Properties

Label 784.6.a.ba.1.2
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,6,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.54138\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0828 q^{3} -79.8276 q^{5} -141.338 q^{9} +O(q^{10})\) \(q+10.0828 q^{3} -79.8276 q^{5} -141.338 q^{9} -351.904 q^{11} -291.683 q^{13} -804.883 q^{15} -370.075 q^{17} -1504.93 q^{19} +425.711 q^{23} +3247.45 q^{25} -3875.19 q^{27} -7783.93 q^{29} +2575.18 q^{31} -3548.16 q^{33} +739.618 q^{37} -2940.97 q^{39} +7029.84 q^{41} -1835.23 q^{43} +11282.7 q^{45} +1532.68 q^{47} -3731.38 q^{51} -9537.46 q^{53} +28091.6 q^{55} -15173.8 q^{57} +29674.1 q^{59} -46510.8 q^{61} +23284.3 q^{65} -26746.1 q^{67} +4292.34 q^{69} +14388.8 q^{71} -70095.1 q^{73} +32743.3 q^{75} +27085.8 q^{79} -4727.49 q^{81} +79755.4 q^{83} +29542.2 q^{85} -78483.6 q^{87} +43577.3 q^{89} +25964.9 q^{93} +120135. q^{95} +103374. q^{97} +49737.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9} - 424 q^{11} - 924 q^{13} - 892 q^{15} - 2346 q^{17} + 360 q^{19} + 12 q^{23} + 1872 q^{25} - 2872 q^{27} - 7052 q^{29} - 3548 q^{31} - 3398 q^{33} + 11090 q^{37} - 1624 q^{39} + 3500 q^{41} + 12680 q^{43} + 1300 q^{45} + 22956 q^{47} + 384 q^{51} + 3042 q^{53} + 25076 q^{55} - 19058 q^{57} + 65808 q^{59} - 42486 q^{61} - 3164 q^{65} - 42312 q^{67} + 5154 q^{69} + 2208 q^{71} - 50506 q^{73} + 35608 q^{75} - 9004 q^{79} + 51178 q^{81} + 104328 q^{83} - 53106 q^{85} - 80008 q^{87} - 26666 q^{89} + 38718 q^{93} + 198140 q^{95} + 209132 q^{97} + 66944 q^{99}+O(q^{100}) \) 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
\(3\) 10.0828 0.646810 0.323405 0.946261i \(-0.395172\pi\)
0.323405 + 0.946261i \(0.395172\pi\)
\(4\) 0 0
\(5\) −79.8276 −1.42800 −0.714000 0.700146i \(-0.753118\pi\)
−0.714000 + 0.700146i \(0.753118\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −141.338 −0.581637
\(10\) 0 0
\(11\) −351.904 −0.876884 −0.438442 0.898760i \(-0.644469\pi\)
−0.438442 + 0.898760i \(0.644469\pi\)
\(12\) 0 0
\(13\) −291.683 −0.478688 −0.239344 0.970935i \(-0.576932\pi\)
−0.239344 + 0.970935i \(0.576932\pi\)
\(14\) 0 0
\(15\) −804.883 −0.923644
\(16\) 0 0
\(17\) −370.075 −0.310576 −0.155288 0.987869i \(-0.549631\pi\)
−0.155288 + 0.987869i \(0.549631\pi\)
\(18\) 0 0
\(19\) −1504.93 −0.956381 −0.478190 0.878256i \(-0.658707\pi\)
−0.478190 + 0.878256i \(0.658707\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 425.711 0.167801 0.0839006 0.996474i \(-0.473262\pi\)
0.0839006 + 0.996474i \(0.473262\pi\)
\(24\) 0 0
\(25\) 3247.45 1.03918
\(26\) 0 0
\(27\) −3875.19 −1.02302
\(28\) 0 0
\(29\) −7783.93 −1.71872 −0.859358 0.511374i \(-0.829137\pi\)
−0.859358 + 0.511374i \(0.829137\pi\)
\(30\) 0 0
\(31\) 2575.18 0.481285 0.240643 0.970614i \(-0.422642\pi\)
0.240643 + 0.970614i \(0.422642\pi\)
\(32\) 0 0
\(33\) −3548.16 −0.567177
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 739.618 0.0888184 0.0444092 0.999013i \(-0.485859\pi\)
0.0444092 + 0.999013i \(0.485859\pi\)
\(38\) 0 0
\(39\) −2940.97 −0.309620
\(40\) 0 0
\(41\) 7029.84 0.653109 0.326554 0.945178i \(-0.394112\pi\)
0.326554 + 0.945178i \(0.394112\pi\)
\(42\) 0 0
\(43\) −1835.23 −0.151363 −0.0756816 0.997132i \(-0.524113\pi\)
−0.0756816 + 0.997132i \(0.524113\pi\)
\(44\) 0 0
\(45\) 11282.7 0.830578
\(46\) 0 0
\(47\) 1532.68 0.101206 0.0506032 0.998719i \(-0.483886\pi\)
0.0506032 + 0.998719i \(0.483886\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3731.38 −0.200883
\(52\) 0 0
\(53\) −9537.46 −0.466383 −0.233192 0.972431i \(-0.574917\pi\)
−0.233192 + 0.972431i \(0.574917\pi\)
\(54\) 0 0
\(55\) 28091.6 1.25219
\(56\) 0 0
\(57\) −15173.8 −0.618596
\(58\) 0 0
\(59\) 29674.1 1.10981 0.554903 0.831915i \(-0.312755\pi\)
0.554903 + 0.831915i \(0.312755\pi\)
\(60\) 0 0
\(61\) −46510.8 −1.60040 −0.800201 0.599732i \(-0.795274\pi\)
−0.800201 + 0.599732i \(0.795274\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23284.3 0.683566
\(66\) 0 0
\(67\) −26746.1 −0.727902 −0.363951 0.931418i \(-0.618572\pi\)
−0.363951 + 0.931418i \(0.618572\pi\)
\(68\) 0 0
\(69\) 4292.34 0.108535
\(70\) 0 0
\(71\) 14388.8 0.338748 0.169374 0.985552i \(-0.445825\pi\)
0.169374 + 0.985552i \(0.445825\pi\)
\(72\) 0 0
\(73\) −70095.1 −1.53950 −0.769752 0.638343i \(-0.779620\pi\)
−0.769752 + 0.638343i \(0.779620\pi\)
\(74\) 0 0
\(75\) 32743.3 0.672154
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 27085.8 0.488285 0.244143 0.969739i \(-0.421493\pi\)
0.244143 + 0.969739i \(0.421493\pi\)
\(80\) 0 0
\(81\) −4727.49 −0.0800604
\(82\) 0 0
\(83\) 79755.4 1.27076 0.635382 0.772198i \(-0.280843\pi\)
0.635382 + 0.772198i \(0.280843\pi\)
\(84\) 0 0
\(85\) 29542.2 0.443502
\(86\) 0 0
\(87\) −78483.6 −1.11168
\(88\) 0 0
\(89\) 43577.3 0.583157 0.291579 0.956547i \(-0.405820\pi\)
0.291579 + 0.956547i \(0.405820\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 25964.9 0.311300
\(94\) 0 0
\(95\) 120135. 1.36571
\(96\) 0 0
\(97\) 103374. 1.11553 0.557765 0.829999i \(-0.311659\pi\)
0.557765 + 0.829999i \(0.311659\pi\)
\(98\) 0 0
\(99\) 49737.3 0.510028
\(100\) 0 0
\(101\) 28700.2 0.279951 0.139975 0.990155i \(-0.455298\pi\)
0.139975 + 0.990155i \(0.455298\pi\)
\(102\) 0 0
\(103\) −29227.9 −0.271459 −0.135730 0.990746i \(-0.543338\pi\)
−0.135730 + 0.990746i \(0.543338\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −87858.4 −0.741863 −0.370932 0.928660i \(-0.620962\pi\)
−0.370932 + 0.928660i \(0.620962\pi\)
\(108\) 0 0
\(109\) 220628. 1.77867 0.889333 0.457260i \(-0.151169\pi\)
0.889333 + 0.457260i \(0.151169\pi\)
\(110\) 0 0
\(111\) 7457.39 0.0574486
\(112\) 0 0
\(113\) 39665.6 0.292225 0.146113 0.989268i \(-0.453324\pi\)
0.146113 + 0.989268i \(0.453324\pi\)
\(114\) 0 0
\(115\) −33983.5 −0.239620
\(116\) 0 0
\(117\) 41225.8 0.278423
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −37214.9 −0.231075
\(122\) 0 0
\(123\) 70880.2 0.422437
\(124\) 0 0
\(125\) −9774.87 −0.0559546
\(126\) 0 0
\(127\) −51740.3 −0.284655 −0.142328 0.989820i \(-0.545459\pi\)
−0.142328 + 0.989820i \(0.545459\pi\)
\(128\) 0 0
\(129\) −18504.2 −0.0979031
\(130\) 0 0
\(131\) 166674. 0.848572 0.424286 0.905528i \(-0.360525\pi\)
0.424286 + 0.905528i \(0.360525\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 309347. 1.46087
\(136\) 0 0
\(137\) 28259.4 0.128636 0.0643178 0.997929i \(-0.479513\pi\)
0.0643178 + 0.997929i \(0.479513\pi\)
\(138\) 0 0
\(139\) −336393. −1.47676 −0.738380 0.674384i \(-0.764409\pi\)
−0.738380 + 0.674384i \(0.764409\pi\)
\(140\) 0 0
\(141\) 15453.7 0.0654612
\(142\) 0 0
\(143\) 102644. 0.419753
\(144\) 0 0
\(145\) 621373. 2.45433
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −355381. −1.31138 −0.655691 0.755030i \(-0.727622\pi\)
−0.655691 + 0.755030i \(0.727622\pi\)
\(150\) 0 0
\(151\) 358797. 1.28058 0.640290 0.768133i \(-0.278814\pi\)
0.640290 + 0.768133i \(0.278814\pi\)
\(152\) 0 0
\(153\) 52305.7 0.180643
\(154\) 0 0
\(155\) −205570. −0.687275
\(156\) 0 0
\(157\) 458911. 1.48586 0.742932 0.669367i \(-0.233435\pi\)
0.742932 + 0.669367i \(0.233435\pi\)
\(158\) 0 0
\(159\) −96164.0 −0.301661
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −502441. −1.48121 −0.740603 0.671943i \(-0.765460\pi\)
−0.740603 + 0.671943i \(0.765460\pi\)
\(164\) 0 0
\(165\) 283241. 0.809928
\(166\) 0 0
\(167\) 676652. 1.87748 0.938738 0.344632i \(-0.111996\pi\)
0.938738 + 0.344632i \(0.111996\pi\)
\(168\) 0 0
\(169\) −286214. −0.770858
\(170\) 0 0
\(171\) 212703. 0.556267
\(172\) 0 0
\(173\) −249160. −0.632941 −0.316470 0.948602i \(-0.602498\pi\)
−0.316470 + 0.948602i \(0.602498\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 299196. 0.717833
\(178\) 0 0
\(179\) −139258. −0.324853 −0.162427 0.986721i \(-0.551932\pi\)
−0.162427 + 0.986721i \(0.551932\pi\)
\(180\) 0 0
\(181\) 306246. 0.694823 0.347412 0.937713i \(-0.387061\pi\)
0.347412 + 0.937713i \(0.387061\pi\)
\(182\) 0 0
\(183\) −468957. −1.03516
\(184\) 0 0
\(185\) −59041.9 −0.126833
\(186\) 0 0
\(187\) 130231. 0.272339
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −227494. −0.451218 −0.225609 0.974218i \(-0.572437\pi\)
−0.225609 + 0.974218i \(0.572437\pi\)
\(192\) 0 0
\(193\) −672374. −1.29933 −0.649663 0.760223i \(-0.725090\pi\)
−0.649663 + 0.760223i \(0.725090\pi\)
\(194\) 0 0
\(195\) 234770. 0.442137
\(196\) 0 0
\(197\) −1282.76 −0.00235493 −0.00117747 0.999999i \(-0.500375\pi\)
−0.00117747 + 0.999999i \(0.500375\pi\)
\(198\) 0 0
\(199\) −368898. −0.660349 −0.330175 0.943920i \(-0.607108\pi\)
−0.330175 + 0.943920i \(0.607108\pi\)
\(200\) 0 0
\(201\) −269674. −0.470814
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −561175. −0.932640
\(206\) 0 0
\(207\) −60169.0 −0.0975994
\(208\) 0 0
\(209\) 529589. 0.838635
\(210\) 0 0
\(211\) −502168. −0.776503 −0.388251 0.921553i \(-0.626921\pi\)
−0.388251 + 0.921553i \(0.626921\pi\)
\(212\) 0 0
\(213\) 145078. 0.219106
\(214\) 0 0
\(215\) 146502. 0.216147
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −706753. −0.995765
\(220\) 0 0
\(221\) 107945. 0.148669
\(222\) 0 0
\(223\) 1.17328e6 1.57993 0.789967 0.613149i \(-0.210098\pi\)
0.789967 + 0.613149i \(0.210098\pi\)
\(224\) 0 0
\(225\) −458988. −0.604428
\(226\) 0 0
\(227\) 910159. 1.17234 0.586168 0.810189i \(-0.300636\pi\)
0.586168 + 0.810189i \(0.300636\pi\)
\(228\) 0 0
\(229\) 521924. 0.657686 0.328843 0.944385i \(-0.393341\pi\)
0.328843 + 0.944385i \(0.393341\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.04279e6 −1.25836 −0.629182 0.777258i \(-0.716610\pi\)
−0.629182 + 0.777258i \(0.716610\pi\)
\(234\) 0 0
\(235\) −122350. −0.144523
\(236\) 0 0
\(237\) 273100. 0.315828
\(238\) 0 0
\(239\) 1.53447e6 1.73766 0.868830 0.495110i \(-0.164872\pi\)
0.868830 + 0.495110i \(0.164872\pi\)
\(240\) 0 0
\(241\) −1.00758e6 −1.11747 −0.558735 0.829346i \(-0.688713\pi\)
−0.558735 + 0.829346i \(0.688713\pi\)
\(242\) 0 0
\(243\) 894004. 0.971234
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 438961. 0.457808
\(248\) 0 0
\(249\) 804155. 0.821942
\(250\) 0 0
\(251\) −8511.89 −0.00852789 −0.00426394 0.999991i \(-0.501357\pi\)
−0.00426394 + 0.999991i \(0.501357\pi\)
\(252\) 0 0
\(253\) −149809. −0.147142
\(254\) 0 0
\(255\) 297867. 0.286862
\(256\) 0 0
\(257\) −527532. −0.498214 −0.249107 0.968476i \(-0.580137\pi\)
−0.249107 + 0.968476i \(0.580137\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.10016e6 0.999670
\(262\) 0 0
\(263\) 352085. 0.313876 0.156938 0.987608i \(-0.449838\pi\)
0.156938 + 0.987608i \(0.449838\pi\)
\(264\) 0 0
\(265\) 761353. 0.665996
\(266\) 0 0
\(267\) 439380. 0.377192
\(268\) 0 0
\(269\) 479540. 0.404058 0.202029 0.979380i \(-0.435246\pi\)
0.202029 + 0.979380i \(0.435246\pi\)
\(270\) 0 0
\(271\) 977611. 0.808617 0.404308 0.914623i \(-0.367512\pi\)
0.404308 + 0.914623i \(0.367512\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.14279e6 −0.911243
\(276\) 0 0
\(277\) 968723. 0.758578 0.379289 0.925278i \(-0.376169\pi\)
0.379289 + 0.925278i \(0.376169\pi\)
\(278\) 0 0
\(279\) −363970. −0.279934
\(280\) 0 0
\(281\) −318333. −0.240501 −0.120250 0.992744i \(-0.538370\pi\)
−0.120250 + 0.992744i \(0.538370\pi\)
\(282\) 0 0
\(283\) −1.77210e6 −1.31529 −0.657646 0.753327i \(-0.728448\pi\)
−0.657646 + 0.753327i \(0.728448\pi\)
\(284\) 0 0
\(285\) 1.21129e6 0.883356
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.28290e6 −0.903543
\(290\) 0 0
\(291\) 1.04229e6 0.721535
\(292\) 0 0
\(293\) 1.64148e6 1.11703 0.558516 0.829494i \(-0.311371\pi\)
0.558516 + 0.829494i \(0.311371\pi\)
\(294\) 0 0
\(295\) −2.36881e6 −1.58480
\(296\) 0 0
\(297\) 1.36369e6 0.897068
\(298\) 0 0
\(299\) −124172. −0.0803243
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 289377. 0.181075
\(304\) 0 0
\(305\) 3.71285e6 2.28537
\(306\) 0 0
\(307\) 466930. 0.282752 0.141376 0.989956i \(-0.454847\pi\)
0.141376 + 0.989956i \(0.454847\pi\)
\(308\) 0 0
\(309\) −294698. −0.175582
\(310\) 0 0
\(311\) 2.43796e6 1.42931 0.714654 0.699478i \(-0.246584\pi\)
0.714654 + 0.699478i \(0.246584\pi\)
\(312\) 0 0
\(313\) 2.42094e6 1.39676 0.698381 0.715726i \(-0.253904\pi\)
0.698381 + 0.715726i \(0.253904\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.87611e6 1.04860 0.524301 0.851533i \(-0.324327\pi\)
0.524301 + 0.851533i \(0.324327\pi\)
\(318\) 0 0
\(319\) 2.73919e6 1.50711
\(320\) 0 0
\(321\) −885855. −0.479844
\(322\) 0 0
\(323\) 556936. 0.297029
\(324\) 0 0
\(325\) −947225. −0.497445
\(326\) 0 0
\(327\) 2.22454e6 1.15046
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.08310e6 0.543373 0.271686 0.962386i \(-0.412419\pi\)
0.271686 + 0.962386i \(0.412419\pi\)
\(332\) 0 0
\(333\) −104536. −0.0516601
\(334\) 0 0
\(335\) 2.13507e6 1.03944
\(336\) 0 0
\(337\) −2.59465e6 −1.24453 −0.622263 0.782809i \(-0.713786\pi\)
−0.622263 + 0.782809i \(0.713786\pi\)
\(338\) 0 0
\(339\) 399939. 0.189014
\(340\) 0 0
\(341\) −906213. −0.422031
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −342647. −0.154989
\(346\) 0 0
\(347\) 1.87051e6 0.833943 0.416972 0.908920i \(-0.363091\pi\)
0.416972 + 0.908920i \(0.363091\pi\)
\(348\) 0 0
\(349\) −1.61685e6 −0.710568 −0.355284 0.934758i \(-0.615616\pi\)
−0.355284 + 0.934758i \(0.615616\pi\)
\(350\) 0 0
\(351\) 1.13033e6 0.489706
\(352\) 0 0
\(353\) −578305. −0.247013 −0.123507 0.992344i \(-0.539414\pi\)
−0.123507 + 0.992344i \(0.539414\pi\)
\(354\) 0 0
\(355\) −1.14862e6 −0.483733
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.96818e6 0.805988 0.402994 0.915203i \(-0.367970\pi\)
0.402994 + 0.915203i \(0.367970\pi\)
\(360\) 0 0
\(361\) −211299. −0.0853355
\(362\) 0 0
\(363\) −375229. −0.149462
\(364\) 0 0
\(365\) 5.59553e6 2.19841
\(366\) 0 0
\(367\) −2.17452e6 −0.842749 −0.421375 0.906887i \(-0.638452\pi\)
−0.421375 + 0.906887i \(0.638452\pi\)
\(368\) 0 0
\(369\) −993583. −0.379873
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.38476e6 0.515349 0.257675 0.966232i \(-0.417044\pi\)
0.257675 + 0.966232i \(0.417044\pi\)
\(374\) 0 0
\(375\) −98557.7 −0.0361920
\(376\) 0 0
\(377\) 2.27044e6 0.822728
\(378\) 0 0
\(379\) −3.37190e6 −1.20580 −0.602902 0.797815i \(-0.705989\pi\)
−0.602902 + 0.797815i \(0.705989\pi\)
\(380\) 0 0
\(381\) −521685. −0.184118
\(382\) 0 0
\(383\) −3.28060e6 −1.14276 −0.571382 0.820685i \(-0.693592\pi\)
−0.571382 + 0.820685i \(0.693592\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 259388. 0.0880385
\(388\) 0 0
\(389\) −2.94810e6 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(390\) 0 0
\(391\) −157545. −0.0521150
\(392\) 0 0
\(393\) 1.68053e6 0.548865
\(394\) 0 0
\(395\) −2.16219e6 −0.697271
\(396\) 0 0
\(397\) −69270.2 −0.0220582 −0.0110291 0.999939i \(-0.503511\pi\)
−0.0110291 + 0.999939i \(0.503511\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.34786e6 −1.03970 −0.519848 0.854259i \(-0.674012\pi\)
−0.519848 + 0.854259i \(0.674012\pi\)
\(402\) 0 0
\(403\) −751134. −0.230385
\(404\) 0 0
\(405\) 377384. 0.114326
\(406\) 0 0
\(407\) −260274. −0.0778834
\(408\) 0 0
\(409\) 2.91217e6 0.860812 0.430406 0.902636i \(-0.358370\pi\)
0.430406 + 0.902636i \(0.358370\pi\)
\(410\) 0 0
\(411\) 284933. 0.0832028
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.36669e6 −1.81465
\(416\) 0 0
\(417\) −3.39177e6 −0.955183
\(418\) 0 0
\(419\) 4.62361e6 1.28661 0.643304 0.765611i \(-0.277563\pi\)
0.643304 + 0.765611i \(0.277563\pi\)
\(420\) 0 0
\(421\) −2.63042e6 −0.723303 −0.361652 0.932313i \(-0.617787\pi\)
−0.361652 + 0.932313i \(0.617787\pi\)
\(422\) 0 0
\(423\) −216626. −0.0588654
\(424\) 0 0
\(425\) −1.20180e6 −0.322746
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.03494e6 0.271500
\(430\) 0 0
\(431\) −7.54128e6 −1.95547 −0.977736 0.209837i \(-0.932707\pi\)
−0.977736 + 0.209837i \(0.932707\pi\)
\(432\) 0 0
\(433\) 5.83558e6 1.49577 0.747883 0.663830i \(-0.231070\pi\)
0.747883 + 0.663830i \(0.231070\pi\)
\(434\) 0 0
\(435\) 6.26516e6 1.58748
\(436\) 0 0
\(437\) −640663. −0.160482
\(438\) 0 0
\(439\) −168104. −0.0416310 −0.0208155 0.999783i \(-0.506626\pi\)
−0.0208155 + 0.999783i \(0.506626\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.84151e6 0.687924 0.343962 0.938984i \(-0.388231\pi\)
0.343962 + 0.938984i \(0.388231\pi\)
\(444\) 0 0
\(445\) −3.47867e6 −0.832748
\(446\) 0 0
\(447\) −3.58323e6 −0.848214
\(448\) 0 0
\(449\) −1.41567e6 −0.331396 −0.165698 0.986177i \(-0.552988\pi\)
−0.165698 + 0.986177i \(0.552988\pi\)
\(450\) 0 0
\(451\) −2.47382e6 −0.572701
\(452\) 0 0
\(453\) 3.61767e6 0.828291
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.55727e6 0.348799 0.174399 0.984675i \(-0.444202\pi\)
0.174399 + 0.984675i \(0.444202\pi\)
\(458\) 0 0
\(459\) 1.43411e6 0.317725
\(460\) 0 0
\(461\) −4.45345e6 −0.975987 −0.487994 0.872847i \(-0.662271\pi\)
−0.487994 + 0.872847i \(0.662271\pi\)
\(462\) 0 0
\(463\) −4.92263e6 −1.06720 −0.533599 0.845738i \(-0.679161\pi\)
−0.533599 + 0.845738i \(0.679161\pi\)
\(464\) 0 0
\(465\) −2.07271e6 −0.444536
\(466\) 0 0
\(467\) −5.09090e6 −1.08020 −0.540098 0.841602i \(-0.681613\pi\)
−0.540098 + 0.841602i \(0.681613\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.62709e6 0.961071
\(472\) 0 0
\(473\) 645825. 0.132728
\(474\) 0 0
\(475\) −4.88717e6 −0.993856
\(476\) 0 0
\(477\) 1.34800e6 0.271266
\(478\) 0 0
\(479\) 8.30085e6 1.65304 0.826521 0.562907i \(-0.190317\pi\)
0.826521 + 0.562907i \(0.190317\pi\)
\(480\) 0 0
\(481\) −215734. −0.0425163
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.25208e6 −1.59298
\(486\) 0 0
\(487\) −8.63401e6 −1.64964 −0.824822 0.565392i \(-0.808725\pi\)
−0.824822 + 0.565392i \(0.808725\pi\)
\(488\) 0 0
\(489\) −5.06599e6 −0.958059
\(490\) 0 0
\(491\) −95039.5 −0.0177910 −0.00889550 0.999960i \(-0.502832\pi\)
−0.00889550 + 0.999960i \(0.502832\pi\)
\(492\) 0 0
\(493\) 2.88064e6 0.533792
\(494\) 0 0
\(495\) −3.97041e6 −0.728320
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.14203e6 −0.385101 −0.192551 0.981287i \(-0.561676\pi\)
−0.192551 + 0.981287i \(0.561676\pi\)
\(500\) 0 0
\(501\) 6.82252e6 1.21437
\(502\) 0 0
\(503\) −5.24794e6 −0.924844 −0.462422 0.886660i \(-0.653019\pi\)
−0.462422 + 0.886660i \(0.653019\pi\)
\(504\) 0 0
\(505\) −2.29107e6 −0.399769
\(506\) 0 0
\(507\) −2.88583e6 −0.498598
\(508\) 0 0
\(509\) −1.05891e7 −1.81160 −0.905802 0.423702i \(-0.860730\pi\)
−0.905802 + 0.423702i \(0.860730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.83187e6 0.978395
\(514\) 0 0
\(515\) 2.33319e6 0.387644
\(516\) 0 0
\(517\) −539357. −0.0887462
\(518\) 0 0
\(519\) −2.51222e6 −0.409392
\(520\) 0 0
\(521\) −4.54465e6 −0.733510 −0.366755 0.930318i \(-0.619531\pi\)
−0.366755 + 0.930318i \(0.619531\pi\)
\(522\) 0 0
\(523\) 5.27197e6 0.842789 0.421394 0.906877i \(-0.361541\pi\)
0.421394 + 0.906877i \(0.361541\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −953009. −0.149476
\(528\) 0 0
\(529\) −6.25511e6 −0.971843
\(530\) 0 0
\(531\) −4.19407e6 −0.645504
\(532\) 0 0
\(533\) −2.05048e6 −0.312635
\(534\) 0 0
\(535\) 7.01353e6 1.05938
\(536\) 0 0
\(537\) −1.40410e6 −0.210118
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.93445e6 0.871741 0.435871 0.900009i \(-0.356441\pi\)
0.435871 + 0.900009i \(0.356441\pi\)
\(542\) 0 0
\(543\) 3.08781e6 0.449418
\(544\) 0 0
\(545\) −1.76122e7 −2.53994
\(546\) 0 0
\(547\) 8.82017e6 1.26040 0.630200 0.776433i \(-0.282973\pi\)
0.630200 + 0.776433i \(0.282973\pi\)
\(548\) 0 0
\(549\) 6.57374e6 0.930854
\(550\) 0 0
\(551\) 1.17142e7 1.64375
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −595306. −0.0820366
\(556\) 0 0
\(557\) −1.18224e6 −0.161461 −0.0807304 0.996736i \(-0.525725\pi\)
−0.0807304 + 0.996736i \(0.525725\pi\)
\(558\) 0 0
\(559\) 535306. 0.0724556
\(560\) 0 0
\(561\) 1.31309e6 0.176151
\(562\) 0 0
\(563\) −4.07741e6 −0.542142 −0.271071 0.962559i \(-0.587378\pi\)
−0.271071 + 0.962559i \(0.587378\pi\)
\(564\) 0 0
\(565\) −3.16641e6 −0.417298
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.17615e6 1.05869 0.529344 0.848407i \(-0.322438\pi\)
0.529344 + 0.848407i \(0.322438\pi\)
\(570\) 0 0
\(571\) −3.30615e6 −0.424357 −0.212179 0.977231i \(-0.568056\pi\)
−0.212179 + 0.977231i \(0.568056\pi\)
\(572\) 0 0
\(573\) −2.29377e6 −0.291852
\(574\) 0 0
\(575\) 1.38247e6 0.174376
\(576\) 0 0
\(577\) −7.14994e6 −0.894052 −0.447026 0.894521i \(-0.647517\pi\)
−0.447026 + 0.894521i \(0.647517\pi\)
\(578\) 0 0
\(579\) −6.77939e6 −0.840416
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.35627e6 0.408964
\(584\) 0 0
\(585\) −3.29096e6 −0.397588
\(586\) 0 0
\(587\) −9.69191e6 −1.16095 −0.580476 0.814277i \(-0.697133\pi\)
−0.580476 + 0.814277i \(0.697133\pi\)
\(588\) 0 0
\(589\) −3.87545e6 −0.460292
\(590\) 0 0
\(591\) −12933.7 −0.00152319
\(592\) 0 0
\(593\) 6.63960e6 0.775363 0.387682 0.921793i \(-0.373276\pi\)
0.387682 + 0.921793i \(0.373276\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.71951e6 −0.427120
\(598\) 0 0
\(599\) 3.24191e6 0.369177 0.184588 0.982816i \(-0.440905\pi\)
0.184588 + 0.982816i \(0.440905\pi\)
\(600\) 0 0
\(601\) −5.65076e6 −0.638147 −0.319074 0.947730i \(-0.603372\pi\)
−0.319074 + 0.947730i \(0.603372\pi\)
\(602\) 0 0
\(603\) 3.78023e6 0.423375
\(604\) 0 0
\(605\) 2.97078e6 0.329975
\(606\) 0 0
\(607\) −235674. −0.0259621 −0.0129811 0.999916i \(-0.504132\pi\)
−0.0129811 + 0.999916i \(0.504132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −447057. −0.0484462
\(612\) 0 0
\(613\) 788877. 0.0847926 0.0423963 0.999101i \(-0.486501\pi\)
0.0423963 + 0.999101i \(0.486501\pi\)
\(614\) 0 0
\(615\) −5.65820e6 −0.603240
\(616\) 0 0
\(617\) 1.67739e7 1.77387 0.886935 0.461894i \(-0.152830\pi\)
0.886935 + 0.461894i \(0.152830\pi\)
\(618\) 0 0
\(619\) −8.22300e6 −0.862588 −0.431294 0.902211i \(-0.641943\pi\)
−0.431294 + 0.902211i \(0.641943\pi\)
\(620\) 0 0
\(621\) −1.64971e6 −0.171664
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.36798e6 −0.959281
\(626\) 0 0
\(627\) 5.33972e6 0.542437
\(628\) 0 0
\(629\) −273714. −0.0275849
\(630\) 0 0
\(631\) 5.94507e6 0.594406 0.297203 0.954814i \(-0.403946\pi\)
0.297203 + 0.954814i \(0.403946\pi\)
\(632\) 0 0
\(633\) −5.06324e6 −0.502249
\(634\) 0 0
\(635\) 4.13030e6 0.406488
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.03368e6 −0.197029
\(640\) 0 0
\(641\) −1.06761e7 −1.02628 −0.513141 0.858304i \(-0.671518\pi\)
−0.513141 + 0.858304i \(0.671518\pi\)
\(642\) 0 0
\(643\) 3.13159e6 0.298701 0.149351 0.988784i \(-0.452282\pi\)
0.149351 + 0.988784i \(0.452282\pi\)
\(644\) 0 0
\(645\) 1.47715e6 0.139806
\(646\) 0 0
\(647\) 4.93457e6 0.463435 0.231717 0.972783i \(-0.425566\pi\)
0.231717 + 0.972783i \(0.425566\pi\)
\(648\) 0 0
\(649\) −1.04424e7 −0.973170
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.72224e6 0.525150 0.262575 0.964912i \(-0.415428\pi\)
0.262575 + 0.964912i \(0.415428\pi\)
\(654\) 0 0
\(655\) −1.33052e7 −1.21176
\(656\) 0 0
\(657\) 9.90710e6 0.895433
\(658\) 0 0
\(659\) −362477. −0.0325137 −0.0162569 0.999868i \(-0.505175\pi\)
−0.0162569 + 0.999868i \(0.505175\pi\)
\(660\) 0 0
\(661\) −1.91211e7 −1.70219 −0.851096 0.525011i \(-0.824061\pi\)
−0.851096 + 0.525011i \(0.824061\pi\)
\(662\) 0 0
\(663\) 1.08838e6 0.0961604
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.31370e6 −0.288403
\(668\) 0 0
\(669\) 1.18299e7 1.02192
\(670\) 0 0
\(671\) 1.63673e7 1.40337
\(672\) 0 0
\(673\) −573374. −0.0487978 −0.0243989 0.999702i \(-0.507767\pi\)
−0.0243989 + 0.999702i \(0.507767\pi\)
\(674\) 0 0
\(675\) −1.25845e7 −1.06310
\(676\) 0 0
\(677\) 1.16903e7 0.980291 0.490146 0.871641i \(-0.336944\pi\)
0.490146 + 0.871641i \(0.336944\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.17691e6 0.758279
\(682\) 0 0
\(683\) −1.83674e7 −1.50659 −0.753297 0.657681i \(-0.771538\pi\)
−0.753297 + 0.657681i \(0.771538\pi\)
\(684\) 0 0
\(685\) −2.25588e6 −0.183692
\(686\) 0 0
\(687\) 5.26244e6 0.425398
\(688\) 0 0
\(689\) 2.78191e6 0.223252
\(690\) 0 0
\(691\) −2.35611e7 −1.87716 −0.938579 0.345066i \(-0.887857\pi\)
−0.938579 + 0.345066i \(0.887857\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.68535e7 2.10881
\(696\) 0 0
\(697\) −2.60157e6 −0.202840
\(698\) 0 0
\(699\) −1.05142e7 −0.813921
\(700\) 0 0
\(701\) 1.32980e7 1.02210 0.511048 0.859552i \(-0.329257\pi\)
0.511048 + 0.859552i \(0.329257\pi\)
\(702\) 0 0
\(703\) −1.11307e6 −0.0849442
\(704\) 0 0
\(705\) −1.23363e6 −0.0934786
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.85353e6 −0.512034 −0.256017 0.966672i \(-0.582410\pi\)
−0.256017 + 0.966672i \(0.582410\pi\)
\(710\) 0 0
\(711\) −3.82825e6 −0.284005
\(712\) 0 0
\(713\) 1.09628e6 0.0807602
\(714\) 0 0
\(715\) −8.19384e6 −0.599408
\(716\) 0 0
\(717\) 1.54717e7 1.12394
\(718\) 0 0
\(719\) −2.65729e7 −1.91698 −0.958490 0.285127i \(-0.907964\pi\)
−0.958490 + 0.285127i \(0.907964\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.01592e7 −0.722791
\(724\) 0 0
\(725\) −2.52779e7 −1.78606
\(726\) 0 0
\(727\) −2.16991e6 −0.152267 −0.0761335 0.997098i \(-0.524258\pi\)
−0.0761335 + 0.997098i \(0.524258\pi\)
\(728\) 0 0
\(729\) 1.01628e7 0.708264
\(730\) 0 0
\(731\) 679174. 0.0470097
\(732\) 0 0
\(733\) −1.74653e7 −1.20065 −0.600324 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.41203e6 0.638285
\(738\) 0 0
\(739\) 1.36461e7 0.919172 0.459586 0.888133i \(-0.347998\pi\)
0.459586 + 0.888133i \(0.347998\pi\)
\(740\) 0 0
\(741\) 4.42594e6 0.296114
\(742\) 0 0
\(743\) −1.48965e7 −0.989944 −0.494972 0.868909i \(-0.664822\pi\)
−0.494972 + 0.868909i \(0.664822\pi\)
\(744\) 0 0
\(745\) 2.83692e7 1.87265
\(746\) 0 0
\(747\) −1.12725e7 −0.739124
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.53463e7 1.63989 0.819944 0.572443i \(-0.194004\pi\)
0.819944 + 0.572443i \(0.194004\pi\)
\(752\) 0 0
\(753\) −85823.3 −0.00551592
\(754\) 0 0
\(755\) −2.86419e7 −1.82867
\(756\) 0 0
\(757\) −2.66725e7 −1.69170 −0.845852 0.533417i \(-0.820908\pi\)
−0.845852 + 0.533417i \(0.820908\pi\)
\(758\) 0 0
\(759\) −1.51049e6 −0.0951729
\(760\) 0 0
\(761\) 579829. 0.0362943 0.0181471 0.999835i \(-0.494223\pi\)
0.0181471 + 0.999835i \(0.494223\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.17544e6 −0.257958
\(766\) 0 0
\(767\) −8.65541e6 −0.531250
\(768\) 0 0
\(769\) 1.52438e7 0.929562 0.464781 0.885426i \(-0.346133\pi\)
0.464781 + 0.885426i \(0.346133\pi\)
\(770\) 0 0
\(771\) −5.31898e6 −0.322250
\(772\) 0 0
\(773\) −1.94926e7 −1.17333 −0.586665 0.809830i \(-0.699559\pi\)
−0.586665 + 0.809830i \(0.699559\pi\)
\(774\) 0 0
\(775\) 8.36275e6 0.500144
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.05794e7 −0.624621
\(780\) 0 0
\(781\) −5.06345e6 −0.297043
\(782\) 0 0
\(783\) 3.01642e7 1.75828
\(784\) 0 0
\(785\) −3.66337e7 −2.12181
\(786\) 0 0
\(787\) −1.36277e7 −0.784305 −0.392153 0.919900i \(-0.628270\pi\)
−0.392153 + 0.919900i \(0.628270\pi\)
\(788\) 0 0
\(789\) 3.54999e6 0.203018
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.35664e7 0.766093
\(794\) 0 0
\(795\) 7.67654e6 0.430772
\(796\) 0 0
\(797\) 3.62853e6 0.202341 0.101171 0.994869i \(-0.467741\pi\)
0.101171 + 0.994869i \(0.467741\pi\)
\(798\) 0 0
\(799\) −567208. −0.0314323
\(800\) 0 0
\(801\) −6.15913e6 −0.339186
\(802\) 0 0
\(803\) 2.46667e7 1.34997
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.83509e6 0.261349
\(808\) 0 0
\(809\) 2.98780e7 1.60502 0.802509 0.596640i \(-0.203498\pi\)
0.802509 + 0.596640i \(0.203498\pi\)
\(810\) 0 0
\(811\) 1.02643e6 0.0547995 0.0273998 0.999625i \(-0.491277\pi\)
0.0273998 + 0.999625i \(0.491277\pi\)
\(812\) 0 0
\(813\) 9.85702e6 0.523021
\(814\) 0 0
\(815\) 4.01086e7 2.11516
\(816\) 0 0
\(817\) 2.76189e6 0.144761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.15062e7 −0.595766 −0.297883 0.954602i \(-0.596281\pi\)
−0.297883 + 0.954602i \(0.596281\pi\)
\(822\) 0 0
\(823\) 2.51210e7 1.29282 0.646408 0.762992i \(-0.276270\pi\)
0.646408 + 0.762992i \(0.276270\pi\)
\(824\) 0 0
\(825\) −1.15225e7 −0.589401
\(826\) 0 0
\(827\) 2.14447e6 0.109032 0.0545162 0.998513i \(-0.482638\pi\)
0.0545162 + 0.998513i \(0.482638\pi\)
\(828\) 0 0
\(829\) 818065. 0.0413430 0.0206715 0.999786i \(-0.493420\pi\)
0.0206715 + 0.999786i \(0.493420\pi\)
\(830\) 0 0
\(831\) 9.76741e6 0.490656
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −5.40155e7 −2.68104
\(836\) 0 0
\(837\) −9.97929e6 −0.492364
\(838\) 0 0
\(839\) 2.11279e7 1.03622 0.518110 0.855314i \(-0.326636\pi\)
0.518110 + 0.855314i \(0.326636\pi\)
\(840\) 0 0
\(841\) 4.00785e7 1.95398
\(842\) 0 0
\(843\) −3.20968e6 −0.155558
\(844\) 0 0
\(845\) 2.28478e7 1.10079
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.78677e7 −0.850744
\(850\) 0 0
\(851\) 314863. 0.0149038
\(852\) 0 0
\(853\) −1.89000e7 −0.889386 −0.444693 0.895683i \(-0.646687\pi\)
−0.444693 + 0.895683i \(0.646687\pi\)
\(854\) 0 0
\(855\) −1.69796e7 −0.794349
\(856\) 0 0
\(857\) 3.72286e7 1.73151 0.865753 0.500471i \(-0.166840\pi\)
0.865753 + 0.500471i \(0.166840\pi\)
\(858\) 0 0
\(859\) −3.02064e6 −0.139674 −0.0698371 0.997558i \(-0.522248\pi\)
−0.0698371 + 0.997558i \(0.522248\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.71843e7 1.24248 0.621242 0.783619i \(-0.286629\pi\)
0.621242 + 0.783619i \(0.286629\pi\)
\(864\) 0 0
\(865\) 1.98899e7 0.903839
\(866\) 0 0
\(867\) −1.29352e7 −0.584420
\(868\) 0 0
\(869\) −9.53158e6 −0.428169
\(870\) 0 0
\(871\) 7.80136e6 0.348438
\(872\) 0 0
\(873\) −1.46106e7 −0.648834
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.73868e7 0.763344 0.381672 0.924298i \(-0.375348\pi\)
0.381672 + 0.924298i \(0.375348\pi\)
\(878\) 0 0
\(879\) 1.65506e7 0.722507
\(880\) 0 0
\(881\) −8.14472e6 −0.353538 −0.176769 0.984252i \(-0.556565\pi\)
−0.176769 + 0.984252i \(0.556565\pi\)
\(882\) 0 0
\(883\) 3.10298e7 1.33930 0.669649 0.742678i \(-0.266445\pi\)
0.669649 + 0.742678i \(0.266445\pi\)
\(884\) 0 0
\(885\) −2.38841e7 −1.02507
\(886\) 0 0
\(887\) 1.47028e7 0.627465 0.313733 0.949511i \(-0.398420\pi\)
0.313733 + 0.949511i \(0.398420\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.66362e6 0.0702037
\(892\) 0 0
\(893\) −2.30657e6 −0.0967918
\(894\) 0 0
\(895\) 1.11166e7 0.463890
\(896\) 0 0
\(897\) −1.25200e6 −0.0519545
\(898\) 0 0
\(899\) −2.00450e7 −0.827193
\(900\) 0 0
\(901\) 3.52958e6 0.144848
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.44469e7 −0.992208
\(906\) 0 0
\(907\) −1.30940e7 −0.528512 −0.264256 0.964453i \(-0.585126\pi\)
−0.264256 + 0.964453i \(0.585126\pi\)
\(908\) 0 0
\(909\) −4.05643e6 −0.162830
\(910\) 0 0
\(911\) −2.68695e7 −1.07266 −0.536332 0.844007i \(-0.680191\pi\)
−0.536332 + 0.844007i \(0.680191\pi\)
\(912\) 0 0
\(913\) −2.80662e7 −1.11431
\(914\) 0 0
\(915\) 3.74357e7 1.47820
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.27317e6 −0.0497277 −0.0248638 0.999691i \(-0.507915\pi\)
−0.0248638 + 0.999691i \(0.507915\pi\)
\(920\) 0 0
\(921\) 4.70795e6 0.182887
\(922\) 0 0
\(923\) −4.19695e6 −0.162155
\(924\) 0 0
\(925\) 2.40187e6 0.0922986
\(926\) 0 0
\(927\) 4.13101e6 0.157891
\(928\) 0 0
\(929\) −9.31705e6 −0.354192 −0.177096 0.984194i \(-0.556670\pi\)
−0.177096 + 0.984194i \(0.556670\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.45814e7 0.924490
\(934\) 0 0
\(935\) −1.03960e7 −0.388900
\(936\) 0 0
\(937\) 1.18158e7 0.439657 0.219829 0.975539i \(-0.429450\pi\)
0.219829 + 0.975539i \(0.429450\pi\)
\(938\) 0 0
\(939\) 2.44097e7 0.903439
\(940\) 0 0
\(941\) −2.53529e7 −0.933371 −0.466685 0.884423i \(-0.654552\pi\)
−0.466685 + 0.884423i \(0.654552\pi\)
\(942\) 0 0
\(943\) 2.99268e6 0.109592
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.64941e7 −0.960008 −0.480004 0.877266i \(-0.659365\pi\)
−0.480004 + 0.877266i \(0.659365\pi\)
\(948\) 0 0
\(949\) 2.04455e7 0.736941
\(950\) 0 0
\(951\) 1.89164e7 0.678246
\(952\) 0 0
\(953\) −1.88335e7 −0.671735 −0.335868 0.941909i \(-0.609030\pi\)
−0.335868 + 0.941909i \(0.609030\pi\)
\(954\) 0 0
\(955\) 1.81603e7 0.644339
\(956\) 0 0
\(957\) 2.76186e7 0.974816
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.19976e7 −0.768365
\(962\) 0 0
\(963\) 1.24177e7 0.431495
\(964\) 0 0
\(965\) 5.36740e7 1.85544
\(966\) 0 0
\(967\) 3.14956e7 1.08314 0.541569 0.840656i \(-0.317831\pi\)
0.541569 + 0.840656i \(0.317831\pi\)
\(968\) 0 0
\(969\) 5.61545e6 0.192121
\(970\) 0 0
\(971\) 6.85669e6 0.233381 0.116691 0.993168i \(-0.462771\pi\)
0.116691 + 0.993168i \(0.462771\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −9.55064e6 −0.321752
\(976\) 0 0
\(977\) 2.81471e7 0.943402 0.471701 0.881759i \(-0.343640\pi\)
0.471701 + 0.881759i \(0.343640\pi\)
\(978\) 0 0
\(979\) −1.53350e7 −0.511361
\(980\) 0 0
\(981\) −3.11831e7 −1.03454
\(982\) 0 0
\(983\) −2.34916e7 −0.775406 −0.387703 0.921784i \(-0.626731\pi\)
−0.387703 + 0.921784i \(0.626731\pi\)
\(984\) 0 0
\(985\) 102399. 0.00336285
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −781278. −0.0253989
\(990\) 0 0
\(991\) 2.14412e6 0.0693530 0.0346765 0.999399i \(-0.488960\pi\)
0.0346765 + 0.999399i \(0.488960\pi\)
\(992\) 0 0
\(993\) 1.09206e7 0.351459
\(994\) 0 0
\(995\) 2.94483e7 0.942979
\(996\) 0 0
\(997\) 2.50872e7 0.799307 0.399654 0.916666i \(-0.369130\pi\)
0.399654 + 0.916666i \(0.369130\pi\)
\(998\) 0 0
\(999\) −2.86616e6 −0.0908628
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 784.6.a.ba.1.2 2
4.3 odd 2 49.6.a.d.1.2 2
7.2 even 3 112.6.i.c.81.1 4
7.4 even 3 112.6.i.c.65.1 4
7.6 odd 2 784.6.a.t.1.1 2
12.11 even 2 441.6.a.n.1.1 2
28.3 even 6 49.6.c.f.30.1 4
28.11 odd 6 7.6.c.a.2.1 4
28.19 even 6 49.6.c.f.18.1 4
28.23 odd 6 7.6.c.a.4.1 yes 4
28.27 even 2 49.6.a.e.1.2 2
84.11 even 6 63.6.e.d.37.2 4
84.23 even 6 63.6.e.d.46.2 4
84.83 odd 2 441.6.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.1 4 28.11 odd 6
7.6.c.a.4.1 yes 4 28.23 odd 6
49.6.a.d.1.2 2 4.3 odd 2
49.6.a.e.1.2 2 28.27 even 2
49.6.c.f.18.1 4 28.19 even 6
49.6.c.f.30.1 4 28.3 even 6
63.6.e.d.37.2 4 84.11 even 6
63.6.e.d.46.2 4 84.23 even 6
112.6.i.c.65.1 4 7.4 even 3
112.6.i.c.81.1 4 7.2 even 3
441.6.a.m.1.1 2 84.83 odd 2
441.6.a.n.1.1 2 12.11 even 2
784.6.a.t.1.1 2 7.6 odd 2
784.6.a.ba.1.2 2 1.1 even 1 trivial