Properties

Label 1008.2.r.i.337.1
Level $1008$
Weight $2$
Character 1008.337
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 1008.337
Dual form 1008.2.r.i.673.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11334 - 1.32683i) q^{3} +(1.43969 - 2.49362i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\) \(q+(-1.11334 - 1.32683i) q^{3} +(1.43969 - 2.49362i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-0.520945 + 2.95442i) q^{9} +(-0.592396 - 1.02606i) q^{11} +(2.37939 - 4.12122i) q^{13} +(-4.91147 + 0.866025i) q^{15} +5.41147 q^{17} -1.10607 q^{19} +(0.592396 - 1.62760i) q^{21} +(2.95084 - 5.11100i) q^{23} +(-1.64543 - 2.84997i) q^{25} +(4.50000 - 2.59808i) q^{27} +(-2.49273 - 4.31753i) q^{29} +(-2.78699 + 4.82721i) q^{31} +(-0.701867 + 1.92836i) q^{33} +2.87939 q^{35} -2.42602 q^{37} +(-8.11721 + 1.43128i) q^{39} +(-3.81908 + 6.61484i) q^{41} +(-3.78699 - 6.55926i) q^{43} +(6.61721 + 5.55250i) q^{45} +(0.141559 + 0.245188i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-6.02481 - 7.18009i) q^{51} -4.22668 q^{53} -3.41147 q^{55} +(1.23143 + 1.46756i) q^{57} +(5.86231 - 10.1538i) q^{59} +(-0.992726 - 1.71945i) q^{61} +(-2.81908 + 1.02606i) q^{63} +(-6.85117 - 11.8666i) q^{65} +(-6.76991 + 11.7258i) q^{67} +(-10.0667 + 1.77503i) q^{69} +5.11381 q^{71} -0.327696 q^{73} +(-1.94949 + 5.35619i) q^{75} +(0.592396 - 1.02606i) q^{77} +(-5.24763 - 9.08916i) q^{79} +(-8.45723 - 3.07818i) q^{81} +(3.62701 + 6.28217i) q^{83} +(7.79086 - 13.4942i) q^{85} +(-2.95336 + 8.11430i) q^{87} -14.7023 q^{89} +4.75877 q^{91} +(9.50774 - 1.67647i) q^{93} +(-1.59240 + 2.75811i) q^{95} +(-9.04710 - 15.6700i) q^{97} +(3.34002 - 1.21567i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} + 3 q^{7} + 3 q^{13} - 9 q^{15} + 12 q^{17} + 18 q^{19} + 6 q^{23} + 6 q^{25} + 27 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} + 6 q^{35} - 30 q^{37} - 18 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 9 q^{47} - 3 q^{49} - 9 q^{51} - 12 q^{53} + 27 q^{57} + 3 q^{59} + 12 q^{61} - 15 q^{65} - 12 q^{67} - 27 q^{69} - 42 q^{71} + 6 q^{73} - 9 q^{75} - 15 q^{79} - 6 q^{83} + 15 q^{85} + 9 q^{87} - 36 q^{89} + 6 q^{91} + 9 q^{93} - 6 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) −1.11334 1.32683i −0.642788 0.766044i
\(4\) 0 0
\(5\) 1.43969 2.49362i 0.643850 1.11518i −0.340716 0.940166i \(-0.610669\pi\)
0.984566 0.175015i \(-0.0559973\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) −0.520945 + 2.95442i −0.173648 + 0.984808i
\(10\) 0 0
\(11\) −0.592396 1.02606i −0.178614 0.309369i 0.762792 0.646644i \(-0.223828\pi\)
−0.941406 + 0.337275i \(0.890495\pi\)
\(12\) 0 0
\(13\) 2.37939 4.12122i 0.659923 1.14302i −0.320713 0.947177i \(-0.603922\pi\)
0.980635 0.195843i \(-0.0627443\pi\)
\(14\) 0 0
\(15\) −4.91147 + 0.866025i −1.26814 + 0.223607i
\(16\) 0 0
\(17\) 5.41147 1.31248 0.656238 0.754554i \(-0.272147\pi\)
0.656238 + 0.754554i \(0.272147\pi\)
\(18\) 0 0
\(19\) −1.10607 −0.253749 −0.126875 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(20\) 0 0
\(21\) 0.592396 1.62760i 0.129271 0.355170i
\(22\) 0 0
\(23\) 2.95084 5.11100i 0.615292 1.06572i −0.375041 0.927008i \(-0.622371\pi\)
0.990333 0.138709i \(-0.0442952\pi\)
\(24\) 0 0
\(25\) −1.64543 2.84997i −0.329086 0.569994i
\(26\) 0 0
\(27\) 4.50000 2.59808i 0.866025 0.500000i
\(28\) 0 0
\(29\) −2.49273 4.31753i −0.462888 0.801745i 0.536216 0.844081i \(-0.319853\pi\)
−0.999103 + 0.0423361i \(0.986520\pi\)
\(30\) 0 0
\(31\) −2.78699 + 4.82721i −0.500558 + 0.866992i 0.499442 + 0.866347i \(0.333538\pi\)
−1.00000 0.000644439i \(0.999795\pi\)
\(32\) 0 0
\(33\) −0.701867 + 1.92836i −0.122179 + 0.335685i
\(34\) 0 0
\(35\) 2.87939 0.486705
\(36\) 0 0
\(37\) −2.42602 −0.398836 −0.199418 0.979915i \(-0.563905\pi\)
−0.199418 + 0.979915i \(0.563905\pi\)
\(38\) 0 0
\(39\) −8.11721 + 1.43128i −1.29979 + 0.229189i
\(40\) 0 0
\(41\) −3.81908 + 6.61484i −0.596440 + 1.03306i 0.396902 + 0.917861i \(0.370085\pi\)
−0.993342 + 0.115203i \(0.963248\pi\)
\(42\) 0 0
\(43\) −3.78699 6.55926i −0.577510 1.00028i −0.995764 0.0919470i \(-0.970691\pi\)
0.418253 0.908330i \(-0.362642\pi\)
\(44\) 0 0
\(45\) 6.61721 + 5.55250i 0.986436 + 0.827718i
\(46\) 0 0
\(47\) 0.141559 + 0.245188i 0.0206485 + 0.0357643i 0.876165 0.482011i \(-0.160094\pi\)
−0.855516 + 0.517776i \(0.826760\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) −6.02481 7.18009i −0.843643 1.00541i
\(52\) 0 0
\(53\) −4.22668 −0.580579 −0.290290 0.956939i \(-0.593752\pi\)
−0.290290 + 0.956939i \(0.593752\pi\)
\(54\) 0 0
\(55\) −3.41147 −0.460003
\(56\) 0 0
\(57\) 1.23143 + 1.46756i 0.163107 + 0.194383i
\(58\) 0 0
\(59\) 5.86231 10.1538i 0.763208 1.32191i −0.177981 0.984034i \(-0.556957\pi\)
0.941189 0.337881i \(-0.109710\pi\)
\(60\) 0 0
\(61\) −0.992726 1.71945i −0.127106 0.220153i 0.795448 0.606021i \(-0.207235\pi\)
−0.922554 + 0.385868i \(0.873902\pi\)
\(62\) 0 0
\(63\) −2.81908 + 1.02606i −0.355170 + 0.129271i
\(64\) 0 0
\(65\) −6.85117 11.8666i −0.849783 1.47187i
\(66\) 0 0
\(67\) −6.76991 + 11.7258i −0.827077 + 1.43254i 0.0732450 + 0.997314i \(0.476664\pi\)
−0.900322 + 0.435225i \(0.856669\pi\)
\(68\) 0 0
\(69\) −10.0667 + 1.77503i −1.21189 + 0.213689i
\(70\) 0 0
\(71\) 5.11381 0.606897 0.303449 0.952848i \(-0.401862\pi\)
0.303449 + 0.952848i \(0.401862\pi\)
\(72\) 0 0
\(73\) −0.327696 −0.0383539 −0.0191770 0.999816i \(-0.506105\pi\)
−0.0191770 + 0.999816i \(0.506105\pi\)
\(74\) 0 0
\(75\) −1.94949 + 5.35619i −0.225108 + 0.618479i
\(76\) 0 0
\(77\) 0.592396 1.02606i 0.0675098 0.116930i
\(78\) 0 0
\(79\) −5.24763 9.08916i −0.590404 1.02261i −0.994178 0.107751i \(-0.965635\pi\)
0.403774 0.914859i \(-0.367698\pi\)
\(80\) 0 0
\(81\) −8.45723 3.07818i −0.939693 0.342020i
\(82\) 0 0
\(83\) 3.62701 + 6.28217i 0.398116 + 0.689558i 0.993494 0.113889i \(-0.0363307\pi\)
−0.595377 + 0.803446i \(0.702997\pi\)
\(84\) 0 0
\(85\) 7.79086 13.4942i 0.845037 1.46365i
\(86\) 0 0
\(87\) −2.95336 + 8.11430i −0.316634 + 0.869944i
\(88\) 0 0
\(89\) −14.7023 −1.55844 −0.779222 0.626748i \(-0.784386\pi\)
−0.779222 + 0.626748i \(0.784386\pi\)
\(90\) 0 0
\(91\) 4.75877 0.498855
\(92\) 0 0
\(93\) 9.50774 1.67647i 0.985907 0.173842i
\(94\) 0 0
\(95\) −1.59240 + 2.75811i −0.163376 + 0.282976i
\(96\) 0 0
\(97\) −9.04710 15.6700i −0.918594 1.59105i −0.801552 0.597925i \(-0.795992\pi\)
−0.117042 0.993127i \(-0.537341\pi\)
\(98\) 0 0
\(99\) 3.34002 1.21567i 0.335685 0.122179i
\(100\) 0 0
\(101\) 8.84389 + 15.3181i 0.880000 + 1.52421i 0.851339 + 0.524616i \(0.175791\pi\)
0.0286612 + 0.999589i \(0.490876\pi\)
\(102\) 0 0
\(103\) 6.58899 11.4125i 0.649233 1.12450i −0.334074 0.942547i \(-0.608423\pi\)
0.983306 0.181957i \(-0.0582432\pi\)
\(104\) 0 0
\(105\) −3.20574 3.82045i −0.312848 0.372838i
\(106\) 0 0
\(107\) 13.9513 1.34872 0.674362 0.738401i \(-0.264419\pi\)
0.674362 + 0.738401i \(0.264419\pi\)
\(108\) 0 0
\(109\) −10.1702 −0.974133 −0.487066 0.873365i \(-0.661933\pi\)
−0.487066 + 0.873365i \(0.661933\pi\)
\(110\) 0 0
\(111\) 2.70099 + 3.21891i 0.256367 + 0.305526i
\(112\) 0 0
\(113\) 2.83750 4.91469i 0.266929 0.462335i −0.701138 0.713026i \(-0.747324\pi\)
0.968067 + 0.250691i \(0.0806577\pi\)
\(114\) 0 0
\(115\) −8.49660 14.7165i −0.792312 1.37232i
\(116\) 0 0
\(117\) 10.9363 + 9.17664i 1.01106 + 0.848380i
\(118\) 0 0
\(119\) 2.70574 + 4.68647i 0.248035 + 0.429608i
\(120\) 0 0
\(121\) 4.79813 8.31061i 0.436194 0.755510i
\(122\) 0 0
\(123\) 13.0287 2.29731i 1.17476 0.207141i
\(124\) 0 0
\(125\) 4.92127 0.440172
\(126\) 0 0
\(127\) 20.0915 1.78283 0.891417 0.453184i \(-0.149712\pi\)
0.891417 + 0.453184i \(0.149712\pi\)
\(128\) 0 0
\(129\) −4.48680 + 12.3274i −0.395040 + 1.08536i
\(130\) 0 0
\(131\) 3.84002 6.65111i 0.335504 0.581111i −0.648077 0.761575i \(-0.724427\pi\)
0.983582 + 0.180464i \(0.0577600\pi\)
\(132\) 0 0
\(133\) −0.553033 0.957882i −0.0479541 0.0830589i
\(134\) 0 0
\(135\) 14.9617i 1.28770i
\(136\) 0 0
\(137\) 7.13429 + 12.3569i 0.609523 + 1.05573i 0.991319 + 0.131478i \(0.0419724\pi\)
−0.381796 + 0.924247i \(0.624694\pi\)
\(138\) 0 0
\(139\) −7.63950 + 13.2320i −0.647974 + 1.12232i 0.335632 + 0.941993i \(0.391050\pi\)
−0.983606 + 0.180331i \(0.942283\pi\)
\(140\) 0 0
\(141\) 0.167718 0.460802i 0.0141244 0.0388066i
\(142\) 0 0
\(143\) −5.63816 −0.471486
\(144\) 0 0
\(145\) −14.3550 −1.19212
\(146\) 0 0
\(147\) 1.70574 0.300767i 0.140687 0.0248069i
\(148\) 0 0
\(149\) 7.20826 12.4851i 0.590524 1.02282i −0.403638 0.914919i \(-0.632254\pi\)
0.994162 0.107899i \(-0.0344122\pi\)
\(150\) 0 0
\(151\) 5.90420 + 10.2264i 0.480477 + 0.832211i 0.999749 0.0223984i \(-0.00713022\pi\)
−0.519272 + 0.854609i \(0.673797\pi\)
\(152\) 0 0
\(153\) −2.81908 + 15.9878i −0.227909 + 1.29254i
\(154\) 0 0
\(155\) 8.02481 + 13.8994i 0.644569 + 1.11643i
\(156\) 0 0
\(157\) −6.76011 + 11.7089i −0.539516 + 0.934469i 0.459414 + 0.888222i \(0.348059\pi\)
−0.998930 + 0.0462468i \(0.985274\pi\)
\(158\) 0 0
\(159\) 4.70574 + 5.60808i 0.373189 + 0.444750i
\(160\) 0 0
\(161\) 5.90167 0.465117
\(162\) 0 0
\(163\) −23.9641 −1.87701 −0.938506 0.345262i \(-0.887790\pi\)
−0.938506 + 0.345262i \(0.887790\pi\)
\(164\) 0 0
\(165\) 3.79813 + 4.52644i 0.295684 + 0.352383i
\(166\) 0 0
\(167\) −2.48545 + 4.30493i −0.192330 + 0.333125i −0.946022 0.324102i \(-0.894938\pi\)
0.753692 + 0.657228i \(0.228271\pi\)
\(168\) 0 0
\(169\) −4.82295 8.35359i −0.370996 0.642584i
\(170\) 0 0
\(171\) 0.576199 3.26779i 0.0440631 0.249894i
\(172\) 0 0
\(173\) 12.6099 + 21.8411i 0.958716 + 1.66054i 0.725626 + 0.688089i \(0.241550\pi\)
0.233090 + 0.972455i \(0.425116\pi\)
\(174\) 0 0
\(175\) 1.64543 2.84997i 0.124383 0.215437i
\(176\) 0 0
\(177\) −19.9991 + 3.52638i −1.50323 + 0.265059i
\(178\) 0 0
\(179\) 9.27362 0.693143 0.346572 0.938024i \(-0.387346\pi\)
0.346572 + 0.938024i \(0.387346\pi\)
\(180\) 0 0
\(181\) 13.3327 0.991015 0.495508 0.868604i \(-0.334982\pi\)
0.495508 + 0.868604i \(0.334982\pi\)
\(182\) 0 0
\(183\) −1.17617 + 3.23151i −0.0869453 + 0.238880i
\(184\) 0 0
\(185\) −3.49273 + 6.04958i −0.256790 + 0.444774i
\(186\) 0 0
\(187\) −3.20574 5.55250i −0.234427 0.406039i
\(188\) 0 0
\(189\) 4.50000 + 2.59808i 0.327327 + 0.188982i
\(190\) 0 0
\(191\) 4.38326 + 7.59202i 0.317161 + 0.549339i 0.979895 0.199516i \(-0.0639370\pi\)
−0.662733 + 0.748856i \(0.730604\pi\)
\(192\) 0 0
\(193\) −7.62108 + 13.2001i −0.548577 + 0.950164i 0.449795 + 0.893132i \(0.351497\pi\)
−0.998372 + 0.0570321i \(0.981836\pi\)
\(194\) 0 0
\(195\) −8.11721 + 22.3019i −0.581286 + 1.59707i
\(196\) 0 0
\(197\) −8.64496 −0.615928 −0.307964 0.951398i \(-0.599648\pi\)
−0.307964 + 0.951398i \(0.599648\pi\)
\(198\) 0 0
\(199\) 13.3746 0.948103 0.474051 0.880497i \(-0.342791\pi\)
0.474051 + 0.880497i \(0.342791\pi\)
\(200\) 0 0
\(201\) 23.0954 4.07234i 1.62902 0.287241i
\(202\) 0 0
\(203\) 2.49273 4.31753i 0.174955 0.303031i
\(204\) 0 0
\(205\) 10.9966 + 19.0467i 0.768036 + 1.33028i
\(206\) 0 0
\(207\) 13.5628 + 11.3806i 0.942682 + 0.791004i
\(208\) 0 0
\(209\) 0.655230 + 1.13489i 0.0453232 + 0.0785021i
\(210\) 0 0
\(211\) −3.64290 + 6.30969i −0.250788 + 0.434377i −0.963743 0.266832i \(-0.914023\pi\)
0.712955 + 0.701210i \(0.247356\pi\)
\(212\) 0 0
\(213\) −5.69341 6.78514i −0.390106 0.464910i
\(214\) 0 0
\(215\) −21.8084 −1.48732
\(216\) 0 0
\(217\) −5.57398 −0.378386
\(218\) 0 0
\(219\) 0.364837 + 0.434796i 0.0246534 + 0.0293808i
\(220\) 0 0
\(221\) 12.8760 22.3019i 0.866132 1.50019i
\(222\) 0 0
\(223\) −0.295607 0.512007i −0.0197953 0.0342865i 0.855958 0.517045i \(-0.172968\pi\)
−0.875753 + 0.482759i \(0.839635\pi\)
\(224\) 0 0
\(225\) 9.27719 3.37662i 0.618479 0.225108i
\(226\) 0 0
\(227\) −1.74170 3.01671i −0.115600 0.200226i 0.802419 0.596761i \(-0.203546\pi\)
−0.918020 + 0.396535i \(0.870213\pi\)
\(228\) 0 0
\(229\) 12.3007 21.3054i 0.812850 1.40790i −0.0980107 0.995185i \(-0.531248\pi\)
0.910861 0.412713i \(-0.135419\pi\)
\(230\) 0 0
\(231\) −2.02094 + 0.356347i −0.132968 + 0.0234459i
\(232\) 0 0
\(233\) 11.4165 0.747922 0.373961 0.927445i \(-0.377999\pi\)
0.373961 + 0.927445i \(0.377999\pi\)
\(234\) 0 0
\(235\) 0.815207 0.0531783
\(236\) 0 0
\(237\) −6.21735 + 17.0820i −0.403860 + 1.10960i
\(238\) 0 0
\(239\) 2.04664 3.54488i 0.132386 0.229299i −0.792210 0.610249i \(-0.791070\pi\)
0.924596 + 0.380950i \(0.124403\pi\)
\(240\) 0 0
\(241\) 3.60354 + 6.24152i 0.232124 + 0.402051i 0.958433 0.285317i \(-0.0920990\pi\)
−0.726309 + 0.687369i \(0.758766\pi\)
\(242\) 0 0
\(243\) 5.33157 + 14.6484i 0.342020 + 0.939693i
\(244\) 0 0
\(245\) 1.43969 + 2.49362i 0.0919786 + 0.159312i
\(246\) 0 0
\(247\) −2.63176 + 4.55834i −0.167455 + 0.290040i
\(248\) 0 0
\(249\) 4.29726 11.8066i 0.272328 0.748214i
\(250\) 0 0
\(251\) −5.19934 −0.328179 −0.164090 0.986445i \(-0.552469\pi\)
−0.164090 + 0.986445i \(0.552469\pi\)
\(252\) 0 0
\(253\) −6.99226 −0.439600
\(254\) 0 0
\(255\) −26.5783 + 4.68647i −1.66440 + 0.293478i
\(256\) 0 0
\(257\) 6.71688 11.6340i 0.418988 0.725708i −0.576850 0.816850i \(-0.695718\pi\)
0.995838 + 0.0911421i \(0.0290518\pi\)
\(258\) 0 0
\(259\) −1.21301 2.10100i −0.0753728 0.130550i
\(260\) 0 0
\(261\) 14.0544 5.11538i 0.869944 0.316634i
\(262\) 0 0
\(263\) −5.53849 9.59294i −0.341518 0.591526i 0.643197 0.765701i \(-0.277608\pi\)
−0.984715 + 0.174175i \(0.944274\pi\)
\(264\) 0 0
\(265\) −6.08512 + 10.5397i −0.373806 + 0.647451i
\(266\) 0 0
\(267\) 16.3687 + 19.5075i 1.00175 + 1.19384i
\(268\) 0 0
\(269\) 18.1489 1.10656 0.553279 0.832996i \(-0.313376\pi\)
0.553279 + 0.832996i \(0.313376\pi\)
\(270\) 0 0
\(271\) 13.1557 0.799152 0.399576 0.916700i \(-0.369157\pi\)
0.399576 + 0.916700i \(0.369157\pi\)
\(272\) 0 0
\(273\) −5.29813 6.31407i −0.320658 0.382145i
\(274\) 0 0
\(275\) −1.94949 + 3.37662i −0.117559 + 0.203618i
\(276\) 0 0
\(277\) 11.8983 + 20.6084i 0.714898 + 1.23824i 0.962999 + 0.269505i \(0.0868601\pi\)
−0.248101 + 0.968734i \(0.579807\pi\)
\(278\) 0 0
\(279\) −12.8097 10.7487i −0.766899 0.643505i
\(280\) 0 0
\(281\) −5.99660 10.3864i −0.357727 0.619601i 0.629854 0.776714i \(-0.283115\pi\)
−0.987581 + 0.157112i \(0.949781\pi\)
\(282\) 0 0
\(283\) 7.63223 13.2194i 0.453689 0.785812i −0.544923 0.838486i \(-0.683441\pi\)
0.998612 + 0.0526741i \(0.0167745\pi\)
\(284\) 0 0
\(285\) 5.43242 0.957882i 0.321789 0.0567400i
\(286\) 0 0
\(287\) −7.63816 −0.450866
\(288\) 0 0
\(289\) 12.2841 0.722591
\(290\) 0 0
\(291\) −10.7189 + 29.4500i −0.628355 + 1.72639i
\(292\) 0 0
\(293\) 2.50727 4.34273i 0.146477 0.253705i −0.783446 0.621459i \(-0.786540\pi\)
0.929923 + 0.367755i \(0.119873\pi\)
\(294\) 0 0
\(295\) −16.8799 29.2368i −0.982783 1.70223i
\(296\) 0 0
\(297\) −5.33157 3.07818i −0.309369 0.178614i
\(298\) 0 0
\(299\) −14.0424 24.3221i −0.812090 1.40658i
\(300\) 0 0
\(301\) 3.78699 6.55926i 0.218278 0.378069i
\(302\) 0 0
\(303\) 10.4782 28.7886i 0.601956 1.65386i
\(304\) 0 0
\(305\) −5.71688 −0.327348
\(306\) 0 0
\(307\) −15.1310 −0.863574 −0.431787 0.901976i \(-0.642117\pi\)
−0.431787 + 0.901976i \(0.642117\pi\)
\(308\) 0 0
\(309\) −22.4782 + 3.96351i −1.27874 + 0.225476i
\(310\) 0 0
\(311\) −2.49407 + 4.31986i −0.141426 + 0.244957i −0.928034 0.372496i \(-0.878502\pi\)
0.786608 + 0.617453i \(0.211835\pi\)
\(312\) 0 0
\(313\) 12.7640 + 22.1079i 0.721463 + 1.24961i 0.960413 + 0.278579i \(0.0898634\pi\)
−0.238950 + 0.971032i \(0.576803\pi\)
\(314\) 0 0
\(315\) −1.50000 + 8.50692i −0.0845154 + 0.479311i
\(316\) 0 0
\(317\) 2.60694 + 4.51536i 0.146421 + 0.253608i 0.929902 0.367807i \(-0.119891\pi\)
−0.783481 + 0.621415i \(0.786558\pi\)
\(318\) 0 0
\(319\) −2.95336 + 5.11538i −0.165357 + 0.286406i
\(320\) 0 0
\(321\) −15.5326 18.5110i −0.866943 1.03318i
\(322\) 0 0
\(323\) −5.98545 −0.333039
\(324\) 0 0
\(325\) −15.6604 −0.868685
\(326\) 0 0
\(327\) 11.3229 + 13.4942i 0.626160 + 0.746229i
\(328\) 0 0
\(329\) −0.141559 + 0.245188i −0.00780442 + 0.0135176i
\(330\) 0 0
\(331\) −2.07263 3.58991i −0.113922 0.197319i 0.803426 0.595404i \(-0.203008\pi\)
−0.917348 + 0.398085i \(0.869675\pi\)
\(332\) 0 0
\(333\) 1.26382 7.16750i 0.0692571 0.392776i
\(334\) 0 0
\(335\) 19.4932 + 33.7632i 1.06503 + 1.84468i
\(336\) 0 0
\(337\) −1.96064 + 3.39592i −0.106803 + 0.184988i −0.914473 0.404646i \(-0.867395\pi\)
0.807671 + 0.589634i \(0.200728\pi\)
\(338\) 0 0
\(339\) −9.68004 + 1.70685i −0.525748 + 0.0927035i
\(340\) 0 0
\(341\) 6.60401 0.357627
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −10.0667 + 27.6580i −0.541973 + 1.48906i
\(346\) 0 0
\(347\) 6.22534 10.7826i 0.334194 0.578840i −0.649136 0.760672i \(-0.724869\pi\)
0.983330 + 0.181832i \(0.0582027\pi\)
\(348\) 0 0
\(349\) 18.1976 + 31.5191i 0.974094 + 1.68718i 0.682892 + 0.730520i \(0.260722\pi\)
0.291203 + 0.956661i \(0.405945\pi\)
\(350\) 0 0
\(351\) 24.7273i 1.31985i
\(352\) 0 0
\(353\) 14.9217 + 25.8452i 0.794204 + 1.37560i 0.923343 + 0.383975i \(0.125445\pi\)
−0.129139 + 0.991626i \(0.541221\pi\)
\(354\) 0 0
\(355\) 7.36231 12.7519i 0.390751 0.676800i
\(356\) 0 0
\(357\) 3.20574 8.80769i 0.169666 0.466152i
\(358\) 0 0
\(359\) 17.9540 0.947575 0.473788 0.880639i \(-0.342886\pi\)
0.473788 + 0.880639i \(0.342886\pi\)
\(360\) 0 0
\(361\) −17.7766 −0.935611
\(362\) 0 0
\(363\) −16.3687 + 2.88624i −0.859134 + 0.151489i
\(364\) 0 0
\(365\) −0.471782 + 0.817150i −0.0246942 + 0.0427716i
\(366\) 0 0
\(367\) 10.5089 + 18.2020i 0.548561 + 0.950136i 0.998373 + 0.0570129i \(0.0181576\pi\)
−0.449812 + 0.893123i \(0.648509\pi\)
\(368\) 0 0
\(369\) −17.5535 14.7291i −0.913799 0.766768i
\(370\) 0 0
\(371\) −2.11334 3.66041i −0.109719 0.190039i
\(372\) 0 0
\(373\) 14.9376 25.8727i 0.773441 1.33964i −0.162225 0.986754i \(-0.551867\pi\)
0.935666 0.352886i \(-0.114800\pi\)
\(374\) 0 0
\(375\) −5.47906 6.52968i −0.282937 0.337191i
\(376\) 0 0
\(377\) −23.7246 −1.22188
\(378\) 0 0
\(379\) −0.985452 −0.0506193 −0.0253096 0.999680i \(-0.508057\pi\)
−0.0253096 + 0.999680i \(0.508057\pi\)
\(380\) 0 0
\(381\) −22.3687 26.6580i −1.14598 1.36573i
\(382\) 0 0
\(383\) −0.994070 + 1.72178i −0.0507946 + 0.0879789i −0.890305 0.455365i \(-0.849509\pi\)
0.839510 + 0.543344i \(0.182842\pi\)
\(384\) 0 0
\(385\) −1.70574 2.95442i −0.0869324 0.150571i
\(386\) 0 0
\(387\) 21.3516 7.77136i 1.08536 0.395040i
\(388\) 0 0
\(389\) −7.04323 12.1992i −0.357106 0.618526i 0.630370 0.776295i \(-0.282903\pi\)
−0.987476 + 0.157769i \(0.949570\pi\)
\(390\) 0 0
\(391\) 15.9684 27.6580i 0.807556 1.39873i
\(392\) 0 0
\(393\) −13.1001 + 2.30991i −0.660815 + 0.116519i
\(394\) 0 0
\(395\) −30.2199 −1.52053
\(396\) 0 0
\(397\) −17.3874 −0.872650 −0.436325 0.899789i \(-0.643720\pi\)
−0.436325 + 0.899789i \(0.643720\pi\)
\(398\) 0 0
\(399\) −0.655230 + 1.80023i −0.0328025 + 0.0901242i
\(400\) 0 0
\(401\) 5.86871 10.1649i 0.293069 0.507611i −0.681465 0.731851i \(-0.738657\pi\)
0.974534 + 0.224240i \(0.0719900\pi\)
\(402\) 0 0
\(403\) 13.2626 + 22.9716i 0.660659 + 1.14430i
\(404\) 0 0
\(405\) −19.8516 + 16.6575i −0.986436 + 0.827718i
\(406\) 0 0
\(407\) 1.43717 + 2.48925i 0.0712377 + 0.123387i
\(408\) 0 0
\(409\) 2.26352 3.92053i 0.111924 0.193858i −0.804622 0.593787i \(-0.797632\pi\)
0.916546 + 0.399930i \(0.130965\pi\)
\(410\) 0 0
\(411\) 8.45265 23.2235i 0.416938 1.14553i
\(412\) 0 0
\(413\) 11.7246 0.576931
\(414\) 0 0
\(415\) 20.8871 1.02531
\(416\) 0 0
\(417\) 26.0620 4.59543i 1.27626 0.225039i
\(418\) 0 0
\(419\) −7.44609 + 12.8970i −0.363765 + 0.630060i −0.988577 0.150715i \(-0.951842\pi\)
0.624812 + 0.780775i \(0.285176\pi\)
\(420\) 0 0
\(421\) −13.7875 23.8806i −0.671959 1.16387i −0.977348 0.211640i \(-0.932120\pi\)
0.305388 0.952228i \(-0.401214\pi\)
\(422\) 0 0
\(423\) −0.798133 + 0.290497i −0.0388066 + 0.0141244i
\(424\) 0 0
\(425\) −8.90420 15.4225i −0.431917 0.748102i
\(426\) 0 0
\(427\) 0.992726 1.71945i 0.0480414 0.0832101i
\(428\) 0 0
\(429\) 6.27719 + 7.48086i 0.303066 + 0.361179i
\(430\) 0 0
\(431\) −25.5827 −1.23227 −0.616137 0.787639i \(-0.711303\pi\)
−0.616137 + 0.787639i \(0.711303\pi\)
\(432\) 0 0
\(433\) −19.5425 −0.939153 −0.469577 0.882892i \(-0.655593\pi\)
−0.469577 + 0.882892i \(0.655593\pi\)
\(434\) 0 0
\(435\) 15.9820 + 19.0467i 0.766281 + 0.913218i
\(436\) 0 0
\(437\) −3.26382 + 5.65311i −0.156130 + 0.270425i
\(438\) 0 0
\(439\) 4.78359 + 8.28541i 0.228308 + 0.395441i 0.957307 0.289074i \(-0.0933473\pi\)
−0.728999 + 0.684515i \(0.760014\pi\)
\(440\) 0 0
\(441\) −2.29813 1.92836i −0.109435 0.0918268i
\(442\) 0 0
\(443\) −7.92902 13.7335i −0.376719 0.652496i 0.613864 0.789412i \(-0.289614\pi\)
−0.990583 + 0.136916i \(0.956281\pi\)
\(444\) 0 0
\(445\) −21.1668 + 36.6620i −1.00340 + 1.73795i
\(446\) 0 0
\(447\) −24.5908 + 4.33602i −1.16311 + 0.205087i
\(448\) 0 0
\(449\) 30.0223 1.41684 0.708420 0.705791i \(-0.249408\pi\)
0.708420 + 0.705791i \(0.249408\pi\)
\(450\) 0 0
\(451\) 9.04963 0.426130
\(452\) 0 0
\(453\) 6.99525 19.2193i 0.328666 0.903001i
\(454\) 0 0
\(455\) 6.85117 11.8666i 0.321188 0.556313i
\(456\) 0 0
\(457\) 8.71554 + 15.0958i 0.407696 + 0.706150i 0.994631 0.103484i \(-0.0329991\pi\)
−0.586935 + 0.809634i \(0.699666\pi\)
\(458\) 0 0
\(459\) 24.3516 14.0594i 1.13664 0.656238i
\(460\) 0 0
\(461\) −5.16684 8.94923i −0.240644 0.416807i 0.720254 0.693710i \(-0.244025\pi\)
−0.960898 + 0.276903i \(0.910692\pi\)
\(462\) 0 0
\(463\) −4.67958 + 8.10527i −0.217478 + 0.376684i −0.954036 0.299691i \(-0.903117\pi\)
0.736558 + 0.676374i \(0.236450\pi\)
\(464\) 0 0
\(465\) 9.50774 26.1223i 0.440911 1.21139i
\(466\) 0 0
\(467\) −24.7273 −1.14424 −0.572122 0.820169i \(-0.693880\pi\)
−0.572122 + 0.820169i \(0.693880\pi\)
\(468\) 0 0
\(469\) −13.5398 −0.625211
\(470\) 0 0
\(471\) 23.0620 4.06645i 1.06264 0.187372i
\(472\) 0 0
\(473\) −4.48680 + 7.77136i −0.206303 + 0.357327i
\(474\) 0 0
\(475\) 1.81996 + 3.15225i 0.0835053 + 0.144635i
\(476\) 0 0
\(477\) 2.20187 12.4874i 0.100817 0.571759i
\(478\) 0 0
\(479\) 19.1425 + 33.1558i 0.874643 + 1.51493i 0.857143 + 0.515079i \(0.172237\pi\)
0.0175001 + 0.999847i \(0.494429\pi\)
\(480\) 0 0
\(481\) −5.77244 + 9.99816i −0.263201 + 0.455877i
\(482\) 0 0
\(483\) −6.57057 7.83051i −0.298971 0.356300i
\(484\) 0 0
\(485\) −52.1002 −2.36575
\(486\) 0 0
\(487\) −1.29498 −0.0586811 −0.0293405 0.999569i \(-0.509341\pi\)
−0.0293405 + 0.999569i \(0.509341\pi\)
\(488\) 0 0
\(489\) 26.6802 + 31.7962i 1.20652 + 1.43788i
\(490\) 0 0
\(491\) 11.7160 20.2927i 0.528736 0.915797i −0.470703 0.882292i \(-0.656000\pi\)
0.999439 0.0335055i \(-0.0106671\pi\)
\(492\) 0 0
\(493\) −13.4893 23.3642i −0.607529 1.05227i
\(494\) 0 0
\(495\) 1.77719 10.0789i 0.0798787 0.453015i
\(496\) 0 0
\(497\) 2.55690 + 4.42869i 0.114693 + 0.198654i
\(498\) 0 0
\(499\) −17.3773 + 30.0984i −0.777916 + 1.34739i 0.155225 + 0.987879i \(0.450390\pi\)
−0.933141 + 0.359511i \(0.882944\pi\)
\(500\) 0 0
\(501\) 8.47906 1.49509i 0.378816 0.0667955i
\(502\) 0 0
\(503\) −8.61318 −0.384043 −0.192021 0.981391i \(-0.561504\pi\)
−0.192021 + 0.981391i \(0.561504\pi\)
\(504\) 0 0
\(505\) 50.9299 2.26635
\(506\) 0 0
\(507\) −5.71419 + 15.6996i −0.253776 + 0.697244i
\(508\) 0 0
\(509\) 4.58125 7.93496i 0.203060 0.351711i −0.746453 0.665439i \(-0.768245\pi\)
0.949513 + 0.313728i \(0.101578\pi\)
\(510\) 0 0
\(511\) −0.163848 0.283793i −0.00724821 0.0125543i
\(512\) 0 0
\(513\) −4.97730 + 2.87365i −0.219753 + 0.126875i
\(514\) 0 0
\(515\) −18.9722 32.8609i −0.836017 1.44802i
\(516\) 0 0
\(517\) 0.167718 0.290497i 0.00737625 0.0127760i
\(518\) 0 0
\(519\) 14.9402 41.0478i 0.655800 1.80180i
\(520\) 0 0
\(521\) −9.13846 −0.400363 −0.200182 0.979759i \(-0.564153\pi\)
−0.200182 + 0.979759i \(0.564153\pi\)
\(522\) 0 0
\(523\) 11.8821 0.519567 0.259783 0.965667i \(-0.416349\pi\)
0.259783 + 0.965667i \(0.416349\pi\)
\(524\) 0 0
\(525\) −5.61334 + 0.989783i −0.244986 + 0.0431977i
\(526\) 0 0
\(527\) −15.0817 + 26.1223i −0.656970 + 1.13791i
\(528\) 0 0
\(529\) −5.91488 10.2449i −0.257169 0.445429i
\(530\) 0 0
\(531\) 26.9447 + 22.6093i 1.16930 + 0.981161i
\(532\) 0 0
\(533\) 18.1741 + 31.4785i 0.787208 + 1.36348i
\(534\) 0 0
\(535\) 20.0856 34.7893i 0.868376 1.50407i
\(536\) 0 0
\(537\) −10.3247 12.3045i −0.445544 0.530978i
\(538\) 0 0
\(539\) 1.18479 0.0510326
\(540\) 0 0
\(541\) 16.9763 0.729867 0.364934 0.931034i \(-0.381092\pi\)
0.364934 + 0.931034i \(0.381092\pi\)
\(542\) 0 0
\(543\) −14.8439 17.6903i −0.637012 0.759162i
\(544\) 0 0
\(545\) −14.6420 + 25.3607i −0.627195 + 1.08633i
\(546\) 0 0
\(547\) 0.830222 + 1.43799i 0.0354977 + 0.0614839i 0.883228 0.468943i \(-0.155365\pi\)
−0.847731 + 0.530427i \(0.822032\pi\)
\(548\) 0 0
\(549\) 5.59714 2.03719i 0.238880 0.0869453i
\(550\) 0 0
\(551\) 2.75712 + 4.77547i 0.117457 + 0.203442i
\(552\) 0 0
\(553\) 5.24763 9.08916i 0.223152 0.386510i
\(554\) 0 0
\(555\) 11.9153 2.10100i 0.505778 0.0891823i
\(556\) 0 0
\(557\) −23.0455 −0.976470 −0.488235 0.872712i \(-0.662359\pi\)
−0.488235 + 0.872712i \(0.662359\pi\)
\(558\) 0 0
\(559\) −36.0428 −1.52445
\(560\) 0 0
\(561\) −3.79813 + 10.4353i −0.160357 + 0.440578i
\(562\) 0 0
\(563\) 13.6985 23.7264i 0.577321 0.999950i −0.418464 0.908233i \(-0.637431\pi\)
0.995785 0.0917165i \(-0.0292354\pi\)
\(564\) 0 0
\(565\) −8.17024 14.1513i −0.343725 0.595349i
\(566\) 0 0
\(567\) −1.56283 8.86327i −0.0656328 0.372222i
\(568\) 0 0
\(569\) −20.1732 34.9411i −0.845706 1.46481i −0.885007 0.465579i \(-0.845846\pi\)
0.0393005 0.999227i \(-0.487487\pi\)
\(570\) 0 0
\(571\) 2.52182 4.36792i 0.105535 0.182792i −0.808422 0.588604i \(-0.799678\pi\)
0.913957 + 0.405812i \(0.133011\pi\)
\(572\) 0 0
\(573\) 5.19325 14.2683i 0.216951 0.596068i
\(574\) 0 0
\(575\) −19.4216 −0.809936
\(576\) 0 0
\(577\) 8.95037 0.372609 0.186304 0.982492i \(-0.440349\pi\)
0.186304 + 0.982492i \(0.440349\pi\)
\(578\) 0 0
\(579\) 25.9991 4.58435i 1.08049 0.190519i
\(580\) 0 0
\(581\) −3.62701 + 6.28217i −0.150474 + 0.260628i
\(582\) 0 0
\(583\) 2.50387 + 4.33683i 0.103700 + 0.179613i
\(584\) 0 0
\(585\) 38.6279 14.0594i 1.59707 0.581286i
\(586\) 0 0
\(587\) 21.4572 + 37.1650i 0.885635 + 1.53396i 0.844984 + 0.534791i \(0.179610\pi\)
0.0406505 + 0.999173i \(0.487057\pi\)
\(588\) 0 0
\(589\) 3.08260 5.33921i 0.127016 0.219998i
\(590\) 0 0
\(591\) 9.62479 + 11.4704i 0.395911 + 0.471828i
\(592\) 0 0
\(593\) −20.6081 −0.846274 −0.423137 0.906066i \(-0.639071\pi\)
−0.423137 + 0.906066i \(0.639071\pi\)
\(594\) 0 0
\(595\) 15.5817 0.638788
\(596\) 0 0
\(597\) −14.8905 17.7458i −0.609429 0.726289i
\(598\) 0 0
\(599\) −10.5321 + 18.2421i −0.430329 + 0.745353i −0.996902 0.0786597i \(-0.974936\pi\)
0.566572 + 0.824012i \(0.308269\pi\)
\(600\) 0 0
\(601\) −8.95424 15.5092i −0.365251 0.632633i 0.623565 0.781771i \(-0.285684\pi\)
−0.988816 + 0.149138i \(0.952350\pi\)
\(602\) 0 0
\(603\) −31.1163 26.1097i −1.26716 1.06327i
\(604\) 0 0
\(605\) −13.8157 23.9294i −0.561687 0.972870i
\(606\) 0 0
\(607\) −17.7618 + 30.7643i −0.720928 + 1.24868i 0.239701 + 0.970847i \(0.422951\pi\)
−0.960628 + 0.277837i \(0.910383\pi\)
\(608\) 0 0
\(609\) −8.50387 + 1.49946i −0.344594 + 0.0607613i
\(610\) 0 0
\(611\) 1.34730 0.0545058
\(612\) 0 0
\(613\) −2.19253 −0.0885556 −0.0442778 0.999019i \(-0.514099\pi\)
−0.0442778 + 0.999019i \(0.514099\pi\)
\(614\) 0 0
\(615\) 13.0287 35.7960i 0.525367 1.44343i
\(616\) 0 0
\(617\) 11.7258 20.3097i 0.472063 0.817637i −0.527426 0.849601i \(-0.676843\pi\)
0.999489 + 0.0319637i \(0.0101761\pi\)
\(618\) 0 0
\(619\) −7.52481 13.0334i −0.302448 0.523855i 0.674242 0.738510i \(-0.264470\pi\)
−0.976690 + 0.214655i \(0.931137\pi\)
\(620\) 0 0
\(621\) 30.6660i 1.23058i
\(622\) 0 0
\(623\) −7.35117 12.7326i −0.294518 0.510121i
\(624\) 0 0
\(625\) 15.3123 26.5216i 0.612491 1.06087i
\(626\) 0 0
\(627\) 0.776311 2.13290i 0.0310029 0.0851797i
\(628\) 0 0
\(629\) −13.1284 −0.523462
\(630\) 0 0
\(631\) −0.854408 −0.0340135 −0.0170067 0.999855i \(-0.505414\pi\)
−0.0170067 + 0.999855i \(0.505414\pi\)
\(632\) 0 0
\(633\) 12.4277 2.19133i 0.493956 0.0870977i
\(634\) 0 0
\(635\) 28.9256 50.1006i 1.14788 1.98818i
\(636\) 0 0
\(637\) 2.37939 + 4.12122i 0.0942747 + 0.163289i
\(638\) 0 0
\(639\) −2.66401 + 15.1084i −0.105387 + 0.597677i
\(640\) 0 0
\(641\) −9.58630 16.6040i −0.378636 0.655817i 0.612228 0.790681i \(-0.290274\pi\)
−0.990864 + 0.134864i \(0.956940\pi\)
\(642\) 0 0
\(643\) 16.7562 29.0227i 0.660802 1.14454i −0.319604 0.947551i \(-0.603550\pi\)
0.980405 0.196991i \(-0.0631169\pi\)
\(644\) 0 0
\(645\) 24.2802 + 28.9360i 0.956031 + 1.13935i
\(646\) 0 0
\(647\) 20.0966 0.790078 0.395039 0.918664i \(-0.370731\pi\)
0.395039 + 0.918664i \(0.370731\pi\)
\(648\) 0 0
\(649\) −13.8912 −0.545279
\(650\) 0 0
\(651\) 6.20574 + 7.39571i 0.243222 + 0.289861i
\(652\) 0 0
\(653\) −9.16179 + 15.8687i −0.358528 + 0.620990i −0.987715 0.156265i \(-0.950055\pi\)
0.629187 + 0.777254i \(0.283388\pi\)
\(654\) 0 0
\(655\) −11.0569 19.1511i −0.432029 0.748296i
\(656\) 0 0
\(657\) 0.170711 0.968153i 0.00666009 0.0377712i
\(658\) 0 0
\(659\) 6.45976 + 11.1886i 0.251637 + 0.435847i 0.963977 0.265987i \(-0.0856979\pi\)
−0.712340 + 0.701835i \(0.752365\pi\)
\(660\) 0 0
\(661\) −1.91101 + 3.30996i −0.0743296 + 0.128743i −0.900795 0.434246i \(-0.857015\pi\)
0.826465 + 0.562988i \(0.190348\pi\)
\(662\) 0 0
\(663\) −43.9261 + 7.74535i −1.70595 + 0.300805i
\(664\) 0 0
\(665\) −3.18479 −0.123501
\(666\) 0 0
\(667\) −29.4225 −1.13924
\(668\) 0 0
\(669\) −0.350233 + 0.962258i −0.0135408 + 0.0372030i
\(670\) 0 0
\(671\) −1.17617 + 2.03719i −0.0454057 + 0.0786450i
\(672\) 0 0
\(673\) 19.8045 + 34.3025i 0.763409 + 1.32226i 0.941084 + 0.338173i \(0.109809\pi\)
−0.177675 + 0.984089i \(0.556858\pi\)
\(674\) 0 0
\(675\) −14.8089 8.54990i −0.569994 0.329086i
\(676\) 0 0
\(677\) 13.1600 + 22.7937i 0.505779 + 0.876035i 0.999978 + 0.00668595i \(0.00212822\pi\)
−0.494199 + 0.869349i \(0.664538\pi\)
\(678\) 0 0
\(679\) 9.04710 15.6700i 0.347196 0.601361i
\(680\) 0 0
\(681\) −2.06355 + 5.66955i −0.0790754 + 0.217258i
\(682\) 0 0
\(683\) −25.2080 −0.964558 −0.482279 0.876018i \(-0.660191\pi\)
−0.482279 + 0.876018i \(0.660191\pi\)
\(684\) 0 0
\(685\) 41.0847 1.56977
\(686\) 0 0
\(687\) −41.9634 + 7.39928i −1.60100 + 0.282300i
\(688\) 0 0
\(689\) −10.0569 + 17.4191i −0.383138 + 0.663614i
\(690\) 0 0
\(691\) 9.29473 + 16.0989i 0.353588 + 0.612433i 0.986875 0.161484i \(-0.0516281\pi\)
−0.633287 + 0.773917i \(0.718295\pi\)
\(692\) 0 0
\(693\) 2.72281 + 2.28471i 0.103431 + 0.0867890i
\(694\) 0 0
\(695\) 21.9971 + 38.1000i 0.834396 + 1.44522i
\(696\) 0 0
\(697\) −20.6668 + 35.7960i −0.782812 + 1.35587i
\(698\) 0 0
\(699\) −12.7105 15.1478i −0.480755 0.572941i
\(700\) 0 0
\(701\) −47.8580 −1.80757 −0.903786 0.427984i \(-0.859224\pi\)
−0.903786 + 0.427984i \(0.859224\pi\)
\(702\) 0 0
\(703\) 2.68334 0.101204
\(704\) 0 0
\(705\) −0.907604 1.08164i −0.0341823 0.0407369i
\(706\) 0 0
\(707\) −8.84389 + 15.3181i −0.332609 + 0.576095i
\(708\) 0 0
\(709\) 4.69965 + 8.14002i 0.176499 + 0.305705i 0.940679 0.339298i \(-0.110189\pi\)
−0.764180 + 0.645003i \(0.776856\pi\)
\(710\) 0 0
\(711\) 29.5869 10.7688i 1.10960 0.403860i
\(712\) 0 0
\(713\) 16.4479 + 28.4886i 0.615979 + 1.06691i
\(714\) 0 0
\(715\) −8.11721 + 14.0594i −0.303566 + 0.525793i
\(716\) 0 0
\(717\) −6.98205 + 1.23112i −0.260749 + 0.0459772i
\(718\) 0 0
\(719\) −10.6895 −0.398653 −0.199326 0.979933i \(-0.563875\pi\)
−0.199326 + 0.979933i \(0.563875\pi\)
\(720\) 0 0
\(721\) 13.1780 0.490774
\(722\) 0 0
\(723\) 4.26945 11.7302i 0.158782 0.436251i
\(724\) 0 0
\(725\) −8.20321 + 14.2084i −0.304660 + 0.527686i
\(726\) 0 0
\(727\) 25.5458 + 44.2466i 0.947440 + 1.64101i 0.750791 + 0.660540i \(0.229673\pi\)
0.196649 + 0.980474i \(0.436994\pi\)
\(728\) 0 0
\(729\) 13.5000 23.3827i 0.500000 0.866025i
\(730\) 0 0
\(731\) −20.4932 35.4953i −0.757968 1.31284i
\(732\) 0 0
\(733\) −17.3949 + 30.1288i −0.642494 + 1.11283i 0.342380 + 0.939562i \(0.388767\pi\)
−0.984874 + 0.173271i \(0.944566\pi\)
\(734\) 0 0
\(735\) 1.70574 4.68647i 0.0629171 0.172863i
\(736\) 0 0
\(737\) 16.0419 0.590911
\(738\) 0 0
\(739\) −41.9469 −1.54304 −0.771520 0.636205i \(-0.780503\pi\)
−0.771520 + 0.636205i \(0.780503\pi\)
\(740\) 0 0
\(741\) 8.97818 1.58310i 0.329822 0.0581564i
\(742\) 0 0
\(743\) −8.05809 + 13.9570i −0.295622 + 0.512033i −0.975130 0.221636i \(-0.928860\pi\)
0.679507 + 0.733669i \(0.262194\pi\)
\(744\) 0 0
\(745\) −20.7554 35.9494i −0.760418 1.31708i
\(746\) 0 0
\(747\) −20.4497 + 7.44307i −0.748214 + 0.272328i
\(748\) 0 0
\(749\) 6.97565 + 12.0822i 0.254885 + 0.441473i
\(750\) 0 0
\(751\) −25.6746 + 44.4697i −0.936879 + 1.62272i −0.165630 + 0.986188i \(0.552966\pi\)
−0.771249 + 0.636534i \(0.780368\pi\)
\(752\) 0 0
\(753\) 5.78864 + 6.89863i 0.210950 + 0.251400i
\(754\) 0 0
\(755\) 34.0009 1.23742
\(756\) 0 0
\(757\) 16.8553 0.612618 0.306309 0.951932i \(-0.400906\pi\)
0.306309 + 0.951932i \(0.400906\pi\)
\(758\) 0 0
\(759\) 7.78477 + 9.27752i 0.282569 + 0.336753i
\(760\) 0 0
\(761\) 1.42333 2.46529i 0.0515958 0.0893666i −0.839074 0.544017i \(-0.816903\pi\)
0.890670 + 0.454651i \(0.150236\pi\)
\(762\) 0 0
\(763\) −5.08512 8.80769i −0.184094 0.318860i
\(764\) 0 0
\(765\) 35.8089 + 30.0472i 1.29467 + 1.08636i
\(766\) 0 0
\(767\) −27.8974 48.3197i −1.00732 1.74472i
\(768\) 0 0
\(769\) 18.9149 32.7616i 0.682090 1.18141i −0.292252 0.956341i \(-0.594405\pi\)
0.974342 0.225073i \(-0.0722621\pi\)
\(770\) 0 0
\(771\) −22.9145 + 4.04044i −0.825244 + 0.145513i
\(772\) 0 0
\(773\) 2.87702 0.103479 0.0517396 0.998661i \(-0.483523\pi\)
0.0517396 + 0.998661i \(0.483523\pi\)
\(774\) 0 0
\(775\) 18.3432 0.658906
\(776\) 0 0
\(777\) −1.43717 + 3.94858i −0.0515581 + 0.141655i
\(778\) 0 0
\(779\) 4.22416 7.31645i 0.151346 0.262139i
\(780\) 0 0
\(781\) −3.02940 5.24708i −0.108400 0.187755i
\(782\) 0 0
\(783\) −22.4345 12.9526i −0.801745 0.462888i
\(784\) 0 0
\(785\) 19.4650 + 33.7143i 0.694735 + 1.20332i
\(786\) 0 0
\(787\) −13.0710 + 22.6397i −0.465932 + 0.807018i −0.999243 0.0389010i \(-0.987614\pi\)
0.533311 + 0.845919i \(0.320948\pi\)
\(788\) 0 0
\(789\) −6.56196 + 18.0288i −0.233612 + 0.641843i
\(790\) 0 0
\(791\) 5.67499 0.201779
\(792\) 0 0
\(793\) −9.44831 −0.335519
\(794\) 0 0
\(795\) 20.7592 3.66041i 0.736254 0.129821i
\(796\) 0 0
\(797\) 13.4568 23.3078i 0.476663 0.825605i −0.522979 0.852345i \(-0.675179\pi\)
0.999642 + 0.0267406i \(0.00851282\pi\)
\(798\) 0 0
\(799\) 0.766044 + 1.32683i 0.0271007 + 0.0469398i
\(800\) 0 0
\(801\) 7.65910 43.4369i 0.270621 1.53477i
\(802\) 0 0
\(803\) 0.194126 + 0.336236i 0.00685055 + 0.0118655i
\(804\) 0 0
\(805\) 8.49660 14.7165i 0.299466 0.518690i
\(806\) 0 0
\(807\) −20.2059 24.0805i −0.711281 0.847672i
\(808\) 0 0
\(809\) −32.4228 −1.13993 −0.569963 0.821670i \(-0.693043\pi\)
−0.569963 + 0.821670i \(0.693043\pi\)
\(810\) 0 0
\(811\) −21.2754 −0.747080 −0.373540 0.927614i \(-0.621856\pi\)
−0.373540 + 0.927614i \(0.621856\pi\)
\(812\) 0 0
\(813\) −14.6468 17.4553i −0.513685 0.612186i
\(814\) 0 0
\(815\) −34.5009 + 59.7574i −1.20851 + 2.09321i
\(816\) 0 0
\(817\) 4.18866 + 7.25498i 0.146543 + 0.253820i
\(818\) 0 0
\(819\) −2.47906 + 14.0594i −0.0866252 + 0.491276i
\(820\) 0 0
\(821\) −24.9354 43.1894i −0.870252 1.50732i −0.861736 0.507357i \(-0.830623\pi\)
−0.00851573 0.999964i \(-0.502711\pi\)
\(822\) 0 0
\(823\) −9.47296 + 16.4077i −0.330207 + 0.571935i −0.982552 0.185987i \(-0.940452\pi\)
0.652345 + 0.757922i \(0.273785\pi\)
\(824\) 0 0
\(825\) 6.65064 1.17269i 0.231546 0.0408278i
\(826\) 0 0
\(827\) 12.1489 0.422458 0.211229 0.977437i \(-0.432253\pi\)
0.211229 + 0.977437i \(0.432253\pi\)
\(828\) 0 0
\(829\) 11.2189 0.389650 0.194825 0.980838i \(-0.437586\pi\)
0.194825 + 0.980838i \(0.437586\pi\)
\(830\) 0 0
\(831\) 14.0970 38.7311i 0.489019 1.34357i
\(832\) 0 0
\(833\) −2.70574 + 4.68647i −0.0937482 + 0.162377i
\(834\) 0 0
\(835\) 7.15657 + 12.3955i 0.247663 + 0.428966i
\(836\) 0 0
\(837\) 28.9632i 1.00112i
\(838\) 0 0
\(839\) −2.63516 4.56424i −0.0909759 0.157575i 0.816946 0.576714i \(-0.195665\pi\)
−0.907922 + 0.419139i \(0.862332\pi\)
\(840\) 0 0
\(841\) 2.07263 3.58991i 0.0714701 0.123790i
\(842\) 0 0
\(843\) −7.10472 + 19.5201i −0.244700 + 0.672307i
\(844\) 0 0
\(845\) −27.7743 −0.955463
\(846\) 0 0
\(847\) 9.59627 0.329732
\(848\) 0 0
\(849\) −26.0371 + 4.59105i −0.893592 + 0.157564i
\(850\) 0 0
\(851\) −7.15880 + 12.3994i −0.245400 + 0.425046i
\(852\) 0 0
\(853\) 1.94222 + 3.36402i 0.0665003 + 0.115182i 0.897359 0.441302i \(-0.145483\pi\)
−0.830858 + 0.556484i \(0.812150\pi\)
\(854\) 0 0
\(855\) −7.31908 6.14144i −0.250307 0.210033i
\(856\) 0 0
\(857\) −9.50253 16.4589i −0.324600 0.562224i 0.656831 0.754038i \(-0.271896\pi\)
−0.981431 + 0.191814i \(0.938563\pi\)
\(858\) 0 0
\(859\) −25.4513 + 44.0830i −0.868387 + 1.50409i −0.00474324 + 0.999989i \(0.501510\pi\)
−0.863644 + 0.504102i \(0.831824\pi\)
\(860\) 0 0
\(861\) 8.50387 + 10.1345i 0.289811 + 0.345383i
\(862\) 0 0
\(863\) −12.7615 −0.434405 −0.217203 0.976127i \(-0.569693\pi\)
−0.217203 + 0.976127i \(0.569693\pi\)
\(864\) 0 0
\(865\) 72.6177 2.46908
\(866\) 0 0
\(867\) −13.6763 16.2988i −0.464473 0.553537i
\(868\) 0 0
\(869\) −6.21735 + 10.7688i −0.210909 + 0.365305i
\(870\) 0 0
\(871\) 32.2165 + 55.8006i 1.09161 + 1.89073i
\(872\) 0 0
\(873\) 51.0090 18.5658i 1.72639 0.628355i
\(874\) 0 0
\(875\) 2.46064 + 4.26195i 0.0831847 + 0.144080i
\(876\) 0 0
\(877\) −7.12226 + 12.3361i −0.240502 + 0.416561i −0.960857 0.277044i \(-0.910645\pi\)
0.720356 + 0.693605i \(0.243979\pi\)
\(878\) 0 0
\(879\) −8.55350 + 1.50821i −0.288502 + 0.0508708i
\(880\) 0 0
\(881\) 38.3286 1.29132 0.645662 0.763623i \(-0.276581\pi\)
0.645662 + 0.763623i \(0.276581\pi\)
\(882\) 0 0
\(883\) 42.6483 1.43523 0.717614 0.696441i \(-0.245234\pi\)
0.717614 + 0.696441i \(0.245234\pi\)
\(884\) 0 0
\(885\) −19.9991 + 54.9471i −0.672263 + 1.84703i
\(886\) 0 0
\(887\) 11.2117 19.4192i 0.376451 0.652032i −0.614092 0.789234i \(-0.710478\pi\)
0.990543 + 0.137202i \(0.0438110\pi\)
\(888\) 0 0
\(889\) 10.0458 + 17.3998i 0.336924 + 0.583569i
\(890\) 0 0
\(891\) 1.85163 + 10.5011i 0.0620321 + 0.351801i
\(892\) 0 0
\(893\) −0.156574 0.271194i −0.00523955 0.00907517i
\(894\) 0 0
\(895\) 13.3512 23.1249i 0.446280 0.772980i
\(896\) 0 0
\(897\) −16.6373 + 45.7105i −0.555503 + 1.52623i
\(898\) 0 0
\(899\) 27.7888 0.926808
\(900\) 0 0
\(901\) −22.8726 −0.761996
\(902\) 0 0
\(903\) −12.9192 + 2.27801i −0.429925 + 0.0758073i
\(904\) 0 0
\(905\) 19.1951 33.2468i 0.638065 1.10516i
\(906\) 0 0
\(907\) 21.6584 + 37.5134i 0.719155 + 1.24561i 0.961335 + 0.275381i \(0.0888042\pi\)
−0.242180 + 0.970231i \(0.577863\pi\)
\(908\) 0 0
\(909\) −49.8632 + 18.1487i −1.65386 + 0.601956i
\(910\) 0 0
\(911\) 4.44310 + 7.69567i 0.147206 + 0.254969i 0.930194 0.367068i \(-0.119639\pi\)
−0.782988 + 0.622037i \(0.786305\pi\)
\(912\) 0 0
\(913\) 4.29726 7.44307i 0.142218 0.246330i
\(914\) 0 0
\(915\) 6.36484 + 7.58532i 0.210415 + 0.250763i
\(916\) 0 0
\(917\) 7.68004 0.253617
\(918\) 0 0
\(919\) 8.57398 0.282829 0.141415 0.989950i \(-0.454835\pi\)
0.141415 + 0.989950i \(0.454835\pi\)
\(920\) 0 0
\(921\) 16.8460 + 20.0763i 0.555095 + 0.661536i
\(922\) 0 0
\(923\) 12.1677 21.0751i 0.400505 0.693696i
\(924\) 0 0
\(925\) 3.99185 + 6.91408i 0.131251 + 0.227334i
\(926\) 0 0
\(927\) 30.2848 + 25.4119i 0.994682 + 0.834638i
\(928\) 0 0
\(929\) −0.825474 1.42976i −0.0270829 0.0469090i 0.852166 0.523271i \(-0.175288\pi\)
−0.879249 + 0.476362i \(0.841955\pi\)
\(930\) 0 0
\(931\) 0.553033 0.957882i 0.0181249 0.0313933i
\(932\) 0 0
\(933\) 8.50846 1.50027i 0.278554 0.0491166i
\(934\) 0 0
\(935\) −18.4611 −0.603743
\(936\) 0 0
\(937\) 47.4953 1.55160 0.775801 0.630978i \(-0.217346\pi\)
0.775801 + 0.630978i \(0.217346\pi\)
\(938\) 0 0
\(939\) 15.1227 41.5492i 0.493510 1.35591i
\(940\) 0 0
\(941\) 11.6964 20.2588i 0.381292 0.660417i −0.609955 0.792436i \(-0.708813\pi\)
0.991247 + 0.132019i \(0.0421460\pi\)
\(942\) 0 0
\(943\) 22.5390 + 39.0386i 0.733969 + 1.27127i
\(944\) 0 0
\(945\) 12.9572 7.48086i 0.421499 0.243352i
\(946\) 0 0
\(947\) 1.96363 + 3.40111i 0.0638094 + 0.110521i 0.896165 0.443720i \(-0.146342\pi\)
−0.832356 + 0.554242i \(0.813008\pi\)
\(948\) 0 0
\(949\) −0.779715 + 1.35051i −0.0253106 + 0.0438393i
\(950\) 0 0
\(951\) 3.08869 8.48610i 0.100158 0.275181i
\(952\) 0 0
\(953\) 2.42427 0.0785297 0.0392649 0.999229i \(-0.487498\pi\)
0.0392649 + 0.999229i \(0.487498\pi\)
\(954\) 0 0
\(955\) 25.2422 0.816817
\(956\) 0 0
\(957\) 10.0753 1.77655i 0.325689 0.0574277i
\(958\) 0 0
\(959\) −7.13429 + 12.3569i −0.230378 + 0.399027i
\(960\) 0 0
\(961\) −0.0346151 0.0599551i −0.00111662 0.00193404i
\(962\) 0 0
\(963\) −7.26786 + 41.2181i −0.234203 + 1.32823i
\(964\) 0 0
\(965\) 21.9440 + 38.0082i 0.706403 + 1.22353i
\(966\) 0 0
\(967\) 6.35844 11.0131i 0.204474 0.354159i −0.745491 0.666515i \(-0.767785\pi\)
0.949965 + 0.312357i \(0.101118\pi\)
\(968\) 0 0
\(969\) 6.66385 + 7.94166i 0.214074 + 0.255123i
\(970\) 0 0
\(971\) 25.6236 0.822301 0.411150 0.911568i \(-0.365127\pi\)
0.411150 + 0.911568i \(0.365127\pi\)
\(972\) 0 0
\(973\) −15.2790 −0.489822
\(974\) 0 0
\(975\) 17.4354 + 20.7787i 0.558380 + 0.665451i
\(976\) 0 0
\(977\) −22.9979 + 39.8336i −0.735769 + 1.27439i 0.218616 + 0.975811i \(0.429846\pi\)
−0.954385 + 0.298579i \(0.903487\pi\)
\(978\) 0 0
\(979\) 8.70961 + 15.0855i 0.278360 + 0.482134i
\(980\) 0 0
\(981\) 5.29813 30.0472i 0.169156 0.959333i
\(982\) 0 0
\(983\) 1.36231 + 2.35959i 0.0434510 + 0.0752593i 0.886933 0.461898i \(-0.152831\pi\)
−0.843482 + 0.537157i \(0.819498\pi\)
\(984\) 0 0
\(985\) −12.4461 + 21.5573i −0.396565 + 0.686871i
\(986\) 0 0
\(987\) 0.482926 0.0851529i 0.0153717 0.00271045i
\(988\) 0 0
\(989\) −44.6991 −1.42135
\(990\) 0 0
\(991\) 50.2303 1.59562 0.797809 0.602910i \(-0.205992\pi\)
0.797809 + 0.602910i \(0.205992\pi\)
\(992\) 0 0
\(993\) −2.45564 + 6.74682i −0.0779274 + 0.214104i
\(994\) 0 0
\(995\) 19.2554 33.3513i 0.610436 1.05731i
\(996\) 0 0
\(997\) −21.6263 37.4578i −0.684912 1.18630i −0.973465 0.228838i \(-0.926507\pi\)
0.288553 0.957464i \(-0.406826\pi\)
\(998\) 0 0
\(999\) −10.9171 + 6.30299i −0.345402 + 0.199418i
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 1008.2.r.i.337.1 6
3.2 odd 2 3024.2.r.h.1009.1 6
4.3 odd 2 504.2.r.c.337.3 yes 6
9.2 odd 6 3024.2.r.h.2017.1 6
9.4 even 3 9072.2.a.br.1.1 3
9.5 odd 6 9072.2.a.cc.1.3 3
9.7 even 3 inner 1008.2.r.i.673.1 6
12.11 even 2 1512.2.r.c.1009.1 6
36.7 odd 6 504.2.r.c.169.3 6
36.11 even 6 1512.2.r.c.505.1 6
36.23 even 6 4536.2.a.v.1.3 3
36.31 odd 6 4536.2.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.3 6 36.7 odd 6
504.2.r.c.337.3 yes 6 4.3 odd 2
1008.2.r.i.337.1 6 1.1 even 1 trivial
1008.2.r.i.673.1 6 9.7 even 3 inner
1512.2.r.c.505.1 6 36.11 even 6
1512.2.r.c.1009.1 6 12.11 even 2
3024.2.r.h.1009.1 6 3.2 odd 2
3024.2.r.h.2017.1 6 9.2 odd 6
4536.2.a.s.1.1 3 36.31 odd 6
4536.2.a.v.1.3 3 36.23 even 6
9072.2.a.br.1.1 3 9.4 even 3
9072.2.a.cc.1.3 3 9.5 odd 6