Properties

Label 450.6.c.g.199.1
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} -16.0000 q^{4} -79.0000i q^{7} +64.0000i q^{8} +O(q^{10})\) \(q-4.00000i q^{2} -16.0000 q^{4} -79.0000i q^{7} +64.0000i q^{8} -150.000 q^{11} -137.000i q^{13} -316.000 q^{14} +256.000 q^{16} +2034.00i q^{17} +1969.00 q^{19} +600.000i q^{22} -1350.00i q^{23} -548.000 q^{26} +1264.00i q^{28} -2946.00 q^{29} +713.000 q^{31} -1024.00i q^{32} +8136.00 q^{34} -3238.00i q^{37} -7876.00i q^{38} -6564.00 q^{41} +19579.0i q^{43} +2400.00 q^{44} -5400.00 q^{46} +21150.0i q^{47} +10566.0 q^{49} +2192.00i q^{52} -25896.0i q^{53} +5056.00 q^{56} +11784.0i q^{58} +25350.0 q^{59} +50615.0 q^{61} -2852.00i q^{62} -4096.00 q^{64} -22519.0i q^{67} -32544.0i q^{68} -33900.0 q^{71} -82442.0i q^{73} -12952.0 q^{74} -31504.0 q^{76} +11850.0i q^{77} +81472.0 q^{79} +26256.0i q^{82} +25782.0i q^{83} +78316.0 q^{86} -9600.00i q^{88} +103728. q^{89} -10823.0 q^{91} +21600.0i q^{92} +84600.0 q^{94} -57343.0i q^{97} -42264.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 300 q^{11} - 632 q^{14} + 512 q^{16} + 3938 q^{19} - 1096 q^{26} - 5892 q^{29} + 1426 q^{31} + 16272 q^{34} - 13128 q^{41} + 4800 q^{44} - 10800 q^{46} + 21132 q^{49} + 10112 q^{56} + 50700 q^{59} + 101230 q^{61} - 8192 q^{64} - 67800 q^{71} - 25904 q^{74} - 63008 q^{76} + 162944 q^{79} + 156632 q^{86} + 207456 q^{89} - 21646 q^{91} + 169200 q^{94}+O(q^{100}) \) 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
\(7\) − 79.0000i − 0.609371i −0.952453 0.304686i \(-0.901449\pi\)
0.952453 0.304686i \(-0.0985514\pi\)
\(8\) 64.0000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −150.000 −0.373774 −0.186887 0.982381i \(-0.559840\pi\)
−0.186887 + 0.982381i \(0.559840\pi\)
\(12\) 0 0
\(13\) − 137.000i − 0.224834i −0.993661 0.112417i \(-0.964141\pi\)
0.993661 0.112417i \(-0.0358593\pi\)
\(14\) −316.000 −0.430891
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 2034.00i 1.70698i 0.521109 + 0.853490i \(0.325519\pi\)
−0.521109 + 0.853490i \(0.674481\pi\)
\(18\) 0 0
\(19\) 1969.00 1.25130 0.625650 0.780104i \(-0.284834\pi\)
0.625650 + 0.780104i \(0.284834\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 600.000i 0.264298i
\(23\) − 1350.00i − 0.532126i −0.963956 0.266063i \(-0.914277\pi\)
0.963956 0.266063i \(-0.0857229\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −548.000 −0.158982
\(27\) 0 0
\(28\) 1264.00i 0.304686i
\(29\) −2946.00 −0.650486 −0.325243 0.945631i \(-0.605446\pi\)
−0.325243 + 0.945631i \(0.605446\pi\)
\(30\) 0 0
\(31\) 713.000 0.133256 0.0666278 0.997778i \(-0.478776\pi\)
0.0666278 + 0.997778i \(0.478776\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 0 0
\(34\) 8136.00 1.20702
\(35\) 0 0
\(36\) 0 0
\(37\) − 3238.00i − 0.388841i −0.980918 0.194421i \(-0.937717\pi\)
0.980918 0.194421i \(-0.0622827\pi\)
\(38\) − 7876.00i − 0.884803i
\(39\) 0 0
\(40\) 0 0
\(41\) −6564.00 −0.609830 −0.304915 0.952380i \(-0.598628\pi\)
−0.304915 + 0.952380i \(0.598628\pi\)
\(42\) 0 0
\(43\) 19579.0i 1.61480i 0.590003 + 0.807401i \(0.299127\pi\)
−0.590003 + 0.807401i \(0.700873\pi\)
\(44\) 2400.00 0.186887
\(45\) 0 0
\(46\) −5400.00 −0.376270
\(47\) 21150.0i 1.39658i 0.715815 + 0.698290i \(0.246055\pi\)
−0.715815 + 0.698290i \(0.753945\pi\)
\(48\) 0 0
\(49\) 10566.0 0.628667
\(50\) 0 0
\(51\) 0 0
\(52\) 2192.00i 0.112417i
\(53\) − 25896.0i − 1.26632i −0.774021 0.633159i \(-0.781758\pi\)
0.774021 0.633159i \(-0.218242\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5056.00 0.215445
\(57\) 0 0
\(58\) 11784.0i 0.459963i
\(59\) 25350.0 0.948086 0.474043 0.880502i \(-0.342794\pi\)
0.474043 + 0.880502i \(0.342794\pi\)
\(60\) 0 0
\(61\) 50615.0 1.74163 0.870813 0.491615i \(-0.163593\pi\)
0.870813 + 0.491615i \(0.163593\pi\)
\(62\) − 2852.00i − 0.0942259i
\(63\) 0 0
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 22519.0i − 0.612861i −0.951893 0.306431i \(-0.900865\pi\)
0.951893 0.306431i \(-0.0991347\pi\)
\(68\) − 32544.0i − 0.853490i
\(69\) 0 0
\(70\) 0 0
\(71\) −33900.0 −0.798094 −0.399047 0.916931i \(-0.630659\pi\)
−0.399047 + 0.916931i \(0.630659\pi\)
\(72\) 0 0
\(73\) − 82442.0i − 1.81068i −0.424689 0.905339i \(-0.639617\pi\)
0.424689 0.905339i \(-0.360383\pi\)
\(74\) −12952.0 −0.274952
\(75\) 0 0
\(76\) −31504.0 −0.625650
\(77\) 11850.0i 0.227767i
\(78\) 0 0
\(79\) 81472.0 1.46873 0.734363 0.678757i \(-0.237481\pi\)
0.734363 + 0.678757i \(0.237481\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 26256.0i 0.431215i
\(83\) 25782.0i 0.410791i 0.978679 + 0.205396i \(0.0658481\pi\)
−0.978679 + 0.205396i \(0.934152\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 78316.0 1.14184
\(87\) 0 0
\(88\) − 9600.00i − 0.132149i
\(89\) 103728. 1.38810 0.694050 0.719926i \(-0.255824\pi\)
0.694050 + 0.719926i \(0.255824\pi\)
\(90\) 0 0
\(91\) −10823.0 −0.137007
\(92\) 21600.0i 0.266063i
\(93\) 0 0
\(94\) 84600.0 0.987531
\(95\) 0 0
\(96\) 0 0
\(97\) − 57343.0i − 0.618801i −0.950932 0.309401i \(-0.899872\pi\)
0.950932 0.309401i \(-0.100128\pi\)
\(98\) − 42264.0i − 0.444534i
\(99\) 0 0
\(100\) 0 0
\(101\) 91032.0 0.887954 0.443977 0.896038i \(-0.353567\pi\)
0.443977 + 0.896038i \(0.353567\pi\)
\(102\) 0 0
\(103\) − 191636.i − 1.77985i −0.456104 0.889926i \(-0.650756\pi\)
0.456104 0.889926i \(-0.349244\pi\)
\(104\) 8768.00 0.0794909
\(105\) 0 0
\(106\) −103584. −0.895423
\(107\) − 9288.00i − 0.0784265i −0.999231 0.0392132i \(-0.987515\pi\)
0.999231 0.0392132i \(-0.0124852\pi\)
\(108\) 0 0
\(109\) 20635.0 0.166356 0.0831780 0.996535i \(-0.473493\pi\)
0.0831780 + 0.996535i \(0.473493\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 20224.0i − 0.152343i
\(113\) 176892.i 1.30320i 0.758561 + 0.651602i \(0.225903\pi\)
−0.758561 + 0.651602i \(0.774097\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 47136.0 0.325243
\(117\) 0 0
\(118\) − 101400.i − 0.670398i
\(119\) 160686. 1.04019
\(120\) 0 0
\(121\) −138551. −0.860293
\(122\) − 202460.i − 1.23151i
\(123\) 0 0
\(124\) −11408.0 −0.0666278
\(125\) 0 0
\(126\) 0 0
\(127\) − 256480.i − 1.41106i −0.708682 0.705528i \(-0.750710\pi\)
0.708682 0.705528i \(-0.249290\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −177600. −0.904200 −0.452100 0.891967i \(-0.649325\pi\)
−0.452100 + 0.891967i \(0.649325\pi\)
\(132\) 0 0
\(133\) − 155551.i − 0.762507i
\(134\) −90076.0 −0.433358
\(135\) 0 0
\(136\) −130176. −0.603509
\(137\) − 260886.i − 1.18754i −0.804634 0.593772i \(-0.797638\pi\)
0.804634 0.593772i \(-0.202362\pi\)
\(138\) 0 0
\(139\) 217684. 0.955629 0.477815 0.878461i \(-0.341429\pi\)
0.477815 + 0.878461i \(0.341429\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 135600.i 0.564337i
\(143\) 20550.0i 0.0840372i
\(144\) 0 0
\(145\) 0 0
\(146\) −329768. −1.28034
\(147\) 0 0
\(148\) 51808.0i 0.194421i
\(149\) 421794. 1.55645 0.778224 0.627987i \(-0.216121\pi\)
0.778224 + 0.627987i \(0.216121\pi\)
\(150\) 0 0
\(151\) −101917. −0.363751 −0.181876 0.983322i \(-0.558217\pi\)
−0.181876 + 0.983322i \(0.558217\pi\)
\(152\) 126016.i 0.442402i
\(153\) 0 0
\(154\) 47400.0 0.161056
\(155\) 0 0
\(156\) 0 0
\(157\) − 101773.i − 0.329521i −0.986334 0.164761i \(-0.947315\pi\)
0.986334 0.164761i \(-0.0526852\pi\)
\(158\) − 325888.i − 1.03855i
\(159\) 0 0
\(160\) 0 0
\(161\) −106650. −0.324262
\(162\) 0 0
\(163\) 202249.i 0.596235i 0.954529 + 0.298117i \(0.0963587\pi\)
−0.954529 + 0.298117i \(0.903641\pi\)
\(164\) 105024. 0.304915
\(165\) 0 0
\(166\) 103128. 0.290473
\(167\) − 231600.i − 0.642610i −0.946976 0.321305i \(-0.895879\pi\)
0.946976 0.321305i \(-0.104121\pi\)
\(168\) 0 0
\(169\) 352524. 0.949450
\(170\) 0 0
\(171\) 0 0
\(172\) − 313264.i − 0.807401i
\(173\) − 174222.i − 0.442576i −0.975209 0.221288i \(-0.928974\pi\)
0.975209 0.221288i \(-0.0710261\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −38400.0 −0.0934436
\(177\) 0 0
\(178\) − 414912.i − 0.981535i
\(179\) −642066. −1.49778 −0.748888 0.662696i \(-0.769412\pi\)
−0.748888 + 0.662696i \(0.769412\pi\)
\(180\) 0 0
\(181\) −56071.0 −0.127216 −0.0636080 0.997975i \(-0.520261\pi\)
−0.0636080 + 0.997975i \(0.520261\pi\)
\(182\) 43292.0i 0.0968789i
\(183\) 0 0
\(184\) 86400.0 0.188135
\(185\) 0 0
\(186\) 0 0
\(187\) − 305100.i − 0.638026i
\(188\) − 338400.i − 0.698290i
\(189\) 0 0
\(190\) 0 0
\(191\) 209694. 0.415913 0.207957 0.978138i \(-0.433319\pi\)
0.207957 + 0.978138i \(0.433319\pi\)
\(192\) 0 0
\(193\) 715597.i 1.38285i 0.722448 + 0.691425i \(0.243017\pi\)
−0.722448 + 0.691425i \(0.756983\pi\)
\(194\) −229372. −0.437558
\(195\) 0 0
\(196\) −169056. −0.314333
\(197\) − 508854.i − 0.934174i −0.884211 0.467087i \(-0.845304\pi\)
0.884211 0.467087i \(-0.154696\pi\)
\(198\) 0 0
\(199\) 986017. 1.76503 0.882514 0.470286i \(-0.155849\pi\)
0.882514 + 0.470286i \(0.155849\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 364128.i − 0.627879i
\(203\) 232734.i 0.396387i
\(204\) 0 0
\(205\) 0 0
\(206\) −766544. −1.25855
\(207\) 0 0
\(208\) − 35072.0i − 0.0562085i
\(209\) −295350. −0.467704
\(210\) 0 0
\(211\) 119495. 0.184775 0.0923876 0.995723i \(-0.470550\pi\)
0.0923876 + 0.995723i \(0.470550\pi\)
\(212\) 414336.i 0.633159i
\(213\) 0 0
\(214\) −37152.0 −0.0554559
\(215\) 0 0
\(216\) 0 0
\(217\) − 56327.0i − 0.0812021i
\(218\) − 82540.0i − 0.117631i
\(219\) 0 0
\(220\) 0 0
\(221\) 278658. 0.383788
\(222\) 0 0
\(223\) − 48545.0i − 0.0653706i −0.999466 0.0326853i \(-0.989594\pi\)
0.999466 0.0326853i \(-0.0104059\pi\)
\(224\) −80896.0 −0.107723
\(225\) 0 0
\(226\) 707568. 0.921504
\(227\) 287652.i 0.370512i 0.982690 + 0.185256i \(0.0593115\pi\)
−0.982690 + 0.185256i \(0.940689\pi\)
\(228\) 0 0
\(229\) −72065.0 −0.0908104 −0.0454052 0.998969i \(-0.514458\pi\)
−0.0454052 + 0.998969i \(0.514458\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 188544.i − 0.229981i
\(233\) − 569148.i − 0.686808i −0.939188 0.343404i \(-0.888420\pi\)
0.939188 0.343404i \(-0.111580\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −405600. −0.474043
\(237\) 0 0
\(238\) − 642744.i − 0.735522i
\(239\) 696504. 0.788731 0.394365 0.918954i \(-0.370964\pi\)
0.394365 + 0.918954i \(0.370964\pi\)
\(240\) 0 0
\(241\) 576137. 0.638974 0.319487 0.947591i \(-0.396489\pi\)
0.319487 + 0.947591i \(0.396489\pi\)
\(242\) 554204.i 0.608319i
\(243\) 0 0
\(244\) −809840. −0.870813
\(245\) 0 0
\(246\) 0 0
\(247\) − 269753.i − 0.281335i
\(248\) 45632.0i 0.0471130i
\(249\) 0 0
\(250\) 0 0
\(251\) 1.22492e6 1.22723 0.613613 0.789607i \(-0.289715\pi\)
0.613613 + 0.789607i \(0.289715\pi\)
\(252\) 0 0
\(253\) 202500.i 0.198895i
\(254\) −1.02592e6 −0.997767
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 972132.i − 0.918105i −0.888409 0.459053i \(-0.848189\pi\)
0.888409 0.459053i \(-0.151811\pi\)
\(258\) 0 0
\(259\) −255802. −0.236949
\(260\) 0 0
\(261\) 0 0
\(262\) 710400.i 0.639366i
\(263\) 1.76190e6i 1.57070i 0.619055 + 0.785348i \(0.287516\pi\)
−0.619055 + 0.785348i \(0.712484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −622204. −0.539174
\(267\) 0 0
\(268\) 360304.i 0.306431i
\(269\) 1.74383e6 1.46935 0.734674 0.678421i \(-0.237335\pi\)
0.734674 + 0.678421i \(0.237335\pi\)
\(270\) 0 0
\(271\) −1.70426e6 −1.40965 −0.704826 0.709381i \(-0.748975\pi\)
−0.704826 + 0.709381i \(0.748975\pi\)
\(272\) 520704.i 0.426745i
\(273\) 0 0
\(274\) −1.04354e6 −0.839720
\(275\) 0 0
\(276\) 0 0
\(277\) 1.46972e6i 1.15089i 0.817840 + 0.575446i \(0.195171\pi\)
−0.817840 + 0.575446i \(0.804829\pi\)
\(278\) − 870736.i − 0.675732i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.22530e6 0.925715 0.462858 0.886433i \(-0.346824\pi\)
0.462858 + 0.886433i \(0.346824\pi\)
\(282\) 0 0
\(283\) 637333.i 0.473043i 0.971626 + 0.236521i \(0.0760073\pi\)
−0.971626 + 0.236521i \(0.923993\pi\)
\(284\) 542400. 0.399047
\(285\) 0 0
\(286\) 82200.0 0.0594233
\(287\) 518556.i 0.371613i
\(288\) 0 0
\(289\) −2.71730e6 −1.91378
\(290\) 0 0
\(291\) 0 0
\(292\) 1.31907e6i 0.905339i
\(293\) 35094.0i 0.0238816i 0.999929 + 0.0119408i \(0.00380097\pi\)
−0.999929 + 0.0119408i \(0.996199\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 207232. 0.137476
\(297\) 0 0
\(298\) − 1.68718e6i − 1.10058i
\(299\) −184950. −0.119640
\(300\) 0 0
\(301\) 1.54674e6 0.984014
\(302\) 407668.i 0.257211i
\(303\) 0 0
\(304\) 504064. 0.312825
\(305\) 0 0
\(306\) 0 0
\(307\) 2.88040e6i 1.74424i 0.489293 + 0.872120i \(0.337255\pi\)
−0.489293 + 0.872120i \(0.662745\pi\)
\(308\) − 189600.i − 0.113884i
\(309\) 0 0
\(310\) 0 0
\(311\) 2.38305e6 1.39712 0.698558 0.715554i \(-0.253826\pi\)
0.698558 + 0.715554i \(0.253826\pi\)
\(312\) 0 0
\(313\) 1.93081e6i 1.11399i 0.830517 + 0.556993i \(0.188045\pi\)
−0.830517 + 0.556993i \(0.811955\pi\)
\(314\) −407092. −0.233007
\(315\) 0 0
\(316\) −1.30355e6 −0.734363
\(317\) 2.18995e6i 1.22401i 0.790852 + 0.612007i \(0.209638\pi\)
−0.790852 + 0.612007i \(0.790362\pi\)
\(318\) 0 0
\(319\) 441900. 0.243135
\(320\) 0 0
\(321\) 0 0
\(322\) 426600.i 0.229288i
\(323\) 4.00495e6i 2.13595i
\(324\) 0 0
\(325\) 0 0
\(326\) 808996. 0.421602
\(327\) 0 0
\(328\) − 420096.i − 0.215608i
\(329\) 1.67085e6 0.851036
\(330\) 0 0
\(331\) 1.41429e6 0.709527 0.354764 0.934956i \(-0.384561\pi\)
0.354764 + 0.934956i \(0.384561\pi\)
\(332\) − 412512.i − 0.205396i
\(333\) 0 0
\(334\) −926400. −0.454394
\(335\) 0 0
\(336\) 0 0
\(337\) 1.38208e6i 0.662916i 0.943470 + 0.331458i \(0.107541\pi\)
−0.943470 + 0.331458i \(0.892459\pi\)
\(338\) − 1.41010e6i − 0.671362i
\(339\) 0 0
\(340\) 0 0
\(341\) −106950. −0.0498075
\(342\) 0 0
\(343\) − 2.16247e6i − 0.992463i
\(344\) −1.25306e6 −0.570919
\(345\) 0 0
\(346\) −696888. −0.312948
\(347\) 283758.i 0.126510i 0.997997 + 0.0632549i \(0.0201481\pi\)
−0.997997 + 0.0632549i \(0.979852\pi\)
\(348\) 0 0
\(349\) −2.13809e6 −0.939642 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 153600.i 0.0660746i
\(353\) 3.16015e6i 1.34980i 0.737908 + 0.674901i \(0.235814\pi\)
−0.737908 + 0.674901i \(0.764186\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.65965e6 −0.694050
\(357\) 0 0
\(358\) 2.56826e6i 1.05909i
\(359\) −3.59564e6 −1.47245 −0.736225 0.676737i \(-0.763393\pi\)
−0.736225 + 0.676737i \(0.763393\pi\)
\(360\) 0 0
\(361\) 1.40086e6 0.565754
\(362\) 224284.i 0.0899553i
\(363\) 0 0
\(364\) 173168. 0.0685037
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.62043e6i − 1.79068i −0.445385 0.895339i \(-0.646933\pi\)
0.445385 0.895339i \(-0.353067\pi\)
\(368\) − 345600.i − 0.133031i
\(369\) 0 0
\(370\) 0 0
\(371\) −2.04578e6 −0.771658
\(372\) 0 0
\(373\) 3.51983e6i 1.30993i 0.755657 + 0.654967i \(0.227318\pi\)
−0.755657 + 0.654967i \(0.772682\pi\)
\(374\) −1.22040e6 −0.451152
\(375\) 0 0
\(376\) −1.35360e6 −0.493765
\(377\) 403602.i 0.146251i
\(378\) 0 0
\(379\) −595061. −0.212796 −0.106398 0.994324i \(-0.533932\pi\)
−0.106398 + 0.994324i \(0.533932\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 838776.i − 0.294095i
\(383\) − 1.30050e6i − 0.453016i −0.974009 0.226508i \(-0.927269\pi\)
0.974009 0.226508i \(-0.0727309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.86239e6 0.977823
\(387\) 0 0
\(388\) 917488.i 0.309401i
\(389\) 620724. 0.207981 0.103991 0.994578i \(-0.466839\pi\)
0.103991 + 0.994578i \(0.466839\pi\)
\(390\) 0 0
\(391\) 2.74590e6 0.908328
\(392\) 676224.i 0.222267i
\(393\) 0 0
\(394\) −2.03542e6 −0.660561
\(395\) 0 0
\(396\) 0 0
\(397\) 1.18622e6i 0.377737i 0.982002 + 0.188869i \(0.0604821\pi\)
−0.982002 + 0.188869i \(0.939518\pi\)
\(398\) − 3.94407e6i − 1.24806i
\(399\) 0 0
\(400\) 0 0
\(401\) −3.13334e6 −0.973077 −0.486538 0.873659i \(-0.661741\pi\)
−0.486538 + 0.873659i \(0.661741\pi\)
\(402\) 0 0
\(403\) − 97681.0i − 0.0299604i
\(404\) −1.45651e6 −0.443977
\(405\) 0 0
\(406\) 930936. 0.280288
\(407\) 485700.i 0.145339i
\(408\) 0 0
\(409\) −567581. −0.167772 −0.0838860 0.996475i \(-0.526733\pi\)
−0.0838860 + 0.996475i \(0.526733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.06618e6i 0.889926i
\(413\) − 2.00265e6i − 0.577737i
\(414\) 0 0
\(415\) 0 0
\(416\) −140288. −0.0397454
\(417\) 0 0
\(418\) 1.18140e6i 0.330717i
\(419\) −6.09000e6 −1.69466 −0.847329 0.531068i \(-0.821791\pi\)
−0.847329 + 0.531068i \(0.821791\pi\)
\(420\) 0 0
\(421\) 3.07139e6 0.844558 0.422279 0.906466i \(-0.361230\pi\)
0.422279 + 0.906466i \(0.361230\pi\)
\(422\) − 477980.i − 0.130656i
\(423\) 0 0
\(424\) 1.65734e6 0.447711
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.99858e6i − 1.06130i
\(428\) 148608.i 0.0392132i
\(429\) 0 0
\(430\) 0 0
\(431\) 669150. 0.173512 0.0867562 0.996230i \(-0.472350\pi\)
0.0867562 + 0.996230i \(0.472350\pi\)
\(432\) 0 0
\(433\) 3.25439e6i 0.834161i 0.908870 + 0.417080i \(0.136947\pi\)
−0.908870 + 0.417080i \(0.863053\pi\)
\(434\) −225308. −0.0574186
\(435\) 0 0
\(436\) −330160. −0.0831780
\(437\) − 2.65815e6i − 0.665849i
\(438\) 0 0
\(439\) −1.58759e6 −0.393168 −0.196584 0.980487i \(-0.562985\pi\)
−0.196584 + 0.980487i \(0.562985\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 1.11463e6i − 0.271379i
\(443\) − 213888.i − 0.0517818i −0.999665 0.0258909i \(-0.991758\pi\)
0.999665 0.0258909i \(-0.00824225\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −194180. −0.0462240
\(447\) 0 0
\(448\) 323584.i 0.0761714i
\(449\) 3.70724e6 0.867831 0.433916 0.900953i \(-0.357132\pi\)
0.433916 + 0.900953i \(0.357132\pi\)
\(450\) 0 0
\(451\) 984600. 0.227939
\(452\) − 2.83027e6i − 0.651602i
\(453\) 0 0
\(454\) 1.15061e6 0.261992
\(455\) 0 0
\(456\) 0 0
\(457\) 3.83193e6i 0.858275i 0.903239 + 0.429138i \(0.141182\pi\)
−0.903239 + 0.429138i \(0.858818\pi\)
\(458\) 288260.i 0.0642127i
\(459\) 0 0
\(460\) 0 0
\(461\) −5.58672e6 −1.22435 −0.612174 0.790723i \(-0.709705\pi\)
−0.612174 + 0.790723i \(0.709705\pi\)
\(462\) 0 0
\(463\) 4.40142e6i 0.954203i 0.878848 + 0.477101i \(0.158313\pi\)
−0.878848 + 0.477101i \(0.841687\pi\)
\(464\) −754176. −0.162621
\(465\) 0 0
\(466\) −2.27659e6 −0.485647
\(467\) − 3.35225e6i − 0.711287i −0.934622 0.355643i \(-0.884262\pi\)
0.934622 0.355643i \(-0.115738\pi\)
\(468\) 0 0
\(469\) −1.77900e6 −0.373460
\(470\) 0 0
\(471\) 0 0
\(472\) 1.62240e6i 0.335199i
\(473\) − 2.93685e6i − 0.603572i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.57098e6 −0.520093
\(477\) 0 0
\(478\) − 2.78602e6i − 0.557717i
\(479\) 909966. 0.181212 0.0906059 0.995887i \(-0.471120\pi\)
0.0906059 + 0.995887i \(0.471120\pi\)
\(480\) 0 0
\(481\) −443606. −0.0874248
\(482\) − 2.30455e6i − 0.451823i
\(483\) 0 0
\(484\) 2.21682e6 0.430146
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.03579e6i − 0.771093i −0.922689 0.385546i \(-0.874013\pi\)
0.922689 0.385546i \(-0.125987\pi\)
\(488\) 3.23936e6i 0.615757i
\(489\) 0 0
\(490\) 0 0
\(491\) −379944. −0.0711239 −0.0355620 0.999367i \(-0.511322\pi\)
−0.0355620 + 0.999367i \(0.511322\pi\)
\(492\) 0 0
\(493\) − 5.99216e6i − 1.11037i
\(494\) −1.07901e6 −0.198934
\(495\) 0 0
\(496\) 182528. 0.0333139
\(497\) 2.67810e6i 0.486335i
\(498\) 0 0
\(499\) 7.42433e6 1.33477 0.667384 0.744714i \(-0.267414\pi\)
0.667384 + 0.744714i \(0.267414\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.89970e6i − 0.867780i
\(503\) 2.29988e6i 0.405309i 0.979250 + 0.202654i \(0.0649568\pi\)
−0.979250 + 0.202654i \(0.935043\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 810000. 0.140640
\(507\) 0 0
\(508\) 4.10368e6i 0.705528i
\(509\) 6.73721e6 1.15262 0.576310 0.817231i \(-0.304492\pi\)
0.576310 + 0.817231i \(0.304492\pi\)
\(510\) 0 0
\(511\) −6.51292e6 −1.10338
\(512\) − 262144.i − 0.0441942i
\(513\) 0 0
\(514\) −3.88853e6 −0.649198
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.17250e6i − 0.522006i
\(518\) 1.02321e6i 0.167548i
\(519\) 0 0
\(520\) 0 0
\(521\) 9.26806e6 1.49587 0.747936 0.663770i \(-0.231045\pi\)
0.747936 + 0.663770i \(0.231045\pi\)
\(522\) 0 0
\(523\) − 1.10620e7i − 1.76839i −0.467113 0.884197i \(-0.654706\pi\)
0.467113 0.884197i \(-0.345294\pi\)
\(524\) 2.84160e6 0.452100
\(525\) 0 0
\(526\) 7.04760e6 1.11065
\(527\) 1.45024e6i 0.227465i
\(528\) 0 0
\(529\) 4.61384e6 0.716842
\(530\) 0 0
\(531\) 0 0
\(532\) 2.48882e6i 0.381253i
\(533\) 899268.i 0.137111i
\(534\) 0 0
\(535\) 0 0
\(536\) 1.44122e6 0.216679
\(537\) 0 0
\(538\) − 6.97534e6i − 1.03899i
\(539\) −1.58490e6 −0.234979
\(540\) 0 0
\(541\) −1.30754e7 −1.92072 −0.960358 0.278771i \(-0.910073\pi\)
−0.960358 + 0.278771i \(0.910073\pi\)
\(542\) 6.81702e6i 0.996774i
\(543\) 0 0
\(544\) 2.08282e6 0.301754
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.84605e6i − 0.263800i −0.991263 0.131900i \(-0.957892\pi\)
0.991263 0.131900i \(-0.0421078\pi\)
\(548\) 4.17418e6i 0.593772i
\(549\) 0 0
\(550\) 0 0
\(551\) −5.80067e6 −0.813953
\(552\) 0 0
\(553\) − 6.43629e6i − 0.894999i
\(554\) 5.87887e6 0.813803
\(555\) 0 0
\(556\) −3.48294e6 −0.477815
\(557\) − 6.48782e6i − 0.886055i −0.896508 0.443027i \(-0.853904\pi\)
0.896508 0.443027i \(-0.146096\pi\)
\(558\) 0 0
\(559\) 2.68232e6 0.363063
\(560\) 0 0
\(561\) 0 0
\(562\) − 4.90121e6i − 0.654579i
\(563\) 3.01538e6i 0.400932i 0.979701 + 0.200466i \(0.0642456\pi\)
−0.979701 + 0.200466i \(0.935754\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.54933e6 0.334492
\(567\) 0 0
\(568\) − 2.16960e6i − 0.282169i
\(569\) −3.49810e6 −0.452952 −0.226476 0.974017i \(-0.572720\pi\)
−0.226476 + 0.974017i \(0.572720\pi\)
\(570\) 0 0
\(571\) −1.23749e7 −1.58837 −0.794185 0.607676i \(-0.792102\pi\)
−0.794185 + 0.607676i \(0.792102\pi\)
\(572\) − 328800.i − 0.0420186i
\(573\) 0 0
\(574\) 2.07422e6 0.262770
\(575\) 0 0
\(576\) 0 0
\(577\) 6.80687e6i 0.851153i 0.904922 + 0.425577i \(0.139929\pi\)
−0.904922 + 0.425577i \(0.860071\pi\)
\(578\) 1.08692e7i 1.35325i
\(579\) 0 0
\(580\) 0 0
\(581\) 2.03678e6 0.250325
\(582\) 0 0
\(583\) 3.88440e6i 0.473317i
\(584\) 5.27629e6 0.640172
\(585\) 0 0
\(586\) 140376. 0.0168869
\(587\) − 7.77113e6i − 0.930870i −0.885082 0.465435i \(-0.845898\pi\)
0.885082 0.465435i \(-0.154102\pi\)
\(588\) 0 0
\(589\) 1.40390e6 0.166743
\(590\) 0 0
\(591\) 0 0
\(592\) − 828928.i − 0.0972104i
\(593\) − 1.51222e7i − 1.76595i −0.469418 0.882976i \(-0.655536\pi\)
0.469418 0.882976i \(-0.344464\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.74870e6 −0.778224
\(597\) 0 0
\(598\) 739800.i 0.0845982i
\(599\) −1.67012e6 −0.190187 −0.0950937 0.995468i \(-0.530315\pi\)
−0.0950937 + 0.995468i \(0.530315\pi\)
\(600\) 0 0
\(601\) 4.12220e6 0.465525 0.232763 0.972534i \(-0.425223\pi\)
0.232763 + 0.972534i \(0.425223\pi\)
\(602\) − 6.18696e6i − 0.695803i
\(603\) 0 0
\(604\) 1.63067e6 0.181876
\(605\) 0 0
\(606\) 0 0
\(607\) 6.81870e6i 0.751155i 0.926791 + 0.375578i \(0.122556\pi\)
−0.926791 + 0.375578i \(0.877444\pi\)
\(608\) − 2.01626e6i − 0.221201i
\(609\) 0 0
\(610\) 0 0
\(611\) 2.89755e6 0.313999
\(612\) 0 0
\(613\) − 1.59466e7i − 1.71402i −0.515300 0.857010i \(-0.672319\pi\)
0.515300 0.857010i \(-0.327681\pi\)
\(614\) 1.15216e7 1.23336
\(615\) 0 0
\(616\) −758400. −0.0805279
\(617\) 8.21952e6i 0.869228i 0.900617 + 0.434614i \(0.143115\pi\)
−0.900617 + 0.434614i \(0.856885\pi\)
\(618\) 0 0
\(619\) 1.23323e7 1.29366 0.646828 0.762636i \(-0.276095\pi\)
0.646828 + 0.762636i \(0.276095\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 9.53220e6i − 0.987910i
\(623\) − 8.19451e6i − 0.845869i
\(624\) 0 0
\(625\) 0 0
\(626\) 7.72325e6 0.787706
\(627\) 0 0
\(628\) 1.62837e6i 0.164761i
\(629\) 6.58609e6 0.663745
\(630\) 0 0
\(631\) −3.01912e6 −0.301861 −0.150931 0.988544i \(-0.548227\pi\)
−0.150931 + 0.988544i \(0.548227\pi\)
\(632\) 5.21421e6i 0.519273i
\(633\) 0 0
\(634\) 8.75981e6 0.865509
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.44754e6i − 0.141346i
\(638\) − 1.76760e6i − 0.171922i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.18052e7 1.13482 0.567411 0.823435i \(-0.307945\pi\)
0.567411 + 0.823435i \(0.307945\pi\)
\(642\) 0 0
\(643\) 2.85360e6i 0.272186i 0.990696 + 0.136093i \(0.0434546\pi\)
−0.990696 + 0.136093i \(0.956545\pi\)
\(644\) 1.70640e6 0.162131
\(645\) 0 0
\(646\) 1.60198e7 1.51034
\(647\) 3.40192e6i 0.319494i 0.987158 + 0.159747i \(0.0510679\pi\)
−0.987158 + 0.159747i \(0.948932\pi\)
\(648\) 0 0
\(649\) −3.80250e6 −0.354370
\(650\) 0 0
\(651\) 0 0
\(652\) − 3.23598e6i − 0.298117i
\(653\) − 9.39166e6i − 0.861905i −0.902375 0.430953i \(-0.858178\pi\)
0.902375 0.430953i \(-0.141822\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.68038e6 −0.152458
\(657\) 0 0
\(658\) − 6.68340e6i − 0.601773i
\(659\) 9.36659e6 0.840171 0.420086 0.907484i \(-0.362000\pi\)
0.420086 + 0.907484i \(0.362000\pi\)
\(660\) 0 0
\(661\) 1.57748e7 1.40430 0.702150 0.712029i \(-0.252223\pi\)
0.702150 + 0.712029i \(0.252223\pi\)
\(662\) − 5.65717e6i − 0.501712i
\(663\) 0 0
\(664\) −1.65005e6 −0.145237
\(665\) 0 0
\(666\) 0 0
\(667\) 3.97710e6i 0.346140i
\(668\) 3.70560e6i 0.321305i
\(669\) 0 0
\(670\) 0 0
\(671\) −7.59225e6 −0.650975
\(672\) 0 0
\(673\) 6.64515e6i 0.565545i 0.959187 + 0.282773i \(0.0912542\pi\)
−0.959187 + 0.282773i \(0.908746\pi\)
\(674\) 5.52832e6 0.468753
\(675\) 0 0
\(676\) −5.64038e6 −0.474725
\(677\) 1.57780e7i 1.32306i 0.749917 + 0.661532i \(0.230094\pi\)
−0.749917 + 0.661532i \(0.769906\pi\)
\(678\) 0 0
\(679\) −4.53010e6 −0.377080
\(680\) 0 0
\(681\) 0 0
\(682\) 427800.i 0.0352192i
\(683\) − 1.96654e7i − 1.61306i −0.591190 0.806532i \(-0.701342\pi\)
0.591190 0.806532i \(-0.298658\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.64987e6 −0.701777
\(687\) 0 0
\(688\) 5.01222e6i 0.403701i
\(689\) −3.54775e6 −0.284712
\(690\) 0 0
\(691\) 990464. 0.0789121 0.0394560 0.999221i \(-0.487437\pi\)
0.0394560 + 0.999221i \(0.487437\pi\)
\(692\) 2.78755e6i 0.221288i
\(693\) 0 0
\(694\) 1.13503e6 0.0894560
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.33512e7i − 1.04097i
\(698\) 8.55236e6i 0.664427i
\(699\) 0 0
\(700\) 0 0
\(701\) −6.52919e6 −0.501839 −0.250920 0.968008i \(-0.580733\pi\)
−0.250920 + 0.968008i \(0.580733\pi\)
\(702\) 0 0
\(703\) − 6.37562e6i − 0.486558i
\(704\) 614400. 0.0467218
\(705\) 0 0
\(706\) 1.26406e7 0.954455
\(707\) − 7.19153e6i − 0.541094i
\(708\) 0 0
\(709\) 2.02106e7 1.50996 0.754979 0.655749i \(-0.227647\pi\)
0.754979 + 0.655749i \(0.227647\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.63859e6i 0.490768i
\(713\) − 962550.i − 0.0709087i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.02731e7 0.748888
\(717\) 0 0
\(718\) 1.43826e7i 1.04118i
\(719\) −2.27720e7 −1.64277 −0.821387 0.570371i \(-0.806800\pi\)
−0.821387 + 0.570371i \(0.806800\pi\)
\(720\) 0 0
\(721\) −1.51392e7 −1.08459
\(722\) − 5.60345e6i − 0.400048i
\(723\) 0 0
\(724\) 897136. 0.0636080
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.09186e6i − 0.497650i −0.968548 0.248825i \(-0.919956\pi\)
0.968548 0.248825i \(-0.0800445\pi\)
\(728\) − 692672.i − 0.0484394i
\(729\) 0 0
\(730\) 0 0
\(731\) −3.98237e7 −2.75644
\(732\) 0 0
\(733\) 1.38982e7i 0.955432i 0.878514 + 0.477716i \(0.158535\pi\)
−0.878514 + 0.477716i \(0.841465\pi\)
\(734\) −1.84817e7 −1.26620
\(735\) 0 0
\(736\) −1.38240e6 −0.0940674
\(737\) 3.37785e6i 0.229072i
\(738\) 0 0
\(739\) −1.54857e7 −1.04309 −0.521543 0.853225i \(-0.674643\pi\)
−0.521543 + 0.853225i \(0.674643\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.18314e6i 0.545645i
\(743\) 159276.i 0.0105847i 0.999986 + 0.00529235i \(0.00168461\pi\)
−0.999986 + 0.00529235i \(0.998315\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.40793e7 0.926263
\(747\) 0 0
\(748\) 4.88160e6i 0.319013i
\(749\) −733752. −0.0477908
\(750\) 0 0
\(751\) 1.15554e7 0.747625 0.373812 0.927504i \(-0.378050\pi\)
0.373812 + 0.927504i \(0.378050\pi\)
\(752\) 5.41440e6i 0.349145i
\(753\) 0 0
\(754\) 1.61441e6 0.103415
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.34402e6i − 0.402369i −0.979553 0.201185i \(-0.935521\pi\)
0.979553 0.201185i \(-0.0644792\pi\)
\(758\) 2.38024e6i 0.150469i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.17748e7 0.737040 0.368520 0.929620i \(-0.379865\pi\)
0.368520 + 0.929620i \(0.379865\pi\)
\(762\) 0 0
\(763\) − 1.63016e6i − 0.101373i
\(764\) −3.35510e6 −0.207957
\(765\) 0 0
\(766\) −5.20200e6 −0.320331
\(767\) − 3.47295e6i − 0.213162i
\(768\) 0 0
\(769\) 2.47869e7 1.51149 0.755745 0.654865i \(-0.227275\pi\)
0.755745 + 0.654865i \(0.227275\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1.14496e7i − 0.691425i
\(773\) − 1.38017e7i − 0.830774i −0.909645 0.415387i \(-0.863646\pi\)
0.909645 0.415387i \(-0.136354\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.66995e6 0.218779
\(777\) 0 0
\(778\) − 2.48290e6i − 0.147065i
\(779\) −1.29245e7 −0.763081
\(780\) 0 0
\(781\) 5.08500e6 0.298307
\(782\) − 1.09836e7i − 0.642285i
\(783\) 0 0
\(784\) 2.70490e6 0.157167
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.53795e7i − 0.885129i −0.896737 0.442565i \(-0.854069\pi\)
0.896737 0.442565i \(-0.145931\pi\)
\(788\) 8.14166e6i 0.467087i
\(789\) 0 0
\(790\) 0 0
\(791\) 1.39745e7 0.794135
\(792\) 0 0
\(793\) − 6.93426e6i − 0.391577i
\(794\) 4.74489e6 0.267101
\(795\) 0 0
\(796\) −1.57763e7 −0.882514
\(797\) 2.33978e7i 1.30475i 0.757894 + 0.652377i \(0.226228\pi\)
−0.757894 + 0.652377i \(0.773772\pi\)
\(798\) 0 0
\(799\) −4.30191e7 −2.38393
\(800\) 0 0
\(801\) 0 0
\(802\) 1.25334e7i 0.688069i
\(803\) 1.23663e7i 0.676785i
\(804\) 0 0
\(805\) 0 0
\(806\) −390724. −0.0211852
\(807\) 0 0
\(808\) 5.82605e6i 0.313939i
\(809\) −2.80188e7 −1.50515 −0.752573 0.658509i \(-0.771187\pi\)
−0.752573 + 0.658509i \(0.771187\pi\)
\(810\) 0 0
\(811\) −1.65641e6 −0.0884332 −0.0442166 0.999022i \(-0.514079\pi\)
−0.0442166 + 0.999022i \(0.514079\pi\)
\(812\) − 3.72374e6i − 0.198194i
\(813\) 0 0
\(814\) 1.94280e6 0.102770
\(815\) 0 0
\(816\) 0 0
\(817\) 3.85511e7i 2.02060i
\(818\) 2.27032e6i 0.118633i
\(819\) 0 0
\(820\) 0 0
\(821\) −2.97382e7 −1.53978 −0.769888 0.638179i \(-0.779688\pi\)
−0.769888 + 0.638179i \(0.779688\pi\)
\(822\) 0 0
\(823\) 1.60108e7i 0.823972i 0.911190 + 0.411986i \(0.135165\pi\)
−0.911190 + 0.411986i \(0.864835\pi\)
\(824\) 1.22647e7 0.629273
\(825\) 0 0
\(826\) −8.01060e6 −0.408522
\(827\) − 2.54748e7i − 1.29523i −0.761969 0.647614i \(-0.775767\pi\)
0.761969 0.647614i \(-0.224233\pi\)
\(828\) 0 0
\(829\) −1.89971e7 −0.960064 −0.480032 0.877251i \(-0.659375\pi\)
−0.480032 + 0.877251i \(0.659375\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 561152.i 0.0281043i
\(833\) 2.14912e7i 1.07312i
\(834\) 0 0
\(835\) 0 0
\(836\) 4.72560e6 0.233852
\(837\) 0 0
\(838\) 2.43600e7i 1.19830i
\(839\) 3.98539e7 1.95463 0.977317 0.211782i \(-0.0679267\pi\)
0.977317 + 0.211782i \(0.0679267\pi\)
\(840\) 0 0
\(841\) −1.18322e7 −0.576868
\(842\) − 1.22856e7i − 0.597193i
\(843\) 0 0
\(844\) −1.91192e6 −0.0923876
\(845\) 0 0
\(846\) 0 0
\(847\) 1.09455e7i 0.524238i
\(848\) − 6.62938e6i − 0.316580i
\(849\) 0 0
\(850\) 0 0
\(851\) −4.37130e6 −0.206912
\(852\) 0 0
\(853\) − 2.19179e7i − 1.03140i −0.856769 0.515700i \(-0.827532\pi\)
0.856769 0.515700i \(-0.172468\pi\)
\(854\) −1.59943e7 −0.750450
\(855\) 0 0
\(856\) 594432. 0.0277280
\(857\) 2.72213e7i 1.26607i 0.774125 + 0.633033i \(0.218190\pi\)
−0.774125 + 0.633033i \(0.781810\pi\)
\(858\) 0 0
\(859\) −3.76214e7 −1.73961 −0.869805 0.493396i \(-0.835755\pi\)
−0.869805 + 0.493396i \(0.835755\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 2.67660e6i − 0.122692i
\(863\) 2.51995e7i 1.15177i 0.817531 + 0.575885i \(0.195342\pi\)
−0.817531 + 0.575885i \(0.804658\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.30176e7 0.589841
\(867\) 0 0
\(868\) 901232.i 0.0406011i
\(869\) −1.22208e7 −0.548972
\(870\) 0 0
\(871\) −3.08510e6 −0.137792
\(872\) 1.32064e6i 0.0588157i
\(873\) 0 0
\(874\) −1.06326e7 −0.470826
\(875\) 0 0
\(876\) 0 0
\(877\) 2.77307e6i 0.121748i 0.998145 + 0.0608739i \(0.0193888\pi\)
−0.998145 + 0.0608739i \(0.980611\pi\)
\(878\) 6.35037e6i 0.278012i
\(879\) 0 0
\(880\) 0 0
\(881\) 1.37827e7 0.598268 0.299134 0.954211i \(-0.403302\pi\)
0.299134 + 0.954211i \(0.403302\pi\)
\(882\) 0 0
\(883\) 4.96322e6i 0.214221i 0.994247 + 0.107110i \(0.0341598\pi\)
−0.994247 + 0.107110i \(0.965840\pi\)
\(884\) −4.45853e6 −0.191894
\(885\) 0 0
\(886\) −855552. −0.0366153
\(887\) − 4.12640e7i − 1.76101i −0.474037 0.880505i \(-0.657204\pi\)
0.474037 0.880505i \(-0.342796\pi\)
\(888\) 0 0
\(889\) −2.02619e7 −0.859857
\(890\) 0 0
\(891\) 0 0
\(892\) 776720.i 0.0326853i
\(893\) 4.16444e7i 1.74754i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.29434e6 0.0538613
\(897\) 0 0
\(898\) − 1.48290e7i − 0.613649i
\(899\) −2.10050e6 −0.0866808
\(900\) 0 0
\(901\) 5.26725e7 2.16158
\(902\) − 3.93840e6i − 0.161177i
\(903\) 0 0
\(904\) −1.13211e7 −0.460752
\(905\) 0 0
\(906\) 0 0
\(907\) 1.66308e7i 0.671268i 0.941992 + 0.335634i \(0.108951\pi\)
−0.941992 + 0.335634i \(0.891049\pi\)
\(908\) − 4.60243e6i − 0.185256i
\(909\) 0 0
\(910\) 0 0
\(911\) −3.64995e7 −1.45711 −0.728554 0.684989i \(-0.759807\pi\)
−0.728554 + 0.684989i \(0.759807\pi\)
\(912\) 0 0
\(913\) − 3.86730e6i − 0.153543i
\(914\) 1.53277e7 0.606892
\(915\) 0 0
\(916\) 1.15304e6 0.0454052
\(917\) 1.40304e7i 0.550994i
\(918\) 0 0
\(919\) −6.83190e6 −0.266841 −0.133421 0.991060i \(-0.542596\pi\)
−0.133421 + 0.991060i \(0.542596\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.23469e7i 0.865744i
\(923\) 4.64430e6i 0.179439i
\(924\) 0 0
\(925\) 0 0
\(926\) 1.76057e7 0.674723
\(927\) 0 0
\(928\) 3.01670e6i 0.114991i
\(929\) 5.05115e6 0.192022 0.0960111 0.995380i \(-0.469392\pi\)
0.0960111 + 0.995380i \(0.469392\pi\)
\(930\) 0 0
\(931\) 2.08045e7 0.786651
\(932\) 9.10637e6i 0.343404i
\(933\) 0 0
\(934\) −1.34090e7 −0.502956
\(935\) 0 0
\(936\) 0 0
\(937\) 8.30023e6i 0.308845i 0.988005 + 0.154423i \(0.0493518\pi\)
−0.988005 + 0.154423i \(0.950648\pi\)
\(938\) 7.11600e6i 0.264076i
\(939\) 0 0
\(940\) 0 0
\(941\) 4.87332e7 1.79412 0.897059 0.441910i \(-0.145699\pi\)
0.897059 + 0.441910i \(0.145699\pi\)
\(942\) 0 0
\(943\) 8.86140e6i 0.324506i
\(944\) 6.48960e6 0.237022
\(945\) 0 0
\(946\) −1.17474e7 −0.426790
\(947\) 7.34777e6i 0.266245i 0.991100 + 0.133122i \(0.0425003\pi\)
−0.991100 + 0.133122i \(0.957500\pi\)
\(948\) 0 0
\(949\) −1.12946e7 −0.407102
\(950\) 0 0
\(951\) 0 0
\(952\) 1.02839e7i 0.367761i
\(953\) − 3.51437e7i − 1.25347i −0.779231 0.626737i \(-0.784390\pi\)
0.779231 0.626737i \(-0.215610\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.11441e7 −0.394365
\(957\) 0 0
\(958\) − 3.63986e6i − 0.128136i
\(959\) −2.06100e7 −0.723655
\(960\) 0 0
\(961\) −2.81208e7 −0.982243
\(962\) 1.77442e6i 0.0618187i
\(963\) 0 0
\(964\) −9.21819e6 −0.319487
\(965\) 0 0
\(966\) 0 0
\(967\) 2.31186e6i 0.0795050i 0.999210 + 0.0397525i \(0.0126570\pi\)
−0.999210 + 0.0397525i \(0.987343\pi\)
\(968\) − 8.86726e6i − 0.304159i
\(969\) 0 0
\(970\) 0 0
\(971\) 3.73588e7 1.27158 0.635791 0.771861i \(-0.280674\pi\)
0.635791 + 0.771861i \(0.280674\pi\)
\(972\) 0 0
\(973\) − 1.71970e7i − 0.582333i
\(974\) −1.61432e7 −0.545245
\(975\) 0 0
\(976\) 1.29574e7 0.435406
\(977\) 1.02666e7i 0.344104i 0.985088 + 0.172052i \(0.0550397\pi\)
−0.985088 + 0.172052i \(0.944960\pi\)
\(978\) 0 0
\(979\) −1.55592e7 −0.518837
\(980\) 0 0
\(981\) 0 0
\(982\) 1.51978e6i 0.0502922i
\(983\) 2.98404e7i 0.984966i 0.870322 + 0.492483i \(0.163911\pi\)
−0.870322 + 0.492483i \(0.836089\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.39687e7 −0.785148
\(987\) 0 0
\(988\) 4.31605e6i 0.140668i
\(989\) 2.64316e7 0.859278
\(990\) 0 0
\(991\) −2.34222e7 −0.757605 −0.378803 0.925477i \(-0.623664\pi\)
−0.378803 + 0.925477i \(0.623664\pi\)
\(992\) − 730112.i − 0.0235565i
\(993\) 0 0
\(994\) 1.07124e7 0.343891
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.60709e7i − 0.830651i −0.909673 0.415326i \(-0.863668\pi\)
0.909673 0.415326i \(-0.136332\pi\)
\(998\) − 2.96973e7i − 0.943824i
\(999\) 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.g.199.1 2
3.2 odd 2 150.6.c.c.49.2 2
5.2 odd 4 450.6.a.t.1.1 1
5.3 odd 4 450.6.a.e.1.1 1
5.4 even 2 inner 450.6.c.g.199.2 2
15.2 even 4 150.6.a.g.1.1 1
15.8 even 4 150.6.a.i.1.1 yes 1
15.14 odd 2 150.6.c.c.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.g.1.1 1 15.2 even 4
150.6.a.i.1.1 yes 1 15.8 even 4
150.6.c.c.49.1 2 15.14 odd 2
150.6.c.c.49.2 2 3.2 odd 2
450.6.a.e.1.1 1 5.3 odd 4
450.6.a.t.1.1 1 5.2 odd 4
450.6.c.g.199.1 2 1.1 even 1 trivial
450.6.c.g.199.2 2 5.4 even 2 inner