Properties

Label 1386.2.a.m.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

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: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.23607 q^{10} -1.00000 q^{11} +1.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.47214 q^{17} -2.76393 q^{19} -3.23607 q^{20} +1.00000 q^{22} -4.00000 q^{23} +5.47214 q^{25} -1.23607 q^{26} +1.00000 q^{28} +4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} -6.47214 q^{34} -3.23607 q^{35} -10.9443 q^{37} +2.76393 q^{38} +3.23607 q^{40} -6.47214 q^{41} -1.52786 q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} -5.47214 q^{50} +1.23607 q^{52} +0.472136 q^{53} +3.23607 q^{55} -1.00000 q^{56} -4.47214 q^{58} -7.23607 q^{59} -5.23607 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -15.4164 q^{67} +6.47214 q^{68} +3.23607 q^{70} +2.47214 q^{71} -4.94427 q^{73} +10.9443 q^{74} -2.76393 q^{76} -1.00000 q^{77} -3.23607 q^{80} +6.47214 q^{82} -10.1803 q^{83} -20.9443 q^{85} +1.52786 q^{86} +1.00000 q^{88} -10.0000 q^{89} +1.23607 q^{91} -4.00000 q^{92} -2.00000 q^{94} +8.94427 q^{95} +3.52786 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 10 q^{38} + 2 q^{40} - 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} - 2 q^{56} - 10 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 4 q^{68} + 2 q^{70} - 4 q^{71} + 8 q^{73} + 4 q^{74} - 10 q^{76} - 2 q^{77} - 2 q^{80} + 4 q^{82} + 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} - 20 q^{89} - 2 q^{91} - 8 q^{92} - 4 q^{94} + 16 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.23607 1.02333
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) 0 0
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) −3.23607 −0.723607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) −1.23607 −0.242413
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.47214 −1.10996
\(35\) −3.23607 −0.546995
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 2.76393 0.448369
\(39\) 0 0
\(40\) 3.23607 0.511667
\(41\) −6.47214 −1.01078 −0.505389 0.862892i \(-0.668651\pi\)
−0.505389 + 0.862892i \(0.668651\pi\)
\(42\) 0 0
\(43\) −1.52786 −0.232997 −0.116499 0.993191i \(-0.537167\pi\)
−0.116499 + 0.993191i \(0.537167\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.47214 −0.773877
\(51\) 0 0
\(52\) 1.23607 0.171412
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 0 0
\(55\) 3.23607 0.436351
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −4.47214 −0.587220
\(59\) −7.23607 −0.942056 −0.471028 0.882118i \(-0.656117\pi\)
−0.471028 + 0.882118i \(0.656117\pi\)
\(60\) 0 0
\(61\) −5.23607 −0.670410 −0.335205 0.942145i \(-0.608806\pi\)
−0.335205 + 0.942145i \(0.608806\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −15.4164 −1.88341 −0.941707 0.336434i \(-0.890779\pi\)
−0.941707 + 0.336434i \(0.890779\pi\)
\(68\) 6.47214 0.784862
\(69\) 0 0
\(70\) 3.23607 0.386784
\(71\) 2.47214 0.293389 0.146694 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(72\) 0 0
\(73\) −4.94427 −0.578683 −0.289342 0.957226i \(-0.593436\pi\)
−0.289342 + 0.957226i \(0.593436\pi\)
\(74\) 10.9443 1.27225
\(75\) 0 0
\(76\) −2.76393 −0.317045
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −3.23607 −0.361803
\(81\) 0 0
\(82\) 6.47214 0.714728
\(83\) −10.1803 −1.11744 −0.558719 0.829357i \(-0.688707\pi\)
−0.558719 + 0.829357i \(0.688707\pi\)
\(84\) 0 0
\(85\) −20.9443 −2.27173
\(86\) 1.52786 0.164754
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 8.94427 0.917663
\(96\) 0 0
\(97\) 3.52786 0.358200 0.179100 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 5.47214 0.547214
\(101\) 14.1803 1.41100 0.705498 0.708712i \(-0.250723\pi\)
0.705498 + 0.708712i \(0.250723\pi\)
\(102\) 0 0
\(103\) 2.94427 0.290108 0.145054 0.989424i \(-0.453664\pi\)
0.145054 + 0.989424i \(0.453664\pi\)
\(104\) −1.23607 −0.121206
\(105\) 0 0
\(106\) −0.472136 −0.0458579
\(107\) 6.47214 0.625685 0.312842 0.949805i \(-0.398719\pi\)
0.312842 + 0.949805i \(0.398719\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −3.23607 −0.308547
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −8.47214 −0.796992 −0.398496 0.917170i \(-0.630468\pi\)
−0.398496 + 0.917170i \(0.630468\pi\)
\(114\) 0 0
\(115\) 12.9443 1.20706
\(116\) 4.47214 0.415227
\(117\) 0 0
\(118\) 7.23607 0.666134
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.23607 0.474051
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −9.23607 −0.806959 −0.403480 0.914989i \(-0.632199\pi\)
−0.403480 + 0.914989i \(0.632199\pi\)
\(132\) 0 0
\(133\) −2.76393 −0.239663
\(134\) 15.4164 1.33177
\(135\) 0 0
\(136\) −6.47214 −0.554981
\(137\) −15.8885 −1.35745 −0.678725 0.734393i \(-0.737467\pi\)
−0.678725 + 0.734393i \(0.737467\pi\)
\(138\) 0 0
\(139\) −8.29180 −0.703301 −0.351650 0.936131i \(-0.614379\pi\)
−0.351650 + 0.936131i \(0.614379\pi\)
\(140\) −3.23607 −0.273498
\(141\) 0 0
\(142\) −2.47214 −0.207457
\(143\) −1.23607 −0.103365
\(144\) 0 0
\(145\) −14.4721 −1.20185
\(146\) 4.94427 0.409191
\(147\) 0 0
\(148\) −10.9443 −0.899614
\(149\) −22.3607 −1.83186 −0.915929 0.401340i \(-0.868545\pi\)
−0.915929 + 0.401340i \(0.868545\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 2.76393 0.224184
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −6.47214 −0.519854
\(156\) 0 0
\(157\) 18.6525 1.48863 0.744315 0.667829i \(-0.232776\pi\)
0.744315 + 0.667829i \(0.232776\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 3.23607 0.255834
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 7.41641 0.580898 0.290449 0.956890i \(-0.406195\pi\)
0.290449 + 0.956890i \(0.406195\pi\)
\(164\) −6.47214 −0.505389
\(165\) 0 0
\(166\) 10.1803 0.790148
\(167\) 15.4164 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 20.9443 1.60635
\(171\) 0 0
\(172\) −1.52786 −0.116499
\(173\) −1.23607 −0.0939765 −0.0469883 0.998895i \(-0.514962\pi\)
−0.0469883 + 0.998895i \(0.514962\pi\)
\(174\) 0 0
\(175\) 5.47214 0.413655
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) 4.76393 0.354100 0.177050 0.984202i \(-0.443345\pi\)
0.177050 + 0.984202i \(0.443345\pi\)
\(182\) −1.23607 −0.0916235
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 35.4164 2.60387
\(186\) 0 0
\(187\) −6.47214 −0.473289
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) −8.94427 −0.648886
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) 2.94427 0.211933 0.105967 0.994370i \(-0.466206\pi\)
0.105967 + 0.994370i \(0.466206\pi\)
\(194\) −3.52786 −0.253286
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 1.05573 0.0748386 0.0374193 0.999300i \(-0.488086\pi\)
0.0374193 + 0.999300i \(0.488086\pi\)
\(200\) −5.47214 −0.386938
\(201\) 0 0
\(202\) −14.1803 −0.997725
\(203\) 4.47214 0.313882
\(204\) 0 0
\(205\) 20.9443 1.46281
\(206\) −2.94427 −0.205137
\(207\) 0 0
\(208\) 1.23607 0.0857059
\(209\) 2.76393 0.191185
\(210\) 0 0
\(211\) −22.4721 −1.54705 −0.773523 0.633768i \(-0.781507\pi\)
−0.773523 + 0.633768i \(0.781507\pi\)
\(212\) 0.472136 0.0324264
\(213\) 0 0
\(214\) −6.47214 −0.442426
\(215\) 4.94427 0.337197
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 3.23607 0.218176
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 8.47214 0.567336 0.283668 0.958923i \(-0.408449\pi\)
0.283668 + 0.958923i \(0.408449\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 8.47214 0.563558
\(227\) 14.7639 0.979917 0.489958 0.871746i \(-0.337012\pi\)
0.489958 + 0.871746i \(0.337012\pi\)
\(228\) 0 0
\(229\) 12.7639 0.843464 0.421732 0.906720i \(-0.361422\pi\)
0.421732 + 0.906720i \(0.361422\pi\)
\(230\) −12.9443 −0.853520
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) −2.94427 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(234\) 0 0
\(235\) −6.47214 −0.422196
\(236\) −7.23607 −0.471028
\(237\) 0 0
\(238\) −6.47214 −0.419526
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −11.4164 −0.735395 −0.367698 0.929945i \(-0.619854\pi\)
−0.367698 + 0.929945i \(0.619854\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −5.23607 −0.335205
\(245\) −3.23607 −0.206745
\(246\) 0 0
\(247\) −3.41641 −0.217381
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 1.52786 0.0966306
\(251\) −24.7639 −1.56309 −0.781543 0.623852i \(-0.785567\pi\)
−0.781543 + 0.623852i \(0.785567\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.9443 0.682685 0.341342 0.939939i \(-0.389118\pi\)
0.341342 + 0.939939i \(0.389118\pi\)
\(258\) 0 0
\(259\) −10.9443 −0.680044
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 9.23607 0.570606
\(263\) −12.9443 −0.798178 −0.399089 0.916912i \(-0.630674\pi\)
−0.399089 + 0.916912i \(0.630674\pi\)
\(264\) 0 0
\(265\) −1.52786 −0.0938559
\(266\) 2.76393 0.169468
\(267\) 0 0
\(268\) −15.4164 −0.941707
\(269\) 27.2361 1.66061 0.830306 0.557307i \(-0.188166\pi\)
0.830306 + 0.557307i \(0.188166\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 6.47214 0.392431
\(273\) 0 0
\(274\) 15.8885 0.959862
\(275\) −5.47214 −0.329982
\(276\) 0 0
\(277\) 12.4721 0.749378 0.374689 0.927151i \(-0.377749\pi\)
0.374689 + 0.927151i \(0.377749\pi\)
\(278\) 8.29180 0.497309
\(279\) 0 0
\(280\) 3.23607 0.193392
\(281\) 24.8328 1.48140 0.740701 0.671835i \(-0.234494\pi\)
0.740701 + 0.671835i \(0.234494\pi\)
\(282\) 0 0
\(283\) −16.6525 −0.989887 −0.494943 0.868925i \(-0.664811\pi\)
−0.494943 + 0.868925i \(0.664811\pi\)
\(284\) 2.47214 0.146694
\(285\) 0 0
\(286\) 1.23607 0.0730902
\(287\) −6.47214 −0.382038
\(288\) 0 0
\(289\) 24.8885 1.46403
\(290\) 14.4721 0.849833
\(291\) 0 0
\(292\) −4.94427 −0.289342
\(293\) −4.65248 −0.271801 −0.135900 0.990723i \(-0.543393\pi\)
−0.135900 + 0.990723i \(0.543393\pi\)
\(294\) 0 0
\(295\) 23.4164 1.36336
\(296\) 10.9443 0.636123
\(297\) 0 0
\(298\) 22.3607 1.29532
\(299\) −4.94427 −0.285935
\(300\) 0 0
\(301\) −1.52786 −0.0880646
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) −2.76393 −0.158522
\(305\) 16.9443 0.970226
\(306\) 0 0
\(307\) 32.0689 1.83027 0.915134 0.403150i \(-0.132085\pi\)
0.915134 + 0.403150i \(0.132085\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 6.47214 0.367593
\(311\) −5.41641 −0.307136 −0.153568 0.988138i \(-0.549076\pi\)
−0.153568 + 0.988138i \(0.549076\pi\)
\(312\) 0 0
\(313\) 28.4721 1.60934 0.804670 0.593722i \(-0.202342\pi\)
0.804670 + 0.593722i \(0.202342\pi\)
\(314\) −18.6525 −1.05262
\(315\) 0 0
\(316\) 0 0
\(317\) 13.0557 0.733283 0.366641 0.930362i \(-0.380508\pi\)
0.366641 + 0.930362i \(0.380508\pi\)
\(318\) 0 0
\(319\) −4.47214 −0.250392
\(320\) −3.23607 −0.180902
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −17.8885 −0.995345
\(324\) 0 0
\(325\) 6.76393 0.375195
\(326\) −7.41641 −0.410757
\(327\) 0 0
\(328\) 6.47214 0.357364
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 0.944272 0.0519019 0.0259509 0.999663i \(-0.491739\pi\)
0.0259509 + 0.999663i \(0.491739\pi\)
\(332\) −10.1803 −0.558719
\(333\) 0 0
\(334\) −15.4164 −0.843548
\(335\) 49.8885 2.72570
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 11.4721 0.624002
\(339\) 0 0
\(340\) −20.9443 −1.13586
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 1.52786 0.0823769
\(345\) 0 0
\(346\) 1.23607 0.0664514
\(347\) 6.47214 0.347442 0.173721 0.984795i \(-0.444421\pi\)
0.173721 + 0.984795i \(0.444421\pi\)
\(348\) 0 0
\(349\) 8.29180 0.443850 0.221925 0.975064i \(-0.428766\pi\)
0.221925 + 0.975064i \(0.428766\pi\)
\(350\) −5.47214 −0.292498
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 34.9443 1.85990 0.929948 0.367691i \(-0.119852\pi\)
0.929948 + 0.367691i \(0.119852\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 8.94427 0.472719
\(359\) 26.8328 1.41618 0.708091 0.706121i \(-0.249557\pi\)
0.708091 + 0.706121i \(0.249557\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) −4.76393 −0.250387
\(363\) 0 0
\(364\) 1.23607 0.0647876
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 21.4164 1.11793 0.558964 0.829192i \(-0.311199\pi\)
0.558964 + 0.829192i \(0.311199\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −35.4164 −1.84121
\(371\) 0.472136 0.0245121
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 6.47214 0.334666
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 5.52786 0.284699
\(378\) 0 0
\(379\) 5.52786 0.283947 0.141974 0.989870i \(-0.454655\pi\)
0.141974 + 0.989870i \(0.454655\pi\)
\(380\) 8.94427 0.458831
\(381\) 0 0
\(382\) 6.47214 0.331143
\(383\) −11.8885 −0.607476 −0.303738 0.952756i \(-0.598235\pi\)
−0.303738 + 0.952756i \(0.598235\pi\)
\(384\) 0 0
\(385\) 3.23607 0.164925
\(386\) −2.94427 −0.149859
\(387\) 0 0
\(388\) 3.52786 0.179100
\(389\) −6.58359 −0.333801 −0.166901 0.985974i \(-0.553376\pi\)
−0.166901 + 0.985974i \(0.553376\pi\)
\(390\) 0 0
\(391\) −25.8885 −1.30924
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −10.2918 −0.516530 −0.258265 0.966074i \(-0.583151\pi\)
−0.258265 + 0.966074i \(0.583151\pi\)
\(398\) −1.05573 −0.0529189
\(399\) 0 0
\(400\) 5.47214 0.273607
\(401\) 30.3607 1.51614 0.758070 0.652173i \(-0.226143\pi\)
0.758070 + 0.652173i \(0.226143\pi\)
\(402\) 0 0
\(403\) 2.47214 0.123146
\(404\) 14.1803 0.705498
\(405\) 0 0
\(406\) −4.47214 −0.221948
\(407\) 10.9443 0.542487
\(408\) 0 0
\(409\) −23.4164 −1.15787 −0.578933 0.815375i \(-0.696531\pi\)
−0.578933 + 0.815375i \(0.696531\pi\)
\(410\) −20.9443 −1.03436
\(411\) 0 0
\(412\) 2.94427 0.145054
\(413\) −7.23607 −0.356064
\(414\) 0 0
\(415\) 32.9443 1.61717
\(416\) −1.23607 −0.0606032
\(417\) 0 0
\(418\) −2.76393 −0.135188
\(419\) −12.7639 −0.623559 −0.311779 0.950155i \(-0.600925\pi\)
−0.311779 + 0.950155i \(0.600925\pi\)
\(420\) 0 0
\(421\) 7.52786 0.366886 0.183443 0.983030i \(-0.441276\pi\)
0.183443 + 0.983030i \(0.441276\pi\)
\(422\) 22.4721 1.09393
\(423\) 0 0
\(424\) −0.472136 −0.0229289
\(425\) 35.4164 1.71795
\(426\) 0 0
\(427\) −5.23607 −0.253391
\(428\) 6.47214 0.312842
\(429\) 0 0
\(430\) −4.94427 −0.238434
\(431\) −40.9443 −1.97222 −0.986108 0.166105i \(-0.946881\pi\)
−0.986108 + 0.166105i \(0.946881\pi\)
\(432\) 0 0
\(433\) 19.5279 0.938449 0.469225 0.883079i \(-0.344533\pi\)
0.469225 + 0.883079i \(0.344533\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 11.0557 0.528867
\(438\) 0 0
\(439\) −8.94427 −0.426887 −0.213443 0.976955i \(-0.568468\pi\)
−0.213443 + 0.976955i \(0.568468\pi\)
\(440\) −3.23607 −0.154273
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 7.05573 0.335228 0.167614 0.985853i \(-0.446394\pi\)
0.167614 + 0.985853i \(0.446394\pi\)
\(444\) 0 0
\(445\) 32.3607 1.53404
\(446\) −8.47214 −0.401167
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −1.05573 −0.0498229 −0.0249114 0.999690i \(-0.507930\pi\)
−0.0249114 + 0.999690i \(0.507930\pi\)
\(450\) 0 0
\(451\) 6.47214 0.304761
\(452\) −8.47214 −0.398496
\(453\) 0 0
\(454\) −14.7639 −0.692906
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 9.05573 0.423609 0.211805 0.977312i \(-0.432066\pi\)
0.211805 + 0.977312i \(0.432066\pi\)
\(458\) −12.7639 −0.596419
\(459\) 0 0
\(460\) 12.9443 0.603530
\(461\) −29.2361 −1.36166 −0.680830 0.732442i \(-0.738381\pi\)
−0.680830 + 0.732442i \(0.738381\pi\)
\(462\) 0 0
\(463\) −21.5279 −1.00048 −0.500242 0.865885i \(-0.666756\pi\)
−0.500242 + 0.865885i \(0.666756\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) 2.94427 0.136391
\(467\) −13.1246 −0.607335 −0.303667 0.952778i \(-0.598211\pi\)
−0.303667 + 0.952778i \(0.598211\pi\)
\(468\) 0 0
\(469\) −15.4164 −0.711864
\(470\) 6.47214 0.298537
\(471\) 0 0
\(472\) 7.23607 0.333067
\(473\) 1.52786 0.0702513
\(474\) 0 0
\(475\) −15.1246 −0.693965
\(476\) 6.47214 0.296650
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) 32.3607 1.47860 0.739299 0.673378i \(-0.235157\pi\)
0.739299 + 0.673378i \(0.235157\pi\)
\(480\) 0 0
\(481\) −13.5279 −0.616818
\(482\) 11.4164 0.520003
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −11.4164 −0.518392
\(486\) 0 0
\(487\) −0.944272 −0.0427890 −0.0213945 0.999771i \(-0.506811\pi\)
−0.0213945 + 0.999771i \(0.506811\pi\)
\(488\) 5.23607 0.237026
\(489\) 0 0
\(490\) 3.23607 0.146191
\(491\) −0.944272 −0.0426144 −0.0213072 0.999773i \(-0.506783\pi\)
−0.0213072 + 0.999773i \(0.506783\pi\)
\(492\) 0 0
\(493\) 28.9443 1.30358
\(494\) 3.41641 0.153711
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 2.47214 0.110890
\(498\) 0 0
\(499\) 12.3607 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(500\) −1.52786 −0.0683282
\(501\) 0 0
\(502\) 24.7639 1.10527
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −45.8885 −2.04201
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 34.0689 1.51008 0.755038 0.655681i \(-0.227618\pi\)
0.755038 + 0.655681i \(0.227618\pi\)
\(510\) 0 0
\(511\) −4.94427 −0.218722
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −10.9443 −0.482731
\(515\) −9.52786 −0.419848
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) 10.9443 0.480864
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −34.3607 −1.50537 −0.752684 0.658382i \(-0.771241\pi\)
−0.752684 + 0.658382i \(0.771241\pi\)
\(522\) 0 0
\(523\) −27.7082 −1.21160 −0.605798 0.795619i \(-0.707146\pi\)
−0.605798 + 0.795619i \(0.707146\pi\)
\(524\) −9.23607 −0.403480
\(525\) 0 0
\(526\) 12.9443 0.564397
\(527\) 12.9443 0.563861
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 1.52786 0.0663662
\(531\) 0 0
\(532\) −2.76393 −0.119832
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −20.9443 −0.905500
\(536\) 15.4164 0.665887
\(537\) 0 0
\(538\) −27.2361 −1.17423
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −9.05573 −0.389336 −0.194668 0.980869i \(-0.562363\pi\)
−0.194668 + 0.980869i \(0.562363\pi\)
\(542\) 16.9443 0.727819
\(543\) 0 0
\(544\) −6.47214 −0.277491
\(545\) 32.3607 1.38618
\(546\) 0 0
\(547\) 16.9443 0.724485 0.362242 0.932084i \(-0.382011\pi\)
0.362242 + 0.932084i \(0.382011\pi\)
\(548\) −15.8885 −0.678725
\(549\) 0 0
\(550\) 5.47214 0.233333
\(551\) −12.3607 −0.526583
\(552\) 0 0
\(553\) 0 0
\(554\) −12.4721 −0.529890
\(555\) 0 0
\(556\) −8.29180 −0.351650
\(557\) 28.8328 1.22169 0.610843 0.791752i \(-0.290831\pi\)
0.610843 + 0.791752i \(0.290831\pi\)
\(558\) 0 0
\(559\) −1.88854 −0.0798769
\(560\) −3.23607 −0.136749
\(561\) 0 0
\(562\) −24.8328 −1.04751
\(563\) −26.7639 −1.12797 −0.563983 0.825787i \(-0.690732\pi\)
−0.563983 + 0.825787i \(0.690732\pi\)
\(564\) 0 0
\(565\) 27.4164 1.15342
\(566\) 16.6525 0.699956
\(567\) 0 0
\(568\) −2.47214 −0.103729
\(569\) −16.8328 −0.705668 −0.352834 0.935686i \(-0.614782\pi\)
−0.352834 + 0.935686i \(0.614782\pi\)
\(570\) 0 0
\(571\) −45.8885 −1.92038 −0.960188 0.279355i \(-0.909879\pi\)
−0.960188 + 0.279355i \(0.909879\pi\)
\(572\) −1.23607 −0.0516826
\(573\) 0 0
\(574\) 6.47214 0.270142
\(575\) −21.8885 −0.912815
\(576\) 0 0
\(577\) 9.05573 0.376995 0.188497 0.982074i \(-0.439638\pi\)
0.188497 + 0.982074i \(0.439638\pi\)
\(578\) −24.8885 −1.03523
\(579\) 0 0
\(580\) −14.4721 −0.600923
\(581\) −10.1803 −0.422352
\(582\) 0 0
\(583\) −0.472136 −0.0195539
\(584\) 4.94427 0.204595
\(585\) 0 0
\(586\) 4.65248 0.192192
\(587\) 28.1803 1.16313 0.581564 0.813501i \(-0.302441\pi\)
0.581564 + 0.813501i \(0.302441\pi\)
\(588\) 0 0
\(589\) −5.52786 −0.227772
\(590\) −23.4164 −0.964038
\(591\) 0 0
\(592\) −10.9443 −0.449807
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) −20.9443 −0.858631
\(596\) −22.3607 −0.915929
\(597\) 0 0
\(598\) 4.94427 0.202186
\(599\) −12.3607 −0.505044 −0.252522 0.967591i \(-0.581260\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(600\) 0 0
\(601\) 18.8328 0.768207 0.384103 0.923290i \(-0.374511\pi\)
0.384103 + 0.923290i \(0.374511\pi\)
\(602\) 1.52786 0.0622711
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −3.23607 −0.131565
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 2.76393 0.112092
\(609\) 0 0
\(610\) −16.9443 −0.686054
\(611\) 2.47214 0.100012
\(612\) 0 0
\(613\) 19.5279 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(614\) −32.0689 −1.29419
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 5.41641 0.218056 0.109028 0.994039i \(-0.465226\pi\)
0.109028 + 0.994039i \(0.465226\pi\)
\(618\) 0 0
\(619\) −48.5410 −1.95103 −0.975514 0.219937i \(-0.929415\pi\)
−0.975514 + 0.219937i \(0.929415\pi\)
\(620\) −6.47214 −0.259927
\(621\) 0 0
\(622\) 5.41641 0.217178
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) −28.4721 −1.13798
\(627\) 0 0
\(628\) 18.6525 0.744315
\(629\) −70.8328 −2.82429
\(630\) 0 0
\(631\) −4.58359 −0.182470 −0.0912350 0.995829i \(-0.529081\pi\)
−0.0912350 + 0.995829i \(0.529081\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −13.0557 −0.518509
\(635\) 38.8328 1.54103
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) 4.47214 0.177054
\(639\) 0 0
\(640\) 3.23607 0.127917
\(641\) −36.4721 −1.44056 −0.720281 0.693682i \(-0.755987\pi\)
−0.720281 + 0.693682i \(0.755987\pi\)
\(642\) 0 0
\(643\) −23.2361 −0.916341 −0.458171 0.888864i \(-0.651495\pi\)
−0.458171 + 0.888864i \(0.651495\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 17.8885 0.703815
\(647\) −24.8328 −0.976279 −0.488139 0.872766i \(-0.662324\pi\)
−0.488139 + 0.872766i \(0.662324\pi\)
\(648\) 0 0
\(649\) 7.23607 0.284041
\(650\) −6.76393 −0.265303
\(651\) 0 0
\(652\) 7.41641 0.290449
\(653\) −1.63932 −0.0641516 −0.0320758 0.999485i \(-0.510212\pi\)
−0.0320758 + 0.999485i \(0.510212\pi\)
\(654\) 0 0
\(655\) 29.8885 1.16784
\(656\) −6.47214 −0.252694
\(657\) 0 0
\(658\) −2.00000 −0.0779681
\(659\) −43.4164 −1.69126 −0.845632 0.533767i \(-0.820776\pi\)
−0.845632 + 0.533767i \(0.820776\pi\)
\(660\) 0 0
\(661\) 37.1246 1.44398 0.721990 0.691903i \(-0.243228\pi\)
0.721990 + 0.691903i \(0.243228\pi\)
\(662\) −0.944272 −0.0367002
\(663\) 0 0
\(664\) 10.1803 0.395074
\(665\) 8.94427 0.346844
\(666\) 0 0
\(667\) −17.8885 −0.692647
\(668\) 15.4164 0.596479
\(669\) 0 0
\(670\) −49.8885 −1.92736
\(671\) 5.23607 0.202136
\(672\) 0 0
\(673\) 31.8885 1.22921 0.614607 0.788834i \(-0.289315\pi\)
0.614607 + 0.788834i \(0.289315\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −11.4721 −0.441236
\(677\) −32.0689 −1.23251 −0.616254 0.787548i \(-0.711350\pi\)
−0.616254 + 0.787548i \(0.711350\pi\)
\(678\) 0 0
\(679\) 3.52786 0.135387
\(680\) 20.9443 0.803176
\(681\) 0 0
\(682\) 2.00000 0.0765840
\(683\) −15.0557 −0.576091 −0.288046 0.957617i \(-0.593006\pi\)
−0.288046 + 0.957617i \(0.593006\pi\)
\(684\) 0 0
\(685\) 51.4164 1.96452
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −1.52786 −0.0582493
\(689\) 0.583592 0.0222331
\(690\) 0 0
\(691\) −18.6525 −0.709574 −0.354787 0.934947i \(-0.615447\pi\)
−0.354787 + 0.934947i \(0.615447\pi\)
\(692\) −1.23607 −0.0469883
\(693\) 0 0
\(694\) −6.47214 −0.245679
\(695\) 26.8328 1.01783
\(696\) 0 0
\(697\) −41.8885 −1.58664
\(698\) −8.29180 −0.313849
\(699\) 0 0
\(700\) 5.47214 0.206827
\(701\) −46.7214 −1.76464 −0.882321 0.470649i \(-0.844020\pi\)
−0.882321 + 0.470649i \(0.844020\pi\)
\(702\) 0 0
\(703\) 30.2492 1.14087
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −34.9443 −1.31515
\(707\) 14.1803 0.533307
\(708\) 0 0
\(709\) −4.47214 −0.167955 −0.0839773 0.996468i \(-0.526762\pi\)
−0.0839773 + 0.996468i \(0.526762\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −8.94427 −0.334263
\(717\) 0 0
\(718\) −26.8328 −1.00139
\(719\) 36.8328 1.37363 0.686816 0.726831i \(-0.259008\pi\)
0.686816 + 0.726831i \(0.259008\pi\)
\(720\) 0 0
\(721\) 2.94427 0.109650
\(722\) 11.3607 0.422801
\(723\) 0 0
\(724\) 4.76393 0.177050
\(725\) 24.4721 0.908872
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) −1.23607 −0.0458117
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) −9.88854 −0.365741
\(732\) 0 0
\(733\) 8.87539 0.327820 0.163910 0.986475i \(-0.447589\pi\)
0.163910 + 0.986475i \(0.447589\pi\)
\(734\) −21.4164 −0.790494
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 15.4164 0.567871
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 35.4164 1.30193
\(741\) 0 0
\(742\) −0.472136 −0.0173327
\(743\) 13.8885 0.509521 0.254761 0.967004i \(-0.418003\pi\)
0.254761 + 0.967004i \(0.418003\pi\)
\(744\) 0 0
\(745\) 72.3607 2.65109
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) −6.47214 −0.236645
\(749\) 6.47214 0.236487
\(750\) 0 0
\(751\) 0.944272 0.0344570 0.0172285 0.999852i \(-0.494516\pi\)
0.0172285 + 0.999852i \(0.494516\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) −5.52786 −0.201313
\(755\) −38.8328 −1.41327
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) −5.52786 −0.200781
\(759\) 0 0
\(760\) −8.94427 −0.324443
\(761\) −15.4164 −0.558844 −0.279422 0.960168i \(-0.590143\pi\)
−0.279422 + 0.960168i \(0.590143\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) −6.47214 −0.234154
\(765\) 0 0
\(766\) 11.8885 0.429551
\(767\) −8.94427 −0.322959
\(768\) 0 0
\(769\) −16.5836 −0.598020 −0.299010 0.954250i \(-0.596656\pi\)
−0.299010 + 0.954250i \(0.596656\pi\)
\(770\) −3.23607 −0.116620
\(771\) 0 0
\(772\) 2.94427 0.105967
\(773\) −2.29180 −0.0824302 −0.0412151 0.999150i \(-0.513123\pi\)
−0.0412151 + 0.999150i \(0.513123\pi\)
\(774\) 0 0
\(775\) 10.9443 0.393130
\(776\) −3.52786 −0.126643
\(777\) 0 0
\(778\) 6.58359 0.236033
\(779\) 17.8885 0.640924
\(780\) 0 0
\(781\) −2.47214 −0.0884600
\(782\) 25.8885 0.925772
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −60.3607 −2.15437
\(786\) 0 0
\(787\) −5.81966 −0.207448 −0.103724 0.994606i \(-0.533076\pi\)
−0.103724 + 0.994606i \(0.533076\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) −8.47214 −0.301234
\(792\) 0 0
\(793\) −6.47214 −0.229832
\(794\) 10.2918 0.365242
\(795\) 0 0
\(796\) 1.05573 0.0374193
\(797\) −7.59675 −0.269091 −0.134545 0.990907i \(-0.542957\pi\)
−0.134545 + 0.990907i \(0.542957\pi\)
\(798\) 0 0
\(799\) 12.9443 0.457935
\(800\) −5.47214 −0.193469
\(801\) 0 0
\(802\) −30.3607 −1.07207
\(803\) 4.94427 0.174480
\(804\) 0 0
\(805\) 12.9443 0.456226
\(806\) −2.47214 −0.0870773
\(807\) 0 0
\(808\) −14.1803 −0.498863
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 0 0
\(811\) 9.23607 0.324322 0.162161 0.986764i \(-0.448154\pi\)
0.162161 + 0.986764i \(0.448154\pi\)
\(812\) 4.47214 0.156941
\(813\) 0 0
\(814\) −10.9443 −0.383597
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 4.22291 0.147741
\(818\) 23.4164 0.818736
\(819\) 0 0
\(820\) 20.9443 0.731406
\(821\) −25.4164 −0.887039 −0.443519 0.896265i \(-0.646270\pi\)
−0.443519 + 0.896265i \(0.646270\pi\)
\(822\) 0 0
\(823\) 34.2492 1.19385 0.596926 0.802296i \(-0.296388\pi\)
0.596926 + 0.802296i \(0.296388\pi\)
\(824\) −2.94427 −0.102569
\(825\) 0 0
\(826\) 7.23607 0.251775
\(827\) 0.944272 0.0328356 0.0164178 0.999865i \(-0.494774\pi\)
0.0164178 + 0.999865i \(0.494774\pi\)
\(828\) 0 0
\(829\) −1.70820 −0.0593284 −0.0296642 0.999560i \(-0.509444\pi\)
−0.0296642 + 0.999560i \(0.509444\pi\)
\(830\) −32.9443 −1.14351
\(831\) 0 0
\(832\) 1.23607 0.0428529
\(833\) 6.47214 0.224246
\(834\) 0 0
\(835\) −49.8885 −1.72646
\(836\) 2.76393 0.0955926
\(837\) 0 0
\(838\) 12.7639 0.440923
\(839\) 36.8328 1.27161 0.635805 0.771850i \(-0.280668\pi\)
0.635805 + 0.771850i \(0.280668\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −7.52786 −0.259427
\(843\) 0 0
\(844\) −22.4721 −0.773523
\(845\) 37.1246 1.27713
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0.472136 0.0162132
\(849\) 0 0
\(850\) −35.4164 −1.21477
\(851\) 43.7771 1.50066
\(852\) 0 0
\(853\) −34.5410 −1.18266 −0.591331 0.806429i \(-0.701397\pi\)
−0.591331 + 0.806429i \(0.701397\pi\)
\(854\) 5.23607 0.179175
\(855\) 0 0
\(856\) −6.47214 −0.221213
\(857\) 37.5279 1.28193 0.640964 0.767571i \(-0.278535\pi\)
0.640964 + 0.767571i \(0.278535\pi\)
\(858\) 0 0
\(859\) −25.1246 −0.857241 −0.428620 0.903485i \(-0.641000\pi\)
−0.428620 + 0.903485i \(0.641000\pi\)
\(860\) 4.94427 0.168598
\(861\) 0 0
\(862\) 40.9443 1.39457
\(863\) −27.4164 −0.933265 −0.466633 0.884451i \(-0.654533\pi\)
−0.466633 + 0.884451i \(0.654533\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) −19.5279 −0.663584
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −19.0557 −0.645679
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −11.0557 −0.373966
\(875\) −1.52786 −0.0516512
\(876\) 0 0
\(877\) 26.9443 0.909843 0.454922 0.890531i \(-0.349667\pi\)
0.454922 + 0.890531i \(0.349667\pi\)
\(878\) 8.94427 0.301855
\(879\) 0 0
\(880\) 3.23607 0.109088
\(881\) 24.8328 0.836639 0.418319 0.908300i \(-0.362619\pi\)
0.418319 + 0.908300i \(0.362619\pi\)
\(882\) 0 0
\(883\) 50.8328 1.71066 0.855330 0.518083i \(-0.173354\pi\)
0.855330 + 0.518083i \(0.173354\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −7.05573 −0.237042
\(887\) −0.360680 −0.0121104 −0.00605522 0.999982i \(-0.501927\pi\)
−0.00605522 + 0.999982i \(0.501927\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) −32.3607 −1.08473
\(891\) 0 0
\(892\) 8.47214 0.283668
\(893\) −5.52786 −0.184983
\(894\) 0 0
\(895\) 28.9443 0.967500
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 1.05573 0.0352301
\(899\) 8.94427 0.298308
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) −6.47214 −0.215499
\(903\) 0 0
\(904\) 8.47214 0.281779
\(905\) −15.4164 −0.512459
\(906\) 0 0
\(907\) 20.3607 0.676065 0.338033 0.941134i \(-0.390239\pi\)
0.338033 + 0.941134i \(0.390239\pi\)
\(908\) 14.7639 0.489958
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 10.1803 0.336920
\(914\) −9.05573 −0.299537
\(915\) 0 0
\(916\) 12.7639 0.421732
\(917\) −9.23607 −0.305002
\(918\) 0 0
\(919\) −57.8885 −1.90957 −0.954783 0.297302i \(-0.903913\pi\)
−0.954783 + 0.297302i \(0.903913\pi\)
\(920\) −12.9443 −0.426760
\(921\) 0 0
\(922\) 29.2361 0.962839
\(923\) 3.05573 0.100581
\(924\) 0 0
\(925\) −59.8885 −1.96912
\(926\) 21.5279 0.707450
\(927\) 0 0
\(928\) −4.47214 −0.146805
\(929\) 40.2492 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(930\) 0 0
\(931\) −2.76393 −0.0905842
\(932\) −2.94427 −0.0964428
\(933\) 0 0
\(934\) 13.1246 0.429450
\(935\) 20.9443 0.684951
\(936\) 0 0
\(937\) −20.9443 −0.684220 −0.342110 0.939660i \(-0.611141\pi\)
−0.342110 + 0.939660i \(0.611141\pi\)
\(938\) 15.4164 0.503364
\(939\) 0 0
\(940\) −6.47214 −0.211098
\(941\) 34.1803 1.11425 0.557124 0.830430i \(-0.311905\pi\)
0.557124 + 0.830430i \(0.311905\pi\)
\(942\) 0 0
\(943\) 25.8885 0.843047
\(944\) −7.23607 −0.235514
\(945\) 0 0
\(946\) −1.52786 −0.0496751
\(947\) 0.944272 0.0306847 0.0153424 0.999882i \(-0.495116\pi\)
0.0153424 + 0.999882i \(0.495116\pi\)
\(948\) 0 0
\(949\) −6.11146 −0.198386
\(950\) 15.1246 0.490707
\(951\) 0 0
\(952\) −6.47214 −0.209763
\(953\) −5.05573 −0.163771 −0.0818855 0.996642i \(-0.526094\pi\)
−0.0818855 + 0.996642i \(0.526094\pi\)
\(954\) 0 0
\(955\) 20.9443 0.677741
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −32.3607 −1.04553
\(959\) −15.8885 −0.513068
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 13.5279 0.436156
\(963\) 0 0
\(964\) −11.4164 −0.367698
\(965\) −9.52786 −0.306713
\(966\) 0 0
\(967\) 10.1115 0.325163 0.162581 0.986695i \(-0.448018\pi\)
0.162581 + 0.986695i \(0.448018\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 11.4164 0.366559
\(971\) 16.5410 0.530827 0.265413 0.964135i \(-0.414492\pi\)
0.265413 + 0.964135i \(0.414492\pi\)
\(972\) 0 0
\(973\) −8.29180 −0.265823
\(974\) 0.944272 0.0302564
\(975\) 0 0
\(976\) −5.23607 −0.167602
\(977\) −24.8328 −0.794472 −0.397236 0.917716i \(-0.630031\pi\)
−0.397236 + 0.917716i \(0.630031\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) −3.23607 −0.103372
\(981\) 0 0
\(982\) 0.944272 0.0301329
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) 0 0
\(985\) 58.2492 1.85597
\(986\) −28.9443 −0.921773
\(987\) 0 0
\(988\) −3.41641 −0.108690
\(989\) 6.11146 0.194333
\(990\) 0 0
\(991\) 44.3607 1.40916 0.704582 0.709623i \(-0.251135\pi\)
0.704582 + 0.709623i \(0.251135\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) −2.47214 −0.0784114
\(995\) −3.41641 −0.108307
\(996\) 0 0
\(997\) 1.81966 0.0576292 0.0288146 0.999585i \(-0.490827\pi\)
0.0288146 + 0.999585i \(0.490827\pi\)
\(998\) −12.3607 −0.391270
\(999\) 0 0
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 1386.2.a.m.1.1 2
3.2 odd 2 154.2.a.d.1.1 2
7.6 odd 2 9702.2.a.cu.1.2 2
12.11 even 2 1232.2.a.p.1.2 2
15.2 even 4 3850.2.c.q.1849.4 4
15.8 even 4 3850.2.c.q.1849.1 4
15.14 odd 2 3850.2.a.bj.1.2 2
21.2 odd 6 1078.2.e.q.67.2 4
21.5 even 6 1078.2.e.n.67.1 4
21.11 odd 6 1078.2.e.q.177.2 4
21.17 even 6 1078.2.e.n.177.1 4
21.20 even 2 1078.2.a.w.1.2 2
24.5 odd 2 4928.2.a.bt.1.2 2
24.11 even 2 4928.2.a.bk.1.1 2
33.32 even 2 1694.2.a.l.1.1 2
84.83 odd 2 8624.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 2 3.2 odd 2
1078.2.a.w.1.2 2 21.20 even 2
1078.2.e.n.67.1 4 21.5 even 6
1078.2.e.n.177.1 4 21.17 even 6
1078.2.e.q.67.2 4 21.2 odd 6
1078.2.e.q.177.2 4 21.11 odd 6
1232.2.a.p.1.2 2 12.11 even 2
1386.2.a.m.1.1 2 1.1 even 1 trivial
1694.2.a.l.1.1 2 33.32 even 2
3850.2.a.bj.1.2 2 15.14 odd 2
3850.2.c.q.1849.1 4 15.8 even 4
3850.2.c.q.1849.4 4 15.2 even 4
4928.2.a.bk.1.1 2 24.11 even 2
4928.2.a.bt.1.2 2 24.5 odd 2
8624.2.a.bf.1.1 2 84.83 odd 2
9702.2.a.cu.1.2 2 7.6 odd 2