Properties

Label 84.4.i.a.25.2
Level $84$
Weight $4$
Character 84.25
Analytic conductor $4.956$
Analytic rank $0$
Dimension $4$
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] = [84,4,Mod(25,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("84.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.i (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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.2
Root \(3.72311 + 6.44862i\) of defining polynomial
Character \(\chi\) \(=\) 84.25
Dual form 84.4.i.a.37.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{3} +(0.723111 + 1.25246i) q^{5} +(-12.3924 + 13.7633i) q^{7} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{3} +(0.723111 + 1.25246i) q^{5} +(-12.3924 + 13.7633i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(-23.0618 + 39.9442i) q^{11} -32.2311 q^{13} -4.33867 q^{15} +(-38.8924 + 67.3637i) q^{17} +(-6.33067 - 10.9650i) q^{19} +(-17.1693 - 52.8414i) q^{21} +(50.4622 + 87.4031i) q^{23} +(61.4542 - 106.442i) q^{25} +27.0000 q^{27} +213.908 q^{29} +(-21.0378 + 36.4385i) q^{31} +(-69.1853 - 119.833i) q^{33} +(-26.1991 - 5.56874i) q^{35} +(-155.040 - 268.537i) q^{37} +(48.3467 - 83.7389i) q^{39} +44.0320 q^{41} +381.339 q^{43} +(6.50800 - 11.2722i) q^{45} +(179.032 + 310.093i) q^{47} +(-35.8547 - 341.121i) q^{49} +(-116.677 - 202.091i) q^{51} +(-92.4920 + 160.201i) q^{53} -66.7049 q^{55} +37.9840 q^{57} +(-227.325 + 393.738i) q^{59} +(-5.92444 - 10.2614i) q^{61} +(163.040 + 34.6549i) q^{63} +(-23.3067 - 40.3683i) q^{65} +(-295.180 + 511.266i) q^{67} -302.773 q^{69} +494.366 q^{71} +(-487.825 + 844.937i) q^{73} +(184.363 + 319.325i) q^{75} +(-263.970 - 812.411i) q^{77} +(-149.667 - 259.231i) q^{79} +(-40.5000 + 70.1481i) q^{81} -1406.07 q^{83} -112.494 q^{85} +(-320.863 + 555.750i) q^{87} +(347.629 + 602.112i) q^{89} +(399.422 - 443.605i) q^{91} +(-63.1133 - 109.316i) q^{93} +(9.15555 - 15.8579i) q^{95} +481.940 q^{97} +415.112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 11 q^{5} + 6 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 11 q^{5} + 6 q^{7} - 18 q^{9} + 5 q^{11} + 10 q^{13} + 66 q^{15} - 100 q^{17} - 67 q^{19} - 27 q^{21} - 76 q^{23} + 93 q^{25} + 108 q^{27} + 550 q^{29} - 362 q^{31} + 15 q^{33} - 466 q^{35} + 5 q^{37} - 15 q^{39} - 324 q^{41} + 1442 q^{43} - 99 q^{45} + 216 q^{47} + 190 q^{49} - 300 q^{51} - 495 q^{53} - 1406 q^{55} + 402 q^{57} - 173 q^{59} + 532 q^{61} + 27 q^{63} - 510 q^{65} - 111 q^{67} + 456 q^{69} + 3200 q^{71} - 1215 q^{73} + 279 q^{75} - 653 q^{77} - 1460 q^{79} - 162 q^{81} - 2818 q^{83} + 328 q^{85} - 825 q^{87} + 1974 q^{89} + 1945 q^{91} - 1086 q^{93} - 658 q^{95} + 1122 q^{97} - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0.723111 + 1.25246i 0.0646770 + 0.112024i 0.896551 0.442941i \(-0.146065\pi\)
−0.831874 + 0.554965i \(0.812732\pi\)
\(6\) 0 0
\(7\) −12.3924 + 13.7633i −0.669129 + 0.743146i
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.166667 0.288675i
\(10\) 0 0
\(11\) −23.0618 + 39.9442i −0.632126 + 1.09487i 0.354990 + 0.934870i \(0.384484\pi\)
−0.987116 + 0.160005i \(0.948849\pi\)
\(12\) 0 0
\(13\) −32.2311 −0.687639 −0.343819 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(14\) 0 0
\(15\) −4.33867 −0.0746826
\(16\) 0 0
\(17\) −38.8924 + 67.3637i −0.554871 + 0.961064i 0.443043 + 0.896500i \(0.353899\pi\)
−0.997914 + 0.0645639i \(0.979434\pi\)
\(18\) 0 0
\(19\) −6.33067 10.9650i −0.0764397 0.132397i 0.825272 0.564736i \(-0.191022\pi\)
−0.901711 + 0.432338i \(0.857689\pi\)
\(20\) 0 0
\(21\) −17.1693 52.8414i −0.178412 0.549092i
\(22\) 0 0
\(23\) 50.4622 + 87.4031i 0.457483 + 0.792383i 0.998827 0.0484177i \(-0.0154179\pi\)
−0.541345 + 0.840801i \(0.682085\pi\)
\(24\) 0 0
\(25\) 61.4542 106.442i 0.491634 0.851535i
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 213.908 1.36972 0.684859 0.728676i \(-0.259864\pi\)
0.684859 + 0.728676i \(0.259864\pi\)
\(30\) 0 0
\(31\) −21.0378 + 36.4385i −0.121887 + 0.211114i −0.920512 0.390715i \(-0.872228\pi\)
0.798625 + 0.601829i \(0.205561\pi\)
\(32\) 0 0
\(33\) −69.1853 119.833i −0.364958 0.632126i
\(34\) 0 0
\(35\) −26.1991 5.56874i −0.126527 0.0268940i
\(36\) 0 0
\(37\) −155.040 268.537i −0.688876 1.19317i −0.972202 0.234145i \(-0.924771\pi\)
0.283326 0.959024i \(-0.408562\pi\)
\(38\) 0 0
\(39\) 48.3467 83.7389i 0.198504 0.343819i
\(40\) 0 0
\(41\) 44.0320 0.167723 0.0838615 0.996477i \(-0.473275\pi\)
0.0838615 + 0.996477i \(0.473275\pi\)
\(42\) 0 0
\(43\) 381.339 1.35241 0.676205 0.736714i \(-0.263624\pi\)
0.676205 + 0.736714i \(0.263624\pi\)
\(44\) 0 0
\(45\) 6.50800 11.2722i 0.0215590 0.0373413i
\(46\) 0 0
\(47\) 179.032 + 310.093i 0.555628 + 0.962375i 0.997854 + 0.0654721i \(0.0208553\pi\)
−0.442227 + 0.896903i \(0.645811\pi\)
\(48\) 0 0
\(49\) −35.8547 341.121i −0.104533 0.994521i
\(50\) 0 0
\(51\) −116.677 202.091i −0.320355 0.554871i
\(52\) 0 0
\(53\) −92.4920 + 160.201i −0.239712 + 0.415194i −0.960632 0.277825i \(-0.910386\pi\)
0.720919 + 0.693019i \(0.243720\pi\)
\(54\) 0 0
\(55\) −66.7049 −0.163536
\(56\) 0 0
\(57\) 37.9840 0.0882650
\(58\) 0 0
\(59\) −227.325 + 393.738i −0.501613 + 0.868820i 0.498385 + 0.866956i \(0.333927\pi\)
−0.999998 + 0.00186377i \(0.999407\pi\)
\(60\) 0 0
\(61\) −5.92444 10.2614i −0.0124352 0.0215384i 0.859741 0.510731i \(-0.170625\pi\)
−0.872176 + 0.489192i \(0.837292\pi\)
\(62\) 0 0
\(63\) 163.040 + 34.6549i 0.326049 + 0.0693032i
\(64\) 0 0
\(65\) −23.3067 40.3683i −0.0444744 0.0770319i
\(66\) 0 0
\(67\) −295.180 + 511.266i −0.538238 + 0.932255i 0.460761 + 0.887524i \(0.347576\pi\)
−0.998999 + 0.0447309i \(0.985757\pi\)
\(68\) 0 0
\(69\) −302.773 −0.528255
\(70\) 0 0
\(71\) 494.366 0.826345 0.413172 0.910653i \(-0.364421\pi\)
0.413172 + 0.910653i \(0.364421\pi\)
\(72\) 0 0
\(73\) −487.825 + 844.937i −0.782131 + 1.35469i 0.148567 + 0.988902i \(0.452534\pi\)
−0.930698 + 0.365789i \(0.880799\pi\)
\(74\) 0 0
\(75\) 184.363 + 319.325i 0.283845 + 0.491634i
\(76\) 0 0
\(77\) −263.970 812.411i −0.390678 1.20237i
\(78\) 0 0
\(79\) −149.667 259.231i −0.213150 0.369187i 0.739549 0.673103i \(-0.235039\pi\)
−0.952699 + 0.303916i \(0.901706\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −1406.07 −1.85947 −0.929735 0.368229i \(-0.879964\pi\)
−0.929735 + 0.368229i \(0.879964\pi\)
\(84\) 0 0
\(85\) −112.494 −0.143550
\(86\) 0 0
\(87\) −320.863 + 555.750i −0.395403 + 0.684859i
\(88\) 0 0
\(89\) 347.629 + 602.112i 0.414030 + 0.717120i 0.995326 0.0965715i \(-0.0307877\pi\)
−0.581296 + 0.813692i \(0.697454\pi\)
\(90\) 0 0
\(91\) 399.422 443.605i 0.460119 0.511016i
\(92\) 0 0
\(93\) −63.1133 109.316i −0.0703715 0.121887i
\(94\) 0 0
\(95\) 9.15555 15.8579i 0.00988779 0.0171261i
\(96\) 0 0
\(97\) 481.940 0.504470 0.252235 0.967666i \(-0.418834\pi\)
0.252235 + 0.967666i \(0.418834\pi\)
\(98\) 0 0
\(99\) 415.112 0.421417
\(100\) 0 0
\(101\) 592.052 1025.46i 0.583281 1.01027i −0.411806 0.911272i \(-0.635102\pi\)
0.995087 0.0990014i \(-0.0315648\pi\)
\(102\) 0 0
\(103\) 641.765 + 1111.57i 0.613932 + 1.06336i 0.990571 + 0.137001i \(0.0437465\pi\)
−0.376639 + 0.926360i \(0.622920\pi\)
\(104\) 0 0
\(105\) 53.7667 59.7142i 0.0499723 0.0555001i
\(106\) 0 0
\(107\) −806.684 1397.22i −0.728833 1.26238i −0.957377 0.288842i \(-0.906730\pi\)
0.228544 0.973533i \(-0.426603\pi\)
\(108\) 0 0
\(109\) 76.9164 133.223i 0.0675895 0.117069i −0.830250 0.557391i \(-0.811803\pi\)
0.897840 + 0.440322i \(0.145136\pi\)
\(110\) 0 0
\(111\) 930.240 0.795446
\(112\) 0 0
\(113\) 1581.08 1.31625 0.658123 0.752910i \(-0.271350\pi\)
0.658123 + 0.752910i \(0.271350\pi\)
\(114\) 0 0
\(115\) −72.9796 + 126.404i −0.0591772 + 0.102498i
\(116\) 0 0
\(117\) 145.040 + 251.217i 0.114606 + 0.198504i
\(118\) 0 0
\(119\) −445.172 1370.09i −0.342931 1.05543i
\(120\) 0 0
\(121\) −398.191 689.687i −0.299167 0.518172i
\(122\) 0 0
\(123\) −66.0480 + 114.398i −0.0484175 + 0.0838615i
\(124\) 0 0
\(125\) 358.531 0.256544
\(126\) 0 0
\(127\) 1916.30 1.33893 0.669465 0.742844i \(-0.266523\pi\)
0.669465 + 0.742844i \(0.266523\pi\)
\(128\) 0 0
\(129\) −572.008 + 990.747i −0.390407 + 0.676205i
\(130\) 0 0
\(131\) −1250.24 2165.48i −0.833849 1.44427i −0.894964 0.446137i \(-0.852799\pi\)
0.0611158 0.998131i \(-0.480534\pi\)
\(132\) 0 0
\(133\) 229.367 + 48.7530i 0.149539 + 0.0317851i
\(134\) 0 0
\(135\) 19.5240 + 33.8166i 0.0124471 + 0.0215590i
\(136\) 0 0
\(137\) −145.446 + 251.920i −0.0907030 + 0.157102i −0.907807 0.419388i \(-0.862245\pi\)
0.817104 + 0.576490i \(0.195578\pi\)
\(138\) 0 0
\(139\) −1348.77 −0.823028 −0.411514 0.911403i \(-0.635000\pi\)
−0.411514 + 0.911403i \(0.635000\pi\)
\(140\) 0 0
\(141\) −1074.19 −0.641584
\(142\) 0 0
\(143\) 743.307 1287.44i 0.434674 0.752878i
\(144\) 0 0
\(145\) 154.680 + 267.913i 0.0885892 + 0.153441i
\(146\) 0 0
\(147\) 940.040 + 418.528i 0.527437 + 0.234827i
\(148\) 0 0
\(149\) 1424.40 + 2467.14i 0.783165 + 1.35648i 0.930089 + 0.367334i \(0.119729\pi\)
−0.146924 + 0.989148i \(0.546937\pi\)
\(150\) 0 0
\(151\) 744.656 1289.78i 0.401320 0.695106i −0.592566 0.805522i \(-0.701885\pi\)
0.993885 + 0.110416i \(0.0352183\pi\)
\(152\) 0 0
\(153\) 700.064 0.369914
\(154\) 0 0
\(155\) −60.8506 −0.0315331
\(156\) 0 0
\(157\) −1821.69 + 3155.26i −0.926030 + 1.60393i −0.136134 + 0.990690i \(0.543468\pi\)
−0.789896 + 0.613241i \(0.789865\pi\)
\(158\) 0 0
\(159\) −277.476 480.603i −0.138398 0.239712i
\(160\) 0 0
\(161\) −1828.30 388.614i −0.894971 0.190230i
\(162\) 0 0
\(163\) −384.581 666.114i −0.184802 0.320087i 0.758708 0.651431i \(-0.225831\pi\)
−0.943510 + 0.331345i \(0.892498\pi\)
\(164\) 0 0
\(165\) 100.057 173.304i 0.0472088 0.0817681i
\(166\) 0 0
\(167\) −2399.78 −1.11198 −0.555991 0.831188i \(-0.687661\pi\)
−0.555991 + 0.831188i \(0.687661\pi\)
\(168\) 0 0
\(169\) −1158.16 −0.527153
\(170\) 0 0
\(171\) −56.9760 + 98.6853i −0.0254799 + 0.0441325i
\(172\) 0 0
\(173\) 1668.32 + 2889.62i 0.733181 + 1.26991i 0.955517 + 0.294937i \(0.0952986\pi\)
−0.222335 + 0.974970i \(0.571368\pi\)
\(174\) 0 0
\(175\) 703.419 + 2164.88i 0.303848 + 0.935142i
\(176\) 0 0
\(177\) −681.975 1181.21i −0.289607 0.501613i
\(178\) 0 0
\(179\) −1230.89 + 2131.96i −0.513970 + 0.890223i 0.485898 + 0.874015i \(0.338493\pi\)
−0.999869 + 0.0162074i \(0.994841\pi\)
\(180\) 0 0
\(181\) 1316.74 0.540732 0.270366 0.962758i \(-0.412855\pi\)
0.270366 + 0.962758i \(0.412855\pi\)
\(182\) 0 0
\(183\) 35.5466 0.0143589
\(184\) 0 0
\(185\) 224.222 388.364i 0.0891089 0.154341i
\(186\) 0 0
\(187\) −1793.86 3107.05i −0.701497 1.21503i
\(188\) 0 0
\(189\) −334.596 + 371.608i −0.128774 + 0.143019i
\(190\) 0 0
\(191\) 1633.98 + 2830.14i 0.619010 + 1.07216i 0.989667 + 0.143387i \(0.0457994\pi\)
−0.370657 + 0.928770i \(0.620867\pi\)
\(192\) 0 0
\(193\) −116.836 + 202.366i −0.0435753 + 0.0754747i −0.886990 0.461788i \(-0.847208\pi\)
0.843415 + 0.537262i \(0.180542\pi\)
\(194\) 0 0
\(195\) 139.840 0.0513546
\(196\) 0 0
\(197\) −31.8632 −0.0115236 −0.00576182 0.999983i \(-0.501834\pi\)
−0.00576182 + 0.999983i \(0.501834\pi\)
\(198\) 0 0
\(199\) 739.451 1280.77i 0.263408 0.456237i −0.703737 0.710461i \(-0.748487\pi\)
0.967145 + 0.254224i \(0.0818200\pi\)
\(200\) 0 0
\(201\) −885.539 1533.80i −0.310752 0.538238i
\(202\) 0 0
\(203\) −2650.85 + 2944.08i −0.916518 + 1.01790i
\(204\) 0 0
\(205\) 31.8400 + 55.1485i 0.0108478 + 0.0187890i
\(206\) 0 0
\(207\) 454.160 786.628i 0.152494 0.264128i
\(208\) 0 0
\(209\) 583.986 0.193278
\(210\) 0 0
\(211\) −4498.67 −1.46778 −0.733889 0.679269i \(-0.762297\pi\)
−0.733889 + 0.679269i \(0.762297\pi\)
\(212\) 0 0
\(213\) −741.549 + 1284.40i −0.238545 + 0.413172i
\(214\) 0 0
\(215\) 275.750 + 477.613i 0.0874698 + 0.151502i
\(216\) 0 0
\(217\) −240.803 741.111i −0.0753308 0.231843i
\(218\) 0 0
\(219\) −1463.47 2534.81i −0.451564 0.782131i
\(220\) 0 0
\(221\) 1253.55 2171.21i 0.381551 0.660865i
\(222\) 0 0
\(223\) 5382.75 1.61639 0.808196 0.588913i \(-0.200444\pi\)
0.808196 + 0.588913i \(0.200444\pi\)
\(224\) 0 0
\(225\) −1106.18 −0.327756
\(226\) 0 0
\(227\) 2712.78 4698.68i 0.793188 1.37384i −0.130795 0.991409i \(-0.541753\pi\)
0.923983 0.382433i \(-0.124914\pi\)
\(228\) 0 0
\(229\) −994.873 1723.17i −0.287088 0.497250i 0.686026 0.727577i \(-0.259354\pi\)
−0.973113 + 0.230327i \(0.926020\pi\)
\(230\) 0 0
\(231\) 2506.66 + 532.802i 0.713966 + 0.151757i
\(232\) 0 0
\(233\) 3148.83 + 5453.93i 0.885351 + 1.53347i 0.845311 + 0.534274i \(0.179415\pi\)
0.0400396 + 0.999198i \(0.487252\pi\)
\(234\) 0 0
\(235\) −258.920 + 448.463i −0.0718727 + 0.124487i
\(236\) 0 0
\(237\) 898.003 0.246125
\(238\) 0 0
\(239\) 3395.77 0.919054 0.459527 0.888164i \(-0.348019\pi\)
0.459527 + 0.888164i \(0.348019\pi\)
\(240\) 0 0
\(241\) 3186.97 5519.99i 0.851829 1.47541i −0.0277273 0.999616i \(-0.508827\pi\)
0.879556 0.475795i \(-0.157840\pi\)
\(242\) 0 0
\(243\) −121.500 210.444i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 401.315 291.575i 0.104649 0.0760328i
\(246\) 0 0
\(247\) 204.044 + 353.415i 0.0525629 + 0.0910416i
\(248\) 0 0
\(249\) 2109.10 3653.07i 0.536783 0.929735i
\(250\) 0 0
\(251\) −2650.91 −0.666630 −0.333315 0.942815i \(-0.608167\pi\)
−0.333315 + 0.942815i \(0.608167\pi\)
\(252\) 0 0
\(253\) −4654.99 −1.15675
\(254\) 0 0
\(255\) 168.741 292.269i 0.0414392 0.0717748i
\(256\) 0 0
\(257\) 936.035 + 1621.26i 0.227192 + 0.393507i 0.956975 0.290171i \(-0.0937123\pi\)
−0.729783 + 0.683679i \(0.760379\pi\)
\(258\) 0 0
\(259\) 5617.27 + 1193.98i 1.34765 + 0.286448i
\(260\) 0 0
\(261\) −962.588 1667.25i −0.228286 0.395403i
\(262\) 0 0
\(263\) −3063.05 + 5305.35i −0.718158 + 1.24389i 0.243570 + 0.969883i \(0.421681\pi\)
−0.961729 + 0.274003i \(0.911652\pi\)
\(264\) 0 0
\(265\) −267.528 −0.0620155
\(266\) 0 0
\(267\) −2085.78 −0.478080
\(268\) 0 0
\(269\) 4252.06 7364.78i 0.963764 1.66929i 0.250866 0.968022i \(-0.419285\pi\)
0.712899 0.701267i \(-0.247382\pi\)
\(270\) 0 0
\(271\) −1060.77 1837.31i −0.237776 0.411840i 0.722300 0.691580i \(-0.243085\pi\)
−0.960076 + 0.279740i \(0.909752\pi\)
\(272\) 0 0
\(273\) 553.387 + 1703.14i 0.122683 + 0.377577i
\(274\) 0 0
\(275\) 2834.49 + 4909.48i 0.621549 + 1.07655i
\(276\) 0 0
\(277\) 4010.68 6946.71i 0.869959 1.50681i 0.00792096 0.999969i \(-0.497479\pi\)
0.862038 0.506844i \(-0.169188\pi\)
\(278\) 0 0
\(279\) 378.680 0.0812580
\(280\) 0 0
\(281\) −8244.17 −1.75020 −0.875100 0.483943i \(-0.839204\pi\)
−0.875100 + 0.483943i \(0.839204\pi\)
\(282\) 0 0
\(283\) −3050.64 + 5283.86i −0.640784 + 1.10987i 0.344475 + 0.938796i \(0.388057\pi\)
−0.985258 + 0.171074i \(0.945276\pi\)
\(284\) 0 0
\(285\) 27.4666 + 47.5736i 0.00570872 + 0.00988779i
\(286\) 0 0
\(287\) −545.664 + 606.024i −0.112228 + 0.124643i
\(288\) 0 0
\(289\) −568.744 985.094i −0.115763 0.200508i
\(290\) 0 0
\(291\) −722.911 + 1252.12i −0.145628 + 0.252235i
\(292\) 0 0
\(293\) −1965.98 −0.391993 −0.195996 0.980605i \(-0.562794\pi\)
−0.195996 + 0.980605i \(0.562794\pi\)
\(294\) 0 0
\(295\) −657.524 −0.129771
\(296\) 0 0
\(297\) −622.668 + 1078.49i −0.121653 + 0.210709i
\(298\) 0 0
\(299\) −1626.45 2817.10i −0.314583 0.544873i
\(300\) 0 0
\(301\) −4725.72 + 5248.46i −0.904936 + 1.00504i
\(302\) 0 0
\(303\) 1776.16 + 3076.39i 0.336758 + 0.583281i
\(304\) 0 0
\(305\) 8.56806 14.8403i 0.00160854 0.00278608i
\(306\) 0 0
\(307\) 997.810 0.185498 0.0927492 0.995690i \(-0.470435\pi\)
0.0927492 + 0.995690i \(0.470435\pi\)
\(308\) 0 0
\(309\) −3850.59 −0.708908
\(310\) 0 0
\(311\) 3450.72 5976.82i 0.629171 1.08976i −0.358547 0.933512i \(-0.616728\pi\)
0.987718 0.156245i \(-0.0499388\pi\)
\(312\) 0 0
\(313\) 3170.86 + 5492.08i 0.572612 + 0.991792i 0.996297 + 0.0859827i \(0.0274030\pi\)
−0.423685 + 0.905810i \(0.639264\pi\)
\(314\) 0 0
\(315\) 74.4920 + 229.261i 0.0133243 + 0.0410076i
\(316\) 0 0
\(317\) −612.487 1060.86i −0.108519 0.187961i 0.806651 0.591028i \(-0.201278\pi\)
−0.915171 + 0.403067i \(0.867944\pi\)
\(318\) 0 0
\(319\) −4933.11 + 8544.40i −0.865834 + 1.49967i
\(320\) 0 0
\(321\) 4840.10 0.841583
\(322\) 0 0
\(323\) 984.860 0.169657
\(324\) 0 0
\(325\) −1980.74 + 3430.74i −0.338066 + 0.585548i
\(326\) 0 0
\(327\) 230.749 + 399.670i 0.0390228 + 0.0675895i
\(328\) 0 0
\(329\) −6486.53 1378.74i −1.08697 0.231041i
\(330\) 0 0
\(331\) −1868.15 3235.73i −0.310220 0.537317i 0.668190 0.743991i \(-0.267069\pi\)
−0.978410 + 0.206674i \(0.933736\pi\)
\(332\) 0 0
\(333\) −1395.36 + 2416.83i −0.229625 + 0.397723i
\(334\) 0 0
\(335\) −853.790 −0.139246
\(336\) 0 0
\(337\) −3928.18 −0.634960 −0.317480 0.948265i \(-0.602837\pi\)
−0.317480 + 0.948265i \(0.602837\pi\)
\(338\) 0 0
\(339\) −2371.63 + 4107.78i −0.379968 + 0.658123i
\(340\) 0 0
\(341\) −970.337 1680.67i −0.154096 0.266902i
\(342\) 0 0
\(343\) 5139.26 + 3733.84i 0.809021 + 0.587780i
\(344\) 0 0
\(345\) −218.939 379.213i −0.0341660 0.0591772i
\(346\) 0 0
\(347\) −271.003 + 469.390i −0.0419256 + 0.0726173i −0.886227 0.463252i \(-0.846683\pi\)
0.844301 + 0.535869i \(0.180016\pi\)
\(348\) 0 0
\(349\) 2331.24 0.357561 0.178780 0.983889i \(-0.442785\pi\)
0.178780 + 0.983889i \(0.442785\pi\)
\(350\) 0 0
\(351\) −870.240 −0.132336
\(352\) 0 0
\(353\) 1280.15 2217.28i 0.193018 0.334318i −0.753231 0.657756i \(-0.771506\pi\)
0.946249 + 0.323439i \(0.104839\pi\)
\(354\) 0 0
\(355\) 357.482 + 619.176i 0.0534455 + 0.0925703i
\(356\) 0 0
\(357\) 4227.35 + 898.542i 0.626709 + 0.133210i
\(358\) 0 0
\(359\) 2284.27 + 3956.46i 0.335819 + 0.581655i 0.983642 0.180136i \(-0.0576537\pi\)
−0.647823 + 0.761791i \(0.724320\pi\)
\(360\) 0 0
\(361\) 3349.35 5801.24i 0.488314 0.845785i
\(362\) 0 0
\(363\) 2389.15 0.345448
\(364\) 0 0
\(365\) −1411.01 −0.202344
\(366\) 0 0
\(367\) 3190.91 5526.82i 0.453854 0.786098i −0.544768 0.838587i \(-0.683382\pi\)
0.998621 + 0.0524893i \(0.0167155\pi\)
\(368\) 0 0
\(369\) −198.144 343.195i −0.0279538 0.0484175i
\(370\) 0 0
\(371\) −1058.68 3258.27i −0.148151 0.455960i
\(372\) 0 0
\(373\) 3186.01 + 5518.33i 0.442266 + 0.766027i 0.997857 0.0654285i \(-0.0208414\pi\)
−0.555591 + 0.831456i \(0.687508\pi\)
\(374\) 0 0
\(375\) −537.796 + 931.490i −0.0740578 + 0.128272i
\(376\) 0 0
\(377\) −6894.51 −0.941870
\(378\) 0 0
\(379\) 1494.59 0.202564 0.101282 0.994858i \(-0.467706\pi\)
0.101282 + 0.994858i \(0.467706\pi\)
\(380\) 0 0
\(381\) −2874.45 + 4978.69i −0.386516 + 0.669465i
\(382\) 0 0
\(383\) −4410.13 7638.57i −0.588374 1.01909i −0.994446 0.105252i \(-0.966435\pi\)
0.406072 0.913841i \(-0.366898\pi\)
\(384\) 0 0
\(385\) 826.637 918.077i 0.109427 0.121531i
\(386\) 0 0
\(387\) −1716.02 2972.24i −0.225402 0.390407i
\(388\) 0 0
\(389\) 4939.24 8555.01i 0.643777 1.11505i −0.340806 0.940134i \(-0.610700\pi\)
0.984583 0.174921i \(-0.0559669\pi\)
\(390\) 0 0
\(391\) −7850.40 −1.01537
\(392\) 0 0
\(393\) 7501.45 0.962845
\(394\) 0 0
\(395\) 216.452 374.906i 0.0275718 0.0477558i
\(396\) 0 0
\(397\) −970.898 1681.64i −0.122740 0.212593i 0.798107 0.602516i \(-0.205835\pi\)
−0.920847 + 0.389923i \(0.872502\pi\)
\(398\) 0 0
\(399\) −470.715 + 522.784i −0.0590607 + 0.0655938i
\(400\) 0 0
\(401\) −3792.48 6568.77i −0.472288 0.818026i 0.527209 0.849735i \(-0.323238\pi\)
−0.999497 + 0.0317090i \(0.989905\pi\)
\(402\) 0 0
\(403\) 678.071 1174.45i 0.0838142 0.145170i
\(404\) 0 0
\(405\) −117.144 −0.0143727
\(406\) 0 0
\(407\) 14302.0 1.74183
\(408\) 0 0
\(409\) −4353.90 + 7541.18i −0.526373 + 0.911705i 0.473155 + 0.880979i \(0.343115\pi\)
−0.999528 + 0.0307253i \(0.990218\pi\)
\(410\) 0 0
\(411\) −436.339 755.761i −0.0523674 0.0907030i
\(412\) 0 0
\(413\) −2602.01 8008.11i −0.310016 0.954124i
\(414\) 0 0
\(415\) −1016.74 1761.05i −0.120265 0.208305i
\(416\) 0 0
\(417\) 2023.15 3504.20i 0.237588 0.411514i
\(418\) 0 0
\(419\) 6647.96 0.775117 0.387558 0.921845i \(-0.373319\pi\)
0.387558 + 0.921845i \(0.373319\pi\)
\(420\) 0 0
\(421\) 11670.6 1.35105 0.675524 0.737338i \(-0.263917\pi\)
0.675524 + 0.737338i \(0.263917\pi\)
\(422\) 0 0
\(423\) 1611.29 2790.83i 0.185209 0.320792i
\(424\) 0 0
\(425\) 4780.21 + 8279.57i 0.545586 + 0.944983i
\(426\) 0 0
\(427\) 214.649 + 45.6246i 0.0243269 + 0.00517079i
\(428\) 0 0
\(429\) 2229.92 + 3862.33i 0.250959 + 0.434674i
\(430\) 0 0
\(431\) 782.821 1355.89i 0.0874876 0.151533i −0.818961 0.573849i \(-0.805450\pi\)
0.906449 + 0.422316i \(0.138783\pi\)
\(432\) 0 0
\(433\) 15446.4 1.71434 0.857168 0.515037i \(-0.172222\pi\)
0.857168 + 0.515037i \(0.172222\pi\)
\(434\) 0 0
\(435\) −928.077 −0.102294
\(436\) 0 0
\(437\) 638.919 1106.64i 0.0699397 0.121139i
\(438\) 0 0
\(439\) −5348.70 9264.21i −0.581502 1.00719i −0.995302 0.0968229i \(-0.969132\pi\)
0.413800 0.910368i \(-0.364201\pi\)
\(440\) 0 0
\(441\) −2497.43 + 1814.50i −0.269672 + 0.195930i
\(442\) 0 0
\(443\) 526.981 + 912.758i 0.0565183 + 0.0978926i 0.892900 0.450254i \(-0.148667\pi\)
−0.836382 + 0.548147i \(0.815333\pi\)
\(444\) 0 0
\(445\) −502.749 + 870.787i −0.0535564 + 0.0927624i
\(446\) 0 0
\(447\) −8546.42 −0.904321
\(448\) 0 0
\(449\) 1139.33 0.119752 0.0598759 0.998206i \(-0.480930\pi\)
0.0598759 + 0.998206i \(0.480930\pi\)
\(450\) 0 0
\(451\) −1015.46 + 1758.82i −0.106022 + 0.183636i
\(452\) 0 0
\(453\) 2233.97 + 3869.35i 0.231702 + 0.401320i
\(454\) 0 0
\(455\) 844.427 + 179.487i 0.0870051 + 0.0184933i
\(456\) 0 0
\(457\) −2852.08 4939.95i −0.291936 0.505648i 0.682332 0.731043i \(-0.260966\pi\)
−0.974267 + 0.225395i \(0.927633\pi\)
\(458\) 0 0
\(459\) −1050.10 + 1818.82i −0.106785 + 0.184957i
\(460\) 0 0
\(461\) −6476.39 −0.654307 −0.327154 0.944971i \(-0.606089\pi\)
−0.327154 + 0.944971i \(0.606089\pi\)
\(462\) 0 0
\(463\) −232.366 −0.0233239 −0.0116619 0.999932i \(-0.503712\pi\)
−0.0116619 + 0.999932i \(0.503712\pi\)
\(464\) 0 0
\(465\) 91.2759 158.094i 0.00910284 0.0157666i
\(466\) 0 0
\(467\) 518.583 + 898.212i 0.0513858 + 0.0890028i 0.890574 0.454838i \(-0.150303\pi\)
−0.839188 + 0.543841i \(0.816970\pi\)
\(468\) 0 0
\(469\) −3378.69 10398.5i −0.332651 1.02379i
\(470\) 0 0
\(471\) −5465.07 9465.78i −0.534644 0.926030i
\(472\) 0 0
\(473\) −8794.35 + 15232.3i −0.854893 + 1.48072i
\(474\) 0 0
\(475\) −1556.18 −0.150321
\(476\) 0 0
\(477\) 1664.86 0.159808
\(478\) 0 0
\(479\) −5142.32 + 8906.77i −0.490519 + 0.849605i −0.999940 0.0109129i \(-0.996526\pi\)
0.509421 + 0.860517i \(0.329860\pi\)
\(480\) 0 0
\(481\) 4997.11 + 8655.25i 0.473698 + 0.820469i
\(482\) 0 0
\(483\) 3752.10 4167.15i 0.353471 0.392571i
\(484\) 0 0
\(485\) 348.496 + 603.614i 0.0326276 + 0.0565127i
\(486\) 0 0
\(487\) −5422.92 + 9392.77i −0.504591 + 0.873977i 0.495395 + 0.868668i \(0.335023\pi\)
−0.999986 + 0.00530928i \(0.998310\pi\)
\(488\) 0 0
\(489\) 2307.49 0.213391
\(490\) 0 0
\(491\) 8442.11 0.775941 0.387971 0.921672i \(-0.373176\pi\)
0.387971 + 0.921672i \(0.373176\pi\)
\(492\) 0 0
\(493\) −8319.42 + 14409.7i −0.760016 + 1.31639i
\(494\) 0 0
\(495\) 300.172 + 519.913i 0.0272560 + 0.0472088i
\(496\) 0 0
\(497\) −6126.41 + 6804.09i −0.552931 + 0.614095i
\(498\) 0 0
\(499\) −4905.42 8496.45i −0.440074 0.762231i 0.557620 0.830096i \(-0.311714\pi\)
−0.997694 + 0.0678654i \(0.978381\pi\)
\(500\) 0 0
\(501\) 3599.68 6234.82i 0.321001 0.555991i
\(502\) 0 0
\(503\) −6433.96 −0.570330 −0.285165 0.958478i \(-0.592048\pi\)
−0.285165 + 0.958478i \(0.592048\pi\)
\(504\) 0 0
\(505\) 1712.48 0.150900
\(506\) 0 0
\(507\) 1737.23 3008.98i 0.152176 0.263577i
\(508\) 0 0
\(509\) 10280.3 + 17806.0i 0.895220 + 1.55057i 0.833532 + 0.552471i \(0.186315\pi\)
0.0616885 + 0.998095i \(0.480351\pi\)
\(510\) 0 0
\(511\) −5583.75 17184.9i −0.483387 1.48770i
\(512\) 0 0
\(513\) −170.928 296.056i −0.0147108 0.0254799i
\(514\) 0 0
\(515\) −928.135 + 1607.58i −0.0794146 + 0.137550i
\(516\) 0 0
\(517\) −16515.2 −1.40491
\(518\) 0 0
\(519\) −10009.9 −0.846605
\(520\) 0 0
\(521\) −9563.06 + 16563.7i −0.804156 + 1.39284i 0.112703 + 0.993629i \(0.464049\pi\)
−0.916859 + 0.399211i \(0.869284\pi\)
\(522\) 0 0
\(523\) 1522.33 + 2636.75i 0.127279 + 0.220454i 0.922621 0.385707i \(-0.126042\pi\)
−0.795343 + 0.606160i \(0.792709\pi\)
\(524\) 0 0
\(525\) −6679.66 1419.79i −0.555285 0.118028i
\(526\) 0 0
\(527\) −1636.42 2834.36i −0.135263 0.234282i
\(528\) 0 0
\(529\) 990.629 1715.82i 0.0814193 0.141022i
\(530\) 0 0
\(531\) 4091.85 0.334409
\(532\) 0 0
\(533\) −1419.20 −0.115333
\(534\) 0 0
\(535\) 1166.64 2020.69i 0.0942774 0.163293i
\(536\) 0 0
\(537\) −3692.66 6395.87i −0.296741 0.513970i
\(538\) 0 0
\(539\) 14452.7 + 6434.67i 1.15495 + 0.514213i
\(540\) 0 0
\(541\) 2947.55 + 5105.30i 0.234242 + 0.405719i 0.959052 0.283230i \(-0.0914058\pi\)
−0.724810 + 0.688949i \(0.758072\pi\)
\(542\) 0 0
\(543\) −1975.11 + 3420.99i −0.156096 + 0.270366i
\(544\) 0 0
\(545\) 222.476 0.0174860
\(546\) 0 0
\(547\) 11151.1 0.871641 0.435821 0.900034i \(-0.356458\pi\)
0.435821 + 0.900034i \(0.356458\pi\)
\(548\) 0 0
\(549\) −53.3200 + 92.3529i −0.00414506 + 0.00717946i
\(550\) 0 0
\(551\) −1354.18 2345.51i −0.104701 0.181347i
\(552\) 0 0
\(553\) 5422.61 + 1152.60i 0.416985 + 0.0886320i
\(554\) 0 0
\(555\) 672.667 + 1165.09i 0.0514471 + 0.0891089i
\(556\) 0 0
\(557\) −872.173 + 1510.65i −0.0663468 + 0.114916i −0.897291 0.441440i \(-0.854468\pi\)
0.830944 + 0.556356i \(0.187801\pi\)
\(558\) 0 0
\(559\) −12291.0 −0.929969
\(560\) 0 0
\(561\) 10763.1 0.810019
\(562\) 0 0
\(563\) −5762.19 + 9980.41i −0.431345 + 0.747112i −0.996989 0.0775374i \(-0.975294\pi\)
0.565644 + 0.824650i \(0.308628\pi\)
\(564\) 0 0
\(565\) 1143.30 + 1980.25i 0.0851309 + 0.147451i
\(566\) 0 0
\(567\) −463.572 1426.72i −0.0343354 0.105673i
\(568\) 0 0
\(569\) −1411.38 2444.58i −0.103986 0.180110i 0.809337 0.587344i \(-0.199826\pi\)
−0.913324 + 0.407235i \(0.866493\pi\)
\(570\) 0 0
\(571\) 2583.82 4475.31i 0.189369 0.327996i −0.755671 0.654951i \(-0.772689\pi\)
0.945040 + 0.326955i \(0.106023\pi\)
\(572\) 0 0
\(573\) −9803.90 −0.714771
\(574\) 0 0
\(575\) 12404.5 0.899656
\(576\) 0 0
\(577\) 7357.61 12743.7i 0.530851 0.919461i −0.468501 0.883463i \(-0.655206\pi\)
0.999352 0.0359981i \(-0.0114610\pi\)
\(578\) 0 0
\(579\) −350.508 607.097i −0.0251582 0.0435753i
\(580\) 0 0
\(581\) 17424.6 19352.1i 1.24423 1.38186i
\(582\) 0 0
\(583\) −4266.06 7389.03i −0.303057 0.524910i
\(584\) 0 0
\(585\) −209.760 + 363.315i −0.0148248 + 0.0256773i
\(586\) 0 0
\(587\) 9981.64 0.701851 0.350925 0.936403i \(-0.385867\pi\)
0.350925 + 0.936403i \(0.385867\pi\)
\(588\) 0 0
\(589\) 532.733 0.0372680
\(590\) 0 0
\(591\) 47.7947 82.7829i 0.00332659 0.00576182i
\(592\) 0 0
\(593\) 837.605 + 1450.78i 0.0580039 + 0.100466i 0.893569 0.448925i \(-0.148193\pi\)
−0.835565 + 0.549391i \(0.814860\pi\)
\(594\) 0 0
\(595\) 1394.08 1548.29i 0.0960532 0.106678i
\(596\) 0 0
\(597\) 2218.35 + 3842.30i 0.152079 + 0.263408i
\(598\) 0 0
\(599\) −519.955 + 900.588i −0.0354671 + 0.0614308i −0.883214 0.468970i \(-0.844625\pi\)
0.847747 + 0.530401i \(0.177959\pi\)
\(600\) 0 0
\(601\) 4472.61 0.303563 0.151782 0.988414i \(-0.451499\pi\)
0.151782 + 0.988414i \(0.451499\pi\)
\(602\) 0 0
\(603\) 5313.23 0.358825
\(604\) 0 0
\(605\) 575.873 997.441i 0.0386984 0.0670277i
\(606\) 0 0
\(607\) 8895.38 + 15407.3i 0.594814 + 1.03025i 0.993573 + 0.113193i \(0.0361078\pi\)
−0.398759 + 0.917056i \(0.630559\pi\)
\(608\) 0 0
\(609\) −3672.67 11303.2i −0.244374 0.752101i
\(610\) 0 0
\(611\) −5770.40 9994.63i −0.382071 0.661766i
\(612\) 0 0
\(613\) 2163.37 3747.06i 0.142541 0.246888i −0.785912 0.618339i \(-0.787806\pi\)
0.928453 + 0.371450i \(0.121139\pi\)
\(614\) 0 0
\(615\) −191.040 −0.0125260
\(616\) 0 0
\(617\) −18866.2 −1.23100 −0.615498 0.788139i \(-0.711045\pi\)
−0.615498 + 0.788139i \(0.711045\pi\)
\(618\) 0 0
\(619\) 5089.69 8815.60i 0.330488 0.572422i −0.652120 0.758116i \(-0.726120\pi\)
0.982608 + 0.185694i \(0.0594534\pi\)
\(620\) 0 0
\(621\) 1362.48 + 2359.88i 0.0880426 + 0.152494i
\(622\) 0 0
\(623\) −12595.0 2677.12i −0.809965 0.172162i
\(624\) 0 0
\(625\) −7422.52 12856.2i −0.475041 0.822796i
\(626\) 0 0
\(627\) −875.979 + 1517.24i −0.0557946 + 0.0966391i
\(628\) 0 0
\(629\) 24119.5 1.52895
\(630\) 0 0
\(631\) −1661.72 −0.104837 −0.0524184 0.998625i \(-0.516693\pi\)
−0.0524184 + 0.998625i \(0.516693\pi\)
\(632\) 0 0
\(633\) 6748.01 11687.9i 0.423711 0.733889i
\(634\) 0 0
\(635\) 1385.70 + 2400.10i 0.0865980 + 0.149992i
\(636\) 0 0
\(637\) 1155.64 + 10994.7i 0.0718806 + 0.683871i
\(638\) 0 0
\(639\) −2224.65 3853.20i −0.137724 0.238545i
\(640\) 0 0
\(641\) 1138.82 1972.50i 0.0701728 0.121543i −0.828804 0.559539i \(-0.810978\pi\)
0.898977 + 0.437996i \(0.144312\pi\)
\(642\) 0 0
\(643\) 1217.38 0.0746638 0.0373319 0.999303i \(-0.488114\pi\)
0.0373319 + 0.999303i \(0.488114\pi\)
\(644\) 0 0
\(645\) −1654.50 −0.101001
\(646\) 0 0
\(647\) −5365.90 + 9294.01i −0.326052 + 0.564738i −0.981725 0.190307i \(-0.939052\pi\)
0.655673 + 0.755045i \(0.272385\pi\)
\(648\) 0 0
\(649\) −10485.0 18160.6i −0.634166 1.09841i
\(650\) 0 0
\(651\) 2286.67 + 486.041i 0.137667 + 0.0292618i
\(652\) 0 0
\(653\) −7614.03 13187.9i −0.456294 0.790324i 0.542468 0.840077i \(-0.317490\pi\)
−0.998762 + 0.0497526i \(0.984157\pi\)
\(654\) 0 0
\(655\) 1808.13 3131.77i 0.107862 0.186822i
\(656\) 0 0
\(657\) 8780.85 0.521421
\(658\) 0 0
\(659\) 10590.8 0.626038 0.313019 0.949747i \(-0.398660\pi\)
0.313019 + 0.949747i \(0.398660\pi\)
\(660\) 0 0
\(661\) 1934.15 3350.04i 0.113812 0.197128i −0.803492 0.595315i \(-0.797027\pi\)
0.917304 + 0.398187i \(0.130361\pi\)
\(662\) 0 0
\(663\) 3760.64 + 6513.62i 0.220288 + 0.381551i
\(664\) 0 0
\(665\) 104.796 + 322.528i 0.00611103 + 0.0188077i
\(666\) 0 0
\(667\) 10794.3 + 18696.3i 0.626622 + 1.08534i
\(668\) 0 0
\(669\) −8074.12 + 13984.8i −0.466612 + 0.808196i
\(670\) 0 0
\(671\) 546.512 0.0314424
\(672\) 0 0
\(673\) 11028.7 0.631687 0.315843 0.948811i \(-0.397713\pi\)
0.315843 + 0.948811i \(0.397713\pi\)
\(674\) 0 0
\(675\) 1659.26 2873.93i 0.0946150 0.163878i
\(676\) 0 0
\(677\) −9396.24 16274.8i −0.533422 0.923914i −0.999238 0.0390325i \(-0.987572\pi\)
0.465816 0.884882i \(-0.345761\pi\)
\(678\) 0 0
\(679\) −5972.42 + 6633.07i −0.337556 + 0.374895i
\(680\) 0 0
\(681\) 8138.35 + 14096.0i 0.457947 + 0.793188i
\(682\) 0 0
\(683\) 10121.9 17531.6i 0.567062 0.982181i −0.429792 0.902928i \(-0.641413\pi\)
0.996855 0.0792531i \(-0.0252535\pi\)
\(684\) 0 0
\(685\) −420.695 −0.0234656
\(686\) 0 0
\(687\) 5969.24 0.331500
\(688\) 0 0
\(689\) 2981.12 5163.45i 0.164835 0.285503i
\(690\) 0 0
\(691\) 9601.45 + 16630.2i 0.528591 + 0.915546i 0.999444 + 0.0333346i \(0.0106127\pi\)
−0.470854 + 0.882211i \(0.656054\pi\)
\(692\) 0 0
\(693\) −5144.25 + 5713.29i −0.281983 + 0.313175i
\(694\) 0 0
\(695\) −975.308 1689.28i −0.0532310 0.0921988i
\(696\) 0 0
\(697\) −1712.51 + 2966.16i −0.0930646 + 0.161193i
\(698\) 0 0
\(699\) −18893.0 −1.02231
\(700\) 0 0
\(701\) −22156.9 −1.19380 −0.596900 0.802316i \(-0.703601\pi\)
−0.596900 + 0.802316i \(0.703601\pi\)
\(702\) 0 0
\(703\) −1963.01 + 3400.04i −0.105315 + 0.182411i
\(704\) 0 0
\(705\) −776.760 1345.39i −0.0414957 0.0718727i
\(706\) 0 0
\(707\) 6776.76 + 20856.6i 0.360490 + 1.10947i
\(708\) 0 0
\(709\) 13604.5 + 23563.8i 0.720634 + 1.24817i 0.960746 + 0.277429i \(0.0894823\pi\)
−0.240112 + 0.970745i \(0.577184\pi\)
\(710\) 0 0
\(711\) −1347.00 + 2333.08i −0.0710501 + 0.123062i
\(712\) 0 0
\(713\) −4246.45 −0.223045
\(714\) 0 0
\(715\) 2149.97 0.112454
\(716\) 0 0
\(717\) −5093.65 + 8822.46i −0.265308 + 0.459527i
\(718\) 0 0
\(719\) −10776.2 18664.9i −0.558947 0.968125i −0.997585 0.0694605i \(-0.977872\pi\)
0.438638 0.898664i \(-0.355461\pi\)
\(720\) 0 0
\(721\) −23251.9 4942.29i −1.20103 0.255285i
\(722\) 0 0
\(723\) 9560.91 + 16560.0i 0.491804 + 0.851829i
\(724\) 0 0
\(725\) 13145.6 22768.8i 0.673399 1.16636i
\(726\) 0 0
\(727\) −20599.1 −1.05086 −0.525431 0.850836i \(-0.676096\pi\)
−0.525431 + 0.850836i \(0.676096\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −14831.2 + 25688.4i −0.750412 + 1.29975i
\(732\) 0 0
\(733\) −9880.58 17113.7i −0.497882 0.862357i 0.502115 0.864801i \(-0.332555\pi\)
−0.999997 + 0.00244415i \(0.999222\pi\)
\(734\) 0 0
\(735\) 155.561 + 1480.01i 0.00780676 + 0.0742734i
\(736\) 0 0
\(737\) −13614.7 23581.4i −0.680468 1.17861i
\(738\) 0 0
\(739\) −16704.4 + 28933.0i −0.831506 + 1.44021i 0.0653376 + 0.997863i \(0.479188\pi\)
−0.896844 + 0.442348i \(0.854146\pi\)
\(740\) 0 0
\(741\) −1224.27 −0.0606944
\(742\) 0 0
\(743\) −36225.4 −1.78867 −0.894335 0.447398i \(-0.852351\pi\)
−0.894335 + 0.447398i \(0.852351\pi\)
\(744\) 0 0
\(745\) −2060.00 + 3568.03i −0.101306 + 0.175466i
\(746\) 0 0
\(747\) 6327.31 + 10959.2i 0.309912 + 0.536783i
\(748\) 0 0
\(749\) 29227.1 + 6212.34i 1.42581 + 0.303063i
\(750\) 0 0
\(751\) 13660.2 + 23660.2i 0.663740 + 1.14963i 0.979625 + 0.200834i \(0.0643651\pi\)
−0.315885 + 0.948797i \(0.602302\pi\)
\(752\) 0 0
\(753\) 3976.37 6887.28i 0.192440 0.333315i
\(754\) 0 0
\(755\) 2153.88 0.103825
\(756\) 0 0
\(757\) −9918.43 −0.476211 −0.238105 0.971239i \(-0.576526\pi\)
−0.238105 + 0.971239i \(0.576526\pi\)
\(758\) 0 0
\(759\) 6982.49 12094.0i 0.333924 0.578373i
\(760\) 0 0
\(761\) 2817.52 + 4880.08i 0.134211 + 0.232461i 0.925296 0.379246i \(-0.123817\pi\)
−0.791084 + 0.611707i \(0.790483\pi\)
\(762\) 0 0
\(763\) 880.403 + 2709.58i 0.0417729 + 0.128563i
\(764\) 0 0
\(765\) 506.224 + 876.806i 0.0239249 + 0.0414392i
\(766\) 0 0
\(767\) 7326.93 12690.6i 0.344929 0.597434i
\(768\) 0 0
\(769\) 12089.3 0.566907 0.283453 0.958986i \(-0.408520\pi\)
0.283453 + 0.958986i \(0.408520\pi\)
\(770\) 0 0
\(771\) −5616.21 −0.262338
\(772\) 0 0
\(773\) 7085.68 12272.8i 0.329695 0.571049i −0.652756 0.757568i \(-0.726387\pi\)
0.982451 + 0.186519i \(0.0597207\pi\)
\(774\) 0 0
\(775\) 2585.72 + 4478.60i 0.119848 + 0.207582i
\(776\) 0 0
\(777\) −11527.9 + 12803.1i −0.532256 + 0.591132i
\(778\) 0 0
\(779\) −278.752 482.812i −0.0128207 0.0222061i
\(780\) 0 0
\(781\) −11401.0 + 19747.0i −0.522354 + 0.904744i
\(782\) 0 0
\(783\) 5775.53 0.263602
\(784\) 0 0
\(785\) −5269.14 −0.239571
\(786\) 0 0
\(787\) 12853.8 22263.5i 0.582197 1.00840i −0.413021 0.910721i \(-0.635526\pi\)
0.995218 0.0976740i \(-0.0311403\pi\)
\(788\) 0 0
\(789\) −9189.14 15916.1i −0.414629 0.718158i
\(790\) 0 0
\(791\) −19593.5 + 21760.9i −0.880739 + 0.978164i
\(792\) 0 0
\(793\) 190.951 + 330.737i 0.00855092 + 0.0148106i
\(794\) 0 0
\(795\) 401.292 695.058i 0.0179023 0.0310078i
\(796\) 0 0
\(797\) −6194.68 −0.275316 −0.137658 0.990480i \(-0.543957\pi\)
−0.137658 + 0.990480i \(0.543957\pi\)
\(798\) 0 0
\(799\) −27852.0 −1.23321
\(800\) 0 0
\(801\) 3128.66 5419.01i 0.138010 0.239040i
\(802\) 0 0
\(803\) −22500.2 38971.5i −0.988811 1.71267i
\(804\) 0 0
\(805\) −835.340 2570.90i −0.0365738 0.112562i
\(806\) 0 0
\(807\) 12756.2 + 22094.3i 0.556430 + 0.963764i
\(808\) 0 0
\(809\) −15407.5 + 26686.7i −0.669593 + 1.15977i 0.308425 + 0.951249i \(0.400198\pi\)
−0.978018 + 0.208520i \(0.933135\pi\)
\(810\) 0 0
\(811\) 43024.1 1.86286 0.931431 0.363917i \(-0.118561\pi\)
0.931431 + 0.363917i \(0.118561\pi\)
\(812\) 0 0
\(813\) 6364.63 0.274560
\(814\) 0 0
\(815\) 556.190 963.349i 0.0239049 0.0414045i
\(816\) 0 0
\(817\) −2414.13 4181.39i −0.103378 0.179056i
\(818\) 0 0
\(819\) −5254.96 1116.96i −0.224204 0.0476556i
\(820\) 0 0
\(821\) 5576.12 + 9658.12i 0.237038 + 0.410561i 0.959863 0.280470i \(-0.0904902\pi\)
−0.722825 + 0.691031i \(0.757157\pi\)
\(822\) 0 0
\(823\) 22456.4 38895.6i 0.951130 1.64741i 0.208145 0.978098i \(-0.433257\pi\)
0.742985 0.669308i \(-0.233409\pi\)
\(824\) 0 0
\(825\) −17006.9 −0.717703
\(826\) 0 0
\(827\) −3213.42 −0.135117 −0.0675584 0.997715i \(-0.521521\pi\)
−0.0675584 + 0.997715i \(0.521521\pi\)
\(828\) 0 0
\(829\) 4397.45 7616.61i 0.184234 0.319102i −0.759084 0.650992i \(-0.774353\pi\)
0.943318 + 0.331890i \(0.107686\pi\)
\(830\) 0 0
\(831\) 12032.0 + 20840.1i 0.502271 + 0.869959i
\(832\) 0 0
\(833\) 24373.6 + 10851.7i 1.01380 + 0.451368i
\(834\) 0 0
\(835\) −1735.31 3005.65i −0.0719197 0.124568i
\(836\) 0 0
\(837\) −568.020 + 983.840i −0.0234572 + 0.0406290i
\(838\) 0 0
\(839\) 45817.0 1.88532 0.942658 0.333761i \(-0.108318\pi\)
0.942658 + 0.333761i \(0.108318\pi\)
\(840\) 0 0
\(841\) 21367.8 0.876125
\(842\) 0 0
\(843\) 12366.3 21419.0i 0.505239 0.875100i
\(844\) 0 0
\(845\) −837.475 1450.55i −0.0340947 0.0590537i
\(846\) 0 0
\(847\) 14426.9 + 3066.50i 0.585259 + 0.124399i
\(848\) 0 0
\(849\) −9151.92 15851.6i −0.369957 0.640784i
\(850\) 0 0
\(851\) 15647.3 27102.0i 0.630298 1.09171i
\(852\) 0 0
\(853\) −16373.6 −0.657234 −0.328617 0.944463i \(-0.606583\pi\)
−0.328617 + 0.944463i \(0.606583\pi\)
\(854\) 0 0
\(855\) −164.800 −0.00659186
\(856\) 0 0
\(857\) 2215.10 3836.67i 0.0882922 0.152927i −0.818497 0.574511i \(-0.805192\pi\)
0.906789 + 0.421584i \(0.138526\pi\)
\(858\) 0 0
\(859\) −2472.19 4281.97i −0.0981958 0.170080i 0.812742 0.582624i \(-0.197974\pi\)
−0.910938 + 0.412544i \(0.864640\pi\)
\(860\) 0 0
\(861\) −756.000 2326.71i −0.0299238 0.0920954i
\(862\) 0 0
\(863\) −6589.31 11413.0i −0.259911 0.450178i 0.706307 0.707905i \(-0.250360\pi\)
−0.966218 + 0.257727i \(0.917026\pi\)
\(864\) 0 0
\(865\) −2412.77 + 4179.04i −0.0948399 + 0.164268i
\(866\) 0 0
\(867\) 3412.47 0.133672
\(868\) 0 0
\(869\) 13806.4 0.538951
\(870\) 0 0
\(871\) 9513.96 16478.7i 0.370113 0.641054i
\(872\) 0 0
\(873\) −2168.73 3756.35i −0.0840784 0.145628i
\(874\) 0 0
\(875\) −4443.07 + 4934.55i −0.171661 + 0.190649i
\(876\) 0 0
\(877\) 6492.95 + 11246.1i 0.250001 + 0.433015i 0.963526 0.267615i \(-0.0862356\pi\)
−0.713524 + 0.700630i \(0.752902\pi\)
\(878\) 0 0
\(879\) 2948.97 5107.77i 0.113159 0.195996i
\(880\) 0 0
\(881\) 36877.4 1.41025 0.705126 0.709082i \(-0.250890\pi\)
0.705126 + 0.709082i \(0.250890\pi\)
\(882\) 0 0
\(883\) −24874.5 −0.948012 −0.474006 0.880522i \(-0.657192\pi\)
−0.474006 + 0.880522i \(0.657192\pi\)
\(884\) 0 0
\(885\) 986.287 1708.30i 0.0374618 0.0648857i
\(886\) 0 0
\(887\) −2205.32 3819.72i −0.0834806 0.144593i 0.821262 0.570551i \(-0.193270\pi\)
−0.904743 + 0.425958i \(0.859937\pi\)
\(888\) 0 0
\(889\) −23747.6 + 26374.5i −0.895917 + 0.995021i
\(890\) 0 0
\(891\) −1868.00 3235.48i −0.0702362 0.121653i
\(892\) 0 0
\(893\) 2266.78 3926.18i 0.0849440 0.147127i
\(894\) 0 0
\(895\) −3560.27 −0.132968
\(896\) 0 0
\(897\) 9758.72 0.363249
\(898\) 0 0
\(899\) −4500.16 + 7794.50i −0.166951 + 0.289167i
\(900\) 0 0
\(901\) −7194.48 12461.2i −0.266019 0.460758i
\(902\) 0 0
\(903\) −6547.33 20150.5i −0.241286 0.742598i
\(904\) 0 0
\(905\) 952.149 + 1649.17i 0.0349729 + 0.0605749i
\(906\) 0 0
\(907\) 3637.00 6299.46i 0.133147 0.230618i −0.791741 0.610857i \(-0.790825\pi\)
0.924888 + 0.380239i \(0.124158\pi\)
\(908\) 0 0
\(909\) −10656.9 −0.388854
\(910\) 0 0
\(911\) −49491.9 −1.79993 −0.899967 0.435957i \(-0.856410\pi\)
−0.899967 + 0.435957i \(0.856410\pi\)
\(912\) 0 0
\(913\) 32426.4 56164.2i 1.17542 2.03589i
\(914\) 0 0
\(915\) 25.7042 + 44.5209i 0.000928692 + 0.00160854i
\(916\) 0 0
\(917\) 45297.7 + 9628.22i 1.63125 + 0.346730i
\(918\) 0 0
\(919\) 2507.95 + 4343.89i 0.0900213 + 0.155922i 0.907520 0.420009i \(-0.137973\pi\)
−0.817499 + 0.575931i \(0.804640\pi\)
\(920\) 0 0
\(921\) −1496.71 + 2592.39i −0.0535488 + 0.0927492i
\(922\) 0 0
\(923\) −15934.0 −0.568227
\(924\) 0 0
\(925\) −38111.4 −1.35470
\(926\) 0 0
\(927\) 5775.89 10004.1i 0.204644 0.354454i
\(928\) 0 0
\(929\) 26079.1 + 45170.4i 0.921021 + 1.59525i 0.797839 + 0.602870i \(0.205976\pi\)
0.123181 + 0.992384i \(0.460690\pi\)
\(930\) 0 0
\(931\) −3513.42 + 2552.67i −0.123682 + 0.0898608i
\(932\) 0 0
\(933\) 10352.1 + 17930.4i 0.363252 + 0.629171i
\(934\) 0 0
\(935\) 2594.32 4493.49i 0.0907414 0.157169i
\(936\) 0 0
\(937\) −14821.9 −0.516766 −0.258383 0.966042i \(-0.583190\pi\)
−0.258383 + 0.966042i \(0.583190\pi\)
\(938\) 0 0
\(939\) −19025.1 −0.661195
\(940\) 0 0
\(941\) −13246.1 + 22943.0i −0.458886 + 0.794814i −0.998902 0.0468405i \(-0.985085\pi\)
0.540016 + 0.841655i \(0.318418\pi\)
\(942\) 0 0
\(943\) 2221.95 + 3848.53i 0.0767304 + 0.132901i
\(944\) 0 0
\(945\) −707.376 150.356i −0.0243502 0.00517574i
\(946\) 0 0
\(947\) −3205.12 5551.44i −0.109982 0.190494i 0.805781 0.592214i \(-0.201746\pi\)
−0.915763 + 0.401720i \(0.868413\pi\)
\(948\) 0 0
\(949\) 15723.1 27233.3i 0.537824 0.931538i
\(950\) 0 0
\(951\) 3674.92 0.125307
\(952\) 0 0
\(953\) −25108.5 −0.853458 −0.426729 0.904380i \(-0.640334\pi\)
−0.426729 + 0.904380i \(0.640334\pi\)
\(954\) 0 0
\(955\) −2363.10 + 4093.02i −0.0800715 + 0.138688i
\(956\) 0 0
\(957\) −14799.3 25633.2i −0.499890 0.865834i
\(958\) 0 0
\(959\) −1664.81 5123.72i −0.0560579 0.172527i
\(960\) 0 0
\(961\) 14010.3 + 24266.6i 0.470287 + 0.814561i
\(962\) 0 0
\(963\) −7260.16 + 12575.0i −0.242944 + 0.420792i
\(964\) 0 0
\(965\) −337.941 −0.0112733
\(966\) 0 0
\(967\) 32928.1 1.09503 0.547516 0.836795i \(-0.315574\pi\)
0.547516 + 0.836795i \(0.315574\pi\)
\(968\) 0 0
\(969\) −1477.29 + 2558.74i −0.0489757 + 0.0848283i
\(970\) 0 0
\(971\) 15771.6 + 27317.2i 0.521251 + 0.902834i 0.999695 + 0.0247155i \(0.00786798\pi\)
−0.478443 + 0.878119i \(0.658799\pi\)
\(972\) 0 0
\(973\) 16714.5 18563.4i 0.550712 0.611630i
\(974\) 0 0
\(975\) −5942.21 10292.2i −0.195183 0.338066i
\(976\) 0 0
\(977\) −26940.9 + 46663.0i −0.882206 + 1.52803i −0.0333230 + 0.999445i \(0.510609\pi\)
−0.848883 + 0.528581i \(0.822724\pi\)
\(978\) 0 0
\(979\) −32067.8 −1.04688
\(980\) 0 0
\(981\) −1384.50 −0.0450597
\(982\) 0 0
\(983\) −17844.3 + 30907.3i −0.578988 + 1.00284i 0.416608 + 0.909086i \(0.363219\pi\)
−0.995596 + 0.0937503i \(0.970114\pi\)
\(984\) 0 0
\(985\) −23.0406 39.9075i −0.000745314 0.00129092i
\(986\) 0 0
\(987\) 13311.9 14784.4i 0.429302 0.476790i
\(988\) 0 0
\(989\) 19243.2 + 33330.2i 0.618704 + 1.07163i
\(990\) 0 0
\(991\) −7419.86 + 12851.6i −0.237840 + 0.411951i −0.960094 0.279676i \(-0.909773\pi\)
0.722254 + 0.691628i \(0.243106\pi\)
\(992\) 0 0
\(993\) 11208.9 0.358211
\(994\) 0 0
\(995\) 2138.82 0.0681459
\(996\) 0 0
\(997\) −8686.03 + 15044.7i −0.275917 + 0.477903i −0.970366 0.241639i \(-0.922315\pi\)
0.694449 + 0.719542i \(0.255648\pi\)
\(998\) 0 0
\(999\) −4186.08 7250.50i −0.132574 0.229625i
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 84.4.i.a.25.2 4
3.2 odd 2 252.4.k.f.109.1 4
4.3 odd 2 336.4.q.i.193.2 4
7.2 even 3 inner 84.4.i.a.37.2 yes 4
7.3 odd 6 588.4.a.f.1.2 2
7.4 even 3 588.4.a.i.1.1 2
7.5 odd 6 588.4.i.j.373.1 4
7.6 odd 2 588.4.i.j.361.1 4
21.2 odd 6 252.4.k.f.37.1 4
21.5 even 6 1764.4.k.q.1549.2 4
21.11 odd 6 1764.4.a.o.1.2 2
21.17 even 6 1764.4.a.y.1.1 2
21.20 even 2 1764.4.k.q.361.2 4
28.3 even 6 2352.4.a.bx.1.2 2
28.11 odd 6 2352.4.a.bt.1.1 2
28.23 odd 6 336.4.q.i.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.2 4 1.1 even 1 trivial
84.4.i.a.37.2 yes 4 7.2 even 3 inner
252.4.k.f.37.1 4 21.2 odd 6
252.4.k.f.109.1 4 3.2 odd 2
336.4.q.i.193.2 4 4.3 odd 2
336.4.q.i.289.2 4 28.23 odd 6
588.4.a.f.1.2 2 7.3 odd 6
588.4.a.i.1.1 2 7.4 even 3
588.4.i.j.361.1 4 7.6 odd 2
588.4.i.j.373.1 4 7.5 odd 6
1764.4.a.o.1.2 2 21.11 odd 6
1764.4.a.y.1.1 2 21.17 even 6
1764.4.k.q.361.2 4 21.20 even 2
1764.4.k.q.1549.2 4 21.5 even 6
2352.4.a.bt.1.1 2 28.11 odd 6
2352.4.a.bx.1.2 2 28.3 even 6