Properties

Label 245.6.a.c.1.2
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
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] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(2\)
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 \(4.53113\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.53113 q^{2} +10.5934 q^{3} -11.4689 q^{4} +25.0000 q^{5} +48.0000 q^{6} -196.963 q^{8} -130.780 q^{9} +O(q^{10})\) \(q+4.53113 q^{2} +10.5934 q^{3} -11.4689 q^{4} +25.0000 q^{5} +48.0000 q^{6} -196.963 q^{8} -130.780 q^{9} +113.278 q^{10} +90.5195 q^{11} -121.494 q^{12} +74.8502 q^{13} +264.835 q^{15} -525.461 q^{16} -1032.31 q^{17} -592.582 q^{18} +31.6771 q^{19} -286.722 q^{20} +410.156 q^{22} -3857.08 q^{23} -2086.51 q^{24} +625.000 q^{25} +339.156 q^{26} -3959.60 q^{27} -866.917 q^{29} +1200.00 q^{30} -3526.76 q^{31} +3921.89 q^{32} +958.908 q^{33} -4677.54 q^{34} +1499.90 q^{36} -9531.22 q^{37} +143.533 q^{38} +792.917 q^{39} -4924.08 q^{40} +14503.9 q^{41} -9844.30 q^{43} -1038.16 q^{44} -3269.50 q^{45} -17476.9 q^{46} +16993.5 q^{47} -5566.41 q^{48} +2831.96 q^{50} -10935.7 q^{51} -858.447 q^{52} -29621.1 q^{53} -17941.4 q^{54} +2262.99 q^{55} +335.568 q^{57} -3928.11 q^{58} -50697.4 q^{59} -3037.35 q^{60} +2921.38 q^{61} -15980.2 q^{62} +34585.3 q^{64} +1871.25 q^{65} +4344.94 q^{66} -41086.2 q^{67} +11839.5 q^{68} -40859.5 q^{69} +61753.1 q^{71} +25758.9 q^{72} +23664.5 q^{73} -43187.2 q^{74} +6620.87 q^{75} -363.300 q^{76} +3592.81 q^{78} +45191.5 q^{79} -13136.5 q^{80} -10166.0 q^{81} +65718.8 q^{82} -39095.9 q^{83} -25807.8 q^{85} -44605.8 q^{86} -9183.58 q^{87} -17829.0 q^{88} +41891.1 q^{89} -14814.5 q^{90} +44236.3 q^{92} -37360.4 q^{93} +76999.9 q^{94} +791.927 q^{95} +41546.1 q^{96} -8036.53 q^{97} -11838.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9} + 25 q^{10} - 601 q^{11} + 144 q^{12} + 577 q^{13} - 75 q^{15} - 543 q^{16} - 41 q^{17} - 387 q^{18} - 630 q^{19} - 775 q^{20} + 2852 q^{22} - 442 q^{23} - 4560 q^{24} + 1250 q^{25} - 1434 q^{26} + 135 q^{27} + 5885 q^{29} + 2400 q^{30} + 396 q^{31} - 1839 q^{32} + 10359 q^{33} - 8178 q^{34} + 2637 q^{36} - 8904 q^{37} + 2480 q^{38} - 6033 q^{39} - 375 q^{40} - 1774 q^{41} - 27122 q^{43} + 12468 q^{44} - 4725 q^{45} - 29536 q^{46} + 21289 q^{47} - 5328 q^{48} + 625 q^{50} - 24411 q^{51} - 10666 q^{52} - 55582 q^{53} - 32400 q^{54} - 15025 q^{55} + 9330 q^{57} - 27770 q^{58} - 59600 q^{59} + 3600 q^{60} + 51846 q^{61} - 29832 q^{62} + 55489 q^{64} + 14425 q^{65} - 28848 q^{66} - 45344 q^{67} - 7522 q^{68} - 87282 q^{69} + 80744 q^{71} + 15165 q^{72} + 13532 q^{73} - 45402 q^{74} - 1875 q^{75} + 12560 q^{76} + 27696 q^{78} - 51795 q^{79} - 13575 q^{80} - 51678 q^{81} + 123198 q^{82} - 109828 q^{83} - 1025 q^{85} + 16404 q^{86} - 100965 q^{87} - 143660 q^{88} + 37650 q^{89} - 9675 q^{90} - 22464 q^{92} - 90684 q^{93} + 61832 q^{94} - 15750 q^{95} + 119856 q^{96} + 96339 q^{97} + 28422 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.53113 0.800998 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(3\) 10.5934 0.679566 0.339783 0.940504i \(-0.389646\pi\)
0.339783 + 0.940504i \(0.389646\pi\)
\(4\) −11.4689 −0.358402
\(5\) 25.0000 0.447214
\(6\) 48.0000 0.544331
\(7\) 0 0
\(8\) −196.963 −1.08808
\(9\) −130.780 −0.538190
\(10\) 113.278 0.358217
\(11\) 90.5195 0.225559 0.112780 0.993620i \(-0.464025\pi\)
0.112780 + 0.993620i \(0.464025\pi\)
\(12\) −121.494 −0.243558
\(13\) 74.8502 0.122838 0.0614192 0.998112i \(-0.480437\pi\)
0.0614192 + 0.998112i \(0.480437\pi\)
\(14\) 0 0
\(15\) 264.835 0.303911
\(16\) −525.461 −0.513146
\(17\) −1032.31 −0.866342 −0.433171 0.901312i \(-0.642605\pi\)
−0.433171 + 0.901312i \(0.642605\pi\)
\(18\) −592.582 −0.431089
\(19\) 31.6771 0.0201308 0.0100654 0.999949i \(-0.496796\pi\)
0.0100654 + 0.999949i \(0.496796\pi\)
\(20\) −286.722 −0.160282
\(21\) 0 0
\(22\) 410.156 0.180672
\(23\) −3857.08 −1.52033 −0.760167 0.649728i \(-0.774883\pi\)
−0.760167 + 0.649728i \(0.774883\pi\)
\(24\) −2086.51 −0.739421
\(25\) 625.000 0.200000
\(26\) 339.156 0.0983934
\(27\) −3959.60 −1.04530
\(28\) 0 0
\(29\) −866.917 −0.191418 −0.0957089 0.995409i \(-0.530512\pi\)
−0.0957089 + 0.995409i \(0.530512\pi\)
\(30\) 1200.00 0.243432
\(31\) −3526.76 −0.659131 −0.329566 0.944133i \(-0.606902\pi\)
−0.329566 + 0.944133i \(0.606902\pi\)
\(32\) 3921.89 0.677049
\(33\) 958.908 0.153282
\(34\) −4677.54 −0.693938
\(35\) 0 0
\(36\) 1499.90 0.192888
\(37\) −9531.22 −1.14458 −0.572288 0.820053i \(-0.693944\pi\)
−0.572288 + 0.820053i \(0.693944\pi\)
\(38\) 143.533 0.0161247
\(39\) 792.917 0.0834769
\(40\) −4924.08 −0.486603
\(41\) 14503.9 1.34748 0.673742 0.738967i \(-0.264686\pi\)
0.673742 + 0.738967i \(0.264686\pi\)
\(42\) 0 0
\(43\) −9844.30 −0.811921 −0.405960 0.913891i \(-0.633063\pi\)
−0.405960 + 0.913891i \(0.633063\pi\)
\(44\) −1038.16 −0.0808409
\(45\) −3269.50 −0.240686
\(46\) −17476.9 −1.21778
\(47\) 16993.5 1.12212 0.561059 0.827775i \(-0.310394\pi\)
0.561059 + 0.827775i \(0.310394\pi\)
\(48\) −5566.41 −0.348716
\(49\) 0 0
\(50\) 2831.96 0.160200
\(51\) −10935.7 −0.588736
\(52\) −858.447 −0.0440256
\(53\) −29621.1 −1.44848 −0.724239 0.689549i \(-0.757809\pi\)
−0.724239 + 0.689549i \(0.757809\pi\)
\(54\) −17941.4 −0.837285
\(55\) 2262.99 0.100873
\(56\) 0 0
\(57\) 335.568 0.0136802
\(58\) −3928.11 −0.153325
\(59\) −50697.4 −1.89607 −0.948037 0.318159i \(-0.896935\pi\)
−0.948037 + 0.318159i \(0.896935\pi\)
\(60\) −3037.35 −0.108922
\(61\) 2921.38 0.100522 0.0502612 0.998736i \(-0.483995\pi\)
0.0502612 + 0.998736i \(0.483995\pi\)
\(62\) −15980.2 −0.527963
\(63\) 0 0
\(64\) 34585.3 1.05546
\(65\) 1871.25 0.0549350
\(66\) 4344.94 0.122779
\(67\) −41086.2 −1.11817 −0.559086 0.829109i \(-0.688848\pi\)
−0.559086 + 0.829109i \(0.688848\pi\)
\(68\) 11839.5 0.310499
\(69\) −40859.5 −1.03317
\(70\) 0 0
\(71\) 61753.1 1.45383 0.726914 0.686729i \(-0.240954\pi\)
0.726914 + 0.686729i \(0.240954\pi\)
\(72\) 25758.9 0.585592
\(73\) 23664.5 0.519745 0.259872 0.965643i \(-0.416320\pi\)
0.259872 + 0.965643i \(0.416320\pi\)
\(74\) −43187.2 −0.916802
\(75\) 6620.87 0.135913
\(76\) −363.300 −0.00721493
\(77\) 0 0
\(78\) 3592.81 0.0668648
\(79\) 45191.5 0.814683 0.407341 0.913276i \(-0.366456\pi\)
0.407341 + 0.913276i \(0.366456\pi\)
\(80\) −13136.5 −0.229486
\(81\) −10166.0 −0.172162
\(82\) 65718.8 1.07933
\(83\) −39095.9 −0.622925 −0.311462 0.950259i \(-0.600819\pi\)
−0.311462 + 0.950259i \(0.600819\pi\)
\(84\) 0 0
\(85\) −25807.8 −0.387440
\(86\) −44605.8 −0.650347
\(87\) −9183.58 −0.130081
\(88\) −17829.0 −0.245426
\(89\) 41891.1 0.560592 0.280296 0.959914i \(-0.409567\pi\)
0.280296 + 0.959914i \(0.409567\pi\)
\(90\) −14814.5 −0.192789
\(91\) 0 0
\(92\) 44236.3 0.544891
\(93\) −37360.4 −0.447923
\(94\) 76999.9 0.898815
\(95\) 791.927 0.00900277
\(96\) 41546.1 0.460099
\(97\) −8036.53 −0.0867240 −0.0433620 0.999059i \(-0.513807\pi\)
−0.0433620 + 0.999059i \(0.513807\pi\)
\(98\) 0 0
\(99\) −11838.2 −0.121394
\(100\) −7168.04 −0.0716804
\(101\) −124979. −1.21908 −0.609542 0.792754i \(-0.708647\pi\)
−0.609542 + 0.792754i \(0.708647\pi\)
\(102\) −49551.0 −0.471577
\(103\) 76635.1 0.711762 0.355881 0.934531i \(-0.384181\pi\)
0.355881 + 0.934531i \(0.384181\pi\)
\(104\) −14742.7 −0.133658
\(105\) 0 0
\(106\) −134217. −1.16023
\(107\) 135296. 1.14242 0.571209 0.820804i \(-0.306474\pi\)
0.571209 + 0.820804i \(0.306474\pi\)
\(108\) 45412.1 0.374638
\(109\) −199508. −1.60840 −0.804202 0.594356i \(-0.797407\pi\)
−0.804202 + 0.594356i \(0.797407\pi\)
\(110\) 10253.9 0.0807992
\(111\) −100968. −0.777814
\(112\) 0 0
\(113\) 122569. 0.902991 0.451496 0.892273i \(-0.350891\pi\)
0.451496 + 0.892273i \(0.350891\pi\)
\(114\) 1520.50 0.0109578
\(115\) −96427.0 −0.679914
\(116\) 9942.56 0.0686046
\(117\) −9788.92 −0.0661104
\(118\) −229716. −1.51875
\(119\) 0 0
\(120\) −52162.6 −0.330679
\(121\) −152857. −0.949123
\(122\) 13237.1 0.0805183
\(123\) 153645. 0.915704
\(124\) 40448.0 0.236234
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 40046.2 0.220319 0.110160 0.993914i \(-0.464864\pi\)
0.110160 + 0.993914i \(0.464864\pi\)
\(128\) 31210.2 0.168373
\(129\) −104284. −0.551754
\(130\) 8478.89 0.0440029
\(131\) −211963. −1.07915 −0.539576 0.841937i \(-0.681415\pi\)
−0.539576 + 0.841937i \(0.681415\pi\)
\(132\) −10997.6 −0.0549367
\(133\) 0 0
\(134\) −186167. −0.895654
\(135\) −98989.9 −0.467473
\(136\) 203328. 0.942647
\(137\) 40980.5 0.186542 0.0932709 0.995641i \(-0.470268\pi\)
0.0932709 + 0.995641i \(0.470268\pi\)
\(138\) −185140. −0.827565
\(139\) 418948. 1.83917 0.919587 0.392886i \(-0.128523\pi\)
0.919587 + 0.392886i \(0.128523\pi\)
\(140\) 0 0
\(141\) 180019. 0.762554
\(142\) 279811. 1.16451
\(143\) 6775.40 0.0277073
\(144\) 68719.9 0.276170
\(145\) −21672.9 −0.0856047
\(146\) 107227. 0.416314
\(147\) 0 0
\(148\) 109312. 0.410218
\(149\) −64962.7 −0.239717 −0.119858 0.992791i \(-0.538244\pi\)
−0.119858 + 0.992791i \(0.538244\pi\)
\(150\) 30000.0 0.108866
\(151\) 379801. 1.35554 0.677772 0.735272i \(-0.262946\pi\)
0.677772 + 0.735272i \(0.262946\pi\)
\(152\) −6239.22 −0.0219039
\(153\) 135006. 0.466256
\(154\) 0 0
\(155\) −88169.1 −0.294773
\(156\) −9093.86 −0.0299183
\(157\) 281546. 0.911590 0.455795 0.890085i \(-0.349355\pi\)
0.455795 + 0.890085i \(0.349355\pi\)
\(158\) 204768. 0.652559
\(159\) −313788. −0.984337
\(160\) 98047.1 0.302785
\(161\) 0 0
\(162\) −46063.3 −0.137901
\(163\) −382587. −1.12788 −0.563938 0.825817i \(-0.690714\pi\)
−0.563938 + 0.825817i \(0.690714\pi\)
\(164\) −166343. −0.482941
\(165\) 23972.7 0.0685499
\(166\) −177148. −0.498961
\(167\) 388635. 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(168\) 0 0
\(169\) −365690. −0.984911
\(170\) −116939. −0.310339
\(171\) −4142.73 −0.0108342
\(172\) 112903. 0.290994
\(173\) 389693. 0.989936 0.494968 0.868911i \(-0.335180\pi\)
0.494968 + 0.868911i \(0.335180\pi\)
\(174\) −41612.0 −0.104195
\(175\) 0 0
\(176\) −47564.5 −0.115745
\(177\) −537057. −1.28851
\(178\) 189814. 0.449033
\(179\) −149812. −0.349473 −0.174737 0.984615i \(-0.555907\pi\)
−0.174737 + 0.984615i \(0.555907\pi\)
\(180\) 37497.5 0.0862623
\(181\) 821019. 1.86276 0.931380 0.364049i \(-0.118606\pi\)
0.931380 + 0.364049i \(0.118606\pi\)
\(182\) 0 0
\(183\) 30947.3 0.0683117
\(184\) 759702. 1.65424
\(185\) −238281. −0.511870
\(186\) −169285. −0.358786
\(187\) −93444.5 −0.195411
\(188\) −194897. −0.402170
\(189\) 0 0
\(190\) 3588.32 0.00721120
\(191\) 286619. 0.568487 0.284244 0.958752i \(-0.408258\pi\)
0.284244 + 0.958752i \(0.408258\pi\)
\(192\) 366376. 0.717255
\(193\) 993573. 1.92002 0.960012 0.279959i \(-0.0903209\pi\)
0.960012 + 0.279959i \(0.0903209\pi\)
\(194\) −36414.6 −0.0694657
\(195\) 19822.9 0.0373320
\(196\) 0 0
\(197\) −38209.0 −0.0701456 −0.0350728 0.999385i \(-0.511166\pi\)
−0.0350728 + 0.999385i \(0.511166\pi\)
\(198\) −53640.2 −0.0972361
\(199\) −730487. −1.30761 −0.653807 0.756661i \(-0.726829\pi\)
−0.653807 + 0.756661i \(0.726829\pi\)
\(200\) −123102. −0.217615
\(201\) −435242. −0.759872
\(202\) −566296. −0.976484
\(203\) 0 0
\(204\) 125420. 0.211004
\(205\) 362596. 0.602613
\(206\) 347244. 0.570120
\(207\) 504429. 0.818228
\(208\) −39330.9 −0.0630340
\(209\) 2867.39 0.00454069
\(210\) 0 0
\(211\) 194286. 0.300424 0.150212 0.988654i \(-0.452004\pi\)
0.150212 + 0.988654i \(0.452004\pi\)
\(212\) 339721. 0.519138
\(213\) 654175. 0.987972
\(214\) 613043. 0.915075
\(215\) −246107. −0.363102
\(216\) 779894. 1.13737
\(217\) 0 0
\(218\) −903999. −1.28833
\(219\) 250687. 0.353201
\(220\) −25953.9 −0.0361531
\(221\) −77268.8 −0.106420
\(222\) −457499. −0.623028
\(223\) −1.13569e6 −1.52931 −0.764656 0.644438i \(-0.777091\pi\)
−0.764656 + 0.644438i \(0.777091\pi\)
\(224\) 0 0
\(225\) −81737.6 −0.107638
\(226\) 555375. 0.723294
\(227\) 143806. 0.185231 0.0926155 0.995702i \(-0.470477\pi\)
0.0926155 + 0.995702i \(0.470477\pi\)
\(228\) −3848.58 −0.00490302
\(229\) 3832.50 0.00482940 0.00241470 0.999997i \(-0.499231\pi\)
0.00241470 + 0.999997i \(0.499231\pi\)
\(230\) −436923. −0.544610
\(231\) 0 0
\(232\) 170751. 0.208277
\(233\) −1.35599e6 −1.63631 −0.818157 0.574995i \(-0.805004\pi\)
−0.818157 + 0.574995i \(0.805004\pi\)
\(234\) −44354.8 −0.0529543
\(235\) 424838. 0.501827
\(236\) 581442. 0.679557
\(237\) 478731. 0.553631
\(238\) 0 0
\(239\) −478372. −0.541716 −0.270858 0.962619i \(-0.587307\pi\)
−0.270858 + 0.962619i \(0.587307\pi\)
\(240\) −139160. −0.155951
\(241\) 1.31082e6 1.45379 0.726894 0.686750i \(-0.240963\pi\)
0.726894 + 0.686750i \(0.240963\pi\)
\(242\) −692616. −0.760246
\(243\) 854490. 0.928307
\(244\) −33504.9 −0.0360275
\(245\) 0 0
\(246\) 696185. 0.733477
\(247\) 2371.04 0.00247284
\(248\) 694642. 0.717186
\(249\) −414157. −0.423318
\(250\) 70798.9 0.0716434
\(251\) −340113. −0.340753 −0.170376 0.985379i \(-0.554498\pi\)
−0.170376 + 0.985379i \(0.554498\pi\)
\(252\) 0 0
\(253\) −349141. −0.342925
\(254\) 181454. 0.176475
\(255\) −273392. −0.263291
\(256\) −965313. −0.920594
\(257\) 58839.6 0.0555696 0.0277848 0.999614i \(-0.491155\pi\)
0.0277848 + 0.999614i \(0.491155\pi\)
\(258\) −472526. −0.441954
\(259\) 0 0
\(260\) −21461.2 −0.0196888
\(261\) 113376. 0.103019
\(262\) −960433. −0.864398
\(263\) −1380.79 −0.00123095 −0.000615473 1.00000i \(-0.500196\pi\)
−0.000615473 1.00000i \(0.500196\pi\)
\(264\) −188869. −0.166783
\(265\) −740528. −0.647779
\(266\) 0 0
\(267\) 443769. 0.380959
\(268\) 471212. 0.400756
\(269\) −886839. −0.747247 −0.373624 0.927580i \(-0.621885\pi\)
−0.373624 + 0.927580i \(0.621885\pi\)
\(270\) −448536. −0.374445
\(271\) 376955. 0.311793 0.155897 0.987773i \(-0.450173\pi\)
0.155897 + 0.987773i \(0.450173\pi\)
\(272\) 542441. 0.444559
\(273\) 0 0
\(274\) 185688. 0.149420
\(275\) 56574.7 0.0451118
\(276\) 468613. 0.370289
\(277\) −880363. −0.689386 −0.344693 0.938716i \(-0.612017\pi\)
−0.344693 + 0.938716i \(0.612017\pi\)
\(278\) 1.89831e6 1.47317
\(279\) 461231. 0.354738
\(280\) 0 0
\(281\) 2.08492e6 1.57515 0.787577 0.616217i \(-0.211335\pi\)
0.787577 + 0.616217i \(0.211335\pi\)
\(282\) 815689. 0.610804
\(283\) −692401. −0.513915 −0.256958 0.966423i \(-0.582720\pi\)
−0.256958 + 0.966423i \(0.582720\pi\)
\(284\) −708238. −0.521055
\(285\) 8389.19 0.00611798
\(286\) 30700.2 0.0221935
\(287\) 0 0
\(288\) −512905. −0.364381
\(289\) −354186. −0.249452
\(290\) −98202.8 −0.0685692
\(291\) −85134.1 −0.0589347
\(292\) −271405. −0.186278
\(293\) −2.27624e6 −1.54899 −0.774495 0.632580i \(-0.781996\pi\)
−0.774495 + 0.632580i \(0.781996\pi\)
\(294\) 0 0
\(295\) −1.26743e6 −0.847950
\(296\) 1.87730e6 1.24539
\(297\) −358421. −0.235777
\(298\) −294354. −0.192013
\(299\) −288703. −0.186755
\(300\) −75933.9 −0.0487116
\(301\) 0 0
\(302\) 1.72093e6 1.08579
\(303\) −1.32395e6 −0.828448
\(304\) −16645.1 −0.0103300
\(305\) 73034.5 0.0449550
\(306\) 611730. 0.373470
\(307\) 2.65187e6 1.60586 0.802928 0.596076i \(-0.203274\pi\)
0.802928 + 0.596076i \(0.203274\pi\)
\(308\) 0 0
\(309\) 811825. 0.483689
\(310\) −399505. −0.236112
\(311\) −2.57626e6 −1.51039 −0.755195 0.655501i \(-0.772458\pi\)
−0.755195 + 0.655501i \(0.772458\pi\)
\(312\) −156175. −0.0908293
\(313\) −2.51549e6 −1.45131 −0.725657 0.688057i \(-0.758464\pi\)
−0.725657 + 0.688057i \(0.758464\pi\)
\(314\) 1.27572e6 0.730182
\(315\) 0 0
\(316\) −518295. −0.291984
\(317\) −2.30397e6 −1.28774 −0.643870 0.765135i \(-0.722672\pi\)
−0.643870 + 0.765135i \(0.722672\pi\)
\(318\) −1.42181e6 −0.788452
\(319\) −78472.9 −0.0431760
\(320\) 864633. 0.472016
\(321\) 1.43324e6 0.776349
\(322\) 0 0
\(323\) −32700.7 −0.0174402
\(324\) 116592. 0.0617031
\(325\) 46781.4 0.0245677
\(326\) −1.73355e6 −0.903427
\(327\) −2.11347e6 −1.09302
\(328\) −2.85672e6 −1.46617
\(329\) 0 0
\(330\) 108623. 0.0549084
\(331\) 697305. 0.349827 0.174913 0.984584i \(-0.444035\pi\)
0.174913 + 0.984584i \(0.444035\pi\)
\(332\) 448385. 0.223258
\(333\) 1.24649e6 0.615999
\(334\) 1.76096e6 0.863739
\(335\) −1.02715e6 −0.500062
\(336\) 0 0
\(337\) −1.14848e6 −0.550868 −0.275434 0.961320i \(-0.588822\pi\)
−0.275434 + 0.961320i \(0.588822\pi\)
\(338\) −1.65699e6 −0.788911
\(339\) 1.29842e6 0.613642
\(340\) 295987. 0.138859
\(341\) −319241. −0.148673
\(342\) −18771.3 −0.00867817
\(343\) 0 0
\(344\) 1.93896e6 0.883433
\(345\) −1.02149e6 −0.462046
\(346\) 1.76575e6 0.792937
\(347\) −2.43716e6 −1.08658 −0.543289 0.839546i \(-0.682821\pi\)
−0.543289 + 0.839546i \(0.682821\pi\)
\(348\) 105325. 0.0466213
\(349\) −896801. −0.394124 −0.197062 0.980391i \(-0.563140\pi\)
−0.197062 + 0.980391i \(0.563140\pi\)
\(350\) 0 0
\(351\) −296377. −0.128403
\(352\) 355007. 0.152715
\(353\) −4.33422e6 −1.85129 −0.925644 0.378396i \(-0.876476\pi\)
−0.925644 + 0.378396i \(0.876476\pi\)
\(354\) −2.43347e6 −1.03209
\(355\) 1.54383e6 0.650172
\(356\) −480444. −0.200917
\(357\) 0 0
\(358\) −678817. −0.279927
\(359\) −3.59167e6 −1.47082 −0.735411 0.677621i \(-0.763011\pi\)
−0.735411 + 0.677621i \(0.763011\pi\)
\(360\) 643972. 0.261885
\(361\) −2.47510e6 −0.999595
\(362\) 3.72014e6 1.49207
\(363\) −1.61928e6 −0.644992
\(364\) 0 0
\(365\) 591612. 0.232437
\(366\) 140226. 0.0547175
\(367\) −361791. −0.140215 −0.0701073 0.997539i \(-0.522334\pi\)
−0.0701073 + 0.997539i \(0.522334\pi\)
\(368\) 2.02674e6 0.780152
\(369\) −1.89682e6 −0.725202
\(370\) −1.07968e6 −0.410006
\(371\) 0 0
\(372\) 428481. 0.160537
\(373\) −3.08325e6 −1.14746 −0.573729 0.819045i \(-0.694504\pi\)
−0.573729 + 0.819045i \(0.694504\pi\)
\(374\) −423409. −0.156524
\(375\) 165522. 0.0607822
\(376\) −3.34710e6 −1.22095
\(377\) −64888.9 −0.0235135
\(378\) 0 0
\(379\) −5.04227e6 −1.80313 −0.901567 0.432639i \(-0.857582\pi\)
−0.901567 + 0.432639i \(0.857582\pi\)
\(380\) −9082.51 −0.00322661
\(381\) 424225. 0.149721
\(382\) 1.29871e6 0.455357
\(383\) 3.11928e6 1.08657 0.543285 0.839548i \(-0.317180\pi\)
0.543285 + 0.839548i \(0.317180\pi\)
\(384\) 330622. 0.114420
\(385\) 0 0
\(386\) 4.50201e6 1.53794
\(387\) 1.28744e6 0.436968
\(388\) 92169.9 0.0310821
\(389\) −1.64560e6 −0.551378 −0.275689 0.961247i \(-0.588906\pi\)
−0.275689 + 0.961247i \(0.588906\pi\)
\(390\) 89820.2 0.0299028
\(391\) 3.98171e6 1.31713
\(392\) 0 0
\(393\) −2.24541e6 −0.733355
\(394\) −173130. −0.0561865
\(395\) 1.12979e6 0.364337
\(396\) 135770. 0.0435078
\(397\) −3.72727e6 −1.18690 −0.593451 0.804870i \(-0.702235\pi\)
−0.593451 + 0.804870i \(0.702235\pi\)
\(398\) −3.30993e6 −1.04740
\(399\) 0 0
\(400\) −328413. −0.102629
\(401\) 4.84820e6 1.50564 0.752818 0.658229i \(-0.228694\pi\)
0.752818 + 0.658229i \(0.228694\pi\)
\(402\) −1.97214e6 −0.608656
\(403\) −263979. −0.0809667
\(404\) 1.43337e6 0.436923
\(405\) −254149. −0.0769930
\(406\) 0 0
\(407\) −862761. −0.258169
\(408\) 2.15393e6 0.640591
\(409\) 3.03126e6 0.896014 0.448007 0.894030i \(-0.352134\pi\)
0.448007 + 0.894030i \(0.352134\pi\)
\(410\) 1.64297e6 0.482692
\(411\) 434123. 0.126768
\(412\) −878918. −0.255097
\(413\) 0 0
\(414\) 2.28563e6 0.655399
\(415\) −977396. −0.278580
\(416\) 293554. 0.0831676
\(417\) 4.43808e6 1.24984
\(418\) 12992.5 0.00363708
\(419\) −1.66905e6 −0.464444 −0.232222 0.972663i \(-0.574600\pi\)
−0.232222 + 0.972663i \(0.574600\pi\)
\(420\) 0 0
\(421\) −1.76031e6 −0.484043 −0.242022 0.970271i \(-0.577810\pi\)
−0.242022 + 0.970271i \(0.577810\pi\)
\(422\) 880335. 0.240639
\(423\) −2.22242e6 −0.603913
\(424\) 5.83427e6 1.57606
\(425\) −645196. −0.173268
\(426\) 2.96415e6 0.791364
\(427\) 0 0
\(428\) −1.55169e6 −0.409445
\(429\) 71774.4 0.0188290
\(430\) −1.11514e6 −0.290844
\(431\) −648832. −0.168244 −0.0841219 0.996455i \(-0.526809\pi\)
−0.0841219 + 0.996455i \(0.526809\pi\)
\(432\) 2.08061e6 0.536392
\(433\) 360993. 0.0925293 0.0462646 0.998929i \(-0.485268\pi\)
0.0462646 + 0.998929i \(0.485268\pi\)
\(434\) 0 0
\(435\) −229590. −0.0581740
\(436\) 2.28814e6 0.576455
\(437\) −122181. −0.0306055
\(438\) 1.13590e6 0.282913
\(439\) −1.07021e6 −0.265038 −0.132519 0.991180i \(-0.542307\pi\)
−0.132519 + 0.991180i \(0.542307\pi\)
\(440\) −445725. −0.109758
\(441\) 0 0
\(442\) −350115. −0.0852423
\(443\) −797145. −0.192987 −0.0964935 0.995334i \(-0.530763\pi\)
−0.0964935 + 0.995334i \(0.530763\pi\)
\(444\) 1.15799e6 0.278770
\(445\) 1.04728e6 0.250704
\(446\) −5.14594e6 −1.22498
\(447\) −688175. −0.162903
\(448\) 0 0
\(449\) 6.40103e6 1.49842 0.749211 0.662331i \(-0.230433\pi\)
0.749211 + 0.662331i \(0.230433\pi\)
\(450\) −370364. −0.0862178
\(451\) 1.31288e6 0.303937
\(452\) −1.40572e6 −0.323634
\(453\) 4.02338e6 0.921181
\(454\) 651605. 0.148370
\(455\) 0 0
\(456\) −66094.4 −0.0148851
\(457\) 5.91843e6 1.32561 0.662806 0.748791i \(-0.269366\pi\)
0.662806 + 0.748791i \(0.269366\pi\)
\(458\) 17365.5 0.00386834
\(459\) 4.08755e6 0.905588
\(460\) 1.10591e6 0.243683
\(461\) 8.84337e6 1.93805 0.969026 0.246960i \(-0.0794315\pi\)
0.969026 + 0.246960i \(0.0794315\pi\)
\(462\) 0 0
\(463\) −6.76858e6 −1.46739 −0.733695 0.679479i \(-0.762206\pi\)
−0.733695 + 0.679479i \(0.762206\pi\)
\(464\) 455531. 0.0982252
\(465\) −934009. −0.200317
\(466\) −6.14416e6 −1.31068
\(467\) 1.04740e6 0.222239 0.111120 0.993807i \(-0.464556\pi\)
0.111120 + 0.993807i \(0.464556\pi\)
\(468\) 112268. 0.0236941
\(469\) 0 0
\(470\) 1.92500e6 0.401962
\(471\) 2.98252e6 0.619486
\(472\) 9.98551e6 2.06308
\(473\) −891101. −0.183136
\(474\) 2.16919e6 0.443457
\(475\) 19798.2 0.00402616
\(476\) 0 0
\(477\) 3.87386e6 0.779556
\(478\) −2.16757e6 −0.433913
\(479\) 9.37725e6 1.86740 0.933699 0.358060i \(-0.116562\pi\)
0.933699 + 0.358060i \(0.116562\pi\)
\(480\) 1.03865e6 0.205763
\(481\) −713414. −0.140598
\(482\) 5.93950e6 1.16448
\(483\) 0 0
\(484\) 1.75310e6 0.340168
\(485\) −200913. −0.0387841
\(486\) 3.87181e6 0.743572
\(487\) −2.88960e6 −0.552098 −0.276049 0.961144i \(-0.589025\pi\)
−0.276049 + 0.961144i \(0.589025\pi\)
\(488\) −575404. −0.109376
\(489\) −4.05290e6 −0.766467
\(490\) 0 0
\(491\) −1.04845e6 −0.196265 −0.0981324 0.995173i \(-0.531287\pi\)
−0.0981324 + 0.995173i \(0.531287\pi\)
\(492\) −1.76213e6 −0.328191
\(493\) 894930. 0.165833
\(494\) 10743.5 0.00198074
\(495\) −295954. −0.0542889
\(496\) 1.85318e6 0.338230
\(497\) 0 0
\(498\) −1.87660e6 −0.339077
\(499\) 6.38524e6 1.14796 0.573979 0.818870i \(-0.305399\pi\)
0.573979 + 0.818870i \(0.305399\pi\)
\(500\) −179201. −0.0320565
\(501\) 4.11696e6 0.732795
\(502\) −1.54110e6 −0.272942
\(503\) −1.35497e6 −0.238787 −0.119394 0.992847i \(-0.538095\pi\)
−0.119394 + 0.992847i \(0.538095\pi\)
\(504\) 0 0
\(505\) −3.12448e6 −0.545191
\(506\) −1.58200e6 −0.274682
\(507\) −3.87390e6 −0.669312
\(508\) −459285. −0.0789628
\(509\) −6.43410e6 −1.10076 −0.550381 0.834914i \(-0.685518\pi\)
−0.550381 + 0.834914i \(0.685518\pi\)
\(510\) −1.23878e6 −0.210896
\(511\) 0 0
\(512\) −5.37268e6 −0.905767
\(513\) −125429. −0.0210428
\(514\) 266610. 0.0445111
\(515\) 1.91588e6 0.318310
\(516\) 1.19603e6 0.197750
\(517\) 1.53825e6 0.253104
\(518\) 0 0
\(519\) 4.12817e6 0.672727
\(520\) −368568. −0.0597736
\(521\) −6.16552e6 −0.995120 −0.497560 0.867430i \(-0.665771\pi\)
−0.497560 + 0.867430i \(0.665771\pi\)
\(522\) 513719. 0.0825181
\(523\) −7.12400e6 −1.13886 −0.569429 0.822040i \(-0.692836\pi\)
−0.569429 + 0.822040i \(0.692836\pi\)
\(524\) 2.43098e6 0.386770
\(525\) 0 0
\(526\) −6256.55 −0.000985985 0
\(527\) 3.64072e6 0.571033
\(528\) −503869. −0.0786562
\(529\) 8.44071e6 1.31141
\(530\) −3.35543e6 −0.518870
\(531\) 6.63021e6 1.02045
\(532\) 0 0
\(533\) 1.08562e6 0.165523
\(534\) 2.01077e6 0.305148
\(535\) 3.38240e6 0.510905
\(536\) 8.09246e6 1.21666
\(537\) −1.58702e6 −0.237490
\(538\) −4.01838e6 −0.598544
\(539\) 0 0
\(540\) 1.13530e6 0.167543
\(541\) −9.51129e6 −1.39716 −0.698580 0.715532i \(-0.746184\pi\)
−0.698580 + 0.715532i \(0.746184\pi\)
\(542\) 1.70803e6 0.249746
\(543\) 8.69737e6 1.26587
\(544\) −4.04862e6 −0.586556
\(545\) −4.98771e6 −0.719300
\(546\) 0 0
\(547\) 1.48406e6 0.212071 0.106036 0.994362i \(-0.466184\pi\)
0.106036 + 0.994362i \(0.466184\pi\)
\(548\) −470001. −0.0668570
\(549\) −382058. −0.0541002
\(550\) 256347. 0.0361345
\(551\) −27461.4 −0.00385340
\(552\) 8.04782e6 1.12417
\(553\) 0 0
\(554\) −3.98904e6 −0.552197
\(555\) −2.52420e6 −0.347849
\(556\) −4.80486e6 −0.659164
\(557\) −7.72192e6 −1.05460 −0.527299 0.849680i \(-0.676795\pi\)
−0.527299 + 0.849680i \(0.676795\pi\)
\(558\) 2.08990e6 0.284144
\(559\) −736847. −0.0997351
\(560\) 0 0
\(561\) −989894. −0.132795
\(562\) 9.44703e6 1.26169
\(563\) −3.41747e6 −0.454395 −0.227198 0.973849i \(-0.572956\pi\)
−0.227198 + 0.973849i \(0.572956\pi\)
\(564\) −2.06461e6 −0.273301
\(565\) 3.06422e6 0.403830
\(566\) −3.13736e6 −0.411645
\(567\) 0 0
\(568\) −1.21631e7 −1.58188
\(569\) −2.42117e6 −0.313505 −0.156752 0.987638i \(-0.550102\pi\)
−0.156752 + 0.987638i \(0.550102\pi\)
\(570\) 38012.5 0.00490049
\(571\) 2.31941e6 0.297705 0.148853 0.988859i \(-0.452442\pi\)
0.148853 + 0.988859i \(0.452442\pi\)
\(572\) −77706.2 −0.00993037
\(573\) 3.03626e6 0.386325
\(574\) 0 0
\(575\) −2.41067e6 −0.304067
\(576\) −4.52307e6 −0.568038
\(577\) −2.56645e6 −0.320917 −0.160459 0.987043i \(-0.551297\pi\)
−0.160459 + 0.987043i \(0.551297\pi\)
\(578\) −1.60486e6 −0.199811
\(579\) 1.05253e7 1.30478
\(580\) 248564. 0.0306809
\(581\) 0 0
\(582\) −385753. −0.0472066
\(583\) −2.68129e6 −0.326718
\(584\) −4.66103e6 −0.565522
\(585\) −244723. −0.0295655
\(586\) −1.03139e7 −1.24074
\(587\) −2.09851e6 −0.251371 −0.125685 0.992070i \(-0.540113\pi\)
−0.125685 + 0.992070i \(0.540113\pi\)
\(588\) 0 0
\(589\) −111718. −0.0132688
\(590\) −5.74291e6 −0.679207
\(591\) −404763. −0.0476686
\(592\) 5.00829e6 0.587334
\(593\) 5.11831e6 0.597709 0.298854 0.954299i \(-0.403396\pi\)
0.298854 + 0.954299i \(0.403396\pi\)
\(594\) −1.62405e6 −0.188857
\(595\) 0 0
\(596\) 745048. 0.0859150
\(597\) −7.73833e6 −0.888610
\(598\) −1.30815e6 −0.149591
\(599\) −2.61775e6 −0.298099 −0.149049 0.988830i \(-0.547621\pi\)
−0.149049 + 0.988830i \(0.547621\pi\)
\(600\) −1.30407e6 −0.147884
\(601\) −9.49925e6 −1.07276 −0.536381 0.843976i \(-0.680209\pi\)
−0.536381 + 0.843976i \(0.680209\pi\)
\(602\) 0 0
\(603\) 5.37326e6 0.601789
\(604\) −4.35589e6 −0.485830
\(605\) −3.82143e6 −0.424461
\(606\) −5.99899e6 −0.663586
\(607\) −6.07366e6 −0.669081 −0.334541 0.942381i \(-0.608581\pi\)
−0.334541 + 0.942381i \(0.608581\pi\)
\(608\) 124234. 0.0136295
\(609\) 0 0
\(610\) 330929. 0.0360089
\(611\) 1.27197e6 0.137839
\(612\) −1.54837e6 −0.167107
\(613\) −9.20907e6 −0.989839 −0.494920 0.868939i \(-0.664802\pi\)
−0.494920 + 0.868939i \(0.664802\pi\)
\(614\) 1.20160e7 1.28629
\(615\) 3.84112e6 0.409515
\(616\) 0 0
\(617\) 1.37224e7 1.45117 0.725585 0.688133i \(-0.241569\pi\)
0.725585 + 0.688133i \(0.241569\pi\)
\(618\) 3.67848e6 0.387434
\(619\) 6.80356e6 0.713690 0.356845 0.934164i \(-0.383852\pi\)
0.356845 + 0.934164i \(0.383852\pi\)
\(620\) 1.01120e6 0.105647
\(621\) 1.52725e7 1.58921
\(622\) −1.16734e7 −1.20982
\(623\) 0 0
\(624\) −416647. −0.0428358
\(625\) 390625. 0.0400000
\(626\) −1.13980e7 −1.16250
\(627\) 30375.4 0.00308570
\(628\) −3.22901e6 −0.326716
\(629\) 9.83921e6 0.991593
\(630\) 0 0
\(631\) −2.80897e6 −0.280850 −0.140425 0.990091i \(-0.544847\pi\)
−0.140425 + 0.990091i \(0.544847\pi\)
\(632\) −8.90105e6 −0.886438
\(633\) 2.05815e6 0.204158
\(634\) −1.04396e7 −1.03148
\(635\) 1.00115e6 0.0985297
\(636\) 3.59880e6 0.352788
\(637\) 0 0
\(638\) −355571. −0.0345839
\(639\) −8.07608e6 −0.782435
\(640\) 780256. 0.0752986
\(641\) −3.83494e6 −0.368649 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(642\) 6.49420e6 0.621854
\(643\) −1.59562e7 −1.52195 −0.760977 0.648779i \(-0.775280\pi\)
−0.760977 + 0.648779i \(0.775280\pi\)
\(644\) 0 0
\(645\) −2.60711e6 −0.246752
\(646\) −148171. −0.0139695
\(647\) 7.48025e6 0.702514 0.351257 0.936279i \(-0.385754\pi\)
0.351257 + 0.936279i \(0.385754\pi\)
\(648\) 2.00232e6 0.187325
\(649\) −4.58910e6 −0.427677
\(650\) 211972. 0.0196787
\(651\) 0 0
\(652\) 4.38785e6 0.404234
\(653\) 2.62102e6 0.240540 0.120270 0.992741i \(-0.461624\pi\)
0.120270 + 0.992741i \(0.461624\pi\)
\(654\) −9.57641e6 −0.875504
\(655\) −5.29908e6 −0.482611
\(656\) −7.62121e6 −0.691456
\(657\) −3.09485e6 −0.279721
\(658\) 0 0
\(659\) −8.01276e6 −0.718734 −0.359367 0.933196i \(-0.617007\pi\)
−0.359367 + 0.933196i \(0.617007\pi\)
\(660\) −274940. −0.0245685
\(661\) 7.21439e6 0.642238 0.321119 0.947039i \(-0.395941\pi\)
0.321119 + 0.947039i \(0.395941\pi\)
\(662\) 3.15958e6 0.280210
\(663\) −818539. −0.0723195
\(664\) 7.70044e6 0.677790
\(665\) 0 0
\(666\) 5.64803e6 0.493414
\(667\) 3.34377e6 0.291019
\(668\) −4.45721e6 −0.386475
\(669\) −1.20308e7 −1.03927
\(670\) −4.65417e6 −0.400549
\(671\) 264442. 0.0226738
\(672\) 0 0
\(673\) 1.43323e7 1.21977 0.609887 0.792488i \(-0.291215\pi\)
0.609887 + 0.792488i \(0.291215\pi\)
\(674\) −5.20390e6 −0.441244
\(675\) −2.47475e6 −0.209060
\(676\) 4.19406e6 0.352994
\(677\) 4.94781e6 0.414898 0.207449 0.978246i \(-0.433484\pi\)
0.207449 + 0.978246i \(0.433484\pi\)
\(678\) 5.88330e6 0.491526
\(679\) 0 0
\(680\) 5.08319e6 0.421565
\(681\) 1.52340e6 0.125877
\(682\) −1.44652e6 −0.119087
\(683\) 7.77331e6 0.637608 0.318804 0.947821i \(-0.396719\pi\)
0.318804 + 0.947821i \(0.396719\pi\)
\(684\) 47512.5 0.00388300
\(685\) 1.02451e6 0.0834241
\(686\) 0 0
\(687\) 40599.1 0.00328190
\(688\) 5.17280e6 0.416634
\(689\) −2.21715e6 −0.177929
\(690\) −4.62849e6 −0.370098
\(691\) 7.69151e6 0.612797 0.306398 0.951903i \(-0.400876\pi\)
0.306398 + 0.951903i \(0.400876\pi\)
\(692\) −4.46934e6 −0.354795
\(693\) 0 0
\(694\) −1.10431e7 −0.870347
\(695\) 1.04737e7 0.822504
\(696\) 1.80883e6 0.141538
\(697\) −1.49725e7 −1.16738
\(698\) −4.06352e6 −0.315692
\(699\) −1.43645e7 −1.11198
\(700\) 0 0
\(701\) 1.53139e7 1.17704 0.588521 0.808482i \(-0.299711\pi\)
0.588521 + 0.808482i \(0.299711\pi\)
\(702\) −1.34292e6 −0.102851
\(703\) −301921. −0.0230412
\(704\) 3.13065e6 0.238069
\(705\) 4.50048e6 0.341025
\(706\) −1.96389e7 −1.48288
\(707\) 0 0
\(708\) 6.15944e6 0.461804
\(709\) 1.84381e7 1.37753 0.688764 0.724986i \(-0.258154\pi\)
0.688764 + 0.724986i \(0.258154\pi\)
\(710\) 6.99528e6 0.520786
\(711\) −5.91015e6 −0.438454
\(712\) −8.25100e6 −0.609968
\(713\) 1.36030e7 1.00210
\(714\) 0 0
\(715\) 169385. 0.0123911
\(716\) 1.71817e6 0.125252
\(717\) −5.06758e6 −0.368132
\(718\) −1.62743e7 −1.17813
\(719\) 99951.2 0.00721051 0.00360525 0.999994i \(-0.498852\pi\)
0.00360525 + 0.999994i \(0.498852\pi\)
\(720\) 1.71800e6 0.123507
\(721\) 0 0
\(722\) −1.12150e7 −0.800673
\(723\) 1.38860e7 0.987945
\(724\) −9.41616e6 −0.667617
\(725\) −541823. −0.0382836
\(726\) −7.33715e6 −0.516637
\(727\) −1.25198e7 −0.878538 −0.439269 0.898356i \(-0.644762\pi\)
−0.439269 + 0.898356i \(0.644762\pi\)
\(728\) 0 0
\(729\) 1.15223e7 0.803007
\(730\) 2.68067e6 0.186181
\(731\) 1.01624e7 0.703401
\(732\) −354931. −0.0244831
\(733\) 1.72341e7 1.18476 0.592378 0.805660i \(-0.298189\pi\)
0.592378 + 0.805660i \(0.298189\pi\)
\(734\) −1.63932e6 −0.112312
\(735\) 0 0
\(736\) −1.51270e7 −1.02934
\(737\) −3.71910e6 −0.252214
\(738\) −8.59472e6 −0.580886
\(739\) −1.83842e7 −1.23832 −0.619160 0.785265i \(-0.712527\pi\)
−0.619160 + 0.785265i \(0.712527\pi\)
\(740\) 2.73281e6 0.183455
\(741\) 25117.3 0.00168046
\(742\) 0 0
\(743\) 2.79507e7 1.85746 0.928732 0.370752i \(-0.120900\pi\)
0.928732 + 0.370752i \(0.120900\pi\)
\(744\) 7.35861e6 0.487375
\(745\) −1.62407e6 −0.107205
\(746\) −1.39706e7 −0.919111
\(747\) 5.11296e6 0.335252
\(748\) 1.07170e6 0.0700358
\(749\) 0 0
\(750\) 750000. 0.0486864
\(751\) 1.70136e7 1.10077 0.550383 0.834912i \(-0.314482\pi\)
0.550383 + 0.834912i \(0.314482\pi\)
\(752\) −8.92944e6 −0.575810
\(753\) −3.60295e6 −0.231564
\(754\) −294020. −0.0188342
\(755\) 9.49502e6 0.606218
\(756\) 0 0
\(757\) −1.91600e7 −1.21522 −0.607611 0.794234i \(-0.707872\pi\)
−0.607611 + 0.794234i \(0.707872\pi\)
\(758\) −2.28472e7 −1.44431
\(759\) −3.69858e6 −0.233040
\(760\) −155980. −0.00979571
\(761\) 2.06404e7 1.29198 0.645992 0.763344i \(-0.276444\pi\)
0.645992 + 0.763344i \(0.276444\pi\)
\(762\) 1.92222e6 0.119926
\(763\) 0 0
\(764\) −3.28719e6 −0.203747
\(765\) 3.37515e6 0.208516
\(766\) 1.41339e7 0.870341
\(767\) −3.79471e6 −0.232911
\(768\) −1.02259e7 −0.625605
\(769\) −1.11748e7 −0.681432 −0.340716 0.940166i \(-0.610669\pi\)
−0.340716 + 0.940166i \(0.610669\pi\)
\(770\) 0 0
\(771\) 623311. 0.0377632
\(772\) −1.13952e7 −0.688141
\(773\) 3.07555e7 1.85129 0.925645 0.378394i \(-0.123523\pi\)
0.925645 + 0.378394i \(0.123523\pi\)
\(774\) 5.83355e6 0.350010
\(775\) −2.20423e6 −0.131826
\(776\) 1.58290e6 0.0943624
\(777\) 0 0
\(778\) −7.45641e6 −0.441653
\(779\) 459440. 0.0271259
\(780\) −227347. −0.0133799
\(781\) 5.58986e6 0.327924
\(782\) 1.80417e7 1.05502
\(783\) 3.43264e6 0.200089
\(784\) 0 0
\(785\) 7.03864e6 0.407675
\(786\) −1.01742e7 −0.587416
\(787\) −3.52357e6 −0.202790 −0.101395 0.994846i \(-0.532331\pi\)
−0.101395 + 0.994846i \(0.532331\pi\)
\(788\) 438214. 0.0251403
\(789\) −14627.3 −0.000836509 0
\(790\) 5.11921e6 0.291833
\(791\) 0 0
\(792\) 2.33168e6 0.132086
\(793\) 218666. 0.0123480
\(794\) −1.68887e7 −0.950706
\(795\) −7.84470e6 −0.440209
\(796\) 8.37786e6 0.468652
\(797\) 3.57469e6 0.199339 0.0996695 0.995021i \(-0.468221\pi\)
0.0996695 + 0.995021i \(0.468221\pi\)
\(798\) 0 0
\(799\) −1.75426e7 −0.972139
\(800\) 2.45118e6 0.135410
\(801\) −5.47853e6 −0.301705
\(802\) 2.19678e7 1.20601
\(803\) 2.14210e6 0.117233
\(804\) 4.99173e6 0.272340
\(805\) 0 0
\(806\) −1.19612e6 −0.0648542
\(807\) −9.39463e6 −0.507804
\(808\) 2.46163e7 1.32646
\(809\) 3.38466e7 1.81821 0.909104 0.416569i \(-0.136767\pi\)
0.909104 + 0.416569i \(0.136767\pi\)
\(810\) −1.15158e6 −0.0616712
\(811\) −3.21058e7 −1.71408 −0.857042 0.515247i \(-0.827700\pi\)
−0.857042 + 0.515247i \(0.827700\pi\)
\(812\) 0 0
\(813\) 3.99323e6 0.211884
\(814\) −3.90928e6 −0.206793
\(815\) −9.56468e6 −0.504402
\(816\) 5.74628e6 0.302108
\(817\) −311839. −0.0163446
\(818\) 1.37350e7 0.717706
\(819\) 0 0
\(820\) −4.15857e6 −0.215978
\(821\) 3.04076e6 0.157443 0.0787216 0.996897i \(-0.474916\pi\)
0.0787216 + 0.996897i \(0.474916\pi\)
\(822\) 1.96707e6 0.101541
\(823\) 3.72147e6 0.191520 0.0957601 0.995404i \(-0.469472\pi\)
0.0957601 + 0.995404i \(0.469472\pi\)
\(824\) −1.50943e7 −0.774452
\(825\) 599318. 0.0306565
\(826\) 0 0
\(827\) −8.77895e6 −0.446354 −0.223177 0.974778i \(-0.571643\pi\)
−0.223177 + 0.974778i \(0.571643\pi\)
\(828\) −5.78523e6 −0.293255
\(829\) −6.61553e6 −0.334332 −0.167166 0.985929i \(-0.553462\pi\)
−0.167166 + 0.985929i \(0.553462\pi\)
\(830\) −4.42871e6 −0.223142
\(831\) −9.32602e6 −0.468483
\(832\) 2.58872e6 0.129651
\(833\) 0 0
\(834\) 2.01095e7 1.00112
\(835\) 9.71588e6 0.482243
\(836\) −32885.8 −0.00162739
\(837\) 1.39646e7 0.688991
\(838\) −7.56267e6 −0.372019
\(839\) 4.67110e6 0.229094 0.114547 0.993418i \(-0.463458\pi\)
0.114547 + 0.993418i \(0.463458\pi\)
\(840\) 0 0
\(841\) −1.97596e7 −0.963359
\(842\) −7.97619e6 −0.387718
\(843\) 2.20863e7 1.07042
\(844\) −2.22824e6 −0.107673
\(845\) −9.14226e6 −0.440465
\(846\) −1.00701e7 −0.483733
\(847\) 0 0
\(848\) 1.55648e7 0.743280
\(849\) −7.33487e6 −0.349239
\(850\) −2.92347e6 −0.138788
\(851\) 3.67627e7 1.74014
\(852\) −7.50264e6 −0.354091
\(853\) 3.66146e7 1.72299 0.861494 0.507768i \(-0.169529\pi\)
0.861494 + 0.507768i \(0.169529\pi\)
\(854\) 0 0
\(855\) −103568. −0.00484520
\(856\) −2.66483e7 −1.24304
\(857\) −1.15520e7 −0.537288 −0.268644 0.963240i \(-0.586575\pi\)
−0.268644 + 0.963240i \(0.586575\pi\)
\(858\) 325219. 0.0150820
\(859\) −3.53878e7 −1.63633 −0.818165 0.574983i \(-0.805009\pi\)
−0.818165 + 0.574983i \(0.805009\pi\)
\(860\) 2.82257e6 0.130137
\(861\) 0 0
\(862\) −2.93994e6 −0.134763
\(863\) 1.07433e7 0.491034 0.245517 0.969392i \(-0.421042\pi\)
0.245517 + 0.969392i \(0.421042\pi\)
\(864\) −1.55291e7 −0.707720
\(865\) 9.74232e6 0.442713
\(866\) 1.63571e6 0.0741158
\(867\) −3.75203e6 −0.169519
\(868\) 0 0
\(869\) 4.09071e6 0.183759
\(870\) −1.04030e6 −0.0465973
\(871\) −3.07531e6 −0.137355
\(872\) 3.92958e7 1.75007
\(873\) 1.05102e6 0.0466740
\(874\) −553618. −0.0245150
\(875\) 0 0
\(876\) −2.87510e6 −0.126588
\(877\) 4.58293e6 0.201208 0.100604 0.994927i \(-0.467923\pi\)
0.100604 + 0.994927i \(0.467923\pi\)
\(878\) −4.84927e6 −0.212295
\(879\) −2.41131e7 −1.05264
\(880\) −1.18911e6 −0.0517626
\(881\) −1.68070e7 −0.729543 −0.364772 0.931097i \(-0.618853\pi\)
−0.364772 + 0.931097i \(0.618853\pi\)
\(882\) 0 0
\(883\) −5.28260e6 −0.228006 −0.114003 0.993480i \(-0.536367\pi\)
−0.114003 + 0.993480i \(0.536367\pi\)
\(884\) 886186. 0.0381412
\(885\) −1.34264e7 −0.576238
\(886\) −3.61197e6 −0.154582
\(887\) −4.43107e7 −1.89104 −0.945518 0.325569i \(-0.894444\pi\)
−0.945518 + 0.325569i \(0.894444\pi\)
\(888\) 1.98870e7 0.846322
\(889\) 0 0
\(890\) 4.74535e6 0.200814
\(891\) −920219. −0.0388326
\(892\) 1.30250e7 0.548109
\(893\) 538305. 0.0225892
\(894\) −3.11821e6 −0.130485
\(895\) −3.74530e6 −0.156289
\(896\) 0 0
\(897\) −3.05834e6 −0.126913
\(898\) 2.90039e7 1.20023
\(899\) 3.05741e6 0.126170
\(900\) 937438. 0.0385777
\(901\) 3.05783e7 1.25488
\(902\) 5.94883e6 0.243453
\(903\) 0 0
\(904\) −2.41415e7 −0.982524
\(905\) 2.05255e7 0.833051
\(906\) 1.82304e7 0.737865
\(907\) −3.95608e7 −1.59679 −0.798394 0.602135i \(-0.794317\pi\)
−0.798394 + 0.602135i \(0.794317\pi\)
\(908\) −1.64930e6 −0.0663872
\(909\) 1.63448e7 0.656099
\(910\) 0 0
\(911\) −2.40982e7 −0.962030 −0.481015 0.876712i \(-0.659732\pi\)
−0.481015 + 0.876712i \(0.659732\pi\)
\(912\) −176328. −0.00701994
\(913\) −3.53894e6 −0.140506
\(914\) 2.68172e7 1.06181
\(915\) 773682. 0.0305499
\(916\) −43954.4 −0.00173087
\(917\) 0 0
\(918\) 1.85212e7 0.725375
\(919\) −2.24660e7 −0.877480 −0.438740 0.898614i \(-0.644575\pi\)
−0.438740 + 0.898614i \(0.644575\pi\)
\(920\) 1.89925e7 0.739799
\(921\) 2.80923e7 1.09129
\(922\) 4.00704e7 1.55238
\(923\) 4.62223e6 0.178586
\(924\) 0 0
\(925\) −5.95701e6 −0.228915
\(926\) −3.06693e7 −1.17538
\(927\) −1.00224e7 −0.383063
\(928\) −3.39995e6 −0.129599
\(929\) 1.16280e7 0.442043 0.221021 0.975269i \(-0.429061\pi\)
0.221021 + 0.975269i \(0.429061\pi\)
\(930\) −4.23212e6 −0.160454
\(931\) 0 0
\(932\) 1.55517e7 0.586458
\(933\) −2.72913e7 −1.02641
\(934\) 4.74591e6 0.178013
\(935\) −2.33611e6 −0.0873906
\(936\) 1.92806e6 0.0719333
\(937\) 2.40956e7 0.896580 0.448290 0.893888i \(-0.352033\pi\)
0.448290 + 0.893888i \(0.352033\pi\)
\(938\) 0 0
\(939\) −2.66475e7 −0.986264
\(940\) −4.87241e6 −0.179856
\(941\) 567590. 0.0208959 0.0104479 0.999945i \(-0.496674\pi\)
0.0104479 + 0.999945i \(0.496674\pi\)
\(942\) 1.35142e7 0.496207
\(943\) −5.59425e7 −2.04863
\(944\) 2.66395e7 0.972962
\(945\) 0 0
\(946\) −4.03769e6 −0.146692
\(947\) 9.04501e6 0.327743 0.163872 0.986482i \(-0.447602\pi\)
0.163872 + 0.986482i \(0.447602\pi\)
\(948\) −5.49050e6 −0.198423
\(949\) 1.77129e6 0.0638446
\(950\) 89708.1 0.00322495
\(951\) −2.44068e7 −0.875105
\(952\) 0 0
\(953\) −1.48053e7 −0.528063 −0.264032 0.964514i \(-0.585052\pi\)
−0.264032 + 0.964514i \(0.585052\pi\)
\(954\) 1.75529e7 0.624423
\(955\) 7.16546e6 0.254235
\(956\) 5.48639e6 0.194152
\(957\) −831294. −0.0293410
\(958\) 4.24895e7 1.49578
\(959\) 0 0
\(960\) 9.15939e6 0.320766
\(961\) −1.61911e7 −0.565546
\(962\) −3.23257e6 −0.112619
\(963\) −1.76940e7 −0.614838
\(964\) −1.50336e7 −0.521041
\(965\) 2.48393e7 0.858661
\(966\) 0 0
\(967\) −2.02304e7 −0.695726 −0.347863 0.937545i \(-0.613093\pi\)
−0.347863 + 0.937545i \(0.613093\pi\)
\(968\) 3.01072e7 1.03272
\(969\) −346411. −0.0118517
\(970\) −910364. −0.0310660
\(971\) 255216. 0.00868679 0.00434340 0.999991i \(-0.498617\pi\)
0.00434340 + 0.999991i \(0.498617\pi\)
\(972\) −9.80004e6 −0.332707
\(973\) 0 0
\(974\) −1.30932e7 −0.442229
\(975\) 495573. 0.0166954
\(976\) −1.53507e6 −0.0515827
\(977\) 3.15610e7 1.05783 0.528914 0.848675i \(-0.322599\pi\)
0.528914 + 0.848675i \(0.322599\pi\)
\(978\) −1.83642e7 −0.613938
\(979\) 3.79196e6 0.126447
\(980\) 0 0
\(981\) 2.60917e7 0.865627
\(982\) −4.75064e6 −0.157208
\(983\) 5.24054e7 1.72978 0.864892 0.501958i \(-0.167387\pi\)
0.864892 + 0.501958i \(0.167387\pi\)
\(984\) −3.02624e7 −0.996357
\(985\) −955225. −0.0313701
\(986\) 4.05504e6 0.132832
\(987\) 0 0
\(988\) −27193.1 −0.000886270 0
\(989\) 3.79702e7 1.23439
\(990\) −1.34101e6 −0.0434853
\(991\) 3.73103e7 1.20683 0.603413 0.797429i \(-0.293807\pi\)
0.603413 + 0.797429i \(0.293807\pi\)
\(992\) −1.38316e7 −0.446264
\(993\) 7.38682e6 0.237730
\(994\) 0 0
\(995\) −1.82622e7 −0.584783
\(996\) 4.74992e6 0.151718
\(997\) 2.27918e7 0.726174 0.363087 0.931755i \(-0.381723\pi\)
0.363087 + 0.931755i \(0.381723\pi\)
\(998\) 2.89323e7 0.919512
\(999\) 3.77398e7 1.19643
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 245.6.a.c.1.2 2
7.6 odd 2 35.6.a.b.1.2 2
21.20 even 2 315.6.a.c.1.1 2
28.27 even 2 560.6.a.l.1.2 2
35.13 even 4 175.6.b.d.99.1 4
35.27 even 4 175.6.b.d.99.4 4
35.34 odd 2 175.6.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.a.b.1.2 2 7.6 odd 2
175.6.a.d.1.1 2 35.34 odd 2
175.6.b.d.99.1 4 35.13 even 4
175.6.b.d.99.4 4 35.27 even 4
245.6.a.c.1.2 2 1.1 even 1 trivial
315.6.a.c.1.1 2 21.20 even 2
560.6.a.l.1.2 2 28.27 even 2