Properties

Label 600.3.l.f.401.5
Level $600$
Weight $3$
Character 600.401
Analytic conductor $16.349$
Analytic rank $0$
Dimension $8$
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] = [600,3,Mod(401,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (of order \(2\), degree \(1\), 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: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.681615360000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.5
Root \(-0.542939 - 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 600.401
Dual form 600.3.l.f.401.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.40140 - 1.79813i) q^{3} +10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} +O(q^{10})\) \(q+(2.40140 - 1.79813i) q^{3} +10.2132 q^{7} +(2.53346 - 8.63606i) q^{9} -8.19300i q^{11} +13.5822 q^{13} +15.4710i q^{17} -25.4934 q^{19} +(24.5261 - 18.3647i) q^{21} -17.9156i q^{23} +(-9.44491 - 25.2941i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(-14.7321 - 19.6747i) q^{33} -11.8387 q^{37} +(32.6163 - 24.4225i) q^{39} -46.3781i q^{41} +54.0181 q^{43} +43.0955i q^{47} +55.3102 q^{49} +(27.8188 + 37.1521i) q^{51} -82.7421i q^{53} +(-61.2199 + 45.8404i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(25.8748 - 88.2022i) q^{63} -34.4995 q^{67} +(-32.2147 - 43.0227i) q^{69} -68.0061i q^{71} +44.7191 q^{73} -83.6770i q^{77} -11.7499 q^{79} +(-68.1632 - 43.7582i) q^{81} +144.436i q^{83} +(75.5255 + 100.864i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(92.4247 - 69.2061i) q^{93} -63.9013 q^{97} +(-70.7552 - 20.7566i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + 8 q^{13} - 8 q^{19} + 28 q^{21} - 20 q^{27} + 120 q^{31} + 112 q^{33} - 8 q^{37} - 72 q^{39} + 328 q^{43} + 64 q^{49} + 64 q^{51} - 72 q^{57} + 8 q^{61} - 88 q^{63} - 152 q^{67} + 100 q^{69} - 32 q^{73} + 88 q^{79} + 224 q^{81} + 152 q^{87} + 560 q^{91} + 368 q^{93} - 144 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 2.40140 1.79813i 0.800467 0.599377i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 10.2132 1.45903 0.729517 0.683963i \(-0.239745\pi\)
0.729517 + 0.683963i \(0.239745\pi\)
\(8\) 0 0
\(9\) 2.53346 8.63606i 0.281496 0.959563i
\(10\) 0 0
\(11\) 8.19300i 0.744818i −0.928069 0.372409i \(-0.878532\pi\)
0.928069 0.372409i \(-0.121468\pi\)
\(12\) 0 0
\(13\) 13.5822 1.04478 0.522392 0.852705i \(-0.325040\pi\)
0.522392 + 0.852705i \(0.325040\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.4710i 0.910058i 0.890477 + 0.455029i \(0.150371\pi\)
−0.890477 + 0.455029i \(0.849629\pi\)
\(18\) 0 0
\(19\) −25.4934 −1.34176 −0.670879 0.741567i \(-0.734083\pi\)
−0.670879 + 0.741567i \(0.734083\pi\)
\(20\) 0 0
\(21\) 24.5261 18.3647i 1.16791 0.874511i
\(22\) 0 0
\(23\) 17.9156i 0.778941i −0.921039 0.389471i \(-0.872658\pi\)
0.921039 0.389471i \(-0.127342\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −9.44491 25.2941i −0.349811 0.936820i
\(28\) 0 0
\(29\) 42.0022i 1.44835i 0.689615 + 0.724177i \(0.257780\pi\)
−0.689615 + 0.724177i \(0.742220\pi\)
\(30\) 0 0
\(31\) 38.4878 1.24154 0.620771 0.783992i \(-0.286820\pi\)
0.620771 + 0.783992i \(0.286820\pi\)
\(32\) 0 0
\(33\) −14.7321 19.6747i −0.446426 0.596202i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.8387 −0.319965 −0.159982 0.987120i \(-0.551144\pi\)
−0.159982 + 0.987120i \(0.551144\pi\)
\(38\) 0 0
\(39\) 32.6163 24.4225i 0.836315 0.626219i
\(40\) 0 0
\(41\) 46.3781i 1.13117i −0.824689 0.565587i \(-0.808650\pi\)
0.824689 0.565587i \(-0.191350\pi\)
\(42\) 0 0
\(43\) 54.0181 1.25623 0.628117 0.778119i \(-0.283826\pi\)
0.628117 + 0.778119i \(0.283826\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 43.0955i 0.916925i 0.888714 + 0.458462i \(0.151600\pi\)
−0.888714 + 0.458462i \(0.848400\pi\)
\(48\) 0 0
\(49\) 55.3102 1.12878
\(50\) 0 0
\(51\) 27.8188 + 37.1521i 0.545468 + 0.728472i
\(52\) 0 0
\(53\) 82.7421i 1.56117i −0.625049 0.780586i \(-0.714921\pi\)
0.625049 0.780586i \(-0.285079\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −61.2199 + 45.8404i −1.07403 + 0.804218i
\(58\) 0 0
\(59\) 45.8928i 0.777844i −0.921271 0.388922i \(-0.872848\pi\)
0.921271 0.388922i \(-0.127152\pi\)
\(60\) 0 0
\(61\) −93.6873 −1.53586 −0.767929 0.640535i \(-0.778713\pi\)
−0.767929 + 0.640535i \(0.778713\pi\)
\(62\) 0 0
\(63\) 25.8748 88.2022i 0.410712 1.40003i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −34.4995 −0.514917 −0.257459 0.966289i \(-0.582885\pi\)
−0.257459 + 0.966289i \(0.582885\pi\)
\(68\) 0 0
\(69\) −32.2147 43.0227i −0.466879 0.623517i
\(70\) 0 0
\(71\) 68.0061i 0.957832i −0.877861 0.478916i \(-0.841030\pi\)
0.877861 0.478916i \(-0.158970\pi\)
\(72\) 0 0
\(73\) 44.7191 0.612591 0.306295 0.951937i \(-0.400911\pi\)
0.306295 + 0.951937i \(0.400911\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 83.6770i 1.08671i
\(78\) 0 0
\(79\) −11.7499 −0.148733 −0.0743665 0.997231i \(-0.523693\pi\)
−0.0743665 + 0.997231i \(0.523693\pi\)
\(80\) 0 0
\(81\) −68.1632 43.7582i −0.841521 0.540225i
\(82\) 0 0
\(83\) 144.436i 1.74020i 0.492877 + 0.870099i \(0.335945\pi\)
−0.492877 + 0.870099i \(0.664055\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 75.5255 + 100.864i 0.868109 + 1.15936i
\(88\) 0 0
\(89\) 63.7094i 0.715836i 0.933753 + 0.357918i \(0.116513\pi\)
−0.933753 + 0.357918i \(0.883487\pi\)
\(90\) 0 0
\(91\) 138.718 1.52438
\(92\) 0 0
\(93\) 92.4247 69.2061i 0.993814 0.744151i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −63.9013 −0.658776 −0.329388 0.944195i \(-0.606842\pi\)
−0.329388 + 0.944195i \(0.606842\pi\)
\(98\) 0 0
\(99\) −70.7552 20.7566i −0.714699 0.209663i
\(100\) 0 0
\(101\) 50.8769i 0.503731i −0.967762 0.251866i \(-0.918956\pi\)
0.967762 0.251866i \(-0.0810441\pi\)
\(102\) 0 0
\(103\) −45.9491 −0.446108 −0.223054 0.974806i \(-0.571603\pi\)
−0.223054 + 0.974806i \(0.571603\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 110.249i 1.03036i 0.857082 + 0.515181i \(0.172275\pi\)
−0.857082 + 0.515181i \(0.827725\pi\)
\(108\) 0 0
\(109\) −49.2797 −0.452107 −0.226054 0.974115i \(-0.572582\pi\)
−0.226054 + 0.974115i \(0.572582\pi\)
\(110\) 0 0
\(111\) −28.4295 + 21.2875i −0.256121 + 0.191779i
\(112\) 0 0
\(113\) 157.450i 1.39337i 0.717379 + 0.696683i \(0.245342\pi\)
−0.717379 + 0.696683i \(0.754658\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 34.4099 117.297i 0.294102 1.00254i
\(118\) 0 0
\(119\) 158.009i 1.32781i
\(120\) 0 0
\(121\) 53.8748 0.445246
\(122\) 0 0
\(123\) −83.3939 111.373i −0.677999 0.905468i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 135.065 1.06351 0.531753 0.846900i \(-0.321534\pi\)
0.531753 + 0.846900i \(0.321534\pi\)
\(128\) 0 0
\(129\) 129.719 97.1315i 1.00557 0.752957i
\(130\) 0 0
\(131\) 150.653i 1.15002i 0.818146 + 0.575010i \(0.195002\pi\)
−0.818146 + 0.575010i \(0.804998\pi\)
\(132\) 0 0
\(133\) −260.370 −1.95767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 139.086i 1.01523i −0.861585 0.507613i \(-0.830528\pi\)
0.861585 0.507613i \(-0.169472\pi\)
\(138\) 0 0
\(139\) 110.296 0.793496 0.396748 0.917928i \(-0.370139\pi\)
0.396748 + 0.917928i \(0.370139\pi\)
\(140\) 0 0
\(141\) 77.4912 + 103.490i 0.549583 + 0.733968i
\(142\) 0 0
\(143\) 111.279i 0.778174i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 132.822 99.4549i 0.903551 0.676564i
\(148\) 0 0
\(149\) 90.8076i 0.609447i 0.952441 + 0.304723i \(0.0985640\pi\)
−0.952441 + 0.304723i \(0.901436\pi\)
\(150\) 0 0
\(151\) 111.286 0.736992 0.368496 0.929629i \(-0.379873\pi\)
0.368496 + 0.929629i \(0.379873\pi\)
\(152\) 0 0
\(153\) 133.608 + 39.1951i 0.873258 + 0.256177i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −142.755 −0.909269 −0.454635 0.890678i \(-0.650230\pi\)
−0.454635 + 0.890678i \(0.650230\pi\)
\(158\) 0 0
\(159\) −148.781 198.697i −0.935730 1.24967i
\(160\) 0 0
\(161\) 182.977i 1.13650i
\(162\) 0 0
\(163\) 170.179 1.04405 0.522023 0.852932i \(-0.325178\pi\)
0.522023 + 0.852932i \(0.325178\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.38780i 0.0382503i 0.999817 + 0.0191251i \(0.00608809\pi\)
−0.999817 + 0.0191251i \(0.993912\pi\)
\(168\) 0 0
\(169\) 15.4759 0.0915734
\(170\) 0 0
\(171\) −64.5865 + 220.163i −0.377699 + 1.28750i
\(172\) 0 0
\(173\) 238.403i 1.37805i 0.724736 + 0.689027i \(0.241962\pi\)
−0.724736 + 0.689027i \(0.758038\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −82.5212 110.207i −0.466221 0.622638i
\(178\) 0 0
\(179\) 243.541i 1.36057i 0.732950 + 0.680283i \(0.238143\pi\)
−0.732950 + 0.680283i \(0.761857\pi\)
\(180\) 0 0
\(181\) −325.449 −1.79806 −0.899030 0.437887i \(-0.855727\pi\)
−0.899030 + 0.437887i \(0.855727\pi\)
\(182\) 0 0
\(183\) −224.981 + 168.462i −1.22940 + 0.920557i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 126.754 0.677828
\(188\) 0 0
\(189\) −96.4631 258.335i −0.510387 1.36685i
\(190\) 0 0
\(191\) 166.001i 0.869113i 0.900645 + 0.434556i \(0.143095\pi\)
−0.900645 + 0.434556i \(0.856905\pi\)
\(192\) 0 0
\(193\) −239.408 −1.24045 −0.620227 0.784422i \(-0.712959\pi\)
−0.620227 + 0.784422i \(0.712959\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 257.518i 1.30720i 0.756840 + 0.653600i \(0.226742\pi\)
−0.756840 + 0.653600i \(0.773258\pi\)
\(198\) 0 0
\(199\) −78.4735 −0.394339 −0.197170 0.980369i \(-0.563175\pi\)
−0.197170 + 0.980369i \(0.563175\pi\)
\(200\) 0 0
\(201\) −82.8471 + 62.0345i −0.412174 + 0.308629i
\(202\) 0 0
\(203\) 428.979i 2.11320i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −154.721 45.3886i −0.747443 0.219268i
\(208\) 0 0
\(209\) 208.867i 0.999365i
\(210\) 0 0
\(211\) −112.724 −0.534237 −0.267119 0.963664i \(-0.586072\pi\)
−0.267119 + 0.963664i \(0.586072\pi\)
\(212\) 0 0
\(213\) −122.284 163.310i −0.574102 0.766713i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 393.085 1.81145
\(218\) 0 0
\(219\) 107.389 80.4108i 0.490359 0.367172i
\(220\) 0 0
\(221\) 210.130i 0.950814i
\(222\) 0 0
\(223\) 204.686 0.917872 0.458936 0.888469i \(-0.348231\pi\)
0.458936 + 0.888469i \(0.348231\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 115.071i 0.506920i −0.967346 0.253460i \(-0.918431\pi\)
0.967346 0.253460i \(-0.0815685\pi\)
\(228\) 0 0
\(229\) −129.458 −0.565317 −0.282659 0.959221i \(-0.591216\pi\)
−0.282659 + 0.959221i \(0.591216\pi\)
\(230\) 0 0
\(231\) −150.462 200.942i −0.651351 0.869879i
\(232\) 0 0
\(233\) 250.150i 1.07361i 0.843708 + 0.536803i \(0.180368\pi\)
−0.843708 + 0.536803i \(0.819632\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −28.2162 + 21.1278i −0.119056 + 0.0891470i
\(238\) 0 0
\(239\) 49.2556i 0.206090i 0.994677 + 0.103045i \(0.0328586\pi\)
−0.994677 + 0.103045i \(0.967141\pi\)
\(240\) 0 0
\(241\) −457.672 −1.89905 −0.949526 0.313688i \(-0.898435\pi\)
−0.949526 + 0.313688i \(0.898435\pi\)
\(242\) 0 0
\(243\) −242.370 + 17.4851i −0.997408 + 0.0719551i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −346.256 −1.40185
\(248\) 0 0
\(249\) 259.715 + 346.850i 1.04303 + 1.39297i
\(250\) 0 0
\(251\) 119.717i 0.476959i −0.971148 0.238479i \(-0.923351\pi\)
0.971148 0.238479i \(-0.0766489\pi\)
\(252\) 0 0
\(253\) −146.783 −0.580169
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 111.805i 0.435041i 0.976056 + 0.217520i \(0.0697969\pi\)
−0.976056 + 0.217520i \(0.930203\pi\)
\(258\) 0 0
\(259\) −120.911 −0.466839
\(260\) 0 0
\(261\) 362.734 + 106.411i 1.38979 + 0.407705i
\(262\) 0 0
\(263\) 55.3652i 0.210514i 0.994445 + 0.105257i \(0.0335665\pi\)
−0.994445 + 0.105257i \(0.966433\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 114.558 + 152.992i 0.429055 + 0.573003i
\(268\) 0 0
\(269\) 309.427i 1.15028i −0.818053 0.575142i \(-0.804947\pi\)
0.818053 0.575142i \(-0.195053\pi\)
\(270\) 0 0
\(271\) −194.062 −0.716096 −0.358048 0.933703i \(-0.616558\pi\)
−0.358048 + 0.933703i \(0.616558\pi\)
\(272\) 0 0
\(273\) 333.118 249.433i 1.22021 0.913675i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.20416 −0.0187876 −0.00939379 0.999956i \(-0.502990\pi\)
−0.00939379 + 0.999956i \(0.502990\pi\)
\(278\) 0 0
\(279\) 97.5073 332.383i 0.349489 1.19134i
\(280\) 0 0
\(281\) 240.004i 0.854105i 0.904227 + 0.427053i \(0.140448\pi\)
−0.904227 + 0.427053i \(0.859552\pi\)
\(282\) 0 0
\(283\) −153.109 −0.541022 −0.270511 0.962717i \(-0.587193\pi\)
−0.270511 + 0.962717i \(0.587193\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 473.671i 1.65042i
\(288\) 0 0
\(289\) 49.6485 0.171794
\(290\) 0 0
\(291\) −153.453 + 114.903i −0.527329 + 0.394855i
\(292\) 0 0
\(293\) 169.252i 0.577653i −0.957381 0.288827i \(-0.906735\pi\)
0.957381 0.288827i \(-0.0932651\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −207.235 + 77.3821i −0.697760 + 0.260546i
\(298\) 0 0
\(299\) 243.334i 0.813825i
\(300\) 0 0
\(301\) 551.699 1.83289
\(302\) 0 0
\(303\) −91.4832 122.176i −0.301925 0.403220i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −330.401 −1.07623 −0.538113 0.842873i \(-0.680863\pi\)
−0.538113 + 0.842873i \(0.680863\pi\)
\(308\) 0 0
\(309\) −110.342 + 82.6224i −0.357095 + 0.267387i
\(310\) 0 0
\(311\) 133.596i 0.429570i −0.976661 0.214785i \(-0.931095\pi\)
0.976661 0.214785i \(-0.0689050\pi\)
\(312\) 0 0
\(313\) 459.981 1.46959 0.734794 0.678290i \(-0.237279\pi\)
0.734794 + 0.678290i \(0.237279\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 298.150i 0.940536i −0.882524 0.470268i \(-0.844157\pi\)
0.882524 0.470268i \(-0.155843\pi\)
\(318\) 0 0
\(319\) 344.124 1.07876
\(320\) 0 0
\(321\) 198.241 + 264.751i 0.617574 + 0.824771i
\(322\) 0 0
\(323\) 394.408i 1.22108i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −118.340 + 88.6112i −0.361897 + 0.270982i
\(328\) 0 0
\(329\) 440.144i 1.33782i
\(330\) 0 0
\(331\) 397.599 1.20120 0.600602 0.799548i \(-0.294927\pi\)
0.600602 + 0.799548i \(0.294927\pi\)
\(332\) 0 0
\(333\) −29.9929 + 102.240i −0.0900687 + 0.307026i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 427.942 1.26986 0.634929 0.772571i \(-0.281030\pi\)
0.634929 + 0.772571i \(0.281030\pi\)
\(338\) 0 0
\(339\) 283.116 + 378.101i 0.835151 + 1.11534i
\(340\) 0 0
\(341\) 315.331i 0.924723i
\(342\) 0 0
\(343\) 64.4477 0.187894
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 138.137i 0.398090i 0.979990 + 0.199045i \(0.0637840\pi\)
−0.979990 + 0.199045i \(0.936216\pi\)
\(348\) 0 0
\(349\) −301.421 −0.863669 −0.431835 0.901953i \(-0.642134\pi\)
−0.431835 + 0.901953i \(0.642134\pi\)
\(350\) 0 0
\(351\) −128.283 343.550i −0.365477 0.978775i
\(352\) 0 0
\(353\) 142.622i 0.404028i −0.979383 0.202014i \(-0.935251\pi\)
0.979383 0.202014i \(-0.0647487\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 284.120 + 379.443i 0.795856 + 1.06286i
\(358\) 0 0
\(359\) 3.05663i 0.00851429i 0.999991 + 0.00425715i \(0.00135510\pi\)
−0.999991 + 0.00425715i \(0.998645\pi\)
\(360\) 0 0
\(361\) 288.913 0.800313
\(362\) 0 0
\(363\) 129.375 96.8739i 0.356405 0.266870i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 370.072 1.00837 0.504186 0.863595i \(-0.331793\pi\)
0.504186 + 0.863595i \(0.331793\pi\)
\(368\) 0 0
\(369\) −400.525 117.497i −1.08543 0.318420i
\(370\) 0 0
\(371\) 845.065i 2.27780i
\(372\) 0 0
\(373\) −455.556 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 570.482i 1.51322i
\(378\) 0 0
\(379\) −380.051 −1.00277 −0.501387 0.865223i \(-0.667177\pi\)
−0.501387 + 0.865223i \(0.667177\pi\)
\(380\) 0 0
\(381\) 324.346 242.865i 0.851301 0.637440i
\(382\) 0 0
\(383\) 164.078i 0.428402i −0.976790 0.214201i \(-0.931285\pi\)
0.976790 0.214201i \(-0.0687147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 136.853 466.503i 0.353624 1.20544i
\(388\) 0 0
\(389\) 562.622i 1.44633i −0.690676 0.723164i \(-0.742687\pi\)
0.690676 0.723164i \(-0.257313\pi\)
\(390\) 0 0
\(391\) 277.173 0.708882
\(392\) 0 0
\(393\) 270.893 + 361.778i 0.689295 + 0.920553i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 93.4668 0.235433 0.117716 0.993047i \(-0.462443\pi\)
0.117716 + 0.993047i \(0.462443\pi\)
\(398\) 0 0
\(399\) −625.253 + 468.179i −1.56705 + 1.17338i
\(400\) 0 0
\(401\) 23.4831i 0.0585613i −0.999571 0.0292807i \(-0.990678\pi\)
0.999571 0.0292807i \(-0.00932166\pi\)
\(402\) 0 0
\(403\) 522.749 1.29714
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 96.9944i 0.238316i
\(408\) 0 0
\(409\) 697.472 1.70531 0.852655 0.522474i \(-0.174991\pi\)
0.852655 + 0.522474i \(0.174991\pi\)
\(410\) 0 0
\(411\) −250.095 334.001i −0.608503 0.812656i
\(412\) 0 0
\(413\) 468.714i 1.13490i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 264.865 198.326i 0.635167 0.475603i
\(418\) 0 0
\(419\) 115.537i 0.275744i −0.990450 0.137872i \(-0.955974\pi\)
0.990450 0.137872i \(-0.0440262\pi\)
\(420\) 0 0
\(421\) 117.396 0.278850 0.139425 0.990233i \(-0.455475\pi\)
0.139425 + 0.990233i \(0.455475\pi\)
\(422\) 0 0
\(423\) 372.175 + 109.181i 0.879847 + 0.258110i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −956.851 −2.24087
\(428\) 0 0
\(429\) −200.094 267.225i −0.466419 0.622903i
\(430\) 0 0
\(431\) 673.123i 1.56177i 0.624675 + 0.780885i \(0.285232\pi\)
−0.624675 + 0.780885i \(0.714768\pi\)
\(432\) 0 0
\(433\) 392.655 0.906824 0.453412 0.891301i \(-0.350207\pi\)
0.453412 + 0.891301i \(0.350207\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 456.731i 1.04515i
\(438\) 0 0
\(439\) 409.353 0.932468 0.466234 0.884661i \(-0.345611\pi\)
0.466234 + 0.884661i \(0.345611\pi\)
\(440\) 0 0
\(441\) 140.126 477.662i 0.317746 1.08313i
\(442\) 0 0
\(443\) 355.935i 0.803464i −0.915757 0.401732i \(-0.868408\pi\)
0.915757 0.401732i \(-0.131592\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 163.284 + 218.065i 0.365288 + 0.487842i
\(448\) 0 0
\(449\) 294.636i 0.656205i −0.944642 0.328103i \(-0.893591\pi\)
0.944642 0.328103i \(-0.106409\pi\)
\(450\) 0 0
\(451\) −379.976 −0.842519
\(452\) 0 0
\(453\) 267.242 200.106i 0.589938 0.441736i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −491.814 −1.07618 −0.538090 0.842887i \(-0.680854\pi\)
−0.538090 + 0.842887i \(0.680854\pi\)
\(458\) 0 0
\(459\) 391.325 146.122i 0.852561 0.318349i
\(460\) 0 0
\(461\) 42.6832i 0.0925883i 0.998928 + 0.0462942i \(0.0147412\pi\)
−0.998928 + 0.0462942i \(0.985259\pi\)
\(462\) 0 0
\(463\) 422.582 0.912705 0.456352 0.889799i \(-0.349156\pi\)
0.456352 + 0.889799i \(0.349156\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 203.497i 0.435754i 0.975976 + 0.217877i \(0.0699131\pi\)
−0.975976 + 0.217877i \(0.930087\pi\)
\(468\) 0 0
\(469\) −352.351 −0.751282
\(470\) 0 0
\(471\) −342.813 + 256.693i −0.727840 + 0.544995i
\(472\) 0 0
\(473\) 442.570i 0.935666i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −714.566 209.624i −1.49804 0.439463i
\(478\) 0 0
\(479\) 485.253i 1.01305i 0.862224 + 0.506527i \(0.169071\pi\)
−0.862224 + 0.506527i \(0.830929\pi\)
\(480\) 0 0
\(481\) −160.795 −0.334294
\(482\) 0 0
\(483\) −329.016 439.401i −0.681192 0.909732i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −140.013 −0.287500 −0.143750 0.989614i \(-0.545916\pi\)
−0.143750 + 0.989614i \(0.545916\pi\)
\(488\) 0 0
\(489\) 408.669 306.005i 0.835724 0.625776i
\(490\) 0 0
\(491\) 20.4450i 0.0416396i −0.999783 0.0208198i \(-0.993372\pi\)
0.999783 0.0208198i \(-0.00662763\pi\)
\(492\) 0 0
\(493\) −649.816 −1.31809
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 694.562i 1.39751i
\(498\) 0 0
\(499\) 49.2051 0.0986074 0.0493037 0.998784i \(-0.484300\pi\)
0.0493037 + 0.998784i \(0.484300\pi\)
\(500\) 0 0
\(501\) 11.4861 + 15.3397i 0.0229263 + 0.0306181i
\(502\) 0 0
\(503\) 149.060i 0.296343i −0.988962 0.148171i \(-0.952661\pi\)
0.988962 0.148171i \(-0.0473387\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 37.1639 27.8277i 0.0733015 0.0548870i
\(508\) 0 0
\(509\) 655.375i 1.28757i −0.765205 0.643787i \(-0.777362\pi\)
0.765205 0.643787i \(-0.222638\pi\)
\(510\) 0 0
\(511\) 456.727 0.893790
\(512\) 0 0
\(513\) 240.783 + 644.834i 0.469362 + 1.25699i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 353.081 0.682942
\(518\) 0 0
\(519\) 428.680 + 572.502i 0.825973 + 1.10309i
\(520\) 0 0
\(521\) 419.868i 0.805890i 0.915224 + 0.402945i \(0.132013\pi\)
−0.915224 + 0.402945i \(0.867987\pi\)
\(522\) 0 0
\(523\) −146.678 −0.280454 −0.140227 0.990119i \(-0.544783\pi\)
−0.140227 + 0.990119i \(0.544783\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 595.444i 1.12988i
\(528\) 0 0
\(529\) 208.030 0.393251
\(530\) 0 0
\(531\) −396.333 116.268i −0.746390 0.218960i
\(532\) 0 0
\(533\) 629.917i 1.18183i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 437.919 + 584.840i 0.815491 + 1.08909i
\(538\) 0 0
\(539\) 453.156i 0.840735i
\(540\) 0 0
\(541\) 7.07300 0.0130739 0.00653697 0.999979i \(-0.497919\pi\)
0.00653697 + 0.999979i \(0.497919\pi\)
\(542\) 0 0
\(543\) −781.534 + 585.199i −1.43929 + 1.07772i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.46995 −0.00634360 −0.00317180 0.999995i \(-0.501010\pi\)
−0.00317180 + 0.999995i \(0.501010\pi\)
\(548\) 0 0
\(549\) −237.353 + 809.090i −0.432337 + 1.47375i
\(550\) 0 0
\(551\) 1070.78i 1.94334i
\(552\) 0 0
\(553\) −120.005 −0.217006
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 654.467i 1.17499i −0.809230 0.587493i \(-0.800115\pi\)
0.809230 0.587493i \(-0.199885\pi\)
\(558\) 0 0
\(559\) 733.684 1.31249
\(560\) 0 0
\(561\) 304.387 227.920i 0.542579 0.406274i
\(562\) 0 0
\(563\) 546.529i 0.970745i −0.874307 0.485373i \(-0.838684\pi\)
0.874307 0.485373i \(-0.161316\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −696.167 446.913i −1.22781 0.788207i
\(568\) 0 0
\(569\) 746.774i 1.31243i 0.754573 + 0.656216i \(0.227844\pi\)
−0.754573 + 0.656216i \(0.772156\pi\)
\(570\) 0 0
\(571\) −230.358 −0.403430 −0.201715 0.979444i \(-0.564651\pi\)
−0.201715 + 0.979444i \(0.564651\pi\)
\(572\) 0 0
\(573\) 298.491 + 398.634i 0.520926 + 0.695696i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −525.585 −0.910893 −0.455446 0.890263i \(-0.650520\pi\)
−0.455446 + 0.890263i \(0.650520\pi\)
\(578\) 0 0
\(579\) −574.914 + 430.486i −0.992943 + 0.743499i
\(580\) 0 0
\(581\) 1475.16i 2.53901i
\(582\) 0 0
\(583\) −677.906 −1.16279
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 75.3619i 0.128385i −0.997938 0.0641924i \(-0.979553\pi\)
0.997938 0.0641924i \(-0.0204471\pi\)
\(588\) 0 0
\(589\) −981.185 −1.66585
\(590\) 0 0
\(591\) 463.051 + 618.405i 0.783505 + 1.04637i
\(592\) 0 0
\(593\) 1142.87i 1.92727i 0.267230 + 0.963633i \(0.413892\pi\)
−0.267230 + 0.963633i \(0.586108\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −188.446 + 141.106i −0.315656 + 0.236358i
\(598\) 0 0
\(599\) 1060.02i 1.76965i −0.465927 0.884823i \(-0.654279\pi\)
0.465927 0.884823i \(-0.345721\pi\)
\(600\) 0 0
\(601\) −530.900 −0.883361 −0.441681 0.897172i \(-0.645618\pi\)
−0.441681 + 0.897172i \(0.645618\pi\)
\(602\) 0 0
\(603\) −87.4030 + 297.940i −0.144947 + 0.494095i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 719.995 1.18615 0.593077 0.805146i \(-0.297913\pi\)
0.593077 + 0.805146i \(0.297913\pi\)
\(608\) 0 0
\(609\) 771.360 + 1030.15i 1.26660 + 1.69154i
\(610\) 0 0
\(611\) 585.331i 0.957988i
\(612\) 0 0
\(613\) 216.348 0.352932 0.176466 0.984307i \(-0.443533\pi\)
0.176466 + 0.984307i \(0.443533\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 231.190i 0.374701i 0.982293 + 0.187350i \(0.0599900\pi\)
−0.982293 + 0.187350i \(0.940010\pi\)
\(618\) 0 0
\(619\) −689.633 −1.11411 −0.557054 0.830476i \(-0.688068\pi\)
−0.557054 + 0.830476i \(0.688068\pi\)
\(620\) 0 0
\(621\) −453.161 + 169.212i −0.729728 + 0.272482i
\(622\) 0 0
\(623\) 650.679i 1.04443i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 375.570 + 501.574i 0.598996 + 0.799959i
\(628\) 0 0
\(629\) 183.156i 0.291187i
\(630\) 0 0
\(631\) −616.973 −0.977770 −0.488885 0.872348i \(-0.662596\pi\)
−0.488885 + 0.872348i \(0.662596\pi\)
\(632\) 0 0
\(633\) −270.696 + 202.692i −0.427639 + 0.320209i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 751.234 1.17933
\(638\) 0 0
\(639\) −587.305 172.291i −0.919100 0.269626i
\(640\) 0 0
\(641\) 1131.85i 1.76576i −0.469598 0.882880i \(-0.655601\pi\)
0.469598 0.882880i \(-0.344399\pi\)
\(642\) 0 0
\(643\) 637.481 0.991418 0.495709 0.868489i \(-0.334908\pi\)
0.495709 + 0.868489i \(0.334908\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 800.782i 1.23769i −0.785515 0.618843i \(-0.787602\pi\)
0.785515 0.618843i \(-0.212398\pi\)
\(648\) 0 0
\(649\) −375.999 −0.579352
\(650\) 0 0
\(651\) 943.955 706.818i 1.45001 1.08574i
\(652\) 0 0
\(653\) 313.185i 0.479610i −0.970821 0.239805i \(-0.922916\pi\)
0.970821 0.239805i \(-0.0770835\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 113.294 386.197i 0.172442 0.587819i
\(658\) 0 0
\(659\) 878.311i 1.33279i −0.745597 0.666397i \(-0.767836\pi\)
0.745597 0.666397i \(-0.232164\pi\)
\(660\) 0 0
\(661\) 200.421 0.303208 0.151604 0.988441i \(-0.451556\pi\)
0.151604 + 0.988441i \(0.451556\pi\)
\(662\) 0 0
\(663\) 377.841 + 504.606i 0.569896 + 0.761096i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 752.497 1.12818
\(668\) 0 0
\(669\) 491.532 368.051i 0.734727 0.550151i
\(670\) 0 0
\(671\) 767.580i 1.14393i
\(672\) 0 0
\(673\) 60.8622 0.0904341 0.0452171 0.998977i \(-0.485602\pi\)
0.0452171 + 0.998977i \(0.485602\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 415.595i 0.613877i −0.951729 0.306939i \(-0.900695\pi\)
0.951729 0.306939i \(-0.0993047\pi\)
\(678\) 0 0
\(679\) −652.639 −0.961177
\(680\) 0 0
\(681\) −206.912 276.331i −0.303836 0.405772i
\(682\) 0 0
\(683\) 434.504i 0.636170i 0.948062 + 0.318085i \(0.103040\pi\)
−0.948062 + 0.318085i \(0.896960\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −310.880 + 232.782i −0.452518 + 0.338838i
\(688\) 0 0
\(689\) 1123.82i 1.63109i
\(690\) 0 0
\(691\) −166.630 −0.241144 −0.120572 0.992705i \(-0.538473\pi\)
−0.120572 + 0.992705i \(0.538473\pi\)
\(692\) 0 0
\(693\) −722.640 211.992i −1.04277 0.305905i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 717.516 1.02943
\(698\) 0 0
\(699\) 449.802 + 600.711i 0.643494 + 0.859386i
\(700\) 0 0
\(701\) 248.345i 0.354272i 0.984186 + 0.177136i \(0.0566833\pi\)
−0.984186 + 0.177136i \(0.943317\pi\)
\(702\) 0 0
\(703\) 301.809 0.429315
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 519.617i 0.734961i
\(708\) 0 0
\(709\) −24.1911 −0.0341200 −0.0170600 0.999854i \(-0.505431\pi\)
−0.0170600 + 0.999854i \(0.505431\pi\)
\(710\) 0 0
\(711\) −29.7679 + 101.473i −0.0418677 + 0.142719i
\(712\) 0 0
\(713\) 689.534i 0.967088i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 88.5679 + 118.282i 0.123526 + 0.164969i
\(718\) 0 0
\(719\) 323.584i 0.450048i −0.974353 0.225024i \(-0.927754\pi\)
0.974353 0.225024i \(-0.0722460\pi\)
\(720\) 0 0
\(721\) −469.289 −0.650886
\(722\) 0 0
\(723\) −1099.05 + 822.953i −1.52013 + 1.13825i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 454.949 0.625789 0.312895 0.949788i \(-0.398701\pi\)
0.312895 + 0.949788i \(0.398701\pi\)
\(728\) 0 0
\(729\) −550.588 + 477.802i −0.755264 + 0.655421i
\(730\) 0 0
\(731\) 835.713i 1.14325i
\(732\) 0 0
\(733\) −131.588 −0.179520 −0.0897600 0.995963i \(-0.528610\pi\)
−0.0897600 + 0.995963i \(0.528610\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 282.654i 0.383520i
\(738\) 0 0
\(739\) 1033.59 1.39864 0.699319 0.714810i \(-0.253487\pi\)
0.699319 + 0.714810i \(0.253487\pi\)
\(740\) 0 0
\(741\) −831.500 + 622.613i −1.12213 + 0.840234i
\(742\) 0 0
\(743\) 439.535i 0.591569i 0.955255 + 0.295784i \(0.0955810\pi\)
−0.955255 + 0.295784i \(0.904419\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1247.36 + 365.924i 1.66983 + 0.489858i
\(748\) 0 0
\(749\) 1126.00i 1.50333i
\(750\) 0 0
\(751\) 496.343 0.660909 0.330455 0.943822i \(-0.392798\pi\)
0.330455 + 0.943822i \(0.392798\pi\)
\(752\) 0 0
\(753\) −215.266 287.488i −0.285878 0.381790i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −468.694 −0.619147 −0.309574 0.950875i \(-0.600186\pi\)
−0.309574 + 0.950875i \(0.600186\pi\)
\(758\) 0 0
\(759\) −352.485 + 263.935i −0.464406 + 0.347740i
\(760\) 0 0
\(761\) 156.940i 0.206229i −0.994669 0.103115i \(-0.967119\pi\)
0.994669 0.103115i \(-0.0328808\pi\)
\(762\) 0 0
\(763\) −503.305 −0.659640
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 623.325i 0.812679i
\(768\) 0 0
\(769\) 1188.63 1.54569 0.772844 0.634596i \(-0.218834\pi\)
0.772844 + 0.634596i \(0.218834\pi\)
\(770\) 0 0
\(771\) 201.041 + 268.490i 0.260753 + 0.348236i
\(772\) 0 0
\(773\) 950.291i 1.22935i 0.788779 + 0.614677i \(0.210714\pi\)
−0.788779 + 0.614677i \(0.789286\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −290.357 + 217.414i −0.373690 + 0.279813i
\(778\) 0 0
\(779\) 1182.34i 1.51776i
\(780\) 0 0
\(781\) −557.174 −0.713411
\(782\) 0 0
\(783\) 1062.41 396.707i 1.35685 0.506650i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1056.60 −1.34257 −0.671284 0.741200i \(-0.734257\pi\)
−0.671284 + 0.741200i \(0.734257\pi\)
\(788\) 0 0
\(789\) 99.5538 + 132.954i 0.126177 + 0.168510i
\(790\) 0 0
\(791\) 1608.08i 2.03297i
\(792\) 0 0
\(793\) −1272.48 −1.60464
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 78.7126i 0.0987612i −0.998780 0.0493806i \(-0.984275\pi\)
0.998780 0.0493806i \(-0.0157247\pi\)
\(798\) 0 0
\(799\) −666.730 −0.834455
\(800\) 0 0
\(801\) 550.198 + 161.405i 0.686889 + 0.201505i
\(802\) 0 0
\(803\) 366.384i 0.456268i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −556.389 743.058i −0.689454 0.920765i
\(808\) 0 0
\(809\) 176.199i 0.217799i −0.994053 0.108899i \(-0.965267\pi\)
0.994053 0.108899i \(-0.0347327\pi\)
\(810\) 0 0
\(811\) 1103.89 1.36115 0.680574 0.732679i \(-0.261730\pi\)
0.680574 + 0.732679i \(0.261730\pi\)
\(812\) 0 0
\(813\) −466.021 + 348.949i −0.573211 + 0.429211i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1377.10 −1.68556
\(818\) 0 0
\(819\) 351.437 1197.98i 0.429105 1.46273i
\(820\) 0 0
\(821\) 1321.32i 1.60940i −0.593679 0.804702i \(-0.702325\pi\)
0.593679 0.804702i \(-0.297675\pi\)
\(822\) 0 0
\(823\) −760.814 −0.924439 −0.462220 0.886765i \(-0.652947\pi\)
−0.462220 + 0.886765i \(0.652947\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 610.599i 0.738331i 0.929364 + 0.369165i \(0.120356\pi\)
−0.929364 + 0.369165i \(0.879644\pi\)
\(828\) 0 0
\(829\) 265.766 0.320586 0.160293 0.987069i \(-0.448756\pi\)
0.160293 + 0.987069i \(0.448756\pi\)
\(830\) 0 0
\(831\) −12.4973 + 9.35776i −0.0150388 + 0.0112608i
\(832\) 0 0
\(833\) 855.704i 1.02726i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −363.514 973.516i −0.434306 1.16310i
\(838\) 0 0
\(839\) 48.5299i 0.0578425i −0.999582 0.0289213i \(-0.990793\pi\)
0.999582 0.0289213i \(-0.00920721\pi\)
\(840\) 0 0
\(841\) −923.188 −1.09773
\(842\) 0 0
\(843\) 431.558 + 576.345i 0.511931 + 0.683683i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 550.236 0.649629
\(848\) 0 0
\(849\) −367.677 + 275.310i −0.433071 + 0.324276i
\(850\) 0 0
\(851\) 212.098i 0.249234i
\(852\) 0 0
\(853\) 331.180 0.388253 0.194127 0.980976i \(-0.437813\pi\)
0.194127 + 0.980976i \(0.437813\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 159.395i 0.185992i 0.995666 + 0.0929961i \(0.0296444\pi\)
−0.995666 + 0.0929961i \(0.970356\pi\)
\(858\) 0 0
\(859\) −1635.01 −1.90339 −0.951694 0.307049i \(-0.900658\pi\)
−0.951694 + 0.307049i \(0.900658\pi\)
\(860\) 0 0
\(861\) −851.722 1137.47i −0.989224 1.32111i
\(862\) 0 0
\(863\) 152.237i 0.176405i −0.996103 0.0882025i \(-0.971888\pi\)
0.996103 0.0882025i \(-0.0281123\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 119.226 89.2744i 0.137516 0.102969i
\(868\) 0 0
\(869\) 96.2669i 0.110779i
\(870\) 0 0
\(871\) −468.578 −0.537977
\(872\) 0 0
\(873\) −161.891 + 551.856i −0.185443 + 0.632137i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −521.588 −0.594742 −0.297371 0.954762i \(-0.596110\pi\)
−0.297371 + 0.954762i \(0.596110\pi\)
\(878\) 0 0
\(879\) −304.338 406.443i −0.346232 0.462392i
\(880\) 0 0
\(881\) 821.701i 0.932692i 0.884603 + 0.466346i \(0.154430\pi\)
−0.884603 + 0.466346i \(0.845570\pi\)
\(882\) 0 0
\(883\) −747.791 −0.846876 −0.423438 0.905925i \(-0.639177\pi\)
−0.423438 + 0.905925i \(0.639177\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1160.21i 1.30801i 0.756490 + 0.654006i \(0.226913\pi\)
−0.756490 + 0.654006i \(0.773087\pi\)
\(888\) 0 0
\(889\) 1379.45 1.55169
\(890\) 0 0
\(891\) −358.511 + 558.461i −0.402369 + 0.626780i
\(892\) 0 0
\(893\) 1098.65i 1.23029i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −437.546 584.342i −0.487788 0.651440i
\(898\) 0 0
\(899\) 1616.57i 1.79819i
\(900\) 0 0
\(901\) 1280.10 1.42076
\(902\) 0 0
\(903\) 1324.85 992.027i 1.46717 1.09859i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 354.207 0.390526 0.195263 0.980751i \(-0.437444\pi\)
0.195263 + 0.980751i \(0.437444\pi\)
\(908\) 0 0
\(909\) −439.376 128.894i −0.483362 0.141798i
\(910\) 0 0
\(911\) 1461.82i 1.60463i −0.596900 0.802316i \(-0.703601\pi\)
0.596900 0.802316i \(-0.296399\pi\)
\(912\) 0 0
\(913\) 1183.37 1.29613
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1538.65i 1.67792i
\(918\) 0 0
\(919\) 600.104 0.652997 0.326499 0.945198i \(-0.394131\pi\)
0.326499 + 0.945198i \(0.394131\pi\)
\(920\) 0 0
\(921\) −793.427 + 594.105i −0.861484 + 0.645065i
\(922\) 0 0
\(923\) 923.672i 1.00073i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −116.410 + 396.819i −0.125577 + 0.428068i
\(928\) 0 0
\(929\) 243.828i 0.262462i −0.991352 0.131231i \(-0.958107\pi\)
0.991352 0.131231i \(-0.0418930\pi\)
\(930\) 0 0
\(931\) −1410.05 −1.51455
\(932\) 0 0
\(933\) −240.223 320.818i −0.257474 0.343856i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −894.637 −0.954789 −0.477395 0.878689i \(-0.658419\pi\)
−0.477395 + 0.878689i \(0.658419\pi\)
\(938\) 0 0
\(939\) 1104.60 827.105i 1.17636 0.880837i
\(940\) 0 0
\(941\) 592.336i 0.629475i −0.949179 0.314737i \(-0.898084\pi\)
0.949179 0.314737i \(-0.101916\pi\)
\(942\) 0 0
\(943\) −830.894 −0.881118
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 526.526i 0.555994i −0.960582 0.277997i \(-0.910330\pi\)
0.960582 0.277997i \(-0.0896705\pi\)
\(948\) 0 0
\(949\) 607.383 0.640025
\(950\) 0 0
\(951\) −536.112 715.977i −0.563735 0.752868i
\(952\) 0 0
\(953\) 117.158i 0.122936i −0.998109 0.0614678i \(-0.980422\pi\)
0.998109 0.0614678i \(-0.0195781\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 826.381 618.780i 0.863512 0.646583i
\(958\) 0 0
\(959\) 1420.52i 1.48125i
\(960\) 0 0
\(961\) 520.311 0.541427
\(962\) 0 0
\(963\) 952.114 + 279.311i 0.988696 + 0.290042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 811.294 0.838981 0.419490 0.907760i \(-0.362209\pi\)
0.419490 + 0.907760i \(0.362209\pi\)
\(968\) 0 0
\(969\) −709.197 947.132i −0.731885 0.977432i
\(970\) 0 0
\(971\) 1052.66i 1.08410i 0.840346 + 0.542050i \(0.182352\pi\)
−0.840346 + 0.542050i \(0.817648\pi\)
\(972\) 0 0
\(973\) 1126.48 1.15774
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 830.832i 0.850391i −0.905102 0.425195i \(-0.860205\pi\)
0.905102 0.425195i \(-0.139795\pi\)
\(978\) 0 0
\(979\) 521.971 0.533167
\(980\) 0 0
\(981\) −124.848 + 425.582i −0.127266 + 0.433825i
\(982\) 0 0
\(983\) 1056.30i 1.07457i −0.843402 0.537283i \(-0.819451\pi\)
0.843402 0.537283i \(-0.180549\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 791.436 + 1056.96i 0.801861 + 1.07088i
\(988\) 0 0
\(989\) 967.768i 0.978532i
\(990\) 0 0
\(991\) 1592.65 1.60712 0.803559 0.595225i \(-0.202937\pi\)
0.803559 + 0.595225i \(0.202937\pi\)
\(992\) 0 0
\(993\) 954.795 714.934i 0.961525 0.719974i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −978.023 −0.980966 −0.490483 0.871451i \(-0.663180\pi\)
−0.490483 + 0.871451i \(0.663180\pi\)
\(998\) 0 0
\(999\) 111.815 + 299.450i 0.111927 + 0.299749i
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 600.3.l.f.401.5 8
3.2 odd 2 inner 600.3.l.f.401.6 8
4.3 odd 2 1200.3.l.x.401.4 8
5.2 odd 4 600.3.c.d.449.3 16
5.3 odd 4 600.3.c.d.449.14 16
5.4 even 2 120.3.l.a.41.4 yes 8
12.11 even 2 1200.3.l.x.401.3 8
15.2 even 4 600.3.c.d.449.13 16
15.8 even 4 600.3.c.d.449.4 16
15.14 odd 2 120.3.l.a.41.3 8
20.3 even 4 1200.3.c.m.449.3 16
20.7 even 4 1200.3.c.m.449.14 16
20.19 odd 2 240.3.l.d.161.5 8
40.19 odd 2 960.3.l.g.641.4 8
40.29 even 2 960.3.l.h.641.5 8
60.23 odd 4 1200.3.c.m.449.13 16
60.47 odd 4 1200.3.c.m.449.4 16
60.59 even 2 240.3.l.d.161.6 8
120.29 odd 2 960.3.l.h.641.6 8
120.59 even 2 960.3.l.g.641.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.3 8 15.14 odd 2
120.3.l.a.41.4 yes 8 5.4 even 2
240.3.l.d.161.5 8 20.19 odd 2
240.3.l.d.161.6 8 60.59 even 2
600.3.c.d.449.3 16 5.2 odd 4
600.3.c.d.449.4 16 15.8 even 4
600.3.c.d.449.13 16 15.2 even 4
600.3.c.d.449.14 16 5.3 odd 4
600.3.l.f.401.5 8 1.1 even 1 trivial
600.3.l.f.401.6 8 3.2 odd 2 inner
960.3.l.g.641.3 8 120.59 even 2
960.3.l.g.641.4 8 40.19 odd 2
960.3.l.h.641.5 8 40.29 even 2
960.3.l.h.641.6 8 120.29 odd 2
1200.3.c.m.449.3 16 20.3 even 4
1200.3.c.m.449.4 16 60.47 odd 4
1200.3.c.m.449.13 16 60.23 odd 4
1200.3.c.m.449.14 16 20.7 even 4
1200.3.l.x.401.3 8 12.11 even 2
1200.3.l.x.401.4 8 4.3 odd 2