Properties

Label 2550.2.d.a.2449.2
Level $2550$
Weight $2$
Character 2550.2449
Analytic conductor $20.362$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

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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: \(1\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -5.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -3.00000 q^{21} -5.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} -10.0000 q^{29} +5.00000 q^{31} +1.00000i q^{32} -5.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} -1.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} -3.00000i q^{42} +1.00000i q^{43} +5.00000 q^{44} +6.00000 q^{46} +3.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -1.00000 q^{51} -2.00000i q^{52} +1.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} -1.00000i q^{57} -10.0000i q^{58} -8.00000 q^{59} -2.00000 q^{61} +5.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +5.00000 q^{66} -11.0000i q^{67} -1.00000i q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} +12.0000i q^{73} +3.00000 q^{74} +1.00000 q^{76} -15.0000i q^{77} -2.00000i q^{78} -5.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -18.0000i q^{83} +3.00000 q^{84} -1.00000 q^{86} -10.0000i q^{87} +5.00000i q^{88} -12.0000 q^{89} -6.00000 q^{91} +6.00000i q^{92} +5.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -2.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 10 q^{11} - 6 q^{14} + 2 q^{16} - 2 q^{19} - 6 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{29} + 10 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} + 10 q^{44} + 12 q^{46} - 4 q^{49} - 2 q^{51} + 2 q^{54} + 6 q^{56} - 16 q^{59} - 4 q^{61} - 2 q^{64} + 10 q^{66} + 12 q^{69} + 12 q^{71} + 6 q^{74} + 2 q^{76} - 10 q^{79} + 2 q^{81} + 6 q^{84} - 2 q^{86} - 24 q^{89} - 12 q^{91} - 6 q^{94} - 2 q^{96} + 10 q^{99}+O(q^{100}) \) 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
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000i 0.242536i
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) − 5.00000i − 1.06600i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 3.00000i − 0.566947i
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 5.00000i − 0.870388i
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 3.00000i − 0.462910i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) − 2.00000i − 0.277350i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) − 1.00000i − 0.132453i
\(58\) − 10.0000i − 1.31306i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 5.00000i 0.635001i
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) − 11.0000i − 1.34386i −0.740613 0.671932i \(-0.765465\pi\)
0.740613 0.671932i \(-0.234535\pi\)
\(68\) − 1.00000i − 0.121268i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) − 15.0000i − 1.70941i
\(78\) − 2.00000i − 0.226455i
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 18.0000i − 1.97576i −0.155230 0.987878i \(-0.549612\pi\)
0.155230 0.987878i \(-0.450388\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) − 10.0000i − 1.07211i
\(88\) 5.00000i 0.533002i
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 6.00000i 0.625543i
\(93\) 5.00000i 0.518476i
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 11.0000 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(102\) − 1.00000i − 0.0990148i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) − 5.00000i − 0.483368i −0.970355 0.241684i \(-0.922300\pi\)
0.970355 0.241684i \(-0.0776998\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 3.00000i 0.283473i
\(113\) − 1.00000i − 0.0940721i −0.998893 0.0470360i \(-0.985022\pi\)
0.998893 0.0470360i \(-0.0149776\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) − 2.00000i − 0.184900i
\(118\) − 8.00000i − 0.736460i
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) − 2.00000i − 0.181071i
\(123\) 6.00000i 0.541002i
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 5.00000i 0.435194i
\(133\) − 3.00000i − 0.260133i
\(134\) 11.0000 0.950255
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 16.0000i 1.36697i 0.729964 + 0.683486i \(0.239537\pi\)
−0.729964 + 0.683486i \(0.760463\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 6.00000i 0.503509i
\(143\) − 10.0000i − 0.836242i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) − 2.00000i − 0.164957i
\(148\) 3.00000i 0.246598i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 1.00000i − 0.0808452i
\(154\) 15.0000 1.20873
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) − 5.00000i − 0.397779i
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 1.00000i 0.0785674i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 3.00000i 0.231455i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 1.00000i − 0.0762493i
\(173\) − 20.0000i − 1.52057i −0.649589 0.760286i \(-0.725059\pi\)
0.649589 0.760286i \(-0.274941\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) − 8.00000i − 0.601317i
\(178\) − 12.0000i − 0.899438i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) − 2.00000i − 0.147844i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) −5.00000 −0.366618
\(187\) − 5.00000i − 0.365636i
\(188\) − 3.00000i − 0.218797i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 5.00000i 0.355335i
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 11.0000i 0.773957i
\(203\) − 30.0000i − 2.10559i
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 6.00000i 0.417029i
\(208\) 2.00000i 0.138675i
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) − 1.00000i − 0.0686803i
\(213\) 6.00000i 0.411113i
\(214\) 5.00000 0.341793
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 15.0000i 1.01827i
\(218\) − 5.00000i − 0.338643i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 3.00000i 0.201347i
\(223\) − 18.0000i − 1.20537i −0.797980 0.602685i \(-0.794098\pi\)
0.797980 0.602685i \(-0.205902\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 1.00000 0.0665190
\(227\) − 27.0000i − 1.79205i −0.444001 0.896026i \(-0.646441\pi\)
0.444001 0.896026i \(-0.353559\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 10.0000i 0.656532i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) − 5.00000i − 0.324785i
\(238\) − 3.00000i − 0.194461i
\(239\) −13.0000 −0.840900 −0.420450 0.907316i \(-0.638128\pi\)
−0.420450 + 0.907316i \(0.638128\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 14.0000i 0.899954i
\(243\) 1.00000i 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) − 2.00000i − 0.127257i
\(248\) − 5.00000i − 0.317500i
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 30.0000i 1.88608i
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) − 1.00000i − 0.0622573i
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) − 20.0000i − 1.23560i
\(263\) − 3.00000i − 0.184988i −0.995713 0.0924940i \(-0.970516\pi\)
0.995713 0.0924940i \(-0.0294839\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) − 12.0000i − 0.734388i
\(268\) 11.0000i 0.671932i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 1.00000i 0.0606339i
\(273\) − 6.00000i − 0.363137i
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) − 31.0000i − 1.86261i −0.364241 0.931305i \(-0.618672\pi\)
0.364241 0.931305i \(-0.381328\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) − 3.00000i − 0.178647i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) 18.0000i 1.06251i
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) − 12.0000i − 0.702247i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −3.00000 −0.174371
\(297\) 5.00000i 0.290129i
\(298\) 6.00000i 0.347571i
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 8.00000i 0.460348i
\(303\) 11.0000i 0.631933i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 15.0000i 0.854704i
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 34.0000i 1.92179i 0.276907 + 0.960897i \(0.410691\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) − 1.00000i − 0.0560772i
\(319\) 50.0000 2.79946
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) 18.0000i 1.00310i
\(323\) − 1.00000i − 0.0556415i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 0 0
\(327\) − 5.00000i − 0.276501i
\(328\) − 6.00000i − 0.331295i
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 18.0000i 0.987878i
\(333\) 3.00000i 0.164399i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 12.0000i 0.653682i 0.945079 + 0.326841i \(0.105984\pi\)
−0.945079 + 0.326841i \(0.894016\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) −25.0000 −1.35383
\(342\) 1.00000i 0.0540738i
\(343\) 15.0000i 0.809924i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 20.0000 1.07521
\(347\) − 23.0000i − 1.23470i −0.786687 0.617352i \(-0.788205\pi\)
0.786687 0.617352i \(-0.211795\pi\)
\(348\) 10.0000i 0.536056i
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) − 5.00000i − 0.266501i
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) − 3.00000i − 0.158777i
\(358\) 12.0000i 0.634220i
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 17.0000i − 0.893500i
\(363\) 14.0000i 0.734809i
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 5.00000i 0.260998i 0.991448 + 0.130499i \(0.0416579\pi\)
−0.991448 + 0.130499i \(0.958342\pi\)
\(368\) − 6.00000i − 0.312772i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) − 5.00000i − 0.259238i
\(373\) 18.0000i 0.932005i 0.884783 + 0.466002i \(0.154306\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) − 20.0000i − 1.03005i
\(378\) 3.00000i 0.154303i
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 3.00000i 0.153493i
\(383\) 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
\(386\) −24.0000 −1.22157
\(387\) − 1.00000i − 0.0508329i
\(388\) 14.0000i 0.710742i
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 2.00000i 0.101015i
\(393\) − 20.0000i − 1.00887i
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 21.0000i 1.05396i 0.849878 + 0.526980i \(0.176676\pi\)
−0.849878 + 0.526980i \(0.823324\pi\)
\(398\) − 17.0000i − 0.852133i
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 11.0000i 0.548630i
\(403\) 10.0000i 0.498135i
\(404\) −11.0000 −0.547270
\(405\) 0 0
\(406\) 30.0000 1.48888
\(407\) 15.0000i 0.743522i
\(408\) 1.00000i 0.0495074i
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) 4.00000i 0.197066i
\(413\) − 24.0000i − 1.18096i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 8.00000i − 0.391762i
\(418\) 5.00000i 0.244558i
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 18.0000i 0.876226i
\(423\) − 3.00000i − 0.145865i
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) − 6.00000i − 0.290360i
\(428\) 5.00000i 0.241684i
\(429\) 10.0000 0.482805
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 21.0000i − 1.00920i −0.863355 0.504598i \(-0.831641\pi\)
0.863355 0.504598i \(-0.168359\pi\)
\(434\) −15.0000 −0.720023
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) 6.00000i 0.287019i
\(438\) − 12.0000i − 0.573382i
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) − 2.00000i − 0.0951303i
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) −3.00000 −0.142374
\(445\) 0 0
\(446\) 18.0000 0.852325
\(447\) 6.00000i 0.283790i
\(448\) − 3.00000i − 0.141737i
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) 1.00000i 0.0470360i
\(453\) 8.00000i 0.375873i
\(454\) 27.0000 1.26717
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 25.0000i − 1.16945i −0.811231 0.584725i \(-0.801202\pi\)
0.811231 0.584725i \(-0.198798\pi\)
\(458\) − 24.0000i − 1.12145i
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 15.0000i 0.697863i
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 33.0000 1.52380
\(470\) 0 0
\(471\) 0 0
\(472\) 8.00000i 0.368230i
\(473\) − 5.00000i − 0.229900i
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) − 1.00000i − 0.0457869i
\(478\) − 13.0000i − 0.594606i
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 18.0000i 0.819028i
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 0 0
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) − 10.0000i − 0.450377i
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 18.0000i 0.807410i
\(498\) 18.0000i 0.806599i
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) − 18.0000i − 0.803379i
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −30.0000 −1.33366
\(507\) 9.00000i 0.399704i
\(508\) − 12.0000i − 0.532414i
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) 1.00000i 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) − 15.0000i − 0.659699i
\(518\) 9.00000i 0.395437i
\(519\) 20.0000 0.877903
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 10.0000i 0.437688i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 3.00000 0.130806
\(527\) 5.00000i 0.217803i
\(528\) − 5.00000i − 0.217597i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 3.00000i 0.130066i
\(533\) 12.0000i 0.519778i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) 12.0000i 0.517838i
\(538\) 6.00000i 0.258678i
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) −41.0000 −1.76273 −0.881364 0.472438i \(-0.843374\pi\)
−0.881364 + 0.472438i \(0.843374\pi\)
\(542\) 14.0000i 0.601351i
\(543\) − 17.0000i − 0.729540i
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) 22.0000i 0.940652i 0.882493 + 0.470326i \(0.155864\pi\)
−0.882493 + 0.470326i \(0.844136\pi\)
\(548\) − 16.0000i − 0.683486i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) − 6.00000i − 0.255377i
\(553\) − 15.0000i − 0.637865i
\(554\) 31.0000 1.31706
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 17.0000i 0.720313i 0.932892 + 0.360157i \(0.117277\pi\)
−0.932892 + 0.360157i \(0.882723\pi\)
\(558\) − 5.00000i − 0.211667i
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) − 30.0000i − 1.26547i
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 3.00000i 0.125988i
\(568\) − 6.00000i − 0.251754i
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 10.0000i 0.418121i
\(573\) 3.00000i 0.125327i
\(574\) −18.0000 −0.751305
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 33.0000i 1.37381i 0.726748 + 0.686904i \(0.241031\pi\)
−0.726748 + 0.686904i \(0.758969\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) 54.0000 2.24030
\(582\) 14.0000i 0.580319i
\(583\) − 5.00000i − 0.207079i
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) − 3.00000i − 0.123299i
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 17.0000i − 0.695764i
\(598\) 12.0000i 0.490716i
\(599\) −35.0000 −1.43006 −0.715031 0.699093i \(-0.753587\pi\)
−0.715031 + 0.699093i \(0.753587\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) − 3.00000i − 0.122271i
\(603\) 11.0000i 0.447955i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −11.0000 −0.446844
\(607\) 20.0000i 0.811775i 0.913923 + 0.405887i \(0.133038\pi\)
−0.913923 + 0.405887i \(0.866962\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 1.00000i 0.0404226i
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −15.0000 −0.604367
\(617\) − 13.0000i − 0.523360i −0.965155 0.261680i \(-0.915723\pi\)
0.965155 0.261680i \(-0.0842766\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) − 2.00000i − 0.0801927i
\(623\) − 36.0000i − 1.44231i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −34.0000 −1.35891
\(627\) 5.00000i 0.199681i
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 5.00000i 0.198889i
\(633\) 18.0000i 0.715436i
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) − 4.00000i − 0.158486i
\(638\) 50.0000i 1.97952i
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 5.00000i 0.197334i
\(643\) − 34.0000i − 1.34083i −0.741987 0.670415i \(-0.766116\pi\)
0.741987 0.670415i \(-0.233884\pi\)
\(644\) −18.0000 −0.709299
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 0 0
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) 5.00000 0.195515
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 12.0000i − 0.468165i
\(658\) − 9.00000i − 0.350857i
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) − 25.0000i − 0.971653i
\(663\) − 2.00000i − 0.0776736i
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) 60.0000i 2.32321i
\(668\) 18.0000i 0.696441i
\(669\) 18.0000 0.695920
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) − 3.00000i − 0.115728i
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) −12.0000 −0.462223
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 1.00000i 0.0384048i
\(679\) 42.0000 1.61181
\(680\) 0 0
\(681\) 27.0000 1.03464
\(682\) − 25.0000i − 0.957299i
\(683\) 8.00000i 0.306111i 0.988218 + 0.153056i \(0.0489114\pi\)
−0.988218 + 0.153056i \(0.951089\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) − 24.0000i − 0.915657i
\(688\) 1.00000i 0.0381246i
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 20.0000i 0.760286i
\(693\) 15.0000i 0.569803i
\(694\) 23.0000 0.873068
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 6.00000i 0.227266i
\(698\) 14.0000i 0.529908i
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 3.00000i 0.113147i
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 33.0000i 1.24109i
\(708\) 8.00000i 0.300658i
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 5.00000 0.187515
\(712\) 12.0000i 0.449719i
\(713\) − 30.0000i − 1.12351i
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 13.0000i − 0.485494i
\(718\) − 15.0000i − 0.559795i
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) − 18.0000i − 0.669891i
\(723\) 0 0
\(724\) 17.0000 0.631800
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) 2.00000i 0.0741759i 0.999312 + 0.0370879i \(0.0118082\pi\)
−0.999312 + 0.0370879i \(0.988192\pi\)
\(728\) 6.00000i 0.222375i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −1.00000 −0.0369863
\(732\) 2.00000i 0.0739221i
\(733\) 16.0000i 0.590973i 0.955347 + 0.295487i \(0.0954818\pi\)
−0.955347 + 0.295487i \(0.904518\pi\)
\(734\) −5.00000 −0.184553
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 55.0000i 2.02595i
\(738\) − 6.00000i − 0.220863i
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) − 3.00000i − 0.110133i
\(743\) − 18.0000i − 0.660356i −0.943919 0.330178i \(-0.892891\pi\)
0.943919 0.330178i \(-0.107109\pi\)
\(744\) 5.00000 0.183309
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) 18.0000i 0.658586i
\(748\) 5.00000i 0.182818i
\(749\) 15.0000 0.548088
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 3.00000i 0.109399i
\(753\) − 18.0000i − 0.655956i
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 14.0000i 0.508839i 0.967094 + 0.254419i \(0.0818843\pi\)
−0.967094 + 0.254419i \(0.918116\pi\)
\(758\) 8.00000i 0.290573i
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) − 15.0000i − 0.543036i
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) − 16.0000i − 0.577727i
\(768\) 1.00000i 0.0360844i
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) − 24.0000i − 0.863779i
\(773\) 34.0000i 1.22290i 0.791285 + 0.611448i \(0.209412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 9.00000i 0.322873i
\(778\) 5.00000i 0.179259i
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 6.00000i 0.214560i
\(783\) 10.0000i 0.357371i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) 18.0000i 0.641631i 0.947142 + 0.320815i \(0.103957\pi\)
−0.947142 + 0.320815i \(0.896043\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) 3.00000 0.106803
\(790\) 0 0
\(791\) 3.00000 0.106668
\(792\) − 5.00000i − 0.177667i
\(793\) − 4.00000i − 0.142044i
\(794\) −21.0000 −0.745262
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) − 11.0000i − 0.389640i −0.980839 0.194820i \(-0.937588\pi\)
0.980839 0.194820i \(-0.0624123\pi\)
\(798\) 3.00000i 0.106199i
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) − 18.0000i − 0.635602i
\(803\) − 60.0000i − 2.11735i
\(804\) −11.0000 −0.387940
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 6.00000i 0.211210i
\(808\) − 11.0000i − 0.386979i
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) 30.0000i 1.05279i
\(813\) 14.0000i 0.491001i
\(814\) −15.0000 −0.525750
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) − 1.00000i − 0.0349856i
\(818\) 2.00000i 0.0699284i
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) − 16.0000i − 0.558064i
\(823\) − 4.00000i − 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 53.0000i − 1.84299i −0.388390 0.921495i \(-0.626968\pi\)
0.388390 0.921495i \(-0.373032\pi\)
\(828\) − 6.00000i − 0.208514i
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 31.0000 1.07538
\(832\) − 2.00000i − 0.0693375i
\(833\) − 2.00000i − 0.0692959i
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) − 5.00000i − 0.172825i
\(838\) − 24.0000i − 0.829066i
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 18.0000i 0.620321i
\(843\) − 30.0000i − 1.03325i
\(844\) −18.0000 −0.619586
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 42.0000i 1.44314i
\(848\) 1.00000i 0.0343401i
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) − 6.00000i − 0.205557i
\(853\) 37.0000i 1.26686i 0.773802 + 0.633428i \(0.218353\pi\)
−0.773802 + 0.633428i \(0.781647\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −5.00000 −0.170896
\(857\) − 21.0000i − 0.717346i −0.933463 0.358673i \(-0.883229\pi\)
0.933463 0.358673i \(-0.116771\pi\)
\(858\) 10.0000i 0.341394i
\(859\) 23.0000 0.784750 0.392375 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) 33.0000i 1.12333i 0.827364 + 0.561667i \(0.189840\pi\)
−0.827364 + 0.561667i \(0.810160\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 21.0000 0.713609
\(867\) − 1.00000i − 0.0339618i
\(868\) − 15.0000i − 0.509133i
\(869\) 25.0000 0.848067
\(870\) 0 0
\(871\) 22.0000 0.745442
\(872\) 5.00000i 0.169321i
\(873\) 14.0000i 0.473828i
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) − 16.0000i − 0.539974i
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 2.00000i 0.0673435i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) − 3.00000i − 0.100673i
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 18.0000i 0.602685i
\(893\) − 3.00000i − 0.100391i
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 12.0000i 0.400668i
\(898\) − 21.0000i − 0.700779i
\(899\) −50.0000 −1.66759
\(900\) 0 0
\(901\) −1.00000 −0.0333148
\(902\) − 30.0000i − 0.998891i
\(903\) − 3.00000i − 0.0998337i
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 30.0000i − 0.996134i −0.867139 0.498067i \(-0.834043\pi\)
0.867139 0.498067i \(-0.165957\pi\)
\(908\) 27.0000i 0.896026i
\(909\) −11.0000 −0.364847
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 90.0000i 2.97857i
\(914\) 25.0000 0.826927
\(915\) 0 0
\(916\) 24.0000 0.792982
\(917\) − 60.0000i − 1.98137i
\(918\) 1.00000i 0.0330049i
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) − 15.0000i − 0.493999i
\(923\) 12.0000i 0.394985i
\(924\) −15.0000 −0.493464
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00000i 0.131377i
\(928\) − 10.0000i − 0.328266i
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) − 10.0000i − 0.327561i
\(933\) − 2.00000i − 0.0654771i
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 33.0000i 1.07749i
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) − 36.0000i − 1.17232i
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 5.00000 0.162564
\(947\) 61.0000i 1.98223i 0.132994 + 0.991117i \(0.457541\pi\)
−0.132994 + 0.991117i \(0.542459\pi\)
\(948\) 5.00000i 0.162392i
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 3.00000i 0.0972306i
\(953\) 14.0000i 0.453504i 0.973952 + 0.226752i \(0.0728108\pi\)
−0.973952 + 0.226752i \(0.927189\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) 13.0000 0.420450
\(957\) 50.0000i 1.61627i
\(958\) − 26.0000i − 0.840022i
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 6.00000i 0.193448i
\(963\) 5.00000i 0.161123i
\(964\) 0 0
\(965\) 0 0
\(966\) −18.0000 −0.579141
\(967\) − 36.0000i − 1.15768i −0.815440 0.578841i \(-0.803505\pi\)
0.815440 0.578841i \(-0.196495\pi\)
\(968\) − 14.0000i − 0.449977i
\(969\) 1.00000 0.0321246
\(970\) 0 0
\(971\) −38.0000 −1.21948 −0.609739 0.792602i \(-0.708726\pi\)
−0.609739 + 0.792602i \(0.708726\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 24.0000i − 0.769405i
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 10.0000i − 0.319928i −0.987123 0.159964i \(-0.948862\pi\)
0.987123 0.159964i \(-0.0511379\pi\)
\(978\) 0 0
\(979\) 60.0000 1.91761
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) 4.00000i 0.127645i
\(983\) 6.00000i 0.191370i 0.995412 + 0.0956851i \(0.0305042\pi\)
−0.995412 + 0.0956851i \(0.969496\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 10.0000 0.318465
\(987\) − 9.00000i − 0.286473i
\(988\) 2.00000i 0.0636285i
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 5.00000i 0.158750i
\(993\) − 25.0000i − 0.793351i
\(994\) −18.0000 −0.570925
\(995\) 0 0
\(996\) −18.0000 −0.570352
\(997\) − 43.0000i − 1.36182i −0.732365 0.680912i \(-0.761584\pi\)
0.732365 0.680912i \(-0.238416\pi\)
\(998\) − 22.0000i − 0.696398i
\(999\) −3.00000 −0.0949158
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.a.2449.2 2
5.2 odd 4 2550.2.a.h.1.1 1
5.3 odd 4 2550.2.a.z.1.1 yes 1
5.4 even 2 inner 2550.2.d.a.2449.1 2
15.2 even 4 7650.2.a.bl.1.1 1
15.8 even 4 7650.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.h.1.1 1 5.2 odd 4
2550.2.a.z.1.1 yes 1 5.3 odd 4
2550.2.d.a.2449.1 2 5.4 even 2 inner
2550.2.d.a.2449.2 2 1.1 even 1 trivial
7650.2.a.be.1.1 1 15.8 even 4
7650.2.a.bl.1.1 1 15.2 even 4