Properties

Label 175.6.a.d.1.2
Level $175$
Weight $6$
Character 175.1
Self dual yes
Analytic conductor $28.067$
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] = [175,6,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, 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("175.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.53113 q^{2} -13.5934 q^{3} -19.5311 q^{4} -48.0000 q^{6} +49.0000 q^{7} -181.963 q^{8} -58.2198 q^{9} +O(q^{10})\) \(q+3.53113 q^{2} -13.5934 q^{3} -19.5311 q^{4} -48.0000 q^{6} +49.0000 q^{7} -181.963 q^{8} -58.2198 q^{9} -691.520 q^{11} +265.494 q^{12} +502.150 q^{13} +173.025 q^{14} -17.5389 q^{16} +991.313 q^{17} -205.582 q^{18} +661.677 q^{19} -666.076 q^{21} -2441.84 q^{22} -3415.08 q^{23} +2473.49 q^{24} +1773.16 q^{26} +4094.60 q^{27} -957.025 q^{28} +6751.92 q^{29} -3922.76 q^{31} +5760.89 q^{32} +9400.09 q^{33} +3500.46 q^{34} +1137.10 q^{36} -627.222 q^{37} +2336.47 q^{38} -6825.92 q^{39} +16277.9 q^{41} -2352.00 q^{42} +17277.7 q^{43} +13506.2 q^{44} -12059.1 q^{46} +4295.47 q^{47} +238.413 q^{48} +2401.00 q^{49} -13475.3 q^{51} -9807.55 q^{52} +25960.9 q^{53} +14458.6 q^{54} -8916.19 q^{56} -8994.43 q^{57} +23841.9 q^{58} +8902.63 q^{59} -48924.6 q^{61} -13851.8 q^{62} -2852.77 q^{63} +20903.7 q^{64} +33192.9 q^{66} +4257.80 q^{67} -19361.5 q^{68} +46422.5 q^{69} +18990.9 q^{71} +10593.9 q^{72} -10132.5 q^{73} -2214.80 q^{74} -12923.3 q^{76} -33884.5 q^{77} -24103.2 q^{78} -96986.5 q^{79} -41512.0 q^{81} +57479.2 q^{82} -70732.1 q^{83} +13009.2 q^{84} +61009.8 q^{86} -91781.4 q^{87} +125831. q^{88} +4241.12 q^{89} +24605.3 q^{91} +66700.3 q^{92} +53323.6 q^{93} +15167.9 q^{94} -78309.9 q^{96} +104376. q^{97} +8478.24 q^{98} +40260.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{3} - 31 q^{4} - 96 q^{6} + 98 q^{7} + 15 q^{8} - 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 3 q^{3} - 31 q^{4} - 96 q^{6} + 98 q^{7} + 15 q^{8} - 189 q^{9} - 601 q^{11} + 144 q^{12} + 577 q^{13} - 49 q^{14} - 543 q^{16} - 41 q^{17} + 387 q^{18} + 630 q^{19} - 147 q^{21} - 2852 q^{22} + 442 q^{23} + 4560 q^{24} + 1434 q^{26} + 135 q^{27} - 1519 q^{28} + 5885 q^{29} - 396 q^{31} + 1839 q^{32} + 10359 q^{33} + 8178 q^{34} + 2637 q^{36} + 8904 q^{37} + 2480 q^{38} - 6033 q^{39} + 1774 q^{41} - 4704 q^{42} + 27122 q^{43} + 12468 q^{44} - 29536 q^{46} + 21289 q^{47} - 5328 q^{48} + 4802 q^{49} - 24411 q^{51} - 10666 q^{52} + 55582 q^{53} + 32400 q^{54} + 735 q^{56} - 9330 q^{57} + 27770 q^{58} + 59600 q^{59} - 51846 q^{61} - 29832 q^{62} - 9261 q^{63} + 55489 q^{64} + 28848 q^{66} + 45344 q^{67} - 7522 q^{68} + 87282 q^{69} + 80744 q^{71} - 15165 q^{72} + 13532 q^{73} - 45402 q^{74} - 12560 q^{76} - 29449 q^{77} - 27696 q^{78} - 51795 q^{79} - 51678 q^{81} + 123198 q^{82} - 109828 q^{83} + 7056 q^{84} + 16404 q^{86} - 100965 q^{87} + 143660 q^{88} - 37650 q^{89} + 28273 q^{91} + 22464 q^{92} + 90684 q^{93} - 61832 q^{94} - 119856 q^{96} + 96339 q^{97} - 2401 q^{98} + 28422 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.53113 0.624221 0.312111 0.950046i \(-0.398964\pi\)
0.312111 + 0.950046i \(0.398964\pi\)
\(3\) −13.5934 −0.872016 −0.436008 0.899943i \(-0.643608\pi\)
−0.436008 + 0.899943i \(0.643608\pi\)
\(4\) −19.5311 −0.610348
\(5\) 0 0
\(6\) −48.0000 −0.544331
\(7\) 49.0000 0.377964
\(8\) −181.963 −1.00521
\(9\) −58.2198 −0.239588
\(10\) 0 0
\(11\) −691.520 −1.72315 −0.861574 0.507632i \(-0.830521\pi\)
−0.861574 + 0.507632i \(0.830521\pi\)
\(12\) 265.494 0.532233
\(13\) 502.150 0.824091 0.412045 0.911163i \(-0.364815\pi\)
0.412045 + 0.911163i \(0.364815\pi\)
\(14\) 173.025 0.235933
\(15\) 0 0
\(16\) −17.5389 −0.0171278
\(17\) 991.313 0.831934 0.415967 0.909380i \(-0.363443\pi\)
0.415967 + 0.909380i \(0.363443\pi\)
\(18\) −205.582 −0.149556
\(19\) 661.677 0.420496 0.210248 0.977648i \(-0.432573\pi\)
0.210248 + 0.977648i \(0.432573\pi\)
\(20\) 0 0
\(21\) −666.076 −0.329591
\(22\) −2441.84 −1.07563
\(23\) −3415.08 −1.34611 −0.673056 0.739592i \(-0.735019\pi\)
−0.673056 + 0.739592i \(0.735019\pi\)
\(24\) 2473.49 0.876562
\(25\) 0 0
\(26\) 1773.16 0.514415
\(27\) 4094.60 1.08094
\(28\) −957.025 −0.230690
\(29\) 6751.92 1.49084 0.745422 0.666593i \(-0.232248\pi\)
0.745422 + 0.666593i \(0.232248\pi\)
\(30\) 0 0
\(31\) −3922.76 −0.733142 −0.366571 0.930390i \(-0.619468\pi\)
−0.366571 + 0.930390i \(0.619468\pi\)
\(32\) 5760.89 0.994522
\(33\) 9400.09 1.50261
\(34\) 3500.46 0.519311
\(35\) 0 0
\(36\) 1137.10 0.146232
\(37\) −627.222 −0.0753212 −0.0376606 0.999291i \(-0.511991\pi\)
−0.0376606 + 0.999291i \(0.511991\pi\)
\(38\) 2336.47 0.262483
\(39\) −6825.92 −0.718620
\(40\) 0 0
\(41\) 16277.9 1.51230 0.756149 0.654399i \(-0.227078\pi\)
0.756149 + 0.654399i \(0.227078\pi\)
\(42\) −2352.00 −0.205738
\(43\) 17277.7 1.42500 0.712500 0.701672i \(-0.247563\pi\)
0.712500 + 0.701672i \(0.247563\pi\)
\(44\) 13506.2 1.05172
\(45\) 0 0
\(46\) −12059.1 −0.840272
\(47\) 4295.47 0.283639 0.141820 0.989893i \(-0.454705\pi\)
0.141820 + 0.989893i \(0.454705\pi\)
\(48\) 238.413 0.0149357
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −13475.3 −0.725460
\(52\) −9807.55 −0.502982
\(53\) 25960.9 1.26949 0.634745 0.772721i \(-0.281105\pi\)
0.634745 + 0.772721i \(0.281105\pi\)
\(54\) 14458.6 0.674746
\(55\) 0 0
\(56\) −8916.19 −0.379935
\(57\) −8994.43 −0.366679
\(58\) 23841.9 0.930616
\(59\) 8902.63 0.332957 0.166479 0.986045i \(-0.446760\pi\)
0.166479 + 0.986045i \(0.446760\pi\)
\(60\) 0 0
\(61\) −48924.6 −1.68346 −0.841730 0.539898i \(-0.818463\pi\)
−0.841730 + 0.539898i \(0.818463\pi\)
\(62\) −13851.8 −0.457643
\(63\) −2852.77 −0.0905557
\(64\) 20903.7 0.637930
\(65\) 0 0
\(66\) 33192.9 0.937963
\(67\) 4257.80 0.115877 0.0579387 0.998320i \(-0.481547\pi\)
0.0579387 + 0.998320i \(0.481547\pi\)
\(68\) −19361.5 −0.507769
\(69\) 46422.5 1.17383
\(70\) 0 0
\(71\) 18990.9 0.447095 0.223547 0.974693i \(-0.428236\pi\)
0.223547 + 0.974693i \(0.428236\pi\)
\(72\) 10593.9 0.240837
\(73\) −10132.5 −0.222541 −0.111270 0.993790i \(-0.535492\pi\)
−0.111270 + 0.993790i \(0.535492\pi\)
\(74\) −2214.80 −0.0470171
\(75\) 0 0
\(76\) −12923.3 −0.256649
\(77\) −33884.5 −0.651289
\(78\) −24103.2 −0.448578
\(79\) −96986.5 −1.74841 −0.874205 0.485557i \(-0.838617\pi\)
−0.874205 + 0.485557i \(0.838617\pi\)
\(80\) 0 0
\(81\) −41512.0 −0.703010
\(82\) 57479.2 0.944009
\(83\) −70732.1 −1.12699 −0.563497 0.826118i \(-0.690544\pi\)
−0.563497 + 0.826118i \(0.690544\pi\)
\(84\) 13009.2 0.201165
\(85\) 0 0
\(86\) 61009.8 0.889515
\(87\) −91781.4 −1.30004
\(88\) 125831. 1.73213
\(89\) 4241.12 0.0567552 0.0283776 0.999597i \(-0.490966\pi\)
0.0283776 + 0.999597i \(0.490966\pi\)
\(90\) 0 0
\(91\) 24605.3 0.311477
\(92\) 66700.3 0.821596
\(93\) 53323.6 0.639311
\(94\) 15167.9 0.177054
\(95\) 0 0
\(96\) −78309.9 −0.867239
\(97\) 104376. 1.12634 0.563170 0.826341i \(-0.309582\pi\)
0.563170 + 0.826341i \(0.309582\pi\)
\(98\) 8478.24 0.0891745
\(99\) 40260.2 0.412845
\(100\) 0 0
\(101\) −45715.1 −0.445919 −0.222959 0.974828i \(-0.571572\pi\)
−0.222959 + 0.974828i \(0.571572\pi\)
\(102\) −47583.0 −0.452847
\(103\) −89278.1 −0.829186 −0.414593 0.910007i \(-0.636076\pi\)
−0.414593 + 0.910007i \(0.636076\pi\)
\(104\) −91372.7 −0.828387
\(105\) 0 0
\(106\) 91671.2 0.792443
\(107\) 106330. 0.897834 0.448917 0.893573i \(-0.351810\pi\)
0.448917 + 0.893573i \(0.351810\pi\)
\(108\) −79972.1 −0.659750
\(109\) −49816.5 −0.401613 −0.200806 0.979631i \(-0.564356\pi\)
−0.200806 + 0.979631i \(0.564356\pi\)
\(110\) 0 0
\(111\) 8526.08 0.0656813
\(112\) −859.405 −0.00647370
\(113\) 37160.7 0.273771 0.136886 0.990587i \(-0.456291\pi\)
0.136886 + 0.990587i \(0.456291\pi\)
\(114\) −31760.5 −0.228889
\(115\) 0 0
\(116\) −131873. −0.909933
\(117\) −29235.1 −0.197442
\(118\) 31436.3 0.207839
\(119\) 48574.4 0.314441
\(120\) 0 0
\(121\) 317148. 1.96924
\(122\) −172759. −1.05085
\(123\) −221271. −1.31875
\(124\) 76616.0 0.447471
\(125\) 0 0
\(126\) −10073.5 −0.0565268
\(127\) 46510.2 0.255882 0.127941 0.991782i \(-0.459163\pi\)
0.127941 + 0.991782i \(0.459163\pi\)
\(128\) −110535. −0.596313
\(129\) −234862. −1.24262
\(130\) 0 0
\(131\) 381771. 1.94368 0.971839 0.235646i \(-0.0757206\pi\)
0.971839 + 0.235646i \(0.0757206\pi\)
\(132\) −183594. −0.917117
\(133\) 32422.2 0.158933
\(134\) 15034.9 0.0723331
\(135\) 0 0
\(136\) −180382. −0.836271
\(137\) 1894.54 0.00862389 0.00431194 0.999991i \(-0.498627\pi\)
0.00431194 + 0.999991i \(0.498627\pi\)
\(138\) 163924. 0.732730
\(139\) 201798. 0.885889 0.442944 0.896549i \(-0.353934\pi\)
0.442944 + 0.896549i \(0.353934\pi\)
\(140\) 0 0
\(141\) −58390.0 −0.247338
\(142\) 67059.3 0.279086
\(143\) −347246. −1.42003
\(144\) 1021.11 0.00410362
\(145\) 0 0
\(146\) −35779.1 −0.138915
\(147\) −32637.7 −0.124574
\(148\) 12250.4 0.0459721
\(149\) −466237. −1.72045 −0.860224 0.509917i \(-0.829676\pi\)
−0.860224 + 0.509917i \(0.829676\pi\)
\(150\) 0 0
\(151\) −122212. −0.436185 −0.218093 0.975928i \(-0.569983\pi\)
−0.218093 + 0.975928i \(0.569983\pi\)
\(152\) −120401. −0.422688
\(153\) −57714.1 −0.199321
\(154\) −119650. −0.406548
\(155\) 0 0
\(156\) 133318. 0.438608
\(157\) 410638. 1.32957 0.664784 0.747036i \(-0.268524\pi\)
0.664784 + 0.747036i \(0.268524\pi\)
\(158\) −342472. −1.09139
\(159\) −352896. −1.10702
\(160\) 0 0
\(161\) −167339. −0.508782
\(162\) −146584. −0.438834
\(163\) −78525.4 −0.231495 −0.115747 0.993279i \(-0.536926\pi\)
−0.115747 + 0.993279i \(0.536926\pi\)
\(164\) −317925. −0.923028
\(165\) 0 0
\(166\) −249764. −0.703494
\(167\) 597714. 1.65845 0.829224 0.558916i \(-0.188783\pi\)
0.829224 + 0.558916i \(0.188783\pi\)
\(168\) 121201. 0.331309
\(169\) −119139. −0.320875
\(170\) 0 0
\(171\) −38522.7 −0.100746
\(172\) −337453. −0.869745
\(173\) 59874.0 0.152098 0.0760490 0.997104i \(-0.475769\pi\)
0.0760490 + 0.997104i \(0.475769\pi\)
\(174\) −324092. −0.811512
\(175\) 0 0
\(176\) 12128.5 0.0295138
\(177\) −121017. −0.290344
\(178\) 14975.9 0.0354278
\(179\) 616812. 1.43887 0.719433 0.694562i \(-0.244402\pi\)
0.719433 + 0.694562i \(0.244402\pi\)
\(180\) 0 0
\(181\) −37287.0 −0.0845981 −0.0422990 0.999105i \(-0.513468\pi\)
−0.0422990 + 0.999105i \(0.513468\pi\)
\(182\) 86884.6 0.194431
\(183\) 665051. 1.46800
\(184\) 621418. 1.35313
\(185\) 0 0
\(186\) 188293. 0.399072
\(187\) −685513. −1.43355
\(188\) −83895.4 −0.173119
\(189\) 200635. 0.408557
\(190\) 0 0
\(191\) 326760. 0.648106 0.324053 0.946039i \(-0.394954\pi\)
0.324053 + 0.946039i \(0.394954\pi\)
\(192\) −284152. −0.556285
\(193\) 265735. 0.513518 0.256759 0.966475i \(-0.417345\pi\)
0.256759 + 0.966475i \(0.417345\pi\)
\(194\) 368563. 0.703085
\(195\) 0 0
\(196\) −46894.2 −0.0871925
\(197\) 517865. 0.950716 0.475358 0.879792i \(-0.342318\pi\)
0.475358 + 0.879792i \(0.342318\pi\)
\(198\) 142164. 0.257707
\(199\) −148687. −0.266158 −0.133079 0.991105i \(-0.542486\pi\)
−0.133079 + 0.991105i \(0.542486\pi\)
\(200\) 0 0
\(201\) −57878.0 −0.101047
\(202\) −161426. −0.278352
\(203\) 330844. 0.563486
\(204\) 263188. 0.442783
\(205\) 0 0
\(206\) −315252. −0.517595
\(207\) 198825. 0.322512
\(208\) −8807.15 −0.0141149
\(209\) −457563. −0.724577
\(210\) 0 0
\(211\) 7443.09 0.0115093 0.00575463 0.999983i \(-0.498168\pi\)
0.00575463 + 0.999983i \(0.498168\pi\)
\(212\) −507045. −0.774831
\(213\) −258151. −0.389874
\(214\) 375465. 0.560447
\(215\) 0 0
\(216\) −745066. −1.08658
\(217\) −192215. −0.277101
\(218\) −175909. −0.250695
\(219\) 137735. 0.194059
\(220\) 0 0
\(221\) 497788. 0.685589
\(222\) 30106.7 0.0409997
\(223\) −119157. −0.160456 −0.0802280 0.996777i \(-0.525565\pi\)
−0.0802280 + 0.996777i \(0.525565\pi\)
\(224\) 282283. 0.375894
\(225\) 0 0
\(226\) 131219. 0.170894
\(227\) 388843. 0.500852 0.250426 0.968136i \(-0.419429\pi\)
0.250426 + 0.968136i \(0.419429\pi\)
\(228\) 175671. 0.223802
\(229\) 732622. 0.923191 0.461595 0.887091i \(-0.347277\pi\)
0.461595 + 0.887091i \(0.347277\pi\)
\(230\) 0 0
\(231\) 460605. 0.567934
\(232\) −1.22860e6 −1.49862
\(233\) 1.12639e6 1.35925 0.679626 0.733559i \(-0.262142\pi\)
0.679626 + 0.733559i \(0.262142\pi\)
\(234\) −103233. −0.123248
\(235\) 0 0
\(236\) −173878. −0.203220
\(237\) 1.31837e6 1.52464
\(238\) 171522. 0.196281
\(239\) 772317. 0.874583 0.437292 0.899320i \(-0.355938\pi\)
0.437292 + 0.899320i \(0.355938\pi\)
\(240\) 0 0
\(241\) 1.40297e6 1.55598 0.777991 0.628275i \(-0.216239\pi\)
0.777991 + 0.628275i \(0.216239\pi\)
\(242\) 1.11989e6 1.22924
\(243\) −430698. −0.467905
\(244\) 955553. 1.02750
\(245\) 0 0
\(246\) −781337. −0.823191
\(247\) 332261. 0.346527
\(248\) 713798. 0.736964
\(249\) 961489. 0.982757
\(250\) 0 0
\(251\) −1.63922e6 −1.64230 −0.821151 0.570712i \(-0.806667\pi\)
−0.821151 + 0.570712i \(0.806667\pi\)
\(252\) 55717.9 0.0552705
\(253\) 2.36159e6 2.31955
\(254\) 164234. 0.159727
\(255\) 0 0
\(256\) −1.05923e6 −1.01016
\(257\) 223664. 0.211234 0.105617 0.994407i \(-0.466318\pi\)
0.105617 + 0.994407i \(0.466318\pi\)
\(258\) −829330. −0.775672
\(259\) −30733.9 −0.0284687
\(260\) 0 0
\(261\) −393096. −0.357188
\(262\) 1.34808e6 1.21329
\(263\) −299519. −0.267014 −0.133507 0.991048i \(-0.542624\pi\)
−0.133507 + 0.991048i \(0.542624\pi\)
\(264\) −1.71047e6 −1.51045
\(265\) 0 0
\(266\) 114487. 0.0992091
\(267\) −57651.2 −0.0494914
\(268\) −83159.7 −0.0707255
\(269\) 134341. 0.113195 0.0565974 0.998397i \(-0.481975\pi\)
0.0565974 + 0.998397i \(0.481975\pi\)
\(270\) 0 0
\(271\) 1.93414e6 1.59980 0.799898 0.600136i \(-0.204887\pi\)
0.799898 + 0.600136i \(0.204887\pi\)
\(272\) −17386.5 −0.0142492
\(273\) −334470. −0.271613
\(274\) 6689.88 0.00538321
\(275\) 0 0
\(276\) −906683. −0.716445
\(277\) −177599. −0.139072 −0.0695362 0.997579i \(-0.522152\pi\)
−0.0695362 + 0.997579i \(0.522152\pi\)
\(278\) 712574. 0.552991
\(279\) 228383. 0.175652
\(280\) 0 0
\(281\) 1.85131e6 1.39867 0.699333 0.714796i \(-0.253481\pi\)
0.699333 + 0.714796i \(0.253481\pi\)
\(282\) −206183. −0.154394
\(283\) −2.39851e6 −1.78023 −0.890114 0.455737i \(-0.849376\pi\)
−0.890114 + 0.455737i \(0.849376\pi\)
\(284\) −370914. −0.272883
\(285\) 0 0
\(286\) −1.22617e6 −0.886413
\(287\) 797615. 0.571595
\(288\) −335398. −0.238275
\(289\) −437155. −0.307887
\(290\) 0 0
\(291\) −1.41882e6 −0.982186
\(292\) 197899. 0.135827
\(293\) −2.49922e6 −1.70073 −0.850364 0.526196i \(-0.823618\pi\)
−0.850364 + 0.526196i \(0.823618\pi\)
\(294\) −115248. −0.0777616
\(295\) 0 0
\(296\) 114131. 0.0757139
\(297\) −2.83149e6 −1.86262
\(298\) −1.64634e6 −1.07394
\(299\) −1.71488e6 −1.10932
\(300\) 0 0
\(301\) 846607. 0.538599
\(302\) −431546. −0.272276
\(303\) 621422. 0.388848
\(304\) −11605.1 −0.00720218
\(305\) 0 0
\(306\) −203796. −0.124421
\(307\) −3.07195e6 −1.86024 −0.930119 0.367258i \(-0.880297\pi\)
−0.930119 + 0.367258i \(0.880297\pi\)
\(308\) 661802. 0.397513
\(309\) 1.21359e6 0.723063
\(310\) 0 0
\(311\) 661233. 0.387662 0.193831 0.981035i \(-0.437909\pi\)
0.193831 + 0.981035i \(0.437909\pi\)
\(312\) 1.24206e6 0.722367
\(313\) 3.29393e6 1.90043 0.950217 0.311588i \(-0.100861\pi\)
0.950217 + 0.311588i \(0.100861\pi\)
\(314\) 1.45002e6 0.829944
\(315\) 0 0
\(316\) 1.89426e6 1.06714
\(317\) −639724. −0.357556 −0.178778 0.983889i \(-0.557214\pi\)
−0.178778 + 0.983889i \(0.557214\pi\)
\(318\) −1.24612e6 −0.691023
\(319\) −4.66908e6 −2.56894
\(320\) 0 0
\(321\) −1.44538e6 −0.782926
\(322\) −590895. −0.317593
\(323\) 655929. 0.349825
\(324\) 810777. 0.429081
\(325\) 0 0
\(326\) −277283. −0.144504
\(327\) 677175. 0.350213
\(328\) −2.96197e6 −1.52018
\(329\) 210478. 0.107206
\(330\) 0 0
\(331\) −1.13876e6 −0.571298 −0.285649 0.958334i \(-0.592209\pi\)
−0.285649 + 0.958334i \(0.592209\pi\)
\(332\) 1.38148e6 0.687858
\(333\) 36516.8 0.0180460
\(334\) 2.11060e6 1.03524
\(335\) 0 0
\(336\) 11682.2 0.00564518
\(337\) −685493. −0.328797 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(338\) −420694. −0.200297
\(339\) −505140. −0.238733
\(340\) 0 0
\(341\) 2.71267e6 1.26331
\(342\) −136029. −0.0628877
\(343\) 117649. 0.0539949
\(344\) −3.14390e6 −1.43243
\(345\) 0 0
\(346\) 211423. 0.0949428
\(347\) 1.25151e6 0.557970 0.278985 0.960295i \(-0.410002\pi\)
0.278985 + 0.960295i \(0.410002\pi\)
\(348\) 1.79259e6 0.793476
\(349\) −3.16606e6 −1.39141 −0.695706 0.718327i \(-0.744908\pi\)
−0.695706 + 0.718327i \(0.744908\pi\)
\(350\) 0 0
\(351\) 2.05610e6 0.890793
\(352\) −3.98376e6 −1.71371
\(353\) 2.43368e6 1.03951 0.519754 0.854316i \(-0.326024\pi\)
0.519754 + 0.854316i \(0.326024\pi\)
\(354\) −427326. −0.181239
\(355\) 0 0
\(356\) −82833.8 −0.0346404
\(357\) −660290. −0.274198
\(358\) 2.17804e6 0.898171
\(359\) −2.13021e6 −0.872341 −0.436170 0.899864i \(-0.643665\pi\)
−0.436170 + 0.899864i \(0.643665\pi\)
\(360\) 0 0
\(361\) −2.03828e6 −0.823183
\(362\) −131665. −0.0528079
\(363\) −4.31112e6 −1.71721
\(364\) −480570. −0.190109
\(365\) 0 0
\(366\) 2.34838e6 0.916360
\(367\) 3.10976e6 1.20521 0.602604 0.798041i \(-0.294130\pi\)
0.602604 + 0.798041i \(0.294130\pi\)
\(368\) 59896.7 0.0230559
\(369\) −947694. −0.362328
\(370\) 0 0
\(371\) 1.27208e6 0.479822
\(372\) −1.04147e6 −0.390202
\(373\) 3.15189e6 1.17300 0.586502 0.809948i \(-0.300505\pi\)
0.586502 + 0.809948i \(0.300505\pi\)
\(374\) −2.42063e6 −0.894849
\(375\) 0 0
\(376\) −781617. −0.285118
\(377\) 3.39047e6 1.22859
\(378\) 708469. 0.255030
\(379\) 342350. 0.122426 0.0612129 0.998125i \(-0.480503\pi\)
0.0612129 + 0.998125i \(0.480503\pi\)
\(380\) 0 0
\(381\) −632231. −0.223133
\(382\) 1.15383e6 0.404562
\(383\) −3.69387e6 −1.28672 −0.643361 0.765563i \(-0.722461\pi\)
−0.643361 + 0.765563i \(0.722461\pi\)
\(384\) 1.50254e6 0.519994
\(385\) 0 0
\(386\) 938345. 0.320549
\(387\) −1.00590e6 −0.341413
\(388\) −2.03857e6 −0.687459
\(389\) 2.05313e6 0.687928 0.343964 0.938983i \(-0.388230\pi\)
0.343964 + 0.938983i \(0.388230\pi\)
\(390\) 0 0
\(391\) −3.38541e6 −1.11988
\(392\) −436893. −0.143602
\(393\) −5.18956e6 −1.69492
\(394\) 1.82865e6 0.593457
\(395\) 0 0
\(396\) −786326. −0.251979
\(397\) 2.28107e6 0.726377 0.363189 0.931716i \(-0.381688\pi\)
0.363189 + 0.931716i \(0.381688\pi\)
\(398\) −525031. −0.166141
\(399\) −440727. −0.138592
\(400\) 0 0
\(401\) 3.32082e6 1.03130 0.515649 0.856800i \(-0.327551\pi\)
0.515649 + 0.856800i \(0.327551\pi\)
\(402\) −204375. −0.0630756
\(403\) −1.96981e6 −0.604175
\(404\) 892867. 0.272166
\(405\) 0 0
\(406\) 1.16825e6 0.351740
\(407\) 433737. 0.129790
\(408\) 2.45201e6 0.729242
\(409\) 5.15938e6 1.52507 0.762534 0.646948i \(-0.223955\pi\)
0.762534 + 0.646948i \(0.223955\pi\)
\(410\) 0 0
\(411\) −25753.3 −0.00752017
\(412\) 1.74370e6 0.506092
\(413\) 436229. 0.125846
\(414\) 702078. 0.201319
\(415\) 0 0
\(416\) 2.89283e6 0.819576
\(417\) −2.74311e6 −0.772509
\(418\) −1.61571e6 −0.452297
\(419\) −4.85187e6 −1.35012 −0.675062 0.737761i \(-0.735883\pi\)
−0.675062 + 0.737761i \(0.735883\pi\)
\(420\) 0 0
\(421\) −6.14767e6 −1.69046 −0.845231 0.534401i \(-0.820537\pi\)
−0.845231 + 0.534401i \(0.820537\pi\)
\(422\) 26282.5 0.00718432
\(423\) −250082. −0.0679565
\(424\) −4.72392e6 −1.27611
\(425\) 0 0
\(426\) −911563. −0.243368
\(427\) −2.39731e6 −0.636288
\(428\) −2.07674e6 −0.547991
\(429\) 4.72025e6 1.23829
\(430\) 0 0
\(431\) 3.55411e6 0.921590 0.460795 0.887507i \(-0.347564\pi\)
0.460795 + 0.887507i \(0.347564\pi\)
\(432\) −71814.7 −0.0185141
\(433\) −2.82650e6 −0.724485 −0.362243 0.932084i \(-0.617989\pi\)
−0.362243 + 0.932084i \(0.617989\pi\)
\(434\) −678737. −0.172973
\(435\) 0 0
\(436\) 972973. 0.245123
\(437\) −2.25968e6 −0.566035
\(438\) 486360. 0.121136
\(439\) 4.64410e6 1.15011 0.575056 0.818114i \(-0.304980\pi\)
0.575056 + 0.818114i \(0.304980\pi\)
\(440\) 0 0
\(441\) −139786. −0.0342268
\(442\) 1.75775e6 0.427959
\(443\) 6.15534e6 1.49019 0.745097 0.666957i \(-0.232403\pi\)
0.745097 + 0.666957i \(0.232403\pi\)
\(444\) −166524. −0.0400884
\(445\) 0 0
\(446\) −420757. −0.100160
\(447\) 6.33774e6 1.50026
\(448\) 1.02428e6 0.241115
\(449\) 3.67035e6 0.859196 0.429598 0.903020i \(-0.358655\pi\)
0.429598 + 0.903020i \(0.358655\pi\)
\(450\) 0 0
\(451\) −1.12565e7 −2.60591
\(452\) −725791. −0.167096
\(453\) 1.66127e6 0.380360
\(454\) 1.37305e6 0.312642
\(455\) 0 0
\(456\) 1.63665e6 0.368591
\(457\) 866327. 0.194040 0.0970201 0.995282i \(-0.469069\pi\)
0.0970201 + 0.995282i \(0.469069\pi\)
\(458\) 2.58698e6 0.576275
\(459\) 4.05903e6 0.899271
\(460\) 0 0
\(461\) −1.88572e6 −0.413261 −0.206631 0.978419i \(-0.566250\pi\)
−0.206631 + 0.978419i \(0.566250\pi\)
\(462\) 1.62645e6 0.354517
\(463\) 8.49017e6 1.84062 0.920309 0.391192i \(-0.127937\pi\)
0.920309 + 0.391192i \(0.127937\pi\)
\(464\) −118421. −0.0255349
\(465\) 0 0
\(466\) 3.97744e6 0.848474
\(467\) −1.82738e6 −0.387737 −0.193868 0.981028i \(-0.562104\pi\)
−0.193868 + 0.981028i \(0.562104\pi\)
\(468\) 570994. 0.120508
\(469\) 208632. 0.0437975
\(470\) 0 0
\(471\) −5.58197e6 −1.15940
\(472\) −1.61995e6 −0.334693
\(473\) −1.19479e7 −2.45549
\(474\) 4.65535e6 0.951714
\(475\) 0 0
\(476\) −948712. −0.191919
\(477\) −1.51144e6 −0.304155
\(478\) 2.72715e6 0.545933
\(479\) 4.68744e6 0.933463 0.466731 0.884399i \(-0.345432\pi\)
0.466731 + 0.884399i \(0.345432\pi\)
\(480\) 0 0
\(481\) −314960. −0.0620715
\(482\) 4.95405e6 0.971277
\(483\) 2.27470e6 0.443666
\(484\) −6.19426e6 −1.20192
\(485\) 0 0
\(486\) −1.52085e6 −0.292076
\(487\) −4.59651e6 −0.878225 −0.439112 0.898432i \(-0.644707\pi\)
−0.439112 + 0.898432i \(0.644707\pi\)
\(488\) 8.90247e6 1.69224
\(489\) 1.06743e6 0.201867
\(490\) 0 0
\(491\) 6.62099e6 1.23942 0.619711 0.784830i \(-0.287250\pi\)
0.619711 + 0.784830i \(0.287250\pi\)
\(492\) 4.32167e6 0.804895
\(493\) 6.69327e6 1.24028
\(494\) 1.17326e6 0.216310
\(495\) 0 0
\(496\) 68800.9 0.0125571
\(497\) 930554. 0.168986
\(498\) 3.39514e6 0.613458
\(499\) −4.52632e6 −0.813756 −0.406878 0.913482i \(-0.633383\pi\)
−0.406878 + 0.913482i \(0.633383\pi\)
\(500\) 0 0
\(501\) −8.12495e6 −1.44619
\(502\) −5.78829e6 −1.02516
\(503\) −3.83316e6 −0.675518 −0.337759 0.941233i \(-0.609669\pi\)
−0.337759 + 0.941233i \(0.609669\pi\)
\(504\) 519099. 0.0910278
\(505\) 0 0
\(506\) 8.33909e6 1.44791
\(507\) 1.61950e6 0.279808
\(508\) −908397. −0.156177
\(509\) −3.25460e6 −0.556806 −0.278403 0.960464i \(-0.589805\pi\)
−0.278403 + 0.960464i \(0.589805\pi\)
\(510\) 0 0
\(511\) −496492. −0.0841124
\(512\) −203165. −0.0342511
\(513\) 2.70930e6 0.454531
\(514\) 789788. 0.131857
\(515\) 0 0
\(516\) 4.58713e6 0.758432
\(517\) −2.97040e6 −0.488752
\(518\) −108525. −0.0177708
\(519\) −813891. −0.132632
\(520\) 0 0
\(521\) −1.07842e6 −0.174057 −0.0870287 0.996206i \(-0.527737\pi\)
−0.0870287 + 0.996206i \(0.527737\pi\)
\(522\) −1.38807e6 −0.222964
\(523\) −408626. −0.0653238 −0.0326619 0.999466i \(-0.510398\pi\)
−0.0326619 + 0.999466i \(0.510398\pi\)
\(524\) −7.45641e6 −1.18632
\(525\) 0 0
\(526\) −1.05764e6 −0.166676
\(527\) −3.88869e6 −0.609925
\(528\) −164867. −0.0257365
\(529\) 5.22642e6 0.812016
\(530\) 0 0
\(531\) −518310. −0.0797724
\(532\) −633242. −0.0970042
\(533\) 8.17392e6 1.24627
\(534\) −203574. −0.0308936
\(535\) 0 0
\(536\) −774763. −0.116481
\(537\) −8.38456e6 −1.25471
\(538\) 474374. 0.0706586
\(539\) −1.66034e6 −0.246164
\(540\) 0 0
\(541\) 1.13918e7 1.67340 0.836701 0.547659i \(-0.184481\pi\)
0.836701 + 0.547659i \(0.184481\pi\)
\(542\) 6.82970e6 0.998627
\(543\) 506856. 0.0737709
\(544\) 5.71084e6 0.827376
\(545\) 0 0
\(546\) −1.18106e6 −0.169547
\(547\) 3.44866e6 0.492813 0.246406 0.969167i \(-0.420750\pi\)
0.246406 + 0.969167i \(0.420750\pi\)
\(548\) −37002.6 −0.00526357
\(549\) 2.84838e6 0.403337
\(550\) 0 0
\(551\) 4.46759e6 0.626894
\(552\) −8.44718e6 −1.17995
\(553\) −4.75234e6 −0.660837
\(554\) −627124. −0.0868119
\(555\) 0 0
\(556\) −3.94134e6 −0.540700
\(557\) 4.69772e6 0.641577 0.320789 0.947151i \(-0.396052\pi\)
0.320789 + 0.947151i \(0.396052\pi\)
\(558\) 806449. 0.109646
\(559\) 8.67599e6 1.17433
\(560\) 0 0
\(561\) 9.31844e6 1.25007
\(562\) 6.53722e6 0.873077
\(563\) 561642. 0.0746772 0.0373386 0.999303i \(-0.488112\pi\)
0.0373386 + 0.999303i \(0.488112\pi\)
\(564\) 1.14042e6 0.150962
\(565\) 0 0
\(566\) −8.46945e6 −1.11126
\(567\) −2.03409e6 −0.265713
\(568\) −3.45564e6 −0.449426
\(569\) 5.19019e6 0.672051 0.336026 0.941853i \(-0.390917\pi\)
0.336026 + 0.941853i \(0.390917\pi\)
\(570\) 0 0
\(571\) −5.20274e6 −0.667793 −0.333897 0.942610i \(-0.608364\pi\)
−0.333897 + 0.942610i \(0.608364\pi\)
\(572\) 6.78211e6 0.866712
\(573\) −4.44178e6 −0.565159
\(574\) 2.81648e6 0.356802
\(575\) 0 0
\(576\) −1.21701e6 −0.152840
\(577\) −8.70662e6 −1.08870 −0.544352 0.838857i \(-0.683225\pi\)
−0.544352 + 0.838857i \(0.683225\pi\)
\(578\) −1.54365e6 −0.192189
\(579\) −3.61224e6 −0.447796
\(580\) 0 0
\(581\) −3.46588e6 −0.425964
\(582\) −5.01003e6 −0.613102
\(583\) −1.79524e7 −2.18752
\(584\) 1.84374e6 0.223701
\(585\) 0 0
\(586\) −8.82505e6 −1.06163
\(587\) 4.35717e6 0.521926 0.260963 0.965349i \(-0.415960\pi\)
0.260963 + 0.965349i \(0.415960\pi\)
\(588\) 637452. 0.0760333
\(589\) −2.59560e6 −0.308283
\(590\) 0 0
\(591\) −7.03954e6 −0.829040
\(592\) 11000.8 0.00129009
\(593\) 3.22991e6 0.377184 0.188592 0.982055i \(-0.439608\pi\)
0.188592 + 0.982055i \(0.439608\pi\)
\(594\) −9.99837e6 −1.16269
\(595\) 0 0
\(596\) 9.10614e6 1.05007
\(597\) 2.02115e6 0.232094
\(598\) −6.05547e6 −0.692460
\(599\) −7.23988e6 −0.824450 −0.412225 0.911082i \(-0.635248\pi\)
−0.412225 + 0.911082i \(0.635248\pi\)
\(600\) 0 0
\(601\) −1.06837e7 −1.20652 −0.603262 0.797543i \(-0.706133\pi\)
−0.603262 + 0.797543i \(0.706133\pi\)
\(602\) 2.98948e6 0.336205
\(603\) −247889. −0.0277628
\(604\) 2.38693e6 0.266225
\(605\) 0 0
\(606\) 2.19432e6 0.242727
\(607\) −2.51528e6 −0.277086 −0.138543 0.990356i \(-0.544242\pi\)
−0.138543 + 0.990356i \(0.544242\pi\)
\(608\) 3.81185e6 0.418193
\(609\) −4.49729e6 −0.491369
\(610\) 0 0
\(611\) 2.15697e6 0.233744
\(612\) 1.12722e6 0.121655
\(613\) −213999. −0.0230017 −0.0115009 0.999934i \(-0.503661\pi\)
−0.0115009 + 0.999934i \(0.503661\pi\)
\(614\) −1.08475e7 −1.16120
\(615\) 0 0
\(616\) 6.16572e6 0.654684
\(617\) 127951. 0.0135310 0.00676552 0.999977i \(-0.497846\pi\)
0.00676552 + 0.999977i \(0.497846\pi\)
\(618\) 4.28535e6 0.451352
\(619\) −1.23980e7 −1.30054 −0.650272 0.759701i \(-0.725345\pi\)
−0.650272 + 0.759701i \(0.725345\pi\)
\(620\) 0 0
\(621\) −1.39834e7 −1.45507
\(622\) 2.33490e6 0.241987
\(623\) 207815. 0.0214514
\(624\) 119719. 0.0123084
\(625\) 0 0
\(626\) 1.16313e7 1.18629
\(627\) 6.21983e6 0.631843
\(628\) −8.02023e6 −0.811499
\(629\) −621774. −0.0626622
\(630\) 0 0
\(631\) −5.87683e6 −0.587584 −0.293792 0.955869i \(-0.594917\pi\)
−0.293792 + 0.955869i \(0.594917\pi\)
\(632\) 1.76480e7 1.75753
\(633\) −101177. −0.0100363
\(634\) −2.25895e6 −0.223194
\(635\) 0 0
\(636\) 6.89246e6 0.675665
\(637\) 1.20566e6 0.117727
\(638\) −1.64871e7 −1.60359
\(639\) −1.10565e6 −0.107118
\(640\) 0 0
\(641\) −2.60122e6 −0.250053 −0.125026 0.992153i \(-0.539902\pi\)
−0.125026 + 0.992153i \(0.539902\pi\)
\(642\) −5.10384e6 −0.488719
\(643\) −1.31345e7 −1.25282 −0.626408 0.779495i \(-0.715476\pi\)
−0.626408 + 0.779495i \(0.715476\pi\)
\(644\) 3.26832e6 0.310534
\(645\) 0 0
\(646\) 2.31617e6 0.218368
\(647\) −5.54662e6 −0.520916 −0.260458 0.965485i \(-0.583874\pi\)
−0.260458 + 0.965485i \(0.583874\pi\)
\(648\) 7.55366e6 0.706675
\(649\) −6.15634e6 −0.573734
\(650\) 0 0
\(651\) 2.61286e6 0.241637
\(652\) 1.53369e6 0.141292
\(653\) 1.54993e7 1.42242 0.711212 0.702978i \(-0.248147\pi\)
0.711212 + 0.702978i \(0.248147\pi\)
\(654\) 2.39119e6 0.218610
\(655\) 0 0
\(656\) −285495. −0.0259024
\(657\) 589912. 0.0533180
\(658\) 743225. 0.0669200
\(659\) −4.30145e6 −0.385835 −0.192917 0.981215i \(-0.561795\pi\)
−0.192917 + 0.981215i \(0.561795\pi\)
\(660\) 0 0
\(661\) 861980. 0.0767350 0.0383675 0.999264i \(-0.487784\pi\)
0.0383675 + 0.999264i \(0.487784\pi\)
\(662\) −4.02111e6 −0.356616
\(663\) −6.76662e6 −0.597844
\(664\) 1.28706e7 1.13287
\(665\) 0 0
\(666\) 128945. 0.0112647
\(667\) −2.30583e7 −2.00684
\(668\) −1.16740e7 −1.01223
\(669\) 1.61974e6 0.139920
\(670\) 0 0
\(671\) 3.38323e7 2.90085
\(672\) −3.83719e6 −0.327786
\(673\) 953818. 0.0811760 0.0405880 0.999176i \(-0.487077\pi\)
0.0405880 + 0.999176i \(0.487077\pi\)
\(674\) −2.42056e6 −0.205242
\(675\) 0 0
\(676\) 2.32691e6 0.195845
\(677\) 1.48606e7 1.24613 0.623065 0.782170i \(-0.285887\pi\)
0.623065 + 0.782170i \(0.285887\pi\)
\(678\) −1.78371e6 −0.149022
\(679\) 5.11440e6 0.425716
\(680\) 0 0
\(681\) −5.28569e6 −0.436751
\(682\) 9.57878e6 0.788586
\(683\) 1.57605e7 1.29276 0.646381 0.763015i \(-0.276282\pi\)
0.646381 + 0.763015i \(0.276282\pi\)
\(684\) 752392. 0.0614900
\(685\) 0 0
\(686\) 415434. 0.0337048
\(687\) −9.95882e6 −0.805037
\(688\) −303032. −0.0244071
\(689\) 1.30362e7 1.04618
\(690\) 0 0
\(691\) 1.59740e7 1.27267 0.636337 0.771411i \(-0.280449\pi\)
0.636337 + 0.771411i \(0.280449\pi\)
\(692\) −1.16941e6 −0.0928326
\(693\) 1.97275e6 0.156041
\(694\) 4.41925e6 0.348297
\(695\) 0 0
\(696\) 1.67008e7 1.30682
\(697\) 1.61365e7 1.25813
\(698\) −1.11798e7 −0.868549
\(699\) −1.53115e7 −1.18529
\(700\) 0 0
\(701\) −1.85736e7 −1.42758 −0.713790 0.700360i \(-0.753023\pi\)
−0.713790 + 0.700360i \(0.753023\pi\)
\(702\) 7.26036e6 0.556052
\(703\) −415019. −0.0316723
\(704\) −1.44553e7 −1.09925
\(705\) 0 0
\(706\) 8.59365e6 0.648883
\(707\) −2.24004e6 −0.168541
\(708\) 2.36360e6 0.177211
\(709\) −1.71029e7 −1.27778 −0.638888 0.769300i \(-0.720605\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(710\) 0 0
\(711\) 5.64654e6 0.418898
\(712\) −771727. −0.0570511
\(713\) 1.33965e7 0.986890
\(714\) −2.33157e6 −0.171160
\(715\) 0 0
\(716\) −1.20470e7 −0.878208
\(717\) −1.04984e7 −0.762651
\(718\) −7.52204e6 −0.544534
\(719\) −1.40945e7 −1.01678 −0.508390 0.861127i \(-0.669759\pi\)
−0.508390 + 0.861127i \(0.669759\pi\)
\(720\) 0 0
\(721\) −4.37463e6 −0.313403
\(722\) −7.19744e6 −0.513848
\(723\) −1.90711e7 −1.35684
\(724\) 728256. 0.0516343
\(725\) 0 0
\(726\) −1.52231e7 −1.07192
\(727\) −2.24196e7 −1.57323 −0.786616 0.617443i \(-0.788169\pi\)
−0.786616 + 0.617443i \(0.788169\pi\)
\(728\) −4.47726e6 −0.313101
\(729\) 1.59421e7 1.11103
\(730\) 0 0
\(731\) 1.71276e7 1.18551
\(732\) −1.29892e7 −0.895993
\(733\) 8.02464e6 0.551653 0.275826 0.961208i \(-0.411049\pi\)
0.275826 + 0.961208i \(0.411049\pi\)
\(734\) 1.09810e7 0.752316
\(735\) 0 0
\(736\) −1.96739e7 −1.33874
\(737\) −2.94435e6 −0.199674
\(738\) −3.34643e6 −0.226173
\(739\) 1.15678e7 0.779181 0.389591 0.920988i \(-0.372616\pi\)
0.389591 + 0.920988i \(0.372616\pi\)
\(740\) 0 0
\(741\) −4.51655e6 −0.302177
\(742\) 4.49189e6 0.299515
\(743\) −3.79387e6 −0.252122 −0.126061 0.992023i \(-0.540233\pi\)
−0.126061 + 0.992023i \(0.540233\pi\)
\(744\) −9.70293e6 −0.642644
\(745\) 0 0
\(746\) 1.11297e7 0.732214
\(747\) 4.11801e6 0.270014
\(748\) 1.33888e7 0.874961
\(749\) 5.21017e6 0.339349
\(750\) 0 0
\(751\) −9.07884e6 −0.587395 −0.293698 0.955898i \(-0.594886\pi\)
−0.293698 + 0.955898i \(0.594886\pi\)
\(752\) −75337.8 −0.00485812
\(753\) 2.22825e7 1.43211
\(754\) 1.19722e7 0.766912
\(755\) 0 0
\(756\) −3.91863e6 −0.249362
\(757\) 2.33210e7 1.47913 0.739565 0.673085i \(-0.235031\pi\)
0.739565 + 0.673085i \(0.235031\pi\)
\(758\) 1.20888e6 0.0764208
\(759\) −3.21020e7 −2.02269
\(760\) 0 0
\(761\) 1.44859e7 0.906745 0.453373 0.891321i \(-0.350221\pi\)
0.453373 + 0.891321i \(0.350221\pi\)
\(762\) −2.23249e6 −0.139284
\(763\) −2.44101e6 −0.151795
\(764\) −6.38200e6 −0.395570
\(765\) 0 0
\(766\) −1.30435e7 −0.803200
\(767\) 4.47045e6 0.274387
\(768\) 1.43985e7 0.880876
\(769\) 1.59424e7 0.972162 0.486081 0.873914i \(-0.338426\pi\)
0.486081 + 0.873914i \(0.338426\pi\)
\(770\) 0 0
\(771\) −3.04036e6 −0.184200
\(772\) −5.19011e6 −0.313425
\(773\) 3.36066e6 0.202290 0.101145 0.994872i \(-0.467749\pi\)
0.101145 + 0.994872i \(0.467749\pi\)
\(774\) −3.55198e6 −0.213117
\(775\) 0 0
\(776\) −1.89925e7 −1.13221
\(777\) 417778. 0.0248252
\(778\) 7.24988e6 0.429419
\(779\) 1.07707e7 0.635916
\(780\) 0 0
\(781\) −1.31326e7 −0.770411
\(782\) −1.19543e7 −0.699050
\(783\) 2.76464e7 1.61151
\(784\) −42110.9 −0.00244683
\(785\) 0 0
\(786\) −1.83250e7 −1.05800
\(787\) 3.28918e6 0.189300 0.0946500 0.995511i \(-0.469827\pi\)
0.0946500 + 0.995511i \(0.469827\pi\)
\(788\) −1.01145e7 −0.580268
\(789\) 4.07147e6 0.232841
\(790\) 0 0
\(791\) 1.82088e6 0.103476
\(792\) −7.32586e6 −0.414998
\(793\) −2.45675e7 −1.38732
\(794\) 8.05475e6 0.453420
\(795\) 0 0
\(796\) 2.90402e6 0.162449
\(797\) 2.51939e6 0.140492 0.0702458 0.997530i \(-0.477622\pi\)
0.0702458 + 0.997530i \(0.477622\pi\)
\(798\) −1.55626e6 −0.0865120
\(799\) 4.25816e6 0.235969
\(800\) 0 0
\(801\) −246917. −0.0135978
\(802\) 1.17262e7 0.643758
\(803\) 7.00682e6 0.383470
\(804\) 1.13042e6 0.0616738
\(805\) 0 0
\(806\) −6.95567e6 −0.377139
\(807\) −1.82614e6 −0.0987077
\(808\) 8.31845e6 0.448244
\(809\) 8.93808e6 0.480146 0.240073 0.970755i \(-0.422829\pi\)
0.240073 + 0.970755i \(0.422829\pi\)
\(810\) 0 0
\(811\) 3.01341e7 1.60881 0.804406 0.594080i \(-0.202484\pi\)
0.804406 + 0.594080i \(0.202484\pi\)
\(812\) −6.46176e6 −0.343922
\(813\) −2.62915e7 −1.39505
\(814\) 1.53158e6 0.0810174
\(815\) 0 0
\(816\) 236342. 0.0124255
\(817\) 1.14323e7 0.599207
\(818\) 1.82184e7 0.951980
\(819\) −1.43252e6 −0.0746261
\(820\) 0 0
\(821\) −3.25440e7 −1.68505 −0.842524 0.538658i \(-0.818931\pi\)
−0.842524 + 0.538658i \(0.818931\pi\)
\(822\) −90938.1 −0.00469425
\(823\) 499640. 0.0257133 0.0128566 0.999917i \(-0.495907\pi\)
0.0128566 + 0.999917i \(0.495907\pi\)
\(824\) 1.62453e7 0.833509
\(825\) 0 0
\(826\) 1.54038e6 0.0785557
\(827\) 4.64084e6 0.235957 0.117978 0.993016i \(-0.462359\pi\)
0.117978 + 0.993016i \(0.462359\pi\)
\(828\) −3.88328e6 −0.196844
\(829\) −3.08794e7 −1.56057 −0.780283 0.625427i \(-0.784925\pi\)
−0.780283 + 0.625427i \(0.784925\pi\)
\(830\) 0 0
\(831\) 2.41417e6 0.121273
\(832\) 1.04968e7 0.525712
\(833\) 2.38014e6 0.118848
\(834\) −9.68629e6 −0.482217
\(835\) 0 0
\(836\) 8.93671e6 0.442244
\(837\) −1.60621e7 −0.792482
\(838\) −1.71326e7 −0.842777
\(839\) 1.93485e7 0.948948 0.474474 0.880269i \(-0.342638\pi\)
0.474474 + 0.880269i \(0.342638\pi\)
\(840\) 0 0
\(841\) 2.50772e7 1.22261
\(842\) −2.17082e7 −1.05522
\(843\) −2.51656e7 −1.21966
\(844\) −145372. −0.00702465
\(845\) 0 0
\(846\) −883071. −0.0424199
\(847\) 1.55403e7 0.744303
\(848\) −455325. −0.0217436
\(849\) 3.26039e7 1.55239
\(850\) 0 0
\(851\) 2.14201e6 0.101391
\(852\) 5.04197e6 0.237959
\(853\) 3.05998e7 1.43994 0.719972 0.694003i \(-0.244155\pi\)
0.719972 + 0.694003i \(0.244155\pi\)
\(854\) −8.46520e6 −0.397185
\(855\) 0 0
\(856\) −1.93481e7 −0.902515
\(857\) 1.33039e7 0.618766 0.309383 0.950937i \(-0.399877\pi\)
0.309383 + 0.950937i \(0.399877\pi\)
\(858\) 1.66678e7 0.772967
\(859\) −1.27708e7 −0.590520 −0.295260 0.955417i \(-0.595406\pi\)
−0.295260 + 0.955417i \(0.595406\pi\)
\(860\) 0 0
\(861\) −1.08423e7 −0.498440
\(862\) 1.25500e7 0.575276
\(863\) −1.37987e7 −0.630684 −0.315342 0.948978i \(-0.602119\pi\)
−0.315342 + 0.948978i \(0.602119\pi\)
\(864\) 2.35885e7 1.07502
\(865\) 0 0
\(866\) −9.98074e6 −0.452239
\(867\) 5.94241e6 0.268482
\(868\) 3.75418e6 0.169128
\(869\) 6.70680e7 3.01277
\(870\) 0 0
\(871\) 2.13806e6 0.0954934
\(872\) 9.06477e6 0.403706
\(873\) −6.07673e6 −0.269857
\(874\) −7.97922e6 −0.353331
\(875\) 0 0
\(876\) −2.69012e6 −0.118443
\(877\) −6.68992e6 −0.293712 −0.146856 0.989158i \(-0.546915\pi\)
−0.146856 + 0.989158i \(0.546915\pi\)
\(878\) 1.63989e7 0.717924
\(879\) 3.39728e7 1.48306
\(880\) 0 0
\(881\) −7.25051e6 −0.314723 −0.157362 0.987541i \(-0.550299\pi\)
−0.157362 + 0.987541i \(0.550299\pi\)
\(882\) −493602. −0.0213651
\(883\) 2.55944e7 1.10470 0.552348 0.833613i \(-0.313732\pi\)
0.552348 + 0.833613i \(0.313732\pi\)
\(884\) −9.72236e6 −0.418447
\(885\) 0 0
\(886\) 2.17353e7 0.930210
\(887\) 2.33204e7 0.995240 0.497620 0.867395i \(-0.334207\pi\)
0.497620 + 0.867395i \(0.334207\pi\)
\(888\) −1.55143e6 −0.0660237
\(889\) 2.27900e6 0.0967141
\(890\) 0 0
\(891\) 2.87064e7 1.21139
\(892\) 2.32726e6 0.0979340
\(893\) 2.84222e6 0.119269
\(894\) 2.23794e7 0.936493
\(895\) 0 0
\(896\) −5.41620e6 −0.225385
\(897\) 2.33110e7 0.967343
\(898\) 1.29605e7 0.536328
\(899\) −2.64862e7 −1.09300
\(900\) 0 0
\(901\) 2.57354e7 1.05613
\(902\) −3.97480e7 −1.62667
\(903\) −1.15083e7 −0.469667
\(904\) −6.76188e6 −0.275199
\(905\) 0 0
\(906\) 5.86617e6 0.237429
\(907\) −3.71721e7 −1.50037 −0.750187 0.661226i \(-0.770036\pi\)
−0.750187 + 0.661226i \(0.770036\pi\)
\(908\) −7.59453e6 −0.305694
\(909\) 2.66152e6 0.106837
\(910\) 0 0
\(911\) −2.89923e7 −1.15741 −0.578704 0.815537i \(-0.696441\pi\)
−0.578704 + 0.815537i \(0.696441\pi\)
\(912\) 157752. 0.00628042
\(913\) 4.89127e7 1.94198
\(914\) 3.05911e6 0.121124
\(915\) 0 0
\(916\) −1.43089e7 −0.563468
\(917\) 1.87068e7 0.734641
\(918\) 1.43330e7 0.561344
\(919\) −1.25020e6 −0.0488306 −0.0244153 0.999702i \(-0.507772\pi\)
−0.0244153 + 0.999702i \(0.507772\pi\)
\(920\) 0 0
\(921\) 4.17582e7 1.62216
\(922\) −6.65872e6 −0.257966
\(923\) 9.53627e6 0.368447
\(924\) −8.99613e6 −0.346638
\(925\) 0 0
\(926\) 2.99799e7 1.14895
\(927\) 5.19776e6 0.198663
\(928\) 3.88970e7 1.48268
\(929\) 6.87875e6 0.261499 0.130749 0.991415i \(-0.458262\pi\)
0.130749 + 0.991415i \(0.458262\pi\)
\(930\) 0 0
\(931\) 1.58869e6 0.0600709
\(932\) −2.19997e7 −0.829616
\(933\) −8.98840e6 −0.338048
\(934\) −6.45272e6 −0.242034
\(935\) 0 0
\(936\) 5.31971e6 0.198471
\(937\) −2.18600e7 −0.813396 −0.406698 0.913563i \(-0.633320\pi\)
−0.406698 + 0.913563i \(0.633320\pi\)
\(938\) 736708. 0.0273393
\(939\) −4.47756e7 −1.65721
\(940\) 0 0
\(941\) 2.26450e6 0.0833679 0.0416840 0.999131i \(-0.486728\pi\)
0.0416840 + 0.999131i \(0.486728\pi\)
\(942\) −1.97106e7 −0.723725
\(943\) −5.55901e7 −2.03572
\(944\) −156142. −0.00570283
\(945\) 0 0
\(946\) −4.21895e7 −1.53277
\(947\) 2.45846e7 0.890815 0.445408 0.895328i \(-0.353059\pi\)
0.445408 + 0.895328i \(0.353059\pi\)
\(948\) −2.57493e7 −0.930562
\(949\) −5.08803e6 −0.183394
\(950\) 0 0
\(951\) 8.69601e6 0.311795
\(952\) −8.83874e6 −0.316081
\(953\) −4.59681e7 −1.63955 −0.819774 0.572687i \(-0.805901\pi\)
−0.819774 + 0.572687i \(0.805901\pi\)
\(954\) −5.33708e6 −0.189860
\(955\) 0 0
\(956\) −1.50842e7 −0.533800
\(957\) 6.34686e7 2.24016
\(958\) 1.65520e7 0.582687
\(959\) 92832.6 0.00325952
\(960\) 0 0
\(961\) −1.32411e7 −0.462503
\(962\) −1.11216e6 −0.0387463
\(963\) −6.19051e6 −0.215110
\(964\) −2.74015e7 −0.949690
\(965\) 0 0
\(966\) 8.03226e6 0.276946
\(967\) 5.30202e7 1.82337 0.911686 0.410888i \(-0.134781\pi\)
0.911686 + 0.410888i \(0.134781\pi\)
\(968\) −5.77093e7 −1.97951
\(969\) −8.91630e6 −0.305053
\(970\) 0 0
\(971\) −5.43523e7 −1.84999 −0.924996 0.379976i \(-0.875932\pi\)
−0.924996 + 0.379976i \(0.875932\pi\)
\(972\) 8.41202e6 0.285585
\(973\) 9.88809e6 0.334835
\(974\) −1.62309e7 −0.548207
\(975\) 0 0
\(976\) 858083. 0.0288340
\(977\) −1.08929e7 −0.365097 −0.182549 0.983197i \(-0.558435\pi\)
−0.182549 + 0.983197i \(0.558435\pi\)
\(978\) 3.76922e6 0.126010
\(979\) −2.93282e6 −0.0977976
\(980\) 0 0
\(981\) 2.90031e6 0.0962215
\(982\) 2.33796e7 0.773674
\(983\) 3.42136e7 1.12932 0.564658 0.825325i \(-0.309008\pi\)
0.564658 + 0.825325i \(0.309008\pi\)
\(984\) 4.02632e7 1.32562
\(985\) 0 0
\(986\) 2.36348e7 0.774211
\(987\) −2.86111e6 −0.0934850
\(988\) −6.48943e6 −0.211502
\(989\) −5.90047e7 −1.91821
\(990\) 0 0
\(991\) 4.22117e6 0.136537 0.0682683 0.997667i \(-0.478253\pi\)
0.0682683 + 0.997667i \(0.478253\pi\)
\(992\) −2.25986e7 −0.729125
\(993\) 1.54796e7 0.498181
\(994\) 3.28591e6 0.105485
\(995\) 0 0
\(996\) −1.87790e7 −0.599824
\(997\) −4.30323e7 −1.37106 −0.685530 0.728044i \(-0.740429\pi\)
−0.685530 + 0.728044i \(0.740429\pi\)
\(998\) −1.59830e7 −0.507964
\(999\) −2.56822e6 −0.0814177
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 175.6.a.d.1.2 2
5.2 odd 4 175.6.b.d.99.3 4
5.3 odd 4 175.6.b.d.99.2 4
5.4 even 2 35.6.a.b.1.1 2
15.14 odd 2 315.6.a.c.1.2 2
20.19 odd 2 560.6.a.l.1.1 2
35.34 odd 2 245.6.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.b.1.1 2 5.4 even 2
175.6.a.d.1.2 2 1.1 even 1 trivial
175.6.b.d.99.2 4 5.3 odd 4
175.6.b.d.99.3 4 5.2 odd 4
245.6.a.c.1.1 2 35.34 odd 2
315.6.a.c.1.2 2 15.14 odd 2
560.6.a.l.1.1 2 20.19 odd 2