Properties

Label 882.6.a.bk.1.1
Level $882$
Weight $6$
Character 882.1
Self dual yes
Analytic conductor $141.459$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,6,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, 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("882.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(141.458529075\)
Analytic rank: \(1\)
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: no (minimal twist has level 294)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} -413.990 q^{10} -240.191 q^{11} +805.477 q^{13} +256.000 q^{16} +1293.27 q^{17} -275.377 q^{19} -1655.96 q^{20} -960.764 q^{22} -3796.57 q^{23} +7586.73 q^{25} +3221.91 q^{26} -1227.52 q^{29} +5624.86 q^{31} +1024.00 q^{32} +5173.06 q^{34} -9078.49 q^{37} -1101.51 q^{38} -6623.84 q^{40} +18207.4 q^{41} -11708.2 q^{43} -3843.05 q^{44} -15186.3 q^{46} +23048.6 q^{47} +30346.9 q^{50} +12887.6 q^{52} -17662.8 q^{53} +24859.2 q^{55} -4910.07 q^{58} -18376.3 q^{59} +11324.1 q^{61} +22499.5 q^{62} +4096.00 q^{64} -83364.9 q^{65} +36079.3 q^{67} +20692.3 q^{68} +63434.2 q^{71} -52982.4 q^{73} -36314.0 q^{74} -4406.03 q^{76} -48564.0 q^{79} -26495.4 q^{80} +72829.7 q^{82} -113161. q^{83} -133850. q^{85} -46832.9 q^{86} -15372.2 q^{88} -108366. q^{89} -60745.2 q^{92} +92194.4 q^{94} +28500.8 q^{95} +99641.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8} - 432 q^{10} - 124 q^{11} + 720 q^{13} + 512 q^{16} + 1260 q^{17} + 360 q^{19} - 1728 q^{20} - 496 q^{22} - 6524 q^{23} + 4482 q^{25} + 2880 q^{26} - 7088 q^{29} + 5904 q^{31} + 2048 q^{32} + 5040 q^{34} - 6040 q^{37} + 1440 q^{38} - 6912 q^{40} + 17388 q^{41} - 608 q^{43} - 1984 q^{44} - 26096 q^{46} + 30456 q^{47} + 17928 q^{50} + 11520 q^{52} - 3964 q^{53} + 24336 q^{55} - 28352 q^{58} - 40752 q^{59} - 1368 q^{61} + 23616 q^{62} + 8192 q^{64} - 82980 q^{65} - 16224 q^{67} + 20160 q^{68} + 3204 q^{71} + 23976 q^{73} - 24160 q^{74} + 5760 q^{76} - 82160 q^{79} - 27648 q^{80} + 69552 q^{82} - 173736 q^{83} - 133700 q^{85} - 2432 q^{86} - 7936 q^{88} - 200556 q^{89} - 104384 q^{92} + 121824 q^{94} + 25640 q^{95} + 251928 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −103.497 −1.85142 −0.925710 0.378235i \(-0.876531\pi\)
−0.925710 + 0.378235i \(0.876531\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) −413.990 −1.30915
\(11\) −240.191 −0.598515 −0.299257 0.954172i \(-0.596739\pi\)
−0.299257 + 0.954172i \(0.596739\pi\)
\(12\) 0 0
\(13\) 805.477 1.32189 0.660944 0.750435i \(-0.270156\pi\)
0.660944 + 0.750435i \(0.270156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1293.27 1.08534 0.542670 0.839946i \(-0.317414\pi\)
0.542670 + 0.839946i \(0.317414\pi\)
\(18\) 0 0
\(19\) −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 \(-0.769102\pi\)
−0.748242 + 0.663426i \(0.769102\pi\)
\(24\) 0 0
\(25\) 7586.73 2.42775
\(26\) 3221.91 0.934717
\(27\) 0 0
\(28\) 0 0
\(29\) −1227.52 −0.271040 −0.135520 0.990775i \(-0.543270\pi\)
−0.135520 + 0.990775i \(0.543270\pi\)
\(30\) 0 0
\(31\) 5624.86 1.05125 0.525627 0.850715i \(-0.323831\pi\)
0.525627 + 0.850715i \(0.323831\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 5173.06 0.767451
\(35\) 0 0
\(36\) 0 0
\(37\) −9078.49 −1.09021 −0.545104 0.838368i \(-0.683510\pi\)
−0.545104 + 0.838368i \(0.683510\pi\)
\(38\) −1101.51 −0.123745
\(39\) 0 0
\(40\) −6623.84 −0.654576
\(41\) 18207.4 1.69156 0.845782 0.533528i \(-0.179134\pi\)
0.845782 + 0.533528i \(0.179134\pi\)
\(42\) 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
\(45\) 0 0
\(46\) −15186.3 −1.05817
\(47\) 23048.6 1.52195 0.760974 0.648782i \(-0.224721\pi\)
0.760974 + 0.648782i \(0.224721\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 30346.9 1.71668
\(51\) 0 0
\(52\) 12887.6 0.660944
\(53\) −17662.8 −0.863714 −0.431857 0.901942i \(-0.642141\pi\)
−0.431857 + 0.901942i \(0.642141\pi\)
\(54\) 0 0
\(55\) 24859.2 1.10810
\(56\) 0 0
\(57\) 0 0
\(58\) −4910.07 −0.191654
\(59\) −18376.3 −0.687271 −0.343636 0.939103i \(-0.611658\pi\)
−0.343636 + 0.939103i \(0.611658\pi\)
\(60\) 0 0
\(61\) 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
\(66\) 0 0
\(67\) 36079.3 0.981910 0.490955 0.871185i \(-0.336648\pi\)
0.490955 + 0.871185i \(0.336648\pi\)
\(68\) 20692.3 0.542670
\(69\) 0 0
\(70\) 0 0
\(71\) 63434.2 1.49341 0.746703 0.665158i \(-0.231636\pi\)
0.746703 + 0.665158i \(0.231636\pi\)
\(72\) 0 0
\(73\) −52982.4 −1.16366 −0.581828 0.813312i \(-0.697662\pi\)
−0.581828 + 0.813312i \(0.697662\pi\)
\(74\) −36314.0 −0.770893
\(75\) 0 0
\(76\) −4406.03 −0.0875011
\(77\) 0 0
\(78\) 0 0
\(79\) −48564.0 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(80\) −26495.4 −0.462855
\(81\) 0 0
\(82\) 72829.7 1.19612
\(83\) −113161. −1.80303 −0.901513 0.432753i \(-0.857542\pi\)
−0.901513 + 0.432753i \(0.857542\pi\)
\(84\) 0 0
\(85\) −133850. −2.00942
\(86\) −46832.9 −0.682818
\(87\) 0 0
\(88\) −15372.2 −0.211607
\(89\) −108366. −1.45017 −0.725083 0.688662i \(-0.758199\pi\)
−0.725083 + 0.688662i \(0.758199\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −60745.2 −0.748242
\(93\) 0 0
\(94\) 92194.4 1.07618
\(95\) 28500.8 0.324002
\(96\) 0 0
\(97\) 99641.2 1.07525 0.537625 0.843184i \(-0.319321\pi\)
0.537625 + 0.843184i \(0.319321\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 121388. 1.21388
\(101\) −165200. −1.61141 −0.805705 0.592317i \(-0.798213\pi\)
−0.805705 + 0.592317i \(0.798213\pi\)
\(102\) 0 0
\(103\) −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 \(-0.365071\pi\)
0.411310 + 0.911496i \(0.365071\pi\)
\(108\) 0 0
\(109\) 59307.2 0.478125 0.239062 0.971004i \(-0.423160\pi\)
0.239062 + 0.971004i \(0.423160\pi\)
\(110\) 99436.6 0.783546
\(111\) 0 0
\(112\) 0 0
\(113\) −157268. −1.15863 −0.579313 0.815105i \(-0.696679\pi\)
−0.579313 + 0.815105i \(0.696679\pi\)
\(114\) 0 0
\(115\) 392936. 2.77062
\(116\) −19640.3 −0.135520
\(117\) 0 0
\(118\) −73505.2 −0.485974
\(119\) 0 0
\(120\) 0 0
\(121\) −103359. −0.641780
\(122\) 45296.3 0.275527
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) −333459. −1.73055
\(131\) 72328.4 0.368240 0.184120 0.982904i \(-0.441057\pi\)
0.184120 + 0.982904i \(0.441057\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 144317. 0.694315
\(135\) 0 0
\(136\) 82769.0 0.383725
\(137\) −225662. −1.02720 −0.513602 0.858028i \(-0.671689\pi\)
−0.513602 + 0.858028i \(0.671689\pi\)
\(138\) 0 0
\(139\) −327971. −1.43979 −0.719894 0.694084i \(-0.755810\pi\)
−0.719894 + 0.694084i \(0.755810\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 253737. 1.05600
\(143\) −193468. −0.791170
\(144\) 0 0
\(145\) 127045. 0.501808
\(146\) −211930. −0.822829
\(147\) 0 0
\(148\) −145256. −0.545104
\(149\) −164011. −0.605211 −0.302606 0.953116i \(-0.597856\pi\)
−0.302606 + 0.953116i \(0.597856\pi\)
\(150\) 0 0
\(151\) 62308.8 0.222386 0.111193 0.993799i \(-0.464533\pi\)
0.111193 + 0.993799i \(0.464533\pi\)
\(152\) −17624.1 −0.0618726
\(153\) 0 0
\(154\) 0 0
\(155\) −582159. −1.94631
\(156\) 0 0
\(157\) −239807. −0.776447 −0.388224 0.921565i \(-0.626911\pi\)
−0.388224 + 0.921565i \(0.626911\pi\)
\(158\) −194256. −0.619059
\(159\) 0 0
\(160\) −105981. −0.327288
\(161\) 0 0
\(162\) 0 0
\(163\) −436795. −1.28768 −0.643841 0.765160i \(-0.722660\pi\)
−0.643841 + 0.765160i \(0.722660\pi\)
\(164\) 291319. 0.845782
\(165\) 0 0
\(166\) −452644. −1.27493
\(167\) −38846.7 −0.107786 −0.0538931 0.998547i \(-0.517163\pi\)
−0.0538931 + 0.998547i \(0.517163\pi\)
\(168\) 0 0
\(169\) 277501. 0.747390
\(170\) −535399. −1.42087
\(171\) 0 0
\(172\) −187331. −0.482825
\(173\) −396277. −1.00666 −0.503331 0.864093i \(-0.667892\pi\)
−0.503331 + 0.864093i \(0.667892\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −61488.9 −0.149629
\(177\) 0 0
\(178\) −433464. −1.02542
\(179\) −243883. −0.568918 −0.284459 0.958688i \(-0.591814\pi\)
−0.284459 + 0.958688i \(0.591814\pi\)
\(180\) 0 0
\(181\) −299273. −0.679002 −0.339501 0.940606i \(-0.610258\pi\)
−0.339501 + 0.940606i \(0.610258\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −242981. −0.529087
\(185\) 939601. 2.01843
\(186\) 0 0
\(187\) −310631. −0.649592
\(188\) 368778. 0.760974
\(189\) 0 0
\(190\) 114003. 0.229104
\(191\) 776996. 1.54112 0.770558 0.637369i \(-0.219977\pi\)
0.770558 + 0.637369i \(0.219977\pi\)
\(192\) 0 0
\(193\) 276421. 0.534168 0.267084 0.963673i \(-0.413940\pi\)
0.267084 + 0.963673i \(0.413940\pi\)
\(194\) 398565. 0.760317
\(195\) 0 0
\(196\) 0 0
\(197\) 131615. 0.241624 0.120812 0.992675i \(-0.461450\pi\)
0.120812 + 0.992675i \(0.461450\pi\)
\(198\) 0 0
\(199\) 564960. 1.01131 0.505656 0.862735i \(-0.331250\pi\)
0.505656 + 0.862735i \(0.331250\pi\)
\(200\) 485551. 0.858340
\(201\) 0 0
\(202\) −660799. −1.13944
\(203\) 0 0
\(204\) 0 0
\(205\) −1.88442e6 −3.13180
\(206\) −428989. −0.704333
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 389689. 0.581680
\(215\) 1.21177e6 1.78782
\(216\) 0 0
\(217\) 0 0
\(218\) 237229. 0.338085
\(219\) 0 0
\(220\) 397746. 0.554051
\(221\) 1.04170e6 1.43470
\(222\) 0 0
\(223\) −279483. −0.376351 −0.188176 0.982135i \(-0.560257\pi\)
−0.188176 + 0.982135i \(0.560257\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −629071. −0.819273
\(227\) 1.22351e6 1.57595 0.787974 0.615709i \(-0.211130\pi\)
0.787974 + 0.615709i \(0.211130\pi\)
\(228\) 0 0
\(229\) 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 \(-0.648653\pi\)
−0.450215 + 0.892920i \(0.648653\pi\)
\(234\) 0 0
\(235\) −2.38547e6 −2.81776
\(236\) −294021. −0.343636
\(237\) 0 0
\(238\) 0 0
\(239\) −321036. −0.363546 −0.181773 0.983341i \(-0.558184\pi\)
−0.181773 + 0.983341i \(0.558184\pi\)
\(240\) 0 0
\(241\) 1.24254e6 1.37805 0.689027 0.724735i \(-0.258038\pi\)
0.689027 + 0.724735i \(0.258038\pi\)
\(242\) −413437. −0.453807
\(243\) 0 0
\(244\) 181185. 0.194827
\(245\) 0 0
\(246\) 0 0
\(247\) −221810. −0.231333
\(248\) 359991. 0.371675
\(249\) 0 0
\(250\) −1.84711e6 −1.86914
\(251\) −690235. −0.691532 −0.345766 0.938321i \(-0.612381\pi\)
−0.345766 + 0.938321i \(0.612381\pi\)
\(252\) 0 0
\(253\) 911902. 0.895668
\(254\) −369909. −0.359758
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −13786.7 −0.0130205 −0.00651024 0.999979i \(-0.502072\pi\)
−0.00651024 + 0.999979i \(0.502072\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.33384e6 −1.22369
\(261\) 0 0
\(262\) 289314. 0.260385
\(263\) −574833. −0.512451 −0.256226 0.966617i \(-0.582479\pi\)
−0.256226 + 0.966617i \(0.582479\pi\)
\(264\) 0 0
\(265\) 1.82806e6 1.59910
\(266\) 0 0
\(267\) 0 0
\(268\) 577270. 0.490955
\(269\) 1.11893e6 0.942804 0.471402 0.881918i \(-0.343748\pi\)
0.471402 + 0.881918i \(0.343748\pi\)
\(270\) 0 0
\(271\) 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
\(276\) 0 0
\(277\) −837654. −0.655941 −0.327971 0.944688i \(-0.606365\pi\)
−0.327971 + 0.944688i \(0.606365\pi\)
\(278\) −1.31188e6 −1.01808
\(279\) 0 0
\(280\) 0 0
\(281\) 917687. 0.693312 0.346656 0.937992i \(-0.387317\pi\)
0.346656 + 0.937992i \(0.387317\pi\)
\(282\) 0 0
\(283\) −415263. −0.308218 −0.154109 0.988054i \(-0.549251\pi\)
−0.154109 + 0.988054i \(0.549251\pi\)
\(284\) 1.01495e6 0.746703
\(285\) 0 0
\(286\) −773873. −0.559442
\(287\) 0 0
\(288\) 0 0
\(289\) 252680. 0.177962
\(290\) 508180. 0.354832
\(291\) 0 0
\(292\) −847718. −0.581828
\(293\) 1.21491e6 0.826752 0.413376 0.910561i \(-0.364350\pi\)
0.413376 + 0.910561i \(0.364350\pi\)
\(294\) 0 0
\(295\) 1.90190e6 1.27243
\(296\) −581023. −0.385447
\(297\) 0 0
\(298\) −656043. −0.427949
\(299\) −3.05805e6 −1.97819
\(300\) 0 0
\(301\) 0 0
\(302\) 249235. 0.157250
\(303\) 0 0
\(304\) −70496.5 −0.0437505
\(305\) −1.17201e6 −0.721412
\(306\) 0 0
\(307\) −2.60119e6 −1.57517 −0.787584 0.616207i \(-0.788668\pi\)
−0.787584 + 0.616207i \(0.788668\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.32864e6 −1.37625
\(311\) 819164. 0.480253 0.240126 0.970742i \(-0.422811\pi\)
0.240126 + 0.970742i \(0.422811\pi\)
\(312\) 0 0
\(313\) −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 \(-0.352007\pi\)
0.448363 + 0.893852i \(0.352007\pi\)
\(318\) 0 0
\(319\) 294839. 0.162221
\(320\) −423926. −0.231427
\(321\) 0 0
\(322\) 0 0
\(323\) −356135. −0.189937
\(324\) 0 0
\(325\) 6.11094e6 3.20922
\(326\) −1.74718e6 −0.910528
\(327\) 0 0
\(328\) 1.16527e6 0.598058
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00308e6 0.503229 0.251614 0.967828i \(-0.419039\pi\)
0.251614 + 0.967828i \(0.419039\pi\)
\(332\) −1.81058e6 −0.901513
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) −2.14160e6 −1.00471
\(341\) −1.35104e6 −0.629191
\(342\) 0 0
\(343\) 0 0
\(344\) −749326. −0.341409
\(345\) 0 0
\(346\) −1.58511e6 −0.711818
\(347\) −226234. −0.100863 −0.0504317 0.998728i \(-0.516060\pi\)
−0.0504317 + 0.998728i \(0.516060\pi\)
\(348\) 0 0
\(349\) −1.46280e6 −0.642866 −0.321433 0.946932i \(-0.604165\pi\)
−0.321433 + 0.946932i \(0.604165\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −245955. −0.105803
\(353\) −2.80957e6 −1.20006 −0.600031 0.799977i \(-0.704845\pi\)
−0.600031 + 0.799977i \(0.704845\pi\)
\(354\) 0 0
\(355\) −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 \(-0.226475\pi\)
0.757388 + 0.652965i \(0.226475\pi\)
\(360\) 0 0
\(361\) −2.40027e6 −0.969374
\(362\) −1.19709e6 −0.480127
\(363\) 0 0
\(364\) 0 0
\(365\) 5.48354e6 2.15441
\(366\) 0 0
\(367\) 1.16925e6 0.453148 0.226574 0.973994i \(-0.427247\pi\)
0.226574 + 0.973994i \(0.427247\pi\)
\(368\) −971923. −0.374121
\(369\) 0 0
\(370\) 3.75840e6 1.42725
\(371\) 0 0
\(372\) 0 0
\(373\) −4.57690e6 −1.70333 −0.851666 0.524084i \(-0.824408\pi\)
−0.851666 + 0.524084i \(0.824408\pi\)
\(374\) −1.24252e6 −0.459331
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 3.10799e6 1.08973
\(383\) −3.03946e6 −1.05876 −0.529382 0.848384i \(-0.677576\pi\)
−0.529382 + 0.848384i \(0.677576\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.10568e6 0.377714
\(387\) 0 0
\(388\) 1.59426e6 0.537625
\(389\) 459273. 0.153885 0.0769426 0.997036i \(-0.475484\pi\)
0.0769426 + 0.997036i \(0.475484\pi\)
\(390\) 0 0
\(391\) −4.90998e6 −1.62419
\(392\) 0 0
\(393\) 0 0
\(394\) 526461. 0.170854
\(395\) 5.02625e6 1.62088
\(396\) 0 0
\(397\) 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 \(-0.606318\pi\)
−0.327833 + 0.944736i \(0.606318\pi\)
\(402\) 0 0
\(403\) 4.53070e6 1.38964
\(404\) −2.64320e6 −0.805705
\(405\) 0 0
\(406\) 0 0
\(407\) 2.18057e6 0.652506
\(408\) 0 0
\(409\) 2.55745e6 0.755959 0.377980 0.925814i \(-0.376619\pi\)
0.377980 + 0.925814i \(0.376619\pi\)
\(410\) −7.53769e6 −2.21451
\(411\) 0 0
\(412\) −1.71595e6 −0.498038
\(413\) 0 0
\(414\) 0 0
\(415\) 1.17119e7 3.33816
\(416\) 824809. 0.233679
\(417\) 0 0
\(418\) 264572. 0.0740633
\(419\) −2.17401e6 −0.604959 −0.302480 0.953156i \(-0.597814\pi\)
−0.302480 + 0.953156i \(0.597814\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −1.13042e6 −0.305369
\(425\) 9.81166e6 2.63494
\(426\) 0 0
\(427\) 0 0
\(428\) 1.55876e6 0.411310
\(429\) 0 0
\(430\) 4.84708e6 1.26418
\(431\) 6.16874e6 1.59957 0.799785 0.600286i \(-0.204947\pi\)
0.799785 + 0.600286i \(0.204947\pi\)
\(432\) 0 0
\(433\) 4.89596e6 1.25493 0.627463 0.778647i \(-0.284093\pi\)
0.627463 + 0.778647i \(0.284093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 948915. 0.239062
\(437\) 1.04549e6 0.261888
\(438\) 0 0
\(439\) −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 \(-0.761943\pi\)
−0.733134 + 0.680084i \(0.761943\pi\)
\(444\) 0 0
\(445\) 1.12156e7 2.68486
\(446\) −1.11793e6 −0.266121
\(447\) 0 0
\(448\) 0 0
\(449\) 1.20801e6 0.282783 0.141392 0.989954i \(-0.454842\pi\)
0.141392 + 0.989954i \(0.454842\pi\)
\(450\) 0 0
\(451\) −4.37326e6 −1.01243
\(452\) −2.51628e6 −0.579313
\(453\) 0 0
\(454\) 4.89403e6 1.11436
\(455\) 0 0
\(456\) 0 0
\(457\) −5.46550e6 −1.22416 −0.612082 0.790794i \(-0.709668\pi\)
−0.612082 + 0.790794i \(0.709668\pi\)
\(458\) 1.01910e6 0.227014
\(459\) 0 0
\(460\) 6.28697e6 1.38531
\(461\) −1.15956e6 −0.254121 −0.127061 0.991895i \(-0.540554\pi\)
−0.127061 + 0.991895i \(0.540554\pi\)
\(462\) 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
\(465\) 0 0
\(466\) −2.98469e6 −0.636700
\(467\) 5.98923e6 1.27081 0.635403 0.772181i \(-0.280834\pi\)
0.635403 + 0.772181i \(0.280834\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −9.54189e6 −1.99246
\(471\) 0 0
\(472\) −1.17608e6 −0.242987
\(473\) 2.81221e6 0.577956
\(474\) 0 0
\(475\) −2.08921e6 −0.424862
\(476\) 0 0
\(477\) 0 0
\(478\) −1.28415e6 −0.257066
\(479\) −3.54652e6 −0.706258 −0.353129 0.935575i \(-0.614882\pi\)
−0.353129 + 0.935575i \(0.614882\pi\)
\(480\) 0 0
\(481\) −7.31252e6 −1.44113
\(482\) 4.97015e6 0.974432
\(483\) 0 0
\(484\) −1.65375e6 −0.320890
\(485\) −1.03126e7 −1.99074
\(486\) 0 0
\(487\) −8.78320e6 −1.67815 −0.839074 0.544017i \(-0.816903\pi\)
−0.839074 + 0.544017i \(0.816903\pi\)
\(488\) 724742. 0.137763
\(489\) 0 0
\(490\) 0 0
\(491\) 586608. 0.109811 0.0549053 0.998492i \(-0.482514\pi\)
0.0549053 + 0.998492i \(0.482514\pi\)
\(492\) 0 0
\(493\) −1.58751e6 −0.294170
\(494\) −887239. −0.163577
\(495\) 0 0
\(496\) 1.43997e6 0.262814
\(497\) 0 0
\(498\) 0 0
\(499\) −9.59772e6 −1.72551 −0.862753 0.505626i \(-0.831262\pi\)
−0.862753 + 0.505626i \(0.831262\pi\)
\(500\) −7.38844e6 −1.32168
\(501\) 0 0
\(502\) −2.76094e6 −0.488987
\(503\) 7.21481e6 1.27147 0.635733 0.771909i \(-0.280698\pi\)
0.635733 + 0.771909i \(0.280698\pi\)
\(504\) 0 0
\(505\) 1.70978e7 2.98340
\(506\) 3.64761e6 0.633333
\(507\) 0 0
\(508\) −1.47964e6 −0.254387
\(509\) −4.38419e6 −0.750058 −0.375029 0.927013i \(-0.622367\pi\)
−0.375029 + 0.927013i \(0.622367\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) −55146.8 −0.00920688
\(515\) 1.10998e7 1.84416
\(516\) 0 0
\(517\) −5.53606e6 −0.910909
\(518\) 0 0
\(519\) 0 0
\(520\) −5.33535e6 −0.865276
\(521\) −2.06581e6 −0.333424 −0.166712 0.986006i \(-0.553315\pi\)
−0.166712 + 0.986006i \(0.553315\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) 7.97762e6 1.23946
\(530\) 7.31222e6 1.13073
\(531\) 0 0
\(532\) 0 0
\(533\) 1.46657e7 2.23606
\(534\) 0 0
\(535\) −1.00830e7 −1.52301
\(536\) 2.30908e6 0.347158
\(537\) 0 0
\(538\) 4.47571e6 0.666663
\(539\) 0 0
\(540\) 0 0
\(541\) 8.23423e6 1.20957 0.604784 0.796390i \(-0.293260\pi\)
0.604784 + 0.796390i \(0.293260\pi\)
\(542\) 2.77948e6 0.406411
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) −7.28905e6 −1.02746
\(551\) 338030. 0.0474325
\(552\) 0 0
\(553\) 0 0
\(554\) −3.35061e6 −0.463821
\(555\) 0 0
\(556\) −5.24754e6 −0.719894
\(557\) 4.09165e6 0.558806 0.279403 0.960174i \(-0.409864\pi\)
0.279403 + 0.960174i \(0.409864\pi\)
\(558\) 0 0
\(559\) −9.43070e6 −1.27648
\(560\) 0 0
\(561\) 0 0
\(562\) 3.67075e6 0.490246
\(563\) 3.59052e6 0.477404 0.238702 0.971093i \(-0.423278\pi\)
0.238702 + 0.971093i \(0.423278\pi\)
\(564\) 0 0
\(565\) 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 \(-0.718709\pi\)
−0.634293 + 0.773093i \(0.718709\pi\)
\(570\) 0 0
\(571\) −2.21804e6 −0.284694 −0.142347 0.989817i \(-0.545465\pi\)
−0.142347 + 0.989817i \(0.545465\pi\)
\(572\) −3.09549e6 −0.395585
\(573\) 0 0
\(574\) 0 0
\(575\) −2.88036e7 −3.63309
\(576\) 0 0
\(577\) −2.65534e6 −0.332032 −0.166016 0.986123i \(-0.553090\pi\)
−0.166016 + 0.986123i \(0.553090\pi\)
\(578\) 1.01072e6 0.125838
\(579\) 0 0
\(580\) 2.03272e6 0.250904
\(581\) 0 0
\(582\) 0 0
\(583\) 4.24244e6 0.516945
\(584\) −3.39087e6 −0.411414
\(585\) 0 0
\(586\) 4.85964e6 0.584602
\(587\) 2.62690e6 0.314665 0.157332 0.987546i \(-0.449711\pi\)
0.157332 + 0.987546i \(0.449711\pi\)
\(588\) 0 0
\(589\) −1.54896e6 −0.183972
\(590\) 7.60760e6 0.899742
\(591\) 0 0
\(592\) −2.32409e6 −0.272552
\(593\) 1.15062e7 1.34368 0.671839 0.740697i \(-0.265505\pi\)
0.671839 + 0.740697i \(0.265505\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.62417e6 −0.302606
\(597\) 0 0
\(598\) −1.22322e7 −1.39879
\(599\) −6.46264e6 −0.735941 −0.367970 0.929838i \(-0.619947\pi\)
−0.367970 + 0.929838i \(0.619947\pi\)
\(600\) 0 0
\(601\) 8.55332e6 0.965937 0.482968 0.875638i \(-0.339559\pi\)
0.482968 + 0.875638i \(0.339559\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 996941. 0.111193
\(605\) 1.06974e7 1.18820
\(606\) 0 0
\(607\) 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
\(612\) 0 0
\(613\) −1.77145e6 −0.190405 −0.0952023 0.995458i \(-0.530350\pi\)
−0.0952023 + 0.995458i \(0.530350\pi\)
\(614\) −1.04048e7 −1.11381
\(615\) 0 0
\(616\) 0 0
\(617\) −1.40871e7 −1.48974 −0.744869 0.667210i \(-0.767488\pi\)
−0.744869 + 0.667210i \(0.767488\pi\)
\(618\) 0 0
\(619\) −1.13430e7 −1.18988 −0.594940 0.803770i \(-0.702824\pi\)
−0.594940 + 0.803770i \(0.702824\pi\)
\(620\) −9.31455e6 −0.973156
\(621\) 0 0
\(622\) 3.27666e6 0.339590
\(623\) 0 0
\(624\) 0 0
\(625\) 2.40843e7 2.46623
\(626\) −3.52776e6 −0.359801
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 6.41753e6 0.634081
\(635\) 9.57116e6 0.941956
\(636\) 0 0
\(637\) 0 0
\(638\) 1.17935e6 0.114708
\(639\) 0 0
\(640\) −1.69570e6 −0.163644
\(641\) 2.56385e6 0.246461 0.123230 0.992378i \(-0.460675\pi\)
0.123230 + 0.992378i \(0.460675\pi\)
\(642\) 0 0
\(643\) −5.95343e6 −0.567858 −0.283929 0.958845i \(-0.591638\pi\)
−0.283929 + 0.958845i \(0.591638\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.42454e6 −0.134306
\(647\) 1.53522e6 0.144182 0.0720909 0.997398i \(-0.477033\pi\)
0.0720909 + 0.997398i \(0.477033\pi\)
\(648\) 0 0
\(649\) 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 \(-0.131546\pi\)
0.915815 + 0.401600i \(0.131546\pi\)
\(654\) 0 0
\(655\) −7.48581e6 −0.681766
\(656\) 4.66110e6 0.422891
\(657\) 0 0
\(658\) 0 0
\(659\) 1.05477e7 0.946119 0.473060 0.881030i \(-0.343150\pi\)
0.473060 + 0.881030i \(0.343150\pi\)
\(660\) 0 0
\(661\) −1.22044e7 −1.08646 −0.543229 0.839585i \(-0.682798\pi\)
−0.543229 + 0.839585i \(0.682798\pi\)
\(662\) 4.01232e6 0.355837
\(663\) 0 0
\(664\) −7.24231e6 −0.637466
\(665\) 0 0
\(666\) 0 0
\(667\) 4.66036e6 0.405607
\(668\) −621548. −0.0538931
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 4.44001e6 0.373695
\(677\) −5.08381e6 −0.426302 −0.213151 0.977019i \(-0.568373\pi\)
−0.213151 + 0.977019i \(0.568373\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.56639e6 −0.710437
\(681\) 0 0
\(682\) −5.40416e6 −0.444905
\(683\) 1.60331e7 1.31512 0.657559 0.753403i \(-0.271589\pi\)
0.657559 + 0.753403i \(0.271589\pi\)
\(684\) 0 0
\(685\) 2.33554e7 1.90179
\(686\) 0 0
\(687\) 0 0
\(688\) −2.99730e6 −0.241412
\(689\) −1.42270e7 −1.14173
\(690\) 0 0
\(691\) −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
\(696\) 0 0
\(697\) 2.35470e7 1.83592
\(698\) −5.85119e6 −0.454575
\(699\) 0 0
\(700\) 0 0
\(701\) 3.10660e6 0.238776 0.119388 0.992848i \(-0.461907\pi\)
0.119388 + 0.992848i \(0.461907\pi\)
\(702\) 0 0
\(703\) 2.50001e6 0.190789
\(704\) −983822. −0.0748143
\(705\) 0 0
\(706\) −1.12383e7 −0.848571
\(707\) 0 0
\(708\) 0 0
\(709\) −1.38036e7 −1.03128 −0.515640 0.856805i \(-0.672446\pi\)
−0.515640 + 0.856805i \(0.672446\pi\)
\(710\) −2.62611e7 −1.95509
\(711\) 0 0
\(712\) −6.93542e6 −0.512711
\(713\) −2.13552e7 −1.57319
\(714\) 0 0
\(715\) 2.00235e7 1.46479
\(716\) −3.90214e6 −0.284459
\(717\) 0 0
\(718\) 1.47960e7 1.07111
\(719\) 1.29855e7 0.936777 0.468389 0.883523i \(-0.344835\pi\)
0.468389 + 0.883523i \(0.344835\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −9.60107e6 −0.685451
\(723\) 0 0
\(724\) −4.78837e6 −0.339501
\(725\) −9.31285e6 −0.658017
\(726\) 0 0
\(727\) −8.61224e6 −0.604338 −0.302169 0.953254i \(-0.597711\pi\)
−0.302169 + 0.953254i \(0.597711\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.19342e7 1.52340
\(731\) −1.51418e7 −1.04806
\(732\) 0 0
\(733\) 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
\(738\) 0 0
\(739\) −93692.5 −0.00631094 −0.00315547 0.999995i \(-0.501004\pi\)
−0.00315547 + 0.999995i \(0.501004\pi\)
\(740\) 1.50336e7 1.00922
\(741\) 0 0
\(742\) 0 0
\(743\) 1.29874e7 0.863081 0.431541 0.902094i \(-0.357970\pi\)
0.431541 + 0.902094i \(0.357970\pi\)
\(744\) 0 0
\(745\) 1.69747e7 1.12050
\(746\) −1.83076e7 −1.20444
\(747\) 0 0
\(748\) −4.97009e6 −0.324796
\(749\) 0 0
\(750\) 0 0
\(751\) −637441. −0.0412420 −0.0206210 0.999787i \(-0.506564\pi\)
−0.0206210 + 0.999787i \(0.506564\pi\)
\(752\) 5.90044e6 0.380487
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 1.82405e6 0.114552
\(761\) −5.81787e6 −0.364169 −0.182084 0.983283i \(-0.558284\pi\)
−0.182084 + 0.983283i \(0.558284\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.24319e7 0.770558
\(765\) 0 0
\(766\) −1.21578e7 −0.748659
\(767\) −1.48017e7 −0.908496
\(768\) 0 0
\(769\) 1.94310e7 1.18490 0.592448 0.805609i \(-0.298161\pi\)
0.592448 + 0.805609i \(0.298161\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.42274e6 0.267084
\(773\) −8.24179e6 −0.496104 −0.248052 0.968747i \(-0.579790\pi\)
−0.248052 + 0.968747i \(0.579790\pi\)
\(774\) 0 0
\(775\) 4.26743e7 2.55219
\(776\) 6.37704e6 0.380159
\(777\) 0 0
\(778\) 1.83709e6 0.108813
\(779\) −5.01390e6 −0.296027
\(780\) 0 0
\(781\) −1.52363e7 −0.893826
\(782\) −1.96399e7 −1.14848
\(783\) 0 0
\(784\) 0 0
\(785\) 2.48194e7 1.43753
\(786\) 0 0
\(787\) −1.47863e7 −0.850986 −0.425493 0.904962i \(-0.639899\pi\)
−0.425493 + 0.904962i \(0.639899\pi\)
\(788\) 2.10585e6 0.120812
\(789\) 0 0
\(790\) 2.01050e7 1.14614
\(791\) 0 0
\(792\) 0 0
\(793\) 9.12129e6 0.515079
\(794\) 2.17000e7 1.22154
\(795\) 0 0
\(796\) 9.03937e6 0.505656
\(797\) 1.58158e7 0.881956 0.440978 0.897518i \(-0.354632\pi\)
0.440978 + 0.897518i \(0.354632\pi\)
\(798\) 0 0
\(799\) 2.98080e7 1.65183
\(800\) 7.76881e6 0.429170
\(801\) 0 0
\(802\) −8.44508e6 −0.463626
\(803\) 1.27259e7 0.696465
\(804\) 0 0
\(805\) 0 0
\(806\) 1.81228e7 0.982625
\(807\) 0 0
\(808\) −1.05728e7 −0.569720
\(809\) −1.19304e7 −0.640888 −0.320444 0.947267i \(-0.603832\pi\)
−0.320444 + 0.947267i \(0.603832\pi\)
\(810\) 0 0
\(811\) 1.89393e7 1.01114 0.505571 0.862785i \(-0.331282\pi\)
0.505571 + 0.862785i \(0.331282\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8.72228e6 0.461391
\(815\) 4.52071e7 2.38404
\(816\) 0 0
\(817\) 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 \(-0.810900\pi\)
−0.828667 + 0.559741i \(0.810900\pi\)
\(822\) 0 0
\(823\) 3.61430e7 1.86005 0.930025 0.367495i \(-0.119785\pi\)
0.930025 + 0.367495i \(0.119785\pi\)
\(824\) −6.86382e6 −0.352166
\(825\) 0 0
\(826\) 0 0
\(827\) 2.45831e7 1.24989 0.624946 0.780668i \(-0.285121\pi\)
0.624946 + 0.780668i \(0.285121\pi\)
\(828\) 0 0
\(829\) 2.10424e7 1.06343 0.531715 0.846923i \(-0.321548\pi\)
0.531715 + 0.846923i \(0.321548\pi\)
\(830\) 4.68475e7 2.36043
\(831\) 0 0
\(832\) 3.29923e6 0.165236
\(833\) 0 0
\(834\) 0 0
\(835\) 4.02054e6 0.199557
\(836\) 1.05829e6 0.0523707
\(837\) 0 0
\(838\) −8.69604e6 −0.427771
\(839\) −1.12403e7 −0.551283 −0.275642 0.961260i \(-0.588890\pi\)
−0.275642 + 0.961260i \(0.588890\pi\)
\(840\) 0 0
\(841\) −1.90043e7 −0.926537
\(842\) −2.19172e7 −1.06538
\(843\) 0 0
\(844\) 5.01820e6 0.242489
\(845\) −2.87206e7 −1.38373
\(846\) 0 0
\(847\) 0 0
\(848\) −4.52168e6 −0.215928
\(849\) 0 0
\(850\) 3.92466e7 1.86318
\(851\) 3.44672e7 1.63148
\(852\) 0 0
\(853\) −2.63779e7 −1.24127 −0.620636 0.784099i \(-0.713126\pi\)
−0.620636 + 0.784099i \(0.713126\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.23503e6 0.290840
\(857\) −4.26663e7 −1.98442 −0.992208 0.124595i \(-0.960237\pi\)
−0.992208 + 0.124595i \(0.960237\pi\)
\(858\) 0 0
\(859\) 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 \(-0.603239\pi\)
−0.318680 + 0.947862i \(0.603239\pi\)
\(864\) 0 0
\(865\) 4.10137e7 1.86375
\(866\) 1.95838e7 0.887366
\(867\) 0 0
\(868\) 0 0
\(869\) 1.16646e7 0.523988
\(870\) 0 0
\(871\) 2.90611e7 1.29798
\(872\) 3.79566e6 0.169043
\(873\) 0 0
\(874\) 4.18195e6 0.185183
\(875\) 0 0
\(876\) 0 0
\(877\) 3.40772e6 0.149611 0.0748057 0.997198i \(-0.476166\pi\)
0.0748057 + 0.997198i \(0.476166\pi\)
\(878\) −1.41835e7 −0.620937
\(879\) 0 0
\(880\) 6.36394e6 0.277025
\(881\) 3.51391e6 0.152528 0.0762641 0.997088i \(-0.475701\pi\)
0.0762641 + 0.997088i \(0.475701\pi\)
\(882\) 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
\(885\) 0 0
\(886\) −2.42261e7 −1.03681
\(887\) −2.45357e7 −1.04710 −0.523550 0.851995i \(-0.675393\pi\)
−0.523550 + 0.851995i \(0.675393\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4.48624e7 1.89849
\(891\) 0 0
\(892\) −4.47173e6 −0.188176
\(893\) −6.34705e6 −0.266344
\(894\) 0 0
\(895\) 2.52413e7 1.05331
\(896\) 0 0
\(897\) 0 0
\(898\) 4.83203e6 0.199958
\(899\) −6.90462e6 −0.284932
\(900\) 0 0
\(901\) −2.28427e7 −0.937423
\(902\) −1.74930e7 −0.715894
\(903\) 0 0
\(904\) −1.00651e7 −0.409636
\(905\) 3.09740e7 1.25712
\(906\) 0 0
\(907\) −8.14146e6 −0.328613 −0.164306 0.986409i \(-0.552539\pi\)
−0.164306 + 0.986409i \(0.552539\pi\)
\(908\) 1.95761e7 0.787974
\(909\) 0 0
\(910\) 0 0
\(911\) −1.01384e7 −0.404739 −0.202370 0.979309i \(-0.564864\pi\)
−0.202370 + 0.979309i \(0.564864\pi\)
\(912\) 0 0
\(913\) 2.71803e7 1.07914
\(914\) −2.18620e7 −0.865615
\(915\) 0 0
\(916\) 4.07639e6 0.160523
\(917\) 0 0
\(918\) 0 0
\(919\) −1.01176e7 −0.395175 −0.197587 0.980285i \(-0.563311\pi\)
−0.197587 + 0.980285i \(0.563311\pi\)
\(920\) 2.51479e7 0.979562
\(921\) 0 0
\(922\) −4.63824e6 −0.179691
\(923\) 5.10948e7 1.97412
\(924\) 0 0
\(925\) −6.88760e7 −2.64676
\(926\) −1.72534e7 −0.661221
\(927\) 0 0
\(928\) −1.25698e6 −0.0479135
\(929\) −2.05111e7 −0.779739 −0.389870 0.920870i \(-0.627480\pi\)
−0.389870 + 0.920870i \(0.627480\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.19388e7 −0.450215
\(933\) 0 0
\(934\) 2.39569e7 0.898595
\(935\) 3.21495e7 1.20267
\(936\) 0 0
\(937\) −1.46527e7 −0.545218 −0.272609 0.962125i \(-0.587886\pi\)
−0.272609 + 0.962125i \(0.587886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.81676e7 −1.40888
\(941\) −2.59341e7 −0.954765 −0.477382 0.878696i \(-0.658414\pi\)
−0.477382 + 0.878696i \(0.658414\pi\)
\(942\) 0 0
\(943\) −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 \(-0.560624\pi\)
−0.189307 + 0.981918i \(0.560624\pi\)
\(948\) 0 0
\(949\) −4.26761e7 −1.53822
\(950\) −8.35683e6 −0.300423
\(951\) 0 0
\(952\) 0 0
\(953\) −1.15666e6 −0.0412548 −0.0206274 0.999787i \(-0.506566\pi\)
−0.0206274 + 0.999787i \(0.506566\pi\)
\(954\) 0 0
\(955\) −8.04172e7 −2.85325
\(956\) −5.13658e6 −0.181773
\(957\) 0 0
\(958\) −1.41861e7 −0.499400
\(959\) 0 0
\(960\) 0 0
\(961\) 3.00994e6 0.105135
\(962\) −2.92501e7 −1.01904
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) −4.12505e7 −1.40767
\(971\) −6.10209e6 −0.207697 −0.103849 0.994593i \(-0.533116\pi\)
−0.103849 + 0.994593i \(0.533116\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −3.51328e7 −1.18663
\(975\) 0 0
\(976\) 2.89897e6 0.0974134
\(977\) −1.36743e7 −0.458320 −0.229160 0.973389i \(-0.573598\pi\)
−0.229160 + 0.973389i \(0.573598\pi\)
\(978\) 0 0
\(979\) 2.60285e7 0.867945
\(980\) 0 0
\(981\) 0 0
\(982\) 2.34643e6 0.0776478
\(983\) 1.67033e7 0.551338 0.275669 0.961253i \(-0.411101\pi\)
0.275669 + 0.961253i \(0.411101\pi\)
\(984\) 0 0
\(985\) −1.36219e7 −0.447348
\(986\) −6.35003e6 −0.208010
\(987\) 0 0
\(988\) −3.54896e6 −0.115667
\(989\) 4.44511e7 1.44508
\(990\) 0 0
\(991\) 5.74906e7 1.85957 0.929785 0.368102i \(-0.119992\pi\)
0.929785 + 0.368102i \(0.119992\pi\)
\(992\) 5.75986e6 0.185837
\(993\) 0 0
\(994\) 0 0
\(995\) −5.84720e7 −1.87236
\(996\) 0 0
\(997\) 3.94489e7 1.25689 0.628444 0.777855i \(-0.283692\pi\)
0.628444 + 0.777855i \(0.283692\pi\)
\(998\) −3.83909e7 −1.22012
\(999\) 0 0
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 882.6.a.bk.1.1 2
3.2 odd 2 294.6.a.q.1.2 yes 2
7.6 odd 2 882.6.a.bu.1.2 2
21.2 odd 6 294.6.e.x.67.1 4
21.5 even 6 294.6.e.z.67.2 4
21.11 odd 6 294.6.e.x.79.1 4
21.17 even 6 294.6.e.z.79.2 4
21.20 even 2 294.6.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.6.a.n.1.1 2 21.20 even 2
294.6.a.q.1.2 yes 2 3.2 odd 2
294.6.e.x.67.1 4 21.2 odd 6
294.6.e.x.79.1 4 21.11 odd 6
294.6.e.z.67.2 4 21.5 even 6
294.6.e.z.79.2 4 21.17 even 6
882.6.a.bk.1.1 2 1.1 even 1 trivial
882.6.a.bu.1.2 2 7.6 odd 2