Properties

Label 245.6.a.d.1.2
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
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] = [245,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.577880.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 98x - 232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.53323\) 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)
 
\(f(q)\) \(=\) \(q-4.53323 q^{2} +15.7249 q^{3} -11.4498 q^{4} +25.0000 q^{5} -71.2847 q^{6} +196.968 q^{8} +4.27317 q^{9} +O(q^{10})\) \(q-4.53323 q^{2} +15.7249 q^{3} -11.4498 q^{4} +25.0000 q^{5} -71.2847 q^{6} +196.968 q^{8} +4.27317 q^{9} -113.331 q^{10} +126.340 q^{11} -180.048 q^{12} -833.688 q^{13} +393.123 q^{15} -526.506 q^{16} -157.465 q^{17} -19.3713 q^{18} +1005.50 q^{19} -286.246 q^{20} -572.729 q^{22} +73.5368 q^{23} +3097.31 q^{24} +625.000 q^{25} +3779.30 q^{26} -3753.96 q^{27} -3691.42 q^{29} -1782.12 q^{30} -4804.05 q^{31} -3916.21 q^{32} +1986.69 q^{33} +713.825 q^{34} -48.9272 q^{36} +14269.4 q^{37} -4558.15 q^{38} -13109.7 q^{39} +4924.20 q^{40} -18405.0 q^{41} +16338.0 q^{43} -1446.58 q^{44} +106.829 q^{45} -333.359 q^{46} +3364.00 q^{47} -8279.27 q^{48} -2833.27 q^{50} -2476.12 q^{51} +9545.60 q^{52} -16706.5 q^{53} +17017.6 q^{54} +3158.51 q^{55} +15811.4 q^{57} +16734.0 q^{58} +17999.6 q^{59} -4501.20 q^{60} -28042.0 q^{61} +21777.8 q^{62} +34601.2 q^{64} -20842.2 q^{65} -9006.12 q^{66} -47357.0 q^{67} +1802.95 q^{68} +1156.36 q^{69} -49174.1 q^{71} +841.679 q^{72} -80842.8 q^{73} -64686.5 q^{74} +9828.08 q^{75} -11512.8 q^{76} +59429.2 q^{78} -27299.5 q^{79} -13162.7 q^{80} -60069.1 q^{81} +83434.1 q^{82} +195.977 q^{83} -3936.62 q^{85} -74063.9 q^{86} -58047.3 q^{87} +24885.0 q^{88} -3602.48 q^{89} -484.282 q^{90} -841.985 q^{92} -75543.3 q^{93} -15249.8 q^{94} +25137.4 q^{95} -61582.0 q^{96} -158739. q^{97} +539.874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 26 q^{3} + 112 q^{4} + 75 q^{5} + 96 q^{6} - 120 q^{8} + 489 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} - 26 q^{3} + 112 q^{4} + 75 q^{5} + 96 q^{6} - 120 q^{8} + 489 q^{9} - 150 q^{10} - 194 q^{11} - 2956 q^{12} - 1892 q^{13} - 650 q^{15} + 1496 q^{16} + 184 q^{17} - 6078 q^{18} - 1212 q^{19} + 2800 q^{20} - 872 q^{22} + 3188 q^{23} + 13920 q^{24} + 1875 q^{25} + 9836 q^{26} - 7910 q^{27} - 11332 q^{29} + 2400 q^{30} - 9200 q^{31} - 20256 q^{32} + 8278 q^{33} + 18492 q^{34} + 38236 q^{36} + 6042 q^{37} - 25520 q^{38} + 12842 q^{39} - 3000 q^{40} - 10442 q^{41} + 28112 q^{43} - 20436 q^{44} + 12225 q^{45} - 41856 q^{46} - 2330 q^{47} - 68808 q^{48} - 3750 q^{50} + 3622 q^{51} - 63524 q^{52} + 7190 q^{53} + 138960 q^{54} - 4850 q^{55} + 45508 q^{57} + 37260 q^{58} - 23760 q^{59} - 73900 q^{60} - 8722 q^{61} - 24848 q^{62} + 4976 q^{64} - 47300 q^{65} - 24368 q^{66} - 97572 q^{67} - 3564 q^{68} - 92596 q^{69} - 52816 q^{71} - 251640 q^{72} - 34870 q^{73} - 241892 q^{74} - 16250 q^{75} - 115256 q^{76} - 142096 q^{78} - 71546 q^{79} + 37400 q^{80} - 3957 q^{81} - 241908 q^{82} + 20920 q^{83} + 4600 q^{85} + 266024 q^{86} + 112298 q^{87} + 59200 q^{88} + 192622 q^{89} - 151950 q^{90} + 248952 q^{92} - 20392 q^{93} - 360128 q^{94} - 30300 q^{95} + 296256 q^{96} - 116320 q^{97} - 60752 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.53323 −0.801369 −0.400685 0.916216i \(-0.631228\pi\)
−0.400685 + 0.916216i \(0.631228\pi\)
\(3\) 15.7249 1.00875 0.504377 0.863483i \(-0.331722\pi\)
0.504377 + 0.863483i \(0.331722\pi\)
\(4\) −11.4498 −0.357808
\(5\) 25.0000 0.447214
\(6\) −71.2847 −0.808384
\(7\) 0 0
\(8\) 196.968 1.08811
\(9\) 4.27317 0.0175851
\(10\) −113.331 −0.358383
\(11\) 126.340 0.314818 0.157409 0.987533i \(-0.449686\pi\)
0.157409 + 0.987533i \(0.449686\pi\)
\(12\) −180.048 −0.360940
\(13\) −833.688 −1.36819 −0.684093 0.729395i \(-0.739802\pi\)
−0.684093 + 0.729395i \(0.739802\pi\)
\(14\) 0 0
\(15\) 393.123 0.451129
\(16\) −526.506 −0.514166
\(17\) −157.465 −0.132148 −0.0660742 0.997815i \(-0.521047\pi\)
−0.0660742 + 0.997815i \(0.521047\pi\)
\(18\) −19.3713 −0.0140921
\(19\) 1005.50 0.638994 0.319497 0.947587i \(-0.396486\pi\)
0.319497 + 0.947587i \(0.396486\pi\)
\(20\) −286.246 −0.160016
\(21\) 0 0
\(22\) −572.729 −0.252286
\(23\) 73.5368 0.0289858 0.0144929 0.999895i \(-0.495387\pi\)
0.0144929 + 0.999895i \(0.495387\pi\)
\(24\) 3097.31 1.09763
\(25\) 625.000 0.200000
\(26\) 3779.30 1.09642
\(27\) −3753.96 −0.991015
\(28\) 0 0
\(29\) −3691.42 −0.815076 −0.407538 0.913188i \(-0.633613\pi\)
−0.407538 + 0.913188i \(0.633613\pi\)
\(30\) −1782.12 −0.361521
\(31\) −4804.05 −0.897848 −0.448924 0.893570i \(-0.648193\pi\)
−0.448924 + 0.893570i \(0.648193\pi\)
\(32\) −3916.21 −0.676068
\(33\) 1986.69 0.317574
\(34\) 713.825 0.105900
\(35\) 0 0
\(36\) −48.9272 −0.00629207
\(37\) 14269.4 1.71357 0.856784 0.515675i \(-0.172459\pi\)
0.856784 + 0.515675i \(0.172459\pi\)
\(38\) −4558.15 −0.512070
\(39\) −13109.7 −1.38016
\(40\) 4924.20 0.486615
\(41\) −18405.0 −1.70992 −0.854962 0.518691i \(-0.826419\pi\)
−0.854962 + 0.518691i \(0.826419\pi\)
\(42\) 0 0
\(43\) 16338.0 1.34750 0.673749 0.738960i \(-0.264683\pi\)
0.673749 + 0.738960i \(0.264683\pi\)
\(44\) −1446.58 −0.112644
\(45\) 106.829 0.00786428
\(46\) −333.359 −0.0232283
\(47\) 3364.00 0.222132 0.111066 0.993813i \(-0.464574\pi\)
0.111066 + 0.993813i \(0.464574\pi\)
\(48\) −8279.27 −0.518667
\(49\) 0 0
\(50\) −2833.27 −0.160274
\(51\) −2476.12 −0.133305
\(52\) 9545.60 0.489547
\(53\) −16706.5 −0.816949 −0.408475 0.912770i \(-0.633939\pi\)
−0.408475 + 0.912770i \(0.633939\pi\)
\(54\) 17017.6 0.794169
\(55\) 3158.51 0.140791
\(56\) 0 0
\(57\) 15811.4 0.644588
\(58\) 16734.0 0.653177
\(59\) 17999.6 0.673184 0.336592 0.941651i \(-0.390726\pi\)
0.336592 + 0.941651i \(0.390726\pi\)
\(60\) −4501.20 −0.161417
\(61\) −28042.0 −0.964903 −0.482452 0.875923i \(-0.660254\pi\)
−0.482452 + 0.875923i \(0.660254\pi\)
\(62\) 21777.8 0.719508
\(63\) 0 0
\(64\) 34601.2 1.05595
\(65\) −20842.2 −0.611871
\(66\) −9006.12 −0.254494
\(67\) −47357.0 −1.28883 −0.644417 0.764674i \(-0.722900\pi\)
−0.644417 + 0.764674i \(0.722900\pi\)
\(68\) 1802.95 0.0472837
\(69\) 1156.36 0.0292395
\(70\) 0 0
\(71\) −49174.1 −1.15768 −0.578842 0.815440i \(-0.696495\pi\)
−0.578842 + 0.815440i \(0.696495\pi\)
\(72\) 841.679 0.0191344
\(73\) −80842.8 −1.77555 −0.887777 0.460273i \(-0.847751\pi\)
−0.887777 + 0.460273i \(0.847751\pi\)
\(74\) −64686.5 −1.37320
\(75\) 9828.08 0.201751
\(76\) −11512.8 −0.228637
\(77\) 0 0
\(78\) 59429.2 1.10602
\(79\) −27299.5 −0.492138 −0.246069 0.969252i \(-0.579139\pi\)
−0.246069 + 0.969252i \(0.579139\pi\)
\(80\) −13162.7 −0.229942
\(81\) −60069.1 −1.01728
\(82\) 83434.1 1.37028
\(83\) 195.977 0.00312256 0.00156128 0.999999i \(-0.499503\pi\)
0.00156128 + 0.999999i \(0.499503\pi\)
\(84\) 0 0
\(85\) −3936.62 −0.0590985
\(86\) −74063.9 −1.07984
\(87\) −58047.3 −0.822212
\(88\) 24885.0 0.342555
\(89\) −3602.48 −0.0482088 −0.0241044 0.999709i \(-0.507673\pi\)
−0.0241044 + 0.999709i \(0.507673\pi\)
\(90\) −484.282 −0.00630219
\(91\) 0 0
\(92\) −841.985 −0.0103713
\(93\) −75543.3 −0.905708
\(94\) −15249.8 −0.178010
\(95\) 25137.4 0.285767
\(96\) −61582.0 −0.681987
\(97\) −158739. −1.71299 −0.856494 0.516157i \(-0.827362\pi\)
−0.856494 + 0.516157i \(0.827362\pi\)
\(98\) 0 0
\(99\) 539.874 0.00553611
\(100\) −7156.15 −0.0715615
\(101\) −159763. −1.55838 −0.779190 0.626787i \(-0.784369\pi\)
−0.779190 + 0.626787i \(0.784369\pi\)
\(102\) 11224.8 0.106827
\(103\) 53989.3 0.501435 0.250717 0.968060i \(-0.419334\pi\)
0.250717 + 0.968060i \(0.419334\pi\)
\(104\) −164210. −1.48873
\(105\) 0 0
\(106\) 75734.2 0.654678
\(107\) 99584.5 0.840877 0.420438 0.907321i \(-0.361876\pi\)
0.420438 + 0.907321i \(0.361876\pi\)
\(108\) 42982.3 0.354593
\(109\) 120574. 0.972044 0.486022 0.873947i \(-0.338448\pi\)
0.486022 + 0.873947i \(0.338448\pi\)
\(110\) −14318.2 −0.112826
\(111\) 224385. 1.72857
\(112\) 0 0
\(113\) −259644. −1.91285 −0.956426 0.291975i \(-0.905688\pi\)
−0.956426 + 0.291975i \(0.905688\pi\)
\(114\) −71676.5 −0.516553
\(115\) 1838.42 0.0129628
\(116\) 42266.2 0.291641
\(117\) −3562.49 −0.0240597
\(118\) −81596.5 −0.539469
\(119\) 0 0
\(120\) 77432.7 0.490875
\(121\) −145089. −0.900889
\(122\) 127121. 0.773244
\(123\) −289417. −1.72489
\(124\) 55005.6 0.321257
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −118169. −0.650121 −0.325061 0.945693i \(-0.605385\pi\)
−0.325061 + 0.945693i \(0.605385\pi\)
\(128\) −31536.8 −0.170134
\(129\) 256914. 1.35929
\(130\) 94482.4 0.490335
\(131\) −34613.9 −0.176227 −0.0881136 0.996110i \(-0.528084\pi\)
−0.0881136 + 0.996110i \(0.528084\pi\)
\(132\) −22747.3 −0.113631
\(133\) 0 0
\(134\) 214680. 1.03283
\(135\) −93849.0 −0.443195
\(136\) −31015.6 −0.143791
\(137\) 22671.4 0.103199 0.0515997 0.998668i \(-0.483568\pi\)
0.0515997 + 0.998668i \(0.483568\pi\)
\(138\) −5242.05 −0.0234317
\(139\) −125034. −0.548899 −0.274449 0.961602i \(-0.588496\pi\)
−0.274449 + 0.961602i \(0.588496\pi\)
\(140\) 0 0
\(141\) 52898.6 0.224077
\(142\) 222917. 0.927732
\(143\) −105328. −0.430730
\(144\) −2249.85 −0.00904165
\(145\) −92285.5 −0.364513
\(146\) 366479. 1.42287
\(147\) 0 0
\(148\) −163382. −0.613128
\(149\) 390090. 1.43946 0.719729 0.694255i \(-0.244266\pi\)
0.719729 + 0.694255i \(0.244266\pi\)
\(150\) −44552.9 −0.161677
\(151\) −71978.1 −0.256896 −0.128448 0.991716i \(-0.541000\pi\)
−0.128448 + 0.991716i \(0.541000\pi\)
\(152\) 198051. 0.695293
\(153\) −672.875 −0.00232384
\(154\) 0 0
\(155\) −120101. −0.401530
\(156\) 150104. 0.493833
\(157\) 88329.4 0.285994 0.142997 0.989723i \(-0.454326\pi\)
0.142997 + 0.989723i \(0.454326\pi\)
\(158\) 123755. 0.394384
\(159\) −262708. −0.824101
\(160\) −97905.1 −0.302347
\(161\) 0 0
\(162\) 272307. 0.815213
\(163\) 438774. 1.29352 0.646758 0.762695i \(-0.276124\pi\)
0.646758 + 0.762695i \(0.276124\pi\)
\(164\) 210735. 0.611824
\(165\) 49667.3 0.142024
\(166\) −888.410 −0.00250232
\(167\) 391291. 1.08570 0.542848 0.839831i \(-0.317346\pi\)
0.542848 + 0.839831i \(0.317346\pi\)
\(168\) 0 0
\(169\) 323743. 0.871933
\(170\) 17845.6 0.0473597
\(171\) 4296.66 0.0112368
\(172\) −187068. −0.482145
\(173\) −387003. −0.983103 −0.491551 0.870849i \(-0.663570\pi\)
−0.491551 + 0.870849i \(0.663570\pi\)
\(174\) 263141. 0.658895
\(175\) 0 0
\(176\) −66518.9 −0.161869
\(177\) 283043. 0.679077
\(178\) 16330.9 0.0386330
\(179\) 838441. 1.95587 0.977935 0.208911i \(-0.0669917\pi\)
0.977935 + 0.208911i \(0.0669917\pi\)
\(180\) −1223.18 −0.00281390
\(181\) 355885. 0.807447 0.403723 0.914881i \(-0.367716\pi\)
0.403723 + 0.914881i \(0.367716\pi\)
\(182\) 0 0
\(183\) −440958. −0.973350
\(184\) 14484.4 0.0315396
\(185\) 356735. 0.766331
\(186\) 342455. 0.725807
\(187\) −19894.2 −0.0416027
\(188\) −38517.3 −0.0794805
\(189\) 0 0
\(190\) −113954. −0.229005
\(191\) −176395. −0.349866 −0.174933 0.984580i \(-0.555971\pi\)
−0.174933 + 0.984580i \(0.555971\pi\)
\(192\) 544102. 1.06519
\(193\) 697832. 1.34852 0.674261 0.738493i \(-0.264462\pi\)
0.674261 + 0.738493i \(0.264462\pi\)
\(194\) 719600. 1.37274
\(195\) −327742. −0.617228
\(196\) 0 0
\(197\) 478810. 0.879017 0.439509 0.898238i \(-0.355153\pi\)
0.439509 + 0.898238i \(0.355153\pi\)
\(198\) −2447.37 −0.00443646
\(199\) 904995. 1.61999 0.809997 0.586434i \(-0.199469\pi\)
0.809997 + 0.586434i \(0.199469\pi\)
\(200\) 123105. 0.217621
\(201\) −744685. −1.30012
\(202\) 724243. 1.24884
\(203\) 0 0
\(204\) 28351.2 0.0476976
\(205\) −460125. −0.764701
\(206\) −244746. −0.401834
\(207\) 314.236 0.000509717 0
\(208\) 438942. 0.703475
\(209\) 127035. 0.201167
\(210\) 0 0
\(211\) −843592. −1.30445 −0.652223 0.758027i \(-0.726164\pi\)
−0.652223 + 0.758027i \(0.726164\pi\)
\(212\) 191286. 0.292311
\(213\) −773258. −1.16782
\(214\) −451439. −0.673853
\(215\) 408450. 0.602619
\(216\) −739410. −1.07833
\(217\) 0 0
\(218\) −546587. −0.778966
\(219\) −1.27125e6 −1.79110
\(220\) −36164.4 −0.0503761
\(221\) 131277. 0.180803
\(222\) −1.01719e6 −1.38522
\(223\) −584986. −0.787741 −0.393871 0.919166i \(-0.628864\pi\)
−0.393871 + 0.919166i \(0.628864\pi\)
\(224\) 0 0
\(225\) 2670.73 0.00351701
\(226\) 1.17702e6 1.53290
\(227\) −849739. −1.09451 −0.547257 0.836965i \(-0.684328\pi\)
−0.547257 + 0.836965i \(0.684328\pi\)
\(228\) −181038. −0.230639
\(229\) −193287. −0.243564 −0.121782 0.992557i \(-0.538861\pi\)
−0.121782 + 0.992557i \(0.538861\pi\)
\(230\) −8333.98 −0.0103880
\(231\) 0 0
\(232\) −727091. −0.886889
\(233\) −1.12669e6 −1.35961 −0.679803 0.733395i \(-0.737935\pi\)
−0.679803 + 0.733395i \(0.737935\pi\)
\(234\) 16149.6 0.0192807
\(235\) 84099.9 0.0993404
\(236\) −206093. −0.240870
\(237\) −429283. −0.496446
\(238\) 0 0
\(239\) −639017. −0.723632 −0.361816 0.932250i \(-0.617843\pi\)
−0.361816 + 0.932250i \(0.617843\pi\)
\(240\) −206982. −0.231955
\(241\) 256273. 0.284224 0.142112 0.989851i \(-0.454611\pi\)
0.142112 + 0.989851i \(0.454611\pi\)
\(242\) 657722. 0.721945
\(243\) −32369.8 −0.0351661
\(244\) 321076. 0.345250
\(245\) 0 0
\(246\) 1.31200e6 1.38228
\(247\) −838271. −0.874263
\(248\) −946244. −0.976953
\(249\) 3081.73 0.00314989
\(250\) −70831.7 −0.0716766
\(251\) 1.70726e6 1.71047 0.855236 0.518239i \(-0.173412\pi\)
0.855236 + 0.518239i \(0.173412\pi\)
\(252\) 0 0
\(253\) 9290.66 0.00912526
\(254\) 535687. 0.520987
\(255\) −61903.1 −0.0596159
\(256\) −964277. −0.919606
\(257\) −1.45103e6 −1.37039 −0.685193 0.728362i \(-0.740282\pi\)
−0.685193 + 0.728362i \(0.740282\pi\)
\(258\) −1.16465e6 −1.08930
\(259\) 0 0
\(260\) 238640. 0.218932
\(261\) −15774.1 −0.0143332
\(262\) 156913. 0.141223
\(263\) 1.64253e6 1.46428 0.732142 0.681152i \(-0.238521\pi\)
0.732142 + 0.681152i \(0.238521\pi\)
\(264\) 391315. 0.345554
\(265\) −417662. −0.365351
\(266\) 0 0
\(267\) −56648.7 −0.0486308
\(268\) 542230. 0.461155
\(269\) 608608. 0.512811 0.256405 0.966569i \(-0.417462\pi\)
0.256405 + 0.966569i \(0.417462\pi\)
\(270\) 425439. 0.355163
\(271\) 804294. 0.665260 0.332630 0.943057i \(-0.392064\pi\)
0.332630 + 0.943057i \(0.392064\pi\)
\(272\) 82906.3 0.0679462
\(273\) 0 0
\(274\) −102775. −0.0827008
\(275\) 78962.7 0.0629637
\(276\) −13240.2 −0.0104621
\(277\) −820333. −0.642378 −0.321189 0.947015i \(-0.604082\pi\)
−0.321189 + 0.947015i \(0.604082\pi\)
\(278\) 566809. 0.439870
\(279\) −20528.5 −0.0157887
\(280\) 0 0
\(281\) 1.17132e6 0.884934 0.442467 0.896785i \(-0.354103\pi\)
0.442467 + 0.896785i \(0.354103\pi\)
\(282\) −239801. −0.179568
\(283\) −1.99761e6 −1.48267 −0.741335 0.671136i \(-0.765807\pi\)
−0.741335 + 0.671136i \(0.765807\pi\)
\(284\) 563035. 0.414228
\(285\) 395284. 0.288269
\(286\) 477477. 0.345174
\(287\) 0 0
\(288\) −16734.6 −0.0118887
\(289\) −1.39506e6 −0.982537
\(290\) 418351. 0.292110
\(291\) −2.49616e6 −1.72798
\(292\) 925637. 0.635307
\(293\) −174337. −0.118637 −0.0593184 0.998239i \(-0.518893\pi\)
−0.0593184 + 0.998239i \(0.518893\pi\)
\(294\) 0 0
\(295\) 449991. 0.301057
\(296\) 2.81062e6 1.86454
\(297\) −474276. −0.311990
\(298\) −1.76837e6 −1.15354
\(299\) −61306.8 −0.0396580
\(300\) −112530. −0.0721880
\(301\) 0 0
\(302\) 326293. 0.205869
\(303\) −2.51227e6 −1.57202
\(304\) −529400. −0.328549
\(305\) −701049. −0.431518
\(306\) 3050.30 0.00186225
\(307\) 987671. 0.598089 0.299045 0.954239i \(-0.403332\pi\)
0.299045 + 0.954239i \(0.403332\pi\)
\(308\) 0 0
\(309\) 848977. 0.505825
\(310\) 544446. 0.321774
\(311\) 2.40496e6 1.40996 0.704981 0.709226i \(-0.250956\pi\)
0.704981 + 0.709226i \(0.250956\pi\)
\(312\) −2.58219e6 −1.50176
\(313\) 1.89499e6 1.09332 0.546658 0.837356i \(-0.315900\pi\)
0.546658 + 0.837356i \(0.315900\pi\)
\(314\) −400417. −0.229186
\(315\) 0 0
\(316\) 312575. 0.176091
\(317\) −737403. −0.412151 −0.206076 0.978536i \(-0.566069\pi\)
−0.206076 + 0.978536i \(0.566069\pi\)
\(318\) 1.19092e6 0.660409
\(319\) −466375. −0.256601
\(320\) 865031. 0.472234
\(321\) 1.56596e6 0.848238
\(322\) 0 0
\(323\) −158331. −0.0844420
\(324\) 687782. 0.363989
\(325\) −521055. −0.273637
\(326\) −1.98906e6 −1.03658
\(327\) 1.89601e6 0.980553
\(328\) −3.62520e6 −1.86058
\(329\) 0 0
\(330\) −225153. −0.113813
\(331\) 3.68290e6 1.84765 0.923826 0.382813i \(-0.125045\pi\)
0.923826 + 0.382813i \(0.125045\pi\)
\(332\) −2243.91 −0.00111727
\(333\) 60975.6 0.0301332
\(334\) −1.77381e6 −0.870044
\(335\) −1.18392e6 −0.576384
\(336\) 0 0
\(337\) 564674. 0.270846 0.135423 0.990788i \(-0.456761\pi\)
0.135423 + 0.990788i \(0.456761\pi\)
\(338\) −1.46760e6 −0.698740
\(339\) −4.08287e6 −1.92960
\(340\) 45073.7 0.0211459
\(341\) −606945. −0.282659
\(342\) −19477.8 −0.00900479
\(343\) 0 0
\(344\) 3.21807e6 1.46622
\(345\) 28909.0 0.0130763
\(346\) 1.75437e6 0.787828
\(347\) 1.74360e6 0.777361 0.388681 0.921373i \(-0.372931\pi\)
0.388681 + 0.921373i \(0.372931\pi\)
\(348\) 664632. 0.294194
\(349\) −1.75374e6 −0.770727 −0.385363 0.922765i \(-0.625924\pi\)
−0.385363 + 0.922765i \(0.625924\pi\)
\(350\) 0 0
\(351\) 3.12963e6 1.35589
\(352\) −494774. −0.212839
\(353\) 823880. 0.351906 0.175953 0.984399i \(-0.443699\pi\)
0.175953 + 0.984399i \(0.443699\pi\)
\(354\) −1.28310e6 −0.544192
\(355\) −1.22935e6 −0.517732
\(356\) 41247.8 0.0172495
\(357\) 0 0
\(358\) −3.80084e6 −1.56737
\(359\) 3.56016e6 1.45792 0.728959 0.684557i \(-0.240004\pi\)
0.728959 + 0.684557i \(0.240004\pi\)
\(360\) 21042.0 0.00855717
\(361\) −1.46507e6 −0.591686
\(362\) −1.61331e6 −0.647063
\(363\) −2.28152e6 −0.908776
\(364\) 0 0
\(365\) −2.02107e6 −0.794052
\(366\) 1.99896e6 0.780013
\(367\) −3.14208e6 −1.21773 −0.608867 0.793272i \(-0.708376\pi\)
−0.608867 + 0.793272i \(0.708376\pi\)
\(368\) −38717.6 −0.0149035
\(369\) −78647.8 −0.0300691
\(370\) −1.61716e6 −0.614114
\(371\) 0 0
\(372\) 864959. 0.324069
\(373\) −2.64076e6 −0.982782 −0.491391 0.870939i \(-0.663511\pi\)
−0.491391 + 0.870939i \(0.663511\pi\)
\(374\) 90184.8 0.0333391
\(375\) 245702. 0.0902257
\(376\) 662600. 0.241703
\(377\) 3.07749e6 1.11518
\(378\) 0 0
\(379\) −2.62170e6 −0.937531 −0.468766 0.883323i \(-0.655301\pi\)
−0.468766 + 0.883323i \(0.655301\pi\)
\(380\) −287820. −0.102250
\(381\) −1.85820e6 −0.655813
\(382\) 799637. 0.280372
\(383\) 2.15463e6 0.750543 0.375272 0.926915i \(-0.377549\pi\)
0.375272 + 0.926915i \(0.377549\pi\)
\(384\) −495913. −0.171624
\(385\) 0 0
\(386\) −3.16343e6 −1.08066
\(387\) 69815.2 0.0236958
\(388\) 1.81754e6 0.612920
\(389\) −551834. −0.184899 −0.0924495 0.995717i \(-0.529470\pi\)
−0.0924495 + 0.995717i \(0.529470\pi\)
\(390\) 1.48573e6 0.494627
\(391\) −11579.5 −0.00383042
\(392\) 0 0
\(393\) −544301. −0.177770
\(394\) −2.17055e6 −0.704417
\(395\) −682488. −0.220091
\(396\) −6181.47 −0.00198086
\(397\) −721152. −0.229642 −0.114821 0.993386i \(-0.536629\pi\)
−0.114821 + 0.993386i \(0.536629\pi\)
\(398\) −4.10255e6 −1.29821
\(399\) 0 0
\(400\) −329066. −0.102833
\(401\) −5.40555e6 −1.67872 −0.839361 0.543575i \(-0.817070\pi\)
−0.839361 + 0.543575i \(0.817070\pi\)
\(402\) 3.37583e6 1.04187
\(403\) 4.00508e6 1.22842
\(404\) 1.82926e6 0.557601
\(405\) −1.50173e6 −0.454940
\(406\) 0 0
\(407\) 1.80280e6 0.539463
\(408\) −487717. −0.145050
\(409\) 756615. 0.223649 0.111824 0.993728i \(-0.464331\pi\)
0.111824 + 0.993728i \(0.464331\pi\)
\(410\) 2.08585e6 0.612808
\(411\) 356506. 0.104103
\(412\) −618169. −0.179417
\(413\) 0 0
\(414\) −1424.50 −0.000408472 0
\(415\) 4899.43 0.00139645
\(416\) 3.26489e6 0.924987
\(417\) −1.96615e6 −0.553704
\(418\) −575878. −0.161209
\(419\) 4.63489e6 1.28975 0.644873 0.764290i \(-0.276910\pi\)
0.644873 + 0.764290i \(0.276910\pi\)
\(420\) 0 0
\(421\) 6.13555e6 1.68713 0.843564 0.537028i \(-0.180453\pi\)
0.843564 + 0.537028i \(0.180453\pi\)
\(422\) 3.82419e6 1.04534
\(423\) 14374.9 0.00390621
\(424\) −3.29064e6 −0.888927
\(425\) −98415.6 −0.0264297
\(426\) 3.50536e6 0.935854
\(427\) 0 0
\(428\) −1.14023e6 −0.300872
\(429\) −1.65628e6 −0.434501
\(430\) −1.85160e6 −0.482920
\(431\) 2.11928e6 0.549534 0.274767 0.961511i \(-0.411399\pi\)
0.274767 + 0.961511i \(0.411399\pi\)
\(432\) 1.97648e6 0.509546
\(433\) −4.34108e6 −1.11270 −0.556350 0.830948i \(-0.687799\pi\)
−0.556350 + 0.830948i \(0.687799\pi\)
\(434\) 0 0
\(435\) −1.45118e6 −0.367704
\(436\) −1.38055e6 −0.347805
\(437\) 73941.1 0.0185218
\(438\) 5.76285e6 1.43533
\(439\) −6.74539e6 −1.67050 −0.835248 0.549873i \(-0.814676\pi\)
−0.835248 + 0.549873i \(0.814676\pi\)
\(440\) 622125. 0.153195
\(441\) 0 0
\(442\) −595107. −0.144890
\(443\) −4.42765e6 −1.07192 −0.535962 0.844242i \(-0.680051\pi\)
−0.535962 + 0.844242i \(0.680051\pi\)
\(444\) −2.56918e6 −0.618495
\(445\) −90062.0 −0.0215596
\(446\) 2.65188e6 0.631272
\(447\) 6.13413e6 1.45206
\(448\) 0 0
\(449\) −5.84556e6 −1.36839 −0.684196 0.729299i \(-0.739847\pi\)
−0.684196 + 0.729299i \(0.739847\pi\)
\(450\) −12107.0 −0.00281843
\(451\) −2.32529e6 −0.538315
\(452\) 2.97288e6 0.684433
\(453\) −1.13185e6 −0.259145
\(454\) 3.85206e6 0.877109
\(455\) 0 0
\(456\) 3.11433e6 0.701380
\(457\) 3.33675e6 0.747366 0.373683 0.927556i \(-0.378095\pi\)
0.373683 + 0.927556i \(0.378095\pi\)
\(458\) 876213. 0.195185
\(459\) 591117. 0.130961
\(460\) −21049.6 −0.00463820
\(461\) 7.61305e6 1.66842 0.834212 0.551444i \(-0.185923\pi\)
0.834212 + 0.551444i \(0.185923\pi\)
\(462\) 0 0
\(463\) −2.82147e6 −0.611679 −0.305839 0.952083i \(-0.598937\pi\)
−0.305839 + 0.952083i \(0.598937\pi\)
\(464\) 1.94355e6 0.419085
\(465\) −1.88858e6 −0.405045
\(466\) 5.10753e6 1.08955
\(467\) −2.61119e6 −0.554046 −0.277023 0.960863i \(-0.589348\pi\)
−0.277023 + 0.960863i \(0.589348\pi\)
\(468\) 40790.0 0.00860873
\(469\) 0 0
\(470\) −381244. −0.0796084
\(471\) 1.38897e6 0.288497
\(472\) 3.54535e6 0.732495
\(473\) 2.06415e6 0.424217
\(474\) 1.94604e6 0.397837
\(475\) 628436. 0.127799
\(476\) 0 0
\(477\) −71389.6 −0.0143661
\(478\) 2.89681e6 0.579896
\(479\) −817014. −0.162701 −0.0813506 0.996686i \(-0.525923\pi\)
−0.0813506 + 0.996686i \(0.525923\pi\)
\(480\) −1.53955e6 −0.304994
\(481\) −1.18962e7 −2.34448
\(482\) −1.16174e6 −0.227768
\(483\) 0 0
\(484\) 1.66125e6 0.322345
\(485\) −3.96848e6 −0.766072
\(486\) 146740. 0.0281810
\(487\) −2.11091e6 −0.403317 −0.201659 0.979456i \(-0.564633\pi\)
−0.201659 + 0.979456i \(0.564633\pi\)
\(488\) −5.52337e6 −1.04992
\(489\) 6.89968e6 1.30484
\(490\) 0 0
\(491\) 1.71666e6 0.321352 0.160676 0.987007i \(-0.448633\pi\)
0.160676 + 0.987007i \(0.448633\pi\)
\(492\) 3.31378e6 0.617180
\(493\) 581269. 0.107711
\(494\) 3.80007e6 0.700607
\(495\) 13496.8 0.00247582
\(496\) 2.52936e6 0.461643
\(497\) 0 0
\(498\) −13970.2 −0.00252423
\(499\) 7.64091e6 1.37371 0.686853 0.726797i \(-0.258992\pi\)
0.686853 + 0.726797i \(0.258992\pi\)
\(500\) −178904. −0.0320033
\(501\) 6.15302e6 1.09520
\(502\) −7.73941e6 −1.37072
\(503\) −2.90660e6 −0.512231 −0.256115 0.966646i \(-0.582443\pi\)
−0.256115 + 0.966646i \(0.582443\pi\)
\(504\) 0 0
\(505\) −3.99408e6 −0.696929
\(506\) −42116.7 −0.00731270
\(507\) 5.09083e6 0.879566
\(508\) 1.35302e6 0.232618
\(509\) −602745. −0.103119 −0.0515595 0.998670i \(-0.516419\pi\)
−0.0515595 + 0.998670i \(0.516419\pi\)
\(510\) 280621. 0.0477743
\(511\) 0 0
\(512\) 5.38046e6 0.907078
\(513\) −3.77460e6 −0.633253
\(514\) 6.57784e6 1.09818
\(515\) 1.34973e6 0.224248
\(516\) −2.94162e6 −0.486366
\(517\) 425008. 0.0699312
\(518\) 0 0
\(519\) −6.08559e6 −0.991709
\(520\) −4.10525e6 −0.665780
\(521\) 611890. 0.0987595 0.0493797 0.998780i \(-0.484276\pi\)
0.0493797 + 0.998780i \(0.484276\pi\)
\(522\) 71507.5 0.0114862
\(523\) 9.02368e6 1.44255 0.721273 0.692651i \(-0.243557\pi\)
0.721273 + 0.692651i \(0.243557\pi\)
\(524\) 396324. 0.0630554
\(525\) 0 0
\(526\) −7.44598e6 −1.17343
\(527\) 756469. 0.118649
\(528\) −1.04600e6 −0.163286
\(529\) −6.43094e6 −0.999160
\(530\) 1.89336e6 0.292781
\(531\) 76915.6 0.0118380
\(532\) 0 0
\(533\) 1.53440e7 2.33949
\(534\) 256801. 0.0389712
\(535\) 2.48961e6 0.376052
\(536\) −9.32781e6 −1.40239
\(537\) 1.31844e7 1.97299
\(538\) −2.75896e6 −0.410951
\(539\) 0 0
\(540\) 1.07456e6 0.158579
\(541\) 2.23016e6 0.327600 0.163800 0.986494i \(-0.447625\pi\)
0.163800 + 0.986494i \(0.447625\pi\)
\(542\) −3.64605e6 −0.533119
\(543\) 5.59627e6 0.814515
\(544\) 616665. 0.0893413
\(545\) 3.01434e6 0.434711
\(546\) 0 0
\(547\) −1.20299e7 −1.71907 −0.859535 0.511077i \(-0.829247\pi\)
−0.859535 + 0.511077i \(0.829247\pi\)
\(548\) −259584. −0.0369255
\(549\) −119828. −0.0169679
\(550\) −357956. −0.0504571
\(551\) −3.71171e6 −0.520829
\(552\) 227766. 0.0318157
\(553\) 0 0
\(554\) 3.71876e6 0.514782
\(555\) 5.60963e6 0.773040
\(556\) 1.43162e6 0.196400
\(557\) 4.42512e6 0.604348 0.302174 0.953253i \(-0.402288\pi\)
0.302174 + 0.953253i \(0.402288\pi\)
\(558\) 93060.5 0.0126526
\(559\) −1.36208e7 −1.84363
\(560\) 0 0
\(561\) −312834. −0.0419669
\(562\) −5.30987e6 −0.709159
\(563\) 7.64181e6 1.01607 0.508037 0.861335i \(-0.330371\pi\)
0.508037 + 0.861335i \(0.330371\pi\)
\(564\) −605681. −0.0801763
\(565\) −6.49109e6 −0.855453
\(566\) 9.05561e6 1.18817
\(567\) 0 0
\(568\) −9.68572e6 −1.25968
\(569\) 1.63458e6 0.211653 0.105826 0.994385i \(-0.466251\pi\)
0.105826 + 0.994385i \(0.466251\pi\)
\(570\) −1.79191e6 −0.231010
\(571\) −4.20562e6 −0.539809 −0.269905 0.962887i \(-0.586992\pi\)
−0.269905 + 0.962887i \(0.586992\pi\)
\(572\) 1.20599e6 0.154119
\(573\) −2.77379e6 −0.352929
\(574\) 0 0
\(575\) 45960.5 0.00579716
\(576\) 147857. 0.0185689
\(577\) 9.07317e6 1.13454 0.567270 0.823532i \(-0.308000\pi\)
0.567270 + 0.823532i \(0.308000\pi\)
\(578\) 6.32413e6 0.787375
\(579\) 1.09734e7 1.36033
\(580\) 1.05665e6 0.130426
\(581\) 0 0
\(582\) 1.13157e7 1.38475
\(583\) −2.11070e6 −0.257191
\(584\) −1.59234e7 −1.93199
\(585\) −89062.3 −0.0107598
\(586\) 790308. 0.0950719
\(587\) −8.91120e6 −1.06743 −0.533717 0.845663i \(-0.679205\pi\)
−0.533717 + 0.845663i \(0.679205\pi\)
\(588\) 0 0
\(589\) −4.83046e6 −0.573720
\(590\) −2.03991e6 −0.241258
\(591\) 7.52924e6 0.886712
\(592\) −7.51293e6 −0.881059
\(593\) 9.36933e6 1.09414 0.547069 0.837088i \(-0.315744\pi\)
0.547069 + 0.837088i \(0.315744\pi\)
\(594\) 2.15000e6 0.250019
\(595\) 0 0
\(596\) −4.46647e6 −0.515049
\(597\) 1.42310e7 1.63418
\(598\) 277918. 0.0317807
\(599\) 3.91658e6 0.446005 0.223003 0.974818i \(-0.428414\pi\)
0.223003 + 0.974818i \(0.428414\pi\)
\(600\) 1.93582e6 0.219526
\(601\) −9.01467e6 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(602\) 0 0
\(603\) −202365. −0.0226642
\(604\) 824138. 0.0919195
\(605\) −3.62723e6 −0.402890
\(606\) 1.13887e7 1.25977
\(607\) 1.47500e6 0.162488 0.0812439 0.996694i \(-0.474111\pi\)
0.0812439 + 0.996694i \(0.474111\pi\)
\(608\) −3.93773e6 −0.432004
\(609\) 0 0
\(610\) 3.17801e6 0.345805
\(611\) −2.80452e6 −0.303918
\(612\) 7704.31 0.000831487 0
\(613\) −4.14590e6 −0.445623 −0.222811 0.974862i \(-0.571523\pi\)
−0.222811 + 0.974862i \(0.571523\pi\)
\(614\) −4.47734e6 −0.479290
\(615\) −7.23544e6 −0.771395
\(616\) 0 0
\(617\) 2.99859e6 0.317105 0.158553 0.987351i \(-0.449317\pi\)
0.158553 + 0.987351i \(0.449317\pi\)
\(618\) −3.84861e6 −0.405352
\(619\) 9.80100e6 1.02812 0.514060 0.857754i \(-0.328141\pi\)
0.514060 + 0.857754i \(0.328141\pi\)
\(620\) 1.37514e6 0.143671
\(621\) −276054. −0.0287254
\(622\) −1.09022e7 −1.12990
\(623\) 0 0
\(624\) 6.90232e6 0.709633
\(625\) 390625. 0.0400000
\(626\) −8.59041e6 −0.876149
\(627\) 1.99761e6 0.202928
\(628\) −1.01136e6 −0.102331
\(629\) −2.24693e6 −0.226445
\(630\) 0 0
\(631\) 465144. 0.0465065 0.0232533 0.999730i \(-0.492598\pi\)
0.0232533 + 0.999730i \(0.492598\pi\)
\(632\) −5.37713e6 −0.535498
\(633\) −1.32654e7 −1.31587
\(634\) 3.34281e6 0.330285
\(635\) −2.95423e6 −0.290743
\(636\) 3.00797e6 0.294870
\(637\) 0 0
\(638\) 2.11418e6 0.205632
\(639\) −210129. −0.0203580
\(640\) −788419. −0.0760864
\(641\) −1.42324e7 −1.36815 −0.684075 0.729412i \(-0.739794\pi\)
−0.684075 + 0.729412i \(0.739794\pi\)
\(642\) −7.09885e6 −0.679752
\(643\) 7.46870e6 0.712390 0.356195 0.934412i \(-0.384074\pi\)
0.356195 + 0.934412i \(0.384074\pi\)
\(644\) 0 0
\(645\) 6.42285e6 0.607895
\(646\) 717749. 0.0676692
\(647\) −1.00619e7 −0.944972 −0.472486 0.881338i \(-0.656643\pi\)
−0.472486 + 0.881338i \(0.656643\pi\)
\(648\) −1.18317e7 −1.10690
\(649\) 2.27408e6 0.211931
\(650\) 2.36206e6 0.219284
\(651\) 0 0
\(652\) −5.02389e6 −0.462830
\(653\) 1.14074e7 1.04690 0.523450 0.852056i \(-0.324645\pi\)
0.523450 + 0.852056i \(0.324645\pi\)
\(654\) −8.59504e6 −0.785785
\(655\) −865348. −0.0788112
\(656\) 9.69035e6 0.879184
\(657\) −345455. −0.0312233
\(658\) 0 0
\(659\) 9.91403e6 0.889276 0.444638 0.895710i \(-0.353332\pi\)
0.444638 + 0.895710i \(0.353332\pi\)
\(660\) −568683. −0.0508171
\(661\) −151734. −0.0135076 −0.00675382 0.999977i \(-0.502150\pi\)
−0.00675382 + 0.999977i \(0.502150\pi\)
\(662\) −1.66954e7 −1.48065
\(663\) 2.06431e6 0.182386
\(664\) 38601.3 0.00339767
\(665\) 0 0
\(666\) −276416. −0.0241478
\(667\) −271455. −0.0236256
\(668\) −4.48022e6 −0.388471
\(669\) −9.19887e6 −0.794637
\(670\) 5.36700e6 0.461896
\(671\) −3.54283e6 −0.303769
\(672\) 0 0
\(673\) −4.84679e6 −0.412493 −0.206247 0.978500i \(-0.566125\pi\)
−0.206247 + 0.978500i \(0.566125\pi\)
\(674\) −2.55980e6 −0.217048
\(675\) −2.34623e6 −0.198203
\(676\) −3.70680e6 −0.311984
\(677\) −4.35480e6 −0.365172 −0.182586 0.983190i \(-0.558447\pi\)
−0.182586 + 0.983190i \(0.558447\pi\)
\(678\) 1.85086e7 1.54632
\(679\) 0 0
\(680\) −775389. −0.0643054
\(681\) −1.33621e7 −1.10410
\(682\) 2.75142e6 0.226514
\(683\) −2.22591e7 −1.82581 −0.912904 0.408174i \(-0.866166\pi\)
−0.912904 + 0.408174i \(0.866166\pi\)
\(684\) −49196.1 −0.00402060
\(685\) 566785. 0.0461522
\(686\) 0 0
\(687\) −3.03942e6 −0.245696
\(688\) −8.60206e6 −0.692838
\(689\) 1.39280e7 1.11774
\(690\) −131051. −0.0104790
\(691\) −148279. −0.0118137 −0.00590685 0.999983i \(-0.501880\pi\)
−0.00590685 + 0.999983i \(0.501880\pi\)
\(692\) 4.43112e6 0.351762
\(693\) 0 0
\(694\) −7.90413e6 −0.622953
\(695\) −3.12586e6 −0.245475
\(696\) −1.14335e7 −0.894653
\(697\) 2.89815e6 0.225963
\(698\) 7.95008e6 0.617637
\(699\) −1.77171e7 −1.37151
\(700\) 0 0
\(701\) 1.93655e7 1.48845 0.744223 0.667931i \(-0.232820\pi\)
0.744223 + 0.667931i \(0.232820\pi\)
\(702\) −1.41873e7 −1.08657
\(703\) 1.43478e7 1.09496
\(704\) 4.37153e6 0.332431
\(705\) 1.32247e6 0.100210
\(706\) −3.73484e6 −0.282007
\(707\) 0 0
\(708\) −3.24080e6 −0.242979
\(709\) 8.30106e6 0.620180 0.310090 0.950707i \(-0.399641\pi\)
0.310090 + 0.950707i \(0.399641\pi\)
\(710\) 5.57293e6 0.414895
\(711\) −116656. −0.00865429
\(712\) −709573. −0.0524562
\(713\) −353274. −0.0260249
\(714\) 0 0
\(715\) −2.63321e6 −0.192628
\(716\) −9.60002e6 −0.699825
\(717\) −1.00485e7 −0.729967
\(718\) −1.61390e7 −1.16833
\(719\) −7.65756e6 −0.552418 −0.276209 0.961098i \(-0.589078\pi\)
−0.276209 + 0.961098i \(0.589078\pi\)
\(720\) −56246.3 −0.00404355
\(721\) 0 0
\(722\) 6.64151e6 0.474159
\(723\) 4.02987e6 0.286712
\(724\) −4.07483e6 −0.288911
\(725\) −2.30714e6 −0.163015
\(726\) 1.03426e7 0.728265
\(727\) 1.09211e7 0.766353 0.383177 0.923675i \(-0.374830\pi\)
0.383177 + 0.923675i \(0.374830\pi\)
\(728\) 0 0
\(729\) 1.40878e7 0.981802
\(730\) 9.16197e6 0.636329
\(731\) −2.57266e6 −0.178070
\(732\) 5.04890e6 0.348272
\(733\) −2.23972e7 −1.53969 −0.769847 0.638229i \(-0.779667\pi\)
−0.769847 + 0.638229i \(0.779667\pi\)
\(734\) 1.42438e7 0.975854
\(735\) 0 0
\(736\) −287985. −0.0195964
\(737\) −5.98310e6 −0.405749
\(738\) 356529. 0.0240965
\(739\) 5.17870e6 0.348827 0.174413 0.984673i \(-0.444197\pi\)
0.174413 + 0.984673i \(0.444197\pi\)
\(740\) −4.08456e6 −0.274199
\(741\) −1.31817e7 −0.881916
\(742\) 0 0
\(743\) −2.99391e6 −0.198961 −0.0994804 0.995040i \(-0.531718\pi\)
−0.0994804 + 0.995040i \(0.531718\pi\)
\(744\) −1.48796e7 −0.985506
\(745\) 9.75225e6 0.643745
\(746\) 1.19712e7 0.787571
\(747\) 837.445 5.49104e−5 0
\(748\) 227785. 0.0148858
\(749\) 0 0
\(750\) −1.11382e6 −0.0723041
\(751\) 1.32983e7 0.860391 0.430195 0.902736i \(-0.358445\pi\)
0.430195 + 0.902736i \(0.358445\pi\)
\(752\) −1.77117e6 −0.114213
\(753\) 2.68466e7 1.72545
\(754\) −1.39510e7 −0.893667
\(755\) −1.79945e6 −0.114888
\(756\) 0 0
\(757\) −3.09742e6 −0.196454 −0.0982269 0.995164i \(-0.531317\pi\)
−0.0982269 + 0.995164i \(0.531317\pi\)
\(758\) 1.18848e7 0.751308
\(759\) 146095. 0.00920515
\(760\) 4.95127e6 0.310944
\(761\) 2.52491e7 1.58046 0.790231 0.612809i \(-0.209961\pi\)
0.790231 + 0.612809i \(0.209961\pi\)
\(762\) 8.42364e6 0.525548
\(763\) 0 0
\(764\) 2.01969e6 0.125185
\(765\) −16821.9 −0.00103925
\(766\) −9.76743e6 −0.601462
\(767\) −1.50061e7 −0.921041
\(768\) −1.51632e7 −0.927656
\(769\) −807096. −0.0492164 −0.0246082 0.999697i \(-0.507834\pi\)
−0.0246082 + 0.999697i \(0.507834\pi\)
\(770\) 0 0
\(771\) −2.28173e7 −1.38238
\(772\) −7.99007e6 −0.482511
\(773\) −1.97132e7 −1.18661 −0.593306 0.804977i \(-0.702178\pi\)
−0.593306 + 0.804977i \(0.702178\pi\)
\(774\) −316488. −0.0189891
\(775\) −3.00253e6 −0.179570
\(776\) −3.12665e7 −1.86391
\(777\) 0 0
\(778\) 2.50159e6 0.148172
\(779\) −1.85062e7 −1.09263
\(780\) 3.75259e6 0.220849
\(781\) −6.21266e6 −0.364460
\(782\) 52492.4 0.00306958
\(783\) 1.38574e7 0.807753
\(784\) 0 0
\(785\) 2.20824e6 0.127900
\(786\) 2.46744e6 0.142459
\(787\) −6.51938e6 −0.375206 −0.187603 0.982245i \(-0.560072\pi\)
−0.187603 + 0.982245i \(0.560072\pi\)
\(788\) −5.48230e6 −0.314519
\(789\) 2.58287e7 1.47710
\(790\) 3.09387e6 0.176374
\(791\) 0 0
\(792\) 106338. 0.00602386
\(793\) 2.33782e7 1.32017
\(794\) 3.26915e6 0.184028
\(795\) −6.56770e6 −0.368549
\(796\) −1.03621e7 −0.579646
\(797\) 1.71577e7 0.956780 0.478390 0.878147i \(-0.341220\pi\)
0.478390 + 0.878147i \(0.341220\pi\)
\(798\) 0 0
\(799\) −529712. −0.0293544
\(800\) −2.44763e6 −0.135214
\(801\) −15394.0 −0.000847755 0
\(802\) 2.45046e7 1.34528
\(803\) −1.02137e7 −0.558977
\(804\) 8.52653e6 0.465192
\(805\) 0 0
\(806\) −1.81559e7 −0.984421
\(807\) 9.57032e6 0.517300
\(808\) −3.14683e7 −1.69568
\(809\) 9.99850e6 0.537111 0.268555 0.963264i \(-0.413454\pi\)
0.268555 + 0.963264i \(0.413454\pi\)
\(810\) 6.80768e6 0.364575
\(811\) −6.02515e6 −0.321674 −0.160837 0.986981i \(-0.551419\pi\)
−0.160837 + 0.986981i \(0.551419\pi\)
\(812\) 0 0
\(813\) 1.26475e7 0.671084
\(814\) −8.17251e6 −0.432309
\(815\) 1.09693e7 0.578478
\(816\) 1.30369e6 0.0685410
\(817\) 1.64278e7 0.861043
\(818\) −3.42991e6 −0.179225
\(819\) 0 0
\(820\) 5.26836e6 0.273616
\(821\) 1.01366e7 0.524848 0.262424 0.964953i \(-0.415478\pi\)
0.262424 + 0.964953i \(0.415478\pi\)
\(822\) −1.61612e6 −0.0834248
\(823\) −3.17893e7 −1.63599 −0.817995 0.575225i \(-0.804915\pi\)
−0.817995 + 0.575225i \(0.804915\pi\)
\(824\) 1.06342e7 0.545614
\(825\) 1.24168e6 0.0635149
\(826\) 0 0
\(827\) −1.48323e7 −0.754125 −0.377062 0.926188i \(-0.623066\pi\)
−0.377062 + 0.926188i \(0.623066\pi\)
\(828\) −3597.95 −0.000182381 0
\(829\) 2.25871e7 1.14149 0.570747 0.821126i \(-0.306654\pi\)
0.570747 + 0.821126i \(0.306654\pi\)
\(830\) −22210.2 −0.00111907
\(831\) −1.28997e7 −0.648002
\(832\) −2.88466e7 −1.44473
\(833\) 0 0
\(834\) 8.91303e6 0.443721
\(835\) 9.78227e6 0.485538
\(836\) −1.45453e6 −0.0719791
\(837\) 1.80342e7 0.889781
\(838\) −2.10110e7 −1.03356
\(839\) −9.13225e6 −0.447891 −0.223946 0.974602i \(-0.571894\pi\)
−0.223946 + 0.974602i \(0.571894\pi\)
\(840\) 0 0
\(841\) −6.88458e6 −0.335651
\(842\) −2.78138e7 −1.35201
\(843\) 1.84190e7 0.892681
\(844\) 9.65900e6 0.466741
\(845\) 8.09356e6 0.389940
\(846\) −65164.9 −0.00313031
\(847\) 0 0
\(848\) 8.79606e6 0.420048
\(849\) −3.14122e7 −1.49565
\(850\) 446140. 0.0211799
\(851\) 1.04933e6 0.0496692
\(852\) 8.85369e6 0.417855
\(853\) 7.84791e6 0.369302 0.184651 0.982804i \(-0.440885\pi\)
0.184651 + 0.982804i \(0.440885\pi\)
\(854\) 0 0
\(855\) 107417. 0.00502523
\(856\) 1.96150e7 0.914962
\(857\) −3.07813e7 −1.43164 −0.715822 0.698283i \(-0.753948\pi\)
−0.715822 + 0.698283i \(0.753948\pi\)
\(858\) 7.50830e6 0.348196
\(859\) −962472. −0.0445046 −0.0222523 0.999752i \(-0.507084\pi\)
−0.0222523 + 0.999752i \(0.507084\pi\)
\(860\) −4.67669e6 −0.215622
\(861\) 0 0
\(862\) −9.60717e6 −0.440380
\(863\) 8.60992e6 0.393525 0.196762 0.980451i \(-0.436957\pi\)
0.196762 + 0.980451i \(0.436957\pi\)
\(864\) 1.47013e7 0.669994
\(865\) −9.67507e6 −0.439657
\(866\) 1.96791e7 0.891684
\(867\) −2.19372e7 −0.991138
\(868\) 0 0
\(869\) −3.44903e6 −0.154934
\(870\) 6.57854e6 0.294667
\(871\) 3.94809e7 1.76336
\(872\) 2.37491e7 1.05769
\(873\) −678319. −0.0301230
\(874\) −335192. −0.0148428
\(875\) 0 0
\(876\) 1.45556e7 0.640869
\(877\) −7.93397e6 −0.348331 −0.174165 0.984716i \(-0.555723\pi\)
−0.174165 + 0.984716i \(0.555723\pi\)
\(878\) 3.05784e7 1.33868
\(879\) −2.74143e6 −0.119675
\(880\) −1.66297e6 −0.0723900
\(881\) −1.44812e7 −0.628586 −0.314293 0.949326i \(-0.601767\pi\)
−0.314293 + 0.949326i \(0.601767\pi\)
\(882\) 0 0
\(883\) −3.51794e6 −0.151840 −0.0759200 0.997114i \(-0.524189\pi\)
−0.0759200 + 0.997114i \(0.524189\pi\)
\(884\) −1.50310e6 −0.0646929
\(885\) 7.07608e6 0.303693
\(886\) 2.00716e7 0.859007
\(887\) −2.82971e6 −0.120763 −0.0603814 0.998175i \(-0.519232\pi\)
−0.0603814 + 0.998175i \(0.519232\pi\)
\(888\) 4.41967e7 1.88087
\(889\) 0 0
\(890\) 408271. 0.0172772
\(891\) −7.58915e6 −0.320257
\(892\) 6.69800e6 0.281860
\(893\) 3.38249e6 0.141941
\(894\) −2.78074e7 −1.16364
\(895\) 2.09610e7 0.874691
\(896\) 0 0
\(897\) −964044. −0.0400051
\(898\) 2.64993e7 1.09659
\(899\) 1.77337e7 0.731815
\(900\) −30579.5 −0.00125841
\(901\) 2.63068e6 0.107958
\(902\) 1.05411e7 0.431389
\(903\) 0 0
\(904\) −5.11415e7 −2.08138
\(905\) 8.89714e6 0.361101
\(906\) 5.13094e6 0.207671
\(907\) 1.46684e7 0.592059 0.296030 0.955179i \(-0.404337\pi\)
0.296030 + 0.955179i \(0.404337\pi\)
\(908\) 9.72938e6 0.391625
\(909\) −682696. −0.0274042
\(910\) 0 0
\(911\) −87366.6 −0.00348778 −0.00174389 0.999998i \(-0.500555\pi\)
−0.00174389 + 0.999998i \(0.500555\pi\)
\(912\) −8.32478e6 −0.331425
\(913\) 24759.8 0.000983039 0
\(914\) −1.51263e7 −0.598916
\(915\) −1.10239e7 −0.435295
\(916\) 2.21310e6 0.0871491
\(917\) 0 0
\(918\) −2.67967e6 −0.104948
\(919\) 2.94662e7 1.15090 0.575448 0.817838i \(-0.304828\pi\)
0.575448 + 0.817838i \(0.304828\pi\)
\(920\) 362110. 0.0141049
\(921\) 1.55310e7 0.603325
\(922\) −3.45117e7 −1.33702
\(923\) 4.09958e7 1.58393
\(924\) 0 0
\(925\) 8.91838e6 0.342714
\(926\) 1.27904e7 0.490180
\(927\) 230706. 0.00881777
\(928\) 1.44564e7 0.551047
\(929\) 3.07331e7 1.16834 0.584168 0.811633i \(-0.301421\pi\)
0.584168 + 0.811633i \(0.301421\pi\)
\(930\) 8.56137e6 0.324591
\(931\) 0 0
\(932\) 1.29004e7 0.486478
\(933\) 3.78179e7 1.42231
\(934\) 1.18371e7 0.443996
\(935\) −497354. −0.0186053
\(936\) −701697. −0.0261794
\(937\) −1.87911e7 −0.699203 −0.349602 0.936898i \(-0.613683\pi\)
−0.349602 + 0.936898i \(0.613683\pi\)
\(938\) 0 0
\(939\) 2.97985e7 1.10289
\(940\) −962931. −0.0355448
\(941\) −3.59306e7 −1.32279 −0.661394 0.750039i \(-0.730035\pi\)
−0.661394 + 0.750039i \(0.730035\pi\)
\(942\) −6.29653e6 −0.231193
\(943\) −1.35345e6 −0.0495635
\(944\) −9.47692e6 −0.346128
\(945\) 0 0
\(946\) −9.35726e6 −0.339954
\(947\) 4.56598e7 1.65447 0.827235 0.561856i \(-0.189912\pi\)
0.827235 + 0.561856i \(0.189912\pi\)
\(948\) 4.91522e6 0.177632
\(949\) 6.73976e7 2.42929
\(950\) −2.84884e6 −0.102414
\(951\) −1.15956e7 −0.415759
\(952\) 0 0
\(953\) 2.96804e7 1.05861 0.529307 0.848430i \(-0.322452\pi\)
0.529307 + 0.848430i \(0.322452\pi\)
\(954\) 323626. 0.0115126
\(955\) −4.40987e6 −0.156465
\(956\) 7.31665e6 0.258921
\(957\) −7.33371e6 −0.258847
\(958\) 3.70371e6 0.130384
\(959\) 0 0
\(960\) 1.36025e7 0.476368
\(961\) −5.55028e6 −0.193868
\(962\) 5.39283e7 1.87879
\(963\) 425542. 0.0147869
\(964\) −2.93429e6 −0.101697
\(965\) 1.74458e7 0.603077
\(966\) 0 0
\(967\) −5.33565e7 −1.83494 −0.917468 0.397810i \(-0.869770\pi\)
−0.917468 + 0.397810i \(0.869770\pi\)
\(968\) −2.85779e7 −0.980262
\(969\) −2.48974e6 −0.0851812
\(970\) 1.79900e7 0.613906
\(971\) −3.10322e7 −1.05624 −0.528122 0.849169i \(-0.677104\pi\)
−0.528122 + 0.849169i \(0.677104\pi\)
\(972\) 370629. 0.0125827
\(973\) 0 0
\(974\) 9.56922e6 0.323206
\(975\) −8.19355e6 −0.276033
\(976\) 1.47643e7 0.496120
\(977\) −328987. −0.0110266 −0.00551330 0.999985i \(-0.501755\pi\)
−0.00551330 + 0.999985i \(0.501755\pi\)
\(978\) −3.12778e7 −1.04566
\(979\) −455138. −0.0151770
\(980\) 0 0
\(981\) 515232. 0.0170935
\(982\) −7.78202e6 −0.257522
\(983\) −2.90474e7 −0.958790 −0.479395 0.877599i \(-0.659144\pi\)
−0.479395 + 0.877599i \(0.659144\pi\)
\(984\) −5.70060e7 −1.87686
\(985\) 1.19702e7 0.393108
\(986\) −2.63503e6 −0.0863162
\(987\) 0 0
\(988\) 9.59807e6 0.312818
\(989\) 1.20145e6 0.0390583
\(990\) −61184.3 −0.00198405
\(991\) −2.77388e7 −0.897230 −0.448615 0.893725i \(-0.648083\pi\)
−0.448615 + 0.893725i \(0.648083\pi\)
\(992\) 1.88136e7 0.607007
\(993\) 5.79133e7 1.86383
\(994\) 0 0
\(995\) 2.26249e7 0.724483
\(996\) −35285.3 −0.00112706
\(997\) −4.43451e7 −1.41289 −0.706444 0.707769i \(-0.749702\pi\)
−0.706444 + 0.707769i \(0.749702\pi\)
\(998\) −3.46380e7 −1.10085
\(999\) −5.35668e7 −1.69817
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 245.6.a.d.1.2 3
7.6 odd 2 35.6.a.c.1.2 3
21.20 even 2 315.6.a.i.1.2 3
28.27 even 2 560.6.a.q.1.3 3
35.13 even 4 175.6.b.e.99.4 6
35.27 even 4 175.6.b.e.99.3 6
35.34 odd 2 175.6.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.c.1.2 3 7.6 odd 2
175.6.a.e.1.2 3 35.34 odd 2
175.6.b.e.99.3 6 35.27 even 4
175.6.b.e.99.4 6 35.13 even 4
245.6.a.d.1.2 3 1.1 even 1 trivial
315.6.a.i.1.2 3 21.20 even 2
560.6.a.q.1.3 3 28.27 even 2