Properties

Label 950.2.b.d.799.1
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
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] = [950,2,Mod(799,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("950.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (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: \(7.58578819202\)
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 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.d.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{12} +3.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} +1.00000 q^{21} +5.00000i q^{23} -1.00000 q^{24} +3.00000 q^{26} +5.00000i q^{27} +1.00000i q^{28} +5.00000 q^{29} +10.0000 q^{31} -1.00000i q^{32} -7.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} -3.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} -6.00000i q^{43} +5.00000 q^{46} +1.00000i q^{48} +6.00000 q^{49} +7.00000 q^{51} -3.00000i q^{52} -9.00000i q^{53} +5.00000 q^{54} +1.00000 q^{56} +1.00000i q^{57} -5.00000i q^{58} +7.00000 q^{59} -4.00000 q^{61} -10.0000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +7.00000i q^{67} +7.00000i q^{68} -5.00000 q^{69} +2.00000i q^{72} +9.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} +3.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +2.00000i q^{83} -1.00000 q^{84} -6.00000 q^{86} +5.00000i q^{87} +10.0000 q^{89} +3.00000 q^{91} -5.00000i q^{92} +10.0000i q^{93} +1.00000 q^{96} -18.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} - 2 q^{14} + 2 q^{16} + 2 q^{19} + 2 q^{21} - 2 q^{24} + 6 q^{26} + 10 q^{29} + 20 q^{31} - 14 q^{34} - 4 q^{36} - 6 q^{39} + 4 q^{41} + 10 q^{46} + 12 q^{49} + 14 q^{51} + 10 q^{54} + 2 q^{56} + 14 q^{59} - 8 q^{61} - 2 q^{64} - 10 q^{69} + 4 q^{74} - 2 q^{76} + 20 q^{79} + 2 q^{81} - 2 q^{84} - 12 q^{86} + 20 q^{89} + 6 q^{91} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 5.00000i 1.04257i 0.853382 + 0.521286i \(0.174548\pi\)
−0.853382 + 0.521286i \(0.825452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 3.00000 0.588348
\(27\) 5.00000i 0.962250i
\(28\) 1.00000i 0.188982i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) − 3.00000i − 0.416025i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000i 0.132453i
\(58\) − 5.00000i − 0.656532i
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 7.00000i 0.848875i
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000i 0.235702i
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 3.00000i 0.339683i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 5.00000i 0.536056i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) − 5.00000i − 0.521286i
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) 0 0
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) − 7.00000i − 0.693103i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) − 8.00000i − 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) 6.00000i 0.554700i
\(118\) − 7.00000i − 0.644402i
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 4.00000i 0.362143i
\(123\) 2.00000i 0.180334i
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) − 1.00000i − 0.0867110i
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 5.00000i 0.425628i
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) 6.00000i 0.494872i
\(148\) − 2.00000i − 0.164399i
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 14.0000i − 1.13183i
\(154\) 0 0
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) − 1.00000i − 0.0785674i
\(163\) 18.0000i 1.40987i 0.709273 + 0.704934i \(0.249024\pi\)
−0.709273 + 0.704934i \(0.750976\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 14.0000i 1.08335i 0.840587 + 0.541676i \(0.182210\pi\)
−0.840587 + 0.541676i \(0.817790\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 6.00000i 0.457496i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 0 0
\(177\) 7.00000i 0.526152i
\(178\) − 10.0000i − 0.749532i
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) − 3.00000i − 0.222375i
\(183\) − 4.00000i − 0.295689i
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 4.00000i 0.281439i
\(203\) − 5.00000i − 0.350931i
\(204\) −7.00000 −0.490098
\(205\) 0 0
\(206\) 0 0
\(207\) 10.0000i 0.695048i
\(208\) 3.00000i 0.208013i
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 0 0
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) −5.00000 −0.340207
\(217\) − 10.0000i − 0.678844i
\(218\) 13.0000i 0.880471i
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 21.0000 1.41261
\(222\) 2.00000i 0.134231i
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) − 25.0000i − 1.65931i −0.558278 0.829654i \(-0.688538\pi\)
0.558278 0.829654i \(-0.311462\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.00000i 0.328266i
\(233\) − 26.0000i − 1.70332i −0.524097 0.851658i \(-0.675597\pi\)
0.524097 0.851658i \(-0.324403\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 10.0000i 0.649570i
\(238\) 7.00000i 0.453743i
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 16.0000i 1.02640i
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 3.00000i 0.190885i
\(248\) 10.0000i 0.635001i
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0 0
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) − 6.00000i − 0.373544i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 20.0000i 1.23560i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.00000 −0.0613139
\(267\) 10.0000i 0.611990i
\(268\) − 7.00000i − 0.427593i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 5.00000 0.303728 0.151864 0.988401i \(-0.451472\pi\)
0.151864 + 0.988401i \(0.451472\pi\)
\(272\) − 7.00000i − 0.424437i
\(273\) 3.00000i 0.181568i
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 5.00000 0.300965
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 20.0000 1.19737
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 26.0000i 1.54554i 0.634686 + 0.772770i \(0.281129\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.00000i − 0.118056i
\(288\) − 2.00000i − 0.117851i
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) − 9.00000i − 0.526685i
\(293\) − 31.0000i − 1.81104i −0.424304 0.905520i \(-0.639481\pi\)
0.424304 0.905520i \(-0.360519\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 4.00000i 0.231714i
\(299\) −15.0000 −0.867472
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 6.00000i 0.345261i
\(303\) − 4.00000i − 0.229794i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −14.0000 −0.800327
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 33.0000 1.87126 0.935629 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) − 3.00000i − 0.169842i
\(313\) 29.0000i 1.63918i 0.572953 + 0.819588i \(0.305798\pi\)
−0.572953 + 0.819588i \(0.694202\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 3.00000i 0.168497i 0.996445 + 0.0842484i \(0.0268489\pi\)
−0.996445 + 0.0842484i \(0.973151\pi\)
\(318\) − 9.00000i − 0.504695i
\(319\) 0 0
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) − 5.00000i − 0.278639i
\(323\) − 7.00000i − 0.389490i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 18.0000 0.996928
\(327\) − 13.0000i − 0.718902i
\(328\) 2.00000i 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) − 2.00000i − 0.109764i
\(333\) 4.00000i 0.219199i
\(334\) 14.0000 0.766046
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) − 4.00000i − 0.217571i
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 0 0
\(342\) − 2.00000i − 0.108148i
\(343\) − 13.0000i − 0.701934i
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 18.0000i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) − 5.00000i − 0.268028i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −15.0000 −0.800641
\(352\) 0 0
\(353\) − 27.0000i − 1.43706i −0.695493 0.718532i \(-0.744814\pi\)
0.695493 0.718532i \(-0.255186\pi\)
\(354\) 7.00000 0.372046
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) − 7.00000i − 0.370479i
\(358\) 24.0000i 1.26844i
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 10.0000i 0.525588i
\(363\) − 11.0000i − 0.577350i
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 5.00000i 0.260643i
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) − 10.0000i − 0.518476i
\(373\) − 3.00000i − 0.155334i −0.996979 0.0776671i \(-0.975253\pi\)
0.996979 0.0776671i \(-0.0247471\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000i 0.772539i
\(378\) − 5.00000i − 0.257172i
\(379\) 7.00000 0.359566 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 7.00000i 0.358151i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 12.0000i − 0.609994i
\(388\) 18.0000i 0.913812i
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 35.0000 1.77003
\(392\) 6.00000i 0.303046i
\(393\) − 20.0000i − 1.00887i
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.0000i − 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) 17.0000i 0.852133i
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 7.00000i 0.349128i
\(403\) 30.0000i 1.49441i
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) 0 0
\(408\) 7.00000i 0.346552i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) − 7.00000i − 0.344447i
\(414\) 10.0000 0.491473
\(415\) 0 0
\(416\) 3.00000 0.147087
\(417\) − 12.0000i − 0.587643i
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 31.0000 1.51085 0.755424 0.655237i \(-0.227431\pi\)
0.755424 + 0.655237i \(0.227431\pi\)
\(422\) − 5.00000i − 0.243396i
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) − 9.00000i − 0.435031i
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 5.00000i 0.240563i
\(433\) − 12.0000i − 0.576683i −0.957528 0.288342i \(-0.906896\pi\)
0.957528 0.288342i \(-0.0931039\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) 13.0000 0.622587
\(437\) 5.00000i 0.239182i
\(438\) 9.00000i 0.430037i
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) − 21.0000i − 0.998868i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) − 4.00000i − 0.189194i
\(448\) 1.00000i 0.0472456i
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.00000i 0.376288i
\(453\) − 6.00000i − 0.281905i
\(454\) −25.0000 −1.17331
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 41.0000i − 1.91790i −0.283577 0.958950i \(-0.591521\pi\)
0.283577 0.958950i \(-0.408479\pi\)
\(458\) − 18.0000i − 0.841085i
\(459\) 35.0000 1.63366
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 7.00000i 0.322201i
\(473\) 0 0
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) 7.00000 0.320844
\(477\) − 18.0000i − 0.824163i
\(478\) − 27.0000i − 1.23495i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 28.0000i 1.27537i
\(483\) 5.00000i 0.227508i
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 16.0000 0.725775
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 4.00000i − 0.181071i
\(489\) −18.0000 −0.813988
\(490\) 0 0
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) − 35.0000i − 1.57632i
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 2.00000i 0.0896221i
\(499\) −42.0000 −1.88018 −0.940089 0.340929i \(-0.889258\pi\)
−0.940089 + 0.340929i \(0.889258\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) − 4.00000i − 0.178529i
\(503\) 17.0000i 0.757993i 0.925398 + 0.378996i \(0.123731\pi\)
−0.925398 + 0.378996i \(0.876269\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 4.00000i 0.177646i
\(508\) 6.00000i 0.266207i
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) − 1.00000i − 0.0441942i
\(513\) 5.00000i 0.220755i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 0 0
\(518\) − 2.00000i − 0.0878750i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) − 10.0000i − 0.437688i
\(523\) 19.0000i 0.830812i 0.909636 + 0.415406i \(0.136360\pi\)
−0.909636 + 0.415406i \(0.863640\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) − 70.0000i − 3.04925i
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 14.0000 0.607548
\(532\) 1.00000i 0.0433555i
\(533\) 6.00000i 0.259889i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) − 24.0000i − 1.03568i
\(538\) − 10.0000i − 0.431131i
\(539\) 0 0
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) − 5.00000i − 0.214768i
\(543\) − 10.0000i − 0.429141i
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 3.00000i 0.128154i
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 5.00000 0.213007
\(552\) − 5.00000i − 0.212814i
\(553\) − 10.0000i − 0.425243i
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 8.00000i 0.338971i 0.985533 + 0.169485i \(0.0542106\pi\)
−0.985533 + 0.169485i \(0.945789\pi\)
\(558\) − 20.0000i − 0.846668i
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 26.0000i 1.09674i
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 26.0000 1.09286
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) − 7.00000i − 0.292429i
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) − 11.0000i − 0.457936i −0.973434 0.228968i \(-0.926465\pi\)
0.973434 0.228968i \(-0.0735351\pi\)
\(578\) 32.0000i 1.33102i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) − 18.0000i − 0.746124i
\(583\) 0 0
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −31.0000 −1.28060
\(587\) 30.0000i 1.23823i 0.785299 + 0.619116i \(0.212509\pi\)
−0.785299 + 0.619116i \(0.787491\pi\)
\(588\) − 6.00000i − 0.247436i
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 2.00000i 0.0821995i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) − 17.0000i − 0.695764i
\(598\) 15.0000i 0.613396i
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 6.00000i 0.244542i
\(603\) 14.0000i 0.570124i
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) −4.00000 −0.162489
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 5.00000 0.202610
\(610\) 0 0
\(611\) 0 0
\(612\) 14.0000i 0.565916i
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) −25.0000 −1.00322
\(622\) − 33.0000i − 1.32318i
\(623\) − 10.0000i − 0.400642i
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) 29.0000 1.15907
\(627\) 0 0
\(628\) − 18.0000i − 0.718278i
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 5.00000i 0.198732i
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 9.00000i 0.355202i
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) −7.00000 −0.275411
\(647\) − 9.00000i − 0.353827i −0.984226 0.176913i \(-0.943389\pi\)
0.984226 0.176913i \(-0.0566112\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) − 18.0000i − 0.704934i
\(653\) 12.0000i 0.469596i 0.972044 + 0.234798i \(0.0754429\pi\)
−0.972044 + 0.234798i \(0.924557\pi\)
\(654\) −13.0000 −0.508340
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 18.0000i 0.702247i
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 17.0000i 0.660724i
\(663\) 21.0000i 0.815572i
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 25.0000i 0.968004i
\(668\) − 14.0000i − 0.541676i
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) 0 0
\(672\) − 1.00000i − 0.0385758i
\(673\) 28.0000i 1.07932i 0.841883 + 0.539660i \(0.181447\pi\)
−0.841883 + 0.539660i \(0.818553\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 19.0000i 0.730229i 0.930963 + 0.365115i \(0.118970\pi\)
−0.930963 + 0.365115i \(0.881030\pi\)
\(678\) − 8.00000i − 0.307238i
\(679\) −18.0000 −0.690777
\(680\) 0 0
\(681\) 25.0000 0.958002
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 18.0000i 0.686743i
\(688\) − 6.00000i − 0.228748i
\(689\) 27.0000 1.02862
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) − 14.0000i − 0.530288i
\(698\) 10.0000i 0.378506i
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 15.0000i 0.566139i
\(703\) 2.00000i 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) −27.0000 −1.01616
\(707\) 4.00000i 0.150435i
\(708\) − 7.00000i − 0.263076i
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 10.0000i 0.374766i
\(713\) 50.0000i 1.87251i
\(714\) −7.00000 −0.261968
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 27.0000i 1.00833i
\(718\) 27.0000i 1.00763i
\(719\) −29.0000 −1.08152 −0.540759 0.841178i \(-0.681863\pi\)
−0.540759 + 0.841178i \(0.681863\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 0.0372161i
\(723\) − 28.0000i − 1.04133i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 11.0000i 0.407967i 0.978974 + 0.203984i \(0.0653890\pi\)
−0.978974 + 0.203984i \(0.934611\pi\)
\(728\) 3.00000i 0.111187i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −42.0000 −1.55343
\(732\) 4.00000i 0.147844i
\(733\) − 24.0000i − 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) 0 0
\(738\) − 4.00000i − 0.147242i
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 0 0
\(741\) −3.00000 −0.110208
\(742\) 9.00000i 0.330400i
\(743\) − 34.0000i − 1.24734i −0.781688 0.623670i \(-0.785641\pi\)
0.781688 0.623670i \(-0.214359\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) −3.00000 −0.109838
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 0 0
\(753\) 4.00000i 0.145768i
\(754\) 15.0000 0.546268
\(755\) 0 0
\(756\) −5.00000 −0.181848
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) − 7.00000i − 0.254251i
\(759\) 0 0
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) − 6.00000i − 0.217357i
\(763\) 13.0000i 0.470632i
\(764\) 7.00000 0.253251
\(765\) 0 0
\(766\) 0 0
\(767\) 21.0000i 0.758266i
\(768\) 1.00000i 0.0360844i
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 2.00000i − 0.0719816i
\(773\) − 15.0000i − 0.539513i −0.962929 0.269756i \(-0.913057\pi\)
0.962929 0.269756i \(-0.0869431\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 2.00000i 0.0717496i
\(778\) 24.0000i 0.860442i
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 0 0
\(782\) − 35.0000i − 1.25160i
\(783\) 25.0000i 0.893427i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) 11.0000i 0.392108i 0.980593 + 0.196054i \(0.0628127\pi\)
−0.980593 + 0.196054i \(0.937187\pi\)
\(788\) 10.0000i 0.356235i
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) − 12.0000i − 0.426132i
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) 5.00000i 0.177109i 0.996071 + 0.0885545i \(0.0282248\pi\)
−0.996071 + 0.0885545i \(0.971775\pi\)
\(798\) − 1.00000i − 0.0353996i
\(799\) 0 0
\(800\) 0 0
\(801\) 20.0000 0.706665
\(802\) − 18.0000i − 0.635602i
\(803\) 0 0
\(804\) 7.00000 0.246871
\(805\) 0 0
\(806\) 30.0000 1.05670
\(807\) 10.0000i 0.352017i
\(808\) − 4.00000i − 0.140720i
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) 5.00000i 0.175466i
\(813\) 5.00000i 0.175358i
\(814\) 0 0
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) − 6.00000i − 0.209913i
\(818\) 26.0000i 0.909069i
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) −56.0000 −1.95441 −0.977207 0.212290i \(-0.931908\pi\)
−0.977207 + 0.212290i \(0.931908\pi\)
\(822\) − 3.00000i − 0.104637i
\(823\) 31.0000i 1.08059i 0.841475 + 0.540296i \(0.181688\pi\)
−0.841475 + 0.540296i \(0.818312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −7.00000 −0.243561
\(827\) − 9.00000i − 0.312961i −0.987681 0.156480i \(-0.949985\pi\)
0.987681 0.156480i \(-0.0500148\pi\)
\(828\) − 10.0000i − 0.347524i
\(829\) 51.0000 1.77130 0.885652 0.464350i \(-0.153712\pi\)
0.885652 + 0.464350i \(0.153712\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) − 3.00000i − 0.104006i
\(833\) − 42.0000i − 1.45521i
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 50.0000i 1.72825i
\(838\) − 26.0000i − 0.898155i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) − 31.0000i − 1.06833i
\(843\) − 26.0000i − 0.895488i
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0000i 0.377964i
\(848\) − 9.00000i − 0.309061i
\(849\) −26.0000 −0.892318
\(850\) 0 0
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 24.0000i 0.819824i 0.912125 + 0.409912i \(0.134441\pi\)
−0.912125 + 0.409912i \(0.865559\pi\)
\(858\) 0 0
\(859\) −56.0000 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) − 12.0000i − 0.408722i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −12.0000 −0.407777
\(867\) − 32.0000i − 1.08678i
\(868\) 10.0000i 0.339422i
\(869\) 0 0
\(870\) 0 0
\(871\) −21.0000 −0.711558
\(872\) − 13.0000i − 0.440236i
\(873\) − 36.0000i − 1.21842i
\(874\) 5.00000 0.169128
\(875\) 0 0
\(876\) 9.00000 0.304082
\(877\) − 5.00000i − 0.168838i −0.996430 0.0844190i \(-0.973097\pi\)
0.996430 0.0844190i \(-0.0269034\pi\)
\(878\) 10.0000i 0.337484i
\(879\) 31.0000 1.04560
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) − 12.0000i − 0.404061i
\(883\) − 26.0000i − 0.874970i −0.899226 0.437485i \(-0.855869\pi\)
0.899226 0.437485i \(-0.144131\pi\)
\(884\) −21.0000 −0.706306
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 52.0000i 1.74599i 0.487730 + 0.872995i \(0.337825\pi\)
−0.487730 + 0.872995i \(0.662175\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) − 22.0000i − 0.736614i
\(893\) 0 0
\(894\) −4.00000 −0.133780
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 15.0000i − 0.500835i
\(898\) 14.0000i 0.467186i
\(899\) 50.0000 1.66759
\(900\) 0 0
\(901\) −63.0000 −2.09883
\(902\) 0 0
\(903\) − 6.00000i − 0.199667i
\(904\) 8.00000 0.266076
\(905\) 0 0
\(906\) −6.00000 −0.199337
\(907\) 5.00000i 0.166022i 0.996549 + 0.0830111i \(0.0264537\pi\)
−0.996549 + 0.0830111i \(0.973546\pi\)
\(908\) 25.0000i 0.829654i
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 0 0
\(914\) −41.0000 −1.35616
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) 20.0000i 0.660458i
\(918\) − 35.0000i − 1.15517i
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 10.0000i 0.329332i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) − 5.00000i − 0.164133i
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 26.0000i 0.851658i
\(933\) 33.0000i 1.08037i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) − 13.0000i − 0.424691i −0.977195 0.212346i \(-0.931890\pi\)
0.977195 0.212346i \(-0.0681103\pi\)
\(938\) − 7.00000i − 0.228558i
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 18.0000i 0.586472i
\(943\) 10.0000i 0.325645i
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) 0 0
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) − 10.0000i − 0.324785i
\(949\) −27.0000 −0.876457
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) − 7.00000i − 0.226871i
\(953\) − 44.0000i − 1.42530i −0.701520 0.712650i \(-0.747495\pi\)
0.701520 0.712650i \(-0.252505\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) −27.0000 −0.873242
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 6.00000i 0.193448i
\(963\) 18.0000i 0.580042i
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) 5.00000 0.160872
\(967\) 12.0000i 0.385894i 0.981209 + 0.192947i \(0.0618045\pi\)
−0.981209 + 0.192947i \(0.938195\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 7.00000 0.224872
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) − 16.0000i − 0.513200i
\(973\) 12.0000i 0.384702i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 18.0000i 0.575577i
\(979\) 0 0
\(980\) 0 0
\(981\) −26.0000 −0.830116
\(982\) − 14.0000i − 0.446758i
\(983\) 18.0000i 0.574111i 0.957914 + 0.287055i \(0.0926764\pi\)
−0.957914 + 0.287055i \(0.907324\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −35.0000 −1.11463
\(987\) 0 0
\(988\) − 3.00000i − 0.0954427i
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −46.0000 −1.46124 −0.730619 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) − 10.0000i − 0.317500i
\(993\) − 17.0000i − 0.539479i
\(994\) 0 0
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 42.0000i 1.32949i
\(999\) −10.0000 −0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 950.2.b.d.799.1 2
5.2 odd 4 950.2.a.e.1.1 1
5.3 odd 4 190.2.a.a.1.1 1
5.4 even 2 inner 950.2.b.d.799.2 2
15.2 even 4 8550.2.a.l.1.1 1
15.8 even 4 1710.2.a.r.1.1 1
20.3 even 4 1520.2.a.g.1.1 1
20.7 even 4 7600.2.a.g.1.1 1
35.13 even 4 9310.2.a.i.1.1 1
40.3 even 4 6080.2.a.i.1.1 1
40.13 odd 4 6080.2.a.r.1.1 1
95.18 even 4 3610.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.a.1.1 1 5.3 odd 4
950.2.a.e.1.1 1 5.2 odd 4
950.2.b.d.799.1 2 1.1 even 1 trivial
950.2.b.d.799.2 2 5.4 even 2 inner
1520.2.a.g.1.1 1 20.3 even 4
1710.2.a.r.1.1 1 15.8 even 4
3610.2.a.h.1.1 1 95.18 even 4
6080.2.a.i.1.1 1 40.3 even 4
6080.2.a.r.1.1 1 40.13 odd 4
7600.2.a.g.1.1 1 20.7 even 4
8550.2.a.l.1.1 1 15.2 even 4
9310.2.a.i.1.1 1 35.13 even 4