Properties

Label 74.6.a.e.1.1
Level $74$
Weight $6$
Character 74.1
Self dual yes
Analytic conductor $11.868$
Analytic rank $0$
Dimension $5$
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] = [74,6,Mod(1,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.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: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 870x^{3} - 2235x^{2} + 121361x + 481504 \) 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 1.1
Root \(28.3792\) of defining polynomial
Character \(\chi\) \(=\) 74.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -24.3792 q^{3} +16.0000 q^{4} +44.9778 q^{5} -97.5168 q^{6} -145.948 q^{7} +64.0000 q^{8} +351.346 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -24.3792 q^{3} +16.0000 q^{4} +44.9778 q^{5} -97.5168 q^{6} -145.948 q^{7} +64.0000 q^{8} +351.346 q^{9} +179.911 q^{10} +419.587 q^{11} -390.067 q^{12} +373.322 q^{13} -583.793 q^{14} -1096.52 q^{15} +256.000 q^{16} +1931.97 q^{17} +1405.38 q^{18} +1577.61 q^{19} +719.645 q^{20} +3558.10 q^{21} +1678.35 q^{22} +1370.65 q^{23} -1560.27 q^{24} -1101.99 q^{25} +1493.29 q^{26} -2641.38 q^{27} -2335.17 q^{28} +5141.18 q^{29} -4386.10 q^{30} -10547.9 q^{31} +1024.00 q^{32} -10229.2 q^{33} +7727.86 q^{34} -6564.43 q^{35} +5621.53 q^{36} -1369.00 q^{37} +6310.46 q^{38} -9101.29 q^{39} +2878.58 q^{40} +3806.91 q^{41} +14232.4 q^{42} -5509.91 q^{43} +6713.40 q^{44} +15802.8 q^{45} +5482.60 q^{46} +7069.09 q^{47} -6241.08 q^{48} +4493.87 q^{49} -4407.98 q^{50} -47099.8 q^{51} +5973.15 q^{52} +8514.77 q^{53} -10565.5 q^{54} +18872.1 q^{55} -9340.68 q^{56} -38461.0 q^{57} +20564.7 q^{58} +29846.0 q^{59} -17544.4 q^{60} +35774.7 q^{61} -42191.6 q^{62} -51278.2 q^{63} +4096.00 q^{64} +16791.2 q^{65} -40916.8 q^{66} -62617.1 q^{67} +30911.4 q^{68} -33415.4 q^{69} -26257.7 q^{70} +21558.4 q^{71} +22486.1 q^{72} -26703.8 q^{73} -5476.00 q^{74} +26865.7 q^{75} +25241.8 q^{76} -61238.0 q^{77} -36405.2 q^{78} +107249. q^{79} +11514.3 q^{80} -20982.3 q^{81} +15227.7 q^{82} -57311.6 q^{83} +56929.6 q^{84} +86895.6 q^{85} -22039.7 q^{86} -125338. q^{87} +26853.6 q^{88} +47134.6 q^{89} +63211.1 q^{90} -54485.7 q^{91} +21930.4 q^{92} +257149. q^{93} +28276.3 q^{94} +70957.7 q^{95} -24964.3 q^{96} +40076.6 q^{97} +17975.5 q^{98} +147420. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 20 q^{2} + 19 q^{3} + 80 q^{4} + 95 q^{5} + 76 q^{6} + 170 q^{7} + 320 q^{8} + 598 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 20 q^{2} + 19 q^{3} + 80 q^{4} + 95 q^{5} + 76 q^{6} + 170 q^{7} + 320 q^{8} + 598 q^{9} + 380 q^{10} + 903 q^{11} + 304 q^{12} + 1371 q^{13} + 680 q^{14} - 8 q^{15} + 1280 q^{16} + 2070 q^{17} + 2392 q^{18} + 2358 q^{19} + 1520 q^{20} + 2832 q^{21} + 3612 q^{22} + 2097 q^{23} + 1216 q^{24} - 390 q^{25} + 5484 q^{26} + 2614 q^{27} + 2720 q^{28} + 2927 q^{29} - 32 q^{30} - 5849 q^{31} + 5120 q^{32} - 11002 q^{33} + 8280 q^{34} - 10928 q^{35} + 9568 q^{36} - 6845 q^{37} + 9432 q^{38} - 9945 q^{39} + 6080 q^{40} - 14427 q^{41} + 11328 q^{42} - 6972 q^{43} + 14448 q^{44} - 6816 q^{45} + 8388 q^{46} - 18962 q^{47} + 4864 q^{48} - 11957 q^{49} - 1560 q^{50} - 67946 q^{51} + 21936 q^{52} - 23576 q^{53} + 10456 q^{54} - 31415 q^{55} + 10880 q^{56} - 70522 q^{57} + 11708 q^{58} - 18316 q^{59} - 128 q^{60} - 18695 q^{61} - 23396 q^{62} - 88094 q^{63} + 20480 q^{64} - 40706 q^{65} - 44008 q^{66} - 85273 q^{67} + 33120 q^{68} - 87171 q^{69} - 43712 q^{70} + 8760 q^{71} + 38272 q^{72} - 10425 q^{73} - 27380 q^{74} - 10542 q^{75} + 37728 q^{76} + 17238 q^{77} - 39780 q^{78} + 48425 q^{79} + 24320 q^{80} + 33449 q^{81} - 57708 q^{82} + 27704 q^{83} + 45312 q^{84} + 139062 q^{85} - 27888 q^{86} + 6227 q^{87} + 57792 q^{88} + 233646 q^{89} - 27264 q^{90} + 146434 q^{91} + 33552 q^{92} + 301866 q^{93} - 75848 q^{94} + 189498 q^{95} + 19456 q^{96} + 251694 q^{97} - 47828 q^{98} + 182486 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −24.3792 −1.56393 −0.781963 0.623324i \(-0.785782\pi\)
−0.781963 + 0.623324i \(0.785782\pi\)
\(4\) 16.0000 0.500000
\(5\) 44.9778 0.804588 0.402294 0.915511i \(-0.368213\pi\)
0.402294 + 0.915511i \(0.368213\pi\)
\(6\) −97.5168 −1.10586
\(7\) −145.948 −1.12578 −0.562890 0.826532i \(-0.690311\pi\)
−0.562890 + 0.826532i \(0.690311\pi\)
\(8\) 64.0000 0.353553
\(9\) 351.346 1.44587
\(10\) 179.911 0.568930
\(11\) 419.587 1.04554 0.522770 0.852474i \(-0.324899\pi\)
0.522770 + 0.852474i \(0.324899\pi\)
\(12\) −390.067 −0.781963
\(13\) 373.322 0.612668 0.306334 0.951924i \(-0.400898\pi\)
0.306334 + 0.951924i \(0.400898\pi\)
\(14\) −583.793 −0.796047
\(15\) −1096.52 −1.25832
\(16\) 256.000 0.250000
\(17\) 1931.97 1.62135 0.810676 0.585496i \(-0.199100\pi\)
0.810676 + 0.585496i \(0.199100\pi\)
\(18\) 1405.38 1.02238
\(19\) 1577.61 1.00257 0.501287 0.865281i \(-0.332860\pi\)
0.501287 + 0.865281i \(0.332860\pi\)
\(20\) 719.645 0.402294
\(21\) 3558.10 1.76064
\(22\) 1678.35 0.739309
\(23\) 1370.65 0.540265 0.270133 0.962823i \(-0.412932\pi\)
0.270133 + 0.962823i \(0.412932\pi\)
\(24\) −1560.27 −0.552932
\(25\) −1101.99 −0.352638
\(26\) 1493.29 0.433222
\(27\) −2641.38 −0.697303
\(28\) −2335.17 −0.562890
\(29\) 5141.18 1.13519 0.567594 0.823309i \(-0.307874\pi\)
0.567594 + 0.823309i \(0.307874\pi\)
\(30\) −4386.10 −0.889764
\(31\) −10547.9 −1.97134 −0.985671 0.168681i \(-0.946049\pi\)
−0.985671 + 0.168681i \(0.946049\pi\)
\(32\) 1024.00 0.176777
\(33\) −10229.2 −1.63515
\(34\) 7727.86 1.14647
\(35\) −6564.43 −0.905789
\(36\) 5621.53 0.722933
\(37\) −1369.00 −0.164399
\(38\) 6310.46 0.708928
\(39\) −9101.29 −0.958168
\(40\) 2878.58 0.284465
\(41\) 3806.91 0.353682 0.176841 0.984239i \(-0.443412\pi\)
0.176841 + 0.984239i \(0.443412\pi\)
\(42\) 14232.4 1.24496
\(43\) −5509.91 −0.454437 −0.227219 0.973844i \(-0.572963\pi\)
−0.227219 + 0.973844i \(0.572963\pi\)
\(44\) 6713.40 0.522770
\(45\) 15802.8 1.16333
\(46\) 5482.60 0.382025
\(47\) 7069.09 0.466787 0.233393 0.972382i \(-0.425017\pi\)
0.233393 + 0.972382i \(0.425017\pi\)
\(48\) −6241.08 −0.390982
\(49\) 4493.87 0.267381
\(50\) −4407.98 −0.249353
\(51\) −47099.8 −2.53567
\(52\) 5973.15 0.306334
\(53\) 8514.77 0.416374 0.208187 0.978089i \(-0.433244\pi\)
0.208187 + 0.978089i \(0.433244\pi\)
\(54\) −10565.5 −0.493067
\(55\) 18872.1 0.841229
\(56\) −9340.68 −0.398023
\(57\) −38461.0 −1.56795
\(58\) 20564.7 0.802699
\(59\) 29846.0 1.11624 0.558119 0.829761i \(-0.311523\pi\)
0.558119 + 0.829761i \(0.311523\pi\)
\(60\) −17544.4 −0.629158
\(61\) 35774.7 1.23098 0.615491 0.788144i \(-0.288958\pi\)
0.615491 + 0.788144i \(0.288958\pi\)
\(62\) −42191.6 −1.39395
\(63\) −51278.2 −1.62773
\(64\) 4096.00 0.125000
\(65\) 16791.2 0.492945
\(66\) −40916.8 −1.15622
\(67\) −62617.1 −1.70414 −0.852071 0.523426i \(-0.824654\pi\)
−0.852071 + 0.523426i \(0.824654\pi\)
\(68\) 30911.4 0.810676
\(69\) −33415.4 −0.844935
\(70\) −26257.7 −0.640490
\(71\) 21558.4 0.507541 0.253770 0.967265i \(-0.418329\pi\)
0.253770 + 0.967265i \(0.418329\pi\)
\(72\) 22486.1 0.511191
\(73\) −26703.8 −0.586498 −0.293249 0.956036i \(-0.594737\pi\)
−0.293249 + 0.956036i \(0.594737\pi\)
\(74\) −5476.00 −0.116248
\(75\) 26865.7 0.551500
\(76\) 25241.8 0.501287
\(77\) −61238.0 −1.17705
\(78\) −36405.2 −0.677527
\(79\) 107249. 1.93342 0.966711 0.255870i \(-0.0823620\pi\)
0.966711 + 0.255870i \(0.0823620\pi\)
\(80\) 11514.3 0.201147
\(81\) −20982.3 −0.355336
\(82\) 15227.7 0.250091
\(83\) −57311.6 −0.913161 −0.456580 0.889682i \(-0.650926\pi\)
−0.456580 + 0.889682i \(0.650926\pi\)
\(84\) 56929.6 0.880319
\(85\) 86895.6 1.30452
\(86\) −22039.7 −0.321335
\(87\) −125338. −1.77535
\(88\) 26853.6 0.369654
\(89\) 47134.6 0.630761 0.315381 0.948965i \(-0.397868\pi\)
0.315381 + 0.948965i \(0.397868\pi\)
\(90\) 63211.1 0.822596
\(91\) −54485.7 −0.689730
\(92\) 21930.4 0.270133
\(93\) 257149. 3.08303
\(94\) 28276.3 0.330068
\(95\) 70957.7 0.806660
\(96\) −24964.3 −0.276466
\(97\) 40076.6 0.432475 0.216238 0.976341i \(-0.430621\pi\)
0.216238 + 0.976341i \(0.430621\pi\)
\(98\) 17975.5 0.189067
\(99\) 147420. 1.51171
\(100\) −17631.9 −0.176319
\(101\) 91615.6 0.893647 0.446823 0.894622i \(-0.352555\pi\)
0.446823 + 0.894622i \(0.352555\pi\)
\(102\) −188399. −1.79299
\(103\) −34348.8 −0.319020 −0.159510 0.987196i \(-0.550991\pi\)
−0.159510 + 0.987196i \(0.550991\pi\)
\(104\) 23892.6 0.216611
\(105\) 160036. 1.41659
\(106\) 34059.1 0.294421
\(107\) −22234.1 −0.187741 −0.0938705 0.995584i \(-0.529924\pi\)
−0.0938705 + 0.995584i \(0.529924\pi\)
\(108\) −42262.1 −0.348651
\(109\) −231071. −1.86285 −0.931427 0.363927i \(-0.881436\pi\)
−0.931427 + 0.363927i \(0.881436\pi\)
\(110\) 75488.5 0.594839
\(111\) 33375.1 0.257108
\(112\) −37362.7 −0.281445
\(113\) 20790.5 0.153169 0.0765843 0.997063i \(-0.475599\pi\)
0.0765843 + 0.997063i \(0.475599\pi\)
\(114\) −153844. −1.10871
\(115\) 61648.9 0.434691
\(116\) 82258.8 0.567594
\(117\) 131165. 0.885836
\(118\) 119384. 0.789299
\(119\) −281967. −1.82528
\(120\) −70177.5 −0.444882
\(121\) 15002.6 0.0931545
\(122\) 143099. 0.870436
\(123\) −92809.5 −0.553133
\(124\) −168766. −0.985671
\(125\) −190121. −1.08832
\(126\) −205113. −1.15098
\(127\) −83413.4 −0.458909 −0.229455 0.973319i \(-0.573694\pi\)
−0.229455 + 0.973319i \(0.573694\pi\)
\(128\) 16384.0 0.0883883
\(129\) 134327. 0.710706
\(130\) 67164.9 0.348565
\(131\) 346823. 1.76575 0.882877 0.469604i \(-0.155603\pi\)
0.882877 + 0.469604i \(0.155603\pi\)
\(132\) −163667. −0.817574
\(133\) −230250. −1.12868
\(134\) −250468. −1.20501
\(135\) −118803. −0.561041
\(136\) 123646. 0.573234
\(137\) −169756. −0.772723 −0.386362 0.922347i \(-0.626268\pi\)
−0.386362 + 0.922347i \(0.626268\pi\)
\(138\) −133661. −0.597460
\(139\) 201516. 0.884654 0.442327 0.896854i \(-0.354153\pi\)
0.442327 + 0.896854i \(0.354153\pi\)
\(140\) −105031. −0.452895
\(141\) −172339. −0.730020
\(142\) 86233.6 0.358885
\(143\) 156641. 0.640569
\(144\) 89944.5 0.361467
\(145\) 231239. 0.913358
\(146\) −106815. −0.414717
\(147\) −109557. −0.418164
\(148\) −21904.0 −0.0821995
\(149\) −92365.5 −0.340835 −0.170417 0.985372i \(-0.554512\pi\)
−0.170417 + 0.985372i \(0.554512\pi\)
\(150\) 107463. 0.389970
\(151\) 540948. 1.93069 0.965346 0.260975i \(-0.0840440\pi\)
0.965346 + 0.260975i \(0.0840440\pi\)
\(152\) 100967. 0.354464
\(153\) 678788. 2.34426
\(154\) −244952. −0.832299
\(155\) −474422. −1.58612
\(156\) −145621. −0.479084
\(157\) −179937. −0.582603 −0.291301 0.956631i \(-0.594088\pi\)
−0.291301 + 0.956631i \(0.594088\pi\)
\(158\) 428997. 1.36714
\(159\) −207583. −0.651178
\(160\) 46057.3 0.142232
\(161\) −200044. −0.608220
\(162\) −83929.1 −0.251261
\(163\) −538231. −1.58672 −0.793359 0.608754i \(-0.791670\pi\)
−0.793359 + 0.608754i \(0.791670\pi\)
\(164\) 60910.6 0.176841
\(165\) −460088. −1.31562
\(166\) −229246. −0.645702
\(167\) −211189. −0.585977 −0.292988 0.956116i \(-0.594650\pi\)
−0.292988 + 0.956116i \(0.594650\pi\)
\(168\) 227718. 0.622479
\(169\) −231924. −0.624638
\(170\) 347583. 0.922435
\(171\) 554288. 1.44959
\(172\) −88158.6 −0.227219
\(173\) 246945. 0.627315 0.313657 0.949536i \(-0.398446\pi\)
0.313657 + 0.949536i \(0.398446\pi\)
\(174\) −501351. −1.25536
\(175\) 160834. 0.396993
\(176\) 107414. 0.261385
\(177\) −727622. −1.74571
\(178\) 188538. 0.446015
\(179\) −307183. −0.716579 −0.358289 0.933611i \(-0.616640\pi\)
−0.358289 + 0.933611i \(0.616640\pi\)
\(180\) 252844. 0.581663
\(181\) 480322. 1.08977 0.544886 0.838510i \(-0.316573\pi\)
0.544886 + 0.838510i \(0.316573\pi\)
\(182\) −217943. −0.487712
\(183\) −872159. −1.92517
\(184\) 87721.6 0.191013
\(185\) −61574.7 −0.132273
\(186\) 1.02860e6 2.18003
\(187\) 810628. 1.69519
\(188\) 113105. 0.233393
\(189\) 385504. 0.785009
\(190\) 283831. 0.570395
\(191\) −644592. −1.27850 −0.639252 0.768998i \(-0.720756\pi\)
−0.639252 + 0.768998i \(0.720756\pi\)
\(192\) −99857.2 −0.195491
\(193\) 214964. 0.415405 0.207703 0.978192i \(-0.433401\pi\)
0.207703 + 0.978192i \(0.433401\pi\)
\(194\) 160306. 0.305806
\(195\) −409357. −0.770930
\(196\) 71901.9 0.133690
\(197\) 18456.5 0.0338832 0.0169416 0.999856i \(-0.494607\pi\)
0.0169416 + 0.999856i \(0.494607\pi\)
\(198\) 589681. 1.06894
\(199\) −476731. −0.853377 −0.426689 0.904399i \(-0.640320\pi\)
−0.426689 + 0.904399i \(0.640320\pi\)
\(200\) −70527.6 −0.124676
\(201\) 1.52655e6 2.66515
\(202\) 366462. 0.631904
\(203\) −750345. −1.27797
\(204\) −753596. −1.26784
\(205\) 171227. 0.284569
\(206\) −137395. −0.225581
\(207\) 481572. 0.781152
\(208\) 95570.5 0.153167
\(209\) 661947. 1.04823
\(210\) 640143. 1.00168
\(211\) 321403. 0.496986 0.248493 0.968634i \(-0.420065\pi\)
0.248493 + 0.968634i \(0.420065\pi\)
\(212\) 136236. 0.208187
\(213\) −525577. −0.793756
\(214\) −88936.2 −0.132753
\(215\) −247824. −0.365635
\(216\) −169048. −0.246534
\(217\) 1.53945e6 2.21930
\(218\) −924284. −1.31724
\(219\) 651019. 0.917240
\(220\) 301954. 0.420615
\(221\) 721245. 0.993350
\(222\) 133501. 0.181803
\(223\) −319949. −0.430843 −0.215421 0.976521i \(-0.569112\pi\)
−0.215421 + 0.976521i \(0.569112\pi\)
\(224\) −149451. −0.199012
\(225\) −387181. −0.509868
\(226\) 83162.2 0.108307
\(227\) −1.29834e6 −1.67234 −0.836170 0.548471i \(-0.815210\pi\)
−0.836170 + 0.548471i \(0.815210\pi\)
\(228\) −615376. −0.783977
\(229\) −1.08171e6 −1.36308 −0.681541 0.731780i \(-0.738690\pi\)
−0.681541 + 0.731780i \(0.738690\pi\)
\(230\) 246596. 0.307373
\(231\) 1.49293e6 1.84082
\(232\) 329035. 0.401349
\(233\) −1.37325e6 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(234\) 524660. 0.626381
\(235\) 317952. 0.375571
\(236\) 477536. 0.558119
\(237\) −2.61465e6 −3.02373
\(238\) −1.12787e6 −1.29067
\(239\) −1.31403e6 −1.48802 −0.744012 0.668166i \(-0.767080\pi\)
−0.744012 + 0.668166i \(0.767080\pi\)
\(240\) −280710. −0.314579
\(241\) 1.40950e6 1.56322 0.781612 0.623765i \(-0.214398\pi\)
0.781612 + 0.623765i \(0.214398\pi\)
\(242\) 60010.5 0.0658702
\(243\) 1.15339e6 1.25302
\(244\) 572396. 0.615491
\(245\) 202125. 0.215131
\(246\) −371238. −0.391124
\(247\) 588958. 0.614246
\(248\) −675066. −0.696974
\(249\) 1.39721e6 1.42812
\(250\) −760484. −0.769556
\(251\) −935529. −0.937288 −0.468644 0.883387i \(-0.655257\pi\)
−0.468644 + 0.883387i \(0.655257\pi\)
\(252\) −820452. −0.813864
\(253\) 575108. 0.564869
\(254\) −333654. −0.324498
\(255\) −2.11845e6 −2.04017
\(256\) 65536.0 0.0625000
\(257\) 252017. 0.238011 0.119005 0.992894i \(-0.462029\pi\)
0.119005 + 0.992894i \(0.462029\pi\)
\(258\) 537309. 0.502545
\(259\) 199803. 0.185077
\(260\) 268660. 0.246473
\(261\) 1.80633e6 1.64133
\(262\) 1.38729e6 1.24858
\(263\) 1.87392e6 1.67056 0.835281 0.549823i \(-0.185305\pi\)
0.835281 + 0.549823i \(0.185305\pi\)
\(264\) −654669. −0.578112
\(265\) 382976. 0.335009
\(266\) −921000. −0.798097
\(267\) −1.14910e6 −0.986464
\(268\) −1.00187e6 −0.852071
\(269\) 212512. 0.179062 0.0895309 0.995984i \(-0.471463\pi\)
0.0895309 + 0.995984i \(0.471463\pi\)
\(270\) −475214. −0.396716
\(271\) 1.24620e6 1.03078 0.515389 0.856956i \(-0.327647\pi\)
0.515389 + 0.856956i \(0.327647\pi\)
\(272\) 494583. 0.405338
\(273\) 1.32832e6 1.07869
\(274\) −679025. −0.546398
\(275\) −462383. −0.368697
\(276\) −534646. −0.422468
\(277\) −2.40327e6 −1.88193 −0.940963 0.338508i \(-0.890078\pi\)
−0.940963 + 0.338508i \(0.890078\pi\)
\(278\) 806065. 0.625545
\(279\) −3.70596e6 −2.85030
\(280\) −420124. −0.320245
\(281\) −1.01520e6 −0.766980 −0.383490 0.923545i \(-0.625278\pi\)
−0.383490 + 0.923545i \(0.625278\pi\)
\(282\) −689355. −0.516202
\(283\) 449107. 0.333337 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(284\) 344935. 0.253770
\(285\) −1.72989e6 −1.26156
\(286\) 626565. 0.452951
\(287\) −555612. −0.398169
\(288\) 359778. 0.255596
\(289\) 2.31263e6 1.62878
\(290\) 924956. 0.645842
\(291\) −977035. −0.676360
\(292\) −427262. −0.293249
\(293\) 1.50375e6 1.02331 0.511654 0.859191i \(-0.329033\pi\)
0.511654 + 0.859191i \(0.329033\pi\)
\(294\) −438228. −0.295687
\(295\) 1.34241e6 0.898111
\(296\) −87616.0 −0.0581238
\(297\) −1.10829e6 −0.729058
\(298\) −369462. −0.241007
\(299\) 511694. 0.331003
\(300\) 429852. 0.275750
\(301\) 804162. 0.511596
\(302\) 2.16379e6 1.36520
\(303\) −2.23351e6 −1.39760
\(304\) 403869. 0.250644
\(305\) 1.60907e6 0.990434
\(306\) 2.71515e6 1.65764
\(307\) −1.58527e6 −0.959970 −0.479985 0.877277i \(-0.659358\pi\)
−0.479985 + 0.877277i \(0.659358\pi\)
\(308\) −979808. −0.588524
\(309\) 837395. 0.498924
\(310\) −1.89769e6 −1.12155
\(311\) 2.13422e6 1.25124 0.625618 0.780130i \(-0.284847\pi\)
0.625618 + 0.780130i \(0.284847\pi\)
\(312\) −582483. −0.338764
\(313\) 508004. 0.293093 0.146547 0.989204i \(-0.453184\pi\)
0.146547 + 0.989204i \(0.453184\pi\)
\(314\) −719750. −0.411962
\(315\) −2.30638e6 −1.30965
\(316\) 1.71599e6 0.966711
\(317\) −2.67588e6 −1.49561 −0.747805 0.663919i \(-0.768892\pi\)
−0.747805 + 0.663919i \(0.768892\pi\)
\(318\) −830333. −0.460452
\(319\) 2.15717e6 1.18688
\(320\) 184229. 0.100573
\(321\) 542049. 0.293613
\(322\) −800176. −0.430076
\(323\) 3.04790e6 1.62553
\(324\) −335716. −0.177668
\(325\) −411399. −0.216050
\(326\) −2.15292e6 −1.12198
\(327\) 5.63332e6 2.91337
\(328\) 243642. 0.125046
\(329\) −1.03172e6 −0.525499
\(330\) −1.84035e6 −0.930284
\(331\) −3.23173e6 −1.62131 −0.810654 0.585525i \(-0.800888\pi\)
−0.810654 + 0.585525i \(0.800888\pi\)
\(332\) −916986. −0.456580
\(333\) −480992. −0.237699
\(334\) −844757. −0.414348
\(335\) −2.81638e6 −1.37113
\(336\) 910874. 0.440159
\(337\) 3.56562e6 1.71025 0.855127 0.518419i \(-0.173479\pi\)
0.855127 + 0.518419i \(0.173479\pi\)
\(338\) −927695. −0.441686
\(339\) −506857. −0.239544
\(340\) 1.39033e6 0.652260
\(341\) −4.42577e6 −2.06112
\(342\) 2.21715e6 1.02501
\(343\) 1.79708e6 0.824768
\(344\) −352635. −0.160668
\(345\) −1.50295e6 −0.679825
\(346\) 987781. 0.443579
\(347\) 2.38403e6 1.06289 0.531444 0.847093i \(-0.321649\pi\)
0.531444 + 0.847093i \(0.321649\pi\)
\(348\) −2.00540e6 −0.887675
\(349\) −1.22091e6 −0.536563 −0.268281 0.963341i \(-0.586456\pi\)
−0.268281 + 0.963341i \(0.586456\pi\)
\(350\) 643336. 0.280716
\(351\) −986085. −0.427215
\(352\) 429658. 0.184827
\(353\) −1.43335e6 −0.612232 −0.306116 0.951994i \(-0.599030\pi\)
−0.306116 + 0.951994i \(0.599030\pi\)
\(354\) −2.91049e6 −1.23441
\(355\) 969650. 0.408361
\(356\) 754154. 0.315381
\(357\) 6.87413e6 2.85461
\(358\) −1.22873e6 −0.506698
\(359\) 1.56017e6 0.638906 0.319453 0.947602i \(-0.396501\pi\)
0.319453 + 0.947602i \(0.396501\pi\)
\(360\) 1.01138e6 0.411298
\(361\) 12767.9 0.00515646
\(362\) 1.92129e6 0.770585
\(363\) −365752. −0.145687
\(364\) −871771. −0.344865
\(365\) −1.20108e6 −0.471889
\(366\) −3.48864e6 −1.36130
\(367\) 1.72338e6 0.667907 0.333953 0.942590i \(-0.391617\pi\)
0.333953 + 0.942590i \(0.391617\pi\)
\(368\) 350886. 0.135066
\(369\) 1.33754e6 0.511378
\(370\) −246299. −0.0935315
\(371\) −1.24272e6 −0.468745
\(372\) 4.11439e6 1.54152
\(373\) 2.06259e6 0.767612 0.383806 0.923414i \(-0.374613\pi\)
0.383806 + 0.923414i \(0.374613\pi\)
\(374\) 3.24251e6 1.19868
\(375\) 4.63500e6 1.70205
\(376\) 452422. 0.165034
\(377\) 1.91931e6 0.695493
\(378\) 1.54202e6 0.555085
\(379\) −2.51548e6 −0.899546 −0.449773 0.893143i \(-0.648495\pi\)
−0.449773 + 0.893143i \(0.648495\pi\)
\(380\) 1.13532e6 0.403330
\(381\) 2.03355e6 0.717700
\(382\) −2.57837e6 −0.904038
\(383\) 5.45752e6 1.90107 0.950536 0.310615i \(-0.100535\pi\)
0.950536 + 0.310615i \(0.100535\pi\)
\(384\) −399429. −0.138233
\(385\) −2.75435e6 −0.947039
\(386\) 859855. 0.293736
\(387\) −1.93588e6 −0.657055
\(388\) 641225. 0.216238
\(389\) 1.22639e6 0.410917 0.205459 0.978666i \(-0.434131\pi\)
0.205459 + 0.978666i \(0.434131\pi\)
\(390\) −1.63743e6 −0.545130
\(391\) 2.64805e6 0.875960
\(392\) 287608. 0.0945334
\(393\) −8.45528e6 −2.76151
\(394\) 73826.1 0.0239590
\(395\) 4.82384e6 1.55561
\(396\) 2.35872e6 0.755856
\(397\) −3.62294e6 −1.15368 −0.576840 0.816857i \(-0.695714\pi\)
−0.576840 + 0.816857i \(0.695714\pi\)
\(398\) −1.90693e6 −0.603429
\(399\) 5.61331e6 1.76517
\(400\) −282111. −0.0881595
\(401\) −887350. −0.275571 −0.137786 0.990462i \(-0.543999\pi\)
−0.137786 + 0.990462i \(0.543999\pi\)
\(402\) 6.10622e6 1.88455
\(403\) −3.93776e6 −1.20778
\(404\) 1.46585e6 0.446823
\(405\) −943737. −0.285899
\(406\) −3.00138e6 −0.903662
\(407\) −574415. −0.171886
\(408\) −3.01439e6 −0.896496
\(409\) −5.25309e6 −1.55277 −0.776384 0.630261i \(-0.782948\pi\)
−0.776384 + 0.630261i \(0.782948\pi\)
\(410\) 684907. 0.201220
\(411\) 4.13852e6 1.20848
\(412\) −549580. −0.159510
\(413\) −4.35597e6 −1.25664
\(414\) 1.92629e6 0.552358
\(415\) −2.57775e6 −0.734718
\(416\) 382282. 0.108305
\(417\) −4.91281e6 −1.38353
\(418\) 2.64779e6 0.741212
\(419\) −5.44336e6 −1.51472 −0.757359 0.652999i \(-0.773511\pi\)
−0.757359 + 0.652999i \(0.773511\pi\)
\(420\) 2.56057e6 0.708294
\(421\) −2.17294e6 −0.597507 −0.298753 0.954330i \(-0.596571\pi\)
−0.298753 + 0.954330i \(0.596571\pi\)
\(422\) 1.28561e6 0.351422
\(423\) 2.48369e6 0.674912
\(424\) 544945. 0.147210
\(425\) −2.12901e6 −0.571750
\(426\) −2.10231e6 −0.561270
\(427\) −5.22126e6 −1.38582
\(428\) −355745. −0.0938705
\(429\) −3.81879e6 −1.00180
\(430\) −991296. −0.258543
\(431\) −1.98969e6 −0.515932 −0.257966 0.966154i \(-0.583052\pi\)
−0.257966 + 0.966154i \(0.583052\pi\)
\(432\) −676193. −0.174326
\(433\) −2.42522e6 −0.621630 −0.310815 0.950470i \(-0.600602\pi\)
−0.310815 + 0.950470i \(0.600602\pi\)
\(434\) 6.15779e6 1.56928
\(435\) −5.63742e6 −1.42843
\(436\) −3.69713e6 −0.931427
\(437\) 2.16236e6 0.541656
\(438\) 2.60407e6 0.648587
\(439\) −40746.7 −0.0100909 −0.00504547 0.999987i \(-0.501606\pi\)
−0.00504547 + 0.999987i \(0.501606\pi\)
\(440\) 1.20782e6 0.297419
\(441\) 1.57890e6 0.386597
\(442\) 2.88498e6 0.702405
\(443\) 3.59743e6 0.870930 0.435465 0.900206i \(-0.356584\pi\)
0.435465 + 0.900206i \(0.356584\pi\)
\(444\) 534002. 0.128554
\(445\) 2.12001e6 0.507503
\(446\) −1.27980e6 −0.304652
\(447\) 2.25180e6 0.533041
\(448\) −597804. −0.140723
\(449\) −1.33796e6 −0.313204 −0.156602 0.987662i \(-0.550054\pi\)
−0.156602 + 0.987662i \(0.550054\pi\)
\(450\) −1.54872e6 −0.360531
\(451\) 1.59733e6 0.369789
\(452\) 332649. 0.0765843
\(453\) −1.31879e7 −3.01946
\(454\) −5.19337e6 −1.18252
\(455\) −2.45065e6 −0.554948
\(456\) −2.46150e6 −0.554355
\(457\) −7.80585e6 −1.74835 −0.874177 0.485607i \(-0.838599\pi\)
−0.874177 + 0.485607i \(0.838599\pi\)
\(458\) −4.32684e6 −0.963845
\(459\) −5.10305e6 −1.13057
\(460\) 986382. 0.217345
\(461\) −7.78534e6 −1.70618 −0.853091 0.521762i \(-0.825275\pi\)
−0.853091 + 0.521762i \(0.825275\pi\)
\(462\) 5.97174e6 1.30165
\(463\) −3.89653e6 −0.844746 −0.422373 0.906422i \(-0.638803\pi\)
−0.422373 + 0.906422i \(0.638803\pi\)
\(464\) 1.31614e6 0.283797
\(465\) 1.15660e7 2.48057
\(466\) −5.49300e6 −1.17178
\(467\) 2.89728e6 0.614750 0.307375 0.951588i \(-0.400549\pi\)
0.307375 + 0.951588i \(0.400549\pi\)
\(468\) 2.09864e6 0.442918
\(469\) 9.13885e6 1.91849
\(470\) 1.27181e6 0.265569
\(471\) 4.38673e6 0.911148
\(472\) 1.91015e6 0.394650
\(473\) −2.31189e6 −0.475132
\(474\) −1.04586e7 −2.13810
\(475\) −1.73852e6 −0.353546
\(476\) −4.51147e6 −0.912642
\(477\) 2.99163e6 0.602021
\(478\) −5.25611e6 −1.05219
\(479\) 591289. 0.117750 0.0588751 0.998265i \(-0.481249\pi\)
0.0588751 + 0.998265i \(0.481249\pi\)
\(480\) −1.12284e6 −0.222441
\(481\) −511078. −0.100722
\(482\) 5.63799e6 1.10537
\(483\) 4.87691e6 0.951211
\(484\) 240042. 0.0465772
\(485\) 1.80256e6 0.347964
\(486\) 4.61354e6 0.886021
\(487\) 522072. 0.0997488 0.0498744 0.998755i \(-0.484118\pi\)
0.0498744 + 0.998755i \(0.484118\pi\)
\(488\) 2.28958e6 0.435218
\(489\) 1.31216e7 2.48151
\(490\) 808498. 0.152121
\(491\) 1.97110e6 0.368981 0.184491 0.982834i \(-0.440936\pi\)
0.184491 + 0.982834i \(0.440936\pi\)
\(492\) −1.48495e6 −0.276567
\(493\) 9.93258e6 1.84054
\(494\) 2.35583e6 0.434337
\(495\) 6.63064e6 1.21631
\(496\) −2.70026e6 −0.492835
\(497\) −3.14641e6 −0.571379
\(498\) 5.58884e6 1.00983
\(499\) 3.81878e6 0.686553 0.343276 0.939234i \(-0.388463\pi\)
0.343276 + 0.939234i \(0.388463\pi\)
\(500\) −3.04194e6 −0.544158
\(501\) 5.14862e6 0.916425
\(502\) −3.74212e6 −0.662763
\(503\) 5.51646e6 0.972167 0.486083 0.873912i \(-0.338425\pi\)
0.486083 + 0.873912i \(0.338425\pi\)
\(504\) −3.28181e6 −0.575489
\(505\) 4.12067e6 0.719017
\(506\) 2.30043e6 0.399423
\(507\) 5.65411e6 0.976888
\(508\) −1.33461e6 −0.229455
\(509\) 5.41413e6 0.926263 0.463132 0.886289i \(-0.346726\pi\)
0.463132 + 0.886289i \(0.346726\pi\)
\(510\) −8.47378e6 −1.44262
\(511\) 3.89738e6 0.660268
\(512\) 262144. 0.0441942
\(513\) −4.16708e6 −0.699098
\(514\) 1.00807e6 0.168299
\(515\) −1.54493e6 −0.256680
\(516\) 2.14924e6 0.355353
\(517\) 2.96610e6 0.488045
\(518\) 799212. 0.130869
\(519\) −6.02033e6 −0.981075
\(520\) 1.07464e6 0.174283
\(521\) −786128. −0.126882 −0.0634408 0.997986i \(-0.520207\pi\)
−0.0634408 + 0.997986i \(0.520207\pi\)
\(522\) 7.22532e6 1.16060
\(523\) −3.43138e6 −0.548549 −0.274274 0.961651i \(-0.588438\pi\)
−0.274274 + 0.961651i \(0.588438\pi\)
\(524\) 5.54918e6 0.882877
\(525\) −3.92101e6 −0.620868
\(526\) 7.49570e6 1.18127
\(527\) −2.03782e7 −3.19624
\(528\) −2.61868e6 −0.408787
\(529\) −4.55766e6 −0.708113
\(530\) 1.53190e6 0.236887
\(531\) 1.04863e7 1.61393
\(532\) −3.68400e6 −0.564339
\(533\) 1.42121e6 0.216690
\(534\) −4.59642e6 −0.697535
\(535\) −1.00004e6 −0.151054
\(536\) −4.00749e6 −0.602505
\(537\) 7.48887e6 1.12068
\(538\) 850048. 0.126616
\(539\) 1.88557e6 0.279557
\(540\) −1.90086e6 −0.280521
\(541\) −5.88057e6 −0.863826 −0.431913 0.901915i \(-0.642161\pi\)
−0.431913 + 0.901915i \(0.642161\pi\)
\(542\) 4.98481e6 0.728870
\(543\) −1.17099e7 −1.70432
\(544\) 1.97833e6 0.286617
\(545\) −1.03931e7 −1.49883
\(546\) 5.31327e6 0.762747
\(547\) 1.26599e7 1.80910 0.904549 0.426370i \(-0.140208\pi\)
0.904549 + 0.426370i \(0.140208\pi\)
\(548\) −2.71610e6 −0.386362
\(549\) 1.25693e7 1.77984
\(550\) −1.84953e6 −0.260708
\(551\) 8.11079e6 1.13811
\(552\) −2.13858e6 −0.298730
\(553\) −1.56528e7 −2.17661
\(554\) −9.61307e6 −1.33072
\(555\) 1.50114e6 0.206866
\(556\) 3.22426e6 0.442327
\(557\) 5.97765e6 0.816381 0.408190 0.912897i \(-0.366160\pi\)
0.408190 + 0.912897i \(0.366160\pi\)
\(558\) −1.48238e7 −2.01546
\(559\) −2.05697e6 −0.278419
\(560\) −1.68049e6 −0.226447
\(561\) −1.97625e7 −2.65115
\(562\) −4.06078e6 −0.542337
\(563\) 9.55818e6 1.27088 0.635440 0.772151i \(-0.280819\pi\)
0.635440 + 0.772151i \(0.280819\pi\)
\(564\) −2.75742e6 −0.365010
\(565\) 935114. 0.123238
\(566\) 1.79643e6 0.235705
\(567\) 3.06232e6 0.400031
\(568\) 1.37974e6 0.179443
\(569\) 1.32282e7 1.71285 0.856426 0.516270i \(-0.172680\pi\)
0.856426 + 0.516270i \(0.172680\pi\)
\(570\) −6.91957e6 −0.892055
\(571\) 1.05447e7 1.35346 0.676730 0.736232i \(-0.263397\pi\)
0.676730 + 0.736232i \(0.263397\pi\)
\(572\) 2.50626e6 0.320285
\(573\) 1.57146e7 1.99949
\(574\) −2.22245e6 −0.281548
\(575\) −1.51045e6 −0.190518
\(576\) 1.43911e6 0.180733
\(577\) −1.37791e7 −1.72298 −0.861491 0.507772i \(-0.830469\pi\)
−0.861491 + 0.507772i \(0.830469\pi\)
\(578\) 9.25053e6 1.15172
\(579\) −5.24065e6 −0.649663
\(580\) 3.69982e6 0.456679
\(581\) 8.36452e6 1.02802
\(582\) −3.90814e6 −0.478258
\(583\) 3.57269e6 0.435336
\(584\) −1.70905e6 −0.207358
\(585\) 5.89952e6 0.712733
\(586\) 6.01500e6 0.723588
\(587\) −1.39841e7 −1.67509 −0.837547 0.546365i \(-0.816011\pi\)
−0.837547 + 0.546365i \(0.816011\pi\)
\(588\) −1.75291e6 −0.209082
\(589\) −1.66405e7 −1.97642
\(590\) 5.36964e6 0.635061
\(591\) −449955. −0.0529908
\(592\) −350464. −0.0410997
\(593\) 4.24514e6 0.495742 0.247871 0.968793i \(-0.420269\pi\)
0.247871 + 0.968793i \(0.420269\pi\)
\(594\) −4.43316e6 −0.515522
\(595\) −1.26823e7 −1.46860
\(596\) −1.47785e6 −0.170417
\(597\) 1.16223e7 1.33462
\(598\) 2.04678e6 0.234055
\(599\) 1.20353e7 1.37054 0.685268 0.728291i \(-0.259685\pi\)
0.685268 + 0.728291i \(0.259685\pi\)
\(600\) 1.71941e6 0.194985
\(601\) 9.83468e6 1.11064 0.555321 0.831636i \(-0.312595\pi\)
0.555321 + 0.831636i \(0.312595\pi\)
\(602\) 3.21665e6 0.361753
\(603\) −2.20002e7 −2.46396
\(604\) 8.65516e6 0.965346
\(605\) 674786. 0.0749510
\(606\) −8.93406e6 −0.988251
\(607\) −2.24274e6 −0.247062 −0.123531 0.992341i \(-0.539422\pi\)
−0.123531 + 0.992341i \(0.539422\pi\)
\(608\) 1.61548e6 0.177232
\(609\) 1.82928e7 1.99865
\(610\) 6.43628e6 0.700342
\(611\) 2.63905e6 0.285985
\(612\) 1.08606e7 1.17213
\(613\) 1.47635e6 0.158686 0.0793430 0.996847i \(-0.474718\pi\)
0.0793430 + 0.996847i \(0.474718\pi\)
\(614\) −6.34108e6 −0.678801
\(615\) −4.17437e6 −0.445044
\(616\) −3.91923e6 −0.416149
\(617\) 2.92711e6 0.309547 0.154773 0.987950i \(-0.450535\pi\)
0.154773 + 0.987950i \(0.450535\pi\)
\(618\) 3.34958e6 0.352792
\(619\) −4.45422e6 −0.467245 −0.233623 0.972327i \(-0.575058\pi\)
−0.233623 + 0.972327i \(0.575058\pi\)
\(620\) −7.59075e6 −0.793059
\(621\) −3.62041e6 −0.376728
\(622\) 8.53689e6 0.884757
\(623\) −6.87921e6 −0.710098
\(624\) −2.32993e6 −0.239542
\(625\) −5.10750e6 −0.523008
\(626\) 2.03201e6 0.207248
\(627\) −1.61377e7 −1.63936
\(628\) −2.87900e6 −0.291301
\(629\) −2.64486e6 −0.266548
\(630\) −9.22554e6 −0.926063
\(631\) 4.76316e6 0.476235 0.238118 0.971236i \(-0.423470\pi\)
0.238118 + 0.971236i \(0.423470\pi\)
\(632\) 6.86395e6 0.683568
\(633\) −7.83556e6 −0.777250
\(634\) −1.07035e7 −1.05756
\(635\) −3.75175e6 −0.369233
\(636\) −3.32133e6 −0.325589
\(637\) 1.67766e6 0.163816
\(638\) 8.62869e6 0.839254
\(639\) 7.57445e6 0.733836
\(640\) 736917. 0.0711162
\(641\) −1.96386e6 −0.188784 −0.0943921 0.995535i \(-0.530091\pi\)
−0.0943921 + 0.995535i \(0.530091\pi\)
\(642\) 2.16819e6 0.207616
\(643\) −3.04689e6 −0.290622 −0.145311 0.989386i \(-0.546418\pi\)
−0.145311 + 0.989386i \(0.546418\pi\)
\(644\) −3.20070e6 −0.304110
\(645\) 6.04175e6 0.571826
\(646\) 1.21916e7 1.14942
\(647\) 687296. 0.0645480 0.0322740 0.999479i \(-0.489725\pi\)
0.0322740 + 0.999479i \(0.489725\pi\)
\(648\) −1.34286e6 −0.125630
\(649\) 1.25230e7 1.16707
\(650\) −1.64560e6 −0.152771
\(651\) −3.75305e7 −3.47082
\(652\) −8.61170e6 −0.793359
\(653\) −6.72763e6 −0.617418 −0.308709 0.951157i \(-0.599897\pi\)
−0.308709 + 0.951157i \(0.599897\pi\)
\(654\) 2.25333e7 2.06006
\(655\) 1.55994e7 1.42070
\(656\) 974570. 0.0884206
\(657\) −9.38228e6 −0.847998
\(658\) −4.12688e6 −0.371584
\(659\) −1.31090e7 −1.17586 −0.587931 0.808911i \(-0.700057\pi\)
−0.587931 + 0.808911i \(0.700057\pi\)
\(660\) −7.36140e6 −0.657810
\(661\) 3.19740e6 0.284638 0.142319 0.989821i \(-0.454544\pi\)
0.142319 + 0.989821i \(0.454544\pi\)
\(662\) −1.29269e7 −1.14644
\(663\) −1.75834e7 −1.55353
\(664\) −3.66794e6 −0.322851
\(665\) −1.03561e7 −0.908121
\(666\) −1.92397e6 −0.168079
\(667\) 7.04676e6 0.613302
\(668\) −3.37903e6 −0.292988
\(669\) 7.80010e6 0.673806
\(670\) −1.12655e7 −0.969537
\(671\) 1.50106e7 1.28704
\(672\) 3.64349e6 0.311240
\(673\) −2.22874e7 −1.89680 −0.948398 0.317081i \(-0.897297\pi\)
−0.948398 + 0.317081i \(0.897297\pi\)
\(674\) 1.42625e7 1.20933
\(675\) 2.91078e6 0.245895
\(676\) −3.71078e6 −0.312319
\(677\) −7.64388e6 −0.640976 −0.320488 0.947252i \(-0.603847\pi\)
−0.320488 + 0.947252i \(0.603847\pi\)
\(678\) −2.02743e6 −0.169383
\(679\) −5.84910e6 −0.486872
\(680\) 5.56132e6 0.461217
\(681\) 3.16525e7 2.61542
\(682\) −1.77031e7 −1.45743
\(683\) 1.99971e6 0.164027 0.0820133 0.996631i \(-0.473865\pi\)
0.0820133 + 0.996631i \(0.473865\pi\)
\(684\) 8.86860e6 0.724795
\(685\) −7.63526e6 −0.621724
\(686\) 7.18832e6 0.583199
\(687\) 2.63712e7 2.13176
\(688\) −1.41054e6 −0.113609
\(689\) 3.17875e6 0.255099
\(690\) −6.01180e6 −0.480709
\(691\) 1.26728e6 0.100967 0.0504833 0.998725i \(-0.483924\pi\)
0.0504833 + 0.998725i \(0.483924\pi\)
\(692\) 3.95113e6 0.313657
\(693\) −2.15157e7 −1.70185
\(694\) 9.53612e6 0.751576
\(695\) 9.06377e6 0.711782
\(696\) −8.02162e6 −0.627681
\(697\) 7.35483e6 0.573443
\(698\) −4.88364e6 −0.379407
\(699\) 3.34788e7 2.59165
\(700\) 2.57334e6 0.198497
\(701\) −6.80193e6 −0.522802 −0.261401 0.965230i \(-0.584185\pi\)
−0.261401 + 0.965230i \(0.584185\pi\)
\(702\) −3.94434e6 −0.302087
\(703\) −2.15975e6 −0.164822
\(704\) 1.71863e6 0.130693
\(705\) −7.75142e6 −0.587366
\(706\) −5.73341e6 −0.432914
\(707\) −1.33711e7 −1.00605
\(708\) −1.16420e7 −0.872857
\(709\) −7.37523e6 −0.551011 −0.275505 0.961300i \(-0.588845\pi\)
−0.275505 + 0.961300i \(0.588845\pi\)
\(710\) 3.87860e6 0.288755
\(711\) 3.76816e7 2.79547
\(712\) 3.01661e6 0.223008
\(713\) −1.44575e7 −1.06505
\(714\) 2.74965e7 2.01852
\(715\) 7.04538e6 0.515394
\(716\) −4.91492e6 −0.358289
\(717\) 3.20350e7 2.32716
\(718\) 6.24069e6 0.451775
\(719\) −1.30200e7 −0.939264 −0.469632 0.882862i \(-0.655613\pi\)
−0.469632 + 0.882862i \(0.655613\pi\)
\(720\) 4.04551e6 0.290832
\(721\) 5.01314e6 0.359146
\(722\) 51071.6 0.00364617
\(723\) −3.43624e7 −2.44477
\(724\) 7.68515e6 0.544886
\(725\) −5.66555e6 −0.400310
\(726\) −1.46301e6 −0.103016
\(727\) 1.90995e7 1.34025 0.670125 0.742248i \(-0.266241\pi\)
0.670125 + 0.742248i \(0.266241\pi\)
\(728\) −3.48708e6 −0.243856
\(729\) −2.30199e7 −1.60430
\(730\) −4.80432e6 −0.333676
\(731\) −1.06450e7 −0.736802
\(732\) −1.39546e7 −0.962583
\(733\) −4.61791e6 −0.317458 −0.158729 0.987322i \(-0.550740\pi\)
−0.158729 + 0.987322i \(0.550740\pi\)
\(734\) 6.89352e6 0.472281
\(735\) −4.92764e6 −0.336450
\(736\) 1.40355e6 0.0955063
\(737\) −2.62733e7 −1.78175
\(738\) 5.35017e6 0.361599
\(739\) 1.88679e6 0.127090 0.0635450 0.997979i \(-0.479759\pi\)
0.0635450 + 0.997979i \(0.479759\pi\)
\(740\) −985195. −0.0661367
\(741\) −1.43583e7 −0.960635
\(742\) −4.97086e6 −0.331453
\(743\) −4.80888e6 −0.319574 −0.159787 0.987151i \(-0.551081\pi\)
−0.159787 + 0.987151i \(0.551081\pi\)
\(744\) 1.64576e7 1.09002
\(745\) −4.15440e6 −0.274232
\(746\) 8.25038e6 0.542784
\(747\) −2.01362e7 −1.32031
\(748\) 1.29701e7 0.847594
\(749\) 3.24502e6 0.211355
\(750\) 1.85400e7 1.20353
\(751\) −3.06925e6 −0.198579 −0.0992894 0.995059i \(-0.531657\pi\)
−0.0992894 + 0.995059i \(0.531657\pi\)
\(752\) 1.80969e6 0.116697
\(753\) 2.28075e7 1.46585
\(754\) 7.67726e6 0.491788
\(755\) 2.43307e7 1.55341
\(756\) 6.16807e6 0.392505
\(757\) 3.63729e6 0.230695 0.115348 0.993325i \(-0.463202\pi\)
0.115348 + 0.993325i \(0.463202\pi\)
\(758\) −1.00619e7 −0.636075
\(759\) −1.40207e7 −0.883414
\(760\) 4.54129e6 0.285197
\(761\) 2.89664e7 1.81315 0.906574 0.422046i \(-0.138688\pi\)
0.906574 + 0.422046i \(0.138688\pi\)
\(762\) 8.13421e6 0.507491
\(763\) 3.37244e7 2.09716
\(764\) −1.03135e7 −0.639252
\(765\) 3.05304e7 1.88616
\(766\) 2.18301e7 1.34426
\(767\) 1.11422e7 0.683883
\(768\) −1.59772e6 −0.0977454
\(769\) 2.17703e6 0.132754 0.0663770 0.997795i \(-0.478856\pi\)
0.0663770 + 0.997795i \(0.478856\pi\)
\(770\) −1.10174e7 −0.669658
\(771\) −6.14397e6 −0.372232
\(772\) 3.43942e6 0.207703
\(773\) 6.60733e6 0.397720 0.198860 0.980028i \(-0.436276\pi\)
0.198860 + 0.980028i \(0.436276\pi\)
\(774\) −7.74354e6 −0.464608
\(775\) 1.16237e7 0.695170
\(776\) 2.56490e6 0.152903
\(777\) −4.87104e6 −0.289447
\(778\) 4.90556e6 0.290562
\(779\) 6.00584e6 0.354593
\(780\) −6.54970e6 −0.385465
\(781\) 9.04564e6 0.530654
\(782\) 1.05922e7 0.619397
\(783\) −1.35798e7 −0.791569
\(784\) 1.15043e6 0.0668452
\(785\) −8.09320e6 −0.468755
\(786\) −3.38211e7 −1.95268
\(787\) −2.32249e7 −1.33665 −0.668323 0.743871i \(-0.732988\pi\)
−0.668323 + 0.743871i \(0.732988\pi\)
\(788\) 295304. 0.0169416
\(789\) −4.56848e7 −2.61264
\(790\) 1.92954e7 1.09998
\(791\) −3.03434e6 −0.172434
\(792\) 9.43489e6 0.534471
\(793\) 1.33555e7 0.754184
\(794\) −1.44918e7 −0.815775
\(795\) −9.33665e6 −0.523930
\(796\) −7.62770e6 −0.426689
\(797\) 5.59531e6 0.312017 0.156009 0.987756i \(-0.450137\pi\)
0.156009 + 0.987756i \(0.450137\pi\)
\(798\) 2.24532e7 1.24816
\(799\) 1.36572e7 0.756825
\(800\) −1.12844e6 −0.0623382
\(801\) 1.65605e7 0.911996
\(802\) −3.54940e6 −0.194858
\(803\) −1.12046e7 −0.613207
\(804\) 2.44249e7 1.33258
\(805\) −8.99754e6 −0.489366
\(806\) −1.57511e7 −0.854028
\(807\) −5.18087e6 −0.280039
\(808\) 5.86340e6 0.315952
\(809\) −2.24334e7 −1.20510 −0.602550 0.798081i \(-0.705849\pi\)
−0.602550 + 0.798081i \(0.705849\pi\)
\(810\) −3.77495e6 −0.202161
\(811\) 6.24934e6 0.333643 0.166821 0.985987i \(-0.446650\pi\)
0.166821 + 0.985987i \(0.446650\pi\)
\(812\) −1.20055e7 −0.638986
\(813\) −3.03814e7 −1.61206
\(814\) −2.29766e6 −0.121542
\(815\) −2.42085e7 −1.27665
\(816\) −1.20575e7 −0.633919
\(817\) −8.69252e6 −0.455607
\(818\) −2.10123e7 −1.09797
\(819\) −1.91433e7 −0.997257
\(820\) 2.73963e6 0.142284
\(821\) 3.21770e7 1.66605 0.833024 0.553237i \(-0.186608\pi\)
0.833024 + 0.553237i \(0.186608\pi\)
\(822\) 1.65541e7 0.854526
\(823\) −1.60157e7 −0.824227 −0.412114 0.911132i \(-0.635209\pi\)
−0.412114 + 0.911132i \(0.635209\pi\)
\(824\) −2.19832e6 −0.112791
\(825\) 1.12725e7 0.576616
\(826\) −1.74239e7 −0.888577
\(827\) −1.11834e7 −0.568602 −0.284301 0.958735i \(-0.591762\pi\)
−0.284301 + 0.958735i \(0.591762\pi\)
\(828\) 7.70515e6 0.390576
\(829\) −1.20543e7 −0.609193 −0.304597 0.952481i \(-0.598522\pi\)
−0.304597 + 0.952481i \(0.598522\pi\)
\(830\) −1.03110e7 −0.519524
\(831\) 5.85898e7 2.94320
\(832\) 1.52913e6 0.0765835
\(833\) 8.68200e6 0.433518
\(834\) −1.96512e7 −0.978306
\(835\) −9.49883e6 −0.471470
\(836\) 1.05912e7 0.524116
\(837\) 2.78610e7 1.37462
\(838\) −2.17734e7 −1.07107
\(839\) 1.09546e7 0.537270 0.268635 0.963242i \(-0.413427\pi\)
0.268635 + 0.963242i \(0.413427\pi\)
\(840\) 1.02423e7 0.500839
\(841\) 5.92055e6 0.288650
\(842\) −8.69177e6 −0.422501
\(843\) 2.47497e7 1.19950
\(844\) 5.14246e6 0.248493
\(845\) −1.04314e7 −0.502576
\(846\) 9.93477e6 0.477235
\(847\) −2.18961e6 −0.104871
\(848\) 2.17978e6 0.104093
\(849\) −1.09489e7 −0.521314
\(850\) −8.51606e6 −0.404288
\(851\) −1.87642e6 −0.0888191
\(852\) −8.40923e6 −0.396878
\(853\) 2.61056e7 1.22846 0.614231 0.789126i \(-0.289466\pi\)
0.614231 + 0.789126i \(0.289466\pi\)
\(854\) −2.08850e7 −0.979919
\(855\) 2.49307e7 1.16632
\(856\) −1.42298e6 −0.0663765
\(857\) 3.72343e7 1.73177 0.865887 0.500239i \(-0.166755\pi\)
0.865887 + 0.500239i \(0.166755\pi\)
\(858\) −1.52752e7 −0.708382
\(859\) 5.97467e6 0.276268 0.138134 0.990414i \(-0.455889\pi\)
0.138134 + 0.990414i \(0.455889\pi\)
\(860\) −3.96518e6 −0.182817
\(861\) 1.35454e7 0.622706
\(862\) −7.95877e6 −0.364819
\(863\) −2.75432e7 −1.25889 −0.629443 0.777046i \(-0.716717\pi\)
−0.629443 + 0.777046i \(0.716717\pi\)
\(864\) −2.70477e6 −0.123267
\(865\) 1.11071e7 0.504730
\(866\) −9.70089e6 −0.439559
\(867\) −5.63802e7 −2.54729
\(868\) 2.46312e7 1.10965
\(869\) 4.50005e7 2.02147
\(870\) −2.25497e7 −1.01005
\(871\) −2.33763e7 −1.04407
\(872\) −1.47885e7 −0.658619
\(873\) 1.40807e7 0.625301
\(874\) 8.64943e6 0.383009
\(875\) 2.77478e7 1.22520
\(876\) 1.04163e7 0.458620
\(877\) 2.75263e7 1.20851 0.604254 0.796792i \(-0.293471\pi\)
0.604254 + 0.796792i \(0.293471\pi\)
\(878\) −162987. −0.00713537
\(879\) −3.66602e7 −1.60038
\(880\) 4.83127e6 0.210307
\(881\) 8.47736e6 0.367977 0.183989 0.982928i \(-0.441099\pi\)
0.183989 + 0.982928i \(0.441099\pi\)
\(882\) 6.31561e6 0.273365
\(883\) −1.14175e7 −0.492799 −0.246399 0.969168i \(-0.579247\pi\)
−0.246399 + 0.969168i \(0.579247\pi\)
\(884\) 1.15399e7 0.496675
\(885\) −3.27269e7 −1.40458
\(886\) 1.43897e7 0.615840
\(887\) 1.59521e7 0.680782 0.340391 0.940284i \(-0.389441\pi\)
0.340391 + 0.940284i \(0.389441\pi\)
\(888\) 2.13601e6 0.0909014
\(889\) 1.21740e7 0.516631
\(890\) 8.48005e6 0.358859
\(891\) −8.80389e6 −0.371519
\(892\) −5.11919e6 −0.215421
\(893\) 1.11523e7 0.467989
\(894\) 9.00719e6 0.376917
\(895\) −1.38164e7 −0.576551
\(896\) −2.39121e6 −0.0995058
\(897\) −1.24747e7 −0.517665
\(898\) −5.35183e6 −0.221468
\(899\) −5.42286e7 −2.23784
\(900\) −6.19489e6 −0.254934
\(901\) 1.64502e7 0.675088
\(902\) 6.38933e6 0.261480
\(903\) −1.96048e7 −0.800099
\(904\) 1.33059e6 0.0541533
\(905\) 2.16038e7 0.876818
\(906\) −5.27515e7 −2.13508
\(907\) 4.89568e7 1.97604 0.988018 0.154336i \(-0.0493238\pi\)
0.988018 + 0.154336i \(0.0493238\pi\)
\(908\) −2.07735e7 −0.836170
\(909\) 3.21887e7 1.29209
\(910\) −9.80259e6 −0.392408
\(911\) −2.39778e7 −0.957225 −0.478613 0.878026i \(-0.658860\pi\)
−0.478613 + 0.878026i \(0.658860\pi\)
\(912\) −9.84601e6 −0.391988
\(913\) −2.40472e7 −0.954747
\(914\) −3.12234e7 −1.23627
\(915\) −3.92278e7 −1.54897
\(916\) −1.73074e7 −0.681541
\(917\) −5.06182e7 −1.98785
\(918\) −2.04122e7 −0.799435
\(919\) −3.64828e7 −1.42495 −0.712474 0.701699i \(-0.752425\pi\)
−0.712474 + 0.701699i \(0.752425\pi\)
\(920\) 3.94553e6 0.153686
\(921\) 3.86476e7 1.50132
\(922\) −3.11414e7 −1.20645
\(923\) 8.04823e6 0.310954
\(924\) 2.38869e7 0.920409
\(925\) 1.50863e6 0.0579734
\(926\) −1.55861e7 −0.597325
\(927\) −1.20683e7 −0.461260
\(928\) 5.26457e6 0.200675
\(929\) 6.67014e6 0.253569 0.126784 0.991930i \(-0.459534\pi\)
0.126784 + 0.991930i \(0.459534\pi\)
\(930\) 4.62641e7 1.75403
\(931\) 7.08959e6 0.268069
\(932\) −2.19720e7 −0.828572
\(933\) −5.20307e7 −1.95684
\(934\) 1.15891e7 0.434694
\(935\) 3.64603e7 1.36393
\(936\) 8.39456e6 0.313190
\(937\) −3.29325e7 −1.22539 −0.612697 0.790318i \(-0.709915\pi\)
−0.612697 + 0.790318i \(0.709915\pi\)
\(938\) 3.65554e7 1.35658
\(939\) −1.23847e7 −0.458376
\(940\) 5.08724e6 0.187786
\(941\) 4.00487e7 1.47440 0.737198 0.675677i \(-0.236148\pi\)
0.737198 + 0.675677i \(0.236148\pi\)
\(942\) 1.75469e7 0.644279
\(943\) 5.21795e6 0.191082
\(944\) 7.64058e6 0.279059
\(945\) 1.73392e7 0.631609
\(946\) −9.24756e6 −0.335969
\(947\) 6.91973e6 0.250735 0.125367 0.992110i \(-0.459989\pi\)
0.125367 + 0.992110i \(0.459989\pi\)
\(948\) −4.18344e7 −1.51187
\(949\) −9.96914e6 −0.359329
\(950\) −6.95409e6 −0.249995
\(951\) 6.52358e7 2.33902
\(952\) −1.80459e7 −0.645336
\(953\) 2.39076e7 0.852715 0.426357 0.904555i \(-0.359797\pi\)
0.426357 + 0.904555i \(0.359797\pi\)
\(954\) 1.19665e7 0.425693
\(955\) −2.89924e7 −1.02867
\(956\) −2.10245e7 −0.744012
\(957\) −5.25902e7 −1.85620
\(958\) 2.36516e6 0.0832619
\(959\) 2.47756e7 0.869917
\(960\) −4.49136e6 −0.157290
\(961\) 8.26291e7 2.88619
\(962\) −2.04431e6 −0.0712212
\(963\) −7.81184e6 −0.271449
\(964\) 2.25519e7 0.781612
\(965\) 9.66861e6 0.334230
\(966\) 1.95076e7 0.672608
\(967\) −1.85809e7 −0.638998 −0.319499 0.947587i \(-0.603515\pi\)
−0.319499 + 0.947587i \(0.603515\pi\)
\(968\) 960168. 0.0329351
\(969\) −7.43053e7 −2.54220
\(970\) 7.21023e6 0.246048
\(971\) −4.82454e7 −1.64213 −0.821065 0.570834i \(-0.806620\pi\)
−0.821065 + 0.570834i \(0.806620\pi\)
\(972\) 1.84542e7 0.626511
\(973\) −2.94109e7 −0.995926
\(974\) 2.08829e6 0.0705330
\(975\) 1.00296e7 0.337887
\(976\) 9.15833e6 0.307746
\(977\) −1.99798e7 −0.669662 −0.334831 0.942278i \(-0.608679\pi\)
−0.334831 + 0.942278i \(0.608679\pi\)
\(978\) 5.24866e7 1.75469
\(979\) 1.97771e7 0.659486
\(980\) 3.23399e6 0.107566
\(981\) −8.11857e7 −2.69344
\(982\) 7.88439e6 0.260909
\(983\) 2.38930e7 0.788653 0.394327 0.918970i \(-0.370978\pi\)
0.394327 + 0.918970i \(0.370978\pi\)
\(984\) −5.93981e6 −0.195562
\(985\) 830134. 0.0272620
\(986\) 3.97303e7 1.30146
\(987\) 2.51525e7 0.821843
\(988\) 9.42333e6 0.307123
\(989\) −7.55217e6 −0.245517
\(990\) 2.65226e7 0.860058
\(991\) −1.94203e6 −0.0628163 −0.0314082 0.999507i \(-0.509999\pi\)
−0.0314082 + 0.999507i \(0.509999\pi\)
\(992\) −1.08011e7 −0.348487
\(993\) 7.87871e7 2.53561
\(994\) −1.25856e7 −0.404026
\(995\) −2.14423e7 −0.686617
\(996\) 2.23554e7 0.714058
\(997\) −1.18559e7 −0.377744 −0.188872 0.982002i \(-0.560483\pi\)
−0.188872 + 0.982002i \(0.560483\pi\)
\(998\) 1.52751e7 0.485466
\(999\) 3.61605e6 0.114636
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 74.6.a.e.1.1 5
3.2 odd 2 666.6.a.l.1.3 5
4.3 odd 2 592.6.a.e.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.6.a.e.1.1 5 1.1 even 1 trivial
592.6.a.e.1.5 5 4.3 odd 2
666.6.a.l.1.3 5 3.2 odd 2