Properties

Label 684.6.k.f.577.6
Level $684$
Weight $6$
Character 684.577
Analytic conductor $109.703$
Analytic rank $0$
Dimension $18$
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] = [684,6,Mod(505,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.505");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\), degree \(2\), 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: \(109.702532752\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + \cdots + 55\!\cdots\!41 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.6
Root \(19.0963 - 31.3437i\) of defining polynomial
Character \(\chi\) \(=\) 684.577
Dual form 684.6.k.f.505.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(18.0963 + 31.3437i) q^{5} -208.885 q^{7} +O(q^{10})\) \(q+(18.0963 + 31.3437i) q^{5} -208.885 q^{7} +204.444 q^{11} +(-146.736 + 254.155i) q^{13} +(850.109 + 1472.43i) q^{17} +(1329.36 + 841.960i) q^{19} +(1768.22 - 3062.65i) q^{23} +(907.547 - 1571.92i) q^{25} +(3204.96 - 5551.15i) q^{29} -2731.24 q^{31} +(-3780.05 - 6547.24i) q^{35} -5427.82 q^{37} +(3692.75 + 6396.03i) q^{41} +(7870.13 + 13631.5i) q^{43} +(-9723.97 + 16842.4i) q^{47} +26826.0 q^{49} +(4290.51 - 7431.39i) q^{53} +(3699.69 + 6408.05i) q^{55} +(-13969.1 - 24195.1i) q^{59} +(-15366.2 + 26615.0i) q^{61} -10621.5 q^{65} +(-24126.9 + 41788.9i) q^{67} +(3430.92 + 5942.53i) q^{71} +(19300.4 + 33429.3i) q^{73} -42705.4 q^{77} +(-32155.8 - 55695.5i) q^{79} -71525.2 q^{83} +(-30767.7 + 53291.2i) q^{85} +(23256.6 - 40281.7i) q^{89} +(30651.0 - 53089.2i) q^{91} +(-2333.64 + 56903.5i) q^{95} +(-45206.3 - 78299.6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 11 q^{5} + 336 q^{7} + 320 q^{11} + 227 q^{13} - 179 q^{17} - 868 q^{19} + 3425 q^{23} - 7054 q^{25} + 7349 q^{29} - 9960 q^{31} - 15888 q^{35} + 26444 q^{37} + 7311 q^{41} - 8283 q^{43} - 37603 q^{47} + 124738 q^{49} + 20337 q^{53} + 716 q^{55} + 74455 q^{59} - 7569 q^{61} - 188998 q^{65} - 26177 q^{67} + 53463 q^{71} - 14103 q^{73} + 31960 q^{77} + 31825 q^{79} - 82600 q^{83} - 50787 q^{85} + 155197 q^{89} - 2800 q^{91} - 49315 q^{95} + 111241 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 18.0963 + 31.3437i 0.323717 + 0.560694i 0.981252 0.192730i \(-0.0617342\pi\)
−0.657535 + 0.753424i \(0.728401\pi\)
\(6\) 0 0
\(7\) −208.885 −1.61125 −0.805624 0.592427i \(-0.798170\pi\)
−0.805624 + 0.592427i \(0.798170\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 204.444 0.509440 0.254720 0.967015i \(-0.418017\pi\)
0.254720 + 0.967015i \(0.418017\pi\)
\(12\) 0 0
\(13\) −146.736 + 254.155i −0.240813 + 0.417100i −0.960946 0.276736i \(-0.910747\pi\)
0.720133 + 0.693836i \(0.244081\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 850.109 + 1472.43i 0.713432 + 1.23570i 0.963561 + 0.267488i \(0.0861935\pi\)
−0.250130 + 0.968212i \(0.580473\pi\)
\(18\) 0 0
\(19\) 1329.36 + 841.960i 0.844810 + 0.535066i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1768.22 3062.65i 0.696975 1.20720i −0.272535 0.962146i \(-0.587862\pi\)
0.969510 0.245051i \(-0.0788046\pi\)
\(24\) 0 0
\(25\) 907.547 1571.92i 0.290415 0.503013i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3204.96 5551.15i 0.707664 1.22571i −0.258058 0.966130i \(-0.583082\pi\)
0.965722 0.259580i \(-0.0835842\pi\)
\(30\) 0 0
\(31\) −2731.24 −0.510452 −0.255226 0.966881i \(-0.582150\pi\)
−0.255226 + 0.966881i \(0.582150\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3780.05 6547.24i −0.521588 0.903417i
\(36\) 0 0
\(37\) −5427.82 −0.651811 −0.325905 0.945402i \(-0.605669\pi\)
−0.325905 + 0.945402i \(0.605669\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3692.75 + 6396.03i 0.343076 + 0.594225i 0.985002 0.172541i \(-0.0551979\pi\)
−0.641926 + 0.766766i \(0.721865\pi\)
\(42\) 0 0
\(43\) 7870.13 + 13631.5i 0.649099 + 1.12427i 0.983339 + 0.181784i \(0.0581871\pi\)
−0.334240 + 0.942488i \(0.608480\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9723.97 + 16842.4i −0.642094 + 1.11214i 0.342870 + 0.939383i \(0.388601\pi\)
−0.984964 + 0.172757i \(0.944732\pi\)
\(48\) 0 0
\(49\) 26826.0 1.59612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4290.51 7431.39i 0.209807 0.363396i −0.741847 0.670569i \(-0.766050\pi\)
0.951654 + 0.307173i \(0.0993832\pi\)
\(54\) 0 0
\(55\) 3699.69 + 6408.05i 0.164914 + 0.285640i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13969.1 24195.1i −0.522441 0.904894i −0.999659 0.0261092i \(-0.991688\pi\)
0.477218 0.878785i \(-0.341645\pi\)
\(60\) 0 0
\(61\) −15366.2 + 26615.0i −0.528739 + 0.915803i 0.470700 + 0.882294i \(0.344002\pi\)
−0.999438 + 0.0335090i \(0.989332\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10621.5 −0.311820
\(66\) 0 0
\(67\) −24126.9 + 41788.9i −0.656619 + 1.13730i 0.324866 + 0.945760i \(0.394681\pi\)
−0.981485 + 0.191538i \(0.938652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3430.92 + 5942.53i 0.0807727 + 0.139902i 0.903582 0.428415i \(-0.140928\pi\)
−0.822809 + 0.568317i \(0.807595\pi\)
\(72\) 0 0
\(73\) 19300.4 + 33429.3i 0.423897 + 0.734210i 0.996317 0.0857501i \(-0.0273287\pi\)
−0.572420 + 0.819961i \(0.693995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −42705.4 −0.820834
\(78\) 0 0
\(79\) −32155.8 55695.5i −0.579684 1.00404i −0.995515 0.0946008i \(-0.969843\pi\)
0.415831 0.909442i \(-0.363491\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −71525.2 −1.13963 −0.569815 0.821773i \(-0.692985\pi\)
−0.569815 + 0.821773i \(0.692985\pi\)
\(84\) 0 0
\(85\) −30767.7 + 53291.2i −0.461900 + 0.800034i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 23256.6 40281.7i 0.311223 0.539054i −0.667404 0.744696i \(-0.732595\pi\)
0.978627 + 0.205641i \(0.0659280\pi\)
\(90\) 0 0
\(91\) 30651.0 53089.2i 0.388009 0.672051i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2333.64 + 56903.5i −0.0265292 + 0.646890i
\(96\) 0 0
\(97\) −45206.3 78299.6i −0.487831 0.844949i 0.512071 0.858943i \(-0.328879\pi\)
−0.999902 + 0.0139945i \(0.995545\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −73036.4 + 126503.i −0.712420 + 1.23395i 0.251527 + 0.967850i \(0.419067\pi\)
−0.963946 + 0.266097i \(0.914266\pi\)
\(102\) 0 0
\(103\) −85248.6 −0.791761 −0.395880 0.918302i \(-0.629561\pi\)
−0.395880 + 0.918302i \(0.629561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 28226.4 0.238339 0.119170 0.992874i \(-0.461977\pi\)
0.119170 + 0.992874i \(0.461977\pi\)
\(108\) 0 0
\(109\) 109018. + 188825.i 0.878888 + 1.52228i 0.852563 + 0.522625i \(0.175047\pi\)
0.0263250 + 0.999653i \(0.491620\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 174546. 1.28592 0.642958 0.765901i \(-0.277707\pi\)
0.642958 + 0.765901i \(0.277707\pi\)
\(114\) 0 0
\(115\) 127993. 0.902490
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −177575. 307569.i −1.14952 1.99102i
\(120\) 0 0
\(121\) −119254. −0.740471
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 178795. 1.02348
\(126\) 0 0
\(127\) −71846.8 + 124442.i −0.395274 + 0.684634i −0.993136 0.116965i \(-0.962684\pi\)
0.597862 + 0.801599i \(0.296017\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2867.13 4966.01i −0.0145972 0.0252830i 0.858635 0.512588i \(-0.171313\pi\)
−0.873232 + 0.487305i \(0.837980\pi\)
\(132\) 0 0
\(133\) −277684. 175873.i −1.36120 0.862125i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −150187. + 260131.i −0.683645 + 1.18411i 0.290216 + 0.956961i \(0.406273\pi\)
−0.973861 + 0.227146i \(0.927061\pi\)
\(138\) 0 0
\(139\) 218624. 378668.i 0.959757 1.66235i 0.236670 0.971590i \(-0.423944\pi\)
0.723087 0.690757i \(-0.242723\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −29999.4 + 51960.5i −0.122680 + 0.212487i
\(144\) 0 0
\(145\) 231992. 0.916331
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4540.46 7864.31i −0.0167546 0.0290198i 0.857527 0.514440i \(-0.172000\pi\)
−0.874281 + 0.485420i \(0.838667\pi\)
\(150\) 0 0
\(151\) 43328.1 0.154642 0.0773209 0.997006i \(-0.475363\pi\)
0.0773209 + 0.997006i \(0.475363\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −49425.3 85607.2i −0.165242 0.286208i
\(156\) 0 0
\(157\) −50402.1 87299.0i −0.163192 0.282657i 0.772820 0.634626i \(-0.218846\pi\)
−0.936012 + 0.351968i \(0.885512\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −369355. + 639742.i −1.12300 + 1.94509i
\(162\) 0 0
\(163\) −315707. −0.930712 −0.465356 0.885124i \(-0.654074\pi\)
−0.465356 + 0.885124i \(0.654074\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −280911. + 486551.i −0.779430 + 1.35001i 0.152841 + 0.988251i \(0.451158\pi\)
−0.932271 + 0.361761i \(0.882176\pi\)
\(168\) 0 0
\(169\) 142583. + 246962.i 0.384019 + 0.665140i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 323128. + 559674.i 0.820842 + 1.42174i 0.905056 + 0.425292i \(0.139829\pi\)
−0.0842143 + 0.996448i \(0.526838\pi\)
\(174\) 0 0
\(175\) −189573. + 328350.i −0.467930 + 0.810479i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −391686. −0.913704 −0.456852 0.889543i \(-0.651023\pi\)
−0.456852 + 0.889543i \(0.651023\pi\)
\(180\) 0 0
\(181\) −356595. + 617641.i −0.809057 + 1.40133i 0.104461 + 0.994529i \(0.466688\pi\)
−0.913518 + 0.406798i \(0.866645\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −98223.6 170128.i −0.211002 0.365466i
\(186\) 0 0
\(187\) 173800. + 301030.i 0.363451 + 0.629515i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −73696.8 −0.146172 −0.0730862 0.997326i \(-0.523285\pi\)
−0.0730862 + 0.997326i \(0.523285\pi\)
\(192\) 0 0
\(193\) −281706. 487929.i −0.544381 0.942896i −0.998646 0.0520289i \(-0.983431\pi\)
0.454264 0.890867i \(-0.349902\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −227957. −0.418491 −0.209246 0.977863i \(-0.567101\pi\)
−0.209246 + 0.977863i \(0.567101\pi\)
\(198\) 0 0
\(199\) −305231. + 528676.i −0.546382 + 0.946361i 0.452137 + 0.891949i \(0.350662\pi\)
−0.998519 + 0.0544126i \(0.982671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −669468. + 1.15955e6i −1.14022 + 1.97492i
\(204\) 0 0
\(205\) −133650. + 231489.i −0.222119 + 0.384721i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 271780. + 172134.i 0.430380 + 0.272584i
\(210\) 0 0
\(211\) −340640. 590006.i −0.526732 0.912327i −0.999515 0.0311480i \(-0.990084\pi\)
0.472782 0.881179i \(-0.343250\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −284841. + 493359.i −0.420248 + 0.727891i
\(216\) 0 0
\(217\) 570515. 0.822466
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −498968. −0.687214
\(222\) 0 0
\(223\) −464769. 805003.i −0.625856 1.08402i −0.988375 0.152038i \(-0.951416\pi\)
0.362518 0.931977i \(-0.381917\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 836852. 1.07791 0.538957 0.842333i \(-0.318819\pi\)
0.538957 + 0.842333i \(0.318819\pi\)
\(228\) 0 0
\(229\) −43953.2 −0.0553862 −0.0276931 0.999616i \(-0.508816\pi\)
−0.0276931 + 0.999616i \(0.508816\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19649.6 34034.0i −0.0237117 0.0410699i 0.853926 0.520394i \(-0.174215\pi\)
−0.877638 + 0.479324i \(0.840882\pi\)
\(234\) 0 0
\(235\) −703872. −0.831427
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.14239e6 −1.29365 −0.646827 0.762637i \(-0.723904\pi\)
−0.646827 + 0.762637i \(0.723904\pi\)
\(240\) 0 0
\(241\) −142528. + 246866.i −0.158073 + 0.273790i −0.934174 0.356819i \(-0.883861\pi\)
0.776101 + 0.630609i \(0.217195\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 485452. + 840827.i 0.516691 + 0.894935i
\(246\) 0 0
\(247\) −409054. + 214317.i −0.426617 + 0.223519i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 787154. 1.36339e6i 0.788634 1.36595i −0.138169 0.990409i \(-0.544122\pi\)
0.926804 0.375546i \(-0.122545\pi\)
\(252\) 0 0
\(253\) 361503. 626141.i 0.355067 0.614994i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −172908. + 299486.i −0.163299 + 0.282842i −0.936050 0.351867i \(-0.885547\pi\)
0.772751 + 0.634709i \(0.218880\pi\)
\(258\) 0 0
\(259\) 1.13379e6 1.05023
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −494023. 855673.i −0.440411 0.762814i 0.557309 0.830305i \(-0.311834\pi\)
−0.997720 + 0.0674913i \(0.978501\pi\)
\(264\) 0 0
\(265\) 310570. 0.271672
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −418483. 724833.i −0.352612 0.610741i 0.634094 0.773256i \(-0.281373\pi\)
−0.986706 + 0.162514i \(0.948040\pi\)
\(270\) 0 0
\(271\) 1.04254e6 + 1.80574e6i 0.862325 + 1.49359i 0.869679 + 0.493618i \(0.164326\pi\)
−0.00735345 + 0.999973i \(0.502341\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 185543. 321369.i 0.147949 0.256255i
\(276\) 0 0
\(277\) −654742. −0.512709 −0.256355 0.966583i \(-0.582521\pi\)
−0.256355 + 0.966583i \(0.582521\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −432664. + 749397.i −0.326878 + 0.566169i −0.981891 0.189449i \(-0.939330\pi\)
0.655013 + 0.755618i \(0.272663\pi\)
\(282\) 0 0
\(283\) 840551. + 1.45588e6i 0.623876 + 1.08058i 0.988757 + 0.149530i \(0.0477761\pi\)
−0.364882 + 0.931054i \(0.618891\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −771360. 1.33604e6i −0.552780 0.957444i
\(288\) 0 0
\(289\) −735443. + 1.27382e6i −0.517970 + 0.897150i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −495850. −0.337428 −0.168714 0.985665i \(-0.553961\pi\)
−0.168714 + 0.985665i \(0.553961\pi\)
\(294\) 0 0
\(295\) 505577. 875685.i 0.338246 0.585859i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 518925. + 898805.i 0.335681 + 0.581416i
\(300\) 0 0
\(301\) −1.64395e6 2.84741e6i −1.04586 1.81148i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.11228e6 −0.684647
\(306\) 0 0
\(307\) 1.22956e6 + 2.12967e6i 0.744569 + 1.28963i 0.950396 + 0.311043i \(0.100678\pi\)
−0.205826 + 0.978589i \(0.565988\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.82000e6 −1.06702 −0.533508 0.845795i \(-0.679127\pi\)
−0.533508 + 0.845795i \(0.679127\pi\)
\(312\) 0 0
\(313\) −847350. + 1.46765e6i −0.488880 + 0.846765i −0.999918 0.0127931i \(-0.995928\pi\)
0.511038 + 0.859558i \(0.329261\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 785090. 1.35982e6i 0.438805 0.760033i −0.558793 0.829307i \(-0.688735\pi\)
0.997598 + 0.0692749i \(0.0220686\pi\)
\(318\) 0 0
\(319\) 655235. 1.13490e6i 0.360512 0.624426i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −109627. + 2.67315e6i −0.0584671 + 1.42567i
\(324\) 0 0
\(325\) 266340. + 461315.i 0.139871 + 0.242264i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.03119e6 3.51813e6i 1.03457 1.79193i
\(330\) 0 0
\(331\) −1.47267e6 −0.738816 −0.369408 0.929267i \(-0.620440\pi\)
−0.369408 + 0.929267i \(0.620440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.74643e6 −0.850235
\(336\) 0 0
\(337\) −1.04852e6 1.81609e6i −0.502922 0.871087i −0.999994 0.00337788i \(-0.998925\pi\)
0.497072 0.867709i \(-0.334409\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −558386. −0.260045
\(342\) 0 0
\(343\) −2.09282e6 −0.960498
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.08983e6 + 1.88764e6i 0.485887 + 0.841582i 0.999868 0.0162199i \(-0.00516317\pi\)
−0.513981 + 0.857802i \(0.671830\pi\)
\(348\) 0 0
\(349\) 1.80723e6 0.794238 0.397119 0.917767i \(-0.370010\pi\)
0.397119 + 0.917767i \(0.370010\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.11807e6 0.904696 0.452348 0.891842i \(-0.350586\pi\)
0.452348 + 0.891842i \(0.350586\pi\)
\(354\) 0 0
\(355\) −124174. + 215076.i −0.0522950 + 0.0905775i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.04610e6 + 1.81190e6i 0.428388 + 0.741990i 0.996730 0.0808022i \(-0.0257482\pi\)
−0.568342 + 0.822793i \(0.692415\pi\)
\(360\) 0 0
\(361\) 1.05830e6 + 2.23854e6i 0.427408 + 0.904059i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −698534. + 1.20990e6i −0.274445 + 0.475352i
\(366\) 0 0
\(367\) −1.45084e6 + 2.51293e6i −0.562282 + 0.973901i 0.435015 + 0.900423i \(0.356743\pi\)
−0.997297 + 0.0734776i \(0.976590\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −896224. + 1.55231e6i −0.338051 + 0.585521i
\(372\) 0 0
\(373\) −3.15657e6 −1.17474 −0.587372 0.809317i \(-0.699837\pi\)
−0.587372 + 0.809317i \(0.699837\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 940567. + 1.62911e6i 0.340829 + 0.590333i
\(378\) 0 0
\(379\) 3.95334e6 1.41373 0.706864 0.707350i \(-0.250109\pi\)
0.706864 + 0.707350i \(0.250109\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.93528e6 3.35201e6i −0.674136 1.16764i −0.976720 0.214516i \(-0.931183\pi\)
0.302584 0.953123i \(-0.402151\pi\)
\(384\) 0 0
\(385\) −772810. 1.33855e6i −0.265718 0.460237i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.11143e6 + 1.92505e6i −0.372399 + 0.645013i −0.989934 0.141530i \(-0.954798\pi\)
0.617535 + 0.786543i \(0.288131\pi\)
\(390\) 0 0
\(391\) 6.01273e6 1.98898
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.16380e6 2.01577e6i 0.375307 0.650051i
\(396\) 0 0
\(397\) −207721. 359784.i −0.0661462 0.114569i 0.831056 0.556189i \(-0.187737\pi\)
−0.897202 + 0.441621i \(0.854404\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.50770e6 2.61141e6i −0.468224 0.810988i 0.531116 0.847299i \(-0.321773\pi\)
−0.999341 + 0.0363107i \(0.988439\pi\)
\(402\) 0 0
\(403\) 400772. 694157.i 0.122923 0.212910i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.10969e6 −0.332058
\(408\) 0 0
\(409\) −1.89456e6 + 3.28148e6i −0.560016 + 0.969976i 0.437478 + 0.899229i \(0.355872\pi\)
−0.997494 + 0.0707474i \(0.977462\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.91793e6 + 5.05400e6i 0.841782 + 1.45801i
\(414\) 0 0
\(415\) −1.29434e6 2.24187e6i −0.368917 0.638983i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −359202. −0.0999549 −0.0499774 0.998750i \(-0.515915\pi\)
−0.0499774 + 0.998750i \(0.515915\pi\)
\(420\) 0 0
\(421\) −272896. 472669.i −0.0750398 0.129973i 0.826064 0.563577i \(-0.190575\pi\)
−0.901104 + 0.433604i \(0.857242\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.08606e6 0.828765
\(426\) 0 0
\(427\) 3.20977e6 5.55948e6i 0.851930 1.47559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.14797e6 1.98834e6i 0.297671 0.515581i −0.677932 0.735125i \(-0.737124\pi\)
0.975603 + 0.219544i \(0.0704569\pi\)
\(432\) 0 0
\(433\) −2.52128e6 + 4.36698e6i −0.646251 + 1.11934i 0.337760 + 0.941232i \(0.390331\pi\)
−0.984011 + 0.178107i \(0.943003\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.92924e6 2.58260e6i 1.23474 0.646924i
\(438\) 0 0
\(439\) −3.91644e6 6.78347e6i −0.969907 1.67993i −0.695810 0.718226i \(-0.744955\pi\)
−0.274097 0.961702i \(-0.588379\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.75654e6 4.77446e6i 0.667351 1.15589i −0.311291 0.950315i \(-0.600761\pi\)
0.978642 0.205571i \(-0.0659052\pi\)
\(444\) 0 0
\(445\) 1.68344e6 0.402993
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.99055e6 −1.40233 −0.701166 0.712998i \(-0.747337\pi\)
−0.701166 + 0.712998i \(0.747337\pi\)
\(450\) 0 0
\(451\) 754961. + 1.30763e6i 0.174777 + 0.302722i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.21868e6 0.502420
\(456\) 0 0
\(457\) 289587. 0.0648617 0.0324308 0.999474i \(-0.489675\pi\)
0.0324308 + 0.999474i \(0.489675\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.79574e6 + 3.11031e6i 0.393542 + 0.681635i 0.992914 0.118836i \(-0.0379163\pi\)
−0.599372 + 0.800471i \(0.704583\pi\)
\(462\) 0 0
\(463\) 108518. 0.0235260 0.0117630 0.999931i \(-0.496256\pi\)
0.0117630 + 0.999931i \(0.496256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.47855e6 0.950267 0.475133 0.879914i \(-0.342400\pi\)
0.475133 + 0.879914i \(0.342400\pi\)
\(468\) 0 0
\(469\) 5.03974e6 8.72909e6i 1.05798 1.83247i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.60900e6 + 2.78687e6i 0.330677 + 0.572749i
\(474\) 0 0
\(475\) 2.52995e6 1.32553e6i 0.514491 0.269560i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.21830e6 5.57425e6i 0.640896 1.11006i −0.344337 0.938846i \(-0.611896\pi\)
0.985233 0.171218i \(-0.0547703\pi\)
\(480\) 0 0
\(481\) 796459. 1.37951e6i 0.156964 0.271870i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.63614e6 2.83387e6i 0.315838 0.547048i
\(486\) 0 0
\(487\) −983792. −0.187967 −0.0939833 0.995574i \(-0.529960\pi\)
−0.0939833 + 0.995574i \(0.529960\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.28089e6 5.68267e6i −0.614170 1.06377i −0.990530 0.137300i \(-0.956158\pi\)
0.376360 0.926474i \(-0.377176\pi\)
\(492\) 0 0
\(493\) 1.08983e7 2.01948
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −716668. 1.24131e6i −0.130145 0.225418i
\(498\) 0 0
\(499\) −2.23268e6 3.86711e6i −0.401398 0.695241i 0.592497 0.805572i \(-0.298142\pi\)
−0.993895 + 0.110331i \(0.964809\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.44875e6 2.50931e6i 0.255313 0.442215i −0.709667 0.704537i \(-0.751155\pi\)
0.964981 + 0.262322i \(0.0844881\pi\)
\(504\) 0 0
\(505\) −5.28676e6 −0.922489
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.19919e6 + 5.54116e6i −0.547325 + 0.947995i 0.451131 + 0.892458i \(0.351021\pi\)
−0.998457 + 0.0555374i \(0.982313\pi\)
\(510\) 0 0
\(511\) −4.03157e6 6.98289e6i −0.683003 1.18300i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.54269e6 2.67201e6i −0.256306 0.443936i
\(516\) 0 0
\(517\) −1.98801e6 + 3.44333e6i −0.327109 + 0.566569i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.13178e7 1.82671 0.913353 0.407168i \(-0.133484\pi\)
0.913353 + 0.407168i \(0.133484\pi\)
\(522\) 0 0
\(523\) −214962. + 372326.i −0.0343643 + 0.0595208i −0.882696 0.469944i \(-0.844274\pi\)
0.848332 + 0.529465i \(0.177607\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.32185e6 4.02156e6i −0.364173 0.630766i
\(528\) 0 0
\(529\) −3.03505e6 5.25687e6i −0.471549 0.816747i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.16744e6 −0.330468
\(534\) 0 0
\(535\) 510794. + 884721.i 0.0771545 + 0.133635i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.48442e6 0.813128
\(540\) 0 0
\(541\) 4.05574e6 7.02475e6i 0.595768 1.03190i −0.397670 0.917529i \(-0.630181\pi\)
0.993438 0.114372i \(-0.0364855\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.94566e6 + 6.83409e6i −0.569021 + 0.985574i
\(546\) 0 0
\(547\) −4.15311e6 + 7.19340e6i −0.593479 + 1.02794i 0.400281 + 0.916393i \(0.368913\pi\)
−0.993760 + 0.111543i \(0.964421\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.93439e6 4.68103e6i 1.25368 0.656845i
\(552\) 0 0
\(553\) 6.71687e6 + 1.16340e7i 0.934015 + 1.61776i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.99228e6 3.45072e6i 0.272089 0.471272i −0.697307 0.716772i \(-0.745619\pi\)
0.969397 + 0.245500i \(0.0789521\pi\)
\(558\) 0 0
\(559\) −4.61934e6 −0.625245
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.11824e6 −0.813496 −0.406748 0.913540i \(-0.633337\pi\)
−0.406748 + 0.913540i \(0.633337\pi\)
\(564\) 0 0
\(565\) 3.15863e6 + 5.47091e6i 0.416273 + 0.721005i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −908888. −0.117687 −0.0588436 0.998267i \(-0.518741\pi\)
−0.0588436 + 0.998267i \(0.518741\pi\)
\(570\) 0 0
\(571\) 1.83881e6 0.236019 0.118010 0.993012i \(-0.462349\pi\)
0.118010 + 0.993012i \(0.462349\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.20949e6 5.55900e6i −0.404824 0.701176i
\(576\) 0 0
\(577\) 5.44901e6 0.681362 0.340681 0.940179i \(-0.389342\pi\)
0.340681 + 0.940179i \(0.389342\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.49405e7 1.83623
\(582\) 0 0
\(583\) 877170. 1.51930e6i 0.106884 0.185128i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.89239e6 + 3.27772e6i 0.226681 + 0.392623i 0.956823 0.290673i \(-0.0938791\pi\)
−0.730141 + 0.683296i \(0.760546\pi\)
\(588\) 0 0
\(589\) −3.63080e6 2.29959e6i −0.431235 0.273126i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.72339e6 + 9.91320e6i −0.668369 + 1.15765i 0.309990 + 0.950740i \(0.399674\pi\)
−0.978360 + 0.206910i \(0.933659\pi\)
\(594\) 0 0
\(595\) 6.42691e6 1.11317e7i 0.744235 1.28905i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.14807e6 5.45262e6i 0.358491 0.620924i −0.629218 0.777229i \(-0.716625\pi\)
0.987709 + 0.156305i \(0.0499582\pi\)
\(600\) 0 0
\(601\) −3.03647e6 −0.342913 −0.171456 0.985192i \(-0.554847\pi\)
−0.171456 + 0.985192i \(0.554847\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.15805e6 3.73785e6i −0.239703 0.415177i
\(606\) 0 0
\(607\) −1.08106e7 −1.19091 −0.595454 0.803389i \(-0.703028\pi\)
−0.595454 + 0.803389i \(0.703028\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.85372e6 4.94279e6i −0.309249 0.535635i
\(612\) 0 0
\(613\) 4.94957e6 + 8.57291e6i 0.532006 + 0.921461i 0.999302 + 0.0373600i \(0.0118948\pi\)
−0.467296 + 0.884101i \(0.654772\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −973579. + 1.68629e6i −0.102958 + 0.178328i −0.912902 0.408179i \(-0.866164\pi\)
0.809944 + 0.586507i \(0.199497\pi\)
\(618\) 0 0
\(619\) 3.48976e6 0.366074 0.183037 0.983106i \(-0.441407\pi\)
0.183037 + 0.983106i \(0.441407\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.85797e6 + 8.41424e6i −0.501458 + 0.868550i
\(624\) 0 0
\(625\) 399448. + 691864.i 0.0409034 + 0.0708468i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.61424e6 7.99210e6i −0.465022 0.805442i
\(630\) 0 0
\(631\) −4.65587e6 + 8.06421e6i −0.465509 + 0.806285i −0.999224 0.0393793i \(-0.987462\pi\)
0.533716 + 0.845664i \(0.320795\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.20065e6 −0.511827
\(636\) 0 0
\(637\) −3.93635e6 + 6.81796e6i −0.384366 + 0.665741i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.98479e6 5.16981e6i −0.286925 0.496969i 0.686149 0.727461i \(-0.259300\pi\)
−0.973074 + 0.230492i \(0.925966\pi\)
\(642\) 0 0
\(643\) 9.07253e6 + 1.57141e7i 0.865368 + 1.49886i 0.866681 + 0.498863i \(0.166249\pi\)
−0.00131239 + 0.999999i \(0.500418\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.03554e7 −0.972537 −0.486269 0.873809i \(-0.661642\pi\)
−0.486269 + 0.873809i \(0.661642\pi\)
\(648\) 0 0
\(649\) −2.85589e6 4.94655e6i −0.266152 0.460989i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.38119e6 −0.402077 −0.201038 0.979583i \(-0.564432\pi\)
−0.201038 + 0.979583i \(0.564432\pi\)
\(654\) 0 0
\(655\) 103769. 179733.i 0.00945069 0.0163691i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.34358e6 1.27195e7i 0.658710 1.14092i −0.322239 0.946658i \(-0.604436\pi\)
0.980950 0.194262i \(-0.0622311\pi\)
\(660\) 0 0
\(661\) −1.90277e6 + 3.29569e6i −0.169388 + 0.293388i −0.938205 0.346081i \(-0.887512\pi\)
0.768817 + 0.639469i \(0.220846\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 487462. 1.18863e7i 0.0427452 1.04230i
\(666\) 0 0
\(667\) −1.13342e7 1.96313e7i −0.986449 1.70858i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.14153e6 + 5.44128e6i −0.269361 + 0.466547i
\(672\) 0 0
\(673\) 2.15454e7 1.83365 0.916824 0.399291i \(-0.130744\pi\)
0.916824 + 0.399291i \(0.130744\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 182950. 0.0153412 0.00767061 0.999971i \(-0.497558\pi\)
0.00767061 + 0.999971i \(0.497558\pi\)
\(678\) 0 0
\(679\) 9.44293e6 + 1.63556e7i 0.786017 + 1.36142i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.90678e7 1.56404 0.782021 0.623252i \(-0.214189\pi\)
0.782021 + 0.623252i \(0.214189\pi\)
\(684\) 0 0
\(685\) −1.08713e7 −0.885229
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.25915e6 + 2.18091e6i 0.101048 + 0.175021i
\(690\) 0 0
\(691\) 1.57718e7 1.25657 0.628285 0.777983i \(-0.283757\pi\)
0.628285 + 0.777983i \(0.283757\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.58252e7 1.24276
\(696\) 0 0
\(697\) −6.27848e6 + 1.08746e7i −0.489522 + 0.847878i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.89878e6 + 1.54131e7i 0.683967 + 1.18467i 0.973760 + 0.227577i \(0.0730802\pi\)
−0.289793 + 0.957089i \(0.593586\pi\)
\(702\) 0 0
\(703\) −7.21554e6 4.57001e6i −0.550656 0.348762i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.52562e7 2.64245e7i 1.14788 1.98819i
\(708\) 0 0
\(709\) −1.02786e6 + 1.78031e6i −0.0767924 + 0.133008i −0.901864 0.432019i \(-0.857801\pi\)
0.825072 + 0.565028i \(0.191135\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.82944e6 + 8.36483e6i −0.355773 + 0.616217i
\(714\) 0 0
\(715\) −2.17151e6 −0.158854
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.09362e7 1.89420e7i −0.788940 1.36648i −0.926617 0.376007i \(-0.877297\pi\)
0.137676 0.990477i \(-0.456037\pi\)
\(720\) 0 0
\(721\) 1.78072e7 1.27572
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.81729e6 1.00758e7i −0.411032 0.711929i
\(726\) 0 0
\(727\) 1.17837e6 + 2.04099e6i 0.0826884 + 0.143221i 0.904404 0.426677i \(-0.140316\pi\)
−0.821715 + 0.569898i \(0.806983\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.33809e7 + 2.31765e7i −0.926175 + 1.60418i
\(732\) 0 0
\(733\) 2.16077e7 1.48541 0.742707 0.669616i \(-0.233541\pi\)
0.742707 + 0.669616i \(0.233541\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.93260e6 + 8.54351e6i −0.334508 + 0.579385i
\(738\) 0 0
\(739\) −2.23082e6 3.86389e6i −0.150263 0.260264i 0.781061 0.624455i \(-0.214679\pi\)
−0.931324 + 0.364191i \(0.881345\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.70913e6 + 1.33526e7i 0.512311 + 0.887348i 0.999898 + 0.0142742i \(0.00454377\pi\)
−0.487587 + 0.873074i \(0.662123\pi\)
\(744\) 0 0
\(745\) 164331. 284630.i 0.0108475 0.0187884i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.89607e6 −0.384024
\(750\) 0 0
\(751\) −1.64269e6 + 2.84522e6i −0.106281 + 0.184084i −0.914261 0.405126i \(-0.867228\pi\)
0.807980 + 0.589210i \(0.200561\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 784079. + 1.35806e6i 0.0500602 + 0.0867068i
\(756\) 0 0
\(757\) −5.40092e6 9.35466e6i −0.342553 0.593319i 0.642353 0.766409i \(-0.277958\pi\)
−0.984906 + 0.173090i \(0.944625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.38280e6 0.587315 0.293657 0.955911i \(-0.405128\pi\)
0.293657 + 0.955911i \(0.405128\pi\)
\(762\) 0 0
\(763\) −2.27723e7 3.94428e7i −1.41611 2.45277i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.19907e6 0.503241
\(768\) 0 0
\(769\) 2.58311e6 4.47407e6i 0.157517 0.272827i −0.776456 0.630172i \(-0.782985\pi\)
0.933973 + 0.357345i \(0.116318\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.46904e6 + 4.27650e6i −0.148621 + 0.257419i −0.930718 0.365738i \(-0.880817\pi\)
0.782097 + 0.623156i \(0.214150\pi\)
\(774\) 0 0
\(775\) −2.47873e6 + 4.29328e6i −0.148243 + 0.256764i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −476204. + 1.16118e7i −0.0281157 + 0.685575i
\(780\) 0 0
\(781\) 701432. + 1.21492e6i 0.0411489 + 0.0712719i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.82419e6 3.15958e6i 0.105656 0.183002i
\(786\) 0 0
\(787\) −7.65272e6 −0.440432 −0.220216 0.975451i \(-0.570676\pi\)
−0.220216 + 0.975451i \(0.570676\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.64600e7 −2.07193
\(792\) 0 0
\(793\) −4.50955e6 7.81077e6i −0.254654 0.441074i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.09086e7 0.608306 0.304153 0.952623i \(-0.401627\pi\)
0.304153 + 0.952623i \(0.401627\pi\)
\(798\) 0 0
\(799\) −3.30657e7 −1.83236
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.94586e6 + 6.83443e6i 0.215950 + 0.374036i
\(804\) 0 0
\(805\) −2.67359e7 −1.45414
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.87426e7 −1.54403 −0.772013 0.635606i \(-0.780750\pi\)
−0.772013 + 0.635606i \(0.780750\pi\)
\(810\) 0 0
\(811\) −9.14408e6 + 1.58380e7i −0.488189 + 0.845568i −0.999908 0.0135852i \(-0.995676\pi\)
0.511719 + 0.859153i \(0.329009\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.71314e6 9.89544e6i −0.301287 0.521845i
\(816\) 0 0
\(817\) −1.01490e6 + 2.47475e7i −0.0531949 + 1.29711i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.63449e6 + 1.32233e7i −0.395296 + 0.684672i −0.993139 0.116941i \(-0.962691\pi\)
0.597843 + 0.801613i \(0.296025\pi\)
\(822\) 0 0
\(823\) 3.66967e6 6.35605e6i 0.188854 0.327105i −0.756014 0.654555i \(-0.772856\pi\)
0.944869 + 0.327450i \(0.106189\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.35237e6 + 1.61988e7i −0.475508 + 0.823605i −0.999606 0.0280533i \(-0.991069\pi\)
0.524098 + 0.851658i \(0.324403\pi\)
\(828\) 0 0
\(829\) 1.92855e7 0.974641 0.487320 0.873223i \(-0.337974\pi\)
0.487320 + 0.873223i \(0.337974\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.28050e7 + 3.94995e7i 1.13872 + 1.97233i
\(834\) 0 0
\(835\) −2.03338e7 −1.00926
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6.24428e6 1.08154e7i −0.306251 0.530442i 0.671288 0.741196i \(-0.265741\pi\)
−0.977539 + 0.210754i \(0.932408\pi\)
\(840\) 0 0
\(841\) −1.02879e7 1.78192e7i −0.501576 0.868756i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.16047e6 + 8.93819e6i −0.248626 + 0.430634i
\(846\) 0 0
\(847\) 2.49103e7 1.19308
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −9.59760e6 + 1.66235e7i −0.454296 + 0.786864i
\(852\) 0 0
\(853\) 3.42614e6 + 5.93425e6i 0.161225 + 0.279250i 0.935308 0.353834i \(-0.115122\pi\)
−0.774083 + 0.633084i \(0.781789\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.61768e6 + 9.73011e6i 0.261279 + 0.452549i 0.966582 0.256357i \(-0.0825223\pi\)
−0.705303 + 0.708906i \(0.749189\pi\)
\(858\) 0 0
\(859\) 1.89450e6 3.28138e6i 0.0876017 0.151731i −0.818895 0.573943i \(-0.805413\pi\)
0.906497 + 0.422212i \(0.138746\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.08965e7 1.41215 0.706077 0.708135i \(-0.250463\pi\)
0.706077 + 0.708135i \(0.250463\pi\)
\(864\) 0 0
\(865\) −1.16949e7 + 2.02561e7i −0.531441 + 0.920482i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.57407e6 1.13866e7i −0.295314 0.511500i
\(870\) 0 0
\(871\) −7.08057e6 1.22639e7i −0.316244 0.547752i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.73476e7 −1.64908
\(876\) 0 0
\(877\) 1.06989e7 + 1.85310e7i 0.469719 + 0.813577i 0.999401 0.0346194i \(-0.0110219\pi\)
−0.529682 + 0.848197i \(0.677689\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.47758e6 −0.150951 −0.0754756 0.997148i \(-0.524048\pi\)
−0.0754756 + 0.997148i \(0.524048\pi\)
\(882\) 0 0
\(883\) −7.52985e6 + 1.30421e7i −0.325001 + 0.562918i −0.981513 0.191398i \(-0.938698\pi\)
0.656512 + 0.754316i \(0.272031\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.37599e7 + 2.38328e7i −0.587225 + 1.01710i 0.407369 + 0.913264i \(0.366446\pi\)
−0.994594 + 0.103840i \(0.966887\pi\)
\(888\) 0 0
\(889\) 1.50077e7 2.59941e7i 0.636884 1.10312i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.71073e7 + 1.42025e7i −1.13752 + 0.595984i
\(894\) 0 0
\(895\) −7.08807e6 1.22769e7i −0.295781 0.512308i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −8.75350e6 + 1.51615e7i −0.361229 + 0.625667i
\(900\) 0 0
\(901\) 1.45896e7 0.598731
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.58122e7 −1.04762
\(906\) 0 0
\(907\) −1.31824e7 2.28326e7i −0.532079 0.921588i −0.999299 0.0374469i \(-0.988077\pi\)
0.467219 0.884141i \(-0.345256\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00749e6 0.239827 0.119913 0.992784i \(-0.461738\pi\)
0.119913 + 0.992784i \(0.461738\pi\)
\(912\) 0 0
\(913\) −1.46229e7 −0.580573
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 598900. + 1.03733e6i 0.0235197 + 0.0407372i
\(918\) 0 0
\(919\) 1.21396e7 0.474150 0.237075 0.971491i \(-0.423811\pi\)
0.237075 + 0.971491i \(0.423811\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.01376e6 −0.0778044
\(924\) 0 0
\(925\) −4.92600e6 + 8.53209e6i −0.189296 + 0.327869i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.94616e6 + 8.56699e6i 0.188031 + 0.325679i 0.944594 0.328242i \(-0.106456\pi\)
−0.756563 + 0.653921i \(0.773123\pi\)
\(930\) 0 0
\(931\) 3.56614e7 + 2.25864e7i 1.34842 + 0.854030i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.29028e6 + 1.08951e7i −0.235310 + 0.407569i
\(936\) 0 0
\(937\) −1.02564e7 + 1.77646e7i −0.381632 + 0.661007i −0.991296 0.131654i \(-0.957971\pi\)
0.609663 + 0.792660i \(0.291305\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.28320e7 + 3.95462e7i −0.840562 + 1.45590i 0.0488582 + 0.998806i \(0.484442\pi\)
−0.889420 + 0.457090i \(0.848892\pi\)
\(942\) 0 0
\(943\) 2.61184e7 0.956462
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.04770e6 + 8.74287e6i 0.182902 + 0.316796i 0.942868 0.333168i \(-0.108118\pi\)
−0.759966 + 0.649963i \(0.774784\pi\)
\(948\) 0 0
\(949\) −1.13283e7 −0.408319
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.33676e7 + 2.31533e7i 0.476782 + 0.825811i 0.999646 0.0266050i \(-0.00846964\pi\)
−0.522864 + 0.852416i \(0.675136\pi\)
\(954\) 0 0
\(955\) −1.33364e6 2.30993e6i −0.0473184 0.0819579i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.13718e7 5.43375e7i 1.10152 1.90789i
\(960\) 0 0
\(961\) −2.11695e7 −0.739438
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.01957e7 1.76594e7i 0.352451 0.610462i
\(966\) 0 0
\(967\) 4.40549e6 + 7.63054e6i 0.151505 + 0.262415i 0.931781 0.363021i \(-0.118255\pi\)
−0.780276 + 0.625436i \(0.784921\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.90262e7 3.29543e7i −0.647594 1.12167i −0.983696 0.179840i \(-0.942442\pi\)
0.336102 0.941826i \(-0.390891\pi\)
\(972\) 0 0
\(973\) −4.56673e7 + 7.90981e7i −1.54641 + 2.67845i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.16722e7 1.39672 0.698361 0.715746i \(-0.253913\pi\)
0.698361 + 0.715746i \(0.253913\pi\)
\(978\) 0 0
\(979\) 4.75469e6 8.23536e6i 0.158550 0.274616i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.16371e7 3.74766e7i −0.714194 1.23702i −0.963270 0.268536i \(-0.913460\pi\)
0.249076 0.968484i \(-0.419873\pi\)
\(984\) 0 0
\(985\) −4.12517e6 7.14501e6i −0.135473 0.234646i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.56646e7 1.80962
\(990\) 0 0
\(991\) 1.59147e7 + 2.75651e7i 0.514772 + 0.891611i 0.999853 + 0.0171418i \(0.00545668\pi\)
−0.485081 + 0.874469i \(0.661210\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.20942e7 −0.707492
\(996\) 0 0
\(997\) 2.67257e7 4.62903e7i 0.851514 1.47487i −0.0283272 0.999599i \(-0.509018\pi\)
0.879841 0.475267i \(-0.157649\pi\)
\(998\) 0 0
\(999\) 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 684.6.k.f.577.6 18
3.2 odd 2 76.6.e.a.45.9 18
12.11 even 2 304.6.i.d.273.1 18
19.11 even 3 inner 684.6.k.f.505.6 18
57.11 odd 6 76.6.e.a.49.9 yes 18
228.11 even 6 304.6.i.d.49.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.9 18 3.2 odd 2
76.6.e.a.49.9 yes 18 57.11 odd 6
304.6.i.d.49.1 18 228.11 even 6
304.6.i.d.273.1 18 12.11 even 2
684.6.k.f.505.6 18 19.11 even 3 inner
684.6.k.f.577.6 18 1.1 even 1 trivial