Properties

Label 115.6.a.c.1.3
Level $115$
Weight $6$
Character 115.1
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $7$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, 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("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.33855\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.33855 q^{2} +13.0297 q^{3} -20.8541 q^{4} -25.0000 q^{5} -43.5004 q^{6} -186.649 q^{7} +176.456 q^{8} -73.2260 q^{9} +O(q^{10})\) \(q-3.33855 q^{2} +13.0297 q^{3} -20.8541 q^{4} -25.0000 q^{5} -43.5004 q^{6} -186.649 q^{7} +176.456 q^{8} -73.2260 q^{9} +83.4637 q^{10} +162.809 q^{11} -271.723 q^{12} +709.202 q^{13} +623.137 q^{14} -325.743 q^{15} +78.2248 q^{16} -12.3180 q^{17} +244.468 q^{18} +1435.83 q^{19} +521.353 q^{20} -2431.99 q^{21} -543.545 q^{22} +529.000 q^{23} +2299.17 q^{24} +625.000 q^{25} -2367.71 q^{26} -4120.34 q^{27} +3892.40 q^{28} +736.021 q^{29} +1087.51 q^{30} +4916.72 q^{31} -5907.75 q^{32} +2121.36 q^{33} +41.1243 q^{34} +4666.23 q^{35} +1527.06 q^{36} +8462.08 q^{37} -4793.59 q^{38} +9240.72 q^{39} -4411.40 q^{40} +10691.2 q^{41} +8119.31 q^{42} +7549.04 q^{43} -3395.23 q^{44} +1830.65 q^{45} -1766.09 q^{46} +2074.84 q^{47} +1019.25 q^{48} +18030.9 q^{49} -2086.59 q^{50} -160.501 q^{51} -14789.8 q^{52} +13916.4 q^{53} +13756.0 q^{54} -4070.22 q^{55} -32935.3 q^{56} +18708.5 q^{57} -2457.24 q^{58} +6566.95 q^{59} +6793.09 q^{60} +28883.3 q^{61} -16414.7 q^{62} +13667.6 q^{63} +17220.1 q^{64} -17730.1 q^{65} -7082.25 q^{66} +1767.29 q^{67} +256.881 q^{68} +6892.73 q^{69} -15578.4 q^{70} -48021.1 q^{71} -12921.2 q^{72} -73940.0 q^{73} -28251.1 q^{74} +8143.59 q^{75} -29943.0 q^{76} -30388.1 q^{77} -30850.6 q^{78} -82849.5 q^{79} -1955.62 q^{80} -35893.1 q^{81} -35693.0 q^{82} +46175.7 q^{83} +50716.9 q^{84} +307.951 q^{85} -25202.8 q^{86} +9590.17 q^{87} +28728.6 q^{88} +31871.7 q^{89} -6111.71 q^{90} -132372. q^{91} -11031.8 q^{92} +64063.6 q^{93} -6926.96 q^{94} -35895.8 q^{95} -76976.4 q^{96} -21882.2 q^{97} -60196.9 q^{98} -11921.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.33855 −0.590177 −0.295089 0.955470i \(-0.595349\pi\)
−0.295089 + 0.955470i \(0.595349\pi\)
\(3\) 13.0297 0.835858 0.417929 0.908480i \(-0.362756\pi\)
0.417929 + 0.908480i \(0.362756\pi\)
\(4\) −20.8541 −0.651691
\(5\) −25.0000 −0.447214
\(6\) −43.5004 −0.493304
\(7\) −186.649 −1.43973 −0.719864 0.694115i \(-0.755796\pi\)
−0.719864 + 0.694115i \(0.755796\pi\)
\(8\) 176.456 0.974790
\(9\) −73.2260 −0.301341
\(10\) 83.4637 0.263935
\(11\) 162.809 0.405692 0.202846 0.979211i \(-0.434981\pi\)
0.202846 + 0.979211i \(0.434981\pi\)
\(12\) −271.723 −0.544721
\(13\) 709.202 1.16389 0.581945 0.813228i \(-0.302292\pi\)
0.581945 + 0.813228i \(0.302292\pi\)
\(14\) 623.137 0.849695
\(15\) −325.743 −0.373807
\(16\) 78.2248 0.0763914
\(17\) −12.3180 −0.0103376 −0.00516879 0.999987i \(-0.501645\pi\)
−0.00516879 + 0.999987i \(0.501645\pi\)
\(18\) 244.468 0.177845
\(19\) 1435.83 0.912472 0.456236 0.889859i \(-0.349197\pi\)
0.456236 + 0.889859i \(0.349197\pi\)
\(20\) 521.353 0.291445
\(21\) −2431.99 −1.20341
\(22\) −543.545 −0.239430
\(23\) 529.000 0.208514
\(24\) 2299.17 0.814786
\(25\) 625.000 0.200000
\(26\) −2367.71 −0.686901
\(27\) −4120.34 −1.08774
\(28\) 3892.40 0.938258
\(29\) 736.021 0.162516 0.0812579 0.996693i \(-0.474106\pi\)
0.0812579 + 0.996693i \(0.474106\pi\)
\(30\) 1087.51 0.220612
\(31\) 4916.72 0.918907 0.459454 0.888202i \(-0.348045\pi\)
0.459454 + 0.888202i \(0.348045\pi\)
\(32\) −5907.75 −1.01987
\(33\) 2121.36 0.339101
\(34\) 41.1243 0.00610101
\(35\) 4666.23 0.643866
\(36\) 1527.06 0.196381
\(37\) 8462.08 1.01619 0.508093 0.861302i \(-0.330351\pi\)
0.508093 + 0.861302i \(0.330351\pi\)
\(38\) −4793.59 −0.538520
\(39\) 9240.72 0.972846
\(40\) −4411.40 −0.435940
\(41\) 10691.2 0.993267 0.496633 0.867960i \(-0.334569\pi\)
0.496633 + 0.867960i \(0.334569\pi\)
\(42\) 8119.31 0.710225
\(43\) 7549.04 0.622617 0.311308 0.950309i \(-0.399233\pi\)
0.311308 + 0.950309i \(0.399233\pi\)
\(44\) −3395.23 −0.264385
\(45\) 1830.65 0.134764
\(46\) −1766.09 −0.123060
\(47\) 2074.84 0.137006 0.0685031 0.997651i \(-0.478178\pi\)
0.0685031 + 0.997651i \(0.478178\pi\)
\(48\) 1019.25 0.0638524
\(49\) 18030.9 1.07282
\(50\) −2086.59 −0.118035
\(51\) −160.501 −0.00864075
\(52\) −14789.8 −0.758496
\(53\) 13916.4 0.680515 0.340257 0.940332i \(-0.389486\pi\)
0.340257 + 0.940332i \(0.389486\pi\)
\(54\) 13756.0 0.641958
\(55\) −4070.22 −0.181431
\(56\) −32935.3 −1.40343
\(57\) 18708.5 0.762697
\(58\) −2457.24 −0.0959131
\(59\) 6566.95 0.245603 0.122802 0.992431i \(-0.460812\pi\)
0.122802 + 0.992431i \(0.460812\pi\)
\(60\) 6793.09 0.243607
\(61\) 28883.3 0.993852 0.496926 0.867793i \(-0.334462\pi\)
0.496926 + 0.867793i \(0.334462\pi\)
\(62\) −16414.7 −0.542318
\(63\) 13667.6 0.433850
\(64\) 17220.1 0.525516
\(65\) −17730.1 −0.520507
\(66\) −7082.25 −0.200130
\(67\) 1767.29 0.0480974 0.0240487 0.999711i \(-0.492344\pi\)
0.0240487 + 0.999711i \(0.492344\pi\)
\(68\) 256.881 0.00673690
\(69\) 6892.73 0.174288
\(70\) −15578.4 −0.379995
\(71\) −48021.1 −1.13054 −0.565271 0.824906i \(-0.691228\pi\)
−0.565271 + 0.824906i \(0.691228\pi\)
\(72\) −12921.2 −0.293745
\(73\) −73940.0 −1.62395 −0.811975 0.583693i \(-0.801607\pi\)
−0.811975 + 0.583693i \(0.801607\pi\)
\(74\) −28251.1 −0.599730
\(75\) 8143.59 0.167172
\(76\) −29943.0 −0.594650
\(77\) −30388.1 −0.584086
\(78\) −30850.6 −0.574152
\(79\) −82849.5 −1.49356 −0.746779 0.665072i \(-0.768401\pi\)
−0.746779 + 0.665072i \(0.768401\pi\)
\(80\) −1955.62 −0.0341633
\(81\) −35893.1 −0.607852
\(82\) −35693.0 −0.586204
\(83\) 46175.7 0.735730 0.367865 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(84\) 50716.9 0.784250
\(85\) 307.951 0.00462311
\(86\) −25202.8 −0.367454
\(87\) 9590.17 0.135840
\(88\) 28728.6 0.395464
\(89\) 31871.7 0.426511 0.213255 0.976996i \(-0.431593\pi\)
0.213255 + 0.976996i \(0.431593\pi\)
\(90\) −6111.71 −0.0795346
\(91\) −132372. −1.67569
\(92\) −11031.8 −0.135887
\(93\) 64063.6 0.768076
\(94\) −6926.96 −0.0808580
\(95\) −35895.8 −0.408070
\(96\) −76976.4 −0.852471
\(97\) −21882.2 −0.236136 −0.118068 0.993006i \(-0.537670\pi\)
−0.118068 + 0.993006i \(0.537670\pi\)
\(98\) −60196.9 −0.633153
\(99\) −11921.8 −0.122252
\(100\) −13033.8 −0.130338
\(101\) 71523.2 0.697659 0.348830 0.937186i \(-0.386579\pi\)
0.348830 + 0.937186i \(0.386579\pi\)
\(102\) 535.839 0.00509958
\(103\) 170512. 1.58366 0.791831 0.610741i \(-0.209128\pi\)
0.791831 + 0.610741i \(0.209128\pi\)
\(104\) 125143. 1.13455
\(105\) 60799.7 0.538181
\(106\) −46460.6 −0.401624
\(107\) 169593. 1.43201 0.716007 0.698093i \(-0.245968\pi\)
0.716007 + 0.698093i \(0.245968\pi\)
\(108\) 85926.0 0.708868
\(109\) −143948. −1.16049 −0.580243 0.814443i \(-0.697043\pi\)
−0.580243 + 0.814443i \(0.697043\pi\)
\(110\) 13588.6 0.107076
\(111\) 110259. 0.849387
\(112\) −14600.6 −0.109983
\(113\) 127662. 0.940513 0.470257 0.882530i \(-0.344161\pi\)
0.470257 + 0.882530i \(0.344161\pi\)
\(114\) −62459.3 −0.450127
\(115\) −13225.0 −0.0932505
\(116\) −15349.1 −0.105910
\(117\) −51932.0 −0.350728
\(118\) −21924.1 −0.144949
\(119\) 2299.15 0.0148833
\(120\) −57479.4 −0.364384
\(121\) −134544. −0.835414
\(122\) −96428.2 −0.586549
\(123\) 139303. 0.830230
\(124\) −102534. −0.598843
\(125\) −15625.0 −0.0894427
\(126\) −45629.8 −0.256048
\(127\) 314163. 1.72841 0.864203 0.503143i \(-0.167823\pi\)
0.864203 + 0.503143i \(0.167823\pi\)
\(128\) 131558. 0.709727
\(129\) 98362.0 0.520419
\(130\) 59192.6 0.307192
\(131\) −291984. −1.48656 −0.743278 0.668982i \(-0.766730\pi\)
−0.743278 + 0.668982i \(0.766730\pi\)
\(132\) −44239.0 −0.220989
\(133\) −267997. −1.31371
\(134\) −5900.19 −0.0283860
\(135\) 103009. 0.486451
\(136\) −2173.59 −0.0100770
\(137\) 126342. 0.575103 0.287552 0.957765i \(-0.407159\pi\)
0.287552 + 0.957765i \(0.407159\pi\)
\(138\) −23011.7 −0.102861
\(139\) −40849.9 −0.179331 −0.0896653 0.995972i \(-0.528580\pi\)
−0.0896653 + 0.995972i \(0.528580\pi\)
\(140\) −97310.0 −0.419602
\(141\) 27034.7 0.114518
\(142\) 160321. 0.667220
\(143\) 115464. 0.472180
\(144\) −5728.09 −0.0230199
\(145\) −18400.5 −0.0726793
\(146\) 246852. 0.958418
\(147\) 234937. 0.896724
\(148\) −176469. −0.662239
\(149\) −23567.4 −0.0869654 −0.0434827 0.999054i \(-0.513845\pi\)
−0.0434827 + 0.999054i \(0.513845\pi\)
\(150\) −27187.7 −0.0986609
\(151\) −126514. −0.451542 −0.225771 0.974180i \(-0.572490\pi\)
−0.225771 + 0.974180i \(0.572490\pi\)
\(152\) 253361. 0.889469
\(153\) 901.999 0.00311514
\(154\) 101452. 0.344714
\(155\) −122918. −0.410948
\(156\) −192707. −0.633995
\(157\) 440167. 1.42518 0.712588 0.701583i \(-0.247523\pi\)
0.712588 + 0.701583i \(0.247523\pi\)
\(158\) 276597. 0.881464
\(159\) 181327. 0.568814
\(160\) 147694. 0.456102
\(161\) −98737.3 −0.300204
\(162\) 119831. 0.358741
\(163\) −547085. −1.61282 −0.806410 0.591356i \(-0.798593\pi\)
−0.806410 + 0.591356i \(0.798593\pi\)
\(164\) −222955. −0.647303
\(165\) −53033.9 −0.151650
\(166\) −154160. −0.434211
\(167\) 439343. 1.21903 0.609513 0.792776i \(-0.291365\pi\)
0.609513 + 0.792776i \(0.291365\pi\)
\(168\) −429139. −1.17307
\(169\) 131675. 0.354639
\(170\) −1028.11 −0.00272845
\(171\) −105140. −0.274966
\(172\) −157428. −0.405753
\(173\) −544396. −1.38293 −0.691464 0.722410i \(-0.743034\pi\)
−0.691464 + 0.722410i \(0.743034\pi\)
\(174\) −32017.2 −0.0801697
\(175\) −116656. −0.287946
\(176\) 12735.7 0.0309914
\(177\) 85565.6 0.205289
\(178\) −106405. −0.251717
\(179\) 434241. 1.01297 0.506486 0.862248i \(-0.330944\pi\)
0.506486 + 0.862248i \(0.330944\pi\)
\(180\) −38176.5 −0.0878244
\(181\) 244202. 0.554054 0.277027 0.960862i \(-0.410651\pi\)
0.277027 + 0.960862i \(0.410651\pi\)
\(182\) 441930. 0.988951
\(183\) 376341. 0.830719
\(184\) 93345.2 0.203258
\(185\) −211552. −0.454452
\(186\) −213879. −0.453301
\(187\) −2005.48 −0.00419387
\(188\) −43269.0 −0.0892857
\(189\) 769058. 1.56605
\(190\) 119840. 0.240834
\(191\) 149293. 0.296112 0.148056 0.988979i \(-0.452698\pi\)
0.148056 + 0.988979i \(0.452698\pi\)
\(192\) 224373. 0.439257
\(193\) 652578. 1.26107 0.630535 0.776161i \(-0.282836\pi\)
0.630535 + 0.776161i \(0.282836\pi\)
\(194\) 73054.7 0.139362
\(195\) −231018. −0.435070
\(196\) −376017. −0.699146
\(197\) 30264.8 0.0555613 0.0277806 0.999614i \(-0.491156\pi\)
0.0277806 + 0.999614i \(0.491156\pi\)
\(198\) 39801.6 0.0721502
\(199\) 334387. 0.598572 0.299286 0.954163i \(-0.403251\pi\)
0.299286 + 0.954163i \(0.403251\pi\)
\(200\) 110285. 0.194958
\(201\) 23027.4 0.0402026
\(202\) −238784. −0.411743
\(203\) −137378. −0.233979
\(204\) 3347.10 0.00563110
\(205\) −267279. −0.444202
\(206\) −569263. −0.934641
\(207\) −38736.5 −0.0628340
\(208\) 55477.2 0.0889112
\(209\) 233766. 0.370182
\(210\) −202983. −0.317622
\(211\) 303955. 0.470006 0.235003 0.971995i \(-0.424490\pi\)
0.235003 + 0.971995i \(0.424490\pi\)
\(212\) −290214. −0.443485
\(213\) −625703. −0.944972
\(214\) −566193. −0.845142
\(215\) −188726. −0.278443
\(216\) −727059. −1.06032
\(217\) −917702. −1.32298
\(218\) 480578. 0.684893
\(219\) −963419. −1.35739
\(220\) 84880.8 0.118237
\(221\) −8735.97 −0.0120318
\(222\) −368104. −0.501289
\(223\) 117587. 0.158343 0.0791713 0.996861i \(-0.474773\pi\)
0.0791713 + 0.996861i \(0.474773\pi\)
\(224\) 1.10268e6 1.46834
\(225\) −45766.2 −0.0602683
\(226\) −426205. −0.555070
\(227\) −57825.3 −0.0744824 −0.0372412 0.999306i \(-0.511857\pi\)
−0.0372412 + 0.999306i \(0.511857\pi\)
\(228\) −390149. −0.497043
\(229\) −502150. −0.632769 −0.316384 0.948631i \(-0.602469\pi\)
−0.316384 + 0.948631i \(0.602469\pi\)
\(230\) 44152.3 0.0550343
\(231\) −395949. −0.488213
\(232\) 129875. 0.158419
\(233\) −163481. −0.197277 −0.0986385 0.995123i \(-0.531449\pi\)
−0.0986385 + 0.995123i \(0.531449\pi\)
\(234\) 173377. 0.206992
\(235\) −51871.1 −0.0612711
\(236\) −136948. −0.160057
\(237\) −1.07951e6 −1.24840
\(238\) −7675.82 −0.00878379
\(239\) 1.40221e6 1.58788 0.793941 0.607994i \(-0.208026\pi\)
0.793941 + 0.607994i \(0.208026\pi\)
\(240\) −25481.2 −0.0285556
\(241\) −868432. −0.963149 −0.481574 0.876405i \(-0.659935\pi\)
−0.481574 + 0.876405i \(0.659935\pi\)
\(242\) 449183. 0.493043
\(243\) 533566. 0.579659
\(244\) −602335. −0.647684
\(245\) −450772. −0.479779
\(246\) −465071. −0.489983
\(247\) 1.01830e6 1.06202
\(248\) 867585. 0.895742
\(249\) 601658. 0.614966
\(250\) 52164.8 0.0527871
\(251\) 88429.2 0.0885954 0.0442977 0.999018i \(-0.485895\pi\)
0.0442977 + 0.999018i \(0.485895\pi\)
\(252\) −285025. −0.282736
\(253\) 86125.8 0.0845926
\(254\) −1.04885e6 −1.02007
\(255\) 4012.52 0.00386426
\(256\) −990255. −0.944381
\(257\) 1.57515e6 1.48761 0.743806 0.668396i \(-0.233019\pi\)
0.743806 + 0.668396i \(0.233019\pi\)
\(258\) −328386. −0.307140
\(259\) −1.57944e6 −1.46303
\(260\) 369744. 0.339210
\(261\) −53895.9 −0.0489727
\(262\) 974804. 0.877332
\(263\) −268088. −0.238994 −0.119497 0.992835i \(-0.538128\pi\)
−0.119497 + 0.992835i \(0.538128\pi\)
\(264\) 374326. 0.330552
\(265\) −347910. −0.304335
\(266\) 894720. 0.775323
\(267\) 415280. 0.356503
\(268\) −36855.3 −0.0313446
\(269\) 1.08318e6 0.912684 0.456342 0.889804i \(-0.349159\pi\)
0.456342 + 0.889804i \(0.349159\pi\)
\(270\) −343899. −0.287092
\(271\) 1.77658e6 1.46947 0.734735 0.678354i \(-0.237307\pi\)
0.734735 + 0.678354i \(0.237307\pi\)
\(272\) −963.575 −0.000789702 0
\(273\) −1.72477e6 −1.40063
\(274\) −421798. −0.339413
\(275\) 101755. 0.0811383
\(276\) −143742. −0.113582
\(277\) −1.69141e6 −1.32449 −0.662247 0.749285i \(-0.730397\pi\)
−0.662247 + 0.749285i \(0.730397\pi\)
\(278\) 136379. 0.105837
\(279\) −360032. −0.276905
\(280\) 823383. 0.627635
\(281\) 2.59696e6 1.96200 0.981002 0.193997i \(-0.0621451\pi\)
0.981002 + 0.193997i \(0.0621451\pi\)
\(282\) −90256.5 −0.0675858
\(283\) −917799. −0.681211 −0.340605 0.940206i \(-0.610632\pi\)
−0.340605 + 0.940206i \(0.610632\pi\)
\(284\) 1.00144e6 0.736763
\(285\) −467713. −0.341089
\(286\) −385483. −0.278670
\(287\) −1.99550e6 −1.43003
\(288\) 432600. 0.307330
\(289\) −1.41971e6 −0.999893
\(290\) 61431.1 0.0428936
\(291\) −285119. −0.197376
\(292\) 1.54195e6 1.05831
\(293\) −15515.7 −0.0105585 −0.00527925 0.999986i \(-0.501680\pi\)
−0.00527925 + 0.999986i \(0.501680\pi\)
\(294\) −784350. −0.529226
\(295\) −164174. −0.109837
\(296\) 1.49318e6 0.990568
\(297\) −670828. −0.441286
\(298\) 78681.0 0.0513250
\(299\) 375168. 0.242688
\(300\) −169827. −0.108944
\(301\) −1.40902e6 −0.896399
\(302\) 422374. 0.266490
\(303\) 931928. 0.583144
\(304\) 112318. 0.0697050
\(305\) −722082. −0.444464
\(306\) −3011.37 −0.00183849
\(307\) −1.84319e6 −1.11615 −0.558076 0.829790i \(-0.688460\pi\)
−0.558076 + 0.829790i \(0.688460\pi\)
\(308\) 633717. 0.380643
\(309\) 2.22173e6 1.32372
\(310\) 410368. 0.242532
\(311\) −513865. −0.301265 −0.150632 0.988590i \(-0.548131\pi\)
−0.150632 + 0.988590i \(0.548131\pi\)
\(312\) 1.63058e6 0.948321
\(313\) 2.01753e6 1.16402 0.582008 0.813183i \(-0.302267\pi\)
0.582008 + 0.813183i \(0.302267\pi\)
\(314\) −1.46952e6 −0.841107
\(315\) −341689. −0.194024
\(316\) 1.72775e6 0.973338
\(317\) 2.13136e6 1.19127 0.595634 0.803256i \(-0.296901\pi\)
0.595634 + 0.803256i \(0.296901\pi\)
\(318\) −605369. −0.335701
\(319\) 119831. 0.0659313
\(320\) −430502. −0.235018
\(321\) 2.20975e6 1.19696
\(322\) 329639. 0.177174
\(323\) −17686.6 −0.00943276
\(324\) 748517. 0.396131
\(325\) 443251. 0.232778
\(326\) 1.82647e6 0.951850
\(327\) −1.87561e6 −0.970002
\(328\) 1.88652e6 0.968227
\(329\) −387267. −0.197252
\(330\) 177056. 0.0895006
\(331\) −2.10092e6 −1.05400 −0.527000 0.849866i \(-0.676683\pi\)
−0.527000 + 0.849866i \(0.676683\pi\)
\(332\) −962954. −0.479469
\(333\) −619644. −0.306219
\(334\) −1.46677e6 −0.719441
\(335\) −44182.3 −0.0215098
\(336\) −190242. −0.0919301
\(337\) 3.01255e6 1.44497 0.722485 0.691386i \(-0.243000\pi\)
0.722485 + 0.691386i \(0.243000\pi\)
\(338\) −439603. −0.209300
\(339\) 1.66340e6 0.786136
\(340\) −6422.04 −0.00301284
\(341\) 800486. 0.372793
\(342\) 351015. 0.162278
\(343\) −228433. −0.104839
\(344\) 1.33207e6 0.606921
\(345\) −172318. −0.0779442
\(346\) 1.81749e6 0.816173
\(347\) −1.63533e6 −0.729093 −0.364546 0.931185i \(-0.618776\pi\)
−0.364546 + 0.931185i \(0.618776\pi\)
\(348\) −199994. −0.0885257
\(349\) 2.68603e6 1.18045 0.590224 0.807239i \(-0.299039\pi\)
0.590224 + 0.807239i \(0.299039\pi\)
\(350\) 389460. 0.169939
\(351\) −2.92216e6 −1.26601
\(352\) −961833. −0.413755
\(353\) 2.79958e6 1.19579 0.597896 0.801574i \(-0.296003\pi\)
0.597896 + 0.801574i \(0.296003\pi\)
\(354\) −285665. −0.121157
\(355\) 1.20053e6 0.505593
\(356\) −664656. −0.277953
\(357\) 29957.3 0.0124403
\(358\) −1.44973e6 −0.597834
\(359\) −2.68841e6 −1.10093 −0.550464 0.834859i \(-0.685549\pi\)
−0.550464 + 0.834859i \(0.685549\pi\)
\(360\) 323029. 0.131367
\(361\) −414485. −0.167394
\(362\) −815278. −0.326990
\(363\) −1.75308e6 −0.698288
\(364\) 2.76050e6 1.09203
\(365\) 1.84850e6 0.726252
\(366\) −1.25643e6 −0.490272
\(367\) −1.75027e6 −0.678327 −0.339163 0.940728i \(-0.610144\pi\)
−0.339163 + 0.940728i \(0.610144\pi\)
\(368\) 41380.9 0.0159287
\(369\) −782872. −0.299312
\(370\) 706277. 0.268207
\(371\) −2.59748e6 −0.979756
\(372\) −1.33599e6 −0.500548
\(373\) 2.73765e6 1.01884 0.509419 0.860518i \(-0.329860\pi\)
0.509419 + 0.860518i \(0.329860\pi\)
\(374\) 6695.40 0.00247513
\(375\) −203590. −0.0747614
\(376\) 366118. 0.133552
\(377\) 521988. 0.189150
\(378\) −2.56754e6 −0.924245
\(379\) −3.18685e6 −1.13963 −0.569814 0.821773i \(-0.692985\pi\)
−0.569814 + 0.821773i \(0.692985\pi\)
\(380\) 748575. 0.265935
\(381\) 4.09346e6 1.44470
\(382\) −498422. −0.174759
\(383\) 5.39336e6 1.87872 0.939361 0.342929i \(-0.111419\pi\)
0.939361 + 0.342929i \(0.111419\pi\)
\(384\) 1.71416e6 0.593231
\(385\) 759703. 0.261211
\(386\) −2.17866e6 −0.744255
\(387\) −552786. −0.187620
\(388\) 456333. 0.153887
\(389\) −2.27625e6 −0.762685 −0.381343 0.924434i \(-0.624538\pi\)
−0.381343 + 0.924434i \(0.624538\pi\)
\(390\) 771265. 0.256769
\(391\) −6516.24 −0.00215553
\(392\) 3.18165e6 1.04577
\(393\) −3.80448e6 −1.24255
\(394\) −101040. −0.0327910
\(395\) 2.07124e6 0.667939
\(396\) 248619. 0.0796703
\(397\) 1.18059e6 0.375942 0.187971 0.982175i \(-0.439809\pi\)
0.187971 + 0.982175i \(0.439809\pi\)
\(398\) −1.11637e6 −0.353264
\(399\) −3.49193e6 −1.09808
\(400\) 48890.5 0.0152783
\(401\) 299520. 0.0930174 0.0465087 0.998918i \(-0.485190\pi\)
0.0465087 + 0.998918i \(0.485190\pi\)
\(402\) −76877.9 −0.0237267
\(403\) 3.48695e6 1.06951
\(404\) −1.49155e6 −0.454658
\(405\) 897326. 0.271840
\(406\) 458642. 0.138089
\(407\) 1.37770e6 0.412258
\(408\) −28321.3 −0.00842292
\(409\) −6.33364e6 −1.87217 −0.936084 0.351776i \(-0.885578\pi\)
−0.936084 + 0.351776i \(0.885578\pi\)
\(410\) 892325. 0.262158
\(411\) 1.64620e6 0.480705
\(412\) −3.55588e6 −1.03206
\(413\) −1.22571e6 −0.353602
\(414\) 129324. 0.0370832
\(415\) −1.15439e6 −0.329029
\(416\) −4.18979e6 −1.18702
\(417\) −532264. −0.149895
\(418\) −780439. −0.218473
\(419\) −6.58614e6 −1.83272 −0.916359 0.400357i \(-0.868886\pi\)
−0.916359 + 0.400357i \(0.868886\pi\)
\(420\) −1.26792e6 −0.350727
\(421\) 1.01603e6 0.279385 0.139692 0.990195i \(-0.455389\pi\)
0.139692 + 0.990195i \(0.455389\pi\)
\(422\) −1.01477e6 −0.277387
\(423\) −151932. −0.0412857
\(424\) 2.45563e6 0.663359
\(425\) −7698.77 −0.00206752
\(426\) 2.08894e6 0.557701
\(427\) −5.39103e6 −1.43088
\(428\) −3.53670e6 −0.933230
\(429\) 1.50447e6 0.394676
\(430\) 630071. 0.164331
\(431\) −4.79699e6 −1.24387 −0.621936 0.783068i \(-0.713653\pi\)
−0.621936 + 0.783068i \(0.713653\pi\)
\(432\) −322313. −0.0830937
\(433\) −2.75283e6 −0.705601 −0.352800 0.935699i \(-0.614771\pi\)
−0.352800 + 0.935699i \(0.614771\pi\)
\(434\) 3.06379e6 0.780791
\(435\) −239754. −0.0607495
\(436\) 3.00191e6 0.756279
\(437\) 759555. 0.190264
\(438\) 3.21642e6 0.801102
\(439\) −1.72837e6 −0.428031 −0.214016 0.976830i \(-0.568654\pi\)
−0.214016 + 0.976830i \(0.568654\pi\)
\(440\) −718214. −0.176857
\(441\) −1.32033e6 −0.323285
\(442\) 29165.5 0.00710090
\(443\) −7.23089e6 −1.75058 −0.875291 0.483597i \(-0.839330\pi\)
−0.875291 + 0.483597i \(0.839330\pi\)
\(444\) −2.29935e6 −0.553538
\(445\) −796792. −0.190741
\(446\) −392570. −0.0934502
\(447\) −307077. −0.0726907
\(448\) −3.21411e6 −0.756600
\(449\) 203430. 0.0476210 0.0238105 0.999716i \(-0.492420\pi\)
0.0238105 + 0.999716i \(0.492420\pi\)
\(450\) 152793. 0.0355690
\(451\) 1.74062e6 0.402960
\(452\) −2.66227e6 −0.612924
\(453\) −1.64845e6 −0.377425
\(454\) 193053. 0.0439578
\(455\) 3.30930e6 0.749389
\(456\) 3.30123e6 0.743470
\(457\) −7.78163e6 −1.74293 −0.871466 0.490456i \(-0.836830\pi\)
−0.871466 + 0.490456i \(0.836830\pi\)
\(458\) 1.67645e6 0.373446
\(459\) 50754.5 0.0112446
\(460\) 275795. 0.0607705
\(461\) 7.61973e6 1.66989 0.834944 0.550335i \(-0.185500\pi\)
0.834944 + 0.550335i \(0.185500\pi\)
\(462\) 1.32189e6 0.288132
\(463\) 6.13256e6 1.32950 0.664751 0.747065i \(-0.268538\pi\)
0.664751 + 0.747065i \(0.268538\pi\)
\(464\) 57575.1 0.0124148
\(465\) −1.60159e6 −0.343494
\(466\) 545788. 0.116428
\(467\) −5.77261e6 −1.22484 −0.612421 0.790532i \(-0.709804\pi\)
−0.612421 + 0.790532i \(0.709804\pi\)
\(468\) 1.08300e6 0.228566
\(469\) −329863. −0.0692472
\(470\) 173174. 0.0361608
\(471\) 5.73526e6 1.19125
\(472\) 1.15878e6 0.239411
\(473\) 1.22905e6 0.252590
\(474\) 3.60399e6 0.736779
\(475\) 897395. 0.182494
\(476\) −47946.7 −0.00969932
\(477\) −1.01904e6 −0.205067
\(478\) −4.68135e6 −0.937132
\(479\) 7.98550e6 1.59024 0.795121 0.606450i \(-0.207407\pi\)
0.795121 + 0.606450i \(0.207407\pi\)
\(480\) 1.92441e6 0.381236
\(481\) 6.00133e6 1.18273
\(482\) 2.89930e6 0.568429
\(483\) −1.28652e6 −0.250928
\(484\) 2.80580e6 0.544432
\(485\) 547055. 0.105603
\(486\) −1.78133e6 −0.342101
\(487\) −8.56097e6 −1.63569 −0.817844 0.575440i \(-0.804831\pi\)
−0.817844 + 0.575440i \(0.804831\pi\)
\(488\) 5.09662e6 0.968797
\(489\) −7.12838e6 −1.34809
\(490\) 1.50492e6 0.283155
\(491\) −9.04796e6 −1.69374 −0.846870 0.531800i \(-0.821516\pi\)
−0.846870 + 0.531800i \(0.821516\pi\)
\(492\) −2.90504e6 −0.541053
\(493\) −9066.33 −0.00168002
\(494\) −3.39963e6 −0.626778
\(495\) 298046. 0.0546726
\(496\) 384610. 0.0701966
\(497\) 8.96310e6 1.62767
\(498\) −2.00866e6 −0.362939
\(499\) 4.87106e6 0.875734 0.437867 0.899040i \(-0.355734\pi\)
0.437867 + 0.899040i \(0.355734\pi\)
\(500\) 325845. 0.0582890
\(501\) 5.72453e6 1.01893
\(502\) −295225. −0.0522870
\(503\) −1.68291e6 −0.296579 −0.148290 0.988944i \(-0.547377\pi\)
−0.148290 + 0.988944i \(0.547377\pi\)
\(504\) 2.41172e6 0.422913
\(505\) −1.78808e6 −0.312003
\(506\) −287535. −0.0499246
\(507\) 1.71569e6 0.296428
\(508\) −6.55159e6 −1.12639
\(509\) 3.19707e6 0.546963 0.273482 0.961877i \(-0.411825\pi\)
0.273482 + 0.961877i \(0.411825\pi\)
\(510\) −13396.0 −0.00228060
\(511\) 1.38008e7 2.33805
\(512\) −903836. −0.152375
\(513\) −5.91612e6 −0.992529
\(514\) −5.25872e6 −0.877955
\(515\) −4.26280e6 −0.708235
\(516\) −2.05125e6 −0.339152
\(517\) 337803. 0.0555823
\(518\) 5.27304e6 0.863448
\(519\) −7.09334e6 −1.15593
\(520\) −3.12857e6 −0.507385
\(521\) 6.19108e6 0.999244 0.499622 0.866243i \(-0.333472\pi\)
0.499622 + 0.866243i \(0.333472\pi\)
\(522\) 179934. 0.0289026
\(523\) 2.79677e6 0.447098 0.223549 0.974693i \(-0.428236\pi\)
0.223549 + 0.974693i \(0.428236\pi\)
\(524\) 6.08907e6 0.968775
\(525\) −1.51999e6 −0.240682
\(526\) 895023. 0.141049
\(527\) −60564.4 −0.00949928
\(528\) 165943. 0.0259044
\(529\) 279841. 0.0434783
\(530\) 1.16151e6 0.179612
\(531\) −480871. −0.0740103
\(532\) 5.58883e6 0.856134
\(533\) 7.58221e6 1.15605
\(534\) −1.38643e6 −0.210400
\(535\) −4.23981e6 −0.640416
\(536\) 311849. 0.0468849
\(537\) 5.65804e6 0.846702
\(538\) −3.61625e6 −0.538645
\(539\) 2.93558e6 0.435234
\(540\) −2.14815e6 −0.317015
\(541\) −1.43341e6 −0.210561 −0.105281 0.994443i \(-0.533574\pi\)
−0.105281 + 0.994443i \(0.533574\pi\)
\(542\) −5.93118e6 −0.867248
\(543\) 3.18188e6 0.463110
\(544\) 72771.8 0.0105430
\(545\) 3.59871e6 0.518986
\(546\) 5.75823e6 0.826623
\(547\) −1.91785e6 −0.274061 −0.137030 0.990567i \(-0.543756\pi\)
−0.137030 + 0.990567i \(0.543756\pi\)
\(548\) −2.63475e6 −0.374790
\(549\) −2.11500e6 −0.299489
\(550\) −339715. −0.0478860
\(551\) 1.05680e6 0.148291
\(552\) 1.21626e6 0.169895
\(553\) 1.54638e7 2.15032
\(554\) 5.64686e6 0.781687
\(555\) −2.75647e6 −0.379857
\(556\) 851888. 0.116868
\(557\) 1.13283e7 1.54713 0.773565 0.633717i \(-0.218472\pi\)
0.773565 + 0.633717i \(0.218472\pi\)
\(558\) 1.20198e6 0.163423
\(559\) 5.35380e6 0.724657
\(560\) 365015. 0.0491858
\(561\) −26130.9 −0.00350548
\(562\) −8.67008e6 −1.15793
\(563\) 1.28831e7 1.71297 0.856484 0.516173i \(-0.172644\pi\)
0.856484 + 0.516173i \(0.172644\pi\)
\(564\) −563783. −0.0746302
\(565\) −3.19155e6 −0.420610
\(566\) 3.06412e6 0.402035
\(567\) 6.69940e6 0.875142
\(568\) −8.47361e6 −1.10204
\(569\) −8.42700e6 −1.09117 −0.545585 0.838056i \(-0.683692\pi\)
−0.545585 + 0.838056i \(0.683692\pi\)
\(570\) 1.56148e6 0.201303
\(571\) −8.33201e6 −1.06945 −0.534724 0.845027i \(-0.679584\pi\)
−0.534724 + 0.845027i \(0.679584\pi\)
\(572\) −2.40791e6 −0.307715
\(573\) 1.94525e6 0.247508
\(574\) 6.66207e6 0.843974
\(575\) 330625. 0.0417029
\(576\) −1.26096e6 −0.158360
\(577\) −1.35151e7 −1.68997 −0.844985 0.534790i \(-0.820391\pi\)
−0.844985 + 0.534790i \(0.820391\pi\)
\(578\) 4.73975e6 0.590114
\(579\) 8.50292e6 1.05408
\(580\) 383727. 0.0473644
\(581\) −8.61866e6 −1.05925
\(582\) 951884. 0.116487
\(583\) 2.26571e6 0.276079
\(584\) −1.30472e7 −1.58301
\(585\) 1.29830e6 0.156850
\(586\) 51799.9 0.00623139
\(587\) −1.70048e6 −0.203693 −0.101846 0.994800i \(-0.532475\pi\)
−0.101846 + 0.994800i \(0.532475\pi\)
\(588\) −4.89941e6 −0.584387
\(589\) 7.05959e6 0.838477
\(590\) 548102. 0.0648233
\(591\) 394343. 0.0464414
\(592\) 661945. 0.0776279
\(593\) 2.07309e6 0.242092 0.121046 0.992647i \(-0.461375\pi\)
0.121046 + 0.992647i \(0.461375\pi\)
\(594\) 2.23959e6 0.260437
\(595\) −57478.7 −0.00665602
\(596\) 491477. 0.0566745
\(597\) 4.35697e6 0.500321
\(598\) −1.25252e6 −0.143229
\(599\) −5.36195e6 −0.610598 −0.305299 0.952257i \(-0.598756\pi\)
−0.305299 + 0.952257i \(0.598756\pi\)
\(600\) 1.43698e6 0.162957
\(601\) 8.71485e6 0.984179 0.492089 0.870545i \(-0.336233\pi\)
0.492089 + 0.870545i \(0.336233\pi\)
\(602\) 4.70408e6 0.529034
\(603\) −129412. −0.0144937
\(604\) 2.63834e6 0.294265
\(605\) 3.36361e6 0.373609
\(606\) −3.11129e6 −0.344158
\(607\) −1.38917e7 −1.53032 −0.765162 0.643838i \(-0.777341\pi\)
−0.765162 + 0.643838i \(0.777341\pi\)
\(608\) −8.48253e6 −0.930608
\(609\) −1.79000e6 −0.195573
\(610\) 2.41070e6 0.262313
\(611\) 1.47148e6 0.159460
\(612\) −18810.4 −0.00203011
\(613\) −1.14337e7 −1.22896 −0.614478 0.788934i \(-0.710634\pi\)
−0.614478 + 0.788934i \(0.710634\pi\)
\(614\) 6.15357e6 0.658728
\(615\) −3.48258e6 −0.371290
\(616\) −5.36216e6 −0.569361
\(617\) 1.17072e7 1.23806 0.619028 0.785369i \(-0.287527\pi\)
0.619028 + 0.785369i \(0.287527\pi\)
\(618\) −7.41735e6 −0.781227
\(619\) −1.19037e7 −1.24869 −0.624345 0.781149i \(-0.714634\pi\)
−0.624345 + 0.781149i \(0.714634\pi\)
\(620\) 2.56335e6 0.267811
\(621\) −2.17966e6 −0.226809
\(622\) 1.71556e6 0.177800
\(623\) −5.94882e6 −0.614060
\(624\) 722853. 0.0743171
\(625\) 390625. 0.0400000
\(626\) −6.73561e6 −0.686975
\(627\) 3.04591e6 0.309420
\(628\) −9.17929e6 −0.928774
\(629\) −104236. −0.0105049
\(630\) 1.14074e6 0.114508
\(631\) 1.44181e6 0.144156 0.0720782 0.997399i \(-0.477037\pi\)
0.0720782 + 0.997399i \(0.477037\pi\)
\(632\) −1.46193e7 −1.45591
\(633\) 3.96046e6 0.392859
\(634\) −7.11566e6 −0.703059
\(635\) −7.85408e6 −0.772967
\(636\) −3.78141e6 −0.370691
\(637\) 1.27875e7 1.24864
\(638\) −400061. −0.0389112
\(639\) 3.51639e6 0.340679
\(640\) −3.28894e6 −0.317400
\(641\) −1.88460e7 −1.81165 −0.905825 0.423652i \(-0.860748\pi\)
−0.905825 + 0.423652i \(0.860748\pi\)
\(642\) −7.37734e6 −0.706419
\(643\) 1.75602e7 1.67495 0.837476 0.546474i \(-0.184030\pi\)
0.837476 + 0.546474i \(0.184030\pi\)
\(644\) 2.05908e6 0.195640
\(645\) −2.45905e6 −0.232738
\(646\) 59047.6 0.00556700
\(647\) −7.21298e6 −0.677414 −0.338707 0.940892i \(-0.609990\pi\)
−0.338707 + 0.940892i \(0.609990\pi\)
\(648\) −6.33354e6 −0.592528
\(649\) 1.06916e6 0.0996391
\(650\) −1.47982e6 −0.137380
\(651\) −1.19574e7 −1.10582
\(652\) 1.14090e7 1.05106
\(653\) −8.59454e6 −0.788751 −0.394375 0.918949i \(-0.629039\pi\)
−0.394375 + 0.918949i \(0.629039\pi\)
\(654\) 6.26181e6 0.572473
\(655\) 7.29961e6 0.664808
\(656\) 836315. 0.0758771
\(657\) 5.41433e6 0.489363
\(658\) 1.29291e6 0.116414
\(659\) −1.40742e7 −1.26244 −0.631221 0.775603i \(-0.717446\pi\)
−0.631221 + 0.775603i \(0.717446\pi\)
\(660\) 1.10597e6 0.0988292
\(661\) 6.03899e6 0.537601 0.268801 0.963196i \(-0.413373\pi\)
0.268801 + 0.963196i \(0.413373\pi\)
\(662\) 7.01403e6 0.622046
\(663\) −113827. −0.0100569
\(664\) 8.14798e6 0.717183
\(665\) 6.69992e6 0.587510
\(666\) 2.06871e6 0.180723
\(667\) 389355. 0.0338869
\(668\) −9.16211e6 −0.794428
\(669\) 1.53213e6 0.132352
\(670\) 147505. 0.0126946
\(671\) 4.70245e6 0.403197
\(672\) 1.43676e7 1.22733
\(673\) 1.93262e7 1.64478 0.822390 0.568924i \(-0.192640\pi\)
0.822390 + 0.568924i \(0.192640\pi\)
\(674\) −1.00575e7 −0.852789
\(675\) −2.57521e6 −0.217547
\(676\) −2.74596e6 −0.231115
\(677\) 1.28390e7 1.07661 0.538306 0.842749i \(-0.319064\pi\)
0.538306 + 0.842749i \(0.319064\pi\)
\(678\) −5.55334e6 −0.463959
\(679\) 4.08429e6 0.339971
\(680\) 54339.7 0.00450656
\(681\) −753449. −0.0622567
\(682\) −2.67246e6 −0.220014
\(683\) −8.22141e6 −0.674364 −0.337182 0.941439i \(-0.609474\pi\)
−0.337182 + 0.941439i \(0.609474\pi\)
\(684\) 2.19260e6 0.179193
\(685\) −3.15855e6 −0.257194
\(686\) 762635. 0.0618737
\(687\) −6.54289e6 −0.528905
\(688\) 590522. 0.0475626
\(689\) 9.86955e6 0.792044
\(690\) 575293. 0.0460009
\(691\) 5.35847e6 0.426919 0.213459 0.976952i \(-0.431527\pi\)
0.213459 + 0.976952i \(0.431527\pi\)
\(692\) 1.13529e7 0.901242
\(693\) 2.22520e6 0.176009
\(694\) 5.45964e6 0.430294
\(695\) 1.02125e6 0.0801991
\(696\) 1.69224e6 0.132416
\(697\) −131694. −0.0102680
\(698\) −8.96743e6 −0.696674
\(699\) −2.13011e6 −0.164896
\(700\) 2.43275e6 0.187652
\(701\) −1.03936e6 −0.0798863 −0.0399432 0.999202i \(-0.512718\pi\)
−0.0399432 + 0.999202i \(0.512718\pi\)
\(702\) 9.75575e6 0.747168
\(703\) 1.21501e7 0.927241
\(704\) 2.80358e6 0.213197
\(705\) −675866. −0.0512139
\(706\) −9.34652e6 −0.705730
\(707\) −1.33497e7 −1.00444
\(708\) −1.78439e6 −0.133785
\(709\) 6.18292e6 0.461932 0.230966 0.972962i \(-0.425811\pi\)
0.230966 + 0.972962i \(0.425811\pi\)
\(710\) −4.00802e6 −0.298390
\(711\) 6.06673e6 0.450071
\(712\) 5.62395e6 0.415759
\(713\) 2.60095e6 0.191605
\(714\) −100014. −0.00734200
\(715\) −2.88661e6 −0.211165
\(716\) −9.05570e6 −0.660145
\(717\) 1.82704e7 1.32724
\(718\) 8.97537e6 0.649742
\(719\) −1.84088e7 −1.32802 −0.664010 0.747724i \(-0.731147\pi\)
−0.664010 + 0.747724i \(0.731147\pi\)
\(720\) 143202. 0.0102948
\(721\) −3.18259e7 −2.28004
\(722\) 1.38378e6 0.0987924
\(723\) −1.13154e7 −0.805056
\(724\) −5.09260e6 −0.361072
\(725\) 460013. 0.0325031
\(726\) 5.85273e6 0.412114
\(727\) −3.73339e6 −0.261979 −0.130990 0.991384i \(-0.541815\pi\)
−0.130990 + 0.991384i \(0.541815\pi\)
\(728\) −2.33578e7 −1.63344
\(729\) 1.56742e7 1.09236
\(730\) −6.17131e6 −0.428618
\(731\) −92989.3 −0.00643635
\(732\) −7.84826e6 −0.541372
\(733\) −1.37233e7 −0.943407 −0.471704 0.881757i \(-0.656361\pi\)
−0.471704 + 0.881757i \(0.656361\pi\)
\(734\) 5.84334e6 0.400333
\(735\) −5.87344e6 −0.401027
\(736\) −3.12520e6 −0.212659
\(737\) 287731. 0.0195127
\(738\) 2.61365e6 0.176647
\(739\) −7.99846e6 −0.538760 −0.269380 0.963034i \(-0.586819\pi\)
−0.269380 + 0.963034i \(0.586819\pi\)
\(740\) 4.41173e6 0.296162
\(741\) 1.32681e7 0.887695
\(742\) 8.67182e6 0.578230
\(743\) −2.47504e7 −1.64479 −0.822395 0.568917i \(-0.807363\pi\)
−0.822395 + 0.568917i \(0.807363\pi\)
\(744\) 1.13044e7 0.748713
\(745\) 589186. 0.0388921
\(746\) −9.13977e6 −0.601296
\(747\) −3.38126e6 −0.221706
\(748\) 41822.5 0.00273311
\(749\) −3.16543e7 −2.06171
\(750\) 679694. 0.0441225
\(751\) −1.28933e7 −0.834188 −0.417094 0.908863i \(-0.636951\pi\)
−0.417094 + 0.908863i \(0.636951\pi\)
\(752\) 162304. 0.0104661
\(753\) 1.15221e6 0.0740532
\(754\) −1.74268e6 −0.111632
\(755\) 3.16286e6 0.201936
\(756\) −1.60380e7 −1.02058
\(757\) 9.39689e6 0.595998 0.297999 0.954566i \(-0.403681\pi\)
0.297999 + 0.954566i \(0.403681\pi\)
\(758\) 1.06394e7 0.672583
\(759\) 1.12220e6 0.0707074
\(760\) −6.33403e6 −0.397783
\(761\) 2.72564e7 1.70611 0.853053 0.521824i \(-0.174748\pi\)
0.853053 + 0.521824i \(0.174748\pi\)
\(762\) −1.36662e7 −0.852631
\(763\) 2.68678e7 1.67079
\(764\) −3.11337e6 −0.192973
\(765\) −22550.0 −0.00139313
\(766\) −1.80060e7 −1.10878
\(767\) 4.65730e6 0.285855
\(768\) −1.29028e7 −0.789368
\(769\) 1.11517e7 0.680026 0.340013 0.940421i \(-0.389568\pi\)
0.340013 + 0.940421i \(0.389568\pi\)
\(770\) −2.53630e6 −0.154161
\(771\) 2.05238e7 1.24343
\(772\) −1.36089e7 −0.821827
\(773\) 1.54026e7 0.927141 0.463571 0.886060i \(-0.346568\pi\)
0.463571 + 0.886060i \(0.346568\pi\)
\(774\) 1.84550e6 0.110729
\(775\) 3.07295e6 0.183781
\(776\) −3.86124e6 −0.230183
\(777\) −2.05797e7 −1.22289
\(778\) 7.59936e6 0.450120
\(779\) 1.53507e7 0.906328
\(780\) 4.81767e6 0.283531
\(781\) −7.81826e6 −0.458651
\(782\) 21754.8 0.00127215
\(783\) −3.03266e6 −0.176774
\(784\) 1.41046e6 0.0819541
\(785\) −1.10042e7 −0.637358
\(786\) 1.27014e7 0.733325
\(787\) 3.15787e7 1.81743 0.908714 0.417419i \(-0.137065\pi\)
0.908714 + 0.417419i \(0.137065\pi\)
\(788\) −631145. −0.0362088
\(789\) −3.49311e6 −0.199765
\(790\) −6.91492e6 −0.394203
\(791\) −2.38280e7 −1.35408
\(792\) −2.10368e6 −0.119170
\(793\) 2.04841e7 1.15673
\(794\) −3.94144e6 −0.221873
\(795\) −4.53318e6 −0.254381
\(796\) −6.97334e6 −0.390084
\(797\) −6.58456e6 −0.367181 −0.183591 0.983003i \(-0.558772\pi\)
−0.183591 + 0.983003i \(0.558772\pi\)
\(798\) 1.16580e7 0.648060
\(799\) −25558.0 −0.00141631
\(800\) −3.69234e6 −0.203975
\(801\) −2.33384e6 −0.128525
\(802\) −999960. −0.0548968
\(803\) −1.20381e7 −0.658823
\(804\) −480215. −0.0261997
\(805\) 2.46843e6 0.134255
\(806\) −1.16414e7 −0.631199
\(807\) 1.41136e7 0.762874
\(808\) 1.26207e7 0.680072
\(809\) −3.33210e6 −0.178997 −0.0894986 0.995987i \(-0.528526\pi\)
−0.0894986 + 0.995987i \(0.528526\pi\)
\(810\) −2.99577e6 −0.160434
\(811\) −2.74681e7 −1.46648 −0.733240 0.679970i \(-0.761993\pi\)
−0.733240 + 0.679970i \(0.761993\pi\)
\(812\) 2.86489e6 0.152482
\(813\) 2.31483e7 1.22827
\(814\) −4.59952e6 −0.243305
\(815\) 1.36771e7 0.721275
\(816\) −12555.1 −0.000660079 0
\(817\) 1.08392e7 0.568120
\(818\) 2.11451e7 1.10491
\(819\) 9.69306e6 0.504953
\(820\) 5.57387e6 0.289483
\(821\) −4.05029e6 −0.209714 −0.104857 0.994487i \(-0.533439\pi\)
−0.104857 + 0.994487i \(0.533439\pi\)
\(822\) −5.49592e6 −0.283701
\(823\) −3.82361e6 −0.196777 −0.0983884 0.995148i \(-0.531369\pi\)
−0.0983884 + 0.995148i \(0.531369\pi\)
\(824\) 3.00879e7 1.54374
\(825\) 1.32585e6 0.0678201
\(826\) 4.09211e6 0.208688
\(827\) 2.56165e7 1.30244 0.651218 0.758890i \(-0.274258\pi\)
0.651218 + 0.758890i \(0.274258\pi\)
\(828\) 807815. 0.0409483
\(829\) 1.08107e7 0.546344 0.273172 0.961965i \(-0.411927\pi\)
0.273172 + 0.961965i \(0.411927\pi\)
\(830\) 3.85400e6 0.194185
\(831\) −2.20387e7 −1.10709
\(832\) 1.22125e7 0.611642
\(833\) −222105. −0.0110904
\(834\) 1.77699e6 0.0884646
\(835\) −1.09836e7 −0.545165
\(836\) −4.87498e6 −0.241244
\(837\) −2.02586e7 −0.999529
\(838\) 2.19881e7 1.08163
\(839\) 2.32555e6 0.114057 0.0570283 0.998373i \(-0.481837\pi\)
0.0570283 + 0.998373i \(0.481837\pi\)
\(840\) 1.07285e7 0.524613
\(841\) −1.99694e7 −0.973589
\(842\) −3.39208e6 −0.164887
\(843\) 3.38377e7 1.63996
\(844\) −6.33872e6 −0.306299
\(845\) −3.29187e6 −0.158599
\(846\) 507233. 0.0243659
\(847\) 2.51126e7 1.20277
\(848\) 1.08861e6 0.0519855
\(849\) −1.19587e7 −0.569396
\(850\) 25702.7 0.00122020
\(851\) 4.47644e6 0.211889
\(852\) 1.30485e7 0.615829
\(853\) −5.36926e6 −0.252663 −0.126331 0.991988i \(-0.540320\pi\)
−0.126331 + 0.991988i \(0.540320\pi\)
\(854\) 1.79982e7 0.844471
\(855\) 2.62850e6 0.122968
\(856\) 2.99256e7 1.39591
\(857\) 1.99665e7 0.928647 0.464323 0.885666i \(-0.346298\pi\)
0.464323 + 0.885666i \(0.346298\pi\)
\(858\) −5.02275e6 −0.232929
\(859\) −2.23216e7 −1.03215 −0.516074 0.856544i \(-0.672607\pi\)
−0.516074 + 0.856544i \(0.672607\pi\)
\(860\) 3.93571e6 0.181458
\(861\) −2.60008e7 −1.19531
\(862\) 1.60150e7 0.734105
\(863\) −9.96718e6 −0.455560 −0.227780 0.973713i \(-0.573147\pi\)
−0.227780 + 0.973713i \(0.573147\pi\)
\(864\) 2.43419e7 1.10936
\(865\) 1.36099e7 0.618465
\(866\) 9.19044e6 0.416430
\(867\) −1.84984e7 −0.835769
\(868\) 1.91378e7 0.862172
\(869\) −1.34886e7 −0.605924
\(870\) 800431. 0.0358530
\(871\) 1.25337e6 0.0559800
\(872\) −2.54005e7 −1.13123
\(873\) 1.60234e6 0.0711574
\(874\) −2.53581e6 −0.112289
\(875\) 2.91639e6 0.128773
\(876\) 2.00912e7 0.884599
\(877\) −1.05077e7 −0.461328 −0.230664 0.973033i \(-0.574090\pi\)
−0.230664 + 0.973033i \(0.574090\pi\)
\(878\) 5.77025e6 0.252614
\(879\) −202166. −0.00882541
\(880\) −318392. −0.0138598
\(881\) −868216. −0.0376867 −0.0188433 0.999822i \(-0.505998\pi\)
−0.0188433 + 0.999822i \(0.505998\pi\)
\(882\) 4.40797e6 0.190795
\(883\) −8.60019e6 −0.371199 −0.185599 0.982625i \(-0.559423\pi\)
−0.185599 + 0.982625i \(0.559423\pi\)
\(884\) 182181. 0.00784101
\(885\) −2.13914e6 −0.0918081
\(886\) 2.41407e7 1.03315
\(887\) 683430. 0.0291666 0.0145833 0.999894i \(-0.495358\pi\)
0.0145833 + 0.999894i \(0.495358\pi\)
\(888\) 1.94558e7 0.827974
\(889\) −5.86382e7 −2.48844
\(890\) 2.66013e6 0.112571
\(891\) −5.84370e6 −0.246601
\(892\) −2.45218e6 −0.103190
\(893\) 2.97913e6 0.125014
\(894\) 1.02519e6 0.0429004
\(895\) −1.08560e7 −0.453015
\(896\) −2.45551e7 −1.02181
\(897\) 4.88834e6 0.202853
\(898\) −679160. −0.0281049
\(899\) 3.61881e6 0.149337
\(900\) 954413. 0.0392763
\(901\) −171423. −0.00703488
\(902\) −5.81113e6 −0.237818
\(903\) −1.83592e7 −0.749262
\(904\) 2.25267e7 0.916803
\(905\) −6.10504e6 −0.247780
\(906\) 5.50343e6 0.222747
\(907\) −9.60321e6 −0.387613 −0.193806 0.981040i \(-0.562083\pi\)
−0.193806 + 0.981040i \(0.562083\pi\)
\(908\) 1.20590e6 0.0485395
\(909\) −5.23735e6 −0.210234
\(910\) −1.10482e7 −0.442273
\(911\) 4.49021e7 1.79255 0.896274 0.443500i \(-0.146263\pi\)
0.896274 + 0.443500i \(0.146263\pi\)
\(912\) 1.46347e6 0.0582635
\(913\) 7.51782e6 0.298480
\(914\) 2.59794e7 1.02864
\(915\) −9.40853e6 −0.371509
\(916\) 1.04719e7 0.412370
\(917\) 5.44986e7 2.14024
\(918\) −169446. −0.00663629
\(919\) 2.33413e7 0.911668 0.455834 0.890065i \(-0.349341\pi\)
0.455834 + 0.890065i \(0.349341\pi\)
\(920\) −2.33363e6 −0.0908997
\(921\) −2.40162e7 −0.932945
\(922\) −2.54388e7 −0.985530
\(923\) −3.40567e7 −1.31583
\(924\) 8.25716e6 0.318164
\(925\) 5.28880e6 0.203237
\(926\) −2.04738e7 −0.784642
\(927\) −1.24859e7 −0.477223
\(928\) −4.34823e6 −0.165746
\(929\) −2.13234e7 −0.810621 −0.405311 0.914179i \(-0.632837\pi\)
−0.405311 + 0.914179i \(0.632837\pi\)
\(930\) 5.34699e6 0.202722
\(931\) 2.58893e7 0.978917
\(932\) 3.40924e6 0.128564
\(933\) −6.69553e6 −0.251815
\(934\) 1.92721e7 0.722874
\(935\) 50137.1 0.00187556
\(936\) −9.16371e6 −0.341886
\(937\) −1.44458e7 −0.537516 −0.268758 0.963208i \(-0.586613\pi\)
−0.268758 + 0.963208i \(0.586613\pi\)
\(938\) 1.10126e6 0.0408681
\(939\) 2.62879e7 0.972951
\(940\) 1.08172e6 0.0399298
\(941\) 6.83636e6 0.251681 0.125841 0.992050i \(-0.459837\pi\)
0.125841 + 0.992050i \(0.459837\pi\)
\(942\) −1.91474e7 −0.703046
\(943\) 5.65563e6 0.207110
\(944\) 513698. 0.0187620
\(945\) −1.92264e7 −0.700357
\(946\) −4.10324e6 −0.149073
\(947\) −4.93062e7 −1.78660 −0.893298 0.449465i \(-0.851615\pi\)
−0.893298 + 0.449465i \(0.851615\pi\)
\(948\) 2.25122e7 0.813572
\(949\) −5.24384e7 −1.89010
\(950\) −2.99600e6 −0.107704
\(951\) 2.77711e7 0.995731
\(952\) 405698. 0.0145081
\(953\) 693828. 0.0247468 0.0123734 0.999923i \(-0.496061\pi\)
0.0123734 + 0.999923i \(0.496061\pi\)
\(954\) 3.40212e6 0.121026
\(955\) −3.73232e6 −0.132425
\(956\) −2.92418e7 −1.03481
\(957\) 1.56136e6 0.0551092
\(958\) −2.66600e7 −0.938525
\(959\) −2.35816e7 −0.827993
\(960\) −5.60933e6 −0.196441
\(961\) −4.45497e6 −0.155610
\(962\) −2.00357e7 −0.698019
\(963\) −1.24186e7 −0.431525
\(964\) 1.81104e7 0.627675
\(965\) −1.63144e7 −0.563968
\(966\) 4.29511e6 0.148092
\(967\) 1.31567e7 0.452461 0.226231 0.974074i \(-0.427360\pi\)
0.226231 + 0.974074i \(0.427360\pi\)
\(968\) −2.37411e7 −0.814354
\(969\) −230452. −0.00788444
\(970\) −1.82637e6 −0.0623245
\(971\) −2.22126e6 −0.0756052 −0.0378026 0.999285i \(-0.512036\pi\)
−0.0378026 + 0.999285i \(0.512036\pi\)
\(972\) −1.11270e7 −0.377758
\(973\) 7.62460e6 0.258187
\(974\) 2.85812e7 0.965346
\(975\) 5.77545e6 0.194569
\(976\) 2.25939e6 0.0759218
\(977\) −4.18113e7 −1.40138 −0.700692 0.713464i \(-0.747125\pi\)
−0.700692 + 0.713464i \(0.747125\pi\)
\(978\) 2.37984e7 0.795612
\(979\) 5.18899e6 0.173032
\(980\) 9.40044e6 0.312668
\(981\) 1.05408e7 0.349703
\(982\) 3.02070e7 0.999607
\(983\) −3.79365e7 −1.25220 −0.626099 0.779743i \(-0.715350\pi\)
−0.626099 + 0.779743i \(0.715350\pi\)
\(984\) 2.45809e7 0.809300
\(985\) −756620. −0.0248478
\(986\) 30268.4 0.000991510 0
\(987\) −5.04599e6 −0.164875
\(988\) −2.12356e7 −0.692106
\(989\) 3.99344e6 0.129825
\(990\) −995040. −0.0322665
\(991\) 1.98381e6 0.0641676 0.0320838 0.999485i \(-0.489786\pi\)
0.0320838 + 0.999485i \(0.489786\pi\)
\(992\) −2.90468e7 −0.937170
\(993\) −2.73745e7 −0.880994
\(994\) −2.99237e7 −0.960616
\(995\) −8.35967e6 −0.267690
\(996\) −1.25470e7 −0.400768
\(997\) 1.17260e7 0.373605 0.186803 0.982397i \(-0.440187\pi\)
0.186803 + 0.982397i \(0.440187\pi\)
\(998\) −1.62623e7 −0.516838
\(999\) −3.48667e7 −1.10534
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 115.6.a.c.1.3 7
3.2 odd 2 1035.6.a.b.1.5 7
5.4 even 2 575.6.a.d.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.3 7 1.1 even 1 trivial
575.6.a.d.1.5 7 5.4 even 2
1035.6.a.b.1.5 7 3.2 odd 2