Properties

Label 1664.2.n.b.417.8
Level $1664$
Weight $2$
Character 1664.417
Analytic conductor $13.287$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1664,2,Mod(417,1664)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1664, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1664.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1664.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2871068963\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 208)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 417.8
Character \(\chi\) \(=\) 1664.417
Dual form 1664.2.n.b.1249.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.902776 - 0.902776i) q^{3} +(1.29480 - 1.29480i) q^{5} -3.20594i q^{7} -1.36999i q^{9} +O(q^{10})\) \(q+(-0.902776 - 0.902776i) q^{3} +(1.29480 - 1.29480i) q^{5} -3.20594i q^{7} -1.36999i q^{9} +(2.72143 - 2.72143i) q^{11} +(-0.707107 - 0.707107i) q^{13} -2.33784 q^{15} -6.03671 q^{17} +(-1.89947 - 1.89947i) q^{19} +(-2.89424 + 2.89424i) q^{21} -8.67448i q^{23} +1.64697i q^{25} +(-3.94512 + 3.94512i) q^{27} +(6.82640 + 6.82640i) q^{29} +5.23082 q^{31} -4.91369 q^{33} +(-4.15106 - 4.15106i) q^{35} +(0.208536 - 0.208536i) q^{37} +1.27672i q^{39} +9.29774i q^{41} +(-0.0416722 + 0.0416722i) q^{43} +(-1.77387 - 1.77387i) q^{45} +2.16694 q^{47} -3.27803 q^{49} +(5.44980 + 5.44980i) q^{51} +(6.38002 - 6.38002i) q^{53} -7.04745i q^{55} +3.42959i q^{57} +(-3.38796 + 3.38796i) q^{59} +(-1.72743 - 1.72743i) q^{61} -4.39210 q^{63} -1.83113 q^{65} +(1.56033 + 1.56033i) q^{67} +(-7.83111 + 7.83111i) q^{69} +9.73843i q^{71} +6.08728i q^{73} +(1.48684 - 1.48684i) q^{75} +(-8.72475 - 8.72475i) q^{77} -5.31445 q^{79} +3.01316 q^{81} +(-9.71028 - 9.71028i) q^{83} +(-7.81635 + 7.81635i) q^{85} -12.3254i q^{87} -5.94615i q^{89} +(-2.26694 + 2.26694i) q^{91} +(-4.72226 - 4.72226i) q^{93} -4.91887 q^{95} -9.79055 q^{97} +(-3.72834 - 3.72834i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 8 q^{11} - 16 q^{15} + 16 q^{19} + 16 q^{29} + 24 q^{31} - 24 q^{35} + 16 q^{37} + 8 q^{43} - 40 q^{47} - 48 q^{49} + 24 q^{51} - 16 q^{53} - 32 q^{61} + 40 q^{63} - 16 q^{67} - 32 q^{69} + 40 q^{75} - 16 q^{77} + 32 q^{79} - 48 q^{81} - 40 q^{83} + 32 q^{85} - 48 q^{95} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) −0.902776 0.902776i −0.521218 0.521218i 0.396721 0.917939i \(-0.370148\pi\)
−0.917939 + 0.396721i \(0.870148\pi\)
\(4\) 0 0
\(5\) 1.29480 1.29480i 0.579054 0.579054i −0.355589 0.934643i \(-0.615720\pi\)
0.934643 + 0.355589i \(0.115720\pi\)
\(6\) 0 0
\(7\) 3.20594i 1.21173i −0.795567 0.605865i \(-0.792827\pi\)
0.795567 0.605865i \(-0.207173\pi\)
\(8\) 0 0
\(9\) 1.36999i 0.456663i
\(10\) 0 0
\(11\) 2.72143 2.72143i 0.820543 0.820543i −0.165642 0.986186i \(-0.552970\pi\)
0.986186 + 0.165642i \(0.0529698\pi\)
\(12\) 0 0
\(13\) −0.707107 0.707107i −0.196116 0.196116i
\(14\) 0 0
\(15\) −2.33784 −0.603627
\(16\) 0 0
\(17\) −6.03671 −1.46412 −0.732058 0.681242i \(-0.761440\pi\)
−0.732058 + 0.681242i \(0.761440\pi\)
\(18\) 0 0
\(19\) −1.89947 1.89947i −0.435768 0.435768i 0.454817 0.890585i \(-0.349705\pi\)
−0.890585 + 0.454817i \(0.849705\pi\)
\(20\) 0 0
\(21\) −2.89424 + 2.89424i −0.631576 + 0.631576i
\(22\) 0 0
\(23\) 8.67448i 1.80875i −0.426734 0.904377i \(-0.640336\pi\)
0.426734 0.904377i \(-0.359664\pi\)
\(24\) 0 0
\(25\) 1.64697i 0.329393i
\(26\) 0 0
\(27\) −3.94512 + 3.94512i −0.759239 + 0.759239i
\(28\) 0 0
\(29\) 6.82640 + 6.82640i 1.26763 + 1.26763i 0.947309 + 0.320321i \(0.103791\pi\)
0.320321 + 0.947309i \(0.396209\pi\)
\(30\) 0 0
\(31\) 5.23082 0.939484 0.469742 0.882804i \(-0.344347\pi\)
0.469742 + 0.882804i \(0.344347\pi\)
\(32\) 0 0
\(33\) −4.91369 −0.855364
\(34\) 0 0
\(35\) −4.15106 4.15106i −0.701657 0.701657i
\(36\) 0 0
\(37\) 0.208536 0.208536i 0.0342830 0.0342830i −0.689757 0.724041i \(-0.742283\pi\)
0.724041 + 0.689757i \(0.242283\pi\)
\(38\) 0 0
\(39\) 1.27672i 0.204439i
\(40\) 0 0
\(41\) 9.29774i 1.45206i 0.687662 + 0.726031i \(0.258637\pi\)
−0.687662 + 0.726031i \(0.741363\pi\)
\(42\) 0 0
\(43\) −0.0416722 + 0.0416722i −0.00635495 + 0.00635495i −0.710277 0.703922i \(-0.751430\pi\)
0.703922 + 0.710277i \(0.251430\pi\)
\(44\) 0 0
\(45\) −1.77387 1.77387i −0.264433 0.264433i
\(46\) 0 0
\(47\) 2.16694 0.316081 0.158040 0.987433i \(-0.449482\pi\)
0.158040 + 0.987433i \(0.449482\pi\)
\(48\) 0 0
\(49\) −3.27803 −0.468290
\(50\) 0 0
\(51\) 5.44980 + 5.44980i 0.763124 + 0.763124i
\(52\) 0 0
\(53\) 6.38002 6.38002i 0.876363 0.876363i −0.116793 0.993156i \(-0.537261\pi\)
0.993156 + 0.116793i \(0.0372614\pi\)
\(54\) 0 0
\(55\) 7.04745i 0.950278i
\(56\) 0 0
\(57\) 3.42959i 0.454260i
\(58\) 0 0
\(59\) −3.38796 + 3.38796i −0.441075 + 0.441075i −0.892373 0.451298i \(-0.850961\pi\)
0.451298 + 0.892373i \(0.350961\pi\)
\(60\) 0 0
\(61\) −1.72743 1.72743i −0.221175 0.221175i 0.587818 0.808993i \(-0.299987\pi\)
−0.808993 + 0.587818i \(0.799987\pi\)
\(62\) 0 0
\(63\) −4.39210 −0.553353
\(64\) 0 0
\(65\) −1.83113 −0.227124
\(66\) 0 0
\(67\) 1.56033 + 1.56033i 0.190624 + 0.190624i 0.795966 0.605342i \(-0.206963\pi\)
−0.605342 + 0.795966i \(0.706963\pi\)
\(68\) 0 0
\(69\) −7.83111 + 7.83111i −0.942755 + 0.942755i
\(70\) 0 0
\(71\) 9.73843i 1.15574i 0.816129 + 0.577869i \(0.196116\pi\)
−0.816129 + 0.577869i \(0.803884\pi\)
\(72\) 0 0
\(73\) 6.08728i 0.712462i 0.934398 + 0.356231i \(0.115938\pi\)
−0.934398 + 0.356231i \(0.884062\pi\)
\(74\) 0 0
\(75\) 1.48684 1.48684i 0.171686 0.171686i
\(76\) 0 0
\(77\) −8.72475 8.72475i −0.994277 0.994277i
\(78\) 0 0
\(79\) −5.31445 −0.597922 −0.298961 0.954265i \(-0.596640\pi\)
−0.298961 + 0.954265i \(0.596640\pi\)
\(80\) 0 0
\(81\) 3.01316 0.334795
\(82\) 0 0
\(83\) −9.71028 9.71028i −1.06584 1.06584i −0.997674 0.0681682i \(-0.978285\pi\)
−0.0681682 0.997674i \(-0.521715\pi\)
\(84\) 0 0
\(85\) −7.81635 + 7.81635i −0.847802 + 0.847802i
\(86\) 0 0
\(87\) 12.3254i 1.32142i
\(88\) 0 0
\(89\) 5.94615i 0.630291i −0.949043 0.315145i \(-0.897947\pi\)
0.949043 0.315145i \(-0.102053\pi\)
\(90\) 0 0
\(91\) −2.26694 + 2.26694i −0.237640 + 0.237640i
\(92\) 0 0
\(93\) −4.72226 4.72226i −0.489676 0.489676i
\(94\) 0 0
\(95\) −4.91887 −0.504666
\(96\) 0 0
\(97\) −9.79055 −0.994080 −0.497040 0.867728i \(-0.665580\pi\)
−0.497040 + 0.867728i \(0.665580\pi\)
\(98\) 0 0
\(99\) −3.72834 3.72834i −0.374712 0.374712i
\(100\) 0 0
\(101\) 4.75506 4.75506i 0.473146 0.473146i −0.429785 0.902931i \(-0.641411\pi\)
0.902931 + 0.429785i \(0.141411\pi\)
\(102\) 0 0
\(103\) 2.88844i 0.284607i −0.989823 0.142303i \(-0.954549\pi\)
0.989823 0.142303i \(-0.0454509\pi\)
\(104\) 0 0
\(105\) 7.49496i 0.731433i
\(106\) 0 0
\(107\) −3.70795 + 3.70795i −0.358461 + 0.358461i −0.863245 0.504785i \(-0.831572\pi\)
0.504785 + 0.863245i \(0.331572\pi\)
\(108\) 0 0
\(109\) −2.76524 2.76524i −0.264862 0.264862i 0.562164 0.827026i \(-0.309969\pi\)
−0.827026 + 0.562164i \(0.809969\pi\)
\(110\) 0 0
\(111\) −0.376522 −0.0357379
\(112\) 0 0
\(113\) 9.12934 0.858816 0.429408 0.903111i \(-0.358722\pi\)
0.429408 + 0.903111i \(0.358722\pi\)
\(114\) 0 0
\(115\) −11.2317 11.2317i −1.04737 1.04737i
\(116\) 0 0
\(117\) −0.968729 + 0.968729i −0.0895591 + 0.0895591i
\(118\) 0 0
\(119\) 19.3533i 1.77411i
\(120\) 0 0
\(121\) 3.81241i 0.346583i
\(122\) 0 0
\(123\) 8.39378 8.39378i 0.756841 0.756841i
\(124\) 0 0
\(125\) 8.60652 + 8.60652i 0.769790 + 0.769790i
\(126\) 0 0
\(127\) 12.7097 1.12780 0.563901 0.825842i \(-0.309300\pi\)
0.563901 + 0.825842i \(0.309300\pi\)
\(128\) 0 0
\(129\) 0.0752414 0.00662464
\(130\) 0 0
\(131\) −6.39425 6.39425i −0.558668 0.558668i 0.370260 0.928928i \(-0.379269\pi\)
−0.928928 + 0.370260i \(0.879269\pi\)
\(132\) 0 0
\(133\) −6.08957 + 6.08957i −0.528033 + 0.528033i
\(134\) 0 0
\(135\) 10.2163i 0.879281i
\(136\) 0 0
\(137\) 19.8173i 1.69310i −0.532306 0.846552i \(-0.678674\pi\)
0.532306 0.846552i \(-0.321326\pi\)
\(138\) 0 0
\(139\) 7.81457 7.81457i 0.662823 0.662823i −0.293221 0.956045i \(-0.594727\pi\)
0.956045 + 0.293221i \(0.0947273\pi\)
\(140\) 0 0
\(141\) −1.95626 1.95626i −0.164747 0.164747i
\(142\) 0 0
\(143\) −3.84869 −0.321844
\(144\) 0 0
\(145\) 17.6777 1.46805
\(146\) 0 0
\(147\) 2.95933 + 2.95933i 0.244081 + 0.244081i
\(148\) 0 0
\(149\) −6.86136 + 6.86136i −0.562104 + 0.562104i −0.929905 0.367800i \(-0.880111\pi\)
0.367800 + 0.929905i \(0.380111\pi\)
\(150\) 0 0
\(151\) 12.0002i 0.976560i −0.872687 0.488280i \(-0.837624\pi\)
0.872687 0.488280i \(-0.162376\pi\)
\(152\) 0 0
\(153\) 8.27023i 0.668608i
\(154\) 0 0
\(155\) 6.77289 6.77289i 0.544012 0.544012i
\(156\) 0 0
\(157\) −1.81214 1.81214i −0.144625 0.144625i 0.631087 0.775712i \(-0.282609\pi\)
−0.775712 + 0.631087i \(0.782609\pi\)
\(158\) 0 0
\(159\) −11.5195 −0.913553
\(160\) 0 0
\(161\) −27.8098 −2.19172
\(162\) 0 0
\(163\) 13.8786 + 13.8786i 1.08706 + 1.08706i 0.995830 + 0.0912264i \(0.0290787\pi\)
0.0912264 + 0.995830i \(0.470921\pi\)
\(164\) 0 0
\(165\) −6.36227 + 6.36227i −0.495302 + 0.495302i
\(166\) 0 0
\(167\) 3.99276i 0.308969i 0.987995 + 0.154485i \(0.0493717\pi\)
−0.987995 + 0.154485i \(0.950628\pi\)
\(168\) 0 0
\(169\) 1.00000i 0.0769231i
\(170\) 0 0
\(171\) −2.60225 + 2.60225i −0.198999 + 0.198999i
\(172\) 0 0
\(173\) −12.5447 12.5447i −0.953756 0.953756i 0.0452212 0.998977i \(-0.485601\pi\)
−0.998977 + 0.0452212i \(0.985601\pi\)
\(174\) 0 0
\(175\) 5.28007 0.399136
\(176\) 0 0
\(177\) 6.11715 0.459793
\(178\) 0 0
\(179\) 6.56516 + 6.56516i 0.490703 + 0.490703i 0.908528 0.417825i \(-0.137207\pi\)
−0.417825 + 0.908528i \(0.637207\pi\)
\(180\) 0 0
\(181\) −6.91102 + 6.91102i −0.513692 + 0.513692i −0.915656 0.401964i \(-0.868328\pi\)
0.401964 + 0.915656i \(0.368328\pi\)
\(182\) 0 0
\(183\) 3.11897i 0.230561i
\(184\) 0 0
\(185\) 0.540025i 0.0397035i
\(186\) 0 0
\(187\) −16.4285 + 16.4285i −1.20137 + 1.20137i
\(188\) 0 0
\(189\) 12.6478 + 12.6478i 0.919993 + 0.919993i
\(190\) 0 0
\(191\) −11.2566 −0.814502 −0.407251 0.913316i \(-0.633513\pi\)
−0.407251 + 0.913316i \(0.633513\pi\)
\(192\) 0 0
\(193\) −2.54609 −0.183272 −0.0916358 0.995793i \(-0.529210\pi\)
−0.0916358 + 0.995793i \(0.529210\pi\)
\(194\) 0 0
\(195\) 1.65310 + 1.65310i 0.118381 + 0.118381i
\(196\) 0 0
\(197\) −4.30038 + 4.30038i −0.306389 + 0.306389i −0.843507 0.537118i \(-0.819513\pi\)
0.537118 + 0.843507i \(0.319513\pi\)
\(198\) 0 0
\(199\) 26.2885i 1.86354i −0.363048 0.931771i \(-0.618264\pi\)
0.363048 0.931771i \(-0.381736\pi\)
\(200\) 0 0
\(201\) 2.81725i 0.198714i
\(202\) 0 0
\(203\) 21.8850 21.8850i 1.53603 1.53603i
\(204\) 0 0
\(205\) 12.0387 + 12.0387i 0.840823 + 0.840823i
\(206\) 0 0
\(207\) −11.8840 −0.825992
\(208\) 0 0
\(209\) −10.3385 −0.715132
\(210\) 0 0
\(211\) 7.51991 + 7.51991i 0.517692 + 0.517692i 0.916872 0.399180i \(-0.130705\pi\)
−0.399180 + 0.916872i \(0.630705\pi\)
\(212\) 0 0
\(213\) 8.79162 8.79162i 0.602392 0.602392i
\(214\) 0 0
\(215\) 0.107915i 0.00735972i
\(216\) 0 0
\(217\) 16.7697i 1.13840i
\(218\) 0 0
\(219\) 5.49545 5.49545i 0.371348 0.371348i
\(220\) 0 0
\(221\) 4.26860 + 4.26860i 0.287137 + 0.287137i
\(222\) 0 0
\(223\) −18.7318 −1.25438 −0.627188 0.778868i \(-0.715794\pi\)
−0.627188 + 0.778868i \(0.715794\pi\)
\(224\) 0 0
\(225\) 2.25633 0.150422
\(226\) 0 0
\(227\) −0.552713 0.552713i −0.0366849 0.0366849i 0.688526 0.725211i \(-0.258258\pi\)
−0.725211 + 0.688526i \(0.758258\pi\)
\(228\) 0 0
\(229\) 14.9481 14.9481i 0.987798 0.987798i −0.0121281 0.999926i \(-0.503861\pi\)
0.999926 + 0.0121281i \(0.00386060\pi\)
\(230\) 0 0
\(231\) 15.7530i 1.03647i
\(232\) 0 0
\(233\) 23.9526i 1.56919i −0.620009 0.784595i \(-0.712871\pi\)
0.620009 0.784595i \(-0.287129\pi\)
\(234\) 0 0
\(235\) 2.80576 2.80576i 0.183028 0.183028i
\(236\) 0 0
\(237\) 4.79776 + 4.79776i 0.311648 + 0.311648i
\(238\) 0 0
\(239\) 1.49140 0.0964709 0.0482354 0.998836i \(-0.484640\pi\)
0.0482354 + 0.998836i \(0.484640\pi\)
\(240\) 0 0
\(241\) −0.0970048 −0.00624863 −0.00312431 0.999995i \(-0.500995\pi\)
−0.00312431 + 0.999995i \(0.500995\pi\)
\(242\) 0 0
\(243\) 9.11516 + 9.11516i 0.584738 + 0.584738i
\(244\) 0 0
\(245\) −4.24441 + 4.24441i −0.271165 + 0.271165i
\(246\) 0 0
\(247\) 2.68625i 0.170922i
\(248\) 0 0
\(249\) 17.5324i 1.11107i
\(250\) 0 0
\(251\) −7.32996 + 7.32996i −0.462663 + 0.462663i −0.899527 0.436864i \(-0.856089\pi\)
0.436864 + 0.899527i \(0.356089\pi\)
\(252\) 0 0
\(253\) −23.6070 23.6070i −1.48416 1.48416i
\(254\) 0 0
\(255\) 14.1128 0.883780
\(256\) 0 0
\(257\) 28.9029 1.80291 0.901457 0.432869i \(-0.142499\pi\)
0.901457 + 0.432869i \(0.142499\pi\)
\(258\) 0 0
\(259\) −0.668552 0.668552i −0.0415418 0.0415418i
\(260\) 0 0
\(261\) 9.35210 9.35210i 0.578880 0.578880i
\(262\) 0 0
\(263\) 17.4287i 1.07470i 0.843360 + 0.537349i \(0.180574\pi\)
−0.843360 + 0.537349i \(0.819426\pi\)
\(264\) 0 0
\(265\) 16.5217i 1.01492i
\(266\) 0 0
\(267\) −5.36804 + 5.36804i −0.328519 + 0.328519i
\(268\) 0 0
\(269\) 12.0262 + 12.0262i 0.733248 + 0.733248i 0.971262 0.238013i \(-0.0764962\pi\)
−0.238013 + 0.971262i \(0.576496\pi\)
\(270\) 0 0
\(271\) 1.79480 0.109026 0.0545131 0.998513i \(-0.482639\pi\)
0.0545131 + 0.998513i \(0.482639\pi\)
\(272\) 0 0
\(273\) 4.09308 0.247724
\(274\) 0 0
\(275\) 4.48211 + 4.48211i 0.270282 + 0.270282i
\(276\) 0 0
\(277\) 1.31019 1.31019i 0.0787215 0.0787215i −0.666650 0.745371i \(-0.732272\pi\)
0.745371 + 0.666650i \(0.232272\pi\)
\(278\) 0 0
\(279\) 7.16618i 0.429028i
\(280\) 0 0
\(281\) 26.8404i 1.60116i 0.599225 + 0.800581i \(0.295475\pi\)
−0.599225 + 0.800581i \(0.704525\pi\)
\(282\) 0 0
\(283\) −1.50326 + 1.50326i −0.0893595 + 0.0893595i −0.750374 0.661014i \(-0.770126\pi\)
0.661014 + 0.750374i \(0.270126\pi\)
\(284\) 0 0
\(285\) 4.44064 + 4.44064i 0.263041 + 0.263041i
\(286\) 0 0
\(287\) 29.8080 1.75951
\(288\) 0 0
\(289\) 19.4418 1.14364
\(290\) 0 0
\(291\) 8.83868 + 8.83868i 0.518133 + 0.518133i
\(292\) 0 0
\(293\) 4.58243 4.58243i 0.267709 0.267709i −0.560468 0.828176i \(-0.689379\pi\)
0.828176 + 0.560468i \(0.189379\pi\)
\(294\) 0 0
\(295\) 8.77350i 0.510813i
\(296\) 0 0
\(297\) 21.4728i 1.24598i
\(298\) 0 0
\(299\) −6.13378 + 6.13378i −0.354726 + 0.354726i
\(300\) 0 0
\(301\) 0.133599 + 0.133599i 0.00770049 + 0.00770049i
\(302\) 0 0
\(303\) −8.58551 −0.493224
\(304\) 0 0
\(305\) −4.47337 −0.256144
\(306\) 0 0
\(307\) −13.7315 13.7315i −0.783699 0.783699i 0.196754 0.980453i \(-0.436960\pi\)
−0.980453 + 0.196754i \(0.936960\pi\)
\(308\) 0 0
\(309\) −2.60762 + 2.60762i −0.148342 + 0.148342i
\(310\) 0 0
\(311\) 1.60913i 0.0912455i 0.998959 + 0.0456228i \(0.0145272\pi\)
−0.998959 + 0.0456228i \(0.985473\pi\)
\(312\) 0 0
\(313\) 14.3857i 0.813126i −0.913623 0.406563i \(-0.866727\pi\)
0.913623 0.406563i \(-0.133273\pi\)
\(314\) 0 0
\(315\) −5.68691 + 5.68691i −0.320421 + 0.320421i
\(316\) 0 0
\(317\) 12.4126 + 12.4126i 0.697160 + 0.697160i 0.963797 0.266637i \(-0.0859124\pi\)
−0.266637 + 0.963797i \(0.585912\pi\)
\(318\) 0 0
\(319\) 37.1552 2.08029
\(320\) 0 0
\(321\) 6.69489 0.373673
\(322\) 0 0
\(323\) 11.4665 + 11.4665i 0.638015 + 0.638015i
\(324\) 0 0
\(325\) 1.16458 1.16458i 0.0645994 0.0645994i
\(326\) 0 0
\(327\) 4.99279i 0.276102i
\(328\) 0 0
\(329\) 6.94708i 0.383005i
\(330\) 0 0
\(331\) 17.1517 17.1517i 0.942742 0.942742i −0.0557049 0.998447i \(-0.517741\pi\)
0.998447 + 0.0557049i \(0.0177406\pi\)
\(332\) 0 0
\(333\) −0.285692 0.285692i −0.0156558 0.0156558i
\(334\) 0 0
\(335\) 4.04064 0.220764
\(336\) 0 0
\(337\) 30.8004 1.67781 0.838903 0.544281i \(-0.183198\pi\)
0.838903 + 0.544281i \(0.183198\pi\)
\(338\) 0 0
\(339\) −8.24175 8.24175i −0.447631 0.447631i
\(340\) 0 0
\(341\) 14.2353 14.2353i 0.770887 0.770887i
\(342\) 0 0
\(343\) 11.9324i 0.644289i
\(344\) 0 0
\(345\) 20.2795i 1.09181i
\(346\) 0 0
\(347\) −22.1815 + 22.1815i −1.19076 + 1.19076i −0.213910 + 0.976853i \(0.568620\pi\)
−0.976853 + 0.213910i \(0.931380\pi\)
\(348\) 0 0
\(349\) −15.7484 15.7484i −0.842990 0.842990i 0.146256 0.989247i \(-0.453278\pi\)
−0.989247 + 0.146256i \(0.953278\pi\)
\(350\) 0 0
\(351\) 5.57925 0.297798
\(352\) 0 0
\(353\) 28.1232 1.49685 0.748424 0.663220i \(-0.230811\pi\)
0.748424 + 0.663220i \(0.230811\pi\)
\(354\) 0 0
\(355\) 12.6094 + 12.6094i 0.669235 + 0.669235i
\(356\) 0 0
\(357\) 17.4717 17.4717i 0.924701 0.924701i
\(358\) 0 0
\(359\) 2.91495i 0.153845i −0.997037 0.0769225i \(-0.975491\pi\)
0.997037 0.0769225i \(-0.0245094\pi\)
\(360\) 0 0
\(361\) 11.7841i 0.620213i
\(362\) 0 0
\(363\) −3.44176 + 3.44176i −0.180645 + 0.180645i
\(364\) 0 0
\(365\) 7.88183 + 7.88183i 0.412554 + 0.412554i
\(366\) 0 0
\(367\) −6.29896 −0.328803 −0.164402 0.986394i \(-0.552569\pi\)
−0.164402 + 0.986394i \(0.552569\pi\)
\(368\) 0 0
\(369\) 12.7378 0.663104
\(370\) 0 0
\(371\) −20.4539 20.4539i −1.06192 1.06192i
\(372\) 0 0
\(373\) 12.7757 12.7757i 0.661498 0.661498i −0.294235 0.955733i \(-0.595065\pi\)
0.955733 + 0.294235i \(0.0950648\pi\)
\(374\) 0 0
\(375\) 15.5395i 0.802457i
\(376\) 0 0
\(377\) 9.65398i 0.497205i
\(378\) 0 0
\(379\) 17.8540 17.8540i 0.917097 0.917097i −0.0797198 0.996817i \(-0.525403\pi\)
0.996817 + 0.0797198i \(0.0254026\pi\)
\(380\) 0 0
\(381\) −11.4740 11.4740i −0.587831 0.587831i
\(382\) 0 0
\(383\) 9.20840 0.470527 0.235264 0.971932i \(-0.424405\pi\)
0.235264 + 0.971932i \(0.424405\pi\)
\(384\) 0 0
\(385\) −22.5937 −1.15148
\(386\) 0 0
\(387\) 0.0570905 + 0.0570905i 0.00290208 + 0.00290208i
\(388\) 0 0
\(389\) 19.1892 19.1892i 0.972930 0.972930i −0.0267131 0.999643i \(-0.508504\pi\)
0.999643 + 0.0267131i \(0.00850406\pi\)
\(390\) 0 0
\(391\) 52.3653i 2.64823i
\(392\) 0 0
\(393\) 11.5452i 0.582376i
\(394\) 0 0
\(395\) −6.88117 + 6.88117i −0.346229 + 0.346229i
\(396\) 0 0
\(397\) 15.2018 + 15.2018i 0.762956 + 0.762956i 0.976856 0.213900i \(-0.0686165\pi\)
−0.213900 + 0.976856i \(0.568617\pi\)
\(398\) 0 0
\(399\) 10.9950 0.550441
\(400\) 0 0
\(401\) −18.1785 −0.907793 −0.453897 0.891054i \(-0.649966\pi\)
−0.453897 + 0.891054i \(0.649966\pi\)
\(402\) 0 0
\(403\) −3.69875 3.69875i −0.184248 0.184248i
\(404\) 0 0
\(405\) 3.90145 3.90145i 0.193864 0.193864i
\(406\) 0 0
\(407\) 1.13503i 0.0562615i
\(408\) 0 0
\(409\) 5.29435i 0.261789i 0.991396 + 0.130894i \(0.0417849\pi\)
−0.991396 + 0.130894i \(0.958215\pi\)
\(410\) 0 0
\(411\) −17.8906 + 17.8906i −0.882477 + 0.882477i
\(412\) 0 0
\(413\) 10.8616 + 10.8616i 0.534464 + 0.534464i
\(414\) 0 0
\(415\) −25.1458 −1.23436
\(416\) 0 0
\(417\) −14.1096 −0.690951
\(418\) 0 0
\(419\) 1.93469 + 1.93469i 0.0945158 + 0.0945158i 0.752784 0.658268i \(-0.228711\pi\)
−0.658268 + 0.752784i \(0.728711\pi\)
\(420\) 0 0
\(421\) 7.75321 7.75321i 0.377868 0.377868i −0.492464 0.870333i \(-0.663904\pi\)
0.870333 + 0.492464i \(0.163904\pi\)
\(422\) 0 0
\(423\) 2.96869i 0.144343i
\(424\) 0 0
\(425\) 9.94226i 0.482270i
\(426\) 0 0
\(427\) −5.53804 + 5.53804i −0.268004 + 0.268004i
\(428\) 0 0
\(429\) 3.47451 + 3.47451i 0.167751 + 0.167751i
\(430\) 0 0
\(431\) −28.1933 −1.35802 −0.679011 0.734128i \(-0.737591\pi\)
−0.679011 + 0.734128i \(0.737591\pi\)
\(432\) 0 0
\(433\) −28.5535 −1.37219 −0.686097 0.727511i \(-0.740677\pi\)
−0.686097 + 0.727511i \(0.740677\pi\)
\(434\) 0 0
\(435\) −15.9590 15.9590i −0.765175 0.765175i
\(436\) 0 0
\(437\) −16.4769 + 16.4769i −0.788196 + 0.788196i
\(438\) 0 0
\(439\) 0.436290i 0.0208230i −0.999946 0.0104115i \(-0.996686\pi\)
0.999946 0.0104115i \(-0.00331414\pi\)
\(440\) 0 0
\(441\) 4.49087i 0.213851i
\(442\) 0 0
\(443\) −22.4448 + 22.4448i −1.06638 + 1.06638i −0.0687507 + 0.997634i \(0.521901\pi\)
−0.997634 + 0.0687507i \(0.978099\pi\)
\(444\) 0 0
\(445\) −7.69910 7.69910i −0.364972 0.364972i
\(446\) 0 0
\(447\) 12.3885 0.585958
\(448\) 0 0
\(449\) 1.83048 0.0863856 0.0431928 0.999067i \(-0.486247\pi\)
0.0431928 + 0.999067i \(0.486247\pi\)
\(450\) 0 0
\(451\) 25.3032 + 25.3032i 1.19148 + 1.19148i
\(452\) 0 0
\(453\) −10.8335 + 10.8335i −0.509001 + 0.509001i
\(454\) 0 0
\(455\) 5.87048i 0.275213i
\(456\) 0 0
\(457\) 18.3262i 0.857264i −0.903479 0.428632i \(-0.858996\pi\)
0.903479 0.428632i \(-0.141004\pi\)
\(458\) 0 0
\(459\) 23.8156 23.8156i 1.11161 1.11161i
\(460\) 0 0
\(461\) 10.1809 + 10.1809i 0.474171 + 0.474171i 0.903262 0.429090i \(-0.141166\pi\)
−0.429090 + 0.903262i \(0.641166\pi\)
\(462\) 0 0
\(463\) 28.8403 1.34032 0.670162 0.742215i \(-0.266225\pi\)
0.670162 + 0.742215i \(0.266225\pi\)
\(464\) 0 0
\(465\) −12.2288 −0.567097
\(466\) 0 0
\(467\) 11.4431 + 11.4431i 0.529526 + 0.529526i 0.920431 0.390905i \(-0.127838\pi\)
−0.390905 + 0.920431i \(0.627838\pi\)
\(468\) 0 0
\(469\) 5.00231 5.00231i 0.230985 0.230985i
\(470\) 0 0
\(471\) 3.27192i 0.150762i
\(472\) 0 0
\(473\) 0.226816i 0.0104290i
\(474\) 0 0
\(475\) 3.12836 3.12836i 0.143539 0.143539i
\(476\) 0 0
\(477\) −8.74057 8.74057i −0.400203 0.400203i
\(478\) 0 0
\(479\) 26.9400 1.23092 0.615459 0.788169i \(-0.288971\pi\)
0.615459 + 0.788169i \(0.288971\pi\)
\(480\) 0 0
\(481\) −0.294914 −0.0134469
\(482\) 0 0
\(483\) 25.1061 + 25.1061i 1.14237 + 1.14237i
\(484\) 0 0
\(485\) −12.6768 + 12.6768i −0.575626 + 0.575626i
\(486\) 0 0
\(487\) 10.4127i 0.471844i −0.971772 0.235922i \(-0.924189\pi\)
0.971772 0.235922i \(-0.0758110\pi\)
\(488\) 0 0
\(489\) 25.0586i 1.13319i
\(490\) 0 0
\(491\) −3.32818 + 3.32818i −0.150199 + 0.150199i −0.778207 0.628008i \(-0.783870\pi\)
0.628008 + 0.778207i \(0.283870\pi\)
\(492\) 0 0
\(493\) −41.2090 41.2090i −1.85596 1.85596i
\(494\) 0 0
\(495\) −9.65493 −0.433957
\(496\) 0 0
\(497\) 31.2208 1.40044
\(498\) 0 0
\(499\) −12.7043 12.7043i −0.568720 0.568720i 0.363049 0.931770i \(-0.381736\pi\)
−0.931770 + 0.363049i \(0.881736\pi\)
\(500\) 0 0
\(501\) 3.60457 3.60457i 0.161040 0.161040i
\(502\) 0 0
\(503\) 4.37422i 0.195037i 0.995234 + 0.0975184i \(0.0310905\pi\)
−0.995234 + 0.0975184i \(0.968910\pi\)
\(504\) 0 0
\(505\) 12.3137i 0.547954i
\(506\) 0 0
\(507\) 0.902776 0.902776i 0.0400937 0.0400937i
\(508\) 0 0
\(509\) 19.4429 + 19.4429i 0.861791 + 0.861791i 0.991546 0.129755i \(-0.0414192\pi\)
−0.129755 + 0.991546i \(0.541419\pi\)
\(510\) 0 0
\(511\) 19.5154 0.863312
\(512\) 0 0
\(513\) 14.9873 0.661704
\(514\) 0 0
\(515\) −3.73997 3.73997i −0.164803 0.164803i
\(516\) 0 0
\(517\) 5.89719 5.89719i 0.259358 0.259358i
\(518\) 0 0
\(519\) 22.6501i 0.994230i
\(520\) 0 0
\(521\) 36.9872i 1.62044i 0.586127 + 0.810219i \(0.300652\pi\)
−0.586127 + 0.810219i \(0.699348\pi\)
\(522\) 0 0
\(523\) 1.80040 1.80040i 0.0787260 0.0787260i −0.666647 0.745373i \(-0.732271\pi\)
0.745373 + 0.666647i \(0.232271\pi\)
\(524\) 0 0
\(525\) −4.76672 4.76672i −0.208037 0.208037i
\(526\) 0 0
\(527\) −31.5770 −1.37551
\(528\) 0 0
\(529\) −52.2466 −2.27159
\(530\) 0 0
\(531\) 4.64148 + 4.64148i 0.201423 + 0.201423i
\(532\) 0 0
\(533\) 6.57449 6.57449i 0.284773 0.284773i
\(534\) 0 0
\(535\) 9.60213i 0.415136i
\(536\) 0 0
\(537\) 11.8537i 0.511527i
\(538\) 0 0
\(539\) −8.92095 + 8.92095i −0.384253 + 0.384253i
\(540\) 0 0
\(541\) −13.6616 13.6616i −0.587357 0.587357i 0.349558 0.936915i \(-0.386332\pi\)
−0.936915 + 0.349558i \(0.886332\pi\)
\(542\) 0 0
\(543\) 12.4782 0.535491
\(544\) 0 0
\(545\) −7.16089 −0.306739
\(546\) 0 0
\(547\) 5.89436 + 5.89436i 0.252025 + 0.252025i 0.821800 0.569776i \(-0.192970\pi\)
−0.569776 + 0.821800i \(0.692970\pi\)
\(548\) 0 0
\(549\) −2.36656 + 2.36656i −0.101003 + 0.101003i
\(550\) 0 0
\(551\) 25.9330i 1.10478i
\(552\) 0 0
\(553\) 17.0378i 0.724520i
\(554\) 0 0
\(555\) −0.487522 + 0.487522i −0.0206942 + 0.0206942i
\(556\) 0 0
\(557\) −9.07239 9.07239i −0.384409 0.384409i 0.488278 0.872688i \(-0.337625\pi\)
−0.872688 + 0.488278i \(0.837625\pi\)
\(558\) 0 0
\(559\) 0.0589334 0.00249262
\(560\) 0 0
\(561\) 29.6625 1.25235
\(562\) 0 0
\(563\) −17.9168 17.9168i −0.755104 0.755104i 0.220323 0.975427i \(-0.429289\pi\)
−0.975427 + 0.220323i \(0.929289\pi\)
\(564\) 0 0
\(565\) 11.8207 11.8207i 0.497301 0.497301i
\(566\) 0 0
\(567\) 9.65999i 0.405681i
\(568\) 0 0
\(569\) 15.4121i 0.646109i 0.946380 + 0.323054i \(0.104710\pi\)
−0.946380 + 0.323054i \(0.895290\pi\)
\(570\) 0 0
\(571\) −15.6916 + 15.6916i −0.656673 + 0.656673i −0.954591 0.297919i \(-0.903708\pi\)
0.297919 + 0.954591i \(0.403708\pi\)
\(572\) 0 0
\(573\) 10.1622 + 10.1622i 0.424533 + 0.424533i
\(574\) 0 0
\(575\) 14.2866 0.595792
\(576\) 0 0
\(577\) −17.3626 −0.722815 −0.361407 0.932408i \(-0.617704\pi\)
−0.361407 + 0.932408i \(0.617704\pi\)
\(578\) 0 0
\(579\) 2.29855 + 2.29855i 0.0955244 + 0.0955244i
\(580\) 0 0
\(581\) −31.1306 + 31.1306i −1.29151 + 1.29151i
\(582\) 0 0
\(583\) 34.7256i 1.43819i
\(584\) 0 0
\(585\) 2.50863i 0.103719i
\(586\) 0 0
\(587\) −15.0345 + 15.0345i −0.620541 + 0.620541i −0.945670 0.325129i \(-0.894592\pi\)
0.325129 + 0.945670i \(0.394592\pi\)
\(588\) 0 0
\(589\) −9.93577 9.93577i −0.409396 0.409396i
\(590\) 0 0
\(591\) 7.76456 0.319391
\(592\) 0 0
\(593\) −42.6498 −1.75142 −0.875709 0.482840i \(-0.839605\pi\)
−0.875709 + 0.482840i \(0.839605\pi\)
\(594\) 0 0
\(595\) 25.0587 + 25.0587i 1.02731 + 1.02731i
\(596\) 0 0
\(597\) −23.7326 + 23.7326i −0.971311 + 0.971311i
\(598\) 0 0
\(599\) 26.7066i 1.09120i 0.838045 + 0.545602i \(0.183699\pi\)
−0.838045 + 0.545602i \(0.816301\pi\)
\(600\) 0 0
\(601\) 8.30933i 0.338944i 0.985535 + 0.169472i \(0.0542063\pi\)
−0.985535 + 0.169472i \(0.945794\pi\)
\(602\) 0 0
\(603\) 2.13763 2.13763i 0.0870512 0.0870512i
\(604\) 0 0
\(605\) −4.93633 4.93633i −0.200690 0.200690i
\(606\) 0 0
\(607\) −22.9913 −0.933190 −0.466595 0.884471i \(-0.654519\pi\)
−0.466595 + 0.884471i \(0.654519\pi\)
\(608\) 0 0
\(609\) −39.5145 −1.60121
\(610\) 0 0
\(611\) −1.53226 1.53226i −0.0619886 0.0619886i
\(612\) 0 0
\(613\) 6.71226 6.71226i 0.271106 0.271106i −0.558440 0.829545i \(-0.688600\pi\)
0.829545 + 0.558440i \(0.188600\pi\)
\(614\) 0 0
\(615\) 21.7366i 0.876504i
\(616\) 0 0
\(617\) 10.3150i 0.415266i 0.978207 + 0.207633i \(0.0665760\pi\)
−0.978207 + 0.207633i \(0.933424\pi\)
\(618\) 0 0
\(619\) 6.94177 6.94177i 0.279013 0.279013i −0.553702 0.832715i \(-0.686785\pi\)
0.832715 + 0.553702i \(0.186785\pi\)
\(620\) 0 0
\(621\) 34.2219 + 34.2219i 1.37328 + 1.37328i
\(622\) 0 0
\(623\) −19.0630 −0.763743
\(624\) 0 0
\(625\) 14.0527 0.562107
\(626\) 0 0
\(627\) 9.33340 + 9.33340i 0.372740 + 0.372740i
\(628\) 0 0
\(629\) −1.25887 + 1.25887i −0.0501944 + 0.0501944i
\(630\) 0 0
\(631\) 11.6081i 0.462113i −0.972940 0.231056i \(-0.925782\pi\)
0.972940 0.231056i \(-0.0742182\pi\)
\(632\) 0 0
\(633\) 13.5776i 0.539661i
\(634\) 0 0
\(635\) 16.4565 16.4565i 0.653058 0.653058i
\(636\) 0 0
\(637\) 2.31792 + 2.31792i 0.0918393 + 0.0918393i
\(638\) 0 0
\(639\) 13.3416 0.527784
\(640\) 0 0
\(641\) 27.0384 1.06795 0.533977 0.845499i \(-0.320697\pi\)
0.533977 + 0.845499i \(0.320697\pi\)
\(642\) 0 0
\(643\) 12.7661 + 12.7661i 0.503444 + 0.503444i 0.912507 0.409062i \(-0.134144\pi\)
−0.409062 + 0.912507i \(0.634144\pi\)
\(644\) 0 0
\(645\) 0.0974228 0.0974228i 0.00383602 0.00383602i
\(646\) 0 0
\(647\) 10.0134i 0.393667i 0.980437 + 0.196833i \(0.0630658\pi\)
−0.980437 + 0.196833i \(0.936934\pi\)
\(648\) 0 0
\(649\) 18.4402i 0.723843i
\(650\) 0 0
\(651\) −15.1393 + 15.1393i −0.593355 + 0.593355i
\(652\) 0 0
\(653\) −9.13213 9.13213i −0.357368 0.357368i 0.505474 0.862842i \(-0.331318\pi\)
−0.862842 + 0.505474i \(0.831318\pi\)
\(654\) 0 0
\(655\) −16.5586 −0.646998
\(656\) 0 0
\(657\) 8.33951 0.325355
\(658\) 0 0
\(659\) −26.5843 26.5843i −1.03558 1.03558i −0.999343 0.0362320i \(-0.988464\pi\)
−0.0362320 0.999343i \(-0.511536\pi\)
\(660\) 0 0
\(661\) −1.76639 + 1.76639i −0.0687044 + 0.0687044i −0.740624 0.671920i \(-0.765470\pi\)
0.671920 + 0.740624i \(0.265470\pi\)
\(662\) 0 0
\(663\) 7.70718i 0.299322i
\(664\) 0 0
\(665\) 15.7696i 0.611519i
\(666\) 0 0
\(667\) 59.2154 59.2154i 2.29283 2.29283i
\(668\) 0 0
\(669\) 16.9107 + 16.9107i 0.653804 + 0.653804i
\(670\) 0 0
\(671\) −9.40218 −0.362967
\(672\) 0 0
\(673\) 39.1888 1.51062 0.755308 0.655370i \(-0.227487\pi\)
0.755308 + 0.655370i \(0.227487\pi\)
\(674\) 0 0
\(675\) −6.49749 6.49749i −0.250088 0.250088i
\(676\) 0 0
\(677\) 13.8583 13.8583i 0.532616 0.532616i −0.388734 0.921350i \(-0.627087\pi\)
0.921350 + 0.388734i \(0.127087\pi\)
\(678\) 0 0
\(679\) 31.3879i 1.20456i
\(680\) 0 0
\(681\) 0.997953i 0.0382416i
\(682\) 0 0
\(683\) 8.54143 8.54143i 0.326829 0.326829i −0.524550 0.851379i \(-0.675767\pi\)
0.851379 + 0.524550i \(0.175767\pi\)
\(684\) 0 0
\(685\) −25.6595 25.6595i −0.980399 0.980399i
\(686\) 0 0
\(687\) −26.9896 −1.02972
\(688\) 0 0
\(689\) −9.02271 −0.343738
\(690\) 0 0
\(691\) 5.55830 + 5.55830i 0.211448 + 0.211448i 0.804882 0.593435i \(-0.202228\pi\)
−0.593435 + 0.804882i \(0.702228\pi\)
\(692\) 0 0
\(693\) −11.9528 + 11.9528i −0.454050 + 0.454050i
\(694\) 0 0
\(695\) 20.2367i 0.767621i
\(696\) 0 0
\(697\) 56.1277i 2.12599i
\(698\) 0 0
\(699\) −21.6239 + 21.6239i −0.817890 + 0.817890i
\(700\) 0 0
\(701\) 30.3550 + 30.3550i 1.14649 + 1.14649i 0.987238 + 0.159253i \(0.0509086\pi\)
0.159253 + 0.987238i \(0.449091\pi\)
\(702\) 0 0
\(703\) −0.792213 −0.0298789
\(704\) 0 0
\(705\) −5.06595 −0.190795
\(706\) 0 0
\(707\) −15.2444 15.2444i −0.573325 0.573325i
\(708\) 0 0
\(709\) 33.0689 33.0689i 1.24193 1.24193i 0.282730 0.959199i \(-0.408760\pi\)
0.959199 0.282730i \(-0.0912401\pi\)
\(710\) 0 0
\(711\) 7.28074i 0.273049i
\(712\) 0 0
\(713\) 45.3747i 1.69929i
\(714\) 0 0
\(715\) −4.98330 + 4.98330i −0.186365 + 0.186365i
\(716\) 0 0
\(717\) −1.34640 1.34640i −0.0502824 0.0502824i
\(718\) 0 0
\(719\) −24.1360 −0.900121 −0.450060 0.892998i \(-0.648598\pi\)
−0.450060 + 0.892998i \(0.648598\pi\)
\(720\) 0 0
\(721\) −9.26017 −0.344867
\(722\) 0 0
\(723\) 0.0875736 + 0.0875736i 0.00325690 + 0.00325690i
\(724\) 0 0
\(725\) −11.2428 + 11.2428i −0.417549 + 0.417549i
\(726\) 0 0
\(727\) 25.4413i 0.943564i −0.881715 0.471782i \(-0.843611\pi\)
0.881715 0.471782i \(-0.156389\pi\)
\(728\) 0 0
\(729\) 25.4974i 0.944347i
\(730\) 0 0
\(731\) 0.251563 0.251563i 0.00930440 0.00930440i
\(732\) 0 0
\(733\) −1.11861 1.11861i −0.0413170 0.0413170i 0.686146 0.727463i \(-0.259301\pi\)
−0.727463 + 0.686146i \(0.759301\pi\)
\(734\) 0 0
\(735\) 7.66350 0.282673
\(736\) 0 0
\(737\) 8.49266 0.312831
\(738\) 0 0
\(739\) −27.9487 27.9487i −1.02811 1.02811i −0.999593 0.0285153i \(-0.990922\pi\)
−0.0285153 0.999593i \(-0.509078\pi\)
\(740\) 0 0
\(741\) 2.42508 2.42508i 0.0890877 0.0890877i
\(742\) 0 0
\(743\) 6.84243i 0.251024i −0.992092 0.125512i \(-0.959943\pi\)
0.992092 0.125512i \(-0.0400574\pi\)
\(744\) 0 0
\(745\) 17.7682i 0.650977i
\(746\) 0 0
\(747\) −13.3030 + 13.3030i −0.486731 + 0.486731i
\(748\) 0 0
\(749\) 11.8874 + 11.8874i 0.434358 + 0.434358i
\(750\) 0 0
\(751\) −23.3074 −0.850500 −0.425250 0.905076i \(-0.639814\pi\)
−0.425250 + 0.905076i \(0.639814\pi\)
\(752\) 0 0
\(753\) 13.2346 0.482297
\(754\) 0 0
\(755\) −15.5379 15.5379i −0.565481 0.565481i
\(756\) 0 0
\(757\) −10.0617 + 10.0617i −0.365700 + 0.365700i −0.865906 0.500206i \(-0.833258\pi\)
0.500206 + 0.865906i \(0.333258\pi\)
\(758\) 0 0
\(759\) 42.6237i 1.54714i
\(760\) 0 0
\(761\) 10.4743i 0.379695i 0.981814 + 0.189847i \(0.0607993\pi\)
−0.981814 + 0.189847i \(0.939201\pi\)
\(762\) 0 0
\(763\) −8.86519 + 8.86519i −0.320941 + 0.320941i
\(764\) 0 0
\(765\) 10.7083 + 10.7083i 0.387160 + 0.387160i
\(766\) 0 0
\(767\) 4.79131 0.173004
\(768\) 0 0
\(769\) 39.4816 1.42374 0.711871 0.702310i \(-0.247848\pi\)
0.711871 + 0.702310i \(0.247848\pi\)
\(770\) 0 0
\(771\) −26.0928 26.0928i −0.939711 0.939711i
\(772\) 0 0
\(773\) −22.4954 + 22.4954i −0.809104 + 0.809104i −0.984498 0.175394i \(-0.943880\pi\)
0.175394 + 0.984498i \(0.443880\pi\)
\(774\) 0 0
\(775\) 8.61499i 0.309460i
\(776\) 0 0
\(777\) 1.20711i 0.0433047i
\(778\) 0 0
\(779\) 17.6607 17.6607i 0.632762 0.632762i
\(780\) 0 0
\(781\) 26.5025 + 26.5025i 0.948334 + 0.948334i
\(782\) 0 0
\(783\) −53.8619 −1.92487
\(784\) 0 0
\(785\) −4.69274 −0.167491
\(786\) 0 0
\(787\) −16.2640 16.2640i −0.579750 0.579750i 0.355085 0.934834i \(-0.384452\pi\)
−0.934834 + 0.355085i \(0.884452\pi\)
\(788\) 0 0
\(789\) 15.7342 15.7342i 0.560152 0.560152i
\(790\) 0 0
\(791\) 29.2681i 1.04065i
\(792\) 0 0
\(793\) 2.44296i 0.0867520i
\(794\) 0 0
\(795\) −14.9154 + 14.9154i −0.528996 + 0.528996i
\(796\) 0 0
\(797\) −27.5649 27.5649i −0.976398 0.976398i 0.0233299 0.999728i \(-0.492573\pi\)
−0.999728 + 0.0233299i \(0.992573\pi\)
\(798\) 0 0
\(799\) −13.0812 −0.462779
\(800\) 0 0
\(801\) −8.14617 −0.287831
\(802\) 0 0
\(803\) 16.5661 + 16.5661i 0.584606 + 0.584606i
\(804\) 0 0
\(805\) −36.0083 + 36.0083i −1.26913 + 1.26913i
\(806\) 0 0
\(807\) 21.7139i 0.764365i
\(808\) 0 0
\(809\) 5.17865i 0.182072i −0.995848 0.0910358i \(-0.970982\pi\)
0.995848 0.0910358i \(-0.0290178\pi\)
\(810\) 0 0
\(811\) 32.6428 32.6428i 1.14625 1.14625i 0.158960 0.987285i \(-0.449186\pi\)
0.987285 0.158960i \(-0.0508141\pi\)
\(812\) 0 0
\(813\) −1.62030 1.62030i −0.0568264 0.0568264i
\(814\) 0 0
\(815\) 35.9401 1.25893
\(816\) 0 0
\(817\) 0.158310 0.00553857
\(818\) 0 0
\(819\) 3.10569 + 3.10569i 0.108521 + 0.108521i
\(820\) 0 0
\(821\) −12.1496 + 12.1496i −0.424025 + 0.424025i −0.886587 0.462562i \(-0.846930\pi\)
0.462562 + 0.886587i \(0.346930\pi\)
\(822\) 0 0
\(823\) 27.6976i 0.965476i −0.875765 0.482738i \(-0.839642\pi\)
0.875765 0.482738i \(-0.160358\pi\)
\(824\) 0 0
\(825\) 8.09269i 0.281751i
\(826\) 0 0
\(827\) 21.8635 21.8635i 0.760269 0.760269i −0.216102 0.976371i \(-0.569334\pi\)
0.976371 + 0.216102i \(0.0693343\pi\)
\(828\) 0 0
\(829\) −14.7756 14.7756i −0.513178 0.513178i 0.402320 0.915499i \(-0.368204\pi\)
−0.915499 + 0.402320i \(0.868204\pi\)
\(830\) 0 0
\(831\) −2.36561 −0.0820622
\(832\) 0 0
\(833\) 19.7885 0.685632
\(834\) 0 0
\(835\) 5.16984 + 5.16984i 0.178910 + 0.178910i
\(836\) 0 0
\(837\) −20.6362 + 20.6362i −0.713293 + 0.713293i
\(838\) 0 0
\(839\) 0.625398i 0.0215912i −0.999942 0.0107956i \(-0.996564\pi\)
0.999942 0.0107956i \(-0.00343641\pi\)
\(840\) 0 0
\(841\) 64.1994i 2.21377i
\(842\) 0 0
\(843\) 24.2308 24.2308i 0.834554 0.834554i
\(844\) 0 0
\(845\) 1.29480 + 1.29480i 0.0445426 + 0.0445426i
\(846\) 0 0
\(847\) −12.2224 −0.419965
\(848\) 0 0
\(849\) 2.71421 0.0931516
\(850\) 0 0
\(851\) −1.80894 1.80894i −0.0620096 0.0620096i
\(852\) 0 0
\(853\) −14.2267 + 14.2267i −0.487114 + 0.487114i −0.907394 0.420280i \(-0.861932\pi\)
0.420280 + 0.907394i \(0.361932\pi\)
\(854\) 0 0
\(855\) 6.73881i 0.230462i
\(856\) 0 0
\(857\) 36.8671i 1.25936i 0.776857 + 0.629678i \(0.216813\pi\)
−0.776857 + 0.629678i \(0.783187\pi\)
\(858\) 0 0
\(859\) 11.0958 11.0958i 0.378584 0.378584i −0.492007 0.870591i \(-0.663737\pi\)
0.870591 + 0.492007i \(0.163737\pi\)
\(860\) 0 0
\(861\) −26.9099 26.9099i −0.917088 0.917088i
\(862\) 0 0
\(863\) 5.79818 0.197372 0.0986862 0.995119i \(-0.468536\pi\)
0.0986862 + 0.995119i \(0.468536\pi\)
\(864\) 0 0
\(865\) −32.4858 −1.10455
\(866\) 0 0
\(867\) −17.5516 17.5516i −0.596085 0.596085i
\(868\) 0 0
\(869\) −14.4629 + 14.4629i −0.490621 + 0.490621i
\(870\) 0 0
\(871\) 2.20664i 0.0747690i
\(872\) 0 0
\(873\) 13.4130i 0.453960i
\(874\) 0 0
\(875\) 27.5920 27.5920i 0.932778 0.932778i
\(876\) 0 0
\(877\) 11.4174 + 11.4174i 0.385537 + 0.385537i 0.873092 0.487555i \(-0.162111\pi\)
−0.487555 + 0.873092i \(0.662111\pi\)
\(878\) 0 0
\(879\) −8.27382 −0.279069
\(880\) 0 0
\(881\) 21.6667 0.729971 0.364985 0.931013i \(-0.381074\pi\)
0.364985 + 0.931013i \(0.381074\pi\)
\(882\) 0 0
\(883\) −3.37331 3.37331i −0.113521 0.113521i 0.648064 0.761585i \(-0.275579\pi\)
−0.761585 + 0.648064i \(0.775579\pi\)
\(884\) 0 0
\(885\) 7.92050 7.92050i 0.266245 0.266245i
\(886\) 0 0
\(887\) 10.0852i 0.338628i 0.985562 + 0.169314i \(0.0541552\pi\)
−0.985562 + 0.169314i \(0.945845\pi\)
\(888\) 0 0
\(889\) 40.7465i 1.36659i
\(890\) 0 0
\(891\) 8.20011 8.20011i 0.274714 0.274714i
\(892\) 0 0
\(893\) −4.11603 4.11603i −0.137738 0.137738i
\(894\) 0 0
\(895\) 17.0012 0.568287
\(896\) 0 0
\(897\) 11.0749 0.369779
\(898\) 0 0
\(899\) 35.7077 + 35.7077i 1.19092 + 1.19092i
\(900\) 0 0
\(901\) −38.5143 + 38.5143i −1.28310 + 1.28310i
\(902\) 0 0
\(903\) 0.241219i 0.00802727i
\(904\) 0 0
\(905\) 17.8968i 0.594910i
\(906\) 0 0
\(907\) −14.1727 + 14.1727i −0.470597 + 0.470597i −0.902108 0.431511i \(-0.857981\pi\)
0.431511 + 0.902108i \(0.357981\pi\)
\(908\) 0 0
\(909\) −6.51438 6.51438i −0.216068 0.216068i
\(910\) 0 0
\(911\) 5.12861 0.169919 0.0849593 0.996384i \(-0.472924\pi\)
0.0849593 + 0.996384i \(0.472924\pi\)
\(912\) 0 0
\(913\) −52.8518 −1.74914
\(914\) 0 0
\(915\) 4.03845 + 4.03845i 0.133507 + 0.133507i
\(916\) 0 0
\(917\) −20.4996 + 20.4996i −0.676955 + 0.676955i
\(918\) 0 0
\(919\) 1.74928i 0.0577034i 0.999584 + 0.0288517i \(0.00918506\pi\)
−0.999584 + 0.0288517i \(0.990815\pi\)
\(920\) 0 0
\(921\) 24.7930i 0.816956i
\(922\) 0 0
\(923\) 6.88611 6.88611i 0.226659 0.226659i
\(924\) 0 0
\(925\) 0.343451 + 0.343451i 0.0112926 + 0.0112926i
\(926\) 0 0
\(927\) −3.95714 −0.129969
\(928\) 0 0
\(929\) −28.8593 −0.946844 −0.473422 0.880836i \(-0.656981\pi\)
−0.473422 + 0.880836i \(0.656981\pi\)
\(930\) 0 0
\(931\) 6.22652 + 6.22652i 0.204066 + 0.204066i
\(932\) 0 0
\(933\) 1.45269 1.45269i 0.0475588 0.0475588i
\(934\) 0 0
\(935\) 42.5434i 1.39132i
\(936\) 0 0
\(937\) 12.4959i 0.408223i 0.978948 + 0.204111i \(0.0654304\pi\)
−0.978948 + 0.204111i \(0.934570\pi\)
\(938\) 0 0
\(939\) −12.9870 + 12.9870i −0.423816 + 0.423816i
\(940\) 0 0
\(941\) 3.93712 + 3.93712i 0.128347 + 0.128347i 0.768362 0.640015i \(-0.221072\pi\)
−0.640015 + 0.768362i \(0.721072\pi\)
\(942\) 0 0
\(943\) 80.6530 2.62642
\(944\) 0 0
\(945\) 32.7529 1.06545
\(946\) 0 0
\(947\) −31.9342 31.9342i −1.03772 1.03772i −0.999260 0.0384618i \(-0.987754\pi\)
−0.0384618 0.999260i \(-0.512246\pi\)
\(948\) 0 0
\(949\) 4.30436 4.30436i 0.139725 0.139725i
\(950\) 0 0
\(951\) 22.4116i 0.726745i
\(952\) 0 0
\(953\) 28.7558i 0.931492i −0.884918 0.465746i \(-0.845786\pi\)
0.884918 0.465746i \(-0.154214\pi\)
\(954\) 0 0
\(955\) −14.5751 + 14.5751i −0.471640 + 0.471640i
\(956\) 0 0
\(957\) −33.5428 33.5428i −1.08429 1.08429i
\(958\) 0 0
\(959\) −63.5329 −2.05159
\(960\) 0 0
\(961\) −3.63849 −0.117371
\(962\) 0 0
\(963\) 5.07985 + 5.07985i 0.163696 + 0.163696i
\(964\) 0 0
\(965\) −3.29669 + 3.29669i −0.106124 + 0.106124i
\(966\) 0 0
\(967\) 5.77351i 0.185664i 0.995682 + 0.0928318i \(0.0295919\pi\)
−0.995682 + 0.0928318i \(0.970408\pi\)
\(968\) 0 0
\(969\) 20.7034i 0.665090i
\(970\) 0 0
\(971\) 17.9091 17.9091i 0.574729 0.574729i −0.358717 0.933446i \(-0.616786\pi\)
0.933446 + 0.358717i \(0.116786\pi\)
\(972\) 0 0
\(973\) −25.0530 25.0530i −0.803163 0.803163i
\(974\) 0 0
\(975\) −2.10271 −0.0673407
\(976\) 0 0
\(977\) −3.82926 −0.122509 −0.0612544 0.998122i \(-0.519510\pi\)
−0.0612544 + 0.998122i \(0.519510\pi\)
\(978\) 0 0
\(979\) −16.1821 16.1821i −0.517181 0.517181i
\(980\) 0 0
\(981\) −3.78835 + 3.78835i −0.120953 + 0.120953i
\(982\) 0 0
\(983\) 42.4136i 1.35279i −0.736541 0.676393i \(-0.763542\pi\)
0.736541 0.676393i \(-0.236458\pi\)
\(984\) 0 0
\(985\) 11.1363i 0.354832i
\(986\) 0 0
\(987\) −6.27166 + 6.27166i −0.199629 + 0.199629i
\(988\) 0 0
\(989\) 0.361485 + 0.361485i 0.0114946 + 0.0114946i
\(990\) 0 0
\(991\) 59.1239 1.87813 0.939066 0.343736i \(-0.111693\pi\)
0.939066 + 0.343736i \(0.111693\pi\)
\(992\) 0 0
\(993\) −30.9683 −0.982749
\(994\) 0 0
\(995\) −34.0384 34.0384i −1.07909 1.07909i
\(996\) 0 0
\(997\) 35.9832 35.9832i 1.13960 1.13960i 0.151079 0.988522i \(-0.451725\pi\)
0.988522 0.151079i \(-0.0482748\pi\)
\(998\) 0 0
\(999\) 1.64540i 0.0520581i
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 1664.2.n.b.417.8 48
4.3 odd 2 1664.2.n.a.417.17 48
8.3 odd 2 832.2.n.a.209.8 48
8.5 even 2 208.2.n.a.157.24 yes 48
16.3 odd 4 832.2.n.a.625.8 48
16.5 even 4 inner 1664.2.n.b.1249.8 48
16.11 odd 4 1664.2.n.a.1249.17 48
16.13 even 4 208.2.n.a.53.24 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
208.2.n.a.53.24 48 16.13 even 4
208.2.n.a.157.24 yes 48 8.5 even 2
832.2.n.a.209.8 48 8.3 odd 2
832.2.n.a.625.8 48 16.3 odd 4
1664.2.n.a.417.17 48 4.3 odd 2
1664.2.n.a.1249.17 48 16.11 odd 4
1664.2.n.b.417.8 48 1.1 even 1 trivial
1664.2.n.b.1249.8 48 16.5 even 4 inner