Properties

Label 900.3.p.b.101.1
Level $900$
Weight $3$
Character 900.101
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,3,Mod(101,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.101"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 101.1
Root \(1.93649 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 900.101
Dual form 900.3.p.b.401.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} +(-5.87298 + 10.1723i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(13.1190 + 7.57423i) q^{11} +(-8.87298 - 15.3685i) q^{13} -15.1485i q^{17} -11.2540 q^{19} +(17.6190 + 30.5169i) q^{21} +(-29.2379 + 16.8805i) q^{23} -27.0000 q^{27} +(8.23790 + 4.75615i) q^{29} +(-28.1109 - 48.6895i) q^{31} +(39.3569 - 22.7227i) q^{33} -14.0000 q^{37} -53.2379 q^{39} +(-22.5000 + 12.9904i) q^{41} +(-9.99193 + 17.3065i) q^{43} +(-62.6190 - 36.1531i) q^{47} +(-44.4839 - 77.0483i) q^{49} +(-39.3569 - 22.7227i) q^{51} -37.6651i q^{53} +(-16.8810 + 29.2388i) q^{57} +(-55.1190 + 31.8229i) q^{59} +(-0.618950 + 1.07205i) q^{61} +105.714 q^{63} +(42.9758 + 74.4363i) q^{67} +101.283i q^{69} -22.1046i q^{71} +60.2379 q^{73} +(-154.095 + 88.9666i) q^{77} +(-51.6190 + 89.4066i) q^{79} +(-40.5000 + 70.1481i) q^{81} +(78.0000 + 45.0333i) q^{83} +(24.7137 - 14.2685i) q^{87} -12.0964i q^{89} +208.444 q^{91} -168.665 q^{93} +(49.8488 - 86.3406i) q^{97} -136.336i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 8 q^{7} - 18 q^{9} + 6 q^{11} - 20 q^{13} - 76 q^{19} + 24 q^{21} - 24 q^{23} - 108 q^{27} - 60 q^{29} - 4 q^{31} + 18 q^{33} - 56 q^{37} - 120 q^{39} - 90 q^{41} + 22 q^{43} - 204 q^{47}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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
\(3\) 1.50000 2.59808i 0.500000 0.866025i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −5.87298 + 10.1723i −0.838998 + 1.45319i 0.0517360 + 0.998661i \(0.483525\pi\)
−0.890734 + 0.454526i \(0.849809\pi\)
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.500000 0.866025i
\(10\) 0 0
\(11\) 13.1190 + 7.57423i 1.19263 + 0.688566i 0.958903 0.283735i \(-0.0915737\pi\)
0.233729 + 0.972302i \(0.424907\pi\)
\(12\) 0 0
\(13\) −8.87298 15.3685i −0.682537 1.18219i −0.974204 0.225669i \(-0.927543\pi\)
0.291667 0.956520i \(-0.405790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.1485i 0.891086i −0.895261 0.445543i \(-0.853011\pi\)
0.895261 0.445543i \(-0.146989\pi\)
\(18\) 0 0
\(19\) −11.2540 −0.592318 −0.296159 0.955139i \(-0.595706\pi\)
−0.296159 + 0.955139i \(0.595706\pi\)
\(20\) 0 0
\(21\) 17.6190 + 30.5169i 0.838998 + 1.45319i
\(22\) 0 0
\(23\) −29.2379 + 16.8805i −1.27121 + 0.733935i −0.975216 0.221254i \(-0.928985\pi\)
−0.295997 + 0.955189i \(0.595652\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) 8.23790 + 4.75615i 0.284066 + 0.164005i 0.635262 0.772296i \(-0.280892\pi\)
−0.351197 + 0.936302i \(0.614225\pi\)
\(30\) 0 0
\(31\) −28.1109 48.6895i −0.906803 1.57063i −0.818479 0.574537i \(-0.805182\pi\)
−0.0883237 0.996092i \(-0.528151\pi\)
\(32\) 0 0
\(33\) 39.3569 22.7227i 1.19263 0.688566i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −14.0000 −0.378378 −0.189189 0.981941i \(-0.560586\pi\)
−0.189189 + 0.981941i \(0.560586\pi\)
\(38\) 0 0
\(39\) −53.2379 −1.36507
\(40\) 0 0
\(41\) −22.5000 + 12.9904i −0.548780 + 0.316839i −0.748630 0.662988i \(-0.769288\pi\)
0.199849 + 0.979827i \(0.435955\pi\)
\(42\) 0 0
\(43\) −9.99193 + 17.3065i −0.232371 + 0.402478i −0.958505 0.285075i \(-0.907982\pi\)
0.726135 + 0.687552i \(0.241315\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −62.6190 36.1531i −1.33232 0.769214i −0.346664 0.937990i \(-0.612685\pi\)
−0.985655 + 0.168775i \(0.946019\pi\)
\(48\) 0 0
\(49\) −44.4839 77.0483i −0.907834 1.57241i
\(50\) 0 0
\(51\) −39.3569 22.7227i −0.771703 0.445543i
\(52\) 0 0
\(53\) 37.6651i 0.710663i −0.934740 0.355331i \(-0.884368\pi\)
0.934740 0.355331i \(-0.115632\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −16.8810 + 29.2388i −0.296159 + 0.512962i
\(58\) 0 0
\(59\) −55.1190 + 31.8229i −0.934219 + 0.539372i −0.888144 0.459566i \(-0.848005\pi\)
−0.0460759 + 0.998938i \(0.514672\pi\)
\(60\) 0 0
\(61\) −0.618950 + 1.07205i −0.0101467 + 0.0175746i −0.871054 0.491187i \(-0.836563\pi\)
0.860907 + 0.508762i \(0.169897\pi\)
\(62\) 0 0
\(63\) 105.714 1.67800
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 42.9758 + 74.4363i 0.641430 + 1.11099i 0.985114 + 0.171904i \(0.0549918\pi\)
−0.343684 + 0.939085i \(0.611675\pi\)
\(68\) 0 0
\(69\) 101.283i 1.46787i
\(70\) 0 0
\(71\) 22.1046i 0.311332i −0.987810 0.155666i \(-0.950248\pi\)
0.987810 0.155666i \(-0.0497524\pi\)
\(72\) 0 0
\(73\) 60.2379 0.825177 0.412588 0.910918i \(-0.364625\pi\)
0.412588 + 0.910918i \(0.364625\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −154.095 + 88.9666i −2.00123 + 1.15541i
\(78\) 0 0
\(79\) −51.6190 + 89.4066i −0.653404 + 1.13173i 0.328887 + 0.944369i \(0.393327\pi\)
−0.982291 + 0.187360i \(0.940007\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 78.0000 + 45.0333i 0.939759 + 0.542570i 0.889885 0.456185i \(-0.150785\pi\)
0.0498743 + 0.998756i \(0.484118\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 24.7137 14.2685i 0.284066 0.164005i
\(88\) 0 0
\(89\) 12.0964i 0.135915i −0.997688 0.0679574i \(-0.978352\pi\)
0.997688 0.0679574i \(-0.0216482\pi\)
\(90\) 0 0
\(91\) 208.444 2.29059
\(92\) 0 0
\(93\) −168.665 −1.81361
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 49.8488 86.3406i 0.513905 0.890110i −0.485965 0.873978i \(-0.661532\pi\)
0.999870 0.0161312i \(-0.00513495\pi\)
\(98\) 0 0
\(99\) 136.336i 1.37713i
\(100\) 0 0
\(101\) −134.238 77.5023i −1.32909 0.767349i −0.343929 0.938995i \(-0.611758\pi\)
−0.985159 + 0.171646i \(0.945091\pi\)
\(102\) 0 0
\(103\) 77.8569 + 134.852i 0.755892 + 1.30924i 0.944930 + 0.327273i \(0.106130\pi\)
−0.189038 + 0.981970i \(0.560537\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 84.8705i 0.793182i −0.917995 0.396591i \(-0.870193\pi\)
0.917995 0.396591i \(-0.129807\pi\)
\(108\) 0 0
\(109\) −38.0323 −0.348920 −0.174460 0.984664i \(-0.555818\pi\)
−0.174460 + 0.984664i \(0.555818\pi\)
\(110\) 0 0
\(111\) −21.0000 + 36.3731i −0.189189 + 0.327685i
\(112\) 0 0
\(113\) −6.00000 + 3.46410i −0.0530973 + 0.0306558i −0.526314 0.850290i \(-0.676426\pi\)
0.473216 + 0.880946i \(0.343093\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −79.8569 + 138.316i −0.682537 + 1.18219i
\(118\) 0 0
\(119\) 154.095 + 88.9666i 1.29491 + 0.747619i
\(120\) 0 0
\(121\) 54.2379 + 93.9428i 0.448247 + 0.776387i
\(122\) 0 0
\(123\) 77.9423i 0.633677i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −145.903 −1.14884 −0.574422 0.818559i \(-0.694773\pi\)
−0.574422 + 0.818559i \(0.694773\pi\)
\(128\) 0 0
\(129\) 29.9758 + 51.9196i 0.232371 + 0.402478i
\(130\) 0 0
\(131\) 57.7137 33.3210i 0.440563 0.254359i −0.263274 0.964721i \(-0.584802\pi\)
0.703836 + 0.710362i \(0.251469\pi\)
\(132\) 0 0
\(133\) 66.0948 114.479i 0.496953 0.860748i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −63.4052 36.6070i −0.462812 0.267205i 0.250414 0.968139i \(-0.419433\pi\)
−0.713226 + 0.700934i \(0.752767\pi\)
\(138\) 0 0
\(139\) 53.3569 + 92.4168i 0.383862 + 0.664869i 0.991611 0.129261i \(-0.0412604\pi\)
−0.607748 + 0.794130i \(0.707927\pi\)
\(140\) 0 0
\(141\) −187.857 + 108.459i −1.33232 + 0.769214i
\(142\) 0 0
\(143\) 268.824i 1.87989i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −266.903 −1.81567
\(148\) 0 0
\(149\) −144.333 + 83.3305i −0.968676 + 0.559265i −0.898832 0.438293i \(-0.855583\pi\)
−0.0698433 + 0.997558i \(0.522250\pi\)
\(150\) 0 0
\(151\) −1.71370 + 2.96822i −0.0113490 + 0.0196571i −0.871644 0.490139i \(-0.836946\pi\)
0.860295 + 0.509796i \(0.170279\pi\)
\(152\) 0 0
\(153\) −118.071 + 68.1681i −0.771703 + 0.445543i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −33.9677 58.8338i −0.216355 0.374738i 0.737336 0.675526i \(-0.236084\pi\)
−0.953691 + 0.300788i \(0.902750\pi\)
\(158\) 0 0
\(159\) −97.8569 56.4977i −0.615452 0.355331i
\(160\) 0 0
\(161\) 396.556i 2.46308i
\(162\) 0 0
\(163\) −76.4113 −0.468781 −0.234390 0.972143i \(-0.575309\pi\)
−0.234390 + 0.972143i \(0.575309\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 245.522 141.752i 1.47019 0.848816i 0.470752 0.882266i \(-0.343983\pi\)
0.999440 + 0.0334495i \(0.0106493\pi\)
\(168\) 0 0
\(169\) −72.9597 + 126.370i −0.431714 + 0.747751i
\(170\) 0 0
\(171\) 50.6431 + 87.7165i 0.296159 + 0.512962i
\(172\) 0 0
\(173\) 116.903 + 67.4941i 0.675741 + 0.390139i 0.798248 0.602328i \(-0.205760\pi\)
−0.122507 + 0.992468i \(0.539094\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 190.938i 1.07874i
\(178\) 0 0
\(179\) 171.445i 0.957794i −0.877871 0.478897i \(-0.841037\pi\)
0.877871 0.478897i \(-0.158963\pi\)
\(180\) 0 0
\(181\) 120.794 0.667372 0.333686 0.942684i \(-0.391707\pi\)
0.333686 + 0.942684i \(0.391707\pi\)
\(182\) 0 0
\(183\) 1.85685 + 3.21616i 0.0101467 + 0.0175746i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 114.738 198.732i 0.613572 1.06274i
\(188\) 0 0
\(189\) 158.571 274.652i 0.838998 1.45319i
\(190\) 0 0
\(191\) −241.808 139.608i −1.26601 0.730933i −0.291782 0.956485i \(-0.594248\pi\)
−0.974231 + 0.225552i \(0.927581\pi\)
\(192\) 0 0
\(193\) −52.6431 91.1806i −0.272762 0.472438i 0.696806 0.717260i \(-0.254604\pi\)
−0.969568 + 0.244822i \(0.921271\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 57.9538i 0.294182i 0.989123 + 0.147091i \(0.0469910\pi\)
−0.989123 + 0.147091i \(0.953009\pi\)
\(198\) 0 0
\(199\) −25.2702 −0.126986 −0.0634929 0.997982i \(-0.520224\pi\)
−0.0634929 + 0.997982i \(0.520224\pi\)
\(200\) 0 0
\(201\) 257.855 1.28286
\(202\) 0 0
\(203\) −96.7621 + 55.8656i −0.476661 + 0.275200i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 263.141 + 151.925i 1.27121 + 0.733935i
\(208\) 0 0
\(209\) −147.641 85.2406i −0.706417 0.407850i
\(210\) 0 0
\(211\) 52.0161 + 90.0946i 0.246522 + 0.426989i 0.962558 0.271074i \(-0.0873789\pi\)
−0.716036 + 0.698063i \(0.754046\pi\)
\(212\) 0 0
\(213\) −57.4294 33.1569i −0.269622 0.155666i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 660.379 3.04322
\(218\) 0 0
\(219\) 90.3569 156.503i 0.412588 0.714624i
\(220\) 0 0
\(221\) −232.808 + 134.412i −1.05343 + 0.608199i
\(222\) 0 0
\(223\) 123.681 214.223i 0.554625 0.960639i −0.443307 0.896370i \(-0.646195\pi\)
0.997933 0.0642694i \(-0.0204717\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −261.738 151.114i −1.15303 0.665702i −0.203407 0.979094i \(-0.565201\pi\)
−0.949624 + 0.313392i \(0.898535\pi\)
\(228\) 0 0
\(229\) −133.111 230.555i −0.581270 1.00679i −0.995329 0.0965395i \(-0.969223\pi\)
0.414059 0.910250i \(-0.364111\pi\)
\(230\) 0 0
\(231\) 533.800i 2.31082i
\(232\) 0 0
\(233\) 176.969i 0.759526i 0.925084 + 0.379763i \(0.123994\pi\)
−0.925084 + 0.379763i \(0.876006\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 154.857 + 268.220i 0.653404 + 1.13173i
\(238\) 0 0
\(239\) 68.2863 39.4251i 0.285717 0.164959i −0.350292 0.936641i \(-0.613918\pi\)
0.636009 + 0.771682i \(0.280584\pi\)
\(240\) 0 0
\(241\) 110.246 190.952i 0.457452 0.792330i −0.541373 0.840782i \(-0.682095\pi\)
0.998826 + 0.0484519i \(0.0154288\pi\)
\(242\) 0 0
\(243\) 121.500 + 210.444i 0.500000 + 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 99.8569 + 172.957i 0.404279 + 0.700231i
\(248\) 0 0
\(249\) 234.000 135.100i 0.939759 0.542570i
\(250\) 0 0
\(251\) 277.373i 1.10507i 0.833490 + 0.552535i \(0.186339\pi\)
−0.833490 + 0.552535i \(0.813661\pi\)
\(252\) 0 0
\(253\) −511.427 −2.02145
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 131.692 76.0322i 0.512418 0.295845i −0.221409 0.975181i \(-0.571065\pi\)
0.733827 + 0.679336i \(0.237732\pi\)
\(258\) 0 0
\(259\) 82.2218 142.412i 0.317459 0.549854i
\(260\) 0 0
\(261\) 85.6108i 0.328011i
\(262\) 0 0
\(263\) 4.95365 + 2.85999i 0.0188352 + 0.0108745i 0.509388 0.860537i \(-0.329872\pi\)
−0.490553 + 0.871411i \(0.663205\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −31.4274 18.1446i −0.117706 0.0679574i
\(268\) 0 0
\(269\) 361.531i 1.34398i 0.740560 + 0.671990i \(0.234560\pi\)
−0.740560 + 0.671990i \(0.765440\pi\)
\(270\) 0 0
\(271\) 507.427 1.87243 0.936213 0.351433i \(-0.114306\pi\)
0.936213 + 0.351433i \(0.114306\pi\)
\(272\) 0 0
\(273\) 312.665 541.552i 1.14529 1.98371i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 111.778 193.606i 0.403532 0.698937i −0.590618 0.806951i \(-0.701116\pi\)
0.994149 + 0.108014i \(0.0344492\pi\)
\(278\) 0 0
\(279\) −252.998 + 438.205i −0.906803 + 1.57063i
\(280\) 0 0
\(281\) −39.8105 22.9846i −0.141674 0.0817957i 0.427487 0.904021i \(-0.359399\pi\)
−0.569162 + 0.822226i \(0.692732\pi\)
\(282\) 0 0
\(283\) 72.0161 + 124.736i 0.254474 + 0.440762i 0.964753 0.263159i \(-0.0847643\pi\)
−0.710279 + 0.703921i \(0.751431\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 305.169i 1.06331i
\(288\) 0 0
\(289\) 59.5242 0.205966
\(290\) 0 0
\(291\) −149.546 259.022i −0.513905 0.890110i
\(292\) 0 0
\(293\) −361.476 + 208.698i −1.23371 + 0.712280i −0.967800 0.251719i \(-0.919004\pi\)
−0.265905 + 0.963999i \(0.585671\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −354.212 204.504i −1.19263 0.688566i
\(298\) 0 0
\(299\) 518.855 + 299.561i 1.73530 + 1.00188i
\(300\) 0 0
\(301\) −117.365 203.282i −0.389917 0.675355i
\(302\) 0 0
\(303\) −402.714 + 232.507i −1.32909 + 0.767349i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 310.048 1.00993 0.504965 0.863140i \(-0.331505\pi\)
0.504965 + 0.863140i \(0.331505\pi\)
\(308\) 0 0
\(309\) 467.141 1.51178
\(310\) 0 0
\(311\) −242.238 + 139.856i −0.778900 + 0.449698i −0.836040 0.548668i \(-0.815135\pi\)
0.0571403 + 0.998366i \(0.481802\pi\)
\(312\) 0 0
\(313\) −14.5786 + 25.2509i −0.0465771 + 0.0806738i −0.888374 0.459120i \(-0.848165\pi\)
0.841797 + 0.539794i \(0.181498\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −109.095 62.9859i −0.344147 0.198694i 0.317957 0.948105i \(-0.397003\pi\)
−0.662105 + 0.749411i \(0.730337\pi\)
\(318\) 0 0
\(319\) 72.0484 + 124.791i 0.225857 + 0.391196i
\(320\) 0 0
\(321\) −220.500 127.306i −0.686916 0.396591i
\(322\) 0 0
\(323\) 170.481i 0.527806i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −57.0484 + 98.8107i −0.174460 + 0.302173i
\(328\) 0 0
\(329\) 735.520 424.653i 2.23562 1.29074i
\(330\) 0 0
\(331\) 232.903 403.400i 0.703635 1.21873i −0.263547 0.964647i \(-0.584892\pi\)
0.967182 0.254085i \(-0.0817743\pi\)
\(332\) 0 0
\(333\) 63.0000 + 109.119i 0.189189 + 0.327685i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 175.087 + 303.259i 0.519545 + 0.899878i 0.999742 + 0.0227177i \(0.00723189\pi\)
−0.480197 + 0.877161i \(0.659435\pi\)
\(338\) 0 0
\(339\) 20.7846i 0.0613115i
\(340\) 0 0
\(341\) 851.673i 2.49758i
\(342\) 0 0
\(343\) 469.460 1.36869
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −71.1169 + 41.0594i −0.204948 + 0.118327i −0.598961 0.800778i \(-0.704420\pi\)
0.394013 + 0.919105i \(0.371086\pi\)
\(348\) 0 0
\(349\) −22.7621 + 39.4251i −0.0652209 + 0.112966i −0.896792 0.442452i \(-0.854109\pi\)
0.831571 + 0.555418i \(0.187442\pi\)
\(350\) 0 0
\(351\) 239.571 + 414.948i 0.682537 + 1.18219i
\(352\) 0 0
\(353\) −453.877 262.046i −1.28577 0.742340i −0.307873 0.951427i \(-0.599617\pi\)
−0.977897 + 0.209087i \(0.932951\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 462.284 266.900i 1.29491 0.747619i
\(358\) 0 0
\(359\) 295.601i 0.823401i 0.911319 + 0.411701i \(0.135065\pi\)
−0.911319 + 0.411701i \(0.864935\pi\)
\(360\) 0 0
\(361\) −234.347 −0.649160
\(362\) 0 0
\(363\) 325.427 0.896494
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.2379 + 28.1249i −0.0442450 + 0.0766345i −0.887300 0.461193i \(-0.847422\pi\)
0.843055 + 0.537828i \(0.180755\pi\)
\(368\) 0 0
\(369\) 202.500 + 116.913i 0.548780 + 0.316839i
\(370\) 0 0
\(371\) 383.141 + 221.207i 1.03273 + 0.596244i
\(372\) 0 0
\(373\) −123.746 214.334i −0.331759 0.574623i 0.651098 0.758994i \(-0.274309\pi\)
−0.982857 + 0.184371i \(0.940975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 168.805i 0.447759i
\(378\) 0 0
\(379\) 300.746 0.793525 0.396762 0.917921i \(-0.370134\pi\)
0.396762 + 0.917921i \(0.370134\pi\)
\(380\) 0 0
\(381\) −218.855 + 379.068i −0.574422 + 0.994928i
\(382\) 0 0
\(383\) 183.665 106.039i 0.479544 0.276865i −0.240683 0.970604i \(-0.577371\pi\)
0.720226 + 0.693739i \(0.244038\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 179.855 0.464741
\(388\) 0 0
\(389\) −190.760 110.135i −0.490386 0.283124i 0.234349 0.972153i \(-0.424704\pi\)
−0.724734 + 0.689028i \(0.758038\pi\)
\(390\) 0 0
\(391\) 255.714 + 442.909i 0.653999 + 1.13276i
\(392\) 0 0
\(393\) 199.926i 0.508718i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 97.4919 0.245572 0.122786 0.992433i \(-0.460817\pi\)
0.122786 + 0.992433i \(0.460817\pi\)
\(398\) 0 0
\(399\) −198.284 343.438i −0.496953 0.860748i
\(400\) 0 0
\(401\) −66.8347 + 38.5870i −0.166670 + 0.0962270i −0.581015 0.813893i \(-0.697344\pi\)
0.414345 + 0.910120i \(0.364011\pi\)
\(402\) 0 0
\(403\) −498.855 + 864.042i −1.23785 + 2.14402i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −183.665 106.039i −0.451266 0.260539i
\(408\) 0 0
\(409\) 97.4355 + 168.763i 0.238229 + 0.412624i 0.960206 0.279293i \(-0.0900999\pi\)
−0.721978 + 0.691917i \(0.756767\pi\)
\(410\) 0 0
\(411\) −190.216 + 109.821i −0.462812 + 0.267205i
\(412\) 0 0
\(413\) 747.582i 1.81013i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 320.141 0.767724
\(418\) 0 0
\(419\) 196.476 113.435i 0.468916 0.270729i −0.246870 0.969049i \(-0.579402\pi\)
0.715786 + 0.698320i \(0.246069\pi\)
\(420\) 0 0
\(421\) 150.857 261.292i 0.358330 0.620645i −0.629352 0.777120i \(-0.716680\pi\)
0.987682 + 0.156475i \(0.0500130\pi\)
\(422\) 0 0
\(423\) 650.755i 1.53843i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.27017 12.5923i −0.0170262 0.0294902i
\(428\) 0 0
\(429\) −698.425 403.236i −1.62803 0.939944i
\(430\) 0 0
\(431\) 707.661i 1.64191i −0.570996 0.820953i \(-0.693443\pi\)
0.570996 0.820953i \(-0.306557\pi\)
\(432\) 0 0
\(433\) 669.883 1.54707 0.773537 0.633751i \(-0.218486\pi\)
0.773537 + 0.633751i \(0.218486\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 329.044 189.974i 0.752962 0.434723i
\(438\) 0 0
\(439\) 38.3327 66.3941i 0.0873181 0.151239i −0.819059 0.573710i \(-0.805504\pi\)
0.906377 + 0.422471i \(0.138837\pi\)
\(440\) 0 0
\(441\) −400.355 + 693.435i −0.907834 + 1.57241i
\(442\) 0 0
\(443\) 590.117 + 340.704i 1.33209 + 0.769084i 0.985620 0.168976i \(-0.0540462\pi\)
0.346472 + 0.938060i \(0.387379\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 499.983i 1.11853i
\(448\) 0 0
\(449\) 630.327i 1.40385i 0.712253 + 0.701923i \(0.247675\pi\)
−0.712253 + 0.701923i \(0.752325\pi\)
\(450\) 0 0
\(451\) −393.569 −0.872657
\(452\) 0 0
\(453\) 5.14110 + 8.90465i 0.0113490 + 0.0196571i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −286.498 + 496.229i −0.626910 + 1.08584i 0.361258 + 0.932466i \(0.382347\pi\)
−0.988168 + 0.153374i \(0.950986\pi\)
\(458\) 0 0
\(459\) 409.008i 0.891086i
\(460\) 0 0
\(461\) 615.950 + 355.619i 1.33612 + 0.771407i 0.986229 0.165385i \(-0.0528866\pi\)
0.349887 + 0.936792i \(0.386220\pi\)
\(462\) 0 0
\(463\) 321.190 + 556.317i 0.693714 + 1.20155i 0.970612 + 0.240648i \(0.0773601\pi\)
−0.276899 + 0.960899i \(0.589307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 506.331i 1.08422i 0.840307 + 0.542111i \(0.182375\pi\)
−0.840307 + 0.542111i \(0.817625\pi\)
\(468\) 0 0
\(469\) −1009.58 −2.15263
\(470\) 0 0
\(471\) −203.806 −0.432710
\(472\) 0 0
\(473\) −262.167 + 151.362i −0.554265 + 0.320005i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −293.571 + 169.493i −0.615452 + 0.355331i
\(478\) 0 0
\(479\) −644.044 371.839i −1.34456 0.776282i −0.357087 0.934071i \(-0.616230\pi\)
−0.987473 + 0.157789i \(0.949563\pi\)
\(480\) 0 0
\(481\) 124.222 + 215.158i 0.258257 + 0.447315i
\(482\) 0 0
\(483\) −1030.28 594.834i −2.13309 1.23154i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −154.065 −0.316354 −0.158177 0.987411i \(-0.550562\pi\)
−0.158177 + 0.987411i \(0.550562\pi\)
\(488\) 0 0
\(489\) −114.617 + 198.522i −0.234390 + 0.405976i
\(490\) 0 0
\(491\) −145.167 + 83.8124i −0.295657 + 0.170697i −0.640490 0.767967i \(-0.721269\pi\)
0.344833 + 0.938664i \(0.387935\pi\)
\(492\) 0 0
\(493\) 72.0484 124.791i 0.146143 0.253127i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 224.855 + 129.820i 0.452424 + 0.261207i
\(498\) 0 0
\(499\) −247.308 428.351i −0.495608 0.858418i 0.504379 0.863482i \(-0.331721\pi\)
−0.999987 + 0.00506391i \(0.998388\pi\)
\(500\) 0 0
\(501\) 850.514i 1.69763i
\(502\) 0 0
\(503\) 711.349i 1.41421i −0.707107 0.707106i \(-0.750000\pi\)
0.707107 0.707106i \(-0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 218.879 + 379.110i 0.431714 + 0.747751i
\(508\) 0 0
\(509\) −717.714 + 414.372i −1.41005 + 0.814091i −0.995392 0.0958882i \(-0.969431\pi\)
−0.414654 + 0.909979i \(0.636098\pi\)
\(510\) 0 0
\(511\) −353.776 + 612.758i −0.692321 + 1.19914i
\(512\) 0 0
\(513\) 303.859 0.592318
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −547.663 948.581i −1.05931 1.83478i
\(518\) 0 0
\(519\) 350.710 202.482i 0.675741 0.390139i
\(520\) 0 0
\(521\) 622.686i 1.19518i −0.801804 0.597588i \(-0.796126\pi\)
0.801804 0.597588i \(-0.203874\pi\)
\(522\) 0 0
\(523\) 89.4274 0.170989 0.0854946 0.996339i \(-0.472753\pi\)
0.0854946 + 0.996339i \(0.472753\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −737.571 + 425.837i −1.39956 + 0.808039i
\(528\) 0 0
\(529\) 305.403 528.974i 0.577322 0.999951i
\(530\) 0 0
\(531\) 496.071 + 286.406i 0.934219 + 0.539372i
\(532\) 0 0
\(533\) 399.284 + 230.527i 0.749126 + 0.432508i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −445.427 257.168i −0.829474 0.478897i
\(538\) 0 0
\(539\) 1347.72i 2.50042i
\(540\) 0 0
\(541\) 834.629 1.54275 0.771376 0.636380i \(-0.219569\pi\)
0.771376 + 0.636380i \(0.219569\pi\)
\(542\) 0 0
\(543\) 181.192 313.833i 0.333686 0.577961i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 441.782 765.189i 0.807646 1.39888i −0.106845 0.994276i \(-0.534075\pi\)
0.914490 0.404608i \(-0.132592\pi\)
\(548\) 0 0
\(549\) 11.1411 0.0202934
\(550\) 0 0
\(551\) −92.7096 53.5259i −0.168257 0.0971432i
\(552\) 0 0
\(553\) −606.314 1050.17i −1.09641 1.89904i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 256.448i 0.460410i −0.973142 0.230205i \(-0.926060\pi\)
0.973142 0.230205i \(-0.0739396\pi\)
\(558\) 0 0
\(559\) 354.633 0.634406
\(560\) 0 0
\(561\) −344.214 596.196i −0.613572 1.06274i
\(562\) 0 0
\(563\) 180.024 103.937i 0.319759 0.184613i −0.331526 0.943446i \(-0.607564\pi\)
0.651285 + 0.758833i \(0.274230\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −475.712 823.957i −0.838998 1.45319i
\(568\) 0 0
\(569\) 430.923 + 248.794i 0.757334 + 0.437247i 0.828338 0.560229i \(-0.189287\pi\)
−0.0710034 + 0.997476i \(0.522620\pi\)
\(570\) 0 0
\(571\) −453.817 786.033i −0.794775 1.37659i −0.922982 0.384843i \(-0.874256\pi\)
0.128207 0.991747i \(-0.459078\pi\)
\(572\) 0 0
\(573\) −725.425 + 418.825i −1.26601 + 0.730933i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −758.810 −1.31510 −0.657548 0.753413i \(-0.728406\pi\)
−0.657548 + 0.753413i \(0.728406\pi\)
\(578\) 0 0
\(579\) −315.859 −0.545525
\(580\) 0 0
\(581\) −916.185 + 528.960i −1.57691 + 0.910430i
\(582\) 0 0
\(583\) 285.284 494.127i 0.489338 0.847559i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −656.117 378.809i −1.11775 0.645331i −0.176921 0.984225i \(-0.556614\pi\)
−0.940825 + 0.338894i \(0.889947\pi\)
\(588\) 0 0
\(589\) 316.361 + 547.953i 0.537115 + 0.930311i
\(590\) 0 0
\(591\) 150.569 + 86.9308i 0.254769 + 0.147091i
\(592\) 0 0
\(593\) 258.648i 0.436169i 0.975930 + 0.218084i \(0.0699808\pi\)
−0.975930 + 0.218084i \(0.930019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −37.9052 + 65.6538i −0.0634929 + 0.109973i
\(598\) 0 0
\(599\) −495.665 + 286.172i −0.827488 + 0.477750i −0.852992 0.521924i \(-0.825214\pi\)
0.0255038 + 0.999675i \(0.491881\pi\)
\(600\) 0 0
\(601\) −68.8186 + 119.197i −0.114507 + 0.198332i −0.917582 0.397545i \(-0.869862\pi\)
0.803076 + 0.595877i \(0.203195\pi\)
\(602\) 0 0
\(603\) 386.782 669.926i 0.641430 1.11099i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 9.23790 + 16.0005i 0.0152189 + 0.0263600i 0.873535 0.486762i \(-0.161822\pi\)
−0.858316 + 0.513122i \(0.828489\pi\)
\(608\) 0 0
\(609\) 335.194i 0.550400i
\(610\) 0 0
\(611\) 1283.14i 2.10007i
\(612\) 0 0
\(613\) −392.569 −0.640405 −0.320203 0.947349i \(-0.603751\pi\)
−0.320203 + 0.947349i \(0.603751\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −230.353 + 132.994i −0.373343 + 0.215550i −0.674918 0.737893i \(-0.735821\pi\)
0.301575 + 0.953443i \(0.402488\pi\)
\(618\) 0 0
\(619\) −416.736 + 721.808i −0.673240 + 1.16609i 0.303739 + 0.952755i \(0.401765\pi\)
−0.976980 + 0.213332i \(0.931569\pi\)
\(620\) 0 0
\(621\) 789.423 455.774i 1.27121 0.733935i
\(622\) 0 0
\(623\) 123.048 + 71.0420i 0.197509 + 0.114032i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −442.923 + 255.722i −0.706417 + 0.407850i
\(628\) 0 0
\(629\) 212.078i 0.337168i
\(630\) 0 0
\(631\) −351.875 −0.557647 −0.278823 0.960342i \(-0.589944\pi\)
−0.278823 + 0.960342i \(0.589944\pi\)
\(632\) 0 0
\(633\) 312.097 0.493044
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −789.409 + 1367.30i −1.23926 + 2.14646i
\(638\) 0 0
\(639\) −172.288 + 99.4707i −0.269622 + 0.155666i
\(640\) 0 0
\(641\) 355.645 + 205.332i 0.554829 + 0.320331i 0.751067 0.660226i \(-0.229539\pi\)
−0.196239 + 0.980556i \(0.562873\pi\)
\(642\) 0 0
\(643\) −110.593 191.552i −0.171995 0.297904i 0.767122 0.641501i \(-0.221688\pi\)
−0.939117 + 0.343597i \(0.888355\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1095.26i 1.69282i 0.532529 + 0.846412i \(0.321242\pi\)
−0.532529 + 0.846412i \(0.678758\pi\)
\(648\) 0 0
\(649\) −964.137 −1.48557
\(650\) 0 0
\(651\) 990.569 1715.71i 1.52161 2.63551i
\(652\) 0 0
\(653\) −894.048 + 516.179i −1.36914 + 0.790473i −0.990818 0.135201i \(-0.956832\pi\)
−0.378322 + 0.925674i \(0.623499\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −271.071 469.508i −0.412588 0.714624i
\(658\) 0 0
\(659\) −988.851 570.913i −1.50053 0.866333i −1.00000 0.000614829i \(-0.999804\pi\)
−0.500532 0.865718i \(-0.666862\pi\)
\(660\) 0 0
\(661\) 513.282 + 889.031i 0.776524 + 1.34498i 0.933934 + 0.357445i \(0.116352\pi\)
−0.157410 + 0.987533i \(0.550315\pi\)
\(662\) 0 0
\(663\) 806.472i 1.21640i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −321.145 −0.481477
\(668\) 0 0
\(669\) −371.044 642.668i −0.554625 0.960639i
\(670\) 0 0
\(671\) −16.2399 + 9.37614i −0.0242026 + 0.0139734i
\(672\) 0 0
\(673\) −139.952 + 242.403i −0.207952 + 0.360183i −0.951069 0.308978i \(-0.900013\pi\)
0.743117 + 0.669161i \(0.233346\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 249.520 + 144.060i 0.368567 + 0.212792i 0.672832 0.739795i \(-0.265078\pi\)
−0.304265 + 0.952587i \(0.598411\pi\)
\(678\) 0 0
\(679\) 585.522 + 1014.15i 0.862330 + 1.49360i
\(680\) 0 0
\(681\) −785.214 + 453.343i −1.15303 + 0.665702i
\(682\) 0 0
\(683\) 283.805i 0.415527i 0.978179 + 0.207763i \(0.0666184\pi\)
−0.978179 + 0.207763i \(0.933382\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −798.665 −1.16254
\(688\) 0 0
\(689\) −578.855 + 334.202i −0.840138 + 0.485054i
\(690\) 0 0
\(691\) −440.331 + 762.675i −0.637237 + 1.10373i 0.348800 + 0.937197i \(0.386589\pi\)
−0.986037 + 0.166529i \(0.946744\pi\)
\(692\) 0 0
\(693\) 1386.85 + 800.700i 2.00123 + 1.15541i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 196.784 + 340.840i 0.282330 + 0.489011i
\(698\) 0 0
\(699\) 459.780 + 265.454i 0.657768 + 0.379763i
\(700\) 0 0
\(701\) 680.500i 0.970757i 0.874304 + 0.485378i \(0.161318\pi\)
−0.874304 + 0.485378i \(0.838682\pi\)
\(702\) 0 0
\(703\) 157.556 0.224120
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1576.75 910.339i 2.23020 1.28761i
\(708\) 0 0
\(709\) 289.696 501.767i 0.408597 0.707711i −0.586135 0.810213i \(-0.699351\pi\)
0.994733 + 0.102502i \(0.0326847\pi\)
\(710\) 0 0
\(711\) 929.141 1.30681
\(712\) 0 0
\(713\) 1643.81 + 949.052i 2.30548 + 1.33107i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 236.551i 0.329917i
\(718\) 0 0
\(719\) 597.306i 0.830746i −0.909651 0.415373i \(-0.863651\pi\)
0.909651 0.415373i \(-0.136349\pi\)
\(720\) 0 0
\(721\) −1829.01 −2.53677
\(722\) 0 0
\(723\) −330.738 572.855i −0.457452 0.792330i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −336.905 + 583.537i −0.463419 + 0.802664i −0.999129 0.0417376i \(-0.986711\pi\)
0.535710 + 0.844402i \(0.320044\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 262.167 + 151.362i 0.358642 + 0.207062i
\(732\) 0 0
\(733\) −390.760 676.816i −0.533097 0.923351i −0.999253 0.0386484i \(-0.987695\pi\)
0.466156 0.884703i \(-0.345639\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1302.03i 1.76667i
\(738\) 0 0
\(739\) 232.335 0.314391 0.157195 0.987568i \(-0.449755\pi\)
0.157195 + 0.987568i \(0.449755\pi\)
\(740\) 0 0
\(741\) 599.141 0.808557
\(742\) 0 0
\(743\) −434.806 + 251.036i −0.585204 + 0.337868i −0.763199 0.646164i \(-0.776372\pi\)
0.177995 + 0.984031i \(0.443039\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 810.600i 1.08514i
\(748\) 0 0
\(749\) 863.329 + 498.443i 1.15264 + 0.665478i
\(750\) 0 0
\(751\) −477.190 826.516i −0.635405 1.10055i −0.986429 0.164188i \(-0.947500\pi\)
0.351024 0.936367i \(-0.385834\pi\)
\(752\) 0 0
\(753\) 720.635 + 416.059i 0.957019 + 0.552535i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 266.379 0.351888 0.175944 0.984400i \(-0.443702\pi\)
0.175944 + 0.984400i \(0.443702\pi\)
\(758\) 0 0
\(759\) −767.141 + 1328.73i −1.01073 + 1.75063i
\(760\) 0 0
\(761\) 294.665 170.125i 0.387208 0.223555i −0.293742 0.955885i \(-0.594901\pi\)
0.680950 + 0.732330i \(0.261567\pi\)
\(762\) 0 0
\(763\) 223.363 386.876i 0.292743 0.507046i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 978.139 + 564.729i 1.27528 + 0.736283i
\(768\) 0 0
\(769\) −644.552 1116.40i −0.838170 1.45175i −0.891423 0.453171i \(-0.850293\pi\)
0.0532540 0.998581i \(-0.483041\pi\)
\(770\) 0 0
\(771\) 456.193i 0.591690i
\(772\) 0 0
\(773\) 919.244i 1.18919i −0.804025 0.594595i \(-0.797312\pi\)
0.804025 0.594595i \(-0.202688\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −246.665 427.237i −0.317459 0.549854i
\(778\) 0 0
\(779\) 253.216 146.194i 0.325052 0.187669i
\(780\) 0 0
\(781\) 167.425 289.989i 0.214373 0.371305i
\(782\) 0 0
\(783\) −222.423 128.416i −0.284066 0.164005i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −631.855 1094.40i −0.802865 1.39060i −0.917723 0.397221i \(-0.869975\pi\)
0.114858 0.993382i \(-0.463359\pi\)
\(788\) 0 0
\(789\) 14.8609 8.57997i 0.0188352 0.0108745i
\(790\) 0 0
\(791\) 81.3784i 0.102880i
\(792\) 0 0
\(793\) 21.9677 0.0277021
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 530.806 306.461i 0.666006 0.384518i −0.128556 0.991702i \(-0.541034\pi\)
0.794561 + 0.607184i \(0.207701\pi\)
\(798\) 0 0
\(799\) −547.663 + 948.581i −0.685436 + 1.18721i
\(800\) 0 0
\(801\) −94.2822 + 54.4339i −0.117706 + 0.0679574i
\(802\) 0 0
\(803\) 790.258 + 456.256i 0.984132 + 0.568189i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 939.284 + 542.296i 1.16392 + 0.671990i
\(808\) 0 0
\(809\) 273.189i 0.337687i −0.985643 0.168844i \(-0.945997\pi\)
0.985643 0.168844i \(-0.0540033\pi\)
\(810\) 0 0
\(811\) −1446.49 −1.78359 −0.891793 0.452444i \(-0.850552\pi\)
−0.891793 + 0.452444i \(0.850552\pi\)
\(812\) 0 0
\(813\) 761.141 1318.34i 0.936213 1.62157i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 112.450 194.768i 0.137637 0.238395i
\(818\) 0 0
\(819\) −937.996 1624.66i −1.14529 1.98371i
\(820\) 0 0
\(821\) 221.813 + 128.064i 0.270174 + 0.155985i 0.628967 0.777432i \(-0.283478\pi\)
−0.358793 + 0.933417i \(0.616812\pi\)
\(822\) 0 0
\(823\) 615.218 + 1065.59i 0.747531 + 1.29476i 0.949003 + 0.315267i \(0.102094\pi\)
−0.201473 + 0.979494i \(0.564573\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 726.246i 0.878169i −0.898446 0.439085i \(-0.855303\pi\)
0.898446 0.439085i \(-0.144697\pi\)
\(828\) 0 0
\(829\) 1439.23 1.73610 0.868052 0.496474i \(-0.165372\pi\)
0.868052 + 0.496474i \(0.165372\pi\)
\(830\) 0 0
\(831\) −335.335 580.817i −0.403532 0.698937i
\(832\) 0 0
\(833\) −1167.16 + 673.862i −1.40116 + 0.808958i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 758.994 + 1314.62i 0.906803 + 1.57063i
\(838\) 0 0
\(839\) 968.135 + 558.953i 1.15392 + 0.666213i 0.949838 0.312742i \(-0.101247\pi\)
0.204077 + 0.978955i \(0.434581\pi\)
\(840\) 0 0
\(841\) −375.258 649.966i −0.446205 0.772849i
\(842\) 0 0
\(843\) −119.431 + 68.9538i −0.141674 + 0.0817957i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1274.15 −1.50431
\(848\) 0 0
\(849\) 432.097 0.508948
\(850\) 0 0
\(851\) 409.331 236.327i 0.481000 0.277705i
\(852\) 0 0
\(853\) −348.046 + 602.834i −0.408026 + 0.706722i −0.994669 0.103124i \(-0.967116\pi\)
0.586642 + 0.809846i \(0.300449\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −745.040 430.149i −0.869358 0.501924i −0.00222347 0.999998i \(-0.500708\pi\)
−0.867135 + 0.498073i \(0.834041\pi\)
\(858\) 0 0
\(859\) 321.216 + 556.362i 0.373942 + 0.647686i 0.990168 0.139883i \(-0.0446727\pi\)
−0.616226 + 0.787569i \(0.711339\pi\)
\(860\) 0 0
\(861\) −792.853 457.754i −0.920851 0.531654i
\(862\) 0 0
\(863\) 1466.58i 1.69940i −0.527269 0.849698i \(-0.676784\pi\)
0.527269 0.849698i \(-0.323216\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 89.2863 154.648i 0.102983 0.178372i
\(868\) 0 0
\(869\) −1354.37 + 781.948i −1.55854 + 0.899825i
\(870\) 0 0
\(871\) 762.647 1320.94i 0.875599 1.51658i
\(872\) 0 0
\(873\) −897.278 −1.02781
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −96.6835 167.461i −0.110243 0.190947i 0.805625 0.592426i \(-0.201830\pi\)
−0.915868 + 0.401479i \(0.868496\pi\)
\(878\) 0 0
\(879\) 1252.19i 1.42456i
\(880\) 0 0
\(881\) 554.368i 0.629249i 0.949216 + 0.314624i \(0.101879\pi\)
−0.949216 + 0.314624i \(0.898121\pi\)
\(882\) 0 0
\(883\) −352.460 −0.399162 −0.199581 0.979881i \(-0.563958\pi\)
−0.199581 + 0.979881i \(0.563958\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −932.087 + 538.140i −1.05083 + 0.606697i −0.922882 0.385084i \(-0.874173\pi\)
−0.127949 + 0.991781i \(0.540839\pi\)
\(888\) 0 0
\(889\) 856.887 1484.17i 0.963877 1.66948i
\(890\) 0 0
\(891\) −1062.63 + 613.513i −1.19263 + 0.688566i
\(892\) 0 0
\(893\) 704.716 + 406.868i 0.789155 + 0.455619i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1556.56 898.683i 1.73530 1.00188i
\(898\) 0 0
\(899\) 534.799i 0.594882i
\(900\) 0 0
\(901\) −570.569 −0.633261
\(902\) 0 0
\(903\) −704.190 −0.779833
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −110.496 + 191.385i −0.121826 + 0.211008i −0.920488 0.390772i \(-0.872208\pi\)
0.798662 + 0.601780i \(0.205542\pi\)
\(908\) 0 0
\(909\) 1395.04i 1.53470i
\(910\) 0 0
\(911\) 1535.28 + 886.394i 1.68527 + 0.972991i 0.958057 + 0.286578i \(0.0925177\pi\)
0.727212 + 0.686413i \(0.240816\pi\)
\(912\) 0 0
\(913\) 682.185 + 1181.58i 0.747191 + 1.29417i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 782.775i 0.853626i
\(918\) 0 0
\(919\) 1006.85 1.09560 0.547799 0.836610i \(-0.315466\pi\)
0.547799 + 0.836610i \(0.315466\pi\)
\(920\) 0 0
\(921\) 465.073 805.529i 0.504965 0.874625i
\(922\) 0 0
\(923\) −339.714 + 196.134i −0.368054 + 0.212496i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 700.712 1213.67i 0.755892 1.30924i
\(928\) 0 0
\(929\) −974.758 562.777i −1.04926 0.605788i −0.126814 0.991926i \(-0.540475\pi\)
−0.922441 + 0.386139i \(0.873809\pi\)
\(930\) 0 0
\(931\) 500.623 + 867.104i 0.537726 + 0.931369i
\(932\) 0 0
\(933\) 839.137i 0.899396i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1343.20 −1.43351 −0.716756 0.697324i \(-0.754374\pi\)
−0.716756 + 0.697324i \(0.754374\pi\)
\(938\) 0 0
\(939\) 43.7359 + 75.7527i 0.0465771 + 0.0806738i
\(940\) 0 0
\(941\) 618.375 357.019i 0.657147 0.379404i −0.134042 0.990976i \(-0.542796\pi\)
0.791189 + 0.611572i \(0.209462\pi\)
\(942\) 0 0
\(943\) 438.569 759.623i 0.465078 0.805539i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 620.456 + 358.220i 0.655180 + 0.378268i 0.790438 0.612542i \(-0.209853\pi\)
−0.135258 + 0.990810i \(0.543186\pi\)
\(948\) 0 0
\(949\) −534.490 925.764i −0.563214 0.975515i
\(950\) 0 0
\(951\) −327.284 + 188.958i −0.344147 + 0.198694i
\(952\) 0 0
\(953\) 793.300i 0.832424i −0.909268 0.416212i \(-0.863357\pi\)
0.909268 0.416212i \(-0.136643\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 432.290 0.451714
\(958\) 0 0
\(959\) 744.756 429.985i 0.776596 0.448368i
\(960\) 0 0
\(961\) −1099.94 + 1905.16i −1.14458 + 1.98247i
\(962\) 0 0
\(963\) −661.500 + 381.917i −0.686916 + 0.396591i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 914.331 + 1583.67i 0.945533 + 1.63771i 0.754680 + 0.656093i \(0.227792\pi\)
0.190853 + 0.981619i \(0.438875\pi\)
\(968\) 0 0
\(969\) 442.923 + 255.722i 0.457093 + 0.263903i
\(970\) 0 0
\(971\) 1843.33i 1.89839i −0.314691 0.949194i \(-0.601901\pi\)
0.314691 0.949194i \(-0.398099\pi\)
\(972\) 0 0
\(973\) −1253.46 −1.28824
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −289.591 + 167.195i −0.296408 + 0.171131i −0.640828 0.767684i \(-0.721409\pi\)
0.344420 + 0.938816i \(0.388076\pi\)
\(978\) 0 0
\(979\) 91.6210 158.692i 0.0935863 0.162096i
\(980\) 0 0
\(981\) 171.145 + 296.432i 0.174460 + 0.302173i
\(982\) 0 0
\(983\) −1604.71 926.480i −1.63246 0.942502i −0.983331 0.181825i \(-0.941800\pi\)
−0.649131 0.760677i \(-0.724867\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2547.92i 2.58148i
\(988\) 0 0
\(989\) 674.676i 0.682180i
\(990\) 0 0
\(991\) 415.419 0.419192 0.209596 0.977788i \(-0.432785\pi\)
0.209596 + 0.977788i \(0.432785\pi\)
\(992\) 0 0
\(993\) −698.710 1210.20i −0.703635 1.21873i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 596.808 1033.70i 0.598604 1.03681i −0.394423 0.918929i \(-0.629056\pi\)
0.993027 0.117884i \(-0.0376111\pi\)
\(998\) 0 0
\(999\) 378.000 0.378378
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 900.3.p.b.101.1 4
3.2 odd 2 2700.3.p.a.1601.1 4
5.2 odd 4 900.3.u.b.749.1 8
5.3 odd 4 900.3.u.b.749.4 8
5.4 even 2 180.3.o.a.101.2 yes 4
9.4 even 3 2700.3.p.a.2501.1 4
9.5 odd 6 inner 900.3.p.b.401.1 4
15.2 even 4 2700.3.u.a.2249.1 8
15.8 even 4 2700.3.u.a.2249.4 8
15.14 odd 2 540.3.o.a.521.1 4
20.19 odd 2 720.3.bs.a.641.2 4
45.4 even 6 540.3.o.a.341.1 4
45.13 odd 12 2700.3.u.a.449.1 8
45.14 odd 6 180.3.o.a.41.2 4
45.22 odd 12 2700.3.u.a.449.4 8
45.23 even 12 900.3.u.b.149.1 8
45.29 odd 6 1620.3.g.a.161.1 4
45.32 even 12 900.3.u.b.149.4 8
45.34 even 6 1620.3.g.a.161.3 4
60.59 even 2 2160.3.bs.a.1601.1 4
180.59 even 6 720.3.bs.a.401.2 4
180.139 odd 6 2160.3.bs.a.881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.o.a.41.2 4 45.14 odd 6
180.3.o.a.101.2 yes 4 5.4 even 2
540.3.o.a.341.1 4 45.4 even 6
540.3.o.a.521.1 4 15.14 odd 2
720.3.bs.a.401.2 4 180.59 even 6
720.3.bs.a.641.2 4 20.19 odd 2
900.3.p.b.101.1 4 1.1 even 1 trivial
900.3.p.b.401.1 4 9.5 odd 6 inner
900.3.u.b.149.1 8 45.23 even 12
900.3.u.b.149.4 8 45.32 even 12
900.3.u.b.749.1 8 5.2 odd 4
900.3.u.b.749.4 8 5.3 odd 4
1620.3.g.a.161.1 4 45.29 odd 6
1620.3.g.a.161.3 4 45.34 even 6
2160.3.bs.a.881.1 4 180.139 odd 6
2160.3.bs.a.1601.1 4 60.59 even 2
2700.3.p.a.1601.1 4 3.2 odd 2
2700.3.p.a.2501.1 4 9.4 even 3
2700.3.u.a.449.1 8 45.13 odd 12
2700.3.u.a.449.4 8 45.22 odd 12
2700.3.u.a.2249.1 8 15.2 even 4
2700.3.u.a.2249.4 8 15.8 even 4