Properties

Label 560.2.x.b.127.1
Level $560$
Weight $2$
Character 560.127
Analytic conductor $4.472$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(127,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("560.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.x (of order \(4\), degree \(2\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.1
Character \(\chi\) \(=\) 560.127
Dual form 560.2.x.b.463.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.31224 - 2.31224i) q^{3} +(2.03291 + 0.931271i) q^{5} +(0.707107 - 0.707107i) q^{7} +7.69286i q^{9} +O(q^{10})\) \(q+(-2.31224 - 2.31224i) q^{3} +(2.03291 + 0.931271i) q^{5} +(0.707107 - 0.707107i) q^{7} +7.69286i q^{9} +6.09628i q^{11} +(-1.41796 + 1.41796i) q^{13} +(-2.54726 - 6.85389i) q^{15} +(0.291645 + 0.291645i) q^{17} -3.26091 q^{19} -3.26999 q^{21} +(5.29531 + 5.29531i) q^{23} +(3.26547 + 3.78638i) q^{25} +(10.8510 - 10.8510i) q^{27} -2.37807i q^{29} +3.95390i q^{31} +(14.0960 - 14.0960i) q^{33} +(2.09599 - 0.778979i) q^{35} +(-4.09050 - 4.09050i) q^{37} +6.55732 q^{39} +7.03921 q^{41} +(1.83184 + 1.83184i) q^{43} +(-7.16414 + 15.6389i) q^{45} +(3.50821 - 3.50821i) q^{47} -1.00000i q^{49} -1.34870i q^{51} +(-5.49795 + 5.49795i) q^{53} +(-5.67728 + 12.3932i) q^{55} +(7.53999 + 7.53999i) q^{57} -5.02998 q^{59} +4.51192 q^{61} +(5.43968 + 5.43968i) q^{63} +(-4.20309 + 1.56209i) q^{65} +(4.04639 - 4.04639i) q^{67} -24.4880i q^{69} -0.733929i q^{71} +(-2.94871 + 2.94871i) q^{73} +(1.20448 - 16.3055i) q^{75} +(4.31072 + 4.31072i) q^{77} -1.28925 q^{79} -27.1016 q^{81} +(-6.84891 - 6.84891i) q^{83} +(0.321288 + 0.864488i) q^{85} +(-5.49865 + 5.49865i) q^{87} -4.21572i q^{89} +2.00530i q^{91} +(9.14235 - 9.14235i) q^{93} +(-6.62914 - 3.03679i) q^{95} +(2.88225 + 2.88225i) q^{97} -46.8978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{13} + 8 q^{17} - 8 q^{21} + 32 q^{25} + 24 q^{33} - 16 q^{37} + 32 q^{41} - 24 q^{45} + 8 q^{53} + 40 q^{57} + 16 q^{61} - 16 q^{73} + 16 q^{77} - 104 q^{81} - 8 q^{85} - 8 q^{93} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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
\(3\) −2.31224 2.31224i −1.33497 1.33497i −0.900860 0.434110i \(-0.857063\pi\)
−0.434110 0.900860i \(-0.642937\pi\)
\(4\) 0 0
\(5\) 2.03291 + 0.931271i 0.909146 + 0.416477i
\(6\) 0 0
\(7\) 0.707107 0.707107i 0.267261 0.267261i
\(8\) 0 0
\(9\) 7.69286i 2.56429i
\(10\) 0 0
\(11\) 6.09628i 1.83810i 0.394144 + 0.919049i \(0.371041\pi\)
−0.394144 + 0.919049i \(0.628959\pi\)
\(12\) 0 0
\(13\) −1.41796 + 1.41796i −0.393271 + 0.393271i −0.875852 0.482580i \(-0.839700\pi\)
0.482580 + 0.875852i \(0.339700\pi\)
\(14\) 0 0
\(15\) −2.54726 6.85389i −0.657699 1.76967i
\(16\) 0 0
\(17\) 0.291645 + 0.291645i 0.0707342 + 0.0707342i 0.741589 0.670855i \(-0.234073\pi\)
−0.670855 + 0.741589i \(0.734073\pi\)
\(18\) 0 0
\(19\) −3.26091 −0.748104 −0.374052 0.927408i \(-0.622032\pi\)
−0.374052 + 0.927408i \(0.622032\pi\)
\(20\) 0 0
\(21\) −3.26999 −0.713571
\(22\) 0 0
\(23\) 5.29531 + 5.29531i 1.10415 + 1.10415i 0.993905 + 0.110243i \(0.0351630\pi\)
0.110243 + 0.993905i \(0.464837\pi\)
\(24\) 0 0
\(25\) 3.26547 + 3.78638i 0.653094 + 0.757277i
\(26\) 0 0
\(27\) 10.8510 10.8510i 2.08828 2.08828i
\(28\) 0 0
\(29\) 2.37807i 0.441596i −0.975320 0.220798i \(-0.929134\pi\)
0.975320 0.220798i \(-0.0708661\pi\)
\(30\) 0 0
\(31\) 3.95390i 0.710142i 0.934839 + 0.355071i \(0.115543\pi\)
−0.934839 + 0.355071i \(0.884457\pi\)
\(32\) 0 0
\(33\) 14.0960 14.0960i 2.45380 2.45380i
\(34\) 0 0
\(35\) 2.09599 0.778979i 0.354288 0.131671i
\(36\) 0 0
\(37\) −4.09050 4.09050i −0.672474 0.672474i 0.285812 0.958286i \(-0.407737\pi\)
−0.958286 + 0.285812i \(0.907737\pi\)
\(38\) 0 0
\(39\) 6.55732 1.05001
\(40\) 0 0
\(41\) 7.03921 1.09934 0.549670 0.835382i \(-0.314754\pi\)
0.549670 + 0.835382i \(0.314754\pi\)
\(42\) 0 0
\(43\) 1.83184 + 1.83184i 0.279352 + 0.279352i 0.832850 0.553498i \(-0.186707\pi\)
−0.553498 + 0.832850i \(0.686707\pi\)
\(44\) 0 0
\(45\) −7.16414 + 15.6389i −1.06797 + 2.33131i
\(46\) 0 0
\(47\) 3.50821 3.50821i 0.511725 0.511725i −0.403330 0.915055i \(-0.632147\pi\)
0.915055 + 0.403330i \(0.132147\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 1.34870i 0.188856i
\(52\) 0 0
\(53\) −5.49795 + 5.49795i −0.755202 + 0.755202i −0.975445 0.220243i \(-0.929315\pi\)
0.220243 + 0.975445i \(0.429315\pi\)
\(54\) 0 0
\(55\) −5.67728 + 12.3932i −0.765525 + 1.67110i
\(56\) 0 0
\(57\) 7.53999 + 7.53999i 0.998696 + 0.998696i
\(58\) 0 0
\(59\) −5.02998 −0.654847 −0.327424 0.944878i \(-0.606180\pi\)
−0.327424 + 0.944878i \(0.606180\pi\)
\(60\) 0 0
\(61\) 4.51192 0.577693 0.288846 0.957375i \(-0.406728\pi\)
0.288846 + 0.957375i \(0.406728\pi\)
\(62\) 0 0
\(63\) 5.43968 + 5.43968i 0.685335 + 0.685335i
\(64\) 0 0
\(65\) −4.20309 + 1.56209i −0.521330 + 0.193753i
\(66\) 0 0
\(67\) 4.04639 4.04639i 0.494345 0.494345i −0.415327 0.909672i \(-0.636333\pi\)
0.909672 + 0.415327i \(0.136333\pi\)
\(68\) 0 0
\(69\) 24.4880i 2.94801i
\(70\) 0 0
\(71\) 0.733929i 0.0871014i −0.999051 0.0435507i \(-0.986133\pi\)
0.999051 0.0435507i \(-0.0138670\pi\)
\(72\) 0 0
\(73\) −2.94871 + 2.94871i −0.345120 + 0.345120i −0.858288 0.513168i \(-0.828472\pi\)
0.513168 + 0.858288i \(0.328472\pi\)
\(74\) 0 0
\(75\) 1.20448 16.3055i 0.139081 1.88280i
\(76\) 0 0
\(77\) 4.31072 + 4.31072i 0.491252 + 0.491252i
\(78\) 0 0
\(79\) −1.28925 −0.145052 −0.0725262 0.997367i \(-0.523106\pi\)
−0.0725262 + 0.997367i \(0.523106\pi\)
\(80\) 0 0
\(81\) −27.1016 −3.01128
\(82\) 0 0
\(83\) −6.84891 6.84891i −0.751766 0.751766i 0.223043 0.974809i \(-0.428401\pi\)
−0.974809 + 0.223043i \(0.928401\pi\)
\(84\) 0 0
\(85\) 0.321288 + 0.864488i 0.0348486 + 0.0937669i
\(86\) 0 0
\(87\) −5.49865 + 5.49865i −0.589517 + 0.589517i
\(88\) 0 0
\(89\) 4.21572i 0.446866i −0.974719 0.223433i \(-0.928274\pi\)
0.974719 0.223433i \(-0.0717264\pi\)
\(90\) 0 0
\(91\) 2.00530i 0.210212i
\(92\) 0 0
\(93\) 9.14235 9.14235i 0.948018 0.948018i
\(94\) 0 0
\(95\) −6.62914 3.03679i −0.680136 0.311568i
\(96\) 0 0
\(97\) 2.88225 + 2.88225i 0.292648 + 0.292648i 0.838126 0.545477i \(-0.183652\pi\)
−0.545477 + 0.838126i \(0.683652\pi\)
\(98\) 0 0
\(99\) −46.8978 −4.71341
\(100\) 0 0
\(101\) 9.03254 0.898772 0.449386 0.893338i \(-0.351643\pi\)
0.449386 + 0.893338i \(0.351643\pi\)
\(102\) 0 0
\(103\) 2.57683 + 2.57683i 0.253903 + 0.253903i 0.822569 0.568666i \(-0.192540\pi\)
−0.568666 + 0.822569i \(0.692540\pi\)
\(104\) 0 0
\(105\) −6.64761 3.04525i −0.648741 0.297186i
\(106\) 0 0
\(107\) −9.11214 + 9.11214i −0.880904 + 0.880904i −0.993626 0.112723i \(-0.964043\pi\)
0.112723 + 0.993626i \(0.464043\pi\)
\(108\) 0 0
\(109\) 15.0904i 1.44540i 0.691164 + 0.722698i \(0.257098\pi\)
−0.691164 + 0.722698i \(0.742902\pi\)
\(110\) 0 0
\(111\) 18.9164i 1.79546i
\(112\) 0 0
\(113\) 5.53162 5.53162i 0.520371 0.520371i −0.397313 0.917683i \(-0.630057\pi\)
0.917683 + 0.397313i \(0.130057\pi\)
\(114\) 0 0
\(115\) 5.83353 + 15.6963i 0.543980 + 1.46368i
\(116\) 0 0
\(117\) −10.9082 10.9082i −1.00846 1.00846i
\(118\) 0 0
\(119\) 0.412448 0.0378090
\(120\) 0 0
\(121\) −26.1646 −2.37860
\(122\) 0 0
\(123\) −16.2763 16.2763i −1.46758 1.46758i
\(124\) 0 0
\(125\) 3.11227 + 10.7384i 0.278370 + 0.960474i
\(126\) 0 0
\(127\) −2.18696 + 2.18696i −0.194061 + 0.194061i −0.797448 0.603387i \(-0.793817\pi\)
0.603387 + 0.797448i \(0.293817\pi\)
\(128\) 0 0
\(129\) 8.47127i 0.745854i
\(130\) 0 0
\(131\) 10.1306i 0.885116i 0.896740 + 0.442558i \(0.145929\pi\)
−0.896740 + 0.442558i \(0.854071\pi\)
\(132\) 0 0
\(133\) −2.30581 + 2.30581i −0.199939 + 0.199939i
\(134\) 0 0
\(135\) 32.1644 11.9539i 2.76827 1.02883i
\(136\) 0 0
\(137\) −8.91649 8.91649i −0.761787 0.761787i 0.214858 0.976645i \(-0.431071\pi\)
−0.976645 + 0.214858i \(0.931071\pi\)
\(138\) 0 0
\(139\) 21.5075 1.82424 0.912122 0.409918i \(-0.134443\pi\)
0.912122 + 0.409918i \(0.134443\pi\)
\(140\) 0 0
\(141\) −16.2236 −1.36627
\(142\) 0 0
\(143\) −8.64428 8.64428i −0.722871 0.722871i
\(144\) 0 0
\(145\) 2.21462 4.83440i 0.183914 0.401475i
\(146\) 0 0
\(147\) −2.31224 + 2.31224i −0.190710 + 0.190710i
\(148\) 0 0
\(149\) 11.9408i 0.978229i −0.872220 0.489115i \(-0.837320\pi\)
0.872220 0.489115i \(-0.162680\pi\)
\(150\) 0 0
\(151\) 11.2898i 0.918747i 0.888243 + 0.459374i \(0.151926\pi\)
−0.888243 + 0.459374i \(0.848074\pi\)
\(152\) 0 0
\(153\) −2.24358 + 2.24358i −0.181383 + 0.181383i
\(154\) 0 0
\(155\) −3.68215 + 8.03794i −0.295758 + 0.645623i
\(156\) 0 0
\(157\) −6.47755 6.47755i −0.516965 0.516965i 0.399687 0.916652i \(-0.369119\pi\)
−0.916652 + 0.399687i \(0.869119\pi\)
\(158\) 0 0
\(159\) 25.4251 2.01634
\(160\) 0 0
\(161\) 7.48870 0.590192
\(162\) 0 0
\(163\) 0.712128 + 0.712128i 0.0557782 + 0.0557782i 0.734446 0.678667i \(-0.237442\pi\)
−0.678667 + 0.734446i \(0.737442\pi\)
\(164\) 0 0
\(165\) 41.7832 15.5288i 3.25282 1.20891i
\(166\) 0 0
\(167\) 7.07849 7.07849i 0.547750 0.547750i −0.378039 0.925790i \(-0.623402\pi\)
0.925790 + 0.378039i \(0.123402\pi\)
\(168\) 0 0
\(169\) 8.97878i 0.690675i
\(170\) 0 0
\(171\) 25.0857i 1.91835i
\(172\) 0 0
\(173\) 2.94380 2.94380i 0.223813 0.223813i −0.586289 0.810102i \(-0.699412\pi\)
0.810102 + 0.586289i \(0.199412\pi\)
\(174\) 0 0
\(175\) 4.98641 + 0.368342i 0.376937 + 0.0278440i
\(176\) 0 0
\(177\) 11.6305 + 11.6305i 0.874201 + 0.874201i
\(178\) 0 0
\(179\) 13.5919 1.01591 0.507953 0.861385i \(-0.330402\pi\)
0.507953 + 0.861385i \(0.330402\pi\)
\(180\) 0 0
\(181\) −14.0560 −1.04477 −0.522386 0.852709i \(-0.674958\pi\)
−0.522386 + 0.852709i \(0.674958\pi\)
\(182\) 0 0
\(183\) −10.4326 10.4326i −0.771202 0.771202i
\(184\) 0 0
\(185\) −4.50627 12.1250i −0.331307 0.891447i
\(186\) 0 0
\(187\) −1.77795 + 1.77795i −0.130016 + 0.130016i
\(188\) 0 0
\(189\) 15.3456i 1.11623i
\(190\) 0 0
\(191\) 2.06875i 0.149689i 0.997195 + 0.0748446i \(0.0238461\pi\)
−0.997195 + 0.0748446i \(0.976154\pi\)
\(192\) 0 0
\(193\) 11.8858 11.8858i 0.855561 0.855561i −0.135250 0.990811i \(-0.543184\pi\)
0.990811 + 0.135250i \(0.0431838\pi\)
\(194\) 0 0
\(195\) 13.3305 + 6.10664i 0.954613 + 0.437305i
\(196\) 0 0
\(197\) 19.3267 + 19.3267i 1.37697 + 1.37697i 0.849689 + 0.527285i \(0.176790\pi\)
0.527285 + 0.849689i \(0.323210\pi\)
\(198\) 0 0
\(199\) −4.86247 −0.344691 −0.172346 0.985037i \(-0.555135\pi\)
−0.172346 + 0.985037i \(0.555135\pi\)
\(200\) 0 0
\(201\) −18.7124 −1.31987
\(202\) 0 0
\(203\) −1.68155 1.68155i −0.118021 0.118021i
\(204\) 0 0
\(205\) 14.3101 + 6.55540i 0.999460 + 0.457849i
\(206\) 0 0
\(207\) −40.7361 + 40.7361i −2.83135 + 2.83135i
\(208\) 0 0
\(209\) 19.8794i 1.37509i
\(210\) 0 0
\(211\) 9.21039i 0.634070i −0.948414 0.317035i \(-0.897313\pi\)
0.948414 0.317035i \(-0.102687\pi\)
\(212\) 0 0
\(213\) −1.69702 + 1.69702i −0.116278 + 0.116278i
\(214\) 0 0
\(215\) 2.01803 + 5.42990i 0.137628 + 0.370316i
\(216\) 0 0
\(217\) 2.79583 + 2.79583i 0.189793 + 0.189793i
\(218\) 0 0
\(219\) 13.6362 0.921449
\(220\) 0 0
\(221\) −0.827081 −0.0556355
\(222\) 0 0
\(223\) −1.93354 1.93354i −0.129479 0.129479i 0.639397 0.768877i \(-0.279184\pi\)
−0.768877 + 0.639397i \(0.779184\pi\)
\(224\) 0 0
\(225\) −29.1281 + 25.1208i −1.94188 + 1.67472i
\(226\) 0 0
\(227\) 7.89388 7.89388i 0.523935 0.523935i −0.394822 0.918758i \(-0.629194\pi\)
0.918758 + 0.394822i \(0.129194\pi\)
\(228\) 0 0
\(229\) 18.1386i 1.19863i 0.800513 + 0.599316i \(0.204561\pi\)
−0.800513 + 0.599316i \(0.795439\pi\)
\(230\) 0 0
\(231\) 19.9348i 1.31161i
\(232\) 0 0
\(233\) 11.5097 11.5097i 0.754027 0.754027i −0.221201 0.975228i \(-0.570998\pi\)
0.975228 + 0.221201i \(0.0709978\pi\)
\(234\) 0 0
\(235\) 10.3990 3.86479i 0.678354 0.252111i
\(236\) 0 0
\(237\) 2.98106 + 2.98106i 0.193641 + 0.193641i
\(238\) 0 0
\(239\) 6.71436 0.434316 0.217158 0.976136i \(-0.430321\pi\)
0.217158 + 0.976136i \(0.430321\pi\)
\(240\) 0 0
\(241\) −9.79406 −0.630891 −0.315446 0.948944i \(-0.602154\pi\)
−0.315446 + 0.948944i \(0.602154\pi\)
\(242\) 0 0
\(243\) 30.1122 + 30.1122i 1.93170 + 1.93170i
\(244\) 0 0
\(245\) 0.931271 2.03291i 0.0594967 0.129878i
\(246\) 0 0
\(247\) 4.62384 4.62384i 0.294208 0.294208i
\(248\) 0 0
\(249\) 31.6726i 2.00717i
\(250\) 0 0
\(251\) 26.7582i 1.68896i −0.535587 0.844480i \(-0.679909\pi\)
0.535587 0.844480i \(-0.320091\pi\)
\(252\) 0 0
\(253\) −32.2817 + 32.2817i −2.02953 + 2.02953i
\(254\) 0 0
\(255\) 1.25601 2.74179i 0.0786542 0.171698i
\(256\) 0 0
\(257\) −7.67234 7.67234i −0.478588 0.478588i 0.426092 0.904680i \(-0.359890\pi\)
−0.904680 + 0.426092i \(0.859890\pi\)
\(258\) 0 0
\(259\) −5.78484 −0.359452
\(260\) 0 0
\(261\) 18.2941 1.13238
\(262\) 0 0
\(263\) −16.7919 16.7919i −1.03543 1.03543i −0.999349 0.0360845i \(-0.988511\pi\)
−0.0360845 0.999349i \(-0.511489\pi\)
\(264\) 0 0
\(265\) −16.2969 + 6.05678i −1.00111 + 0.372065i
\(266\) 0 0
\(267\) −9.74775 + 9.74775i −0.596552 + 0.596552i
\(268\) 0 0
\(269\) 21.5442i 1.31357i −0.754076 0.656787i \(-0.771915\pi\)
0.754076 0.656787i \(-0.228085\pi\)
\(270\) 0 0
\(271\) 22.3853i 1.35981i 0.733300 + 0.679905i \(0.237979\pi\)
−0.733300 + 0.679905i \(0.762021\pi\)
\(272\) 0 0
\(273\) 4.63672 4.63672i 0.280627 0.280627i
\(274\) 0 0
\(275\) −23.0829 + 19.9072i −1.39195 + 1.20045i
\(276\) 0 0
\(277\) 6.40580 + 6.40580i 0.384887 + 0.384887i 0.872859 0.487972i \(-0.162263\pi\)
−0.487972 + 0.872859i \(0.662263\pi\)
\(278\) 0 0
\(279\) −30.4168 −1.82101
\(280\) 0 0
\(281\) 24.1228 1.43904 0.719522 0.694469i \(-0.244361\pi\)
0.719522 + 0.694469i \(0.244361\pi\)
\(282\) 0 0
\(283\) −10.7620 10.7620i −0.639734 0.639734i 0.310755 0.950490i \(-0.399418\pi\)
−0.950490 + 0.310755i \(0.899418\pi\)
\(284\) 0 0
\(285\) 8.30637 + 22.3499i 0.492027 + 1.32389i
\(286\) 0 0
\(287\) 4.97747 4.97747i 0.293811 0.293811i
\(288\) 0 0
\(289\) 16.8299i 0.989993i
\(290\) 0 0
\(291\) 13.3289i 0.781353i
\(292\) 0 0
\(293\) −10.7195 + 10.7195i −0.626238 + 0.626238i −0.947119 0.320881i \(-0.896021\pi\)
0.320881 + 0.947119i \(0.396021\pi\)
\(294\) 0 0
\(295\) −10.2255 4.68427i −0.595352 0.272729i
\(296\) 0 0
\(297\) 66.1507 + 66.1507i 3.83846 + 3.83846i
\(298\) 0 0
\(299\) −15.0171 −0.868460
\(300\) 0 0
\(301\) 2.59061 0.149320
\(302\) 0 0
\(303\) −20.8854 20.8854i −1.19983 1.19983i
\(304\) 0 0
\(305\) 9.17235 + 4.20182i 0.525207 + 0.240596i
\(306\) 0 0
\(307\) 11.8605 11.8605i 0.676914 0.676914i −0.282386 0.959301i \(-0.591126\pi\)
0.959301 + 0.282386i \(0.0911260\pi\)
\(308\) 0 0
\(309\) 11.9165i 0.677905i
\(310\) 0 0
\(311\) 2.75781i 0.156381i −0.996938 0.0781905i \(-0.975086\pi\)
0.996938 0.0781905i \(-0.0249142\pi\)
\(312\) 0 0
\(313\) −19.5624 + 19.5624i −1.10573 + 1.10573i −0.112030 + 0.993705i \(0.535735\pi\)
−0.993705 + 0.112030i \(0.964265\pi\)
\(314\) 0 0
\(315\) 5.99258 + 16.1242i 0.337644 + 0.908496i
\(316\) 0 0
\(317\) −3.66348 3.66348i −0.205761 0.205761i 0.596702 0.802463i \(-0.296478\pi\)
−0.802463 + 0.596702i \(0.796478\pi\)
\(318\) 0 0
\(319\) 14.4974 0.811696
\(320\) 0 0
\(321\) 42.1388 2.35196
\(322\) 0 0
\(323\) −0.951027 0.951027i −0.0529165 0.0529165i
\(324\) 0 0
\(325\) −9.99925 0.738635i −0.554659 0.0409721i
\(326\) 0 0
\(327\) 34.8925 34.8925i 1.92956 1.92956i
\(328\) 0 0
\(329\) 4.96136i 0.273528i
\(330\) 0 0
\(331\) 25.8355i 1.42005i −0.704178 0.710024i \(-0.748684\pi\)
0.704178 0.710024i \(-0.251316\pi\)
\(332\) 0 0
\(333\) 31.4677 31.4677i 1.72442 1.72442i
\(334\) 0 0
\(335\) 11.9942 4.45768i 0.655316 0.243549i
\(336\) 0 0
\(337\) −13.8537 13.8537i −0.754661 0.754661i 0.220684 0.975345i \(-0.429171\pi\)
−0.975345 + 0.220684i \(0.929171\pi\)
\(338\) 0 0
\(339\) −25.5808 −1.38936
\(340\) 0 0
\(341\) −24.1041 −1.30531
\(342\) 0 0
\(343\) −0.707107 0.707107i −0.0381802 0.0381802i
\(344\) 0 0
\(345\) 22.8049 49.7820i 1.22778 2.68017i
\(346\) 0 0
\(347\) 7.81196 7.81196i 0.419368 0.419368i −0.465618 0.884986i \(-0.654168\pi\)
0.884986 + 0.465618i \(0.154168\pi\)
\(348\) 0 0
\(349\) 16.0157i 0.857302i 0.903470 + 0.428651i \(0.141011\pi\)
−0.903470 + 0.428651i \(0.858989\pi\)
\(350\) 0 0
\(351\) 30.7726i 1.64252i
\(352\) 0 0
\(353\) −15.3102 + 15.3102i −0.814881 + 0.814881i −0.985361 0.170480i \(-0.945468\pi\)
0.170480 + 0.985361i \(0.445468\pi\)
\(354\) 0 0
\(355\) 0.683487 1.49201i 0.0362757 0.0791879i
\(356\) 0 0
\(357\) −0.953677 0.953677i −0.0504739 0.0504739i
\(358\) 0 0
\(359\) 20.1586 1.06393 0.531964 0.846767i \(-0.321454\pi\)
0.531964 + 0.846767i \(0.321454\pi\)
\(360\) 0 0
\(361\) −8.36647 −0.440341
\(362\) 0 0
\(363\) 60.4987 + 60.4987i 3.17536 + 3.17536i
\(364\) 0 0
\(365\) −8.74051 + 3.24842i −0.457499 + 0.170030i
\(366\) 0 0
\(367\) −12.8443 + 12.8443i −0.670466 + 0.670466i −0.957823 0.287358i \(-0.907223\pi\)
0.287358 + 0.957823i \(0.407223\pi\)
\(368\) 0 0
\(369\) 54.1516i 2.81902i
\(370\) 0 0
\(371\) 7.77528i 0.403672i
\(372\) 0 0
\(373\) −21.4076 + 21.4076i −1.10845 + 1.10845i −0.115090 + 0.993355i \(0.536716\pi\)
−0.993355 + 0.115090i \(0.963284\pi\)
\(374\) 0 0
\(375\) 17.6335 32.0261i 0.910588 1.65382i
\(376\) 0 0
\(377\) 3.37200 + 3.37200i 0.173667 + 0.173667i
\(378\) 0 0
\(379\) 22.3326 1.14715 0.573574 0.819154i \(-0.305556\pi\)
0.573574 + 0.819154i \(0.305556\pi\)
\(380\) 0 0
\(381\) 10.1135 0.518132
\(382\) 0 0
\(383\) −4.72177 4.72177i −0.241271 0.241271i 0.576105 0.817376i \(-0.304572\pi\)
−0.817376 + 0.576105i \(0.804572\pi\)
\(384\) 0 0
\(385\) 4.74887 + 12.7778i 0.242025 + 0.651215i
\(386\) 0 0
\(387\) −14.0921 + 14.0921i −0.716340 + 0.716340i
\(388\) 0 0
\(389\) 7.78307i 0.394617i 0.980341 + 0.197309i \(0.0632201\pi\)
−0.980341 + 0.197309i \(0.936780\pi\)
\(390\) 0 0
\(391\) 3.08870i 0.156202i
\(392\) 0 0
\(393\) 23.4244 23.4244i 1.18160 1.18160i
\(394\) 0 0
\(395\) −2.62094 1.20064i −0.131874 0.0604110i
\(396\) 0 0
\(397\) −23.6270 23.6270i −1.18581 1.18581i −0.978216 0.207590i \(-0.933438\pi\)
−0.207590 0.978216i \(-0.566562\pi\)
\(398\) 0 0
\(399\) 10.6632 0.533825
\(400\) 0 0
\(401\) 13.9410 0.696180 0.348090 0.937461i \(-0.386830\pi\)
0.348090 + 0.937461i \(0.386830\pi\)
\(402\) 0 0
\(403\) −5.60648 5.60648i −0.279279 0.279279i
\(404\) 0 0
\(405\) −55.0951 25.2389i −2.73770 1.25413i
\(406\) 0 0
\(407\) 24.9368 24.9368i 1.23607 1.23607i
\(408\) 0 0
\(409\) 32.1994i 1.59216i −0.605194 0.796078i \(-0.706904\pi\)
0.605194 0.796078i \(-0.293096\pi\)
\(410\) 0 0
\(411\) 41.2340i 2.03392i
\(412\) 0 0
\(413\) −3.55673 + 3.55673i −0.175015 + 0.175015i
\(414\) 0 0
\(415\) −7.54505 20.3014i −0.370372 0.996558i
\(416\) 0 0
\(417\) −49.7305 49.7305i −2.43531 2.43531i
\(418\) 0 0
\(419\) −27.5442 −1.34562 −0.672810 0.739815i \(-0.734913\pi\)
−0.672810 + 0.739815i \(0.734913\pi\)
\(420\) 0 0
\(421\) 30.1609 1.46995 0.734977 0.678092i \(-0.237193\pi\)
0.734977 + 0.678092i \(0.237193\pi\)
\(422\) 0 0
\(423\) 26.9882 + 26.9882i 1.31221 + 1.31221i
\(424\) 0 0
\(425\) −0.151922 + 2.05664i −0.00736929 + 0.0997615i
\(426\) 0 0
\(427\) 3.19041 3.19041i 0.154395 0.154395i
\(428\) 0 0
\(429\) 39.9752i 1.93002i
\(430\) 0 0
\(431\) 39.7177i 1.91313i −0.291515 0.956566i \(-0.594159\pi\)
0.291515 0.956566i \(-0.405841\pi\)
\(432\) 0 0
\(433\) 26.2101 26.2101i 1.25958 1.25958i 0.308280 0.951296i \(-0.400247\pi\)
0.951296 0.308280i \(-0.0997534\pi\)
\(434\) 0 0
\(435\) −16.2990 + 6.05754i −0.781477 + 0.290437i
\(436\) 0 0
\(437\) −17.2675 17.2675i −0.826017 0.826017i
\(438\) 0 0
\(439\) 27.0808 1.29249 0.646247 0.763128i \(-0.276338\pi\)
0.646247 + 0.763128i \(0.276338\pi\)
\(440\) 0 0
\(441\) 7.69286 0.366327
\(442\) 0 0
\(443\) −0.742047 0.742047i −0.0352557 0.0352557i 0.689259 0.724515i \(-0.257936\pi\)
−0.724515 + 0.689259i \(0.757936\pi\)
\(444\) 0 0
\(445\) 3.92598 8.57020i 0.186109 0.406267i
\(446\) 0 0
\(447\) −27.6100 + 27.6100i −1.30591 + 1.30591i
\(448\) 0 0
\(449\) 15.7917i 0.745258i 0.927980 + 0.372629i \(0.121544\pi\)
−0.927980 + 0.372629i \(0.878456\pi\)
\(450\) 0 0
\(451\) 42.9130i 2.02069i
\(452\) 0 0
\(453\) 26.1046 26.1046i 1.22650 1.22650i
\(454\) 0 0
\(455\) −1.86748 + 4.07660i −0.0875486 + 0.191114i
\(456\) 0 0
\(457\) −14.5908 14.5908i −0.682527 0.682527i 0.278042 0.960569i \(-0.410314\pi\)
−0.960569 + 0.278042i \(0.910314\pi\)
\(458\) 0 0
\(459\) 6.32928 0.295425
\(460\) 0 0
\(461\) 9.77291 0.455170 0.227585 0.973758i \(-0.426917\pi\)
0.227585 + 0.973758i \(0.426917\pi\)
\(462\) 0 0
\(463\) 2.32678 + 2.32678i 0.108135 + 0.108135i 0.759104 0.650969i \(-0.225637\pi\)
−0.650969 + 0.759104i \(0.725637\pi\)
\(464\) 0 0
\(465\) 27.0996 10.0716i 1.25671 0.467059i
\(466\) 0 0
\(467\) 19.7558 19.7558i 0.914192 0.914192i −0.0824071 0.996599i \(-0.526261\pi\)
0.996599 + 0.0824071i \(0.0262608\pi\)
\(468\) 0 0
\(469\) 5.72246i 0.264239i
\(470\) 0 0
\(471\) 29.9552i 1.38026i
\(472\) 0 0
\(473\) −11.1674 + 11.1674i −0.513477 + 0.513477i
\(474\) 0 0
\(475\) −10.6484 12.3471i −0.488582 0.566522i
\(476\) 0 0
\(477\) −42.2950 42.2950i −1.93655 1.93655i
\(478\) 0 0
\(479\) −36.9223 −1.68702 −0.843511 0.537112i \(-0.819515\pi\)
−0.843511 + 0.537112i \(0.819515\pi\)
\(480\) 0 0
\(481\) 11.6003 0.528930
\(482\) 0 0
\(483\) −17.3156 17.3156i −0.787888 0.787888i
\(484\) 0 0
\(485\) 3.17521 + 8.54353i 0.144179 + 0.387942i
\(486\) 0 0
\(487\) −16.1499 + 16.1499i −0.731822 + 0.731822i −0.970980 0.239159i \(-0.923128\pi\)
0.239159 + 0.970980i \(0.423128\pi\)
\(488\) 0 0
\(489\) 3.29321i 0.148924i
\(490\) 0 0
\(491\) 0.901752i 0.0406955i −0.999793 0.0203477i \(-0.993523\pi\)
0.999793 0.0203477i \(-0.00647734\pi\)
\(492\) 0 0
\(493\) 0.693550 0.693550i 0.0312359 0.0312359i
\(494\) 0 0
\(495\) −95.3392 43.6746i −4.28518 1.96303i
\(496\) 0 0
\(497\) −0.518966 0.518966i −0.0232788 0.0232788i
\(498\) 0 0
\(499\) 24.1739 1.08217 0.541085 0.840968i \(-0.318014\pi\)
0.541085 + 0.840968i \(0.318014\pi\)
\(500\) 0 0
\(501\) −32.7343 −1.46246
\(502\) 0 0
\(503\) 5.72002 + 5.72002i 0.255043 + 0.255043i 0.823034 0.567991i \(-0.192279\pi\)
−0.567991 + 0.823034i \(0.692279\pi\)
\(504\) 0 0
\(505\) 18.3624 + 8.41174i 0.817115 + 0.374318i
\(506\) 0 0
\(507\) 20.7610 20.7610i 0.922030 0.922030i
\(508\) 0 0
\(509\) 27.3014i 1.21011i −0.796183 0.605056i \(-0.793151\pi\)
0.796183 0.605056i \(-0.206849\pi\)
\(510\) 0 0
\(511\) 4.17010i 0.184474i
\(512\) 0 0
\(513\) −35.3841 + 35.3841i −1.56225 + 1.56225i
\(514\) 0 0
\(515\) 2.83875 + 7.63821i 0.125090 + 0.336580i
\(516\) 0 0
\(517\) 21.3870 + 21.3870i 0.940600 + 0.940600i
\(518\) 0 0
\(519\) −13.6135 −0.597567
\(520\) 0 0
\(521\) −10.5026 −0.460129 −0.230065 0.973175i \(-0.573894\pi\)
−0.230065 + 0.973175i \(0.573894\pi\)
\(522\) 0 0
\(523\) −11.7721 11.7721i −0.514757 0.514757i 0.401223 0.915980i \(-0.368585\pi\)
−0.915980 + 0.401223i \(0.868585\pi\)
\(524\) 0 0
\(525\) −10.6781 12.3815i −0.466029 0.540371i
\(526\) 0 0
\(527\) −1.15313 + 1.15313i −0.0502313 + 0.0502313i
\(528\) 0 0
\(529\) 33.0806i 1.43829i
\(530\) 0 0
\(531\) 38.6949i 1.67922i
\(532\) 0 0
\(533\) −9.98131 + 9.98131i −0.432339 + 0.432339i
\(534\) 0 0
\(535\) −27.0101 + 10.0383i −1.16775 + 0.433994i
\(536\) 0 0
\(537\) −31.4276 31.4276i −1.35620 1.35620i
\(538\) 0 0
\(539\) 6.09628 0.262585
\(540\) 0 0
\(541\) −23.3649 −1.00453 −0.502267 0.864712i \(-0.667501\pi\)
−0.502267 + 0.864712i \(0.667501\pi\)
\(542\) 0 0
\(543\) 32.5007 + 32.5007i 1.39474 + 1.39474i
\(544\) 0 0
\(545\) −14.0532 + 30.6774i −0.601974 + 1.31408i
\(546\) 0 0
\(547\) 25.8564 25.8564i 1.10554 1.10554i 0.111808 0.993730i \(-0.464336\pi\)
0.993730 0.111808i \(-0.0356642\pi\)
\(548\) 0 0
\(549\) 34.7096i 1.48137i
\(550\) 0 0
\(551\) 7.75466i 0.330359i
\(552\) 0 0
\(553\) −0.911640 + 0.911640i −0.0387669 + 0.0387669i
\(554\) 0 0
\(555\) −17.6163 + 38.4554i −0.747769 + 1.63234i
\(556\) 0 0
\(557\) 30.7534 + 30.7534i 1.30307 + 1.30307i 0.926315 + 0.376750i \(0.122959\pi\)
0.376750 + 0.926315i \(0.377041\pi\)
\(558\) 0 0
\(559\) −5.19494 −0.219723
\(560\) 0 0
\(561\) 8.22206 0.347136
\(562\) 0 0
\(563\) 17.2257 + 17.2257i 0.725978 + 0.725978i 0.969816 0.243838i \(-0.0784064\pi\)
−0.243838 + 0.969816i \(0.578406\pi\)
\(564\) 0 0
\(565\) 16.3967 6.09386i 0.689815 0.256371i
\(566\) 0 0
\(567\) −19.1637 + 19.1637i −0.804800 + 0.804800i
\(568\) 0 0
\(569\) 7.71562i 0.323455i −0.986835 0.161728i \(-0.948293\pi\)
0.986835 0.161728i \(-0.0517066\pi\)
\(570\) 0 0
\(571\) 34.7991i 1.45630i 0.685419 + 0.728149i \(0.259619\pi\)
−0.685419 + 0.728149i \(0.740381\pi\)
\(572\) 0 0
\(573\) 4.78343 4.78343i 0.199831 0.199831i
\(574\) 0 0
\(575\) −2.75840 + 37.3417i −0.115033 + 1.55726i
\(576\) 0 0
\(577\) −9.91727 9.91727i −0.412861 0.412861i 0.469873 0.882734i \(-0.344300\pi\)
−0.882734 + 0.469873i \(0.844300\pi\)
\(578\) 0 0
\(579\) −54.9657 −2.28430
\(580\) 0 0
\(581\) −9.68582 −0.401836
\(582\) 0 0
\(583\) −33.5170 33.5170i −1.38813 1.38813i
\(584\) 0 0
\(585\) −12.0169 32.3338i −0.496838 1.33684i
\(586\) 0 0
\(587\) 0.719263 0.719263i 0.0296872 0.0296872i −0.692107 0.721795i \(-0.743318\pi\)
0.721795 + 0.692107i \(0.243318\pi\)
\(588\) 0 0
\(589\) 12.8933i 0.531260i
\(590\) 0 0
\(591\) 89.3760i 3.67644i
\(592\) 0 0
\(593\) 7.54021 7.54021i 0.309639 0.309639i −0.535130 0.844770i \(-0.679737\pi\)
0.844770 + 0.535130i \(0.179737\pi\)
\(594\) 0 0
\(595\) 0.838471 + 0.384101i 0.0343739 + 0.0157466i
\(596\) 0 0
\(597\) 11.2432 + 11.2432i 0.460152 + 0.460152i
\(598\) 0 0
\(599\) 5.70994 0.233302 0.116651 0.993173i \(-0.462784\pi\)
0.116651 + 0.993173i \(0.462784\pi\)
\(600\) 0 0
\(601\) 5.67836 0.231625 0.115813 0.993271i \(-0.463053\pi\)
0.115813 + 0.993271i \(0.463053\pi\)
\(602\) 0 0
\(603\) 31.1283 + 31.1283i 1.26764 + 1.26764i
\(604\) 0 0
\(605\) −53.1904 24.3663i −2.16250 0.990632i
\(606\) 0 0
\(607\) −21.5239 + 21.5239i −0.873627 + 0.873627i −0.992866 0.119239i \(-0.961955\pi\)
0.119239 + 0.992866i \(0.461955\pi\)
\(608\) 0 0
\(609\) 7.77626i 0.315110i
\(610\) 0 0
\(611\) 9.94900i 0.402493i
\(612\) 0 0
\(613\) −4.14168 + 4.14168i −0.167281 + 0.167281i −0.785783 0.618502i \(-0.787740\pi\)
0.618502 + 0.785783i \(0.287740\pi\)
\(614\) 0 0
\(615\) −17.9307 48.2459i −0.723034 1.94546i
\(616\) 0 0
\(617\) −15.9214 15.9214i −0.640973 0.640973i 0.309822 0.950795i \(-0.399731\pi\)
−0.950795 + 0.309822i \(0.899731\pi\)
\(618\) 0 0
\(619\) 19.5060 0.784011 0.392006 0.919963i \(-0.371781\pi\)
0.392006 + 0.919963i \(0.371781\pi\)
\(620\) 0 0
\(621\) 114.919 4.61153
\(622\) 0 0
\(623\) −2.98097 2.98097i −0.119430 0.119430i
\(624\) 0 0
\(625\) −3.67341 + 24.7286i −0.146936 + 0.989146i
\(626\) 0 0
\(627\) −45.9659 + 45.9659i −1.83570 + 1.83570i
\(628\) 0 0
\(629\) 2.38594i 0.0951338i
\(630\) 0 0
\(631\) 31.3766i 1.24908i −0.780991 0.624542i \(-0.785286\pi\)
0.780991 0.624542i \(-0.214714\pi\)
\(632\) 0 0
\(633\) −21.2966 + 21.2966i −0.846464 + 0.846464i
\(634\) 0 0
\(635\) −6.48255 + 2.40925i −0.257252 + 0.0956080i
\(636\) 0 0
\(637\) 1.41796 + 1.41796i 0.0561816 + 0.0561816i
\(638\) 0 0
\(639\) 5.64602 0.223353
\(640\) 0 0
\(641\) 19.8373 0.783525 0.391762 0.920066i \(-0.371865\pi\)
0.391762 + 0.920066i \(0.371865\pi\)
\(642\) 0 0
\(643\) −18.4092 18.4092i −0.725988 0.725988i 0.243830 0.969818i \(-0.421596\pi\)
−0.969818 + 0.243830i \(0.921596\pi\)
\(644\) 0 0
\(645\) 7.88905 17.2214i 0.310631 0.678090i
\(646\) 0 0
\(647\) −12.9741 + 12.9741i −0.510065 + 0.510065i −0.914546 0.404481i \(-0.867452\pi\)
0.404481 + 0.914546i \(0.367452\pi\)
\(648\) 0 0
\(649\) 30.6641i 1.20367i
\(650\) 0 0
\(651\) 12.9292i 0.506737i
\(652\) 0 0
\(653\) 13.9963 13.9963i 0.547717 0.547717i −0.378063 0.925780i \(-0.623410\pi\)
0.925780 + 0.378063i \(0.123410\pi\)
\(654\) 0 0
\(655\) −9.43435 + 20.5947i −0.368630 + 0.804700i
\(656\) 0 0
\(657\) −22.6840 22.6840i −0.884987 0.884987i
\(658\) 0 0
\(659\) 9.44981 0.368112 0.184056 0.982916i \(-0.441077\pi\)
0.184056 + 0.982916i \(0.441077\pi\)
\(660\) 0 0
\(661\) 8.58645 0.333974 0.166987 0.985959i \(-0.446596\pi\)
0.166987 + 0.985959i \(0.446596\pi\)
\(662\) 0 0
\(663\) 1.91241 + 1.91241i 0.0742717 + 0.0742717i
\(664\) 0 0
\(665\) −6.83485 + 2.54018i −0.265044 + 0.0985039i
\(666\) 0 0
\(667\) 12.5926 12.5926i 0.487587 0.487587i
\(668\) 0 0
\(669\) 8.94158i 0.345702i
\(670\) 0 0
\(671\) 27.5059i 1.06186i
\(672\) 0 0
\(673\) 2.63136 2.63136i 0.101432 0.101432i −0.654570 0.756001i \(-0.727150\pi\)
0.756001 + 0.654570i \(0.227150\pi\)
\(674\) 0 0
\(675\) 76.5197 + 5.65244i 2.94524 + 0.217562i
\(676\) 0 0
\(677\) 33.8779 + 33.8779i 1.30203 + 1.30203i 0.927020 + 0.375012i \(0.122361\pi\)
0.375012 + 0.927020i \(0.377639\pi\)
\(678\) 0 0
\(679\) 4.07612 0.156427
\(680\) 0 0
\(681\) −36.5050 −1.39888
\(682\) 0 0
\(683\) 21.6544 + 21.6544i 0.828582 + 0.828582i 0.987321 0.158738i \(-0.0507427\pi\)
−0.158738 + 0.987321i \(0.550743\pi\)
\(684\) 0 0
\(685\) −9.82278 26.4301i −0.375309 1.00984i
\(686\) 0 0
\(687\) 41.9407 41.9407i 1.60014 1.60014i
\(688\) 0 0
\(689\) 15.5918i 0.593999i
\(690\) 0 0
\(691\) 2.95056i 0.112245i −0.998424 0.0561224i \(-0.982126\pi\)
0.998424 0.0561224i \(-0.0178737\pi\)
\(692\) 0 0
\(693\) −33.1618 + 33.1618i −1.25971 + 1.25971i
\(694\) 0 0
\(695\) 43.7229 + 20.0293i 1.65851 + 0.759756i
\(696\) 0 0
\(697\) 2.05295 + 2.05295i 0.0777609 + 0.0777609i
\(698\) 0 0
\(699\) −53.2263 −2.01321
\(700\) 0 0
\(701\) 36.0970 1.36336 0.681682 0.731649i \(-0.261249\pi\)
0.681682 + 0.731649i \(0.261249\pi\)
\(702\) 0 0
\(703\) 13.3387 + 13.3387i 0.503080 + 0.503080i
\(704\) 0 0
\(705\) −32.9812 15.1086i −1.24214 0.569021i
\(706\) 0 0
\(707\) 6.38697 6.38697i 0.240207 0.240207i
\(708\) 0 0
\(709\) 34.6727i 1.30216i −0.759008 0.651081i \(-0.774316\pi\)
0.759008 0.651081i \(-0.225684\pi\)
\(710\) 0 0
\(711\) 9.91805i 0.371956i
\(712\) 0 0
\(713\) −20.9371 + 20.9371i −0.784102 + 0.784102i
\(714\) 0 0
\(715\) −9.52291 25.6232i −0.356137 0.958255i
\(716\) 0 0
\(717\) −15.5252 15.5252i −0.579798 0.579798i
\(718\) 0 0
\(719\) 7.21998 0.269260 0.134630 0.990896i \(-0.457015\pi\)
0.134630 + 0.990896i \(0.457015\pi\)
\(720\) 0 0
\(721\) 3.64419 0.135717
\(722\) 0 0
\(723\) 22.6462 + 22.6462i 0.842221 + 0.842221i
\(724\) 0 0
\(725\) 9.00427 7.76550i 0.334410 0.288404i
\(726\) 0 0
\(727\) 25.0772 25.0772i 0.930061 0.930061i −0.0676479 0.997709i \(-0.521549\pi\)
0.997709 + 0.0676479i \(0.0215494\pi\)
\(728\) 0 0
\(729\) 57.9482i 2.14623i
\(730\) 0 0
\(731\) 1.06849i 0.0395196i
\(732\) 0 0
\(733\) 16.7234 16.7234i 0.617692 0.617692i −0.327247 0.944939i \(-0.606121\pi\)
0.944939 + 0.327247i \(0.106121\pi\)
\(734\) 0 0
\(735\) −6.85389 + 2.54726i −0.252810 + 0.0939570i
\(736\) 0 0
\(737\) 24.6679 + 24.6679i 0.908655 + 0.908655i
\(738\) 0 0
\(739\) 18.4665 0.679300 0.339650 0.940552i \(-0.389691\pi\)
0.339650 + 0.940552i \(0.389691\pi\)
\(740\) 0 0
\(741\) −21.3828 −0.785517
\(742\) 0 0
\(743\) −10.9865 10.9865i −0.403054 0.403054i 0.476254 0.879308i \(-0.341994\pi\)
−0.879308 + 0.476254i \(0.841994\pi\)
\(744\) 0 0
\(745\) 11.1201 24.2746i 0.407410 0.889353i
\(746\) 0 0
\(747\) 52.6877 52.6877i 1.92774 1.92774i
\(748\) 0 0
\(749\) 12.8865i 0.470863i
\(750\) 0 0
\(751\) 3.02969i 0.110555i 0.998471 + 0.0552775i \(0.0176044\pi\)
−0.998471 + 0.0552775i \(0.982396\pi\)
\(752\) 0 0
\(753\) −61.8712 + 61.8712i −2.25471 + 2.25471i
\(754\) 0 0
\(755\) −10.5138 + 22.9511i −0.382637 + 0.835276i
\(756\) 0 0
\(757\) 21.4638 + 21.4638i 0.780114 + 0.780114i 0.979850 0.199736i \(-0.0640085\pi\)
−0.199736 + 0.979850i \(0.564009\pi\)
\(758\) 0 0
\(759\) 149.286 5.41872
\(760\) 0 0
\(761\) 16.4590 0.596640 0.298320 0.954466i \(-0.403574\pi\)
0.298320 + 0.954466i \(0.403574\pi\)
\(762\) 0 0
\(763\) 10.6705 + 10.6705i 0.386299 + 0.386299i
\(764\) 0 0
\(765\) −6.65039 + 2.47163i −0.240445 + 0.0893618i
\(766\) 0 0
\(767\) 7.13231 7.13231i 0.257533 0.257533i
\(768\) 0 0
\(769\) 41.4933i 1.49629i 0.663537 + 0.748143i \(0.269054\pi\)
−0.663537 + 0.748143i \(0.730946\pi\)
\(770\) 0 0
\(771\) 35.4805i 1.27780i
\(772\) 0 0
\(773\) −10.2071 + 10.2071i −0.367125 + 0.367125i −0.866428 0.499303i \(-0.833590\pi\)
0.499303 + 0.866428i \(0.333590\pi\)
\(774\) 0 0
\(775\) −14.9710 + 12.9114i −0.537774 + 0.463789i
\(776\) 0 0
\(777\) 13.3759 + 13.3759i 0.479858 + 0.479858i
\(778\) 0 0
\(779\) −22.9542 −0.822420
\(780\) 0 0
\(781\) 4.47424 0.160101
\(782\) 0 0
\(783\) −25.8044 25.8044i −0.922174 0.922174i
\(784\) 0 0
\(785\) −7.13595 19.2007i −0.254693 0.685301i
\(786\) 0 0
\(787\) −25.4080 + 25.4080i −0.905697 + 0.905697i −0.995921 0.0902245i \(-0.971242\pi\)
0.0902245 + 0.995921i \(0.471242\pi\)
\(788\) 0 0
\(789\) 77.6537i 2.76454i
\(790\) 0 0
\(791\) 7.82289i 0.278150i
\(792\) 0 0
\(793\) −6.39773 + 6.39773i −0.227190 + 0.227190i
\(794\) 0 0
\(795\) 51.6871 + 23.6777i 1.83315 + 0.839760i
\(796\) 0 0
\(797\) 1.41505 + 1.41505i 0.0501236 + 0.0501236i 0.731724 0.681601i \(-0.238716\pi\)
−0.681601 + 0.731724i \(0.738716\pi\)
\(798\) 0 0
\(799\) 2.04630 0.0723929
\(800\) 0 0
\(801\) 32.4310 1.14589
\(802\) 0 0
\(803\) −17.9761 17.9761i −0.634364 0.634364i
\(804\) 0 0
\(805\) 15.2239 + 6.97400i 0.536571 + 0.245801i
\(806\) 0 0
\(807\) −49.8153 + 49.8153i −1.75358 + 1.75358i
\(808\) 0 0
\(809\) 29.5078i 1.03744i −0.854945 0.518719i \(-0.826409\pi\)
0.854945 0.518719i \(-0.173591\pi\)
\(810\) 0 0
\(811\) 2.08089i 0.0730699i −0.999332 0.0365350i \(-0.988368\pi\)
0.999332 0.0365350i \(-0.0116320\pi\)
\(812\) 0 0
\(813\) 51.7601 51.7601i 1.81530 1.81530i
\(814\) 0 0
\(815\) 0.784510 + 2.11088i 0.0274802 + 0.0739408i
\(816\) 0 0
\(817\) −5.97345 5.97345i −0.208985 0.208985i
\(818\) 0 0
\(819\) −15.4265 −0.539045
\(820\) 0 0
\(821\) 4.51361 0.157526 0.0787631 0.996893i \(-0.474903\pi\)
0.0787631 + 0.996893i \(0.474903\pi\)
\(822\) 0 0
\(823\) −19.8753 19.8753i −0.692811 0.692811i 0.270038 0.962850i \(-0.412964\pi\)
−0.962850 + 0.270038i \(0.912964\pi\)
\(824\) 0 0
\(825\) 99.4031 + 7.34282i 3.46077 + 0.255644i
\(826\) 0 0
\(827\) −17.6393 + 17.6393i −0.613377 + 0.613377i −0.943824 0.330447i \(-0.892800\pi\)
0.330447 + 0.943824i \(0.392800\pi\)
\(828\) 0 0
\(829\) 39.8366i 1.38358i 0.722097 + 0.691791i \(0.243178\pi\)
−0.722097 + 0.691791i \(0.756822\pi\)
\(830\) 0 0
\(831\) 29.6234i 1.02763i
\(832\) 0 0
\(833\) 0.291645 0.291645i 0.0101049 0.0101049i
\(834\) 0 0
\(835\) 20.9820 7.79797i 0.726110 0.269860i
\(836\) 0 0
\(837\) 42.9038 + 42.9038i 1.48297 + 1.48297i
\(838\) 0 0
\(839\) 9.58183 0.330802 0.165401 0.986226i \(-0.447108\pi\)
0.165401 + 0.986226i \(0.447108\pi\)
\(840\) 0 0
\(841\) 23.3448 0.804993
\(842\) 0 0
\(843\) −55.7776 55.7776i −1.92108 1.92108i
\(844\) 0 0
\(845\) −8.36167 + 18.2531i −0.287650 + 0.627925i
\(846\) 0 0
\(847\) −18.5012 + 18.5012i −0.635708 + 0.635708i
\(848\) 0 0
\(849\) 49.7685i 1.70805i
\(850\) 0 0
\(851\) 43.3209i 1.48502i
\(852\) 0 0
\(853\) 27.7835 27.7835i 0.951290 0.951290i −0.0475780 0.998868i \(-0.515150\pi\)
0.998868 + 0.0475780i \(0.0151503\pi\)
\(854\) 0 0
\(855\) 23.3616 50.9971i 0.798950 1.74406i
\(856\) 0 0
\(857\) −10.8089 10.8089i −0.369224 0.369224i 0.497970 0.867194i \(-0.334079\pi\)
−0.867194 + 0.497970i \(0.834079\pi\)
\(858\) 0 0
\(859\) −45.0374 −1.53666 −0.768328 0.640057i \(-0.778911\pi\)
−0.768328 + 0.640057i \(0.778911\pi\)
\(860\) 0 0
\(861\) −23.0182 −0.784457
\(862\) 0 0
\(863\) −6.21367 6.21367i −0.211516 0.211516i 0.593395 0.804911i \(-0.297787\pi\)
−0.804911 + 0.593395i \(0.797787\pi\)
\(864\) 0 0
\(865\) 8.72596 3.24301i 0.296691 0.110266i
\(866\) 0 0
\(867\) −38.9147 + 38.9147i −1.32161 + 1.32161i
\(868\) 0 0
\(869\) 7.85965i 0.266620i
\(870\) 0 0
\(871\) 11.4752i 0.388824i
\(872\) 0 0
\(873\) −22.1728 + 22.1728i −0.750435 + 0.750435i
\(874\) 0 0
\(875\) 9.79392 + 5.39251i 0.331095 + 0.182300i
\(876\) 0 0
\(877\) −25.5695 25.5695i −0.863419 0.863419i 0.128315 0.991734i \(-0.459043\pi\)
−0.991734 + 0.128315i \(0.959043\pi\)
\(878\) 0 0
\(879\) 49.5719 1.67202
\(880\) 0 0
\(881\) 16.3716 0.551572 0.275786 0.961219i \(-0.411062\pi\)
0.275786 + 0.961219i \(0.411062\pi\)
\(882\) 0 0
\(883\) 28.1512 + 28.1512i 0.947365 + 0.947365i 0.998682 0.0513175i \(-0.0163420\pi\)
−0.0513175 + 0.998682i \(0.516342\pi\)
\(884\) 0 0
\(885\) 12.8126 + 34.4749i 0.430692 + 1.15886i
\(886\) 0 0
\(887\) −20.5121 + 20.5121i −0.688730 + 0.688730i −0.961951 0.273221i \(-0.911911\pi\)
0.273221 + 0.961951i \(0.411911\pi\)
\(888\) 0 0
\(889\) 3.09283i 0.103730i
\(890\) 0 0
\(891\) 165.219i 5.53503i
\(892\) 0 0
\(893\) −11.4399 + 11.4399i −0.382823 + 0.382823i
\(894\) 0 0
\(895\) 27.6311 + 12.6577i 0.923607 + 0.423101i
\(896\) 0 0
\(897\) 34.7230 + 34.7230i 1.15937 + 1.15937i
\(898\) 0 0
\(899\) 9.40264 0.313596
\(900\) 0 0
\(901\) −3.20690 −0.106837
\(902\) 0 0
\(903\) −5.99009 5.99009i −0.199338 0.199338i
\(904\) 0 0
\(905\) −28.5746 13.0899i −0.949851 0.435124i
\(906\) 0 0
\(907\) 7.17842 7.17842i 0.238356 0.238356i −0.577813 0.816169i \(-0.696094\pi\)
0.816169 + 0.577813i \(0.196094\pi\)
\(908\) 0 0
\(909\) 69.4861i 2.30471i
\(910\) 0 0
\(911\) 36.6686i 1.21488i 0.794364 + 0.607442i \(0.207804\pi\)
−0.794364 + 0.607442i \(0.792196\pi\)
\(912\) 0 0
\(913\) 41.7529 41.7529i 1.38182 1.38182i
\(914\) 0 0
\(915\) −11.4930 30.9242i −0.379948 1.02232i
\(916\) 0 0
\(917\) 7.16343 + 7.16343i 0.236557 + 0.236557i
\(918\) 0 0
\(919\) −57.9984 −1.91319 −0.956595 0.291420i \(-0.905872\pi\)
−0.956595 + 0.291420i \(0.905872\pi\)
\(920\) 0 0
\(921\) −54.8485 −1.80732
\(922\) 0 0
\(923\) 1.04068 + 1.04068i 0.0342545 + 0.0342545i
\(924\) 0 0
\(925\) 2.13080 28.8456i 0.0700602 0.948438i
\(926\) 0 0
\(927\) −19.8232 + 19.8232i −0.651080 + 0.651080i
\(928\) 0 0
\(929\) 18.4743i 0.606122i −0.952971 0.303061i \(-0.901991\pi\)
0.952971 0.303061i \(-0.0980086\pi\)
\(930\) 0 0
\(931\) 3.26091i 0.106872i
\(932\) 0 0
\(933\) −6.37670 + 6.37670i −0.208764 + 0.208764i
\(934\) 0 0
\(935\) −5.27016 + 1.95866i −0.172353 + 0.0640551i
\(936\) 0 0
\(937\) 10.7475 + 10.7475i 0.351107 + 0.351107i 0.860521 0.509415i \(-0.170138\pi\)
−0.509415 + 0.860521i \(0.670138\pi\)
\(938\) 0 0
\(939\) 90.4659 2.95224
\(940\) 0 0
\(941\) −34.6701 −1.13021 −0.565106 0.825018i \(-0.691165\pi\)
−0.565106 + 0.825018i \(0.691165\pi\)
\(942\) 0 0
\(943\) 37.2748 + 37.2748i 1.21383 + 1.21383i
\(944\) 0 0
\(945\) 14.2909 31.1963i 0.464884 1.01482i
\(946\) 0 0
\(947\) −24.8525 + 24.8525i −0.807598 + 0.807598i −0.984270 0.176672i \(-0.943467\pi\)
0.176672 + 0.984270i \(0.443467\pi\)
\(948\) 0 0
\(949\) 8.36230i 0.271452i
\(950\) 0 0
\(951\) 16.9416i 0.549370i
\(952\) 0 0
\(953\) −29.0869 + 29.0869i −0.942218 + 0.942218i −0.998419 0.0562014i \(-0.982101\pi\)
0.0562014 + 0.998419i \(0.482101\pi\)
\(954\) 0 0
\(955\) −1.92656 + 4.20558i −0.0623421 + 0.136089i
\(956\) 0 0
\(957\) −33.5213 33.5213i −1.08359 1.08359i
\(958\) 0 0
\(959\) −12.6098 −0.407192
\(960\) 0 0
\(961\) 15.3667 0.495698
\(962\) 0 0
\(963\) −70.0984 70.0984i −2.25889 2.25889i
\(964\) 0 0
\(965\) 35.2318 13.0939i 1.13415 0.421509i
\(966\) 0 0
\(967\) −40.9518 + 40.9518i −1.31692 + 1.31692i −0.400718 + 0.916201i \(0.631239\pi\)
−0.916201 + 0.400718i \(0.868761\pi\)
\(968\) 0 0
\(969\) 4.39800i 0.141284i
\(970\) 0 0
\(971\) 38.8745i 1.24754i −0.781608 0.623770i \(-0.785600\pi\)
0.781608 0.623770i \(-0.214400\pi\)
\(972\) 0 0
\(973\) 15.2081 15.2081i 0.487550 0.487550i
\(974\) 0 0
\(975\) 21.4127 + 24.8285i 0.685756 + 0.795149i
\(976\) 0 0
\(977\) 2.22366 + 2.22366i 0.0711412 + 0.0711412i 0.741782 0.670641i \(-0.233981\pi\)
−0.670641 + 0.741782i \(0.733981\pi\)
\(978\) 0 0
\(979\) 25.7002 0.821383
\(980\) 0 0
\(981\) −116.088 −3.70641
\(982\) 0 0
\(983\) 8.13209 + 8.13209i 0.259373 + 0.259373i 0.824799 0.565426i \(-0.191288\pi\)
−0.565426 + 0.824799i \(0.691288\pi\)
\(984\) 0 0
\(985\) 21.2912 + 57.2880i 0.678393 + 1.82535i
\(986\) 0 0
\(987\) −11.4718 + 11.4718i −0.365152 + 0.365152i
\(988\) 0 0
\(989\) 19.4003i 0.616893i
\(990\) 0 0
\(991\) 16.7134i 0.530917i −0.964122 0.265459i \(-0.914477\pi\)
0.964122 0.265459i \(-0.0855234\pi\)
\(992\) 0 0
\(993\) −59.7378 + 59.7378i −1.89572 + 1.89572i
\(994\) 0 0
\(995\) −9.88498 4.52827i −0.313375 0.143556i
\(996\) 0 0
\(997\) 7.52263 + 7.52263i 0.238244 + 0.238244i 0.816123 0.577879i \(-0.196119\pi\)
−0.577879 + 0.816123i \(0.696119\pi\)
\(998\) 0 0
\(999\) −88.7721 −2.80862
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 560.2.x.b.127.1 24
4.3 odd 2 inner 560.2.x.b.127.12 yes 24
5.3 odd 4 inner 560.2.x.b.463.12 yes 24
20.3 even 4 inner 560.2.x.b.463.1 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.x.b.127.1 24 1.1 even 1 trivial
560.2.x.b.127.12 yes 24 4.3 odd 2 inner
560.2.x.b.463.1 yes 24 20.3 even 4 inner
560.2.x.b.463.12 yes 24 5.3 odd 4 inner