Properties

Label 1800.2.d.t.1549.5
Level $1800$
Weight $2$
Character 1800.1549
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
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] = [1800,2,Mod(1549,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.d (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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1549.5
Root \(1.41216 + 0.0762223i\) of defining polynomial
Character \(\chi\) \(=\) 1800.1549
Dual form 1800.2.d.t.1549.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.576222 - 1.29150i) q^{2} +(-1.33594 - 1.48838i) q^{4} +1.97676i q^{7} +(-2.69204 + 0.867721i) q^{8} +O(q^{10})\) \(q+(0.576222 - 1.29150i) q^{2} +(-1.33594 - 1.48838i) q^{4} +1.97676i q^{7} +(-2.69204 + 0.867721i) q^{8} +1.43055i q^{11} -0.241319 q^{13} +(2.55298 + 1.13905i) q^{14} +(-0.430552 + 3.97676i) q^{16} -7.38407i q^{17} -3.04033i q^{19} +(1.84756 + 0.824316i) q^{22} -0.874337i q^{23} +(-0.139054 + 0.311664i) q^{26} +(2.94217 - 2.64082i) q^{28} -9.07918i q^{29} -7.44764 q^{31} +(4.88789 + 2.84756i) q^{32} +(-9.53652 - 4.25487i) q^{34} -8.81463 q^{37} +(-3.92658 - 1.75191i) q^{38} +1.91319 q^{41} -11.2452 q^{43} +(2.12921 - 1.91113i) q^{44} +(-1.12921 - 0.503813i) q^{46} -3.34374i q^{47} +3.09242 q^{49} +(0.322387 + 0.359175i) q^{52} -9.20632 q^{53} +(-1.71528 - 5.32151i) q^{56} +(-11.7258 - 5.23163i) q^{58} -6.43616i q^{59} +4.57331i q^{61} +(-4.29150 + 9.61862i) q^{62} +(6.49412 - 4.67187i) q^{64} -4.86671 q^{67} +(-10.9903 + 9.86465i) q^{68} +8.21808 q^{71} -4.12714i q^{73} +(-5.07918 + 11.3841i) q^{74} +(-4.52517 + 4.06169i) q^{76} -2.82786 q^{77} +13.6757 q^{79} +(1.10242 - 2.47088i) q^{82} +12.3320 q^{83} +(-6.47972 + 14.5231i) q^{86} +(-1.24132 - 3.85110i) q^{88} -8.08066 q^{89} -0.477031i q^{91} +(-1.30135 + 1.16806i) q^{92} +(-4.31844 - 1.92674i) q^{94} -10.6757i q^{97} +(1.78192 - 3.99385i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 4 q^{4} + 8 q^{8} - 6 q^{14} + 8 q^{16} + 20 q^{22} + 2 q^{26} + 24 q^{28} + 8 q^{31} + 12 q^{32} - 12 q^{34} - 14 q^{38} - 8 q^{43} - 12 q^{44} + 20 q^{46} + 24 q^{52} - 8 q^{53} - 8 q^{56} - 20 q^{58} - 26 q^{62} + 32 q^{64} + 24 q^{67} - 36 q^{68} + 40 q^{71} + 8 q^{74} - 20 q^{76} + 24 q^{77} + 16 q^{79} - 16 q^{82} + 32 q^{83} + 18 q^{86} - 8 q^{88} - 28 q^{92} + 4 q^{94} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.576222 1.29150i 0.407451 0.913227i
\(3\) 0 0
\(4\) −1.33594 1.48838i −0.667968 0.744190i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.97676i 0.747145i 0.927601 + 0.373573i \(0.121867\pi\)
−0.927601 + 0.373573i \(0.878133\pi\)
\(8\) −2.69204 + 0.867721i −0.951779 + 0.306786i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.43055i 0.431328i 0.976468 + 0.215664i \(0.0691915\pi\)
−0.976468 + 0.215664i \(0.930808\pi\)
\(12\) 0 0
\(13\) −0.241319 −0.0669300 −0.0334650 0.999440i \(-0.510654\pi\)
−0.0334650 + 0.999440i \(0.510654\pi\)
\(14\) 2.55298 + 1.13905i 0.682313 + 0.304425i
\(15\) 0 0
\(16\) −0.430552 + 3.97676i −0.107638 + 0.994190i
\(17\) 7.38407i 1.79090i −0.445161 0.895450i \(-0.646854\pi\)
0.445161 0.895450i \(-0.353146\pi\)
\(18\) 0 0
\(19\) 3.04033i 0.697500i −0.937216 0.348750i \(-0.886606\pi\)
0.937216 0.348750i \(-0.113394\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.84756 + 0.824316i 0.393900 + 0.175745i
\(23\) 0.874337i 0.182312i −0.995837 0.0911560i \(-0.970944\pi\)
0.995837 0.0911560i \(-0.0290562\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.139054 + 0.311664i −0.0272707 + 0.0611223i
\(27\) 0 0
\(28\) 2.94217 2.64082i 0.556018 0.499069i
\(29\) 9.07918i 1.68596i −0.537943 0.842981i \(-0.680799\pi\)
0.537943 0.842981i \(-0.319201\pi\)
\(30\) 0 0
\(31\) −7.44764 −1.33764 −0.668818 0.743426i \(-0.733200\pi\)
−0.668818 + 0.743426i \(0.733200\pi\)
\(32\) 4.88789 + 2.84756i 0.864064 + 0.503381i
\(33\) 0 0
\(34\) −9.53652 4.25487i −1.63550 0.729704i
\(35\) 0 0
\(36\) 0 0
\(37\) −8.81463 −1.44912 −0.724558 0.689214i \(-0.757956\pi\)
−0.724558 + 0.689214i \(0.757956\pi\)
\(38\) −3.92658 1.75191i −0.636976 0.284197i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.91319 0.298790 0.149395 0.988778i \(-0.452267\pi\)
0.149395 + 0.988778i \(0.452267\pi\)
\(42\) 0 0
\(43\) −11.2452 −1.71487 −0.857437 0.514589i \(-0.827944\pi\)
−0.857437 + 0.514589i \(0.827944\pi\)
\(44\) 2.12921 1.91113i 0.320990 0.288113i
\(45\) 0 0
\(46\) −1.12921 0.503813i −0.166492 0.0742831i
\(47\) 3.34374i 0.487735i −0.969809 0.243867i \(-0.921584\pi\)
0.969809 0.243867i \(-0.0784162\pi\)
\(48\) 0 0
\(49\) 3.09242 0.441774
\(50\) 0 0
\(51\) 0 0
\(52\) 0.322387 + 0.359175i 0.0447071 + 0.0498086i
\(53\) −9.20632 −1.26459 −0.632293 0.774729i \(-0.717886\pi\)
−0.632293 + 0.774729i \(0.717886\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.71528 5.32151i −0.229213 0.711117i
\(57\) 0 0
\(58\) −11.7258 5.23163i −1.53967 0.686946i
\(59\) 6.43616i 0.837917i −0.908005 0.418958i \(-0.862395\pi\)
0.908005 0.418958i \(-0.137605\pi\)
\(60\) 0 0
\(61\) 4.57331i 0.585552i 0.956181 + 0.292776i \(0.0945790\pi\)
−0.956181 + 0.292776i \(0.905421\pi\)
\(62\) −4.29150 + 9.61862i −0.545021 + 1.22157i
\(63\) 0 0
\(64\) 6.49412 4.67187i 0.811765 0.583984i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.86671 −0.594563 −0.297282 0.954790i \(-0.596080\pi\)
−0.297282 + 0.954790i \(0.596080\pi\)
\(68\) −10.9903 + 9.86465i −1.33277 + 1.19626i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.21808 0.975307 0.487653 0.873037i \(-0.337853\pi\)
0.487653 + 0.873037i \(0.337853\pi\)
\(72\) 0 0
\(73\) 4.12714i 0.483045i −0.970395 0.241523i \(-0.922353\pi\)
0.970395 0.241523i \(-0.0776468\pi\)
\(74\) −5.07918 + 11.3841i −0.590443 + 1.32337i
\(75\) 0 0
\(76\) −4.52517 + 4.06169i −0.519072 + 0.465907i
\(77\) −2.82786 −0.322264
\(78\) 0 0
\(79\) 13.6757 1.53864 0.769320 0.638864i \(-0.220595\pi\)
0.769320 + 0.638864i \(0.220595\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.10242 2.47088i 0.121742 0.272863i
\(83\) 12.3320 1.35361 0.676806 0.736162i \(-0.263364\pi\)
0.676806 + 0.736162i \(0.263364\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.47972 + 14.5231i −0.698726 + 1.56607i
\(87\) 0 0
\(88\) −1.24132 3.85110i −0.132325 0.410528i
\(89\) −8.08066 −0.856548 −0.428274 0.903649i \(-0.640878\pi\)
−0.428274 + 0.903649i \(0.640878\pi\)
\(90\) 0 0
\(91\) 0.477031i 0.0500064i
\(92\) −1.30135 + 1.16806i −0.135675 + 0.121779i
\(93\) 0 0
\(94\) −4.31844 1.92674i −0.445413 0.198728i
\(95\) 0 0
\(96\) 0 0
\(97\) 10.6757i 1.08396i −0.840393 0.541978i \(-0.817676\pi\)
0.840393 0.541978i \(-0.182324\pi\)
\(98\) 1.78192 3.99385i 0.180001 0.403440i
\(99\) 0 0
\(100\) 0 0
\(101\) 13.2063i 1.31408i −0.753856 0.657039i \(-0.771809\pi\)
0.753856 0.657039i \(-0.228191\pi\)
\(102\) 0 0
\(103\) 19.4244i 1.91394i 0.290181 + 0.956972i \(0.406284\pi\)
−0.290181 + 0.956972i \(0.593716\pi\)
\(104\) 0.649641 0.209398i 0.0637025 0.0205331i
\(105\) 0 0
\(106\) −5.30489 + 11.8900i −0.515256 + 1.15485i
\(107\) −14.8085 −1.43159 −0.715795 0.698311i \(-0.753935\pi\)
−0.715795 + 0.698311i \(0.753935\pi\)
\(108\) 0 0
\(109\) 15.2296i 1.45873i 0.684126 + 0.729364i \(0.260184\pi\)
−0.684126 + 0.729364i \(0.739816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −7.86110 0.851098i −0.742804 0.0804212i
\(113\) 1.13890i 0.107138i 0.998564 + 0.0535692i \(0.0170598\pi\)
−0.998564 + 0.0535692i \(0.982940\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −13.5133 + 12.1292i −1.25468 + 1.12617i
\(117\) 0 0
\(118\) −8.31229 3.70866i −0.765208 0.341410i
\(119\) 14.5965 1.33806
\(120\) 0 0
\(121\) 8.95352 0.813956
\(122\) 5.90642 + 2.63524i 0.534742 + 0.238583i
\(123\) 0 0
\(124\) 9.94957 + 11.0849i 0.893498 + 0.995456i
\(125\) 0 0
\(126\) 0 0
\(127\) 2.43616i 0.216174i −0.994141 0.108087i \(-0.965527\pi\)
0.994141 0.108087i \(-0.0344725\pi\)
\(128\) −2.29166 11.0792i −0.202556 0.979271i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.90143i 0.602981i 0.953469 + 0.301491i \(0.0974842\pi\)
−0.953469 + 0.301491i \(0.902516\pi\)
\(132\) 0 0
\(133\) 6.01001 0.521134
\(134\) −2.80431 + 6.28535i −0.242255 + 0.542972i
\(135\) 0 0
\(136\) 6.40731 + 19.8782i 0.549423 + 1.70454i
\(137\) 5.39022i 0.460518i −0.973129 0.230259i \(-0.926043\pi\)
0.973129 0.230259i \(-0.0739573\pi\)
\(138\) 0 0
\(139\) 17.4244i 1.47792i −0.673750 0.738959i \(-0.735318\pi\)
0.673750 0.738959i \(-0.264682\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.73544 10.6136i 0.397389 0.890677i
\(143\) 0.345220i 0.0288687i
\(144\) 0 0
\(145\) 0 0
\(146\) −5.33019 2.37815i −0.441130 0.196817i
\(147\) 0 0
\(148\) 11.7758 + 13.1195i 0.967962 + 1.07842i
\(149\) 2.28551i 0.187236i −0.995608 0.0936180i \(-0.970157\pi\)
0.995608 0.0936180i \(-0.0298432\pi\)
\(150\) 0 0
\(151\) 6.66425 0.542329 0.271164 0.962533i \(-0.412591\pi\)
0.271164 + 0.962533i \(0.412591\pi\)
\(152\) 2.63816 + 8.18468i 0.213983 + 0.663865i
\(153\) 0 0
\(154\) −1.62948 + 3.65217i −0.131307 + 0.294301i
\(155\) 0 0
\(156\) 0 0
\(157\) −17.4144 −1.38982 −0.694910 0.719097i \(-0.744556\pi\)
−0.694910 + 0.719097i \(0.744556\pi\)
\(158\) 7.88026 17.6622i 0.626920 1.40513i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.72836 0.136214
\(162\) 0 0
\(163\) 4.66187 0.365145 0.182573 0.983192i \(-0.441557\pi\)
0.182573 + 0.983192i \(0.441557\pi\)
\(164\) −2.55590 2.84756i −0.199582 0.222357i
\(165\) 0 0
\(166\) 7.10597 15.9267i 0.551530 1.23615i
\(167\) 0.137419i 0.0106338i −0.999986 0.00531690i \(-0.998308\pi\)
0.999986 0.00531690i \(-0.00169243\pi\)
\(168\) 0 0
\(169\) −12.9418 −0.995520
\(170\) 0 0
\(171\) 0 0
\(172\) 15.0228 + 16.7371i 1.14548 + 1.27619i
\(173\) 3.96675 0.301587 0.150793 0.988565i \(-0.451817\pi\)
0.150793 + 0.988565i \(0.451817\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.68896 0.615927i −0.428822 0.0464272i
\(177\) 0 0
\(178\) −4.65626 + 10.4362i −0.349001 + 0.782223i
\(179\) 4.68749i 0.350359i 0.984537 + 0.175180i \(0.0560506\pi\)
−0.984537 + 0.175180i \(0.943949\pi\)
\(180\) 0 0
\(181\) 9.10242i 0.676578i 0.941042 + 0.338289i \(0.109848\pi\)
−0.941042 + 0.338289i \(0.890152\pi\)
\(182\) −0.616084 0.274876i −0.0456672 0.0203751i
\(183\) 0 0
\(184\) 0.758681 + 2.35375i 0.0559307 + 0.173521i
\(185\) 0 0
\(186\) 0 0
\(187\) 10.5633 0.772465
\(188\) −4.97676 + 4.46702i −0.362968 + 0.325791i
\(189\) 0 0
\(190\) 0 0
\(191\) −15.2063 −1.10029 −0.550145 0.835069i \(-0.685428\pi\)
−0.550145 + 0.835069i \(0.685428\pi\)
\(192\) 0 0
\(193\) 20.7564i 1.49408i −0.664780 0.747039i \(-0.731475\pi\)
0.664780 0.747039i \(-0.268525\pi\)
\(194\) −13.7877 6.15159i −0.989898 0.441659i
\(195\) 0 0
\(196\) −4.13127 4.60269i −0.295091 0.328764i
\(197\) −23.2508 −1.65655 −0.828275 0.560322i \(-0.810677\pi\)
−0.828275 + 0.560322i \(0.810677\pi\)
\(198\) 0 0
\(199\) 7.21633 0.511552 0.255776 0.966736i \(-0.417669\pi\)
0.255776 + 0.966736i \(0.417669\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −17.0559 7.60978i −1.20005 0.535422i
\(203\) 17.9474 1.25966
\(204\) 0 0
\(205\) 0 0
\(206\) 25.0866 + 11.1928i 1.74787 + 0.779838i
\(207\) 0 0
\(208\) 0.103901 0.959669i 0.00720420 0.0665411i
\(209\) 4.34935 0.300851
\(210\) 0 0
\(211\) 4.38407i 0.301812i −0.988548 0.150906i \(-0.951781\pi\)
0.988548 0.150906i \(-0.0482191\pi\)
\(212\) 12.2991 + 13.7025i 0.844703 + 0.941092i
\(213\) 0 0
\(214\) −8.53298 + 19.1251i −0.583302 + 1.30737i
\(215\) 0 0
\(216\) 0 0
\(217\) 14.7222i 0.999409i
\(218\) 19.6690 + 8.77561i 1.33215 + 0.594360i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.78192i 0.119865i
\(222\) 0 0
\(223\) 4.98852i 0.334056i −0.985952 0.167028i \(-0.946583\pi\)
0.985952 0.167028i \(-0.0534170\pi\)
\(224\) −5.62894 + 9.66218i −0.376099 + 0.645582i
\(225\) 0 0
\(226\) 1.47088 + 0.656257i 0.0978416 + 0.0436536i
\(227\) −11.2569 −0.747149 −0.373574 0.927600i \(-0.621868\pi\)
−0.373574 + 0.927600i \(0.621868\pi\)
\(228\) 0 0
\(229\) 15.8364i 1.04650i −0.852180 0.523249i \(-0.824720\pi\)
0.852180 0.523249i \(-0.175280\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.87820 + 24.4415i 0.517229 + 1.60466i
\(233\) 10.9591i 0.717956i 0.933346 + 0.358978i \(0.116875\pi\)
−0.933346 + 0.358978i \(0.883125\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.57945 + 8.59830i −0.623569 + 0.559701i
\(237\) 0 0
\(238\) 8.41086 18.8514i 0.545195 1.22196i
\(239\) 17.3182 1.12022 0.560111 0.828418i \(-0.310758\pi\)
0.560111 + 0.828418i \(0.310758\pi\)
\(240\) 0 0
\(241\) 4.76869 0.307178 0.153589 0.988135i \(-0.450917\pi\)
0.153589 + 0.988135i \(0.450917\pi\)
\(242\) 5.15922 11.5635i 0.331647 0.743327i
\(243\) 0 0
\(244\) 6.80682 6.10964i 0.435762 0.391130i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.733691i 0.0466836i
\(248\) 20.0493 6.46247i 1.27313 0.410367i
\(249\) 0 0
\(250\) 0 0
\(251\) 6.15837i 0.388713i −0.980931 0.194356i \(-0.937738\pi\)
0.980931 0.194356i \(-0.0622618\pi\)
\(252\) 0 0
\(253\) 1.25079 0.0786362
\(254\) −3.14630 1.40377i −0.197416 0.0880804i
\(255\) 0 0
\(256\) −15.6293 3.42440i −0.976828 0.214025i
\(257\) 14.1584i 0.883175i −0.897218 0.441587i \(-0.854416\pi\)
0.897218 0.441587i \(-0.145584\pi\)
\(258\) 0 0
\(259\) 17.4244i 1.08270i
\(260\) 0 0
\(261\) 0 0
\(262\) 8.91319 + 3.97676i 0.550659 + 0.245685i
\(263\) 15.5960i 0.961691i 0.876805 + 0.480845i \(0.159670\pi\)
−0.876805 + 0.480845i \(0.840330\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.46310 7.76191i 0.212336 0.475913i
\(267\) 0 0
\(268\) 6.50161 + 7.24352i 0.397149 + 0.442468i
\(269\) 11.3182i 0.690084i 0.938587 + 0.345042i \(0.112135\pi\)
−0.938587 + 0.345042i \(0.887865\pi\)
\(270\) 0 0
\(271\) −6.20485 −0.376918 −0.188459 0.982081i \(-0.560349\pi\)
−0.188459 + 0.982081i \(0.560349\pi\)
\(272\) 29.3647 + 3.17923i 1.78050 + 0.192769i
\(273\) 0 0
\(274\) −6.96146 3.10597i −0.420557 0.187638i
\(275\) 0 0
\(276\) 0 0
\(277\) 18.9288 1.13732 0.568661 0.822572i \(-0.307462\pi\)
0.568661 + 0.822572i \(0.307462\pi\)
\(278\) −22.5036 10.0403i −1.34968 0.602179i
\(279\) 0 0
\(280\) 0 0
\(281\) 21.6231 1.28993 0.644963 0.764214i \(-0.276873\pi\)
0.644963 + 0.764214i \(0.276873\pi\)
\(282\) 0 0
\(283\) 29.1522 1.73292 0.866460 0.499247i \(-0.166390\pi\)
0.866460 + 0.499247i \(0.166390\pi\)
\(284\) −10.9788 12.2316i −0.651473 0.725814i
\(285\) 0 0
\(286\) −0.445851 0.198923i −0.0263637 0.0117626i
\(287\) 3.78192i 0.223240i
\(288\) 0 0
\(289\) −37.5245 −2.20733
\(290\) 0 0
\(291\) 0 0
\(292\) −6.14275 + 5.51359i −0.359477 + 0.322659i
\(293\) −2.32427 −0.135785 −0.0678927 0.997693i \(-0.521628\pi\)
−0.0678927 + 0.997693i \(0.521628\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 23.7293 7.64863i 1.37924 0.444568i
\(297\) 0 0
\(298\) −2.95173 1.31696i −0.170989 0.0762895i
\(299\) 0.210995i 0.0122021i
\(300\) 0 0
\(301\) 22.2290i 1.28126i
\(302\) 3.84009 8.60686i 0.220972 0.495269i
\(303\) 0 0
\(304\) 12.0907 + 1.30902i 0.693447 + 0.0750775i
\(305\) 0 0
\(306\) 0 0
\(307\) 3.52297 0.201066 0.100533 0.994934i \(-0.467945\pi\)
0.100533 + 0.994934i \(0.467945\pi\)
\(308\) 3.77784 + 4.20893i 0.215262 + 0.239826i
\(309\) 0 0
\(310\) 0 0
\(311\) 21.6757 1.22912 0.614559 0.788871i \(-0.289334\pi\)
0.614559 + 0.788871i \(0.289334\pi\)
\(312\) 0 0
\(313\) 12.5486i 0.709288i −0.935001 0.354644i \(-0.884602\pi\)
0.935001 0.354644i \(-0.115398\pi\)
\(314\) −10.0346 + 22.4907i −0.566283 + 1.26922i
\(315\) 0 0
\(316\) −18.2699 20.3547i −1.02776 1.14504i
\(317\) 10.8611 0.610020 0.305010 0.952349i \(-0.401340\pi\)
0.305010 + 0.952349i \(0.401340\pi\)
\(318\) 0 0
\(319\) 12.9882 0.727202
\(320\) 0 0
\(321\) 0 0
\(322\) 0.995917 2.23217i 0.0555003 0.124394i
\(323\) −22.4500 −1.24915
\(324\) 0 0
\(325\) 0 0
\(326\) 2.68627 6.02079i 0.148779 0.333461i
\(327\) 0 0
\(328\) −5.15038 + 1.66011i −0.284382 + 0.0916645i
\(329\) 6.60978 0.364409
\(330\) 0 0
\(331\) 1.23185i 0.0677088i 0.999427 + 0.0338544i \(0.0107783\pi\)
−0.999427 + 0.0338544i \(0.989222\pi\)
\(332\) −16.4747 18.3547i −0.904169 1.00734i
\(333\) 0 0
\(334\) −0.177476 0.0791838i −0.00971107 0.00433275i
\(335\) 0 0
\(336\) 0 0
\(337\) 4.13890i 0.225460i −0.993626 0.112730i \(-0.964040\pi\)
0.993626 0.112730i \(-0.0359595\pi\)
\(338\) −7.45733 + 16.7143i −0.405625 + 0.909136i
\(339\) 0 0
\(340\) 0 0
\(341\) 10.6542i 0.576959i
\(342\) 0 0
\(343\) 19.9503i 1.07721i
\(344\) 30.2724 9.75767i 1.63218 0.526098i
\(345\) 0 0
\(346\) 2.28573 5.12306i 0.122882 0.275417i
\(347\) 17.4586 0.937226 0.468613 0.883404i \(-0.344754\pi\)
0.468613 + 0.883404i \(0.344754\pi\)
\(348\) 0 0
\(349\) 21.2196i 1.13586i 0.823078 + 0.567928i \(0.192255\pi\)
−0.823078 + 0.567928i \(0.807745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.07358 + 6.99237i −0.217122 + 0.372695i
\(353\) 21.0398i 1.11984i −0.828548 0.559918i \(-0.810833\pi\)
0.828548 0.559918i \(-0.189167\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.7952 + 12.0271i 0.572147 + 0.637435i
\(357\) 0 0
\(358\) 6.05388 + 2.70103i 0.319957 + 0.142754i
\(359\) 23.5153 1.24109 0.620546 0.784170i \(-0.286911\pi\)
0.620546 + 0.784170i \(0.286911\pi\)
\(360\) 0 0
\(361\) 9.75639 0.513494
\(362\) 11.7558 + 5.24502i 0.617869 + 0.275672i
\(363\) 0 0
\(364\) −0.710003 + 0.637282i −0.0372143 + 0.0334027i
\(365\) 0 0
\(366\) 0 0
\(367\) 25.4012i 1.32593i 0.748650 + 0.662965i \(0.230702\pi\)
−0.748650 + 0.662965i \(0.769298\pi\)
\(368\) 3.47703 + 0.376448i 0.181253 + 0.0196237i
\(369\) 0 0
\(370\) 0 0
\(371\) 18.1987i 0.944829i
\(372\) 0 0
\(373\) 10.0677 0.521286 0.260643 0.965435i \(-0.416065\pi\)
0.260643 + 0.965435i \(0.416065\pi\)
\(374\) 6.08681 13.6425i 0.314741 0.705436i
\(375\) 0 0
\(376\) 2.90143 + 9.00148i 0.149630 + 0.464216i
\(377\) 2.19098i 0.112841i
\(378\) 0 0
\(379\) 18.9674i 0.974289i 0.873321 + 0.487145i \(0.161962\pi\)
−0.873321 + 0.487145i \(0.838038\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.76222 + 19.6389i −0.448314 + 1.00482i
\(383\) 28.7446i 1.46878i 0.678727 + 0.734391i \(0.262532\pi\)
−0.678727 + 0.734391i \(0.737468\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.8068 11.9603i −1.36443 0.608763i
\(387\) 0 0
\(388\) −15.8895 + 14.2621i −0.806669 + 0.724048i
\(389\) 29.8161i 1.51174i 0.654724 + 0.755868i \(0.272785\pi\)
−0.654724 + 0.755868i \(0.727215\pi\)
\(390\) 0 0
\(391\) −6.45617 −0.326503
\(392\) −8.32490 + 2.68335i −0.420471 + 0.135530i
\(393\) 0 0
\(394\) −13.3976 + 30.0283i −0.674962 + 1.51281i
\(395\) 0 0
\(396\) 0 0
\(397\) 2.73167 0.137099 0.0685494 0.997648i \(-0.478163\pi\)
0.0685494 + 0.997648i \(0.478163\pi\)
\(398\) 4.15821 9.31988i 0.208432 0.467163i
\(399\) 0 0
\(400\) 0 0
\(401\) −25.8744 −1.29211 −0.646054 0.763292i \(-0.723582\pi\)
−0.646054 + 0.763292i \(0.723582\pi\)
\(402\) 0 0
\(403\) 1.79726 0.0895279
\(404\) −19.6560 + 17.6428i −0.977924 + 0.877762i
\(405\) 0 0
\(406\) 10.3417 23.1790i 0.513249 1.15035i
\(407\) 12.6098i 0.625044i
\(408\) 0 0
\(409\) −22.4786 −1.11150 −0.555748 0.831351i \(-0.687568\pi\)
−0.555748 + 0.831351i \(0.687568\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 28.9109 25.9498i 1.42434 1.27845i
\(413\) 12.7227 0.626045
\(414\) 0 0
\(415\) 0 0
\(416\) −1.17954 0.687170i −0.0578318 0.0336913i
\(417\) 0 0
\(418\) 2.50619 5.61718i 0.122582 0.274745i
\(419\) 24.5307i 1.19840i −0.800598 0.599201i \(-0.795485\pi\)
0.800598 0.599201i \(-0.204515\pi\)
\(420\) 0 0
\(421\) 33.3856i 1.62712i −0.581483 0.813558i \(-0.697527\pi\)
0.581483 0.813558i \(-0.302473\pi\)
\(422\) −5.66202 2.52620i −0.275623 0.122974i
\(423\) 0 0
\(424\) 24.7838 7.98852i 1.20361 0.387957i
\(425\) 0 0
\(426\) 0 0
\(427\) −9.04033 −0.437492
\(428\) 19.7832 + 22.0406i 0.956256 + 1.06537i
\(429\) 0 0
\(430\) 0 0
\(431\) −11.6548 −0.561391 −0.280696 0.959797i \(-0.590565\pi\)
−0.280696 + 0.959797i \(0.590565\pi\)
\(432\) 0 0
\(433\) 19.7681i 0.949996i 0.879987 + 0.474998i \(0.157551\pi\)
−0.879987 + 0.474998i \(0.842449\pi\)
\(434\) −19.0137 8.48326i −0.912687 0.407210i
\(435\) 0 0
\(436\) 22.6674 20.3457i 1.08557 0.974383i
\(437\) −2.65827 −0.127163
\(438\) 0 0
\(439\) 25.1699 1.20129 0.600646 0.799515i \(-0.294910\pi\)
0.600646 + 0.799515i \(0.294910\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.30135 + 1.02678i 0.109464 + 0.0488390i
\(443\) −19.5515 −0.928922 −0.464461 0.885594i \(-0.653752\pi\)
−0.464461 + 0.885594i \(0.653752\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −6.44266 2.87449i −0.305069 0.136111i
\(447\) 0 0
\(448\) 9.23517 + 12.8373i 0.436321 + 0.606507i
\(449\) −19.9612 −0.942029 −0.471014 0.882125i \(-0.656112\pi\)
−0.471014 + 0.882125i \(0.656112\pi\)
\(450\) 0 0
\(451\) 2.73692i 0.128876i
\(452\) 1.69511 1.52149i 0.0797313 0.0715650i
\(453\) 0 0
\(454\) −6.48650 + 14.5383i −0.304426 + 0.682317i
\(455\) 0 0
\(456\) 0 0
\(457\) 5.01176i 0.234440i 0.993106 + 0.117220i \(0.0373983\pi\)
−0.993106 + 0.117220i \(0.962602\pi\)
\(458\) −20.4527 9.12528i −0.955690 0.426396i
\(459\) 0 0
\(460\) 0 0
\(461\) 5.12566i 0.238726i 0.992851 + 0.119363i \(0.0380852\pi\)
−0.992851 + 0.119363i \(0.961915\pi\)
\(462\) 0 0
\(463\) 5.79515i 0.269324i −0.990892 0.134662i \(-0.957005\pi\)
0.990892 0.134662i \(-0.0429948\pi\)
\(464\) 36.1057 + 3.90906i 1.67617 + 0.181474i
\(465\) 0 0
\(466\) 14.1537 + 6.31490i 0.655657 + 0.292532i
\(467\) 16.8208 0.778373 0.389186 0.921159i \(-0.372756\pi\)
0.389186 + 0.921159i \(0.372756\pi\)
\(468\) 0 0
\(469\) 9.62032i 0.444225i
\(470\) 0 0
\(471\) 0 0
\(472\) 5.58479 + 17.3264i 0.257061 + 0.797511i
\(473\) 16.0868i 0.739672i
\(474\) 0 0
\(475\) 0 0
\(476\) −19.5000 21.7252i −0.893783 0.995773i
\(477\) 0 0
\(478\) 9.97914 22.3664i 0.456435 1.02302i
\(479\) −36.9065 −1.68630 −0.843151 0.537678i \(-0.819302\pi\)
−0.843151 + 0.537678i \(0.819302\pi\)
\(480\) 0 0
\(481\) 2.12714 0.0969892
\(482\) 2.74782 6.15875i 0.125160 0.280523i
\(483\) 0 0
\(484\) −11.9613 13.3262i −0.543697 0.605738i
\(485\) 0 0
\(486\) 0 0
\(487\) 5.14984i 0.233361i 0.993169 + 0.116681i \(0.0372254\pi\)
−0.993169 + 0.116681i \(0.962775\pi\)
\(488\) −3.96835 12.3115i −0.179639 0.557316i
\(489\) 0 0
\(490\) 0 0
\(491\) 23.1154i 1.04318i −0.853195 0.521591i \(-0.825339\pi\)
0.853195 0.521591i \(-0.174661\pi\)
\(492\) 0 0
\(493\) −67.0414 −3.01939
\(494\) 0.947560 + 0.422769i 0.0426328 + 0.0190213i
\(495\) 0 0
\(496\) 3.20660 29.6175i 0.143980 1.32986i
\(497\) 16.2452i 0.728696i
\(498\) 0 0
\(499\) 14.3111i 0.640654i 0.947307 + 0.320327i \(0.103793\pi\)
−0.947307 + 0.320327i \(0.896207\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7.95352 3.54859i −0.354983 0.158381i
\(503\) 15.4224i 0.687650i −0.939034 0.343825i \(-0.888277\pi\)
0.939034 0.343825i \(-0.111723\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.720730 1.61539i 0.0320404 0.0718127i
\(507\) 0 0
\(508\) −3.62593 + 3.25455i −0.160875 + 0.144397i
\(509\) 43.1578i 1.91294i −0.291835 0.956469i \(-0.594266\pi\)
0.291835 0.956469i \(-0.405734\pi\)
\(510\) 0 0
\(511\) 8.15837 0.360905
\(512\) −13.4285 + 18.2119i −0.593463 + 0.804861i
\(513\) 0 0
\(514\) −18.2855 8.15837i −0.806539 0.359850i
\(515\) 0 0
\(516\) 0 0
\(517\) 4.78340 0.210374
\(518\) −22.5036 10.0403i −0.988751 0.441147i
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8232 0.780848 0.390424 0.920635i \(-0.372328\pi\)
0.390424 + 0.920635i \(0.372328\pi\)
\(522\) 0 0
\(523\) 24.7502 1.08225 0.541126 0.840941i \(-0.317998\pi\)
0.541126 + 0.840941i \(0.317998\pi\)
\(524\) 10.2720 9.21987i 0.448733 0.402772i
\(525\) 0 0
\(526\) 20.1422 + 8.98677i 0.878242 + 0.391842i
\(527\) 54.9939i 2.39557i
\(528\) 0 0
\(529\) 22.2355 0.966762
\(530\) 0 0
\(531\) 0 0
\(532\) −8.02898 8.94517i −0.348100 0.387822i
\(533\) −0.461690 −0.0199980
\(534\) 0 0
\(535\) 0 0
\(536\) 13.1014 4.22295i 0.565893 0.182403i
\(537\) 0 0
\(538\) 14.6175 + 6.52181i 0.630203 + 0.281175i
\(539\) 4.42386i 0.190549i
\(540\) 0 0
\(541\) 16.9982i 0.730812i −0.930848 0.365406i \(-0.880930\pi\)
0.930848 0.365406i \(-0.119070\pi\)
\(542\) −3.57537 + 8.01355i −0.153575 + 0.344211i
\(543\) 0 0
\(544\) 21.0266 36.0925i 0.901506 1.54745i
\(545\) 0 0
\(546\) 0 0
\(547\) 37.2385 1.59220 0.796101 0.605163i \(-0.206892\pi\)
0.796101 + 0.605163i \(0.206892\pi\)
\(548\) −8.02270 + 7.20099i −0.342713 + 0.307611i
\(549\) 0 0
\(550\) 0 0
\(551\) −27.6037 −1.17596
\(552\) 0 0
\(553\) 27.0336i 1.14959i
\(554\) 10.9072 24.4465i 0.463403 1.03863i
\(555\) 0 0
\(556\) −25.9341 + 23.2779i −1.09985 + 0.987202i
\(557\) 14.7604 0.625420 0.312710 0.949849i \(-0.398763\pi\)
0.312710 + 0.949849i \(0.398763\pi\)
\(558\) 0 0
\(559\) 2.71368 0.114776
\(560\) 0 0
\(561\) 0 0
\(562\) 12.4597 27.9262i 0.525581 1.17800i
\(563\) 3.00561 0.126671 0.0633356 0.997992i \(-0.479826\pi\)
0.0633356 + 0.997992i \(0.479826\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.7982 37.6500i 0.706079 1.58255i
\(567\) 0 0
\(568\) −22.1234 + 7.13100i −0.928276 + 0.299210i
\(569\) −23.1840 −0.971923 −0.485962 0.873980i \(-0.661531\pi\)
−0.485962 + 0.873980i \(0.661531\pi\)
\(570\) 0 0
\(571\) 0.202739i 0.00848438i −0.999991 0.00424219i \(-0.998650\pi\)
0.999991 0.00424219i \(-0.00135033\pi\)
\(572\) −0.513819 + 0.461192i −0.0214838 + 0.0192834i
\(573\) 0 0
\(574\) 4.88434 + 2.17923i 0.203869 + 0.0909592i
\(575\) 0 0
\(576\) 0 0
\(577\) 21.8023i 0.907643i −0.891093 0.453821i \(-0.850060\pi\)
0.891093 0.453821i \(-0.149940\pi\)
\(578\) −21.6225 + 48.4629i −0.899376 + 2.01579i
\(579\) 0 0
\(580\) 0 0
\(581\) 24.3774i 1.01134i
\(582\) 0 0
\(583\) 13.1701i 0.545451i
\(584\) 3.58120 + 11.1104i 0.148191 + 0.459752i
\(585\) 0 0
\(586\) −1.33930 + 3.00179i −0.0553258 + 0.124003i
\(587\) 36.7126 1.51529 0.757645 0.652667i \(-0.226350\pi\)
0.757645 + 0.652667i \(0.226350\pi\)
\(588\) 0 0
\(589\) 22.6433i 0.933001i
\(590\) 0 0
\(591\) 0 0
\(592\) 3.79515 35.0537i 0.155980 1.44070i
\(593\) 10.6036i 0.435439i −0.976011 0.217719i \(-0.930138\pi\)
0.976011 0.217719i \(-0.0698618\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.40170 + 3.05329i −0.139339 + 0.125068i
\(597\) 0 0
\(598\) 0.272499 + 0.121580i 0.0111433 + 0.00497177i
\(599\) −25.7988 −1.05411 −0.527056 0.849831i \(-0.676704\pi\)
−0.527056 + 0.849831i \(0.676704\pi\)
\(600\) 0 0
\(601\) 18.5021 0.754717 0.377358 0.926067i \(-0.376832\pi\)
0.377358 + 0.926067i \(0.376832\pi\)
\(602\) −28.7087 12.8089i −1.17008 0.522050i
\(603\) 0 0
\(604\) −8.90300 9.91893i −0.362258 0.403596i
\(605\) 0 0
\(606\) 0 0
\(607\) 37.5828i 1.52544i 0.646730 + 0.762719i \(0.276136\pi\)
−0.646730 + 0.762719i \(0.723864\pi\)
\(608\) 8.65751 14.8608i 0.351108 0.602685i
\(609\) 0 0
\(610\) 0 0
\(611\) 0.806910i 0.0326441i
\(612\) 0 0
\(613\) 4.93405 0.199284 0.0996422 0.995023i \(-0.468230\pi\)
0.0996422 + 0.995023i \(0.468230\pi\)
\(614\) 2.03001 4.54991i 0.0819247 0.183619i
\(615\) 0 0
\(616\) 7.61270 2.45379i 0.306724 0.0988661i
\(617\) 6.26043i 0.252035i −0.992028 0.126018i \(-0.959780\pi\)
0.992028 0.126018i \(-0.0402196\pi\)
\(618\) 0 0
\(619\) 8.02562i 0.322577i 0.986907 + 0.161288i \(0.0515649\pi\)
−0.986907 + 0.161288i \(0.948435\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12.4900 27.9942i 0.500805 1.12246i
\(623\) 15.9735i 0.639966i
\(624\) 0 0
\(625\) 0 0
\(626\) −16.2065 7.23078i −0.647741 0.289000i
\(627\) 0 0
\(628\) 23.2645 + 25.9192i 0.928355 + 1.03429i
\(629\) 65.0878i 2.59522i
\(630\) 0 0
\(631\) −26.5248 −1.05594 −0.527968 0.849264i \(-0.677046\pi\)
−0.527968 + 0.849264i \(0.677046\pi\)
\(632\) −36.8156 + 11.8667i −1.46444 + 0.472032i
\(633\) 0 0
\(634\) 6.25841 14.0271i 0.248553 0.557087i
\(635\) 0 0
\(636\) 0 0
\(637\) −0.746260 −0.0295679
\(638\) 7.48412 16.7743i 0.296299 0.664101i
\(639\) 0 0
\(640\) 0 0
\(641\) 26.5863 1.05009 0.525047 0.851073i \(-0.324048\pi\)
0.525047 + 0.851073i \(0.324048\pi\)
\(642\) 0 0
\(643\) −2.89233 −0.114062 −0.0570312 0.998372i \(-0.518163\pi\)
−0.0570312 + 0.998372i \(0.518163\pi\)
\(644\) −2.30897 2.57245i −0.0909862 0.101369i
\(645\) 0 0
\(646\) −12.9362 + 28.9942i −0.508968 + 1.14076i
\(647\) 12.3472i 0.485420i −0.970099 0.242710i \(-0.921964\pi\)
0.970099 0.242710i \(-0.0780363\pi\)
\(648\) 0 0
\(649\) 9.20726 0.361417
\(650\) 0 0
\(651\) 0 0
\(652\) −6.22795 6.93863i −0.243905 0.271738i
\(653\) 39.0507 1.52817 0.764086 0.645114i \(-0.223190\pi\)
0.764086 + 0.645114i \(0.223190\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.823728 + 7.60830i −0.0321612 + 0.297054i
\(657\) 0 0
\(658\) 3.80870 8.53652i 0.148479 0.332788i
\(659\) 23.7738i 0.926094i −0.886334 0.463047i \(-0.846756\pi\)
0.886334 0.463047i \(-0.153244\pi\)
\(660\) 0 0
\(661\) 21.5051i 0.836450i 0.908343 + 0.418225i \(0.137348\pi\)
−0.908343 + 0.418225i \(0.862652\pi\)
\(662\) 1.59094 + 0.709822i 0.0618335 + 0.0275880i
\(663\) 0 0
\(664\) −33.1982 + 10.7007i −1.28834 + 0.415268i
\(665\) 0 0
\(666\) 0 0
\(667\) −7.93827 −0.307371
\(668\) −0.204532 + 0.183583i −0.00791356 + 0.00710303i
\(669\) 0 0
\(670\) 0 0
\(671\) −6.54235 −0.252565
\(672\) 0 0
\(673\) 36.1896i 1.39501i −0.716582 0.697503i \(-0.754294\pi\)
0.716582 0.697503i \(-0.245706\pi\)
\(674\) −5.34538 2.38492i −0.205896 0.0918639i
\(675\) 0 0
\(676\) 17.2894 + 19.2623i 0.664976 + 0.740856i
\(677\) −9.17214 −0.352514 −0.176257 0.984344i \(-0.556399\pi\)
−0.176257 + 0.984344i \(0.556399\pi\)
\(678\) 0 0
\(679\) 21.1034 0.809873
\(680\) 0 0
\(681\) 0 0
\(682\) −13.7599 6.13921i −0.526895 0.235083i
\(683\) −16.3974 −0.627429 −0.313714 0.949517i \(-0.601573\pi\)
−0.313714 + 0.949517i \(0.601573\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 25.7658 + 11.4958i 0.983742 + 0.438912i
\(687\) 0 0
\(688\) 4.84163 44.7194i 0.184586 1.70491i
\(689\) 2.22166 0.0846387
\(690\) 0 0
\(691\) 24.4904i 0.931657i 0.884875 + 0.465828i \(0.154244\pi\)
−0.884875 + 0.465828i \(0.845756\pi\)
\(692\) −5.29933 5.90404i −0.201450 0.224438i
\(693\) 0 0
\(694\) 10.0600 22.5477i 0.381873 0.855900i
\(695\) 0 0
\(696\) 0 0
\(697\) 14.1271i 0.535104i
\(698\) 27.4050 + 12.2272i 1.03730 + 0.462806i
\(699\) 0 0
\(700\) 0 0
\(701\) 12.3887i 0.467916i 0.972247 + 0.233958i \(0.0751679\pi\)
−0.972247 + 0.233958i \(0.924832\pi\)
\(702\) 0 0
\(703\) 26.7994i 1.01076i
\(704\) 6.68335 + 9.29018i 0.251888 + 0.350137i
\(705\) 0 0
\(706\) −27.1729 12.1236i −1.02266 0.456278i
\(707\) 26.1057 0.981807
\(708\) 0 0
\(709\) 33.4144i 1.25490i −0.778655 0.627452i \(-0.784098\pi\)
0.778655 0.627452i \(-0.215902\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 21.7534 7.01176i 0.815244 0.262777i
\(713\) 6.51175i 0.243867i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.97676 6.26218i 0.260734 0.234029i
\(717\) 0 0
\(718\) 13.5501 30.3700i 0.505684 1.13340i
\(719\) −33.8938 −1.26403 −0.632013 0.774958i \(-0.717771\pi\)
−0.632013 + 0.774958i \(0.717771\pi\)
\(720\) 0 0
\(721\) −38.3974 −1.42999
\(722\) 5.62185 12.6004i 0.209224 0.468937i
\(723\) 0 0
\(724\) 13.5479 12.1603i 0.503503 0.451932i
\(725\) 0 0
\(726\) 0 0
\(727\) 14.1846i 0.526076i −0.964785 0.263038i \(-0.915275\pi\)
0.964785 0.263038i \(-0.0847245\pi\)
\(728\) 0.413929 + 1.28418i 0.0153412 + 0.0475950i
\(729\) 0 0
\(730\) 0 0
\(731\) 83.0352i 3.07117i
\(732\) 0 0
\(733\) −8.09296 −0.298920 −0.149460 0.988768i \(-0.547754\pi\)
−0.149460 + 0.988768i \(0.547754\pi\)
\(734\) 32.8056 + 14.6367i 1.21088 + 0.540251i
\(735\) 0 0
\(736\) 2.48972 4.27366i 0.0917725 0.157529i
\(737\) 6.96208i 0.256452i
\(738\) 0 0
\(739\) 22.0919i 0.812663i 0.913726 + 0.406331i \(0.133192\pi\)
−0.913726 + 0.406331i \(0.866808\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −23.5036 10.4865i −0.862844 0.384971i
\(743\) 8.78340i 0.322232i 0.986936 + 0.161116i \(0.0515093\pi\)
−0.986936 + 0.161116i \(0.948491\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 5.80123 13.0024i 0.212398 0.476052i
\(747\) 0 0
\(748\) −14.1119 15.7222i −0.515982 0.574861i
\(749\) 29.2728i 1.06961i
\(750\) 0 0
\(751\) 13.3779 0.488167 0.244084 0.969754i \(-0.421513\pi\)
0.244084 + 0.969754i \(0.421513\pi\)
\(752\) 13.2973 + 1.43965i 0.484901 + 0.0524988i
\(753\) 0 0
\(754\) 2.82965 + 1.26249i 0.103050 + 0.0459773i
\(755\) 0 0
\(756\) 0 0
\(757\) 9.72450 0.353443 0.176721 0.984261i \(-0.443451\pi\)
0.176721 + 0.984261i \(0.443451\pi\)
\(758\) 24.4963 + 10.9294i 0.889747 + 0.396975i
\(759\) 0 0
\(760\) 0 0
\(761\) −33.8835 −1.22828 −0.614138 0.789198i \(-0.710496\pi\)
−0.614138 + 0.789198i \(0.710496\pi\)
\(762\) 0 0
\(763\) −30.1052 −1.08988
\(764\) 20.3147 + 22.6328i 0.734959 + 0.818826i
\(765\) 0 0
\(766\) 37.1236 + 16.5633i 1.34133 + 0.598456i
\(767\) 1.55317i 0.0560817i
\(768\) 0 0
\(769\) −18.7334 −0.675545 −0.337772 0.941228i \(-0.609673\pi\)
−0.337772 + 0.941228i \(0.609673\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −30.8934 + 27.7292i −1.11188 + 0.997996i
\(773\) −22.5006 −0.809292 −0.404646 0.914474i \(-0.632605\pi\)
−0.404646 + 0.914474i \(0.632605\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 9.26355 + 28.7395i 0.332542 + 1.03169i
\(777\) 0 0
\(778\) 38.5074 + 17.1807i 1.38056 + 0.615958i
\(779\) 5.81673i 0.208406i
\(780\) 0 0
\(781\) 11.7564i 0.420677i
\(782\) −3.72019 + 8.33813i −0.133034 + 0.298171i
\(783\) 0 0
\(784\) −1.33145 + 12.2978i −0.0475517 + 0.439207i
\(785\) 0 0
\(786\) 0 0
\(787\) −28.1063 −1.00188 −0.500940 0.865482i \(-0.667012\pi\)
−0.500940 + 0.865482i \(0.667012\pi\)
\(788\) 31.0616 + 34.6060i 1.10652 + 1.23279i
\(789\) 0 0
\(790\) 0 0
\(791\) −2.25133 −0.0800479
\(792\) 0 0
\(793\) 1.10363i 0.0391910i
\(794\) 1.57405 3.52795i 0.0558610 0.125202i
\(795\) 0 0
\(796\) −9.64055 10.7406i −0.341700 0.380692i
\(797\) 38.1461 1.35120 0.675602 0.737267i \(-0.263884\pi\)
0.675602 + 0.737267i \(0.263884\pi\)
\(798\) 0 0
\(799\) −24.6904 −0.873485
\(800\) 0 0
\(801\) 0 0
\(802\) −14.9094 + 33.4168i −0.526470 + 1.17999i
\(803\) 5.90409 0.208351
\(804\) 0 0
\(805\) 0 0
\(806\) 1.03562 2.32116i 0.0364782 0.0817593i
\(807\) 0 0
\(808\) 11.4594 + 35.5519i 0.403140 + 1.25071i
\(809\) −10.0745 −0.354201 −0.177101 0.984193i \(-0.556672\pi\)
−0.177101 + 0.984193i \(0.556672\pi\)
\(810\) 0 0
\(811\) 40.6001i 1.42566i 0.701335 + 0.712832i \(0.252588\pi\)
−0.701335 + 0.712832i \(0.747412\pi\)
\(812\) −23.9765 26.7125i −0.841411 0.937426i
\(813\) 0 0
\(814\) −16.2855 7.26604i −0.570807 0.254674i
\(815\) 0 0
\(816\) 0 0
\(817\) 34.1891i 1.19612i
\(818\) −12.9527 + 29.0311i −0.452879 + 1.01505i
\(819\) 0 0
\(820\) 0 0
\(821\) 10.3397i 0.360858i 0.983588 + 0.180429i \(0.0577486\pi\)
−0.983588 + 0.180429i \(0.942251\pi\)
\(822\) 0 0
\(823\) 28.5013i 0.993493i −0.867896 0.496746i \(-0.834528\pi\)
0.867896 0.496746i \(-0.165472\pi\)
\(824\) −16.8550 52.2912i −0.587170 1.82165i
\(825\) 0 0
\(826\) 7.33113 16.4314i 0.255083 0.571722i
\(827\) −32.1957 −1.11956 −0.559778 0.828643i \(-0.689114\pi\)
−0.559778 + 0.828643i \(0.689114\pi\)
\(828\) 0 0
\(829\) 31.5286i 1.09503i 0.836795 + 0.547516i \(0.184427\pi\)
−0.836795 + 0.547516i \(0.815573\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.56716 + 1.12741i −0.0543314 + 0.0390860i
\(833\) 22.8346i 0.791173i
\(834\) 0 0
\(835\) 0 0
\(836\) −5.81045 6.47349i −0.200959 0.223890i
\(837\) 0 0
\(838\) −31.6813 14.1351i −1.09441 0.488290i
\(839\) 54.9816 1.89818 0.949089 0.315009i \(-0.102008\pi\)
0.949089 + 0.315009i \(0.102008\pi\)
\(840\) 0 0
\(841\) −53.4316 −1.84247
\(842\) −43.1175 19.2376i −1.48593 0.662970i
\(843\) 0 0
\(844\) −6.52517 + 5.85684i −0.224606 + 0.201601i
\(845\) 0 0
\(846\) 0 0
\(847\) 17.6990i 0.608144i
\(848\) 3.96380 36.6113i 0.136117 1.25724i
\(849\) 0 0
\(850\) 0 0
\(851\) 7.70696i 0.264191i
\(852\) 0 0
\(853\) −4.22607 −0.144698 −0.0723489 0.997379i \(-0.523050\pi\)
−0.0723489 + 0.997379i \(0.523050\pi\)
\(854\) −5.20924 + 11.6756i −0.178257 + 0.399530i
\(855\) 0 0
\(856\) 39.8650 12.8496i 1.36256 0.439191i
\(857\) 54.8223i 1.87270i 0.351074 + 0.936348i \(0.385817\pi\)
−0.351074 + 0.936348i \(0.614183\pi\)
\(858\) 0 0
\(859\) 14.2126i 0.484926i −0.970161 0.242463i \(-0.922045\pi\)
0.970161 0.242463i \(-0.0779554\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.71574 + 15.0521i −0.228739 + 0.512678i
\(863\) 40.7446i 1.38696i −0.720474 0.693482i \(-0.756076\pi\)
0.720474 0.693482i \(-0.243924\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25.5305 + 11.3908i 0.867563 + 0.387077i
\(867\) 0 0
\(868\) −21.9122 + 19.6679i −0.743750 + 0.667573i
\(869\) 19.5638i 0.663658i
\(870\) 0 0
\(871\) 1.17443 0.0397941
\(872\) −13.2150 40.9985i −0.447517 1.38839i
\(873\) 0 0
\(874\) −1.53176 + 3.43316i −0.0518125 + 0.116128i
\(875\) 0 0
\(876\) 0 0
\(877\) −34.4568 −1.16352 −0.581762 0.813359i \(-0.697637\pi\)
−0.581762 + 0.813359i \(0.697637\pi\)
\(878\) 14.5034 32.5068i 0.489467 1.09705i
\(879\) 0 0
\(880\) 0 0
\(881\) −23.8528 −0.803623 −0.401811 0.915722i \(-0.631619\pi\)
−0.401811 + 0.915722i \(0.631619\pi\)
\(882\) 0 0
\(883\) 24.2417 0.815798 0.407899 0.913027i \(-0.366262\pi\)
0.407899 + 0.913027i \(0.366262\pi\)
\(884\) 2.65217 2.38053i 0.0892023 0.0800659i
\(885\) 0 0
\(886\) −11.2660 + 25.2508i −0.378490 + 0.848317i
\(887\) 39.1843i 1.31568i −0.753158 0.657840i \(-0.771470\pi\)
0.753158 0.657840i \(-0.228530\pi\)
\(888\) 0 0
\(889\) 4.81571 0.161514
\(890\) 0 0
\(891\) 0 0
\(892\) −7.42481 + 6.66434i −0.248601 + 0.223139i
\(893\) −10.1661 −0.340195
\(894\) 0 0
\(895\) 0 0
\(896\) 21.9009 4.53005i 0.731658 0.151339i
\(897\) 0 0
\(898\) −11.5021 + 25.7799i −0.383830 + 0.860286i
\(899\) 67.6185i 2.25520i
\(900\) 0 0
\(901\) 67.9802i 2.26475i
\(902\) 3.53473 + 1.57707i 0.117693 + 0.0525108i
\(903\) 0 0
\(904\) −0.988244 3.06595i −0.0328685 0.101972i
\(905\) 0 0
\(906\) 0 0
\(907\) −18.9418 −0.628951 −0.314475 0.949266i \(-0.601829\pi\)
−0.314475 + 0.949266i \(0.601829\pi\)
\(908\) 15.0385 + 16.7546i 0.499071 + 0.556021i
\(909\) 0 0
\(910\) 0 0
\(911\) −17.8493 −0.591375 −0.295688 0.955285i \(-0.595549\pi\)
−0.295688 + 0.955285i \(0.595549\pi\)
\(912\) 0 0
\(913\) 17.6415i 0.583850i
\(914\) 6.47267 + 2.88789i 0.214097 + 0.0955228i
\(915\) 0 0
\(916\) −23.5706 + 21.1564i −0.778793 + 0.699027i
\(917\) −13.6425 −0.450515
\(918\) 0 0
\(919\) −24.4983 −0.808126 −0.404063 0.914731i \(-0.632402\pi\)
−0.404063 + 0.914731i \(0.632402\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.61978 + 2.95352i 0.218011 + 0.0972690i
\(923\) −1.98318 −0.0652772
\(924\) 0 0
\(925\) 0 0
\(926\) −7.48443 3.33930i −0.245954 0.109736i
\(927\) 0 0
\(928\) 25.8535 44.3780i 0.848682 1.45678i
\(929\) 11.5577 0.379196 0.189598 0.981862i \(-0.439282\pi\)
0.189598 + 0.981862i \(0.439282\pi\)
\(930\) 0 0
\(931\) 9.40197i 0.308137i
\(932\) 16.3114 14.6407i 0.534296 0.479572i
\(933\) 0 0
\(934\) 9.69251 21.7240i 0.317148 0.710831i
\(935\) 0 0
\(936\) 0 0
\(937\) 59.5587i 1.94570i 0.231440 + 0.972849i \(0.425656\pi\)
−0.231440 + 0.972849i \(0.574344\pi\)
\(938\) −12.4246 5.54345i −0.405679 0.181000i
\(939\) 0 0
\(940\) 0 0
\(941\) 55.9289i 1.82323i 0.411044 + 0.911615i \(0.365164\pi\)
−0.411044 + 0.911615i \(0.634836\pi\)
\(942\) 0 0
\(943\) 1.67277i 0.0544730i
\(944\) 25.5951 + 2.77110i 0.833048 + 0.0901917i
\(945\) 0 0
\(946\) −20.7761 9.26958i −0.675489 0.301380i
\(947\) −30.0945 −0.977941 −0.488970 0.872300i \(-0.662627\pi\)
−0.488970 + 0.872300i \(0.662627\pi\)
\(948\) 0 0
\(949\) 0.995959i 0.0323302i
\(950\) 0 0
\(951\) 0 0
\(952\) −39.2944 + 12.6657i −1.27354 + 0.410498i
\(953\) 12.4097i 0.401989i −0.979592 0.200995i \(-0.935583\pi\)
0.979592 0.200995i \(-0.0644174\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −23.1360 25.7761i −0.748272 0.833658i
\(957\) 0 0
\(958\) −21.2664 + 47.6647i −0.687085 + 1.53998i
\(959\) 10.6552 0.344074
\(960\) 0 0
\(961\) 24.4674 0.789271
\(962\) 1.22571 2.74720i 0.0395183 0.0885732i
\(963\) 0 0
\(964\) −6.37066 7.09762i −0.205185 0.228599i
\(965\) 0 0
\(966\) 0 0
\(967\) 41.8371i 1.34539i −0.739920 0.672694i \(-0.765137\pi\)
0.739920 0.672694i \(-0.234863\pi\)
\(968\) −24.1032 + 7.76916i −0.774706 + 0.249710i
\(969\) 0 0
\(970\) 0 0
\(971\) 40.6875i 1.30572i −0.757477 0.652862i \(-0.773568\pi\)
0.757477 0.652862i \(-0.226432\pi\)
\(972\) 0 0
\(973\) 34.4439 1.10422
\(974\) 6.65101 + 2.96745i 0.213112 + 0.0950833i
\(975\) 0 0
\(976\) −18.1869 1.96905i −0.582150 0.0630276i
\(977\) 1.67923i 0.0537233i 0.999639 + 0.0268617i \(0.00855136\pi\)
−0.999639 + 0.0268617i \(0.991449\pi\)
\(978\) 0 0
\(979\) 11.5598i 0.369453i
\(980\) 0 0
\(981\) 0 0
\(982\) −29.8535 13.3196i −0.952663 0.425046i
\(983\) 13.4944i 0.430404i 0.976570 + 0.215202i \(0.0690411\pi\)
−0.976570 + 0.215202i \(0.930959\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −38.6307 + 86.5838i −1.23025 + 2.75739i
\(987\) 0 0
\(988\) 1.09201 0.980164i 0.0347415 0.0311832i
\(989\) 9.83208i 0.312642i
\(990\) 0 0
\(991\) −10.8502 −0.344667 −0.172333 0.985039i \(-0.555131\pi\)
−0.172333 + 0.985039i \(0.555131\pi\)
\(992\) −36.4032 21.2076i −1.15580 0.673341i
\(993\) 0 0
\(994\) 20.9806 + 9.36083i 0.665465 + 0.296908i
\(995\) 0 0
\(996\) 0 0
\(997\) 39.0972 1.23822 0.619110 0.785304i \(-0.287493\pi\)
0.619110 + 0.785304i \(0.287493\pi\)
\(998\) 18.4828 + 8.24639i 0.585063 + 0.261035i
\(999\) 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 1800.2.d.t.1549.5 8
3.2 odd 2 600.2.d.g.349.4 8
4.3 odd 2 7200.2.d.t.2449.2 8
5.2 odd 4 1800.2.k.t.901.8 8
5.3 odd 4 1800.2.k.q.901.1 8
5.4 even 2 1800.2.d.s.1549.4 8
8.3 odd 2 7200.2.d.s.2449.2 8
8.5 even 2 1800.2.d.s.1549.3 8
12.11 even 2 2400.2.d.h.49.2 8
15.2 even 4 600.2.k.d.301.1 8
15.8 even 4 600.2.k.e.301.8 yes 8
15.14 odd 2 600.2.d.h.349.5 8
20.3 even 4 7200.2.k.s.3601.1 8
20.7 even 4 7200.2.k.r.3601.7 8
20.19 odd 2 7200.2.d.s.2449.7 8
24.5 odd 2 600.2.d.h.349.6 8
24.11 even 2 2400.2.d.g.49.2 8
40.3 even 4 7200.2.k.s.3601.2 8
40.13 odd 4 1800.2.k.q.901.2 8
40.19 odd 2 7200.2.d.t.2449.7 8
40.27 even 4 7200.2.k.r.3601.8 8
40.29 even 2 inner 1800.2.d.t.1549.6 8
40.37 odd 4 1800.2.k.t.901.7 8
60.23 odd 4 2400.2.k.e.1201.5 8
60.47 odd 4 2400.2.k.d.1201.4 8
60.59 even 2 2400.2.d.g.49.7 8
120.29 odd 2 600.2.d.g.349.3 8
120.53 even 4 600.2.k.e.301.7 yes 8
120.59 even 2 2400.2.d.h.49.7 8
120.77 even 4 600.2.k.d.301.2 yes 8
120.83 odd 4 2400.2.k.e.1201.1 8
120.107 odd 4 2400.2.k.d.1201.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.d.g.349.3 8 120.29 odd 2
600.2.d.g.349.4 8 3.2 odd 2
600.2.d.h.349.5 8 15.14 odd 2
600.2.d.h.349.6 8 24.5 odd 2
600.2.k.d.301.1 8 15.2 even 4
600.2.k.d.301.2 yes 8 120.77 even 4
600.2.k.e.301.7 yes 8 120.53 even 4
600.2.k.e.301.8 yes 8 15.8 even 4
1800.2.d.s.1549.3 8 8.5 even 2
1800.2.d.s.1549.4 8 5.4 even 2
1800.2.d.t.1549.5 8 1.1 even 1 trivial
1800.2.d.t.1549.6 8 40.29 even 2 inner
1800.2.k.q.901.1 8 5.3 odd 4
1800.2.k.q.901.2 8 40.13 odd 4
1800.2.k.t.901.7 8 40.37 odd 4
1800.2.k.t.901.8 8 5.2 odd 4
2400.2.d.g.49.2 8 24.11 even 2
2400.2.d.g.49.7 8 60.59 even 2
2400.2.d.h.49.2 8 12.11 even 2
2400.2.d.h.49.7 8 120.59 even 2
2400.2.k.d.1201.4 8 60.47 odd 4
2400.2.k.d.1201.8 8 120.107 odd 4
2400.2.k.e.1201.1 8 120.83 odd 4
2400.2.k.e.1201.5 8 60.23 odd 4
7200.2.d.s.2449.2 8 8.3 odd 2
7200.2.d.s.2449.7 8 20.19 odd 2
7200.2.d.t.2449.2 8 4.3 odd 2
7200.2.d.t.2449.7 8 40.19 odd 2
7200.2.k.r.3601.7 8 20.7 even 4
7200.2.k.r.3601.8 8 40.27 even 4
7200.2.k.s.3601.1 8 20.3 even 4
7200.2.k.s.3601.2 8 40.3 even 4