Properties

Label 464.6.a.i.1.3
Level $464$
Weight $6$
Character 464.1
Self dual yes
Analytic conductor $74.418$
Analytic rank $0$
Dimension $4$
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] = [464,6,Mod(1,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("464.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 464.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: \(74.4180923932\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.10057\) of defining polynomial
Character \(\chi\) \(=\) 464.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+17.5258 q^{3} +32.5670 q^{5} +220.793 q^{7} +64.1549 q^{9} +O(q^{10})\) \(q+17.5258 q^{3} +32.5670 q^{5} +220.793 q^{7} +64.1549 q^{9} +85.7296 q^{11} +1034.02 q^{13} +570.765 q^{15} -313.020 q^{17} -458.534 q^{19} +3869.58 q^{21} +3448.84 q^{23} -2064.39 q^{25} -3134.41 q^{27} -841.000 q^{29} +7983.23 q^{31} +1502.48 q^{33} +7190.58 q^{35} +152.624 q^{37} +18122.0 q^{39} -18492.2 q^{41} -2072.84 q^{43} +2089.33 q^{45} -15845.6 q^{47} +31942.6 q^{49} -5485.94 q^{51} +9240.52 q^{53} +2791.96 q^{55} -8036.19 q^{57} +14323.2 q^{59} -19580.2 q^{61} +14165.0 q^{63} +33674.9 q^{65} +9193.70 q^{67} +60443.8 q^{69} +19374.7 q^{71} -56912.4 q^{73} -36180.1 q^{75} +18928.5 q^{77} -51573.6 q^{79} -70522.8 q^{81} -19978.1 q^{83} -10194.2 q^{85} -14739.2 q^{87} +130663. q^{89} +228304. q^{91} +139913. q^{93} -14933.1 q^{95} +43603.5 q^{97} +5499.97 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 17.5258 1.12428 0.562141 0.827041i \(-0.309978\pi\)
0.562141 + 0.827041i \(0.309978\pi\)
\(4\) 0 0
\(5\) 32.5670 0.582577 0.291289 0.956635i \(-0.405916\pi\)
0.291289 + 0.956635i \(0.405916\pi\)
\(6\) 0 0
\(7\) 220.793 1.70310 0.851551 0.524272i \(-0.175663\pi\)
0.851551 + 0.524272i \(0.175663\pi\)
\(8\) 0 0
\(9\) 64.1549 0.264012
\(10\) 0 0
\(11\) 85.7296 0.213623 0.106812 0.994279i \(-0.465936\pi\)
0.106812 + 0.994279i \(0.465936\pi\)
\(12\) 0 0
\(13\) 1034.02 1.69695 0.848475 0.529235i \(-0.177521\pi\)
0.848475 + 0.529235i \(0.177521\pi\)
\(14\) 0 0
\(15\) 570.765 0.654981
\(16\) 0 0
\(17\) −313.020 −0.262694 −0.131347 0.991336i \(-0.541930\pi\)
−0.131347 + 0.991336i \(0.541930\pi\)
\(18\) 0 0
\(19\) −458.534 −0.291398 −0.145699 0.989329i \(-0.546543\pi\)
−0.145699 + 0.989329i \(0.546543\pi\)
\(20\) 0 0
\(21\) 3869.58 1.91477
\(22\) 0 0
\(23\) 3448.84 1.35942 0.679709 0.733482i \(-0.262106\pi\)
0.679709 + 0.733482i \(0.262106\pi\)
\(24\) 0 0
\(25\) −2064.39 −0.660604
\(26\) 0 0
\(27\) −3134.41 −0.827459
\(28\) 0 0
\(29\) −841.000 −0.185695
\(30\) 0 0
\(31\) 7983.23 1.49202 0.746009 0.665936i \(-0.231967\pi\)
0.746009 + 0.665936i \(0.231967\pi\)
\(32\) 0 0
\(33\) 1502.48 0.240173
\(34\) 0 0
\(35\) 7190.58 0.992188
\(36\) 0 0
\(37\) 152.624 0.0183282 0.00916409 0.999958i \(-0.497083\pi\)
0.00916409 + 0.999958i \(0.497083\pi\)
\(38\) 0 0
\(39\) 18122.0 1.90785
\(40\) 0 0
\(41\) −18492.2 −1.71802 −0.859011 0.511957i \(-0.828921\pi\)
−0.859011 + 0.511957i \(0.828921\pi\)
\(42\) 0 0
\(43\) −2072.84 −0.170960 −0.0854799 0.996340i \(-0.527242\pi\)
−0.0854799 + 0.996340i \(0.527242\pi\)
\(44\) 0 0
\(45\) 2089.33 0.153807
\(46\) 0 0
\(47\) −15845.6 −1.04632 −0.523159 0.852235i \(-0.675247\pi\)
−0.523159 + 0.852235i \(0.675247\pi\)
\(48\) 0 0
\(49\) 31942.6 1.90055
\(50\) 0 0
\(51\) −5485.94 −0.295343
\(52\) 0 0
\(53\) 9240.52 0.451863 0.225931 0.974143i \(-0.427457\pi\)
0.225931 + 0.974143i \(0.427457\pi\)
\(54\) 0 0
\(55\) 2791.96 0.124452
\(56\) 0 0
\(57\) −8036.19 −0.327614
\(58\) 0 0
\(59\) 14323.2 0.535686 0.267843 0.963463i \(-0.413689\pi\)
0.267843 + 0.963463i \(0.413689\pi\)
\(60\) 0 0
\(61\) −19580.2 −0.673739 −0.336870 0.941551i \(-0.609368\pi\)
−0.336870 + 0.941551i \(0.609368\pi\)
\(62\) 0 0
\(63\) 14165.0 0.449639
\(64\) 0 0
\(65\) 33674.9 0.988604
\(66\) 0 0
\(67\) 9193.70 0.250209 0.125105 0.992144i \(-0.460073\pi\)
0.125105 + 0.992144i \(0.460073\pi\)
\(68\) 0 0
\(69\) 60443.8 1.52837
\(70\) 0 0
\(71\) 19374.7 0.456131 0.228066 0.973646i \(-0.426760\pi\)
0.228066 + 0.973646i \(0.426760\pi\)
\(72\) 0 0
\(73\) −56912.4 −1.24997 −0.624985 0.780637i \(-0.714895\pi\)
−0.624985 + 0.780637i \(0.714895\pi\)
\(74\) 0 0
\(75\) −36180.1 −0.742706
\(76\) 0 0
\(77\) 18928.5 0.363822
\(78\) 0 0
\(79\) −51573.6 −0.929736 −0.464868 0.885380i \(-0.653898\pi\)
−0.464868 + 0.885380i \(0.653898\pi\)
\(80\) 0 0
\(81\) −70522.8 −1.19431
\(82\) 0 0
\(83\) −19978.1 −0.318317 −0.159158 0.987253i \(-0.550878\pi\)
−0.159158 + 0.987253i \(0.550878\pi\)
\(84\) 0 0
\(85\) −10194.2 −0.153040
\(86\) 0 0
\(87\) −14739.2 −0.208774
\(88\) 0 0
\(89\) 130663. 1.74855 0.874274 0.485432i \(-0.161338\pi\)
0.874274 + 0.485432i \(0.161338\pi\)
\(90\) 0 0
\(91\) 228304. 2.89008
\(92\) 0 0
\(93\) 139913. 1.67745
\(94\) 0 0
\(95\) −14933.1 −0.169762
\(96\) 0 0
\(97\) 43603.5 0.470535 0.235268 0.971931i \(-0.424403\pi\)
0.235268 + 0.971931i \(0.424403\pi\)
\(98\) 0 0
\(99\) 5499.97 0.0563991
\(100\) 0 0
\(101\) −56686.0 −0.552933 −0.276467 0.961024i \(-0.589164\pi\)
−0.276467 + 0.961024i \(0.589164\pi\)
\(102\) 0 0
\(103\) 111450. 1.03511 0.517556 0.855649i \(-0.326842\pi\)
0.517556 + 0.855649i \(0.326842\pi\)
\(104\) 0 0
\(105\) 126021. 1.11550
\(106\) 0 0
\(107\) −189434. −1.59955 −0.799775 0.600300i \(-0.795048\pi\)
−0.799775 + 0.600300i \(0.795048\pi\)
\(108\) 0 0
\(109\) 100232. 0.808054 0.404027 0.914747i \(-0.367610\pi\)
0.404027 + 0.914747i \(0.367610\pi\)
\(110\) 0 0
\(111\) 2674.87 0.0206061
\(112\) 0 0
\(113\) −103444. −0.762092 −0.381046 0.924556i \(-0.624436\pi\)
−0.381046 + 0.924556i \(0.624436\pi\)
\(114\) 0 0
\(115\) 112318. 0.791966
\(116\) 0 0
\(117\) 66337.2 0.448015
\(118\) 0 0
\(119\) −69112.8 −0.447395
\(120\) 0 0
\(121\) −153701. −0.954365
\(122\) 0 0
\(123\) −324091. −1.93154
\(124\) 0 0
\(125\) −169003. −0.967430
\(126\) 0 0
\(127\) 247538. 1.36186 0.680930 0.732349i \(-0.261576\pi\)
0.680930 + 0.732349i \(0.261576\pi\)
\(128\) 0 0
\(129\) −36328.2 −0.192207
\(130\) 0 0
\(131\) −198458. −1.01040 −0.505198 0.863004i \(-0.668580\pi\)
−0.505198 + 0.863004i \(0.668580\pi\)
\(132\) 0 0
\(133\) −101241. −0.496281
\(134\) 0 0
\(135\) −102078. −0.482059
\(136\) 0 0
\(137\) 141442. 0.643839 0.321920 0.946767i \(-0.395672\pi\)
0.321920 + 0.946767i \(0.395672\pi\)
\(138\) 0 0
\(139\) 325330. 1.42819 0.714097 0.700047i \(-0.246838\pi\)
0.714097 + 0.700047i \(0.246838\pi\)
\(140\) 0 0
\(141\) −277707. −1.17636
\(142\) 0 0
\(143\) 88645.8 0.362508
\(144\) 0 0
\(145\) −27388.9 −0.108182
\(146\) 0 0
\(147\) 559821. 2.13676
\(148\) 0 0
\(149\) −391462. −1.44452 −0.722261 0.691621i \(-0.756897\pi\)
−0.722261 + 0.691621i \(0.756897\pi\)
\(150\) 0 0
\(151\) 216585. 0.773013 0.386507 0.922287i \(-0.373682\pi\)
0.386507 + 0.922287i \(0.373682\pi\)
\(152\) 0 0
\(153\) −20081.8 −0.0693544
\(154\) 0 0
\(155\) 259990. 0.869216
\(156\) 0 0
\(157\) 564396. 1.82740 0.913702 0.406385i \(-0.133211\pi\)
0.913702 + 0.406385i \(0.133211\pi\)
\(158\) 0 0
\(159\) 161948. 0.508022
\(160\) 0 0
\(161\) 761480. 2.31523
\(162\) 0 0
\(163\) −476986. −1.40617 −0.703083 0.711108i \(-0.748194\pi\)
−0.703083 + 0.711108i \(0.748194\pi\)
\(164\) 0 0
\(165\) 48931.4 0.139919
\(166\) 0 0
\(167\) 203397. 0.564357 0.282178 0.959362i \(-0.408943\pi\)
0.282178 + 0.959362i \(0.408943\pi\)
\(168\) 0 0
\(169\) 697898. 1.87964
\(170\) 0 0
\(171\) −29417.2 −0.0769327
\(172\) 0 0
\(173\) −95409.0 −0.242367 −0.121184 0.992630i \(-0.538669\pi\)
−0.121184 + 0.992630i \(0.538669\pi\)
\(174\) 0 0
\(175\) −455803. −1.12508
\(176\) 0 0
\(177\) 251026. 0.602263
\(178\) 0 0
\(179\) 327011. 0.762833 0.381416 0.924403i \(-0.375436\pi\)
0.381416 + 0.924403i \(0.375436\pi\)
\(180\) 0 0
\(181\) 108581. 0.246353 0.123177 0.992385i \(-0.460692\pi\)
0.123177 + 0.992385i \(0.460692\pi\)
\(182\) 0 0
\(183\) −343159. −0.757473
\(184\) 0 0
\(185\) 4970.52 0.0106776
\(186\) 0 0
\(187\) −26835.1 −0.0561176
\(188\) 0 0
\(189\) −692056. −1.40925
\(190\) 0 0
\(191\) 315738. 0.626244 0.313122 0.949713i \(-0.398625\pi\)
0.313122 + 0.949713i \(0.398625\pi\)
\(192\) 0 0
\(193\) 41432.1 0.0800651 0.0400326 0.999198i \(-0.487254\pi\)
0.0400326 + 0.999198i \(0.487254\pi\)
\(194\) 0 0
\(195\) 590180. 1.11147
\(196\) 0 0
\(197\) 354739. 0.651243 0.325622 0.945500i \(-0.394426\pi\)
0.325622 + 0.945500i \(0.394426\pi\)
\(198\) 0 0
\(199\) 949129. 1.69900 0.849498 0.527591i \(-0.176905\pi\)
0.849498 + 0.527591i \(0.176905\pi\)
\(200\) 0 0
\(201\) 161127. 0.281306
\(202\) 0 0
\(203\) −185687. −0.316258
\(204\) 0 0
\(205\) −602236. −1.00088
\(206\) 0 0
\(207\) 221260. 0.358903
\(208\) 0 0
\(209\) −39309.9 −0.0622495
\(210\) 0 0
\(211\) 380574. 0.588481 0.294241 0.955731i \(-0.404933\pi\)
0.294241 + 0.955731i \(0.404933\pi\)
\(212\) 0 0
\(213\) 339558. 0.512820
\(214\) 0 0
\(215\) −67506.2 −0.0995973
\(216\) 0 0
\(217\) 1.76264e6 2.54106
\(218\) 0 0
\(219\) −997437. −1.40532
\(220\) 0 0
\(221\) −323668. −0.445779
\(222\) 0 0
\(223\) 68362.6 0.0920569 0.0460284 0.998940i \(-0.485344\pi\)
0.0460284 + 0.998940i \(0.485344\pi\)
\(224\) 0 0
\(225\) −132441. −0.174407
\(226\) 0 0
\(227\) 1.07914e6 1.39000 0.694998 0.719012i \(-0.255405\pi\)
0.694998 + 0.719012i \(0.255405\pi\)
\(228\) 0 0
\(229\) −724221. −0.912604 −0.456302 0.889825i \(-0.650826\pi\)
−0.456302 + 0.889825i \(0.650826\pi\)
\(230\) 0 0
\(231\) 331738. 0.409039
\(232\) 0 0
\(233\) 1.37311e6 1.65697 0.828486 0.560009i \(-0.189203\pi\)
0.828486 + 0.560009i \(0.189203\pi\)
\(234\) 0 0
\(235\) −516044. −0.609561
\(236\) 0 0
\(237\) −903870. −1.04529
\(238\) 0 0
\(239\) −1.11446e6 −1.26204 −0.631018 0.775769i \(-0.717362\pi\)
−0.631018 + 0.775769i \(0.717362\pi\)
\(240\) 0 0
\(241\) −1.47058e6 −1.63097 −0.815486 0.578776i \(-0.803530\pi\)
−0.815486 + 0.578776i \(0.803530\pi\)
\(242\) 0 0
\(243\) −474309. −0.515283
\(244\) 0 0
\(245\) 1.04028e6 1.10722
\(246\) 0 0
\(247\) −474132. −0.494489
\(248\) 0 0
\(249\) −350133. −0.357878
\(250\) 0 0
\(251\) −375105. −0.375810 −0.187905 0.982187i \(-0.560170\pi\)
−0.187905 + 0.982187i \(0.560170\pi\)
\(252\) 0 0
\(253\) 295667. 0.290404
\(254\) 0 0
\(255\) −178661. −0.172060
\(256\) 0 0
\(257\) −471074. −0.444894 −0.222447 0.974945i \(-0.571404\pi\)
−0.222447 + 0.974945i \(0.571404\pi\)
\(258\) 0 0
\(259\) 33698.4 0.0312147
\(260\) 0 0
\(261\) −53954.3 −0.0490258
\(262\) 0 0
\(263\) −1.11373e6 −0.992865 −0.496433 0.868075i \(-0.665357\pi\)
−0.496433 + 0.868075i \(0.665357\pi\)
\(264\) 0 0
\(265\) 300936. 0.263245
\(266\) 0 0
\(267\) 2.28998e6 1.96586
\(268\) 0 0
\(269\) −381568. −0.321508 −0.160754 0.986995i \(-0.551393\pi\)
−0.160754 + 0.986995i \(0.551393\pi\)
\(270\) 0 0
\(271\) −1.08834e6 −0.900208 −0.450104 0.892976i \(-0.648613\pi\)
−0.450104 + 0.892976i \(0.648613\pi\)
\(272\) 0 0
\(273\) 4.00121e6 3.24927
\(274\) 0 0
\(275\) −176979. −0.141120
\(276\) 0 0
\(277\) −2.18958e6 −1.71459 −0.857295 0.514825i \(-0.827857\pi\)
−0.857295 + 0.514825i \(0.827857\pi\)
\(278\) 0 0
\(279\) 512163. 0.393911
\(280\) 0 0
\(281\) −1.02712e6 −0.775992 −0.387996 0.921661i \(-0.626833\pi\)
−0.387996 + 0.921661i \(0.626833\pi\)
\(282\) 0 0
\(283\) −889156. −0.659952 −0.329976 0.943989i \(-0.607041\pi\)
−0.329976 + 0.943989i \(0.607041\pi\)
\(284\) 0 0
\(285\) −261715. −0.190861
\(286\) 0 0
\(287\) −4.08295e6 −2.92597
\(288\) 0 0
\(289\) −1.32188e6 −0.930992
\(290\) 0 0
\(291\) 764188. 0.529015
\(292\) 0 0
\(293\) −482000. −0.328003 −0.164001 0.986460i \(-0.552440\pi\)
−0.164001 + 0.986460i \(0.552440\pi\)
\(294\) 0 0
\(295\) 466465. 0.312079
\(296\) 0 0
\(297\) −268712. −0.176765
\(298\) 0 0
\(299\) 3.56616e6 2.30687
\(300\) 0 0
\(301\) −457668. −0.291162
\(302\) 0 0
\(303\) −993470. −0.621653
\(304\) 0 0
\(305\) −637668. −0.392505
\(306\) 0 0
\(307\) −229158. −0.138768 −0.0693839 0.997590i \(-0.522103\pi\)
−0.0693839 + 0.997590i \(0.522103\pi\)
\(308\) 0 0
\(309\) 1.95326e6 1.16376
\(310\) 0 0
\(311\) −2.49699e6 −1.46392 −0.731958 0.681350i \(-0.761393\pi\)
−0.731958 + 0.681350i \(0.761393\pi\)
\(312\) 0 0
\(313\) −2.78111e6 −1.60457 −0.802283 0.596944i \(-0.796381\pi\)
−0.802283 + 0.596944i \(0.796381\pi\)
\(314\) 0 0
\(315\) 461311. 0.261949
\(316\) 0 0
\(317\) 1.94375e6 1.08641 0.543203 0.839602i \(-0.317212\pi\)
0.543203 + 0.839602i \(0.317212\pi\)
\(318\) 0 0
\(319\) −72098.6 −0.0396689
\(320\) 0 0
\(321\) −3.31998e6 −1.79835
\(322\) 0 0
\(323\) 143530. 0.0765487
\(324\) 0 0
\(325\) −2.13461e6 −1.12101
\(326\) 0 0
\(327\) 1.75665e6 0.908481
\(328\) 0 0
\(329\) −3.49860e6 −1.78199
\(330\) 0 0
\(331\) 60445.1 0.0303243 0.0151622 0.999885i \(-0.495174\pi\)
0.0151622 + 0.999885i \(0.495174\pi\)
\(332\) 0 0
\(333\) 9791.59 0.00483886
\(334\) 0 0
\(335\) 299412. 0.145766
\(336\) 0 0
\(337\) −1.72999e6 −0.829791 −0.414895 0.909869i \(-0.636182\pi\)
−0.414895 + 0.909869i \(0.636182\pi\)
\(338\) 0 0
\(339\) −1.81294e6 −0.856807
\(340\) 0 0
\(341\) 684398. 0.318730
\(342\) 0 0
\(343\) 3.34184e6 1.53373
\(344\) 0 0
\(345\) 1.96847e6 0.890394
\(346\) 0 0
\(347\) −1.17667e6 −0.524605 −0.262303 0.964986i \(-0.584482\pi\)
−0.262303 + 0.964986i \(0.584482\pi\)
\(348\) 0 0
\(349\) 1.20893e6 0.531297 0.265648 0.964070i \(-0.414414\pi\)
0.265648 + 0.964070i \(0.414414\pi\)
\(350\) 0 0
\(351\) −3.24103e6 −1.40416
\(352\) 0 0
\(353\) 1.00723e6 0.430223 0.215111 0.976589i \(-0.430989\pi\)
0.215111 + 0.976589i \(0.430989\pi\)
\(354\) 0 0
\(355\) 630978. 0.265732
\(356\) 0 0
\(357\) −1.21126e6 −0.502998
\(358\) 0 0
\(359\) −1.10145e6 −0.451055 −0.225527 0.974237i \(-0.572411\pi\)
−0.225527 + 0.974237i \(0.572411\pi\)
\(360\) 0 0
\(361\) −2.26585e6 −0.915087
\(362\) 0 0
\(363\) −2.69375e6 −1.07298
\(364\) 0 0
\(365\) −1.85347e6 −0.728204
\(366\) 0 0
\(367\) −2.99288e6 −1.15991 −0.579955 0.814648i \(-0.696930\pi\)
−0.579955 + 0.814648i \(0.696930\pi\)
\(368\) 0 0
\(369\) −1.18636e6 −0.453578
\(370\) 0 0
\(371\) 2.04024e6 0.769568
\(372\) 0 0
\(373\) 4.23178e6 1.57489 0.787446 0.616384i \(-0.211403\pi\)
0.787446 + 0.616384i \(0.211403\pi\)
\(374\) 0 0
\(375\) −2.96192e6 −1.08766
\(376\) 0 0
\(377\) −869608. −0.315116
\(378\) 0 0
\(379\) −1.06268e6 −0.380018 −0.190009 0.981782i \(-0.560852\pi\)
−0.190009 + 0.981782i \(0.560852\pi\)
\(380\) 0 0
\(381\) 4.33831e6 1.53112
\(382\) 0 0
\(383\) 3.75736e6 1.30884 0.654418 0.756133i \(-0.272913\pi\)
0.654418 + 0.756133i \(0.272913\pi\)
\(384\) 0 0
\(385\) 616445. 0.211955
\(386\) 0 0
\(387\) −132983. −0.0451354
\(388\) 0 0
\(389\) −3.88561e6 −1.30192 −0.650961 0.759112i \(-0.725634\pi\)
−0.650961 + 0.759112i \(0.725634\pi\)
\(390\) 0 0
\(391\) −1.07956e6 −0.357111
\(392\) 0 0
\(393\) −3.47815e6 −1.13597
\(394\) 0 0
\(395\) −1.67960e6 −0.541643
\(396\) 0 0
\(397\) 719537. 0.229127 0.114564 0.993416i \(-0.463453\pi\)
0.114564 + 0.993416i \(0.463453\pi\)
\(398\) 0 0
\(399\) −1.77433e6 −0.557960
\(400\) 0 0
\(401\) −1.57519e6 −0.489185 −0.244592 0.969626i \(-0.578654\pi\)
−0.244592 + 0.969626i \(0.578654\pi\)
\(402\) 0 0
\(403\) 8.25479e6 2.53188
\(404\) 0 0
\(405\) −2.29672e6 −0.695777
\(406\) 0 0
\(407\) 13084.4 0.00391533
\(408\) 0 0
\(409\) −3.35355e6 −0.991279 −0.495639 0.868528i \(-0.665066\pi\)
−0.495639 + 0.868528i \(0.665066\pi\)
\(410\) 0 0
\(411\) 2.47889e6 0.723857
\(412\) 0 0
\(413\) 3.16247e6 0.912328
\(414\) 0 0
\(415\) −650628. −0.185444
\(416\) 0 0
\(417\) 5.70168e6 1.60569
\(418\) 0 0
\(419\) −6.79853e6 −1.89182 −0.945911 0.324427i \(-0.894829\pi\)
−0.945911 + 0.324427i \(0.894829\pi\)
\(420\) 0 0
\(421\) −1.77259e6 −0.487420 −0.243710 0.969848i \(-0.578365\pi\)
−0.243710 + 0.969848i \(0.578365\pi\)
\(422\) 0 0
\(423\) −1.01657e6 −0.276241
\(424\) 0 0
\(425\) 646196. 0.173537
\(426\) 0 0
\(427\) −4.32317e6 −1.14745
\(428\) 0 0
\(429\) 1.55359e6 0.407562
\(430\) 0 0
\(431\) −1.30221e6 −0.337666 −0.168833 0.985645i \(-0.554000\pi\)
−0.168833 + 0.985645i \(0.554000\pi\)
\(432\) 0 0
\(433\) 2.15386e6 0.552074 0.276037 0.961147i \(-0.410979\pi\)
0.276037 + 0.961147i \(0.410979\pi\)
\(434\) 0 0
\(435\) −480013. −0.121627
\(436\) 0 0
\(437\) −1.58141e6 −0.396133
\(438\) 0 0
\(439\) 5.81916e6 1.44112 0.720558 0.693394i \(-0.243886\pi\)
0.720558 + 0.693394i \(0.243886\pi\)
\(440\) 0 0
\(441\) 2.04927e6 0.501769
\(442\) 0 0
\(443\) −4.34332e6 −1.05151 −0.525754 0.850637i \(-0.676217\pi\)
−0.525754 + 0.850637i \(0.676217\pi\)
\(444\) 0 0
\(445\) 4.25531e6 1.01866
\(446\) 0 0
\(447\) −6.86070e6 −1.62405
\(448\) 0 0
\(449\) 2.68108e6 0.627615 0.313807 0.949487i \(-0.398395\pi\)
0.313807 + 0.949487i \(0.398395\pi\)
\(450\) 0 0
\(451\) −1.58533e6 −0.367010
\(452\) 0 0
\(453\) 3.79584e6 0.869086
\(454\) 0 0
\(455\) 7.43518e6 1.68369
\(456\) 0 0
\(457\) −4.29834e6 −0.962743 −0.481371 0.876517i \(-0.659861\pi\)
−0.481371 + 0.876517i \(0.659861\pi\)
\(458\) 0 0
\(459\) 981134. 0.217369
\(460\) 0 0
\(461\) 2.28441e6 0.500636 0.250318 0.968164i \(-0.419465\pi\)
0.250318 + 0.968164i \(0.419465\pi\)
\(462\) 0 0
\(463\) −1.51960e6 −0.329440 −0.164720 0.986340i \(-0.552672\pi\)
−0.164720 + 0.986340i \(0.552672\pi\)
\(464\) 0 0
\(465\) 4.55654e6 0.977244
\(466\) 0 0
\(467\) 4.71846e6 1.00117 0.500585 0.865687i \(-0.333118\pi\)
0.500585 + 0.865687i \(0.333118\pi\)
\(468\) 0 0
\(469\) 2.02991e6 0.426132
\(470\) 0 0
\(471\) 9.89151e6 2.05452
\(472\) 0 0
\(473\) −177703. −0.0365210
\(474\) 0 0
\(475\) 946591. 0.192499
\(476\) 0 0
\(477\) 592824. 0.119297
\(478\) 0 0
\(479\) −5.85157e6 −1.16529 −0.582644 0.812727i \(-0.697982\pi\)
−0.582644 + 0.812727i \(0.697982\pi\)
\(480\) 0 0
\(481\) 157816. 0.0311020
\(482\) 0 0
\(483\) 1.33456e7 2.60297
\(484\) 0 0
\(485\) 1.42004e6 0.274123
\(486\) 0 0
\(487\) −1.83575e6 −0.350745 −0.175372 0.984502i \(-0.556113\pi\)
−0.175372 + 0.984502i \(0.556113\pi\)
\(488\) 0 0
\(489\) −8.35958e6 −1.58093
\(490\) 0 0
\(491\) 5.75635e6 1.07757 0.538783 0.842445i \(-0.318884\pi\)
0.538783 + 0.842445i \(0.318884\pi\)
\(492\) 0 0
\(493\) 263250. 0.0487811
\(494\) 0 0
\(495\) 179118. 0.0328568
\(496\) 0 0
\(497\) 4.27781e6 0.776838
\(498\) 0 0
\(499\) −4.10166e6 −0.737409 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(500\) 0 0
\(501\) 3.56471e6 0.634497
\(502\) 0 0
\(503\) 1.12157e6 0.197654 0.0988270 0.995105i \(-0.468491\pi\)
0.0988270 + 0.995105i \(0.468491\pi\)
\(504\) 0 0
\(505\) −1.84610e6 −0.322126
\(506\) 0 0
\(507\) 1.22312e7 2.11325
\(508\) 0 0
\(509\) −2.68902e6 −0.460045 −0.230023 0.973185i \(-0.573880\pi\)
−0.230023 + 0.973185i \(0.573880\pi\)
\(510\) 0 0
\(511\) −1.25659e7 −2.12883
\(512\) 0 0
\(513\) 1.43723e6 0.241120
\(514\) 0 0
\(515\) 3.62960e6 0.603033
\(516\) 0 0
\(517\) −1.35844e6 −0.223518
\(518\) 0 0
\(519\) −1.67212e6 −0.272489
\(520\) 0 0
\(521\) 2.88911e6 0.466305 0.233152 0.972440i \(-0.425096\pi\)
0.233152 + 0.972440i \(0.425096\pi\)
\(522\) 0 0
\(523\) 1.40417e6 0.224474 0.112237 0.993681i \(-0.464198\pi\)
0.112237 + 0.993681i \(0.464198\pi\)
\(524\) 0 0
\(525\) −7.98832e6 −1.26490
\(526\) 0 0
\(527\) −2.49891e6 −0.391945
\(528\) 0 0
\(529\) 5.45814e6 0.848019
\(530\) 0 0
\(531\) 918904. 0.141428
\(532\) 0 0
\(533\) −1.91212e7 −2.91540
\(534\) 0 0
\(535\) −6.16930e6 −0.931861
\(536\) 0 0
\(537\) 5.73113e6 0.857640
\(538\) 0 0
\(539\) 2.73843e6 0.406003
\(540\) 0 0
\(541\) −211154. −0.0310174 −0.0155087 0.999880i \(-0.504937\pi\)
−0.0155087 + 0.999880i \(0.504937\pi\)
\(542\) 0 0
\(543\) 1.90298e6 0.276971
\(544\) 0 0
\(545\) 3.26426e6 0.470754
\(546\) 0 0
\(547\) 4.61571e6 0.659584 0.329792 0.944054i \(-0.393021\pi\)
0.329792 + 0.944054i \(0.393021\pi\)
\(548\) 0 0
\(549\) −1.25616e6 −0.177875
\(550\) 0 0
\(551\) 385627. 0.0541113
\(552\) 0 0
\(553\) −1.13871e7 −1.58343
\(554\) 0 0
\(555\) 87112.5 0.0120046
\(556\) 0 0
\(557\) 1.01209e7 1.38223 0.691114 0.722746i \(-0.257120\pi\)
0.691114 + 0.722746i \(0.257120\pi\)
\(558\) 0 0
\(559\) −2.14335e6 −0.290110
\(560\) 0 0
\(561\) −470308. −0.0630921
\(562\) 0 0
\(563\) −3.84774e6 −0.511604 −0.255802 0.966729i \(-0.582340\pi\)
−0.255802 + 0.966729i \(0.582340\pi\)
\(564\) 0 0
\(565\) −3.36885e6 −0.443978
\(566\) 0 0
\(567\) −1.55709e7 −2.03403
\(568\) 0 0
\(569\) 4.38443e6 0.567718 0.283859 0.958866i \(-0.408385\pi\)
0.283859 + 0.958866i \(0.408385\pi\)
\(570\) 0 0
\(571\) −2.61241e6 −0.335313 −0.167656 0.985845i \(-0.553620\pi\)
−0.167656 + 0.985845i \(0.553620\pi\)
\(572\) 0 0
\(573\) 5.53357e6 0.704075
\(574\) 0 0
\(575\) −7.11974e6 −0.898037
\(576\) 0 0
\(577\) −7.06124e6 −0.882961 −0.441480 0.897271i \(-0.645547\pi\)
−0.441480 + 0.897271i \(0.645547\pi\)
\(578\) 0 0
\(579\) 726132. 0.0900159
\(580\) 0 0
\(581\) −4.41103e6 −0.542125
\(582\) 0 0
\(583\) 792186. 0.0965285
\(584\) 0 0
\(585\) 2.16041e6 0.261003
\(586\) 0 0
\(587\) 8.18957e6 0.980993 0.490496 0.871443i \(-0.336815\pi\)
0.490496 + 0.871443i \(0.336815\pi\)
\(588\) 0 0
\(589\) −3.66058e6 −0.434772
\(590\) 0 0
\(591\) 6.21710e6 0.732182
\(592\) 0 0
\(593\) −9.78828e6 −1.14306 −0.571531 0.820580i \(-0.693650\pi\)
−0.571531 + 0.820580i \(0.693650\pi\)
\(594\) 0 0
\(595\) −2.25080e6 −0.260642
\(596\) 0 0
\(597\) 1.66343e7 1.91015
\(598\) 0 0
\(599\) −4.44247e6 −0.505892 −0.252946 0.967480i \(-0.581400\pi\)
−0.252946 + 0.967480i \(0.581400\pi\)
\(600\) 0 0
\(601\) 248973. 0.0281168 0.0140584 0.999901i \(-0.495525\pi\)
0.0140584 + 0.999901i \(0.495525\pi\)
\(602\) 0 0
\(603\) 589821. 0.0660582
\(604\) 0 0
\(605\) −5.00560e6 −0.555991
\(606\) 0 0
\(607\) 5.61487e6 0.618540 0.309270 0.950974i \(-0.399915\pi\)
0.309270 + 0.950974i \(0.399915\pi\)
\(608\) 0 0
\(609\) −3.25432e6 −0.355563
\(610\) 0 0
\(611\) −1.63846e7 −1.77555
\(612\) 0 0
\(613\) 7.11213e6 0.764449 0.382225 0.924069i \(-0.375158\pi\)
0.382225 + 0.924069i \(0.375158\pi\)
\(614\) 0 0
\(615\) −1.05547e7 −1.12527
\(616\) 0 0
\(617\) 872669. 0.0922862 0.0461431 0.998935i \(-0.485307\pi\)
0.0461431 + 0.998935i \(0.485307\pi\)
\(618\) 0 0
\(619\) 9.40244e6 0.986312 0.493156 0.869941i \(-0.335843\pi\)
0.493156 + 0.869941i \(0.335843\pi\)
\(620\) 0 0
\(621\) −1.08101e7 −1.12486
\(622\) 0 0
\(623\) 2.88495e7 2.97796
\(624\) 0 0
\(625\) 947282. 0.0970017
\(626\) 0 0
\(627\) −688939. −0.0699861
\(628\) 0 0
\(629\) −47774.5 −0.00481470
\(630\) 0 0
\(631\) 1.90990e7 1.90957 0.954786 0.297293i \(-0.0960838\pi\)
0.954786 + 0.297293i \(0.0960838\pi\)
\(632\) 0 0
\(633\) 6.66987e6 0.661619
\(634\) 0 0
\(635\) 8.06158e6 0.793388
\(636\) 0 0
\(637\) 3.30292e7 3.22515
\(638\) 0 0
\(639\) 1.24298e6 0.120424
\(640\) 0 0
\(641\) −4.06757e6 −0.391012 −0.195506 0.980703i \(-0.562635\pi\)
−0.195506 + 0.980703i \(0.562635\pi\)
\(642\) 0 0
\(643\) 1.00899e6 0.0962410 0.0481205 0.998842i \(-0.484677\pi\)
0.0481205 + 0.998842i \(0.484677\pi\)
\(644\) 0 0
\(645\) −1.18310e6 −0.111975
\(646\) 0 0
\(647\) −1.99986e7 −1.87819 −0.939095 0.343657i \(-0.888334\pi\)
−0.939095 + 0.343657i \(0.888334\pi\)
\(648\) 0 0
\(649\) 1.22792e6 0.114435
\(650\) 0 0
\(651\) 3.08918e7 2.85687
\(652\) 0 0
\(653\) −1.16599e7 −1.07007 −0.535035 0.844830i \(-0.679702\pi\)
−0.535035 + 0.844830i \(0.679702\pi\)
\(654\) 0 0
\(655\) −6.46320e6 −0.588633
\(656\) 0 0
\(657\) −3.65121e6 −0.330007
\(658\) 0 0
\(659\) −1.40204e7 −1.25761 −0.628806 0.777562i \(-0.716456\pi\)
−0.628806 + 0.777562i \(0.716456\pi\)
\(660\) 0 0
\(661\) 5.39932e6 0.480657 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(662\) 0 0
\(663\) −5.67256e6 −0.501182
\(664\) 0 0
\(665\) −3.29712e6 −0.289122
\(666\) 0 0
\(667\) −2.90047e6 −0.252438
\(668\) 0 0
\(669\) 1.19811e6 0.103498
\(670\) 0 0
\(671\) −1.67860e6 −0.143926
\(672\) 0 0
\(673\) 2.37236e6 0.201903 0.100951 0.994891i \(-0.467811\pi\)
0.100951 + 0.994891i \(0.467811\pi\)
\(674\) 0 0
\(675\) 6.47064e6 0.546623
\(676\) 0 0
\(677\) 1.48279e7 1.24339 0.621695 0.783260i \(-0.286444\pi\)
0.621695 + 0.783260i \(0.286444\pi\)
\(678\) 0 0
\(679\) 9.62736e6 0.801369
\(680\) 0 0
\(681\) 1.89128e7 1.56275
\(682\) 0 0
\(683\) 1.45710e7 1.19519 0.597594 0.801799i \(-0.296123\pi\)
0.597594 + 0.801799i \(0.296123\pi\)
\(684\) 0 0
\(685\) 4.60635e6 0.375086
\(686\) 0 0
\(687\) −1.26926e7 −1.02603
\(688\) 0 0
\(689\) 9.55485e6 0.766789
\(690\) 0 0
\(691\) 2.39101e6 0.190496 0.0952480 0.995454i \(-0.469636\pi\)
0.0952480 + 0.995454i \(0.469636\pi\)
\(692\) 0 0
\(693\) 1.21436e6 0.0960534
\(694\) 0 0
\(695\) 1.05950e7 0.832033
\(696\) 0 0
\(697\) 5.78844e6 0.451315
\(698\) 0 0
\(699\) 2.40649e7 1.86291
\(700\) 0 0
\(701\) −2.67685e6 −0.205745 −0.102872 0.994695i \(-0.532803\pi\)
−0.102872 + 0.994695i \(0.532803\pi\)
\(702\) 0 0
\(703\) −69983.4 −0.00534080
\(704\) 0 0
\(705\) −9.04411e6 −0.685319
\(706\) 0 0
\(707\) −1.25159e7 −0.941701
\(708\) 0 0
\(709\) 1.61177e7 1.20417 0.602085 0.798432i \(-0.294337\pi\)
0.602085 + 0.798432i \(0.294337\pi\)
\(710\) 0 0
\(711\) −3.30870e6 −0.245461
\(712\) 0 0
\(713\) 2.75329e7 2.02828
\(714\) 0 0
\(715\) 2.88693e6 0.211189
\(716\) 0 0
\(717\) −1.95319e7 −1.41888
\(718\) 0 0
\(719\) −2.74012e7 −1.97673 −0.988363 0.152111i \(-0.951393\pi\)
−0.988363 + 0.152111i \(0.951393\pi\)
\(720\) 0 0
\(721\) 2.46074e7 1.76290
\(722\) 0 0
\(723\) −2.57732e7 −1.83367
\(724\) 0 0
\(725\) 1.73615e6 0.122671
\(726\) 0 0
\(727\) 1.44905e7 1.01683 0.508413 0.861113i \(-0.330232\pi\)
0.508413 + 0.861113i \(0.330232\pi\)
\(728\) 0 0
\(729\) 8.82438e6 0.614986
\(730\) 0 0
\(731\) 648840. 0.0449101
\(732\) 0 0
\(733\) 9.24243e6 0.635370 0.317685 0.948196i \(-0.397095\pi\)
0.317685 + 0.948196i \(0.397095\pi\)
\(734\) 0 0
\(735\) 1.82317e7 1.24483
\(736\) 0 0
\(737\) 788172. 0.0534505
\(738\) 0 0
\(739\) 270924. 0.0182489 0.00912445 0.999958i \(-0.497096\pi\)
0.00912445 + 0.999958i \(0.497096\pi\)
\(740\) 0 0
\(741\) −8.30955e6 −0.555945
\(742\) 0 0
\(743\) −1.56128e7 −1.03755 −0.518776 0.854910i \(-0.673612\pi\)
−0.518776 + 0.854910i \(0.673612\pi\)
\(744\) 0 0
\(745\) −1.27488e7 −0.841545
\(746\) 0 0
\(747\) −1.28169e6 −0.0840394
\(748\) 0 0
\(749\) −4.18257e7 −2.72420
\(750\) 0 0
\(751\) −2.81796e6 −0.182320 −0.0911600 0.995836i \(-0.529057\pi\)
−0.0911600 + 0.995836i \(0.529057\pi\)
\(752\) 0 0
\(753\) −6.57403e6 −0.422517
\(754\) 0 0
\(755\) 7.05355e6 0.450340
\(756\) 0 0
\(757\) 1.47622e6 0.0936293 0.0468146 0.998904i \(-0.485093\pi\)
0.0468146 + 0.998904i \(0.485093\pi\)
\(758\) 0 0
\(759\) 5.18182e6 0.326496
\(760\) 0 0
\(761\) −1.59710e7 −0.999700 −0.499850 0.866112i \(-0.666612\pi\)
−0.499850 + 0.866112i \(0.666612\pi\)
\(762\) 0 0
\(763\) 2.21305e7 1.37620
\(764\) 0 0
\(765\) −654005. −0.0404043
\(766\) 0 0
\(767\) 1.48104e7 0.909033
\(768\) 0 0
\(769\) −2.69038e7 −1.64058 −0.820291 0.571947i \(-0.806188\pi\)
−0.820291 + 0.571947i \(0.806188\pi\)
\(770\) 0 0
\(771\) −8.25597e6 −0.500187
\(772\) 0 0
\(773\) 2.10506e7 1.26711 0.633556 0.773697i \(-0.281595\pi\)
0.633556 + 0.773697i \(0.281595\pi\)
\(774\) 0 0
\(775\) −1.64805e7 −0.985633
\(776\) 0 0
\(777\) 590592. 0.0350942
\(778\) 0 0
\(779\) 8.47930e6 0.500629
\(780\) 0 0
\(781\) 1.66099e6 0.0974403
\(782\) 0 0
\(783\) 2.63604e6 0.153655
\(784\) 0 0
\(785\) 1.83807e7 1.06460
\(786\) 0 0
\(787\) 1.46428e7 0.842728 0.421364 0.906892i \(-0.361552\pi\)
0.421364 + 0.906892i \(0.361552\pi\)
\(788\) 0 0
\(789\) −1.95190e7 −1.11626
\(790\) 0 0
\(791\) −2.28396e7 −1.29792
\(792\) 0 0
\(793\) −2.02462e7 −1.14330
\(794\) 0 0
\(795\) 5.27416e6 0.295962
\(796\) 0 0
\(797\) −1.53622e7 −0.856660 −0.428330 0.903622i \(-0.640898\pi\)
−0.428330 + 0.903622i \(0.640898\pi\)
\(798\) 0 0
\(799\) 4.96000e6 0.274862
\(800\) 0 0
\(801\) 8.38267e6 0.461638
\(802\) 0 0
\(803\) −4.87907e6 −0.267023
\(804\) 0 0
\(805\) 2.47991e7 1.34880
\(806\) 0 0
\(807\) −6.68730e6 −0.361466
\(808\) 0 0
\(809\) −7.26215e6 −0.390116 −0.195058 0.980792i \(-0.562490\pi\)
−0.195058 + 0.980792i \(0.562490\pi\)
\(810\) 0 0
\(811\) −4.17728e6 −0.223019 −0.111510 0.993763i \(-0.535569\pi\)
−0.111510 + 0.993763i \(0.535569\pi\)
\(812\) 0 0
\(813\) −1.90741e7 −1.01209
\(814\) 0 0
\(815\) −1.55340e7 −0.819200
\(816\) 0 0
\(817\) 950466. 0.0498174
\(818\) 0 0
\(819\) 1.46468e7 0.763015
\(820\) 0 0
\(821\) 2.70285e7 1.39947 0.699736 0.714402i \(-0.253301\pi\)
0.699736 + 0.714402i \(0.253301\pi\)
\(822\) 0 0
\(823\) 6.95032e6 0.357689 0.178844 0.983877i \(-0.442764\pi\)
0.178844 + 0.983877i \(0.442764\pi\)
\(824\) 0 0
\(825\) −3.10171e6 −0.158659
\(826\) 0 0
\(827\) 370060. 0.0188152 0.00940759 0.999956i \(-0.497005\pi\)
0.00940759 + 0.999956i \(0.497005\pi\)
\(828\) 0 0
\(829\) 1.38451e7 0.699698 0.349849 0.936806i \(-0.386233\pi\)
0.349849 + 0.936806i \(0.386233\pi\)
\(830\) 0 0
\(831\) −3.83741e7 −1.92768
\(832\) 0 0
\(833\) −9.99869e6 −0.499264
\(834\) 0 0
\(835\) 6.62405e6 0.328781
\(836\) 0 0
\(837\) −2.50227e7 −1.23458
\(838\) 0 0
\(839\) −1.86174e7 −0.913089 −0.456544 0.889701i \(-0.650913\pi\)
−0.456544 + 0.889701i \(0.650913\pi\)
\(840\) 0 0
\(841\) 707281. 0.0344828
\(842\) 0 0
\(843\) −1.80012e7 −0.872435
\(844\) 0 0
\(845\) 2.27285e7 1.09504
\(846\) 0 0
\(847\) −3.39362e7 −1.62538
\(848\) 0 0
\(849\) −1.55832e7 −0.741972
\(850\) 0 0
\(851\) 526376. 0.0249157
\(852\) 0 0
\(853\) 2.15389e6 0.101356 0.0506782 0.998715i \(-0.483862\pi\)
0.0506782 + 0.998715i \(0.483862\pi\)
\(854\) 0 0
\(855\) −958031. −0.0448192
\(856\) 0 0
\(857\) −3.33513e7 −1.55118 −0.775588 0.631239i \(-0.782547\pi\)
−0.775588 + 0.631239i \(0.782547\pi\)
\(858\) 0 0
\(859\) 3.11061e7 1.43834 0.719171 0.694833i \(-0.244522\pi\)
0.719171 + 0.694833i \(0.244522\pi\)
\(860\) 0 0
\(861\) −7.15571e7 −3.28961
\(862\) 0 0
\(863\) 1.59387e7 0.728494 0.364247 0.931302i \(-0.381326\pi\)
0.364247 + 0.931302i \(0.381326\pi\)
\(864\) 0 0
\(865\) −3.10719e6 −0.141198
\(866\) 0 0
\(867\) −2.31670e7 −1.04670
\(868\) 0 0
\(869\) −4.42138e6 −0.198613
\(870\) 0 0
\(871\) 9.50644e6 0.424593
\(872\) 0 0
\(873\) 2.79738e6 0.124227
\(874\) 0 0
\(875\) −3.73147e7 −1.64763
\(876\) 0 0
\(877\) 666578. 0.0292652 0.0146326 0.999893i \(-0.495342\pi\)
0.0146326 + 0.999893i \(0.495342\pi\)
\(878\) 0 0
\(879\) −8.44745e6 −0.368768
\(880\) 0 0
\(881\) −2.30910e7 −1.00231 −0.501155 0.865358i \(-0.667091\pi\)
−0.501155 + 0.865358i \(0.667091\pi\)
\(882\) 0 0
\(883\) −3.85060e6 −0.166198 −0.0830992 0.996541i \(-0.526482\pi\)
−0.0830992 + 0.996541i \(0.526482\pi\)
\(884\) 0 0
\(885\) 8.17519e6 0.350864
\(886\) 0 0
\(887\) 4.14426e7 1.76863 0.884316 0.466889i \(-0.154625\pi\)
0.884316 + 0.466889i \(0.154625\pi\)
\(888\) 0 0
\(889\) 5.46547e7 2.31939
\(890\) 0 0
\(891\) −6.04589e6 −0.255133
\(892\) 0 0
\(893\) 7.26574e6 0.304896
\(894\) 0 0
\(895\) 1.06498e7 0.444409
\(896\) 0 0
\(897\) 6.24999e7 2.59357
\(898\) 0 0
\(899\) −6.71389e6 −0.277061
\(900\) 0 0
\(901\) −2.89247e6 −0.118702
\(902\) 0 0
\(903\) −8.02102e6 −0.327348
\(904\) 0 0
\(905\) 3.53617e6 0.143520
\(906\) 0 0
\(907\) 4.63163e7 1.86946 0.934729 0.355363i \(-0.115643\pi\)
0.934729 + 0.355363i \(0.115643\pi\)
\(908\) 0 0
\(909\) −3.63669e6 −0.145981
\(910\) 0 0
\(911\) −2.69515e6 −0.107594 −0.0537969 0.998552i \(-0.517132\pi\)
−0.0537969 + 0.998552i \(0.517132\pi\)
\(912\) 0 0
\(913\) −1.71272e6 −0.0679999
\(914\) 0 0
\(915\) −1.11757e7 −0.441286
\(916\) 0 0
\(917\) −4.38183e7 −1.72081
\(918\) 0 0
\(919\) −1.39792e7 −0.546000 −0.273000 0.962014i \(-0.588016\pi\)
−0.273000 + 0.962014i \(0.588016\pi\)
\(920\) 0 0
\(921\) −4.01618e6 −0.156014
\(922\) 0 0
\(923\) 2.00338e7 0.774032
\(924\) 0 0
\(925\) −315076. −0.0121077
\(926\) 0 0
\(927\) 7.15007e6 0.273282
\(928\) 0 0
\(929\) −7.25580e6 −0.275833 −0.137916 0.990444i \(-0.544041\pi\)
−0.137916 + 0.990444i \(0.544041\pi\)
\(930\) 0 0
\(931\) −1.46468e7 −0.553819
\(932\) 0 0
\(933\) −4.37619e7 −1.64586
\(934\) 0 0
\(935\) −873940. −0.0326928
\(936\) 0 0
\(937\) 3.95850e6 0.147293 0.0736464 0.997284i \(-0.476536\pi\)
0.0736464 + 0.997284i \(0.476536\pi\)
\(938\) 0 0
\(939\) −4.87413e7 −1.80399
\(940\) 0 0
\(941\) −4.07672e7 −1.50085 −0.750425 0.660956i \(-0.770151\pi\)
−0.750425 + 0.660956i \(0.770151\pi\)
\(942\) 0 0
\(943\) −6.37766e7 −2.33551
\(944\) 0 0
\(945\) −2.25382e7 −0.820994
\(946\) 0 0
\(947\) 2.67878e7 0.970650 0.485325 0.874334i \(-0.338701\pi\)
0.485325 + 0.874334i \(0.338701\pi\)
\(948\) 0 0
\(949\) −5.88484e7 −2.12114
\(950\) 0 0
\(951\) 3.40658e7 1.22143
\(952\) 0 0
\(953\) −1.81430e7 −0.647107 −0.323553 0.946210i \(-0.604877\pi\)
−0.323553 + 0.946210i \(0.604877\pi\)
\(954\) 0 0
\(955\) 1.02827e7 0.364835
\(956\) 0 0
\(957\) −1.26359e6 −0.0445990
\(958\) 0 0
\(959\) 3.12294e7 1.09652
\(960\) 0 0
\(961\) 3.51028e7 1.22612
\(962\) 0 0
\(963\) −1.21531e7 −0.422300
\(964\) 0 0
\(965\) 1.34932e6 0.0466441
\(966\) 0 0
\(967\) 1.40529e7 0.483282 0.241641 0.970366i \(-0.422314\pi\)
0.241641 + 0.970366i \(0.422314\pi\)
\(968\) 0 0
\(969\) 2.51549e6 0.0860624
\(970\) 0 0
\(971\) −1.77622e7 −0.604573 −0.302287 0.953217i \(-0.597750\pi\)
−0.302287 + 0.953217i \(0.597750\pi\)
\(972\) 0 0
\(973\) 7.18306e7 2.43236
\(974\) 0 0
\(975\) −3.74108e7 −1.26034
\(976\) 0 0
\(977\) −1.66625e7 −0.558474 −0.279237 0.960222i \(-0.590082\pi\)
−0.279237 + 0.960222i \(0.590082\pi\)
\(978\) 0 0
\(979\) 1.12017e7 0.373531
\(980\) 0 0
\(981\) 6.43037e6 0.213336
\(982\) 0 0
\(983\) 2.28571e7 0.754461 0.377231 0.926119i \(-0.376876\pi\)
0.377231 + 0.926119i \(0.376876\pi\)
\(984\) 0 0
\(985\) 1.15528e7 0.379399
\(986\) 0 0
\(987\) −6.13159e7 −2.00346
\(988\) 0 0
\(989\) −7.14888e6 −0.232406
\(990\) 0 0
\(991\) 2.52864e7 0.817904 0.408952 0.912556i \(-0.365894\pi\)
0.408952 + 0.912556i \(0.365894\pi\)
\(992\) 0 0
\(993\) 1.05935e6 0.0340931
\(994\) 0 0
\(995\) 3.09103e7 0.989796
\(996\) 0 0
\(997\) 4.44014e7 1.41468 0.707340 0.706873i \(-0.249895\pi\)
0.707340 + 0.706873i \(0.249895\pi\)
\(998\) 0 0
\(999\) −478387. −0.0151658
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 464.6.a.i.1.3 4
4.3 odd 2 29.6.a.a.1.3 4
12.11 even 2 261.6.a.a.1.2 4
20.19 odd 2 725.6.a.a.1.2 4
116.115 odd 2 841.6.a.a.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.3 4 4.3 odd 2
261.6.a.a.1.2 4 12.11 even 2
464.6.a.i.1.3 4 1.1 even 1 trivial
725.6.a.a.1.2 4 20.19 odd 2
841.6.a.a.1.2 4 116.115 odd 2