Properties

Label 1800.2.k.t.901.4
Level $1800$
Weight $2$
Character 1800.901
Analytic conductor $14.373$
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] = [1800,2,Mod(901,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 901.4
Root \(-1.08003 + 0.912978i\) of defining polynomial
Character \(\chi\) \(=\) 1800.901
Dual form 1800.2.k.t.901.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0591148 + 1.41298i) q^{2} +(-1.99301 - 0.167056i) q^{4} +1.33411 q^{7} +(0.353863 - 2.80620i) q^{8} +O(q^{10})\) \(q+(-0.0591148 + 1.41298i) q^{2} +(-1.99301 - 0.167056i) q^{4} +1.33411 q^{7} +(0.353863 - 2.80620i) q^{8} -2.94418i q^{11} -2.04184i q^{13} +(-0.0788658 + 1.88507i) q^{14} +(3.94418 + 0.665888i) q^{16} -3.61241 q^{17} +5.35964i q^{19} +(4.16007 + 0.174045i) q^{22} -8.59609 q^{23} +(2.88507 + 0.120703i) q^{26} +(-2.65890 - 0.222871i) q^{28} -5.26432i q^{29} -2.08134 q^{31} +(-1.17404 + 5.53368i) q^{32} +(0.213547 - 5.10425i) q^{34} +6.55659i q^{37} +(-7.57304 - 0.316834i) q^{38} -7.02786 q^{41} -8.50078i q^{43} +(-0.491843 + 5.86779i) q^{44} +(0.508157 - 12.1461i) q^{46} -9.97204 q^{47} -5.22015 q^{49} +(-0.341101 + 4.06940i) q^{52} +6.12318i q^{53} +(0.472092 - 3.74379i) q^{56} +(7.43836 + 0.311199i) q^{58} -4.75190i q^{59} -8.51476i q^{61} +(0.123038 - 2.94089i) q^{62} +(-7.74956 - 1.98602i) q^{64} +10.6961i q^{67} +(7.19957 + 0.603474i) q^{68} +2.62405 q^{71} -15.3875 q^{73} +(-9.26432 - 0.387592i) q^{74} +(0.895358 - 10.6818i) q^{76} -3.92787i q^{77} +10.4450 q^{79} +(0.415451 - 9.93021i) q^{82} -1.52708i q^{83} +(12.0114 + 0.502522i) q^{86} +(-8.26198 - 1.04184i) q^{88} +12.7193 q^{89} -2.72404i q^{91} +(17.1321 + 1.43603i) q^{92} +(0.589496 - 14.0903i) q^{94} -13.4450 q^{97} +(0.308588 - 7.37595i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 4 q^{4} + 8 q^{7} - 4 q^{8} + 6 q^{14} + 8 q^{16} + 12 q^{22} - 8 q^{23} + 2 q^{26} - 4 q^{28} + 8 q^{31} - 28 q^{32} + 12 q^{34} - 30 q^{38} + 12 q^{44} + 20 q^{46} - 20 q^{52} - 8 q^{56} + 12 q^{58} - 30 q^{62} - 32 q^{64} + 28 q^{68} + 40 q^{71} - 16 q^{73} - 8 q^{74} - 20 q^{76} - 16 q^{79} - 24 q^{82} + 18 q^{86} - 8 q^{88} + 36 q^{92} - 4 q^{94} - 8 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0591148 + 1.41298i −0.0418005 + 0.999126i
\(3\) 0 0
\(4\) −1.99301 0.167056i −0.996505 0.0835279i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.33411 0.504247 0.252123 0.967695i \(-0.418871\pi\)
0.252123 + 0.967695i \(0.418871\pi\)
\(8\) 0.353863 2.80620i 0.125109 0.992143i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.94418i 0.887705i −0.896100 0.443853i \(-0.853611\pi\)
0.896100 0.443853i \(-0.146389\pi\)
\(12\) 0 0
\(13\) 2.04184i 0.566304i −0.959075 0.283152i \(-0.908620\pi\)
0.959075 0.283152i \(-0.0913800\pi\)
\(14\) −0.0788658 + 1.88507i −0.0210778 + 0.503806i
\(15\) 0 0
\(16\) 3.94418 + 0.665888i 0.986046 + 0.166472i
\(17\) −3.61241 −0.876138 −0.438069 0.898941i \(-0.644337\pi\)
−0.438069 + 0.898941i \(0.644337\pi\)
\(18\) 0 0
\(19\) 5.35964i 1.22958i 0.788689 + 0.614792i \(0.210760\pi\)
−0.788689 + 0.614792i \(0.789240\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.16007 + 0.174045i 0.886929 + 0.0371065i
\(23\) −8.59609 −1.79241 −0.896205 0.443641i \(-0.853687\pi\)
−0.896205 + 0.443641i \(0.853687\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.88507 + 0.120703i 0.565809 + 0.0236718i
\(27\) 0 0
\(28\) −2.65890 0.222871i −0.502485 0.0421187i
\(29\) 5.26432i 0.977559i −0.872407 0.488780i \(-0.837442\pi\)
0.872407 0.488780i \(-0.162558\pi\)
\(30\) 0 0
\(31\) −2.08134 −0.373820 −0.186910 0.982377i \(-0.559847\pi\)
−0.186910 + 0.982377i \(0.559847\pi\)
\(32\) −1.17404 + 5.53368i −0.207544 + 0.978226i
\(33\) 0 0
\(34\) 0.213547 5.10425i 0.0366230 0.875372i
\(35\) 0 0
\(36\) 0 0
\(37\) 6.55659i 1.07790i 0.842339 + 0.538949i \(0.181178\pi\)
−0.842339 + 0.538949i \(0.818822\pi\)
\(38\) −7.57304 0.316834i −1.22851 0.0513973i
\(39\) 0 0
\(40\) 0 0
\(41\) −7.02786 −1.09757 −0.548784 0.835964i \(-0.684909\pi\)
−0.548784 + 0.835964i \(0.684909\pi\)
\(42\) 0 0
\(43\) 8.50078i 1.29636i −0.761489 0.648178i \(-0.775531\pi\)
0.761489 0.648178i \(-0.224469\pi\)
\(44\) −0.491843 + 5.86779i −0.0741482 + 0.884603i
\(45\) 0 0
\(46\) 0.508157 12.1461i 0.0749236 1.79084i
\(47\) −9.97204 −1.45457 −0.727286 0.686334i \(-0.759219\pi\)
−0.727286 + 0.686334i \(0.759219\pi\)
\(48\) 0 0
\(49\) −5.22015 −0.745735
\(50\) 0 0
\(51\) 0 0
\(52\) −0.341101 + 4.06940i −0.0473022 + 0.564325i
\(53\) 6.12318i 0.841083i 0.907273 + 0.420541i \(0.138160\pi\)
−0.907273 + 0.420541i \(0.861840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.472092 3.74379i 0.0630860 0.500285i
\(57\) 0 0
\(58\) 7.43836 + 0.311199i 0.976705 + 0.0408625i
\(59\) 4.75190i 0.618644i −0.950957 0.309322i \(-0.899898\pi\)
0.950957 0.309322i \(-0.100102\pi\)
\(60\) 0 0
\(61\) 8.51476i 1.09020i −0.838370 0.545101i \(-0.816491\pi\)
0.838370 0.545101i \(-0.183509\pi\)
\(62\) 0.123038 2.94089i 0.0156258 0.373493i
\(63\) 0 0
\(64\) −7.74956 1.98602i −0.968695 0.248253i
\(65\) 0 0
\(66\) 0 0
\(67\) 10.6961i 1.30673i 0.757041 + 0.653367i \(0.226644\pi\)
−0.757041 + 0.653367i \(0.773356\pi\)
\(68\) 7.19957 + 0.603474i 0.873076 + 0.0731820i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.62405 0.311418 0.155709 0.987803i \(-0.450234\pi\)
0.155709 + 0.987803i \(0.450234\pi\)
\(72\) 0 0
\(73\) −15.3875 −1.80097 −0.900485 0.434887i \(-0.856788\pi\)
−0.900485 + 0.434887i \(0.856788\pi\)
\(74\) −9.26432 0.387592i −1.07696 0.0450566i
\(75\) 0 0
\(76\) 0.895358 10.6818i 0.102705 1.22529i
\(77\) 3.92787i 0.447622i
\(78\) 0 0
\(79\) 10.4450 1.17515 0.587575 0.809170i \(-0.300083\pi\)
0.587575 + 0.809170i \(0.300083\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.415451 9.93021i 0.0458789 1.09661i
\(83\) 1.52708i 0.167619i −0.996482 0.0838095i \(-0.973291\pi\)
0.996482 0.0838095i \(-0.0267087\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0114 + 0.502522i 1.29522 + 0.0541883i
\(87\) 0 0
\(88\) −8.26198 1.04184i −0.880730 0.111060i
\(89\) 12.7193 1.34824 0.674120 0.738622i \(-0.264523\pi\)
0.674120 + 0.738622i \(0.264523\pi\)
\(90\) 0 0
\(91\) 2.72404i 0.285557i
\(92\) 17.1321 + 1.43603i 1.78615 + 0.149716i
\(93\) 0 0
\(94\) 0.589496 14.0903i 0.0608018 1.45330i
\(95\) 0 0
\(96\) 0 0
\(97\) −13.4450 −1.36513 −0.682565 0.730825i \(-0.739135\pi\)
−0.682565 + 0.730825i \(0.739135\pi\)
\(98\) 0.308588 7.37595i 0.0311721 0.745083i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.1232i 1.00729i −0.863910 0.503647i \(-0.831991\pi\)
0.863910 0.503647i \(-0.168009\pi\)
\(102\) 0 0
\(103\) 10.7472 1.05896 0.529478 0.848324i \(-0.322388\pi\)
0.529478 + 0.848324i \(0.322388\pi\)
\(104\) −5.72981 0.722530i −0.561854 0.0708499i
\(105\) 0 0
\(106\) −8.65191 0.361971i −0.840348 0.0351577i
\(107\) 4.86518i 0.470335i 0.971955 + 0.235167i \(0.0755638\pi\)
−0.971955 + 0.235167i \(0.924436\pi\)
\(108\) 0 0
\(109\) 15.4573i 1.48054i −0.672310 0.740270i \(-0.734698\pi\)
0.672310 0.740270i \(-0.265302\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 5.26198 + 0.888369i 0.497211 + 0.0839430i
\(113\) 9.88837 0.930220 0.465110 0.885253i \(-0.346015\pi\)
0.465110 + 0.885253i \(0.346015\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.879435 + 10.4918i −0.0816535 + 0.974143i
\(117\) 0 0
\(118\) 6.71432 + 0.280908i 0.618104 + 0.0258596i
\(119\) −4.81936 −0.441790
\(120\) 0 0
\(121\) 2.33178 0.211980
\(122\) 12.0312 + 0.503348i 1.08925 + 0.0455710i
\(123\) 0 0
\(124\) 4.14813 + 0.347700i 0.372513 + 0.0312244i
\(125\) 0 0
\(126\) 0 0
\(127\) −8.75190 −0.776605 −0.388303 0.921532i \(-0.626938\pi\)
−0.388303 + 0.921532i \(0.626938\pi\)
\(128\) 3.26432 10.8326i 0.288528 0.957472i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.471266i 0.0411747i 0.999788 + 0.0205874i \(0.00655362\pi\)
−0.999788 + 0.0205874i \(0.993446\pi\)
\(132\) 0 0
\(133\) 7.15035i 0.620014i
\(134\) −15.1133 0.632297i −1.30559 0.0546222i
\(135\) 0 0
\(136\) −1.27830 + 10.1372i −0.109613 + 0.869254i
\(137\) −1.30382 −0.111393 −0.0556964 0.998448i \(-0.517738\pi\)
−0.0556964 + 0.998448i \(0.517738\pi\)
\(138\) 0 0
\(139\) 8.74723i 0.741930i 0.928647 + 0.370965i \(0.120973\pi\)
−0.928647 + 0.370965i \(0.879027\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.155120 + 3.70773i −0.0130174 + 0.311145i
\(143\) −6.01155 −0.502711
\(144\) 0 0
\(145\) 0 0
\(146\) 0.909629 21.7422i 0.0752814 1.79940i
\(147\) 0 0
\(148\) 1.09532 13.0674i 0.0900345 1.07413i
\(149\) 15.1411i 1.24041i −0.784439 0.620205i \(-0.787049\pi\)
0.784439 0.620205i \(-0.212951\pi\)
\(150\) 0 0
\(151\) −23.2782 −1.89435 −0.947176 0.320713i \(-0.896078\pi\)
−0.947176 + 0.320713i \(0.896078\pi\)
\(152\) 15.0402 + 1.89657i 1.21992 + 0.153833i
\(153\) 0 0
\(154\) 5.54999 + 0.232195i 0.447231 + 0.0187108i
\(155\) 0 0
\(156\) 0 0
\(157\) 21.8976i 1.74762i −0.486270 0.873809i \(-0.661643\pi\)
0.486270 0.873809i \(-0.338357\pi\)
\(158\) −0.617452 + 14.7585i −0.0491219 + 1.17412i
\(159\) 0 0
\(160\) 0 0
\(161\) −11.4682 −0.903817
\(162\) 0 0
\(163\) 11.1643i 0.874458i −0.899350 0.437229i \(-0.855960\pi\)
0.899350 0.437229i \(-0.144040\pi\)
\(164\) 14.0066 + 1.17404i 1.09373 + 0.0916775i
\(165\) 0 0
\(166\) 2.15773 + 0.0902732i 0.167472 + 0.00700656i
\(167\) −10.0952 −0.781192 −0.390596 0.920562i \(-0.627731\pi\)
−0.390596 + 0.920562i \(0.627731\pi\)
\(168\) 0 0
\(169\) 8.83090 0.679300
\(170\) 0 0
\(171\) 0 0
\(172\) −1.42010 + 16.9421i −0.108282 + 1.29183i
\(173\) 13.8162i 1.05043i −0.850970 0.525215i \(-0.823985\pi\)
0.850970 0.525215i \(-0.176015\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.96050 11.6124i 0.147778 0.875318i
\(177\) 0 0
\(178\) −0.751898 + 17.9720i −0.0563571 + 1.34706i
\(179\) 21.9441i 1.64018i 0.572236 + 0.820089i \(0.306076\pi\)
−0.572236 + 0.820089i \(0.693924\pi\)
\(180\) 0 0
\(181\) 1.93021i 0.143471i −0.997424 0.0717356i \(-0.977146\pi\)
0.997424 0.0717356i \(-0.0228538\pi\)
\(182\) 3.84901 + 0.161031i 0.285307 + 0.0119364i
\(183\) 0 0
\(184\) −3.04184 + 24.1224i −0.224247 + 1.77833i
\(185\) 0 0
\(186\) 0 0
\(187\) 10.6356i 0.777752i
\(188\) 19.8744 + 1.66589i 1.44949 + 0.121497i
\(189\) 0 0
\(190\) 0 0
\(191\) −12.1232 −0.877202 −0.438601 0.898682i \(-0.644526\pi\)
−0.438601 + 0.898682i \(0.644526\pi\)
\(192\) 0 0
\(193\) −1.27431 −0.0917267 −0.0458634 0.998948i \(-0.514604\pi\)
−0.0458634 + 0.998948i \(0.514604\pi\)
\(194\) 0.794797 18.9974i 0.0570631 1.36394i
\(195\) 0 0
\(196\) 10.4038 + 0.872056i 0.743129 + 0.0622897i
\(197\) 3.30849i 0.235720i 0.993030 + 0.117860i \(0.0376034\pi\)
−0.993030 + 0.117860i \(0.962397\pi\)
\(198\) 0 0
\(199\) 9.02718 0.639920 0.319960 0.947431i \(-0.396331\pi\)
0.319960 + 0.947431i \(0.396331\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.3038 + 0.598430i 1.00641 + 0.0421054i
\(203\) 7.02319i 0.492931i
\(204\) 0 0
\(205\) 0 0
\(206\) −0.635321 + 15.1856i −0.0442649 + 1.05803i
\(207\) 0 0
\(208\) 1.35964 8.05338i 0.0942738 0.558402i
\(209\) 15.7798 1.09151
\(210\) 0 0
\(211\) 6.61241i 0.455217i 0.973753 + 0.227608i \(0.0730906\pi\)
−0.973753 + 0.227608i \(0.926909\pi\)
\(212\) 1.02291 12.2036i 0.0702539 0.838144i
\(213\) 0 0
\(214\) −6.87439 0.287604i −0.469924 0.0196602i
\(215\) 0 0
\(216\) 0 0
\(217\) −2.77674 −0.188497
\(218\) 21.8408 + 0.913755i 1.47925 + 0.0618873i
\(219\) 0 0
\(220\) 0 0
\(221\) 7.37595i 0.496160i
\(222\) 0 0
\(223\) 0.833237 0.0557976 0.0278988 0.999611i \(-0.491118\pi\)
0.0278988 + 0.999611i \(0.491118\pi\)
\(224\) −1.56631 + 7.38255i −0.104653 + 0.493267i
\(225\) 0 0
\(226\) −0.584549 + 13.9720i −0.0388836 + 0.929407i
\(227\) 10.9999i 0.730089i 0.930990 + 0.365045i \(0.118946\pi\)
−0.930990 + 0.365045i \(0.881054\pi\)
\(228\) 0 0
\(229\) 15.2061i 1.00485i −0.864622 0.502423i \(-0.832442\pi\)
0.864622 0.502423i \(-0.167558\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −14.7728 1.86285i −0.969879 0.122302i
\(233\) −2.47594 −0.162204 −0.0811020 0.996706i \(-0.525844\pi\)
−0.0811020 + 0.996706i \(0.525844\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.793832 + 9.47058i −0.0516741 + 0.616482i
\(237\) 0 0
\(238\) 0.284895 6.80964i 0.0184670 0.441404i
\(239\) 21.0737 1.36314 0.681572 0.731751i \(-0.261297\pi\)
0.681572 + 0.731751i \(0.261297\pi\)
\(240\) 0 0
\(241\) −6.10852 −0.393484 −0.196742 0.980455i \(-0.563036\pi\)
−0.196742 + 0.980455i \(0.563036\pi\)
\(242\) −0.137843 + 3.29475i −0.00886086 + 0.211794i
\(243\) 0 0
\(244\) −1.42244 + 16.9700i −0.0910624 + 1.08639i
\(245\) 0 0
\(246\) 0 0
\(247\) 10.9435 0.696318
\(248\) −0.736508 + 5.84066i −0.0467683 + 0.370882i
\(249\) 0 0
\(250\) 0 0
\(251\) 22.5286i 1.42199i 0.703195 + 0.710997i \(0.251756\pi\)
−0.703195 + 0.710997i \(0.748244\pi\)
\(252\) 0 0
\(253\) 25.3085i 1.59113i
\(254\) 0.517367 12.3662i 0.0324625 0.775927i
\(255\) 0 0
\(256\) 15.1132 + 5.25277i 0.944574 + 0.328298i
\(257\) −14.5286 −0.906271 −0.453136 0.891442i \(-0.649695\pi\)
−0.453136 + 0.891442i \(0.649695\pi\)
\(258\) 0 0
\(259\) 8.74723i 0.543526i
\(260\) 0 0
\(261\) 0 0
\(262\) −0.665888 0.0278588i −0.0411387 0.00172112i
\(263\) −5.29694 −0.326624 −0.163312 0.986575i \(-0.552218\pi\)
−0.163312 + 0.986575i \(0.552218\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.1033 0.422692i −0.619472 0.0259169i
\(267\) 0 0
\(268\) 1.78684 21.3174i 0.109149 1.30217i
\(269\) 27.0737i 1.65071i 0.564613 + 0.825356i \(0.309025\pi\)
−0.564613 + 0.825356i \(0.690975\pi\)
\(270\) 0 0
\(271\) 15.8604 0.963451 0.481726 0.876322i \(-0.340010\pi\)
0.481726 + 0.876322i \(0.340010\pi\)
\(272\) −14.2480 2.40546i −0.863912 0.145852i
\(273\) 0 0
\(274\) 0.0770751 1.84227i 0.00465628 0.111296i
\(275\) 0 0
\(276\) 0 0
\(277\) 9.98592i 0.599996i −0.953940 0.299998i \(-0.903014\pi\)
0.953940 0.299998i \(-0.0969860\pi\)
\(278\) −12.3596 0.517091i −0.741282 0.0310130i
\(279\) 0 0
\(280\) 0 0
\(281\) −13.4218 −0.800676 −0.400338 0.916368i \(-0.631107\pi\)
−0.400338 + 0.916368i \(0.631107\pi\)
\(282\) 0 0
\(283\) 3.83722i 0.228099i 0.993475 + 0.114050i \(0.0363823\pi\)
−0.993475 + 0.114050i \(0.963618\pi\)
\(284\) −5.22976 0.438363i −0.310329 0.0260121i
\(285\) 0 0
\(286\) 0.355372 8.49418i 0.0210136 0.502271i
\(287\) −9.37595 −0.553445
\(288\) 0 0
\(289\) −3.95051 −0.232383
\(290\) 0 0
\(291\) 0 0
\(292\) 30.6674 + 2.57057i 1.79468 + 0.150431i
\(293\) 26.4450i 1.54493i 0.635057 + 0.772466i \(0.280977\pi\)
−0.635057 + 0.772466i \(0.719023\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 18.3991 + 2.32013i 1.06943 + 0.134855i
\(297\) 0 0
\(298\) 21.3941 + 0.895066i 1.23933 + 0.0518498i
\(299\) 17.5518i 1.01505i
\(300\) 0 0
\(301\) 11.3410i 0.653684i
\(302\) 1.37609 32.8916i 0.0791849 1.89270i
\(303\) 0 0
\(304\) −3.56892 + 21.1394i −0.204692 + 1.21243i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.27596i 0.0728230i 0.999337 + 0.0364115i \(0.0115927\pi\)
−0.999337 + 0.0364115i \(0.988407\pi\)
\(308\) −0.656174 + 7.82829i −0.0373890 + 0.446058i
\(309\) 0 0
\(310\) 0 0
\(311\) −2.44496 −0.138641 −0.0693205 0.997594i \(-0.522083\pi\)
−0.0693205 + 0.997594i \(0.522083\pi\)
\(312\) 0 0
\(313\) 22.8325 1.29057 0.645283 0.763943i \(-0.276739\pi\)
0.645283 + 0.763943i \(0.276739\pi\)
\(314\) 30.9408 + 1.29447i 1.74609 + 0.0730513i
\(315\) 0 0
\(316\) −20.8169 1.74489i −1.17104 0.0981579i
\(317\) 2.11163i 0.118601i 0.998240 + 0.0593005i \(0.0188870\pi\)
−0.998240 + 0.0593005i \(0.981113\pi\)
\(318\) 0 0
\(319\) −15.4991 −0.867784
\(320\) 0 0
\(321\) 0 0
\(322\) 0.677938 16.2042i 0.0377800 0.903027i
\(323\) 19.3612i 1.07729i
\(324\) 0 0
\(325\) 0 0
\(326\) 15.7749 + 0.659978i 0.873694 + 0.0365528i
\(327\) 0 0
\(328\) −2.48690 + 19.7216i −0.137316 + 1.08894i
\(329\) −13.3038 −0.733463
\(330\) 0 0
\(331\) 23.2248i 1.27655i 0.769808 + 0.638276i \(0.220352\pi\)
−0.769808 + 0.638276i \(0.779648\pi\)
\(332\) −0.255108 + 3.04349i −0.0140009 + 0.167033i
\(333\) 0 0
\(334\) 0.596777 14.2643i 0.0326542 0.780509i
\(335\) 0 0
\(336\) 0 0
\(337\) 12.8884 0.702074 0.351037 0.936362i \(-0.385829\pi\)
0.351037 + 0.936362i \(0.385829\pi\)
\(338\) −0.522037 + 12.4779i −0.0283951 + 0.678706i
\(339\) 0 0
\(340\) 0 0
\(341\) 6.12785i 0.331841i
\(342\) 0 0
\(343\) −16.3030 −0.880281
\(344\) −23.8549 3.00811i −1.28617 0.162186i
\(345\) 0 0
\(346\) 19.5220 + 0.816745i 1.04951 + 0.0439085i
\(347\) 6.79827i 0.364951i 0.983210 + 0.182475i \(0.0584109\pi\)
−0.983210 + 0.182475i \(0.941589\pi\)
\(348\) 0 0
\(349\) 34.6076i 1.85250i −0.376904 0.926252i \(-0.623011\pi\)
0.376904 0.926252i \(-0.376989\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.2922 + 3.45661i 0.868376 + 0.184238i
\(353\) −12.2433 −0.651647 −0.325823 0.945431i \(-0.605642\pi\)
−0.325823 + 0.945431i \(0.605642\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −25.3496 2.12483i −1.34353 0.112616i
\(357\) 0 0
\(358\) −31.0065 1.29722i −1.63874 0.0685603i
\(359\) 2.01622 0.106412 0.0532059 0.998584i \(-0.483056\pi\)
0.0532059 + 0.998584i \(0.483056\pi\)
\(360\) 0 0
\(361\) −9.72569 −0.511878
\(362\) 2.72734 + 0.114104i 0.143346 + 0.00599716i
\(363\) 0 0
\(364\) −0.455067 + 5.42904i −0.0238520 + 0.284559i
\(365\) 0 0
\(366\) 0 0
\(367\) −13.4131 −0.700159 −0.350079 0.936720i \(-0.613845\pi\)
−0.350079 + 0.936720i \(0.613845\pi\)
\(368\) −33.9046 5.72404i −1.76740 0.298386i
\(369\) 0 0
\(370\) 0 0
\(371\) 8.16900i 0.424113i
\(372\) 0 0
\(373\) 10.0976i 0.522832i 0.965226 + 0.261416i \(0.0841894\pi\)
−0.965226 + 0.261416i \(0.915811\pi\)
\(374\) −15.0279 0.628722i −0.777072 0.0325104i
\(375\) 0 0
\(376\) −3.52873 + 27.9836i −0.181981 + 1.44314i
\(377\) −10.7489 −0.553595
\(378\) 0 0
\(379\) 18.2775i 0.938853i 0.882972 + 0.469426i \(0.155539\pi\)
−0.882972 + 0.469426i \(0.844461\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.716660 17.1298i 0.0366675 0.876436i
\(383\) 11.7734 0.601594 0.300797 0.953688i \(-0.402747\pi\)
0.300797 + 0.953688i \(0.402747\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.0753305 1.80057i 0.00383422 0.0916466i
\(387\) 0 0
\(388\) 26.7960 + 2.24606i 1.36036 + 0.114026i
\(389\) 33.4270i 1.69482i −0.530942 0.847408i \(-0.678162\pi\)
0.530942 0.847408i \(-0.321838\pi\)
\(390\) 0 0
\(391\) 31.0526 1.57040
\(392\) −1.84721 + 14.6488i −0.0932984 + 0.739876i
\(393\) 0 0
\(394\) −4.67482 0.195581i −0.235514 0.00985322i
\(395\) 0 0
\(396\) 0 0
\(397\) 39.0434i 1.95953i −0.200147 0.979766i \(-0.564142\pi\)
0.200147 0.979766i \(-0.435858\pi\)
\(398\) −0.533640 + 12.7552i −0.0267490 + 0.639360i
\(399\) 0 0
\(400\) 0 0
\(401\) 24.6140 1.22916 0.614581 0.788853i \(-0.289325\pi\)
0.614581 + 0.788853i \(0.289325\pi\)
\(402\) 0 0
\(403\) 4.24976i 0.211695i
\(404\) −1.69114 + 20.1756i −0.0841372 + 1.00377i
\(405\) 0 0
\(406\) 9.92361 + 0.415175i 0.492500 + 0.0206048i
\(407\) 19.3038 0.956855
\(408\) 0 0
\(409\) −14.5024 −0.717099 −0.358550 0.933511i \(-0.616729\pi\)
−0.358550 + 0.933511i \(0.616729\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −21.4193 1.79539i −1.05526 0.0884524i
\(413\) 6.33956i 0.311949i
\(414\) 0 0
\(415\) 0 0
\(416\) 11.2989 + 2.39721i 0.553973 + 0.117533i
\(417\) 0 0
\(418\) −0.932818 + 22.2964i −0.0456256 + 1.09055i
\(419\) 12.6419i 0.617598i −0.951127 0.308799i \(-0.900073\pi\)
0.951127 0.308799i \(-0.0999271\pi\)
\(420\) 0 0
\(421\) 16.8389i 0.820677i 0.911933 + 0.410338i \(0.134589\pi\)
−0.911933 + 0.410338i \(0.865411\pi\)
\(422\) −9.34318 0.390891i −0.454819 0.0190283i
\(423\) 0 0
\(424\) 17.1829 + 2.16676i 0.834474 + 0.105227i
\(425\) 0 0
\(426\) 0 0
\(427\) 11.3596i 0.549731i
\(428\) 0.812757 9.69636i 0.0392861 0.468691i
\(429\) 0 0
\(430\) 0 0
\(431\) −5.98845 −0.288454 −0.144227 0.989545i \(-0.546070\pi\)
−0.144227 + 0.989545i \(0.546070\pi\)
\(432\) 0 0
\(433\) −2.22482 −0.106918 −0.0534589 0.998570i \(-0.517025\pi\)
−0.0534589 + 0.998570i \(0.517025\pi\)
\(434\) 0.164146 3.92347i 0.00787928 0.188333i
\(435\) 0 0
\(436\) −2.58223 + 30.8065i −0.123666 + 1.47537i
\(437\) 46.0719i 2.20392i
\(438\) 0 0
\(439\) −2.30460 −0.109993 −0.0549963 0.998487i \(-0.517515\pi\)
−0.0549963 + 0.998487i \(0.517515\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −10.4220 0.436028i −0.495726 0.0207397i
\(443\) 22.1347i 1.05165i 0.850592 + 0.525826i \(0.176244\pi\)
−0.850592 + 0.525826i \(0.823756\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.0492566 + 1.17734i −0.00233237 + 0.0557489i
\(447\) 0 0
\(448\) −10.3388 2.64957i −0.488462 0.125181i
\(449\) −21.5861 −1.01871 −0.509356 0.860556i \(-0.670116\pi\)
−0.509356 + 0.860556i \(0.670116\pi\)
\(450\) 0 0
\(451\) 20.6913i 0.974316i
\(452\) −19.7076 1.65191i −0.926969 0.0776993i
\(453\) 0 0
\(454\) −15.5426 0.650257i −0.729451 0.0305181i
\(455\) 0 0
\(456\) 0 0
\(457\) −2.50088 −0.116986 −0.0584930 0.998288i \(-0.518630\pi\)
−0.0584930 + 0.998288i \(0.518630\pi\)
\(458\) 21.4858 + 0.898904i 1.00397 + 0.0420030i
\(459\) 0 0
\(460\) 0 0
\(461\) 2.59609i 0.120912i −0.998171 0.0604561i \(-0.980744\pi\)
0.998171 0.0604561i \(-0.0192555\pi\)
\(462\) 0 0
\(463\) −27.8604 −1.29478 −0.647392 0.762158i \(-0.724140\pi\)
−0.647392 + 0.762158i \(0.724140\pi\)
\(464\) 3.50545 20.7634i 0.162736 0.963919i
\(465\) 0 0
\(466\) 0.146365 3.49844i 0.00678021 0.162062i
\(467\) 5.75200i 0.266171i 0.991105 + 0.133085i \(0.0424884\pi\)
−0.991105 + 0.133085i \(0.957512\pi\)
\(468\) 0 0
\(469\) 14.2698i 0.658917i
\(470\) 0 0
\(471\) 0 0
\(472\) −13.3348 1.68152i −0.613784 0.0773982i
\(473\) −25.0279 −1.15078
\(474\) 0 0
\(475\) 0 0
\(476\) 9.60503 + 0.805102i 0.440246 + 0.0369018i
\(477\) 0 0
\(478\) −1.24577 + 29.7766i −0.0569801 + 1.36195i
\(479\) 12.5473 0.573299 0.286649 0.958036i \(-0.407459\pi\)
0.286649 + 0.958036i \(0.407459\pi\)
\(480\) 0 0
\(481\) 13.3875 0.610417
\(482\) 0.361104 8.63119i 0.0164478 0.393140i
\(483\) 0 0
\(484\) −4.64726 0.389537i −0.211239 0.0177062i
\(485\) 0 0
\(486\) 0 0
\(487\) −8.60530 −0.389944 −0.194972 0.980809i \(-0.562462\pi\)
−0.194972 + 0.980809i \(0.562462\pi\)
\(488\) −23.8941 3.01305i −1.08164 0.136395i
\(489\) 0 0
\(490\) 0 0
\(491\) 36.8866i 1.66467i −0.554273 0.832335i \(-0.687004\pi\)
0.554273 0.832335i \(-0.312996\pi\)
\(492\) 0 0
\(493\) 19.0169i 0.856477i
\(494\) −0.646923 + 15.4629i −0.0291065 + 0.695710i
\(495\) 0 0
\(496\) −8.20919 1.38594i −0.368603 0.0622305i
\(497\) 3.50078 0.157031
\(498\) 0 0
\(499\) 36.2496i 1.62275i 0.584524 + 0.811377i \(0.301281\pi\)
−0.584524 + 0.811377i \(0.698719\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −31.8325 1.33178i −1.42075 0.0594401i
\(503\) 23.3527 1.04124 0.520622 0.853787i \(-0.325700\pi\)
0.520622 + 0.853787i \(0.325700\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −35.7603 1.49611i −1.58974 0.0665101i
\(507\) 0 0
\(508\) 17.4426 + 1.46206i 0.773891 + 0.0648682i
\(509\) 3.35506i 0.148711i 0.997232 + 0.0743553i \(0.0236899\pi\)
−0.997232 + 0.0743553i \(0.976310\pi\)
\(510\) 0 0
\(511\) −20.5286 −0.908133
\(512\) −8.31546 + 21.0441i −0.367495 + 0.930025i
\(513\) 0 0
\(514\) 0.858858 20.5286i 0.0378826 0.905479i
\(515\) 0 0
\(516\) 0 0
\(517\) 29.3595i 1.29123i
\(518\) −12.3596 0.517091i −0.543051 0.0227197i
\(519\) 0 0
\(520\) 0 0
\(521\) 33.6029 1.47217 0.736084 0.676890i \(-0.236673\pi\)
0.736084 + 0.676890i \(0.236673\pi\)
\(522\) 0 0
\(523\) 0.965721i 0.0422280i −0.999777 0.0211140i \(-0.993279\pi\)
0.999777 0.0211140i \(-0.00672130\pi\)
\(524\) 0.0787277 0.939238i 0.00343924 0.0410308i
\(525\) 0 0
\(526\) 0.313128 7.48446i 0.0136530 0.326338i
\(527\) 7.51865 0.327517
\(528\) 0 0
\(529\) 50.8928 2.21273
\(530\) 0 0
\(531\) 0 0
\(532\) 1.19451 14.2507i 0.0517885 0.617848i
\(533\) 14.3497i 0.621556i
\(534\) 0 0
\(535\) 0 0
\(536\) 30.0154 + 3.78494i 1.29647 + 0.163485i
\(537\) 0 0
\(538\) −38.2545 1.60046i −1.64927 0.0690006i
\(539\) 15.3691i 0.661993i
\(540\) 0 0
\(541\) 6.34877i 0.272955i −0.990643 0.136478i \(-0.956422\pi\)
0.990643 0.136478i \(-0.0435781\pi\)
\(542\) −0.937586 + 22.4104i −0.0402728 + 0.962609i
\(543\) 0 0
\(544\) 4.24113 19.9899i 0.181837 0.857060i
\(545\) 0 0
\(546\) 0 0
\(547\) 2.07433i 0.0886921i 0.999016 + 0.0443460i \(0.0141204\pi\)
−0.999016 + 0.0443460i \(0.985880\pi\)
\(548\) 2.59853 + 0.217811i 0.111004 + 0.00930442i
\(549\) 0 0
\(550\) 0 0
\(551\) 28.2148 1.20199
\(552\) 0 0
\(553\) 13.9347 0.592566
\(554\) 14.1099 + 0.590316i 0.599472 + 0.0250801i
\(555\) 0 0
\(556\) 1.46128 17.4333i 0.0619719 0.739337i
\(557\) 27.6931i 1.17339i 0.809807 + 0.586696i \(0.199572\pi\)
−0.809807 + 0.586696i \(0.800428\pi\)
\(558\) 0 0
\(559\) −17.3572 −0.734131
\(560\) 0 0
\(561\) 0 0
\(562\) 0.793426 18.9647i 0.0334687 0.799976i
\(563\) 3.80771i 0.160476i 0.996776 + 0.0802380i \(0.0255680\pi\)
−0.996776 + 0.0802380i \(0.974432\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.42191 0.226837i −0.227900 0.00953467i
\(567\) 0 0
\(568\) 0.928554 7.36362i 0.0389613 0.308971i
\(569\) −38.6371 −1.61975 −0.809875 0.586603i \(-0.800465\pi\)
−0.809875 + 0.586603i \(0.800465\pi\)
\(570\) 0 0
\(571\) 6.24976i 0.261544i −0.991412 0.130772i \(-0.958254\pi\)
0.991412 0.130772i \(-0.0417456\pi\)
\(572\) 11.9811 + 1.00426i 0.500954 + 0.0419904i
\(573\) 0 0
\(574\) 0.554258 13.2480i 0.0231343 0.552961i
\(575\) 0 0
\(576\) 0 0
\(577\) −2.17377 −0.0904952 −0.0452476 0.998976i \(-0.514408\pi\)
−0.0452476 + 0.998976i \(0.514408\pi\)
\(578\) 0.233534 5.58198i 0.00971372 0.232180i
\(579\) 0 0
\(580\) 0 0
\(581\) 2.03730i 0.0845213i
\(582\) 0 0
\(583\) 18.0278 0.746634
\(584\) −5.44506 + 43.1804i −0.225318 + 1.78682i
\(585\) 0 0
\(586\) −37.3661 1.56329i −1.54358 0.0645789i
\(587\) 34.1688i 1.41030i −0.709059 0.705149i \(-0.750880\pi\)
0.709059 0.705149i \(-0.249120\pi\)
\(588\) 0 0
\(589\) 11.1552i 0.459643i
\(590\) 0 0
\(591\) 0 0
\(592\) −4.36596 + 25.8604i −0.179440 + 1.06286i
\(593\) −12.9952 −0.533650 −0.266825 0.963745i \(-0.585975\pi\)
−0.266825 + 0.963745i \(0.585975\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.52942 + 30.1765i −0.103609 + 1.23608i
\(597\) 0 0
\(598\) −24.8003 1.03757i −1.01416 0.0424295i
\(599\) −47.2572 −1.93088 −0.965439 0.260628i \(-0.916071\pi\)
−0.965439 + 0.260628i \(0.916071\pi\)
\(600\) 0 0
\(601\) −23.5007 −0.958613 −0.479306 0.877648i \(-0.659112\pi\)
−0.479306 + 0.877648i \(0.659112\pi\)
\(602\) 16.0246 + 0.670421i 0.653112 + 0.0273243i
\(603\) 0 0
\(604\) 46.3937 + 3.88876i 1.88773 + 0.158231i
\(605\) 0 0
\(606\) 0 0
\(607\) −0.218591 −0.00887233 −0.00443617 0.999990i \(-0.501412\pi\)
−0.00443617 + 0.999990i \(0.501412\pi\)
\(608\) −29.6585 6.29245i −1.20281 0.255193i
\(609\) 0 0
\(610\) 0 0
\(611\) 20.3613i 0.823730i
\(612\) 0 0
\(613\) 35.7488i 1.44388i −0.691956 0.721940i \(-0.743251\pi\)
0.691956 0.721940i \(-0.256749\pi\)
\(614\) −1.80290 0.0754282i −0.0727593 0.00304404i
\(615\) 0 0
\(616\) −11.0224 1.38993i −0.444105 0.0560018i
\(617\) 33.0836 1.33189 0.665947 0.745999i \(-0.268028\pi\)
0.665947 + 0.745999i \(0.268028\pi\)
\(618\) 0 0
\(619\) 25.1084i 1.00919i 0.863355 + 0.504596i \(0.168359\pi\)
−0.863355 + 0.504596i \(0.831641\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.144534 3.45468i 0.00579527 0.138520i
\(623\) 16.9689 0.679846
\(624\) 0 0
\(625\) 0 0
\(626\) −1.34974 + 32.2617i −0.0539463 + 1.28944i
\(627\) 0 0
\(628\) −3.65812 + 43.6421i −0.145975 + 1.74151i
\(629\) 23.6851i 0.944386i
\(630\) 0 0
\(631\) 23.2829 0.926876 0.463438 0.886129i \(-0.346616\pi\)
0.463438 + 0.886129i \(0.346616\pi\)
\(632\) 3.69608 29.3107i 0.147022 1.16592i
\(633\) 0 0
\(634\) −2.98369 0.124829i −0.118497 0.00495758i
\(635\) 0 0
\(636\) 0 0
\(637\) 10.6587i 0.422313i
\(638\) 0.916228 21.8999i 0.0362738 0.867026i
\(639\) 0 0
\(640\) 0 0
\(641\) 38.3021 1.51284 0.756420 0.654086i \(-0.226946\pi\)
0.756420 + 0.654086i \(0.226946\pi\)
\(642\) 0 0
\(643\) 45.8045i 1.80635i −0.429269 0.903177i \(-0.641229\pi\)
0.429269 0.903177i \(-0.358771\pi\)
\(644\) 22.8561 + 1.91582i 0.900658 + 0.0754940i
\(645\) 0 0
\(646\) 27.3569 + 1.14453i 1.07634 + 0.0450311i
\(647\) 48.1114 1.89146 0.945728 0.324960i \(-0.105351\pi\)
0.945728 + 0.324960i \(0.105351\pi\)
\(648\) 0 0
\(649\) −13.9905 −0.549174
\(650\) 0 0
\(651\) 0 0
\(652\) −1.86507 + 22.2506i −0.0730417 + 0.871402i
\(653\) 38.3331i 1.50009i 0.661386 + 0.750046i \(0.269969\pi\)
−0.661386 + 0.750046i \(0.730031\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −27.7192 4.67977i −1.08225 0.182714i
\(657\) 0 0
\(658\) 0.786453 18.7980i 0.0306591 0.732822i
\(659\) 5.03253i 0.196040i −0.995184 0.0980198i \(-0.968749\pi\)
0.995184 0.0980198i \(-0.0312508\pi\)
\(660\) 0 0
\(661\) 17.4665i 0.679368i 0.940540 + 0.339684i \(0.110320\pi\)
−0.940540 + 0.339684i \(0.889680\pi\)
\(662\) −32.8161 1.37293i −1.27544 0.0533605i
\(663\) 0 0
\(664\) −4.28530 0.540377i −0.166302 0.0209707i
\(665\) 0 0
\(666\) 0 0
\(667\) 45.2526i 1.75219i
\(668\) 20.1199 + 1.68647i 0.778462 + 0.0652513i
\(669\) 0 0
\(670\) 0 0
\(671\) −25.0690 −0.967779
\(672\) 0 0
\(673\) 32.4448 1.25065 0.625327 0.780363i \(-0.284966\pi\)
0.625327 + 0.780363i \(0.284966\pi\)
\(674\) −0.761894 + 18.2110i −0.0293471 + 0.701461i
\(675\) 0 0
\(676\) −17.6001 1.47525i −0.676926 0.0567405i
\(677\) 8.07213i 0.310237i −0.987896 0.155119i \(-0.950424\pi\)
0.987896 0.155119i \(-0.0495760\pi\)
\(678\) 0 0
\(679\) −17.9371 −0.688362
\(680\) 0 0
\(681\) 0 0
\(682\) −8.65851 0.362247i −0.331551 0.0138711i
\(683\) 36.3380i 1.39043i −0.718799 0.695217i \(-0.755308\pi\)
0.718799 0.695217i \(-0.244692\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.963751 23.0358i 0.0367962 0.879512i
\(687\) 0 0
\(688\) 5.66057 33.5286i 0.215807 1.27827i
\(689\) 12.5025 0.476308
\(690\) 0 0
\(691\) 15.0016i 0.570686i −0.958425 0.285343i \(-0.907892\pi\)
0.958425 0.285343i \(-0.0921075\pi\)
\(692\) −2.30808 + 27.5359i −0.0877402 + 1.04676i
\(693\) 0 0
\(694\) −9.60581 0.401879i −0.364632 0.0152551i
\(695\) 0 0
\(696\) 0 0
\(697\) 25.3875 0.961620
\(698\) 48.8998 + 2.04582i 1.85089 + 0.0774356i
\(699\) 0 0
\(700\) 0 0
\(701\) 13.2874i 0.501859i −0.968005 0.250929i \(-0.919264\pi\)
0.968005 0.250929i \(-0.0807362\pi\)
\(702\) 0 0
\(703\) −35.1409 −1.32537
\(704\) −5.84721 + 22.8161i −0.220375 + 0.859916i
\(705\) 0 0
\(706\) 0.723763 17.2996i 0.0272392 0.651077i
\(707\) 13.5054i 0.507925i
\(708\) 0 0
\(709\) 37.8976i 1.42327i 0.702548 + 0.711637i \(0.252046\pi\)
−0.702548 + 0.711637i \(0.747954\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.50088 35.6929i 0.168677 1.33765i
\(713\) 17.8914 0.670038
\(714\) 0 0
\(715\) 0 0
\(716\) 3.66589 43.7348i 0.137001 1.63445i
\(717\) 0 0
\(718\) −0.119188 + 2.84887i −0.00444807 + 0.106319i
\(719\) 4.17909 0.155854 0.0779269 0.996959i \(-0.475170\pi\)
0.0779269 + 0.996959i \(0.475170\pi\)
\(720\) 0 0
\(721\) 14.3380 0.533975
\(722\) 0.574933 13.7422i 0.0213968 0.511431i
\(723\) 0 0
\(724\) −0.322452 + 3.84692i −0.0119838 + 0.142970i
\(725\) 0 0
\(726\) 0 0
\(727\) 26.7727 0.992943 0.496471 0.868053i \(-0.334629\pi\)
0.496471 + 0.868053i \(0.334629\pi\)
\(728\) −7.64421 0.963936i −0.283313 0.0357258i
\(729\) 0 0
\(730\) 0 0
\(731\) 30.7083i 1.13579i
\(732\) 0 0
\(733\) 21.3364i 0.788080i 0.919093 + 0.394040i \(0.128923\pi\)
−0.919093 + 0.394040i \(0.871077\pi\)
\(734\) 0.792914 18.9524i 0.0292670 0.699547i
\(735\) 0 0
\(736\) 10.0922 47.5680i 0.372003 1.75338i
\(737\) 31.4912 1.15999
\(738\) 0 0
\(739\) 13.1038i 0.482033i −0.970521 0.241016i \(-0.922519\pi\)
0.970521 0.241016i \(-0.0774807\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.5426 0.482909i −0.423743 0.0177282i
\(743\) 33.3595 1.22384 0.611921 0.790919i \(-0.290397\pi\)
0.611921 + 0.790919i \(0.290397\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.2676 0.596915i −0.522375 0.0218546i
\(747\) 0 0
\(748\) 1.77674 21.1969i 0.0649640 0.775034i
\(749\) 6.49069i 0.237165i
\(750\) 0 0
\(751\) −1.92100 −0.0700981 −0.0350491 0.999386i \(-0.511159\pi\)
−0.0350491 + 0.999386i \(0.511159\pi\)
\(752\) −39.3316 6.64027i −1.43428 0.242146i
\(753\) 0 0
\(754\) 0.635418 15.1879i 0.0231406 0.553112i
\(755\) 0 0
\(756\) 0 0
\(757\) 13.9908i 0.508504i 0.967138 + 0.254252i \(0.0818292\pi\)
−0.967138 + 0.254252i \(0.918171\pi\)
\(758\) −25.8257 1.08047i −0.938032 0.0392445i
\(759\) 0 0
\(760\) 0 0
\(761\) −25.6618 −0.930240 −0.465120 0.885248i \(-0.653989\pi\)
−0.465120 + 0.885248i \(0.653989\pi\)
\(762\) 0 0
\(763\) 20.6217i 0.746557i
\(764\) 24.1616 + 2.02525i 0.874137 + 0.0732709i
\(765\) 0 0
\(766\) −0.695985 + 16.6356i −0.0251469 + 0.601069i
\(767\) −9.70260 −0.350341
\(768\) 0 0
\(769\) −12.3922 −0.446873 −0.223436 0.974719i \(-0.571728\pi\)
−0.223436 + 0.974719i \(0.571728\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.53971 + 0.212881i 0.0914062 + 0.00766174i
\(773\) 38.4843i 1.38418i −0.721810 0.692091i \(-0.756689\pi\)
0.721810 0.692091i \(-0.243311\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.75767 + 37.7293i −0.170790 + 1.35440i
\(777\) 0 0
\(778\) 47.2316 + 1.97603i 1.69333 + 0.0708442i
\(779\) 37.6668i 1.34955i
\(780\) 0 0
\(781\) 7.72569i 0.276447i
\(782\) −1.83567 + 43.8766i −0.0656434 + 1.56903i
\(783\) 0 0
\(784\) −20.5892 3.47603i −0.735329 0.124144i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.389147i 0.0138716i 0.999976 + 0.00693579i \(0.00220775\pi\)
−0.999976 + 0.00693579i \(0.997792\pi\)
\(788\) 0.552703 6.59386i 0.0196892 0.234896i
\(789\) 0 0
\(790\) 0 0
\(791\) 13.1922 0.469060
\(792\) 0 0
\(793\) −17.3857 −0.617386
\(794\) 55.1674 + 2.30804i 1.95782 + 0.0819094i
\(795\) 0 0
\(796\) −17.9913 1.50804i −0.637683 0.0534512i
\(797\) 0.854188i 0.0302569i 0.999886 + 0.0151284i \(0.00481572\pi\)
−0.999886 + 0.0151284i \(0.995184\pi\)
\(798\) 0 0
\(799\) 36.0231 1.27441
\(800\) 0 0
\(801\) 0 0
\(802\) −1.45505 + 34.7790i −0.0513796 + 1.22809i
\(803\) 45.3036i 1.59873i
\(804\) 0 0
\(805\) 0 0
\(806\) −6.00481 0.251224i −0.211510 0.00884897i
\(807\) 0 0
\(808\) −28.4077 3.58221i −0.999379 0.126022i
\(809\) 10.4107 0.366020 0.183010 0.983111i \(-0.441416\pi\)
0.183010 + 0.983111i \(0.441416\pi\)
\(810\) 0 0
\(811\) 6.08825i 0.213787i −0.994270 0.106894i \(-0.965910\pi\)
0.994270 0.106894i \(-0.0340904\pi\)
\(812\) −1.17326 + 13.9973i −0.0411735 + 0.491209i
\(813\) 0 0
\(814\) −1.14114 + 27.2759i −0.0399970 + 0.956019i
\(815\) 0 0
\(816\) 0 0
\(817\) 45.5611 1.59398
\(818\) 0.857309 20.4916i 0.0299751 0.716472i
\(819\) 0 0
\(820\) 0 0
\(821\) 35.3908i 1.23515i −0.786513 0.617574i \(-0.788116\pi\)
0.786513 0.617574i \(-0.211884\pi\)
\(822\) 0 0
\(823\) 16.2846 0.567646 0.283823 0.958877i \(-0.408397\pi\)
0.283823 + 0.958877i \(0.408397\pi\)
\(824\) 3.80304 30.1589i 0.132485 1.05064i
\(825\) 0 0
\(826\) 8.95766 + 0.374762i 0.311677 + 0.0130396i
\(827\) 32.1362i 1.11748i 0.829341 + 0.558742i \(0.188716\pi\)
−0.829341 + 0.558742i \(0.811284\pi\)
\(828\) 0 0
\(829\) 22.4682i 0.780355i −0.920740 0.390177i \(-0.872414\pi\)
0.920740 0.390177i \(-0.127586\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.05513 + 15.8233i −0.140586 + 0.548576i
\(833\) 18.8573 0.653367
\(834\) 0 0
\(835\) 0 0
\(836\) −31.4492 2.63610i −1.08769 0.0911715i
\(837\) 0 0
\(838\) 17.8627 + 0.747325i 0.617058 + 0.0258159i
\(839\) 16.1358 0.557070 0.278535 0.960426i \(-0.410151\pi\)
0.278535 + 0.960426i \(0.410151\pi\)
\(840\) 0 0
\(841\) 1.28695 0.0443777
\(842\) −23.7930 0.995427i −0.819959 0.0343047i
\(843\) 0 0
\(844\) 1.10464 13.1786i 0.0380233 0.453626i
\(845\) 0 0
\(846\) 0 0
\(847\) 3.11085 0.106890
\(848\) −4.07735 + 24.1509i −0.140017 + 0.829347i
\(849\) 0 0
\(850\) 0 0
\(851\) 56.3611i 1.93203i
\(852\) 0 0
\(853\) 44.6262i 1.52797i −0.645233 0.763986i \(-0.723240\pi\)
0.645233 0.763986i \(-0.276760\pi\)
\(854\) 16.0509 + 0.671523i 0.549251 + 0.0229790i
\(855\) 0 0
\(856\) 13.6527 + 1.72161i 0.466639 + 0.0588433i
\(857\) −4.52553 −0.154589 −0.0772945 0.997008i \(-0.524628\pi\)
−0.0772945 + 0.997008i \(0.524628\pi\)
\(858\) 0 0
\(859\) 42.7783i 1.45958i −0.683673 0.729788i \(-0.739619\pi\)
0.683673 0.729788i \(-0.260381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.354006 8.46155i 0.0120575 0.288202i
\(863\) −23.7734 −0.809257 −0.404629 0.914481i \(-0.632599\pi\)
−0.404629 + 0.914481i \(0.632599\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.131520 3.14362i 0.00446922 0.106824i
\(867\) 0 0
\(868\) 5.53407 + 0.463870i 0.187839 + 0.0157448i
\(869\) 30.7519i 1.04319i
\(870\) 0 0
\(871\) 21.8397 0.740009
\(872\) −43.3763 5.46976i −1.46891 0.185229i
\(873\) 0 0
\(874\) 65.0986 + 2.72353i 2.20199 + 0.0921249i
\(875\) 0 0
\(876\) 0 0
\(877\) 13.1470i 0.443944i −0.975053 0.221972i \(-0.928751\pi\)
0.975053 0.221972i \(-0.0712494\pi\)
\(878\) 0.136236 3.25635i 0.00459774 0.109896i
\(879\) 0 0
\(880\) 0 0
\(881\) −38.9132 −1.31102 −0.655510 0.755187i \(-0.727546\pi\)
−0.655510 + 0.755187i \(0.727546\pi\)
\(882\) 0 0
\(883\) 44.5843i 1.50038i 0.661223 + 0.750190i \(0.270038\pi\)
−0.661223 + 0.750190i \(0.729962\pi\)
\(884\) 1.23220 14.7003i 0.0414432 0.494426i
\(885\) 0 0
\(886\) −31.2759 1.30849i −1.05073 0.0439596i
\(887\) 32.3240 1.08533 0.542667 0.839948i \(-0.317415\pi\)
0.542667 + 0.839948i \(0.317415\pi\)
\(888\) 0 0
\(889\) −11.6760 −0.391601
\(890\) 0 0
\(891\) 0 0
\(892\) −1.66065 0.139197i −0.0556027 0.00466066i
\(893\) 53.4465i 1.78852i
\(894\) 0 0
\(895\) 0 0
\(896\) 4.35497 14.4518i 0.145489 0.482802i
\(897\) 0 0
\(898\) 1.27606 30.5007i 0.0425826 1.01782i
\(899\) 10.9568i 0.365431i
\(900\) 0 0
\(901\) 22.1194i 0.736904i
\(902\) −29.2364 1.22316i −0.973464 0.0407269i
\(903\) 0 0
\(904\) 3.49912 27.7488i 0.116379 0.922911i
\(905\) 0 0
\(906\) 0 0
\(907\) 14.8309i 0.492452i −0.969212 0.246226i \(-0.920809\pi\)
0.969212 0.246226i \(-0.0791905\pi\)
\(908\) 1.83760 21.9229i 0.0609828 0.727538i
\(909\) 0 0
\(910\) 0 0
\(911\) −11.6108 −0.384681 −0.192341 0.981328i \(-0.561608\pi\)
−0.192341 + 0.981328i \(0.561608\pi\)
\(912\) 0 0
\(913\) −4.49601 −0.148796
\(914\) 0.147839 3.53368i 0.00489007 0.116884i
\(915\) 0 0
\(916\) −2.54026 + 30.3059i −0.0839327 + 1.00133i
\(917\) 0.628722i 0.0207622i
\(918\) 0 0
\(919\) −58.2518 −1.92155 −0.960775 0.277330i \(-0.910550\pi\)
−0.960775 + 0.277330i \(0.910550\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.66822 + 0.153468i 0.120807 + 0.00505419i
\(923\) 5.35789i 0.176357i
\(924\) 0 0
\(925\) 0 0
\(926\) 1.64696 39.3661i 0.0541226 1.29365i
\(927\) 0 0
\(928\) 29.1311 + 6.18055i 0.956274 + 0.202886i
\(929\) −18.4433 −0.605105 −0.302553 0.953133i \(-0.597839\pi\)
−0.302553 + 0.953133i \(0.597839\pi\)
\(930\) 0 0
\(931\) 27.9781i 0.916944i
\(932\) 4.93457 + 0.413620i 0.161637 + 0.0135486i
\(933\) 0 0
\(934\) −8.12744 0.340028i −0.265938 0.0111261i
\(935\) 0 0
\(936\) 0 0
\(937\) −16.1005 −0.525982 −0.262991 0.964798i \(-0.584709\pi\)
−0.262991 + 0.964798i \(0.584709\pi\)
\(938\) −20.1629 0.843555i −0.658341 0.0275430i
\(939\) 0 0
\(940\) 0 0
\(941\) 32.0974i 1.04635i 0.852226 + 0.523173i \(0.175252\pi\)
−0.852226 + 0.523173i \(0.824748\pi\)
\(942\) 0 0
\(943\) 60.4121 1.96729
\(944\) 3.16423 18.7424i 0.102987 0.610012i
\(945\) 0 0
\(946\) 1.47952 35.3638i 0.0481033 1.14978i
\(947\) 4.10998i 0.133556i −0.997768 0.0667782i \(-0.978728\pi\)
0.997768 0.0667782i \(-0.0212720\pi\)
\(948\) 0 0
\(949\) 31.4188i 1.01990i
\(950\) 0 0
\(951\) 0 0
\(952\) −1.70539 + 13.5241i −0.0552720 + 0.438318i
\(953\) 31.7208 1.02754 0.513769 0.857928i \(-0.328249\pi\)
0.513769 + 0.857928i \(0.328249\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −42.0001 3.52048i −1.35838 0.113861i
\(957\) 0 0
\(958\) −0.741729 + 17.7290i −0.0239642 + 0.572798i
\(959\) −1.73944 −0.0561695
\(960\) 0 0
\(961\) −26.6680 −0.860259
\(962\) −0.791399 + 18.9162i −0.0255157 + 0.609884i
\(963\) 0 0
\(964\) 12.1743 + 1.02046i 0.392109 + 0.0328669i
\(965\) 0 0
\(966\) 0 0
\(967\) 26.9936 0.868055 0.434027 0.900900i \(-0.357092\pi\)
0.434027 + 0.900900i \(0.357092\pi\)
\(968\) 0.825129 6.54344i 0.0265206 0.210314i
\(969\) 0 0
\(970\) 0 0
\(971\) 14.0559i 0.451076i −0.974234 0.225538i \(-0.927586\pi\)
0.974234 0.225538i \(-0.0724139\pi\)
\(972\) 0 0
\(973\) 11.6698i 0.374116i
\(974\) 0.508701 12.1591i 0.0162998 0.389603i
\(975\) 0 0
\(976\) 5.66988 33.5838i 0.181488 1.07499i
\(977\) 1.14251 0.0365520 0.0182760 0.999833i \(-0.494182\pi\)
0.0182760 + 0.999833i \(0.494182\pi\)
\(978\) 0 0
\(979\) 37.4479i 1.19684i
\(980\) 0 0
\(981\) 0 0
\(982\) 52.1200 + 2.18055i 1.66321 + 0.0695840i
\(983\) 6.41720 0.204677 0.102338 0.994750i \(-0.467368\pi\)
0.102338 + 0.994750i \(0.467368\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −26.8704 1.12418i −0.855728 0.0358011i
\(987\) 0 0
\(988\) −21.8105 1.82818i −0.693885 0.0581620i
\(989\) 73.0735i 2.32360i
\(990\) 0 0
\(991\) −7.39470 −0.234900 −0.117450 0.993079i \(-0.537472\pi\)
−0.117450 + 0.993079i \(0.537472\pi\)
\(992\) 2.44359 11.5175i 0.0775839 0.365680i
\(993\) 0 0
\(994\) −0.206948 + 4.94652i −0.00656399 + 0.156894i
\(995\) 0 0
\(996\) 0 0
\(997\) 31.6649i 1.00284i −0.865205 0.501419i \(-0.832812\pi\)
0.865205 0.501419i \(-0.167188\pi\)
\(998\) −51.2198 2.14289i −1.62133 0.0678319i
\(999\) 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 1800.2.k.t.901.4 8
3.2 odd 2 600.2.k.d.301.5 8
4.3 odd 2 7200.2.k.r.3601.4 8
5.2 odd 4 1800.2.d.s.1549.1 8
5.3 odd 4 1800.2.d.t.1549.8 8
5.4 even 2 1800.2.k.q.901.5 8
8.3 odd 2 7200.2.k.r.3601.3 8
8.5 even 2 inner 1800.2.k.t.901.3 8
12.11 even 2 2400.2.k.d.1201.2 8
15.2 even 4 600.2.d.h.349.8 8
15.8 even 4 600.2.d.g.349.1 8
15.14 odd 2 600.2.k.e.301.4 yes 8
20.3 even 4 7200.2.d.t.2449.6 8
20.7 even 4 7200.2.d.s.2449.3 8
20.19 odd 2 7200.2.k.s.3601.6 8
24.5 odd 2 600.2.k.d.301.6 yes 8
24.11 even 2 2400.2.k.d.1201.6 8
40.3 even 4 7200.2.d.s.2449.6 8
40.13 odd 4 1800.2.d.s.1549.2 8
40.19 odd 2 7200.2.k.s.3601.5 8
40.27 even 4 7200.2.d.t.2449.3 8
40.29 even 2 1800.2.k.q.901.6 8
40.37 odd 4 1800.2.d.t.1549.7 8
60.23 odd 4 2400.2.d.h.49.6 8
60.47 odd 4 2400.2.d.g.49.3 8
60.59 even 2 2400.2.k.e.1201.7 8
120.29 odd 2 600.2.k.e.301.3 yes 8
120.53 even 4 600.2.d.h.349.7 8
120.59 even 2 2400.2.k.e.1201.3 8
120.77 even 4 600.2.d.g.349.2 8
120.83 odd 4 2400.2.d.g.49.6 8
120.107 odd 4 2400.2.d.h.49.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.d.g.349.1 8 15.8 even 4
600.2.d.g.349.2 8 120.77 even 4
600.2.d.h.349.7 8 120.53 even 4
600.2.d.h.349.8 8 15.2 even 4
600.2.k.d.301.5 8 3.2 odd 2
600.2.k.d.301.6 yes 8 24.5 odd 2
600.2.k.e.301.3 yes 8 120.29 odd 2
600.2.k.e.301.4 yes 8 15.14 odd 2
1800.2.d.s.1549.1 8 5.2 odd 4
1800.2.d.s.1549.2 8 40.13 odd 4
1800.2.d.t.1549.7 8 40.37 odd 4
1800.2.d.t.1549.8 8 5.3 odd 4
1800.2.k.q.901.5 8 5.4 even 2
1800.2.k.q.901.6 8 40.29 even 2
1800.2.k.t.901.3 8 8.5 even 2 inner
1800.2.k.t.901.4 8 1.1 even 1 trivial
2400.2.d.g.49.3 8 60.47 odd 4
2400.2.d.g.49.6 8 120.83 odd 4
2400.2.d.h.49.3 8 120.107 odd 4
2400.2.d.h.49.6 8 60.23 odd 4
2400.2.k.d.1201.2 8 12.11 even 2
2400.2.k.d.1201.6 8 24.11 even 2
2400.2.k.e.1201.3 8 120.59 even 2
2400.2.k.e.1201.7 8 60.59 even 2
7200.2.d.s.2449.3 8 20.7 even 4
7200.2.d.s.2449.6 8 40.3 even 4
7200.2.d.t.2449.3 8 40.27 even 4
7200.2.d.t.2449.6 8 20.3 even 4
7200.2.k.r.3601.3 8 8.3 odd 2
7200.2.k.r.3601.4 8 4.3 odd 2
7200.2.k.s.3601.5 8 40.19 odd 2
7200.2.k.s.3601.6 8 20.19 odd 2