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Properties

Label 4225.2.a.n.1.1
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.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: \(33.7367948540\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 325)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +5.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +5.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{12} +5.00000 q^{14} -1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +10.0000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -6.00000 q^{24} -4.00000 q^{27} -5.00000 q^{28} -1.00000 q^{29} -1.00000 q^{31} +5.00000 q^{32} -6.00000 q^{33} +5.00000 q^{34} -1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +8.00000 q^{41} +10.0000 q^{42} -4.00000 q^{43} +3.00000 q^{44} +4.00000 q^{46} +7.00000 q^{47} -2.00000 q^{48} +18.0000 q^{49} +10.0000 q^{51} +3.00000 q^{53} -4.00000 q^{54} -15.0000 q^{56} +8.00000 q^{57} -1.00000 q^{58} +3.00000 q^{59} +1.00000 q^{61} -1.00000 q^{62} +5.00000 q^{63} +7.00000 q^{64} -6.00000 q^{66} +3.00000 q^{67} -5.00000 q^{68} +8.00000 q^{69} -8.00000 q^{71} -3.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} -4.00000 q^{76} -15.0000 q^{77} +10.0000 q^{79} -11.0000 q^{81} +8.00000 q^{82} -9.00000 q^{83} -10.0000 q^{84} -4.00000 q^{86} -2.00000 q^{87} +9.00000 q^{88} +18.0000 q^{89} -4.00000 q^{92} -2.00000 q^{93} +7.00000 q^{94} +10.0000 q^{96} -14.0000 q^{97} +18.0000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 5.00000 1.88982 0.944911 − 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 10.0000 2.18218
\(22\) −3.00000 −0.639602
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −6.00000 −1.22474
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) −5.00000 −0.944911
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 5.00000 0.883883
\(33\) −6.00000 −1.04447
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 10.0000 1.54303
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 7.00000 1.02105 0.510527 − 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) −2.00000 −0.288675
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 10.0000 1.40028
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −15.0000 −2.00446
\(57\) 8.00000 1.05963
\(58\) −1.00000 −0.131306
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −1.00000 −0.127000
\(63\) 5.00000 0.629941
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −5.00000 −0.606339
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.00000 −0.353553
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 8.00000 0.883452
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −10.0000 −1.09109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 9.00000 0.959403
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 10.0000 1.02062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 18.0000 1.81827
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 − 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 10.0000 0.990148
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 16.0000 1.54678 0.773389 − 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −5.00000 −0.472456
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 3.00000 0.276172
\(119\) 25.0000 2.29175
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 1.00000 0.0905357
\(123\) 16.0000 1.44267
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 5.00000 0.445435
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −3.00000 −0.265165
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 6.00000 0.522233
\(133\) 20.0000 1.73422
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −15.0000 −1.28624
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 8.00000 0.681005
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 14.0000 1.17901
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 36.0000 2.96923
\(148\) 4.00000 0.328798
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 − 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) −12.0000 −0.973329
\(153\) 5.00000 0.404226
\(154\) −15.0000 −1.20873
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 − 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 10.0000 0.795557
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) −11.0000 −0.864242
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) −30.0000 −2.31455
\(169\) 0 0
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 13.0000 0.988372 0.494186 − 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 6.00000 0.450988
\(178\) 18.0000 1.34916
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −3.00000 −0.222988 −0.111494 − 0.993765i \(-0.535564\pi\)
−0.111494 + 0.993765i \(0.535564\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) −15.0000 −1.09691
\(188\) −7.00000 −0.510527
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 − 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 14.0000 1.01036
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −18.0000 −1.28571
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −3.00000 −0.213201
\(199\) 6.00000 0.425329 0.212664 − 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) −1.00000 −0.0703598
\(203\) −5.00000 −0.350931
\(204\) −10.0000 −0.700140
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −3.00000 −0.206041
\(213\) −16.0000 −1.09630
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 12.0000 0.816497
\(217\) −5.00000 −0.339422
\(218\) 2.00000 0.135457
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 25.0000 1.67038
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) 9.00000 0.597351 0.298675 − 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) −8.00000 −0.529813
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −30.0000 −1.97386
\(232\) 3.00000 0.196960
\(233\) 14.0000 0.917170 0.458585 − 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 20.0000 1.29914
\(238\) 25.0000 1.62051
\(239\) −5.00000 −0.323423 −0.161712 − 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −2.00000 −0.128565
\(243\) −10.0000 −0.641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −5.00000 −0.314970
\(253\) −12.0000 −0.754434
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 15.0000 0.935674 0.467837 − 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) −8.00000 −0.498058
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 18.0000 1.11204
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 18.0000 1.10782
\(265\) 0 0
\(266\) 20.0000 1.22628
\(267\) 36.0000 2.20316
\(268\) −3.00000 −0.183254
\(269\) −17.0000 −1.03651 −0.518254 − 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −29.0000 −1.76162 −0.880812 − 0.473466i \(-0.843003\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 4.00000 0.239904
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 14.0000 0.833688
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) 5.00000 0.294628
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) 4.00000 0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 36.0000 2.09956
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 12.0000 0.696311
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −13.0000 −0.748066
\(303\) −2.00000 −0.114897
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 5.00000 0.285831
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 15.0000 0.854704
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 − 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 32.0000 1.78607
\(322\) 20.0000 1.11456
\(323\) 20.0000 1.11283
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 4.00000 0.221201
\(328\) −24.0000 −1.32518
\(329\) 35.0000 1.92961
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 9.00000 0.493939
\(333\) −4.00000 −0.219199
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) −10.0000 −0.545545
\(337\) −3.00000 −0.163420 −0.0817102 − 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 4.00000 0.216295
\(343\) 55.0000 2.96972
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 2.00000 0.107211
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.0000 −0.799503
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 50.0000 2.64628
\(358\) −12.0000 −0.634220
\(359\) −23.0000 −1.21389 −0.606947 − 0.794742i \(-0.707606\pi\)
−0.606947 + 0.794742i \(0.707606\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −3.00000 −0.157676
\(363\) −4.00000 −0.209946
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 18.0000 0.939592 0.469796 − 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −4.00000 −0.208514
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 15.0000 0.778761
\(372\) 2.00000 0.103695
\(373\) −31.0000 −1.60512 −0.802560 − 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) −15.0000 −0.775632
\(375\) 0 0
\(376\) −21.0000 −1.08299
\(377\) 0 0
\(378\) −20.0000 −1.02869
\(379\) −27.0000 −1.38690 −0.693448 − 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(380\) 0 0
\(381\) 36.0000 1.84434
\(382\) 10.0000 0.511645
\(383\) −32.0000 −1.63512 −0.817562 − 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) −6.00000 −0.306186
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −34.0000 −1.72387 −0.861934 − 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) −54.0000 −2.72741
\(393\) 36.0000 1.81596
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −8.00000 −0.401508 −0.200754 − 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 6.00000 0.300753
\(399\) 40.0000 2.00250
\(400\) 0 0
\(401\) 26.0000 1.29838 0.649189 − 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 6.00000 0.299253
\(403\) 0 0
\(404\) 1.00000 0.0497519
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) 12.0000 0.594818
\(408\) −30.0000 −1.48522
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) −4.00000 −0.197066
\(413\) 15.0000 0.738102
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) −12.0000 −0.586939
\(419\) −18.0000 −0.879358 −0.439679 − 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 − 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −12.0000 −0.584151
\(423\) 7.00000 0.340352
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 5.00000 0.241967
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 16.0000 0.765384
\(438\) −8.00000 −0.382255
\(439\) −22.0000 −1.05000 −0.525001 − 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −20.0000 −0.945968
\(448\) 35.0000 1.65359
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 10.0000 0.470360
\(453\) −26.0000 −1.22159
\(454\) 9.00000 0.422391
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) −20.0000 −0.933520
\(460\) 0 0
\(461\) −26.0000 −1.21094 −0.605470 − 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) −30.0000 −1.39573
\(463\) 19.0000 0.883005 0.441502 − 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) −9.00000 −0.414259
\(473\) 12.0000 0.551761
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) −25.0000 −1.14587
\(477\) 3.00000 0.137361
\(478\) −5.00000 −0.228695
\(479\) 3.00000 0.137073 0.0685367 − 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 40.0000 1.82006
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −15.0000 −0.679715 −0.339857 − 0.940477i \(-0.610379\pi\)
−0.339857 + 0.940477i \(0.610379\pi\)
\(488\) −3.00000 −0.135804
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −16.0000 −0.721336
\(493\) −5.00000 −0.225189
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −40.0000 −1.79425
\(498\) −18.0000 −0.806599
\(499\) 19.0000 0.850557 0.425278 − 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) −32.0000 −1.42965
\(502\) −12.0000 −0.535586
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −15.0000 −0.668153
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) −18.0000 −0.798621
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) −11.0000 −0.486136
\(513\) −16.0000 −0.706417
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −21.0000 −0.923579
\(518\) −20.0000 −0.878750
\(519\) 26.0000 1.14127
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) −5.00000 −0.217803
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) −20.0000 −0.867110
\(533\) 0 0
\(534\) 36.0000 1.55787
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) −24.0000 −1.03568
\(538\) −17.0000 −0.732922
\(539\) −54.0000 −2.32594
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 − 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −29.0000 −1.24566
\(543\) −6.00000 −0.257485
\(544\) 25.0000 1.07187
\(545\) 0 0
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 − 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) 18.0000 0.768922
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) −24.0000 −1.02151
\(553\) 50.0000 2.12622
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 0 0
\(560\) 0 0
\(561\) −30.0000 −1.26660
\(562\) 18.0000 0.759284
\(563\) −30.0000 −1.26435 −0.632175 − 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) −14.0000 −0.589506
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) −55.0000 −2.30978
\(568\) 24.0000 1.00702
\(569\) 33.0000 1.38343 0.691716 − 0.722170i \(-0.256855\pi\)
0.691716 + 0.722170i \(0.256855\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 − 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 8.00000 0.332756
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) −45.0000 −1.86691
\(582\) −28.0000 −1.16064
\(583\) −9.00000 −0.372742
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 25.0000 1.03186 0.515930 − 0.856631i \(-0.327446\pi\)
0.515930 + 0.856631i \(0.327446\pi\)
\(588\) −36.0000 −1.48461
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 4.00000 0.164399
\(593\) −8.00000 −0.328521 −0.164260 − 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) 12.0000 0.492366
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 − 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) 29.0000 1.18293 0.591467 − 0.806329i \(-0.298549\pi\)
0.591467 + 0.806329i \(0.298549\pi\)
\(602\) −20.0000 −0.815139
\(603\) 3.00000 0.122169
\(604\) 13.0000 0.528962
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 20.0000 0.811107
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) 0 0
\(612\) −5.00000 −0.202113
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 45.0000 1.81310
\(617\) −4.00000 −0.161034 −0.0805170 − 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) 8.00000 0.321807
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) −4.00000 −0.160385
\(623\) 90.0000 3.60577
\(624\) 0 0
\(625\) 0 0
\(626\) −17.0000 −0.679457
\(627\) −24.0000 −0.958468
\(628\) −13.0000 −0.518756
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −30.0000 −1.19334
\(633\) −24.0000 −0.953914
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 − 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 32.0000 1.26294
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −20.0000 −0.788110
\(645\) 0 0
\(646\) 20.0000 0.786889
\(647\) −14.0000 −0.550397 −0.275198 − 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) 33.0000 1.29636
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 12.0000 0.469956
\(653\) −31.0000 −1.21312 −0.606562 − 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) −4.00000 −0.156055
\(658\) 35.0000 1.36444
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 27.0000 1.04780
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −4.00000 −0.154881
\(668\) 16.0000 0.619059
\(669\) −32.0000 −1.23719
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 50.0000 1.92879
\(673\) −23.0000 −0.886585 −0.443292 − 0.896377i \(-0.646190\pi\)
−0.443292 + 0.896377i \(0.646190\pi\)
\(674\) −3.00000 −0.115556
\(675\) 0 0
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −20.0000 −0.768095
\(679\) −70.0000 −2.68635
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 3.00000 0.114876
\(683\) −41.0000 −1.56882 −0.784411 − 0.620242i \(-0.787034\pi\)
−0.784411 + 0.620242i \(0.787034\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 55.0000 2.09991
\(687\) 4.00000 0.152610
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 − 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) −13.0000 −0.494186
\(693\) −15.0000 −0.569803
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 40.0000 1.51511
\(698\) −2.00000 −0.0757011
\(699\) 28.0000 1.05906
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 − 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) −21.0000 −0.791467
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −5.00000 −0.188044
\(708\) −6.00000 −0.225494
\(709\) −8.00000 −0.300446 −0.150223 − 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) −54.0000 −2.02374
\(713\) −4.00000 −0.149801
\(714\) 50.0000 1.87120
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −10.0000 −0.373457
\(718\) −23.0000 −0.858352
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 3.00000 0.111494
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) −2.00000 −0.0739221
\(733\) −42.0000 −1.55131 −0.775653 − 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) −9.00000 −0.331519
\(738\) 8.00000 0.294484
\(739\) −9.00000 −0.331070 −0.165535 − 0.986204i \(-0.552935\pi\)
−0.165535 + 0.986204i \(0.552935\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15.0000 0.550667
\(743\) −41.0000 −1.50414 −0.752072 − 0.659081i \(-0.770945\pi\)
−0.752072 + 0.659081i \(0.770945\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −31.0000 −1.13499
\(747\) −9.00000 −0.329293
\(748\) 15.0000 0.548454
\(749\) 80.0000 2.92314
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −7.00000 −0.255264
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) 0 0
\(756\) 20.0000 0.727393
\(757\) −5.00000 −0.181728 −0.0908640 − 0.995863i \(-0.528963\pi\)
−0.0908640 + 0.995863i \(0.528963\pi\)
\(758\) −27.0000 −0.980684
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 36.0000 1.30414
\(763\) 10.0000 0.362024
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) 0 0
\(768\) −34.0000 −1.22687
\(769\) −40.0000 −1.44244 −0.721218 − 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 4.00000 0.143963
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 42.0000 1.50771
\(777\) −40.0000 −1.43499
\(778\) −34.0000 −1.21896
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 20.0000 0.715199
\(783\) 4.00000 0.142948
\(784\) −18.0000 −0.642857
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) −43.0000 −1.53278 −0.766392 − 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) 8.00000 0.284988
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −50.0000 −1.77780
\(792\) 9.00000 0.319801
\(793\) 0 0
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) −53.0000 −1.87736 −0.938678 − 0.344795i \(-0.887949\pi\)
−0.938678 + 0.344795i \(0.887949\pi\)
\(798\) 40.0000 1.41598
\(799\) 35.0000 1.23821
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 26.0000 0.918092
\(803\) 12.0000 0.423471
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) 0 0
\(807\) −34.0000 −1.19686
\(808\) 3.00000 0.105540
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 − 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 5.00000 0.175466
\(813\) −58.0000 −2.03415
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) −10.0000 −0.350070
\(817\) −16.0000 −0.559769
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −36.0000 −1.25564
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 15.0000 0.521917
\(827\) 19.0000 0.660695 0.330347 − 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) −4.00000 −0.139010
\(829\) 11.0000 0.382046 0.191023 − 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 0 0
\(833\) 90.0000 3.11832
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 4.00000 0.138260
\(838\) −18.0000 −0.621800
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 4.00000 0.137849
\(843\) 36.0000 1.23991
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) −10.0000 −0.343604
\(848\) −3.00000 −0.103020
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 16.0000 0.548151
\(853\) −54.0000 −1.84892 −0.924462 − 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −48.0000 −1.64061
\(857\) 14.0000 0.478231 0.239115 − 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 80.0000 2.72639
\(862\) 0 0
\(863\) −33.0000 −1.12333 −0.561667 − 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) −20.0000 −0.680414
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) 16.0000 0.543388
\(868\) 5.00000 0.169711
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) −14.0000 −0.473828
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −22.0000 −0.742464
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 − 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 18.0000 0.606092
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 22.0000 0.738688 0.369344 − 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) 24.0000 0.805387
\(889\) 90.0000 3.01850
\(890\) 0 0
\(891\) 33.0000 1.10554
\(892\) 16.0000 0.535720
\(893\) 28.0000 0.936984
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) −15.0000 −0.501115
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 1.00000 0.0333519
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) −24.0000 −0.799113
\(903\) −40.0000 −1.33112
\(904\) 30.0000 0.997785
\(905\) 0 0
\(906\) −26.0000 −0.863792
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −9.00000 −0.298675
\(909\) −1.00000 −0.0331679
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 − 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) −8.00000 −0.264906
\(913\) 27.0000 0.893570
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 90.0000 2.97206
\(918\) −20.0000 −0.660098
\(919\) 46.0000 1.51740 0.758700 − 0.651440i \(-0.225835\pi\)
0.758700 + 0.651440i \(0.225835\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) −26.0000 −0.856264
\(923\) 0 0
\(924\) 30.0000 0.986928
\(925\) 0 0
\(926\) 19.0000 0.624379
\(927\) 4.00000 0.131377
\(928\) −5.00000 −0.164133
\(929\) 56.0000 1.83730 0.918650 − 0.395072i \(-0.129280\pi\)
0.918650 + 0.395072i \(0.129280\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) −14.0000 −0.458585
\(933\) −8.00000 −0.261908
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) 0 0
\(937\) 1.00000 0.0326686 0.0163343 − 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(938\) 15.0000 0.489767
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 − 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 26.0000 0.847126
\(943\) 32.0000 1.04206
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 3.00000 0.0974869 0.0487435 − 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) −20.0000 −0.649570
\(949\) 0 0
\(950\) 0 0
\(951\) 44.0000 1.42680
\(952\) −75.0000 −2.43076
\(953\) −9.00000 −0.291539 −0.145769 − 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 5.00000 0.161712
\(957\) 6.00000 0.193952
\(958\) 3.00000 0.0969256
\(959\) −90.0000 −2.90625
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) 0 0
\(966\) 40.0000 1.28698
\(967\) 37.0000 1.18984 0.594920 − 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) 6.00000 0.192847
\(969\) 40.0000 1.28499
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 − 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 10.0000 0.320750
\(973\) 20.0000 0.641171
\(974\) −15.0000 −0.480631
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 46.0000 1.47167 0.735835 − 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) −24.0000 −0.767435
\(979\) −54.0000 −1.72585
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 28.0000 0.893516
\(983\) −11.0000 −0.350846 −0.175423 − 0.984493i \(-0.556129\pi\)
−0.175423 + 0.984493i \(0.556129\pi\)
\(984\) −48.0000 −1.53018
\(985\) 0 0
\(986\) −5.00000 −0.159232
\(987\) 70.0000 2.22812
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −5.00000 −0.158750
\(993\) 40.0000 1.26936
\(994\) −40.0000 −1.26872
\(995\) 0 0
\(996\) 18.0000 0.570352
\(997\) −45.0000 −1.42516 −0.712582 − 0.701589i \(-0.752474\pi\)
−0.712582 + 0.701589i \(0.752474\pi\)
\(998\) 19.0000 0.601434
\(999\) 16.0000 0.506218
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 4225.2.a.n.1.1 1
5.4 even 2 4225.2.a.d.1.1 1
13.5 odd 4 325.2.c.f.51.1 yes 2
13.8 odd 4 325.2.c.f.51.2 yes 2
13.12 even 2 4225.2.a.f.1.1 1
65.8 even 4 325.2.d.c.324.2 2
65.18 even 4 325.2.d.b.324.2 2
65.34 odd 4 325.2.c.a.51.1 ✓ 2
65.44 odd 4 325.2.c.a.51.2 yes 2
65.47 even 4 325.2.d.b.324.1 2
65.57 even 4 325.2.d.c.324.1 2
65.64 even 2 4225.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.2.c.a.51.1 ✓ 2 65.34 odd 4
325.2.c.a.51.2 yes 2 65.44 odd 4
325.2.c.f.51.1 yes 2 13.5 odd 4
325.2.c.f.51.2 yes 2 13.8 odd 4
325.2.d.b.324.1 2 65.47 even 4
325.2.d.b.324.2 2 65.18 even 4
325.2.d.c.324.1 2 65.57 even 4
325.2.d.c.324.2 2 65.8 even 4
4225.2.a.d.1.1 1 5.4 even 2
4225.2.a.f.1.1 1 13.12 even 2
4225.2.a.l.1.1 1 65.64 even 2
4225.2.a.n.1.1 1 1.1 even 1 trivial