Properties

Label 99.3.c.c.10.1
Level $99$
Weight $3$
Character 99.10
Analytic conductor $2.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,3,Mod(10,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 99.c (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: \(2.69755461717\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{46})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 46x^{2} + 2116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 10.1
Root \(3.39116 + 5.87367i\) of defining polynomial
Character \(\chi\) \(=\) 99.10
Dual form 99.3.c.c.10.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} +1.00000 q^{4} -6.78233 q^{5} -11.7473i q^{7} -8.66025i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} +1.00000 q^{4} -6.78233 q^{5} -11.7473i q^{7} -8.66025i q^{8} +11.7473i q^{10} +(6.78233 - 8.66025i) q^{11} +11.7473i q^{13} -20.3470 q^{14} -11.0000 q^{16} +10.3923i q^{17} -6.78233 q^{20} +(-15.0000 - 11.7473i) q^{22} +33.9116 q^{23} +21.0000 q^{25} +20.3470 q^{26} -11.7473i q^{28} +34.6410i q^{29} -10.0000 q^{31} -15.5885i q^{32} +18.0000 q^{34} +79.6743i q^{35} +50.0000 q^{37} +58.7367i q^{40} -34.6410i q^{41} -46.9894i q^{43} +(6.78233 - 8.66025i) q^{44} -58.7367i q^{46} +33.9116 q^{47} -89.0000 q^{49} -36.3731i q^{50} +11.7473i q^{52} +33.9116 q^{53} +(-46.0000 + 58.7367i) q^{55} -101.735 q^{56} +60.0000 q^{58} -67.8233 q^{59} +58.7367i q^{61} +17.3205i q^{62} -71.0000 q^{64} -79.6743i q^{65} -10.0000 q^{67} +10.3923i q^{68} +138.000 q^{70} +33.9116 q^{71} -70.4840i q^{73} -86.6025i q^{74} +(-101.735 - 79.6743i) q^{77} +58.7367i q^{79} +74.6056 q^{80} -60.0000 q^{82} +76.2102i q^{83} -70.4840i q^{85} -81.3880 q^{86} +(-75.0000 - 58.7367i) q^{88} +13.5647 q^{89} +138.000 q^{91} +33.9116 q^{92} -58.7367i q^{94} -40.0000 q^{97} +154.153i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 44 q^{16} - 60 q^{22} + 84 q^{25} - 40 q^{31} + 72 q^{34} + 200 q^{37} - 356 q^{49} - 184 q^{55} + 240 q^{58} - 284 q^{64} - 40 q^{67} + 552 q^{70} - 240 q^{82} - 300 q^{88} + 552 q^{91} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 0.866025i −0.901388 0.433013i \(-0.857451\pi\)
0.901388 0.433013i \(-0.142549\pi\)
\(3\) 0 0
\(4\) 1.00000 0.250000
\(5\) −6.78233 −1.35647 −0.678233 0.734847i \(-0.737254\pi\)
−0.678233 + 0.734847i \(0.737254\pi\)
\(6\) 0 0
\(7\) 11.7473i 1.67819i −0.543984 0.839096i \(-0.683085\pi\)
0.543984 0.839096i \(-0.316915\pi\)
\(8\) 8.66025i 1.08253i
\(9\) 0 0
\(10\) 11.7473i 1.17473i
\(11\) 6.78233 8.66025i 0.616575 0.787296i
\(12\) 0 0
\(13\) 11.7473i 0.903642i 0.892109 + 0.451821i \(0.149225\pi\)
−0.892109 + 0.451821i \(0.850775\pi\)
\(14\) −20.3470 −1.45336
\(15\) 0 0
\(16\) −11.0000 −0.687500
\(17\) 10.3923i 0.611312i 0.952142 + 0.305656i \(0.0988758\pi\)
−0.952142 + 0.305656i \(0.901124\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −6.78233 −0.339116
\(21\) 0 0
\(22\) −15.0000 11.7473i −0.681818 0.533970i
\(23\) 33.9116 1.47442 0.737210 0.675664i \(-0.236143\pi\)
0.737210 + 0.675664i \(0.236143\pi\)
\(24\) 0 0
\(25\) 21.0000 0.840000
\(26\) 20.3470 0.782577
\(27\) 0 0
\(28\) 11.7473i 0.419548i
\(29\) 34.6410i 1.19452i 0.802049 + 0.597259i \(0.203744\pi\)
−0.802049 + 0.597259i \(0.796256\pi\)
\(30\) 0 0
\(31\) −10.0000 −0.322581 −0.161290 0.986907i \(-0.551566\pi\)
−0.161290 + 0.986907i \(0.551566\pi\)
\(32\) 15.5885i 0.487139i
\(33\) 0 0
\(34\) 18.0000 0.529412
\(35\) 79.6743i 2.27641i
\(36\) 0 0
\(37\) 50.0000 1.35135 0.675676 0.737199i \(-0.263852\pi\)
0.675676 + 0.737199i \(0.263852\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 58.7367i 1.46842i
\(41\) 34.6410i 0.844903i −0.906386 0.422451i \(-0.861170\pi\)
0.906386 0.422451i \(-0.138830\pi\)
\(42\) 0 0
\(43\) 46.9894i 1.09278i −0.837532 0.546388i \(-0.816002\pi\)
0.837532 0.546388i \(-0.183998\pi\)
\(44\) 6.78233 8.66025i 0.154144 0.196824i
\(45\) 0 0
\(46\) 58.7367i 1.27688i
\(47\) 33.9116 0.721524 0.360762 0.932658i \(-0.382517\pi\)
0.360762 + 0.932658i \(0.382517\pi\)
\(48\) 0 0
\(49\) −89.0000 −1.81633
\(50\) 36.3731i 0.727461i
\(51\) 0 0
\(52\) 11.7473i 0.225910i
\(53\) 33.9116 0.639842 0.319921 0.947444i \(-0.396344\pi\)
0.319921 + 0.947444i \(0.396344\pi\)
\(54\) 0 0
\(55\) −46.0000 + 58.7367i −0.836364 + 1.06794i
\(56\) −101.735 −1.81670
\(57\) 0 0
\(58\) 60.0000 1.03448
\(59\) −67.8233 −1.14955 −0.574774 0.818312i \(-0.694910\pi\)
−0.574774 + 0.818312i \(0.694910\pi\)
\(60\) 0 0
\(61\) 58.7367i 0.962897i 0.876474 + 0.481448i \(0.159889\pi\)
−0.876474 + 0.481448i \(0.840111\pi\)
\(62\) 17.3205i 0.279363i
\(63\) 0 0
\(64\) −71.0000 −1.10938
\(65\) 79.6743i 1.22576i
\(66\) 0 0
\(67\) −10.0000 −0.149254 −0.0746269 0.997212i \(-0.523777\pi\)
−0.0746269 + 0.997212i \(0.523777\pi\)
\(68\) 10.3923i 0.152828i
\(69\) 0 0
\(70\) 138.000 1.97143
\(71\) 33.9116 0.477629 0.238814 0.971065i \(-0.423241\pi\)
0.238814 + 0.971065i \(0.423241\pi\)
\(72\) 0 0
\(73\) 70.4840i 0.965535i −0.875749 0.482767i \(-0.839632\pi\)
0.875749 0.482767i \(-0.160368\pi\)
\(74\) 86.6025i 1.17030i
\(75\) 0 0
\(76\) 0 0
\(77\) −101.735 79.6743i −1.32123 1.03473i
\(78\) 0 0
\(79\) 58.7367i 0.743503i 0.928332 + 0.371751i \(0.121243\pi\)
−0.928332 + 0.371751i \(0.878757\pi\)
\(80\) 74.6056 0.932570
\(81\) 0 0
\(82\) −60.0000 −0.731707
\(83\) 76.2102i 0.918196i 0.888386 + 0.459098i \(0.151827\pi\)
−0.888386 + 0.459098i \(0.848173\pi\)
\(84\) 0 0
\(85\) 70.4840i 0.829224i
\(86\) −81.3880 −0.946372
\(87\) 0 0
\(88\) −75.0000 58.7367i −0.852273 0.667463i
\(89\) 13.5647 0.152412 0.0762060 0.997092i \(-0.475719\pi\)
0.0762060 + 0.997092i \(0.475719\pi\)
\(90\) 0 0
\(91\) 138.000 1.51648
\(92\) 33.9116 0.368605
\(93\) 0 0
\(94\) 58.7367i 0.624859i
\(95\) 0 0
\(96\) 0 0
\(97\) −40.0000 −0.412371 −0.206186 0.978513i \(-0.566105\pi\)
−0.206186 + 0.978513i \(0.566105\pi\)
\(98\) 154.153i 1.57298i
\(99\) 0 0
\(100\) 21.0000 0.210000
\(101\) 69.2820i 0.685961i −0.939343 0.342980i \(-0.888564\pi\)
0.939343 0.342980i \(-0.111436\pi\)
\(102\) 0 0
\(103\) 170.000 1.65049 0.825243 0.564778i \(-0.191038\pi\)
0.825243 + 0.564778i \(0.191038\pi\)
\(104\) 101.735 0.978221
\(105\) 0 0
\(106\) 58.7367i 0.554120i
\(107\) 93.5307i 0.874119i 0.899433 + 0.437060i \(0.143980\pi\)
−0.899433 + 0.437060i \(0.856020\pi\)
\(108\) 0 0
\(109\) 176.210i 1.61661i 0.588766 + 0.808303i \(0.299614\pi\)
−0.588766 + 0.808303i \(0.700386\pi\)
\(110\) 101.735 + 79.6743i 0.924863 + 0.724312i
\(111\) 0 0
\(112\) 129.221i 1.15376i
\(113\) −67.8233 −0.600206 −0.300103 0.953907i \(-0.597021\pi\)
−0.300103 + 0.953907i \(0.597021\pi\)
\(114\) 0 0
\(115\) −230.000 −2.00000
\(116\) 34.6410i 0.298629i
\(117\) 0 0
\(118\) 117.473i 0.995537i
\(119\) 122.082 1.02590
\(120\) 0 0
\(121\) −29.0000 117.473i −0.239669 0.970855i
\(122\) 101.735 0.833893
\(123\) 0 0
\(124\) −10.0000 −0.0806452
\(125\) 27.1293 0.217035
\(126\) 0 0
\(127\) 105.726i 0.832489i −0.909253 0.416244i \(-0.863346\pi\)
0.909253 0.416244i \(-0.136654\pi\)
\(128\) 60.6218i 0.473608i
\(129\) 0 0
\(130\) −138.000 −1.06154
\(131\) 34.6410i 0.264435i −0.991221 0.132218i \(-0.957790\pi\)
0.991221 0.132218i \(-0.0422098\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 17.3205i 0.129258i
\(135\) 0 0
\(136\) 90.0000 0.661765
\(137\) −271.293 −1.98024 −0.990121 0.140214i \(-0.955221\pi\)
−0.990121 + 0.140214i \(0.955221\pi\)
\(138\) 0 0
\(139\) 117.473i 0.845132i 0.906332 + 0.422566i \(0.138871\pi\)
−0.906332 + 0.422566i \(0.861129\pi\)
\(140\) 79.6743i 0.569102i
\(141\) 0 0
\(142\) 58.7367i 0.413639i
\(143\) 101.735 + 79.6743i 0.711433 + 0.557163i
\(144\) 0 0
\(145\) 234.947i 1.62032i
\(146\) −122.082 −0.836178
\(147\) 0 0
\(148\) 50.0000 0.337838
\(149\) 138.564i 0.929960i −0.885321 0.464980i \(-0.846061\pi\)
0.885321 0.464980i \(-0.153939\pi\)
\(150\) 0 0
\(151\) 58.7367i 0.388985i 0.980904 + 0.194492i \(0.0623060\pi\)
−0.980904 + 0.194492i \(0.937694\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −138.000 + 176.210i −0.896104 + 1.14422i
\(155\) 67.8233 0.437570
\(156\) 0 0
\(157\) 170.000 1.08280 0.541401 0.840764i \(-0.317894\pi\)
0.541401 + 0.840764i \(0.317894\pi\)
\(158\) 101.735 0.643892
\(159\) 0 0
\(160\) 105.726i 0.660788i
\(161\) 398.372i 2.47436i
\(162\) 0 0
\(163\) 140.000 0.858896 0.429448 0.903092i \(-0.358708\pi\)
0.429448 + 0.903092i \(0.358708\pi\)
\(164\) 34.6410i 0.211226i
\(165\) 0 0
\(166\) 132.000 0.795181
\(167\) 200.918i 1.20310i −0.798835 0.601551i \(-0.794550\pi\)
0.798835 0.601551i \(-0.205450\pi\)
\(168\) 0 0
\(169\) 31.0000 0.183432
\(170\) −122.082 −0.718129
\(171\) 0 0
\(172\) 46.9894i 0.273194i
\(173\) 304.841i 1.76209i 0.473036 + 0.881043i \(0.343158\pi\)
−0.473036 + 0.881043i \(0.656842\pi\)
\(174\) 0 0
\(175\) 246.694i 1.40968i
\(176\) −74.6056 + 95.2628i −0.423896 + 0.541266i
\(177\) 0 0
\(178\) 23.4947i 0.131993i
\(179\) −149.211 −0.833582 −0.416791 0.909002i \(-0.636845\pi\)
−0.416791 + 0.909002i \(0.636845\pi\)
\(180\) 0 0
\(181\) −130.000 −0.718232 −0.359116 0.933293i \(-0.616922\pi\)
−0.359116 + 0.933293i \(0.616922\pi\)
\(182\) 239.023i 1.31331i
\(183\) 0 0
\(184\) 293.684i 1.59611i
\(185\) −339.116 −1.83306
\(186\) 0 0
\(187\) 90.0000 + 70.4840i 0.481283 + 0.376920i
\(188\) 33.9116 0.180381
\(189\) 0 0
\(190\) 0 0
\(191\) −88.1703 −0.461625 −0.230812 0.972998i \(-0.574138\pi\)
−0.230812 + 0.972998i \(0.574138\pi\)
\(192\) 0 0
\(193\) 187.957i 0.973873i 0.873437 + 0.486936i \(0.161886\pi\)
−0.873437 + 0.486936i \(0.838114\pi\)
\(194\) 69.2820i 0.357124i
\(195\) 0 0
\(196\) −89.0000 −0.454082
\(197\) 41.5692i 0.211011i −0.994419 0.105506i \(-0.966354\pi\)
0.994419 0.105506i \(-0.0336461\pi\)
\(198\) 0 0
\(199\) −202.000 −1.01508 −0.507538 0.861630i \(-0.669444\pi\)
−0.507538 + 0.861630i \(0.669444\pi\)
\(200\) 181.865i 0.909327i
\(201\) 0 0
\(202\) −120.000 −0.594059
\(203\) 406.940 2.00463
\(204\) 0 0
\(205\) 234.947i 1.14608i
\(206\) 294.449i 1.42936i
\(207\) 0 0
\(208\) 129.221i 0.621254i
\(209\) 0 0
\(210\) 0 0
\(211\) 117.473i 0.556746i 0.960473 + 0.278373i \(0.0897951\pi\)
−0.960473 + 0.278373i \(0.910205\pi\)
\(212\) 33.9116 0.159961
\(213\) 0 0
\(214\) 162.000 0.757009
\(215\) 318.697i 1.48231i
\(216\) 0 0
\(217\) 117.473i 0.541352i
\(218\) 305.205 1.40002
\(219\) 0 0
\(220\) −46.0000 + 58.7367i −0.209091 + 0.266985i
\(221\) −122.082 −0.552407
\(222\) 0 0
\(223\) −250.000 −1.12108 −0.560538 0.828129i \(-0.689406\pi\)
−0.560538 + 0.828129i \(0.689406\pi\)
\(224\) −183.123 −0.817513
\(225\) 0 0
\(226\) 117.473i 0.519794i
\(227\) 45.0333i 0.198385i 0.995068 + 0.0991923i \(0.0316259\pi\)
−0.995068 + 0.0991923i \(0.968374\pi\)
\(228\) 0 0
\(229\) 158.000 0.689956 0.344978 0.938611i \(-0.387886\pi\)
0.344978 + 0.938611i \(0.387886\pi\)
\(230\) 398.372i 1.73205i
\(231\) 0 0
\(232\) 300.000 1.29310
\(233\) 62.3538i 0.267613i −0.991007 0.133807i \(-0.957280\pi\)
0.991007 0.133807i \(-0.0427201\pi\)
\(234\) 0 0
\(235\) −230.000 −0.978723
\(236\) −67.8233 −0.287387
\(237\) 0 0
\(238\) 211.452i 0.888454i
\(239\) 381.051i 1.59436i 0.603744 + 0.797178i \(0.293675\pi\)
−0.603744 + 0.797178i \(0.706325\pi\)
\(240\) 0 0
\(241\) 234.947i 0.974883i 0.873156 + 0.487441i \(0.162070\pi\)
−0.873156 + 0.487441i \(0.837930\pi\)
\(242\) −203.470 + 50.2295i −0.840785 + 0.207560i
\(243\) 0 0
\(244\) 58.7367i 0.240724i
\(245\) 603.627 2.46379
\(246\) 0 0
\(247\) 0 0
\(248\) 86.6025i 0.349204i
\(249\) 0 0
\(250\) 46.9894i 0.187957i
\(251\) 135.647 0.540425 0.270212 0.962801i \(-0.412906\pi\)
0.270212 + 0.962801i \(0.412906\pi\)
\(252\) 0 0
\(253\) 230.000 293.684i 0.909091 1.16080i
\(254\) −183.123 −0.720956
\(255\) 0 0
\(256\) −179.000 −0.699219
\(257\) 339.116 1.31952 0.659760 0.751477i \(-0.270658\pi\)
0.659760 + 0.751477i \(0.270658\pi\)
\(258\) 0 0
\(259\) 587.367i 2.26783i
\(260\) 79.6743i 0.306440i
\(261\) 0 0
\(262\) −60.0000 −0.229008
\(263\) 96.9948i 0.368802i 0.982851 + 0.184401i \(0.0590345\pi\)
−0.982851 + 0.184401i \(0.940966\pi\)
\(264\) 0 0
\(265\) −230.000 −0.867925
\(266\) 0 0
\(267\) 0 0
\(268\) −10.0000 −0.0373134
\(269\) 33.9116 0.126066 0.0630328 0.998011i \(-0.479923\pi\)
0.0630328 + 0.998011i \(0.479923\pi\)
\(270\) 0 0
\(271\) 176.210i 0.650222i −0.945676 0.325111i \(-0.894598\pi\)
0.945676 0.325111i \(-0.105402\pi\)
\(272\) 114.315i 0.420277i
\(273\) 0 0
\(274\) 469.894i 1.71494i
\(275\) 142.429 181.865i 0.517923 0.661328i
\(276\) 0 0
\(277\) 11.7473i 0.0424092i −0.999775 0.0212046i \(-0.993250\pi\)
0.999775 0.0212046i \(-0.00675013\pi\)
\(278\) 203.470 0.731906
\(279\) 0 0
\(280\) 690.000 2.46429
\(281\) 190.526i 0.678027i 0.940782 + 0.339014i \(0.110093\pi\)
−0.940782 + 0.339014i \(0.889907\pi\)
\(282\) 0 0
\(283\) 399.410i 1.41134i 0.708540 + 0.705671i \(0.249354\pi\)
−0.708540 + 0.705671i \(0.750646\pi\)
\(284\) 33.9116 0.119407
\(285\) 0 0
\(286\) 138.000 176.210i 0.482517 0.616119i
\(287\) −406.940 −1.41791
\(288\) 0 0
\(289\) 181.000 0.626298
\(290\) −406.940 −1.40324
\(291\) 0 0
\(292\) 70.4840i 0.241384i
\(293\) 422.620i 1.44239i 0.692732 + 0.721195i \(0.256407\pi\)
−0.692732 + 0.721195i \(0.743593\pi\)
\(294\) 0 0
\(295\) 460.000 1.55932
\(296\) 433.013i 1.46288i
\(297\) 0 0
\(298\) −240.000 −0.805369
\(299\) 398.372i 1.33235i
\(300\) 0 0
\(301\) −552.000 −1.83389
\(302\) 101.735 0.336871
\(303\) 0 0
\(304\) 0 0
\(305\) 398.372i 1.30614i
\(306\) 0 0
\(307\) 70.4840i 0.229590i −0.993389 0.114795i \(-0.963379\pi\)
0.993389 0.114795i \(-0.0366211\pi\)
\(308\) −101.735 79.6743i −0.330308 0.258683i
\(309\) 0 0
\(310\) 117.473i 0.378946i
\(311\) −373.028 −1.19945 −0.599724 0.800207i \(-0.704723\pi\)
−0.599724 + 0.800207i \(0.704723\pi\)
\(312\) 0 0
\(313\) −70.0000 −0.223642 −0.111821 0.993728i \(-0.535668\pi\)
−0.111821 + 0.993728i \(0.535668\pi\)
\(314\) 294.449i 0.937735i
\(315\) 0 0
\(316\) 58.7367i 0.185876i
\(317\) −169.558 −0.534884 −0.267442 0.963574i \(-0.586178\pi\)
−0.267442 + 0.963574i \(0.586178\pi\)
\(318\) 0 0
\(319\) 300.000 + 234.947i 0.940439 + 0.736510i
\(320\) 481.545 1.50483
\(321\) 0 0
\(322\) −690.000 −2.14286
\(323\) 0 0
\(324\) 0 0
\(325\) 246.694i 0.759059i
\(326\) 242.487i 0.743826i
\(327\) 0 0
\(328\) −300.000 −0.914634
\(329\) 398.372i 1.21086i
\(330\) 0 0
\(331\) −40.0000 −0.120846 −0.0604230 0.998173i \(-0.519245\pi\)
−0.0604230 + 0.998173i \(0.519245\pi\)
\(332\) 76.2102i 0.229549i
\(333\) 0 0
\(334\) −348.000 −1.04192
\(335\) 67.8233 0.202458
\(336\) 0 0
\(337\) 164.463i 0.488020i −0.969773 0.244010i \(-0.921537\pi\)
0.969773 0.244010i \(-0.0784630\pi\)
\(338\) 53.6936i 0.158857i
\(339\) 0 0
\(340\) 70.4840i 0.207306i
\(341\) −67.8233 + 86.6025i −0.198895 + 0.253966i
\(342\) 0 0
\(343\) 469.894i 1.36995i
\(344\) −406.940 −1.18296
\(345\) 0 0
\(346\) 528.000 1.52601
\(347\) 491.902i 1.41759i −0.705416 0.708793i \(-0.749240\pi\)
0.705416 0.708793i \(-0.250760\pi\)
\(348\) 0 0
\(349\) 646.104i 1.85130i −0.378381 0.925650i \(-0.623519\pi\)
0.378381 0.925650i \(-0.376481\pi\)
\(350\) −427.287 −1.22082
\(351\) 0 0
\(352\) −135.000 105.726i −0.383523 0.300358i
\(353\) 135.647 0.384268 0.192134 0.981369i \(-0.438459\pi\)
0.192134 + 0.981369i \(0.438459\pi\)
\(354\) 0 0
\(355\) −230.000 −0.647887
\(356\) 13.5647 0.0381030
\(357\) 0 0
\(358\) 258.441i 0.721904i
\(359\) 103.923i 0.289479i −0.989470 0.144740i \(-0.953766\pi\)
0.989470 0.144740i \(-0.0462345\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 225.167i 0.622007i
\(363\) 0 0
\(364\) 138.000 0.379121
\(365\) 478.046i 1.30972i
\(366\) 0 0
\(367\) −130.000 −0.354223 −0.177112 0.984191i \(-0.556675\pi\)
−0.177112 + 0.984191i \(0.556675\pi\)
\(368\) −373.028 −1.01366
\(369\) 0 0
\(370\) 587.367i 1.58748i
\(371\) 398.372i 1.07378i
\(372\) 0 0
\(373\) 481.641i 1.29126i −0.763649 0.645631i \(-0.776594\pi\)
0.763649 0.645631i \(-0.223406\pi\)
\(374\) 122.082 155.885i 0.326422 0.416804i
\(375\) 0 0
\(376\) 293.684i 0.781073i
\(377\) −406.940 −1.07942
\(378\) 0 0
\(379\) 308.000 0.812665 0.406332 0.913725i \(-0.366807\pi\)
0.406332 + 0.913725i \(0.366807\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 152.715i 0.399779i
\(383\) 237.382 0.619795 0.309898 0.950770i \(-0.399705\pi\)
0.309898 + 0.950770i \(0.399705\pi\)
\(384\) 0 0
\(385\) 690.000 + 540.378i 1.79221 + 1.40358i
\(386\) 325.552 0.843399
\(387\) 0 0
\(388\) −40.0000 −0.103093
\(389\) 644.321 1.65635 0.828177 0.560467i \(-0.189378\pi\)
0.828177 + 0.560467i \(0.189378\pi\)
\(390\) 0 0
\(391\) 352.420i 0.901330i
\(392\) 770.763i 1.96623i
\(393\) 0 0
\(394\) −72.0000 −0.182741
\(395\) 398.372i 1.00854i
\(396\) 0 0
\(397\) −310.000 −0.780856 −0.390428 0.920633i \(-0.627673\pi\)
−0.390428 + 0.920633i \(0.627673\pi\)
\(398\) 349.874i 0.879081i
\(399\) 0 0
\(400\) −231.000 −0.577500
\(401\) −556.151 −1.38691 −0.693455 0.720500i \(-0.743912\pi\)
−0.693455 + 0.720500i \(0.743912\pi\)
\(402\) 0 0
\(403\) 117.473i 0.291497i
\(404\) 69.2820i 0.171490i
\(405\) 0 0
\(406\) 704.840i 1.73606i
\(407\) 339.116 433.013i 0.833210 1.06391i
\(408\) 0 0
\(409\) 587.367i 1.43611i −0.695989 0.718053i \(-0.745034\pi\)
0.695989 0.718053i \(-0.254966\pi\)
\(410\) 406.940 0.992536
\(411\) 0 0
\(412\) 170.000 0.412621
\(413\) 796.743i 1.92916i
\(414\) 0 0
\(415\) 516.883i 1.24550i
\(416\) 183.123 0.440199
\(417\) 0 0
\(418\) 0 0
\(419\) 54.2586 0.129496 0.0647478 0.997902i \(-0.479376\pi\)
0.0647478 + 0.997902i \(0.479376\pi\)
\(420\) 0 0
\(421\) −310.000 −0.736342 −0.368171 0.929758i \(-0.620016\pi\)
−0.368171 + 0.929758i \(0.620016\pi\)
\(422\) 203.470 0.482156
\(423\) 0 0
\(424\) 293.684i 0.692650i
\(425\) 218.238i 0.513502i
\(426\) 0 0
\(427\) 690.000 1.61593
\(428\) 93.5307i 0.218530i
\(429\) 0 0
\(430\) 552.000 1.28372
\(431\) 103.923i 0.241121i 0.992706 + 0.120560i \(0.0384691\pi\)
−0.992706 + 0.120560i \(0.961531\pi\)
\(432\) 0 0
\(433\) −610.000 −1.40878 −0.704388 0.709815i \(-0.748778\pi\)
−0.704388 + 0.709815i \(0.748778\pi\)
\(434\) 203.470 0.468825
\(435\) 0 0
\(436\) 176.210i 0.404152i
\(437\) 0 0
\(438\) 0 0
\(439\) 411.157i 0.936576i 0.883576 + 0.468288i \(0.155129\pi\)
−0.883576 + 0.468288i \(0.844871\pi\)
\(440\) 508.675 + 398.372i 1.15608 + 0.905390i
\(441\) 0 0
\(442\) 211.452i 0.478398i
\(443\) −271.293 −0.612400 −0.306200 0.951967i \(-0.599058\pi\)
−0.306200 + 0.951967i \(0.599058\pi\)
\(444\) 0 0
\(445\) −92.0000 −0.206742
\(446\) 433.013i 0.970880i
\(447\) 0 0
\(448\) 834.061i 1.86174i
\(449\) 217.035 0.483373 0.241687 0.970354i \(-0.422299\pi\)
0.241687 + 0.970354i \(0.422299\pi\)
\(450\) 0 0
\(451\) −300.000 234.947i −0.665188 0.520946i
\(452\) −67.8233 −0.150052
\(453\) 0 0
\(454\) 78.0000 0.171806
\(455\) −935.962 −2.05706
\(456\) 0 0
\(457\) 164.463i 0.359875i 0.983678 + 0.179937i \(0.0575895\pi\)
−0.983678 + 0.179937i \(0.942410\pi\)
\(458\) 273.664i 0.597520i
\(459\) 0 0
\(460\) −230.000 −0.500000
\(461\) 658.179i 1.42772i 0.700288 + 0.713860i \(0.253055\pi\)
−0.700288 + 0.713860i \(0.746945\pi\)
\(462\) 0 0
\(463\) −670.000 −1.44708 −0.723542 0.690280i \(-0.757487\pi\)
−0.723542 + 0.690280i \(0.757487\pi\)
\(464\) 381.051i 0.821231i
\(465\) 0 0
\(466\) −108.000 −0.231760
\(467\) 339.116 0.726160 0.363080 0.931758i \(-0.381725\pi\)
0.363080 + 0.931758i \(0.381725\pi\)
\(468\) 0 0
\(469\) 117.473i 0.250476i
\(470\) 398.372i 0.847599i
\(471\) 0 0
\(472\) 587.367i 1.24442i
\(473\) −406.940 318.697i −0.860338 0.673779i
\(474\) 0 0
\(475\) 0 0
\(476\) 122.082 0.256475
\(477\) 0 0
\(478\) 660.000 1.38075
\(479\) 484.974i 1.01247i −0.862395 0.506236i \(-0.831036\pi\)
0.862395 0.506236i \(-0.168964\pi\)
\(480\) 0 0
\(481\) 587.367i 1.22114i
\(482\) 406.940 0.844273
\(483\) 0 0
\(484\) −29.0000 117.473i −0.0599174 0.242714i
\(485\) 271.293 0.559367
\(486\) 0 0
\(487\) 230.000 0.472279 0.236140 0.971719i \(-0.424118\pi\)
0.236140 + 0.971719i \(0.424118\pi\)
\(488\) 508.675 1.04237
\(489\) 0 0
\(490\) 1045.51i 2.13370i
\(491\) 17.3205i 0.0352760i 0.999844 + 0.0176380i \(0.00561464\pi\)
−0.999844 + 0.0176380i \(0.994385\pi\)
\(492\) 0 0
\(493\) −360.000 −0.730223
\(494\) 0 0
\(495\) 0 0
\(496\) 110.000 0.221774
\(497\) 398.372i 0.801553i
\(498\) 0 0
\(499\) 152.000 0.304609 0.152305 0.988334i \(-0.451331\pi\)
0.152305 + 0.988334i \(0.451331\pi\)
\(500\) 27.1293 0.0542586
\(501\) 0 0
\(502\) 234.947i 0.468022i
\(503\) 769.031i 1.52889i −0.644690 0.764444i \(-0.723014\pi\)
0.644690 0.764444i \(-0.276986\pi\)
\(504\) 0 0
\(505\) 469.894i 0.930482i
\(506\) −508.675 398.372i −1.00529 0.787296i
\(507\) 0 0
\(508\) 105.726i 0.208122i
\(509\) 644.321 1.26586 0.632929 0.774210i \(-0.281853\pi\)
0.632929 + 0.774210i \(0.281853\pi\)
\(510\) 0 0
\(511\) −828.000 −1.62035
\(512\) 552.524i 1.07915i
\(513\) 0 0
\(514\) 587.367i 1.14274i
\(515\) −1153.00 −2.23883
\(516\) 0 0
\(517\) 230.000 293.684i 0.444874 0.568053i
\(518\) −1017.35 −1.96400
\(519\) 0 0
\(520\) −690.000 −1.32692
\(521\) −149.211 −0.286394 −0.143197 0.989694i \(-0.545738\pi\)
−0.143197 + 0.989694i \(0.545738\pi\)
\(522\) 0 0
\(523\) 281.936i 0.539075i 0.962990 + 0.269537i \(0.0868708\pi\)
−0.962990 + 0.269537i \(0.913129\pi\)
\(524\) 34.6410i 0.0661088i
\(525\) 0 0
\(526\) 168.000 0.319392
\(527\) 103.923i 0.197197i
\(528\) 0 0
\(529\) 621.000 1.17391
\(530\) 398.372i 0.751645i
\(531\) 0 0
\(532\) 0 0
\(533\) 406.940 0.763489
\(534\) 0 0
\(535\) 634.356i 1.18571i
\(536\) 86.6025i 0.161572i
\(537\) 0 0
\(538\) 58.7367i 0.109176i
\(539\) −603.627 + 770.763i −1.11990 + 1.42999i
\(540\) 0 0
\(541\) 881.051i 1.62856i 0.580473 + 0.814280i \(0.302868\pi\)
−0.580473 + 0.814280i \(0.697132\pi\)
\(542\) −305.205 −0.563109
\(543\) 0 0
\(544\) 162.000 0.297794
\(545\) 1195.12i 2.19287i
\(546\) 0 0
\(547\) 187.957i 0.343615i −0.985131 0.171808i \(-0.945039\pi\)
0.985131 0.171808i \(-0.0549607\pi\)
\(548\) −271.293 −0.495061
\(549\) 0 0
\(550\) −315.000 246.694i −0.572727 0.448535i
\(551\) 0 0
\(552\) 0 0
\(553\) 690.000 1.24774
\(554\) −20.3470 −0.0367274
\(555\) 0 0
\(556\) 117.473i 0.211283i
\(557\) 478.046i 0.858251i −0.903245 0.429126i \(-0.858822\pi\)
0.903245 0.429126i \(-0.141178\pi\)
\(558\) 0 0
\(559\) 552.000 0.987478
\(560\) 876.418i 1.56503i
\(561\) 0 0
\(562\) 330.000 0.587189
\(563\) 1084.26i 1.92587i −0.269737 0.962934i \(-0.586937\pi\)
0.269737 0.962934i \(-0.413063\pi\)
\(564\) 0 0
\(565\) 460.000 0.814159
\(566\) 691.798 1.22226
\(567\) 0 0
\(568\) 293.684i 0.517048i
\(569\) 606.218i 1.06541i 0.846301 + 0.532705i \(0.178824\pi\)
−0.846301 + 0.532705i \(0.821176\pi\)
\(570\) 0 0
\(571\) 117.473i 0.205733i 0.994695 + 0.102866i \(0.0328014\pi\)
−0.994695 + 0.102866i \(0.967199\pi\)
\(572\) 101.735 + 79.6743i 0.177858 + 0.139291i
\(573\) 0 0
\(574\) 704.840i 1.22794i
\(575\) 712.145 1.23851
\(576\) 0 0
\(577\) −400.000 −0.693241 −0.346620 0.938005i \(-0.612671\pi\)
−0.346620 + 0.938005i \(0.612671\pi\)
\(578\) 313.501i 0.542390i
\(579\) 0 0
\(580\) 234.947i 0.405081i
\(581\) 895.268 1.54091
\(582\) 0 0
\(583\) 230.000 293.684i 0.394511 0.503745i
\(584\) −610.410 −1.04522
\(585\) 0 0
\(586\) 732.000 1.24915
\(587\) −881.703 −1.50205 −0.751025 0.660274i \(-0.770440\pi\)
−0.751025 + 0.660274i \(0.770440\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 796.743i 1.35041i
\(591\) 0 0
\(592\) −550.000 −0.929054
\(593\) 613.146i 1.03397i 0.855994 + 0.516986i \(0.172946\pi\)
−0.855994 + 0.516986i \(0.827054\pi\)
\(594\) 0 0
\(595\) −828.000 −1.39160
\(596\) 138.564i 0.232490i
\(597\) 0 0
\(598\) 690.000 1.15385
\(599\) −1064.83 −1.77767 −0.888836 0.458225i \(-0.848485\pi\)
−0.888836 + 0.458225i \(0.848485\pi\)
\(600\) 0 0
\(601\) 117.473i 0.195463i −0.995213 0.0977316i \(-0.968841\pi\)
0.995213 0.0977316i \(-0.0311587\pi\)
\(602\) 956.092i 1.58819i
\(603\) 0 0
\(604\) 58.7367i 0.0972462i
\(605\) 196.688 + 796.743i 0.325103 + 1.31693i
\(606\) 0 0
\(607\) 693.093i 1.14183i −0.821008 0.570917i \(-0.806588\pi\)
0.821008 0.570917i \(-0.193412\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −690.000 −1.13115
\(611\) 398.372i 0.651999i
\(612\) 0 0
\(613\) 105.726i 0.172473i −0.996275 0.0862366i \(-0.972516\pi\)
0.996275 0.0862366i \(-0.0274841\pi\)
\(614\) −122.082 −0.198831
\(615\) 0 0
\(616\) −690.000 + 881.051i −1.12013 + 1.43028i
\(617\) −67.8233 −0.109924 −0.0549622 0.998488i \(-0.517504\pi\)
−0.0549622 + 0.998488i \(0.517504\pi\)
\(618\) 0 0
\(619\) −490.000 −0.791599 −0.395800 0.918337i \(-0.629533\pi\)
−0.395800 + 0.918337i \(0.629533\pi\)
\(620\) 67.8233 0.109392
\(621\) 0 0
\(622\) 646.104i 1.03875i
\(623\) 159.349i 0.255776i
\(624\) 0 0
\(625\) −709.000 −1.13440
\(626\) 121.244i 0.193680i
\(627\) 0 0
\(628\) 170.000 0.270701
\(629\) 519.615i 0.826097i
\(630\) 0 0
\(631\) 338.000 0.535658 0.267829 0.963467i \(-0.413694\pi\)
0.267829 + 0.963467i \(0.413694\pi\)
\(632\) 508.675 0.804865
\(633\) 0 0
\(634\) 293.684i 0.463223i
\(635\) 717.069i 1.12924i
\(636\) 0 0
\(637\) 1045.51i 1.64131i
\(638\) 406.940 519.615i 0.637837 0.814444i
\(639\) 0 0
\(640\) 411.157i 0.642433i
\(641\) 257.729 0.402073 0.201036 0.979584i \(-0.435569\pi\)
0.201036 + 0.979584i \(0.435569\pi\)
\(642\) 0 0
\(643\) −130.000 −0.202177 −0.101089 0.994877i \(-0.532233\pi\)
−0.101089 + 0.994877i \(0.532233\pi\)
\(644\) 398.372i 0.618590i
\(645\) 0 0
\(646\) 0 0
\(647\) 1051.26 1.62482 0.812412 0.583084i \(-0.198154\pi\)
0.812412 + 0.583084i \(0.198154\pi\)
\(648\) 0 0
\(649\) −460.000 + 587.367i −0.708783 + 0.905034i
\(650\) 427.287 0.657364
\(651\) 0 0
\(652\) 140.000 0.214724
\(653\) 237.382 0.363525 0.181762 0.983342i \(-0.441820\pi\)
0.181762 + 0.983342i \(0.441820\pi\)
\(654\) 0 0
\(655\) 234.947i 0.358697i
\(656\) 381.051i 0.580871i
\(657\) 0 0
\(658\) −690.000 −1.04863
\(659\) 398.372i 0.604509i 0.953227 + 0.302255i \(0.0977393\pi\)
−0.953227 + 0.302255i \(0.902261\pi\)
\(660\) 0 0
\(661\) 278.000 0.420575 0.210287 0.977640i \(-0.432560\pi\)
0.210287 + 0.977640i \(0.432560\pi\)
\(662\) 69.2820i 0.104656i
\(663\) 0 0
\(664\) 660.000 0.993976
\(665\) 0 0
\(666\) 0 0
\(667\) 1174.73i 1.76122i
\(668\) 200.918i 0.300775i
\(669\) 0 0
\(670\) 117.473i 0.175333i
\(671\) 508.675 + 398.372i 0.758085 + 0.593698i
\(672\) 0 0
\(673\) 540.378i 0.802939i −0.915872 0.401469i \(-0.868500\pi\)
0.915872 0.401469i \(-0.131500\pi\)
\(674\) −284.858 −0.422638
\(675\) 0 0
\(676\) 31.0000 0.0458580
\(677\) 491.902i 0.726591i 0.931674 + 0.363296i \(0.118349\pi\)
−0.931674 + 0.363296i \(0.881651\pi\)
\(678\) 0 0
\(679\) 469.894i 0.692038i
\(680\) −610.410 −0.897661
\(681\) 0 0
\(682\) 150.000 + 117.473i 0.219941 + 0.172248i
\(683\) 949.526 1.39023 0.695114 0.718899i \(-0.255354\pi\)
0.695114 + 0.718899i \(0.255354\pi\)
\(684\) 0 0
\(685\) 1840.00 2.68613
\(686\) 813.880 1.18641
\(687\) 0 0
\(688\) 516.883i 0.751283i
\(689\) 398.372i 0.578188i
\(690\) 0 0
\(691\) 482.000 0.697540 0.348770 0.937208i \(-0.386599\pi\)
0.348770 + 0.937208i \(0.386599\pi\)
\(692\) 304.841i 0.440522i
\(693\) 0 0
\(694\) −852.000 −1.22767
\(695\) 796.743i 1.14639i
\(696\) 0 0
\(697\) 360.000 0.516499
\(698\) −1119.08 −1.60327
\(699\) 0 0
\(700\) 246.694i 0.352420i
\(701\) 207.846i 0.296499i 0.988950 + 0.148250i \(0.0473639\pi\)
−0.988950 + 0.148250i \(0.952636\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −481.545 + 614.878i −0.684013 + 0.873406i
\(705\) 0 0
\(706\) 234.947i 0.332786i
\(707\) −813.880 −1.15117
\(708\) 0 0
\(709\) −1210.00 −1.70663 −0.853315 0.521397i \(-0.825411\pi\)
−0.853315 + 0.521397i \(0.825411\pi\)
\(710\) 398.372i 0.561087i
\(711\) 0 0
\(712\) 117.473i 0.164991i
\(713\) −339.116 −0.475619
\(714\) 0 0
\(715\) −690.000 540.378i −0.965035 0.755773i
\(716\) −149.211 −0.208396
\(717\) 0 0
\(718\) −180.000 −0.250696
\(719\) 1132.65 1.57531 0.787656 0.616115i \(-0.211295\pi\)
0.787656 + 0.616115i \(0.211295\pi\)
\(720\) 0 0
\(721\) 1997.05i 2.76983i
\(722\) 625.270i 0.866025i
\(723\) 0 0
\(724\) −130.000 −0.179558
\(725\) 727.461i 1.00339i
\(726\) 0 0
\(727\) 1130.00 1.55433 0.777166 0.629295i \(-0.216656\pi\)
0.777166 + 0.629295i \(0.216656\pi\)
\(728\) 1195.12i 1.64164i
\(729\) 0 0
\(730\) 828.000 1.13425
\(731\) 488.328 0.668027
\(732\) 0 0
\(733\) 928.040i 1.26608i 0.774117 + 0.633042i \(0.218194\pi\)
−0.774117 + 0.633042i \(0.781806\pi\)
\(734\) 225.167i 0.306766i
\(735\) 0 0
\(736\) 528.630i 0.718248i
\(737\) −67.8233 + 86.6025i −0.0920262 + 0.117507i
\(738\) 0 0
\(739\) 1057.26i 1.43066i −0.698785 0.715332i \(-0.746276\pi\)
0.698785 0.715332i \(-0.253724\pi\)
\(740\) −339.116 −0.458266
\(741\) 0 0
\(742\) −690.000 −0.929919
\(743\) 131.636i 0.177168i 0.996069 + 0.0885840i \(0.0282342\pi\)
−0.996069 + 0.0885840i \(0.971766\pi\)
\(744\) 0 0
\(745\) 939.787i 1.26146i
\(746\) −834.227 −1.11827
\(747\) 0 0
\(748\) 90.0000 + 70.4840i 0.120321 + 0.0942300i
\(749\) 1098.74 1.46694
\(750\) 0 0
\(751\) −298.000 −0.396804 −0.198402 0.980121i \(-0.563575\pi\)
−0.198402 + 0.980121i \(0.563575\pi\)
\(752\) −373.028 −0.496048
\(753\) 0 0
\(754\) 704.840i 0.934802i
\(755\) 398.372i 0.527645i
\(756\) 0 0
\(757\) −490.000 −0.647292 −0.323646 0.946178i \(-0.604909\pi\)
−0.323646 + 0.946178i \(0.604909\pi\)
\(758\) 533.472i 0.703788i
\(759\) 0 0
\(760\) 0 0
\(761\) 502.295i 0.660046i −0.943973 0.330023i \(-0.892944\pi\)
0.943973 0.330023i \(-0.107056\pi\)
\(762\) 0 0
\(763\) 2070.00 2.71298
\(764\) −88.1703 −0.115406
\(765\) 0 0
\(766\) 411.157i 0.536758i
\(767\) 796.743i 1.03878i
\(768\) 0 0
\(769\) 234.947i 0.305522i −0.988263 0.152761i \(-0.951183\pi\)
0.988263 0.152761i \(-0.0488166\pi\)
\(770\) 935.962 1195.12i 1.21553 1.55210i
\(771\) 0 0
\(772\) 187.957i 0.243468i
\(773\) −1390.38 −1.79868 −0.899339 0.437253i \(-0.855952\pi\)
−0.899339 + 0.437253i \(0.855952\pi\)
\(774\) 0 0
\(775\) −210.000 −0.270968
\(776\) 346.410i 0.446405i
\(777\) 0 0
\(778\) 1116.00i 1.43444i
\(779\) 0 0
\(780\) 0 0
\(781\) 230.000 293.684i 0.294494 0.376035i
\(782\) 610.410 0.780575
\(783\) 0 0
\(784\) 979.000 1.24872
\(785\) −1153.00 −1.46878
\(786\) 0 0
\(787\) 187.957i 0.238828i 0.992845 + 0.119414i \(0.0381015\pi\)
−0.992845 + 0.119414i \(0.961898\pi\)
\(788\) 41.5692i 0.0527528i
\(789\) 0 0
\(790\) −690.000 −0.873418
\(791\) 796.743i 1.00726i
\(792\) 0 0
\(793\) −690.000 −0.870113
\(794\) 536.936i 0.676241i
\(795\) 0 0
\(796\) −202.000 −0.253769
\(797\) 237.382 0.297844 0.148922 0.988849i \(-0.452420\pi\)
0.148922 + 0.988849i \(0.452420\pi\)
\(798\) 0 0
\(799\) 352.420i 0.441077i
\(800\) 327.358i 0.409197i
\(801\) 0 0
\(802\) 963.282i 1.20110i
\(803\) −610.410 478.046i −0.760162 0.595325i
\(804\) 0 0
\(805\) 2701.89i 3.35638i
\(806\) −203.470 −0.252444
\(807\) 0 0
\(808\) −600.000 −0.742574
\(809\) 1143.15i 1.41305i −0.707691 0.706523i \(-0.750263\pi\)
0.707691 0.706523i \(-0.249737\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 406.940 0.501157
\(813\) 0 0
\(814\) −750.000 587.367i −0.921376 0.721581i
\(815\) −949.526 −1.16506
\(816\) 0 0
\(817\) 0 0
\(818\) −1017.35 −1.24370
\(819\) 0 0
\(820\) 234.947i 0.286520i
\(821\) 969.948i 1.18142i 0.806883 + 0.590712i \(0.201153\pi\)
−0.806883 + 0.590712i \(0.798847\pi\)
\(822\) 0 0
\(823\) −310.000 −0.376671 −0.188335 0.982105i \(-0.560309\pi\)
−0.188335 + 0.982105i \(0.560309\pi\)
\(824\) 1472.24i 1.78670i
\(825\) 0 0
\(826\) 1380.00 1.67070
\(827\) 893.738i 1.08070i 0.841441 + 0.540350i \(0.181708\pi\)
−0.841441 + 0.540350i \(0.818292\pi\)
\(828\) 0 0
\(829\) −850.000 −1.02533 −0.512666 0.858588i \(-0.671342\pi\)
−0.512666 + 0.858588i \(0.671342\pi\)
\(830\) −895.268 −1.07864
\(831\) 0 0
\(832\) 834.061i 1.00248i
\(833\) 924.915i 1.11034i
\(834\) 0 0
\(835\) 1362.69i 1.63197i
\(836\) 0 0
\(837\) 0 0
\(838\) 93.9787i 0.112146i
\(839\) 440.851 0.525449 0.262724 0.964871i \(-0.415379\pi\)
0.262724 + 0.964871i \(0.415379\pi\)
\(840\) 0 0
\(841\) −359.000 −0.426873
\(842\) 536.936i 0.637691i
\(843\) 0 0
\(844\) 117.473i 0.139186i
\(845\) −210.252 −0.248819
\(846\) 0 0
\(847\) −1380.00 + 340.673i −1.62928 + 0.402211i
\(848\) −373.028 −0.439892
\(849\) 0 0
\(850\) 378.000 0.444706
\(851\) 1695.58 1.99246
\(852\) 0 0
\(853\) 1069.01i 1.25323i −0.779328 0.626617i \(-0.784439\pi\)
0.779328 0.626617i \(-0.215561\pi\)
\(854\) 1195.12i 1.39943i
\(855\) 0 0
\(856\) 810.000 0.946262
\(857\) 963.020i 1.12371i −0.827235 0.561855i \(-0.810088\pi\)
0.827235 0.561855i \(-0.189912\pi\)
\(858\) 0 0
\(859\) −820.000 −0.954598 −0.477299 0.878741i \(-0.658384\pi\)
−0.477299 + 0.878741i \(0.658384\pi\)
\(860\) 318.697i 0.370578i
\(861\) 0 0
\(862\) 180.000 0.208817
\(863\) −779.968 −0.903787 −0.451893 0.892072i \(-0.649251\pi\)
−0.451893 + 0.892072i \(0.649251\pi\)
\(864\) 0 0
\(865\) 2067.53i 2.39021i
\(866\) 1056.55i 1.22004i
\(867\) 0 0
\(868\) 117.473i 0.135338i
\(869\) 508.675 + 398.372i 0.585356 + 0.458425i
\(870\) 0 0
\(871\) 117.473i 0.134872i
\(872\) 1526.02 1.75003
\(873\) 0 0
\(874\) 0 0
\(875\) 318.697i 0.364226i
\(876\) 0 0
\(877\) 693.093i 0.790300i −0.918617 0.395150i \(-0.870693\pi\)
0.918617 0.395150i \(-0.129307\pi\)
\(878\) 712.145 0.811099
\(879\) 0 0
\(880\) 506.000 646.104i 0.575000 0.734209i
\(881\) −1492.11 −1.69366 −0.846829 0.531865i \(-0.821491\pi\)
−0.846829 + 0.531865i \(0.821491\pi\)
\(882\) 0 0
\(883\) 230.000 0.260476 0.130238 0.991483i \(-0.458426\pi\)
0.130238 + 0.991483i \(0.458426\pi\)
\(884\) −122.082 −0.138102
\(885\) 0 0
\(886\) 469.894i 0.530354i
\(887\) 547.328i 0.617055i 0.951215 + 0.308528i \(0.0998362\pi\)
−0.951215 + 0.308528i \(0.900164\pi\)
\(888\) 0 0
\(889\) −1242.00 −1.39708
\(890\) 159.349i 0.179043i
\(891\) 0 0
\(892\) −250.000 −0.280269
\(893\) 0 0
\(894\) 0 0
\(895\) 1012.00 1.13073
\(896\) 712.145 0.794804
\(897\) 0 0
\(898\) 375.915i 0.418613i
\(899\) 346.410i 0.385328i
\(900\) 0 0
\(901\) 352.420i 0.391143i
\(902\) −406.940 + 519.615i −0.451153 + 0.576070i
\(903\) 0 0
\(904\) 587.367i 0.649742i
\(905\) 881.703 0.974257
\(906\) 0 0
\(907\) −430.000 −0.474090 −0.237045 0.971499i \(-0.576179\pi\)
−0.237045 + 0.971499i \(0.576179\pi\)
\(908\) 45.0333i 0.0495962i
\(909\) 0 0
\(910\) 1621.13i 1.78146i
\(911\) 155.994 0.171233 0.0856167 0.996328i \(-0.472714\pi\)
0.0856167 + 0.996328i \(0.472714\pi\)
\(912\) 0 0
\(913\) 660.000 + 516.883i 0.722892 + 0.566137i
\(914\) 284.858 0.311661
\(915\) 0 0
\(916\) 158.000 0.172489
\(917\) −406.940 −0.443773
\(918\) 0 0
\(919\) 1233.47i 1.34219i 0.741372 + 0.671094i \(0.234175\pi\)
−0.741372 + 0.671094i \(0.765825\pi\)
\(920\) 1991.86i 2.16506i
\(921\) 0 0
\(922\) 1140.00 1.23644
\(923\) 398.372i 0.431605i
\(924\) 0 0
\(925\) 1050.00 1.13514
\(926\) 1160.47i 1.25321i
\(927\) 0 0
\(928\) 540.000 0.581897
\(929\) 949.526 1.02209 0.511047 0.859552i \(-0.329258\pi\)
0.511047 + 0.859552i \(0.329258\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 62.3538i 0.0669033i
\(933\) 0 0
\(934\) 587.367i 0.628873i
\(935\) −610.410 478.046i −0.652845 0.511279i
\(936\) 0 0
\(937\) 986.777i 1.05312i −0.850137 0.526562i \(-0.823481\pi\)
0.850137 0.526562i \(-0.176519\pi\)
\(938\) 203.470 0.216919
\(939\) 0 0
\(940\) −230.000 −0.244681
\(941\) 34.6410i 0.0368130i −0.999831 0.0184065i \(-0.994141\pi\)
0.999831 0.0184065i \(-0.00585930\pi\)
\(942\) 0 0
\(943\) 1174.73i 1.24574i
\(944\) 746.056 0.790314
\(945\) 0 0
\(946\) −552.000 + 704.840i −0.583510 + 0.745074i
\(947\) −271.293 −0.286476 −0.143238 0.989688i \(-0.545752\pi\)
−0.143238 + 0.989688i \(0.545752\pi\)
\(948\) 0 0
\(949\) 828.000 0.872497
\(950\) 0 0
\(951\) 0 0
\(952\) 1057.26i 1.11057i
\(953\) 145.492i 0.152668i −0.997082 0.0763338i \(-0.975679\pi\)
0.997082 0.0763338i \(-0.0243215\pi\)
\(954\) 0 0
\(955\) 598.000 0.626178
\(956\) 381.051i 0.398589i
\(957\) 0 0
\(958\) −840.000 −0.876827
\(959\) 3186.97i 3.32323i
\(960\) 0 0
\(961\) −861.000 −0.895942
\(962\) 1017.35 1.05754
\(963\) 0 0
\(964\) 234.947i 0.243721i
\(965\) 1274.79i 1.32103i
\(966\) 0 0
\(967\) 693.093i 0.716746i −0.933579 0.358373i \(-0.883332\pi\)
0.933579 0.358373i \(-0.116668\pi\)
\(968\) −1017.35 + 251.147i −1.05098 + 0.259450i
\(969\) 0 0
\(970\) 469.894i 0.484426i
\(971\) 868.138 0.894066 0.447033 0.894517i \(-0.352481\pi\)
0.447033 + 0.894517i \(0.352481\pi\)
\(972\) 0 0
\(973\) 1380.00 1.41829
\(974\) 398.372i 0.409006i
\(975\) 0 0
\(976\) 646.104i 0.661992i
\(977\) −881.703 −0.902459 −0.451230 0.892408i \(-0.649014\pi\)
−0.451230 + 0.892408i \(0.649014\pi\)
\(978\) 0 0
\(979\) 92.0000 117.473i 0.0939734 0.119993i
\(980\) 603.627 0.615946
\(981\) 0 0
\(982\) 30.0000 0.0305499
\(983\) 33.9116 0.0344981 0.0172491 0.999851i \(-0.494509\pi\)
0.0172491 + 0.999851i \(0.494509\pi\)
\(984\) 0 0
\(985\) 281.936i 0.286230i
\(986\) 623.538i 0.632392i
\(987\) 0 0
\(988\) 0 0
\(989\) 1593.49i 1.61121i
\(990\) 0 0
\(991\) 830.000 0.837538 0.418769 0.908093i \(-0.362462\pi\)
0.418769 + 0.908093i \(0.362462\pi\)
\(992\) 155.885i 0.157142i
\(993\) 0 0
\(994\) −690.000 −0.694165
\(995\) 1370.03 1.37692
\(996\) 0 0
\(997\) 1186.48i 1.19005i 0.803707 + 0.595026i \(0.202858\pi\)
−0.803707 + 0.595026i \(0.797142\pi\)
\(998\) 263.272i 0.263799i
\(999\) 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 99.3.c.c.10.1 4
3.2 odd 2 inner 99.3.c.c.10.4 yes 4
4.3 odd 2 1584.3.j.i.1297.2 4
11.10 odd 2 inner 99.3.c.c.10.3 yes 4
12.11 even 2 1584.3.j.i.1297.4 4
33.32 even 2 inner 99.3.c.c.10.2 yes 4
44.43 even 2 1584.3.j.i.1297.1 4
132.131 odd 2 1584.3.j.i.1297.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.c.c.10.1 4 1.1 even 1 trivial
99.3.c.c.10.2 yes 4 33.32 even 2 inner
99.3.c.c.10.3 yes 4 11.10 odd 2 inner
99.3.c.c.10.4 yes 4 3.2 odd 2 inner
1584.3.j.i.1297.1 4 44.43 even 2
1584.3.j.i.1297.2 4 4.3 odd 2
1584.3.j.i.1297.3 4 132.131 odd 2
1584.3.j.i.1297.4 4 12.11 even 2