Properties

Label 245.4.a.n.1.4
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $5$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.84623\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.84623 q^{2} +8.96795 q^{3} +6.79345 q^{4} +5.00000 q^{5} +34.4927 q^{6} -4.64067 q^{8} +53.4241 q^{9} +O(q^{10})\) \(q+3.84623 q^{2} +8.96795 q^{3} +6.79345 q^{4} +5.00000 q^{5} +34.4927 q^{6} -4.64067 q^{8} +53.4241 q^{9} +19.2311 q^{10} +23.1120 q^{11} +60.9233 q^{12} -61.0473 q^{13} +44.8397 q^{15} -72.1966 q^{16} -0.688309 q^{17} +205.481 q^{18} -63.2247 q^{19} +33.9672 q^{20} +88.8938 q^{22} +124.502 q^{23} -41.6172 q^{24} +25.0000 q^{25} -234.802 q^{26} +236.970 q^{27} +104.167 q^{29} +172.464 q^{30} -280.022 q^{31} -240.559 q^{32} +207.267 q^{33} -2.64739 q^{34} +362.934 q^{36} -263.871 q^{37} -243.176 q^{38} -547.469 q^{39} -23.2033 q^{40} +243.366 q^{41} +172.541 q^{43} +157.010 q^{44} +267.120 q^{45} +478.861 q^{46} +107.381 q^{47} -647.456 q^{48} +96.1556 q^{50} -6.17272 q^{51} -414.722 q^{52} +44.8086 q^{53} +911.440 q^{54} +115.560 q^{55} -566.996 q^{57} +400.650 q^{58} -457.246 q^{59} +304.617 q^{60} -473.802 q^{61} -1077.03 q^{62} -347.672 q^{64} -305.237 q^{65} +797.195 q^{66} -229.454 q^{67} -4.67599 q^{68} +1116.52 q^{69} +407.688 q^{71} -247.923 q^{72} +348.475 q^{73} -1014.91 q^{74} +224.199 q^{75} -429.514 q^{76} -2105.69 q^{78} +840.135 q^{79} -360.983 q^{80} +682.683 q^{81} +936.040 q^{82} +885.652 q^{83} -3.44154 q^{85} +663.632 q^{86} +934.164 q^{87} -107.255 q^{88} +856.096 q^{89} +1027.41 q^{90} +845.795 q^{92} -2511.22 q^{93} +413.013 q^{94} -316.124 q^{95} -2157.32 q^{96} -189.436 q^{97} +1234.74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} + 8 q^{3} + 35 q^{4} + 25 q^{5} - 16 q^{6} + 33 q^{8} + 81 q^{9} + 5 q^{10} + 47 q^{11} + 98 q^{12} - q^{13} + 40 q^{15} + 171 q^{16} + 2 q^{17} - 51 q^{18} + 21 q^{19} + 175 q^{20} + 523 q^{22} + 201 q^{23} - 848 q^{24} + 125 q^{25} + 47 q^{26} + 518 q^{27} + 190 q^{29} - 80 q^{30} - 388 q^{31} - 95 q^{32} + 262 q^{33} - 130 q^{34} + 1229 q^{36} - 145 q^{37} - 835 q^{38} + 14 q^{39} + 165 q^{40} + 281 q^{41} + 568 q^{43} + 1091 q^{44} + 405 q^{45} + 337 q^{46} + 473 q^{47} - 70 q^{48} + 25 q^{50} + 732 q^{51} + 379 q^{52} + 351 q^{53} + 774 q^{54} + 235 q^{55} + 954 q^{57} + 1818 q^{58} - 708 q^{59} + 490 q^{60} - 1944 q^{61} - 448 q^{62} - 125 q^{64} - 5 q^{65} - 1482 q^{66} + 1118 q^{67} + 3118 q^{68} - 374 q^{69} + 864 q^{71} - 2219 q^{72} + 1652 q^{73} - 3285 q^{74} + 200 q^{75} - 691 q^{76} - 5574 q^{78} + 218 q^{79} + 855 q^{80} - 455 q^{81} - 1027 q^{82} + 1502 q^{83} + 10 q^{85} - 4264 q^{86} - 390 q^{87} + 2131 q^{88} - 2322 q^{89} - 255 q^{90} - 2957 q^{92} - 2288 q^{93} + 2677 q^{94} + 105 q^{95} - 4592 q^{96} - 598 q^{97} + 35 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.84623 1.35985 0.679923 0.733284i \(-0.262013\pi\)
0.679923 + 0.733284i \(0.262013\pi\)
\(3\) 8.96795 1.72588 0.862941 0.505304i \(-0.168620\pi\)
0.862941 + 0.505304i \(0.168620\pi\)
\(4\) 6.79345 0.849181
\(5\) 5.00000 0.447214
\(6\) 34.4927 2.34693
\(7\) 0 0
\(8\) −4.64067 −0.205090
\(9\) 53.4241 1.97867
\(10\) 19.2311 0.608142
\(11\) 23.1120 0.633501 0.316751 0.948509i \(-0.397408\pi\)
0.316751 + 0.948509i \(0.397408\pi\)
\(12\) 60.9233 1.46559
\(13\) −61.0473 −1.30242 −0.651211 0.758897i \(-0.725739\pi\)
−0.651211 + 0.758897i \(0.725739\pi\)
\(14\) 0 0
\(15\) 44.8397 0.771838
\(16\) −72.1966 −1.12807
\(17\) −0.688309 −0.00981996 −0.00490998 0.999988i \(-0.501563\pi\)
−0.00490998 + 0.999988i \(0.501563\pi\)
\(18\) 205.481 2.69069
\(19\) −63.2247 −0.763407 −0.381704 0.924285i \(-0.624663\pi\)
−0.381704 + 0.924285i \(0.624663\pi\)
\(20\) 33.9672 0.379765
\(21\) 0 0
\(22\) 88.8938 0.861464
\(23\) 124.502 1.12871 0.564356 0.825531i \(-0.309125\pi\)
0.564356 + 0.825531i \(0.309125\pi\)
\(24\) −41.6172 −0.353962
\(25\) 25.0000 0.200000
\(26\) −234.802 −1.77109
\(27\) 236.970 1.68907
\(28\) 0 0
\(29\) 104.167 0.667011 0.333506 0.942748i \(-0.391768\pi\)
0.333506 + 0.942748i \(0.391768\pi\)
\(30\) 172.464 1.04958
\(31\) −280.022 −1.62237 −0.811185 0.584790i \(-0.801177\pi\)
−0.811185 + 0.584790i \(0.801177\pi\)
\(32\) −240.559 −1.32891
\(33\) 207.267 1.09335
\(34\) −2.64739 −0.0133536
\(35\) 0 0
\(36\) 362.934 1.68025
\(37\) −263.871 −1.17244 −0.586219 0.810153i \(-0.699384\pi\)
−0.586219 + 0.810153i \(0.699384\pi\)
\(38\) −243.176 −1.03812
\(39\) −547.469 −2.24783
\(40\) −23.2033 −0.0917192
\(41\) 243.366 0.927009 0.463505 0.886095i \(-0.346592\pi\)
0.463505 + 0.886095i \(0.346592\pi\)
\(42\) 0 0
\(43\) 172.541 0.611913 0.305956 0.952046i \(-0.401024\pi\)
0.305956 + 0.952046i \(0.401024\pi\)
\(44\) 157.010 0.537957
\(45\) 267.120 0.884888
\(46\) 478.861 1.53487
\(47\) 107.381 0.333260 0.166630 0.986020i \(-0.446712\pi\)
0.166630 + 0.986020i \(0.446712\pi\)
\(48\) −647.456 −1.94692
\(49\) 0 0
\(50\) 96.1556 0.271969
\(51\) −6.17272 −0.0169481
\(52\) −414.722 −1.10599
\(53\) 44.8086 0.116131 0.0580654 0.998313i \(-0.481507\pi\)
0.0580654 + 0.998313i \(0.481507\pi\)
\(54\) 911.440 2.29687
\(55\) 115.560 0.283310
\(56\) 0 0
\(57\) −566.996 −1.31755
\(58\) 400.650 0.907033
\(59\) −457.246 −1.00895 −0.504477 0.863425i \(-0.668315\pi\)
−0.504477 + 0.863425i \(0.668315\pi\)
\(60\) 304.617 0.655430
\(61\) −473.802 −0.994493 −0.497247 0.867609i \(-0.665656\pi\)
−0.497247 + 0.867609i \(0.665656\pi\)
\(62\) −1077.03 −2.20617
\(63\) 0 0
\(64\) −347.672 −0.679047
\(65\) −305.237 −0.582461
\(66\) 797.195 1.48679
\(67\) −229.454 −0.418392 −0.209196 0.977874i \(-0.567085\pi\)
−0.209196 + 0.977874i \(0.567085\pi\)
\(68\) −4.67599 −0.00833893
\(69\) 1116.52 1.94802
\(70\) 0 0
\(71\) 407.688 0.681460 0.340730 0.940161i \(-0.389326\pi\)
0.340730 + 0.940161i \(0.389326\pi\)
\(72\) −247.923 −0.405806
\(73\) 348.475 0.558712 0.279356 0.960188i \(-0.409879\pi\)
0.279356 + 0.960188i \(0.409879\pi\)
\(74\) −1014.91 −1.59433
\(75\) 224.199 0.345176
\(76\) −429.514 −0.648271
\(77\) 0 0
\(78\) −2105.69 −3.05670
\(79\) 840.135 1.19649 0.598244 0.801314i \(-0.295865\pi\)
0.598244 + 0.801314i \(0.295865\pi\)
\(80\) −360.983 −0.504489
\(81\) 682.683 0.936465
\(82\) 936.040 1.26059
\(83\) 885.652 1.17124 0.585620 0.810586i \(-0.300851\pi\)
0.585620 + 0.810586i \(0.300851\pi\)
\(84\) 0 0
\(85\) −3.44154 −0.00439162
\(86\) 663.632 0.832107
\(87\) 934.164 1.15118
\(88\) −107.255 −0.129925
\(89\) 856.096 1.01962 0.509809 0.860288i \(-0.329716\pi\)
0.509809 + 0.860288i \(0.329716\pi\)
\(90\) 1027.41 1.20331
\(91\) 0 0
\(92\) 845.795 0.958481
\(93\) −2511.22 −2.80002
\(94\) 413.013 0.453182
\(95\) −316.124 −0.341406
\(96\) −2157.32 −2.29355
\(97\) −189.436 −0.198292 −0.0991459 0.995073i \(-0.531611\pi\)
−0.0991459 + 0.995073i \(0.531611\pi\)
\(98\) 0 0
\(99\) 1234.74 1.25349
\(100\) 169.836 0.169836
\(101\) −795.803 −0.784013 −0.392007 0.919962i \(-0.628219\pi\)
−0.392007 + 0.919962i \(0.628219\pi\)
\(102\) −23.7417 −0.0230468
\(103\) −1940.71 −1.85654 −0.928269 0.371909i \(-0.878703\pi\)
−0.928269 + 0.371909i \(0.878703\pi\)
\(104\) 283.300 0.267114
\(105\) 0 0
\(106\) 172.344 0.157920
\(107\) 488.688 0.441526 0.220763 0.975327i \(-0.429145\pi\)
0.220763 + 0.975327i \(0.429145\pi\)
\(108\) 1609.84 1.43433
\(109\) 1630.05 1.43239 0.716197 0.697899i \(-0.245881\pi\)
0.716197 + 0.697899i \(0.245881\pi\)
\(110\) 444.469 0.385259
\(111\) −2366.38 −2.02349
\(112\) 0 0
\(113\) 345.925 0.287981 0.143991 0.989579i \(-0.454007\pi\)
0.143991 + 0.989579i \(0.454007\pi\)
\(114\) −2180.79 −1.79167
\(115\) 622.508 0.504775
\(116\) 707.653 0.566414
\(117\) −3261.40 −2.57706
\(118\) −1758.67 −1.37202
\(119\) 0 0
\(120\) −208.086 −0.158297
\(121\) −796.838 −0.598676
\(122\) −1822.35 −1.35236
\(123\) 2182.49 1.59991
\(124\) −1902.32 −1.37769
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1665.24 −1.16351 −0.581757 0.813362i \(-0.697635\pi\)
−0.581757 + 0.813362i \(0.697635\pi\)
\(128\) 587.249 0.405516
\(129\) 1547.34 1.05609
\(130\) −1174.01 −0.792057
\(131\) −676.747 −0.451356 −0.225678 0.974202i \(-0.572460\pi\)
−0.225678 + 0.974202i \(0.572460\pi\)
\(132\) 1408.06 0.928451
\(133\) 0 0
\(134\) −882.531 −0.568948
\(135\) 1184.85 0.755375
\(136\) 3.19421 0.00201398
\(137\) −1034.23 −0.644967 −0.322484 0.946575i \(-0.604518\pi\)
−0.322484 + 0.946575i \(0.604518\pi\)
\(138\) 4294.40 2.64901
\(139\) −435.826 −0.265944 −0.132972 0.991120i \(-0.542452\pi\)
−0.132972 + 0.991120i \(0.542452\pi\)
\(140\) 0 0
\(141\) 962.992 0.575167
\(142\) 1568.06 0.926681
\(143\) −1410.92 −0.825086
\(144\) −3857.04 −2.23208
\(145\) 520.835 0.298297
\(146\) 1340.31 0.759762
\(147\) 0 0
\(148\) −1792.60 −0.995612
\(149\) 2119.95 1.16559 0.582795 0.812619i \(-0.301959\pi\)
0.582795 + 0.812619i \(0.301959\pi\)
\(150\) 862.319 0.469387
\(151\) −2400.40 −1.29365 −0.646827 0.762637i \(-0.723904\pi\)
−0.646827 + 0.762637i \(0.723904\pi\)
\(152\) 293.405 0.156568
\(153\) −36.7723 −0.0194305
\(154\) 0 0
\(155\) −1400.11 −0.725546
\(156\) −3719.21 −1.90881
\(157\) 506.188 0.257313 0.128657 0.991689i \(-0.458933\pi\)
0.128657 + 0.991689i \(0.458933\pi\)
\(158\) 3231.35 1.62704
\(159\) 401.841 0.200428
\(160\) −1202.80 −0.594309
\(161\) 0 0
\(162\) 2625.75 1.27345
\(163\) 2600.95 1.24983 0.624913 0.780694i \(-0.285134\pi\)
0.624913 + 0.780694i \(0.285134\pi\)
\(164\) 1653.29 0.787199
\(165\) 1036.33 0.488961
\(166\) 3406.42 1.59271
\(167\) −546.077 −0.253034 −0.126517 0.991964i \(-0.540380\pi\)
−0.126517 + 0.991964i \(0.540380\pi\)
\(168\) 0 0
\(169\) 1529.78 0.696303
\(170\) −13.2370 −0.00597193
\(171\) −3377.72 −1.51053
\(172\) 1172.15 0.519625
\(173\) 3020.32 1.32735 0.663673 0.748023i \(-0.268997\pi\)
0.663673 + 0.748023i \(0.268997\pi\)
\(174\) 3593.01 1.56543
\(175\) 0 0
\(176\) −1668.61 −0.714636
\(177\) −4100.55 −1.74134
\(178\) 3292.74 1.38652
\(179\) 1999.17 0.834775 0.417387 0.908729i \(-0.362946\pi\)
0.417387 + 0.908729i \(0.362946\pi\)
\(180\) 1814.67 0.751430
\(181\) 681.145 0.279719 0.139859 0.990171i \(-0.455335\pi\)
0.139859 + 0.990171i \(0.455335\pi\)
\(182\) 0 0
\(183\) −4249.03 −1.71638
\(184\) −577.770 −0.231488
\(185\) −1319.36 −0.524330
\(186\) −9658.73 −3.80759
\(187\) −15.9082 −0.00622096
\(188\) 729.491 0.282998
\(189\) 0 0
\(190\) −1215.88 −0.464260
\(191\) −1612.76 −0.610970 −0.305485 0.952197i \(-0.598819\pi\)
−0.305485 + 0.952197i \(0.598819\pi\)
\(192\) −3117.90 −1.17195
\(193\) 808.921 0.301697 0.150848 0.988557i \(-0.451800\pi\)
0.150848 + 0.988557i \(0.451800\pi\)
\(194\) −728.613 −0.269646
\(195\) −2737.35 −1.00526
\(196\) 0 0
\(197\) 3704.23 1.33967 0.669836 0.742509i \(-0.266364\pi\)
0.669836 + 0.742509i \(0.266364\pi\)
\(198\) 4749.07 1.70455
\(199\) 367.928 0.131064 0.0655319 0.997850i \(-0.479126\pi\)
0.0655319 + 0.997850i \(0.479126\pi\)
\(200\) −116.017 −0.0410181
\(201\) −2057.73 −0.722095
\(202\) −3060.84 −1.06614
\(203\) 0 0
\(204\) −41.9340 −0.0143920
\(205\) 1216.83 0.414571
\(206\) −7464.39 −2.52461
\(207\) 6651.39 2.23335
\(208\) 4407.41 1.46923
\(209\) −1461.25 −0.483620
\(210\) 0 0
\(211\) −1073.36 −0.350205 −0.175102 0.984550i \(-0.556026\pi\)
−0.175102 + 0.984550i \(0.556026\pi\)
\(212\) 304.405 0.0986161
\(213\) 3656.13 1.17612
\(214\) 1879.61 0.600407
\(215\) 862.705 0.273656
\(216\) −1099.70 −0.346412
\(217\) 0 0
\(218\) 6269.55 1.94783
\(219\) 3125.11 0.964271
\(220\) 785.049 0.240582
\(221\) 42.0194 0.0127897
\(222\) −9101.65 −2.75163
\(223\) 1725.16 0.518050 0.259025 0.965871i \(-0.416599\pi\)
0.259025 + 0.965871i \(0.416599\pi\)
\(224\) 0 0
\(225\) 1335.60 0.395734
\(226\) 1330.50 0.391610
\(227\) 3616.99 1.05757 0.528784 0.848757i \(-0.322648\pi\)
0.528784 + 0.848757i \(0.322648\pi\)
\(228\) −3851.86 −1.11884
\(229\) −1461.79 −0.421823 −0.210912 0.977505i \(-0.567643\pi\)
−0.210912 + 0.977505i \(0.567643\pi\)
\(230\) 2394.31 0.686417
\(231\) 0 0
\(232\) −483.404 −0.136798
\(233\) 2135.32 0.600383 0.300192 0.953879i \(-0.402949\pi\)
0.300192 + 0.953879i \(0.402949\pi\)
\(234\) −12544.1 −3.50441
\(235\) 536.907 0.149038
\(236\) −3106.27 −0.856785
\(237\) 7534.29 2.06500
\(238\) 0 0
\(239\) 5952.35 1.61099 0.805493 0.592605i \(-0.201900\pi\)
0.805493 + 0.592605i \(0.201900\pi\)
\(240\) −3237.28 −0.870689
\(241\) 3847.56 1.02839 0.514197 0.857672i \(-0.328090\pi\)
0.514197 + 0.857672i \(0.328090\pi\)
\(242\) −3064.82 −0.814107
\(243\) −275.920 −0.0728407
\(244\) −3218.75 −0.844505
\(245\) 0 0
\(246\) 8394.36 2.17563
\(247\) 3859.70 0.994279
\(248\) 1299.49 0.332732
\(249\) 7942.48 2.02142
\(250\) 480.778 0.121628
\(251\) 1731.69 0.435470 0.217735 0.976008i \(-0.430133\pi\)
0.217735 + 0.976008i \(0.430133\pi\)
\(252\) 0 0
\(253\) 2877.47 0.715041
\(254\) −6404.90 −1.58220
\(255\) −30.8636 −0.00757942
\(256\) 5040.07 1.23049
\(257\) 4054.43 0.984078 0.492039 0.870573i \(-0.336252\pi\)
0.492039 + 0.870573i \(0.336252\pi\)
\(258\) 5951.41 1.43612
\(259\) 0 0
\(260\) −2073.61 −0.494615
\(261\) 5565.03 1.31980
\(262\) −2602.92 −0.613775
\(263\) −6335.82 −1.48549 −0.742744 0.669576i \(-0.766476\pi\)
−0.742744 + 0.669576i \(0.766476\pi\)
\(264\) −961.856 −0.224235
\(265\) 224.043 0.0519353
\(266\) 0 0
\(267\) 7677.42 1.75974
\(268\) −1558.78 −0.355290
\(269\) 1321.63 0.299559 0.149780 0.988719i \(-0.452144\pi\)
0.149780 + 0.988719i \(0.452144\pi\)
\(270\) 4557.20 1.02719
\(271\) 2336.00 0.523624 0.261812 0.965119i \(-0.415680\pi\)
0.261812 + 0.965119i \(0.415680\pi\)
\(272\) 49.6936 0.0110776
\(273\) 0 0
\(274\) −3977.90 −0.877056
\(275\) 577.799 0.126700
\(276\) 7585.05 1.65423
\(277\) −7085.23 −1.53686 −0.768430 0.639934i \(-0.778962\pi\)
−0.768430 + 0.639934i \(0.778962\pi\)
\(278\) −1676.28 −0.361643
\(279\) −14959.9 −3.21013
\(280\) 0 0
\(281\) 2123.17 0.450738 0.225369 0.974273i \(-0.427641\pi\)
0.225369 + 0.974273i \(0.427641\pi\)
\(282\) 3703.88 0.782138
\(283\) −5891.63 −1.23753 −0.618765 0.785576i \(-0.712367\pi\)
−0.618765 + 0.785576i \(0.712367\pi\)
\(284\) 2769.61 0.578683
\(285\) −2834.98 −0.589227
\(286\) −5426.73 −1.12199
\(287\) 0 0
\(288\) −12851.7 −2.62948
\(289\) −4912.53 −0.999904
\(290\) 2003.25 0.405637
\(291\) −1698.85 −0.342228
\(292\) 2367.35 0.474448
\(293\) 8137.75 1.62257 0.811284 0.584652i \(-0.198769\pi\)
0.811284 + 0.584652i \(0.198769\pi\)
\(294\) 0 0
\(295\) −2286.23 −0.451218
\(296\) 1224.54 0.240456
\(297\) 5476.84 1.07003
\(298\) 8153.80 1.58502
\(299\) −7600.49 −1.47006
\(300\) 1523.08 0.293117
\(301\) 0 0
\(302\) −9232.47 −1.75917
\(303\) −7136.72 −1.35312
\(304\) 4564.61 0.861179
\(305\) −2369.01 −0.444751
\(306\) −141.434 −0.0264224
\(307\) 4797.24 0.891833 0.445917 0.895075i \(-0.352878\pi\)
0.445917 + 0.895075i \(0.352878\pi\)
\(308\) 0 0
\(309\) −17404.1 −3.20417
\(310\) −5385.14 −0.986631
\(311\) −5189.53 −0.946210 −0.473105 0.881006i \(-0.656867\pi\)
−0.473105 + 0.881006i \(0.656867\pi\)
\(312\) 2540.62 0.461008
\(313\) 2245.24 0.405459 0.202729 0.979235i \(-0.435019\pi\)
0.202729 + 0.979235i \(0.435019\pi\)
\(314\) 1946.91 0.349907
\(315\) 0 0
\(316\) 5707.42 1.01604
\(317\) −9742.27 −1.72612 −0.863060 0.505101i \(-0.831455\pi\)
−0.863060 + 0.505101i \(0.831455\pi\)
\(318\) 1545.57 0.272551
\(319\) 2407.50 0.422553
\(320\) −1738.36 −0.303679
\(321\) 4382.53 0.762022
\(322\) 0 0
\(323\) 43.5181 0.00749663
\(324\) 4637.77 0.795229
\(325\) −1526.18 −0.260484
\(326\) 10003.8 1.69957
\(327\) 14618.2 2.47214
\(328\) −1129.38 −0.190121
\(329\) 0 0
\(330\) 3985.97 0.664911
\(331\) 11330.8 1.88156 0.940778 0.339023i \(-0.110096\pi\)
0.940778 + 0.339023i \(0.110096\pi\)
\(332\) 6016.63 0.994595
\(333\) −14097.1 −2.31987
\(334\) −2100.33 −0.344087
\(335\) −1147.27 −0.187110
\(336\) 0 0
\(337\) 3637.35 0.587950 0.293975 0.955813i \(-0.405022\pi\)
0.293975 + 0.955813i \(0.405022\pi\)
\(338\) 5883.87 0.946865
\(339\) 3102.23 0.497021
\(340\) −23.3800 −0.00372928
\(341\) −6471.86 −1.02777
\(342\) −12991.5 −2.05409
\(343\) 0 0
\(344\) −800.705 −0.125497
\(345\) 5582.62 0.871183
\(346\) 11616.8 1.80498
\(347\) −4030.12 −0.623482 −0.311741 0.950167i \(-0.600912\pi\)
−0.311741 + 0.950167i \(0.600912\pi\)
\(348\) 6346.20 0.977563
\(349\) −2887.69 −0.442906 −0.221453 0.975171i \(-0.571080\pi\)
−0.221453 + 0.975171i \(0.571080\pi\)
\(350\) 0 0
\(351\) −14466.4 −2.19988
\(352\) −5559.79 −0.841869
\(353\) 524.918 0.0791461 0.0395731 0.999217i \(-0.487400\pi\)
0.0395731 + 0.999217i \(0.487400\pi\)
\(354\) −15771.7 −2.36795
\(355\) 2038.44 0.304758
\(356\) 5815.84 0.865840
\(357\) 0 0
\(358\) 7689.25 1.13517
\(359\) −3285.31 −0.482987 −0.241493 0.970402i \(-0.577637\pi\)
−0.241493 + 0.970402i \(0.577637\pi\)
\(360\) −1239.62 −0.181482
\(361\) −2861.64 −0.417209
\(362\) 2619.84 0.380374
\(363\) −7146.00 −1.03324
\(364\) 0 0
\(365\) 1742.38 0.249864
\(366\) −16342.7 −2.33401
\(367\) −9053.95 −1.28777 −0.643886 0.765121i \(-0.722679\pi\)
−0.643886 + 0.765121i \(0.722679\pi\)
\(368\) −8988.60 −1.27327
\(369\) 13001.6 1.83424
\(370\) −5074.54 −0.713008
\(371\) 0 0
\(372\) −17059.9 −2.37772
\(373\) −5209.10 −0.723102 −0.361551 0.932352i \(-0.617753\pi\)
−0.361551 + 0.932352i \(0.617753\pi\)
\(374\) −61.1864 −0.00845955
\(375\) 1120.99 0.154368
\(376\) −498.322 −0.0683483
\(377\) −6359.12 −0.868730
\(378\) 0 0
\(379\) 3200.68 0.433794 0.216897 0.976194i \(-0.430406\pi\)
0.216897 + 0.976194i \(0.430406\pi\)
\(380\) −2147.57 −0.289916
\(381\) −14933.8 −2.00809
\(382\) −6203.04 −0.830825
\(383\) 2202.81 0.293886 0.146943 0.989145i \(-0.453057\pi\)
0.146943 + 0.989145i \(0.453057\pi\)
\(384\) 5266.42 0.699872
\(385\) 0 0
\(386\) 3111.29 0.410261
\(387\) 9217.85 1.21077
\(388\) −1286.92 −0.168386
\(389\) −6241.59 −0.813525 −0.406762 0.913534i \(-0.633342\pi\)
−0.406762 + 0.913534i \(0.633342\pi\)
\(390\) −10528.5 −1.36700
\(391\) −85.6955 −0.0110839
\(392\) 0 0
\(393\) −6069.04 −0.778988
\(394\) 14247.3 1.82175
\(395\) 4200.68 0.535086
\(396\) 8388.11 1.06444
\(397\) −10407.2 −1.31567 −0.657834 0.753163i \(-0.728527\pi\)
−0.657834 + 0.753163i \(0.728527\pi\)
\(398\) 1415.13 0.178227
\(399\) 0 0
\(400\) −1804.92 −0.225615
\(401\) −14973.3 −1.86466 −0.932330 0.361608i \(-0.882228\pi\)
−0.932330 + 0.361608i \(0.882228\pi\)
\(402\) −7914.49 −0.981938
\(403\) 17094.6 2.11301
\(404\) −5406.25 −0.665769
\(405\) 3413.42 0.418800
\(406\) 0 0
\(407\) −6098.58 −0.742741
\(408\) 28.6455 0.00347589
\(409\) 7059.10 0.853422 0.426711 0.904388i \(-0.359672\pi\)
0.426711 + 0.904388i \(0.359672\pi\)
\(410\) 4680.20 0.563753
\(411\) −9274.95 −1.11314
\(412\) −13184.1 −1.57654
\(413\) 0 0
\(414\) 25582.7 3.03701
\(415\) 4428.26 0.523795
\(416\) 14685.5 1.73081
\(417\) −3908.46 −0.458989
\(418\) −5620.28 −0.657648
\(419\) −928.543 −0.108263 −0.0541316 0.998534i \(-0.517239\pi\)
−0.0541316 + 0.998534i \(0.517239\pi\)
\(420\) 0 0
\(421\) 7105.06 0.822517 0.411258 0.911519i \(-0.365089\pi\)
0.411258 + 0.911519i \(0.365089\pi\)
\(422\) −4128.39 −0.476225
\(423\) 5736.76 0.659411
\(424\) −207.942 −0.0238173
\(425\) −17.2077 −0.00196399
\(426\) 14062.3 1.59934
\(427\) 0 0
\(428\) 3319.88 0.374936
\(429\) −12653.1 −1.42400
\(430\) 3318.16 0.372130
\(431\) −4319.27 −0.482719 −0.241360 0.970436i \(-0.577593\pi\)
−0.241360 + 0.970436i \(0.577593\pi\)
\(432\) −17108.4 −1.90539
\(433\) −4600.60 −0.510602 −0.255301 0.966862i \(-0.582175\pi\)
−0.255301 + 0.966862i \(0.582175\pi\)
\(434\) 0 0
\(435\) 4670.82 0.514825
\(436\) 11073.7 1.21636
\(437\) −7871.58 −0.861667
\(438\) 12019.9 1.31126
\(439\) −1923.74 −0.209146 −0.104573 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(440\) −536.274 −0.0581042
\(441\) 0 0
\(442\) 161.616 0.0173921
\(443\) −4534.87 −0.486362 −0.243181 0.969981i \(-0.578191\pi\)
−0.243181 + 0.969981i \(0.578191\pi\)
\(444\) −16075.9 −1.71831
\(445\) 4280.48 0.455987
\(446\) 6635.35 0.704468
\(447\) 19011.6 2.01167
\(448\) 0 0
\(449\) −17431.9 −1.83221 −0.916105 0.400937i \(-0.868684\pi\)
−0.916105 + 0.400937i \(0.868684\pi\)
\(450\) 5137.03 0.538137
\(451\) 5624.66 0.587262
\(452\) 2350.02 0.244548
\(453\) −21526.6 −2.23269
\(454\) 13911.7 1.43813
\(455\) 0 0
\(456\) 2631.24 0.270217
\(457\) −1962.32 −0.200861 −0.100431 0.994944i \(-0.532022\pi\)
−0.100431 + 0.994944i \(0.532022\pi\)
\(458\) −5622.36 −0.573614
\(459\) −163.108 −0.0165866
\(460\) 4228.98 0.428646
\(461\) −14344.4 −1.44920 −0.724602 0.689167i \(-0.757976\pi\)
−0.724602 + 0.689167i \(0.757976\pi\)
\(462\) 0 0
\(463\) 11657.8 1.17016 0.585081 0.810975i \(-0.301063\pi\)
0.585081 + 0.810975i \(0.301063\pi\)
\(464\) −7520.51 −0.752437
\(465\) −12556.1 −1.25221
\(466\) 8212.91 0.816429
\(467\) 16513.8 1.63633 0.818164 0.574985i \(-0.194992\pi\)
0.818164 + 0.574985i \(0.194992\pi\)
\(468\) −22156.1 −2.18839
\(469\) 0 0
\(470\) 2065.07 0.202669
\(471\) 4539.47 0.444093
\(472\) 2121.92 0.206927
\(473\) 3987.76 0.387648
\(474\) 28978.6 2.80808
\(475\) −1580.62 −0.152681
\(476\) 0 0
\(477\) 2393.86 0.229785
\(478\) 22894.1 2.19069
\(479\) 15620.1 1.48998 0.744988 0.667077i \(-0.232455\pi\)
0.744988 + 0.667077i \(0.232455\pi\)
\(480\) −10786.6 −1.02571
\(481\) 16108.6 1.52701
\(482\) 14798.6 1.39846
\(483\) 0 0
\(484\) −5413.28 −0.508384
\(485\) −947.180 −0.0886788
\(486\) −1061.25 −0.0990521
\(487\) 1901.38 0.176920 0.0884598 0.996080i \(-0.471806\pi\)
0.0884598 + 0.996080i \(0.471806\pi\)
\(488\) 2198.76 0.203961
\(489\) 23325.1 2.15705
\(490\) 0 0
\(491\) −15964.1 −1.46731 −0.733654 0.679524i \(-0.762186\pi\)
−0.733654 + 0.679524i \(0.762186\pi\)
\(492\) 14826.7 1.35861
\(493\) −71.6991 −0.00655003
\(494\) 14845.3 1.35207
\(495\) 6173.68 0.560578
\(496\) 20216.7 1.83015
\(497\) 0 0
\(498\) 30548.6 2.74882
\(499\) −6849.18 −0.614451 −0.307226 0.951637i \(-0.599401\pi\)
−0.307226 + 0.951637i \(0.599401\pi\)
\(500\) 849.181 0.0759531
\(501\) −4897.19 −0.436707
\(502\) 6660.45 0.592173
\(503\) 766.190 0.0679179 0.0339590 0.999423i \(-0.489188\pi\)
0.0339590 + 0.999423i \(0.489188\pi\)
\(504\) 0 0
\(505\) −3979.02 −0.350621
\(506\) 11067.4 0.972345
\(507\) 13719.0 1.20174
\(508\) −11312.7 −0.988035
\(509\) −13604.7 −1.18471 −0.592356 0.805676i \(-0.701802\pi\)
−0.592356 + 0.805676i \(0.701802\pi\)
\(510\) −118.708 −0.0103068
\(511\) 0 0
\(512\) 14687.2 1.26776
\(513\) −14982.4 −1.28945
\(514\) 15594.2 1.33820
\(515\) −9703.53 −0.830269
\(516\) 10511.8 0.896812
\(517\) 2481.80 0.211120
\(518\) 0 0
\(519\) 27086.1 2.29084
\(520\) 1416.50 0.119457
\(521\) −18428.3 −1.54963 −0.774817 0.632185i \(-0.782158\pi\)
−0.774817 + 0.632185i \(0.782158\pi\)
\(522\) 21404.4 1.79472
\(523\) −4340.13 −0.362869 −0.181435 0.983403i \(-0.558074\pi\)
−0.181435 + 0.983403i \(0.558074\pi\)
\(524\) −4597.45 −0.383283
\(525\) 0 0
\(526\) −24369.0 −2.02003
\(527\) 192.742 0.0159316
\(528\) −14964.0 −1.23338
\(529\) 3333.65 0.273991
\(530\) 861.720 0.0706240
\(531\) −24427.9 −1.99639
\(532\) 0 0
\(533\) −14856.8 −1.20736
\(534\) 29529.1 2.39298
\(535\) 2443.44 0.197456
\(536\) 1064.82 0.0858081
\(537\) 17928.4 1.44072
\(538\) 5083.30 0.407354
\(539\) 0 0
\(540\) 8049.21 0.641450
\(541\) 12616.2 1.00262 0.501308 0.865269i \(-0.332852\pi\)
0.501308 + 0.865269i \(0.332852\pi\)
\(542\) 8984.79 0.712047
\(543\) 6108.47 0.482762
\(544\) 165.579 0.0130499
\(545\) 8150.27 0.640586
\(546\) 0 0
\(547\) 20677.8 1.61630 0.808151 0.588975i \(-0.200468\pi\)
0.808151 + 0.588975i \(0.200468\pi\)
\(548\) −7026.01 −0.547694
\(549\) −25312.4 −1.96777
\(550\) 2222.34 0.172293
\(551\) −6585.93 −0.509201
\(552\) −5181.41 −0.399521
\(553\) 0 0
\(554\) −27251.4 −2.08989
\(555\) −11831.9 −0.904932
\(556\) −2960.76 −0.225835
\(557\) −1015.17 −0.0772248 −0.0386124 0.999254i \(-0.512294\pi\)
−0.0386124 + 0.999254i \(0.512294\pi\)
\(558\) −57539.3 −4.36529
\(559\) −10533.2 −0.796969
\(560\) 0 0
\(561\) −142.664 −0.0107366
\(562\) 8166.17 0.612935
\(563\) −5296.64 −0.396495 −0.198248 0.980152i \(-0.563525\pi\)
−0.198248 + 0.980152i \(0.563525\pi\)
\(564\) 6542.04 0.488421
\(565\) 1729.62 0.128789
\(566\) −22660.5 −1.68285
\(567\) 0 0
\(568\) −1891.94 −0.139761
\(569\) 14741.2 1.08609 0.543044 0.839704i \(-0.317272\pi\)
0.543044 + 0.839704i \(0.317272\pi\)
\(570\) −10904.0 −0.801258
\(571\) 13661.4 1.00125 0.500624 0.865665i \(-0.333104\pi\)
0.500624 + 0.865665i \(0.333104\pi\)
\(572\) −9585.04 −0.700648
\(573\) −14463.2 −1.05446
\(574\) 0 0
\(575\) 3112.54 0.225742
\(576\) −18574.1 −1.34361
\(577\) −1078.61 −0.0778220 −0.0389110 0.999243i \(-0.512389\pi\)
−0.0389110 + 0.999243i \(0.512389\pi\)
\(578\) −18894.7 −1.35971
\(579\) 7254.37 0.520693
\(580\) 3538.27 0.253308
\(581\) 0 0
\(582\) −6534.17 −0.465378
\(583\) 1035.61 0.0735690
\(584\) −1617.16 −0.114586
\(585\) −16307.0 −1.15250
\(586\) 31299.6 2.20644
\(587\) −3863.36 −0.271649 −0.135824 0.990733i \(-0.543368\pi\)
−0.135824 + 0.990733i \(0.543368\pi\)
\(588\) 0 0
\(589\) 17704.3 1.23853
\(590\) −8793.35 −0.613587
\(591\) 33219.3 2.31212
\(592\) 19050.6 1.32259
\(593\) 9440.81 0.653774 0.326887 0.945063i \(-0.394000\pi\)
0.326887 + 0.945063i \(0.394000\pi\)
\(594\) 21065.1 1.45507
\(595\) 0 0
\(596\) 14401.8 0.989797
\(597\) 3299.56 0.226201
\(598\) −29233.2 −1.99905
\(599\) −21368.9 −1.45761 −0.728807 0.684719i \(-0.759925\pi\)
−0.728807 + 0.684719i \(0.759925\pi\)
\(600\) −1040.43 −0.0707924
\(601\) −5801.37 −0.393748 −0.196874 0.980429i \(-0.563079\pi\)
−0.196874 + 0.980429i \(0.563079\pi\)
\(602\) 0 0
\(603\) −12258.4 −0.827859
\(604\) −16307.0 −1.09855
\(605\) −3984.19 −0.267736
\(606\) −27449.4 −1.84003
\(607\) −26740.0 −1.78805 −0.894023 0.448021i \(-0.852129\pi\)
−0.894023 + 0.448021i \(0.852129\pi\)
\(608\) 15209.3 1.01450
\(609\) 0 0
\(610\) −9111.74 −0.604793
\(611\) −6555.36 −0.434045
\(612\) −249.811 −0.0165000
\(613\) −8339.75 −0.549493 −0.274746 0.961517i \(-0.588594\pi\)
−0.274746 + 0.961517i \(0.588594\pi\)
\(614\) 18451.3 1.21276
\(615\) 10912.5 0.715501
\(616\) 0 0
\(617\) −26639.2 −1.73818 −0.869088 0.494658i \(-0.835293\pi\)
−0.869088 + 0.494658i \(0.835293\pi\)
\(618\) −66940.3 −4.35717
\(619\) −27860.3 −1.80905 −0.904523 0.426424i \(-0.859773\pi\)
−0.904523 + 0.426424i \(0.859773\pi\)
\(620\) −9511.58 −0.616120
\(621\) 29503.1 1.90647
\(622\) −19960.1 −1.28670
\(623\) 0 0
\(624\) 39525.5 2.53571
\(625\) 625.000 0.0400000
\(626\) 8635.71 0.551361
\(627\) −13104.4 −0.834671
\(628\) 3438.76 0.218506
\(629\) 181.625 0.0115133
\(630\) 0 0
\(631\) −10886.7 −0.686833 −0.343417 0.939183i \(-0.611584\pi\)
−0.343417 + 0.939183i \(0.611584\pi\)
\(632\) −3898.79 −0.245388
\(633\) −9625.85 −0.604413
\(634\) −37470.9 −2.34726
\(635\) −8326.21 −0.520340
\(636\) 2729.89 0.170200
\(637\) 0 0
\(638\) 9259.80 0.574607
\(639\) 21780.4 1.34839
\(640\) 2936.25 0.181352
\(641\) −16851.5 −1.03837 −0.519183 0.854663i \(-0.673764\pi\)
−0.519183 + 0.854663i \(0.673764\pi\)
\(642\) 16856.2 1.03623
\(643\) −4793.72 −0.294006 −0.147003 0.989136i \(-0.546963\pi\)
−0.147003 + 0.989136i \(0.546963\pi\)
\(644\) 0 0
\(645\) 7736.69 0.472298
\(646\) 167.380 0.0101943
\(647\) 26684.5 1.62145 0.810725 0.585428i \(-0.199073\pi\)
0.810725 + 0.585428i \(0.199073\pi\)
\(648\) −3168.10 −0.192060
\(649\) −10567.8 −0.639174
\(650\) −5870.05 −0.354219
\(651\) 0 0
\(652\) 17669.4 1.06133
\(653\) 17541.9 1.05125 0.525625 0.850716i \(-0.323831\pi\)
0.525625 + 0.850716i \(0.323831\pi\)
\(654\) 56225.0 3.36173
\(655\) −3383.74 −0.201853
\(656\) −17570.2 −1.04573
\(657\) 18617.0 1.10551
\(658\) 0 0
\(659\) −26285.3 −1.55377 −0.776883 0.629645i \(-0.783200\pi\)
−0.776883 + 0.629645i \(0.783200\pi\)
\(660\) 7040.28 0.415216
\(661\) −25044.3 −1.47369 −0.736845 0.676062i \(-0.763685\pi\)
−0.736845 + 0.676062i \(0.763685\pi\)
\(662\) 43580.7 2.55863
\(663\) 376.828 0.0220736
\(664\) −4110.02 −0.240210
\(665\) 0 0
\(666\) −54220.6 −3.15466
\(667\) 12969.0 0.752864
\(668\) −3709.75 −0.214872
\(669\) 15471.1 0.894094
\(670\) −4412.66 −0.254441
\(671\) −10950.5 −0.630013
\(672\) 0 0
\(673\) 20799.4 1.19132 0.595658 0.803238i \(-0.296891\pi\)
0.595658 + 0.803238i \(0.296891\pi\)
\(674\) 13990.1 0.799521
\(675\) 5924.25 0.337814
\(676\) 10392.5 0.591288
\(677\) −29772.5 −1.69018 −0.845089 0.534625i \(-0.820453\pi\)
−0.845089 + 0.534625i \(0.820453\pi\)
\(678\) 11931.9 0.675873
\(679\) 0 0
\(680\) 15.9711 0.000900679 0
\(681\) 32437.0 1.82524
\(682\) −24892.2 −1.39761
\(683\) 23890.3 1.33842 0.669208 0.743075i \(-0.266633\pi\)
0.669208 + 0.743075i \(0.266633\pi\)
\(684\) −22946.4 −1.28271
\(685\) −5171.17 −0.288438
\(686\) 0 0
\(687\) −13109.2 −0.728017
\(688\) −12456.9 −0.690282
\(689\) −2735.45 −0.151251
\(690\) 21472.0 1.18467
\(691\) 25297.6 1.39271 0.696356 0.717696i \(-0.254803\pi\)
0.696356 + 0.717696i \(0.254803\pi\)
\(692\) 20518.4 1.12716
\(693\) 0 0
\(694\) −15500.8 −0.847839
\(695\) −2179.13 −0.118934
\(696\) −4335.14 −0.236097
\(697\) −167.511 −0.00910320
\(698\) −11106.7 −0.602284
\(699\) 19149.4 1.03619
\(700\) 0 0
\(701\) 18659.1 1.00534 0.502670 0.864478i \(-0.332351\pi\)
0.502670 + 0.864478i \(0.332351\pi\)
\(702\) −55641.0 −2.99150
\(703\) 16683.2 0.895047
\(704\) −8035.38 −0.430177
\(705\) 4814.96 0.257222
\(706\) 2018.95 0.107627
\(707\) 0 0
\(708\) −27856.9 −1.47871
\(709\) −25694.8 −1.36106 −0.680528 0.732722i \(-0.738249\pi\)
−0.680528 + 0.732722i \(0.738249\pi\)
\(710\) 7840.30 0.414424
\(711\) 44883.5 2.36746
\(712\) −3972.85 −0.209114
\(713\) −34863.2 −1.83119
\(714\) 0 0
\(715\) −7054.62 −0.368990
\(716\) 13581.2 0.708875
\(717\) 53380.4 2.78037
\(718\) −12636.1 −0.656788
\(719\) 27254.7 1.41367 0.706834 0.707379i \(-0.250123\pi\)
0.706834 + 0.707379i \(0.250123\pi\)
\(720\) −19285.2 −0.998218
\(721\) 0 0
\(722\) −11006.5 −0.567340
\(723\) 34504.7 1.77489
\(724\) 4627.32 0.237532
\(725\) 2604.18 0.133402
\(726\) −27485.1 −1.40505
\(727\) 13194.3 0.673110 0.336555 0.941664i \(-0.390738\pi\)
0.336555 + 0.941664i \(0.390738\pi\)
\(728\) 0 0
\(729\) −20906.9 −1.06218
\(730\) 6701.57 0.339776
\(731\) −118.761 −0.00600896
\(732\) −28865.6 −1.45752
\(733\) −5808.95 −0.292713 −0.146356 0.989232i \(-0.546755\pi\)
−0.146356 + 0.989232i \(0.546755\pi\)
\(734\) −34823.5 −1.75117
\(735\) 0 0
\(736\) −29950.0 −1.49996
\(737\) −5303.13 −0.265052
\(738\) 50007.1 2.49429
\(739\) 13936.7 0.693734 0.346867 0.937914i \(-0.387246\pi\)
0.346867 + 0.937914i \(0.387246\pi\)
\(740\) −8962.98 −0.445251
\(741\) 34613.6 1.71601
\(742\) 0 0
\(743\) 32424.2 1.60098 0.800489 0.599347i \(-0.204573\pi\)
0.800489 + 0.599347i \(0.204573\pi\)
\(744\) 11653.7 0.574257
\(745\) 10599.7 0.521268
\(746\) −20035.4 −0.983308
\(747\) 47315.2 2.31750
\(748\) −108.071 −0.00528272
\(749\) 0 0
\(750\) 4311.59 0.209916
\(751\) −35288.3 −1.71463 −0.857316 0.514791i \(-0.827870\pi\)
−0.857316 + 0.514791i \(0.827870\pi\)
\(752\) −7752.58 −0.375941
\(753\) 15529.7 0.751571
\(754\) −24458.6 −1.18134
\(755\) −12002.0 −0.578539
\(756\) 0 0
\(757\) 2079.86 0.0998595 0.0499298 0.998753i \(-0.484100\pi\)
0.0499298 + 0.998753i \(0.484100\pi\)
\(758\) 12310.5 0.589893
\(759\) 25805.0 1.23408
\(760\) 1467.02 0.0700191
\(761\) −12251.4 −0.583593 −0.291797 0.956480i \(-0.594253\pi\)
−0.291797 + 0.956480i \(0.594253\pi\)
\(762\) −57438.8 −2.73069
\(763\) 0 0
\(764\) −10956.2 −0.518824
\(765\) −183.861 −0.00868957
\(766\) 8472.50 0.399639
\(767\) 27913.6 1.31408
\(768\) 45199.1 2.12367
\(769\) −4335.87 −0.203323 −0.101661 0.994819i \(-0.532416\pi\)
−0.101661 + 0.994819i \(0.532416\pi\)
\(770\) 0 0
\(771\) 36359.9 1.69840
\(772\) 5495.37 0.256195
\(773\) −30399.5 −1.41448 −0.707240 0.706973i \(-0.750060\pi\)
−0.707240 + 0.706973i \(0.750060\pi\)
\(774\) 35453.9 1.64647
\(775\) −7000.55 −0.324474
\(776\) 879.109 0.0406677
\(777\) 0 0
\(778\) −24006.6 −1.10627
\(779\) −15386.7 −0.707686
\(780\) −18596.0 −0.853647
\(781\) 9422.47 0.431706
\(782\) −329.604 −0.0150724
\(783\) 24684.4 1.12663
\(784\) 0 0
\(785\) 2530.94 0.115074
\(786\) −23342.9 −1.05930
\(787\) −35274.2 −1.59770 −0.798850 0.601530i \(-0.794558\pi\)
−0.798850 + 0.601530i \(0.794558\pi\)
\(788\) 25164.5 1.13762
\(789\) −56819.3 −2.56378
\(790\) 16156.8 0.727635
\(791\) 0 0
\(792\) −5729.99 −0.257079
\(793\) 28924.3 1.29525
\(794\) −40028.3 −1.78911
\(795\) 2009.21 0.0896342
\(796\) 2499.50 0.111297
\(797\) 27524.0 1.22328 0.611638 0.791138i \(-0.290511\pi\)
0.611638 + 0.791138i \(0.290511\pi\)
\(798\) 0 0
\(799\) −73.9116 −0.00327260
\(800\) −6013.98 −0.265783
\(801\) 45736.1 2.01749
\(802\) −57590.5 −2.53565
\(803\) 8053.95 0.353945
\(804\) −13979.1 −0.613189
\(805\) 0 0
\(806\) 65749.7 2.87337
\(807\) 11852.3 0.517004
\(808\) 3693.06 0.160794
\(809\) −7067.35 −0.307138 −0.153569 0.988138i \(-0.549077\pi\)
−0.153569 + 0.988138i \(0.549077\pi\)
\(810\) 13128.8 0.569503
\(811\) 37757.5 1.63483 0.817413 0.576052i \(-0.195407\pi\)
0.817413 + 0.576052i \(0.195407\pi\)
\(812\) 0 0
\(813\) 20949.1 0.903713
\(814\) −23456.5 −1.01001
\(815\) 13004.7 0.558940
\(816\) 445.649 0.0191187
\(817\) −10908.9 −0.467139
\(818\) 27150.9 1.16052
\(819\) 0 0
\(820\) 8266.47 0.352046
\(821\) −25307.4 −1.07580 −0.537902 0.843008i \(-0.680783\pi\)
−0.537902 + 0.843008i \(0.680783\pi\)
\(822\) −35673.6 −1.51370
\(823\) −25484.8 −1.07940 −0.539699 0.841858i \(-0.681462\pi\)
−0.539699 + 0.841858i \(0.681462\pi\)
\(824\) 9006.17 0.380758
\(825\) 5181.67 0.218670
\(826\) 0 0
\(827\) 34985.7 1.47107 0.735534 0.677488i \(-0.236931\pi\)
0.735534 + 0.677488i \(0.236931\pi\)
\(828\) 45185.8 1.89652
\(829\) −3898.09 −0.163313 −0.0816564 0.996661i \(-0.526021\pi\)
−0.0816564 + 0.996661i \(0.526021\pi\)
\(830\) 17032.1 0.712280
\(831\) −63540.0 −2.65244
\(832\) 21224.4 0.884405
\(833\) 0 0
\(834\) −15032.8 −0.624154
\(835\) −2730.38 −0.113160
\(836\) −9926.90 −0.410681
\(837\) −66356.8 −2.74030
\(838\) −3571.38 −0.147221
\(839\) −22118.8 −0.910161 −0.455081 0.890450i \(-0.650390\pi\)
−0.455081 + 0.890450i \(0.650390\pi\)
\(840\) 0 0
\(841\) −13538.2 −0.555096
\(842\) 27327.7 1.11850
\(843\) 19040.4 0.777921
\(844\) −7291.83 −0.297387
\(845\) 7648.89 0.311396
\(846\) 22064.9 0.896697
\(847\) 0 0
\(848\) −3235.03 −0.131004
\(849\) −52835.8 −2.13583
\(850\) −66.1848 −0.00267073
\(851\) −32852.4 −1.32334
\(852\) 24837.7 0.998739
\(853\) −33152.4 −1.33074 −0.665368 0.746516i \(-0.731725\pi\)
−0.665368 + 0.746516i \(0.731725\pi\)
\(854\) 0 0
\(855\) −16888.6 −0.675530
\(856\) −2267.84 −0.0905527
\(857\) 10346.2 0.412393 0.206197 0.978511i \(-0.433891\pi\)
0.206197 + 0.978511i \(0.433891\pi\)
\(858\) −48666.6 −1.93642
\(859\) 11905.6 0.472893 0.236447 0.971644i \(-0.424017\pi\)
0.236447 + 0.971644i \(0.424017\pi\)
\(860\) 5860.74 0.232383
\(861\) 0 0
\(862\) −16612.9 −0.656424
\(863\) 30802.3 1.21497 0.607487 0.794329i \(-0.292177\pi\)
0.607487 + 0.794329i \(0.292177\pi\)
\(864\) −57005.3 −2.24463
\(865\) 15101.6 0.593607
\(866\) −17694.9 −0.694340
\(867\) −44055.3 −1.72572
\(868\) 0 0
\(869\) 19417.2 0.757977
\(870\) 17965.0 0.700082
\(871\) 14007.5 0.544923
\(872\) −7564.53 −0.293770
\(873\) −10120.4 −0.392354
\(874\) −30275.9 −1.17173
\(875\) 0 0
\(876\) 21230.3 0.818841
\(877\) −21693.3 −0.835268 −0.417634 0.908615i \(-0.637141\pi\)
−0.417634 + 0.908615i \(0.637141\pi\)
\(878\) −7399.13 −0.284406
\(879\) 72978.9 2.80036
\(880\) −8343.03 −0.319595
\(881\) 29963.1 1.14584 0.572918 0.819612i \(-0.305811\pi\)
0.572918 + 0.819612i \(0.305811\pi\)
\(882\) 0 0
\(883\) −4090.61 −0.155900 −0.0779501 0.996957i \(-0.524837\pi\)
−0.0779501 + 0.996957i \(0.524837\pi\)
\(884\) 285.457 0.0108608
\(885\) −20502.8 −0.778749
\(886\) −17442.1 −0.661377
\(887\) 28819.4 1.09094 0.545468 0.838132i \(-0.316352\pi\)
0.545468 + 0.838132i \(0.316352\pi\)
\(888\) 10981.6 0.414998
\(889\) 0 0
\(890\) 16463.7 0.620072
\(891\) 15778.1 0.593252
\(892\) 11719.8 0.439918
\(893\) −6789.16 −0.254413
\(894\) 73122.8 2.73556
\(895\) 9995.83 0.373323
\(896\) 0 0
\(897\) −68160.8 −2.53715
\(898\) −67047.1 −2.49152
\(899\) −29169.1 −1.08214
\(900\) 9073.35 0.336050
\(901\) −30.8421 −0.00114040
\(902\) 21633.7 0.798585
\(903\) 0 0
\(904\) −1605.32 −0.0590621
\(905\) 3405.72 0.125094
\(906\) −82796.3 −3.03612
\(907\) −20638.8 −0.755567 −0.377784 0.925894i \(-0.623314\pi\)
−0.377784 + 0.925894i \(0.623314\pi\)
\(908\) 24571.8 0.898067
\(909\) −42515.1 −1.55130
\(910\) 0 0
\(911\) −15080.1 −0.548435 −0.274218 0.961668i \(-0.588419\pi\)
−0.274218 + 0.961668i \(0.588419\pi\)
\(912\) 40935.2 1.48629
\(913\) 20469.1 0.741982
\(914\) −7547.54 −0.273141
\(915\) −21245.1 −0.767588
\(916\) −9930.56 −0.358204
\(917\) 0 0
\(918\) −627.352 −0.0225552
\(919\) −2164.29 −0.0776858 −0.0388429 0.999245i \(-0.512367\pi\)
−0.0388429 + 0.999245i \(0.512367\pi\)
\(920\) −2888.85 −0.103525
\(921\) 43021.4 1.53920
\(922\) −55171.6 −1.97069
\(923\) −24888.3 −0.887549
\(924\) 0 0
\(925\) −6596.78 −0.234487
\(926\) 44838.6 1.59124
\(927\) −103680. −3.67348
\(928\) −25058.3 −0.886401
\(929\) −13639.1 −0.481684 −0.240842 0.970564i \(-0.577424\pi\)
−0.240842 + 0.970564i \(0.577424\pi\)
\(930\) −48293.7 −1.70281
\(931\) 0 0
\(932\) 14506.2 0.509834
\(933\) −46539.4 −1.63305
\(934\) 63515.6 2.22515
\(935\) −79.5408 −0.00278210
\(936\) 15135.1 0.528531
\(937\) −6568.49 −0.229011 −0.114505 0.993423i \(-0.536528\pi\)
−0.114505 + 0.993423i \(0.536528\pi\)
\(938\) 0 0
\(939\) 20135.2 0.699774
\(940\) 3647.45 0.126560
\(941\) −9454.51 −0.327533 −0.163766 0.986499i \(-0.552364\pi\)
−0.163766 + 0.986499i \(0.552364\pi\)
\(942\) 17459.8 0.603898
\(943\) 30299.4 1.04633
\(944\) 33011.6 1.13817
\(945\) 0 0
\(946\) 15337.8 0.527141
\(947\) 34076.7 1.16932 0.584660 0.811279i \(-0.301228\pi\)
0.584660 + 0.811279i \(0.301228\pi\)
\(948\) 51183.8 1.75356
\(949\) −21273.5 −0.727679
\(950\) −6079.41 −0.207623
\(951\) −87368.1 −2.97908
\(952\) 0 0
\(953\) −32862.3 −1.11702 −0.558508 0.829499i \(-0.688626\pi\)
−0.558508 + 0.829499i \(0.688626\pi\)
\(954\) 9207.32 0.312472
\(955\) −8063.81 −0.273234
\(956\) 40437.0 1.36802
\(957\) 21590.4 0.729276
\(958\) 60078.3 2.02614
\(959\) 0 0
\(960\) −15589.5 −0.524114
\(961\) 48621.4 1.63208
\(962\) 61957.5 2.07650
\(963\) 26107.7 0.873634
\(964\) 26138.2 0.873292
\(965\) 4044.61 0.134923
\(966\) 0 0
\(967\) 57597.8 1.91543 0.957715 0.287720i \(-0.0928972\pi\)
0.957715 + 0.287720i \(0.0928972\pi\)
\(968\) 3697.86 0.122783
\(969\) 390.268 0.0129383
\(970\) −3643.07 −0.120590
\(971\) 28185.7 0.931536 0.465768 0.884907i \(-0.345778\pi\)
0.465768 + 0.884907i \(0.345778\pi\)
\(972\) −1874.45 −0.0618549
\(973\) 0 0
\(974\) 7313.15 0.240583
\(975\) −13686.7 −0.449565
\(976\) 34206.9 1.12186
\(977\) 44263.6 1.44945 0.724727 0.689036i \(-0.241966\pi\)
0.724727 + 0.689036i \(0.241966\pi\)
\(978\) 89713.8 2.93326
\(979\) 19786.0 0.645929
\(980\) 0 0
\(981\) 87084.1 2.83423
\(982\) −61401.4 −1.99531
\(983\) −147.400 −0.00478262 −0.00239131 0.999997i \(-0.500761\pi\)
−0.00239131 + 0.999997i \(0.500761\pi\)
\(984\) −10128.2 −0.328126
\(985\) 18521.1 0.599119
\(986\) −275.771 −0.00890703
\(987\) 0 0
\(988\) 26220.7 0.844323
\(989\) 21481.6 0.690674
\(990\) 23745.3 0.762300
\(991\) 12486.8 0.400260 0.200130 0.979769i \(-0.435864\pi\)
0.200130 + 0.979769i \(0.435864\pi\)
\(992\) 67361.9 2.15599
\(993\) 101614. 3.24734
\(994\) 0 0
\(995\) 1839.64 0.0586135
\(996\) 53956.9 1.71655
\(997\) 6621.15 0.210325 0.105162 0.994455i \(-0.466464\pi\)
0.105162 + 0.994455i \(0.466464\pi\)
\(998\) −26343.5 −0.835559
\(999\) −62529.6 −1.98033
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 245.4.a.n.1.4 5
3.2 odd 2 2205.4.a.bt.1.2 5
5.4 even 2 1225.4.a.bf.1.2 5
7.2 even 3 245.4.e.o.116.2 10
7.3 odd 6 35.4.e.c.16.2 yes 10
7.4 even 3 245.4.e.o.226.2 10
7.5 odd 6 35.4.e.c.11.2 10
7.6 odd 2 245.4.a.m.1.4 5
21.5 even 6 315.4.j.g.46.4 10
21.17 even 6 315.4.j.g.226.4 10
21.20 even 2 2205.4.a.bu.1.2 5
28.3 even 6 560.4.q.n.401.1 10
28.19 even 6 560.4.q.n.81.1 10
35.3 even 12 175.4.k.d.149.8 20
35.12 even 12 175.4.k.d.74.8 20
35.17 even 12 175.4.k.d.149.3 20
35.19 odd 6 175.4.e.d.151.4 10
35.24 odd 6 175.4.e.d.51.4 10
35.33 even 12 175.4.k.d.74.3 20
35.34 odd 2 1225.4.a.bg.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.e.c.11.2 10 7.5 odd 6
35.4.e.c.16.2 yes 10 7.3 odd 6
175.4.e.d.51.4 10 35.24 odd 6
175.4.e.d.151.4 10 35.19 odd 6
175.4.k.d.74.3 20 35.33 even 12
175.4.k.d.74.8 20 35.12 even 12
175.4.k.d.149.3 20 35.17 even 12
175.4.k.d.149.8 20 35.3 even 12
245.4.a.m.1.4 5 7.6 odd 2
245.4.a.n.1.4 5 1.1 even 1 trivial
245.4.e.o.116.2 10 7.2 even 3
245.4.e.o.226.2 10 7.4 even 3
315.4.j.g.46.4 10 21.5 even 6
315.4.j.g.226.4 10 21.17 even 6
560.4.q.n.81.1 10 28.19 even 6
560.4.q.n.401.1 10 28.3 even 6
1225.4.a.bf.1.2 5 5.4 even 2
1225.4.a.bg.1.2 5 35.34 odd 2
2205.4.a.bt.1.2 5 3.2 odd 2
2205.4.a.bu.1.2 5 21.20 even 2