Properties

Label 2550.2.d.b.2449.1
Level $2550$
Weight $2$
Character 2550.2449
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
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] = [2550,2,Mod(2449,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, 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("2550.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2550.2449
Dual form 2550.2.d.b.2449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} -4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} -14.0000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -16.0000i q^{77} +2.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -4.00000i q^{83} -4.00000 q^{84} -12.0000 q^{86} +2.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +8.00000 q^{91} +4.00000i q^{92} -4.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} - 4 q^{29} + 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 4 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49} - 2 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} - 8 q^{71} + 12 q^{74} - 8 q^{76} + 24 q^{79} + 2 q^{81} - 8 q^{84} - 24 q^{86} - 20 q^{89} + 16 q^{91} - 16 q^{94} - 2 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.00000i − 0.242536i
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000i 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 2.00000i 0.277350i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) − 4.00000i − 0.529813i
\(58\) 2.00000i 0.262613i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 1.00000i 0.121268i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 16.0000i − 1.82337i
\(78\) 2.00000i 0.226455i
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 2.00000i 0.214423i
\(88\) − 4.00000i − 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 4.00000i 0.417029i
\(93\) − 4.00000i − 0.414781i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 1.00000i 0.0990148i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000i 0.377964i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) − 2.00000i − 0.180334i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 16.0000i 1.38738i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 4.00000i 0.335673i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 9.00000i 0.742307i
\(148\) − 6.00000i − 0.493197i
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 1.00000i 0.0808452i
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) − 1.00000i − 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 4.00000i 0.308607i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 12.0000i 0.914991i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000i 0.901975i
\(178\) 10.0000i 0.749532i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) − 2.00000i − 0.147844i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 4.00000i 0.292509i
\(188\) 8.00000i 0.583460i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 18.0000i 1.29567i 0.761781 + 0.647834i \(0.224325\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 10.0000i 0.703598i
\(203\) − 8.00000i − 0.561490i
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) − 2.00000i − 0.138675i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 4.00000i 0.274075i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.0000i 1.08615i
\(218\) 10.0000i 0.677285i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) − 6.00000i − 0.402694i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) − 2.00000i − 0.131306i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) − 12.0000i − 0.779484i
\(238\) − 4.00000i − 0.259281i
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 8.00000i − 0.509028i
\(248\) 4.00000i 0.254000i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 16.0000i 1.00591i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 12.0000i 0.747087i
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 20.0000i − 1.23560i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 10.0000i 0.611990i
\(268\) 4.00000i 0.244339i
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) − 8.00000i − 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000i 0.476393i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 8.00000i 0.472225i
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 14.0000i 0.819288i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) − 4.00000i − 0.232104i
\(298\) − 2.00000i − 0.115857i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) 8.00000i 0.460348i
\(303\) 10.0000i 0.574485i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 16.0000i 0.911685i
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) − 10.0000i − 0.561656i −0.959758 0.280828i \(-0.909391\pi\)
0.959758 0.280828i \(-0.0906090\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 16.0000i − 0.891645i
\(323\) − 4.00000i − 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 10.0000i 0.553001i
\(328\) 2.00000i 0.110432i
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000i 0.219529i
\(333\) − 6.00000i − 0.328798i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 4.00000i 0.216295i
\(343\) − 8.00000i − 0.431959i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000i 0.213201i
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) − 4.00000i − 0.211702i
\(358\) 12.0000i 0.634220i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 22.0000i 1.15629i
\(363\) − 5.00000i − 0.262432i
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 4.00000i 0.207390i
\(373\) 38.0000i 1.96757i 0.179364 + 0.983783i \(0.442596\pi\)
−0.179364 + 0.983783i \(0.557404\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 4.00000i 0.206010i
\(378\) 4.00000i 0.205738i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000i 0.409316i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 12.0000i 0.609994i
\(388\) 18.0000i 0.913812i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) − 9.00000i − 0.454569i
\(393\) − 20.0000i − 1.00887i
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 30.0000i 1.50566i 0.658217 + 0.752828i \(0.271311\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 8.00000i − 0.398508i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) − 24.0000i − 1.18964i
\(408\) − 1.00000i − 0.0495074i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) − 48.0000i − 2.36193i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 20.0000i − 0.979404i
\(418\) 16.0000i 0.782586i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 8.00000i 0.388973i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 8.00000i 0.387147i
\(428\) 12.0000i 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) − 16.0000i − 0.765384i
\(438\) 14.0000i 0.668946i
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 2.00000i 0.0951303i
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 2.00000i − 0.0945968i
\(448\) − 4.00000i − 0.188982i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 6.00000i 0.282216i
\(453\) 8.00000i 0.375873i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 6.00000i 0.280362i
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 16.0000i 0.744387i
\(463\) 40.0000i 1.85896i 0.368875 + 0.929479i \(0.379743\pi\)
−0.368875 + 0.929479i \(0.620257\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) − 12.0000i − 0.552345i
\(473\) 48.0000i 2.20704i
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 2.00000i 0.0915737i
\(478\) − 24.0000i − 1.09773i
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) − 10.0000i − 0.455488i
\(483\) − 16.0000i − 0.728025i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 2.00000i 0.0900755i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) − 16.0000i − 0.717698i
\(498\) 4.00000i 0.179244i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 28.0000i 1.24970i
\(503\) 44.0000i 1.96186i 0.194354 + 0.980932i \(0.437739\pi\)
−0.194354 + 0.980932i \(0.562261\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) − 9.00000i − 0.399704i
\(508\) − 16.0000i − 0.709885i
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 32.0000i 1.40736i
\(518\) 24.0000i 1.05450i
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) − 4.00000i − 0.174243i
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) − 16.0000i − 0.693688i
\(533\) − 4.00000i − 0.173259i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 12.0000i 0.517838i
\(538\) − 30.0000i − 1.29339i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 22.0000i 0.944110i
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) − 4.00000i − 0.170251i
\(553\) 48.0000i 2.04117i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) 4.00000i 0.169334i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) − 10.0000i − 0.421825i
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 4.00000i 0.167984i
\(568\) − 4.00000i − 0.167836i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) 8.00000i 0.334205i
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 18.0000i 0.746124i
\(583\) 8.00000i 0.331326i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 44.0000i − 1.81607i −0.418890 0.908037i \(-0.637581\pi\)
0.418890 0.908037i \(-0.362419\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 6.00000i 0.246598i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 4.00000i 0.163709i
\(598\) 8.00000i 0.327144i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) − 48.0000i − 1.95633i
\(603\) 4.00000i 0.162893i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) − 1.00000i − 0.0404226i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 20.0000i − 0.801927i
\(623\) − 40.0000i − 1.60257i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 16.0000i 0.638978i
\(628\) − 10.0000i − 0.399043i
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 12.0000i 0.477334i
\(633\) − 4.00000i − 0.158986i
\(634\) −10.0000 −0.397151
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 18.0000i 0.713186i
\(638\) − 8.00000i − 0.316723i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 16.0000i 0.629025i 0.949253 + 0.314512i \(0.101841\pi\)
−0.949253 + 0.314512i \(0.898159\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 20.0000i 0.783260i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 14.0000i 0.546192i
\(658\) − 32.0000i − 1.24749i
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 2.00000i 0.0776736i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 8.00000i 0.309761i
\(668\) 12.0000i 0.464294i
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) − 4.00000i − 0.154303i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 72.0000 2.76311
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 16.0000i 0.612672i
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 6.00000i 0.228914i
\(688\) − 12.0000i − 0.457496i
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 16.0000i 0.607790i
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) − 2.00000i − 0.0757554i
\(698\) 22.0000i 0.832712i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 24.0000i 0.905177i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) − 40.0000i − 1.50435i
\(708\) − 12.0000i − 0.450988i
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) − 10.0000i − 0.374766i
\(713\) − 16.0000i − 0.599205i
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 24.0000i − 0.896296i
\(718\) − 24.0000i − 0.895672i
\(719\) 44.0000 1.64092 0.820462 0.571702i \(-0.193717\pi\)
0.820462 + 0.571702i \(0.193717\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) − 10.0000i − 0.371904i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 8.00000i 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 2.00000i 0.0739221i
\(733\) − 18.0000i − 0.664845i −0.943131 0.332423i \(-0.892134\pi\)
0.943131 0.332423i \(-0.107866\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 16.0000i 0.589368i
\(738\) 2.00000i 0.0736210i
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 8.00000i − 0.293689i
\(743\) 28.0000i 1.02722i 0.858024 + 0.513610i \(0.171692\pi\)
−0.858024 + 0.513610i \(0.828308\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 4.00000i 0.146352i
\(748\) − 4.00000i − 0.146254i
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 28.0000i 1.02038i
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 34.0000i 1.23575i 0.786276 + 0.617876i \(0.212006\pi\)
−0.786276 + 0.617876i \(0.787994\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) − 40.0000i − 1.44810i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) − 18.0000i − 0.647834i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 24.0000i 0.860995i
\(778\) − 18.0000i − 0.645331i
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 4.00000i 0.143040i
\(783\) − 2.00000i − 0.0714742i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) − 36.0000i − 1.28326i −0.767014 0.641631i \(-0.778258\pi\)
0.767014 0.641631i \(-0.221742\pi\)
\(788\) − 14.0000i − 0.498729i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 4.00000i 0.142134i
\(793\) − 4.00000i − 0.142044i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 6.00000i 0.211867i
\(803\) 56.0000i 1.97620i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) − 30.0000i − 1.05605i
\(808\) − 10.0000i − 0.351799i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 8.00000i 0.280745i
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) − 48.0000i − 1.67931i
\(818\) 10.0000i 0.349642i
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) − 44.0000i − 1.53003i −0.644013 0.765015i \(-0.722732\pi\)
0.644013 0.765015i \(-0.277268\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 2.00000i 0.0693375i
\(833\) 9.00000i 0.311832i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 4.00000i 0.138260i
\(838\) 12.0000i 0.414533i
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000i 0.620321i
\(843\) − 10.0000i − 0.344418i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 20.0000i 0.687208i
\(848\) − 2.00000i − 0.0686803i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) − 4.00000i − 0.137038i
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 4.00000i 0.136241i
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 1.00000i 0.0339618i
\(868\) − 16.0000i − 0.543075i
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 10.0000i − 0.338643i
\(873\) 18.0000i 0.609208i
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 12.0000i − 0.404980i
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 6.00000i 0.201347i
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000i 0.267860i
\(893\) − 32.0000i − 1.07084i
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 8.00000i 0.267112i
\(898\) − 6.00000i − 0.200223i
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 8.00000i 0.266371i
\(903\) − 48.0000i − 1.59734i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 16.0000i 0.529523i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 80.0000i 2.64183i
\(918\) − 1.00000i − 0.0330049i
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) − 14.0000i − 0.461065i
\(923\) 8.00000i 0.263323i
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) 2.00000i 0.0656532i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) − 18.0000i − 0.589610i
\(933\) − 20.0000i − 0.654771i
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) − 10.0000i − 0.325818i
\(943\) − 8.00000i − 0.260516i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 12.0000i 0.389742i
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 4.00000i 0.129641i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) − 8.00000i − 0.258603i
\(958\) 20.0000i 0.646171i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 12.0000i − 0.386896i
\(963\) 12.0000i 0.386695i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 80.0000i 2.56468i
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 20.0000i 0.639529i
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 12.0000i 0.382935i
\(983\) − 28.0000i − 0.893061i −0.894768 0.446531i \(-0.852659\pi\)
0.894768 0.446531i \(-0.147341\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) − 32.0000i − 1.01857i
\(988\) 8.00000i 0.254514i
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 28.0000i − 0.888553i
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 4.00000i 0.126618i
\(999\) −6.00000 −0.189832
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 2550.2.d.b.2449.1 2
5.2 odd 4 510.2.a.c.1.1 1
5.3 odd 4 2550.2.a.n.1.1 1
5.4 even 2 inner 2550.2.d.b.2449.2 2
15.2 even 4 1530.2.a.d.1.1 1
15.8 even 4 7650.2.a.cn.1.1 1
20.7 even 4 4080.2.a.x.1.1 1
85.67 odd 4 8670.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 1 5.2 odd 4
1530.2.a.d.1.1 1 15.2 even 4
2550.2.a.n.1.1 1 5.3 odd 4
2550.2.d.b.2449.1 2 1.1 even 1 trivial
2550.2.d.b.2449.2 2 5.4 even 2 inner
4080.2.a.x.1.1 1 20.7 even 4
7650.2.a.cn.1.1 1 15.8 even 4
8670.2.a.bb.1.1 1 85.67 odd 4