Properties

Label 825.4.a.q.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $3$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.76300\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.76300 q^{2} -3.00000 q^{3} +6.16019 q^{4} +11.2890 q^{6} +23.6321 q^{7} +6.92320 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.76300 q^{2} -3.00000 q^{3} +6.16019 q^{4} +11.2890 q^{6} +23.6321 q^{7} +6.92320 q^{8} +9.00000 q^{9} +11.0000 q^{11} -18.4806 q^{12} -7.39511 q^{13} -88.9277 q^{14} -75.3336 q^{16} -8.68828 q^{17} -33.8670 q^{18} -69.7002 q^{19} -70.8963 q^{21} -41.3930 q^{22} +10.1643 q^{23} -20.7696 q^{24} +27.8278 q^{26} -27.0000 q^{27} +145.578 q^{28} -73.2803 q^{29} -290.970 q^{31} +228.095 q^{32} -33.0000 q^{33} +32.6940 q^{34} +55.4417 q^{36} +105.618 q^{37} +262.282 q^{38} +22.1853 q^{39} +40.5108 q^{41} +266.783 q^{42} +77.3133 q^{43} +67.7621 q^{44} -38.2485 q^{46} +472.981 q^{47} +226.001 q^{48} +215.477 q^{49} +26.0648 q^{51} -45.5553 q^{52} -205.479 q^{53} +101.601 q^{54} +163.610 q^{56} +209.101 q^{57} +275.754 q^{58} +330.261 q^{59} -931.991 q^{61} +1094.92 q^{62} +212.689 q^{63} -255.653 q^{64} +124.179 q^{66} +418.252 q^{67} -53.5215 q^{68} -30.4930 q^{69} -506.329 q^{71} +62.3088 q^{72} -612.497 q^{73} -397.441 q^{74} -429.367 q^{76} +259.953 q^{77} -83.4835 q^{78} +54.6657 q^{79} +81.0000 q^{81} -152.442 q^{82} -538.964 q^{83} -436.735 q^{84} -290.930 q^{86} +219.841 q^{87} +76.1552 q^{88} +781.086 q^{89} -174.762 q^{91} +62.6144 q^{92} +872.909 q^{93} -1779.83 q^{94} -684.285 q^{96} +531.762 q^{97} -810.839 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 9 q^{3} + 7 q^{4} - 3 q^{6} + 16 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 9 q^{3} + 7 q^{4} - 3 q^{6} + 16 q^{7} - 3 q^{8} + 27 q^{9} + 33 q^{11} - 21 q^{12} + 45 q^{13} - 116 q^{14} - 85 q^{16} - 58 q^{17} + 9 q^{18} - 169 q^{19} - 48 q^{21} + 11 q^{22} + 155 q^{23} + 9 q^{24} + 167 q^{26} - 81 q^{27} + 100 q^{28} - 277 q^{29} - 173 q^{31} + 97 q^{32} - 99 q^{33} - 146 q^{34} + 63 q^{36} + 60 q^{37} + 169 q^{38} - 135 q^{39} + 44 q^{41} + 348 q^{42} - 109 q^{43} + 77 q^{44} + 425 q^{46} + 270 q^{47} + 255 q^{48} - 427 q^{49} + 174 q^{51} + 45 q^{52} - 148 q^{53} - 27 q^{54} + 168 q^{56} + 507 q^{57} - 783 q^{58} - 684 q^{59} - 1038 q^{61} + 953 q^{62} + 144 q^{63} - 1129 q^{64} - 33 q^{66} + 314 q^{67} - 366 q^{68} - 465 q^{69} - 1459 q^{71} - 27 q^{72} - 1170 q^{73} - 1764 q^{74} + 211 q^{76} + 176 q^{77} - 501 q^{78} - 506 q^{79} + 243 q^{81} - 1040 q^{82} - 347 q^{83} - 300 q^{84} - 527 q^{86} + 831 q^{87} - 33 q^{88} - 607 q^{89} - 398 q^{91} + 687 q^{92} + 519 q^{93} - 1822 q^{94} - 291 q^{96} + 1263 q^{97} - 2273 q^{98} + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.76300 −1.33042 −0.665211 0.746655i \(-0.731659\pi\)
−0.665211 + 0.746655i \(0.731659\pi\)
\(3\) −3.00000 −0.577350
\(4\) 6.16019 0.770024
\(5\) 0 0
\(6\) 11.2890 0.768120
\(7\) 23.6321 1.27601 0.638007 0.770031i \(-0.279759\pi\)
0.638007 + 0.770031i \(0.279759\pi\)
\(8\) 6.92320 0.305965
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) −18.4806 −0.444574
\(13\) −7.39511 −0.157772 −0.0788860 0.996884i \(-0.525136\pi\)
−0.0788860 + 0.996884i \(0.525136\pi\)
\(14\) −88.9277 −1.69764
\(15\) 0 0
\(16\) −75.3336 −1.17709
\(17\) −8.68828 −0.123954 −0.0619770 0.998078i \(-0.519741\pi\)
−0.0619770 + 0.998078i \(0.519741\pi\)
\(18\) −33.8670 −0.443474
\(19\) −69.7002 −0.841597 −0.420798 0.907154i \(-0.638250\pi\)
−0.420798 + 0.907154i \(0.638250\pi\)
\(20\) 0 0
\(21\) −70.8963 −0.736707
\(22\) −41.3930 −0.401137
\(23\) 10.1643 0.0921484 0.0460742 0.998938i \(-0.485329\pi\)
0.0460742 + 0.998938i \(0.485329\pi\)
\(24\) −20.7696 −0.176649
\(25\) 0 0
\(26\) 27.8278 0.209903
\(27\) −27.0000 −0.192450
\(28\) 145.578 0.982562
\(29\) −73.2803 −0.469235 −0.234617 0.972088i \(-0.575384\pi\)
−0.234617 + 0.972088i \(0.575384\pi\)
\(30\) 0 0
\(31\) −290.970 −1.68580 −0.842898 0.538073i \(-0.819153\pi\)
−0.842898 + 0.538073i \(0.819153\pi\)
\(32\) 228.095 1.26006
\(33\) −33.0000 −0.174078
\(34\) 32.6940 0.164911
\(35\) 0 0
\(36\) 55.4417 0.256675
\(37\) 105.618 0.469284 0.234642 0.972082i \(-0.424608\pi\)
0.234642 + 0.972082i \(0.424608\pi\)
\(38\) 262.282 1.11968
\(39\) 22.1853 0.0910897
\(40\) 0 0
\(41\) 40.5108 0.154310 0.0771551 0.997019i \(-0.475416\pi\)
0.0771551 + 0.997019i \(0.475416\pi\)
\(42\) 266.783 0.980132
\(43\) 77.3133 0.274190 0.137095 0.990558i \(-0.456223\pi\)
0.137095 + 0.990558i \(0.456223\pi\)
\(44\) 67.7621 0.232171
\(45\) 0 0
\(46\) −38.2485 −0.122596
\(47\) 472.981 1.46790 0.733951 0.679202i \(-0.237674\pi\)
0.733951 + 0.679202i \(0.237674\pi\)
\(48\) 226.001 0.679591
\(49\) 215.477 0.628212
\(50\) 0 0
\(51\) 26.0648 0.0715648
\(52\) −45.5553 −0.121488
\(53\) −205.479 −0.532541 −0.266271 0.963898i \(-0.585792\pi\)
−0.266271 + 0.963898i \(0.585792\pi\)
\(54\) 101.601 0.256040
\(55\) 0 0
\(56\) 163.610 0.390416
\(57\) 209.101 0.485896
\(58\) 275.754 0.624281
\(59\) 330.261 0.728751 0.364376 0.931252i \(-0.381282\pi\)
0.364376 + 0.931252i \(0.381282\pi\)
\(60\) 0 0
\(61\) −931.991 −1.95622 −0.978109 0.208094i \(-0.933274\pi\)
−0.978109 + 0.208094i \(0.933274\pi\)
\(62\) 1094.92 2.24282
\(63\) 212.689 0.425338
\(64\) −255.653 −0.499323
\(65\) 0 0
\(66\) 124.179 0.231597
\(67\) 418.252 0.762652 0.381326 0.924441i \(-0.375468\pi\)
0.381326 + 0.924441i \(0.375468\pi\)
\(68\) −53.5215 −0.0954475
\(69\) −30.4930 −0.0532019
\(70\) 0 0
\(71\) −506.329 −0.846340 −0.423170 0.906050i \(-0.639083\pi\)
−0.423170 + 0.906050i \(0.639083\pi\)
\(72\) 62.3088 0.101988
\(73\) −612.497 −0.982018 −0.491009 0.871154i \(-0.663372\pi\)
−0.491009 + 0.871154i \(0.663372\pi\)
\(74\) −397.441 −0.624346
\(75\) 0 0
\(76\) −429.367 −0.648050
\(77\) 259.953 0.384733
\(78\) −83.4835 −0.121188
\(79\) 54.6657 0.0778528 0.0389264 0.999242i \(-0.487606\pi\)
0.0389264 + 0.999242i \(0.487606\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −152.442 −0.205298
\(83\) −538.964 −0.712759 −0.356379 0.934341i \(-0.615989\pi\)
−0.356379 + 0.934341i \(0.615989\pi\)
\(84\) −436.735 −0.567282
\(85\) 0 0
\(86\) −290.930 −0.364788
\(87\) 219.841 0.270913
\(88\) 76.1552 0.0922519
\(89\) 781.086 0.930280 0.465140 0.885237i \(-0.346004\pi\)
0.465140 + 0.885237i \(0.346004\pi\)
\(90\) 0 0
\(91\) −174.762 −0.201319
\(92\) 62.6144 0.0709565
\(93\) 872.909 0.973295
\(94\) −1779.83 −1.95293
\(95\) 0 0
\(96\) −684.285 −0.727495
\(97\) 531.762 0.556621 0.278311 0.960491i \(-0.410226\pi\)
0.278311 + 0.960491i \(0.410226\pi\)
\(98\) −810.839 −0.835787
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1898.16 1.87004 0.935022 0.354591i \(-0.115380\pi\)
0.935022 + 0.354591i \(0.115380\pi\)
\(102\) −98.0820 −0.0952115
\(103\) −1308.66 −1.25190 −0.625952 0.779861i \(-0.715290\pi\)
−0.625952 + 0.779861i \(0.715290\pi\)
\(104\) −51.1978 −0.0482727
\(105\) 0 0
\(106\) 773.218 0.708505
\(107\) −206.655 −0.186711 −0.0933554 0.995633i \(-0.529759\pi\)
−0.0933554 + 0.995633i \(0.529759\pi\)
\(108\) −166.325 −0.148191
\(109\) 595.701 0.523466 0.261733 0.965140i \(-0.415706\pi\)
0.261733 + 0.965140i \(0.415706\pi\)
\(110\) 0 0
\(111\) −316.854 −0.270941
\(112\) −1780.29 −1.50198
\(113\) −1513.00 −1.25957 −0.629785 0.776769i \(-0.716857\pi\)
−0.629785 + 0.776769i \(0.716857\pi\)
\(114\) −786.847 −0.646447
\(115\) 0 0
\(116\) −451.421 −0.361322
\(117\) −66.5560 −0.0525907
\(118\) −1242.77 −0.969547
\(119\) −205.322 −0.158167
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 3507.09 2.60260
\(123\) −121.532 −0.0890911
\(124\) −1792.43 −1.29810
\(125\) 0 0
\(126\) −800.349 −0.565879
\(127\) −77.9297 −0.0544499 −0.0272250 0.999629i \(-0.508667\pi\)
−0.0272250 + 0.999629i \(0.508667\pi\)
\(128\) −862.735 −0.595748
\(129\) −231.940 −0.158304
\(130\) 0 0
\(131\) −8.60213 −0.00573719 −0.00286859 0.999996i \(-0.500913\pi\)
−0.00286859 + 0.999996i \(0.500913\pi\)
\(132\) −203.286 −0.134044
\(133\) −1647.16 −1.07389
\(134\) −1573.89 −1.01465
\(135\) 0 0
\(136\) −60.1507 −0.0379256
\(137\) 665.161 0.414807 0.207403 0.978255i \(-0.433499\pi\)
0.207403 + 0.978255i \(0.433499\pi\)
\(138\) 114.745 0.0707810
\(139\) 745.658 0.455006 0.227503 0.973777i \(-0.426944\pi\)
0.227503 + 0.973777i \(0.426944\pi\)
\(140\) 0 0
\(141\) −1418.94 −0.847494
\(142\) 1905.32 1.12599
\(143\) −81.3462 −0.0475700
\(144\) −678.002 −0.392362
\(145\) 0 0
\(146\) 2304.83 1.30650
\(147\) −646.430 −0.362698
\(148\) 650.628 0.361360
\(149\) −2398.49 −1.31874 −0.659370 0.751819i \(-0.729177\pi\)
−0.659370 + 0.751819i \(0.729177\pi\)
\(150\) 0 0
\(151\) −3254.28 −1.75384 −0.876920 0.480637i \(-0.840405\pi\)
−0.876920 + 0.480637i \(0.840405\pi\)
\(152\) −482.549 −0.257499
\(153\) −78.1945 −0.0413180
\(154\) −978.205 −0.511857
\(155\) 0 0
\(156\) 136.666 0.0701413
\(157\) −2492.15 −1.26685 −0.633424 0.773805i \(-0.718351\pi\)
−0.633424 + 0.773805i \(0.718351\pi\)
\(158\) −205.707 −0.103577
\(159\) 616.437 0.307463
\(160\) 0 0
\(161\) 240.205 0.117583
\(162\) −304.803 −0.147825
\(163\) 807.550 0.388050 0.194025 0.980997i \(-0.437846\pi\)
0.194025 + 0.980997i \(0.437846\pi\)
\(164\) 249.554 0.118823
\(165\) 0 0
\(166\) 2028.12 0.948270
\(167\) −1833.28 −0.849484 −0.424742 0.905314i \(-0.639635\pi\)
−0.424742 + 0.905314i \(0.639635\pi\)
\(168\) −490.829 −0.225407
\(169\) −2142.31 −0.975108
\(170\) 0 0
\(171\) −627.302 −0.280532
\(172\) 476.265 0.211133
\(173\) 1046.74 0.460011 0.230005 0.973189i \(-0.426126\pi\)
0.230005 + 0.973189i \(0.426126\pi\)
\(174\) −827.262 −0.360429
\(175\) 0 0
\(176\) −828.669 −0.354905
\(177\) −990.783 −0.420745
\(178\) −2939.23 −1.23767
\(179\) 1880.80 0.785348 0.392674 0.919678i \(-0.371550\pi\)
0.392674 + 0.919678i \(0.371550\pi\)
\(180\) 0 0
\(181\) −1921.82 −0.789216 −0.394608 0.918850i \(-0.629120\pi\)
−0.394608 + 0.918850i \(0.629120\pi\)
\(182\) 657.630 0.267840
\(183\) 2795.97 1.12942
\(184\) 70.3698 0.0281942
\(185\) 0 0
\(186\) −3284.76 −1.29489
\(187\) −95.5711 −0.0373735
\(188\) 2913.66 1.13032
\(189\) −638.067 −0.245569
\(190\) 0 0
\(191\) 203.716 0.0771747 0.0385874 0.999255i \(-0.487714\pi\)
0.0385874 + 0.999255i \(0.487714\pi\)
\(192\) 766.960 0.288284
\(193\) −2858.36 −1.06606 −0.533030 0.846096i \(-0.678947\pi\)
−0.533030 + 0.846096i \(0.678947\pi\)
\(194\) −2001.02 −0.740542
\(195\) 0 0
\(196\) 1327.38 0.483738
\(197\) 1140.57 0.412499 0.206250 0.978499i \(-0.433874\pi\)
0.206250 + 0.978499i \(0.433874\pi\)
\(198\) −372.537 −0.133712
\(199\) −3223.56 −1.14830 −0.574151 0.818749i \(-0.694668\pi\)
−0.574151 + 0.818749i \(0.694668\pi\)
\(200\) 0 0
\(201\) −1254.76 −0.440317
\(202\) −7142.80 −2.48795
\(203\) −1731.77 −0.598750
\(204\) 160.564 0.0551067
\(205\) 0 0
\(206\) 4924.50 1.66556
\(207\) 91.4791 0.0307161
\(208\) 557.100 0.185711
\(209\) −766.703 −0.253751
\(210\) 0 0
\(211\) 1679.38 0.547930 0.273965 0.961740i \(-0.411665\pi\)
0.273965 + 0.961740i \(0.411665\pi\)
\(212\) −1265.79 −0.410070
\(213\) 1518.99 0.488635
\(214\) 777.642 0.248404
\(215\) 0 0
\(216\) −186.926 −0.0588830
\(217\) −6876.23 −2.15110
\(218\) −2241.62 −0.696431
\(219\) 1837.49 0.566969
\(220\) 0 0
\(221\) 64.2508 0.0195565
\(222\) 1192.32 0.360466
\(223\) 6121.75 1.83831 0.919154 0.393898i \(-0.128874\pi\)
0.919154 + 0.393898i \(0.128874\pi\)
\(224\) 5390.36 1.60785
\(225\) 0 0
\(226\) 5693.44 1.67576
\(227\) 1760.23 0.514671 0.257335 0.966322i \(-0.417155\pi\)
0.257335 + 0.966322i \(0.417155\pi\)
\(228\) 1288.10 0.374152
\(229\) −5780.58 −1.66808 −0.834042 0.551700i \(-0.813979\pi\)
−0.834042 + 0.551700i \(0.813979\pi\)
\(230\) 0 0
\(231\) −779.860 −0.222126
\(232\) −507.334 −0.143569
\(233\) −5433.94 −1.52785 −0.763925 0.645305i \(-0.776730\pi\)
−0.763925 + 0.645305i \(0.776730\pi\)
\(234\) 250.451 0.0699678
\(235\) 0 0
\(236\) 2034.47 0.561156
\(237\) −163.997 −0.0449484
\(238\) 772.629 0.210429
\(239\) 583.827 0.158011 0.0790055 0.996874i \(-0.474826\pi\)
0.0790055 + 0.996874i \(0.474826\pi\)
\(240\) 0 0
\(241\) 4217.86 1.12737 0.563686 0.825989i \(-0.309383\pi\)
0.563686 + 0.825989i \(0.309383\pi\)
\(242\) −455.323 −0.120948
\(243\) −243.000 −0.0641500
\(244\) −5741.25 −1.50633
\(245\) 0 0
\(246\) 457.327 0.118529
\(247\) 515.441 0.132780
\(248\) −2014.44 −0.515795
\(249\) 1616.89 0.411511
\(250\) 0 0
\(251\) −4836.75 −1.21631 −0.608153 0.793820i \(-0.708089\pi\)
−0.608153 + 0.793820i \(0.708089\pi\)
\(252\) 1310.21 0.327521
\(253\) 111.808 0.0277838
\(254\) 293.250 0.0724414
\(255\) 0 0
\(256\) 5291.70 1.29192
\(257\) 4654.14 1.12964 0.564819 0.825215i \(-0.308946\pi\)
0.564819 + 0.825215i \(0.308946\pi\)
\(258\) 872.790 0.210611
\(259\) 2495.98 0.598813
\(260\) 0 0
\(261\) −659.523 −0.156412
\(262\) 32.3698 0.00763288
\(263\) −5102.11 −1.19624 −0.598118 0.801408i \(-0.704084\pi\)
−0.598118 + 0.801408i \(0.704084\pi\)
\(264\) −228.466 −0.0532617
\(265\) 0 0
\(266\) 6198.28 1.42873
\(267\) −2343.26 −0.537097
\(268\) 2576.52 0.587260
\(269\) 4475.15 1.01433 0.507165 0.861849i \(-0.330694\pi\)
0.507165 + 0.861849i \(0.330694\pi\)
\(270\) 0 0
\(271\) −7829.40 −1.75499 −0.877495 0.479586i \(-0.840787\pi\)
−0.877495 + 0.479586i \(0.840787\pi\)
\(272\) 654.519 0.145905
\(273\) 524.286 0.116232
\(274\) −2503.00 −0.551868
\(275\) 0 0
\(276\) −187.843 −0.0409668
\(277\) −3692.51 −0.800943 −0.400472 0.916309i \(-0.631154\pi\)
−0.400472 + 0.916309i \(0.631154\pi\)
\(278\) −2805.91 −0.605351
\(279\) −2618.73 −0.561932
\(280\) 0 0
\(281\) −1296.21 −0.275179 −0.137589 0.990489i \(-0.543935\pi\)
−0.137589 + 0.990489i \(0.543935\pi\)
\(282\) 5339.49 1.12752
\(283\) −1168.50 −0.245442 −0.122721 0.992441i \(-0.539162\pi\)
−0.122721 + 0.992441i \(0.539162\pi\)
\(284\) −3119.08 −0.651703
\(285\) 0 0
\(286\) 306.106 0.0632882
\(287\) 957.355 0.196902
\(288\) 2052.85 0.420019
\(289\) −4837.51 −0.984635
\(290\) 0 0
\(291\) −1595.29 −0.321365
\(292\) −3773.10 −0.756178
\(293\) 8879.43 1.77045 0.885224 0.465164i \(-0.154005\pi\)
0.885224 + 0.465164i \(0.154005\pi\)
\(294\) 2432.52 0.482542
\(295\) 0 0
\(296\) 731.215 0.143584
\(297\) −297.000 −0.0580259
\(298\) 9025.54 1.75448
\(299\) −75.1665 −0.0145384
\(300\) 0 0
\(301\) 1827.08 0.349870
\(302\) 12245.9 2.33335
\(303\) −5694.49 −1.07967
\(304\) 5250.77 0.990632
\(305\) 0 0
\(306\) 294.246 0.0549704
\(307\) 4724.14 0.878244 0.439122 0.898427i \(-0.355290\pi\)
0.439122 + 0.898427i \(0.355290\pi\)
\(308\) 1601.36 0.296253
\(309\) 3925.98 0.722788
\(310\) 0 0
\(311\) 53.7156 0.00979399 0.00489700 0.999988i \(-0.498441\pi\)
0.00489700 + 0.999988i \(0.498441\pi\)
\(312\) 153.593 0.0278703
\(313\) 4663.77 0.842210 0.421105 0.907012i \(-0.361642\pi\)
0.421105 + 0.907012i \(0.361642\pi\)
\(314\) 9377.96 1.68544
\(315\) 0 0
\(316\) 336.751 0.0599486
\(317\) −8664.97 −1.53525 −0.767624 0.640901i \(-0.778561\pi\)
−0.767624 + 0.640901i \(0.778561\pi\)
\(318\) −2319.65 −0.409056
\(319\) −806.084 −0.141480
\(320\) 0 0
\(321\) 619.964 0.107798
\(322\) −903.892 −0.156435
\(323\) 605.575 0.104319
\(324\) 498.976 0.0855582
\(325\) 0 0
\(326\) −3038.81 −0.516271
\(327\) −1787.10 −0.302223
\(328\) 280.464 0.0472135
\(329\) 11177.5 1.87306
\(330\) 0 0
\(331\) −3422.54 −0.568337 −0.284169 0.958774i \(-0.591718\pi\)
−0.284169 + 0.958774i \(0.591718\pi\)
\(332\) −3320.12 −0.548841
\(333\) 950.563 0.156428
\(334\) 6898.66 1.13017
\(335\) 0 0
\(336\) 5340.87 0.867168
\(337\) −118.863 −0.0192134 −0.00960668 0.999954i \(-0.503058\pi\)
−0.00960668 + 0.999954i \(0.503058\pi\)
\(338\) 8061.53 1.29731
\(339\) 4539.01 0.727213
\(340\) 0 0
\(341\) −3200.67 −0.508287
\(342\) 2360.54 0.373226
\(343\) −3013.65 −0.474407
\(344\) 535.255 0.0838925
\(345\) 0 0
\(346\) −3938.87 −0.612009
\(347\) 10434.7 1.61431 0.807154 0.590341i \(-0.201007\pi\)
0.807154 + 0.590341i \(0.201007\pi\)
\(348\) 1354.26 0.208610
\(349\) 7528.91 1.15477 0.577383 0.816473i \(-0.304074\pi\)
0.577383 + 0.816473i \(0.304074\pi\)
\(350\) 0 0
\(351\) 199.668 0.0303632
\(352\) 2509.04 0.379922
\(353\) 215.454 0.0324857 0.0162428 0.999868i \(-0.494830\pi\)
0.0162428 + 0.999868i \(0.494830\pi\)
\(354\) 3728.32 0.559768
\(355\) 0 0
\(356\) 4811.64 0.716338
\(357\) 615.967 0.0913177
\(358\) −7077.45 −1.04485
\(359\) 9105.02 1.33856 0.669282 0.743008i \(-0.266602\pi\)
0.669282 + 0.743008i \(0.266602\pi\)
\(360\) 0 0
\(361\) −2000.88 −0.291715
\(362\) 7231.83 1.04999
\(363\) −363.000 −0.0524864
\(364\) −1076.57 −0.155021
\(365\) 0 0
\(366\) −10521.3 −1.50261
\(367\) −3763.06 −0.535232 −0.267616 0.963526i \(-0.586236\pi\)
−0.267616 + 0.963526i \(0.586236\pi\)
\(368\) −765.717 −0.108467
\(369\) 364.597 0.0514368
\(370\) 0 0
\(371\) −4855.90 −0.679530
\(372\) 5377.29 0.749461
\(373\) −375.989 −0.0521929 −0.0260965 0.999659i \(-0.508308\pi\)
−0.0260965 + 0.999659i \(0.508308\pi\)
\(374\) 359.634 0.0497226
\(375\) 0 0
\(376\) 3274.54 0.449127
\(377\) 541.916 0.0740321
\(378\) 2401.05 0.326711
\(379\) 2967.48 0.402187 0.201094 0.979572i \(-0.435550\pi\)
0.201094 + 0.979572i \(0.435550\pi\)
\(380\) 0 0
\(381\) 233.789 0.0314367
\(382\) −766.584 −0.102675
\(383\) 1550.93 0.206916 0.103458 0.994634i \(-0.467009\pi\)
0.103458 + 0.994634i \(0.467009\pi\)
\(384\) 2588.20 0.343955
\(385\) 0 0
\(386\) 10756.0 1.41831
\(387\) 695.820 0.0913966
\(388\) 3275.76 0.428612
\(389\) −7511.63 −0.979061 −0.489530 0.871986i \(-0.662832\pi\)
−0.489530 + 0.871986i \(0.662832\pi\)
\(390\) 0 0
\(391\) −88.3107 −0.0114222
\(392\) 1491.79 0.192211
\(393\) 25.8064 0.00331237
\(394\) −4291.97 −0.548798
\(395\) 0 0
\(396\) 609.859 0.0773903
\(397\) −11125.1 −1.40643 −0.703217 0.710975i \(-0.748254\pi\)
−0.703217 + 0.710975i \(0.748254\pi\)
\(398\) 12130.3 1.52773
\(399\) 4941.49 0.620010
\(400\) 0 0
\(401\) −13478.0 −1.67845 −0.839225 0.543784i \(-0.816991\pi\)
−0.839225 + 0.543784i \(0.816991\pi\)
\(402\) 4721.66 0.585808
\(403\) 2151.75 0.265971
\(404\) 11693.1 1.43998
\(405\) 0 0
\(406\) 6516.65 0.796591
\(407\) 1161.80 0.141494
\(408\) 180.452 0.0218963
\(409\) 12911.4 1.56095 0.780474 0.625188i \(-0.214978\pi\)
0.780474 + 0.625188i \(0.214978\pi\)
\(410\) 0 0
\(411\) −1995.48 −0.239489
\(412\) −8061.61 −0.963997
\(413\) 7804.76 0.929897
\(414\) −344.236 −0.0408654
\(415\) 0 0
\(416\) −1686.79 −0.198802
\(417\) −2236.97 −0.262698
\(418\) 2885.10 0.337596
\(419\) −8906.41 −1.03844 −0.519220 0.854640i \(-0.673778\pi\)
−0.519220 + 0.854640i \(0.673778\pi\)
\(420\) 0 0
\(421\) 1852.36 0.214439 0.107219 0.994235i \(-0.465805\pi\)
0.107219 + 0.994235i \(0.465805\pi\)
\(422\) −6319.51 −0.728978
\(423\) 4256.83 0.489301
\(424\) −1422.57 −0.162939
\(425\) 0 0
\(426\) −5715.95 −0.650091
\(427\) −22024.9 −2.49616
\(428\) −1273.03 −0.143772
\(429\) 244.039 0.0274646
\(430\) 0 0
\(431\) −4090.74 −0.457178 −0.228589 0.973523i \(-0.573411\pi\)
−0.228589 + 0.973523i \(0.573411\pi\)
\(432\) 2034.01 0.226530
\(433\) −8253.94 −0.916071 −0.458036 0.888934i \(-0.651447\pi\)
−0.458036 + 0.888934i \(0.651447\pi\)
\(434\) 25875.3 2.86187
\(435\) 0 0
\(436\) 3669.63 0.403082
\(437\) −708.458 −0.0775518
\(438\) −6914.48 −0.754308
\(439\) 979.240 0.106461 0.0532307 0.998582i \(-0.483048\pi\)
0.0532307 + 0.998582i \(0.483048\pi\)
\(440\) 0 0
\(441\) 1939.29 0.209404
\(442\) −241.776 −0.0260183
\(443\) −89.5194 −0.00960089 −0.00480045 0.999988i \(-0.501528\pi\)
−0.00480045 + 0.999988i \(0.501528\pi\)
\(444\) −1951.88 −0.208631
\(445\) 0 0
\(446\) −23036.2 −2.44573
\(447\) 7195.48 0.761375
\(448\) −6041.63 −0.637143
\(449\) −17119.1 −1.79933 −0.899666 0.436579i \(-0.856190\pi\)
−0.899666 + 0.436579i \(0.856190\pi\)
\(450\) 0 0
\(451\) 445.619 0.0465263
\(452\) −9320.40 −0.969899
\(453\) 9762.85 1.01258
\(454\) −6623.74 −0.684730
\(455\) 0 0
\(456\) 1447.65 0.148667
\(457\) 1780.97 0.182298 0.0911492 0.995837i \(-0.470946\pi\)
0.0911492 + 0.995837i \(0.470946\pi\)
\(458\) 21752.3 2.21926
\(459\) 234.583 0.0238549
\(460\) 0 0
\(461\) −7216.14 −0.729043 −0.364522 0.931195i \(-0.618767\pi\)
−0.364522 + 0.931195i \(0.618767\pi\)
\(462\) 2934.61 0.295521
\(463\) −9451.28 −0.948679 −0.474339 0.880342i \(-0.657313\pi\)
−0.474339 + 0.880342i \(0.657313\pi\)
\(464\) 5520.47 0.552330
\(465\) 0 0
\(466\) 20447.9 2.03269
\(467\) −7188.38 −0.712289 −0.356144 0.934431i \(-0.615909\pi\)
−0.356144 + 0.934431i \(0.615909\pi\)
\(468\) −409.998 −0.0404961
\(469\) 9884.19 0.973154
\(470\) 0 0
\(471\) 7476.44 0.731415
\(472\) 2286.46 0.222972
\(473\) 850.446 0.0826714
\(474\) 617.122 0.0598003
\(475\) 0 0
\(476\) −1264.83 −0.121792
\(477\) −1849.31 −0.177514
\(478\) −2196.94 −0.210221
\(479\) −8224.85 −0.784558 −0.392279 0.919846i \(-0.628313\pi\)
−0.392279 + 0.919846i \(0.628313\pi\)
\(480\) 0 0
\(481\) −781.058 −0.0740399
\(482\) −15871.8 −1.49988
\(483\) −720.615 −0.0678864
\(484\) 745.383 0.0700022
\(485\) 0 0
\(486\) 914.410 0.0853466
\(487\) 4817.49 0.448257 0.224129 0.974560i \(-0.428046\pi\)
0.224129 + 0.974560i \(0.428046\pi\)
\(488\) −6452.36 −0.598534
\(489\) −2422.65 −0.224041
\(490\) 0 0
\(491\) −4944.04 −0.454422 −0.227211 0.973846i \(-0.572961\pi\)
−0.227211 + 0.973846i \(0.572961\pi\)
\(492\) −748.663 −0.0686023
\(493\) 636.680 0.0581635
\(494\) −1939.61 −0.176654
\(495\) 0 0
\(496\) 21919.8 1.98433
\(497\) −11965.6 −1.07994
\(498\) −6084.37 −0.547484
\(499\) −3616.97 −0.324485 −0.162242 0.986751i \(-0.551873\pi\)
−0.162242 + 0.986751i \(0.551873\pi\)
\(500\) 0 0
\(501\) 5499.85 0.490450
\(502\) 18200.7 1.61820
\(503\) 1176.71 0.104308 0.0521540 0.998639i \(-0.483391\pi\)
0.0521540 + 0.998639i \(0.483391\pi\)
\(504\) 1472.49 0.130139
\(505\) 0 0
\(506\) −420.733 −0.0369642
\(507\) 6426.94 0.562979
\(508\) −480.062 −0.0419278
\(509\) −14606.6 −1.27196 −0.635981 0.771705i \(-0.719404\pi\)
−0.635981 + 0.771705i \(0.719404\pi\)
\(510\) 0 0
\(511\) −14474.6 −1.25307
\(512\) −13010.8 −1.12305
\(513\) 1881.91 0.161965
\(514\) −17513.5 −1.50290
\(515\) 0 0
\(516\) −1428.79 −0.121898
\(517\) 5202.79 0.442589
\(518\) −9392.38 −0.796674
\(519\) −3140.21 −0.265587
\(520\) 0 0
\(521\) 4175.52 0.351119 0.175559 0.984469i \(-0.443827\pi\)
0.175559 + 0.984469i \(0.443827\pi\)
\(522\) 2481.79 0.208094
\(523\) −7136.72 −0.596686 −0.298343 0.954459i \(-0.596434\pi\)
−0.298343 + 0.954459i \(0.596434\pi\)
\(524\) −52.9908 −0.00441777
\(525\) 0 0
\(526\) 19199.3 1.59150
\(527\) 2528.03 0.208961
\(528\) 2486.01 0.204905
\(529\) −12063.7 −0.991509
\(530\) 0 0
\(531\) 2972.35 0.242917
\(532\) −10146.8 −0.826920
\(533\) −299.582 −0.0243458
\(534\) 8817.68 0.714566
\(535\) 0 0
\(536\) 2895.64 0.233345
\(537\) −5642.39 −0.453421
\(538\) −16840.0 −1.34949
\(539\) 2370.24 0.189413
\(540\) 0 0
\(541\) 18984.0 1.50866 0.754329 0.656496i \(-0.227962\pi\)
0.754329 + 0.656496i \(0.227962\pi\)
\(542\) 29462.1 2.33488
\(543\) 5765.47 0.455654
\(544\) −1981.75 −0.156189
\(545\) 0 0
\(546\) −1972.89 −0.154637
\(547\) −13084.1 −1.02273 −0.511366 0.859363i \(-0.670860\pi\)
−0.511366 + 0.859363i \(0.670860\pi\)
\(548\) 4097.52 0.319411
\(549\) −8387.92 −0.652073
\(550\) 0 0
\(551\) 5107.66 0.394907
\(552\) −211.109 −0.0162779
\(553\) 1291.87 0.0993413
\(554\) 13894.9 1.06559
\(555\) 0 0
\(556\) 4593.40 0.350366
\(557\) 11677.1 0.888284 0.444142 0.895956i \(-0.353509\pi\)
0.444142 + 0.895956i \(0.353509\pi\)
\(558\) 9854.28 0.747607
\(559\) −571.740 −0.0432595
\(560\) 0 0
\(561\) 286.713 0.0215776
\(562\) 4877.63 0.366104
\(563\) −13214.1 −0.989180 −0.494590 0.869127i \(-0.664682\pi\)
−0.494590 + 0.869127i \(0.664682\pi\)
\(564\) −8740.97 −0.652591
\(565\) 0 0
\(566\) 4397.06 0.326541
\(567\) 1914.20 0.141779
\(568\) −3505.41 −0.258951
\(569\) −22207.0 −1.63614 −0.818070 0.575118i \(-0.804956\pi\)
−0.818070 + 0.575118i \(0.804956\pi\)
\(570\) 0 0
\(571\) −7251.49 −0.531463 −0.265731 0.964047i \(-0.585613\pi\)
−0.265731 + 0.964047i \(0.585613\pi\)
\(572\) −501.109 −0.0366301
\(573\) −611.148 −0.0445569
\(574\) −3602.53 −0.261963
\(575\) 0 0
\(576\) −2300.88 −0.166441
\(577\) 8190.57 0.590949 0.295475 0.955351i \(-0.404522\pi\)
0.295475 + 0.955351i \(0.404522\pi\)
\(578\) 18203.6 1.30998
\(579\) 8575.09 0.615490
\(580\) 0 0
\(581\) −12736.9 −0.909490
\(582\) 6003.07 0.427552
\(583\) −2260.27 −0.160567
\(584\) −4240.44 −0.300463
\(585\) 0 0
\(586\) −33413.3 −2.35545
\(587\) −9678.56 −0.680540 −0.340270 0.940328i \(-0.610518\pi\)
−0.340270 + 0.940328i \(0.610518\pi\)
\(588\) −3982.13 −0.279286
\(589\) 20280.7 1.41876
\(590\) 0 0
\(591\) −3421.71 −0.238156
\(592\) −7956.59 −0.552388
\(593\) 12938.6 0.895994 0.447997 0.894035i \(-0.352137\pi\)
0.447997 + 0.894035i \(0.352137\pi\)
\(594\) 1117.61 0.0771989
\(595\) 0 0
\(596\) −14775.2 −1.01546
\(597\) 9670.69 0.662973
\(598\) 282.852 0.0193423
\(599\) −11678.3 −0.796601 −0.398300 0.917255i \(-0.630400\pi\)
−0.398300 + 0.917255i \(0.630400\pi\)
\(600\) 0 0
\(601\) 17335.0 1.17655 0.588277 0.808660i \(-0.299807\pi\)
0.588277 + 0.808660i \(0.299807\pi\)
\(602\) −6875.29 −0.465475
\(603\) 3764.27 0.254217
\(604\) −20047.0 −1.35050
\(605\) 0 0
\(606\) 21428.4 1.43642
\(607\) −10259.9 −0.686057 −0.343028 0.939325i \(-0.611453\pi\)
−0.343028 + 0.939325i \(0.611453\pi\)
\(608\) −15898.3 −1.06046
\(609\) 5195.31 0.345689
\(610\) 0 0
\(611\) −3497.75 −0.231594
\(612\) −481.693 −0.0318158
\(613\) −8691.75 −0.572686 −0.286343 0.958127i \(-0.592440\pi\)
−0.286343 + 0.958127i \(0.592440\pi\)
\(614\) −17777.0 −1.16844
\(615\) 0 0
\(616\) 1799.71 0.117715
\(617\) −7438.86 −0.485377 −0.242688 0.970104i \(-0.578029\pi\)
−0.242688 + 0.970104i \(0.578029\pi\)
\(618\) −14773.5 −0.961613
\(619\) −8091.72 −0.525418 −0.262709 0.964875i \(-0.584616\pi\)
−0.262709 + 0.964875i \(0.584616\pi\)
\(620\) 0 0
\(621\) −274.437 −0.0177340
\(622\) −202.132 −0.0130301
\(623\) 18458.7 1.18705
\(624\) −1671.30 −0.107220
\(625\) 0 0
\(626\) −17549.8 −1.12050
\(627\) 2300.11 0.146503
\(628\) −15352.1 −0.975503
\(629\) −917.639 −0.0581696
\(630\) 0 0
\(631\) 10374.5 0.654518 0.327259 0.944935i \(-0.393875\pi\)
0.327259 + 0.944935i \(0.393875\pi\)
\(632\) 378.462 0.0238202
\(633\) −5038.14 −0.316347
\(634\) 32606.3 2.04253
\(635\) 0 0
\(636\) 3797.37 0.236754
\(637\) −1593.47 −0.0991142
\(638\) 3033.29 0.188228
\(639\) −4556.96 −0.282113
\(640\) 0 0
\(641\) 25560.8 1.57502 0.787511 0.616300i \(-0.211369\pi\)
0.787511 + 0.616300i \(0.211369\pi\)
\(642\) −2332.93 −0.143416
\(643\) 22746.6 1.39508 0.697541 0.716544i \(-0.254277\pi\)
0.697541 + 0.716544i \(0.254277\pi\)
\(644\) 1479.71 0.0905415
\(645\) 0 0
\(646\) −2278.78 −0.138789
\(647\) −24638.3 −1.49711 −0.748555 0.663073i \(-0.769252\pi\)
−0.748555 + 0.663073i \(0.769252\pi\)
\(648\) 560.779 0.0339961
\(649\) 3632.87 0.219727
\(650\) 0 0
\(651\) 20628.7 1.24194
\(652\) 4974.66 0.298808
\(653\) 17926.1 1.07428 0.537139 0.843493i \(-0.319505\pi\)
0.537139 + 0.843493i \(0.319505\pi\)
\(654\) 6724.87 0.402085
\(655\) 0 0
\(656\) −3051.82 −0.181637
\(657\) −5512.47 −0.327339
\(658\) −42061.1 −2.49197
\(659\) −28662.2 −1.69426 −0.847132 0.531382i \(-0.821673\pi\)
−0.847132 + 0.531382i \(0.821673\pi\)
\(660\) 0 0
\(661\) 13447.8 0.791312 0.395656 0.918399i \(-0.370517\pi\)
0.395656 + 0.918399i \(0.370517\pi\)
\(662\) 12879.0 0.756129
\(663\) −192.752 −0.0112909
\(664\) −3731.35 −0.218079
\(665\) 0 0
\(666\) −3576.97 −0.208115
\(667\) −744.847 −0.0432392
\(668\) −11293.4 −0.654123
\(669\) −18365.3 −1.06135
\(670\) 0 0
\(671\) −10251.9 −0.589822
\(672\) −16171.1 −0.928294
\(673\) 9621.01 0.551059 0.275529 0.961293i \(-0.411147\pi\)
0.275529 + 0.961293i \(0.411147\pi\)
\(674\) 447.283 0.0255619
\(675\) 0 0
\(676\) −13197.1 −0.750857
\(677\) −15901.5 −0.902723 −0.451361 0.892341i \(-0.649061\pi\)
−0.451361 + 0.892341i \(0.649061\pi\)
\(678\) −17080.3 −0.967501
\(679\) 12566.7 0.710257
\(680\) 0 0
\(681\) −5280.68 −0.297145
\(682\) 12044.1 0.676236
\(683\) −611.084 −0.0342350 −0.0171175 0.999853i \(-0.505449\pi\)
−0.0171175 + 0.999853i \(0.505449\pi\)
\(684\) −3864.30 −0.216017
\(685\) 0 0
\(686\) 11340.4 0.631162
\(687\) 17341.7 0.963069
\(688\) −5824.29 −0.322745
\(689\) 1519.54 0.0840201
\(690\) 0 0
\(691\) 24067.1 1.32497 0.662487 0.749074i \(-0.269501\pi\)
0.662487 + 0.749074i \(0.269501\pi\)
\(692\) 6448.09 0.354219
\(693\) 2339.58 0.128244
\(694\) −39265.9 −2.14771
\(695\) 0 0
\(696\) 1522.00 0.0828899
\(697\) −351.969 −0.0191274
\(698\) −28331.3 −1.53633
\(699\) 16301.8 0.882105
\(700\) 0 0
\(701\) 10093.5 0.543834 0.271917 0.962321i \(-0.412342\pi\)
0.271917 + 0.962321i \(0.412342\pi\)
\(702\) −751.352 −0.0403959
\(703\) −7361.61 −0.394948
\(704\) −2812.19 −0.150551
\(705\) 0 0
\(706\) −810.753 −0.0432197
\(707\) 44857.6 2.38620
\(708\) −6103.41 −0.323984
\(709\) −4970.98 −0.263313 −0.131657 0.991295i \(-0.542030\pi\)
−0.131657 + 0.991295i \(0.542030\pi\)
\(710\) 0 0
\(711\) 491.991 0.0259509
\(712\) 5407.61 0.284633
\(713\) −2957.52 −0.155343
\(714\) −2317.89 −0.121491
\(715\) 0 0
\(716\) 11586.1 0.604737
\(717\) −1751.48 −0.0912277
\(718\) −34262.2 −1.78086
\(719\) −35411.6 −1.83676 −0.918381 0.395698i \(-0.870503\pi\)
−0.918381 + 0.395698i \(0.870503\pi\)
\(720\) 0 0
\(721\) −30926.4 −1.59745
\(722\) 7529.30 0.388105
\(723\) −12653.6 −0.650888
\(724\) −11838.8 −0.607715
\(725\) 0 0
\(726\) 1365.97 0.0698291
\(727\) −34598.6 −1.76505 −0.882525 0.470266i \(-0.844158\pi\)
−0.882525 + 0.470266i \(0.844158\pi\)
\(728\) −1209.91 −0.0615966
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −671.719 −0.0339869
\(732\) 17223.7 0.869683
\(733\) 32108.2 1.61793 0.808966 0.587856i \(-0.200028\pi\)
0.808966 + 0.587856i \(0.200028\pi\)
\(734\) 14160.4 0.712085
\(735\) 0 0
\(736\) 2318.44 0.116112
\(737\) 4600.78 0.229948
\(738\) −1371.98 −0.0684326
\(739\) −27536.9 −1.37072 −0.685359 0.728206i \(-0.740355\pi\)
−0.685359 + 0.728206i \(0.740355\pi\)
\(740\) 0 0
\(741\) −1546.32 −0.0766608
\(742\) 18272.8 0.904062
\(743\) −2437.44 −0.120351 −0.0601757 0.998188i \(-0.519166\pi\)
−0.0601757 + 0.998188i \(0.519166\pi\)
\(744\) 6043.32 0.297794
\(745\) 0 0
\(746\) 1414.85 0.0694386
\(747\) −4850.67 −0.237586
\(748\) −588.736 −0.0287785
\(749\) −4883.69 −0.238246
\(750\) 0 0
\(751\) −29534.7 −1.43507 −0.717534 0.696523i \(-0.754729\pi\)
−0.717534 + 0.696523i \(0.754729\pi\)
\(752\) −35631.4 −1.72785
\(753\) 14510.3 0.702235
\(754\) −2039.23 −0.0984940
\(755\) 0 0
\(756\) −3930.62 −0.189094
\(757\) −747.476 −0.0358883 −0.0179442 0.999839i \(-0.505712\pi\)
−0.0179442 + 0.999839i \(0.505712\pi\)
\(758\) −11166.6 −0.535079
\(759\) −335.423 −0.0160410
\(760\) 0 0
\(761\) −21031.2 −1.00181 −0.500906 0.865502i \(-0.667000\pi\)
−0.500906 + 0.865502i \(0.667000\pi\)
\(762\) −879.749 −0.0418241
\(763\) 14077.7 0.667950
\(764\) 1254.93 0.0594264
\(765\) 0 0
\(766\) −5836.15 −0.275286
\(767\) −2442.32 −0.114976
\(768\) −15875.1 −0.745890
\(769\) 23107.4 1.08358 0.541790 0.840514i \(-0.317747\pi\)
0.541790 + 0.840514i \(0.317747\pi\)
\(770\) 0 0
\(771\) −13962.4 −0.652197
\(772\) −17608.1 −0.820892
\(773\) 24367.7 1.13382 0.566912 0.823778i \(-0.308138\pi\)
0.566912 + 0.823778i \(0.308138\pi\)
\(774\) −2618.37 −0.121596
\(775\) 0 0
\(776\) 3681.49 0.170307
\(777\) −7487.94 −0.345725
\(778\) 28266.3 1.30256
\(779\) −2823.61 −0.129867
\(780\) 0 0
\(781\) −5569.62 −0.255181
\(782\) 332.313 0.0151963
\(783\) 1978.57 0.0903043
\(784\) −16232.6 −0.739460
\(785\) 0 0
\(786\) −97.1095 −0.00440685
\(787\) 11308.1 0.512188 0.256094 0.966652i \(-0.417564\pi\)
0.256094 + 0.966652i \(0.417564\pi\)
\(788\) 7026.14 0.317634
\(789\) 15306.3 0.690647
\(790\) 0 0
\(791\) −35755.5 −1.60723
\(792\) 685.397 0.0307506
\(793\) 6892.18 0.308636
\(794\) 41863.9 1.87115
\(795\) 0 0
\(796\) −19857.8 −0.884221
\(797\) 43620.5 1.93867 0.969333 0.245752i \(-0.0790350\pi\)
0.969333 + 0.245752i \(0.0790350\pi\)
\(798\) −18594.8 −0.824875
\(799\) −4109.39 −0.181952
\(800\) 0 0
\(801\) 7029.77 0.310093
\(802\) 50717.7 2.23305
\(803\) −6737.47 −0.296090
\(804\) −7729.55 −0.339055
\(805\) 0 0
\(806\) −8097.06 −0.353854
\(807\) −13425.4 −0.585623
\(808\) 13141.4 0.572168
\(809\) −7491.05 −0.325551 −0.162776 0.986663i \(-0.552045\pi\)
−0.162776 + 0.986663i \(0.552045\pi\)
\(810\) 0 0
\(811\) −7335.75 −0.317624 −0.158812 0.987309i \(-0.550766\pi\)
−0.158812 + 0.987309i \(0.550766\pi\)
\(812\) −10668.0 −0.461052
\(813\) 23488.2 1.01324
\(814\) −4371.85 −0.188247
\(815\) 0 0
\(816\) −1963.56 −0.0842380
\(817\) −5388.76 −0.230757
\(818\) −48585.7 −2.07672
\(819\) −1572.86 −0.0671064
\(820\) 0 0
\(821\) −37482.9 −1.59338 −0.796689 0.604390i \(-0.793417\pi\)
−0.796689 + 0.604390i \(0.793417\pi\)
\(822\) 7509.01 0.318621
\(823\) 1213.87 0.0514128 0.0257064 0.999670i \(-0.491816\pi\)
0.0257064 + 0.999670i \(0.491816\pi\)
\(824\) −9060.12 −0.383039
\(825\) 0 0
\(826\) −29369.4 −1.23716
\(827\) −7999.10 −0.336343 −0.168172 0.985758i \(-0.553786\pi\)
−0.168172 + 0.985758i \(0.553786\pi\)
\(828\) 563.529 0.0236522
\(829\) −14809.5 −0.620451 −0.310225 0.950663i \(-0.600405\pi\)
−0.310225 + 0.950663i \(0.600405\pi\)
\(830\) 0 0
\(831\) 11077.5 0.462425
\(832\) 1890.58 0.0787791
\(833\) −1872.12 −0.0778693
\(834\) 8417.74 0.349499
\(835\) 0 0
\(836\) −4723.04 −0.195394
\(837\) 7856.18 0.324432
\(838\) 33514.8 1.38156
\(839\) −44598.1 −1.83516 −0.917578 0.397555i \(-0.869859\pi\)
−0.917578 + 0.397555i \(0.869859\pi\)
\(840\) 0 0
\(841\) −19019.0 −0.779819
\(842\) −6970.45 −0.285294
\(843\) 3888.62 0.158875
\(844\) 10345.3 0.421919
\(845\) 0 0
\(846\) −16018.5 −0.650977
\(847\) 2859.49 0.116001
\(848\) 15479.5 0.626848
\(849\) 3505.50 0.141706
\(850\) 0 0
\(851\) 1073.54 0.0432438
\(852\) 9357.25 0.376261
\(853\) 28250.2 1.13396 0.566980 0.823732i \(-0.308112\pi\)
0.566980 + 0.823732i \(0.308112\pi\)
\(854\) 82879.9 3.32095
\(855\) 0 0
\(856\) −1430.71 −0.0571270
\(857\) 28018.4 1.11679 0.558395 0.829575i \(-0.311417\pi\)
0.558395 + 0.829575i \(0.311417\pi\)
\(858\) −918.319 −0.0365395
\(859\) 17367.5 0.689837 0.344919 0.938633i \(-0.387906\pi\)
0.344919 + 0.938633i \(0.387906\pi\)
\(860\) 0 0
\(861\) −2872.07 −0.113681
\(862\) 15393.4 0.608240
\(863\) −49064.8 −1.93532 −0.967662 0.252252i \(-0.918829\pi\)
−0.967662 + 0.252252i \(0.918829\pi\)
\(864\) −6158.56 −0.242498
\(865\) 0 0
\(866\) 31059.6 1.21876
\(867\) 14512.5 0.568480
\(868\) −42358.9 −1.65640
\(869\) 601.323 0.0234735
\(870\) 0 0
\(871\) −3093.02 −0.120325
\(872\) 4124.15 0.160162
\(873\) 4785.86 0.185540
\(874\) 2665.93 0.103177
\(875\) 0 0
\(876\) 11319.3 0.436579
\(877\) 42355.0 1.63082 0.815409 0.578885i \(-0.196512\pi\)
0.815409 + 0.578885i \(0.196512\pi\)
\(878\) −3684.89 −0.141639
\(879\) −26638.3 −1.02217
\(880\) 0 0
\(881\) −36424.8 −1.39294 −0.696472 0.717584i \(-0.745248\pi\)
−0.696472 + 0.717584i \(0.745248\pi\)
\(882\) −7297.55 −0.278596
\(883\) −27158.6 −1.03506 −0.517532 0.855664i \(-0.673149\pi\)
−0.517532 + 0.855664i \(0.673149\pi\)
\(884\) 395.797 0.0150589
\(885\) 0 0
\(886\) 336.862 0.0127732
\(887\) 45832.2 1.73494 0.867472 0.497486i \(-0.165743\pi\)
0.867472 + 0.497486i \(0.165743\pi\)
\(888\) −2193.64 −0.0828985
\(889\) −1841.64 −0.0694789
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 37711.2 1.41554
\(893\) −32966.9 −1.23538
\(894\) −27076.6 −1.01295
\(895\) 0 0
\(896\) −20388.2 −0.760183
\(897\) 225.500 0.00839377
\(898\) 64419.2 2.39387
\(899\) 21322.4 0.791035
\(900\) 0 0
\(901\) 1785.26 0.0660106
\(902\) −1676.86 −0.0618996
\(903\) −5481.23 −0.201998
\(904\) −10474.8 −0.385384
\(905\) 0 0
\(906\) −36737.6 −1.34716
\(907\) 42596.1 1.55940 0.779702 0.626151i \(-0.215371\pi\)
0.779702 + 0.626151i \(0.215371\pi\)
\(908\) 10843.3 0.396309
\(909\) 17083.5 0.623348
\(910\) 0 0
\(911\) 45531.0 1.65588 0.827941 0.560816i \(-0.189512\pi\)
0.827941 + 0.560816i \(0.189512\pi\)
\(912\) −15752.3 −0.571942
\(913\) −5928.60 −0.214905
\(914\) −6701.80 −0.242534
\(915\) 0 0
\(916\) −35609.5 −1.28447
\(917\) −203.286 −0.00732073
\(918\) −882.738 −0.0317372
\(919\) 14353.6 0.515215 0.257607 0.966250i \(-0.417066\pi\)
0.257607 + 0.966250i \(0.417066\pi\)
\(920\) 0 0
\(921\) −14172.4 −0.507055
\(922\) 27154.3 0.969936
\(923\) 3744.36 0.133529
\(924\) −4804.09 −0.171042
\(925\) 0 0
\(926\) 35565.2 1.26214
\(927\) −11777.9 −0.417302
\(928\) −16714.9 −0.591263
\(929\) 38477.7 1.35890 0.679448 0.733724i \(-0.262219\pi\)
0.679448 + 0.733724i \(0.262219\pi\)
\(930\) 0 0
\(931\) −15018.8 −0.528701
\(932\) −33474.1 −1.17648
\(933\) −161.147 −0.00565456
\(934\) 27049.9 0.947645
\(935\) 0 0
\(936\) −460.780 −0.0160909
\(937\) −6917.67 −0.241185 −0.120592 0.992702i \(-0.538479\pi\)
−0.120592 + 0.992702i \(0.538479\pi\)
\(938\) −37194.2 −1.29471
\(939\) −13991.3 −0.486250
\(940\) 0 0
\(941\) 41270.2 1.42972 0.714862 0.699265i \(-0.246489\pi\)
0.714862 + 0.699265i \(0.246489\pi\)
\(942\) −28133.9 −0.973091
\(943\) 411.766 0.0142194
\(944\) −24879.7 −0.857803
\(945\) 0 0
\(946\) −3200.23 −0.109988
\(947\) −7099.57 −0.243617 −0.121808 0.992554i \(-0.538869\pi\)
−0.121808 + 0.992554i \(0.538869\pi\)
\(948\) −1010.25 −0.0346113
\(949\) 4529.48 0.154935
\(950\) 0 0
\(951\) 25994.9 0.886375
\(952\) −1421.49 −0.0483935
\(953\) 27983.8 0.951191 0.475595 0.879664i \(-0.342233\pi\)
0.475595 + 0.879664i \(0.342233\pi\)
\(954\) 6958.96 0.236168
\(955\) 0 0
\(956\) 3596.49 0.121672
\(957\) 2418.25 0.0816833
\(958\) 30950.2 1.04379
\(959\) 15719.2 0.529299
\(960\) 0 0
\(961\) 54872.3 1.84191
\(962\) 2939.12 0.0985043
\(963\) −1859.89 −0.0622370
\(964\) 25982.9 0.868103
\(965\) 0 0
\(966\) 2711.68 0.0903176
\(967\) 22702.4 0.754972 0.377486 0.926015i \(-0.376789\pi\)
0.377486 + 0.926015i \(0.376789\pi\)
\(968\) 837.707 0.0278150
\(969\) −1816.73 −0.0602287
\(970\) 0 0
\(971\) 11642.7 0.384791 0.192396 0.981317i \(-0.438374\pi\)
0.192396 + 0.981317i \(0.438374\pi\)
\(972\) −1496.93 −0.0493971
\(973\) 17621.5 0.580595
\(974\) −18128.2 −0.596372
\(975\) 0 0
\(976\) 70210.2 2.30264
\(977\) −6166.32 −0.201922 −0.100961 0.994890i \(-0.532192\pi\)
−0.100961 + 0.994890i \(0.532192\pi\)
\(978\) 9116.44 0.298069
\(979\) 8591.94 0.280490
\(980\) 0 0
\(981\) 5361.31 0.174489
\(982\) 18604.4 0.604573
\(983\) 18059.0 0.585953 0.292976 0.956120i \(-0.405354\pi\)
0.292976 + 0.956120i \(0.405354\pi\)
\(984\) −841.392 −0.0272587
\(985\) 0 0
\(986\) −2395.83 −0.0773821
\(987\) −33532.6 −1.08141
\(988\) 3175.22 0.102244
\(989\) 785.839 0.0252662
\(990\) 0 0
\(991\) −62330.7 −1.99798 −0.998991 0.0449065i \(-0.985701\pi\)
−0.998991 + 0.0449065i \(0.985701\pi\)
\(992\) −66368.7 −2.12420
\(993\) 10267.6 0.328130
\(994\) 45026.7 1.43678
\(995\) 0 0
\(996\) 9960.36 0.316874
\(997\) 13560.3 0.430752 0.215376 0.976531i \(-0.430902\pi\)
0.215376 + 0.976531i \(0.430902\pi\)
\(998\) 13610.7 0.431702
\(999\) −2851.69 −0.0903138
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 825.4.a.q.1.1 yes 3
3.2 odd 2 2475.4.a.v.1.3 3
5.2 odd 4 825.4.c.n.199.2 6
5.3 odd 4 825.4.c.n.199.5 6
5.4 even 2 825.4.a.o.1.3 3
15.14 odd 2 2475.4.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.o.1.3 3 5.4 even 2
825.4.a.q.1.1 yes 3 1.1 even 1 trivial
825.4.c.n.199.2 6 5.2 odd 4
825.4.c.n.199.5 6 5.3 odd 4
2475.4.a.v.1.3 3 3.2 odd 2
2475.4.a.y.1.1 3 15.14 odd 2