Properties

Label 168.4.q.a.121.1
Level $168$
Weight $4$
Character 168.121
Analytic conductor $9.912$
Analytic rank $0$
Dimension $2$
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] = [168,4,Mod(25,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), 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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 168.121
Dual form 168.4.q.a.25.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{3} +(-3.50000 + 6.06218i) q^{5} +(-17.5000 - 6.06218i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{3} +(-3.50000 + 6.06218i) q^{5} +(-17.5000 - 6.06218i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-3.50000 - 6.06218i) q^{11} -52.0000 q^{13} -21.0000 q^{15} +(-36.0000 - 62.3538i) q^{17} +(-10.0000 + 17.3205i) q^{19} +(-10.5000 - 54.5596i) q^{21} +(24.0000 - 41.5692i) q^{23} +(38.0000 + 65.8179i) q^{25} -27.0000 q^{27} -243.000 q^{29} +(-47.5000 - 82.2724i) q^{31} +(10.5000 - 18.1865i) q^{33} +(98.0000 - 84.8705i) q^{35} +(-176.000 + 304.841i) q^{37} +(-78.0000 - 135.100i) q^{39} -296.000 q^{41} +158.000 q^{43} +(-31.5000 - 54.5596i) q^{45} +(71.0000 - 122.976i) q^{47} +(269.500 + 212.176i) q^{49} +(108.000 - 187.061i) q^{51} +(187.500 + 324.760i) q^{53} +49.0000 q^{55} -60.0000 q^{57} +(-139.500 - 241.621i) q^{59} +(-123.000 + 213.042i) q^{61} +(126.000 - 109.119i) q^{63} +(182.000 - 315.233i) q^{65} +(365.000 + 632.199i) q^{67} +144.000 q^{69} +338.000 q^{71} +(271.000 + 469.386i) q^{73} +(-114.000 + 197.454i) q^{75} +(24.5000 + 127.306i) q^{77} +(152.500 - 264.138i) q^{79} +(-40.5000 - 70.1481i) q^{81} +1123.00 q^{83} +504.000 q^{85} +(-364.500 - 631.333i) q^{87} +(213.000 - 368.927i) q^{89} +(910.000 + 315.233i) q^{91} +(142.500 - 246.817i) q^{93} +(-70.0000 - 121.244i) q^{95} -369.000 q^{97} +63.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 7 q^{5} - 35 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 7 q^{5} - 35 q^{7} - 9 q^{9} - 7 q^{11} - 104 q^{13} - 42 q^{15} - 72 q^{17} - 20 q^{19} - 21 q^{21} + 48 q^{23} + 76 q^{25} - 54 q^{27} - 486 q^{29} - 95 q^{31} + 21 q^{33} + 196 q^{35} - 352 q^{37} - 156 q^{39} - 592 q^{41} + 316 q^{43} - 63 q^{45} + 142 q^{47} + 539 q^{49} + 216 q^{51} + 375 q^{53} + 98 q^{55} - 120 q^{57} - 279 q^{59} - 246 q^{61} + 252 q^{63} + 364 q^{65} + 730 q^{67} + 288 q^{69} + 676 q^{71} + 542 q^{73} - 228 q^{75} + 49 q^{77} + 305 q^{79} - 81 q^{81} + 2246 q^{83} + 1008 q^{85} - 729 q^{87} + 426 q^{89} + 1820 q^{91} + 285 q^{93} - 140 q^{95} - 738 q^{97} + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 1.50000 + 2.59808i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −3.50000 + 6.06218i −0.313050 + 0.542218i −0.979021 0.203760i \(-0.934684\pi\)
0.665971 + 0.745977i \(0.268017\pi\)
\(6\) 0 0
\(7\) −17.5000 6.06218i −0.944911 0.327327i
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −3.50000 6.06218i −0.0959354 0.166165i 0.814063 0.580776i \(-0.197251\pi\)
−0.909999 + 0.414611i \(0.863918\pi\)
\(12\) 0 0
\(13\) −52.0000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −21.0000 −0.361478
\(16\) 0 0
\(17\) −36.0000 62.3538i −0.513605 0.889590i −0.999875 0.0157814i \(-0.994976\pi\)
0.486271 0.873808i \(-0.338357\pi\)
\(18\) 0 0
\(19\) −10.0000 + 17.3205i −0.120745 + 0.209137i −0.920062 0.391773i \(-0.871862\pi\)
0.799317 + 0.600910i \(0.205195\pi\)
\(20\) 0 0
\(21\) −10.5000 54.5596i −0.109109 0.566947i
\(22\) 0 0
\(23\) 24.0000 41.5692i 0.217580 0.376860i −0.736487 0.676451i \(-0.763517\pi\)
0.954068 + 0.299591i \(0.0968503\pi\)
\(24\) 0 0
\(25\) 38.0000 + 65.8179i 0.304000 + 0.526543i
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −243.000 −1.55600 −0.777999 0.628265i \(-0.783765\pi\)
−0.777999 + 0.628265i \(0.783765\pi\)
\(30\) 0 0
\(31\) −47.5000 82.2724i −0.275202 0.476663i 0.694984 0.719025i \(-0.255411\pi\)
−0.970186 + 0.242362i \(0.922078\pi\)
\(32\) 0 0
\(33\) 10.5000 18.1865i 0.0553883 0.0959354i
\(34\) 0 0
\(35\) 98.0000 84.8705i 0.473286 0.409878i
\(36\) 0 0
\(37\) −176.000 + 304.841i −0.782006 + 1.35447i 0.148765 + 0.988873i \(0.452470\pi\)
−0.930771 + 0.365602i \(0.880863\pi\)
\(38\) 0 0
\(39\) −78.0000 135.100i −0.320256 0.554700i
\(40\) 0 0
\(41\) −296.000 −1.12750 −0.563749 0.825946i \(-0.690642\pi\)
−0.563749 + 0.825946i \(0.690642\pi\)
\(42\) 0 0
\(43\) 158.000 0.560344 0.280172 0.959950i \(-0.409609\pi\)
0.280172 + 0.959950i \(0.409609\pi\)
\(44\) 0 0
\(45\) −31.5000 54.5596i −0.104350 0.180739i
\(46\) 0 0
\(47\) 71.0000 122.976i 0.220349 0.381656i −0.734565 0.678539i \(-0.762614\pi\)
0.954914 + 0.296882i \(0.0959470\pi\)
\(48\) 0 0
\(49\) 269.500 + 212.176i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 108.000 187.061i 0.296530 0.513605i
\(52\) 0 0
\(53\) 187.500 + 324.760i 0.485945 + 0.841682i 0.999870 0.0161535i \(-0.00514205\pi\)
−0.513924 + 0.857836i \(0.671809\pi\)
\(54\) 0 0
\(55\) 49.0000 0.120130
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) −139.500 241.621i −0.307820 0.533159i 0.670066 0.742302i \(-0.266266\pi\)
−0.977885 + 0.209143i \(0.932933\pi\)
\(60\) 0 0
\(61\) −123.000 + 213.042i −0.258173 + 0.447168i −0.965752 0.259465i \(-0.916454\pi\)
0.707580 + 0.706634i \(0.249787\pi\)
\(62\) 0 0
\(63\) 126.000 109.119i 0.251976 0.218218i
\(64\) 0 0
\(65\) 182.000 315.233i 0.347297 0.601536i
\(66\) 0 0
\(67\) 365.000 + 632.199i 0.665550 + 1.15277i 0.979136 + 0.203207i \(0.0651363\pi\)
−0.313586 + 0.949560i \(0.601530\pi\)
\(68\) 0 0
\(69\) 144.000 0.251240
\(70\) 0 0
\(71\) 338.000 0.564975 0.282487 0.959271i \(-0.408840\pi\)
0.282487 + 0.959271i \(0.408840\pi\)
\(72\) 0 0
\(73\) 271.000 + 469.386i 0.434495 + 0.752568i 0.997254 0.0740532i \(-0.0235935\pi\)
−0.562759 + 0.826621i \(0.690260\pi\)
\(74\) 0 0
\(75\) −114.000 + 197.454i −0.175514 + 0.304000i
\(76\) 0 0
\(77\) 24.5000 + 127.306i 0.0362602 + 0.188413i
\(78\) 0 0
\(79\) 152.500 264.138i 0.217185 0.376175i −0.736761 0.676153i \(-0.763646\pi\)
0.953946 + 0.299978i \(0.0969793\pi\)
\(80\) 0 0
\(81\) −40.5000 70.1481i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 1123.00 1.48512 0.742562 0.669778i \(-0.233611\pi\)
0.742562 + 0.669778i \(0.233611\pi\)
\(84\) 0 0
\(85\) 504.000 0.643135
\(86\) 0 0
\(87\) −364.500 631.333i −0.449178 0.777999i
\(88\) 0 0
\(89\) 213.000 368.927i 0.253685 0.439395i −0.710853 0.703341i \(-0.751691\pi\)
0.964537 + 0.263946i \(0.0850240\pi\)
\(90\) 0 0
\(91\) 910.000 + 315.233i 1.04828 + 0.363137i
\(92\) 0 0
\(93\) 142.500 246.817i 0.158888 0.275202i
\(94\) 0 0
\(95\) −70.0000 121.244i −0.0755984 0.130940i
\(96\) 0 0
\(97\) −369.000 −0.386250 −0.193125 0.981174i \(-0.561862\pi\)
−0.193125 + 0.981174i \(0.561862\pi\)
\(98\) 0 0
\(99\) 63.0000 0.0639570
\(100\) 0 0
\(101\) −635.000 1099.85i −0.625593 1.08356i −0.988426 0.151704i \(-0.951524\pi\)
0.362833 0.931854i \(-0.381809\pi\)
\(102\) 0 0
\(103\) −916.000 + 1586.56i −0.876273 + 1.51775i −0.0208734 + 0.999782i \(0.506645\pi\)
−0.855400 + 0.517968i \(0.826689\pi\)
\(104\) 0 0
\(105\) 367.500 + 127.306i 0.341565 + 0.118322i
\(106\) 0 0
\(107\) 499.500 865.159i 0.451294 0.781665i −0.547172 0.837020i \(-0.684296\pi\)
0.998467 + 0.0553553i \(0.0176291\pi\)
\(108\) 0 0
\(109\) 833.000 + 1442.80i 0.731990 + 1.26784i 0.956031 + 0.293264i \(0.0947417\pi\)
−0.224041 + 0.974580i \(0.571925\pi\)
\(110\) 0 0
\(111\) −1056.00 −0.902983
\(112\) 0 0
\(113\) −1832.00 −1.52513 −0.762567 0.646910i \(-0.776061\pi\)
−0.762567 + 0.646910i \(0.776061\pi\)
\(114\) 0 0
\(115\) 168.000 + 290.985i 0.136227 + 0.235952i
\(116\) 0 0
\(117\) 234.000 405.300i 0.184900 0.320256i
\(118\) 0 0
\(119\) 252.000 + 1309.43i 0.194124 + 1.00870i
\(120\) 0 0
\(121\) 641.000 1110.24i 0.481593 0.834143i
\(122\) 0 0
\(123\) −444.000 769.031i −0.325481 0.563749i
\(124\) 0 0
\(125\) −1407.00 −1.00677
\(126\) 0 0
\(127\) −1931.00 −1.34920 −0.674601 0.738183i \(-0.735684\pi\)
−0.674601 + 0.738183i \(0.735684\pi\)
\(128\) 0 0
\(129\) 237.000 + 410.496i 0.161757 + 0.280172i
\(130\) 0 0
\(131\) −489.500 + 847.839i −0.326472 + 0.565466i −0.981809 0.189871i \(-0.939193\pi\)
0.655337 + 0.755336i \(0.272527\pi\)
\(132\) 0 0
\(133\) 280.000 242.487i 0.182549 0.158092i
\(134\) 0 0
\(135\) 94.5000 163.679i 0.0602464 0.104350i
\(136\) 0 0
\(137\) 243.000 + 420.888i 0.151539 + 0.262474i 0.931794 0.362989i \(-0.118244\pi\)
−0.780254 + 0.625463i \(0.784910\pi\)
\(138\) 0 0
\(139\) −2630.00 −1.60485 −0.802423 0.596755i \(-0.796456\pi\)
−0.802423 + 0.596755i \(0.796456\pi\)
\(140\) 0 0
\(141\) 426.000 0.254437
\(142\) 0 0
\(143\) 182.000 + 315.233i 0.106431 + 0.184344i
\(144\) 0 0
\(145\) 850.500 1473.11i 0.487105 0.843690i
\(146\) 0 0
\(147\) −147.000 + 1018.45i −0.0824786 + 0.571429i
\(148\) 0 0
\(149\) 939.000 1626.40i 0.516281 0.894225i −0.483540 0.875322i \(-0.660649\pi\)
0.999821 0.0189029i \(-0.00601734\pi\)
\(150\) 0 0
\(151\) −1051.50 1821.25i −0.566688 0.981532i −0.996890 0.0787997i \(-0.974891\pi\)
0.430203 0.902732i \(-0.358442\pi\)
\(152\) 0 0
\(153\) 648.000 0.342403
\(154\) 0 0
\(155\) 665.000 0.344607
\(156\) 0 0
\(157\) −702.000 1215.90i −0.356852 0.618085i 0.630581 0.776123i \(-0.282816\pi\)
−0.987433 + 0.158038i \(0.949483\pi\)
\(158\) 0 0
\(159\) −562.500 + 974.279i −0.280561 + 0.485945i
\(160\) 0 0
\(161\) −672.000 + 581.969i −0.328950 + 0.284879i
\(162\) 0 0
\(163\) 768.000 1330.22i 0.369045 0.639205i −0.620371 0.784308i \(-0.713018\pi\)
0.989417 + 0.145103i \(0.0463513\pi\)
\(164\) 0 0
\(165\) 73.5000 + 127.306i 0.0346786 + 0.0600651i
\(166\) 0 0
\(167\) −3634.00 −1.68388 −0.841938 0.539574i \(-0.818585\pi\)
−0.841938 + 0.539574i \(0.818585\pi\)
\(168\) 0 0
\(169\) 507.000 0.230769
\(170\) 0 0
\(171\) −90.0000 155.885i −0.0402484 0.0697122i
\(172\) 0 0
\(173\) −429.000 + 743.050i −0.188533 + 0.326549i −0.944761 0.327759i \(-0.893707\pi\)
0.756228 + 0.654308i \(0.227040\pi\)
\(174\) 0 0
\(175\) −266.000 1382.18i −0.114901 0.597044i
\(176\) 0 0
\(177\) 418.500 724.863i 0.177720 0.307820i
\(178\) 0 0
\(179\) 1210.00 + 2095.78i 0.505249 + 0.875118i 0.999982 + 0.00607215i \(0.00193284\pi\)
−0.494732 + 0.869046i \(0.664734\pi\)
\(180\) 0 0
\(181\) −2672.00 −1.09728 −0.548641 0.836058i \(-0.684855\pi\)
−0.548641 + 0.836058i \(0.684855\pi\)
\(182\) 0 0
\(183\) −738.000 −0.298112
\(184\) 0 0
\(185\) −1232.00 2133.89i −0.489613 0.848035i
\(186\) 0 0
\(187\) −252.000 + 436.477i −0.0985458 + 0.170686i
\(188\) 0 0
\(189\) 472.500 + 163.679i 0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 856.000 1482.64i 0.324283 0.561674i −0.657084 0.753817i \(-0.728210\pi\)
0.981367 + 0.192143i \(0.0615438\pi\)
\(192\) 0 0
\(193\) 1697.50 + 2940.16i 0.633102 + 1.09657i 0.986914 + 0.161248i \(0.0515521\pi\)
−0.353812 + 0.935317i \(0.615115\pi\)
\(194\) 0 0
\(195\) 1092.00 0.401024
\(196\) 0 0
\(197\) 510.000 0.184447 0.0922233 0.995738i \(-0.470603\pi\)
0.0922233 + 0.995738i \(0.470603\pi\)
\(198\) 0 0
\(199\) −138.000 239.023i −0.0491586 0.0851452i 0.840399 0.541968i \(-0.182321\pi\)
−0.889558 + 0.456823i \(0.848987\pi\)
\(200\) 0 0
\(201\) −1095.00 + 1896.60i −0.384255 + 0.665550i
\(202\) 0 0
\(203\) 4252.50 + 1473.11i 1.47028 + 0.509320i
\(204\) 0 0
\(205\) 1036.00 1794.40i 0.352963 0.611350i
\(206\) 0 0
\(207\) 216.000 + 374.123i 0.0725268 + 0.125620i
\(208\) 0 0
\(209\) 140.000 0.0463349
\(210\) 0 0
\(211\) 3198.00 1.04341 0.521705 0.853126i \(-0.325296\pi\)
0.521705 + 0.853126i \(0.325296\pi\)
\(212\) 0 0
\(213\) 507.000 + 878.150i 0.163094 + 0.282487i
\(214\) 0 0
\(215\) −553.000 + 957.824i −0.175415 + 0.303828i
\(216\) 0 0
\(217\) 332.500 + 1727.72i 0.104016 + 0.540485i
\(218\) 0 0
\(219\) −813.000 + 1408.16i −0.250856 + 0.434495i
\(220\) 0 0
\(221\) 1872.00 + 3242.40i 0.569793 + 0.986911i
\(222\) 0 0
\(223\) −5091.00 −1.52878 −0.764391 0.644752i \(-0.776960\pi\)
−0.764391 + 0.644752i \(0.776960\pi\)
\(224\) 0 0
\(225\) −684.000 −0.202667
\(226\) 0 0
\(227\) 1947.50 + 3373.17i 0.569428 + 0.986278i 0.996623 + 0.0821183i \(0.0261685\pi\)
−0.427195 + 0.904160i \(0.640498\pi\)
\(228\) 0 0
\(229\) 3214.00 5566.81i 0.927455 1.60640i 0.139889 0.990167i \(-0.455325\pi\)
0.787565 0.616231i \(-0.211341\pi\)
\(230\) 0 0
\(231\) −294.000 + 254.611i −0.0837393 + 0.0725204i
\(232\) 0 0
\(233\) 1912.00 3311.68i 0.537593 0.931139i −0.461439 0.887172i \(-0.652667\pi\)
0.999033 0.0439676i \(-0.0139998\pi\)
\(234\) 0 0
\(235\) 497.000 + 860.829i 0.137960 + 0.238955i
\(236\) 0 0
\(237\) 915.000 0.250783
\(238\) 0 0
\(239\) 568.000 0.153727 0.0768637 0.997042i \(-0.475509\pi\)
0.0768637 + 0.997042i \(0.475509\pi\)
\(240\) 0 0
\(241\) −1767.50 3061.40i −0.472426 0.818266i 0.527076 0.849818i \(-0.323288\pi\)
−0.999502 + 0.0315522i \(0.989955\pi\)
\(242\) 0 0
\(243\) 121.500 210.444i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) −2229.50 + 891.140i −0.581378 + 0.232379i
\(246\) 0 0
\(247\) 520.000 900.666i 0.133955 0.232016i
\(248\) 0 0
\(249\) 1684.50 + 2917.64i 0.428718 + 0.742562i
\(250\) 0 0
\(251\) 4335.00 1.09013 0.545065 0.838394i \(-0.316505\pi\)
0.545065 + 0.838394i \(0.316505\pi\)
\(252\) 0 0
\(253\) −336.000 −0.0834946
\(254\) 0 0
\(255\) 756.000 + 1309.43i 0.185657 + 0.321568i
\(256\) 0 0
\(257\) −1787.00 + 3095.17i −0.433735 + 0.751252i −0.997191 0.0748943i \(-0.976138\pi\)
0.563456 + 0.826146i \(0.309471\pi\)
\(258\) 0 0
\(259\) 4928.00 4267.77i 1.18228 1.02389i
\(260\) 0 0
\(261\) 1093.50 1894.00i 0.259333 0.449178i
\(262\) 0 0
\(263\) −2733.00 4733.69i −0.640776 1.10986i −0.985260 0.171064i \(-0.945280\pi\)
0.344484 0.938792i \(-0.388054\pi\)
\(264\) 0 0
\(265\) −2625.00 −0.608500
\(266\) 0 0
\(267\) 1278.00 0.292930
\(268\) 0 0
\(269\) 2907.50 + 5035.94i 0.659009 + 1.14144i 0.980873 + 0.194651i \(0.0623574\pi\)
−0.321864 + 0.946786i \(0.604309\pi\)
\(270\) 0 0
\(271\) 1618.50 2803.32i 0.362793 0.628376i −0.625626 0.780123i \(-0.715157\pi\)
0.988419 + 0.151747i \(0.0484899\pi\)
\(272\) 0 0
\(273\) 546.000 + 2837.10i 0.121046 + 0.628971i
\(274\) 0 0
\(275\) 266.000 460.726i 0.0583287 0.101028i
\(276\) 0 0
\(277\) 1588.00 + 2750.50i 0.344454 + 0.596611i 0.985254 0.171096i \(-0.0547308\pi\)
−0.640801 + 0.767707i \(0.721397\pi\)
\(278\) 0 0
\(279\) 855.000 0.183468
\(280\) 0 0
\(281\) −3282.00 −0.696753 −0.348377 0.937355i \(-0.613267\pi\)
−0.348377 + 0.937355i \(0.613267\pi\)
\(282\) 0 0
\(283\) −1091.00 1889.67i −0.229163 0.396923i 0.728397 0.685155i \(-0.240266\pi\)
−0.957560 + 0.288233i \(0.906932\pi\)
\(284\) 0 0
\(285\) 210.000 363.731i 0.0436468 0.0755984i
\(286\) 0 0
\(287\) 5180.00 + 1794.40i 1.06539 + 0.369060i
\(288\) 0 0
\(289\) −135.500 + 234.693i −0.0275799 + 0.0477698i
\(290\) 0 0
\(291\) −553.500 958.690i −0.111501 0.193125i
\(292\) 0 0
\(293\) 3021.00 0.602351 0.301175 0.953569i \(-0.402621\pi\)
0.301175 + 0.953569i \(0.402621\pi\)
\(294\) 0 0
\(295\) 1953.00 0.385451
\(296\) 0 0
\(297\) 94.5000 + 163.679i 0.0184628 + 0.0319785i
\(298\) 0 0
\(299\) −1248.00 + 2161.60i −0.241384 + 0.418089i
\(300\) 0 0
\(301\) −2765.00 957.824i −0.529475 0.183415i
\(302\) 0 0
\(303\) 1905.00 3299.56i 0.361186 0.625593i
\(304\) 0 0
\(305\) −861.000 1491.30i −0.161642 0.279972i
\(306\) 0 0
\(307\) −1304.00 −0.242421 −0.121210 0.992627i \(-0.538678\pi\)
−0.121210 + 0.992627i \(0.538678\pi\)
\(308\) 0 0
\(309\) −5496.00 −1.01183
\(310\) 0 0
\(311\) −3104.00 5376.29i −0.565954 0.980261i −0.996960 0.0779121i \(-0.975175\pi\)
0.431006 0.902349i \(-0.358159\pi\)
\(312\) 0 0
\(313\) −1668.50 + 2889.93i −0.301307 + 0.521880i −0.976432 0.215824i \(-0.930756\pi\)
0.675125 + 0.737703i \(0.264090\pi\)
\(314\) 0 0
\(315\) 220.500 + 1145.75i 0.0394405 + 0.204939i
\(316\) 0 0
\(317\) −4318.50 + 7479.86i −0.765146 + 1.32527i 0.175024 + 0.984564i \(0.444000\pi\)
−0.940170 + 0.340707i \(0.889334\pi\)
\(318\) 0 0
\(319\) 850.500 + 1473.11i 0.149275 + 0.258553i
\(320\) 0 0
\(321\) 2997.00 0.521110
\(322\) 0 0
\(323\) 1440.00 0.248061
\(324\) 0 0
\(325\) −1976.00 3422.53i −0.337258 0.584148i
\(326\) 0 0
\(327\) −2499.00 + 4328.39i −0.422615 + 0.731990i
\(328\) 0 0
\(329\) −1988.00 + 1721.66i −0.333137 + 0.288505i
\(330\) 0 0
\(331\) −970.000 + 1680.09i −0.161076 + 0.278991i −0.935255 0.353975i \(-0.884830\pi\)
0.774179 + 0.632967i \(0.218163\pi\)
\(332\) 0 0
\(333\) −1584.00 2743.57i −0.260669 0.451491i
\(334\) 0 0
\(335\) −5110.00 −0.833400
\(336\) 0 0
\(337\) −5527.00 −0.893397 −0.446699 0.894684i \(-0.647400\pi\)
−0.446699 + 0.894684i \(0.647400\pi\)
\(338\) 0 0
\(339\) −2748.00 4759.68i −0.440268 0.762567i
\(340\) 0 0
\(341\) −332.500 + 575.907i −0.0528032 + 0.0914578i
\(342\) 0 0
\(343\) −3430.00 5346.84i −0.539949 0.841698i
\(344\) 0 0
\(345\) −504.000 + 872.954i −0.0786506 + 0.136227i
\(346\) 0 0
\(347\) −4782.00 8282.67i −0.739802 1.28137i −0.952584 0.304275i \(-0.901586\pi\)
0.212783 0.977100i \(-0.431747\pi\)
\(348\) 0 0
\(349\) −918.000 −0.140801 −0.0704003 0.997519i \(-0.522428\pi\)
−0.0704003 + 0.997519i \(0.522428\pi\)
\(350\) 0 0
\(351\) 1404.00 0.213504
\(352\) 0 0
\(353\) −5740.00 9941.97i −0.865466 1.49903i −0.866584 0.499031i \(-0.833689\pi\)
0.00111843 0.999999i \(-0.499644\pi\)
\(354\) 0 0
\(355\) −1183.00 + 2049.02i −0.176865 + 0.306339i
\(356\) 0 0
\(357\) −3024.00 + 2618.86i −0.448311 + 0.388249i
\(358\) 0 0
\(359\) −1827.00 + 3164.46i −0.268594 + 0.465219i −0.968499 0.249017i \(-0.919892\pi\)
0.699905 + 0.714236i \(0.253226\pi\)
\(360\) 0 0
\(361\) 3229.50 + 5593.66i 0.470841 + 0.815521i
\(362\) 0 0
\(363\) 3846.00 0.556095
\(364\) 0 0
\(365\) −3794.00 −0.544074
\(366\) 0 0
\(367\) 5033.50 + 8718.28i 0.715931 + 1.24003i 0.962599 + 0.270928i \(0.0873307\pi\)
−0.246669 + 0.969100i \(0.579336\pi\)
\(368\) 0 0
\(369\) 1332.00 2307.09i 0.187916 0.325481i
\(370\) 0 0
\(371\) −1312.50 6819.95i −0.183670 0.954378i
\(372\) 0 0
\(373\) −4400.00 + 7621.02i −0.610786 + 1.05791i 0.380322 + 0.924854i \(0.375813\pi\)
−0.991108 + 0.133059i \(0.957520\pi\)
\(374\) 0 0
\(375\) −2110.50 3655.49i −0.290629 0.503384i
\(376\) 0 0
\(377\) 12636.0 1.72623
\(378\) 0 0
\(379\) 1136.00 0.153964 0.0769821 0.997032i \(-0.475472\pi\)
0.0769821 + 0.997032i \(0.475472\pi\)
\(380\) 0 0
\(381\) −2896.50 5016.89i −0.389481 0.674601i
\(382\) 0 0
\(383\) −2145.00 + 3715.25i −0.286173 + 0.495667i −0.972893 0.231256i \(-0.925717\pi\)
0.686720 + 0.726922i \(0.259050\pi\)
\(384\) 0 0
\(385\) −857.500 297.047i −0.113512 0.0393218i
\(386\) 0 0
\(387\) −711.000 + 1231.49i −0.0933906 + 0.161757i
\(388\) 0 0
\(389\) 431.000 + 746.514i 0.0561763 + 0.0973001i 0.892746 0.450560i \(-0.148776\pi\)
−0.836570 + 0.547861i \(0.815442\pi\)
\(390\) 0 0
\(391\) −3456.00 −0.447001
\(392\) 0 0
\(393\) −2937.00 −0.376977
\(394\) 0 0
\(395\) 1067.50 + 1848.96i 0.135979 + 0.235523i
\(396\) 0 0
\(397\) 4570.00 7915.47i 0.577737 1.00067i −0.418001 0.908447i \(-0.637269\pi\)
0.995738 0.0922238i \(-0.0293975\pi\)
\(398\) 0 0
\(399\) 1050.00 + 363.731i 0.131744 + 0.0456374i
\(400\) 0 0
\(401\) 7116.00 12325.3i 0.886175 1.53490i 0.0418135 0.999125i \(-0.486686\pi\)
0.844361 0.535774i \(-0.179980\pi\)
\(402\) 0 0
\(403\) 2470.00 + 4278.17i 0.305309 + 0.528810i
\(404\) 0 0
\(405\) 567.000 0.0695666
\(406\) 0 0
\(407\) 2464.00 0.300088
\(408\) 0 0
\(409\) 5000.50 + 8661.12i 0.604545 + 1.04710i 0.992123 + 0.125266i \(0.0399783\pi\)
−0.387578 + 0.921837i \(0.626688\pi\)
\(410\) 0 0
\(411\) −729.000 + 1262.67i −0.0874913 + 0.151539i
\(412\) 0 0
\(413\) 976.500 + 5074.04i 0.116345 + 0.604546i
\(414\) 0 0
\(415\) −3930.50 + 6807.83i −0.464917 + 0.805260i
\(416\) 0 0
\(417\) −3945.00 6832.94i −0.463279 0.802423i
\(418\) 0 0
\(419\) −10256.0 −1.19580 −0.597898 0.801572i \(-0.703997\pi\)
−0.597898 + 0.801572i \(0.703997\pi\)
\(420\) 0 0
\(421\) −4502.00 −0.521174 −0.260587 0.965450i \(-0.583916\pi\)
−0.260587 + 0.965450i \(0.583916\pi\)
\(422\) 0 0
\(423\) 639.000 + 1106.78i 0.0734497 + 0.127219i
\(424\) 0 0
\(425\) 2736.00 4738.89i 0.312272 0.540871i
\(426\) 0 0
\(427\) 3444.00 2982.59i 0.390320 0.338027i
\(428\) 0 0
\(429\) −546.000 + 945.700i −0.0614479 + 0.106431i
\(430\) 0 0
\(431\) −6178.00 10700.6i −0.690450 1.19589i −0.971691 0.236257i \(-0.924079\pi\)
0.281241 0.959637i \(-0.409254\pi\)
\(432\) 0 0
\(433\) 862.000 0.0956699 0.0478350 0.998855i \(-0.484768\pi\)
0.0478350 + 0.998855i \(0.484768\pi\)
\(434\) 0 0
\(435\) 5103.00 0.562460
\(436\) 0 0
\(437\) 480.000 + 831.384i 0.0525435 + 0.0910080i
\(438\) 0 0
\(439\) 682.500 1182.12i 0.0742003 0.128519i −0.826538 0.562881i \(-0.809693\pi\)
0.900738 + 0.434362i \(0.143026\pi\)
\(440\) 0 0
\(441\) −2866.50 + 1145.75i −0.309524 + 0.123718i
\(442\) 0 0
\(443\) −4465.50 + 7734.47i −0.478922 + 0.829517i −0.999708 0.0241705i \(-0.992306\pi\)
0.520786 + 0.853687i \(0.325639\pi\)
\(444\) 0 0
\(445\) 1491.00 + 2582.49i 0.158832 + 0.275105i
\(446\) 0 0
\(447\) 5634.00 0.596150
\(448\) 0 0
\(449\) 11228.0 1.18014 0.590069 0.807353i \(-0.299100\pi\)
0.590069 + 0.807353i \(0.299100\pi\)
\(450\) 0 0
\(451\) 1036.00 + 1794.40i 0.108167 + 0.187351i
\(452\) 0 0
\(453\) 3154.50 5463.75i 0.327177 0.566688i
\(454\) 0 0
\(455\) −5096.00 + 4413.27i −0.525064 + 0.454719i
\(456\) 0 0
\(457\) −6075.50 + 10523.1i −0.621882 + 1.07713i 0.367253 + 0.930121i \(0.380298\pi\)
−0.989135 + 0.147010i \(0.953035\pi\)
\(458\) 0 0
\(459\) 972.000 + 1683.55i 0.0988433 + 0.171202i
\(460\) 0 0
\(461\) 18534.0 1.87248 0.936241 0.351358i \(-0.114280\pi\)
0.936241 + 0.351358i \(0.114280\pi\)
\(462\) 0 0
\(463\) 17096.0 1.71602 0.858011 0.513631i \(-0.171700\pi\)
0.858011 + 0.513631i \(0.171700\pi\)
\(464\) 0 0
\(465\) 997.500 + 1727.72i 0.0994795 + 0.172304i
\(466\) 0 0
\(467\) 1070.00 1853.29i 0.106025 0.183641i −0.808131 0.589002i \(-0.799521\pi\)
0.914157 + 0.405361i \(0.132854\pi\)
\(468\) 0 0
\(469\) −2555.00 13276.2i −0.251554 1.30711i
\(470\) 0 0
\(471\) 2106.00 3647.70i 0.206028 0.356852i
\(472\) 0 0
\(473\) −553.000 957.824i −0.0537568 0.0931095i
\(474\) 0 0
\(475\) −1520.00 −0.146826
\(476\) 0 0
\(477\) −3375.00 −0.323964
\(478\) 0 0
\(479\) 6633.00 + 11488.7i 0.632713 + 1.09589i 0.986995 + 0.160752i \(0.0513920\pi\)
−0.354282 + 0.935139i \(0.615275\pi\)
\(480\) 0 0
\(481\) 9152.00 15851.7i 0.867558 1.50265i
\(482\) 0 0
\(483\) −2520.00 872.954i −0.237400 0.0822376i
\(484\) 0 0
\(485\) 1291.50 2236.94i 0.120915 0.209432i
\(486\) 0 0
\(487\) 9591.50 + 16613.0i 0.892469 + 1.54580i 0.836906 + 0.547346i \(0.184362\pi\)
0.0555627 + 0.998455i \(0.482305\pi\)
\(488\) 0 0
\(489\) 4608.00 0.426137
\(490\) 0 0
\(491\) −21693.0 −1.99387 −0.996936 0.0782185i \(-0.975077\pi\)
−0.996936 + 0.0782185i \(0.975077\pi\)
\(492\) 0 0
\(493\) 8748.00 + 15152.0i 0.799169 + 1.38420i
\(494\) 0 0
\(495\) −220.500 + 381.917i −0.0200217 + 0.0346786i
\(496\) 0 0
\(497\) −5915.00 2049.02i −0.533851 0.184931i
\(498\) 0 0
\(499\) −9829.00 + 17024.3i −0.881776 + 1.52728i −0.0324122 + 0.999475i \(0.510319\pi\)
−0.849364 + 0.527807i \(0.823014\pi\)
\(500\) 0 0
\(501\) −5451.00 9441.41i −0.486093 0.841938i
\(502\) 0 0
\(503\) 19436.0 1.72288 0.861440 0.507860i \(-0.169563\pi\)
0.861440 + 0.507860i \(0.169563\pi\)
\(504\) 0 0
\(505\) 8890.00 0.783366
\(506\) 0 0
\(507\) 760.500 + 1317.22i 0.0666173 + 0.115385i
\(508\) 0 0
\(509\) −1544.50 + 2675.15i −0.134497 + 0.232955i −0.925405 0.378980i \(-0.876275\pi\)
0.790908 + 0.611934i \(0.209608\pi\)
\(510\) 0 0
\(511\) −1897.00 9857.10i −0.164224 0.853332i
\(512\) 0 0
\(513\) 270.000 467.654i 0.0232374 0.0402484i
\(514\) 0 0
\(515\) −6412.00 11105.9i −0.548634 0.950262i
\(516\) 0 0
\(517\) −994.000 −0.0845572
\(518\) 0 0
\(519\) −2574.00 −0.217700
\(520\) 0 0
\(521\) −975.000 1688.75i −0.0819876 0.142007i 0.822116 0.569320i \(-0.192793\pi\)
−0.904104 + 0.427313i \(0.859460\pi\)
\(522\) 0 0
\(523\) −6566.00 + 11372.6i −0.548970 + 0.950843i 0.449376 + 0.893343i \(0.351646\pi\)
−0.998345 + 0.0575005i \(0.981687\pi\)
\(524\) 0 0
\(525\) 3192.00 2764.35i 0.265353 0.229802i
\(526\) 0 0
\(527\) −3420.00 + 5923.61i −0.282690 + 0.489633i
\(528\) 0 0
\(529\) 4931.50 + 8541.61i 0.405318 + 0.702031i
\(530\) 0 0
\(531\) 2511.00 0.205213
\(532\) 0 0
\(533\) 15392.0 1.25085
\(534\) 0 0
\(535\) 3496.50 + 6056.12i 0.282555 + 0.489399i
\(536\) 0 0
\(537\) −3630.00 + 6287.34i −0.291706 + 0.505249i
\(538\) 0 0
\(539\) 343.000 2376.37i 0.0274101 0.189903i
\(540\) 0 0
\(541\) −1465.00 + 2537.45i −0.116424 + 0.201652i −0.918348 0.395774i \(-0.870476\pi\)
0.801924 + 0.597426i \(0.203810\pi\)
\(542\) 0 0
\(543\) −4008.00 6942.06i −0.316758 0.548641i
\(544\) 0 0
\(545\) −11662.0 −0.916597
\(546\) 0 0
\(547\) 19824.0 1.54957 0.774783 0.632227i \(-0.217859\pi\)
0.774783 + 0.632227i \(0.217859\pi\)
\(548\) 0 0
\(549\) −1107.00 1917.38i −0.0860576 0.149056i
\(550\) 0 0
\(551\) 2430.00 4208.88i 0.187879 0.325416i
\(552\) 0 0
\(553\) −4270.00 + 3697.93i −0.328352 + 0.284362i
\(554\) 0 0
\(555\) 3696.00 6401.66i 0.282678 0.489613i
\(556\) 0 0
\(557\) 9584.50 + 16600.8i 0.729099 + 1.26284i 0.957264 + 0.289215i \(0.0933941\pi\)
−0.228165 + 0.973622i \(0.573273\pi\)
\(558\) 0 0
\(559\) −8216.00 −0.621645
\(560\) 0 0
\(561\) −1512.00 −0.113791
\(562\) 0 0
\(563\) −3633.50 6293.41i −0.271996 0.471111i 0.697377 0.716705i \(-0.254350\pi\)
−0.969373 + 0.245594i \(0.921017\pi\)
\(564\) 0 0
\(565\) 6412.00 11105.9i 0.477442 0.826954i
\(566\) 0 0
\(567\) 283.500 + 1473.11i 0.0209980 + 0.109109i
\(568\) 0 0
\(569\) 2446.00 4236.60i 0.180214 0.312139i −0.761739 0.647883i \(-0.775654\pi\)
0.941953 + 0.335744i \(0.108988\pi\)
\(570\) 0 0
\(571\) −11063.0 19161.7i −0.810809 1.40436i −0.912298 0.409526i \(-0.865694\pi\)
0.101489 0.994837i \(-0.467639\pi\)
\(572\) 0 0
\(573\) 5136.00 0.374449
\(574\) 0 0
\(575\) 3648.00 0.264578
\(576\) 0 0
\(577\) 548.500 + 950.030i 0.0395743 + 0.0685446i 0.885134 0.465336i \(-0.154066\pi\)
−0.845560 + 0.533881i \(0.820733\pi\)
\(578\) 0 0
\(579\) −5092.50 + 8820.47i −0.365522 + 0.633102i
\(580\) 0 0
\(581\) −19652.5 6807.83i −1.40331 0.486121i
\(582\) 0 0
\(583\) 1312.50 2273.32i 0.0932388 0.161494i
\(584\) 0 0
\(585\) 1638.00 + 2837.10i 0.115766 + 0.200512i
\(586\) 0 0
\(587\) 9969.00 0.700962 0.350481 0.936570i \(-0.386018\pi\)
0.350481 + 0.936570i \(0.386018\pi\)
\(588\) 0 0
\(589\) 1900.00 0.132917
\(590\) 0 0
\(591\) 765.000 + 1325.02i 0.0532452 + 0.0922233i
\(592\) 0 0
\(593\) 3212.00 5563.35i 0.222430 0.385260i −0.733115 0.680104i \(-0.761934\pi\)
0.955545 + 0.294844i \(0.0952678\pi\)
\(594\) 0 0
\(595\) −8820.00 3055.34i −0.607705 0.210515i
\(596\) 0 0
\(597\) 414.000 717.069i 0.0283817 0.0491586i
\(598\) 0 0
\(599\) −8523.00 14762.3i −0.581370 1.00696i −0.995317 0.0966609i \(-0.969184\pi\)
0.413948 0.910301i \(-0.364150\pi\)
\(600\) 0 0
\(601\) −18877.0 −1.28121 −0.640606 0.767869i \(-0.721317\pi\)
−0.640606 + 0.767869i \(0.721317\pi\)
\(602\) 0 0
\(603\) −6570.00 −0.443700
\(604\) 0 0
\(605\) 4487.00 + 7771.71i 0.301525 + 0.522256i
\(606\) 0 0
\(607\) 6891.50 11936.4i 0.460819 0.798163i −0.538183 0.842828i \(-0.680889\pi\)
0.999002 + 0.0446657i \(0.0142223\pi\)
\(608\) 0 0
\(609\) 2551.50 + 13258.0i 0.169773 + 0.882168i
\(610\) 0 0
\(611\) −3692.00 + 6394.73i −0.244456 + 0.423409i
\(612\) 0 0
\(613\) −9096.00 15754.7i −0.599321 1.03806i −0.992921 0.118773i \(-0.962104\pi\)
0.393600 0.919282i \(-0.371229\pi\)
\(614\) 0 0
\(615\) 6216.00 0.407566
\(616\) 0 0
\(617\) −13054.0 −0.851757 −0.425879 0.904780i \(-0.640035\pi\)
−0.425879 + 0.904780i \(0.640035\pi\)
\(618\) 0 0
\(619\) −3623.00 6275.22i −0.235251 0.407468i 0.724094 0.689701i \(-0.242258\pi\)
−0.959346 + 0.282234i \(0.908925\pi\)
\(620\) 0 0
\(621\) −648.000 + 1122.37i −0.0418733 + 0.0725268i
\(622\) 0 0
\(623\) −5964.00 + 5164.98i −0.383535 + 0.332151i
\(624\) 0 0
\(625\) 174.500 302.243i 0.0111680 0.0193435i
\(626\) 0 0
\(627\) 210.000 + 363.731i 0.0133757 + 0.0231675i
\(628\) 0 0
\(629\) 25344.0 1.60657
\(630\) 0 0
\(631\) 2817.00 0.177723 0.0888613 0.996044i \(-0.471677\pi\)
0.0888613 + 0.996044i \(0.471677\pi\)
\(632\) 0 0
\(633\) 4797.00 + 8308.65i 0.301206 + 0.521705i
\(634\) 0 0
\(635\) 6758.50 11706.1i 0.422367 0.731561i
\(636\) 0 0
\(637\) −14014.0 11033.2i −0.871672 0.686264i
\(638\) 0 0
\(639\) −1521.00 + 2634.45i −0.0941625 + 0.163094i
\(640\) 0 0
\(641\) −7893.00 13671.1i −0.486357 0.842395i 0.513520 0.858078i \(-0.328341\pi\)
−0.999877 + 0.0156827i \(0.995008\pi\)
\(642\) 0 0
\(643\) −17426.0 −1.06876 −0.534381 0.845244i \(-0.679455\pi\)
−0.534381 + 0.845244i \(0.679455\pi\)
\(644\) 0 0
\(645\) −3318.00 −0.202552
\(646\) 0 0
\(647\) 12917.0 + 22372.9i 0.784884 + 1.35946i 0.929069 + 0.369907i \(0.120611\pi\)
−0.144185 + 0.989551i \(0.546056\pi\)
\(648\) 0 0
\(649\) −976.500 + 1691.35i −0.0590616 + 0.102298i
\(650\) 0 0
\(651\) −3990.00 + 3455.44i −0.240216 + 0.208033i
\(652\) 0 0
\(653\) −13991.5 + 24234.0i −0.838483 + 1.45230i 0.0526789 + 0.998612i \(0.483224\pi\)
−0.891162 + 0.453684i \(0.850109\pi\)
\(654\) 0 0
\(655\) −3426.50 5934.87i −0.204404 0.354038i
\(656\) 0 0
\(657\) −4878.00 −0.289663
\(658\) 0 0
\(659\) −28296.0 −1.67262 −0.836309 0.548258i \(-0.815291\pi\)
−0.836309 + 0.548258i \(0.815291\pi\)
\(660\) 0 0
\(661\) −10627.0 18406.5i −0.625329 1.08310i −0.988477 0.151370i \(-0.951631\pi\)
0.363148 0.931731i \(-0.381702\pi\)
\(662\) 0 0
\(663\) −5616.00 + 9727.20i −0.328970 + 0.569793i
\(664\) 0 0
\(665\) 490.000 + 2546.11i 0.0285735 + 0.148472i
\(666\) 0 0
\(667\) −5832.00 + 10101.3i −0.338555 + 0.586394i
\(668\) 0 0
\(669\) −7636.50 13226.8i −0.441322 0.764391i
\(670\) 0 0
\(671\) 1722.00 0.0990716
\(672\) 0 0
\(673\) −28259.0 −1.61858 −0.809290 0.587409i \(-0.800148\pi\)
−0.809290 + 0.587409i \(0.800148\pi\)
\(674\) 0 0
\(675\) −1026.00 1777.08i −0.0585048 0.101333i
\(676\) 0 0
\(677\) −15889.5 + 27521.4i −0.902043 + 1.56238i −0.0772049 + 0.997015i \(0.524600\pi\)
−0.824838 + 0.565369i \(0.808734\pi\)
\(678\) 0 0
\(679\) 6457.50 + 2236.94i 0.364972 + 0.126430i
\(680\) 0 0
\(681\) −5842.50 + 10119.5i −0.328759 + 0.569428i
\(682\) 0 0
\(683\) 4873.50 + 8441.15i 0.273030 + 0.472901i 0.969636 0.244552i \(-0.0786409\pi\)
−0.696606 + 0.717453i \(0.745308\pi\)
\(684\) 0 0
\(685\) −3402.00 −0.189757
\(686\) 0 0
\(687\) 19284.0 1.07093
\(688\) 0 0
\(689\) −9750.00 16887.5i −0.539108 0.933762i
\(690\) 0 0
\(691\) 12256.0 21228.0i 0.674733 1.16867i −0.301814 0.953367i \(-0.597592\pi\)
0.976547 0.215304i \(-0.0690744\pi\)
\(692\) 0 0
\(693\) −1102.50 381.917i −0.0604336 0.0209348i
\(694\) 0 0
\(695\) 9205.00 15943.5i 0.502396 0.870176i
\(696\) 0 0
\(697\) 10656.0 + 18456.7i 0.579089 + 1.00301i
\(698\) 0 0
\(699\) 11472.0 0.620760
\(700\) 0 0
\(701\) 14325.0 0.771823 0.385911 0.922536i \(-0.373887\pi\)
0.385911 + 0.922536i \(0.373887\pi\)
\(702\) 0 0
\(703\) −3520.00 6096.82i −0.188847 0.327092i
\(704\) 0 0
\(705\) −1491.00 + 2582.49i −0.0796515 + 0.137960i
\(706\) 0 0
\(707\) 4445.00 + 23096.9i 0.236452 + 1.22864i
\(708\) 0 0
\(709\) −6039.00 + 10459.9i −0.319886 + 0.554059i −0.980464 0.196698i \(-0.936978\pi\)
0.660578 + 0.750758i \(0.270311\pi\)
\(710\) 0 0
\(711\) 1372.50 + 2377.24i 0.0723949 + 0.125392i
\(712\) 0 0
\(713\) −4560.00 −0.239514
\(714\) 0 0
\(715\) −2548.00 −0.133272
\(716\) 0 0
\(717\) 852.000 + 1475.71i 0.0443773 + 0.0768637i
\(718\) 0 0
\(719\) 5329.00 9230.10i 0.276409 0.478755i −0.694081 0.719897i \(-0.744189\pi\)
0.970490 + 0.241143i \(0.0775222\pi\)
\(720\) 0 0
\(721\) 25648.0 22211.8i 1.32480 1.14731i
\(722\) 0 0
\(723\) 5302.50 9184.20i 0.272755 0.472426i
\(724\) 0 0
\(725\) −9234.00 15993.8i −0.473024 0.819301i
\(726\) 0 0
\(727\) 19099.0 0.974337 0.487168 0.873308i \(-0.338030\pi\)
0.487168 + 0.873308i \(0.338030\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −5688.00 9851.90i −0.287795 0.498476i
\(732\) 0 0
\(733\) −703.000 + 1217.63i −0.0354241 + 0.0613564i −0.883194 0.469008i \(-0.844612\pi\)
0.847770 + 0.530364i \(0.177945\pi\)
\(734\) 0 0
\(735\) −5659.50 4455.70i −0.284019 0.223607i
\(736\) 0 0
\(737\) 2555.00 4425.39i 0.127700 0.221182i
\(738\) 0 0
\(739\) −3365.00 5828.35i −0.167501 0.290121i 0.770039 0.637996i \(-0.220237\pi\)
−0.937541 + 0.347875i \(0.886903\pi\)
\(740\) 0 0
\(741\) 3120.00 0.154678
\(742\) 0 0
\(743\) −1166.00 −0.0575725 −0.0287863 0.999586i \(-0.509164\pi\)
−0.0287863 + 0.999586i \(0.509164\pi\)
\(744\) 0 0
\(745\) 6573.00 + 11384.8i 0.323243 + 0.559873i
\(746\) 0 0
\(747\) −5053.50 + 8752.92i −0.247521 + 0.428718i
\(748\) 0 0
\(749\) −13986.0 + 12112.2i −0.682293 + 0.590883i
\(750\) 0 0
\(751\) 8483.50 14693.9i 0.412207 0.713963i −0.582924 0.812527i \(-0.698091\pi\)
0.995131 + 0.0985636i \(0.0314248\pi\)
\(752\) 0 0
\(753\) 6502.50 + 11262.7i 0.314694 + 0.545065i
\(754\) 0 0
\(755\) 14721.0 0.709605
\(756\) 0 0
\(757\) 11878.0 0.570295 0.285147 0.958484i \(-0.407957\pi\)
0.285147 + 0.958484i \(0.407957\pi\)
\(758\) 0 0
\(759\) −504.000 872.954i −0.0241028 0.0417473i
\(760\) 0 0
\(761\) −13662.0 + 23663.3i −0.650785 + 1.12719i 0.332148 + 0.943227i \(0.392227\pi\)
−0.982933 + 0.183965i \(0.941107\pi\)
\(762\) 0 0
\(763\) −5831.00 30298.8i −0.276666 1.43760i
\(764\) 0 0
\(765\) −2268.00 + 3928.29i −0.107189 + 0.185657i
\(766\) 0 0
\(767\) 7254.00 + 12564.3i 0.341495 + 0.591487i
\(768\) 0 0
\(769\) −11971.0 −0.561359 −0.280680 0.959802i \(-0.590560\pi\)
−0.280680 + 0.959802i \(0.590560\pi\)
\(770\) 0 0
\(771\) −10722.0 −0.500834
\(772\) 0 0
\(773\) 169.000 + 292.717i 0.00786353 + 0.0136200i 0.869930 0.493175i \(-0.164164\pi\)
−0.862067 + 0.506795i \(0.830830\pi\)
\(774\) 0 0
\(775\) 3610.00 6252.70i 0.167323 0.289811i
\(776\) 0 0
\(777\) 18480.0 + 6401.66i 0.853238 + 0.295570i
\(778\) 0 0
\(779\) 2960.00 5126.87i 0.136140 0.235801i
\(780\) 0 0
\(781\) −1183.00 2049.02i −0.0542011 0.0938791i
\(782\) 0 0
\(783\) 6561.00 0.299452
\(784\) 0 0
\(785\) 9828.00 0.446849
\(786\) 0 0
\(787\) −7057.00 12223.1i −0.319638 0.553629i 0.660775 0.750584i \(-0.270228\pi\)
−0.980412 + 0.196956i \(0.936895\pi\)
\(788\) 0 0
\(789\) 8199.00 14201.1i 0.369952 0.640776i
\(790\) 0 0
\(791\) 32060.0 + 11105.9i 1.44112 + 0.499217i
\(792\) 0 0
\(793\) 6396.00 11078.2i 0.286417 0.496089i
\(794\) 0 0
\(795\) −3937.50 6819.95i −0.175659 0.304250i
\(796\) 0 0
\(797\) 1563.00 0.0694659 0.0347329 0.999397i \(-0.488942\pi\)
0.0347329 + 0.999397i \(0.488942\pi\)
\(798\) 0 0
\(799\) −10224.0 −0.452690
\(800\) 0 0
\(801\) 1917.00 + 3320.34i 0.0845616 + 0.146465i
\(802\) 0 0
\(803\) 1897.00 3285.70i 0.0833670 0.144396i
\(804\) 0 0
\(805\) −1176.00 6110.68i −0.0514889 0.267544i
\(806\) 0 0
\(807\) −8722.50 + 15107.8i −0.380479 + 0.659009i
\(808\) 0 0
\(809\) 15784.0 + 27338.7i 0.685953 + 1.18811i 0.973136 + 0.230230i \(0.0739477\pi\)
−0.287183 + 0.957876i \(0.592719\pi\)
\(810\) 0 0
\(811\) 2626.00 0.113701 0.0568504 0.998383i \(-0.481894\pi\)
0.0568504 + 0.998383i \(0.481894\pi\)
\(812\) 0 0
\(813\) 9711.00 0.418917
\(814\) 0 0
\(815\) 5376.00 + 9311.51i 0.231059 + 0.400206i
\(816\) 0 0
\(817\) −1580.00 + 2736.64i −0.0676588 + 0.117188i
\(818\) 0 0
\(819\) −6552.00 + 5674.20i −0.279543 + 0.242091i
\(820\) 0 0
\(821\) −180.500 + 312.635i −0.00767295 + 0.0132899i −0.869836 0.493340i \(-0.835776\pi\)
0.862163 + 0.506630i \(0.169109\pi\)
\(822\) 0 0
\(823\) 14772.0 + 25585.9i 0.625662 + 1.08368i 0.988413 + 0.151792i \(0.0485043\pi\)
−0.362751 + 0.931886i \(0.618162\pi\)
\(824\) 0 0
\(825\) 1596.00 0.0673522
\(826\) 0 0
\(827\) 24163.0 1.01600 0.507999 0.861358i \(-0.330385\pi\)
0.507999 + 0.861358i \(0.330385\pi\)
\(828\) 0 0
\(829\) 19598.0 + 33944.7i 0.821070 + 1.42213i 0.904886 + 0.425653i \(0.139956\pi\)
−0.0838169 + 0.996481i \(0.526711\pi\)
\(830\) 0 0
\(831\) −4764.00 + 8251.49i −0.198870 + 0.344454i
\(832\) 0 0
\(833\) 3528.00 24442.7i 0.146744 1.01667i
\(834\) 0 0
\(835\) 12719.0 22030.0i 0.527137 0.913028i
\(836\) 0 0
\(837\) 1282.50 + 2221.36i 0.0529626 + 0.0917339i
\(838\) 0 0
\(839\) 6528.00 0.268619 0.134310 0.990939i \(-0.457118\pi\)
0.134310 + 0.990939i \(0.457118\pi\)
\(840\) 0 0
\(841\) 34660.0 1.42113
\(842\) 0 0
\(843\) −4923.00 8526.89i −0.201135 0.348377i
\(844\) 0 0
\(845\) −1774.50 + 3073.52i −0.0722422 + 0.125127i
\(846\) 0 0
\(847\) −17948.0 + 15543.4i −0.728100 + 0.630553i
\(848\) 0 0
\(849\) 3273.00 5669.00i 0.132308 0.229163i
\(850\) 0 0
\(851\) 8448.00 + 14632.4i 0.340298 + 0.589414i
\(852\) 0 0
\(853\) 1046.00 0.0419864 0.0209932 0.999780i \(-0.493317\pi\)
0.0209932 + 0.999780i \(0.493317\pi\)
\(854\) 0 0
\(855\) 1260.00 0.0503989
\(856\) 0 0
\(857\) −16455.0 28500.9i −0.655883 1.13602i −0.981672 0.190581i \(-0.938963\pi\)
0.325788 0.945443i \(-0.394370\pi\)
\(858\) 0 0
\(859\) −11341.0 + 19643.2i −0.450466 + 0.780229i −0.998415 0.0562823i \(-0.982075\pi\)
0.547949 + 0.836512i \(0.315409\pi\)
\(860\) 0 0
\(861\) 3108.00 + 16149.6i 0.123020 + 0.639231i
\(862\) 0 0
\(863\) 3725.00 6451.89i 0.146930 0.254490i −0.783161 0.621818i \(-0.786394\pi\)
0.930091 + 0.367328i \(0.119728\pi\)
\(864\) 0 0
\(865\) −3003.00 5201.35i −0.118041 0.204452i
\(866\) 0 0
\(867\) −813.000 −0.0318465
\(868\) 0 0
\(869\) −2135.00 −0.0833428
\(870\) 0 0
\(871\) −18980.0 32874.3i −0.738361 1.27888i
\(872\) 0 0
\(873\) 1660.50 2876.07i 0.0643750 0.111501i
\(874\) 0 0
\(875\) 24622.5 + 8529.48i 0.951306 + 0.329542i
\(876\) 0 0
\(877\) −5040.00 + 8729.54i −0.194058 + 0.336118i −0.946591 0.322436i \(-0.895498\pi\)
0.752533 + 0.658554i \(0.228832\pi\)
\(878\) 0 0
\(879\) 4531.50 + 7848.79i 0.173884 + 0.301175i
\(880\) 0 0
\(881\) −32958.0 −1.26037 −0.630183 0.776446i \(-0.717020\pi\)
−0.630183 + 0.776446i \(0.717020\pi\)
\(882\) 0 0
\(883\) −19784.0 −0.754003 −0.377001 0.926213i \(-0.623045\pi\)
−0.377001 + 0.926213i \(0.623045\pi\)
\(884\) 0 0
\(885\) 2929.50 + 5074.04i 0.111270 + 0.192726i
\(886\) 0 0
\(887\) 5740.00 9941.97i 0.217283 0.376346i −0.736693 0.676227i \(-0.763614\pi\)
0.953976 + 0.299881i \(0.0969471\pi\)
\(888\) 0 0
\(889\) 33792.5 + 11706.1i 1.27488 + 0.441630i
\(890\) 0 0
\(891\) −283.500 + 491.036i −0.0106595 + 0.0184628i
\(892\) 0 0
\(893\) 1420.00 + 2459.51i 0.0532122 + 0.0921662i
\(894\) 0 0
\(895\) −16940.0 −0.632672
\(896\) 0 0
\(897\) −7488.00 −0.278726
\(898\) 0 0
\(899\) 11542.5 + 19992.2i 0.428213 + 0.741688i
\(900\) 0 0
\(901\) 13500.0 23382.7i 0.499168 0.864584i
\(902\) 0 0
\(903\) −1659.00 8620.42i −0.0611385 0.317685i
\(904\) 0 0
\(905\) 9352.00 16198.1i 0.343504 0.594966i
\(906\) 0 0
\(907\) 22676.0 + 39276.0i 0.830148 + 1.43786i 0.897920 + 0.440158i \(0.145077\pi\)
−0.0677725 + 0.997701i \(0.521589\pi\)
\(908\) 0 0
\(909\) 11430.0 0.417062
\(910\) 0 0
\(911\) −39906.0 −1.45131 −0.725656 0.688058i \(-0.758463\pi\)
−0.725656 + 0.688058i \(0.758463\pi\)
\(912\) 0 0
\(913\) −3930.50 6807.83i −0.142476 0.246776i
\(914\) 0 0
\(915\) 2583.00 4473.89i 0.0933239 0.161642i
\(916\) 0 0
\(917\) 13706.0 11869.7i 0.493579 0.427452i
\(918\) 0 0
\(919\) −8764.00 + 15179.7i −0.314579 + 0.544866i −0.979348 0.202183i \(-0.935197\pi\)
0.664769 + 0.747049i \(0.268530\pi\)
\(920\) 0 0
\(921\) −1956.00 3387.89i −0.0699809 0.121210i
\(922\) 0 0
\(923\) −17576.0 −0.626783
\(924\) 0 0
\(925\) −26752.0 −0.950919
\(926\) 0 0
\(927\) −8244.00 14279.0i −0.292091 0.505917i
\(928\) 0 0
\(929\) −8533.00 + 14779.6i −0.301355 + 0.521962i −0.976443 0.215775i \(-0.930772\pi\)
0.675088 + 0.737737i \(0.264106\pi\)
\(930\) 0 0
\(931\) −6370.00 + 2546.11i −0.224241 + 0.0896300i
\(932\) 0 0
\(933\) 9312.00 16128.9i 0.326754 0.565954i
\(934\) 0 0
\(935\) −1764.00 3055.34i −0.0616994 0.106867i
\(936\) 0 0
\(937\) −30821.0 −1.07458 −0.537288 0.843399i \(-0.680551\pi\)
−0.537288 + 0.843399i \(0.680551\pi\)
\(938\) 0 0
\(939\) −10011.0 −0.347920
\(940\) 0 0
\(941\) −2089.50 3619.12i −0.0723866 0.125377i 0.827560 0.561377i \(-0.189728\pi\)
−0.899947 + 0.436000i \(0.856395\pi\)
\(942\) 0 0
\(943\) −7104.00 + 12304.5i −0.245321 + 0.424909i
\(944\) 0 0
\(945\) −2646.00 + 2291.50i −0.0910840 + 0.0788811i
\(946\) 0 0
\(947\) −4888.00 + 8466.26i −0.167728 + 0.290514i −0.937621 0.347660i \(-0.886976\pi\)
0.769893 + 0.638174i \(0.220310\pi\)
\(948\) 0 0
\(949\) −14092.0 24408.1i −0.482029 0.834899i
\(950\) 0 0
\(951\) −25911.0 −0.883514
\(952\) 0 0
\(953\) 14778.0 0.502315 0.251158 0.967946i \(-0.419189\pi\)
0.251158 + 0.967946i \(0.419189\pi\)
\(954\) 0 0
\(955\) 5992.00 + 10378.4i 0.203033 + 0.351664i
\(956\) 0 0
\(957\) −2551.50 + 4419.33i −0.0861842 + 0.149275i
\(958\) 0 0
\(959\) −1701.00 8838.66i −0.0572765 0.297617i
\(960\) 0 0
\(961\) 10383.0 17983.9i 0.348528 0.603668i
\(962\) 0 0
\(963\) 4495.50 + 7786.43i 0.150431 + 0.260555i
\(964\) 0 0
\(965\) −23765.0 −0.792769
\(966\) 0 0
\(967\) 8759.00 0.291283 0.145641 0.989337i \(-0.453475\pi\)
0.145641 + 0.989337i \(0.453475\pi\)
\(968\) 0 0
\(969\) 2160.00 + 3741.23i 0.0716091 + 0.124031i
\(970\) 0 0
\(971\) 16789.5 29080.3i 0.554893 0.961102i −0.443019 0.896512i \(-0.646093\pi\)
0.997912 0.0645901i \(-0.0205740\pi\)
\(972\) 0 0
\(973\) 46025.0 + 15943.5i 1.51644 + 0.525309i
\(974\) 0 0
\(975\) 5928.00 10267.6i 0.194716 0.337258i
\(976\) 0 0
\(977\) −21317.0 36922.1i −0.698046 1.20905i −0.969143 0.246500i \(-0.920720\pi\)
0.271097 0.962552i \(-0.412614\pi\)
\(978\) 0 0
\(979\) −2982.00 −0.0973495
\(980\) 0 0
\(981\) −14994.0 −0.487993
\(982\) 0 0
\(983\) −14574.0 25242.9i −0.472877 0.819048i 0.526641 0.850088i \(-0.323451\pi\)
−0.999518 + 0.0310404i \(0.990118\pi\)
\(984\) 0 0
\(985\) −1785.00 + 3091.71i −0.0577409 + 0.100010i
\(986\) 0 0
\(987\) −7455.00 2582.49i −0.240421 0.0832842i
\(988\) 0 0
\(989\) 3792.00 6567.94i 0.121920 0.211171i
\(990\) 0 0
\(991\) 22398.5 + 38795.3i 0.717974 + 1.24357i 0.961801 + 0.273748i \(0.0882635\pi\)
−0.243828 + 0.969819i \(0.578403\pi\)
\(992\) 0 0
\(993\) −5820.00 −0.185994
\(994\) 0 0
\(995\) 1932.00 0.0615563
\(996\) 0 0
\(997\) −5327.00 9226.63i −0.169215 0.293090i 0.768929 0.639334i \(-0.220790\pi\)
−0.938144 + 0.346245i \(0.887457\pi\)
\(998\) 0 0
\(999\) 4752.00 8230.71i 0.150497 0.260669i
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 168.4.q.a.121.1 yes 2
3.2 odd 2 504.4.s.e.289.1 2
4.3 odd 2 336.4.q.a.289.1 2
7.2 even 3 1176.4.a.f.1.1 1
7.4 even 3 inner 168.4.q.a.25.1 2
7.5 odd 6 1176.4.a.i.1.1 1
21.11 odd 6 504.4.s.e.361.1 2
28.11 odd 6 336.4.q.a.193.1 2
28.19 even 6 2352.4.a.c.1.1 1
28.23 odd 6 2352.4.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.a.25.1 2 7.4 even 3 inner
168.4.q.a.121.1 yes 2 1.1 even 1 trivial
336.4.q.a.193.1 2 28.11 odd 6
336.4.q.a.289.1 2 4.3 odd 2
504.4.s.e.289.1 2 3.2 odd 2
504.4.s.e.361.1 2 21.11 odd 6
1176.4.a.f.1.1 1 7.2 even 3
1176.4.a.i.1.1 1 7.5 odd 6
2352.4.a.c.1.1 1 28.19 even 6
2352.4.a.bg.1.1 1 28.23 odd 6