i
\(285\) 0 0
\(286\) −11.0469 + 26.9717i −0.0386254 + 0.0943065i
\(287\) −69.6274 −0.242604
\(288\) 0 0
\(289\) 192.837 0.667258
\(290\) 0 0
\(291\) 0 0
\(292\) −195.869 + 199.002i −0.670784 + 0.681515i
\(293\) 448.994i 1.53240i 0.642601 + 0.766201i \(0.277855\pi\)
−0.642601 + 0.766201i \(0.722145\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 141.105 + 329.509i 0.476705 + 1.11320i
\(297\) 0 0
\(298\) −84.0418 + 205.194i −0.282020 + 0.688570i
\(299\) 271.716i 0.908749i
\(300\) 0 0
\(301\) −224.837 −0.746968
\(302\) −262.373 107.461i −0.868785 0.355831i
\(303\) 0 0
\(304\) 437.840 6.94899i 1.44026 0.0228585i
\(305\) 0 0
\(306\) 0 0
\(307\) −10.6681 −0.0347495 −0.0173747 0.999849i \(-0.505531\pi\)
−0.0173747 + 0.999849i \(0.505531\pi\)
\(308\) −11.3112 + 11.4922i −0.0367248 + 0.0373123i
\(309\) 0 0
\(310\) 0 0
\(311\) 370.172i 1.19026i 0.803628 + 0.595132i \(0.202900\pi\)
−0.803628 + 0.595132i \(0.797100\pi\)
\(312\) 0 0
\(313\) 42.7125i 0.136462i −0.997670 0.0682309i \(-0.978265\pi\)
0.997670 0.0682309i \(-0.0217354\pi\)
\(314\) −330.989 135.564i −1.05411 0.431733i
\(315\) 0 0
\(316\) 90.6015 92.0509i 0.286714 0.291300i
\(317\) 143.100i 0.451420i −0.974195 0.225710i \(-0.927530\pi\)
0.974195 0.225710i \(-0.0724701\pi\)
\(318\) 0 0
\(319\) 2.76206i 0.00865851i
\(320\) 0 0
\(321\) 0 0
\(322\) −56.9754 + 139.109i −0.176942 + 0.432017i
\(323\) 268.382 0.830903
\(324\) 0 0
\(325\) 0 0
\(326\) 168.688 411.864i 0.517449 1.26339i
\(327\) 0 0
\(328\) 125.019 53.5367i 0.381156 0.163222i
\(329\) −240.962 −0.732409
\(330\) 0 0
\(331\) 52.1655i 0.157600i 0.996890 + 0.0787999i \(0.0251088\pi\)
−0.996890 + 0.0787999i \(0.974891\pi\)
\(332\) 194.329 + 191.270i 0.585329 + 0.576113i
\(333\) 0 0
\(334\) 161.372 394.000i 0.483149 1.17964i
\(335\) 0 0
\(336\) 0 0
\(337\) 441.900i 1.31128i −0.755075 0.655638i \(-0.772400\pi\)
0.755075 0.655638i \(-0.227600\pi\)
\(338\) −38.0721 + 92.9555i −0.112639 + 0.275016i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.93752 −0.00568186
\(342\) 0 0
\(343\) −332.676 −0.969900
\(344\) 403.706 172.878i 1.17356 0.502553i
\(345\) 0 0
\(346\) 38.4500 + 15.7481i 0.111127 + 0.0455147i
\(347\) −85.8196 −0.247319 −0.123659 0.992325i \(-0.539463\pi\)
−0.123659 + 0.992325i \(0.539463\pi\)
\(348\) 0 0
\(349\) 408.931 1.17172 0.585861 0.810411i \(-0.300756\pi\)
0.585861 + 0.810411i \(0.300756\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.4735 29.3320i 0.0325951 0.0833296i
\(353\) 441.163i 1.24975i −0.780724 0.624876i \(-0.785149\pi\)
0.780724 0.624876i \(-0.214851\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 193.301 + 190.257i 0.542980 + 0.534430i
\(357\) 0 0
\(358\) −68.8694 28.2070i −0.192373 0.0787906i
\(359\) 430.814i 1.20004i −0.799985 0.600020i \(-0.795159\pi\)
0.799985 0.600020i \(-0.204841\pi\)
\(360\) 0 0
\(361\) −388.031 −1.07488
\(362\) 241.148 588.780i 0.666155 1.62646i
\(363\) 0 0
\(364\) −170.156 + 172.878i −0.467462 + 0.474940i
\(365\) 0 0
\(366\) 0 0
\(367\) 346.709 0.944710 0.472355 0.881408i \(-0.343404\pi\)
0.472355 + 0.881408i \(0.343404\pi\)
\(368\) −4.65953 293.586i −0.0126618 0.797788i
\(369\) 0 0
\(370\) 0 0
\(371\) 365.442i 0.985018i
\(372\) 0 0
\(373\) 577.287i 1.54769i 0.633376 + 0.773844i \(0.281668\pi\)
−0.633376 + 0.773844i \(0.718332\pi\)
\(374\) 7.31639 17.8635i 0.0195625 0.0477633i
\(375\) 0 0
\(376\) 432.659 185.277i 1.15069 0.492757i
\(377\) 41.5500i 0.110212i
\(378\) 0 0
\(379\) 500.315i 1.32009i −0.751225 0.660046i \(-0.770537\pi\)
0.751225 0.660046i \(-0.229463\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −477.384 195.523i −1.24970 0.511841i
\(383\) 393.159 1.02653 0.513263 0.858231i \(-0.328437\pi\)
0.513263 + 0.858231i \(0.328437\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −236.657 96.9283i −0.613101 0.251110i
\(387\) 0 0
\(388\) 3.26188 3.31406i 0.00840690 0.00854139i
\(389\) −98.7750 −0.253920 −0.126960 0.991908i \(-0.540522\pi\)
−0.126960 + 0.991908i \(0.540522\pi\)
\(390\) 0 0
\(391\) 179.959i 0.460252i
\(392\) 236.985 101.484i 0.604554 0.258887i
\(393\) 0 0
\(394\) −180.058 73.7468i −0.457000 0.187175i
\(395\) 0 0
\(396\) 0 0
\(397\) 199.225i 0.501826i 0.968010 + 0.250913i \(0.0807308\pi\)
−0.968010 + 0.250913i \(0.919269\pi\)
\(398\) 36.4328 + 14.9219i 0.0915397 + 0.0374922i
\(399\) 0 0
\(400\) 0 0
\(401\) 105.094 0.262079 0.131040 0.991377i \(-0.458169\pi\)
0.131040 + 0.991377i \(0.458169\pi\)
\(402\) 0 0
\(403\) −29.1462 −0.0723232
\(404\) −376.392 370.466i −0.931664 0.916995i
\(405\) 0 0
\(406\) −8.71252 + 21.2722i −0.0214594 + 0.0523946i
\(407\) −44.1008 −0.108356
\(408\) 0 0
\(409\) −368.163 −0.900153 −0.450076 0.892990i \(-0.648603\pi\)
−0.450076 + 0.892990i \(0.648603\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 318.150 + 313.141i 0.772209 + 0.760050i
\(413\) 275.813i 0.667827i
\(414\) 0 0
\(415\) 0 0
\(416\) 172.597 441.245i 0.414896 1.06068i
\(417\) 0 0
\(418\) −20.4194 + 49.8554i −0.0488503 + 0.119271i
\(419\) 47.0535i 0.112300i 0.998422 + 0.0561498i \(0.0178824\pi\)
−0.998422 + 0.0561498i \(0.982118\pi\)
\(420\) 0 0
\(421\) 7.81255 0.0185571 0.00927856 0.999957i \(-0.497047\pi\)
0.00927856 + 0.999957i \(0.497047\pi\)
\(422\) 435.486 + 178.363i 1.03196 + 0.422662i
\(423\) 0 0
\(424\) 280.989 + 656.167i 0.662710 + 1.54756i
\(425\) 0 0
\(426\) 0 0
\(427\) 147.574 0.345607
\(428\) −177.494 174.699i −0.414705 0.408176i
\(429\) 0 0
\(430\) 0 0
\(431\) 681.258i 1.58065i 0.612691 + 0.790323i \(0.290087\pi\)
−0.612691 + 0.790323i \(0.709913\pi\)
\(432\) 0 0
\(433\) 4.06878i 0.00939673i 0.999989 + 0.00469836i \(0.00149554\pi\)
−0.999989 + 0.00469836i \(0.998504\pi\)
\(434\) −14.9219 6.11161i −0.0343822 0.0140820i
\(435\) 0 0
\(436\) 248.570 + 244.657i 0.570115 + 0.561139i
\(437\) 502.250i 1.14931i
\(438\) 0 0
\(439\) 176.815i 0.402768i −0.979512 0.201384i \(-0.935456\pi\)
0.979512 0.201384i \(-0.0645439\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 110.061 268.722i 0.249007 0.607968i
\(443\) −469.454 −1.05972 −0.529858 0.848087i \(-0.677755\pi\)
−0.529858 + 0.848087i \(0.677755\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.5969 + 60.0549i −0.0551499 + 0.134652i
\(447\) 0 0
\(448\) 180.887 189.711i 0.403766 0.423462i
\(449\) 514.444 1.14575 0.572877 0.819641i \(-0.305827\pi\)
0.572877 + 0.819641i \(0.305827\pi\)
\(450\) 0 0
\(451\) 16.7323i 0.0371005i
\(452\) −236.326 + 240.107i −0.522846 + 0.531210i
\(453\) 0 0
\(454\) −219.519 + 535.970i −0.483522 + 1.18055i
\(455\) 0 0
\(456\) 0 0
\(457\) 684.537i 1.49789i 0.662630 + 0.748947i \(0.269440\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(458\) −75.2629 + 183.759i −0.164329 + 0.401221i
\(459\) 0 0
\(460\) 0 0
\(461\) −204.250 −0.443058 −0.221529 0.975154i \(-0.571105\pi\)
−0.221529 + 0.975154i \(0.571105\pi\)
\(462\) 0 0
\(463\) 550.353 1.18867 0.594334 0.804218i \(-0.297416\pi\)
0.594334 + 0.804218i \(0.297416\pi\)
\(464\) −0.712521 44.8943i −0.00153561 0.0967550i
\(465\) 0 0
\(466\) 93.9734 + 38.4890i 0.201660 + 0.0825944i
\(467\) −10.0013 −0.0214160 −0.0107080 0.999943i \(-0.503409\pi\)
−0.0107080 + 0.999943i \(0.503409\pi\)
\(468\) 0 0
\(469\) 104.681 0.223201
\(470\) 0 0
\(471\) 0 0
\(472\) −212.073 495.234i −0.449307 1.04923i
\(473\) 54.0312i 0.114231i
\(474\) 0 0
\(475\) 0 0
\(476\) 112.695 114.498i 0.236755 0.240542i
\(477\) 0 0
\(478\) −576.515 236.125i −1.20610 0.493985i
\(479\) 548.574i 1.14525i 0.819818 + 0.572625i \(0.194075\pi\)
−0.819818 + 0.572625i \(0.805925\pi\)
\(480\) 0 0
\(481\) −663.412 −1.37924
\(482\) 241.883 590.573i 0.501831 1.22525i
\(483\) 0 0
\(484\) −342.183 336.795i −0.706989 0.695858i
\(485\) 0 0
\(486\) 0 0
\(487\) 429.005 0.880913 0.440457 0.897774i \(-0.354817\pi\)
0.440457 + 0.897774i \(0.354817\pi\)
\(488\) −264.976 + 113.470i −0.542985 + 0.232521i
\(489\) 0 0
\(490\) 0 0
\(491\) 380.015i 0.773961i −0.922088 0.386980i \(-0.873518\pi\)
0.922088 0.386980i \(-0.126482\pi\)
\(492\) 0 0
\(493\) 27.5188i 0.0558190i
\(494\) −307.172 + 749.981i −0.621805 + 1.51818i
\(495\) 0 0
\(496\) 31.4922 0.499815i 0.0634923 0.00100769i
\(497\) 491.938i 0.989814i
\(498\) 0 0
\(499\) 395.763i 0.793112i 0.918011 + 0.396556i \(0.129795\pi\)
−0.918011 + 0.396556i \(0.870205\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −162.183 66.4257i −0.323073 0.132322i
\(503\) 325.436 0.646991 0.323495 0.946230i \(-0.395142\pi\)
0.323495 + 0.946230i \(0.395142\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 33.4297 + 13.6919i 0.0660666 + 0.0270591i
\(507\) 0 0
\(508\) 213.971 + 210.602i 0.421202 + 0.414570i
\(509\) 1.22499 0.00240667 0.00120333 0.999999i \(-0.499617\pi\)
0.00120333 + 0.999999i \(0.499617\pi\)
\(510\) 0 0
\(511\) 285.908i 0.559506i
\(512\) −178.922 + 479.720i −0.349458 + 0.936952i
\(513\) 0 0
\(514\) −217.073 88.9074i −0.422322 0.172972i
\(515\) 0 0
\(516\) 0 0
\(517\) 57.9063i 0.112004i
\(518\) −339.645 139.109i −0.655685 0.268551i
\(519\) 0 0
\(520\) 0 0
\(521\) −865.375 −1.66099 −0.830494 0.557027i \(-0.811942\pi\)
−0.830494 + 0.557027i \(0.811942\pi\)
\(522\) 0 0
\(523\) 295.370 0.564761 0.282380 0.959302i \(-0.408876\pi\)
0.282380 + 0.959302i \(0.408876\pi\)
\(524\) −392.250 + 398.525i −0.748569 + 0.760543i
\(525\) 0 0
\(526\) 36.2453 88.4953i 0.0689074 0.168242i
\(527\) 19.3037 0.0366294
\(528\) 0 0
\(529\) −192.225 −0.363374
\(530\) 0 0
\(531\) 0 0
\(532\) −314.523 + 319.555i −0.591209 + 0.600667i
\(533\) 251.706i 0.472244i
\(534\) 0 0
\(535\) 0 0
\(536\) −187.960 + 80.4897i −0.350672 + 0.150167i
\(537\) 0 0
\(538\) 363.314 887.056i 0.675305 1.64880i
\(539\) 31.7176i 0.0588453i
\(540\) 0 0
\(541\) −200.250 −0.370148 −0.185074 0.982725i \(-0.559252\pi\)
−0.185074 + 0.982725i \(0.559252\pi\)
\(542\) −448.237 183.586i −0.827006 0.338719i
\(543\) 0 0
\(544\) −114.312 + 292.238i −0.210132 + 0.537203i
\(545\) 0 0
\(546\) 0 0
\(547\) 989.581 1.80911 0.904553 0.426361i \(-0.140205\pi\)
0.904553 + 0.426361i \(0.140205\pi\)
\(548\) −157.586 + 160.107i −0.287566 + 0.292166i
\(549\) 0 0
\(550\) 0 0
\(551\) 76.8026i 0.139388i
\(552\) 0 0
\(553\) 132.250i 0.239150i
\(554\) 61.5500 + 25.2092i 0.111101 + 0.0455040i
\(555\) 0 0
\(556\) 545.301 554.024i 0.980757 0.996446i
\(557\) 641.319i 1.15138i −0.817668 0.575690i \(-0.804733\pi\)
0.817668 0.575690i \(-0.195267\pi\)
\(558\) 0 0
\(559\) 812.798i 1.45402i
\(560\) 0 0
\(561\) 0 0
\(562\) 186.523 455.408i 0.331891 0.810334i
\(563\) −382.237 −0.678928 −0.339464 0.940619i \(-0.610246\pi\)
−0.339464 + 0.940619i \(0.610246\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.15552 + 2.82127i −0.00204155 + 0.00498458i
\(567\) 0 0
\(568\) −378.252 883.297i −0.665937 1.55510i
\(569\) −576.025 −1.01235 −0.506173 0.862432i \(-0.668940\pi\)
−0.506173 + 0.862432i \(0.668940\pi\)
\(570\) 0 0
\(571\) 1100.67i 1.92762i 0.266582 + 0.963812i \(0.414106\pi\)
−0.266582 + 0.963812i \(0.585894\pi\)
\(572\) 41.5448 + 40.8907i 0.0726307 + 0.0714872i
\(573\) 0 0
\(574\) −52.7797 + 128.865i −0.0919506 + 0.224504i
\(575\) 0 0
\(576\) 0 0
\(577\) 326.475i 0.565814i 0.959147 + 0.282907i \(0.0912989\pi\)
−0.959147 + 0.282907i \(0.908701\pi\)
\(578\) 146.177 356.900i 0.252901 0.617474i
\(579\) 0 0
\(580\) 0 0
\(581\) −279.194 −0.480540
\(582\) 0 0
\(583\) −87.8201 −0.150635
\(584\) 219.835 + 513.360i 0.376430 + 0.879042i
\(585\) 0 0
\(586\) 830.989 + 340.351i 1.41807 + 0.580803i
\(587\) −90.5502 −0.154259 −0.0771296 0.997021i \(-0.524576\pi\)
−0.0771296 + 0.997021i \(0.524576\pi\)
\(588\) 0 0
\(589\) −53.8750 −0.0914686
\(590\) 0 0
\(591\) 0 0
\(592\) 716.810 11.3765i 1.21083 0.0192171i
\(593\) 896.163i 1.51124i −0.655013 0.755618i \(-0.727337\pi\)
0.655013 0.755618i \(-0.272663\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 316.062 + 311.086i 0.530306 + 0.521956i
\(597\) 0 0
\(598\) 502.887 + 205.969i 0.840947 + 0.344429i
\(599\) 1157.76i 1.93282i 0.257001 + 0.966411i \(0.417266\pi\)
−0.257001 + 0.966411i \(0.582734\pi\)
\(600\) 0 0
\(601\) −438.969 −0.730397 −0.365199 0.930930i \(-0.618999\pi\)
−0.365199 + 0.930930i \(0.618999\pi\)
\(602\) −170.434 + 416.125i −0.283112 + 0.691237i
\(603\) 0 0
\(604\) −397.773 + 404.137i −0.658565 + 0.669100i
\(605\) 0 0
\(606\) 0 0
\(607\) −1156.84 −1.90583 −0.952915 0.303237i \(-0.901933\pi\)
−0.952915 + 0.303237i \(0.901933\pi\)
\(608\) 319.035 815.613i 0.524728 1.34147i
\(609\) 0 0
\(610\) 0 0
\(611\) 871.090i 1.42568i
\(612\) 0 0
\(613\) 409.225i 0.667577i 0.942648 + 0.333789i \(0.108327\pi\)
−0.942648 + 0.333789i \(0.891673\pi\)
\(614\) −8.08673 + 19.7443i −0.0131706 + 0.0321568i
\(615\) 0 0
\(616\) 12.6953 + 29.6461i 0.0206092 + 0.0481267i
\(617\) 999.150i 1.61937i 0.586866 + 0.809684i \(0.300361\pi\)
−0.586866 + 0.809684i \(0.699639\pi\)
\(618\) 0 0
\(619\) 266.604i 0.430701i −0.976537 0.215350i \(-0.930911\pi\)
0.976537 0.215350i \(-0.0690894\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 685.107 + 280.602i 1.10146 + 0.451128i
\(623\) −277.716 −0.445772
\(624\) 0 0
\(625\) 0 0
\(626\) −79.0515 32.3774i −0.126280 0.0517210i
\(627\) 0 0
\(628\) −501.799 + 509.827i −0.799043 + 0.811826i
\(629\) 439.381 0.698539
\(630\) 0 0
\(631\) 1053.40i 1.66941i 0.550696 + 0.834706i \(0.314362\pi\)
−0.550696 + 0.834706i \(0.685638\pi\)
\(632\) −101.687 237.461i −0.160898 0.375729i
\(633\) 0 0
\(634\) −264.847 108.474i −0.417739 0.171095i
\(635\) 0 0
\(636\) 0 0
\(637\) 477.131i 0.749029i
\(638\) 5.11198 + 2.09373i 0.00801250 + 0.00328170i
\(639\) 0 0
\(640\) 0 0
\(641\) −686.187 −1.07050 −0.535248 0.844695i \(-0.679782\pi\)
−0.535248 + 0.844695i \(0.679782\pi\)
\(642\) 0 0
\(643\) 454.690 0.707138 0.353569 0.935408i \(-0.384968\pi\)
0.353569 + 0.935408i \(0.384968\pi\)
\(644\) 214.272 + 210.898i 0.332720 + 0.327482i
\(645\) 0 0
\(646\) 203.441 496.716i 0.314925 0.768910i
\(647\) 928.430 1.43498 0.717489 0.696570i \(-0.245292\pi\)
0.717489 + 0.696570i \(0.245292\pi\)
\(648\) 0 0
\(649\) 66.2812 0.102128
\(650\) 0 0
\(651\) 0 0
\(652\) −634.399 624.410i −0.973004 0.957684i
\(653\) 536.744i 0.821966i 0.911643 + 0.410983i \(0.134814\pi\)
−0.911643 + 0.410983i \(0.865186\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.31639 271.966i −0.00657986 0.414582i
\(657\) 0 0
\(658\) −182.657 + 445.969i −0.277594 + 0.677764i
\(659\) 967.387i 1.46796i −0.679170 0.733981i \(-0.737660\pi\)
0.679170 0.733981i \(-0.262340\pi\)
\(660\) 0 0
\(661\) −534.281 −0.808292 −0.404146 0.914694i \(-0.632431\pi\)
−0.404146 + 0.914694i \(0.632431\pi\)
\(662\) 96.5469 + 39.5430i 0.145841 + 0.0597327i
\(663\) 0 0
\(664\) 501.305 214.673i 0.754978 0.323303i
\(665\) 0 0
\(666\) 0 0
\(667\) 51.4987 0.0772094
\(668\) −606.884 597.328i −0.908508 0.894204i
\(669\) 0 0
\(670\) 0 0
\(671\) 35.4639i 0.0528523i
\(672\) 0 0
\(673\) 129.225i 0.192013i 0.995381 + 0.0960067i \(0.0306070\pi\)
−0.995381 + 0.0960067i \(0.969393\pi\)
\(674\) −817.860 334.973i −1.21344 0.496993i
\(675\) 0 0
\(676\) 143.180 + 140.926i 0.211805 + 0.208471i
\(677\) 111.163i 0.164199i −0.996624 0.0820993i \(-0.973838\pi\)
0.996624 0.0820993i \(-0.0261625\pi\)
\(678\) 0 0
\(679\) 4.76132i 0.00701226i
\(680\) 0 0
\(681\) 0 0
\(682\) −1.46869 + 3.58592i −0.00215351 + 0.00525794i
\(683\) −710.467 −1.04022 −0.520108 0.854101i \(-0.674108\pi\)
−0.520108 + 0.854101i \(0.674108\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −252.178 + 615.710i −0.367607 + 0.897536i
\(687\) 0 0
\(688\) −13.9383 878.219i −0.0202591 1.27648i
\(689\) −1321.09 −1.91740
\(690\) 0 0
\(691\) 64.0073i 0.0926300i −0.998927 0.0463150i \(-0.985252\pi\)
0.998927 0.0463150i \(-0.0147478\pi\)
\(692\) 58.2925 59.2250i 0.0842377 0.0855853i
\(693\) 0 0
\(694\) −65.0539 + 158.833i −0.0937375 + 0.228867i
\(695\) 0 0
\(696\) 0 0
\(697\) 166.706i 0.239177i
\(698\) 309.982 756.842i 0.444100 1.08430i
\(699\) 0 0
\(700\) 0 0
\(701\) 122.031 0.174082 0.0870408 0.996205i \(-0.472259\pi\)
0.0870408 + 0.996205i \(0.472259\pi\)
\(702\) 0 0
\(703\) −1226.28 −1.74435
\(704\) −45.5899 43.4695i −0.0647584 0.0617464i
\(705\) 0 0
\(706\) −816.495 334.414i −1.15651 0.473675i
\(707\) 540.764 0.764872
\(708\) 0 0
\(709\) 1349.09 1.90280 0.951402 0.307953i \(-0.0996440\pi\)
0.951402 + 0.307953i \(0.0996440\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 498.652 213.537i 0.700354 0.299911i
\(713\) 36.1250i 0.0506662i
\(714\) 0 0
\(715\) 0 0
\(716\) −104.410 + 106.080i −0.145824 + 0.148157i
\(717\) 0 0
\(718\) −797.343 326.570i −1.11051 0.454833i
\(719\) 1101.82i 1.53243i −0.642584 0.766216i \(-0.722138\pi\)
0.642584 0.766216i \(-0.277862\pi\)
\(720\) 0 0
\(721\) −457.087 −0.633963
\(722\) −294.139 + 718.161i −0.407395 + 0.994683i
\(723\) 0 0
\(724\) −906.905 892.625i −1.25263 1.23291i
\(725\) 0 0
\(726\) 0 0
\(727\) −31.7188 −0.0436298 −0.0218149 0.999762i \(-0.506944\pi\)
−0.0218149 + 0.999762i \(0.506944\pi\)
\(728\) 190.976 + 445.969i 0.262330 + 0.612594i
\(729\) 0 0
\(730\) 0 0
\(731\) 538.320i 0.736415i
\(732\) 0 0
\(733\) 121.319i 0.165510i 0.996570 + 0.0827549i \(0.0263719\pi\)
−0.996570 + 0.0827549i \(0.973628\pi\)
\(734\) 262.816 641.682i 0.358059 0.874226i
\(735\) 0 0
\(736\) −546.895 213.923i −0.743064 0.290656i
\(737\) 25.1562i 0.0341333i
\(738\) 0 0
\(739\) 311.498i 0.421513i 0.977539 + 0.210757i \(0.0675927\pi\)
−0.977539 + 0.210757i \(0.932407\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −676.352 277.016i −0.911526 0.373336i
\(743\) 298.259 0.401425 0.200712 0.979650i \(-0.435674\pi\)
0.200712 + 0.979650i \(0.435674\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1068.43 + 437.601i 1.43222 + 0.586597i
\(747\) 0 0
\(748\) −27.5153 27.0821i −0.0367852 0.0362060i
\(749\) 255.006 0.340462
\(750\) 0 0
\(751\) 580.864i 0.773454i −0.922194 0.386727i \(-0.873606\pi\)
0.922194 0.386727i \(-0.126394\pi\)
\(752\) −14.9379 941.203i −0.0198642 1.25160i
\(753\) 0 0
\(754\) 76.9000 + 31.4962i 0.101989 + 0.0417721i
\(755\) 0 0
\(756\) 0 0
\(757\) 507.600i 0.670541i 0.942122 + 0.335271i \(0.108828\pi\)
−0.942122 + 0.335271i \(0.891172\pi\)
\(758\) −925.974 379.254i −1.22160 0.500335i
\(759\) 0 0
\(760\) 0 0
\(761\) −240.875 −0.316524 −0.158262 0.987397i \(-0.550589\pi\)
−0.158262 + 0.987397i \(0.550589\pi\)
\(762\) 0 0
\(763\) −357.122 −0.468050
\(764\) −723.742 + 735.320i −0.947306 + 0.962461i
\(765\) 0 0
\(766\) 298.027 727.652i 0.389069 0.949937i
\(767\) 997.075 1.29997
\(768\) 0 0
\(769\) 161.681 0.210249 0.105124 0.994459i \(-0.466476\pi\)
0.105124 + 0.994459i \(0.466476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −358.786 + 364.526i −0.464749 + 0.472184i
\(773\) 545.506i 0.705700i −0.935680 0.352850i \(-0.885213\pi\)
0.935680 0.352850i \(-0.114787\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.66100 8.54918i −0.00471778 0.0110170i
\(777\) 0 0
\(778\) −74.8744 + 182.811i −0.0962396 + 0.234975i
\(779\) 465.263i 0.597257i
\(780\) 0 0
\(781\) 118.219 0.151368
\(782\) −333.064 136.414i −0.425913 0.174443i
\(783\) 0 0
\(784\) −8.18210 515.535i −0.0104363 0.657570i
\(785\) 0 0
\(786\) 0 0
\(787\) −953.227 −1.21122 −0.605608 0.795763i \(-0.707070\pi\)
−0.605608 + 0.795763i \(0.707070\pi\)
\(788\) −272.978 + 277.345i −0.346419 + 0.351961i
\(789\) 0 0
\(790\) 0 0
\(791\) 344.963i 0.436110i
\(792\) 0 0
\(793\) 533.488i 0.672746i
\(794\) 368.722 + 151.019i 0.464385 + 0.190200i
\(795\) 0 0
\(796\) 55.2343 56.1179i 0.0693898 0.0704999i
\(797\) 456.975i 0.573369i −0.958025 0.286684i \(-0.907447\pi\)
0.958025 0.286684i \(-0.0925531\pi\)
\(798\) 0 0
\(799\) 576.927i 0.722061i
\(800\) 0 0
\(801\) 0 0
\(802\) 79.6642 194.505i 0.0993319 0.242526i
\(803\) −68.7071 −0.0855631
\(804\) 0 0
\(805\) 0 0
\(806\) −22.0937 + 53.9433i −0.0274116 + 0.0669272i
\(807\) 0 0
\(808\) −970.968 + 415.795i −1.20169 + 0.514598i
\(809\) 1073.79 1.32730 0.663651 0.748042i \(-0.269006\pi\)
0.663651 + 0.748042i \(0.269006\pi\)
\(810\) 0 0
\(811\) 108.299i 0.133537i 0.997768 + 0.0667687i \(0.0212690\pi\)
−0.997768 + 0.0667687i \(0.978731\pi\)
\(812\) 32.7658 + 32.2499i 0.0403520 + 0.0397167i
\(813\) 0 0
\(814\) −33.4297 + 81.6209i −0.0410684 + 0.100271i
\(815\) 0 0
\(816\) 0 0
\(817\) 1502.41i 1.83893i
\(818\) −279.078 + 681.388i −0.341171 + 0.832993i
\(819\) 0 0
\(820\) 0 0
\(821\) 1262.19 1.53738 0.768689 0.639623i \(-0.220909\pi\)
0.768689 + 0.639623i \(0.220909\pi\)
\(822\) 0 0
\(823\) 862.264 1.04771 0.523854 0.851808i \(-0.324494\pi\)
0.523854 + 0.851808i \(0.324494\pi\)
\(824\) 820.722 351.456i 0.996022 0.426524i
\(825\) 0 0
\(826\) 510.469 + 209.074i 0.618001 + 0.253116i
\(827\) −720.596 −0.871338 −0.435669 0.900107i \(-0.643488\pi\)
−0.435669 + 0.900107i \(0.643488\pi\)
\(828\) 0 0
\(829\) −385.569 −0.465101 −0.232550 0.972584i \(-0.574707\pi\)
−0.232550 + 0.972584i \(0.574707\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −685.813 653.916i −0.824295 0.785956i
\(833\) 316.006i 0.379359i
\(834\) 0 0
\(835\) 0 0
\(836\) 76.7930 + 75.5838i 0.0918576 + 0.0904113i
\(837\) 0 0
\(838\) 87.0858 + 35.6680i 0.103921 + 0.0425632i
\(839\) 561.530i 0.669284i −0.942345 0.334642i \(-0.891385\pi\)
0.942345 0.334642i \(-0.108615\pi\)
\(840\) 0 0
\(841\) −833.125 −0.990636
\(842\) 5.92214 14.4593i 0.00703343 0.0171726i
\(843\) 0 0
\(844\) 660.223 670.784i 0.782254 0.794768i
\(845\) 0 0
\(846\) 0 0
\(847\) 491.616 0.580420
\(848\) 1427.42 22.6547i 1.68328 0.0267154i
\(849\) 0 0
\(850\) 0 0
\(851\) 822.259i 0.966227i
\(852\) 0 0
\(853\) 151.087i 0.177125i −0.996071 0.0885624i \(-0.971773\pi\)
0.996071 0.0885624i \(-0.0282273\pi\)
\(854\) 111.866 273.128i 0.130990 0.319822i
\(855\) 0 0
\(856\) −457.876 + 196.075i −0.534902 + 0.229060i
\(857\) 444.225i 0.518349i −0.965831 0.259174i \(-0.916550\pi\)
0.965831 0.259174i \(-0.0834505\pi\)
\(858\) 0 0
\(859\) 731.867i 0.851999i −0.904723 0.426000i \(-0.859923\pi\)
0.904723 0.426000i \(-0.140077\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1260.86 + 516.414i 1.46271 + 0.599088i
\(863\) −965.801 −1.11912 −0.559560 0.828790i \(-0.689030\pi\)
−0.559560 + 0.828790i \(0.689030\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 7.53042 + 3.08426i 0.00869564 + 0.00356150i
\(867\) 0 0
\(868\) −22.6225 + 22.9844i −0.0260628 + 0.0264797i
\(869\) 31.7813 0.0365723
\(870\) 0 0
\(871\) 378.428i 0.434475i
\(872\) 641.229 274.592i 0.735355 0.314899i
\(873\) 0 0
\(874\) 929.555 + 380.721i 1.06356 + 0.435607i
\(875\) 0 0
\(876\) 0 0
\(877\) 1440.78i 1.64284i −0.570320 0.821422i \(-0.693181\pi\)
0.570320 0.821422i \(-0.306819\pi\)
\(878\) −327.246 134.031i −0.372718 0.152655i
\(879\) 0 0
\(880\) 0 0
\(881\) −1538.44 −1.74624 −0.873120 0.487505i \(-0.837907\pi\)
−0.873120 + 0.487505i \(0.837907\pi\)
\(882\) 0 0
\(883\) 229.616 0.260040 0.130020 0.991511i \(-0.458496\pi\)
0.130020 + 0.991511i \(0.458496\pi\)
\(884\) −413.916 407.398i −0.468230 0.460858i
\(885\) 0 0
\(886\) −355.860 + 868.856i −0.401648 + 0.980650i
\(887\) −456.945 −0.515158 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(888\) 0 0
\(889\) −307.412 −0.345796
\(890\) 0 0
\(891\) 0 0
\(892\) 92.5033 + 91.0469i 0.103703 + 0.102070i
\(893\) 1610.16i 1.80309i
\(894\) 0 0
\(895\) 0 0
\(896\) −213.995 478.589i −0.238834 0.534140i
\(897\) 0 0
\(898\) 389.964 952.123i 0.434258 1.06027i
\(899\) 5.52413i 0.00614475i
\(900\) 0 0
\(901\) 874.962 0.971102
\(902\) 30.9679 + 12.6836i 0.0343325 + 0.0140617i
\(903\) 0 0
\(904\) 265.243 + 619.397i 0.293410 + 0.685174i
\(905\) 0 0
\(906\) 0 0
\(907\) −797.272 −0.879021 −0.439511 0.898237i \(-0.644848\pi\)
−0.439511 + 0.898237i \(0.644848\pi\)
\(908\) 825.561 + 812.562i 0.909208 + 0.894892i
\(909\) 0 0
\(910\) 0 0
\(911\) 531.621i 0.583557i 0.956486 + 0.291779i \(0.0942470\pi\)
−0.956486 + 0.291779i \(0.905753\pi\)
\(912\) 0 0
\(913\) 67.0937i 0.0734871i
\(914\) 1266.93 + 518.900i 1.38614 + 0.567724i
\(915\) 0 0
\(916\) 283.047 + 278.590i 0.309003 + 0.304138i
\(917\) 572.562i 0.624386i
\(918\) 0 0
\(919\) 593.469i 0.645777i −0.946437 0.322888i \(-0.895346\pi\)
0.946437 0.322888i \(-0.104654\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −154.828 + 378.022i −0.167926 + 0.410002i
\(923\) 1778.38 1.92673
\(924\) 0 0
\(925\) 0 0
\(926\) 417.184 1018.58i 0.450523 1.09998i
\(927\) 0 0
\(928\) −83.6297 32.7125i −0.0901182 0.0352506i
\(929\) −198.775 −0.213967 −0.106983 0.994261i \(-0.534119\pi\)
−0.106983 + 0.994261i \(0.534119\pi\)
\(930\) 0 0
\(931\) 881.948i 0.947312i
\(932\) 142.469 144.748i 0.152864 0.155309i
\(933\) 0 0
\(934\) −7.58125 + 18.5101i −0.00811697 + 0.0198181i
\(935\) 0 0
\(936\) 0 0
\(937\) 1200.43i 1.28114i −0.767898 0.640572i \(-0.778697\pi\)
0.767898 0.640572i \(-0.221303\pi\)
\(938\) 79.3515 193.742i 0.0845965 0.206548i
\(939\) 0 0
\(940\) 0 0
\(941\) 1338.16 1.42206 0.711029 0.703163i \(-0.248230\pi\)
0.711029 + 0.703163i \(0.248230\pi\)
\(942\) 0 0
\(943\) 311.974 0.330832
\(944\) −1077.33 + 17.0984i −1.14124 + 0.0181127i
\(945\) 0 0
\(946\) 100.000 + 40.9573i 0.105708 + 0.0432952i
\(947\) −586.516 −0.619341 −0.309671 0.950844i \(-0.600219\pi\)
−0.309671 + 0.950844i \(0.600219\pi\)
\(948\) 0 0
\(949\) −1033.57 −1.08911
\(950\) 0 0
\(951\) 0 0
\(952\) −126.484 295.367i −0.132862 0.310260i
\(953\) 1598.72i 1.67757i −0.544462 0.838785i \(-0.683266\pi\)
0.544462 0.838785i \(-0.316734\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −874.031 + 888.013i −0.914259 + 0.928884i
\(957\) 0 0
\(958\) 1015.29 + 415.836i 1.05980 + 0.434067i
\(959\) 230.026i 0.239861i
\(960\) 0 0
\(961\) 957.125 0.995968
\(962\) −502.887 + 1227.83i −0.522751 + 1.27633i
\(963\) 0 0
\(964\) −909.666 895.344i −0.943637 0.928780i
\(965\) 0 0
\(966\) 0 0
\(967\) 320.675 0.331619 0.165809 0.986158i \(-0.446976\pi\)
0.165809 + 0.986158i \(0.446976\pi\)
\(968\) −882.719 + 378.005i −0.911900 + 0.390501i
\(969\) 0 0
\(970\) 0 0
\(971\) 1402.93i 1.44483i −0.691459 0.722416i \(-0.743032\pi\)
0.691459 0.722416i \(-0.256968\pi\)
\(972\) 0 0
\(973\) 795.969i 0.818056i
\(974\) 325.198 793.994i 0.333879 0.815189i
\(975\) 0 0
\(976\) 9.14852 + 576.427i 0.00937349 + 0.590602i
\(977\) 1226.24i 1.25511i 0.778572 + 0.627556i \(0.215944\pi\)
−0.778572 + 0.627556i \(0.784056\pi\)
\(978\) 0 0
\(979\) 66.7386i 0.0681702i
\(980\) 0 0
\(981\) 0 0
\(982\) −703.324 288.062i −0.716216 0.293343i
\(983\) 1372.96 1.39670 0.698352 0.715754i \(-0.253917\pi\)
0.698352 + 0.715754i \(0.253917\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −50.9312 20.8601i −0.0516544 0.0211562i
\(987\) 0 0
\(988\) 1155.20 + 1137.02i 1.16924 + 1.15083i
\(989\) 1007.41 1.01862
\(990\) 0 0
\(991\) 362.298i 0.365588i −0.983151 0.182794i \(-0.941486\pi\)
0.983151 0.182794i \(-0.0585142\pi\)
\(992\) 22.9470 58.6640i 0.0231320 0.0591371i
\(993\) 0 0
\(994\) 910.469 + 372.903i 0.915964 + 0.375154i
\(995\) 0 0
\(996\) 0 0
\(997\) 941.087i 0.943919i −0.881620 0.471960i \(-0.843547\pi\)
0.881620 0.471960i \(-0.156453\pi\)
\(998\) 732.470 + 300.000i 0.733938 + 0.300601i
\(999\) 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 900.3.f.d.199.5 8
3.2 odd 2 100.3.d.a.99.4 8
4.3 odd 2 inner 900.3.f.d.199.3 8
5.2 odd 4 900.3.c.m.451.4 4
5.3 odd 4 900.3.c.l.451.1 4
5.4 even 2 inner 900.3.f.d.199.4 8
12.11 even 2 100.3.d.a.99.6 8
15.2 even 4 100.3.b.d.51.1 4
15.8 even 4 100.3.b.e.51.4 yes 4
15.14 odd 2 100.3.d.a.99.5 8
20.3 even 4 900.3.c.l.451.2 4
20.7 even 4 900.3.c.m.451.3 4
20.19 odd 2 inner 900.3.f.d.199.6 8
24.5 odd 2 1600.3.h.o.1599.7 8
24.11 even 2 1600.3.h.o.1599.2 8
60.23 odd 4 100.3.b.e.51.3 yes 4
60.47 odd 4 100.3.b.d.51.2 yes 4
60.59 even 2 100.3.d.a.99.3 8
120.29 odd 2 1600.3.h.o.1599.1 8
120.53 even 4 1600.3.b.o.1151.4 4
120.59 even 2 1600.3.h.o.1599.8 8
120.77 even 4 1600.3.b.p.1151.1 4
120.83 odd 4 1600.3.b.o.1151.1 4
120.107 odd 4 1600.3.b.p.1151.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.3.b.d.51.1 4 15.2 even 4
100.3.b.d.51.2 yes 4 60.47 odd 4
100.3.b.e.51.3 yes 4 60.23 odd 4
100.3.b.e.51.4 yes 4 15.8 even 4
100.3.d.a.99.3 8 60.59 even 2
100.3.d.a.99.4 8 3.2 odd 2
100.3.d.a.99.5 8 15.14 odd 2
100.3.d.a.99.6 8 12.11 even 2
900.3.c.l.451.1 4 5.3 odd 4
900.3.c.l.451.2 4 20.3 even 4
900.3.c.m.451.3 4 20.7 even 4
900.3.c.m.451.4 4 5.2 odd 4
900.3.f.d.199.3 8 4.3 odd 2 inner
900.3.f.d.199.4 8 5.4 even 2 inner
900.3.f.d.199.5 8 1.1 even 1 trivial
900.3.f.d.199.6 8 20.19 odd 2 inner
1600.3.b.o.1151.1 4 120.83 odd 4
1600.3.b.o.1151.4 4 120.53 even 4
1600.3.b.p.1151.1 4 120.77 even 4
1600.3.b.p.1151.4 4 120.107 odd 4
1600.3.h.o.1599.1 8 120.29 odd 2
1600.3.h.o.1599.2 8 24.11 even 2
1600.3.h.o.1599.7 8 24.5 odd 2
1600.3.h.o.1599.8 8 120.59 even 2