Properties

Label 507.4.b.k.337.6
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.6
Root \(-5.06791i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.82093i q^{2} +3.00000 q^{3} +0.0423641 q^{4} -3.41089i q^{5} -8.46278i q^{6} -13.3442i q^{7} -22.6869i q^{8} +9.00000 q^{9} -9.62187 q^{10} -35.4529i q^{11} +0.127092 q^{12} -37.6430 q^{14} -10.2327i q^{15} -63.6593 q^{16} -69.6526 q^{17} -25.3884i q^{18} +12.4014i q^{19} -0.144499i q^{20} -40.0325i q^{21} -100.010 q^{22} +126.251 q^{23} -68.0608i q^{24} +113.366 q^{25} +27.0000 q^{27} -0.565314i q^{28} -179.060 q^{29} -28.8656 q^{30} +255.935i q^{31} -1.91716i q^{32} -106.359i q^{33} +196.485i q^{34} -45.5155 q^{35} +0.381277 q^{36} -207.235i q^{37} +34.9833 q^{38} -77.3825 q^{40} -117.701i q^{41} -112.929 q^{42} -553.224 q^{43} -1.50193i q^{44} -30.6980i q^{45} -356.146i q^{46} -62.9185i q^{47} -190.978 q^{48} +164.933 q^{49} -319.797i q^{50} -208.958 q^{51} -147.031 q^{53} -76.1651i q^{54} -120.926 q^{55} -302.738 q^{56} +37.2041i q^{57} +505.115i q^{58} -274.087i q^{59} -0.433498i q^{60} +603.039 q^{61} +721.974 q^{62} -120.098i q^{63} -514.683 q^{64} -300.031 q^{66} -741.019i q^{67} -2.95077 q^{68} +378.754 q^{69} +128.396i q^{70} +572.574i q^{71} -204.182i q^{72} -26.7155i q^{73} -584.595 q^{74} +340.098 q^{75} +0.525372i q^{76} -473.090 q^{77} -207.798 q^{79} +217.135i q^{80} +81.0000 q^{81} -332.026 q^{82} -1031.37i q^{83} -1.69594i q^{84} +237.577i q^{85} +1560.60i q^{86} -537.179 q^{87} -804.318 q^{88} -1229.66i q^{89} -86.5968 q^{90} +5.34852 q^{92} +767.805i q^{93} -177.488 q^{94} +42.2996 q^{95} -5.75148i q^{96} +1795.12i q^{97} -465.264i q^{98} -319.076i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.82093i − 0.997349i −0.866789 0.498674i \(-0.833820\pi\)
0.866789 0.498674i \(-0.166180\pi\)
\(3\) 3.00000 0.577350
\(4\) 0.0423641 0.00529552
\(5\) − 3.41089i − 0.305079i −0.988297 0.152539i \(-0.951255\pi\)
0.988297 0.152539i \(-0.0487451\pi\)
\(6\) − 8.46278i − 0.575820i
\(7\) − 13.3442i − 0.720518i −0.932852 0.360259i \(-0.882688\pi\)
0.932852 0.360259i \(-0.117312\pi\)
\(8\) − 22.6869i − 1.00263i
\(9\) 9.00000 0.333333
\(10\) −9.62187 −0.304270
\(11\) − 35.4529i − 0.971769i −0.874023 0.485885i \(-0.838498\pi\)
0.874023 0.485885i \(-0.161502\pi\)
\(12\) 0.127092 0.00305737
\(13\) 0 0
\(14\) −37.6430 −0.718607
\(15\) − 10.2327i − 0.176137i
\(16\) −63.6593 −0.994676
\(17\) −69.6526 −0.993720 −0.496860 0.867831i \(-0.665514\pi\)
−0.496860 + 0.867831i \(0.665514\pi\)
\(18\) − 25.3884i − 0.332450i
\(19\) 12.4014i 0.149740i 0.997193 + 0.0748701i \(0.0238542\pi\)
−0.997193 + 0.0748701i \(0.976146\pi\)
\(20\) − 0.144499i − 0.00161555i
\(21\) − 40.0325i − 0.415991i
\(22\) −100.010 −0.969193
\(23\) 126.251 1.14457 0.572287 0.820053i \(-0.306056\pi\)
0.572287 + 0.820053i \(0.306056\pi\)
\(24\) − 68.0608i − 0.578869i
\(25\) 113.366 0.906927
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 0.565314i − 0.00381551i
\(29\) −179.060 −1.14657 −0.573286 0.819356i \(-0.694331\pi\)
−0.573286 + 0.819356i \(0.694331\pi\)
\(30\) −28.8656 −0.175670
\(31\) 255.935i 1.48281i 0.671055 + 0.741407i \(0.265841\pi\)
−0.671055 + 0.741407i \(0.734159\pi\)
\(32\) − 1.91716i − 0.0105909i
\(33\) − 106.359i − 0.561051i
\(34\) 196.485i 0.991085i
\(35\) −45.5155 −0.219815
\(36\) 0.381277 0.00176517
\(37\) − 207.235i − 0.920790i −0.887714 0.460395i \(-0.847708\pi\)
0.887714 0.460395i \(-0.152292\pi\)
\(38\) 34.9833 0.149343
\(39\) 0 0
\(40\) −77.3825 −0.305881
\(41\) − 117.701i − 0.448337i −0.974550 0.224169i \(-0.928033\pi\)
0.974550 0.224169i \(-0.0719666\pi\)
\(42\) −112.929 −0.414888
\(43\) −553.224 −1.96200 −0.980998 0.194016i \(-0.937849\pi\)
−0.980998 + 0.194016i \(0.937849\pi\)
\(44\) − 1.50193i − 0.00514602i
\(45\) − 30.6980i − 0.101693i
\(46\) − 356.146i − 1.14154i
\(47\) − 62.9185i − 0.195268i −0.995222 0.0976341i \(-0.968873\pi\)
0.995222 0.0976341i \(-0.0311275\pi\)
\(48\) −190.978 −0.574277
\(49\) 164.933 0.480854
\(50\) − 319.797i − 0.904522i
\(51\) −208.958 −0.573724
\(52\) 0 0
\(53\) −147.031 −0.381061 −0.190531 0.981681i \(-0.561021\pi\)
−0.190531 + 0.981681i \(0.561021\pi\)
\(54\) − 76.1651i − 0.191940i
\(55\) −120.926 −0.296466
\(56\) −302.738 −0.722413
\(57\) 37.2041i 0.0864526i
\(58\) 505.115i 1.14353i
\(59\) − 274.087i − 0.604798i −0.953181 0.302399i \(-0.902213\pi\)
0.953181 0.302399i \(-0.0977875\pi\)
\(60\) − 0.433498i 0 0.000932738i
\(61\) 603.039 1.26576 0.632879 0.774251i \(-0.281873\pi\)
0.632879 + 0.774251i \(0.281873\pi\)
\(62\) 721.974 1.47888
\(63\) − 120.098i − 0.240173i
\(64\) −514.683 −1.00524
\(65\) 0 0
\(66\) −300.031 −0.559564
\(67\) − 741.019i − 1.35119i −0.737272 0.675596i \(-0.763887\pi\)
0.737272 0.675596i \(-0.236113\pi\)
\(68\) −2.95077 −0.00526226
\(69\) 378.754 0.660820
\(70\) 128.396i 0.219232i
\(71\) 572.574i 0.957071i 0.878068 + 0.478536i \(0.158832\pi\)
−0.878068 + 0.478536i \(0.841168\pi\)
\(72\) − 204.182i − 0.334210i
\(73\) − 26.7155i − 0.0428330i −0.999771 0.0214165i \(-0.993182\pi\)
0.999771 0.0214165i \(-0.00681761\pi\)
\(74\) −584.595 −0.918349
\(75\) 340.098 0.523614
\(76\) 0.525372i 0 0.000792952i
\(77\) −473.090 −0.700177
\(78\) 0 0
\(79\) −207.798 −0.295938 −0.147969 0.988992i \(-0.547274\pi\)
−0.147969 + 0.988992i \(0.547274\pi\)
\(80\) 217.135i 0.303455i
\(81\) 81.0000 0.111111
\(82\) −332.026 −0.447149
\(83\) − 1031.37i − 1.36395i −0.731374 0.681976i \(-0.761121\pi\)
0.731374 0.681976i \(-0.238879\pi\)
\(84\) − 1.69594i − 0.00220289i
\(85\) 237.577i 0.303163i
\(86\) 1560.60i 1.95679i
\(87\) −537.179 −0.661973
\(88\) −804.318 −0.974325
\(89\) − 1229.66i − 1.46453i −0.681019 0.732266i \(-0.738463\pi\)
0.681019 0.732266i \(-0.261537\pi\)
\(90\) −86.5968 −0.101423
\(91\) 0 0
\(92\) 5.34852 0.00606111
\(93\) 767.805i 0.856103i
\(94\) −177.488 −0.194750
\(95\) 42.2996 0.0456826
\(96\) − 5.75148i − 0.00611467i
\(97\) 1795.12i 1.87903i 0.342502 + 0.939517i \(0.388726\pi\)
−0.342502 + 0.939517i \(0.611274\pi\)
\(98\) − 465.264i − 0.479579i
\(99\) − 319.076i − 0.323923i
\(100\) 4.80265 0.00480265
\(101\) 769.697 0.758295 0.379147 0.925336i \(-0.376217\pi\)
0.379147 + 0.925336i \(0.376217\pi\)
\(102\) 589.455i 0.572203i
\(103\) −1543.66 −1.47671 −0.738357 0.674410i \(-0.764398\pi\)
−0.738357 + 0.674410i \(0.764398\pi\)
\(104\) 0 0
\(105\) −136.546 −0.126910
\(106\) 414.764i 0.380051i
\(107\) 2027.64 1.83196 0.915978 0.401228i \(-0.131417\pi\)
0.915978 + 0.401228i \(0.131417\pi\)
\(108\) 1.14383 0.00101912
\(109\) 1644.97i 1.44550i 0.691108 + 0.722751i \(0.257123\pi\)
−0.691108 + 0.722751i \(0.742877\pi\)
\(110\) 341.123i 0.295680i
\(111\) − 621.705i − 0.531618i
\(112\) 849.481i 0.716682i
\(113\) −654.514 −0.544880 −0.272440 0.962173i \(-0.587831\pi\)
−0.272440 + 0.962173i \(0.587831\pi\)
\(114\) 104.950 0.0862234
\(115\) − 430.629i − 0.349186i
\(116\) −7.58571 −0.00607169
\(117\) 0 0
\(118\) −773.179 −0.603194
\(119\) 929.456i 0.715993i
\(120\) −232.148 −0.176601
\(121\) 74.0896 0.0556646
\(122\) − 1701.13i − 1.26240i
\(123\) − 353.103i − 0.258848i
\(124\) 10.8425i 0.00785227i
\(125\) − 813.039i − 0.581763i
\(126\) −338.787 −0.239536
\(127\) −1325.54 −0.926161 −0.463080 0.886316i \(-0.653256\pi\)
−0.463080 + 0.886316i \(0.653256\pi\)
\(128\) 1436.55i 0.991983i
\(129\) −1659.67 −1.13276
\(130\) 0 0
\(131\) 930.114 0.620339 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(132\) − 4.50580i − 0.00297106i
\(133\) 165.486 0.107891
\(134\) −2090.36 −1.34761
\(135\) − 92.0939i − 0.0587125i
\(136\) 1580.20i 0.996333i
\(137\) 2751.03i 1.71559i 0.513989 + 0.857797i \(0.328167\pi\)
−0.513989 + 0.857797i \(0.671833\pi\)
\(138\) − 1068.44i − 0.659068i
\(139\) 2436.65 1.48686 0.743432 0.668812i \(-0.233197\pi\)
0.743432 + 0.668812i \(0.233197\pi\)
\(140\) −1.92822 −0.00116403
\(141\) − 188.755i − 0.112738i
\(142\) 1615.19 0.954534
\(143\) 0 0
\(144\) −572.934 −0.331559
\(145\) 610.753i 0.349795i
\(146\) −75.3624 −0.0427194
\(147\) 494.799 0.277621
\(148\) − 8.77933i − 0.00487606i
\(149\) − 1517.58i − 0.834397i −0.908815 0.417198i \(-0.863012\pi\)
0.908815 0.417198i \(-0.136988\pi\)
\(150\) − 959.391i − 0.522226i
\(151\) − 583.642i − 0.314544i −0.987555 0.157272i \(-0.949730\pi\)
0.987555 0.157272i \(-0.0502699\pi\)
\(152\) 281.349 0.150134
\(153\) −626.873 −0.331240
\(154\) 1334.55i 0.698321i
\(155\) 872.965 0.452376
\(156\) 0 0
\(157\) 26.0932 0.0132641 0.00663206 0.999978i \(-0.497889\pi\)
0.00663206 + 0.999978i \(0.497889\pi\)
\(158\) 586.184i 0.295154i
\(159\) −441.093 −0.220006
\(160\) −6.53922 −0.00323107
\(161\) − 1684.72i − 0.824686i
\(162\) − 228.495i − 0.110817i
\(163\) − 17.0905i − 0.00821246i −0.999992 0.00410623i \(-0.998693\pi\)
0.999992 0.00410623i \(-0.00130706\pi\)
\(164\) − 4.98631i − 0.00237418i
\(165\) −362.778 −0.171165
\(166\) −2909.43 −1.36034
\(167\) − 3919.48i − 1.81616i −0.418799 0.908079i \(-0.637549\pi\)
0.418799 0.908079i \(-0.362451\pi\)
\(168\) −908.215 −0.417085
\(169\) 0 0
\(170\) 670.188 0.302359
\(171\) 111.612i 0.0499134i
\(172\) −23.4368 −0.0103898
\(173\) 1976.35 0.868551 0.434276 0.900780i \(-0.357004\pi\)
0.434276 + 0.900780i \(0.357004\pi\)
\(174\) 1515.34i 0.660218i
\(175\) − 1512.77i − 0.653457i
\(176\) 2256.91i 0.966596i
\(177\) − 822.260i − 0.349180i
\(178\) −3468.77 −1.46065
\(179\) 2784.34 1.16263 0.581316 0.813678i \(-0.302538\pi\)
0.581316 + 0.813678i \(0.302538\pi\)
\(180\) − 1.30049i 0 0.000538517i
\(181\) 1886.53 0.774723 0.387361 0.921928i \(-0.373387\pi\)
0.387361 + 0.921928i \(0.373387\pi\)
\(182\) 0 0
\(183\) 1809.12 0.730786
\(184\) − 2864.25i − 1.14758i
\(185\) −706.855 −0.280914
\(186\) 2165.92 0.853834
\(187\) 2469.39i 0.965666i
\(188\) − 2.66549i − 0.00103405i
\(189\) − 360.293i − 0.138664i
\(190\) − 119.324i − 0.0455615i
\(191\) 2447.94 0.927364 0.463682 0.886002i \(-0.346528\pi\)
0.463682 + 0.886002i \(0.346528\pi\)
\(192\) −1544.05 −0.580375
\(193\) 1925.66i 0.718196i 0.933300 + 0.359098i \(0.116916\pi\)
−0.933300 + 0.359098i \(0.883084\pi\)
\(194\) 5063.89 1.87405
\(195\) 0 0
\(196\) 6.98724 0.00254637
\(197\) − 1819.95i − 0.658203i −0.944294 0.329102i \(-0.893254\pi\)
0.944294 0.329102i \(-0.106746\pi\)
\(198\) −900.092 −0.323064
\(199\) 1273.97 0.453816 0.226908 0.973916i \(-0.427138\pi\)
0.226908 + 0.973916i \(0.427138\pi\)
\(200\) − 2571.92i − 0.909312i
\(201\) − 2223.06i − 0.780111i
\(202\) − 2171.26i − 0.756284i
\(203\) 2389.40i 0.826125i
\(204\) −8.85231 −0.00303817
\(205\) −401.465 −0.136778
\(206\) 4354.56i 1.47280i
\(207\) 1136.26 0.381525
\(208\) 0 0
\(209\) 439.664 0.145513
\(210\) 385.188i 0.126574i
\(211\) −1319.44 −0.430492 −0.215246 0.976560i \(-0.569055\pi\)
−0.215246 + 0.976560i \(0.569055\pi\)
\(212\) −6.22884 −0.00201792
\(213\) 1717.72i 0.552565i
\(214\) − 5719.83i − 1.82710i
\(215\) 1886.98i 0.598564i
\(216\) − 612.547i − 0.192956i
\(217\) 3415.24 1.06839
\(218\) 4640.35 1.44167
\(219\) − 80.1464i − 0.0247296i
\(220\) −5.12292 −0.00156994
\(221\) 0 0
\(222\) −1753.79 −0.530209
\(223\) 1203.45i 0.361384i 0.983540 + 0.180692i \(0.0578338\pi\)
−0.983540 + 0.180692i \(0.942166\pi\)
\(224\) −25.5829 −0.00763095
\(225\) 1020.29 0.302309
\(226\) 1846.34i 0.543436i
\(227\) − 4361.23i − 1.27518i −0.770377 0.637589i \(-0.779932\pi\)
0.770377 0.637589i \(-0.220068\pi\)
\(228\) 1.57612i 0 0.000457811i
\(229\) − 5384.29i − 1.55373i −0.629669 0.776864i \(-0.716809\pi\)
0.629669 0.776864i \(-0.283191\pi\)
\(230\) −1214.77 −0.348260
\(231\) −1419.27 −0.404247
\(232\) 4062.32i 1.14959i
\(233\) 4913.60 1.38155 0.690774 0.723070i \(-0.257270\pi\)
0.690774 + 0.723070i \(0.257270\pi\)
\(234\) 0 0
\(235\) −214.608 −0.0595722
\(236\) − 11.6114i − 0.00320272i
\(237\) −623.394 −0.170860
\(238\) 2621.93 0.714094
\(239\) 963.718i 0.260827i 0.991460 + 0.130414i \(0.0416306\pi\)
−0.991460 + 0.130414i \(0.958369\pi\)
\(240\) 651.404i 0.175200i
\(241\) − 1544.48i − 0.412817i −0.978466 0.206409i \(-0.933822\pi\)
0.978466 0.206409i \(-0.0661776\pi\)
\(242\) − 209.001i − 0.0555170i
\(243\) 243.000 0.0641500
\(244\) 25.5472 0.00670284
\(245\) − 562.568i − 0.146698i
\(246\) −996.079 −0.258161
\(247\) 0 0
\(248\) 5806.38 1.48671
\(249\) − 3094.12i − 0.787478i
\(250\) −2293.52 −0.580221
\(251\) −3768.17 −0.947588 −0.473794 0.880636i \(-0.657116\pi\)
−0.473794 + 0.880636i \(0.657116\pi\)
\(252\) − 5.08783i − 0.00127184i
\(253\) − 4475.98i − 1.11226i
\(254\) 3739.25i 0.923705i
\(255\) 712.731i 0.175031i
\(256\) −65.0695 −0.0158861
\(257\) −1282.68 −0.311327 −0.155664 0.987810i \(-0.549752\pi\)
−0.155664 + 0.987810i \(0.549752\pi\)
\(258\) 4681.81i 1.12976i
\(259\) −2765.38 −0.663445
\(260\) 0 0
\(261\) −1611.54 −0.382190
\(262\) − 2623.78i − 0.618694i
\(263\) 5029.74 1.17927 0.589633 0.807671i \(-0.299272\pi\)
0.589633 + 0.807671i \(0.299272\pi\)
\(264\) −2412.95 −0.562527
\(265\) 501.506i 0.116254i
\(266\) − 466.824i − 0.107604i
\(267\) − 3688.97i − 0.845548i
\(268\) − 31.3926i − 0.00715526i
\(269\) 5628.46 1.27574 0.637868 0.770146i \(-0.279816\pi\)
0.637868 + 0.770146i \(0.279816\pi\)
\(270\) −259.790 −0.0585568
\(271\) − 3368.39i − 0.755037i −0.926002 0.377518i \(-0.876778\pi\)
0.926002 0.377518i \(-0.123222\pi\)
\(272\) 4434.03 0.988429
\(273\) 0 0
\(274\) 7760.45 1.71104
\(275\) − 4019.15i − 0.881324i
\(276\) 16.0456 0.00349938
\(277\) 6507.75 1.41160 0.705799 0.708412i \(-0.250588\pi\)
0.705799 + 0.708412i \(0.250588\pi\)
\(278\) − 6873.62i − 1.48292i
\(279\) 2303.41i 0.494272i
\(280\) 1032.61i 0.220393i
\(281\) 3625.51i 0.769680i 0.922983 + 0.384840i \(0.125743\pi\)
−0.922983 + 0.384840i \(0.874257\pi\)
\(282\) −532.465 −0.112439
\(283\) 4635.41 0.973662 0.486831 0.873496i \(-0.338153\pi\)
0.486831 + 0.873496i \(0.338153\pi\)
\(284\) 24.2566i 0.00506819i
\(285\) 126.899 0.0263749
\(286\) 0 0
\(287\) −1570.62 −0.323035
\(288\) − 17.2545i − 0.00353031i
\(289\) −61.5178 −0.0125214
\(290\) 1722.89 0.348867
\(291\) 5385.35i 1.08486i
\(292\) − 1.13178i 0 0.000226823i
\(293\) 2907.07i 0.579634i 0.957082 + 0.289817i \(0.0935945\pi\)
−0.957082 + 0.289817i \(0.906406\pi\)
\(294\) − 1395.79i − 0.276885i
\(295\) −934.879 −0.184511
\(296\) −4701.53 −0.923212
\(297\) − 957.229i − 0.187017i
\(298\) −4280.99 −0.832185
\(299\) 0 0
\(300\) 14.4079 0.00277281
\(301\) 7382.32i 1.41365i
\(302\) −1646.41 −0.313710
\(303\) 2309.09 0.437802
\(304\) − 789.461i − 0.148943i
\(305\) − 2056.90i − 0.386156i
\(306\) 1768.36i 0.330362i
\(307\) − 933.950i − 0.173627i −0.996225 0.0868133i \(-0.972332\pi\)
0.996225 0.0868133i \(-0.0276683\pi\)
\(308\) −20.0421 −0.00370780
\(309\) −4630.98 −0.852581
\(310\) − 2462.57i − 0.451176i
\(311\) 3633.55 0.662508 0.331254 0.943542i \(-0.392528\pi\)
0.331254 + 0.943542i \(0.392528\pi\)
\(312\) 0 0
\(313\) −7507.26 −1.35570 −0.677852 0.735198i \(-0.737089\pi\)
−0.677852 + 0.735198i \(0.737089\pi\)
\(314\) − 73.6071i − 0.0132289i
\(315\) −409.639 −0.0732716
\(316\) −8.80319 −0.00156715
\(317\) 1214.60i 0.215202i 0.994194 + 0.107601i \(0.0343168\pi\)
−0.994194 + 0.107601i \(0.965683\pi\)
\(318\) 1244.29i 0.219423i
\(319\) 6348.19i 1.11420i
\(320\) 1755.52i 0.306677i
\(321\) 6082.92 1.05768
\(322\) −4752.47 −0.822500
\(323\) − 863.786i − 0.148800i
\(324\) 3.43149 0.000588391 0
\(325\) 0 0
\(326\) −48.2111 −0.00819069
\(327\) 4934.92i 0.834561i
\(328\) −2670.28 −0.449517
\(329\) −839.595 −0.140694
\(330\) 1023.37i 0.170711i
\(331\) 2443.20i 0.405712i 0.979209 + 0.202856i \(0.0650223\pi\)
−0.979209 + 0.202856i \(0.934978\pi\)
\(332\) − 43.6933i − 0.00722283i
\(333\) − 1865.12i − 0.306930i
\(334\) −11056.6 −1.81134
\(335\) −2527.53 −0.412220
\(336\) 2548.44i 0.413777i
\(337\) −1890.96 −0.305660 −0.152830 0.988253i \(-0.548839\pi\)
−0.152830 + 0.988253i \(0.548839\pi\)
\(338\) 0 0
\(339\) −1963.54 −0.314587
\(340\) 10.0647i 0.00160540i
\(341\) 9073.64 1.44095
\(342\) 314.850 0.0497811
\(343\) − 6777.95i − 1.06698i
\(344\) 12551.0i 1.96716i
\(345\) − 1291.89i − 0.201602i
\(346\) − 5575.15i − 0.866248i
\(347\) −2791.57 −0.431872 −0.215936 0.976408i \(-0.569280\pi\)
−0.215936 + 0.976408i \(0.569280\pi\)
\(348\) −22.7571 −0.00350549
\(349\) 6917.33i 1.06096i 0.847696 + 0.530482i \(0.177989\pi\)
−0.847696 + 0.530482i \(0.822011\pi\)
\(350\) −4267.43 −0.651724
\(351\) 0 0
\(352\) −67.9690 −0.0102919
\(353\) 7638.37i 1.15170i 0.817556 + 0.575849i \(0.195328\pi\)
−0.817556 + 0.575849i \(0.804672\pi\)
\(354\) −2319.54 −0.348254
\(355\) 1952.99 0.291982
\(356\) − 52.0933i − 0.00775545i
\(357\) 2788.37i 0.413379i
\(358\) − 7854.42i − 1.15955i
\(359\) 5490.97i 0.807249i 0.914925 + 0.403625i \(0.132250\pi\)
−0.914925 + 0.403625i \(0.867750\pi\)
\(360\) −696.443 −0.101960
\(361\) 6705.21 0.977578
\(362\) − 5321.77i − 0.772669i
\(363\) 222.269 0.0321380
\(364\) 0 0
\(365\) −91.1234 −0.0130674
\(366\) − 5103.39i − 0.728848i
\(367\) 6737.43 0.958287 0.479143 0.877737i \(-0.340947\pi\)
0.479143 + 0.877737i \(0.340947\pi\)
\(368\) −8037.07 −1.13848
\(369\) − 1059.31i − 0.149446i
\(370\) 1993.99i 0.280169i
\(371\) 1962.01i 0.274561i
\(372\) 32.5274i 0.00453351i
\(373\) −3930.42 −0.545602 −0.272801 0.962070i \(-0.587950\pi\)
−0.272801 + 0.962070i \(0.587950\pi\)
\(374\) 6965.97 0.963106
\(375\) − 2439.12i − 0.335881i
\(376\) −1427.43 −0.195782
\(377\) 0 0
\(378\) −1016.36 −0.138296
\(379\) 11897.7i 1.61252i 0.591563 + 0.806258i \(0.298511\pi\)
−0.591563 + 0.806258i \(0.701489\pi\)
\(380\) 1.79199 0.000241913 0
\(381\) −3976.61 −0.534719
\(382\) − 6905.46i − 0.924906i
\(383\) 4748.31i 0.633492i 0.948510 + 0.316746i \(0.102590\pi\)
−0.948510 + 0.316746i \(0.897410\pi\)
\(384\) 4309.64i 0.572722i
\(385\) 1613.66i 0.213609i
\(386\) 5432.14 0.716292
\(387\) −4979.02 −0.653999
\(388\) 76.0485i 0.00995046i
\(389\) −11299.5 −1.47277 −0.736386 0.676561i \(-0.763469\pi\)
−0.736386 + 0.676561i \(0.763469\pi\)
\(390\) 0 0
\(391\) −8793.73 −1.13739
\(392\) − 3741.82i − 0.482119i
\(393\) 2790.34 0.358153
\(394\) −5133.95 −0.656458
\(395\) 708.776i 0.0902845i
\(396\) − 13.5174i − 0.00171534i
\(397\) − 13853.0i − 1.75129i −0.482953 0.875646i \(-0.660436\pi\)
0.482953 0.875646i \(-0.339564\pi\)
\(398\) − 3593.78i − 0.452613i
\(399\) 496.457 0.0622906
\(400\) −7216.79 −0.902099
\(401\) − 5434.64i − 0.676790i −0.941004 0.338395i \(-0.890116\pi\)
0.941004 0.338395i \(-0.109884\pi\)
\(402\) −6271.08 −0.778042
\(403\) 0 0
\(404\) 32.6076 0.00401556
\(405\) − 276.282i − 0.0338977i
\(406\) 6740.34 0.823935
\(407\) −7347.09 −0.894795
\(408\) 4740.61i 0.575233i
\(409\) − 3472.43i − 0.419806i −0.977722 0.209903i \(-0.932685\pi\)
0.977722 0.209903i \(-0.0673148\pi\)
\(410\) 1132.50i 0.136416i
\(411\) 8253.09i 0.990498i
\(412\) −65.3959 −0.00781996
\(413\) −3657.46 −0.435767
\(414\) − 3205.31i − 0.380513i
\(415\) −3517.90 −0.416113
\(416\) 0 0
\(417\) 7309.95 0.858441
\(418\) − 1240.26i − 0.145127i
\(419\) −4261.65 −0.496886 −0.248443 0.968647i \(-0.579919\pi\)
−0.248443 + 0.968647i \(0.579919\pi\)
\(420\) −5.78467 −0.000672055 0
\(421\) 6235.06i 0.721801i 0.932604 + 0.360900i \(0.117531\pi\)
−0.932604 + 0.360900i \(0.882469\pi\)
\(422\) 3722.04i 0.429351i
\(423\) − 566.266i − 0.0650894i
\(424\) 3335.68i 0.382064i
\(425\) −7896.22 −0.901231
\(426\) 4845.57 0.551100
\(427\) − 8047.06i − 0.912001i
\(428\) 85.8992 0.00970115
\(429\) 0 0
\(430\) 5323.05 0.596977
\(431\) 2520.58i 0.281698i 0.990031 + 0.140849i \(0.0449833\pi\)
−0.990031 + 0.140849i \(0.955017\pi\)
\(432\) −1718.80 −0.191426
\(433\) 1283.89 0.142494 0.0712468 0.997459i \(-0.477302\pi\)
0.0712468 + 0.997459i \(0.477302\pi\)
\(434\) − 9634.15i − 1.06556i
\(435\) 1832.26i 0.201954i
\(436\) 69.6878i 0.00765468i
\(437\) 1565.69i 0.171389i
\(438\) −226.087 −0.0246641
\(439\) 16942.0 1.84190 0.920951 0.389677i \(-0.127414\pi\)
0.920951 + 0.389677i \(0.127414\pi\)
\(440\) 2743.44i 0.297246i
\(441\) 1484.40 0.160285
\(442\) 0 0
\(443\) −2163.00 −0.231980 −0.115990 0.993250i \(-0.537004\pi\)
−0.115990 + 0.993250i \(0.537004\pi\)
\(444\) − 26.3380i − 0.00281519i
\(445\) −4194.22 −0.446798
\(446\) 3394.83 0.360426
\(447\) − 4552.75i − 0.481739i
\(448\) 6868.01i 0.724293i
\(449\) 1673.35i 0.175880i 0.996126 + 0.0879400i \(0.0280284\pi\)
−0.996126 + 0.0879400i \(0.971972\pi\)
\(450\) − 2878.17i − 0.301507i
\(451\) −4172.85 −0.435680
\(452\) −27.7279 −0.00288542
\(453\) − 1750.92i − 0.181602i
\(454\) −12302.7 −1.27180
\(455\) 0 0
\(456\) 844.046 0.0866800
\(457\) − 4652.38i − 0.476212i −0.971239 0.238106i \(-0.923473\pi\)
0.971239 0.238106i \(-0.0765266\pi\)
\(458\) −15188.7 −1.54961
\(459\) −1880.62 −0.191241
\(460\) − 18.2432i − 0.00184912i
\(461\) 3460.07i 0.349570i 0.984607 + 0.174785i \(0.0559230\pi\)
−0.984607 + 0.174785i \(0.944077\pi\)
\(462\) 4003.66i 0.403176i
\(463\) 2105.64i 0.211355i 0.994400 + 0.105677i \(0.0337011\pi\)
−0.994400 + 0.105677i \(0.966299\pi\)
\(464\) 11398.8 1.14047
\(465\) 2618.89 0.261179
\(466\) − 13860.9i − 1.37789i
\(467\) 1415.88 0.140298 0.0701489 0.997537i \(-0.477653\pi\)
0.0701489 + 0.997537i \(0.477653\pi\)
\(468\) 0 0
\(469\) −9888.28 −0.973557
\(470\) 605.393i 0.0594143i
\(471\) 78.2797 0.00765804
\(472\) −6218.19 −0.606388
\(473\) 19613.4i 1.90661i
\(474\) 1758.55i 0.170407i
\(475\) 1405.89i 0.135803i
\(476\) 39.3756i 0.00379155i
\(477\) −1323.28 −0.127020
\(478\) 2718.58 0.260136
\(479\) 985.594i 0.0940145i 0.998895 + 0.0470073i \(0.0149684\pi\)
−0.998895 + 0.0470073i \(0.985032\pi\)
\(480\) −19.6177 −0.00186546
\(481\) 0 0
\(482\) −4356.88 −0.411723
\(483\) − 5054.16i − 0.476133i
\(484\) 3.13874 0.000294773 0
\(485\) 6122.93 0.573254
\(486\) − 685.486i − 0.0639800i
\(487\) 14724.7i 1.37011i 0.728494 + 0.685053i \(0.240221\pi\)
−0.728494 + 0.685053i \(0.759779\pi\)
\(488\) − 13681.1i − 1.26909i
\(489\) − 51.2715i − 0.00474147i
\(490\) −1586.96 −0.146310
\(491\) 16301.2 1.49830 0.749149 0.662401i \(-0.230463\pi\)
0.749149 + 0.662401i \(0.230463\pi\)
\(492\) − 14.9589i − 0.00137073i
\(493\) 12472.0 1.13937
\(494\) 0 0
\(495\) −1088.33 −0.0988221
\(496\) − 16292.6i − 1.47492i
\(497\) 7640.53 0.689587
\(498\) −8728.30 −0.785391
\(499\) 3230.42i 0.289806i 0.989446 + 0.144903i \(0.0462870\pi\)
−0.989446 + 0.144903i \(0.953713\pi\)
\(500\) − 34.4437i − 0.00308074i
\(501\) − 11758.4i − 1.04856i
\(502\) 10629.7i 0.945076i
\(503\) −3577.57 −0.317129 −0.158565 0.987349i \(-0.550687\pi\)
−0.158565 + 0.987349i \(0.550687\pi\)
\(504\) −2724.65 −0.240804
\(505\) − 2625.35i − 0.231340i
\(506\) −12626.4 −1.10931
\(507\) 0 0
\(508\) −56.1552 −0.00490450
\(509\) − 13864.7i − 1.20735i −0.797231 0.603674i \(-0.793703\pi\)
0.797231 0.603674i \(-0.206297\pi\)
\(510\) 2010.56 0.174567
\(511\) −356.496 −0.0308619
\(512\) 11675.9i 1.00783i
\(513\) 334.836i 0.0288175i
\(514\) 3618.34i 0.310502i
\(515\) 5265.25i 0.450514i
\(516\) −70.3105 −0.00599854
\(517\) −2230.64 −0.189756
\(518\) 7800.94i 0.661686i
\(519\) 5929.06 0.501458
\(520\) 0 0
\(521\) −9554.66 −0.803449 −0.401725 0.915760i \(-0.631589\pi\)
−0.401725 + 0.915760i \(0.631589\pi\)
\(522\) 4546.03i 0.381177i
\(523\) −19809.1 −1.65620 −0.828098 0.560583i \(-0.810577\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(524\) 39.4035 0.00328502
\(525\) − 4538.32i − 0.377274i
\(526\) − 14188.5i − 1.17614i
\(527\) − 17826.5i − 1.47350i
\(528\) 6770.73i 0.558064i
\(529\) 3772.38 0.310050
\(530\) 1414.71 0.115946
\(531\) − 2466.78i − 0.201599i
\(532\) 7.01066 0.000571336 0
\(533\) 0 0
\(534\) −10406.3 −0.843306
\(535\) − 6916.05i − 0.558891i
\(536\) −16811.4 −1.35475
\(537\) 8353.02 0.671246
\(538\) − 15877.5i − 1.27235i
\(539\) − 5847.36i − 0.467279i
\(540\) − 3.90148i 0 0.000310913i
\(541\) 1011.14i 0.0803551i 0.999193 + 0.0401775i \(0.0127923\pi\)
−0.999193 + 0.0401775i \(0.987208\pi\)
\(542\) −9501.98 −0.753035
\(543\) 5659.59 0.447286
\(544\) 133.535i 0.0105244i
\(545\) 5610.81 0.440992
\(546\) 0 0
\(547\) 15745.2 1.23074 0.615372 0.788237i \(-0.289006\pi\)
0.615372 + 0.788237i \(0.289006\pi\)
\(548\) 116.545i 0.00908495i
\(549\) 5427.35 0.421919
\(550\) −11337.7 −0.878987
\(551\) − 2220.58i − 0.171688i
\(552\) − 8592.76i − 0.662558i
\(553\) 2772.89i 0.213229i
\(554\) − 18357.9i − 1.40785i
\(555\) −2120.57 −0.162186
\(556\) 103.227 0.00787371
\(557\) − 8510.94i − 0.647433i −0.946154 0.323716i \(-0.895068\pi\)
0.946154 0.323716i \(-0.104932\pi\)
\(558\) 6497.76 0.492961
\(559\) 0 0
\(560\) 2897.48 0.218645
\(561\) 7408.16i 0.557528i
\(562\) 10227.3 0.767639
\(563\) −16764.1 −1.25493 −0.627463 0.778646i \(-0.715907\pi\)
−0.627463 + 0.778646i \(0.715907\pi\)
\(564\) − 7.99646i 0 0.000597006i
\(565\) 2232.47i 0.166232i
\(566\) − 13076.1i − 0.971080i
\(567\) − 1080.88i − 0.0800575i
\(568\) 12990.0 0.959589
\(569\) −20755.4 −1.52919 −0.764597 0.644508i \(-0.777062\pi\)
−0.764597 + 0.644508i \(0.777062\pi\)
\(570\) − 357.972i − 0.0263049i
\(571\) 23408.6 1.71562 0.857810 0.513968i \(-0.171825\pi\)
0.857810 + 0.513968i \(0.171825\pi\)
\(572\) 0 0
\(573\) 7343.81 0.535414
\(574\) 4430.62i 0.322179i
\(575\) 14312.6 1.03805
\(576\) −4632.14 −0.335080
\(577\) 10387.2i 0.749436i 0.927139 + 0.374718i \(0.122261\pi\)
−0.927139 + 0.374718i \(0.877739\pi\)
\(578\) 173.537i 0.0124882i
\(579\) 5776.97i 0.414650i
\(580\) 25.8740i 0.00185234i
\(581\) −13762.8 −0.982752
\(582\) 15191.7 1.08198
\(583\) 5212.68i 0.370304i
\(584\) −606.092 −0.0429457
\(585\) 0 0
\(586\) 8200.63 0.578097
\(587\) − 2898.42i − 0.203800i −0.994795 0.101900i \(-0.967508\pi\)
0.994795 0.101900i \(-0.0324921\pi\)
\(588\) 20.9617 0.00147015
\(589\) −3173.94 −0.222037
\(590\) 2637.23i 0.184022i
\(591\) − 5459.85i − 0.380014i
\(592\) 13192.4i 0.915888i
\(593\) − 8805.33i − 0.609766i −0.952390 0.304883i \(-0.901383\pi\)
0.952390 0.304883i \(-0.0986174\pi\)
\(594\) −2700.27 −0.186521
\(595\) 3170.27 0.218434
\(596\) − 64.2910i − 0.00441856i
\(597\) 3821.91 0.262011
\(598\) 0 0
\(599\) −20236.8 −1.38039 −0.690194 0.723624i \(-0.742475\pi\)
−0.690194 + 0.723624i \(0.742475\pi\)
\(600\) − 7715.77i − 0.524992i
\(601\) −9884.92 −0.670906 −0.335453 0.942057i \(-0.608889\pi\)
−0.335453 + 0.942057i \(0.608889\pi\)
\(602\) 20825.0 1.40991
\(603\) − 6669.17i − 0.450397i
\(604\) − 24.7255i − 0.00166567i
\(605\) − 252.711i − 0.0169821i
\(606\) − 6513.78i − 0.436641i
\(607\) 22919.3 1.53256 0.766281 0.642505i \(-0.222105\pi\)
0.766281 + 0.642505i \(0.222105\pi\)
\(608\) 23.7754 0.00158589
\(609\) 7168.21i 0.476963i
\(610\) −5802.36 −0.385132
\(611\) 0 0
\(612\) −26.5569 −0.00175409
\(613\) 916.052i 0.0603572i 0.999545 + 0.0301786i \(0.00960761\pi\)
−0.999545 + 0.0301786i \(0.990392\pi\)
\(614\) −2634.61 −0.173166
\(615\) −1204.40 −0.0789690
\(616\) 10733.0i 0.702019i
\(617\) − 30179.4i − 1.96917i −0.174915 0.984584i \(-0.555965\pi\)
0.174915 0.984584i \(-0.444035\pi\)
\(618\) 13063.7i 0.850320i
\(619\) − 5571.12i − 0.361748i −0.983506 0.180874i \(-0.942107\pi\)
0.983506 0.180874i \(-0.0578927\pi\)
\(620\) 36.9824 0.00239556
\(621\) 3408.78 0.220273
\(622\) − 10250.0i − 0.660751i
\(623\) −16408.8 −1.05522
\(624\) 0 0
\(625\) 11397.5 0.729443
\(626\) 21177.5i 1.35211i
\(627\) 1318.99 0.0840120
\(628\) 1.10542 7.02403e−5 0
\(629\) 14434.5i 0.915007i
\(630\) 1155.56i 0.0730773i
\(631\) 11360.1i 0.716703i 0.933587 + 0.358352i \(0.116661\pi\)
−0.933587 + 0.358352i \(0.883339\pi\)
\(632\) 4714.30i 0.296717i
\(633\) −3958.32 −0.248545
\(634\) 3426.31 0.214631
\(635\) 4521.26i 0.282552i
\(636\) −18.6865 −0.00116504
\(637\) 0 0
\(638\) 17907.8 1.11125
\(639\) 5153.17i 0.319024i
\(640\) 4899.89 0.302633
\(641\) 24709.8 1.52259 0.761293 0.648408i \(-0.224565\pi\)
0.761293 + 0.648408i \(0.224565\pi\)
\(642\) − 17159.5i − 1.05488i
\(643\) 15226.6i 0.933869i 0.884292 + 0.466934i \(0.154642\pi\)
−0.884292 + 0.466934i \(0.845358\pi\)
\(644\) − 71.3717i − 0.00436714i
\(645\) 5660.95i 0.345581i
\(646\) −2436.68 −0.148405
\(647\) −16706.6 −1.01515 −0.507576 0.861607i \(-0.669458\pi\)
−0.507576 + 0.861607i \(0.669458\pi\)
\(648\) − 1837.64i − 0.111403i
\(649\) −9717.18 −0.587724
\(650\) 0 0
\(651\) 10245.7 0.616838
\(652\) − 0.724024i 0 4.34892e-5i
\(653\) −28205.8 −1.69032 −0.845158 0.534516i \(-0.820494\pi\)
−0.845158 + 0.534516i \(0.820494\pi\)
\(654\) 13921.0 0.832349
\(655\) − 3172.51i − 0.189252i
\(656\) 7492.77i 0.445951i
\(657\) − 240.439i − 0.0142777i
\(658\) 2368.44i 0.140321i
\(659\) −11424.2 −0.675299 −0.337649 0.941272i \(-0.609632\pi\)
−0.337649 + 0.941272i \(0.609632\pi\)
\(660\) −15.3688 −0.000906407 0
\(661\) 26518.2i 1.56042i 0.625516 + 0.780211i \(0.284888\pi\)
−0.625516 + 0.780211i \(0.715112\pi\)
\(662\) 6892.10 0.404636
\(663\) 0 0
\(664\) −23398.7 −1.36754
\(665\) − 564.453i − 0.0329151i
\(666\) −5261.36 −0.306116
\(667\) −22606.5 −1.31234
\(668\) − 166.045i − 0.00961749i
\(669\) 3610.34i 0.208645i
\(670\) 7129.98i 0.411127i
\(671\) − 21379.5i − 1.23002i
\(672\) −76.7488 −0.00440573
\(673\) −22816.8 −1.30687 −0.653434 0.756984i \(-0.726672\pi\)
−0.653434 + 0.756984i \(0.726672\pi\)
\(674\) 5334.27i 0.304849i
\(675\) 3060.88 0.174538
\(676\) 0 0
\(677\) 17737.6 1.00696 0.503478 0.864008i \(-0.332053\pi\)
0.503478 + 0.864008i \(0.332053\pi\)
\(678\) 5539.01i 0.313753i
\(679\) 23954.3 1.35388
\(680\) 5389.89 0.303960
\(681\) − 13083.7i − 0.736224i
\(682\) − 25596.1i − 1.43713i
\(683\) 18878.4i 1.05763i 0.848737 + 0.528816i \(0.177364\pi\)
−0.848737 + 0.528816i \(0.822636\pi\)
\(684\) 4.72835i 0 0.000264317i
\(685\) 9383.45 0.523391
\(686\) −19120.1 −1.06415
\(687\) − 16152.9i − 0.897045i
\(688\) 35217.8 1.95155
\(689\) 0 0
\(690\) −3644.32 −0.201068
\(691\) 11692.8i 0.643726i 0.946786 + 0.321863i \(0.104309\pi\)
−0.946786 + 0.321863i \(0.895691\pi\)
\(692\) 83.7265 0.00459943
\(693\) −4257.81 −0.233392
\(694\) 7874.82i 0.430726i
\(695\) − 8311.14i − 0.453611i
\(696\) 12186.9i 0.663714i
\(697\) 8198.19i 0.445522i
\(698\) 19513.3 1.05815
\(699\) 14740.8 0.797638
\(700\) − 64.0873i − 0.00346039i
\(701\) −20658.1 −1.11304 −0.556522 0.830833i \(-0.687865\pi\)
−0.556522 + 0.830833i \(0.687865\pi\)
\(702\) 0 0
\(703\) 2569.99 0.137879
\(704\) 18247.0i 0.976861i
\(705\) −643.823 −0.0343940
\(706\) 21547.3 1.14864
\(707\) − 10271.0i − 0.546365i
\(708\) − 34.8343i − 0.00184909i
\(709\) 20695.8i 1.09626i 0.836394 + 0.548128i \(0.184659\pi\)
−0.836394 + 0.548128i \(0.815341\pi\)
\(710\) − 5509.23i − 0.291208i
\(711\) −1870.18 −0.0986461
\(712\) −27897.1 −1.46838
\(713\) 32312.1i 1.69719i
\(714\) 7865.79 0.412283
\(715\) 0 0
\(716\) 117.956 0.00615674
\(717\) 2891.16i 0.150589i
\(718\) 15489.6 0.805109
\(719\) 26299.2 1.36411 0.682053 0.731302i \(-0.261087\pi\)
0.682053 + 0.731302i \(0.261087\pi\)
\(720\) 1954.21i 0.101152i
\(721\) 20598.9i 1.06400i
\(722\) − 18914.9i − 0.974986i
\(723\) − 4633.45i − 0.238340i
\(724\) 79.9213 0.00410256
\(725\) −20299.3 −1.03986
\(726\) − 627.004i − 0.0320528i
\(727\) 16915.6 0.862952 0.431476 0.902125i \(-0.357993\pi\)
0.431476 + 0.902125i \(0.357993\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 257.053i 0.0130328i
\(731\) 38533.5 1.94967
\(732\) 76.6417 0.00386989
\(733\) − 13203.1i − 0.665302i −0.943050 0.332651i \(-0.892057\pi\)
0.943050 0.332651i \(-0.107943\pi\)
\(734\) − 19005.8i − 0.955746i
\(735\) − 1687.70i − 0.0846964i
\(736\) − 242.044i − 0.0121221i
\(737\) −26271.3 −1.31305
\(738\) −2988.24 −0.149050
\(739\) 16813.9i 0.836954i 0.908227 + 0.418477i \(0.137436\pi\)
−0.908227 + 0.418477i \(0.862564\pi\)
\(740\) −29.9453 −0.00148758
\(741\) 0 0
\(742\) 5534.68 0.273834
\(743\) − 32682.1i − 1.61371i −0.590747 0.806857i \(-0.701167\pi\)
0.590747 0.806857i \(-0.298833\pi\)
\(744\) 17419.1 0.858355
\(745\) −5176.30 −0.254557
\(746\) 11087.4i 0.544156i
\(747\) − 9282.37i − 0.454651i
\(748\) 104.613i 0.00511370i
\(749\) − 27057.2i − 1.31996i
\(750\) −6880.57 −0.334991
\(751\) −2157.76 −0.104844 −0.0524219 0.998625i \(-0.516694\pi\)
−0.0524219 + 0.998625i \(0.516694\pi\)
\(752\) 4005.35i 0.194229i
\(753\) −11304.5 −0.547090
\(754\) 0 0
\(755\) −1990.74 −0.0959606
\(756\) − 15.2635i 0 0.000734296i
\(757\) 13769.6 0.661117 0.330558 0.943786i \(-0.392763\pi\)
0.330558 + 0.943786i \(0.392763\pi\)
\(758\) 33562.6 1.60824
\(759\) − 13427.9i − 0.642165i
\(760\) − 959.648i − 0.0458028i
\(761\) − 16905.2i − 0.805272i −0.915360 0.402636i \(-0.868094\pi\)
0.915360 0.402636i \(-0.131906\pi\)
\(762\) 11217.7i 0.533302i
\(763\) 21950.8 1.04151
\(764\) 103.705 0.00491087
\(765\) 2138.19i 0.101054i
\(766\) 13394.7 0.631813
\(767\) 0 0
\(768\) −195.208 −0.00917185
\(769\) − 1153.35i − 0.0540845i −0.999634 0.0270422i \(-0.991391\pi\)
0.999634 0.0270422i \(-0.00860886\pi\)
\(770\) 4552.01 0.213043
\(771\) −3848.03 −0.179745
\(772\) 81.5787i 0.00380322i
\(773\) 20713.9i 0.963811i 0.876223 + 0.481905i \(0.160055\pi\)
−0.876223 + 0.481905i \(0.839945\pi\)
\(774\) 14045.4i 0.652265i
\(775\) 29014.3i 1.34480i
\(776\) 40725.7 1.88398
\(777\) −8296.14 −0.383040
\(778\) 31875.1i 1.46887i
\(779\) 1459.65 0.0671341
\(780\) 0 0
\(781\) 20299.4 0.930053
\(782\) 24806.5i 1.13437i
\(783\) −4834.61 −0.220658
\(784\) −10499.5 −0.478294
\(785\) − 89.0010i − 0.00404660i
\(786\) − 7871.35i − 0.357203i
\(787\) 10870.2i 0.492353i 0.969225 + 0.246177i \(0.0791743\pi\)
−0.969225 + 0.246177i \(0.920826\pi\)
\(788\) − 77.1006i − 0.00348553i
\(789\) 15089.2 0.680850
\(790\) 1999.41 0.0900451
\(791\) 8733.95i 0.392596i
\(792\) −7238.86 −0.324775
\(793\) 0 0
\(794\) −39078.4 −1.74665
\(795\) 1504.52i 0.0671192i
\(796\) 53.9706 0.00240319
\(797\) −12026.6 −0.534511 −0.267255 0.963626i \(-0.586117\pi\)
−0.267255 + 0.963626i \(0.586117\pi\)
\(798\) − 1400.47i − 0.0621255i
\(799\) 4382.43i 0.194042i
\(800\) − 217.341i − 0.00960519i
\(801\) − 11066.9i − 0.488177i
\(802\) −15330.7 −0.674996
\(803\) −947.142 −0.0416238
\(804\) − 94.1778i − 0.00413109i
\(805\) −5746.39 −0.251594
\(806\) 0 0
\(807\) 16885.4 0.736547
\(808\) − 17462.1i − 0.760289i
\(809\) −32384.0 −1.40737 −0.703685 0.710512i \(-0.748463\pi\)
−0.703685 + 0.710512i \(0.748463\pi\)
\(810\) −779.371 −0.0338078
\(811\) − 42506.8i − 1.84046i −0.391375 0.920231i \(-0.628000\pi\)
0.391375 0.920231i \(-0.372000\pi\)
\(812\) 101.225i 0.00437476i
\(813\) − 10105.2i − 0.435921i
\(814\) 20725.6i 0.892423i
\(815\) −58.2937 −0.00250545
\(816\) 13302.1 0.570670
\(817\) − 6860.72i − 0.293790i
\(818\) −9795.47 −0.418693
\(819\) 0 0
\(820\) −17.0077 −0.000724312 0
\(821\) 20052.6i 0.852427i 0.904623 + 0.426213i \(0.140153\pi\)
−0.904623 + 0.426213i \(0.859847\pi\)
\(822\) 23281.4 0.987872
\(823\) 31625.5 1.33948 0.669742 0.742594i \(-0.266405\pi\)
0.669742 + 0.742594i \(0.266405\pi\)
\(824\) 35020.9i 1.48060i
\(825\) − 12057.5i − 0.508832i
\(826\) 10317.4i 0.434612i
\(827\) 5440.97i 0.228780i 0.993436 + 0.114390i \(0.0364913\pi\)
−0.993436 + 0.114390i \(0.963509\pi\)
\(828\) 48.1367 0.00202037
\(829\) 24845.8 1.04093 0.520465 0.853883i \(-0.325759\pi\)
0.520465 + 0.853883i \(0.325759\pi\)
\(830\) 9923.75i 0.415010i
\(831\) 19523.2 0.814986
\(832\) 0 0
\(833\) −11488.0 −0.477834
\(834\) − 20620.9i − 0.856165i
\(835\) −13368.9 −0.554071
\(836\) 18.6260 0.000770566 0
\(837\) 6910.24i 0.285368i
\(838\) 12021.8i 0.495568i
\(839\) 45887.8i 1.88823i 0.329621 + 0.944113i \(0.393079\pi\)
−0.329621 + 0.944113i \(0.606921\pi\)
\(840\) 3097.82i 0.127244i
\(841\) 7673.40 0.314625
\(842\) 17588.7 0.719887
\(843\) 10876.5i 0.444375i
\(844\) −55.8969 −0.00227968
\(845\) 0 0
\(846\) −1597.40 −0.0649168
\(847\) − 988.665i − 0.0401073i
\(848\) 9359.88 0.379033
\(849\) 13906.2 0.562144
\(850\) 22274.7i 0.898842i
\(851\) − 26163.7i − 1.05391i
\(852\) 72.7698i 0.00292612i
\(853\) − 33655.3i − 1.35092i −0.737397 0.675460i \(-0.763945\pi\)
0.737397 0.675460i \(-0.236055\pi\)
\(854\) −22700.2 −0.909583
\(855\) 380.696 0.0152275
\(856\) − 46000.9i − 1.83677i
\(857\) −41365.6 −1.64880 −0.824399 0.566009i \(-0.808487\pi\)
−0.824399 + 0.566009i \(0.808487\pi\)
\(858\) 0 0
\(859\) 14866.0 0.590480 0.295240 0.955423i \(-0.404600\pi\)
0.295240 + 0.955423i \(0.404600\pi\)
\(860\) 79.9404i 0.00316970i
\(861\) −4711.87 −0.186504
\(862\) 7110.37 0.280952
\(863\) − 15469.1i − 0.610167i −0.952326 0.305083i \(-0.901316\pi\)
0.952326 0.305083i \(-0.0986843\pi\)
\(864\) − 51.7634i − 0.00203822i
\(865\) − 6741.12i − 0.264977i
\(866\) − 3621.76i − 0.142116i
\(867\) −184.553 −0.00722925
\(868\) 144.684 0.00565770
\(869\) 7367.05i 0.287584i
\(870\) 5168.67 0.201419
\(871\) 0 0
\(872\) 37319.4 1.44930
\(873\) 16156.0i 0.626345i
\(874\) 4416.69 0.170934
\(875\) −10849.3 −0.419171
\(876\) − 3.39533i 0 0.000130956i
\(877\) − 38100.7i − 1.46701i −0.679684 0.733505i \(-0.737883\pi\)
0.679684 0.733505i \(-0.262117\pi\)
\(878\) − 47792.0i − 1.83702i
\(879\) 8721.21i 0.334652i
\(880\) 7698.06 0.294888
\(881\) −3879.84 −0.148371 −0.0741857 0.997244i \(-0.523636\pi\)
−0.0741857 + 0.997244i \(0.523636\pi\)
\(882\) − 4187.38i − 0.159860i
\(883\) −13046.5 −0.497225 −0.248613 0.968603i \(-0.579975\pi\)
−0.248613 + 0.968603i \(0.579975\pi\)
\(884\) 0 0
\(885\) −2804.64 −0.106528
\(886\) 6101.66i 0.231365i
\(887\) −32833.2 −1.24288 −0.621438 0.783463i \(-0.713451\pi\)
−0.621438 + 0.783463i \(0.713451\pi\)
\(888\) −14104.6 −0.533017
\(889\) 17688.2i 0.667315i
\(890\) 11831.6i 0.445613i
\(891\) − 2871.69i − 0.107974i
\(892\) 50.9829i 0.00191372i
\(893\) 780.274 0.0292395
\(894\) −12843.0 −0.480462
\(895\) − 9497.06i − 0.354695i
\(896\) 19169.5 0.714741
\(897\) 0 0
\(898\) 4720.39 0.175414
\(899\) − 45827.6i − 1.70015i
\(900\) 43.2238 0.00160088
\(901\) 10241.1 0.378668
\(902\) 11771.3i 0.434525i
\(903\) 22147.0i 0.816173i
\(904\) 14848.9i 0.546314i
\(905\) − 6434.74i − 0.236352i
\(906\) −4939.23 −0.181120
\(907\) −21346.3 −0.781468 −0.390734 0.920504i \(-0.627779\pi\)
−0.390734 + 0.920504i \(0.627779\pi\)
\(908\) − 184.760i − 0.00675272i
\(909\) 6927.28 0.252765
\(910\) 0 0
\(911\) 7792.71 0.283407 0.141704 0.989909i \(-0.454742\pi\)
0.141704 + 0.989909i \(0.454742\pi\)
\(912\) − 2368.38i − 0.0859923i
\(913\) −36565.2 −1.32545
\(914\) −13124.0 −0.474950
\(915\) − 6170.69i − 0.222947i
\(916\) − 228.101i − 0.00822779i
\(917\) − 12411.6i − 0.446965i
\(918\) 5305.09i 0.190734i
\(919\) −31352.0 −1.12536 −0.562681 0.826674i \(-0.690230\pi\)
−0.562681 + 0.826674i \(0.690230\pi\)
\(920\) −9769.64 −0.350104
\(921\) − 2801.85i − 0.100243i
\(922\) 9760.61 0.348643
\(923\) 0 0
\(924\) −60.1262 −0.00214070
\(925\) − 23493.4i − 0.835089i
\(926\) 5939.85 0.210794
\(927\) −13893.0 −0.492238
\(928\) 343.286i 0.0121432i
\(929\) 12675.3i 0.447646i 0.974630 + 0.223823i \(0.0718537\pi\)
−0.974630 + 0.223823i \(0.928146\pi\)
\(930\) − 7387.71i − 0.260487i
\(931\) 2045.39i 0.0720032i
\(932\) 208.161 0.00731601
\(933\) 10900.7 0.382499
\(934\) − 3994.09i − 0.139926i
\(935\) 8422.80 0.294604
\(936\) 0 0
\(937\) −12308.5 −0.429138 −0.214569 0.976709i \(-0.568835\pi\)
−0.214569 + 0.976709i \(0.568835\pi\)
\(938\) 27894.1i 0.970976i
\(939\) −22521.8 −0.782717
\(940\) −9.09167 −0.000315466 0
\(941\) 20465.2i 0.708978i 0.935060 + 0.354489i \(0.115345\pi\)
−0.935060 + 0.354489i \(0.884655\pi\)
\(942\) − 220.821i − 0.00763774i
\(943\) − 14859.9i − 0.513155i
\(944\) 17448.2i 0.601578i
\(945\) −1228.92 −0.0423034
\(946\) 55328.0 1.90155
\(947\) − 11170.8i − 0.383318i −0.981462 0.191659i \(-0.938613\pi\)
0.981462 0.191659i \(-0.0613867\pi\)
\(948\) −26.4096 −0.000904792 0
\(949\) 0 0
\(950\) 3965.91 0.135443
\(951\) 3643.81i 0.124247i
\(952\) 21086.5 0.717876
\(953\) −14109.4 −0.479587 −0.239794 0.970824i \(-0.577080\pi\)
−0.239794 + 0.970824i \(0.577080\pi\)
\(954\) 3732.87i 0.126684i
\(955\) − 8349.64i − 0.282919i
\(956\) 40.8271i 0.00138122i
\(957\) 19044.6i 0.643285i
\(958\) 2780.29 0.0937653
\(959\) 36710.2 1.23612
\(960\) 5266.57i 0.177060i
\(961\) −35711.6 −1.19874
\(962\) 0 0
\(963\) 18248.8 0.610652
\(964\) − 65.4307i − 0.00218608i
\(965\) 6568.19 0.219106
\(966\) −14257.4 −0.474870
\(967\) − 40785.8i − 1.35634i −0.734904 0.678171i \(-0.762773\pi\)
0.734904 0.678171i \(-0.237227\pi\)
\(968\) − 1680.87i − 0.0558110i
\(969\) − 2591.36i − 0.0859096i
\(970\) − 17272.4i − 0.571734i
\(971\) 4230.29 0.139811 0.0699055 0.997554i \(-0.477730\pi\)
0.0699055 + 0.997554i \(0.477730\pi\)
\(972\) 10.2945 0.000339708 0
\(973\) − 32515.1i − 1.07131i
\(974\) 41537.4 1.36647
\(975\) 0 0
\(976\) −38389.0 −1.25902
\(977\) − 59925.0i − 1.96230i −0.193242 0.981151i \(-0.561900\pi\)
0.193242 0.981151i \(-0.438100\pi\)
\(978\) −144.633 −0.00472890
\(979\) −43594.9 −1.42319
\(980\) − 23.8327i 0 0.000776844i
\(981\) 14804.8i 0.481834i
\(982\) − 45984.6i − 1.49433i
\(983\) 23333.3i 0.757087i 0.925584 + 0.378543i \(0.123575\pi\)
−0.925584 + 0.378543i \(0.876425\pi\)
\(984\) −8010.83 −0.259528
\(985\) −6207.64 −0.200804
\(986\) − 35182.5i − 1.13635i
\(987\) −2518.79 −0.0812298
\(988\) 0 0
\(989\) −69845.2 −2.24565
\(990\) 3070.11i 0.0985601i
\(991\) −32523.5 −1.04253 −0.521263 0.853396i \(-0.674539\pi\)
−0.521263 + 0.853396i \(0.674539\pi\)
\(992\) 490.668 0.0157044
\(993\) 7329.61i 0.234238i
\(994\) − 21553.4i − 0.687759i
\(995\) − 4345.37i − 0.138450i
\(996\) − 131.080i − 0.00417010i
\(997\) −7114.35 −0.225992 −0.112996 0.993595i \(-0.536045\pi\)
−0.112996 + 0.993595i \(0.536045\pi\)
\(998\) 9112.77 0.289038
\(999\) − 5595.35i − 0.177206i
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 507.4.b.k.337.6 18
13.5 odd 4 507.4.a.p.1.3 yes 9
13.8 odd 4 507.4.a.o.1.7 9
13.12 even 2 inner 507.4.b.k.337.13 18
39.5 even 4 1521.4.a.bf.1.7 9
39.8 even 4 1521.4.a.bi.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.7 9 13.8 odd 4
507.4.a.p.1.3 yes 9 13.5 odd 4
507.4.b.k.337.6 18 1.1 even 1 trivial
507.4.b.k.337.13 18 13.12 even 2 inner
1521.4.a.bf.1.7 9 39.5 even 4
1521.4.a.bi.1.3 9 39.8 even 4