Properties

Label 128.4.g.a.113.7
Level $128$
Weight $4$
Character 128.113
Analytic conductor $7.552$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,4,Mod(17,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.g (of order \(8\), degree \(4\), 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: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 113.7
Character \(\chi\) \(=\) 128.113
Dual form 128.4.g.a.17.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.998206 + 2.40988i) q^{3} +(17.4005 + 7.20752i) q^{5} +(-4.37099 - 4.37099i) q^{7} +(14.2808 - 14.2808i) q^{9} +O(q^{10})\) \(q+(0.998206 + 2.40988i) q^{3} +(17.4005 + 7.20752i) q^{5} +(-4.37099 - 4.37099i) q^{7} +(14.2808 - 14.2808i) q^{9} +(-11.7977 + 28.4821i) q^{11} +(-12.9230 + 5.35288i) q^{13} +49.1278i q^{15} +72.9239i q^{17} +(143.136 - 59.2889i) q^{19} +(6.17043 - 14.8967i) q^{21} +(-83.6411 + 83.6411i) q^{23} +(162.441 + 162.441i) q^{25} +(113.737 + 47.1114i) q^{27} +(-39.6554 - 95.7367i) q^{29} +29.0324 q^{31} -80.4149 q^{33} +(-44.5534 - 107.562i) q^{35} +(-267.681 - 110.877i) q^{37} +(-25.7996 - 25.7996i) q^{39} +(-124.918 + 124.918i) q^{41} +(27.0156 - 65.2215i) q^{43} +(351.421 - 145.564i) q^{45} -282.627i q^{47} -304.789i q^{49} +(-175.738 + 72.7931i) q^{51} +(-51.4343 + 124.173i) q^{53} +(-410.570 + 410.570i) q^{55} +(285.759 + 285.759i) q^{57} +(222.476 + 92.1524i) q^{59} +(-226.809 - 547.566i) q^{61} -124.842 q^{63} -263.447 q^{65} +(-356.015 - 859.496i) q^{67} +(-285.056 - 118.074i) q^{69} +(-690.837 - 690.837i) q^{71} +(223.345 - 223.345i) q^{73} +(-229.314 + 553.612i) q^{75} +(176.062 - 72.9275i) q^{77} +698.000i q^{79} -224.174i q^{81} +(915.116 - 379.053i) q^{83} +(-525.601 + 1268.91i) q^{85} +(191.130 - 191.130i) q^{87} +(-163.738 - 163.738i) q^{89} +(79.8837 + 33.0889i) q^{91} +(28.9803 + 69.9647i) q^{93} +2917.97 q^{95} -839.460 q^{97} +(238.266 + 575.225i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\)

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
\(3\) 0.998206 + 2.40988i 0.192105 + 0.463782i 0.990357 0.138542i \(-0.0442415\pi\)
−0.798252 + 0.602324i \(0.794242\pi\)
\(4\) 0 0
\(5\) 17.4005 + 7.20752i 1.55635 + 0.644661i 0.984450 0.175664i \(-0.0562073\pi\)
0.571898 + 0.820325i \(0.306207\pi\)
\(6\) 0 0
\(7\) −4.37099 4.37099i −0.236012 0.236012i 0.579185 0.815196i \(-0.303371\pi\)
−0.815196 + 0.579185i \(0.803371\pi\)
\(8\) 0 0
\(9\) 14.2808 14.2808i 0.528917 0.528917i
\(10\) 0 0
\(11\) −11.7977 + 28.4821i −0.323375 + 0.780697i 0.675678 + 0.737197i \(0.263851\pi\)
−0.999053 + 0.0435002i \(0.986149\pi\)
\(12\) 0 0
\(13\) −12.9230 + 5.35288i −0.275707 + 0.114202i −0.516253 0.856436i \(-0.672674\pi\)
0.240546 + 0.970638i \(0.422674\pi\)
\(14\) 0 0
\(15\) 49.1278i 0.845649i
\(16\) 0 0
\(17\) 72.9239i 1.04039i 0.854047 + 0.520195i \(0.174141\pi\)
−0.854047 + 0.520195i \(0.825859\pi\)
\(18\) 0 0
\(19\) 143.136 59.2889i 1.72830 0.715884i 0.728783 0.684745i \(-0.240086\pi\)
0.999515 0.0311397i \(-0.00991368\pi\)
\(20\) 0 0
\(21\) 6.17043 14.8967i 0.0641190 0.154797i
\(22\) 0 0
\(23\) −83.6411 + 83.6411i −0.758278 + 0.758278i −0.976009 0.217731i \(-0.930134\pi\)
0.217731 + 0.976009i \(0.430134\pi\)
\(24\) 0 0
\(25\) 162.441 + 162.441i 1.29953 + 1.29953i
\(26\) 0 0
\(27\) 113.737 + 47.1114i 0.810692 + 0.335800i
\(28\) 0 0
\(29\) −39.6554 95.7367i −0.253925 0.613030i 0.744589 0.667523i \(-0.232646\pi\)
−0.998514 + 0.0544937i \(0.982646\pi\)
\(30\) 0 0
\(31\) 29.0324 0.168206 0.0841029 0.996457i \(-0.473198\pi\)
0.0841029 + 0.996457i \(0.473198\pi\)
\(32\) 0 0
\(33\) −80.4149 −0.424195
\(34\) 0 0
\(35\) −44.5534 107.562i −0.215169 0.519463i
\(36\) 0 0
\(37\) −267.681 110.877i −1.18936 0.492650i −0.301815 0.953366i \(-0.597593\pi\)
−0.887547 + 0.460716i \(0.847593\pi\)
\(38\) 0 0
\(39\) −25.7996 25.7996i −0.105929 0.105929i
\(40\) 0 0
\(41\) −124.918 + 124.918i −0.475827 + 0.475827i −0.903794 0.427968i \(-0.859230\pi\)
0.427968 + 0.903794i \(0.359230\pi\)
\(42\) 0 0
\(43\) 27.0156 65.2215i 0.0958103 0.231306i −0.868707 0.495326i \(-0.835048\pi\)
0.964517 + 0.264020i \(0.0850485\pi\)
\(44\) 0 0
\(45\) 351.421 145.564i 1.16415 0.482207i
\(46\) 0 0
\(47\) 282.627i 0.877136i −0.898698 0.438568i \(-0.855486\pi\)
0.898698 0.438568i \(-0.144514\pi\)
\(48\) 0 0
\(49\) 304.789i 0.888597i
\(50\) 0 0
\(51\) −175.738 + 72.7931i −0.482515 + 0.199864i
\(52\) 0 0
\(53\) −51.4343 + 124.173i −0.133303 + 0.321821i −0.976410 0.215924i \(-0.930724\pi\)
0.843107 + 0.537745i \(0.180724\pi\)
\(54\) 0 0
\(55\) −410.570 + 410.570i −1.00657 + 1.00657i
\(56\) 0 0
\(57\) 285.759 + 285.759i 0.664029 + 0.664029i
\(58\) 0 0
\(59\) 222.476 + 92.1524i 0.490913 + 0.203343i 0.614387 0.789005i \(-0.289404\pi\)
−0.123474 + 0.992348i \(0.539404\pi\)
\(60\) 0 0
\(61\) −226.809 547.566i −0.476065 1.14932i −0.961440 0.275016i \(-0.911317\pi\)
0.485374 0.874306i \(-0.338683\pi\)
\(62\) 0 0
\(63\) −124.842 −0.249661
\(64\) 0 0
\(65\) −263.447 −0.502717
\(66\) 0 0
\(67\) −356.015 859.496i −0.649166 1.56723i −0.813975 0.580900i \(-0.802701\pi\)
0.164808 0.986326i \(-0.447299\pi\)
\(68\) 0 0
\(69\) −285.056 118.074i −0.497344 0.206007i
\(70\) 0 0
\(71\) −690.837 690.837i −1.15475 1.15475i −0.985588 0.169163i \(-0.945894\pi\)
−0.169163 0.985588i \(-0.554106\pi\)
\(72\) 0 0
\(73\) 223.345 223.345i 0.358089 0.358089i −0.505019 0.863108i \(-0.668515\pi\)
0.863108 + 0.505019i \(0.168515\pi\)
\(74\) 0 0
\(75\) −229.314 + 553.612i −0.353052 + 0.852342i
\(76\) 0 0
\(77\) 176.062 72.9275i 0.260574 0.107933i
\(78\) 0 0
\(79\) 698.000i 0.994065i 0.867732 + 0.497033i \(0.165577\pi\)
−0.867732 + 0.497033i \(0.834423\pi\)
\(80\) 0 0
\(81\) 224.174i 0.307509i
\(82\) 0 0
\(83\) 915.116 379.053i 1.21020 0.501283i 0.315922 0.948785i \(-0.397686\pi\)
0.894283 + 0.447502i \(0.147686\pi\)
\(84\) 0 0
\(85\) −525.601 + 1268.91i −0.670699 + 1.61921i
\(86\) 0 0
\(87\) 191.130 191.130i 0.235532 0.235532i
\(88\) 0 0
\(89\) −163.738 163.738i −0.195014 0.195014i 0.602845 0.797859i \(-0.294034\pi\)
−0.797859 + 0.602845i \(0.794034\pi\)
\(90\) 0 0
\(91\) 79.8837 + 33.0889i 0.0920229 + 0.0381171i
\(92\) 0 0
\(93\) 28.9803 + 69.9647i 0.0323131 + 0.0780108i
\(94\) 0 0
\(95\) 2917.97 3.15134
\(96\) 0 0
\(97\) −839.460 −0.878704 −0.439352 0.898315i \(-0.644792\pi\)
−0.439352 + 0.898315i \(0.644792\pi\)
\(98\) 0 0
\(99\) 238.266 + 575.225i 0.241885 + 0.583963i
\(100\) 0 0
\(101\) 1561.02 + 646.594i 1.53789 + 0.637015i 0.981076 0.193622i \(-0.0620235\pi\)
0.556814 + 0.830637i \(0.312024\pi\)
\(102\) 0 0
\(103\) 80.1481 + 80.1481i 0.0766721 + 0.0766721i 0.744403 0.667731i \(-0.232734\pi\)
−0.667731 + 0.744403i \(0.732734\pi\)
\(104\) 0 0
\(105\) 214.737 214.737i 0.199583 0.199583i
\(106\) 0 0
\(107\) −552.814 + 1334.61i −0.499463 + 1.20581i 0.450310 + 0.892872i \(0.351313\pi\)
−0.949773 + 0.312939i \(0.898687\pi\)
\(108\) 0 0
\(109\) 490.666 203.241i 0.431168 0.178596i −0.156535 0.987672i \(-0.550032\pi\)
0.587703 + 0.809077i \(0.300032\pi\)
\(110\) 0 0
\(111\) 755.757i 0.646246i
\(112\) 0 0
\(113\) 109.197i 0.0909060i −0.998966 0.0454530i \(-0.985527\pi\)
0.998966 0.0454530i \(-0.0144731\pi\)
\(114\) 0 0
\(115\) −2058.24 + 852.552i −1.66898 + 0.691312i
\(116\) 0 0
\(117\) −108.107 + 260.993i −0.0854230 + 0.206229i
\(118\) 0 0
\(119\) 318.750 318.750i 0.245544 0.245544i
\(120\) 0 0
\(121\) 269.116 + 269.116i 0.202191 + 0.202191i
\(122\) 0 0
\(123\) −425.731 176.344i −0.312088 0.129271i
\(124\) 0 0
\(125\) 754.814 + 1822.28i 0.540101 + 1.30392i
\(126\) 0 0
\(127\) −1519.49 −1.06167 −0.530837 0.847474i \(-0.678122\pi\)
−0.530837 + 0.847474i \(0.678122\pi\)
\(128\) 0 0
\(129\) 184.143 0.125681
\(130\) 0 0
\(131\) −386.047 932.000i −0.257474 0.621597i 0.741296 0.671178i \(-0.234211\pi\)
−0.998770 + 0.0495811i \(0.984211\pi\)
\(132\) 0 0
\(133\) −884.798 366.495i −0.576855 0.238941i
\(134\) 0 0
\(135\) 1639.52 + 1639.52i 1.04524 + 1.04524i
\(136\) 0 0
\(137\) 121.798 121.798i 0.0759557 0.0759557i −0.668108 0.744064i \(-0.732896\pi\)
0.744064 + 0.668108i \(0.232896\pi\)
\(138\) 0 0
\(139\) 2.36667 5.71364i 0.00144416 0.00348651i −0.923156 0.384426i \(-0.874399\pi\)
0.924600 + 0.380939i \(0.124399\pi\)
\(140\) 0 0
\(141\) 681.098 282.120i 0.406800 0.168502i
\(142\) 0 0
\(143\) 431.225i 0.252174i
\(144\) 0 0
\(145\) 1951.68i 1.11778i
\(146\) 0 0
\(147\) 734.505 304.242i 0.412115 0.170704i
\(148\) 0 0
\(149\) 311.513 752.060i 0.171276 0.413497i −0.814811 0.579727i \(-0.803159\pi\)
0.986087 + 0.166229i \(0.0531592\pi\)
\(150\) 0 0
\(151\) −571.254 + 571.254i −0.307868 + 0.307868i −0.844082 0.536214i \(-0.819854\pi\)
0.536214 + 0.844082i \(0.319854\pi\)
\(152\) 0 0
\(153\) 1041.41 + 1041.41i 0.550281 + 0.550281i
\(154\) 0 0
\(155\) 505.179 + 209.252i 0.261787 + 0.108436i
\(156\) 0 0
\(157\) −219.641 530.260i −0.111651 0.269550i 0.858171 0.513365i \(-0.171601\pi\)
−0.969822 + 0.243814i \(0.921601\pi\)
\(158\) 0 0
\(159\) −350.585 −0.174863
\(160\) 0 0
\(161\) 731.190 0.357924
\(162\) 0 0
\(163\) 570.638 + 1377.64i 0.274207 + 0.661995i 0.999655 0.0262826i \(-0.00836698\pi\)
−0.725447 + 0.688278i \(0.758367\pi\)
\(164\) 0 0
\(165\) −1399.26 579.593i −0.660196 0.273462i
\(166\) 0 0
\(167\) −642.374 642.374i −0.297655 0.297655i 0.542440 0.840095i \(-0.317501\pi\)
−0.840095 + 0.542440i \(0.817501\pi\)
\(168\) 0 0
\(169\) −1415.16 + 1415.16i −0.644134 + 0.644134i
\(170\) 0 0
\(171\) 1197.40 2890.78i 0.535483 1.29277i
\(172\) 0 0
\(173\) −1861.21 + 770.937i −0.817948 + 0.338805i −0.752120 0.659026i \(-0.770969\pi\)
−0.0658275 + 0.997831i \(0.520969\pi\)
\(174\) 0 0
\(175\) 1420.06i 0.613406i
\(176\) 0 0
\(177\) 628.127i 0.266740i
\(178\) 0 0
\(179\) 317.221 131.397i 0.132459 0.0548664i −0.315469 0.948936i \(-0.602162\pi\)
0.447928 + 0.894069i \(0.352162\pi\)
\(180\) 0 0
\(181\) 532.578 1285.76i 0.218708 0.528008i −0.776002 0.630731i \(-0.782755\pi\)
0.994710 + 0.102722i \(0.0327553\pi\)
\(182\) 0 0
\(183\) 1093.17 1093.17i 0.441581 0.441581i
\(184\) 0 0
\(185\) −3858.63 3858.63i −1.53347 1.53347i
\(186\) 0 0
\(187\) −2077.02 860.331i −0.812230 0.336437i
\(188\) 0 0
\(189\) −291.220 703.067i −0.112080 0.270585i
\(190\) 0 0
\(191\) −3161.63 −1.19773 −0.598867 0.800848i \(-0.704382\pi\)
−0.598867 + 0.800848i \(0.704382\pi\)
\(192\) 0 0
\(193\) −1428.55 −0.532796 −0.266398 0.963863i \(-0.585834\pi\)
−0.266398 + 0.963863i \(0.585834\pi\)
\(194\) 0 0
\(195\) −262.975 634.877i −0.0965744 0.233151i
\(196\) 0 0
\(197\) 1377.48 + 570.573i 0.498181 + 0.206353i 0.617603 0.786490i \(-0.288104\pi\)
−0.119421 + 0.992844i \(0.538104\pi\)
\(198\) 0 0
\(199\) −1207.72 1207.72i −0.430217 0.430217i 0.458485 0.888702i \(-0.348392\pi\)
−0.888702 + 0.458485i \(0.848392\pi\)
\(200\) 0 0
\(201\) 1715.91 1715.91i 0.602143 0.602143i
\(202\) 0 0
\(203\) −245.131 + 591.798i −0.0847528 + 0.204611i
\(204\) 0 0
\(205\) −3073.98 + 1273.28i −1.04730 + 0.433805i
\(206\) 0 0
\(207\) 2388.92i 0.802132i
\(208\) 0 0
\(209\) 4776.28i 1.58078i
\(210\) 0 0
\(211\) 627.654 259.983i 0.204784 0.0848245i −0.277934 0.960600i \(-0.589650\pi\)
0.482718 + 0.875776i \(0.339650\pi\)
\(212\) 0 0
\(213\) 975.239 2354.43i 0.313720 0.757386i
\(214\) 0 0
\(215\) 940.170 940.170i 0.298228 0.298228i
\(216\) 0 0
\(217\) −126.901 126.901i −0.0396985 0.0396985i
\(218\) 0 0
\(219\) 761.178 + 315.290i 0.234866 + 0.0972846i
\(220\) 0 0
\(221\) −390.353 942.394i −0.118814 0.286843i
\(222\) 0 0
\(223\) −1684.85 −0.505946 −0.252973 0.967473i \(-0.581408\pi\)
−0.252973 + 0.967473i \(0.581408\pi\)
\(224\) 0 0
\(225\) 4639.56 1.37468
\(226\) 0 0
\(227\) 1751.96 + 4229.60i 0.512254 + 1.23669i 0.942569 + 0.334011i \(0.108402\pi\)
−0.430315 + 0.902679i \(0.641598\pi\)
\(228\) 0 0
\(229\) 1072.62 + 444.292i 0.309521 + 0.128208i 0.532037 0.846721i \(-0.321427\pi\)
−0.222515 + 0.974929i \(0.571427\pi\)
\(230\) 0 0
\(231\) 351.493 + 351.493i 0.100115 + 0.100115i
\(232\) 0 0
\(233\) −224.008 + 224.008i −0.0629840 + 0.0629840i −0.737897 0.674913i \(-0.764181\pi\)
0.674913 + 0.737897i \(0.264181\pi\)
\(234\) 0 0
\(235\) 2037.04 4917.85i 0.565455 1.36513i
\(236\) 0 0
\(237\) −1682.10 + 696.748i −0.461030 + 0.190965i
\(238\) 0 0
\(239\) 5695.35i 1.54143i 0.637181 + 0.770714i \(0.280101\pi\)
−0.637181 + 0.770714i \(0.719899\pi\)
\(240\) 0 0
\(241\) 5864.62i 1.56752i 0.621061 + 0.783762i \(0.286702\pi\)
−0.621061 + 0.783762i \(0.713298\pi\)
\(242\) 0 0
\(243\) 3611.13 1495.78i 0.953309 0.394874i
\(244\) 0 0
\(245\) 2196.77 5303.48i 0.572844 1.38297i
\(246\) 0 0
\(247\) −1532.38 + 1532.38i −0.394749 + 0.394749i
\(248\) 0 0
\(249\) 1826.95 + 1826.95i 0.464972 + 0.464972i
\(250\) 0 0
\(251\) −1284.55 532.079i −0.323029 0.133803i 0.215276 0.976553i \(-0.430935\pi\)
−0.538305 + 0.842750i \(0.680935\pi\)
\(252\) 0 0
\(253\) −1395.50 3369.04i −0.346777 0.837193i
\(254\) 0 0
\(255\) −3582.59 −0.879805
\(256\) 0 0
\(257\) 6068.35 1.47289 0.736446 0.676497i \(-0.236503\pi\)
0.736446 + 0.676497i \(0.236503\pi\)
\(258\) 0 0
\(259\) 685.388 + 1654.67i 0.164432 + 0.396974i
\(260\) 0 0
\(261\) −1933.50 800.883i −0.458547 0.189937i
\(262\) 0 0
\(263\) 1919.51 + 1919.51i 0.450046 + 0.450046i 0.895370 0.445324i \(-0.146911\pi\)
−0.445324 + 0.895370i \(0.646911\pi\)
\(264\) 0 0
\(265\) −1789.97 + 1789.97i −0.414931 + 0.414931i
\(266\) 0 0
\(267\) 231.146 558.035i 0.0529808 0.127907i
\(268\) 0 0
\(269\) 3969.40 1644.18i 0.899697 0.372667i 0.115593 0.993297i \(-0.463123\pi\)
0.784103 + 0.620630i \(0.213123\pi\)
\(270\) 0 0
\(271\) 4538.69i 1.01736i −0.860954 0.508682i \(-0.830133\pi\)
0.860954 0.508682i \(-0.169867\pi\)
\(272\) 0 0
\(273\) 225.540i 0.0500011i
\(274\) 0 0
\(275\) −6543.07 + 2710.23i −1.43477 + 0.594301i
\(276\) 0 0
\(277\) −1889.91 + 4562.63i −0.409940 + 0.989683i 0.575213 + 0.818004i \(0.304919\pi\)
−0.985153 + 0.171679i \(0.945081\pi\)
\(278\) 0 0
\(279\) 414.605 414.605i 0.0889669 0.0889669i
\(280\) 0 0
\(281\) −30.7223 30.7223i −0.00652220 0.00652220i 0.703838 0.710360i \(-0.251468\pi\)
−0.710360 + 0.703838i \(0.751468\pi\)
\(282\) 0 0
\(283\) 5549.45 + 2298.66i 1.16565 + 0.482830i 0.879754 0.475429i \(-0.157707\pi\)
0.285901 + 0.958259i \(0.407707\pi\)
\(284\) 0 0
\(285\) 2912.73 + 7031.95i 0.605387 + 1.46153i
\(286\) 0 0
\(287\) 1092.03 0.224601
\(288\) 0 0
\(289\) −404.895 −0.0824130
\(290\) 0 0
\(291\) −837.954 2023.00i −0.168803 0.407527i
\(292\) 0 0
\(293\) 2663.79 + 1103.38i 0.531128 + 0.220000i 0.632097 0.774889i \(-0.282194\pi\)
−0.100969 + 0.994890i \(0.532194\pi\)
\(294\) 0 0
\(295\) 3207.00 + 3207.00i 0.632944 + 0.632944i
\(296\) 0 0
\(297\) −2683.66 + 2683.66i −0.524316 + 0.524316i
\(298\) 0 0
\(299\) 633.173 1528.61i 0.122466 0.295659i
\(300\) 0 0
\(301\) −403.168 + 166.998i −0.0772033 + 0.0319787i
\(302\) 0 0
\(303\) 4407.30i 0.835620i
\(304\) 0 0
\(305\) 11162.7i 2.09565i
\(306\) 0 0
\(307\) −401.539 + 166.323i −0.0746484 + 0.0309204i −0.419695 0.907665i \(-0.637863\pi\)
0.345047 + 0.938585i \(0.387863\pi\)
\(308\) 0 0
\(309\) −113.143 + 273.152i −0.0208301 + 0.0502883i
\(310\) 0 0
\(311\) 5345.11 5345.11i 0.974577 0.974577i −0.0251081 0.999685i \(-0.507993\pi\)
0.999685 + 0.0251081i \(0.00799300\pi\)
\(312\) 0 0
\(313\) −1787.53 1787.53i −0.322803 0.322803i 0.527039 0.849841i \(-0.323302\pi\)
−0.849841 + 0.527039i \(0.823302\pi\)
\(314\) 0 0
\(315\) −2172.32 899.804i −0.388560 0.160947i
\(316\) 0 0
\(317\) 2370.77 + 5723.55i 0.420050 + 1.01409i 0.982332 + 0.187145i \(0.0599234\pi\)
−0.562282 + 0.826945i \(0.690077\pi\)
\(318\) 0 0
\(319\) 3194.62 0.560703
\(320\) 0 0
\(321\) −3768.08 −0.655183
\(322\) 0 0
\(323\) 4323.58 + 10438.0i 0.744800 + 1.79811i
\(324\) 0 0
\(325\) −2968.74 1229.69i −0.506696 0.209880i
\(326\) 0 0
\(327\) 979.572 + 979.572i 0.165659 + 0.165659i
\(328\) 0 0
\(329\) −1235.36 + 1235.36i −0.207014 + 0.207014i
\(330\) 0 0
\(331\) −1696.86 + 4096.57i −0.281775 + 0.680265i −0.999877 0.0156724i \(-0.995011\pi\)
0.718102 + 0.695938i \(0.245011\pi\)
\(332\) 0 0
\(333\) −5406.09 + 2239.28i −0.889646 + 0.368503i
\(334\) 0 0
\(335\) 17521.6i 2.85764i
\(336\) 0 0
\(337\) 927.470i 0.149918i 0.997187 + 0.0749592i \(0.0238826\pi\)
−0.997187 + 0.0749592i \(0.976117\pi\)
\(338\) 0 0
\(339\) 263.152 109.001i 0.0421606 0.0174635i
\(340\) 0 0
\(341\) −342.515 + 826.904i −0.0543936 + 0.131318i
\(342\) 0 0
\(343\) −2831.48 + 2831.48i −0.445731 + 0.445731i
\(344\) 0 0
\(345\) −4109.10 4109.10i −0.641237 0.641237i
\(346\) 0 0
\(347\) −3765.13 1559.57i −0.582487 0.241274i 0.0719277 0.997410i \(-0.477085\pi\)
−0.654415 + 0.756136i \(0.727085\pi\)
\(348\) 0 0
\(349\) −1631.14 3937.93i −0.250181 0.603990i 0.748037 0.663657i \(-0.230996\pi\)
−0.998218 + 0.0596662i \(0.980996\pi\)
\(350\) 0 0
\(351\) −1722.00 −0.261862
\(352\) 0 0
\(353\) −11289.2 −1.70216 −0.851080 0.525037i \(-0.824052\pi\)
−0.851080 + 0.525037i \(0.824052\pi\)
\(354\) 0 0
\(355\) −7041.69 17000.1i −1.05277 2.54162i
\(356\) 0 0
\(357\) 1086.33 + 449.972i 0.161049 + 0.0667088i
\(358\) 0 0
\(359\) −2970.97 2970.97i −0.436774 0.436774i 0.454151 0.890925i \(-0.349943\pi\)
−0.890925 + 0.454151i \(0.849943\pi\)
\(360\) 0 0
\(361\) 12122.7 12122.7i 1.76742 1.76742i
\(362\) 0 0
\(363\) −379.904 + 917.170i −0.0549306 + 0.132614i
\(364\) 0 0
\(365\) 5496.07 2276.55i 0.788157 0.326465i
\(366\) 0 0
\(367\) 12266.5i 1.74470i 0.488882 + 0.872350i \(0.337405\pi\)
−0.488882 + 0.872350i \(0.662595\pi\)
\(368\) 0 0
\(369\) 3567.84i 0.503346i
\(370\) 0 0
\(371\) 767.581 317.942i 0.107415 0.0444926i
\(372\) 0 0
\(373\) −905.804 + 2186.80i −0.125739 + 0.303561i −0.974196 0.225703i \(-0.927532\pi\)
0.848457 + 0.529265i \(0.177532\pi\)
\(374\) 0 0
\(375\) −3638.03 + 3638.03i −0.500979 + 0.500979i
\(376\) 0 0
\(377\) 1024.93 + 1024.93i 0.140018 + 0.140018i
\(378\) 0 0
\(379\) −6881.62 2850.46i −0.932679 0.386328i −0.135985 0.990711i \(-0.543420\pi\)
−0.796694 + 0.604383i \(0.793420\pi\)
\(380\) 0 0
\(381\) −1516.76 3661.78i −0.203953 0.492385i
\(382\) 0 0
\(383\) 2433.37 0.324645 0.162323 0.986738i \(-0.448101\pi\)
0.162323 + 0.986738i \(0.448101\pi\)
\(384\) 0 0
\(385\) 3589.20 0.475124
\(386\) 0 0
\(387\) −545.609 1317.22i −0.0716663 0.173018i
\(388\) 0 0
\(389\) 4253.32 + 1761.78i 0.554375 + 0.229630i 0.642241 0.766503i \(-0.278005\pi\)
−0.0878663 + 0.996132i \(0.528005\pi\)
\(390\) 0 0
\(391\) −6099.44 6099.44i −0.788905 0.788905i
\(392\) 0 0
\(393\) 1860.66 1860.66i 0.238824 0.238824i
\(394\) 0 0
\(395\) −5030.85 + 12145.6i −0.640835 + 1.54711i
\(396\) 0 0
\(397\) −306.541 + 126.974i −0.0387528 + 0.0160519i −0.401976 0.915650i \(-0.631676\pi\)
0.363223 + 0.931702i \(0.381676\pi\)
\(398\) 0 0
\(399\) 2498.10i 0.313437i
\(400\) 0 0
\(401\) 10975.6i 1.36682i −0.730036 0.683409i \(-0.760497\pi\)
0.730036 0.683409i \(-0.239503\pi\)
\(402\) 0 0
\(403\) −375.186 + 155.407i −0.0463755 + 0.0192094i
\(404\) 0 0
\(405\) 1615.74 3900.74i 0.198239 0.478591i
\(406\) 0 0
\(407\) 6316.01 6316.01i 0.769221 0.769221i
\(408\) 0 0
\(409\) 8384.88 + 8384.88i 1.01371 + 1.01371i 0.999905 + 0.0138007i \(0.00439304\pi\)
0.0138007 + 0.999905i \(0.495607\pi\)
\(410\) 0 0
\(411\) 415.100 + 171.940i 0.0498184 + 0.0206355i
\(412\) 0 0
\(413\) −569.642 1375.24i −0.0678698 0.163852i
\(414\) 0 0
\(415\) 18655.5 2.20666
\(416\) 0 0
\(417\) 16.1316 0.00189441
\(418\) 0 0
\(419\) −533.639 1288.32i −0.0622195 0.150211i 0.889712 0.456522i \(-0.150905\pi\)
−0.951931 + 0.306311i \(0.900905\pi\)
\(420\) 0 0
\(421\) 1963.49 + 813.303i 0.227303 + 0.0941520i 0.493428 0.869786i \(-0.335744\pi\)
−0.266125 + 0.963938i \(0.585744\pi\)
\(422\) 0 0
\(423\) −4036.13 4036.13i −0.463932 0.463932i
\(424\) 0 0
\(425\) −11845.8 + 11845.8i −1.35201 + 1.35201i
\(426\) 0 0
\(427\) −1402.03 + 3384.79i −0.158897 + 0.383610i
\(428\) 0 0
\(429\) 1039.20 430.451i 0.116954 0.0484438i
\(430\) 0 0
\(431\) 4521.29i 0.505297i 0.967558 + 0.252649i \(0.0813016\pi\)
−0.967558 + 0.252649i \(0.918698\pi\)
\(432\) 0 0
\(433\) 2122.80i 0.235601i −0.993037 0.117801i \(-0.962416\pi\)
0.993037 0.117801i \(-0.0375844\pi\)
\(434\) 0 0
\(435\) 4703.33 1948.18i 0.518408 0.214732i
\(436\) 0 0
\(437\) −7013.07 + 16931.1i −0.767691 + 1.85337i
\(438\) 0 0
\(439\) −8028.12 + 8028.12i −0.872805 + 0.872805i −0.992777 0.119972i \(-0.961719\pi\)
0.119972 + 0.992777i \(0.461719\pi\)
\(440\) 0 0
\(441\) −4352.62 4352.62i −0.469994 0.469994i
\(442\) 0 0
\(443\) −3136.09 1299.01i −0.336343 0.139318i 0.208119 0.978104i \(-0.433266\pi\)
−0.544462 + 0.838786i \(0.683266\pi\)
\(444\) 0 0
\(445\) −1668.98 4029.28i −0.177792 0.429227i
\(446\) 0 0
\(447\) 2123.33 0.224676
\(448\) 0 0
\(449\) −16955.9 −1.78218 −0.891088 0.453832i \(-0.850057\pi\)
−0.891088 + 0.453832i \(0.850057\pi\)
\(450\) 0 0
\(451\) −2084.18 5031.66i −0.217606 0.525347i
\(452\) 0 0
\(453\) −1946.88 806.426i −0.201926 0.0836406i
\(454\) 0 0
\(455\) 1151.53 + 1151.53i 0.118647 + 0.118647i
\(456\) 0 0
\(457\) 6223.46 6223.46i 0.637027 0.637027i −0.312794 0.949821i \(-0.601265\pi\)
0.949821 + 0.312794i \(0.101265\pi\)
\(458\) 0 0
\(459\) −3435.55 + 8294.14i −0.349363 + 0.843437i
\(460\) 0 0
\(461\) 10872.5 4503.54i 1.09844 0.454991i 0.241501 0.970401i \(-0.422360\pi\)
0.856944 + 0.515410i \(0.172360\pi\)
\(462\) 0 0
\(463\) 13182.5i 1.32320i −0.749858 0.661599i \(-0.769878\pi\)
0.749858 0.661599i \(-0.230122\pi\)
\(464\) 0 0
\(465\) 1426.30i 0.142243i
\(466\) 0 0
\(467\) −7043.25 + 2917.41i −0.697907 + 0.289083i −0.703290 0.710903i \(-0.748287\pi\)
0.00538305 + 0.999986i \(0.498287\pi\)
\(468\) 0 0
\(469\) −2200.71 + 5312.99i −0.216673 + 0.523094i
\(470\) 0 0
\(471\) 1058.62 1058.62i 0.103564 0.103564i
\(472\) 0 0
\(473\) 1538.92 + 1538.92i 0.149598 + 0.149598i
\(474\) 0 0
\(475\) 32882.1 + 13620.2i 3.17628 + 1.31566i
\(476\) 0 0
\(477\) 1038.77 + 2507.81i 0.0997107 + 0.240723i
\(478\) 0 0
\(479\) −4612.37 −0.439968 −0.219984 0.975503i \(-0.570601\pi\)
−0.219984 + 0.975503i \(0.570601\pi\)
\(480\) 0 0
\(481\) 4052.74 0.384177
\(482\) 0 0
\(483\) 729.878 + 1762.08i 0.0687590 + 0.165999i
\(484\) 0 0
\(485\) −14607.0 6050.43i −1.36757 0.566466i
\(486\) 0 0
\(487\) 10988.9 + 10988.9i 1.02249 + 1.02249i 0.999741 + 0.0227515i \(0.00724266\pi\)
0.0227515 + 0.999741i \(0.492757\pi\)
\(488\) 0 0
\(489\) −2750.34 + 2750.34i −0.254345 + 0.254345i
\(490\) 0 0
\(491\) 7836.91 18920.0i 0.720316 1.73900i 0.0478645 0.998854i \(-0.484758\pi\)
0.672451 0.740142i \(-0.265242\pi\)
\(492\) 0 0
\(493\) 6981.49 2891.83i 0.637790 0.264181i
\(494\) 0 0
\(495\) 11726.5i 1.06478i
\(496\) 0 0
\(497\) 6039.29i 0.545069i
\(498\) 0 0
\(499\) 6404.45 2652.81i 0.574554 0.237988i −0.0764356 0.997075i \(-0.524354\pi\)
0.650990 + 0.759086i \(0.274354\pi\)
\(500\) 0 0
\(501\) 906.824 2189.27i 0.0808661 0.195228i
\(502\) 0 0
\(503\) −2553.60 + 2553.60i −0.226361 + 0.226361i −0.811171 0.584810i \(-0.801169\pi\)
0.584810 + 0.811171i \(0.301169\pi\)
\(504\) 0 0
\(505\) 22502.1 + 22502.1i 1.98283 + 1.98283i
\(506\) 0 0
\(507\) −4823.00 1997.75i −0.422479 0.174997i
\(508\) 0 0
\(509\) 3141.82 + 7585.02i 0.273593 + 0.660511i 0.999632 0.0271425i \(-0.00864080\pi\)
−0.726039 + 0.687654i \(0.758641\pi\)
\(510\) 0 0
\(511\) −1952.48 −0.169026
\(512\) 0 0
\(513\) 19073.0 1.64151
\(514\) 0 0
\(515\) 816.948 + 1972.29i 0.0699010 + 0.168756i
\(516\) 0 0
\(517\) 8049.80 + 3334.34i 0.684777 + 0.283644i
\(518\) 0 0
\(519\) −3715.74 3715.74i −0.314263 0.314263i
\(520\) 0 0
\(521\) −7065.05 + 7065.05i −0.594099 + 0.594099i −0.938736 0.344637i \(-0.888002\pi\)
0.344637 + 0.938736i \(0.388002\pi\)
\(522\) 0 0
\(523\) 111.041 268.077i 0.00928391 0.0224134i −0.919170 0.393862i \(-0.871139\pi\)
0.928454 + 0.371449i \(0.121139\pi\)
\(524\) 0 0
\(525\) 3422.17 1417.51i 0.284487 0.117838i
\(526\) 0 0
\(527\) 2117.16i 0.175000i
\(528\) 0 0
\(529\) 1824.68i 0.149970i
\(530\) 0 0
\(531\) 4493.13 1861.11i 0.367204 0.152101i
\(532\) 0 0
\(533\) 945.642 2282.98i 0.0768486 0.185529i
\(534\) 0 0
\(535\) −19238.5 + 19238.5i −1.55468 + 1.55468i
\(536\) 0 0
\(537\) 633.304 + 633.304i 0.0508921 + 0.0508921i
\(538\) 0 0
\(539\) 8681.02 + 3595.79i 0.693725 + 0.287350i
\(540\) 0 0
\(541\) −8420.37 20328.6i −0.669168 1.61552i −0.783005 0.622016i \(-0.786314\pi\)
0.113837 0.993499i \(-0.463686\pi\)
\(542\) 0 0
\(543\) 3630.14 0.286896
\(544\) 0 0
\(545\) 10002.7 0.786181
\(546\) 0 0
\(547\) −646.357 1560.44i −0.0505233 0.121974i 0.896603 0.442836i \(-0.146028\pi\)
−0.947126 + 0.320862i \(0.896028\pi\)
\(548\) 0 0
\(549\) −11058.7 4580.65i −0.859696 0.356098i
\(550\) 0 0
\(551\) −11352.2 11352.2i −0.877717 0.877717i
\(552\) 0 0
\(553\) 3050.95 3050.95i 0.234611 0.234611i
\(554\) 0 0
\(555\) 5447.14 13150.5i 0.416609 1.00578i
\(556\) 0 0
\(557\) −6482.01 + 2684.93i −0.493091 + 0.204245i −0.615351 0.788253i \(-0.710986\pi\)
0.122260 + 0.992498i \(0.460986\pi\)
\(558\) 0 0
\(559\) 987.467i 0.0747145i
\(560\) 0 0
\(561\) 5864.17i 0.441329i
\(562\) 0 0
\(563\) 945.289 391.551i 0.0707623 0.0293107i −0.347022 0.937857i \(-0.612807\pi\)
0.417784 + 0.908546i \(0.362807\pi\)
\(564\) 0 0
\(565\) 787.039 1900.08i 0.0586035 0.141481i
\(566\) 0 0
\(567\) −979.862 + 979.862i −0.0725756 + 0.0725756i
\(568\) 0 0
\(569\) −3760.45 3760.45i −0.277058 0.277058i 0.554875 0.831934i \(-0.312766\pi\)
−0.831934 + 0.554875i \(0.812766\pi\)
\(570\) 0 0
\(571\) −2056.20 851.705i −0.150699 0.0624216i 0.306059 0.952013i \(-0.400990\pi\)
−0.456758 + 0.889591i \(0.650990\pi\)
\(572\) 0 0
\(573\) −3155.95 7619.15i −0.230091 0.555488i
\(574\) 0 0
\(575\) −27173.5 −1.97080
\(576\) 0 0
\(577\) −23621.3 −1.70427 −0.852137 0.523318i \(-0.824694\pi\)
−0.852137 + 0.523318i \(0.824694\pi\)
\(578\) 0 0
\(579\) −1425.99 3442.65i −0.102353 0.247101i
\(580\) 0 0
\(581\) −5656.80 2343.13i −0.403931 0.167314i
\(582\) 0 0
\(583\) −2929.91 2929.91i −0.208138 0.208138i
\(584\) 0 0
\(585\) −3762.23 + 3762.23i −0.265896 + 0.265896i
\(586\) 0 0
\(587\) −7316.12 + 17662.7i −0.514427 + 1.24194i 0.426857 + 0.904319i \(0.359621\pi\)
−0.941283 + 0.337617i \(0.890379\pi\)
\(588\) 0 0
\(589\) 4155.59 1721.30i 0.290710 0.120416i
\(590\) 0 0
\(591\) 3889.13i 0.270689i
\(592\) 0 0
\(593\) 16542.0i 1.14553i 0.819720 + 0.572764i \(0.194129\pi\)
−0.819720 + 0.572764i \(0.805871\pi\)
\(594\) 0 0
\(595\) 7843.81 3249.01i 0.540445 0.223860i
\(596\) 0 0
\(597\) 1704.91 4116.02i 0.116880 0.282174i
\(598\) 0 0
\(599\) 8579.81 8579.81i 0.585244 0.585244i −0.351095 0.936340i \(-0.614191\pi\)
0.936340 + 0.351095i \(0.114191\pi\)
\(600\) 0 0
\(601\) −13633.8 13633.8i −0.925346 0.925346i 0.0720542 0.997401i \(-0.477045\pi\)
−0.997401 + 0.0720542i \(0.977045\pi\)
\(602\) 0 0
\(603\) −17358.4 7190.09i −1.17229 0.485578i
\(604\) 0 0
\(605\) 2743.09 + 6622.41i 0.184335 + 0.445023i
\(606\) 0 0
\(607\) 23984.1 1.60376 0.801881 0.597484i \(-0.203833\pi\)
0.801881 + 0.597484i \(0.203833\pi\)
\(608\) 0 0
\(609\) −1670.86 −0.111176
\(610\) 0 0
\(611\) 1512.87 + 3652.38i 0.100170 + 0.241833i
\(612\) 0 0
\(613\) 18072.9 + 7486.04i 1.19080 + 0.493244i 0.888014 0.459816i \(-0.152085\pi\)
0.302782 + 0.953060i \(0.402085\pi\)
\(614\) 0 0
\(615\) −6136.93 6136.93i −0.402382 0.402382i
\(616\) 0 0
\(617\) 18674.2 18674.2i 1.21847 1.21847i 0.250301 0.968168i \(-0.419470\pi\)
0.968168 0.250301i \(-0.0805297\pi\)
\(618\) 0 0
\(619\) −3214.09 + 7759.51i −0.208700 + 0.503846i −0.993219 0.116258i \(-0.962910\pi\)
0.784519 + 0.620105i \(0.212910\pi\)
\(620\) 0 0
\(621\) −13453.5 + 5572.64i −0.869359 + 0.360100i
\(622\) 0 0
\(623\) 1431.40i 0.0920511i
\(624\) 0 0
\(625\) 8433.25i 0.539728i
\(626\) 0 0
\(627\) −11510.3 + 4767.71i −0.733136 + 0.303675i
\(628\) 0 0
\(629\) 8085.58 19520.3i 0.512549 1.23740i
\(630\) 0 0
\(631\) 20023.0 20023.0i 1.26324 1.26324i 0.313722 0.949515i \(-0.398424\pi\)
0.949515 0.313722i \(-0.101576\pi\)
\(632\) 0 0
\(633\) 1253.06 + 1253.06i 0.0786801 + 0.0786801i
\(634\) 0 0
\(635\) −26439.8 10951.7i −1.65233 0.684419i
\(636\) 0 0
\(637\) 1631.50 + 3938.78i 0.101479 + 0.244992i
\(638\) 0 0
\(639\) −19731.4 −1.22154
\(640\) 0 0
\(641\) 8637.88 0.532255 0.266128 0.963938i \(-0.414256\pi\)
0.266128 + 0.963938i \(0.414256\pi\)
\(642\) 0 0
\(643\) −5274.73 12734.3i −0.323507 0.781015i −0.999045 0.0436901i \(-0.986089\pi\)
0.675538 0.737325i \(-0.263911\pi\)
\(644\) 0 0
\(645\) 3204.18 + 1327.22i 0.195604 + 0.0810219i
\(646\) 0 0
\(647\) 14785.3 + 14785.3i 0.898406 + 0.898406i 0.995295 0.0968891i \(-0.0308892\pi\)
−0.0968891 + 0.995295i \(0.530889\pi\)
\(648\) 0 0
\(649\) −5249.38 + 5249.38i −0.317498 + 0.317498i
\(650\) 0 0
\(651\) 179.143 432.488i 0.0107852 0.0260377i
\(652\) 0 0
\(653\) −25046.7 + 10374.7i −1.50100 + 0.621733i −0.973677 0.227933i \(-0.926803\pi\)
−0.527321 + 0.849666i \(0.676803\pi\)
\(654\) 0 0
\(655\) 18999.7i 1.13340i
\(656\) 0 0
\(657\) 6379.06i 0.378799i
\(658\) 0 0
\(659\) 23962.3 9925.50i 1.41645 0.586711i 0.462481 0.886629i \(-0.346959\pi\)
0.953965 + 0.299918i \(0.0969594\pi\)
\(660\) 0 0
\(661\) −398.063 + 961.010i −0.0234234 + 0.0565491i −0.935158 0.354230i \(-0.884743\pi\)
0.911735 + 0.410779i \(0.134743\pi\)
\(662\) 0 0
\(663\) 1881.41 1881.41i 0.110208 0.110208i
\(664\) 0 0
\(665\) −12754.4 12754.4i −0.743752 0.743752i
\(666\) 0 0
\(667\) 11324.4 + 4690.70i 0.657392 + 0.272301i
\(668\) 0 0
\(669\) −1681.83 4060.30i −0.0971948 0.234649i
\(670\) 0 0
\(671\) 18271.6 1.05122
\(672\) 0 0
\(673\) −15307.7 −0.876775 −0.438387 0.898786i \(-0.644450\pi\)
−0.438387 + 0.898786i \(0.644450\pi\)
\(674\) 0 0
\(675\) 10822.7 + 26128.3i 0.617135 + 1.48990i
\(676\) 0 0
\(677\) −1043.10 432.068i −0.0592168 0.0245284i 0.352878 0.935669i \(-0.385203\pi\)
−0.412095 + 0.911141i \(0.635203\pi\)
\(678\) 0 0
\(679\) 3669.28 + 3669.28i 0.207384 + 0.207384i
\(680\) 0 0
\(681\) −8444.03 + 8444.03i −0.475148 + 0.475148i
\(682\) 0 0
\(683\) 8946.40 21598.5i 0.501207 1.21002i −0.447619 0.894224i \(-0.647728\pi\)
0.948827 0.315797i \(-0.102272\pi\)
\(684\) 0 0
\(685\) 2997.22 1241.49i 0.167179 0.0692479i
\(686\) 0 0
\(687\) 3028.37i 0.168180i
\(688\) 0 0
\(689\) 1880.01i 0.103952i
\(690\) 0 0
\(691\) −19676.4 + 8150.22i −1.08325 + 0.448696i −0.851647 0.524115i \(-0.824396\pi\)
−0.231600 + 0.972811i \(0.574396\pi\)
\(692\) 0 0
\(693\) 1472.85 3555.77i 0.0807342 0.194910i
\(694\) 0 0
\(695\) 82.3624 82.3624i 0.00449523 0.00449523i
\(696\) 0 0
\(697\) −9109.50 9109.50i −0.495046 0.495046i
\(698\) 0 0
\(699\) −763.440 316.227i −0.0413104 0.0171113i
\(700\) 0 0
\(701\) 2739.27 + 6613.19i 0.147591 + 0.356315i 0.980334 0.197343i \(-0.0632313\pi\)
−0.832744 + 0.553658i \(0.813231\pi\)
\(702\) 0 0
\(703\) −44888.5 −2.40825
\(704\) 0 0
\(705\) 13884.8 0.741749
\(706\) 0 0
\(707\) −3996.93 9649.45i −0.212617 0.513303i
\(708\) 0 0
\(709\) −2388.26 989.248i −0.126506 0.0524005i 0.318532 0.947912i \(-0.396810\pi\)
−0.445039 + 0.895511i \(0.646810\pi\)
\(710\) 0 0
\(711\) 9967.97 + 9967.97i 0.525778 + 0.525778i
\(712\) 0 0
\(713\) −2428.31 + 2428.31i −0.127547 + 0.127547i
\(714\) 0 0
\(715\) 3108.06 7503.53i 0.162566 0.392470i
\(716\) 0 0
\(717\) −13725.1 + 5685.13i −0.714887 + 0.296116i
\(718\) 0 0
\(719\) 11843.1i 0.614288i 0.951663 + 0.307144i \(0.0993734\pi\)
−0.951663 + 0.307144i \(0.900627\pi\)
\(720\) 0 0
\(721\) 700.654i 0.0361910i
\(722\) 0 0
\(723\) −14133.0 + 5854.10i −0.726990 + 0.301129i
\(724\) 0 0
\(725\) 9109.88 21993.2i 0.466665 1.12663i
\(726\) 0 0
\(727\) 4691.36 4691.36i 0.239330 0.239330i −0.577243 0.816573i \(-0.695871\pi\)
0.816573 + 0.577243i \(0.195871\pi\)
\(728\) 0 0
\(729\) 2929.40 + 2929.40i 0.148829 + 0.148829i
\(730\) 0 0
\(731\) 4756.20 + 1970.08i 0.240649 + 0.0996801i
\(732\) 0 0
\(733\) 13224.6 + 31927.0i 0.666386 + 1.60880i 0.787610 + 0.616174i \(0.211318\pi\)
−0.121224 + 0.992625i \(0.538682\pi\)
\(734\) 0 0
\(735\) 14973.6 0.751441
\(736\) 0 0
\(737\) 28680.4 1.43345
\(738\) 0 0
\(739\) 13199.0 + 31865.1i 0.657012 + 1.58617i 0.802396 + 0.596793i \(0.203558\pi\)
−0.145383 + 0.989375i \(0.546442\pi\)
\(740\) 0 0
\(741\) −5222.48 2163.22i −0.258911 0.107244i
\(742\) 0 0
\(743\) 14307.6 + 14307.6i 0.706452 + 0.706452i 0.965787 0.259335i \(-0.0835035\pi\)
−0.259335 + 0.965787i \(0.583503\pi\)
\(744\) 0 0
\(745\) 10841.0 10841.0i 0.533131 0.533131i
\(746\) 0 0
\(747\) 7655.38 18481.7i 0.374961 0.905235i
\(748\) 0 0
\(749\) 8249.93 3417.23i 0.402464 0.166706i
\(750\) 0 0
\(751\) 15781.7i 0.766820i −0.923578 0.383410i \(-0.874750\pi\)
0.923578 0.383410i \(-0.125250\pi\)
\(752\) 0 0
\(753\) 3626.75i 0.175519i
\(754\) 0 0
\(755\) −14057.4 + 5822.78i −0.677619 + 0.280679i
\(756\) 0 0
\(757\) −15069.9 + 36382.0i −0.723548 + 1.74680i −0.0605650 + 0.998164i \(0.519290\pi\)
−0.662983 + 0.748635i \(0.730710\pi\)
\(758\) 0 0
\(759\) 6726.00 6726.00i 0.321658 0.321658i
\(760\) 0 0
\(761\) −12120.7 12120.7i −0.577363 0.577363i 0.356813 0.934176i \(-0.383863\pi\)
−0.934176 + 0.356813i \(0.883863\pi\)
\(762\) 0 0
\(763\) −3033.06 1256.34i −0.143911 0.0596100i
\(764\) 0 0
\(765\) 10615.1 + 25627.0i 0.501684 + 1.21117i
\(766\) 0 0
\(767\) −3368.33 −0.158570
\(768\) 0 0
\(769\) 5213.88 0.244496 0.122248 0.992500i \(-0.460990\pi\)
0.122248 + 0.992500i \(0.460990\pi\)
\(770\) 0 0
\(771\) 6057.46 + 14624.0i 0.282950 + 0.683101i
\(772\) 0 0
\(773\) 25960.5 + 10753.2i 1.20793 + 0.500343i 0.893555 0.448954i \(-0.148203\pi\)
0.314380 + 0.949297i \(0.398203\pi\)
\(774\) 0 0
\(775\) 4716.05 + 4716.05i 0.218588 + 0.218588i
\(776\) 0 0
\(777\) −3303.41 + 3303.41i −0.152521 + 0.152521i
\(778\) 0 0
\(779\) −10474.0 + 25286.5i −0.481733 + 1.16301i
\(780\) 0 0
\(781\) 27826.7 11526.2i 1.27493 0.528093i
\(782\) 0 0
\(783\) 12757.0i 0.582246i
\(784\) 0 0
\(785\) 10809.9i 0.491491i
\(786\) 0 0
\(787\) −31369.3 + 12993.6i −1.42083 + 0.588528i −0.955069 0.296382i \(-0.904220\pi\)
−0.465762 + 0.884910i \(0.654220\pi\)
\(788\) 0 0
\(789\) −2709.73 + 6541.86i −0.122267 + 0.295179i
\(790\) 0 0
\(791\) −477.299 + 477.299i −0.0214549 + 0.0214549i
\(792\) 0 0
\(793\) 5862.11 + 5862.11i 0.262509 + 0.262509i
\(794\) 0 0
\(795\) −6100.36 2526.85i −0.272148 0.112727i
\(796\) 0 0
\(797\) 8986.32 + 21694.9i 0.399387 + 0.964207i 0.987812 + 0.155654i \(0.0497485\pi\)
−0.588424 + 0.808552i \(0.700251\pi\)
\(798\) 0 0
\(799\) 20610.3 0.912564
\(800\) 0 0
\(801\) −4676.62 −0.206292
\(802\) 0 0
\(803\) 3726.37 + 8996.26i 0.163762 + 0.395356i
\(804\) 0 0
\(805\) 12723.1 + 5270.07i 0.557055 + 0.230740i
\(806\) 0 0
\(807\) 7924.55 + 7924.55i 0.345672 + 0.345672i
\(808\) 0 0
\(809\) −13193.6 + 13193.6i −0.573376 + 0.573376i −0.933070 0.359694i \(-0.882881\pi\)
0.359694 + 0.933070i \(0.382881\pi\)
\(810\) 0 0
\(811\) −140.367 + 338.877i −0.00607764 + 0.0146727i −0.926889 0.375335i \(-0.877528\pi\)
0.920812 + 0.390008i \(0.127528\pi\)
\(812\) 0 0
\(813\) 10937.7 4530.55i 0.471835 0.195441i
\(814\) 0 0
\(815\) 28084.5i 1.20707i
\(816\) 0 0
\(817\) 10937.3i 0.468356i
\(818\) 0 0
\(819\) 1613.33 668.265i 0.0688333 0.0285117i
\(820\) 0 0
\(821\) 6304.96 15221.5i 0.268020 0.647058i −0.731370 0.681981i \(-0.761119\pi\)
0.999390 + 0.0349231i \(0.0111186\pi\)
\(822\) 0 0
\(823\) 769.291 769.291i 0.0325830 0.0325830i −0.690628 0.723211i \(-0.742666\pi\)
0.723211 + 0.690628i \(0.242666\pi\)
\(824\) 0 0
\(825\) −13062.7 13062.7i −0.551253 0.551253i
\(826\) 0 0
\(827\) −22898.4 9484.82i −0.962823 0.398815i −0.154788 0.987948i \(-0.549469\pi\)
−0.808036 + 0.589133i \(0.799469\pi\)
\(828\) 0 0
\(829\) −10844.8 26181.7i −0.454350 1.09690i −0.970651 0.240492i \(-0.922691\pi\)
0.516301 0.856407i \(-0.327309\pi\)
\(830\) 0 0
\(831\) −12881.9 −0.537749
\(832\) 0 0
\(833\) 22226.4 0.924488
\(834\) 0 0
\(835\) −6547.70 15807.5i −0.271368 0.655141i
\(836\) 0 0
\(837\) 3302.06 + 1367.76i 0.136363 + 0.0564834i
\(838\) 0 0
\(839\) −23945.5 23945.5i −0.985327 0.985327i 0.0145666 0.999894i \(-0.495363\pi\)
−0.999894 + 0.0145666i \(0.995363\pi\)
\(840\) 0 0
\(841\) 9652.67 9652.67i 0.395779 0.395779i
\(842\) 0 0
\(843\) 43.3699 104.704i 0.00177193 0.00427783i
\(844\) 0 0
\(845\) −34824.4 + 14424.7i −1.41775 + 0.587249i
\(846\) 0 0
\(847\) 2352.61i 0.0954386i
\(848\) 0 0
\(849\) 15668.0i 0.633364i
\(850\) 0 0
\(851\) 31663.0 13115.2i 1.27543 0.528302i
\(852\) 0 0
\(853\) 8663.21 20914.8i 0.347740 0.839520i −0.649146 0.760664i \(-0.724873\pi\)
0.996886 0.0788554i \(-0.0251266\pi\)
\(854\) 0 0
\(855\) 41670.8 41670.8i 1.66680 1.66680i
\(856\) 0 0
\(857\) −9119.55 9119.55i −0.363498 0.363498i 0.501601 0.865099i \(-0.332745\pi\)
−0.865099 + 0.501601i \(0.832745\pi\)
\(858\) 0 0
\(859\) 30582.3 + 12667.6i 1.21473 + 0.503159i 0.895732 0.444595i \(-0.146652\pi\)
0.319002 + 0.947754i \(0.396652\pi\)
\(860\) 0 0
\(861\) 1090.07 + 2631.66i 0.0431470 + 0.104166i
\(862\) 0 0
\(863\) −1210.85 −0.0477612 −0.0238806 0.999715i \(-0.507602\pi\)
−0.0238806 + 0.999715i \(0.507602\pi\)
\(864\) 0 0
\(865\) −37942.5 −1.49143
\(866\) 0 0
\(867\) −404.169 975.749i −0.0158319 0.0382217i
\(868\) 0 0
\(869\) −19880.5 8234.77i −0.776064 0.321456i
\(870\) 0 0
\(871\) 9201.55 + 9201.55i 0.357959 + 0.357959i
\(872\) 0 0
\(873\) −11988.1 + 11988.1i −0.464762 + 0.464762i
\(874\) 0 0
\(875\) 4665.90 11264.5i 0.180270 0.435210i
\(876\) 0 0
\(877\) −36421.4 + 15086.2i −1.40235 + 0.580873i −0.950361 0.311150i \(-0.899286\pi\)
−0.451990 + 0.892023i \(0.649286\pi\)
\(878\) 0 0
\(879\) 7520.83i 0.288591i
\(880\) 0 0
\(881\) 39286.1i 1.50236i 0.660095 + 0.751182i \(0.270516\pi\)
−0.660095 + 0.751182i \(0.729484\pi\)
\(882\) 0 0
\(883\) 234.092 96.9642i 0.00892167 0.00369548i −0.378218 0.925717i \(-0.623463\pi\)
0.387140 + 0.922021i \(0.373463\pi\)
\(884\) 0 0
\(885\) −4527.24 + 10929.7i −0.171957 + 0.415140i
\(886\) 0 0
\(887\) −13552.8 + 13552.8i −0.513032 + 0.513032i −0.915454 0.402422i \(-0.868168\pi\)
0.402422 + 0.915454i \(0.368168\pi\)
\(888\) 0 0
\(889\) 6641.66 + 6641.66i 0.250567 + 0.250567i
\(890\) 0 0
\(891\) 6384.93 + 2644.73i 0.240071 + 0.0994407i
\(892\) 0 0
\(893\) −16756.6 40454.1i −0.627928 1.51595i
\(894\) 0 0
\(895\) 6466.85 0.241523
\(896\) 0 0
\(897\) 4315.82 0.160648
\(898\) 0 0
\(899\) −1151.29 2779.47i −0.0427117 0.103115i
\(900\) 0 0
\(901\) −9055.21 3750.79i −0.334820 0.138687i
\(902\) 0 0
\(903\) −804.889 804.889i −0.0296623 0.0296623i
\(904\) 0 0
\(905\) 18534.2 18534.2i 0.680772 0.680772i
\(906\) 0 0
\(907\) 2600.00 6276.95i 0.0951835 0.229793i −0.869115 0.494609i \(-0.835311\pi\)
0.964299 + 0.264816i \(0.0853112\pi\)
\(908\) 0 0
\(909\) 31526.4 13058.6i 1.15034 0.476488i
\(910\) 0 0
\(911\) 22984.1i 0.835891i 0.908472 + 0.417946i \(0.137250\pi\)
−0.908472 + 0.417946i \(0.862750\pi\)
\(912\) 0 0
\(913\) 30536.3i 1.10691i
\(914\) 0 0
\(915\) 26900.7 11142.6i 0.971924 0.402584i
\(916\) 0 0
\(917\) −2386.36 + 5761.17i −0.0859372 + 0.207471i
\(918\) 0 0
\(919\) −38048.0 + 38048.0i −1.36571 + 1.36571i −0.499258 + 0.866453i \(0.666394\pi\)
−0.866453 + 0.499258i \(0.833606\pi\)
\(920\) 0 0
\(921\) −801.638 801.638i −0.0286806 0.0286806i
\(922\) 0 0
\(923\) 12625.6 + 5229.71i 0.450247 + 0.186499i
\(924\) 0 0
\(925\) −25471.3 61493.2i −0.905396 2.18582i
\(926\) 0 0
\(927\) 2289.15 0.0811064
\(928\) 0 0
\(929\) 28435.1 1.00423 0.502113 0.864802i \(-0.332556\pi\)
0.502113 + 0.864802i \(0.332556\pi\)
\(930\) 0 0
\(931\) −18070.6 43626.3i −0.636133 1.53576i
\(932\) 0 0
\(933\) 18216.6 + 7545.57i 0.639212 + 0.264770i
\(934\) 0 0
\(935\) −29940.4 29940.4i −1.04723 1.04723i
\(936\) 0 0
\(937\) 10599.3 10599.3i 0.369546 0.369546i −0.497766 0.867311i \(-0.665846\pi\)
0.867311 + 0.497766i \(0.165846\pi\)
\(938\) 0 0
\(939\) 2523.42 6092.07i 0.0876982 0.211722i
\(940\) 0 0
\(941\) −17446.0 + 7226.38i −0.604383 + 0.250344i −0.663825 0.747888i \(-0.731068\pi\)
0.0594418 + 0.998232i \(0.481068\pi\)
\(942\) 0 0
\(943\) 20896.5i 0.721617i
\(944\) 0 0
\(945\) 14332.7i 0.493378i
\(946\) 0 0
\(947\) 8125.64 3365.75i 0.278826 0.115493i −0.238889 0.971047i \(-0.576783\pi\)
0.517714 + 0.855554i \(0.326783\pi\)
\(948\) 0 0
\(949\) −1690.74 + 4081.81i −0.0578333 + 0.139622i
\(950\) 0 0
\(951\) −11426.6 + 11426.6i −0.389623 + 0.389623i
\(952\) 0 0
\(953\) 20053.9 + 20053.9i 0.681648 + 0.681648i 0.960371 0.278723i \(-0.0899113\pi\)
−0.278723 + 0.960371i \(0.589911\pi\)
\(954\) 0 0
\(955\) −55013.9 22787.5i −1.86409 0.772132i
\(956\) 0 0
\(957\) 3188.89 + 7698.66i 0.107714 + 0.260044i
\(958\) 0 0
\(959\) −1064.76 −0.0358529
\(960\) 0 0
\(961\) −28948.1 −0.971707
\(962\) 0 0
\(963\) 11164.7 + 26953.9i 0.373599 + 0.901949i
\(964\) 0 0
\(965\) −24857.6 10296.3i −0.829216 0.343473i
\(966\) 0 0
\(967\) −17483.6 17483.6i −0.581421 0.581421i 0.353872 0.935294i \(-0.384865\pi\)
−0.935294 + 0.353872i \(0.884865\pi\)
\(968\) 0 0
\(969\) −20838.6 + 20838.6i −0.690849 + 0.690849i
\(970\) 0 0
\(971\) −2567.45 + 6198.36i −0.0848540 + 0.204856i −0.960611 0.277896i \(-0.910363\pi\)
0.875757 + 0.482752i \(0.160363\pi\)
\(972\) 0 0
\(973\) −35.3190 + 14.6296i −0.00116369 + 0.000482018i
\(974\) 0 0
\(975\) 8381.81i 0.275316i
\(976\) 0 0
\(977\) 31111.6i 1.01878i −0.860536 0.509390i \(-0.829871\pi\)
0.860536 0.509390i \(-0.170129\pi\)
\(978\) 0 0
\(979\) 6595.34 2731.88i 0.215309 0.0891841i
\(980\) 0 0
\(981\) 4104.66 9909.52i 0.133590 0.322514i
\(982\) 0 0
\(983\) −17048.2 + 17048.2i −0.553156 + 0.553156i −0.927350 0.374194i \(-0.877919\pi\)
0.374194 + 0.927350i \(0.377919\pi\)
\(984\) 0 0
\(985\) 19856.5 + 19856.5i 0.642316 + 0.642316i
\(986\) 0 0
\(987\) −4210.22 1743.93i −0.135778 0.0562410i
\(988\) 0 0
\(989\) 3195.58 + 7714.81i 0.102744 + 0.248045i
\(990\) 0 0
\(991\) −32857.3 −1.05323 −0.526613 0.850105i \(-0.676538\pi\)
−0.526613 + 0.850105i \(0.676538\pi\)
\(992\) 0 0
\(993\) −11566.1 −0.369625
\(994\) 0 0
\(995\) −12310.3 29719.7i −0.392223 0.946911i
\(996\) 0 0
\(997\) −30160.2 12492.8i −0.958057 0.396840i −0.151804 0.988411i \(-0.548508\pi\)
−0.806253 + 0.591570i \(0.798508\pi\)
\(998\) 0 0
\(999\) −25221.6 25221.6i −0.798775 0.798775i
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 128.4.g.a.113.7 44
4.3 odd 2 32.4.g.a.5.2 44
8.3 odd 2 256.4.g.b.225.7 44
8.5 even 2 256.4.g.a.225.5 44
32.3 odd 8 256.4.g.b.33.7 44
32.13 even 8 inner 128.4.g.a.17.7 44
32.19 odd 8 32.4.g.a.13.2 yes 44
32.29 even 8 256.4.g.a.33.5 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.5.2 44 4.3 odd 2
32.4.g.a.13.2 yes 44 32.19 odd 8
128.4.g.a.17.7 44 32.13 even 8 inner
128.4.g.a.113.7 44 1.1 even 1 trivial
256.4.g.a.33.5 44 32.29 even 8
256.4.g.a.225.5 44 8.5 even 2
256.4.g.b.33.7 44 32.3 odd 8
256.4.g.b.225.7 44 8.3 odd 2