Properties

Label 450.6.c.a.199.2
Level $450$
Weight $6$
Character 450.199
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,6,Mod(199,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.6.c.a.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{2} -16.0000 q^{4} -142.000i q^{7} -64.0000i q^{8} +O(q^{10})\) \(q+4.00000i q^{2} -16.0000 q^{4} -142.000i q^{7} -64.0000i q^{8} -777.000 q^{11} -884.000i q^{13} +568.000 q^{14} +256.000 q^{16} +27.0000i q^{17} -1145.00 q^{19} -3108.00i q^{22} +1854.00i q^{23} +3536.00 q^{26} +2272.00i q^{28} -4920.00 q^{29} +1802.00 q^{31} +1024.00i q^{32} -108.000 q^{34} +13178.0i q^{37} -4580.00i q^{38} +15123.0 q^{41} -7844.00i q^{43} +12432.0 q^{44} -7416.00 q^{46} +6732.00i q^{47} -3357.00 q^{49} +14144.0i q^{52} +3414.00i q^{53} -9088.00 q^{56} -19680.0i q^{58} +33960.0 q^{59} +47402.0 q^{61} +7208.00i q^{62} -4096.00 q^{64} -13177.0i q^{67} -432.000i q^{68} +7548.00 q^{71} +59821.0i q^{73} -52712.0 q^{74} +18320.0 q^{76} +110334. i q^{77} -75830.0 q^{79} +60492.0i q^{82} +46299.0i q^{83} +31376.0 q^{86} +49728.0i q^{88} -30585.0 q^{89} -125528. q^{91} -29664.0i q^{92} -26928.0 q^{94} +104018. i q^{97} -13428.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 1554 q^{11} + 1136 q^{14} + 512 q^{16} - 2290 q^{19} + 7072 q^{26} - 9840 q^{29} + 3604 q^{31} - 216 q^{34} + 30246 q^{41} + 24864 q^{44} - 14832 q^{46} - 6714 q^{49} - 18176 q^{56} + 67920 q^{59} + 94804 q^{61} - 8192 q^{64} + 15096 q^{71} - 105424 q^{74} + 36640 q^{76} - 151660 q^{79} + 62752 q^{86} - 61170 q^{89} - 251056 q^{91} - 53856 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 0.707107i
\(3\) 0 0
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 142.000i − 1.09533i −0.836699 0.547663i \(-0.815518\pi\)
0.836699 0.547663i \(-0.184482\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −777.000 −1.93615 −0.968076 0.250658i \(-0.919353\pi\)
−0.968076 + 0.250658i \(0.919353\pi\)
\(12\) 0 0
\(13\) − 884.000i − 1.45075i −0.688352 0.725377i \(-0.741665\pi\)
0.688352 0.725377i \(-0.258335\pi\)
\(14\) 568.000 0.774512
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 27.0000i 0.0226590i 0.999936 + 0.0113295i \(0.00360637\pi\)
−0.999936 + 0.0113295i \(0.996394\pi\)
\(18\) 0 0
\(19\) −1145.00 −0.727648 −0.363824 0.931468i \(-0.618529\pi\)
−0.363824 + 0.931468i \(0.618529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 3108.00i − 1.36907i
\(23\) 1854.00i 0.730786i 0.930853 + 0.365393i \(0.119065\pi\)
−0.930853 + 0.365393i \(0.880935\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3536.00 1.02584
\(27\) 0 0
\(28\) 2272.00i 0.547663i
\(29\) −4920.00 −1.08635 −0.543175 0.839619i \(-0.682778\pi\)
−0.543175 + 0.839619i \(0.682778\pi\)
\(30\) 0 0
\(31\) 1802.00 0.336783 0.168392 0.985720i \(-0.446143\pi\)
0.168392 + 0.985720i \(0.446143\pi\)
\(32\) 1024.00i 0.176777i
\(33\) 0 0
\(34\) −108.000 −0.0160224
\(35\) 0 0
\(36\) 0 0
\(37\) 13178.0i 1.58251i 0.611489 + 0.791253i \(0.290571\pi\)
−0.611489 + 0.791253i \(0.709429\pi\)
\(38\) − 4580.00i − 0.514525i
\(39\) 0 0
\(40\) 0 0
\(41\) 15123.0 1.40501 0.702503 0.711681i \(-0.252066\pi\)
0.702503 + 0.711681i \(0.252066\pi\)
\(42\) 0 0
\(43\) − 7844.00i − 0.646944i −0.946238 0.323472i \(-0.895150\pi\)
0.946238 0.323472i \(-0.104850\pi\)
\(44\) 12432.0 0.968076
\(45\) 0 0
\(46\) −7416.00 −0.516744
\(47\) 6732.00i 0.444528i 0.974986 + 0.222264i \(0.0713447\pi\)
−0.974986 + 0.222264i \(0.928655\pi\)
\(48\) 0 0
\(49\) −3357.00 −0.199738
\(50\) 0 0
\(51\) 0 0
\(52\) 14144.0i 0.725377i
\(53\) 3414.00i 0.166945i 0.996510 + 0.0834726i \(0.0266011\pi\)
−0.996510 + 0.0834726i \(0.973399\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9088.00 −0.387256
\(57\) 0 0
\(58\) − 19680.0i − 0.768166i
\(59\) 33960.0 1.27010 0.635050 0.772471i \(-0.280980\pi\)
0.635050 + 0.772471i \(0.280980\pi\)
\(60\) 0 0
\(61\) 47402.0 1.63107 0.815534 0.578709i \(-0.196443\pi\)
0.815534 + 0.578709i \(0.196443\pi\)
\(62\) 7208.00i 0.238142i
\(63\) 0 0
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 13177.0i − 0.358616i −0.983793 0.179308i \(-0.942614\pi\)
0.983793 0.179308i \(-0.0573858\pi\)
\(68\) − 432.000i − 0.0113295i
\(69\) 0 0
\(70\) 0 0
\(71\) 7548.00 0.177699 0.0888497 0.996045i \(-0.471681\pi\)
0.0888497 + 0.996045i \(0.471681\pi\)
\(72\) 0 0
\(73\) 59821.0i 1.31385i 0.753955 + 0.656926i \(0.228144\pi\)
−0.753955 + 0.656926i \(0.771856\pi\)
\(74\) −52712.0 −1.11900
\(75\) 0 0
\(76\) 18320.0 0.363824
\(77\) 110334.i 2.12072i
\(78\) 0 0
\(79\) −75830.0 −1.36702 −0.683508 0.729943i \(-0.739546\pi\)
−0.683508 + 0.729943i \(0.739546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 60492.0i 0.993490i
\(83\) 46299.0i 0.737694i 0.929490 + 0.368847i \(0.120247\pi\)
−0.929490 + 0.368847i \(0.879753\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 31376.0 0.457458
\(87\) 0 0
\(88\) 49728.0i 0.684533i
\(89\) −30585.0 −0.409292 −0.204646 0.978836i \(-0.565604\pi\)
−0.204646 + 0.978836i \(0.565604\pi\)
\(90\) 0 0
\(91\) −125528. −1.58905
\(92\) − 29664.0i − 0.365393i
\(93\) 0 0
\(94\) −26928.0 −0.314329
\(95\) 0 0
\(96\) 0 0
\(97\) 104018.i 1.12248i 0.827653 + 0.561241i \(0.189676\pi\)
−0.827653 + 0.561241i \(0.810324\pi\)
\(98\) − 13428.0i − 0.141236i
\(99\) 0 0
\(100\) 0 0
\(101\) 23898.0 0.233109 0.116554 0.993184i \(-0.462815\pi\)
0.116554 + 0.993184i \(0.462815\pi\)
\(102\) 0 0
\(103\) 22636.0i 0.210236i 0.994460 + 0.105118i \(0.0335220\pi\)
−0.994460 + 0.105118i \(0.966478\pi\)
\(104\) −56576.0 −0.512919
\(105\) 0 0
\(106\) −13656.0 −0.118048
\(107\) − 60633.0i − 0.511976i −0.966680 0.255988i \(-0.917599\pi\)
0.966680 0.255988i \(-0.0824008\pi\)
\(108\) 0 0
\(109\) 7090.00 0.0571584 0.0285792 0.999592i \(-0.490902\pi\)
0.0285792 + 0.999592i \(0.490902\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 36352.0i − 0.273831i
\(113\) − 128841.i − 0.949201i −0.880201 0.474600i \(-0.842593\pi\)
0.880201 0.474600i \(-0.157407\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 78720.0 0.543175
\(117\) 0 0
\(118\) 135840.i 0.898096i
\(119\) 3834.00 0.0248190
\(120\) 0 0
\(121\) 442678. 2.74868
\(122\) 189608.i 1.15334i
\(123\) 0 0
\(124\) −28832.0 −0.168392
\(125\) 0 0
\(126\) 0 0
\(127\) 141338.i 0.777588i 0.921325 + 0.388794i \(0.127108\pi\)
−0.921325 + 0.388794i \(0.872892\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −80052.0 −0.407562 −0.203781 0.979016i \(-0.565323\pi\)
−0.203781 + 0.979016i \(0.565323\pi\)
\(132\) 0 0
\(133\) 162590.i 0.797012i
\(134\) 52708.0 0.253580
\(135\) 0 0
\(136\) 1728.00 0.00801118
\(137\) − 32253.0i − 0.146814i −0.997302 0.0734072i \(-0.976613\pi\)
0.997302 0.0734072i \(-0.0233873\pi\)
\(138\) 0 0
\(139\) −394865. −1.73345 −0.866726 0.498785i \(-0.833780\pi\)
−0.866726 + 0.498785i \(0.833780\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 30192.0i 0.125652i
\(143\) 686868.i 2.80888i
\(144\) 0 0
\(145\) 0 0
\(146\) −239284. −0.929034
\(147\) 0 0
\(148\) − 210848.i − 0.791253i
\(149\) −491400. −1.81330 −0.906650 0.421884i \(-0.861369\pi\)
−0.906650 + 0.421884i \(0.861369\pi\)
\(150\) 0 0
\(151\) 200402. 0.715253 0.357626 0.933865i \(-0.383586\pi\)
0.357626 + 0.933865i \(0.383586\pi\)
\(152\) 73280.0i 0.257263i
\(153\) 0 0
\(154\) −441336. −1.49957
\(155\) 0 0
\(156\) 0 0
\(157\) − 22942.0i − 0.0742818i −0.999310 0.0371409i \(-0.988175\pi\)
0.999310 0.0371409i \(-0.0118250\pi\)
\(158\) − 303320.i − 0.966626i
\(159\) 0 0
\(160\) 0 0
\(161\) 263268. 0.800448
\(162\) 0 0
\(163\) 336241.i 0.991246i 0.868538 + 0.495623i \(0.165060\pi\)
−0.868538 + 0.495623i \(0.834940\pi\)
\(164\) −241968. −0.702503
\(165\) 0 0
\(166\) −185196. −0.521629
\(167\) − 59748.0i − 0.165780i −0.996559 0.0828900i \(-0.973585\pi\)
0.996559 0.0828900i \(-0.0264150\pi\)
\(168\) 0 0
\(169\) −410163. −1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) 125504.i 0.323472i
\(173\) − 60696.0i − 0.154186i −0.997024 0.0770930i \(-0.975436\pi\)
0.997024 0.0770930i \(-0.0245638\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −198912. −0.484038
\(177\) 0 0
\(178\) − 122340.i − 0.289413i
\(179\) −7995.00 −0.0186503 −0.00932515 0.999957i \(-0.502968\pi\)
−0.00932515 + 0.999957i \(0.502968\pi\)
\(180\) 0 0
\(181\) −454798. −1.03186 −0.515932 0.856630i \(-0.672554\pi\)
−0.515932 + 0.856630i \(0.672554\pi\)
\(182\) − 502112.i − 1.12363i
\(183\) 0 0
\(184\) 118656. 0.258372
\(185\) 0 0
\(186\) 0 0
\(187\) − 20979.0i − 0.0438713i
\(188\) − 107712.i − 0.222264i
\(189\) 0 0
\(190\) 0 0
\(191\) 428298. 0.849499 0.424749 0.905311i \(-0.360362\pi\)
0.424749 + 0.905311i \(0.360362\pi\)
\(192\) 0 0
\(193\) 835531.i 1.61462i 0.590130 + 0.807308i \(0.299076\pi\)
−0.590130 + 0.807308i \(0.700924\pi\)
\(194\) −416072. −0.793714
\(195\) 0 0
\(196\) 53712.0 0.0998691
\(197\) − 678318.i − 1.24528i −0.782508 0.622641i \(-0.786060\pi\)
0.782508 0.622641i \(-0.213940\pi\)
\(198\) 0 0
\(199\) 31900.0 0.0571029 0.0285514 0.999592i \(-0.490911\pi\)
0.0285514 + 0.999592i \(0.490911\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 95592.0i 0.164833i
\(203\) 698640.i 1.18991i
\(204\) 0 0
\(205\) 0 0
\(206\) −90544.0 −0.148659
\(207\) 0 0
\(208\) − 226304.i − 0.362689i
\(209\) 889665. 1.40884
\(210\) 0 0
\(211\) −423673. −0.655126 −0.327563 0.944829i \(-0.606227\pi\)
−0.327563 + 0.944829i \(0.606227\pi\)
\(212\) − 54624.0i − 0.0834726i
\(213\) 0 0
\(214\) 242532. 0.362022
\(215\) 0 0
\(216\) 0 0
\(217\) − 255884.i − 0.368887i
\(218\) 28360.0i 0.0404171i
\(219\) 0 0
\(220\) 0 0
\(221\) 23868.0 0.0328727
\(222\) 0 0
\(223\) − 398204.i − 0.536221i −0.963388 0.268110i \(-0.913601\pi\)
0.963388 0.268110i \(-0.0863992\pi\)
\(224\) 145408. 0.193628
\(225\) 0 0
\(226\) 515364. 0.671186
\(227\) 1.25761e6i 1.61988i 0.586515 + 0.809938i \(0.300500\pi\)
−0.586515 + 0.809938i \(0.699500\pi\)
\(228\) 0 0
\(229\) −203780. −0.256787 −0.128393 0.991723i \(-0.540982\pi\)
−0.128393 + 0.991723i \(0.540982\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 314880.i 0.384083i
\(233\) 823974.i 0.994314i 0.867661 + 0.497157i \(0.165623\pi\)
−0.867661 + 0.497157i \(0.834377\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −543360. −0.635050
\(237\) 0 0
\(238\) 15336.0i 0.0175497i
\(239\) −555960. −0.629577 −0.314788 0.949162i \(-0.601934\pi\)
−0.314788 + 0.949162i \(0.601934\pi\)
\(240\) 0 0
\(241\) 523577. 0.580681 0.290341 0.956923i \(-0.406231\pi\)
0.290341 + 0.956923i \(0.406231\pi\)
\(242\) 1.77071e6i 1.94361i
\(243\) 0 0
\(244\) −758432. −0.815534
\(245\) 0 0
\(246\) 0 0
\(247\) 1.01218e6i 1.05564i
\(248\) − 115328.i − 0.119071i
\(249\) 0 0
\(250\) 0 0
\(251\) −113127. −0.113340 −0.0566698 0.998393i \(-0.518048\pi\)
−0.0566698 + 0.998393i \(0.518048\pi\)
\(252\) 0 0
\(253\) − 1.44056e6i − 1.41491i
\(254\) −565352. −0.549838
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 872958.i − 0.824443i −0.911084 0.412221i \(-0.864753\pi\)
0.911084 0.412221i \(-0.135247\pi\)
\(258\) 0 0
\(259\) 1.87128e6 1.73336
\(260\) 0 0
\(261\) 0 0
\(262\) − 320208.i − 0.288190i
\(263\) − 1.64647e6i − 1.46779i −0.679264 0.733894i \(-0.737701\pi\)
0.679264 0.733894i \(-0.262299\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −650360. −0.563572
\(267\) 0 0
\(268\) 210832.i 0.179308i
\(269\) 1.78872e6 1.50717 0.753584 0.657352i \(-0.228323\pi\)
0.753584 + 0.657352i \(0.228323\pi\)
\(270\) 0 0
\(271\) 1.12140e6 0.927552 0.463776 0.885953i \(-0.346494\pi\)
0.463776 + 0.885953i \(0.346494\pi\)
\(272\) 6912.00i 0.00566476i
\(273\) 0 0
\(274\) 129012. 0.103813
\(275\) 0 0
\(276\) 0 0
\(277\) − 598312.i − 0.468520i −0.972174 0.234260i \(-0.924733\pi\)
0.972174 0.234260i \(-0.0752667\pi\)
\(278\) − 1.57946e6i − 1.22574i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.53050e6 1.15629 0.578145 0.815934i \(-0.303777\pi\)
0.578145 + 0.815934i \(0.303777\pi\)
\(282\) 0 0
\(283\) 1.79700e6i 1.33377i 0.745159 + 0.666887i \(0.232374\pi\)
−0.745159 + 0.666887i \(0.767626\pi\)
\(284\) −120768. −0.0888497
\(285\) 0 0
\(286\) −2.74747e6 −1.98618
\(287\) − 2.14747e6i − 1.53894i
\(288\) 0 0
\(289\) 1.41913e6 0.999487
\(290\) 0 0
\(291\) 0 0
\(292\) − 957136.i − 0.656926i
\(293\) 754494.i 0.513437i 0.966486 + 0.256718i \(0.0826412\pi\)
−0.966486 + 0.256718i \(0.917359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 843392. 0.559500
\(297\) 0 0
\(298\) − 1.96560e6i − 1.28220i
\(299\) 1.63894e6 1.06019
\(300\) 0 0
\(301\) −1.11385e6 −0.708614
\(302\) 801608.i 0.505760i
\(303\) 0 0
\(304\) −293120. −0.181912
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.96627e6i − 1.19068i −0.803472 0.595342i \(-0.797017\pi\)
0.803472 0.595342i \(-0.202983\pi\)
\(308\) − 1.76534e6i − 1.06036i
\(309\) 0 0
\(310\) 0 0
\(311\) 599298. 0.351352 0.175676 0.984448i \(-0.443789\pi\)
0.175676 + 0.984448i \(0.443789\pi\)
\(312\) 0 0
\(313\) 721366.i 0.416193i 0.978108 + 0.208097i \(0.0667268\pi\)
−0.978108 + 0.208097i \(0.933273\pi\)
\(314\) 91768.0 0.0525251
\(315\) 0 0
\(316\) 1.21328e6 0.683508
\(317\) − 102348.i − 0.0572046i −0.999591 0.0286023i \(-0.990894\pi\)
0.999591 0.0286023i \(-0.00910564\pi\)
\(318\) 0 0
\(319\) 3.82284e6 2.10334
\(320\) 0 0
\(321\) 0 0
\(322\) 1.05307e6i 0.566003i
\(323\) − 30915.0i − 0.0164878i
\(324\) 0 0
\(325\) 0 0
\(326\) −1.34496e6 −0.700917
\(327\) 0 0
\(328\) − 967872.i − 0.496745i
\(329\) 955944. 0.486903
\(330\) 0 0
\(331\) 1.31048e6 0.657445 0.328722 0.944427i \(-0.393382\pi\)
0.328722 + 0.944427i \(0.393382\pi\)
\(332\) − 740784.i − 0.368847i
\(333\) 0 0
\(334\) 238992. 0.117224
\(335\) 0 0
\(336\) 0 0
\(337\) − 804397.i − 0.385830i −0.981215 0.192915i \(-0.938206\pi\)
0.981215 0.192915i \(-0.0617941\pi\)
\(338\) − 1.64065e6i − 0.781133i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.40015e6 −0.652063
\(342\) 0 0
\(343\) − 1.90990e6i − 0.876547i
\(344\) −502016. −0.228729
\(345\) 0 0
\(346\) 242784. 0.109026
\(347\) 2.88321e6i 1.28544i 0.766101 + 0.642720i \(0.222194\pi\)
−0.766101 + 0.642720i \(0.777806\pi\)
\(348\) 0 0
\(349\) −1.27355e6 −0.559696 −0.279848 0.960044i \(-0.590284\pi\)
−0.279848 + 0.960044i \(0.590284\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 795648.i − 0.342266i
\(353\) 2.83061e6i 1.20905i 0.796587 + 0.604524i \(0.206637\pi\)
−0.796587 + 0.604524i \(0.793363\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 489360. 0.204646
\(357\) 0 0
\(358\) − 31980.0i − 0.0131878i
\(359\) −981090. −0.401766 −0.200883 0.979615i \(-0.564381\pi\)
−0.200883 + 0.979615i \(0.564381\pi\)
\(360\) 0 0
\(361\) −1.16507e6 −0.470528
\(362\) − 1.81919e6i − 0.729637i
\(363\) 0 0
\(364\) 2.00845e6 0.794524
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.19105e6i − 1.62427i −0.583470 0.812134i \(-0.698306\pi\)
0.583470 0.812134i \(-0.301694\pi\)
\(368\) 474624.i 0.182696i
\(369\) 0 0
\(370\) 0 0
\(371\) 484788. 0.182859
\(372\) 0 0
\(373\) − 3.23455e6i − 1.20377i −0.798584 0.601883i \(-0.794417\pi\)
0.798584 0.601883i \(-0.205583\pi\)
\(374\) 83916.0 0.0310217
\(375\) 0 0
\(376\) 430848. 0.157165
\(377\) 4.34928e6i 1.57603i
\(378\) 0 0
\(379\) −1.39036e6 −0.497196 −0.248598 0.968607i \(-0.579970\pi\)
−0.248598 + 0.968607i \(0.579970\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.71319e6i 0.600686i
\(383\) 1.14197e6i 0.397795i 0.980020 + 0.198897i \(0.0637361\pi\)
−0.980020 + 0.198897i \(0.936264\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.34212e6 −1.14171
\(387\) 0 0
\(388\) − 1.66429e6i − 0.561241i
\(389\) 3.46299e6 1.16032 0.580159 0.814503i \(-0.302990\pi\)
0.580159 + 0.814503i \(0.302990\pi\)
\(390\) 0 0
\(391\) −50058.0 −0.0165589
\(392\) 214848.i 0.0706181i
\(393\) 0 0
\(394\) 2.71327e6 0.880547
\(395\) 0 0
\(396\) 0 0
\(397\) 5.94007e6i 1.89154i 0.324839 + 0.945769i \(0.394690\pi\)
−0.324839 + 0.945769i \(0.605310\pi\)
\(398\) 127600.i 0.0403778i
\(399\) 0 0
\(400\) 0 0
\(401\) 2.27412e6 0.706241 0.353121 0.935578i \(-0.385121\pi\)
0.353121 + 0.935578i \(0.385121\pi\)
\(402\) 0 0
\(403\) − 1.59297e6i − 0.488590i
\(404\) −382368. −0.116554
\(405\) 0 0
\(406\) −2.79456e6 −0.841392
\(407\) − 1.02393e7i − 3.06397i
\(408\) 0 0
\(409\) 4.29552e6 1.26972 0.634859 0.772628i \(-0.281058\pi\)
0.634859 + 0.772628i \(0.281058\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 362176.i − 0.105118i
\(413\) − 4.82232e6i − 1.39117i
\(414\) 0 0
\(415\) 0 0
\(416\) 905216. 0.256460
\(417\) 0 0
\(418\) 3.55866e6i 0.996198i
\(419\) 1.79705e6 0.500062 0.250031 0.968238i \(-0.419559\pi\)
0.250031 + 0.968238i \(0.419559\pi\)
\(420\) 0 0
\(421\) −257548. −0.0708195 −0.0354098 0.999373i \(-0.511274\pi\)
−0.0354098 + 0.999373i \(0.511274\pi\)
\(422\) − 1.69469e6i − 0.463244i
\(423\) 0 0
\(424\) 218496. 0.0590240
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.73108e6i − 1.78655i
\(428\) 970128.i 0.255988i
\(429\) 0 0
\(430\) 0 0
\(431\) −2.22910e6 −0.578012 −0.289006 0.957327i \(-0.593325\pi\)
−0.289006 + 0.957327i \(0.593325\pi\)
\(432\) 0 0
\(433\) 4.20585e6i 1.07804i 0.842294 + 0.539019i \(0.181205\pi\)
−0.842294 + 0.539019i \(0.818795\pi\)
\(434\) 1.02354e6 0.260843
\(435\) 0 0
\(436\) −113440. −0.0285792
\(437\) − 2.12283e6i − 0.531755i
\(438\) 0 0
\(439\) −352640. −0.0873314 −0.0436657 0.999046i \(-0.513904\pi\)
−0.0436657 + 0.999046i \(0.513904\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 95472.0i 0.0232445i
\(443\) 1.28362e6i 0.310761i 0.987855 + 0.155381i \(0.0496604\pi\)
−0.987855 + 0.155381i \(0.950340\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.59282e6 0.379165
\(447\) 0 0
\(448\) 581632.i 0.136916i
\(449\) −2.10398e6 −0.492521 −0.246260 0.969204i \(-0.579202\pi\)
−0.246260 + 0.969204i \(0.579202\pi\)
\(450\) 0 0
\(451\) −1.17506e7 −2.72031
\(452\) 2.06146e6i 0.474600i
\(453\) 0 0
\(454\) −5.03045e6 −1.14543
\(455\) 0 0
\(456\) 0 0
\(457\) 825233.i 0.184836i 0.995720 + 0.0924179i \(0.0294596\pi\)
−0.995720 + 0.0924179i \(0.970540\pi\)
\(458\) − 815120.i − 0.181576i
\(459\) 0 0
\(460\) 0 0
\(461\) −4.50145e6 −0.986507 −0.493254 0.869886i \(-0.664193\pi\)
−0.493254 + 0.869886i \(0.664193\pi\)
\(462\) 0 0
\(463\) 1.44212e6i 0.312642i 0.987706 + 0.156321i \(0.0499635\pi\)
−0.987706 + 0.156321i \(0.950037\pi\)
\(464\) −1.25952e6 −0.271588
\(465\) 0 0
\(466\) −3.29590e6 −0.703086
\(467\) − 393348.i − 0.0834612i −0.999129 0.0417306i \(-0.986713\pi\)
0.999129 0.0417306i \(-0.0132871\pi\)
\(468\) 0 0
\(469\) −1.87113e6 −0.392801
\(470\) 0 0
\(471\) 0 0
\(472\) − 2.17344e6i − 0.449048i
\(473\) 6.09479e6i 1.25258i
\(474\) 0 0
\(475\) 0 0
\(476\) −61344.0 −0.0124095
\(477\) 0 0
\(478\) − 2.22384e6i − 0.445178i
\(479\) −9.17697e6 −1.82751 −0.913757 0.406262i \(-0.866832\pi\)
−0.913757 + 0.406262i \(0.866832\pi\)
\(480\) 0 0
\(481\) 1.16494e7 2.29583
\(482\) 2.09431e6i 0.410604i
\(483\) 0 0
\(484\) −7.08285e6 −1.37434
\(485\) 0 0
\(486\) 0 0
\(487\) 6.60598e6i 1.26216i 0.775717 + 0.631080i \(0.217388\pi\)
−0.775717 + 0.631080i \(0.782612\pi\)
\(488\) − 3.03373e6i − 0.576670i
\(489\) 0 0
\(490\) 0 0
\(491\) −38052.0 −0.00712318 −0.00356159 0.999994i \(-0.501134\pi\)
−0.00356159 + 0.999994i \(0.501134\pi\)
\(492\) 0 0
\(493\) − 132840.i − 0.0246157i
\(494\) −4.04872e6 −0.746449
\(495\) 0 0
\(496\) 461312. 0.0841958
\(497\) − 1.07182e6i − 0.194639i
\(498\) 0 0
\(499\) −6.85670e6 −1.23272 −0.616359 0.787465i \(-0.711393\pi\)
−0.616359 + 0.787465i \(0.711393\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 452508.i − 0.0801433i
\(503\) 8.20016e6i 1.44512i 0.691311 + 0.722558i \(0.257034\pi\)
−0.691311 + 0.722558i \(0.742966\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5.76223e6 1.00049
\(507\) 0 0
\(508\) − 2.26141e6i − 0.388794i
\(509\) 4.06581e6 0.695589 0.347794 0.937571i \(-0.386931\pi\)
0.347794 + 0.937571i \(0.386931\pi\)
\(510\) 0 0
\(511\) 8.49458e6 1.43910
\(512\) 262144.i 0.0441942i
\(513\) 0 0
\(514\) 3.49183e6 0.582969
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.23076e6i − 0.860674i
\(518\) 7.48510e6i 1.22567i
\(519\) 0 0
\(520\) 0 0
\(521\) −5.28408e6 −0.852854 −0.426427 0.904522i \(-0.640228\pi\)
−0.426427 + 0.904522i \(0.640228\pi\)
\(522\) 0 0
\(523\) − 2.53383e6i − 0.405063i −0.979276 0.202532i \(-0.935083\pi\)
0.979276 0.202532i \(-0.0649169\pi\)
\(524\) 1.28083e6 0.203781
\(525\) 0 0
\(526\) 6.58586e6 1.03788
\(527\) 48654.0i 0.00763119i
\(528\) 0 0
\(529\) 2.99903e6 0.465952
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.60144e6i − 0.398506i
\(533\) − 1.33687e7i − 2.03832i
\(534\) 0 0
\(535\) 0 0
\(536\) −843328. −0.126790
\(537\) 0 0
\(538\) 7.15488e6i 1.06573i
\(539\) 2.60839e6 0.386723
\(540\) 0 0
\(541\) 498752. 0.0732641 0.0366321 0.999329i \(-0.488337\pi\)
0.0366321 + 0.999329i \(0.488337\pi\)
\(542\) 4.48561e6i 0.655878i
\(543\) 0 0
\(544\) −27648.0 −0.00400559
\(545\) 0 0
\(546\) 0 0
\(547\) 3.00269e6i 0.429084i 0.976715 + 0.214542i \(0.0688259\pi\)
−0.976715 + 0.214542i \(0.931174\pi\)
\(548\) 516048.i 0.0734072i
\(549\) 0 0
\(550\) 0 0
\(551\) 5.63340e6 0.790481
\(552\) 0 0
\(553\) 1.07679e7i 1.49733i
\(554\) 2.39325e6 0.331294
\(555\) 0 0
\(556\) 6.31784e6 0.866726
\(557\) 1.27373e7i 1.73956i 0.493441 + 0.869779i \(0.335739\pi\)
−0.493441 + 0.869779i \(0.664261\pi\)
\(558\) 0 0
\(559\) −6.93410e6 −0.938556
\(560\) 0 0
\(561\) 0 0
\(562\) 6.12199e6i 0.817621i
\(563\) − 5.97082e6i − 0.793894i −0.917841 0.396947i \(-0.870070\pi\)
0.917841 0.396947i \(-0.129930\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7.18800e6 −0.943121
\(567\) 0 0
\(568\) − 483072.i − 0.0628262i
\(569\) −9.26906e6 −1.20020 −0.600102 0.799924i \(-0.704873\pi\)
−0.600102 + 0.799924i \(0.704873\pi\)
\(570\) 0 0
\(571\) −3.89535e6 −0.499984 −0.249992 0.968248i \(-0.580428\pi\)
−0.249992 + 0.968248i \(0.580428\pi\)
\(572\) − 1.09899e7i − 1.40444i
\(573\) 0 0
\(574\) 8.58986e6 1.08819
\(575\) 0 0
\(576\) 0 0
\(577\) 7.29416e6i 0.912086i 0.889958 + 0.456043i \(0.150734\pi\)
−0.889958 + 0.456043i \(0.849266\pi\)
\(578\) 5.67651e6i 0.706744i
\(579\) 0 0
\(580\) 0 0
\(581\) 6.57446e6 0.808015
\(582\) 0 0
\(583\) − 2.65268e6i − 0.323231i
\(584\) 3.82854e6 0.464517
\(585\) 0 0
\(586\) −3.01798e6 −0.363054
\(587\) 8.72820e6i 1.04551i 0.852482 + 0.522756i \(0.175096\pi\)
−0.852482 + 0.522756i \(0.824904\pi\)
\(588\) 0 0
\(589\) −2.06329e6 −0.245060
\(590\) 0 0
\(591\) 0 0
\(592\) 3.37357e6i 0.395626i
\(593\) − 1.30963e7i − 1.52937i −0.644407 0.764683i \(-0.722896\pi\)
0.644407 0.764683i \(-0.277104\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.86240e6 0.906650
\(597\) 0 0
\(598\) 6.55574e6i 0.749668i
\(599\) −1.30168e7 −1.48231 −0.741155 0.671334i \(-0.765721\pi\)
−0.741155 + 0.671334i \(0.765721\pi\)
\(600\) 0 0
\(601\) −9.93997e6 −1.12253 −0.561266 0.827635i \(-0.689686\pi\)
−0.561266 + 0.827635i \(0.689686\pi\)
\(602\) − 4.45539e6i − 0.501066i
\(603\) 0 0
\(604\) −3.20643e6 −0.357626
\(605\) 0 0
\(606\) 0 0
\(607\) 1.56438e7i 1.72334i 0.507470 + 0.861670i \(0.330581\pi\)
−0.507470 + 0.861670i \(0.669419\pi\)
\(608\) − 1.17248e6i − 0.128631i
\(609\) 0 0
\(610\) 0 0
\(611\) 5.95109e6 0.644901
\(612\) 0 0
\(613\) − 9.33793e6i − 1.00369i −0.864958 0.501845i \(-0.832655\pi\)
0.864958 0.501845i \(-0.167345\pi\)
\(614\) 7.86507e6 0.841941
\(615\) 0 0
\(616\) 7.06138e6 0.749786
\(617\) 5.06680e6i 0.535823i 0.963444 + 0.267911i \(0.0863334\pi\)
−0.963444 + 0.267911i \(0.913667\pi\)
\(618\) 0 0
\(619\) −1.37670e7 −1.44415 −0.722077 0.691813i \(-0.756812\pi\)
−0.722077 + 0.691813i \(0.756812\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.39719e6i 0.248443i
\(623\) 4.34307e6i 0.448308i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.88546e6 −0.294293
\(627\) 0 0
\(628\) 367072.i 0.0371409i
\(629\) −355806. −0.0358580
\(630\) 0 0
\(631\) 2.07060e6 0.207025 0.103513 0.994628i \(-0.466992\pi\)
0.103513 + 0.994628i \(0.466992\pi\)
\(632\) 4.85312e6i 0.483313i
\(633\) 0 0
\(634\) 409392. 0.0404498
\(635\) 0 0
\(636\) 0 0
\(637\) 2.96759e6i 0.289771i
\(638\) 1.52914e7i 1.48729i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.79114e7 1.72181 0.860903 0.508768i \(-0.169899\pi\)
0.860903 + 0.508768i \(0.169899\pi\)
\(642\) 0 0
\(643\) 1.71414e7i 1.63500i 0.575929 + 0.817500i \(0.304641\pi\)
−0.575929 + 0.817500i \(0.695359\pi\)
\(644\) −4.21229e6 −0.400224
\(645\) 0 0
\(646\) 123660. 0.0116586
\(647\) 8.48773e6i 0.797133i 0.917139 + 0.398567i \(0.130492\pi\)
−0.917139 + 0.398567i \(0.869508\pi\)
\(648\) 0 0
\(649\) −2.63869e7 −2.45910
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.37986e6i − 0.495623i
\(653\) − 2.45479e6i − 0.225284i −0.993636 0.112642i \(-0.964069\pi\)
0.993636 0.112642i \(-0.0359313\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.87149e6 0.351252
\(657\) 0 0
\(658\) 3.82378e6i 0.344293i
\(659\) −5.91557e6 −0.530619 −0.265309 0.964163i \(-0.585474\pi\)
−0.265309 + 0.964163i \(0.585474\pi\)
\(660\) 0 0
\(661\) 4.33095e6 0.385549 0.192775 0.981243i \(-0.438251\pi\)
0.192775 + 0.981243i \(0.438251\pi\)
\(662\) 5.24191e6i 0.464884i
\(663\) 0 0
\(664\) 2.96314e6 0.260814
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.12168e6i − 0.793890i
\(668\) 955968.i 0.0828900i
\(669\) 0 0
\(670\) 0 0
\(671\) −3.68314e7 −3.15799
\(672\) 0 0
\(673\) 9.13985e6i 0.777860i 0.921267 + 0.388930i \(0.127155\pi\)
−0.921267 + 0.388930i \(0.872845\pi\)
\(674\) 3.21759e6 0.272823
\(675\) 0 0
\(676\) 6.56261e6 0.552344
\(677\) − 4.57229e6i − 0.383409i −0.981453 0.191704i \(-0.938599\pi\)
0.981453 0.191704i \(-0.0614015\pi\)
\(678\) 0 0
\(679\) 1.47706e7 1.22948
\(680\) 0 0
\(681\) 0 0
\(682\) − 5.60062e6i − 0.461078i
\(683\) 1.53221e7i 1.25681i 0.777888 + 0.628403i \(0.216291\pi\)
−0.777888 + 0.628403i \(0.783709\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7.63960e6 0.619813
\(687\) 0 0
\(688\) − 2.00806e6i − 0.161736i
\(689\) 3.01798e6 0.242196
\(690\) 0 0
\(691\) 7.02548e6 0.559733 0.279866 0.960039i \(-0.409710\pi\)
0.279866 + 0.960039i \(0.409710\pi\)
\(692\) 971136.i 0.0770930i
\(693\) 0 0
\(694\) −1.15328e7 −0.908944
\(695\) 0 0
\(696\) 0 0
\(697\) 408321.i 0.0318361i
\(698\) − 5.09420e6i − 0.395765i
\(699\) 0 0
\(700\) 0 0
\(701\) −7.91125e6 −0.608065 −0.304033 0.952662i \(-0.598333\pi\)
−0.304033 + 0.952662i \(0.598333\pi\)
\(702\) 0 0
\(703\) − 1.50888e7i − 1.15151i
\(704\) 3.18259e6 0.242019
\(705\) 0 0
\(706\) −1.13225e7 −0.854927
\(707\) − 3.39352e6i − 0.255330i
\(708\) 0 0
\(709\) 1.54485e7 1.15418 0.577088 0.816682i \(-0.304189\pi\)
0.577088 + 0.816682i \(0.304189\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.95744e6i 0.144707i
\(713\) 3.34091e6i 0.246116i
\(714\) 0 0
\(715\) 0 0
\(716\) 127920. 0.00932515
\(717\) 0 0
\(718\) − 3.92436e6i − 0.284091i
\(719\) 2.30544e7 1.66315 0.831574 0.555414i \(-0.187440\pi\)
0.831574 + 0.555414i \(0.187440\pi\)
\(720\) 0 0
\(721\) 3.21431e6 0.230277
\(722\) − 4.66030e6i − 0.332714i
\(723\) 0 0
\(724\) 7.27677e6 0.515932
\(725\) 0 0
\(726\) 0 0
\(727\) 1.62905e7i 1.14314i 0.820555 + 0.571568i \(0.193665\pi\)
−0.820555 + 0.571568i \(0.806335\pi\)
\(728\) 8.03379e6i 0.561813i
\(729\) 0 0
\(730\) 0 0
\(731\) 211788. 0.0146591
\(732\) 0 0
\(733\) − 1.28279e7i − 0.881853i −0.897543 0.440927i \(-0.854650\pi\)
0.897543 0.440927i \(-0.145350\pi\)
\(734\) 1.67642e7 1.14853
\(735\) 0 0
\(736\) −1.89850e6 −0.129186
\(737\) 1.02385e7i 0.694335i
\(738\) 0 0
\(739\) 1.24535e7 0.838840 0.419420 0.907792i \(-0.362234\pi\)
0.419420 + 0.907792i \(0.362234\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.93915e6i 0.129301i
\(743\) − 2.63247e7i − 1.74941i −0.484656 0.874705i \(-0.661055\pi\)
0.484656 0.874705i \(-0.338945\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.29382e7 0.851192
\(747\) 0 0
\(748\) 335664.i 0.0219357i
\(749\) −8.60989e6 −0.560780
\(750\) 0 0
\(751\) −1.74994e7 −1.13220 −0.566102 0.824335i \(-0.691549\pi\)
−0.566102 + 0.824335i \(0.691549\pi\)
\(752\) 1.72339e6i 0.111132i
\(753\) 0 0
\(754\) −1.73971e7 −1.11442
\(755\) 0 0
\(756\) 0 0
\(757\) 3.46381e6i 0.219692i 0.993949 + 0.109846i \(0.0350358\pi\)
−0.993949 + 0.109846i \(0.964964\pi\)
\(758\) − 5.56142e6i − 0.351571i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.26175e7 0.789792 0.394896 0.918726i \(-0.370781\pi\)
0.394896 + 0.918726i \(0.370781\pi\)
\(762\) 0 0
\(763\) − 1.00678e6i − 0.0626070i
\(764\) −6.85277e6 −0.424749
\(765\) 0 0
\(766\) −4.56790e6 −0.281284
\(767\) − 3.00206e7i − 1.84260i
\(768\) 0 0
\(769\) −5.70804e6 −0.348074 −0.174037 0.984739i \(-0.555681\pi\)
−0.174037 + 0.984739i \(0.555681\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1.33685e7i − 0.807308i
\(773\) 1.20827e7i 0.727303i 0.931535 + 0.363652i \(0.118470\pi\)
−0.931535 + 0.363652i \(0.881530\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.65715e6 0.396857
\(777\) 0 0
\(778\) 1.38520e7i 0.820469i
\(779\) −1.73158e7 −1.02235
\(780\) 0 0
\(781\) −5.86480e6 −0.344053
\(782\) − 200232.i − 0.0117089i
\(783\) 0 0
\(784\) −859392. −0.0499346
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.37636e7i − 0.792126i −0.918223 0.396063i \(-0.870376\pi\)
0.918223 0.396063i \(-0.129624\pi\)
\(788\) 1.08531e7i 0.622641i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.82954e7 −1.03968
\(792\) 0 0
\(793\) − 4.19034e7i − 2.36628i
\(794\) −2.37603e7 −1.33752
\(795\) 0 0
\(796\) −510400. −0.0285514
\(797\) 8.77738e6i 0.489462i 0.969591 + 0.244731i \(0.0786997\pi\)
−0.969591 + 0.244731i \(0.921300\pi\)
\(798\) 0 0
\(799\) −181764. −0.0100726
\(800\) 0 0
\(801\) 0 0
\(802\) 9.09649e6i 0.499388i
\(803\) − 4.64809e7i − 2.54382i
\(804\) 0 0
\(805\) 0 0
\(806\) 6.37187e6 0.345485
\(807\) 0 0
\(808\) − 1.52947e6i − 0.0824163i
\(809\) −1.02046e7 −0.548181 −0.274091 0.961704i \(-0.588377\pi\)
−0.274091 + 0.961704i \(0.588377\pi\)
\(810\) 0 0
\(811\) −1.17375e6 −0.0626647 −0.0313323 0.999509i \(-0.509975\pi\)
−0.0313323 + 0.999509i \(0.509975\pi\)
\(812\) − 1.11782e7i − 0.594954i
\(813\) 0 0
\(814\) 4.09572e7 2.16655
\(815\) 0 0
\(816\) 0 0
\(817\) 8.98138e6i 0.470747i
\(818\) 1.71821e7i 0.897826i
\(819\) 0 0
\(820\) 0 0
\(821\) 1.98062e7 1.02552 0.512759 0.858533i \(-0.328623\pi\)
0.512759 + 0.858533i \(0.328623\pi\)
\(822\) 0 0
\(823\) 3.06722e7i 1.57850i 0.614070 + 0.789251i \(0.289531\pi\)
−0.614070 + 0.789251i \(0.710469\pi\)
\(824\) 1.44870e6 0.0743296
\(825\) 0 0
\(826\) 1.92893e7 0.983707
\(827\) − 2.55520e7i − 1.29915i −0.760296 0.649577i \(-0.774946\pi\)
0.760296 0.649577i \(-0.225054\pi\)
\(828\) 0 0
\(829\) 9.19402e6 0.464643 0.232321 0.972639i \(-0.425368\pi\)
0.232321 + 0.972639i \(0.425368\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.62086e6i 0.181344i
\(833\) − 90639.0i − 0.00452588i
\(834\) 0 0
\(835\) 0 0
\(836\) −1.42346e7 −0.704419
\(837\) 0 0
\(838\) 7.18818e6i 0.353597i
\(839\) −1.56910e7 −0.769564 −0.384782 0.923008i \(-0.625723\pi\)
−0.384782 + 0.923008i \(0.625723\pi\)
\(840\) 0 0
\(841\) 3.69525e6 0.180158
\(842\) − 1.03019e6i − 0.0500770i
\(843\) 0 0
\(844\) 6.77877e6 0.327563
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.28603e7i − 3.01070i
\(848\) 873984.i 0.0417363i
\(849\) 0 0
\(850\) 0 0
\(851\) −2.44320e7 −1.15647
\(852\) 0 0
\(853\) − 1.60111e6i − 0.0753442i −0.999290 0.0376721i \(-0.988006\pi\)
0.999290 0.0376721i \(-0.0119942\pi\)
\(854\) 2.69243e7 1.26328
\(855\) 0 0
\(856\) −3.88051e6 −0.181011
\(857\) 1.64613e7i 0.765616i 0.923828 + 0.382808i \(0.125043\pi\)
−0.923828 + 0.382808i \(0.874957\pi\)
\(858\) 0 0
\(859\) −1.96736e7 −0.909705 −0.454853 0.890567i \(-0.650308\pi\)
−0.454853 + 0.890567i \(0.650308\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 8.91641e6i − 0.408716i
\(863\) 3.68068e7i 1.68229i 0.540810 + 0.841145i \(0.318118\pi\)
−0.540810 + 0.841145i \(0.681882\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.68234e7 −0.762288
\(867\) 0 0
\(868\) 4.09414e6i 0.184444i
\(869\) 5.89199e7 2.64675
\(870\) 0 0
\(871\) −1.16485e7 −0.520264
\(872\) − 453760.i − 0.0202085i
\(873\) 0 0
\(874\) 8.49132e6 0.376008
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.69596e7i − 1.18363i −0.806075 0.591813i \(-0.798412\pi\)
0.806075 0.591813i \(-0.201588\pi\)
\(878\) − 1.41056e6i − 0.0617526i
\(879\) 0 0
\(880\) 0 0
\(881\) −3.47335e7 −1.50768 −0.753839 0.657059i \(-0.771800\pi\)
−0.753839 + 0.657059i \(0.771800\pi\)
\(882\) 0 0
\(883\) − 2.16187e7i − 0.933101i −0.884494 0.466551i \(-0.845497\pi\)
0.884494 0.466551i \(-0.154503\pi\)
\(884\) −381888. −0.0164363
\(885\) 0 0
\(886\) −5.13448e6 −0.219741
\(887\) − 4.48163e6i − 0.191261i −0.995417 0.0956306i \(-0.969513\pi\)
0.995417 0.0956306i \(-0.0304867\pi\)
\(888\) 0 0
\(889\) 2.00700e7 0.851712
\(890\) 0 0
\(891\) 0 0
\(892\) 6.37126e6i 0.268110i
\(893\) − 7.70814e6i − 0.323460i
\(894\) 0 0
\(895\) 0 0
\(896\) −2.32653e6 −0.0968140
\(897\) 0 0
\(898\) − 8.41590e6i − 0.348265i
\(899\) −8.86584e6 −0.365865
\(900\) 0 0
\(901\) −92178.0 −0.00378282
\(902\) − 4.70023e7i − 1.92355i
\(903\) 0 0
\(904\) −8.24582e6 −0.335593
\(905\) 0 0
\(906\) 0 0
\(907\) − 3.36639e7i − 1.35877i −0.733782 0.679385i \(-0.762247\pi\)
0.733782 0.679385i \(-0.237753\pi\)
\(908\) − 2.01218e7i − 0.809938i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.03175e6 0.0411887 0.0205943 0.999788i \(-0.493444\pi\)
0.0205943 + 0.999788i \(0.493444\pi\)
\(912\) 0 0
\(913\) − 3.59743e7i − 1.42829i
\(914\) −3.30093e6 −0.130699
\(915\) 0 0
\(916\) 3.26048e6 0.128393
\(917\) 1.13674e7i 0.446413i
\(918\) 0 0
\(919\) −4.10147e6 −0.160196 −0.0800978 0.996787i \(-0.525523\pi\)
−0.0800978 + 0.996787i \(0.525523\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 1.80058e7i − 0.697566i
\(923\) − 6.67243e6i − 0.257798i
\(924\) 0 0
\(925\) 0 0
\(926\) −5.76846e6 −0.221071
\(927\) 0 0
\(928\) − 5.03808e6i − 0.192042i
\(929\) 7.71603e6 0.293329 0.146664 0.989186i \(-0.453146\pi\)
0.146664 + 0.989186i \(0.453146\pi\)
\(930\) 0 0
\(931\) 3.84376e6 0.145339
\(932\) − 1.31836e7i − 0.497157i
\(933\) 0 0
\(934\) 1.57339e6 0.0590160
\(935\) 0 0
\(936\) 0 0
\(937\) 4.38458e7i 1.63147i 0.578426 + 0.815735i \(0.303667\pi\)
−0.578426 + 0.815735i \(0.696333\pi\)
\(938\) − 7.48454e6i − 0.277752i
\(939\) 0 0
\(940\) 0 0
\(941\) 1.00215e7 0.368944 0.184472 0.982838i \(-0.440942\pi\)
0.184472 + 0.982838i \(0.440942\pi\)
\(942\) 0 0
\(943\) 2.80380e7i 1.02676i
\(944\) 8.69376e6 0.317525
\(945\) 0 0
\(946\) −2.43792e7 −0.885708
\(947\) − 2.79530e7i − 1.01287i −0.862279 0.506434i \(-0.830963\pi\)
0.862279 0.506434i \(-0.169037\pi\)
\(948\) 0 0
\(949\) 5.28818e7 1.90608
\(950\) 0 0
\(951\) 0 0
\(952\) − 245376.i − 0.00877485i
\(953\) − 2.31811e7i − 0.826803i −0.910549 0.413401i \(-0.864341\pi\)
0.910549 0.413401i \(-0.135659\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.89536e6 0.314788
\(957\) 0 0
\(958\) − 3.67079e7i − 1.29225i
\(959\) −4.57993e6 −0.160810
\(960\) 0 0
\(961\) −2.53819e7 −0.886577
\(962\) 4.65974e7i 1.62339i
\(963\) 0 0
\(964\) −8.37723e6 −0.290341
\(965\) 0 0
\(966\) 0 0
\(967\) − 1.58435e6i − 0.0544861i −0.999629 0.0272430i \(-0.991327\pi\)
0.999629 0.0272430i \(-0.00867280\pi\)
\(968\) − 2.83314e7i − 0.971806i
\(969\) 0 0
\(970\) 0 0
\(971\) −3.44552e7 −1.17275 −0.586376 0.810039i \(-0.699446\pi\)
−0.586376 + 0.810039i \(0.699446\pi\)
\(972\) 0 0
\(973\) 5.60708e7i 1.89869i
\(974\) −2.64239e7 −0.892483
\(975\) 0 0
\(976\) 1.21349e7 0.407767
\(977\) 2.93599e7i 0.984052i 0.870581 + 0.492026i \(0.163743\pi\)
−0.870581 + 0.492026i \(0.836257\pi\)
\(978\) 0 0
\(979\) 2.37645e7 0.792452
\(980\) 0 0
\(981\) 0 0
\(982\) − 152208.i − 0.00503685i
\(983\) 8.93957e6i 0.295075i 0.989056 + 0.147538i \(0.0471348\pi\)
−0.989056 + 0.147538i \(0.952865\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 531360. 0.0174059
\(987\) 0 0
\(988\) − 1.61949e7i − 0.527819i
\(989\) 1.45428e7 0.472777
\(990\) 0 0
\(991\) −1.78899e7 −0.578660 −0.289330 0.957229i \(-0.593433\pi\)
−0.289330 + 0.957229i \(0.593433\pi\)
\(992\) 1.84525e6i 0.0595354i
\(993\) 0 0
\(994\) 4.28726e6 0.137630
\(995\) 0 0
\(996\) 0 0
\(997\) 3.30517e7i 1.05307i 0.850155 + 0.526533i \(0.176508\pi\)
−0.850155 + 0.526533i \(0.823492\pi\)
\(998\) − 2.74268e7i − 0.871663i
\(999\) 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 450.6.c.a.199.2 2
3.2 odd 2 50.6.b.c.49.1 2
5.2 odd 4 450.6.a.j.1.1 1
5.3 odd 4 450.6.a.n.1.1 1
5.4 even 2 inner 450.6.c.a.199.1 2
12.11 even 2 400.6.c.g.49.1 2
15.2 even 4 50.6.a.f.1.1 yes 1
15.8 even 4 50.6.a.a.1.1 1
15.14 odd 2 50.6.b.c.49.2 2
60.23 odd 4 400.6.a.j.1.1 1
60.47 odd 4 400.6.a.e.1.1 1
60.59 even 2 400.6.c.g.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.6.a.a.1.1 1 15.8 even 4
50.6.a.f.1.1 yes 1 15.2 even 4
50.6.b.c.49.1 2 3.2 odd 2
50.6.b.c.49.2 2 15.14 odd 2
400.6.a.e.1.1 1 60.47 odd 4
400.6.a.j.1.1 1 60.23 odd 4
400.6.c.g.49.1 2 12.11 even 2
400.6.c.g.49.2 2 60.59 even 2
450.6.a.j.1.1 1 5.2 odd 4
450.6.a.n.1.1 1 5.3 odd 4
450.6.c.a.199.1 2 5.4 even 2 inner
450.6.c.a.199.2 2 1.1 even 1 trivial