Properties

Label 2550.2.d.l.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: yes
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.l.2449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -5.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} +5.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} +4.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -5.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -9.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} -10.0000 q^{41} +3.00000i q^{42} -11.0000i q^{43} +5.00000 q^{44} +4.00000 q^{46} +9.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} +4.00000i q^{52} -3.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} +1.00000i q^{57} -4.00000i q^{58} +8.00000 q^{59} -14.0000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} -7.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} +14.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -9.00000 q^{74} -1.00000 q^{76} -15.0000i q^{77} -4.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -8.00000i q^{83} +3.00000 q^{84} -11.0000 q^{86} +4.00000i q^{87} -5.00000i q^{88} +2.00000 q^{89} +12.0000 q^{91} -4.00000i q^{92} -1.00000i q^{93} +9.00000 q^{94} +1.00000 q^{96} -14.0000i q^{97} +2.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 10 q^{11} + 6 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{21} - 2 q^{24} - 8 q^{26} + 8 q^{29} - 2 q^{31} - 2 q^{34} + 2 q^{36} + 8 q^{39} - 20 q^{41} + 10 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} - 6 q^{56} + 16 q^{59} - 28 q^{61} - 2 q^{64} - 10 q^{66} - 8 q^{69} + 28 q^{71} - 18 q^{74} - 2 q^{76} + 10 q^{79} + 2 q^{81} + 6 q^{84} - 22 q^{86} + 4 q^{89} + 24 q^{91} + 18 q^{94} + 2 q^{96} + 10 q^{99}+O(q^{100}) \) 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
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 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 \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 5.00000i 1.06600i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) − 3.00000i − 0.566947i
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 5.00000i − 0.870388i
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 9.00000i − 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 3.00000i 0.462910i
\(43\) − 11.0000i − 1.67748i −0.544529 0.838742i \(-0.683292\pi\)
0.544529 0.838742i \(-0.316708\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 4.00000i 0.554700i
\(53\) − 3.00000i − 0.412082i −0.978543 0.206041i \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 1.00000i 0.132453i
\(58\) − 4.00000i − 0.525226i
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 1.00000i 0.121268i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 15.0000i − 1.70941i
\(78\) − 4.00000i − 0.452911i
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 4.00000i 0.428845i
\(88\) − 5.00000i − 0.533002i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) − 4.00000i − 0.417029i
\(93\) − 1.00000i − 0.103695i
\(94\) 9.00000 0.928279
\(95\) 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
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) − 1.00000i − 0.0990148i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) − 15.0000i − 1.45010i −0.688694 0.725052i \(-0.741816\pi\)
0.688694 0.725052i \(-0.258184\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 3.00000i 0.283473i
\(113\) 3.00000i 0.282216i 0.989994 + 0.141108i \(0.0450665\pi\)
−0.989994 + 0.141108i \(0.954933\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 4.00000i 0.369800i
\(118\) − 8.00000i − 0.736460i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 14.0000i 1.26750i
\(123\) − 10.0000i − 0.901670i
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 5.00000i 0.435194i
\(133\) 3.00000i 0.260133i
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 4.00000i 0.340503i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) − 14.0000i − 1.17485i
\(143\) 20.0000i 1.67248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 2.00000i − 0.164957i
\(148\) 9.00000i 0.739795i
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 1.00000i 0.0808452i
\(154\) −15.0000 −1.20873
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) − 24.0000i − 1.91541i −0.287754 0.957704i \(-0.592909\pi\)
0.287754 0.957704i \(-0.407091\pi\)
\(158\) − 5.00000i − 0.397779i
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) − 1.00000i − 0.0785674i
\(163\) 18.0000i 1.40987i 0.709273 + 0.704934i \(0.249024\pi\)
−0.709273 + 0.704934i \(0.750976\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 11.0000i 0.838742i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 8.00000i 0.601317i
\(178\) − 2.00000i − 0.149906i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) − 12.0000i − 0.889499i
\(183\) − 14.0000i − 1.03491i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 5.00000i 0.365636i
\(188\) − 9.00000i − 0.656392i
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) − 5.00000i − 0.355335i
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) − 9.00000i − 0.633238i
\(203\) 12.0000i 0.842235i
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) − 4.00000i − 0.278019i
\(208\) − 4.00000i − 0.277350i
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 3.00000i 0.206041i
\(213\) 14.0000i 0.959264i
\(214\) −15.0000 −1.02538
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 3.00000i − 0.203653i
\(218\) 1.00000i 0.0677285i
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) − 9.00000i − 0.604040i
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) − 21.0000i − 1.39382i −0.717159 0.696909i \(-0.754558\pi\)
0.717159 0.696909i \(-0.245442\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 4.00000i 0.262613i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 5.00000i 0.324785i
\(238\) − 3.00000i − 0.194461i
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) − 14.0000i − 0.899954i
\(243\) 1.00000i 0.0641500i
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) − 4.00000i − 0.254514i
\(248\) − 1.00000i − 0.0635001i
\(249\) 8.00000 0.506979
\(250\) 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
\(253\) − 20.0000i − 1.25739i
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 12.0000i − 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) − 11.0000i − 0.684830i
\(259\) 27.0000 1.67770
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) − 4.00000i − 0.247121i
\(263\) 7.00000i 0.431638i 0.976433 + 0.215819i \(0.0692422\pi\)
−0.976433 + 0.215819i \(0.930758\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) 2.00000i 0.122398i
\(268\) 7.00000i 0.427593i
\(269\) −28.0000 −1.70719 −0.853595 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) 12.0000i 0.726273i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 13.0000i − 0.781094i −0.920583 0.390547i \(-0.872286\pi\)
0.920583 0.390547i \(-0.127714\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 9.00000i 0.535942i
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) 20.0000 1.18262
\(287\) − 30.0000i − 1.77084i
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 2.00000i 0.117041i
\(293\) 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
\(296\) 9.00000 0.523114
\(297\) 5.00000i 0.290129i
\(298\) 14.0000i 0.810998i
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 33.0000 1.90209
\(302\) 24.0000i 1.38104i
\(303\) 9.00000i 0.517036i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 15.0000i 0.854704i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 4.00000i 0.226455i
\(313\) 16.0000i 0.904373i 0.891923 + 0.452187i \(0.149356\pi\)
−0.891923 + 0.452187i \(0.850644\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) 4.00000i 0.224662i 0.993671 + 0.112331i \(0.0358318\pi\)
−0.993671 + 0.112331i \(0.964168\pi\)
\(318\) − 3.00000i − 0.168232i
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 12.0000i 0.668734i
\(323\) − 1.00000i − 0.0556415i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 18.0000 0.996928
\(327\) − 1.00000i − 0.0553001i
\(328\) − 10.0000i − 0.552158i
\(329\) −27.0000 −1.48856
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 8.00000i 0.439057i
\(333\) 9.00000i 0.493197i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) 24.0000i 1.30736i 0.756770 + 0.653682i \(0.226776\pi\)
−0.756770 + 0.653682i \(0.773224\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 1.00000i 0.0540738i
\(343\) 15.0000i 0.809924i
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 11.0000i 0.590511i 0.955418 + 0.295255i \(0.0954048\pi\)
−0.955418 + 0.295255i \(0.904595\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 5.00000i 0.266501i
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 3.00000i 0.158777i
\(358\) − 12.0000i − 0.634220i
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 5.00000i 0.262794i
\(363\) 14.0000i 0.734809i
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) 25.0000i 1.30499i 0.757793 + 0.652495i \(0.226278\pi\)
−0.757793 + 0.652495i \(0.773722\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 1.00000i 0.0518476i
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) − 16.0000i − 0.824042i
\(378\) − 3.00000i − 0.154303i
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 9.00000i 0.460480i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 11.0000i 0.559161i
\(388\) 14.0000i 0.710742i
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) − 2.00000i − 0.101015i
\(393\) 4.00000i 0.201773i
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 15.0000i 0.752828i 0.926451 + 0.376414i \(0.122843\pi\)
−0.926451 + 0.376414i \(0.877157\pi\)
\(398\) − 1.00000i − 0.0501255i
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) − 7.00000i − 0.349128i
\(403\) 4.00000i 0.199254i
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 45.0000i 2.23057i
\(408\) 1.00000i 0.0495074i
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 14.0000i 0.689730i
\(413\) 24.0000i 1.18096i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 8.00000i − 0.391762i
\(418\) 5.00000i 0.244558i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 12.0000i 0.584151i
\(423\) − 9.00000i − 0.437595i
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 14.0000 0.678302
\(427\) − 42.0000i − 2.03252i
\(428\) 15.0000i 0.725052i
\(429\) −20.0000 −0.965609
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 27.0000i − 1.29754i −0.760986 0.648769i \(-0.775284\pi\)
0.760986 0.648769i \(-0.224716\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 4.00000i 0.191346i
\(438\) − 2.00000i − 0.0955637i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 4.00000i 0.190261i
\(443\) 38.0000i 1.80543i 0.430234 + 0.902717i \(0.358431\pi\)
−0.430234 + 0.902717i \(0.641569\pi\)
\(444\) −9.00000 −0.427121
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) − 14.0000i − 0.662177i
\(448\) − 3.00000i − 0.141737i
\(449\) 41.0000 1.93491 0.967455 0.253044i \(-0.0814317\pi\)
0.967455 + 0.253044i \(0.0814317\pi\)
\(450\) 0 0
\(451\) 50.0000 2.35441
\(452\) − 3.00000i − 0.141108i
\(453\) − 24.0000i − 1.12762i
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 37.0000i 1.73079i 0.501093 + 0.865393i \(0.332931\pi\)
−0.501093 + 0.865393i \(0.667069\pi\)
\(458\) − 12.0000i − 0.560723i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −5.00000 −0.232873 −0.116437 0.993198i \(-0.537147\pi\)
−0.116437 + 0.993198i \(0.537147\pi\)
\(462\) − 15.0000i − 0.697863i
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 6.00000i − 0.277647i −0.990317 0.138823i \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 8.00000i 0.368230i
\(473\) 55.0000i 2.52890i
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 3.00000i 0.137361i
\(478\) 5.00000i 0.228695i
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 8.00000i 0.364390i
\(483\) − 12.0000i − 0.546019i
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) −18.0000 −0.813988
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 10.0000i 0.450835i
\(493\) − 4.00000i − 0.180151i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 42.0000i 1.88396i
\(498\) − 8.00000i − 0.358489i
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 18.0000i 0.803379i
\(503\) − 34.0000i − 1.51599i −0.652263 0.757993i \(-0.726180\pi\)
0.652263 0.757993i \(-0.273820\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) −20.0000 −0.889108
\(507\) − 3.00000i − 0.133235i
\(508\) 2.00000i 0.0887357i
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −11.0000 −0.484248
\(517\) − 45.0000i − 1.97910i
\(518\) − 27.0000i − 1.18631i
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 7.00000 0.305215
\(527\) 1.00000i 0.0435607i
\(528\) − 5.00000i − 0.217597i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) − 3.00000i − 0.130066i
\(533\) 40.0000i 1.73259i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 12.0000i 0.517838i
\(538\) 28.0000i 1.20717i
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) − 4.00000i − 0.171815i
\(543\) − 5.00000i − 0.214571i
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) − 4.00000i − 0.170251i
\(553\) 15.0000i 0.637865i
\(554\) −13.0000 −0.552317
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 9.00000i 0.381342i 0.981654 + 0.190671i \(0.0610664\pi\)
−0.981654 + 0.190671i \(0.938934\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) −44.0000 −1.86100
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) 18.0000i 0.759284i
\(563\) − 16.0000i − 0.674320i −0.941447 0.337160i \(-0.890534\pi\)
0.941447 0.337160i \(-0.109466\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 3.00000i 0.125988i
\(568\) 14.0000i 0.587427i
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) − 20.0000i − 0.836242i
\(573\) − 9.00000i − 0.375980i
\(574\) −30.0000 −1.25218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 45.0000i − 1.87337i −0.350167 0.936687i \(-0.613875\pi\)
0.350167 0.936687i \(-0.386125\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) − 14.0000i − 0.580319i
\(583\) 15.0000i 0.621237i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 6.00000i 0.247647i 0.992304 + 0.123823i \(0.0395156\pi\)
−0.992304 + 0.123823i \(0.960484\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) −1.00000 −0.0412043
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) − 9.00000i − 0.369898i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 1.00000i 0.0409273i
\(598\) − 16.0000i − 0.654289i
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) − 33.0000i − 1.34498i
\(603\) 7.00000i 0.285062i
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) − 1.00000i − 0.0404226i
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) 19.0000i 0.764911i 0.923974 + 0.382456i \(0.124922\pi\)
−0.923974 + 0.382456i \(0.875078\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 10.0000i − 0.400963i
\(623\) 6.00000i 0.240385i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 16.0000 0.639489
\(627\) − 5.00000i − 0.199681i
\(628\) 24.0000i 0.957704i
\(629\) −9.00000 −0.358854
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 5.00000i 0.198889i
\(633\) − 12.0000i − 0.476957i
\(634\) 4.00000 0.158860
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 8.00000i 0.316972i
\(638\) 20.0000i 0.791808i
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) − 15.0000i − 0.592003i
\(643\) − 44.0000i − 1.73519i −0.497271 0.867595i \(-0.665665\pi\)
0.497271 0.867595i \(-0.334335\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −1.00000 −0.0393445
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) − 18.0000i − 0.704934i
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) −1.00000 −0.0391031
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 2.00000i 0.0780274i
\(658\) 27.0000i 1.05257i
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) − 5.00000i − 0.194331i
\(663\) − 4.00000i − 0.155347i
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 9.00000 0.348743
\(667\) 16.0000i 0.619522i
\(668\) − 18.0000i − 0.696441i
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) 70.0000 2.70232
\(672\) 3.00000i 0.115728i
\(673\) − 4.00000i − 0.154189i −0.997024 0.0770943i \(-0.975436\pi\)
0.997024 0.0770943i \(-0.0245643\pi\)
\(674\) 24.0000 0.924445
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 3.00000i 0.115214i
\(679\) 42.0000 1.61181
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) − 5.00000i − 0.191460i
\(683\) 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
\(687\) 12.0000i 0.457829i
\(688\) − 11.0000i − 0.419371i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 15.0000i 0.569803i
\(694\) 11.0000 0.417554
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) 10.0000i 0.378777i
\(698\) − 10.0000i − 0.378506i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 4.00000i 0.150970i
\(703\) − 9.00000i − 0.339441i
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 27.0000i 1.01544i
\(708\) − 8.00000i − 0.300658i
\(709\) −49.0000 −1.84023 −0.920117 0.391644i \(-0.871906\pi\)
−0.920117 + 0.391644i \(0.871906\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 2.00000i 0.0749532i
\(713\) − 4.00000i − 0.149801i
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 5.00000i − 0.186728i
\(718\) − 25.0000i − 0.932992i
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 18.0000i 0.669891i
\(723\) − 8.00000i − 0.297523i
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) 12.0000i 0.444750i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −11.0000 −0.406850
\(732\) 14.0000i 0.517455i
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 25.0000 0.922767
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 35.0000i 1.28924i
\(738\) − 10.0000i − 0.368105i
\(739\) 17.0000 0.625355 0.312678 0.949859i \(-0.398774\pi\)
0.312678 + 0.949859i \(0.398774\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) − 9.00000i − 0.330400i
\(743\) − 38.0000i − 1.39408i −0.717030 0.697042i \(-0.754499\pi\)
0.717030 0.697042i \(-0.245501\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 8.00000i 0.292705i
\(748\) − 5.00000i − 0.182818i
\(749\) 45.0000 1.64426
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 9.00000i 0.328196i
\(753\) − 18.0000i − 0.655956i
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) − 46.0000i − 1.67190i −0.548807 0.835949i \(-0.684918\pi\)
0.548807 0.835949i \(-0.315082\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) − 2.00000i − 0.0724524i
\(763\) − 3.00000i − 0.108607i
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) − 32.0000i − 1.15545i
\(768\) 1.00000i 0.0360844i
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 6.00000i 0.215945i
\(773\) − 22.0000i − 0.791285i −0.918405 0.395643i \(-0.870522\pi\)
0.918405 0.395643i \(-0.129478\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 27.0000i 0.968620i
\(778\) 21.0000i 0.752886i
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −70.0000 −2.50480
\(782\) − 4.00000i − 0.143040i
\(783\) − 4.00000i − 0.142948i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) 50.0000i 1.78231i 0.453701 + 0.891154i \(0.350103\pi\)
−0.453701 + 0.891154i \(0.649897\pi\)
\(788\) 12.0000i 0.427482i
\(789\) −7.00000 −0.249207
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 5.00000i 0.177667i
\(793\) 56.0000i 1.98862i
\(794\) 15.0000 0.532330
\(795\) 0 0
\(796\) −1.00000 −0.0354441
\(797\) − 39.0000i − 1.38145i −0.723117 0.690725i \(-0.757291\pi\)
0.723117 0.690725i \(-0.242709\pi\)
\(798\) 3.00000i 0.106199i
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 30.0000i 1.05934i
\(803\) 10.0000i 0.352892i
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) − 28.0000i − 0.985647i
\(808\) 9.00000i 0.316619i
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 4.00000i 0.140286i
\(814\) 45.0000 1.57725
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) − 11.0000i − 0.384841i
\(818\) − 26.0000i − 0.909069i
\(819\) −12.0000 −0.419314
\(820\) 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
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 39.0000i − 1.35616i −0.734987 0.678081i \(-0.762812\pi\)
0.734987 0.678081i \(-0.237188\pi\)
\(828\) 4.00000i 0.139010i
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 4.00000i 0.138675i
\(833\) 2.00000i 0.0692959i
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) 1.00000i 0.0345651i
\(838\) 28.0000i 0.967244i
\(839\) −10.0000 −0.345238 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) − 24.0000i − 0.827095i
\(843\) − 18.0000i − 0.619953i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 42.0000i 1.44314i
\(848\) − 3.00000i − 0.103020i
\(849\) 22.0000 0.755038
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) − 14.0000i − 0.479632i
\(853\) − 9.00000i − 0.308154i −0.988059 0.154077i \(-0.950760\pi\)
0.988059 0.154077i \(-0.0492404\pi\)
\(854\) −42.0000 −1.43721
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) − 5.00000i − 0.170797i −0.996347 0.0853984i \(-0.972784\pi\)
0.996347 0.0853984i \(-0.0272163\pi\)
\(858\) 20.0000i 0.682789i
\(859\) 25.0000 0.852989 0.426494 0.904490i \(-0.359748\pi\)
0.426494 + 0.904490i \(0.359748\pi\)
\(860\) 0 0
\(861\) 30.0000 1.02240
\(862\) − 14.0000i − 0.476842i
\(863\) − 17.0000i − 0.578687i −0.957225 0.289343i \(-0.906563\pi\)
0.957225 0.289343i \(-0.0934369\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −27.0000 −0.917497
\(867\) − 1.00000i − 0.0339618i
\(868\) 3.00000i 0.101827i
\(869\) −25.0000 −0.848067
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) − 1.00000i − 0.0338643i
\(873\) 14.0000i 0.473828i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 46.0000i 1.55331i 0.629926 + 0.776655i \(0.283085\pi\)
−0.629926 + 0.776655i \(0.716915\pi\)
\(878\) 20.0000i 0.674967i
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) − 2.00000i − 0.0673435i
\(883\) − 48.0000i − 1.61533i −0.589643 0.807664i \(-0.700731\pi\)
0.589643 0.807664i \(-0.299269\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 38.0000 1.27663
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 9.00000i 0.302020i
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 6.00000i 0.200895i
\(893\) 9.00000i 0.301174i
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 16.0000i 0.534224i
\(898\) − 41.0000i − 1.36819i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) − 50.0000i − 1.66482i
\(903\) 33.0000i 1.09817i
\(904\) −3.00000 −0.0997785
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) 24.0000i 0.796907i 0.917189 + 0.398453i \(0.130453\pi\)
−0.917189 + 0.398453i \(0.869547\pi\)
\(908\) 21.0000i 0.696909i
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 40.0000i 1.32381i
\(914\) 37.0000 1.22385
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 12.0000i 0.396275i
\(918\) 1.00000i 0.0330049i
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 5.00000i 0.164666i
\(923\) − 56.0000i − 1.84326i
\(924\) −15.0000 −0.493464
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 14.0000i 0.459820i
\(928\) − 4.00000i − 0.131306i
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) − 6.00000i − 0.196537i
\(933\) 10.0000i 0.327385i
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) − 21.0000i − 0.685674i
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) − 24.0000i − 0.781962i
\(943\) − 40.0000i − 1.30258i
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 55.0000 1.78820
\(947\) − 29.0000i − 0.942373i −0.882034 0.471187i \(-0.843826\pi\)
0.882034 0.471187i \(-0.156174\pi\)
\(948\) − 5.00000i − 0.162392i
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 3.00000i 0.0972306i
\(953\) − 12.0000i − 0.388718i −0.980930 0.194359i \(-0.937737\pi\)
0.980930 0.194359i \(-0.0622627\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 5.00000 0.161712
\(957\) − 20.0000i − 0.646508i
\(958\) 30.0000i 0.969256i
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 36.0000i 1.16069i
\(963\) 15.0000i 0.483368i
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) − 24.0000i − 0.771788i −0.922543 0.385894i \(-0.873893\pi\)
0.922543 0.385894i \(-0.126107\pi\)
\(968\) 14.0000i 0.449977i
\(969\) 1.00000 0.0321246
\(970\) 0 0
\(971\) −10.0000 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 24.0000i − 0.769405i
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) − 48.0000i − 1.53566i −0.640656 0.767828i \(-0.721338\pi\)
0.640656 0.767828i \(-0.278662\pi\)
\(978\) 18.0000i 0.575577i
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 30.0000i 0.957338i
\(983\) 28.0000i 0.893061i 0.894768 + 0.446531i \(0.147341\pi\)
−0.894768 + 0.446531i \(0.852659\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) − 27.0000i − 0.859419i
\(988\) 4.00000i 0.127257i
\(989\) 44.0000 1.39912
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 5.00000i 0.158670i
\(994\) 42.0000 1.33216
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) − 25.0000i − 0.791758i −0.918303 0.395879i \(-0.870440\pi\)
0.918303 0.395879i \(-0.129560\pi\)
\(998\) − 38.0000i − 1.20287i
\(999\) −9.00000 −0.284747
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.l.2449.1 2
5.2 odd 4 2550.2.a.bb.1.1 yes 1
5.3 odd 4 2550.2.a.f.1.1 1
5.4 even 2 inner 2550.2.d.l.2449.2 2
15.2 even 4 7650.2.a.f.1.1 1
15.8 even 4 7650.2.a.ch.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.f.1.1 1 5.3 odd 4
2550.2.a.bb.1.1 yes 1 5.2 odd 4
2550.2.d.l.2449.1 2 1.1 even 1 trivial
2550.2.d.l.2449.2 2 5.4 even 2 inner
7650.2.a.f.1.1 1 15.2 even 4
7650.2.a.ch.1.1 1 15.8 even 4