Properties

Label 960.3.i.a.929.24
Level $960$
Weight $3$
Character 960.929
Analytic conductor $26.158$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(929,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.929");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.i (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.24
Character \(\chi\) \(=\) 960.929
Dual form 960.3.i.a.929.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.44379 + 2.62973i) q^{3} +(2.67751 + 4.22267i) q^{5} +1.30019i q^{7} +(-4.83095 + 7.59354i) q^{9} +O(q^{10})\) \(q+(1.44379 + 2.62973i) q^{3} +(2.67751 + 4.22267i) q^{5} +1.30019i q^{7} +(-4.83095 + 7.59354i) q^{9} -9.69683 q^{11} -15.3824 q^{13} +(-7.23873 + 13.1378i) q^{15} +9.62982 q^{17} -9.66190i q^{19} +(-3.41913 + 1.87719i) q^{21} +16.1769 q^{23} +(-10.6619 + 22.6125i) q^{25} +(-26.9438 - 1.74063i) q^{27} -47.2899 q^{29} -40.6635 q^{31} +(-14.0002 - 25.5000i) q^{33} +(-5.49025 + 3.48126i) q^{35} -43.5469 q^{37} +(-22.2090 - 40.4516i) q^{39} +4.20658i q^{41} +53.8877 q^{43} +(-44.9999 - 0.0677491i) q^{45} +48.5306 q^{47} +47.3095 q^{49} +(13.9034 + 25.3238i) q^{51} +9.87301i q^{53} +(-25.9633 - 40.9465i) q^{55} +(25.4082 - 13.9497i) q^{57} +70.4452 q^{59} +92.5902i q^{61} +(-9.87301 - 6.28113i) q^{63} +(-41.1866 - 64.9549i) q^{65} -25.9882 q^{67} +(23.3559 + 42.5407i) q^{69} -41.0296i q^{71} -17.3255i q^{73} +(-74.8582 + 4.60969i) q^{75} -12.6077i q^{77} -92.5902 q^{79} +(-34.3238 - 73.3681i) q^{81} -54.2976i q^{83} +(25.7839 + 40.6635i) q^{85} +(-68.2766 - 124.360i) q^{87} +102.103i q^{89} -20.0000i q^{91} +(-58.7095 - 106.934i) q^{93} +(40.7990 - 25.8698i) q^{95} +97.2013i q^{97} +(46.8449 - 73.6333i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 32 q^{9} + 32 q^{25} - 352 q^{49} - 352 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.44379 + 2.62973i 0.481263 + 0.876576i
\(4\) 0 0
\(5\) 2.67751 + 4.22267i 0.535502 + 0.844534i
\(6\) 0 0
\(7\) 1.30019i 0.185741i 0.995678 + 0.0928704i \(0.0296042\pi\)
−0.995678 + 0.0928704i \(0.970396\pi\)
\(8\) 0 0
\(9\) −4.83095 + 7.59354i −0.536772 + 0.843727i
\(10\) 0 0
\(11\) −9.69683 −0.881530 −0.440765 0.897622i \(-0.645293\pi\)
−0.440765 + 0.897622i \(0.645293\pi\)
\(12\) 0 0
\(13\) −15.3824 −1.18326 −0.591632 0.806208i \(-0.701516\pi\)
−0.591632 + 0.806208i \(0.701516\pi\)
\(14\) 0 0
\(15\) −7.23873 + 13.1378i −0.482582 + 0.875851i
\(16\) 0 0
\(17\) 9.62982 0.566460 0.283230 0.959052i \(-0.408594\pi\)
0.283230 + 0.959052i \(0.408594\pi\)
\(18\) 0 0
\(19\) 9.66190i 0.508521i −0.967136 0.254261i \(-0.918168\pi\)
0.967136 0.254261i \(-0.0818321\pi\)
\(20\) 0 0
\(21\) −3.41913 + 1.87719i −0.162816 + 0.0893901i
\(22\) 0 0
\(23\) 16.1769 0.703341 0.351671 0.936124i \(-0.385614\pi\)
0.351671 + 0.936124i \(0.385614\pi\)
\(24\) 0 0
\(25\) −10.6619 + 22.6125i −0.426476 + 0.904499i
\(26\) 0 0
\(27\) −26.9438 1.74063i −0.997920 0.0644677i
\(28\) 0 0
\(29\) −47.2899 −1.63069 −0.815343 0.578978i \(-0.803452\pi\)
−0.815343 + 0.578978i \(0.803452\pi\)
\(30\) 0 0
\(31\) −40.6635 −1.31173 −0.655864 0.754879i \(-0.727695\pi\)
−0.655864 + 0.754879i \(0.727695\pi\)
\(32\) 0 0
\(33\) −14.0002 25.5000i −0.424248 0.772729i
\(34\) 0 0
\(35\) −5.49025 + 3.48126i −0.156864 + 0.0994644i
\(36\) 0 0
\(37\) −43.5469 −1.17694 −0.588472 0.808518i \(-0.700270\pi\)
−0.588472 + 0.808518i \(0.700270\pi\)
\(38\) 0 0
\(39\) −22.2090 40.4516i −0.569461 1.03722i
\(40\) 0 0
\(41\) 4.20658i 0.102600i 0.998683 + 0.0512998i \(0.0163364\pi\)
−0.998683 + 0.0512998i \(0.983664\pi\)
\(42\) 0 0
\(43\) 53.8877 1.25320 0.626601 0.779340i \(-0.284446\pi\)
0.626601 + 0.779340i \(0.284446\pi\)
\(44\) 0 0
\(45\) −44.9999 0.0677491i −0.999999 0.00150554i
\(46\) 0 0
\(47\) 48.5306 1.03256 0.516282 0.856418i \(-0.327315\pi\)
0.516282 + 0.856418i \(0.327315\pi\)
\(48\) 0 0
\(49\) 47.3095 0.965500
\(50\) 0 0
\(51\) 13.9034 + 25.3238i 0.272616 + 0.496545i
\(52\) 0 0
\(53\) 9.87301i 0.186283i 0.995653 + 0.0931416i \(0.0296909\pi\)
−0.995653 + 0.0931416i \(0.970309\pi\)
\(54\) 0 0
\(55\) −25.9633 40.9465i −0.472061 0.744483i
\(56\) 0 0
\(57\) 25.4082 13.9497i 0.445758 0.244732i
\(58\) 0 0
\(59\) 70.4452 1.19399 0.596993 0.802246i \(-0.296362\pi\)
0.596993 + 0.802246i \(0.296362\pi\)
\(60\) 0 0
\(61\) 92.5902i 1.51787i 0.651165 + 0.758936i \(0.274281\pi\)
−0.651165 + 0.758936i \(0.725719\pi\)
\(62\) 0 0
\(63\) −9.87301 6.28113i −0.156714 0.0997005i
\(64\) 0 0
\(65\) −41.1866 64.9549i −0.633639 0.999307i
\(66\) 0 0
\(67\) −25.9882 −0.387883 −0.193942 0.981013i \(-0.562127\pi\)
−0.193942 + 0.981013i \(0.562127\pi\)
\(68\) 0 0
\(69\) 23.3559 + 42.5407i 0.338492 + 0.616532i
\(70\) 0 0
\(71\) 41.0296i 0.577882i −0.957347 0.288941i \(-0.906697\pi\)
0.957347 0.288941i \(-0.0933031\pi\)
\(72\) 0 0
\(73\) 17.3255i 0.237335i −0.992934 0.118668i \(-0.962138\pi\)
0.992934 0.118668i \(-0.0378623\pi\)
\(74\) 0 0
\(75\) −74.8582 + 4.60969i −0.998109 + 0.0614626i
\(76\) 0 0
\(77\) 12.6077i 0.163736i
\(78\) 0 0
\(79\) −92.5902 −1.17203 −0.586014 0.810301i \(-0.699304\pi\)
−0.586014 + 0.810301i \(0.699304\pi\)
\(80\) 0 0
\(81\) −34.3238 73.3681i −0.423751 0.905779i
\(82\) 0 0
\(83\) 54.2976i 0.654188i −0.944992 0.327094i \(-0.893931\pi\)
0.944992 0.327094i \(-0.106069\pi\)
\(84\) 0 0
\(85\) 25.7839 + 40.6635i 0.303340 + 0.478395i
\(86\) 0 0
\(87\) −68.2766 124.360i −0.784788 1.42942i
\(88\) 0 0
\(89\) 102.103i 1.14723i 0.819127 + 0.573613i \(0.194458\pi\)
−0.819127 + 0.573613i \(0.805542\pi\)
\(90\) 0 0
\(91\) 20.0000i 0.219780i
\(92\) 0 0
\(93\) −58.7095 106.934i −0.631285 1.14983i
\(94\) 0 0
\(95\) 40.7990 25.8698i 0.429464 0.272314i
\(96\) 0 0
\(97\) 97.2013i 1.00208i 0.865425 + 0.501038i \(0.167048\pi\)
−0.865425 + 0.501038i \(0.832952\pi\)
\(98\) 0 0
\(99\) 46.8449 73.6333i 0.473181 0.743771i
\(100\) 0 0
\(101\) −100.840 −0.998416 −0.499208 0.866482i \(-0.666376\pi\)
−0.499208 + 0.866482i \(0.666376\pi\)
\(102\) 0 0
\(103\) 162.926i 1.58180i −0.611944 0.790901i \(-0.709612\pi\)
0.611944 0.790901i \(-0.290388\pi\)
\(104\) 0 0
\(105\) −17.0815 9.41169i −0.162681 0.0896351i
\(106\) 0 0
\(107\) 129.708i 1.21223i 0.795378 + 0.606113i \(0.207272\pi\)
−0.795378 + 0.606113i \(0.792728\pi\)
\(108\) 0 0
\(109\) 173.917i 1.59557i 0.602941 + 0.797786i \(0.293995\pi\)
−0.602941 + 0.797786i \(0.706005\pi\)
\(110\) 0 0
\(111\) −62.8725 114.517i −0.566419 1.03168i
\(112\) 0 0
\(113\) −2.66730 −0.0236045 −0.0118022 0.999930i \(-0.503757\pi\)
−0.0118022 + 0.999930i \(0.503757\pi\)
\(114\) 0 0
\(115\) 43.3136 + 68.3095i 0.376640 + 0.593996i
\(116\) 0 0
\(117\) 74.3118 116.807i 0.635143 0.998351i
\(118\) 0 0
\(119\) 12.5205i 0.105215i
\(120\) 0 0
\(121\) −26.9714 −0.222904
\(122\) 0 0
\(123\) −11.0622 + 6.07341i −0.0899363 + 0.0493773i
\(124\) 0 0
\(125\) −124.032 + 15.5234i −0.992259 + 0.124187i
\(126\) 0 0
\(127\) 14.3020i 0.112614i 0.998413 + 0.0563072i \(0.0179326\pi\)
−0.998413 + 0.0563072i \(0.982067\pi\)
\(128\) 0 0
\(129\) 77.8024 + 141.710i 0.603119 + 1.09853i
\(130\) 0 0
\(131\) −189.375 −1.44561 −0.722804 0.691054i \(-0.757147\pi\)
−0.722804 + 0.691054i \(0.757147\pi\)
\(132\) 0 0
\(133\) 12.5623 0.0944531
\(134\) 0 0
\(135\) −64.7922 118.435i −0.479942 0.877300i
\(136\) 0 0
\(137\) 134.229 0.979773 0.489887 0.871786i \(-0.337038\pi\)
0.489887 + 0.871786i \(0.337038\pi\)
\(138\) 0 0
\(139\) 86.2809i 0.620726i −0.950618 0.310363i \(-0.899549\pi\)
0.950618 0.310363i \(-0.100451\pi\)
\(140\) 0 0
\(141\) 70.0678 + 127.622i 0.496935 + 0.905122i
\(142\) 0 0
\(143\) 149.161 1.04308
\(144\) 0 0
\(145\) −126.619 199.690i −0.873235 1.37717i
\(146\) 0 0
\(147\) 68.3049 + 124.411i 0.464659 + 0.846335i
\(148\) 0 0
\(149\) 14.2545 0.0956680 0.0478340 0.998855i \(-0.484768\pi\)
0.0478340 + 0.998855i \(0.484768\pi\)
\(150\) 0 0
\(151\) −11.2631 −0.0745904 −0.0372952 0.999304i \(-0.511874\pi\)
−0.0372952 + 0.999304i \(0.511874\pi\)
\(152\) 0 0
\(153\) −46.5212 + 73.1244i −0.304060 + 0.477937i
\(154\) 0 0
\(155\) −108.877 171.709i −0.702432 1.10780i
\(156\) 0 0
\(157\) −145.803 −0.928684 −0.464342 0.885656i \(-0.653709\pi\)
−0.464342 + 0.885656i \(0.653709\pi\)
\(158\) 0 0
\(159\) −25.9633 + 14.2545i −0.163291 + 0.0896512i
\(160\) 0 0
\(161\) 21.0329i 0.130639i
\(162\) 0 0
\(163\) −200.137 −1.22783 −0.613916 0.789372i \(-0.710407\pi\)
−0.613916 + 0.789372i \(0.710407\pi\)
\(164\) 0 0
\(165\) 70.1927 127.395i 0.425411 0.772089i
\(166\) 0 0
\(167\) 152.730 0.914551 0.457275 0.889325i \(-0.348825\pi\)
0.457275 + 0.889325i \(0.348825\pi\)
\(168\) 0 0
\(169\) 67.6190 0.400113
\(170\) 0 0
\(171\) 73.3681 + 46.6762i 0.429053 + 0.272960i
\(172\) 0 0
\(173\) 146.426i 0.846394i 0.906038 + 0.423197i \(0.139092\pi\)
−0.906038 + 0.423197i \(0.860908\pi\)
\(174\) 0 0
\(175\) −29.4004 13.8624i −0.168002 0.0792140i
\(176\) 0 0
\(177\) 101.708 + 185.252i 0.574621 + 1.04662i
\(178\) 0 0
\(179\) 261.103 1.45868 0.729339 0.684152i \(-0.239828\pi\)
0.729339 + 0.684152i \(0.239828\pi\)
\(180\) 0 0
\(181\) 185.180i 1.02310i −0.859255 0.511548i \(-0.829072\pi\)
0.859255 0.511548i \(-0.170928\pi\)
\(182\) 0 0
\(183\) −243.487 + 133.681i −1.33053 + 0.730496i
\(184\) 0 0
\(185\) −116.597 183.884i −0.630255 0.993969i
\(186\) 0 0
\(187\) −93.3787 −0.499351
\(188\) 0 0
\(189\) 2.26314 35.0320i 0.0119743 0.185354i
\(190\) 0 0
\(191\) 242.710i 1.27073i 0.772211 + 0.635366i \(0.219151\pi\)
−0.772211 + 0.635366i \(0.780849\pi\)
\(192\) 0 0
\(193\) 17.3255i 0.0897692i −0.998992 0.0448846i \(-0.985708\pi\)
0.998992 0.0448846i \(-0.0142920\pi\)
\(194\) 0 0
\(195\) 111.349 202.091i 0.571022 1.03636i
\(196\) 0 0
\(197\) 353.156i 1.79267i 0.443377 + 0.896335i \(0.353780\pi\)
−0.443377 + 0.896335i \(0.646220\pi\)
\(198\) 0 0
\(199\) 63.1898 0.317537 0.158768 0.987316i \(-0.449248\pi\)
0.158768 + 0.987316i \(0.449248\pi\)
\(200\) 0 0
\(201\) −37.5214 68.3419i −0.186674 0.340009i
\(202\) 0 0
\(203\) 61.4856i 0.302885i
\(204\) 0 0
\(205\) −17.7630 + 11.2631i −0.0866488 + 0.0549422i
\(206\) 0 0
\(207\) −78.1496 + 122.840i −0.377534 + 0.593428i
\(208\) 0 0
\(209\) 93.6899i 0.448277i
\(210\) 0 0
\(211\) 122.900i 0.582464i 0.956652 + 0.291232i \(0.0940652\pi\)
−0.956652 + 0.291232i \(0.905935\pi\)
\(212\) 0 0
\(213\) 107.897 59.2381i 0.506558 0.278113i
\(214\) 0 0
\(215\) 144.285 + 227.550i 0.671091 + 1.05837i
\(216\) 0 0
\(217\) 52.8701i 0.243641i
\(218\) 0 0
\(219\) 45.5613 25.0143i 0.208042 0.114220i
\(220\) 0 0
\(221\) −148.130 −0.670271
\(222\) 0 0
\(223\) 272.544i 1.22217i 0.791566 + 0.611084i \(0.209266\pi\)
−0.791566 + 0.611084i \(0.790734\pi\)
\(224\) 0 0
\(225\) −120.202 190.201i −0.534229 0.845339i
\(226\) 0 0
\(227\) 305.124i 1.34416i 0.740480 + 0.672079i \(0.234598\pi\)
−0.740480 + 0.672079i \(0.765402\pi\)
\(228\) 0 0
\(229\) 429.162i 1.87407i −0.349237 0.937034i \(-0.613559\pi\)
0.349237 0.937034i \(-0.386441\pi\)
\(230\) 0 0
\(231\) 33.1548 18.2028i 0.143527 0.0788001i
\(232\) 0 0
\(233\) −281.343 −1.20748 −0.603741 0.797180i \(-0.706324\pi\)
−0.603741 + 0.797180i \(0.706324\pi\)
\(234\) 0 0
\(235\) 129.941 + 204.929i 0.552940 + 0.872036i
\(236\) 0 0
\(237\) −133.681 243.487i −0.564054 1.02737i
\(238\) 0 0
\(239\) 94.5798i 0.395731i 0.980229 + 0.197866i \(0.0634010\pi\)
−0.980229 + 0.197866i \(0.936599\pi\)
\(240\) 0 0
\(241\) 129.662 0.538016 0.269008 0.963138i \(-0.413304\pi\)
0.269008 + 0.963138i \(0.413304\pi\)
\(242\) 0 0
\(243\) 143.382 196.190i 0.590049 0.807367i
\(244\) 0 0
\(245\) 126.672 + 199.773i 0.517027 + 0.815398i
\(246\) 0 0
\(247\) 148.624i 0.601715i
\(248\) 0 0
\(249\) 142.788 78.3943i 0.573446 0.314836i
\(250\) 0 0
\(251\) 172.548 0.687443 0.343722 0.939072i \(-0.388312\pi\)
0.343722 + 0.939072i \(0.388312\pi\)
\(252\) 0 0
\(253\) −156.864 −0.620017
\(254\) 0 0
\(255\) −69.7076 + 126.514i −0.273363 + 0.496134i
\(256\) 0 0
\(257\) 466.977 1.81703 0.908516 0.417850i \(-0.137216\pi\)
0.908516 + 0.417850i \(0.137216\pi\)
\(258\) 0 0
\(259\) 56.6190i 0.218606i
\(260\) 0 0
\(261\) 228.455 359.098i 0.875307 1.37585i
\(262\) 0 0
\(263\) 276.675 1.05200 0.525999 0.850485i \(-0.323692\pi\)
0.525999 + 0.850485i \(0.323692\pi\)
\(264\) 0 0
\(265\) −41.6905 + 26.4351i −0.157323 + 0.0997550i
\(266\) 0 0
\(267\) −268.503 + 147.415i −1.00563 + 0.552117i
\(268\) 0 0
\(269\) −282.005 −1.04835 −0.524173 0.851612i \(-0.675626\pi\)
−0.524173 + 0.851612i \(0.675626\pi\)
\(270\) 0 0
\(271\) 173.917 0.641761 0.320881 0.947120i \(-0.396021\pi\)
0.320881 + 0.947120i \(0.396021\pi\)
\(272\) 0 0
\(273\) 52.5946 28.8758i 0.192654 0.105772i
\(274\) 0 0
\(275\) 103.387 219.269i 0.375952 0.797343i
\(276\) 0 0
\(277\) 514.542 1.85755 0.928776 0.370641i \(-0.120862\pi\)
0.928776 + 0.370641i \(0.120862\pi\)
\(278\) 0 0
\(279\) 196.444 308.780i 0.704099 1.10674i
\(280\) 0 0
\(281\) 139.962i 0.498086i −0.968492 0.249043i \(-0.919884\pi\)
0.968492 0.249043i \(-0.0801161\pi\)
\(282\) 0 0
\(283\) 338.740 1.19696 0.598481 0.801137i \(-0.295771\pi\)
0.598481 + 0.801137i \(0.295771\pi\)
\(284\) 0 0
\(285\) 126.936 + 69.9399i 0.445389 + 0.245403i
\(286\) 0 0
\(287\) −5.46933 −0.0190569
\(288\) 0 0
\(289\) −196.267 −0.679123
\(290\) 0 0
\(291\) −255.613 + 140.338i −0.878396 + 0.482261i
\(292\) 0 0
\(293\) 346.018i 1.18095i 0.807057 + 0.590474i \(0.201059\pi\)
−0.807057 + 0.590474i \(0.798941\pi\)
\(294\) 0 0
\(295\) 188.618 + 297.467i 0.639381 + 1.00836i
\(296\) 0 0
\(297\) 261.270 + 16.8786i 0.879697 + 0.0568302i
\(298\) 0 0
\(299\) −248.839 −0.832238
\(300\) 0 0
\(301\) 70.0639i 0.232771i
\(302\) 0 0
\(303\) −145.592 265.182i −0.480500 0.875188i
\(304\) 0 0
\(305\) −390.978 + 247.911i −1.28190 + 0.812823i
\(306\) 0 0
\(307\) −511.995 −1.66774 −0.833868 0.551964i \(-0.813878\pi\)
−0.833868 + 0.551964i \(0.813878\pi\)
\(308\) 0 0
\(309\) 428.450 235.230i 1.38657 0.761262i
\(310\) 0 0
\(311\) 296.260i 0.952604i 0.879282 + 0.476302i \(0.158023\pi\)
−0.879282 + 0.476302i \(0.841977\pi\)
\(312\) 0 0
\(313\) 162.681i 0.519746i −0.965643 0.259873i \(-0.916319\pi\)
0.965643 0.259873i \(-0.0836807\pi\)
\(314\) 0 0
\(315\) 0.0880864 58.5083i 0.000279639 0.185741i
\(316\) 0 0
\(317\) 178.780i 0.563974i 0.959418 + 0.281987i \(0.0909936\pi\)
−0.959418 + 0.281987i \(0.909006\pi\)
\(318\) 0 0
\(319\) 458.562 1.43750
\(320\) 0 0
\(321\) −341.098 + 187.271i −1.06261 + 0.583400i
\(322\) 0 0
\(323\) 93.0424i 0.288057i
\(324\) 0 0
\(325\) 164.006 347.835i 0.504634 1.07026i
\(326\) 0 0
\(327\) −457.355 + 251.100i −1.39864 + 0.767889i
\(328\) 0 0
\(329\) 63.0987i 0.191789i
\(330\) 0 0
\(331\) 636.138i 1.92187i −0.276779 0.960934i \(-0.589267\pi\)
0.276779 0.960934i \(-0.410733\pi\)
\(332\) 0 0
\(333\) 210.373 330.675i 0.631751 0.993019i
\(334\) 0 0
\(335\) −69.5836 109.740i −0.207712 0.327581i
\(336\) 0 0
\(337\) 503.332i 1.49357i 0.665067 + 0.746783i \(0.268403\pi\)
−0.665067 + 0.746783i \(0.731597\pi\)
\(338\) 0 0
\(339\) −3.85102 7.01429i −0.0113599 0.0206911i
\(340\) 0 0
\(341\) 394.308 1.15633
\(342\) 0 0
\(343\) 125.220i 0.365073i
\(344\) 0 0
\(345\) −117.100 + 212.528i −0.339420 + 0.616022i
\(346\) 0 0
\(347\) 508.389i 1.46510i 0.680714 + 0.732549i \(0.261670\pi\)
−0.680714 + 0.732549i \(0.738330\pi\)
\(348\) 0 0
\(349\) 81.3271i 0.233029i 0.993189 + 0.116514i \(0.0371721\pi\)
−0.993189 + 0.116514i \(0.962828\pi\)
\(350\) 0 0
\(351\) 414.462 + 26.7751i 1.18080 + 0.0762823i
\(352\) 0 0
\(353\) −69.0366 −0.195571 −0.0977856 0.995208i \(-0.531176\pi\)
−0.0977856 + 0.995208i \(0.531176\pi\)
\(354\) 0 0
\(355\) 173.255 109.857i 0.488041 0.309457i
\(356\) 0 0
\(357\) −32.9256 + 18.0770i −0.0922287 + 0.0506359i
\(358\) 0 0
\(359\) 646.070i 1.79964i 0.436264 + 0.899819i \(0.356301\pi\)
−0.436264 + 0.899819i \(0.643699\pi\)
\(360\) 0 0
\(361\) 267.648 0.741406
\(362\) 0 0
\(363\) −38.9410 70.9275i −0.107276 0.195393i
\(364\) 0 0
\(365\) 73.1597 46.3890i 0.200438 0.127093i
\(366\) 0 0
\(367\) 337.553i 0.919762i −0.887980 0.459881i \(-0.847892\pi\)
0.887980 0.459881i \(-0.152108\pi\)
\(368\) 0 0
\(369\) −31.9428 20.3218i −0.0865660 0.0550726i
\(370\) 0 0
\(371\) −12.8367 −0.0346004
\(372\) 0 0
\(373\) 561.349 1.50496 0.752478 0.658617i \(-0.228858\pi\)
0.752478 + 0.658617i \(0.228858\pi\)
\(374\) 0 0
\(375\) −219.899 303.759i −0.586396 0.810024i
\(376\) 0 0
\(377\) 727.433 1.92953
\(378\) 0 0
\(379\) 439.519i 1.15968i 0.814730 + 0.579840i \(0.196885\pi\)
−0.814730 + 0.579840i \(0.803115\pi\)
\(380\) 0 0
\(381\) −37.6105 + 20.6491i −0.0987152 + 0.0541971i
\(382\) 0 0
\(383\) −456.521 −1.19196 −0.595980 0.802999i \(-0.703236\pi\)
−0.595980 + 0.802999i \(0.703236\pi\)
\(384\) 0 0
\(385\) 53.2381 33.7572i 0.138281 0.0876809i
\(386\) 0 0
\(387\) −260.329 + 409.198i −0.672684 + 1.05736i
\(388\) 0 0
\(389\) 190.894 0.490729 0.245364 0.969431i \(-0.421092\pi\)
0.245364 + 0.969431i \(0.421092\pi\)
\(390\) 0 0
\(391\) 155.780 0.398415
\(392\) 0 0
\(393\) −273.417 498.004i −0.695717 1.26719i
\(394\) 0 0
\(395\) −247.911 390.978i −0.627623 0.989818i
\(396\) 0 0
\(397\) 247.218 0.622715 0.311357 0.950293i \(-0.399216\pi\)
0.311357 + 0.950293i \(0.399216\pi\)
\(398\) 0 0
\(399\) 18.1372 + 33.0353i 0.0454568 + 0.0827954i
\(400\) 0 0
\(401\) 14.2590i 0.0355585i −0.999842 0.0177793i \(-0.994340\pi\)
0.999842 0.0177793i \(-0.00565961\pi\)
\(402\) 0 0
\(403\) 625.504 1.55212
\(404\) 0 0
\(405\) 217.907 341.382i 0.538042 0.842918i
\(406\) 0 0
\(407\) 422.267 1.03751
\(408\) 0 0
\(409\) −166.957 −0.408208 −0.204104 0.978949i \(-0.565428\pi\)
−0.204104 + 0.978949i \(0.565428\pi\)
\(410\) 0 0
\(411\) 193.798 + 352.986i 0.471528 + 0.858846i
\(412\) 0 0
\(413\) 91.5918i 0.221772i
\(414\) 0 0
\(415\) 229.281 145.382i 0.552485 0.350319i
\(416\) 0 0
\(417\) 226.896 124.571i 0.544114 0.298732i
\(418\) 0 0
\(419\) −369.052 −0.880793 −0.440396 0.897803i \(-0.645162\pi\)
−0.440396 + 0.897803i \(0.645162\pi\)
\(420\) 0 0
\(421\) 417.899i 0.992633i −0.868142 0.496317i \(-0.834686\pi\)
0.868142 0.496317i \(-0.165314\pi\)
\(422\) 0 0
\(423\) −234.449 + 368.519i −0.554252 + 0.871203i
\(424\) 0 0
\(425\) −102.672 + 217.754i −0.241582 + 0.512362i
\(426\) 0 0
\(427\) −120.384 −0.281931
\(428\) 0 0
\(429\) 215.357 + 392.253i 0.501997 + 0.914342i
\(430\) 0 0
\(431\) 419.349i 0.972967i −0.873690 0.486483i \(-0.838279\pi\)
0.873690 0.486483i \(-0.161721\pi\)
\(432\) 0 0
\(433\) 208.799i 0.482215i −0.970498 0.241108i \(-0.922489\pi\)
0.970498 0.241108i \(-0.0775107\pi\)
\(434\) 0 0
\(435\) 342.319 621.283i 0.786939 1.42824i
\(436\) 0 0
\(437\) 156.299i 0.357664i
\(438\) 0 0
\(439\) −121.991 −0.277883 −0.138941 0.990301i \(-0.544370\pi\)
−0.138941 + 0.990301i \(0.544370\pi\)
\(440\) 0 0
\(441\) −228.550 + 359.247i −0.518254 + 0.814619i
\(442\) 0 0
\(443\) 413.268i 0.932885i −0.884551 0.466443i \(-0.845535\pi\)
0.884551 0.466443i \(-0.154465\pi\)
\(444\) 0 0
\(445\) −431.148 + 273.382i −0.968871 + 0.614341i
\(446\) 0 0
\(447\) 20.5805 + 37.4856i 0.0460415 + 0.0838603i
\(448\) 0 0
\(449\) 836.712i 1.86350i −0.363099 0.931751i \(-0.618281\pi\)
0.363099 0.931751i \(-0.381719\pi\)
\(450\) 0 0
\(451\) 40.7905i 0.0904446i
\(452\) 0 0
\(453\) −16.2616 29.6190i −0.0358976 0.0653842i
\(454\) 0 0
\(455\) 84.4534 53.5502i 0.185612 0.117693i
\(456\) 0 0
\(457\) 868.060i 1.89948i 0.313046 + 0.949738i \(0.398650\pi\)
−0.313046 + 0.949738i \(0.601350\pi\)
\(458\) 0 0
\(459\) −259.464 16.7619i −0.565281 0.0365184i
\(460\) 0 0
\(461\) −532.709 −1.15555 −0.577776 0.816196i \(-0.696079\pi\)
−0.577776 + 0.816196i \(0.696079\pi\)
\(462\) 0 0
\(463\) 215.775i 0.466037i −0.972472 0.233018i \(-0.925140\pi\)
0.972472 0.233018i \(-0.0748602\pi\)
\(464\) 0 0
\(465\) 294.352 534.228i 0.633016 1.14888i
\(466\) 0 0
\(467\) 545.281i 1.16762i 0.811889 + 0.583812i \(0.198440\pi\)
−0.811889 + 0.583812i \(0.801560\pi\)
\(468\) 0 0
\(469\) 33.7894i 0.0720457i
\(470\) 0 0
\(471\) −210.509 383.423i −0.446941 0.814062i
\(472\) 0 0
\(473\) −522.540 −1.10474
\(474\) 0 0
\(475\) 218.480 + 103.014i 0.459957 + 0.216872i
\(476\) 0 0
\(477\) −74.9711 47.6960i −0.157172 0.0999917i
\(478\) 0 0
\(479\) 160.650i 0.335387i −0.985839 0.167694i \(-0.946368\pi\)
0.985839 0.167694i \(-0.0536319\pi\)
\(480\) 0 0
\(481\) 669.857 1.39263
\(482\) 0 0
\(483\) −55.3108 + 30.3670i −0.114515 + 0.0628717i
\(484\) 0 0
\(485\) −410.449 + 260.257i −0.846287 + 0.536613i
\(486\) 0 0
\(487\) 26.4247i 0.0542602i −0.999632 0.0271301i \(-0.991363\pi\)
0.999632 0.0271301i \(-0.00863684\pi\)
\(488\) 0 0
\(489\) −288.955 526.305i −0.590909 1.07629i
\(490\) 0 0
\(491\) 893.826 1.82042 0.910210 0.414147i \(-0.135920\pi\)
0.910210 + 0.414147i \(0.135920\pi\)
\(492\) 0 0
\(493\) −455.393 −0.923718
\(494\) 0 0
\(495\) 436.357 + 0.656952i 0.881529 + 0.00132718i
\(496\) 0 0
\(497\) 53.3461 0.107336
\(498\) 0 0
\(499\) 156.929i 0.314486i 0.987560 + 0.157243i \(0.0502606\pi\)
−0.987560 + 0.157243i \(0.949739\pi\)
\(500\) 0 0
\(501\) 220.510 + 401.639i 0.440139 + 0.801674i
\(502\) 0 0
\(503\) 709.881 1.41129 0.705647 0.708563i \(-0.250656\pi\)
0.705647 + 0.708563i \(0.250656\pi\)
\(504\) 0 0
\(505\) −270.000 425.814i −0.534653 0.843197i
\(506\) 0 0
\(507\) 97.6276 + 177.820i 0.192559 + 0.350729i
\(508\) 0 0
\(509\) 138.402 0.271909 0.135954 0.990715i \(-0.456590\pi\)
0.135954 + 0.990715i \(0.456590\pi\)
\(510\) 0 0
\(511\) 22.5263 0.0440828
\(512\) 0 0
\(513\) −16.8178 + 260.329i −0.0327832 + 0.507463i
\(514\) 0 0
\(515\) 687.981 436.234i 1.33589 0.847057i
\(516\) 0 0
\(517\) −470.593 −0.910237
\(518\) 0 0
\(519\) −385.061 + 211.408i −0.741929 + 0.407338i
\(520\) 0 0
\(521\) 104.670i 0.200903i 0.994942 + 0.100451i \(0.0320287\pi\)
−0.994942 + 0.100451i \(0.967971\pi\)
\(522\) 0 0
\(523\) −617.735 −1.18114 −0.590569 0.806987i \(-0.701097\pi\)
−0.590569 + 0.806987i \(0.701097\pi\)
\(524\) 0 0
\(525\) −5.99346 97.3295i −0.0114161 0.185390i
\(526\) 0 0
\(527\) −391.582 −0.743041
\(528\) 0 0
\(529\) −267.310 −0.505311
\(530\) 0 0
\(531\) −340.317 + 534.929i −0.640899 + 1.00740i
\(532\) 0 0
\(533\) 64.7074i 0.121402i
\(534\) 0 0
\(535\) −547.715 + 347.295i −1.02377 + 0.649149i
\(536\) 0 0
\(537\) 376.978 + 686.631i 0.702007 + 1.27864i
\(538\) 0 0
\(539\) −458.753 −0.851118
\(540\) 0 0
\(541\) 415.414i 0.767862i 0.923362 + 0.383931i \(0.125430\pi\)
−0.923362 + 0.383931i \(0.874570\pi\)
\(542\) 0 0
\(543\) 486.975 267.361i 0.896822 0.492378i
\(544\) 0 0
\(545\) −734.396 + 465.665i −1.34752 + 0.854431i
\(546\) 0 0
\(547\) 218.356 0.399188 0.199594 0.979879i \(-0.436038\pi\)
0.199594 + 0.979879i \(0.436038\pi\)
\(548\) 0 0
\(549\) −703.088 447.299i −1.28067 0.814752i
\(550\) 0 0
\(551\) 456.910i 0.829238i
\(552\) 0 0
\(553\) 120.384i 0.217693i
\(554\) 0 0
\(555\) 315.224 572.109i 0.567972 1.03083i
\(556\) 0 0
\(557\) 8.20400i 0.0147289i −0.999973 0.00736445i \(-0.997656\pi\)
0.999973 0.00736445i \(-0.00234420\pi\)
\(558\) 0 0
\(559\) −828.923 −1.48287
\(560\) 0 0
\(561\) −134.819 245.561i −0.240319 0.437720i
\(562\) 0 0
\(563\) 222.751i 0.395649i −0.980237 0.197825i \(-0.936612\pi\)
0.980237 0.197825i \(-0.0633877\pi\)
\(564\) 0 0
\(565\) −7.14173 11.2631i −0.0126402 0.0199348i
\(566\) 0 0
\(567\) 95.3921 44.6273i 0.168240 0.0787078i
\(568\) 0 0
\(569\) 196.287i 0.344968i 0.985012 + 0.172484i \(0.0551794\pi\)
−0.985012 + 0.172484i \(0.944821\pi\)
\(570\) 0 0
\(571\) 830.167i 1.45388i −0.686700 0.726941i \(-0.740941\pi\)
0.686700 0.726941i \(-0.259059\pi\)
\(572\) 0 0
\(573\) −638.261 + 350.421i −1.11389 + 0.611556i
\(574\) 0 0
\(575\) −172.476 + 365.799i −0.299958 + 0.636171i
\(576\) 0 0
\(577\) 574.421i 0.995531i −0.867312 0.497765i \(-0.834154\pi\)
0.867312 0.497765i \(-0.165846\pi\)
\(578\) 0 0
\(579\) 45.5613 25.0143i 0.0786896 0.0432026i
\(580\) 0 0
\(581\) 70.5970 0.121509
\(582\) 0 0
\(583\) 95.7369i 0.164214i
\(584\) 0 0
\(585\) 692.208 + 1.04215i 1.18326 + 0.00178145i
\(586\) 0 0
\(587\) 962.030i 1.63889i 0.573156 + 0.819446i \(0.305719\pi\)
−0.573156 + 0.819446i \(0.694281\pi\)
\(588\) 0 0
\(589\) 392.887i 0.667041i
\(590\) 0 0
\(591\) −928.705 + 509.882i −1.57141 + 0.862745i
\(592\) 0 0
\(593\) 621.054 1.04731 0.523655 0.851931i \(-0.324568\pi\)
0.523655 + 0.851931i \(0.324568\pi\)
\(594\) 0 0
\(595\) −52.8701 + 33.5238i −0.0888574 + 0.0563426i
\(596\) 0 0
\(597\) 91.2327 + 166.172i 0.152819 + 0.278345i
\(598\) 0 0
\(599\) 958.318i 1.59986i −0.600091 0.799932i \(-0.704869\pi\)
0.600091 0.799932i \(-0.295131\pi\)
\(600\) 0 0
\(601\) 356.424 0.593051 0.296526 0.955025i \(-0.404172\pi\)
0.296526 + 0.955025i \(0.404172\pi\)
\(602\) 0 0
\(603\) 125.548 197.342i 0.208205 0.327268i
\(604\) 0 0
\(605\) −72.2162 113.891i −0.119366 0.188250i
\(606\) 0 0
\(607\) 303.308i 0.499684i 0.968287 + 0.249842i \(0.0803787\pi\)
−0.968287 + 0.249842i \(0.919621\pi\)
\(608\) 0 0
\(609\) 161.690 88.7722i 0.265502 0.145767i
\(610\) 0 0
\(611\) −746.518 −1.22180
\(612\) 0 0
\(613\) −547.468 −0.893096 −0.446548 0.894760i \(-0.647347\pi\)
−0.446548 + 0.894760i \(0.647347\pi\)
\(614\) 0 0
\(615\) −55.2650 30.4503i −0.0898619 0.0495127i
\(616\) 0 0
\(617\) 707.134 1.14608 0.573042 0.819526i \(-0.305763\pi\)
0.573042 + 0.819526i \(0.305763\pi\)
\(618\) 0 0
\(619\) 979.405i 1.58224i 0.611663 + 0.791119i \(0.290501\pi\)
−0.611663 + 0.791119i \(0.709499\pi\)
\(620\) 0 0
\(621\) −435.866 28.1579i −0.701878 0.0453428i
\(622\) 0 0
\(623\) −132.753 −0.213086
\(624\) 0 0
\(625\) −397.648 482.184i −0.636236 0.771494i
\(626\) 0 0
\(627\) −246.379 + 135.268i −0.392949 + 0.215739i
\(628\) 0 0
\(629\) −419.349 −0.666691
\(630\) 0 0
\(631\) −596.205 −0.944857 −0.472429 0.881369i \(-0.656623\pi\)
−0.472429 + 0.881369i \(0.656623\pi\)
\(632\) 0 0
\(633\) −323.194 + 177.442i −0.510575 + 0.280318i
\(634\) 0 0
\(635\) −60.3928 + 38.2938i −0.0951068 + 0.0603052i
\(636\) 0 0
\(637\) −727.735 −1.14244
\(638\) 0 0
\(639\) 311.560 + 198.212i 0.487575 + 0.310191i
\(640\) 0 0
\(641\) 249.767i 0.389653i 0.980838 + 0.194826i \(0.0624143\pi\)
−0.980838 + 0.194826i \(0.937586\pi\)
\(642\) 0 0
\(643\) −280.012 −0.435478 −0.217739 0.976007i \(-0.569868\pi\)
−0.217739 + 0.976007i \(0.569868\pi\)
\(644\) 0 0
\(645\) −390.078 + 707.963i −0.604772 + 1.09762i
\(646\) 0 0
\(647\) 496.013 0.766635 0.383318 0.923617i \(-0.374781\pi\)
0.383318 + 0.923617i \(0.374781\pi\)
\(648\) 0 0
\(649\) −683.095 −1.05253
\(650\) 0 0
\(651\) 139.034 76.3333i 0.213570 0.117255i
\(652\) 0 0
\(653\) 800.639i 1.22609i −0.790047 0.613046i \(-0.789944\pi\)
0.790047 0.613046i \(-0.210056\pi\)
\(654\) 0 0
\(655\) −507.052 799.666i −0.774125 1.22086i
\(656\) 0 0
\(657\) 131.562 + 83.6984i 0.200246 + 0.127395i
\(658\) 0 0
\(659\) −270.228 −0.410057 −0.205029 0.978756i \(-0.565729\pi\)
−0.205029 + 0.978756i \(0.565729\pi\)
\(660\) 0 0
\(661\) 521.752i 0.789337i 0.918824 + 0.394669i \(0.129141\pi\)
−0.918824 + 0.394669i \(0.870859\pi\)
\(662\) 0 0
\(663\) −213.868 389.542i −0.322577 0.587544i
\(664\) 0 0
\(665\) 33.6356 + 53.0463i 0.0505798 + 0.0797689i
\(666\) 0 0
\(667\) −765.001 −1.14693
\(668\) 0 0
\(669\) −716.716 + 393.495i −1.07132 + 0.588184i
\(670\) 0 0
\(671\) 897.832i 1.33805i
\(672\) 0 0
\(673\) 895.960i 1.33129i 0.746267 + 0.665646i \(0.231844\pi\)
−0.746267 + 0.665646i \(0.768156\pi\)
\(674\) 0 0
\(675\) 326.632 590.708i 0.483900 0.875123i
\(676\) 0 0
\(677\) 119.542i 0.176576i −0.996095 0.0882879i \(-0.971860\pi\)
0.996095 0.0882879i \(-0.0281395\pi\)
\(678\) 0 0
\(679\) −126.380 −0.186126
\(680\) 0 0
\(681\) −802.393 + 440.534i −1.17826 + 0.646893i
\(682\) 0 0
\(683\) 259.642i 0.380149i −0.981770 0.190075i \(-0.939127\pi\)
0.981770 0.190075i \(-0.0608730\pi\)
\(684\) 0 0
\(685\) 359.399 + 566.805i 0.524670 + 0.827452i
\(686\) 0 0
\(687\) 1128.58 619.619i 1.64276 0.901919i
\(688\) 0 0
\(689\) 151.871i 0.220422i
\(690\) 0 0
\(691\) 609.662i 0.882289i −0.897436 0.441145i \(-0.854573\pi\)
0.897436 0.441145i \(-0.145427\pi\)
\(692\) 0 0
\(693\) 95.7369 + 60.9071i 0.138149 + 0.0878890i
\(694\) 0 0
\(695\) 364.336 231.018i 0.524225 0.332400i
\(696\) 0 0
\(697\) 40.5086i 0.0581185i
\(698\) 0 0
\(699\) −406.200 739.857i −0.581116 1.05845i
\(700\) 0 0
\(701\) 51.8162 0.0739175 0.0369587 0.999317i \(-0.488233\pi\)
0.0369587 + 0.999317i \(0.488233\pi\)
\(702\) 0 0
\(703\) 420.746i 0.598501i
\(704\) 0 0
\(705\) −351.299 + 637.583i −0.498297 + 0.904373i
\(706\) 0 0
\(707\) 131.111i 0.185447i
\(708\) 0 0
\(709\) 614.342i 0.866491i 0.901276 + 0.433246i \(0.142632\pi\)
−0.901276 + 0.433246i \(0.857368\pi\)
\(710\) 0 0
\(711\) 447.299 703.088i 0.629113 0.988872i
\(712\) 0 0
\(713\) −657.808 −0.922592
\(714\) 0 0
\(715\) 399.379 + 629.857i 0.558572 + 0.880919i
\(716\) 0 0
\(717\) −248.719 + 136.553i −0.346889 + 0.190451i
\(718\) 0 0
\(719\) 822.709i 1.14424i −0.820170 0.572120i \(-0.806121\pi\)
0.820170 0.572120i \(-0.193879\pi\)
\(720\) 0 0
\(721\) 211.833 0.293805
\(722\) 0 0
\(723\) 187.204 + 340.976i 0.258927 + 0.471612i
\(724\) 0 0
\(725\) 504.200 1069.34i 0.695449 1.47495i
\(726\) 0 0
\(727\) 406.884i 0.559675i −0.960047 0.279837i \(-0.909719\pi\)
0.960047 0.279837i \(-0.0902806\pi\)
\(728\) 0 0
\(729\) 722.940 + 93.7984i 0.991688 + 0.128667i
\(730\) 0 0
\(731\) 518.928 0.709888
\(732\) 0 0
\(733\) 757.218 1.03304 0.516520 0.856275i \(-0.327227\pi\)
0.516520 + 0.856275i \(0.327227\pi\)
\(734\) 0 0
\(735\) −342.461 + 621.541i −0.465933 + 0.845634i
\(736\) 0 0
\(737\) 252.003 0.341931
\(738\) 0 0
\(739\) 36.3095i 0.0491333i −0.999698 0.0245667i \(-0.992179\pi\)
0.999698 0.0245667i \(-0.00782060\pi\)
\(740\) 0 0
\(741\) −390.840 + 214.581i −0.527449 + 0.289583i
\(742\) 0 0
\(743\) −769.119 −1.03515 −0.517577 0.855637i \(-0.673166\pi\)
−0.517577 + 0.855637i \(0.673166\pi\)
\(744\) 0 0
\(745\) 38.1666 + 60.1922i 0.0512304 + 0.0807949i
\(746\) 0 0
\(747\) 412.311 + 262.309i 0.551956 + 0.351150i
\(748\) 0 0
\(749\) −168.645 −0.225160
\(750\) 0 0
\(751\) −907.765 −1.20874 −0.604371 0.796703i \(-0.706575\pi\)
−0.604371 + 0.796703i \(0.706575\pi\)
\(752\) 0 0
\(753\) 249.123 + 453.755i 0.330841 + 0.602596i
\(754\) 0 0
\(755\) −30.1572 47.5606i −0.0399433 0.0629941i
\(756\) 0 0
\(757\) −114.599 −0.151386 −0.0756928 0.997131i \(-0.524117\pi\)
−0.0756928 + 0.997131i \(0.524117\pi\)
\(758\) 0 0
\(759\) −226.479 412.510i −0.298391 0.543492i
\(760\) 0 0
\(761\) 200.928i 0.264031i −0.991248 0.132016i \(-0.957855\pi\)
0.991248 0.132016i \(-0.0421449\pi\)
\(762\) 0 0
\(763\) −226.125 −0.296363
\(764\) 0 0
\(765\) −433.341 0.652411i −0.566459 0.000852825i
\(766\) 0 0
\(767\) −1083.62 −1.41280
\(768\) 0 0
\(769\) −285.267 −0.370958 −0.185479 0.982648i \(-0.559384\pi\)
−0.185479 + 0.982648i \(0.559384\pi\)
\(770\) 0 0
\(771\) 674.216 + 1228.02i 0.874470 + 1.59277i
\(772\) 0 0
\(773\) 229.211i 0.296521i −0.988948 0.148260i \(-0.952633\pi\)
0.988948 0.148260i \(-0.0473674\pi\)
\(774\) 0 0
\(775\) 433.551 919.503i 0.559420 1.18646i
\(776\) 0 0
\(777\) 148.893 81.7459i 0.191625 0.105207i
\(778\) 0 0
\(779\) 40.6436 0.0521740
\(780\) 0 0
\(781\) 397.857i 0.509420i
\(782\) 0 0
\(783\) 1274.17 + 82.3141i 1.62729 + 0.105127i
\(784\) 0 0
\(785\) −390.390 615.680i −0.497312 0.784305i
\(786\) 0 0
\(787\) −453.267 −0.575943 −0.287971 0.957639i \(-0.592981\pi\)
−0.287971 + 0.957639i \(0.592981\pi\)
\(788\) 0 0
\(789\) 399.461 + 727.582i 0.506287 + 0.922157i
\(790\) 0 0
\(791\) 3.46799i 0.00438431i
\(792\) 0 0
\(793\) 1424.26i 1.79604i
\(794\) 0 0
\(795\) −129.709 71.4680i −0.163156 0.0898969i
\(796\) 0 0
\(797\) 1276.67i 1.60185i −0.598765 0.800925i \(-0.704342\pi\)
0.598765 0.800925i \(-0.295658\pi\)
\(798\) 0 0
\(799\) 467.340 0.584906
\(800\) 0 0
\(801\) −775.324 493.255i −0.967945 0.615799i
\(802\) 0 0
\(803\) 168.002i 0.209218i
\(804\) 0 0
\(805\) −88.8150 + 56.3157i −0.110329 + 0.0699575i
\(806\) 0 0
\(807\) −407.156 741.598i −0.504530 0.918956i
\(808\) 0 0
\(809\) 797.708i 0.986042i −0.870018 0.493021i \(-0.835893\pi\)
0.870018 0.493021i \(-0.164107\pi\)
\(810\) 0 0
\(811\) 449.833i 0.554665i −0.960774 0.277333i \(-0.910550\pi\)
0.960774 0.277333i \(-0.0894504\pi\)
\(812\) 0 0
\(813\) 251.100 + 457.355i 0.308856 + 0.562553i
\(814\) 0 0
\(815\) −535.867 845.111i −0.657506 1.03695i
\(816\) 0 0
\(817\) 520.657i 0.637280i
\(818\) 0 0
\(819\) 151.871 + 96.6190i 0.185435 + 0.117972i
\(820\) 0 0
\(821\) −42.7636 −0.0520872 −0.0260436 0.999661i \(-0.508291\pi\)
−0.0260436 + 0.999661i \(0.508291\pi\)
\(822\) 0 0
\(823\) 1434.25i 1.74271i −0.490657 0.871353i \(-0.663243\pi\)
0.490657 0.871353i \(-0.336757\pi\)
\(824\) 0 0
\(825\) 725.888 44.6994i 0.879864 0.0541811i
\(826\) 0 0
\(827\) 679.647i 0.821822i −0.911675 0.410911i \(-0.865211\pi\)
0.911675 0.410911i \(-0.134789\pi\)
\(828\) 0 0
\(829\) 336.572i 0.405997i −0.979179 0.202998i \(-0.934931\pi\)
0.979179 0.202998i \(-0.0650687\pi\)
\(830\) 0 0
\(831\) 742.890 + 1353.11i 0.893971 + 1.62829i
\(832\) 0 0
\(833\) 455.582 0.546917
\(834\) 0 0
\(835\) 408.936 + 644.929i 0.489743 + 0.772370i
\(836\) 0 0
\(837\) 1095.63 + 70.7801i 1.30900 + 0.0845640i
\(838\) 0 0
\(839\) 835.229i 0.995506i 0.867319 + 0.497753i \(0.165841\pi\)
−0.867319 + 0.497753i \(0.834159\pi\)
\(840\) 0 0
\(841\) 1395.33 1.65914
\(842\) 0 0
\(843\) 368.063 202.076i 0.436611 0.239710i
\(844\) 0 0
\(845\) 181.051 + 285.533i 0.214261 + 0.337909i
\(846\) 0 0
\(847\) 35.0678i 0.0414024i
\(848\) 0 0
\(849\) 489.069 + 890.795i 0.576053 + 1.04923i
\(850\) 0 0
\(851\) −704.452 −0.827793
\(852\) 0 0
\(853\) −131.483 −0.154142 −0.0770708 0.997026i \(-0.524557\pi\)
−0.0770708 + 0.997026i \(0.524557\pi\)
\(854\) 0 0
\(855\) −0.654585 + 434.785i −0.000765597 + 0.508521i
\(856\) 0 0
\(857\) 126.816 0.147976 0.0739880 0.997259i \(-0.476427\pi\)
0.0739880 + 0.997259i \(0.476427\pi\)
\(858\) 0 0
\(859\) 63.1858i 0.0735573i 0.999323 + 0.0367787i \(0.0117097\pi\)
−0.999323 + 0.0367787i \(0.988290\pi\)
\(860\) 0 0
\(861\) −7.89656 14.3829i −0.00917138 0.0167048i
\(862\) 0 0
\(863\) −742.235 −0.860064 −0.430032 0.902814i \(-0.641498\pi\)
−0.430032 + 0.902814i \(0.641498\pi\)
\(864\) 0 0
\(865\) −618.310 + 392.057i −0.714809 + 0.453245i
\(866\) 0 0
\(867\) −283.367 516.128i −0.326837 0.595304i
\(868\) 0 0
\(869\) 897.832 1.03318
\(870\) 0 0
\(871\) 399.761 0.458968
\(872\) 0 0
\(873\) −738.102 469.575i −0.845478 0.537886i
\(874\) 0 0
\(875\) −20.1832 161.265i −0.0230665 0.184303i
\(876\) 0 0
\(877\) 253.700 0.289282 0.144641 0.989484i \(-0.453797\pi\)
0.144641 + 0.989484i \(0.453797\pi\)
\(878\) 0 0
\(879\) −909.933 + 499.576i −1.03519 + 0.568346i
\(880\) 0 0
\(881\) 536.960i 0.609489i 0.952434 + 0.304745i \(0.0985711\pi\)
−0.952434 + 0.304745i \(0.901429\pi\)
\(882\) 0 0
\(883\) 1329.87 1.50608 0.753039 0.657976i \(-0.228587\pi\)
0.753039 + 0.657976i \(0.228587\pi\)
\(884\) 0 0
\(885\) −509.934 + 925.492i −0.576196 + 1.04575i
\(886\) 0 0
\(887\) −1035.09 −1.16695 −0.583476 0.812130i \(-0.698308\pi\)
−0.583476 + 0.812130i \(0.698308\pi\)
\(888\) 0 0
\(889\) −18.5953 −0.0209171
\(890\) 0 0
\(891\) 332.832 + 711.438i 0.373549 + 0.798472i
\(892\) 0 0
\(893\) 468.898i 0.525081i
\(894\) 0 0
\(895\) 699.106 + 1102.55i 0.781124 + 1.23190i
\(896\) 0 0
\(897\) −359.271 654.380i −0.400525 0.729520i
\(898\) 0 0
\(899\) 1922.97 2.13901
\(900\) 0 0
\(901\) 95.0753i 0.105522i
\(902\) 0 0
\(903\) −184.249 + 101.157i −0.204041 + 0.112024i
\(904\) 0 0
\(905\) 781.956 495.822i 0.864040 0.547870i
\(906\) 0 0
\(907\) −172.237 −0.189898 −0.0949488 0.995482i \(-0.530269\pi\)
−0.0949488 + 0.995482i \(0.530269\pi\)
\(908\) 0 0
\(909\) 487.153 765.733i 0.535922 0.842391i
\(910\) 0 0
\(911\) 790.732i 0.867982i −0.900917 0.433991i \(-0.857105\pi\)
0.900917 0.433991i \(-0.142895\pi\)
\(912\) 0 0
\(913\) 526.515i 0.576687i
\(914\) 0 0
\(915\) −1216.43 670.236i −1.32943 0.732498i
\(916\) 0 0
\(917\) 246.222i 0.268508i
\(918\) 0 0
\(919\) 187.666 0.204206 0.102103 0.994774i \(-0.467443\pi\)
0.102103 + 0.994774i \(0.467443\pi\)
\(920\) 0 0
\(921\) −739.212 1346.41i −0.802619 1.46190i
\(922\) 0 0
\(923\) 631.135i 0.683787i
\(924\) 0 0
\(925\) 464.293 984.703i 0.501938 1.06454i
\(926\) 0 0
\(927\) 1237.18 + 787.086i 1.33461 + 0.849067i
\(928\) 0 0
\(929\) 1157.28i 1.24573i 0.782331 + 0.622863i \(0.214031\pi\)
−0.782331 + 0.622863i \(0.785969\pi\)
\(930\) 0 0
\(931\) 457.100i 0.490977i
\(932\) 0 0
\(933\) −779.083 + 427.736i −0.835030 + 0.458453i
\(934\) 0 0
\(935\) −250.022 394.308i −0.267403 0.421719i
\(936\) 0 0
\(937\) 1321.20i 1.41004i 0.709190 + 0.705018i \(0.249061\pi\)
−0.709190 + 0.705018i \(0.750939\pi\)
\(938\) 0 0
\(939\) 427.806 234.876i 0.455597 0.250134i
\(940\) 0 0
\(941\) 1594.66 1.69464 0.847322 0.531080i \(-0.178213\pi\)
0.847322 + 0.531080i \(0.178213\pi\)
\(942\) 0 0
\(943\) 68.0492i 0.0721625i
\(944\) 0 0
\(945\) 153.988 84.2419i 0.162950 0.0891449i
\(946\) 0 0
\(947\) 368.963i 0.389613i 0.980842 + 0.194806i \(0.0624079\pi\)
−0.980842 + 0.194806i \(0.937592\pi\)
\(948\) 0 0
\(949\) 266.508i 0.280830i
\(950\) 0 0
\(951\) −470.143 + 258.120i −0.494367 + 0.271420i
\(952\) 0 0
\(953\) −1432.04 −1.50266 −0.751331 0.659925i \(-0.770588\pi\)
−0.751331 + 0.659925i \(0.770588\pi\)
\(954\) 0 0
\(955\) −1024.88 + 649.857i −1.07318 + 0.680479i
\(956\) 0 0
\(957\) 662.067 + 1205.89i 0.691815 + 1.26008i
\(958\) 0 0
\(959\) 174.522i 0.181984i
\(960\) 0 0
\(961\) 692.524 0.720628
\(962\) 0 0
\(963\) −984.945 626.614i −1.02279 0.650690i
\(964\) 0 0
\(965\) 73.1597 46.3890i 0.0758132 0.0480716i
\(966\) 0 0
\(967\) 729.658i 0.754558i 0.926100 + 0.377279i \(0.123140\pi\)
−0.926100 + 0.377279i \(0.876860\pi\)
\(968\) 0 0
\(969\) 244.676 134.333i 0.252504 0.138631i
\(970\) 0 0
\(971\) −959.275 −0.987925 −0.493963 0.869483i \(-0.664452\pi\)
−0.493963 + 0.869483i \(0.664452\pi\)
\(972\) 0 0
\(973\) 112.181 0.115294
\(974\) 0 0
\(975\) 1151.50 70.9083i 1.18103 0.0727265i
\(976\) 0 0
\(977\) −901.809 −0.923039 −0.461520 0.887130i \(-0.652696\pi\)
−0.461520 + 0.887130i \(0.652696\pi\)
\(978\) 0 0
\(979\) 990.076i 1.01131i
\(980\) 0 0
\(981\) −1320.65 840.186i −1.34623 0.856459i
\(982\) 0 0
\(983\) 477.936 0.486201 0.243101 0.970001i \(-0.421835\pi\)
0.243101 + 0.970001i \(0.421835\pi\)
\(984\) 0 0
\(985\) −1491.26 + 945.578i −1.51397 + 0.959978i
\(986\) 0 0
\(987\) −165.932 + 91.1011i −0.168118 + 0.0923011i
\(988\) 0 0
\(989\) 871.733 0.881428
\(990\) 0 0
\(991\) 1099.82 1.10981 0.554904 0.831914i \(-0.312755\pi\)
0.554904 + 0.831914i \(0.312755\pi\)
\(992\) 0 0
\(993\) 1672.87 918.449i 1.68466 0.924923i
\(994\) 0 0
\(995\) 169.191 + 266.830i 0.170042 + 0.268171i
\(996\) 0 0
\(997\) −884.637 −0.887298 −0.443649 0.896201i \(-0.646316\pi\)
−0.443649 + 0.896201i \(0.646316\pi\)
\(998\) 0 0
\(999\) 1173.32 + 75.7990i 1.17450 + 0.0758748i
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 960.3.i.a.929.24 yes 32
3.2 odd 2 inner 960.3.i.a.929.17 yes 32
4.3 odd 2 inner 960.3.i.a.929.12 yes 32
5.4 even 2 inner 960.3.i.a.929.11 yes 32
8.3 odd 2 inner 960.3.i.a.929.21 yes 32
8.5 even 2 inner 960.3.i.a.929.9 32
12.11 even 2 inner 960.3.i.a.929.13 yes 32
15.14 odd 2 inner 960.3.i.a.929.14 yes 32
20.19 odd 2 inner 960.3.i.a.929.23 yes 32
24.5 odd 2 inner 960.3.i.a.929.16 yes 32
24.11 even 2 inner 960.3.i.a.929.20 yes 32
40.19 odd 2 inner 960.3.i.a.929.10 yes 32
40.29 even 2 inner 960.3.i.a.929.22 yes 32
60.59 even 2 inner 960.3.i.a.929.18 yes 32
120.29 odd 2 inner 960.3.i.a.929.19 yes 32
120.59 even 2 inner 960.3.i.a.929.15 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.3.i.a.929.9 32 8.5 even 2 inner
960.3.i.a.929.10 yes 32 40.19 odd 2 inner
960.3.i.a.929.11 yes 32 5.4 even 2 inner
960.3.i.a.929.12 yes 32 4.3 odd 2 inner
960.3.i.a.929.13 yes 32 12.11 even 2 inner
960.3.i.a.929.14 yes 32 15.14 odd 2 inner
960.3.i.a.929.15 yes 32 120.59 even 2 inner
960.3.i.a.929.16 yes 32 24.5 odd 2 inner
960.3.i.a.929.17 yes 32 3.2 odd 2 inner
960.3.i.a.929.18 yes 32 60.59 even 2 inner
960.3.i.a.929.19 yes 32 120.29 odd 2 inner
960.3.i.a.929.20 yes 32 24.11 even 2 inner
960.3.i.a.929.21 yes 32 8.3 odd 2 inner
960.3.i.a.929.22 yes 32 40.29 even 2 inner
960.3.i.a.929.23 yes 32 20.19 odd 2 inner
960.3.i.a.929.24 yes 32 1.1 even 1 trivial