Properties

Label 684.6.k.f.505.5
Level $684$
Weight $6$
Character 684.505
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 505.5
Root \(16.4645 + 26.7852i\) of defining polynomial
Character \(\chi\) \(=\) 684.505
Dual form 684.6.k.f.577.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(15.4645 - 26.7852i) q^{5} +132.225 q^{7} +O(q^{10})\) \(q+(15.4645 - 26.7852i) q^{5} +132.225 q^{7} -670.612 q^{11} +(-411.330 - 712.445i) q^{13} +(-731.251 + 1266.56i) q^{17} +(-1573.26 - 30.7850i) q^{19} +(1145.60 + 1984.23i) q^{23} +(1084.20 + 1877.89i) q^{25} +(1381.18 + 2392.27i) q^{29} +10591.8 q^{31} +(2044.79 - 3541.69i) q^{35} +4815.22 q^{37} +(7285.18 - 12618.3i) q^{41} +(-5150.23 + 8920.46i) q^{43} +(2555.09 + 4425.55i) q^{47} +676.549 q^{49} +(3724.23 + 6450.55i) q^{53} +(-10370.7 + 17962.5i) q^{55} +(17150.1 - 29704.9i) q^{59} +(20608.6 + 35695.1i) q^{61} -25444.0 q^{65} +(-19625.6 - 33992.5i) q^{67} +(-5848.69 + 10130.2i) q^{71} +(-13696.8 + 23723.6i) q^{73} -88671.9 q^{77} +(-2583.48 + 4474.71i) q^{79} +86152.6 q^{83} +(22616.8 + 39173.5i) q^{85} +(-25384.0 - 43966.4i) q^{89} +(-54388.3 - 94203.3i) q^{91} +(-25154.2 + 41664.1i) q^{95} +(-13304.7 + 23044.5i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 15.4645 26.7852i 0.276637 0.479149i −0.693910 0.720062i \(-0.744113\pi\)
0.970547 + 0.240913i \(0.0774468\pi\)
\(6\) 0 0
\(7\) 132.225 1.01993 0.509964 0.860196i \(-0.329659\pi\)
0.509964 + 0.860196i \(0.329659\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −670.612 −1.67105 −0.835525 0.549452i \(-0.814837\pi\)
−0.835525 + 0.549452i \(0.814837\pi\)
\(12\) 0 0
\(13\) −411.330 712.445i −0.675044 1.16921i −0.976456 0.215717i \(-0.930791\pi\)
0.301412 0.953494i \(-0.402542\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −731.251 + 1266.56i −0.613683 + 1.06293i 0.376931 + 0.926241i \(0.376979\pi\)
−0.990614 + 0.136689i \(0.956354\pi\)
\(18\) 0 0
\(19\) −1573.26 30.7850i −0.999809 0.0195639i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1145.60 + 1984.23i 0.451556 + 0.782118i 0.998483 0.0550625i \(-0.0175358\pi\)
−0.546927 + 0.837180i \(0.684202\pi\)
\(24\) 0 0
\(25\) 1084.20 + 1877.89i 0.346944 + 0.600925i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1381.18 + 2392.27i 0.304968 + 0.528221i 0.977254 0.212071i \(-0.0680208\pi\)
−0.672286 + 0.740292i \(0.734687\pi\)
\(30\) 0 0
\(31\) 10591.8 1.97955 0.989776 0.142628i \(-0.0455552\pi\)
0.989776 + 0.142628i \(0.0455552\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2044.79 3541.69i 0.282150 0.488698i
\(36\) 0 0
\(37\) 4815.22 0.578245 0.289122 0.957292i \(-0.406637\pi\)
0.289122 + 0.957292i \(0.406637\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7285.18 12618.3i 0.676832 1.17231i −0.299098 0.954222i \(-0.596686\pi\)
0.975930 0.218085i \(-0.0699810\pi\)
\(42\) 0 0
\(43\) −5150.23 + 8920.46i −0.424772 + 0.735726i −0.996399 0.0847874i \(-0.972979\pi\)
0.571628 + 0.820513i \(0.306312\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2555.09 + 4425.55i 0.168718 + 0.292228i 0.937969 0.346718i \(-0.112704\pi\)
−0.769251 + 0.638946i \(0.779371\pi\)
\(48\) 0 0
\(49\) 676.549 0.0402540
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3724.23 + 6450.55i 0.182115 + 0.315433i 0.942601 0.333922i \(-0.108372\pi\)
−0.760485 + 0.649355i \(0.775039\pi\)
\(54\) 0 0
\(55\) −10370.7 + 17962.5i −0.462274 + 0.800682i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 17150.1 29704.9i 0.641412 1.11096i −0.343705 0.939077i \(-0.611682\pi\)
0.985118 0.171881i \(-0.0549845\pi\)
\(60\) 0 0
\(61\) 20608.6 + 35695.1i 0.709126 + 1.22824i 0.965182 + 0.261580i \(0.0842434\pi\)
−0.256056 + 0.966662i \(0.582423\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25444.0 −0.746968
\(66\) 0 0
\(67\) −19625.6 33992.5i −0.534116 0.925116i −0.999206 0.0398521i \(-0.987311\pi\)
0.465090 0.885263i \(-0.346022\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5848.69 + 10130.2i −0.137693 + 0.238492i −0.926623 0.375992i \(-0.877302\pi\)
0.788930 + 0.614483i \(0.210635\pi\)
\(72\) 0 0
\(73\) −13696.8 + 23723.6i −0.300824 + 0.521043i −0.976323 0.216318i \(-0.930595\pi\)
0.675498 + 0.737361i \(0.263928\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −88671.9 −1.70435
\(78\) 0 0
\(79\) −2583.48 + 4474.71i −0.0465733 + 0.0806673i −0.888372 0.459124i \(-0.848163\pi\)
0.841799 + 0.539791i \(0.181497\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 86152.6 1.37269 0.686346 0.727275i \(-0.259214\pi\)
0.686346 + 0.727275i \(0.259214\pi\)
\(84\) 0 0
\(85\) 22616.8 + 39173.5i 0.339535 + 0.588091i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −25384.0 43966.4i −0.339692 0.588364i 0.644683 0.764450i \(-0.276989\pi\)
−0.984375 + 0.176087i \(0.943656\pi\)
\(90\) 0 0
\(91\) −54388.3 94203.3i −0.688497 1.19251i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −25154.2 + 41664.1i −0.285958 + 0.473645i
\(96\) 0 0
\(97\) −13304.7 + 23044.5i −0.143574 + 0.248678i −0.928840 0.370481i \(-0.879193\pi\)
0.785266 + 0.619159i \(0.212526\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −32085.7 55574.0i −0.312974 0.542086i 0.666031 0.745924i \(-0.267992\pi\)
−0.979005 + 0.203838i \(0.934658\pi\)
\(102\) 0 0
\(103\) 66441.0 0.617082 0.308541 0.951211i \(-0.400159\pi\)
0.308541 + 0.951211i \(0.400159\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 187958. 1.58709 0.793546 0.608510i \(-0.208232\pi\)
0.793546 + 0.608510i \(0.208232\pi\)
\(108\) 0 0
\(109\) 89626.3 155237.i 0.722552 1.25150i −0.237422 0.971407i \(-0.576302\pi\)
0.959974 0.280090i \(-0.0903643\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 59332.4 0.437115 0.218557 0.975824i \(-0.429865\pi\)
0.218557 + 0.975824i \(0.429865\pi\)
\(114\) 0 0
\(115\) 70864.1 0.499668
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −96689.9 + 167472.i −0.625913 + 1.08411i
\(120\) 0 0
\(121\) 288669. 1.79241
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 163719. 0.937184
\(126\) 0 0
\(127\) −33405.1 57859.3i −0.183782 0.318320i 0.759383 0.650643i \(-0.225501\pi\)
−0.943165 + 0.332324i \(0.892167\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 175425. 303846.i 0.893129 1.54695i 0.0570264 0.998373i \(-0.481838\pi\)
0.836103 0.548573i \(-0.184829\pi\)
\(132\) 0 0
\(133\) −208025. 4070.56i −1.01973 0.0199538i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 124390. + 215450.i 0.566219 + 0.980719i 0.996935 + 0.0782321i \(0.0249275\pi\)
−0.430717 + 0.902487i \(0.641739\pi\)
\(138\) 0 0
\(139\) 221518. + 383680.i 0.972459 + 1.68435i 0.688078 + 0.725637i \(0.258455\pi\)
0.284381 + 0.958711i \(0.408212\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 275843. + 477774.i 1.12803 + 1.95381i
\(144\) 0 0
\(145\) 85436.8 0.337462
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 120138. 208085.i 0.443317 0.767847i −0.554617 0.832106i \(-0.687135\pi\)
0.997933 + 0.0642591i \(0.0204684\pi\)
\(150\) 0 0
\(151\) −166073. −0.592730 −0.296365 0.955075i \(-0.595774\pi\)
−0.296365 + 0.955075i \(0.595774\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 163797. 283705.i 0.547617 0.948501i
\(156\) 0 0
\(157\) −50926.7 + 88207.7i −0.164891 + 0.285599i −0.936617 0.350356i \(-0.886061\pi\)
0.771726 + 0.635956i \(0.219394\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 151477. + 262365.i 0.460555 + 0.797704i
\(162\) 0 0
\(163\) −256024. −0.754764 −0.377382 0.926058i \(-0.623176\pi\)
−0.377382 + 0.926058i \(0.623176\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −53946.9 93438.8i −0.149684 0.259260i 0.781427 0.623997i \(-0.214492\pi\)
−0.931111 + 0.364737i \(0.881159\pi\)
\(168\) 0 0
\(169\) −152738. + 264551.i −0.411369 + 0.712512i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −197251. + 341649.i −0.501077 + 0.867890i 0.498922 + 0.866647i \(0.333729\pi\)
−0.999999 + 0.00124378i \(0.999604\pi\)
\(174\) 0 0
\(175\) 143359. + 248305.i 0.353858 + 0.612900i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 229285. 0.534863 0.267432 0.963577i \(-0.413825\pi\)
0.267432 + 0.963577i \(0.413825\pi\)
\(180\) 0 0
\(181\) −269882. 467449.i −0.612318 1.06057i −0.990849 0.134977i \(-0.956904\pi\)
0.378531 0.925589i \(-0.376429\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 74464.8 128977.i 0.159964 0.277065i
\(186\) 0 0
\(187\) 490386. 849373.i 1.02550 1.77621i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −448848. −0.890259 −0.445129 0.895466i \(-0.646842\pi\)
−0.445129 + 0.895466i \(0.646842\pi\)
\(192\) 0 0
\(193\) 23904.7 41404.2i 0.0461945 0.0800113i −0.842004 0.539472i \(-0.818624\pi\)
0.888198 + 0.459461i \(0.151957\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −43802.6 −0.0804145 −0.0402072 0.999191i \(-0.512802\pi\)
−0.0402072 + 0.999191i \(0.512802\pi\)
\(198\) 0 0
\(199\) 300067. + 519730.i 0.537137 + 0.930348i 0.999057 + 0.0434264i \(0.0138274\pi\)
−0.461920 + 0.886922i \(0.652839\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 182627. + 316319.i 0.311046 + 0.538747i
\(204\) 0 0
\(205\) −225323. 390271.i −0.374473 0.648607i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.05505e6 + 20644.8i 1.67073 + 0.0326923i
\(210\) 0 0
\(211\) 239624. 415040.i 0.370530 0.641777i −0.619117 0.785299i \(-0.712509\pi\)
0.989647 + 0.143522i \(0.0458427\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 159291. + 275900.i 0.235015 + 0.407058i
\(216\) 0 0
\(217\) 1.40051e6 2.01900
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.20314e6 1.65705
\(222\) 0 0
\(223\) −93000.5 + 161082.i −0.125234 + 0.216912i −0.921824 0.387608i \(-0.873302\pi\)
0.796590 + 0.604520i \(0.206635\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −131418. −0.169274 −0.0846371 0.996412i \(-0.526973\pi\)
−0.0846371 + 0.996412i \(0.526973\pi\)
\(228\) 0 0
\(229\) 105774. 0.133288 0.0666440 0.997777i \(-0.478771\pi\)
0.0666440 + 0.997777i \(0.478771\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −504870. + 874460.i −0.609241 + 1.05524i 0.382124 + 0.924111i \(0.375193\pi\)
−0.991366 + 0.131126i \(0.958141\pi\)
\(234\) 0 0
\(235\) 158053. 0.186695
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 538983. 0.610351 0.305176 0.952296i \(-0.401285\pi\)
0.305176 + 0.952296i \(0.401285\pi\)
\(240\) 0 0
\(241\) 618134. + 1.07064e6i 0.685551 + 1.18741i 0.973263 + 0.229692i \(0.0737720\pi\)
−0.287712 + 0.957717i \(0.592895\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10462.5 18121.5i 0.0111357 0.0192877i
\(246\) 0 0
\(247\) 625197. + 1.13352e6i 0.652040 + 1.18219i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 665047. + 1.15190e6i 0.666298 + 1.15406i 0.978932 + 0.204188i \(0.0654553\pi\)
−0.312634 + 0.949874i \(0.601211\pi\)
\(252\) 0 0
\(253\) −768250. 1.33065e6i −0.754573 1.30696i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −471458. 816590.i −0.445257 0.771207i 0.552813 0.833305i \(-0.313554\pi\)
−0.998070 + 0.0620979i \(0.980221\pi\)
\(258\) 0 0
\(259\) 636694. 0.589768
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 848058. 1.46888e6i 0.756025 1.30947i −0.188838 0.982008i \(-0.560472\pi\)
0.944863 0.327465i \(-0.106194\pi\)
\(264\) 0 0
\(265\) 230373. 0.201519
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −403755. + 699325.i −0.340203 + 0.589248i −0.984470 0.175552i \(-0.943829\pi\)
0.644268 + 0.764800i \(0.277162\pi\)
\(270\) 0 0
\(271\) −1.02987e6 + 1.78379e6i −0.851845 + 1.47544i 0.0276959 + 0.999616i \(0.491183\pi\)
−0.879541 + 0.475823i \(0.842150\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −727078. 1.25934e6i −0.579761 1.00418i
\(276\) 0 0
\(277\) −1.06729e6 −0.835761 −0.417881 0.908502i \(-0.637227\pi\)
−0.417881 + 0.908502i \(0.637227\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 352862. + 611176.i 0.266587 + 0.461743i 0.967978 0.251034i \(-0.0807706\pi\)
−0.701391 + 0.712777i \(0.747437\pi\)
\(282\) 0 0
\(283\) 190924. 330690.i 0.141708 0.245446i −0.786432 0.617677i \(-0.788074\pi\)
0.928140 + 0.372231i \(0.121407\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 963286. 1.66846e6i 0.690320 1.19567i
\(288\) 0 0
\(289\) −359527. 622720.i −0.253214 0.438579i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.08651e6 1.41988 0.709941 0.704261i \(-0.248722\pi\)
0.709941 + 0.704261i \(0.248722\pi\)
\(294\) 0 0
\(295\) −530435. 918740.i −0.354876 0.614664i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 942435. 1.63235e6i 0.609640 1.05593i
\(300\) 0 0
\(301\) −680991. + 1.17951e6i −0.433237 + 0.750388i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.27480e6 0.784681
\(306\) 0 0
\(307\) 805874. 1.39582e6i 0.488002 0.845244i −0.511903 0.859043i \(-0.671059\pi\)
0.999905 + 0.0137995i \(0.00439266\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −785721. −0.460646 −0.230323 0.973114i \(-0.573978\pi\)
−0.230323 + 0.973114i \(0.573978\pi\)
\(312\) 0 0
\(313\) 1.34106e6 + 2.32278e6i 0.773724 + 1.34013i 0.935509 + 0.353303i \(0.114942\pi\)
−0.161785 + 0.986826i \(0.551725\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.30544e6 + 2.26109e6i 0.729642 + 1.26378i 0.957034 + 0.289974i \(0.0936467\pi\)
−0.227392 + 0.973803i \(0.573020\pi\)
\(318\) 0 0
\(319\) −926235. 1.60429e6i −0.509618 0.882684i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.18944e6 1.97012e6i 0.634361 1.05072i
\(324\) 0 0
\(325\) 891929. 1.54487e6i 0.468405 0.811302i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 337848. + 585170.i 0.172080 + 0.298052i
\(330\) 0 0
\(331\) 2.09316e6 1.05010 0.525052 0.851070i \(-0.324046\pi\)
0.525052 + 0.851070i \(0.324046\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.21400e6 −0.591024
\(336\) 0 0
\(337\) −1.63109e6 + 2.82513e6i −0.782353 + 1.35508i 0.148214 + 0.988955i \(0.452647\pi\)
−0.930568 + 0.366120i \(0.880686\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.10301e6 −3.30793
\(342\) 0 0
\(343\) −2.13285e6 −0.978872
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −602372. + 1.04334e6i −0.268560 + 0.465159i −0.968490 0.249052i \(-0.919881\pi\)
0.699930 + 0.714211i \(0.253214\pi\)
\(348\) 0 0
\(349\) 2.97327e6 1.30668 0.653342 0.757063i \(-0.273366\pi\)
0.653342 + 0.757063i \(0.273366\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.66290e6 −1.13741 −0.568706 0.822541i \(-0.692556\pi\)
−0.568706 + 0.822541i \(0.692556\pi\)
\(354\) 0 0
\(355\) 180894. + 313317.i 0.0761820 + 0.131951i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −415704. + 720020.i −0.170235 + 0.294855i −0.938502 0.345274i \(-0.887786\pi\)
0.768267 + 0.640129i \(0.221119\pi\)
\(360\) 0 0
\(361\) 2.47420e6 + 96865.8i 0.999235 + 0.0391203i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 423629. + 733746.i 0.166438 + 0.288279i
\(366\) 0 0
\(367\) 543635. + 941604.i 0.210689 + 0.364925i 0.951930 0.306314i \(-0.0990958\pi\)
−0.741241 + 0.671239i \(0.765763\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 492438. + 852927.i 0.185745 + 0.321719i
\(372\) 0 0
\(373\) 3.00979e6 1.12012 0.560059 0.828452i \(-0.310778\pi\)
0.560059 + 0.828452i \(0.310778\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.13624e6 1.96803e6i 0.411734 0.713145i
\(378\) 0 0
\(379\) −1.58924e6 −0.568317 −0.284158 0.958777i \(-0.591714\pi\)
−0.284158 + 0.958777i \(0.591714\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.14784e6 + 1.98811e6i −0.399837 + 0.692538i −0.993705 0.112024i \(-0.964267\pi\)
0.593869 + 0.804562i \(0.297600\pi\)
\(384\) 0 0
\(385\) −1.37126e6 + 2.37510e6i −0.471486 + 0.816638i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.12902e6 + 3.68757e6i 0.713355 + 1.23557i 0.963590 + 0.267383i \(0.0861588\pi\)
−0.250235 + 0.968185i \(0.580508\pi\)
\(390\) 0 0
\(391\) −3.35087e6 −1.10845
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 79904.2 + 138398.i 0.0257678 + 0.0446311i
\(396\) 0 0
\(397\) −173073. + 299772.i −0.0551130 + 0.0954585i −0.892266 0.451511i \(-0.850885\pi\)
0.837153 + 0.546969i \(0.184219\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −777399. + 1.34649e6i −0.241425 + 0.418161i −0.961121 0.276129i \(-0.910948\pi\)
0.719695 + 0.694290i \(0.244282\pi\)
\(402\) 0 0
\(403\) −4.35674e6 7.54610e6i −1.33629 2.31451i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.22914e6 −0.966276
\(408\) 0 0
\(409\) −796042. 1.37878e6i −0.235303 0.407557i 0.724058 0.689739i \(-0.242275\pi\)
−0.959361 + 0.282183i \(0.908942\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.26768e6 3.92774e6i 0.654195 1.13310i
\(414\) 0 0
\(415\) 1.33230e6 2.30762e6i 0.379737 0.657724i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.11283e6 1.14447 0.572236 0.820089i \(-0.306076\pi\)
0.572236 + 0.820089i \(0.306076\pi\)
\(420\) 0 0
\(421\) 1.57964e6 2.73602e6i 0.434364 0.752341i −0.562879 0.826539i \(-0.690306\pi\)
0.997244 + 0.0741981i \(0.0236397\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.17129e6 −0.851655
\(426\) 0 0
\(427\) 2.72498e6 + 4.71980e6i 0.723257 + 1.25272i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −508299. 880400.i −0.131803 0.228290i 0.792568 0.609783i \(-0.208743\pi\)
−0.924372 + 0.381493i \(0.875410\pi\)
\(432\) 0 0
\(433\) −1.49339e6 2.58663e6i −0.382784 0.663001i 0.608675 0.793419i \(-0.291701\pi\)
−0.991459 + 0.130419i \(0.958368\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.74124e6 3.15698e6i −0.436168 0.790802i
\(438\) 0 0
\(439\) −3.05381e6 + 5.28936e6i −0.756277 + 1.30991i 0.188459 + 0.982081i \(0.439651\pi\)
−0.944737 + 0.327830i \(0.893683\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.29170e6 3.96935e6i −0.554816 0.960970i −0.997918 0.0644983i \(-0.979455\pi\)
0.443102 0.896471i \(-0.353878\pi\)
\(444\) 0 0
\(445\) −1.57020e6 −0.375885
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.65968e6 0.622607 0.311304 0.950310i \(-0.399234\pi\)
0.311304 + 0.950310i \(0.399234\pi\)
\(450\) 0 0
\(451\) −4.88553e6 + 8.46199e6i −1.13102 + 1.95899i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.36434e6 −0.761854
\(456\) 0 0
\(457\) 5.09945e6 1.14217 0.571087 0.820889i \(-0.306522\pi\)
0.571087 + 0.820889i \(0.306522\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.18373e6 + 5.51438e6i −0.697725 + 1.20849i 0.271529 + 0.962430i \(0.412471\pi\)
−0.969254 + 0.246064i \(0.920863\pi\)
\(462\) 0 0
\(463\) −5.07803e6 −1.10089 −0.550443 0.834873i \(-0.685541\pi\)
−0.550443 + 0.834873i \(0.685541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.58423e6 −0.760507 −0.380254 0.924882i \(-0.624163\pi\)
−0.380254 + 0.924882i \(0.624163\pi\)
\(468\) 0 0
\(469\) −2.59500e6 4.49467e6i −0.544760 0.943552i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.45381e6 5.98217e6i 0.709815 1.22944i
\(474\) 0 0
\(475\) −1.64792e6 2.98779e6i −0.335121 0.607598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.65416e6 + 6.32919e6i 0.727694 + 1.26040i 0.957855 + 0.287251i \(0.0927414\pi\)
−0.230161 + 0.973153i \(0.573925\pi\)
\(480\) 0 0
\(481\) −1.98064e6 3.43058e6i −0.390341 0.676090i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 411501. + 712740.i 0.0794358 + 0.137587i
\(486\) 0 0
\(487\) 5.63917e6 1.07744 0.538719 0.842485i \(-0.318908\pi\)
0.538719 + 0.842485i \(0.318908\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.98866e6 6.90857e6i 0.746661 1.29325i −0.202754 0.979230i \(-0.564989\pi\)
0.949415 0.314025i \(-0.101678\pi\)
\(492\) 0 0
\(493\) −4.03995e6 −0.748616
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −773345. + 1.33947e6i −0.140437 + 0.243244i
\(498\) 0 0
\(499\) 2.71915e6 4.70971e6i 0.488858 0.846727i −0.511060 0.859545i \(-0.670747\pi\)
0.999918 + 0.0128184i \(0.00408034\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.08679e6 7.07853e6i −0.720216 1.24745i −0.960913 0.276850i \(-0.910709\pi\)
0.240697 0.970600i \(-0.422624\pi\)
\(504\) 0 0
\(505\) −1.98475e6 −0.346320
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.63560e6 + 9.76115e6i 0.964153 + 1.66996i 0.711873 + 0.702308i \(0.247847\pi\)
0.252280 + 0.967654i \(0.418820\pi\)
\(510\) 0 0
\(511\) −1.81107e6 + 3.13686e6i −0.306819 + 0.531427i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.02747e6 1.77964e6i 0.170708 0.295674i
\(516\) 0 0
\(517\) −1.71348e6 2.96783e6i −0.281937 0.488329i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.41682e6 −1.03568 −0.517840 0.855477i \(-0.673264\pi\)
−0.517840 + 0.855477i \(0.673264\pi\)
\(522\) 0 0
\(523\) −1.77525e6 3.07483e6i −0.283796 0.491549i 0.688520 0.725217i \(-0.258261\pi\)
−0.972317 + 0.233668i \(0.924927\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.74529e6 + 1.34152e7i −1.21482 + 2.10413i
\(528\) 0 0
\(529\) 593395. 1.02779e6i 0.0921945 0.159686i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.19865e7 −1.82757
\(534\) 0 0
\(535\) 2.90668e6 5.03451e6i 0.439048 0.760454i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −453702. −0.0672665
\(540\) 0 0
\(541\) 2.87812e6 + 4.98505e6i 0.422781 + 0.732279i 0.996210 0.0869765i \(-0.0277205\pi\)
−0.573429 + 0.819255i \(0.694387\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.77204e6 4.80132e6i −0.399769 0.692420i
\(546\) 0 0
\(547\) 561663. + 972828.i 0.0802615 + 0.139017i 0.903362 0.428878i \(-0.141091\pi\)
−0.823101 + 0.567895i \(0.807758\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.09931e6 3.80619e6i −0.294576 0.534086i
\(552\) 0 0
\(553\) −341601. + 591671.i −0.0475014 + 0.0822749i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.85326e6 4.94199e6i −0.389676 0.674938i 0.602730 0.797945i \(-0.294080\pi\)
−0.992406 + 0.123007i \(0.960746\pi\)
\(558\) 0 0
\(559\) 8.47378e6 1.14696
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.30560e7 −1.73596 −0.867980 0.496599i \(-0.834582\pi\)
−0.867980 + 0.496599i \(0.834582\pi\)
\(564\) 0 0
\(565\) 917543. 1.58923e6i 0.120922 0.209443i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.04953e6 −1.04229 −0.521147 0.853467i \(-0.674496\pi\)
−0.521147 + 0.853467i \(0.674496\pi\)
\(570\) 0 0
\(571\) −604331. −0.0775683 −0.0387842 0.999248i \(-0.512348\pi\)
−0.0387842 + 0.999248i \(0.512348\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.48411e6 + 4.30260e6i −0.313329 + 0.542702i
\(576\) 0 0
\(577\) 3.91150e6 0.489107 0.244554 0.969636i \(-0.421359\pi\)
0.244554 + 0.969636i \(0.421359\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.13916e7 1.40005
\(582\) 0 0
\(583\) −2.49751e6 4.32582e6i −0.304324 0.527105i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 720645. 1.24819e6i 0.0863229 0.149516i −0.819631 0.572891i \(-0.805822\pi\)
0.905954 + 0.423376i \(0.139155\pi\)
\(588\) 0 0
\(589\) −1.66637e7 326070.i −1.97917 0.0387278i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.01304e6 + 1.75464e6i 0.118302 + 0.204904i 0.919095 0.394037i \(-0.128922\pi\)
−0.800793 + 0.598941i \(0.795588\pi\)
\(594\) 0 0
\(595\) 2.99052e6 + 5.17973e6i 0.346301 + 0.599811i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −697546. 1.20819e6i −0.0794339 0.137584i 0.823572 0.567212i \(-0.191978\pi\)
−0.903006 + 0.429628i \(0.858645\pi\)
\(600\) 0 0
\(601\) −1.12873e7 −1.27469 −0.637346 0.770578i \(-0.719968\pi\)
−0.637346 + 0.770578i \(0.719968\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.46412e6 7.73208e6i 0.495847 0.858832i
\(606\) 0 0
\(607\) 1.45297e7 1.60061 0.800306 0.599591i \(-0.204670\pi\)
0.800306 + 0.599591i \(0.204670\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.10197e6 3.64072e6i 0.227784 0.394534i
\(612\) 0 0
\(613\) −7.68601e6 + 1.33126e7i −0.826133 + 1.43090i 0.0749172 + 0.997190i \(0.476131\pi\)
−0.901050 + 0.433715i \(0.857203\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.39641e6 2.41866e6i −0.147673 0.255777i 0.782694 0.622407i \(-0.213845\pi\)
−0.930367 + 0.366630i \(0.880512\pi\)
\(618\) 0 0
\(619\) −8.26283e6 −0.866766 −0.433383 0.901210i \(-0.642680\pi\)
−0.433383 + 0.901210i \(0.642680\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.35641e6 5.81347e6i −0.346461 0.600089i
\(624\) 0 0
\(625\) −856296. + 1.48315e6i −0.0876848 + 0.151874i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.52113e6 + 6.09878e6i −0.354859 + 0.614634i
\(630\) 0 0
\(631\) −329136. 570081.i −0.0329081 0.0569985i 0.849102 0.528228i \(-0.177144\pi\)
−0.882010 + 0.471230i \(0.843810\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.06637e6 −0.203364
\(636\) 0 0
\(637\) −278285. 482004.i −0.0271732 0.0470654i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 626336. 1.08485e6i 0.0602091 0.104285i −0.834350 0.551236i \(-0.814157\pi\)
0.894559 + 0.446950i \(0.147490\pi\)
\(642\) 0 0
\(643\) 7.69468e6 1.33276e7i 0.733945 1.27123i −0.221240 0.975219i \(-0.571010\pi\)
0.955185 0.296010i \(-0.0956562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.31905e6 −0.123880 −0.0619398 0.998080i \(-0.519729\pi\)
−0.0619398 + 0.998080i \(0.519729\pi\)
\(648\) 0 0
\(649\) −1.15011e7 + 1.99205e7i −1.07183 + 1.85647i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.35430e6 0.216062 0.108031 0.994148i \(-0.465545\pi\)
0.108031 + 0.994148i \(0.465545\pi\)
\(654\) 0 0
\(655\) −5.42572e6 9.39762e6i −0.494145 0.855884i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.16918e6 2.02508e6i −0.104874 0.181647i 0.808813 0.588066i \(-0.200111\pi\)
−0.913687 + 0.406419i \(0.866777\pi\)
\(660\) 0 0
\(661\) 4.27183e6 + 7.39902e6i 0.380286 + 0.658674i 0.991103 0.133098i \(-0.0424924\pi\)
−0.610817 + 0.791772i \(0.709159\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.32603e6 + 5.50905e6i −0.291657 + 0.483084i
\(666\) 0 0
\(667\) −3.16454e6 + 5.48115e6i −0.275421 + 0.477042i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.38204e7 2.39376e7i −1.18498 2.05245i
\(672\) 0 0
\(673\) 7.81154e6 0.664813 0.332406 0.943136i \(-0.392139\pi\)
0.332406 + 0.943136i \(0.392139\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.38338e6 0.786842 0.393421 0.919358i \(-0.371291\pi\)
0.393421 + 0.919358i \(0.371291\pi\)
\(678\) 0 0
\(679\) −1.75922e6 + 3.04706e6i −0.146435 + 0.253634i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.64103e7 1.34606 0.673031 0.739614i \(-0.264992\pi\)
0.673031 + 0.739614i \(0.264992\pi\)
\(684\) 0 0
\(685\) 7.69450e6 0.626547
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.06377e6 5.30661e6i 0.245872 0.425862i
\(690\) 0 0
\(691\) −1.44148e7 −1.14846 −0.574228 0.818695i \(-0.694698\pi\)
−0.574228 + 0.818695i \(0.694698\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.37026e7 1.07607
\(696\) 0 0
\(697\) 1.06546e7 + 1.84543e7i 0.830721 + 1.43885i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.22639e6 + 1.59806e7i −0.709148 + 1.22828i 0.256025 + 0.966670i \(0.417587\pi\)
−0.965174 + 0.261610i \(0.915746\pi\)
\(702\) 0 0
\(703\) −7.57559e6 148237.i −0.578134 0.0113127i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.24254e6 7.34829e6i −0.319211 0.552889i
\(708\) 0 0
\(709\) 2.36759e6 + 4.10078e6i 0.176885 + 0.306373i 0.940812 0.338929i \(-0.110065\pi\)
−0.763927 + 0.645302i \(0.776731\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.21340e7 + 2.10166e7i 0.893879 + 1.54824i
\(714\) 0 0
\(715\) 1.70631e7 1.24822
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.63001e6 2.82326e6i 0.117589 0.203671i −0.801222 0.598367i \(-0.795817\pi\)
0.918812 + 0.394696i \(0.129150\pi\)
\(720\) 0 0
\(721\) 8.78518e6 0.629379
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.99495e6 + 5.18740e6i −0.211614 + 0.366526i
\(726\) 0 0
\(727\) 1.20435e7 2.08600e7i 0.845117 1.46379i −0.0404018 0.999184i \(-0.512864\pi\)
0.885519 0.464603i \(-0.153803\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.53222e6 1.30462e7i −0.521350 0.903005i
\(732\) 0 0
\(733\) 8.14935e6 0.560225 0.280113 0.959967i \(-0.409628\pi\)
0.280113 + 0.959967i \(0.409628\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.31611e7 + 2.27958e7i 0.892535 + 1.54592i
\(738\) 0 0
\(739\) −3.07531e6 + 5.32659e6i −0.207146 + 0.358788i −0.950814 0.309761i \(-0.899751\pi\)
0.743668 + 0.668549i \(0.233084\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.28020e6 1.08776e7i 0.417351 0.722873i −0.578321 0.815809i \(-0.696292\pi\)
0.995672 + 0.0929364i \(0.0296253\pi\)
\(744\) 0 0
\(745\) −3.71573e6 6.43584e6i −0.245275 0.424829i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.48529e7 1.61872
\(750\) 0 0
\(751\) −6.87642e6 1.19103e7i −0.444900 0.770590i 0.553145 0.833085i \(-0.313428\pi\)
−0.998045 + 0.0624953i \(0.980094\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.56823e6 + 4.44831e6i −0.163971 + 0.284006i
\(756\) 0 0
\(757\) −3.85263e6 + 6.67296e6i −0.244353 + 0.423232i −0.961950 0.273227i \(-0.911909\pi\)
0.717596 + 0.696459i \(0.245242\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.30446e7 1.44247 0.721236 0.692689i \(-0.243574\pi\)
0.721236 + 0.692689i \(0.243574\pi\)
\(762\) 0 0
\(763\) 1.18509e7 2.05263e7i 0.736951 1.27644i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.82174e7 −1.73193
\(768\) 0 0
\(769\) 8.54029e6 + 1.47922e7i 0.520783 + 0.902023i 0.999708 + 0.0241667i \(0.00769325\pi\)
−0.478925 + 0.877856i \(0.658973\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.20473e6 + 1.24790e7i 0.433680 + 0.751155i 0.997187 0.0749561i \(-0.0238817\pi\)
−0.563507 + 0.826111i \(0.690548\pi\)
\(774\) 0 0
\(775\) 1.14837e7 + 1.98903e7i 0.686794 + 1.18956i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.18500e7 + 1.96276e7i −0.699637 + 1.15884i
\(780\) 0 0
\(781\) 3.92220e6 6.79345e6i 0.230092 0.398532i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.57511e6 + 2.72817e6i 0.0912298 + 0.158015i
\(786\) 0 0
\(787\) −2.96282e7 −1.70518 −0.852588 0.522584i \(-0.824968\pi\)
−0.852588 + 0.522584i \(0.824968\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.84525e6 0.445826
\(792\) 0 0
\(793\) 1.69538e7 2.93649e7i 0.957382 1.65823i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.29223e7 −1.27824 −0.639121 0.769107i \(-0.720702\pi\)
−0.639121 + 0.769107i \(0.720702\pi\)
\(798\) 0 0
\(799\) −7.47365e6 −0.414158
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.18526e6 1.59093e7i 0.502693 0.870690i
\(804\) 0 0
\(805\) 9.37003e6 0.509625
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.65984e6 −0.250322 −0.125161 0.992136i \(-0.539945\pi\)
−0.125161 + 0.992136i \(0.539945\pi\)
\(810\) 0 0
\(811\) 848246. + 1.46921e6i 0.0452866 + 0.0784387i 0.887780 0.460268i \(-0.152247\pi\)
−0.842494 + 0.538706i \(0.818913\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.95927e6 + 6.85766e6i −0.208795 + 0.361644i
\(816\) 0 0
\(817\) 8.37727e6 1.38757e7i 0.439084 0.727275i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −469873. 813845.i −0.0243289 0.0421389i 0.853605 0.520921i \(-0.174412\pi\)
−0.877934 + 0.478783i \(0.841078\pi\)
\(822\) 0 0
\(823\) −1.57070e7 2.72053e7i −0.808338 1.40008i −0.914014 0.405682i \(-0.867034\pi\)
0.105676 0.994401i \(-0.466299\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.21728e6 7.30454e6i −0.214421 0.371389i 0.738672 0.674065i \(-0.235453\pi\)
−0.953093 + 0.302676i \(0.902120\pi\)
\(828\) 0 0
\(829\) −9.81671e6 −0.496112 −0.248056 0.968746i \(-0.579792\pi\)
−0.248056 + 0.968746i \(0.579792\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −494727. + 856892.i −0.0247032 + 0.0427872i
\(834\) 0 0
\(835\) −3.33704e6 −0.165632
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.20733e7 2.09116e7i 0.592135 1.02561i −0.401809 0.915723i \(-0.631618\pi\)
0.993944 0.109885i \(-0.0350482\pi\)
\(840\) 0 0
\(841\) 6.44027e6 1.11549e7i 0.313989 0.543844i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.72403e6 + 8.18227e6i 0.227599 + 0.394214i
\(846\) 0 0
\(847\) 3.81694e7 1.82813
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.51629e6 + 9.55449e6i 0.261110 + 0.452255i
\(852\) 0 0
\(853\) −1.09389e7 + 1.89467e7i −0.514754 + 0.891580i 0.485099 + 0.874459i \(0.338783\pi\)
−0.999853 + 0.0171210i \(0.994550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −664571. + 1.15107e6i −0.0309093 + 0.0535365i −0.881066 0.472993i \(-0.843174\pi\)
0.850157 + 0.526529i \(0.176507\pi\)
\(858\) 0 0
\(859\) −8.07444e6 1.39853e7i −0.373362 0.646681i 0.616719 0.787184i \(-0.288462\pi\)
−0.990080 + 0.140502i \(0.955128\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.37431e7 0.628141 0.314071 0.949400i \(-0.398307\pi\)
0.314071 + 0.949400i \(0.398307\pi\)
\(864\) 0 0
\(865\) 6.10077e6 + 1.05668e7i 0.277233 + 0.480181i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.73251e6 3.00080e6i 0.0778263 0.134799i
\(870\) 0 0
\(871\) −1.61452e7 + 2.79643e7i −0.721103 + 1.24899i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.16478e7 0.955860
\(876\) 0 0
\(877\) −1.51904e7 + 2.63106e7i −0.666916 + 1.15513i 0.311846 + 0.950133i \(0.399053\pi\)
−0.978762 + 0.205000i \(0.934281\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.96130e7 0.851341 0.425671 0.904878i \(-0.360038\pi\)
0.425671 + 0.904878i \(0.360038\pi\)
\(882\) 0 0
\(883\) −9.76660e6 1.69162e7i −0.421543 0.730133i 0.574548 0.818471i \(-0.305178\pi\)
−0.996091 + 0.0883376i \(0.971845\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.00975e6 + 6.94508e6i 0.171123 + 0.296393i 0.938813 0.344428i \(-0.111927\pi\)
−0.767690 + 0.640822i \(0.778594\pi\)
\(888\) 0 0
\(889\) −4.41700e6 7.65047e6i −0.187445 0.324663i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.88359e6 7.04120e6i −0.162969 0.295473i
\(894\) 0 0
\(895\) 3.54577e6 6.14145e6i 0.147963 0.256279i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.46292e7 + 2.53386e7i 0.603701 + 1.04564i
\(900\) 0 0
\(901\) −1.08934e7 −0.447045
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.66943e7 −0.677558
\(906\) 0 0
\(907\) −1.11897e6 + 1.93811e6i −0.0451647 + 0.0782276i −0.887724 0.460376i \(-0.847715\pi\)
0.842559 + 0.538604i \(0.181048\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.98221e7 1.58975 0.794874 0.606774i \(-0.207537\pi\)
0.794874 + 0.606774i \(0.207537\pi\)
\(912\) 0 0
\(913\) −5.77750e7 −2.29384
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.31957e7 4.01761e7i 0.910928 1.57777i
\(918\) 0 0
\(919\) 8.32391e6 0.325116 0.162558 0.986699i \(-0.448025\pi\)
0.162558 + 0.986699i \(0.448025\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.62297e6 0.371796
\(924\) 0 0
\(925\) 5.22066e6 + 9.04245e6i 0.200619 + 0.347482i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.49343e6 + 7.78285e6i −0.170820 + 0.295869i −0.938707 0.344717i \(-0.887975\pi\)
0.767887 + 0.640586i \(0.221308\pi\)
\(930\) 0 0
\(931\) −1.06439e6 20827.6i −0.0402463 0.000787526i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.51671e7 2.62702e7i −0.567380 0.982730i
\(936\) 0 0
\(937\) −623801. 1.08045e6i −0.0232112 0.0402029i 0.854187 0.519967i \(-0.174056\pi\)
−0.877398 + 0.479764i \(0.840722\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.30000e6 + 2.25167e6i 0.0478596 + 0.0828953i 0.888963 0.457979i \(-0.151427\pi\)
−0.841103 + 0.540875i \(0.818093\pi\)
\(942\) 0 0
\(943\) 3.33835e7 1.22251
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.03456e7 + 1.79192e7i −0.374871 + 0.649296i −0.990308 0.138891i \(-0.955646\pi\)
0.615437 + 0.788186i \(0.288980\pi\)
\(948\) 0 0
\(949\) 2.25357e7 0.812279
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.50162e7 + 2.60089e7i −0.535586 + 0.927662i 0.463549 + 0.886071i \(0.346576\pi\)
−0.999135 + 0.0415907i \(0.986757\pi\)
\(954\) 0 0
\(955\) −6.94120e6 + 1.20225e7i −0.246278 + 0.426567i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.64475e7 + 2.84879e7i 0.577502 + 1.00026i
\(960\) 0 0
\(961\) 8.35579e7 2.91863
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −739348. 1.28059e6i −0.0255582 0.0442681i
\(966\) 0 0
\(967\) 7.38839e6 1.27971e7i 0.254088 0.440093i −0.710560 0.703637i \(-0.751558\pi\)
0.964647 + 0.263544i \(0.0848915\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.30678e6 9.19161e6i 0.180627 0.312855i −0.761467 0.648203i \(-0.775521\pi\)
0.942094 + 0.335348i \(0.108854\pi\)
\(972\) 0 0
\(973\) 2.92903e7 + 5.07322e7i 0.991839 + 1.71791i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.66481e7 1.22833 0.614165 0.789177i \(-0.289493\pi\)
0.614165 + 0.789177i \(0.289493\pi\)
\(978\) 0 0
\(979\) 1.70228e7 + 2.94844e7i 0.567642 + 0.983186i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.80511e6 1.17868e7i 0.224621 0.389056i −0.731584 0.681751i \(-0.761219\pi\)
0.956206 + 0.292695i \(0.0945521\pi\)
\(984\) 0 0
\(985\) −677384. + 1.17326e6i −0.0222456 + 0.0385305i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.36003e7 −0.767232
\(990\) 0 0
\(991\) 1.59467e6 2.76205e6i 0.0515806 0.0893402i −0.839082 0.544005i \(-0.816907\pi\)
0.890663 + 0.454664i \(0.150241\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.85615e7 0.594367
\(996\) 0 0
\(997\) −1.71729e7 2.97444e7i −0.547150 0.947692i −0.998468 0.0553288i \(-0.982379\pi\)
0.451318 0.892363i \(-0.350954\pi\)
\(998\) 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.505.5 18
3.2 odd 2 76.6.e.a.49.7 yes 18
12.11 even 2 304.6.i.d.49.3 18
19.7 even 3 inner 684.6.k.f.577.5 18
57.26 odd 6 76.6.e.a.45.7 18
228.83 even 6 304.6.i.d.273.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.7 18 57.26 odd 6
76.6.e.a.49.7 yes 18 3.2 odd 2
304.6.i.d.49.3 18 12.11 even 2
304.6.i.d.273.3 18 228.83 even 6
684.6.k.f.505.5 18 1.1 even 1 trivial
684.6.k.f.577.5 18 19.7 even 3 inner