Properties

Label 136.4.k.a.89.3
Level $136$
Weight $4$
Character 136.89
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
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] = [136,4,Mod(81,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.k (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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 89.3
Root \(-1.84168i\) of defining polynomial
Character \(\chi\) \(=\) 136.89
Dual form 136.4.k.a.81.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.84168 + 1.84168i) q^{3} +(4.90158 - 4.90158i) q^{5} +(6.14401 + 6.14401i) q^{7} +20.2164i q^{9} +O(q^{10})\) \(q+(-1.84168 + 1.84168i) q^{3} +(4.90158 - 4.90158i) q^{5} +(6.14401 + 6.14401i) q^{7} +20.2164i q^{9} +(-19.4554 - 19.4554i) q^{11} +64.7611 q^{13} +18.0543i q^{15} +(69.5389 - 8.79432i) q^{17} +162.870i q^{19} -22.6306 q^{21} +(114.732 + 114.732i) q^{23} +76.9491i q^{25} +(-86.9575 - 86.9575i) q^{27} +(21.2323 - 21.2323i) q^{29} +(-11.9156 + 11.9156i) q^{31} +71.6612 q^{33} +60.2306 q^{35} +(206.020 - 206.020i) q^{37} +(-119.269 + 119.269i) q^{39} +(-164.183 - 164.183i) q^{41} -150.537i q^{43} +(99.0925 + 99.0925i) q^{45} -371.156 q^{47} -267.502i q^{49} +(-111.872 + 144.265i) q^{51} -55.4454i q^{53} -190.724 q^{55} +(-299.954 - 299.954i) q^{57} -15.6670i q^{59} +(127.371 + 127.371i) q^{61} +(-124.210 + 124.210i) q^{63} +(317.432 - 317.432i) q^{65} +159.183 q^{67} -422.597 q^{69} +(27.7245 - 27.7245i) q^{71} +(-525.063 + 525.063i) q^{73} +(-141.715 - 141.715i) q^{75} -239.068i q^{77} +(-526.679 - 526.679i) q^{79} -225.549 q^{81} -768.177i q^{83} +(297.744 - 383.956i) q^{85} +78.2062i q^{87} +1576.01 q^{89} +(397.893 + 397.893i) q^{91} -43.8895i q^{93} +(798.320 + 798.320i) q^{95} +(-1259.75 + 1259.75i) q^{97} +(393.319 - 393.319i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7} + 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} + 246 q^{23} + 572 q^{27} + 260 q^{29} + 566 q^{31} - 904 q^{33} - 804 q^{35} - 28 q^{37} + 476 q^{39} - 786 q^{41} - 604 q^{45} + 448 q^{47} - 380 q^{51} + 1612 q^{55} + 1004 q^{57} + 660 q^{61} - 2250 q^{63} - 796 q^{65} - 284 q^{67} - 2532 q^{69} + 326 q^{71} + 730 q^{73} - 864 q^{75} + 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} - 3112 q^{91} + 3356 q^{95} - 1194 q^{97} + 2876 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.84168 + 1.84168i −0.354431 + 0.354431i −0.861755 0.507324i \(-0.830635\pi\)
0.507324 + 0.861755i \(0.330635\pi\)
\(4\) 0 0
\(5\) 4.90158 4.90158i 0.438410 0.438410i −0.453066 0.891477i \(-0.649670\pi\)
0.891477 + 0.453066i \(0.149670\pi\)
\(6\) 0 0
\(7\) 6.14401 + 6.14401i 0.331745 + 0.331745i 0.853249 0.521504i \(-0.174629\pi\)
−0.521504 + 0.853249i \(0.674629\pi\)
\(8\) 0 0
\(9\) 20.2164i 0.748757i
\(10\) 0 0
\(11\) −19.4554 19.4554i −0.533275 0.533275i 0.388270 0.921546i \(-0.373073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(12\) 0 0
\(13\) 64.7611 1.38165 0.690827 0.723020i \(-0.257247\pi\)
0.690827 + 0.723020i \(0.257247\pi\)
\(14\) 0 0
\(15\) 18.0543i 0.310773i
\(16\) 0 0
\(17\) 69.5389 8.79432i 0.992098 0.125467i
\(18\) 0 0
\(19\) 162.870i 1.96658i 0.182058 + 0.983288i \(0.441724\pi\)
−0.182058 + 0.983288i \(0.558276\pi\)
\(20\) 0 0
\(21\) −22.6306 −0.235162
\(22\) 0 0
\(23\) 114.732 + 114.732i 1.04014 + 1.04014i 0.999160 + 0.0409792i \(0.0130478\pi\)
0.0409792 + 0.999160i \(0.486952\pi\)
\(24\) 0 0
\(25\) 76.9491i 0.615593i
\(26\) 0 0
\(27\) −86.9575 86.9575i −0.619814 0.619814i
\(28\) 0 0
\(29\) 21.2323 21.2323i 0.135957 0.135957i −0.635853 0.771810i \(-0.719352\pi\)
0.771810 + 0.635853i \(0.219352\pi\)
\(30\) 0 0
\(31\) −11.9156 + 11.9156i −0.0690358 + 0.0690358i −0.740782 0.671746i \(-0.765545\pi\)
0.671746 + 0.740782i \(0.265545\pi\)
\(32\) 0 0
\(33\) 71.6612 0.378019
\(34\) 0 0
\(35\) 60.2306 0.290881
\(36\) 0 0
\(37\) 206.020 206.020i 0.915392 0.915392i −0.0812980 0.996690i \(-0.525907\pi\)
0.996690 + 0.0812980i \(0.0259065\pi\)
\(38\) 0 0
\(39\) −119.269 + 119.269i −0.489701 + 0.489701i
\(40\) 0 0
\(41\) −164.183 164.183i −0.625393 0.625393i 0.321513 0.946905i \(-0.395809\pi\)
−0.946905 + 0.321513i \(0.895809\pi\)
\(42\) 0 0
\(43\) 150.537i 0.533877i −0.963714 0.266938i \(-0.913988\pi\)
0.963714 0.266938i \(-0.0860120\pi\)
\(44\) 0 0
\(45\) 99.0925 + 99.0925i 0.328263 + 0.328263i
\(46\) 0 0
\(47\) −371.156 −1.15189 −0.575944 0.817489i \(-0.695365\pi\)
−0.575944 + 0.817489i \(0.695365\pi\)
\(48\) 0 0
\(49\) 267.502i 0.779890i
\(50\) 0 0
\(51\) −111.872 + 144.265i −0.307161 + 0.396100i
\(52\) 0 0
\(53\) 55.4454i 0.143698i −0.997416 0.0718492i \(-0.977110\pi\)
0.997416 0.0718492i \(-0.0228900\pi\)
\(54\) 0 0
\(55\) −190.724 −0.467587
\(56\) 0 0
\(57\) −299.954 299.954i −0.697015 0.697015i
\(58\) 0 0
\(59\) 15.6670i 0.0345706i −0.999851 0.0172853i \(-0.994498\pi\)
0.999851 0.0172853i \(-0.00550236\pi\)
\(60\) 0 0
\(61\) 127.371 + 127.371i 0.267348 + 0.267348i 0.828031 0.560683i \(-0.189461\pi\)
−0.560683 + 0.828031i \(0.689461\pi\)
\(62\) 0 0
\(63\) −124.210 + 124.210i −0.248397 + 0.248397i
\(64\) 0 0
\(65\) 317.432 317.432i 0.605731 0.605731i
\(66\) 0 0
\(67\) 159.183 0.290258 0.145129 0.989413i \(-0.453640\pi\)
0.145129 + 0.989413i \(0.453640\pi\)
\(68\) 0 0
\(69\) −422.597 −0.737315
\(70\) 0 0
\(71\) 27.7245 27.7245i 0.0463422 0.0463422i −0.683556 0.729898i \(-0.739567\pi\)
0.729898 + 0.683556i \(0.239567\pi\)
\(72\) 0 0
\(73\) −525.063 + 525.063i −0.841834 + 0.841834i −0.989097 0.147263i \(-0.952954\pi\)
0.147263 + 0.989097i \(0.452954\pi\)
\(74\) 0 0
\(75\) −141.715 141.715i −0.218185 0.218185i
\(76\) 0 0
\(77\) 239.068i 0.353823i
\(78\) 0 0
\(79\) −526.679 526.679i −0.750076 0.750076i 0.224417 0.974493i \(-0.427952\pi\)
−0.974493 + 0.224417i \(0.927952\pi\)
\(80\) 0 0
\(81\) −225.549 −0.309395
\(82\) 0 0
\(83\) 768.177i 1.01588i −0.861391 0.507942i \(-0.830406\pi\)
0.861391 0.507942i \(-0.169594\pi\)
\(84\) 0 0
\(85\) 297.744 383.956i 0.379940 0.489952i
\(86\) 0 0
\(87\) 78.2062i 0.0963745i
\(88\) 0 0
\(89\) 1576.01 1.87704 0.938520 0.345224i \(-0.112197\pi\)
0.938520 + 0.345224i \(0.112197\pi\)
\(90\) 0 0
\(91\) 397.893 + 397.893i 0.458357 + 0.458357i
\(92\) 0 0
\(93\) 43.8895i 0.0489368i
\(94\) 0 0
\(95\) 798.320 + 798.320i 0.862167 + 0.862167i
\(96\) 0 0
\(97\) −1259.75 + 1259.75i −1.31864 + 1.31864i −0.403782 + 0.914855i \(0.632305\pi\)
−0.914855 + 0.403782i \(0.867695\pi\)
\(98\) 0 0
\(99\) 393.319 393.319i 0.399294 0.399294i
\(100\) 0 0
\(101\) −661.687 −0.651884 −0.325942 0.945390i \(-0.605681\pi\)
−0.325942 + 0.945390i \(0.605681\pi\)
\(102\) 0 0
\(103\) −930.870 −0.890499 −0.445249 0.895407i \(-0.646885\pi\)
−0.445249 + 0.895407i \(0.646885\pi\)
\(104\) 0 0
\(105\) −110.925 + 110.925i −0.103097 + 0.103097i
\(106\) 0 0
\(107\) 993.949 993.949i 0.898025 0.898025i −0.0972362 0.995261i \(-0.531000\pi\)
0.995261 + 0.0972362i \(0.0310002\pi\)
\(108\) 0 0
\(109\) 459.889 + 459.889i 0.404123 + 0.404123i 0.879683 0.475560i \(-0.157755\pi\)
−0.475560 + 0.879683i \(0.657755\pi\)
\(110\) 0 0
\(111\) 758.845i 0.648887i
\(112\) 0 0
\(113\) 46.1480 + 46.1480i 0.0384181 + 0.0384181i 0.726055 0.687637i \(-0.241352\pi\)
−0.687637 + 0.726055i \(0.741352\pi\)
\(114\) 0 0
\(115\) 1124.73 0.912016
\(116\) 0 0
\(117\) 1309.24i 1.03452i
\(118\) 0 0
\(119\) 481.280 + 373.215i 0.370747 + 0.287501i
\(120\) 0 0
\(121\) 573.973i 0.431235i
\(122\) 0 0
\(123\) 604.745 0.443317
\(124\) 0 0
\(125\) 989.869 + 989.869i 0.708293 + 0.708293i
\(126\) 0 0
\(127\) 1959.28i 1.36896i 0.729031 + 0.684480i \(0.239971\pi\)
−0.729031 + 0.684480i \(0.760029\pi\)
\(128\) 0 0
\(129\) 277.241 + 277.241i 0.189222 + 0.189222i
\(130\) 0 0
\(131\) 1104.58 1104.58i 0.736697 0.736697i −0.235240 0.971937i \(-0.575588\pi\)
0.971937 + 0.235240i \(0.0755878\pi\)
\(132\) 0 0
\(133\) −1000.67 + 1000.67i −0.652402 + 0.652402i
\(134\) 0 0
\(135\) −852.458 −0.543466
\(136\) 0 0
\(137\) −446.567 −0.278487 −0.139244 0.990258i \(-0.544467\pi\)
−0.139244 + 0.990258i \(0.544467\pi\)
\(138\) 0 0
\(139\) 1738.60 1738.60i 1.06091 1.06091i 0.0628879 0.998021i \(-0.479969\pi\)
0.998021 0.0628879i \(-0.0200310\pi\)
\(140\) 0 0
\(141\) 683.550 683.550i 0.408265 0.408265i
\(142\) 0 0
\(143\) −1259.95 1259.95i −0.736802 0.736802i
\(144\) 0 0
\(145\) 208.144i 0.119210i
\(146\) 0 0
\(147\) 492.653 + 492.653i 0.276417 + 0.276417i
\(148\) 0 0
\(149\) −1504.06 −0.826961 −0.413481 0.910513i \(-0.635687\pi\)
−0.413481 + 0.910513i \(0.635687\pi\)
\(150\) 0 0
\(151\) 1081.49i 0.582852i 0.956593 + 0.291426i \(0.0941297\pi\)
−0.956593 + 0.291426i \(0.905870\pi\)
\(152\) 0 0
\(153\) 177.790 + 1405.83i 0.0939442 + 0.742840i
\(154\) 0 0
\(155\) 116.811i 0.0605320i
\(156\) 0 0
\(157\) −3618.49 −1.83941 −0.919705 0.392611i \(-0.871572\pi\)
−0.919705 + 0.392611i \(0.871572\pi\)
\(158\) 0 0
\(159\) 102.113 + 102.113i 0.0509311 + 0.0509311i
\(160\) 0 0
\(161\) 1409.82i 0.690122i
\(162\) 0 0
\(163\) 641.962 + 641.962i 0.308480 + 0.308480i 0.844320 0.535839i \(-0.180005\pi\)
−0.535839 + 0.844320i \(0.680005\pi\)
\(164\) 0 0
\(165\) 351.253 351.253i 0.165727 0.165727i
\(166\) 0 0
\(167\) 2397.82 2397.82i 1.11107 1.11107i 0.118064 0.993006i \(-0.462331\pi\)
0.993006 0.118064i \(-0.0376688\pi\)
\(168\) 0 0
\(169\) 1997.00 0.908967
\(170\) 0 0
\(171\) −3292.65 −1.47249
\(172\) 0 0
\(173\) 2381.19 2381.19i 1.04647 1.04647i 0.0476001 0.998866i \(-0.484843\pi\)
0.998866 0.0476001i \(-0.0151573\pi\)
\(174\) 0 0
\(175\) −472.776 + 472.776i −0.204220 + 0.204220i
\(176\) 0 0
\(177\) 28.8535 + 28.8535i 0.0122529 + 0.0122529i
\(178\) 0 0
\(179\) 2021.08i 0.843924i −0.906613 0.421962i \(-0.861341\pi\)
0.906613 0.421962i \(-0.138659\pi\)
\(180\) 0 0
\(181\) −1828.47 1828.47i −0.750878 0.750878i 0.223765 0.974643i \(-0.428165\pi\)
−0.974643 + 0.223765i \(0.928165\pi\)
\(182\) 0 0
\(183\) −469.154 −0.189513
\(184\) 0 0
\(185\) 2019.65i 0.802635i
\(186\) 0 0
\(187\) −1524.01 1181.81i −0.595970 0.462153i
\(188\) 0 0
\(189\) 1068.53i 0.411240i
\(190\) 0 0
\(191\) −3474.47 −1.31625 −0.658125 0.752909i \(-0.728650\pi\)
−0.658125 + 0.752909i \(0.728650\pi\)
\(192\) 0 0
\(193\) −2144.34 2144.34i −0.799755 0.799755i 0.183302 0.983057i \(-0.441321\pi\)
−0.983057 + 0.183302i \(0.941321\pi\)
\(194\) 0 0
\(195\) 1169.21i 0.429380i
\(196\) 0 0
\(197\) 1178.53 + 1178.53i 0.426226 + 0.426226i 0.887341 0.461114i \(-0.152550\pi\)
−0.461114 + 0.887341i \(0.652550\pi\)
\(198\) 0 0
\(199\) −1157.98 + 1157.98i −0.412497 + 0.412497i −0.882607 0.470111i \(-0.844214\pi\)
0.470111 + 0.882607i \(0.344214\pi\)
\(200\) 0 0
\(201\) −293.164 + 293.164i −0.102877 + 0.102877i
\(202\) 0 0
\(203\) 260.903 0.0902059
\(204\) 0 0
\(205\) −1609.51 −0.548357
\(206\) 0 0
\(207\) −2319.47 + 2319.47i −0.778812 + 0.778812i
\(208\) 0 0
\(209\) 3168.70 3168.70i 1.04873 1.04873i
\(210\) 0 0
\(211\) 694.783 + 694.783i 0.226687 + 0.226687i 0.811307 0.584620i \(-0.198757\pi\)
−0.584620 + 0.811307i \(0.698757\pi\)
\(212\) 0 0
\(213\) 102.119i 0.0328502i
\(214\) 0 0
\(215\) −737.869 737.869i −0.234057 0.234057i
\(216\) 0 0
\(217\) −146.419 −0.0458046
\(218\) 0 0
\(219\) 1933.99i 0.596745i
\(220\) 0 0
\(221\) 4503.42 569.530i 1.37074 0.173352i
\(222\) 0 0
\(223\) 665.828i 0.199942i −0.994990 0.0999712i \(-0.968125\pi\)
0.994990 0.0999712i \(-0.0318751\pi\)
\(224\) 0 0
\(225\) −1555.64 −0.460929
\(226\) 0 0
\(227\) −1410.05 1410.05i −0.412283 0.412283i 0.470250 0.882533i \(-0.344164\pi\)
−0.882533 + 0.470250i \(0.844164\pi\)
\(228\) 0 0
\(229\) 4743.56i 1.36884i 0.729090 + 0.684418i \(0.239944\pi\)
−0.729090 + 0.684418i \(0.760056\pi\)
\(230\) 0 0
\(231\) 440.287 + 440.287i 0.125406 + 0.125406i
\(232\) 0 0
\(233\) −2734.00 + 2734.00i −0.768713 + 0.768713i −0.977880 0.209167i \(-0.932925\pi\)
0.209167 + 0.977880i \(0.432925\pi\)
\(234\) 0 0
\(235\) −1819.25 + 1819.25i −0.504999 + 0.504999i
\(236\) 0 0
\(237\) 1939.95 0.531700
\(238\) 0 0
\(239\) −2739.49 −0.741435 −0.370718 0.928746i \(-0.620888\pi\)
−0.370718 + 0.928746i \(0.620888\pi\)
\(240\) 0 0
\(241\) 3780.15 3780.15i 1.01038 1.01038i 0.0104308 0.999946i \(-0.496680\pi\)
0.999946 0.0104308i \(-0.00332030\pi\)
\(242\) 0 0
\(243\) 2763.24 2763.24i 0.729473 0.729473i
\(244\) 0 0
\(245\) −1311.18 1311.18i −0.341912 0.341912i
\(246\) 0 0
\(247\) 10547.6i 2.71713i
\(248\) 0 0
\(249\) 1414.73 + 1414.73i 0.360061 + 0.360061i
\(250\) 0 0
\(251\) 2741.32 0.689364 0.344682 0.938719i \(-0.387987\pi\)
0.344682 + 0.938719i \(0.387987\pi\)
\(252\) 0 0
\(253\) 4464.30i 1.10936i
\(254\) 0 0
\(255\) 158.775 + 1255.47i 0.0389916 + 0.308317i
\(256\) 0 0
\(257\) 1407.95i 0.341733i −0.985294 0.170866i \(-0.945343\pi\)
0.985294 0.170866i \(-0.0546567\pi\)
\(258\) 0 0
\(259\) 2531.58 0.607354
\(260\) 0 0
\(261\) 429.242 + 429.242i 0.101799 + 0.101799i
\(262\) 0 0
\(263\) 88.8255i 0.0208259i −0.999946 0.0104130i \(-0.996685\pi\)
0.999946 0.0104130i \(-0.00331461\pi\)
\(264\) 0 0
\(265\) −271.770 271.770i −0.0629988 0.0629988i
\(266\) 0 0
\(267\) −2902.50 + 2902.50i −0.665281 + 0.665281i
\(268\) 0 0
\(269\) −1308.48 + 1308.48i −0.296578 + 0.296578i −0.839672 0.543094i \(-0.817253\pi\)
0.543094 + 0.839672i \(0.317253\pi\)
\(270\) 0 0
\(271\) 3712.58 0.832190 0.416095 0.909321i \(-0.363398\pi\)
0.416095 + 0.909321i \(0.363398\pi\)
\(272\) 0 0
\(273\) −1465.58 −0.324912
\(274\) 0 0
\(275\) 1497.08 1497.08i 0.328280 0.328280i
\(276\) 0 0
\(277\) 660.849 660.849i 0.143345 0.143345i −0.631793 0.775138i \(-0.717681\pi\)
0.775138 + 0.631793i \(0.217681\pi\)
\(278\) 0 0
\(279\) −240.892 240.892i −0.0516910 0.0516910i
\(280\) 0 0
\(281\) 4243.00i 0.900769i 0.892835 + 0.450385i \(0.148713\pi\)
−0.892835 + 0.450385i \(0.851287\pi\)
\(282\) 0 0
\(283\) 767.424 + 767.424i 0.161197 + 0.161197i 0.783097 0.621900i \(-0.213639\pi\)
−0.621900 + 0.783097i \(0.713639\pi\)
\(284\) 0 0
\(285\) −2940.50 −0.611158
\(286\) 0 0
\(287\) 2017.48i 0.414942i
\(288\) 0 0
\(289\) 4758.32 1223.09i 0.968516 0.248951i
\(290\) 0 0
\(291\) 4640.09i 0.934732i
\(292\) 0 0
\(293\) 4696.58 0.936441 0.468220 0.883612i \(-0.344895\pi\)
0.468220 + 0.883612i \(0.344895\pi\)
\(294\) 0 0
\(295\) −76.7929 76.7929i −0.0151561 0.0151561i
\(296\) 0 0
\(297\) 3383.59i 0.661063i
\(298\) 0 0
\(299\) 7430.15 + 7430.15i 1.43711 + 1.43711i
\(300\) 0 0
\(301\) 924.901 924.901i 0.177111 0.177111i
\(302\) 0 0
\(303\) 1218.61 1218.61i 0.231048 0.231048i
\(304\) 0 0
\(305\) 1248.64 0.234417
\(306\) 0 0
\(307\) −7273.04 −1.35210 −0.676050 0.736856i \(-0.736310\pi\)
−0.676050 + 0.736856i \(0.736310\pi\)
\(308\) 0 0
\(309\) 1714.36 1714.36i 0.315620 0.315620i
\(310\) 0 0
\(311\) 546.061 546.061i 0.0995635 0.0995635i −0.655570 0.755134i \(-0.727572\pi\)
0.755134 + 0.655570i \(0.227572\pi\)
\(312\) 0 0
\(313\) −2165.60 2165.60i −0.391076 0.391076i 0.483995 0.875071i \(-0.339185\pi\)
−0.875071 + 0.483995i \(0.839185\pi\)
\(314\) 0 0
\(315\) 1217.65i 0.217799i
\(316\) 0 0
\(317\) 1458.79 + 1458.79i 0.258466 + 0.258466i 0.824430 0.565964i \(-0.191496\pi\)
−0.565964 + 0.824430i \(0.691496\pi\)
\(318\) 0 0
\(319\) −826.167 −0.145005
\(320\) 0 0
\(321\) 3661.07i 0.636576i
\(322\) 0 0
\(323\) 1432.33 + 11325.8i 0.246740 + 1.95104i
\(324\) 0 0
\(325\) 4983.31i 0.850536i
\(326\) 0 0
\(327\) −1693.94 −0.286468
\(328\) 0 0
\(329\) −2280.39 2280.39i −0.382133 0.382133i
\(330\) 0 0
\(331\) 1122.37i 0.186378i −0.995648 0.0931890i \(-0.970294\pi\)
0.995648 0.0931890i \(-0.0297061\pi\)
\(332\) 0 0
\(333\) 4164.99 + 4164.99i 0.685406 + 0.685406i
\(334\) 0 0
\(335\) 780.248 780.248i 0.127252 0.127252i
\(336\) 0 0
\(337\) −1267.19 + 1267.19i −0.204832 + 0.204832i −0.802067 0.597235i \(-0.796266\pi\)
0.597235 + 0.802067i \(0.296266\pi\)
\(338\) 0 0
\(339\) −169.980 −0.0272331
\(340\) 0 0
\(341\) 463.647 0.0736302
\(342\) 0 0
\(343\) 3750.93 3750.93i 0.590470 0.590470i
\(344\) 0 0
\(345\) −2071.39 + 2071.39i −0.323247 + 0.323247i
\(346\) 0 0
\(347\) −7844.57 7844.57i −1.21360 1.21360i −0.969835 0.243763i \(-0.921618\pi\)
−0.243763 0.969835i \(-0.578382\pi\)
\(348\) 0 0
\(349\) 753.380i 0.115552i −0.998330 0.0577758i \(-0.981599\pi\)
0.998330 0.0577758i \(-0.0184008\pi\)
\(350\) 0 0
\(351\) −5631.46 5631.46i −0.856368 0.856368i
\(352\) 0 0
\(353\) 6089.17 0.918113 0.459057 0.888407i \(-0.348187\pi\)
0.459057 + 0.888407i \(0.348187\pi\)
\(354\) 0 0
\(355\) 271.788i 0.0406338i
\(356\) 0 0
\(357\) −1573.70 + 199.020i −0.233303 + 0.0295050i
\(358\) 0 0
\(359\) 10897.8i 1.60213i −0.598579 0.801064i \(-0.704268\pi\)
0.598579 0.801064i \(-0.295732\pi\)
\(360\) 0 0
\(361\) −19667.6 −2.86742
\(362\) 0 0
\(363\) 1057.07 + 1057.07i 0.152843 + 0.152843i
\(364\) 0 0
\(365\) 5147.27i 0.738138i
\(366\) 0 0
\(367\) 7319.04 + 7319.04i 1.04101 + 1.04101i 0.999122 + 0.0418881i \(0.0133373\pi\)
0.0418881 + 0.999122i \(0.486663\pi\)
\(368\) 0 0
\(369\) 3319.20 3319.20i 0.468267 0.468267i
\(370\) 0 0
\(371\) 340.657 340.657i 0.0476712 0.0476712i
\(372\) 0 0
\(373\) −3532.25 −0.490329 −0.245164 0.969481i \(-0.578842\pi\)
−0.245164 + 0.969481i \(0.578842\pi\)
\(374\) 0 0
\(375\) −3646.04 −0.502082
\(376\) 0 0
\(377\) 1375.03 1375.03i 0.187845 0.187845i
\(378\) 0 0
\(379\) 7321.72 7321.72i 0.992326 0.992326i −0.00764499 0.999971i \(-0.502433\pi\)
0.999971 + 0.00764499i \(0.00243350\pi\)
\(380\) 0 0
\(381\) −3608.36 3608.36i −0.485202 0.485202i
\(382\) 0 0
\(383\) 8011.16i 1.06880i −0.845231 0.534401i \(-0.820537\pi\)
0.845231 0.534401i \(-0.179463\pi\)
\(384\) 0 0
\(385\) −1171.81 1171.81i −0.155120 0.155120i
\(386\) 0 0
\(387\) 3043.33 0.399744
\(388\) 0 0
\(389\) 10247.3i 1.33563i −0.744329 0.667814i \(-0.767230\pi\)
0.744329 0.667814i \(-0.232770\pi\)
\(390\) 0 0
\(391\) 8987.30 + 6969.33i 1.16242 + 0.901417i
\(392\) 0 0
\(393\) 4068.55i 0.522216i
\(394\) 0 0
\(395\) −5163.12 −0.657682
\(396\) 0 0
\(397\) −8066.34 8066.34i −1.01974 1.01974i −0.999801 0.0199422i \(-0.993652\pi\)
−0.0199422 0.999801i \(-0.506348\pi\)
\(398\) 0 0
\(399\) 3685.84i 0.462463i
\(400\) 0 0
\(401\) −4181.06 4181.06i −0.520679 0.520679i 0.397098 0.917776i \(-0.370017\pi\)
−0.917776 + 0.397098i \(0.870017\pi\)
\(402\) 0 0
\(403\) −771.669 + 771.669i −0.0953835 + 0.0953835i
\(404\) 0 0
\(405\) −1105.54 + 1105.54i −0.135642 + 0.135642i
\(406\) 0 0
\(407\) −8016.42 −0.976312
\(408\) 0 0
\(409\) −335.990 −0.0406201 −0.0203100 0.999794i \(-0.506465\pi\)
−0.0203100 + 0.999794i \(0.506465\pi\)
\(410\) 0 0
\(411\) 822.432 822.432i 0.0987046 0.0987046i
\(412\) 0 0
\(413\) 96.2580 96.2580i 0.0114686 0.0114686i
\(414\) 0 0
\(415\) −3765.28 3765.28i −0.445374 0.445374i
\(416\) 0 0
\(417\) 6403.89i 0.752038i
\(418\) 0 0
\(419\) −5848.32 5848.32i −0.681883 0.681883i 0.278541 0.960424i \(-0.410149\pi\)
−0.960424 + 0.278541i \(0.910149\pi\)
\(420\) 0 0
\(421\) −2751.51 −0.318528 −0.159264 0.987236i \(-0.550912\pi\)
−0.159264 + 0.987236i \(0.550912\pi\)
\(422\) 0 0
\(423\) 7503.46i 0.862484i
\(424\) 0 0
\(425\) 676.715 + 5350.95i 0.0772364 + 0.610728i
\(426\) 0 0
\(427\) 1565.14i 0.177383i
\(428\) 0 0
\(429\) 4640.86 0.522291
\(430\) 0 0
\(431\) 7731.11 + 7731.11i 0.864025 + 0.864025i 0.991803 0.127778i \(-0.0407845\pi\)
−0.127778 + 0.991803i \(0.540784\pi\)
\(432\) 0 0
\(433\) 4563.47i 0.506481i 0.967403 + 0.253240i \(0.0814964\pi\)
−0.967403 + 0.253240i \(0.918504\pi\)
\(434\) 0 0
\(435\) 383.334 + 383.334i 0.0422516 + 0.0422516i
\(436\) 0 0
\(437\) −18686.3 + 18686.3i −2.04551 + 2.04551i
\(438\) 0 0
\(439\) 6386.15 6386.15i 0.694292 0.694292i −0.268881 0.963173i \(-0.586654\pi\)
0.963173 + 0.268881i \(0.0866539\pi\)
\(440\) 0 0
\(441\) 5407.95 0.583949
\(442\) 0 0
\(443\) −5568.83 −0.597253 −0.298626 0.954370i \(-0.596528\pi\)
−0.298626 + 0.954370i \(0.596528\pi\)
\(444\) 0 0
\(445\) 7724.93 7724.93i 0.822914 0.822914i
\(446\) 0 0
\(447\) 2769.99 2769.99i 0.293101 0.293101i
\(448\) 0 0
\(449\) −2068.57 2068.57i −0.217421 0.217421i 0.589990 0.807411i \(-0.299132\pi\)
−0.807411 + 0.589990i \(0.799132\pi\)
\(450\) 0 0
\(451\) 6388.50i 0.667013i
\(452\) 0 0
\(453\) −1991.76 1991.76i −0.206581 0.206581i
\(454\) 0 0
\(455\) 3900.60 0.401897
\(456\) 0 0
\(457\) 7185.44i 0.735494i −0.929926 0.367747i \(-0.880129\pi\)
0.929926 0.367747i \(-0.119871\pi\)
\(458\) 0 0
\(459\) −6811.66 5282.20i −0.692682 0.537150i
\(460\) 0 0
\(461\) 3480.28i 0.351612i 0.984425 + 0.175806i \(0.0562531\pi\)
−0.984425 + 0.175806i \(0.943747\pi\)
\(462\) 0 0
\(463\) 15913.5 1.59732 0.798662 0.601779i \(-0.205541\pi\)
0.798662 + 0.601779i \(0.205541\pi\)
\(464\) 0 0
\(465\) −215.128 215.128i −0.0214544 0.0214544i
\(466\) 0 0
\(467\) 15651.4i 1.55088i 0.631422 + 0.775439i \(0.282472\pi\)
−0.631422 + 0.775439i \(0.717528\pi\)
\(468\) 0 0
\(469\) 978.022 + 978.022i 0.0962918 + 0.0962918i
\(470\) 0 0
\(471\) 6664.10 6664.10i 0.651944 0.651944i
\(472\) 0 0
\(473\) −2928.76 + 2928.76i −0.284703 + 0.284703i
\(474\) 0 0
\(475\) −12532.7 −1.21061
\(476\) 0 0
\(477\) 1120.91 0.107595
\(478\) 0 0
\(479\) 599.977 599.977i 0.0572310 0.0572310i −0.677912 0.735143i \(-0.737115\pi\)
0.735143 + 0.677912i \(0.237115\pi\)
\(480\) 0 0
\(481\) 13342.1 13342.1i 1.26475 1.26475i
\(482\) 0 0
\(483\) −2596.44 2596.44i −0.244601 0.244601i
\(484\) 0 0
\(485\) 12349.5i 1.15621i
\(486\) 0 0
\(487\) 7737.55 + 7737.55i 0.719963 + 0.719963i 0.968597 0.248635i \(-0.0799819\pi\)
−0.248635 + 0.968597i \(0.579982\pi\)
\(488\) 0 0
\(489\) −2364.57 −0.218670
\(490\) 0 0
\(491\) 18450.4i 1.69583i 0.530130 + 0.847916i \(0.322143\pi\)
−0.530130 + 0.847916i \(0.677857\pi\)
\(492\) 0 0
\(493\) 1289.75 1663.20i 0.117824 0.151940i
\(494\) 0 0
\(495\) 3855.77i 0.350109i
\(496\) 0 0
\(497\) 340.679 0.0307476
\(498\) 0 0
\(499\) −2322.23 2322.23i −0.208331 0.208331i 0.595227 0.803558i \(-0.297062\pi\)
−0.803558 + 0.595227i \(0.797062\pi\)
\(500\) 0 0
\(501\) 8832.01i 0.787595i
\(502\) 0 0
\(503\) 518.275 + 518.275i 0.0459418 + 0.0459418i 0.729704 0.683763i \(-0.239658\pi\)
−0.683763 + 0.729704i \(0.739658\pi\)
\(504\) 0 0
\(505\) −3243.31 + 3243.31i −0.285793 + 0.285793i
\(506\) 0 0
\(507\) −3677.83 + 3677.83i −0.322166 + 0.322166i
\(508\) 0 0
\(509\) 22300.1 1.94192 0.970958 0.239248i \(-0.0769010\pi\)
0.970958 + 0.239248i \(0.0769010\pi\)
\(510\) 0 0
\(511\) −6451.97 −0.558549
\(512\) 0 0
\(513\) 14162.8 14162.8i 1.21891 1.21891i
\(514\) 0 0
\(515\) −4562.73 + 4562.73i −0.390404 + 0.390404i
\(516\) 0 0
\(517\) 7221.00 + 7221.00i 0.614273 + 0.614273i
\(518\) 0 0
\(519\) 8770.78i 0.741800i
\(520\) 0 0
\(521\) −2962.20 2962.20i −0.249091 0.249091i 0.571507 0.820598i \(-0.306359\pi\)
−0.820598 + 0.571507i \(0.806359\pi\)
\(522\) 0 0
\(523\) 6211.85 0.519360 0.259680 0.965695i \(-0.416383\pi\)
0.259680 + 0.965695i \(0.416383\pi\)
\(524\) 0 0
\(525\) 1741.40i 0.144764i
\(526\) 0 0
\(527\) −723.810 + 933.389i −0.0598285 + 0.0771519i
\(528\) 0 0
\(529\) 14159.7i 1.16378i
\(530\) 0 0
\(531\) 316.731 0.0258850
\(532\) 0 0
\(533\) −10632.7 10632.7i −0.864076 0.864076i
\(534\) 0 0
\(535\) 9743.84i 0.787407i
\(536\) 0 0
\(537\) 3722.17 + 3722.17i 0.299113 + 0.299113i
\(538\) 0 0
\(539\) −5204.37 + 5204.37i −0.415896 + 0.415896i
\(540\) 0 0
\(541\) 11931.1 11931.1i 0.948166 0.948166i −0.0505554 0.998721i \(-0.516099\pi\)
0.998721 + 0.0505554i \(0.0160991\pi\)
\(542\) 0 0
\(543\) 6734.90 0.532269
\(544\) 0 0
\(545\) 4508.37 0.354344
\(546\) 0 0
\(547\) −15281.2 + 15281.2i −1.19447 + 1.19447i −0.218677 + 0.975797i \(0.570174\pi\)
−0.975797 + 0.218677i \(0.929826\pi\)
\(548\) 0 0
\(549\) −2575.00 + 2575.00i −0.200179 + 0.200179i
\(550\) 0 0
\(551\) 3458.11 + 3458.11i 0.267369 + 0.267369i
\(552\) 0 0
\(553\) 6471.84i 0.497668i
\(554\) 0 0
\(555\) 3719.54 + 3719.54i 0.284479 + 0.284479i
\(556\) 0 0
\(557\) −15096.4 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(558\) 0 0
\(559\) 9748.95i 0.737633i
\(560\) 0 0
\(561\) 4983.24 630.212i 0.375032 0.0474288i
\(562\) 0 0
\(563\) 11050.9i 0.827244i −0.910449 0.413622i \(-0.864264\pi\)
0.910449 0.413622i \(-0.135736\pi\)
\(564\) 0 0
\(565\) 452.396 0.0336858
\(566\) 0 0
\(567\) −1385.77 1385.77i −0.102640 0.102640i
\(568\) 0 0
\(569\) 16142.9i 1.18936i 0.803963 + 0.594680i \(0.202721\pi\)
−0.803963 + 0.594680i \(0.797279\pi\)
\(570\) 0 0
\(571\) 16389.9 + 16389.9i 1.20122 + 1.20122i 0.973796 + 0.227424i \(0.0730304\pi\)
0.227424 + 0.973796i \(0.426970\pi\)
\(572\) 0 0
\(573\) 6398.85 6398.85i 0.466520 0.466520i
\(574\) 0 0
\(575\) −8828.49 + 8828.49i −0.640302 + 0.640302i
\(576\) 0 0
\(577\) 15499.1 1.11826 0.559130 0.829080i \(-0.311135\pi\)
0.559130 + 0.829080i \(0.311135\pi\)
\(578\) 0 0
\(579\) 7898.35 0.566916
\(580\) 0 0
\(581\) 4719.68 4719.68i 0.337014 0.337014i
\(582\) 0 0
\(583\) −1078.71 + 1078.71i −0.0766308 + 0.0766308i
\(584\) 0 0
\(585\) 6417.34 + 6417.34i 0.453546 + 0.453546i
\(586\) 0 0
\(587\) 11464.9i 0.806145i −0.915168 0.403072i \(-0.867942\pi\)
0.915168 0.403072i \(-0.132058\pi\)
\(588\) 0 0
\(589\) −1940.70 1940.70i −0.135764 0.135764i
\(590\) 0 0
\(591\) −4340.93 −0.302136
\(592\) 0 0
\(593\) 10536.2i 0.729632i −0.931080 0.364816i \(-0.881132\pi\)
0.931080 0.364816i \(-0.118868\pi\)
\(594\) 0 0
\(595\) 4188.37 529.687i 0.288582 0.0364959i
\(596\) 0 0
\(597\) 4265.24i 0.292403i
\(598\) 0 0
\(599\) −19638.2 −1.33956 −0.669779 0.742561i \(-0.733611\pi\)
−0.669779 + 0.742561i \(0.733611\pi\)
\(600\) 0 0
\(601\) −2479.04 2479.04i −0.168257 0.168257i 0.617956 0.786213i \(-0.287961\pi\)
−0.786213 + 0.617956i \(0.787961\pi\)
\(602\) 0 0
\(603\) 3218.12i 0.217333i
\(604\) 0 0
\(605\) −2813.37 2813.37i −0.189058 0.189058i
\(606\) 0 0
\(607\) −5773.82 + 5773.82i −0.386083 + 0.386083i −0.873288 0.487205i \(-0.838016\pi\)
0.487205 + 0.873288i \(0.338016\pi\)
\(608\) 0 0
\(609\) −480.499 + 480.499i −0.0319718 + 0.0319718i
\(610\) 0 0
\(611\) −24036.5 −1.59151
\(612\) 0 0
\(613\) −23768.5 −1.56607 −0.783036 0.621976i \(-0.786330\pi\)
−0.783036 + 0.621976i \(0.786330\pi\)
\(614\) 0 0
\(615\) 2964.20 2964.20i 0.194355 0.194355i
\(616\) 0 0
\(617\) −13670.6 + 13670.6i −0.891990 + 0.891990i −0.994710 0.102720i \(-0.967245\pi\)
0.102720 + 0.994710i \(0.467245\pi\)
\(618\) 0 0
\(619\) −7930.48 7930.48i −0.514948 0.514948i 0.401091 0.916038i \(-0.368631\pi\)
−0.916038 + 0.401091i \(0.868631\pi\)
\(620\) 0 0
\(621\) 19953.5i 1.28939i
\(622\) 0 0
\(623\) 9683.01 + 9683.01i 0.622699 + 0.622699i
\(624\) 0 0
\(625\) 85.2057 0.00545316
\(626\) 0 0
\(627\) 11671.5i 0.743402i
\(628\) 0 0
\(629\) 12514.6 16138.2i 0.793307 1.02301i
\(630\) 0 0
\(631\) 17029.3i 1.07437i −0.843465 0.537184i \(-0.819488\pi\)
0.843465 0.537184i \(-0.180512\pi\)
\(632\) 0 0
\(633\) −2559.13 −0.160689
\(634\) 0 0
\(635\) 9603.56 + 9603.56i 0.600166 + 0.600166i
\(636\) 0 0
\(637\) 17323.8i 1.07754i
\(638\) 0 0
\(639\) 560.491 + 560.491i 0.0346990 + 0.0346990i
\(640\) 0 0
\(641\) 8695.98 8695.98i 0.535835 0.535835i −0.386468 0.922303i \(-0.626305\pi\)
0.922303 + 0.386468i \(0.126305\pi\)
\(642\) 0 0
\(643\) 11119.5 11119.5i 0.681973 0.681973i −0.278472 0.960444i \(-0.589828\pi\)
0.960444 + 0.278472i \(0.0898278\pi\)
\(644\) 0 0
\(645\) 2717.84 0.165914
\(646\) 0 0
\(647\) −16389.0 −0.995856 −0.497928 0.867218i \(-0.665906\pi\)
−0.497928 + 0.867218i \(0.665906\pi\)
\(648\) 0 0
\(649\) −304.808 + 304.808i −0.0184357 + 0.0184357i
\(650\) 0 0
\(651\) 269.657 269.657i 0.0162346 0.0162346i
\(652\) 0 0
\(653\) 16983.6 + 16983.6i 1.01779 + 1.01779i 0.999839 + 0.0179542i \(0.00571530\pi\)
0.0179542 + 0.999839i \(0.494285\pi\)
\(654\) 0 0
\(655\) 10828.3i 0.645951i
\(656\) 0 0
\(657\) −10614.9 10614.9i −0.630330 0.630330i
\(658\) 0 0
\(659\) −10440.0 −0.617122 −0.308561 0.951205i \(-0.599847\pi\)
−0.308561 + 0.951205i \(0.599847\pi\)
\(660\) 0 0
\(661\) 29766.6i 1.75157i 0.482702 + 0.875784i \(0.339655\pi\)
−0.482702 + 0.875784i \(0.660345\pi\)
\(662\) 0 0
\(663\) −7244.95 + 9342.73i −0.424390 + 0.547272i
\(664\) 0 0
\(665\) 9809.76i 0.572039i
\(666\) 0 0
\(667\) 4872.04 0.282828
\(668\) 0 0
\(669\) 1226.24 + 1226.24i 0.0708658 + 0.0708658i
\(670\) 0 0
\(671\) 4956.13i 0.285141i
\(672\) 0 0
\(673\) −21398.0 21398.0i −1.22560 1.22560i −0.965611 0.259992i \(-0.916280\pi\)
−0.259992 0.965611i \(-0.583720\pi\)
\(674\) 0 0
\(675\) 6691.30 6691.30i 0.381553 0.381553i
\(676\) 0 0
\(677\) −22225.8 + 22225.8i −1.26176 + 1.26176i −0.311515 + 0.950241i \(0.600836\pi\)
−0.950241 + 0.311515i \(0.899164\pi\)
\(678\) 0 0
\(679\) −15479.8 −0.874903
\(680\) 0 0
\(681\) 5193.72 0.292252
\(682\) 0 0
\(683\) −2456.51 + 2456.51i −0.137622 + 0.137622i −0.772562 0.634940i \(-0.781025\pi\)
0.634940 + 0.772562i \(0.281025\pi\)
\(684\) 0 0
\(685\) −2188.88 + 2188.88i −0.122092 + 0.122092i
\(686\) 0 0
\(687\) −8736.11 8736.11i −0.485158 0.485158i
\(688\) 0 0
\(689\) 3590.71i 0.198541i
\(690\) 0 0
\(691\) −12190.7 12190.7i −0.671136 0.671136i 0.286842 0.957978i \(-0.407395\pi\)
−0.957978 + 0.286842i \(0.907395\pi\)
\(692\) 0 0
\(693\) 4833.11 0.264928
\(694\) 0 0
\(695\) 17043.8i 0.930227i
\(696\) 0 0
\(697\) −12861.0 9973.24i −0.698917 0.541985i
\(698\) 0 0
\(699\) 10070.3i 0.544912i
\(700\) 0 0
\(701\) −14126.4 −0.761121 −0.380561 0.924756i \(-0.624269\pi\)
−0.380561 + 0.924756i \(0.624269\pi\)
\(702\) 0 0
\(703\) 33554.5 + 33554.5i 1.80019 + 1.80019i
\(704\) 0 0
\(705\) 6700.95i 0.357975i
\(706\) 0 0
\(707\) −4065.41 4065.41i −0.216259 0.216259i
\(708\) 0 0
\(709\) 13811.0 13811.0i 0.731570 0.731570i −0.239361 0.970931i \(-0.576938\pi\)
0.970931 + 0.239361i \(0.0769379\pi\)
\(710\) 0 0
\(711\) 10647.6 10647.6i 0.561625 0.561625i
\(712\) 0 0
\(713\) −2734.20 −0.143614
\(714\) 0 0
\(715\) −12351.5 −0.646043
\(716\) 0 0
\(717\) 5045.26 5045.26i 0.262788 0.262788i
\(718\) 0 0
\(719\) 23222.3 23222.3i 1.20451 1.20451i 0.231732 0.972780i \(-0.425561\pi\)
0.972780 0.231732i \(-0.0744394\pi\)
\(720\) 0 0
\(721\) −5719.27 5719.27i −0.295419 0.295419i
\(722\) 0 0
\(723\) 13923.6i 0.716218i
\(724\) 0 0
\(725\) 1633.81 + 1633.81i 0.0836939 + 0.0836939i
\(726\) 0 0
\(727\) 11849.0 0.604479 0.302240 0.953232i \(-0.402266\pi\)
0.302240 + 0.953232i \(0.402266\pi\)
\(728\) 0 0
\(729\) 4088.18i 0.207701i
\(730\) 0 0
\(731\) −1323.87 10468.2i −0.0669838 0.529658i
\(732\) 0 0
\(733\) 6968.81i 0.351158i 0.984465 + 0.175579i \(0.0561797\pi\)
−0.984465 + 0.175579i \(0.943820\pi\)
\(734\) 0 0
\(735\) 4829.56 0.242368
\(736\) 0 0
\(737\) −3096.97 3096.97i −0.154788 0.154788i
\(738\) 0 0
\(739\) 12473.0i 0.620877i −0.950593 0.310439i \(-0.899524\pi\)
0.950593 0.310439i \(-0.100476\pi\)
\(740\) 0 0
\(741\) −19425.4 19425.4i −0.963034 0.963034i
\(742\) 0 0
\(743\) −7331.57 + 7331.57i −0.362004 + 0.362004i −0.864550 0.502546i \(-0.832397\pi\)
0.502546 + 0.864550i \(0.332397\pi\)
\(744\) 0 0
\(745\) −7372.26 + 7372.26i −0.362548 + 0.362548i
\(746\) 0 0
\(747\) 15529.8 0.760650
\(748\) 0 0
\(749\) 12213.7 0.595831
\(750\) 0 0
\(751\) −14291.7 + 14291.7i −0.694421 + 0.694421i −0.963201 0.268780i \(-0.913379\pi\)
0.268780 + 0.963201i \(0.413379\pi\)
\(752\) 0 0
\(753\) −5048.62 + 5048.62i −0.244332 + 0.244332i
\(754\) 0 0
\(755\) 5301.02 + 5301.02i 0.255528 + 0.255528i
\(756\) 0 0
\(757\) 27310.5i 1.31125i 0.755087 + 0.655624i \(0.227594\pi\)
−0.755087 + 0.655624i \(0.772406\pi\)
\(758\) 0 0
\(759\) 8221.81 + 8221.81i 0.393192 + 0.393192i
\(760\) 0 0
\(761\) −12317.3 −0.586730 −0.293365 0.956001i \(-0.594775\pi\)
−0.293365 + 0.956001i \(0.594775\pi\)
\(762\) 0 0
\(763\) 5651.13i 0.268132i
\(764\) 0 0
\(765\) 7762.23 + 6019.33i 0.366855 + 0.284483i
\(766\) 0 0
\(767\) 1014.61i 0.0477646i
\(768\) 0 0
\(769\) 23448.3 1.09957 0.549784 0.835307i \(-0.314710\pi\)
0.549784 + 0.835307i \(0.314710\pi\)
\(770\) 0 0
\(771\) 2592.99 + 2592.99i 0.121121 + 0.121121i
\(772\) 0 0
\(773\) 8920.95i 0.415090i −0.978225 0.207545i \(-0.933453\pi\)
0.978225 0.207545i \(-0.0665473\pi\)
\(774\) 0 0
\(775\) −916.896 916.896i −0.0424979 0.0424979i
\(776\) 0 0
\(777\) −4662.35 + 4662.35i −0.215265 + 0.215265i
\(778\) 0 0
\(779\) 26740.5 26740.5i 1.22988 1.22988i
\(780\) 0 0
\(781\) −1078.78 −0.0494263
\(782\) 0 0
\(783\) −3692.62 −0.168536
\(784\) 0 0
\(785\) −17736.3 + 17736.3i −0.806416 + 0.806416i
\(786\) 0 0
\(787\) −1469.14 + 1469.14i −0.0665429 + 0.0665429i −0.739595 0.673052i \(-0.764983\pi\)
0.673052 + 0.739595i \(0.264983\pi\)
\(788\) 0 0
\(789\) 163.588 + 163.588i 0.00738135 + 0.00738135i
\(790\) 0 0
\(791\) 567.067i 0.0254900i
\(792\) 0 0
\(793\) 8248.72 + 8248.72i 0.369383 + 0.369383i
\(794\) 0 0
\(795\) 1001.03 0.0446575
\(796\) 0 0
\(797\) 17841.0i 0.792926i 0.918051 + 0.396463i \(0.129762\pi\)
−0.918051 + 0.396463i \(0.870238\pi\)
\(798\) 0 0
\(799\) −25809.8 + 3264.07i −1.14279 + 0.144524i
\(800\) 0 0
\(801\) 31861.3i 1.40545i
\(802\) 0 0
\(803\) 20430.6 0.897859
\(804\) 0 0
\(805\) 6910.36 + 6910.36i 0.302557 + 0.302557i
\(806\) 0 0
\(807\) 4819.60i 0.210233i
\(808\) 0 0
\(809\) 16280.0 + 16280.0i 0.707510 + 0.707510i 0.966011 0.258501i \(-0.0832287\pi\)
−0.258501 + 0.966011i \(0.583229\pi\)
\(810\) 0 0
\(811\) 1081.03 1081.03i 0.0468067 0.0468067i −0.683316 0.730123i \(-0.739463\pi\)
0.730123 + 0.683316i \(0.239463\pi\)
\(812\) 0 0
\(813\) −6837.38 + 6837.38i −0.294954 + 0.294954i
\(814\) 0 0
\(815\) 6293.25 0.270482
\(816\) 0 0
\(817\) 24518.0 1.04991
\(818\) 0 0
\(819\) −8043.97 + 8043.97i −0.343198 + 0.343198i
\(820\) 0 0
\(821\) −20742.1 + 20742.1i −0.881736 + 0.881736i −0.993711 0.111975i \(-0.964282\pi\)
0.111975 + 0.993711i \(0.464282\pi\)
\(822\) 0 0
\(823\) 12843.2 + 12843.2i 0.543969 + 0.543969i 0.924690 0.380721i \(-0.124324\pi\)
−0.380721 + 0.924690i \(0.624324\pi\)
\(824\) 0 0
\(825\) 5514.26i 0.232706i
\(826\) 0 0
\(827\) 15185.3 + 15185.3i 0.638507 + 0.638507i 0.950187 0.311680i \(-0.100892\pi\)
−0.311680 + 0.950187i \(0.600892\pi\)
\(828\) 0 0
\(829\) 16515.8 0.691938 0.345969 0.938246i \(-0.387550\pi\)
0.345969 + 0.938246i \(0.387550\pi\)
\(830\) 0 0
\(831\) 2434.14i 0.101612i
\(832\) 0 0
\(833\) −2352.50 18601.8i −0.0978504 0.773728i
\(834\) 0 0
\(835\) 23506.2i 0.974209i
\(836\) 0 0
\(837\) 2072.30 0.0855787
\(838\) 0 0
\(839\) 17119.0 + 17119.0i 0.704428 + 0.704428i 0.965358 0.260930i \(-0.0840292\pi\)
−0.260930 + 0.965358i \(0.584029\pi\)
\(840\) 0 0
\(841\) 23487.4i 0.963032i
\(842\) 0 0
\(843\) −7814.24 7814.24i −0.319261 0.319261i
\(844\) 0 0
\(845\) 9788.46 9788.46i 0.398501 0.398501i
\(846\) 0 0
\(847\) 3526.50 3526.50i 0.143060 0.143060i
\(848\) 0 0
\(849\) −2826.70 −0.114266
\(850\) 0 0
\(851\) 47274.1 1.90427
\(852\) 0 0
\(853\) 2845.44 2845.44i 0.114216 0.114216i −0.647689 0.761905i \(-0.724264\pi\)
0.761905 + 0.647689i \(0.224264\pi\)
\(854\) 0 0
\(855\) −16139.2 + 16139.2i −0.645554 + 0.645554i
\(856\) 0 0
\(857\) 8967.32 + 8967.32i 0.357430 + 0.357430i 0.862865 0.505435i \(-0.168668\pi\)
−0.505435 + 0.862865i \(0.668668\pi\)
\(858\) 0 0
\(859\) 39576.5i 1.57198i −0.618238 0.785991i \(-0.712153\pi\)
0.618238 0.785991i \(-0.287847\pi\)
\(860\) 0 0
\(861\) 3715.55 + 3715.55i 0.147068 + 0.147068i
\(862\) 0 0
\(863\) 3357.47 0.132433 0.0662164 0.997805i \(-0.478907\pi\)
0.0662164 + 0.997805i \(0.478907\pi\)
\(864\) 0 0
\(865\) 23343.2i 0.917564i
\(866\) 0 0
\(867\) −6510.74 + 11015.8i −0.255036 + 0.431508i
\(868\) 0 0
\(869\) 20493.5i 0.799994i
\(870\) 0 0
\(871\) 10308.9 0.401037
\(872\) 0 0
\(873\) −25467.6 25467.6i −0.987339 0.987339i
\(874\) 0 0
\(875\) 12163.5i 0.469945i
\(876\) 0 0
\(877\) 8148.69 + 8148.69i 0.313754 + 0.313754i 0.846362 0.532608i \(-0.178788\pi\)
−0.532608 + 0.846362i \(0.678788\pi\)
\(878\) 0 0
\(879\) −8649.59 + 8649.59i −0.331904 + 0.331904i
\(880\) 0 0
\(881\) 27477.3 27477.3i 1.05078 1.05078i 0.0521358 0.998640i \(-0.483397\pi\)
0.998640 0.0521358i \(-0.0166029\pi\)
\(882\) 0 0
\(883\) −27805.7 −1.05972 −0.529862 0.848084i \(-0.677756\pi\)
−0.529862 + 0.848084i \(0.677756\pi\)
\(884\) 0 0
\(885\) 282.856 0.0107436
\(886\) 0 0
\(887\) −35707.9 + 35707.9i −1.35169 + 1.35169i −0.467927 + 0.883767i \(0.654999\pi\)
−0.883767 + 0.467927i \(0.845001\pi\)
\(888\) 0 0
\(889\) −12037.8 + 12037.8i −0.454146 + 0.454146i
\(890\) 0 0
\(891\) 4388.15 + 4388.15i 0.164993 + 0.164993i
\(892\) 0 0
\(893\) 60450.2i 2.26527i
\(894\) 0 0
\(895\) −9906.47 9906.47i −0.369985 0.369985i
\(896\) 0 0
\(897\) −27367.9 −1.01871
\(898\) 0 0
\(899\) 505.993i 0.0187717i
\(900\) 0 0
\(901\) −487.605 3855.61i −0.0180294 0.142563i
\(902\) 0 0
\(903\) 3406.74i 0.125547i
\(904\) 0 0
\(905\) −17924.8 −0.658386
\(906\) 0 0
\(907\) −11594.5 11594.5i −0.424464 0.424464i 0.462274 0.886737i \(-0.347034\pi\)
−0.886737 + 0.462274i \(0.847034\pi\)
\(908\) 0 0
\(909\) 13377.0i 0.488103i
\(910\) 0 0
\(911\) −11849.5 11849.5i −0.430945 0.430945i 0.458005 0.888950i \(-0.348564\pi\)
−0.888950 + 0.458005i \(0.848564\pi\)
\(912\) 0 0
\(913\) −14945.2 + 14945.2i −0.541746 + 0.541746i
\(914\) 0 0
\(915\) −2299.60 + 2299.60i −0.0830845 + 0.0830845i
\(916\) 0 0
\(917\) 13573.0 0.488791
\(918\) 0 0
\(919\) 34873.5 1.25176 0.625882 0.779918i \(-0.284739\pi\)
0.625882 + 0.779918i \(0.284739\pi\)
\(920\) 0 0
\(921\) 13394.6 13394.6i 0.479226 0.479226i
\(922\) 0 0
\(923\) 1795.47 1795.47i 0.0640288 0.0640288i
\(924\) 0 0
\(925\) 15853.1 + 15853.1i 0.563508 + 0.563508i
\(926\) 0 0
\(927\) 18818.9i 0.666767i
\(928\) 0 0
\(929\) 20106.2 + 20106.2i 0.710077 + 0.710077i 0.966551 0.256474i \(-0.0825608\pi\)
−0.256474 + 0.966551i \(0.582561\pi\)
\(930\) 0 0
\(931\) 43568.1 1.53371
\(932\) 0 0
\(933\) 2011.34i 0.0705768i
\(934\) 0 0
\(935\) −13262.8 + 1677.29i −0.463892 + 0.0586667i
\(936\) 0 0
\(937\) 14175.0i 0.494212i 0.968988 + 0.247106i \(0.0794796\pi\)
−0.968988 + 0.247106i \(0.920520\pi\)
\(938\) 0 0
\(939\) 7976.66 0.277219
\(940\) 0 0
\(941\) −33557.7 33557.7i −1.16254 1.16254i −0.983918 0.178620i \(-0.942837\pi\)
−0.178620 0.983918i \(-0.557163\pi\)
\(942\) 0 0
\(943\) 37674.0i 1.30099i
\(944\) 0 0
\(945\) −5237.50 5237.50i −0.180292 0.180292i
\(946\) 0 0
\(947\) 37237.0 37237.0i 1.27776 1.27776i 0.335842 0.941918i \(-0.390979\pi\)
0.941918 0.335842i \(-0.109021\pi\)
\(948\) 0 0
\(949\) −34003.6 + 34003.6i −1.16312 + 1.16312i
\(950\) 0 0
\(951\) −5373.24 −0.183217
\(952\) 0 0
\(953\) 15886.0 0.539978 0.269989 0.962863i \(-0.412980\pi\)
0.269989 + 0.962863i \(0.412980\pi\)
\(954\) 0 0
\(955\) −17030.4 + 17030.4i −0.577058 + 0.577058i
\(956\) 0 0
\(957\) 1521.53 1521.53i 0.0513942 0.0513942i
\(958\) 0 0
\(959\) −2743.71 2743.71i −0.0923869 0.0923869i
\(960\) 0 0
\(961\) 29507.0i 0.990468i
\(962\) 0 0
\(963\) 20094.1 + 20094.1i 0.672403 + 0.672403i
\(964\) 0 0
\(965\) −21021.3 −0.701242
\(966\) 0 0
\(967\) 22253.3i 0.740040i 0.929024 + 0.370020i \(0.120649\pi\)
−0.929024 + 0.370020i \(0.879351\pi\)
\(968\) 0 0
\(969\) −23496.4 18220.6i −0.778960 0.604055i
\(970\) 0 0
\(971\) 10677.3i 0.352886i −0.984311 0.176443i \(-0.943541\pi\)
0.984311 0.176443i \(-0.0564591\pi\)
\(972\) 0 0
\(973\) 21364.0 0.703902
\(974\) 0 0
\(975\) −9177.65 9177.65i −0.301456 0.301456i
\(976\) 0 0
\(977\) 41948.5i 1.37364i −0.726826 0.686822i \(-0.759005\pi\)
0.726826 0.686822i \(-0.240995\pi\)
\(978\) 0 0
\(979\) −30661.9 30661.9i −1.00098 1.00098i
\(980\) 0 0
\(981\) −9297.33 + 9297.33i −0.302590 + 0.302590i
\(982\) 0 0
\(983\) −30718.5 + 30718.5i −0.996713 + 0.996713i −0.999995 0.00328142i \(-0.998955\pi\)
0.00328142 + 0.999995i \(0.498955\pi\)
\(984\) 0 0
\(985\) 11553.3 0.373724
\(986\) 0 0
\(987\) 8399.47 0.270880
\(988\) 0 0
\(989\) 17271.4 17271.4i 0.555306 0.555306i
\(990\) 0 0
\(991\) −42517.1 + 42517.1i −1.36287 + 1.36287i −0.492622 + 0.870243i \(0.663961\pi\)
−0.870243 + 0.492622i \(0.836039\pi\)
\(992\) 0 0
\(993\) 2067.05 + 2067.05i 0.0660582 + 0.0660582i
\(994\) 0 0
\(995\) 11351.8i 0.361686i
\(996\) 0 0
\(997\) −42498.1 42498.1i −1.34998 1.34998i −0.885677 0.464301i \(-0.846306\pi\)
−0.464301 0.885677i \(-0.653694\pi\)
\(998\) 0 0
\(999\) −35830.0 −1.13475
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 136.4.k.a.89.3 yes 14
4.3 odd 2 272.4.o.g.225.5 14
17.8 even 8 2312.4.a.m.1.9 14
17.9 even 8 2312.4.a.m.1.6 14
17.13 even 4 inner 136.4.k.a.81.3 14
68.47 odd 4 272.4.o.g.81.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.k.a.81.3 14 17.13 even 4 inner
136.4.k.a.89.3 yes 14 1.1 even 1 trivial
272.4.o.g.81.5 14 68.47 odd 4
272.4.o.g.225.5 14 4.3 odd 2
2312.4.a.m.1.6 14 17.9 even 8
2312.4.a.m.1.9 14 17.8 even 8