Properties

Label 1950.2.f.m.649.1
Level $1950$
Weight $2$
Character 1950.649
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1950,2,Mod(649,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.f (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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1950.649
Dual form 1950.2.f.m.649.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} -2.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000i q^{12} -3.60555i q^{13} +2.60555 q^{14} +1.00000 q^{16} +2.60555i q^{17} +1.00000 q^{18} +2.60555i q^{19} +2.60555i q^{21} -8.60555i q^{23} +1.00000i q^{24} +3.60555i q^{26} +1.00000i q^{27} -2.60555 q^{28} +2.60555 q^{29} -6.00000i q^{31} -1.00000 q^{32} -2.60555i q^{34} -1.00000 q^{36} -5.21110 q^{37} -2.60555i q^{38} -3.60555 q^{39} +11.2111i q^{41} -2.60555i q^{42} +8.00000i q^{43} +8.60555i q^{46} -5.21110 q^{47} -1.00000i q^{48} -0.211103 q^{49} +2.60555 q^{51} -3.60555i q^{52} +6.00000i q^{53} -1.00000i q^{54} +2.60555 q^{56} +2.60555 q^{57} -2.60555 q^{58} +5.21110i q^{59} +3.21110 q^{61} +6.00000i q^{62} +2.60555 q^{63} +1.00000 q^{64} -11.2111 q^{67} +2.60555i q^{68} -8.60555 q^{69} +5.21110i q^{71} +1.00000 q^{72} -8.60555 q^{73} +5.21110 q^{74} +2.60555i q^{76} +3.60555 q^{78} -14.4222 q^{79} +1.00000 q^{81} -11.2111i q^{82} +17.2111 q^{83} +2.60555i q^{84} -8.00000i q^{86} -2.60555i q^{87} +0.788897i q^{89} +9.39445i q^{91} -8.60555i q^{92} -6.00000 q^{93} +5.21110 q^{94} +1.00000i q^{96} -8.60555 q^{97} +0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{14} + 4 q^{16} + 4 q^{18} + 4 q^{28} - 4 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{37} + 8 q^{47} + 28 q^{49} - 4 q^{51} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 16 q^{61} - 4 q^{63} + 4 q^{64} - 16 q^{67} - 20 q^{69} + 4 q^{72} - 20 q^{73} - 8 q^{74} + 4 q^{81} + 40 q^{83} - 24 q^{93} - 8 q^{94} - 20 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000i 0.408248i
\(7\) −2.60555 −0.984806 −0.492403 0.870367i \(-0.663881\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 3.60555i − 1.00000i
\(14\) 2.60555 0.696363
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.60555i 0.631939i 0.948769 + 0.315970i \(0.102330\pi\)
−0.948769 + 0.315970i \(0.897670\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.60555i 0.597754i 0.954292 + 0.298877i \(0.0966121\pi\)
−0.954292 + 0.298877i \(0.903388\pi\)
\(20\) 0 0
\(21\) 2.60555i 0.568578i
\(22\) 0 0
\(23\) − 8.60555i − 1.79438i −0.441643 0.897191i \(-0.645604\pi\)
0.441643 0.897191i \(-0.354396\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 0 0
\(26\) 3.60555i 0.707107i
\(27\) 1.00000i 0.192450i
\(28\) −2.60555 −0.492403
\(29\) 2.60555 0.483839 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) − 6.00000i − 1.07763i −0.842424 0.538816i \(-0.818872\pi\)
0.842424 0.538816i \(-0.181128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) − 2.60555i − 0.446848i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −5.21110 −0.856700 −0.428350 0.903613i \(-0.640905\pi\)
−0.428350 + 0.903613i \(0.640905\pi\)
\(38\) − 2.60555i − 0.422676i
\(39\) −3.60555 −0.577350
\(40\) 0 0
\(41\) 11.2111i 1.75088i 0.483327 + 0.875440i \(0.339428\pi\)
−0.483327 + 0.875440i \(0.660572\pi\)
\(42\) − 2.60555i − 0.402045i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.60555i 1.26882i
\(47\) −5.21110 −0.760117 −0.380059 0.924962i \(-0.624096\pi\)
−0.380059 + 0.924962i \(0.624096\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) 2.60555 0.364850
\(52\) − 3.60555i − 0.500000i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 0 0
\(56\) 2.60555 0.348181
\(57\) 2.60555 0.345114
\(58\) −2.60555 −0.342126
\(59\) 5.21110i 0.678428i 0.940709 + 0.339214i \(0.110161\pi\)
−0.940709 + 0.339214i \(0.889839\pi\)
\(60\) 0 0
\(61\) 3.21110 0.411140 0.205570 0.978642i \(-0.434095\pi\)
0.205570 + 0.978642i \(0.434095\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 2.60555 0.328269
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −11.2111 −1.36965 −0.684827 0.728706i \(-0.740122\pi\)
−0.684827 + 0.728706i \(0.740122\pi\)
\(68\) 2.60555i 0.315970i
\(69\) −8.60555 −1.03599
\(70\) 0 0
\(71\) 5.21110i 0.618444i 0.950990 + 0.309222i \(0.100069\pi\)
−0.950990 + 0.309222i \(0.899931\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.60555 −1.00720 −0.503602 0.863936i \(-0.667992\pi\)
−0.503602 + 0.863936i \(0.667992\pi\)
\(74\) 5.21110 0.605778
\(75\) 0 0
\(76\) 2.60555i 0.298877i
\(77\) 0 0
\(78\) 3.60555 0.408248
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 11.2111i − 1.23806i
\(83\) 17.2111 1.88916 0.944582 0.328276i \(-0.106467\pi\)
0.944582 + 0.328276i \(0.106467\pi\)
\(84\) 2.60555i 0.284289i
\(85\) 0 0
\(86\) − 8.00000i − 0.862662i
\(87\) − 2.60555i − 0.279344i
\(88\) 0 0
\(89\) 0.788897i 0.0836230i 0.999126 + 0.0418115i \(0.0133129\pi\)
−0.999126 + 0.0418115i \(0.986687\pi\)
\(90\) 0 0
\(91\) 9.39445i 0.984806i
\(92\) − 8.60555i − 0.897191i
\(93\) −6.00000 −0.622171
\(94\) 5.21110 0.537484
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) −8.60555 −0.873761 −0.436881 0.899519i \(-0.643917\pi\)
−0.436881 + 0.899519i \(0.643917\pi\)
\(98\) 0.211103 0.0213246
\(99\) 0 0
\(100\) 0 0
\(101\) −14.6056 −1.45331 −0.726653 0.687004i \(-0.758925\pi\)
−0.726653 + 0.687004i \(0.758925\pi\)
\(102\) −2.60555 −0.257988
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 3.60555i 0.353553i
\(105\) 0 0
\(106\) − 6.00000i − 0.582772i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 8.60555i 0.824262i 0.911125 + 0.412131i \(0.135215\pi\)
−0.911125 + 0.412131i \(0.864785\pi\)
\(110\) 0 0
\(111\) 5.21110i 0.494616i
\(112\) −2.60555 −0.246201
\(113\) 7.81665i 0.735329i 0.929959 + 0.367664i \(0.119843\pi\)
−0.929959 + 0.367664i \(0.880157\pi\)
\(114\) −2.60555 −0.244032
\(115\) 0 0
\(116\) 2.60555 0.241919
\(117\) 3.60555i 0.333333i
\(118\) − 5.21110i − 0.479721i
\(119\) − 6.78890i − 0.622337i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) −3.21110 −0.290720
\(123\) 11.2111 1.01087
\(124\) − 6.00000i − 0.538816i
\(125\) 0 0
\(126\) −2.60555 −0.232121
\(127\) 13.2111i 1.17230i 0.810204 + 0.586148i \(0.199356\pi\)
−0.810204 + 0.586148i \(0.800644\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 15.3944 1.34502 0.672510 0.740088i \(-0.265216\pi\)
0.672510 + 0.740088i \(0.265216\pi\)
\(132\) 0 0
\(133\) − 6.78890i − 0.588672i
\(134\) 11.2111 0.968492
\(135\) 0 0
\(136\) − 2.60555i − 0.223424i
\(137\) 11.2111 0.957829 0.478915 0.877862i \(-0.341030\pi\)
0.478915 + 0.877862i \(0.341030\pi\)
\(138\) 8.60555 0.732553
\(139\) −2.78890 −0.236551 −0.118276 0.992981i \(-0.537737\pi\)
−0.118276 + 0.992981i \(0.537737\pi\)
\(140\) 0 0
\(141\) 5.21110i 0.438854i
\(142\) − 5.21110i − 0.437306i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 8.60555 0.712200
\(147\) 0.211103i 0.0174114i
\(148\) −5.21110 −0.428350
\(149\) 0.788897i 0.0646290i 0.999478 + 0.0323145i \(0.0102878\pi\)
−0.999478 + 0.0323145i \(0.989712\pi\)
\(150\) 0 0
\(151\) − 6.00000i − 0.488273i −0.969741 0.244137i \(-0.921495\pi\)
0.969741 0.244137i \(-0.0785045\pi\)
\(152\) − 2.60555i − 0.211338i
\(153\) − 2.60555i − 0.210646i
\(154\) 0 0
\(155\) 0 0
\(156\) −3.60555 −0.288675
\(157\) 8.42221i 0.672165i 0.941833 + 0.336083i \(0.109102\pi\)
−0.941833 + 0.336083i \(0.890898\pi\)
\(158\) 14.4222 1.14737
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 22.4222i 1.76712i
\(162\) −1.00000 −0.0785674
\(163\) −4.42221 −0.346374 −0.173187 0.984889i \(-0.555406\pi\)
−0.173187 + 0.984889i \(0.555406\pi\)
\(164\) 11.2111i 0.875440i
\(165\) 0 0
\(166\) −17.2111 −1.33584
\(167\) 5.21110 0.403247 0.201624 0.979463i \(-0.435378\pi\)
0.201624 + 0.979463i \(0.435378\pi\)
\(168\) − 2.60555i − 0.201023i
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) − 2.60555i − 0.199251i
\(172\) 8.00000i 0.609994i
\(173\) 16.4222i 1.24856i 0.781202 + 0.624279i \(0.214607\pi\)
−0.781202 + 0.624279i \(0.785393\pi\)
\(174\) 2.60555i 0.197526i
\(175\) 0 0
\(176\) 0 0
\(177\) 5.21110 0.391690
\(178\) − 0.788897i − 0.0591304i
\(179\) −1.81665 −0.135783 −0.0678915 0.997693i \(-0.521627\pi\)
−0.0678915 + 0.997693i \(0.521627\pi\)
\(180\) 0 0
\(181\) −20.4222 −1.51797 −0.758985 0.651108i \(-0.774305\pi\)
−0.758985 + 0.651108i \(0.774305\pi\)
\(182\) − 9.39445i − 0.696363i
\(183\) − 3.21110i − 0.237372i
\(184\) 8.60555i 0.634410i
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) −5.21110 −0.380059
\(189\) − 2.60555i − 0.189526i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) −13.8167 −0.994545 −0.497272 0.867595i \(-0.665665\pi\)
−0.497272 + 0.867595i \(0.665665\pi\)
\(194\) 8.60555 0.617843
\(195\) 0 0
\(196\) −0.211103 −0.0150788
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 6.42221 0.455258 0.227629 0.973748i \(-0.426903\pi\)
0.227629 + 0.973748i \(0.426903\pi\)
\(200\) 0 0
\(201\) 11.2111i 0.790770i
\(202\) 14.6056 1.02764
\(203\) −6.78890 −0.476487
\(204\) 2.60555 0.182425
\(205\) 0 0
\(206\) 4.00000i 0.278693i
\(207\) 8.60555i 0.598127i
\(208\) − 3.60555i − 0.250000i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.78890 −0.191996 −0.0959978 0.995382i \(-0.530604\pi\)
−0.0959978 + 0.995382i \(0.530604\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 5.21110 0.357059
\(214\) 0 0
\(215\) 0 0
\(216\) − 1.00000i − 0.0680414i
\(217\) 15.6333i 1.06126i
\(218\) − 8.60555i − 0.582841i
\(219\) 8.60555i 0.581509i
\(220\) 0 0
\(221\) 9.39445 0.631939
\(222\) − 5.21110i − 0.349746i
\(223\) −19.8167 −1.32702 −0.663511 0.748167i \(-0.730934\pi\)
−0.663511 + 0.748167i \(0.730934\pi\)
\(224\) 2.60555 0.174091
\(225\) 0 0
\(226\) − 7.81665i − 0.519956i
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 2.60555 0.172557
\(229\) 1.81665i 0.120048i 0.998197 + 0.0600239i \(0.0191177\pi\)
−0.998197 + 0.0600239i \(0.980882\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.60555 −0.171063
\(233\) 19.8167i 1.29823i 0.760689 + 0.649116i \(0.224861\pi\)
−0.760689 + 0.649116i \(0.775139\pi\)
\(234\) − 3.60555i − 0.235702i
\(235\) 0 0
\(236\) 5.21110i 0.339214i
\(237\) 14.4222i 0.936823i
\(238\) 6.78890i 0.440059i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) − 22.4222i − 1.44434i −0.691715 0.722171i \(-0.743145\pi\)
0.691715 0.722171i \(-0.256855\pi\)
\(242\) −11.0000 −0.707107
\(243\) − 1.00000i − 0.0641500i
\(244\) 3.21110 0.205570
\(245\) 0 0
\(246\) −11.2111 −0.714794
\(247\) 9.39445 0.597754
\(248\) 6.00000i 0.381000i
\(249\) − 17.2111i − 1.09071i
\(250\) 0 0
\(251\) −20.6056 −1.30061 −0.650305 0.759673i \(-0.725359\pi\)
−0.650305 + 0.759673i \(0.725359\pi\)
\(252\) 2.60555 0.164134
\(253\) 0 0
\(254\) − 13.2111i − 0.828938i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.3944i 1.33455i 0.744812 + 0.667275i \(0.232539\pi\)
−0.744812 + 0.667275i \(0.767461\pi\)
\(258\) −8.00000 −0.498058
\(259\) 13.5778 0.843683
\(260\) 0 0
\(261\) −2.60555 −0.161280
\(262\) −15.3944 −0.951072
\(263\) 13.8167i 0.851971i 0.904730 + 0.425986i \(0.140073\pi\)
−0.904730 + 0.425986i \(0.859927\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.78890i 0.416254i
\(267\) 0.788897 0.0482797
\(268\) −11.2111 −0.684827
\(269\) −4.18335 −0.255063 −0.127532 0.991835i \(-0.540705\pi\)
−0.127532 + 0.991835i \(0.540705\pi\)
\(270\) 0 0
\(271\) − 28.4222i − 1.72653i −0.504754 0.863263i \(-0.668417\pi\)
0.504754 0.863263i \(-0.331583\pi\)
\(272\) 2.60555i 0.157985i
\(273\) 9.39445 0.568578
\(274\) −11.2111 −0.677287
\(275\) 0 0
\(276\) −8.60555 −0.517993
\(277\) − 12.4222i − 0.746378i −0.927755 0.373189i \(-0.878264\pi\)
0.927755 0.373189i \(-0.121736\pi\)
\(278\) 2.78890 0.167267
\(279\) 6.00000i 0.359211i
\(280\) 0 0
\(281\) − 12.7889i − 0.762922i −0.924385 0.381461i \(-0.875421\pi\)
0.924385 0.381461i \(-0.124579\pi\)
\(282\) − 5.21110i − 0.310317i
\(283\) − 18.4222i − 1.09509i −0.836777 0.547543i \(-0.815563\pi\)
0.836777 0.547543i \(-0.184437\pi\)
\(284\) 5.21110i 0.309222i
\(285\) 0 0
\(286\) 0 0
\(287\) − 29.2111i − 1.72428i
\(288\) 1.00000 0.0589256
\(289\) 10.2111 0.600653
\(290\) 0 0
\(291\) 8.60555i 0.504466i
\(292\) −8.60555 −0.503602
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) − 0.211103i − 0.0123118i
\(295\) 0 0
\(296\) 5.21110 0.302889
\(297\) 0 0
\(298\) − 0.788897i − 0.0456996i
\(299\) −31.0278 −1.79438
\(300\) 0 0
\(301\) − 20.8444i − 1.20145i
\(302\) 6.00000i 0.345261i
\(303\) 14.6056i 0.839067i
\(304\) 2.60555i 0.149439i
\(305\) 0 0
\(306\) 2.60555i 0.148949i
\(307\) 23.2111 1.32473 0.662364 0.749182i \(-0.269553\pi\)
0.662364 + 0.749182i \(0.269553\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 3.60555 0.204124
\(313\) 32.4222i 1.83261i 0.400480 + 0.916306i \(0.368844\pi\)
−0.400480 + 0.916306i \(0.631156\pi\)
\(314\) − 8.42221i − 0.475293i
\(315\) 0 0
\(316\) −14.4222 −0.811312
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) − 22.4222i − 1.24954i
\(323\) −6.78890 −0.377744
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.42221 0.244923
\(327\) 8.60555 0.475888
\(328\) − 11.2111i − 0.619030i
\(329\) 13.5778 0.748568
\(330\) 0 0
\(331\) 9.39445i 0.516366i 0.966096 + 0.258183i \(0.0831237\pi\)
−0.966096 + 0.258183i \(0.916876\pi\)
\(332\) 17.2111 0.944582
\(333\) 5.21110 0.285567
\(334\) −5.21110 −0.285139
\(335\) 0 0
\(336\) 2.60555i 0.142144i
\(337\) 29.6333i 1.61423i 0.590395 + 0.807115i \(0.298972\pi\)
−0.590395 + 0.807115i \(0.701028\pi\)
\(338\) 13.0000 0.707107
\(339\) 7.81665 0.424542
\(340\) 0 0
\(341\) 0 0
\(342\) 2.60555i 0.140892i
\(343\) 18.7889 1.01451
\(344\) − 8.00000i − 0.431331i
\(345\) 0 0
\(346\) − 16.4222i − 0.882863i
\(347\) − 15.6333i − 0.839240i −0.907700 0.419620i \(-0.862163\pi\)
0.907700 0.419620i \(-0.137837\pi\)
\(348\) − 2.60555i − 0.139672i
\(349\) 13.8167i 0.739589i 0.929114 + 0.369794i \(0.120572\pi\)
−0.929114 + 0.369794i \(0.879428\pi\)
\(350\) 0 0
\(351\) 3.60555 0.192450
\(352\) 0 0
\(353\) −23.2111 −1.23540 −0.617701 0.786413i \(-0.711936\pi\)
−0.617701 + 0.786413i \(0.711936\pi\)
\(354\) −5.21110 −0.276967
\(355\) 0 0
\(356\) 0.788897i 0.0418115i
\(357\) −6.78890 −0.359307
\(358\) 1.81665 0.0960131
\(359\) − 27.6333i − 1.45843i −0.684285 0.729215i \(-0.739885\pi\)
0.684285 0.729215i \(-0.260115\pi\)
\(360\) 0 0
\(361\) 12.2111 0.642690
\(362\) 20.4222 1.07337
\(363\) − 11.0000i − 0.577350i
\(364\) 9.39445i 0.492403i
\(365\) 0 0
\(366\) 3.21110i 0.167847i
\(367\) − 23.6333i − 1.23365i −0.787101 0.616824i \(-0.788419\pi\)
0.787101 0.616824i \(-0.211581\pi\)
\(368\) − 8.60555i − 0.448595i
\(369\) − 11.2111i − 0.583627i
\(370\) 0 0
\(371\) − 15.6333i − 0.811641i
\(372\) −6.00000 −0.311086
\(373\) 8.42221i 0.436085i 0.975939 + 0.218043i \(0.0699672\pi\)
−0.975939 + 0.218043i \(0.930033\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 5.21110 0.268742
\(377\) − 9.39445i − 0.483839i
\(378\) 2.60555i 0.134015i
\(379\) − 1.02776i − 0.0527923i −0.999652 0.0263961i \(-0.991597\pi\)
0.999652 0.0263961i \(-0.00840313\pi\)
\(380\) 0 0
\(381\) 13.2111 0.676825
\(382\) 12.0000 0.613973
\(383\) 15.6333 0.798825 0.399412 0.916771i \(-0.369214\pi\)
0.399412 + 0.916771i \(0.369214\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) 13.8167 0.703249
\(387\) − 8.00000i − 0.406663i
\(388\) −8.60555 −0.436881
\(389\) 2.60555 0.132107 0.0660533 0.997816i \(-0.478959\pi\)
0.0660533 + 0.997816i \(0.478959\pi\)
\(390\) 0 0
\(391\) 22.4222 1.13394
\(392\) 0.211103 0.0106623
\(393\) − 15.3944i − 0.776547i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −39.6333 −1.98914 −0.994569 0.104076i \(-0.966811\pi\)
−0.994569 + 0.104076i \(0.966811\pi\)
\(398\) −6.42221 −0.321916
\(399\) −6.78890 −0.339870
\(400\) 0 0
\(401\) − 23.2111i − 1.15911i −0.814934 0.579554i \(-0.803227\pi\)
0.814934 0.579554i \(-0.196773\pi\)
\(402\) − 11.2111i − 0.559159i
\(403\) −21.6333 −1.07763
\(404\) −14.6056 −0.726653
\(405\) 0 0
\(406\) 6.78890 0.336927
\(407\) 0 0
\(408\) −2.60555 −0.128994
\(409\) − 29.2111i − 1.44440i −0.691686 0.722198i \(-0.743132\pi\)
0.691686 0.722198i \(-0.256868\pi\)
\(410\) 0 0
\(411\) − 11.2111i − 0.553003i
\(412\) − 4.00000i − 0.197066i
\(413\) − 13.5778i − 0.668120i
\(414\) − 8.60555i − 0.422940i
\(415\) 0 0
\(416\) 3.60555i 0.176777i
\(417\) 2.78890i 0.136573i
\(418\) 0 0
\(419\) −25.8167 −1.26123 −0.630613 0.776097i \(-0.717196\pi\)
−0.630613 + 0.776097i \(0.717196\pi\)
\(420\) 0 0
\(421\) − 1.81665i − 0.0885383i −0.999020 0.0442691i \(-0.985904\pi\)
0.999020 0.0442691i \(-0.0140959\pi\)
\(422\) 2.78890 0.135761
\(423\) 5.21110 0.253372
\(424\) − 6.00000i − 0.291386i
\(425\) 0 0
\(426\) −5.21110 −0.252479
\(427\) −8.36669 −0.404893
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 12.0000i − 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 4.78890i 0.230140i 0.993357 + 0.115070i \(0.0367092\pi\)
−0.993357 + 0.115070i \(0.963291\pi\)
\(434\) − 15.6333i − 0.750423i
\(435\) 0 0
\(436\) 8.60555i 0.412131i
\(437\) 22.4222 1.07260
\(438\) − 8.60555i − 0.411189i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0.211103 0.0100525
\(442\) −9.39445 −0.446848
\(443\) − 27.6333i − 1.31290i −0.754370 0.656449i \(-0.772058\pi\)
0.754370 0.656449i \(-0.227942\pi\)
\(444\) 5.21110i 0.247308i
\(445\) 0 0
\(446\) 19.8167 0.938346
\(447\) 0.788897 0.0373136
\(448\) −2.60555 −0.123101
\(449\) − 9.63331i − 0.454624i −0.973822 0.227312i \(-0.927006\pi\)
0.973822 0.227312i \(-0.0729937\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.81665i 0.367664i
\(453\) −6.00000 −0.281905
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −2.60555 −0.122016
\(457\) 12.2389 0.572510 0.286255 0.958154i \(-0.407590\pi\)
0.286255 + 0.958154i \(0.407590\pi\)
\(458\) − 1.81665i − 0.0848867i
\(459\) −2.60555 −0.121617
\(460\) 0 0
\(461\) − 9.63331i − 0.448668i −0.974512 0.224334i \(-0.927979\pi\)
0.974512 0.224334i \(-0.0720206\pi\)
\(462\) 0 0
\(463\) −38.6056 −1.79415 −0.897076 0.441876i \(-0.854313\pi\)
−0.897076 + 0.441876i \(0.854313\pi\)
\(464\) 2.60555 0.120960
\(465\) 0 0
\(466\) − 19.8167i − 0.917989i
\(467\) 1.57779i 0.0730116i 0.999333 + 0.0365058i \(0.0116227\pi\)
−0.999333 + 0.0365058i \(0.988377\pi\)
\(468\) 3.60555i 0.166667i
\(469\) 29.2111 1.34884
\(470\) 0 0
\(471\) 8.42221 0.388075
\(472\) − 5.21110i − 0.239860i
\(473\) 0 0
\(474\) − 14.4222i − 0.662434i
\(475\) 0 0
\(476\) − 6.78890i − 0.311169i
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 34.4222i 1.57279i 0.617724 + 0.786395i \(0.288055\pi\)
−0.617724 + 0.786395i \(0.711945\pi\)
\(480\) 0 0
\(481\) 18.7889i 0.856700i
\(482\) 22.4222i 1.02130i
\(483\) 22.4222 1.02025
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) 37.0278 1.67789 0.838944 0.544218i \(-0.183174\pi\)
0.838944 + 0.544218i \(0.183174\pi\)
\(488\) −3.21110 −0.145360
\(489\) 4.42221i 0.199979i
\(490\) 0 0
\(491\) −13.8167 −0.623537 −0.311768 0.950158i \(-0.600921\pi\)
−0.311768 + 0.950158i \(0.600921\pi\)
\(492\) 11.2111 0.505436
\(493\) 6.78890i 0.305757i
\(494\) −9.39445 −0.422676
\(495\) 0 0
\(496\) − 6.00000i − 0.269408i
\(497\) − 13.5778i − 0.609047i
\(498\) 17.2111i 0.771248i
\(499\) − 13.0278i − 0.583202i −0.956540 0.291601i \(-0.905812\pi\)
0.956540 0.291601i \(-0.0941880\pi\)
\(500\) 0 0
\(501\) − 5.21110i − 0.232815i
\(502\) 20.6056 0.919671
\(503\) − 41.4500i − 1.84816i −0.382196 0.924081i \(-0.624832\pi\)
0.382196 0.924081i \(-0.375168\pi\)
\(504\) −2.60555 −0.116060
\(505\) 0 0
\(506\) 0 0
\(507\) 13.0000i 0.577350i
\(508\) 13.2111i 0.586148i
\(509\) 9.63331i 0.426989i 0.976944 + 0.213494i \(0.0684845\pi\)
−0.976944 + 0.213494i \(0.931515\pi\)
\(510\) 0 0
\(511\) 22.4222 0.991900
\(512\) −1.00000 −0.0441942
\(513\) −2.60555 −0.115038
\(514\) − 21.3944i − 0.943669i
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −13.5778 −0.596574
\(519\) 16.4222 0.720855
\(520\) 0 0
\(521\) −21.6333 −0.947772 −0.473886 0.880586i \(-0.657149\pi\)
−0.473886 + 0.880586i \(0.657149\pi\)
\(522\) 2.60555 0.114042
\(523\) − 24.8444i − 1.08637i −0.839613 0.543185i \(-0.817218\pi\)
0.839613 0.543185i \(-0.182782\pi\)
\(524\) 15.3944 0.672510
\(525\) 0 0
\(526\) − 13.8167i − 0.602435i
\(527\) 15.6333 0.680998
\(528\) 0 0
\(529\) −51.0555 −2.21980
\(530\) 0 0
\(531\) − 5.21110i − 0.226143i
\(532\) − 6.78890i − 0.294336i
\(533\) 40.4222 1.75088
\(534\) −0.788897 −0.0341389
\(535\) 0 0
\(536\) 11.2111 0.484246
\(537\) 1.81665i 0.0783944i
\(538\) 4.18335 0.180357
\(539\) 0 0
\(540\) 0 0
\(541\) − 43.0278i − 1.84991i −0.380079 0.924954i \(-0.624103\pi\)
0.380079 0.924954i \(-0.375897\pi\)
\(542\) 28.4222i 1.22084i
\(543\) 20.4222i 0.876401i
\(544\) − 2.60555i − 0.111712i
\(545\) 0 0
\(546\) −9.39445 −0.402045
\(547\) 14.4222i 0.616649i 0.951281 + 0.308324i \(0.0997682\pi\)
−0.951281 + 0.308324i \(0.900232\pi\)
\(548\) 11.2111 0.478915
\(549\) −3.21110 −0.137047
\(550\) 0 0
\(551\) 6.78890i 0.289217i
\(552\) 8.60555 0.366277
\(553\) 37.5778 1.59797
\(554\) 12.4222i 0.527769i
\(555\) 0 0
\(556\) −2.78890 −0.118276
\(557\) −40.4222 −1.71274 −0.856372 0.516360i \(-0.827287\pi\)
−0.856372 + 0.516360i \(0.827287\pi\)
\(558\) − 6.00000i − 0.254000i
\(559\) 28.8444 1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) 12.7889i 0.539467i
\(563\) − 38.0555i − 1.60385i −0.597426 0.801924i \(-0.703810\pi\)
0.597426 0.801924i \(-0.296190\pi\)
\(564\) 5.21110i 0.219427i
\(565\) 0 0
\(566\) 18.4222i 0.774343i
\(567\) −2.60555 −0.109423
\(568\) − 5.21110i − 0.218653i
\(569\) 9.63331 0.403849 0.201925 0.979401i \(-0.435280\pi\)
0.201925 + 0.979401i \(0.435280\pi\)
\(570\) 0 0
\(571\) 42.0555 1.75997 0.879984 0.475003i \(-0.157553\pi\)
0.879984 + 0.475003i \(0.157553\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 29.2111i 1.21925i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 44.6056 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(578\) −10.2111 −0.424726
\(579\) 13.8167i 0.574201i
\(580\) 0 0
\(581\) −44.8444 −1.86046
\(582\) − 8.60555i − 0.356712i
\(583\) 0 0
\(584\) 8.60555 0.356100
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −22.4222 −0.925463 −0.462732 0.886498i \(-0.653131\pi\)
−0.462732 + 0.886498i \(0.653131\pi\)
\(588\) 0.211103i 0.00870572i
\(589\) 15.6333 0.644159
\(590\) 0 0
\(591\) − 6.00000i − 0.246807i
\(592\) −5.21110 −0.214175
\(593\) 4.42221 0.181598 0.0907991 0.995869i \(-0.471058\pi\)
0.0907991 + 0.995869i \(0.471058\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.788897i 0.0323145i
\(597\) − 6.42221i − 0.262843i
\(598\) 31.0278 1.26882
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −41.6333 −1.69826 −0.849129 0.528185i \(-0.822873\pi\)
−0.849129 + 0.528185i \(0.822873\pi\)
\(602\) 20.8444i 0.849555i
\(603\) 11.2111 0.456551
\(604\) − 6.00000i − 0.244137i
\(605\) 0 0
\(606\) − 14.6056i − 0.593310i
\(607\) − 2.78890i − 0.113198i −0.998397 0.0565989i \(-0.981974\pi\)
0.998397 0.0565989i \(-0.0180256\pi\)
\(608\) − 2.60555i − 0.105669i
\(609\) 6.78890i 0.275100i
\(610\) 0 0
\(611\) 18.7889i 0.760117i
\(612\) − 2.60555i − 0.105323i
\(613\) −18.7889 −0.758876 −0.379438 0.925217i \(-0.623883\pi\)
−0.379438 + 0.925217i \(0.623883\pi\)
\(614\) −23.2111 −0.936724
\(615\) 0 0
\(616\) 0 0
\(617\) 16.4222 0.661133 0.330567 0.943783i \(-0.392760\pi\)
0.330567 + 0.943783i \(0.392760\pi\)
\(618\) 4.00000 0.160904
\(619\) 4.18335i 0.168143i 0.996460 + 0.0840714i \(0.0267924\pi\)
−0.996460 + 0.0840714i \(0.973208\pi\)
\(620\) 0 0
\(621\) 8.60555 0.345329
\(622\) −12.0000 −0.481156
\(623\) − 2.05551i − 0.0823524i
\(624\) −3.60555 −0.144338
\(625\) 0 0
\(626\) − 32.4222i − 1.29585i
\(627\) 0 0
\(628\) 8.42221i 0.336083i
\(629\) − 13.5778i − 0.541382i
\(630\) 0 0
\(631\) 11.2111i 0.446307i 0.974783 + 0.223153i \(0.0716351\pi\)
−0.974783 + 0.223153i \(0.928365\pi\)
\(632\) 14.4222 0.573685
\(633\) 2.78890i 0.110849i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0.761141i 0.0301575i
\(638\) 0 0
\(639\) − 5.21110i − 0.206148i
\(640\) 0 0
\(641\) −28.4222 −1.12261 −0.561305 0.827609i \(-0.689700\pi\)
−0.561305 + 0.827609i \(0.689700\pi\)
\(642\) 0 0
\(643\) −33.6333 −1.32637 −0.663184 0.748456i \(-0.730795\pi\)
−0.663184 + 0.748456i \(0.730795\pi\)
\(644\) 22.4222i 0.883559i
\(645\) 0 0
\(646\) 6.78890 0.267106
\(647\) 27.3944i 1.07699i 0.842630 + 0.538493i \(0.181006\pi\)
−0.842630 + 0.538493i \(0.818994\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 15.6333 0.612718
\(652\) −4.42221 −0.173187
\(653\) 24.7889i 0.970065i 0.874496 + 0.485032i \(0.161192\pi\)
−0.874496 + 0.485032i \(0.838808\pi\)
\(654\) −8.60555 −0.336504
\(655\) 0 0
\(656\) 11.2111i 0.437720i
\(657\) 8.60555 0.335735
\(658\) −13.5778 −0.529318
\(659\) −24.2389 −0.944212 −0.472106 0.881542i \(-0.656506\pi\)
−0.472106 + 0.881542i \(0.656506\pi\)
\(660\) 0 0
\(661\) 0.238859i 0.00929054i 0.999989 + 0.00464527i \(0.00147864\pi\)
−0.999989 + 0.00464527i \(0.998521\pi\)
\(662\) − 9.39445i − 0.365126i
\(663\) − 9.39445i − 0.364850i
\(664\) −17.2111 −0.667920
\(665\) 0 0
\(666\) −5.21110 −0.201926
\(667\) − 22.4222i − 0.868191i
\(668\) 5.21110 0.201624
\(669\) 19.8167i 0.766156i
\(670\) 0 0
\(671\) 0 0
\(672\) − 2.60555i − 0.100511i
\(673\) 5.63331i 0.217148i 0.994088 + 0.108574i \(0.0346284\pi\)
−0.994088 + 0.108574i \(0.965372\pi\)
\(674\) − 29.6333i − 1.14143i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) − 44.0555i − 1.69319i −0.532237 0.846595i \(-0.678648\pi\)
0.532237 0.846595i \(-0.321352\pi\)
\(678\) −7.81665 −0.300197
\(679\) 22.4222 0.860485
\(680\) 0 0
\(681\) 24.0000i 0.919682i
\(682\) 0 0
\(683\) −5.21110 −0.199397 −0.0996986 0.995018i \(-0.531788\pi\)
−0.0996986 + 0.995018i \(0.531788\pi\)
\(684\) − 2.60555i − 0.0996257i
\(685\) 0 0
\(686\) −18.7889 −0.717363
\(687\) 1.81665 0.0693097
\(688\) 8.00000i 0.304997i
\(689\) 21.6333 0.824163
\(690\) 0 0
\(691\) 30.2389i 1.15034i 0.818034 + 0.575170i \(0.195064\pi\)
−0.818034 + 0.575170i \(0.804936\pi\)
\(692\) 16.4222i 0.624279i
\(693\) 0 0
\(694\) 15.6333i 0.593432i
\(695\) 0 0
\(696\) 2.60555i 0.0987632i
\(697\) −29.2111 −1.10645
\(698\) − 13.8167i − 0.522968i
\(699\) 19.8167 0.749535
\(700\) 0 0
\(701\) 10.9722 0.414416 0.207208 0.978297i \(-0.433562\pi\)
0.207208 + 0.978297i \(0.433562\pi\)
\(702\) −3.60555 −0.136083
\(703\) − 13.5778i − 0.512096i
\(704\) 0 0
\(705\) 0 0
\(706\) 23.2111 0.873561
\(707\) 38.0555 1.43122
\(708\) 5.21110 0.195845
\(709\) − 8.60555i − 0.323188i −0.986857 0.161594i \(-0.948336\pi\)
0.986857 0.161594i \(-0.0516635\pi\)
\(710\) 0 0
\(711\) 14.4222 0.540875
\(712\) − 0.788897i − 0.0295652i
\(713\) −51.6333 −1.93368
\(714\) 6.78890 0.254068
\(715\) 0 0
\(716\) −1.81665 −0.0678915
\(717\) 0 0
\(718\) 27.6333i 1.03127i
\(719\) 8.36669 0.312025 0.156012 0.987755i \(-0.450136\pi\)
0.156012 + 0.987755i \(0.450136\pi\)
\(720\) 0 0
\(721\) 10.4222i 0.388143i
\(722\) −12.2111 −0.454450
\(723\) −22.4222 −0.833891
\(724\) −20.4222 −0.758985
\(725\) 0 0
\(726\) 11.0000i 0.408248i
\(727\) − 14.4222i − 0.534890i −0.963573 0.267445i \(-0.913821\pi\)
0.963573 0.267445i \(-0.0861794\pi\)
\(728\) − 9.39445i − 0.348181i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −20.8444 −0.770958
\(732\) − 3.21110i − 0.118686i
\(733\) 38.0555 1.40561 0.702806 0.711381i \(-0.251930\pi\)
0.702806 + 0.711381i \(0.251930\pi\)
\(734\) 23.6333i 0.872321i
\(735\) 0 0
\(736\) 8.60555i 0.317205i
\(737\) 0 0
\(738\) 11.2111i 0.412686i
\(739\) − 30.2389i − 1.11235i −0.831064 0.556177i \(-0.812268\pi\)
0.831064 0.556177i \(-0.187732\pi\)
\(740\) 0 0
\(741\) − 9.39445i − 0.345114i
\(742\) 15.6333i 0.573917i
\(743\) −20.8444 −0.764707 −0.382354 0.924016i \(-0.624886\pi\)
−0.382354 + 0.924016i \(0.624886\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) − 8.42221i − 0.308359i
\(747\) −17.2111 −0.629721
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.4222 0.672236 0.336118 0.941820i \(-0.390886\pi\)
0.336118 + 0.941820i \(0.390886\pi\)
\(752\) −5.21110 −0.190029
\(753\) 20.6056i 0.750908i
\(754\) 9.39445i 0.342126i
\(755\) 0 0
\(756\) − 2.60555i − 0.0947630i
\(757\) − 27.2111i − 0.989004i −0.869176 0.494502i \(-0.835350\pi\)
0.869176 0.494502i \(-0.164650\pi\)
\(758\) 1.02776i 0.0373298i
\(759\) 0 0
\(760\) 0 0
\(761\) 9.63331i 0.349207i 0.984639 + 0.174604i \(0.0558644\pi\)
−0.984639 + 0.174604i \(0.944136\pi\)
\(762\) −13.2111 −0.478588
\(763\) − 22.4222i − 0.811738i
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −15.6333 −0.564854
\(767\) 18.7889 0.678428
\(768\) − 1.00000i − 0.0360844i
\(769\) 44.8444i 1.61713i 0.588406 + 0.808565i \(0.299756\pi\)
−0.588406 + 0.808565i \(0.700244\pi\)
\(770\) 0 0
\(771\) 21.3944 0.770502
\(772\) −13.8167 −0.497272
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 8.00000i 0.287554i
\(775\) 0 0
\(776\) 8.60555 0.308921
\(777\) − 13.5778i − 0.487101i
\(778\) −2.60555 −0.0934135
\(779\) −29.2111 −1.04660
\(780\) 0 0
\(781\) 0 0
\(782\) −22.4222 −0.801816
\(783\) 2.60555i 0.0931148i
\(784\) −0.211103 −0.00753938
\(785\) 0 0
\(786\) 15.3944i 0.549102i
\(787\) 37.2666 1.32841 0.664206 0.747550i \(-0.268770\pi\)
0.664206 + 0.747550i \(0.268770\pi\)
\(788\) 6.00000 0.213741
\(789\) 13.8167 0.491886
\(790\) 0 0
\(791\) − 20.3667i − 0.724156i
\(792\) 0 0
\(793\) − 11.5778i − 0.411140i
\(794\) 39.6333 1.40653
\(795\) 0 0
\(796\) 6.42221 0.227629
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 6.78890 0.240324
\(799\) − 13.5778i − 0.480348i
\(800\) 0 0
\(801\) − 0.788897i − 0.0278743i
\(802\) 23.2111i 0.819613i
\(803\) 0 0
\(804\) 11.2111i 0.395385i
\(805\) 0 0
\(806\) 21.6333 0.762001
\(807\) 4.18335i 0.147261i
\(808\) 14.6056 0.513822
\(809\) −50.8444 −1.78759 −0.893797 0.448471i \(-0.851969\pi\)
−0.893797 + 0.448471i \(0.851969\pi\)
\(810\) 0 0
\(811\) 18.2389i 0.640453i 0.947341 + 0.320226i \(0.103759\pi\)
−0.947341 + 0.320226i \(0.896241\pi\)
\(812\) −6.78890 −0.238244
\(813\) −28.4222 −0.996810
\(814\) 0 0
\(815\) 0 0
\(816\) 2.60555 0.0912125
\(817\) −20.8444 −0.729254
\(818\) 29.2111i 1.02134i
\(819\) − 9.39445i − 0.328269i
\(820\) 0 0
\(821\) − 11.2111i − 0.391270i −0.980677 0.195635i \(-0.937323\pi\)
0.980677 0.195635i \(-0.0626768\pi\)
\(822\) 11.2111i 0.391032i
\(823\) − 4.00000i − 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) 4.00000i 0.139347i
\(825\) 0 0
\(826\) 13.5778i 0.472432i
\(827\) −15.6333 −0.543623 −0.271812 0.962350i \(-0.587623\pi\)
−0.271812 + 0.962350i \(0.587623\pi\)
\(828\) 8.60555i 0.299064i
\(829\) 10.8444 0.376642 0.188321 0.982108i \(-0.439695\pi\)
0.188321 + 0.982108i \(0.439695\pi\)
\(830\) 0 0
\(831\) −12.4222 −0.430922
\(832\) − 3.60555i − 0.125000i
\(833\) − 0.550039i − 0.0190577i
\(834\) − 2.78890i − 0.0965716i
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 25.8167 0.891822
\(839\) 10.4222i 0.359814i 0.983684 + 0.179907i \(0.0575798\pi\)
−0.983684 + 0.179907i \(0.942420\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 1.81665i 0.0626060i
\(843\) −12.7889 −0.440473
\(844\) −2.78890 −0.0959978
\(845\) 0 0
\(846\) −5.21110 −0.179161
\(847\) −28.6611 −0.984806
\(848\) 6.00000i 0.206041i
\(849\) −18.4222 −0.632248
\(850\) 0 0
\(851\) 44.8444i 1.53725i
\(852\) 5.21110 0.178529
\(853\) −29.2111 −1.00017 −0.500085 0.865977i \(-0.666698\pi\)
−0.500085 + 0.865977i \(0.666698\pi\)
\(854\) 8.36669 0.286302
\(855\) 0 0
\(856\) 0 0
\(857\) − 13.0278i − 0.445020i −0.974930 0.222510i \(-0.928575\pi\)
0.974930 0.222510i \(-0.0714249\pi\)
\(858\) 0 0
\(859\) −10.7889 −0.368112 −0.184056 0.982916i \(-0.558923\pi\)
−0.184056 + 0.982916i \(0.558923\pi\)
\(860\) 0 0
\(861\) −29.2111 −0.995512
\(862\) 12.0000i 0.408722i
\(863\) 8.36669 0.284806 0.142403 0.989809i \(-0.454517\pi\)
0.142403 + 0.989809i \(0.454517\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 0 0
\(866\) − 4.78890i − 0.162733i
\(867\) − 10.2111i − 0.346787i
\(868\) 15.6333i 0.530629i
\(869\) 0 0
\(870\) 0 0
\(871\) 40.4222i 1.36965i
\(872\) − 8.60555i − 0.291421i
\(873\) 8.60555 0.291254
\(874\) −22.4222 −0.758442
\(875\) 0 0
\(876\) 8.60555i 0.290755i
\(877\) −32.8444 −1.10908 −0.554538 0.832158i \(-0.687105\pi\)
−0.554538 + 0.832158i \(0.687105\pi\)
\(878\) 8.00000 0.269987
\(879\) − 18.0000i − 0.607125i
\(880\) 0 0
\(881\) 24.7889 0.835159 0.417580 0.908640i \(-0.362878\pi\)
0.417580 + 0.908640i \(0.362878\pi\)
\(882\) −0.211103 −0.00710819
\(883\) 38.4222i 1.29301i 0.762910 + 0.646505i \(0.223770\pi\)
−0.762910 + 0.646505i \(0.776230\pi\)
\(884\) 9.39445 0.315970
\(885\) 0 0
\(886\) 27.6333i 0.928359i
\(887\) − 43.0278i − 1.44473i −0.691512 0.722365i \(-0.743055\pi\)
0.691512 0.722365i \(-0.256945\pi\)
\(888\) − 5.21110i − 0.174873i
\(889\) − 34.4222i − 1.15448i
\(890\) 0 0
\(891\) 0 0
\(892\) −19.8167 −0.663511
\(893\) − 13.5778i − 0.454364i
\(894\) −0.788897 −0.0263847
\(895\) 0 0
\(896\) 2.60555 0.0870454
\(897\) 31.0278i 1.03599i
\(898\) 9.63331i 0.321468i
\(899\) − 15.6333i − 0.521400i
\(900\) 0 0
\(901\) −15.6333 −0.520821
\(902\) 0 0
\(903\) −20.8444 −0.693659
\(904\) − 7.81665i − 0.259978i
\(905\) 0 0
\(906\) 6.00000 0.199337
\(907\) − 50.4222i − 1.67424i −0.547018 0.837121i \(-0.684237\pi\)
0.547018 0.837121i \(-0.315763\pi\)
\(908\) −24.0000 −0.796468
\(909\) 14.6056 0.484436
\(910\) 0 0
\(911\) 15.6333 0.517955 0.258977 0.965883i \(-0.416615\pi\)
0.258977 + 0.965883i \(0.416615\pi\)
\(912\) 2.60555 0.0862784
\(913\) 0 0
\(914\) −12.2389 −0.404825
\(915\) 0 0
\(916\) 1.81665i 0.0600239i
\(917\) −40.1110 −1.32458
\(918\) 2.60555 0.0859960
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) − 23.2111i − 0.764832i
\(922\) 9.63331i 0.317256i
\(923\) 18.7889 0.618444
\(924\) 0 0
\(925\) 0 0
\(926\) 38.6056 1.26866
\(927\) 4.00000i 0.131377i
\(928\) −2.60555 −0.0855314
\(929\) 24.7889i 0.813297i 0.913585 + 0.406649i \(0.133303\pi\)
−0.913585 + 0.406649i \(0.866697\pi\)
\(930\) 0 0
\(931\) − 0.550039i − 0.0180268i
\(932\) 19.8167i 0.649116i
\(933\) − 12.0000i − 0.392862i
\(934\) − 1.57779i − 0.0516270i
\(935\) 0 0
\(936\) − 3.60555i − 0.117851i
\(937\) 53.6333i 1.75212i 0.482199 + 0.876062i \(0.339838\pi\)
−0.482199 + 0.876062i \(0.660162\pi\)
\(938\) −29.2111 −0.953776
\(939\) 32.4222 1.05806
\(940\) 0 0
\(941\) 54.0000i 1.76035i 0.474650 + 0.880175i \(0.342575\pi\)
−0.474650 + 0.880175i \(0.657425\pi\)
\(942\) −8.42221 −0.274410
\(943\) 96.4777 3.14175
\(944\) 5.21110i 0.169607i
\(945\) 0 0
\(946\) 0 0
\(947\) −27.6333 −0.897962 −0.448981 0.893541i \(-0.648213\pi\)
−0.448981 + 0.893541i \(0.648213\pi\)
\(948\) 14.4222i 0.468411i
\(949\) 31.0278i 1.00720i
\(950\) 0 0
\(951\) − 18.0000i − 0.583690i
\(952\) 6.78890i 0.220029i
\(953\) 30.2389i 0.979533i 0.871854 + 0.489766i \(0.162918\pi\)
−0.871854 + 0.489766i \(0.837082\pi\)
\(954\) 6.00000i 0.194257i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) − 34.4222i − 1.11213i
\(959\) −29.2111 −0.943276
\(960\) 0 0
\(961\) −5.00000 −0.161290
\(962\) − 18.7889i − 0.605778i
\(963\) 0 0
\(964\) − 22.4222i − 0.722171i
\(965\) 0 0
\(966\) −22.4222 −0.721423
\(967\) −42.2389 −1.35831 −0.679155 0.733995i \(-0.737654\pi\)
−0.679155 + 0.733995i \(0.737654\pi\)
\(968\) −11.0000 −0.353553
\(969\) 6.78890i 0.218091i
\(970\) 0 0
\(971\) −16.9722 −0.544665 −0.272333 0.962203i \(-0.587795\pi\)
−0.272333 + 0.962203i \(0.587795\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 7.26662 0.232957
\(974\) −37.0278 −1.18645
\(975\) 0 0
\(976\) 3.21110 0.102785
\(977\) 38.8444 1.24274 0.621371 0.783516i \(-0.286576\pi\)
0.621371 + 0.783516i \(0.286576\pi\)
\(978\) − 4.42221i − 0.141407i
\(979\) 0 0
\(980\) 0 0
\(981\) − 8.60555i − 0.274754i
\(982\) 13.8167 0.440907
\(983\) −13.5778 −0.433064 −0.216532 0.976275i \(-0.569475\pi\)
−0.216532 + 0.976275i \(0.569475\pi\)
\(984\) −11.2111 −0.357397
\(985\) 0 0
\(986\) − 6.78890i − 0.216203i
\(987\) − 13.5778i − 0.432186i
\(988\) 9.39445 0.298877
\(989\) 68.8444 2.18912
\(990\) 0 0
\(991\) 6.42221 0.204008 0.102004 0.994784i \(-0.467475\pi\)
0.102004 + 0.994784i \(0.467475\pi\)
\(992\) 6.00000i 0.190500i
\(993\) 9.39445 0.298124
\(994\) 13.5778i 0.430662i
\(995\) 0 0
\(996\) − 17.2111i − 0.545355i
\(997\) − 12.4222i − 0.393415i −0.980462 0.196708i \(-0.936975\pi\)
0.980462 0.196708i \(-0.0630250\pi\)
\(998\) 13.0278i 0.412386i
\(999\) − 5.21110i − 0.164872i
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 1950.2.f.m.649.1 4
5.2 odd 4 390.2.b.c.181.1 4
5.3 odd 4 1950.2.b.k.1351.4 4
5.4 even 2 1950.2.f.n.649.4 4
13.12 even 2 1950.2.f.n.649.2 4
15.2 even 4 1170.2.b.d.181.3 4
20.7 even 4 3120.2.g.q.961.2 4
65.12 odd 4 390.2.b.c.181.4 yes 4
65.38 odd 4 1950.2.b.k.1351.1 4
65.47 even 4 5070.2.a.bf.1.1 2
65.57 even 4 5070.2.a.z.1.2 2
65.64 even 2 inner 1950.2.f.m.649.3 4
195.77 even 4 1170.2.b.d.181.2 4
260.207 even 4 3120.2.g.q.961.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.1 4 5.2 odd 4
390.2.b.c.181.4 yes 4 65.12 odd 4
1170.2.b.d.181.2 4 195.77 even 4
1170.2.b.d.181.3 4 15.2 even 4
1950.2.b.k.1351.1 4 65.38 odd 4
1950.2.b.k.1351.4 4 5.3 odd 4
1950.2.f.m.649.1 4 1.1 even 1 trivial
1950.2.f.m.649.3 4 65.64 even 2 inner
1950.2.f.n.649.2 4 13.12 even 2
1950.2.f.n.649.4 4 5.4 even 2
3120.2.g.q.961.2 4 20.7 even 4
3120.2.g.q.961.3 4 260.207 even 4
5070.2.a.z.1.2 2 65.57 even 4
5070.2.a.bf.1.1 2 65.47 even 4