Properties

Label 650.6.b.j.599.2
Level $650$
Weight $6$
Character 650.599
Analytic conductor $104.249$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(2.14150 - 2.14150i\) of defining polynomial
Character \(\chi\) \(=\) 650.599
Dual form 650.6.b.j.599.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} +3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} -10.2576i q^{7} +64.0000i q^{8} +229.184 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} +3.71700i q^{3} -16.0000 q^{4} +14.8680 q^{6} -10.2576i q^{7} +64.0000i q^{8} +229.184 q^{9} +197.274 q^{11} -59.4720i q^{12} -169.000i q^{13} -41.0305 q^{14} +256.000 q^{16} -949.722i q^{17} -916.736i q^{18} -2233.30 q^{19} +38.1276 q^{21} -789.095i q^{22} +367.911i q^{23} -237.888 q^{24} -676.000 q^{26} +1755.11i q^{27} +164.122i q^{28} +6602.16 q^{29} -9911.27 q^{31} -1024.00i q^{32} +733.267i q^{33} -3798.89 q^{34} -3666.94 q^{36} -9397.70i q^{37} +8933.22i q^{38} +628.173 q^{39} +20588.3 q^{41} -152.510i q^{42} +16341.8i q^{43} -3156.38 q^{44} +1471.64 q^{46} -13538.5i q^{47} +951.552i q^{48} +16701.8 q^{49} +3530.12 q^{51} +2704.00i q^{52} +35049.6i q^{53} +7020.43 q^{54} +656.488 q^{56} -8301.20i q^{57} -26408.6i q^{58} +6170.98 q^{59} +13746.1 q^{61} +39645.1i q^{62} -2350.88i q^{63} -4096.00 q^{64} +2933.07 q^{66} -62136.4i q^{67} +15195.6i q^{68} -1367.52 q^{69} -38355.6 q^{71} +14667.8i q^{72} -59967.3i q^{73} -37590.8 q^{74} +35732.9 q^{76} -2023.56i q^{77} -2512.69i q^{78} -89670.2 q^{79} +49167.9 q^{81} -82353.1i q^{82} +38134.8i q^{83} -610.042 q^{84} +65367.1 q^{86} +24540.2i q^{87} +12625.5i q^{88} +78278.2 q^{89} -1733.54 q^{91} -5886.57i q^{92} -36840.2i q^{93} -54154.1 q^{94} +3806.21 q^{96} +80567.1i q^{97} -66807.1i q^{98} +45212.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39} - 31812 q^{41} - 1536 q^{44} + 18960 q^{46} - 15534 q^{49} - 67944 q^{51} - 81344 q^{54} + 29952 q^{56} + 155520 q^{59} + 35964 q^{61} - 24576 q^{64} + 180576 q^{66} + 63720 q^{69} - 9888 q^{71} - 202896 q^{74} - 11520 q^{76} - 84240 q^{79} + 293038 q^{81} - 220928 q^{84} + 314448 q^{86} - 39228 q^{89} - 79092 q^{91} - 142224 q^{94} + 45056 q^{96} - 700680 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 3.71700i 0.238446i 0.992868 + 0.119223i \(0.0380403\pi\)
−0.992868 + 0.119223i \(0.961960\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 14.8680 0.168607
\(7\) − 10.2576i − 0.0791228i −0.999217 0.0395614i \(-0.987404\pi\)
0.999217 0.0395614i \(-0.0125961\pi\)
\(8\) 64.0000i 0.353553i
\(9\) 229.184 0.943144
\(10\) 0 0
\(11\) 197.274 0.491572 0.245786 0.969324i \(-0.420954\pi\)
0.245786 + 0.969324i \(0.420954\pi\)
\(12\) − 59.4720i − 0.119223i
\(13\) − 169.000i − 0.277350i
\(14\) −41.0305 −0.0559483
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 949.722i − 0.797029i −0.917162 0.398515i \(-0.869526\pi\)
0.917162 0.398515i \(-0.130474\pi\)
\(18\) − 916.736i − 0.666903i
\(19\) −2233.30 −1.41927 −0.709633 0.704571i \(-0.751139\pi\)
−0.709633 + 0.704571i \(0.751139\pi\)
\(20\) 0 0
\(21\) 38.1276 0.0188665
\(22\) − 789.095i − 0.347594i
\(23\) 367.911i 0.145018i 0.997368 + 0.0725092i \(0.0231007\pi\)
−0.997368 + 0.0725092i \(0.976899\pi\)
\(24\) −237.888 −0.0843033
\(25\) 0 0
\(26\) −676.000 −0.196116
\(27\) 1755.11i 0.463334i
\(28\) 164.122i 0.0395614i
\(29\) 6602.16 1.45778 0.728888 0.684633i \(-0.240037\pi\)
0.728888 + 0.684633i \(0.240037\pi\)
\(30\) 0 0
\(31\) −9911.27 −1.85236 −0.926179 0.377084i \(-0.876927\pi\)
−0.926179 + 0.377084i \(0.876927\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 733.267i 0.117213i
\(34\) −3798.89 −0.563585
\(35\) 0 0
\(36\) −3666.94 −0.471572
\(37\) − 9397.70i − 1.12854i −0.825590 0.564271i \(-0.809158\pi\)
0.825590 0.564271i \(-0.190842\pi\)
\(38\) 8933.22i 1.00357i
\(39\) 628.173 0.0661330
\(40\) 0 0
\(41\) 20588.3 1.91276 0.956380 0.292126i \(-0.0943626\pi\)
0.956380 + 0.292126i \(0.0943626\pi\)
\(42\) − 152.510i − 0.0133406i
\(43\) 16341.8i 1.34781i 0.738819 + 0.673904i \(0.235384\pi\)
−0.738819 + 0.673904i \(0.764616\pi\)
\(44\) −3156.38 −0.245786
\(45\) 0 0
\(46\) 1471.64 0.102543
\(47\) − 13538.5i − 0.893977i −0.894540 0.446989i \(-0.852496\pi\)
0.894540 0.446989i \(-0.147504\pi\)
\(48\) 951.552i 0.0596114i
\(49\) 16701.8 0.993740
\(50\) 0 0
\(51\) 3530.12 0.190048
\(52\) 2704.00i 0.138675i
\(53\) 35049.6i 1.71393i 0.515374 + 0.856965i \(0.327653\pi\)
−0.515374 + 0.856965i \(0.672347\pi\)
\(54\) 7020.43 0.327627
\(55\) 0 0
\(56\) 656.488 0.0279741
\(57\) − 8301.20i − 0.338418i
\(58\) − 26408.6i − 1.03080i
\(59\) 6170.98 0.230794 0.115397 0.993319i \(-0.463186\pi\)
0.115397 + 0.993319i \(0.463186\pi\)
\(60\) 0 0
\(61\) 13746.1 0.472993 0.236496 0.971632i \(-0.424001\pi\)
0.236496 + 0.971632i \(0.424001\pi\)
\(62\) 39645.1i 1.30981i
\(63\) − 2350.88i − 0.0746242i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 2933.07 0.0828824
\(67\) − 62136.4i − 1.69106i −0.533927 0.845530i \(-0.679284\pi\)
0.533927 0.845530i \(-0.320716\pi\)
\(68\) 15195.6i 0.398515i
\(69\) −1367.52 −0.0345790
\(70\) 0 0
\(71\) −38355.6 −0.902989 −0.451495 0.892274i \(-0.649109\pi\)
−0.451495 + 0.892274i \(0.649109\pi\)
\(72\) 14667.8i 0.333452i
\(73\) − 59967.3i − 1.31707i −0.752552 0.658533i \(-0.771177\pi\)
0.752552 0.658533i \(-0.228823\pi\)
\(74\) −37590.8 −0.797999
\(75\) 0 0
\(76\) 35732.9 0.709633
\(77\) − 2023.56i − 0.0388946i
\(78\) − 2512.69i − 0.0467631i
\(79\) −89670.2 −1.61652 −0.808259 0.588827i \(-0.799590\pi\)
−0.808259 + 0.588827i \(0.799590\pi\)
\(80\) 0 0
\(81\) 49167.9 0.832664
\(82\) − 82353.1i − 1.35253i
\(83\) 38134.8i 0.607611i 0.952734 + 0.303806i \(0.0982573\pi\)
−0.952734 + 0.303806i \(0.901743\pi\)
\(84\) −610.042 −0.00943325
\(85\) 0 0
\(86\) 65367.1 0.953044
\(87\) 24540.2i 0.347601i
\(88\) 12625.5i 0.173797i
\(89\) 78278.2 1.04753 0.523764 0.851863i \(-0.324527\pi\)
0.523764 + 0.851863i \(0.324527\pi\)
\(90\) 0 0
\(91\) −1733.54 −0.0219447
\(92\) − 5886.57i − 0.0725092i
\(93\) − 36840.2i − 0.441687i
\(94\) −54154.1 −0.632137
\(95\) 0 0
\(96\) 3806.21 0.0421517
\(97\) 80567.1i 0.869417i 0.900571 + 0.434709i \(0.143149\pi\)
−0.900571 + 0.434709i \(0.856851\pi\)
\(98\) − 66807.1i − 0.702680i
\(99\) 45212.0 0.463623
\(100\) 0 0
\(101\) −10770.8 −0.105062 −0.0525309 0.998619i \(-0.516729\pi\)
−0.0525309 + 0.998619i \(0.516729\pi\)
\(102\) − 14120.5i − 0.134384i
\(103\) − 44993.4i − 0.417884i −0.977928 0.208942i \(-0.932998\pi\)
0.977928 0.208942i \(-0.0670020\pi\)
\(104\) 10816.0 0.0980581
\(105\) 0 0
\(106\) 140198. 1.21193
\(107\) − 194227.i − 1.64003i −0.572345 0.820013i \(-0.693966\pi\)
0.572345 0.820013i \(-0.306034\pi\)
\(108\) − 28081.7i − 0.231667i
\(109\) −216396. −1.74455 −0.872275 0.489016i \(-0.837356\pi\)
−0.872275 + 0.489016i \(0.837356\pi\)
\(110\) 0 0
\(111\) 34931.3 0.269096
\(112\) − 2625.95i − 0.0197807i
\(113\) − 141318.i − 1.04112i −0.853824 0.520562i \(-0.825722\pi\)
0.853824 0.520562i \(-0.174278\pi\)
\(114\) −33204.8 −0.239298
\(115\) 0 0
\(116\) −105635. −0.728888
\(117\) − 38732.1i − 0.261581i
\(118\) − 24683.9i − 0.163196i
\(119\) −9741.89 −0.0630632
\(120\) 0 0
\(121\) −122134. −0.758356
\(122\) − 54984.3i − 0.334456i
\(123\) 76526.7i 0.456089i
\(124\) 158580. 0.926179
\(125\) 0 0
\(126\) −9403.53 −0.0527673
\(127\) − 188285.i − 1.03587i −0.855420 0.517935i \(-0.826701\pi\)
0.855420 0.517935i \(-0.173299\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) −60742.4 −0.321379
\(130\) 0 0
\(131\) 263916. 1.34365 0.671826 0.740709i \(-0.265510\pi\)
0.671826 + 0.740709i \(0.265510\pi\)
\(132\) − 11732.3i − 0.0586067i
\(133\) 22908.4i 0.112296i
\(134\) −248546. −1.19576
\(135\) 0 0
\(136\) 60782.2 0.281792
\(137\) − 337417.i − 1.53591i −0.640504 0.767955i \(-0.721275\pi\)
0.640504 0.767955i \(-0.278725\pi\)
\(138\) 5470.10i 0.0244510i
\(139\) 86203.3 0.378431 0.189216 0.981936i \(-0.439405\pi\)
0.189216 + 0.981936i \(0.439405\pi\)
\(140\) 0 0
\(141\) 50322.7 0.213165
\(142\) 153422.i 0.638510i
\(143\) − 33339.3i − 0.136338i
\(144\) 58671.1 0.235786
\(145\) 0 0
\(146\) −239869. −0.931306
\(147\) 62080.5i 0.236953i
\(148\) 150363.i 0.564271i
\(149\) 290191. 1.07082 0.535412 0.844591i \(-0.320156\pi\)
0.535412 + 0.844591i \(0.320156\pi\)
\(150\) 0 0
\(151\) 129889. 0.463585 0.231793 0.972765i \(-0.425541\pi\)
0.231793 + 0.972765i \(0.425541\pi\)
\(152\) − 142932.i − 0.501787i
\(153\) − 217661.i − 0.751713i
\(154\) −8094.24 −0.0275026
\(155\) 0 0
\(156\) −10050.8 −0.0330665
\(157\) − 46206.2i − 0.149607i −0.997198 0.0748033i \(-0.976167\pi\)
0.997198 0.0748033i \(-0.0238329\pi\)
\(158\) 358681.i 1.14305i
\(159\) −130279. −0.408679
\(160\) 0 0
\(161\) 3773.89 0.0114743
\(162\) − 196672.i − 0.588782i
\(163\) − 379076.i − 1.11753i −0.829328 0.558763i \(-0.811276\pi\)
0.829328 0.558763i \(-0.188724\pi\)
\(164\) −329412. −0.956380
\(165\) 0 0
\(166\) 152539. 0.429646
\(167\) − 569176.i − 1.57927i −0.613580 0.789633i \(-0.710271\pi\)
0.613580 0.789633i \(-0.289729\pi\)
\(168\) 2440.17i 0.00667031i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) −511838. −1.33857
\(172\) − 261468.i − 0.673904i
\(173\) 113968.i 0.289513i 0.989467 + 0.144757i \(0.0462399\pi\)
−0.989467 + 0.144757i \(0.953760\pi\)
\(174\) 98160.9 0.245791
\(175\) 0 0
\(176\) 50502.1 0.122893
\(177\) 22937.5i 0.0550318i
\(178\) − 313113.i − 0.740714i
\(179\) −320294. −0.747165 −0.373582 0.927597i \(-0.621871\pi\)
−0.373582 + 0.927597i \(0.621871\pi\)
\(180\) 0 0
\(181\) 451978. 1.02547 0.512733 0.858548i \(-0.328633\pi\)
0.512733 + 0.858548i \(0.328633\pi\)
\(182\) 6934.15i 0.0155173i
\(183\) 51094.2i 0.112783i
\(184\) −23546.3 −0.0512717
\(185\) 0 0
\(186\) −147361. −0.312320
\(187\) − 187355.i − 0.391798i
\(188\) 216616.i 0.446989i
\(189\) 18003.2 0.0366603
\(190\) 0 0
\(191\) 182918. 0.362805 0.181402 0.983409i \(-0.441936\pi\)
0.181402 + 0.983409i \(0.441936\pi\)
\(192\) − 15224.8i − 0.0298057i
\(193\) − 508696.i − 0.983026i −0.870870 0.491513i \(-0.836444\pi\)
0.870870 0.491513i \(-0.163556\pi\)
\(194\) 322268. 0.614771
\(195\) 0 0
\(196\) −267228. −0.496870
\(197\) 111666.i 0.205000i 0.994733 + 0.102500i \(0.0326842\pi\)
−0.994733 + 0.102500i \(0.967316\pi\)
\(198\) − 180848.i − 0.327831i
\(199\) −521218. −0.933011 −0.466505 0.884518i \(-0.654487\pi\)
−0.466505 + 0.884518i \(0.654487\pi\)
\(200\) 0 0
\(201\) 230961. 0.403226
\(202\) 43083.2i 0.0742899i
\(203\) − 67722.4i − 0.115343i
\(204\) −56481.9 −0.0950241
\(205\) 0 0
\(206\) −179974. −0.295489
\(207\) 84319.2i 0.136773i
\(208\) − 43264.0i − 0.0693375i
\(209\) −440572. −0.697672
\(210\) 0 0
\(211\) −543969. −0.841140 −0.420570 0.907260i \(-0.638170\pi\)
−0.420570 + 0.907260i \(0.638170\pi\)
\(212\) − 560793.i − 0.856965i
\(213\) − 142568.i − 0.215314i
\(214\) −776909. −1.15967
\(215\) 0 0
\(216\) −112327. −0.163813
\(217\) 101666.i 0.146564i
\(218\) 865585.i 1.23358i
\(219\) 222899. 0.314049
\(220\) 0 0
\(221\) −160503. −0.221056
\(222\) − 139725.i − 0.190279i
\(223\) − 114372.i − 0.154013i −0.997031 0.0770063i \(-0.975464\pi\)
0.997031 0.0770063i \(-0.0245362\pi\)
\(224\) −10503.8 −0.0139871
\(225\) 0 0
\(226\) −565274. −0.736186
\(227\) − 213250.i − 0.274678i −0.990524 0.137339i \(-0.956145\pi\)
0.990524 0.137339i \(-0.0438550\pi\)
\(228\) 132819.i 0.169209i
\(229\) −1.18502e6 −1.49326 −0.746631 0.665239i \(-0.768330\pi\)
−0.746631 + 0.665239i \(0.768330\pi\)
\(230\) 0 0
\(231\) 7521.58 0.00927425
\(232\) 422538.i 0.515402i
\(233\) − 1.37128e6i − 1.65476i −0.561641 0.827381i \(-0.689830\pi\)
0.561641 0.827381i \(-0.310170\pi\)
\(234\) −154928. −0.184966
\(235\) 0 0
\(236\) −98735.7 −0.115397
\(237\) − 333304.i − 0.385452i
\(238\) 38967.6i 0.0445924i
\(239\) 1.09024e6 1.23460 0.617301 0.786727i \(-0.288226\pi\)
0.617301 + 0.786727i \(0.288226\pi\)
\(240\) 0 0
\(241\) 84897.3 0.0941567 0.0470784 0.998891i \(-0.485009\pi\)
0.0470784 + 0.998891i \(0.485009\pi\)
\(242\) 488536.i 0.536239i
\(243\) 609249.i 0.661879i
\(244\) −219937. −0.236496
\(245\) 0 0
\(246\) 306107. 0.322504
\(247\) 377429.i 0.393634i
\(248\) − 634321.i − 0.654907i
\(249\) −141747. −0.144882
\(250\) 0 0
\(251\) 425949. 0.426750 0.213375 0.976970i \(-0.431554\pi\)
0.213375 + 0.976970i \(0.431554\pi\)
\(252\) 37614.1i 0.0373121i
\(253\) 72579.1i 0.0712870i
\(254\) −753139. −0.732471
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 250941.i 0.236995i 0.992954 + 0.118497i \(0.0378077\pi\)
−0.992954 + 0.118497i \(0.962192\pi\)
\(258\) 242969.i 0.227249i
\(259\) −96398.1 −0.0892933
\(260\) 0 0
\(261\) 1.51311e6 1.37489
\(262\) − 1.05566e6i − 0.950106i
\(263\) 1.48299e6i 1.32205i 0.750363 + 0.661026i \(0.229879\pi\)
−0.750363 + 0.661026i \(0.770121\pi\)
\(264\) −46929.1 −0.0414412
\(265\) 0 0
\(266\) 91633.6 0.0794055
\(267\) 290960.i 0.249779i
\(268\) 994183.i 0.845530i
\(269\) −398773. −0.336004 −0.168002 0.985787i \(-0.553732\pi\)
−0.168002 + 0.985787i \(0.553732\pi\)
\(270\) 0 0
\(271\) 182496. 0.150949 0.0754743 0.997148i \(-0.475953\pi\)
0.0754743 + 0.997148i \(0.475953\pi\)
\(272\) − 243129.i − 0.199257i
\(273\) − 6443.57i − 0.00523262i
\(274\) −1.34967e6 −1.08605
\(275\) 0 0
\(276\) 21880.4 0.0172895
\(277\) − 79202.9i − 0.0620215i −0.999519 0.0310107i \(-0.990127\pi\)
0.999519 0.0310107i \(-0.00987260\pi\)
\(278\) − 344813.i − 0.267591i
\(279\) −2.27150e6 −1.74704
\(280\) 0 0
\(281\) −1.51199e6 −1.14231 −0.571154 0.820843i \(-0.693504\pi\)
−0.571154 + 0.820843i \(0.693504\pi\)
\(282\) − 201291.i − 0.150730i
\(283\) − 2.50781e6i − 1.86135i −0.365848 0.930675i \(-0.619221\pi\)
0.365848 0.930675i \(-0.380779\pi\)
\(284\) 613689. 0.451495
\(285\) 0 0
\(286\) −133357. −0.0964053
\(287\) − 211187.i − 0.151343i
\(288\) − 234684.i − 0.166726i
\(289\) 517885. 0.364744
\(290\) 0 0
\(291\) −299468. −0.207309
\(292\) 959477.i 0.658533i
\(293\) − 1.75384e6i − 1.19349i −0.802429 0.596747i \(-0.796459\pi\)
0.802429 0.596747i \(-0.203541\pi\)
\(294\) 248322. 0.167551
\(295\) 0 0
\(296\) 601453. 0.399000
\(297\) 346237.i 0.227762i
\(298\) − 1.16076e6i − 0.757187i
\(299\) 62176.9 0.0402208
\(300\) 0 0
\(301\) 167628. 0.106642
\(302\) − 519556.i − 0.327804i
\(303\) − 40035.1i − 0.0250515i
\(304\) −571726. −0.354817
\(305\) 0 0
\(306\) −870644. −0.531541
\(307\) − 1.55298e6i − 0.940414i −0.882556 0.470207i \(-0.844179\pi\)
0.882556 0.470207i \(-0.155821\pi\)
\(308\) 32377.0i 0.0194473i
\(309\) 167240. 0.0996426
\(310\) 0 0
\(311\) −188125. −0.110292 −0.0551462 0.998478i \(-0.517562\pi\)
−0.0551462 + 0.998478i \(0.517562\pi\)
\(312\) 40203.1i 0.0233815i
\(313\) − 846464.i − 0.488369i −0.969729 0.244184i \(-0.921480\pi\)
0.969729 0.244184i \(-0.0785202\pi\)
\(314\) −184825. −0.105788
\(315\) 0 0
\(316\) 1.43472e6 0.808259
\(317\) − 303332.i − 0.169539i −0.996401 0.0847696i \(-0.972985\pi\)
0.996401 0.0847696i \(-0.0270154\pi\)
\(318\) 521117.i 0.288980i
\(319\) 1.30243e6 0.716603
\(320\) 0 0
\(321\) 721943. 0.391057
\(322\) − 15095.6i − 0.00811352i
\(323\) 2.12102e6i 1.13120i
\(324\) −786687. −0.416332
\(325\) 0 0
\(326\) −1.51630e6 −0.790210
\(327\) − 804345.i − 0.415980i
\(328\) 1.31765e6i 0.676263i
\(329\) −138873. −0.0707340
\(330\) 0 0
\(331\) −3.36224e6 −1.68678 −0.843390 0.537301i \(-0.819444\pi\)
−0.843390 + 0.537301i \(0.819444\pi\)
\(332\) − 610156.i − 0.303806i
\(333\) − 2.15380e6i − 1.06438i
\(334\) −2.27670e6 −1.11671
\(335\) 0 0
\(336\) 9760.67 0.00471662
\(337\) 2.42001e6i 1.16076i 0.814346 + 0.580380i \(0.197096\pi\)
−0.814346 + 0.580380i \(0.802904\pi\)
\(338\) 114244.i 0.0543928i
\(339\) 525281. 0.248252
\(340\) 0 0
\(341\) −1.95523e6 −0.910568
\(342\) 2.04735e6i 0.946514i
\(343\) − 343720.i − 0.157750i
\(344\) −1.04587e6 −0.476522
\(345\) 0 0
\(346\) 455872. 0.204717
\(347\) − 208658.i − 0.0930274i −0.998918 0.0465137i \(-0.985189\pi\)
0.998918 0.0465137i \(-0.0148111\pi\)
\(348\) − 392644.i − 0.173800i
\(349\) 2.56139e6 1.12567 0.562837 0.826568i \(-0.309710\pi\)
0.562837 + 0.826568i \(0.309710\pi\)
\(350\) 0 0
\(351\) 296613. 0.128506
\(352\) − 202008.i − 0.0868986i
\(353\) 2.22758e6i 0.951472i 0.879588 + 0.475736i \(0.157818\pi\)
−0.879588 + 0.475736i \(0.842182\pi\)
\(354\) 91750.2 0.0389134
\(355\) 0 0
\(356\) −1.25245e6 −0.523764
\(357\) − 36210.6i − 0.0150372i
\(358\) 1.28118e6i 0.528325i
\(359\) 4.69711e6 1.92351 0.961756 0.273908i \(-0.0883164\pi\)
0.961756 + 0.273908i \(0.0883164\pi\)
\(360\) 0 0
\(361\) 2.51155e6 1.01432
\(362\) − 1.80791e6i − 0.725113i
\(363\) − 453973.i − 0.180827i
\(364\) 27736.6 0.0109724
\(365\) 0 0
\(366\) 204377. 0.0797497
\(367\) − 3.22496e6i − 1.24985i −0.780684 0.624926i \(-0.785129\pi\)
0.780684 0.624926i \(-0.214871\pi\)
\(368\) 94185.1i 0.0362546i
\(369\) 4.71850e6 1.80401
\(370\) 0 0
\(371\) 359525. 0.135611
\(372\) 589443.i 0.220843i
\(373\) 3.09204e6i 1.15073i 0.817898 + 0.575364i \(0.195140\pi\)
−0.817898 + 0.575364i \(0.804860\pi\)
\(374\) −749421. −0.277043
\(375\) 0 0
\(376\) 866465. 0.316069
\(377\) − 1.11576e6i − 0.404314i
\(378\) − 72013.0i − 0.0259228i
\(379\) 3.63851e6 1.30115 0.650573 0.759444i \(-0.274529\pi\)
0.650573 + 0.759444i \(0.274529\pi\)
\(380\) 0 0
\(381\) 699854. 0.246999
\(382\) − 731672.i − 0.256542i
\(383\) − 1.89242e6i − 0.659204i −0.944120 0.329602i \(-0.893085\pi\)
0.944120 0.329602i \(-0.106915\pi\)
\(384\) −60899.4 −0.0210758
\(385\) 0 0
\(386\) −2.03478e6 −0.695104
\(387\) 3.74527e6i 1.27118i
\(388\) − 1.28907e6i − 0.434709i
\(389\) −79129.5 −0.0265133 −0.0132567 0.999912i \(-0.504220\pi\)
−0.0132567 + 0.999912i \(0.504220\pi\)
\(390\) 0 0
\(391\) 349413. 0.115584
\(392\) 1.06891e6i 0.351340i
\(393\) 980975.i 0.320388i
\(394\) 446663. 0.144957
\(395\) 0 0
\(396\) −723391. −0.231812
\(397\) − 1.82235e6i − 0.580305i −0.956980 0.290153i \(-0.906294\pi\)
0.956980 0.290153i \(-0.0937061\pi\)
\(398\) 2.08487e6i 0.659738i
\(399\) −85150.6 −0.0267766
\(400\) 0 0
\(401\) −117774. −0.0365754 −0.0182877 0.999833i \(-0.505821\pi\)
−0.0182877 + 0.999833i \(0.505821\pi\)
\(402\) − 923845.i − 0.285124i
\(403\) 1.67500e6i 0.513752i
\(404\) 172333. 0.0525309
\(405\) 0 0
\(406\) −270890. −0.0815601
\(407\) − 1.85392e6i − 0.554760i
\(408\) 225928.i 0.0671922i
\(409\) −3.73101e6 −1.10285 −0.551427 0.834223i \(-0.685917\pi\)
−0.551427 + 0.834223i \(0.685917\pi\)
\(410\) 0 0
\(411\) 1.25418e6 0.366231
\(412\) 719894.i 0.208942i
\(413\) − 63299.6i − 0.0182610i
\(414\) 337277. 0.0967132
\(415\) 0 0
\(416\) −173056. −0.0490290
\(417\) 320418.i 0.0902353i
\(418\) 1.76229e6i 0.493329i
\(419\) 2.69321e6 0.749438 0.374719 0.927138i \(-0.377739\pi\)
0.374719 + 0.927138i \(0.377739\pi\)
\(420\) 0 0
\(421\) 5.56789e6 1.53103 0.765517 0.643415i \(-0.222483\pi\)
0.765517 + 0.643415i \(0.222483\pi\)
\(422\) 2.17588e6i 0.594776i
\(423\) − 3.10281e6i − 0.843149i
\(424\) −2.24317e6 −0.605966
\(425\) 0 0
\(426\) −570271. −0.152250
\(427\) − 141002.i − 0.0374245i
\(428\) 3.10763e6i 0.820013i
\(429\) 123922. 0.0325091
\(430\) 0 0
\(431\) −3.75659e6 −0.974093 −0.487047 0.873376i \(-0.661926\pi\)
−0.487047 + 0.873376i \(0.661926\pi\)
\(432\) 449308.i 0.115834i
\(433\) − 1.60978e6i − 0.412615i −0.978487 0.206308i \(-0.933855\pi\)
0.978487 0.206308i \(-0.0661448\pi\)
\(434\) 406664. 0.103636
\(435\) 0 0
\(436\) 3.46234e6 0.872275
\(437\) − 821657.i − 0.205820i
\(438\) − 891595.i − 0.222066i
\(439\) 1.92307e6 0.476248 0.238124 0.971235i \(-0.423468\pi\)
0.238124 + 0.971235i \(0.423468\pi\)
\(440\) 0 0
\(441\) 3.82778e6 0.937239
\(442\) 642012.i 0.156310i
\(443\) 1.43060e6i 0.346346i 0.984891 + 0.173173i \(0.0554019\pi\)
−0.984891 + 0.173173i \(0.944598\pi\)
\(444\) −558900. −0.134548
\(445\) 0 0
\(446\) −457487. −0.108903
\(447\) 1.07864e6i 0.255333i
\(448\) 42015.2i 0.00989035i
\(449\) 3.30270e6 0.773131 0.386565 0.922262i \(-0.373661\pi\)
0.386565 + 0.922262i \(0.373661\pi\)
\(450\) 0 0
\(451\) 4.06153e6 0.940260
\(452\) 2.26110e6i 0.520562i
\(453\) 482797.i 0.110540i
\(454\) −853000. −0.194227
\(455\) 0 0
\(456\) 531277. 0.119649
\(457\) 6.85658e6i 1.53574i 0.640606 + 0.767869i \(0.278683\pi\)
−0.640606 + 0.767869i \(0.721317\pi\)
\(458\) 4.74007e6i 1.05590i
\(459\) 1.66687e6 0.369291
\(460\) 0 0
\(461\) 8.33435e6 1.82650 0.913250 0.407400i \(-0.133565\pi\)
0.913250 + 0.407400i \(0.133565\pi\)
\(462\) − 30086.3i − 0.00655789i
\(463\) 2.87024e6i 0.622250i 0.950369 + 0.311125i \(0.100706\pi\)
−0.950369 + 0.311125i \(0.899294\pi\)
\(464\) 1.69015e6 0.364444
\(465\) 0 0
\(466\) −5.48511e6 −1.17009
\(467\) 6.61814e6i 1.40425i 0.712055 + 0.702124i \(0.247765\pi\)
−0.712055 + 0.702124i \(0.752235\pi\)
\(468\) 619713.i 0.130790i
\(469\) −637372. −0.133801
\(470\) 0 0
\(471\) 171748. 0.0356731
\(472\) 394943.i 0.0815979i
\(473\) 3.22380e6i 0.662545i
\(474\) −1.33322e6 −0.272556
\(475\) 0 0
\(476\) 155870. 0.0315316
\(477\) 8.03280e6i 1.61648i
\(478\) − 4.36096e6i − 0.872995i
\(479\) 1.47435e6 0.293603 0.146802 0.989166i \(-0.453102\pi\)
0.146802 + 0.989166i \(0.453102\pi\)
\(480\) 0 0
\(481\) −1.58821e6 −0.313001
\(482\) − 339589.i − 0.0665789i
\(483\) 14027.6i 0.00273599i
\(484\) 1.95415e6 0.379178
\(485\) 0 0
\(486\) 2.43699e6 0.468019
\(487\) 2.63151e6i 0.502786i 0.967885 + 0.251393i \(0.0808886\pi\)
−0.967885 + 0.251393i \(0.919111\pi\)
\(488\) 879749.i 0.167228i
\(489\) 1.40903e6 0.266469
\(490\) 0 0
\(491\) 4.70461e6 0.880684 0.440342 0.897830i \(-0.354857\pi\)
0.440342 + 0.897830i \(0.354857\pi\)
\(492\) − 1.22443e6i − 0.228045i
\(493\) − 6.27022e6i − 1.16189i
\(494\) 1.50971e6 0.278341
\(495\) 0 0
\(496\) −2.53728e6 −0.463089
\(497\) 393437.i 0.0714471i
\(498\) 566988.i 0.102447i
\(499\) −409148. −0.0735578 −0.0367789 0.999323i \(-0.511710\pi\)
−0.0367789 + 0.999323i \(0.511710\pi\)
\(500\) 0 0
\(501\) 2.11563e6 0.376569
\(502\) − 1.70380e6i − 0.301758i
\(503\) − 9.62836e6i − 1.69681i −0.529350 0.848404i \(-0.677564\pi\)
0.529350 0.848404i \(-0.322436\pi\)
\(504\) 150456. 0.0263836
\(505\) 0 0
\(506\) 290317. 0.0504075
\(507\) − 106161.i − 0.0183420i
\(508\) 3.01255e6i 0.517935i
\(509\) 1.24824e6 0.213552 0.106776 0.994283i \(-0.465947\pi\)
0.106776 + 0.994283i \(0.465947\pi\)
\(510\) 0 0
\(511\) −615122. −0.104210
\(512\) − 262144.i − 0.0441942i
\(513\) − 3.91969e6i − 0.657595i
\(514\) 1.00376e6 0.167581
\(515\) 0 0
\(516\) 971878. 0.160689
\(517\) − 2.67079e6i − 0.439455i
\(518\) 385592.i 0.0631399i
\(519\) −423620. −0.0690332
\(520\) 0 0
\(521\) −7.13050e6 −1.15087 −0.575434 0.817848i \(-0.695167\pi\)
−0.575434 + 0.817848i \(0.695167\pi\)
\(522\) − 6.05243e6i − 0.972196i
\(523\) − 4.85487e6i − 0.776110i −0.921636 0.388055i \(-0.873147\pi\)
0.921636 0.388055i \(-0.126853\pi\)
\(524\) −4.22265e6 −0.671826
\(525\) 0 0
\(526\) 5.93196e6 0.934833
\(527\) 9.41295e6i 1.47638i
\(528\) 187716.i 0.0293033i
\(529\) 6.30098e6 0.978970
\(530\) 0 0
\(531\) 1.41429e6 0.217672
\(532\) − 366534.i − 0.0561482i
\(533\) − 3.47942e6i − 0.530504i
\(534\) 1.16384e6 0.176620
\(535\) 0 0
\(536\) 3.97673e6 0.597880
\(537\) − 1.19053e6i − 0.178158i
\(538\) 1.59509e6i 0.237591i
\(539\) 3.29482e6 0.488495
\(540\) 0 0
\(541\) −1.13451e7 −1.66653 −0.833266 0.552872i \(-0.813532\pi\)
−0.833266 + 0.552872i \(0.813532\pi\)
\(542\) − 729983.i − 0.106737i
\(543\) 1.68000e6i 0.244518i
\(544\) −972516. −0.140896
\(545\) 0 0
\(546\) −25774.3 −0.00370002
\(547\) 401928.i 0.0574355i 0.999588 + 0.0287177i \(0.00914240\pi\)
−0.999588 + 0.0287177i \(0.990858\pi\)
\(548\) 5.39867e6i 0.767955i
\(549\) 3.15038e6 0.446100
\(550\) 0 0
\(551\) −1.47446e7 −2.06897
\(552\) − 87521.6i − 0.0122255i
\(553\) 919803.i 0.127903i
\(554\) −316812. −0.0438558
\(555\) 0 0
\(556\) −1.37925e6 −0.189216
\(557\) 7.58501e6i 1.03590i 0.855411 + 0.517950i \(0.173305\pi\)
−0.855411 + 0.517950i \(0.826695\pi\)
\(558\) 9.08601e6i 1.23534i
\(559\) 2.76176e6 0.373814
\(560\) 0 0
\(561\) 696400. 0.0934225
\(562\) 6.04796e6i 0.807733i
\(563\) 9.36555e6i 1.24527i 0.782514 + 0.622633i \(0.213937\pi\)
−0.782514 + 0.622633i \(0.786063\pi\)
\(564\) −805163. −0.106583
\(565\) 0 0
\(566\) −1.00312e7 −1.31617
\(567\) − 504346.i − 0.0658827i
\(568\) − 2.45476e6i − 0.319255i
\(569\) −1.36255e7 −1.76430 −0.882150 0.470969i \(-0.843904\pi\)
−0.882150 + 0.470969i \(0.843904\pi\)
\(570\) 0 0
\(571\) 1.29016e7 1.65598 0.827988 0.560746i \(-0.189485\pi\)
0.827988 + 0.560746i \(0.189485\pi\)
\(572\) 533428.i 0.0681688i
\(573\) 679907.i 0.0865093i
\(574\) −844747. −0.107016
\(575\) 0 0
\(576\) −938737. −0.117893
\(577\) 3.12690e6i 0.390998i 0.980704 + 0.195499i \(0.0626327\pi\)
−0.980704 + 0.195499i \(0.937367\pi\)
\(578\) − 2.07154e6i − 0.257913i
\(579\) 1.89082e6 0.234398
\(580\) 0 0
\(581\) 391172. 0.0480759
\(582\) 1.19787e6i 0.146590i
\(583\) 6.91436e6i 0.842521i
\(584\) 3.83791e6 0.465653
\(585\) 0 0
\(586\) −7.01535e6 −0.843928
\(587\) − 1.61598e7i − 1.93571i −0.251511 0.967854i \(-0.580928\pi\)
0.251511 0.967854i \(-0.419072\pi\)
\(588\) − 993289.i − 0.118476i
\(589\) 2.21349e7 2.62899
\(590\) 0 0
\(591\) −415062. −0.0488814
\(592\) − 2.40581e6i − 0.282135i
\(593\) 5.02852e6i 0.587223i 0.955925 + 0.293612i \(0.0948572\pi\)
−0.955925 + 0.293612i \(0.905143\pi\)
\(594\) 1.38495e6 0.161052
\(595\) 0 0
\(596\) −4.64305e6 −0.535412
\(597\) − 1.93737e6i − 0.222472i
\(598\) − 248708.i − 0.0284404i
\(599\) −6.83761e6 −0.778642 −0.389321 0.921102i \(-0.627290\pi\)
−0.389321 + 0.921102i \(0.627290\pi\)
\(600\) 0 0
\(601\) −8.01912e6 −0.905608 −0.452804 0.891610i \(-0.649576\pi\)
−0.452804 + 0.891610i \(0.649576\pi\)
\(602\) − 670511.i − 0.0754075i
\(603\) − 1.42407e7i − 1.59491i
\(604\) −2.07822e6 −0.231793
\(605\) 0 0
\(606\) −160140. −0.0177141
\(607\) 7.83186e6i 0.862766i 0.902169 + 0.431383i \(0.141974\pi\)
−0.902169 + 0.431383i \(0.858026\pi\)
\(608\) 2.28690e6i 0.250893i
\(609\) 251724. 0.0275031
\(610\) 0 0
\(611\) −2.28801e6 −0.247945
\(612\) 3.48258e6i 0.375857i
\(613\) 1.57922e7i 1.69743i 0.528849 + 0.848716i \(0.322624\pi\)
−0.528849 + 0.848716i \(0.677376\pi\)
\(614\) −6.21191e6 −0.664973
\(615\) 0 0
\(616\) 129508. 0.0137513
\(617\) 8.38747e6i 0.886989i 0.896277 + 0.443494i \(0.146261\pi\)
−0.896277 + 0.443494i \(0.853739\pi\)
\(618\) − 668962.i − 0.0704580i
\(619\) −1.35269e6 −0.141896 −0.0709481 0.997480i \(-0.522602\pi\)
−0.0709481 + 0.997480i \(0.522602\pi\)
\(620\) 0 0
\(621\) −645723. −0.0671920
\(622\) 752500.i 0.0779884i
\(623\) − 802948.i − 0.0828834i
\(624\) 160812. 0.0165332
\(625\) 0 0
\(626\) −3.38586e6 −0.345329
\(627\) − 1.63761e6i − 0.166357i
\(628\) 739299.i 0.0748033i
\(629\) −8.92521e6 −0.899480
\(630\) 0 0
\(631\) −7.20270e6 −0.720149 −0.360074 0.932924i \(-0.617249\pi\)
−0.360074 + 0.932924i \(0.617249\pi\)
\(632\) − 5.73889e6i − 0.571525i
\(633\) − 2.02193e6i − 0.200566i
\(634\) −1.21333e6 −0.119882
\(635\) 0 0
\(636\) 2.08447e6 0.204340
\(637\) − 2.82260e6i − 0.275614i
\(638\) − 5.20973e6i − 0.506715i
\(639\) −8.79048e6 −0.851649
\(640\) 0 0
\(641\) 2.63844e6 0.253631 0.126815 0.991926i \(-0.459524\pi\)
0.126815 + 0.991926i \(0.459524\pi\)
\(642\) − 2.88777e6i − 0.276519i
\(643\) − 6.76792e6i − 0.645547i −0.946476 0.322774i \(-0.895385\pi\)
0.946476 0.322774i \(-0.104615\pi\)
\(644\) −60382.2 −0.00573713
\(645\) 0 0
\(646\) 8.48408e6 0.799877
\(647\) 1.01251e7i 0.950907i 0.879741 + 0.475453i \(0.157716\pi\)
−0.879741 + 0.475453i \(0.842284\pi\)
\(648\) 3.14675e6i 0.294391i
\(649\) 1.21737e6 0.113452
\(650\) 0 0
\(651\) −377893. −0.0349475
\(652\) 6.06522e6i 0.558763i
\(653\) 8.17558e6i 0.750301i 0.926964 + 0.375150i \(0.122409\pi\)
−0.926964 + 0.375150i \(0.877591\pi\)
\(654\) −3.21738e6 −0.294143
\(655\) 0 0
\(656\) 5.27060e6 0.478190
\(657\) − 1.37435e7i − 1.24218i
\(658\) 555492.i 0.0500165i
\(659\) 1.24909e7 1.12042 0.560211 0.828350i \(-0.310720\pi\)
0.560211 + 0.828350i \(0.310720\pi\)
\(660\) 0 0
\(661\) 5.43688e6 0.484001 0.242001 0.970276i \(-0.422196\pi\)
0.242001 + 0.970276i \(0.422196\pi\)
\(662\) 1.34490e7i 1.19273i
\(663\) − 596590.i − 0.0527099i
\(664\) −2.44062e6 −0.214823
\(665\) 0 0
\(666\) −8.61521e6 −0.752628
\(667\) 2.42900e6i 0.211404i
\(668\) 9.10681e6i 0.789633i
\(669\) 425120. 0.0367236
\(670\) 0 0
\(671\) 2.71174e6 0.232510
\(672\) − 39042.7i − 0.00333516i
\(673\) − 9.60528e6i − 0.817471i −0.912653 0.408736i \(-0.865970\pi\)
0.912653 0.408736i \(-0.134030\pi\)
\(674\) 9.68004e6 0.820782
\(675\) 0 0
\(676\) 456976. 0.0384615
\(677\) 5.99107e6i 0.502380i 0.967938 + 0.251190i \(0.0808220\pi\)
−0.967938 + 0.251190i \(0.919178\pi\)
\(678\) − 2.10112e6i − 0.175541i
\(679\) 826427. 0.0687907
\(680\) 0 0
\(681\) 792650. 0.0654958
\(682\) 7.82093e6i 0.643869i
\(683\) − 1.49302e7i − 1.22465i −0.790605 0.612327i \(-0.790234\pi\)
0.790605 0.612327i \(-0.209766\pi\)
\(684\) 8.18940e6 0.669286
\(685\) 0 0
\(686\) −1.37488e6 −0.111546
\(687\) − 4.40471e6i − 0.356062i
\(688\) 4.18349e6i 0.336952i
\(689\) 5.92338e6 0.475359
\(690\) 0 0
\(691\) −4.19350e6 −0.334104 −0.167052 0.985948i \(-0.553425\pi\)
−0.167052 + 0.985948i \(0.553425\pi\)
\(692\) − 1.82349e6i − 0.144757i
\(693\) − 463767.i − 0.0366832i
\(694\) −834631. −0.0657803
\(695\) 0 0
\(696\) −1.57057e6 −0.122895
\(697\) − 1.95531e7i − 1.52453i
\(698\) − 1.02456e7i − 0.795972i
\(699\) 5.09704e6 0.394571
\(700\) 0 0
\(701\) −1.88086e7 −1.44564 −0.722820 0.691036i \(-0.757155\pi\)
−0.722820 + 0.691036i \(0.757155\pi\)
\(702\) − 1.18645e6i − 0.0908673i
\(703\) 2.09879e7i 1.60170i
\(704\) −808033. −0.0614466
\(705\) 0 0
\(706\) 8.91031e6 0.672792
\(707\) 110483.i 0.00831278i
\(708\) − 367001.i − 0.0275159i
\(709\) −1.59830e7 −1.19411 −0.597054 0.802201i \(-0.703662\pi\)
−0.597054 + 0.802201i \(0.703662\pi\)
\(710\) 0 0
\(711\) −2.05510e7 −1.52461
\(712\) 5.00980e6i 0.370357i
\(713\) − 3.64646e6i − 0.268626i
\(714\) −144843. −0.0106329
\(715\) 0 0
\(716\) 5.12471e6 0.373582
\(717\) 4.05242e6i 0.294386i
\(718\) − 1.87885e7i − 1.36013i
\(719\) −1.01853e7 −0.734771 −0.367386 0.930069i \(-0.619747\pi\)
−0.367386 + 0.930069i \(0.619747\pi\)
\(720\) 0 0
\(721\) −461525. −0.0330641
\(722\) − 1.00462e7i − 0.717231i
\(723\) 315564.i 0.0224513i
\(724\) −7.23165e6 −0.512733
\(725\) 0 0
\(726\) −1.81589e6 −0.127864
\(727\) − 4.80166e6i − 0.336942i −0.985707 0.168471i \(-0.946117\pi\)
0.985707 0.168471i \(-0.0538829\pi\)
\(728\) − 110946.i − 0.00775863i
\(729\) 9.68323e6 0.674841
\(730\) 0 0
\(731\) 1.55201e7 1.07424
\(732\) − 817507.i − 0.0563915i
\(733\) − 4.57724e6i − 0.314662i −0.987546 0.157331i \(-0.949711\pi\)
0.987546 0.157331i \(-0.0502889\pi\)
\(734\) −1.28998e7 −0.883779
\(735\) 0 0
\(736\) 376741. 0.0256359
\(737\) − 1.22579e7i − 0.831279i
\(738\) − 1.88740e7i − 1.27563i
\(739\) −2.54398e7 −1.71358 −0.856788 0.515669i \(-0.827543\pi\)
−0.856788 + 0.515669i \(0.827543\pi\)
\(740\) 0 0
\(741\) −1.40290e6 −0.0938603
\(742\) − 1.43810e6i − 0.0958914i
\(743\) 1.06776e7i 0.709583i 0.934946 + 0.354791i \(0.115448\pi\)
−0.934946 + 0.354791i \(0.884552\pi\)
\(744\) 2.35777e6 0.156160
\(745\) 0 0
\(746\) 1.23681e7 0.813687
\(747\) 8.73987e6i 0.573065i
\(748\) 2.99768e6i 0.195899i
\(749\) −1.99231e6 −0.129763
\(750\) 0 0
\(751\) 1.24759e7 0.807184 0.403592 0.914939i \(-0.367762\pi\)
0.403592 + 0.914939i \(0.367762\pi\)
\(752\) − 3.46586e6i − 0.223494i
\(753\) 1.58325e6i 0.101757i
\(754\) −4.46306e6 −0.285893
\(755\) 0 0
\(756\) −288052. −0.0183302
\(757\) 1.89987e6i 0.120499i 0.998183 + 0.0602495i \(0.0191896\pi\)
−0.998183 + 0.0602495i \(0.980810\pi\)
\(758\) − 1.45541e7i − 0.920049i
\(759\) −269777. −0.0169981
\(760\) 0 0
\(761\) 1.16295e7 0.727949 0.363975 0.931409i \(-0.381420\pi\)
0.363975 + 0.931409i \(0.381420\pi\)
\(762\) − 2.79942e6i − 0.174655i
\(763\) 2.21971e6i 0.138034i
\(764\) −2.92669e6 −0.181402
\(765\) 0 0
\(766\) −7.56967e6 −0.466128
\(767\) − 1.04290e6i − 0.0640107i
\(768\) 243597.i 0.0149029i
\(769\) 2.05493e7 1.25309 0.626544 0.779386i \(-0.284469\pi\)
0.626544 + 0.779386i \(0.284469\pi\)
\(770\) 0 0
\(771\) −932748. −0.0565104
\(772\) 8.13913e6i 0.491513i
\(773\) 5.50647e6i 0.331455i 0.986172 + 0.165727i \(0.0529972\pi\)
−0.986172 + 0.165727i \(0.947003\pi\)
\(774\) 1.49811e7 0.898857
\(775\) 0 0
\(776\) −5.15629e6 −0.307385
\(777\) − 358312.i − 0.0212916i
\(778\) 316518.i 0.0187478i
\(779\) −4.59799e7 −2.71472
\(780\) 0 0
\(781\) −7.56655e6 −0.443885
\(782\) − 1.39765e6i − 0.0817301i
\(783\) 1.15875e7i 0.675438i
\(784\) 4.27566e6 0.248435
\(785\) 0 0
\(786\) 3.92390e6 0.226549
\(787\) 1.66836e7i 0.960182i 0.877219 + 0.480091i \(0.159396\pi\)
−0.877219 + 0.480091i \(0.840604\pi\)
\(788\) − 1.78665e6i − 0.102500i
\(789\) −5.51228e6 −0.315238
\(790\) 0 0
\(791\) −1.44959e6 −0.0823767
\(792\) 2.89357e6i 0.163916i
\(793\) − 2.32309e6i − 0.131185i
\(794\) −7.28942e6 −0.410338
\(795\) 0 0
\(796\) 8.33949e6 0.466505
\(797\) − 1.53978e7i − 0.858643i −0.903152 0.429321i \(-0.858753\pi\)
0.903152 0.429321i \(-0.141247\pi\)
\(798\) 340602.i 0.0189339i
\(799\) −1.28578e7 −0.712526
\(800\) 0 0
\(801\) 1.79401e7 0.987969
\(802\) 471097.i 0.0258627i
\(803\) − 1.18300e7i − 0.647434i
\(804\) −3.69538e6 −0.201613
\(805\) 0 0
\(806\) 6.70002e6 0.363277
\(807\) − 1.48224e6i − 0.0801188i
\(808\) − 689332.i − 0.0371450i
\(809\) 2.01482e7 1.08234 0.541171 0.840912i \(-0.317981\pi\)
0.541171 + 0.840912i \(0.317981\pi\)
\(810\) 0 0
\(811\) −3.88568e6 −0.207451 −0.103725 0.994606i \(-0.533076\pi\)
−0.103725 + 0.994606i \(0.533076\pi\)
\(812\) 1.08356e6i 0.0576717i
\(813\) 678337.i 0.0359931i
\(814\) −7.41568e6 −0.392274
\(815\) 0 0
\(816\) 903710. 0.0475121
\(817\) − 3.64961e7i − 1.91290i
\(818\) 1.49240e7i 0.779836i
\(819\) −397299. −0.0206970
\(820\) 0 0
\(821\) 1.69007e7 0.875079 0.437539 0.899199i \(-0.355850\pi\)
0.437539 + 0.899199i \(0.355850\pi\)
\(822\) − 5.01672e6i − 0.258964i
\(823\) − 166131.i − 0.00854971i −0.999991 0.00427485i \(-0.998639\pi\)
0.999991 0.00427485i \(-0.00136073\pi\)
\(824\) 2.87958e6 0.147744
\(825\) 0 0
\(826\) −253198. −0.0129125
\(827\) − 1.64293e7i − 0.835324i −0.908602 0.417662i \(-0.862850\pi\)
0.908602 0.417662i \(-0.137150\pi\)
\(828\) − 1.34911e6i − 0.0683866i
\(829\) −811068. −0.0409894 −0.0204947 0.999790i \(-0.506524\pi\)
−0.0204947 + 0.999790i \(0.506524\pi\)
\(830\) 0 0
\(831\) 294397. 0.0147888
\(832\) 692224.i 0.0346688i
\(833\) − 1.58621e7i − 0.792040i
\(834\) 1.28167e6 0.0638060
\(835\) 0 0
\(836\) 7.04916e6 0.348836
\(837\) − 1.73953e7i − 0.858261i
\(838\) − 1.07729e7i − 0.529933i
\(839\) −1.09462e7 −0.536855 −0.268427 0.963300i \(-0.586504\pi\)
−0.268427 + 0.963300i \(0.586504\pi\)
\(840\) 0 0
\(841\) 2.30773e7 1.12511
\(842\) − 2.22715e7i − 1.08261i
\(843\) − 5.62007e6i − 0.272378i
\(844\) 8.70351e6 0.420570
\(845\) 0 0
\(846\) −1.24112e7 −0.596196
\(847\) 1.25281e6i 0.0600033i
\(848\) 8.97269e6i 0.428483i
\(849\) 9.32152e6 0.443831
\(850\) 0 0
\(851\) 3.45752e6 0.163659
\(852\) 2.28108e6i 0.107657i
\(853\) − 1.24011e7i − 0.583564i −0.956485 0.291782i \(-0.905752\pi\)
0.956485 0.291782i \(-0.0942482\pi\)
\(854\) −564009. −0.0264631
\(855\) 0 0
\(856\) 1.24305e7 0.579837
\(857\) 1.78406e7i 0.829768i 0.909874 + 0.414884i \(0.136178\pi\)
−0.909874 + 0.414884i \(0.863822\pi\)
\(858\) − 495688.i − 0.0229874i
\(859\) −1.43038e7 −0.661408 −0.330704 0.943735i \(-0.607286\pi\)
−0.330704 + 0.943735i \(0.607286\pi\)
\(860\) 0 0
\(861\) 784982. 0.0360871
\(862\) 1.50264e7i 0.688788i
\(863\) − 2.13905e7i − 0.977673i −0.872376 0.488836i \(-0.837422\pi\)
0.872376 0.488836i \(-0.162578\pi\)
\(864\) 1.79723e6 0.0819067
\(865\) 0 0
\(866\) −6.43910e6 −0.291763
\(867\) 1.92498e6i 0.0869717i
\(868\) − 1.62666e6i − 0.0732819i
\(869\) −1.76896e7 −0.794636
\(870\) 0 0
\(871\) −1.05011e7 −0.469016
\(872\) − 1.38494e7i − 0.616791i
\(873\) 1.84647e7i 0.819985i
\(874\) −3.28663e6 −0.145536
\(875\) 0 0
\(876\) −3.56638e6 −0.157024
\(877\) − 3.28786e7i − 1.44349i −0.692159 0.721745i \(-0.743340\pi\)
0.692159 0.721745i \(-0.256660\pi\)
\(878\) − 7.69227e6i − 0.336758i
\(879\) 6.51902e6 0.284584
\(880\) 0 0
\(881\) −2.63729e7 −1.14477 −0.572384 0.819986i \(-0.693981\pi\)
−0.572384 + 0.819986i \(0.693981\pi\)
\(882\) − 1.53111e7i − 0.662728i
\(883\) 1.89435e7i 0.817634i 0.912616 + 0.408817i \(0.134059\pi\)
−0.912616 + 0.408817i \(0.865941\pi\)
\(884\) 2.56805e6 0.110528
\(885\) 0 0
\(886\) 5.72241e6 0.244903
\(887\) 1.00174e7i 0.427509i 0.976887 + 0.213755i \(0.0685693\pi\)
−0.976887 + 0.213755i \(0.931431\pi\)
\(888\) 2.23560e6i 0.0951397i
\(889\) −1.93135e6 −0.0819610
\(890\) 0 0
\(891\) 9.69955e6 0.409314
\(892\) 1.82995e6i 0.0770063i
\(893\) 3.02356e7i 1.26879i
\(894\) 4.31456e6 0.180548
\(895\) 0 0
\(896\) 168061. 0.00699353
\(897\) 231112.i 0.00959049i
\(898\) − 1.32108e7i − 0.546686i
\(899\) −6.54357e7 −2.70032
\(900\) 0 0
\(901\) 3.32874e7 1.36605
\(902\) − 1.62461e7i − 0.664864i
\(903\) 623072.i 0.0254284i
\(904\) 9.04438e6 0.368093
\(905\) 0 0
\(906\) 1.93119e6 0.0781636
\(907\) − 2.20644e7i − 0.890581i −0.895386 0.445290i \(-0.853100\pi\)
0.895386 0.445290i \(-0.146900\pi\)
\(908\) 3.41200e6i 0.137339i
\(909\) −2.46850e6 −0.0990884
\(910\) 0 0
\(911\) −2.02691e7 −0.809167 −0.404583 0.914501i \(-0.632583\pi\)
−0.404583 + 0.914501i \(0.632583\pi\)
\(912\) − 2.12511e6i − 0.0846045i
\(913\) 7.52299e6i 0.298685i
\(914\) 2.74263e7 1.08593
\(915\) 0 0
\(916\) 1.89603e7 0.746631
\(917\) − 2.70715e6i − 0.106314i
\(918\) − 6.66746e6i − 0.261128i
\(919\) 1.79004e7 0.699154 0.349577 0.936908i \(-0.386325\pi\)
0.349577 + 0.936908i \(0.386325\pi\)
\(920\) 0 0
\(921\) 5.77242e6 0.224238
\(922\) − 3.33374e7i − 1.29153i
\(923\) 6.48209e6i 0.250444i
\(924\) −120345. −0.00463713
\(925\) 0 0
\(926\) 1.14809e7 0.439997
\(927\) − 1.03118e7i − 0.394124i
\(928\) − 6.76061e6i − 0.257701i
\(929\) −9.25686e6 −0.351904 −0.175952 0.984399i \(-0.556300\pi\)
−0.175952 + 0.984399i \(0.556300\pi\)
\(930\) 0 0
\(931\) −3.73002e7 −1.41038
\(932\) 2.19404e7i 0.827381i
\(933\) − 699260.i − 0.0262987i
\(934\) 2.64726e7 0.992953
\(935\) 0 0
\(936\) 2.47885e6 0.0924828
\(937\) − 8.47287e6i − 0.315269i −0.987498 0.157635i \(-0.949613\pi\)
0.987498 0.157635i \(-0.0503868\pi\)
\(938\) 2.54949e6i 0.0946119i
\(939\) 3.14631e6 0.116449
\(940\) 0 0
\(941\) −1.22548e7 −0.451161 −0.225580 0.974225i \(-0.572428\pi\)
−0.225580 + 0.974225i \(0.572428\pi\)
\(942\) − 686994.i − 0.0252247i
\(943\) 7.57465e6i 0.277385i
\(944\) 1.57977e6 0.0576984
\(945\) 0 0
\(946\) 1.28952e7 0.468490
\(947\) 2.12327e7i 0.769363i 0.923049 + 0.384681i \(0.125689\pi\)
−0.923049 + 0.384681i \(0.874311\pi\)
\(948\) 5.33287e6i 0.192726i
\(949\) −1.01345e7 −0.365288
\(950\) 0 0
\(951\) 1.12749e6 0.0404259
\(952\) − 623481.i − 0.0222962i
\(953\) 2.31217e7i 0.824684i 0.911029 + 0.412342i \(0.135289\pi\)
−0.911029 + 0.412342i \(0.864711\pi\)
\(954\) 3.21312e7 1.14303
\(955\) 0 0
\(956\) −1.74438e7 −0.617301
\(957\) 4.84114e6i 0.170871i
\(958\) − 5.89739e6i − 0.207609i
\(959\) −3.46110e6 −0.121525
\(960\) 0 0
\(961\) 6.96040e7 2.43123
\(962\) 6.35285e6i 0.221325i
\(963\) − 4.45137e7i − 1.54678i
\(964\) −1.35836e6 −0.0470784
\(965\) 0 0
\(966\) 56110.2 0.00193464
\(967\) 1.92947e7i 0.663548i 0.943359 + 0.331774i \(0.107647\pi\)
−0.943359 + 0.331774i \(0.892353\pi\)
\(968\) − 7.81658e6i − 0.268120i
\(969\) −7.88383e6 −0.269729
\(970\) 0 0
\(971\) 1.63863e7 0.557743 0.278871 0.960328i \(-0.410040\pi\)
0.278871 + 0.960328i \(0.410040\pi\)
\(972\) − 9.74798e6i − 0.330940i
\(973\) − 884241.i − 0.0299425i
\(974\) 1.05260e7 0.355523
\(975\) 0 0
\(976\) 3.51900e6 0.118248
\(977\) 3.83317e6i 0.128476i 0.997935 + 0.0642380i \(0.0204617\pi\)
−0.997935 + 0.0642380i \(0.979538\pi\)
\(978\) − 5.63611e6i − 0.188422i
\(979\) 1.54422e7 0.514936
\(980\) 0 0
\(981\) −4.95945e7 −1.64536
\(982\) − 1.88185e7i − 0.622738i
\(983\) − 2.80937e7i − 0.927310i −0.886016 0.463655i \(-0.846538\pi\)
0.886016 0.463655i \(-0.153462\pi\)
\(984\) −4.89771e6 −0.161252
\(985\) 0 0
\(986\) −2.50809e7 −0.821581
\(987\) − 516191.i − 0.0168662i
\(988\) − 6.03886e6i − 0.196817i
\(989\) −6.01231e6 −0.195457
\(990\) 0 0
\(991\) −4.83597e7 −1.56423 −0.782113 0.623137i \(-0.785858\pi\)
−0.782113 + 0.623137i \(0.785858\pi\)
\(992\) 1.01491e7i 0.327454i
\(993\) − 1.24974e7i − 0.402206i
\(994\) 1.57375e6 0.0505207
\(995\) 0 0
\(996\) 2.26795e6 0.0724411
\(997\) − 4.50212e7i − 1.43443i −0.696852 0.717215i \(-0.745416\pi\)
0.696852 0.717215i \(-0.254584\pi\)
\(998\) 1.63659e6i 0.0520132i
\(999\) 1.64940e7 0.522892
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 650.6.b.j.599.2 6
5.2 odd 4 650.6.a.j.1.2 3
5.3 odd 4 130.6.a.f.1.2 3
5.4 even 2 inner 650.6.b.j.599.5 6
20.3 even 4 1040.6.a.l.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.6.a.f.1.2 3 5.3 odd 4
650.6.a.j.1.2 3 5.2 odd 4
650.6.b.j.599.2 6 1.1 even 1 trivial
650.6.b.j.599.5 6 5.4 even 2 inner
1040.6.a.l.1.2 3 20.3 even 4