Properties

Label 155.2.a.b.1.1
Level $155$
Weight $2$
Character 155.1
Self dual yes
Analytic conductor $1.238$
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] = [155,2,Mod(1,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, 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("155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.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: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -2.00000 q^{12} -4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +8.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} +6.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} +1.00000 q^{31} -5.00000 q^{32} +8.00000 q^{33} +8.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} -8.00000 q^{42} -6.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -2.00000 q^{46} +8.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -16.0000 q^{51} -12.0000 q^{53} +4.00000 q^{54} -4.00000 q^{55} +12.0000 q^{56} +8.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +10.0000 q^{61} -1.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} -8.00000 q^{66} +8.00000 q^{67} +8.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} +3.00000 q^{72} -4.00000 q^{73} +4.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +2.00000 q^{83} -8.00000 q^{84} +8.00000 q^{85} +6.00000 q^{86} -12.0000 q^{87} +12.0000 q^{88} +14.0000 q^{89} +1.00000 q^{90} -2.00000 q^{92} +2.00000 q^{93} -8.00000 q^{94} -4.00000 q^{95} -10.0000 q^{96} -18.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −4.00000 −1.06904
\(15\) −2.00000 −0.516398
\(16\) −1.00000 −0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 8.00000 1.74574
\(22\) −4.00000 −0.852803
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 6.00000 1.22474
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) 1.00000 0.179605
\(32\) −5.00000 −0.883883
\(33\) 8.00000 1.39262
\(34\) 8.00000 1.37199
\(35\) −4.00000 −0.676123
\(36\) −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
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −8.00000 −1.23443
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) −2.00000 −0.294884
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −2.00000 −0.288675
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −16.0000 −2.24045
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 4.00000 0.544331
\(55\) −4.00000 −0.539360
\(56\) 12.0000 1.60357
\(57\) 8.00000 1.05963
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −1.00000 −0.127000
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −8.00000 −0.984732
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 8.00000 0.970143
\(69\) 4.00000 0.481543
\(70\) 4.00000 0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 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
\(75\) 2.00000 0.230940
\(76\) −4.00000 −0.458831
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) −8.00000 −0.872872
\(85\) 8.00000 0.867722
\(86\) 6.00000 0.646997
\(87\) −12.0000 −1.28654
\(88\) 12.0000 1.27920
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 2.00000 0.207390
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) −10.0000 −1.02062
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −9.00000 −0.909137
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 16.0000 1.58424
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 12.0000 1.16554
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 4.00000 0.384900
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 4.00000 0.381385
\(111\) −8.00000 −0.759326
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −8.00000 −0.749269
\(115\) −2.00000 −0.186501
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −32.0000 −2.93344
\(120\) −6.00000 −0.547723
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −12.0000 −1.08200
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) −4.00000 −0.356348
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 3.00000 0.265165
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −8.00000 −0.696311
\(133\) 16.0000 1.38738
\(134\) −8.00000 −0.691095
\(135\) 4.00000 0.344265
\(136\) −24.0000 −2.05798
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −4.00000 −0.340503
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 4.00000 0.338062
\(141\) 16.0000 1.34744
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 6.00000 0.498273
\(146\) 4.00000 0.331042
\(147\) 18.0000 1.48461
\(148\) 4.00000 0.328798
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −2.00000 −0.163299
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 12.0000 0.973329
\(153\) −8.00000 −0.646762
\(154\) −16.0000 −1.28932
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −24.0000 −1.90332
\(160\) 5.00000 0.395285
\(161\) 8.00000 0.630488
\(162\) 11.0000 0.864242
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) −8.00000 −0.622799
\(166\) −2.00000 −0.155230
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 24.0000 1.85164
\(169\) −13.0000 −1.00000
\(170\) −8.00000 −0.613572
\(171\) 4.00000 0.305888
\(172\) 6.00000 0.457496
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 12.0000 0.909718
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) −8.00000 −0.601317
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 20.0000 1.47844
\(184\) 6.00000 0.442326
\(185\) 4.00000 0.294086
\(186\) −2.00000 −0.146647
\(187\) −32.0000 −2.34007
\(188\) −8.00000 −0.583460
\(189\) −16.0000 −1.16383
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 14.0000 1.01036
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −4.00000 −0.284268
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 3.00000 0.212132
\(201\) 16.0000 1.12855
\(202\) −10.0000 −0.703598
\(203\) −24.0000 −1.68447
\(204\) 16.0000 1.12022
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 8.00000 0.552052
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 6.00000 0.409197
\(216\) −12.0000 −0.816497
\(217\) 4.00000 0.271538
\(218\) −10.0000 −0.677285
\(219\) −8.00000 −0.540590
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) −20.0000 −1.33631
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −8.00000 −0.529813
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 2.00000 0.131876
\(231\) 32.0000 2.10545
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 32.0000 2.07425
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −5.00000 −0.321412
\(243\) −10.0000 −0.641500
\(244\) −10.0000 −0.640184
\(245\) −9.00000 −0.574989
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) −4.00000 −0.251976
\(253\) 8.00000 0.502956
\(254\) −2.00000 −0.125491
\(255\) 16.0000 1.00196
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 12.0000 0.747087
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 4.00000 0.247121
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 24.0000 1.47710
\(265\) 12.0000 0.737154
\(266\) −16.0000 −0.981023
\(267\) 28.0000 1.71357
\(268\) −8.00000 −0.488678
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −4.00000 −0.243432
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 4.00000 0.241209
\(276\) −4.00000 −0.240772
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) −12.0000 −0.717137
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −16.0000 −0.952786
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) −5.00000 −0.294628
\(289\) 47.0000 2.76471
\(290\) −6.00000 −0.352332
\(291\) −36.0000 −2.11036
\(292\) 4.00000 0.234082
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) −18.0000 −1.04978
\(295\) 4.00000 0.232889
\(296\) −12.0000 −0.697486
\(297\) −16.0000 −0.928414
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) −4.00000 −0.229416
\(305\) −10.0000 −0.572598
\(306\) 8.00000 0.457330
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −16.0000 −0.911685
\(309\) 16.0000 0.910208
\(310\) 1.00000 0.0567962
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −14.0000 −0.790066
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 24.0000 1.34585
\(319\) −24.0000 −1.34374
\(320\) −7.00000 −0.391312
\(321\) −8.00000 −0.446516
\(322\) −8.00000 −0.445823
\(323\) −32.0000 −1.78053
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 20.0000 1.10600
\(328\) −18.0000 −0.993884
\(329\) 32.0000 1.76422
\(330\) 8.00000 0.440386
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −2.00000 −0.109764
\(333\) −4.00000 −0.219199
\(334\) −14.0000 −0.766046
\(335\) −8.00000 −0.437087
\(336\) −8.00000 −0.436436
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 13.0000 0.707107
\(339\) −36.0000 −1.95525
\(340\) −8.00000 −0.433861
\(341\) 4.00000 0.216612
\(342\) −4.00000 −0.216295
\(343\) 8.00000 0.431959
\(344\) −18.0000 −0.970495
\(345\) −4.00000 −0.215353
\(346\) −6.00000 −0.322562
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 12.0000 0.643268
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) −64.0000 −3.38724
\(358\) −4.00000 −0.211407
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 18.0000 0.946059
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −20.0000 −1.04542
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −2.00000 −0.104257
\(369\) −6.00000 −0.312348
\(370\) −4.00000 −0.207950
\(371\) −48.0000 −2.49204
\(372\) −2.00000 −0.103695
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 32.0000 1.65468
\(375\) −2.00000 −0.103280
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 16.0000 0.822951
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) 4.00000 0.204926
\(382\) −8.00000 −0.409316
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 6.00000 0.306186
\(385\) −16.0000 −0.815436
\(386\) 18.0000 0.916176
\(387\) −6.00000 −0.304997
\(388\) 18.0000 0.913812
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 27.0000 1.36371
\(393\) −8.00000 −0.403547
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 20.0000 1.00251
\(399\) 32.0000 1.60200
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −16.0000 −0.798007
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 11.0000 0.546594
\(406\) 24.0000 1.19110
\(407\) −16.0000 −0.793091
\(408\) −48.0000 −2.37635
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) −6.00000 −0.296319
\(411\) 24.0000 1.18383
\(412\) −8.00000 −0.394132
\(413\) −16.0000 −0.787309
\(414\) −2.00000 −0.0982946
\(415\) −2.00000 −0.0981761
\(416\) 0 0
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 8.00000 0.390360
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) −36.0000 −1.74831
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 4.00000 0.192450
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) −4.00000 −0.192006
\(435\) 12.0000 0.575356
\(436\) −10.0000 −0.478913
\(437\) 8.00000 0.382692
\(438\) 8.00000 0.382255
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −12.0000 −0.572078
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 8.00000 0.379663
\(445\) −14.0000 −0.663664
\(446\) 26.0000 1.23114
\(447\) 12.0000 0.567581
\(448\) 28.0000 1.32288
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) −20.0000 −0.935561 −0.467780 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) −18.0000 −0.841085
\(459\) 32.0000 1.49363
\(460\) 2.00000 0.0932505
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −32.0000 −1.48877
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 6.00000 0.278543
\(465\) −2.00000 −0.0927478
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 8.00000 0.369012
\(471\) 28.0000 1.29017
\(472\) −12.0000 −0.552345
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 32.0000 1.46672
\(477\) −12.0000 −0.549442
\(478\) −20.0000 −0.914779
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 10.0000 0.456435
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 16.0000 0.728025
\(484\) −5.00000 −0.227273
\(485\) 18.0000 0.817338
\(486\) 10.0000 0.453609
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 30.0000 1.35804
\(489\) −32.0000 −1.44709
\(490\) 9.00000 0.406579
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 12.0000 0.541002
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 1.00000 0.0447214
\(501\) 28.0000 1.25095
\(502\) −16.0000 −0.714115
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 12.0000 0.534522
\(505\) −10.0000 −0.444994
\(506\) −8.00000 −0.355643
\(507\) −26.0000 −1.15470
\(508\) −2.00000 −0.0887357
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −16.0000 −0.708492
\(511\) −16.0000 −0.707798
\(512\) 11.0000 0.486136
\(513\) −16.0000 −0.706417
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 12.0000 0.528271
\(517\) 32.0000 1.40736
\(518\) 16.0000 0.703000
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 6.00000 0.262613
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 4.00000 0.174741
\(525\) 8.00000 0.349149
\(526\) −6.00000 −0.261612
\(527\) −8.00000 −0.348485
\(528\) −8.00000 −0.348155
\(529\) −19.0000 −0.826087
\(530\) −12.0000 −0.521247
\(531\) −4.00000 −0.173585
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) −28.0000 −1.21168
\(535\) 4.00000 0.172935
\(536\) 24.0000 1.03664
\(537\) 8.00000 0.345225
\(538\) −6.00000 −0.258678
\(539\) 36.0000 1.55063
\(540\) −4.00000 −0.172133
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 4.00000 0.171815
\(543\) −36.0000 −1.54491
\(544\) 40.0000 1.71499
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −12.0000 −0.512615
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) −24.0000 −1.02243
\(552\) 12.0000 0.510754
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) −64.0000 −2.70208
\(562\) −6.00000 −0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −16.0000 −0.673722
\(565\) 18.0000 0.757266
\(566\) −16.0000 −0.672530
\(567\) −44.0000 −1.84783
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 8.00000 0.335083
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 24.0000 1.00174
\(575\) 2.00000 0.0834058
\(576\) 7.00000 0.291667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −47.0000 −1.95494
\(579\) −36.0000 −1.49611
\(580\) −6.00000 −0.249136
\(581\) 8.00000 0.331896
\(582\) 36.0000 1.49225
\(583\) −48.0000 −1.98796
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) −18.0000 −0.742307
\(589\) 4.00000 0.164817
\(590\) −4.00000 −0.164677
\(591\) −16.0000 −0.658152
\(592\) 4.00000 0.164399
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 16.0000 0.656488
\(595\) 32.0000 1.31187
\(596\) −6.00000 −0.245770
\(597\) −40.0000 −1.63709
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 6.00000 0.244949
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 24.0000 0.978167
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) −20.0000 −0.812444
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −20.0000 −0.811107
\(609\) −48.0000 −1.94506
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 8.00000 0.323381
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) −28.0000 −1.12999
\(615\) 12.0000 0.483887
\(616\) 48.0000 1.93398
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −16.0000 −0.643614
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 1.00000 0.0401610
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) 56.0000 2.24359
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 32.0000 1.27796
\(628\) −14.0000 −0.558661
\(629\) 32.0000 1.27592
\(630\) 4.00000 0.159364
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 40.0000 1.58986
\(634\) 6.00000 0.238290
\(635\) −2.00000 −0.0793676
\(636\) 24.0000 0.951662
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 8.00000 0.315735
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) −8.00000 −0.315244
\(645\) 12.0000 0.472500
\(646\) 32.0000 1.25902
\(647\) −14.0000 −0.550397 −0.275198 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) −33.0000 −1.29636
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 16.0000 0.626608
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) −20.0000 −0.782062
\(655\) 4.00000 0.156293
\(656\) 6.00000 0.234261
\(657\) −4.00000 −0.156055
\(658\) −32.0000 −1.24749
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 8.00000 0.311400
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −16.0000 −0.620453
\(666\) 4.00000 0.154997
\(667\) −12.0000 −0.464642
\(668\) −14.0000 −0.541676
\(669\) −52.0000 −2.01044
\(670\) 8.00000 0.309067
\(671\) 40.0000 1.54418
\(672\) −40.0000 −1.54303
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 8.00000 0.308148
\(675\) −4.00000 −0.153960
\(676\) 13.0000 0.500000
\(677\) −16.0000 −0.614930 −0.307465 0.951559i \(-0.599481\pi\)
−0.307465 + 0.951559i \(0.599481\pi\)
\(678\) 36.0000 1.38257
\(679\) −72.0000 −2.76311
\(680\) 24.0000 0.920358
\(681\) 16.0000 0.613121
\(682\) −4.00000 −0.153168
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −4.00000 −0.152944
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) 36.0000 1.37349
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 4.00000 0.152277
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −6.00000 −0.228086
\(693\) 16.0000 0.607790
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) −36.0000 −1.36458
\(697\) 48.0000 1.81813
\(698\) −22.0000 −0.832712
\(699\) 12.0000 0.453882
\(700\) −4.00000 −0.151186
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) 28.0000 1.05529
\(705\) −16.0000 −0.602595
\(706\) 16.0000 0.602168
\(707\) 40.0000 1.50435
\(708\) 8.00000 0.300658
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 42.0000 1.57402
\(713\) 2.00000 0.0749006
\(714\) 64.0000 2.39514
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 40.0000 1.49383
\(718\) 8.00000 0.298557
\(719\) −52.0000 −1.93927 −0.969636 0.244551i \(-0.921359\pi\)
−0.969636 + 0.244551i \(0.921359\pi\)
\(720\) 1.00000 0.0372678
\(721\) 32.0000 1.19174
\(722\) 3.00000 0.111648
\(723\) −20.0000 −0.743808
\(724\) 18.0000 0.668965
\(725\) −6.00000 −0.222834
\(726\) −10.0000 −0.371135
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −4.00000 −0.148047
\(731\) 48.0000 1.77534
\(732\) −20.0000 −0.739221
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −22.0000 −0.812035
\(735\) −18.0000 −0.663940
\(736\) −10.0000 −0.368605
\(737\) 32.0000 1.17874
\(738\) 6.00000 0.220863
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 48.0000 1.76214
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 6.00000 0.219971
\(745\) −6.00000 −0.219823
\(746\) −14.0000 −0.512576
\(747\) 2.00000 0.0731762
\(748\) 32.0000 1.17004
\(749\) −16.0000 −0.584627
\(750\) 2.00000 0.0730297
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) 32.0000 1.16614
\(754\) 0 0
\(755\) 0 0
\(756\) 16.0000 0.581914
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −28.0000 −1.01701
\(759\) 16.0000 0.580763
\(760\) −12.0000 −0.435286
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −4.00000 −0.144905
\(763\) 40.0000 1.44810
\(764\) −8.00000 −0.289430
\(765\) 8.00000 0.289241
\(766\) −18.0000 −0.650366
\(767\) 0 0
\(768\) −34.0000 −1.22687
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 16.0000 0.576600
\(771\) −36.0000 −1.29651
\(772\) 18.0000 0.647834
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 6.00000 0.215666
\(775\) 1.00000 0.0359211
\(776\) −54.0000 −1.93849
\(777\) −32.0000 −1.14799
\(778\) 10.0000 0.358517
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 24.0000 0.857690
\(784\) −9.00000 −0.321429
\(785\) −14.0000 −0.499681
\(786\) 8.00000 0.285351
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 8.00000 0.284988
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −72.0000 −2.56003
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) −26.0000 −0.922705
\(795\) 24.0000 0.851192
\(796\) 20.0000 0.708881
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) −32.0000 −1.13279
\(799\) −64.0000 −2.26416
\(800\) −5.00000 −0.176777
\(801\) 14.0000 0.494666
\(802\) −2.00000 −0.0706225
\(803\) −16.0000 −0.564628
\(804\) −16.0000 −0.564276
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 30.0000 1.05540
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −11.0000 −0.386501
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 24.0000 0.842235
\(813\) −8.00000 −0.280572
\(814\) 16.0000 0.560800
\(815\) 16.0000 0.560456
\(816\) 16.0000 0.560112
\(817\) −24.0000 −0.839654
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −24.0000 −0.837096
\(823\) −38.0000 −1.32460 −0.662298 0.749240i \(-0.730419\pi\)
−0.662298 + 0.749240i \(0.730419\pi\)
\(824\) 24.0000 0.836080
\(825\) 8.00000 0.278524
\(826\) 16.0000 0.556711
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 2.00000 0.0694210
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) −72.0000 −2.49465
\(834\) 0 0
\(835\) −14.0000 −0.484490
\(836\) −16.0000 −0.553372
\(837\) −4.00000 −0.138260
\(838\) −4.00000 −0.138178
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) −24.0000 −0.828079
\(841\) 7.00000 0.241379
\(842\) 18.0000 0.620321
\(843\) 12.0000 0.413302
\(844\) −20.0000 −0.688428
\(845\) 13.0000 0.447214
\(846\) −8.00000 −0.275046
\(847\) 20.0000 0.687208
\(848\) 12.0000 0.412082
\(849\) 32.0000 1.09824
\(850\) 8.00000 0.274398
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) −40.0000 −1.36877
\(855\) −4.00000 −0.136797
\(856\) −12.0000 −0.410152
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −6.00000 −0.204598
\(861\) −48.0000 −1.63584
\(862\) −40.0000 −1.36241
\(863\) −38.0000 −1.29354 −0.646768 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(864\) 20.0000 0.680414
\(865\) −6.00000 −0.204006
\(866\) −24.0000 −0.815553
\(867\) 94.0000 3.19241
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 0 0
\(872\) 30.0000 1.01593
\(873\) −18.0000 −0.609208
\(874\) −8.00000 −0.270604
\(875\) −4.00000 −0.135225
\(876\) 8.00000 0.270295
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 0 0
\(879\) 44.0000 1.48408
\(880\) 4.00000 0.134840
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) −9.00000 −0.303046
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 4.00000 0.134383
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −24.0000 −0.805387
\(889\) 8.00000 0.268311
\(890\) 14.0000 0.469281
\(891\) −44.0000 −1.47406
\(892\) 26.0000 0.870544
\(893\) 32.0000 1.07084
\(894\) −12.0000 −0.401340
\(895\) −4.00000 −0.133705
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) −6.00000 −0.200111
\(900\) −1.00000 −0.0333333
\(901\) 96.0000 3.19822
\(902\) 24.0000 0.799113
\(903\) −48.0000 −1.59734
\(904\) −54.0000 −1.79601
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) −8.00000 −0.265489
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.00000 −0.264906
\(913\) 8.00000 0.264761
\(914\) 20.0000 0.661541
\(915\) −20.0000 −0.661180
\(916\) −18.0000 −0.594737
\(917\) −16.0000 −0.528367
\(918\) −32.0000 −1.05616
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −6.00000 −0.197814
\(921\) 56.0000 1.84526
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) −32.0000 −1.05272
\(925\) −4.00000 −0.131519
\(926\) −2.00000 −0.0657241
\(927\) 8.00000 0.262754
\(928\) 30.0000 0.984798
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 2.00000 0.0655826
\(931\) 36.0000 1.17985
\(932\) −6.00000 −0.196537
\(933\) 48.0000 1.57145
\(934\) 28.0000 0.916188
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −32.0000 −1.04484
\(939\) −16.0000 −0.522140
\(940\) 8.00000 0.260931
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −28.0000 −0.912289
\(943\) −12.0000 −0.390774
\(944\) 4.00000 0.130189
\(945\) 16.0000 0.520480
\(946\) 24.0000 0.780307
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) −12.0000 −0.389127
\(952\) −96.0000 −3.11138
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 12.0000 0.388514
\(955\) −8.00000 −0.258874
\(956\) −20.0000 −0.646846
\(957\) −48.0000 −1.55162
\(958\) −16.0000 −0.516937
\(959\) 48.0000 1.55000
\(960\) −14.0000 −0.451848
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) 18.0000 0.579441
\(966\) −16.0000 −0.514792
\(967\) −54.0000 −1.73652 −0.868261 0.496107i \(-0.834762\pi\)
−0.868261 + 0.496107i \(0.834762\pi\)
\(968\) 15.0000 0.482118
\(969\) −64.0000 −2.05598
\(970\) −18.0000 −0.577945
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 32.0000 1.02325
\(979\) 56.0000 1.78977
\(980\) 9.00000 0.287494
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) −36.0000 −1.14764
\(985\) 8.00000 0.254901
\(986\) −48.0000 −1.52863
\(987\) 64.0000 2.03714
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 4.00000 0.127128
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −5.00000 −0.158750
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) −4.00000 −0.126745
\(997\) 30.0000 0.950110 0.475055 0.879956i \(-0.342428\pi\)
0.475055 + 0.879956i \(0.342428\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 155.2.a.b.1.1 1
3.2 odd 2 1395.2.a.d.1.1 1
4.3 odd 2 2480.2.a.b.1.1 1
5.2 odd 4 775.2.b.b.249.1 2
5.3 odd 4 775.2.b.b.249.2 2
5.4 even 2 775.2.a.b.1.1 1
7.6 odd 2 7595.2.a.c.1.1 1
8.3 odd 2 9920.2.a.bd.1.1 1
8.5 even 2 9920.2.a.g.1.1 1
15.14 odd 2 6975.2.a.d.1.1 1
31.30 odd 2 4805.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
155.2.a.b.1.1 1 1.1 even 1 trivial
775.2.a.b.1.1 1 5.4 even 2
775.2.b.b.249.1 2 5.2 odd 4
775.2.b.b.249.2 2 5.3 odd 4
1395.2.a.d.1.1 1 3.2 odd 2
2480.2.a.b.1.1 1 4.3 odd 2
4805.2.a.d.1.1 1 31.30 odd 2
6975.2.a.d.1.1 1 15.14 odd 2
7595.2.a.c.1.1 1 7.6 odd 2
9920.2.a.g.1.1 1 8.5 even 2
9920.2.a.bd.1.1 1 8.3 odd 2