Properties

Label 1305.2.c.l.784.8
Level $1305$
Weight $2$
Character 1305.784
Analytic conductor $10.420$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(784,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 148x^{8} + 502x^{6} + 792x^{4} + 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 784.8
Root \(1.27263i\) of defining polynomial
Character \(\chi\) \(=\) 1305.784
Dual form 1305.2.c.l.784.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.27263i q^{2} +0.380419 q^{4} +(-1.10723 - 1.94269i) q^{5} +0.255813i q^{7} +3.02939i q^{8} +(2.47232 - 1.40909i) q^{10} -4.63446 q^{11} -5.02700i q^{13} -0.325555 q^{14} -3.09444 q^{16} -0.336444i q^{17} +2.91437 q^{19} +(-0.421212 - 0.739035i) q^{20} -5.89795i q^{22} -8.65656i q^{23} +(-2.54807 + 4.30201i) q^{25} +6.39750 q^{26} +0.0973162i q^{28} +1.00000 q^{29} -3.26943 q^{31} +2.12070i q^{32} +0.428167 q^{34} +(0.496966 - 0.283245i) q^{35} -3.86954i q^{37} +3.70891i q^{38} +(5.88515 - 3.35424i) q^{40} -5.71649 q^{41} -6.98619i q^{43} -1.76304 q^{44} +11.0166 q^{46} -0.336444i q^{47} +6.93456 q^{49} +(-5.47486 - 3.24275i) q^{50} -1.91237i q^{52} -6.01484i q^{53} +(5.13143 + 9.00332i) q^{55} -0.774958 q^{56} +1.27263i q^{58} -13.2799 q^{59} -7.77792 q^{61} -4.16077i q^{62} -8.88775 q^{64} +(-9.76589 + 5.56606i) q^{65} -11.2605i q^{67} -0.127989i q^{68} +(0.360465 + 0.632452i) q^{70} +13.9668 q^{71} -8.46426i q^{73} +4.92448 q^{74} +1.10868 q^{76} -1.18556i q^{77} +15.3102 q^{79} +(3.42627 + 6.01154i) q^{80} -7.27496i q^{82} -7.60521i q^{83} +(-0.653605 + 0.372521i) q^{85} +8.89082 q^{86} -14.0396i q^{88} -13.0383 q^{89} +1.28597 q^{91} -3.29312i q^{92} +0.428167 q^{94} +(-3.22689 - 5.66171i) q^{95} +11.5445i q^{97} +8.82511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 10 q^{10} + 12 q^{11} - 16 q^{14} + 16 q^{16} + 20 q^{19} - 14 q^{20} + 8 q^{25} + 56 q^{26} + 12 q^{29} - 16 q^{31} - 4 q^{34} + 16 q^{35} + 16 q^{40} + 32 q^{41} - 68 q^{44} + 20 q^{46}+ \cdots + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27263i 0.899884i 0.893058 + 0.449942i \(0.148555\pi\)
−0.893058 + 0.449942i \(0.851445\pi\)
\(3\) 0 0
\(4\) 0.380419 0.190209
\(5\) −1.10723 1.94269i −0.495169 0.868796i
\(6\) 0 0
\(7\) 0.255813i 0.0966884i 0.998831 + 0.0483442i \(0.0153944\pi\)
−0.998831 + 0.0483442i \(0.984606\pi\)
\(8\) 3.02939i 1.07105i
\(9\) 0 0
\(10\) 2.47232 1.40909i 0.781816 0.445595i
\(11\) −4.63446 −1.39734 −0.698672 0.715442i \(-0.746225\pi\)
−0.698672 + 0.715442i \(0.746225\pi\)
\(12\) 0 0
\(13\) 5.02700i 1.39424i −0.716955 0.697120i \(-0.754465\pi\)
0.716955 0.697120i \(-0.245535\pi\)
\(14\) −0.325555 −0.0870083
\(15\) 0 0
\(16\) −3.09444 −0.773611
\(17\) 0.336444i 0.0815995i −0.999167 0.0407998i \(-0.987009\pi\)
0.999167 0.0407998i \(-0.0129906\pi\)
\(18\) 0 0
\(19\) 2.91437 0.668602 0.334301 0.942466i \(-0.391500\pi\)
0.334301 + 0.942466i \(0.391500\pi\)
\(20\) −0.421212 0.739035i −0.0941859 0.165253i
\(21\) 0 0
\(22\) 5.89795i 1.25745i
\(23\) 8.65656i 1.80502i −0.430671 0.902509i \(-0.641723\pi\)
0.430671 0.902509i \(-0.358277\pi\)
\(24\) 0 0
\(25\) −2.54807 + 4.30201i −0.509614 + 0.860403i
\(26\) 6.39750 1.25465
\(27\) 0 0
\(28\) 0.0973162i 0.0183910i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.26943 −0.587207 −0.293603 0.955927i \(-0.594854\pi\)
−0.293603 + 0.955927i \(0.594854\pi\)
\(32\) 2.12070i 0.374890i
\(33\) 0 0
\(34\) 0.428167 0.0734301
\(35\) 0.496966 0.283245i 0.0840025 0.0478771i
\(36\) 0 0
\(37\) 3.86954i 0.636148i −0.948066 0.318074i \(-0.896964\pi\)
0.948066 0.318074i \(-0.103036\pi\)
\(38\) 3.70891i 0.601664i
\(39\) 0 0
\(40\) 5.88515 3.35424i 0.930524 0.530351i
\(41\) −5.71649 −0.892765 −0.446383 0.894842i \(-0.647288\pi\)
−0.446383 + 0.894842i \(0.647288\pi\)
\(42\) 0 0
\(43\) 6.98619i 1.06538i −0.846309 0.532692i \(-0.821180\pi\)
0.846309 0.532692i \(-0.178820\pi\)
\(44\) −1.76304 −0.265788
\(45\) 0 0
\(46\) 11.0166 1.62431
\(47\) 0.336444i 0.0490753i −0.999699 0.0245377i \(-0.992189\pi\)
0.999699 0.0245377i \(-0.00781137\pi\)
\(48\) 0 0
\(49\) 6.93456 0.990651
\(50\) −5.47486 3.24275i −0.774263 0.458594i
\(51\) 0 0
\(52\) 1.91237i 0.265197i
\(53\) 6.01484i 0.826202i −0.910685 0.413101i \(-0.864446\pi\)
0.910685 0.413101i \(-0.135554\pi\)
\(54\) 0 0
\(55\) 5.13143 + 9.00332i 0.691922 + 1.21401i
\(56\) −0.774958 −0.103558
\(57\) 0 0
\(58\) 1.27263i 0.167104i
\(59\) −13.2799 −1.72890 −0.864451 0.502718i \(-0.832334\pi\)
−0.864451 + 0.502718i \(0.832334\pi\)
\(60\) 0 0
\(61\) −7.77792 −0.995861 −0.497930 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(62\) 4.16077i 0.528418i
\(63\) 0 0
\(64\) −8.88775 −1.11097
\(65\) −9.76589 + 5.56606i −1.21131 + 0.690385i
\(66\) 0 0
\(67\) 11.2605i 1.37569i −0.725856 0.687847i \(-0.758556\pi\)
0.725856 0.687847i \(-0.241444\pi\)
\(68\) 0.127989i 0.0155210i
\(69\) 0 0
\(70\) 0.360465 + 0.632452i 0.0430839 + 0.0755925i
\(71\) 13.9668 1.65756 0.828778 0.559578i \(-0.189037\pi\)
0.828778 + 0.559578i \(0.189037\pi\)
\(72\) 0 0
\(73\) 8.46426i 0.990667i −0.868703 0.495333i \(-0.835046\pi\)
0.868703 0.495333i \(-0.164954\pi\)
\(74\) 4.92448 0.572459
\(75\) 0 0
\(76\) 1.10868 0.127174
\(77\) 1.18556i 0.135107i
\(78\) 0 0
\(79\) 15.3102 1.72253 0.861267 0.508153i \(-0.169672\pi\)
0.861267 + 0.508153i \(0.169672\pi\)
\(80\) 3.42627 + 6.01154i 0.383069 + 0.672111i
\(81\) 0 0
\(82\) 7.27496i 0.803385i
\(83\) 7.60521i 0.834781i −0.908727 0.417390i \(-0.862945\pi\)
0.908727 0.417390i \(-0.137055\pi\)
\(84\) 0 0
\(85\) −0.653605 + 0.372521i −0.0708934 + 0.0404056i
\(86\) 8.89082 0.958722
\(87\) 0 0
\(88\) 14.0396i 1.49662i
\(89\) −13.0383 −1.38206 −0.691029 0.722827i \(-0.742843\pi\)
−0.691029 + 0.722827i \(0.742843\pi\)
\(90\) 0 0
\(91\) 1.28597 0.134807
\(92\) 3.29312i 0.343331i
\(93\) 0 0
\(94\) 0.428167 0.0441621
\(95\) −3.22689 5.66171i −0.331071 0.580879i
\(96\) 0 0
\(97\) 11.5445i 1.17216i 0.810253 + 0.586081i \(0.199330\pi\)
−0.810253 + 0.586081i \(0.800670\pi\)
\(98\) 8.82511i 0.891471i
\(99\) 0 0
\(100\) −0.969334 + 1.63657i −0.0969334 + 0.163657i
\(101\) 8.19832 0.815763 0.407882 0.913035i \(-0.366268\pi\)
0.407882 + 0.913035i \(0.366268\pi\)
\(102\) 0 0
\(103\) 1.73014i 0.170476i 0.996361 + 0.0852380i \(0.0271651\pi\)
−0.996361 + 0.0852380i \(0.972835\pi\)
\(104\) 15.2287 1.49330
\(105\) 0 0
\(106\) 7.65465 0.743486
\(107\) 5.38921i 0.520995i 0.965475 + 0.260497i \(0.0838865\pi\)
−0.965475 + 0.260497i \(0.916113\pi\)
\(108\) 0 0
\(109\) −14.2391 −1.36386 −0.681929 0.731418i \(-0.738859\pi\)
−0.681929 + 0.731418i \(0.738859\pi\)
\(110\) −11.4579 + 6.53040i −1.09247 + 0.622649i
\(111\) 0 0
\(112\) 0.791600i 0.0747992i
\(113\) 9.89466i 0.930811i 0.885098 + 0.465406i \(0.154091\pi\)
−0.885098 + 0.465406i \(0.845909\pi\)
\(114\) 0 0
\(115\) −16.8170 + 9.58483i −1.56819 + 0.893790i
\(116\) 0.380419 0.0353210
\(117\) 0 0
\(118\) 16.9004i 1.55581i
\(119\) 0.0860668 0.00788973
\(120\) 0 0
\(121\) 10.4783 0.952569
\(122\) 9.89840i 0.896159i
\(123\) 0 0
\(124\) −1.24375 −0.111692
\(125\) 11.1788 + 0.186778i 0.999860 + 0.0167059i
\(126\) 0 0
\(127\) 12.2226i 1.08458i 0.840191 + 0.542291i \(0.182443\pi\)
−0.840191 + 0.542291i \(0.817557\pi\)
\(128\) 7.06940i 0.624852i
\(129\) 0 0
\(130\) −7.08352 12.4283i −0.621266 1.09004i
\(131\) 12.0464 1.05250 0.526250 0.850330i \(-0.323598\pi\)
0.526250 + 0.850330i \(0.323598\pi\)
\(132\) 0 0
\(133\) 0.745535i 0.0646461i
\(134\) 14.3305 1.23796
\(135\) 0 0
\(136\) 1.01922 0.0873972
\(137\) 5.47512i 0.467771i −0.972264 0.233886i \(-0.924856\pi\)
0.972264 0.233886i \(-0.0751441\pi\)
\(138\) 0 0
\(139\) −2.71403 −0.230201 −0.115100 0.993354i \(-0.536719\pi\)
−0.115100 + 0.993354i \(0.536719\pi\)
\(140\) 0.189055 0.107752i 0.0159781 0.00910668i
\(141\) 0 0
\(142\) 17.7745i 1.49161i
\(143\) 23.2975i 1.94823i
\(144\) 0 0
\(145\) −1.10723 1.94269i −0.0919507 0.161331i
\(146\) 10.7719 0.891485
\(147\) 0 0
\(148\) 1.47204i 0.121001i
\(149\) −3.70395 −0.303439 −0.151720 0.988424i \(-0.548481\pi\)
−0.151720 + 0.988424i \(0.548481\pi\)
\(150\) 0 0
\(151\) 15.3481 1.24901 0.624505 0.781021i \(-0.285301\pi\)
0.624505 + 0.781021i \(0.285301\pi\)
\(152\) 8.82875i 0.716107i
\(153\) 0 0
\(154\) 1.50877 0.121580
\(155\) 3.62002 + 6.35148i 0.290767 + 0.510163i
\(156\) 0 0
\(157\) 21.8231i 1.74168i 0.491571 + 0.870838i \(0.336423\pi\)
−0.491571 + 0.870838i \(0.663577\pi\)
\(158\) 19.4842i 1.55008i
\(159\) 0 0
\(160\) 4.11985 2.34811i 0.325703 0.185634i
\(161\) 2.21447 0.174524
\(162\) 0 0
\(163\) 3.08019i 0.241259i 0.992698 + 0.120630i \(0.0384914\pi\)
−0.992698 + 0.120630i \(0.961509\pi\)
\(164\) −2.17466 −0.169812
\(165\) 0 0
\(166\) 9.67861 0.751206
\(167\) 8.55017i 0.661632i 0.943695 + 0.330816i \(0.107324\pi\)
−0.943695 + 0.330816i \(0.892676\pi\)
\(168\) 0 0
\(169\) −12.2707 −0.943903
\(170\) −0.474081 0.831796i −0.0363603 0.0637958i
\(171\) 0 0
\(172\) 2.65768i 0.202646i
\(173\) 5.76340i 0.438183i −0.975704 0.219092i \(-0.929691\pi\)
0.975704 0.219092i \(-0.0703093\pi\)
\(174\) 0 0
\(175\) −1.10051 0.651831i −0.0831910 0.0492738i
\(176\) 14.3411 1.08100
\(177\) 0 0
\(178\) 16.5929i 1.24369i
\(179\) 14.1430 1.05710 0.528548 0.848903i \(-0.322736\pi\)
0.528548 + 0.848903i \(0.322736\pi\)
\(180\) 0 0
\(181\) −15.0322 −1.11734 −0.558668 0.829391i \(-0.688688\pi\)
−0.558668 + 0.829391i \(0.688688\pi\)
\(182\) 1.63657i 0.121310i
\(183\) 0 0
\(184\) 26.2241 1.93326
\(185\) −7.51731 + 4.28448i −0.552683 + 0.315001i
\(186\) 0 0
\(187\) 1.55924i 0.114023i
\(188\) 0.127989i 0.00933459i
\(189\) 0 0
\(190\) 7.20525 4.10662i 0.522724 0.297926i
\(191\) 12.2634 0.887346 0.443673 0.896189i \(-0.353675\pi\)
0.443673 + 0.896189i \(0.353675\pi\)
\(192\) 0 0
\(193\) 8.34203i 0.600472i −0.953865 0.300236i \(-0.902935\pi\)
0.953865 0.300236i \(-0.0970655\pi\)
\(194\) −14.6918 −1.05481
\(195\) 0 0
\(196\) 2.63804 0.188431
\(197\) 6.69263i 0.476830i 0.971163 + 0.238415i \(0.0766279\pi\)
−0.971163 + 0.238415i \(0.923372\pi\)
\(198\) 0 0
\(199\) −12.5583 −0.890235 −0.445117 0.895472i \(-0.646838\pi\)
−0.445117 + 0.895472i \(0.646838\pi\)
\(200\) −13.0325 7.71910i −0.921535 0.545822i
\(201\) 0 0
\(202\) 10.4334i 0.734092i
\(203\) 0.255813i 0.0179546i
\(204\) 0 0
\(205\) 6.32948 + 11.1054i 0.442070 + 0.775631i
\(206\) −2.20183 −0.153409
\(207\) 0 0
\(208\) 15.5558i 1.07860i
\(209\) −13.5065 −0.934267
\(210\) 0 0
\(211\) −13.6479 −0.939563 −0.469782 0.882783i \(-0.655667\pi\)
−0.469782 + 0.882783i \(0.655667\pi\)
\(212\) 2.28816i 0.157151i
\(213\) 0 0
\(214\) −6.85846 −0.468835
\(215\) −13.5720 + 7.73534i −0.925602 + 0.527546i
\(216\) 0 0
\(217\) 0.836364i 0.0567761i
\(218\) 18.1211i 1.22731i
\(219\) 0 0
\(220\) 1.95209 + 3.42503i 0.131610 + 0.230916i
\(221\) −1.69130 −0.113769
\(222\) 0 0
\(223\) 13.4721i 0.902158i −0.892484 0.451079i \(-0.851039\pi\)
0.892484 0.451079i \(-0.148961\pi\)
\(224\) −0.542503 −0.0362475
\(225\) 0 0
\(226\) −12.5922 −0.837622
\(227\) 1.54298i 0.102411i −0.998688 0.0512057i \(-0.983694\pi\)
0.998688 0.0512057i \(-0.0163064\pi\)
\(228\) 0 0
\(229\) −7.89526 −0.521734 −0.260867 0.965375i \(-0.584008\pi\)
−0.260867 + 0.965375i \(0.584008\pi\)
\(230\) −12.1979 21.4018i −0.804307 1.41119i
\(231\) 0 0
\(232\) 3.02939i 0.198889i
\(233\) 28.4782i 1.86567i −0.360301 0.932836i \(-0.617326\pi\)
0.360301 0.932836i \(-0.382674\pi\)
\(234\) 0 0
\(235\) −0.653605 + 0.372521i −0.0426365 + 0.0243006i
\(236\) −5.05194 −0.328853
\(237\) 0 0
\(238\) 0.109531i 0.00709984i
\(239\) 8.88508 0.574728 0.287364 0.957821i \(-0.407221\pi\)
0.287364 + 0.957821i \(0.407221\pi\)
\(240\) 0 0
\(241\) −29.8901 −1.92539 −0.962695 0.270589i \(-0.912782\pi\)
−0.962695 + 0.270589i \(0.912782\pi\)
\(242\) 13.3349i 0.857202i
\(243\) 0 0
\(244\) −2.95887 −0.189422
\(245\) −7.67817 13.4717i −0.490540 0.860674i
\(246\) 0 0
\(247\) 14.6505i 0.932192i
\(248\) 9.90437i 0.628928i
\(249\) 0 0
\(250\) −0.237699 + 14.2264i −0.0150334 + 0.899758i
\(251\) 15.9551 1.00708 0.503539 0.863973i \(-0.332031\pi\)
0.503539 + 0.863973i \(0.332031\pi\)
\(252\) 0 0
\(253\) 40.1185i 2.52223i
\(254\) −15.5548 −0.975998
\(255\) 0 0
\(256\) −8.77878 −0.548674
\(257\) 17.1593i 1.07037i −0.844735 0.535185i \(-0.820242\pi\)
0.844735 0.535185i \(-0.179758\pi\)
\(258\) 0 0
\(259\) 0.989880 0.0615081
\(260\) −3.71513 + 2.11743i −0.230403 + 0.131318i
\(261\) 0 0
\(262\) 15.3306i 0.947128i
\(263\) 1.72781i 0.106542i −0.998580 0.0532708i \(-0.983035\pi\)
0.998580 0.0532708i \(-0.0169647\pi\)
\(264\) 0 0
\(265\) −11.6850 + 6.65983i −0.717801 + 0.409110i
\(266\) −0.948789 −0.0581740
\(267\) 0 0
\(268\) 4.28372i 0.261670i
\(269\) −4.97931 −0.303594 −0.151797 0.988412i \(-0.548506\pi\)
−0.151797 + 0.988412i \(0.548506\pi\)
\(270\) 0 0
\(271\) 13.0881 0.795047 0.397524 0.917592i \(-0.369870\pi\)
0.397524 + 0.917592i \(0.369870\pi\)
\(272\) 1.04111i 0.0631263i
\(273\) 0 0
\(274\) 6.96779 0.420939
\(275\) 11.8089 19.9375i 0.712106 1.20228i
\(276\) 0 0
\(277\) 20.8147i 1.25063i 0.780371 + 0.625316i \(0.215030\pi\)
−0.780371 + 0.625316i \(0.784970\pi\)
\(278\) 3.45394i 0.207154i
\(279\) 0 0
\(280\) 0.858059 + 1.50550i 0.0512788 + 0.0899709i
\(281\) −3.93338 −0.234646 −0.117323 0.993094i \(-0.537431\pi\)
−0.117323 + 0.993094i \(0.537431\pi\)
\(282\) 0 0
\(283\) 5.47745i 0.325600i 0.986659 + 0.162800i \(0.0520526\pi\)
−0.986659 + 0.162800i \(0.947947\pi\)
\(284\) 5.31324 0.315282
\(285\) 0 0
\(286\) −29.6490 −1.75318
\(287\) 1.46235i 0.0863201i
\(288\) 0 0
\(289\) 16.8868 0.993342
\(290\) 2.47232 1.40909i 0.145180 0.0827449i
\(291\) 0 0
\(292\) 3.21996i 0.188434i
\(293\) 11.9974i 0.700893i 0.936583 + 0.350446i \(0.113970\pi\)
−0.936583 + 0.350446i \(0.886030\pi\)
\(294\) 0 0
\(295\) 14.7040 + 25.7988i 0.856099 + 1.50206i
\(296\) 11.7223 0.681347
\(297\) 0 0
\(298\) 4.71375i 0.273060i
\(299\) −43.5166 −2.51663
\(300\) 0 0
\(301\) 1.78716 0.103010
\(302\) 19.5324i 1.12396i
\(303\) 0 0
\(304\) −9.01836 −0.517238
\(305\) 8.61197 + 15.1101i 0.493120 + 0.865200i
\(306\) 0 0
\(307\) 26.4807i 1.51133i −0.654957 0.755666i \(-0.727313\pi\)
0.654957 0.755666i \(-0.272687\pi\)
\(308\) 0.451009i 0.0256986i
\(309\) 0 0
\(310\) −8.08307 + 4.60694i −0.459088 + 0.261656i
\(311\) 1.71530 0.0972660 0.0486330 0.998817i \(-0.484514\pi\)
0.0486330 + 0.998817i \(0.484514\pi\)
\(312\) 0 0
\(313\) 33.3782i 1.88665i 0.331871 + 0.943325i \(0.392320\pi\)
−0.331871 + 0.943325i \(0.607680\pi\)
\(314\) −27.7727 −1.56731
\(315\) 0 0
\(316\) 5.82429 0.327642
\(317\) 17.9712i 1.00936i −0.863305 0.504682i \(-0.831610\pi\)
0.863305 0.504682i \(-0.168390\pi\)
\(318\) 0 0
\(319\) −4.63446 −0.259480
\(320\) 9.84080 + 17.2661i 0.550118 + 0.965205i
\(321\) 0 0
\(322\) 2.81819i 0.157052i
\(323\) 0.980521i 0.0545577i
\(324\) 0 0
\(325\) 21.6262 + 12.8092i 1.19961 + 0.710524i
\(326\) −3.91994 −0.217105
\(327\) 0 0
\(328\) 17.3175i 0.956196i
\(329\) 0.0860668 0.00474501
\(330\) 0 0
\(331\) 6.49393 0.356939 0.178469 0.983945i \(-0.442885\pi\)
0.178469 + 0.983945i \(0.442885\pi\)
\(332\) 2.89317i 0.158783i
\(333\) 0 0
\(334\) −10.8812 −0.595392
\(335\) −21.8757 + 12.4680i −1.19520 + 0.681202i
\(336\) 0 0
\(337\) 6.36156i 0.346536i 0.984875 + 0.173268i \(0.0554327\pi\)
−0.984875 + 0.173268i \(0.944567\pi\)
\(338\) 15.6161i 0.849403i
\(339\) 0 0
\(340\) −0.248643 + 0.141714i −0.0134846 + 0.00768552i
\(341\) 15.1521 0.820530
\(342\) 0 0
\(343\) 3.56465i 0.192473i
\(344\) 21.1639 1.14108
\(345\) 0 0
\(346\) 7.33466 0.394314
\(347\) 21.6036i 1.15974i 0.814708 + 0.579871i \(0.196897\pi\)
−0.814708 + 0.579871i \(0.803103\pi\)
\(348\) 0 0
\(349\) 4.12603 0.220861 0.110431 0.993884i \(-0.464777\pi\)
0.110431 + 0.993884i \(0.464777\pi\)
\(350\) 0.829538 1.40054i 0.0443407 0.0748622i
\(351\) 0 0
\(352\) 9.82830i 0.523850i
\(353\) 7.07724i 0.376684i −0.982104 0.188342i \(-0.939689\pi\)
0.982104 0.188342i \(-0.0603113\pi\)
\(354\) 0 0
\(355\) −15.4645 27.1331i −0.820771 1.44008i
\(356\) −4.96002 −0.262881
\(357\) 0 0
\(358\) 17.9988i 0.951264i
\(359\) −14.4359 −0.761900 −0.380950 0.924596i \(-0.624403\pi\)
−0.380950 + 0.924596i \(0.624403\pi\)
\(360\) 0 0
\(361\) −10.5064 −0.552971
\(362\) 19.1304i 1.00547i
\(363\) 0 0
\(364\) 0.489209 0.0256415
\(365\) −16.4434 + 9.37190i −0.860688 + 0.490548i
\(366\) 0 0
\(367\) 15.4145i 0.804630i 0.915501 + 0.402315i \(0.131794\pi\)
−0.915501 + 0.402315i \(0.868206\pi\)
\(368\) 26.7873i 1.39638i
\(369\) 0 0
\(370\) −5.45255 9.56673i −0.283464 0.497351i
\(371\) 1.53868 0.0798841
\(372\) 0 0
\(373\) 29.8574i 1.54596i −0.634432 0.772979i \(-0.718766\pi\)
0.634432 0.772979i \(-0.281234\pi\)
\(374\) −1.98433 −0.102607
\(375\) 0 0
\(376\) 1.01922 0.0525621
\(377\) 5.02700i 0.258904i
\(378\) 0 0
\(379\) −37.3451 −1.91829 −0.959144 0.282920i \(-0.908697\pi\)
−0.959144 + 0.282920i \(0.908697\pi\)
\(380\) −1.22757 2.15382i −0.0629729 0.110489i
\(381\) 0 0
\(382\) 15.6067i 0.798508i
\(383\) 25.8159i 1.31913i 0.751647 + 0.659566i \(0.229260\pi\)
−0.751647 + 0.659566i \(0.770740\pi\)
\(384\) 0 0
\(385\) −2.30317 + 1.31269i −0.117380 + 0.0669008i
\(386\) 10.6163 0.540355
\(387\) 0 0
\(388\) 4.39173i 0.222956i
\(389\) 20.6935 1.04920 0.524600 0.851349i \(-0.324215\pi\)
0.524600 + 0.851349i \(0.324215\pi\)
\(390\) 0 0
\(391\) −2.91245 −0.147289
\(392\) 21.0075i 1.06104i
\(393\) 0 0
\(394\) −8.51723 −0.429092
\(395\) −16.9520 29.7430i −0.852946 1.49653i
\(396\) 0 0
\(397\) 2.38045i 0.119471i 0.998214 + 0.0597357i \(0.0190258\pi\)
−0.998214 + 0.0597357i \(0.980974\pi\)
\(398\) 15.9820i 0.801108i
\(399\) 0 0
\(400\) 7.88487 13.3123i 0.394243 0.665617i
\(401\) 32.0936 1.60268 0.801339 0.598211i \(-0.204121\pi\)
0.801339 + 0.598211i \(0.204121\pi\)
\(402\) 0 0
\(403\) 16.4354i 0.818707i
\(404\) 3.11879 0.155166
\(405\) 0 0
\(406\) −0.325555 −0.0161570
\(407\) 17.9332i 0.888918i
\(408\) 0 0
\(409\) 31.5011 1.55763 0.778815 0.627253i \(-0.215821\pi\)
0.778815 + 0.627253i \(0.215821\pi\)
\(410\) −14.1330 + 8.05507i −0.697978 + 0.397812i
\(411\) 0 0
\(412\) 0.658179i 0.0324261i
\(413\) 3.39719i 0.167165i
\(414\) 0 0
\(415\) −14.7746 + 8.42074i −0.725255 + 0.413358i
\(416\) 10.6608 0.522686
\(417\) 0 0
\(418\) 17.1888i 0.840732i
\(419\) −3.15481 −0.154123 −0.0770613 0.997026i \(-0.524554\pi\)
−0.0770613 + 0.997026i \(0.524554\pi\)
\(420\) 0 0
\(421\) −11.9631 −0.583048 −0.291524 0.956564i \(-0.594162\pi\)
−0.291524 + 0.956564i \(0.594162\pi\)
\(422\) 17.3687i 0.845497i
\(423\) 0 0
\(424\) 18.2213 0.884904
\(425\) 1.44739 + 0.857282i 0.0702085 + 0.0415843i
\(426\) 0 0
\(427\) 1.98970i 0.0962882i
\(428\) 2.05016i 0.0990981i
\(429\) 0 0
\(430\) −9.84420 17.2721i −0.474730 0.832934i
\(431\) 31.2658 1.50602 0.753011 0.658008i \(-0.228601\pi\)
0.753011 + 0.658008i \(0.228601\pi\)
\(432\) 0 0
\(433\) 30.6459i 1.47275i 0.676575 + 0.736373i \(0.263463\pi\)
−0.676575 + 0.736373i \(0.736537\pi\)
\(434\) 1.06438 0.0510919
\(435\) 0 0
\(436\) −5.41682 −0.259419
\(437\) 25.2284i 1.20684i
\(438\) 0 0
\(439\) −0.717425 −0.0342408 −0.0171204 0.999853i \(-0.505450\pi\)
−0.0171204 + 0.999853i \(0.505450\pi\)
\(440\) −27.2745 + 15.5451i −1.30026 + 0.741083i
\(441\) 0 0
\(442\) 2.15240i 0.102379i
\(443\) 27.0649i 1.28589i 0.765911 + 0.642947i \(0.222288\pi\)
−0.765911 + 0.642947i \(0.777712\pi\)
\(444\) 0 0
\(445\) 14.4365 + 25.3294i 0.684353 + 1.20073i
\(446\) 17.1450 0.811838
\(447\) 0 0
\(448\) 2.27361i 0.107418i
\(449\) −30.3663 −1.43307 −0.716536 0.697550i \(-0.754273\pi\)
−0.716536 + 0.697550i \(0.754273\pi\)
\(450\) 0 0
\(451\) 26.4929 1.24750
\(452\) 3.76411i 0.177049i
\(453\) 0 0
\(454\) 1.96364 0.0921584
\(455\) −1.42387 2.49825i −0.0667522 0.117120i
\(456\) 0 0
\(457\) 2.43167i 0.113749i −0.998381 0.0568743i \(-0.981887\pi\)
0.998381 0.0568743i \(-0.0181134\pi\)
\(458\) 10.0477i 0.469500i
\(459\) 0 0
\(460\) −6.39750 + 3.64625i −0.298285 + 0.170007i
\(461\) −8.28439 −0.385842 −0.192921 0.981214i \(-0.561796\pi\)
−0.192921 + 0.981214i \(0.561796\pi\)
\(462\) 0 0
\(463\) 7.05654i 0.327945i 0.986465 + 0.163973i \(0.0524309\pi\)
−0.986465 + 0.163973i \(0.947569\pi\)
\(464\) −3.09444 −0.143656
\(465\) 0 0
\(466\) 36.2422 1.67889
\(467\) 6.07165i 0.280963i −0.990083 0.140481i \(-0.955135\pi\)
0.990083 0.140481i \(-0.0448650\pi\)
\(468\) 0 0
\(469\) 2.88060 0.133014
\(470\) −0.474081 0.831796i −0.0218677 0.0383679i
\(471\) 0 0
\(472\) 40.2301i 1.85174i
\(473\) 32.3773i 1.48871i
\(474\) 0 0
\(475\) −7.42603 + 12.5377i −0.340729 + 0.575267i
\(476\) 0.0327414 0.00150070
\(477\) 0 0
\(478\) 11.3074i 0.517188i
\(479\) −29.7675 −1.36011 −0.680055 0.733161i \(-0.738044\pi\)
−0.680055 + 0.733161i \(0.738044\pi\)
\(480\) 0 0
\(481\) −19.4522 −0.886943
\(482\) 38.0390i 1.73263i
\(483\) 0 0
\(484\) 3.98613 0.181188
\(485\) 22.4273 12.7824i 1.01837 0.580419i
\(486\) 0 0
\(487\) 41.9050i 1.89890i −0.313923 0.949448i \(-0.601643\pi\)
0.313923 0.949448i \(-0.398357\pi\)
\(488\) 23.5623i 1.06662i
\(489\) 0 0
\(490\) 17.1444 9.77145i 0.774507 0.441429i
\(491\) 7.22435 0.326030 0.163015 0.986624i \(-0.447878\pi\)
0.163015 + 0.986624i \(0.447878\pi\)
\(492\) 0 0
\(493\) 0.336444i 0.0151527i
\(494\) 18.6447 0.838864
\(495\) 0 0
\(496\) 10.1171 0.454270
\(497\) 3.57290i 0.160266i
\(498\) 0 0
\(499\) 15.5607 0.696593 0.348296 0.937384i \(-0.386760\pi\)
0.348296 + 0.937384i \(0.386760\pi\)
\(500\) 4.25262 + 0.0710538i 0.190183 + 0.00317762i
\(501\) 0 0
\(502\) 20.3049i 0.906253i
\(503\) 38.5096i 1.71706i −0.512765 0.858529i \(-0.671379\pi\)
0.512765 0.858529i \(-0.328621\pi\)
\(504\) 0 0
\(505\) −9.07745 15.9268i −0.403941 0.708732i
\(506\) −51.0560 −2.26971
\(507\) 0 0
\(508\) 4.64971i 0.206298i
\(509\) 31.6646 1.40351 0.701754 0.712419i \(-0.252400\pi\)
0.701754 + 0.712419i \(0.252400\pi\)
\(510\) 0 0
\(511\) 2.16527 0.0957860
\(512\) 25.3109i 1.11860i
\(513\) 0 0
\(514\) 21.8375 0.963209
\(515\) 3.36113 1.91567i 0.148109 0.0844145i
\(516\) 0 0
\(517\) 1.55924i 0.0685751i
\(518\) 1.25975i 0.0553502i
\(519\) 0 0
\(520\) −16.8617 29.5847i −0.739437 1.29737i
\(521\) −20.2023 −0.885079 −0.442539 0.896749i \(-0.645922\pi\)
−0.442539 + 0.896749i \(0.645922\pi\)
\(522\) 0 0
\(523\) 12.1052i 0.529325i −0.964341 0.264663i \(-0.914739\pi\)
0.964341 0.264663i \(-0.0852606\pi\)
\(524\) 4.58268 0.200195
\(525\) 0 0
\(526\) 2.19886 0.0958751
\(527\) 1.09998i 0.0479158i
\(528\) 0 0
\(529\) −51.9361 −2.25809
\(530\) −8.47548 14.8706i −0.368151 0.645938i
\(531\) 0 0
\(532\) 0.283615i 0.0122963i
\(533\) 28.7368i 1.24473i
\(534\) 0 0
\(535\) 10.4696 5.96711i 0.452638 0.257981i
\(536\) 34.1125 1.47344
\(537\) 0 0
\(538\) 6.33681i 0.273199i
\(539\) −32.1380 −1.38428
\(540\) 0 0
\(541\) −14.4279 −0.620304 −0.310152 0.950687i \(-0.600380\pi\)
−0.310152 + 0.950687i \(0.600380\pi\)
\(542\) 16.6563i 0.715450i
\(543\) 0 0
\(544\) 0.713495 0.0305909
\(545\) 15.7660 + 27.6621i 0.675341 + 1.18492i
\(546\) 0 0
\(547\) 41.9178i 1.79228i −0.443774 0.896139i \(-0.646361\pi\)
0.443774 0.896139i \(-0.353639\pi\)
\(548\) 2.08284i 0.0889744i
\(549\) 0 0
\(550\) 25.3731 + 15.0284i 1.08191 + 0.640813i
\(551\) 2.91437 0.124156
\(552\) 0 0
\(553\) 3.91656i 0.166549i
\(554\) −26.4893 −1.12542
\(555\) 0 0
\(556\) −1.03247 −0.0437863
\(557\) 3.60324i 0.152674i 0.997082 + 0.0763372i \(0.0243225\pi\)
−0.997082 + 0.0763372i \(0.975677\pi\)
\(558\) 0 0
\(559\) −35.1196 −1.48540
\(560\) −1.53783 + 0.876486i −0.0649853 + 0.0370383i
\(561\) 0 0
\(562\) 5.00573i 0.211154i
\(563\) 15.5829i 0.656741i −0.944549 0.328370i \(-0.893501\pi\)
0.944549 0.328370i \(-0.106499\pi\)
\(564\) 0 0
\(565\) 19.2222 10.9557i 0.808685 0.460909i
\(566\) −6.97075 −0.293003
\(567\) 0 0
\(568\) 42.3109i 1.77532i
\(569\) 36.9718 1.54994 0.774970 0.631998i \(-0.217765\pi\)
0.774970 + 0.631998i \(0.217765\pi\)
\(570\) 0 0
\(571\) 29.9692 1.25417 0.627086 0.778950i \(-0.284248\pi\)
0.627086 + 0.778950i \(0.284248\pi\)
\(572\) 8.86279i 0.370572i
\(573\) 0 0
\(574\) 1.86103 0.0776780
\(575\) 37.2407 + 22.0575i 1.55304 + 0.919863i
\(576\) 0 0
\(577\) 7.12327i 0.296546i −0.988946 0.148273i \(-0.952629\pi\)
0.988946 0.148273i \(-0.0473714\pi\)
\(578\) 21.4906i 0.893892i
\(579\) 0 0
\(580\) −0.421212 0.739035i −0.0174899 0.0306867i
\(581\) 1.94552 0.0807136
\(582\) 0 0
\(583\) 27.8756i 1.15449i
\(584\) 25.6415 1.06105
\(585\) 0 0
\(586\) −15.2682 −0.630722
\(587\) 7.87237i 0.324928i −0.986715 0.162464i \(-0.948056\pi\)
0.986715 0.162464i \(-0.0519441\pi\)
\(588\) 0 0
\(589\) −9.52833 −0.392608
\(590\) −32.8322 + 18.7127i −1.35168 + 0.770390i
\(591\) 0 0
\(592\) 11.9741i 0.492131i
\(593\) 17.9188i 0.735836i 0.929858 + 0.367918i \(0.119929\pi\)
−0.929858 + 0.367918i \(0.880071\pi\)
\(594\) 0 0
\(595\) −0.0952959 0.167201i −0.00390675 0.00685457i
\(596\) −1.40905 −0.0577170
\(597\) 0 0
\(598\) 55.3804i 2.26467i
\(599\) 20.1934 0.825078 0.412539 0.910940i \(-0.364642\pi\)
0.412539 + 0.910940i \(0.364642\pi\)
\(600\) 0 0
\(601\) 23.2527 0.948496 0.474248 0.880391i \(-0.342720\pi\)
0.474248 + 0.880391i \(0.342720\pi\)
\(602\) 2.27439i 0.0926973i
\(603\) 0 0
\(604\) 5.83870 0.237573
\(605\) −11.6019 20.3560i −0.471683 0.827589i
\(606\) 0 0
\(607\) 17.6177i 0.715082i −0.933897 0.357541i \(-0.883615\pi\)
0.933897 0.357541i \(-0.116385\pi\)
\(608\) 6.18050i 0.250652i
\(609\) 0 0
\(610\) −19.2295 + 10.9598i −0.778580 + 0.443750i
\(611\) −1.69130 −0.0684228
\(612\) 0 0
\(613\) 13.4161i 0.541873i −0.962597 0.270937i \(-0.912667\pi\)
0.962597 0.270937i \(-0.0873334\pi\)
\(614\) 33.7001 1.36002
\(615\) 0 0
\(616\) 3.59151 0.144706
\(617\) 7.99166i 0.321732i 0.986976 + 0.160866i \(0.0514287\pi\)
−0.986976 + 0.160866i \(0.948571\pi\)
\(618\) 0 0
\(619\) 15.0770 0.605996 0.302998 0.952991i \(-0.402012\pi\)
0.302998 + 0.952991i \(0.402012\pi\)
\(620\) 1.37712 + 2.41622i 0.0553066 + 0.0970378i
\(621\) 0 0
\(622\) 2.18294i 0.0875281i
\(623\) 3.33538i 0.133629i
\(624\) 0 0
\(625\) −12.0147 21.9237i −0.480586 0.876947i
\(626\) −42.4781 −1.69776
\(627\) 0 0
\(628\) 8.30193i 0.331283i
\(629\) −1.30188 −0.0519094
\(630\) 0 0
\(631\) 10.5154 0.418611 0.209306 0.977850i \(-0.432880\pi\)
0.209306 + 0.977850i \(0.432880\pi\)
\(632\) 46.3805i 1.84492i
\(633\) 0 0
\(634\) 22.8707 0.908311
\(635\) 23.7447 13.5333i 0.942281 0.537052i
\(636\) 0 0
\(637\) 34.8600i 1.38121i
\(638\) 5.89795i 0.233502i
\(639\) 0 0
\(640\) −13.7336 + 7.82747i −0.542870 + 0.309408i
\(641\) 27.2479 1.07623 0.538113 0.842873i \(-0.319137\pi\)
0.538113 + 0.842873i \(0.319137\pi\)
\(642\) 0 0
\(643\) 36.3023i 1.43162i −0.698293 0.715812i \(-0.746057\pi\)
0.698293 0.715812i \(-0.253943\pi\)
\(644\) 0.842424 0.0331962
\(645\) 0 0
\(646\) 1.24784 0.0490955
\(647\) 8.77892i 0.345135i 0.984998 + 0.172567i \(0.0552063\pi\)
−0.984998 + 0.172567i \(0.944794\pi\)
\(648\) 0 0
\(649\) 61.5454 2.41587
\(650\) −16.3013 + 27.5221i −0.639389 + 1.07951i
\(651\) 0 0
\(652\) 1.17176i 0.0458898i
\(653\) 44.1466i 1.72759i −0.503845 0.863794i \(-0.668082\pi\)
0.503845 0.863794i \(-0.331918\pi\)
\(654\) 0 0
\(655\) −13.3382 23.4024i −0.521166 0.914408i
\(656\) 17.6894 0.690653
\(657\) 0 0
\(658\) 0.109531i 0.00426996i
\(659\) 22.7653 0.886810 0.443405 0.896321i \(-0.353770\pi\)
0.443405 + 0.896321i \(0.353770\pi\)
\(660\) 0 0
\(661\) 29.3474 1.14148 0.570740 0.821131i \(-0.306656\pi\)
0.570740 + 0.821131i \(0.306656\pi\)
\(662\) 8.26435i 0.321203i
\(663\) 0 0
\(664\) 23.0391 0.894092
\(665\) 1.44834 0.825481i 0.0561643 0.0320108i
\(666\) 0 0
\(667\) 8.65656i 0.335183i
\(668\) 3.25264i 0.125849i
\(669\) 0 0
\(670\) −15.8672 27.8397i −0.613002 1.07554i
\(671\) 36.0465 1.39156
\(672\) 0 0
\(673\) 42.2267i 1.62772i 0.581060 + 0.813860i \(0.302638\pi\)
−0.581060 + 0.813860i \(0.697362\pi\)
\(674\) −8.09590 −0.311842
\(675\) 0 0
\(676\) −4.66802 −0.179539
\(677\) 19.9803i 0.767907i −0.923353 0.383953i \(-0.874562\pi\)
0.923353 0.383953i \(-0.125438\pi\)
\(678\) 0 0
\(679\) −2.95323 −0.113334
\(680\) −1.12851 1.98002i −0.0432764 0.0759304i
\(681\) 0 0
\(682\) 19.2829i 0.738381i
\(683\) 23.0491i 0.881950i −0.897519 0.440975i \(-0.854633\pi\)
0.897519 0.440975i \(-0.145367\pi\)
\(684\) 0 0
\(685\) −10.6364 + 6.06223i −0.406398 + 0.231626i
\(686\) −4.53647 −0.173203
\(687\) 0 0
\(688\) 21.6184i 0.824193i
\(689\) −30.2366 −1.15192
\(690\) 0 0
\(691\) 4.01208 0.152627 0.0763134 0.997084i \(-0.475685\pi\)
0.0763134 + 0.997084i \(0.475685\pi\)
\(692\) 2.19250i 0.0833465i
\(693\) 0 0
\(694\) −27.4933 −1.04363
\(695\) 3.00506 + 5.27250i 0.113988 + 0.199997i
\(696\) 0 0
\(697\) 1.92328i 0.0728493i
\(698\) 5.25090i 0.198750i
\(699\) 0 0
\(700\) −0.418656 0.247969i −0.0158237 0.00937234i
\(701\) −4.49887 −0.169920 −0.0849599 0.996384i \(-0.527076\pi\)
−0.0849599 + 0.996384i \(0.527076\pi\)
\(702\) 0 0
\(703\) 11.2773i 0.425330i
\(704\) 41.1900 1.55240
\(705\) 0 0
\(706\) 9.00670 0.338971
\(707\) 2.09724i 0.0788748i
\(708\) 0 0
\(709\) −36.7777 −1.38121 −0.690607 0.723230i \(-0.742657\pi\)
−0.690607 + 0.723230i \(0.742657\pi\)
\(710\) 34.5304 19.6806i 1.29590 0.738598i
\(711\) 0 0
\(712\) 39.4981i 1.48025i
\(713\) 28.3020i 1.05992i
\(714\) 0 0
\(715\) 45.2597 25.7957i 1.69262 0.964705i
\(716\) 5.38026 0.201070
\(717\) 0 0
\(718\) 18.3716i 0.685621i
\(719\) −25.4279 −0.948300 −0.474150 0.880444i \(-0.657245\pi\)
−0.474150 + 0.880444i \(0.657245\pi\)
\(720\) 0 0
\(721\) −0.442594 −0.0164831
\(722\) 13.3708i 0.497609i
\(723\) 0 0
\(724\) −5.71854 −0.212528
\(725\) −2.54807 + 4.30201i −0.0946330 + 0.159773i
\(726\) 0 0
\(727\) 11.6265i 0.431205i 0.976481 + 0.215602i \(0.0691715\pi\)
−0.976481 + 0.215602i \(0.930829\pi\)
\(728\) 3.89571i 0.144385i
\(729\) 0 0
\(730\) −11.9269 20.9263i −0.441436 0.774519i
\(731\) −2.35046 −0.0869349
\(732\) 0 0
\(733\) 25.6926i 0.948978i −0.880261 0.474489i \(-0.842633\pi\)
0.880261 0.474489i \(-0.157367\pi\)
\(734\) −19.6169 −0.724073
\(735\) 0 0
\(736\) 18.3580 0.676683
\(737\) 52.1866i 1.92232i
\(738\) 0 0
\(739\) 33.3612 1.22721 0.613606 0.789612i \(-0.289718\pi\)
0.613606 + 0.789612i \(0.289718\pi\)
\(740\) −2.85972 + 1.62990i −0.105126 + 0.0599162i
\(741\) 0 0
\(742\) 1.95816i 0.0718864i
\(743\) 25.6430i 0.940750i −0.882467 0.470375i \(-0.844119\pi\)
0.882467 0.470375i \(-0.155881\pi\)
\(744\) 0 0
\(745\) 4.10113 + 7.19562i 0.150254 + 0.263627i
\(746\) 37.9973 1.39118
\(747\) 0 0
\(748\) 0.593162i 0.0216882i
\(749\) −1.37863 −0.0503741
\(750\) 0 0
\(751\) 46.6411 1.70196 0.850979 0.525200i \(-0.176009\pi\)
0.850979 + 0.525200i \(0.176009\pi\)
\(752\) 1.04111i 0.0379652i
\(753\) 0 0
\(754\) 6.39750 0.232983
\(755\) −16.9939 29.8165i −0.618472 1.08514i
\(756\) 0 0
\(757\) 2.69262i 0.0978650i 0.998802 + 0.0489325i \(0.0155819\pi\)
−0.998802 + 0.0489325i \(0.984418\pi\)
\(758\) 47.5264i 1.72624i
\(759\) 0 0
\(760\) 17.1515 9.77549i 0.622151 0.354594i
\(761\) 0.471501 0.0170919 0.00854595 0.999963i \(-0.497280\pi\)
0.00854595 + 0.999963i \(0.497280\pi\)
\(762\) 0 0
\(763\) 3.64255i 0.131869i
\(764\) 4.66521 0.168781
\(765\) 0 0
\(766\) −32.8540 −1.18706
\(767\) 66.7583i 2.41050i
\(768\) 0 0
\(769\) −9.04380 −0.326128 −0.163064 0.986616i \(-0.552138\pi\)
−0.163064 + 0.986616i \(0.552138\pi\)
\(770\) −1.67056 2.93108i −0.0602029 0.105629i
\(771\) 0 0
\(772\) 3.17346i 0.114215i
\(773\) 9.91726i 0.356699i 0.983967 + 0.178350i \(0.0570758\pi\)
−0.983967 + 0.178350i \(0.942924\pi\)
\(774\) 0 0
\(775\) 8.33074 14.0651i 0.299249 0.505234i
\(776\) −34.9726 −1.25544
\(777\) 0 0
\(778\) 26.3351i 0.944159i
\(779\) −16.6600 −0.596905
\(780\) 0 0
\(781\) −64.7287 −2.31617
\(782\) 3.70646i 0.132543i
\(783\) 0 0
\(784\) −21.4586 −0.766379
\(785\) 42.3955 24.1633i 1.51316 0.862425i
\(786\) 0 0
\(787\) 8.53609i 0.304279i 0.988359 + 0.152139i \(0.0486162\pi\)
−0.988359 + 0.152139i \(0.951384\pi\)
\(788\) 2.54600i 0.0906976i
\(789\) 0 0
\(790\) 37.8517 21.5735i 1.34670 0.767552i
\(791\) −2.53119 −0.0899986
\(792\) 0 0
\(793\) 39.0996i 1.38847i
\(794\) −3.02943 −0.107510
\(795\) 0 0
\(796\) −4.77741 −0.169331
\(797\) 12.5563i 0.444768i −0.974959 0.222384i \(-0.928616\pi\)
0.974959 0.222384i \(-0.0713838\pi\)
\(798\) 0 0
\(799\) −0.113194 −0.00400453
\(800\) −9.12327 5.40369i −0.322556 0.191049i
\(801\) 0 0
\(802\) 40.8432i 1.44222i
\(803\) 39.2273i 1.38430i
\(804\) 0 0
\(805\) −2.45193 4.30201i −0.0864191 0.151626i
\(806\) −20.9162 −0.736741
\(807\) 0 0
\(808\) 24.8359i 0.873723i
\(809\) 16.4082 0.576883 0.288441 0.957498i \(-0.406863\pi\)
0.288441 + 0.957498i \(0.406863\pi\)
\(810\) 0 0
\(811\) 29.4688 1.03479 0.517394 0.855747i \(-0.326902\pi\)
0.517394 + 0.855747i \(0.326902\pi\)
\(812\) 0.0973162i 0.00341513i
\(813\) 0 0
\(814\) −22.8223 −0.799923
\(815\) 5.98385 3.41049i 0.209605 0.119464i
\(816\) 0 0
\(817\) 20.3603i 0.712318i
\(818\) 40.0892i 1.40169i
\(819\) 0 0
\(820\) 2.40785 + 4.22468i 0.0840859 + 0.147532i
\(821\) 28.5209 0.995385 0.497693 0.867353i \(-0.334181\pi\)
0.497693 + 0.867353i \(0.334181\pi\)
\(822\) 0 0
\(823\) 16.7546i 0.584028i −0.956414 0.292014i \(-0.905675\pi\)
0.956414 0.292014i \(-0.0943253\pi\)
\(824\) −5.24127 −0.182588
\(825\) 0 0
\(826\) 4.32336 0.150429
\(827\) 3.38369i 0.117662i 0.998268 + 0.0588312i \(0.0187374\pi\)
−0.998268 + 0.0588312i \(0.981263\pi\)
\(828\) 0 0
\(829\) 14.4434 0.501639 0.250819 0.968034i \(-0.419300\pi\)
0.250819 + 0.968034i \(0.419300\pi\)
\(830\) −10.7165 18.8025i −0.371974 0.652645i
\(831\) 0 0
\(832\) 44.6787i 1.54896i
\(833\) 2.33309i 0.0808367i
\(834\) 0 0
\(835\) 16.6103 9.46702i 0.574823 0.327620i
\(836\) −5.13814 −0.177706
\(837\) 0 0
\(838\) 4.01490i 0.138692i
\(839\) 27.6642 0.955075 0.477538 0.878611i \(-0.341529\pi\)
0.477538 + 0.878611i \(0.341529\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 15.2246i 0.524675i
\(843\) 0 0
\(844\) −5.19193 −0.178714
\(845\) 13.5866 + 23.8382i 0.467392 + 0.820060i
\(846\) 0 0
\(847\) 2.68048i 0.0921024i
\(848\) 18.6126i 0.639159i
\(849\) 0 0
\(850\) −1.09100 + 1.84198i −0.0374210 + 0.0631795i
\(851\) −33.4969 −1.14826
\(852\) 0 0
\(853\) 4.26068i 0.145883i −0.997336 0.0729415i \(-0.976761\pi\)
0.997336 0.0729415i \(-0.0232386\pi\)
\(854\) 2.53214 0.0866481
\(855\) 0 0
\(856\) −16.3260 −0.558011
\(857\) 28.7104i 0.980729i −0.871518 0.490364i \(-0.836864\pi\)
0.871518 0.490364i \(-0.163136\pi\)
\(858\) 0 0
\(859\) 39.3734 1.34340 0.671701 0.740822i \(-0.265564\pi\)
0.671701 + 0.740822i \(0.265564\pi\)
\(860\) −5.16304 + 2.94267i −0.176058 + 0.100344i
\(861\) 0 0
\(862\) 39.7898i 1.35525i
\(863\) 22.1235i 0.753094i 0.926398 + 0.376547i \(0.122889\pi\)
−0.926398 + 0.376547i \(0.877111\pi\)
\(864\) 0 0
\(865\) −11.1965 + 6.38142i −0.380692 + 0.216975i
\(866\) −39.0008 −1.32530
\(867\) 0 0
\(868\) 0.318168i 0.0107993i
\(869\) −70.9546 −2.40697
\(870\) 0 0
\(871\) −56.6068 −1.91805
\(872\) 43.1357i 1.46076i
\(873\) 0 0
\(874\) 32.1064 1.08602
\(875\) −0.0477803 + 2.85968i −0.00161527 + 0.0966749i
\(876\) 0 0
\(877\) 32.3439i 1.09217i −0.837728 0.546087i \(-0.816117\pi\)
0.837728 0.546087i \(-0.183883\pi\)
\(878\) 0.913015i 0.0308128i
\(879\) 0 0
\(880\) −15.8789 27.8603i −0.535278 0.939169i
\(881\) −54.6109 −1.83989 −0.919944 0.392051i \(-0.871766\pi\)
−0.919944 + 0.392051i \(0.871766\pi\)
\(882\) 0 0
\(883\) 47.5511i 1.60022i −0.599852 0.800111i \(-0.704774\pi\)
0.599852 0.800111i \(-0.295226\pi\)
\(884\) −0.643403 −0.0216400
\(885\) 0 0
\(886\) −34.4436 −1.15715
\(887\) 10.4630i 0.351312i −0.984452 0.175656i \(-0.943795\pi\)
0.984452 0.175656i \(-0.0562047\pi\)
\(888\) 0 0
\(889\) −3.12671 −0.104866
\(890\) −32.2349 + 18.3722i −1.08052 + 0.615838i
\(891\) 0 0
\(892\) 5.12504i 0.171599i
\(893\) 0.980521i 0.0328119i
\(894\) 0 0
\(895\) −15.6596 27.4754i −0.523442 0.918402i
\(896\) 1.80845 0.0604160
\(897\) 0 0
\(898\) 38.6449i 1.28960i
\(899\) −3.26943 −0.109042
\(900\) 0 0
\(901\) −2.02365 −0.0674177
\(902\) 33.7156i 1.12261i
\(903\) 0 0
\(904\) −29.9747 −0.996945
\(905\) 16.6442 + 29.2029i 0.553271 + 0.970738i
\(906\) 0 0
\(907\) 10.7211i 0.355989i 0.984032 + 0.177994i \(0.0569609\pi\)
−0.984032 + 0.177994i \(0.943039\pi\)
\(908\) 0.586980i 0.0194796i
\(909\) 0 0
\(910\) 3.17934 1.81206i 0.105394 0.0600692i
\(911\) −3.57634 −0.118489 −0.0592447 0.998243i \(-0.518869\pi\)
−0.0592447 + 0.998243i \(0.518869\pi\)
\(912\) 0 0
\(913\) 35.2461i 1.16648i
\(914\) 3.09461 0.102361
\(915\) 0 0
\(916\) −3.00351 −0.0992386
\(917\) 3.08163i 0.101765i
\(918\) 0 0
\(919\) 22.2776 0.734870 0.367435 0.930049i \(-0.380236\pi\)
0.367435 + 0.930049i \(0.380236\pi\)
\(920\) −29.0362 50.9452i −0.957294 1.67961i
\(921\) 0 0
\(922\) 10.5429i 0.347213i
\(923\) 70.2112i 2.31103i
\(924\) 0 0
\(925\) 16.6468 + 9.85986i 0.547344 + 0.324190i
\(926\) −8.98035 −0.295113
\(927\) 0 0
\(928\) 2.12070i 0.0696153i
\(929\) 21.1779 0.694826 0.347413 0.937712i \(-0.387060\pi\)
0.347413 + 0.937712i \(0.387060\pi\)
\(930\) 0 0
\(931\) 20.2099 0.662352
\(932\) 10.8337i 0.354868i
\(933\) 0 0
\(934\) 7.72695 0.252834
\(935\) 3.02911 1.72644i 0.0990624 0.0564605i
\(936\) 0 0
\(937\) 34.4462i 1.12531i −0.826693 0.562654i \(-0.809780\pi\)
0.826693 0.562654i \(-0.190220\pi\)
\(938\) 3.66593i 0.119697i
\(939\) 0 0
\(940\) −0.248643 + 0.141714i −0.00810986 + 0.00462220i
\(941\) 0.296167 0.00965477 0.00482738 0.999988i \(-0.498463\pi\)
0.00482738 + 0.999988i \(0.498463\pi\)
\(942\) 0 0
\(943\) 49.4851i 1.61146i
\(944\) 41.0940 1.33750
\(945\) 0 0
\(946\) −41.2042 −1.33966
\(947\) 46.9467i 1.52556i −0.646656 0.762782i \(-0.723833\pi\)
0.646656 0.762782i \(-0.276167\pi\)
\(948\) 0 0
\(949\) −42.5498 −1.38123
\(950\) −15.9558 9.45057i −0.517674 0.306617i
\(951\) 0 0
\(952\) 0.260730i 0.00845029i
\(953\) 1.91375i 0.0619926i −0.999519 0.0309963i \(-0.990132\pi\)
0.999519 0.0309963i \(-0.00986801\pi\)
\(954\) 0 0
\(955\) −13.5784 23.8239i −0.439386 0.770923i
\(956\) 3.38005 0.109319
\(957\) 0 0
\(958\) 37.8829i 1.22394i
\(959\) 1.40061 0.0452280
\(960\) 0 0
\(961\) −20.3108 −0.655188
\(962\) 24.7554i 0.798145i
\(963\) 0 0
\(964\) −11.3707 −0.366227
\(965\) −16.2060 + 9.23656i −0.521688 + 0.297335i
\(966\) 0 0
\(967\) 4.01326i 0.129058i 0.997916 + 0.0645288i \(0.0205544\pi\)
−0.997916 + 0.0645288i \(0.979446\pi\)
\(968\) 31.7427i 1.02025i
\(969\) 0 0
\(970\) 16.2672 + 28.5416i 0.522309 + 0.916414i
\(971\) −4.54751 −0.145937 −0.0729683 0.997334i \(-0.523247\pi\)
−0.0729683 + 0.997334i \(0.523247\pi\)
\(972\) 0 0
\(973\) 0.694284i 0.0222577i
\(974\) 53.3295 1.70879
\(975\) 0 0
\(976\) 24.0683 0.770409
\(977\) 44.2333i 1.41515i 0.706639 + 0.707574i \(0.250211\pi\)
−0.706639 + 0.707574i \(0.749789\pi\)
\(978\) 0 0
\(979\) 60.4256 1.93121
\(980\) −2.92092 5.12488i −0.0933053 0.163708i
\(981\) 0 0
\(982\) 9.19391i 0.293389i
\(983\) 34.2083i 1.09107i −0.838087 0.545537i \(-0.816326\pi\)
0.838087 0.545537i \(-0.183674\pi\)
\(984\) 0 0
\(985\) 13.0017 7.41030i 0.414269 0.236112i
\(986\) 0.428167 0.0136356
\(987\) 0 0
\(988\) 5.57334i 0.177312i
\(989\) −60.4764 −1.92304
\(990\) 0 0
\(991\) −1.76770 −0.0561528 −0.0280764 0.999606i \(-0.508938\pi\)
−0.0280764 + 0.999606i \(0.508938\pi\)
\(992\) 6.93347i 0.220138i
\(993\) 0 0
\(994\) −4.54697 −0.144221
\(995\) 13.9050 + 24.3969i 0.440817 + 0.773433i
\(996\) 0 0
\(997\) 23.7398i 0.751848i 0.926650 + 0.375924i \(0.122675\pi\)
−0.926650 + 0.375924i \(0.877325\pi\)
\(998\) 19.8030i 0.626853i
\(999\) 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 1305.2.c.l.784.8 yes 12
3.2 odd 2 1305.2.c.k.784.5 12
5.2 odd 4 6525.2.a.cf.1.5 12
5.3 odd 4 6525.2.a.cf.1.8 12
5.4 even 2 inner 1305.2.c.l.784.5 yes 12
15.2 even 4 6525.2.a.ce.1.8 12
15.8 even 4 6525.2.a.ce.1.5 12
15.14 odd 2 1305.2.c.k.784.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.5 12 3.2 odd 2
1305.2.c.k.784.8 yes 12 15.14 odd 2
1305.2.c.l.784.5 yes 12 5.4 even 2 inner
1305.2.c.l.784.8 yes 12 1.1 even 1 trivial
6525.2.a.ce.1.5 12 15.8 even 4
6525.2.a.ce.1.8 12 15.2 even 4
6525.2.a.cf.1.5 12 5.2 odd 4
6525.2.a.cf.1.8 12 5.3 odd 4