Properties

Label 130.6.a.a.1.2
Level $130$
Weight $6$
Character 130.1
Self dual yes
Analytic conductor $20.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,6,Mod(1,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, 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("130.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 130.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: \(20.8498965757\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.16228\) of defining polynomial
Character \(\chi\) \(=\) 130.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +13.8114 q^{3} +16.0000 q^{4} +25.0000 q^{5} -55.2456 q^{6} +213.246 q^{7} -64.0000 q^{8} -52.2456 q^{9} -100.000 q^{10} +587.285 q^{11} +220.982 q^{12} -169.000 q^{13} -852.982 q^{14} +345.285 q^{15} +256.000 q^{16} +162.605 q^{17} +208.982 q^{18} -81.3914 q^{19} +400.000 q^{20} +2945.22 q^{21} -2349.14 q^{22} -2948.84 q^{23} -883.929 q^{24} +625.000 q^{25} +676.000 q^{26} -4077.75 q^{27} +3411.93 q^{28} +6254.43 q^{29} -1381.14 q^{30} -4034.62 q^{31} -1024.00 q^{32} +8111.22 q^{33} -650.420 q^{34} +5331.14 q^{35} -835.929 q^{36} +7617.22 q^{37} +325.566 q^{38} -2334.12 q^{39} -1600.00 q^{40} +958.121 q^{41} -11780.9 q^{42} +169.655 q^{43} +9396.56 q^{44} -1306.14 q^{45} +11795.4 q^{46} +21612.6 q^{47} +3535.72 q^{48} +28666.7 q^{49} -2500.00 q^{50} +2245.80 q^{51} -2704.00 q^{52} +24789.1 q^{53} +16311.0 q^{54} +14682.1 q^{55} -13647.7 q^{56} -1124.13 q^{57} -25017.7 q^{58} +40109.2 q^{59} +5524.56 q^{60} -16343.0 q^{61} +16138.5 q^{62} -11141.1 q^{63} +4096.00 q^{64} -4225.00 q^{65} -32444.9 q^{66} +18362.1 q^{67} +2601.68 q^{68} -40727.6 q^{69} -21324.6 q^{70} -77846.0 q^{71} +3343.72 q^{72} -62130.2 q^{73} -30468.9 q^{74} +8632.12 q^{75} -1302.26 q^{76} +125236. q^{77} +9336.50 q^{78} -60599.1 q^{79} +6400.00 q^{80} -43623.7 q^{81} -3832.48 q^{82} -2654.46 q^{83} +47123.5 q^{84} +4065.12 q^{85} -678.619 q^{86} +86382.4 q^{87} -37586.2 q^{88} +8229.77 q^{89} +5224.56 q^{90} -36038.5 q^{91} -47181.5 q^{92} -55723.7 q^{93} -86450.3 q^{94} -2034.79 q^{95} -14142.9 q^{96} -53844.0 q^{97} -114667. q^{98} -30683.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 4 q^{3} + 32 q^{4} + 50 q^{5} + 16 q^{6} + 300 q^{7} - 128 q^{8} + 22 q^{9} - 200 q^{10} + 384 q^{11} - 64 q^{12} - 338 q^{13} - 1200 q^{14} - 100 q^{15} + 512 q^{16} - 244 q^{17} - 88 q^{18}+ \cdots - 45776 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 13.8114 0.886001 0.443000 0.896521i \(-0.353914\pi\)
0.443000 + 0.896521i \(0.353914\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) −55.2456 −0.626497
\(7\) 213.246 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(8\) −64.0000 −0.353553
\(9\) −52.2456 −0.215002
\(10\) −100.000 −0.316228
\(11\) 587.285 1.46341 0.731707 0.681620i \(-0.238724\pi\)
0.731707 + 0.681620i \(0.238724\pi\)
\(12\) 220.982 0.443000
\(13\) −169.000 −0.277350
\(14\) −852.982 −1.16311
\(15\) 345.285 0.396232
\(16\) 256.000 0.250000
\(17\) 162.605 0.136462 0.0682310 0.997670i \(-0.478265\pi\)
0.0682310 + 0.997670i \(0.478265\pi\)
\(18\) 208.982 0.152030
\(19\) −81.3914 −0.0517243 −0.0258622 0.999666i \(-0.508233\pi\)
−0.0258622 + 0.999666i \(0.508233\pi\)
\(20\) 400.000 0.223607
\(21\) 2945.22 1.45737
\(22\) −2349.14 −1.03479
\(23\) −2948.84 −1.16234 −0.581169 0.813783i \(-0.697404\pi\)
−0.581169 + 0.813783i \(0.697404\pi\)
\(24\) −883.929 −0.313249
\(25\) 625.000 0.200000
\(26\) 676.000 0.196116
\(27\) −4077.75 −1.07649
\(28\) 3411.93 0.822441
\(29\) 6254.43 1.38100 0.690499 0.723333i \(-0.257391\pi\)
0.690499 + 0.723333i \(0.257391\pi\)
\(30\) −1381.14 −0.280178
\(31\) −4034.62 −0.754048 −0.377024 0.926204i \(-0.623052\pi\)
−0.377024 + 0.926204i \(0.623052\pi\)
\(32\) −1024.00 −0.176777
\(33\) 8111.22 1.29659
\(34\) −650.420 −0.0964932
\(35\) 5331.14 0.735614
\(36\) −835.929 −0.107501
\(37\) 7617.22 0.914728 0.457364 0.889280i \(-0.348794\pi\)
0.457364 + 0.889280i \(0.348794\pi\)
\(38\) 325.566 0.0365746
\(39\) −2334.12 −0.245732
\(40\) −1600.00 −0.158114
\(41\) 958.121 0.0890145 0.0445072 0.999009i \(-0.485828\pi\)
0.0445072 + 0.999009i \(0.485828\pi\)
\(42\) −11780.9 −1.03051
\(43\) 169.655 0.0139925 0.00699624 0.999976i \(-0.497773\pi\)
0.00699624 + 0.999976i \(0.497773\pi\)
\(44\) 9396.56 0.731707
\(45\) −1306.14 −0.0961519
\(46\) 11795.4 0.821897
\(47\) 21612.6 1.42712 0.713562 0.700592i \(-0.247081\pi\)
0.713562 + 0.700592i \(0.247081\pi\)
\(48\) 3535.72 0.221500
\(49\) 28666.7 1.70564
\(50\) −2500.00 −0.141421
\(51\) 2245.80 0.120905
\(52\) −2704.00 −0.138675
\(53\) 24789.1 1.21219 0.606096 0.795392i \(-0.292735\pi\)
0.606096 + 0.795392i \(0.292735\pi\)
\(54\) 16311.0 0.761196
\(55\) 14682.1 0.654458
\(56\) −13647.7 −0.581554
\(57\) −1124.13 −0.0458278
\(58\) −25017.7 −0.976513
\(59\) 40109.2 1.50008 0.750040 0.661392i \(-0.230034\pi\)
0.750040 + 0.661392i \(0.230034\pi\)
\(60\) 5524.56 0.198116
\(61\) −16343.0 −0.562351 −0.281176 0.959656i \(-0.590724\pi\)
−0.281176 + 0.959656i \(0.590724\pi\)
\(62\) 16138.5 0.533192
\(63\) −11141.1 −0.353653
\(64\) 4096.00 0.125000
\(65\) −4225.00 −0.124035
\(66\) −32444.9 −0.916824
\(67\) 18362.1 0.499731 0.249866 0.968281i \(-0.419614\pi\)
0.249866 + 0.968281i \(0.419614\pi\)
\(68\) 2601.68 0.0682310
\(69\) −40727.6 −1.02983
\(70\) −21324.6 −0.520158
\(71\) −77846.0 −1.83270 −0.916348 0.400382i \(-0.868877\pi\)
−0.916348 + 0.400382i \(0.868877\pi\)
\(72\) 3343.72 0.0760148
\(73\) −62130.2 −1.36457 −0.682285 0.731087i \(-0.739014\pi\)
−0.682285 + 0.731087i \(0.739014\pi\)
\(74\) −30468.9 −0.646810
\(75\) 8632.12 0.177200
\(76\) −1302.26 −0.0258622
\(77\) 125236. 2.40714
\(78\) 9336.50 0.173759
\(79\) −60599.1 −1.09244 −0.546221 0.837641i \(-0.683934\pi\)
−0.546221 + 0.837641i \(0.683934\pi\)
\(80\) 6400.00 0.111803
\(81\) −43623.7 −0.738772
\(82\) −3832.48 −0.0629427
\(83\) −2654.46 −0.0422941 −0.0211471 0.999776i \(-0.506732\pi\)
−0.0211471 + 0.999776i \(0.506732\pi\)
\(84\) 47123.5 0.728684
\(85\) 4065.12 0.0610276
\(86\) −678.619 −0.00989418
\(87\) 86382.4 1.22357
\(88\) −37586.2 −0.517395
\(89\) 8229.77 0.110132 0.0550659 0.998483i \(-0.482463\pi\)
0.0550659 + 0.998483i \(0.482463\pi\)
\(90\) 5224.56 0.0679897
\(91\) −36038.5 −0.456208
\(92\) −47181.5 −0.581169
\(93\) −55723.7 −0.668087
\(94\) −86450.3 −1.00913
\(95\) −2034.79 −0.0231318
\(96\) −14142.9 −0.156624
\(97\) −53844.0 −0.581042 −0.290521 0.956869i \(-0.593829\pi\)
−0.290521 + 0.956869i \(0.593829\pi\)
\(98\) −114667. −1.20607
\(99\) −30683.0 −0.314637
\(100\) 10000.0 0.100000
\(101\) −167556. −1.63439 −0.817196 0.576360i \(-0.804473\pi\)
−0.817196 + 0.576360i \(0.804473\pi\)
\(102\) −8983.20 −0.0854930
\(103\) 30973.8 0.287675 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(104\) 10816.0 0.0980581
\(105\) 73630.4 0.651755
\(106\) −99156.4 −0.857149
\(107\) −97343.8 −0.821956 −0.410978 0.911645i \(-0.634813\pi\)
−0.410978 + 0.911645i \(0.634813\pi\)
\(108\) −65244.0 −0.538247
\(109\) 16696.8 0.134607 0.0673035 0.997733i \(-0.478560\pi\)
0.0673035 + 0.997733i \(0.478560\pi\)
\(110\) −58728.5 −0.462772
\(111\) 105204. 0.810450
\(112\) 54590.9 0.411221
\(113\) 219357. 1.61605 0.808027 0.589145i \(-0.200535\pi\)
0.808027 + 0.589145i \(0.200535\pi\)
\(114\) 4496.51 0.0324051
\(115\) −73721.1 −0.519813
\(116\) 100071. 0.690499
\(117\) 8829.50 0.0596309
\(118\) −160437. −1.06072
\(119\) 34674.8 0.224464
\(120\) −22098.2 −0.140089
\(121\) 183852. 1.14158
\(122\) 65372.1 0.397642
\(123\) 13233.0 0.0788669
\(124\) −64554.0 −0.377024
\(125\) 15625.0 0.0894427
\(126\) 44564.5 0.250071
\(127\) 158753. 0.873399 0.436700 0.899607i \(-0.356147\pi\)
0.436700 + 0.899607i \(0.356147\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 2343.17 0.0123974
\(130\) 16900.0 0.0877058
\(131\) 160566. 0.817477 0.408739 0.912651i \(-0.365969\pi\)
0.408739 + 0.912651i \(0.365969\pi\)
\(132\) 129779. 0.648293
\(133\) −17356.4 −0.0850804
\(134\) −73448.6 −0.353363
\(135\) −101944. −0.481422
\(136\) −10406.7 −0.0482466
\(137\) −295668. −1.34587 −0.672934 0.739702i \(-0.734966\pi\)
−0.672934 + 0.739702i \(0.734966\pi\)
\(138\) 162910. 0.728201
\(139\) 6197.57 0.0272072 0.0136036 0.999907i \(-0.495670\pi\)
0.0136036 + 0.999907i \(0.495670\pi\)
\(140\) 85298.2 0.367807
\(141\) 298500. 1.26443
\(142\) 311384. 1.29591
\(143\) −99251.1 −0.405878
\(144\) −13374.9 −0.0537506
\(145\) 156361. 0.617601
\(146\) 248521. 0.964896
\(147\) 395926. 1.51120
\(148\) 121875. 0.457364
\(149\) −371033. −1.36914 −0.684569 0.728948i \(-0.740010\pi\)
−0.684569 + 0.728948i \(0.740010\pi\)
\(150\) −34528.5 −0.125299
\(151\) −508361. −1.81439 −0.907193 0.420714i \(-0.861780\pi\)
−0.907193 + 0.420714i \(0.861780\pi\)
\(152\) 5209.05 0.0182873
\(153\) −8495.39 −0.0293396
\(154\) −500943. −1.70211
\(155\) −100866. −0.337220
\(156\) −37346.0 −0.122866
\(157\) 60014.9 0.194317 0.0971583 0.995269i \(-0.469025\pi\)
0.0971583 + 0.995269i \(0.469025\pi\)
\(158\) 242397. 0.772474
\(159\) 342372. 1.07400
\(160\) −25600.0 −0.0790569
\(161\) −628828. −1.91191
\(162\) 174495. 0.522391
\(163\) −365913. −1.07872 −0.539360 0.842075i \(-0.681334\pi\)
−0.539360 + 0.842075i \(0.681334\pi\)
\(164\) 15329.9 0.0445072
\(165\) 202780. 0.579851
\(166\) 10617.8 0.0299065
\(167\) 67392.0 0.186990 0.0934948 0.995620i \(-0.470196\pi\)
0.0934948 + 0.995620i \(0.470196\pi\)
\(168\) −188494. −0.515257
\(169\) 28561.0 0.0769231
\(170\) −16260.5 −0.0431531
\(171\) 4252.34 0.0111208
\(172\) 2714.48 0.00699624
\(173\) 518921. 1.31821 0.659107 0.752049i \(-0.270934\pi\)
0.659107 + 0.752049i \(0.270934\pi\)
\(174\) −345530. −0.865191
\(175\) 133278. 0.328977
\(176\) 150345. 0.365853
\(177\) 553964. 1.32907
\(178\) −32919.1 −0.0778750
\(179\) −89728.6 −0.209314 −0.104657 0.994508i \(-0.533374\pi\)
−0.104657 + 0.994508i \(0.533374\pi\)
\(180\) −20898.2 −0.0480760
\(181\) −376272. −0.853700 −0.426850 0.904322i \(-0.640377\pi\)
−0.426850 + 0.904322i \(0.640377\pi\)
\(182\) 144154. 0.322588
\(183\) −225720. −0.498244
\(184\) 188726. 0.410948
\(185\) 190430. 0.409079
\(186\) 222895. 0.472409
\(187\) 95495.4 0.199700
\(188\) 345801. 0.713562
\(189\) −869562. −1.77070
\(190\) 8139.14 0.0163567
\(191\) 206748. 0.410070 0.205035 0.978755i \(-0.434269\pi\)
0.205035 + 0.978755i \(0.434269\pi\)
\(192\) 56571.4 0.110750
\(193\) −413088. −0.798269 −0.399135 0.916892i \(-0.630689\pi\)
−0.399135 + 0.916892i \(0.630689\pi\)
\(194\) 215376. 0.410859
\(195\) −58353.1 −0.109895
\(196\) 458667. 0.852819
\(197\) 538825. 0.989196 0.494598 0.869122i \(-0.335315\pi\)
0.494598 + 0.869122i \(0.335315\pi\)
\(198\) 122732. 0.222482
\(199\) −147336. −0.263740 −0.131870 0.991267i \(-0.542098\pi\)
−0.131870 + 0.991267i \(0.542098\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 253607. 0.442762
\(202\) 670223. 1.15569
\(203\) 1.33373e6 2.27158
\(204\) 35932.8 0.0604527
\(205\) 23953.0 0.0398085
\(206\) −123895. −0.203417
\(207\) 154064. 0.249905
\(208\) −43264.0 −0.0693375
\(209\) −47799.9 −0.0756940
\(210\) −294522. −0.460860
\(211\) −1.14921e6 −1.77703 −0.888515 0.458847i \(-0.848262\pi\)
−0.888515 + 0.458847i \(0.848262\pi\)
\(212\) 396626. 0.606096
\(213\) −1.07516e6 −1.62377
\(214\) 389375. 0.581211
\(215\) 4241.37 0.00625763
\(216\) 260976. 0.380598
\(217\) −860365. −1.24032
\(218\) −66787.3 −0.0951815
\(219\) −858104. −1.20901
\(220\) 234914. 0.327229
\(221\) −27480.2 −0.0378477
\(222\) −420817. −0.573075
\(223\) −1.39430e6 −1.87756 −0.938782 0.344513i \(-0.888044\pi\)
−0.938782 + 0.344513i \(0.888044\pi\)
\(224\) −218363. −0.290777
\(225\) −32653.5 −0.0430005
\(226\) −877429. −1.14272
\(227\) 1.40291e6 1.80703 0.903515 0.428556i \(-0.140977\pi\)
0.903515 + 0.428556i \(0.140977\pi\)
\(228\) −17986.1 −0.0229139
\(229\) −796626. −1.00384 −0.501922 0.864913i \(-0.667373\pi\)
−0.501922 + 0.864913i \(0.667373\pi\)
\(230\) 294884. 0.367563
\(231\) 1.72968e6 2.13273
\(232\) −400284. −0.488257
\(233\) 1.15975e6 1.39951 0.699753 0.714385i \(-0.253293\pi\)
0.699753 + 0.714385i \(0.253293\pi\)
\(234\) −35318.0 −0.0421654
\(235\) 540314. 0.638230
\(236\) 641748. 0.750040
\(237\) −836958. −0.967905
\(238\) −138699. −0.158720
\(239\) −687529. −0.778568 −0.389284 0.921118i \(-0.627278\pi\)
−0.389284 + 0.921118i \(0.627278\pi\)
\(240\) 88392.9 0.0990579
\(241\) −391345. −0.434028 −0.217014 0.976169i \(-0.569632\pi\)
−0.217014 + 0.976169i \(0.569632\pi\)
\(242\) −735409. −0.807218
\(243\) 388389. 0.421941
\(244\) −261488. −0.281176
\(245\) 716667. 0.762785
\(246\) −52931.9 −0.0557673
\(247\) 13755.2 0.0143457
\(248\) 258216. 0.266596
\(249\) −36661.7 −0.0374727
\(250\) −62500.0 −0.0632456
\(251\) 944723. 0.946499 0.473249 0.880928i \(-0.343081\pi\)
0.473249 + 0.880928i \(0.343081\pi\)
\(252\) −178258. −0.176827
\(253\) −1.73181e6 −1.70098
\(254\) −635012. −0.617587
\(255\) 56145.0 0.0540705
\(256\) 65536.0 0.0625000
\(257\) 297219. 0.280701 0.140351 0.990102i \(-0.455177\pi\)
0.140351 + 0.990102i \(0.455177\pi\)
\(258\) −9372.67 −0.00876626
\(259\) 1.62434e6 1.50462
\(260\) −67600.0 −0.0620174
\(261\) −326766. −0.296918
\(262\) −642264. −0.578044
\(263\) −1.53097e6 −1.36483 −0.682414 0.730966i \(-0.739070\pi\)
−0.682414 + 0.730966i \(0.739070\pi\)
\(264\) −519118. −0.458412
\(265\) 619728. 0.542109
\(266\) 69425.4 0.0601609
\(267\) 113665. 0.0975769
\(268\) 293794. 0.249866
\(269\) −1.00878e6 −0.849994 −0.424997 0.905195i \(-0.639725\pi\)
−0.424997 + 0.905195i \(0.639725\pi\)
\(270\) 407775. 0.340417
\(271\) 2.19122e6 1.81244 0.906220 0.422806i \(-0.138955\pi\)
0.906220 + 0.422806i \(0.138955\pi\)
\(272\) 41626.9 0.0341155
\(273\) −497742. −0.404201
\(274\) 1.18267e6 0.951673
\(275\) 367053. 0.292683
\(276\) −651642. −0.514916
\(277\) −17924.0 −0.0140357 −0.00701786 0.999975i \(-0.502234\pi\)
−0.00701786 + 0.999975i \(0.502234\pi\)
\(278\) −24790.3 −0.0192384
\(279\) 210791. 0.162122
\(280\) −341193. −0.260079
\(281\) −2.40663e6 −1.81821 −0.909106 0.416566i \(-0.863234\pi\)
−0.909106 + 0.416566i \(0.863234\pi\)
\(282\) −1.19400e6 −0.894090
\(283\) −1.36921e6 −1.01626 −0.508128 0.861281i \(-0.669662\pi\)
−0.508128 + 0.861281i \(0.669662\pi\)
\(284\) −1.24554e6 −0.916348
\(285\) −28103.2 −0.0204948
\(286\) 397004. 0.286999
\(287\) 204315. 0.146418
\(288\) 53499.4 0.0380074
\(289\) −1.39342e6 −0.981378
\(290\) −625443. −0.436710
\(291\) −743660. −0.514804
\(292\) −994083. −0.682285
\(293\) −390649. −0.265838 −0.132919 0.991127i \(-0.542435\pi\)
−0.132919 + 0.991127i \(0.542435\pi\)
\(294\) −1.58371e6 −1.06858
\(295\) 1.00273e6 0.670856
\(296\) −487502. −0.323405
\(297\) −2.39480e6 −1.57535
\(298\) 1.48413e6 0.968127
\(299\) 498355. 0.322374
\(300\) 138114. 0.0886001
\(301\) 36178.1 0.0230160
\(302\) 2.03344e6 1.28297
\(303\) −2.31418e6 −1.44807
\(304\) −20836.2 −0.0129311
\(305\) −408575. −0.251491
\(306\) 33981.6 0.0207463
\(307\) 566546. 0.343075 0.171537 0.985178i \(-0.445127\pi\)
0.171537 + 0.985178i \(0.445127\pi\)
\(308\) 2.00377e6 1.20357
\(309\) 427791. 0.254880
\(310\) 403462. 0.238451
\(311\) 881239. 0.516646 0.258323 0.966059i \(-0.416830\pi\)
0.258323 + 0.966059i \(0.416830\pi\)
\(312\) 149384. 0.0868795
\(313\) −916421. −0.528731 −0.264365 0.964423i \(-0.585162\pi\)
−0.264365 + 0.964423i \(0.585162\pi\)
\(314\) −240060. −0.137403
\(315\) −278528. −0.158159
\(316\) −969586. −0.546221
\(317\) 2.01084e6 1.12390 0.561951 0.827170i \(-0.310051\pi\)
0.561951 + 0.827170i \(0.310051\pi\)
\(318\) −1.36949e6 −0.759435
\(319\) 3.67313e6 2.02097
\(320\) 102400. 0.0559017
\(321\) −1.34445e6 −0.728254
\(322\) 2.51531e6 1.35192
\(323\) −13234.7 −0.00705840
\(324\) −697980. −0.369386
\(325\) −105625. −0.0554700
\(326\) 1.46365e6 0.762770
\(327\) 230606. 0.119262
\(328\) −61319.7 −0.0314714
\(329\) 4.60879e6 2.34745
\(330\) −811122. −0.410016
\(331\) 2.47715e6 1.24275 0.621373 0.783515i \(-0.286575\pi\)
0.621373 + 0.783515i \(0.286575\pi\)
\(332\) −42471.3 −0.0211471
\(333\) −397966. −0.196669
\(334\) −269568. −0.132222
\(335\) 459054. 0.223487
\(336\) 753976. 0.364342
\(337\) 2.55550e6 1.22575 0.612873 0.790181i \(-0.290014\pi\)
0.612873 + 0.790181i \(0.290014\pi\)
\(338\) −114244. −0.0543928
\(339\) 3.02963e6 1.43183
\(340\) 65042.0 0.0305138
\(341\) −2.36947e6 −1.10348
\(342\) −17009.4 −0.00786362
\(343\) 2.52902e6 1.16069
\(344\) −10857.9 −0.00494709
\(345\) −1.01819e6 −0.460555
\(346\) −2.07568e6 −0.932118
\(347\) −475000. −0.211773 −0.105886 0.994378i \(-0.533768\pi\)
−0.105886 + 0.994378i \(0.533768\pi\)
\(348\) 1.38212e6 0.611783
\(349\) 615277. 0.270400 0.135200 0.990818i \(-0.456832\pi\)
0.135200 + 0.990818i \(0.456832\pi\)
\(350\) −533114. −0.232622
\(351\) 689140. 0.298565
\(352\) −601380. −0.258697
\(353\) −4.37400e6 −1.86828 −0.934140 0.356908i \(-0.883831\pi\)
−0.934140 + 0.356908i \(0.883831\pi\)
\(354\) −2.21586e6 −0.939796
\(355\) −1.94615e6 −0.819607
\(356\) 131676. 0.0550659
\(357\) 478907. 0.198875
\(358\) 358914. 0.148007
\(359\) 4.59562e6 1.88195 0.940975 0.338476i \(-0.109911\pi\)
0.940975 + 0.338476i \(0.109911\pi\)
\(360\) 83592.9 0.0339948
\(361\) −2.46947e6 −0.997325
\(362\) 1.50509e6 0.603657
\(363\) 2.53926e6 1.01144
\(364\) −576616. −0.228104
\(365\) −1.55326e6 −0.610254
\(366\) 902879. 0.352312
\(367\) −4.59362e6 −1.78029 −0.890143 0.455681i \(-0.849396\pi\)
−0.890143 + 0.455681i \(0.849396\pi\)
\(368\) −754904. −0.290584
\(369\) −50057.6 −0.0191383
\(370\) −761722. −0.289262
\(371\) 5.28617e6 1.99391
\(372\) −891580. −0.334043
\(373\) 1.74314e6 0.648724 0.324362 0.945933i \(-0.394850\pi\)
0.324362 + 0.945933i \(0.394850\pi\)
\(374\) −381982. −0.141209
\(375\) 215803. 0.0792463
\(376\) −1.38320e6 −0.504565
\(377\) −1.05700e6 −0.383020
\(378\) 3.47825e6 1.25208
\(379\) −4.42529e6 −1.58250 −0.791250 0.611493i \(-0.790569\pi\)
−0.791250 + 0.611493i \(0.790569\pi\)
\(380\) −32556.6 −0.0115659
\(381\) 2.19260e6 0.773833
\(382\) −826992. −0.289963
\(383\) 1.39773e6 0.486884 0.243442 0.969915i \(-0.421723\pi\)
0.243442 + 0.969915i \(0.421723\pi\)
\(384\) −226286. −0.0783122
\(385\) 3.13090e6 1.07651
\(386\) 1.65235e6 0.564462
\(387\) −8863.71 −0.00300842
\(388\) −861504. −0.290521
\(389\) 1.63535e6 0.547946 0.273973 0.961737i \(-0.411662\pi\)
0.273973 + 0.961737i \(0.411662\pi\)
\(390\) 233412. 0.0777074
\(391\) −479497. −0.158615
\(392\) −1.83467e6 −0.603034
\(393\) 2.21764e6 0.724286
\(394\) −2.15530e6 −0.699468
\(395\) −1.51498e6 −0.488555
\(396\) −490928. −0.157319
\(397\) 4.90653e6 1.56242 0.781210 0.624268i \(-0.214603\pi\)
0.781210 + 0.624268i \(0.214603\pi\)
\(398\) 589344. 0.186493
\(399\) −239715. −0.0753813
\(400\) 160000. 0.0500000
\(401\) −951184. −0.295395 −0.147698 0.989033i \(-0.547186\pi\)
−0.147698 + 0.989033i \(0.547186\pi\)
\(402\) −1.01443e6 −0.313080
\(403\) 681851. 0.209135
\(404\) −2.68089e6 −0.817196
\(405\) −1.09059e6 −0.330389
\(406\) −5.33492e6 −1.60625
\(407\) 4.47348e6 1.33863
\(408\) −143731. −0.0427465
\(409\) 4.49464e6 1.32858 0.664289 0.747476i \(-0.268735\pi\)
0.664289 + 0.747476i \(0.268735\pi\)
\(410\) −95812.1 −0.0281488
\(411\) −4.08358e6 −1.19244
\(412\) 495581. 0.143837
\(413\) 8.55312e6 2.46746
\(414\) −616256. −0.176710
\(415\) −66361.4 −0.0189145
\(416\) 173056. 0.0490290
\(417\) 85597.0 0.0241056
\(418\) 191200. 0.0535238
\(419\) −828906. −0.230659 −0.115329 0.993327i \(-0.536792\pi\)
−0.115329 + 0.993327i \(0.536792\pi\)
\(420\) 1.17809e6 0.325877
\(421\) 744813. 0.204806 0.102403 0.994743i \(-0.467347\pi\)
0.102403 + 0.994743i \(0.467347\pi\)
\(422\) 4.59686e6 1.25655
\(423\) −1.12916e6 −0.306835
\(424\) −1.58650e6 −0.428574
\(425\) 101628. 0.0272924
\(426\) 4.30065e6 1.14818
\(427\) −3.48508e6 −0.925002
\(428\) −1.55750e6 −0.410978
\(429\) −1.37080e6 −0.359608
\(430\) −16965.5 −0.00442481
\(431\) −3.89529e6 −1.01006 −0.505030 0.863102i \(-0.668519\pi\)
−0.505030 + 0.863102i \(0.668519\pi\)
\(432\) −1.04390e6 −0.269123
\(433\) −5.88297e6 −1.50792 −0.753958 0.656923i \(-0.771858\pi\)
−0.753958 + 0.656923i \(0.771858\pi\)
\(434\) 3.44146e6 0.877038
\(435\) 2.15956e6 0.547195
\(436\) 267149. 0.0673035
\(437\) 240011. 0.0601211
\(438\) 3.43242e6 0.854899
\(439\) 1.73024e6 0.428495 0.214248 0.976779i \(-0.431270\pi\)
0.214248 + 0.976779i \(0.431270\pi\)
\(440\) −939656. −0.231386
\(441\) −1.49771e6 −0.366716
\(442\) 109921. 0.0267624
\(443\) −106432. −0.0257670 −0.0128835 0.999917i \(-0.504101\pi\)
−0.0128835 + 0.999917i \(0.504101\pi\)
\(444\) 1.68327e6 0.405225
\(445\) 205744. 0.0492524
\(446\) 5.57721e6 1.32764
\(447\) −5.12449e6 −1.21306
\(448\) 873454. 0.205610
\(449\) −4.89220e6 −1.14522 −0.572609 0.819828i \(-0.694069\pi\)
−0.572609 + 0.819828i \(0.694069\pi\)
\(450\) 130614. 0.0304059
\(451\) 562690. 0.130265
\(452\) 3.50972e6 0.808027
\(453\) −7.02117e6 −1.60755
\(454\) −5.61165e6 −1.27776
\(455\) −900962. −0.204023
\(456\) 71944.2 0.0162026
\(457\) 177252. 0.0397008 0.0198504 0.999803i \(-0.493681\pi\)
0.0198504 + 0.999803i \(0.493681\pi\)
\(458\) 3.18651e6 0.709824
\(459\) −663063. −0.146900
\(460\) −1.17954e6 −0.259906
\(461\) 7.15676e6 1.56843 0.784213 0.620491i \(-0.213067\pi\)
0.784213 + 0.620491i \(0.213067\pi\)
\(462\) −6.91872e6 −1.50807
\(463\) 3.44228e6 0.746267 0.373133 0.927778i \(-0.378283\pi\)
0.373133 + 0.927778i \(0.378283\pi\)
\(464\) 1.60114e6 0.345249
\(465\) −1.39309e6 −0.298778
\(466\) −4.63900e6 −0.989600
\(467\) −9.22486e6 −1.95735 −0.978673 0.205423i \(-0.934143\pi\)
−0.978673 + 0.205423i \(0.934143\pi\)
\(468\) 141272. 0.0298155
\(469\) 3.91565e6 0.821999
\(470\) −2.16126e6 −0.451296
\(471\) 828889. 0.172165
\(472\) −2.56699e6 −0.530358
\(473\) 99635.7 0.0204768
\(474\) 3.34783e6 0.684412
\(475\) −50869.6 −0.0103449
\(476\) 554797. 0.112232
\(477\) −1.29512e6 −0.260624
\(478\) 2.75012e6 0.550531
\(479\) 1.23331e6 0.245602 0.122801 0.992431i \(-0.460812\pi\)
0.122801 + 0.992431i \(0.460812\pi\)
\(480\) −353572. −0.0700445
\(481\) −1.28731e6 −0.253700
\(482\) 1.56538e6 0.306904
\(483\) −8.68498e6 −1.69395
\(484\) 2.94164e6 0.570789
\(485\) −1.34610e6 −0.259850
\(486\) −1.55356e6 −0.298357
\(487\) −6.21276e6 −1.18703 −0.593516 0.804823i \(-0.702260\pi\)
−0.593516 + 0.804823i \(0.702260\pi\)
\(488\) 1.04595e6 0.198821
\(489\) −5.05376e6 −0.955747
\(490\) −2.86667e6 −0.539370
\(491\) −7.05221e6 −1.32014 −0.660072 0.751202i \(-0.729474\pi\)
−0.660072 + 0.751202i \(0.729474\pi\)
\(492\) 211728. 0.0394335
\(493\) 1.01700e6 0.188454
\(494\) −55020.6 −0.0101440
\(495\) −767075. −0.140710
\(496\) −1.03286e6 −0.188512
\(497\) −1.66003e7 −3.01457
\(498\) 146647. 0.0264972
\(499\) 5.63009e6 1.01220 0.506098 0.862476i \(-0.331088\pi\)
0.506098 + 0.862476i \(0.331088\pi\)
\(500\) 250000. 0.0447214
\(501\) 930778. 0.165673
\(502\) −3.77889e6 −0.669276
\(503\) 9.65803e6 1.70204 0.851018 0.525137i \(-0.175986\pi\)
0.851018 + 0.525137i \(0.175986\pi\)
\(504\) 713032. 0.125035
\(505\) −4.18890e6 −0.730922
\(506\) 6.92724e6 1.20277
\(507\) 394467. 0.0681539
\(508\) 2.54005e6 0.436700
\(509\) −4.39209e6 −0.751409 −0.375705 0.926739i \(-0.622599\pi\)
−0.375705 + 0.926739i \(0.622599\pi\)
\(510\) −224580. −0.0382337
\(511\) −1.32490e7 −2.24456
\(512\) −262144. −0.0441942
\(513\) 331894. 0.0556809
\(514\) −1.18888e6 −0.198486
\(515\) 774345. 0.128652
\(516\) 37490.7 0.00619868
\(517\) 1.26927e7 2.08847
\(518\) −6.49735e6 −1.06393
\(519\) 7.16702e6 1.16794
\(520\) 270400. 0.0438529
\(521\) −8.38443e6 −1.35325 −0.676627 0.736326i \(-0.736559\pi\)
−0.676627 + 0.736326i \(0.736559\pi\)
\(522\) 1.30707e6 0.209953
\(523\) 2.54489e6 0.406831 0.203416 0.979092i \(-0.434796\pi\)
0.203416 + 0.979092i \(0.434796\pi\)
\(524\) 2.56906e6 0.408739
\(525\) 1.84076e6 0.291474
\(526\) 6.12389e6 0.965080
\(527\) −656050. −0.102899
\(528\) 2.07647e6 0.324146
\(529\) 2.25933e6 0.351028
\(530\) −2.47891e6 −0.383329
\(531\) −2.09553e6 −0.322521
\(532\) −277702. −0.0425402
\(533\) −161922. −0.0246882
\(534\) −454658. −0.0689973
\(535\) −2.43359e6 −0.367590
\(536\) −1.17518e6 −0.176682
\(537\) −1.23928e6 −0.185452
\(538\) 4.03512e6 0.601036
\(539\) 1.68355e7 2.49605
\(540\) −1.63110e6 −0.240711
\(541\) −3.83465e6 −0.563290 −0.281645 0.959519i \(-0.590880\pi\)
−0.281645 + 0.959519i \(0.590880\pi\)
\(542\) −8.76490e6 −1.28159
\(543\) −5.19684e6 −0.756379
\(544\) −166508. −0.0241233
\(545\) 417421. 0.0601981
\(546\) 1.99097e6 0.285813
\(547\) −2.61787e6 −0.374093 −0.187047 0.982351i \(-0.559892\pi\)
−0.187047 + 0.982351i \(0.559892\pi\)
\(548\) −4.73068e6 −0.672934
\(549\) 853850. 0.120907
\(550\) −1.46821e6 −0.206958
\(551\) −509057. −0.0714312
\(552\) 2.60657e6 0.364101
\(553\) −1.29225e7 −1.79694
\(554\) 71695.9 0.00992476
\(555\) 2.63011e6 0.362444
\(556\) 99161.1 0.0136036
\(557\) 5.40419e6 0.738061 0.369031 0.929417i \(-0.379690\pi\)
0.369031 + 0.929417i \(0.379690\pi\)
\(558\) −843164. −0.114638
\(559\) −28671.7 −0.00388082
\(560\) 1.36477e6 0.183903
\(561\) 1.31892e6 0.176935
\(562\) 9.62654e6 1.28567
\(563\) 904604. 0.120278 0.0601392 0.998190i \(-0.480846\pi\)
0.0601392 + 0.998190i \(0.480846\pi\)
\(564\) 4.77599e6 0.632217
\(565\) 5.48393e6 0.722722
\(566\) 5.47683e6 0.718602
\(567\) −9.30257e6 −1.21519
\(568\) 4.98215e6 0.647956
\(569\) −6.69355e6 −0.866714 −0.433357 0.901222i \(-0.642671\pi\)
−0.433357 + 0.901222i \(0.642671\pi\)
\(570\) 112413. 0.0144920
\(571\) 8.22981e6 1.05633 0.528165 0.849142i \(-0.322880\pi\)
0.528165 + 0.849142i \(0.322880\pi\)
\(572\) −1.58802e6 −0.202939
\(573\) 2.85548e6 0.363322
\(574\) −817260. −0.103533
\(575\) −1.84303e6 −0.232467
\(576\) −213998. −0.0268753
\(577\) 3.07720e6 0.384783 0.192391 0.981318i \(-0.438376\pi\)
0.192391 + 0.981318i \(0.438376\pi\)
\(578\) 5.57367e6 0.693939
\(579\) −5.70532e6 −0.707267
\(580\) 2.50177e6 0.308801
\(581\) −566051. −0.0695689
\(582\) 2.97464e6 0.364021
\(583\) 1.45583e7 1.77394
\(584\) 3.97633e6 0.482448
\(585\) 220737. 0.0266678
\(586\) 1.56260e6 0.187976
\(587\) 2.64175e6 0.316443 0.158222 0.987404i \(-0.449424\pi\)
0.158222 + 0.987404i \(0.449424\pi\)
\(588\) 6.33482e6 0.755599
\(589\) 328384. 0.0390026
\(590\) −4.01092e6 −0.474367
\(591\) 7.44193e6 0.876429
\(592\) 1.95001e6 0.228682
\(593\) −5.06641e6 −0.591649 −0.295824 0.955242i \(-0.595594\pi\)
−0.295824 + 0.955242i \(0.595594\pi\)
\(594\) 9.57920e6 1.11394
\(595\) 866870. 0.100383
\(596\) −5.93654e6 −0.684569
\(597\) −2.03492e6 −0.233674
\(598\) −1.99342e6 −0.227953
\(599\) 1.31136e7 1.49332 0.746662 0.665204i \(-0.231655\pi\)
0.746662 + 0.665204i \(0.231655\pi\)
\(600\) −552456. −0.0626497
\(601\) 7.53425e6 0.850852 0.425426 0.904993i \(-0.360124\pi\)
0.425426 + 0.904993i \(0.360124\pi\)
\(602\) −144713. −0.0162748
\(603\) −959341. −0.107443
\(604\) −8.13378e6 −0.907193
\(605\) 4.59631e6 0.510529
\(606\) 9.25672e6 1.02394
\(607\) 8.76101e6 0.965123 0.482561 0.875862i \(-0.339707\pi\)
0.482561 + 0.875862i \(0.339707\pi\)
\(608\) 83344.8 0.00914365
\(609\) 1.84207e7 2.01262
\(610\) 1.63430e6 0.177831
\(611\) −3.65253e6 −0.395813
\(612\) −135926. −0.0146698
\(613\) −469316. −0.0504445 −0.0252223 0.999682i \(-0.508029\pi\)
−0.0252223 + 0.999682i \(0.508029\pi\)
\(614\) −2.26618e6 −0.242591
\(615\) 330824. 0.0352704
\(616\) −8.01509e6 −0.851054
\(617\) −4.14114e6 −0.437933 −0.218966 0.975732i \(-0.570268\pi\)
−0.218966 + 0.975732i \(0.570268\pi\)
\(618\) −1.71116e6 −0.180227
\(619\) −1.31422e7 −1.37861 −0.689304 0.724472i \(-0.742084\pi\)
−0.689304 + 0.724472i \(0.742084\pi\)
\(620\) −1.61385e6 −0.168610
\(621\) 1.20246e7 1.25125
\(622\) −3.52496e6 −0.365324
\(623\) 1.75496e6 0.181154
\(624\) −597536. −0.0614331
\(625\) 390625. 0.0400000
\(626\) 3.66569e6 0.373869
\(627\) −660184. −0.0670650
\(628\) 960239. 0.0971583
\(629\) 1.23860e6 0.124826
\(630\) 1.11411e6 0.111835
\(631\) 9.79229e6 0.979063 0.489532 0.871986i \(-0.337168\pi\)
0.489532 + 0.871986i \(0.337168\pi\)
\(632\) 3.87834e6 0.386237
\(633\) −1.58722e7 −1.57445
\(634\) −8.04335e6 −0.794719
\(635\) 3.96883e6 0.390596
\(636\) 5.47795e6 0.537001
\(637\) −4.84467e6 −0.473059
\(638\) −1.46925e7 −1.42904
\(639\) 4.06711e6 0.394034
\(640\) −409600. −0.0395285
\(641\) 3.38154e6 0.325064 0.162532 0.986703i \(-0.448034\pi\)
0.162532 + 0.986703i \(0.448034\pi\)
\(642\) 5.37781e6 0.514953
\(643\) −3.04366e6 −0.290314 −0.145157 0.989409i \(-0.546369\pi\)
−0.145157 + 0.989409i \(0.546369\pi\)
\(644\) −1.00612e7 −0.955954
\(645\) 58579.2 0.00554427
\(646\) 52938.6 0.00499104
\(647\) 8.51735e6 0.799915 0.399957 0.916534i \(-0.369025\pi\)
0.399957 + 0.916534i \(0.369025\pi\)
\(648\) 2.79192e6 0.261195
\(649\) 2.35555e7 2.19524
\(650\) 422500. 0.0392232
\(651\) −1.18828e7 −1.09892
\(652\) −5.85461e6 −0.539360
\(653\) −8.83998e6 −0.811275 −0.405638 0.914034i \(-0.632950\pi\)
−0.405638 + 0.914034i \(0.632950\pi\)
\(654\) −922425. −0.0843309
\(655\) 4.01415e6 0.365587
\(656\) 245279. 0.0222536
\(657\) 3.24603e6 0.293386
\(658\) −1.84351e7 −1.65990
\(659\) −2.97807e6 −0.267129 −0.133564 0.991040i \(-0.542642\pi\)
−0.133564 + 0.991040i \(0.542642\pi\)
\(660\) 3.24449e6 0.289925
\(661\) −7.90309e6 −0.703548 −0.351774 0.936085i \(-0.614421\pi\)
−0.351774 + 0.936085i \(0.614421\pi\)
\(662\) −9.90860e6 −0.878754
\(663\) −379540. −0.0335331
\(664\) 169885. 0.0149532
\(665\) −433909. −0.0380491
\(666\) 1.59186e6 0.139066
\(667\) −1.84433e7 −1.60519
\(668\) 1.07827e6 0.0934948
\(669\) −1.92572e7 −1.66352
\(670\) −1.83621e6 −0.158029
\(671\) −9.59800e6 −0.822952
\(672\) −3.01590e6 −0.257629
\(673\) 1.82522e7 1.55338 0.776688 0.629885i \(-0.216898\pi\)
0.776688 + 0.629885i \(0.216898\pi\)
\(674\) −1.02220e7 −0.866734
\(675\) −2.54859e6 −0.215299
\(676\) 456976. 0.0384615
\(677\) 5.39380e6 0.452296 0.226148 0.974093i \(-0.427387\pi\)
0.226148 + 0.974093i \(0.427387\pi\)
\(678\) −1.21185e7 −1.01245
\(679\) −1.14820e7 −0.955746
\(680\) −260168. −0.0215765
\(681\) 1.93762e7 1.60103
\(682\) 9.47789e6 0.780280
\(683\) 2.30171e7 1.88799 0.943994 0.329963i \(-0.107036\pi\)
0.943994 + 0.329963i \(0.107036\pi\)
\(684\) 68037.4 0.00556042
\(685\) −7.39169e6 −0.601891
\(686\) −1.01161e7 −0.820733
\(687\) −1.10025e7 −0.889406
\(688\) 43431.6 0.00349812
\(689\) −4.18936e6 −0.336201
\(690\) 4.07276e6 0.325661
\(691\) 4.87910e6 0.388727 0.194364 0.980930i \(-0.437736\pi\)
0.194364 + 0.980930i \(0.437736\pi\)
\(692\) 8.30274e6 0.659107
\(693\) −6.54302e6 −0.517541
\(694\) 1.90000e6 0.149746
\(695\) 154939. 0.0121674
\(696\) −5.52847e6 −0.432596
\(697\) 155795. 0.0121471
\(698\) −2.46111e6 −0.191202
\(699\) 1.60178e7 1.23996
\(700\) 2.13246e6 0.164488
\(701\) 1.47820e7 1.13615 0.568077 0.822976i \(-0.307688\pi\)
0.568077 + 0.822976i \(0.307688\pi\)
\(702\) −2.75656e6 −0.211118
\(703\) −619976. −0.0473137
\(704\) 2.40552e6 0.182927
\(705\) 7.46249e6 0.565472
\(706\) 1.74960e7 1.32107
\(707\) −3.57305e7 −2.68838
\(708\) 8.86343e6 0.664536
\(709\) −2.75364e6 −0.205727 −0.102863 0.994695i \(-0.532800\pi\)
−0.102863 + 0.994695i \(0.532800\pi\)
\(710\) 7.78460e6 0.579550
\(711\) 3.16603e6 0.234878
\(712\) −526705. −0.0389375
\(713\) 1.18975e7 0.876457
\(714\) −1.91563e6 −0.140626
\(715\) −2.48128e6 −0.181514
\(716\) −1.43566e6 −0.104657
\(717\) −9.49574e6 −0.689812
\(718\) −1.83825e7 −1.33074
\(719\) −2.10250e7 −1.51675 −0.758373 0.651820i \(-0.774006\pi\)
−0.758373 + 0.651820i \(0.774006\pi\)
\(720\) −334372. −0.0240380
\(721\) 6.60502e6 0.473191
\(722\) 9.87790e6 0.705215
\(723\) −5.40502e6 −0.384549
\(724\) −6.02035e6 −0.426850
\(725\) 3.90902e6 0.276200
\(726\) −1.01570e7 −0.715196
\(727\) 2.21381e7 1.55347 0.776737 0.629825i \(-0.216873\pi\)
0.776737 + 0.629825i \(0.216873\pi\)
\(728\) 2.30646e6 0.161294
\(729\) 1.59648e7 1.11261
\(730\) 6.21302e6 0.431515
\(731\) 27586.7 0.00190944
\(732\) −3.61152e6 −0.249122
\(733\) 1.23177e7 0.846775 0.423388 0.905949i \(-0.360841\pi\)
0.423388 + 0.905949i \(0.360841\pi\)
\(734\) 1.83745e7 1.25885
\(735\) 9.89816e6 0.675828
\(736\) 3.01962e6 0.205474
\(737\) 1.07838e7 0.731313
\(738\) 200230. 0.0135328
\(739\) −1.79039e7 −1.20597 −0.602986 0.797752i \(-0.706023\pi\)
−0.602986 + 0.797752i \(0.706023\pi\)
\(740\) 3.04689e6 0.204539
\(741\) 189978. 0.0127103
\(742\) −2.11447e7 −1.40991
\(743\) −9.82085e6 −0.652645 −0.326323 0.945258i \(-0.605810\pi\)
−0.326323 + 0.945258i \(0.605810\pi\)
\(744\) 3.56632e6 0.236204
\(745\) −9.27584e6 −0.612297
\(746\) −6.97256e6 −0.458717
\(747\) 138684. 0.00909334
\(748\) 1.52793e6 0.0998501
\(749\) −2.07581e7 −1.35202
\(750\) −863212. −0.0560356
\(751\) 1.04747e7 0.677705 0.338852 0.940840i \(-0.389961\pi\)
0.338852 + 0.940840i \(0.389961\pi\)
\(752\) 5.53282e6 0.356781
\(753\) 1.30479e7 0.838599
\(754\) 4.22800e6 0.270836
\(755\) −1.27090e7 −0.811418
\(756\) −1.39130e7 −0.885352
\(757\) 2.20487e7 1.39844 0.699219 0.714908i \(-0.253531\pi\)
0.699219 + 0.714908i \(0.253531\pi\)
\(758\) 1.77012e7 1.11900
\(759\) −2.39187e7 −1.50707
\(760\) 130226. 0.00817833
\(761\) −2.75293e7 −1.72319 −0.861596 0.507594i \(-0.830535\pi\)
−0.861596 + 0.507594i \(0.830535\pi\)
\(762\) −8.77040e6 −0.547182
\(763\) 3.56052e6 0.221413
\(764\) 3.30797e6 0.205035
\(765\) −212385. −0.0131211
\(766\) −5.59091e6 −0.344279
\(767\) −6.77846e6 −0.416047
\(768\) 905143. 0.0553751
\(769\) 1.50464e7 0.917525 0.458763 0.888559i \(-0.348293\pi\)
0.458763 + 0.888559i \(0.348293\pi\)
\(770\) −1.25236e7 −0.761205
\(771\) 4.10501e6 0.248701
\(772\) −6.60941e6 −0.399135
\(773\) 2.28606e6 0.137606 0.0688031 0.997630i \(-0.478082\pi\)
0.0688031 + 0.997630i \(0.478082\pi\)
\(774\) 35454.8 0.00212727
\(775\) −2.52164e6 −0.150810
\(776\) 3.44601e6 0.205430
\(777\) 2.24344e7 1.33310
\(778\) −6.54142e6 −0.387456
\(779\) −77982.8 −0.00460421
\(780\) −933650. −0.0549474
\(781\) −4.57178e7 −2.68199
\(782\) 1.91799e6 0.112158
\(783\) −2.55040e7 −1.48663
\(784\) 7.33867e6 0.426410
\(785\) 1.50037e6 0.0869011
\(786\) −8.87056e6 −0.512147
\(787\) 9.89052e6 0.569223 0.284611 0.958643i \(-0.408135\pi\)
0.284611 + 0.958643i \(0.408135\pi\)
\(788\) 8.62121e6 0.494598
\(789\) −2.11449e7 −1.20924
\(790\) 6.05991e6 0.345461
\(791\) 4.67770e7 2.65822
\(792\) 1.96371e6 0.111241
\(793\) 2.76197e6 0.155968
\(794\) −1.96261e7 −1.10480
\(795\) 8.55930e6 0.480309
\(796\) −2.35738e6 −0.131870
\(797\) 2.93049e7 1.63416 0.817080 0.576525i \(-0.195592\pi\)
0.817080 + 0.576525i \(0.195592\pi\)
\(798\) 958862. 0.0533026
\(799\) 3.51431e6 0.194748
\(800\) −640000. −0.0353553
\(801\) −429969. −0.0236786
\(802\) 3.80474e6 0.208876
\(803\) −3.64881e7 −1.99693
\(804\) 4.05771e6 0.221381
\(805\) −1.57207e7 −0.855031
\(806\) −2.72741e6 −0.147881
\(807\) −1.39327e7 −0.753095
\(808\) 1.07236e7 0.577845
\(809\) 1.02027e7 0.548078 0.274039 0.961719i \(-0.411640\pi\)
0.274039 + 0.961719i \(0.411640\pi\)
\(810\) 4.36237e6 0.233620
\(811\) −1.26432e7 −0.675001 −0.337501 0.941325i \(-0.609582\pi\)
−0.337501 + 0.941325i \(0.609582\pi\)
\(812\) 2.13397e7 1.13579
\(813\) 3.02639e7 1.60582
\(814\) −1.78939e7 −0.946551
\(815\) −9.14782e6 −0.482418
\(816\) 574925. 0.0302264
\(817\) −13808.4 −0.000723752 0
\(818\) −1.79786e7 −0.939446
\(819\) 1.88285e6 0.0980858
\(820\) 383248. 0.0199042
\(821\) −1.56881e7 −0.812294 −0.406147 0.913808i \(-0.633128\pi\)
−0.406147 + 0.913808i \(0.633128\pi\)
\(822\) 1.63343e7 0.843183
\(823\) 2.45799e7 1.26497 0.632487 0.774571i \(-0.282034\pi\)
0.632487 + 0.774571i \(0.282034\pi\)
\(824\) −1.98232e6 −0.101708
\(825\) 5.06951e6 0.259317
\(826\) −3.42125e7 −1.74475
\(827\) 3.62365e7 1.84239 0.921197 0.389098i \(-0.127213\pi\)
0.921197 + 0.389098i \(0.127213\pi\)
\(828\) 2.46502e6 0.124953
\(829\) −2.81815e7 −1.42422 −0.712110 0.702068i \(-0.752260\pi\)
−0.712110 + 0.702068i \(0.752260\pi\)
\(830\) 265446. 0.0133746
\(831\) −247555. −0.0124357
\(832\) −692224. −0.0346688
\(833\) 4.66134e6 0.232755
\(834\) −342388. −0.0170453
\(835\) 1.68480e6 0.0836243
\(836\) −764799. −0.0378470
\(837\) 1.64522e7 0.811727
\(838\) 3.31562e6 0.163100
\(839\) −8.42806e6 −0.413354 −0.206677 0.978409i \(-0.566265\pi\)
−0.206677 + 0.978409i \(0.566265\pi\)
\(840\) −4.71235e6 −0.230430
\(841\) 1.86068e7 0.907155
\(842\) −2.97925e6 −0.144820
\(843\) −3.32390e7 −1.61094
\(844\) −1.83874e7 −0.888515
\(845\) 714025. 0.0344010
\(846\) 4.51664e6 0.216965
\(847\) 3.92057e7 1.87776
\(848\) 6.34601e6 0.303048
\(849\) −1.89107e7 −0.900404
\(850\) −406512. −0.0192986
\(851\) −2.24620e7 −1.06322
\(852\) −1.72026e7 −0.811886
\(853\) −3.10763e7 −1.46237 −0.731184 0.682180i \(-0.761032\pi\)
−0.731184 + 0.682180i \(0.761032\pi\)
\(854\) 1.39403e7 0.654075
\(855\) 106309. 0.00497339
\(856\) 6.23000e6 0.290605
\(857\) −5.72064e6 −0.266068 −0.133034 0.991111i \(-0.542472\pi\)
−0.133034 + 0.991111i \(0.542472\pi\)
\(858\) 5.48318e6 0.254281
\(859\) −6.04140e6 −0.279354 −0.139677 0.990197i \(-0.544606\pi\)
−0.139677 + 0.990197i \(0.544606\pi\)
\(860\) 67861.9 0.00312882
\(861\) 2.82187e6 0.129727
\(862\) 1.55812e7 0.714220
\(863\) 2.09227e7 0.956292 0.478146 0.878280i \(-0.341309\pi\)
0.478146 + 0.878280i \(0.341309\pi\)
\(864\) 4.17562e6 0.190299
\(865\) 1.29730e7 0.589523
\(866\) 2.35319e7 1.06626
\(867\) −1.92450e7 −0.869502
\(868\) −1.37658e7 −0.620160
\(869\) −3.55889e7 −1.59869
\(870\) −8.63824e6 −0.386925
\(871\) −3.10320e6 −0.138601
\(872\) −1.06860e6 −0.0475908
\(873\) 2.81311e6 0.124925
\(874\) −960042. −0.0425120
\(875\) 3.33196e6 0.147123
\(876\) −1.37297e7 −0.604505
\(877\) −5.01539e6 −0.220194 −0.110097 0.993921i \(-0.535116\pi\)
−0.110097 + 0.993921i \(0.535116\pi\)
\(878\) −6.92097e6 −0.302992
\(879\) −5.39541e6 −0.235533
\(880\) 3.75862e6 0.163615
\(881\) −2.14791e7 −0.932346 −0.466173 0.884694i \(-0.654368\pi\)
−0.466173 + 0.884694i \(0.654368\pi\)
\(882\) 5.99082e6 0.259307
\(883\) 996549. 0.0430127 0.0215064 0.999769i \(-0.493154\pi\)
0.0215064 + 0.999769i \(0.493154\pi\)
\(884\) −439684. −0.0189239
\(885\) 1.38491e7 0.594379
\(886\) 425728. 0.0182200
\(887\) 4.67336e6 0.199444 0.0997219 0.995015i \(-0.468205\pi\)
0.0997219 + 0.995015i \(0.468205\pi\)
\(888\) −6.73308e6 −0.286537
\(889\) 3.38534e7 1.43664
\(890\) −822977. −0.0348267
\(891\) −2.56196e7 −1.08113
\(892\) −2.23088e7 −0.938782
\(893\) −1.75908e6 −0.0738170
\(894\) 2.04980e7 0.857762
\(895\) −2.24321e6 −0.0936080
\(896\) −3.49382e6 −0.145388
\(897\) 6.88297e6 0.285624
\(898\) 1.95688e7 0.809792
\(899\) −2.52343e7 −1.04134
\(900\) −522456. −0.0215002
\(901\) 4.03083e6 0.165418
\(902\) −2.25076e6 −0.0921112
\(903\) 499670. 0.0203922
\(904\) −1.40389e7 −0.571362
\(905\) −9.40680e6 −0.381786
\(906\) 2.80847e7 1.13671
\(907\) 1.18130e7 0.476807 0.238403 0.971166i \(-0.423376\pi\)
0.238403 + 0.971166i \(0.423376\pi\)
\(908\) 2.24466e7 0.903515
\(909\) 8.75405e6 0.351398
\(910\) 3.60385e6 0.144266
\(911\) 3.55560e7 1.41944 0.709721 0.704483i \(-0.248821\pi\)
0.709721 + 0.704483i \(0.248821\pi\)
\(912\) −287777. −0.0114569
\(913\) −1.55892e6 −0.0618938
\(914\) −709006. −0.0280727
\(915\) −5.64299e6 −0.222821
\(916\) −1.27460e7 −0.501922
\(917\) 3.42400e7 1.34465
\(918\) 2.65225e6 0.103874
\(919\) 2.66902e7 1.04247 0.521235 0.853413i \(-0.325472\pi\)
0.521235 + 0.853413i \(0.325472\pi\)
\(920\) 4.71815e6 0.183782
\(921\) 7.82478e6 0.303965
\(922\) −2.86270e7 −1.10905
\(923\) 1.31560e7 0.508299
\(924\) 2.76749e7 1.06637
\(925\) 4.76076e6 0.182946
\(926\) −1.37691e7 −0.527690
\(927\) −1.61824e6 −0.0618507
\(928\) −6.40454e6 −0.244128
\(929\) −3.08229e7 −1.17175 −0.585874 0.810402i \(-0.699249\pi\)
−0.585874 + 0.810402i \(0.699249\pi\)
\(930\) 5.57237e6 0.211268
\(931\) −2.33322e6 −0.0882230
\(932\) 1.85560e7 0.699753
\(933\) 1.21711e7 0.457749
\(934\) 3.68994e7 1.38405
\(935\) 2.38739e6 0.0893087
\(936\) −565088. −0.0210827
\(937\) 2.78462e7 1.03613 0.518067 0.855340i \(-0.326652\pi\)
0.518067 + 0.855340i \(0.326652\pi\)
\(938\) −1.56626e7 −0.581241
\(939\) −1.26571e7 −0.468456
\(940\) 8.64503e6 0.319115
\(941\) 2.80836e7 1.03390 0.516950 0.856016i \(-0.327067\pi\)
0.516950 + 0.856016i \(0.327067\pi\)
\(942\) −3.31556e6 −0.121739
\(943\) −2.82535e6 −0.103465
\(944\) 1.02680e7 0.375020
\(945\) −2.17391e7 −0.791883
\(946\) −398543. −0.0144793
\(947\) −4.49006e7 −1.62696 −0.813481 0.581592i \(-0.802430\pi\)
−0.813481 + 0.581592i \(0.802430\pi\)
\(948\) −1.33913e7 −0.483953
\(949\) 1.05000e7 0.378463
\(950\) 203479. 0.00731492
\(951\) 2.77724e7 0.995779
\(952\) −2.21919e6 −0.0793600
\(953\) −1.13992e7 −0.406577 −0.203289 0.979119i \(-0.565163\pi\)
−0.203289 + 0.979119i \(0.565163\pi\)
\(954\) 5.18048e6 0.184289
\(955\) 5.16870e6 0.183389
\(956\) −1.10005e7 −0.389284
\(957\) 5.07311e7 1.79058
\(958\) −4.93322e6 −0.173667
\(959\) −6.30498e7 −2.21380
\(960\) 1.41429e6 0.0495290
\(961\) −1.23510e7 −0.431412
\(962\) 5.14924e6 0.179393
\(963\) 5.08578e6 0.176722
\(964\) −6.26152e6 −0.217014
\(965\) −1.03272e7 −0.356997
\(966\) 3.47399e7 1.19781
\(967\) −5.54751e7 −1.90780 −0.953898 0.300130i \(-0.902970\pi\)
−0.953898 + 0.300130i \(0.902970\pi\)
\(968\) −1.17665e7 −0.403609
\(969\) −182789. −0.00625375
\(970\) 5.38440e6 0.183742
\(971\) −2.45562e6 −0.0835819 −0.0417910 0.999126i \(-0.513306\pi\)
−0.0417910 + 0.999126i \(0.513306\pi\)
\(972\) 6.21423e6 0.210970
\(973\) 1.32160e6 0.0447527
\(974\) 2.48510e7 0.839358
\(975\) −1.45883e6 −0.0491465
\(976\) −4.18381e6 −0.140588
\(977\) 3.31344e7 1.11056 0.555280 0.831663i \(-0.312611\pi\)
0.555280 + 0.831663i \(0.312611\pi\)
\(978\) 2.02151e7 0.675815
\(979\) 4.83322e6 0.161168
\(980\) 1.14667e7 0.381392
\(981\) −872335. −0.0289408
\(982\) 2.82088e7 0.933483
\(983\) −4.46461e7 −1.47367 −0.736834 0.676073i \(-0.763680\pi\)
−0.736834 + 0.676073i \(0.763680\pi\)
\(984\) −846911. −0.0278837
\(985\) 1.34706e7 0.442382
\(986\) −4.06801e6 −0.133257
\(987\) 6.36537e7 2.07985
\(988\) 220082. 0.00717287
\(989\) −500285. −0.0162640
\(990\) 3.06830e6 0.0994970
\(991\) 4.99797e7 1.61662 0.808312 0.588754i \(-0.200381\pi\)
0.808312 + 0.588754i \(0.200381\pi\)
\(992\) 4.13145e6 0.133298
\(993\) 3.42129e7 1.10107
\(994\) 6.64013e7 2.13162
\(995\) −3.68340e6 −0.117948
\(996\) −586588. −0.0187363
\(997\) −1.25647e7 −0.400326 −0.200163 0.979763i \(-0.564147\pi\)
−0.200163 + 0.979763i \(0.564147\pi\)
\(998\) −2.25204e7 −0.715730
\(999\) −3.10611e7 −0.984699
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 130.6.a.a.1.2 2
4.3 odd 2 1040.6.a.e.1.1 2
5.2 odd 4 650.6.b.g.599.1 4
5.3 odd 4 650.6.b.g.599.4 4
5.4 even 2 650.6.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.6.a.a.1.2 2 1.1 even 1 trivial
650.6.a.h.1.1 2 5.4 even 2
650.6.b.g.599.1 4 5.2 odd 4
650.6.b.g.599.4 4 5.3 odd 4
1040.6.a.e.1.1 2 4.3 odd 2