Properties

Label 1050.3.c.b.449.8
Level $1050$
Weight $3$
Character 1050.449
Analytic conductor $28.610$
Analytic rank $0$
Dimension $32$
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] = [1050,3,Mod(449,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.c (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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.8
Character \(\chi\) \(=\) 1050.449
Dual form 1050.3.c.b.449.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +(-0.236961 + 2.99063i) q^{3} +2.00000 q^{4} +(0.335113 - 4.22939i) q^{6} +2.64575i q^{7} -2.82843 q^{8} +(-8.88770 - 1.41732i) q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +(-0.236961 + 2.99063i) q^{3} +2.00000 q^{4} +(0.335113 - 4.22939i) q^{6} +2.64575i q^{7} -2.82843 q^{8} +(-8.88770 - 1.41732i) q^{9} -9.89436i q^{11} +(-0.473921 + 5.98125i) q^{12} +6.33097i q^{13} -3.74166i q^{14} +4.00000 q^{16} -5.17268 q^{17} +(12.5691 + 2.00439i) q^{18} -10.2878 q^{19} +(-7.91246 - 0.626939i) q^{21} +13.9927i q^{22} +15.2580 q^{23} +(0.670226 - 8.45877i) q^{24} -8.95334i q^{26} +(6.34471 - 26.2439i) q^{27} +5.29150i q^{28} -15.1060i q^{29} -13.0380 q^{31} -5.65685 q^{32} +(29.5904 + 2.34457i) q^{33} +7.31527 q^{34} +(-17.7754 - 2.83464i) q^{36} -35.8403i q^{37} +14.5492 q^{38} +(-18.9336 - 1.50019i) q^{39} -63.0675i q^{41} +(11.1899 + 0.886625i) q^{42} +19.0241i q^{43} -19.7887i q^{44} -21.5780 q^{46} -10.0367 q^{47} +(-0.947842 + 11.9625i) q^{48} -7.00000 q^{49} +(1.22572 - 15.4696i) q^{51} +12.6619i q^{52} +72.2509 q^{53} +(-8.97278 + 37.1145i) q^{54} -7.48331i q^{56} +(2.43781 - 30.7670i) q^{57} +21.3632i q^{58} -0.266801i q^{59} -17.6626 q^{61} +18.4385 q^{62} +(3.74988 - 23.5146i) q^{63} +8.00000 q^{64} +(-41.8471 - 3.31573i) q^{66} +5.55641i q^{67} -10.3454 q^{68} +(-3.61554 + 45.6309i) q^{69} -139.562i q^{71} +(25.1382 + 4.00879i) q^{72} +32.7875i q^{73} +50.6858i q^{74} -20.5757 q^{76} +26.1780 q^{77} +(26.7761 + 2.12159i) q^{78} -2.91874 q^{79} +(76.9824 + 25.1934i) q^{81} +89.1909i q^{82} -88.5027 q^{83} +(-15.8249 - 1.25388i) q^{84} -26.9042i q^{86} +(45.1766 + 3.57954i) q^{87} +27.9855i q^{88} +146.550i q^{89} -16.7502 q^{91} +30.5159 q^{92} +(3.08949 - 38.9918i) q^{93} +14.1941 q^{94} +(1.34045 - 16.9175i) q^{96} -167.535i q^{97} +9.89949 q^{98} +(-14.0235 + 87.9381i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9} + 128 q^{16} + 48 q^{19} + 56 q^{21} - 32 q^{24} + 48 q^{31} + 256 q^{34} - 32 q^{36} + 192 q^{39} + 160 q^{46} - 224 q^{49} + 288 q^{51} - 80 q^{54} - 112 q^{61} + 256 q^{64} - 192 q^{66} + 344 q^{69} + 96 q^{76} - 256 q^{79} + 160 q^{81} + 112 q^{84} - 448 q^{91} + 416 q^{94} - 64 q^{96} - 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) −0.236961 + 2.99063i −0.0789868 + 0.996876i
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 0.335113 4.22939i 0.0558521 0.704898i
\(7\) 2.64575i 0.377964i
\(8\) −2.82843 −0.353553
\(9\) −8.88770 1.41732i −0.987522 0.157480i
\(10\) 0 0
\(11\) 9.89436i 0.899488i −0.893158 0.449744i \(-0.851515\pi\)
0.893158 0.449744i \(-0.148485\pi\)
\(12\) −0.473921 + 5.98125i −0.0394934 + 0.498438i
\(13\) 6.33097i 0.486998i 0.969901 + 0.243499i \(0.0782952\pi\)
−0.969901 + 0.243499i \(0.921705\pi\)
\(14\) 3.74166i 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −5.17268 −0.304275 −0.152138 0.988359i \(-0.548616\pi\)
−0.152138 + 0.988359i \(0.548616\pi\)
\(18\) 12.5691 + 2.00439i 0.698284 + 0.111355i
\(19\) −10.2878 −0.541464 −0.270732 0.962655i \(-0.587266\pi\)
−0.270732 + 0.962655i \(0.587266\pi\)
\(20\) 0 0
\(21\) −7.91246 0.626939i −0.376784 0.0298542i
\(22\) 13.9927i 0.636034i
\(23\) 15.2580 0.663390 0.331695 0.943387i \(-0.392380\pi\)
0.331695 + 0.943387i \(0.392380\pi\)
\(24\) 0.670226 8.45877i 0.0279261 0.352449i
\(25\) 0 0
\(26\) 8.95334i 0.344359i
\(27\) 6.34471 26.2439i 0.234989 0.971998i
\(28\) 5.29150i 0.188982i
\(29\) 15.1060i 0.520898i −0.965488 0.260449i \(-0.916129\pi\)
0.965488 0.260449i \(-0.0838706\pi\)
\(30\) 0 0
\(31\) −13.0380 −0.420581 −0.210290 0.977639i \(-0.567441\pi\)
−0.210290 + 0.977639i \(0.567441\pi\)
\(32\) −5.65685 −0.176777
\(33\) 29.5904 + 2.34457i 0.896677 + 0.0710477i
\(34\) 7.31527 0.215155
\(35\) 0 0
\(36\) −17.7754 2.83464i −0.493761 0.0787401i
\(37\) 35.8403i 0.968657i −0.874886 0.484328i \(-0.839064\pi\)
0.874886 0.484328i \(-0.160936\pi\)
\(38\) 14.5492 0.382873
\(39\) −18.9336 1.50019i −0.485476 0.0384664i
\(40\) 0 0
\(41\) 63.0675i 1.53823i −0.639109 0.769116i \(-0.720697\pi\)
0.639109 0.769116i \(-0.279303\pi\)
\(42\) 11.1899 + 0.886625i 0.266426 + 0.0211101i
\(43\) 19.0241i 0.442422i 0.975226 + 0.221211i \(0.0710009\pi\)
−0.975226 + 0.221211i \(0.928999\pi\)
\(44\) 19.7887i 0.449744i
\(45\) 0 0
\(46\) −21.5780 −0.469087
\(47\) −10.0367 −0.213547 −0.106774 0.994283i \(-0.534052\pi\)
−0.106774 + 0.994283i \(0.534052\pi\)
\(48\) −0.947842 + 11.9625i −0.0197467 + 0.249219i
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 1.22572 15.4696i 0.0240337 0.303325i
\(52\) 12.6619i 0.243499i
\(53\) 72.2509 1.36323 0.681613 0.731713i \(-0.261279\pi\)
0.681613 + 0.731713i \(0.261279\pi\)
\(54\) −8.97278 + 37.1145i −0.166163 + 0.687306i
\(55\) 0 0
\(56\) 7.48331i 0.133631i
\(57\) 2.43781 30.7670i 0.0427686 0.539773i
\(58\) 21.3632i 0.368331i
\(59\) 0.266801i 0.00452205i −0.999997 0.00226102i \(-0.999280\pi\)
0.999997 0.00226102i \(-0.000719707\pi\)
\(60\) 0 0
\(61\) −17.6626 −0.289551 −0.144775 0.989465i \(-0.546246\pi\)
−0.144775 + 0.989465i \(0.546246\pi\)
\(62\) 18.4385 0.297395
\(63\) 3.74988 23.5146i 0.0595219 0.373248i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) −41.8471 3.31573i −0.634047 0.0502383i
\(67\) 5.55641i 0.0829315i 0.999140 + 0.0414657i \(0.0132027\pi\)
−0.999140 + 0.0414657i \(0.986797\pi\)
\(68\) −10.3454 −0.152138
\(69\) −3.61554 + 45.6309i −0.0523991 + 0.661317i
\(70\) 0 0
\(71\) 139.562i 1.96566i −0.184505 0.982832i \(-0.559068\pi\)
0.184505 0.982832i \(-0.440932\pi\)
\(72\) 25.1382 + 4.00879i 0.349142 + 0.0556776i
\(73\) 32.7875i 0.449144i 0.974458 + 0.224572i \(0.0720984\pi\)
−0.974458 + 0.224572i \(0.927902\pi\)
\(74\) 50.6858i 0.684944i
\(75\) 0 0
\(76\) −20.5757 −0.270732
\(77\) 26.1780 0.339974
\(78\) 26.7761 + 2.12159i 0.343283 + 0.0271999i
\(79\) −2.91874 −0.0369461 −0.0184730 0.999829i \(-0.505880\pi\)
−0.0184730 + 0.999829i \(0.505880\pi\)
\(80\) 0 0
\(81\) 76.9824 + 25.1934i 0.950400 + 0.311030i
\(82\) 89.1909i 1.08769i
\(83\) −88.5027 −1.06630 −0.533149 0.846021i \(-0.678991\pi\)
−0.533149 + 0.846021i \(0.678991\pi\)
\(84\) −15.8249 1.25388i −0.188392 0.0149271i
\(85\) 0 0
\(86\) 26.9042i 0.312839i
\(87\) 45.1766 + 3.57954i 0.519271 + 0.0411441i
\(88\) 27.9855i 0.318017i
\(89\) 146.550i 1.64663i 0.567588 + 0.823313i \(0.307877\pi\)
−0.567588 + 0.823313i \(0.692123\pi\)
\(90\) 0 0
\(91\) −16.7502 −0.184068
\(92\) 30.5159 0.331695
\(93\) 3.08949 38.9918i 0.0332203 0.419267i
\(94\) 14.1941 0.151001
\(95\) 0 0
\(96\) 1.34045 16.9175i 0.0139630 0.176224i
\(97\) 167.535i 1.72716i −0.504210 0.863581i \(-0.668216\pi\)
0.504210 0.863581i \(-0.331784\pi\)
\(98\) 9.89949 0.101015
\(99\) −14.0235 + 87.9381i −0.141651 + 0.888264i
\(100\) 0 0
\(101\) 77.9574i 0.771856i −0.922529 0.385928i \(-0.873881\pi\)
0.922529 0.385928i \(-0.126119\pi\)
\(102\) −1.73343 + 21.8773i −0.0169944 + 0.214483i
\(103\) 17.0241i 0.165283i 0.996579 + 0.0826415i \(0.0263356\pi\)
−0.996579 + 0.0826415i \(0.973664\pi\)
\(104\) 17.9067i 0.172180i
\(105\) 0 0
\(106\) −102.178 −0.963946
\(107\) 189.549 1.77149 0.885745 0.464173i \(-0.153648\pi\)
0.885745 + 0.464173i \(0.153648\pi\)
\(108\) 12.6894 52.4879i 0.117495 0.485999i
\(109\) 77.0791 0.707148 0.353574 0.935407i \(-0.384966\pi\)
0.353574 + 0.935407i \(0.384966\pi\)
\(110\) 0 0
\(111\) 107.185 + 8.49274i 0.965631 + 0.0765112i
\(112\) 10.5830i 0.0944911i
\(113\) −22.2664 −0.197048 −0.0985239 0.995135i \(-0.531412\pi\)
−0.0985239 + 0.995135i \(0.531412\pi\)
\(114\) −3.44758 + 43.5112i −0.0302419 + 0.381677i
\(115\) 0 0
\(116\) 30.2121i 0.260449i
\(117\) 8.97302 56.2678i 0.0766925 0.480921i
\(118\) 0.377313i 0.00319757i
\(119\) 13.6856i 0.115005i
\(120\) 0 0
\(121\) 23.1016 0.190922
\(122\) 24.9787 0.204743
\(123\) 188.611 + 14.9445i 1.53343 + 0.121500i
\(124\) −26.0760 −0.210290
\(125\) 0 0
\(126\) −5.30313 + 33.2547i −0.0420883 + 0.263926i
\(127\) 128.979i 1.01559i 0.861479 + 0.507793i \(0.169538\pi\)
−0.861479 + 0.507793i \(0.830462\pi\)
\(128\) −11.3137 −0.0883883
\(129\) −56.8941 4.50797i −0.441040 0.0349455i
\(130\) 0 0
\(131\) 38.6800i 0.295267i −0.989042 0.147634i \(-0.952834\pi\)
0.989042 0.147634i \(-0.0471657\pi\)
\(132\) 59.1807 + 4.68915i 0.448339 + 0.0355238i
\(133\) 27.2190i 0.204654i
\(134\) 7.85795i 0.0586414i
\(135\) 0 0
\(136\) 14.6305 0.107578
\(137\) 187.436 1.36814 0.684072 0.729414i \(-0.260207\pi\)
0.684072 + 0.729414i \(0.260207\pi\)
\(138\) 5.11314 64.5318i 0.0370517 0.467622i
\(139\) 199.427 1.43472 0.717362 0.696701i \(-0.245350\pi\)
0.717362 + 0.696701i \(0.245350\pi\)
\(140\) 0 0
\(141\) 2.37831 30.0161i 0.0168674 0.212880i
\(142\) 197.371i 1.38993i
\(143\) 62.6409 0.438048
\(144\) −35.5508 5.66928i −0.246881 0.0393700i
\(145\) 0 0
\(146\) 46.3685i 0.317593i
\(147\) 1.65872 20.9344i 0.0112838 0.142411i
\(148\) 71.6806i 0.484328i
\(149\) 178.869i 1.20047i −0.799825 0.600233i \(-0.795075\pi\)
0.799825 0.600233i \(-0.204925\pi\)
\(150\) 0 0
\(151\) 237.977 1.57600 0.788002 0.615672i \(-0.211115\pi\)
0.788002 + 0.615672i \(0.211115\pi\)
\(152\) 29.0984 0.191437
\(153\) 45.9732 + 7.33135i 0.300479 + 0.0479173i
\(154\) −37.0213 −0.240398
\(155\) 0 0
\(156\) −37.8671 3.00038i −0.242738 0.0192332i
\(157\) 213.542i 1.36014i −0.733148 0.680069i \(-0.761950\pi\)
0.733148 0.680069i \(-0.238050\pi\)
\(158\) 4.12772 0.0261248
\(159\) −17.1206 + 216.076i −0.107677 + 1.35897i
\(160\) 0 0
\(161\) 40.3688i 0.250738i
\(162\) −108.870 35.6289i −0.672034 0.219932i
\(163\) 68.2021i 0.418418i 0.977871 + 0.209209i \(0.0670888\pi\)
−0.977871 + 0.209209i \(0.932911\pi\)
\(164\) 126.135i 0.769116i
\(165\) 0 0
\(166\) 125.162 0.753986
\(167\) 75.9689 0.454903 0.227452 0.973789i \(-0.426961\pi\)
0.227452 + 0.973789i \(0.426961\pi\)
\(168\) 22.3798 + 1.77325i 0.133213 + 0.0105551i
\(169\) 128.919 0.762833
\(170\) 0 0
\(171\) 91.4351 + 14.5812i 0.534708 + 0.0852699i
\(172\) 38.0483i 0.221211i
\(173\) −182.168 −1.05299 −0.526497 0.850177i \(-0.676495\pi\)
−0.526497 + 0.850177i \(0.676495\pi\)
\(174\) −63.8893 5.06223i −0.367180 0.0290933i
\(175\) 0 0
\(176\) 39.5775i 0.224872i
\(177\) 0.797901 + 0.0632212i 0.00450792 + 0.000357182i
\(178\) 207.253i 1.16434i
\(179\) 253.745i 1.41757i −0.705424 0.708786i \(-0.749243\pi\)
0.705424 0.708786i \(-0.250757\pi\)
\(180\) 0 0
\(181\) −260.351 −1.43840 −0.719202 0.694801i \(-0.755493\pi\)
−0.719202 + 0.694801i \(0.755493\pi\)
\(182\) 23.6883 0.130156
\(183\) 4.18534 52.8223i 0.0228707 0.288646i
\(184\) −43.1560 −0.234544
\(185\) 0 0
\(186\) −4.36920 + 55.1427i −0.0234903 + 0.296466i
\(187\) 51.1804i 0.273692i
\(188\) −20.0734 −0.106774
\(189\) 69.4350 + 16.7865i 0.367381 + 0.0888176i
\(190\) 0 0
\(191\) 48.4377i 0.253600i −0.991928 0.126800i \(-0.959529\pi\)
0.991928 0.126800i \(-0.0404707\pi\)
\(192\) −1.89568 + 23.9250i −0.00987336 + 0.124609i
\(193\) 103.284i 0.535149i 0.963537 + 0.267574i \(0.0862221\pi\)
−0.963537 + 0.267574i \(0.913778\pi\)
\(194\) 236.930i 1.22129i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) −330.940 −1.67990 −0.839950 0.542664i \(-0.817416\pi\)
−0.839950 + 0.542664i \(0.817416\pi\)
\(198\) 19.8322 124.363i 0.100163 0.628097i
\(199\) 254.131 1.27704 0.638521 0.769605i \(-0.279547\pi\)
0.638521 + 0.769605i \(0.279547\pi\)
\(200\) 0 0
\(201\) −16.6172 1.31665i −0.0826724 0.00655050i
\(202\) 110.248i 0.545785i
\(203\) 39.9669 0.196881
\(204\) 2.45144 30.9391i 0.0120169 0.151662i
\(205\) 0 0
\(206\) 24.0758i 0.116873i
\(207\) −135.608 21.6254i −0.655112 0.104471i
\(208\) 25.3239i 0.121749i
\(209\) 101.791i 0.487041i
\(210\) 0 0
\(211\) 115.082 0.545413 0.272707 0.962097i \(-0.412081\pi\)
0.272707 + 0.962097i \(0.412081\pi\)
\(212\) 144.502 0.681613
\(213\) 417.378 + 33.0707i 1.95952 + 0.155262i
\(214\) −268.063 −1.25263
\(215\) 0 0
\(216\) −17.9456 + 74.2291i −0.0830813 + 0.343653i
\(217\) 34.4953i 0.158965i
\(218\) −109.006 −0.500029
\(219\) −98.0552 7.76934i −0.447741 0.0354765i
\(220\) 0 0
\(221\) 32.7481i 0.148181i
\(222\) −151.582 12.0105i −0.682804 0.0541016i
\(223\) 304.920i 1.36735i −0.729784 0.683677i \(-0.760380\pi\)
0.729784 0.683677i \(-0.239620\pi\)
\(224\) 14.9666i 0.0668153i
\(225\) 0 0
\(226\) 31.4894 0.139334
\(227\) −205.433 −0.904990 −0.452495 0.891767i \(-0.649466\pi\)
−0.452495 + 0.891767i \(0.649466\pi\)
\(228\) 4.87562 61.5341i 0.0213843 0.269886i
\(229\) 313.595 1.36941 0.684706 0.728820i \(-0.259931\pi\)
0.684706 + 0.728820i \(0.259931\pi\)
\(230\) 0 0
\(231\) −6.20316 + 78.2887i −0.0268535 + 0.338912i
\(232\) 42.7264i 0.184165i
\(233\) −252.829 −1.08510 −0.542552 0.840022i \(-0.682542\pi\)
−0.542552 + 0.840022i \(0.682542\pi\)
\(234\) −12.6898 + 79.5746i −0.0542298 + 0.340063i
\(235\) 0 0
\(236\) 0.533601i 0.00226102i
\(237\) 0.691627 8.72887i 0.00291826 0.0368307i
\(238\) 19.3544i 0.0813210i
\(239\) 197.651i 0.826990i −0.910506 0.413495i \(-0.864308\pi\)
0.910506 0.413495i \(-0.135692\pi\)
\(240\) 0 0
\(241\) −330.829 −1.37274 −0.686368 0.727254i \(-0.740796\pi\)
−0.686368 + 0.727254i \(0.740796\pi\)
\(242\) −32.6706 −0.135002
\(243\) −93.5860 + 224.256i −0.385128 + 0.922863i
\(244\) −35.3252 −0.144775
\(245\) 0 0
\(246\) −266.737 21.1347i −1.08430 0.0859136i
\(247\) 65.1319i 0.263692i
\(248\) 36.8770 0.148698
\(249\) 20.9717 264.679i 0.0842235 1.06297i
\(250\) 0 0
\(251\) 17.1119i 0.0681750i 0.999419 + 0.0340875i \(0.0108525\pi\)
−0.999419 + 0.0340875i \(0.989148\pi\)
\(252\) 7.49976 47.0293i 0.0297609 0.186624i
\(253\) 150.968i 0.596711i
\(254\) 182.404i 0.718127i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −43.8481 −0.170615 −0.0853075 0.996355i \(-0.527187\pi\)
−0.0853075 + 0.996355i \(0.527187\pi\)
\(258\) 80.4604 + 6.37523i 0.311862 + 0.0247102i
\(259\) 94.8245 0.366118
\(260\) 0 0
\(261\) −21.4101 + 134.258i −0.0820311 + 0.514399i
\(262\) 54.7018i 0.208786i
\(263\) −244.864 −0.931042 −0.465521 0.885037i \(-0.654133\pi\)
−0.465521 + 0.885037i \(0.654133\pi\)
\(264\) −83.6942 6.63146i −0.317023 0.0251191i
\(265\) 0 0
\(266\) 38.4935i 0.144712i
\(267\) −438.275 34.7265i −1.64148 0.130062i
\(268\) 11.1128i 0.0414657i
\(269\) 250.644i 0.931762i −0.884847 0.465881i \(-0.845737\pi\)
0.884847 0.465881i \(-0.154263\pi\)
\(270\) 0 0
\(271\) 274.065 1.01131 0.505656 0.862735i \(-0.331251\pi\)
0.505656 + 0.862735i \(0.331251\pi\)
\(272\) −20.6907 −0.0760688
\(273\) 3.96913 50.0935i 0.0145389 0.183493i
\(274\) −265.074 −0.967424
\(275\) 0 0
\(276\) −7.23107 + 91.2618i −0.0261995 + 0.330659i
\(277\) 208.404i 0.752361i 0.926546 + 0.376181i \(0.122763\pi\)
−0.926546 + 0.376181i \(0.877237\pi\)
\(278\) −282.032 −1.01450
\(279\) 115.878 + 18.4790i 0.415333 + 0.0662331i
\(280\) 0 0
\(281\) 239.837i 0.853514i −0.904366 0.426757i \(-0.859656\pi\)
0.904366 0.426757i \(-0.140344\pi\)
\(282\) −3.36343 + 42.4491i −0.0119271 + 0.150529i
\(283\) 306.227i 1.08207i 0.840999 + 0.541036i \(0.181968\pi\)
−0.840999 + 0.541036i \(0.818032\pi\)
\(284\) 279.124i 0.982832i
\(285\) 0 0
\(286\) −88.5876 −0.309747
\(287\) 166.861 0.581397
\(288\) 50.2764 + 8.01758i 0.174571 + 0.0278388i
\(289\) −262.243 −0.907417
\(290\) 0 0
\(291\) 501.034 + 39.6991i 1.72177 + 0.136423i
\(292\) 65.5750i 0.224572i
\(293\) −409.157 −1.39644 −0.698220 0.715883i \(-0.746024\pi\)
−0.698220 + 0.715883i \(0.746024\pi\)
\(294\) −2.34579 + 29.6057i −0.00797888 + 0.100700i
\(295\) 0 0
\(296\) 101.372i 0.342472i
\(297\) −259.667 62.7769i −0.874300 0.211370i
\(298\) 252.959i 0.848857i
\(299\) 96.5977i 0.323069i
\(300\) 0 0
\(301\) −50.3331 −0.167220
\(302\) −336.550 −1.11440
\(303\) 233.142 + 18.4728i 0.769444 + 0.0609665i
\(304\) −41.1513 −0.135366
\(305\) 0 0
\(306\) −65.0159 10.3681i −0.212470 0.0338826i
\(307\) 61.0085i 0.198725i −0.995051 0.0993623i \(-0.968320\pi\)
0.995051 0.0993623i \(-0.0316803\pi\)
\(308\) 52.3561 0.169987
\(309\) −50.9129 4.03405i −0.164767 0.0130552i
\(310\) 0 0
\(311\) 332.103i 1.06786i 0.845530 + 0.533928i \(0.179285\pi\)
−0.845530 + 0.533928i \(0.820715\pi\)
\(312\) 53.5522 + 4.24318i 0.171642 + 0.0135999i
\(313\) 269.682i 0.861603i 0.902447 + 0.430802i \(0.141769\pi\)
−0.902447 + 0.430802i \(0.858231\pi\)
\(314\) 301.993i 0.961762i
\(315\) 0 0
\(316\) −5.83748 −0.0184730
\(317\) 221.234 0.697899 0.348949 0.937142i \(-0.386539\pi\)
0.348949 + 0.937142i \(0.386539\pi\)
\(318\) 24.2122 305.577i 0.0761390 0.960934i
\(319\) −149.465 −0.468542
\(320\) 0 0
\(321\) −44.9157 + 566.871i −0.139924 + 1.76595i
\(322\) 57.0901i 0.177298i
\(323\) 53.2156 0.164754
\(324\) 153.965 + 50.3869i 0.475200 + 0.155515i
\(325\) 0 0
\(326\) 96.4523i 0.295866i
\(327\) −18.2647 + 230.515i −0.0558553 + 0.704938i
\(328\) 178.382i 0.543847i
\(329\) 26.5547i 0.0807132i
\(330\) 0 0
\(331\) 151.066 0.456391 0.228196 0.973615i \(-0.426717\pi\)
0.228196 + 0.973615i \(0.426717\pi\)
\(332\) −177.005 −0.533149
\(333\) −50.7972 + 318.538i −0.152544 + 0.956570i
\(334\) −107.436 −0.321665
\(335\) 0 0
\(336\) −31.6498 2.50775i −0.0941959 0.00746356i
\(337\) 386.598i 1.14717i −0.819144 0.573587i \(-0.805551\pi\)
0.819144 0.573587i \(-0.194449\pi\)
\(338\) −182.319 −0.539405
\(339\) 5.27626 66.5905i 0.0155642 0.196432i
\(340\) 0 0
\(341\) 129.003i 0.378307i
\(342\) −129.309 20.6209i −0.378096 0.0602949i
\(343\) 18.5203i 0.0539949i
\(344\) 53.8084i 0.156420i
\(345\) 0 0
\(346\) 257.624 0.744579
\(347\) −101.455 −0.292378 −0.146189 0.989257i \(-0.546701\pi\)
−0.146189 + 0.989257i \(0.546701\pi\)
\(348\) 90.3531 + 7.15907i 0.259635 + 0.0205721i
\(349\) −201.227 −0.576581 −0.288290 0.957543i \(-0.593087\pi\)
−0.288290 + 0.957543i \(0.593087\pi\)
\(350\) 0 0
\(351\) 166.150 + 40.1682i 0.473361 + 0.114439i
\(352\) 55.9710i 0.159008i
\(353\) −324.605 −0.919560 −0.459780 0.888033i \(-0.652072\pi\)
−0.459780 + 0.888033i \(0.652072\pi\)
\(354\) −1.12840 0.0894083i −0.00318758 0.000252566i
\(355\) 0 0
\(356\) 293.099i 0.823313i
\(357\) 40.9286 + 3.24295i 0.114646 + 0.00908390i
\(358\) 358.850i 1.00237i
\(359\) 71.5032i 0.199173i −0.995029 0.0995866i \(-0.968248\pi\)
0.995029 0.0995866i \(-0.0317520\pi\)
\(360\) 0 0
\(361\) −255.161 −0.706816
\(362\) 368.192 1.01711
\(363\) −5.47416 + 69.0882i −0.0150803 + 0.190326i
\(364\) −33.5003 −0.0920339
\(365\) 0 0
\(366\) −5.91896 + 74.7020i −0.0161720 + 0.204104i
\(367\) 120.662i 0.328780i −0.986395 0.164390i \(-0.947434\pi\)
0.986395 0.164390i \(-0.0525656\pi\)
\(368\) 61.0319 0.165847
\(369\) −89.3869 + 560.525i −0.242241 + 1.51904i
\(370\) 0 0
\(371\) 191.158i 0.515251i
\(372\) 6.17898 77.9836i 0.0166102 0.209633i
\(373\) 336.650i 0.902547i 0.892386 + 0.451273i \(0.149030\pi\)
−0.892386 + 0.451273i \(0.850970\pi\)
\(374\) 72.3800i 0.193529i
\(375\) 0 0
\(376\) 28.3881 0.0755003
\(377\) 95.6359 0.253676
\(378\) −98.1958 23.7397i −0.259777 0.0628035i
\(379\) 190.311 0.502141 0.251071 0.967969i \(-0.419217\pi\)
0.251071 + 0.967969i \(0.419217\pi\)
\(380\) 0 0
\(381\) −385.729 30.5630i −1.01241 0.0802179i
\(382\) 68.5012i 0.179322i
\(383\) −608.518 −1.58882 −0.794409 0.607383i \(-0.792219\pi\)
−0.794409 + 0.607383i \(0.792219\pi\)
\(384\) 2.68090 33.8351i 0.00698152 0.0881122i
\(385\) 0 0
\(386\) 146.065i 0.378407i
\(387\) 26.9633 169.081i 0.0696726 0.436901i
\(388\) 335.069i 0.863581i
\(389\) 72.7694i 0.187068i 0.995616 + 0.0935339i \(0.0298164\pi\)
−0.995616 + 0.0935339i \(0.970184\pi\)
\(390\) 0 0
\(391\) −78.9245 −0.201853
\(392\) 19.7990 0.0505076
\(393\) 115.678 + 9.16564i 0.294345 + 0.0233222i
\(394\) 468.020 1.18787
\(395\) 0 0
\(396\) −28.0470 + 175.876i −0.0708257 + 0.444132i
\(397\) 188.473i 0.474744i −0.971419 0.237372i \(-0.923714\pi\)
0.971419 0.237372i \(-0.0762860\pi\)
\(398\) −359.396 −0.903004
\(399\) 81.4020 + 6.44984i 0.204015 + 0.0161650i
\(400\) 0 0
\(401\) 675.476i 1.68448i 0.539104 + 0.842240i \(0.318763\pi\)
−0.539104 + 0.842240i \(0.681237\pi\)
\(402\) 23.5002 + 1.86202i 0.0584582 + 0.00463190i
\(403\) 82.5432i 0.204822i
\(404\) 155.915i 0.385928i
\(405\) 0 0
\(406\) −56.5217 −0.139216
\(407\) −354.617 −0.871295
\(408\) −3.46686 + 43.7545i −0.00849721 + 0.107241i
\(409\) −658.803 −1.61077 −0.805383 0.592755i \(-0.798040\pi\)
−0.805383 + 0.592755i \(0.798040\pi\)
\(410\) 0 0
\(411\) −44.4149 + 560.550i −0.108065 + 1.36387i
\(412\) 34.0483i 0.0826415i
\(413\) 0.705888 0.00170917
\(414\) 191.779 + 30.5830i 0.463234 + 0.0738719i
\(415\) 0 0
\(416\) 35.8134i 0.0860898i
\(417\) −47.2562 + 596.410i −0.113324 + 1.43024i
\(418\) 143.955i 0.344390i
\(419\) 499.544i 1.19223i 0.802899 + 0.596115i \(0.203290\pi\)
−0.802899 + 0.596115i \(0.796710\pi\)
\(420\) 0 0
\(421\) −322.054 −0.764974 −0.382487 0.923961i \(-0.624932\pi\)
−0.382487 + 0.923961i \(0.624932\pi\)
\(422\) −162.751 −0.385665
\(423\) 89.2033 + 14.2252i 0.210883 + 0.0336294i
\(424\) −204.357 −0.481973
\(425\) 0 0
\(426\) −590.262 46.7690i −1.38559 0.109786i
\(427\) 46.7309i 0.109440i
\(428\) 379.099 0.885745
\(429\) −14.8434 + 187.336i −0.0346001 + 0.436680i
\(430\) 0 0
\(431\) 188.011i 0.436221i −0.975924 0.218110i \(-0.930011\pi\)
0.975924 0.218110i \(-0.0699893\pi\)
\(432\) 25.3789 104.976i 0.0587473 0.242999i
\(433\) 450.861i 1.04125i −0.853785 0.520625i \(-0.825699\pi\)
0.853785 0.520625i \(-0.174301\pi\)
\(434\) 48.7837i 0.112405i
\(435\) 0 0
\(436\) 154.158 0.353574
\(437\) −156.971 −0.359202
\(438\) 138.671 + 10.9875i 0.316600 + 0.0250856i
\(439\) 25.6728 0.0584801 0.0292400 0.999572i \(-0.490691\pi\)
0.0292400 + 0.999572i \(0.490691\pi\)
\(440\) 0 0
\(441\) 62.2139 + 9.92125i 0.141075 + 0.0224972i
\(442\) 46.3128i 0.104780i
\(443\) −128.613 −0.290322 −0.145161 0.989408i \(-0.546370\pi\)
−0.145161 + 0.989408i \(0.546370\pi\)
\(444\) 214.370 + 16.9855i 0.482815 + 0.0382556i
\(445\) 0 0
\(446\) 431.222i 0.966866i
\(447\) 534.931 + 42.3850i 1.19671 + 0.0948210i
\(448\) 21.1660i 0.0472456i
\(449\) 746.161i 1.66183i −0.556401 0.830914i \(-0.687818\pi\)
0.556401 0.830914i \(-0.312182\pi\)
\(450\) 0 0
\(451\) −624.013 −1.38362
\(452\) −44.5328 −0.0985239
\(453\) −56.3911 + 711.700i −0.124484 + 1.57108i
\(454\) 290.526 0.639925
\(455\) 0 0
\(456\) −6.89516 + 87.0224i −0.0151210 + 0.190838i
\(457\) 861.637i 1.88542i −0.333612 0.942710i \(-0.608268\pi\)
0.333612 0.942710i \(-0.391732\pi\)
\(458\) −443.491 −0.968320
\(459\) −32.8192 + 135.752i −0.0715014 + 0.295755i
\(460\) 0 0
\(461\) 119.284i 0.258750i 0.991596 + 0.129375i \(0.0412972\pi\)
−0.991596 + 0.129375i \(0.958703\pi\)
\(462\) 8.77259 110.717i 0.0189883 0.239647i
\(463\) 815.164i 1.76061i −0.474406 0.880306i \(-0.657337\pi\)
0.474406 0.880306i \(-0.342663\pi\)
\(464\) 60.4242i 0.130225i
\(465\) 0 0
\(466\) 357.555 0.767285
\(467\) 396.290 0.848587 0.424293 0.905525i \(-0.360523\pi\)
0.424293 + 0.905525i \(0.360523\pi\)
\(468\) 17.9460 112.536i 0.0383462 0.240460i
\(469\) −14.7009 −0.0313452
\(470\) 0 0
\(471\) 638.623 + 50.6009i 1.35589 + 0.107433i
\(472\) 0.754626i 0.00159878i
\(473\) 188.232 0.397953
\(474\) −0.978108 + 12.3445i −0.00206352 + 0.0260432i
\(475\) 0 0
\(476\) 27.3712i 0.0575026i
\(477\) −642.145 102.403i −1.34622 0.214681i
\(478\) 279.520i 0.584770i
\(479\) 757.324i 1.58105i −0.612429 0.790526i \(-0.709807\pi\)
0.612429 0.790526i \(-0.290193\pi\)
\(480\) 0 0
\(481\) 226.904 0.471734
\(482\) 467.864 0.970671
\(483\) −120.728 9.56581i −0.249954 0.0198050i
\(484\) 46.2031 0.0954610
\(485\) 0 0
\(486\) 132.351 317.146i 0.272326 0.652563i
\(487\) 286.196i 0.587672i 0.955856 + 0.293836i \(0.0949319\pi\)
−0.955856 + 0.293836i \(0.905068\pi\)
\(488\) 49.9574 0.102372
\(489\) −203.967 16.1612i −0.417110 0.0330495i
\(490\) 0 0
\(491\) 825.348i 1.68095i −0.541848 0.840476i \(-0.682275\pi\)
0.541848 0.840476i \(-0.317725\pi\)
\(492\) 377.223 + 29.8890i 0.766713 + 0.0607501i
\(493\) 78.1387i 0.158496i
\(494\) 92.1104i 0.186458i
\(495\) 0 0
\(496\) −52.1520 −0.105145
\(497\) 369.247 0.742951
\(498\) −29.6584 + 374.312i −0.0595550 + 0.751631i
\(499\) −384.354 −0.770249 −0.385125 0.922865i \(-0.625842\pi\)
−0.385125 + 0.922865i \(0.625842\pi\)
\(500\) 0 0
\(501\) −18.0016 + 227.195i −0.0359314 + 0.453482i
\(502\) 24.1999i 0.0482070i
\(503\) 84.4100 0.167813 0.0839066 0.996474i \(-0.473260\pi\)
0.0839066 + 0.996474i \(0.473260\pi\)
\(504\) −10.6063 + 66.5095i −0.0210442 + 0.131963i
\(505\) 0 0
\(506\) 213.501i 0.421938i
\(507\) −30.5487 + 385.548i −0.0602538 + 0.760450i
\(508\) 257.959i 0.507793i
\(509\) 111.260i 0.218585i −0.994010 0.109293i \(-0.965141\pi\)
0.994010 0.109293i \(-0.0348585\pi\)
\(510\) 0 0
\(511\) −86.7476 −0.169760
\(512\) −22.6274 −0.0441942
\(513\) −65.2733 + 269.993i −0.127238 + 0.526302i
\(514\) 62.0105 0.120643
\(515\) 0 0
\(516\) −113.788 9.01594i −0.220520 0.0174727i
\(517\) 99.3069i 0.192083i
\(518\) −134.102 −0.258884
\(519\) 43.1666 544.796i 0.0831726 1.04970i
\(520\) 0 0
\(521\) 227.355i 0.436383i −0.975906 0.218191i \(-0.929984\pi\)
0.975906 0.218191i \(-0.0700157\pi\)
\(522\) 30.2785 189.870i 0.0580048 0.363735i
\(523\) 957.094i 1.83001i −0.403446 0.915004i \(-0.632188\pi\)
0.403446 0.915004i \(-0.367812\pi\)
\(524\) 77.3601i 0.147634i
\(525\) 0 0
\(526\) 346.290 0.658346
\(527\) 67.4414 0.127972
\(528\) 118.361 + 9.37829i 0.224169 + 0.0177619i
\(529\) −296.195 −0.559914
\(530\) 0 0
\(531\) −0.378142 + 2.37124i −0.000712132 + 0.00446562i
\(532\) 54.4381i 0.102327i
\(533\) 399.279 0.749116
\(534\) 619.815 + 49.1107i 1.16070 + 0.0919676i
\(535\) 0 0
\(536\) 15.7159i 0.0293207i
\(537\) 758.857 + 60.1276i 1.41314 + 0.111969i
\(538\) 354.464i 0.658855i
\(539\) 69.2605i 0.128498i
\(540\) 0 0
\(541\) −179.389 −0.331587 −0.165793 0.986161i \(-0.553019\pi\)
−0.165793 + 0.986161i \(0.553019\pi\)
\(542\) −387.587 −0.715105
\(543\) 61.6930 778.614i 0.113615 1.43391i
\(544\) 29.2611 0.0537888
\(545\) 0 0
\(546\) −5.61320 + 70.8429i −0.0102806 + 0.129749i
\(547\) 182.706i 0.334014i 0.985956 + 0.167007i \(0.0534102\pi\)
−0.985956 + 0.167007i \(0.946590\pi\)
\(548\) 374.872 0.684072
\(549\) 156.980 + 25.0336i 0.285938 + 0.0455985i
\(550\) 0 0
\(551\) 155.408i 0.282048i
\(552\) 10.2263 129.064i 0.0185259 0.233811i
\(553\) 7.72226i 0.0139643i
\(554\) 294.728i 0.532000i
\(555\) 0 0
\(556\) 398.853 0.717362
\(557\) −591.471 −1.06189 −0.530943 0.847407i \(-0.678162\pi\)
−0.530943 + 0.847407i \(0.678162\pi\)
\(558\) −163.876 26.1333i −0.293685 0.0468339i
\(559\) −120.441 −0.215458
\(560\) 0 0
\(561\) −153.061 12.1277i −0.272837 0.0216181i
\(562\) 339.181i 0.603526i
\(563\) −97.5678 −0.173300 −0.0866499 0.996239i \(-0.527616\pi\)
−0.0866499 + 0.996239i \(0.527616\pi\)
\(564\) 4.75661 60.0321i 0.00843371 0.106440i
\(565\) 0 0
\(566\) 433.070i 0.765141i
\(567\) −66.6556 + 203.676i −0.117558 + 0.359217i
\(568\) 394.741i 0.694967i
\(569\) 604.562i 1.06250i 0.847215 + 0.531250i \(0.178277\pi\)
−0.847215 + 0.531250i \(0.821723\pi\)
\(570\) 0 0
\(571\) 651.161 1.14039 0.570194 0.821510i \(-0.306868\pi\)
0.570194 + 0.821510i \(0.306868\pi\)
\(572\) 125.282 0.219024
\(573\) 144.859 + 11.4778i 0.252808 + 0.0200311i
\(574\) −235.977 −0.411110
\(575\) 0 0
\(576\) −71.1016 11.3386i −0.123440 0.0196850i
\(577\) 18.4946i 0.0320531i −0.999872 0.0160265i \(-0.994898\pi\)
0.999872 0.0160265i \(-0.00510162\pi\)
\(578\) 370.868 0.641640
\(579\) −308.883 24.4742i −0.533477 0.0422697i
\(580\) 0 0
\(581\) 234.156i 0.403023i
\(582\) −708.569 56.1430i −1.21747 0.0964657i
\(583\) 714.877i 1.22620i
\(584\) 92.7371i 0.158796i
\(585\) 0 0
\(586\) 578.635 0.987432
\(587\) 704.219 1.19969 0.599846 0.800115i \(-0.295228\pi\)
0.599846 + 0.800115i \(0.295228\pi\)
\(588\) 3.31745 41.8688i 0.00564192 0.0712054i
\(589\) 134.133 0.227729
\(590\) 0 0
\(591\) 78.4198 989.719i 0.132690 1.67465i
\(592\) 143.361i 0.242164i
\(593\) 730.470 1.23182 0.615911 0.787816i \(-0.288788\pi\)
0.615911 + 0.787816i \(0.288788\pi\)
\(594\) 367.225 + 88.7799i 0.618224 + 0.149461i
\(595\) 0 0
\(596\) 357.739i 0.600233i
\(597\) −60.2191 + 760.012i −0.100869 + 1.27305i
\(598\) 136.610i 0.228444i
\(599\) 486.764i 0.812627i 0.913734 + 0.406314i \(0.133186\pi\)
−0.913734 + 0.406314i \(0.866814\pi\)
\(600\) 0 0
\(601\) 486.097 0.808814 0.404407 0.914579i \(-0.367478\pi\)
0.404407 + 0.914579i \(0.367478\pi\)
\(602\) 71.1818 0.118242
\(603\) 7.87522 49.3837i 0.0130601 0.0818967i
\(604\) 475.953 0.788002
\(605\) 0 0
\(606\) −329.712 26.1245i −0.544079 0.0431098i
\(607\) 815.911i 1.34417i 0.740474 + 0.672085i \(0.234601\pi\)
−0.740474 + 0.672085i \(0.765399\pi\)
\(608\) 58.1967 0.0957183
\(609\) −9.47057 + 119.526i −0.0155510 + 0.196266i
\(610\) 0 0
\(611\) 63.5421i 0.103997i
\(612\) 91.9464 + 14.6627i 0.150239 + 0.0239587i
\(613\) 387.354i 0.631899i 0.948776 + 0.315949i \(0.102323\pi\)
−0.948776 + 0.315949i \(0.897677\pi\)
\(614\) 86.2790i 0.140520i
\(615\) 0 0
\(616\) −74.0426 −0.120199
\(617\) −1021.58 −1.65572 −0.827860 0.560935i \(-0.810442\pi\)
−0.827860 + 0.560935i \(0.810442\pi\)
\(618\) 72.0017 + 5.70501i 0.116508 + 0.00923141i
\(619\) 529.302 0.855091 0.427546 0.903994i \(-0.359378\pi\)
0.427546 + 0.903994i \(0.359378\pi\)
\(620\) 0 0
\(621\) 96.8074 400.429i 0.155890 0.644813i
\(622\) 469.665i 0.755089i
\(623\) −387.734 −0.622366
\(624\) −75.7343 6.00076i −0.121369 0.00961660i
\(625\) 0 0
\(626\) 381.388i 0.609246i
\(627\) −304.420 24.1206i −0.485519 0.0384698i
\(628\) 427.083i 0.680069i
\(629\) 185.390i 0.294738i
\(630\) 0 0
\(631\) −504.590 −0.799667 −0.399833 0.916588i \(-0.630932\pi\)
−0.399833 + 0.916588i \(0.630932\pi\)
\(632\) 8.25545 0.0130624
\(633\) −27.2699 + 344.168i −0.0430805 + 0.543709i
\(634\) −312.872 −0.493489
\(635\) 0 0
\(636\) −34.2412 + 432.151i −0.0538384 + 0.679483i
\(637\) 44.3168i 0.0695711i
\(638\) 211.375 0.331309
\(639\) −197.804 + 1240.39i −0.309553 + 1.94114i
\(640\) 0 0
\(641\) 1067.75i 1.66575i −0.553459 0.832877i \(-0.686692\pi\)
0.553459 0.832877i \(-0.313308\pi\)
\(642\) 63.5204 801.677i 0.0989414 1.24872i
\(643\) 575.185i 0.894534i −0.894401 0.447267i \(-0.852397\pi\)
0.894401 0.447267i \(-0.147603\pi\)
\(644\) 80.7376i 0.125369i
\(645\) 0 0
\(646\) −75.2583 −0.116499
\(647\) −998.983 −1.54402 −0.772012 0.635608i \(-0.780749\pi\)
−0.772012 + 0.635608i \(0.780749\pi\)
\(648\) −217.739 71.2578i −0.336017 0.109966i
\(649\) −2.63982 −0.00406752
\(650\) 0 0
\(651\) 103.163 + 8.17402i 0.158468 + 0.0125561i
\(652\) 136.404i 0.209209i
\(653\) 473.061 0.724442 0.362221 0.932092i \(-0.382019\pi\)
0.362221 + 0.932092i \(0.382019\pi\)
\(654\) 25.8302 325.997i 0.0394957 0.498467i
\(655\) 0 0
\(656\) 252.270i 0.384558i
\(657\) 46.4704 291.405i 0.0707312 0.443539i
\(658\) 37.5539i 0.0570729i
\(659\) 732.225i 1.11111i −0.831478 0.555557i \(-0.812505\pi\)
0.831478 0.555557i \(-0.187495\pi\)
\(660\) 0 0
\(661\) −682.586 −1.03266 −0.516328 0.856391i \(-0.672701\pi\)
−0.516328 + 0.856391i \(0.672701\pi\)
\(662\) −213.639 −0.322717
\(663\) 97.9373 + 7.76000i 0.147718 + 0.0117044i
\(664\) 250.324 0.376993
\(665\) 0 0
\(666\) 71.8381 450.481i 0.107865 0.676397i
\(667\) 230.488i 0.345559i
\(668\) 151.938 0.227452
\(669\) 911.902 + 72.2540i 1.36308 + 0.108003i
\(670\) 0 0
\(671\) 174.760i 0.260447i
\(672\) 44.7596 + 3.54650i 0.0666066 + 0.00527753i
\(673\) 621.860i 0.924012i −0.886877 0.462006i \(-0.847130\pi\)
0.886877 0.462006i \(-0.152870\pi\)
\(674\) 546.732i 0.811175i
\(675\) 0 0
\(676\) 257.838 0.381417
\(677\) 1249.46 1.84559 0.922794 0.385293i \(-0.125900\pi\)
0.922794 + 0.385293i \(0.125900\pi\)
\(678\) −7.46175 + 94.1732i −0.0110055 + 0.138898i
\(679\) 443.255 0.652806
\(680\) 0 0
\(681\) 48.6795 614.373i 0.0714823 0.902163i
\(682\) 182.437i 0.267503i
\(683\) 161.483 0.236432 0.118216 0.992988i \(-0.462282\pi\)
0.118216 + 0.992988i \(0.462282\pi\)
\(684\) 182.870 + 29.1623i 0.267354 + 0.0426349i
\(685\) 0 0
\(686\) 26.1916i 0.0381802i
\(687\) −74.3097 + 937.846i −0.108165 + 1.36513i
\(688\) 76.0965i 0.110605i
\(689\) 457.419i 0.663888i
\(690\) 0 0
\(691\) −520.487 −0.753237 −0.376619 0.926368i \(-0.622913\pi\)
−0.376619 + 0.926368i \(0.622913\pi\)
\(692\) −364.336 −0.526497
\(693\) −232.662 37.1027i −0.335732 0.0535392i
\(694\) 143.479 0.206742
\(695\) 0 0
\(696\) −127.779 10.1245i −0.183590 0.0145466i
\(697\) 326.228i 0.468046i
\(698\) 284.578 0.407704
\(699\) 59.9106 756.118i 0.0857090 1.08171i
\(700\) 0 0
\(701\) 1017.26i 1.45115i −0.688142 0.725576i \(-0.741573\pi\)
0.688142 0.725576i \(-0.258427\pi\)
\(702\) −234.971 56.8064i −0.334717 0.0809208i
\(703\) 368.719i 0.524493i
\(704\) 79.1549i 0.112436i
\(705\) 0 0
\(706\) 459.060 0.650227
\(707\) 206.256 0.291734
\(708\) 1.59580 + 0.126442i 0.00225396 + 0.000178591i
\(709\) −1317.44 −1.85817 −0.929086 0.369863i \(-0.879405\pi\)
−0.929086 + 0.369863i \(0.879405\pi\)
\(710\) 0 0
\(711\) 25.9409 + 4.13679i 0.0364851 + 0.00581828i
\(712\) 414.505i 0.582170i
\(713\) −198.933 −0.279009
\(714\) −57.8818 4.58623i −0.0810669 0.00642329i
\(715\) 0 0
\(716\) 507.491i 0.708786i
\(717\) 591.099 + 46.8354i 0.824406 + 0.0653213i
\(718\) 101.121i 0.140837i
\(719\) 8.42284i 0.0117147i 0.999983 + 0.00585733i \(0.00186446\pi\)
−0.999983 + 0.00585733i \(0.998136\pi\)
\(720\) 0 0
\(721\) −45.0417 −0.0624711
\(722\) 360.852 0.499795
\(723\) 78.3935 989.388i 0.108428 1.36845i
\(724\) −520.703 −0.719202
\(725\) 0 0
\(726\) 7.74163 97.7054i 0.0106634 0.134580i
\(727\) 540.554i 0.743541i 0.928325 + 0.371770i \(0.121249\pi\)
−0.928325 + 0.371770i \(0.878751\pi\)
\(728\) 47.3766 0.0650778
\(729\) −648.489 333.021i −0.889560 0.456818i
\(730\) 0 0
\(731\) 98.4058i 0.134618i
\(732\) 8.37068 105.645i 0.0114354 0.144323i
\(733\) 581.326i 0.793078i 0.918018 + 0.396539i \(0.129789\pi\)
−0.918018 + 0.396539i \(0.870211\pi\)
\(734\) 170.642i 0.232483i
\(735\) 0 0
\(736\) −86.3121 −0.117272
\(737\) 54.9771 0.0745959
\(738\) 126.412 792.702i 0.171290 1.07412i
\(739\) 886.532 1.19964 0.599819 0.800136i \(-0.295239\pi\)
0.599819 + 0.800136i \(0.295239\pi\)
\(740\) 0 0
\(741\) 194.785 + 15.4337i 0.262868 + 0.0208282i
\(742\) 270.338i 0.364337i
\(743\) 1466.84 1.97421 0.987106 0.160066i \(-0.0511706\pi\)
0.987106 + 0.160066i \(0.0511706\pi\)
\(744\) −8.73840 + 110.285i −0.0117452 + 0.148233i
\(745\) 0 0
\(746\) 476.095i 0.638197i
\(747\) 786.586 + 125.437i 1.05299 + 0.167921i
\(748\) 102.361i 0.136846i
\(749\) 501.500i 0.669560i
\(750\) 0 0
\(751\) −578.835 −0.770752 −0.385376 0.922760i \(-0.625928\pi\)
−0.385376 + 0.922760i \(0.625928\pi\)
\(752\) −40.1469 −0.0533868
\(753\) −51.1754 4.05485i −0.0679620 0.00538493i
\(754\) −135.250 −0.179376
\(755\) 0 0
\(756\) 138.870 + 33.5731i 0.183690 + 0.0444088i
\(757\) 318.935i 0.421314i −0.977560 0.210657i \(-0.932440\pi\)
0.977560 0.210657i \(-0.0675603\pi\)
\(758\) −269.141 −0.355067
\(759\) 451.488 + 35.7734i 0.594846 + 0.0471323i
\(760\) 0 0
\(761\) 1038.56i 1.36473i 0.731010 + 0.682367i \(0.239049\pi\)
−0.731010 + 0.682367i \(0.760951\pi\)
\(762\) 545.503 + 43.2226i 0.715884 + 0.0567226i
\(763\) 203.932i 0.267277i
\(764\) 96.8753i 0.126800i
\(765\) 0 0
\(766\) 860.574 1.12346
\(767\) 1.68911 0.00220223
\(768\) −3.79137 + 47.8500i −0.00493668 + 0.0623047i
\(769\) −1061.13 −1.37989 −0.689943 0.723863i \(-0.742365\pi\)
−0.689943 + 0.723863i \(0.742365\pi\)
\(770\) 0 0
\(771\) 10.3903 131.133i 0.0134763 0.170082i
\(772\) 206.567i 0.267574i
\(773\) 413.194 0.534533 0.267266 0.963623i \(-0.413880\pi\)
0.267266 + 0.963623i \(0.413880\pi\)
\(774\) −38.1319 + 239.116i −0.0492660 + 0.308936i
\(775\) 0 0
\(776\) 473.860i 0.610644i
\(777\) −22.4697 + 283.585i −0.0289185 + 0.364974i
\(778\) 102.911i 0.132277i
\(779\) 648.828i 0.832898i
\(780\) 0 0
\(781\) −1380.88 −1.76809
\(782\) 111.616 0.142732
\(783\) −396.442 95.8435i −0.506312 0.122406i
\(784\) −28.0000 −0.0357143
\(785\) 0 0
\(786\) −163.593 12.9622i −0.208133 0.0164913i
\(787\) 470.548i 0.597901i 0.954269 + 0.298951i \(0.0966366\pi\)
−0.954269 + 0.298951i \(0.903363\pi\)
\(788\) −661.881 −0.839950
\(789\) 58.0231 732.297i 0.0735401 0.928133i
\(790\) 0 0
\(791\) 58.9113i 0.0744770i
\(792\) 39.6644 248.727i 0.0500813 0.314049i
\(793\) 111.821i 0.141011i
\(794\) 266.542i 0.335695i
\(795\) 0 0
\(796\) 508.262 0.638521
\(797\) 1377.25 1.72805 0.864023 0.503453i \(-0.167937\pi\)
0.864023 + 0.503453i \(0.167937\pi\)
\(798\) −115.120 9.12144i −0.144260 0.0114304i
\(799\) 51.9167 0.0649771
\(800\) 0 0
\(801\) 207.708 1302.49i 0.259311 1.62608i
\(802\) 955.267i 1.19111i
\(803\) 324.411 0.403999
\(804\) −33.2343 2.63330i −0.0413362 0.00327525i
\(805\) 0 0
\(806\) 116.734i 0.144831i
\(807\) 749.583 + 59.3927i 0.928851 + 0.0735969i
\(808\) 220.497i 0.272892i
\(809\) 48.7456i 0.0602541i −0.999546 0.0301271i \(-0.990409\pi\)
0.999546 0.0301271i \(-0.00959119\pi\)
\(810\) 0 0
\(811\) −154.380 −0.190357 −0.0951786 0.995460i \(-0.530342\pi\)
−0.0951786 + 0.995460i \(0.530342\pi\)
\(812\) 79.9337 0.0984405
\(813\) −64.9427 + 819.627i −0.0798803 + 1.00815i
\(814\) 501.504 0.616099
\(815\) 0 0
\(816\) 4.90288 61.8782i 0.00600844 0.0758311i
\(817\) 195.717i 0.239556i
\(818\) 931.688 1.13898
\(819\) 148.870 + 23.7404i 0.181771 + 0.0289870i
\(820\) 0 0
\(821\) 882.281i 1.07464i 0.843378 + 0.537321i \(0.180564\pi\)
−0.843378 + 0.537321i \(0.819436\pi\)
\(822\) 62.8121 792.738i 0.0764138 0.964402i
\(823\) 342.633i 0.416323i 0.978095 + 0.208161i \(0.0667479\pi\)
−0.978095 + 0.208161i \(0.933252\pi\)
\(824\) 48.1516i 0.0584364i
\(825\) 0 0
\(826\) −0.998277 −0.00120857
\(827\) −380.344 −0.459908 −0.229954 0.973202i \(-0.573858\pi\)
−0.229954 + 0.973202i \(0.573858\pi\)
\(828\) −271.216 43.2509i −0.327556 0.0522353i
\(829\) 273.732 0.330196 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(830\) 0 0
\(831\) −623.259 49.3835i −0.750010 0.0594266i
\(832\) 50.6478i 0.0608747i
\(833\) 36.2088 0.0434679
\(834\) 66.8304 843.452i 0.0801324 1.01133i
\(835\) 0 0
\(836\) 203.583i 0.243520i
\(837\) −82.7223 + 342.168i −0.0988320 + 0.408803i
\(838\) 706.463i 0.843034i
\(839\) 1377.92i 1.64234i 0.570683 + 0.821171i \(0.306679\pi\)
−0.570683 + 0.821171i \(0.693321\pi\)
\(840\) 0 0
\(841\) 612.807 0.728665
\(842\) 455.453 0.540918
\(843\) 717.264 + 56.8320i 0.850847 + 0.0674164i
\(844\) 230.164 0.272707
\(845\) 0 0
\(846\) −126.153 20.1175i −0.149116 0.0237796i
\(847\) 61.1210i 0.0721618i
\(848\) 289.004 0.340806
\(849\) −915.809 72.5636i −1.07869 0.0854695i
\(850\) 0 0
\(851\) 546.850i 0.642597i
\(852\) 834.756 + 66.1414i 0.979761 + 0.0776308i
\(853\) 580.625i 0.680685i 0.940301 + 0.340343i \(0.110543\pi\)
−0.940301 + 0.340343i \(0.889457\pi\)
\(854\) 66.0874i 0.0773857i
\(855\) 0 0
\(856\) −536.126 −0.626316
\(857\) −714.979 −0.834281 −0.417141 0.908842i \(-0.636968\pi\)
−0.417141 + 0.908842i \(0.636968\pi\)
\(858\) 20.9918 264.933i 0.0244659 0.308779i
\(859\) 975.849 1.13603 0.568015 0.823019i \(-0.307712\pi\)
0.568015 + 0.823019i \(0.307712\pi\)
\(860\) 0 0
\(861\) −39.5395 + 499.019i −0.0459227 + 0.579581i
\(862\) 265.888i 0.308455i
\(863\) −485.722 −0.562830 −0.281415 0.959586i \(-0.590804\pi\)
−0.281415 + 0.959586i \(0.590804\pi\)
\(864\) −35.8911 + 148.458i −0.0415406 + 0.171827i
\(865\) 0 0
\(866\) 637.614i 0.736275i
\(867\) 62.1413 784.272i 0.0716740 0.904581i
\(868\) 68.9906i 0.0794823i
\(869\) 28.8791i 0.0332326i
\(870\) 0 0
\(871\) −35.1775 −0.0403874
\(872\) −218.013 −0.250014
\(873\) −237.450 + 1489.00i −0.271994 + 1.70561i
\(874\) 221.991 0.253994
\(875\) 0 0
\(876\) −196.110 15.5387i −0.223870 0.0177382i
\(877\) 499.530i 0.569589i 0.958589 + 0.284795i \(0.0919254\pi\)
−0.958589 + 0.284795i \(0.908075\pi\)
\(878\) −36.3068 −0.0413517
\(879\) 96.9540 1223.64i 0.110300 1.39208i
\(880\) 0 0
\(881\) 74.9416i 0.0850643i −0.999095 0.0425321i \(-0.986458\pi\)
0.999095 0.0425321i \(-0.0135425\pi\)
\(882\) −87.9837 14.0308i −0.0997548 0.0159079i
\(883\) 493.010i 0.558335i −0.960242 0.279168i \(-0.909941\pi\)
0.960242 0.279168i \(-0.0900585\pi\)
\(884\) 65.4962i 0.0740907i
\(885\) 0 0
\(886\) 181.886 0.205289
\(887\) 1423.57 1.60492 0.802461 0.596705i \(-0.203524\pi\)
0.802461 + 0.596705i \(0.203524\pi\)
\(888\) −303.165 24.0211i −0.341402 0.0270508i
\(889\) −341.247 −0.383855
\(890\) 0 0
\(891\) 249.273 761.692i 0.279768 0.854873i
\(892\) 609.840i 0.683677i
\(893\) 103.256 0.115628
\(894\) −756.507 59.9414i −0.846205 0.0670485i
\(895\) 0 0
\(896\) 29.9333i 0.0334077i
\(897\) −288.888 22.8898i −0.322060 0.0255182i
\(898\) 1055.23i 1.17509i
\(899\) 196.953i 0.219080i
\(900\) 0 0
\(901\) −373.731 −0.414796
\(902\) 882.488 0.978368
\(903\) 11.9270 150.528i 0.0132082 0.166697i
\(904\) 62.9789 0.0696669
\(905\) 0 0
\(906\) 79.7490 1006.50i 0.0880232 1.11092i
\(907\) 1109.55i 1.22331i −0.791123 0.611657i \(-0.790503\pi\)
0.791123 0.611657i \(-0.209497\pi\)
\(908\) −410.866 −0.452495
\(909\) −110.491 + 692.862i −0.121552 + 0.762225i
\(910\) 0 0
\(911\) 201.238i 0.220898i −0.993882 0.110449i \(-0.964771\pi\)
0.993882 0.110449i \(-0.0352288\pi\)
\(912\) 9.75123 123.068i 0.0106921 0.134943i
\(913\) 875.678i 0.959122i
\(914\) 1218.54i 1.33319i
\(915\) 0 0
\(916\) 627.190 0.684706
\(917\) 102.338 0.111601
\(918\) 46.4133 191.982i 0.0505592 0.209130i
\(919\) 267.388 0.290956 0.145478 0.989362i \(-0.453528\pi\)
0.145478 + 0.989362i \(0.453528\pi\)
\(920\) 0 0
\(921\) 182.454 + 14.4566i 0.198104 + 0.0156966i
\(922\) 168.693i 0.182964i
\(923\) 883.563 0.957273
\(924\) −12.4063 + 156.577i −0.0134268 + 0.169456i
\(925\) 0 0
\(926\) 1152.82i 1.24494i
\(927\) 24.1287 151.306i 0.0260288 0.163221i
\(928\) 85.4527i 0.0920827i
\(929\) 599.465i 0.645280i −0.946522 0.322640i \(-0.895430\pi\)
0.946522 0.322640i \(-0.104570\pi\)
\(930\) 0 0
\(931\) 72.0148 0.0773521
\(932\) −505.659 −0.542552
\(933\) −993.198 78.6954i −1.06452 0.0843466i
\(934\) −560.439 −0.600041
\(935\) 0 0
\(936\) −25.3795 + 159.149i −0.0271149 + 0.170031i
\(937\) 804.479i 0.858569i −0.903169 0.429285i \(-0.858766\pi\)
0.903169 0.429285i \(-0.141234\pi\)
\(938\) 20.7902 0.0221644
\(939\) −806.518 63.9040i −0.858912 0.0680553i
\(940\) 0 0
\(941\) 832.160i 0.884336i 0.896932 + 0.442168i \(0.145791\pi\)
−0.896932 + 0.442168i \(0.854209\pi\)
\(942\) −903.149 71.5605i −0.958757 0.0759666i
\(943\) 962.282i 1.02045i
\(944\) 1.06720i 0.00113051i
\(945\) 0 0
\(946\) −266.200 −0.281395
\(947\) 293.644 0.310078 0.155039 0.987908i \(-0.450450\pi\)
0.155039 + 0.987908i \(0.450450\pi\)
\(948\) 1.38325 17.4577i 0.00145913 0.0184153i
\(949\) −207.577 −0.218732
\(950\) 0 0
\(951\) −52.4237 + 661.628i −0.0551248 + 0.695718i
\(952\) 38.7088i 0.0406605i
\(953\) 1309.64 1.37422 0.687112 0.726551i \(-0.258878\pi\)
0.687112 + 0.726551i \(0.258878\pi\)
\(954\) 908.130 + 144.819i 0.951918 + 0.151802i
\(955\) 0 0
\(956\) 395.301i 0.413495i
\(957\) 35.4172 446.993i 0.0370086 0.467078i
\(958\) 1071.02i 1.11797i
\(959\) 495.908i 0.517110i
\(960\) 0 0
\(961\) −791.011 −0.823112
\(962\) −320.891 −0.333566
\(963\) −1684.66 268.652i −1.74938 0.278974i
\(964\) −661.659 −0.686368
\(965\) 0 0
\(966\) 170.735 + 13.5281i 0.176744 + 0.0140042i
\(967\) 1512.04i 1.56364i 0.623503 + 0.781821i \(0.285709\pi\)
−0.623503 + 0.781821i \(0.714291\pi\)
\(968\) −65.3411 −0.0675011
\(969\) −12.6100 + 159.148i −0.0130134 + 0.164239i
\(970\) 0 0
\(971\) 1169.63i 1.20456i 0.798283 + 0.602282i \(0.205742\pi\)
−0.798283 + 0.602282i \(0.794258\pi\)
\(972\) −187.172 + 448.512i −0.192564 + 0.461432i
\(973\) 527.633i 0.542274i
\(974\) 404.742i 0.415547i
\(975\) 0 0
\(976\) −70.6504 −0.0723877
\(977\) 497.435 0.509145 0.254573 0.967054i \(-0.418065\pi\)
0.254573 + 0.967054i \(0.418065\pi\)
\(978\) 288.453 + 22.8554i 0.294942 + 0.0233695i
\(979\) 1450.02 1.48112
\(980\) 0 0
\(981\) −685.056 109.246i −0.698324 0.111362i
\(982\) 1167.22i 1.18861i
\(983\) −634.120 −0.645087 −0.322543 0.946555i \(-0.604538\pi\)
−0.322543 + 0.946555i \(0.604538\pi\)
\(984\) −533.474 42.2695i −0.542148 0.0429568i
\(985\) 0 0
\(986\) 110.505i 0.112074i
\(987\) 79.4151 + 6.29240i 0.0804611 + 0.00637528i
\(988\) 130.264i 0.131846i
\(989\) 290.270i 0.293498i
\(990\) 0 0
\(991\) −1453.55 −1.46675 −0.733375 0.679824i \(-0.762056\pi\)
−0.733375 + 0.679824i \(0.762056\pi\)
\(992\) 73.7540 0.0743488
\(993\) −35.7966 + 451.781i −0.0360489 + 0.454965i
\(994\) −522.194 −0.525346
\(995\) 0 0
\(996\) 41.9433 529.357i 0.0421118 0.531483i
\(997\) 1390.83i 1.39501i 0.716579 + 0.697506i \(0.245707\pi\)
−0.716579 + 0.697506i \(0.754293\pi\)
\(998\) 543.559 0.544649
\(999\) −940.591 227.396i −0.941533 0.227624i
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 1050.3.c.b.449.8 32
3.2 odd 2 inner 1050.3.c.b.449.26 32
5.2 odd 4 1050.3.e.c.701.8 yes 16
5.3 odd 4 1050.3.e.b.701.9 yes 16
5.4 even 2 inner 1050.3.c.b.449.25 32
15.2 even 4 1050.3.e.c.701.16 yes 16
15.8 even 4 1050.3.e.b.701.1 16
15.14 odd 2 inner 1050.3.c.b.449.7 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.3.c.b.449.7 32 15.14 odd 2 inner
1050.3.c.b.449.8 32 1.1 even 1 trivial
1050.3.c.b.449.25 32 5.4 even 2 inner
1050.3.c.b.449.26 32 3.2 odd 2 inner
1050.3.e.b.701.1 16 15.8 even 4
1050.3.e.b.701.9 yes 16 5.3 odd 4
1050.3.e.c.701.8 yes 16 5.2 odd 4
1050.3.e.c.701.16 yes 16 15.2 even 4