Properties

Label 294.6.a.q.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: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) 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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +103.497 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +103.497 q^{5} -36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} -413.990 q^{10} +240.191 q^{11} +144.000 q^{12} +805.477 q^{13} +931.477 q^{15} +256.000 q^{16} -1293.27 q^{17} -324.000 q^{18} -275.377 q^{19} +1655.96 q^{20} -960.764 q^{22} +3796.57 q^{23} -576.000 q^{24} +7586.73 q^{25} -3221.91 q^{26} +729.000 q^{27} +1227.52 q^{29} -3725.91 q^{30} +5624.86 q^{31} -1024.00 q^{32} +2161.72 q^{33} +5173.06 q^{34} +1296.00 q^{36} -9078.49 q^{37} +1101.51 q^{38} +7249.30 q^{39} -6623.84 q^{40} -18207.4 q^{41} -11708.2 q^{43} +3843.05 q^{44} +8383.30 q^{45} -15186.3 q^{46} -23048.6 q^{47} +2304.00 q^{48} -30346.9 q^{50} -11639.4 q^{51} +12887.6 q^{52} +17662.8 q^{53} -2916.00 q^{54} +24859.2 q^{55} -2478.39 q^{57} -4910.07 q^{58} +18376.3 q^{59} +14903.6 q^{60} +11324.1 q^{61} -22499.5 q^{62} +4096.00 q^{64} +83364.9 q^{65} -8646.87 q^{66} +36079.3 q^{67} -20692.3 q^{68} +34169.2 q^{69} -63434.2 q^{71} -5184.00 q^{72} -52982.4 q^{73} +36314.0 q^{74} +68280.5 q^{75} -4406.03 q^{76} -28997.2 q^{78} -48564.0 q^{79} +26495.4 q^{80} +6561.00 q^{81} +72829.7 q^{82} +113161. q^{83} -133850. q^{85} +46832.9 q^{86} +11047.7 q^{87} -15372.2 q^{88} +108366. q^{89} -33533.2 q^{90} +60745.2 q^{92} +50623.8 q^{93} +92194.4 q^{94} -28500.8 q^{95} -9216.00 q^{96} +99641.2 q^{97} +19455.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} + 108 q^{5} - 72 q^{6} - 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} + 108 q^{5} - 72 q^{6} - 128 q^{8} + 162 q^{9} - 432 q^{10} + 124 q^{11} + 288 q^{12} + 720 q^{13} + 972 q^{15} + 512 q^{16} - 1260 q^{17} - 648 q^{18} + 360 q^{19} + 1728 q^{20} - 496 q^{22} + 6524 q^{23} - 1152 q^{24} + 4482 q^{25} - 2880 q^{26} + 1458 q^{27} + 7088 q^{29} - 3888 q^{30} + 5904 q^{31} - 2048 q^{32} + 1116 q^{33} + 5040 q^{34} + 2592 q^{36} - 6040 q^{37} - 1440 q^{38} + 6480 q^{39} - 6912 q^{40} - 17388 q^{41} - 608 q^{43} + 1984 q^{44} + 8748 q^{45} - 26096 q^{46} - 30456 q^{47} + 4608 q^{48} - 17928 q^{50} - 11340 q^{51} + 11520 q^{52} + 3964 q^{53} - 5832 q^{54} + 24336 q^{55} + 3240 q^{57} - 28352 q^{58} + 40752 q^{59} + 15552 q^{60} - 1368 q^{61} - 23616 q^{62} + 8192 q^{64} + 82980 q^{65} - 4464 q^{66} - 16224 q^{67} - 20160 q^{68} + 58716 q^{69} - 3204 q^{71} - 10368 q^{72} + 23976 q^{73} + 24160 q^{74} + 40338 q^{75} + 5760 q^{76} - 25920 q^{78} - 82160 q^{79} + 27648 q^{80} + 13122 q^{81} + 69552 q^{82} + 173736 q^{83} - 133700 q^{85} + 2432 q^{86} + 63792 q^{87} - 7936 q^{88} + 200556 q^{89} - 34992 q^{90} + 104384 q^{92} + 53136 q^{93} + 121824 q^{94} - 25640 q^{95} - 18432 q^{96} + 251928 q^{97} + 10044 q^{99}+O(q^{100}) \) 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
\(5\) 103.497 1.85142 0.925710 0.378235i \(-0.123469\pi\)
0.925710 + 0.378235i \(0.123469\pi\)
\(6\) −36.0000 −0.408248
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) −413.990 −1.30915
\(11\) 240.191 0.598515 0.299257 0.954172i \(-0.403261\pi\)
0.299257 + 0.954172i \(0.403261\pi\)
\(12\) 144.000 0.288675
\(13\) 805.477 1.32189 0.660944 0.750435i \(-0.270156\pi\)
0.660944 + 0.750435i \(0.270156\pi\)
\(14\) 0 0
\(15\) 931.477 1.06892
\(16\) 256.000 0.250000
\(17\) −1293.27 −1.08534 −0.542670 0.839946i \(-0.682586\pi\)
−0.542670 + 0.839946i \(0.682586\pi\)
\(18\) −324.000 −0.235702
\(19\) −275.377 −0.175002 −0.0875011 0.996164i \(-0.527888\pi\)
−0.0875011 + 0.996164i \(0.527888\pi\)
\(20\) 1655.96 0.925710
\(21\) 0 0
\(22\) −960.764 −0.423214
\(23\) 3796.57 1.49648 0.748242 0.663426i \(-0.230898\pi\)
0.748242 + 0.663426i \(0.230898\pi\)
\(24\) −576.000 −0.204124
\(25\) 7586.73 2.42775
\(26\) −3221.91 −0.934717
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 1227.52 0.271040 0.135520 0.990775i \(-0.456730\pi\)
0.135520 + 0.990775i \(0.456730\pi\)
\(30\) −3725.91 −0.755839
\(31\) 5624.86 1.05125 0.525627 0.850715i \(-0.323831\pi\)
0.525627 + 0.850715i \(0.323831\pi\)
\(32\) −1024.00 −0.176777
\(33\) 2161.72 0.345553
\(34\) 5173.06 0.767451
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −9078.49 −1.09021 −0.545104 0.838368i \(-0.683510\pi\)
−0.545104 + 0.838368i \(0.683510\pi\)
\(38\) 1101.51 0.123745
\(39\) 7249.30 0.763193
\(40\) −6623.84 −0.654576
\(41\) −18207.4 −1.69156 −0.845782 0.533528i \(-0.820866\pi\)
−0.845782 + 0.533528i \(0.820866\pi\)
\(42\) 0 0
\(43\) −11708.2 −0.965650 −0.482825 0.875717i \(-0.660389\pi\)
−0.482825 + 0.875717i \(0.660389\pi\)
\(44\) 3843.05 0.299257
\(45\) 8383.30 0.617140
\(46\) −15186.3 −1.05817
\(47\) −23048.6 −1.52195 −0.760974 0.648782i \(-0.775279\pi\)
−0.760974 + 0.648782i \(0.775279\pi\)
\(48\) 2304.00 0.144338
\(49\) 0 0
\(50\) −30346.9 −1.71668
\(51\) −11639.4 −0.626621
\(52\) 12887.6 0.660944
\(53\) 17662.8 0.863714 0.431857 0.901942i \(-0.357859\pi\)
0.431857 + 0.901942i \(0.357859\pi\)
\(54\) −2916.00 −0.136083
\(55\) 24859.2 1.10810
\(56\) 0 0
\(57\) −2478.39 −0.101038
\(58\) −4910.07 −0.191654
\(59\) 18376.3 0.687271 0.343636 0.939103i \(-0.388342\pi\)
0.343636 + 0.939103i \(0.388342\pi\)
\(60\) 14903.6 0.534459
\(61\) 11324.1 0.389654 0.194827 0.980838i \(-0.437586\pi\)
0.194827 + 0.980838i \(0.437586\pi\)
\(62\) −22499.5 −0.743349
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 83364.9 2.44737
\(66\) −8646.87 −0.244343
\(67\) 36079.3 0.981910 0.490955 0.871185i \(-0.336648\pi\)
0.490955 + 0.871185i \(0.336648\pi\)
\(68\) −20692.3 −0.542670
\(69\) 34169.2 0.863996
\(70\) 0 0
\(71\) −63434.2 −1.49341 −0.746703 0.665158i \(-0.768364\pi\)
−0.746703 + 0.665158i \(0.768364\pi\)
\(72\) −5184.00 −0.117851
\(73\) −52982.4 −1.16366 −0.581828 0.813312i \(-0.697662\pi\)
−0.581828 + 0.813312i \(0.697662\pi\)
\(74\) 36314.0 0.770893
\(75\) 68280.5 1.40166
\(76\) −4406.03 −0.0875011
\(77\) 0 0
\(78\) −28997.2 −0.539659
\(79\) −48564.0 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(80\) 26495.4 0.462855
\(81\) 6561.00 0.111111
\(82\) 72829.7 1.19612
\(83\) 113161. 1.80303 0.901513 0.432753i \(-0.142458\pi\)
0.901513 + 0.432753i \(0.142458\pi\)
\(84\) 0 0
\(85\) −133850. −2.00942
\(86\) 46832.9 0.682818
\(87\) 11047.7 0.156485
\(88\) −15372.2 −0.211607
\(89\) 108366. 1.45017 0.725083 0.688662i \(-0.241801\pi\)
0.725083 + 0.688662i \(0.241801\pi\)
\(90\) −33533.2 −0.436384
\(91\) 0 0
\(92\) 60745.2 0.748242
\(93\) 50623.8 0.606942
\(94\) 92194.4 1.07618
\(95\) −28500.8 −0.324002
\(96\) −9216.00 −0.102062
\(97\) 99641.2 1.07525 0.537625 0.843184i \(-0.319321\pi\)
0.537625 + 0.843184i \(0.319321\pi\)
\(98\) 0 0
\(99\) 19455.5 0.199505
\(100\) 121388. 1.21388
\(101\) 165200. 1.61141 0.805705 0.592317i \(-0.201787\pi\)
0.805705 + 0.592317i \(0.201787\pi\)
\(102\) 46557.6 0.443088
\(103\) −107247. −0.996077 −0.498038 0.867155i \(-0.665946\pi\)
−0.498038 + 0.867155i \(0.665946\pi\)
\(104\) −51550.5 −0.467358
\(105\) 0 0
\(106\) −70651.2 −0.610738
\(107\) −97422.3 −0.822620 −0.411310 0.911496i \(-0.634929\pi\)
−0.411310 + 0.911496i \(0.634929\pi\)
\(108\) 11664.0 0.0962250
\(109\) 59307.2 0.478125 0.239062 0.971004i \(-0.423160\pi\)
0.239062 + 0.971004i \(0.423160\pi\)
\(110\) −99436.6 −0.783546
\(111\) −81706.4 −0.629432
\(112\) 0 0
\(113\) 157268. 1.15863 0.579313 0.815105i \(-0.303321\pi\)
0.579313 + 0.815105i \(0.303321\pi\)
\(114\) 9913.56 0.0714443
\(115\) 392936. 2.77062
\(116\) 19640.3 0.135520
\(117\) 65243.7 0.440630
\(118\) −73505.2 −0.485974
\(119\) 0 0
\(120\) −59614.5 −0.377919
\(121\) −103359. −0.641780
\(122\) −45296.3 −0.275527
\(123\) −163867. −0.976625
\(124\) 89997.8 0.525627
\(125\) 461778. 2.64337
\(126\) 0 0
\(127\) −92477.3 −0.508775 −0.254387 0.967102i \(-0.581874\pi\)
−0.254387 + 0.967102i \(0.581874\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −105374. −0.557518
\(130\) −333459. −1.73055
\(131\) −72328.4 −0.368240 −0.184120 0.982904i \(-0.558943\pi\)
−0.184120 + 0.982904i \(0.558943\pi\)
\(132\) 34587.5 0.172776
\(133\) 0 0
\(134\) −144317. −0.694315
\(135\) 75449.7 0.356306
\(136\) 82769.0 0.383725
\(137\) 225662. 1.02720 0.513602 0.858028i \(-0.328311\pi\)
0.513602 + 0.858028i \(0.328311\pi\)
\(138\) −136677. −0.610937
\(139\) −327971. −1.43979 −0.719894 0.694084i \(-0.755810\pi\)
−0.719894 + 0.694084i \(0.755810\pi\)
\(140\) 0 0
\(141\) −207437. −0.878697
\(142\) 253737. 1.05600
\(143\) 193468. 0.791170
\(144\) 20736.0 0.0833333
\(145\) 127045. 0.501808
\(146\) 211930. 0.822829
\(147\) 0 0
\(148\) −145256. −0.545104
\(149\) 164011. 0.605211 0.302606 0.953116i \(-0.402144\pi\)
0.302606 + 0.953116i \(0.402144\pi\)
\(150\) −273122. −0.991126
\(151\) 62308.8 0.222386 0.111193 0.993799i \(-0.464533\pi\)
0.111193 + 0.993799i \(0.464533\pi\)
\(152\) 17624.1 0.0618726
\(153\) −104755. −0.361780
\(154\) 0 0
\(155\) 582159. 1.94631
\(156\) 115989. 0.381596
\(157\) −239807. −0.776447 −0.388224 0.921565i \(-0.626911\pi\)
−0.388224 + 0.921565i \(0.626911\pi\)
\(158\) 194256. 0.619059
\(159\) 158965. 0.498665
\(160\) −105981. −0.327288
\(161\) 0 0
\(162\) −26244.0 −0.0785674
\(163\) −436795. −1.28768 −0.643841 0.765160i \(-0.722660\pi\)
−0.643841 + 0.765160i \(0.722660\pi\)
\(164\) −291319. −0.845782
\(165\) 223732. 0.639763
\(166\) −452644. −1.27493
\(167\) 38846.7 0.107786 0.0538931 0.998547i \(-0.482837\pi\)
0.0538931 + 0.998547i \(0.482837\pi\)
\(168\) 0 0
\(169\) 277501. 0.747390
\(170\) 535399. 1.42087
\(171\) −22305.5 −0.0583340
\(172\) −187331. −0.482825
\(173\) 396277. 1.00666 0.503331 0.864093i \(-0.332108\pi\)
0.503331 + 0.864093i \(0.332108\pi\)
\(174\) −44190.7 −0.110652
\(175\) 0 0
\(176\) 61488.9 0.149629
\(177\) 165387. 0.396796
\(178\) −433464. −1.02542
\(179\) 243883. 0.568918 0.284459 0.958688i \(-0.408186\pi\)
0.284459 + 0.958688i \(0.408186\pi\)
\(180\) 134133. 0.308570
\(181\) −299273. −0.679002 −0.339501 0.940606i \(-0.610258\pi\)
−0.339501 + 0.940606i \(0.610258\pi\)
\(182\) 0 0
\(183\) 101917. 0.224967
\(184\) −242981. −0.529087
\(185\) −939601. −2.01843
\(186\) −202495. −0.429173
\(187\) −310631. −0.649592
\(188\) −368778. −0.760974
\(189\) 0 0
\(190\) 114003. 0.229104
\(191\) −776996. −1.54112 −0.770558 0.637369i \(-0.780023\pi\)
−0.770558 + 0.637369i \(0.780023\pi\)
\(192\) 36864.0 0.0721688
\(193\) 276421. 0.534168 0.267084 0.963673i \(-0.413940\pi\)
0.267084 + 0.963673i \(0.413940\pi\)
\(194\) −398565. −0.760317
\(195\) 750284. 1.41299
\(196\) 0 0
\(197\) −131615. −0.241624 −0.120812 0.992675i \(-0.538550\pi\)
−0.120812 + 0.992675i \(0.538550\pi\)
\(198\) −77821.9 −0.141071
\(199\) 564960. 1.01131 0.505656 0.862735i \(-0.331250\pi\)
0.505656 + 0.862735i \(0.331250\pi\)
\(200\) −485551. −0.858340
\(201\) 324714. 0.566906
\(202\) −660799. −1.13944
\(203\) 0 0
\(204\) −186230. −0.313311
\(205\) −1.88442e6 −3.13180
\(206\) 428989. 0.704333
\(207\) 307522. 0.498828
\(208\) 206202. 0.330472
\(209\) −66143.0 −0.104741
\(210\) 0 0
\(211\) 313637. 0.484977 0.242489 0.970154i \(-0.422036\pi\)
0.242489 + 0.970154i \(0.422036\pi\)
\(212\) 282605. 0.431857
\(213\) −570908. −0.862218
\(214\) 389689. 0.581680
\(215\) −1.21177e6 −1.78782
\(216\) −46656.0 −0.0680414
\(217\) 0 0
\(218\) −237229. −0.338085
\(219\) −476841. −0.671837
\(220\) 397746. 0.554051
\(221\) −1.04170e6 −1.43470
\(222\) 326826. 0.445076
\(223\) −279483. −0.376351 −0.188176 0.982135i \(-0.560257\pi\)
−0.188176 + 0.982135i \(0.560257\pi\)
\(224\) 0 0
\(225\) 614525. 0.809251
\(226\) −629071. −0.819273
\(227\) −1.22351e6 −1.57595 −0.787974 0.615709i \(-0.788870\pi\)
−0.787974 + 0.615709i \(0.788870\pi\)
\(228\) −39654.3 −0.0505188
\(229\) 254775. 0.321046 0.160523 0.987032i \(-0.448682\pi\)
0.160523 + 0.987032i \(0.448682\pi\)
\(230\) −1.57174e6 −1.95912
\(231\) 0 0
\(232\) −78561.2 −0.0958270
\(233\) 746173. 0.900429 0.450215 0.892920i \(-0.351347\pi\)
0.450215 + 0.892920i \(0.351347\pi\)
\(234\) −260975. −0.311572
\(235\) −2.38547e6 −2.81776
\(236\) 294021. 0.343636
\(237\) −437076. −0.505459
\(238\) 0 0
\(239\) 321036. 0.363546 0.181773 0.983341i \(-0.441816\pi\)
0.181773 + 0.983341i \(0.441816\pi\)
\(240\) 238458. 0.267229
\(241\) 1.24254e6 1.37805 0.689027 0.724735i \(-0.258038\pi\)
0.689027 + 0.724735i \(0.258038\pi\)
\(242\) 413437. 0.453807
\(243\) 59049.0 0.0641500
\(244\) 181185. 0.194827
\(245\) 0 0
\(246\) 655467. 0.690578
\(247\) −221810. −0.231333
\(248\) −359991. −0.371675
\(249\) 1.01845e6 1.04098
\(250\) −1.84711e6 −1.86914
\(251\) 690235. 0.691532 0.345766 0.938321i \(-0.387619\pi\)
0.345766 + 0.938321i \(0.387619\pi\)
\(252\) 0 0
\(253\) 911902. 0.895668
\(254\) 369909. 0.359758
\(255\) −1.20465e6 −1.16014
\(256\) 65536.0 0.0625000
\(257\) 13786.7 0.0130205 0.00651024 0.999979i \(-0.497928\pi\)
0.00651024 + 0.999979i \(0.497928\pi\)
\(258\) 421496. 0.394225
\(259\) 0 0
\(260\) 1.33384e6 1.22369
\(261\) 99429.0 0.0903466
\(262\) 289314. 0.260385
\(263\) 574833. 0.512451 0.256226 0.966617i \(-0.417521\pi\)
0.256226 + 0.966617i \(0.417521\pi\)
\(264\) −138350. −0.122171
\(265\) 1.82806e6 1.59910
\(266\) 0 0
\(267\) 975293. 0.837253
\(268\) 577270. 0.490955
\(269\) −1.11893e6 −0.942804 −0.471402 0.881918i \(-0.656252\pi\)
−0.471402 + 0.881918i \(0.656252\pi\)
\(270\) −301799. −0.251946
\(271\) 694869. 0.574751 0.287376 0.957818i \(-0.407217\pi\)
0.287376 + 0.957818i \(0.407217\pi\)
\(272\) −331076. −0.271335
\(273\) 0 0
\(274\) −902648. −0.726343
\(275\) 1.82226e6 1.45305
\(276\) 546706. 0.431998
\(277\) −837654. −0.655941 −0.327971 0.944688i \(-0.606365\pi\)
−0.327971 + 0.944688i \(0.606365\pi\)
\(278\) 1.31188e6 1.01808
\(279\) 455614. 0.350418
\(280\) 0 0
\(281\) −917687. −0.693312 −0.346656 0.937992i \(-0.612683\pi\)
−0.346656 + 0.937992i \(0.612683\pi\)
\(282\) 829750. 0.621333
\(283\) −415263. −0.308218 −0.154109 0.988054i \(-0.549251\pi\)
−0.154109 + 0.988054i \(0.549251\pi\)
\(284\) −1.01495e6 −0.746703
\(285\) −256507. −0.187063
\(286\) −773873. −0.559442
\(287\) 0 0
\(288\) −82944.0 −0.0589256
\(289\) 252680. 0.177962
\(290\) −508180. −0.354832
\(291\) 896771. 0.620796
\(292\) −847718. −0.581828
\(293\) −1.21491e6 −0.826752 −0.413376 0.910561i \(-0.635650\pi\)
−0.413376 + 0.910561i \(0.635650\pi\)
\(294\) 0 0
\(295\) 1.90190e6 1.27243
\(296\) 581023. 0.385447
\(297\) 175099. 0.115184
\(298\) −656043. −0.427949
\(299\) 3.05805e6 1.97819
\(300\) 1.09249e6 0.700832
\(301\) 0 0
\(302\) −249235. −0.157250
\(303\) 1.48680e6 0.930348
\(304\) −70496.5 −0.0437505
\(305\) 1.17201e6 0.721412
\(306\) 419018. 0.255817
\(307\) −2.60119e6 −1.57517 −0.787584 0.616207i \(-0.788668\pi\)
−0.787584 + 0.616207i \(0.788668\pi\)
\(308\) 0 0
\(309\) −965225. −0.575085
\(310\) −2.32864e6 −1.37625
\(311\) −819164. −0.480253 −0.240126 0.970742i \(-0.577189\pi\)
−0.240126 + 0.970742i \(0.577189\pi\)
\(312\) −463955. −0.269829
\(313\) −881939. −0.508836 −0.254418 0.967094i \(-0.581884\pi\)
−0.254418 + 0.967094i \(0.581884\pi\)
\(314\) 959226. 0.549031
\(315\) 0 0
\(316\) −777024. −0.437741
\(317\) −1.60438e6 −0.896726 −0.448363 0.893852i \(-0.647993\pi\)
−0.448363 + 0.893852i \(0.647993\pi\)
\(318\) −635861. −0.352610
\(319\) 294839. 0.162221
\(320\) 423926. 0.231427
\(321\) −876801. −0.474940
\(322\) 0 0
\(323\) 356135. 0.189937
\(324\) 104976. 0.0555556
\(325\) 6.11094e6 3.20922
\(326\) 1.74718e6 0.910528
\(327\) 533765. 0.276045
\(328\) 1.16527e6 0.598058
\(329\) 0 0
\(330\) −894929. −0.452381
\(331\) 1.00308e6 0.503229 0.251614 0.967828i \(-0.419039\pi\)
0.251614 + 0.967828i \(0.419039\pi\)
\(332\) 1.81058e6 0.901513
\(333\) −735358. −0.363403
\(334\) −155387. −0.0762164
\(335\) 3.73412e6 1.81793
\(336\) 0 0
\(337\) −2.62317e6 −1.25821 −0.629103 0.777322i \(-0.716578\pi\)
−0.629103 + 0.777322i \(0.716578\pi\)
\(338\) −1.11000e6 −0.528484
\(339\) 1.41541e6 0.668933
\(340\) −2.14160e6 −1.00471
\(341\) 1.35104e6 0.629191
\(342\) 89222.1 0.0412484
\(343\) 0 0
\(344\) 749326. 0.341409
\(345\) 3.53642e6 1.59962
\(346\) −1.58511e6 −0.711818
\(347\) 226234. 0.100863 0.0504317 0.998728i \(-0.483940\pi\)
0.0504317 + 0.998728i \(0.483940\pi\)
\(348\) 176763. 0.0782424
\(349\) −1.46280e6 −0.642866 −0.321433 0.946932i \(-0.604165\pi\)
−0.321433 + 0.946932i \(0.604165\pi\)
\(350\) 0 0
\(351\) 587193. 0.254398
\(352\) −245955. −0.105803
\(353\) 2.80957e6 1.20006 0.600031 0.799977i \(-0.295155\pi\)
0.600031 + 0.799977i \(0.295155\pi\)
\(354\) −661547. −0.280577
\(355\) −6.56528e6 −2.76492
\(356\) 1.73385e6 0.725083
\(357\) 0 0
\(358\) −975534. −0.402286
\(359\) −3.69900e6 −1.51478 −0.757388 0.652965i \(-0.773525\pi\)
−0.757388 + 0.652965i \(0.773525\pi\)
\(360\) −536531. −0.218192
\(361\) −2.40027e6 −0.969374
\(362\) 1.19709e6 0.480127
\(363\) −930234. −0.370532
\(364\) 0 0
\(365\) −5.48354e6 −2.15441
\(366\) −407667. −0.159075
\(367\) 1.16925e6 0.453148 0.226574 0.973994i \(-0.427247\pi\)
0.226574 + 0.973994i \(0.427247\pi\)
\(368\) 971923. 0.374121
\(369\) −1.47480e6 −0.563855
\(370\) 3.75840e6 1.42725
\(371\) 0 0
\(372\) 809980. 0.303471
\(373\) −4.57690e6 −1.70333 −0.851666 0.524084i \(-0.824408\pi\)
−0.851666 + 0.524084i \(0.824408\pi\)
\(374\) 1.24252e6 0.459331
\(375\) 4.15600e6 1.52615
\(376\) 1.47511e6 0.538090
\(377\) 988738. 0.358284
\(378\) 0 0
\(379\) −2.35520e6 −0.842229 −0.421114 0.907007i \(-0.638361\pi\)
−0.421114 + 0.907007i \(0.638361\pi\)
\(380\) −456013. −0.162001
\(381\) −832295. −0.293741
\(382\) 3.10799e6 1.08973
\(383\) 3.03946e6 1.05876 0.529382 0.848384i \(-0.322424\pi\)
0.529382 + 0.848384i \(0.322424\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −1.10568e6 −0.377714
\(387\) −948366. −0.321883
\(388\) 1.59426e6 0.537625
\(389\) −459273. −0.153885 −0.0769426 0.997036i \(-0.524516\pi\)
−0.0769426 + 0.997036i \(0.524516\pi\)
\(390\) −3.00114e6 −0.999135
\(391\) −4.90998e6 −1.62419
\(392\) 0 0
\(393\) −650956. −0.212603
\(394\) 526461. 0.170854
\(395\) −5.02625e6 −1.62088
\(396\) 311287. 0.0997525
\(397\) 5.42500e6 1.72752 0.863761 0.503901i \(-0.168102\pi\)
0.863761 + 0.503901i \(0.168102\pi\)
\(398\) −2.25984e6 −0.715106
\(399\) 0 0
\(400\) 1.94220e6 0.606938
\(401\) 2.11127e6 0.655667 0.327833 0.944736i \(-0.393682\pi\)
0.327833 + 0.944736i \(0.393682\pi\)
\(402\) −1.29886e6 −0.400863
\(403\) 4.53070e6 1.38964
\(404\) 2.64320e6 0.805705
\(405\) 679047. 0.205713
\(406\) 0 0
\(407\) −2.18057e6 −0.652506
\(408\) 744921. 0.221544
\(409\) 2.55745e6 0.755959 0.377980 0.925814i \(-0.376619\pi\)
0.377980 + 0.925814i \(0.376619\pi\)
\(410\) 7.53769e6 2.21451
\(411\) 2.03096e6 0.593057
\(412\) −1.71595e6 −0.498038
\(413\) 0 0
\(414\) −1.23009e6 −0.352725
\(415\) 1.17119e7 3.33816
\(416\) −824809. −0.233679
\(417\) −2.95174e6 −0.831262
\(418\) 264572. 0.0740633
\(419\) 2.17401e6 0.604959 0.302480 0.953156i \(-0.402186\pi\)
0.302480 + 0.953156i \(0.402186\pi\)
\(420\) 0 0
\(421\) −5.47930e6 −1.50668 −0.753338 0.657633i \(-0.771558\pi\)
−0.753338 + 0.657633i \(0.771558\pi\)
\(422\) −1.25455e6 −0.342931
\(423\) −1.86694e6 −0.507316
\(424\) −1.13042e6 −0.305369
\(425\) −9.81166e6 −2.63494
\(426\) 2.28363e6 0.609680
\(427\) 0 0
\(428\) −1.55876e6 −0.411310
\(429\) 1.74121e6 0.456782
\(430\) 4.84708e6 1.26418
\(431\) −6.16874e6 −1.59957 −0.799785 0.600286i \(-0.795053\pi\)
−0.799785 + 0.600286i \(0.795053\pi\)
\(432\) 186624. 0.0481125
\(433\) 4.89596e6 1.25493 0.627463 0.778647i \(-0.284093\pi\)
0.627463 + 0.778647i \(0.284093\pi\)
\(434\) 0 0
\(435\) 1.14341e6 0.289719
\(436\) 948915. 0.239062
\(437\) −1.04549e6 −0.261888
\(438\) 1.90737e6 0.475060
\(439\) −3.54588e6 −0.878137 −0.439069 0.898454i \(-0.644691\pi\)
−0.439069 + 0.898454i \(0.644691\pi\)
\(440\) −1.59099e6 −0.391773
\(441\) 0 0
\(442\) 4.16679e6 1.01448
\(443\) 6.05652e6 1.46627 0.733134 0.680084i \(-0.238057\pi\)
0.733134 + 0.680084i \(0.238057\pi\)
\(444\) −1.30730e6 −0.314716
\(445\) 1.12156e7 2.68486
\(446\) 1.11793e6 0.266121
\(447\) 1.47610e6 0.349419
\(448\) 0 0
\(449\) −1.20801e6 −0.282783 −0.141392 0.989954i \(-0.545158\pi\)
−0.141392 + 0.989954i \(0.545158\pi\)
\(450\) −2.45810e6 −0.572227
\(451\) −4.37326e6 −1.01243
\(452\) 2.51628e6 0.579313
\(453\) 560779. 0.128394
\(454\) 4.89403e6 1.11436
\(455\) 0 0
\(456\) 158617. 0.0357222
\(457\) −5.46550e6 −1.22416 −0.612082 0.790794i \(-0.709668\pi\)
−0.612082 + 0.790794i \(0.709668\pi\)
\(458\) −1.01910e6 −0.227014
\(459\) −942791. −0.208874
\(460\) 6.28697e6 1.38531
\(461\) 1.15956e6 0.254121 0.127061 0.991895i \(-0.459446\pi\)
0.127061 + 0.991895i \(0.459446\pi\)
\(462\) 0 0
\(463\) −4.31335e6 −0.935108 −0.467554 0.883964i \(-0.654865\pi\)
−0.467554 + 0.883964i \(0.654865\pi\)
\(464\) 314245. 0.0677599
\(465\) 5.23943e6 1.12370
\(466\) −2.98469e6 −0.636700
\(467\) −5.98923e6 −1.27081 −0.635403 0.772181i \(-0.719166\pi\)
−0.635403 + 0.772181i \(0.719166\pi\)
\(468\) 1.04390e6 0.220315
\(469\) 0 0
\(470\) 9.54189e6 1.99246
\(471\) −2.15826e6 −0.448282
\(472\) −1.17608e6 −0.242987
\(473\) −2.81221e6 −0.577956
\(474\) 1.74830e6 0.357414
\(475\) −2.08921e6 −0.424862
\(476\) 0 0
\(477\) 1.43069e6 0.287905
\(478\) −1.28415e6 −0.257066
\(479\) 3.54652e6 0.706258 0.353129 0.935575i \(-0.385118\pi\)
0.353129 + 0.935575i \(0.385118\pi\)
\(480\) −953833. −0.188960
\(481\) −7.31252e6 −1.44113
\(482\) −4.97015e6 −0.974432
\(483\) 0 0
\(484\) −1.65375e6 −0.320890
\(485\) 1.03126e7 1.99074
\(486\) −236196. −0.0453609
\(487\) −8.78320e6 −1.67815 −0.839074 0.544017i \(-0.816903\pi\)
−0.839074 + 0.544017i \(0.816903\pi\)
\(488\) −724742. −0.137763
\(489\) −3.93115e6 −0.743443
\(490\) 0 0
\(491\) −586608. −0.109811 −0.0549053 0.998492i \(-0.517486\pi\)
−0.0549053 + 0.998492i \(0.517486\pi\)
\(492\) −2.62187e6 −0.488313
\(493\) −1.58751e6 −0.294170
\(494\) 887239. 0.163577
\(495\) 2.01359e6 0.369367
\(496\) 1.43997e6 0.262814
\(497\) 0 0
\(498\) −4.07380e6 −0.736082
\(499\) −9.59772e6 −1.72551 −0.862753 0.505626i \(-0.831262\pi\)
−0.862753 + 0.505626i \(0.831262\pi\)
\(500\) 7.38844e6 1.32168
\(501\) 349620. 0.0622304
\(502\) −2.76094e6 −0.488987
\(503\) −7.21481e6 −1.27147 −0.635733 0.771909i \(-0.719302\pi\)
−0.635733 + 0.771909i \(0.719302\pi\)
\(504\) 0 0
\(505\) 1.70978e7 2.98340
\(506\) −3.64761e6 −0.633333
\(507\) 2.49751e6 0.431506
\(508\) −1.47964e6 −0.254387
\(509\) 4.38419e6 0.750058 0.375029 0.927013i \(-0.377633\pi\)
0.375029 + 0.927013i \(0.377633\pi\)
\(510\) 4.81859e6 0.820342
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) −200750. −0.0336792
\(514\) −55146.8 −0.00920688
\(515\) −1.10998e7 −1.84416
\(516\) −1.68598e6 −0.278759
\(517\) −5.53606e6 −0.910909
\(518\) 0 0
\(519\) 3.56650e6 0.581197
\(520\) −5.33535e6 −0.865276
\(521\) 2.06581e6 0.333424 0.166712 0.986006i \(-0.446685\pi\)
0.166712 + 0.986006i \(0.446685\pi\)
\(522\) −397716. −0.0638847
\(523\) 7.96570e6 1.27341 0.636707 0.771106i \(-0.280296\pi\)
0.636707 + 0.771106i \(0.280296\pi\)
\(524\) −1.15725e6 −0.184120
\(525\) 0 0
\(526\) −2.29933e6 −0.362358
\(527\) −7.27445e6 −1.14097
\(528\) 553400. 0.0863882
\(529\) 7.97762e6 1.23946
\(530\) −7.31222e6 −1.13073
\(531\) 1.48848e6 0.229090
\(532\) 0 0
\(533\) −1.46657e7 −2.23606
\(534\) −3.90117e6 −0.592028
\(535\) −1.00830e7 −1.52301
\(536\) −2.30908e6 −0.347158
\(537\) 2.19495e6 0.328465
\(538\) 4.47571e6 0.666663
\(539\) 0 0
\(540\) 1.20719e6 0.178153
\(541\) 8.23423e6 1.20957 0.604784 0.796390i \(-0.293260\pi\)
0.604784 + 0.796390i \(0.293260\pi\)
\(542\) −2.77948e6 −0.406411
\(543\) −2.69346e6 −0.392022
\(544\) 1.32430e6 0.191863
\(545\) 6.13815e6 0.885209
\(546\) 0 0
\(547\) 5.20365e6 0.743601 0.371800 0.928313i \(-0.378741\pi\)
0.371800 + 0.928313i \(0.378741\pi\)
\(548\) 3.61059e6 0.513602
\(549\) 917251. 0.129885
\(550\) −7.28905e6 −1.02746
\(551\) −338030. −0.0474325
\(552\) −2.18683e6 −0.305469
\(553\) 0 0
\(554\) 3.35061e6 0.463821
\(555\) −8.45641e6 −1.16534
\(556\) −5.24754e6 −0.719894
\(557\) −4.09165e6 −0.558806 −0.279403 0.960174i \(-0.590136\pi\)
−0.279403 + 0.960174i \(0.590136\pi\)
\(558\) −1.82246e6 −0.247783
\(559\) −9.43070e6 −1.27648
\(560\) 0 0
\(561\) −2.79568e6 −0.375042
\(562\) 3.67075e6 0.490246
\(563\) −3.59052e6 −0.477404 −0.238702 0.971093i \(-0.576722\pi\)
−0.238702 + 0.971093i \(0.576722\pi\)
\(564\) −3.31900e6 −0.439349
\(565\) 1.62768e7 2.14510
\(566\) 1.66105e6 0.217943
\(567\) 0 0
\(568\) 4.05979e6 0.527999
\(569\) 9.79717e6 1.26859 0.634293 0.773093i \(-0.281291\pi\)
0.634293 + 0.773093i \(0.281291\pi\)
\(570\) 1.02603e6 0.132273
\(571\) −2.21804e6 −0.284694 −0.142347 0.989817i \(-0.545465\pi\)
−0.142347 + 0.989817i \(0.545465\pi\)
\(572\) 3.09549e6 0.395585
\(573\) −6.99297e6 −0.889764
\(574\) 0 0
\(575\) 2.88036e7 3.63309
\(576\) 331776. 0.0416667
\(577\) −2.65534e6 −0.332032 −0.166016 0.986123i \(-0.553090\pi\)
−0.166016 + 0.986123i \(0.553090\pi\)
\(578\) −1.01072e6 −0.125838
\(579\) 2.48779e6 0.308402
\(580\) 2.03272e6 0.250904
\(581\) 0 0
\(582\) −3.58708e6 −0.438969
\(583\) 4.24244e6 0.516945
\(584\) 3.39087e6 0.411414
\(585\) 6.75255e6 0.815790
\(586\) 4.85964e6 0.584602
\(587\) −2.62690e6 −0.314665 −0.157332 0.987546i \(-0.550289\pi\)
−0.157332 + 0.987546i \(0.550289\pi\)
\(588\) 0 0
\(589\) −1.54896e6 −0.183972
\(590\) −7.60760e6 −0.899742
\(591\) −1.18454e6 −0.139502
\(592\) −2.32409e6 −0.272552
\(593\) −1.15062e7 −1.34368 −0.671839 0.740697i \(-0.734495\pi\)
−0.671839 + 0.740697i \(0.734495\pi\)
\(594\) −700397. −0.0814475
\(595\) 0 0
\(596\) 2.62417e6 0.302606
\(597\) 5.08464e6 0.583881
\(598\) −1.22322e7 −1.39879
\(599\) 6.46264e6 0.735941 0.367970 0.929838i \(-0.380053\pi\)
0.367970 + 0.929838i \(0.380053\pi\)
\(600\) −4.36995e6 −0.495563
\(601\) 8.55332e6 0.965937 0.482968 0.875638i \(-0.339559\pi\)
0.482968 + 0.875638i \(0.339559\pi\)
\(602\) 0 0
\(603\) 2.92243e6 0.327303
\(604\) 996941. 0.111193
\(605\) −1.06974e7 −1.18820
\(606\) −5.94719e6 −0.657856
\(607\) 2.99042e6 0.329428 0.164714 0.986341i \(-0.447330\pi\)
0.164714 + 0.986341i \(0.447330\pi\)
\(608\) 281986. 0.0309363
\(609\) 0 0
\(610\) −4.68806e6 −0.510115
\(611\) −1.85651e7 −2.01185
\(612\) −1.67607e6 −0.180890
\(613\) −1.77145e6 −0.190405 −0.0952023 0.995458i \(-0.530350\pi\)
−0.0952023 + 0.995458i \(0.530350\pi\)
\(614\) 1.04048e7 1.11381
\(615\) −1.69598e7 −1.80814
\(616\) 0 0
\(617\) 1.40871e7 1.48974 0.744869 0.667210i \(-0.232512\pi\)
0.744869 + 0.667210i \(0.232512\pi\)
\(618\) 3.86090e6 0.406647
\(619\) −1.13430e7 −1.18988 −0.594940 0.803770i \(-0.702824\pi\)
−0.594940 + 0.803770i \(0.702824\pi\)
\(620\) 9.31455e6 0.973156
\(621\) 2.76770e6 0.287999
\(622\) 3.27666e6 0.339590
\(623\) 0 0
\(624\) 1.85582e6 0.190798
\(625\) 2.40843e7 2.46623
\(626\) 3.52776e6 0.359801
\(627\) −595287. −0.0604724
\(628\) −3.83690e6 −0.388224
\(629\) 1.17409e7 1.18325
\(630\) 0 0
\(631\) −2.32628e6 −0.232588 −0.116294 0.993215i \(-0.537102\pi\)
−0.116294 + 0.993215i \(0.537102\pi\)
\(632\) 3.10810e6 0.309529
\(633\) 2.82273e6 0.280002
\(634\) 6.41753e6 0.634081
\(635\) −9.57116e6 −0.941956
\(636\) 2.54344e6 0.249333
\(637\) 0 0
\(638\) −1.17935e6 −0.114708
\(639\) −5.13817e6 −0.497802
\(640\) −1.69570e6 −0.163644
\(641\) −2.56385e6 −0.246461 −0.123230 0.992378i \(-0.539325\pi\)
−0.123230 + 0.992378i \(0.539325\pi\)
\(642\) 3.50720e6 0.335833
\(643\) −5.95343e6 −0.567858 −0.283929 0.958845i \(-0.591638\pi\)
−0.283929 + 0.958845i \(0.591638\pi\)
\(644\) 0 0
\(645\) −1.09059e7 −1.03220
\(646\) −1.42454e6 −0.134306
\(647\) −1.53522e6 −0.144182 −0.0720909 0.997398i \(-0.522967\pi\)
−0.0720909 + 0.997398i \(0.522967\pi\)
\(648\) −419904. −0.0392837
\(649\) 4.41382e6 0.411342
\(650\) −2.44437e7 −2.26926
\(651\) 0 0
\(652\) −6.98871e6 −0.643841
\(653\) −1.99582e7 −1.83163 −0.915815 0.401600i \(-0.868454\pi\)
−0.915815 + 0.401600i \(0.868454\pi\)
\(654\) −2.13506e6 −0.195194
\(655\) −7.48581e6 −0.681766
\(656\) −4.66110e6 −0.422891
\(657\) −4.29157e6 −0.387885
\(658\) 0 0
\(659\) −1.05477e7 −0.946119 −0.473060 0.881030i \(-0.656850\pi\)
−0.473060 + 0.881030i \(0.656850\pi\)
\(660\) 3.57972e6 0.319881
\(661\) −1.22044e7 −1.08646 −0.543229 0.839585i \(-0.682798\pi\)
−0.543229 + 0.839585i \(0.682798\pi\)
\(662\) −4.01232e6 −0.355837
\(663\) −9.37527e6 −0.828323
\(664\) −7.24231e6 −0.637466
\(665\) 0 0
\(666\) 2.94143e6 0.256964
\(667\) 4.66036e6 0.405607
\(668\) 621548. 0.0538931
\(669\) −2.51535e6 −0.217287
\(670\) −1.49365e7 −1.28547
\(671\) 2.71994e6 0.233213
\(672\) 0 0
\(673\) −7.97959e6 −0.679114 −0.339557 0.940585i \(-0.610277\pi\)
−0.339557 + 0.940585i \(0.610277\pi\)
\(674\) 1.04927e7 0.889686
\(675\) 5.53072e6 0.467221
\(676\) 4.44001e6 0.373695
\(677\) 5.08381e6 0.426302 0.213151 0.977019i \(-0.431627\pi\)
0.213151 + 0.977019i \(0.431627\pi\)
\(678\) −5.66164e6 −0.473007
\(679\) 0 0
\(680\) 8.56639e6 0.710437
\(681\) −1.10116e7 −0.909874
\(682\) −5.40416e6 −0.444905
\(683\) −1.60331e7 −1.31512 −0.657559 0.753403i \(-0.728411\pi\)
−0.657559 + 0.753403i \(0.728411\pi\)
\(684\) −356888. −0.0291670
\(685\) 2.33554e7 1.90179
\(686\) 0 0
\(687\) 2.29297e6 0.185356
\(688\) −2.99730e6 −0.241412
\(689\) 1.42270e7 1.14173
\(690\) −1.41457e7 −1.13110
\(691\) −1.10131e7 −0.877438 −0.438719 0.898624i \(-0.644568\pi\)
−0.438719 + 0.898624i \(0.644568\pi\)
\(692\) 6.34044e6 0.503331
\(693\) 0 0
\(694\) −904935. −0.0713212
\(695\) −3.39442e7 −2.66565
\(696\) −707050. −0.0553258
\(697\) 2.35470e7 1.83592
\(698\) 5.85119e6 0.454575
\(699\) 6.71555e6 0.519863
\(700\) 0 0
\(701\) −3.10660e6 −0.238776 −0.119388 0.992848i \(-0.538093\pi\)
−0.119388 + 0.992848i \(0.538093\pi\)
\(702\) −2.34877e6 −0.179886
\(703\) 2.50001e6 0.190789
\(704\) 983822. 0.0748143
\(705\) −2.14692e7 −1.62684
\(706\) −1.12383e7 −0.848571
\(707\) 0 0
\(708\) 2.64619e6 0.198398
\(709\) −1.38036e7 −1.03128 −0.515640 0.856805i \(-0.672446\pi\)
−0.515640 + 0.856805i \(0.672446\pi\)
\(710\) 2.62611e7 1.95509
\(711\) −3.93369e6 −0.291827
\(712\) −6.93542e6 −0.512711
\(713\) 2.13552e7 1.57319
\(714\) 0 0
\(715\) 2.00235e7 1.46479
\(716\) 3.90214e6 0.284459
\(717\) 2.88933e6 0.209893
\(718\) 1.47960e7 1.07111
\(719\) −1.29855e7 −0.936777 −0.468389 0.883523i \(-0.655165\pi\)
−0.468389 + 0.883523i \(0.655165\pi\)
\(720\) 2.14612e6 0.154285
\(721\) 0 0
\(722\) 9.60107e6 0.685451
\(723\) 1.11828e7 0.795620
\(724\) −4.78837e6 −0.339501
\(725\) 9.31285e6 0.658017
\(726\) 3.72094e6 0.262006
\(727\) −8.61224e6 −0.604338 −0.302169 0.953254i \(-0.597711\pi\)
−0.302169 + 0.953254i \(0.597711\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 2.19342e7 1.52340
\(731\) 1.51418e7 1.04806
\(732\) 1.63067e6 0.112483
\(733\) 6.02001e6 0.413844 0.206922 0.978357i \(-0.433655\pi\)
0.206922 + 0.978357i \(0.433655\pi\)
\(734\) −4.67698e6 −0.320424
\(735\) 0 0
\(736\) −3.88769e6 −0.264544
\(737\) 8.66593e6 0.587688
\(738\) 5.89920e6 0.398706
\(739\) −93692.5 −0.00631094 −0.00315547 0.999995i \(-0.501004\pi\)
−0.00315547 + 0.999995i \(0.501004\pi\)
\(740\) −1.50336e7 −1.00922
\(741\) −1.99629e6 −0.133560
\(742\) 0 0
\(743\) −1.29874e7 −0.863081 −0.431541 0.902094i \(-0.642030\pi\)
−0.431541 + 0.902094i \(0.642030\pi\)
\(744\) −3.23992e6 −0.214586
\(745\) 1.69747e7 1.12050
\(746\) 1.83076e7 1.20444
\(747\) 9.16605e6 0.601008
\(748\) −4.97009e6 −0.324796
\(749\) 0 0
\(750\) −1.66240e7 −1.07915
\(751\) −637441. −0.0412420 −0.0206210 0.999787i \(-0.506564\pi\)
−0.0206210 + 0.999787i \(0.506564\pi\)
\(752\) −5.90044e6 −0.380487
\(753\) 6.21211e6 0.399256
\(754\) −3.95495e6 −0.253345
\(755\) 6.44880e6 0.411729
\(756\) 0 0
\(757\) −4.86768e6 −0.308732 −0.154366 0.988014i \(-0.549334\pi\)
−0.154366 + 0.988014i \(0.549334\pi\)
\(758\) 9.42081e6 0.595546
\(759\) 8.20712e6 0.517114
\(760\) 1.82405e6 0.114552
\(761\) 5.81787e6 0.364169 0.182084 0.983283i \(-0.441716\pi\)
0.182084 + 0.983283i \(0.441716\pi\)
\(762\) 3.32918e6 0.207706
\(763\) 0 0
\(764\) −1.24319e7 −0.770558
\(765\) −1.08418e7 −0.669806
\(766\) −1.21578e7 −0.748659
\(767\) 1.48017e7 0.908496
\(768\) 589824. 0.0360844
\(769\) 1.94310e7 1.18490 0.592448 0.805609i \(-0.298161\pi\)
0.592448 + 0.805609i \(0.298161\pi\)
\(770\) 0 0
\(771\) 124080. 0.00751738
\(772\) 4.42274e6 0.267084
\(773\) 8.24179e6 0.496104 0.248052 0.968747i \(-0.420210\pi\)
0.248052 + 0.968747i \(0.420210\pi\)
\(774\) 3.79346e6 0.227606
\(775\) 4.26743e7 2.55219
\(776\) −6.37704e6 −0.380159
\(777\) 0 0
\(778\) 1.83709e6 0.108813
\(779\) 5.01390e6 0.296027
\(780\) 1.20045e7 0.706495
\(781\) −1.52363e7 −0.893826
\(782\) 1.96399e7 1.14848
\(783\) 894861. 0.0521616
\(784\) 0 0
\(785\) −2.48194e7 −1.43753
\(786\) 2.60382e6 0.150333
\(787\) −1.47863e7 −0.850986 −0.425493 0.904962i \(-0.639899\pi\)
−0.425493 + 0.904962i \(0.639899\pi\)
\(788\) −2.10585e6 −0.120812
\(789\) 5.17350e6 0.295864
\(790\) 2.01050e7 1.14614
\(791\) 0 0
\(792\) −1.24515e6 −0.0705356
\(793\) 9.12129e6 0.515079
\(794\) −2.17000e7 −1.22154
\(795\) 1.64525e7 0.923239
\(796\) 9.03937e6 0.505656
\(797\) −1.58158e7 −0.881956 −0.440978 0.897518i \(-0.645368\pi\)
−0.440978 + 0.897518i \(0.645368\pi\)
\(798\) 0 0
\(799\) 2.98080e7 1.65183
\(800\) −7.76881e6 −0.429170
\(801\) 8.77764e6 0.483389
\(802\) −8.44508e6 −0.463626
\(803\) −1.27259e7 −0.696465
\(804\) 5.19543e6 0.283453
\(805\) 0 0
\(806\) −1.81228e7 −0.982625
\(807\) −1.00704e7 −0.544328
\(808\) −1.05728e7 −0.569720
\(809\) 1.19304e7 0.640888 0.320444 0.947267i \(-0.396168\pi\)
0.320444 + 0.947267i \(0.396168\pi\)
\(810\) −2.71619e6 −0.145461
\(811\) 1.89393e7 1.01114 0.505571 0.862785i \(-0.331282\pi\)
0.505571 + 0.862785i \(0.331282\pi\)
\(812\) 0 0
\(813\) 6.25382e6 0.331833
\(814\) 8.72228e6 0.461391
\(815\) −4.52071e7 −2.38404
\(816\) −2.97969e6 −0.156655
\(817\) 3.22417e6 0.168991
\(818\) −1.02298e7 −0.534544
\(819\) 0 0
\(820\) −3.01507e7 −1.56590
\(821\) 3.20087e7 1.65733 0.828667 0.559741i \(-0.189100\pi\)
0.828667 + 0.559741i \(0.189100\pi\)
\(822\) −8.12383e6 −0.419354
\(823\) 3.61430e7 1.86005 0.930025 0.367495i \(-0.119785\pi\)
0.930025 + 0.367495i \(0.119785\pi\)
\(824\) 6.86382e6 0.352166
\(825\) 1.64004e7 0.838916
\(826\) 0 0
\(827\) −2.45831e7 −1.24989 −0.624946 0.780668i \(-0.714879\pi\)
−0.624946 + 0.780668i \(0.714879\pi\)
\(828\) 4.92036e6 0.249414
\(829\) 2.10424e7 1.06343 0.531715 0.846923i \(-0.321548\pi\)
0.531715 + 0.846923i \(0.321548\pi\)
\(830\) −4.68475e7 −2.36043
\(831\) −7.53888e6 −0.378708
\(832\) 3.29923e6 0.165236
\(833\) 0 0
\(834\) 1.18070e7 0.587791
\(835\) 4.02054e6 0.199557
\(836\) −1.05829e6 −0.0523707
\(837\) 4.10053e6 0.202314
\(838\) −8.69604e6 −0.427771
\(839\) 1.12403e7 0.551283 0.275642 0.961260i \(-0.411110\pi\)
0.275642 + 0.961260i \(0.411110\pi\)
\(840\) 0 0
\(841\) −1.90043e7 −0.926537
\(842\) 2.19172e7 1.06538
\(843\) −8.25918e6 −0.400284
\(844\) 5.01820e6 0.242489
\(845\) 2.87206e7 1.38373
\(846\) 7.46775e6 0.358727
\(847\) 0 0
\(848\) 4.52168e6 0.215928
\(849\) −3.73737e6 −0.177950
\(850\) 3.92466e7 1.86318
\(851\) −3.44672e7 −1.63148
\(852\) −9.13453e6 −0.431109
\(853\) −2.63779e7 −1.24127 −0.620636 0.784099i \(-0.713126\pi\)
−0.620636 + 0.784099i \(0.713126\pi\)
\(854\) 0 0
\(855\) −2.30856e6 −0.108001
\(856\) 6.23503e6 0.290840
\(857\) 4.26663e7 1.98442 0.992208 0.124595i \(-0.0397632\pi\)
0.992208 + 0.124595i \(0.0397632\pi\)
\(858\) −6.96486e6 −0.322994
\(859\) 2.92617e7 1.35306 0.676530 0.736415i \(-0.263483\pi\)
0.676530 + 0.736415i \(0.263483\pi\)
\(860\) −1.93883e7 −0.893911
\(861\) 0 0
\(862\) 2.46750e7 1.13107
\(863\) 1.39448e7 0.637359 0.318680 0.947862i \(-0.396761\pi\)
0.318680 + 0.947862i \(0.396761\pi\)
\(864\) −746496. −0.0340207
\(865\) 4.10137e7 1.86375
\(866\) −1.95838e7 −0.887366
\(867\) 2.27412e6 0.102746
\(868\) 0 0
\(869\) −1.16646e7 −0.523988
\(870\) −4.57362e6 −0.204862
\(871\) 2.90611e7 1.29798
\(872\) −3.79566e6 −0.169043
\(873\) 8.07094e6 0.358417
\(874\) 4.18195e6 0.185183
\(875\) 0 0
\(876\) −7.62946e6 −0.335918
\(877\) 3.40772e6 0.149611 0.0748057 0.997198i \(-0.476166\pi\)
0.0748057 + 0.997198i \(0.476166\pi\)
\(878\) 1.41835e7 0.620937
\(879\) −1.09342e7 −0.477325
\(880\) 6.36394e6 0.277025
\(881\) −3.51391e6 −0.152528 −0.0762641 0.997088i \(-0.524299\pi\)
−0.0762641 + 0.997088i \(0.524299\pi\)
\(882\) 0 0
\(883\) 2.14895e7 0.927524 0.463762 0.885960i \(-0.346499\pi\)
0.463762 + 0.885960i \(0.346499\pi\)
\(884\) −1.66671e7 −0.717349
\(885\) 1.71171e7 0.734636
\(886\) −2.42261e7 −1.03681
\(887\) 2.45357e7 1.04710 0.523550 0.851995i \(-0.324607\pi\)
0.523550 + 0.851995i \(0.324607\pi\)
\(888\) 5.22921e6 0.222538
\(889\) 0 0
\(890\) −4.48624e7 −1.89849
\(891\) 1.57589e6 0.0665016
\(892\) −4.47173e6 −0.188176
\(893\) 6.34705e6 0.266344
\(894\) −5.90439e6 −0.247076
\(895\) 2.52413e7 1.05331
\(896\) 0 0
\(897\) 2.75225e7 1.14211
\(898\) 4.83203e6 0.199958
\(899\) 6.90462e6 0.284932
\(900\) 9.83240e6 0.404625
\(901\) −2.28427e7 −0.937423
\(902\) 1.74930e7 0.715894
\(903\) 0 0
\(904\) −1.00651e7 −0.409636
\(905\) −3.09740e7 −1.25712
\(906\) −2.24312e6 −0.0907886
\(907\) −8.14146e6 −0.328613 −0.164306 0.986409i \(-0.552539\pi\)
−0.164306 + 0.986409i \(0.552539\pi\)
\(908\) −1.95761e7 −0.787974
\(909\) 1.33812e7 0.537137
\(910\) 0 0
\(911\) 1.01384e7 0.404739 0.202370 0.979309i \(-0.435136\pi\)
0.202370 + 0.979309i \(0.435136\pi\)
\(912\) −634468. −0.0252594
\(913\) 2.71803e7 1.07914
\(914\) 2.18620e7 0.865615
\(915\) 1.05481e7 0.416507
\(916\) 4.07639e6 0.160523
\(917\) 0 0
\(918\) 3.77116e6 0.147696
\(919\) −1.01176e7 −0.395175 −0.197587 0.980285i \(-0.563311\pi\)
−0.197587 + 0.980285i \(0.563311\pi\)
\(920\) −2.51479e7 −0.979562
\(921\) −2.34108e7 −0.909424
\(922\) −4.63824e6 −0.179691
\(923\) −5.10948e7 −1.97412
\(924\) 0 0
\(925\) −6.88760e7 −2.64676
\(926\) 1.72534e7 0.661221
\(927\) −8.68702e6 −0.332026
\(928\) −1.25698e6 −0.0479135
\(929\) 2.05111e7 0.779739 0.389870 0.920870i \(-0.372520\pi\)
0.389870 + 0.920870i \(0.372520\pi\)
\(930\) −2.09577e7 −0.794579
\(931\) 0 0
\(932\) 1.19388e7 0.450215
\(933\) −7.37248e6 −0.277274
\(934\) 2.39569e7 0.898595
\(935\) −3.21495e7 −1.20267
\(936\) −4.17559e6 −0.155786
\(937\) −1.46527e7 −0.545218 −0.272609 0.962125i \(-0.587886\pi\)
−0.272609 + 0.962125i \(0.587886\pi\)
\(938\) 0 0
\(939\) −7.93745e6 −0.293777
\(940\) −3.81676e7 −1.40888
\(941\) 2.59341e7 0.954765 0.477382 0.878696i \(-0.341586\pi\)
0.477382 + 0.878696i \(0.341586\pi\)
\(942\) 8.63303e6 0.316983
\(943\) −6.91258e7 −2.53140
\(944\) 4.70433e6 0.171818
\(945\) 0 0
\(946\) 1.12488e7 0.408676
\(947\) 1.04490e7 0.378615 0.189307 0.981918i \(-0.439376\pi\)
0.189307 + 0.981918i \(0.439376\pi\)
\(948\) −6.99322e6 −0.252730
\(949\) −4.26761e7 −1.53822
\(950\) 8.35683e6 0.300423
\(951\) −1.44394e7 −0.517725
\(952\) 0 0
\(953\) 1.15666e6 0.0412548 0.0206274 0.999787i \(-0.493434\pi\)
0.0206274 + 0.999787i \(0.493434\pi\)
\(954\) −5.72275e6 −0.203579
\(955\) −8.04172e7 −2.85325
\(956\) 5.13658e6 0.181773
\(957\) 2.65355e6 0.0936585
\(958\) −1.41861e7 −0.499400
\(959\) 0 0
\(960\) 3.81533e6 0.133615
\(961\) 3.00994e6 0.105135
\(962\) 2.92501e7 1.01904
\(963\) −7.89121e6 −0.274207
\(964\) 1.98806e7 0.689027
\(965\) 2.86089e7 0.988969
\(966\) 0 0
\(967\) 2.00516e7 0.689577 0.344789 0.938680i \(-0.387951\pi\)
0.344789 + 0.938680i \(0.387951\pi\)
\(968\) 6.61500e6 0.226904
\(969\) 3.20522e6 0.109660
\(970\) −4.12505e7 −1.40767
\(971\) 6.10209e6 0.207697 0.103849 0.994593i \(-0.466884\pi\)
0.103849 + 0.994593i \(0.466884\pi\)
\(972\) 944784. 0.0320750
\(973\) 0 0
\(974\) 3.51328e7 1.18663
\(975\) 5.49984e7 1.85284
\(976\) 2.89897e6 0.0974134
\(977\) 1.36743e7 0.458320 0.229160 0.973389i \(-0.426402\pi\)
0.229160 + 0.973389i \(0.426402\pi\)
\(978\) 1.57246e7 0.525694
\(979\) 2.60285e7 0.867945
\(980\) 0 0
\(981\) 4.80388e6 0.159375
\(982\) 2.34643e6 0.0776478
\(983\) −1.67033e7 −0.551338 −0.275669 0.961253i \(-0.588899\pi\)
−0.275669 + 0.961253i \(0.588899\pi\)
\(984\) 1.04875e7 0.345289
\(985\) −1.36219e7 −0.447348
\(986\) 6.35003e6 0.208010
\(987\) 0 0
\(988\) −3.54896e6 −0.115667
\(989\) −4.44511e7 −1.44508
\(990\) −8.05437e6 −0.261182
\(991\) 5.74906e7 1.85957 0.929785 0.368102i \(-0.119992\pi\)
0.929785 + 0.368102i \(0.119992\pi\)
\(992\) −5.75986e6 −0.185837
\(993\) 9.02772e6 0.290539
\(994\) 0 0
\(995\) 5.84720e7 1.87236
\(996\) 1.62952e7 0.520489
\(997\) 3.94489e7 1.25689 0.628444 0.777855i \(-0.283692\pi\)
0.628444 + 0.777855i \(0.283692\pi\)
\(998\) 3.83909e7 1.22012
\(999\) −6.61822e6 −0.209811
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.q.1.2 yes 2
3.2 odd 2 882.6.a.bk.1.1 2
7.2 even 3 294.6.e.x.67.1 4
7.3 odd 6 294.6.e.z.79.2 4
7.4 even 3 294.6.e.x.79.1 4
7.5 odd 6 294.6.e.z.67.2 4
7.6 odd 2 294.6.a.n.1.1 2
21.20 even 2 882.6.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.6.a.n.1.1 2 7.6 odd 2
294.6.a.q.1.2 yes 2 1.1 even 1 trivial
294.6.e.x.67.1 4 7.2 even 3
294.6.e.x.79.1 4 7.4 even 3
294.6.e.z.67.2 4 7.5 odd 6
294.6.e.z.79.2 4 7.3 odd 6
882.6.a.bk.1.1 2 3.2 odd 2
882.6.a.bu.1.2 2 21.20 even 2