Properties

Label 1120.3.c.g.209.13
Level $1120$
Weight $3$
Character 1120.209
Analytic conductor $30.518$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,3,Mod(209,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, 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("1120.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1120.c (of order \(2\), degree \(1\), 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: \(30.5177896084\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.13
Character \(\chi\) \(=\) 1120.209
Dual form 1120.3.c.g.209.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.68915i q^{3} +(0.693644 - 4.95165i) q^{5} +(4.51720 + 5.34742i) q^{7} -4.60986 q^{9} +O(q^{10})\) \(q-3.68915i q^{3} +(0.693644 - 4.95165i) q^{5} +(4.51720 + 5.34742i) q^{7} -4.60986 q^{9} -10.2130i q^{11} +14.4358i q^{13} +(-18.2674 - 2.55896i) q^{15} -16.4391 q^{17} +31.0538 q^{19} +(19.7275 - 16.6647i) q^{21} -20.2273i q^{23} +(-24.0377 - 6.86937i) q^{25} -16.1959i q^{27} -56.8127i q^{29} -25.7499i q^{31} -37.6775 q^{33} +(29.6119 - 18.6584i) q^{35} -66.0553 q^{37} +53.2560 q^{39} -0.504264i q^{41} -26.2293 q^{43} +(-3.19760 + 22.8264i) q^{45} -20.2286 q^{47} +(-8.18977 + 48.3107i) q^{49} +60.6465i q^{51} +53.0620 q^{53} +(-50.5714 - 7.08422i) q^{55} -114.562i q^{57} +59.3867 q^{59} -69.5062 q^{61} +(-20.8237 - 24.6509i) q^{63} +(71.4812 + 10.0133i) q^{65} +61.9289 q^{67} -74.6215 q^{69} +25.5504 q^{71} +34.7213 q^{73} +(-25.3422 + 88.6788i) q^{75} +(54.6134 - 46.1344i) q^{77} -64.4865 q^{79} -101.238 q^{81} +9.27026i q^{83} +(-11.4029 + 81.4009i) q^{85} -209.591 q^{87} -22.0650i q^{89} +(-77.1944 + 65.2095i) q^{91} -94.9952 q^{93} +(21.5403 - 153.768i) q^{95} -74.3644 q^{97} +47.0807i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - 224 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) 3.68915i 1.22972i −0.788637 0.614859i \(-0.789213\pi\)
0.788637 0.614859i \(-0.210787\pi\)
\(4\) 0 0
\(5\) 0.693644 4.95165i 0.138729 0.990330i
\(6\) 0 0
\(7\) 4.51720 + 5.34742i 0.645315 + 0.763917i
\(8\) 0 0
\(9\) −4.60986 −0.512207
\(10\) 0 0
\(11\) 10.2130i 0.928458i −0.885715 0.464229i \(-0.846331\pi\)
0.885715 0.464229i \(-0.153669\pi\)
\(12\) 0 0
\(13\) 14.4358i 1.11045i 0.831701 + 0.555224i \(0.187367\pi\)
−0.831701 + 0.555224i \(0.812633\pi\)
\(14\) 0 0
\(15\) −18.2674 2.55896i −1.21783 0.170597i
\(16\) 0 0
\(17\) −16.4391 −0.967008 −0.483504 0.875342i \(-0.660636\pi\)
−0.483504 + 0.875342i \(0.660636\pi\)
\(18\) 0 0
\(19\) 31.0538 1.63441 0.817206 0.576346i \(-0.195522\pi\)
0.817206 + 0.576346i \(0.195522\pi\)
\(20\) 0 0
\(21\) 19.7275 16.6647i 0.939403 0.793555i
\(22\) 0 0
\(23\) 20.2273i 0.879446i −0.898133 0.439723i \(-0.855077\pi\)
0.898133 0.439723i \(-0.144923\pi\)
\(24\) 0 0
\(25\) −24.0377 6.86937i −0.961509 0.274775i
\(26\) 0 0
\(27\) 16.1959i 0.599848i
\(28\) 0 0
\(29\) 56.8127i 1.95906i −0.201302 0.979529i \(-0.564517\pi\)
0.201302 0.979529i \(-0.435483\pi\)
\(30\) 0 0
\(31\) 25.7499i 0.830641i −0.909675 0.415320i \(-0.863669\pi\)
0.909675 0.415320i \(-0.136331\pi\)
\(32\) 0 0
\(33\) −37.6775 −1.14174
\(34\) 0 0
\(35\) 29.6119 18.6584i 0.846054 0.533097i
\(36\) 0 0
\(37\) −66.0553 −1.78528 −0.892640 0.450771i \(-0.851149\pi\)
−0.892640 + 0.450771i \(0.851149\pi\)
\(38\) 0 0
\(39\) 53.2560 1.36554
\(40\) 0 0
\(41\) 0.504264i 0.0122991i −0.999981 0.00614956i \(-0.998043\pi\)
0.999981 0.00614956i \(-0.00195748\pi\)
\(42\) 0 0
\(43\) −26.2293 −0.609983 −0.304992 0.952355i \(-0.598654\pi\)
−0.304992 + 0.952355i \(0.598654\pi\)
\(44\) 0 0
\(45\) −3.19760 + 22.8264i −0.0710578 + 0.507254i
\(46\) 0 0
\(47\) −20.2286 −0.430395 −0.215197 0.976571i \(-0.569039\pi\)
−0.215197 + 0.976571i \(0.569039\pi\)
\(48\) 0 0
\(49\) −8.18977 + 48.3107i −0.167138 + 0.985933i
\(50\) 0 0
\(51\) 60.6465i 1.18915i
\(52\) 0 0
\(53\) 53.0620 1.00117 0.500585 0.865687i \(-0.333118\pi\)
0.500585 + 0.865687i \(0.333118\pi\)
\(54\) 0 0
\(55\) −50.5714 7.08422i −0.919481 0.128804i
\(56\) 0 0
\(57\) 114.562i 2.00987i
\(58\) 0 0
\(59\) 59.3867 1.00655 0.503277 0.864125i \(-0.332127\pi\)
0.503277 + 0.864125i \(0.332127\pi\)
\(60\) 0 0
\(61\) −69.5062 −1.13945 −0.569723 0.821837i \(-0.692950\pi\)
−0.569723 + 0.821837i \(0.692950\pi\)
\(62\) 0 0
\(63\) −20.8237 24.6509i −0.330535 0.391283i
\(64\) 0 0
\(65\) 71.4812 + 10.0133i 1.09971 + 0.154051i
\(66\) 0 0
\(67\) 61.9289 0.924312 0.462156 0.886799i \(-0.347076\pi\)
0.462156 + 0.886799i \(0.347076\pi\)
\(68\) 0 0
\(69\) −74.6215 −1.08147
\(70\) 0 0
\(71\) 25.5504 0.359864 0.179932 0.983679i \(-0.442412\pi\)
0.179932 + 0.983679i \(0.442412\pi\)
\(72\) 0 0
\(73\) 34.7213 0.475634 0.237817 0.971310i \(-0.423568\pi\)
0.237817 + 0.971310i \(0.423568\pi\)
\(74\) 0 0
\(75\) −25.3422 + 88.6788i −0.337895 + 1.18238i
\(76\) 0 0
\(77\) 54.6134 46.1344i 0.709265 0.599148i
\(78\) 0 0
\(79\) −64.4865 −0.816284 −0.408142 0.912918i \(-0.633823\pi\)
−0.408142 + 0.912918i \(0.633823\pi\)
\(80\) 0 0
\(81\) −101.238 −1.24985
\(82\) 0 0
\(83\) 9.27026i 0.111690i 0.998439 + 0.0558449i \(0.0177852\pi\)
−0.998439 + 0.0558449i \(0.982215\pi\)
\(84\) 0 0
\(85\) −11.4029 + 81.4009i −0.134152 + 0.957658i
\(86\) 0 0
\(87\) −209.591 −2.40909
\(88\) 0 0
\(89\) 22.0650i 0.247921i −0.992287 0.123961i \(-0.960440\pi\)
0.992287 0.123961i \(-0.0395597\pi\)
\(90\) 0 0
\(91\) −77.1944 + 65.2095i −0.848290 + 0.716588i
\(92\) 0 0
\(93\) −94.9952 −1.02145
\(94\) 0 0
\(95\) 21.5403 153.768i 0.226740 1.61861i
\(96\) 0 0
\(97\) −74.3644 −0.766643 −0.383321 0.923615i \(-0.625220\pi\)
−0.383321 + 0.923615i \(0.625220\pi\)
\(98\) 0 0
\(99\) 47.0807i 0.475563i
\(100\) 0 0
\(101\) 104.960 1.03920 0.519602 0.854408i \(-0.326080\pi\)
0.519602 + 0.854408i \(0.326080\pi\)
\(102\) 0 0
\(103\) 59.3436 0.576151 0.288076 0.957608i \(-0.406985\pi\)
0.288076 + 0.957608i \(0.406985\pi\)
\(104\) 0 0
\(105\) −68.8337 109.243i −0.655560 1.04041i
\(106\) 0 0
\(107\) −168.968 −1.57914 −0.789571 0.613660i \(-0.789697\pi\)
−0.789571 + 0.613660i \(0.789697\pi\)
\(108\) 0 0
\(109\) 40.5581i 0.372092i 0.982541 + 0.186046i \(0.0595674\pi\)
−0.982541 + 0.186046i \(0.940433\pi\)
\(110\) 0 0
\(111\) 243.688i 2.19539i
\(112\) 0 0
\(113\) 145.160i 1.28460i 0.766453 + 0.642300i \(0.222020\pi\)
−0.766453 + 0.642300i \(0.777980\pi\)
\(114\) 0 0
\(115\) −100.158 14.0305i −0.870942 0.122005i
\(116\) 0 0
\(117\) 66.5472i 0.568779i
\(118\) 0 0
\(119\) −74.2589 87.9070i −0.624024 0.738714i
\(120\) 0 0
\(121\) 16.6937 0.137965
\(122\) 0 0
\(123\) −1.86031 −0.0151245
\(124\) 0 0
\(125\) −50.6883 + 114.262i −0.405507 + 0.914092i
\(126\) 0 0
\(127\) 131.272i 1.03364i −0.856094 0.516819i \(-0.827116\pi\)
0.856094 0.516819i \(-0.172884\pi\)
\(128\) 0 0
\(129\) 96.7639i 0.750108i
\(130\) 0 0
\(131\) −47.1382 −0.359834 −0.179917 0.983682i \(-0.557583\pi\)
−0.179917 + 0.983682i \(0.557583\pi\)
\(132\) 0 0
\(133\) 140.276 + 166.058i 1.05471 + 1.24856i
\(134\) 0 0
\(135\) −80.1965 11.2342i −0.594048 0.0832162i
\(136\) 0 0
\(137\) 48.2151i 0.351935i −0.984396 0.175967i \(-0.943695\pi\)
0.984396 0.175967i \(-0.0563053\pi\)
\(138\) 0 0
\(139\) 89.2672 0.642210 0.321105 0.947044i \(-0.395946\pi\)
0.321105 + 0.947044i \(0.395946\pi\)
\(140\) 0 0
\(141\) 74.6263i 0.529264i
\(142\) 0 0
\(143\) 147.434 1.03100
\(144\) 0 0
\(145\) −281.317 39.4078i −1.94012 0.271778i
\(146\) 0 0
\(147\) 178.226 + 30.2133i 1.21242 + 0.205533i
\(148\) 0 0
\(149\) 49.6284i 0.333077i −0.986035 0.166538i \(-0.946741\pi\)
0.986035 0.166538i \(-0.0532589\pi\)
\(150\) 0 0
\(151\) 65.9244 0.436585 0.218293 0.975883i \(-0.429951\pi\)
0.218293 + 0.975883i \(0.429951\pi\)
\(152\) 0 0
\(153\) 75.7822 0.495308
\(154\) 0 0
\(155\) −127.504 17.8612i −0.822609 0.115234i
\(156\) 0 0
\(157\) 3.29611i 0.0209944i 0.999945 + 0.0104972i \(0.00334142\pi\)
−0.999945 + 0.0104972i \(0.996659\pi\)
\(158\) 0 0
\(159\) 195.754i 1.23116i
\(160\) 0 0
\(161\) 108.164 91.3706i 0.671824 0.567520i
\(162\) 0 0
\(163\) −80.8482 −0.496001 −0.248001 0.968760i \(-0.579774\pi\)
−0.248001 + 0.968760i \(0.579774\pi\)
\(164\) 0 0
\(165\) −26.1348 + 186.566i −0.158393 + 1.13070i
\(166\) 0 0
\(167\) −185.849 −1.11287 −0.556434 0.830892i \(-0.687831\pi\)
−0.556434 + 0.830892i \(0.687831\pi\)
\(168\) 0 0
\(169\) −39.3931 −0.233095
\(170\) 0 0
\(171\) −143.154 −0.837157
\(172\) 0 0
\(173\) 203.853i 1.17834i −0.808009 0.589170i \(-0.799455\pi\)
0.808009 0.589170i \(-0.200545\pi\)
\(174\) 0 0
\(175\) −71.8498 159.570i −0.410570 0.911829i
\(176\) 0 0
\(177\) 219.087i 1.23778i
\(178\) 0 0
\(179\) 252.488i 1.41055i −0.708934 0.705275i \(-0.750824\pi\)
0.708934 0.705275i \(-0.249176\pi\)
\(180\) 0 0
\(181\) 105.014 0.580186 0.290093 0.956998i \(-0.406314\pi\)
0.290093 + 0.956998i \(0.406314\pi\)
\(182\) 0 0
\(183\) 256.419i 1.40120i
\(184\) 0 0
\(185\) −45.8189 + 327.083i −0.247670 + 1.76802i
\(186\) 0 0
\(187\) 167.894i 0.897827i
\(188\) 0 0
\(189\) 86.6063 73.1602i 0.458234 0.387091i
\(190\) 0 0
\(191\) 91.0901 0.476912 0.238456 0.971153i \(-0.423359\pi\)
0.238456 + 0.971153i \(0.423359\pi\)
\(192\) 0 0
\(193\) 126.767i 0.656822i −0.944535 0.328411i \(-0.893487\pi\)
0.944535 0.328411i \(-0.106513\pi\)
\(194\) 0 0
\(195\) 36.9407 263.705i 0.189439 1.35233i
\(196\) 0 0
\(197\) −315.300 −1.60051 −0.800253 0.599663i \(-0.795301\pi\)
−0.800253 + 0.599663i \(0.795301\pi\)
\(198\) 0 0
\(199\) 304.652i 1.53091i 0.643488 + 0.765456i \(0.277487\pi\)
−0.643488 + 0.765456i \(0.722513\pi\)
\(200\) 0 0
\(201\) 228.465i 1.13664i
\(202\) 0 0
\(203\) 303.801 256.634i 1.49656 1.26421i
\(204\) 0 0
\(205\) −2.49694 0.349780i −0.0121802 0.00170624i
\(206\) 0 0
\(207\) 93.2449i 0.450458i
\(208\) 0 0
\(209\) 317.154i 1.51748i
\(210\) 0 0
\(211\) 119.367i 0.565722i −0.959161 0.282861i \(-0.908716\pi\)
0.959161 0.282861i \(-0.0912836\pi\)
\(212\) 0 0
\(213\) 94.2593i 0.442532i
\(214\) 0 0
\(215\) −18.1938 + 129.878i −0.0846223 + 0.604085i
\(216\) 0 0
\(217\) 137.695 116.317i 0.634541 0.536025i
\(218\) 0 0
\(219\) 128.092i 0.584896i
\(220\) 0 0
\(221\) 237.313i 1.07381i
\(222\) 0 0
\(223\) 225.950 1.01323 0.506615 0.862172i \(-0.330896\pi\)
0.506615 + 0.862172i \(0.330896\pi\)
\(224\) 0 0
\(225\) 110.811 + 31.6668i 0.492491 + 0.140741i
\(226\) 0 0
\(227\) 233.683i 1.02944i 0.857358 + 0.514721i \(0.172104\pi\)
−0.857358 + 0.514721i \(0.827896\pi\)
\(228\) 0 0
\(229\) 410.025 1.79050 0.895251 0.445562i \(-0.146996\pi\)
0.895251 + 0.445562i \(0.146996\pi\)
\(230\) 0 0
\(231\) −170.197 201.477i −0.736783 0.872196i
\(232\) 0 0
\(233\) 195.365i 0.838476i 0.907876 + 0.419238i \(0.137703\pi\)
−0.907876 + 0.419238i \(0.862297\pi\)
\(234\) 0 0
\(235\) −14.0314 + 100.165i −0.0597081 + 0.426233i
\(236\) 0 0
\(237\) 237.901i 1.00380i
\(238\) 0 0
\(239\) 1.13036 0.00472952 0.00236476 0.999997i \(-0.499247\pi\)
0.00236476 + 0.999997i \(0.499247\pi\)
\(240\) 0 0
\(241\) 184.725i 0.766495i 0.923646 + 0.383247i \(0.125194\pi\)
−0.923646 + 0.383247i \(0.874806\pi\)
\(242\) 0 0
\(243\) 227.719i 0.937116i
\(244\) 0 0
\(245\) 233.537 + 74.0634i 0.953213 + 0.302299i
\(246\) 0 0
\(247\) 448.288i 1.81493i
\(248\) 0 0
\(249\) 34.1994 0.137347
\(250\) 0 0
\(251\) 161.026 0.641537 0.320769 0.947158i \(-0.396059\pi\)
0.320769 + 0.947158i \(0.396059\pi\)
\(252\) 0 0
\(253\) −206.582 −0.816529
\(254\) 0 0
\(255\) 300.300 + 42.0671i 1.17765 + 0.164969i
\(256\) 0 0
\(257\) 280.740 1.09237 0.546186 0.837664i \(-0.316079\pi\)
0.546186 + 0.837664i \(0.316079\pi\)
\(258\) 0 0
\(259\) −298.385 353.225i −1.15207 1.36380i
\(260\) 0 0
\(261\) 261.899i 1.00344i
\(262\) 0 0
\(263\) 114.489i 0.435318i 0.976025 + 0.217659i \(0.0698421\pi\)
−0.976025 + 0.217659i \(0.930158\pi\)
\(264\) 0 0
\(265\) 36.8061 262.745i 0.138891 0.991489i
\(266\) 0 0
\(267\) −81.4012 −0.304874
\(268\) 0 0
\(269\) 199.064 0.740014 0.370007 0.929029i \(-0.379355\pi\)
0.370007 + 0.929029i \(0.379355\pi\)
\(270\) 0 0
\(271\) 138.300i 0.510333i −0.966897 0.255167i \(-0.917870\pi\)
0.966897 0.255167i \(-0.0821303\pi\)
\(272\) 0 0
\(273\) 240.568 + 284.782i 0.881202 + 1.04316i
\(274\) 0 0
\(275\) −70.1572 + 245.498i −0.255117 + 0.892721i
\(276\) 0 0
\(277\) 347.378 1.25407 0.627035 0.778991i \(-0.284268\pi\)
0.627035 + 0.778991i \(0.284268\pi\)
\(278\) 0 0
\(279\) 118.703i 0.425460i
\(280\) 0 0
\(281\) −261.441 −0.930396 −0.465198 0.885207i \(-0.654017\pi\)
−0.465198 + 0.885207i \(0.654017\pi\)
\(282\) 0 0
\(283\) 88.8373i 0.313913i −0.987606 0.156956i \(-0.949832\pi\)
0.987606 0.156956i \(-0.0501682\pi\)
\(284\) 0 0
\(285\) −567.273 79.4655i −1.99043 0.278826i
\(286\) 0 0
\(287\) 2.69651 2.27786i 0.00939551 0.00793681i
\(288\) 0 0
\(289\) −18.7547 −0.0648952
\(290\) 0 0
\(291\) 274.342i 0.942755i
\(292\) 0 0
\(293\) 159.380i 0.543960i −0.962303 0.271980i \(-0.912322\pi\)
0.962303 0.271980i \(-0.0876784\pi\)
\(294\) 0 0
\(295\) 41.1932 294.062i 0.139638 0.996822i
\(296\) 0 0
\(297\) −165.409 −0.556934
\(298\) 0 0
\(299\) 291.997 0.976579
\(300\) 0 0
\(301\) −118.483 140.259i −0.393631 0.465977i
\(302\) 0 0
\(303\) 387.212i 1.27793i
\(304\) 0 0
\(305\) −48.2126 + 344.171i −0.158074 + 1.12843i
\(306\) 0 0
\(307\) 190.921i 0.621892i −0.950428 0.310946i \(-0.899354\pi\)
0.950428 0.310946i \(-0.100646\pi\)
\(308\) 0 0
\(309\) 218.928i 0.708503i
\(310\) 0 0
\(311\) 97.1437i 0.312359i −0.987729 0.156180i \(-0.950082\pi\)
0.987729 0.156180i \(-0.0499179\pi\)
\(312\) 0 0
\(313\) −131.747 −0.420917 −0.210459 0.977603i \(-0.567496\pi\)
−0.210459 + 0.977603i \(0.567496\pi\)
\(314\) 0 0
\(315\) −136.507 + 86.0127i −0.433355 + 0.273056i
\(316\) 0 0
\(317\) 64.5641 0.203672 0.101836 0.994801i \(-0.467528\pi\)
0.101836 + 0.994801i \(0.467528\pi\)
\(318\) 0 0
\(319\) −580.230 −1.81890
\(320\) 0 0
\(321\) 623.349i 1.94190i
\(322\) 0 0
\(323\) −510.498 −1.58049
\(324\) 0 0
\(325\) 99.1650 347.004i 0.305123 1.06771i
\(326\) 0 0
\(327\) 149.625 0.457569
\(328\) 0 0
\(329\) −91.3765 108.171i −0.277740 0.328786i
\(330\) 0 0
\(331\) 52.7603i 0.159397i −0.996819 0.0796983i \(-0.974604\pi\)
0.996819 0.0796983i \(-0.0253957\pi\)
\(332\) 0 0
\(333\) 304.506 0.914432
\(334\) 0 0
\(335\) 42.9566 306.650i 0.128229 0.915374i
\(336\) 0 0
\(337\) 251.074i 0.745026i 0.928027 + 0.372513i \(0.121504\pi\)
−0.928027 + 0.372513i \(0.878496\pi\)
\(338\) 0 0
\(339\) 535.517 1.57970
\(340\) 0 0
\(341\) −262.984 −0.771215
\(342\) 0 0
\(343\) −295.333 + 174.435i −0.861028 + 0.508558i
\(344\) 0 0
\(345\) −51.7608 + 369.500i −0.150031 + 1.07101i
\(346\) 0 0
\(347\) 555.781 1.60167 0.800837 0.598882i \(-0.204388\pi\)
0.800837 + 0.598882i \(0.204388\pi\)
\(348\) 0 0
\(349\) 202.673 0.580726 0.290363 0.956917i \(-0.406224\pi\)
0.290363 + 0.956917i \(0.406224\pi\)
\(350\) 0 0
\(351\) 233.801 0.666100
\(352\) 0 0
\(353\) 332.885 0.943018 0.471509 0.881861i \(-0.343710\pi\)
0.471509 + 0.881861i \(0.343710\pi\)
\(354\) 0 0
\(355\) 17.7229 126.517i 0.0499236 0.356385i
\(356\) 0 0
\(357\) −324.302 + 273.953i −0.908410 + 0.767374i
\(358\) 0 0
\(359\) 333.845 0.929931 0.464965 0.885329i \(-0.346067\pi\)
0.464965 + 0.885329i \(0.346067\pi\)
\(360\) 0 0
\(361\) 603.340 1.67130
\(362\) 0 0
\(363\) 61.5858i 0.169658i
\(364\) 0 0
\(365\) 24.0842 171.928i 0.0659842 0.471035i
\(366\) 0 0
\(367\) 330.735 0.901184 0.450592 0.892730i \(-0.351213\pi\)
0.450592 + 0.892730i \(0.351213\pi\)
\(368\) 0 0
\(369\) 2.32459i 0.00629970i
\(370\) 0 0
\(371\) 239.692 + 283.745i 0.646070 + 0.764811i
\(372\) 0 0
\(373\) −114.255 −0.306315 −0.153157 0.988202i \(-0.548944\pi\)
−0.153157 + 0.988202i \(0.548944\pi\)
\(374\) 0 0
\(375\) 421.528 + 186.997i 1.12408 + 0.498659i
\(376\) 0 0
\(377\) 820.138 2.17543
\(378\) 0 0
\(379\) 355.076i 0.936877i −0.883496 0.468439i \(-0.844817\pi\)
0.883496 0.468439i \(-0.155183\pi\)
\(380\) 0 0
\(381\) −484.283 −1.27108
\(382\) 0 0
\(383\) 436.582 1.13990 0.569951 0.821679i \(-0.306962\pi\)
0.569951 + 0.821679i \(0.306962\pi\)
\(384\) 0 0
\(385\) −190.559 302.427i −0.494959 0.785526i
\(386\) 0 0
\(387\) 120.913 0.312438
\(388\) 0 0
\(389\) 639.735i 1.64456i −0.569080 0.822282i \(-0.692701\pi\)
0.569080 0.822282i \(-0.307299\pi\)
\(390\) 0 0
\(391\) 332.519i 0.850432i
\(392\) 0 0
\(393\) 173.900i 0.442494i
\(394\) 0 0
\(395\) −44.7307 + 319.315i −0.113242 + 0.808391i
\(396\) 0 0
\(397\) 516.755i 1.30165i −0.759228 0.650825i \(-0.774423\pi\)
0.759228 0.650825i \(-0.225577\pi\)
\(398\) 0 0
\(399\) 612.613 517.501i 1.53537 1.29700i
\(400\) 0 0
\(401\) −261.169 −0.651295 −0.325647 0.945491i \(-0.605582\pi\)
−0.325647 + 0.945491i \(0.605582\pi\)
\(402\) 0 0
\(403\) 371.721 0.922383
\(404\) 0 0
\(405\) −70.2231 + 501.295i −0.173390 + 1.23777i
\(406\) 0 0
\(407\) 674.626i 1.65756i
\(408\) 0 0
\(409\) 94.8415i 0.231886i 0.993256 + 0.115943i \(0.0369890\pi\)
−0.993256 + 0.115943i \(0.963011\pi\)
\(410\) 0 0
\(411\) −177.873 −0.432781
\(412\) 0 0
\(413\) 268.262 + 317.566i 0.649544 + 0.768924i
\(414\) 0 0
\(415\) 45.9031 + 6.43026i 0.110610 + 0.0154946i
\(416\) 0 0
\(417\) 329.320i 0.789737i
\(418\) 0 0
\(419\) 46.0965 0.110016 0.0550078 0.998486i \(-0.482482\pi\)
0.0550078 + 0.998486i \(0.482482\pi\)
\(420\) 0 0
\(421\) 710.750i 1.68824i 0.536153 + 0.844121i \(0.319877\pi\)
−0.536153 + 0.844121i \(0.680123\pi\)
\(422\) 0 0
\(423\) 93.2508 0.220451
\(424\) 0 0
\(425\) 395.159 + 112.926i 0.929787 + 0.265709i
\(426\) 0 0
\(427\) −313.974 371.679i −0.735301 0.870442i
\(428\) 0 0
\(429\) 543.906i 1.26785i
\(430\) 0 0
\(431\) −411.504 −0.954766 −0.477383 0.878695i \(-0.658415\pi\)
−0.477383 + 0.878695i \(0.658415\pi\)
\(432\) 0 0
\(433\) −329.582 −0.761159 −0.380580 0.924748i \(-0.624275\pi\)
−0.380580 + 0.924748i \(0.624275\pi\)
\(434\) 0 0
\(435\) −145.381 + 1037.82i −0.334210 + 2.38579i
\(436\) 0 0
\(437\) 628.134i 1.43738i
\(438\) 0 0
\(439\) 370.448i 0.843844i −0.906632 0.421922i \(-0.861356\pi\)
0.906632 0.421922i \(-0.138644\pi\)
\(440\) 0 0
\(441\) 37.7537 222.706i 0.0856093 0.505002i
\(442\) 0 0
\(443\) 203.437 0.459226 0.229613 0.973282i \(-0.426254\pi\)
0.229613 + 0.973282i \(0.426254\pi\)
\(444\) 0 0
\(445\) −109.258 15.3053i −0.245524 0.0343938i
\(446\) 0 0
\(447\) −183.087 −0.409590
\(448\) 0 0
\(449\) 274.557 0.611485 0.305742 0.952114i \(-0.401095\pi\)
0.305742 + 0.952114i \(0.401095\pi\)
\(450\) 0 0
\(451\) −5.15007 −0.0114192
\(452\) 0 0
\(453\) 243.205i 0.536877i
\(454\) 0 0
\(455\) 269.350 + 427.472i 0.591977 + 0.939499i
\(456\) 0 0
\(457\) 186.534i 0.408170i 0.978953 + 0.204085i \(0.0654219\pi\)
−0.978953 + 0.204085i \(0.934578\pi\)
\(458\) 0 0
\(459\) 266.247i 0.580058i
\(460\) 0 0
\(461\) −501.181 −1.08716 −0.543580 0.839357i \(-0.682932\pi\)
−0.543580 + 0.839357i \(0.682932\pi\)
\(462\) 0 0
\(463\) 160.592i 0.346850i 0.984847 + 0.173425i \(0.0554835\pi\)
−0.984847 + 0.173425i \(0.944517\pi\)
\(464\) 0 0
\(465\) −65.8929 + 470.383i −0.141705 + 1.01158i
\(466\) 0 0
\(467\) 340.763i 0.729686i 0.931069 + 0.364843i \(0.118877\pi\)
−0.931069 + 0.364843i \(0.881123\pi\)
\(468\) 0 0
\(469\) 279.745 + 331.160i 0.596472 + 0.706097i
\(470\) 0 0
\(471\) 12.1599 0.0258171
\(472\) 0 0
\(473\) 267.881i 0.566344i
\(474\) 0 0
\(475\) −746.463 213.320i −1.57150 0.449095i
\(476\) 0 0
\(477\) −244.609 −0.512806
\(478\) 0 0
\(479\) 784.653i 1.63811i 0.573717 + 0.819053i \(0.305501\pi\)
−0.573717 + 0.819053i \(0.694499\pi\)
\(480\) 0 0
\(481\) 953.563i 1.98246i
\(482\) 0 0
\(483\) −337.080 399.032i −0.697889 0.826154i
\(484\) 0 0
\(485\) −51.5824 + 368.226i −0.106355 + 0.759230i
\(486\) 0 0
\(487\) 195.648i 0.401741i −0.979618 0.200871i \(-0.935623\pi\)
0.979618 0.200871i \(-0.0643771\pi\)
\(488\) 0 0
\(489\) 298.262i 0.609942i
\(490\) 0 0
\(491\) 153.533i 0.312694i 0.987702 + 0.156347i \(0.0499718\pi\)
−0.987702 + 0.156347i \(0.950028\pi\)
\(492\) 0 0
\(493\) 933.952i 1.89443i
\(494\) 0 0
\(495\) 233.127 + 32.6573i 0.470964 + 0.0659743i
\(496\) 0 0
\(497\) 115.416 + 136.629i 0.232226 + 0.274907i
\(498\) 0 0
\(499\) 636.233i 1.27502i −0.770443 0.637509i \(-0.779965\pi\)
0.770443 0.637509i \(-0.220035\pi\)
\(500\) 0 0
\(501\) 685.625i 1.36851i
\(502\) 0 0
\(503\) 48.3325 0.0960885 0.0480442 0.998845i \(-0.484701\pi\)
0.0480442 + 0.998845i \(0.484701\pi\)
\(504\) 0 0
\(505\) 72.8046 519.724i 0.144168 1.02916i
\(506\) 0 0
\(507\) 145.327i 0.286641i
\(508\) 0 0
\(509\) −243.531 −0.478449 −0.239225 0.970964i \(-0.576893\pi\)
−0.239225 + 0.970964i \(0.576893\pi\)
\(510\) 0 0
\(511\) 156.843 + 185.669i 0.306934 + 0.363345i
\(512\) 0 0
\(513\) 502.945i 0.980399i
\(514\) 0 0
\(515\) 41.1633 293.849i 0.0799287 0.570580i
\(516\) 0 0
\(517\) 206.595i 0.399604i
\(518\) 0 0
\(519\) −752.044 −1.44903
\(520\) 0 0
\(521\) 630.569i 1.21030i 0.796110 + 0.605152i \(0.206888\pi\)
−0.796110 + 0.605152i \(0.793112\pi\)
\(522\) 0 0
\(523\) 829.843i 1.58670i 0.608767 + 0.793349i \(0.291664\pi\)
−0.608767 + 0.793349i \(0.708336\pi\)
\(524\) 0 0
\(525\) −588.679 + 265.065i −1.12129 + 0.504886i
\(526\) 0 0
\(527\) 423.306i 0.803236i
\(528\) 0 0
\(529\) 119.858 0.226574
\(530\) 0 0
\(531\) −273.765 −0.515564
\(532\) 0 0
\(533\) 7.27947 0.0136575
\(534\) 0 0
\(535\) −117.204 + 836.671i −0.219072 + 1.56387i
\(536\) 0 0
\(537\) −931.469 −1.73458
\(538\) 0 0
\(539\) 493.400 + 83.6425i 0.915398 + 0.155181i
\(540\) 0 0
\(541\) 120.015i 0.221840i −0.993829 0.110920i \(-0.964620\pi\)
0.993829 0.110920i \(-0.0353797\pi\)
\(542\) 0 0
\(543\) 387.412i 0.713465i
\(544\) 0 0
\(545\) 200.829 + 28.1329i 0.368494 + 0.0516199i
\(546\) 0 0
\(547\) 634.970 1.16082 0.580411 0.814324i \(-0.302892\pi\)
0.580411 + 0.814324i \(0.302892\pi\)
\(548\) 0 0
\(549\) 320.414 0.583632
\(550\) 0 0
\(551\) 1764.25i 3.20191i
\(552\) 0 0
\(553\) −291.298 344.836i −0.526760 0.623573i
\(554\) 0 0
\(555\) 1206.66 + 169.033i 2.17416 + 0.304564i
\(556\) 0 0
\(557\) −113.560 −0.203877 −0.101939 0.994791i \(-0.532505\pi\)
−0.101939 + 0.994791i \(0.532505\pi\)
\(558\) 0 0
\(559\) 378.641i 0.677355i
\(560\) 0 0
\(561\) 619.386 1.10407
\(562\) 0 0
\(563\) 964.206i 1.71262i −0.516461 0.856311i \(-0.672751\pi\)
0.516461 0.856311i \(-0.327249\pi\)
\(564\) 0 0
\(565\) 718.781 + 100.689i 1.27218 + 0.178211i
\(566\) 0 0
\(567\) −457.312 541.362i −0.806547 0.954782i
\(568\) 0 0
\(569\) −761.751 −1.33875 −0.669377 0.742923i \(-0.733439\pi\)
−0.669377 + 0.742923i \(0.733439\pi\)
\(570\) 0 0
\(571\) 35.8939i 0.0628614i −0.999506 0.0314307i \(-0.989994\pi\)
0.999506 0.0314307i \(-0.0100064\pi\)
\(572\) 0 0
\(573\) 336.046i 0.586467i
\(574\) 0 0
\(575\) −138.949 + 486.217i −0.241650 + 0.845595i
\(576\) 0 0
\(577\) 714.079 1.23757 0.618786 0.785559i \(-0.287625\pi\)
0.618786 + 0.785559i \(0.287625\pi\)
\(578\) 0 0
\(579\) −467.661 −0.807705
\(580\) 0 0
\(581\) −49.5720 + 41.8756i −0.0853218 + 0.0720751i
\(582\) 0 0
\(583\) 541.925i 0.929545i
\(584\) 0 0
\(585\) −329.518 46.1600i −0.563279 0.0789060i
\(586\) 0 0
\(587\) 785.786i 1.33865i −0.742971 0.669323i \(-0.766584\pi\)
0.742971 0.669323i \(-0.233416\pi\)
\(588\) 0 0
\(589\) 799.632i 1.35761i
\(590\) 0 0
\(591\) 1163.19i 1.96817i
\(592\) 0 0
\(593\) −618.608 −1.04318 −0.521592 0.853195i \(-0.674662\pi\)
−0.521592 + 0.853195i \(0.674662\pi\)
\(594\) 0 0
\(595\) −486.794 + 306.728i −0.818141 + 0.515509i
\(596\) 0 0
\(597\) 1123.91 1.88259
\(598\) 0 0
\(599\) 218.429 0.364656 0.182328 0.983238i \(-0.441637\pi\)
0.182328 + 0.983238i \(0.441637\pi\)
\(600\) 0 0
\(601\) 810.310i 1.34827i 0.738609 + 0.674135i \(0.235483\pi\)
−0.738609 + 0.674135i \(0.764517\pi\)
\(602\) 0 0
\(603\) −285.484 −0.473439
\(604\) 0 0
\(605\) 11.5795 82.6616i 0.0191397 0.136631i
\(606\) 0 0
\(607\) −223.586 −0.368346 −0.184173 0.982894i \(-0.558961\pi\)
−0.184173 + 0.982894i \(0.558961\pi\)
\(608\) 0 0
\(609\) −946.764 1120.77i −1.55462 1.84034i
\(610\) 0 0
\(611\) 292.016i 0.477931i
\(612\) 0 0
\(613\) −416.624 −0.679648 −0.339824 0.940489i \(-0.610368\pi\)
−0.339824 + 0.940489i \(0.610368\pi\)
\(614\) 0 0
\(615\) −1.29039 + 9.21160i −0.00209820 + 0.0149782i
\(616\) 0 0
\(617\) 633.821i 1.02726i 0.858011 + 0.513631i \(0.171700\pi\)
−0.858011 + 0.513631i \(0.828300\pi\)
\(618\) 0 0
\(619\) 891.451 1.44015 0.720073 0.693898i \(-0.244108\pi\)
0.720073 + 0.693898i \(0.244108\pi\)
\(620\) 0 0
\(621\) −327.599 −0.527534
\(622\) 0 0
\(623\) 117.991 99.6721i 0.189391 0.159987i
\(624\) 0 0
\(625\) 530.624 + 330.248i 0.848998 + 0.528397i
\(626\) 0 0
\(627\) −1170.03 −1.86608
\(628\) 0 0
\(629\) 1085.89 1.72638
\(630\) 0 0
\(631\) 139.865 0.221656 0.110828 0.993840i \(-0.464650\pi\)
0.110828 + 0.993840i \(0.464650\pi\)
\(632\) 0 0
\(633\) −440.365 −0.695679
\(634\) 0 0
\(635\) −650.014 91.0561i −1.02364 0.143395i
\(636\) 0 0
\(637\) −697.405 118.226i −1.09483 0.185598i
\(638\) 0 0
\(639\) −117.784 −0.184325
\(640\) 0 0
\(641\) −500.317 −0.780525 −0.390263 0.920704i \(-0.627616\pi\)
−0.390263 + 0.920704i \(0.627616\pi\)
\(642\) 0 0
\(643\) 1189.12i 1.84933i 0.380788 + 0.924663i \(0.375653\pi\)
−0.380788 + 0.924663i \(0.624347\pi\)
\(644\) 0 0
\(645\) 479.141 + 67.1197i 0.742854 + 0.104062i
\(646\) 0 0
\(647\) 1020.39 1.57711 0.788555 0.614964i \(-0.210829\pi\)
0.788555 + 0.614964i \(0.210829\pi\)
\(648\) 0 0
\(649\) 606.519i 0.934544i
\(650\) 0 0
\(651\) −429.113 507.979i −0.659159 0.780306i
\(652\) 0 0
\(653\) 155.938 0.238802 0.119401 0.992846i \(-0.461903\pi\)
0.119401 + 0.992846i \(0.461903\pi\)
\(654\) 0 0
\(655\) −32.6972 + 233.412i −0.0499193 + 0.356354i
\(656\) 0 0
\(657\) −160.060 −0.243623
\(658\) 0 0
\(659\) 220.018i 0.333866i −0.985968 0.166933i \(-0.946614\pi\)
0.985968 0.166933i \(-0.0533864\pi\)
\(660\) 0 0
\(661\) 948.923 1.43559 0.717794 0.696256i \(-0.245152\pi\)
0.717794 + 0.696256i \(0.245152\pi\)
\(662\) 0 0
\(663\) −875.483 −1.32049
\(664\) 0 0
\(665\) 919.562 579.415i 1.38280 0.871301i
\(666\) 0 0
\(667\) −1149.17 −1.72289
\(668\) 0 0
\(669\) 833.566i 1.24599i
\(670\) 0 0
\(671\) 709.870i 1.05793i
\(672\) 0 0
\(673\) 745.489i 1.10771i −0.832613 0.553855i \(-0.813156\pi\)
0.832613 0.553855i \(-0.186844\pi\)
\(674\) 0 0
\(675\) −111.256 + 389.312i −0.164823 + 0.576759i
\(676\) 0 0
\(677\) 8.12211i 0.0119972i 0.999982 + 0.00599860i \(0.00190943\pi\)
−0.999982 + 0.00599860i \(0.998091\pi\)
\(678\) 0 0
\(679\) −335.919 397.657i −0.494726 0.585652i
\(680\) 0 0
\(681\) 862.094 1.26592
\(682\) 0 0
\(683\) −476.643 −0.697867 −0.348934 0.937147i \(-0.613456\pi\)
−0.348934 + 0.937147i \(0.613456\pi\)
\(684\) 0 0
\(685\) −238.744 33.4441i −0.348532 0.0488235i
\(686\) 0 0
\(687\) 1512.65i 2.20181i
\(688\) 0 0
\(689\) 765.994i 1.11175i
\(690\) 0 0
\(691\) −85.4900 −0.123719 −0.0618596 0.998085i \(-0.519703\pi\)
−0.0618596 + 0.998085i \(0.519703\pi\)
\(692\) 0 0
\(693\) −251.760 + 212.673i −0.363290 + 0.306888i
\(694\) 0 0
\(695\) 61.9196 442.020i 0.0890930 0.636000i
\(696\) 0 0
\(697\) 8.28967i 0.0118934i
\(698\) 0 0
\(699\) 720.731 1.03109
\(700\) 0 0
\(701\) 23.5524i 0.0335982i −0.999859 0.0167991i \(-0.994652\pi\)
0.999859 0.0167991i \(-0.00534758\pi\)
\(702\) 0 0
\(703\) −2051.27 −2.91788
\(704\) 0 0
\(705\) 369.523 + 51.7641i 0.524146 + 0.0734242i
\(706\) 0 0
\(707\) 474.124 + 561.263i 0.670614 + 0.793866i
\(708\) 0 0
\(709\) 358.652i 0.505856i 0.967485 + 0.252928i \(0.0813935\pi\)
−0.967485 + 0.252928i \(0.918606\pi\)
\(710\) 0 0
\(711\) 297.274 0.418106
\(712\) 0 0
\(713\) −520.849 −0.730504
\(714\) 0 0
\(715\) 102.267 730.040i 0.143030 1.02104i
\(716\) 0 0
\(717\) 4.17005i 0.00581598i
\(718\) 0 0
\(719\) 209.881i 0.291906i −0.989292 0.145953i \(-0.953375\pi\)
0.989292 0.145953i \(-0.0466249\pi\)
\(720\) 0 0
\(721\) 268.067 + 317.335i 0.371799 + 0.440132i
\(722\) 0 0
\(723\) 681.480 0.942572
\(724\) 0 0
\(725\) −390.267 + 1365.65i −0.538300 + 1.88365i
\(726\) 0 0
\(727\) −1.50873 −0.00207528 −0.00103764 0.999999i \(-0.500330\pi\)
−0.00103764 + 0.999999i \(0.500330\pi\)
\(728\) 0 0
\(729\) −71.0498 −0.0974620
\(730\) 0 0
\(731\) 431.187 0.589859
\(732\) 0 0
\(733\) 530.040i 0.723111i −0.932351 0.361555i \(-0.882246\pi\)
0.932351 0.361555i \(-0.117754\pi\)
\(734\) 0 0
\(735\) 273.231 861.555i 0.371743 1.17218i
\(736\) 0 0
\(737\) 632.482i 0.858185i
\(738\) 0 0
\(739\) 890.676i 1.20524i −0.798026 0.602622i \(-0.794123\pi\)
0.798026 0.602622i \(-0.205877\pi\)
\(740\) 0 0
\(741\) 1653.80 2.23185
\(742\) 0 0
\(743\) 912.740i 1.22845i −0.789130 0.614226i \(-0.789468\pi\)
0.789130 0.614226i \(-0.210532\pi\)
\(744\) 0 0
\(745\) −245.743 34.4245i −0.329856 0.0462073i
\(746\) 0 0
\(747\) 42.7346i 0.0572083i
\(748\) 0 0
\(749\) −763.263 903.543i −1.01904 1.20633i
\(750\) 0 0
\(751\) −910.503 −1.21239 −0.606194 0.795317i \(-0.707304\pi\)
−0.606194 + 0.795317i \(0.707304\pi\)
\(752\) 0 0
\(753\) 594.049i 0.788910i
\(754\) 0 0
\(755\) 45.7281 326.435i 0.0605670 0.432364i
\(756\) 0 0
\(757\) 813.010 1.07399 0.536995 0.843585i \(-0.319559\pi\)
0.536995 + 0.843585i \(0.319559\pi\)
\(758\) 0 0
\(759\) 762.113i 1.00410i
\(760\) 0 0
\(761\) 1222.33i 1.60622i −0.595832 0.803109i \(-0.703178\pi\)
0.595832 0.803109i \(-0.296822\pi\)
\(762\) 0 0
\(763\) −216.881 + 183.209i −0.284248 + 0.240117i
\(764\) 0 0
\(765\) 52.5658 375.247i 0.0687135 0.490519i
\(766\) 0 0
\(767\) 857.296i 1.11773i
\(768\) 0 0
\(769\) 600.119i 0.780388i 0.920733 + 0.390194i \(0.127592\pi\)
−0.920733 + 0.390194i \(0.872408\pi\)
\(770\) 0 0
\(771\) 1035.69i 1.34331i
\(772\) 0 0
\(773\) 308.438i 0.399015i 0.979896 + 0.199507i \(0.0639341\pi\)
−0.979896 + 0.199507i \(0.936066\pi\)
\(774\) 0 0
\(775\) −176.885 + 618.968i −0.228239 + 0.798668i
\(776\) 0 0
\(777\) −1303.10 + 1100.79i −1.67710 + 1.41672i
\(778\) 0 0
\(779\) 15.6593i 0.0201018i
\(780\) 0 0
\(781\) 260.947i 0.334119i
\(782\) 0 0
\(783\) −920.133 −1.17514
\(784\) 0 0
\(785\) 16.3212 + 2.28633i 0.0207913 + 0.00291252i
\(786\) 0 0
\(787\) 670.641i 0.852149i −0.904688 0.426074i \(-0.859896\pi\)
0.904688 0.426074i \(-0.140104\pi\)
\(788\) 0 0
\(789\) 422.366 0.535319
\(790\) 0 0
\(791\) −776.231 + 655.717i −0.981328 + 0.828972i
\(792\) 0 0
\(793\) 1003.38i 1.26530i
\(794\) 0 0
\(795\) −969.306 135.784i −1.21925 0.170797i
\(796\) 0 0
\(797\) 774.242i 0.971445i 0.874113 + 0.485722i \(0.161443\pi\)
−0.874113 + 0.485722i \(0.838557\pi\)
\(798\) 0 0
\(799\) 332.540 0.416195
\(800\) 0 0
\(801\) 101.717i 0.126987i
\(802\) 0 0
\(803\) 354.610i 0.441607i
\(804\) 0 0
\(805\) −377.409 598.967i −0.468830 0.744059i
\(806\) 0 0
\(807\) 734.377i 0.910008i
\(808\) 0 0
\(809\) 1122.72 1.38778 0.693892 0.720079i \(-0.255894\pi\)
0.693892 + 0.720079i \(0.255894\pi\)
\(810\) 0 0
\(811\) −651.714 −0.803593 −0.401796 0.915729i \(-0.631614\pi\)
−0.401796 + 0.915729i \(0.631614\pi\)
\(812\) 0 0
\(813\) −510.211 −0.627566
\(814\) 0 0
\(815\) −56.0799 + 400.332i −0.0688097 + 0.491205i
\(816\) 0 0
\(817\) −814.520 −0.996964
\(818\) 0 0
\(819\) 355.855 300.607i 0.434500 0.367041i
\(820\) 0 0
\(821\) 75.0402i 0.0914009i −0.998955 0.0457005i \(-0.985448\pi\)
0.998955 0.0457005i \(-0.0145520\pi\)
\(822\) 0 0
\(823\) 863.981i 1.04979i −0.851166 0.524897i \(-0.824104\pi\)
0.851166 0.524897i \(-0.175896\pi\)
\(824\) 0 0
\(825\) 905.681 + 258.821i 1.09780 + 0.313722i
\(826\) 0 0
\(827\) 1535.63 1.85687 0.928434 0.371496i \(-0.121155\pi\)
0.928434 + 0.371496i \(0.121155\pi\)
\(828\) 0 0
\(829\) −129.821 −0.156599 −0.0782997 0.996930i \(-0.524949\pi\)
−0.0782997 + 0.996930i \(0.524949\pi\)
\(830\) 0 0
\(831\) 1281.53i 1.54215i
\(832\) 0 0
\(833\) 134.633 794.187i 0.161624 0.953406i
\(834\) 0 0
\(835\) −128.913 + 920.259i −0.154387 + 1.10211i
\(836\) 0 0
\(837\) −417.042 −0.498258
\(838\) 0 0
\(839\) 502.794i 0.599277i −0.954053 0.299639i \(-0.903134\pi\)
0.954053 0.299639i \(-0.0968661\pi\)
\(840\) 0 0
\(841\) −2386.68 −2.83791
\(842\) 0 0
\(843\) 964.497i 1.14412i
\(844\) 0 0
\(845\) −27.3248 + 195.061i −0.0323370 + 0.230841i
\(846\) 0 0
\(847\) 75.4090 + 89.2685i 0.0890307 + 0.105394i
\(848\) 0 0
\(849\) −327.735 −0.386024
\(850\) 0 0
\(851\) 1336.12i 1.57006i
\(852\) 0 0
\(853\) 837.144i 0.981411i −0.871326 0.490705i \(-0.836739\pi\)
0.871326 0.490705i \(-0.163261\pi\)
\(854\) 0 0
\(855\) −99.2978 + 708.848i −0.116138 + 0.829062i
\(856\) 0 0
\(857\) 74.6446 0.0870999 0.0435499 0.999051i \(-0.486133\pi\)
0.0435499 + 0.999051i \(0.486133\pi\)
\(858\) 0 0
\(859\) 985.823 1.14764 0.573820 0.818981i \(-0.305461\pi\)
0.573820 + 0.818981i \(0.305461\pi\)
\(860\) 0 0
\(861\) −8.40339 9.94785i −0.00976003 0.0115538i
\(862\) 0 0
\(863\) 1293.70i 1.49907i 0.661966 + 0.749534i \(0.269722\pi\)
−0.661966 + 0.749534i \(0.730278\pi\)
\(864\) 0 0
\(865\) −1009.41 141.401i −1.16695 0.163470i
\(866\) 0 0
\(867\) 69.1890i 0.0798028i
\(868\) 0 0
\(869\) 658.603i 0.757886i
\(870\) 0 0
\(871\) 893.994i 1.02640i
\(872\) 0 0
\(873\) 342.809 0.392680
\(874\) 0 0
\(875\) −839.974 + 245.091i −0.959970 + 0.280103i
\(876\) 0 0
\(877\) 92.9231 0.105956 0.0529778 0.998596i \(-0.483129\pi\)
0.0529778 + 0.998596i \(0.483129\pi\)
\(878\) 0 0
\(879\) −587.978 −0.668917
\(880\) 0 0
\(881\) 58.1276i 0.0659792i 0.999456 + 0.0329896i \(0.0105028\pi\)
−0.999456 + 0.0329896i \(0.989497\pi\)
\(882\) 0 0
\(883\) 698.151 0.790658 0.395329 0.918540i \(-0.370631\pi\)
0.395329 + 0.918540i \(0.370631\pi\)
\(884\) 0 0
\(885\) −1084.84 151.968i −1.22581 0.171716i
\(886\) 0 0
\(887\) 472.357 0.532534 0.266267 0.963899i \(-0.414210\pi\)
0.266267 + 0.963899i \(0.414210\pi\)
\(888\) 0 0
\(889\) 701.967 592.983i 0.789614 0.667022i
\(890\) 0 0
\(891\) 1033.95i 1.16043i
\(892\) 0 0
\(893\) −628.174 −0.703442
\(894\) 0 0
\(895\) −1250.23 175.137i −1.39691 0.195684i
\(896\) 0 0
\(897\) 1077.22i 1.20092i
\(898\) 0 0
\(899\) −1462.92 −1.62727
\(900\) 0 0
\(901\) −872.294 −0.968140
\(902\) 0 0
\(903\) −517.437 + 437.102i −0.573020 + 0.484055i
\(904\) 0 0
\(905\) 72.8421 519.991i 0.0804885 0.574576i
\(906\) 0 0
\(907\) −497.001 −0.547961 −0.273981 0.961735i \(-0.588340\pi\)
−0.273981 + 0.961735i \(0.588340\pi\)
\(908\) 0 0
\(909\) −483.849 −0.532288
\(910\) 0 0
\(911\) 1091.09 1.19769 0.598844 0.800865i \(-0.295627\pi\)
0.598844 + 0.800865i \(0.295627\pi\)
\(912\) 0 0
\(913\) 94.6776 0.103699
\(914\) 0 0
\(915\) 1269.70 + 177.864i 1.38765 + 0.194387i
\(916\) 0 0
\(917\) −212.933 252.068i −0.232206 0.274883i
\(918\) 0 0
\(919\) 606.413 0.659862 0.329931 0.944005i \(-0.392974\pi\)
0.329931 + 0.944005i \(0.392974\pi\)
\(920\) 0 0
\(921\) −704.336 −0.764752
\(922\) 0 0
\(923\) 368.841i 0.399611i
\(924\) 0 0
\(925\) 1587.82 + 453.758i 1.71656 + 0.490550i
\(926\) 0 0
\(927\) −273.566 −0.295108
\(928\) 0 0
\(929\) 718.552i 0.773468i 0.922191 + 0.386734i \(0.126397\pi\)
−0.922191 + 0.386734i \(0.873603\pi\)
\(930\) 0 0
\(931\) −254.324 + 1500.23i −0.273173 + 1.61142i
\(932\) 0 0
\(933\) −358.378 −0.384114
\(934\) 0 0
\(935\) 831.351 + 116.458i 0.889145 + 0.124554i
\(936\) 0 0
\(937\) −1453.79 −1.55153 −0.775767 0.631020i \(-0.782637\pi\)
−0.775767 + 0.631020i \(0.782637\pi\)
\(938\) 0 0
\(939\) 486.035i 0.517609i
\(940\) 0 0
\(941\) −1458.29 −1.54972 −0.774862 0.632130i \(-0.782181\pi\)
−0.774862 + 0.632130i \(0.782181\pi\)
\(942\) 0 0
\(943\) −10.1999 −0.0108164
\(944\) 0 0
\(945\) −302.190 479.591i −0.319777 0.507504i
\(946\) 0 0
\(947\) −474.327 −0.500873 −0.250437 0.968133i \(-0.580574\pi\)
−0.250437 + 0.968133i \(0.580574\pi\)
\(948\) 0 0
\(949\) 501.231i 0.528167i
\(950\) 0 0
\(951\) 238.187i 0.250460i
\(952\) 0 0
\(953\) 110.230i 0.115666i −0.998326 0.0578331i \(-0.981581\pi\)
0.998326 0.0578331i \(-0.0184191\pi\)
\(954\) 0 0
\(955\) 63.1841 451.047i 0.0661614 0.472300i
\(956\) 0 0
\(957\) 2140.56i 2.23674i
\(958\) 0 0
\(959\) 257.826 217.797i 0.268849 0.227109i
\(960\) 0 0
\(961\) 297.945 0.310036
\(962\) 0 0
\(963\) 778.920 0.808847
\(964\) 0 0
\(965\) −627.704 87.9309i −0.650470 0.0911201i
\(966\) 0 0
\(967\) 1909.96i 1.97514i −0.157174 0.987571i \(-0.550238\pi\)
0.157174 0.987571i \(-0.449762\pi\)
\(968\) 0 0
\(969\) 1883.31i 1.94356i
\(970\) 0 0
\(971\) −1347.96 −1.38822 −0.694111 0.719868i \(-0.744202\pi\)
−0.694111 + 0.719868i \(0.744202\pi\)
\(972\) 0 0
\(973\) 403.238 + 477.349i 0.414427 + 0.490595i
\(974\) 0 0
\(975\) −1280.15 365.835i −1.31298 0.375215i
\(976\) 0 0
\(977\) 1520.98i 1.55678i 0.627780 + 0.778391i \(0.283964\pi\)
−0.627780 + 0.778391i \(0.716036\pi\)
\(978\) 0 0
\(979\) −225.351 −0.230185
\(980\) 0 0
\(981\) 186.967i 0.190588i
\(982\) 0 0
\(983\) 1569.64 1.59678 0.798391 0.602139i \(-0.205685\pi\)
0.798391 + 0.602139i \(0.205685\pi\)
\(984\) 0 0
\(985\) −218.706 + 1561.25i −0.222036 + 1.58503i
\(986\) 0 0
\(987\) −399.058 + 337.102i −0.404314 + 0.341542i
\(988\) 0 0
\(989\) 530.547i 0.536448i
\(990\) 0 0
\(991\) 1546.21 1.56026 0.780128 0.625620i \(-0.215154\pi\)
0.780128 + 0.625620i \(0.215154\pi\)
\(992\) 0 0
\(993\) −194.641 −0.196013
\(994\) 0 0
\(995\) 1508.53 + 211.320i 1.51611 + 0.212382i
\(996\) 0 0
\(997\) 1493.93i 1.49842i 0.662331 + 0.749211i \(0.269567\pi\)
−0.662331 + 0.749211i \(0.730433\pi\)
\(998\) 0 0
\(999\) 1069.83i 1.07090i
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 1120.3.c.g.209.13 80
4.3 odd 2 280.3.c.g.69.6 yes 80
5.4 even 2 inner 1120.3.c.g.209.6 80
7.6 odd 2 inner 1120.3.c.g.209.24 80
8.3 odd 2 280.3.c.g.69.73 yes 80
8.5 even 2 inner 1120.3.c.g.209.2 80
20.19 odd 2 280.3.c.g.69.75 yes 80
28.27 even 2 280.3.c.g.69.5 80
35.34 odd 2 inner 1120.3.c.g.209.1 80
40.19 odd 2 280.3.c.g.69.8 yes 80
40.29 even 2 inner 1120.3.c.g.209.23 80
56.13 odd 2 inner 1120.3.c.g.209.5 80
56.27 even 2 280.3.c.g.69.74 yes 80
140.139 even 2 280.3.c.g.69.76 yes 80
280.69 odd 2 inner 1120.3.c.g.209.14 80
280.139 even 2 280.3.c.g.69.7 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.3.c.g.69.5 80 28.27 even 2
280.3.c.g.69.6 yes 80 4.3 odd 2
280.3.c.g.69.7 yes 80 280.139 even 2
280.3.c.g.69.8 yes 80 40.19 odd 2
280.3.c.g.69.73 yes 80 8.3 odd 2
280.3.c.g.69.74 yes 80 56.27 even 2
280.3.c.g.69.75 yes 80 20.19 odd 2
280.3.c.g.69.76 yes 80 140.139 even 2
1120.3.c.g.209.1 80 35.34 odd 2 inner
1120.3.c.g.209.2 80 8.5 even 2 inner
1120.3.c.g.209.5 80 56.13 odd 2 inner
1120.3.c.g.209.6 80 5.4 even 2 inner
1120.3.c.g.209.13 80 1.1 even 1 trivial
1120.3.c.g.209.14 80 280.69 odd 2 inner
1120.3.c.g.209.23 80 40.29 even 2 inner
1120.3.c.g.209.24 80 7.6 odd 2 inner