Properties

Label 900.3.c.m.451.4
Level $900$
Weight $3$
Character 900.451
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(451,900)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
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.c (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: 4.0.8405.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 8x^{2} - 7x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 451.4
Root \(0.500000 + 2.53999i\) of defining polynomial
Character \(\chi\) \(=\) 900.451
Dual form 900.3.c.m.451.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.85078 + 0.758030i) q^{2} +(2.85078 + 2.80590i) q^{4} +4.09573i q^{7} +(3.14922 + 7.35408i) q^{8} +O(q^{10})\) \(q+(1.85078 + 0.758030i) q^{2} +(2.85078 + 2.80590i) q^{4} +4.09573i q^{7} +(3.14922 + 7.35408i) q^{8} -0.984255i q^{11} -14.8062 q^{13} +(-3.10469 + 7.58030i) q^{14} +(0.253905 + 15.9980i) q^{16} -9.80625 q^{17} +27.3684i q^{19} +(0.746095 - 1.82164i) q^{22} +18.3514i q^{23} +(-27.4031 - 11.2236i) q^{26} +(-11.4922 + 11.6760i) q^{28} +2.80625 q^{29} -1.96851i q^{31} +(-11.6570 + 29.8012i) q^{32} +(-18.1492 - 7.43343i) q^{34} +44.8062 q^{37} +(-20.7461 + 50.6530i) q^{38} -17.0000 q^{41} +54.8956i q^{43} +(2.76172 - 2.80590i) q^{44} +(-13.9109 + 33.9645i) q^{46} -58.8326i q^{47} +32.2250 q^{49} +(-42.2094 - 41.5448i) q^{52} -89.2250 q^{53} +(-30.1203 + 12.8984i) q^{56} +(5.19375 + 2.12722i) q^{58} -67.3415i q^{59} +36.0312 q^{61} +(1.49219 - 3.64328i) q^{62} +(-44.1648 + 46.3192i) q^{64} +25.5586i q^{67} +(-27.9555 - 27.5153i) q^{68} +120.110i q^{71} -69.8062 q^{73} +(82.9266 + 33.9645i) q^{74} +(-76.7930 + 78.0214i) q^{76} +4.03124 q^{77} -32.2897i q^{79} +(-31.4633 - 12.8865i) q^{82} +68.1670i q^{83} +(-41.6125 + 101.600i) q^{86} +(7.23828 - 3.09963i) q^{88} +67.8062 q^{89} -60.6424i q^{91} +(-51.4922 + 52.3159i) q^{92} +(44.5969 - 108.886i) q^{94} -1.16251 q^{97} +(59.6414 + 24.4275i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 5 q^{4} + 19 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 5 q^{4} + 19 q^{8} - 8 q^{13} + 26 q^{14} - 31 q^{16} + 12 q^{17} + 35 q^{22} - 84 q^{26} - 110 q^{28} - 40 q^{29} + 11 q^{32} - 79 q^{34} + 128 q^{37} - 115 q^{38} - 68 q^{41} - 85 q^{44} + 34 q^{46} - 76 q^{49} - 92 q^{52} - 152 q^{53} + 46 q^{56} + 72 q^{58} - 112 q^{61} + 70 q^{62} - 55 q^{64} - 67 q^{68} - 228 q^{73} + 114 q^{74} + 45 q^{76} - 240 q^{77} - 17 q^{82} - 64 q^{86} + 125 q^{88} + 220 q^{89} - 270 q^{92} + 204 q^{94} - 312 q^{97} + 309 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.85078 + 0.758030i 0.925391 + 0.379015i
\(3\) 0 0
\(4\) 2.85078 + 2.80590i 0.712695 + 0.701474i
\(5\) 0 0
\(6\) 0 0
\(7\) 4.09573i 0.585104i 0.956250 + 0.292552i \(0.0945045\pi\)
−0.956250 + 0.292552i \(0.905495\pi\)
\(8\) 3.14922 + 7.35408i 0.393652 + 0.919259i
\(9\) 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
\(13\) −14.8062 −1.13894 −0.569471 0.822011i \(-0.692852\pi\)
−0.569471 + 0.822011i \(0.692852\pi\)
\(14\) −3.10469 + 7.58030i −0.221763 + 0.541450i
\(15\) 0 0
\(16\) 0.253905 + 15.9980i 0.0158691 + 0.999874i
\(17\) −9.80625 −0.576838 −0.288419 0.957504i \(-0.593130\pi\)
−0.288419 + 0.957504i \(0.593130\pi\)
\(18\) 0 0
\(19\) 27.3684i 1.44044i 0.693744 + 0.720222i \(0.255960\pi\)
−0.693744 + 0.720222i \(0.744040\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.746095 1.82164i 0.0339134 0.0828018i
\(23\) 18.3514i 0.797888i 0.916975 + 0.398944i \(0.130623\pi\)
−0.916975 + 0.398944i \(0.869377\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −27.4031 11.2236i −1.05397 0.431676i
\(27\) 0 0
\(28\) −11.4922 + 11.6760i −0.410435 + 0.417001i
\(29\) 2.80625 0.0967672 0.0483836 0.998829i \(-0.484593\pi\)
0.0483836 + 0.998829i \(0.484593\pi\)
\(30\) 0 0
\(31\) 1.96851i 0.0635003i −0.999496 0.0317502i \(-0.989892\pi\)
0.999496 0.0317502i \(-0.0101081\pi\)
\(32\) −11.6570 + 29.8012i −0.364282 + 0.931289i
\(33\) 0 0
\(34\) −18.1492 7.43343i −0.533801 0.218630i
\(35\) 0 0
\(36\) 0 0
\(37\) 44.8062 1.21098 0.605490 0.795853i \(-0.292977\pi\)
0.605490 + 0.795853i \(0.292977\pi\)
\(38\) −20.7461 + 50.6530i −0.545950 + 1.33297i
\(39\) 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
\(43\) 54.8956i 1.27664i 0.769771 + 0.638321i \(0.220371\pi\)
−0.769771 + 0.638321i \(0.779629\pi\)
\(44\) 2.76172 2.80590i 0.0627663 0.0637703i
\(45\) 0 0
\(46\) −13.9109 + 33.9645i −0.302412 + 0.738358i
\(47\) 58.8326i 1.25176i −0.779920 0.625879i \(-0.784740\pi\)
0.779920 0.625879i \(-0.215260\pi\)
\(48\) 0 0
\(49\) 32.2250 0.657653
\(50\) 0 0
\(51\) 0 0
\(52\) −42.2094 41.5448i −0.811719 0.798938i
\(53\) −89.2250 −1.68349 −0.841745 0.539875i \(-0.818472\pi\)
−0.841745 + 0.539875i \(0.818472\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −30.1203 + 12.8984i −0.537863 + 0.230328i
\(57\) 0 0
\(58\) 5.19375 + 2.12722i 0.0895474 + 0.0366762i
\(59\) 67.3415i 1.14138i −0.821165 0.570691i \(-0.806676\pi\)
0.821165 0.570691i \(-0.193324\pi\)
\(60\) 0 0
\(61\) 36.0312 0.590676 0.295338 0.955393i \(-0.404568\pi\)
0.295338 + 0.955393i \(0.404568\pi\)
\(62\) 1.49219 3.64328i 0.0240676 0.0587626i
\(63\) 0 0
\(64\) −44.1648 + 46.3192i −0.690076 + 0.723737i
\(65\) 0 0
\(66\) 0 0
\(67\) 25.5586i 0.381472i 0.981641 + 0.190736i \(0.0610875\pi\)
−0.981641 + 0.190736i \(0.938913\pi\)
\(68\) −27.9555 27.5153i −0.411110 0.404637i
\(69\) 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
\(73\) −69.8062 −0.956250 −0.478125 0.878292i \(-0.658683\pi\)
−0.478125 + 0.878292i \(0.658683\pi\)
\(74\) 82.9266 + 33.9645i 1.12063 + 0.458979i
\(75\) 0 0
\(76\) −76.7930 + 78.0214i −1.01043 + 1.02660i
\(77\) 4.03124 0.0523538
\(78\) 0 0
\(79\) 32.2897i 0.408730i −0.978895 0.204365i \(-0.934487\pi\)
0.978895 0.204365i \(-0.0655130\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −31.4633 12.8865i −0.383699 0.157153i
\(83\) 68.1670i 0.821289i 0.911795 + 0.410645i \(0.134696\pi\)
−0.911795 + 0.410645i \(0.865304\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −41.6125 + 101.600i −0.483866 + 1.18139i
\(87\) 0 0
\(88\) 7.23828 3.09963i 0.0822532 0.0352231i
\(89\) 67.8062 0.761868 0.380934 0.924602i \(-0.375603\pi\)
0.380934 + 0.924602i \(0.375603\pi\)
\(90\) 0 0
\(91\) 60.6424i 0.666400i
\(92\) −51.4922 + 52.3159i −0.559698 + 0.568651i
\(93\) 0 0
\(94\) 44.5969 108.886i 0.474435 1.15836i
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16251 −0.0119846 −0.00599232 0.999982i \(-0.501907\pi\)
−0.00599232 + 0.999982i \(0.501907\pi\)
\(98\) 59.6414 + 24.4275i 0.608586 + 0.249260i
\(99\) 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
\(103\) 111.601i 1.08350i 0.840538 + 0.541752i \(0.182239\pi\)
−0.840538 + 0.541752i \(0.817761\pi\)
\(104\) −46.6281 108.886i −0.448347 1.04698i
\(105\) 0 0
\(106\) −165.136 67.6352i −1.55789 0.638068i
\(107\) 62.2615i 0.581883i 0.956741 + 0.290942i \(0.0939685\pi\)
−0.956741 + 0.290942i \(0.906031\pi\)
\(108\) 0 0
\(109\) 87.1938 0.799943 0.399971 0.916528i \(-0.369020\pi\)
0.399971 + 0.916528i \(0.369020\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −65.5234 + 1.03993i −0.585031 + 0.00928507i
\(113\) −84.2250 −0.745354 −0.372677 0.927961i \(-0.621560\pi\)
−0.372677 + 0.927961i \(0.621560\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.00000 + 7.87404i 0.0689655 + 0.0678796i
\(117\) 0 0
\(118\) 51.0469 124.634i 0.432601 1.05622i
\(119\) 40.1637i 0.337510i
\(120\) 0 0
\(121\) 120.031 0.991994
\(122\) 66.6859 + 27.3128i 0.546606 + 0.223875i
\(123\) 0 0
\(124\) 5.52343 5.61179i 0.0445438 0.0452564i
\(125\) 0 0
\(126\) 0 0
\(127\) 75.0568i 0.590999i −0.955343 0.295499i \(-0.904514\pi\)
0.955343 0.295499i \(-0.0954860\pi\)
\(128\) −116.851 + 52.2484i −0.912897 + 0.408191i
\(129\) 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
\(133\) −112.094 −0.842810
\(134\) −19.3742 + 47.3034i −0.144584 + 0.353011i
\(135\) 0 0
\(136\) −30.8820 72.1159i −0.227074 0.530264i
\(137\) 56.1625 0.409945 0.204973 0.978768i \(-0.434289\pi\)
0.204973 + 0.978768i \(0.434289\pi\)
\(138\) 0 0
\(139\) 194.341i 1.39814i −0.715054 0.699069i \(-0.753598\pi\)
0.715054 0.699069i \(-0.246402\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −91.0469 + 222.297i −0.641175 + 1.56547i
\(143\) 14.5731i 0.101910i
\(144\) 0 0
\(145\) 0 0
\(146\) −129.196 52.9152i −0.884905 0.362433i
\(147\) 0 0
\(148\) 127.733 + 125.722i 0.863059 + 0.849470i
\(149\) 110.869 0.744085 0.372043 0.928216i \(-0.378657\pi\)
0.372043 + 0.928216i \(0.378657\pi\)
\(150\) 0 0
\(151\) 141.763i 0.938831i −0.882977 0.469415i \(-0.844465\pi\)
0.882977 0.469415i \(-0.155535\pi\)
\(152\) −201.270 + 86.1892i −1.32414 + 0.567034i
\(153\) 0 0
\(154\) 7.46095 + 3.05580i 0.0484477 + 0.0198429i
\(155\) 0 0
\(156\) 0 0
\(157\) 178.837 1.13909 0.569546 0.821959i \(-0.307119\pi\)
0.569546 + 0.821959i \(0.307119\pi\)
\(158\) 24.4766 59.7612i 0.154915 0.378235i
\(159\) 0 0
\(160\) 0 0
\(161\) −75.1625 −0.466848
\(162\) 0 0
\(163\) 222.535i 1.36525i −0.730771 0.682623i \(-0.760839\pi\)
0.730771 0.682623i \(-0.239161\pi\)
\(164\) −48.4633 47.7002i −0.295508 0.290855i
\(165\) 0 0
\(166\) −51.6727 + 126.162i −0.311281 + 0.760014i
\(167\) 212.883i 1.27475i 0.770554 + 0.637375i \(0.219980\pi\)
−0.770554 + 0.637375i \(0.780020\pi\)
\(168\) 0 0
\(169\) 50.2250 0.297189
\(170\) 0 0
\(171\) 0 0
\(172\) −154.031 + 156.495i −0.895530 + 0.909856i
\(173\) 20.7750 0.120087 0.0600434 0.998196i \(-0.480876\pi\)
0.0600434 + 0.998196i \(0.480876\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.7461 0.249908i 0.0894664 0.00141993i
\(177\) 0 0
\(178\) 125.495 + 51.3992i 0.705025 + 0.288759i
\(179\) 37.2110i 0.207883i 0.994583 + 0.103941i \(0.0331454\pi\)
−0.994583 + 0.103941i \(0.966855\pi\)
\(180\) 0 0
\(181\) 318.125 1.75760 0.878798 0.477193i \(-0.158346\pi\)
0.878798 + 0.477193i \(0.158346\pi\)
\(182\) 45.9688 112.236i 0.252576 0.616680i
\(183\) 0 0
\(184\) −134.958 + 57.7927i −0.733466 + 0.314091i
\(185\) 0 0
\(186\) 0 0
\(187\) 9.65185i 0.0516142i
\(188\) 165.078 167.719i 0.878075 0.892122i
\(189\) 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
\(193\) −127.869 −0.662532 −0.331266 0.943537i \(-0.607476\pi\)
−0.331266 + 0.943537i \(0.607476\pi\)
\(194\) −2.15155 0.881217i −0.0110905 0.00454235i
\(195\) 0 0
\(196\) 91.8664 + 90.4200i 0.468706 + 0.461326i
\(197\) 97.2875 0.493845 0.246923 0.969035i \(-0.420581\pi\)
0.246923 + 0.969035i \(0.420581\pi\)
\(198\) 0 0
\(199\) 19.6851i 0.0989201i −0.998776 0.0494600i \(-0.984250\pi\)
0.998776 0.0494600i \(-0.0157500\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 244.361 + 100.084i 1.20971 + 0.495464i
\(203\) 11.4936i 0.0566189i
\(204\) 0 0
\(205\) 0 0
\(206\) −84.5969 + 206.549i −0.410664 + 1.00266i
\(207\) 0 0
\(208\) −3.75938 236.870i −0.0180740 1.13880i
\(209\) 26.9375 0.128888
\(210\) 0 0
\(211\) 235.298i 1.11516i 0.830124 + 0.557579i \(0.188270\pi\)
−0.830124 + 0.557579i \(0.811730\pi\)
\(212\) −254.361 250.356i −1.19982 1.18092i
\(213\) 0 0
\(214\) −47.1961 + 115.232i −0.220542 + 0.538469i
\(215\) 0 0
\(216\) 0 0
\(217\) 8.06248 0.0371543
\(218\) 161.377 + 66.0955i 0.740259 + 0.303190i
\(219\) 0 0
\(220\) 0 0
\(221\) 145.194 0.656985
\(222\) 0 0
\(223\) 32.4484i 0.145509i 0.997350 + 0.0727543i \(0.0231789\pi\)
−0.997350 + 0.0727543i \(0.976821\pi\)
\(224\) −122.058 47.7440i −0.544901 0.213143i
\(225\) 0 0
\(226\) −155.882 63.8451i −0.689743 0.282500i
\(227\) 289.591i 1.27573i −0.770147 0.637866i \(-0.779817\pi\)
0.770147 0.637866i \(-0.220183\pi\)
\(228\) 0 0
\(229\) 99.2875 0.433570 0.216785 0.976219i \(-0.430443\pi\)
0.216785 + 0.976219i \(0.430443\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.83749 + 20.6374i 0.0380926 + 0.0889541i
\(233\) 50.7750 0.217918 0.108959 0.994046i \(-0.465248\pi\)
0.108959 + 0.994046i \(0.465248\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 188.953 191.976i 0.800649 0.813457i
\(237\) 0 0
\(238\) 30.4453 74.3343i 0.127922 0.312329i
\(239\) 311.498i 1.30334i 0.758503 + 0.651670i \(0.225931\pi\)
−0.758503 + 0.651670i \(0.774069\pi\)
\(240\) 0 0
\(241\) 319.094 1.32404 0.662020 0.749486i \(-0.269699\pi\)
0.662020 + 0.749486i \(0.269699\pi\)
\(242\) 222.152 + 90.9873i 0.917982 + 0.375981i
\(243\) 0 0
\(244\) 102.717 + 101.100i 0.420972 + 0.414344i
\(245\) 0 0
\(246\) 0 0
\(247\) 405.224i 1.64058i
\(248\) 14.4766 6.19927i 0.0583733 0.0249970i
\(249\) 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
\(253\) 18.0625 0.0713932
\(254\) 56.8953 138.914i 0.223997 0.546904i
\(255\) 0 0
\(256\) −255.871 + 8.12395i −0.999496 + 0.0317342i
\(257\) 117.287 0.456372 0.228186 0.973618i \(-0.426721\pi\)
0.228186 + 0.973618i \(0.426721\pi\)
\(258\) 0 0
\(259\) 183.514i 0.708549i
\(260\) 0 0
\(261\) 0 0
\(262\) 105.969 258.730i 0.404461 0.987518i
\(263\) 47.8151i 0.181806i −0.995860 0.0909032i \(-0.971025\pi\)
0.995860 0.0909032i \(-0.0289754\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −207.461 84.9704i −0.779928 0.319438i
\(267\) 0 0
\(268\) −71.7149 + 72.8621i −0.267593 + 0.271873i
\(269\) −479.287 −1.78174 −0.890869 0.454260i \(-0.849904\pi\)
−0.890869 + 0.454260i \(0.849904\pi\)
\(270\) 0 0
\(271\) 242.188i 0.893683i −0.894613 0.446842i \(-0.852549\pi\)
0.894613 0.446842i \(-0.147451\pi\)
\(272\) −2.48986 156.880i −0.00915389 0.576766i
\(273\) 0 0
\(274\) 103.945 + 42.5729i 0.379360 + 0.155375i
\(275\) 0 0
\(276\) 0 0
\(277\) −33.2562 −0.120059 −0.0600293 0.998197i \(-0.519119\pi\)
−0.0600293 + 0.998197i \(0.519119\pi\)
\(278\) 147.316 359.683i 0.529915 1.29382i
\(279\) 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
\(283\) 1.52437i 0.00538646i 0.999996 + 0.00269323i \(0.000857282\pi\)
−0.999996 + 0.00269323i \(0.999143\pi\)
\(284\) −337.016 + 342.407i −1.18667 + 1.20566i
\(285\) 0 0
\(286\) −11.0469 + 26.9717i −0.0386254 + 0.0943065i
\(287\) 69.6274i 0.242604i
\(288\) 0 0
\(289\) −192.837 −0.667258
\(290\) 0 0
\(291\) 0 0
\(292\) −199.002 195.869i −0.681515 0.670784i
\(293\) 448.994 1.53240 0.766201 0.642601i \(-0.222145\pi\)
0.766201 + 0.642601i \(0.222145\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 141.105 + 329.509i 0.476705 + 1.11320i
\(297\) 0 0
\(298\) 205.194 + 84.0418i 0.688570 + 0.282020i
\(299\) 271.716i 0.908749i
\(300\) 0 0
\(301\) −224.837 −0.746968
\(302\) 107.461 262.373i 0.355831 0.868785i
\(303\) 0 0
\(304\) −437.840 + 6.94899i −1.44026 + 0.0228585i
\(305\) 0 0
\(306\) 0 0
\(307\) 10.6681i 0.0347495i −0.999849 0.0173747i \(-0.994469\pi\)
0.999849 0.0173747i \(-0.00553083\pi\)
\(308\) 11.4922 + 11.3112i 0.0373123 + 0.0367248i
\(309\) 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
\(313\) −42.7125 −0.136462 −0.0682309 0.997670i \(-0.521735\pi\)
−0.0682309 + 0.997670i \(0.521735\pi\)
\(314\) 330.989 + 135.564i 1.05411 + 0.431733i
\(315\) 0 0
\(316\) 90.6015 92.0509i 0.286714 0.291300i
\(317\) 143.100 0.451420 0.225710 0.974195i \(-0.427530\pi\)
0.225710 + 0.974195i \(0.427530\pi\)
\(318\) 0 0
\(319\) 2.76206i 0.00865851i
\(320\) 0 0
\(321\) 0 0
\(322\) −139.109 56.9754i −0.432017 0.176942i
\(323\) 268.382i 0.830903i
\(324\) 0 0
\(325\) 0 0
\(326\) 168.688 411.864i 0.517449 1.26339i
\(327\) 0 0
\(328\) −53.5367 125.019i −0.163222 0.381156i
\(329\) 240.962 0.732409
\(330\) 0 0
\(331\) 52.1655i 0.157600i 0.996890 + 0.0787999i \(0.0251088\pi\)
−0.996890 + 0.0787999i \(0.974891\pi\)
\(332\) −191.270 + 194.329i −0.576113 + 0.585329i
\(333\) 0 0
\(334\) −161.372 + 394.000i −0.483149 + 1.17964i
\(335\) 0 0
\(336\) 0 0
\(337\) 441.900 1.31128 0.655638 0.755075i \(-0.272400\pi\)
0.655638 + 0.755075i \(0.272400\pi\)
\(338\) 92.9555 + 38.0721i 0.275016 + 0.112639i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.93752 −0.00568186
\(342\) 0 0
\(343\) 332.676i 0.969900i
\(344\) −403.706 + 172.878i −1.17356 + 0.502553i
\(345\) 0 0
\(346\) 38.4500 + 15.7481i 0.111127 + 0.0455147i
\(347\) 85.8196i 0.247319i −0.992325 0.123659i \(-0.960537\pi\)
0.992325 0.123659i \(-0.0394630\pi\)
\(348\) 0 0
\(349\) −408.931 −1.17172 −0.585861 0.810411i \(-0.699244\pi\)
−0.585861 + 0.810411i \(0.699244\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 29.3320 + 11.4735i 0.0833296 + 0.0325951i
\(353\) −441.163 −1.24975 −0.624876 0.780724i \(-0.714851\pi\)
−0.624876 + 0.780724i \(0.714851\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 193.301 + 190.257i 0.542980 + 0.534430i
\(357\) 0 0
\(358\) −28.2070 + 68.8694i −0.0787906 + 0.192373i
\(359\) 430.814i 1.20004i 0.799985 + 0.600020i \(0.204841\pi\)
−0.799985 + 0.600020i \(0.795159\pi\)
\(360\) 0 0
\(361\) −388.031 −1.07488
\(362\) 588.780 + 241.148i 1.62646 + 0.666155i
\(363\) 0 0
\(364\) 170.156 172.878i 0.467462 0.474940i
\(365\) 0 0
\(366\) 0 0
\(367\) 346.709i 0.944710i 0.881408 + 0.472355i \(0.156596\pi\)
−0.881408 + 0.472355i \(0.843404\pi\)
\(368\) −293.586 + 4.65953i −0.797788 + 0.0126618i
\(369\) 0 0
\(370\) 0 0
\(371\) 365.442i 0.985018i
\(372\) 0 0
\(373\) 577.287 1.54769 0.773844 0.633376i \(-0.218332\pi\)
0.773844 + 0.633376i \(0.218332\pi\)
\(374\) −7.31639 + 17.8635i −0.0195625 + 0.0477633i
\(375\) 0 0
\(376\) 432.659 185.277i 1.15069 0.492757i
\(377\) −41.5500 −0.110212
\(378\) 0 0
\(379\) 500.315i 1.32009i 0.751225 + 0.660046i \(0.229463\pi\)
−0.751225 + 0.660046i \(0.770537\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 195.523 477.384i 0.511841 1.24970i
\(383\) 393.159i 1.02653i −0.858231 0.513263i \(-0.828437\pi\)
0.858231 0.513263i \(-0.171563\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −236.657 96.9283i −0.613101 0.251110i
\(387\) 0 0
\(388\) −3.31406 3.26188i −0.00854139 0.00840690i
\(389\) 98.7750 0.253920 0.126960 0.991908i \(-0.459478\pi\)
0.126960 + 0.991908i \(0.459478\pi\)
\(390\) 0 0
\(391\) 179.959i 0.460252i
\(392\) 101.484 + 236.985i 0.258887 + 0.604554i
\(393\) 0 0
\(394\) 180.058 + 73.7468i 0.457000 + 0.187175i
\(395\) 0 0
\(396\) 0 0
\(397\) −199.225 −0.501826 −0.250913 0.968010i \(-0.580731\pi\)
−0.250913 + 0.968010i \(0.580731\pi\)
\(398\) 14.9219 36.4328i 0.0374922 0.0915397i
\(399\) 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
\(403\) 29.1462i 0.0723232i
\(404\) 376.392 + 370.466i 0.931664 + 0.916995i
\(405\) 0 0
\(406\) −8.71252 + 21.2722i −0.0214594 + 0.0523946i
\(407\) 44.1008i 0.108356i
\(408\) 0 0
\(409\) 368.163 0.900153 0.450076 0.892990i \(-0.351397\pi\)
0.450076 + 0.892990i \(0.351397\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −313.141 + 318.150i −0.760050 + 0.772209i
\(413\) 275.813 0.667827
\(414\) 0 0
\(415\) 0 0
\(416\) 172.597 441.245i 0.414896 1.06068i
\(417\) 0 0
\(418\) 49.8554 + 20.4194i 0.119271 + 0.0488503i
\(419\) 47.0535i 0.112300i −0.998422 0.0561498i \(-0.982118\pi\)
0.998422 0.0561498i \(-0.0178824\pi\)
\(420\) 0 0
\(421\) 7.81255 0.0185571 0.00927856 0.999957i \(-0.497047\pi\)
0.00927856 + 0.999957i \(0.497047\pi\)
\(422\) −178.363 + 435.486i −0.422662 + 1.03196i
\(423\) 0 0
\(424\) −280.989 656.167i −0.662710 1.54756i
\(425\) 0 0
\(426\) 0 0
\(427\) 147.574i 0.345607i
\(428\) −174.699 + 177.494i −0.408176 + 0.414705i
\(429\) 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
\(433\) 4.06878 0.00939673 0.00469836 0.999989i \(-0.498504\pi\)
0.00469836 + 0.999989i \(0.498504\pi\)
\(434\) 14.9219 + 6.11161i 0.0343822 + 0.0140820i
\(435\) 0 0
\(436\) 248.570 + 244.657i 0.570115 + 0.561139i
\(437\) −502.250 −1.14931
\(438\) 0 0
\(439\) 176.815i 0.402768i 0.979512 + 0.201384i \(0.0645439\pi\)
−0.979512 + 0.201384i \(0.935456\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 268.722 + 110.061i 0.607968 + 0.249007i
\(443\) 469.454i 1.05972i 0.848087 + 0.529858i \(0.177755\pi\)
−0.848087 + 0.529858i \(0.822245\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.5969 + 60.0549i −0.0551499 + 0.134652i
\(447\) 0 0
\(448\) −189.711 180.887i −0.423462 0.403766i
\(449\) −514.444 −1.14575 −0.572877 0.819641i \(-0.694173\pi\)
−0.572877 + 0.819641i \(0.694173\pi\)
\(450\) 0 0
\(451\) 16.7323i 0.0371005i
\(452\) −240.107 236.326i −0.531210 0.522846i
\(453\) 0 0
\(454\) 219.519 535.970i 0.483522 1.18055i
\(455\) 0 0
\(456\) 0 0
\(457\) −684.537 −1.49789 −0.748947 0.662630i \(-0.769440\pi\)
−0.748947 + 0.662630i \(0.769440\pi\)
\(458\) 183.759 + 75.2629i 0.401221 + 0.164329i
\(459\) 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
\(463\) 550.353i 1.18867i −0.804218 0.594334i \(-0.797416\pi\)
0.804218 0.594334i \(-0.202584\pi\)
\(464\) 0.712521 + 44.8943i 0.00153561 + 0.0967550i
\(465\) 0 0
\(466\) 93.9734 + 38.4890i 0.201660 + 0.0825944i
\(467\) 10.0013i 0.0214160i −0.999943 0.0107080i \(-0.996591\pi\)
0.999943 0.0107080i \(-0.00340852\pi\)
\(468\) 0 0
\(469\) −104.681 −0.223201
\(470\) 0 0
\(471\) 0 0
\(472\) 495.234 212.073i 1.04923 0.449307i
\(473\) 54.0312 0.114231
\(474\) 0 0
\(475\) 0 0
\(476\) 112.695 114.498i 0.236755 0.240542i
\(477\) 0 0
\(478\) −236.125 + 576.515i −0.493985 + 1.20610i
\(479\) 548.574i 1.14525i −0.819818 0.572625i \(-0.805925\pi\)
0.819818 0.572625i \(-0.194075\pi\)
\(480\) 0 0
\(481\) −663.412 −1.37924
\(482\) 590.573 + 241.883i 1.22525 + 0.501831i
\(483\) 0 0
\(484\) 342.183 + 336.795i 0.706989 + 0.695858i
\(485\) 0 0
\(486\) 0 0
\(487\) 429.005i 0.880913i 0.897774 + 0.440457i \(0.145183\pi\)
−0.897774 + 0.440457i \(0.854817\pi\)
\(488\) 113.470 + 264.976i 0.232521 + 0.542985i
\(489\) 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
\(493\) −27.5188 −0.0558190
\(494\) 307.172 749.981i 0.621805 1.51818i
\(495\) 0 0
\(496\) 31.4922 0.499815i 0.0634923 0.00100769i
\(497\) −491.938 −0.989814
\(498\) 0 0
\(499\) 395.763i 0.793112i −0.918011 0.396556i \(-0.870205\pi\)
0.918011 0.396556i \(-0.129795\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 66.4257 162.183i 0.132322 0.323073i
\(503\) 325.436i 0.646991i −0.946230 0.323495i \(-0.895142\pi\)
0.946230 0.323495i \(-0.104858\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 33.4297 + 13.6919i 0.0660666 + 0.0270591i
\(507\) 0 0
\(508\) 210.602 213.971i 0.414570 0.421202i
\(509\) −1.22499 −0.00240667 −0.00120333 0.999999i \(-0.500383\pi\)
−0.00120333 + 0.999999i \(0.500383\pi\)
\(510\) 0 0
\(511\) 285.908i 0.559506i
\(512\) −479.720 178.922i −0.936952 0.349458i
\(513\) 0 0
\(514\) 217.073 + 88.9074i 0.422322 + 0.172972i
\(515\) 0 0
\(516\) 0 0
\(517\) −57.9063 −0.112004
\(518\) −139.109 + 339.645i −0.268551 + 0.655685i
\(519\) 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
\(523\) 295.370i 0.564761i −0.959302 0.282380i \(-0.908876\pi\)
0.959302 0.282380i \(-0.0911240\pi\)
\(524\) 392.250 398.525i 0.748569 0.760543i
\(525\) 0 0
\(526\) 36.2453 88.4953i 0.0689074 0.168242i
\(527\) 19.3037i 0.0366294i
\(528\) 0 0
\(529\) 192.225 0.363374
\(530\) 0 0
\(531\) 0 0
\(532\) −319.555 314.523i −0.600667 0.591209i
\(533\) 251.706 0.472244
\(534\) 0 0
\(535\) 0 0
\(536\) −187.960 + 80.4897i −0.350672 + 0.150167i
\(537\) 0 0
\(538\) −887.056 363.314i −1.64880 0.675305i
\(539\) 31.7176i 0.0588453i
\(540\) 0 0
\(541\) −200.250 −0.370148 −0.185074 0.982725i \(-0.559252\pi\)
−0.185074 + 0.982725i \(0.559252\pi\)
\(542\) 183.586 448.237i 0.338719 0.827006i
\(543\) 0 0
\(544\) 114.312 292.238i 0.210132 0.537203i
\(545\) 0 0
\(546\) 0 0
\(547\) 989.581i 1.80911i 0.426361 + 0.904553i \(0.359795\pi\)
−0.426361 + 0.904553i \(0.640205\pi\)
\(548\) 160.107 + 157.586i 0.292166 + 0.287566i
\(549\) 0 0
\(550\) 0 0
\(551\) 76.8026i 0.139388i
\(552\) 0 0
\(553\) 132.250 0.239150
\(554\) −61.5500 25.2092i −0.111101 0.0455040i
\(555\) 0 0
\(556\) 545.301 554.024i 0.980757 0.996446i
\(557\) 641.319 1.15138 0.575690 0.817668i \(-0.304733\pi\)
0.575690 + 0.817668i \(0.304733\pi\)
\(558\) 0 0
\(559\) 812.798i 1.45402i
\(560\) 0 0
\(561\) 0 0
\(562\) 455.408 + 186.523i 0.810334 + 0.331891i
\(563\) 382.237i 0.678928i 0.940619 + 0.339464i \(0.110246\pi\)
−0.940619 + 0.339464i \(0.889754\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.15552 + 2.82127i −0.00204155 + 0.00498458i
\(567\) 0 0
\(568\) −883.297 + 378.252i −1.55510 + 0.665937i
\(569\) 576.025 1.01235 0.506173 0.862432i \(-0.331060\pi\)
0.506173 + 0.862432i \(0.331060\pi\)
\(570\) 0 0
\(571\) 1100.67i 1.92762i 0.266582 + 0.963812i \(0.414106\pi\)
−0.266582 + 0.963812i \(0.585894\pi\)
\(572\) −40.8907 + 41.5448i −0.0714872 + 0.0726307i
\(573\) 0 0
\(574\) 52.7797 128.865i 0.0919506 0.224504i
\(575\) 0 0
\(576\) 0 0
\(577\) −326.475 −0.565814 −0.282907 0.959147i \(-0.591299\pi\)
−0.282907 + 0.959147i \(0.591299\pi\)
\(578\) −356.900 146.177i −0.617474 0.252901i
\(579\) 0 0
\(580\) 0 0
\(581\) −279.194 −0.480540
\(582\) 0 0
\(583\) 87.8201i 0.150635i
\(584\) −219.835 513.360i −0.376430 0.879042i
\(585\) 0 0
\(586\) 830.989 + 340.351i 1.41807 + 0.580803i
\(587\) 90.5502i 0.154259i −0.997021 0.0771296i \(-0.975424\pi\)
0.997021 0.0771296i \(-0.0245755\pi\)
\(588\) 0 0
\(589\) 53.8750 0.0914686
\(590\) 0 0
\(591\) 0 0
\(592\) 11.3765 + 716.810i 0.0192171 + 1.21083i
\(593\) −896.163 −1.51124 −0.755618 0.655013i \(-0.772663\pi\)
−0.755618 + 0.655013i \(0.772663\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 316.062 + 311.086i 0.530306 + 0.521956i
\(597\) 0 0
\(598\) 205.969 502.887i 0.344429 0.840947i
\(599\) 1157.76i 1.93282i −0.257001 0.966411i \(-0.582734\pi\)
0.257001 0.966411i \(-0.417266\pi\)
\(600\) 0 0
\(601\) −438.969 −0.730397 −0.365199 0.930930i \(-0.618999\pi\)
−0.365199 + 0.930930i \(0.618999\pi\)
\(602\) −416.125 170.434i −0.691237 0.283112i
\(603\) 0 0
\(604\) 397.773 404.137i 0.658565 0.669100i
\(605\) 0 0
\(606\) 0 0
\(607\) 1156.84i 1.90583i −0.303237 0.952915i \(-0.598067\pi\)
0.303237 0.952915i \(-0.401933\pi\)
\(608\) −815.613 319.035i −1.34147 0.524728i
\(609\) 0 0
\(610\) 0 0
\(611\) 871.090i 1.42568i
\(612\) 0 0
\(613\) 409.225 0.667577 0.333789 0.942648i \(-0.391673\pi\)
0.333789 + 0.942648i \(0.391673\pi\)
\(614\) 8.08673 19.7443i 0.0131706 0.0321568i
\(615\) 0 0
\(616\) 12.6953 + 29.6461i 0.0206092 + 0.0481267i
\(617\) −999.150 −1.61937 −0.809684 0.586866i \(-0.800361\pi\)
−0.809684 + 0.586866i \(0.800361\pi\)
\(618\) 0 0
\(619\) 266.604i 0.430701i 0.976537 + 0.215350i \(0.0690894\pi\)
−0.976537 + 0.215350i \(0.930911\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −280.602 + 685.107i −0.451128 + 1.10146i
\(623\) 277.716i 0.445772i
\(624\) 0 0
\(625\) 0 0
\(626\) −79.0515 32.3774i −0.126280 0.0517210i
\(627\) 0 0
\(628\) 509.827 + 501.799i 0.811826 + 0.799043i
\(629\) −439.381 −0.698539
\(630\) 0 0
\(631\) 1053.40i 1.66941i 0.550696 + 0.834706i \(0.314362\pi\)
−0.550696 + 0.834706i \(0.685638\pi\)
\(632\) 237.461 101.687i 0.375729 0.160898i
\(633\) 0 0
\(634\) 264.847 + 108.474i 0.417739 + 0.171095i
\(635\) 0 0
\(636\) 0 0
\(637\) −477.131 −0.749029
\(638\) 2.09373 5.11198i 0.00328170 0.00801250i
\(639\) 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
\(643\) 454.690i 0.707138i −0.935408 0.353569i \(-0.884968\pi\)
0.935408 0.353569i \(-0.115032\pi\)
\(644\) −214.272 210.898i −0.332720 0.327482i
\(645\) 0 0
\(646\) 203.441 496.716i 0.314925 0.768910i
\(647\) 928.430i 1.43498i 0.696570 + 0.717489i \(0.254708\pi\)
−0.696570 + 0.717489i \(0.745292\pi\)
\(648\) 0 0
\(649\) −66.2812 −0.102128
\(650\) 0 0
\(651\) 0 0
\(652\) 624.410 634.399i 0.957684 0.973004i
\(653\) 536.744 0.821966 0.410983 0.911643i \(-0.365186\pi\)
0.410983 + 0.911643i \(0.365186\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.31639 271.966i −0.00657986 0.414582i
\(657\) 0 0
\(658\) 445.969 + 182.657i 0.677764 + 0.277594i
\(659\) 967.387i 1.46796i 0.679170 + 0.733981i \(0.262340\pi\)
−0.679170 + 0.733981i \(0.737660\pi\)
\(660\) 0 0
\(661\) −534.281 −0.808292 −0.404146 0.914694i \(-0.632431\pi\)
−0.404146 + 0.914694i \(0.632431\pi\)
\(662\) −39.5430 + 96.5469i −0.0597327 + 0.145841i
\(663\) 0 0
\(664\) −501.305 + 214.673i −0.754978 + 0.323303i
\(665\) 0 0
\(666\) 0 0
\(667\) 51.4987i 0.0772094i
\(668\) −597.328 + 606.884i −0.894204 + 0.908508i
\(669\) 0 0
\(670\) 0 0
\(671\) 35.4639i 0.0528523i
\(672\) 0 0
\(673\) 129.225 0.192013 0.0960067 0.995381i \(-0.469393\pi\)
0.0960067 + 0.995381i \(0.469393\pi\)
\(674\) 817.860 + 334.973i 1.21344 + 0.496993i
\(675\) 0 0
\(676\) 143.180 + 140.926i 0.211805 + 0.208471i
\(677\) 111.163 0.164199 0.0820993 0.996624i \(-0.473838\pi\)
0.0820993 + 0.996624i \(0.473838\pi\)
\(678\) 0 0
\(679\) 4.76132i 0.00701226i
\(680\) 0 0
\(681\) 0 0
\(682\) −3.58592 1.46869i −0.00525794 0.00215351i
\(683\) 710.467i 1.04022i 0.854101 + 0.520108i \(0.174108\pi\)
−0.854101 + 0.520108i \(0.825892\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −252.178 + 615.710i −0.367607 + 0.897536i
\(687\) 0 0
\(688\) −878.219 + 13.9383i −1.27648 + 0.0202591i
\(689\) 1321.09 1.91740
\(690\) 0 0
\(691\) 64.0073i 0.0926300i −0.998927 0.0463150i \(-0.985252\pi\)
0.998927 0.0463150i \(-0.0147478\pi\)
\(692\) 59.2250 + 58.2925i 0.0855853 + 0.0842377i
\(693\) 0 0
\(694\) 65.0539 158.833i 0.0937375 0.228867i
\(695\) 0 0
\(696\) 0 0
\(697\) 166.706 0.239177
\(698\) −756.842 309.982i −1.08430 0.444100i
\(699\) 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
\(703\) 1226.28i 1.74435i
\(704\) 45.5899 + 43.4695i 0.0647584 + 0.0617464i
\(705\) 0 0
\(706\) −816.495 334.414i −1.15651 0.473675i
\(707\) 540.764i 0.764872i
\(708\) 0 0
\(709\) −1349.09 −1.90280 −0.951402 0.307953i \(-0.900356\pi\)
−0.951402 + 0.307953i \(0.900356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 213.537 + 498.652i 0.299911 + 0.700354i
\(713\) 36.1250 0.0506662
\(714\) 0 0
\(715\) 0 0
\(716\) −104.410 + 106.080i −0.145824 + 0.148157i
\(717\) 0 0
\(718\) −326.570 + 797.343i −0.454833 + 1.11051i
\(719\) 1101.82i 1.53243i 0.642584 + 0.766216i \(0.277862\pi\)
−0.642584 + 0.766216i \(0.722138\pi\)
\(720\) 0 0
\(721\) −457.087 −0.633963
\(722\) −718.161 294.139i −0.994683 0.407395i
\(723\) 0 0
\(724\) 906.905 + 892.625i 1.25263 + 1.23291i
\(725\) 0 0
\(726\) 0 0
\(727\) 31.7188i 0.0436298i −0.999762 0.0218149i \(-0.993056\pi\)
0.999762 0.0218149i \(-0.00694445\pi\)
\(728\) 445.969 190.976i 0.612594 0.262330i
\(729\) 0 0
\(730\) 0 0
\(731\) 538.320i 0.736415i
\(732\) 0 0
\(733\) 121.319 0.165510 0.0827549 0.996570i \(-0.473628\pi\)
0.0827549 + 0.996570i \(0.473628\pi\)
\(734\) −262.816 + 641.682i −0.358059 + 0.874226i
\(735\) 0 0
\(736\) −546.895 213.923i −0.743064 0.290656i
\(737\) 25.1562 0.0341333
\(738\) 0 0
\(739\) 311.498i 0.421513i −0.977539 0.210757i \(-0.932407\pi\)
0.977539 0.210757i \(-0.0675927\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 277.016 676.352i 0.373336 0.911526i
\(743\) 298.259i 0.401425i −0.979650 0.200712i \(-0.935674\pi\)
0.979650 0.200712i \(-0.0643257\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1068.43 + 437.601i 1.43222 + 0.586597i
\(747\) 0 0
\(748\) −27.0821 + 27.5153i −0.0362060 + 0.0367852i
\(749\) −255.006 −0.340462
\(750\) 0 0
\(751\) 580.864i 0.773454i −0.922194 0.386727i \(-0.873606\pi\)
0.922194 0.386727i \(-0.126394\pi\)
\(752\) 941.203 14.9379i 1.25160 0.0198642i
\(753\) 0 0
\(754\) −76.9000 31.4962i −0.101989 0.0417721i
\(755\) 0 0
\(756\) 0 0
\(757\) −507.600 −0.670541 −0.335271 0.942122i \(-0.608828\pi\)
−0.335271 + 0.942122i \(0.608828\pi\)
\(758\) −379.254 + 925.974i −0.500335 + 1.22160i
\(759\) 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
\(763\) 357.122i 0.468050i
\(764\) 723.742 735.320i 0.947306 0.962461i
\(765\) 0 0
\(766\) 298.027 727.652i 0.389069 0.949937i
\(767\) 997.075i 1.29997i
\(768\) 0 0
\(769\) −161.681 −0.210249 −0.105124 0.994459i \(-0.533524\pi\)
−0.105124 + 0.994459i \(0.533524\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −364.526 358.786i −0.472184 0.464749i
\(773\) −545.506 −0.705700 −0.352850 0.935680i \(-0.614787\pi\)
−0.352850 + 0.935680i \(0.614787\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.66100 8.54918i −0.00471778 0.0110170i
\(777\) 0 0
\(778\) 182.811 + 74.8744i 0.234975 + 0.0962396i
\(779\) 465.263i 0.597257i
\(780\) 0 0
\(781\) 118.219 0.151368
\(782\) 136.414 333.064i 0.174443 0.425913i
\(783\) 0 0
\(784\) 8.18210 + 515.535i 0.0104363 + 0.657570i
\(785\) 0 0
\(786\) 0 0
\(787\) 953.227i 1.21122i −0.795763 0.605608i \(-0.792930\pi\)
0.795763 0.605608i \(-0.207070\pi\)
\(788\) 277.345 + 272.978i 0.351961 + 0.346419i
\(789\) 0 0
\(790\) 0 0
\(791\) 344.963i 0.436110i
\(792\) 0 0
\(793\) −533.488 −0.672746
\(794\) −368.722 151.019i −0.464385 0.190200i
\(795\) 0 0
\(796\) 55.2343 56.1179i 0.0693898 0.0704999i
\(797\) 456.975 0.573369 0.286684 0.958025i \(-0.407447\pi\)
0.286684 + 0.958025i \(0.407447\pi\)
\(798\) 0 0
\(799\) 576.927i 0.722061i
\(800\) 0 0
\(801\) 0 0
\(802\) 194.505 + 79.6642i 0.242526 + 0.0993319i
\(803\) 68.7071i 0.0855631i
\(804\) 0 0
\(805\) 0 0
\(806\) −22.0937 + 53.9433i −0.0274116 + 0.0669272i
\(807\) 0 0
\(808\) 415.795 + 970.968i 0.514598 + 1.20169i
\(809\) −1073.79 −1.32730 −0.663651 0.748042i \(-0.730994\pi\)
−0.663651 + 0.748042i \(0.730994\pi\)
\(810\) 0 0
\(811\) 108.299i 0.133537i 0.997768 + 0.0667687i \(0.0212690\pi\)
−0.997768 + 0.0667687i \(0.978731\pi\)
\(812\) −32.2499 + 32.7658i −0.0397167 + 0.0403520i
\(813\) 0 0
\(814\) 33.4297 81.6209i 0.0410684 0.100271i
\(815\) 0 0
\(816\) 0 0
\(817\) −1502.41 −1.83893
\(818\) 681.388 + 279.078i 0.832993 + 0.341171i
\(819\) 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
\(823\) 862.264i 1.04771i −0.851808 0.523854i \(-0.824494\pi\)
0.851808 0.523854i \(-0.175506\pi\)
\(824\) −820.722 + 351.456i −0.996022 + 0.426524i
\(825\) 0 0
\(826\) 510.469 + 209.074i 0.618001 + 0.253116i
\(827\) 720.596i 0.871338i −0.900107 0.435669i \(-0.856512\pi\)
0.900107 0.435669i \(-0.143488\pi\)
\(828\) 0 0
\(829\) 385.569 0.465101 0.232550 0.972584i \(-0.425293\pi\)
0.232550 + 0.972584i \(0.425293\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 653.916 685.813i 0.785956 0.824295i
\(833\) −316.006 −0.379359
\(834\) 0 0
\(835\) 0 0
\(836\) 76.7930 + 75.5838i 0.0918576 + 0.0904113i
\(837\) 0 0
\(838\) 35.6680 87.0858i 0.0425632 0.103921i
\(839\) 561.530i 0.669284i 0.942345 + 0.334642i \(0.108615\pi\)
−0.942345 + 0.334642i \(0.891385\pi\)
\(840\) 0 0
\(841\) −833.125 −0.990636
\(842\) 14.4593 + 5.92214i 0.0171726 + 0.00703343i
\(843\) 0 0
\(844\) −660.223 + 670.784i −0.782254 + 0.794768i
\(845\) 0 0
\(846\) 0 0
\(847\) 491.616i 0.580420i
\(848\) −22.6547 1427.42i −0.0267154 1.68328i
\(849\) 0 0
\(850\) 0 0
\(851\) 822.259i 0.966227i
\(852\) 0 0
\(853\) −151.087 −0.177125 −0.0885624 0.996071i \(-0.528227\pi\)
−0.0885624 + 0.996071i \(0.528227\pi\)
\(854\) −111.866 + 273.128i −0.130990 + 0.319822i
\(855\) 0 0
\(856\) −457.876 + 196.075i −0.534902 + 0.229060i
\(857\) 444.225 0.518349 0.259174 0.965831i \(-0.416550\pi\)
0.259174 + 0.965831i \(0.416550\pi\)
\(858\) 0 0
\(859\) 731.867i 0.851999i 0.904723 + 0.426000i \(0.140077\pi\)
−0.904723 + 0.426000i \(0.859923\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −516.414 + 1260.86i −0.599088 + 1.46271i
\(863\) 965.801i 1.11912i 0.828790 + 0.559560i \(0.189030\pi\)
−0.828790 + 0.559560i \(0.810970\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 7.53042 + 3.08426i 0.00869564 + 0.00356150i
\(867\) 0 0
\(868\) 22.9844 + 22.6225i 0.0264797 + 0.0260628i
\(869\) −31.7813 −0.0365723
\(870\) 0 0
\(871\) 378.428i 0.434475i
\(872\) 274.592 + 641.229i 0.314899 + 0.735355i
\(873\) 0 0
\(874\) −929.555 380.721i −1.06356 0.435607i
\(875\) 0 0
\(876\) 0 0
\(877\) 1440.78 1.64284 0.821422 0.570320i \(-0.193181\pi\)
0.821422 + 0.570320i \(0.193181\pi\)
\(878\) −134.031 + 327.246i −0.152655 + 0.372718i
\(879\) 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
\(883\) 229.616i 0.260040i −0.991511 0.130020i \(-0.958496\pi\)
0.991511 0.130020i \(-0.0415042\pi\)
\(884\) 413.916 + 407.398i 0.468230 + 0.460858i
\(885\) 0 0
\(886\) −355.860 + 868.856i −0.401648 + 0.980650i
\(887\) 456.945i 0.515158i −0.966257 0.257579i \(-0.917075\pi\)
0.966257 0.257579i \(-0.0829248\pi\)
\(888\) 0 0
\(889\) 307.412 0.345796
\(890\) 0 0
\(891\) 0 0
\(892\) −91.0469 + 92.5033i −0.102070 + 0.103703i
\(893\) 1610.16 1.80309
\(894\) 0 0
\(895\) 0 0
\(896\) −213.995 478.589i −0.238834 0.534140i
\(897\) 0 0
\(898\) −952.123 389.964i −1.06027 0.434258i
\(899\) 5.52413i 0.00614475i
\(900\) 0 0
\(901\) 874.962 0.971102
\(902\) −12.6836 + 30.9679i −0.0140617 + 0.0343325i
\(903\) 0 0
\(904\) −265.243 619.397i −0.293410 0.685174i
\(905\) 0 0
\(906\) 0 0
\(907\) 797.272i 0.879021i −0.898237 0.439511i \(-0.855152\pi\)
0.898237 0.439511i \(-0.144848\pi\)
\(908\) 812.562 825.561i 0.894892 0.909208i
\(909\) 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
\(913\) 67.0937 0.0734871
\(914\) −1266.93 518.900i −1.38614 0.567724i
\(915\) 0 0
\(916\) 283.047 + 278.590i 0.309003 + 0.304138i
\(917\) 572.562 0.624386
\(918\) 0 0
\(919\) 593.469i 0.645777i 0.946437 + 0.322888i \(0.104654\pi\)
−0.946437 + 0.322888i \(0.895346\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −378.022 154.828i −0.410002 0.167926i
\(923\) 1778.38i 1.92673i
\(924\) 0 0
\(925\) 0 0
\(926\) 417.184 1018.58i 0.450523 1.09998i
\(927\) 0 0
\(928\) −32.7125 + 83.6297i −0.0352506 + 0.0901182i
\(929\) 198.775 0.213967 0.106983 0.994261i \(-0.465881\pi\)
0.106983 + 0.994261i \(0.465881\pi\)
\(930\) 0 0
\(931\) 881.948i 0.947312i
\(932\) 144.748 + 142.469i 0.155309 + 0.152864i
\(933\) 0 0
\(934\) 7.58125 18.5101i 0.00811697 0.0198181i
\(935\) 0 0
\(936\) 0 0
\(937\) 1200.43 1.28114 0.640572 0.767898i \(-0.278697\pi\)
0.640572 + 0.767898i \(0.278697\pi\)
\(938\) −193.742 79.3515i −0.206548 0.0845965i
\(939\) 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
\(943\) 311.974i 0.330832i
\(944\) 1077.33 17.0984i 1.14124 0.0181127i
\(945\) 0 0
\(946\) 100.000 + 40.9573i 0.105708 + 0.0432952i
\(947\) 586.516i 0.619341i −0.950844 0.309671i \(-0.899781\pi\)
0.950844 0.309671i \(-0.100219\pi\)
\(948\) 0 0
\(949\) 1033.57 1.08911
\(950\) 0 0
\(951\) 0 0
\(952\) 295.367 126.484i 0.310260 0.132862i
\(953\) −1598.72 −1.67757 −0.838785 0.544462i \(-0.816734\pi\)
−0.838785 + 0.544462i \(0.816734\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −874.031 + 888.013i −0.914259 + 0.928884i
\(957\) 0 0
\(958\) 415.836 1015.29i 0.434067 1.05980i
\(959\) 230.026i 0.239861i
\(960\) 0 0
\(961\) 957.125 0.995968
\(962\) −1227.83 502.887i −1.27633 0.522751i
\(963\) 0 0
\(964\) 909.666 + 895.344i 0.943637 + 0.928780i
\(965\) 0 0
\(966\) 0 0
\(967\) 320.675i 0.331619i 0.986158 + 0.165809i \(0.0530236\pi\)
−0.986158 + 0.165809i \(0.946976\pi\)
\(968\) 378.005 + 882.719i 0.390501 + 0.911900i
\(969\) 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
\(973\) 795.969 0.818056
\(974\) −325.198 + 793.994i −0.333879 + 0.815189i
\(975\) 0 0
\(976\) 9.14852 + 576.427i 0.00937349 + 0.590602i
\(977\) −1226.24 −1.25511 −0.627556 0.778572i \(-0.715944\pi\)
−0.627556 + 0.778572i \(0.715944\pi\)
\(978\) 0 0
\(979\) 66.7386i 0.0681702i
\(980\) 0 0
\(981\) 0 0
\(982\) 288.062 703.324i 0.293343 0.716216i
\(983\) 1372.96i 1.39670i −0.715754 0.698352i \(-0.753917\pi\)
0.715754 0.698352i \(-0.246083\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −50.9312 20.8601i −0.0516544 0.0211562i
\(987\) 0 0
\(988\) 1137.02 1155.20i 1.15083 1.16924i
\(989\) −1007.41 −1.01862
\(990\) 0 0
\(991\) 362.298i 0.365588i −0.983151 0.182794i \(-0.941486\pi\)
0.983151 0.182794i \(-0.0585142\pi\)
\(992\) 58.6640 + 22.9470i 0.0591371 + 0.0231320i
\(993\) 0 0
\(994\) −910.469 372.903i −0.915964 0.375154i
\(995\) 0 0
\(996\) 0 0
\(997\) 941.087 0.943919 0.471960 0.881620i \(-0.343547\pi\)
0.471960 + 0.881620i \(0.343547\pi\)
\(998\) 300.000 732.470i 0.300601 0.733938i
\(999\) 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.c.m.451.4 4
3.2 odd 2 100.3.b.d.51.1 4
4.3 odd 2 inner 900.3.c.m.451.3 4
5.2 odd 4 900.3.f.d.199.4 8
5.3 odd 4 900.3.f.d.199.5 8
5.4 even 2 900.3.c.l.451.1 4
12.11 even 2 100.3.b.d.51.2 yes 4
15.2 even 4 100.3.d.a.99.5 8
15.8 even 4 100.3.d.a.99.4 8
15.14 odd 2 100.3.b.e.51.4 yes 4
20.3 even 4 900.3.f.d.199.3 8
20.7 even 4 900.3.f.d.199.6 8
20.19 odd 2 900.3.c.l.451.2 4
24.5 odd 2 1600.3.b.p.1151.1 4
24.11 even 2 1600.3.b.p.1151.4 4
60.23 odd 4 100.3.d.a.99.6 8
60.47 odd 4 100.3.d.a.99.3 8
60.59 even 2 100.3.b.e.51.3 yes 4
120.29 odd 2 1600.3.b.o.1151.4 4
120.53 even 4 1600.3.h.o.1599.7 8
120.59 even 2 1600.3.b.o.1151.1 4
120.77 even 4 1600.3.h.o.1599.1 8
120.83 odd 4 1600.3.h.o.1599.2 8
120.107 odd 4 1600.3.h.o.1599.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.3.b.d.51.1 4 3.2 odd 2
100.3.b.d.51.2 yes 4 12.11 even 2
100.3.b.e.51.3 yes 4 60.59 even 2
100.3.b.e.51.4 yes 4 15.14 odd 2
100.3.d.a.99.3 8 60.47 odd 4
100.3.d.a.99.4 8 15.8 even 4
100.3.d.a.99.5 8 15.2 even 4
100.3.d.a.99.6 8 60.23 odd 4
900.3.c.l.451.1 4 5.4 even 2
900.3.c.l.451.2 4 20.19 odd 2
900.3.c.m.451.3 4 4.3 odd 2 inner
900.3.c.m.451.4 4 1.1 even 1 trivial
900.3.f.d.199.3 8 20.3 even 4
900.3.f.d.199.4 8 5.2 odd 4
900.3.f.d.199.5 8 5.3 odd 4
900.3.f.d.199.6 8 20.7 even 4
1600.3.b.o.1151.1 4 120.59 even 2
1600.3.b.o.1151.4 4 120.29 odd 2
1600.3.b.p.1151.1 4 24.5 odd 2
1600.3.b.p.1151.4 4 24.11 even 2
1600.3.h.o.1599.1 8 120.77 even 4
1600.3.h.o.1599.2 8 120.83 odd 4
1600.3.h.o.1599.7 8 120.53 even 4
1600.3.h.o.1599.8 8 120.107 odd 4