Properties

Label 363.4.a.j.1.1
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $2$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +10.1168 q^{12} +15.0217 q^{13} -16.0000 q^{14} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} -30.3505 q^{18} +78.7011 q^{19} -11.7663 q^{20} +14.2337 q^{21} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} -50.6576 q^{26} +27.0000 q^{27} +16.0000 q^{28} -243.125 q^{29} +35.2989 q^{30} +278.717 q^{31} +143.606 q^{32} +246.745 q^{34} -16.5544 q^{35} +30.3505 q^{36} +102.380 q^{37} -265.402 q^{38} +45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -48.0000 q^{42} +280.016 q^{43} -31.4021 q^{45} -377.696 q^{46} -169.870 q^{47} -238.818 q^{48} -320.489 q^{49} +380.481 q^{50} -219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +74.0435 q^{56} +236.103 q^{57} +819.886 q^{58} +196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +42.7011 q^{63} +152.568 q^{64} -52.4128 q^{65} +900.587 q^{67} -246.745 q^{68} +336.000 q^{69} +55.8260 q^{70} +756.500 q^{71} +140.454 q^{72} +1019.81 q^{73} -345.255 q^{74} -338.478 q^{75} +265.402 q^{76} -151.973 q^{78} +327.549 q^{79} +277.755 q^{80} +81.0000 q^{81} -813.581 q^{82} +756.619 q^{83} +48.0000 q^{84} +255.294 q^{85} -944.293 q^{86} -729.375 q^{87} +508.978 q^{89} +105.897 q^{90} +71.2716 q^{91} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} -274.598 q^{95} +430.818 q^{96} +614.358 q^{97} +1080.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20} - 6 q^{21} + 224 q^{23} - 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} + 32 q^{28} - 222 q^{29} + 174 q^{30} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} + 228 q^{39} - 534 q^{40} + 494 q^{41} - 96 q^{42} + 66 q^{43} + 144 q^{45} - 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} + 985 q^{50} + 78 q^{51} - 94 q^{52} - 84 q^{53} - 27 q^{54} + 240 q^{56} + 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} + 1104 q^{61} - 1696 q^{62} - 18 q^{63} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} - 81 q^{72} + 592 q^{73} - 702 q^{74} + 426 q^{75} + 324 q^{76} + 282 q^{78} + 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} - 348 q^{83} + 96 q^{84} + 2188 q^{85} - 1452 q^{86} - 666 q^{87} + 972 q^{89} + 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} + 824 q^{94} - 756 q^{95} + 741 q^{96} - 1184 q^{97} + 375 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.37228 −1.19228 −0.596141 0.802880i \(-0.703300\pi\)
−0.596141 + 0.802880i \(0.703300\pi\)
\(3\) 3.00000 0.577350
\(4\) 3.37228 0.421535
\(5\) −3.48913 −0.312077 −0.156038 0.987751i \(-0.549872\pi\)
−0.156038 + 0.987751i \(0.549872\pi\)
\(6\) −10.1168 −0.688364
\(7\) 4.74456 0.256182 0.128091 0.991762i \(-0.459115\pi\)
0.128091 + 0.991762i \(0.459115\pi\)
\(8\) 15.6060 0.689693
\(9\) 9.00000 0.333333
\(10\) 11.7663 0.372083
\(11\) 0 0
\(12\) 10.1168 0.243373
\(13\) 15.0217 0.320483 0.160242 0.987078i \(-0.448773\pi\)
0.160242 + 0.987078i \(0.448773\pi\)
\(14\) −16.0000 −0.305441
\(15\) −10.4674 −0.180178
\(16\) −79.6060 −1.24384
\(17\) −73.1684 −1.04388 −0.521940 0.852982i \(-0.674791\pi\)
−0.521940 + 0.852982i \(0.674791\pi\)
\(18\) −30.3505 −0.397427
\(19\) 78.7011 0.950277 0.475138 0.879911i \(-0.342398\pi\)
0.475138 + 0.879911i \(0.342398\pi\)
\(20\) −11.7663 −0.131551
\(21\) 14.2337 0.147907
\(22\) 0 0
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 46.8179 0.398194
\(25\) −112.826 −0.902608
\(26\) −50.6576 −0.382106
\(27\) 27.0000 0.192450
\(28\) 16.0000 0.107990
\(29\) −243.125 −1.55680 −0.778399 0.627769i \(-0.783968\pi\)
−0.778399 + 0.627769i \(0.783968\pi\)
\(30\) 35.2989 0.214822
\(31\) 278.717 1.61481 0.807405 0.589998i \(-0.200871\pi\)
0.807405 + 0.589998i \(0.200871\pi\)
\(32\) 143.606 0.793318
\(33\) 0 0
\(34\) 246.745 1.24460
\(35\) −16.5544 −0.0799486
\(36\) 30.3505 0.140512
\(37\) 102.380 0.454898 0.227449 0.973790i \(-0.426961\pi\)
0.227449 + 0.973790i \(0.426961\pi\)
\(38\) −265.402 −1.13300
\(39\) 45.0652 0.185031
\(40\) −54.4512 −0.215237
\(41\) 241.255 0.918970 0.459485 0.888186i \(-0.348034\pi\)
0.459485 + 0.888186i \(0.348034\pi\)
\(42\) −48.0000 −0.176347
\(43\) 280.016 0.993071 0.496536 0.868016i \(-0.334605\pi\)
0.496536 + 0.868016i \(0.334605\pi\)
\(44\) 0 0
\(45\) −31.4021 −0.104026
\(46\) −377.696 −1.21061
\(47\) −169.870 −0.527192 −0.263596 0.964633i \(-0.584909\pi\)
−0.263596 + 0.964633i \(0.584909\pi\)
\(48\) −238.818 −0.718133
\(49\) −320.489 −0.934371
\(50\) 380.481 1.07616
\(51\) −219.505 −0.602684
\(52\) 50.6576 0.135095
\(53\) −409.652 −1.06170 −0.530849 0.847466i \(-0.678127\pi\)
−0.530849 + 0.847466i \(0.678127\pi\)
\(54\) −91.0516 −0.229455
\(55\) 0 0
\(56\) 74.0435 0.176687
\(57\) 236.103 0.548643
\(58\) 819.886 1.85614
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) −35.2989 −0.0759512
\(61\) 701.359 1.47213 0.736064 0.676912i \(-0.236682\pi\)
0.736064 + 0.676912i \(0.236682\pi\)
\(62\) −939.913 −1.92531
\(63\) 42.7011 0.0853941
\(64\) 152.568 0.297984
\(65\) −52.4128 −0.100015
\(66\) 0 0
\(67\) 900.587 1.64215 0.821076 0.570819i \(-0.193374\pi\)
0.821076 + 0.570819i \(0.193374\pi\)
\(68\) −246.745 −0.440032
\(69\) 336.000 0.586227
\(70\) 55.8260 0.0953212
\(71\) 756.500 1.26451 0.632254 0.774762i \(-0.282130\pi\)
0.632254 + 0.774762i \(0.282130\pi\)
\(72\) 140.454 0.229898
\(73\) 1019.81 1.63507 0.817536 0.575877i \(-0.195339\pi\)
0.817536 + 0.575877i \(0.195339\pi\)
\(74\) −345.255 −0.542367
\(75\) −338.478 −0.521121
\(76\) 265.402 0.400575
\(77\) 0 0
\(78\) −151.973 −0.220609
\(79\) 327.549 0.466483 0.233241 0.972419i \(-0.425067\pi\)
0.233241 + 0.972419i \(0.425067\pi\)
\(80\) 277.755 0.388175
\(81\) 81.0000 0.111111
\(82\) −813.581 −1.09567
\(83\) 756.619 1.00060 0.500300 0.865852i \(-0.333223\pi\)
0.500300 + 0.865852i \(0.333223\pi\)
\(84\) 48.0000 0.0623480
\(85\) 255.294 0.325771
\(86\) −944.293 −1.18402
\(87\) −729.375 −0.898818
\(88\) 0 0
\(89\) 508.978 0.606198 0.303099 0.952959i \(-0.401979\pi\)
0.303099 + 0.952959i \(0.401979\pi\)
\(90\) 105.897 0.124028
\(91\) 71.2716 0.0821022
\(92\) 377.696 0.428016
\(93\) 836.152 0.932311
\(94\) 572.848 0.628561
\(95\) −274.598 −0.296559
\(96\) 430.818 0.458023
\(97\) 614.358 0.643079 0.321539 0.946896i \(-0.395800\pi\)
0.321539 + 0.946896i \(0.395800\pi\)
\(98\) 1080.78 1.11403
\(99\) 0 0
\(100\) −380.481 −0.380481
\(101\) 1015.92 1.00087 0.500434 0.865775i \(-0.333174\pi\)
0.500434 + 0.865775i \(0.333174\pi\)
\(102\) 740.234 0.718569
\(103\) 1102.16 1.05436 0.527181 0.849753i \(-0.323249\pi\)
0.527181 + 0.849753i \(0.323249\pi\)
\(104\) 234.429 0.221035
\(105\) −49.6631 −0.0461583
\(106\) 1381.46 1.26584
\(107\) −1377.58 −1.24463 −0.622315 0.782767i \(-0.713808\pi\)
−0.622315 + 0.782767i \(0.713808\pi\)
\(108\) 91.0516 0.0811245
\(109\) −320.217 −0.281388 −0.140694 0.990053i \(-0.544933\pi\)
−0.140694 + 0.990053i \(0.544933\pi\)
\(110\) 0 0
\(111\) 307.141 0.262636
\(112\) −377.696 −0.318651
\(113\) −1629.45 −1.35651 −0.678254 0.734828i \(-0.737263\pi\)
−0.678254 + 0.734828i \(0.737263\pi\)
\(114\) −796.206 −0.654136
\(115\) −390.782 −0.316875
\(116\) −819.886 −0.656245
\(117\) 135.196 0.106828
\(118\) −660.967 −0.515652
\(119\) −347.152 −0.267423
\(120\) −163.354 −0.124267
\(121\) 0 0
\(122\) −2365.18 −1.75519
\(123\) 723.766 0.530568
\(124\) 939.913 0.680699
\(125\) 829.805 0.593760
\(126\) −144.000 −0.101814
\(127\) −2291.26 −1.60091 −0.800457 0.599390i \(-0.795410\pi\)
−0.800457 + 0.599390i \(0.795410\pi\)
\(128\) −1663.35 −1.14860
\(129\) 840.049 0.573350
\(130\) 176.751 0.119247
\(131\) 1147.41 0.765267 0.382633 0.923900i \(-0.375017\pi\)
0.382633 + 0.923900i \(0.375017\pi\)
\(132\) 0 0
\(133\) 373.402 0.243444
\(134\) −3037.03 −1.95791
\(135\) −94.2064 −0.0600592
\(136\) −1141.86 −0.719956
\(137\) 1268.60 0.791121 0.395561 0.918440i \(-0.370550\pi\)
0.395561 + 0.918440i \(0.370550\pi\)
\(138\) −1133.09 −0.698947
\(139\) 486.288 0.296737 0.148368 0.988932i \(-0.452598\pi\)
0.148368 + 0.988932i \(0.452598\pi\)
\(140\) −55.8260 −0.0337011
\(141\) −509.609 −0.304374
\(142\) −2551.13 −1.50765
\(143\) 0 0
\(144\) −716.454 −0.414614
\(145\) 848.293 0.485841
\(146\) −3439.10 −1.94947
\(147\) −961.467 −0.539459
\(148\) 345.255 0.191756
\(149\) −2354.11 −1.29434 −0.647169 0.762346i \(-0.724047\pi\)
−0.647169 + 0.762346i \(0.724047\pi\)
\(150\) 1141.44 0.621323
\(151\) 570.070 0.307229 0.153615 0.988131i \(-0.450909\pi\)
0.153615 + 0.988131i \(0.450909\pi\)
\(152\) 1228.21 0.655399
\(153\) −658.516 −0.347960
\(154\) 0 0
\(155\) −972.479 −0.503945
\(156\) 151.973 0.0779971
\(157\) −2072.67 −1.05361 −0.526807 0.849985i \(-0.676611\pi\)
−0.526807 + 0.849985i \(0.676611\pi\)
\(158\) −1104.59 −0.556179
\(159\) −1228.96 −0.612972
\(160\) −501.059 −0.247576
\(161\) 531.391 0.260121
\(162\) −273.155 −0.132476
\(163\) 2676.51 1.28614 0.643069 0.765808i \(-0.277661\pi\)
0.643069 + 0.765808i \(0.277661\pi\)
\(164\) 813.581 0.387378
\(165\) 0 0
\(166\) −2551.53 −1.19300
\(167\) 1188.12 0.550536 0.275268 0.961368i \(-0.411233\pi\)
0.275268 + 0.961368i \(0.411233\pi\)
\(168\) 222.130 0.102010
\(169\) −1971.35 −0.897290
\(170\) −860.923 −0.388410
\(171\) 708.310 0.316759
\(172\) 944.293 0.418615
\(173\) −807.147 −0.354718 −0.177359 0.984146i \(-0.556755\pi\)
−0.177359 + 0.984146i \(0.556755\pi\)
\(174\) 2459.66 1.07164
\(175\) −535.310 −0.231232
\(176\) 0 0
\(177\) 588.000 0.249699
\(178\) −1716.42 −0.722758
\(179\) −1950.39 −0.814408 −0.407204 0.913337i \(-0.633496\pi\)
−0.407204 + 0.913337i \(0.633496\pi\)
\(180\) −105.897 −0.0438505
\(181\) 1061.61 0.435959 0.217980 0.975953i \(-0.430053\pi\)
0.217980 + 0.975953i \(0.430053\pi\)
\(182\) −240.348 −0.0978889
\(183\) 2104.08 0.849933
\(184\) 1747.87 0.700297
\(185\) −357.218 −0.141963
\(186\) −2819.74 −1.11158
\(187\) 0 0
\(188\) −572.848 −0.222230
\(189\) 128.103 0.0493023
\(190\) 926.021 0.353582
\(191\) 2136.41 0.809348 0.404674 0.914461i \(-0.367385\pi\)
0.404674 + 0.914461i \(0.367385\pi\)
\(192\) 457.704 0.172041
\(193\) −3947.76 −1.47236 −0.736181 0.676784i \(-0.763373\pi\)
−0.736181 + 0.676784i \(0.763373\pi\)
\(194\) −2071.79 −0.766731
\(195\) −157.238 −0.0577439
\(196\) −1080.78 −0.393870
\(197\) −923.886 −0.334133 −0.167066 0.985946i \(-0.553429\pi\)
−0.167066 + 0.985946i \(0.553429\pi\)
\(198\) 0 0
\(199\) −476.152 −0.169616 −0.0848078 0.996397i \(-0.527028\pi\)
−0.0848078 + 0.996397i \(0.527028\pi\)
\(200\) −1760.76 −0.622522
\(201\) 2701.76 0.948097
\(202\) −3425.96 −1.19332
\(203\) −1153.52 −0.398824
\(204\) −740.234 −0.254053
\(205\) −841.770 −0.286789
\(206\) −3716.80 −1.25710
\(207\) 1008.00 0.338458
\(208\) −1195.82 −0.398631
\(209\) 0 0
\(210\) 167.478 0.0550337
\(211\) 4918.24 1.60467 0.802336 0.596872i \(-0.203590\pi\)
0.802336 + 0.596872i \(0.203590\pi\)
\(212\) −1381.46 −0.447543
\(213\) 2269.50 0.730064
\(214\) 4645.57 1.48395
\(215\) −977.012 −0.309915
\(216\) 421.361 0.132731
\(217\) 1322.39 0.413686
\(218\) 1079.86 0.335494
\(219\) 3059.44 0.944010
\(220\) 0 0
\(221\) −1099.12 −0.334546
\(222\) −1035.77 −0.313136
\(223\) 2100.29 0.630700 0.315350 0.948975i \(-0.397878\pi\)
0.315350 + 0.948975i \(0.397878\pi\)
\(224\) 681.348 0.203234
\(225\) −1015.43 −0.300869
\(226\) 5494.95 1.61734
\(227\) 2257.16 0.659970 0.329985 0.943986i \(-0.392956\pi\)
0.329985 + 0.943986i \(0.392956\pi\)
\(228\) 796.206 0.231272
\(229\) −5311.07 −1.53260 −0.766301 0.642482i \(-0.777905\pi\)
−0.766301 + 0.642482i \(0.777905\pi\)
\(230\) 1317.83 0.377804
\(231\) 0 0
\(232\) −3794.20 −1.07371
\(233\) −2466.27 −0.693435 −0.346718 0.937970i \(-0.612704\pi\)
−0.346718 + 0.937970i \(0.612704\pi\)
\(234\) −455.918 −0.127369
\(235\) 592.696 0.164524
\(236\) 660.967 0.182311
\(237\) 982.646 0.269324
\(238\) 1170.70 0.318844
\(239\) −1429.40 −0.386863 −0.193432 0.981114i \(-0.561962\pi\)
−0.193432 + 0.981114i \(0.561962\pi\)
\(240\) 833.266 0.224113
\(241\) 978.989 0.261669 0.130835 0.991404i \(-0.458234\pi\)
0.130835 + 0.991404i \(0.458234\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 2365.18 0.620553
\(245\) 1118.23 0.291595
\(246\) −2440.74 −0.632586
\(247\) 1182.23 0.304548
\(248\) 4349.65 1.11372
\(249\) 2269.86 0.577696
\(250\) −2798.33 −0.707929
\(251\) −6530.63 −1.64227 −0.821135 0.570734i \(-0.806659\pi\)
−0.821135 + 0.570734i \(0.806659\pi\)
\(252\) 144.000 0.0359966
\(253\) 0 0
\(254\) 7726.76 1.90874
\(255\) 765.882 0.188084
\(256\) 4388.74 1.07147
\(257\) 8130.26 1.97335 0.986676 0.162696i \(-0.0520188\pi\)
0.986676 + 0.162696i \(0.0520188\pi\)
\(258\) −2832.88 −0.683595
\(259\) 485.750 0.116537
\(260\) −176.751 −0.0421600
\(261\) −2188.12 −0.518933
\(262\) −3869.40 −0.912414
\(263\) 4549.42 1.06665 0.533326 0.845910i \(-0.320942\pi\)
0.533326 + 0.845910i \(0.320942\pi\)
\(264\) 0 0
\(265\) 1429.33 0.331332
\(266\) −1259.22 −0.290254
\(267\) 1526.93 0.349988
\(268\) 3037.03 0.692225
\(269\) −29.1522 −0.00660760 −0.00330380 0.999995i \(-0.501052\pi\)
−0.00330380 + 0.999995i \(0.501052\pi\)
\(270\) 317.690 0.0716075
\(271\) −7711.22 −1.72850 −0.864250 0.503063i \(-0.832206\pi\)
−0.864250 + 0.503063i \(0.832206\pi\)
\(272\) 5824.64 1.29842
\(273\) 213.815 0.0474017
\(274\) −4278.07 −0.943239
\(275\) 0 0
\(276\) 1133.09 0.247115
\(277\) −1127.52 −0.244571 −0.122286 0.992495i \(-0.539022\pi\)
−0.122286 + 0.992495i \(0.539022\pi\)
\(278\) −1639.90 −0.353794
\(279\) 2508.46 0.538270
\(280\) −258.347 −0.0551400
\(281\) 1872.47 0.397517 0.198758 0.980049i \(-0.436309\pi\)
0.198758 + 0.980049i \(0.436309\pi\)
\(282\) 1718.54 0.362900
\(283\) −2124.48 −0.446245 −0.223123 0.974790i \(-0.571625\pi\)
−0.223123 + 0.974790i \(0.571625\pi\)
\(284\) 2551.13 0.533034
\(285\) −823.794 −0.171219
\(286\) 0 0
\(287\) 1144.65 0.235424
\(288\) 1292.45 0.264439
\(289\) 440.621 0.0896846
\(290\) −2860.68 −0.579259
\(291\) 1843.07 0.371282
\(292\) 3439.10 0.689241
\(293\) 3324.19 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(294\) 3242.34 0.643187
\(295\) −683.869 −0.134971
\(296\) 1597.75 0.313740
\(297\) 0 0
\(298\) 7938.73 1.54322
\(299\) 1682.44 0.325411
\(300\) −1141.44 −0.219671
\(301\) 1328.55 0.254407
\(302\) −1922.44 −0.366304
\(303\) 3047.75 0.577851
\(304\) −6265.07 −1.18200
\(305\) −2447.13 −0.459417
\(306\) 2220.70 0.414866
\(307\) 1698.94 0.315843 0.157921 0.987452i \(-0.449521\pi\)
0.157921 + 0.987452i \(0.449521\pi\)
\(308\) 0 0
\(309\) 3306.49 0.608736
\(310\) 3279.47 0.600844
\(311\) 6928.83 1.26334 0.631668 0.775239i \(-0.282370\pi\)
0.631668 + 0.775239i \(0.282370\pi\)
\(312\) 703.287 0.127615
\(313\) −3560.75 −0.643020 −0.321510 0.946906i \(-0.604190\pi\)
−0.321510 + 0.946906i \(0.604190\pi\)
\(314\) 6989.64 1.25620
\(315\) −148.989 −0.0266495
\(316\) 1104.59 0.196639
\(317\) 332.750 0.0589561 0.0294780 0.999565i \(-0.490615\pi\)
0.0294780 + 0.999565i \(0.490615\pi\)
\(318\) 4144.39 0.730835
\(319\) 0 0
\(320\) −532.329 −0.0929940
\(321\) −4132.73 −0.718587
\(322\) −1792.00 −0.310137
\(323\) −5758.43 −0.991975
\(324\) 273.155 0.0468372
\(325\) −1694.84 −0.289271
\(326\) −9025.94 −1.53344
\(327\) −960.652 −0.162459
\(328\) 3765.02 0.633807
\(329\) −805.957 −0.135057
\(330\) 0 0
\(331\) −541.445 −0.0899108 −0.0449554 0.998989i \(-0.514315\pi\)
−0.0449554 + 0.998989i \(0.514315\pi\)
\(332\) 2551.53 0.421788
\(333\) 921.423 0.151633
\(334\) −4006.67 −0.656393
\(335\) −3142.26 −0.512478
\(336\) −1133.09 −0.183973
\(337\) −816.531 −0.131986 −0.0659930 0.997820i \(-0.521022\pi\)
−0.0659930 + 0.997820i \(0.521022\pi\)
\(338\) 6647.94 1.06982
\(339\) −4888.34 −0.783180
\(340\) 860.923 0.137324
\(341\) 0 0
\(342\) −2388.62 −0.377666
\(343\) −3147.97 −0.495552
\(344\) 4369.92 0.684914
\(345\) −1172.35 −0.182948
\(346\) 2721.93 0.422924
\(347\) −6260.53 −0.968539 −0.484269 0.874919i \(-0.660914\pi\)
−0.484269 + 0.874919i \(0.660914\pi\)
\(348\) −2459.66 −0.378884
\(349\) 12768.5 1.95840 0.979198 0.202906i \(-0.0650386\pi\)
0.979198 + 0.202906i \(0.0650386\pi\)
\(350\) 1805.22 0.275694
\(351\) 405.587 0.0616771
\(352\) 0 0
\(353\) −2649.28 −0.399453 −0.199727 0.979852i \(-0.564005\pi\)
−0.199727 + 0.979852i \(0.564005\pi\)
\(354\) −1982.90 −0.297712
\(355\) −2639.52 −0.394623
\(356\) 1716.42 0.255534
\(357\) −1041.46 −0.154397
\(358\) 6577.27 0.971004
\(359\) 3203.91 0.471020 0.235510 0.971872i \(-0.424324\pi\)
0.235510 + 0.971872i \(0.424324\pi\)
\(360\) −490.061 −0.0717457
\(361\) −665.143 −0.0969737
\(362\) −3580.04 −0.519786
\(363\) 0 0
\(364\) 240.348 0.0346089
\(365\) −3558.26 −0.510268
\(366\) −7095.54 −1.01336
\(367\) −8429.40 −1.19894 −0.599470 0.800397i \(-0.704622\pi\)
−0.599470 + 0.800397i \(0.704622\pi\)
\(368\) −8915.87 −1.26297
\(369\) 2171.30 0.306323
\(370\) 1204.64 0.169260
\(371\) −1943.62 −0.271988
\(372\) 2819.74 0.393002
\(373\) 9388.53 1.30327 0.651635 0.758533i \(-0.274083\pi\)
0.651635 + 0.758533i \(0.274083\pi\)
\(374\) 0 0
\(375\) 2489.41 0.342807
\(376\) −2650.98 −0.363600
\(377\) −3652.16 −0.498928
\(378\) −432.000 −0.0587822
\(379\) −14264.5 −1.93329 −0.966647 0.256112i \(-0.917558\pi\)
−0.966647 + 0.256112i \(0.917558\pi\)
\(380\) −926.021 −0.125010
\(381\) −6873.77 −0.924288
\(382\) −7204.58 −0.964970
\(383\) 13462.2 1.79605 0.898026 0.439942i \(-0.145001\pi\)
0.898026 + 0.439942i \(0.145001\pi\)
\(384\) −4990.05 −0.663144
\(385\) 0 0
\(386\) 13313.0 1.75547
\(387\) 2520.15 0.331024
\(388\) 2071.79 0.271080
\(389\) −941.881 −0.122764 −0.0613821 0.998114i \(-0.519551\pi\)
−0.0613821 + 0.998114i \(0.519551\pi\)
\(390\) 530.252 0.0688470
\(391\) −8194.87 −1.05993
\(392\) −5001.54 −0.644429
\(393\) 3442.24 0.441827
\(394\) 3115.60 0.398380
\(395\) −1142.86 −0.145578
\(396\) 0 0
\(397\) −847.839 −0.107183 −0.0535917 0.998563i \(-0.517067\pi\)
−0.0535917 + 0.998563i \(0.517067\pi\)
\(398\) 1605.72 0.202230
\(399\) 1120.21 0.140553
\(400\) 8981.62 1.12270
\(401\) 12203.6 1.51975 0.759875 0.650069i \(-0.225260\pi\)
0.759875 + 0.650069i \(0.225260\pi\)
\(402\) −9111.10 −1.13040
\(403\) 4186.82 0.517520
\(404\) 3425.96 0.421901
\(405\) −282.619 −0.0346752
\(406\) 3890.00 0.475511
\(407\) 0 0
\(408\) −3425.59 −0.415667
\(409\) −8759.53 −1.05900 −0.529500 0.848310i \(-0.677620\pi\)
−0.529500 + 0.848310i \(0.677620\pi\)
\(410\) 2838.69 0.341934
\(411\) 3805.79 0.456754
\(412\) 3716.80 0.444451
\(413\) 929.934 0.110797
\(414\) −3399.26 −0.403537
\(415\) −2639.94 −0.312264
\(416\) 2157.21 0.254245
\(417\) 1458.86 0.171321
\(418\) 0 0
\(419\) −11188.4 −1.30451 −0.652256 0.757999i \(-0.726177\pi\)
−0.652256 + 0.757999i \(0.726177\pi\)
\(420\) −167.478 −0.0194574
\(421\) −14082.3 −1.63023 −0.815116 0.579298i \(-0.803327\pi\)
−0.815116 + 0.579298i \(0.803327\pi\)
\(422\) −16585.7 −1.91322
\(423\) −1528.83 −0.175731
\(424\) −6393.02 −0.732246
\(425\) 8255.30 0.942214
\(426\) −7653.39 −0.870441
\(427\) 3327.64 0.377133
\(428\) −4645.57 −0.524655
\(429\) 0 0
\(430\) 3294.76 0.369505
\(431\) 5616.05 0.627647 0.313823 0.949481i \(-0.398390\pi\)
0.313823 + 0.949481i \(0.398390\pi\)
\(432\) −2149.36 −0.239378
\(433\) 7195.75 0.798627 0.399314 0.916814i \(-0.369248\pi\)
0.399314 + 0.916814i \(0.369248\pi\)
\(434\) −4459.48 −0.493230
\(435\) 2544.88 0.280500
\(436\) −1079.86 −0.118615
\(437\) 8814.52 0.964887
\(438\) −10317.3 −1.12553
\(439\) −101.959 −0.0110848 −0.00554240 0.999985i \(-0.501764\pi\)
−0.00554240 + 0.999985i \(0.501764\pi\)
\(440\) 0 0
\(441\) −2884.40 −0.311457
\(442\) 3706.53 0.398873
\(443\) 4953.74 0.531285 0.265642 0.964072i \(-0.414416\pi\)
0.265642 + 0.964072i \(0.414416\pi\)
\(444\) 1035.77 0.110710
\(445\) −1775.89 −0.189180
\(446\) −7082.78 −0.751972
\(447\) −7062.34 −0.747287
\(448\) 723.869 0.0763383
\(449\) −11602.0 −1.21945 −0.609723 0.792615i \(-0.708719\pi\)
−0.609723 + 0.792615i \(0.708719\pi\)
\(450\) 3424.33 0.358721
\(451\) 0 0
\(452\) −5494.95 −0.571816
\(453\) 1710.21 0.177379
\(454\) −7611.79 −0.786870
\(455\) −248.676 −0.0256222
\(456\) 3684.62 0.378395
\(457\) 3530.68 0.361397 0.180698 0.983539i \(-0.442164\pi\)
0.180698 + 0.983539i \(0.442164\pi\)
\(458\) 17910.4 1.82729
\(459\) −1975.55 −0.200895
\(460\) −1317.83 −0.133574
\(461\) −11566.3 −1.16854 −0.584271 0.811559i \(-0.698619\pi\)
−0.584271 + 0.811559i \(0.698619\pi\)
\(462\) 0 0
\(463\) 10888.5 1.09294 0.546470 0.837479i \(-0.315971\pi\)
0.546470 + 0.837479i \(0.315971\pi\)
\(464\) 19354.2 1.93641
\(465\) −2917.44 −0.290953
\(466\) 8316.94 0.826770
\(467\) 10688.0 1.05906 0.529529 0.848292i \(-0.322369\pi\)
0.529529 + 0.848292i \(0.322369\pi\)
\(468\) 455.918 0.0450317
\(469\) 4272.89 0.420690
\(470\) −1998.74 −0.196159
\(471\) −6218.02 −0.608304
\(472\) 3058.77 0.298287
\(473\) 0 0
\(474\) −3313.76 −0.321110
\(475\) −8879.53 −0.857728
\(476\) −1170.70 −0.112728
\(477\) −3686.87 −0.353900
\(478\) 4820.35 0.461250
\(479\) −2341.90 −0.223391 −0.111696 0.993742i \(-0.535628\pi\)
−0.111696 + 0.993742i \(0.535628\pi\)
\(480\) −1503.18 −0.142938
\(481\) 1537.93 0.145787
\(482\) −3301.43 −0.311983
\(483\) 1594.17 0.150181
\(484\) 0 0
\(485\) −2143.57 −0.200690
\(486\) −819.464 −0.0764849
\(487\) 6748.91 0.627972 0.313986 0.949428i \(-0.398335\pi\)
0.313986 + 0.949428i \(0.398335\pi\)
\(488\) 10945.4 1.01532
\(489\) 8029.53 0.742552
\(490\) −3770.98 −0.347664
\(491\) −7361.40 −0.676609 −0.338305 0.941037i \(-0.609853\pi\)
−0.338305 + 0.941037i \(0.609853\pi\)
\(492\) 2440.74 0.223653
\(493\) 17789.1 1.62511
\(494\) −3986.80 −0.363107
\(495\) 0 0
\(496\) −22187.6 −2.00857
\(497\) 3589.26 0.323944
\(498\) −7654.60 −0.688777
\(499\) 10381.7 0.931359 0.465680 0.884953i \(-0.345810\pi\)
0.465680 + 0.884953i \(0.345810\pi\)
\(500\) 2798.33 0.250291
\(501\) 3564.36 0.317852
\(502\) 22023.1 1.95805
\(503\) −19149.0 −1.69744 −0.848721 0.528840i \(-0.822627\pi\)
−0.848721 + 0.528840i \(0.822627\pi\)
\(504\) 666.391 0.0588957
\(505\) −3544.67 −0.312348
\(506\) 0 0
\(507\) −5914.04 −0.518051
\(508\) −7726.76 −0.674841
\(509\) 16073.2 1.39967 0.699836 0.714303i \(-0.253256\pi\)
0.699836 + 0.714303i \(0.253256\pi\)
\(510\) −2582.77 −0.224249
\(511\) 4838.58 0.418877
\(512\) −1493.27 −0.128894
\(513\) 2124.93 0.182881
\(514\) −27417.5 −2.35279
\(515\) −3845.58 −0.329042
\(516\) 2832.88 0.241687
\(517\) 0 0
\(518\) −1638.09 −0.138945
\(519\) −2421.44 −0.204797
\(520\) −817.952 −0.0689799
\(521\) −18955.3 −1.59395 −0.796975 0.604012i \(-0.793568\pi\)
−0.796975 + 0.604012i \(0.793568\pi\)
\(522\) 7378.97 0.618714
\(523\) 4442.19 0.371402 0.185701 0.982606i \(-0.440544\pi\)
0.185701 + 0.982606i \(0.440544\pi\)
\(524\) 3869.40 0.322587
\(525\) −1605.93 −0.133502
\(526\) −15341.9 −1.27175
\(527\) −20393.3 −1.68567
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) −4820.09 −0.395041
\(531\) 1764.00 0.144164
\(532\) 1259.22 0.102620
\(533\) 3624.08 0.294515
\(534\) −5149.25 −0.417285
\(535\) 4806.54 0.388420
\(536\) 14054.5 1.13258
\(537\) −5851.17 −0.470199
\(538\) 98.3096 0.00787812
\(539\) 0 0
\(540\) −317.690 −0.0253171
\(541\) −2180.90 −0.173316 −0.0866580 0.996238i \(-0.527619\pi\)
−0.0866580 + 0.996238i \(0.527619\pi\)
\(542\) 26004.4 2.06086
\(543\) 3184.82 0.251701
\(544\) −10507.4 −0.828129
\(545\) 1117.28 0.0878146
\(546\) −721.044 −0.0565162
\(547\) −8225.04 −0.642920 −0.321460 0.946923i \(-0.604174\pi\)
−0.321460 + 0.946923i \(0.604174\pi\)
\(548\) 4278.07 0.333485
\(549\) 6312.23 0.490709
\(550\) 0 0
\(551\) −19134.2 −1.47939
\(552\) 5243.61 0.404316
\(553\) 1554.08 0.119505
\(554\) 3802.32 0.291598
\(555\) −1071.65 −0.0819625
\(556\) 1639.90 0.125085
\(557\) 25181.9 1.91561 0.957804 0.287423i \(-0.0927986\pi\)
0.957804 + 0.287423i \(0.0927986\pi\)
\(558\) −8459.22 −0.641769
\(559\) 4206.33 0.318263
\(560\) 1317.83 0.0994435
\(561\) 0 0
\(562\) −6314.50 −0.473952
\(563\) 4504.50 0.337197 0.168599 0.985685i \(-0.446076\pi\)
0.168599 + 0.985685i \(0.446076\pi\)
\(564\) −1718.54 −0.128304
\(565\) 5685.34 0.423335
\(566\) 7164.36 0.532050
\(567\) 384.310 0.0284647
\(568\) 11805.9 0.872122
\(569\) 13447.0 0.990732 0.495366 0.868684i \(-0.335034\pi\)
0.495366 + 0.868684i \(0.335034\pi\)
\(570\) 2778.06 0.204141
\(571\) 2605.52 0.190959 0.0954795 0.995431i \(-0.469562\pi\)
0.0954795 + 0.995431i \(0.469562\pi\)
\(572\) 0 0
\(573\) 6409.24 0.467277
\(574\) −3860.09 −0.280691
\(575\) −12636.5 −0.916485
\(576\) 1373.11 0.0993281
\(577\) 6339.65 0.457406 0.228703 0.973496i \(-0.426552\pi\)
0.228703 + 0.973496i \(0.426552\pi\)
\(578\) −1485.90 −0.106929
\(579\) −11843.3 −0.850069
\(580\) 2860.68 0.204799
\(581\) 3589.83 0.256336
\(582\) −6215.37 −0.442672
\(583\) 0 0
\(584\) 15915.2 1.12770
\(585\) −471.715 −0.0333385
\(586\) −11210.1 −0.790247
\(587\) −13370.6 −0.940140 −0.470070 0.882629i \(-0.655771\pi\)
−0.470070 + 0.882629i \(0.655771\pi\)
\(588\) −3242.34 −0.227401
\(589\) 21935.3 1.53452
\(590\) 2306.20 0.160923
\(591\) −2771.66 −0.192912
\(592\) −8150.09 −0.565822
\(593\) −14319.3 −0.991608 −0.495804 0.868434i \(-0.665127\pi\)
−0.495804 + 0.868434i \(0.665127\pi\)
\(594\) 0 0
\(595\) 1211.26 0.0834567
\(596\) −7938.73 −0.545609
\(597\) −1428.46 −0.0979276
\(598\) −5673.65 −0.387981
\(599\) −5788.63 −0.394853 −0.197427 0.980318i \(-0.563258\pi\)
−0.197427 + 0.980318i \(0.563258\pi\)
\(600\) −5282.28 −0.359413
\(601\) −23968.1 −1.62675 −0.813375 0.581739i \(-0.802372\pi\)
−0.813375 + 0.581739i \(0.802372\pi\)
\(602\) −4480.26 −0.303325
\(603\) 8105.28 0.547384
\(604\) 1922.44 0.129508
\(605\) 0 0
\(606\) −10277.9 −0.688961
\(607\) 23526.6 1.57317 0.786585 0.617482i \(-0.211847\pi\)
0.786585 + 0.617482i \(0.211847\pi\)
\(608\) 11301.9 0.753872
\(609\) −3460.56 −0.230261
\(610\) 8252.40 0.547754
\(611\) −2551.74 −0.168956
\(612\) −2220.70 −0.146677
\(613\) −1228.07 −0.0809159 −0.0404579 0.999181i \(-0.512882\pi\)
−0.0404579 + 0.999181i \(0.512882\pi\)
\(614\) −5729.31 −0.376573
\(615\) −2525.31 −0.165578
\(616\) 0 0
\(617\) −9844.90 −0.642368 −0.321184 0.947017i \(-0.604081\pi\)
−0.321184 + 0.947017i \(0.604081\pi\)
\(618\) −11150.4 −0.725785
\(619\) −6551.68 −0.425419 −0.212709 0.977115i \(-0.568229\pi\)
−0.212709 + 0.977115i \(0.568229\pi\)
\(620\) −3279.47 −0.212430
\(621\) 3024.00 0.195409
\(622\) −23365.9 −1.50625
\(623\) 2414.88 0.155297
\(624\) −3587.46 −0.230150
\(625\) 11208.0 0.717309
\(626\) 12007.8 0.766661
\(627\) 0 0
\(628\) −6989.64 −0.444135
\(629\) −7491.01 −0.474859
\(630\) 502.434 0.0317737
\(631\) −26440.5 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(632\) 5111.72 0.321730
\(633\) 14754.7 0.926458
\(634\) −1122.13 −0.0702923
\(635\) 7994.48 0.499608
\(636\) −4144.39 −0.258389
\(637\) −4814.31 −0.299450
\(638\) 0 0
\(639\) 6808.50 0.421502
\(640\) 5803.64 0.358451
\(641\) −27927.2 −1.72084 −0.860421 0.509584i \(-0.829799\pi\)
−0.860421 + 0.509584i \(0.829799\pi\)
\(642\) 13936.7 0.856758
\(643\) −16737.7 −1.02655 −0.513274 0.858225i \(-0.671568\pi\)
−0.513274 + 0.858225i \(0.671568\pi\)
\(644\) 1792.00 0.109650
\(645\) −2931.03 −0.178929
\(646\) 19419.1 1.18271
\(647\) 7818.70 0.475092 0.237546 0.971376i \(-0.423657\pi\)
0.237546 + 0.971376i \(0.423657\pi\)
\(648\) 1264.08 0.0766325
\(649\) 0 0
\(650\) 5715.49 0.344892
\(651\) 3967.17 0.238842
\(652\) 9025.94 0.542152
\(653\) 19747.6 1.18344 0.591719 0.806144i \(-0.298450\pi\)
0.591719 + 0.806144i \(0.298450\pi\)
\(654\) 3239.59 0.193697
\(655\) −4003.47 −0.238822
\(656\) −19205.4 −1.14305
\(657\) 9178.33 0.545024
\(658\) 2717.91 0.161026
\(659\) −7867.72 −0.465072 −0.232536 0.972588i \(-0.574702\pi\)
−0.232536 + 0.972588i \(0.574702\pi\)
\(660\) 0 0
\(661\) 4227.41 0.248755 0.124378 0.992235i \(-0.460307\pi\)
0.124378 + 0.992235i \(0.460307\pi\)
\(662\) 1825.90 0.107199
\(663\) −3297.35 −0.193150
\(664\) 11807.8 0.690106
\(665\) −1302.85 −0.0759733
\(666\) −3107.30 −0.180789
\(667\) −27230.0 −1.58073
\(668\) 4006.67 0.232070
\(669\) 6300.88 0.364135
\(670\) 10596.6 0.611018
\(671\) 0 0
\(672\) 2044.04 0.117337
\(673\) −29397.6 −1.68379 −0.841897 0.539638i \(-0.818561\pi\)
−0.841897 + 0.539638i \(0.818561\pi\)
\(674\) 2753.57 0.157364
\(675\) −3046.30 −0.173707
\(676\) −6647.94 −0.378239
\(677\) −5737.14 −0.325696 −0.162848 0.986651i \(-0.552068\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(678\) 16484.8 0.933771
\(679\) 2914.86 0.164745
\(680\) 3984.11 0.224682
\(681\) 6771.49 0.381034
\(682\) 0 0
\(683\) 32097.6 1.79821 0.899107 0.437729i \(-0.144217\pi\)
0.899107 + 0.437729i \(0.144217\pi\)
\(684\) 2388.62 0.133525
\(685\) −4426.30 −0.246891
\(686\) 10615.8 0.590837
\(687\) −15933.2 −0.884848
\(688\) −22291.0 −1.23523
\(689\) −6153.69 −0.340257
\(690\) 3953.48 0.218125
\(691\) −16456.2 −0.905965 −0.452983 0.891519i \(-0.649640\pi\)
−0.452983 + 0.891519i \(0.649640\pi\)
\(692\) −2721.93 −0.149526
\(693\) 0 0
\(694\) 21112.3 1.15477
\(695\) −1696.72 −0.0926047
\(696\) −11382.6 −0.619909
\(697\) −17652.3 −0.959294
\(698\) −43058.9 −2.33496
\(699\) −7398.80 −0.400355
\(700\) −1805.22 −0.0974725
\(701\) −27238.1 −1.46758 −0.733788 0.679379i \(-0.762249\pi\)
−0.733788 + 0.679379i \(0.762249\pi\)
\(702\) −1367.75 −0.0735364
\(703\) 8057.44 0.432279
\(704\) 0 0
\(705\) 1778.09 0.0949882
\(706\) 8934.12 0.476261
\(707\) 4820.09 0.256405
\(708\) 1982.90 0.105257
\(709\) 28761.4 1.52349 0.761747 0.647875i \(-0.224342\pi\)
0.761747 + 0.647875i \(0.224342\pi\)
\(710\) 8901.21 0.470502
\(711\) 2947.94 0.155494
\(712\) 7943.10 0.418090
\(713\) 31216.3 1.63964
\(714\) 3512.09 0.184085
\(715\) 0 0
\(716\) −6577.27 −0.343302
\(717\) −4288.21 −0.223356
\(718\) −10804.5 −0.561588
\(719\) −27272.0 −1.41456 −0.707282 0.706931i \(-0.750079\pi\)
−0.707282 + 0.706931i \(0.750079\pi\)
\(720\) 2499.80 0.129392
\(721\) 5229.28 0.270109
\(722\) 2243.05 0.115620
\(723\) 2936.97 0.151075
\(724\) 3580.04 0.183772
\(725\) 27430.8 1.40518
\(726\) 0 0
\(727\) 3979.75 0.203027 0.101514 0.994834i \(-0.467631\pi\)
0.101514 + 0.994834i \(0.467631\pi\)
\(728\) 1112.26 0.0566253
\(729\) 729.000 0.0370370
\(730\) 11999.5 0.608383
\(731\) −20488.3 −1.03665
\(732\) 7095.54 0.358277
\(733\) −9342.48 −0.470767 −0.235384 0.971903i \(-0.575635\pi\)
−0.235384 + 0.971903i \(0.575635\pi\)
\(734\) 28426.3 1.42947
\(735\) 3354.68 0.168353
\(736\) 16083.9 0.805515
\(737\) 0 0
\(738\) −7322.23 −0.365224
\(739\) 28928.0 1.43997 0.719983 0.693992i \(-0.244150\pi\)
0.719983 + 0.693992i \(0.244150\pi\)
\(740\) −1204.64 −0.0598425
\(741\) 3546.68 0.175831
\(742\) 6554.43 0.324287
\(743\) −4857.04 −0.239822 −0.119911 0.992785i \(-0.538261\pi\)
−0.119911 + 0.992785i \(0.538261\pi\)
\(744\) 13049.0 0.643008
\(745\) 8213.80 0.403933
\(746\) −31660.8 −1.55386
\(747\) 6809.57 0.333533
\(748\) 0 0
\(749\) −6536.00 −0.318852
\(750\) −8395.00 −0.408723
\(751\) 14355.4 0.697517 0.348759 0.937213i \(-0.386603\pi\)
0.348759 + 0.937213i \(0.386603\pi\)
\(752\) 13522.6 0.655744
\(753\) −19591.9 −0.948165
\(754\) 12316.1 0.594863
\(755\) −1989.05 −0.0958792
\(756\) 432.000 0.0207827
\(757\) −17714.9 −0.850538 −0.425269 0.905067i \(-0.639821\pi\)
−0.425269 + 0.905067i \(0.639821\pi\)
\(758\) 48103.9 2.30503
\(759\) 0 0
\(760\) −4285.37 −0.204535
\(761\) 7945.82 0.378497 0.189248 0.981929i \(-0.439395\pi\)
0.189248 + 0.981929i \(0.439395\pi\)
\(762\) 23180.3 1.10201
\(763\) −1519.29 −0.0720866
\(764\) 7204.58 0.341168
\(765\) 2297.64 0.108590
\(766\) −45398.4 −2.14140
\(767\) 2944.26 0.138606
\(768\) 13166.2 0.618613
\(769\) −27308.1 −1.28057 −0.640284 0.768139i \(-0.721183\pi\)
−0.640284 + 0.768139i \(0.721183\pi\)
\(770\) 0 0
\(771\) 24390.8 1.13932
\(772\) −13313.0 −0.620653
\(773\) −18872.6 −0.878136 −0.439068 0.898454i \(-0.644691\pi\)
−0.439068 + 0.898454i \(0.644691\pi\)
\(774\) −8498.64 −0.394674
\(775\) −31446.6 −1.45754
\(776\) 9587.65 0.443527
\(777\) 1457.25 0.0672826
\(778\) 3176.29 0.146369
\(779\) 18987.1 0.873276
\(780\) −530.252 −0.0243411
\(781\) 0 0
\(782\) 27635.4 1.26373
\(783\) −6564.37 −0.299606
\(784\) 25512.8 1.16221
\(785\) 7231.82 0.328808
\(786\) −11608.2 −0.526782
\(787\) 14512.1 0.657307 0.328654 0.944450i \(-0.393405\pi\)
0.328654 + 0.944450i \(0.393405\pi\)
\(788\) −3115.60 −0.140849
\(789\) 13648.3 0.615832
\(790\) 3854.04 0.173570
\(791\) −7731.01 −0.347513
\(792\) 0 0
\(793\) 10535.6 0.471792
\(794\) 2859.15 0.127793
\(795\) 4287.98 0.191294
\(796\) −1605.72 −0.0714989
\(797\) 29108.9 1.29371 0.646856 0.762612i \(-0.276083\pi\)
0.646856 + 0.762612i \(0.276083\pi\)
\(798\) −3777.65 −0.167578
\(799\) 12429.1 0.550325
\(800\) −16202.5 −0.716056
\(801\) 4580.80 0.202066
\(802\) −41154.0 −1.81197
\(803\) 0 0
\(804\) 9111.10 0.399656
\(805\) −1854.09 −0.0811777
\(806\) −14119.1 −0.617029
\(807\) −87.4567 −0.00381490
\(808\) 15854.4 0.690291
\(809\) 3000.83 0.130413 0.0652063 0.997872i \(-0.479229\pi\)
0.0652063 + 0.997872i \(0.479229\pi\)
\(810\) 953.071 0.0413426
\(811\) −6239.39 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(812\) −3890.00 −0.168118
\(813\) −23133.7 −0.997950
\(814\) 0 0
\(815\) −9338.68 −0.401374
\(816\) 17473.9 0.749645
\(817\) 22037.6 0.943693
\(818\) 29539.6 1.26263
\(819\) 641.445 0.0273674
\(820\) −2838.69 −0.120892
\(821\) −14922.4 −0.634342 −0.317171 0.948368i \(-0.602733\pi\)
−0.317171 + 0.948368i \(0.602733\pi\)
\(822\) −12834.2 −0.544580
\(823\) −25737.8 −1.09011 −0.545057 0.838399i \(-0.683492\pi\)
−0.545057 + 0.838399i \(0.683492\pi\)
\(824\) 17200.3 0.727186
\(825\) 0 0
\(826\) −3136.00 −0.132101
\(827\) −27043.4 −1.13711 −0.568555 0.822645i \(-0.692497\pi\)
−0.568555 + 0.822645i \(0.692497\pi\)
\(828\) 3399.26 0.142672
\(829\) −9795.41 −0.410384 −0.205192 0.978722i \(-0.565782\pi\)
−0.205192 + 0.978722i \(0.565782\pi\)
\(830\) 8902.62 0.372306
\(831\) −3382.56 −0.141203
\(832\) 2291.84 0.0954990
\(833\) 23449.7 0.975370
\(834\) −4919.70 −0.204263
\(835\) −4145.50 −0.171809
\(836\) 0 0
\(837\) 7525.37 0.310770
\(838\) 37730.5 1.55535
\(839\) 28875.5 1.18819 0.594095 0.804395i \(-0.297510\pi\)
0.594095 + 0.804395i \(0.297510\pi\)
\(840\) −775.041 −0.0318351
\(841\) 34720.7 1.42362
\(842\) 47489.3 1.94369
\(843\) 5617.41 0.229506
\(844\) 16585.7 0.676426
\(845\) 6878.28 0.280024
\(846\) 5155.63 0.209520
\(847\) 0 0
\(848\) 32610.7 1.32059
\(849\) −6373.45 −0.257640
\(850\) −27839.2 −1.12338
\(851\) 11466.6 0.461892
\(852\) 7653.39 0.307747
\(853\) 47157.1 1.89288 0.946441 0.322878i \(-0.104650\pi\)
0.946441 + 0.322878i \(0.104650\pi\)
\(854\) −11221.7 −0.449649
\(855\) −2471.38 −0.0988531
\(856\) −21498.4 −0.858412
\(857\) −5021.31 −0.200145 −0.100073 0.994980i \(-0.531908\pi\)
−0.100073 + 0.994980i \(0.531908\pi\)
\(858\) 0 0
\(859\) −22921.1 −0.910428 −0.455214 0.890382i \(-0.650437\pi\)
−0.455214 + 0.890382i \(0.650437\pi\)
\(860\) −3294.76 −0.130640
\(861\) 3433.95 0.135922
\(862\) −18938.9 −0.748332
\(863\) −19488.1 −0.768693 −0.384347 0.923189i \(-0.625573\pi\)
−0.384347 + 0.923189i \(0.625573\pi\)
\(864\) 3877.36 0.152674
\(865\) 2816.24 0.110699
\(866\) −24266.1 −0.952188
\(867\) 1321.86 0.0517794
\(868\) 4459.48 0.174383
\(869\) 0 0
\(870\) −8582.05 −0.334435
\(871\) 13528.4 0.526282
\(872\) −4997.30 −0.194071
\(873\) 5529.22 0.214360
\(874\) −29725.0 −1.15042
\(875\) 3937.06 0.152111
\(876\) 10317.3 0.397933
\(877\) 8455.67 0.325573 0.162787 0.986661i \(-0.447952\pi\)
0.162787 + 0.986661i \(0.447952\pi\)
\(878\) 343.834 0.0132162
\(879\) 9972.56 0.382669
\(880\) 0 0
\(881\) −11291.2 −0.431794 −0.215897 0.976416i \(-0.569268\pi\)
−0.215897 + 0.976416i \(0.569268\pi\)
\(882\) 9727.02 0.371344
\(883\) 31818.1 1.21264 0.606322 0.795219i \(-0.292644\pi\)
0.606322 + 0.795219i \(0.292644\pi\)
\(884\) −3706.53 −0.141023
\(885\) −2051.61 −0.0779254
\(886\) −16705.4 −0.633441
\(887\) −17481.1 −0.661732 −0.330866 0.943678i \(-0.607341\pi\)
−0.330866 + 0.943678i \(0.607341\pi\)
\(888\) 4793.24 0.181138
\(889\) −10871.0 −0.410126
\(890\) 5988.80 0.225556
\(891\) 0 0
\(892\) 7082.78 0.265862
\(893\) −13368.9 −0.500978
\(894\) 23816.2 0.890976
\(895\) 6805.16 0.254158
\(896\) −7891.87 −0.294251
\(897\) 5047.31 0.187876
\(898\) 39125.1 1.45392
\(899\) −67763.1 −2.51393
\(900\) −3424.33 −0.126827
\(901\) 29973.6 1.10829
\(902\) 0 0
\(903\) 3985.66 0.146882
\(904\) −25429.1 −0.935574
\(905\) −3704.08 −0.136053
\(906\) −5767.31 −0.211486
\(907\) 10607.4 0.388326 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(908\) 7611.79 0.278201
\(909\) 9143.26 0.333623
\(910\) 838.604 0.0305489
\(911\) −41249.2 −1.50016 −0.750080 0.661347i \(-0.769985\pi\)
−0.750080 + 0.661347i \(0.769985\pi\)
\(912\) −18795.2 −0.682425
\(913\) 0 0
\(914\) −11906.5 −0.430887
\(915\) −7341.38 −0.265244
\(916\) −17910.4 −0.646045
\(917\) 5443.97 0.196048
\(918\) 6662.10 0.239523
\(919\) 13858.1 0.497429 0.248714 0.968577i \(-0.419992\pi\)
0.248714 + 0.968577i \(0.419992\pi\)
\(920\) −6098.53 −0.218546
\(921\) 5096.83 0.182352
\(922\) 39004.9 1.39323
\(923\) 11363.9 0.405253
\(924\) 0 0
\(925\) −11551.2 −0.410595
\(926\) −36719.0 −1.30309
\(927\) 9919.47 0.351454
\(928\) −34914.2 −1.23504
\(929\) 20893.7 0.737890 0.368945 0.929451i \(-0.379719\pi\)
0.368945 + 0.929451i \(0.379719\pi\)
\(930\) 9838.42 0.346897
\(931\) −25222.8 −0.887911
\(932\) −8316.94 −0.292307
\(933\) 20786.5 0.729388
\(934\) −36042.9 −1.26270
\(935\) 0 0
\(936\) 2109.86 0.0736784
\(937\) −3203.52 −0.111691 −0.0558454 0.998439i \(-0.517785\pi\)
−0.0558454 + 0.998439i \(0.517785\pi\)
\(938\) −14409.4 −0.501581
\(939\) −10682.2 −0.371248
\(940\) 1998.74 0.0693528
\(941\) −19951.6 −0.691182 −0.345591 0.938385i \(-0.612322\pi\)
−0.345591 + 0.938385i \(0.612322\pi\)
\(942\) 20968.9 0.725270
\(943\) 27020.6 0.933099
\(944\) −15602.8 −0.537952
\(945\) −446.968 −0.0153861
\(946\) 0 0
\(947\) −38216.7 −1.31138 −0.655689 0.755031i \(-0.727622\pi\)
−0.655689 + 0.755031i \(0.727622\pi\)
\(948\) 3313.76 0.113529
\(949\) 15319.4 0.524014
\(950\) 29944.3 1.02265
\(951\) 998.249 0.0340383
\(952\) −5417.65 −0.184440
\(953\) 47661.4 1.62004 0.810022 0.586399i \(-0.199455\pi\)
0.810022 + 0.586399i \(0.199455\pi\)
\(954\) 12433.2 0.421948
\(955\) −7454.21 −0.252579
\(956\) −4820.35 −0.163077
\(957\) 0 0
\(958\) 7897.56 0.266345
\(959\) 6018.94 0.202671
\(960\) −1596.99 −0.0536901
\(961\) 47892.3 1.60761
\(962\) −5186.34 −0.173819
\(963\) −12398.2 −0.414876
\(964\) 3301.43 0.110303
\(965\) 13774.2 0.459490
\(966\) −5376.00 −0.179058
\(967\) −18933.2 −0.629628 −0.314814 0.949153i \(-0.601942\pi\)
−0.314814 + 0.949153i \(0.601942\pi\)
\(968\) 0 0
\(969\) −17275.3 −0.572717
\(970\) 7228.73 0.239279
\(971\) −40660.3 −1.34382 −0.671911 0.740632i \(-0.734526\pi\)
−0.671911 + 0.740632i \(0.734526\pi\)
\(972\) 819.464 0.0270415
\(973\) 2307.23 0.0760188
\(974\) −22759.2 −0.748720
\(975\) −5084.53 −0.167011
\(976\) −55832.3 −1.83110
\(977\) −22502.8 −0.736876 −0.368438 0.929652i \(-0.620107\pi\)
−0.368438 + 0.929652i \(0.620107\pi\)
\(978\) −27077.8 −0.885331
\(979\) 0 0
\(980\) 3770.98 0.122918
\(981\) −2881.96 −0.0937959
\(982\) 24824.7 0.806709
\(983\) −4435.20 −0.143907 −0.0719536 0.997408i \(-0.522923\pi\)
−0.0719536 + 0.997408i \(0.522923\pi\)
\(984\) 11295.1 0.365929
\(985\) 3223.55 0.104275
\(986\) −59989.8 −1.93759
\(987\) −2417.87 −0.0779753
\(988\) 3986.80 0.128378
\(989\) 31361.8 1.00834
\(990\) 0 0
\(991\) 7362.76 0.236010 0.118005 0.993013i \(-0.462350\pi\)
0.118005 + 0.993013i \(0.462350\pi\)
\(992\) 40025.5 1.28106
\(993\) −1624.33 −0.0519101
\(994\) −12104.0 −0.386233
\(995\) 1661.35 0.0529331
\(996\) 7654.60 0.243519
\(997\) 53480.1 1.69883 0.849413 0.527728i \(-0.176956\pi\)
0.849413 + 0.527728i \(0.176956\pi\)
\(998\) −35010.0 −1.11044
\(999\) 2764.27 0.0875452
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 363.4.a.j.1.1 2
3.2 odd 2 1089.4.a.t.1.2 2
11.10 odd 2 33.4.a.d.1.2 2
33.32 even 2 99.4.a.e.1.1 2
44.43 even 2 528.4.a.o.1.1 2
55.32 even 4 825.4.c.i.199.4 4
55.43 even 4 825.4.c.i.199.1 4
55.54 odd 2 825.4.a.k.1.1 2
77.76 even 2 1617.4.a.j.1.2 2
88.21 odd 2 2112.4.a.ba.1.2 2
88.43 even 2 2112.4.a.bh.1.2 2
132.131 odd 2 1584.4.a.x.1.2 2
165.164 even 2 2475.4.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 11.10 odd 2
99.4.a.e.1.1 2 33.32 even 2
363.4.a.j.1.1 2 1.1 even 1 trivial
528.4.a.o.1.1 2 44.43 even 2
825.4.a.k.1.1 2 55.54 odd 2
825.4.c.i.199.1 4 55.43 even 4
825.4.c.i.199.4 4 55.32 even 4
1089.4.a.t.1.2 2 3.2 odd 2
1584.4.a.x.1.2 2 132.131 odd 2
1617.4.a.j.1.2 2 77.76 even 2
2112.4.a.ba.1.2 2 88.21 odd 2
2112.4.a.bh.1.2 2 88.43 even 2
2475.4.a.o.1.2 2 165.164 even 2