Properties

Label 2850.2.d.v.799.1
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $4$
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] = [2850,2,Mod(799,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2850.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not 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: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.v.799.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.16228i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.16228i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.16228 q^{11} +1.00000i q^{12} +5.16228i q^{13} -3.16228 q^{14} +1.00000 q^{16} -0.837722i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.16228 q^{21} -4.16228i q^{22} -5.32456i q^{23} +1.00000 q^{24} +5.16228 q^{26} +1.00000i q^{27} +3.16228i q^{28} +3.83772 q^{29} +5.32456 q^{31} -1.00000i q^{32} -4.16228i q^{33} -0.837722 q^{34} +1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} +5.16228 q^{39} -6.32456 q^{41} +3.16228i q^{42} -9.16228i q^{43} -4.16228 q^{44} -5.32456 q^{46} -6.00000i q^{47} -1.00000i q^{48} -3.00000 q^{49} -0.837722 q^{51} -5.16228i q^{52} +1.83772i q^{53} +1.00000 q^{54} +3.16228 q^{56} -1.00000i q^{57} -3.83772i q^{58} +7.48683 q^{59} -4.16228 q^{61} -5.32456i q^{62} +3.16228i q^{63} -1.00000 q^{64} -4.16228 q^{66} +6.48683i q^{67} +0.837722i q^{68} -5.32456 q^{69} +1.16228 q^{71} -1.00000i q^{72} -1.00000i q^{73} -10.0000 q^{74} -1.00000 q^{76} -13.1623i q^{77} -5.16228i q^{78} +5.32456 q^{79} +1.00000 q^{81} +6.32456i q^{82} +6.48683i q^{83} +3.16228 q^{84} -9.16228 q^{86} -3.83772i q^{87} +4.16228i q^{88} +7.32456 q^{89} +16.3246 q^{91} +5.32456i q^{92} -5.32456i q^{93} -6.00000 q^{94} -1.00000 q^{96} -0.513167i q^{97} +3.00000i q^{98} -4.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{11} + 4 q^{16} + 4 q^{19} + 4 q^{24} + 8 q^{26} + 28 q^{29} - 4 q^{31} - 16 q^{34} + 4 q^{36} + 8 q^{39} - 4 q^{44} + 4 q^{46} - 12 q^{49} - 16 q^{51} + 4 q^{54} - 8 q^{59} - 4 q^{61} - 4 q^{64} - 4 q^{66} + 4 q^{69} - 8 q^{71} - 40 q^{74} - 4 q^{76} - 4 q^{79} + 4 q^{81} - 24 q^{86} + 4 q^{89} + 40 q^{91} - 24 q^{94} - 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 3.16228i − 1.19523i −0.801784 0.597614i \(-0.796115\pi\)
0.801784 0.597614i \(-0.203885\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.16228 1.25497 0.627487 0.778627i \(-0.284084\pi\)
0.627487 + 0.778627i \(0.284084\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 5.16228i 1.43176i 0.698224 + 0.715879i \(0.253974\pi\)
−0.698224 + 0.715879i \(0.746026\pi\)
\(14\) −3.16228 −0.845154
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.837722i − 0.203178i −0.994826 0.101589i \(-0.967607\pi\)
0.994826 0.101589i \(-0.0323926\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.16228 −0.690066
\(22\) − 4.16228i − 0.887401i
\(23\) − 5.32456i − 1.11025i −0.831768 0.555123i \(-0.812671\pi\)
0.831768 0.555123i \(-0.187329\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 5.16228 1.01241
\(27\) 1.00000i 0.192450i
\(28\) 3.16228i 0.597614i
\(29\) 3.83772 0.712647 0.356324 0.934363i \(-0.384030\pi\)
0.356324 + 0.934363i \(0.384030\pi\)
\(30\) 0 0
\(31\) 5.32456 0.956318 0.478159 0.878273i \(-0.341304\pi\)
0.478159 + 0.878273i \(0.341304\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.16228i − 0.724560i
\(34\) −0.837722 −0.143668
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 5.16228 0.826626
\(40\) 0 0
\(41\) −6.32456 −0.987730 −0.493865 0.869539i \(-0.664416\pi\)
−0.493865 + 0.869539i \(0.664416\pi\)
\(42\) 3.16228i 0.487950i
\(43\) − 9.16228i − 1.39723i −0.715496 0.698617i \(-0.753799\pi\)
0.715496 0.698617i \(-0.246201\pi\)
\(44\) −4.16228 −0.627487
\(45\) 0 0
\(46\) −5.32456 −0.785063
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −0.837722 −0.117305
\(52\) − 5.16228i − 0.715879i
\(53\) 1.83772i 0.252431i 0.992003 + 0.126215i \(0.0402830\pi\)
−0.992003 + 0.126215i \(0.959717\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.16228 0.422577
\(57\) − 1.00000i − 0.132453i
\(58\) − 3.83772i − 0.503918i
\(59\) 7.48683 0.974703 0.487351 0.873206i \(-0.337963\pi\)
0.487351 + 0.873206i \(0.337963\pi\)
\(60\) 0 0
\(61\) −4.16228 −0.532925 −0.266463 0.963845i \(-0.585855\pi\)
−0.266463 + 0.963845i \(0.585855\pi\)
\(62\) − 5.32456i − 0.676219i
\(63\) 3.16228i 0.398410i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.16228 −0.512341
\(67\) 6.48683i 0.792493i 0.918144 + 0.396246i \(0.129687\pi\)
−0.918144 + 0.396246i \(0.870313\pi\)
\(68\) 0.837722i 0.101589i
\(69\) −5.32456 −0.641001
\(70\) 0 0
\(71\) 1.16228 0.137937 0.0689685 0.997619i \(-0.478029\pi\)
0.0689685 + 0.997619i \(0.478029\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 1.00000i − 0.117041i −0.998286 0.0585206i \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 13.1623i − 1.49998i
\(78\) − 5.16228i − 0.584513i
\(79\) 5.32456 0.599059 0.299530 0.954087i \(-0.403170\pi\)
0.299530 + 0.954087i \(0.403170\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.32456i 0.698430i
\(83\) 6.48683i 0.712022i 0.934482 + 0.356011i \(0.115864\pi\)
−0.934482 + 0.356011i \(0.884136\pi\)
\(84\) 3.16228 0.345033
\(85\) 0 0
\(86\) −9.16228 −0.987994
\(87\) − 3.83772i − 0.411447i
\(88\) 4.16228i 0.443700i
\(89\) 7.32456 0.776401 0.388201 0.921575i \(-0.373097\pi\)
0.388201 + 0.921575i \(0.373097\pi\)
\(90\) 0 0
\(91\) 16.3246 1.71128
\(92\) 5.32456i 0.555123i
\(93\) − 5.32456i − 0.552131i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 0.513167i − 0.0521042i −0.999661 0.0260521i \(-0.991706\pi\)
0.999661 0.0260521i \(-0.00829358\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −4.16228 −0.418325
\(100\) 0 0
\(101\) −13.8114 −1.37428 −0.687142 0.726523i \(-0.741135\pi\)
−0.687142 + 0.726523i \(0.741135\pi\)
\(102\) 0.837722i 0.0829469i
\(103\) 15.6491i 1.54195i 0.636864 + 0.770976i \(0.280231\pi\)
−0.636864 + 0.770976i \(0.719769\pi\)
\(104\) −5.16228 −0.506203
\(105\) 0 0
\(106\) 1.83772 0.178495
\(107\) − 12.8377i − 1.24107i −0.784179 0.620535i \(-0.786916\pi\)
0.784179 0.620535i \(-0.213084\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −7.67544 −0.735174 −0.367587 0.929989i \(-0.619816\pi\)
−0.367587 + 0.929989i \(0.619816\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) − 3.16228i − 0.298807i
\(113\) 9.64911i 0.907712i 0.891075 + 0.453856i \(0.149952\pi\)
−0.891075 + 0.453856i \(0.850048\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −3.83772 −0.356324
\(117\) − 5.16228i − 0.477253i
\(118\) − 7.48683i − 0.689219i
\(119\) −2.64911 −0.242844
\(120\) 0 0
\(121\) 6.32456 0.574960
\(122\) 4.16228i 0.376835i
\(123\) 6.32456i 0.570266i
\(124\) −5.32456 −0.478159
\(125\) 0 0
\(126\) 3.16228 0.281718
\(127\) − 15.6491i − 1.38863i −0.719669 0.694317i \(-0.755707\pi\)
0.719669 0.694317i \(-0.244293\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −9.16228 −0.806694
\(130\) 0 0
\(131\) −12.4868 −1.09098 −0.545490 0.838117i \(-0.683656\pi\)
−0.545490 + 0.838117i \(0.683656\pi\)
\(132\) 4.16228i 0.362280i
\(133\) − 3.16228i − 0.274204i
\(134\) 6.48683 0.560377
\(135\) 0 0
\(136\) 0.837722 0.0718341
\(137\) − 2.32456i − 0.198600i −0.995058 0.0993001i \(-0.968340\pi\)
0.995058 0.0993001i \(-0.0316604\pi\)
\(138\) 5.32456i 0.453256i
\(139\) −21.4868 −1.82249 −0.911245 0.411865i \(-0.864877\pi\)
−0.911245 + 0.411865i \(0.864877\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 1.16228i − 0.0975362i
\(143\) 21.4868i 1.79682i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 3.00000i 0.247436i
\(148\) 10.0000i 0.821995i
\(149\) −16.6491 −1.36395 −0.681974 0.731376i \(-0.738878\pi\)
−0.681974 + 0.731376i \(0.738878\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 0.837722i 0.0677258i
\(154\) −13.1623 −1.06065
\(155\) 0 0
\(156\) −5.16228 −0.413313
\(157\) − 18.3246i − 1.46246i −0.682132 0.731229i \(-0.738947\pi\)
0.682132 0.731229i \(-0.261053\pi\)
\(158\) − 5.32456i − 0.423599i
\(159\) 1.83772 0.145741
\(160\) 0 0
\(161\) −16.8377 −1.32700
\(162\) − 1.00000i − 0.0785674i
\(163\) − 22.6491i − 1.77402i −0.461755 0.887008i \(-0.652780\pi\)
0.461755 0.887008i \(-0.347220\pi\)
\(164\) 6.32456 0.493865
\(165\) 0 0
\(166\) 6.48683 0.503476
\(167\) − 2.64911i − 0.204994i −0.994733 0.102497i \(-0.967317\pi\)
0.994733 0.102497i \(-0.0326833\pi\)
\(168\) − 3.16228i − 0.243975i
\(169\) −13.6491 −1.04993
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 9.16228i 0.698617i
\(173\) − 2.16228i − 0.164395i −0.996616 0.0821975i \(-0.973806\pi\)
0.996616 0.0821975i \(-0.0261938\pi\)
\(174\) −3.83772 −0.290937
\(175\) 0 0
\(176\) 4.16228 0.313743
\(177\) − 7.48683i − 0.562745i
\(178\) − 7.32456i − 0.548999i
\(179\) 20.6491 1.54339 0.771693 0.635995i \(-0.219410\pi\)
0.771693 + 0.635995i \(0.219410\pi\)
\(180\) 0 0
\(181\) 9.16228 0.681027 0.340513 0.940240i \(-0.389399\pi\)
0.340513 + 0.940240i \(0.389399\pi\)
\(182\) − 16.3246i − 1.21006i
\(183\) 4.16228i 0.307684i
\(184\) 5.32456 0.392531
\(185\) 0 0
\(186\) −5.32456 −0.390415
\(187\) − 3.48683i − 0.254982i
\(188\) 6.00000i 0.437595i
\(189\) 3.16228 0.230022
\(190\) 0 0
\(191\) 21.6491 1.56647 0.783237 0.621723i \(-0.213567\pi\)
0.783237 + 0.621723i \(0.213567\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 9.16228i − 0.659515i −0.944066 0.329758i \(-0.893033\pi\)
0.944066 0.329758i \(-0.106967\pi\)
\(194\) −0.513167 −0.0368432
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 24.1359i 1.71961i 0.510618 + 0.859807i \(0.329416\pi\)
−0.510618 + 0.859807i \(0.670584\pi\)
\(198\) 4.16228i 0.295800i
\(199\) 18.6491 1.32200 0.661000 0.750386i \(-0.270132\pi\)
0.661000 + 0.750386i \(0.270132\pi\)
\(200\) 0 0
\(201\) 6.48683 0.457546
\(202\) 13.8114i 0.971766i
\(203\) − 12.1359i − 0.851776i
\(204\) 0.837722 0.0586523
\(205\) 0 0
\(206\) 15.6491 1.09033
\(207\) 5.32456i 0.370082i
\(208\) 5.16228i 0.357940i
\(209\) 4.16228 0.287911
\(210\) 0 0
\(211\) −18.8114 −1.29503 −0.647515 0.762053i \(-0.724192\pi\)
−0.647515 + 0.762053i \(0.724192\pi\)
\(212\) − 1.83772i − 0.126215i
\(213\) − 1.16228i − 0.0796380i
\(214\) −12.8377 −0.877569
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 16.8377i − 1.14302i
\(218\) 7.67544i 0.519847i
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) 4.32456 0.290901
\(222\) 10.0000i 0.671156i
\(223\) − 25.6491i − 1.71759i −0.512318 0.858796i \(-0.671213\pi\)
0.512318 0.858796i \(-0.328787\pi\)
\(224\) −3.16228 −0.211289
\(225\) 0 0
\(226\) 9.64911 0.641849
\(227\) 2.00000i 0.132745i 0.997795 + 0.0663723i \(0.0211425\pi\)
−0.997795 + 0.0663723i \(0.978857\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −8.16228 −0.539378 −0.269689 0.962947i \(-0.586921\pi\)
−0.269689 + 0.962947i \(0.586921\pi\)
\(230\) 0 0
\(231\) −13.1623 −0.866014
\(232\) 3.83772i 0.251959i
\(233\) − 11.8114i − 0.773790i −0.922124 0.386895i \(-0.873548\pi\)
0.922124 0.386895i \(-0.126452\pi\)
\(234\) −5.16228 −0.337469
\(235\) 0 0
\(236\) −7.48683 −0.487351
\(237\) − 5.32456i − 0.345867i
\(238\) 2.64911i 0.171716i
\(239\) 24.3246 1.57342 0.786712 0.617320i \(-0.211782\pi\)
0.786712 + 0.617320i \(0.211782\pi\)
\(240\) 0 0
\(241\) 10.5132 0.677213 0.338606 0.940928i \(-0.390045\pi\)
0.338606 + 0.940928i \(0.390045\pi\)
\(242\) − 6.32456i − 0.406558i
\(243\) − 1.00000i − 0.0641500i
\(244\) 4.16228 0.266463
\(245\) 0 0
\(246\) 6.32456 0.403239
\(247\) 5.16228i 0.328468i
\(248\) 5.32456i 0.338110i
\(249\) 6.48683 0.411086
\(250\) 0 0
\(251\) −10.6491 −0.672166 −0.336083 0.941832i \(-0.609102\pi\)
−0.336083 + 0.941832i \(0.609102\pi\)
\(252\) − 3.16228i − 0.199205i
\(253\) − 22.1623i − 1.39333i
\(254\) −15.6491 −0.981913
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 19.3246i 1.20543i 0.797956 + 0.602716i \(0.205915\pi\)
−0.797956 + 0.602716i \(0.794085\pi\)
\(258\) 9.16228i 0.570418i
\(259\) −31.6228 −1.96494
\(260\) 0 0
\(261\) −3.83772 −0.237549
\(262\) 12.4868i 0.771439i
\(263\) 31.9737i 1.97158i 0.167980 + 0.985790i \(0.446275\pi\)
−0.167980 + 0.985790i \(0.553725\pi\)
\(264\) 4.16228 0.256170
\(265\) 0 0
\(266\) −3.16228 −0.193892
\(267\) − 7.32456i − 0.448256i
\(268\) − 6.48683i − 0.396246i
\(269\) −12.3246 −0.751441 −0.375721 0.926733i \(-0.622605\pi\)
−0.375721 + 0.926733i \(0.622605\pi\)
\(270\) 0 0
\(271\) −1.16228 −0.0706033 −0.0353017 0.999377i \(-0.511239\pi\)
−0.0353017 + 0.999377i \(0.511239\pi\)
\(272\) − 0.837722i − 0.0507944i
\(273\) − 16.3246i − 0.988007i
\(274\) −2.32456 −0.140432
\(275\) 0 0
\(276\) 5.32456 0.320501
\(277\) − 24.8114i − 1.49077i −0.666633 0.745386i \(-0.732265\pi\)
0.666633 0.745386i \(-0.267735\pi\)
\(278\) 21.4868i 1.28869i
\(279\) −5.32456 −0.318773
\(280\) 0 0
\(281\) 7.64911 0.456308 0.228154 0.973625i \(-0.426731\pi\)
0.228154 + 0.973625i \(0.426731\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 24.6491i 1.46524i 0.680639 + 0.732619i \(0.261702\pi\)
−0.680639 + 0.732619i \(0.738298\pi\)
\(284\) −1.16228 −0.0689685
\(285\) 0 0
\(286\) 21.4868 1.27054
\(287\) 20.0000i 1.18056i
\(288\) 1.00000i 0.0589256i
\(289\) 16.2982 0.958719
\(290\) 0 0
\(291\) −0.513167 −0.0300824
\(292\) 1.00000i 0.0585206i
\(293\) 16.4868i 0.963171i 0.876399 + 0.481586i \(0.159939\pi\)
−0.876399 + 0.481586i \(0.840061\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 4.16228i 0.241520i
\(298\) 16.6491i 0.964457i
\(299\) 27.4868 1.58960
\(300\) 0 0
\(301\) −28.9737 −1.67001
\(302\) 10.0000i 0.575435i
\(303\) 13.8114i 0.793444i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0.837722 0.0478894
\(307\) − 13.5132i − 0.771237i −0.922658 0.385619i \(-0.873988\pi\)
0.922658 0.385619i \(-0.126012\pi\)
\(308\) 13.1623i 0.749990i
\(309\) 15.6491 0.890247
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 5.16228i 0.292256i
\(313\) − 10.6754i − 0.603412i −0.953401 0.301706i \(-0.902444\pi\)
0.953401 0.301706i \(-0.0975561\pi\)
\(314\) −18.3246 −1.03411
\(315\) 0 0
\(316\) −5.32456 −0.299530
\(317\) − 4.16228i − 0.233777i −0.993145 0.116888i \(-0.962708\pi\)
0.993145 0.116888i \(-0.0372920\pi\)
\(318\) − 1.83772i − 0.103054i
\(319\) 15.9737 0.894354
\(320\) 0 0
\(321\) −12.8377 −0.716532
\(322\) 16.8377i 0.938330i
\(323\) − 0.837722i − 0.0466121i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −22.6491 −1.25442
\(327\) 7.67544i 0.424453i
\(328\) − 6.32456i − 0.349215i
\(329\) −18.9737 −1.04605
\(330\) 0 0
\(331\) −3.18861 −0.175262 −0.0876310 0.996153i \(-0.527930\pi\)
−0.0876310 + 0.996153i \(0.527930\pi\)
\(332\) − 6.48683i − 0.356011i
\(333\) 10.0000i 0.547997i
\(334\) −2.64911 −0.144953
\(335\) 0 0
\(336\) −3.16228 −0.172516
\(337\) 2.97367i 0.161986i 0.996715 + 0.0809930i \(0.0258091\pi\)
−0.996715 + 0.0809930i \(0.974191\pi\)
\(338\) 13.6491i 0.742414i
\(339\) 9.64911 0.524068
\(340\) 0 0
\(341\) 22.1623 1.20015
\(342\) 1.00000i 0.0540738i
\(343\) − 12.6491i − 0.682988i
\(344\) 9.16228 0.493997
\(345\) 0 0
\(346\) −2.16228 −0.116245
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 3.83772i 0.205724i
\(349\) 17.8377 0.954831 0.477416 0.878678i \(-0.341574\pi\)
0.477416 + 0.878678i \(0.341574\pi\)
\(350\) 0 0
\(351\) −5.16228 −0.275542
\(352\) − 4.16228i − 0.221850i
\(353\) 23.8114i 1.26735i 0.773598 + 0.633676i \(0.218455\pi\)
−0.773598 + 0.633676i \(0.781545\pi\)
\(354\) −7.48683 −0.397921
\(355\) 0 0
\(356\) −7.32456 −0.388201
\(357\) 2.64911i 0.140206i
\(358\) − 20.6491i − 1.09134i
\(359\) 11.6754 0.616206 0.308103 0.951353i \(-0.400306\pi\)
0.308103 + 0.951353i \(0.400306\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 9.16228i − 0.481559i
\(363\) − 6.32456i − 0.331953i
\(364\) −16.3246 −0.855639
\(365\) 0 0
\(366\) 4.16228 0.217566
\(367\) 7.48683i 0.390810i 0.980723 + 0.195405i \(0.0626020\pi\)
−0.980723 + 0.195405i \(0.937398\pi\)
\(368\) − 5.32456i − 0.277562i
\(369\) 6.32456 0.329243
\(370\) 0 0
\(371\) 5.81139 0.301712
\(372\) 5.32456i 0.276065i
\(373\) − 16.1359i − 0.835487i −0.908565 0.417744i \(-0.862821\pi\)
0.908565 0.417744i \(-0.137179\pi\)
\(374\) −3.48683 −0.180300
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 19.8114i 1.02034i
\(378\) − 3.16228i − 0.162650i
\(379\) −22.9737 −1.18008 −0.590039 0.807375i \(-0.700888\pi\)
−0.590039 + 0.807375i \(0.700888\pi\)
\(380\) 0 0
\(381\) −15.6491 −0.801728
\(382\) − 21.6491i − 1.10766i
\(383\) − 3.48683i − 0.178169i −0.996024 0.0890844i \(-0.971606\pi\)
0.996024 0.0890844i \(-0.0283941\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −9.16228 −0.466348
\(387\) 9.16228i 0.465745i
\(388\) 0.513167i 0.0260521i
\(389\) 34.6491 1.75678 0.878390 0.477945i \(-0.158618\pi\)
0.878390 + 0.477945i \(0.158618\pi\)
\(390\) 0 0
\(391\) −4.46050 −0.225577
\(392\) − 3.00000i − 0.151523i
\(393\) 12.4868i 0.629877i
\(394\) 24.1359 1.21595
\(395\) 0 0
\(396\) 4.16228 0.209162
\(397\) − 17.1359i − 0.860028i −0.902822 0.430014i \(-0.858509\pi\)
0.902822 0.430014i \(-0.141491\pi\)
\(398\) − 18.6491i − 0.934795i
\(399\) −3.16228 −0.158312
\(400\) 0 0
\(401\) 13.6491 0.681604 0.340802 0.940135i \(-0.389301\pi\)
0.340802 + 0.940135i \(0.389301\pi\)
\(402\) − 6.48683i − 0.323534i
\(403\) 27.4868i 1.36922i
\(404\) 13.8114 0.687142
\(405\) 0 0
\(406\) −12.1359 −0.602297
\(407\) − 41.6228i − 2.06316i
\(408\) − 0.837722i − 0.0414734i
\(409\) 7.67544 0.379526 0.189763 0.981830i \(-0.439228\pi\)
0.189763 + 0.981830i \(0.439228\pi\)
\(410\) 0 0
\(411\) −2.32456 −0.114662
\(412\) − 15.6491i − 0.770976i
\(413\) − 23.6754i − 1.16499i
\(414\) 5.32456 0.261688
\(415\) 0 0
\(416\) 5.16228 0.253101
\(417\) 21.4868i 1.05221i
\(418\) − 4.16228i − 0.203584i
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 18.6491 0.908902 0.454451 0.890772i \(-0.349835\pi\)
0.454451 + 0.890772i \(0.349835\pi\)
\(422\) 18.8114i 0.915724i
\(423\) 6.00000i 0.291730i
\(424\) −1.83772 −0.0892477
\(425\) 0 0
\(426\) −1.16228 −0.0563125
\(427\) 13.1623i 0.636967i
\(428\) 12.8377i 0.620535i
\(429\) 21.4868 1.03739
\(430\) 0 0
\(431\) −0.324555 −0.0156333 −0.00781664 0.999969i \(-0.502488\pi\)
−0.00781664 + 0.999969i \(0.502488\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 28.1359i 1.35213i 0.736843 + 0.676064i \(0.236316\pi\)
−0.736843 + 0.676064i \(0.763684\pi\)
\(434\) −16.8377 −0.808237
\(435\) 0 0
\(436\) 7.67544 0.367587
\(437\) − 5.32456i − 0.254708i
\(438\) 1.00000i 0.0477818i
\(439\) −13.6491 −0.651437 −0.325718 0.945467i \(-0.605606\pi\)
−0.325718 + 0.945467i \(0.605606\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) − 4.32456i − 0.205698i
\(443\) − 11.8377i − 0.562427i −0.959645 0.281214i \(-0.909263\pi\)
0.959645 0.281214i \(-0.0907369\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −25.6491 −1.21452
\(447\) 16.6491i 0.787476i
\(448\) 3.16228i 0.149404i
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 0 0
\(451\) −26.3246 −1.23957
\(452\) − 9.64911i − 0.453856i
\(453\) 10.0000i 0.469841i
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 9.67544i − 0.452598i −0.974058 0.226299i \(-0.927337\pi\)
0.974058 0.226299i \(-0.0726627\pi\)
\(458\) 8.16228i 0.381398i
\(459\) 0.837722 0.0391015
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 13.1623i 0.612365i
\(463\) − 21.6754i − 1.00734i −0.863895 0.503672i \(-0.831982\pi\)
0.863895 0.503672i \(-0.168018\pi\)
\(464\) 3.83772 0.178162
\(465\) 0 0
\(466\) −11.8114 −0.547152
\(467\) 37.1359i 1.71845i 0.511601 + 0.859223i \(0.329053\pi\)
−0.511601 + 0.859223i \(0.670947\pi\)
\(468\) 5.16228i 0.238626i
\(469\) 20.5132 0.947210
\(470\) 0 0
\(471\) −18.3246 −0.844351
\(472\) 7.48683i 0.344609i
\(473\) − 38.1359i − 1.75349i
\(474\) −5.32456 −0.244565
\(475\) 0 0
\(476\) 2.64911 0.121422
\(477\) − 1.83772i − 0.0841435i
\(478\) − 24.3246i − 1.11258i
\(479\) 39.9737 1.82644 0.913222 0.407463i \(-0.133586\pi\)
0.913222 + 0.407463i \(0.133586\pi\)
\(480\) 0 0
\(481\) 51.6228 2.35380
\(482\) − 10.5132i − 0.478862i
\(483\) 16.8377i 0.766143i
\(484\) −6.32456 −0.287480
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 28.6491i 1.29822i 0.760697 + 0.649108i \(0.224857\pi\)
−0.760697 + 0.649108i \(0.775143\pi\)
\(488\) − 4.16228i − 0.188417i
\(489\) −22.6491 −1.02423
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) − 6.32456i − 0.285133i
\(493\) − 3.21495i − 0.144794i
\(494\) 5.16228 0.232262
\(495\) 0 0
\(496\) 5.32456 0.239080
\(497\) − 3.67544i − 0.164866i
\(498\) − 6.48683i − 0.290682i
\(499\) 5.81139 0.260153 0.130077 0.991504i \(-0.458478\pi\)
0.130077 + 0.991504i \(0.458478\pi\)
\(500\) 0 0
\(501\) −2.64911 −0.118354
\(502\) 10.6491i 0.475293i
\(503\) − 29.2982i − 1.30634i −0.757210 0.653172i \(-0.773438\pi\)
0.757210 0.653172i \(-0.226562\pi\)
\(504\) −3.16228 −0.140859
\(505\) 0 0
\(506\) −22.1623 −0.985233
\(507\) 13.6491i 0.606178i
\(508\) 15.6491i 0.694317i
\(509\) −16.8114 −0.745152 −0.372576 0.928002i \(-0.621525\pi\)
−0.372576 + 0.928002i \(0.621525\pi\)
\(510\) 0 0
\(511\) −3.16228 −0.139891
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 19.3246 0.852370
\(515\) 0 0
\(516\) 9.16228 0.403347
\(517\) − 24.9737i − 1.09834i
\(518\) 31.6228i 1.38943i
\(519\) −2.16228 −0.0949135
\(520\) 0 0
\(521\) 19.9737 0.875062 0.437531 0.899203i \(-0.355853\pi\)
0.437531 + 0.899203i \(0.355853\pi\)
\(522\) 3.83772i 0.167973i
\(523\) − 16.6491i − 0.728015i −0.931396 0.364007i \(-0.881408\pi\)
0.931396 0.364007i \(-0.118592\pi\)
\(524\) 12.4868 0.545490
\(525\) 0 0
\(526\) 31.9737 1.39412
\(527\) − 4.46050i − 0.194302i
\(528\) − 4.16228i − 0.181140i
\(529\) −5.35089 −0.232647
\(530\) 0 0
\(531\) −7.48683 −0.324901
\(532\) 3.16228i 0.137102i
\(533\) − 32.6491i − 1.41419i
\(534\) −7.32456 −0.316965
\(535\) 0 0
\(536\) −6.48683 −0.280189
\(537\) − 20.6491i − 0.891075i
\(538\) 12.3246i 0.531349i
\(539\) −12.4868 −0.537846
\(540\) 0 0
\(541\) −0.162278 −0.00697686 −0.00348843 0.999994i \(-0.501110\pi\)
−0.00348843 + 0.999994i \(0.501110\pi\)
\(542\) 1.16228i 0.0499241i
\(543\) − 9.16228i − 0.393191i
\(544\) −0.837722 −0.0359170
\(545\) 0 0
\(546\) −16.3246 −0.698626
\(547\) 16.8114i 0.718803i 0.933183 + 0.359402i \(0.117019\pi\)
−0.933183 + 0.359402i \(0.882981\pi\)
\(548\) 2.32456i 0.0993001i
\(549\) 4.16228 0.177642
\(550\) 0 0
\(551\) 3.83772 0.163492
\(552\) − 5.32456i − 0.226628i
\(553\) − 16.8377i − 0.716013i
\(554\) −24.8114 −1.05413
\(555\) 0 0
\(556\) 21.4868 0.911245
\(557\) 22.8377i 0.967665i 0.875161 + 0.483833i \(0.160756\pi\)
−0.875161 + 0.483833i \(0.839244\pi\)
\(558\) 5.32456i 0.225406i
\(559\) 47.2982 2.00050
\(560\) 0 0
\(561\) −3.48683 −0.147214
\(562\) − 7.64911i − 0.322658i
\(563\) 31.1623i 1.31333i 0.754181 + 0.656667i \(0.228034\pi\)
−0.754181 + 0.656667i \(0.771966\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 24.6491 1.03608
\(567\) − 3.16228i − 0.132803i
\(568\) 1.16228i 0.0487681i
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −15.8114 −0.661686 −0.330843 0.943686i \(-0.607333\pi\)
−0.330843 + 0.943686i \(0.607333\pi\)
\(572\) − 21.4868i − 0.898410i
\(573\) − 21.6491i − 0.904405i
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 16.6754i 0.694208i 0.937827 + 0.347104i \(0.112835\pi\)
−0.937827 + 0.347104i \(0.887165\pi\)
\(578\) − 16.2982i − 0.677917i
\(579\) −9.16228 −0.380771
\(580\) 0 0
\(581\) 20.5132 0.851030
\(582\) 0.513167i 0.0212715i
\(583\) 7.64911i 0.316794i
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 16.4868 0.681065
\(587\) 18.8114i 0.776429i 0.921569 + 0.388215i \(0.126908\pi\)
−0.921569 + 0.388215i \(0.873092\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 5.32456 0.219394
\(590\) 0 0
\(591\) 24.1359 0.992820
\(592\) − 10.0000i − 0.410997i
\(593\) − 2.32456i − 0.0954580i −0.998860 0.0477290i \(-0.984802\pi\)
0.998860 0.0477290i \(-0.0151984\pi\)
\(594\) 4.16228 0.170780
\(595\) 0 0
\(596\) 16.6491 0.681974
\(597\) − 18.6491i − 0.763257i
\(598\) − 27.4868i − 1.12402i
\(599\) 4.18861 0.171142 0.0855710 0.996332i \(-0.472729\pi\)
0.0855710 + 0.996332i \(0.472729\pi\)
\(600\) 0 0
\(601\) 45.8114 1.86869 0.934343 0.356376i \(-0.115988\pi\)
0.934343 + 0.356376i \(0.115988\pi\)
\(602\) 28.9737i 1.18088i
\(603\) − 6.48683i − 0.264164i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 13.8114 0.561049
\(607\) − 2.02633i − 0.0822464i −0.999154 0.0411232i \(-0.986906\pi\)
0.999154 0.0411232i \(-0.0130936\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −12.1359 −0.491773
\(610\) 0 0
\(611\) 30.9737 1.25306
\(612\) − 0.837722i − 0.0338629i
\(613\) − 27.2982i − 1.10256i −0.834319 0.551282i \(-0.814139\pi\)
0.834319 0.551282i \(-0.185861\pi\)
\(614\) −13.5132 −0.545347
\(615\) 0 0
\(616\) 13.1623 0.530323
\(617\) 11.6754i 0.470036i 0.971991 + 0.235018i \(0.0755149\pi\)
−0.971991 + 0.235018i \(0.924485\pi\)
\(618\) − 15.6491i − 0.629500i
\(619\) −10.8377 −0.435605 −0.217802 0.975993i \(-0.569889\pi\)
−0.217802 + 0.975993i \(0.569889\pi\)
\(620\) 0 0
\(621\) 5.32456 0.213667
\(622\) 26.0000i 1.04251i
\(623\) − 23.1623i − 0.927977i
\(624\) 5.16228 0.206656
\(625\) 0 0
\(626\) −10.6754 −0.426677
\(627\) − 4.16228i − 0.166225i
\(628\) 18.3246i 0.731229i
\(629\) −8.37722 −0.334022
\(630\) 0 0
\(631\) 45.2982 1.80329 0.901647 0.432473i \(-0.142359\pi\)
0.901647 + 0.432473i \(0.142359\pi\)
\(632\) 5.32456i 0.211799i
\(633\) 18.8114i 0.747686i
\(634\) −4.16228 −0.165305
\(635\) 0 0
\(636\) −1.83772 −0.0728704
\(637\) − 15.4868i − 0.613611i
\(638\) − 15.9737i − 0.632403i
\(639\) −1.16228 −0.0459790
\(640\) 0 0
\(641\) −37.2982 −1.47319 −0.736596 0.676333i \(-0.763568\pi\)
−0.736596 + 0.676333i \(0.763568\pi\)
\(642\) 12.8377i 0.506664i
\(643\) 6.64911i 0.262215i 0.991368 + 0.131108i \(0.0418534\pi\)
−0.991368 + 0.131108i \(0.958147\pi\)
\(644\) 16.8377 0.663499
\(645\) 0 0
\(646\) −0.837722 −0.0329597
\(647\) 9.00000i 0.353827i 0.984226 + 0.176913i \(0.0566112\pi\)
−0.984226 + 0.176913i \(0.943389\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 31.1623 1.22323
\(650\) 0 0
\(651\) −16.8377 −0.659922
\(652\) 22.6491i 0.887008i
\(653\) 6.97367i 0.272901i 0.990647 + 0.136450i \(0.0435694\pi\)
−0.990647 + 0.136450i \(0.956431\pi\)
\(654\) 7.67544 0.300134
\(655\) 0 0
\(656\) −6.32456 −0.246932
\(657\) 1.00000i 0.0390137i
\(658\) 18.9737i 0.739671i
\(659\) 49.8114 1.94038 0.970188 0.242353i \(-0.0779193\pi\)
0.970188 + 0.242353i \(0.0779193\pi\)
\(660\) 0 0
\(661\) 37.2982 1.45073 0.725366 0.688363i \(-0.241670\pi\)
0.725366 + 0.688363i \(0.241670\pi\)
\(662\) 3.18861i 0.123929i
\(663\) − 4.32456i − 0.167952i
\(664\) −6.48683 −0.251738
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) − 20.4342i − 0.791214i
\(668\) 2.64911i 0.102497i
\(669\) −25.6491 −0.991652
\(670\) 0 0
\(671\) −17.3246 −0.668807
\(672\) 3.16228i 0.121988i
\(673\) − 32.4605i − 1.25126i −0.780120 0.625630i \(-0.784842\pi\)
0.780120 0.625630i \(-0.215158\pi\)
\(674\) 2.97367 0.114541
\(675\) 0 0
\(676\) 13.6491 0.524966
\(677\) − 18.4868i − 0.710507i −0.934770 0.355253i \(-0.884395\pi\)
0.934770 0.355253i \(-0.115605\pi\)
\(678\) − 9.64911i − 0.370572i
\(679\) −1.62278 −0.0622765
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) − 22.1623i − 0.848637i
\(683\) − 37.1096i − 1.41996i −0.704222 0.709980i \(-0.748704\pi\)
0.704222 0.709980i \(-0.251296\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −12.6491 −0.482945
\(687\) 8.16228i 0.311410i
\(688\) − 9.16228i − 0.349309i
\(689\) −9.48683 −0.361420
\(690\) 0 0
\(691\) 13.4868 0.513063 0.256532 0.966536i \(-0.417420\pi\)
0.256532 + 0.966536i \(0.417420\pi\)
\(692\) 2.16228i 0.0821975i
\(693\) 13.1623i 0.499994i
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 3.83772 0.145468
\(697\) 5.29822i 0.200684i
\(698\) − 17.8377i − 0.675168i
\(699\) −11.8114 −0.446748
\(700\) 0 0
\(701\) −30.9737 −1.16986 −0.584930 0.811084i \(-0.698878\pi\)
−0.584930 + 0.811084i \(0.698878\pi\)
\(702\) 5.16228i 0.194838i
\(703\) − 10.0000i − 0.377157i
\(704\) −4.16228 −0.156872
\(705\) 0 0
\(706\) 23.8114 0.896153
\(707\) 43.6754i 1.64258i
\(708\) 7.48683i 0.281372i
\(709\) −23.5132 −0.883056 −0.441528 0.897248i \(-0.645563\pi\)
−0.441528 + 0.897248i \(0.645563\pi\)
\(710\) 0 0
\(711\) −5.32456 −0.199686
\(712\) 7.32456i 0.274499i
\(713\) − 28.3509i − 1.06175i
\(714\) 2.64911 0.0991405
\(715\) 0 0
\(716\) −20.6491 −0.771693
\(717\) − 24.3246i − 0.908417i
\(718\) − 11.6754i − 0.435724i
\(719\) −19.3246 −0.720684 −0.360342 0.932820i \(-0.617340\pi\)
−0.360342 + 0.932820i \(0.617340\pi\)
\(720\) 0 0
\(721\) 49.4868 1.84299
\(722\) − 1.00000i − 0.0372161i
\(723\) − 10.5132i − 0.390989i
\(724\) −9.16228 −0.340513
\(725\) 0 0
\(726\) −6.32456 −0.234726
\(727\) 49.6228i 1.84041i 0.391440 + 0.920203i \(0.371977\pi\)
−0.391440 + 0.920203i \(0.628023\pi\)
\(728\) 16.3246i 0.605028i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −7.67544 −0.283887
\(732\) − 4.16228i − 0.153842i
\(733\) 50.4868i 1.86477i 0.361462 + 0.932387i \(0.382278\pi\)
−0.361462 + 0.932387i \(0.617722\pi\)
\(734\) 7.48683 0.276344
\(735\) 0 0
\(736\) −5.32456 −0.196266
\(737\) 27.0000i 0.994558i
\(738\) − 6.32456i − 0.232810i
\(739\) −6.32456 −0.232653 −0.116326 0.993211i \(-0.537112\pi\)
−0.116326 + 0.993211i \(0.537112\pi\)
\(740\) 0 0
\(741\) 5.16228 0.189641
\(742\) − 5.81139i − 0.213343i
\(743\) 4.64911i 0.170559i 0.996357 + 0.0852797i \(0.0271784\pi\)
−0.996357 + 0.0852797i \(0.972822\pi\)
\(744\) 5.32456 0.195208
\(745\) 0 0
\(746\) −16.1359 −0.590779
\(747\) − 6.48683i − 0.237341i
\(748\) 3.48683i 0.127491i
\(749\) −40.5964 −1.48336
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 10.6491i 0.388075i
\(754\) 19.8114 0.721488
\(755\) 0 0
\(756\) −3.16228 −0.115011
\(757\) 4.86406i 0.176787i 0.996086 + 0.0883936i \(0.0281733\pi\)
−0.996086 + 0.0883936i \(0.971827\pi\)
\(758\) 22.9737i 0.834441i
\(759\) −22.1623 −0.804440
\(760\) 0 0
\(761\) −39.4868 −1.43140 −0.715698 0.698410i \(-0.753891\pi\)
−0.715698 + 0.698410i \(0.753891\pi\)
\(762\) 15.6491i 0.566907i
\(763\) 24.2719i 0.878701i
\(764\) −21.6491 −0.783237
\(765\) 0 0
\(766\) −3.48683 −0.125984
\(767\) 38.6491i 1.39554i
\(768\) − 1.00000i − 0.0360844i
\(769\) −34.6754 −1.25043 −0.625214 0.780453i \(-0.714988\pi\)
−0.625214 + 0.780453i \(0.714988\pi\)
\(770\) 0 0
\(771\) 19.3246 0.695957
\(772\) 9.16228i 0.329758i
\(773\) 8.97367i 0.322760i 0.986892 + 0.161380i \(0.0515945\pi\)
−0.986892 + 0.161380i \(0.948405\pi\)
\(774\) 9.16228 0.329331
\(775\) 0 0
\(776\) 0.513167 0.0184216
\(777\) 31.6228i 1.13446i
\(778\) − 34.6491i − 1.24223i
\(779\) −6.32456 −0.226601
\(780\) 0 0
\(781\) 4.83772 0.173107
\(782\) 4.46050i 0.159507i
\(783\) 3.83772i 0.137149i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 12.4868 0.445391
\(787\) − 34.4868i − 1.22932i −0.788791 0.614661i \(-0.789293\pi\)
0.788791 0.614661i \(-0.210707\pi\)
\(788\) − 24.1359i − 0.859807i
\(789\) 31.9737 1.13829
\(790\) 0 0
\(791\) 30.5132 1.08492
\(792\) − 4.16228i − 0.147900i
\(793\) − 21.4868i − 0.763020i
\(794\) −17.1359 −0.608132
\(795\) 0 0
\(796\) −18.6491 −0.661000
\(797\) − 47.6228i − 1.68689i −0.537219 0.843443i \(-0.680525\pi\)
0.537219 0.843443i \(-0.319475\pi\)
\(798\) 3.16228i 0.111943i
\(799\) −5.02633 −0.177819
\(800\) 0 0
\(801\) −7.32456 −0.258800
\(802\) − 13.6491i − 0.481967i
\(803\) − 4.16228i − 0.146884i
\(804\) −6.48683 −0.228773
\(805\) 0 0
\(806\) 27.4868 0.968182
\(807\) 12.3246i 0.433845i
\(808\) − 13.8114i − 0.485883i
\(809\) 23.1623 0.814342 0.407171 0.913352i \(-0.366515\pi\)
0.407171 + 0.913352i \(0.366515\pi\)
\(810\) 0 0
\(811\) 11.1886 0.392885 0.196443 0.980515i \(-0.437061\pi\)
0.196443 + 0.980515i \(0.437061\pi\)
\(812\) 12.1359i 0.425888i
\(813\) 1.16228i 0.0407629i
\(814\) −41.6228 −1.45888
\(815\) 0 0
\(816\) −0.837722 −0.0293261
\(817\) − 9.16228i − 0.320548i
\(818\) − 7.67544i − 0.268366i
\(819\) −16.3246 −0.570426
\(820\) 0 0
\(821\) 36.8377 1.28565 0.642823 0.766015i \(-0.277763\pi\)
0.642823 + 0.766015i \(0.277763\pi\)
\(822\) 2.32456i 0.0810782i
\(823\) − 27.2982i − 0.951556i −0.879565 0.475778i \(-0.842167\pi\)
0.879565 0.475778i \(-0.157833\pi\)
\(824\) −15.6491 −0.545163
\(825\) 0 0
\(826\) −23.6754 −0.823774
\(827\) − 5.16228i − 0.179510i −0.995964 0.0897550i \(-0.971392\pi\)
0.995964 0.0897550i \(-0.0286084\pi\)
\(828\) − 5.32456i − 0.185041i
\(829\) 22.7851 0.791358 0.395679 0.918389i \(-0.370509\pi\)
0.395679 + 0.918389i \(0.370509\pi\)
\(830\) 0 0
\(831\) −24.8114 −0.860698
\(832\) − 5.16228i − 0.178970i
\(833\) 2.51317i 0.0870761i
\(834\) 21.4868 0.744028
\(835\) 0 0
\(836\) −4.16228 −0.143955
\(837\) 5.32456i 0.184044i
\(838\) 8.00000i 0.276355i
\(839\) 12.8377 0.443207 0.221604 0.975137i \(-0.428871\pi\)
0.221604 + 0.975137i \(0.428871\pi\)
\(840\) 0 0
\(841\) −14.2719 −0.492134
\(842\) − 18.6491i − 0.642691i
\(843\) − 7.64911i − 0.263449i
\(844\) 18.8114 0.647515
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 20.0000i − 0.687208i
\(848\) 1.83772i 0.0631076i
\(849\) 24.6491 0.845955
\(850\) 0 0
\(851\) −53.2456 −1.82523
\(852\) 1.16228i 0.0398190i
\(853\) 32.3246i 1.10677i 0.832925 + 0.553386i \(0.186664\pi\)
−0.832925 + 0.553386i \(0.813336\pi\)
\(854\) 13.1623 0.450404
\(855\) 0 0
\(856\) 12.8377 0.438784
\(857\) − 25.6754i − 0.877056i −0.898717 0.438528i \(-0.855500\pi\)
0.898717 0.438528i \(-0.144500\pi\)
\(858\) − 21.4868i − 0.733548i
\(859\) 5.35089 0.182570 0.0912850 0.995825i \(-0.470903\pi\)
0.0912850 + 0.995825i \(0.470903\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 0.324555i 0.0110544i
\(863\) 54.9737i 1.87133i 0.352896 + 0.935663i \(0.385197\pi\)
−0.352896 + 0.935663i \(0.614803\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 28.1359 0.956098
\(867\) − 16.2982i − 0.553517i
\(868\) 16.8377i 0.571510i
\(869\) 22.1623 0.751804
\(870\) 0 0
\(871\) −33.4868 −1.13466
\(872\) − 7.67544i − 0.259923i
\(873\) 0.513167i 0.0173681i
\(874\) −5.32456 −0.180106
\(875\) 0 0
\(876\) 1.00000 0.0337869
\(877\) − 45.6228i − 1.54057i −0.637699 0.770286i \(-0.720114\pi\)
0.637699 0.770286i \(-0.279886\pi\)
\(878\) 13.6491i 0.460635i
\(879\) 16.4868 0.556087
\(880\) 0 0
\(881\) 14.3246 0.482607 0.241303 0.970450i \(-0.422425\pi\)
0.241303 + 0.970450i \(0.422425\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) − 29.4868i − 0.992311i −0.868234 0.496155i \(-0.834745\pi\)
0.868234 0.496155i \(-0.165255\pi\)
\(884\) −4.32456 −0.145451
\(885\) 0 0
\(886\) −11.8377 −0.397696
\(887\) − 21.6754i − 0.727790i −0.931440 0.363895i \(-0.881447\pi\)
0.931440 0.363895i \(-0.118553\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) −49.4868 −1.65974
\(890\) 0 0
\(891\) 4.16228 0.139442
\(892\) 25.6491i 0.858796i
\(893\) − 6.00000i − 0.200782i
\(894\) 16.6491 0.556830
\(895\) 0 0
\(896\) 3.16228 0.105644
\(897\) − 27.4868i − 0.917759i
\(898\) − 11.0000i − 0.367075i
\(899\) 20.4342 0.681518
\(900\) 0 0
\(901\) 1.53950 0.0512882
\(902\) 26.3246i 0.876512i
\(903\) 28.9737i 0.964183i
\(904\) −9.64911 −0.320925
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) 38.6491i 1.28332i 0.766988 + 0.641661i \(0.221755\pi\)
−0.766988 + 0.641661i \(0.778245\pi\)
\(908\) − 2.00000i − 0.0663723i
\(909\) 13.8114 0.458095
\(910\) 0 0
\(911\) −7.02633 −0.232793 −0.116396 0.993203i \(-0.537134\pi\)
−0.116396 + 0.993203i \(0.537134\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 27.0000i 0.893570i
\(914\) −9.67544 −0.320035
\(915\) 0 0
\(916\) 8.16228 0.269689
\(917\) 39.4868i 1.30397i
\(918\) − 0.837722i − 0.0276490i
\(919\) 18.8377 0.621399 0.310700 0.950508i \(-0.399437\pi\)
0.310700 + 0.950508i \(0.399437\pi\)
\(920\) 0 0
\(921\) −13.5132 −0.445274
\(922\) − 12.0000i − 0.395199i
\(923\) 6.00000i 0.197492i
\(924\) 13.1623 0.433007
\(925\) 0 0
\(926\) −21.6754 −0.712299
\(927\) − 15.6491i − 0.513984i
\(928\) − 3.83772i − 0.125979i
\(929\) 37.2982 1.22371 0.611857 0.790968i \(-0.290423\pi\)
0.611857 + 0.790968i \(0.290423\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 11.8114i 0.386895i
\(933\) 26.0000i 0.851202i
\(934\) 37.1359 1.21513
\(935\) 0 0
\(936\) 5.16228 0.168734
\(937\) 26.6491i 0.870588i 0.900288 + 0.435294i \(0.143356\pi\)
−0.900288 + 0.435294i \(0.856644\pi\)
\(938\) − 20.5132i − 0.669779i
\(939\) −10.6754 −0.348380
\(940\) 0 0
\(941\) 32.8114 1.06962 0.534810 0.844972i \(-0.320383\pi\)
0.534810 + 0.844972i \(0.320383\pi\)
\(942\) 18.3246i 0.597046i
\(943\) 33.6754i 1.09662i
\(944\) 7.48683 0.243676
\(945\) 0 0
\(946\) −38.1359 −1.23991
\(947\) 48.0000i 1.55979i 0.625910 + 0.779895i \(0.284728\pi\)
−0.625910 + 0.779895i \(0.715272\pi\)
\(948\) 5.32456i 0.172934i
\(949\) 5.16228 0.167575
\(950\) 0 0
\(951\) −4.16228 −0.134971
\(952\) − 2.64911i − 0.0858582i
\(953\) − 40.6754i − 1.31761i −0.752315 0.658803i \(-0.771063\pi\)
0.752315 0.658803i \(-0.228937\pi\)
\(954\) −1.83772 −0.0594985
\(955\) 0 0
\(956\) −24.3246 −0.786712
\(957\) − 15.9737i − 0.516355i
\(958\) − 39.9737i − 1.29149i
\(959\) −7.35089 −0.237373
\(960\) 0 0
\(961\) −2.64911 −0.0854552
\(962\) − 51.6228i − 1.66439i
\(963\) 12.8377i 0.413690i
\(964\) −10.5132 −0.338606
\(965\) 0 0
\(966\) 16.8377 0.541745
\(967\) − 24.3246i − 0.782225i −0.920343 0.391112i \(-0.872090\pi\)
0.920343 0.391112i \(-0.127910\pi\)
\(968\) 6.32456i 0.203279i
\(969\) −0.837722 −0.0269115
\(970\) 0 0
\(971\) 12.5132 0.401567 0.200783 0.979636i \(-0.435651\pi\)
0.200783 + 0.979636i \(0.435651\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 67.9473i 2.17829i
\(974\) 28.6491 0.917977
\(975\) 0 0
\(976\) −4.16228 −0.133231
\(977\) 53.9473i 1.72593i 0.505265 + 0.862964i \(0.331395\pi\)
−0.505265 + 0.862964i \(0.668605\pi\)
\(978\) 22.6491i 0.724239i
\(979\) 30.4868 0.974363
\(980\) 0 0
\(981\) 7.67544 0.245058
\(982\) 0 0
\(983\) − 36.1359i − 1.15256i −0.817253 0.576279i \(-0.804504\pi\)
0.817253 0.576279i \(-0.195496\pi\)
\(984\) −6.32456 −0.201619
\(985\) 0 0
\(986\) −3.21495 −0.102385
\(987\) 18.9737i 0.603938i
\(988\) − 5.16228i − 0.164234i
\(989\) −48.7851 −1.55127
\(990\) 0 0
\(991\) 24.2982 0.771858 0.385929 0.922528i \(-0.373881\pi\)
0.385929 + 0.922528i \(0.373881\pi\)
\(992\) − 5.32456i − 0.169055i
\(993\) 3.18861i 0.101188i
\(994\) −3.67544 −0.116578
\(995\) 0 0
\(996\) −6.48683 −0.205543
\(997\) − 13.5132i − 0.427966i −0.976837 0.213983i \(-0.931356\pi\)
0.976837 0.213983i \(-0.0686438\pi\)
\(998\) − 5.81139i − 0.183956i
\(999\) 10.0000 0.316386
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 2850.2.d.v.799.1 4
5.2 odd 4 2850.2.a.bg.1.2 yes 2
5.3 odd 4 2850.2.a.bf.1.1 2
5.4 even 2 inner 2850.2.d.v.799.4 4
15.2 even 4 8550.2.a.bq.1.2 2
15.8 even 4 8550.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bf.1.1 2 5.3 odd 4
2850.2.a.bg.1.2 yes 2 5.2 odd 4
2850.2.d.v.799.1 4 1.1 even 1 trivial
2850.2.d.v.799.4 4 5.4 even 2 inner
8550.2.a.bq.1.2 2 15.2 even 4
8550.2.a.ca.1.1 2 15.8 even 4