Properties

Label 1300.2.bs.d.193.2
Level $1300$
Weight $2$
Character 1300.193
Analytic conductor $10.381$
Analytic rank $0$
Dimension $20$
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] = [1300,2,Mod(193,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bs (of order \(12\), degree \(4\), 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 193.2
Root \(-1.14923i\) of defining polynomial
Character \(\chi\) \(=\) 1300.193
Dual form 1300.2.bs.d.357.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.387600 + 1.44654i) q^{3} +(-2.10545 - 1.21558i) q^{7} +(0.655821 + 0.378639i) q^{9} +O(q^{10})\) \(q+(-0.387600 + 1.44654i) q^{3} +(-2.10545 - 1.21558i) q^{7} +(0.655821 + 0.378639i) q^{9} +(-2.60249 - 0.697334i) q^{11} +(3.57773 - 0.447059i) q^{13} +(-6.05137 + 1.62146i) q^{17} +(-1.00046 - 3.73376i) q^{19} +(2.57447 - 2.57447i) q^{21} +(1.14784 + 0.307564i) q^{23} +(-3.97874 + 3.97874i) q^{27} +(-5.29270 + 3.05574i) q^{29} +(-6.23000 - 6.23000i) q^{31} +(2.01745 - 3.49432i) q^{33} +(6.69111 - 3.86311i) q^{37} +(-0.740038 + 5.34862i) q^{39} +(0.913943 - 3.41088i) q^{41} +(-1.17207 - 4.37424i) q^{43} -6.81282i q^{47} +(-0.544711 - 0.943467i) q^{49} -9.38205i q^{51} +(-4.88693 - 4.88693i) q^{53} +5.78882 q^{57} +(-9.04287 + 2.42303i) q^{59} +(-4.87808 + 8.44908i) q^{61} +(-0.920534 - 1.59441i) q^{63} +(-0.507281 - 0.878637i) q^{67} +(-0.889809 + 1.54120i) q^{69} +(10.5637 - 2.83054i) q^{71} +9.06732 q^{73} +(4.63174 + 4.63174i) q^{77} -11.6191i q^{79} +(-3.07735 - 5.33013i) q^{81} +1.09215i q^{83} +(-2.36881 - 8.84053i) q^{87} +(-1.29560 + 4.83523i) q^{89} +(-8.07618 - 3.40777i) q^{91} +(11.4267 - 6.59721i) q^{93} +(-1.38144 + 2.39272i) q^{97} +(-1.44273 - 1.44273i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9} - 8 q^{13} + 20 q^{19} - 12 q^{21} - 6 q^{23} + 20 q^{27} + 24 q^{29} + 8 q^{31} + 10 q^{33} + 4 q^{39} + 6 q^{41} - 38 q^{43} + 14 q^{49} - 30 q^{53} + 76 q^{57} - 24 q^{59} - 32 q^{61} + 24 q^{63} - 22 q^{67} - 16 q^{69} + 44 q^{73} + 12 q^{77} + 2 q^{81} - 38 q^{87} - 30 q^{89} - 72 q^{91} + 48 q^{93} - 46 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −0.387600 + 1.44654i −0.223781 + 0.835162i 0.759108 + 0.650965i \(0.225635\pi\)
−0.982889 + 0.184198i \(0.941031\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.10545 1.21558i −0.795787 0.459448i 0.0462092 0.998932i \(-0.485286\pi\)
−0.841996 + 0.539484i \(0.818619\pi\)
\(8\) 0 0
\(9\) 0.655821 + 0.378639i 0.218607 + 0.126213i
\(10\) 0 0
\(11\) −2.60249 0.697334i −0.784679 0.210254i −0.155832 0.987784i \(-0.549806\pi\)
−0.628847 + 0.777529i \(0.716473\pi\)
\(12\) 0 0
\(13\) 3.57773 0.447059i 0.992283 0.123992i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.05137 + 1.62146i −1.46767 + 0.393262i −0.902133 0.431459i \(-0.857999\pi\)
−0.565540 + 0.824721i \(0.691332\pi\)
\(18\) 0 0
\(19\) −1.00046 3.73376i −0.229521 0.856583i −0.980543 0.196306i \(-0.937105\pi\)
0.751022 0.660277i \(-0.229561\pi\)
\(20\) 0 0
\(21\) 2.57447 2.57447i 0.561795 0.561795i
\(22\) 0 0
\(23\) 1.14784 + 0.307564i 0.239342 + 0.0641315i 0.376496 0.926418i \(-0.377129\pi\)
−0.137154 + 0.990550i \(0.543796\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.97874 + 3.97874i −0.765710 + 0.765710i
\(28\) 0 0
\(29\) −5.29270 + 3.05574i −0.982830 + 0.567437i −0.903123 0.429382i \(-0.858732\pi\)
−0.0797063 + 0.996818i \(0.525398\pi\)
\(30\) 0 0
\(31\) −6.23000 6.23000i −1.11894 1.11894i −0.991897 0.127043i \(-0.959451\pi\)
−0.127043 0.991897i \(-0.540549\pi\)
\(32\) 0 0
\(33\) 2.01745 3.49432i 0.351193 0.608283i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.69111 3.86311i 1.10001 0.635092i 0.163787 0.986496i \(-0.447629\pi\)
0.936224 + 0.351404i \(0.114296\pi\)
\(38\) 0 0
\(39\) −0.740038 + 5.34862i −0.118501 + 0.856465i
\(40\) 0 0
\(41\) 0.913943 3.41088i 0.142734 0.532690i −0.857112 0.515130i \(-0.827744\pi\)
0.999846 0.0175599i \(-0.00558976\pi\)
\(42\) 0 0
\(43\) −1.17207 4.37424i −0.178740 0.667066i −0.995884 0.0906330i \(-0.971111\pi\)
0.817145 0.576433i \(-0.195556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.81282i 0.993752i −0.867822 0.496876i \(-0.834480\pi\)
0.867822 0.496876i \(-0.165520\pi\)
\(48\) 0 0
\(49\) −0.544711 0.943467i −0.0778159 0.134781i
\(50\) 0 0
\(51\) 9.38205i 1.31375i
\(52\) 0 0
\(53\) −4.88693 4.88693i −0.671272 0.671272i 0.286737 0.958009i \(-0.407429\pi\)
−0.958009 + 0.286737i \(0.907429\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.78882 0.766748
\(58\) 0 0
\(59\) −9.04287 + 2.42303i −1.17728 + 0.315452i −0.793848 0.608116i \(-0.791925\pi\)
−0.383434 + 0.923568i \(0.625259\pi\)
\(60\) 0 0
\(61\) −4.87808 + 8.44908i −0.624574 + 1.08179i 0.364049 + 0.931380i \(0.381394\pi\)
−0.988623 + 0.150415i \(0.951939\pi\)
\(62\) 0 0
\(63\) −0.920534 1.59441i −0.115976 0.200877i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.507281 0.878637i −0.0619743 0.107343i 0.833374 0.552710i \(-0.186406\pi\)
−0.895348 + 0.445368i \(0.853073\pi\)
\(68\) 0 0
\(69\) −0.889809 + 1.54120i −0.107120 + 0.185538i
\(70\) 0 0
\(71\) 10.5637 2.83054i 1.25368 0.335924i 0.429926 0.902864i \(-0.358540\pi\)
0.823759 + 0.566941i \(0.191873\pi\)
\(72\) 0 0
\(73\) 9.06732 1.06125 0.530625 0.847607i \(-0.321957\pi\)
0.530625 + 0.847607i \(0.321957\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.63174 + 4.63174i 0.527836 + 0.527836i
\(78\) 0 0
\(79\) 11.6191i 1.30725i −0.756820 0.653623i \(-0.773248\pi\)
0.756820 0.653623i \(-0.226752\pi\)
\(80\) 0 0
\(81\) −3.07735 5.33013i −0.341928 0.592236i
\(82\) 0 0
\(83\) 1.09215i 0.119879i 0.998202 + 0.0599397i \(0.0190909\pi\)
−0.998202 + 0.0599397i \(0.980909\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.36881 8.84053i −0.253963 0.947804i
\(88\) 0 0
\(89\) −1.29560 + 4.83523i −0.137333 + 0.512533i 0.862644 + 0.505811i \(0.168807\pi\)
−0.999977 + 0.00672267i \(0.997860\pi\)
\(90\) 0 0
\(91\) −8.07618 3.40777i −0.846613 0.357231i
\(92\) 0 0
\(93\) 11.4267 6.59721i 1.18489 0.684099i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.38144 + 2.39272i −0.140264 + 0.242944i −0.927596 0.373585i \(-0.878128\pi\)
0.787332 + 0.616529i \(0.211462\pi\)
\(98\) 0 0
\(99\) −1.44273 1.44273i −0.145000 0.145000i
\(100\) 0 0
\(101\) 11.2722 6.50799i 1.12162 0.647569i 0.179808 0.983702i \(-0.442452\pi\)
0.941815 + 0.336133i \(0.109119\pi\)
\(102\) 0 0
\(103\) 3.42536 3.42536i 0.337511 0.337511i −0.517919 0.855430i \(-0.673293\pi\)
0.855430 + 0.517919i \(0.173293\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.28992 + 0.613583i 0.221375 + 0.0593173i 0.367802 0.929904i \(-0.380111\pi\)
−0.146427 + 0.989222i \(0.546777\pi\)
\(108\) 0 0
\(109\) −2.27846 + 2.27846i −0.218237 + 0.218237i −0.807755 0.589518i \(-0.799318\pi\)
0.589518 + 0.807755i \(0.299318\pi\)
\(110\) 0 0
\(111\) 2.99469 + 11.1763i 0.284243 + 1.06081i
\(112\) 0 0
\(113\) −13.3698 + 3.58242i −1.25772 + 0.337006i −0.825315 0.564672i \(-0.809003\pi\)
−0.432408 + 0.901678i \(0.642336\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.51562 + 1.06148i 0.232570 + 0.0981334i
\(118\) 0 0
\(119\) 14.7119 + 3.94204i 1.34864 + 0.361366i
\(120\) 0 0
\(121\) −3.23962 1.87040i −0.294511 0.170036i
\(122\) 0 0
\(123\) 4.57974 + 2.64412i 0.412942 + 0.238412i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.13358 4.23059i 0.100589 0.375404i −0.897218 0.441587i \(-0.854416\pi\)
0.997807 + 0.0661832i \(0.0210822\pi\)
\(128\) 0 0
\(129\) 6.78183 0.597107
\(130\) 0 0
\(131\) 12.1470 1.06129 0.530643 0.847596i \(-0.321951\pi\)
0.530643 + 0.847596i \(0.321951\pi\)
\(132\) 0 0
\(133\) −2.43228 + 9.07740i −0.210906 + 0.787110i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.3932 7.15524i −1.05883 0.611313i −0.133719 0.991019i \(-0.542692\pi\)
−0.925107 + 0.379706i \(0.876025\pi\)
\(138\) 0 0
\(139\) −13.2937 7.67514i −1.12756 0.650997i −0.184239 0.982881i \(-0.558982\pi\)
−0.943320 + 0.331885i \(0.892315\pi\)
\(140\) 0 0
\(141\) 9.85504 + 2.64065i 0.829944 + 0.222383i
\(142\) 0 0
\(143\) −9.62273 1.33141i −0.804694 0.111338i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.57590 0.422260i 0.129978 0.0348274i
\(148\) 0 0
\(149\) −0.861750 3.21610i −0.0705973 0.263473i 0.921602 0.388137i \(-0.126881\pi\)
−0.992199 + 0.124664i \(0.960215\pi\)
\(150\) 0 0
\(151\) −14.6691 + 14.6691i −1.19375 + 1.19375i −0.217749 + 0.976005i \(0.569871\pi\)
−0.976005 + 0.217749i \(0.930129\pi\)
\(152\) 0 0
\(153\) −4.58257 1.22789i −0.370479 0.0992694i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.7756 + 14.7756i −1.17922 + 1.17922i −0.199281 + 0.979942i \(0.563861\pi\)
−0.979942 + 0.199281i \(0.936139\pi\)
\(158\) 0 0
\(159\) 8.96334 5.17499i 0.710839 0.410403i
\(160\) 0 0
\(161\) −2.04286 2.04286i −0.161000 0.161000i
\(162\) 0 0
\(163\) 4.46942 7.74127i 0.350072 0.606343i −0.636190 0.771533i \(-0.719490\pi\)
0.986262 + 0.165190i \(0.0528237\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.7729 + 9.10649i −1.22054 + 0.704681i −0.965034 0.262125i \(-0.915577\pi\)
−0.255510 + 0.966806i \(0.582243\pi\)
\(168\) 0 0
\(169\) 12.6003 3.19891i 0.969252 0.246070i
\(170\) 0 0
\(171\) 0.757624 2.82749i 0.0579370 0.216224i
\(172\) 0 0
\(173\) 0.469884 + 1.75363i 0.0357246 + 0.133326i 0.981485 0.191539i \(-0.0613478\pi\)
−0.945760 + 0.324865i \(0.894681\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0201i 1.05381i
\(178\) 0 0
\(179\) −9.09987 15.7614i −0.680156 1.17807i −0.974933 0.222499i \(-0.928578\pi\)
0.294776 0.955566i \(-0.404755\pi\)
\(180\) 0 0
\(181\) 17.4351i 1.29594i −0.761667 0.647969i \(-0.775619\pi\)
0.761667 0.647969i \(-0.224381\pi\)
\(182\) 0 0
\(183\) −10.3312 10.3312i −0.763706 0.763706i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.8793 1.23434
\(188\) 0 0
\(189\) 13.2136 3.54056i 0.961145 0.257538i
\(190\) 0 0
\(191\) −10.5403 + 18.2564i −0.762672 + 1.32099i 0.178796 + 0.983886i \(0.442780\pi\)
−0.941469 + 0.337101i \(0.890554\pi\)
\(192\) 0 0
\(193\) 9.68786 + 16.7799i 0.697348 + 1.20784i 0.969383 + 0.245554i \(0.0789700\pi\)
−0.272035 + 0.962287i \(0.587697\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.18595 + 10.7144i 0.440731 + 0.763368i 0.997744 0.0671359i \(-0.0213861\pi\)
−0.557013 + 0.830504i \(0.688053\pi\)
\(198\) 0 0
\(199\) −5.08783 + 8.81238i −0.360667 + 0.624693i −0.988071 0.154001i \(-0.950784\pi\)
0.627404 + 0.778694i \(0.284117\pi\)
\(200\) 0 0
\(201\) 1.46761 0.393245i 0.103517 0.0277373i
\(202\) 0 0
\(203\) 14.8580 1.04283
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.636325 + 0.636325i 0.0442277 + 0.0442277i
\(208\) 0 0
\(209\) 10.4147i 0.720400i
\(210\) 0 0
\(211\) 8.67864 + 15.0319i 0.597463 + 1.03484i 0.993194 + 0.116469i \(0.0371577\pi\)
−0.395732 + 0.918366i \(0.629509\pi\)
\(212\) 0 0
\(213\) 16.3780i 1.12220i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.54388 + 20.6901i 0.376343 + 1.40453i
\(218\) 0 0
\(219\) −3.51450 + 13.1163i −0.237488 + 0.886316i
\(220\) 0 0
\(221\) −20.9253 + 8.50646i −1.40759 + 0.572207i
\(222\) 0 0
\(223\) −19.0202 + 10.9813i −1.27368 + 0.735362i −0.975679 0.219203i \(-0.929654\pi\)
−0.298005 + 0.954564i \(0.596321\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.06760 7.04528i 0.269976 0.467612i −0.698879 0.715239i \(-0.746318\pi\)
0.968855 + 0.247628i \(0.0796510\pi\)
\(228\) 0 0
\(229\) 10.1234 + 10.1234i 0.668973 + 0.668973i 0.957478 0.288505i \(-0.0931582\pi\)
−0.288505 + 0.957478i \(0.593158\pi\)
\(230\) 0 0
\(231\) −8.49528 + 4.90475i −0.558949 + 0.322709i
\(232\) 0 0
\(233\) −1.66935 + 1.66935i −0.109363 + 0.109363i −0.759671 0.650308i \(-0.774640\pi\)
0.650308 + 0.759671i \(0.274640\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.8075 + 4.50355i 1.09176 + 0.292537i
\(238\) 0 0
\(239\) −16.8726 + 16.8726i −1.09140 + 1.09140i −0.0960168 + 0.995380i \(0.530610\pi\)
−0.995380 + 0.0960168i \(0.969390\pi\)
\(240\) 0 0
\(241\) −1.24155 4.63353i −0.0799753 0.298472i 0.914340 0.404947i \(-0.132710\pi\)
−0.994315 + 0.106475i \(0.966043\pi\)
\(242\) 0 0
\(243\) −7.40216 + 1.98340i −0.474849 + 0.127235i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.24858 12.9111i −0.333959 0.821514i
\(248\) 0 0
\(249\) −1.57985 0.423319i −0.100119 0.0268268i
\(250\) 0 0
\(251\) 3.86584 + 2.23194i 0.244009 + 0.140879i 0.617018 0.786949i \(-0.288341\pi\)
−0.373009 + 0.927828i \(0.621674\pi\)
\(252\) 0 0
\(253\) −2.77277 1.60086i −0.174323 0.100645i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.05177 + 7.65731i −0.127986 + 0.477650i −0.999928 0.0119581i \(-0.996194\pi\)
0.871943 + 0.489608i \(0.162860\pi\)
\(258\) 0 0
\(259\) −18.7838 −1.16717
\(260\) 0 0
\(261\) −4.62809 −0.286471
\(262\) 0 0
\(263\) −1.04147 + 3.88683i −0.0642200 + 0.239672i −0.990573 0.136983i \(-0.956259\pi\)
0.926353 + 0.376656i \(0.122926\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.49220 3.74827i −0.397316 0.229391i
\(268\) 0 0
\(269\) 16.5636 + 9.56301i 1.00990 + 0.583067i 0.911163 0.412047i \(-0.135186\pi\)
0.0987386 + 0.995113i \(0.468519\pi\)
\(270\) 0 0
\(271\) 7.39786 + 1.98225i 0.449388 + 0.120413i 0.476413 0.879222i \(-0.341937\pi\)
−0.0270250 + 0.999635i \(0.508603\pi\)
\(272\) 0 0
\(273\) 8.05981 10.3617i 0.487802 0.627118i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.6084 3.91430i 0.877731 0.235187i 0.208303 0.978064i \(-0.433206\pi\)
0.669428 + 0.742877i \(0.266539\pi\)
\(278\) 0 0
\(279\) −1.72685 6.44468i −0.103384 0.385833i
\(280\) 0 0
\(281\) 13.0253 13.0253i 0.777027 0.777027i −0.202297 0.979324i \(-0.564841\pi\)
0.979324 + 0.202297i \(0.0648407\pi\)
\(282\) 0 0
\(283\) −8.11105 2.17335i −0.482152 0.129192i 0.00955242 0.999954i \(-0.496959\pi\)
−0.491705 + 0.870762i \(0.663626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.07048 + 6.07048i −0.358329 + 0.358329i
\(288\) 0 0
\(289\) 19.2675 11.1241i 1.13338 0.654360i
\(290\) 0 0
\(291\) −2.92573 2.92573i −0.171510 0.171510i
\(292\) 0 0
\(293\) −0.870229 + 1.50728i −0.0508393 + 0.0880563i −0.890325 0.455325i \(-0.849523\pi\)
0.839486 + 0.543382i \(0.182856\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.1291 7.58011i 0.761830 0.439843i
\(298\) 0 0
\(299\) 4.24417 + 0.587226i 0.245447 + 0.0339602i
\(300\) 0 0
\(301\) −2.84951 + 10.6345i −0.164243 + 0.612963i
\(302\) 0 0
\(303\) 5.04500 + 18.8282i 0.289827 + 1.08165i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.53050i 0.543934i −0.962307 0.271967i \(-0.912326\pi\)
0.962307 0.271967i \(-0.0876742\pi\)
\(308\) 0 0
\(309\) 3.62726 + 6.28260i 0.206348 + 0.357405i
\(310\) 0 0
\(311\) 19.5311i 1.10751i 0.832680 + 0.553755i \(0.186806\pi\)
−0.832680 + 0.553755i \(0.813194\pi\)
\(312\) 0 0
\(313\) −17.0195 17.0195i −0.962000 0.962000i 0.0373038 0.999304i \(-0.488123\pi\)
−0.999304 + 0.0373038i \(0.988123\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.07976 0.397639 0.198819 0.980036i \(-0.436289\pi\)
0.198819 + 0.980036i \(0.436289\pi\)
\(318\) 0 0
\(319\) 15.9050 4.26174i 0.890512 0.238612i
\(320\) 0 0
\(321\) −1.77515 + 3.07465i −0.0990791 + 0.171610i
\(322\) 0 0
\(323\) 12.1083 + 20.9722i 0.673723 + 1.16692i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.41276 4.17903i −0.133426 0.231101i
\(328\) 0 0
\(329\) −8.28155 + 14.3441i −0.456577 + 0.790814i
\(330\) 0 0
\(331\) 15.2412 4.08386i 0.837730 0.224469i 0.185647 0.982617i \(-0.440562\pi\)
0.652083 + 0.758147i \(0.273895\pi\)
\(332\) 0 0
\(333\) 5.85090 0.320627
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.66232 4.66232i −0.253973 0.253973i 0.568625 0.822597i \(-0.307476\pi\)
−0.822597 + 0.568625i \(0.807476\pi\)
\(338\) 0 0
\(339\) 20.7285i 1.12582i
\(340\) 0 0
\(341\) 11.8691 + 20.5579i 0.642747 + 1.11327i
\(342\) 0 0
\(343\) 19.6667i 1.06190i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.47707 + 24.1728i 0.347707 + 1.29766i 0.889417 + 0.457096i \(0.151111\pi\)
−0.541710 + 0.840566i \(0.682223\pi\)
\(348\) 0 0
\(349\) 7.28299 27.1805i 0.389850 1.45494i −0.440528 0.897739i \(-0.645209\pi\)
0.830378 0.557200i \(-0.188124\pi\)
\(350\) 0 0
\(351\) −12.4561 + 16.0136i −0.664859 + 0.854743i
\(352\) 0 0
\(353\) 8.40368 4.85187i 0.447283 0.258239i −0.259399 0.965770i \(-0.583525\pi\)
0.706682 + 0.707531i \(0.250191\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −11.4047 + 19.7535i −0.603599 + 1.04546i
\(358\) 0 0
\(359\) −2.09591 2.09591i −0.110618 0.110618i 0.649631 0.760249i \(-0.274923\pi\)
−0.760249 + 0.649631i \(0.774923\pi\)
\(360\) 0 0
\(361\) 3.51444 2.02906i 0.184970 0.106793i
\(362\) 0 0
\(363\) 3.96129 3.96129i 0.207914 0.207914i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.1325 + 2.71499i 0.528912 + 0.141721i 0.513386 0.858158i \(-0.328391\pi\)
0.0155260 + 0.999879i \(0.495058\pi\)
\(368\) 0 0
\(369\) 1.89087 1.89087i 0.0984350 0.0984350i
\(370\) 0 0
\(371\) 4.34873 + 16.2297i 0.225775 + 0.842603i
\(372\) 0 0
\(373\) −15.1023 + 4.04666i −0.781969 + 0.209528i −0.627653 0.778493i \(-0.715984\pi\)
−0.154317 + 0.988021i \(0.549318\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.5697 + 13.2988i −0.904888 + 0.684921i
\(378\) 0 0
\(379\) 12.4625 + 3.33930i 0.640153 + 0.171529i 0.564273 0.825588i \(-0.309157\pi\)
0.0758805 + 0.997117i \(0.475823\pi\)
\(380\) 0 0
\(381\) 5.68036 + 3.27956i 0.291014 + 0.168017i
\(382\) 0 0
\(383\) −24.8327 14.3371i −1.26889 0.732594i −0.294112 0.955771i \(-0.595024\pi\)
−0.974778 + 0.223177i \(0.928357\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.887585 3.31251i 0.0451185 0.168385i
\(388\) 0 0
\(389\) 35.3120 1.79039 0.895194 0.445677i \(-0.147037\pi\)
0.895194 + 0.445677i \(0.147037\pi\)
\(390\) 0 0
\(391\) −7.44474 −0.376496
\(392\) 0 0
\(393\) −4.70817 + 17.5711i −0.237496 + 0.886346i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.76940 4.48566i −0.389935 0.225129i 0.292197 0.956358i \(-0.405614\pi\)
−0.682132 + 0.731229i \(0.738947\pi\)
\(398\) 0 0
\(399\) −12.1881 7.03680i −0.610168 0.352281i
\(400\) 0 0
\(401\) −1.05240 0.281990i −0.0525543 0.0140819i 0.232446 0.972609i \(-0.425327\pi\)
−0.285000 + 0.958527i \(0.591994\pi\)
\(402\) 0 0
\(403\) −25.0744 19.5041i −1.24905 0.971566i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.1074 + 5.38776i −0.996686 + 0.267061i
\(408\) 0 0
\(409\) −7.28956 27.2050i −0.360446 1.34520i −0.873491 0.486840i \(-0.838150\pi\)
0.513046 0.858361i \(-0.328517\pi\)
\(410\) 0 0
\(411\) 15.1540 15.1540i 0.747491 0.747491i
\(412\) 0 0
\(413\) 21.9847 + 5.89080i 1.08180 + 0.289867i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.2551 16.2551i 0.796014 0.796014i
\(418\) 0 0
\(419\) −3.96691 + 2.29029i −0.193796 + 0.111888i −0.593758 0.804643i \(-0.702357\pi\)
0.399962 + 0.916532i \(0.369023\pi\)
\(420\) 0 0
\(421\) 16.0987 + 16.0987i 0.784602 + 0.784602i 0.980604 0.196002i \(-0.0627959\pi\)
−0.196002 + 0.980604i \(0.562796\pi\)
\(422\) 0 0
\(423\) 2.57960 4.46799i 0.125424 0.217241i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.5411 11.8594i 0.994056 0.573918i
\(428\) 0 0
\(429\) 5.65571 13.4037i 0.273060 0.647134i
\(430\) 0 0
\(431\) −0.0745698 + 0.278298i −0.00359190 + 0.0134051i −0.967699 0.252110i \(-0.918875\pi\)
0.964107 + 0.265515i \(0.0855421\pi\)
\(432\) 0 0
\(433\) −10.0790 37.6152i −0.484364 1.80767i −0.582909 0.812537i \(-0.698086\pi\)
0.0985457 0.995133i \(-0.468581\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.59348i 0.219736i
\(438\) 0 0
\(439\) 14.3895 + 24.9233i 0.686771 + 1.18952i 0.972877 + 0.231324i \(0.0743059\pi\)
−0.286105 + 0.958198i \(0.592361\pi\)
\(440\) 0 0
\(441\) 0.824995i 0.0392855i
\(442\) 0 0
\(443\) 25.9217 + 25.9217i 1.23158 + 1.23158i 0.963357 + 0.268222i \(0.0864359\pi\)
0.268222 + 0.963357i \(0.413564\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.98624 0.235841
\(448\) 0 0
\(449\) −38.4485 + 10.3023i −1.81450 + 0.486193i −0.996082 0.0884331i \(-0.971814\pi\)
−0.818416 + 0.574626i \(0.805147\pi\)
\(450\) 0 0
\(451\) −4.75705 + 8.23944i −0.224001 + 0.387980i
\(452\) 0 0
\(453\) −15.5337 26.9052i −0.729839 1.26412i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.92971 6.80646i −0.183824 0.318393i 0.759355 0.650676i \(-0.225514\pi\)
−0.943180 + 0.332283i \(0.892181\pi\)
\(458\) 0 0
\(459\) 17.6255 30.5282i 0.822687 1.42494i
\(460\) 0 0
\(461\) −19.4014 + 5.19859i −0.903614 + 0.242123i −0.680568 0.732685i \(-0.738267\pi\)
−0.223046 + 0.974808i \(0.571600\pi\)
\(462\) 0 0
\(463\) 1.60260 0.0744790 0.0372395 0.999306i \(-0.488144\pi\)
0.0372395 + 0.999306i \(0.488144\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.6094 17.6094i −0.814864 0.814864i 0.170495 0.985359i \(-0.445463\pi\)
−0.985359 + 0.170495i \(0.945463\pi\)
\(468\) 0 0
\(469\) 2.46657i 0.113896i
\(470\) 0 0
\(471\) −15.6465 27.1006i −0.720955 1.24873i
\(472\) 0 0
\(473\) 12.2012i 0.561013i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.35457 5.05534i −0.0620216 0.231468i
\(478\) 0 0
\(479\) −5.67830 + 21.1917i −0.259448 + 0.968273i 0.706113 + 0.708099i \(0.250447\pi\)
−0.965562 + 0.260175i \(0.916220\pi\)
\(480\) 0 0
\(481\) 22.2119 16.8125i 1.01278 0.766583i
\(482\) 0 0
\(483\) 3.74690 2.16328i 0.170490 0.0984325i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.85260 11.8691i 0.310521 0.537838i −0.667954 0.744202i \(-0.732830\pi\)
0.978475 + 0.206364i \(0.0661632\pi\)
\(488\) 0 0
\(489\) 9.46573 + 9.46573i 0.428055 + 0.428055i
\(490\) 0 0
\(491\) 20.0133 11.5547i 0.903187 0.521455i 0.0249539 0.999689i \(-0.492056\pi\)
0.878233 + 0.478234i \(0.158723\pi\)
\(492\) 0 0
\(493\) 27.0733 27.0733i 1.21932 1.21932i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.6822 6.88153i −1.15200 0.308679i
\(498\) 0 0
\(499\) −15.3195 + 15.3195i −0.685795 + 0.685795i −0.961300 0.275505i \(-0.911155\pi\)
0.275505 + 0.961300i \(0.411155\pi\)
\(500\) 0 0
\(501\) −7.05936 26.3459i −0.315389 1.17705i
\(502\) 0 0
\(503\) −33.5206 + 8.98181i −1.49461 + 0.400479i −0.911290 0.411765i \(-0.864913\pi\)
−0.583318 + 0.812244i \(0.698246\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.256508 + 19.4667i −0.0113919 + 0.864549i
\(508\) 0 0
\(509\) −17.1190 4.58701i −0.758785 0.203316i −0.141374 0.989956i \(-0.545152\pi\)
−0.617411 + 0.786641i \(0.711819\pi\)
\(510\) 0 0
\(511\) −19.0908 11.0221i −0.844528 0.487589i
\(512\) 0 0
\(513\) 18.8362 + 10.8751i 0.831640 + 0.480148i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.75081 + 17.7303i −0.208940 + 0.779776i
\(518\) 0 0
\(519\) −2.71883 −0.119343
\(520\) 0 0
\(521\) −13.1460 −0.575938 −0.287969 0.957640i \(-0.592980\pi\)
−0.287969 + 0.957640i \(0.592980\pi\)
\(522\) 0 0
\(523\) 5.87752 21.9352i 0.257006 0.959160i −0.709957 0.704245i \(-0.751286\pi\)
0.966963 0.254915i \(-0.0820475\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.8017 + 27.5983i 2.08228 + 1.20220i
\(528\) 0 0
\(529\) −18.6956 10.7939i −0.812854 0.469301i
\(530\) 0 0
\(531\) −6.84796 1.83491i −0.297176 0.0796282i
\(532\) 0 0
\(533\) 1.74497 12.6118i 0.0755832 0.546277i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 26.3267 7.05423i 1.13608 0.304412i
\(538\) 0 0
\(539\) 0.759691 + 2.83521i 0.0327222 + 0.122121i
\(540\) 0 0
\(541\) 12.3506 12.3506i 0.530993 0.530993i −0.389875 0.920868i \(-0.627482\pi\)
0.920868 + 0.389875i \(0.127482\pi\)
\(542\) 0 0
\(543\) 25.2206 + 6.75783i 1.08232 + 0.290006i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.96444 7.96444i 0.340535 0.340535i −0.516034 0.856568i \(-0.672592\pi\)
0.856568 + 0.516034i \(0.172592\pi\)
\(548\) 0 0
\(549\) −6.39830 + 3.69406i −0.273073 + 0.157659i
\(550\) 0 0
\(551\) 16.7045 + 16.7045i 0.711637 + 0.711637i
\(552\) 0 0
\(553\) −14.1239 + 24.4634i −0.600611 + 1.04029i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.7800 13.1520i 0.965219 0.557269i 0.0674437 0.997723i \(-0.478516\pi\)
0.897775 + 0.440454i \(0.145182\pi\)
\(558\) 0 0
\(559\) −6.14891 15.1259i −0.260071 0.639756i
\(560\) 0 0
\(561\) −6.54242 + 24.4167i −0.276221 + 1.03087i
\(562\) 0 0
\(563\) 11.9307 + 44.5258i 0.502817 + 1.87654i 0.480894 + 0.876779i \(0.340312\pi\)
0.0219234 + 0.999760i \(0.493021\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 14.9631i 0.628391i
\(568\) 0 0
\(569\) 15.3819 + 26.6422i 0.644841 + 1.11690i 0.984338 + 0.176290i \(0.0564097\pi\)
−0.339497 + 0.940607i \(0.610257\pi\)
\(570\) 0 0
\(571\) 20.6467i 0.864038i −0.901865 0.432019i \(-0.857801\pi\)
0.901865 0.432019i \(-0.142199\pi\)
\(572\) 0 0
\(573\) −22.3232 22.3232i −0.932567 0.932567i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −20.3744 −0.848198 −0.424099 0.905616i \(-0.639409\pi\)
−0.424099 + 0.905616i \(0.639409\pi\)
\(578\) 0 0
\(579\) −28.0278 + 7.51004i −1.16480 + 0.312106i
\(580\) 0 0
\(581\) 1.32761 2.29948i 0.0550783 0.0953985i
\(582\) 0 0
\(583\) 9.31035 + 16.1260i 0.385595 + 0.667870i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.5802 37.3781i −0.890712 1.54276i −0.839024 0.544095i \(-0.816873\pi\)
−0.0516879 0.998663i \(-0.516460\pi\)
\(588\) 0 0
\(589\) −17.0285 + 29.4942i −0.701645 + 1.21529i
\(590\) 0 0
\(591\) −17.8965 + 4.79535i −0.736163 + 0.197254i
\(592\) 0 0
\(593\) 34.3448 1.41037 0.705186 0.709022i \(-0.250863\pi\)
0.705186 + 0.709022i \(0.250863\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.7754 10.7754i −0.441010 0.441010i
\(598\) 0 0
\(599\) 8.69200i 0.355146i −0.984108 0.177573i \(-0.943175\pi\)
0.984108 0.177573i \(-0.0568245\pi\)
\(600\) 0 0
\(601\) −1.73390 3.00320i −0.0707271 0.122503i 0.828493 0.559999i \(-0.189199\pi\)
−0.899220 + 0.437496i \(0.855865\pi\)
\(602\) 0 0
\(603\) 0.768305i 0.0312878i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.83078 + 18.0287i 0.196075 + 0.731764i 0.991986 + 0.126349i \(0.0403260\pi\)
−0.795910 + 0.605414i \(0.793007\pi\)
\(608\) 0 0
\(609\) −5.75898 + 21.4928i −0.233366 + 0.870932i
\(610\) 0 0
\(611\) −3.04573 24.3744i −0.123217 0.986083i
\(612\) 0 0
\(613\) 20.9846 12.1155i 0.847560 0.489339i −0.0122667 0.999925i \(-0.503905\pi\)
0.859827 + 0.510586i \(0.170571\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.94688 + 13.7644i −0.319930 + 0.554134i −0.980473 0.196654i \(-0.936993\pi\)
0.660544 + 0.750788i \(0.270326\pi\)
\(618\) 0 0
\(619\) 10.3555 + 10.3555i 0.416225 + 0.416225i 0.883900 0.467676i \(-0.154908\pi\)
−0.467676 + 0.883900i \(0.654908\pi\)
\(620\) 0 0
\(621\) −5.79070 + 3.34326i −0.232373 + 0.134160i
\(622\) 0 0
\(623\) 8.60545 8.60545i 0.344770 0.344770i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −15.0653 4.03674i −0.601651 0.161212i
\(628\) 0 0
\(629\) −34.2265 + 34.2265i −1.36470 + 1.36470i
\(630\) 0 0
\(631\) −10.7149 39.9887i −0.426555 1.59193i −0.760503 0.649334i \(-0.775048\pi\)
0.333948 0.942591i \(-0.391619\pi\)
\(632\) 0 0
\(633\) −25.1081 + 6.72769i −0.997956 + 0.267402i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.37061 3.13195i −0.0939271 0.124092i
\(638\) 0 0
\(639\) 7.99967 + 2.14351i 0.316462 + 0.0847958i
\(640\) 0 0
\(641\) −3.26442 1.88471i −0.128937 0.0744417i 0.434144 0.900843i \(-0.357051\pi\)
−0.563081 + 0.826402i \(0.690384\pi\)
\(642\) 0 0
\(643\) −19.8336 11.4509i −0.782159 0.451580i 0.0550358 0.998484i \(-0.482473\pi\)
−0.837195 + 0.546905i \(0.815806\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.33051 23.6258i 0.248878 0.928825i −0.722517 0.691353i \(-0.757015\pi\)
0.971395 0.237471i \(-0.0763186\pi\)
\(648\) 0 0
\(649\) 25.2236 0.990113
\(650\) 0 0
\(651\) −32.0779 −1.25723
\(652\) 0 0
\(653\) 7.56514 28.2335i 0.296047 1.10486i −0.644336 0.764743i \(-0.722866\pi\)
0.940383 0.340119i \(-0.110467\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.94654 + 3.43324i 0.231997 + 0.133943i
\(658\) 0 0
\(659\) −11.0616 6.38643i −0.430900 0.248780i 0.268830 0.963188i \(-0.413363\pi\)
−0.699730 + 0.714407i \(0.746696\pi\)
\(660\) 0 0
\(661\) −16.9009 4.52859i −0.657370 0.176142i −0.0853110 0.996354i \(-0.527188\pi\)
−0.572059 + 0.820213i \(0.693855\pi\)
\(662\) 0 0
\(663\) −4.19433 33.5664i −0.162894 1.30361i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.01503 + 1.87967i −0.271623 + 0.0727812i
\(668\) 0 0
\(669\) −8.51270 31.7698i −0.329120 1.22829i
\(670\) 0 0
\(671\) 18.5870 18.5870i 0.717542 0.717542i
\(672\) 0 0
\(673\) 23.0450 + 6.17489i 0.888320 + 0.238025i 0.673993 0.738738i \(-0.264578\pi\)
0.214326 + 0.976762i \(0.431244\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.9432 + 18.9432i −0.728048 + 0.728048i −0.970231 0.242183i \(-0.922137\pi\)
0.242183 + 0.970231i \(0.422137\pi\)
\(678\) 0 0
\(679\) 5.81711 3.35851i 0.223240 0.128888i
\(680\) 0 0
\(681\) 8.61471 + 8.61471i 0.330116 + 0.330116i
\(682\) 0 0
\(683\) −1.69188 + 2.93042i −0.0647380 + 0.112129i −0.896578 0.442886i \(-0.853954\pi\)
0.831840 + 0.555016i \(0.187288\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −18.5678 + 10.7201i −0.708405 + 0.408998i
\(688\) 0 0
\(689\) −19.6689 15.2994i −0.749324 0.582860i
\(690\) 0 0
\(691\) 12.8180 47.8376i 0.487621 1.81983i −0.0803328 0.996768i \(-0.525598\pi\)
0.567954 0.823060i \(-0.307735\pi\)
\(692\) 0 0
\(693\) 1.28384 + 4.79135i 0.0487690 + 0.182008i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 22.1224i 0.837947i
\(698\) 0 0
\(699\) −1.76775 3.06183i −0.0668625 0.115809i
\(700\) 0 0
\(701\) 5.71523i 0.215861i −0.994158 0.107931i \(-0.965578\pi\)
0.994158 0.107931i \(-0.0344224\pi\)
\(702\) 0 0
\(703\) −21.1181 21.1181i −0.796485 0.796485i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.6440 −1.19010
\(708\) 0 0
\(709\) 9.56101 2.56186i 0.359071 0.0962128i −0.0747734 0.997201i \(-0.523823\pi\)
0.433845 + 0.900988i \(0.357157\pi\)
\(710\) 0 0
\(711\) 4.39942 7.62003i 0.164991 0.285773i
\(712\) 0 0
\(713\) −5.23494 9.06719i −0.196050 0.339569i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −17.8671 30.9467i −0.667259 1.15573i
\(718\) 0 0
\(719\) 5.37245 9.30535i 0.200358 0.347031i −0.748286 0.663377i \(-0.769123\pi\)
0.948644 + 0.316346i \(0.102456\pi\)
\(720\) 0 0
\(721\) −11.3758 + 3.04812i −0.423655 + 0.113518i
\(722\) 0 0
\(723\) 7.18383 0.267169
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −14.2084 14.2084i −0.526961 0.526961i 0.392704 0.919665i \(-0.371540\pi\)
−0.919665 + 0.392704i \(0.871540\pi\)
\(728\) 0 0
\(729\) 29.9404i 1.10890i
\(730\) 0 0
\(731\) 14.1853 + 24.5697i 0.524663 + 0.908743i
\(732\) 0 0
\(733\) 37.5445i 1.38674i 0.720582 + 0.693369i \(0.243875\pi\)
−0.720582 + 0.693369i \(0.756125\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.707489 + 2.64038i 0.0260607 + 0.0972598i
\(738\) 0 0
\(739\) 7.64320 28.5248i 0.281160 1.04930i −0.670440 0.741964i \(-0.733895\pi\)
0.951600 0.307339i \(-0.0994386\pi\)
\(740\) 0 0
\(741\) 20.7108 2.58794i 0.760832 0.0950705i
\(742\) 0 0
\(743\) 33.7356 19.4772i 1.23764 0.714551i 0.269027 0.963133i \(-0.413298\pi\)
0.968611 + 0.248582i \(0.0799645\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.413532 + 0.716258i −0.0151303 + 0.0262065i
\(748\) 0 0
\(749\) −4.07546 4.07546i −0.148914 0.148914i
\(750\) 0 0
\(751\) −22.7530 + 13.1365i −0.830270 + 0.479357i −0.853945 0.520363i \(-0.825797\pi\)
0.0236752 + 0.999720i \(0.492463\pi\)
\(752\) 0 0
\(753\) −4.72700 + 4.72700i −0.172261 + 0.172261i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −26.7365 7.16401i −0.971753 0.260380i −0.262185 0.965018i \(-0.584443\pi\)
−0.709568 + 0.704637i \(0.751110\pi\)
\(758\) 0 0
\(759\) 3.39044 3.39044i 0.123065 0.123065i
\(760\) 0 0
\(761\) −10.2417 38.2227i −0.371263 1.38557i −0.858729 0.512430i \(-0.828745\pi\)
0.487466 0.873142i \(-0.337921\pi\)
\(762\) 0 0
\(763\) 7.56686 2.02753i 0.273939 0.0734017i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.2697 + 12.7116i −1.12908 + 0.458991i
\(768\) 0 0
\(769\) 0.642529 + 0.172165i 0.0231702 + 0.00620843i 0.270386 0.962752i \(-0.412849\pi\)
−0.247215 + 0.968961i \(0.579516\pi\)
\(770\) 0 0
\(771\) −10.2814 5.93595i −0.370275 0.213778i
\(772\) 0 0
\(773\) 13.9854 + 8.07445i 0.503018 + 0.290418i 0.729959 0.683491i \(-0.239539\pi\)
−0.226941 + 0.973909i \(0.572872\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.28059 27.1715i 0.261190 0.974773i
\(778\) 0 0
\(779\) −13.6498 −0.489054
\(780\) 0 0
\(781\) −29.4658 −1.05437
\(782\) 0 0
\(783\) 8.90028 33.2163i 0.318070 1.18705i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.8188 + 16.0612i 0.991633 + 0.572520i 0.905762 0.423787i \(-0.139299\pi\)
0.0858710 + 0.996306i \(0.472633\pi\)
\(788\) 0 0
\(789\) −5.21880 3.01307i −0.185794 0.107268i
\(790\) 0 0
\(791\) 32.5042 + 8.70947i 1.15572 + 0.309673i
\(792\) 0 0
\(793\) −13.6752 + 32.4093i −0.485621 + 1.15089i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.58199 + 0.423892i −0.0560369 + 0.0150150i −0.286729 0.958012i \(-0.592568\pi\)
0.230692 + 0.973027i \(0.425901\pi\)
\(798\) 0 0
\(799\) 11.0467 + 41.2269i 0.390805 + 1.45850i
\(800\) 0 0
\(801\) −2.68048 + 2.68048i −0.0947103 + 0.0947103i
\(802\) 0 0
\(803\) −23.5976 6.32295i −0.832740 0.223132i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20.2534 + 20.2534i −0.712952 + 0.712952i
\(808\) 0 0
\(809\) −9.27418 + 5.35445i −0.326063 + 0.188253i −0.654092 0.756415i \(-0.726949\pi\)
0.328029 + 0.944668i \(0.393616\pi\)
\(810\) 0 0
\(811\) 37.8784 + 37.8784i 1.33009 + 1.33009i 0.905283 + 0.424809i \(0.139659\pi\)
0.424809 + 0.905283i \(0.360341\pi\)
\(812\) 0 0
\(813\) −5.73482 + 9.93301i −0.201129 + 0.348366i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −15.1598 + 8.75249i −0.530373 + 0.306211i
\(818\) 0 0
\(819\) −4.00622 5.29284i −0.139989 0.184947i
\(820\) 0 0
\(821\) 5.96951 22.2785i 0.208337 0.777526i −0.780069 0.625694i \(-0.784816\pi\)
0.988406 0.151832i \(-0.0485173\pi\)
\(822\) 0 0
\(823\) −1.95458 7.29458i −0.0681323 0.254273i 0.923456 0.383704i \(-0.125352\pi\)
−0.991588 + 0.129431i \(0.958685\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.2354i 1.32957i −0.747033 0.664787i \(-0.768522\pi\)
0.747033 0.664787i \(-0.231478\pi\)
\(828\) 0 0
\(829\) −24.4408 42.3328i −0.848865 1.47028i −0.882222 0.470834i \(-0.843953\pi\)
0.0333566 0.999444i \(-0.489380\pi\)
\(830\) 0 0
\(831\) 22.6488i 0.785679i
\(832\) 0 0
\(833\) 4.82604 + 4.82604i 0.167213 + 0.167213i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 49.5751 1.71357
\(838\) 0 0
\(839\) 24.6702 6.61035i 0.851709 0.228215i 0.193547 0.981091i \(-0.438001\pi\)
0.658162 + 0.752876i \(0.271334\pi\)
\(840\) 0 0
\(841\) 4.17511 7.23150i 0.143969 0.249362i
\(842\) 0 0
\(843\) 13.7931 + 23.8904i 0.475060 + 0.822828i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.54725 + 7.87607i 0.156245 + 0.270625i
\(848\) 0 0
\(849\) 6.28769 10.8906i 0.215793 0.373765i
\(850\) 0 0
\(851\) 8.86851 2.37631i 0.304008 0.0814588i
\(852\) 0 0
\(853\) −7.17091 −0.245527 −0.122764 0.992436i \(-0.539176\pi\)
−0.122764 + 0.992436i \(0.539176\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.7248 10.7248i −0.366350 0.366350i 0.499794 0.866144i \(-0.333409\pi\)
−0.866144 + 0.499794i \(0.833409\pi\)
\(858\) 0 0
\(859\) 27.2503i 0.929770i 0.885371 + 0.464885i \(0.153904\pi\)
−0.885371 + 0.464885i \(0.846096\pi\)
\(860\) 0 0
\(861\) −6.42829 11.1341i −0.219076 0.379450i
\(862\) 0 0
\(863\) 8.34796i 0.284168i 0.989855 + 0.142084i \(0.0453803\pi\)
−0.989855 + 0.142084i \(0.954620\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.62342 + 32.1830i 0.292867 + 1.09299i
\(868\) 0 0
\(869\) −8.10236 + 30.2384i −0.274854 + 1.02577i
\(870\) 0 0
\(871\) −2.20772 2.91674i −0.0748056 0.0988299i
\(872\) 0 0
\(873\) −1.81196 + 1.04613i −0.0613254 + 0.0354062i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.3047 26.5085i 0.516803 0.895130i −0.483006 0.875617i \(-0.660455\pi\)
0.999810 0.0195127i \(-0.00621147\pi\)
\(878\) 0 0
\(879\) −1.84305 1.84305i −0.0621644 0.0621644i
\(880\) 0 0
\(881\) −6.20042 + 3.57981i −0.208897 + 0.120607i −0.600799 0.799400i \(-0.705151\pi\)
0.391901 + 0.920007i \(0.371817\pi\)
\(882\) 0 0
\(883\) 31.1878 31.1878i 1.04955 1.04955i 0.0508451 0.998707i \(-0.483809\pi\)
0.998707 0.0508451i \(-0.0161915\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.3613 + 11.3507i 1.42235 + 0.381118i 0.886318 0.463077i \(-0.153255\pi\)
0.536035 + 0.844196i \(0.319921\pi\)
\(888\) 0 0
\(889\) −7.52935 + 7.52935i −0.252526 + 0.252526i
\(890\) 0 0
\(891\) 4.29188 + 16.0175i 0.143783 + 0.536607i
\(892\) 0 0
\(893\) −25.4374 + 6.81594i −0.851231 + 0.228087i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.49449 + 5.91177i −0.0832887 + 0.197388i
\(898\) 0 0
\(899\) 52.0108 + 13.9362i 1.73466 + 0.464800i
\(900\) 0 0
\(901\) 37.4966 + 21.6487i 1.24919 + 0.721222i
\(902\) 0 0
\(903\) −14.2788 8.24388i −0.475169 0.274339i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.524912 + 1.95900i −0.0174294 + 0.0650475i −0.974093 0.226149i \(-0.927387\pi\)
0.956663 + 0.291196i \(0.0940532\pi\)
\(908\) 0 0
\(909\) 9.85671 0.326926
\(910\) 0 0
\(911\) −11.7972 −0.390859 −0.195429 0.980718i \(-0.562610\pi\)
−0.195429 + 0.980718i \(0.562610\pi\)
\(912\) 0 0
\(913\) 0.761596 2.84232i 0.0252052 0.0940669i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −25.5749 14.7657i −0.844557 0.487605i
\(918\) 0 0
\(919\) −32.9872 19.0452i −1.08815 0.628242i −0.155065 0.987904i \(-0.549559\pi\)
−0.933083 + 0.359662i \(0.882892\pi\)
\(920\) 0 0
\(921\) 13.7863 + 3.69402i 0.454274 + 0.121722i
\(922\) 0 0
\(923\) 36.5287 14.8495i 1.20236 0.488778i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.54340 0.949451i 0.116380 0.0311841i
\(928\) 0 0
\(929\) 2.43693 + 9.09474i 0.0799530 + 0.298389i 0.994311 0.106519i \(-0.0339704\pi\)
−0.914358 + 0.404907i \(0.867304\pi\)
\(930\) 0 0
\(931\) −2.97772 + 2.97772i −0.0975908 + 0.0975908i
\(932\) 0 0
\(933\) −28.2527 7.57028i −0.924951 0.247840i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 39.2985 39.2985i 1.28382 1.28382i 0.345351 0.938474i \(-0.387760\pi\)
0.938474 0.345351i \(-0.112240\pi\)
\(938\) 0 0
\(939\) 31.2162 18.0227i 1.01870 0.588149i
\(940\) 0 0
\(941\) −19.4526 19.4526i −0.634137 0.634137i 0.314966 0.949103i \(-0.398007\pi\)
−0.949103 + 0.314966i \(0.898007\pi\)
\(942\) 0 0
\(943\) 2.09813 3.63406i 0.0683244 0.118341i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.1987 8.19763i 0.461396 0.266387i −0.251235 0.967926i \(-0.580837\pi\)
0.712631 + 0.701539i \(0.247503\pi\)
\(948\) 0 0
\(949\) 32.4404 4.05363i 1.05306 0.131586i
\(950\) 0 0
\(951\) −2.74411 + 10.2412i −0.0889840 + 0.332093i
\(952\) 0 0
\(953\) −7.57657 28.2762i −0.245429 0.915955i −0.973167 0.230099i \(-0.926095\pi\)
0.727738 0.685855i \(-0.240572\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.6592i 0.797119i
\(958\) 0 0
\(959\) 17.3956 + 30.1300i 0.561733 + 0.972950i
\(960\) 0 0
\(961\) 46.6257i 1.50406i
\(962\) 0 0
\(963\) 1.26945 + 1.26945i 0.0409076 + 0.0409076i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 50.2534 1.61604 0.808021 0.589154i \(-0.200539\pi\)
0.808021 + 0.589154i \(0.200539\pi\)
\(968\) 0 0
\(969\) −35.0303 + 9.38635i −1.12534 + 0.301533i
\(970\) 0 0
\(971\) 23.9370 41.4601i 0.768175 1.33052i −0.170376 0.985379i \(-0.554498\pi\)
0.938551 0.345139i \(-0.112168\pi\)
\(972\) 0 0
\(973\) 18.6595 + 32.3193i 0.598198 + 1.03611i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.4272 + 28.4527i 0.525552 + 0.910283i 0.999557 + 0.0297606i \(0.00947448\pi\)
−0.474005 + 0.880522i \(0.657192\pi\)
\(978\) 0 0
\(979\) 6.74354 11.6802i 0.215524 0.373299i
\(980\) 0 0
\(981\) −2.35698 + 0.631551i −0.0752526 + 0.0201639i
\(982\) 0 0
\(983\) −23.3463 −0.744631 −0.372315 0.928106i \(-0.621436\pi\)
−0.372315 + 0.928106i \(0.621436\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −17.5394 17.5394i −0.558285 0.558285i
\(988\) 0 0
\(989\) 5.38144i 0.171120i
\(990\) 0 0
\(991\) −20.4952 35.4988i −0.651053 1.12766i −0.982868 0.184312i \(-0.940994\pi\)
0.331815 0.943345i \(-0.392339\pi\)
\(992\) 0 0
\(993\) 23.6299i 0.749873i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.6809 + 39.8616i 0.338267 + 1.26243i 0.900284 + 0.435303i \(0.143359\pi\)
−0.562018 + 0.827125i \(0.689975\pi\)
\(998\) 0 0
\(999\) −11.2519 + 41.9926i −0.355993 + 1.32859i
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 1300.2.bs.d.193.2 20
5.2 odd 4 1300.2.bn.d.557.4 20
5.3 odd 4 260.2.bf.c.37.2 20
5.4 even 2 260.2.bk.c.193.4 yes 20
13.6 odd 12 1300.2.bn.d.1293.4 20
65.19 odd 12 260.2.bf.c.253.2 yes 20
65.32 even 12 inner 1300.2.bs.d.357.2 20
65.58 even 12 260.2.bk.c.97.4 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.bf.c.37.2 20 5.3 odd 4
260.2.bf.c.253.2 yes 20 65.19 odd 12
260.2.bk.c.97.4 yes 20 65.58 even 12
260.2.bk.c.193.4 yes 20 5.4 even 2
1300.2.bn.d.557.4 20 5.2 odd 4
1300.2.bn.d.1293.4 20 13.6 odd 12
1300.2.bs.d.193.2 20 1.1 even 1 trivial
1300.2.bs.d.357.2 20 65.32 even 12 inner