Properties

Label 507.4.a.q.1.9
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(1,507)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

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: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-4.48584\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.48584 q^{2} -3.00000 q^{3} +22.0945 q^{4} +13.3185 q^{5} -16.4575 q^{6} +21.4234 q^{7} +77.3200 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.48584 q^{2} -3.00000 q^{3} +22.0945 q^{4} +13.3185 q^{5} -16.4575 q^{6} +21.4234 q^{7} +77.3200 q^{8} +9.00000 q^{9} +73.0635 q^{10} -19.0520 q^{11} -66.2834 q^{12} +117.525 q^{14} -39.9556 q^{15} +247.410 q^{16} -71.7906 q^{17} +49.3726 q^{18} -102.134 q^{19} +294.266 q^{20} -64.2701 q^{21} -104.516 q^{22} -37.8302 q^{23} -231.960 q^{24} +52.3838 q^{25} -27.0000 q^{27} +473.338 q^{28} +40.8605 q^{29} -219.190 q^{30} -6.05542 q^{31} +738.690 q^{32} +57.1559 q^{33} -393.832 q^{34} +285.328 q^{35} +198.850 q^{36} -285.682 q^{37} -560.293 q^{38} +1029.79 q^{40} -342.705 q^{41} -352.576 q^{42} +306.458 q^{43} -420.943 q^{44} +119.867 q^{45} -207.530 q^{46} +346.863 q^{47} -742.229 q^{48} +115.961 q^{49} +287.369 q^{50} +215.372 q^{51} +398.219 q^{53} -148.118 q^{54} -253.745 q^{55} +1656.46 q^{56} +306.403 q^{57} +224.155 q^{58} -208.497 q^{59} -882.799 q^{60} +546.936 q^{61} -33.2191 q^{62} +192.810 q^{63} +2073.06 q^{64} +313.548 q^{66} -678.268 q^{67} -1586.18 q^{68} +113.490 q^{69} +1565.27 q^{70} +957.777 q^{71} +695.880 q^{72} +270.360 q^{73} -1567.21 q^{74} -157.151 q^{75} -2256.60 q^{76} -408.157 q^{77} -1032.86 q^{79} +3295.14 q^{80} +81.0000 q^{81} -1880.03 q^{82} -1065.90 q^{83} -1420.01 q^{84} -956.147 q^{85} +1681.18 q^{86} -122.582 q^{87} -1473.10 q^{88} +427.185 q^{89} +657.571 q^{90} -835.837 q^{92} +18.1663 q^{93} +1902.83 q^{94} -1360.28 q^{95} -2216.07 q^{96} +698.084 q^{97} +636.144 q^{98} -171.468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.48584 1.93954 0.969769 0.244025i \(-0.0784677\pi\)
0.969769 + 0.244025i \(0.0784677\pi\)
\(3\) −3.00000 −0.577350
\(4\) 22.0945 2.76181
\(5\) 13.3185 1.19125 0.595624 0.803264i \(-0.296905\pi\)
0.595624 + 0.803264i \(0.296905\pi\)
\(6\) −16.4575 −1.11979
\(7\) 21.4234 1.15675 0.578377 0.815770i \(-0.303686\pi\)
0.578377 + 0.815770i \(0.303686\pi\)
\(8\) 77.3200 3.41709
\(9\) 9.00000 0.333333
\(10\) 73.0635 2.31047
\(11\) −19.0520 −0.522217 −0.261108 0.965310i \(-0.584088\pi\)
−0.261108 + 0.965310i \(0.584088\pi\)
\(12\) −66.2834 −1.59453
\(13\) 0 0
\(14\) 117.525 2.24357
\(15\) −39.9556 −0.687767
\(16\) 247.410 3.86578
\(17\) −71.7906 −1.02422 −0.512111 0.858919i \(-0.671136\pi\)
−0.512111 + 0.858919i \(0.671136\pi\)
\(18\) 49.3726 0.646513
\(19\) −102.134 −1.23322 −0.616611 0.787268i \(-0.711495\pi\)
−0.616611 + 0.787268i \(0.711495\pi\)
\(20\) 294.266 3.29000
\(21\) −64.2701 −0.667852
\(22\) −104.516 −1.01286
\(23\) −37.8302 −0.342962 −0.171481 0.985187i \(-0.554855\pi\)
−0.171481 + 0.985187i \(0.554855\pi\)
\(24\) −231.960 −1.97286
\(25\) 52.3838 0.419070
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 473.338 3.19473
\(29\) 40.8605 0.261642 0.130821 0.991406i \(-0.458239\pi\)
0.130821 + 0.991406i \(0.458239\pi\)
\(30\) −219.190 −1.33395
\(31\) −6.05542 −0.0350834 −0.0175417 0.999846i \(-0.505584\pi\)
−0.0175417 + 0.999846i \(0.505584\pi\)
\(32\) 738.690 4.08073
\(33\) 57.1559 0.301502
\(34\) −393.832 −1.98652
\(35\) 285.328 1.37798
\(36\) 198.850 0.920603
\(37\) −285.682 −1.26935 −0.634673 0.772781i \(-0.718865\pi\)
−0.634673 + 0.772781i \(0.718865\pi\)
\(38\) −560.293 −2.39188
\(39\) 0 0
\(40\) 1029.79 4.07060
\(41\) −342.705 −1.30540 −0.652702 0.757615i \(-0.726364\pi\)
−0.652702 + 0.757615i \(0.726364\pi\)
\(42\) −352.576 −1.29532
\(43\) 306.458 1.08685 0.543424 0.839458i \(-0.317128\pi\)
0.543424 + 0.839458i \(0.317128\pi\)
\(44\) −420.943 −1.44226
\(45\) 119.867 0.397082
\(46\) −207.530 −0.665188
\(47\) 346.863 1.07649 0.538246 0.842788i \(-0.319087\pi\)
0.538246 + 0.842788i \(0.319087\pi\)
\(48\) −742.229 −2.23191
\(49\) 115.961 0.338079
\(50\) 287.369 0.812802
\(51\) 215.372 0.591335
\(52\) 0 0
\(53\) 398.219 1.03207 0.516034 0.856568i \(-0.327408\pi\)
0.516034 + 0.856568i \(0.327408\pi\)
\(54\) −148.118 −0.373264
\(55\) −253.745 −0.622089
\(56\) 1656.46 3.95274
\(57\) 306.403 0.712001
\(58\) 224.155 0.507464
\(59\) −208.497 −0.460067 −0.230034 0.973183i \(-0.573884\pi\)
−0.230034 + 0.973183i \(0.573884\pi\)
\(60\) −882.799 −1.89948
\(61\) 546.936 1.14800 0.574000 0.818855i \(-0.305391\pi\)
0.574000 + 0.818855i \(0.305391\pi\)
\(62\) −33.2191 −0.0680456
\(63\) 192.810 0.385585
\(64\) 2073.06 4.04895
\(65\) 0 0
\(66\) 313.548 0.584774
\(67\) −678.268 −1.23677 −0.618386 0.785875i \(-0.712213\pi\)
−0.618386 + 0.785875i \(0.712213\pi\)
\(68\) −1586.18 −2.82871
\(69\) 113.490 0.198009
\(70\) 1565.27 2.67264
\(71\) 957.777 1.60095 0.800474 0.599368i \(-0.204581\pi\)
0.800474 + 0.599368i \(0.204581\pi\)
\(72\) 695.880 1.13903
\(73\) 270.360 0.433469 0.216734 0.976231i \(-0.430459\pi\)
0.216734 + 0.976231i \(0.430459\pi\)
\(74\) −1567.21 −2.46195
\(75\) −157.151 −0.241950
\(76\) −2256.60 −3.40592
\(77\) −408.157 −0.604076
\(78\) 0 0
\(79\) −1032.86 −1.47096 −0.735482 0.677544i \(-0.763044\pi\)
−0.735482 + 0.677544i \(0.763044\pi\)
\(80\) 3295.14 4.60510
\(81\) 81.0000 0.111111
\(82\) −1880.03 −2.53188
\(83\) −1065.90 −1.40961 −0.704806 0.709400i \(-0.748966\pi\)
−0.704806 + 0.709400i \(0.748966\pi\)
\(84\) −1420.01 −1.84448
\(85\) −956.147 −1.22010
\(86\) 1681.18 2.10798
\(87\) −122.582 −0.151059
\(88\) −1473.10 −1.78446
\(89\) 427.185 0.508782 0.254391 0.967101i \(-0.418125\pi\)
0.254391 + 0.967101i \(0.418125\pi\)
\(90\) 657.571 0.770156
\(91\) 0 0
\(92\) −835.837 −0.947196
\(93\) 18.1663 0.0202554
\(94\) 1902.83 2.08790
\(95\) −1360.28 −1.46907
\(96\) −2216.07 −2.35601
\(97\) 698.084 0.730718 0.365359 0.930867i \(-0.380946\pi\)
0.365359 + 0.930867i \(0.380946\pi\)
\(98\) 636.144 0.655717
\(99\) −171.468 −0.174072
\(100\) 1157.39 1.15739
\(101\) 88.7364 0.0874218 0.0437109 0.999044i \(-0.486082\pi\)
0.0437109 + 0.999044i \(0.486082\pi\)
\(102\) 1181.50 1.14692
\(103\) −1427.61 −1.36570 −0.682849 0.730560i \(-0.739259\pi\)
−0.682849 + 0.730560i \(0.739259\pi\)
\(104\) 0 0
\(105\) −855.985 −0.795577
\(106\) 2184.57 2.00173
\(107\) −15.0233 −0.0135735 −0.00678673 0.999977i \(-0.502160\pi\)
−0.00678673 + 0.999977i \(0.502160\pi\)
\(108\) −596.551 −0.531510
\(109\) 2053.56 1.80455 0.902274 0.431163i \(-0.141896\pi\)
0.902274 + 0.431163i \(0.141896\pi\)
\(110\) −1392.00 −1.20657
\(111\) 857.046 0.732857
\(112\) 5300.35 4.47175
\(113\) 717.456 0.597280 0.298640 0.954366i \(-0.403467\pi\)
0.298640 + 0.954366i \(0.403467\pi\)
\(114\) 1680.88 1.38095
\(115\) −503.843 −0.408553
\(116\) 902.792 0.722605
\(117\) 0 0
\(118\) −1143.78 −0.892318
\(119\) −1538.00 −1.18477
\(120\) −3089.37 −2.35016
\(121\) −968.023 −0.727290
\(122\) 3000.41 2.22659
\(123\) 1028.12 0.753675
\(124\) −133.791 −0.0968937
\(125\) −967.143 −0.692031
\(126\) 1057.73 0.747856
\(127\) −117.640 −0.0821955 −0.0410978 0.999155i \(-0.513086\pi\)
−0.0410978 + 0.999155i \(0.513086\pi\)
\(128\) 5462.96 3.77236
\(129\) −919.375 −0.627492
\(130\) 0 0
\(131\) −262.376 −0.174991 −0.0874957 0.996165i \(-0.527886\pi\)
−0.0874957 + 0.996165i \(0.527886\pi\)
\(132\) 1262.83 0.832690
\(133\) −2188.06 −1.42653
\(134\) −3720.87 −2.39876
\(135\) −359.601 −0.229256
\(136\) −5550.85 −3.49986
\(137\) −1317.41 −0.821563 −0.410782 0.911734i \(-0.634744\pi\)
−0.410782 + 0.911734i \(0.634744\pi\)
\(138\) 622.591 0.384047
\(139\) −6.43478 −0.00392656 −0.00196328 0.999998i \(-0.500625\pi\)
−0.00196328 + 0.999998i \(0.500625\pi\)
\(140\) 6304.18 3.80572
\(141\) −1040.59 −0.621513
\(142\) 5254.22 3.10510
\(143\) 0 0
\(144\) 2226.69 1.28859
\(145\) 544.203 0.311680
\(146\) 1483.15 0.840729
\(147\) −347.883 −0.195190
\(148\) −6311.99 −3.50569
\(149\) 548.312 0.301473 0.150736 0.988574i \(-0.451836\pi\)
0.150736 + 0.988574i \(0.451836\pi\)
\(150\) −862.107 −0.469272
\(151\) 1485.90 0.800799 0.400399 0.916341i \(-0.368871\pi\)
0.400399 + 0.916341i \(0.368871\pi\)
\(152\) −7897.03 −4.21404
\(153\) −646.116 −0.341408
\(154\) −2239.09 −1.17163
\(155\) −80.6494 −0.0417930
\(156\) 0 0
\(157\) 2132.36 1.08396 0.541978 0.840393i \(-0.317676\pi\)
0.541978 + 0.840393i \(0.317676\pi\)
\(158\) −5666.13 −2.85299
\(159\) −1194.66 −0.595865
\(160\) 9838.28 4.86115
\(161\) −810.450 −0.396723
\(162\) 444.353 0.215504
\(163\) 480.633 0.230957 0.115479 0.993310i \(-0.463160\pi\)
0.115479 + 0.993310i \(0.463160\pi\)
\(164\) −7571.88 −3.60527
\(165\) 761.234 0.359163
\(166\) −5847.36 −2.73400
\(167\) −2919.71 −1.35290 −0.676448 0.736490i \(-0.736482\pi\)
−0.676448 + 0.736490i \(0.736482\pi\)
\(168\) −4969.37 −2.28211
\(169\) 0 0
\(170\) −5245.27 −2.36643
\(171\) −919.209 −0.411074
\(172\) 6771.04 3.00167
\(173\) −2339.62 −1.02820 −0.514098 0.857731i \(-0.671873\pi\)
−0.514098 + 0.857731i \(0.671873\pi\)
\(174\) −672.464 −0.292985
\(175\) 1122.24 0.484761
\(176\) −4713.64 −2.01877
\(177\) 625.490 0.265620
\(178\) 2343.47 0.986802
\(179\) −4558.71 −1.90354 −0.951772 0.306808i \(-0.900739\pi\)
−0.951772 + 0.306808i \(0.900739\pi\)
\(180\) 2648.40 1.09667
\(181\) −3524.56 −1.44739 −0.723696 0.690118i \(-0.757558\pi\)
−0.723696 + 0.690118i \(0.757558\pi\)
\(182\) 0 0
\(183\) −1640.81 −0.662798
\(184\) −2925.03 −1.17193
\(185\) −3804.87 −1.51211
\(186\) 99.6573 0.0392862
\(187\) 1367.75 0.534866
\(188\) 7663.75 2.97306
\(189\) −578.431 −0.222617
\(190\) −7462.29 −2.84932
\(191\) 3343.27 1.26655 0.633273 0.773929i \(-0.281711\pi\)
0.633273 + 0.773929i \(0.281711\pi\)
\(192\) −6219.18 −2.33766
\(193\) 3341.33 1.24619 0.623094 0.782147i \(-0.285876\pi\)
0.623094 + 0.782147i \(0.285876\pi\)
\(194\) 3829.58 1.41726
\(195\) 0 0
\(196\) 2562.10 0.933709
\(197\) 3653.95 1.32149 0.660745 0.750611i \(-0.270241\pi\)
0.660745 + 0.750611i \(0.270241\pi\)
\(198\) −940.645 −0.337620
\(199\) 1126.55 0.401301 0.200651 0.979663i \(-0.435694\pi\)
0.200651 + 0.979663i \(0.435694\pi\)
\(200\) 4050.31 1.43200
\(201\) 2034.81 0.714050
\(202\) 486.794 0.169558
\(203\) 875.371 0.302655
\(204\) 4758.53 1.63315
\(205\) −4564.33 −1.55506
\(206\) −7831.66 −2.64882
\(207\) −340.471 −0.114321
\(208\) 0 0
\(209\) 1945.86 0.644009
\(210\) −4695.80 −1.54305
\(211\) −74.5243 −0.0243150 −0.0121575 0.999926i \(-0.503870\pi\)
−0.0121575 + 0.999926i \(0.503870\pi\)
\(212\) 8798.44 2.85037
\(213\) −2873.33 −0.924307
\(214\) −82.4156 −0.0263262
\(215\) 4081.58 1.29471
\(216\) −2087.64 −0.657620
\(217\) −129.728 −0.0405829
\(218\) 11265.5 3.49999
\(219\) −811.079 −0.250263
\(220\) −5606.35 −1.71809
\(221\) 0 0
\(222\) 4701.62 1.42140
\(223\) −3178.62 −0.954512 −0.477256 0.878764i \(-0.658369\pi\)
−0.477256 + 0.878764i \(0.658369\pi\)
\(224\) 15825.2 4.72039
\(225\) 471.454 0.139690
\(226\) 3935.85 1.15845
\(227\) 1349.76 0.394655 0.197327 0.980338i \(-0.436774\pi\)
0.197327 + 0.980338i \(0.436774\pi\)
\(228\) 6769.81 1.96641
\(229\) 4821.46 1.39131 0.695657 0.718374i \(-0.255113\pi\)
0.695657 + 0.718374i \(0.255113\pi\)
\(230\) −2764.00 −0.792404
\(231\) 1224.47 0.348763
\(232\) 3159.34 0.894055
\(233\) −2400.88 −0.675050 −0.337525 0.941317i \(-0.609590\pi\)
−0.337525 + 0.941317i \(0.609590\pi\)
\(234\) 0 0
\(235\) 4619.71 1.28237
\(236\) −4606.63 −1.27062
\(237\) 3098.59 0.849262
\(238\) −8437.21 −2.29791
\(239\) 1880.85 0.509047 0.254523 0.967067i \(-0.418081\pi\)
0.254523 + 0.967067i \(0.418081\pi\)
\(240\) −9885.41 −2.65875
\(241\) −5435.01 −1.45270 −0.726349 0.687327i \(-0.758784\pi\)
−0.726349 + 0.687327i \(0.758784\pi\)
\(242\) −5310.42 −1.41061
\(243\) −243.000 −0.0641500
\(244\) 12084.3 3.17056
\(245\) 1544.43 0.402736
\(246\) 5640.08 1.46178
\(247\) 0 0
\(248\) −468.205 −0.119883
\(249\) 3197.70 0.813840
\(250\) −5305.59 −1.34222
\(251\) 1256.70 0.316024 0.158012 0.987437i \(-0.449492\pi\)
0.158012 + 0.987437i \(0.449492\pi\)
\(252\) 4260.04 1.06491
\(253\) 720.739 0.179101
\(254\) −645.353 −0.159421
\(255\) 2868.44 0.704426
\(256\) 13384.5 3.26769
\(257\) 5504.87 1.33612 0.668062 0.744105i \(-0.267124\pi\)
0.668062 + 0.744105i \(0.267124\pi\)
\(258\) −5043.55 −1.21705
\(259\) −6120.27 −1.46832
\(260\) 0 0
\(261\) 367.745 0.0872139
\(262\) −1439.35 −0.339402
\(263\) 2032.44 0.476522 0.238261 0.971201i \(-0.423423\pi\)
0.238261 + 0.971201i \(0.423423\pi\)
\(264\) 4419.29 1.03026
\(265\) 5303.70 1.22945
\(266\) −12003.4 −2.76682
\(267\) −1281.56 −0.293745
\(268\) −14986.0 −3.41572
\(269\) 5523.82 1.25202 0.626009 0.779816i \(-0.284687\pi\)
0.626009 + 0.779816i \(0.284687\pi\)
\(270\) −1972.71 −0.444650
\(271\) −3918.69 −0.878389 −0.439194 0.898392i \(-0.644736\pi\)
−0.439194 + 0.898392i \(0.644736\pi\)
\(272\) −17761.7 −3.95941
\(273\) 0 0
\(274\) −7227.12 −1.59345
\(275\) −998.013 −0.218845
\(276\) 2507.51 0.546864
\(277\) −2944.56 −0.638705 −0.319353 0.947636i \(-0.603465\pi\)
−0.319353 + 0.947636i \(0.603465\pi\)
\(278\) −35.3002 −0.00761570
\(279\) −54.4988 −0.0116945
\(280\) 22061.6 4.70869
\(281\) 5048.76 1.07183 0.535914 0.844273i \(-0.319967\pi\)
0.535914 + 0.844273i \(0.319967\pi\)
\(282\) −5708.50 −1.20545
\(283\) 4491.66 0.943469 0.471734 0.881741i \(-0.343628\pi\)
0.471734 + 0.881741i \(0.343628\pi\)
\(284\) 21161.6 4.42151
\(285\) 4080.84 0.848169
\(286\) 0 0
\(287\) −7341.90 −1.51003
\(288\) 6648.21 1.36024
\(289\) 240.893 0.0490318
\(290\) 2985.41 0.604515
\(291\) −2094.25 −0.421880
\(292\) 5973.45 1.19716
\(293\) 4754.86 0.948061 0.474031 0.880508i \(-0.342799\pi\)
0.474031 + 0.880508i \(0.342799\pi\)
\(294\) −1908.43 −0.378578
\(295\) −2776.88 −0.548054
\(296\) −22088.9 −4.33747
\(297\) 514.403 0.100501
\(298\) 3007.95 0.584718
\(299\) 0 0
\(300\) −3472.17 −0.668220
\(301\) 6565.38 1.25722
\(302\) 8151.40 1.55318
\(303\) −266.209 −0.0504730
\(304\) −25269.0 −4.76736
\(305\) 7284.40 1.36755
\(306\) −3544.49 −0.662173
\(307\) 1623.31 0.301783 0.150891 0.988550i \(-0.451786\pi\)
0.150891 + 0.988550i \(0.451786\pi\)
\(308\) −9018.02 −1.66834
\(309\) 4282.84 0.788486
\(310\) −442.430 −0.0810592
\(311\) 6683.49 1.21860 0.609302 0.792938i \(-0.291450\pi\)
0.609302 + 0.792938i \(0.291450\pi\)
\(312\) 0 0
\(313\) 1584.55 0.286147 0.143073 0.989712i \(-0.454301\pi\)
0.143073 + 0.989712i \(0.454301\pi\)
\(314\) 11697.8 2.10237
\(315\) 2567.95 0.459327
\(316\) −22820.6 −4.06252
\(317\) 787.932 0.139605 0.0698023 0.997561i \(-0.477763\pi\)
0.0698023 + 0.997561i \(0.477763\pi\)
\(318\) −6553.70 −1.15570
\(319\) −778.474 −0.136634
\(320\) 27610.2 4.82330
\(321\) 45.0700 0.00783664
\(322\) −4446.00 −0.769459
\(323\) 7332.28 1.26309
\(324\) 1789.65 0.306868
\(325\) 0 0
\(326\) 2636.67 0.447951
\(327\) −6160.69 −1.04186
\(328\) −26498.0 −4.46069
\(329\) 7430.97 1.24524
\(330\) 4176.01 0.696611
\(331\) −9493.56 −1.57647 −0.788237 0.615371i \(-0.789006\pi\)
−0.788237 + 0.615371i \(0.789006\pi\)
\(332\) −23550.5 −3.89308
\(333\) −2571.14 −0.423115
\(334\) −16017.1 −2.62399
\(335\) −9033.55 −1.47330
\(336\) −15901.1 −2.58177
\(337\) −4123.06 −0.666461 −0.333230 0.942845i \(-0.608139\pi\)
−0.333230 + 0.942845i \(0.608139\pi\)
\(338\) 0 0
\(339\) −2152.37 −0.344840
\(340\) −21125.6 −3.36969
\(341\) 115.368 0.0183211
\(342\) −5042.63 −0.797294
\(343\) −4863.94 −0.765680
\(344\) 23695.4 3.71386
\(345\) 1511.53 0.235878
\(346\) −12834.8 −1.99423
\(347\) 2320.76 0.359034 0.179517 0.983755i \(-0.442546\pi\)
0.179517 + 0.983755i \(0.442546\pi\)
\(348\) −2708.38 −0.417196
\(349\) 3818.85 0.585726 0.292863 0.956154i \(-0.405392\pi\)
0.292863 + 0.956154i \(0.405392\pi\)
\(350\) 6156.41 0.940212
\(351\) 0 0
\(352\) −14073.5 −2.13102
\(353\) −6065.53 −0.914549 −0.457274 0.889326i \(-0.651174\pi\)
−0.457274 + 0.889326i \(0.651174\pi\)
\(354\) 3431.34 0.515180
\(355\) 12756.2 1.90712
\(356\) 9438.43 1.40516
\(357\) 4613.99 0.684029
\(358\) −25008.4 −3.69199
\(359\) 7406.02 1.08879 0.544394 0.838830i \(-0.316760\pi\)
0.544394 + 0.838830i \(0.316760\pi\)
\(360\) 9268.11 1.35687
\(361\) 3572.42 0.520836
\(362\) −19335.2 −2.80727
\(363\) 2904.07 0.419901
\(364\) 0 0
\(365\) 3600.80 0.516368
\(366\) −9001.22 −1.28552
\(367\) −4754.23 −0.676209 −0.338105 0.941109i \(-0.609786\pi\)
−0.338105 + 0.941109i \(0.609786\pi\)
\(368\) −9359.55 −1.32582
\(369\) −3084.35 −0.435134
\(370\) −20872.9 −2.93279
\(371\) 8531.20 1.19385
\(372\) 401.374 0.0559416
\(373\) −4684.13 −0.650229 −0.325114 0.945675i \(-0.605403\pi\)
−0.325114 + 0.945675i \(0.605403\pi\)
\(374\) 7503.27 1.03739
\(375\) 2901.43 0.399544
\(376\) 26819.4 3.67847
\(377\) 0 0
\(378\) −3173.18 −0.431775
\(379\) 12103.9 1.64047 0.820233 0.572030i \(-0.193844\pi\)
0.820233 + 0.572030i \(0.193844\pi\)
\(380\) −30054.7 −4.05730
\(381\) 352.919 0.0474556
\(382\) 18340.6 2.45651
\(383\) −5601.56 −0.747327 −0.373663 0.927564i \(-0.621898\pi\)
−0.373663 + 0.927564i \(0.621898\pi\)
\(384\) −16388.9 −2.17797
\(385\) −5436.06 −0.719604
\(386\) 18330.0 2.41703
\(387\) 2758.13 0.362283
\(388\) 15423.8 2.01810
\(389\) 9450.46 1.23177 0.615883 0.787837i \(-0.288799\pi\)
0.615883 + 0.787837i \(0.288799\pi\)
\(390\) 0 0
\(391\) 2715.85 0.351270
\(392\) 8966.11 1.15525
\(393\) 787.127 0.101031
\(394\) 20045.0 2.56308
\(395\) −13756.2 −1.75228
\(396\) −3788.49 −0.480754
\(397\) −2723.98 −0.344364 −0.172182 0.985065i \(-0.555082\pi\)
−0.172182 + 0.985065i \(0.555082\pi\)
\(398\) 6180.07 0.778339
\(399\) 6564.18 0.823610
\(400\) 12960.2 1.62003
\(401\) 4680.46 0.582870 0.291435 0.956591i \(-0.405867\pi\)
0.291435 + 0.956591i \(0.405867\pi\)
\(402\) 11162.6 1.38493
\(403\) 0 0
\(404\) 1960.58 0.241442
\(405\) 1078.80 0.132361
\(406\) 4802.15 0.587011
\(407\) 5442.80 0.662874
\(408\) 16652.6 2.02065
\(409\) −3272.70 −0.395659 −0.197830 0.980236i \(-0.563389\pi\)
−0.197830 + 0.980236i \(0.563389\pi\)
\(410\) −25039.2 −3.01609
\(411\) 3952.24 0.474330
\(412\) −31542.3 −3.77179
\(413\) −4466.71 −0.532185
\(414\) −1867.77 −0.221729
\(415\) −14196.2 −1.67920
\(416\) 0 0
\(417\) 19.3043 0.00226700
\(418\) 10674.7 1.24908
\(419\) 1501.48 0.175065 0.0875323 0.996162i \(-0.472102\pi\)
0.0875323 + 0.996162i \(0.472102\pi\)
\(420\) −18912.5 −2.19723
\(421\) −16578.1 −1.91916 −0.959580 0.281436i \(-0.909189\pi\)
−0.959580 + 0.281436i \(0.909189\pi\)
\(422\) −408.828 −0.0471598
\(423\) 3121.76 0.358831
\(424\) 30790.3 3.52667
\(425\) −3760.66 −0.429221
\(426\) −15762.6 −1.79273
\(427\) 11717.2 1.32795
\(428\) −331.933 −0.0374873
\(429\) 0 0
\(430\) 22390.9 2.51113
\(431\) −3776.55 −0.422065 −0.211032 0.977479i \(-0.567683\pi\)
−0.211032 + 0.977479i \(0.567683\pi\)
\(432\) −6680.06 −0.743969
\(433\) −709.953 −0.0787948 −0.0393974 0.999224i \(-0.512544\pi\)
−0.0393974 + 0.999224i \(0.512544\pi\)
\(434\) −711.665 −0.0787120
\(435\) −1632.61 −0.179949
\(436\) 45372.4 4.98382
\(437\) 3863.76 0.422949
\(438\) −4449.45 −0.485395
\(439\) −16262.2 −1.76800 −0.884002 0.467484i \(-0.845161\pi\)
−0.884002 + 0.467484i \(0.845161\pi\)
\(440\) −19619.5 −2.12574
\(441\) 1043.65 0.112693
\(442\) 0 0
\(443\) 2899.45 0.310964 0.155482 0.987839i \(-0.450307\pi\)
0.155482 + 0.987839i \(0.450307\pi\)
\(444\) 18936.0 2.02401
\(445\) 5689.49 0.606085
\(446\) −17437.4 −1.85131
\(447\) −1644.94 −0.174055
\(448\) 44412.0 4.68363
\(449\) 7817.78 0.821701 0.410851 0.911703i \(-0.365232\pi\)
0.410851 + 0.911703i \(0.365232\pi\)
\(450\) 2586.32 0.270934
\(451\) 6529.20 0.681703
\(452\) 15851.8 1.64957
\(453\) −4457.69 −0.462341
\(454\) 7404.57 0.765448
\(455\) 0 0
\(456\) 23691.1 2.43297
\(457\) 14452.9 1.47939 0.739694 0.672943i \(-0.234970\pi\)
0.739694 + 0.672943i \(0.234970\pi\)
\(458\) 26449.8 2.69851
\(459\) 1938.35 0.197112
\(460\) −11132.1 −1.12834
\(461\) −2891.46 −0.292123 −0.146061 0.989276i \(-0.546660\pi\)
−0.146061 + 0.989276i \(0.546660\pi\)
\(462\) 6717.26 0.676440
\(463\) −9223.83 −0.925848 −0.462924 0.886398i \(-0.653200\pi\)
−0.462924 + 0.886398i \(0.653200\pi\)
\(464\) 10109.3 1.01145
\(465\) 241.948 0.0241292
\(466\) −13170.8 −1.30929
\(467\) −7988.25 −0.791546 −0.395773 0.918348i \(-0.629523\pi\)
−0.395773 + 0.918348i \(0.629523\pi\)
\(468\) 0 0
\(469\) −14530.8 −1.43064
\(470\) 25343.0 2.48720
\(471\) −6397.09 −0.625822
\(472\) −16121.0 −1.57209
\(473\) −5838.64 −0.567570
\(474\) 16998.4 1.64718
\(475\) −5350.18 −0.516806
\(476\) −33981.2 −3.27212
\(477\) 3583.97 0.344023
\(478\) 10318.1 0.987315
\(479\) 6908.64 0.659006 0.329503 0.944155i \(-0.393119\pi\)
0.329503 + 0.944155i \(0.393119\pi\)
\(480\) −29514.8 −2.80659
\(481\) 0 0
\(482\) −29815.6 −2.81756
\(483\) 2431.35 0.229048
\(484\) −21387.9 −2.00863
\(485\) 9297.47 0.870466
\(486\) −1333.06 −0.124421
\(487\) −13455.0 −1.25196 −0.625978 0.779841i \(-0.715300\pi\)
−0.625978 + 0.779841i \(0.715300\pi\)
\(488\) 42289.1 3.92282
\(489\) −1441.90 −0.133343
\(490\) 8472.52 0.781121
\(491\) −1044.02 −0.0959589 −0.0479795 0.998848i \(-0.515278\pi\)
−0.0479795 + 0.998848i \(0.515278\pi\)
\(492\) 22715.7 2.08151
\(493\) −2933.40 −0.267979
\(494\) 0 0
\(495\) −2283.70 −0.207363
\(496\) −1498.17 −0.135625
\(497\) 20518.8 1.85190
\(498\) 17542.1 1.57847
\(499\) 16190.7 1.45249 0.726246 0.687435i \(-0.241263\pi\)
0.726246 + 0.687435i \(0.241263\pi\)
\(500\) −21368.5 −1.91126
\(501\) 8759.12 0.781095
\(502\) 6894.04 0.612940
\(503\) −11854.8 −1.05086 −0.525429 0.850838i \(-0.676095\pi\)
−0.525429 + 0.850838i \(0.676095\pi\)
\(504\) 14908.1 1.31758
\(505\) 1181.84 0.104141
\(506\) 3953.86 0.347372
\(507\) 0 0
\(508\) −2599.19 −0.227008
\(509\) −6647.07 −0.578834 −0.289417 0.957203i \(-0.593461\pi\)
−0.289417 + 0.957203i \(0.593461\pi\)
\(510\) 15735.8 1.36626
\(511\) 5792.02 0.501416
\(512\) 29721.4 2.56545
\(513\) 2757.63 0.237334
\(514\) 30198.8 2.59146
\(515\) −19013.7 −1.62688
\(516\) −20313.1 −1.73301
\(517\) −6608.42 −0.562162
\(518\) −33574.8 −2.84786
\(519\) 7018.86 0.593630
\(520\) 0 0
\(521\) 16839.6 1.41604 0.708019 0.706193i \(-0.249589\pi\)
0.708019 + 0.706193i \(0.249589\pi\)
\(522\) 2017.39 0.169155
\(523\) 10414.1 0.870701 0.435351 0.900261i \(-0.356624\pi\)
0.435351 + 0.900261i \(0.356624\pi\)
\(524\) −5797.05 −0.483293
\(525\) −3366.71 −0.279877
\(526\) 11149.6 0.924233
\(527\) 434.723 0.0359332
\(528\) 14140.9 1.16554
\(529\) −10735.9 −0.882377
\(530\) 29095.3 2.38456
\(531\) −1876.47 −0.153356
\(532\) −48344.1 −3.93981
\(533\) 0 0
\(534\) −7030.42 −0.569730
\(535\) −200.089 −0.0161694
\(536\) −52443.7 −4.22616
\(537\) 13676.1 1.09901
\(538\) 30302.8 2.42834
\(539\) −2209.29 −0.176550
\(540\) −7945.19 −0.633160
\(541\) 5464.03 0.434227 0.217114 0.976146i \(-0.430336\pi\)
0.217114 + 0.976146i \(0.430336\pi\)
\(542\) −21497.3 −1.70367
\(543\) 10573.7 0.835653
\(544\) −53031.0 −4.17957
\(545\) 27350.5 2.14966
\(546\) 0 0
\(547\) 24902.3 1.94652 0.973261 0.229702i \(-0.0737753\pi\)
0.973261 + 0.229702i \(0.0737753\pi\)
\(548\) −29107.5 −2.26900
\(549\) 4922.43 0.382667
\(550\) −5474.94 −0.424459
\(551\) −4173.26 −0.322662
\(552\) 8775.08 0.676617
\(553\) −22127.4 −1.70154
\(554\) −16153.4 −1.23879
\(555\) 11414.6 0.873014
\(556\) −142.173 −0.0108444
\(557\) 9817.71 0.746840 0.373420 0.927662i \(-0.378185\pi\)
0.373420 + 0.927662i \(0.378185\pi\)
\(558\) −298.972 −0.0226819
\(559\) 0 0
\(560\) 70593.0 5.32696
\(561\) −4103.26 −0.308805
\(562\) 27696.7 2.07885
\(563\) −23333.7 −1.74671 −0.873356 0.487083i \(-0.838061\pi\)
−0.873356 + 0.487083i \(0.838061\pi\)
\(564\) −22991.2 −1.71650
\(565\) 9555.48 0.711508
\(566\) 24640.6 1.82989
\(567\) 1735.29 0.128528
\(568\) 74055.4 5.47059
\(569\) 24543.9 1.80832 0.904160 0.427194i \(-0.140498\pi\)
0.904160 + 0.427194i \(0.140498\pi\)
\(570\) 22386.9 1.64506
\(571\) −10562.5 −0.774126 −0.387063 0.922053i \(-0.626510\pi\)
−0.387063 + 0.922053i \(0.626510\pi\)
\(572\) 0 0
\(573\) −10029.8 −0.731241
\(574\) −40276.5 −2.92876
\(575\) −1981.69 −0.143725
\(576\) 18657.5 1.34965
\(577\) 18922.5 1.36526 0.682629 0.730765i \(-0.260837\pi\)
0.682629 + 0.730765i \(0.260837\pi\)
\(578\) 1321.50 0.0950991
\(579\) −10024.0 −0.719486
\(580\) 12023.9 0.860801
\(581\) −22835.2 −1.63057
\(582\) −11488.7 −0.818253
\(583\) −7586.85 −0.538963
\(584\) 20904.2 1.48120
\(585\) 0 0
\(586\) 26084.4 1.83880
\(587\) 7989.65 0.561786 0.280893 0.959739i \(-0.409369\pi\)
0.280893 + 0.959739i \(0.409369\pi\)
\(588\) −7686.29 −0.539077
\(589\) 618.466 0.0432656
\(590\) −15233.5 −1.06297
\(591\) −10961.9 −0.762962
\(592\) −70680.4 −4.90701
\(593\) 25239.2 1.74781 0.873905 0.486097i \(-0.161580\pi\)
0.873905 + 0.486097i \(0.161580\pi\)
\(594\) 2821.93 0.194925
\(595\) −20483.9 −1.41136
\(596\) 12114.7 0.832610
\(597\) −3379.65 −0.231691
\(598\) 0 0
\(599\) −7412.19 −0.505599 −0.252800 0.967519i \(-0.581351\pi\)
−0.252800 + 0.967519i \(0.581351\pi\)
\(600\) −12150.9 −0.826767
\(601\) −21459.2 −1.45647 −0.728236 0.685327i \(-0.759659\pi\)
−0.728236 + 0.685327i \(0.759659\pi\)
\(602\) 36016.6 2.43842
\(603\) −6104.42 −0.412257
\(604\) 32830.1 2.21165
\(605\) −12892.7 −0.866382
\(606\) −1460.38 −0.0978944
\(607\) −18246.7 −1.22011 −0.610057 0.792358i \(-0.708853\pi\)
−0.610057 + 0.792358i \(0.708853\pi\)
\(608\) −75445.6 −5.03244
\(609\) −2626.11 −0.174738
\(610\) 39961.1 2.65242
\(611\) 0 0
\(612\) −14275.6 −0.942902
\(613\) 24087.5 1.58709 0.793545 0.608512i \(-0.208233\pi\)
0.793545 + 0.608512i \(0.208233\pi\)
\(614\) 8905.23 0.585319
\(615\) 13693.0 0.897813
\(616\) −31558.7 −2.06418
\(617\) 14400.6 0.939621 0.469811 0.882767i \(-0.344322\pi\)
0.469811 + 0.882767i \(0.344322\pi\)
\(618\) 23495.0 1.52930
\(619\) −25635.8 −1.66460 −0.832302 0.554322i \(-0.812978\pi\)
−0.832302 + 0.554322i \(0.812978\pi\)
\(620\) −1781.91 −0.115424
\(621\) 1021.41 0.0660031
\(622\) 36664.6 2.36353
\(623\) 9151.76 0.588535
\(624\) 0 0
\(625\) −19428.9 −1.24345
\(626\) 8692.58 0.554993
\(627\) −5837.58 −0.371819
\(628\) 47113.4 2.99368
\(629\) 20509.3 1.30009
\(630\) 14087.4 0.890881
\(631\) 2757.17 0.173948 0.0869741 0.996211i \(-0.472280\pi\)
0.0869741 + 0.996211i \(0.472280\pi\)
\(632\) −79861.0 −5.02643
\(633\) 223.573 0.0140383
\(634\) 4322.47 0.270768
\(635\) −1566.79 −0.0979152
\(636\) −26395.3 −1.64566
\(637\) 0 0
\(638\) −4270.58 −0.265006
\(639\) 8620.00 0.533649
\(640\) 72758.7 4.49382
\(641\) 14317.0 0.882196 0.441098 0.897459i \(-0.354589\pi\)
0.441098 + 0.897459i \(0.354589\pi\)
\(642\) 247.247 0.0151995
\(643\) 51.9171 0.00318415 0.00159208 0.999999i \(-0.499493\pi\)
0.00159208 + 0.999999i \(0.499493\pi\)
\(644\) −17906.5 −1.09567
\(645\) −12244.7 −0.747498
\(646\) 40223.8 2.44982
\(647\) −9735.07 −0.591538 −0.295769 0.955260i \(-0.595576\pi\)
−0.295769 + 0.955260i \(0.595576\pi\)
\(648\) 6262.92 0.379677
\(649\) 3972.27 0.240255
\(650\) 0 0
\(651\) 389.183 0.0234305
\(652\) 10619.3 0.637860
\(653\) 10842.6 0.649778 0.324889 0.945752i \(-0.394673\pi\)
0.324889 + 0.945752i \(0.394673\pi\)
\(654\) −33796.6 −2.02072
\(655\) −3494.46 −0.208458
\(656\) −84788.5 −5.04640
\(657\) 2433.24 0.144490
\(658\) 40765.1 2.41518
\(659\) 28940.9 1.71074 0.855369 0.518020i \(-0.173331\pi\)
0.855369 + 0.518020i \(0.173331\pi\)
\(660\) 16819.0 0.991940
\(661\) 3108.58 0.182919 0.0914597 0.995809i \(-0.470847\pi\)
0.0914597 + 0.995809i \(0.470847\pi\)
\(662\) −52080.2 −3.05763
\(663\) 0 0
\(664\) −82415.4 −4.81677
\(665\) −29141.8 −1.69935
\(666\) −14104.8 −0.820648
\(667\) −1545.76 −0.0897333
\(668\) −64509.4 −3.73644
\(669\) 9535.86 0.551088
\(670\) −49556.6 −2.85752
\(671\) −10420.2 −0.599505
\(672\) −47475.7 −2.72532
\(673\) −9950.09 −0.569908 −0.284954 0.958541i \(-0.591978\pi\)
−0.284954 + 0.958541i \(0.591978\pi\)
\(674\) −22618.4 −1.29263
\(675\) −1414.36 −0.0806501
\(676\) 0 0
\(677\) −3483.60 −0.197763 −0.0988816 0.995099i \(-0.531527\pi\)
−0.0988816 + 0.995099i \(0.531527\pi\)
\(678\) −11807.6 −0.668830
\(679\) 14955.3 0.845261
\(680\) −73929.3 −4.16920
\(681\) −4049.28 −0.227854
\(682\) 632.889 0.0355346
\(683\) −16811.4 −0.941829 −0.470915 0.882179i \(-0.656076\pi\)
−0.470915 + 0.882179i \(0.656076\pi\)
\(684\) −20309.4 −1.13531
\(685\) −17546.0 −0.978685
\(686\) −26682.8 −1.48506
\(687\) −14464.4 −0.803276
\(688\) 75820.8 4.20151
\(689\) 0 0
\(690\) 8292.01 0.457495
\(691\) −3078.78 −0.169497 −0.0847484 0.996402i \(-0.527009\pi\)
−0.0847484 + 0.996402i \(0.527009\pi\)
\(692\) −51692.6 −2.83968
\(693\) −3673.42 −0.201359
\(694\) 12731.3 0.696361
\(695\) −85.7020 −0.00467750
\(696\) −9478.01 −0.516183
\(697\) 24603.0 1.33702
\(698\) 20949.6 1.13604
\(699\) 7202.63 0.389740
\(700\) 24795.2 1.33882
\(701\) −18608.6 −1.00262 −0.501311 0.865267i \(-0.667149\pi\)
−0.501311 + 0.865267i \(0.667149\pi\)
\(702\) 0 0
\(703\) 29177.9 1.56539
\(704\) −39495.9 −2.11443
\(705\) −13859.1 −0.740376
\(706\) −33274.5 −1.77380
\(707\) 1901.03 0.101126
\(708\) 13819.9 0.733591
\(709\) 19046.2 1.00888 0.504439 0.863448i \(-0.331699\pi\)
0.504439 + 0.863448i \(0.331699\pi\)
\(710\) 69978.5 3.69894
\(711\) −9295.77 −0.490322
\(712\) 33030.0 1.73855
\(713\) 229.078 0.0120323
\(714\) 25311.6 1.32670
\(715\) 0 0
\(716\) −100722. −5.25722
\(717\) −5642.55 −0.293898
\(718\) 40628.2 2.11174
\(719\) −14013.4 −0.726859 −0.363430 0.931622i \(-0.618394\pi\)
−0.363430 + 0.931622i \(0.618394\pi\)
\(720\) 29656.2 1.53503
\(721\) −30584.3 −1.57978
\(722\) 19597.7 1.01018
\(723\) 16305.0 0.838715
\(724\) −77873.2 −3.99742
\(725\) 2140.43 0.109646
\(726\) 15931.3 0.814414
\(727\) 2578.98 0.131567 0.0657834 0.997834i \(-0.479045\pi\)
0.0657834 + 0.997834i \(0.479045\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 19753.4 1.00152
\(731\) −22000.8 −1.11317
\(732\) −36252.8 −1.83052
\(733\) 5029.12 0.253417 0.126709 0.991940i \(-0.459559\pi\)
0.126709 + 0.991940i \(0.459559\pi\)
\(734\) −26081.0 −1.31153
\(735\) −4633.30 −0.232520
\(736\) −27944.8 −1.39954
\(737\) 12922.3 0.645863
\(738\) −16920.2 −0.843960
\(739\) −23856.4 −1.18751 −0.593757 0.804644i \(-0.702356\pi\)
−0.593757 + 0.804644i \(0.702356\pi\)
\(740\) −84066.5 −4.17614
\(741\) 0 0
\(742\) 46800.8 2.31551
\(743\) 4887.62 0.241332 0.120666 0.992693i \(-0.461497\pi\)
0.120666 + 0.992693i \(0.461497\pi\)
\(744\) 1404.62 0.0692147
\(745\) 7302.72 0.359129
\(746\) −25696.4 −1.26114
\(747\) −9593.10 −0.469870
\(748\) 30219.8 1.47720
\(749\) −321.851 −0.0157012
\(750\) 15916.8 0.774932
\(751\) 24814.3 1.20571 0.602854 0.797851i \(-0.294030\pi\)
0.602854 + 0.797851i \(0.294030\pi\)
\(752\) 85817.2 4.16148
\(753\) −3770.09 −0.182456
\(754\) 0 0
\(755\) 19790.0 0.953949
\(756\) −12780.1 −0.614826
\(757\) −6552.91 −0.314623 −0.157311 0.987549i \(-0.550283\pi\)
−0.157311 + 0.987549i \(0.550283\pi\)
\(758\) 66400.2 3.18175
\(759\) −2162.22 −0.103404
\(760\) −105177. −5.01996
\(761\) 20351.4 0.969431 0.484715 0.874672i \(-0.338923\pi\)
0.484715 + 0.874672i \(0.338923\pi\)
\(762\) 1936.06 0.0920420
\(763\) 43994.3 2.08742
\(764\) 73867.7 3.49796
\(765\) −8605.32 −0.406701
\(766\) −30729.3 −1.44947
\(767\) 0 0
\(768\) −40153.4 −1.88660
\(769\) −22209.7 −1.04148 −0.520742 0.853714i \(-0.674345\pi\)
−0.520742 + 0.853714i \(0.674345\pi\)
\(770\) −29821.4 −1.39570
\(771\) −16514.6 −0.771412
\(772\) 73824.9 3.44173
\(773\) 28496.6 1.32594 0.662971 0.748645i \(-0.269295\pi\)
0.662971 + 0.748645i \(0.269295\pi\)
\(774\) 15130.6 0.702661
\(775\) −317.206 −0.0147024
\(776\) 53975.9 2.49693
\(777\) 18360.8 0.847735
\(778\) 51843.7 2.38906
\(779\) 35001.9 1.60985
\(780\) 0 0
\(781\) −18247.5 −0.836041
\(782\) 14898.7 0.681301
\(783\) −1103.23 −0.0503530
\(784\) 28689.9 1.30694
\(785\) 28400.0 1.29126
\(786\) 4318.05 0.195954
\(787\) −2394.07 −0.108436 −0.0542181 0.998529i \(-0.517267\pi\)
−0.0542181 + 0.998529i \(0.517267\pi\)
\(788\) 80732.2 3.64970
\(789\) −6097.31 −0.275120
\(790\) −75464.6 −3.39862
\(791\) 15370.3 0.690906
\(792\) −13257.9 −0.594821
\(793\) 0 0
\(794\) −14943.3 −0.667907
\(795\) −15911.1 −0.709822
\(796\) 24890.5 1.10832
\(797\) −37996.7 −1.68872 −0.844362 0.535773i \(-0.820020\pi\)
−0.844362 + 0.535773i \(0.820020\pi\)
\(798\) 36010.1 1.59742
\(799\) −24901.5 −1.10257
\(800\) 38695.4 1.71011
\(801\) 3844.67 0.169594
\(802\) 25676.3 1.13050
\(803\) −5150.88 −0.226364
\(804\) 44957.9 1.97207
\(805\) −10794.0 −0.472595
\(806\) 0 0
\(807\) −16571.4 −0.722853
\(808\) 6861.10 0.298729
\(809\) −25993.4 −1.12964 −0.564821 0.825213i \(-0.691055\pi\)
−0.564821 + 0.825213i \(0.691055\pi\)
\(810\) 5918.14 0.256719
\(811\) −15524.4 −0.672176 −0.336088 0.941831i \(-0.609104\pi\)
−0.336088 + 0.941831i \(0.609104\pi\)
\(812\) 19340.9 0.835875
\(813\) 11756.1 0.507138
\(814\) 29858.3 1.28567
\(815\) 6401.33 0.275127
\(816\) 53285.1 2.28597
\(817\) −31299.9 −1.34033
\(818\) −17953.5 −0.767396
\(819\) 0 0
\(820\) −100847. −4.29477
\(821\) −2029.40 −0.0862684 −0.0431342 0.999069i \(-0.513734\pi\)
−0.0431342 + 0.999069i \(0.513734\pi\)
\(822\) 21681.3 0.919980
\(823\) −42010.4 −1.77933 −0.889667 0.456610i \(-0.849064\pi\)
−0.889667 + 0.456610i \(0.849064\pi\)
\(824\) −110383. −4.66672
\(825\) 2994.04 0.126350
\(826\) −24503.6 −1.03219
\(827\) 3941.24 0.165720 0.0828599 0.996561i \(-0.473595\pi\)
0.0828599 + 0.996561i \(0.473595\pi\)
\(828\) −7522.53 −0.315732
\(829\) 11264.7 0.471941 0.235970 0.971760i \(-0.424173\pi\)
0.235970 + 0.971760i \(0.424173\pi\)
\(830\) −77878.4 −3.25686
\(831\) 8833.68 0.368757
\(832\) 0 0
\(833\) −8324.92 −0.346268
\(834\) 105.901 0.00439693
\(835\) −38886.3 −1.61163
\(836\) 42992.7 1.77863
\(837\) 163.496 0.00675181
\(838\) 8236.88 0.339544
\(839\) 35221.1 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(840\) −66184.8 −2.71856
\(841\) −22719.4 −0.931544
\(842\) −90944.8 −3.72228
\(843\) −15146.3 −0.618820
\(844\) −1646.57 −0.0671533
\(845\) 0 0
\(846\) 17125.5 0.695966
\(847\) −20738.3 −0.841295
\(848\) 98523.2 3.98974
\(849\) −13475.0 −0.544712
\(850\) −20630.4 −0.832490
\(851\) 10807.4 0.435338
\(852\) −63484.7 −2.55276
\(853\) −37662.7 −1.51178 −0.755888 0.654701i \(-0.772795\pi\)
−0.755888 + 0.654701i \(0.772795\pi\)
\(854\) 64278.8 2.57562
\(855\) −12242.5 −0.489691
\(856\) −1161.60 −0.0463818
\(857\) 800.368 0.0319020 0.0159510 0.999873i \(-0.494922\pi\)
0.0159510 + 0.999873i \(0.494922\pi\)
\(858\) 0 0
\(859\) −8800.69 −0.349564 −0.174782 0.984607i \(-0.555922\pi\)
−0.174782 + 0.984607i \(0.555922\pi\)
\(860\) 90180.4 3.57573
\(861\) 22025.7 0.871816
\(862\) −20717.6 −0.818611
\(863\) 17991.0 0.709641 0.354821 0.934934i \(-0.384542\pi\)
0.354821 + 0.934934i \(0.384542\pi\)
\(864\) −19944.6 −0.785336
\(865\) −31160.3 −1.22484
\(866\) −3894.69 −0.152826
\(867\) −722.680 −0.0283085
\(868\) −2866.26 −0.112082
\(869\) 19678.1 0.768162
\(870\) −8956.24 −0.349017
\(871\) 0 0
\(872\) 158782. 6.16631
\(873\) 6282.76 0.243573
\(874\) 21196.0 0.820325
\(875\) −20719.5 −0.800510
\(876\) −17920.4 −0.691179
\(877\) 44304.5 1.70588 0.852940 0.522008i \(-0.174817\pi\)
0.852940 + 0.522008i \(0.174817\pi\)
\(878\) −89212.0 −3.42911
\(879\) −14264.6 −0.547363
\(880\) −62778.8 −2.40486
\(881\) 10814.6 0.413567 0.206783 0.978387i \(-0.433700\pi\)
0.206783 + 0.978387i \(0.433700\pi\)
\(882\) 5725.30 0.218572
\(883\) 14530.8 0.553793 0.276897 0.960900i \(-0.410694\pi\)
0.276897 + 0.960900i \(0.410694\pi\)
\(884\) 0 0
\(885\) 8330.63 0.316419
\(886\) 15905.9 0.603126
\(887\) −3385.82 −0.128168 −0.0640838 0.997945i \(-0.520412\pi\)
−0.0640838 + 0.997945i \(0.520412\pi\)
\(888\) 66266.8 2.50424
\(889\) −2520.24 −0.0950800
\(890\) 31211.7 1.17552
\(891\) −1543.21 −0.0580241
\(892\) −70229.9 −2.63618
\(893\) −35426.6 −1.32755
\(894\) −9023.86 −0.337587
\(895\) −60715.4 −2.26759
\(896\) 117035. 4.36369
\(897\) 0 0
\(898\) 42887.1 1.59372
\(899\) −247.428 −0.00917929
\(900\) 10416.5 0.385797
\(901\) −28588.4 −1.05707
\(902\) 35818.2 1.32219
\(903\) −19696.1 −0.725854
\(904\) 55473.7 2.04096
\(905\) −46942.0 −1.72420
\(906\) −24454.2 −0.896729
\(907\) 38174.2 1.39752 0.698762 0.715354i \(-0.253735\pi\)
0.698762 + 0.715354i \(0.253735\pi\)
\(908\) 29822.2 1.08996
\(909\) 798.628 0.0291406
\(910\) 0 0
\(911\) −11699.5 −0.425489 −0.212745 0.977108i \(-0.568240\pi\)
−0.212745 + 0.977108i \(0.568240\pi\)
\(912\) 75807.0 2.75244
\(913\) 20307.5 0.736123
\(914\) 79286.6 2.86933
\(915\) −21853.2 −0.789557
\(916\) 106528. 3.84254
\(917\) −5620.97 −0.202422
\(918\) 10633.5 0.382306
\(919\) 21615.6 0.775879 0.387939 0.921685i \(-0.373187\pi\)
0.387939 + 0.921685i \(0.373187\pi\)
\(920\) −38957.1 −1.39606
\(921\) −4869.93 −0.174234
\(922\) −15862.1 −0.566583
\(923\) 0 0
\(924\) 27054.1 0.963218
\(925\) −14965.1 −0.531945
\(926\) −50600.5 −1.79572
\(927\) −12848.5 −0.455232
\(928\) 30183.3 1.06769
\(929\) 22325.9 0.788471 0.394236 0.919009i \(-0.371009\pi\)
0.394236 + 0.919009i \(0.371009\pi\)
\(930\) 1327.29 0.0467995
\(931\) −11843.6 −0.416926
\(932\) −53046.1 −1.86436
\(933\) −20050.5 −0.703561
\(934\) −43822.3 −1.53523
\(935\) 18216.5 0.637158
\(936\) 0 0
\(937\) −29401.5 −1.02509 −0.512543 0.858661i \(-0.671297\pi\)
−0.512543 + 0.858661i \(0.671297\pi\)
\(938\) −79713.7 −2.77478
\(939\) −4753.65 −0.165207
\(940\) 102070. 3.54166
\(941\) 30280.7 1.04902 0.524508 0.851406i \(-0.324249\pi\)
0.524508 + 0.851406i \(0.324249\pi\)
\(942\) −35093.4 −1.21381
\(943\) 12964.6 0.447704
\(944\) −51584.1 −1.77852
\(945\) −7703.86 −0.265192
\(946\) −32029.8 −1.10082
\(947\) 25226.1 0.865617 0.432809 0.901486i \(-0.357523\pi\)
0.432809 + 0.901486i \(0.357523\pi\)
\(948\) 68461.7 2.34550
\(949\) 0 0
\(950\) −29350.2 −1.00237
\(951\) −2363.80 −0.0806008
\(952\) −118918. −4.04848
\(953\) −11893.7 −0.404275 −0.202137 0.979357i \(-0.564789\pi\)
−0.202137 + 0.979357i \(0.564789\pi\)
\(954\) 19661.1 0.667245
\(955\) 44527.5 1.50877
\(956\) 41556.4 1.40589
\(957\) 2335.42 0.0788855
\(958\) 37899.7 1.27817
\(959\) −28223.4 −0.950346
\(960\) −82830.5 −2.78473
\(961\) −29754.3 −0.998769
\(962\) 0 0
\(963\) −135.210 −0.00452449
\(964\) −120084. −4.01207
\(965\) 44501.7 1.48452
\(966\) 13338.0 0.444247
\(967\) 8534.72 0.283824 0.141912 0.989879i \(-0.454675\pi\)
0.141912 + 0.989879i \(0.454675\pi\)
\(968\) −74847.5 −2.48522
\(969\) −21996.9 −0.729247
\(970\) 51004.4 1.68830
\(971\) 25615.3 0.846585 0.423292 0.905993i \(-0.360874\pi\)
0.423292 + 0.905993i \(0.360874\pi\)
\(972\) −5368.95 −0.177170
\(973\) −137.855 −0.00454206
\(974\) −73811.8 −2.42822
\(975\) 0 0
\(976\) 135317. 4.43791
\(977\) −22995.5 −0.753011 −0.376506 0.926414i \(-0.622874\pi\)
−0.376506 + 0.926414i \(0.622874\pi\)
\(978\) −7910.02 −0.258624
\(979\) −8138.72 −0.265694
\(980\) 34123.4 1.11228
\(981\) 18482.1 0.601516
\(982\) −5727.31 −0.186116
\(983\) −20592.1 −0.668144 −0.334072 0.942548i \(-0.608423\pi\)
−0.334072 + 0.942548i \(0.608423\pi\)
\(984\) 79493.9 2.57538
\(985\) 48665.4 1.57422
\(986\) −16092.2 −0.519756
\(987\) −22292.9 −0.718937
\(988\) 0 0
\(989\) −11593.4 −0.372748
\(990\) −12528.0 −0.402189
\(991\) −11557.9 −0.370485 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(992\) −4473.08 −0.143166
\(993\) 28480.7 0.910178
\(994\) 112563. 3.59183
\(995\) 15004.0 0.478049
\(996\) 70651.5 2.24767
\(997\) −15481.5 −0.491778 −0.245889 0.969298i \(-0.579080\pi\)
−0.245889 + 0.969298i \(0.579080\pi\)
\(998\) 88819.4 2.81716
\(999\) 7713.41 0.244286
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.a.q.1.9 yes 9
3.2 odd 2 1521.4.a.be.1.1 9
13.5 odd 4 507.4.b.j.337.1 18
13.8 odd 4 507.4.b.j.337.18 18
13.12 even 2 507.4.a.n.1.1 9
39.38 odd 2 1521.4.a.bj.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.n.1.1 9 13.12 even 2
507.4.a.q.1.9 yes 9 1.1 even 1 trivial
507.4.b.j.337.1 18 13.5 odd 4
507.4.b.j.337.18 18 13.8 odd 4
1521.4.a.be.1.1 9 3.2 odd 2
1521.4.a.bj.1.9 9 39.38 odd 2