Properties

Label 294.6.a.r.1.2
Level $294$
Weight $6$
Character 294.1
Self dual yes
Analytic conductor $47.153$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, 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("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.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: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9601}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-48.4923\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +22.4923 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +22.4923 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +89.9694 q^{10} +340.462 q^{11} -144.000 q^{12} -728.416 q^{13} -202.431 q^{15} +256.000 q^{16} +809.847 q^{17} +324.000 q^{18} +1026.20 q^{19} +359.878 q^{20} +1361.85 q^{22} +1422.03 q^{23} -576.000 q^{24} -2619.09 q^{25} -2913.66 q^{26} -729.000 q^{27} +5218.03 q^{29} -809.724 q^{30} -7037.74 q^{31} +1024.00 q^{32} -3064.16 q^{33} +3239.39 q^{34} +1296.00 q^{36} +12792.1 q^{37} +4104.81 q^{38} +6555.74 q^{39} +1439.51 q^{40} +1173.51 q^{41} +3664.17 q^{43} +5447.39 q^{44} +1821.88 q^{45} +5688.12 q^{46} +9313.19 q^{47} -2304.00 q^{48} -10476.4 q^{50} -7288.62 q^{51} -11654.7 q^{52} +35642.8 q^{53} -2916.00 q^{54} +7657.78 q^{55} -9235.81 q^{57} +20872.1 q^{58} -30376.2 q^{59} -3238.90 q^{60} +32186.2 q^{61} -28151.0 q^{62} +4096.00 q^{64} -16383.8 q^{65} -12256.6 q^{66} +21351.1 q^{67} +12957.6 q^{68} -12798.3 q^{69} +61153.7 q^{71} +5184.00 q^{72} +41267.8 q^{73} +51168.4 q^{74} +23571.8 q^{75} +16419.2 q^{76} +26223.0 q^{78} -35000.5 q^{79} +5758.04 q^{80} +6561.00 q^{81} +4694.02 q^{82} +86193.0 q^{83} +18215.4 q^{85} +14656.7 q^{86} -46962.3 q^{87} +21789.6 q^{88} +77992.6 q^{89} +7287.52 q^{90} +22752.5 q^{92} +63339.7 q^{93} +37252.8 q^{94} +23081.7 q^{95} -9216.00 q^{96} -161765. q^{97} +27577.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} - 53 q^{5} - 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} - 53 q^{5} - 72 q^{6} + 128 q^{8} + 162 q^{9} - 212 q^{10} + 191 q^{11} - 288 q^{12} - 379 q^{13} + 477 q^{15} + 512 q^{16} - 340 q^{17} + 648 q^{18} - 1769 q^{19} - 848 q^{20} + 764 q^{22} + 3236 q^{23} - 1152 q^{24} - 45 q^{25} - 1516 q^{26} - 1458 q^{27} + 4459 q^{29} + 1908 q^{30} + 1994 q^{31} + 2048 q^{32} - 1719 q^{33} - 1360 q^{34} + 2592 q^{36} + 20587 q^{37} - 7076 q^{38} + 3411 q^{39} - 3392 q^{40} + 8814 q^{41} + 15853 q^{43} + 3056 q^{44} - 4293 q^{45} + 12944 q^{46} + 33912 q^{47} - 4608 q^{48} - 180 q^{50} + 3060 q^{51} - 6064 q^{52} + 49239 q^{53} - 5832 q^{54} + 18941 q^{55} + 15921 q^{57} + 17836 q^{58} - 56735 q^{59} + 7632 q^{60} + 67508 q^{61} + 7976 q^{62} + 8192 q^{64} - 42762 q^{65} - 6876 q^{66} + 75723 q^{67} - 5440 q^{68} - 29124 q^{69} - 8992 q^{71} + 10368 q^{72} - 3201 q^{73} + 82348 q^{74} + 405 q^{75} - 28304 q^{76} + 13644 q^{78} + 26612 q^{79} - 13568 q^{80} + 13122 q^{81} + 35256 q^{82} - 949 q^{83} + 105020 q^{85} + 63412 q^{86} - 40131 q^{87} + 12224 q^{88} + 176562 q^{89} - 17172 q^{90} + 51776 q^{92} - 17946 q^{93} + 135648 q^{94} + 234098 q^{95} - 18432 q^{96} - 129423 q^{97} + 15471 q^{99}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) 22.4923 0.402355 0.201178 0.979555i \(-0.435523\pi\)
0.201178 + 0.979555i \(0.435523\pi\)
\(6\) −36.0000 −0.408248
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 89.9694 0.284508
\(11\) 340.462 0.848373 0.424186 0.905575i \(-0.360560\pi\)
0.424186 + 0.905575i \(0.360560\pi\)
\(12\) −144.000 −0.288675
\(13\) −728.416 −1.19542 −0.597711 0.801712i \(-0.703923\pi\)
−0.597711 + 0.801712i \(0.703923\pi\)
\(14\) 0 0
\(15\) −202.431 −0.232300
\(16\) 256.000 0.250000
\(17\) 809.847 0.679643 0.339821 0.940490i \(-0.389633\pi\)
0.339821 + 0.940490i \(0.389633\pi\)
\(18\) 324.000 0.235702
\(19\) 1026.20 0.652152 0.326076 0.945344i \(-0.394274\pi\)
0.326076 + 0.945344i \(0.394274\pi\)
\(20\) 359.878 0.201178
\(21\) 0 0
\(22\) 1361.85 0.599890
\(23\) 1422.03 0.560518 0.280259 0.959924i \(-0.409580\pi\)
0.280259 + 0.959924i \(0.409580\pi\)
\(24\) −576.000 −0.204124
\(25\) −2619.09 −0.838110
\(26\) −2913.66 −0.845290
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 5218.03 1.15216 0.576079 0.817394i \(-0.304582\pi\)
0.576079 + 0.817394i \(0.304582\pi\)
\(30\) −809.724 −0.164261
\(31\) −7037.74 −1.31531 −0.657657 0.753318i \(-0.728452\pi\)
−0.657657 + 0.753318i \(0.728452\pi\)
\(32\) 1024.00 0.176777
\(33\) −3064.16 −0.489808
\(34\) 3239.39 0.480580
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 12792.1 1.53616 0.768082 0.640351i \(-0.221211\pi\)
0.768082 + 0.640351i \(0.221211\pi\)
\(38\) 4104.81 0.461141
\(39\) 6555.74 0.690177
\(40\) 1439.51 0.142254
\(41\) 1173.51 0.109025 0.0545124 0.998513i \(-0.482640\pi\)
0.0545124 + 0.998513i \(0.482640\pi\)
\(42\) 0 0
\(43\) 3664.17 0.302207 0.151103 0.988518i \(-0.451717\pi\)
0.151103 + 0.988518i \(0.451717\pi\)
\(44\) 5447.39 0.424186
\(45\) 1821.88 0.134118
\(46\) 5688.12 0.396346
\(47\) 9313.19 0.614970 0.307485 0.951553i \(-0.400513\pi\)
0.307485 + 0.951553i \(0.400513\pi\)
\(48\) −2304.00 −0.144338
\(49\) 0 0
\(50\) −10476.4 −0.592633
\(51\) −7288.62 −0.392392
\(52\) −11654.7 −0.597711
\(53\) 35642.8 1.74294 0.871469 0.490451i \(-0.163168\pi\)
0.871469 + 0.490451i \(0.163168\pi\)
\(54\) −2916.00 −0.136083
\(55\) 7657.78 0.341347
\(56\) 0 0
\(57\) −9235.81 −0.376520
\(58\) 20872.1 0.814698
\(59\) −30376.2 −1.13607 −0.568033 0.823006i \(-0.692295\pi\)
−0.568033 + 0.823006i \(0.692295\pi\)
\(60\) −3238.90 −0.116150
\(61\) 32186.2 1.10751 0.553753 0.832681i \(-0.313195\pi\)
0.553753 + 0.832681i \(0.313195\pi\)
\(62\) −28151.0 −0.930067
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −16383.8 −0.480984
\(66\) −12256.6 −0.346347
\(67\) 21351.1 0.581076 0.290538 0.956863i \(-0.406166\pi\)
0.290538 + 0.956863i \(0.406166\pi\)
\(68\) 12957.6 0.339821
\(69\) −12798.3 −0.323615
\(70\) 0 0
\(71\) 61153.7 1.43972 0.719859 0.694121i \(-0.244207\pi\)
0.719859 + 0.694121i \(0.244207\pi\)
\(72\) 5184.00 0.117851
\(73\) 41267.8 0.906367 0.453184 0.891417i \(-0.350288\pi\)
0.453184 + 0.891417i \(0.350288\pi\)
\(74\) 51168.4 1.08623
\(75\) 23571.8 0.483883
\(76\) 16419.2 0.326076
\(77\) 0 0
\(78\) 26223.0 0.488029
\(79\) −35000.5 −0.630966 −0.315483 0.948931i \(-0.602167\pi\)
−0.315483 + 0.948931i \(0.602167\pi\)
\(80\) 5758.04 0.100589
\(81\) 6561.00 0.111111
\(82\) 4694.02 0.0770922
\(83\) 86193.0 1.37334 0.686668 0.726972i \(-0.259073\pi\)
0.686668 + 0.726972i \(0.259073\pi\)
\(84\) 0 0
\(85\) 18215.4 0.273458
\(86\) 14656.7 0.213692
\(87\) −46962.3 −0.665198
\(88\) 21789.6 0.299945
\(89\) 77992.6 1.04371 0.521853 0.853035i \(-0.325241\pi\)
0.521853 + 0.853035i \(0.325241\pi\)
\(90\) 7287.52 0.0948361
\(91\) 0 0
\(92\) 22752.5 0.280259
\(93\) 63339.7 0.759397
\(94\) 37252.8 0.434850
\(95\) 23081.7 0.262397
\(96\) −9216.00 −0.102062
\(97\) −161765. −1.74565 −0.872823 0.488037i \(-0.837713\pi\)
−0.872823 + 0.488037i \(0.837713\pi\)
\(98\) 0 0
\(99\) 27577.4 0.282791
\(100\) −41905.5 −0.419055
\(101\) −65854.4 −0.642364 −0.321182 0.947017i \(-0.604080\pi\)
−0.321182 + 0.947017i \(0.604080\pi\)
\(102\) −29154.5 −0.277463
\(103\) 130198. 1.20924 0.604619 0.796515i \(-0.293326\pi\)
0.604619 + 0.796515i \(0.293326\pi\)
\(104\) −46618.6 −0.422645
\(105\) 0 0
\(106\) 142571. 1.23244
\(107\) 2590.95 0.0218776 0.0109388 0.999940i \(-0.496518\pi\)
0.0109388 + 0.999940i \(0.496518\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 110654. 0.892072 0.446036 0.895015i \(-0.352835\pi\)
0.446036 + 0.895015i \(0.352835\pi\)
\(110\) 30631.1 0.241369
\(111\) −115129. −0.886905
\(112\) 0 0
\(113\) −193910. −1.42858 −0.714291 0.699849i \(-0.753251\pi\)
−0.714291 + 0.699849i \(0.753251\pi\)
\(114\) −36943.3 −0.266240
\(115\) 31984.8 0.225527
\(116\) 83488.5 0.576079
\(117\) −59001.7 −0.398474
\(118\) −121505. −0.803319
\(119\) 0 0
\(120\) −12955.6 −0.0821304
\(121\) −45136.8 −0.280264
\(122\) 128745. 0.783124
\(123\) −10561.5 −0.0629455
\(124\) −112604. −0.657657
\(125\) −129198. −0.739573
\(126\) 0 0
\(127\) 27429.2 0.150905 0.0754526 0.997149i \(-0.475960\pi\)
0.0754526 + 0.997149i \(0.475960\pi\)
\(128\) 16384.0 0.0883883
\(129\) −32977.5 −0.174479
\(130\) −65535.1 −0.340107
\(131\) 150376. 0.765597 0.382798 0.923832i \(-0.374960\pi\)
0.382798 + 0.923832i \(0.374960\pi\)
\(132\) −49026.5 −0.244904
\(133\) 0 0
\(134\) 85404.3 0.410883
\(135\) −16396.9 −0.0774333
\(136\) 51830.2 0.240290
\(137\) 96102.0 0.437453 0.218726 0.975786i \(-0.429810\pi\)
0.218726 + 0.975786i \(0.429810\pi\)
\(138\) −51193.1 −0.228830
\(139\) 100854. 0.442749 0.221375 0.975189i \(-0.428946\pi\)
0.221375 + 0.975189i \(0.428946\pi\)
\(140\) 0 0
\(141\) −83818.7 −0.355053
\(142\) 244615. 1.01803
\(143\) −247998. −1.01416
\(144\) 20736.0 0.0833333
\(145\) 117366. 0.463577
\(146\) 165071. 0.640898
\(147\) 0 0
\(148\) 204674. 0.768082
\(149\) −360944. −1.33191 −0.665954 0.745993i \(-0.731975\pi\)
−0.665954 + 0.745993i \(0.731975\pi\)
\(150\) 94287.4 0.342157
\(151\) −434056. −1.54918 −0.774592 0.632461i \(-0.782045\pi\)
−0.774592 + 0.632461i \(0.782045\pi\)
\(152\) 65676.9 0.230570
\(153\) 65597.6 0.226548
\(154\) 0 0
\(155\) −158295. −0.529223
\(156\) 104892. 0.345088
\(157\) −511596. −1.65645 −0.828225 0.560396i \(-0.810649\pi\)
−0.828225 + 0.560396i \(0.810649\pi\)
\(158\) −140002. −0.446160
\(159\) −320785. −1.00629
\(160\) 23032.2 0.0711270
\(161\) 0 0
\(162\) 26244.0 0.0785674
\(163\) −251269. −0.740748 −0.370374 0.928883i \(-0.620770\pi\)
−0.370374 + 0.928883i \(0.620770\pi\)
\(164\) 18776.1 0.0545124
\(165\) −68920.0 −0.197077
\(166\) 344772. 0.971095
\(167\) 419277. 1.16335 0.581674 0.813422i \(-0.302398\pi\)
0.581674 + 0.813422i \(0.302398\pi\)
\(168\) 0 0
\(169\) 159297. 0.429032
\(170\) 72861.4 0.193364
\(171\) 83122.3 0.217384
\(172\) 58626.7 0.151103
\(173\) −461376. −1.17203 −0.586017 0.810299i \(-0.699305\pi\)
−0.586017 + 0.810299i \(0.699305\pi\)
\(174\) −187849. −0.470366
\(175\) 0 0
\(176\) 87158.2 0.212093
\(177\) 273386. 0.655907
\(178\) 311970. 0.738012
\(179\) 741706. 1.73021 0.865105 0.501590i \(-0.167251\pi\)
0.865105 + 0.501590i \(0.167251\pi\)
\(180\) 29150.1 0.0670592
\(181\) 301371. 0.683762 0.341881 0.939743i \(-0.388936\pi\)
0.341881 + 0.939743i \(0.388936\pi\)
\(182\) 0 0
\(183\) −289676. −0.639418
\(184\) 91010.0 0.198173
\(185\) 287725. 0.618084
\(186\) 253359. 0.536974
\(187\) 275722. 0.576590
\(188\) 149011. 0.307485
\(189\) 0 0
\(190\) 92326.7 0.185542
\(191\) 263183. 0.522004 0.261002 0.965338i \(-0.415947\pi\)
0.261002 + 0.965338i \(0.415947\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −831127. −1.60611 −0.803053 0.595908i \(-0.796792\pi\)
−0.803053 + 0.595908i \(0.796792\pi\)
\(194\) −647061. −1.23436
\(195\) 147454. 0.277696
\(196\) 0 0
\(197\) −1.05421e6 −1.93536 −0.967681 0.252178i \(-0.918853\pi\)
−0.967681 + 0.252178i \(0.918853\pi\)
\(198\) 110310. 0.199963
\(199\) 698568. 1.25048 0.625239 0.780434i \(-0.285002\pi\)
0.625239 + 0.780434i \(0.285002\pi\)
\(200\) −167622. −0.296317
\(201\) −192160. −0.335484
\(202\) −263418. −0.454220
\(203\) 0 0
\(204\) −116618. −0.196196
\(205\) 26394.9 0.0438667
\(206\) 520792. 0.855060
\(207\) 115184. 0.186839
\(208\) −186474. −0.298855
\(209\) 349382. 0.553268
\(210\) 0 0
\(211\) −99693.6 −0.154156 −0.0770781 0.997025i \(-0.524559\pi\)
−0.0770781 + 0.997025i \(0.524559\pi\)
\(212\) 570284. 0.871469
\(213\) −550384. −0.831221
\(214\) 10363.8 0.0154698
\(215\) 82415.7 0.121594
\(216\) −46656.0 −0.0680414
\(217\) 0 0
\(218\) 442615. 0.630790
\(219\) −371410. −0.523291
\(220\) 122525. 0.170674
\(221\) −589905. −0.812459
\(222\) −460516. −0.627137
\(223\) 526194. 0.708572 0.354286 0.935137i \(-0.384724\pi\)
0.354286 + 0.935137i \(0.384724\pi\)
\(224\) 0 0
\(225\) −212147. −0.279370
\(226\) −775642. −1.01016
\(227\) −981605. −1.26436 −0.632182 0.774820i \(-0.717840\pi\)
−0.632182 + 0.774820i \(0.717840\pi\)
\(228\) −147773. −0.188260
\(229\) −85081.0 −0.107212 −0.0536061 0.998562i \(-0.517072\pi\)
−0.0536061 + 0.998562i \(0.517072\pi\)
\(230\) 127939. 0.159472
\(231\) 0 0
\(232\) 333954. 0.407349
\(233\) 113123. 0.136509 0.0682544 0.997668i \(-0.478257\pi\)
0.0682544 + 0.997668i \(0.478257\pi\)
\(234\) −236007. −0.281763
\(235\) 209476. 0.247436
\(236\) −486019. −0.568033
\(237\) 315004. 0.364288
\(238\) 0 0
\(239\) −895100. −1.01362 −0.506812 0.862057i \(-0.669176\pi\)
−0.506812 + 0.862057i \(0.669176\pi\)
\(240\) −51822.4 −0.0580750
\(241\) 527715. 0.585270 0.292635 0.956224i \(-0.405468\pi\)
0.292635 + 0.956224i \(0.405468\pi\)
\(242\) −180547. −0.198177
\(243\) −59049.0 −0.0641500
\(244\) 514980. 0.553753
\(245\) 0 0
\(246\) −42246.2 −0.0445092
\(247\) −747501. −0.779596
\(248\) −450416. −0.465034
\(249\) −775737. −0.792895
\(250\) −516793. −0.522957
\(251\) −1.19293e6 −1.19517 −0.597584 0.801806i \(-0.703873\pi\)
−0.597584 + 0.801806i \(0.703873\pi\)
\(252\) 0 0
\(253\) 484147. 0.475528
\(254\) 109717. 0.106706
\(255\) −163938. −0.157881
\(256\) 65536.0 0.0625000
\(257\) −1.19492e6 −1.12851 −0.564257 0.825599i \(-0.690837\pi\)
−0.564257 + 0.825599i \(0.690837\pi\)
\(258\) −131910. −0.123375
\(259\) 0 0
\(260\) −262140. −0.240492
\(261\) 422661. 0.384052
\(262\) 601504. 0.541359
\(263\) −2.12230e6 −1.89199 −0.945993 0.324186i \(-0.894910\pi\)
−0.945993 + 0.324186i \(0.894910\pi\)
\(264\) −196106. −0.173173
\(265\) 801690. 0.701280
\(266\) 0 0
\(267\) −701933. −0.602584
\(268\) 341617. 0.290538
\(269\) 154977. 0.130583 0.0652913 0.997866i \(-0.479202\pi\)
0.0652913 + 0.997866i \(0.479202\pi\)
\(270\) −65587.7 −0.0547536
\(271\) 1.95888e6 1.62026 0.810129 0.586252i \(-0.199397\pi\)
0.810129 + 0.586252i \(0.199397\pi\)
\(272\) 207321. 0.169911
\(273\) 0 0
\(274\) 384408. 0.309326
\(275\) −891701. −0.711030
\(276\) −204772. −0.161808
\(277\) −1.74953e6 −1.37000 −0.685001 0.728542i \(-0.740198\pi\)
−0.685001 + 0.728542i \(0.740198\pi\)
\(278\) 403418. 0.313071
\(279\) −570057. −0.438438
\(280\) 0 0
\(281\) 1.40665e6 1.06272 0.531361 0.847145i \(-0.321681\pi\)
0.531361 + 0.847145i \(0.321681\pi\)
\(282\) −335275. −0.251060
\(283\) 1.95707e6 1.45258 0.726291 0.687388i \(-0.241243\pi\)
0.726291 + 0.687388i \(0.241243\pi\)
\(284\) 978460. 0.719859
\(285\) −207735. −0.151495
\(286\) −991991. −0.717121
\(287\) 0 0
\(288\) 82944.0 0.0589256
\(289\) −764005. −0.538086
\(290\) 469463. 0.327798
\(291\) 1.45589e6 1.00785
\(292\) 660285. 0.453184
\(293\) −1.26998e6 −0.864229 −0.432114 0.901819i \(-0.642232\pi\)
−0.432114 + 0.901819i \(0.642232\pi\)
\(294\) 0 0
\(295\) −683232. −0.457102
\(296\) 818695. 0.543116
\(297\) −248197. −0.163269
\(298\) −1.44378e6 −0.941802
\(299\) −1.03583e6 −0.670055
\(300\) 377150. 0.241942
\(301\) 0 0
\(302\) −1.73622e6 −1.09544
\(303\) 592690. 0.370869
\(304\) 262708. 0.163038
\(305\) 723944. 0.445611
\(306\) 262390. 0.160193
\(307\) −41854.4 −0.0253451 −0.0126726 0.999920i \(-0.504034\pi\)
−0.0126726 + 0.999920i \(0.504034\pi\)
\(308\) 0 0
\(309\) −1.17178e6 −0.698154
\(310\) −633182. −0.374217
\(311\) −2.29189e6 −1.34367 −0.671837 0.740699i \(-0.734494\pi\)
−0.671837 + 0.740699i \(0.734494\pi\)
\(312\) 419568. 0.244014
\(313\) 2.27150e6 1.31054 0.655271 0.755394i \(-0.272554\pi\)
0.655271 + 0.755394i \(0.272554\pi\)
\(314\) −2.04639e6 −1.17129
\(315\) 0 0
\(316\) −560007. −0.315483
\(317\) −1.98474e6 −1.10932 −0.554659 0.832078i \(-0.687151\pi\)
−0.554659 + 0.832078i \(0.687151\pi\)
\(318\) −1.28314e6 −0.711551
\(319\) 1.77654e6 0.977459
\(320\) 92128.7 0.0502944
\(321\) −23318.6 −0.0126310
\(322\) 0 0
\(323\) 831066. 0.443230
\(324\) 104976. 0.0555556
\(325\) 1.90779e6 1.00189
\(326\) −1.00508e6 −0.523788
\(327\) −995884. −0.515038
\(328\) 75104.3 0.0385461
\(329\) 0 0
\(330\) −275680. −0.139354
\(331\) 638461. 0.320305 0.160153 0.987092i \(-0.448801\pi\)
0.160153 + 0.987092i \(0.448801\pi\)
\(332\) 1.37909e6 0.686668
\(333\) 1.03616e6 0.512055
\(334\) 1.67711e6 0.822611
\(335\) 480236. 0.233799
\(336\) 0 0
\(337\) 2.72026e6 1.30478 0.652388 0.757886i \(-0.273767\pi\)
0.652388 + 0.757886i \(0.273767\pi\)
\(338\) 637186. 0.303371
\(339\) 1.74519e6 0.824792
\(340\) 291446. 0.136729
\(341\) −2.39608e6 −1.11588
\(342\) 332489. 0.153714
\(343\) 0 0
\(344\) 234507. 0.106846
\(345\) −287863. −0.130208
\(346\) −1.84551e6 −0.828753
\(347\) 2.62076e6 1.16843 0.584217 0.811597i \(-0.301402\pi\)
0.584217 + 0.811597i \(0.301402\pi\)
\(348\) −751397. −0.332599
\(349\) 575592. 0.252960 0.126480 0.991969i \(-0.459632\pi\)
0.126480 + 0.991969i \(0.459632\pi\)
\(350\) 0 0
\(351\) 531015. 0.230059
\(352\) 348633. 0.149972
\(353\) −2.58972e6 −1.10615 −0.553077 0.833130i \(-0.686546\pi\)
−0.553077 + 0.833130i \(0.686546\pi\)
\(354\) 1.09354e6 0.463797
\(355\) 1.37549e6 0.579278
\(356\) 1.24788e6 0.521853
\(357\) 0 0
\(358\) 2.96682e6 1.22344
\(359\) 2.86779e6 1.17439 0.587193 0.809447i \(-0.300233\pi\)
0.587193 + 0.809447i \(0.300233\pi\)
\(360\) 116600. 0.0474180
\(361\) −1.42301e6 −0.574698
\(362\) 1.20548e6 0.483493
\(363\) 406231. 0.161811
\(364\) 0 0
\(365\) 928210. 0.364682
\(366\) −1.15870e6 −0.452137
\(367\) −1.95451e6 −0.757482 −0.378741 0.925503i \(-0.623643\pi\)
−0.378741 + 0.925503i \(0.623643\pi\)
\(368\) 364040. 0.140129
\(369\) 95053.9 0.0363416
\(370\) 1.15090e6 0.437052
\(371\) 0 0
\(372\) 1.01344e6 0.379698
\(373\) 2.21110e6 0.822878 0.411439 0.911437i \(-0.365026\pi\)
0.411439 + 0.911437i \(0.365026\pi\)
\(374\) 1.10289e6 0.407711
\(375\) 1.16278e6 0.426993
\(376\) 596044. 0.217425
\(377\) −3.80090e6 −1.37731
\(378\) 0 0
\(379\) 3.81232e6 1.36330 0.681649 0.731679i \(-0.261263\pi\)
0.681649 + 0.731679i \(0.261263\pi\)
\(380\) 369307. 0.131198
\(381\) −246863. −0.0871252
\(382\) 1.05273e6 0.369112
\(383\) 3.80606e6 1.32580 0.662901 0.748707i \(-0.269325\pi\)
0.662901 + 0.748707i \(0.269325\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −3.32451e6 −1.13569
\(387\) 296797. 0.100736
\(388\) −2.58825e6 −0.872823
\(389\) 3.07524e6 1.03040 0.515200 0.857070i \(-0.327718\pi\)
0.515200 + 0.857070i \(0.327718\pi\)
\(390\) 589816. 0.196361
\(391\) 1.15163e6 0.380952
\(392\) 0 0
\(393\) −1.35338e6 −0.442017
\(394\) −4.21685e6 −1.36851
\(395\) −787242. −0.253873
\(396\) 441238. 0.141395
\(397\) 2.19133e6 0.697800 0.348900 0.937160i \(-0.386555\pi\)
0.348900 + 0.937160i \(0.386555\pi\)
\(398\) 2.79427e6 0.884221
\(399\) 0 0
\(400\) −670488. −0.209528
\(401\) 1.77668e6 0.551757 0.275879 0.961192i \(-0.411031\pi\)
0.275879 + 0.961192i \(0.411031\pi\)
\(402\) −768639. −0.237223
\(403\) 5.12640e6 1.57235
\(404\) −1.05367e6 −0.321182
\(405\) 147572. 0.0447061
\(406\) 0 0
\(407\) 4.35522e6 1.30324
\(408\) −466472. −0.138731
\(409\) 2.36118e6 0.697945 0.348973 0.937133i \(-0.386531\pi\)
0.348973 + 0.937133i \(0.386531\pi\)
\(410\) 105580. 0.0310185
\(411\) −864918. −0.252563
\(412\) 2.08317e6 0.604619
\(413\) 0 0
\(414\) 460738. 0.132115
\(415\) 1.93868e6 0.552569
\(416\) −745898. −0.211323
\(417\) −907690. −0.255621
\(418\) 1.39753e6 0.391219
\(419\) −3.23493e6 −0.900181 −0.450090 0.892983i \(-0.648608\pi\)
−0.450090 + 0.892983i \(0.648608\pi\)
\(420\) 0 0
\(421\) −2.85759e6 −0.785769 −0.392884 0.919588i \(-0.628523\pi\)
−0.392884 + 0.919588i \(0.628523\pi\)
\(422\) −398774. −0.109005
\(423\) 754369. 0.204990
\(424\) 2.28114e6 0.616222
\(425\) −2.12107e6 −0.569615
\(426\) −2.20153e6 −0.587762
\(427\) 0 0
\(428\) 41455.2 0.0109388
\(429\) 2.23198e6 0.585527
\(430\) 329663. 0.0859803
\(431\) −88386.2 −0.0229188 −0.0114594 0.999934i \(-0.503648\pi\)
−0.0114594 + 0.999934i \(0.503648\pi\)
\(432\) −186624. −0.0481125
\(433\) −3.09418e6 −0.793097 −0.396549 0.918014i \(-0.629792\pi\)
−0.396549 + 0.918014i \(0.629792\pi\)
\(434\) 0 0
\(435\) −1.05629e6 −0.267646
\(436\) 1.77046e6 0.446036
\(437\) 1.45929e6 0.365543
\(438\) −1.48564e6 −0.370023
\(439\) 485178. 0.120155 0.0600773 0.998194i \(-0.480865\pi\)
0.0600773 + 0.998194i \(0.480865\pi\)
\(440\) 490098. 0.120684
\(441\) 0 0
\(442\) −2.35962e6 −0.574495
\(443\) 1.27520e6 0.308722 0.154361 0.988015i \(-0.450668\pi\)
0.154361 + 0.988015i \(0.450668\pi\)
\(444\) −1.84206e6 −0.443453
\(445\) 1.75424e6 0.419941
\(446\) 2.10478e6 0.501036
\(447\) 3.24850e6 0.768978
\(448\) 0 0
\(449\) −3.79144e6 −0.887541 −0.443770 0.896141i \(-0.646359\pi\)
−0.443770 + 0.896141i \(0.646359\pi\)
\(450\) −848587. −0.197544
\(451\) 399534. 0.0924937
\(452\) −3.10257e6 −0.714291
\(453\) 3.90650e6 0.894422
\(454\) −3.92642e6 −0.894040
\(455\) 0 0
\(456\) −591092. −0.133120
\(457\) 1.70281e6 0.381395 0.190698 0.981649i \(-0.438925\pi\)
0.190698 + 0.981649i \(0.438925\pi\)
\(458\) −340324. −0.0758104
\(459\) −590378. −0.130797
\(460\) 511757. 0.112764
\(461\) 4.55537e6 0.998323 0.499161 0.866509i \(-0.333642\pi\)
0.499161 + 0.866509i \(0.333642\pi\)
\(462\) 0 0
\(463\) 5.82647e6 1.26314 0.631572 0.775317i \(-0.282410\pi\)
0.631572 + 0.775317i \(0.282410\pi\)
\(464\) 1.33582e6 0.288039
\(465\) 1.42466e6 0.305547
\(466\) 452492. 0.0965263
\(467\) −3.54261e6 −0.751677 −0.375839 0.926685i \(-0.622645\pi\)
−0.375839 + 0.926685i \(0.622645\pi\)
\(468\) −944027. −0.199237
\(469\) 0 0
\(470\) 837902. 0.174964
\(471\) 4.60437e6 0.956352
\(472\) −1.94408e6 −0.401660
\(473\) 1.24751e6 0.256384
\(474\) 1.26002e6 0.257591
\(475\) −2.68772e6 −0.546575
\(476\) 0 0
\(477\) 2.88707e6 0.580979
\(478\) −3.58040e6 −0.716740
\(479\) −4.23542e6 −0.843447 −0.421723 0.906725i \(-0.638575\pi\)
−0.421723 + 0.906725i \(0.638575\pi\)
\(480\) −207289. −0.0410652
\(481\) −9.31797e6 −1.83636
\(482\) 2.11086e6 0.413849
\(483\) 0 0
\(484\) −722189. −0.140132
\(485\) −3.63848e6 −0.702370
\(486\) −236196. −0.0453609
\(487\) −5.65479e6 −1.08042 −0.540212 0.841529i \(-0.681656\pi\)
−0.540212 + 0.841529i \(0.681656\pi\)
\(488\) 2.05992e6 0.391562
\(489\) 2.26142e6 0.427671
\(490\) 0 0
\(491\) 8.33183e6 1.55968 0.779842 0.625976i \(-0.215299\pi\)
0.779842 + 0.625976i \(0.215299\pi\)
\(492\) −168985. −0.0314728
\(493\) 4.22581e6 0.783055
\(494\) −2.99001e6 −0.551258
\(495\) 620280. 0.113782
\(496\) −1.80166e6 −0.328828
\(497\) 0 0
\(498\) −3.10295e6 −0.560662
\(499\) −7.23769e6 −1.30121 −0.650607 0.759415i \(-0.725485\pi\)
−0.650607 + 0.759415i \(0.725485\pi\)
\(500\) −2.06717e6 −0.369787
\(501\) −3.77349e6 −0.671659
\(502\) −4.77170e6 −0.845112
\(503\) −6.16761e6 −1.08692 −0.543459 0.839436i \(-0.682886\pi\)
−0.543459 + 0.839436i \(0.682886\pi\)
\(504\) 0 0
\(505\) −1.48122e6 −0.258459
\(506\) 1.93659e6 0.336249
\(507\) −1.43367e6 −0.247702
\(508\) 438868. 0.0754526
\(509\) 1.83488e6 0.313916 0.156958 0.987605i \(-0.449831\pi\)
0.156958 + 0.987605i \(0.449831\pi\)
\(510\) −655753. −0.111639
\(511\) 0 0
\(512\) 262144. 0.0441942
\(513\) −748101. −0.125507
\(514\) −4.77969e6 −0.797980
\(515\) 2.92846e6 0.486543
\(516\) −527640. −0.0872395
\(517\) 3.17079e6 0.521724
\(518\) 0 0
\(519\) 4.15239e6 0.676674
\(520\) −1.04856e6 −0.170054
\(521\) 5.31040e6 0.857103 0.428551 0.903517i \(-0.359024\pi\)
0.428551 + 0.903517i \(0.359024\pi\)
\(522\) 1.69064e6 0.271566
\(523\) −2.98835e6 −0.477725 −0.238862 0.971053i \(-0.576774\pi\)
−0.238862 + 0.971053i \(0.576774\pi\)
\(524\) 2.40601e6 0.382798
\(525\) 0 0
\(526\) −8.48921e6 −1.33784
\(527\) −5.69950e6 −0.893943
\(528\) −784424. −0.122452
\(529\) −4.41417e6 −0.685820
\(530\) 3.20676e6 0.495880
\(531\) −2.46047e6 −0.378688
\(532\) 0 0
\(533\) −854800. −0.130331
\(534\) −2.80773e6 −0.426091
\(535\) 58276.6 0.00880257
\(536\) 1.36647e6 0.205441
\(537\) −6.67535e6 −0.998938
\(538\) 619907. 0.0923359
\(539\) 0 0
\(540\) −262351. −0.0387167
\(541\) 2.16832e6 0.318516 0.159258 0.987237i \(-0.449090\pi\)
0.159258 + 0.987237i \(0.449090\pi\)
\(542\) 7.83551e6 1.14570
\(543\) −2.71234e6 −0.394770
\(544\) 829283. 0.120145
\(545\) 2.48886e6 0.358930
\(546\) 0 0
\(547\) −9.13272e6 −1.30506 −0.652532 0.757761i \(-0.726293\pi\)
−0.652532 + 0.757761i \(0.726293\pi\)
\(548\) 1.53763e6 0.218726
\(549\) 2.60709e6 0.369168
\(550\) −3.56681e6 −0.502774
\(551\) 5.35475e6 0.751381
\(552\) −819090. −0.114415
\(553\) 0 0
\(554\) −6.99810e6 −0.968737
\(555\) −2.58952e6 −0.356851
\(556\) 1.61367e6 0.221375
\(557\) −9.22028e6 −1.25923 −0.629617 0.776906i \(-0.716788\pi\)
−0.629617 + 0.776906i \(0.716788\pi\)
\(558\) −2.28023e6 −0.310022
\(559\) −2.66904e6 −0.361264
\(560\) 0 0
\(561\) −2.48150e6 −0.332894
\(562\) 5.62660e6 0.751458
\(563\) 8.49277e6 1.12922 0.564610 0.825358i \(-0.309027\pi\)
0.564610 + 0.825358i \(0.309027\pi\)
\(564\) −1.34110e6 −0.177527
\(565\) −4.36150e6 −0.574797
\(566\) 7.82828e6 1.02713
\(567\) 0 0
\(568\) 3.91384e6 0.509017
\(569\) −4.70912e6 −0.609760 −0.304880 0.952391i \(-0.598616\pi\)
−0.304880 + 0.952391i \(0.598616\pi\)
\(570\) −830940. −0.107123
\(571\) −1.72807e6 −0.221805 −0.110902 0.993831i \(-0.535374\pi\)
−0.110902 + 0.993831i \(0.535374\pi\)
\(572\) −3.96796e6 −0.507081
\(573\) −2.36864e6 −0.301379
\(574\) 0 0
\(575\) −3.72443e6 −0.469776
\(576\) 331776. 0.0416667
\(577\) 2.38499e6 0.298227 0.149113 0.988820i \(-0.452358\pi\)
0.149113 + 0.988820i \(0.452358\pi\)
\(578\) −3.05602e6 −0.380484
\(579\) 7.48014e6 0.927285
\(580\) 1.87785e6 0.231788
\(581\) 0 0
\(582\) 5.82355e6 0.712657
\(583\) 1.21350e7 1.47866
\(584\) 2.64114e6 0.320449
\(585\) −1.32709e6 −0.160328
\(586\) −5.07993e6 −0.611102
\(587\) −3.44904e6 −0.413145 −0.206573 0.978431i \(-0.566231\pi\)
−0.206573 + 0.978431i \(0.566231\pi\)
\(588\) 0 0
\(589\) −7.22214e6 −0.857784
\(590\) −2.73293e6 −0.323220
\(591\) 9.48790e6 1.11738
\(592\) 3.27478e6 0.384041
\(593\) 1.13468e6 0.132507 0.0662533 0.997803i \(-0.478895\pi\)
0.0662533 + 0.997803i \(0.478895\pi\)
\(594\) −992786. −0.115449
\(595\) 0 0
\(596\) −5.77511e6 −0.665954
\(597\) −6.28711e6 −0.721963
\(598\) −4.14332e6 −0.473800
\(599\) 7.61865e6 0.867583 0.433792 0.901013i \(-0.357175\pi\)
0.433792 + 0.901013i \(0.357175\pi\)
\(600\) 1.50860e6 0.171079
\(601\) 9.78414e6 1.10493 0.552467 0.833535i \(-0.313686\pi\)
0.552467 + 0.833535i \(0.313686\pi\)
\(602\) 0 0
\(603\) 1.72944e6 0.193692
\(604\) −6.94489e6 −0.774592
\(605\) −1.01523e6 −0.112766
\(606\) 2.37076e6 0.262244
\(607\) 1.85335e6 0.204167 0.102083 0.994776i \(-0.467449\pi\)
0.102083 + 0.994776i \(0.467449\pi\)
\(608\) 1.05083e6 0.115285
\(609\) 0 0
\(610\) 2.89578e6 0.315094
\(611\) −6.78388e6 −0.735148
\(612\) 1.04956e6 0.113274
\(613\) −7.12097e6 −0.765399 −0.382699 0.923873i \(-0.625005\pi\)
−0.382699 + 0.923873i \(0.625005\pi\)
\(614\) −167417. −0.0179217
\(615\) −237554. −0.0253265
\(616\) 0 0
\(617\) −2.75212e6 −0.291041 −0.145521 0.989355i \(-0.546486\pi\)
−0.145521 + 0.989355i \(0.546486\pi\)
\(618\) −4.68713e6 −0.493669
\(619\) −1.30332e7 −1.36718 −0.683588 0.729868i \(-0.739581\pi\)
−0.683588 + 0.729868i \(0.739581\pi\)
\(620\) −2.53273e6 −0.264612
\(621\) −1.03666e6 −0.107872
\(622\) −9.16758e6 −0.950120
\(623\) 0 0
\(624\) 1.67827e6 0.172544
\(625\) 5.27870e6 0.540539
\(626\) 9.08598e6 0.926693
\(627\) −3.14444e6 −0.319429
\(628\) −8.18554e6 −0.828225
\(629\) 1.03597e7 1.04404
\(630\) 0 0
\(631\) −4.40820e6 −0.440745 −0.220373 0.975416i \(-0.570727\pi\)
−0.220373 + 0.975416i \(0.570727\pi\)
\(632\) −2.24003e6 −0.223080
\(633\) 897242. 0.0890021
\(634\) −7.93897e6 −0.784406
\(635\) 616948. 0.0607175
\(636\) −5.13256e6 −0.503143
\(637\) 0 0
\(638\) 7.10616e6 0.691168
\(639\) 4.95345e6 0.479906
\(640\) 368515. 0.0355635
\(641\) −7.98390e6 −0.767485 −0.383742 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(642\) −93274.2 −0.00893149
\(643\) 1.52102e6 0.145080 0.0725398 0.997366i \(-0.476890\pi\)
0.0725398 + 0.997366i \(0.476890\pi\)
\(644\) 0 0
\(645\) −741741. −0.0702026
\(646\) 3.32426e6 0.313411
\(647\) −1.25249e7 −1.17629 −0.588146 0.808755i \(-0.700142\pi\)
−0.588146 + 0.808755i \(0.700142\pi\)
\(648\) 419904. 0.0392837
\(649\) −1.03419e7 −0.963806
\(650\) 7.63116e6 0.708447
\(651\) 0 0
\(652\) −4.02031e6 −0.370374
\(653\) −1.70740e7 −1.56694 −0.783471 0.621428i \(-0.786553\pi\)
−0.783471 + 0.621428i \(0.786553\pi\)
\(654\) −3.98353e6 −0.364187
\(655\) 3.38231e6 0.308042
\(656\) 300417. 0.0272562
\(657\) 3.34269e6 0.302122
\(658\) 0 0
\(659\) −2.12585e6 −0.190686 −0.0953432 0.995444i \(-0.530395\pi\)
−0.0953432 + 0.995444i \(0.530395\pi\)
\(660\) −1.10272e6 −0.0985384
\(661\) −2.60011e6 −0.231466 −0.115733 0.993280i \(-0.536922\pi\)
−0.115733 + 0.993280i \(0.536922\pi\)
\(662\) 2.55384e6 0.226490
\(663\) 5.30915e6 0.469074
\(664\) 5.51635e6 0.485547
\(665\) 0 0
\(666\) 4.14464e6 0.362078
\(667\) 7.42020e6 0.645805
\(668\) 6.70843e6 0.581674
\(669\) −4.73575e6 −0.409094
\(670\) 1.92094e6 0.165321
\(671\) 1.09582e7 0.939577
\(672\) 0 0
\(673\) −1.44746e7 −1.23188 −0.615942 0.787792i \(-0.711224\pi\)
−0.615942 + 0.787792i \(0.711224\pi\)
\(674\) 1.08810e7 0.922615
\(675\) 1.90932e6 0.161294
\(676\) 2.54875e6 0.214516
\(677\) −3.28845e6 −0.275753 −0.137876 0.990449i \(-0.544028\pi\)
−0.137876 + 0.990449i \(0.544028\pi\)
\(678\) 6.98077e6 0.583216
\(679\) 0 0
\(680\) 1.16578e6 0.0966819
\(681\) 8.83444e6 0.729981
\(682\) −9.58433e6 −0.789043
\(683\) 8.81887e6 0.723371 0.361685 0.932300i \(-0.382201\pi\)
0.361685 + 0.932300i \(0.382201\pi\)
\(684\) 1.32996e6 0.108692
\(685\) 2.16156e6 0.176011
\(686\) 0 0
\(687\) 765729. 0.0618989
\(688\) 938026. 0.0755517
\(689\) −2.59628e7 −2.08354
\(690\) −1.15145e6 −0.0920711
\(691\) −8.59617e6 −0.684873 −0.342436 0.939541i \(-0.611252\pi\)
−0.342436 + 0.939541i \(0.611252\pi\)
\(692\) −7.38202e6 −0.586017
\(693\) 0 0
\(694\) 1.04831e7 0.826208
\(695\) 2.26845e6 0.178143
\(696\) −3.00559e6 −0.235183
\(697\) 950360. 0.0740979
\(698\) 2.30237e6 0.178870
\(699\) −1.01811e6 −0.0788134
\(700\) 0 0
\(701\) 2.06437e7 1.58669 0.793347 0.608770i \(-0.208337\pi\)
0.793347 + 0.608770i \(0.208337\pi\)
\(702\) 2.12406e6 0.162676
\(703\) 1.31273e7 1.00181
\(704\) 1.39453e6 0.106047
\(705\) −1.88528e6 −0.142858
\(706\) −1.03589e7 −0.782169
\(707\) 0 0
\(708\) 4.37417e6 0.327954
\(709\) −426882. −0.0318928 −0.0159464 0.999873i \(-0.505076\pi\)
−0.0159464 + 0.999873i \(0.505076\pi\)
\(710\) 5.50196e6 0.409611
\(711\) −2.83504e6 −0.210322
\(712\) 4.99153e6 0.369006
\(713\) −1.00079e7 −0.737257
\(714\) 0 0
\(715\) −5.57805e6 −0.408054
\(716\) 1.18673e7 0.865105
\(717\) 8.05590e6 0.585216
\(718\) 1.14712e7 0.830417
\(719\) 3.80724e6 0.274655 0.137328 0.990526i \(-0.456149\pi\)
0.137328 + 0.990526i \(0.456149\pi\)
\(720\) 466401. 0.0335296
\(721\) 0 0
\(722\) −5.69204e6 −0.406373
\(723\) −4.74943e6 −0.337906
\(724\) 4.82193e6 0.341881
\(725\) −1.36665e7 −0.965635
\(726\) 1.62493e6 0.114417
\(727\) −2.22044e7 −1.55813 −0.779065 0.626943i \(-0.784306\pi\)
−0.779065 + 0.626943i \(0.784306\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 3.71284e6 0.257869
\(731\) 2.96741e6 0.205393
\(732\) −4.63482e6 −0.319709
\(733\) −2.90572e7 −1.99753 −0.998766 0.0496729i \(-0.984182\pi\)
−0.998766 + 0.0496729i \(0.984182\pi\)
\(734\) −7.81804e6 −0.535621
\(735\) 0 0
\(736\) 1.45616e6 0.0990865
\(737\) 7.26923e6 0.492969
\(738\) 380216. 0.0256974
\(739\) 2.13523e7 1.43824 0.719122 0.694883i \(-0.244544\pi\)
0.719122 + 0.694883i \(0.244544\pi\)
\(740\) 4.60359e6 0.309042
\(741\) 6.72751e6 0.450100
\(742\) 0 0
\(743\) 3.91874e6 0.260420 0.130210 0.991486i \(-0.458435\pi\)
0.130210 + 0.991486i \(0.458435\pi\)
\(744\) 4.05374e6 0.268487
\(745\) −8.11848e6 −0.535900
\(746\) 8.84438e6 0.581863
\(747\) 6.98163e6 0.457778
\(748\) 4.41155e6 0.288295
\(749\) 0 0
\(750\) 4.65113e6 0.301930
\(751\) 5.66912e6 0.366789 0.183394 0.983039i \(-0.441291\pi\)
0.183394 + 0.983039i \(0.441291\pi\)
\(752\) 2.38418e6 0.153743
\(753\) 1.07363e7 0.690031
\(754\) −1.52036e7 −0.973908
\(755\) −9.76293e6 −0.623323
\(756\) 0 0
\(757\) −1.91706e7 −1.21590 −0.607949 0.793976i \(-0.708007\pi\)
−0.607949 + 0.793976i \(0.708007\pi\)
\(758\) 1.52493e7 0.963998
\(759\) −4.35732e6 −0.274546
\(760\) 1.47723e6 0.0927712
\(761\) 1.46336e7 0.915987 0.457993 0.888956i \(-0.348568\pi\)
0.457993 + 0.888956i \(0.348568\pi\)
\(762\) −987453. −0.0616068
\(763\) 0 0
\(764\) 4.21092e6 0.261002
\(765\) 1.47544e6 0.0911526
\(766\) 1.52242e7 0.937483
\(767\) 2.21265e7 1.35808
\(768\) −589824. −0.0360844
\(769\) 1.57337e7 0.959432 0.479716 0.877424i \(-0.340740\pi\)
0.479716 + 0.877424i \(0.340740\pi\)
\(770\) 0 0
\(771\) 1.07543e7 0.651548
\(772\) −1.32980e7 −0.803053
\(773\) 4.04904e6 0.243727 0.121864 0.992547i \(-0.461113\pi\)
0.121864 + 0.992547i \(0.461113\pi\)
\(774\) 1.18719e6 0.0712308
\(775\) 1.84325e7 1.10238
\(776\) −1.03530e7 −0.617179
\(777\) 0 0
\(778\) 1.23010e7 0.728602
\(779\) 1.20425e6 0.0711007
\(780\) 2.35926e6 0.138848
\(781\) 2.08205e7 1.22142
\(782\) 4.60651e6 0.269374
\(783\) −3.80395e6 −0.221733
\(784\) 0 0
\(785\) −1.15070e7 −0.666481
\(786\) −5.41353e6 −0.312554
\(787\) −2.16111e7 −1.24377 −0.621886 0.783108i \(-0.713633\pi\)
−0.621886 + 0.783108i \(0.713633\pi\)
\(788\) −1.68674e7 −0.967681
\(789\) 1.91007e7 1.09234
\(790\) −3.14897e6 −0.179515
\(791\) 0 0
\(792\) 1.76495e6 0.0999817
\(793\) −2.34450e7 −1.32394
\(794\) 8.76530e6 0.493419
\(795\) −7.21521e6 −0.404884
\(796\) 1.11771e7 0.625239
\(797\) 1.38162e7 0.770445 0.385222 0.922824i \(-0.374125\pi\)
0.385222 + 0.922824i \(0.374125\pi\)
\(798\) 0 0
\(799\) 7.54226e6 0.417960
\(800\) −2.68195e6 −0.148158
\(801\) 6.31740e6 0.347902
\(802\) 7.10672e6 0.390151
\(803\) 1.40501e7 0.768937
\(804\) −3.07456e6 −0.167742
\(805\) 0 0
\(806\) 2.05056e7 1.11182
\(807\) −1.39479e6 −0.0753919
\(808\) −4.21468e6 −0.227110
\(809\) −3.50756e7 −1.88423 −0.942116 0.335287i \(-0.891167\pi\)
−0.942116 + 0.335287i \(0.891167\pi\)
\(810\) 590289. 0.0316120
\(811\) 6.56287e6 0.350382 0.175191 0.984534i \(-0.443946\pi\)
0.175191 + 0.984534i \(0.443946\pi\)
\(812\) 0 0
\(813\) −1.76299e7 −0.935456
\(814\) 1.74209e7 0.921530
\(815\) −5.65164e6 −0.298044
\(816\) −1.86589e6 −0.0980980
\(817\) 3.76017e6 0.197085
\(818\) 9.44473e6 0.493522
\(819\) 0 0
\(820\) 422318. 0.0219334
\(821\) −1.77894e6 −0.0921093 −0.0460547 0.998939i \(-0.514665\pi\)
−0.0460547 + 0.998939i \(0.514665\pi\)
\(822\) −3.45967e6 −0.178589
\(823\) 1.89586e7 0.975676 0.487838 0.872934i \(-0.337786\pi\)
0.487838 + 0.872934i \(0.337786\pi\)
\(824\) 8.33268e6 0.427530
\(825\) 8.02531e6 0.410513
\(826\) 0 0
\(827\) 2.82110e7 1.43435 0.717174 0.696894i \(-0.245435\pi\)
0.717174 + 0.696894i \(0.245435\pi\)
\(828\) 1.84295e6 0.0934196
\(829\) −2.24021e7 −1.13214 −0.566072 0.824356i \(-0.691538\pi\)
−0.566072 + 0.824356i \(0.691538\pi\)
\(830\) 7.75473e6 0.390725
\(831\) 1.57457e7 0.790971
\(832\) −2.98359e6 −0.149428
\(833\) 0 0
\(834\) −3.63076e6 −0.180752
\(835\) 9.43051e6 0.468079
\(836\) 5.59012e6 0.276634
\(837\) 5.13052e6 0.253132
\(838\) −1.29397e7 −0.636524
\(839\) −1.50303e7 −0.737162 −0.368581 0.929596i \(-0.620156\pi\)
−0.368581 + 0.929596i \(0.620156\pi\)
\(840\) 0 0
\(841\) 6.71672e6 0.327467
\(842\) −1.14304e7 −0.555622
\(843\) −1.26598e7 −0.613563
\(844\) −1.59510e6 −0.0770781
\(845\) 3.58295e6 0.172623
\(846\) 3.01747e6 0.144950
\(847\) 0 0
\(848\) 9.12455e6 0.435734
\(849\) −1.76136e7 −0.838648
\(850\) −8.48426e6 −0.402779
\(851\) 1.81908e7 0.861048
\(852\) −8.80614e6 −0.415611
\(853\) −3.18297e7 −1.49782 −0.748911 0.662670i \(-0.769423\pi\)
−0.748911 + 0.662670i \(0.769423\pi\)
\(854\) 0 0
\(855\) 1.86962e6 0.0874656
\(856\) 165821. 0.00773490
\(857\) 2.17823e7 1.01310 0.506549 0.862211i \(-0.330921\pi\)
0.506549 + 0.862211i \(0.330921\pi\)
\(858\) 8.92792e6 0.414030
\(859\) 3.73349e7 1.72636 0.863181 0.504895i \(-0.168469\pi\)
0.863181 + 0.504895i \(0.168469\pi\)
\(860\) 1.31865e6 0.0607972
\(861\) 0 0
\(862\) −353545. −0.0162060
\(863\) −1.54613e7 −0.706675 −0.353338 0.935496i \(-0.614953\pi\)
−0.353338 + 0.935496i \(0.614953\pi\)
\(864\) −746496. −0.0340207
\(865\) −1.03774e7 −0.471574
\(866\) −1.23767e7 −0.560805
\(867\) 6.87604e6 0.310664
\(868\) 0 0
\(869\) −1.19163e7 −0.535294
\(870\) −4.22517e6 −0.189254
\(871\) −1.55525e7 −0.694630
\(872\) 7.08184e6 0.315395
\(873\) −1.31030e7 −0.581882
\(874\) 5.83716e6 0.258478
\(875\) 0 0
\(876\) −5.94256e6 −0.261646
\(877\) −6.81762e6 −0.299319 −0.149659 0.988738i \(-0.547818\pi\)
−0.149659 + 0.988738i \(0.547818\pi\)
\(878\) 1.94071e6 0.0849621
\(879\) 1.14298e7 0.498963
\(880\) 1.96039e6 0.0853368
\(881\) −1.20903e7 −0.524805 −0.262402 0.964959i \(-0.584515\pi\)
−0.262402 + 0.964959i \(0.584515\pi\)
\(882\) 0 0
\(883\) 1.63966e7 0.707706 0.353853 0.935301i \(-0.384871\pi\)
0.353853 + 0.935301i \(0.384871\pi\)
\(884\) −9.43848e6 −0.406230
\(885\) 6.14909e6 0.263908
\(886\) 5.10078e6 0.218299
\(887\) −2.57006e7 −1.09682 −0.548409 0.836210i \(-0.684766\pi\)
−0.548409 + 0.836210i \(0.684766\pi\)
\(888\) −7.36826e6 −0.313568
\(889\) 0 0
\(890\) 7.01695e6 0.296943
\(891\) 2.23377e6 0.0942636
\(892\) 8.41911e6 0.354286
\(893\) 9.55721e6 0.401054
\(894\) 1.29940e7 0.543749
\(895\) 1.66827e7 0.696160
\(896\) 0 0
\(897\) 9.32247e6 0.386856
\(898\) −1.51658e7 −0.627586
\(899\) −3.67232e7 −1.51545
\(900\) −3.39435e6 −0.139685
\(901\) 2.88652e7 1.18457
\(902\) 1.59813e6 0.0654029
\(903\) 0 0
\(904\) −1.24103e7 −0.505080
\(905\) 6.77854e6 0.275115
\(906\) 1.56260e7 0.632452
\(907\) −2.34481e7 −0.946434 −0.473217 0.880946i \(-0.656907\pi\)
−0.473217 + 0.880946i \(0.656907\pi\)
\(908\) −1.57057e7 −0.632182
\(909\) −5.33421e6 −0.214121
\(910\) 0 0
\(911\) −3.29134e7 −1.31395 −0.656973 0.753914i \(-0.728163\pi\)
−0.656973 + 0.753914i \(0.728163\pi\)
\(912\) −2.36437e6 −0.0941300
\(913\) 2.93454e7 1.16510
\(914\) 6.81123e6 0.269687
\(915\) −6.51550e6 −0.257273
\(916\) −1.36130e6 −0.0536061
\(917\) 0 0
\(918\) −2.36151e6 −0.0924877
\(919\) −2.52264e7 −0.985294 −0.492647 0.870229i \(-0.663971\pi\)
−0.492647 + 0.870229i \(0.663971\pi\)
\(920\) 2.04703e6 0.0797359
\(921\) 376689. 0.0146330
\(922\) 1.82215e7 0.705921
\(923\) −4.45454e7 −1.72107
\(924\) 0 0
\(925\) −3.35037e7 −1.28748
\(926\) 2.33059e7 0.893178
\(927\) 1.05460e7 0.403079
\(928\) 5.34327e6 0.203675
\(929\) 2.51164e7 0.954812 0.477406 0.878683i \(-0.341577\pi\)
0.477406 + 0.878683i \(0.341577\pi\)
\(930\) 5.69863e6 0.216055
\(931\) 0 0
\(932\) 1.80997e6 0.0682544
\(933\) 2.06270e7 0.775770
\(934\) −1.41704e7 −0.531516
\(935\) 6.20163e6 0.231994
\(936\) −3.77611e6 −0.140882
\(937\) 8.25845e6 0.307291 0.153645 0.988126i \(-0.450899\pi\)
0.153645 + 0.988126i \(0.450899\pi\)
\(938\) 0 0
\(939\) −2.04435e7 −0.756642
\(940\) 3.35161e6 0.123718
\(941\) 1.27088e7 0.467875 0.233938 0.972252i \(-0.424839\pi\)
0.233938 + 0.972252i \(0.424839\pi\)
\(942\) 1.84175e7 0.676243
\(943\) 1.66876e6 0.0611103
\(944\) −7.77630e6 −0.284016
\(945\) 0 0
\(946\) 4.99003e6 0.181291
\(947\) 1.06278e7 0.385096 0.192548 0.981288i \(-0.438325\pi\)
0.192548 + 0.981288i \(0.438325\pi\)
\(948\) 5.04007e6 0.182144
\(949\) −3.00601e7 −1.08349
\(950\) −1.07509e7 −0.386487
\(951\) 1.78627e7 0.640465
\(952\) 0 0
\(953\) 749727. 0.0267406 0.0133703 0.999911i \(-0.495744\pi\)
0.0133703 + 0.999911i \(0.495744\pi\)
\(954\) 1.15483e7 0.410814
\(955\) 5.91959e6 0.210031
\(956\) −1.43216e7 −0.506812
\(957\) −1.59889e7 −0.564336
\(958\) −1.69417e7 −0.596407
\(959\) 0 0
\(960\) −829158. −0.0290375
\(961\) 2.09007e7 0.730050
\(962\) −3.72719e7 −1.29851
\(963\) 209867. 0.00729253
\(964\) 8.44343e6 0.292635
\(965\) −1.86940e7 −0.646225
\(966\) 0 0
\(967\) 2.97713e7 1.02384 0.511920 0.859033i \(-0.328934\pi\)
0.511920 + 0.859033i \(0.328934\pi\)
\(968\) −2.88876e6 −0.0990883
\(969\) −7.47960e6 −0.255899
\(970\) −1.45539e7 −0.496651
\(971\) −1.51210e7 −0.514676 −0.257338 0.966321i \(-0.582845\pi\)
−0.257338 + 0.966321i \(0.582845\pi\)
\(972\) −944784. −0.0320750
\(973\) 0 0
\(974\) −2.26192e7 −0.763975
\(975\) −1.71701e7 −0.578444
\(976\) 8.23968e6 0.276876
\(977\) 1.93107e6 0.0647235 0.0323617 0.999476i \(-0.489697\pi\)
0.0323617 + 0.999476i \(0.489697\pi\)
\(978\) 9.04570e6 0.302409
\(979\) 2.65535e7 0.885452
\(980\) 0 0
\(981\) 8.96295e6 0.297357
\(982\) 3.33273e7 1.10286
\(983\) 4.22343e7 1.39406 0.697030 0.717042i \(-0.254504\pi\)
0.697030 + 0.717042i \(0.254504\pi\)
\(984\) −675939. −0.0222546
\(985\) −2.37117e7 −0.778703
\(986\) 1.69032e7 0.553704
\(987\) 0 0
\(988\) −1.19600e7 −0.389798
\(989\) 5.21056e6 0.169392
\(990\) 2.48112e6 0.0804563
\(991\) 5.83480e6 0.188730 0.0943651 0.995538i \(-0.469918\pi\)
0.0943651 + 0.995538i \(0.469918\pi\)
\(992\) −7.20665e6 −0.232517
\(993\) −5.74615e6 −0.184928
\(994\) 0 0
\(995\) 1.57124e7 0.503136
\(996\) −1.24118e7 −0.396448
\(997\) 1.23461e7 0.393360 0.196680 0.980468i \(-0.436984\pi\)
0.196680 + 0.980468i \(0.436984\pi\)
\(998\) −2.89508e7 −0.920097
\(999\) −9.32545e6 −0.295635
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 294.6.a.r.1.2 2
3.2 odd 2 882.6.a.bh.1.1 2
7.2 even 3 42.6.e.c.25.1 4
7.3 odd 6 294.6.e.s.79.2 4
7.4 even 3 42.6.e.c.37.1 yes 4
7.5 odd 6 294.6.e.s.67.2 4
7.6 odd 2 294.6.a.w.1.1 2
21.2 odd 6 126.6.g.h.109.2 4
21.11 odd 6 126.6.g.h.37.2 4
21.20 even 2 882.6.a.bb.1.2 2
28.11 odd 6 336.6.q.f.289.1 4
28.23 odd 6 336.6.q.f.193.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.c.25.1 4 7.2 even 3
42.6.e.c.37.1 yes 4 7.4 even 3
126.6.g.h.37.2 4 21.11 odd 6
126.6.g.h.109.2 4 21.2 odd 6
294.6.a.r.1.2 2 1.1 even 1 trivial
294.6.a.w.1.1 2 7.6 odd 2
294.6.e.s.67.2 4 7.5 odd 6
294.6.e.s.79.2 4 7.3 odd 6
336.6.q.f.193.1 4 28.23 odd 6
336.6.q.f.289.1 4 28.11 odd 6
882.6.a.bb.1.2 2 21.20 even 2
882.6.a.bh.1.1 2 3.2 odd 2